LMFDB - Embedded newform 153.8.d.b.118.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [153,8,Mod(118,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("153.118");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 153 = 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	153.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:	$47.7949088991$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 16832x^{8} + 93191572x^{6} + 192821327856x^{4} + 116860780245888x^{2} + 9421474370420736 $ x^10 + 16832x^8 + 93191572x^6 + 192821327856x^4 + 116860780245888x^2 + 9421474370420736
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			118.7
Root			$9.73524i$ of defining polynomial
Character	$\chi$	$=$	153.118
Dual form			153.8.d.b.118.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.2698 q^{2} -0.992706 q^{4} -181.480i q^{5} -1340.88i q^{7} -1453.72 q^{8} +O(q^{10})$	\(q+11.2698 q^{2} -0.992706 q^{4} -181.480i q^{5} -1340.88i q^{7} -1453.72 q^{8} -2045.24i q^{10} +5783.08i q^{11} +834.906 q^{13} -15111.4i q^{14} -16255.9 q^{16} +(-12887.2 - 15628.8i) q^{17} -9260.11 q^{19} +180.157i q^{20} +65173.9i q^{22} +107751. i q^{23} +45189.9 q^{25} +9409.18 q^{26} +1331.10i q^{28} +101843. i q^{29} +20966.7i q^{31} +2875.12 q^{32} +(-145236. - 176132. i) q^{34} -243344. q^{35} +30107.7i q^{37} -104359. q^{38} +263821. i q^{40} +395730. i q^{41} -564125. q^{43} -5740.90i q^{44} +1.21433e6i q^{46} -502222. q^{47} -974426. q^{49} +509279. q^{50} -828.816 q^{52} -253229. q^{53} +1.04952e6 q^{55} +1.94926e6i q^{56} +1.14775e6i q^{58} -2.58041e6 q^{59} -1.00567e6i q^{61} +236290. i q^{62} +2.11316e6 q^{64} -151519. i q^{65} -3.93496e6 q^{67} +(12793.2 + 15514.8i) q^{68} -2.74243e6 q^{70} -3.95445e6i q^{71} +5.07004e6i q^{73} +339306. i q^{74} +9192.56 q^{76} +7.75444e6 q^{77} -811675. i q^{79} +2.95013e6i q^{80} +4.45977e6i q^{82} +3.58083e6 q^{83} +(-2.83631e6 + 2.33878e6i) q^{85} -6.35754e6 q^{86} -8.40696e6i q^{88} -5.31432e6 q^{89} -1.11951e6i q^{91} -106965. i q^{92} -5.65992e6 q^{94} +1.68053e6i q^{95} +1.41602e7i q^{97} -1.09815e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 18 q^{2} + 690 q^{4} + 3330 q^{8}+O(q^{10}) $ 10 * q + 18 * q^2 + 690 * q^4 + 3330 * q^8	$ 10 q + 18 q^{2} + 690 q^{4} + 3330 q^{8} - 316 q^{13} + 92226 q^{16} + 29454 q^{17} - 90184 q^{19} + 78370 q^{25} + 180420 q^{26} + 1019778 q^{32} + 1123478 q^{34} + 460704 q^{35} + 1583880 q^{38} - 112936 q^{43} - 3214512 q^{47} - 230842 q^{49} + 359514 q^{50} + 4611732 q^{52} + 4396356 q^{53} + 2187488 q^{55} + 6096552 q^{59} + 4541698 q^{64} - 10368488 q^{67} + 712902 q^{68} + 11368032 q^{70} - 14210712 q^{76} - 7163520 q^{77} - 7060104 q^{83} + 8300608 q^{85} - 30033048 q^{86} - 24760956 q^{89} - 54883072 q^{94} - 81770994 q^{98}+O(q^{100}) $ 10 * q + 18 * q^2 + 690 * q^4 + 3330 * q^8 - 316 * q^13 + 92226 * q^16 + 29454 * q^17 - 90184 * q^19 + 78370 * q^25 + 180420 * q^26 + 1019778 * q^32 + 1123478 * q^34 + 460704 * q^35 + 1583880 * q^38 - 112936 * q^43 - 3214512 * q^47 - 230842 * q^49 + 359514 * q^50 + 4611732 * q^52 + 4396356 * q^53 + 2187488 * q^55 + 6096552 * q^59 + 4541698 * q^64 - 10368488 * q^67 + 712902 * q^68 + 11368032 * q^70 - 14210712 * q^76 - 7163520 * q^77 - 7060104 * q^83 + 8300608 * q^85 - 30033048 * q^86 - 24760956 * q^89 - 54883072 * q^94 - 81770994 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/153\mathbb{Z}\right)^\times$.

$n$	$37$	$137$
$\chi(n)$	$-1$	$1$

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 $

$2$	11.2698			0.996115			0.498057	−	0.867144i	$-0.334047\pi$
$2$	11.2698			0.996115			0.498057	+	0.867144i	$0.334047\pi$
$3$		0			0
$3$		0			0
$4$	−0.992706			−0.00775551
$5$		−	181.480i		−	0.649284i	−0.945837	−	0.324642i	$-0.894756\pi$
$5$		−	181.480i		−	0.649284i	0.945837	−	0.324642i	$-0.105244\pi$
$6$		0			0
$7$		−	1340.88i		−	1.47757i	−0.673941	−	0.738785i	$-0.735400\pi$
$7$		−	1340.88i		−	1.47757i	0.673941	−	0.738785i	$-0.264600\pi$
$8$	−1453.72			−1.00384
$9$		0			0
$10$		−	2045.24i		−	0.646761i
$11$			5783.08i			1.31004i	0.755611	+	0.655021i	$0.227340\pi$
$11$			5783.08i			1.31004i	−0.755611	+	0.655021i	$0.772660\pi$
$12$		0			0
$13$	834.906			0.105399			0.0526994	−	0.998610i	$-0.483217\pi$
$13$	834.906			0.105399			0.0526994	+	0.998610i	$0.483217\pi$
$14$		−	15111.4i		−	1.47183i
$15$		0			0
$16$	−16255.9			−0.992184
$17$	−12887.2	−	15628.8i	−0.636192	−	0.771531i
$17$	−12887.2	−	15628.8i	−0.636192	−	0.771531i
$18$		0			0
$19$	−9260.11			−0.309727			−0.154863	−	0.987936i	$-0.549494\pi$
$19$	−9260.11			−0.309727			−0.154863	+	0.987936i	$0.549494\pi$
$20$			180.157i			0.00503553i
$21$		0			0
$22$			65173.9i			1.30495i
$23$			107751.i			1.84661i	0.384071	+	0.923303i	$0.374522\pi$
$23$			107751.i			1.84661i	−0.384071	+	0.923303i	$0.625478\pi$
$24$		0			0
$25$	45189.9			0.578431
$26$	9409.18			0.104989
$27$		0			0
$28$			1331.10i			0.0114593i
$29$			101843.i			0.775422i	0.921781	+	0.387711i	$0.126734\pi$
$29$			101843.i			0.775422i	−0.921781	+	0.387711i	$0.873266\pi$
$30$		0			0
$31$			20966.7i			0.126405i	0.998001	+	0.0632025i	$0.0201314\pi$
$31$			20966.7i			0.126405i	−0.998001	+	0.0632025i	$0.979869\pi$
$32$	2875.12			0.0155107
$33$		0			0
$34$	−145236.	−	176132.i	−0.633720	−	0.768533i
$35$	−243344.			−0.959362
$36$		0			0
$37$			30107.7i			0.0977173i	0.998806	+	0.0488586i	$0.0155584\pi$
$37$			30107.7i			0.0977173i	−0.998806	+	0.0488586i	$0.984442\pi$
$38$	−104359.			−0.308523
$39$		0			0
$40$			263821.i			0.651777i
$41$			395730.i			0.896716i	0.893854	+	0.448358i	$0.147991\pi$
$41$			395730.i			0.896716i	−0.893854	+	0.448358i	$0.852009\pi$
$42$		0			0
$43$	−564125.			−1.08202			−0.541010	−	0.841016i	$-0.681958\pi$
$43$	−564125.			−1.08202			−0.541010	+	0.841016i	$0.681958\pi$
$44$		−	5740.90i		−	0.0101600i
$45$		0			0
$46$			1.21433e6i			1.83943i
$47$	−502222.			−0.705591			−0.352796	−	0.935700i	$-0.614769\pi$
$47$	−502222.			−0.705591			−0.352796	+	0.935700i	$0.614769\pi$
$48$		0			0
$49$	−974426.			−1.18321
$50$	509279.			0.576183
$51$		0			0
$52$	−828.816			−0.000817422
$53$	−253229.			−0.233640			−0.116820	−	0.993153i	$-0.537270\pi$
$53$	−253229.			−0.233640			−0.116820	+	0.993153i	$0.537270\pi$
$54$		0			0
