LMFDB - Embedded newform 6525.2.a.ce.1.3

Error: no document with id 268310344 found in table mf_hecke_traces.

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6525,2,Mod(1,6525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,0,16,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$52.1023873189$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 20x^{10} + 148x^{8} - 502x^{6} + 792x^{4} - 496x^{2} + 45 $ x^12 - 20x^10 + 148x^8 - 502x^6 + 792x^4 - 496*x^2 + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1305)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.78841$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.78841 q^{2} +1.19839 q^{4} -4.04635 q^{7} +1.43360 q^{8} -4.80986 q^{11} +0.533576 q^{13} +7.23651 q^{14} -4.96064 q^{16} +0.299844 q^{17} -6.02908 q^{19} +8.60198 q^{22} -0.379104 q^{23} -0.954250 q^{26} -4.84912 q^{28} +1.00000 q^{29} +9.14506 q^{31} +6.00444 q^{32} -0.536243 q^{34} +8.51769 q^{37} +10.7824 q^{38} +2.24106 q^{41} +6.01126 q^{43} -5.76411 q^{44} +0.677992 q^{46} +0.299844 q^{47} +9.37293 q^{49} +0.639435 q^{52} +11.8608 q^{53} -5.80083 q^{56} -1.78841 q^{58} -6.53090 q^{59} -0.755856 q^{61} -16.3551 q^{62} -0.817104 q^{64} +3.95461 q^{67} +0.359332 q^{68} -10.6771 q^{71} +11.0455 q^{73} -15.2331 q^{74} -7.22522 q^{76} +19.4624 q^{77} +7.01095 q^{79} -4.00793 q^{82} -6.15340 q^{83} -10.7506 q^{86} -6.89539 q^{88} -10.0521 q^{89} -2.15903 q^{91} -0.454316 q^{92} -0.536243 q^{94} +2.41013 q^{97} -16.7626 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{4} - 12 q^{11} - 16 q^{14} + 16 q^{16} - 20 q^{19} - 56 q^{26} + 12 q^{29} - 16 q^{31} + 4 q^{34} - 32 q^{41} - 68 q^{44} + 20 q^{46} - 4 q^{49} - 76 q^{56} - 44 q^{59} - 52 q^{61} + 36 q^{64}+ \cdots + 4 q^{94}+O(q^{100}) $ 12 * q + 16 * q^4 - 12 * q^11 - 16 * q^14 + 16 * q^16 - 20 * q^19 - 56 * q^26 + 12 * q^29 - 16 * q^31 + 4 * q^34 - 32 * q^41 - 68 * q^44 + 20 * q^46 - 4 * q^49 - 76 * q^56 - 44 * q^59 - 52 * q^61 + 36 * q^64 - 20 * q^71 - 20 * q^74 - 28 * q^76 - 4 * q^79 - 36 * q^86 - 68 * q^89 + 48 * q^91 + 4 * q^94

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 $

$2$	−1.78841	−1.26459	−0.632297	−	0.774726i	$-0.717888\pi$
$2$	−1.78841	−1.26459	−0.632297	+	0.774726i	$0.717888\pi$
$3$	0	0
$3$	0	0
$4$	1.19839	0.599197
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.04635	−1.52938	−0.764688	−	0.644401i	$-0.777107\pi$
$7$	−4.04635	−1.52938	−0.764688	+	0.644401i	$0.777107\pi$
$8$	1.43360	0.506853
$9$	0	0
$10$	0	0
$11$	−4.80986	−1.45023	−0.725113	−	0.688630i	$-0.758213\pi$
$11$	−4.80986	−1.45023	−0.725113	+	0.688630i	$0.758213\pi$
$12$	0	0
$13$	0.533576	0.147987	0.0739937	−	0.997259i	$-0.476426\pi$
$13$	0.533576	0.147987	0.0739937	+	0.997259i	$0.476426\pi$
$14$	7.23651	1.93404
$15$	0	0
$16$	−4.96064	−1.24016
$17$	0.299844	0.0727229	0.0363615	−	0.999339i	$-0.488423\pi$
$17$	0.299844	0.0727229	0.0363615	+	0.999339i	$0.488423\pi$
$18$	0	0
$19$	−6.02908	−1.38317	−0.691583	−	0.722297i	$-0.743086\pi$
$19$	−6.02908	−1.38317	−0.691583	+	0.722297i	$0.743086\pi$
$20$	0	0
$21$	0	0
$22$	8.60198	1.83395
$23$	−0.379104	−0.0790487	−0.0395243	−	0.999219i	$-0.512584\pi$
$23$	−0.379104	−0.0790487	−0.0395243	+	0.999219i	$0.512584\pi$
$24$	0	0
$25$	0	0
$26$	−0.954250	−0.187144
$27$	0	0
$28$	−4.84912	−0.916398
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	9.14506	1.64250	0.821251	−	0.570567i	$-0.193277\pi$
$31$	9.14506	1.64250	0.821251	+	0.570567i	$0.193277\pi$
$32$	6.00444	1.06145
$33$	0	0
$34$	−0.536243	−0.0919650
$35$	0	0
$36$	0	0
$37$	8.51769	1.40030	0.700150	−	0.713996i	$-0.253116\pi$
$37$	8.51769	1.40030	0.700150	+	0.713996i	$0.253116\pi$
$38$	10.7824	1.74914
$39$	0	0
$40$	0	0
$41$	2.24106	0.349995	0.174998	−	0.984569i	$-0.444008\pi$
$41$	2.24106	0.349995	0.174998	+	0.984569i	$0.444008\pi$
$42$	0	0
$43$	6.01126	0.916708	0.458354	−	0.888770i	$-0.348439\pi$
$43$	6.01126	0.916708	0.458354	+	0.888770i	$0.348439\pi$
$44$	−5.76411	−0.868972
$45$	0	0
$46$	0.677992	0.0999645
$47$	0.299844	0.0437368	0.0218684	−	0.999761i	$-0.493039\pi$
$47$	0.299844	0.0437368	0.0218684	+	0.999761i	$0.493039\pi$
$48$	0	0
$49$	9.37293	1.33899
$50$	0	0
$51$	0	0
$52$	0.639435	0.0886736
$53$	11.8608	1.62920	0.814601	−	0.580022i	$-0.196956\pi$
$53$	11.8608	1.62920	0.814601	+	0.580022i	$0.196956\pi$
$54$	0	0
$55$	0	0
$56$	−5.80083	−0.775168
$57$	0	0
$58$	−1.78841	−0.234829
$59$	−6.53090	−0.850251	−0.425126	−	0.905134i	$-0.639770\pi$
$59$	−6.53090	−0.850251	−0.425126	+	0.905134i	$0.639770\pi$
$60$	0	0
$61$	−0.755856	−0.0967774	−0.0483887	−	0.998829i	$-0.515409\pi$
$61$	−0.755856	−0.0967774	−0.0483887	+	0.998829i	$0.515409\pi$
$62$	−16.3551	−2.07710
$63$	0	0
$64$	−0.817104	−0.102138
$65$	0	0
$66$	0	0
