LMFDB - Embedded newform 9450.2.a.eg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9450,2,Mod(1,9450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9450.a (trivial)

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:	yes
Analytic conductor:	$75.4586299101$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1890)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.16228$ of defining polynomial
Character	$\chi$	$=$	9450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -5.16228 q^{11} -4.16228 q^{13} -1.00000 q^{14} +1.00000 q^{16} +5.00000 q^{17} -7.16228 q^{19} +5.16228 q^{22} +6.16228 q^{23} +4.16228 q^{26} +1.00000 q^{28} -4.16228 q^{29} +10.4868 q^{31} -1.00000 q^{32} -5.00000 q^{34} -2.83772 q^{37} +7.16228 q^{38} +10.3246 q^{41} -7.00000 q^{43} -5.16228 q^{44} -6.16228 q^{46} -2.00000 q^{47} +1.00000 q^{49} -4.16228 q^{52} +6.48683 q^{53} -1.00000 q^{56} +4.16228 q^{58} +7.00000 q^{59} +4.32456 q^{61} -10.4868 q^{62} +1.00000 q^{64} -2.67544 q^{67} +5.00000 q^{68} +12.4868 q^{71} -10.0000 q^{73} +2.83772 q^{74} -7.16228 q^{76} -5.16228 q^{77} +13.1623 q^{79} -10.3246 q^{82} +4.32456 q^{83} +7.00000 q^{86} +5.16228 q^{88} +2.67544 q^{89} -4.16228 q^{91} +6.16228 q^{92} +2.00000 q^{94} +5.16228 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 4 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{17} - 8 q^{19} + 4 q^{22} + 6 q^{23} + 2 q^{26} + 2 q^{28} - 2 q^{29} + 2 q^{31} - 2 q^{32} - 10 q^{34} - 12 q^{37} + 8 q^{38} + 8 q^{41} - 14 q^{43} - 4 q^{44} - 6 q^{46} - 4 q^{47} + 2 q^{49} - 2 q^{52} - 6 q^{53} - 2 q^{56} + 2 q^{58} + 14 q^{59} - 4 q^{61} - 2 q^{62} + 2 q^{64} - 18 q^{67} + 10 q^{68} + 6 q^{71} - 20 q^{73} + 12 q^{74} - 8 q^{76} - 4 q^{77} + 20 q^{79} - 8 q^{82} - 4 q^{83} + 14 q^{86} + 4 q^{88} + 18 q^{89} - 2 q^{91} + 6 q^{92} + 4 q^{94} + 4 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 4 * q^11 - 2 * q^13 - 2 * q^14 + 2 * q^16 + 10 * q^17 - 8 * q^19 + 4 * q^22 + 6 * q^23 + 2 * q^26 + 2 * q^28 - 2 * q^29 + 2 * q^31 - 2 * q^32 - 10 * q^34 - 12 * q^37 + 8 * q^38 + 8 * q^41 - 14 * q^43 - 4 * q^44 - 6 * q^46 - 4 * q^47 + 2 * q^49 - 2 * q^52 - 6 * q^53 - 2 * q^56 + 2 * q^58 + 14 * q^59 - 4 * q^61 - 2 * q^62 + 2 * q^64 - 18 * q^67 + 10 * q^68 + 6 * q^71 - 20 * q^73 + 12 * q^74 - 8 * q^76 - 4 * q^77 + 20 * q^79 - 8 * q^82 - 4 * q^83 + 14 * q^86 + 4 * q^88 + 18 * q^89 - 2 * q^91 + 6 * q^92 + 4 * q^94 + 4 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−5.16228	−1.55649	−0.778243	−	0.627964i	$-0.783889\pi$
$11$	−5.16228	−1.55649	−0.778243	+	0.627964i	$0.783889\pi$
$12$	0	0
$13$	−4.16228	−1.15441	−0.577204	−	0.816600i	$-0.695856\pi$
$13$	−4.16228	−1.15441	−0.577204	+	0.816600i	$0.695856\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	−7.16228	−1.64314	−0.821570	−	0.570108i	$-0.806901\pi$
$19$	−7.16228	−1.64314	−0.821570	+	0.570108i	$0.806901\pi$
$20$	0	0
$21$	0	0
$22$	5.16228	1.10060
$23$	6.16228	1.28492	0.642462	−	0.766318i	$-0.277913\pi$
$23$	6.16228	1.28492	0.642462	+	0.766318i	$0.277913\pi$
$24$	0	0
$25$	0	0
$26$	4.16228	0.816290
$27$	0	0
$28$	1.00000	0.188982
$29$	−4.16228	−0.772916	−0.386458	−	0.922307i	$-0.626302\pi$
$29$	−4.16228	−0.772916	−0.386458	+	0.922307i	$0.626302\pi$
$30$	0	0
$31$	10.4868	1.88349	0.941745	−	0.336327i	$-0.109185\pi$
$31$	10.4868	1.88349	0.941745	+	0.336327i	$0.109185\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.00000	−0.857493
$35$	0	0
$36$	0	0
$37$	−2.83772	−0.466519	−0.233259	−	0.972415i	$-0.574939\pi$
$37$	−2.83772	−0.466519	−0.233259	+	0.972415i	$0.574939\pi$
$38$	7.16228	1.16187
$39$	0	0
$40$	0	0
$41$	10.3246	1.61242	0.806212	−	0.591626i	$-0.201514\pi$
$41$	10.3246	1.61242	0.806212	+	0.591626i	$0.201514\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	−5.16228	−0.778243
$45$	0	0
$46$	−6.16228	−0.908578
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−4.16228	−0.577204
$53$	6.48683	0.891035	0.445518	−	0.895273i	$-0.353020\pi$
$53$	6.48683	0.891035	0.445518	+	0.895273i	$0.353020\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	4.16228	0.546534
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	0	0
$61$	4.32456	0.553703	0.276851	−	0.960913i	$-0.410709\pi$
$61$	4.32456	0.553703	0.276851	+	0.960913i	$0.410709\pi$
$62$	−10.4868	−1.33183
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−2.67544	−0.326858	−0.163429	−	0.986555i	$-0.552255\pi$
$67$	−2.67544	−0.326858	−0.163429	+	0.986555i	$0.552255\pi$
$68$	5.00000	0.606339
$69$	0	0
$70$	0	0
$71$	12.4868	1.48191	0.740957	−	0.671552i	$-0.234372\pi$
$71$	12.4868	1.48191	0.740957	+	0.671552i	$0.234372\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	2.83772	0.329879
