LMFDB - Embedded newform 72.8.f.a.35.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,8,Mod(35,72)]

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([1, 1, 1]))

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

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

chi := DirichletCharacter("72.35");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	72.f (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:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			35.20
Character	$\chi$	$=$	72.35
Dual form			72.8.f.a.35.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.18397 + 8.74017i) q^{2} +(-24.7812 + 125.578i) q^{4} +166.398 q^{5} -750.222i q^{7} +(-1275.60 + 685.559i) q^{8} +O(q^{10})$	\(q+(7.18397 + 8.74017i) q^{2} +(-24.7812 + 125.578i) q^{4} +166.398 q^{5} -750.222i q^{7} +(-1275.60 + 685.559i) q^{8} +(1195.40 + 1454.35i) q^{10} +8691.94i q^{11} +9209.54i q^{13} +(6557.07 - 5389.57i) q^{14} +(-15155.8 - 6223.95i) q^{16} -6845.42i q^{17} -12465.9 q^{19} +(-4123.53 + 20896.0i) q^{20} +(-75969.1 + 62442.6i) q^{22} +30540.3 q^{23} -50436.7 q^{25} +(-80493.0 + 66161.1i) q^{26} +(94211.6 + 18591.4i) q^{28} -122115. q^{29} +247508. i q^{31} +(-54480.3 - 177177. i) q^{32} +(59830.1 - 49177.3i) q^{34} -124835. i q^{35} +270412. i q^{37} +(-89554.4 - 108954. i) q^{38} +(-212258. + 114076. i) q^{40} -465855. i q^{41} -191804. q^{43} +(-1.09152e6 - 215396. i) q^{44} +(219401. + 266927. i) q^{46} -138358. q^{47} +260710. q^{49} +(-362336. - 440826. i) q^{50} +(-1.15652e6 - 228223. i) q^{52} +1.65255e6 q^{53} +1.44632e6i q^{55} +(514321. + 956985. i) q^{56} +(-877267. - 1.06730e6i) q^{58} -100518. i q^{59} +1.05236e6i q^{61} +(-2.16326e6 + 1.77809e6i) q^{62} +(1.15717e6 - 1.74900e6i) q^{64} +1.53245e6i q^{65} +2.72917e6 q^{67} +(859636. + 169637. i) q^{68} +(1.09108e6 - 896814. i) q^{70} +5.83185e6 q^{71} -4.00385e6 q^{73} +(-2.36345e6 + 1.94263e6i) q^{74} +(308919. - 1.56544e6i) q^{76} +6.52089e6 q^{77} -296509. i q^{79} +(-2.52189e6 - 1.03565e6i) q^{80} +(4.07165e6 - 3.34669e6i) q^{82} +2.86831e6i q^{83} -1.13906e6i q^{85} +(-1.37791e6 - 1.67640e6i) q^{86} +(-5.95883e6 - 1.10875e7i) q^{88} -1.14123e7i q^{89} +6.90921e6 q^{91} +(-756824. + 3.83520e6i) q^{92} +(-993960. - 1.20927e6i) q^{94} -2.07429e6 q^{95} +1.12274e7 q^{97} +(1.87293e6 + 2.27865e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 52 q^{4}+O(q^{10}) $ 28 * q + 52 * q^4	$ 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) $ 28 * q + 52 * q^4 + 10092 * q^10 - 1928 * q^16 - 121168 * q^19 + 59576 * q^22 + 437500 * q^25 + 46872 * q^28 - 114748 * q^34 + 1054752 * q^40 + 1505696 * q^43 - 476184 * q^46 - 2272076 * q^49 + 1468392 * q^52 + 3054996 * q^58 - 4186016 * q^64 - 776272 * q^67 + 3238872 * q^70 - 2534128 * q^73 - 21642832 * q^76 + 10334372 * q^82 + 10834016 * q^88 - 3406992 * q^91 - 22555944 * q^94 - 26311456 * q^97

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 $

$2$	7.18397	+	8.74017i	0.634979	+	0.772529i
$2$	7.18397	+	8.74017i	0.634979	+	0.772529i
$3$		0			0
$3$		0			0
$4$	−24.7812	+	125.578i	−0.193603	+	0.981080i
$5$	166.398			0.595323			0.297662	−	0.954671i	$-0.403793\pi$
$5$	166.398			0.595323			0.297662	+	0.954671i	$0.403793\pi$
$6$		0			0
$7$		−	750.222i		−	0.826698i	−0.910573	−	0.413349i	$-0.864359\pi$
$7$		−	750.222i		−	0.826698i	0.910573	−	0.413349i	$-0.135641\pi$
$8$	−1275.60	+	685.559i	−0.880847	+	0.473401i
$9$		0			0
$10$	1195.40	+	1454.35i	0.378018	+	0.459905i
$11$			8691.94i			1.96899i	0.175427	+	0.984493i	$0.443870\pi$
$11$			8691.94i			1.96899i	−0.175427	+	0.984493i	$0.556130\pi$
$12$		0			0
$13$			9209.54i			1.16262i	0.813683	+	0.581308i	$0.197459\pi$
$13$			9209.54i			1.16262i	−0.813683	+	0.581308i	$0.802541\pi$
$14$	6557.07	−	5389.57i	0.638648	−	0.524936i
$15$		0			0
$16$	−15155.8	−	6223.95i	−0.925036	−	0.379880i
$17$		−	6845.42i		−	0.337932i	−0.985622	−	0.168966i	$-0.945957\pi$
$17$		−	6845.42i		−	0.337932i	0.985622	−	0.168966i	$-0.0540427\pi$
$18$		0			0
$19$	−12465.9			−0.416951			−0.208475	−	0.978028i	$-0.566850\pi$
$19$	−12465.9			−0.416951			−0.208475	+	0.978028i	$0.566850\pi$
$20$	−4123.53	+	20896.0i	−0.115256	+	0.584060i
$21$		0			0
$22$	−75969.1	+	62442.6i	−1.52110	+	1.25026i
$23$	30540.3			0.523391			0.261695	−	0.965151i	$-0.415718\pi$
$23$	30540.3			0.523391			0.261695	+	0.965151i	$0.415718\pi$
$24$		0			0
$25$	−50436.7			−0.645590
$26$	−80493.0	+	66161.1i	−0.898155	+	0.738237i
$27$		0			0
$28$	94211.6	+	18591.4i	0.811057	+	0.160051i
$29$	−122115.			−0.929768			−0.464884	−	0.885372i	$-0.653904\pi$
$29$	−122115.			−0.929768			−0.464884	+	0.885372i	$0.653904\pi$
$30$		0			0
$31$			247508.i			1.49218i	0.665842	+	0.746092i	$0.268072\pi$
$31$			247508.i			1.49218i	−0.665842	+	0.746092i	$0.731928\pi$
$32$	−54480.3	−	177177.i	−0.293910	−	0.955833i
$33$		0			0
$34$	59830.1	−	49177.3i	0.261062	−	0.214580i
$35$		−	124835.i		−	0.492152i
$36$		0			0
$37$			270412.i			0.877648i	0.898573	+	0.438824i	$0.144605\pi$
$37$			270412.i			0.877648i	−0.898573	+	0.438824i	$0.855395\pi$
$38$	−89554.4	−	108954.i	−0.264755	−	0.322107i
$39$		0			0
$40$	−212258.	+	114076.i	−0.524389	+	0.281827i
$41$		−	465855.i		−	1.05562i	−0.849363	−	0.527810i	$-0.823013\pi$
$41$		−	465855.i		−	1.05562i	0.849363	−	0.527810i	$-0.176987\pi$
$42$		0			0
$43$	−191804.			−0.367890			−0.183945	−	0.982937i	$-0.558887\pi$
$43$	−191804.			−0.367890			−0.183945	+	0.982937i	$0.558887\pi$
$44$	−1.09152e6	−	215396.i	−1.93173	−	0.381201i
$45$		0			0
$46$	219401.	+	266927.i	0.332342	+	0.404334i
$47$	−138358.			−0.194385			−0.0971923	−	0.995266i	$-0.530986\pi$
$47$	−138358.			−0.194385			−0.0971923	+	0.995266i	$0.530986\pi$
$48$		0			0
$49$	260710.			0.316571
$50$	−362336.	−	440826.i	−0.409936	−	0.498737i
$51$		0			0
$52$	−1.15652e6	−	228223.i	−1.14062	−	0.225086i
$53$	1.65255e6			1.52471			0.762356	−	0.647158i	$-0.224043\pi$
$53$	1.65255e6			1.52471			0.762356	+	0.647158i	$0.224043\pi$
$54$		0			0
$55$			1.44632e6i			1.17218i
$56$	514321.	+	956985.i	0.391360	+	0.728194i
$57$		0			0
$58$	−877267.	−	1.06730e6i	−0.590383	−	0.718273i
$59$		−	100518.i		−	0.0637182i	−0.999492	−	0.0318591i	$-0.989857\pi$
$59$		−	100518.i		−	0.0637182i	0.999492	−	0.0318591i	$-0.0101428\pi$
$60$		0			0
$61$			1.05236e6i			0.593624i	0.954936	+	0.296812i	$0.0959234\pi$
$61$			1.05236e6i			0.593624i	−0.954936	+	0.296812i	$0.904077\pi$
$62$	−2.16326e6	+	1.77809e6i	−1.15276	+	0.947506i
$63$		0			0
$64$	1.15717e6	−	1.74900e6i	0.551782	−	0.833988i
$65$			1.53245e6i			0.692133i
$66$		0			0
$67$	2.72917e6			1.10858			0.554292	−	0.832322i	$-0.312989\pi$
$67$	2.72917e6			1.10858			0.554292	+	0.832322i	$0.312989\pi$
$68$	859636.	+	169637.i	0.331538	+	0.0654245i
$69$		0			0
$70$	1.09108e6	−	896814.i	0.380202	−	0.312507i
$71$	5.83185e6			1.93376			0.966879	−	0.255235i	$-0.0821528\pi$
$71$	5.83185e6			1.93376			0.966879	+	0.255235i	$0.0821528\pi$
