LMFDB - Embedded newform 7600.2.a.ck.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.66064384.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 13x^{2} - 1 $ x^6 - 9x^4 + 13x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.285442$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.21789 q^{3} +2.59637 q^{7} +7.35482 q^{9} +O(q^{10})$	$q+3.21789 q^{3} +2.59637 q^{7} +7.35482 q^{9} -0.741113 q^{11} +3.78878 q^{13} +3.16725 q^{17} +1.00000 q^{19} +8.35482 q^{21} +0.570885 q^{23} +14.0133 q^{27} +6.00000 q^{29} -5.83705 q^{31} -2.38482 q^{33} +1.40396 q^{37} +12.1919 q^{39} -3.83705 q^{41} -2.59637 q^{43} +5.08247 q^{47} -0.258887 q^{49} +10.1919 q^{51} +0.160905 q^{53} +3.21789 q^{57} -8.35482 q^{59} -8.57816 q^{61} +19.0958 q^{63} -14.8464 q^{67} +1.83705 q^{69} -3.64518 q^{71} +10.8461 q^{73} -1.92420 q^{77} -1.83705 q^{79} +23.0289 q^{81} -4.19876 q^{83} +19.3073 q^{87} -16.9015 q^{89} +9.83705 q^{91} -18.7830 q^{93} +3.78878 q^{97} -5.45075 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{9}+O(q^{10}) $ 6 * q + 14 * q^9	$ 6 q + 14 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 36 q^{29} + 8 q^{39} + 12 q^{41} - 4 q^{49} - 4 q^{51} - 20 q^{59} - 14 q^{61} - 24 q^{69} - 52 q^{71} + 24 q^{79} + 38 q^{81} + 24 q^{89} + 24 q^{91} + 30 q^{99}+O(q^{100}) $ 6 * q + 14 * q^9 - 2 * q^11 + 6 * q^19 + 20 * q^21 + 36 * q^29 + 8 * q^39 + 12 * q^41 - 4 * q^49 - 4 * q^51 - 20 * q^59 - 14 * q^61 - 24 * q^69 - 52 * q^71 + 24 * q^79 + 38 * q^81 + 24 * q^89 + 24 * q^91 + 30 * 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
$3$	3.21789	1.85785	0.928925	−	0.370268i	$-0.120734\pi$
$3$	3.21789	1.85785	0.928925	+	0.370268i	$0.120734\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.59637	0.981334	0.490667	−	0.871347i	$-0.336753\pi$
$7$	2.59637	0.981334	0.490667	+	0.871347i	$0.336753\pi$
$8$	0	0
$9$	7.35482	2.45161
$10$	0	0
$11$	−0.741113	−0.223454	−0.111727	−	0.993739i	$-0.535638\pi$
$11$	−0.741113	−0.223454	−0.111727	+	0.993739i	$0.535638\pi$
$12$	0	0
$13$	3.78878	1.05082	0.525409	−	0.850850i	$-0.323912\pi$
$13$	3.78878	1.05082	0.525409	+	0.850850i	$0.323912\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.16725	0.768171	0.384086	−	0.923298i	$-0.374517\pi$
$17$	3.16725	0.768171	0.384086	+	0.923298i	$0.374517\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	8.35482	1.82317
$22$	0	0
$23$	0.570885	0.119038	0.0595189	−	0.998227i	$-0.481043\pi$
$23$	0.570885	0.119038	0.0595189	+	0.998227i	$0.481043\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	14.0133	2.69687
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−5.83705	−1.04836	−0.524182	−	0.851606i	$-0.675629\pi$
$31$	−5.83705	−1.04836	−0.524182	+	0.851606i	$0.675629\pi$
$32$	0	0
$33$	−2.38482	−0.415144
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.40396	0.230809	0.115404	−	0.993319i	$-0.463184\pi$
$37$	1.40396	0.230809	0.115404	+	0.993319i	$0.463184\pi$
$38$	0	0
$39$	12.1919	1.95226
$40$	0	0
$41$	−3.83705	−0.599246	−0.299623	−	0.954058i	$-0.596861\pi$
$41$	−3.83705	−0.599246	−0.299623	+	0.954058i	$0.596861\pi$
$42$	0	0
$43$	−2.59637	−0.395942	−0.197971	−	0.980208i	$-0.563435\pi$
$43$	−2.59637	−0.395942	−0.197971	+	0.980208i	$0.563435\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.08247	0.741354	0.370677	−	0.928762i	$-0.379126\pi$
$47$	5.08247	0.741354	0.370677	+	0.928762i	$0.379126\pi$
$48$	0	0
$49$	−0.258887	−0.0369839
$50$	0	0
$51$	10.1919	1.42715
$52$	0	0
$53$	0.160905	0.0221020	0.0110510	−	0.999939i	$-0.496482\pi$
$53$	0.160905	0.0221020	0.0110510	+	0.999939i	$0.496482\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.21789	0.426220
$58$	0	0
$59$	−8.35482	−1.08770	−0.543852	−	0.839181i	$-0.683035\pi$
$59$	−8.35482	−1.08770	−0.543852	+	0.839181i	$0.683035\pi$
$60$	0	0
$61$	−8.57816	−1.09832	−0.549160	−	0.835717i	$-0.685052\pi$
$61$	−8.57816	−1.09832	−0.549160	+	0.835717i	$0.685052\pi$
$62$	0	0
$63$	19.0958	2.40584
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−14.8464	−1.81378	−0.906888	−	0.421371i	$-0.861549\pi$
$67$	−14.8464	−1.81378	−0.906888	+	0.421371i	$0.861549\pi$
$68$	0	0
$69$	1.83705	0.221154
$70$	0	0
$71$	−3.64518	−0.432603	−0.216302	−	0.976327i	$-0.569399\pi$
$71$	−3.64518	−0.432603	−0.216302	+	0.976327i	$0.569399\pi$
$72$	0	0
$73$	10.8461	1.26944	0.634719	−	0.772743i	$-0.281116\pi$
$73$	10.8461	1.26944	0.634719	+	0.772743i	$0.281116\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.92420	−0.219283
$78$	0	0
$79$	−1.83705	−0.206684	−0.103342	−	0.994646i	$-0.532954\pi$
$79$	−1.83705	−0.206684	−0.103342	+	0.994646i	$0.532954\pi$
$80$	0	0