$75$	0	0
$76$	−7.16228	−0.821570
$77$	−5.16228	−0.588296
$78$	0	0
$79$	13.1623	1.48087	0.740436	−	0.672127i	$-0.234619\pi$
$79$	13.1623	1.48087	0.740436	+	0.672127i	$0.234619\pi$
$80$	0	0
$81$	0	0
$82$	−10.3246	−1.14016
$83$	4.32456	0.474682	0.237341	−	0.971426i	$-0.423724\pi$
$83$	4.32456	0.474682	0.237341	+	0.971426i	$0.423724\pi$
$84$	0	0
$85$	0	0
$86$	7.00000	0.754829
$87$	0	0
$88$	5.16228	0.550301
$89$	2.67544	0.283597	0.141798	−	0.989896i	$-0.454712\pi$
$89$	2.67544	0.283597	0.141798	+	0.989896i	$0.454712\pi$
$90$	0	0
$91$	−4.16228	−0.436325
$92$	6.16228	0.642462
$93$	0	0
$94$	2.00000	0.206284
$95$	0	0
$96$	0	0
$97$	5.16228	0.524150	0.262075	−	0.965048i	$-0.415593\pi$
$97$	5.16228	0.524150	0.262075	+	0.965048i	$0.415593\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	0	0
$101$	5.16228	0.513666	0.256833	−	0.966456i	$-0.417321\pi$
$101$	5.16228	0.513666	0.256833	+	0.966456i	$0.417321\pi$
$102$	0	0
$103$	−14.4868	−1.42743	−0.713715	−	0.700436i	$-0.752989\pi$
$103$	−14.4868	−1.42743	−0.713715	+	0.700436i	$0.752989\pi$
$104$	4.16228	0.408145
$105$	0	0
$106$	−6.48683	−0.630057
$107$	−1.16228	−0.112362	−0.0561808	−	0.998421i	$-0.517892\pi$
$107$	−1.16228	−0.112362	−0.0561808	+	0.998421i	$0.517892\pi$
$108$	0	0
$109$	−3.16228	−0.302891	−0.151446	−	0.988466i	$-0.548393\pi$
$109$	−3.16228	−0.302891	−0.151446	+	0.988466i	$0.548393\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	−13.1623	−1.23820	−0.619101	−	0.785311i	$-0.712503\pi$
$113$	−13.1623	−1.23820	−0.619101	+	0.785311i	$0.712503\pi$
$114$	0	0
$115$	0	0
$116$	−4.16228	−0.386458
$117$	0	0
$118$	−7.00000	−0.644402
$119$	5.00000	0.458349
$120$	0	0
$121$	15.6491	1.42265
$122$	−4.32456	−0.391527
$123$	0	0
$124$	10.4868	0.941745
$125$	0	0
$126$	0	0
$127$	−18.3246	−1.62604	−0.813021	−	0.582235i	$-0.802178\pi$
$127$	−18.3246	−1.62604	−0.813021	+	0.582235i	$0.802178\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	5.32456	0.465209	0.232604	−	0.972571i	$-0.425275\pi$
$131$	5.32456	0.465209	0.232604	+	0.972571i	$0.425275\pi$
$132$	0	0
$133$	−7.16228	−0.621048
$134$	2.67544	0.231123
$135$	0	0
$136$	−5.00000	−0.428746
$137$	−8.32456	−0.711215	−0.355607	−	0.934635i	$-0.615726\pi$
$137$	−8.32456	−0.711215	−0.355607	+	0.934635i	$0.615726\pi$
$138$	0	0
$139$	−17.4868	−1.48321	−0.741607	−	0.670835i	$-0.765936\pi$
$139$	−17.4868	−1.48321	−0.741607	+	0.670835i	$0.765936\pi$
$140$	0	0
$141$	0	0
$142$	−12.4868	−1.04787
$143$	21.4868	1.79682
$144$	0	0
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	−2.83772	−0.233259
$149$	−10.8114	−0.885703	−0.442852	−	0.896595i	$-0.646033\pi$
$149$	−10.8114	−0.885703	−0.442852	+	0.896595i	$0.646033\pi$
$150$	0	0
$151$	−13.8114	−1.12395	−0.561977	−	0.827153i	$-0.689959\pi$
$151$	−13.8114	−1.12395	−0.561977	+	0.827153i	$0.689959\pi$
$152$	7.16228	0.580937
$153$	0	0
$154$	5.16228	0.415988
$155$	0	0
$156$	0	0
$157$	8.16228	0.651421	0.325710	−	0.945470i	$-0.394397\pi$
$157$	8.16228	0.651421	0.325710	+	0.945470i	$0.394397\pi$
$158$	−13.1623	−1.04713
$159$	0	0
$160$	0	0
$161$	6.16228	0.485656
$162$	0	0
$163$	11.6491	0.912429	0.456214	−	0.889870i	$-0.349205\pi$
$163$	11.6491	0.912429	0.456214	+	0.889870i	$0.349205\pi$
$164$	10.3246	0.806212
$165$	0	0
$166$	−4.32456	−0.335651
$167$	−21.4868	−1.66270	−0.831351	−	0.555748i	$-0.812432\pi$
$167$	−21.4868	−1.66270	−0.831351	+	0.555748i	$0.812432\pi$
$168$	0	0
$169$	4.32456	0.332658
$170$	0	0
$171$	0	0
$172$	−7.00000	−0.533745
$173$	−19.8114	−1.50623	−0.753116	−	0.657888i	$-0.771450\pi$
$173$	−19.8114	−1.50623	−0.753116	+	0.657888i	$0.771450\pi$
$174$	0	0
$175$	0	0
$176$	−5.16228	−0.389121
$177$	0	0
$178$	−2.67544	−0.200533
$179$	−5.67544	−0.424203	−0.212101	−	0.977248i	$-0.568031\pi$
$179$	−5.67544	−0.424203	−0.212101	+	0.977248i	$0.568031\pi$
$180$	0	0
$181$	−4.48683	−0.333504	−0.166752	−	0.985999i	$-0.553328\pi$
$181$	−4.48683	−0.333504	−0.166752	+	0.985999i	$0.553328\pi$
$182$	4.16228	0.308529
$183$	0	0
$184$	−6.16228	−0.454289
$185$	0	0
$186$	0	0
$187$	−25.8114	−1.88752
$188$	−2.00000	−0.145865
$189$	0	0
$190$	0	0
$191$	−26.3246	−1.90478	−0.952389	−	0.304886i	$-0.901382\pi$
$191$	−26.3246	−1.90478	−0.952389	+	0.304886i	$0.901382\pi$
$192$	0	0
$193$	−19.9737	−1.43774	−0.718868	−	0.695147i	$-0.755339\pi$
$193$	−19.9737	−1.43774	−0.718868	+	0.695147i	$0.755339\pi$
$194$	−5.16228	−0.370630
$195$	0	0
$196$	1.00000	0.0714286
$197$	−5.67544	−0.404359	−0.202179	−	0.979349i	$-0.564802\pi$
$197$	−5.67544	−0.404359	−0.202179	+	0.979349i	$0.564802\pi$
$198$	0	0
$199$	−24.4868	−1.73583	−0.867913	−	0.496717i	$-0.834539\pi$
