LMFDB - Embedded newform 72.8.d.b.37.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,8,Mod(37,72)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("72.37");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	72.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.4917218349$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 $ x^6 - 3x^5 - 10x^4 - 24x^3 - 320x^2 - 3072*x + 32768
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.2
Root			$5.57668 + 0.949035i$ of defining polynomial
Character	$\chi$	$=$	72.37
Dual form			72.8.d.b.37.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-11.1534 + 1.89807i) q^{2} +(120.795 - 42.3397i) q^{4} +338.443i q^{5} -438.996 q^{7} +(-1266.90 + 701.506i) q^{8} +O(q^{10})$	\(q+(-11.1534 + 1.89807i) q^{2} +(120.795 - 42.3397i) q^{4} +338.443i q^{5} -438.996 q^{7} +(-1266.90 + 701.506i) q^{8} +(-642.387 - 3774.77i) q^{10} +1966.58i q^{11} +2210.98i q^{13} +(4896.28 - 833.245i) q^{14} +(12798.7 - 10228.8i) q^{16} +12114.9 q^{17} +32872.2i q^{19} +(14329.5 + 40882.1i) q^{20} +(-3732.70 - 21933.9i) q^{22} -19605.1 q^{23} -36418.4 q^{25} +(-4196.59 - 24659.8i) q^{26} +(-53028.4 + 18587.0i) q^{28} -160689. i q^{29} -229270. q^{31} +(-123333. + 138378. i) q^{32} +(-135122. + 22994.9i) q^{34} -148575. i q^{35} -496284. i q^{37} +(-62393.6 - 366635. i) q^{38} +(-237420. - 428774. i) q^{40} -599971. q^{41} +88346.0i q^{43} +(83264.3 + 237552. i) q^{44} +(218662. - 37211.8i) q^{46} -820344. q^{47} -630825. q^{49} +(406187. - 69124.6i) q^{50} +(93612.2 + 267075. i) q^{52} +1.53717e6i q^{53} -665574. q^{55} +(556165. - 307959. i) q^{56} +(304999. + 1.79222e6i) q^{58} -1.82480e6i q^{59} -484582. i q^{61} +(2.55713e6 - 435171. i) q^{62} +(1.11293e6 - 1.77748e6i) q^{64} -748290. q^{65} -79878.2i q^{67} +(1.46341e6 - 512940. i) q^{68} +(282006. + 1.65711e6i) q^{70} -1.27078e6 q^{71} +3.70820e6 q^{73} +(941981. + 5.53523e6i) q^{74} +(1.39180e6 + 3.97078e6i) q^{76} -863321. i q^{77} -2.55846e6 q^{79} +(3.46187e6 + 4.33163e6i) q^{80} +(6.69169e6 - 1.13879e6i) q^{82} -1.53414e6i q^{83} +4.10019e6i q^{85} +(-167687. - 985355. i) q^{86} +(-1.37957e6 - 2.49146e6i) q^{88} -1.99492e6 q^{89} -970612. i q^{91} +(-2.36819e6 + 830073. i) q^{92} +(9.14959e6 - 1.55707e6i) q^{94} -1.11253e7 q^{95} -28917.7 q^{97} +(7.03582e6 - 1.19735e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 116 q^{4} - 688 q^{7} - 1512 q^{8}+O(q^{10}) $ 6 * q - 6 * q^2 + 116 * q^4 - 688 * q^7 - 1512 * q^8	$ 6 q - 6 q^{2} + 116 q^{4} - 688 q^{7} - 1512 q^{8} - 1656 q^{10} - 12048 q^{14} + 35344 q^{16} - 1452 q^{17} + 114768 q^{20} + 152860 q^{22} + 1296 q^{23} - 39314 q^{25} + 316968 q^{26} - 480800 q^{28} - 89280 q^{31} - 817056 q^{32} - 1009108 q^{34} - 974124 q^{38} + 954464 q^{40} - 521244 q^{41} + 1096344 q^{44} + 929840 q^{46} - 1566432 q^{47} - 511050 q^{49} + 148626 q^{50} + 823952 q^{52} - 3270256 q^{55} + 2468928 q^{56} + 3130744 q^{58} + 7055808 q^{62} - 4792768 q^{64} - 1416480 q^{65} - 6608040 q^{68} - 7406912 q^{70} + 7597104 q^{71} + 2089564 q^{73} - 7744200 q^{74} + 9241288 q^{76} + 16015904 q^{79} + 12600384 q^{80} + 10715932 q^{82} + 5639076 q^{86} + 1541200 q^{88} - 2169084 q^{89} - 669600 q^{92} + 15503712 q^{94} - 48537936 q^{95} - 1088308 q^{97} + 14983242 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 116 * q^4 - 688 * q^7 - 1512 * q^8 - 1656 * q^10 - 12048 * q^14 + 35344 * q^16 - 1452 * q^17 + 114768 * q^20 + 152860 * q^22 + 1296 * q^23 - 39314 * q^25 + 316968 * q^26 - 480800 * q^28 - 89280 * q^31 - 817056 * q^32 - 1009108 * q^34 - 974124 * q^38 + 954464 * q^40 - 521244 * q^41 + 1096344 * q^44 + 929840 * q^46 - 1566432 * q^47 - 511050 * q^49 + 148626 * q^50 + 823952 * q^52 - 3270256 * q^55 + 2468928 * q^56 + 3130744 * q^58 + 7055808 * q^62 - 4792768 * q^64 - 1416480 * q^65 - 6608040 * q^68 - 7406912 * q^70 + 7597104 * q^71 + 2089564 * q^73 - 7744200 * q^74 + 9241288 * q^76 + 16015904 * q^79 + 12600384 * q^80 + 10715932 * q^82 + 5639076 * q^86 + 1541200 * q^88 - 2169084 * q^89 - 669600 * q^92 + 15503712 * q^94 - 48537936 * q^95 - 1088308 * q^97 + 14983242 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−11.1534	+	1.89807i	−0.985827	+	0.167767i
$2$	−11.1534	+	1.89807i	−0.985827	+	0.167767i
$3$		0			0
$3$		0			0
$4$	120.795	−	42.3397i	0.943708	−	0.330779i
$5$			338.443i			1.21085i	0.795903	+	0.605425i	$0.206997\pi$
$5$			338.443i			1.21085i	−0.795903	+	0.605425i	$0.793003\pi$
$6$		0			0
$7$	−438.996			−0.483746			−0.241873	−	0.970308i	$-0.577762\pi$
$7$	−438.996			−0.483746			−0.241873	+	0.970308i	$0.577762\pi$
$8$	−1266.90	+	701.506i	−0.874839	+	0.484414i
$9$		0			0
$10$	−642.387	−	3774.77i	−0.203141	−	1.19369i
$11$			1966.58i			0.445489i	0.974877	+	0.222744i	$0.0715015\pi$
$11$			1966.58i			0.445489i	−0.974877	+	0.222744i	$0.928498\pi$
$12$		0			0
$13$			2210.98i			0.279115i	0.990214	+	0.139557i	$0.0445680\pi$
$13$			2210.98i			0.279115i	−0.990214	+	0.139557i	$0.955432\pi$
$14$	4896.28	−	833.245i	0.476890	−	0.0811568i
$15$		0			0
$16$	12798.7	−	10228.8i	0.781171	−	0.624317i
$17$	12114.9			0.598064			0.299032	−	0.954243i	$-0.403336\pi$
$17$	12114.9			0.598064			0.299032	+	0.954243i	$0.403336\pi$
$18$		0			0
$19$			32872.2i			1.09949i	0.835333	+	0.549744i	$0.185275\pi$
$19$			32872.2i			1.09949i	−0.835333	+	0.549744i	$0.814725\pi$
$20$	14329.5	+	40882.1i	0.400523	+	1.14269i
$21$		0			0
$22$	−3732.70	−	21933.9i	−0.0747384	−	0.439175i
$23$	−19605.1			−0.335986			−0.167993	−	0.985788i	$-0.553729\pi$
$23$	−19605.1			−0.335986			−0.167993	+	0.985788i	$0.553729\pi$
$24$		0			0
$25$	−36418.4			−0.466155
$26$	−4196.59	−	24659.8i	−0.0468263	−	0.275159i
$27$		0			0
$28$	−53028.4	+	18587.0i	−0.456515	+	0.160013i
$29$		−	160689.i		−	1.22347i	−0.791063	−	0.611735i	$-0.790472\pi$
$29$		−	160689.i		−	1.22347i	0.791063	−	0.611735i	$-0.209528\pi$
$30$		0			0
$31$	−229270.			−1.38224			−0.691118	−	0.722742i	$-0.742881\pi$
$31$	−229270.			−1.38224			−0.691118	+	0.722742i	$0.742881\pi$
$32$	−123333.	+	138378.i	−0.665359	+	0.746523i
$33$		0			0
$34$	−135122.	+	22994.9i	−0.589588	+	0.100336i
$35$		−	148575.i		−	0.585744i
$36$		0			0
$37$		−	496284.i		−	1.61074i	−0.592775	−	0.805368i	$-0.701968\pi$
$37$		−	496284.i		−	1.61074i	0.592775	−	0.805368i	$-0.298032\pi$
$38$	−62393.6	−	366635.i	−0.184458	−	1.08391i
$39$		0			0
$40$	−237420.	−	428774.i	−0.586552	−	1.05930i
$41$	−599971.			−1.35952			−0.679762	−	0.733433i	$-0.737917\pi$
$41$	−599971.			−1.35952			−0.679762	+	0.733433i	$0.737917\pi$
$42$		0			0
$43$			88346.0i			0.169452i	0.996404	+	0.0847262i	$0.0270015\pi$
$43$			88346.0i			0.169452i	−0.996404	+	0.0847262i	$0.972998\pi$
$44$	83264.3	+	237552.i	0.147358	+	0.420412i
$45$		0			0
$46$	218662.	−	37211.8i	0.331224	−	0.0563674i
$47$	−820344.			−1.15253			−0.576267	−	0.817262i	$-0.695491\pi$
$47$	−820344.			−1.15253			−0.576267	+	0.817262i	$0.695491\pi$
$48$		0			0
$49$	−630825.			−0.765989
$50$	406187.	−	69124.6i	0.459548	−	0.0782056i
$51$		0			0
$52$	93612.2	+	267075.i	0.0923253	+	0.263403i
$53$			1.53717e6i			1.41826i	0.705077	+	0.709131i	$0.250913\pi$
$53$			1.53717e6i			1.41826i	−0.705077	+	0.709131i	$0.749087\pi$
$54$		0			0
$55$	−665574.			−0.539420
