LMFDB - Embedded newform 9200.2.a.cq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.37988$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.95969 q^{3} +2.28394 q^{7} +0.840379 q^{9} +O(q^{10})$	$q-1.95969 q^{3} +2.28394 q^{7} +0.840379 q^{9} -1.12432 q^{11} -5.95969 q^{13} -5.80007 q^{17} +4.08401 q^{19} -4.47581 q^{21} -1.00000 q^{23} +4.23218 q^{27} -0.408263 q^{29} +3.19187 q^{31} +2.20332 q^{33} +9.80345 q^{37} +11.6791 q^{39} +6.27087 q^{41} +7.75474 q^{43} -6.40020 q^{47} -1.78361 q^{49} +11.3663 q^{51} +6.73590 q^{53} -8.00339 q^{57} +4.75976 q^{59} -6.33265 q^{61} +1.91938 q^{63} +0.283942 q^{67} +1.95969 q^{69} +13.9516 q^{71} -9.61659 q^{73} -2.56788 q^{77} -4.48387 q^{79} -10.8149 q^{81} -10.8223 q^{83} +0.800068 q^{87} +5.68414 q^{89} -13.6116 q^{91} -6.25508 q^{93} -11.0676 q^{97} -0.944856 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 6 * q^7 + 4 * q^9	$ 4 q + 2 q^{3} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 14 q^{13} - 14 q^{17} + 4 q^{19} - 2 q^{21} - 4 q^{23} + 14 q^{27} + 4 q^{29} - 14 q^{33} - 2 q^{37} + 8 q^{39} - 8 q^{41} + 4 q^{43} + 2 q^{47} - 10 q^{51} - 4 q^{53} - 8 q^{61} - 12 q^{63} - 2 q^{67} - 2 q^{69} + 24 q^{71} - 18 q^{73} - 4 q^{77} - 24 q^{79} + 8 q^{81} - 6 q^{83} - 6 q^{87} - 8 q^{89} - 26 q^{91} - 2 q^{93} - 34 q^{97} - 36 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 6 * q^7 + 4 * q^9 - 2 * q^11 - 14 * q^13 - 14 * q^17 + 4 * q^19 - 2 * q^21 - 4 * q^23 + 14 * q^27 + 4 * q^29 - 14 * q^33 - 2 * q^37 + 8 * q^39 - 8 * q^41 + 4 * q^43 + 2 * q^47 - 10 * q^51 - 4 * q^53 - 8 * q^61 - 12 * q^63 - 2 * q^67 - 2 * q^69 + 24 * q^71 - 18 * q^73 - 4 * q^77 - 24 * q^79 + 8 * q^81 - 6 * q^83 - 6 * q^87 - 8 * q^89 - 26 * q^91 - 2 * q^93 - 34 * q^97 - 36 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.95969	−1.13143	−0.565713	−	0.824602i	$-0.691399\pi$
$3$	−1.95969	−1.13143	−0.565713	+	0.824602i	$0.691399\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.28394	0.863249	0.431624	−	0.902053i	$-0.357941\pi$
$7$	2.28394	0.863249	0.431624	+	0.902053i	$0.357941\pi$
$8$	0	0
$9$	0.840379	0.280126
$10$	0	0
$11$	−1.12432	−0.338996	−0.169498	−	0.985531i	$-0.554215\pi$
$11$	−1.12432	−0.338996	−0.169498	+	0.985531i	$0.554215\pi$
$12$	0	0
$13$	−5.95969	−1.65292	−0.826460	−	0.562995i	$-0.809649\pi$
$13$	−5.95969	−1.65292	−0.826460	+	0.562995i	$0.809649\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.80007	−1.40672	−0.703362	−	0.710832i	$-0.748318\pi$
$17$	−5.80007	−1.40672	−0.703362	+	0.710832i	$0.748318\pi$
$18$	0	0
$19$	4.08401	0.936936	0.468468	−	0.883480i	$-0.344806\pi$
$19$	4.08401	0.936936	0.468468	+	0.883480i	$0.344806\pi$
$20$	0	0
$21$	−4.47581	−0.976703
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.23218	0.814484
$28$	0	0
$29$	−0.408263	−0.0758125	−0.0379062	−	0.999281i	$-0.512069\pi$
$29$	−0.408263	−0.0758125	−0.0379062	+	0.999281i	$0.512069\pi$
$30$	0	0
$31$	3.19187	0.573277	0.286639	−	0.958039i	$-0.407462\pi$
$31$	3.19187	0.573277	0.286639	+	0.958039i	$0.407462\pi$
$32$	0	0
$33$	2.20332	0.383549
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.80345	1.61168	0.805839	−	0.592135i	$-0.201715\pi$
$37$	9.80345	1.61168	0.805839	+	0.592135i	$0.201715\pi$
$38$	0	0
$39$	11.6791	1.87016
$40$	0	0
$41$	6.27087	0.979345	0.489673	−	0.871906i	$-0.337116\pi$
$41$	6.27087	0.979345	0.489673	+	0.871906i	$0.337116\pi$
$42$	0	0
$43$	7.75474	1.18259	0.591294	−	0.806456i	$-0.298617\pi$
$43$	7.75474	1.18259	0.591294	+	0.806456i	$0.298617\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.40020	−0.933566	−0.466783	−	0.884372i	$-0.654587\pi$
$47$	−6.40020	−0.933566	−0.466783	+	0.884372i	$0.654587\pi$
$48$	0	0
$49$	−1.78361	−0.254801
$50$	0	0
$51$	11.3663	1.59160
$52$	0	0
$53$	6.73590	0.925247	0.462624	−	0.886555i	$-0.346908\pi$
$53$	6.73590	0.925247	0.462624	+	0.886555i	$0.346908\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.00339	−1.06007
$58$	0	0
$59$	4.75976	0.619667	0.309834	−	0.950791i	$-0.399727\pi$
$59$	4.75976	0.619667	0.309834	+	0.950791i	$0.399727\pi$
$60$	0	0
$61$	−6.33265	−0.810813	−0.405406	−	0.914137i	$-0.632870\pi$
$61$	−6.33265	−0.810813	−0.405406	+	0.914137i	$0.632870\pi$
$62$	0	0
$63$	1.91938	0.241819
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.283942	0.0346890	0.0173445	−	0.999850i	$-0.494479\pi$
$67$	0.283942	0.0346890	0.0173445	+	0.999850i	$0.494479\pi$
$68$	0	0
$69$	1.95969	0.235919
$70$	0	0
$71$	13.9516	1.65575	0.827877	−	0.560910i	$-0.189549\pi$
$71$	13.9516	1.65575	0.827877	+	0.560910i	$0.189549\pi$
$72$	0	0
$73$	−9.61659	−1.12554	−0.562769	−	0.826615i	$-0.690264\pi$