$55$	1.04952e6			0.850588
$56$			1.94926e6i			1.48324i
$57$		0			0
$58$			1.14775e6i			0.772409i
$59$	−2.58041e6			−1.63571			−0.817855	−	0.575424i	$-0.804837\pi$
$59$	−2.58041e6			−1.63571			−0.817855	+	0.575424i	$0.804837\pi$
$60$		0			0
$61$		−	1.00567e6i		−	0.567285i	−0.958930	−	0.283642i	$-0.908457\pi$
$61$		−	1.00567e6i		−	0.567285i	0.958930	−	0.283642i	$-0.0915428\pi$
$62$			236290.i			0.125914i
$63$		0			0
$64$	2.11316e6			1.00763
$65$		−	151519.i		−	0.0684337i
$66$		0			0
$67$	−3.93496e6			−1.59837			−0.799187	−	0.601083i	$-0.794736\pi$
$67$	−3.93496e6			−1.59837			−0.799187	+	0.601083i	$0.794736\pi$
$68$	12793.2	+	15514.8i	0.00493400	+	0.00598362i
$69$		0			0
$70$	−2.74243e6			−0.955634
$71$		−	3.95445e6i		−	1.31124i	−0.755092	−	0.655619i	$-0.772408\pi$
$71$		−	3.95445e6i		−	1.31124i	0.755092	−	0.655619i	$-0.227592\pi$
$72$		0			0
$73$			5.07004e6i			1.52539i	0.646758	+	0.762696i	$0.276125\pi$
$73$			5.07004e6i			1.52539i	−0.646758	+	0.762696i	$0.723875\pi$
$74$			339306.i			0.0973376i
$75$		0			0
$76$	9192.56			0.00240209
$77$	7.75444e6			1.93568
$78$		0			0
$79$		−	811675.i		−	0.185220i	−0.995702	−	0.0926099i	$-0.970479\pi$
$79$		−	811675.i		−	0.185220i	0.995702	−	0.0926099i	$-0.0295209\pi$
$80$			2.95013e6i			0.644209i
$81$		0			0
$82$			4.45977e6i			0.893232i
$83$	3.58083e6			0.687401			0.343700	−	0.939079i	$-0.388320\pi$
$83$	3.58083e6			0.687401			0.343700	+	0.939079i	$0.388320\pi$
$84$		0			0
$85$	−2.83631e6	+	2.33878e6i	−0.500942	+	0.413069i
$86$	−6.35754e6			−1.07782
$87$		0			0
$88$		−	8.40696e6i		−	1.31507i
$89$	−5.31432e6			−0.799065			−0.399533	−	0.916719i	$-0.630828\pi$
$89$	−5.31432e6			−0.799065			−0.399533	+	0.916719i	$0.630828\pi$
$90$		0			0
$91$		−	1.11951e6i		−	0.155734i
$92$		−	106965.i		−	0.0143214i
$93$		0			0
$94$	−5.65992e6			−0.702850
$95$			1.68053e6i			0.201100i
$96$		0			0
$97$			1.41602e7i			1.57532i	0.616109	+	0.787661i	$0.288708\pi$
$97$			1.41602e7i			1.57532i	−0.616109	+	0.787661i	$0.711292\pi$
$98$	−1.09815e7			−1.17861
$99$		0			0
$100$	−44860.3			−0.00448603
$101$	1.23888e7			1.19648			0.598238	−	0.801318i	$-0.295868\pi$
$101$	1.23888e7			1.19648			0.598238	+	0.801318i	$0.295868\pi$
$102$		0			0
$103$	1.21917e7			1.09935			0.549674	−	0.835379i	$-0.314752\pi$
$103$	1.21917e7			1.09935			0.549674	+	0.835379i	$0.314752\pi$
$104$	−1.21372e6			−0.105804
$105$		0			0
$106$	−2.85382e6			−0.232732
$107$		−	1.24578e6i		−	0.0983099i	−0.998791	−	0.0491550i	$-0.984347\pi$
$107$		−	1.24578e6i		−	0.0983099i	0.998791	−	0.0491550i	$-0.0156528\pi$
$108$		0			0
$109$		−	1.87674e7i		−	1.38807i	−0.719940	−	0.694037i	$-0.755831\pi$
$109$		−	1.87674e7i		−	1.38807i	0.719940	−	0.694037i	$-0.244169\pi$
$110$	1.18278e7			0.847283
$111$		0			0
$112$			2.17973e7i			1.46602i
$113$		−	1.49197e6i		−	0.0972716i	−0.998817	−	0.0486358i	$-0.984513\pi$
$113$		−	1.49197e6i		−	0.0972716i	0.998817	−	0.0486358i	$-0.0154874\pi$
$114$		0			0
$115$	1.95547e7			1.19897
$116$		−	101100.i		−	0.00601380i
$117$		0			0
$118$	−2.90806e7			−1.62936
$119$	−2.09563e7	+	1.72803e7i	−1.13999	+	0.940018i
$120$		0			0
$121$	−1.39569e7			−0.716208
$122$		−	1.13337e7i		−	0.565081i
$123$		0			0
$124$		−	20813.8i		−	0.000980337i
$125$		−	2.23792e7i		−	1.02485i
$126$		0			0
$127$	−2.30800e7			−0.999823			−0.499912	−	0.866076i	$-0.666634\pi$
$127$	−2.30800e7			−0.999823			−0.499912	+	0.866076i	$0.666634\pi$
$128$	2.34468e7			0.988209
$129$		0			0
$130$		−	1.70758e6i		−	0.0681678i
$131$			2.80146e7i			1.08877i	0.838837	+	0.544383i	$0.183236\pi$
$131$			2.80146e7i			1.08877i	−0.838837	+	0.544383i	$0.816764\pi$
$132$		0			0
$133$			1.24167e7i			0.457642i
$134$	−4.43460e7			−1.59216
$135$		0			0
$136$	1.87344e7	+	2.27198e7i	0.638635	+	0.774493i
$137$	−9.25192e6			−0.307404			−0.153702	−	0.988117i	$-0.549120\pi$
$137$	−9.25192e6			−0.307404			−0.153702	+	0.988117i	$0.549120\pi$
$138$		0			0
$139$		−	2.55874e7i		−	0.808119i	−0.914733	−	0.404059i	$-0.867599\pi$
$139$		−	2.55874e7i		−	0.808119i	0.914733	−	0.404059i	$-0.132401\pi$
$140$	241569.			0.00744034
$141$		0			0
$142$		−	4.45656e7i		−	1.30614i
$143$			4.82833e6i			0.138077i
$144$		0			0
$145$	1.84825e7			0.503469
$146$			5.71381e7i			1.51946i
$147$		0			0
$148$		−	29888.1i		−	0.000757848i
$149$	−2.91086e7			−0.720890			−0.360445	−	0.932780i	$-0.617375\pi$
$149$	−2.91086e7			−0.720890			−0.360445	+	0.932780i	$0.617375\pi$
$150$		0			0
$151$	6.19449e7			1.46415			0.732076	−	0.681223i	$-0.238552\pi$
$151$	6.19449e7			1.46415			0.732076	+	0.681223i	$0.238552\pi$
$152$	1.34616e7			0.310916
$153$		0			0
$154$	8.73906e7			1.92816
$155$	3.80505e6			0.0820727
$156$		0			0
$157$	−6.50819e7			−1.34218			−0.671091	−	0.741375i	$-0.734174\pi$
$157$	−6.50819e7			−1.34218			−0.671091	+	0.741375i	$0.734174\pi$
$158$		−	9.14737e6i		−	0.184500i
$159$		0			0
$160$		−	521777.i		−	0.0100708i
$161$	1.44482e8			2.72849
$162$		0			0
$163$			2.63730e7i			0.476983i	0.971145	+	0.238492i	$0.0766529\pi$
$163$			2.63730e7i			0.476983i	−0.971145	+	0.238492i	$0.923347\pi$
$164$		−	392843.i		−	0.00695449i
$165$		0			0
$166$	4.03550e7			0.684730
$167$		−	7.74984e7i		−	1.28761i	−0.765189	−	0.643806i	$-0.777355\pi$
$167$		−	7.74984e7i		−	1.28761i	0.765189	−	0.643806i	$-0.222645\pi$
$168$		0			0
$169$	−6.20514e7			−0.988891
$170$	−3.19645e7	+	2.63574e7i	−0.498996	+	0.411464i
$171$		0			0
$172$	560010.			0.00839163
$173$		−	8.07573e7i		−	1.18582i	−0.805267	−	0.592912i	$-0.797978\pi$
$173$		−	8.07573e7i		−	1.18582i	0.805267	−	0.592912i	$-0.202022\pi$
$174$		0			0
$175$		−	6.05944e7i		−	0.854672i
$176$		−	9.40095e7i		−	1.29980i
$177$		0			0
$178$	−5.98910e7			−0.795961
$179$	−4.00077e7			−0.521385			−0.260692	−	0.965422i	$-0.583951\pi$
$179$	−4.00077e7			−0.521385			−0.260692	+	0.965422i	$0.583951\pi$
$180$		0			0
$181$			6.80903e6i			0.0853513i	0.999089	+	0.0426756i	$0.0135882\pi$
$181$			6.80903e6i			0.0853513i	−0.999089	+	0.0426756i	$0.986412\pi$
$182$		−	1.26166e7i		−	0.155129i
$183$		0			0
$184$		−	1.56640e8i		−	1.85370i
$185$	5.46395e6			0.0634462
$186$		0			0
$187$	9.03824e7	−	7.45279e7i	1.01074	−	0.833438i
$188$	498559.			0.00547222
$189$		0			0
$190$			1.89391e7i			0.200319i
$191$	−1.06003e7			−0.110078			−0.0550391	−	0.998484i	$-0.517528\pi$