$67$	3.95461	0.483133	0.241567	−	0.970384i	$-0.422339\pi$
$67$	3.95461	0.483133	0.241567	+	0.970384i	$0.422339\pi$
$68$	0.359332	0.0435754
$69$	0	0
$70$	0	0
$71$	−10.6771	−1.26714	−0.633571	−	0.773684i	$-0.718412\pi$
$71$	−10.6771	−1.26714	−0.633571	+	0.773684i	$0.718412\pi$
$72$	0	0
$73$	11.0455	1.29278	0.646389	−	0.763008i	$-0.276278\pi$
$73$	11.0455	1.29278	0.646389	+	0.763008i	$0.276278\pi$
$74$	−15.2331	−1.77081
$75$	0	0
$76$	−7.22522	−0.828790
$77$	19.4624	2.21794
$78$	0	0
$79$	7.01095	0.788793	0.394397	−	0.918940i	$-0.370954\pi$
$79$	7.01095	0.788793	0.394397	+	0.918940i	$0.370954\pi$
$80$	0	0
$81$	0	0
$82$	−4.00793	−0.442602
$83$	−6.15340	−0.675424	−0.337712	−	0.941250i	$-0.609653\pi$
$83$	−6.15340	−0.675424	−0.337712	+	0.941250i	$0.609653\pi$
$84$	0	0
$85$	0	0
$86$	−10.7506	−1.15926
$87$	0	0
$88$	−6.89539	−0.735051
$89$	−10.0521	−1.06552	−0.532760	−	0.846267i	$-0.678845\pi$
$89$	−10.0521	−1.06552	−0.532760	+	0.846267i	$0.678845\pi$
$90$	0	0
$91$	−2.15903	−0.226328
$92$	−0.454316	−0.0473658
$93$	0	0
$94$	−0.536243	−0.0553093
$95$	0	0
$96$	0	0
$97$	2.41013	0.244711	0.122356	−	0.992486i	$-0.460955\pi$
$97$	2.41013	0.244711	0.122356	+	0.992486i	$0.460955\pi$
$98$	−16.7626	−1.69328
$99$	0	0
$100$	0	0
$101$	−16.6038	−1.65214	−0.826071	−	0.563565i	$-0.809429\pi$
$101$	−16.6038	−1.65214	−0.826071	+	0.563565i	$0.809429\pi$
$102$	0	0
$103$	−13.4127	−1.32159	−0.660795	−	0.750566i	$-0.729781\pi$
$103$	−13.4127	−1.32159	−0.660795	+	0.750566i	$0.729781\pi$
$104$	0.764932	0.0750078
$105$	0	0
$106$	−21.2119	−2.06028
$107$	9.12013	0.881676	0.440838	−	0.897587i	$-0.354681\pi$
$107$	9.12013	0.881676	0.440838	+	0.897587i	$0.354681\pi$
$108$	0	0
$109$	12.7379	1.22007	0.610037	−	0.792373i	$-0.291155\pi$
$109$	12.7379	1.22007	0.610037	+	0.792373i	$0.291155\pi$
$110$	0	0
$111$	0	0
$112$	20.0725	1.89667
$113$	13.7766	1.29599	0.647995	−	0.761645i	$-0.275608\pi$
$113$	13.7766	1.29599	0.647995	+	0.761645i	$0.275608\pi$
$114$	0	0
$115$	0	0
$116$	1.19839	0.111268
$117$	0	0
$118$	11.6799	1.07522
$119$	−1.21327	−0.111221
$120$	0	0
$121$	12.1347	1.10316
$122$	1.35178	0.122384
$123$	0	0
$124$	10.9594	0.984183
$125$	0	0
$126$	0	0
$127$	−6.39541	−0.567501	−0.283751	−	0.958898i	$-0.591579\pi$
$127$	−6.39541	−0.567501	−0.283751	+	0.958898i	$0.591579\pi$
$128$	−10.5476	−0.932283
$129$	0	0
$130$	0	0
$131$	15.6797	1.36994	0.684970	−	0.728572i	$-0.259816\pi$
$131$	15.6797	1.36994	0.684970	+	0.728572i	$0.259816\pi$
$132$	0	0
$133$	24.3958	2.11538
$134$	−7.07246	−0.610967
$135$	0	0
$136$	0.429855	0.0368598
$137$	−14.1792	−1.21141	−0.605704	−	0.795690i	$-0.707108\pi$
$137$	−14.1792	−1.21141	−0.605704	+	0.795690i	$0.707108\pi$
$138$	0	0
$139$	6.15903	0.522402	0.261201	−	0.965284i	$-0.415881\pi$
$139$	6.15903	0.522402	0.261201	+	0.965284i	$0.415881\pi$
$140$	0	0
$141$	0	0
$142$	19.0950	1.60242
$143$	−2.56643	−0.214615
$144$	0	0
$145$	0	0
$146$	−19.7538	−1.63484
$147$	0	0
$148$	10.2076	0.839056
$149$	−3.29888	−0.270254	−0.135127	−	0.990828i	$-0.543144\pi$
$149$	−3.29888	−0.270254	−0.135127	+	0.990828i	$0.543144\pi$
$150$	0	0
$151$	−11.7747	−0.958209	−0.479104	−	0.877758i	$-0.659039\pi$
$151$	−11.7747	−0.958209	−0.479104	+	0.877758i	$0.659039\pi$
$152$	−8.64327	−0.701061
$153$	0	0
$154$	−34.8066	−2.80479
$155$	0	0
$156$	0	0
$157$	−20.0255	−1.59821	−0.799103	−	0.601195i	$-0.794692\pi$
$157$	−20.0255	−1.59821	−0.799103	+	0.601195i	$0.794692\pi$
$158$	−12.5384	−0.997503
$159$	0	0
$160$	0	0
$161$	1.53399	0.120895
$162$	0	0
$163$	8.63537	0.676374	0.338187	−	0.941079i	$-0.390186\pi$
$163$	8.63537	0.676374	0.338187	+	0.941079i	$0.390186\pi$
$164$	2.68568	0.209716
$165$	0	0
$166$	11.0048	0.854137
$167$	−5.98646	−0.463246	−0.231623	−	0.972806i	$-0.574404\pi$
$167$	−5.98646	−0.463246	−0.231623	+	0.972806i	$0.574404\pi$
$168$	0	0
$169$	−12.7153	−0.978100
$170$	0	0
$171$	0	0
$172$	7.20386	0.549289
$173$	−7.75331	−0.589473	−0.294737	−	0.955579i	$-0.595232\pi$
$173$	−7.75331	−0.589473	−0.294737	+	0.955579i	$0.595232\pi$
$174$	0	0
$175$	0	0
$176$	23.8600	1.79851
$177$	0	0
$178$	17.9772	1.34745
$179$	−9.21698	−0.688909	−0.344455	−	0.938803i	$-0.611936\pi$
$179$	−9.21698	−0.688909	−0.344455	+	0.938803i	$0.611936\pi$
$180$	0	0
$181$	13.9345	1.03574	0.517871	−	0.855459i	$-0.326725\pi$
$181$	13.9345	1.03574	0.517871	+	0.855459i	$0.326725\pi$
$182$	3.86123	0.286213
$183$	0	0
$184$	−0.543482	−0.0400660
$185$	0	0
$186$	0	0
$187$	−1.44221	−0.105465
$188$	0.359332	0.0262070
$189$	0	0
$190$	0	0
$191$	−15.8529	−1.14708	−0.573538	−	0.819179i	$-0.694430\pi$
$191$	−15.8529	−1.14708	−0.573538	+	0.819179i	$0.694430\pi$
$192$	0	0
$193$	12.3360	0.887963	0.443982	−	0.896036i	$-0.353566\pi$
$193$	12.3360	0.887963	0.443982	+	0.896036i	$0.353566\pi$
$194$	−4.31029	−0.309460
$195$	0	0