$72$		0			0
$73$	−4.00385e6			−1.20461			−0.602307	−	0.798265i	$-0.705752\pi$
$73$	−4.00385e6			−1.20461			−0.602307	+	0.798265i	$0.705752\pi$
$74$	−2.36345e6	+	1.94263e6i	−0.678009	+	0.557288i
$75$		0			0
$76$	308919.	−	1.56544e6i	0.0807229	−	0.409062i
$77$	6.52089e6			1.62776
$78$		0			0
$79$		−	296509.i		−	0.0676618i	−0.999428	−	0.0338309i	$-0.989229\pi$
$79$		−	296509.i		−	0.0676618i	0.999428	−	0.0338309i	$-0.0107708\pi$
$80$	−2.52189e6	−	1.03565e6i	−0.550695	−	0.226151i
$81$		0			0
$82$	4.07165e6	−	3.34669e6i	0.815497	−	0.670296i
$83$			2.86831e6i			0.550621i	0.961355	+	0.275311i	$0.0887807\pi$
$83$			2.86831e6i			0.550621i	−0.961355	+	0.275311i	$0.911219\pi$
$84$		0			0
$85$		−	1.13906e6i		−	0.201179i
$86$	−1.37791e6	−	1.67640e6i	−0.233602	−	0.284206i
$87$		0			0
$88$	−5.95883e6	−	1.10875e7i	−0.932120	−	1.73437i
$89$		−	1.14123e7i		−	1.71596i	−0.513679	−	0.857982i	$-0.671718\pi$
$89$		−	1.14123e7i		−	1.71596i	0.513679	−	0.857982i	$-0.328282\pi$
$90$		0			0
$91$	6.90921e6			0.961132
$92$	−756824.	+	3.83520e6i	−0.101330	+	0.513488i
$93$		0			0
$94$	−993960.	−	1.20927e6i	−0.123430	−	0.150168i
$95$	−2.07429e6			−0.248221
$96$		0			0
$97$	1.12274e7			1.24904			0.624521	−	0.781008i	$-0.285294\pi$
$97$	1.12274e7			1.24904			0.624521	+	0.781008i	$0.285294\pi$
$98$	1.87293e6	+	2.27865e6i	0.201016	+	0.244560i
$99$		0			0
$100$	1.24988e6	−	6.33376e6i	0.124988	−	0.633376i
$101$	1.15791e7			1.11828			0.559138	−	0.829074i	$-0.311132\pi$
$101$	1.15791e7			1.11828			0.559138	+	0.829074i	$0.311132\pi$
$102$		0			0
$103$			1.80373e7i			1.62645i	0.581947	+	0.813227i	$0.302291\pi$
$103$			1.80373e7i			1.62645i	−0.581947	+	0.813227i	$0.697709\pi$
$104$	−6.31368e6	−	1.17477e7i	−0.550384	−	1.02409i
$105$		0			0
$106$	1.18718e7	+	1.44435e7i	0.968160	+	1.17788i
$107$			5.04706e6i			0.398286i	0.979970	+	0.199143i	$0.0638158\pi$
$107$			5.04706e6i			0.398286i	−0.979970	+	0.199143i	$0.936184\pi$
$108$		0			0
$109$		−	1.66178e7i		−	1.22908i	−0.788884	−	0.614542i	$-0.789341\pi$
$109$		−	1.66178e7i		−	1.22908i	0.788884	−	0.614542i	$-0.210659\pi$
$110$	−1.26411e7	+	1.03903e7i	−0.905545	+	0.744312i
$111$		0			0
$112$	−4.66935e6	+	1.13702e7i	−0.314046	+	0.764725i
$113$			4.43084e6i			0.288876i	0.989514	+	0.144438i	$0.0461374\pi$
$113$			4.43084e6i			0.288876i	−0.989514	+	0.144438i	$0.953863\pi$
$114$		0			0
$115$	5.08184e6			0.311587
$116$	3.02614e6	−	1.53349e7i	0.180006	−	0.912177i
$117$		0			0
$118$	878548.	−	722121.i	0.0492242	−	0.0404597i
$119$	−5.13559e6			−0.279367
$120$		0			0
$121$	−5.60627e7			−2.87690
$122$	−9.19783e6	+	7.56014e6i	−0.458592	+	0.376939i
$123$		0			0
$124$	−3.10816e7	−	6.13353e6i	−1.46395	−	0.288891i
$125$	−2.13924e7			−0.979658
$126$		0			0
$127$			2.77155e7i			1.20063i	0.799762	+	0.600317i	$0.204959\pi$
$127$			2.77155e7i			1.20063i	−0.799762	+	0.600317i	$0.795041\pi$
$128$	2.35996e7	−	2.45089e6i	0.994651	−	0.103297i
$129$		0			0
$130$	−1.33939e7	+	1.10091e7i	−0.534693	+	0.439490i
$131$			1.72039e7i			0.668616i	0.942464	+	0.334308i	$0.108503\pi$
$131$			1.72039e7i			0.668616i	−0.942464	+	0.334308i	$0.891497\pi$
$132$		0			0
$133$			9.35217e6i			0.344692i
$134$	1.96063e7	+	2.38534e7i	0.703928	+	0.856414i
$135$		0			0
$136$	4.69294e6	+	8.73203e6i	0.159977	+	0.297666i
$137$		−	3.74030e7i		−	1.24275i	−0.783513	−	0.621376i	$-0.786574\pi$
$137$		−	3.74030e7i		−	1.24275i	0.783513	−	0.621376i	$-0.213426\pi$
$138$		0			0
$139$	2.28530e7			0.721758			0.360879	−	0.932613i	$-0.382477\pi$
$139$	2.28530e7			0.721758			0.360879	+	0.932613i	$0.382477\pi$
$140$	1.56766e7	+	3.09357e6i	0.482841	+	0.0952821i
$141$		0			0
$142$	4.18958e7	+	5.09714e7i	1.22790	+	1.49388i
$143$	−8.00488e7			−2.28917
$144$		0			0
$145$	−2.03196e7			−0.553512
$146$	−2.87635e7	−	3.49943e7i	−0.764904	−	0.930599i
$147$		0			0
$148$	−3.39579e7	−	6.70114e6i	−0.861043	−	0.169915i
$149$	2.50798e7			0.621116			0.310558	−	0.950554i	$-0.399484\pi$
$149$	2.50798e7			0.621116			0.310558	+	0.950554i	$0.399484\pi$
$150$		0			0
$151$		−	3.71651e7i		−	0.878448i	−0.898377	−	0.439224i	$-0.855253\pi$
$151$		−	3.71651e7i		−	0.878448i	0.898377	−	0.439224i	$-0.144747\pi$
$152$	1.59015e7	−	8.54608e6i	0.367270	−	0.197385i
$153$		0			0
$154$	4.68459e7	+	5.69937e7i	1.03359	+	1.25749i
$155$			4.11848e7i			0.888332i
$156$		0			0
$157$		−	7.74558e7i		−	1.59737i	−0.601750	−	0.798685i	$-0.705529\pi$
$157$		−	7.74558e7i		−	1.59737i	0.601750	−	0.798685i	$-0.294471\pi$
$158$	2.59154e6	−	2.13011e6i	0.0522707	−	0.0429638i
$159$		0			0
$160$	−9.06541e6	−	2.94819e7i	−0.174972	−	0.569030i
$161$		−	2.29120e7i		−	0.432686i
$162$		0			0
$163$	7.06104e7			1.27706			0.638531	−	0.769596i	$-0.279542\pi$
$163$	7.06104e7			1.27706			0.638531	+	0.769596i	$0.279542\pi$
$164$	5.85013e7	+	1.15444e7i	1.03565	+	0.204371i
$165$		0			0
$166$	−2.50695e7	+	2.06059e7i	−0.425371	+	0.349633i
$167$	−6.83606e7			−1.13579			−0.567895	−	0.823101i	$-0.692242\pi$
$167$	−6.83606e7			−1.13579			−0.567895	+	0.823101i	$0.692242\pi$
$168$		0			0
$169$	−2.20672e7			−0.351677
$170$	9.95561e6	−	8.18300e6i	0.155416	−	0.127744i
$171$		0			0
$172$	4.75312e6	−	2.40864e7i	0.0712245	−	0.360929i
$173$	−4.63994e7			−0.681319			−0.340660	−	0.940187i	$-0.610650\pi$
$173$	−4.63994e7			−0.681319			−0.340660	+	0.940187i	$0.610650\pi$
$174$		0			0
$175$			3.78388e7i			0.533708i
$176$	5.40982e7	−	1.31733e8i	0.747978	−	1.82138i
$177$		0			0
$178$	9.97455e7	−	8.19856e7i	1.32563	−	1.08960i
$179$			1.11843e7i			0.145755i	0.997341	+	0.0728777i	$0.0232183\pi$
$179$			1.11843e7i			0.145755i	−0.997341	+	0.0728777i	$0.976782\pi$
$180$		0			0
$181$		−	1.42553e8i		−	1.78690i	−0.449162	−	0.893450i	$-0.648277\pi$
$181$		−	1.42553e8i		−	1.78690i	0.449162	−	0.893450i	$-0.351723\pi$
$182$	4.96355e7	+	6.03876e7i	0.610299	+	0.742503i
$183$		0			0
$184$	−3.89573e7	+	2.09372e7i	−0.461027	+	0.247774i
$185$			4.49961e7i			0.522485i
$186$		0			0
$187$	5.95000e7			0.665382
$188$	3.42868e6	−	1.73748e7i	0.0376334	−	0.190707i
$189$		0			0
$190$	−1.49017e7	−	1.81297e7i	−0.157615	−	0.191758i
$191$	6.46450e7			0.671303			0.335651	−	0.941986i	$-0.391044\pi$
$191$	6.46450e7			0.671303			0.335651	+	0.941986i	$0.391044\pi$
$192$		0			0
$193$	−1.07572e8			−1.07708			−0.538542	−	0.842599i	$-0.681025\pi$
$193$	−1.07572e8			−1.07708			−0.538542	+	0.842599i	$0.681025\pi$
$194$	8.06571e7	+	9.81291e7i	0.793115	+	0.964921i
$195$		0			0
$196$	−6.46069e6	+	3.27394e7i	−0.0612890	+	0.310581i
$197$	7.48526e7			0.697550			0.348775	−	0.937207i	$-0.386598\pi$
$197$	7.48526e7			0.697550			0.348775	+	0.937207i	$0.386598\pi$
$198$		0			0
$199$		−	3.50626e7i		−	0.315398i	−0.987487	−	0.157699i	$-0.949592\pi$
$199$		−	3.50626e7i		−	0.315398i	0.987487	−	0.157699i	$-0.0504075\pi$
$200$	6.43372e7	−	3.45773e7i	0.568666	−	0.305623i
$201$		0			0
$202$	8.31838e7	+	1.01203e8i	0.710082	+	0.863901i
$203$			9.16131e7i			0.768637i
$204$		0			0