$191$	−1.06003e7			−0.110078			−0.0550391	+	0.998484i	$0.517528\pi$
$192$		0			0
$193$		−	1.05434e8i		−	1.05568i	−0.849345	−	0.527839i	$-0.823003\pi$
$193$		−	1.05434e8i		−	1.05568i	0.849345	−	0.527839i	$-0.176997\pi$
$194$			1.59582e8i			1.56920i
$195$		0			0
$196$	967318.			0.00917642
$197$		−	7.34973e7i		−	0.684920i	−0.939532	−	0.342460i	$-0.888740\pi$
$197$		−	7.34973e7i		−	0.684920i	0.939532	−	0.342460i	$-0.111260\pi$
$198$		0			0
$199$			4.61952e7i			0.415538i	0.978178	+	0.207769i	$0.0666203\pi$
$199$			4.61952e7i			0.415538i	−0.978178	+	0.207769i	$0.933380\pi$
$200$	−6.56933e7			−0.580652
$201$		0			0
$202$	1.39619e8			1.19183
$203$	1.36560e8			1.14574
$204$		0			0
$205$	7.18171e7			0.582223
$206$	1.37398e8			1.09508
$207$		0			0
$208$	−1.35722e7			−0.104575
$209$		−	5.35520e7i		−	0.405755i
$210$		0			0
$211$			1.62815e8i			1.19318i	0.802546	+	0.596590i	$0.203478\pi$
$211$			1.62815e8i			1.19318i	−0.802546	+	0.596590i	$0.796522\pi$
$212$	251382.			0.00181200
$213$		0			0
$214$		−	1.40396e7i		−	0.0979280i
$215$			1.02377e8i			0.702538i
$216$		0			0
$217$	2.81139e7			0.186772
$218$		−	2.11504e8i		−	1.38268i
$219$		0			0
$220$	−1.04186e6			−0.00659675
$221$	−1.07596e7	−	1.30485e7i	−0.0670539	−	0.0813184i
$222$		0			0
$223$	−7.65887e7			−0.462485			−0.231243	−	0.972896i	$-0.574279\pi$
$223$	−7.65887e7			−0.462485			−0.231243	+	0.972896i	$0.574279\pi$
$224$		−	3.85520e6i		−	0.0229181i
$225$		0			0
$226$		−	1.68141e7i		−	0.0968937i
$227$		−	6.64444e7i		−	0.377023i	−0.982071	−	0.188512i	$-0.939634\pi$
$227$		−	6.64444e7i		−	0.377023i	0.982071	−	0.188512i	$-0.0603663\pi$
$228$		0			0
$229$	1.54121e8			0.848084			0.424042	−	0.905643i	$-0.360611\pi$
$229$	1.54121e8			0.848084			0.424042	+	0.905643i	$0.360611\pi$
$230$	2.20377e8			1.19431
$231$		0			0
$232$		−	1.48051e8i		−	0.778400i
$233$			1.03207e8i			0.534522i	0.963624	+	0.267261i	$0.0861185\pi$
$233$			1.03207e8i			0.534522i	−0.963624	+	0.267261i	$0.913881\pi$
$234$		0			0
$235$			9.11434e7i			0.458129i
$236$	2.56159e6			0.0126858
$237$		0			0
$238$	−2.36173e8	+	1.94744e8i	−1.13556	+	0.936366i
$239$	−1.00214e7			−0.0474826			−0.0237413	−	0.999718i	$-0.507558\pi$
$239$	−1.00214e7			−0.0474826			−0.0237413	+	0.999718i	$0.507558\pi$
$240$		0			0
$241$			2.58640e8i			1.19024i	0.803636	+	0.595121i	$0.202896\pi$
$241$			2.58640e8i			1.19024i	−0.803636	+	0.595121i	$0.797104\pi$
$242$	−1.57290e8			−0.713425
$243$		0			0
$244$			998335.i			0.00439959i
$245$			1.76839e8i			0.768240i
$246$		0			0
$247$	−7.73132e6			−0.0326448
$248$		−	3.04797e7i		−	0.126890i
$249$		0			0
$250$		−	2.52208e8i		−	1.02087i
$251$	−5.22128e7			−0.208410			−0.104205	−	0.994556i	$-0.533230\pi$
$251$	−5.22128e7			−0.208410			−0.104205	+	0.994556i	$0.533230\pi$
$252$		0			0
$253$	−6.23134e8			−2.41913
$254$	−2.60106e8			−0.995938
$255$		0			0
$256$	−6.24519e6			−0.0232651
$257$	−3.93545e8			−1.44620			−0.723099	−	0.690744i	$-0.757283\pi$
$257$	−3.93545e8			−1.44620			−0.723099	+	0.690744i	$0.757283\pi$
$258$		0			0
$259$	4.03709e7			0.144384
$260$			150414.i			0.000530739i
$261$		0			0
$262$			3.15717e8i			1.08454i
$263$	−4.50386e8			−1.52665			−0.763326	−	0.646013i	$-0.776435\pi$
$263$	−4.50386e8			−1.52665			−0.763326	+	0.646013i	$0.776435\pi$
$264$		0			0
$265$			4.59560e7i			0.151699i
$266$			1.39933e8i			0.455864i
$267$		0			0
$268$	3.90626e6			0.0123962
$269$			5.16491e8i			1.61782i	0.587935	+	0.808909i	$0.299941\pi$
$269$			5.16491e8i			1.61782i	−0.587935	+	0.808909i	$0.700059\pi$
$270$		0			0
$271$	−1.28639e8			−0.392627			−0.196313	−	0.980541i	$-0.562897\pi$
$271$	−1.28639e8			−0.392627			−0.196313	+	0.980541i	$0.562897\pi$
$272$	2.09494e8	+	2.54060e8i	0.631220	+	0.765501i
$273$		0			0
$274$	−1.04267e8			−0.306210
$275$			2.61337e8i			0.757768i
$276$		0			0
$277$		−	5.17053e8i		−	1.46169i	−0.682543	−	0.730845i	$-0.739126\pi$
$277$		−	5.17053e8i		−	1.46169i	0.682543	−	0.730845i	$-0.260874\pi$
$278$		−	2.88364e8i		−	0.804979i
$279$		0			0
$280$	3.53753e8			0.963046
$281$	−2.97615e8			−0.800170			−0.400085	−	0.916478i	$-0.631019\pi$
$281$	−2.97615e8			−0.800170			−0.400085	+	0.916478i	$0.631019\pi$
$282$		0			0
$283$			6.36824e8i			1.67019i	0.550103	+	0.835097i	$0.314588\pi$
$283$			6.36824e8i			1.67019i	−0.550103	+	0.835097i	$0.685412\pi$
$284$			3.92560e6i			0.0101693i
$285$		0			0
$286$			5.44141e7i			0.137540i
$287$	5.30627e8			1.32496
$288$		0			0
$289$	−7.81772e7	+	4.02823e8i	−0.190519	+	0.981684i
$290$	2.08293e8			0.501513
$291$		0			0
$292$		−	5.03306e6i		−	0.0118302i
$293$	−1.47396e8			−0.342333			−0.171167	−	0.985242i	$-0.554754\pi$
$293$	−1.47396e8			−0.342333			−0.171167	+	0.985242i	$0.554754\pi$
$294$		0			0
$295$			4.68293e8i			1.06204i
$296$		−	4.37680e7i		−	0.0980925i
$297$		0			0
$298$	−3.28046e8			−0.718089
$299$			8.99620e7i			0.194630i
$300$		0			0
$301$			7.56425e8i			1.59876i
$302$	6.98103e8			1.45846
$303$		0			0
$304$	1.50532e8			0.307306
$305$	−1.82509e8			−0.368329
$306$		0			0
$307$	−9.32079e8			−1.83852			−0.919260	−	0.393651i	$-0.871212\pi$
$307$	−9.32079e8			−1.83852			−0.919260	+	0.393651i	$0.871212\pi$
$308$	−7.69788e6			−0.0150122
$309$		0			0
$310$	4.28819e7			0.0817539
$311$		−	6.73465e8i		−	1.26956i	−0.772692	−	0.634781i	$-0.781090\pi$
$311$		−	6.73465e8i		−	1.26956i	0.772692	−	0.634781i	$-0.218910\pi$
$312$		0			0
$313$		−	9.22735e8i		−	1.70087i	−0.526079	−	0.850436i	$-0.676338\pi$
$313$		−	9.22735e8i		−	1.70087i	0.526079	−	0.850436i	$-0.323662\pi$
$314$	−7.33457e8			−1.33697
$315$		0			0
$316$			805754.i			0.00143647i
$317$			1.25264e8i			0.220861i	0.993884	+	0.110430i	$0.0352229\pi$
$317$			1.25264e8i			0.220861i	−0.993884	+	0.110430i	$0.964777\pi$
$318$		0			0
$319$	−5.88966e8			−1.01583
$320$		−	3.83497e8i		−	0.654241i
$321$		0			0
$322$	1.62827e9			2.71789
$323$	1.19337e8	+	1.44724e8i	0.197046	+	0.238963i
$324$		0			0
$325$	3.77293e7			0.0609659
$326$			2.97217e8i			0.475130i
$327$		0			0
$328$		−	5.75278e8i		−	0.900160i
$329$			6.73421e8i			1.04256i
$330$		0			0
$331$	−1.08117e8			−0.163868			−0.0819341	−	0.996638i	$-0.526110\pi$
$331$	−1.08117e8			−0.163868			−0.0819341	+	0.996638i	$0.526110\pi$
$332$	−3.55471e6			−0.00533115
$333$		0			0
$334$		−	8.73387e8i		−	1.28261i