$81$	23.0289	2.55877
$82$	0	0
$83$	−4.19876	−0.460873	−0.230437	−	0.973087i	$-0.574015\pi$
$83$	−4.19876	−0.460873	−0.230437	+	0.973087i	$0.574015\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	19.3073	2.06996
$88$	0	0
$89$	−16.9015	−1.79156	−0.895778	−	0.444502i	$-0.853381\pi$
$89$	−16.9015	−1.79156	−0.895778	+	0.444502i	$0.853381\pi$
$90$	0	0
$91$	9.83705	1.03120
$92$	0	0
$93$	−18.7830	−1.94770
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.78878	0.384692	0.192346	−	0.981327i	$-0.438390\pi$
$97$	3.78878	0.384692	0.192346	+	0.981327i	$0.438390\pi$
$98$	0	0
$99$	−5.45075	−0.547821
$100$	0	0
$101$	8.35482	0.831336	0.415668	−	0.909517i	$-0.363548\pi$
$101$	8.35482	0.831336	0.415668	+	0.909517i	$0.363548\pi$
$102$	0	0
$103$	2.07612	0.204566	0.102283	−	0.994755i	$-0.467385\pi$
$103$	2.07612	0.204566	0.102283	+	0.994755i	$0.467385\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.70399	−0.551426	−0.275713	−	0.961240i	$-0.588914\pi$
$107$	−5.70399	−0.551426	−0.275713	+	0.961240i	$0.588914\pi$
$108$	0	0
$109$	1.64518	0.157580	0.0787899	−	0.996891i	$-0.474894\pi$
$109$	1.64518	0.157580	0.0787899	+	0.996891i	$0.474894\pi$
$110$	0	0
$111$	4.51777	0.428808
$112$	0	0
$113$	3.89006	0.365946	0.182973	−	0.983118i	$-0.441428\pi$
$113$	3.89006	0.365946	0.182973	+	0.983118i	$0.441428\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	27.8658	2.57619
$118$	0	0
$119$	8.22334	0.753832
$120$	0	0
$121$	−10.4508	−0.950068
$122$	0	0
$123$	−12.3472	−1.11331
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.4233	−1.27986	−0.639931	−	0.768432i	$-0.721037\pi$
$127$	−14.4233	−1.27986	−0.639931	+	0.768432i	$0.721037\pi$
$128$	0	0
$129$	−8.35482	−0.735601
$130$	0	0
$131$	−9.96853	−0.870954	−0.435477	−	0.900200i	$-0.643420\pi$
$131$	−9.96853	−0.870954	−0.435477	+	0.900200i	$0.643420\pi$
$132$	0	0
$133$	2.59637	0.225133
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.70431	0.829095	0.414548	−	0.910028i	$-0.363940\pi$
$137$	9.70431	0.829095	0.414548	+	0.910028i	$0.363940\pi$
$138$	0	0
$139$	13.4508	1.14088	0.570439	−	0.821340i	$-0.306773\pi$
$139$	13.4508	1.14088	0.570439	+	0.821340i	$0.306773\pi$
$140$	0	0
$141$	16.3548	1.37732
$142$	0	0
$143$	−2.80791	−0.234809
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.833070	−0.0687105
$148$	0	0
$149$	15.0959	1.23671	0.618353	−	0.785900i	$-0.287800\pi$
$149$	15.0959	1.23671	0.618353	+	0.785900i	$0.287800\pi$
$150$	0	0
$151$	−14.1919	−1.15492	−0.577459	−	0.816420i	$-0.695956\pi$
$151$	−14.1919	−1.15492	−0.577459	+	0.816420i	$0.695956\pi$
$152$	0	0
$153$	23.2946	1.88325
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.57755	0.604754	0.302377	−	0.953188i	$-0.402220\pi$
$157$	7.57755	0.604754	0.302377	+	0.953188i	$0.402220\pi$
$158$	0	0
$159$	0.517774	0.0410622
$160$	0	0
$161$	1.48223	0.116816
$162$	0	0
$163$	19.6757	1.54112	0.770559	−	0.637369i	$-0.219977\pi$
$163$	19.6757	1.54112	0.770559	+	0.637369i	$0.219977\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.7954	0.835376	0.417688	−	0.908590i	$-0.362840\pi$
$167$	10.7954	0.835376	0.417688	+	0.908590i	$0.362840\pi$
$168$	0	0
$169$	1.35482	0.104217
$170$	0	0
$171$	7.35482	0.562437
$172$	0	0
$173$	20.3895	1.55018	0.775092	−	0.631848i	$-0.217703\pi$
$173$	20.3895	1.55018	0.775092	+	0.631848i	$0.217703\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−26.8849	−2.02079
$178$	0	0
$179$	−25.0645	−1.87341	−0.936703	−	0.350126i	$-0.886139\pi$
$179$	−25.0645	−1.87341	−0.936703	+	0.350126i	$0.886139\pi$
$180$	0	0
$181$	−19.4193	−1.44342	−0.721712	−	0.692194i	$-0.756644\pi$
$181$	−19.4193	−1.44342	−0.721712	+	0.692194i	$0.756644\pi$
$182$	0	0
$183$	−27.6036	−2.04051
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.34729	−0.171651
$188$	0	0
$189$	36.3837	2.64653
$190$	0	0
$191$	−11.4508	−0.828547	−0.414274	−	0.910152i	$-0.635964\pi$
$191$	−11.4508	−0.828547	−0.414274	+	0.910152i	$0.635964\pi$
$192$	0	0
$193$	3.78878	0.272722	0.136361	−	0.990659i	$-0.456459\pi$
$193$	3.78878	0.272722	0.136361	+	0.990659i	$0.456459\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.28354	0.162695	0.0813477	−	0.996686i	$-0.474078\pi$
$197$	2.28354	0.162695	0.0813477	+	0.996686i	$0.474078\pi$
$198$	0	0
$199$	19.4508	1.37883	0.689414	−	0.724368i	$-0.257868\pi$
$199$	19.4508	1.37883	0.689414	+	0.724368i	$0.257868\pi$
$200$	0	0
$201$	−47.7741	−3.36972
$202$	0	0
$203$	15.5782	1.09337
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.19876	0.291834
$208$	0	0
$209$	−0.741113	−0.0512639
$210$	0	0
$211$	−11.2274	−0.772927	−0.386463	−	0.922305i	$-0.626303\pi$