$56$	556165.	−	307959.i	0.423200	−	0.234333i
$57$		0			0
$58$	304999.	+	1.79222e6i	0.205258	+	1.20613i
$59$		−	1.82480e6i		−	1.15673i	−0.815776	−	0.578367i	$-0.803690\pi$
$59$		−	1.82480e6i		−	1.15673i	0.815776	−	0.578367i	$-0.196310\pi$
$60$		0			0
$61$		−	484582.i		−	0.273346i	−0.990616	−	0.136673i	$-0.956359\pi$
$61$		−	484582.i		−	0.273346i	0.990616	−	0.136673i	$-0.0436410\pi$
$62$	2.55713e6	−	435171.i	1.36264	−	0.231894i
$63$		0			0
$64$	1.11293e6	−	1.77748e6i	0.530687	−	0.847568i
$65$	−748290.			−0.337966
$66$		0			0
$67$		−	79878.2i		−	0.0324464i	−0.999868	−	0.0162232i	$-0.994836\pi$
$67$		−	79878.2i		−	0.0324464i	0.999868	−	0.0162232i	$-0.00516423\pi$
$68$	1.46341e6	−	512940.i	0.564398	−	0.197827i
$69$		0			0
$70$	282006.	+	1.65711e6i	0.0982686	+	0.577442i
$71$	−1.27078e6			−0.421373			−0.210686	−	0.977554i	$-0.567570\pi$
$71$	−1.27078e6			−0.421373			−0.210686	+	0.977554i	$0.567570\pi$
$72$		0			0
$73$	3.70820e6			1.11566			0.557832	−	0.829954i	$-0.311633\pi$
$73$	3.70820e6			1.11566			0.557832	+	0.829954i	$0.311633\pi$
$74$	941981.	+	5.53523e6i	0.270229	+	1.58791i
$75$		0			0
$76$	1.39180e6	+	3.97078e6i	0.363687	+	1.03760i
$77$		−	863321.i		−	0.215504i
$78$		0			0
$79$	−2.55846e6			−0.583827			−0.291914	−	0.956445i	$-0.594292\pi$
$79$	−2.55846e6			−0.583827			−0.291914	+	0.956445i	$0.594292\pi$
$80$	3.46187e6	+	4.33163e6i	0.755954	+	0.945880i
$81$		0			0
$82$	6.69169e6	−	1.13879e6i	1.34025	−	0.228083i
$83$		−	1.53414e6i		−	0.294505i	−0.989099	−	0.147252i	$-0.952957\pi$
$83$		−	1.53414e6i		−	0.294505i	0.989099	−	0.147252i	$-0.0470430\pi$
$84$		0			0
$85$			4.10019e6i			0.724166i
$86$	−167687.	−	985355.i	−0.0284285	−	0.167051i
$87$		0			0
$88$	−1.37957e6	−	2.49146e6i	−0.215801	−	0.389731i
$89$	−1.99492e6			−0.299958			−0.149979	−	0.988689i	$-0.547921\pi$
$89$	−1.99492e6			−0.299958			−0.149979	+	0.988689i	$0.547921\pi$
$90$		0			0
$91$		−	970612.i		−	0.135021i
$92$	−2.36819e6	+	830073.i	−0.317073	+	0.111137i
$93$		0			0
$94$	9.14959e6	−	1.55707e6i	1.13620	−	0.193357i
$95$	−1.11253e7			−1.33131
$96$		0			0
$97$	−28917.7			−0.00321708			−0.00160854	−	0.999999i	$-0.500512\pi$
$97$	−28917.7			−0.00321708			−0.00160854	+	0.999999i	$0.500512\pi$
$98$	7.03582e6	−	1.19735e6i	0.755133	−	0.128508i
$99$		0			0
$100$	−4.39915e6	+	1.54194e6i	−0.439915	+	0.154194i
$101$		−	1.68077e7i		−	1.62324i	−0.584182	−	0.811622i	$-0.698585\pi$
$101$		−	1.68077e7i		−	1.62324i	0.584182	−	0.811622i	$-0.301415\pi$
$102$		0			0
$103$	−1.27746e7			−1.15191			−0.575953	−	0.817483i	$-0.695369\pi$
$103$	−1.27746e7			−1.15191			−0.575953	+	0.817483i	$0.695369\pi$
$104$	−1.55102e6	−	2.80110e6i	−0.135207	−	0.244181i
$105$		0			0
$106$	−2.91766e6	−	1.71446e7i	−0.237938	−	1.39816i
$107$			1.35610e7i			1.07016i	0.844803	+	0.535078i	$0.179718\pi$
$107$			1.35610e7i			1.07016i	−0.844803	+	0.535078i	$0.820282\pi$
$108$		0			0
$109$			4.74206e6i			0.350731i	0.984503	+	0.175366i	$0.0561108\pi$
$109$			4.74206e6i			0.350731i	−0.984503	+	0.175366i	$0.943889\pi$
$110$	7.42338e6	−	1.26331e6i	0.531774	−	0.0904969i
$111$		0			0
$112$	−5.61858e6	+	4.49041e6i	−0.377889	+	0.302011i
$113$	8.06832e6			0.526028			0.263014	−	0.964792i	$-0.415283\pi$
$113$	8.06832e6			0.526028			0.263014	+	0.964792i	$0.415283\pi$
$114$		0			0
$115$		−	6.63520e6i		−	0.406828i
$116$	−6.80352e6	−	1.94104e7i	−0.404698	−	1.15460i
$117$		0			0
$118$	3.46360e6	+	2.03527e7i	0.194062	+	1.14034i
$119$	−5.31839e6			−0.289311
$120$		0			0
$121$	1.56197e7			0.801540
$122$	919770.	+	5.40471e6i	0.0458585	+	0.269472i
$123$		0			0
$124$	−2.76946e7	+	9.70723e6i	−1.30443	+	0.457214i
$125$			1.41153e7i			0.646405i
$126$		0			0
$127$	−1.12410e7			−0.486960			−0.243480	−	0.969906i	$-0.578289\pi$
$127$	−1.12410e7			−0.486960			−0.243480	+	0.969906i	$0.578289\pi$
$128$	−9.03913e6	+	2.19373e7i	−0.380971	+	0.924587i
$129$		0			0
$130$	8.34594e6	−	1.42031e6i	0.333176	−	0.0566996i
$131$			8.81527e6i			0.342599i	0.985219	+	0.171299i	$0.0547966\pi$
$131$			8.81527e6i			0.342599i	−0.985219	+	0.171299i	$0.945203\pi$
$132$		0			0
$133$		−	1.44308e7i		−	0.531874i
$134$	151614.	+	890910.i	0.00544344	+	0.0319865i
$135$		0			0
$136$	−1.53484e7	+	8.49866e6i	−0.523210	+	0.289711i
$137$	3.33729e7			1.10885			0.554424	−	0.832234i	$-0.312939\pi$
$137$	3.33729e7			1.10885			0.554424	+	0.832234i	$0.312939\pi$
$138$		0			0
$139$		−	4.68161e7i		−	1.47857i	−0.673391	−	0.739287i	$-0.735163\pi$
$139$		−	4.68161e7i		−	1.47857i	0.673391	−	0.739287i	$-0.264837\pi$
$140$	−6.29062e6	−	1.79471e7i	−0.193752	−	0.552771i
$141$		0			0
$142$	1.41735e7	−	2.41203e6i	0.415401	−	0.0706926i
$143$	−4.34806e6			−0.124343
$144$		0			0
$145$	5.43840e7			1.48144
$146$	−4.13589e7	+	7.03843e6i	−1.09985	+	0.187172i
$147$		0			0
$148$	−2.10125e7	−	5.99485e7i	−0.532797	−	1.52006i
$149$			3.83709e7i			0.950277i	0.879911	+	0.475139i	$0.157602\pi$
$149$			3.83709e7i			0.950277i	−0.879911	+	0.475139i	$0.842398\pi$
$150$		0			0
$151$	−7.17648e7			−1.69626			−0.848130	−	0.529788i	$-0.822271\pi$
$151$	−7.17648e7			−1.69626			−0.848130	+	0.529788i	$0.822271\pi$
$152$	−2.30600e7	−	4.16458e7i	−0.532607	−	0.961875i
$153$		0			0
$154$	1.63864e6	+	9.62892e6i	0.0361544	+	0.212449i
$155$		−	7.75948e7i		−	1.67368i
$156$		0			0
$157$			4.03778e7i			0.832710i	0.909202	+	0.416355i	$0.136693\pi$
$157$			4.03778e7i			0.832710i	−0.909202	+	0.416355i	$0.863307\pi$
$158$	2.85355e7	−	4.85614e6i	0.575553	−	0.0979471i
$159$		0			0
$160$	−4.68332e7	−	4.17413e7i	−0.903927	−	0.805649i
$161$	8.60656e6			0.162532
$162$		0			0
$163$			9.84512e7i			1.78059i	0.455383	+	0.890296i	$0.349502\pi$
$163$			9.84512e7i			1.78059i	−0.455383	+	0.890296i	$0.650498\pi$
$164$	−7.24733e7	+	2.54026e7i	−1.28299	+	0.449702i
$165$		0			0
$166$	2.91191e6	+	1.71108e7i	0.0494083	+	0.290331i
$167$	4.24811e7			0.705810			0.352905	−	0.935659i	$-0.385194\pi$
$167$	4.24811e7			0.705810			0.352905	+	0.935659i	$0.385194\pi$
$168$		0			0
$169$	5.78601e7			0.922095
$170$	−7.78245e6	−	4.57309e7i	−0.121491	−	0.713902i
$171$		0			0
$172$	3.74054e6	+	1.06717e7i	0.0560512	+	0.159914i
$173$			2.65257e7i			0.389498i	0.980853	+	0.194749i	$0.0623893\pi$
$173$			2.65257e7i			0.389498i	−0.980853	+	0.194749i	$0.937611\pi$
$174$		0			0
$175$	1.59875e7			0.225501
$176$	2.01158e7	+	2.51697e7i	0.278126	+	0.348003i
$177$		0			0
$178$	2.22501e7	−	3.78650e6i	0.295707	−	0.0503231i
$179$			2.42148e7i			0.315570i	0.987474	+	0.157785i	$0.0504353\pi$
$179$			2.42148e7i			0.315570i	−0.987474	+	0.157785i	$0.949565\pi$
$180$		0			0
$181$			2.78961e7i			0.349679i	0.984597	+	0.174839i	$0.0559406\pi$
$181$			2.78961e7i			0.349679i	−0.984597	+	0.174839i	$0.944059\pi$
$182$	1.84229e6	+	1.08256e7i	0.0226521	+	0.133107i
$183$		0			0
$184$	2.48377e7	−	1.37531e7i	0.293934	−	0.162756i
$185$	1.67964e8			1.95036
$186$		0			0
$187$			2.38249e7i			0.266431i
$188$	−9.90932e7	+	3.47331e7i	−1.08766	+	0.381234i
$189$		0			0
$190$	1.24085e8	−	2.11167e7i	1.31245	−	0.223351i
$191$	−1.67596e7			−0.174039			−0.0870195	−	0.996207i	$-0.527734\pi$
$191$	−1.67596e7			−0.174039			−0.0870195	+	0.996207i	$0.527734\pi$
$192$		0			0
$193$	8.75008e7			0.876116			0.438058	−	0.898947i	$-0.355667\pi$
$193$	8.75008e7			0.876116			0.438058	+	0.898947i	$0.355667\pi$