$199$	−24.4868	−1.73583	−0.867913	+	0.496717i	$0.834539\pi$
$200$	0	0
$201$	0	0
$202$	−5.16228	−0.363217
$203$	−4.16228	−0.292135
$204$	0	0
$205$	0	0
$206$	14.4868	1.00935
$207$	0	0
$208$	−4.16228	−0.288602
$209$	36.9737	2.55752
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	6.48683	0.445518
$213$	0	0
$214$	1.16228	0.0794517
$215$	0	0
$216$	0	0
$217$	10.4868	0.711893
$218$	3.16228	0.214176
$219$	0	0
$220$	0	0
$221$	−20.8114	−1.39993
$222$	0	0
$223$	0.649111	0.0434677	0.0217338	−	0.999764i	$-0.493081\pi$
$223$	0.649111	0.0434677	0.0217338	+	0.999764i	$0.493081\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	13.1623	0.875542
$227$	13.3246	0.884382	0.442191	−	0.896921i	$-0.354201\pi$
$227$	13.3246	0.884382	0.442191	+	0.896921i	$0.354201\pi$
$228$	0	0
$229$	5.67544	0.375044	0.187522	−	0.982260i	$-0.439954\pi$
$229$	5.67544	0.375044	0.187522	+	0.982260i	$0.439954\pi$
$230$	0	0
$231$	0	0
$232$	4.16228	0.273267
$233$	−16.1359	−1.05710	−0.528550	−	0.848902i	$-0.677264\pi$
$233$	−16.1359	−1.05710	−0.528550	+	0.848902i	$0.677264\pi$
$234$	0	0
$235$	0	0
$236$	7.00000	0.455661
$237$	0	0
$238$	−5.00000	−0.324102
$239$	−6.32456	−0.409101	−0.204551	−	0.978856i	$-0.565573\pi$
$239$	−6.32456	−0.409101	−0.204551	+	0.978856i	$0.565573\pi$
$240$	0	0
$241$	−0.837722	−0.0539624	−0.0269812	−	0.999636i	$-0.508589\pi$
$241$	−0.837722	−0.0539624	−0.0269812	+	0.999636i	$0.508589\pi$
$242$	−15.6491	−1.00596
$243$	0	0
$244$	4.32456	0.276851
$245$	0	0
$246$	0	0
$247$	29.8114	1.89685
$248$	−10.4868	−0.665915
$249$	0	0
$250$	0	0
$251$	−28.6491	−1.80832	−0.904158	−	0.427198i	$-0.859501\pi$
$251$	−28.6491	−1.80832	−0.904158	+	0.427198i	$0.859501\pi$
$252$	0	0
$253$	−31.8114	−1.99996
$254$	18.3246	1.14978
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−2.83772	−0.176327
$260$	0	0
$261$	0	0
$262$	−5.32456	−0.328952
$263$	−2.16228	−0.133332	−0.0666659	−	0.997775i	$-0.521236\pi$
$263$	−2.16228	−0.133332	−0.0666659	+	0.997775i	$0.521236\pi$
$264$	0	0
$265$	0	0
$266$	7.16228	0.439147
$267$	0	0
$268$	−2.67544	−0.163429
$269$	−9.67544	−0.589922	−0.294961	−	0.955509i	$-0.595307\pi$
$269$	−9.67544	−0.589922	−0.294961	+	0.955509i	$0.595307\pi$
$270$	0	0
$271$	−9.83772	−0.597599	−0.298800	−	0.954316i	$-0.596586\pi$
$271$	−9.83772	−0.597599	−0.298800	+	0.954316i	$0.596586\pi$
$272$	5.00000	0.303170
$273$	0	0
$274$	8.32456	0.502905
$275$	0	0
$276$	0	0
$277$	12.6491	0.760011	0.380006	−	0.924984i	$-0.375922\pi$
$277$	12.6491	0.760011	0.380006	+	0.924984i	$0.375922\pi$
$278$	17.4868	1.04879
$279$	0	0
$280$	0	0
$281$	21.4868	1.28180	0.640898	−	0.767626i	$-0.278562\pi$
$281$	21.4868	1.28180	0.640898	+	0.767626i	$0.278562\pi$
$282$	0	0
$283$	9.81139	0.583226	0.291613	−	0.956536i	$-0.405808\pi$
$283$	9.81139	0.583226	0.291613	+	0.956536i	$0.405808\pi$
$284$	12.4868	0.740957
$285$	0	0
$286$	−21.4868	−1.27054
$287$	10.3246	0.609439
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	−10.0000	−0.585206
$293$	12.9737	0.757930	0.378965	−	0.925411i	$-0.376280\pi$
$293$	12.9737	0.757930	0.378965	+	0.925411i	$0.376280\pi$
$294$	0	0
$295$	0	0
$296$	2.83772	0.164939
$297$	0	0
$298$	10.8114	0.626287
$299$	−25.6491	−1.48333
$300$	0	0
$301$	−7.00000	−0.403473
$302$	13.8114	0.794756
$303$	0	0
$304$	−7.16228	−0.410785
$305$	0	0
$306$	0	0
$307$	−4.83772	−0.276103	−0.138052	−	0.990425i	$-0.544084\pi$
$307$	−4.83772	−0.276103	−0.138052	+	0.990425i	$0.544084\pi$
$308$	−5.16228	−0.294148
$309$	0	0
$310$	0	0
$311$	2.51317	0.142509	0.0712543	−	0.997458i	$-0.477300\pi$
$311$	2.51317	0.142509	0.0712543	+	0.997458i	$0.477300\pi$
$312$	0	0
$313$	24.4605	1.38259	0.691295	−	0.722573i	$-0.257041\pi$
$313$	24.4605	1.38259	0.691295	+	0.722573i	$0.257041\pi$
$314$	−8.16228	−0.460624
$315$	0	0
$316$	13.1623	0.740436
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	21.4868	1.20303
$320$	0	0
$321$	0	0
$322$	−6.16228	−0.343410
$323$	−35.8114	−1.99260
$324$	0	0
$325$	0	0
$326$	−11.6491	−0.645185
$327$	0	0
$328$	−10.3246	−0.570078
$329$	−2.00000	−0.110264
$330$	0	0
$331$	2.35089	0.129217	0.0646083	−	0.997911i	$-0.479420\pi$
$331$	2.35089	0.129217	0.0646083	+	0.997911i	$0.479420\pi$
$332$	4.32456	0.237341
$333$	0	0
$334$	21.4868	1.17571
$335$	0	0
$336$	0	0
$337$	−3.32456	−0.181100	−0.0905500	−	0.995892i	$-0.528863\pi$
$337$	−3.32456	−0.181100	−0.0905500	+	0.995892i	$0.528863\pi$
$338$	−4.32456	−0.235225
$339$	0	0
$340$	0	0
$341$	−54.1359	−2.93163
$342$	0	0
$343$	1.00000	0.0539949
$344$	7.00000	0.377415
$345$	0	0
$346$	19.8114	1.06507
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	8.16228	0.436917	0.218458	−	0.975846i	$-0.429897\pi$
$349$	8.16228	0.436917	0.218458	+	0.975846i	$0.429897\pi$