$211$	−11.2274	−0.772927	−0.386463	+	0.922305i	$0.626303\pi$
$212$	0	0
$213$	−11.7298	−0.803712
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−15.1551	−1.02880
$218$	0	0
$219$	34.9015	2.35843
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	4.03785	0.270394	0.135197	−	0.990819i	$-0.456833\pi$
$223$	4.03785	0.270394	0.135197	+	0.990819i	$0.456833\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.2185	0.744600	0.372300	−	0.928112i	$-0.378569\pi$
$227$	11.2185	0.744600	0.372300	+	0.928112i	$0.378569\pi$
$228$	0	0
$229$	16.1315	1.06600	0.532999	−	0.846116i	$-0.321065\pi$
$229$	16.1315	1.06600	0.532999	+	0.846116i	$0.321065\pi$
$230$	0	0
$231$	−6.19186	−0.407395
$232$	0	0
$233$	−2.12676	−0.139329	−0.0696644	−	0.997570i	$-0.522193\pi$
$233$	−2.12676	−0.139329	−0.0696644	+	0.997570i	$0.522193\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5.91141	−0.383987
$238$	0	0
$239$	14.4152	0.932442	0.466221	−	0.884668i	$-0.345615\pi$
$239$	14.4152	0.932442	0.466221	+	0.884668i	$0.345615\pi$
$240$	0	0
$241$	−0.162955	−0.0104968	−0.00524842	−	0.999986i	$-0.501671\pi$
$241$	−0.162955	−0.0104968	−0.00524842	+	0.999986i	$0.501671\pi$
$242$	0	0
$243$	32.0645	2.05694
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.78878	0.241074
$248$	0	0
$249$	−13.5111	−0.856233
$250$	0	0
$251$	−12.9330	−0.816322	−0.408161	−	0.912910i	$-0.633830\pi$
$251$	−12.9330	−0.816322	−0.408161	+	0.912910i	$0.633830\pi$
$252$	0	0
$253$	−0.423090	−0.0265995
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.0445	−0.688938	−0.344469	−	0.938798i	$-0.611941\pi$
$257$	−11.0445	−0.688938	−0.344469	+	0.938798i	$0.611941\pi$
$258$	0	0
$259$	3.64518	0.226501
$260$	0	0
$261$	44.1289	2.73151
$262$	0	0
$263$	−17.8527	−1.10085	−0.550424	−	0.834885i	$-0.685534\pi$
$263$	−17.8527	−1.10085	−0.550424	+	0.834885i	$0.685534\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−54.3872	−3.32844
$268$	0	0
$269$	24.9934	1.52387	0.761936	−	0.647652i	$-0.224249\pi$
$269$	24.9934	1.52387	0.761936	+	0.647652i	$0.224249\pi$
$270$	0	0
$271$	23.8660	1.44975	0.724877	−	0.688879i	$-0.241897\pi$
$271$	23.8660	1.44975	0.724877	+	0.688879i	$0.241897\pi$
$272$	0	0
$273$	31.6545	1.91582
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−21.2315	−1.27568	−0.637840	−	0.770169i	$-0.720172\pi$
$277$	−21.2315	−1.27568	−0.637840	+	0.770169i	$0.720172\pi$
$278$	0	0
$279$	−42.9304	−2.57018
$280$	0	0
$281$	3.83705	0.228899	0.114449	−	0.993429i	$-0.463490\pi$
$281$	3.83705	0.228899	0.114449	+	0.993429i	$0.463490\pi$
$282$	0	0
$283$	−0.211545	−0.0125751	−0.00628753	−	0.999980i	$-0.502001\pi$
$283$	−0.211545	−0.0125751	−0.00628753	+	0.999980i	$0.502001\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.96237	−0.588060
$288$	0	0
$289$	−6.96853	−0.409913
$290$	0	0
$291$	12.1919	0.714700
$292$	0	0
$293$	−14.9942	−0.875970	−0.437985	−	0.898982i	$-0.644308\pi$
$293$	−14.9942	−0.875970	−0.437985	+	0.898982i	$0.644308\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−10.3855	−0.602626
$298$	0	0
$299$	2.16295	0.125087
$300$	0	0
$301$	−6.74111	−0.388551
$302$	0	0
$303$	26.8849	1.54450
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.65303	−0.0943434	−0.0471717	−	0.998887i	$-0.515021\pi$
$307$	−1.65303	−0.0943434	−0.0471717	+	0.998887i	$0.515021\pi$
$308$	0	0
$309$	6.68073	0.380053
$310$	0	0
$311$	0.741113	0.0420247	0.0210123	−	0.999779i	$-0.493311\pi$
$311$	0.741113	0.0420247	0.0210123	+	0.999779i	$0.493311\pi$
$312$	0	0
$313$	−26.8849	−1.51962	−0.759812	−	0.650143i	$-0.774709\pi$
$313$	−26.8849	−1.51962	−0.759812	+	0.650143i	$0.774709\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.16155	−0.458398	−0.229199	−	0.973380i	$-0.573611\pi$
$317$	−8.16155	−0.458398	−0.229199	+	0.973380i	$0.573611\pi$
$318$	0	0
$319$	−4.44668	−0.248966
$320$	0	0
$321$	−18.3548	−1.02447
$322$	0	0
$323$	3.16725	0.176231
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.29401	0.292759
$328$	0	0
$329$	13.1959	0.727516
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	10.3258	0.565852
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9.90275	−0.539437	−0.269718	−	0.962939i	$-0.586931\pi$
$337$	−9.90275	−0.539437	−0.269718	+	0.962939i	$0.586931\pi$
$338$	0	0
$339$	12.5178	0.679872
$340$	0	0
$341$	4.32591	0.234261
$342$	0	0
$343$	−18.8467	−1.01763
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.2781	1.14227	0.571133	−	0.820858i	$-0.306504\pi$
$347$	21.2781	1.14227	0.571133	+	0.820858i	$0.306504\pi$
$348$	0	0
$349$	−16.4152	−0.878686	−0.439343	−	0.898319i	$-0.644789\pi$
$349$	−16.4152	−0.878686	−0.439343	+	0.898319i	$0.644789\pi$
$350$	0	0
