LMFDB - Embedded newform 6975.2.a.ce.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.3057647616.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 4x^{6} + 20x^{5} + 4x^{4} - 20x^{3} - 4x^{2} + 4x + 1 $ x^8 - 4x^7 - 4x^6 + 20x^5 + 4x^4 - 20x^3 - 4x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1395)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.61132$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93185 q^{2} +1.73205 q^{4} +1.12603 q^{7} +0.517638 q^{8} -1.59245 q^{11} -4.20241 q^{13} -2.17533 q^{14} -4.46410 q^{16} +1.41421 q^{17} -2.00000 q^{19} +3.07638 q^{22} -1.41421 q^{23} +8.11843 q^{26} +1.95035 q^{28} +1.00957 q^{29} +1.00000 q^{31} +7.58871 q^{32} -2.73205 q^{34} +5.02672 q^{37} +3.86370 q^{38} +5.94311 q^{41} -3.07638 q^{43} -2.75821 q^{44} +2.73205 q^{46} +8.10634 q^{47} -5.73205 q^{49} -7.27879 q^{52} +5.27792 q^{53} +0.582877 q^{56} -1.95035 q^{58} +2.17533 q^{59} -10.9282 q^{61} -1.93185 q^{62} -5.73205 q^{64} +12.0038 q^{67} +2.44949 q^{68} +6.95268 q^{71} +5.02672 q^{73} -9.71088 q^{74} -3.46410 q^{76} -1.79315 q^{77} -10.1962 q^{79} -11.4812 q^{82} -1.41421 q^{83} +5.94311 q^{86} -0.824313 q^{88} -11.3033 q^{89} -4.73205 q^{91} -2.44949 q^{92} -15.6603 q^{94} +1.42775 q^{97} +11.0735 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{16} - 16 q^{19} + 8 q^{31} - 8 q^{34} + 8 q^{46} - 32 q^{49} - 32 q^{61} - 32 q^{64} - 40 q^{79} - 24 q^{91} - 56 q^{94}+O(q^{100}) $ 8 * q - 8 * q^16 - 16 * q^19 + 8 * q^31 - 8 * q^34 + 8 * q^46 - 32 * q^49 - 32 * q^61 - 32 * q^64 - 40 * q^79 - 24 * q^91 - 56 * 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 $

$2$	−1.93185	−1.36603	−0.683013	−	0.730406i	$-0.739331\pi$
$2$	−1.93185	−1.36603	−0.683013	+	0.730406i	$0.739331\pi$
$3$	0	0
$3$	0	0
$4$	1.73205	0.866025
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.12603	0.425600	0.212800	−	0.977096i	$-0.431742\pi$
$7$	1.12603	0.425600	0.212800	+	0.977096i	$0.431742\pi$
$8$	0.517638	0.183013
$9$	0	0
$10$	0	0
$11$	−1.59245	−0.480142	−0.240071	−	0.970755i	$-0.577171\pi$
$11$	−1.59245	−0.480142	−0.240071	+	0.970755i	$0.577171\pi$
$12$	0	0
$13$	−4.20241	−1.16554	−0.582769	−	0.812638i	$-0.698031\pi$
$13$	−4.20241	−1.16554	−0.582769	+	0.812638i	$0.698031\pi$
$14$	−2.17533	−0.581381
$15$	0	0
$16$	−4.46410	−1.11603
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	3.07638	0.655886
$23$	−1.41421	−0.294884	−0.147442	−	0.989071i	$-0.547104\pi$