$335$			7.14117e8i			1.03780i
$336$		0			0
$337$			7.83029e8i			1.11448i	0.830351	+	0.557241i	$0.188140\pi$
$337$			7.83029e8i			1.11448i	−0.830351	+	0.557241i	$0.811860\pi$
$338$	−6.99304e8			−0.985049
$339$		0			0
$340$	2.81562e6	−	2.32172e6i	0.00388506	−	0.00320356i
$341$	−1.21252e8			−0.165596
$342$		0			0
$343$			2.02316e8i			0.270708i
$344$	8.20077e8			1.08618
$345$		0			0
$346$		−	9.10114e8i		−	1.18122i
$347$		−	1.37878e9i		−	1.77151i	−0.464156	−	0.885753i	$-0.653642\pi$
$347$		−	1.37878e9i		−	1.77151i	0.464156	−	0.885753i	$-0.346358\pi$
$348$		0			0
$349$	1.28126e9			1.61342			0.806711	−	0.590946i	$-0.201246\pi$
$349$	1.28126e9			1.61342			0.806711	+	0.590946i	$0.201246\pi$
$350$		−	6.82884e8i		−	0.851351i
$351$		0			0
$352$			1.66270e7i			0.0203196i
$353$	9.04425e8			1.09436			0.547181	−	0.837014i	$-0.315701\pi$
$353$	9.04425e8			1.09436			0.547181	+	0.837014i	$0.315701\pi$
$354$		0			0
$355$	−7.17654e8			−0.851365
$356$	5.27555e6			0.00619716
$357$		0			0
$358$	−4.50877e8			−0.519359
$359$	−4.17207e8			−0.475906			−0.237953	−	0.971277i	$-0.576476\pi$
$359$	−4.17207e8			−0.475906			−0.237953	+	0.971277i	$0.576476\pi$
$360$		0			0
$361$	−8.08122e8			−0.904069
$362$			7.67360e7i			0.0850197i
$363$		0			0
$364$			1.11135e6i			0.00120780i
$365$	9.20112e8			0.990412
$366$		0			0
$367$		−	5.32275e8i		−	0.562089i	−0.959695	−	0.281044i	$-0.909319\pi$
$367$		−	5.32275e8i		−	0.562089i	0.959695	−	0.281044i	$-0.0906808\pi$
$368$		−	1.75160e9i		−	1.83217i
$369$		0			0
$370$	6.15774e7			0.0631997
$371$			3.39550e8i			0.345219i
$372$		0			0
$373$	1.26420e9			1.26135			0.630676	−	0.776047i	$-0.282778\pi$
$373$	1.26420e9			1.26135			0.630676	+	0.776047i	$0.282778\pi$
$374$	1.01859e9	−	8.39911e8i	1.00681	−	0.830200i
$375$		0			0
$376$	7.30088e8			0.708301
$377$			8.50293e7i			0.0817286i
$378$		0			0
$379$			1.13321e9i			1.06923i	0.845095	+	0.534617i	$0.179544\pi$
$379$			1.13321e9i			1.06923i	−0.845095	+	0.534617i	$0.820456\pi$
$380$		−	1.66827e6i		−	0.00155964i
$381$		0			0
$382$	−1.19463e8			−0.109650
$383$	−1.84833e9			−1.68106			−0.840529	−	0.541766i	$-0.817756\pi$
$383$	−1.84833e9			−1.68106			−0.840529	+	0.541766i	$0.817756\pi$
$384$		0			0
$385$		−	1.40728e9i		−	1.25680i
$386$		−	1.18822e9i		−	1.05158i
$387$		0			0
$388$		−	1.40569e7i		−	0.0122174i
$389$	1.95204e8			0.168138			0.0840690	−	0.996460i	$-0.473208\pi$
$389$	1.95204e8			0.168138			0.0840690	+	0.996460i	$0.473208\pi$
$390$		0			0
$391$	1.68402e9	−	1.38861e9i	1.42471	−	1.17480i
$392$	1.41654e9			1.18776
$393$		0			0
$394$		−	8.28297e8i		−	0.682259i
$395$	−1.47303e8			−0.120260
$396$		0			0
$397$			1.46433e9i			1.17455i	0.809386	+	0.587276i	$0.199800\pi$
$397$			1.46433e9i			1.17455i	−0.809386	+	0.587276i	$0.800200\pi$
$398$			5.20608e8i			0.413924i
$399$		0			0
$400$	−7.34605e8			−0.573910
$401$			3.82480e8i			0.296212i	0.988971	+	0.148106i	$0.0473177\pi$
$401$			3.82480e8i			0.296212i	−0.988971	+	0.148106i	$0.952682\pi$
$402$		0			0
$403$			1.75052e7i			0.0133229i
$404$	−1.22984e7			−0.00927929
$405$		0			0
$406$	1.53899e9			1.14129
$407$	−1.74115e8			−0.128014
$408$		0			0
$409$	9.94154e7			0.0718492			0.0359246	−	0.999355i	$-0.488562\pi$
$409$	9.94154e7			0.0718492			0.0359246	+	0.999355i	$0.488562\pi$
$410$	8.09361e8			0.579961
$411$		0			0
$412$	−1.21028e7			−0.00852601
$413$			3.46003e9i			2.41688i
$414$		0			0
$415$		−	6.49849e8i		−	0.446318i
$416$	2.40045e6			0.00163481
$417$		0			0
$418$		−	6.03517e8i		−	0.404178i
$419$		−	4.80667e6i		−	0.00319224i	−0.999999	−	0.00159612i	$-0.999492\pi$
$419$		−	4.80667e6i		−	0.00319224i	0.999999	−	0.00159612i	$-0.000508061\pi$
$420$		0			0
$421$	1.43850e8			0.0939558			0.0469779	−	0.998896i	$-0.485041\pi$
$421$	1.43850e8			0.0939558			0.0469779	+	0.998896i	$0.485041\pi$
$422$			1.83489e9i			1.18854i
$423$		0			0
$424$	3.68122e8			0.234537
$425$	−5.82373e8	−	7.06262e8i	−0.367993	−	0.446277i
$426$		0			0
$427$	−1.34849e9			−0.838203
$428$			1.23669e6i			0.000762444i
$429$		0			0
$430$			1.15377e9i			0.699808i
$431$			3.48593e8i			0.209724i	0.994487	+	0.104862i	$0.0334400\pi$
$431$			3.48593e8i			0.209724i	−0.994487	+	0.104862i	$0.966560\pi$
$432$		0			0
$433$	4.06627e8			0.240707			0.120353	−	0.992731i	$-0.461597\pi$
$433$	4.06627e8			0.240707			0.120353	+	0.992731i	$0.461597\pi$
$434$	3.16837e8			0.186047
$435$		0			0
$436$			1.86306e7i			0.0107652i
$437$		−	9.97787e8i		−	0.571943i
$438$		0			0
$439$		−	1.42570e9i		−	0.804270i	−0.915580	−	0.402135i	$-0.868268\pi$
$439$		−	1.42570e9i		−	0.804270i	0.915580	−	0.402135i	$-0.131732\pi$
$440$	−1.52570e9			−0.853855
$441$		0			0
$442$	−1.21258e8	−	1.47054e8i	−0.0667934	−	0.0810025i
$443$	−2.27492e9			−1.24323			−0.621616	−	0.783322i	$-0.713524\pi$
$443$	−2.27492e9			−1.24323			−0.621616	+	0.783322i	$0.713524\pi$
$444$		0			0
$445$			9.64443e8i			0.518820i
$446$	−8.63136e8			−0.460688
$447$		0			0
$448$		−	2.83351e9i		−	1.48885i
$449$		−	2.35939e9i		−	1.23009i	−0.788491	−	0.615047i	$-0.789137\pi$
$449$		−	2.35939e9i		−	1.23009i	0.788491	−	0.615047i	$-0.210863\pi$
$450$		0			0
$451$	−2.28854e9			−1.17474
$452$			1.48109e6i			0.000754391i
$453$		0			0
$454$		−	7.48812e8i		−	0.375558i
$455$	−2.03169e8			−0.101116
$456$		0			0
$457$	−2.50521e9			−1.22783			−0.613913	−	0.789374i	$-0.710405\pi$
$457$	−2.50521e9			−1.22783			−0.613913	+	0.789374i	$0.710405\pi$
$458$	1.73691e9			0.844789
$459$		0			0
$460$	−1.94121e7			−0.00929864
$461$	4.50046e8			0.213946			0.106973	−	0.994262i	$-0.465884\pi$
$461$	4.50046e8			0.213946			0.106973	+	0.994262i	$0.465884\pi$
$462$		0			0
$463$	−2.73938e8			−0.128268			−0.0641340	−	0.997941i	$-0.520429\pi$
$463$	−2.73938e8			−0.128268			−0.0641340	+	0.997941i	$0.520429\pi$
$464$		−	1.65555e9i		−	0.769362i
$465$		0			0
$466$			1.16312e9i			0.532445i
$467$	7.85850e8			0.357051			0.178526	−	0.983935i	$-0.442867\pi$
$467$	7.85850e8			0.357051			0.178526	+	0.983935i	$0.442867\pi$
$468$		0			0
$469$			5.27632e9i			2.36171i
$470$			1.02716e9i			0.456349i
$471$		0			0
$472$	3.75118e9			1.64199
$473$		−	3.26238e9i		−	1.41749i
$474$		0			0
$475$	−4.18463e8			−0.179155
$476$	2.08035e7	−	1.71542e7i	0.00884121	−	0.00729033i
$477$		0			0
$478$	−1.12938e8			−0.0472981