$351$	53.0934	2.83391
$352$	0	0
$353$	23.8744	1.27071	0.635354	−	0.772221i	$-0.280854\pi$
$353$	23.8744	1.27071	0.635354	+	0.772221i	$0.280854\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	26.4618	1.40051
$358$	0	0
$359$	−2.22334	−0.117343	−0.0586717	−	0.998277i	$-0.518687\pi$
$359$	−2.22334	−0.117343	−0.0586717	+	0.998277i	$0.518687\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−33.6294	−1.76508
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.52057	0.235972	0.117986	−	0.993015i	$-0.462356\pi$
$367$	4.52057	0.235972	0.117986	+	0.993015i	$0.462356\pi$
$368$	0	0
$369$	−28.2208	−1.46911
$370$	0	0
$371$	0.417768	0.0216894
$372$	0	0
$373$	15.5186	0.803521	0.401760	−	0.915745i	$-0.368398\pi$
$373$	15.5186	0.803521	0.401760	+	0.915745i	$0.368398\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	22.7327	1.17079
$378$	0	0
$379$	18.9015	0.970905	0.485453	−	0.874263i	$-0.338655\pi$
$379$	18.9015	0.970905	0.485453	+	0.874263i	$0.338655\pi$
$380$	0	0
$381$	−46.4126	−2.37779
$382$	0	0
$383$	13.7046	0.700274	0.350137	−	0.936699i	$-0.386135\pi$
$383$	13.7046	0.700274	0.350137	+	0.936699i	$0.386135\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−19.0958	−0.970694
$388$	0	0
$389$	12.7411	0.646000	0.323000	−	0.946399i	$-0.395309\pi$
$389$	12.7411	0.646000	0.323000	+	0.946399i	$0.395309\pi$
$390$	0	0
$391$	1.80814	0.0914413
$392$	0	0
$393$	−32.0776	−1.61810
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−38.6522	−1.93990	−0.969950	−	0.243306i	$-0.921768\pi$
$397$	−38.6522	−1.93990	−0.969950	+	0.243306i	$0.921768\pi$
$398$	0	0
$399$	8.35482	0.418264
$400$	0	0
$401$	31.8660	1.59131	0.795655	−	0.605750i	$-0.207127\pi$
$401$	31.8660	1.59131	0.795655	+	0.605750i	$0.207127\pi$
$402$	0	0
$403$	−22.1153	−1.10164
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.04049	−0.0515751
$408$	0	0
$409$	11.0645	0.547102	0.273551	−	0.961857i	$-0.411802\pi$
$409$	11.0645	0.547102	0.273551	+	0.961857i	$0.411802\pi$
$410$	0	0
$411$	31.2274	1.54033
$412$	0	0
$413$	−21.6922	−1.06740
$414$	0	0
$415$	0	0
$416$	0	0
$417$	43.2830	2.11958
$418$	0	0
$419$	25.7452	1.25773	0.628867	−	0.777513i	$-0.283519\pi$
$419$	25.7452	1.25773	0.628867	+	0.777513i	$0.283519\pi$
$420$	0	0
$421$	27.4482	1.33774	0.668871	−	0.743378i	$-0.266778\pi$
$421$	27.4482	1.33774	0.668871	+	0.743378i	$0.266778\pi$
$422$	0	0
$423$	37.3806	1.81751
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−22.2720	−1.07782
$428$	0	0
$429$	−9.03555	−0.436240
$430$	0	0
$431$	−1.74519	−0.0840627	−0.0420314	−	0.999116i	$-0.513383\pi$
$431$	−1.74519	−0.0840627	−0.0420314	+	0.999116i	$0.513383\pi$
$432$	0	0
$433$	−18.5208	−0.890052	−0.445026	−	0.895518i	$-0.646806\pi$
$433$	−18.5208	−0.890052	−0.445026	+	0.895518i	$0.646806\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.570885	0.0273091
$438$	0	0
$439$	−29.4482	−1.40549	−0.702743	−	0.711444i	$-0.748041\pi$
$439$	−29.4482	−1.40549	−0.702743	+	0.711444i	$0.748041\pi$
$440$	0	0
$441$	−1.90407	−0.0906699
$442$	0	0
$443$	−11.7388	−0.557726	−0.278863	−	0.960331i	$-0.589958\pi$
$443$	−11.7388	−0.557726	−0.278863	+	0.960331i	$0.589958\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	48.5771	2.29762
$448$	0	0
$449$	−7.06446	−0.333392	−0.166696	−	0.986008i	$-0.553310\pi$
$449$	−7.06446	−0.333392	−0.166696	+	0.986008i	$0.553310\pi$
$450$	0	0
$451$	2.84368	0.133904
$452$	0	0
$453$	−45.6679	−2.14566
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−34.5000	−1.61384	−0.806920	−	0.590660i	$-0.798867\pi$
$457$	−34.5000	−1.61384	−0.806920	+	0.590660i	$0.798867\pi$
$458$	0	0
$459$	44.3837	2.07166
$460$	0	0
$461$	8.03147	0.374063	0.187032	−	0.982354i	$-0.440113\pi$
$461$	8.03147	0.374063	0.187032	+	0.982354i	$0.440113\pi$
$462$	0	0
$463$	25.3290	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$463$	25.3290	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−26.8759	−1.24367	−0.621834	−	0.783149i	$-0.713612\pi$
$467$	−26.8759	−1.24367	−0.621834	+	0.783149i	$0.713612\pi$
$468$	0	0
$469$	−38.5467	−1.77992
$470$	0	0
$471$	24.3837	1.12354
$472$	0	0
$473$	1.92420	0.0884748
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.18343	0.0541854
$478$	0	0
$479$	28.9015	1.32054	0.660272	−	0.751027i	$-0.270441\pi$
$479$	28.9015	1.32054	0.660272	+	0.751027i	$0.270441\pi$
$480$	0	0
$481$	5.31927	0.242538
$482$	0	0
$483$	4.76964	0.217026
$484$	0	0
$485$	0	0
$486$	0	0
$487$	17.7294	0.803395	0.401697	−	0.915773i	$-0.368420\pi$
$487$	17.7294	0.803395	0.401697	+	0.915773i	$0.368420\pi$
$488$	0	0
$489$	63.3141	2.86316
$490$	0	0
$491$	35.1645	1.58695	0.793475	−	0.608603i	$-0.208270\pi$