$194$	322529.	−	54887.7i	0.00317148	−	0.000539721i
$195$		0			0
$196$	−7.62003e7	+	2.67089e7i	−0.722871	+	0.253373i
$197$		−	2.56239e7i		−	0.238789i	−0.992847	−	0.119394i	$-0.961905\pi$
$197$		−	2.56239e7i		−	0.238789i	0.992847	−	0.119394i	$-0.0380953\pi$
$198$		0			0
$199$	5.31884e7			0.478444			0.239222	−	0.970965i	$-0.423108\pi$
$199$	5.31884e7			0.478444			0.239222	+	0.970965i	$0.423108\pi$
$200$	4.61385e7	−	2.55477e7i	0.407811	−	0.225812i
$201$		0			0
$202$	3.19022e7	+	1.87462e8i	0.272327	+	1.60024i
$203$			7.05419e7i			0.591849i
$204$		0			0
$205$		−	2.03056e8i		−	1.64618i
$206$	1.42480e8	−	2.42471e7i	1.13558	−	0.193252i
$207$		0			0
$208$	2.26157e7	+	2.82977e7i	0.174256	+	0.218036i
$209$	−6.46457e7			−0.489810
$210$		0			0
$211$		−	2.01165e7i		−	0.147423i	−0.997280	−	0.0737114i	$-0.976516\pi$
$211$		−	2.01165e7i		−	0.147423i	0.997280	−	0.0737114i	$-0.0234844\pi$
$212$	6.50833e7	+	1.85682e8i	0.469131	+	1.33843i
$213$		0			0
$214$	−2.57396e7	−	1.51250e8i	−0.179537	−	1.05499i
$215$	−2.99001e7			−0.205181
$216$		0			0
$217$	1.00649e8			0.668651
$218$	−9.00076e6	−	5.28899e7i	−0.0588412	−	0.345760i
$219$		0			0
$220$	−8.03978e7	+	2.81802e7i	−0.509055	+	0.178429i
$221$			2.67858e7i			0.166929i
$222$		0			0
$223$	−1.67012e8			−1.00851			−0.504254	−	0.863555i	$-0.668232\pi$
$223$	−1.67012e8			−1.00851			−0.504254	+	0.863555i	$0.668232\pi$
$224$	5.41429e7	−	6.07476e7i	0.321865	−	0.361128i
$225$		0			0
$226$	−8.99889e7	+	1.53142e7i	−0.518572	+	0.0882502i
$227$			1.37308e8i			0.779122i	0.921001	+	0.389561i	$0.127373\pi$
$227$			1.37308e8i			0.779122i	−0.921001	+	0.389561i	$0.872627\pi$
$228$		0			0
$229$			2.67935e8i			1.47436i	0.675694	+	0.737182i	$0.263844\pi$
$229$			2.67935e8i			1.47436i	−0.675694	+	0.737182i	$0.736156\pi$
$230$	1.25941e7	+	7.40047e7i	0.0682525	+	0.401062i
$231$		0			0
$232$	1.12724e8	+	2.03577e8i	0.592665	+	1.07034i
$233$	−1.98032e8			−1.02563			−0.512815	−	0.858499i	$-0.671397\pi$
$233$	−1.98032e8			−1.02563			−0.512815	+	0.858499i	$0.671397\pi$
$234$		0			0
$235$		−	2.77639e8i		−	1.39554i
$236$	−7.72615e7	−	2.20426e8i	−0.382623	−	1.09162i
$237$		0			0
$238$	5.93179e7	−	1.00947e7i	0.285211	−	0.0485370i
$239$	−8.22277e7			−0.389606			−0.194803	−	0.980842i	$-0.562407\pi$
$239$	−8.22277e7			−0.389606			−0.194803	+	0.980842i	$0.562407\pi$
$240$		0			0
$241$	−2.70650e8			−1.24551			−0.622757	−	0.782415i	$-0.713988\pi$
$241$	−2.70650e8			−1.24551			−0.622757	+	0.782415i	$0.713988\pi$
$242$	−1.74213e8	+	2.96473e7i	−0.790179	+	0.134472i
$243$		0			0
$244$	−2.05170e7	−	5.85349e7i	−0.0904171	−	0.257959i
$245$		−	2.13498e8i		−	0.927498i
$246$		0			0
$247$	−7.26797e7			−0.306884
$248$	2.90463e8	−	1.60835e8i	1.20923	−	0.669574i
$249$		0			0
$250$	−2.67918e7	−	1.57433e8i	−0.108446	−	0.637243i
$251$		−	2.90747e8i		−	1.16053i	−0.814427	−	0.580266i	$-0.802949\pi$
$251$		−	2.90747e8i		−	1.16053i	0.814427	−	0.580266i	$-0.197051\pi$
$252$		0			0
$253$		−	3.85549e7i		−	0.149678i
$254$	1.25375e8	−	2.13363e7i	0.480058	−	0.0816959i
$255$		0			0
$256$	5.91782e7	−	2.61831e8i	0.220456	−	0.975397i
$257$	−4.36047e7			−0.160239			−0.0801193	−	0.996785i	$-0.525530\pi$
$257$	−4.36047e7			−0.160239			−0.0801193	+	0.996785i	$0.525530\pi$
$258$		0			0
$259$			2.17867e8i			0.779188i
$260$	−9.03894e7	+	3.16823e7i	−0.318941	+	0.111792i
$261$		0			0
$262$	−1.67320e7	−	9.83198e7i	−0.0574769	−	0.337743i
$263$	−4.27678e8			−1.44968			−0.724840	−	0.688917i	$-0.758086\pi$
$263$	−4.27678e8			−1.44968			−0.724840	+	0.688917i	$0.758086\pi$
$264$		0			0
$265$	−5.20244e8			−1.71730
$266$	2.73906e7	+	1.60951e8i	0.0892309	+	0.524335i
$267$		0			0
$268$	−3.38202e6	−	9.64886e6i	−0.0107326	−	0.0306199i
$269$		−	8.26134e7i		−	0.258772i	−0.991594	−	0.129386i	$-0.958699\pi$
$269$		−	8.26134e7i		−	0.258772i	0.991594	−	0.129386i	$-0.0413007\pi$
$270$		0			0
$271$	6.15189e7			0.187766			0.0938829	−	0.995583i	$-0.470072\pi$
$271$	6.15189e7			0.187766			0.0938829	+	0.995583i	$0.470072\pi$
$272$	1.55055e8	−	1.23921e8i	0.467190	−	0.373382i
$273$		0			0
$274$	−3.72220e8	+	6.33441e7i	−1.09313	+	0.186028i
$275$		−	7.16196e7i		−	0.207667i
$276$		0			0
$277$		−	4.39237e7i		−	0.124171i	−0.998071	−	0.0620854i	$-0.980225\pi$
$277$		−	4.39237e7i		−	0.124171i	0.998071	−	0.0620854i	$-0.0197751\pi$
$278$	8.88601e7	+	5.22156e8i	0.248056	+	1.45762i
$279$		0			0
$280$	1.04226e8	+	1.88230e8i	0.283742	+	0.512432i
$281$	−5.80931e8			−1.56190			−0.780948	−	0.624596i	$-0.785264\pi$
$281$	−5.80931e8			−1.56190			−0.780948	+	0.624596i	$0.785264\pi$
$282$		0			0
$283$		−	6.03790e8i		−	1.58356i	−0.610810	−	0.791778i	$-0.709156\pi$
$283$		−	6.03790e8i		−	1.58356i	0.610810	−	0.791778i	$-0.290844\pi$
$284$	−1.53504e8	+	5.38045e7i	−0.397653	+	0.139381i
$285$		0			0
$286$	4.84955e7	−	8.25293e6i	0.122580	−	0.0208606i
$287$	2.63385e8			0.657665
$288$		0			0
$289$	−2.63568e8			−0.642319
$290$	−6.06564e8	+	1.03225e8i	−1.46044	+	0.248536i
$291$		0			0
$292$	4.47931e8	−	1.57004e8i	1.05286	−	0.369038i
$293$		−	1.10504e8i		−	0.256649i	−0.991732	−	0.128325i	$-0.959040\pi$
$293$		−	1.10504e8i		−	0.256649i	0.991732	−	0.128325i	$-0.0409600\pi$
$294$		0			0
$295$	6.17591e8			1.40063
$296$	3.48146e8	+	6.28743e8i	0.780263	+	1.40913i
$297$		0			0
$298$	−7.28307e7	−	4.27965e8i	−0.159425	−	0.936809i
$299$		−	4.33464e7i		−	0.0937787i
$300$		0			0
$301$		−	3.87836e7i		−	0.0819719i
$302$	8.00419e8	−	1.36215e8i	1.67222	−	0.284577i
$303$		0			0
$304$	3.36243e8	+	4.20721e8i	0.686430	+	0.858888i
$305$	1.64003e8			0.330981
$306$		0			0
$307$			7.03386e8i			1.38742i	0.720252	+	0.693712i	$0.244026\pi$
$307$			7.03386e8i			1.38742i	−0.720252	+	0.693712i	$0.755974\pi$
$308$	−3.65527e7	−	1.04285e8i	−0.0712840	−	0.203373i
$309$		0			0
$310$	1.47280e8	+	8.65443e8i	0.280788	+	1.64996i
$311$	8.61240e8			1.62354			0.811769	−	0.583978i	$-0.198505\pi$
$311$	8.61240e8			1.62354			0.811769	+	0.583978i	$0.198505\pi$
$312$		0			0
$313$	2.42056e8			0.446181			0.223090	−	0.974798i	$-0.428385\pi$
$313$	2.42056e8			0.446181			0.223090	+	0.974798i	$0.428385\pi$
$314$	−7.66398e7	−	4.50348e8i	−0.139701	−	0.820908i
$315$		0			0
$316$	−3.09049e8	+	1.08325e8i	−0.550963	+	0.193118i
$317$			5.38362e8i			0.949221i	0.880196	+	0.474610i	$0.157411\pi$
$317$			5.38362e8i			0.949221i	−0.880196	+	0.474610i	$0.842589\pi$
$318$		0			0
$319$	3.16007e8			0.545042
$320$	6.01575e8	+	3.76663e8i	1.02628	+	0.642581i
$321$		0			0
$322$	−9.59920e7	+	1.63358e7i	−0.160228	+	0.0272675i
$323$			3.98242e8i			0.657565i
$324$		0			0
$325$		−	8.05203e7i		−	0.130111i
$326$	−1.86867e8	−	1.09806e9i	−0.298725	−	1.75535i
$327$		0			0
$328$	7.60105e8	−	4.20883e8i	1.18936	−	0.658572i
$329$	3.60128e8			0.557534
$330$		0			0
$331$			1.05054e9i			1.59226i	0.605124	+	0.796131i	$0.293123\pi$
$331$			1.05054e9i			1.59226i	−0.605124	+	0.796131i	$0.706877\pi$
$332$	−6.49551e7	−	1.85316e8i	−0.0974160	−	0.277927i
$333$		0			0
$334$	−4.73806e8	+	8.06320e7i	−0.695806	+	0.118412i
$335$	2.70342e7			0.0392877
$336$		0			0
$337$	−2.04579e8			−0.291177			−0.145589	−	0.989345i	$-0.546508\pi$
$337$	−2.04579e8			−0.291177			−0.145589	+	0.989345i	$0.546508\pi$
$338$	−6.45334e8	+	1.09822e8i	−0.909026	+	0.154697i
$339$		0			0
$340$	1.73601e8	+	4.95281e8i	0.239539	+	0.683401i