$23$	−1.41421	−0.294884	−0.147442	+	0.989071i	$0.547104\pi$
$24$	0	0
$25$	0	0
$26$	8.11843	1.59216
$27$	0	0
$28$	1.95035	0.368581
$29$	1.00957	0.187473	0.0937365	−	0.995597i	$-0.470119\pi$
$29$	1.00957	0.187473	0.0937365	+	0.995597i	$0.470119\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	7.58871	1.34151
$33$	0	0
$34$	−2.73205	−0.468543
$35$	0	0
$36$	0	0
$37$	5.02672	0.826388	0.413194	−	0.910643i	$-0.364413\pi$
$37$	5.02672	0.826388	0.413194	+	0.910643i	$0.364413\pi$
$38$	3.86370	0.626775
$39$	0	0
$40$	0	0
$41$	5.94311	0.928157	0.464079	−	0.885794i	$-0.346386\pi$
$41$	5.94311	0.928157	0.464079	+	0.885794i	$0.346386\pi$
$42$	0	0
$43$	−3.07638	−0.469143	−0.234572	−	0.972099i	$-0.575369\pi$
$43$	−3.07638	−0.469143	−0.234572	+	0.972099i	$0.575369\pi$
$44$	−2.75821	−0.415815
$45$	0	0
$46$	2.73205	0.402819
$47$	8.10634	1.18243	0.591216	−	0.806513i	$-0.298648\pi$
$47$	8.10634	1.18243	0.591216	+	0.806513i	$0.298648\pi$
$48$	0	0
$49$	−5.73205	−0.818864
$50$	0	0
$51$	0	0
$52$	−7.27879	−1.00939
$53$	5.27792	0.724978	0.362489	−	0.931988i	$-0.381927\pi$
$53$	5.27792	0.724978	0.362489	+	0.931988i	$0.381927\pi$
$54$	0	0
$55$	0	0
$56$	0.582877	0.0778903
$57$	0	0
$58$	−1.95035	−0.256093
$59$	2.17533	0.283204	0.141602	−	0.989924i	$-0.454775\pi$
$59$	2.17533	0.283204	0.141602	+	0.989924i	$0.454775\pi$
$60$	0	0
$61$	−10.9282	−1.39921	−0.699607	−	0.714528i	$-0.746641\pi$
$61$	−10.9282	−1.39921	−0.699607	+	0.714528i	$0.746641\pi$
$62$	−1.93185	−0.245345
$63$	0	0
$64$	−5.73205	−0.716506
$65$	0	0
$66$	0	0
$67$	12.0038	1.46650	0.733248	−	0.679961i	$-0.238003\pi$
$67$	12.0038	1.46650	0.733248	+	0.679961i	$0.238003\pi$
$68$	2.44949	0.297044
$69$	0	0
$70$	0	0
$71$	6.95268	0.825131	0.412566	−	0.910928i	$-0.364633\pi$
$71$	6.95268	0.825131	0.412566	+	0.910928i	$0.364633\pi$
$72$	0	0
$73$	5.02672	0.588333	0.294167	−	0.955754i	$-0.404958\pi$
$73$	5.02672	0.588333	0.294167	+	0.955754i	$0.404958\pi$
$74$	−9.71088	−1.12887
$75$	0	0
$76$	−3.46410	−0.397360
$77$	−1.79315	−0.204349
$78$	0	0
$79$	−10.1962	−1.14716	−0.573578	−	0.819151i	$-0.694445\pi$
$79$	−10.1962	−1.14716	−0.573578	+	0.819151i	$0.694445\pi$
$80$	0	0
$81$	0	0
$82$	−11.4812	−1.26789
$83$	−1.41421	−0.155230	−0.0776151	−	0.996983i	$-0.524731\pi$
$83$	−1.41421	−0.155230	−0.0776151	+	0.996983i	$0.524731\pi$
$84$	0	0
$85$	0	0
$86$	5.94311	0.640862
$87$	0	0
$88$	−0.824313	−0.0878721
$89$	−11.3033	−1.19815	−0.599076	−	0.800693i	$-0.704465\pi$
$89$	−11.3033	−1.19815	−0.599076	+	0.800693i	$0.704465\pi$