$491$	35.1645	1.58695	0.793475	+	0.608603i	$0.208270\pi$
$492$	0	0
$493$	19.0035	0.855875
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.46422	−0.424528
$498$	0	0
$499$	−21.4508	−0.960268	−0.480134	−	0.877195i	$-0.659412\pi$
$499$	−21.4508	−0.960268	−0.480134	+	0.877195i	$0.659412\pi$
$500$	0	0
$501$	34.7385	1.55200
$502$	0	0
$503$	5.34053	0.238122	0.119061	−	0.992887i	$-0.462012\pi$
$503$	5.34053	0.238122	0.119061	+	0.992887i	$0.462012\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.35966	0.193619
$508$	0	0
$509$	36.1919	1.60418	0.802088	−	0.597206i	$-0.203722\pi$
$509$	36.1919	1.60418	0.802088	+	0.597206i	$0.203722\pi$
$510$	0	0
$511$	28.1604	1.24574
$512$	0	0
$513$	14.0133	0.618704
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.76668	−0.165658
$518$	0	0
$519$	65.6111	2.88001
$520$	0	0
$521$	−2.77259	−0.121469	−0.0607346	−	0.998154i	$-0.519344\pi$
$521$	−2.77259	−0.121469	−0.0607346	+	0.998154i	$0.519344\pi$
$522$	0	0
$523$	20.5373	0.898033	0.449016	−	0.893524i	$-0.351774\pi$
$523$	20.5373	0.898033	0.449016	+	0.893524i	$0.351774\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.4874	−0.805323
$528$	0	0
$529$	−22.6741	−0.985830
$530$	0	0
$531$	−61.4482	−2.66662
$532$	0	0
$533$	−14.5377	−0.629698
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−80.6547	−3.48051
$538$	0	0
$539$	0.191865	0.00826419
$540$	0	0
$541$	35.4797	1.52539	0.762695	−	0.646758i	$-0.223876\pi$
$541$	35.4797	1.52539	0.762695	+	0.646758i	$0.223876\pi$
$542$	0	0
$543$	−62.4891	−2.68166
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−43.0756	−1.84178	−0.920890	−	0.389822i	$-0.872537\pi$
$547$	−43.0756	−1.84178	−0.920890	+	0.389822i	$0.872537\pi$
$548$	0	0
$549$	−63.0908	−2.69265
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−4.76964	−0.202826
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.4376	1.71340	0.856698	−	0.515818i	$-0.172512\pi$
$557$	40.4376	1.71340	0.856698	+	0.515818i	$0.172512\pi$
$558$	0	0
$559$	−9.83705	−0.416063
$560$	0	0
$561$	−7.55332	−0.318902
$562$	0	0
$563$	−19.7173	−0.830986	−0.415493	−	0.909596i	$-0.636391\pi$
$563$	−19.7173	−0.830986	−0.415493	+	0.909596i	$0.636391\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	59.7915	2.51101
$568$	0	0
$569$	18.6807	0.783137	0.391568	−	0.920149i	$-0.371933\pi$
$569$	18.6807	0.783137	0.391568	+	0.920149i	$0.371933\pi$
$570$	0	0
$571$	29.9371	1.25283	0.626413	−	0.779491i	$-0.284522\pi$
$571$	29.9371	1.25283	0.626413	+	0.779491i	$0.284522\pi$
$572$	0	0
$573$	−36.8473	−1.53932
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−0.156779	−0.00652679	−0.00326339	−	0.999995i	$-0.501039\pi$
$577$	−0.156779	−0.00652679	−0.00326339	+	0.999995i	$0.501039\pi$
$578$	0	0
$579$	12.1919	0.506677
$580$	0	0
$581$	−10.9015	−0.452271
$582$	0	0
$583$	−0.119249	−0.00493877
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−31.1474	−1.28559	−0.642795	−	0.766038i	$-0.722225\pi$
$587$	−31.1474	−1.28559	−0.642795	+	0.766038i	$0.722225\pi$
$588$	0	0
$589$	−5.83705	−0.240511
$590$	0	0
$591$	7.34818	0.302264
$592$	0	0
$593$	28.8728	1.18567	0.592833	−	0.805326i	$-0.298009\pi$
$593$	28.8728	1.18567	0.592833	+	0.805326i	$0.298009\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	62.5904	2.56165
$598$	0	0
$599$	−25.3274	−1.03485	−0.517425	−	0.855728i	$-0.673109\pi$
$599$	−25.3274	−1.03485	−0.517425	+	0.855728i	$0.673109\pi$
$600$	0	0
$601$	−19.8370	−0.809170	−0.404585	−	0.914500i	$-0.632584\pi$
$601$	−19.8370	−0.809170	−0.404585	+	0.914500i	$0.632584\pi$
$602$	0	0
$603$	−109.193	−4.44667
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.49921	0.101440	0.0507199	−	0.998713i	$-0.483848\pi$
$607$	2.49921	0.101440	0.0507199	+	0.998713i	$0.483848\pi$
$608$	0	0
$609$	50.1289	2.03133
$610$	0	0
$611$	19.2563	0.779027
$612$	0	0
$613$	−0.883711	−0.0356927	−0.0178464	−	0.999841i	$-0.505681\pi$
$613$	−0.883711	−0.0356927	−0.0178464	+	0.999841i	$0.505681\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29.4085	−1.18394	−0.591971	−	0.805959i	$-0.701650\pi$
$617$	−29.4085	−1.18394	−0.591971	+	0.805959i	$0.701650\pi$
$618$	0	0
$619$	−30.3208	−1.21870	−0.609348	−	0.792903i	$-0.708569\pi$
$619$	−30.3208	−1.21870	−0.609348	+	0.792903i	$0.708569\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	−43.8825	−1.75811
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.38482	−0.0952405
$628$	0	0
$629$	4.44668	0.177301
$630$	0	0
$631$	−17.7767	−0.707678	−0.353839	−	0.935306i	$-0.615124\pi$
$631$	−17.7767	−0.707678	−0.353839	+	0.935306i	$0.615124\pi$
$632$	0	0
$633$	−36.1286	−1.43598