$341$		−	4.50878e8i		−	0.615770i
$342$		0			0
$343$	6.38462e8			0.854291
$344$	−6.19753e7	−	1.11926e8i	−0.0820850	−	0.148244i
$345$		0			0
$346$	−5.03476e7	−	2.95851e8i	−0.0653450	−	0.383978i
$347$			6.28852e8i			0.807970i	0.914766	+	0.403985i	$0.132375\pi$
$347$			6.28852e8i			0.807970i	−0.914766	+	0.403985i	$0.867625\pi$
$348$		0			0
$349$		−	9.86717e8i		−	1.24252i	−0.783604	−	0.621260i	$-0.786621\pi$
$349$		−	9.86717e8i		−	1.24252i	0.783604	−	0.621260i	$-0.213379\pi$
$350$	−1.78315e8	+	3.03455e7i	−0.222305	+	0.0378317i
$351$		0			0
$352$	−2.72132e8	−	2.42545e8i	−0.332568	−	0.296410i
$353$	−5.59732e8			−0.677281			−0.338641	−	0.940916i	$-0.609967\pi$
$353$	−5.59732e8			−0.677281			−0.338641	+	0.940916i	$0.609967\pi$
$354$		0			0
$355$		−	4.30087e8i		−	0.510219i
$356$	−2.40976e8	+	8.44643e7i	−0.283073	+	0.0992198i
$357$		0			0
$358$	−4.59614e7	−	2.70076e8i	−0.0529423	−	0.311097i
$359$	−1.42390e8			−0.162424			−0.0812119	−	0.996697i	$-0.525879\pi$
$359$	−1.42390e8			−0.162424			−0.0812119	+	0.996697i	$0.525879\pi$
$360$		0			0
$361$	−1.86707e8			−0.208875
$362$	−5.29488e7	−	3.11136e8i	−0.0586646	−	0.344723i
$363$		0			0
$364$	−4.10954e7	−	1.17245e8i	−0.0446620	−	0.127420i
$365$			1.25501e9i			1.35090i
$366$		0			0
$367$	7.13452e8			0.753414			0.376707	−	0.926333i	$-0.377056\pi$
$367$	7.13452e8			0.753414			0.376707	+	0.926333i	$0.377056\pi$
$368$	−2.50920e8	+	2.00537e8i	−0.262463	+	0.209762i
$369$		0			0
$370$	−1.87336e9	+	3.18807e8i	−1.92271	+	0.327206i
$371$		−	6.74812e8i		−	0.686079i
$372$		0			0
$373$			4.14729e8i			0.413794i	0.978363	+	0.206897i	$0.0663364\pi$
$373$			4.14729e8i			0.413794i	−0.978363	+	0.206897i	$0.933664\pi$
$374$	−4.52212e7	−	2.65727e8i	−0.0446984	−	0.262655i
$375$		0			0
$376$	1.03930e9	−	5.75476e8i	1.00828	−	0.558303i
$377$	3.55280e8			0.341489
$378$		0			0
$379$		−	1.23625e8i		−	0.116646i	−0.998298	−	0.0583229i	$-0.981425\pi$
$379$		−	1.23625e8i		−	0.116646i	0.998298	−	0.0583229i	$-0.0185753\pi$
$380$	−1.34388e9	+	4.71043e8i	−1.25637	+	0.440371i
$381$		0			0
$382$	1.86926e8	−	3.18109e7i	0.171572	−	0.0291980i
$383$	1.35784e9			1.23496			0.617480	−	0.786586i	$-0.288154\pi$
$383$	1.35784e9			1.23496			0.617480	+	0.786586i	$0.288154\pi$
$384$		0			0
$385$	2.92184e8			0.260942
$386$	−9.75927e8	+	1.66083e8i	−0.863698	+	0.146983i
$387$		0			0
$388$	−3.49310e6	+	1.22436e6i	−0.00303599	+	0.00106414i
$389$		−	1.00573e9i		−	0.866281i	−0.901326	−	0.433141i	$-0.857405\pi$
$389$		−	1.00573e9i		−	0.866281i	0.901326	−	0.433141i	$-0.142595\pi$
$390$		0			0
$391$	−2.37513e8			−0.200941
$392$	7.99194e8	−	4.42528e8i	0.670117	−	0.371056i
$393$		0			0
$394$	4.86359e7	+	2.85793e8i	0.0400609	+	0.235404i
$395$		−	8.65893e8i		−	0.706927i
$396$		0			0
$397$		−	2.30080e8i		−	0.184549i	−0.995734	−	0.0922747i	$-0.970586\pi$
$397$		−	2.30080e8i		−	0.184549i	0.995734	−	0.0922747i	$-0.0294138\pi$
$398$	−5.93229e8	+	1.00955e8i	−0.471663	+	0.0802672i
$399$		0			0
$400$	−4.66108e8	+	3.72517e8i	−0.364147	+	0.291029i
$401$	1.24791e9			0.966446			0.483223	−	0.875497i	$-0.339466\pi$
$401$	1.24791e9			0.966446			0.483223	+	0.875497i	$0.339466\pi$
$402$		0			0
$403$		−	5.06912e8i		−	0.385802i
$404$	−7.11633e8	−	2.03028e9i	−0.536935	−	1.53187i
$405$		0			0
$406$	−1.33893e8	−	7.86778e8i	−0.0992928	−	0.583460i
$407$	9.75982e8			0.717565
$408$		0			0
$409$	−1.79923e9			−1.30033			−0.650166	−	0.759792i	$-0.725301\pi$
$409$	−1.79923e9			−1.30033			−0.650166	+	0.759792i	$0.725301\pi$
$410$	3.85414e8	+	2.26475e9i	0.276175	+	1.62285i
$411$		0			0
$412$	−1.54310e9	+	5.40873e8i	−1.08706	+	0.381026i
$413$			8.01081e8i			0.559566i
$414$		0			0
$415$	5.19219e8			0.356601
$416$	−3.05952e8	−	2.72688e8i	−0.208366	−	0.185712i
$417$		0			0
$418$	7.21016e8	−	1.22702e8i	0.482868	−	0.0821740i
$419$			2.66870e8i			0.177235i	0.996066	+	0.0886177i	$0.0282449\pi$
$419$			2.66870e8i			0.177235i	−0.996066	+	0.0886177i	$0.971755\pi$
$420$		0			0
$421$			2.70575e9i			1.76726i	0.468185	+	0.883630i	$0.344908\pi$
$421$			2.70575e9i			1.76726i	−0.468185	+	0.883630i	$0.655092\pi$
$422$	3.81826e7	+	2.24367e8i	0.0247327	+	0.145333i
$423$		0			0
$424$	−1.07833e9	−	1.94745e9i	−0.687026	−	1.24075i
$425$	−4.41204e8			−0.278791
$426$		0			0
$427$			2.12730e8i			0.132230i
$428$	5.74167e8	+	1.63809e9i	0.353985	+	1.00992i
$429$		0			0
$430$	3.33486e8	−	5.67524e7i	0.202273	−	0.0344227i
$431$	−1.78455e9			−1.07364			−0.536820	−	0.843697i	$-0.680374\pi$
$431$	−1.78455e9			−1.07364			−0.536820	+	0.843697i	$0.680374\pi$
$432$		0			0
$433$	1.21276e9			0.717905			0.358953	−	0.933356i	$-0.383134\pi$
$433$	1.21276e9			0.717905			0.358953	+	0.933356i	$0.383134\pi$
$434$	−1.12257e9	+	1.91038e8i	−0.659174	+	0.112178i
$435$		0			0
$436$	2.00777e8	+	5.72816e8i	0.116014	+	0.330988i
$437$		−	6.44461e8i		−	0.369413i
$438$		0			0
$439$	−1.76141e9			−0.993654			−0.496827	−	0.867850i	$-0.665502\pi$
$439$	−1.76141e9			−0.993654			−0.496827	+	0.867850i	$0.665502\pi$
$440$	8.43217e8	−	4.66904e8i	0.471905	−	0.261302i
$441$		0			0
$442$	−5.08412e7	−	2.98751e8i	−0.0280052	−	0.164563i
$443$			8.06208e8i			0.440590i	0.975433	+	0.220295i	$0.0707019\pi$
$443$			8.06208e8i			0.440590i	−0.975433	+	0.220295i	$0.929298\pi$
$444$		0			0
$445$		−	6.75166e8i		−	0.363204i
$446$	1.86274e9	−	3.16999e8i	0.994214	−	0.169195i
$447$		0			0
$448$	−4.88572e8	+	7.80307e8i	−0.256718	+	0.410008i
$449$	−5.85913e8			−0.305472			−0.152736	−	0.988267i	$-0.548808\pi$
$449$	−5.85913e8			−0.305472			−0.152736	+	0.988267i	$0.548808\pi$
$450$		0			0
$451$		−	1.17989e9i		−	0.605653i
$452$	9.74611e8	−	3.41610e8i	0.496417	−	0.173999i
$453$		0			0
$454$	−2.60620e8	−	1.53145e9i	−0.130711	−	0.768079i
$455$	3.28496e8			0.163490
$456$		0			0
$457$	5.52640e8			0.270854			0.135427	−	0.990787i	$-0.456759\pi$
$457$	5.52640e8			0.270854			0.135427	+	0.990787i	$0.456759\pi$
$458$	−5.08559e8	−	2.98837e9i	−0.247350	−	1.45347i
$459$		0			0
$460$	−2.80932e8	−	8.01496e8i	−0.134570	−	0.383927i
$461$		−	1.74101e9i		−	0.827652i	−0.910356	−	0.413826i	$-0.864192\pi$
$461$		−	1.74101e9i		−	0.827652i	0.910356	−	0.413826i	$-0.135808\pi$
$462$		0			0
$463$	−2.84431e9			−1.33181			−0.665906	−	0.746035i	$-0.731955\pi$
$463$	−2.84431e9			−1.33181			−0.665906	+	0.746035i	$0.731955\pi$
$464$	−1.64366e9	−	2.05661e9i	−0.763833	−	0.955739i
$465$		0			0
$466$	2.20873e9	−	3.75879e8i	1.01109	−	0.172067i
$467$			1.63130e9i			0.741183i	0.928796	+	0.370592i	$0.120845\pi$
$467$			1.63130e9i			0.741183i	−0.928796	+	0.370592i	$0.879155\pi$
$468$		0			0
$469$			3.50662e7i			0.0156958i
$470$	5.26979e8	+	3.09661e9i	0.234127	+	1.37576i
$471$		0			0
$472$	1.28011e9	+	2.31185e9i	0.560338	+	1.01196i
$473$	−1.73739e8			−0.0754891
$474$		0			0
$475$		−	1.19715e9i		−	0.512532i
$476$	−6.42433e8	+	2.25179e8i	−0.273026	+	0.0956981i
$477$		0			0
$478$	9.17114e8	−	1.56074e8i	0.384084	−	0.0653630i
$479$	−3.69345e9			−1.53553			−0.767765	−	0.640732i	$-0.778631\pi$
$479$	−3.69345e9			−1.53553			−0.767765	+	0.640732i	$0.778631\pi$
$480$		0			0
$481$	1.09727e9			0.449580
$482$	3.01866e9	−	5.13713e8i	1.22786	−	0.208956i
$483$		0			0
$484$	1.88678e9	−	6.61335e8i	0.756420	−	0.265132i
$485$		−	9.78697e6i		−	0.00389540i
$486$		0			0