$196$	11.2325	0.802319
$197$	6.51599	0.464245	0.232122	−	0.972687i	$-0.425433\pi$
$197$	6.51599	0.464245	0.232122	+	0.972687i	$0.425433\pi$
$198$	0	0
$199$	−8.40626	−0.595904	−0.297952	−	0.954581i	$-0.596304\pi$
$199$	−8.40626	−0.595904	−0.297952	+	0.954581i	$0.596304\pi$
$200$	0	0
$201$	0	0
$202$	29.6944	2.08929
$203$	−4.04635	−0.283998
$204$	0	0
$205$	0	0
$206$	23.9873	1.67128
$207$	0	0
$208$	−2.64688	−0.183528
$209$	28.9990	2.00591
$210$	0	0
$211$	−9.42279	−0.648692	−0.324346	−	0.945939i	$-0.605144\pi$
$211$	−9.42279	−0.648692	−0.324346	+	0.945939i	$0.605144\pi$
$212$	14.2139	0.976214
$213$	0	0
$214$	−16.3105	−1.11496
$215$	0	0
$216$	0	0
$217$	−37.0041	−2.51200
$218$	−22.7806	−1.54290
$219$	0	0
$220$	0	0
$221$	0.159990	0.0107621
$222$	0	0
$223$	18.6498	1.24889	0.624443	−	0.781071i	$-0.285326\pi$
$223$	18.6498	1.24889	0.624443	+	0.781071i	$0.285326\pi$
$224$	−24.2961	−1.62335
$225$	0	0
$226$	−24.6381	−1.63890
$227$	−4.72192	−0.313405	−0.156703	−	0.987646i	$-0.550086\pi$
$227$	−4.72192	−0.313405	−0.156703	+	0.987646i	$0.550086\pi$
$228$	0	0
$229$	23.6102	1.56021	0.780104	−	0.625650i	$-0.215166\pi$
$229$	23.6102	1.56021	0.780104	+	0.625650i	$0.215166\pi$
$230$	0	0
$231$	0	0
$232$	1.43360	0.0941201
$233$	−15.0959	−0.988965	−0.494482	−	0.869188i	$-0.664642\pi$
$233$	−15.0959	−0.988965	−0.494482	+	0.869188i	$0.664642\pi$
$234$	0	0
$235$	0	0
$236$	−7.82660	−0.509468
$237$	0	0
$238$	2.16983	0.140649
$239$	16.5534	1.07075	0.535374	−	0.844615i	$-0.320171\pi$
$239$	16.5534	1.07075	0.535374	+	0.844615i	$0.320171\pi$
$240$	0	0
$241$	14.0704	0.906356	0.453178	−	0.891420i	$-0.350290\pi$
$241$	14.0704	0.906356	0.453178	+	0.891420i	$0.350290\pi$
$242$	−21.7018	−1.39505
$243$	0	0
$244$	−0.905814	−0.0579888
$245$	0	0
$246$	0	0
$247$	−3.21697	−0.204691
$248$	13.1103	0.832506
$249$	0	0
$250$	0	0
$251$	−28.4712	−1.79709	−0.898544	−	0.438883i	$-0.855374\pi$
$251$	−28.4712	−1.79709	−0.898544	+	0.438883i	$0.855374\pi$
$252$	0	0
$253$	1.82344	0.114638
$254$	11.4376	0.717659
$255$	0	0
$256$	20.4976	1.28110
$257$	−16.3705	−1.02117	−0.510583	−	0.859829i	$-0.670570\pi$
$257$	−16.3705	−1.02117	−0.510583	+	0.859829i	$0.670570\pi$
$258$	0	0
$259$	−34.4655	−2.14158
$260$	0	0
$261$	0	0
$262$	−28.0416	−1.73242
$263$	6.95196	0.428676	0.214338	−	0.976760i	$-0.431241\pi$
$263$	6.95196	0.428676	0.214338	+	0.976760i	$0.431241\pi$
$264$	0	0
$265$	0	0
$266$	−43.6295	−2.67510
$267$	0	0
$268$	4.73919	0.289492
$269$	−17.9274	−1.09305	−0.546525	−	0.837443i	$-0.684050\pi$
$269$	−17.9274	−1.09305	−0.546525	+	0.837443i	$0.684050\pi$
$270$	0	0
$271$	−16.2551	−0.987427	−0.493713	−	0.869625i	$-0.664361\pi$
$271$	−16.2551	−0.987427	−0.493713	+	0.869625i	$0.664361\pi$
$272$	−1.48742	−0.0901881
$273$	0	0
$274$	25.3581	1.53194
$275$	0	0
$276$	0	0
$277$	−26.4786	−1.59094	−0.795471	−	0.605992i	$-0.792776\pi$
$277$	−26.4786	−1.59094	−0.795471	+	0.605992i	$0.792776\pi$
$278$	−11.0149	−0.660627
$279$	0	0
$280$	0	0
$281$	9.09714	0.542690	0.271345	−	0.962482i	$-0.412532\pi$
$281$	9.09714	0.542690	0.271345	+	0.962482i	$0.412532\pi$
$282$	0	0
$283$	−6.18547	−0.367688	−0.183844	−	0.982955i	$-0.558854\pi$
$283$	−6.18547	−0.367688	−0.183844	+	0.982955i	$0.558854\pi$
$284$	−12.7954	−0.759269
$285$	0	0
$286$	4.58981	0.271401
$287$	−9.06811	−0.535274
$288$	0	0
$289$	−16.9101	−0.994711
$290$	0	0
$291$	0	0
$292$	13.2369	0.774629
$293$	2.22796	0.130159	0.0650795	−	0.997880i	$-0.479270\pi$
$293$	2.22796	0.130159	0.0650795	+	0.997880i	$0.479270\pi$
$294$	0	0
$295$	0	0
$296$	12.2109	0.709745
$297$	0	0
$298$	5.89973	0.341762
$299$	−0.202281	−0.0116982
$300$	0	0
$301$	−24.3236	−1.40199
$302$	21.0579	1.21175
$303$	0	0
$304$	29.9081	1.71535
$305$	0	0
$306$	0	0
$307$	34.9479	1.99458	0.997291	−	0.0735559i	$-0.0234347\pi$
$307$	34.9479	1.99458	0.997291	+	0.0735559i	$0.0234347\pi$
$308$	23.3236	1.32898
$309$	0	0
$310$	0	0
$311$	−11.8968	−0.674608	−0.337304	−	0.941396i	$-0.609515\pi$
$311$	−11.8968	−0.674608	−0.337304	+	0.941396i	$0.609515\pi$
$312$	0	0
$313$	−2.38938	−0.135056	−0.0675278	−	0.997717i	$-0.521511\pi$
$313$	−2.38938	−0.135056	−0.0675278	+	0.997717i	$0.521511\pi$
$314$	35.8136	2.02108
$315$	0	0
$316$	8.40188	0.472643
$317$	23.9964	1.34777	0.673886	−	0.738836i	$-0.264624\pi$
$317$	23.9964	1.34777	0.673886	+	0.738836i	$0.264624\pi$
$318$	0	0
$319$	−4.80986	−0.269300
$320$	0	0
$321$	0	0
$322$	−2.74339	−0.152883
$323$	−1.80779	−0.100588
$324$	0	0
$325$	0	0
$326$	−15.4435	−0.855339
$327$	0	0
$328$	3.21278	0.177396
$329$	−1.21327	−0.0668900
$330$	0	0
$331$	31.5068	1.73177	0.865885	−	0.500243i	$-0.166756\pi$
$331$	31.5068	1.73177	0.865885	+	0.500243i	$0.166756\pi$
$332$	−7.37421	−0.404712
$333$	0	0
$334$	10.7062	0.585818
$335$	0	0
$336$	0	0