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.980865	−0.0388633
$638$	0	0
$639$	−26.8096	−1.06057
$640$	0	0
$641$	−32.6675	−1.29029	−0.645143	−	0.764062i	$-0.723202\pi$
$641$	−32.6675	−1.29029	−0.645143	+	0.764062i	$0.723202\pi$
$642$	0	0
$643$	−31.8661	−1.25668	−0.628338	−	0.777941i	$-0.716264\pi$
$643$	−31.8661	−1.25668	−0.628338	+	0.777941i	$0.716264\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.2601	−0.835820	−0.417910	−	0.908488i	$-0.637237\pi$
$647$	−21.2601	−0.835820	−0.417910	+	0.908488i	$0.637237\pi$
$648$	0	0
$649$	6.19186	0.243052
$650$	0	0
$651$	−48.7675	−1.91135
$652$	0	0
$653$	12.8340	0.502234	0.251117	−	0.967957i	$-0.419202\pi$
$653$	12.8340	0.502234	0.251117	+	0.967957i	$0.419202\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	79.7710	3.11216
$658$	0	0
$659$	−20.3548	−0.792911	−0.396456	−	0.918054i	$-0.629760\pi$
$659$	−20.3548	−0.792911	−0.396456	+	0.918054i	$0.629760\pi$
$660$	0	0
$661$	−30.7385	−1.19559	−0.597795	−	0.801649i	$-0.703957\pi$
$661$	−30.7385	−1.19559	−0.597795	+	0.801649i	$0.703957\pi$
$662$	0	0
$663$	38.6147	1.49967
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.42531	0.132629
$668$	0	0
$669$	12.9934	0.502352
$670$	0	0
$671$	6.35738	0.245424
$672$	0	0
$673$	21.2094	0.817564	0.408782	−	0.912632i	$-0.365954\pi$
$673$	21.2094	0.817564	0.408782	+	0.912632i	$0.365954\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11.2650	0.432951	0.216475	−	0.976288i	$-0.430544\pi$
$677$	11.2650	0.432951	0.216475	+	0.976288i	$0.430544\pi$
$678$	0	0
$679$	9.83705	0.377511
$680$	0	0
$681$	36.1000	1.38336
$682$	0	0
$683$	12.3603	0.472954	0.236477	−	0.971637i	$-0.424007\pi$
$683$	12.3603	0.472954	0.236477	+	0.971637i	$0.424007\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	51.9093	1.98046
$688$	0	0
$689$	0.609632	0.0232251
$690$	0	0
$691$	−22.7493	−0.865423	−0.432711	−	0.901533i	$-0.642443\pi$
$691$	−22.7493	−0.865423	−0.432711	+	0.901533i	$0.642443\pi$
$692$	0	0
$693$	−14.1521	−0.537595
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−12.1529	−0.460323
$698$	0	0
$699$	−6.84368	−0.258852
$700$	0	0
$701$	−16.0289	−0.605404	−0.302702	−	0.953085i	$-0.597889\pi$
$701$	−16.0289	−0.605404	−0.302702	+	0.953085i	$0.597889\pi$
$702$	0	0
$703$	1.40396	0.0529512
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21.6922	0.815818
$708$	0	0
$709$	31.4193	1.17998	0.589988	−	0.807412i	$-0.299133\pi$
$709$	31.4193	1.17998	0.589988	+	0.807412i	$0.299133\pi$
$710$	0	0
$711$	−13.5111	−0.506707
$712$	0	0
$713$	−3.33228	−0.124795
$714$	0	0
$715$	0	0
$716$	0	0
$717$	46.3865	1.73234
$718$	0	0
$719$	−11.2589	−0.419886	−0.209943	−	0.977714i	$-0.567328\pi$
$719$	−11.2589	−0.419886	−0.209943	+	0.977714i	$0.567328\pi$
$720$	0	0
$721$	5.39037	0.200748
$722$	0	0
$723$	−0.524371	−0.0195016
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−48.9829	−1.81668	−0.908338	−	0.418237i	$-0.862648\pi$
$727$	−48.9829	−1.81668	−0.908338	+	0.418237i	$0.862648\pi$
$728$	0	0
$729$	34.0934	1.26272
$730$	0	0
$731$	−8.22334	−0.304151
$732$	0	0
$733$	−35.9260	−1.32696	−0.663479	−	0.748195i	$-0.730921\pi$
$733$	−35.9260	−1.32696	−0.663479	+	0.748195i	$0.730921\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.0029	0.405296
$738$	0	0
$739$	−14.3523	−0.527956	−0.263978	−	0.964529i	$-0.585035\pi$
$739$	−14.3523	−0.527956	−0.263978	+	0.964529i	$0.585035\pi$
$740$	0	0
$741$	12.1919	0.447879
$742$	0	0
$743$	−12.5629	−0.460887	−0.230443	−	0.973086i	$-0.574018\pi$
$743$	−12.5629	−0.460887	−0.230443	+	0.973086i	$0.574018\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−30.8811	−1.12988
$748$	0	0
$749$	−14.8096	−0.541133
$750$	0	0
$751$	−26.4548	−0.965350	−0.482675	−	0.875799i	$-0.660335\pi$
$751$	−26.4548	−0.965350	−0.482675	+	0.875799i	$0.660335\pi$
$752$	0	0
$753$	−41.6169	−1.51660
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−15.7350	−0.571897	−0.285949	−	0.958245i	$-0.592309\pi$
$757$	−15.7350	−0.571897	−0.285949	+	0.958245i	$0.592309\pi$
$758$	0	0
$759$	−1.36146	−0.0494178
$760$	0	0
$761$	16.9619	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$761$	16.9619	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$762$	0	0
$763$	4.27149	0.154638
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−31.6545	−1.14298
$768$	0	0
$769$	41.9974	1.51447	0.757233	−	0.653145i	$-0.226551\pi$
$769$	41.9974	1.51447	0.757233	+	0.653145i	$0.226551\pi$
$770$	0	0
$771$	−35.5400	−1.27994
$772$	0	0
$773$	−40.9579	−1.47315	−0.736576	−	0.676355i	$-0.763559\pi$
$773$	−40.9579	−1.47315	−0.736576	+	0.676355i	$0.763559\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	11.7298	0.420804