$90$	0	0
$91$	−4.73205	−0.496054
$92$	−2.44949	−0.255377
$93$	0	0
$94$	−15.6603	−1.61523
$95$	0	0
$96$	0	0
$97$	1.42775	0.144966	0.0724831	−	0.997370i	$-0.476908\pi$
$97$	1.42775	0.144966	0.0724831	+	0.997370i	$0.476908\pi$
$98$	11.0735	1.11859
$99$	0	0
$100$	0	0
$101$	−4.77735	−0.475364	−0.237682	−	0.971343i	$-0.576388\pi$
$101$	−4.77735	−0.475364	−0.237682	+	0.971343i	$0.576388\pi$
$102$	0	0
$103$	−0.301719	−0.0297293	−0.0148647	−	0.999890i	$-0.504732\pi$
$103$	−0.301719	−0.0297293	−0.0148647	+	0.999890i	$0.504732\pi$
$104$	−2.17533	−0.213308
$105$	0	0
$106$	−10.1962	−0.990338
$107$	6.31319	0.610319	0.305160	−	0.952301i	$-0.401290\pi$
$107$	6.31319	0.610319	0.305160	+	0.952301i	$0.401290\pi$
$108$	0	0
$109$	0.196152	0.0187880	0.00939400	−	0.999956i	$-0.497010\pi$
$109$	0.196152	0.0187880	0.00939400	+	0.999956i	$0.497010\pi$
$110$	0	0
$111$	0	0
$112$	−5.02672	−0.474981
$113$	−5.65685	−0.532152	−0.266076	−	0.963952i	$-0.585727\pi$
$113$	−5.65685	−0.532152	−0.266076	+	0.963952i	$0.585727\pi$
$114$	0	0
$115$	0	0
$116$	1.74863	0.162356
$117$	0	0
$118$	−4.20241	−0.386863
$119$	1.59245	0.145980
$120$	0	0
$121$	−8.46410	−0.769464
$122$	21.1117	1.91136
$123$	0	0
$124$	1.73205	0.155543
$125$	0	0
$126$	0	0
$127$	−16.8096	−1.49161	−0.745807	−	0.666162i	$-0.767936\pi$
$127$	−16.8096	−1.49161	−0.745807	+	0.666162i	$0.767936\pi$
$128$	−4.10394	−0.362740
$129$	0	0
$130$	0	0
$131$	−7.69174	−0.672030	−0.336015	−	0.941857i	$-0.609079\pi$
$131$	−7.69174	−0.672030	−0.336015	+	0.941857i	$0.609079\pi$
$132$	0	0
$133$	−2.25207	−0.195279
$134$	−23.1895	−2.00327
$135$	0	0
$136$	0.732051	0.0627728
$137$	21.4906	1.83607	0.918033	−	0.396504i	$-0.129777\pi$
$137$	21.4906	1.83607	0.918033	+	0.396504i	$0.129777\pi$
$138$	0	0
$139$	−11.4641	−0.972372	−0.486186	−	0.873855i	$-0.661612\pi$
$139$	−11.4641	−0.972372	−0.486186	+	0.873855i	$0.661612\pi$
$140$	0	0
$141$	0	0
$142$	−13.4315	−1.12715
$143$	6.69213	0.559624
$144$	0	0
$145$	0	0
$146$	−9.71088	−0.803678
$147$	0	0
$148$	8.70654	0.715673
$149$	11.4595	0.938800	0.469400	−	0.882986i	$-0.344470\pi$
$149$	11.4595	0.938800	0.469400	+	0.882986i	$0.344470\pi$
$150$	0	0
$151$	−12.1962	−0.992509	−0.496254	−	0.868177i	$-0.665292\pi$
$151$	−12.1962	−0.992509	−0.496254	+	0.868177i	$0.665292\pi$
$152$	−1.03528	−0.0839720
$153$	0	0
$154$	3.46410	0.279145
$155$	0	0
$156$	0	0
$157$	12.0846	0.964459	0.482230	−	0.876045i	$-0.339827\pi$
$157$	12.0846	0.964459	0.482230	+	0.876045i	$0.339827\pi$