$487$	1.57153e9			0.616554			0.308277	−	0.951297i	$-0.400248\pi$
$487$	1.57153e9			0.616554			0.308277	+	0.951297i	$0.400248\pi$
$488$	3.39937e8	+	6.13918e8i	0.132413	+	0.239134i
$489$		0			0
$490$	4.05234e8	+	2.38122e9i	0.155604	+	0.914352i
$491$			1.63493e9i			0.623323i	0.950193	+	0.311662i	$0.100886\pi$
$491$			1.63493e9i			0.623323i	−0.950193	+	0.311662i	$0.899114\pi$
$492$		0			0
$493$		−	1.94673e9i		−	0.731713i
$494$	8.10622e8	−	1.37951e8i	0.302534	−	0.0514850i
$495$		0			0
$496$	−2.93436e9	+	2.34516e9i	−1.07976	+	0.862953i
$497$	5.57868e8			0.203838
$498$		0			0
$499$		−	6.86870e8i		−	0.247470i	−0.992315	−	0.123735i	$-0.960513\pi$
$499$		−	6.86870e8i		−	0.247470i	0.992315	−	0.123735i	$-0.0394873\pi$
$500$	5.97637e8	+	1.70505e9i	0.213817	+	0.610018i
$501$		0			0
$502$	5.51858e8	+	3.24280e9i	0.194699	+	1.14408i
$503$	−2.33472e9			−0.817990			−0.408995	−	0.912537i	$-0.634121\pi$
$503$	−2.33472e9			−0.817990			−0.408995	+	0.912537i	$0.634121\pi$
$504$		0			0
$505$	5.68845e9			1.96550
$506$	7.31799e7	+	4.30017e8i	0.0251111	+	0.147557i
$507$		0			0
$508$	−1.35786e9	+	4.75942e8i	−0.459548	+	0.161076i
$509$			6.21342e8i			0.208842i	0.994533	+	0.104421i	$0.0332990\pi$
$509$			6.21342e8i			0.208842i	−0.994533	+	0.104421i	$0.966701\pi$
$510$		0			0
$511$	−1.62789e9			−0.539699
$512$	−1.63062e8	+	3.03262e9i	−0.0536917	+	0.998558i
$513$		0			0
$514$	4.86339e8	−	8.27647e7i	0.157968	−	0.0268828i
$515$		−	4.32347e9i		−	1.39478i
$516$		0			0
$517$		−	1.61327e9i		−	0.513441i
$518$	−4.13526e8	−	2.42995e9i	−0.130722	−	0.768144i
$519$		0			0
$520$	9.48010e8	−	5.24930e8i	0.295666	−	0.163715i
$521$	1.65562e9			0.512897			0.256448	−	0.966558i	$-0.417448\pi$
$521$	1.65562e9			0.512897			0.256448	+	0.966558i	$0.417448\pi$
$522$		0			0
$523$			4.42671e9i			1.35309i	0.736403	+	0.676543i	$0.236523\pi$
$523$			4.42671e9i			1.35309i	−0.736403	+	0.676543i	$0.763477\pi$
$524$	3.73236e8	+	1.06484e9i	0.113324	+	0.323313i
$525$		0			0
$526$	4.77005e9	−	8.11763e8i	1.42913	−	0.243209i
$527$	−2.77758e9			−0.826665
$528$		0			0
$529$	−3.02047e9			−0.887113
$530$	5.80247e9	−	9.87459e8i	1.69296	−	0.288107i
$531$		0			0
$532$	−6.10994e8	−	1.74316e9i	−0.175932	−	0.501933i
$533$		−	1.32652e9i		−	0.379463i
$534$		0			0
$535$	−4.58961e9			−1.29580
$536$	5.60350e7	+	1.01198e8i	0.0157175	+	0.0283854i
$537$		0			0
$538$	1.56806e8	+	9.21417e8i	0.0434135	+	0.255105i
$539$		−	1.24057e9i		−	0.341240i
$540$		0			0
$541$			3.71878e9i			1.00974i	0.863195	+	0.504871i	$0.168460\pi$
$541$			3.71878e9i			1.00974i	−0.863195	+	0.504871i	$0.831540\pi$
$542$	−6.86143e8	+	1.16767e8i	−0.185104	+	0.0315009i
$543$		0			0
$544$	−1.49417e9	+	1.67644e9i	−0.397928	+	0.446469i
$545$	−1.60492e9			−0.424683
$546$		0			0
$547$		−	3.20749e9i		−	0.837934i	−0.908002	−	0.418967i	$-0.862392\pi$
$547$		−	3.20749e9i		−	0.837934i	0.908002	−	0.418967i	$-0.137608\pi$
$548$	4.03127e9	−	1.41300e9i	1.04643	−	0.366783i
$549$		0			0
$550$	1.35939e8	+	7.98799e8i	0.0348397	+	0.204724i
$551$	5.28219e9			1.34519
$552$		0			0
$553$	1.12316e9			0.282424
$554$	8.33702e7	+	4.89897e8i	0.0208318	+	0.122411i
$555$		0			0
$556$	−1.98218e9	−	5.65513e9i	−0.489081	−	1.39534i
$557$			4.80739e9i			1.17873i	0.807865	+	0.589367i	$0.200623\pi$
$557$			4.80739e9i			1.17873i	−0.807865	+	0.589367i	$0.799377\pi$
$558$		0			0
$559$	−1.95331e8			−0.0472967
$560$	−1.51975e9	−	1.90157e9i	−0.365690	−	0.457566i
$561$		0			0
$562$	6.47933e9	−	1.10265e9i	1.53976	−	0.262035i
$563$		−	3.77127e9i		−	0.890653i	−0.895368	−	0.445326i	$-0.853088\pi$
$563$		−	3.77127e9i		−	0.890653i	0.895368	−	0.445326i	$-0.146912\pi$
$564$		0			0
$565$			2.73066e9i			0.636940i
$566$	1.14603e9	+	6.73428e9i	0.265669	+	1.56111i
$567$		0			0
$568$	1.60996e9	−	8.91461e8i	0.368633	−	0.204119i
$569$	−2.09341e9			−0.476388			−0.238194	−	0.971218i	$-0.576555\pi$
$569$	−2.09341e9			−0.476388			−0.238194	+	0.971218i	$0.576555\pi$
$570$		0			0
$571$			6.86085e9i			1.54224i	0.636691	+	0.771119i	$0.280303\pi$
$571$			6.86085e9i			1.54224i	−0.636691	+	0.771119i	$0.719697\pi$
$572$	−5.25223e8	+	1.84096e8i	−0.117343	+	0.0411299i
$573$		0			0
$574$	−2.93763e9	+	4.99923e8i	−0.648343	+	0.110335i
$575$	7.13986e8			0.156622
$576$		0			0
$577$	5.70742e9			1.23687			0.618435	−	0.785836i	$-0.287767\pi$
$577$	5.70742e9			1.23687			0.618435	+	0.785836i	$0.287767\pi$
$578$	2.93967e9	−	5.00271e8i	0.633215	−	0.107760i
$579$		0			0
$580$	6.56930e9	−	2.30260e9i	1.39804	−	0.490028i
$581$			6.73483e8i			0.142466i
$582$		0			0
$583$	−3.02297e9			−0.631820
$584$	−4.69793e9	+	2.60133e9i	−0.976027	+	0.540443i
$585$		0			0
$586$	2.09744e8	+	1.23249e9i	0.0430573	+	0.253012i
$587$			1.56510e9i			0.319380i	0.987167	+	0.159690i	$0.0510495\pi$
$587$			1.56510e9i			0.319380i	−0.987167	+	0.159690i	$0.948951\pi$
$588$		0			0
$589$		−	7.53661e9i		−	1.51975i
$590$	−6.88821e9	+	1.17223e9i	−1.38078	+	0.234980i
$591$		0			0
$592$	−5.07640e9	−	6.35179e9i	−1.00561	−	1.25826i
$593$	2.78410e9			0.548268			0.274134	−	0.961692i	$-0.411609\pi$
$593$	2.78410e9			0.548268			0.274134	+	0.961692i	$0.411609\pi$
$594$		0			0
$595$		−	1.79997e9i		−	0.350312i
$596$	1.62461e9	+	4.63500e9i	0.314331	+	0.896784i
$597$		0			0
$598$	8.22745e7	+	4.83458e8i	0.0157330	+	0.0924496i
$599$	−6.18885e9			−1.17657			−0.588283	−	0.808655i	$-0.700196\pi$
$599$	−6.18885e9			−1.17657			−0.588283	+	0.808655i	$0.700196\pi$
$600$		0			0
$601$	−2.42206e9			−0.455118			−0.227559	−	0.973764i	$-0.573074\pi$
$601$	−2.42206e9			−0.455118			−0.227559	+	0.973764i	$0.573074\pi$
$602$	7.36139e7	+	4.32567e8i	0.0137522	+	0.0808101i
$603$		0			0
$604$	−8.66881e9	+	3.03850e9i	−1.60077	+	0.561087i
$605$			5.28639e9i			0.970544i
$606$		0			0
$607$	−4.62465e9			−0.839302			−0.419651	−	0.907686i	$-0.637848\pi$
$607$	−4.62465e9			−0.839302			−0.419651	+	0.907686i	$0.637848\pi$
$608$	−4.54880e9	−	4.05424e9i	−0.820794	−	0.731555i
$609$		0			0
$610$	−1.82919e9	+	3.11289e8i	−0.326290	+	0.0555277i
$611$		−	1.81376e9i		−	0.321689i
$612$		0			0
$613$			5.45433e9i			0.956378i	0.878257	+	0.478189i	$0.158707\pi$
$613$			5.45433e9i			0.956378i	−0.878257	+	0.478189i	$0.841293\pi$
$614$	−1.33508e9	−	7.84511e9i	−0.232764	−	1.36776i
$615$		0			0
$616$	6.05625e8	+	1.09374e9i	0.104393	+	0.188531i
$617$	2.32837e9			0.399074			0.199537	−	0.979890i	$-0.436056\pi$
$617$	2.32837e9			0.399074			0.199537	+	0.979890i	$0.436056\pi$
$618$		0			0
$619$		−	9.58626e9i		−	1.62455i	−0.583278	−	0.812273i	$-0.698230\pi$
$619$		−	9.58626e9i		−	1.62455i	0.583278	−	0.812273i	$-0.301770\pi$
$620$	−3.28534e9	−	9.37304e9i	−0.553617	−	1.57946i
$621$		0			0
$622$	−9.60571e9	+	1.63469e9i	−1.60053	+	0.272377i
$623$	8.75763e8			0.145104
$624$		0			0
$625$	−7.62240e9			−1.24885
$626$	−2.69974e9	+	4.59439e8i	−0.439857	+	0.0748545i
$627$		0			0
$628$	1.70958e9	+	4.87742e9i	0.275443	+	0.785835i
$629$		−	6.01242e9i		−	0.963324i
$630$		0			0
$631$	1.18616e10			1.87949			0.939747	−	0.341870i	$-0.111060\pi$
$631$	1.18616e10			1.87949			0.939747	+	0.341870i	$0.111060\pi$
$632$	3.24132e9	−	1.79478e9i	0.510755	−	0.282814i
$633$		0			0
$634$	−1.02185e9	−	6.00455e9i	−0.159248	−	0.935767i
$635$		−	3.80445e9i		−	0.589635i
$636$		0			0
$637$		−	1.39474e9i		−	0.213799i