$778$	0	0
$779$	−3.83705	−0.137476
$780$	0	0
$781$	2.70149	0.0966669
$782$	0	0
$783$	84.0800	3.00477
$784$	0	0
$785$	0	0
$786$	0	0
$787$	28.5379	1.01727	0.508634	−	0.860983i	$-0.330151\pi$
$787$	28.5379	1.01727	0.508634	+	0.860983i	$0.330151\pi$
$788$	0	0
$789$	−57.4482	−2.04521
$790$	0	0
$791$	10.1000	0.359115
$792$	0	0
$793$	−32.5007	−1.15413
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33.2790	−1.17880	−0.589402	−	0.807840i	$-0.700636\pi$
$797$	−33.2790	−1.17880	−0.589402	+	0.807840i	$0.700636\pi$
$798$	0	0
$799$	16.0974	0.569487
$800$	0	0
$801$	−124.308	−4.39219
$802$	0	0
$803$	−8.03817	−0.283661
$804$	0	0
$805$	0	0
$806$	0	0
$807$	80.4259	2.83113
$808$	0	0
$809$	−34.4234	−1.21026	−0.605130	−	0.796126i	$-0.706879\pi$
$809$	−34.4234	−1.21026	−0.605130	+	0.796126i	$0.706879\pi$
$810$	0	0
$811$	11.2563	0.395263	0.197631	−	0.980276i	$-0.436675\pi$
$811$	11.2563	0.395263	0.197631	+	0.980276i	$0.436675\pi$
$812$	0	0
$813$	76.7980	2.69342
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.59637	−0.0908353
$818$	0	0
$819$	72.3497	2.52810
$820$	0	0
$821$	−31.3167	−1.09296	−0.546480	−	0.837472i	$-0.684033\pi$
$821$	−31.3167	−1.09296	−0.546480	+	0.837472i	$0.684033\pi$
$822$	0	0
$823$	−13.4800	−0.469882	−0.234941	−	0.972010i	$-0.575490\pi$
$823$	−13.4800	−0.469882	−0.234941	+	0.972010i	$0.575490\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.9882	0.555963	0.277982	−	0.960586i	$-0.410335\pi$
$827$	15.9882	0.555963	0.277982	+	0.960586i	$0.410335\pi$
$828$	0	0
$829$	−48.4837	−1.68391	−0.841955	−	0.539548i	$-0.818595\pi$
$829$	−48.4837	−1.68391	−0.841955	+	0.539548i	$0.818595\pi$
$830$	0	0
$831$	−68.3208	−2.37002
$832$	0	0
$833$	−0.819960	−0.0284099
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−81.7965	−2.82730
$838$	0	0
$839$	18.9934	0.655724	0.327862	−	0.944726i	$-0.393672\pi$
$839$	18.9934	0.655724	0.327862	+	0.944726i	$0.393672\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	12.3472	0.425260
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−27.1340	−0.932334
$848$	0	0
$849$	−0.680729	−0.0233626
$850$	0	0
$851$	0.801497	0.0274750
$852$	0	0
$853$	44.1210	1.51067	0.755337	−	0.655337i	$-0.227473\pi$
$853$	44.1210	1.51067	0.755337	+	0.655337i	$0.227473\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.4149	1.10727	0.553635	−	0.832759i	$-0.313240\pi$
$857$	32.4149	1.10727	0.553635	+	0.832759i	$0.313240\pi$
$858$	0	0
$859$	−6.29444	−0.214763	−0.107382	−	0.994218i	$-0.534247\pi$
$859$	−6.29444	−0.214763	−0.107382	+	0.994218i	$0.534247\pi$
$860$	0	0
$861$	−32.0578	−1.09253
$862$	0	0
$863$	17.4338	0.593453	0.296726	−	0.954963i	$-0.404105\pi$
$863$	17.4338	0.593453	0.296726	+	0.954963i	$0.404105\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−22.4240	−0.761557
$868$	0	0
$869$	1.36146	0.0461843
$870$	0	0
$871$	−56.2497	−1.90595
$872$	0	0
$873$	27.8658	0.943113
$874$	0	0
$875$	0	0
$876$	0	0
$877$	23.2904	0.786462	0.393231	−	0.919440i	$-0.371357\pi$
$877$	23.2904	0.786462	0.393231	+	0.919440i	$0.371357\pi$
$878$	0	0
$879$	−48.2497	−1.62742
$880$	0	0
$881$	28.9619	0.975751	0.487875	−	0.872913i	$-0.337772\pi$
$881$	28.9619	0.975751	0.487875	+	0.872913i	$0.337772\pi$
$882$	0	0
$883$	37.3627	1.25735	0.628677	−	0.777667i	$-0.283597\pi$
$883$	37.3627	1.25735	0.628677	+	0.777667i	$0.283597\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.0234	0.773050	0.386525	−	0.922279i	$-0.373675\pi$
$887$	23.0234	0.773050	0.386525	+	0.922279i	$0.373675\pi$
$888$	0	0
$889$	−37.4482	−1.25597
$890$	0	0
$891$	−17.0670	−0.571767
$892$	0	0
$893$	5.08247	0.170078
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.96015	0.232393
$898$	0	0
$899$	−35.0223	−1.16806
$900$	0	0
$901$	0.509626	0.0169781
$902$	0	0
$903$	−21.6922	−0.721870
$904$	0	0
$905$	0	0
$906$	0	0
$907$	35.2693	1.17110	0.585549	−	0.810637i	$-0.300879\pi$
$907$	35.2693	1.17110	0.585549	+	0.810637i	$0.300879\pi$
$908$	0	0
$909$	61.4482	2.03811
$910$	0	0
$911$	4.97260	0.164750	0.0823748	−	0.996601i	$-0.473750\pi$
$911$	4.97260	0.164750	0.0823748	+	0.996601i	$0.473750\pi$
$912$	0	0
$913$	3.11175	0.102984
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−25.8819	−0.854697
$918$	0	0
$919$	48.7096	1.60678	0.803391	−	0.595451i	$-0.203027\pi$
$919$	48.7096	1.60678	0.803391	+	0.595451i	$0.203027\pi$
$920$	0	0
$921$	−5.31927	−0.175276
$922$	0	0
$923$	−13.8108	−0.454587
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15.2695	0.501516
$928$	0	0
$929$	32.5126	1.06671	0.533353	−	0.845893i	$-0.320932\pi$
$929$	32.5126	1.06671	0.533353	+	0.845893i	$0.320932\pi$
$930$	0	0
$931$	−0.258887	−0.00848468