$337$	22.5941	1.23078	0.615389	−	0.788223i	$-0.288999\pi$
$337$	22.5941	1.23078	0.615389	+	0.788223i	$0.288999\pi$
$338$	22.7401	1.23690
$339$	0	0
$340$	0	0
$341$	−43.9865	−2.38200
$342$	0	0
$343$	−9.60169	−0.518443
$344$	8.61771	0.464636
$345$	0	0
$346$	13.8661	0.745444
$347$	25.3237	1.35945	0.679724	−	0.733468i	$-0.262099\pi$
$347$	25.3237	1.35945	0.679724	+	0.733468i	$0.262099\pi$
$348$	0	0
$349$	−11.9657	−0.640511	−0.320255	−	0.947331i	$-0.603769\pi$
$349$	−11.9657	−0.640511	−0.320255	+	0.947331i	$0.603769\pi$
$350$	0	0
$351$	0	0
$352$	−28.8805	−1.53934
$353$	27.8488	1.48224	0.741120	−	0.671372i	$-0.234295\pi$
$353$	27.8488	1.48224	0.741120	+	0.671372i	$0.234295\pi$
$354$	0	0
$355$	0	0
$356$	−12.0464	−0.638456
$357$	0	0
$358$	16.4837	0.871190
$359$	−25.8847	−1.36614	−0.683071	−	0.730352i	$-0.739356\pi$
$359$	−25.8847	−1.36614	−0.683071	+	0.730352i	$0.739356\pi$
$360$	0	0
$361$	17.3498	0.913150
$362$	−24.9205	−1.30979
$363$	0	0
$364$	−2.58738	−0.135615
$365$	0	0
$366$	0	0
$367$	−30.0434	−1.56825	−0.784126	−	0.620602i	$-0.786888\pi$
$367$	−30.0434	−1.56825	−0.784126	+	0.620602i	$0.786888\pi$
$368$	1.88060	0.0980330
$369$	0	0
$370$	0	0
$371$	−47.9928	−2.49166
$372$	0	0
$373$	17.3011	0.895819	0.447909	−	0.894079i	$-0.352169\pi$
$373$	17.3011	0.895819	0.447909	+	0.894079i	$0.352169\pi$
$374$	2.57925	0.133370
$375$	0	0
$376$	0.429855	0.0221681
$377$	0.533576	0.0274806
$378$	0	0
$379$	6.00981	0.308703	0.154351	−	0.988016i	$-0.450671\pi$
$379$	6.00981	0.308703	0.154351	+	0.988016i	$0.450671\pi$
$380$	0	0
$381$	0	0
$382$	28.3514	1.45059
$383$	−15.9914	−0.817124	−0.408562	−	0.912731i	$-0.633970\pi$
$383$	−15.9914	−0.817124	−0.408562	+	0.912731i	$0.633970\pi$
$384$	0	0
$385$	0	0
$386$	−22.0617	−1.12291
$387$	0	0
$388$	2.88828	0.146630
$389$	3.12916	0.158655	0.0793274	−	0.996849i	$-0.474723\pi$
$389$	3.12916	0.158655	0.0793274	+	0.996849i	$0.474723\pi$
$390$	0	0
$391$	−0.113672	−0.00574865
$392$	13.4370	0.678670
$393$	0	0
$394$	−11.6532	−0.587081
$395$	0	0
$396$	0	0
$397$	−39.3500	−1.97492	−0.987461	−	0.157863i	$-0.949540\pi$
$397$	−39.3500	−1.97492	−0.987461	+	0.157863i	$0.949540\pi$
$398$	15.0338	0.753577
$399$	0	0
$400$	0	0
$401$	5.07853	0.253610	0.126805	−	0.991928i	$-0.459528\pi$
$401$	5.07853	0.253610	0.126805	+	0.991928i	$0.459528\pi$
$402$	0	0
$403$	4.87959	0.243070
$404$	−19.8979	−0.989960
$405$	0	0
$406$	7.23651	0.359142
$407$	−40.9689	−2.03075
$408$	0	0
$409$	−4.17655	−0.206517	−0.103259	−	0.994655i	$-0.532927\pi$
$409$	−4.17655	−0.206517	−0.103259	+	0.994655i	$0.532927\pi$
$410$	0	0
$411$	0	0
$412$	−16.0737	−0.791894
$413$	26.4263	1.30035
$414$	0	0
$415$	0	0
$416$	3.20383	0.157081
$417$	0	0
$418$	−51.8620	−2.53666
$419$	−16.1637	−0.789647	−0.394824	−	0.918757i	$-0.629194\pi$
$419$	−16.1637	−0.789647	−0.394824	+	0.918757i	$0.629194\pi$
$420$	0	0
$421$	4.91937	0.239755	0.119878	−	0.992789i	$-0.461750\pi$
$421$	4.91937	0.239755	0.119878	+	0.992789i	$0.461750\pi$
$422$	16.8518	0.820331
$423$	0	0
$424$	17.0035	0.825765
$425$	0	0
$426$	0	0
$427$	3.05846	0.148009
$428$	10.9295	0.528298
$429$	0	0
$430$	0	0
$431$	41.1719	1.98318	0.991591	−	0.129411i	$-0.0413087\pi$
$431$	41.1719	1.98318	0.991591	+	0.129411i	$0.0413087\pi$
$432$	0	0
$433$	−22.9724	−1.10398	−0.551991	−	0.833850i	$-0.686132\pi$
$433$	−22.9724	−1.10398	−0.551991	+	0.833850i	$0.686132\pi$
$434$	66.1783	3.17666
$435$	0	0
$436$	15.2651	0.731065
$437$	2.28565	0.109337
$438$	0	0
$439$	5.53152	0.264005	0.132003	−	0.991249i	$-0.457859\pi$
$439$	5.53152	0.264005	0.132003	+	0.991249i	$0.457859\pi$
$440$	0	0
$441$	0	0
$442$	−0.286127	−0.0136097
$443$	−3.02219	−0.143589	−0.0717944	−	0.997419i	$-0.522873\pi$
$443$	−3.02219	−0.143589	−0.0717944	+	0.997419i	$0.522873\pi$
$444$	0	0
$445$	0	0
$446$	−33.3535	−1.57933
$447$	0	0
$448$	3.30629	0.156207
$449$	7.44290	0.351252	0.175626	−	0.984457i	$-0.443805\pi$
$449$	7.44290	0.351252	0.175626	+	0.984457i	$0.443805\pi$
$450$	0	0
$451$	−10.7792	−0.507572
$452$	16.5098	0.776554
$453$	0	0
$454$	8.44471	0.396330
$455$	0	0
$456$	0	0
$457$	2.81240	0.131558	0.0657792	−	0.997834i	$-0.479047\pi$
$457$	2.81240	0.131558	0.0657792	+	0.997834i	$0.479047\pi$
$458$	−42.2246	−1.97303
$459$	0	0
$460$	0	0
$461$	15.3906	0.716810	0.358405	−	0.933566i	$-0.383321\pi$
$461$	15.3906	0.716810	0.358405	+	0.933566i	$0.383321\pi$
$462$	0	0
$463$	23.7976	1.10597	0.552984	−	0.833192i	$-0.313489\pi$
$463$	23.7976	1.10597	0.552984	+	0.833192i	$0.313489\pi$
$464$	−4.96064	−0.230292
$465$	0	0
$466$	26.9976	1.25064
$467$	−41.1731	−1.90526	−0.952632	−	0.304126i	$-0.901636\pi$
$467$	−41.1731	−1.90526	−0.952632	+	0.304126i	$0.901636\pi$
$468$	0	0
$469$	−16.0017	−0.738892
$470$	0	0
$471$	0	0
$472$	−9.36267	−0.430952
$473$	−28.9133	−1.32943
$474$	0	0
$475$	0	0