$205$		−	7.75173e7i		−	0.628435i
$206$	−1.57649e8	+	1.29580e8i	−1.25648	+	1.03276i
$207$		0			0
$208$	5.73198e7	−	1.39578e8i	0.441655	−	1.07546i
$209$		−	1.08353e8i		−	0.820970i
$210$		0			0
$211$	−6.32841e7			−0.463774			−0.231887	−	0.972743i	$-0.574490\pi$
$211$	−6.32841e7			−0.463774			−0.231887	+	0.972743i	$0.574490\pi$
$212$	−4.09520e7	+	2.07524e8i	−0.295189	+	1.49586i
$213$		0			0
$214$	−4.41121e7	+	3.62579e7i	−0.307688	+	0.252903i
$215$	−3.19158e7			−0.219013
$216$		0			0
$217$	1.85686e8			1.23359
$218$	1.45243e8	−	1.19382e8i	0.949504	−	0.780443i
$219$		0			0
$220$	−1.81626e8	−	3.58415e7i	−1.15000	−	0.226938i
$221$	6.30432e7			0.392885
$222$		0			0
$223$			1.91569e8i			1.15680i	0.815753	+	0.578401i	$0.196323\pi$
$223$			1.91569e8i			1.15680i	−0.815753	+	0.578401i	$0.803677\pi$
$224$	−1.32922e8	+	4.08723e7i	−0.790185	+	0.242975i
$225$		0			0
$226$	−3.87263e7	+	3.18310e7i	−0.223165	+	0.183430i
$227$			1.07263e8i			0.608641i	0.952570	+	0.304320i	$0.0984293\pi$
$227$			1.07263e8i			0.608641i	−0.952570	+	0.304320i	$0.901571\pi$
$228$		0			0
$229$			2.78557e8i			1.53282i	0.642354	+	0.766408i	$0.277958\pi$
$229$			2.78557e8i			1.53282i	−0.642354	+	0.766408i	$0.722042\pi$
$230$	3.65078e7	+	4.44162e7i	0.197851	+	0.240710i
$231$		0			0
$232$	1.55770e8	−	8.37167e7i	0.818983	−	0.440153i
$233$			7.33025e6i			0.0379641i	0.999820	+	0.0189820i	$0.00604254\pi$
$233$			7.33025e6i			0.0379641i	−0.999820	+	0.0189820i	$0.993957\pi$
$234$		0			0
$235$	−2.30225e7			−0.115722
$236$	1.26229e7	+	2.49096e6i	0.0625126	+	0.0123360i
$237$		0			0
$238$	−3.68939e7	−	4.48859e7i	−0.177392	−	0.215819i
$239$	7.15951e7			0.339227			0.169614	−	0.985511i	$-0.445748\pi$
$239$	7.15951e7			0.339227			0.169614	+	0.985511i	$0.445748\pi$
$240$		0			0
$241$	−2.67697e8			−1.23192			−0.615961	−	0.787776i	$-0.711232\pi$
$241$	−2.67697e8			−1.23192			−0.615961	+	0.787776i	$0.711232\pi$
$242$	−4.02753e8	−	4.89997e8i	−1.82677	−	2.22249i
$243$		0			0
$244$	−1.32154e8	−	2.60788e7i	−0.582392	−	0.114927i
$245$	4.33815e7			0.188462
$246$		0			0
$247$		−	1.14805e8i		−	0.484754i
$248$	−1.69681e8	−	3.15721e8i	−0.706403	−	1.31439i
$249$		0			0
$250$	−1.53682e8	−	1.86973e8i	−0.622063	−	0.756815i
$251$		−	4.02680e8i		−	1.60732i	−0.595090	−	0.803659i	$-0.702884\pi$
$251$		−	4.02680e8i		−	1.60732i	0.595090	−	0.803659i	$-0.297116\pi$
$252$		0			0
$253$			2.65454e8i			1.03055i
$254$	−2.42239e8	+	1.99108e8i	−0.927525	+	0.762377i
$255$		0			0
$256$	1.90960e8	+	1.88658e8i	0.711383	+	0.702805i
$257$		−	9.64785e7i		−	0.354539i	−0.984162	−	0.177270i	$-0.943274\pi$
$257$		−	9.64785e7i		−	0.354539i	0.984162	−	0.177270i	$-0.0567265\pi$
$258$		0			0
$259$	2.02869e8			0.725550
$260$	−1.92442e8	−	3.79759e7i	−0.679037	−	0.133999i
$261$		0			0
$262$	−1.50365e8	+	1.23592e8i	−0.516525	+	0.424557i
$263$	4.51392e8			1.53006			0.765031	−	0.643993i	$-0.222724\pi$
$263$	4.51392e8			1.53006			0.765031	+	0.643993i	$0.222724\pi$
$264$		0			0
$265$	2.74980e8			0.907696
$266$	−8.17396e7	+	6.71857e7i	−0.266285	+	0.218872i
$267$		0			0
$268$	−6.76320e7	+	3.42724e8i	−0.214625	+	1.08761i
$269$	−1.28565e8			−0.402709			−0.201354	−	0.979518i	$-0.564534\pi$
$269$	−1.28565e8			−0.402709			−0.201354	+	0.979518i	$0.564534\pi$
$270$		0			0
$271$			2.72095e8i			0.830477i	0.909713	+	0.415238i	$0.136302\pi$
$271$			2.72095e8i			0.830477i	−0.909713	+	0.415238i	$0.863698\pi$
$272$	−4.26055e7	+	1.03748e8i	−0.128373	+	0.312599i
$273$		0			0
$274$	3.26909e8	−	2.68702e8i	0.960062	−	0.789122i
$275$		−	4.38393e8i		−	1.27116i
$276$		0			0
$277$			3.63332e8i			1.02713i	0.858051	+	0.513564i	$0.171675\pi$
$277$			3.63332e8i			1.02713i	−0.858051	+	0.513564i	$0.828325\pi$
$278$	1.64175e8	+	1.99739e8i	0.458301	+	0.557579i
$279$		0			0
$280$	8.55820e7	+	1.59240e8i	0.232986	+	0.433511i
$281$			4.12431e8i			1.10887i	0.832229	+	0.554433i	$0.187065\pi$
$281$			4.12431e8i			1.10887i	−0.832229	+	0.554433i	$0.812935\pi$
$282$		0			0
$283$	−4.48777e8			−1.17700			−0.588502	−	0.808496i	$-0.700282\pi$
$283$	−4.48777e8			−1.17700			−0.588502	+	0.808496i	$0.700282\pi$
$284$	−1.44520e8	+	7.32354e8i	−0.374381	+	1.89717i
$285$		0			0
$286$	−5.75068e8	−	6.99640e8i	−1.45358	−	1.76845i
$287$	−3.49495e8			−0.872678
$288$		0			0
$289$	3.63479e8			0.885802
$290$	−1.45975e8	−	1.77597e8i	−0.351469	−	0.427605i
$291$		0			0
$292$	9.92201e7	−	5.02796e8i	0.233217	−	1.18182i
$293$	2.75769e8			0.640484			0.320242	−	0.947336i	$-0.396236\pi$
$293$	2.75769e8			0.640484			0.320242	+	0.947336i	$0.396236\pi$
$294$		0			0
$295$		−	1.67260e7i		−	0.0379329i
$296$	−1.85384e8	−	3.44939e8i	−0.415480	−	0.773074i
$297$		0			0
$298$	1.80173e8	+	2.19202e8i	0.394396	+	0.479830i
$299$			2.81262e8i			0.608502i
$300$		0			0
$301$			1.43895e8i			0.304134i
$302$	3.24830e8	−	2.66993e8i	0.678627	−	0.557796i
$303$		0			0
$304$	1.88930e8	+	7.75869e7i	0.385694	+	0.158391i
$305$			1.75111e8i			0.353398i
$306$		0			0
$307$	−9.71621e8			−1.91652			−0.958258	−	0.285905i	$-0.907706\pi$
$307$	−9.71621e8			−1.91652			−0.958258	+	0.285905i	$0.907706\pi$
$308$	−1.61595e8	+	8.18882e8i	−0.315138	+	1.59696i
$309$		0			0
$310$	−3.59962e8	+	2.95870e8i	−0.686263	+	0.564073i
$311$	7.54388e8			1.42211			0.711055	−	0.703136i	$-0.248218\pi$
$311$	7.54388e8			1.42211			0.711055	+	0.703136i	$0.248218\pi$
$312$		0			0
$313$	2.49396e8			0.459710			0.229855	−	0.973225i	$-0.426175\pi$
$313$	2.49396e8			0.459710			0.229855	+	0.973225i	$0.426175\pi$
$314$	6.76977e8	−	5.56440e8i	1.23401	−	1.01430i
$315$		0			0
$316$	3.72351e7	+	7.34784e6i	0.0663816	+	0.0130995i
$317$	4.59560e8			0.810280			0.405140	−	0.914255i	$-0.367223\pi$
$317$	4.59560e8			0.810280			0.405140	+	0.914255i	$0.367223\pi$
$318$		0			0
$319$		−	1.06141e9i		−	1.83070i
$320$	1.92551e8	−	2.91030e8i	0.328489	−	0.496493i
$321$		0			0
$322$	2.00255e8	−	1.64599e8i	0.334262	−	0.274746i
$323$			8.53341e7i			0.140901i
$324$		0			0
$325$		−	4.64499e8i		−	0.750574i
$326$	5.07263e8	+	6.17147e8i	0.810908	+	0.986568i
$327$		0			0
$328$	3.19371e8	+	5.94246e8i	0.499732	+	0.929839i
$329$			1.03799e8i			0.160697i
$330$		0			0
$331$	1.69556e8			0.256989			0.128495	−	0.991710i	$-0.458985\pi$
$331$	1.69556e8			0.256989			0.128495	+	0.991710i	$0.458985\pi$
$332$	−3.60197e8	−	7.10801e7i	−0.540203	−	0.106602i
$333$		0			0
$334$	−4.91100e8	−	5.97483e8i	−0.721203	−	0.877431i
$335$	4.54128e8			0.659966
$336$		0			0
$337$	1.10636e9			1.57468			0.787341	−	0.616518i	$-0.211457\pi$
$337$	1.10636e9			1.57468			0.787341	+	0.616518i	$0.211457\pi$
$338$	−1.58530e8	−	1.92871e8i	−0.223307	−	0.271681i
$339$		0			0
$340$	1.43042e8	+	2.82273e7i	0.197372	+	0.0389487i
$341$	−2.15132e9			−2.93809
$342$		0			0
$343$		−	8.13430e8i		−	1.08841i
$344$	2.44665e8	−	1.31493e8i	0.324054	−	0.174160i
$345$		0			0
$346$	−3.33332e8	−	4.05538e8i	−0.432624	−	0.526339i
$347$		−	8.00799e7i		−	0.102889i	−0.998676	−	0.0514447i	$-0.983617\pi$