$73$	−9.61659	−1.12554	−0.562769	+	0.826615i	$0.690264\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.56788	−0.292637
$78$	0	0
$79$	−4.48387	−0.504475	−0.252238	−	0.967665i	$-0.581166\pi$
$79$	−4.48387	−0.504475	−0.252238	+	0.967665i	$0.581166\pi$
$80$	0	0
$81$	−10.8149	−1.20166
$82$	0	0
$83$	−10.8223	−1.18790	−0.593951	−	0.804501i	$-0.702433\pi$
$83$	−10.8223	−1.18790	−0.593951	+	0.804501i	$0.702433\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.800068	0.0857763
$88$	0	0
$89$	5.68414	0.602518	0.301259	−	0.953542i	$-0.402593\pi$
$89$	5.68414	0.602518	0.301259	+	0.953542i	$0.402593\pi$
$90$	0	0
$91$	−13.6116	−1.42688
$92$	0	0
$93$	−6.25508	−0.648621
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.0676	−1.12374	−0.561870	−	0.827226i	$-0.689918\pi$
$97$	−11.0676	−1.12374	−0.561870	+	0.827226i	$0.689918\pi$
$98$	0	0
$99$	−0.944856	−0.0949616
$100$	0	0
$101$	1.60014	0.159219	0.0796097	−	0.996826i	$-0.474633\pi$
$101$	1.60014	0.159219	0.0796097	+	0.996826i	$0.474633\pi$
$102$	0	0
$103$	1.23218	0.121411	0.0607054	−	0.998156i	$-0.480665\pi$
$103$	1.23218	0.121411	0.0607054	+	0.998156i	$0.480665\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.235232	−0.0227408	−0.0113704	−	0.999935i	$-0.503619\pi$
$107$	−0.235232	−0.0227408	−0.0113704	+	0.999935i	$0.503619\pi$
$108$	0	0
$109$	7.43550	0.712192	0.356096	−	0.934449i	$-0.384108\pi$
$109$	7.43550	0.712192	0.356096	+	0.934449i	$0.384108\pi$
$110$	0	0
$111$	−19.2117	−1.82350
$112$	0	0
$113$	2.28394	0.214855	0.107428	−	0.994213i	$-0.465739\pi$
$113$	2.28394	0.214855	0.107428	+	0.994213i	$0.465739\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.00840	−0.463027
$118$	0	0
$119$	−13.2470	−1.21435
$120$	0	0
$121$	−9.73590	−0.885082
$122$	0	0
$123$	−12.2890	−1.10806
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.2151	−0.906444	−0.453222	−	0.891398i	$-0.649725\pi$
$127$	−10.2151	−0.906444	−0.453222	+	0.891398i	$0.649725\pi$
$128$	0	0
$129$	−15.1969	−1.33801
$130$	0	0
$131$	16.2232	1.41742	0.708712	−	0.705498i	$-0.249276\pi$
$131$	16.2232	1.41742	0.708712	+	0.705498i	$0.249276\pi$
$132$	0	0
$133$	9.32764	0.808809
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.89715	0.418392	0.209196	−	0.977874i	$-0.432915\pi$
$137$	4.89715	0.418392	0.209196	+	0.977874i	$0.432915\pi$
$138$	0	0
$139$	−4.43889	−0.376502	−0.188251	−	0.982121i	$-0.560282\pi$
$139$	−4.43889	−0.376502	−0.188251	+	0.982121i	$0.560282\pi$
$140$	0	0
$141$	12.5424	1.05626
$142$	0	0
$143$	6.70060	0.560333
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.49532	0.288289
$148$	0	0
$149$	−2.78361	−0.228042	−0.114021	−	0.993478i	$-0.536373\pi$
$149$	−2.78361	−0.228042	−0.114021	+	0.993478i	$0.536373\pi$
$150$	0	0
$151$	−11.9278	−0.970669	−0.485334	−	0.874329i	$-0.661302\pi$
$151$	−11.9278	−0.970669	−0.485334	+	0.874329i	$0.661302\pi$
$152$	0	0
$153$	−4.87426	−0.394060
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−22.9550	−1.83201	−0.916005	−	0.401167i	$-0.868605\pi$
$157$	−22.9550	−1.83201	−0.916005	+	0.401167i	$0.868605\pi$
$158$	0	0
$159$	−13.2003	−1.04685
$160$	0	0
$161$	−2.28394	−0.180000
$162$	0	0
$163$	−10.2083	−0.799578	−0.399789	−	0.916607i	$-0.630917\pi$
$163$	−10.2083	−0.799578	−0.399789	+	0.916607i	$0.630917\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.84715	0.529849	0.264924	−	0.964269i	$-0.414653\pi$
$167$	6.84715	0.529849	0.264924	+	0.964269i	$0.414653\pi$
$168$	0	0
$169$	22.5179	1.73215
$170$	0	0
$171$	3.43212	0.262461
$172$	0	0
$173$	−4.29539	−0.326572	−0.163286	−	0.986579i	$-0.552209\pi$
$173$	−4.29539	−0.326572	−0.163286	+	0.986579i	$0.552209\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.32764	−0.701108
$178$	0	0
$179$	14.8626	1.11088	0.555442	−	0.831555i	$-0.312549\pi$
$179$	14.8626	1.11088	0.555442	+	0.831555i	$0.312549\pi$
$180$	0	0
$181$	13.1311	0.976027	0.488013	−	0.872836i	$-0.337722\pi$
$181$	13.1311	0.976027	0.488013	+	0.872836i	$0.337722\pi$
$182$	0	0
$183$	12.4100	0.917375
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.52114	0.476873
$188$	0	0
$189$	9.66606	0.703103
$190$	0	0
$191$	−13.9819	−1.01170	−0.505848	−	0.862623i	$-0.668820\pi$
$191$	−13.9819	−1.01170	−0.505848	+	0.862623i	$0.668820\pi$
$192$	0	0
$193$	−8.71267	−0.627152	−0.313576	−	0.949563i	$-0.601527\pi$
$193$	−8.71267	−0.627152	−0.313576	+	0.949563i	$0.601527\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.4876	1.60218	0.801088	−	0.598547i	$-0.204255\pi$
$197$	22.4876	1.60218	0.801088	+	0.598547i	$0.204255\pi$
$198$	0	0
$199$	−9.01078	−0.638757	−0.319379	−	0.947627i	$-0.603474\pi$
$199$	−9.01078	−0.638757	−0.319379	+	0.947627i	$0.603474\pi$
$200$	0	0
$201$	−0.556437	−0.0392481
$202$	0	0
$203$	−0.932448	−0.0654450