$476$	−1.45398	−0.0666431
$477$	0	0
$478$	−29.6041	−1.35406
$479$	21.5362	0.984013	0.492007	−	0.870591i	$-0.336264\pi$
$479$	21.5362	0.984013	0.492007	+	0.870591i	$0.336264\pi$
$480$	0	0
$481$	4.54484	0.207227
$482$	−25.1636	−1.14617
$483$	0	0
$484$	14.5422	0.661009
$485$	0	0
$486$	0	0
$487$	−6.21272	−0.281525	−0.140763	−	0.990043i	$-0.544955\pi$
$487$	−6.21272	−0.281525	−0.140763	+	0.990043i	$0.544955\pi$
$488$	−1.08359	−0.0490519
$489$	0	0
$490$	0	0
$491$	24.2502	1.09439	0.547197	−	0.837004i	$-0.315695\pi$
$491$	24.2502	1.09439	0.547197	+	0.837004i	$0.315695\pi$
$492$	0	0
$493$	0.299844	0.0135043
$494$	5.75326	0.258851
$495$	0	0
$496$	−45.3654	−2.03696
$497$	43.2034	1.93794
$498$	0	0
$499$	35.0007	1.56685	0.783424	−	0.621487i	$-0.213471\pi$
$499$	35.0007	1.56685	0.783424	+	0.621487i	$0.213471\pi$
$500$	0	0
$501$	0	0
$502$	50.9181	2.27259
$503$	−2.11250	−0.0941917	−0.0470958	−	0.998890i	$-0.514997\pi$
$503$	−2.11250	−0.0941917	−0.0470958	+	0.998890i	$0.514997\pi$
$504$	0	0
$505$	0	0
$506$	−3.26104	−0.144971
$507$	0	0
$508$	−7.66423	−0.340045
$509$	−39.5330	−1.75227	−0.876134	−	0.482068i	$-0.839886\pi$
$509$	−39.5330	−1.75227	−0.876134	+	0.482068i	$0.839886\pi$
$510$	0	0
$511$	−44.6939	−1.97714
$512$	−15.5628	−0.687785
$513$	0	0
$514$	29.2772	1.29136
$515$	0	0
$516$	0	0
$517$	−1.44221	−0.0634283
$518$	61.6384	2.70823
$519$	0	0
$520$	0	0
$521$	24.4951	1.07315	0.536575	−	0.843853i	$-0.319718\pi$
$521$	24.4951	1.07315	0.536575	+	0.843853i	$0.319718\pi$
$522$	0	0
$523$	−13.4730	−0.589133	−0.294567	−	0.955631i	$-0.595175\pi$
$523$	−13.4730	−0.589133	−0.294567	+	0.955631i	$0.595175\pi$
$524$	18.7904	0.820864
$525$	0	0
$526$	−12.4329	−0.542101
$527$	2.74210	0.119448
$528$	0	0
$529$	−22.8563	−0.993751
$530$	0	0
$531$	0	0
$532$	29.2358	1.26753
$533$	1.19578	0.0517949
$534$	0	0
$535$	0	0
$536$	5.66932	0.244877
$537$	0	0
$538$	32.0614	1.38226
$539$	−45.0825	−1.94184
$540$	0	0
$541$	−36.4294	−1.56622	−0.783111	−	0.621882i	$-0.786368\pi$
$541$	−36.4294	−1.56622	−0.783111	+	0.621882i	$0.786368\pi$
$542$	29.0707	1.24869
$543$	0	0
$544$	1.80040	0.0771915
$545$	0	0
$546$	0	0
$547$	−27.1863	−1.16240	−0.581201	−	0.813760i	$-0.697417\pi$
$547$	−27.1863	−1.16240	−0.581201	+	0.813760i	$0.697417\pi$
$548$	−16.9922	−0.725872
$549$	0	0
$550$	0	0
$551$	−6.02908	−0.256848
$552$	0	0
$553$	−28.3687	−1.20636
$554$	47.3544	2.01190
$555$	0	0
$556$	7.38095	0.313022
$557$	41.0921	1.74113	0.870564	−	0.492055i	$-0.163754\pi$
$557$	41.0921	1.74113	0.870564	+	0.492055i	$0.163754\pi$
$558$	0	0
$559$	3.20746	0.135661
$560$	0	0
$561$	0	0
$562$	−16.2694	−0.686282
$563$	−26.6574	−1.12348	−0.561738	−	0.827315i	$-0.689867\pi$
$563$	−26.6574	−1.12348	−0.561738	+	0.827315i	$0.689867\pi$
$564$	0	0
$565$	0	0
$566$	11.0621	0.464976
$567$	0	0
$568$	−15.3067	−0.642254
$569$	−17.9467	−0.752363	−0.376181	−	0.926546i	$-0.622763\pi$
$569$	−17.9467	−0.752363	−0.376181	+	0.926546i	$0.622763\pi$
$570$	0	0
$571$	−44.9019	−1.87909	−0.939543	−	0.342432i	$-0.888749\pi$
$571$	−44.9019	−1.87909	−0.939543	+	0.342432i	$0.888749\pi$
$572$	−3.07559	−0.128597
$573$	0	0
$574$	16.2175	0.676904
$575$	0	0
$576$	0	0
$577$	−24.3767	−1.01482	−0.507408	−	0.861706i	$-0.669396\pi$
$577$	−24.3767	−1.01482	−0.507408	+	0.861706i	$0.669396\pi$
$578$	30.2421	1.25791
$579$	0	0
$580$	0	0
$581$	24.8988	1.03298
$582$	0	0
$583$	−57.0486	−2.36271
$584$	15.8348	0.655248
$585$	0	0
$586$	−3.98450	−0.164598
$587$	0.641677	0.0264848	0.0132424	−	0.999912i	$-0.495785\pi$
$587$	0.641677	0.0264848	0.0132424	+	0.999912i	$0.495785\pi$
$588$	0	0
$589$	−55.1363	−2.27185
$590$	0	0
$591$	0	0
$592$	−42.2532	−1.73660
$593$	7.16264	0.294134	0.147067	−	0.989127i	$-0.453017\pi$
$593$	7.16264	0.294134	0.147067	+	0.989127i	$0.453017\pi$
$594$	0	0
$595$	0	0
$596$	−3.95336	−0.161936
$597$	0	0
$598$	0.361760	0.0147935
$599$	−33.7150	−1.37756	−0.688779	−	0.724971i	$-0.741853\pi$
$599$	−33.7150	−1.37756	−0.688779	+	0.724971i	$0.741853\pi$
$600$	0	0
$601$	12.3214	0.502600	0.251300	−	0.967909i	$-0.419142\pi$
$601$	12.3214	0.502600	0.251300	+	0.967909i	$0.419142\pi$
$602$	43.5005	1.77295
$603$	0	0
$604$	−14.1107	−0.574156
$605$	0	0
$606$	0	0
$607$	−3.07175	−0.124678	−0.0623392	−	0.998055i	$-0.519856\pi$
$607$	−3.07175	−0.124678	−0.0623392	+	0.998055i	$0.519856\pi$
$608$	−36.2013	−1.46816
$609$	0	0
$610$	0	0
$611$	0.159990	0.00647249
$612$	0	0
$613$	−35.7184	−1.44265	−0.721327	−	0.692595i	$-0.756468\pi$
$613$	−35.7184	−1.44265	−0.721327	+	0.692595i	$0.756468\pi$
$614$	−62.5010	−2.52234
$615$	0	0
$616$	27.9011	1.12417
$617$	−29.1211	−1.17237	−0.586185	−	0.810177i	$-0.699371\pi$
$617$	−29.1211	−1.17237	−0.586185	+	0.810177i	$0.699371\pi$
$618$	0	0
$619$	11.7565	0.472534	0.236267	−	0.971688i	$-0.424076\pi$