$638$	−3.52454e9	+	5.99804e8i	−0.537317	+	0.0914402i
$639$		0			0
$640$	−7.42451e9	−	3.05923e9i	−1.11954	−	0.461298i
$641$	1.06130e8			0.0159161			0.00795805	−	0.999968i	$-0.497467\pi$
$641$	1.06130e8			0.0159161			0.00795805	+	0.999968i	$0.497467\pi$
$642$		0			0
$643$		−	2.19289e9i		−	0.325296i	−0.986684	−	0.162648i	$-0.947996\pi$
$643$		−	2.19289e9i		−	0.325296i	0.986684	−	0.162648i	$-0.0520035\pi$
$644$	1.03963e9	−	3.64399e8i	0.153383	−	0.0537621i
$645$		0			0
$646$	−7.55891e8	−	4.44174e9i	−0.110318	−	0.648245i
$647$	4.23914e9			0.615337			0.307668	−	0.951494i	$-0.400451\pi$
$647$	4.23914e9			0.615337			0.307668	+	0.951494i	$0.400451\pi$
$648$		0			0
$649$	3.58862e9			0.515312
$650$	1.52833e8	+	8.98072e8i	0.0218283	+	0.128267i
$651$		0			0
$652$	4.16839e9	+	1.18924e10i	0.588982	+	1.68036i
$653$		−	9.93257e9i		−	1.39594i	−0.716129	−	0.697968i	$-0.754088\pi$
$653$		−	9.93257e9i		−	1.39594i	0.716129	−	0.697968i	$-0.245912\pi$
$654$		0			0
$655$	−2.98346e9			−0.414836
$656$	−7.67885e9	+	6.13699e9i	−1.06202	+	0.848774i
$657$		0			0
$658$	−4.01664e9	+	6.83548e8i	−0.549632	+	0.0935359i
$659$			1.36634e10i			1.85977i	0.367852	+	0.929884i	$0.380093\pi$
$659$			1.36634e10i			1.85977i	−0.367852	+	0.929884i	$0.619907\pi$
$660$		0			0
$661$		−	1.03765e10i		−	1.39747i	−0.715378	−	0.698737i	$-0.753746\pi$
$661$		−	1.03765e10i		−	1.39747i	0.715378	−	0.698737i	$-0.246254\pi$
$662$	−1.99400e9	−	1.17170e10i	−0.267129	−	1.56969i
$663$		0			0
$664$	1.07621e9	+	1.94361e9i	0.142662	+	0.257644i
$665$	4.88398e9			0.644019
$666$		0			0
$667$			3.15032e9i			0.411069i
$668$	5.13149e9	−	1.79863e9i	0.666079	−	0.233467i
$669$		0			0
$670$	−3.01522e8	+	5.13127e7i	−0.0387308	+	0.00659118i
$671$	9.52968e8			0.121773
$672$		0			0
$673$	4.70776e9			0.595336			0.297668	−	0.954669i	$-0.403791\pi$
$673$	4.70776e9			0.595336			0.297668	+	0.954669i	$0.403791\pi$
$674$	2.28175e9	−	3.88306e8i	0.287050	−	0.0488500i
$675$		0			0
$676$	6.98919e9	−	2.44978e9i	0.870189	−	0.305009i
$677$			9.55050e9i			1.18295i	0.806324	+	0.591474i	$0.201454\pi$
$677$			9.55050e9i			1.18295i	−0.806324	+	0.591474i	$0.798546\pi$
$678$		0			0
$679$	1.26947e7			0.00155625
$680$	−2.87631e9	−	5.19454e9i	−0.350796	−	0.633528i
$681$		0			0
$682$	8.55798e8	+	5.02880e9i	0.103306	+	0.607043i
$683$			1.06442e10i			1.27832i	0.769073	+	0.639161i	$0.220718\pi$
$683$			1.06442e10i			1.27832i	−0.769073	+	0.639161i	$0.779282\pi$
$684$		0			0
$685$			1.12948e10i			1.34265i
$686$	−7.12100e9	+	1.21185e9i	−0.842183	+	0.143322i
$687$		0			0
$688$	9.03675e8	+	1.13071e9i	0.105792	+	0.132371i
$689$	−3.39865e9			−0.395858
$690$		0			0
$691$			8.41537e9i			0.970287i	0.874435	+	0.485143i	$0.161233\pi$
$691$			8.41537e9i			0.970287i	−0.874435	+	0.485143i	$0.838767\pi$
$692$	1.12309e9	+	3.20416e9i	0.128838	+	0.367573i
$693$		0			0
$694$	−1.19360e9	−	7.01381e9i	−0.135551	−	0.796519i
$695$	1.58445e10			1.79033
$696$		0			0
$697$	−7.26858e9			−0.813083
$698$	1.87286e9	+	1.10052e10i	0.208454	+	1.22491i
$699$		0			0
$700$	1.93121e9	−	6.76907e8i	0.212807	−	0.0745909i
$701$		−	8.95355e9i		−	0.981707i	−0.871242	−	0.490854i	$-0.836685\pi$
$701$		−	8.95355e9i		−	0.981707i	0.871242	−	0.490854i	$-0.163315\pi$
$702$		0			0
$703$	1.63139e10			1.77099
$704$	3.49555e9	+	2.18867e9i	0.377582	+	0.236415i
$705$		0			0
$706$	6.24289e9	−	1.06241e9i	0.667682	−	0.113626i
$707$			7.37853e9i			0.785239i
$708$		0			0
$709$			8.11796e9i			0.855431i	0.903913	+	0.427716i	$0.140682\pi$
$709$			8.11796e9i			0.855431i	−0.903913	+	0.427716i	$0.859318\pi$
$710$	8.16334e8	+	4.79691e9i	0.0855980	+	0.502988i
$711$		0			0
$712$	2.52737e9	−	1.39945e9i	0.262415	−	0.145304i
$713$	4.49486e9			0.464412
$714$		0			0
$715$		−	1.47157e9i		−	0.150560i
$716$	1.02525e9	+	2.92502e9i	0.104384	+	0.297806i
$717$		0			0
$718$	1.58813e9	−	2.70266e8i	0.160122	−	0.0272494i
$719$	−9.54339e8			−0.0957528			−0.0478764	−	0.998853i	$-0.515245\pi$
$719$	−9.54339e8			−0.0957528			−0.0478764	+	0.998853i	$0.515245\pi$
$720$		0			0
$721$	5.60800e9			0.557231
$722$	2.08241e9	−	3.54383e8i	0.205914	−	0.0350423i
$723$		0			0
$724$	1.18111e9	+	3.36971e9i	0.115666	+	0.329995i
$725$			5.85203e9i			0.570327i
$726$		0			0
$727$	1.75084e10			1.68996			0.844979	−	0.534799i	$-0.179613\pi$
$727$	1.75084e10			1.68996			0.844979	+	0.534799i	$0.179613\pi$
$728$	6.80890e8	+	1.22967e9i	0.0654059	+	0.118121i
$729$		0			0
$730$	−2.38210e9	−	1.39976e10i	−0.226637	−	1.33175i
$731$			1.07030e9i			0.101343i
$732$		0			0
$733$			1.16062e10i			1.08849i	0.838925	+	0.544247i	$0.183185\pi$
$733$			1.16062e10i			1.08849i	−0.838925	+	0.544247i	$0.816815\pi$
$734$	−7.95738e9	+	1.35418e9i	−0.742735	+	0.126398i
$735$		0			0
$736$	2.41796e9	−	2.71292e9i	0.223551	−	0.250821i
$737$	1.57087e8			0.0144545
$738$		0			0
$739$			4.74800e9i			0.432768i	0.976308	+	0.216384i	$0.0694263\pi$
$739$			4.74800e9i			0.432768i	−0.976308	+	0.216384i	$0.930574\pi$
$740$	2.02891e10	−	7.11153e9i	1.84057	−	0.645137i
$741$		0			0
$742$	1.28084e9	+	7.52642e9i	0.115102	+	0.676355i
$743$	4.35857e9			0.389837			0.194918	−	0.980819i	$-0.437556\pi$
$743$	4.35857e9			0.389837			0.194918	+	0.980819i	$0.437556\pi$
$744$		0			0
$745$	−1.29864e10			−1.15064
$746$	−7.87185e8	−	4.62562e9i	−0.0694210	−	0.407929i
$747$		0			0
$748$	1.00874e9	+	2.87792e9i	0.0881297	+	0.251433i
$749$		−	5.95321e9i		−	0.517684i
$750$		0			0
$751$	−6.10987e9			−0.526371			−0.263186	−	0.964745i	$-0.584773\pi$
$751$	−6.10987e9			−0.526371			−0.263186	+	0.964745i	$0.584773\pi$
$752$	−1.04993e10	+	8.39115e9i	−0.900326	+	0.719547i
$753$		0			0
$754$	−3.96256e9	+	6.74346e8i	−0.336648	+	0.0572906i
$755$		−	2.42883e10i		−	2.05391i
$756$		0			0
$757$		−	2.29472e10i		−	1.92262i	−0.275460	−	0.961312i	$-0.588830\pi$
$757$		−	2.29472e10i		−	1.92262i	0.275460	−	0.961312i	$-0.411170\pi$
$758$	2.34649e8	+	1.37883e9i	0.0195693	+	0.114993i
$759$		0			0
$760$	1.40947e10	−	7.80449e9i	1.16469	−	0.644907i
$761$	−7.15151e9			−0.588236			−0.294118	−	0.955769i	$-0.595026\pi$
$761$	−7.15151e9			−0.588236			−0.294118	+	0.955769i	$0.595026\pi$
$762$		0			0
$763$		−	2.08175e9i		−	0.169665i
$764$	−2.02447e9	+	7.09596e8i	−0.164242	+	0.0575684i
$765$		0			0
$766$	−1.51445e10	+	2.57728e9i	−1.21746	+	0.207186i
$767$	4.03460e9			0.322862
$768$		0			0
$769$	−1.85381e10			−1.47002			−0.735011	−	0.678055i	$-0.762823\pi$
$769$	−1.85381e10			−1.47002			−0.735011	+	0.678055i	$0.762823\pi$
$770$	−3.25884e9	+	5.54586e8i	−0.257244	+	0.0437776i
$771$		0			0
$772$	1.05696e10	−	3.70476e9i	0.826798	−	0.289800i
$773$			8.25535e9i			0.642846i	0.946936	+	0.321423i	$0.104161\pi$
$773$			8.25535e9i			0.642846i	−0.946936	+	0.321423i	$0.895839\pi$
$774$		0			0
$775$	8.34966e9			0.644336
$776$	3.66358e7	−	2.02859e7i	0.00281443	−	0.00155840i
$777$		0			0
$778$	1.90895e9	+	1.12173e10i	0.145334	+	0.854003i
$779$		−	1.97223e10i		−	1.49478i
$780$		0			0
$781$		−	2.49909e9i		−	0.187717i
$782$	2.64907e9	−	4.50816e8i	0.198093	−	0.0337113i
$783$		0			0
$784$	−8.07375e9	+	6.45259e9i	−0.598369	+	0.478220i
$785$	−1.36656e10			−1.00829
$786$		0			0
$787$			1.53185e9i			0.112023i	0.998430	+	0.0560113i	$0.0178383\pi$
$787$			1.53185e9i			0.112023i	−0.998430	+	0.0560113i	$0.982162\pi$
$788$	−1.08491e9	−	3.09523e9i	−0.0789862	−	0.225347i