$347$		−	8.00799e7i		−	0.102889i	0.998676	−	0.0514447i	$-0.0163826\pi$
$348$		0			0
$349$			6.61204e8i			0.832619i	0.909223	+	0.416309i	$0.136677\pi$
$349$			6.61204e8i			0.832619i	−0.909223	+	0.416309i	$0.863323\pi$
$350$	−3.30717e8	+	2.71832e8i	−0.412305	+	0.338893i
$351$		0			0
$352$	1.54001e9	−	4.73540e8i	1.88202	−	0.578705i
$353$			1.07830e9i			1.30475i	0.757895	+	0.652376i	$0.226228\pi$
$353$			1.07830e9i			1.30475i	−0.757895	+	0.652376i	$0.773772\pi$
$354$		0			0
$355$	9.70408e8			1.15121
$356$	1.43314e9	+	2.82810e8i	1.68350	+	0.332216i
$357$		0			0
$358$	−9.77530e7	+	8.03480e7i	−0.112600	+	0.0925517i
$359$	−2.34950e8			−0.268006			−0.134003	−	0.990981i	$-0.542783\pi$
$359$	−2.34950e8			−0.268006			−0.134003	+	0.990981i	$0.542783\pi$
$360$		0			0
$361$	−7.38474e8			−0.826152
$362$	1.24593e9	−	1.02409e9i	1.38043	−	1.13464i
$363$		0			0
$364$	−1.71218e8	+	8.67646e8i	−0.186078	+	0.942948i
$365$	−6.66232e8			−0.717134
$366$		0			0
$367$		−	5.09803e8i		−	0.538358i	−0.963090	−	0.269179i	$-0.913248\pi$
$367$		−	5.09803e8i		−	0.538358i	0.963090	−	0.269179i	$-0.0867524\pi$
$368$	−4.62862e8	−	1.90081e8i	−0.484155	−	0.198825i
$369$		0			0
$370$	−3.93273e8	+	3.23250e8i	−0.403635	+	0.331767i
$371$		−	1.23978e9i		−	1.26048i
$372$		0			0
$373$		−	1.19562e9i		−	1.19293i	−0.802641	−	0.596463i	$-0.796572\pi$
$373$		−	1.19562e9i		−	1.19293i	0.802641	−	0.596463i	$-0.203428\pi$
$374$	4.27446e8	+	5.20040e8i	0.422504	+	0.514027i
$375$		0			0
$376$	1.76490e8	−	9.48526e7i	0.171223	−	0.0920220i
$377$		−	1.12462e9i		−	1.08096i
$378$		0			0
$379$	−1.27016e9			−1.19845			−0.599226	−	0.800580i	$-0.704525\pi$
$379$	−1.27016e9			−1.19845			−0.599226	+	0.800580i	$0.704525\pi$
$380$	5.14034e7	−	2.60486e8i	0.0480562	−	0.243524i
$381$		0			0
$382$	4.64408e8	+	5.65009e8i	0.426263	+	0.518601i
$383$	−1.06974e9			−0.972933			−0.486467	−	0.873699i	$-0.661714\pi$
$383$	−1.06974e9			−0.972933			−0.486467	+	0.873699i	$0.661714\pi$
$384$		0			0
$385$	1.08506e9			0.969041
$386$	−7.72796e8	−	9.40200e8i	−0.683926	−	0.832079i
$387$		0			0
$388$	−2.78227e8	+	1.40991e9i	−0.241818	+	1.22541i
$389$	4.09168e8			0.352434			0.176217	−	0.984351i	$-0.443614\pi$
$389$	4.09168e8			0.352434			0.176217	+	0.984351i	$0.443614\pi$
$390$		0			0
$391$		−	2.09061e8i		−	0.176870i
$392$	−3.32562e8	+	1.78732e8i	−0.278850	+	0.149865i
$393$		0			0
$394$	5.37739e8	+	6.54225e8i	0.442930	+	0.538878i
$395$		−	4.93385e7i		−	0.0402806i
$396$		0			0
$397$			4.20143e8i			0.337000i	0.985702	+	0.168500i	$0.0538923\pi$
$397$			4.20143e8i			0.337000i	−0.985702	+	0.168500i	$0.946108\pi$
$398$	3.06453e8	−	2.51889e8i	0.243654	−	0.200271i
$399$		0			0
$400$	7.64408e8	+	3.13916e8i	0.597194	+	0.245247i
$401$			9.24601e8i			0.716060i	0.933710	+	0.358030i	$0.116551\pi$
$401$			9.24601e8i			0.716060i	−0.933710	+	0.358030i	$0.883449\pi$
$402$		0			0
$403$	−2.27943e9			−1.73484
$404$	−2.86943e8	+	1.45408e9i	−0.216502	+	1.09712i
$405$		0			0
$406$	−8.00714e8	+	6.58146e8i	−0.593795	+	0.488069i
$407$	−2.35041e9			−1.72808
$408$		0			0
$409$	−1.39824e9			−1.01053			−0.505265	−	0.862964i	$-0.668605\pi$
$409$	−1.39824e9			−1.01053			−0.505265	+	0.862964i	$0.668605\pi$
$410$	6.77515e8	−	5.56882e8i	0.485484	−	0.399043i
$411$		0			0
$412$	−2.26510e9	−	4.46986e8i	−1.59568	−	0.314886i
$413$	−7.54111e7			−0.0526757
$414$		0			0
$415$			4.77281e8i			0.327798i
$416$	1.63172e9	−	5.01739e8i	1.11127	−	0.341705i
$417$		0			0
$418$	9.47020e8	−	7.78402e8i	0.634223	−	0.521299i
$419$			1.83425e9i			1.21818i	0.793102	+	0.609088i	$0.208464\pi$
$419$			1.83425e9i			1.21818i	−0.793102	+	0.609088i	$0.791536\pi$
$420$		0			0
$421$			2.18531e9i			1.42733i	0.700485	+	0.713667i	$0.252967\pi$
$421$			2.18531e9i			1.42733i	−0.700485	+	0.713667i	$0.747033\pi$
$422$	−4.54631e8	−	5.53114e8i	−0.294487	−	0.358279i
$423$		0			0
$424$	−2.10799e9	+	1.13292e9i	−1.34304	+	0.721801i
$425$			3.45261e8i			0.218165i
$426$		0			0
$427$	7.89506e8			0.490747
$428$	−6.33801e8	−	1.25072e8i	−0.390750	−	0.0771093i
$429$		0			0
$430$	−2.29282e8	−	2.78949e8i	−0.139069	−	0.169194i
$431$	3.35011e8			0.201553			0.100776	−	0.994909i	$-0.467867\pi$
$431$	3.35011e8			0.201553			0.100776	+	0.994909i	$0.467867\pi$
$432$		0			0
$433$	−8.65039e8			−0.512068			−0.256034	−	0.966668i	$-0.582416\pi$
$433$	−8.65039e8			−0.512068			−0.256034	+	0.966668i	$0.582416\pi$
$434$	1.33396e9	+	1.62293e9i	0.783301	+	0.952981i
$435$		0			0
$436$	2.08684e9	+	4.11809e8i	1.20583	+	0.237954i
$437$	−3.80711e8			−0.218228
$438$		0			0
$439$			1.73415e9i			0.978273i	0.872207	+	0.489136i	$0.162688\pi$
$439$			1.73415e9i			0.978273i	−0.872207	+	0.489136i	$0.837312\pi$
$440$	−9.91538e8	−	1.84493e9i	−0.554913	−	1.03251i
$441$		0			0
$442$	4.52900e8	+	5.51008e8i	0.249474	+	0.303515i
$443$		−	2.19800e8i		−	0.120120i	−0.998195	−	0.0600598i	$-0.980871\pi$
$443$		−	2.19800e8i		−	0.120120i	0.998195	−	0.0600598i	$-0.0191291\pi$
$444$		0			0
$445$		−	1.89898e9i		−	1.02155i
$446$	−1.67435e9	+	1.37623e9i	−0.893663	+	0.734545i
$447$		0			0
$448$	−1.31214e9	−	8.68135e8i	−0.689456	−	0.456157i
$449$		−	4.50746e8i		−	0.235001i	−0.993073	−	0.117501i	$-0.962512\pi$
$449$		−	4.50746e8i		−	0.235001i	0.993073	−	0.117501i	$-0.0374882\pi$
$450$		0			0
$451$	4.04919e9			2.07850
$452$	−5.56417e8	−	1.09801e8i	−0.283410	−	0.0559272i
$453$		0			0
$454$	−9.37500e8	+	7.70577e8i	−0.470193	+	0.386474i
$455$	1.14968e9			0.572185
$456$		0			0
$457$	2.75170e8			0.134863			0.0674317	−	0.997724i	$-0.478520\pi$
$457$	2.75170e8			0.134863			0.0674317	+	0.997724i	$0.478520\pi$
$458$	−2.43464e9	+	2.00114e9i	−1.18414	+	0.973306i
$459$		0			0
$460$	−1.25934e8	+	6.38169e8i	−0.0603241	+	0.305691i
$461$	7.26285e8			0.345266			0.172633	−	0.984986i	$-0.444773\pi$
$461$	7.26285e8			0.345266			0.172633	+	0.984986i	$0.444773\pi$
$462$		0			0
$463$			8.27651e8i			0.387537i	0.981047	+	0.193769i	$0.0620711\pi$
$463$			8.27651e8i			0.387537i	−0.981047	+	0.193769i	$0.937929\pi$
$464$	1.85074e9	+	7.60035e8i	0.860069	+	0.353200i
$465$		0			0
$466$	−6.40676e7	+	5.26603e7i	−0.0293284	+	0.0241064i
$467$		−	2.25212e9i		−	1.02325i	−0.859208	−	0.511626i	$-0.829043\pi$
$467$		−	2.25212e9i		−	1.02325i	0.859208	−	0.511626i	$-0.170957\pi$
$468$		0			0
$469$		−	2.04748e9i		−	0.916464i
$470$	−1.65393e8	−	2.01221e8i	−0.0734809	−	0.0893984i
$471$		0			0
$472$	6.89112e7	+	1.28221e8i	0.0301643	+	0.0561260i
$473$		−	1.66715e9i		−	0.724369i
$474$		0			0
$475$	6.28738e8			0.269179
$476$	1.27266e8	−	6.44918e8i	0.0540863	−	0.274082i
$477$		0			0
$478$	5.14337e8	+	6.25753e8i	0.215402	+	0.262063i
$479$	3.32761e9			1.38343			0.691717	−	0.722169i	$-0.256855\pi$
$479$	3.32761e9			1.38343			0.691717	+	0.722169i	$0.256855\pi$
$480$		0			0
$481$	−2.49038e9			−1.02037
$482$	−1.92313e9	−	2.33972e9i	−0.782245	−	0.951696i
$483$		0			0
$484$	1.38930e9	−	7.04025e9i	0.556977	−	2.82247i
$485$	1.86821e9			0.743584
$486$		0			0