$350$	0	0
$351$	0	0
$352$	5.16228	0.275150
$353$	−7.32456	−0.389847	−0.194923	−	0.980818i	$-0.562446\pi$
$353$	−7.32456	−0.389847	−0.194923	+	0.980818i	$0.562446\pi$
$354$	0	0
$355$	0	0
$356$	2.67544	0.141798
$357$	0	0
$358$	5.67544	0.299957
$359$	−7.83772	−0.413659	−0.206830	−	0.978377i	$-0.566315\pi$
$359$	−7.83772	−0.413659	−0.206830	+	0.978377i	$0.566315\pi$
$360$	0	0
$361$	32.2982	1.69991
$362$	4.48683	0.235823
$363$	0	0
$364$	−4.16228	−0.218163
$365$	0	0
$366$	0	0
$367$	−30.8114	−1.60834	−0.804171	−	0.594398i	$-0.797390\pi$
$367$	−30.8114	−1.60834	−0.804171	+	0.594398i	$0.797390\pi$
$368$	6.16228	0.321231
$369$	0	0
$370$	0	0
$371$	6.48683	0.336780
$372$	0	0
$373$	−29.8114	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$373$	−29.8114	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$374$	25.8114	1.33468
$375$	0	0
$376$	2.00000	0.103142
$377$	17.3246	0.892260
$378$	0	0
$379$	30.6491	1.57434	0.787170	−	0.616737i	$-0.211546\pi$
$379$	30.6491	1.57434	0.787170	+	0.616737i	$0.211546\pi$
$380$	0	0
$381$	0	0
$382$	26.3246	1.34688
$383$	13.6754	0.698783	0.349391	−	0.936977i	$-0.386388\pi$
$383$	13.6754	0.698783	0.349391	+	0.936977i	$0.386388\pi$
$384$	0	0
$385$	0	0
$386$	19.9737	1.01663
$387$	0	0
$388$	5.16228	0.262075
$389$	34.3246	1.74032	0.870162	−	0.492766i	$-0.164014\pi$
$389$	34.3246	1.74032	0.870162	+	0.492766i	$0.164014\pi$
$390$	0	0
$391$	30.8114	1.55820
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	5.67544	0.285925
$395$	0	0
$396$	0	0
$397$	−18.3246	−0.919683	−0.459842	−	0.888001i	$-0.652094\pi$
$397$	−18.3246	−0.919683	−0.459842	+	0.888001i	$0.652094\pi$
$398$	24.4868	1.22741
$399$	0	0
$400$	0	0
$401$	33.6228	1.67904	0.839521	−	0.543328i	$-0.182836\pi$
$401$	33.6228	1.67904	0.839521	+	0.543328i	$0.182836\pi$
$402$	0	0
$403$	−43.6491	−2.17432
$404$	5.16228	0.256833
$405$	0	0
$406$	4.16228	0.206570
$407$	14.6491	0.726129
$408$	0	0
$409$	−19.2982	−0.954236	−0.477118	−	0.878839i	$-0.658319\pi$
$409$	−19.2982	−0.954236	−0.477118	+	0.878839i	$0.658319\pi$
$410$	0	0
$411$	0	0
$412$	−14.4868	−0.713715
$413$	7.00000	0.344447
$414$	0	0
$415$	0	0
$416$	4.16228	0.204072
$417$	0	0
$418$	−36.9737	−1.80844
$419$	34.6228	1.69143	0.845717	−	0.533632i	$-0.179173\pi$
$419$	34.6228	1.69143	0.845717	+	0.533632i	$0.179173\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	3.00000	0.146038
$423$	0	0
$424$	−6.48683	−0.315028
$425$	0	0
$426$	0	0
$427$	4.32456	0.209280
$428$	−1.16228	−0.0561808
$429$	0	0
$430$	0	0
$431$	−31.2982	−1.50758	−0.753791	−	0.657114i	$-0.771777\pi$
$431$	−31.2982	−1.50758	−0.753791	+	0.657114i	$0.771777\pi$
$432$	0	0
$433$	−37.2982	−1.79244	−0.896219	−	0.443612i	$-0.853697\pi$
$433$	−37.2982	−1.79244	−0.896219	+	0.443612i	$0.853697\pi$
$434$	−10.4868	−0.503384
$435$	0	0
$436$	−3.16228	−0.151446
$437$	−44.1359	−2.11131
$438$	0	0
$439$	−20.8114	−0.993273	−0.496637	−	0.867959i	$-0.665432\pi$
$439$	−20.8114	−0.993273	−0.496637	+	0.867959i	$0.665432\pi$
$440$	0	0
$441$	0	0
$442$	20.8114	0.989897
$443$	9.48683	0.450733	0.225367	−	0.974274i	$-0.427642\pi$
$443$	9.48683	0.450733	0.225367	+	0.974274i	$0.427642\pi$
$444$	0	0
$445$	0	0
$446$	−0.649111	−0.0307363
$447$	0	0
$448$	1.00000	0.0472456
$449$	21.2982	1.00513	0.502563	−	0.864541i	$-0.332391\pi$
$449$	21.2982	1.00513	0.502563	+	0.864541i	$0.332391\pi$
$450$	0	0
$451$	−53.2982	−2.50972
$452$	−13.1623	−0.619101
$453$	0	0
$454$	−13.3246	−0.625352
$455$	0	0
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	−5.67544	−0.265196
$459$	0	0
$460$	0	0
$461$	−15.2982	−0.712509	−0.356255	−	0.934389i	$-0.615946\pi$
$461$	−15.2982	−0.712509	−0.356255	+	0.934389i	$0.615946\pi$
$462$	0	0
$463$	10.8377	0.503672	0.251836	−	0.967770i	$-0.418966\pi$
$463$	10.8377	0.503672	0.251836	+	0.967770i	$0.418966\pi$
$464$	−4.16228	−0.193229
$465$	0	0
$466$	16.1359	0.747483
$467$	10.6491	0.492782	0.246391	−	0.969170i	$-0.420755\pi$
$467$	10.6491	0.492782	0.246391	+	0.969170i	$0.420755\pi$
$468$	0	0
$469$	−2.67544	−0.123541
$470$	0	0
$471$	0	0
$472$	−7.00000	−0.322201
$473$	36.1359	1.66153
$474$	0	0
$475$	0	0
$476$	5.00000	0.229175
$477$	0	0
$478$	6.32456	0.289278
$479$	3.35089	0.153106	0.0765530	−	0.997066i	$-0.475609\pi$
$479$	3.35089	0.153106	0.0765530	+	0.997066i	$0.475609\pi$
$480$	0	0
$481$	11.8114	0.538553
$482$	0.837722	0.0381572
$483$	0	0
$484$	15.6491	0.711323
$485$	0	0
$486$	0	0
$487$	−10.6491	−0.482557	−0.241279	−	0.970456i	$-0.577567\pi$
$487$	−10.6491	−0.482557	−0.241279	+	0.970456i	$0.577567\pi$
$488$	−4.32456	−0.195763
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	−20.8114	−0.937298
$494$	−29.8114	−1.34128