$789$		0			0
$790$	1.64353e9	+	9.65762e9i	0.118599	+	0.696907i
$791$	−3.54196e9			−0.254464
$792$		0			0
$793$	1.07140e9			0.0762950
$794$	4.36708e8	+	2.56617e9i	0.0309613	+	0.181934i
$795$		0			0
$796$	6.42487e9	−	2.25198e9i	0.451511	−	0.158259i
$797$			2.78553e9i			0.194896i	0.995241	+	0.0974480i	$0.0310680\pi$
$797$			2.78553e9i			0.194896i	−0.995241	+	0.0974480i	$0.968932\pi$
$798$		0			0
$799$	−9.93837e9			−0.689289
$800$	4.49161e9	−	5.03952e9i	0.310161	−	0.347996i
$801$		0			0
$802$	−1.39184e10	+	2.36862e9i	−0.952749	+	0.162138i
$803$			7.29247e9i			0.497016i
$804$		0			0
$805$			2.91283e9i			0.196802i
$806$	9.62154e8	+	5.65377e9i	0.0647250	+	0.380334i
$807$		0			0
$808$	1.17907e10	+	2.12937e10i	0.786322	+	1.42008i
$809$	−1.17657e10			−0.781267			−0.390634	−	0.920546i	$-0.627744\pi$
$809$	−1.17657e10			−0.781267			−0.390634	+	0.920546i	$0.627744\pi$
$810$		0			0
$811$		−	6.29491e9i		−	0.414397i	−0.978299	−	0.207198i	$-0.933565\pi$
$811$		−	6.29491e9i		−	0.414397i	0.978299	−	0.207198i	$-0.0664346\pi$
$812$	2.98672e9	+	8.52108e9i	0.195771	+	0.558533i
$813$		0			0
$814$	−1.08855e10	+	1.85248e9i	−0.707395	+	0.120384i
$815$	−3.33201e10			−2.15603
$816$		0			0
$817$	−2.90413e9			−0.186311
$818$	2.00674e10	−	3.41505e9i	1.28190	−	0.218153i
$819$		0			0
$820$	−8.59732e9	−	2.45281e10i	−0.544521	−	1.55351i
$821$		−	4.27400e9i		−	0.269546i	−0.990876	−	0.134773i	$-0.956969\pi$
$821$		−	4.27400e9i		−	0.269546i	0.990876	−	0.134773i	$-0.0430306\pi$
$822$		0			0
$823$	−3.16411e9			−0.197858			−0.0989288	−	0.995095i	$-0.531542\pi$
$823$	−3.16411e9			−0.197858			−0.0989288	+	0.995095i	$0.531542\pi$
$824$	1.61842e10	−	8.96146e9i	1.00773	−	0.557999i
$825$		0			0
$826$	−1.52051e9	−	8.93475e9i	−0.0938768	−	0.551635i
$827$		−	3.47251e9i		−	0.213488i	−0.994287	−	0.106744i	$-0.965957\pi$
$827$		−	3.47251e9i		−	0.213488i	0.994287	−	0.106744i	$-0.0340426\pi$
$828$		0			0
$829$		−	5.21388e9i		−	0.317848i	−0.987291	−	0.158924i	$-0.949197\pi$
$829$		−	5.21388e9i		−	0.317848i	0.987291	−	0.158924i	$-0.0508026\pi$
$830$	−5.79104e9	+	9.85514e8i	−0.351547	+	0.0598259i
$831$		0			0
$832$	3.92997e9	+	2.46067e9i	0.236569	+	0.148123i
$833$	−7.64237e9			−0.458111
$834$		0			0
$835$			1.43774e10i			0.854629i
$836$	−7.80885e9	+	2.73708e9i	−0.462238	+	0.162019i
$837$		0			0
$838$	−5.06538e8	−	2.97650e9i	−0.0297343	−	0.174723i
$839$	2.48652e10			1.45353			0.726766	−	0.686885i	$-0.241023\pi$
$839$	2.48652e10			1.45353			0.726766	+	0.686885i	$0.241023\pi$
$840$		0			0
$841$	−8.57107e9			−0.496877
$842$	−5.13570e9	−	3.01782e10i	−0.296488	−	1.74221i
$843$		0			0
$844$	−8.51728e8	−	2.42997e9i	−0.0487643	−	0.139124i
$845$			1.95823e10i			1.11652i
$846$		0			0
$847$	−6.85701e9			−0.387742
$848$	1.57234e10	+	1.96738e10i	0.885446	+	1.10791i
$849$		0			0
$850$	4.92091e9	−	8.37436e8i	0.274839	−	0.0467720i
$851$			9.72969e9i			0.541185i
$852$		0			0
$853$		−	3.01930e10i		−	1.66565i	−0.553536	−	0.832826i	$-0.686722\pi$
$853$		−	3.01930e10i		−	1.66565i	0.553536	−	0.832826i	$-0.313278\pi$
$854$	−4.03776e8	−	2.37265e9i	−0.0221839	−	0.130356i
$855$		0			0
$856$	−9.51310e9	−	1.71804e10i	−0.518399	−	0.936215i
$857$	−2.57761e10			−1.39889			−0.699446	−	0.714685i	$-0.746570\pi$
$857$	−2.57761e10			−1.39889			−0.699446	+	0.714685i	$0.746570\pi$
$858$		0			0
$859$		−	2.77848e10i		−	1.49565i	−0.663894	−	0.747827i	$-0.731097\pi$
$859$		−	2.77848e10i		−	1.49565i	0.663894	−	0.747827i	$-0.268903\pi$
$860$	−3.61177e9	+	1.26596e9i	−0.193631	+	0.0678696i
$861$		0			0
$862$	1.99037e10	−	3.38720e9i	1.05842	−	0.180121i
$863$	−3.02990e10			−1.60469			−0.802344	−	0.596862i	$-0.796414\pi$
$863$	−3.02990e10			−1.60469			−0.802344	+	0.596862i	$0.796414\pi$
$864$		0			0
$865$	−8.97743e9			−0.471624
$866$	−1.35263e10	+	2.30190e9i	−0.707730	+	0.120441i
$867$		0			0
$868$	1.21578e10	−	4.26144e9i	0.631012	−	0.221176i
$869$		−	5.03142e9i		−	0.260089i
$870$		0			0
$871$	1.76609e8			0.00905627
$872$	−3.32659e9	−	6.00773e9i	−0.169899	−	0.306834i
$873$		0			0
$874$	1.22323e9	+	7.18791e9i	0.0619753	+	0.364177i
$875$		−	6.19656e9i		−	0.312696i
$876$		0			0
$877$		−	2.37410e9i		−	0.118850i	−0.998233	−	0.0594252i	$-0.981073\pi$
$877$		−	2.37410e9i		−	0.118850i	0.998233	−	0.0594252i	$-0.0189268\pi$
$878$	1.96457e10	−	3.34328e9i	0.979570	−	0.166703i
$879$		0			0
$880$	−8.51848e9	+	6.80803e9i	−0.421379	+	0.336769i
$881$	3.28058e10			1.61635			0.808175	−	0.588942i	$-0.200456\pi$
$881$	3.28058e10			1.61635			0.808175	+	0.588942i	$0.200456\pi$
$882$		0			0
$883$			2.14967e10i			1.05078i	0.850863	+	0.525388i	$0.176080\pi$
$883$			2.14967e10i			1.05078i	−0.850863	+	0.525388i	$0.823920\pi$
$884$	1.13410e9	+	3.23558e9i	0.0552164	+	0.157532i
$885$		0			0
$886$	−1.53024e9	−	8.99193e9i	−0.0739165	−	0.434345i
$887$	1.50080e10			0.722089			0.361045	−	0.932549i	$-0.382420\pi$
$887$	1.50080e10			0.722089			0.361045	+	0.932549i	$0.382420\pi$
$888$		0			0
$889$	4.93477e9			0.235565
$890$	1.28151e9	+	7.53037e9i	0.0609337	+	0.358056i
$891$		0			0
$892$	−2.01741e10	+	7.07121e9i	−0.951737	+	0.333593i
$893$		−	2.69665e10i		−	1.26720i
$894$		0			0
$895$	−8.19532e9			−0.382107
$896$	3.96814e9	−	9.63038e9i	0.184293	−	0.447266i
$897$		0			0
$898$	6.53489e9	−	1.11210e9i	0.301142	−	0.0512481i
$899$			3.68412e10i			1.69112i
$900$		0			0
$901$			1.86226e10i			0.848212i
$902$	2.23951e9	+	1.31597e10i	0.101609	+	0.597069i
$903$		0			0
$904$	−1.02218e10	+	5.65998e9i	−0.460190	+	0.254815i
$905$	−9.44124e9			−0.423408
$906$		0			0
$907$			1.57906e10i			0.702704i	0.936243	+	0.351352i	$0.114278\pi$
$907$			1.57906e10i			0.702704i	−0.936243	+	0.351352i	$0.885722\pi$
$908$	5.81358e9	+	1.65861e10i	0.257717	+	0.735264i
$909$		0			0
$910$	−3.66384e9	+	6.23509e8i	−0.161173	+	0.0274282i
$911$	6.82128e9			0.298917			0.149459	−	0.988768i	$-0.452247\pi$
$911$	6.82128e9			0.298917			0.149459	+	0.988768i	$0.452247\pi$
$912$		0			0
$913$	3.01701e9			0.131199
$914$	−6.16379e9	+	1.04895e9i	−0.267015	+	0.0454405i
$915$		0			0
$916$	1.13443e10	+	3.23651e10i	0.487688	+	1.39137i
$917$		−	3.86987e9i		−	0.165731i
$918$		0			0
$919$	−1.08588e10			−0.461506			−0.230753	−	0.973012i	$-0.574119\pi$
$919$	−1.08588e10			−0.461506			−0.230753	+	0.973012i	$0.574119\pi$
$920$	4.65463e9	+	8.40614e9i	0.197073	+	0.355909i
$921$		0			0
$922$	3.30455e9	+	1.94181e10i	0.138853	+	0.815921i
$923$		−	2.80967e9i		−	0.117611i
$924$		0			0
$925$			1.80739e10i			0.750853i
$926$	3.17236e10	−	5.39869e9i	1.31294	−	0.223435i
$927$		0			0
$928$	2.22359e10	+	1.98183e10i	0.913349	+	0.814046i
$929$	2.99248e9			0.122455			0.0612275	−	0.998124i	$-0.480498\pi$
$929$	2.99248e9			0.122455			0.0612275	+	0.998124i	$0.480498\pi$
$930$		0			0
$931$		−	2.07366e10i		−	0.842197i
$932$	−2.39213e10	+	8.38463e9i	−0.967895	+	0.339256i
$933$		0			0
$934$	−3.09633e9	−	1.81945e10i	−0.124346	−	0.730678i
$935$	−8.06335e9			−0.322608
$936$		0			0
$937$	2.39333e10			0.950417			0.475208	−	0.879873i	$-0.342373\pi$
$937$	2.39333e10			0.950417			0.475208	+	0.879873i	$0.342373\pi$
$938$	−6.65581e7	−	3.91106e8i	−0.00263324	−	0.0154734i
$939$		0			0
$940$	−1.17552e10	−	3.35374e10i	−0.461616	−	1.31699i
$941$		−	3.17291e10i		−	1.24135i	−0.784068	−	0.620675i	$-0.786859\pi$