$158$	19.6975	1.56705
$159$	0	0
$160$	0	0
$161$	−1.59245	−0.125503
$162$	0	0
$163$	−9.75173	−0.763814	−0.381907	−	0.924201i	$-0.624733\pi$
$163$	−9.75173	−0.763814	−0.381907	+	0.924201i	$0.624733\pi$
$164$	10.2938	0.803808
$165$	0	0
$166$	2.73205	0.212048
$167$	−8.20788	−0.635145	−0.317572	−	0.948234i	$-0.602868\pi$
$167$	−8.20788	−0.635145	−0.317572	+	0.948234i	$0.602868\pi$
$168$	0	0
$169$	4.66025	0.358481
$170$	0	0
$171$	0	0
$172$	−5.32844	−0.406290
$173$	−10.5558	−0.802545	−0.401273	−	0.915959i	$-0.631432\pi$
$173$	−10.5558	−0.802545	−0.401273	+	0.915959i	$0.631432\pi$
$174$	0	0
$175$	0	0
$176$	7.10886	0.535851
$177$	0	0
$178$	21.8364	1.63670
$179$	10.7205	0.801285	0.400642	−	0.916235i	$-0.368787\pi$
$179$	10.7205	0.801285	0.400642	+	0.916235i	$0.368787\pi$
$180$	0	0
$181$	2.92820	0.217652	0.108826	−	0.994061i	$-0.465291\pi$
$181$	2.92820	0.217652	0.108826	+	0.994061i	$0.465291\pi$
$182$	9.14162	0.677622
$183$	0	0
$184$	−0.732051	−0.0539675
$185$	0	0
$186$	0	0
$187$	−2.25207	−0.164687
$188$	14.0406	1.02402
$189$	0	0
$190$	0	0
$191$	−23.9286	−1.73141	−0.865707	−	0.500552i	$-0.833130\pi$
$191$	−23.9286	−1.73141	−0.865707	+	0.500552i	$0.833130\pi$
$192$	0	0
$193$	−9.83257	−0.707764	−0.353882	−	0.935290i	$-0.615139\pi$
$193$	−9.83257	−0.707764	−0.353882	+	0.935290i	$0.615139\pi$
$194$	−2.75821	−0.198028
$195$	0	0
$196$	−9.92820	−0.709157
$197$	13.0053	0.926591	0.463295	−	0.886204i	$-0.346667\pi$
$197$	13.0053	0.926591	0.463295	+	0.886204i	$0.346667\pi$
$198$	0	0
$199$	0.196152	0.0139049	0.00695244	−	0.999976i	$-0.497787\pi$
$199$	0.196152	0.0139049	0.00695244	+	0.999976i	$0.497787\pi$
$200$	0	0
$201$	0	0
$202$	9.22913	0.649360
$203$	1.13681	0.0797886
$204$	0	0
$205$	0	0
$206$	0.582877	0.0406110
$207$	0	0
$208$	18.7600	1.30077
$209$	3.18490	0.220304
$210$	0	0
$211$	−1.46410	−0.100793	−0.0503965	−	0.998729i	$-0.516048\pi$
$211$	−1.46410	−0.100793	−0.0503965	+	0.998729i	$0.516048\pi$
$212$	9.14162	0.627849
$213$	0	0
$214$	−12.1962	−0.833712
$215$	0	0
$216$	0	0
$217$	1.12603	0.0764401
$218$	−0.378937	−0.0256649
$219$	0	0
$220$	0	0
$221$	−5.94311	−0.399777
$222$	0	0
$223$	15.3819	1.03005	0.515024	−	0.857176i	$-0.327783\pi$
$223$	15.3819	1.03005	0.515024	+	0.857176i	$0.327783\pi$
$224$	8.54513	0.570945
$225$	0	0
$226$	10.9282	0.726933
$227$	4.52004	0.300006	0.150003	−	0.988686i	$-0.452072\pi$
$227$	4.52004	0.300006	0.150003	+	0.988686i	$0.452072\pi$
$228$	0	0
$229$	5.85641	0.387002	0.193501	−	0.981100i	$-0.438016\pi$