$932$	0	0
$933$	2.38482	0.0780755
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.385560	−0.0125957	−0.00629785	−	0.999980i	$-0.502005\pi$
$937$	−0.385560	−0.0125957	−0.00629785	+	0.999980i	$0.502005\pi$
$938$	0	0
$939$	−86.5126	−2.82323
$940$	0	0
$941$	45.3482	1.47831	0.739154	−	0.673536i	$-0.235225\pi$
$941$	45.3482	1.47831	0.739154	+	0.673536i	$0.235225\pi$
$942$	0	0
$943$	−2.19051	−0.0713329
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.61202	−0.312349	−0.156174	−	0.987730i	$-0.549916\pi$
$947$	−9.61202	−0.312349	−0.156174	+	0.987730i	$0.549916\pi$
$948$	0	0
$949$	41.0934	1.33395
$950$	0	0
$951$	−26.2630	−0.851635
$952$	0	0
$953$	−59.8421	−1.93848	−0.969238	−	0.246126i	$-0.920842\pi$
$953$	−59.8421	−1.93848	−0.969238	+	0.246126i	$0.920842\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−14.3089	−0.462542
$958$	0	0
$959$	25.1959	0.813619
$960$	0	0
$961$	3.07110	0.0990676
$962$	0	0
$963$	−41.9518	−1.35188
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0.368324	0.0118445	0.00592225	−	0.999982i	$-0.498115\pi$
$967$	0.368324	0.0118445	0.00592225	+	0.999982i	$0.498115\pi$
$968$	0	0
$969$	10.1919	0.327410
$970$	0	0
$971$	45.8370	1.47098	0.735490	−	0.677535i	$-0.236952\pi$
$971$	45.8370	1.47098	0.735490	+	0.677535i	$0.236952\pi$
$972$	0	0
$973$	34.9231	1.11958
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.0337	0.928872	0.464436	−	0.885607i	$-0.346257\pi$
$977$	29.0337	0.928872	0.464436	+	0.885607i	$0.346257\pi$
$978$	0	0
$979$	12.5259	0.400330
$980$	0	0
$981$	12.1000	0.386323
$982$	0	0
$983$	11.0160	0.351355	0.175677	−	0.984448i	$-0.443788\pi$
$983$	11.0160	0.351355	0.175677	+	0.984448i	$0.443788\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	42.4631	1.35161
$988$	0	0
$989$	−1.48223	−0.0471320
$990$	0	0
$991$	25.6822	0.815823	0.407912	−	0.913021i	$-0.366257\pi$
$991$	25.6822	0.815823	0.407912	+	0.913021i	$0.366257\pi$
$992$	0	0
$993$	−25.7431	−0.816933
$994$	0	0
$995$	0	0
$996$	0	0
$997$	20.3854	0.645611	0.322805	−	0.946465i	$-0.395374\pi$
$997$	20.3854	0.645611	0.322805	+	0.946465i	$0.395374\pi$
$998$	0	0
$999$	19.6741	0.622461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.ck.1.6		6
4.3	odd	2		475.2.a.j.1.4		6
5.2	odd	4		1520.2.d.h.609.1		6
5.3	odd	4		1520.2.d.h.609.6		6
5.4	even	2	inner	7600.2.a.ck.1.1		6
12.11	even	2		4275.2.a.br.1.3		6
20.3	even	4		95.2.b.b.39.3	✓	6
20.7	even	4		95.2.b.b.39.4	yes	6
20.19	odd	2		475.2.a.j.1.3		6
60.23	odd	4		855.2.c.d.514.4		6
60.47	odd	4		855.2.c.d.514.3		6
60.59	even	2		4275.2.a.br.1.4		6
76.75	even	2		9025.2.a.bx.1.3		6
380.227	odd	4		1805.2.b.e.1084.3		6
380.303	odd	4		1805.2.b.e.1084.4		6
380.379	even	2		9025.2.a.bx.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.b.39.3	✓	6	20.3	even	4
95.2.b.b.39.4	yes	6	20.7	even	4
475.2.a.j.1.3		6	20.19	odd	2
475.2.a.j.1.4		6	4.3	odd	2
855.2.c.d.514.3		6	60.47	odd	4
855.2.c.d.514.4		6	60.23	odd	4
1520.2.d.h.609.1		6	5.2	odd	4
1520.2.d.h.609.6		6	5.3	odd	4
1805.2.b.e.1084.3		6	380.227	odd	4
1805.2.b.e.1084.4		6	380.303	odd	4
4275.2.a.br.1.3		6	12.11	even	2
4275.2.a.br.1.4		6	60.59	even	2
7600.2.a.ck.1.1		6	5.4	even	2	inner
7600.2.a.ck.1.6		6	1.1	even	1	trivial
9025.2.a.bx.1.3		6	76.75	even	2
9025.2.a.bx.1.4		6	380.379	even	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+3.21789 q^{3} +2.59637 q^{7} +7.35482 q^{9} +O(q^{10})\)	\(q+3.21789 q^{3} +2.59637 q^{7} +7.35482 q^{9} -0.741113 q^{11} +3.78878 q^{13} +3.16725 q^{17} +1.00000 q^{19} +8.35482 q^{21} +0.570885 q^{23} +14.0133 q^{27} +6.00000 q^{29} -5.83705 q^{31} -2.38482 q^{33} +1.40396 q^{37} +12.1919 q^{39} -3.83705 q^{41} -2.59637 q^{43} +5.08247 q^{47} -0.258887 q^{49} +10.1919 q^{51} +0.160905 q^{53} +3.21789 q^{57} -8.35482 q^{59} -8.57816 q^{61} +19.0958 q^{63} -14.8464 q^{67} +1.83705 q^{69} -3.64518 q^{71} +10.8461 q^{73} -1.92420 q^{77} -1.83705 q^{79} +23.0289 q^{81} -4.19876 q^{83} +19.3073 q^{87} -16.9015 q^{89} +9.83705 q^{91} -18.7830 q^{93} +3.78878 q^{97} -5.45075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{9}+O(q^{10}) \) 6 * q + 14 * q^9	\( 6 q + 14 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 36 q^{29} + 8 q^{39} + 12 q^{41} - 4 q^{49} - 4 q^{51} - 20 q^{59} - 14 q^{61} - 24 q^{69} - 52 q^{71} + 24 q^{79} + 38 q^{81} + 24 q^{89} + 24 q^{91} + 30 q^{99}+O(q^{100}) \) 6 * q + 14 * q^9 - 2 * q^11 + 6 * q^19 + 20 * q^21 + 36 * q^29 + 8 * q^39 + 12 * q^41 - 4 * q^49 - 4 * q^51 - 20 * q^59 - 14 * q^61 - 24 * q^69 - 52 * q^71 + 24 * q^79 + 38 * q^81 + 24 * q^89 + 24 * q^91 + 30 * q^99

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