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.840379	−0.0584104
$208$	0	0
$209$	−4.59174	−0.317617
$210$	0	0
$211$	−5.60014	−0.385529	−0.192765	−	0.981245i	$-0.561745\pi$
$211$	−5.60014	−0.385529	−0.192765	+	0.981245i	$0.561745\pi$
$212$	0	0
$213$	−27.3408	−1.87336
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.29005	0.494881
$218$	0	0
$219$	18.8455	1.27346
$220$	0	0
$221$	34.5666	2.32520
$222$	0	0
$223$	−7.06517	−0.473119	−0.236559	−	0.971617i	$-0.576020\pi$
$223$	−7.06517	−0.473119	−0.236559	+	0.971617i	$0.576020\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.1072	1.66643	0.833213	−	0.552952i	$-0.186499\pi$
$227$	25.1072	1.66643	0.833213	+	0.552952i	$0.186499\pi$
$228$	0	0
$229$	−3.20366	−0.211704	−0.105852	−	0.994382i	$-0.533757\pi$
$229$	−3.20366	−0.211704	−0.105852	+	0.994382i	$0.533757\pi$
$230$	0	0
$231$	5.03225	0.331098
$232$	0	0
$233$	−18.6388	−1.22107	−0.610535	−	0.791989i	$-0.709046\pi$
$233$	−18.6388	−1.22107	−0.610535	+	0.791989i	$0.709046\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.78700	0.570777
$238$	0	0
$239$	−3.18185	−0.205817	−0.102908	−	0.994691i	$-0.532815\pi$
$239$	−3.18185	−0.205817	−0.102908	+	0.994691i	$0.532815\pi$
$240$	0	0
$241$	4.79844	0.309095	0.154547	−	0.987985i	$-0.450608\pi$
$241$	4.79844	0.309095	0.154547	+	0.987985i	$0.450608\pi$
$242$	0	0
$243$	8.49728	0.545101
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−24.3394	−1.54868
$248$	0	0
$249$	21.2083	1.34402
$250$	0	0
$251$	2.24668	0.141809	0.0709045	−	0.997483i	$-0.477411\pi$
$251$	2.24668	0.141809	0.0709045	+	0.997483i	$0.477411\pi$
$252$	0	0
$253$	1.12432	0.0706855
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.0148	−0.687086	−0.343543	−	0.939137i	$-0.611627\pi$
$257$	−11.0148	−0.687086	−0.343543	+	0.939137i	$0.611627\pi$
$258$	0	0
$259$	22.3905	1.39128
$260$	0	0
$261$	−0.343095	−0.0212371
$262$	0	0
$263$	−11.0585	−0.681898	−0.340949	−	0.940082i	$-0.610748\pi$
$263$	−11.0585	−0.681898	−0.340949	+	0.940082i	$0.610748\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−11.1392	−0.681705
$268$	0	0
$269$	−9.90392	−0.603853	−0.301926	−	0.953331i	$-0.597630\pi$
$269$	−9.90392	−0.603853	−0.301926	+	0.953331i	$0.597630\pi$
$270$	0	0
$271$	−4.82655	−0.293192	−0.146596	−	0.989196i	$-0.546832\pi$
$271$	−4.82655	−0.293192	−0.146596	+	0.989196i	$0.546832\pi$
$272$	0	0
$273$	26.6745	1.61441
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.54303	−0.453217	−0.226608	−	0.973986i	$-0.572764\pi$
$277$	−7.54303	−0.453217	−0.226608	+	0.973986i	$0.572764\pi$
$278$	0	0
$279$	2.68238	0.160590
$280$	0	0
$281$	6.01145	0.358613	0.179306	−	0.983793i	$-0.442615\pi$
$281$	6.01145	0.358613	0.179306	+	0.983793i	$0.442615\pi$
$282$	0	0
$283$	−15.6388	−0.929631	−0.464816	−	0.885407i	$-0.653879\pi$
$283$	−15.6388	−0.929631	−0.464816	+	0.885407i	$0.653879\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.3223	0.845419
$288$	0	0
$289$	16.6408	0.978870
$290$	0	0
$291$	21.6890	1.27143
$292$	0	0
$293$	3.21639	0.187904	0.0939518	−	0.995577i	$-0.470050\pi$
$293$	3.21639	0.187904	0.0939518	+	0.995577i	$0.470050\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.75833	−0.276106
$298$	0	0
$299$	5.95969	0.344658
$300$	0	0
$301$	17.7114	1.02087
$302$	0	0
$303$	−3.13577	−0.180145
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−34.4702	−1.96732	−0.983659	−	0.180044i	$-0.942376\pi$
$307$	−34.4702	−1.96732	−0.983659	+	0.180044i	$0.942376\pi$
$308$	0	0
$309$	−2.41470	−0.137367
$310$	0	0
$311$	18.7443	1.06289	0.531446	−	0.847092i	$-0.321649\pi$
$311$	18.7443	1.06289	0.531446	+	0.847092i	$0.321649\pi$
$312$	0	0
$313$	−9.91260	−0.560294	−0.280147	−	0.959957i	$-0.590383\pi$
$313$	−9.91260	−0.560294	−0.280147	+	0.959957i	$0.590383\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.24024	0.518984	0.259492	−	0.965745i	$-0.416445\pi$
$317$	9.24024	0.518984	0.259492	+	0.965745i	$0.416445\pi$
$318$	0	0
$319$	0.459018	0.0257001
$320$	0	0
$321$	0.460982	0.0257295
$322$	0	0
$323$	−23.6875	−1.31801
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−14.5713	−0.805793
$328$	0	0
$329$	−14.6177	−0.805899
$330$	0	0
$331$	23.9684	1.31742	0.658712	−	0.752395i	$-0.271102\pi$
$331$	23.9684	1.31742	0.658712	+	0.752395i	$0.271102\pi$
$332$	0	0
$333$	8.23862	0.451474
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9.76477	−0.531921	−0.265960	−	0.963984i	$-0.585689\pi$
$337$	−9.76477	−0.531921	−0.265960	+	0.963984i	$0.585689\pi$
$338$	0	0
$339$	−4.47581	−0.243093
$340$	0	0
$341$	−3.58869	−0.194338
$342$	0	0
$343$	−20.0613	−1.08321
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.38441	0.0743190	0.0371595	−	0.999309i	$-0.488169\pi$