$479$			2.05967e9i			0.856295i	0.903709	+	0.428147i	$0.140834\pi$
$479$			2.05967e9i			0.856295i	−0.903709	+	0.428147i	$0.859166\pi$
$480$		0			0
$481$			2.51371e7i			0.0102993i
$482$			2.91481e9i			1.18562i
$483$		0			0
$484$	1.38551e7			0.00555456
$485$	2.56980e9			1.02283
$486$		0			0
$487$		−	1.41049e8i		−	0.0553373i	−0.999617	−	0.0276686i	$-0.991192\pi$
$487$		−	1.41049e8i		−	0.0553373i	0.999617	−	0.0276686i	$-0.00880832\pi$
$488$			1.46196e9i			0.569463i
$489$		0			0
$490$			1.99293e9i			0.765255i
$491$	2.96070e9			1.12878			0.564390	−	0.825508i	$-0.309111\pi$
$491$	2.96070e9			1.12878			0.564390	+	0.825508i	$0.309111\pi$
$492$		0			0
$493$	1.59168e9	−	1.31247e9i	0.598262	−	0.493318i
$494$	−8.71300e7			−0.0325180
$495$		0			0
$496$		−	3.40834e8i		−	0.125417i
$497$	−5.30245e9			−1.93744
$498$		0			0
$499$			2.78980e9i			1.00513i	0.864541	+	0.502563i	$0.167610\pi$
$499$			2.78980e9i			1.00513i	−0.864541	+	0.502563i	$0.832390\pi$
$500$			2.22160e7i			0.00794823i
$501$		0			0
$502$	−5.88425e8			−0.207600
$503$			4.97120e9i			1.74170i	0.491548	+	0.870851i	$0.336431\pi$
$503$			4.97120e9i			1.74170i	−0.491548	+	0.870851i	$0.663569\pi$
$504$		0			0
$505$		−	2.24832e9i		−	0.776852i
$506$	−7.02256e9			−2.40973
$507$		0			0
$508$	2.29117e7			0.00775414
$509$	4.95725e9			1.66620			0.833102	−	0.553119i	$-0.186563\pi$
$509$	4.95725e9			1.66620			0.833102	+	0.553119i	$0.186563\pi$
$510$		0			0
$511$	6.79833e9			2.25387
$512$	−3.07157e9			−1.01138
$513$		0			0
$514$	−4.43515e9			−1.44058
$515$		−	2.21256e9i		−	0.713789i
$516$		0			0
$517$		−	2.90439e9i		−	0.924354i
$518$	4.54970e8			0.143823
$519$		0			0
$520$			2.20265e8i			0.0686965i
$521$		−	1.64511e9i		−	0.509639i	−0.966989	−	0.254820i	$-0.917984\pi$
$521$		−	1.64511e9i		−	0.509639i	0.966989	−	0.254820i	$-0.0820161\pi$
$522$		0			0
$523$	2.45495e9			0.750389			0.375194	−	0.926946i	$-0.377576\pi$
$523$	2.45495e9			0.750389			0.375194	+	0.926946i	$0.377576\pi$
$524$		−	2.78102e7i		−	0.00844394i
$525$		0			0
$526$	−5.07574e9			−1.52072
$527$	3.27684e8	−	2.70203e8i	0.0975254	−	0.0804179i
$528$		0			0
$529$	−8.20548e9			−2.40996
$530$			5.17913e8i			0.151109i
$531$		0			0
$532$		−	1.23262e7i		−	0.00354925i
$533$			3.30397e8i			0.0945128i
$534$		0			0
$535$	−2.26084e8			−0.0638310
$536$	5.72031e9			1.60451
$537$		0			0
$538$			5.82072e9i			1.61153i
$539$		−	5.63518e9i		−	1.55006i
$540$		0			0
$541$		−	1.68819e9i		−	0.458384i	−0.973381	−	0.229192i	$-0.926392\pi$
$541$		−	1.68819e9i		−	0.458384i	0.973381	−	0.229192i	$-0.0736084\pi$
$542$	−1.44973e9			−0.391101
$543$		0			0
$544$	−3.70523e7	−	4.49345e7i	−0.00986777	−	0.0119670i
$545$	−3.40592e9			−0.901253
$546$		0			0
$547$			9.53791e8i			0.249171i	0.992209	+	0.124586i	$0.0397601\pi$
$547$			9.53791e8i			0.249171i	−0.992209	+	0.124586i	$0.960240\pi$
$548$	9.18444e6			0.00238408
$549$		0			0
$550$			2.94520e9i			0.754824i
$551$		−	9.43077e8i		−	0.240169i
$552$		0			0
$553$	−1.08836e9			−0.273675
$554$		−	5.82705e9i		−	1.45601i
$555$		0			0
$556$			2.54008e7i			0.00626738i
$557$	2.97677e9			0.729880			0.364940	−	0.931031i	$-0.381089\pi$
$557$	2.97677e9			0.729880			0.364940	+	0.931031i	$0.381089\pi$
$558$		0			0
$559$	−4.70991e8			−0.114044
$560$	3.95579e9			0.951864
$561$		0			0
$562$	−3.35404e9			−0.797061
$563$	2.79052e9			0.659030			0.329515	−	0.944150i	$-0.393115\pi$
$563$	2.79052e9			0.659030			0.329515	+	0.944150i	$0.393115\pi$
$564$		0			0
$565$	−2.70763e8			−0.0631568
$566$			7.17685e9i			1.66370i
$567$		0			0
$568$			5.74864e9i			1.31627i
$569$	−1.84290e8			−0.0419381			−0.0209690	−	0.999780i	$-0.506675\pi$
$569$	−1.84290e8			−0.0419381			−0.0209690	+	0.999780i	$0.506675\pi$
$570$		0			0
$571$		−	4.32214e9i		−	0.971566i	−0.874080	−	0.485783i	$-0.838535\pi$
$571$		−	4.32214e9i		−	0.971566i	0.874080	−	0.485783i	$-0.161465\pi$
$572$		−	4.79311e6i		−	0.00107086i
$573$		0			0
$574$	5.98004e9			1.31981
$575$			4.86926e9i			1.06813i
$576$		0			0
$577$	5.07158e9			1.09908			0.549538	−	0.835469i	$-0.314804\pi$
$577$	5.07158e9			1.09908			0.549538	+	0.835469i	$0.314804\pi$
$578$	−8.81038e8	+	4.53971e9i	−0.189779	+	0.977869i
$579$		0			0
$580$	−1.83477e7			−0.00390466
$581$		−	4.80147e9i		−	1.01568i
$582$		0			0
$583$		−	1.46444e9i		−	0.306078i
$584$		−	7.37040e9i		−	1.53125i
$585$		0			0
$586$	−1.66112e9			−0.341003
$587$	1.06866e9			0.218075			0.109038	−	0.994038i	$-0.465223\pi$
$587$	1.06866e9			0.218075			0.109038	+	0.994038i	$0.465223\pi$
$588$		0			0
$589$		−	1.94154e8i		−	0.0391510i
$590$			5.27755e9i			1.05791i
$591$		0			0
$592$		−	4.89429e8i		−	0.0969536i
$593$	−2.06599e9			−0.406853			−0.203426	−	0.979090i	$-0.565208\pi$
$593$	−2.06599e9			−0.406853			−0.203426	+	0.979090i	$0.565208\pi$
$594$		0			0
$595$	3.13603e9	+	3.80316e9i	0.610338	+	0.740177i
$596$	2.88963e7			0.00559087
$597$		0			0
$598$			1.01385e9i			0.193874i
$599$	−3.49423e9			−0.664289			−0.332145	−	0.943228i	$-0.607772\pi$
$599$	−3.49423e9			−0.664289			−0.332145	+	0.943228i	$0.607772\pi$
$600$		0			0
$601$		−	1.88346e8i		−	0.0353912i	−0.999843	−	0.0176956i	$-0.994367\pi$
$601$		−	1.88346e8i		−	0.0353912i	0.999843	−	0.0176956i	$-0.00563298\pi$
$602$			8.52473e9i			1.59255i
$603$		0			0
$604$	−6.14930e7			−0.0113552
$605$			2.53290e9i			0.465022i
$606$		0			0
$607$		−	1.55905e9i		−	0.282943i	−0.989942	−	0.141472i	$-0.954817\pi$
$607$		−	1.55905e9i		−	0.282943i	0.989942	−	0.141472i	$-0.0451834\pi$
$608$	−2.66239e7			−0.00480407
$609$		0			0
$610$	−2.05683e9			−0.366898
$611$	−4.19308e8			−0.0743685
$612$		0			0
$613$	6.51258e9			1.14193			0.570967	−	0.820973i	$-0.306568\pi$
$613$	6.51258e9			1.14193			0.570967	+	0.820973i	$0.306568\pi$
$614$	−1.05043e10			−1.83138
$615$		0			0
$616$	−1.12728e10			−1.94311
$617$		−	9.31298e9i		−	1.59621i	−0.602516	−	0.798106i	$-0.705835\pi$
$617$		−	9.31298e9i		−	1.59621i	0.602516	−	0.798106i	$-0.294165\pi$
$618$		0			0
$619$			3.90386e9i			0.661572i	0.943706	+	0.330786i	$0.107314\pi$
$619$			3.90386e9i			0.661572i	−0.943706	+	0.330786i	$0.892686\pi$
$620$	−3.77729e6			−0.000636516
$621$		0			0
$622$		−	7.58978e9i		−	1.26463i
$623$			7.12588e9i			1.18067i
$624$		0			0
$625$	−5.30926e8			−0.0869869
$626$		−	1.03990e10i		−	1.69426i
$627$		0			0
$628$	6.46072e7			0.0104093