$619$	11.7565	0.472534	0.236267	+	0.971688i	$0.424076\pi$
$620$	0	0
$621$	0	0
$622$	21.2764	0.853105
$623$	40.6742	1.62958
$624$	0	0
$625$	0	0
$626$	4.27317	0.170790
$627$	0	0
$628$	−23.9984	−0.957640
$629$	2.55398	0.101834
$630$	0	0
$631$	30.6569	1.22043	0.610216	−	0.792235i	$-0.291083\pi$
$631$	30.6569	1.22043	0.610216	+	0.792235i	$0.291083\pi$
$632$	10.0509	0.399802
$633$	0	0
$634$	−42.9153	−1.70438
$635$	0	0
$636$	0	0
$637$	5.00117	0.198154
$638$	8.60198	0.340556
$639$	0	0
$640$	0	0
$641$	45.7844	1.80838	0.904188	−	0.427135i	$-0.140477\pi$
$641$	45.7844	1.80838	0.904188	+	0.427135i	$0.140477\pi$
$642$	0	0
$643$	−43.7394	−1.72491	−0.862456	−	0.506132i	$-0.831075\pi$
$643$	−43.7394	−1.72491	−0.862456	+	0.506132i	$0.831075\pi$
$644$	1.83832	0.0724400
$645$	0	0
$646$	3.23306	0.127203
$647$	10.6969	0.420537	0.210269	−	0.977644i	$-0.432566\pi$
$647$	10.6969	0.420537	0.210269	+	0.977644i	$0.432566\pi$
$648$	0	0
$649$	31.4127	1.23306
$650$	0	0
$651$	0	0
$652$	10.3486	0.405282
$653$	17.4970	0.684709	0.342355	−	0.939571i	$-0.388776\pi$
$653$	17.4970	0.684709	0.342355	+	0.939571i	$0.388776\pi$
$654$	0	0
$655$	0	0
$656$	−11.1171	−0.434050
$657$	0	0
$658$	2.16983	0.0845887
$659$	−38.0331	−1.48156	−0.740779	−	0.671748i	$-0.765544\pi$
$659$	−38.0331	−1.48156	−0.740779	+	0.671748i	$0.765544\pi$
$660$	0	0
$661$	−1.61112	−0.0626654	−0.0313327	−	0.999509i	$-0.509975\pi$
$661$	−1.61112	−0.0626654	−0.0313327	+	0.999509i	$0.509975\pi$
$662$	−56.3469	−2.18999
$663$	0	0
$664$	−8.82149	−0.342340
$665$	0	0
$666$	0	0
$667$	−0.379104	−0.0146790
$668$	−7.17414	−0.277576
$669$	0	0
$670$	0	0
$671$	3.63556	0.140349
$672$	0	0
$673$	−6.25241	−0.241013	−0.120506	−	0.992713i	$-0.538452\pi$
$673$	−6.25241	−0.241013	−0.120506	+	0.992713i	$0.538452\pi$
$674$	−40.4074	−1.55644
$675$	0	0
$676$	−15.2379	−0.586075
$677$	23.4775	0.902312	0.451156	−	0.892445i	$-0.351012\pi$
$677$	23.4775	0.902312	0.451156	+	0.892445i	$0.351012\pi$
$678$	0	0
$679$	−9.75221	−0.374256
$680$	0	0
$681$	0	0
$682$	78.6656	3.01226
$683$	6.95644	0.266181	0.133090	−	0.991104i	$-0.457510\pi$
$683$	6.95644	0.266181	0.133090	+	0.991104i	$0.457510\pi$
$684$	0	0
$685$	0	0
$686$	17.1717	0.655619
$687$	0	0
$688$	−29.8197	−1.13686
$689$	6.32862	0.241101
$690$	0	0
$691$	−50.6725	−1.92767	−0.963836	−	0.266498i	$-0.914133\pi$
$691$	−50.6725	−1.92767	−0.963836	+	0.266498i	$0.914133\pi$
$692$	−9.29153	−0.353211
$693$	0	0
$694$	−45.2891	−1.71915
$695$	0	0
$696$	0	0
$697$	0.671970	0.0254527
$698$	21.3996	0.809986
$699$	0	0
$700$	0	0
$701$	−26.3580	−0.995527	−0.497763	−	0.867313i	$-0.665845\pi$
$701$	−26.3580	−0.995527	−0.497763	+	0.867313i	$0.665845\pi$
$702$	0	0
$703$	−51.3539	−1.93685
$704$	3.93015	0.148123
$705$	0	0
$706$	−49.8049	−1.87443
$707$	67.1849	2.52675
$708$	0	0
$709$	1.33497	0.0501357	0.0250678	−	0.999686i	$-0.492020\pi$
$709$	1.33497	0.0501357	0.0250678	+	0.999686i	$0.492020\pi$
$710$	0	0
$711$	0	0
$712$	−14.4106	−0.540061
$713$	−3.46693	−0.129838
$714$	0	0
$715$	0	0
$716$	−11.0456	−0.412793
$717$	0	0
$718$	46.2923	1.72761
$719$	−5.44486	−0.203059	−0.101530	−	0.994833i	$-0.532374\pi$
$719$	−5.44486	−0.203059	−0.101530	+	0.994833i	$0.532374\pi$
$720$	0	0
$721$	54.2724	2.02121
$722$	−31.0286	−1.15476
$723$	0	0
$724$	16.6990	0.620614
$725$	0	0
$726$	0	0
$727$	32.6745	1.21183	0.605916	−	0.795529i	$-0.292807\pi$
$727$	32.6745	1.21183	0.605916	+	0.795529i	$0.292807\pi$
$728$	−3.09518	−0.114715
$729$	0	0
$730$	0	0
$731$	1.80244	0.0666657
$732$	0	0
$733$	20.9195	0.772678	0.386339	−	0.922357i	$-0.373740\pi$
$733$	20.9195	0.772678	0.386339	+	0.922357i	$0.373740\pi$
$734$	53.7297	1.98320
$735$	0	0
$736$	−2.27631	−0.0839059
$737$	−19.0211	−0.700653
$738$	0	0
$739$	13.2392	0.487011	0.243505	−	0.969900i	$-0.421703\pi$
$739$	13.2392	0.487011	0.243505	+	0.969900i	$0.421703\pi$
$740$	0	0
$741$	0	0
$742$	85.8306	3.15094
$743$	−45.8151	−1.68079	−0.840397	−	0.541971i	$-0.817678\pi$
$743$	−45.8151	−1.68079	−0.840397	+	0.541971i	$0.817678\pi$
$744$	0	0
$745$	0	0
$746$	−30.9414	−1.13285
$747$	0	0
$748$	−1.72834	−0.0631942
$749$	−36.9032	−1.34841
$750$	0	0
$751$	−26.2736	−0.958738	−0.479369	−	0.877614i	$-0.659134\pi$
$751$	−26.2736	−0.958738	−0.479369	+	0.877614i	$0.659134\pi$
$752$	−1.48742	−0.0542406
$753$	0	0
$754$	−0.954250	−0.0347518
$755$	0	0
$756$	0	0
$757$	−19.2465	−0.699527	−0.349763	−	0.936838i	$-0.613738\pi$
$757$	−19.2465	−0.699527	−0.349763	+	0.936838i	$0.613738\pi$
$758$	−10.7480	−0.390384
$759$	0	0
$760$	0	0
$761$	19.9183	0.722037	0.361019	−	0.932559i	$-0.382429\pi$
$761$	19.9183	0.722037	0.361019	+	0.932559i	$0.382429\pi$
$762$	0	0
$763$	−51.5422	−1.86595
$764$	−18.9980	−0.687325
$765$	0	0
$766$	28.5992	1.03333
$767$	−3.48473	−0.125826
$768$	0	0
$769$	13.5530	0.488734	0.244367	−	0.969683i	$-0.421420\pi$