$495$	0	0
$496$	10.4868	0.470873
$497$	12.4868	0.560111
$498$	0	0
$499$	−15.6754	−0.701729	−0.350865	−	0.936426i	$-0.614112\pi$
$499$	−15.6754	−0.701729	−0.350865	+	0.936426i	$0.614112\pi$
$500$	0	0
$501$	0	0
$502$	28.6491	1.27867
$503$	19.1623	0.854404	0.427202	−	0.904156i	$-0.359499\pi$
$503$	19.1623	0.854404	0.427202	+	0.904156i	$0.359499\pi$
$504$	0	0
$505$	0	0
$506$	31.8114	1.41419
$507$	0	0
$508$	−18.3246	−0.813021
$509$	10.8377	0.480374	0.240187	−	0.970727i	$-0.422791\pi$
$509$	10.8377	0.480374	0.240187	+	0.970727i	$0.422791\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−12.0000	−0.529297
$515$	0	0
$516$	0	0
$517$	10.3246	0.454073
$518$	2.83772	0.124682
$519$	0	0
$520$	0	0
$521$	−13.3246	−0.583759	−0.291880	−	0.956455i	$-0.594281\pi$
$521$	−13.3246	−0.583759	−0.291880	+	0.956455i	$0.594281\pi$
$522$	0	0
$523$	−44.0000	−1.92399	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000	−1.92399	−0.961993	+	0.273075i	$0.911959\pi$
$524$	5.32456	0.232604
$525$	0	0
$526$	2.16228	0.0942798
$527$	52.4342	2.28407
$528$	0	0
$529$	14.9737	0.651029
$530$	0	0
$531$	0	0
$532$	−7.16228	−0.310524
$533$	−42.9737	−1.86140
$534$	0	0
$535$	0	0
$536$	2.67544	0.115562
$537$	0	0
$538$	9.67544	0.417138
$539$	−5.16228	−0.222355
$540$	0	0
$541$	8.83772	0.379963	0.189982	−	0.981788i	$-0.439157\pi$
$541$	8.83772	0.379963	0.189982	+	0.981788i	$0.439157\pi$
$542$	9.83772	0.422566
$543$	0	0
$544$	−5.00000	−0.214373
$545$	0	0
$546$	0	0
$547$	−12.3246	−0.526960	−0.263480	−	0.964665i	$-0.584870\pi$
$547$	−12.3246	−0.526960	−0.263480	+	0.964665i	$0.584870\pi$
$548$	−8.32456	−0.355607
$549$	0	0
$550$	0	0
$551$	29.8114	1.27001
$552$	0	0
$553$	13.1623	0.559717
$554$	−12.6491	−0.537409
$555$	0	0
$556$	−17.4868	−0.741607
$557$	26.8114	1.13603	0.568017	−	0.823016i	$-0.307711\pi$
$557$	26.8114	1.13603	0.568017	+	0.823016i	$0.307711\pi$
$558$	0	0
$559$	29.1359	1.23232
$560$	0	0
$561$	0	0
$562$	−21.4868	−0.906367
$563$	−4.02633	−0.169690	−0.0848449	−	0.996394i	$-0.527039\pi$
$563$	−4.02633	−0.169690	−0.0848449	+	0.996394i	$0.527039\pi$
$564$	0	0
$565$	0	0
$566$	−9.81139	−0.412403
$567$	0	0
$568$	−12.4868	−0.523936
$569$	18.8377	0.789718	0.394859	−	0.918742i	$-0.370793\pi$
$569$	18.8377	0.789718	0.394859	+	0.918742i	$0.370793\pi$
$570$	0	0
$571$	−34.6228	−1.44892	−0.724459	−	0.689318i	$-0.757910\pi$
$571$	−34.6228	−1.44892	−0.724459	+	0.689318i	$0.757910\pi$
$572$	21.4868	0.898410
$573$	0	0
$574$	−10.3246	−0.430939
$575$	0	0
$576$	0	0
$577$	−18.5132	−0.770713	−0.385357	−	0.922768i	$-0.625922\pi$
$577$	−18.5132	−0.770713	−0.385357	+	0.922768i	$0.625922\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	0	0
$581$	4.32456	0.179413
$582$	0	0
$583$	−33.4868	−1.38688
$584$	10.0000	0.413803
$585$	0	0
$586$	−12.9737	−0.535937
$587$	−43.2719	−1.78602	−0.893011	−	0.450035i	$-0.851412\pi$
$587$	−43.2719	−1.78602	−0.893011	+	0.450035i	$0.851412\pi$
$588$	0	0
$589$	−75.1096	−3.09484
$590$	0	0
$591$	0	0
$592$	−2.83772	−0.116630
$593$	−18.9737	−0.779155	−0.389578	−	0.920994i	$-0.627379\pi$
$593$	−18.9737	−0.779155	−0.389578	+	0.920994i	$0.627379\pi$
$594$	0	0
$595$	0	0
$596$	−10.8114	−0.442852
$597$	0	0
$598$	25.6491	1.04887
$599$	−11.5132	−0.470415	−0.235208	−	0.971945i	$-0.575577\pi$
$599$	−11.5132	−0.470415	−0.235208	+	0.971945i	$0.575577\pi$
$600$	0	0
$601$	38.9737	1.58977	0.794884	−	0.606761i	$-0.207531\pi$
$601$	38.9737	1.58977	0.794884	+	0.606761i	$0.207531\pi$
$602$	7.00000	0.285299
$603$	0	0
$604$	−13.8114	−0.561977
$605$	0	0
$606$	0	0
$607$	29.8377	1.21108	0.605538	−	0.795816i	$-0.292958\pi$
$607$	29.8377	1.21108	0.605538	+	0.795816i	$0.292958\pi$
$608$	7.16228	0.290469
$609$	0	0
$610$	0	0
$611$	8.32456	0.336775
$612$	0	0
$613$	20.7851	0.839500	0.419750	−	0.907640i	$-0.362118\pi$
$613$	20.7851	0.839500	0.419750	+	0.907640i	$0.362118\pi$
$614$	4.83772	0.195235
$615$	0	0
$616$	5.16228	0.207994
$617$	−24.3246	−0.979270	−0.489635	−	0.871928i	$-0.662870\pi$
$617$	−24.3246	−0.979270	−0.489635	+	0.871928i	$0.662870\pi$
$618$	0	0
$619$	−5.35089	−0.215070	−0.107535	−	0.994201i	$-0.534296\pi$
$619$	−5.35089	−0.215070	−0.107535	+	0.994201i	$0.534296\pi$
$620$	0	0
$621$	0	0
$622$	−2.51317	−0.100769
$623$	2.67544	0.107189
$624$	0	0
$625$	0	0
$626$	−24.4605	−0.977638
$627$	0	0
$628$	8.16228	0.325710
$629$	−14.1886	−0.565737
$630$	0	0
$631$	22.3246	0.888727	0.444363	−	0.895847i	$-0.353430\pi$
$631$	22.3246	0.888727	0.444363	+	0.895847i	$0.353430\pi$
$632$	−13.1623	−0.523567
$633$	0	0
$634$	10.0000	0.397151
$635$	0	0
$636$	0	0
$637$	−4.16228	−0.164915
$638$	−21.4868	−0.850672
$639$	0	0
$640$	0	0
$641$	−12.5132	−0.494240	−0.247120	−	0.968985i	$-0.579484\pi$