$629$	4.70546e8	−	3.88005e8i	0.0753919	−	0.0621670i
$630$		0			0
$631$	−5.56074e9			−0.881109			−0.440554	−	0.897726i	$-0.645218\pi$
$631$	−5.56074e9			−0.881109			−0.440554	+	0.897726i	$0.645218\pi$
$632$			1.17994e9i			0.185931i
$633$		0			0
$634$			1.41169e9i			0.220003i
$635$			4.18857e9i			0.649169i
$636$		0			0
$637$	−8.13553e8			−0.124709
$638$	−6.63750e9			−1.01189
$639$		0			0
$640$		−	4.25513e9i		−	0.641628i
$641$			4.88359e9i			0.732379i	0.930540	+	0.366189i	$0.119338\pi$
$641$			4.88359e9i			0.732379i	−0.930540	+	0.366189i	$0.880662\pi$
$642$		0			0
$643$			7.57678e9i			1.12395i	0.827155	+	0.561974i	$0.189958\pi$
$643$			7.57678e9i			1.12395i	−0.827155	+	0.561974i	$0.810042\pi$
$644$	−1.43428e8			−0.0211608
$645$		0			0
$646$	1.34490e9	+	1.63100e9i	0.196280	+	0.238035i
$647$	2.00383e8			0.0290868			0.0145434	−	0.999894i	$-0.495371\pi$
$647$	2.00383e8			0.0290868			0.0145434	+	0.999894i	$0.495371\pi$
$648$		0			0
$649$		−	1.49227e10i		−	2.14285i
$650$	4.25200e8			0.0607290
$651$		0			0
$652$		−	2.61806e7i		−	0.00369925i
$653$		−	3.52835e8i		−	0.0495879i	−0.999693	−	0.0247940i	$-0.992107\pi$
$653$		−	3.52835e8i		−	0.0495879i	0.999693	−	0.0247940i	$-0.00789297\pi$
$654$		0			0
$655$	5.08409e9			0.706918
$656$		−	6.43296e9i		−	0.889708i
$657$		0			0
$658$			7.58929e9i			1.03851i
$659$	1.26579e10			1.72290			0.861452	−	0.507839i	$-0.169556\pi$
$659$	1.26579e10			1.72290			0.861452	+	0.507839i	$0.169556\pi$
$660$		0			0
$661$	2.37300e9			0.319589			0.159794	−	0.987150i	$-0.448917\pi$
$661$	2.37300e9			0.319589			0.159794	+	0.987150i	$0.448917\pi$
$662$	−1.21845e9			−0.163232
$663$		0			0
$664$	−5.20550e9			−0.690040
$665$	2.25339e9			0.297140
$666$		0			0
$667$	−1.09737e10			−1.43190
$668$			7.69331e7i			0.00998609i
$669$		0			0
$670$			8.04792e9i			1.03377i
$671$	5.81587e9			0.743166
$672$		0			0
$673$		−	5.89136e9i		−	0.745011i	−0.928030	−	0.372506i	$-0.878499\pi$
$673$		−	5.89136e9i		−	0.745011i	0.928030	−	0.372506i	$-0.121501\pi$
$674$			8.82454e9i			1.11015i
$675$		0			0
$676$	6.15988e7			0.00766936
$677$			3.71173e9i			0.459744i	0.973221	+	0.229872i	$0.0738307\pi$
$677$			3.71173e9i			0.459744i	−0.973221	+	0.229872i	$0.926169\pi$
$678$		0			0
$679$	1.89872e10			2.32765
$680$	4.12319e9	−	3.39992e9i	0.502866	−	0.414655i
$681$		0			0
$682$	−1.36648e9			−0.164952
$683$		−	1.14125e10i		−	1.37060i	−0.728263	−	0.685298i	$-0.759672\pi$
$683$		−	1.14125e10i		−	1.37060i	0.728263	−	0.685298i	$-0.240328\pi$
$684$		0			0
$685$			1.67904e9i			0.199593i
$686$			2.28005e9i			0.269656i
$687$		0			0
$688$	9.17038e9			1.07356
$689$	−2.11422e8			−0.0246254
$690$		0			0
$691$			5.70644e9i			0.657948i	0.944339	+	0.328974i	$0.106703\pi$
$691$			5.70644e9i			0.657948i	−0.944339	+	0.328974i	$0.893297\pi$
$692$			8.01682e7i			0.00919668i
$693$		0			0
$694$		−	1.55385e10i		−	1.76462i
$695$	−4.64362e9			−0.524698
$696$		0			0
$697$	6.18476e9	−	5.09986e9i	0.691844	−	0.570484i
$698$	1.44395e10			1.60715
$699$		0			0
$700$			6.01524e7i			0.00662842i
$701$	−1.20522e10			−1.32146			−0.660730	−	0.750624i	$-0.729753\pi$
$701$	−1.20522e10			−1.32146			−0.660730	+	0.750624i	$0.729753\pi$
$702$		0			0
$703$		−	2.78800e8i		−	0.0302656i
$704$			1.22206e10i			1.32004i
$705$		0			0
$706$	1.01926e10			1.09011
$707$		−	1.66119e10i		−	1.76788i
$708$		0			0
$709$		−	3.97961e9i		−	0.419352i	−0.977771	−	0.209676i	$-0.932759\pi$
$709$		−	3.97961e9i		−	0.419352i	0.977771	−	0.209676i	$-0.0672410\pi$
$710$	−8.08778e9			−0.848057
$711$		0			0
$712$	7.72550e9			0.802134
$713$	−2.25919e9			−0.233421
$714$		0			0
$715$	8.76246e8			0.0896510
$716$	3.97159e7			0.00404361
$717$		0			0
$718$	−4.70182e9			−0.474057
$719$			1.26059e10i			1.26480i	0.774642	+	0.632400i	$0.217930\pi$
$719$			1.26059e10i			1.26480i	−0.774642	+	0.632400i	$0.782070\pi$
$720$		0			0
$721$		−	1.63477e10i		−	1.62436i
$722$	−9.10734e9			−0.900557
$723$		0			0
$724$		−	6.75936e6i		−	0.000661943i
$725$			4.60228e9i			0.448528i
$726$		0			0
$727$	−1.31885e10			−1.27299			−0.636494	−	0.771282i	$-0.719616\pi$
$727$	−1.31885e10			−1.27299			−0.636494	+	0.771282i	$0.719616\pi$
$728$			1.62745e9i			0.156332i
$729$		0			0
$730$	1.03694e10			0.986563
$731$	7.27000e9	+	8.81657e9i	0.688373	+	0.834812i
$732$		0			0
$733$	−5.16161e9			−0.484085			−0.242042	−	0.970266i	$-0.577817\pi$
$733$	−5.16161e9			−0.484085			−0.242042	+	0.970266i	$0.577817\pi$
$734$		−	5.99861e9i		−	0.559905i
$735$		0			0
$736$			3.09797e8i			0.0286421i
$737$		−	2.27562e10i		−	2.09394i
$738$		0			0
$739$	1.10100e10			1.00353			0.501766	−	0.865004i	$-0.332684\pi$
$739$	1.10100e10			1.00353			0.501766	+	0.865004i	$0.332684\pi$
$740$	−5.42410e6			−0.000492058
$741$		0			0
$742$			3.82665e9i			0.343878i
$743$		−	1.21999e10i		−	1.09118i	−0.838053	−	0.545588i	$-0.816306\pi$
$743$		−	1.21999e10i		−	1.09118i	0.838053	−	0.545588i	$-0.183694\pi$
$744$		0			0
$745$			5.28263e9i			0.468062i
$746$	1.42473e10			1.25645
$747$		0			0
$748$	−8.97231e7	+	7.39843e7i	−0.00783878	+	0.00646374i
$749$	−1.67044e9			−0.145260
$750$		0			0
$751$		−	3.48173e9i		−	0.299954i	−0.988689	−	0.149977i	$-0.952080\pi$
$751$		−	3.48173e9i		−	0.299954i	0.988689	−	0.149977i	$-0.0479200\pi$
$752$	8.16409e9			0.700077
$753$		0			0
$754$			9.58259e8i			0.0814110i
$755$		−	1.12418e10i		−	0.950649i
$756$		0			0
$757$	4.98292e9			0.417492			0.208746	−	0.977970i	$-0.433062\pi$
$757$	4.98292e9			0.417492			0.208746	+	0.977970i	$0.433062\pi$
$758$			1.27710e10i			1.06508i
$759$		0			0
$760$		−	2.44301e9i		−	0.201873i
$761$	1.21391e10			0.998481			0.499241	−	0.866463i	$-0.333612\pi$
$761$	1.21391e10			0.998481			0.499241	+	0.866463i	$0.333612\pi$
$762$		0			0
$763$	−2.51650e10			−2.05097
$764$	1.05230e7			0.000853713
$765$		0			0
$766$	−2.08302e10			−1.67453
$767$	−2.15440e9			−0.172402
$768$		0			0
$769$	−3.52402e9			−0.279445			−0.139723	−	0.990191i	$-0.544621\pi$
$769$	−3.52402e9			−0.279445			−0.139723	+	0.990191i	$0.544621\pi$
$770$		−	1.58597e10i		−	1.25192i
$771$		0			0
$772$			1.04665e8i			0.00818732i
$773$	−1.50830e10			−1.17452			−0.587258	−	0.809400i	$-0.699793\pi$
$773$	−1.50830e10			−1.17452			−0.587258	+	0.809400i	$0.699793\pi$
$774$		0			0
$775$			9.47484e8i			0.0731166i