$229$	5.85641	0.387002	0.193501	+	0.981100i	$0.438016\pi$
$230$	0	0
$231$	0	0
$232$	0.522594	0.0343099
$233$	−16.5916	−1.08695	−0.543477	−	0.839424i	$-0.682892\pi$
$233$	−16.5916	−1.08695	−0.543477	+	0.839424i	$0.682892\pi$
$234$	0	0
$235$	0	0
$236$	3.76778	0.245261
$237$	0	0
$238$	−3.07638	−0.199412
$239$	25.6772	1.66092	0.830461	−	0.557076i	$-0.188077\pi$
$239$	25.6772	1.66092	0.830461	+	0.557076i	$0.188077\pi$
$240$	0	0
$241$	0.392305	0.0252706	0.0126353	−	0.999920i	$-0.495978\pi$
$241$	0.392305	0.0252706	0.0126353	+	0.999920i	$0.495978\pi$
$242$	16.3514	1.05111
$243$	0	0
$244$	−18.9282	−1.21175
$245$	0	0
$246$	0	0
$247$	8.40482	0.534786
$248$	0.517638	0.0328701
$249$	0	0
$250$	0	0
$251$	15.9245	1.00515	0.502573	−	0.864535i	$-0.332387\pi$
$251$	15.9245	1.00515	0.502573	+	0.864535i	$0.332387\pi$
$252$	0	0
$253$	2.25207	0.141586
$254$	32.4737	2.03758
$255$	0	0
$256$	19.3923	1.21202
$257$	−12.4505	−0.776642	−0.388321	−	0.921524i	$-0.626945\pi$
$257$	−12.4505	−0.776642	−0.388321	+	0.921524i	$0.626945\pi$
$258$	0	0
$259$	5.66025	0.351711
$260$	0	0
$261$	0	0
$262$	14.8593	0.918010
$263$	−18.5606	−1.14450	−0.572249	−	0.820080i	$-0.693929\pi$
$263$	−18.5606	−1.14450	−0.572249	+	0.820080i	$0.693929\pi$
$264$	0	0
$265$	0	0
$266$	4.35066	0.266756
$267$	0	0
$268$	20.7912	1.27002
$269$	−24.7820	−1.51099	−0.755493	−	0.655157i	$-0.772602\pi$
$269$	−24.7820	−1.51099	−0.755493	+	0.655157i	$0.772602\pi$
$270$	0	0
$271$	20.2487	1.23002	0.615011	−	0.788519i	$-0.289152\pi$
$271$	20.2487	1.23002	0.615011	+	0.788519i	$0.289152\pi$
$272$	−6.31319	−0.382794
$273$	0	0
$274$	−41.5167	−2.50811
$275$	0	0
$276$	0	0
$277$	6.23360	0.374541	0.187270	−	0.982308i	$-0.440036\pi$
$277$	6.23360	0.374541	0.187270	+	0.982308i	$0.440036\pi$
$278$	22.1469	1.32829
$279$	0	0
$280$	0	0
$281$	−9.55470	−0.569986	−0.284993	−	0.958530i	$-0.591991\pi$
$281$	−9.55470	−0.569986	−0.284993	+	0.958530i	$0.591991\pi$
$282$	0	0
$283$	−28.5926	−1.69965	−0.849826	−	0.527064i	$-0.823293\pi$
$283$	−28.5926	−1.69965	−0.849826	+	0.527064i	$0.823293\pi$
$284$	12.0424	0.714585
$285$	0	0
$286$	−12.9282	−0.764461
$287$	6.69213	0.395024
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0	0
$291$	0	0
$292$	8.70654	0.509512
$293$	12.4505	0.727367	0.363684	−	0.931523i	$-0.381519\pi$
$293$	12.4505	0.727367	0.363684	+	0.931523i	$0.381519\pi$
$294$	0	0
$295$	0	0
$296$	2.60202	0.151240
$297$	0	0
$298$	−22.1381	−1.28242
$299$	5.94311	0.343699