$347$	1.38441	0.0743190	0.0371595	+	0.999309i	$0.488169\pi$
$348$	0	0
$349$	−32.3106	−1.72954	−0.864772	−	0.502164i	$-0.832537\pi$
$349$	−32.3106	−1.72954	−0.864772	+	0.502164i	$0.832537\pi$
$350$	0	0
$351$	−25.2225	−1.34628
$352$	0	0
$353$	−11.4386	−0.608813	−0.304406	−	0.952542i	$-0.598458\pi$
$353$	−11.4386	−0.608813	−0.304406	+	0.952542i	$0.598458\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	25.9600	1.37395
$358$	0	0
$359$	−9.95568	−0.525441	−0.262720	−	0.964872i	$-0.584620\pi$
$359$	−9.95568	−0.525441	−0.262720	+	0.964872i	$0.584620\pi$
$360$	0	0
$361$	−2.32087	−0.122151
$362$	0	0
$363$	19.0793	1.00141
$364$	0	0
$365$	0	0
$366$	0	0
$367$	27.6785	1.44481	0.722403	−	0.691472i	$-0.243037\pi$
$367$	27.6785	1.44481	0.722403	+	0.691472i	$0.243037\pi$
$368$	0	0
$369$	5.26991	0.274341
$370$	0	0
$371$	15.3844	0.798719
$372$	0	0
$373$	−28.7356	−1.48787	−0.743936	−	0.668251i	$-0.767043\pi$
$373$	−28.7356	−1.48787	−0.743936	+	0.668251i	$0.767043\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.43312	0.125312
$378$	0	0
$379$	18.2715	0.938546	0.469273	−	0.883053i	$-0.344516\pi$
$379$	18.2715	0.938546	0.469273	+	0.883053i	$0.344516\pi$
$380$	0	0
$381$	20.0184	1.02557
$382$	0	0
$383$	−29.0356	−1.48365	−0.741826	−	0.670593i	$-0.766040\pi$
$383$	−29.0356	−1.48365	−0.741826	+	0.670593i	$0.766040\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.51693	0.331274
$388$	0	0
$389$	2.86423	0.145222	0.0726112	−	0.997360i	$-0.476867\pi$
$389$	2.86423	0.145222	0.0726112	+	0.997360i	$0.476867\pi$
$390$	0	0
$391$	5.80007	0.293322
$392$	0	0
$393$	−31.7923	−1.60371
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−16.5592	−0.831080	−0.415540	−	0.909575i	$-0.636407\pi$
$397$	−16.5592	−0.831080	−0.415540	+	0.909575i	$0.636407\pi$
$398$	0	0
$399$	−18.2793	−0.915108
$400$	0	0
$401$	−3.09479	−0.154547	−0.0772733	−	0.997010i	$-0.524621\pi$
$401$	−3.09479	−0.154547	−0.0772733	+	0.997010i	$0.524621\pi$
$402$	0	0
$403$	−19.0226	−0.947582
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.0222	−0.546352
$408$	0	0
$409$	8.93617	0.441865	0.220933	−	0.975289i	$-0.429090\pi$
$409$	8.93617	0.441865	0.220933	+	0.975289i	$0.429090\pi$
$410$	0	0
$411$	−9.59689	−0.473379
$412$	0	0
$413$	10.8710	0.534927
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.69884	0.425984
$418$	0	0
$419$	−24.1237	−1.17852	−0.589260	−	0.807944i	$-0.700581\pi$
$419$	−24.1237	−1.17852	−0.589260	+	0.807944i	$0.700581\pi$
$420$	0	0
$421$	23.8602	1.16288	0.581438	−	0.813591i	$-0.302490\pi$
$421$	23.8602	1.16288	0.581438	+	0.813591i	$0.302490\pi$
$422$	0	0
$423$	−5.37860	−0.261516
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−14.4634	−0.699933
$428$	0	0
$429$	−13.1311	−0.633975
$430$	0	0
$431$	−4.45096	−0.214395	−0.107198	−	0.994238i	$-0.534188\pi$
$431$	−4.45096	−0.214395	−0.107198	+	0.994238i	$0.534188\pi$
$432$	0	0
$433$	−8.10929	−0.389707	−0.194854	−	0.980832i	$-0.562423\pi$
$433$	−8.10929	−0.389707	−0.194854	+	0.980832i	$0.562423\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.08401	−0.195365
$438$	0	0
$439$	−4.47180	−0.213428	−0.106714	−	0.994290i	$-0.534033\pi$
$439$	−4.47180	−0.213428	−0.106714	+	0.994290i	$0.534033\pi$
$440$	0	0
$441$	−1.49891	−0.0713766
$442$	0	0
$443$	9.60047	0.456132	0.228066	−	0.973646i	$-0.426760\pi$
$443$	9.60047	0.456132	0.228066	+	0.973646i	$0.426760\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.45501	0.258013
$448$	0	0
$449$	−6.79944	−0.320886	−0.160443	−	0.987045i	$-0.551292\pi$
$449$	−6.79944	−0.320886	−0.160443	+	0.987045i	$0.551292\pi$
$450$	0	0
$451$	−7.05047	−0.331994
$452$	0	0
$453$	23.3747	1.09824
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.5582	0.914894	0.457447	−	0.889237i	$-0.348764\pi$
$457$	19.5582	0.914894	0.457447	+	0.889237i	$0.348764\pi$
$458$	0	0
$459$	−24.5470	−1.14575
$460$	0	0
$461$	−42.7081	−1.98912	−0.994558	−	0.104184i	$-0.966777\pi$
$461$	−42.7081	−1.98912	−0.994558	+	0.104184i	$0.966777\pi$
$462$	0	0
$463$	−7.42209	−0.344934	−0.172467	−	0.985015i	$-0.555174\pi$
$463$	−7.42209	−0.344934	−0.172467	+	0.985015i	$0.555174\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.5041	−1.04136	−0.520682	−	0.853751i	$-0.674322\pi$
$467$	−22.5041	−1.04136	−0.520682	+	0.853751i	$0.674322\pi$
$468$	0	0
$469$	0.648506	0.0299452
$470$	0	0
$471$	44.9847	2.07278
$472$	0	0
$473$	−8.71882	−0.400892
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.66071	0.259186
$478$	0	0
$479$	10.1667	0.464530	0.232265	−	0.972653i	$-0.425386\pi$
$479$	10.1667	0.464530	0.232265	+	0.972653i	$0.425386\pi$
$480$	0	0
$481$	−58.4255	−2.66398
$482$	0	0
$483$	4.47581	0.203657
$484$	0	0
$485$	0	0
$486$	0	0