$769$	13.5530	0.488734	0.244367	+	0.969683i	$0.421420\pi$
$770$	0	0
$771$	0	0
$772$	14.7834	0.532065
$773$	4.56321	0.164127	0.0820635	−	0.996627i	$-0.473849\pi$
$773$	4.56321	0.164127	0.0820635	+	0.996627i	$0.473849\pi$
$774$	0	0
$775$	0	0
$776$	3.45515	0.124033
$777$	0	0
$778$	−5.59621	−0.200634
$779$	−13.5115	−0.484101
$780$	0	0
$781$	51.3555	1.83764
$782$	0.203292	0.00726971
$783$	0	0
$784$	−46.4957	−1.66056
$785$	0	0
$786$	0	0
$787$	17.3056	0.616878	0.308439	−	0.951244i	$-0.400193\pi$
$787$	17.3056	0.616878	0.308439	+	0.951244i	$0.400193\pi$
$788$	7.80873	0.278174
$789$	0	0
$790$	0	0
$791$	−55.7448	−1.98206
$792$	0	0
$793$	−0.403307	−0.0143218
$794$	70.3738	2.49747
$795$	0	0
$796$	−10.0740	−0.357064
$797$	−25.4230	−0.900531	−0.450265	−	0.892895i	$-0.648671\pi$
$797$	−25.4230	−0.900531	−0.450265	+	0.892895i	$0.648671\pi$
$798$	0	0
$799$	0.0899066	0.00318067
$800$	0	0
$801$	0	0
$802$	−9.08247	−0.320713
$803$	−53.1273	−1.87482
$804$	0	0
$805$	0	0
$806$	−8.72668	−0.307384
$807$	0	0
$808$	−23.8032	−0.837393
$809$	39.8770	1.40200	0.701001	−	0.713160i	$-0.252737\pi$
$809$	39.8770	1.40200	0.701001	+	0.713160i	$0.252737\pi$
$810$	0	0
$811$	37.9902	1.33402	0.667008	−	0.745050i	$-0.267574\pi$
$811$	37.9902	1.33402	0.667008	+	0.745050i	$0.267574\pi$
$812$	−4.84912	−0.170171
$813$	0	0
$814$	73.2690	2.56808
$815$	0	0
$816$	0	0
$817$	−36.2424	−1.26796
$818$	7.46937	0.261160
$819$	0	0
$820$	0	0
$821$	2.02361	0.0706245	0.0353122	−	0.999376i	$-0.488757\pi$
$821$	2.02361	0.0706245	0.0353122	+	0.999376i	$0.488757\pi$
$822$	0	0
$823$	−1.80919	−0.0630642	−0.0315321	−	0.999503i	$-0.510039\pi$
$823$	−1.80919	−0.0630642	−0.0315321	+	0.999503i	$0.510039\pi$
$824$	−19.2284	−0.669851
$825$	0	0
$826$	−47.2610	−1.64442
$827$	−18.8027	−0.653834	−0.326917	−	0.945053i	$-0.606010\pi$
$827$	−18.8027	−0.653834	−0.326917	+	0.945053i	$0.606010\pi$
$828$	0	0
$829$	13.7594	0.477884	0.238942	−	0.971034i	$-0.423199\pi$
$829$	13.7594	0.477884	0.238942	+	0.971034i	$0.423199\pi$
$830$	0	0
$831$	0	0
$832$	−0.435987	−0.0151151
$833$	2.81042	0.0973753
$834$	0	0
$835$	0	0
$836$	34.7523	1.20193
$837$	0	0
$838$	28.9072	0.998583
$839$	14.1913	0.489938	0.244969	−	0.969531i	$-0.421222\pi$
$839$	14.1913	0.489938	0.244969	+	0.969531i	$0.421222\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−8.79782	−0.303193
$843$	0	0
$844$	−11.2922	−0.388694
$845$	0	0
$846$	0	0
$847$	−49.1014	−1.68714
$848$	−58.8370	−2.02047
$849$	0	0
$850$	0	0
$851$	−3.22909	−0.110692
$852$	0	0
$853$	16.7293	0.572801	0.286401	−	0.958110i	$-0.407541\pi$
$853$	16.7293	0.572801	0.286401	+	0.958110i	$0.407541\pi$
$854$	−5.46976	−0.187171
$855$	0	0
$856$	13.0746	0.446880
$857$	−33.2564	−1.13602	−0.568009	−	0.823023i	$-0.692286\pi$
$857$	−33.2564	−1.13602	−0.568009	+	0.823023i	$0.692286\pi$
$858$	0	0
$859$	−56.5319	−1.92884	−0.964422	−	0.264366i	$-0.914837\pi$
$859$	−56.5319	−1.92884	−0.964422	+	0.264366i	$0.914837\pi$
$860$	0	0
$861$	0	0
$862$	−73.6321	−2.50792
$863$	−43.9271	−1.49530	−0.747648	−	0.664096i	$-0.768817\pi$
$863$	−43.9271	−1.49530	−0.747648	+	0.664096i	$0.768817\pi$
$864$	0	0
$865$	0	0
$866$	41.0839	1.39609
$867$	0	0
$868$	−44.3455	−1.50519
$869$	−33.7217	−1.14393
$870$	0	0
$871$	2.11009	0.0714976
$872$	18.2611	0.618397
$873$	0	0
$874$	−4.08767	−0.138267
$875$	0	0
$876$	0	0
$877$	13.5059	0.456063	0.228032	−	0.973654i	$-0.426771\pi$
$877$	13.5059	0.456063	0.228032	+	0.973654i	$0.426771\pi$
$878$	−9.89261	−0.333859
$879$	0	0
$880$	0	0
$881$	−40.0575	−1.34957	−0.674785	−	0.738014i	$-0.735764\pi$
$881$	−40.0575	−1.34957	−0.674785	+	0.738014i	$0.735764\pi$
$882$	0	0
$883$	−16.0102	−0.538786	−0.269393	−	0.963030i	$-0.586823\pi$
$883$	−16.0102	−0.538786	−0.269393	+	0.963030i	$0.586823\pi$
$884$	0.191731	0.00644861
$885$	0	0
$886$	5.40491	0.181581
$887$	−13.8430	−0.464802	−0.232401	−	0.972620i	$-0.574658\pi$
$887$	−13.8430	−0.464802	−0.232401	+	0.972620i	$0.574658\pi$
$888$	0	0
$889$	25.8781	0.867923
$890$	0	0
$891$	0	0
$892$	22.3499	0.748329
$893$	−1.80779	−0.0604953
$894$	0	0
$895$	0	0
$896$	42.6792	1.42581
$897$	0	0
$898$	−13.3109	−0.444191
$899$	9.14506	0.305005
$900$	0	0
$901$	3.55639	0.118480
$902$	19.2776	0.641873
$903$	0	0
$904$	19.7500	0.656876
$905$	0	0
$906$	0	0
$907$	2.47532	0.0821915	0.0410958	−	0.999155i	$-0.486915\pi$
$907$	2.47532	0.0821915	0.0410958	+	0.999155i	$0.486915\pi$
$908$	−5.65873	−0.187791
$909$	0	0
$910$	0	0
$911$	−4.32063	−0.143149	−0.0715744	−	0.997435i	$-0.522802\pi$
$911$	−4.32063	−0.143149	−0.0715744	+	0.997435i	$0.522802\pi$
$912$	0	0
$913$	29.5970	0.979518
$914$	−5.02971	−0.166368
$915$	0	0
$916$	28.2944	0.934872
$917$	−63.4454	−2.09515
$918$	0	0
$919$	34.6470	1.14290	0.571450	−	0.820637i	$-0.306381\pi$