$487$			3.21388e8i			0.126089i	0.998011	+	0.0630446i	$0.0200810\pi$
$487$			3.21388e8i			0.126089i	−0.998011	+	0.0630446i	$0.979919\pi$
$488$	−7.21457e8	−	1.34240e9i	−0.281022	−	0.522892i
$489$		0			0
$490$	3.11652e8	+	3.79162e8i	0.119669	+	0.145592i
$491$			3.60400e9i			1.37404i	0.726639	+	0.687020i	$0.241081\pi$
$491$			3.60400e9i			1.37404i	−0.726639	+	0.687020i	$0.758919\pi$
$492$		0			0
$493$			8.35925e8i			0.314198i
$494$	1.00341e9	−	8.24755e8i	0.374487	−	0.307809i
$495$		0			0
$496$	1.54048e9	−	3.75117e9i	0.566851	−	1.38032i
$497$		−	4.37518e9i		−	1.59863i
$498$		0			0
$499$	−1.80781e8			−0.0651331			−0.0325666	−	0.999470i	$-0.510368\pi$
$499$	−1.80781e8			−0.0651331			−0.0325666	+	0.999470i	$0.510368\pi$
$500$	5.30129e8	−	2.68642e9i	0.189665	−	0.961123i
$501$		0			0
$502$	3.51949e9	−	2.89284e9i	1.24170	−	1.02061i
$503$	−2.24260e9			−0.785711			−0.392856	−	0.919600i	$-0.628513\pi$
$503$	−2.24260e9			−0.785711			−0.392856	+	0.919600i	$0.628513\pi$
$504$		0			0
$505$	1.92673e9			0.665736
$506$	−2.32012e9	+	1.90702e9i	−0.796129	+	0.654377i
$507$		0			0
$508$	−3.48047e9	−	6.86824e8i	−1.17792	−	0.232446i
$509$	4.67698e9			1.57200			0.786002	−	0.618224i	$-0.212148\pi$
$509$	4.67698e9			1.57200			0.786002	+	0.618224i	$0.212148\pi$
$510$		0			0
$511$			3.00378e9i			0.995851i
$512$	−2.77048e8	+	3.02434e9i	−0.0912242	+	0.995830i
$513$		0			0
$514$	8.43238e8	−	6.93098e8i	0.273892	−	0.225125i
$515$			3.00137e9i			0.968266i
$516$		0			0
$517$		−	1.20260e9i		−	0.382741i
$518$	1.45741e9	+	1.77311e9i	0.460709	+	0.560509i
$519$		0			0
$520$	−1.05058e9	−	1.95480e9i	−0.327657	−	0.609663i
$521$			8.70545e8i			0.269687i	0.990867	+	0.134843i	$0.0430531\pi$
$521$			8.70545e8i			0.269687i	−0.990867	+	0.134843i	$0.956947\pi$
$522$		0			0
$523$	2.10726e9			0.644114			0.322057	−	0.946720i	$-0.395626\pi$
$523$	2.10726e9			0.644114			0.322057	+	0.946720i	$0.395626\pi$
$524$	−2.16043e9	−	4.26332e8i	−0.655966	−	0.129446i
$525$		0			0
$526$	3.24279e9	+	3.94525e9i	0.971558	+	1.18202i
$527$	1.69429e9			0.504256
$528$		0			0
$529$	−2.47212e9			−0.726062
$530$	1.97545e9	+	2.40337e9i	0.576368	+	0.701222i
$531$		0			0
$532$	−1.17443e9	−	2.31758e8i	−0.338171	−	0.0667334i
$533$	4.29031e9			1.22728
$534$		0			0
$535$			8.39820e8i			0.237109i
$536$	−3.48133e9	+	1.87101e9i	−0.976493	+	0.524805i
$537$		0			0
$538$	−9.23609e8	−	1.12368e9i	−0.255712	−	0.311104i
$539$			2.26607e9i			0.623323i
$540$		0			0
$541$		−	2.00110e9i		−	0.543347i	−0.962389	−	0.271674i	$-0.912423\pi$
$541$		−	2.00110e9i		−	0.543347i	0.962389	−	0.271674i	$-0.0875771\pi$
$542$	−2.37815e9	+	1.95472e9i	−0.641568	+	0.527336i
$543$		0			0
$544$	−1.21285e9	+	3.72941e8i	−0.323006	+	0.0993215i
$545$		−	2.76517e9i		−	0.731703i
$546$		0			0
$547$	6.96444e9			1.81941			0.909705	−	0.415254i	$-0.136307\pi$
$547$	6.96444e9			1.81941			0.909705	+	0.415254i	$0.136307\pi$
$548$	4.69700e9	+	9.26890e8i	1.21924	+	0.240600i
$549$		0			0
$550$	3.83163e9	−	3.14940e9i	0.982006	−	0.807158i
$551$	1.52226e9			0.387668
$552$		0			0
$553$	−2.22448e8			−0.0559358
$554$	−3.17559e9	+	2.61017e9i	−0.793487	+	0.652205i
$555$		0			0
$556$	−5.66324e8	+	2.86984e9i	−0.139734	+	0.708102i
$557$	−1.51765e8			−0.0372115			−0.0186058	−	0.999827i	$-0.505923\pi$
$557$	−1.51765e8			−0.0372115			−0.0186058	+	0.999827i	$0.505923\pi$
$558$		0			0
$559$		−	1.76643e9i		−	0.427715i
$560$	−7.76970e8	+	1.89198e9i	−0.186959	+	0.455259i
$561$		0			0
$562$	−3.60471e9	+	2.96289e9i	−0.856631	+	0.704106i
$563$		−	4.29891e9i		−	1.01526i	−0.861574	−	0.507632i	$-0.830521\pi$
$563$		−	4.29891e9i		−	1.01526i	0.861574	−	0.507632i	$-0.169479\pi$
$564$		0			0
$565$			7.37282e8i			0.171975i
$566$	−3.22400e9	−	3.92239e9i	−0.747373	−	0.909270i
$567$		0			0
$568$	−7.43912e9	+	3.99808e9i	−1.70334	+	0.915444i
$569$			5.91647e9i			1.34639i	0.739467	+	0.673193i	$0.235078\pi$
$569$			5.91647e9i			1.34639i	−0.739467	+	0.673193i	$0.764922\pi$
$570$		0			0
$571$	−3.96765e9			−0.891881			−0.445941	−	0.895063i	$-0.647131\pi$
$571$	−3.96765e9			−0.891881			−0.445941	+	0.895063i	$0.647131\pi$
$572$	1.98370e9	−	1.00524e10i	0.443191	−	2.24586i
$573$		0			0
$574$	−2.51076e9	−	3.05464e9i	−0.554132	−	0.674169i
$575$	−1.54035e9			−0.337896
$576$		0			0
$577$	−6.30305e9			−1.36595			−0.682976	−	0.730441i	$-0.739315\pi$
$577$	−6.30305e9			−1.36595			−0.682976	+	0.730441i	$0.739315\pi$
$578$	2.61122e9	+	3.17687e9i	0.562466	+	0.684308i
$579$		0			0
$580$	5.03544e8	−	2.55170e9i	0.107162	−	0.543040i
$581$	2.15187e9			0.455197
$582$		0			0
$583$			1.43638e10i			3.00213i
$584$	5.10732e9	−	2.74487e9i	1.06108	−	0.570266i
$585$		0			0
$586$	1.98111e9	+	2.41027e9i	0.406694	+	0.494793i
$587$			2.41073e8i			0.0491944i	0.999697	+	0.0245972i	$0.00783032\pi$
$587$			2.41073e8i			0.0491944i	−0.999697	+	0.0245972i	$0.992170\pi$
$588$		0			0
$589$		−	3.08540e9i		−	0.622168i
$590$	1.46189e8	−	1.20159e8i	0.0293043	−	0.0240866i
$591$		0			0
$592$	1.68303e9	−	4.09831e9i	0.333401	−	0.811856i
$593$			1.32868e9i			0.261655i	0.991405	+	0.130828i	$0.0417635\pi$
$593$			1.32868e9i			0.261655i	−0.991405	+	0.130828i	$0.958237\pi$
$594$		0			0
$595$	−8.54551e8			−0.166314
$596$	−6.21508e8	+	3.14948e9i	−0.120250	+	0.609365i
$597$		0			0
$598$	−2.45828e9	+	2.02058e9i	−0.470086	+	0.386386i
$599$	−7.40415e9			−1.40761			−0.703803	−	0.710395i	$-0.748516\pi$
$599$	−7.40415e9			−1.40761			−0.703803	+	0.710395i	$0.748516\pi$
$600$		0			0
$601$	1.16287e7			0.00218509			0.00109254	−	0.999999i	$-0.499652\pi$
$601$	1.16287e7			0.00218509			0.00109254	+	0.999999i	$0.499652\pi$
$602$	−1.25767e9	+	1.03374e9i	−0.234952	+	0.193119i
$603$		0			0
$604$	4.66713e9	+	9.20995e8i	0.861828	+	0.170070i
$605$	−9.32871e9			−1.71269
$606$		0			0
$607$		−	7.47105e9i		−	1.35588i	−0.735117	−	0.677940i	$-0.762873\pi$
$607$		−	7.47105e9i		−	1.35588i	0.735117	−	0.677940i	$-0.237127\pi$
$608$	6.79144e8	+	2.20866e9i	0.122546	+	0.398535i
$609$		0			0
$610$	−1.53050e9	+	1.25799e9i	−0.273010	+	0.224400i
$611$		−	1.27422e9i		−	0.225995i
$612$		0			0
$613$			3.85770e9i			0.676420i	0.941071	+	0.338210i	$0.109821\pi$
$613$			3.85770e9i			0.676420i	−0.941071	+	0.338210i	$0.890179\pi$
$614$	−6.98009e9	−	8.49213e9i	−1.21695	−	1.48056i
$615$		0			0
$616$	−8.31806e9	+	4.47045e9i	−1.43380	+	0.770582i
$617$		−	6.73937e9i		−	1.15511i	−0.816354	−	0.577553i	$-0.804008\pi$
$617$		−	6.73937e9i		−	1.15511i	0.816354	−	0.577553i	$-0.195992\pi$
$618$		0			0
$619$	2.08224e9			0.352869			0.176434	−	0.984312i	$-0.443544\pi$
$619$	2.08224e9			0.352869			0.176434	+	0.984312i	$0.443544\pi$
$620$	−5.17191e9	−	1.02061e9i	−0.871525	−	0.171984i
$621$		0			0
$622$	5.41950e9	+	6.59348e9i	0.903010	+	1.09862i
$623$	−8.56176e9			−1.41858
$624$		0			0
$625$	3.80718e8			0.0623769
$626$	1.79165e9	+	2.17976e9i	0.291906	+	0.355139i
$627$		0			0
$628$	9.72677e9	+	1.91945e9i	1.56715	+	0.309255i
$629$	1.85109e9			0.296585
$630$		0			0