$487$	34.9917	1.58562	0.792812	−	0.609467i	$-0.208616\pi$
$487$	34.9917	1.58562	0.792812	+	0.609467i	$0.208616\pi$
$488$	0	0
$489$	20.0051	0.904664
$490$	0	0
$491$	2.27087	0.102483	0.0512415	−	0.998686i	$-0.483682\pi$
$491$	2.27087	0.102483	0.0512415	+	0.998686i	$0.483682\pi$
$492$	0	0
$493$	2.36795	0.106647
$494$	0	0
$495$	0	0
$496$	0	0
$497$	31.8647	1.42933
$498$	0	0
$499$	−20.9929	−0.939773	−0.469887	−	0.882727i	$-0.655705\pi$
$499$	−20.9929	−0.939773	−0.469887	+	0.882727i	$0.655705\pi$
$500$	0	0
$501$	−13.4183	−0.599485
$502$	0	0
$503$	−18.5041	−0.825055	−0.412528	−	0.910945i	$-0.635354\pi$
$503$	−18.5041	−0.825055	−0.412528	+	0.910945i	$0.635354\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−44.1280	−1.95980
$508$	0	0
$509$	−41.8472	−1.85484	−0.927421	−	0.374019i	$-0.877980\pi$
$509$	−41.8472	−1.85484	−0.927421	+	0.374019i	$0.877980\pi$
$510$	0	0
$511$	−21.9637	−0.971619
$512$	0	0
$513$	17.2843	0.763120
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.19588	0.316475
$518$	0	0
$519$	8.41762	0.369493
$520$	0	0
$521$	34.1103	1.49440	0.747199	−	0.664600i	$-0.231398\pi$
$521$	34.1103	1.49440	0.747199	+	0.664600i	$0.231398\pi$
$522$	0	0
$523$	36.4755	1.59496	0.797482	−	0.603343i	$-0.206165\pi$
$523$	36.4755	1.59496	0.797482	+	0.603343i	$0.206165\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.5131	−0.806442
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−37.3724	−1.61878
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−29.1261	−1.25688
$538$	0	0
$539$	2.00535	0.0863765
$540$	0	0
$541$	28.2171	1.21315	0.606573	−	0.795028i	$-0.292544\pi$
$541$	28.2171	1.21315	0.606573	+	0.795028i	$0.292544\pi$
$542$	0	0
$543$	−25.7329	−1.10430
$544$	0	0
$545$	0	0
$546$	0	0
$547$	15.1638	0.648356	0.324178	−	0.945996i	$-0.394912\pi$
$547$	15.1638	0.648356	0.324178	+	0.945996i	$0.394912\pi$
$548$	0	0
$549$	−5.32183	−0.227130
$550$	0	0
$551$	−1.66735	−0.0710314
$552$	0	0
$553$	−10.2409	−0.435488
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.52222	0.403469	0.201735	−	0.979440i	$-0.435342\pi$
$557$	9.52222	0.403469	0.201735	+	0.979440i	$0.435342\pi$
$558$	0	0
$559$	−46.2159	−1.95472
$560$	0	0
$561$	−12.7794	−0.539547
$562$	0	0
$563$	32.7494	1.38022	0.690112	−	0.723702i	$-0.257561\pi$
$563$	32.7494	1.38022	0.690112	+	0.723702i	$0.257561\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−24.7006	−1.03733
$568$	0	0
$569$	−26.8926	−1.12740	−0.563698	−	0.825981i	$-0.690622\pi$
$569$	−26.8926	−1.12740	−0.563698	+	0.825981i	$0.690622\pi$
$570$	0	0
$571$	−0.920000	−0.0385008	−0.0192504	−	0.999815i	$-0.506128\pi$
$571$	−0.920000	−0.0385008	−0.0192504	+	0.999815i	$0.506128\pi$
$572$	0	0
$573$	27.4002	1.14466
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−42.5888	−1.77300	−0.886498	−	0.462732i	$-0.846869\pi$
$577$	−42.5888	−1.77300	−0.886498	+	0.462732i	$0.846869\pi$
$578$	0	0
$579$	17.0741	0.709576
$580$	0	0
$581$	−24.7175	−1.02545
$582$	0	0
$583$	−7.57332	−0.313655
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.51212	0.392607	0.196304	−	0.980543i	$-0.437106\pi$
$587$	9.51212	0.392607	0.196304	+	0.980543i	$0.437106\pi$
$588$	0	0
$589$	13.0356	0.537124
$590$	0	0
$591$	−44.0687	−1.81274
$592$	0	0
$593$	31.0719	1.27597	0.637986	−	0.770048i	$-0.279768\pi$
$593$	31.0719	1.27597	0.637986	+	0.770048i	$0.279768\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.6583	0.722707
$598$	0	0
$599$	−2.04065	−0.0833787	−0.0416893	−	0.999131i	$-0.513274\pi$
$599$	−2.04065	−0.0833787	−0.0416893	+	0.999131i	$0.513274\pi$
$600$	0	0
$601$	−25.2377	−1.02947	−0.514733	−	0.857351i	$-0.672109\pi$
$601$	−25.2377	−1.02947	−0.514733	+	0.857351i	$0.672109\pi$
$602$	0	0
$603$	0.238619	0.00971731
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10.4305	0.423361	0.211680	−	0.977339i	$-0.432106\pi$
$607$	10.4305	0.423361	0.211680	+	0.977339i	$0.432106\pi$
$608$	0	0
$609$	1.82731	0.0740463
$610$	0	0
$611$	38.1432	1.54311
$612$	0	0
$613$	4.97745	0.201037	0.100519	−	0.994935i	$-0.467950\pi$
$613$	4.97745	0.201037	0.100519	+	0.994935i	$0.467950\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.5881	−1.23143	−0.615715	−	0.787969i	$-0.711133\pi$
$617$	−30.5881	−1.23143	−0.615715	+	0.787969i	$0.711133\pi$
$618$	0	0
$619$	48.0404	1.93091	0.965453	−	0.260579i	$-0.0839133\pi$
$619$	48.0404	1.93091	0.965453	+	0.260579i	$0.0839133\pi$
$620$	0	0
$621$	−4.23218	−0.169832
$622$	0	0
$623$	12.9823	0.520123
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.99837	0.359360
$628$	0	0
$629$	−56.8607	−2.26718
$630$	0	0
$631$	−10.6157	−0.422605	−0.211303	−	0.977421i	$-0.567771\pi$
$631$	−10.6157	−0.422605	−0.211303	+	0.977421i	$0.567771\pi$