$300$	0	0
$301$	−3.46410	−0.199667
$302$	23.5612	1.35579
$303$	0	0
$304$	8.92820	0.512068
$305$	0	0
$306$	0	0
$307$	−20.1877	−1.15218	−0.576088	−	0.817388i	$-0.695421\pi$
$307$	−20.1877	−1.15218	−0.576088	+	0.817388i	$0.695421\pi$
$308$	−3.10583	−0.176971
$309$	0	0
$310$	0	0
$311$	−29.8717	−1.69387	−0.846935	−	0.531697i	$-0.821555\pi$
$311$	−29.8717	−1.69387	−0.846935	+	0.531697i	$0.821555\pi$
$312$	0	0
$313$	21.6155	1.22178	0.610890	−	0.791716i	$-0.290812\pi$
$313$	21.6155	1.22178	0.610890	+	0.791716i	$0.290812\pi$
$314$	−23.3457	−1.31748
$315$	0	0
$316$	−17.6603	−0.993467
$317$	−27.1475	−1.52475	−0.762377	−	0.647134i	$-0.775968\pi$
$317$	−27.1475	−1.52475	−0.762377	+	0.647134i	$0.775968\pi$
$318$	0	0
$319$	−1.60770	−0.0900136
$320$	0	0
$321$	0	0
$322$	3.07638	0.171440
$323$	−2.82843	−0.157378
$324$	0	0
$325$	0	0
$326$	18.8389	1.04339
$327$	0	0
$328$	3.07638	0.169865
$329$	9.12801	0.503243
$330$	0	0
$331$	14.5885	0.801854	0.400927	−	0.916110i	$-0.368688\pi$
$331$	14.5885	0.801854	0.400927	+	0.916110i	$0.368688\pi$
$332$	−2.44949	−0.134433
$333$	0	0
$334$	15.8564	0.867624
$335$	0	0
$336$	0	0
$337$	−0.522594	−0.0284675	−0.0142337	−	0.999899i	$-0.504531\pi$
$337$	−0.522594	−0.0284675	−0.0142337	+	0.999899i	$0.504531\pi$
$338$	−9.00292	−0.489694
$339$	0	0
$340$	0	0
$341$	−1.59245	−0.0862360
$342$	0	0
$343$	−14.3367	−0.774109
$344$	−1.59245	−0.0858592
$345$	0	0
$346$	20.3923	1.09630
$347$	14.9743	0.803865	0.401932	−	0.915669i	$-0.368339\pi$
$347$	14.9743	0.803865	0.401932	+	0.915669i	$0.368339\pi$
$348$	0	0
$349$	−25.6603	−1.37356	−0.686781	−	0.726864i	$-0.740977\pi$
$349$	−25.6603	−1.37356	−0.686781	+	0.726864i	$0.740977\pi$
$350$	0	0
$351$	0	0
$352$	−12.0846	−0.644113
$353$	−16.1112	−0.857510	−0.428755	−	0.903421i	$-0.641048\pi$
$353$	−16.1112	−0.857510	−0.428755	+	0.903421i	$0.641048\pi$
$354$	0	0
$355$	0	0
$356$	−19.5779	−1.03763
$357$	0	0
$358$	−20.7103	−1.09458
$359$	27.9669	1.47604	0.738018	−	0.674781i	$-0.235762\pi$
$359$	27.9669	1.47604	0.738018	+	0.674781i	$0.235762\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−5.65685	−0.297318
$363$	0	0
$364$	−8.19615	−0.429595
$365$	0	0
$366$	0	0
$367$	32.4124	1.69191	0.845957	−	0.533251i	$-0.179030\pi$
$367$	32.4124	1.69191	0.845957	+	0.533251i	$0.179030\pi$
$368$	6.31319	0.329098
$369$	0	0
$370$	0	0
$371$	5.94311	0.308551
$372$	0	0
$373$	−18.4583	−0.955733	−0.477866	−	0.878433i	$-0.658590\pi$
$373$	−18.4583	−0.955733	−0.477866	+	0.878433i	$0.658590\pi$