$941$		−	3.17291e10i		−	1.24135i	0.784068	−	0.620675i	$-0.213141\pi$
$942$		0			0
$943$	1.17625e10			0.456781
$944$	−1.86656e10	−	2.33551e10i	−0.722169	−	0.903607i
$945$		0			0
$946$	1.93778e9	−	3.29769e8i	0.0744192	−	0.0126646i
$947$		−	4.13161e10i		−	1.58086i	−0.612551	−	0.790431i	$-0.709857\pi$
$947$		−	4.13161e10i		−	1.58086i	0.612551	−	0.790431i	$-0.290143\pi$
$948$		0			0
$949$			8.19876e9i			0.311399i
$950$	2.27228e9	+	1.33523e10i	0.0859861	+	0.505268i
$951$		0			0
$952$	6.73788e9	−	3.73088e9i	0.253101	−	0.140146i
$953$	4.31993e10			1.61678			0.808390	−	0.588647i	$-0.200339\pi$
$953$	4.31993e10			1.61678			0.808390	+	0.588647i	$0.200339\pi$
$954$		0			0
$955$		−	5.67216e9i		−	0.210735i
$956$	−9.93266e9	+	3.48149e9i	−0.367674	+	0.128873i
$957$		0			0
$958$	4.11944e10	−	7.01043e9i	1.51377	−	0.257611i
$959$	−1.46506e10			−0.536401
$960$		0			0
$961$	2.50523e10			0.910574
$962$	−1.22383e10	+	2.08270e9i	−0.443208	+	0.0754248i
$963$		0			0
$964$	−3.26931e10	+	1.14592e10i	−1.17540	+	0.411990i
$965$			2.96140e10i			1.06084i
$966$		0			0
$967$	6.99318e9			0.248703			0.124352	−	0.992238i	$-0.460315\pi$
$967$	6.99318e9			0.248703			0.124352	+	0.992238i	$0.460315\pi$
$968$	−1.97887e10	+	1.09573e10i	−0.701218	+	0.388277i
$969$		0			0
$970$	1.85763e7	+	1.09157e8i	0.000653520	+	0.00384019i
$971$			8.48357e9i			0.297380i	0.988884	+	0.148690i	$0.0475056\pi$
$971$			8.48357e9i			0.297380i	−0.988884	+	0.148690i	$0.952494\pi$
$972$		0			0
$973$			2.05521e10i			0.715255i
$974$	−1.75278e10	+	2.98287e9i	−0.607815	+	0.103437i
$975$		0			0
$976$	−4.95670e9	−	6.20202e9i	−0.170655	−	0.213530i
$977$	1.72197e10			0.590739			0.295370	−	0.955383i	$-0.404557\pi$
$977$	1.72197e10			0.590739			0.295370	+	0.955383i	$0.404557\pi$
$978$		0			0
$979$		−	3.92317e9i		−	0.133628i
$980$	−9.03944e9	−	2.57894e10i	−0.306796	−	0.875287i
$981$		0			0
$982$	−3.10321e9	−	1.82349e10i	−0.104573	−	0.614489i
$983$	3.76267e10			1.26345			0.631725	−	0.775192i	$-0.282347\pi$
$983$	3.76267e10			1.26345			0.631725	+	0.775192i	$0.282347\pi$
$984$		0			0
$985$	8.67222e9			0.289137
$986$	3.69502e9	+	2.17125e10i	0.122757	+	0.721342i
$987$		0			0
$988$	−8.77932e9	+	3.07723e9i	−0.289609	+	0.101511i
$989$		−	1.73203e9i		−	0.0569336i
$990$		0			0
$991$	−3.13036e9			−0.102173			−0.0510866	−	0.998694i	$-0.516268\pi$
$991$	−3.13036e9			−0.102173			−0.0510866	+	0.998694i	$0.516268\pi$
$992$	2.82767e10	−	3.17261e10i	0.919683	−	1.03187i
$993$		0			0
$994$	−6.22210e9	+	1.05887e9i	−0.200949	+	0.0341973i
$995$			1.80012e10i			0.579323i
$996$		0			0
$997$			1.67605e9i			0.0535617i	0.999641	+	0.0267808i	$0.00852563\pi$
$997$			1.67605e9i			0.0535617i	−0.999641	+	0.0267808i	$0.991474\pi$
$998$	1.30373e9	+	7.66090e9i	0.0415173	+	0.243963i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.d.b.37.2		6
3.2	odd	2		8.8.b.a.5.5	✓	6
4.3	odd	2		288.8.d.b.145.6		6
8.3	odd	2		288.8.d.b.145.1		6
8.5	even	2	inner	72.8.d.b.37.1		6
12.11	even	2		32.8.b.a.17.1		6
24.5	odd	2		8.8.b.a.5.6	yes	6
24.11	even	2		32.8.b.a.17.6		6
48.5	odd	4		256.8.a.r.1.6		6
48.11	even	4		256.8.a.q.1.1		6
48.29	odd	4		256.8.a.r.1.1		6
48.35	even	4		256.8.a.q.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.8.b.a.5.5	✓	6	3.2	odd	2
8.8.b.a.5.6	yes	6	24.5	odd	2
32.8.b.a.17.1		6	12.11	even	2
32.8.b.a.17.6		6	24.11	even	2
72.8.d.b.37.1		6	8.5	even	2	inner
72.8.d.b.37.2		6	1.1	even	1	trivial
256.8.a.q.1.1		6	48.11	even	4
256.8.a.q.1.6		6	48.35	even	4
256.8.a.r.1.1		6	48.29	odd	4
256.8.a.r.1.6		6	48.5	odd	4
288.8.d.b.145.1		6	8.3	odd	2
288.8.d.b.145.6		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-11.1534 + 1.89807i) q^{2} +(120.795 - 42.3397i) q^{4} +338.443i q^{5} -438.996 q^{7} +(-1266.90 + 701.506i) q^{8} +O(q^{10})\)	\(q+(-11.1534 + 1.89807i) q^{2} +(120.795 - 42.3397i) q^{4} +338.443i q^{5} -438.996 q^{7} +(-1266.90 + 701.506i) q^{8} +(-642.387 - 3774.77i) q^{10} +1966.58i q^{11} +2210.98i q^{13} +(4896.28 - 833.245i) q^{14} +(12798.7 - 10228.8i) q^{16} +12114.9 q^{17} +32872.2i q^{19} +(14329.5 + 40882.1i) q^{20} +(-3732.70 - 21933.9i) q^{22} -19605.1 q^{23} -36418.4 q^{25} +(-4196.59 - 24659.8i) q^{26} +(-53028.4 + 18587.0i) q^{28} -160689. i q^{29} -229270. q^{31} +(-123333. + 138378. i) q^{32} +(-135122. + 22994.9i) q^{34} -148575. i q^{35} -496284. i q^{37} +(-62393.6 - 366635. i) q^{38} +(-237420. - 428774. i) q^{40} -599971. q^{41} +88346.0i q^{43} +(83264.3 + 237552. i) q^{44} +(218662. - 37211.8i) q^{46} -820344. q^{47} -630825. q^{49} +(406187. - 69124.6i) q^{50} +(93612.2 + 267075. i) q^{52} +1.53717e6i q^{53} -665574. q^{55} +(556165. - 307959. i) q^{56} +(304999. + 1.79222e6i) q^{58} -1.82480e6i q^{59} -484582. i q^{61} +(2.55713e6 - 435171. i) q^{62} +(1.11293e6 - 1.77748e6i) q^{64} -748290. q^{65} -79878.2i q^{67} +(1.46341e6 - 512940. i) q^{68} +(282006. + 1.65711e6i) q^{70} -1.27078e6 q^{71} +3.70820e6 q^{73} +(941981. + 5.53523e6i) q^{74} +(1.39180e6 + 3.97078e6i) q^{76} -863321. i q^{77} -2.55846e6 q^{79} +(3.46187e6 + 4.33163e6i) q^{80} +(6.69169e6 - 1.13879e6i) q^{82} -1.53414e6i q^{83} +4.10019e6i q^{85} +(-167687. - 985355. i) q^{86} +(-1.37957e6 - 2.49146e6i) q^{88} -1.99492e6 q^{89} -970612. i q^{91} +(-2.36819e6 + 830073. i) q^{92} +(9.14959e6 - 1.55707e6i) q^{94} -1.11253e7 q^{95} -28917.7 q^{97} +(7.03582e6 - 1.19735e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 116 q^{4} - 688 q^{7} - 1512 q^{8}+O(q^{10}) \) 6 * q - 6 * q^2 + 116 * q^4 - 688 * q^7 - 1512 * q^8	\( 6 q - 6 q^{2} + 116 q^{4} - 688 q^{7} - 1512 q^{8} - 1656 q^{10} - 12048 q^{14} + 35344 q^{16} - 1452 q^{17} + 114768 q^{20} + 152860 q^{22} + 1296 q^{23} - 39314 q^{25} + 316968 q^{26} - 480800 q^{28} - 89280 q^{31} - 817056 q^{32} - 1009108 q^{34} - 974124 q^{38} + 954464 q^{40} - 521244 q^{41} + 1096344 q^{44} + 929840 q^{46} - 1566432 q^{47} - 511050 q^{49} + 148626 q^{50} + 823952 q^{52} - 3270256 q^{55} + 2468928 q^{56} + 3130744 q^{58} + 7055808 q^{62} - 4792768 q^{64} - 1416480 q^{65} - 6608040 q^{68} - 7406912 q^{70} + 7597104 q^{71} + 2089564 q^{73} - 7744200 q^{74} + 9241288 q^{76} + 16015904 q^{79} + 12600384 q^{80} + 10715932 q^{82} + 5639076 q^{86} + 1541200 q^{88} - 2169084 q^{89} - 669600 q^{92} + 15503712 q^{94} - 48537936 q^{95} - 1088308 q^{97} + 14983242 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 116 * q^4 - 688 * q^7 - 1512 * q^8 - 1656 * q^10 - 12048 * q^14 + 35344 * q^16 - 1452 * q^17 + 114768 * q^20 + 152860 * q^22 + 1296 * q^23 - 39314 * q^25 + 316968 * q^26 - 480800 * q^28 - 89280 * q^31 - 817056 * q^32 - 1009108 * q^34 - 974124 * q^38 + 954464 * q^40 - 521244 * q^41 + 1096344 * q^44 + 929840 * q^46 - 1566432 * q^47 - 511050 * q^49 + 148626 * q^50 + 823952 * q^52 - 3270256 * q^55 + 2468928 * q^56 + 3130744 * q^58 + 7055808 * q^62 - 4792768 * q^64 - 1416480 * q^65 - 6608040 * q^68 - 7406912 * q^70 + 7597104 * q^71 + 2089564 * q^73 - 7744200 * q^74 + 9241288 * q^76 + 16015904 * q^79 + 12600384 * q^80 + 10715932 * q^82 + 5639076 * q^86 + 1541200 * q^88 - 2169084 * q^89 - 669600 * q^92 + 15503712 * q^94 - 48537936 * q^95 - 1088308 * q^97 + 14983242 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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