$632$	0	0
$633$	10.9745	0.436198
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.6298	0.421166
$638$	0	0
$639$	11.7247	0.463820
$640$	0	0
$641$	3.47844	0.137390	0.0686951	−	0.997638i	$-0.478116\pi$
$641$	3.47844	0.137390	0.0686951	+	0.997638i	$0.478116\pi$
$642$	0	0
$643$	2.05348	0.0809813	0.0404906	−	0.999180i	$-0.487108\pi$
$643$	2.05348	0.0809813	0.0404906	+	0.999180i	$0.487108\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.3408	−0.760367	−0.380184	−	0.924911i	$-0.624139\pi$
$647$	−19.3408	−0.760367	−0.380184	+	0.924911i	$0.624139\pi$
$648$	0	0
$649$	−5.35149	−0.210064
$650$	0	0
$651$	−14.2862	−0.559922
$652$	0	0
$653$	−21.9288	−0.858139	−0.429070	−	0.903271i	$-0.641159\pi$
$653$	−21.9288	−0.858139	−0.429070	+	0.903271i	$0.641159\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.08158	−0.315293
$658$	0	0
$659$	−38.1351	−1.48553	−0.742767	−	0.669550i	$-0.766487\pi$
$659$	−38.1351	−1.48553	−0.742767	+	0.669550i	$0.766487\pi$
$660$	0	0
$661$	−28.9007	−1.12411	−0.562053	−	0.827101i	$-0.689988\pi$
$661$	−28.9007	−1.12411	−0.562053	+	0.827101i	$0.689988\pi$
$662$	0	0
$663$	−67.7398	−2.63079
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.408263	0.0158080
$668$	0	0
$669$	13.8455	0.535299
$670$	0	0
$671$	7.11993	0.274862
$672$	0	0
$673$	−49.5051	−1.90828	−0.954140	−	0.299361i	$-0.903226\pi$
$673$	−49.5051	−1.90828	−0.954140	+	0.299361i	$0.903226\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.87426	−0.264199	−0.132100	−	0.991236i	$-0.542172\pi$
$677$	−6.87426	−0.264199	−0.132100	+	0.991236i	$0.542172\pi$
$678$	0	0
$679$	−25.2776	−0.970067
$680$	0	0
$681$	−49.2024	−1.88544
$682$	0	0
$683$	−36.9887	−1.41533	−0.707666	−	0.706547i	$-0.750252\pi$
$683$	−36.9887	−1.41533	−0.707666	+	0.706547i	$0.750252\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.27817	0.239527
$688$	0	0
$689$	−40.1439	−1.52936
$690$	0	0
$691$	−38.5485	−1.46645	−0.733227	−	0.679984i	$-0.761987\pi$
$691$	−38.5485	−1.46645	−0.733227	+	0.679984i	$0.761987\pi$
$692$	0	0
$693$	−2.15800	−0.0819755
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−36.3715	−1.37767
$698$	0	0
$699$	36.5263	1.38155
$700$	0	0
$701$	−36.3146	−1.37158	−0.685791	−	0.727798i	$-0.740544\pi$
$701$	−36.3146	−1.37158	−0.685791	+	0.727798i	$0.740544\pi$
$702$	0	0
$703$	40.0374	1.51004
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.65462	0.137446
$708$	0	0
$709$	−3.24463	−0.121855	−0.0609274	−	0.998142i	$-0.519406\pi$
$709$	−3.24463	−0.121855	−0.0609274	+	0.998142i	$0.519406\pi$
$710$	0	0
$711$	−3.76815	−0.141317
$712$	0	0
$713$	−3.19187	−0.119537
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.23543	0.232867
$718$	0	0
$719$	−10.4214	−0.388654	−0.194327	−	0.980937i	$-0.562252\pi$
$719$	−10.4214	−0.388654	−0.194327	+	0.980937i	$0.562252\pi$
$720$	0	0
$721$	2.81424	0.104808
$722$	0	0
$723$	−9.40345	−0.349718
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−9.67651	−0.358882	−0.179441	−	0.983769i	$-0.557429\pi$
$727$	−9.67651	−0.358882	−0.179441	+	0.983769i	$0.557429\pi$
$728$	0	0
$729$	15.7927	0.584914
$730$	0	0
$731$	−44.9780	−1.66357
$732$	0	0
$733$	−15.5782	−0.575396	−0.287698	−	0.957721i	$-0.592890\pi$
$733$	−15.5782	−0.575396	−0.287698	+	0.957721i	$0.592890\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.319242	−0.0117594
$738$	0	0
$739$	−14.4328	−0.530919	−0.265459	−	0.964122i	$-0.585524\pi$
$739$	−14.4328	−0.530919	−0.265459	+	0.964122i	$0.585524\pi$
$740$	0	0
$741$	47.6977	1.75222
$742$	0	0
$743$	−4.50305	−0.165201	−0.0826005	−	0.996583i	$-0.526323\pi$
$743$	−4.50305	−0.165201	−0.0826005	+	0.996583i	$0.526323\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.09483	−0.332763
$748$	0	0
$749$	−0.537257	−0.0196309
$750$	0	0
$751$	−18.5451	−0.676721	−0.338361	−	0.941017i	$-0.609872\pi$
$751$	−18.5451	−0.676721	−0.338361	+	0.941017i	$0.609872\pi$
$752$	0	0
$753$	−4.40279	−0.160447
$754$	0	0
$755$	0	0
$756$	0	0
$757$	27.0366	0.982663	0.491332	−	0.870973i	$-0.336510\pi$
$757$	27.0366	0.982663	0.491332	+	0.870973i	$0.336510\pi$
$758$	0	0
$759$	−2.20332	−0.0799754
$760$	0	0
$761$	15.4066	0.558490	0.279245	−	0.960220i	$-0.409916\pi$
$761$	15.4066	0.558490	0.279245	+	0.960220i	$0.409916\pi$
$762$	0	0
$763$	16.9823	0.614799
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−28.3667	−1.02426
$768$	0	0
$769$	15.1216	0.545299	0.272650	−	0.962113i	$-0.412100\pi$
$769$	15.1216	0.545299	0.272650	+	0.962113i	$0.412100\pi$
$770$	0	0
$771$	21.5856	0.777388
$772$	0	0
$773$	28.7131	1.03274	0.516370	−	0.856366i	$-0.327283\pi$
$773$	28.7131	1.03274	0.516370	+	0.856366i	$0.327283\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−43.8784	−1.57413