$776$		−	2.05850e10i		−	1.58137i
$777$		0			0
$778$	2.19990e9			0.167485
$779$		−	3.66450e9i		−	0.277737i
$780$		0			0
$781$	2.28689e10			1.71778
$782$	1.89784e10	−	1.56493e10i	1.41918	−	1.17023i
$783$		0			0
$784$	1.58402e10			1.17396
$785$			1.18111e10i			0.871457i
$786$		0			0
$787$		−	1.44317e10i		−	1.05538i	−0.849438	−	0.527688i	$-0.823059\pi$
$787$		−	1.44317e10i		−	1.05538i	0.849438	−	0.527688i	$-0.176941\pi$
$788$			7.29612e7i			0.00531191i
$789$		0			0
$790$	−1.66007e9			−0.119793
$791$	−2.00056e9			−0.143726
$792$		0			0
$793$		−	8.39640e8i		−	0.0597911i
$794$			1.65027e10i			1.16999i
$795$		0			0
$796$		−	4.58582e7i		−	0.00322271i
$797$	2.21165e10			1.54744			0.773718	−	0.633530i	$-0.218395\pi$
$797$	2.21165e10			1.54744			0.773718	+	0.633530i	$0.218395\pi$
$798$		0			0
$799$	6.47225e9	+	7.84910e9i	0.448892	+	0.544385i
$800$	1.29926e8			0.00897186
$801$		0			0
$802$			4.31045e9i			0.295061i
$803$	−2.93205e10			−1.99833
$804$		0			0
$805$		−	2.62206e10i		−	1.77156i
$806$			1.97280e8i			0.0132712i
$807$		0			0
$808$	−1.80098e10			−1.20107
$809$			2.52729e10i			1.67817i	0.544000	+	0.839085i	$0.316909\pi$
$809$			2.52729e10i			1.67817i	−0.544000	+	0.839085i	$0.683091\pi$
$810$		0			0
$811$		−	3.18823e9i		−	0.209882i	−0.994478	−	0.104941i	$-0.966535\pi$
$811$		−	3.18823e9i		−	0.209882i	0.994478	−	0.104941i	$-0.0334655\pi$
$812$	−1.35564e8			−0.00888580
$813$		0			0
$814$	−1.96224e9			−0.127516
$815$	4.78618e9			0.309697
$816$		0			0
$817$	5.22385e9			0.335130
$818$	1.12039e9			0.0715701
$819$		0			0
$820$	−7.12933e7			−0.00451544
$821$		−	1.12444e10i		−	0.709147i	−0.935028	−	0.354574i	$-0.884626\pi$
$821$		−	1.12444e10i		−	0.709147i	0.935028	−	0.354574i	$-0.115374\pi$
$822$		0			0
$823$		−	4.92004e8i		−	0.0307659i	−0.999882	−	0.0153829i	$-0.995103\pi$
$823$		−	4.92004e8i		−	0.0307659i	0.999882	−	0.0153829i	$-0.00489674\pi$
$824$	−1.77233e10			−1.10357
$825$		0			0
$826$			3.89937e10i			2.40749i
$827$			1.50602e10i			0.925896i	0.886385	+	0.462948i	$0.153208\pi$
$827$			1.50602e10i			0.925896i	−0.886385	+	0.462948i	$0.846792\pi$
$828$		0			0
$829$	2.03521e10			1.24070			0.620351	−	0.784324i	$-0.286990\pi$
$829$	2.03521e10			1.24070			0.620351	+	0.784324i	$0.286990\pi$
$830$		−	7.32364e9i		−	0.444584i
$831$		0			0
$832$	1.76429e9			0.106203
$833$	1.25576e10	+	1.52291e10i	0.752750	+	0.912884i
$834$		0			0
$835$	−1.40644e10			−0.836025
$836$			5.31613e7i			0.00314684i
$837$		0			0
$838$		−	5.41700e7i		−	0.00317984i
$839$			9.34797e9i			0.546450i	0.961950	+	0.273225i	$0.0880904\pi$
$839$			9.34797e9i			0.546450i	−0.961950	+	0.273225i	$0.911910\pi$
$840$		0			0
$841$	6.87788e9			0.398721
$842$	1.62116e9			0.0935908
$843$		0			0
$844$		−	1.61628e8i		−	0.00925373i
$845$			1.12611e10i			0.642071i
$846$		0			0
$847$			1.87145e10i			1.05825i
$848$	4.11647e9			0.231814
$849$		0			0
$850$	−6.56320e9	−	7.95940e9i	−0.366563	−	0.444543i
$851$	−3.24414e9			−0.180445
$852$		0			0
$853$			3.01288e10i			1.66211i	0.556189	+	0.831056i	$0.312263\pi$
$853$			3.01288e10i			1.66211i	−0.556189	+	0.831056i	$0.687737\pi$
$854$	−1.51971e10			−0.834946
$855$		0			0
$856$			1.81101e9i			0.0986875i
$857$		−	9.63658e9i		−	0.522986i	−0.965205	−	0.261493i	$-0.915785\pi$
$857$		−	9.63658e9i		−	0.522986i	0.965205	−	0.261493i	$-0.0842149\pi$
$858$		0			0
$859$	−1.94466e10			−1.04681			−0.523404	−	0.852085i	$-0.675338\pi$
$859$	−1.94466e10			−1.04681			−0.523404	+	0.852085i	$0.675338\pi$
$860$		−	1.01631e8i		−	0.00544854i
$861$		0			0
$862$			3.92855e9i			0.208909i
$863$	−4.07602e9			−0.215873			−0.107936	−	0.994158i	$-0.534424\pi$
$863$	−4.07602e9			−0.215873			−0.107936	+	0.994158i	$0.534424\pi$
$864$		0			0
$865$	−1.46559e10			−0.769936
$866$	4.58258e9			0.239771
$867$		0			0
$868$	−2.79089e7			−0.00144852
$869$	4.69398e9			0.242645
$870$		0			0
$871$	−3.28532e9			−0.168467
$872$			2.72825e10i			1.39340i
$873$		0			0
$874$		−	1.12448e10i		−	0.569721i
$875$	−3.00079e10			−1.51429
$876$		0			0
$877$			1.49339e10i			0.747612i	0.927507	+	0.373806i	$0.121947\pi$
$877$			1.49339e10i			0.747612i	−0.927507	+	0.373806i	$0.878053\pi$
$878$		−	1.60673e10i		−	0.801145i
$879$		0			0
$880$	−1.70609e10			−0.843940
$881$			1.55484e10i			0.766073i	0.923733	+	0.383036i	$0.125122\pi$
$881$			1.55484e10i			0.766073i	−0.923733	+	0.383036i	$0.874878\pi$
$882$		0			0
$883$	3.92273e9			0.191746			0.0958730	−	0.995394i	$-0.469436\pi$
$883$	3.92273e9			0.191746			0.0958730	+	0.995394i	$0.469436\pi$
$884$	1.06811e7	+	1.29534e7i	0.000520038	+	0.000630666i
$885$		0			0
$886$	−2.56377e10			−1.23840
$887$			8.20990e9i			0.395007i	0.980302	+	0.197504i	$0.0632834\pi$
$887$			8.20990e9i			0.395007i	−0.980302	+	0.197504i	$0.936717\pi$
$888$		0			0
$889$			3.09476e10i			1.47731i
$890$			1.08690e10i			0.516804i
$891$		0			0
$892$	7.60301e7			0.00358681
$893$	4.65063e9			0.218540
$894$		0			0
$895$			7.26061e9i			0.338526i
$896$		−	3.14394e10i		−	1.46015i
$897$		0			0
$898$		−	2.65898e10i		−	1.22531i
$899$	−2.13531e9			−0.0980173
$900$		0			0
$901$	3.26342e9	+	3.95765e9i	0.148640	+	0.180260i
$902$	−2.57912e10			−1.17017
$903$		0			0
$904$			2.16890e9i			0.0976451i
$905$	1.23570e9			0.0554172
$906$		0			0
$907$			2.87835e9i			0.128091i	0.997947	+	0.0640455i	$0.0204003\pi$
$907$			2.87835e9i			0.128091i	−0.997947	+	0.0640455i	$0.979600\pi$
$908$			6.59598e7i			0.00292401i
$909$		0			0
$910$	−2.28967e9			−0.100723
$911$			8.67655e9i			0.380218i	0.981763	+	0.190109i	$0.0608842\pi$
$911$			8.67655e9i			0.380218i	−0.981763	+	0.190109i	$0.939116\pi$
$912$		0			0
$913$			2.07082e10i			0.900523i
$914$	−2.82330e10			−1.22305
$915$		0			0
$916$	−1.52997e8			−0.00657733
$917$	3.75643e10			1.60873
$918$		0			0
$919$	−3.38295e10			−1.43778			−0.718889	−	0.695125i	$-0.755349\pi$
$919$	−3.38295e10			−1.43778			−0.718889	+	0.695125i	$0.755349\pi$
$920$	−2.84270e10			−1.20358
$921$		0			0
$922$	5.07191e9			0.213115
$923$		−	3.30159e9i		−	0.138203i
$924$		0			0
$925$			1.36056e9i			0.0565227i
$926$	−3.08721e9			−0.127770
$927$		0			0
$928$			2.92811e8i			0.0120273i
$929$		−	2.87996e10i		−	1.17850i	−0.807950	−	0.589252i	$-0.799423\pi$
$929$		−	2.87996e10i		−	1.17850i	0.807950	−	0.589252i	$-0.200577\pi$
$930$		0			0
$931$	9.02329e9			0.366472