$631$		−	1.02675e10i		−	1.62690i	−0.581633	−	0.813451i	$-0.697586\pi$
$631$		−	1.02675e10i		−	1.62690i	0.581633	−	0.813451i	$-0.302414\pi$
$632$	2.03274e8	+	3.78228e8i	0.0320312	+	0.0595996i
$633$		0			0
$634$	3.30147e9	+	4.01663e9i	0.514511	+	0.625965i
$635$			4.61181e9i			0.714765i
$636$		0			0
$637$			2.40102e9i			0.368050i
$638$	9.27693e9	−	7.62516e9i	1.41427	−	1.16246i
$639$		0			0
$640$	3.92693e9	−	4.07823e8i	0.592139	−	0.0614953i
$641$		−	7.12515e9i		−	1.06854i	−0.845314	−	0.534270i	$-0.820586\pi$
$641$		−	7.12515e9i		−	1.06854i	0.845314	−	0.534270i	$-0.179414\pi$
$642$		0			0
$643$	−4.07206e9			−0.604054			−0.302027	−	0.953299i	$-0.597663\pi$
$643$	−4.07206e9			−0.604054			−0.302027	+	0.953299i	$0.597663\pi$
$644$	2.87725e9	+	5.67786e8i	0.424499	+	0.0837692i
$645$		0			0
$646$	−7.45834e8	+	6.13037e8i	−0.108850	+	0.0894691i
$647$	9.52423e9			1.38250			0.691250	−	0.722616i	$-0.257060\pi$
$647$	9.52423e9			1.38250			0.691250	+	0.722616i	$0.257060\pi$
$648$		0			0
$649$	8.73700e8			0.125460
$650$	4.05980e9	−	3.33695e9i	0.579840	−	0.476599i
$651$		0			0
$652$	−1.74981e9	+	8.86713e9i	−0.247243	+	1.25290i
$653$	−9.31382e9			−1.30898			−0.654488	−	0.756072i	$-0.727116\pi$
$653$	−9.31382e9			−1.30898			−0.654488	+	0.756072i	$0.727116\pi$
$654$		0			0
$655$			2.86269e9i			0.398043i
$656$	−2.89946e9	+	7.06040e9i	−0.401008	+	0.976486i
$657$		0			0
$658$	−9.07224e8	+	7.45691e8i	−0.124143	+	0.102040i
$659$		−	3.37116e9i		−	0.458860i	−0.973325	−	0.229430i	$-0.926314\pi$
$659$		−	3.37116e9i		−	0.458860i	0.973325	−	0.229430i	$-0.0736862\pi$
$660$		0			0
$661$		−	4.67132e8i		−	0.0629122i	−0.999505	−	0.0314561i	$-0.989986\pi$
$661$		−	4.67132e8i		−	0.0629122i	0.999505	−	0.0314561i	$-0.0100144\pi$
$662$	1.21808e9	+	1.48195e9i	0.163183	+	0.198532i
$663$		0			0
$664$	−1.96640e9	−	3.65882e9i	−0.260665	−	0.485013i
$665$			1.55618e9i			0.205203i
$666$		0			0
$667$	−3.72942e9			−0.486632
$668$	1.69406e9	−	8.58460e9i	0.219892	−	1.11430i
$669$		0			0
$670$	3.26244e9	+	3.96916e9i	0.419065	+	0.509843i
$671$	−9.14708e9			−1.16884
$672$		0			0
$673$	7.88186e9			0.996727			0.498364	−	0.866968i	$-0.333935\pi$
$673$	7.88186e9			0.996727			0.498364	+	0.866968i	$0.333935\pi$
$674$	7.94807e9	+	9.66979e9i	0.999890	+	1.21649i
$675$		0			0
$676$	5.46851e8	−	2.77116e9i	0.0680857	−	0.345023i
$677$	−1.42433e10			−1.76421			−0.882104	−	0.471054i	$-0.843874\pi$
$677$	−1.42433e10			−1.76421			−0.882104	+	0.471054i	$0.843874\pi$
$678$		0			0
$679$		−	8.42302e9i		−	1.03258i
$680$	7.80895e8	+	1.45299e9i	0.0952382	+	0.177207i
$681$		0			0
$682$	−1.54550e10	−	1.88029e10i	−1.86563	−	2.26976i
$683$			3.97439e9i			0.477308i	0.971105	+	0.238654i	$0.0767061\pi$
$683$			3.97439e9i			0.477308i	−0.971105	+	0.238654i	$0.923294\pi$
$684$		0			0
$685$		−	6.22378e9i		−	0.739839i
$686$	7.10952e9	−	5.84366e9i	0.840826	−	0.691115i
$687$		0			0
$688$	2.90694e9	+	1.19378e9i	0.340311	+	0.139754i
$689$			1.52192e10i			1.77265i
$690$		0			0
$691$	1.14648e9			0.132189			0.0660944	−	0.997813i	$-0.478946\pi$
$691$	1.14648e9			0.132189			0.0660944	+	0.997813i	$0.478946\pi$
$692$	1.14983e9	−	5.82675e9i	0.131905	−	0.668429i
$693$		0			0
$694$	6.99912e8	−	5.75292e8i	0.0794850	−	0.0653326i
$695$	3.80269e9			0.429679
$696$		0			0
$697$	−3.18897e9			−0.356727
$698$	−5.77903e9	+	4.75007e9i	−0.643222	+	0.528696i
$699$		0			0
$700$	−4.75172e9	−	9.37689e8i	−0.523610	−	0.103327i
$701$	5.42915e9			0.595277			0.297638	−	0.954679i	$-0.403801\pi$
$701$	5.42915e9			0.595277			0.297638	+	0.954679i	$0.403801\pi$
$702$		0			0
$703$		−	3.37092e9i		−	0.365936i
$704$	1.52022e10	+	1.00581e10i	1.64211	+	1.08645i
$705$		0			0
$706$	−9.42453e9	+	7.74648e9i	−1.00796	+	0.828491i
$707$		−	8.68688e9i		−	0.924477i
$708$		0			0
$709$		−	2.46125e9i		−	0.259354i	−0.991556	−	0.129677i	$-0.958606\pi$
$709$		−	2.46125e9i		−	0.259354i	0.991556	−	0.129677i	$-0.0413941\pi$
$710$	6.97138e9	+	8.48153e9i	0.730995	+	0.889344i
$711$		0			0
$712$	7.82380e9	+	1.45576e10i	0.812340	+	1.51150i
$713$			7.55896e9i			0.780995i
$714$		0			0
$715$	−1.33200e10			−1.36280
$716$	−1.40451e9	−	2.77161e8i	−0.142998	−	0.0282187i
$717$		0			0
$718$	−1.68787e9	−	2.05350e9i	−0.170179	−	0.207043i
$719$	1.87766e10			1.88393			0.941967	−	0.335705i	$-0.108975\pi$
$719$	1.87766e10			1.88393			0.941967	+	0.335705i	$0.108975\pi$
$720$		0			0
$721$	1.35320e10			1.34459
$722$	−5.30517e9	−	6.45439e9i	−0.524589	−	0.638227i
$723$		0			0
$724$	1.79015e10	+	3.53262e9i	1.75309	+	0.345949i
$725$	6.15906e9			0.600249
$726$		0			0
$727$		−	2.64393e9i		−	0.255200i	−0.991826	−	0.127600i	$-0.959273\pi$
$727$		−	2.64393e9i		−	0.255200i	0.991826	−	0.127600i	$-0.0407273\pi$
$728$	−8.81340e9	+	4.73666e9i	−0.846610	+	0.455002i
$729$		0			0
$730$	−4.78619e9	−	5.82298e9i	−0.455365	−	0.554007i
$731$			1.31298e9i			0.124322i
$732$		0			0
$733$		−	1.00402e10i		−	0.941629i	−0.882232	−	0.470815i	$-0.843960\pi$
$733$		−	1.00402e10i		−	0.941629i	0.882232	−	0.470815i	$-0.156040\pi$
$734$	4.45577e9	−	3.66241e9i	0.415897	−	0.341846i
$735$		0			0
$736$	−1.66385e9	−	5.41103e9i	−0.153830	−	0.500274i
$737$			2.37218e10i			2.18279i
$738$		0			0
$739$	−3.44469e8			−0.0313975			−0.0156987	−	0.999877i	$-0.504997\pi$
$739$	−3.44469e8			−0.0313975			−0.0156987	+	0.999877i	$0.504997\pi$
$740$	−5.65053e9	−	1.11505e9i	−0.512599	−	0.101155i
$741$		0			0
$742$	1.08359e10	−	8.90651e9i	0.973755	−	0.800376i
$743$	−6.52679e9			−0.583766			−0.291883	−	0.956454i	$-0.594282\pi$
$743$	−6.52679e9			−0.583766			−0.291883	+	0.956454i	$0.594282\pi$
$744$		0			0
$745$	4.17323e9			0.369765
$746$	1.04500e10	−	8.58932e9i	0.921570	−	0.757483i
$747$		0			0
$748$	−1.47448e9	+	7.47190e9i	−0.128820	+	0.652793i
$749$	3.78641e9			0.329262
$750$		0			0
$751$		−	5.80249e9i		−	0.499890i	−0.968260	−	0.249945i	$-0.919587\pi$
$751$		−	5.80249e9i		−	0.499890i	0.968260	−	0.249945i	$-0.0804126\pi$
$752$	2.09693e9	+	8.61134e8i	0.179813	+	0.0738428i
$753$		0			0
$754$	9.82937e9	−	8.07923e9i	0.835076	−	0.686389i
$755$		−	6.18420e9i		−	0.522961i
$756$		0			0
$757$			7.76287e9i			0.650409i	0.945644	+	0.325204i	$0.105433\pi$
$757$			7.76287e9i			0.650409i	−0.945644	+	0.325204i	$0.894567\pi$
$758$	−9.12478e9	−	1.11014e10i	−0.760992	−	0.925839i
$759$		0			0
$760$	2.64597e9	−	1.42205e9i	0.218644	−	0.117508i
$761$		−	2.04247e10i		−	1.68000i	−0.542587	−	0.839999i	$-0.682555\pi$
$761$		−	2.04247e10i		−	1.68000i	0.542587	−	0.839999i	$-0.317445\pi$
$762$		0			0
$763$	−1.24671e10			−1.01608
$764$	−1.60198e9	+	8.11801e9i	−0.129966	+	0.658602i
$765$		0			0
$766$	−7.68499e9	−	9.34972e9i	−0.617792	−	0.751619i
$767$	9.25728e8			0.0740798
$768$		0			0
$769$	1.69057e10			1.34058			0.670289	−	0.742100i	$-0.266170\pi$
$769$	1.69057e10			1.34058			0.670289	+	0.742100i	$0.266170\pi$
$770$	7.79505e9	+	9.48363e9i	0.615321	+	0.748612i
$771$		0			0
$772$	2.66577e9	−	1.35087e10i	0.208527	−	1.05671i