$919$	34.6470	1.14290	0.571450	+	0.820637i	$0.306381\pi$
$920$	0	0
$921$	0	0
$922$	−27.5246	−0.906473
$923$	−5.69706	−0.187521
$924$	0	0
$925$	0	0
$926$	−42.5597	−1.39860
$927$	0	0
$928$	6.00444	0.197106
$929$	−9.64854	−0.316558	−0.158279	−	0.987394i	$-0.550595\pi$
$929$	−9.64854	−0.316558	−0.158279	+	0.987394i	$0.550595\pi$
$930$	0	0
$931$	−56.5102	−1.85205
$932$	−18.0908	−0.592585
$933$	0	0
$934$	73.6342	2.40938
$935$	0	0
$936$	0	0
$937$	−5.16386	−0.168696	−0.0843480	−	0.996436i	$-0.526881\pi$
$937$	−5.16386	−0.168696	−0.0843480	+	0.996436i	$0.526881\pi$
$938$	28.6176	0.934398
$939$	0	0
$940$	0	0
$941$	−46.0541	−1.50132	−0.750661	−	0.660688i	$-0.770265\pi$
$941$	−46.0541	−1.50132	−0.750661	+	0.660688i	$0.770265\pi$
$942$	0	0
$943$	−0.849596	−0.0276666
$944$	32.3975	1.05445
$945$	0	0
$946$	51.7087	1.68120
$947$	7.00646	0.227679	0.113840	−	0.993499i	$-0.463685\pi$
$947$	7.00646	0.227679	0.113840	+	0.993499i	$0.463685\pi$
$948$	0	0
$949$	5.89361	0.191315
$950$	0	0
$951$	0	0
$952$	−1.73934	−0.0563725
$953$	48.2817	1.56400	0.781998	−	0.623281i	$-0.214201\pi$
$953$	48.2817	1.56400	0.781998	+	0.623281i	$0.214201\pi$
$954$	0	0
$955$	0	0
$956$	19.8375	0.641589
$957$	0	0
$958$	−38.5154	−1.24438
$959$	57.3738	1.85270
$960$	0	0
$961$	52.6322	1.69781
$962$	−8.12801	−0.262058
$963$	0	0
$964$	16.8619	0.543086
$965$	0	0
$966$	0	0
$967$	19.8989	0.639906	0.319953	−	0.947433i	$-0.396333\pi$
$967$	19.8989	0.639906	0.319953	+	0.947433i	$0.396333\pi$
$968$	17.3963	0.559138
$969$	0	0
$970$	0	0
$971$	28.8071	0.924464	0.462232	−	0.886759i	$-0.347049\pi$
$971$	28.8071	0.924464	0.462232	+	0.886759i	$0.347049\pi$
$972$	0	0
$973$	−24.9216	−0.798950
$974$	11.1109	0.356015
$975$	0	0
$976$	3.74953	0.120019
$977$	−34.4662	−1.10267	−0.551336	−	0.834284i	$-0.685882\pi$
$977$	−34.4662	−1.10267	−0.551336	+	0.834284i	$0.685882\pi$
$978$	0	0
$979$	48.3491	1.54524
$980$	0	0
$981$	0	0
$982$	−43.3691	−1.38396
$983$	−21.8433	−0.696695	−0.348347	−	0.937366i	$-0.613257\pi$
$983$	−21.8433	−0.696695	−0.348347	+	0.937366i	$0.613257\pi$
$984$	0	0
$985$	0	0
$986$	−0.536243	−0.0170775
$987$	0	0
$988$	−3.85521	−0.122650
$989$	−2.27889	−0.0724646
$990$	0	0
$991$	−6.00703	−0.190820	−0.0954099	−	0.995438i	$-0.530416\pi$
$991$	−6.00703	−0.190820	−0.0954099	+	0.995438i	$0.530416\pi$
$992$	54.9110	1.74343
$993$	0	0
$994$	−77.2652	−2.45070
$995$	0	0
$996$	0	0
$997$	−19.9499	−0.631821	−0.315911	−	0.948789i	$-0.602310\pi$
$997$	−19.9499	−0.631821	−0.315911	+	0.948789i	$0.602310\pi$
$998$	−62.5955	−1.98143
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.ce.1.3		12
3.2	odd	2		6525.2.a.cf.1.10		12
5.2	odd	4		1305.2.c.k.784.3	✓	12
5.3	odd	4		1305.2.c.k.784.10	yes	12
5.4	even	2	inner	6525.2.a.ce.1.10		12
15.2	even	4		1305.2.c.l.784.10	yes	12
15.8	even	4		1305.2.c.l.784.3	yes	12
15.14	odd	2		6525.2.a.cf.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1305.2.c.k.784.3	✓	12	5.2	odd	4
1305.2.c.k.784.10	yes	12	5.3	odd	4
1305.2.c.l.784.3	yes	12	15.8	even	4
1305.2.c.l.784.10	yes	12	15.2	even	4
6525.2.a.ce.1.3		12	1.1	even	1	trivial
6525.2.a.ce.1.10		12	5.4	even	2	inner
6525.2.a.cf.1.3		12	15.14	odd	2
6525.2.a.cf.1.10		12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q-1.78841 q^{2} +1.19839 q^{4} -4.04635 q^{7} +1.43360 q^{8} -4.80986 q^{11} +0.533576 q^{13} +7.23651 q^{14} -4.96064 q^{16} +0.299844 q^{17} -6.02908 q^{19} +8.60198 q^{22} -0.379104 q^{23} -0.954250 q^{26} -4.84912 q^{28} +1.00000 q^{29} +9.14506 q^{31} +6.00444 q^{32} -0.536243 q^{34} +8.51769 q^{37} +10.7824 q^{38} +2.24106 q^{41} +6.01126 q^{43} -5.76411 q^{44} +0.677992 q^{46} +0.299844 q^{47} +9.37293 q^{49} +0.639435 q^{52} +11.8608 q^{53} -5.80083 q^{56} -1.78841 q^{58} -6.53090 q^{59} -0.755856 q^{61} -16.3551 q^{62} -0.817104 q^{64} +3.95461 q^{67} +0.359332 q^{68} -10.6771 q^{71} +11.0455 q^{73} -15.2331 q^{74} -7.22522 q^{76} +19.4624 q^{77} +7.01095 q^{79} -4.00793 q^{82} -6.15340 q^{83} -10.7506 q^{86} -6.89539 q^{88} -10.0521 q^{89} -2.15903 q^{91} -0.454316 q^{92} -0.536243 q^{94} +2.41013 q^{97} -16.7626 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{4} - 12 q^{11} - 16 q^{14} + 16 q^{16} - 20 q^{19} - 56 q^{26} + 12 q^{29} - 16 q^{31} + 4 q^{34} - 32 q^{41} - 68 q^{44} + 20 q^{46} - 4 q^{49} - 76 q^{56} - 44 q^{59} - 52 q^{61} + 36 q^{64}+ \cdots + 4 q^{94}+O(q^{100}) \) 12 * q + 16 * q^4 - 12 * q^11 - 16 * q^14 + 16 * q^16 - 20 * q^19 - 56 * q^26 + 12 * q^29 - 16 * q^31 + 4 * q^34 - 32 * q^41 - 68 * q^44 + 20 * q^46 - 4 * q^49 - 76 * q^56 - 44 * q^59 - 52 * q^61 + 36 * q^64 - 20 * q^71 - 20 * q^74 - 28 * q^76 - 4 * q^79 - 36 * q^86 - 68 * q^89 + 48 * q^91 + 4 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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