$641$	−12.5132	−0.494240	−0.247120	+	0.968985i	$0.579484\pi$
$642$	0	0
$643$	36.7851	1.45066	0.725330	−	0.688401i	$-0.241687\pi$
$643$	36.7851	1.45066	0.725330	+	0.688401i	$0.241687\pi$
$644$	6.16228	0.242828
$645$	0	0
$646$	35.8114	1.40898
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	−36.1359	−1.41846
$650$	0	0
$651$	0	0
$652$	11.6491	0.456214
$653$	−3.13594	−0.122719	−0.0613595	−	0.998116i	$-0.519544\pi$
$653$	−3.13594	−0.122719	−0.0613595	+	0.998116i	$0.519544\pi$
$654$	0	0
$655$	0	0
$656$	10.3246	0.403106
$657$	0	0
$658$	2.00000	0.0779681
$659$	−1.81139	−0.0705617	−0.0352808	−	0.999377i	$-0.511233\pi$
$659$	−1.81139	−0.0705617	−0.0352808	+	0.999377i	$0.511233\pi$
$660$	0	0
$661$	−12.3246	−0.479370	−0.239685	−	0.970851i	$-0.577044\pi$
$661$	−12.3246	−0.479370	−0.239685	+	0.970851i	$0.577044\pi$
$662$	−2.35089	−0.0913699
$663$	0	0
$664$	−4.32456	−0.167825
$665$	0	0
$666$	0	0
$667$	−25.6491	−0.993138
$668$	−21.4868	−0.831351
$669$	0	0
$670$	0	0
$671$	−22.3246	−0.861830
$672$	0	0
$673$	1.64911	0.0635685	0.0317843	−	0.999495i	$-0.489881\pi$
$673$	1.64911	0.0635685	0.0317843	+	0.999495i	$0.489881\pi$
$674$	3.32456	0.128057
$675$	0	0
$676$	4.32456	0.166329
$677$	8.00000	0.307465	0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000	0.307465	0.153732	+	0.988113i	$0.450871\pi$
$678$	0	0
$679$	5.16228	0.198110
$680$	0	0
$681$	0	0
$682$	54.1359	2.07297
$683$	−19.4868	−0.745643	−0.372821	−	0.927903i	$-0.621610\pi$
$683$	−19.4868	−0.745643	−0.372821	+	0.927903i	$0.621610\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−7.00000	−0.266872
$689$	−27.0000	−1.02862
$690$	0	0
$691$	−26.9737	−1.02613	−0.513063	−	0.858351i	$-0.671489\pi$
$691$	−26.9737	−1.02613	−0.513063	+	0.858351i	$0.671489\pi$
$692$	−19.8114	−0.753116
$693$	0	0
$694$	−18.0000	−0.683271
$695$	0	0
$696$	0	0
$697$	51.6228	1.95535
$698$	−8.16228	−0.308947
$699$	0	0
$700$	0	0
$701$	−5.35089	−0.202100	−0.101050	−	0.994881i	$-0.532220\pi$
$701$	−5.35089	−0.202100	−0.101050	+	0.994881i	$0.532220\pi$
$702$	0	0
$703$	20.3246	0.766555
$704$	−5.16228	−0.194561
$705$	0	0
$706$	7.32456	0.275663
$707$	5.16228	0.194147
$708$	0	0
$709$	26.5132	0.995723	0.497861	−	0.867257i	$-0.334119\pi$
$709$	26.5132	0.995723	0.497861	+	0.867257i	$0.334119\pi$
$710$	0	0
$711$	0	0
$712$	−2.67544	−0.100267
$713$	64.6228	2.42014
$714$	0	0
$715$	0	0
$716$	−5.67544	−0.212101
$717$	0	0
$718$	7.83772	0.292501
$719$	−32.8377	−1.22464	−0.612320	−	0.790610i	$-0.709764\pi$
$719$	−32.8377	−1.22464	−0.612320	+	0.790610i	$0.709764\pi$
$720$	0	0
$721$	−14.4868	−0.539518
$722$	−32.2982	−1.20202
$723$	0	0
$724$	−4.48683	−0.166752
$725$	0	0
$726$	0	0
$727$	−4.16228	−0.154370	−0.0771852	−	0.997017i	$-0.524593\pi$
$727$	−4.16228	−0.154370	−0.0771852	+	0.997017i	$0.524593\pi$
$728$	4.16228	0.154264
$729$	0	0
$730$	0	0
$731$	−35.0000	−1.29452
$732$	0	0
$733$	36.8114	1.35966	0.679830	−	0.733370i	$-0.262054\pi$
$733$	36.8114	1.35966	0.679830	+	0.733370i	$0.262054\pi$
$734$	30.8114	1.13727
$735$	0	0
$736$	−6.16228	−0.227145
$737$	13.8114	0.508749
$738$	0	0
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	0	0
$741$	0	0
$742$	−6.48683	−0.238139
$743$	−6.48683	−0.237979	−0.118989	−	0.992896i	$-0.537965\pi$
$743$	−6.48683	−0.237979	−0.118989	+	0.992896i	$0.537965\pi$
$744$	0	0
$745$	0	0
$746$	29.8114	1.09147
$747$	0	0
$748$	−25.8114	−0.943758
$749$	−1.16228	−0.0424687
$750$	0	0
$751$	−28.7851	−1.05038	−0.525191	−	0.850985i	$-0.676006\pi$
$751$	−28.7851	−1.05038	−0.525191	+	0.850985i	$0.676006\pi$
$752$	−2.00000	−0.0729325
$753$	0	0
$754$	−17.3246	−0.630923
$755$	0	0
$756$	0	0
$757$	−32.6491	−1.18665	−0.593326	−	0.804962i	$-0.702186\pi$
$757$	−32.6491	−1.18665	−0.593326	+	0.804962i	$0.702186\pi$
$758$	−30.6491	−1.11323
$759$	0	0
$760$	0	0
$761$	1.64911	0.0597802	0.0298901	−	0.999553i	$-0.490484\pi$
$761$	1.64911	0.0597802	0.0298901	+	0.999553i	$0.490484\pi$
$762$	0	0
$763$	−3.16228	−0.114482
$764$	−26.3246	−0.952389
$765$	0	0
$766$	−13.6754	−0.494114
$767$	−29.1359	−1.05204
$768$	0	0
$769$	29.8114	1.07503	0.537513	−	0.843255i	$-0.319364\pi$
$769$	29.8114	1.07503	0.537513	+	0.843255i	$0.319364\pi$
$770$	0	0
$771$	0	0
$772$	−19.9737	−0.718868
$773$	−42.9737	−1.54566	−0.772828	−	0.634616i	$-0.781158\pi$
$773$	−42.9737	−1.54566	−0.772828	+	0.634616i	$0.781158\pi$
$774$	0	0
$775$	0	0
$776$	−5.16228	−0.185315
$777$	0	0
$778$	−34.3246	−1.23059
$779$	−73.9473	−2.64944
$780$	0	0
$781$	−64.4605	−2.30658
$782$	−30.8114	−1.10181
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	−9.62278	−0.343015	−0.171507	−	0.985183i	$-0.554864\pi$