$778$	0	0
$779$	25.6103	0.917584
$780$	0	0
$781$	−15.6861	−0.561293
$782$	0	0
$783$	−1.72784	−0.0617481
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16.3131	−0.581501	−0.290750	−	0.956799i	$-0.593905\pi$
$787$	−16.3131	−0.581501	−0.290750	+	0.956799i	$0.593905\pi$
$788$	0	0
$789$	21.6713	0.771518
$790$	0	0
$791$	5.21639	0.185473
$792$	0	0
$793$	37.7406	1.34021
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.88374	−0.102147	−0.0510736	−	0.998695i	$-0.516264\pi$
$797$	−2.88374	−0.102147	−0.0510736	+	0.998695i	$0.516264\pi$
$798$	0	0
$799$	37.1216	1.31327
$800$	0	0
$801$	4.77684	0.168781
$802$	0	0
$803$	10.8121	0.381552
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.4086	0.683215
$808$	0	0
$809$	39.7878	1.39886	0.699432	−	0.714699i	$-0.253436\pi$
$809$	39.7878	1.39886	0.699432	+	0.714699i	$0.253436\pi$
$810$	0	0
$811$	−38.5454	−1.35351	−0.676756	−	0.736208i	$-0.736615\pi$
$811$	−38.5454	−1.35351	−0.676756	+	0.736208i	$0.736615\pi$
$812$	0	0
$813$	9.45853	0.331725
$814$	0	0
$815$	0	0
$816$	0	0
$817$	31.6705	1.10801
$818$	0	0
$819$	−11.4389	−0.399707
$820$	0	0
$821$	−16.3428	−0.570368	−0.285184	−	0.958473i	$-0.592055\pi$
$821$	−16.3428	−0.570368	−0.285184	+	0.958473i	$0.592055\pi$
$822$	0	0
$823$	38.6446	1.34707	0.673534	−	0.739157i	$-0.264776\pi$
$823$	38.6446	1.34707	0.673534	+	0.739157i	$0.264776\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21.6969	0.754474	0.377237	−	0.926117i	$-0.376874\pi$
$827$	21.6969	0.754474	0.377237	+	0.926117i	$0.376874\pi$
$828$	0	0
$829$	13.9429	0.484259	0.242129	−	0.970244i	$-0.422154\pi$
$829$	13.9429	0.484259	0.242129	+	0.970244i	$0.422154\pi$
$830$	0	0
$831$	14.7820	0.512781
$832$	0	0
$833$	10.3451	0.358435
$834$	0	0
$835$	0	0
$836$	0	0
$837$	13.5086	0.466925
$838$	0	0
$839$	−52.4484	−1.81072	−0.905360	−	0.424644i	$-0.860399\pi$
$839$	−52.4484	−1.81072	−0.905360	+	0.424644i	$0.860399\pi$
$840$	0	0
$841$	−28.8333	−0.994252
$842$	0	0
$843$	−11.7806	−0.405744
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−22.2362	−0.764046
$848$	0	0
$849$	30.6472	1.05181
$850$	0	0
$851$	−9.80345	−0.336058
$852$	0	0
$853$	−30.4634	−1.04305	−0.521524	−	0.853237i	$-0.674636\pi$
$853$	−30.4634	−1.04305	−0.521524	+	0.853237i	$0.674636\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.4911	1.45147	0.725735	−	0.687975i	$-0.241500\pi$
$857$	42.4911	1.45147	0.725735	+	0.687975i	$0.241500\pi$
$858$	0	0
$859$	15.5856	0.531775	0.265888	−	0.964004i	$-0.414335\pi$
$859$	15.5856	0.531775	0.265888	+	0.964004i	$0.414335\pi$
$860$	0	0
$861$	−28.0673	−0.956529
$862$	0	0
$863$	50.8852	1.73215	0.866076	−	0.499913i	$-0.166635\pi$
$863$	50.8852	1.73215	0.866076	+	0.499913i	$0.166635\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−32.6108	−1.10752
$868$	0	0
$869$	5.04131	0.171015
$870$	0	0
$871$	−1.69220	−0.0573382
$872$	0	0
$873$	−9.30094	−0.314789
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.1080	1.01667	0.508337	−	0.861158i	$-0.330260\pi$
$877$	30.1080	1.01667	0.508337	+	0.861158i	$0.330260\pi$
$878$	0	0
$879$	−6.30312	−0.212599
$880$	0	0
$881$	20.2245	0.681379	0.340690	−	0.940176i	$-0.389339\pi$
$881$	20.2245	0.681379	0.340690	+	0.940176i	$0.389339\pi$
$882$	0	0
$883$	−30.6482	−1.03139	−0.515697	−	0.856771i	$-0.672467\pi$
$883$	−30.6482	−1.03139	−0.515697	+	0.856771i	$0.672467\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.0064	0.470290	0.235145	−	0.971960i	$-0.424443\pi$
$887$	14.0064	0.470290	0.235145	+	0.971960i	$0.424443\pi$
$888$	0	0
$889$	−23.3307	−0.782487
$890$	0	0
$891$	12.1594	0.407356
$892$	0	0
$893$	−26.1385	−0.874691
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−11.6791	−0.389955
$898$	0	0
$899$	−1.30312	−0.0434616
$900$	0	0
$901$	−39.0687	−1.30157
$902$	0	0
$903$	−34.7088	−1.15504
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−36.4739	−1.21109	−0.605547	−	0.795809i	$-0.707046\pi$
$907$	−36.4739	−1.21109	−0.605547	+	0.795809i	$0.707046\pi$
$908$	0	0
$909$	1.34472	0.0446016
$910$	0	0
$911$	14.2327	0.471549	0.235775	−	0.971808i	$-0.424237\pi$
$911$	14.2327	0.471549	0.235775	+	0.971808i	$0.424237\pi$
$912$	0	0
$913$	12.1677	0.402693
$914$	0	0
$915$	0	0
$916$	0	0
$917$	37.0528	1.22359
$918$	0	0
$919$	−34.3246	−1.13226	−0.566132	−	0.824315i	$-0.691561\pi$
$919$	−34.3246	−1.13226	−0.566132	+	0.824315i	$0.691561\pi$
$920$	0	0
$921$	67.5508	2.22588
$922$	0	0
$923$	−83.1474	−2.73683
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.03550	0.0340103
$928$	0	0
$929$	6.14579	0.201637	0.100818	−	0.994905i	$-0.467854\pi$
$929$	6.14579	0.201637	0.100818	+	0.994905i	$0.467854\pi$
$930$	0	0
$931$	−7.28428	−0.238733
$932$	0	0
$933$	−36.7330	−1.20258