$773$	−9.71619e9			−0.756603			−0.378301	−	0.925682i	$-0.623492\pi$
$773$	−9.71619e9			−0.756603			−0.378301	+	0.925682i	$0.623492\pi$
$774$		0			0
$775$		−	1.24835e10i		−	0.963340i
$776$	−1.43217e10	+	7.69702e9i	−1.10021	+	0.591298i
$777$		0			0
$778$	2.93945e9	+	3.57620e9i	0.223788	+	0.272266i
$779$			5.80729e9i			0.440141i
$780$		0			0
$781$			5.06901e10i			3.80754i
$782$	1.82723e9	−	1.50189e9i	0.136637	−	0.112309i
$783$		0			0
$784$	−3.95126e9	−	1.62264e9i	−0.292839	−	0.120259i
$785$		−	1.28885e10i		−	0.950951i
$786$		0			0
$787$	2.20410e10			1.61183			0.805915	−	0.592031i	$-0.201674\pi$
$787$	2.20410e10			1.61183			0.805915	+	0.592031i	$0.201674\pi$
$788$	−1.85494e9	+	9.39986e9i	−0.135048	+	0.684352i
$789$		0			0
$790$	4.31227e8	−	3.54446e8i	0.0311180	−	0.0255774i
$791$	3.32411e9			0.238813
$792$		0			0
$793$	−9.69179e9			−0.690157
$794$	−3.67212e9	+	3.01829e9i	−0.260342	+	0.213988i
$795$		0			0
$796$	4.40310e9	+	8.68892e8i	0.309430	+	0.0610619i
$797$	1.13976e10			0.797460			0.398730	−	0.917068i	$-0.369451\pi$
$797$	1.13976e10			0.797460			0.398730	+	0.917068i	$0.369451\pi$
$798$		0			0
$799$			9.47119e8i			0.0656887i
$800$	2.74781e9	+	8.93622e9i	0.189746	+	0.617076i
$801$		0			0
$802$	−8.08117e9	+	6.64231e9i	−0.553177	+	0.454683i
$803$		−	3.48012e10i		−	2.37187i
$804$		0			0
$805$		−	3.81251e9i		−	0.257588i
$806$	−1.63754e10	−	1.99226e10i	−1.10159	−	1.34021i
$807$		0			0
$808$	−1.47703e10	+	7.93814e9i	−0.985030	+	0.529394i
$809$			3.32572e9i			0.220834i	0.993885	+	0.110417i	$0.0352186\pi$
$809$			3.32572e9i			0.220834i	−0.993885	+	0.110417i	$0.964781\pi$
$810$		0			0
$811$	−1.93568e10			−1.27427			−0.637133	−	0.770754i	$-0.719880\pi$
$811$	−1.93568e10			−1.27427			−0.637133	+	0.770754i	$0.719880\pi$
$812$	−1.15046e10	−	2.27028e9i	−0.754095	−	0.148810i
$813$		0			0
$814$	−1.68853e10	−	2.05430e10i	−1.09729	−	1.33499i
$815$	1.17494e10			0.760265
$816$		0			0
$817$	2.39100e9			0.153392
$818$	−1.00449e10	−	1.22208e10i	−0.641665	−	0.780664i
$819$		0			0
$820$	9.73449e9	+	1.92097e9i	0.616545	+	0.121667i
$821$	−1.00753e9			−0.0635417			−0.0317709	−	0.999495i	$-0.510115\pi$
$821$	−1.00753e9			−0.0635417			−0.0317709	+	0.999495i	$0.510115\pi$
$822$		0			0
$823$		−	6.91310e9i		−	0.432288i	−0.976361	−	0.216144i	$-0.930652\pi$
$823$		−	6.91310e9i		−	0.432288i	0.976361	−	0.216144i	$-0.0693481\pi$
$824$	−1.23656e10	−	2.30085e10i	−0.769966	−	1.43266i
$825$		0			0
$826$	−5.41751e8	−	6.59106e8i	−0.0334480	−	0.0406935i
$827$			1.99238e9i			0.122491i	0.998123	+	0.0612454i	$0.0195072\pi$
$827$			1.99238e9i			0.122491i	−0.998123	+	0.0612454i	$0.980493\pi$
$828$		0			0
$829$		−	9.21131e9i		−	0.561540i	−0.959775	−	0.280770i	$-0.909410\pi$
$829$		−	9.21131e9i		−	0.561540i	0.959775	−	0.280770i	$-0.0905898\pi$
$830$	−4.17152e9	+	3.42877e9i	−0.253233	+	0.208145i
$831$		0			0
$832$	1.61075e10	+	1.06570e10i	0.969608	+	0.641511i
$833$		−	1.78467e9i		−	0.106979i
$834$		0			0
$835$	−1.13751e10			−0.676162
$836$	1.36067e10	+	2.68510e9i	0.805437	+	0.158942i
$837$		0			0
$838$	−1.60317e10	+	1.31772e10i	−0.941077	+	0.773517i
$839$	1.95616e10			1.14351			0.571753	−	0.820426i	$-0.306264\pi$
$839$	1.95616e10			1.14351			0.571753	+	0.820426i	$0.306264\pi$
$840$		0			0
$841$	−2.33790e9			−0.135532
$842$	−1.91000e10	+	1.56992e10i	−1.10266	+	0.906328i
$843$		0			0
$844$	1.56825e9	−	7.94711e9i	0.0897879	−	0.454999i
$845$	−3.67194e9			−0.209361
$846$		0			0
$847$			4.20595e10i			2.37833i
$848$	−2.50456e10	−	1.02854e10i	−1.41041	−	0.579207i
$849$		0			0
$850$	−3.01764e9	+	2.48034e9i	−0.168539	+	0.138530i
$851$			8.25847e9i			0.459353i
$852$		0			0
$853$			2.07361e10i			1.14394i	0.820273	+	0.571972i	$0.193821\pi$
$853$			2.07361e10i			1.14394i	−0.820273	+	0.571972i	$0.806179\pi$
$854$	5.67179e9	+	6.90042e9i	0.311614	+	0.379117i
$855$		0			0
$856$	−3.46005e9	−	6.43804e9i	−0.188549	−	0.350829i
$857$			3.15860e10i			1.71420i	0.515151	+	0.857099i	$0.327736\pi$
$857$			3.15860e10i			1.71420i	−0.515151	+	0.857099i	$0.672264\pi$
$858$		0			0
$859$	1.20261e10			0.647362			0.323681	−	0.946166i	$-0.395080\pi$
$859$	1.20261e10			0.647362			0.323681	+	0.946166i	$0.395080\pi$
$860$	7.90910e8	−	4.00792e9i	0.0424016	−	0.214870i
$861$		0			0
$862$	2.40671e9	+	2.92805e9i	0.127982	+	0.155705i
$863$	4.93491e9			0.261362			0.130681	−	0.991424i	$-0.458284\pi$
$863$	4.93491e9			0.261362			0.130681	+	0.991424i	$0.458284\pi$
$864$		0			0
$865$	−7.72076e9			−0.405605
$866$	−6.21441e9	−	7.56059e9i	−0.325153	−	0.395588i
$867$		0			0
$868$	−4.60151e9	+	2.33181e10i	−0.238826	+	1.21025i
$869$	2.57724e9			0.133225
$870$		0			0
$871$			2.51344e10i			1.28886i
$872$	1.13925e10	+	2.11978e10i	0.581850	+	1.08264i
$873$		0			0
$874$	−2.73502e9	−	3.32748e9i	−0.138570	−	0.168588i
$875$			1.60491e10i			0.809881i
$876$		0			0
$877$			6.10257e9i			0.305502i	0.988265	+	0.152751i	$0.0488133\pi$
$877$			6.10257e9i			0.305502i	−0.988265	+	0.152751i	$0.951187\pi$
$878$	−1.51567e10	+	1.24581e10i	−0.755744	+	0.621183i
$879$		0			0
$880$	9.00183e9	−	2.19201e10i	0.445289	−	1.08431i
$881$		−	1.47462e9i		−	0.0726548i	−0.999340	−	0.0363274i	$-0.988434\pi$
$881$		−	1.47462e9i		−	0.0726548i	0.999340	−	0.0363274i	$-0.0115659\pi$
$882$		0			0
$883$	1.11517e10			0.545103			0.272552	−	0.962141i	$-0.412132\pi$
$883$	1.11517e10			0.545103			0.272552	+	0.962141i	$0.412132\pi$
$884$	−1.56228e9	+	7.91685e9i	−0.0760636	+	0.385451i
$885$		0			0
$886$	1.92109e9	−	1.57903e9i	0.0927959	−	0.0762734i
$887$	−8.52333e9			−0.410088			−0.205044	−	0.978753i	$-0.565734\pi$
$887$	−8.52333e9			−0.410088			−0.205044	+	0.978753i	$0.565734\pi$
$888$		0			0
$889$	2.07928e10			0.992561
$890$	1.65974e10	−	1.36422e10i	0.789180	−	0.648665i
$891$		0			0
$892$	−2.40569e10	−	4.74731e9i	−1.13492	−	0.223960i
$893$	1.72475e9			0.0810489
$894$		0			0
$895$			1.86105e9i			0.0867716i
$896$	−1.83871e9	−	1.77050e10i	−0.0853957	−	0.822275i
$897$		0			0
$898$	3.93960e9	−	3.23815e9i	0.181545	−	0.149221i
$899$		−	3.02243e10i		−	1.38739i
$900$		0			0
$901$		−	1.13124e10i		−	0.515248i
$902$	2.90892e10	+	3.53906e10i	1.31980	+	1.60570i
$903$		0			0
$904$	−3.03760e9	−	5.65199e9i	−0.136754	−	0.254455i
$905$		−	2.37205e10i		−	1.06378i
$906$		0			0
$907$	−2.69804e10			−1.20067			−0.600334	−	0.799749i	$-0.704966\pi$
$907$	−2.69804e10			−1.20067			−0.600334	+	0.799749i	$0.704966\pi$
$908$	−1.34699e10	−	2.65811e9i	−0.597125	−	0.117835i
$909$		0			0
$910$	8.25925e9	+	1.00484e10i	0.363325	+	0.442029i
$911$	−1.36401e10			−0.597726			−0.298863	−	0.954296i	$-0.596607\pi$
$911$	−1.36401e10			−0.597726			−0.298863	+	0.954296i	$0.596607\pi$
$912$		0			0
$913$	−2.49312e10			−1.08416
$914$	1.97681e9	+	2.40503e9i	0.0856354	+	0.104186i
$915$		0			0
$916$	−3.49807e10	−	6.90297e9i	−1.50381	−	0.296758i
$917$	1.29067e10			0.552743
$918$		0			0
$919$		−	2.57488e10i		−	1.09434i	−0.837020	−	0.547172i	$-0.815705\pi$
$919$		−	2.57488e10i		−	1.09434i	0.837020	−	0.547172i	$-0.184295\pi$