$787$	−9.62278	−0.343015	−0.171507	+	0.985183i	$0.554864\pi$
$788$	−5.67544	−0.202179
$789$	0	0
$790$	0	0
$791$	−13.1623	−0.467997
$792$	0	0
$793$	−18.0000	−0.639199
$794$	18.3246	0.650314
$795$	0	0
$796$	−24.4868	−0.867913
$797$	−16.9737	−0.601238	−0.300619	−	0.953744i	$-0.597193\pi$
$797$	−16.9737	−0.601238	−0.300619	+	0.953744i	$0.597193\pi$
$798$	0	0
$799$	−10.0000	−0.353775
$800$	0	0
$801$	0	0
$802$	−33.6228	−1.18726
$803$	51.6228	1.82173
$804$	0	0
$805$	0	0
$806$	43.6491	1.53747
$807$	0	0
$808$	−5.16228	−0.181608
$809$	4.64911	0.163454	0.0817270	−	0.996655i	$-0.473956\pi$
$809$	4.64911	0.163454	0.0817270	+	0.996655i	$0.473956\pi$
$810$	0	0
$811$	−11.2982	−0.396734	−0.198367	−	0.980128i	$-0.563564\pi$
$811$	−11.2982	−0.396734	−0.198367	+	0.980128i	$0.563564\pi$
$812$	−4.16228	−0.146067
$813$	0	0
$814$	−14.6491	−0.513451
$815$	0	0
$816$	0	0
$817$	50.1359	1.75403
$818$	19.2982	0.674746
$819$	0	0
$820$	0	0
$821$	7.18861	0.250884	0.125442	−	0.992101i	$-0.459965\pi$
$821$	7.18861	0.250884	0.125442	+	0.992101i	$0.459965\pi$
$822$	0	0
$823$	−3.48683	−0.121543	−0.0607717	−	0.998152i	$-0.519356\pi$
$823$	−3.48683	−0.121543	−0.0607717	+	0.998152i	$0.519356\pi$
$824$	14.4868	0.504673
$825$	0	0
$826$	−7.00000	−0.243561
$827$	−24.3246	−0.845848	−0.422924	−	0.906165i	$-0.638996\pi$
$827$	−24.3246	−0.845848	−0.422924	+	0.906165i	$0.638996\pi$
$828$	0	0
$829$	34.2719	1.19031	0.595156	−	0.803610i	$-0.297090\pi$
$829$	34.2719	1.19031	0.595156	+	0.803610i	$0.297090\pi$
$830$	0	0
$831$	0	0
$832$	−4.16228	−0.144301
$833$	5.00000	0.173240
$834$	0	0
$835$	0	0
$836$	36.9737	1.27876
$837$	0	0
$838$	−34.6228	−1.19602
$839$	−4.51317	−0.155812	−0.0779059	−	0.996961i	$-0.524823\pi$
$839$	−4.51317	−0.155812	−0.0779059	+	0.996961i	$0.524823\pi$
$840$	0	0
$841$	−11.6754	−0.402602
$842$	2.00000	0.0689246
$843$	0	0
$844$	−3.00000	−0.103264
$845$	0	0
$846$	0	0
$847$	15.6491	0.537710
$848$	6.48683	0.222759
$849$	0	0
$850$	0	0
$851$	−17.4868	−0.599441
$852$	0	0
$853$	−6.16228	−0.210992	−0.105496	−	0.994420i	$-0.533643\pi$
$853$	−6.16228	−0.210992	−0.105496	+	0.994420i	$0.533643\pi$
$854$	−4.32456	−0.147983
$855$	0	0
$856$	1.16228	0.0397258
$857$	−12.6754	−0.432985	−0.216492	−	0.976284i	$-0.569462\pi$
$857$	−12.6754	−0.432985	−0.216492	+	0.976284i	$0.569462\pi$
$858$	0	0
$859$	38.7851	1.32333	0.661664	−	0.749800i	$-0.269850\pi$
$859$	38.7851	1.32333	0.661664	+	0.749800i	$0.269850\pi$
$860$	0	0
$861$	0	0
$862$	31.2982	1.06602
$863$	−1.83772	−0.0625568	−0.0312784	−	0.999511i	$-0.509958\pi$
$863$	−1.83772	−0.0625568	−0.0312784	+	0.999511i	$0.509958\pi$
$864$	0	0
$865$	0	0
$866$	37.2982	1.26745
$867$	0	0
$868$	10.4868	0.355946
$869$	−67.9473	−2.30496
$870$	0	0
$871$	11.1359	0.377327
$872$	3.16228	0.107088
$873$	0	0
$874$	44.1359	1.49292
$875$	0	0
$876$	0	0
$877$	−26.9737	−0.910836	−0.455418	−	0.890278i	$-0.650510\pi$
$877$	−26.9737	−0.910836	−0.455418	+	0.890278i	$0.650510\pi$
$878$	20.8114	0.702350
$879$	0	0
$880$	0	0
$881$	−40.9473	−1.37955	−0.689775	−	0.724023i	$-0.742291\pi$
$881$	−40.9473	−1.37955	−0.689775	+	0.724023i	$0.742291\pi$
$882$	0	0
$883$	−5.70178	−0.191880	−0.0959401	−	0.995387i	$-0.530586\pi$
$883$	−5.70178	−0.191880	−0.0959401	+	0.995387i	$0.530586\pi$
$884$	−20.8114	−0.699963
$885$	0	0
$886$	−9.48683	−0.318716
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	−18.3246	−0.614586
$890$	0	0
$891$	0	0
$892$	0.649111	0.0217338
$893$	14.3246	0.479353
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−21.2982	−0.710731
$899$	−43.6491	−1.45578
$900$	0	0
$901$	32.4342	1.08054
$902$	53.2982	1.77464
$903$	0	0
$904$	13.1623	0.437771
$905$	0	0
$906$	0	0
$907$	−21.2982	−0.707196	−0.353598	−	0.935398i	$-0.615042\pi$
$907$	−21.2982	−0.707196	−0.353598	+	0.935398i	$0.615042\pi$
$908$	13.3246	0.442191
$909$	0	0
$910$	0	0
$911$	9.29822	0.308064	0.154032	−	0.988066i	$-0.450774\pi$
$911$	9.29822	0.308064	0.154032	+	0.988066i	$0.450774\pi$
$912$	0	0
$913$	−22.3246	−0.738835
$914$	1.00000	0.0330771
$915$	0	0
$916$	5.67544	0.187522
$917$	5.32456	0.175832
$918$	0	0
$919$	10.6491	0.351282	0.175641	−	0.984454i	$-0.443800\pi$
$919$	10.6491	0.351282	0.175641	+	0.984454i	$0.443800\pi$
$920$	0	0
$921$	0	0
$922$	15.2982	0.503820
$923$	−51.9737	−1.71073
$924$	0	0
$925$	0	0
$926$	−10.8377	−0.356150
$927$	0	0
$928$	4.16228	0.136633
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	−7.16228	−0.234734
$932$	−16.1359	−0.528550
$933$	0	0
$934$	−10.6491	−0.348450
$935$	0	0
$936$	0	0
$937$	−16.5132	−0.539462	−0.269731	−	0.962936i	$-0.586935\pi$
$937$	−16.5132	−0.539462	−0.269731	+	0.962936i	$0.586935\pi$
$938$	2.67544	0.0873564
$939$	0	0
$940$	0	0