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.4581	−0.602998	−0.301499	−	0.953466i	$-0.597487\pi$
$937$	−18.4581	−0.602998	−0.301499	+	0.953466i	$0.597487\pi$
$938$	0	0
$939$	19.4256	0.633931
$940$	0	0
$941$	30.7623	1.00282	0.501411	−	0.865209i	$-0.332815\pi$
$941$	30.7623	1.00282	0.501411	+	0.865209i	$0.332815\pi$
$942$	0	0
$943$	−6.27087	−0.204208
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.4692	0.372698	0.186349	−	0.982484i	$-0.440334\pi$
$947$	11.4692	0.372698	0.186349	+	0.982484i	$0.440334\pi$
$948$	0	0
$949$	57.3119	1.86042
$950$	0	0
$951$	−18.1080	−0.587192
$952$	0	0
$953$	−2.40664	−0.0779586	−0.0389793	−	0.999240i	$-0.512411\pi$
$953$	−2.40664	−0.0779586	−0.0389793	+	0.999240i	$0.512411\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.899533	−0.0290778
$958$	0	0
$959$	11.1848	0.361176
$960$	0	0
$961$	−20.8119	−0.671353
$962$	0	0
$963$	−0.197684	−0.00637029
$964$	0	0
$965$	0	0
$966$	0	0
$967$	57.4766	1.84832	0.924162	−	0.382001i	$-0.124765\pi$
$967$	57.4766	1.84832	0.924162	+	0.382001i	$0.124765\pi$
$968$	0	0
$969$	46.4202	1.49123
$970$	0	0
$971$	−53.1909	−1.70698	−0.853489	−	0.521111i	$-0.825518\pi$
$971$	−53.1909	−1.70698	−0.853489	+	0.521111i	$0.825518\pi$
$972$	0	0
$973$	−10.1382	−0.325015
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−54.4716	−1.74270	−0.871350	−	0.490661i	$-0.836755\pi$
$977$	−54.4716	−1.74270	−0.871350	+	0.490661i	$0.836755\pi$
$978$	0	0
$979$	−6.39080	−0.204251
$980$	0	0
$981$	6.24864	0.199504
$982$	0	0
$983$	37.5908	1.19896	0.599480	−	0.800390i	$-0.295374\pi$
$983$	37.5908	1.19896	0.599480	+	0.800390i	$0.295374\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	28.6461	0.911816
$988$	0	0
$989$	−7.75474	−0.246587
$990$	0	0
$991$	−14.0545	−0.446455	−0.223228	−	0.974766i	$-0.571659\pi$
$991$	−14.0545	−0.446455	−0.223228	+	0.974766i	$0.571659\pi$
$992$	0	0
$993$	−46.9706	−1.49057
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−10.9378	−0.346404	−0.173202	−	0.984886i	$-0.555411\pi$
$997$	−10.9378	−0.346404	−0.173202	+	0.984886i	$0.555411\pi$
$998$	0	0
$999$	41.4900	1.31269

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cq.1.1		4
4.3	odd	2		575.2.a.i.1.1		4
5.2	odd	4		1840.2.e.d.369.7		8
5.3	odd	4		1840.2.e.d.369.2		8
5.4	even	2		9200.2.a.ck.1.4		4
12.11	even	2		5175.2.a.bv.1.4		4
20.3	even	4		115.2.b.b.24.8	yes	8
20.7	even	4		115.2.b.b.24.1	✓	8
20.19	odd	2		575.2.a.j.1.4		4
60.23	odd	4		1035.2.b.e.829.1		8
60.47	odd	4		1035.2.b.e.829.8		8
60.59	even	2		5175.2.a.bw.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.1	✓	8	20.7	even	4
115.2.b.b.24.8	yes	8	20.3	even	4
575.2.a.i.1.1		4	4.3	odd	2
575.2.a.j.1.4		4	20.19	odd	2
1035.2.b.e.829.1		8	60.23	odd	4
1035.2.b.e.829.8		8	60.47	odd	4
1840.2.e.d.369.2		8	5.3	odd	4
1840.2.e.d.369.7		8	5.2	odd	4
5175.2.a.bv.1.4		4	12.11	even	2
5175.2.a.bw.1.1		4	60.59	even	2
9200.2.a.ck.1.4		4	5.4	even	2
9200.2.a.cq.1.1		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q-1.95969 q^{3} +2.28394 q^{7} +0.840379 q^{9} +O(q^{10})\)	\(q-1.95969 q^{3} +2.28394 q^{7} +0.840379 q^{9} -1.12432 q^{11} -5.95969 q^{13} -5.80007 q^{17} +4.08401 q^{19} -4.47581 q^{21} -1.00000 q^{23} +4.23218 q^{27} -0.408263 q^{29} +3.19187 q^{31} +2.20332 q^{33} +9.80345 q^{37} +11.6791 q^{39} +6.27087 q^{41} +7.75474 q^{43} -6.40020 q^{47} -1.78361 q^{49} +11.3663 q^{51} +6.73590 q^{53} -8.00339 q^{57} +4.75976 q^{59} -6.33265 q^{61} +1.91938 q^{63} +0.283942 q^{67} +1.95969 q^{69} +13.9516 q^{71} -9.61659 q^{73} -2.56788 q^{77} -4.48387 q^{79} -10.8149 q^{81} -10.8223 q^{83} +0.800068 q^{87} +5.68414 q^{89} -13.6116 q^{91} -6.25508 q^{93} -11.0676 q^{97} -0.944856 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 6 * q^7 + 4 * q^9	\( 4 q + 2 q^{3} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 14 q^{13} - 14 q^{17} + 4 q^{19} - 2 q^{21} - 4 q^{23} + 14 q^{27} + 4 q^{29} - 14 q^{33} - 2 q^{37} + 8 q^{39} - 8 q^{41} + 4 q^{43} + 2 q^{47} - 10 q^{51} - 4 q^{53} - 8 q^{61} - 12 q^{63} - 2 q^{67} - 2 q^{69} + 24 q^{71} - 18 q^{73} - 4 q^{77} - 24 q^{79} + 8 q^{81} - 6 q^{83} - 6 q^{87} - 8 q^{89} - 26 q^{91} - 2 q^{93} - 34 q^{97} - 36 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 6 * q^7 + 4 * q^9 - 2 * q^11 - 14 * q^13 - 14 * q^17 + 4 * q^19 - 2 * q^21 - 4 * q^23 + 14 * q^27 + 4 * q^29 - 14 * q^33 - 2 * q^37 + 8 * q^39 - 8 * q^41 + 4 * q^43 + 2 * q^47 - 10 * q^51 - 4 * q^53 - 8 * q^61 - 12 * q^63 - 2 * q^67 - 2 * q^69 + 24 * q^71 - 18 * q^73 - 4 * q^77 - 24 * q^79 + 8 * q^81 - 6 * q^83 - 6 * q^87 - 8 * q^89 - 26 * q^91 - 2 * q^93 - 34 * q^97 - 36 * q^99

Introduction

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