$920$	−6.48241e9	+	3.48390e9i	−0.274460	+	0.147506i
$921$		0			0
$922$	5.21761e9	+	6.34785e9i	0.219237	+	0.266728i
$923$			5.37087e10i			2.24822i
$924$		0			0
$925$		−	1.36387e10i		−	0.566601i
$926$	−7.23381e9	+	5.94582e9i	−0.299384	+	0.246078i
$927$		0			0
$928$	6.65284e9	+	2.16359e10i	0.273268	+	0.888703i
$929$		−	3.26697e10i		−	1.33687i	−0.743769	−	0.668436i	$-0.766964\pi$
$929$		−	3.26697e10i		−	1.33687i	0.743769	−	0.668436i	$-0.233036\pi$
$930$		0			0
$931$	−3.24997e9			−0.131994
$932$	−9.20520e8	−	1.81652e8i	−0.0372458	−	0.00734996i
$933$		0			0
$934$	1.96839e10	−	1.61792e10i	0.790492	−	0.649744i
$935$	9.90067e9			0.396118
$936$		0			0
$937$	1.41361e10			0.561358			0.280679	−	0.959802i	$-0.409440\pi$
$937$	1.41361e10			0.561358			0.280679	+	0.959802i	$0.409440\pi$
$938$	1.78954e10	−	1.47091e10i	0.707995	−	0.581936i
$939$		0			0
$940$	5.70524e8	−	2.89112e9i	0.0224041	−	0.113532i
$941$	2.18935e10			0.856548			0.428274	−	0.903649i	$-0.359122\pi$
$941$	2.18935e10			0.856548			0.428274	+	0.903649i	$0.359122\pi$
$942$		0			0
$943$		−	1.42274e10i		−	0.552501i
$944$	−6.25621e8	+	1.52344e9i	−0.0242053	+	0.0589416i
$945$		0			0
$946$	1.45712e10	−	1.19767e10i	0.559596	−	0.459959i
$947$		−	3.93208e9i		−	0.150452i	−0.997167	−	0.0752259i	$-0.976032\pi$
$947$		−	3.93208e9i		−	0.150452i	0.997167	−	0.0752259i	$-0.0239678\pi$
$948$		0			0
$949$		−	3.68736e10i		−	1.40050i
$950$	4.51683e9	+	5.49527e9i	0.170923	+	0.207949i
$951$		0			0
$952$	6.55096e9	−	3.52074e9i	0.246080	−	0.132253i
$953$		−	2.53127e9i		−	0.0947355i	−0.998878	−	0.0473677i	$-0.984917\pi$
$953$		−	2.53127e9i		−	0.0947355i	0.998878	−	0.0473677i	$-0.0150833\pi$
$954$		0			0
$955$	1.07568e10			0.399642
$956$	−1.77421e9	+	8.99078e9i	−0.0656753	+	0.332809i
$957$		0			0
$958$	2.39055e10	+	2.90839e10i	0.878451	+	1.06874i
$959$	−2.80606e10			−1.02738
$960$		0			0
$961$	−3.37474e10			−1.22662
$962$	−1.78908e10	−	2.17663e10i	−0.647913	−	0.788264i
$963$		0			0
$964$	6.63384e9	−	3.36169e10i	0.238504	−	1.20861i
$965$	−1.78998e10			−0.641213
$966$		0			0
$967$		−	1.74142e9i		−	0.0619314i	−0.999520	−	0.0309657i	$-0.990142\pi$
$967$		−	1.74142e9i		−	0.0619314i	0.999520	−	0.0309657i	$-0.00985826\pi$
$968$	7.15137e10	−	3.84342e10i	2.53411	−	1.36193i
$969$		0			0
$970$	1.34212e10	+	1.63285e10i	0.472160	+	0.574440i
$971$		−	5.71812e9i		−	0.200441i	−0.994965	−	0.100220i	$-0.968045\pi$
$971$		−	5.71812e9i		−	0.200441i	0.994965	−	0.100220i	$-0.0319548\pi$
$972$		0			0
$973$		−	1.71448e10i		−	0.596675i
$974$	−2.80898e9	+	2.30884e9i	−0.0974076	+	0.0800640i
$975$		0			0
$976$	6.54986e9	−	1.59494e10i	0.225506	−	0.549123i
$977$		−	1.64895e10i		−	0.565688i	−0.959166	−	0.282844i	$-0.908722\pi$
$977$		−	1.64895e10i		−	0.565688i	0.959166	−	0.282844i	$-0.0912778\pi$
$978$		0			0
$979$	9.91951e10			3.37871
$980$	−1.07504e9	+	5.44778e9i	−0.0364868	+	0.184896i
$981$		0			0
$982$	−3.14995e10	+	2.58910e10i	−1.06149	+	0.872486i
$983$	−2.20432e10			−0.740182			−0.370091	−	0.928996i	$-0.620673\pi$
$983$	−2.20432e10			−0.740182			−0.370091	+	0.928996i	$0.620673\pi$
$984$		0			0
$985$	1.24553e10			0.415268
$986$	−7.30613e9	+	6.00526e9i	−0.242727	+	0.199509i
$987$		0			0
$988$	1.44170e10	+	2.84500e9i	0.475582	+	0.0938498i
$989$	−5.85775e9			−0.192550
$990$		0			0
$991$			7.03533e9i			0.229629i	0.993387	+	0.114814i	$0.0366273\pi$
$991$			7.03533e9i			0.229629i	−0.993387	+	0.114814i	$0.963373\pi$
$992$	4.38526e10	−	1.34843e10i	1.42628	−	0.438568i
$993$		0			0
$994$	3.82399e10	−	3.14312e10i	1.23499	−	1.01510i
$995$		−	5.83434e9i		−	0.187764i
$996$		0			0
$997$		−	2.68981e10i		−	0.859584i	−0.902928	−	0.429792i	$-0.858587\pi$
$997$		−	2.68981e10i		−	0.859584i	0.902928	−	0.429792i	$-0.141413\pi$
$998$	−1.29873e9	−	1.58006e9i	−0.0413582	−	0.0503172i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.f.a.35.20	yes	28
3.2	odd	2	inner	72.8.f.a.35.9	✓	28
4.3	odd	2		288.8.f.a.143.17		28
8.3	odd	2	inner	72.8.f.a.35.10	yes	28
8.5	even	2		288.8.f.a.143.12		28
12.11	even	2		288.8.f.a.143.11		28
24.5	odd	2		288.8.f.a.143.18		28
24.11	even	2	inner	72.8.f.a.35.19	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.9	✓	28	3.2	odd	2	inner
72.8.f.a.35.10	yes	28	8.3	odd	2	inner
72.8.f.a.35.19	yes	28	24.11	even	2	inner
72.8.f.a.35.20	yes	28	1.1	even	1	trivial
288.8.f.a.143.11		28	12.11	even	2
288.8.f.a.143.12		28	8.5	even	2
288.8.f.a.143.17		28	4.3	odd	2
288.8.f.a.143.18		28	24.5	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.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(7.18397 + 8.74017i) q^{2} +(-24.7812 + 125.578i) q^{4} +166.398 q^{5} -750.222i q^{7} +(-1275.60 + 685.559i) q^{8} +O(q^{10})\)	\(q+(7.18397 + 8.74017i) q^{2} +(-24.7812 + 125.578i) q^{4} +166.398 q^{5} -750.222i q^{7} +(-1275.60 + 685.559i) q^{8} +(1195.40 + 1454.35i) q^{10} +8691.94i q^{11} +9209.54i q^{13} +(6557.07 - 5389.57i) q^{14} +(-15155.8 - 6223.95i) q^{16} -6845.42i q^{17} -12465.9 q^{19} +(-4123.53 + 20896.0i) q^{20} +(-75969.1 + 62442.6i) q^{22} +30540.3 q^{23} -50436.7 q^{25} +(-80493.0 + 66161.1i) q^{26} +(94211.6 + 18591.4i) q^{28} -122115. q^{29} +247508. i q^{31} +(-54480.3 - 177177. i) q^{32} +(59830.1 - 49177.3i) q^{34} -124835. i q^{35} +270412. i q^{37} +(-89554.4 - 108954. i) q^{38} +(-212258. + 114076. i) q^{40} -465855. i q^{41} -191804. q^{43} +(-1.09152e6 - 215396. i) q^{44} +(219401. + 266927. i) q^{46} -138358. q^{47} +260710. q^{49} +(-362336. - 440826. i) q^{50} +(-1.15652e6 - 228223. i) q^{52} +1.65255e6 q^{53} +1.44632e6i q^{55} +(514321. + 956985. i) q^{56} +(-877267. - 1.06730e6i) q^{58} -100518. i q^{59} +1.05236e6i q^{61} +(-2.16326e6 + 1.77809e6i) q^{62} +(1.15717e6 - 1.74900e6i) q^{64} +1.53245e6i q^{65} +2.72917e6 q^{67} +(859636. + 169637. i) q^{68} +(1.09108e6 - 896814. i) q^{70} +5.83185e6 q^{71} -4.00385e6 q^{73} +(-2.36345e6 + 1.94263e6i) q^{74} +(308919. - 1.56544e6i) q^{76} +6.52089e6 q^{77} -296509. i q^{79} +(-2.52189e6 - 1.03565e6i) q^{80} +(4.07165e6 - 3.34669e6i) q^{82} +2.86831e6i q^{83} -1.13906e6i q^{85} +(-1.37791e6 - 1.67640e6i) q^{86} +(-5.95883e6 - 1.10875e7i) q^{88} -1.14123e7i q^{89} +6.90921e6 q^{91} +(-756824. + 3.83520e6i) q^{92} +(-993960. - 1.20927e6i) q^{94} -2.07429e6 q^{95} +1.12274e7 q^{97} +(1.87293e6 + 2.27865e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 52 q^{4}+O(q^{10}) \) 28 * q + 52 * q^4	\( 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) \) 28 * q + 52 * q^4 + 10092 * q^10 - 1928 * q^16 - 121168 * q^19 + 59576 * q^22 + 437500 * q^25 + 46872 * q^28 - 114748 * q^34 + 1054752 * q^40 + 1505696 * q^43 - 476184 * q^46 - 2272076 * q^49 + 1468392 * q^52 + 3054996 * q^58 - 4186016 * q^64 - 776272 * q^67 + 3238872 * q^70 - 2534128 * q^73 - 21642832 * q^76 + 10334372 * q^82 + 10834016 * q^88 - 3406992 * q^91 - 22555944 * q^94 - 26311456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more