$932$		−	1.02455e8i		−	0.00414549i
$933$		0			0
$934$	8.85633e9			0.355664
$935$	−1.35253e10	−	1.64026e10i	−0.541138	−	0.656255i
$936$		0			0
$937$	2.44266e9			0.0970005			0.0485002	−	0.998823i	$-0.484556\pi$
$937$	2.44266e9			0.0970005			0.0485002	+	0.998823i	$0.484556\pi$
$938$			5.94628e10i			2.35253i
$939$		0			0
$940$		−	9.04786e7i		−	0.00355303i
$941$			1.68559e10i			0.659459i	0.944075	+	0.329730i	$0.106958\pi$
$941$			1.68559e10i			0.659459i	−0.944075	+	0.329730i	$0.893042\pi$
$942$		0			0
$943$	−4.26403e10			−1.65588
$944$	4.19470e10			1.62293
$945$		0			0
$946$		−	3.67662e10i		−	1.41198i
$947$			3.11941e9i			0.119357i	0.998218	+	0.0596784i	$0.0190075\pi$
$947$			3.11941e9i			0.119357i	−0.998218	+	0.0596784i	$0.980992\pi$
$948$		0			0
$949$			4.23300e9i			0.160774i
$950$	−4.71598e9			−0.178459
$951$		0			0
$952$	3.04646e10	−	2.51206e10i	1.14437	−	0.943628i
$953$	−3.09228e10			−1.15732			−0.578661	−	0.815568i	$-0.696424\pi$
$953$	−3.09228e10			−1.15732			−0.578661	+	0.815568i	$0.696424\pi$
$954$		0			0
$955$			1.92374e9i			0.0714720i
$956$	9.94828e6			0.000368252
$957$		0			0
$958$			2.32120e10i			0.852968i
$959$			1.24058e10i			0.454211i
$960$		0			0
$961$	2.70730e10			0.984022
$962$			2.83289e8i			0.0102593i
$963$		0			0
$964$		−	2.56753e8i		−	0.00923095i
$965$	−1.91342e10			−0.685434
$966$		0			0
$967$	2.19940e10			0.782187			0.391094	−	0.920351i	$-0.372097\pi$
$967$	2.19940e10			0.782187			0.391094	+	0.920351i	$0.372097\pi$
$968$	2.02893e10			0.718958
$969$		0			0
$970$	2.89610e10			1.01886
$971$	−2.37732e10			−0.833336			−0.416668	−	0.909059i	$-0.636802\pi$
$971$	−2.37732e10			−0.833336			−0.416668	+	0.909059i	$0.636802\pi$
$972$		0			0
$973$	−3.43098e10			−1.19405
$974$		−	1.58958e9i		−	0.0551223i
$975$		0			0
$976$			1.63481e10i			0.562851i
$977$	3.19823e10			1.09718			0.548591	−	0.836091i	$-0.315164\pi$
$977$	3.19823e10			1.09718			0.548591	+	0.836091i	$0.315164\pi$
$978$		0			0
$979$		−	3.07331e10i		−	1.04681i
$980$		−	1.75549e8i		−	0.00595810i
$981$		0			0
$982$	3.33664e10			1.12439
$983$			3.41146e10i			1.14552i	0.819722	+	0.572761i	$0.194128\pi$
$983$			3.41146e10i			1.14552i	−0.819722	+	0.572761i	$0.805872\pi$
$984$		0			0
$985$	−1.33383e10			−0.444707
$986$	1.79378e10	−	1.47913e10i	0.595937	−	0.491401i
$987$		0			0
$988$	7.67492e6			0.000253177
$989$		−	6.07851e10i		−	1.99807i
$990$		0			0
$991$		−	5.48617e9i		−	0.179065i	−0.995984	−	0.0895326i	$-0.971463\pi$
$991$		−	5.48617e9i		−	0.179065i	0.995984	−	0.0895326i	$-0.0285373\pi$
$992$			6.02818e7i			0.00196063i
$993$		0			0
$994$	−5.97573e10			−1.92992
$995$	8.38351e9			0.269802
$996$		0			0
$997$		−	3.21399e10i		−	1.02710i	−0.858061	−	0.513548i	$-0.828331\pi$
$997$		−	3.21399e10i		−	1.02710i	0.858061	−	0.513548i	$-0.171669\pi$
$998$			3.14403e10i			1.00122i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.8.d.b.118.7		10
3.2	odd	2		17.8.b.a.16.4	yes	10
12.11	even	2		272.8.b.c.33.5		10
17.16	even	2	inner	153.8.d.b.118.8		10
51.38	odd	4		289.8.a.f.1.8		10
51.47	odd	4		289.8.a.f.1.7		10
51.50	odd	2		17.8.b.a.16.3	✓	10
204.203	even	2		272.8.b.c.33.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.8.b.a.16.3	✓	10	51.50	odd	2
17.8.b.a.16.4	yes	10	3.2	odd	2
153.8.d.b.118.7		10	1.1	even	1	trivial
153.8.d.b.118.8		10	17.16	even	2	inner
272.8.b.c.33.5		10	12.11	even	2
272.8.b.c.33.6		10	204.203	even	2
289.8.a.f.1.7		10	51.47	odd	4
289.8.a.f.1.8		10	51.38	odd	4

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:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	153.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.2698 q^{2} -0.992706 q^{4} -181.480i q^{5} -1340.88i q^{7} -1453.72 q^{8} +O(q^{10})\)	\(q+11.2698 q^{2} -0.992706 q^{4} -181.480i q^{5} -1340.88i q^{7} -1453.72 q^{8} -2045.24i q^{10} +5783.08i q^{11} +834.906 q^{13} -15111.4i q^{14} -16255.9 q^{16} +(-12887.2 - 15628.8i) q^{17} -9260.11 q^{19} +180.157i q^{20} +65173.9i q^{22} +107751. i q^{23} +45189.9 q^{25} +9409.18 q^{26} +1331.10i q^{28} +101843. i q^{29} +20966.7i q^{31} +2875.12 q^{32} +(-145236. - 176132. i) q^{34} -243344. q^{35} +30107.7i q^{37} -104359. q^{38} +263821. i q^{40} +395730. i q^{41} -564125. q^{43} -5740.90i q^{44} +1.21433e6i q^{46} -502222. q^{47} -974426. q^{49} +509279. q^{50} -828.816 q^{52} -253229. q^{53} +1.04952e6 q^{55} +1.94926e6i q^{56} +1.14775e6i q^{58} -2.58041e6 q^{59} -1.00567e6i q^{61} +236290. i q^{62} +2.11316e6 q^{64} -151519. i q^{65} -3.93496e6 q^{67} +(12793.2 + 15514.8i) q^{68} -2.74243e6 q^{70} -3.95445e6i q^{71} +5.07004e6i q^{73} +339306. i q^{74} +9192.56 q^{76} +7.75444e6 q^{77} -811675. i q^{79} +2.95013e6i q^{80} +4.45977e6i q^{82} +3.58083e6 q^{83} +(-2.83631e6 + 2.33878e6i) q^{85} -6.35754e6 q^{86} -8.40696e6i q^{88} -5.31432e6 q^{89} -1.11951e6i q^{91} -106965. i q^{92} -5.65992e6 q^{94} +1.68053e6i q^{95} +1.41602e7i q^{97} -1.09815e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 18 q^{2} + 690 q^{4} + 3330 q^{8}+O(q^{10}) \) 10 * q + 18 * q^2 + 690 * q^4 + 3330 * q^8	\( 10 q + 18 q^{2} + 690 q^{4} + 3330 q^{8} - 316 q^{13} + 92226 q^{16} + 29454 q^{17} - 90184 q^{19} + 78370 q^{25} + 180420 q^{26} + 1019778 q^{32} + 1123478 q^{34} + 460704 q^{35} + 1583880 q^{38} - 112936 q^{43} - 3214512 q^{47} - 230842 q^{49} + 359514 q^{50} + 4611732 q^{52} + 4396356 q^{53} + 2187488 q^{55} + 6096552 q^{59} + 4541698 q^{64} - 10368488 q^{67} + 712902 q^{68} + 11368032 q^{70} - 14210712 q^{76} - 7163520 q^{77} - 7060104 q^{83} + 8300608 q^{85} - 30033048 q^{86} - 24760956 q^{89} - 54883072 q^{94} - 81770994 q^{98}+O(q^{100}) \) 10 * q + 18 * q^2 + 690 * q^4 + 3330 * q^8 - 316 * q^13 + 92226 * q^16 + 29454 * q^17 - 90184 * q^19 + 78370 * q^25 + 180420 * q^26 + 1019778 * q^32 + 1123478 * q^34 + 460704 * q^35 + 1583880 * q^38 - 112936 * q^43 - 3214512 * q^47 - 230842 * q^49 + 359514 * q^50 + 4611732 * q^52 + 4396356 * q^53 + 2187488 * q^55 + 6096552 * q^59 + 4541698 * q^64 - 10368488 * q^67 + 712902 * q^68 + 11368032 * q^70 - 14210712 * q^76 - 7163520 * q^77 - 7060104 * q^83 + 8300608 * q^85 - 30033048 * q^86 - 24760956 * q^89 - 54883072 * q^94 - 81770994 * q^98

Introduction

L-functions

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