$941$	−17.3509	−0.565623	−0.282811	−	0.959176i	$-0.591267\pi$
$941$	−17.3509	−0.565623	−0.282811	+	0.959176i	$0.591267\pi$
$942$	0	0
$943$	63.6228	2.07184
$944$	7.00000	0.227831
$945$	0	0
$946$	−36.1359	−1.17488
$947$	14.5132	0.471615	0.235807	−	0.971800i	$-0.424227\pi$
$947$	14.5132	0.471615	0.235807	+	0.971800i	$0.424227\pi$
$948$	0	0
$949$	41.6228	1.35113
$950$	0	0
$951$	0	0
$952$	−5.00000	−0.162051
$953$	27.6228	0.894790	0.447395	−	0.894337i	$-0.352352\pi$
$953$	27.6228	0.894790	0.447395	+	0.894337i	$0.352352\pi$
$954$	0	0
$955$	0	0
$956$	−6.32456	−0.204551
$957$	0	0
$958$	−3.35089	−0.108262
$959$	−8.32456	−0.268814
$960$	0	0
$961$	78.9737	2.54754
$962$	−11.8114	−0.380814
$963$	0	0
$964$	−0.837722	−0.0269812
$965$	0	0
$966$	0	0
$967$	−43.6228	−1.40281	−0.701407	−	0.712761i	$-0.747444\pi$
$967$	−43.6228	−1.40281	−0.701407	+	0.712761i	$0.747444\pi$
$968$	−15.6491	−0.502981
$969$	0	0
$970$	0	0
$971$	40.9473	1.31406	0.657031	−	0.753863i	$-0.271812\pi$
$971$	40.9473	1.31406	0.657031	+	0.753863i	$0.271812\pi$
$972$	0	0
$973$	−17.4868	−0.560602
$974$	10.6491	0.341220
$975$	0	0
$976$	4.32456	0.138426
$977$	−31.9473	−1.02209	−0.511043	−	0.859555i	$-0.670741\pi$
$977$	−31.9473	−1.02209	−0.511043	+	0.859555i	$0.670741\pi$
$978$	0	0
$979$	−13.8114	−0.441414
$980$	0	0
$981$	0	0
$982$	12.0000	0.382935
$983$	−29.6754	−0.946500	−0.473250	−	0.880928i	$-0.656919\pi$
$983$	−29.6754	−0.946500	−0.473250	+	0.880928i	$0.656919\pi$
$984$	0	0
$985$	0	0
$986$	20.8114	0.662770
$987$	0	0
$988$	29.8114	0.948427
$989$	−43.1359	−1.37164
$990$	0	0
$991$	59.9473	1.90429	0.952145	−	0.305647i	$-0.0988728\pi$
$991$	59.9473	1.90429	0.952145	+	0.305647i	$0.0988728\pi$
$992$	−10.4868	−0.332957
$993$	0	0
$994$	−12.4868	−0.396058
$995$	0	0
$996$	0	0
$997$	−16.1623	−0.511864	−0.255932	−	0.966695i	$-0.582382\pi$
$997$	−16.1623	−0.511864	−0.255932	+	0.966695i	$0.582382\pi$
$998$	15.6754	0.496198
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9450.2.a.eg.1.1		2
3.2	odd	2		9450.2.a.ew.1.2		2
5.2	odd	4		1890.2.g.o.379.2	✓	4
5.3	odd	4		1890.2.g.o.379.4	yes	4
5.4	even	2		9450.2.a.en.1.1		2
15.2	even	4		1890.2.g.p.379.3	yes	4
15.8	even	4		1890.2.g.p.379.1	yes	4
15.14	odd	2		9450.2.a.ef.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1890.2.g.o.379.2	✓	4	5.2	odd	4
1890.2.g.o.379.4	yes	4	5.3	odd	4
1890.2.g.p.379.1	yes	4	15.8	even	4
1890.2.g.p.379.3	yes	4	15.2	even	4
9450.2.a.ef.1.2		2	15.14	odd	2
9450.2.a.eg.1.1		2	1.1	even	1	trivial
9450.2.a.en.1.1		2	5.4	even	2
9450.2.a.ew.1.2		2	3.2	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:	\( N \)	\(=\)	\( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -5.16228 q^{11} -4.16228 q^{13} -1.00000 q^{14} +1.00000 q^{16} +5.00000 q^{17} -7.16228 q^{19} +5.16228 q^{22} +6.16228 q^{23} +4.16228 q^{26} +1.00000 q^{28} -4.16228 q^{29} +10.4868 q^{31} -1.00000 q^{32} -5.00000 q^{34} -2.83772 q^{37} +7.16228 q^{38} +10.3246 q^{41} -7.00000 q^{43} -5.16228 q^{44} -6.16228 q^{46} -2.00000 q^{47} +1.00000 q^{49} -4.16228 q^{52} +6.48683 q^{53} -1.00000 q^{56} +4.16228 q^{58} +7.00000 q^{59} +4.32456 q^{61} -10.4868 q^{62} +1.00000 q^{64} -2.67544 q^{67} +5.00000 q^{68} +12.4868 q^{71} -10.0000 q^{73} +2.83772 q^{74} -7.16228 q^{76} -5.16228 q^{77} +13.1623 q^{79} -10.3246 q^{82} +4.32456 q^{83} +7.00000 q^{86} +5.16228 q^{88} +2.67544 q^{89} -4.16228 q^{91} +6.16228 q^{92} +2.00000 q^{94} +5.16228 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 4 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{17} - 8 q^{19} + 4 q^{22} + 6 q^{23} + 2 q^{26} + 2 q^{28} - 2 q^{29} + 2 q^{31} - 2 q^{32} - 10 q^{34} - 12 q^{37} + 8 q^{38} + 8 q^{41} - 14 q^{43} - 4 q^{44} - 6 q^{46} - 4 q^{47} + 2 q^{49} - 2 q^{52} - 6 q^{53} - 2 q^{56} + 2 q^{58} + 14 q^{59} - 4 q^{61} - 2 q^{62} + 2 q^{64} - 18 q^{67} + 10 q^{68} + 6 q^{71} - 20 q^{73} + 12 q^{74} - 8 q^{76} - 4 q^{77} + 20 q^{79} - 8 q^{82} - 4 q^{83} + 14 q^{86} + 4 q^{88} + 18 q^{89} - 2 q^{91} + 6 q^{92} + 4 q^{94} + 4 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 4 * q^11 - 2 * q^13 - 2 * q^14 + 2 * q^16 + 10 * q^17 - 8 * q^19 + 4 * q^22 + 6 * q^23 + 2 * q^26 + 2 * q^28 - 2 * q^29 + 2 * q^31 - 2 * q^32 - 10 * q^34 - 12 * q^37 + 8 * q^38 + 8 * q^41 - 14 * q^43 - 4 * q^44 - 6 * q^46 - 4 * q^47 + 2 * q^49 - 2 * q^52 - 6 * q^53 - 2 * q^56 + 2 * q^58 + 14 * q^59 - 4 * q^61 - 2 * q^62 + 2 * q^64 - 18 * q^67 + 10 * q^68 + 6 * q^71 - 20 * q^73 + 12 * q^74 - 8 * q^76 - 4 * q^77 + 20 * q^79 - 8 * q^82 - 4 * q^83 + 14 * q^86 + 4 * q^88 + 18 * q^89 - 2 * q^91 + 6 * q^92 + 4 * q^94 + 4 * q^97 - 2 * q^98

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