$374$	4.35066	0.224967
$375$	0	0
$376$	4.19615	0.216400
$377$	−4.24264	−0.218507
$378$	0	0
$379$	11.8564	0.609023	0.304511	−	0.952509i	$-0.401507\pi$
$379$	11.8564	0.609023	0.304511	+	0.952509i	$0.401507\pi$
$380$	0	0
$381$	0	0
$382$	46.2265	2.36515
$383$	0.859411	0.0439138	0.0219569	−	0.999759i	$-0.493010\pi$
$383$	0.859411	0.0439138	0.0219569	+	0.999759i	$0.493010\pi$
$384$	0	0
$385$	0	0
$386$	18.9951	0.966824
$387$	0	0
$388$	2.47294	0.125544
$389$	−27.2278	−1.38051	−0.690253	−	0.723568i	$-0.742501\pi$
$389$	−27.2278	−1.38051	−0.690253	+	0.723568i	$0.742501\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	−2.96713	−0.149863
$393$	0	0
$394$	−25.1244	−1.26575
$395$	0	0
$396$	0	0
$397$	12.3055	0.617596	0.308798	−	0.951128i	$-0.400073\pi$
$397$	12.3055	0.617596	0.308798	+	0.951128i	$0.400073\pi$
$398$	−0.378937	−0.0189944
$399$	0	0
$400$	0	0
$401$	−1.74863	−0.0873225	−0.0436613	−	0.999046i	$-0.513902\pi$
$401$	−1.74863	−0.0873225	−0.0436613	+	0.999046i	$0.513902\pi$
$402$	0	0
$403$	−4.20241	−0.209337
$404$	−8.27462	−0.411677
$405$	0	0
$406$	−2.19615	−0.108993
$407$	−8.00481	−0.396784
$408$	0	0
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	0	0
$412$	−0.522594	−0.0257463
$413$	2.44949	0.120532
$414$	0	0
$415$	0	0
$416$	−31.8909	−1.56358
$417$	0	0
$418$	−6.15276	−0.300941
$419$	18.8389	0.920340	0.460170	−	0.887831i	$-0.347788\pi$
$419$	18.8389	0.920340	0.460170	+	0.887831i	$0.347788\pi$
$420$	0	0
$421$	−30.1962	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$421$	−30.1962	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$422$	2.82843	0.137686
$423$	0	0
$424$	2.73205	0.132680
$425$	0	0
$426$	0	0
$427$	−12.3055	−0.595505
$428$	10.9348	0.528552
$429$	0	0
$430$	0	0
$431$	−8.97183	−0.432158	−0.216079	−	0.976376i	$-0.569327\pi$
$431$	−8.97183	−0.432158	−0.216079	+	0.976376i	$0.569327\pi$
$432$	0	0
$433$	15.0802	0.724707	0.362353	−	0.932041i	$-0.381973\pi$
$433$	15.0802	0.724707	0.362353	+	0.932041i	$0.381973\pi$
$434$	−2.17533	−0.104419
$435$	0	0
$436$	0.339746	0.0162709
$437$	2.82843	0.135302
$438$	0	0
$439$	−14.9282	−0.712484	−0.356242	−	0.934394i	$-0.615942\pi$
$439$	−14.9282	−0.712484	−0.356242	+	0.934394i	$0.615942\pi$
$440$	0	0
$441$	0	0
$442$	11.4812	0.546105
$443$	−2.24642	−0.106731	−0.0533653	−	0.998575i	$-0.516995\pi$
$443$	−2.24642	−0.106731	−0.0533653	+	0.998575i	$0.516995\pi$
$444$	0	0
$445$	0	0
$446$	−29.7155	−1.40707
$447$	0	0
$448$	−6.45448	−0.304945
$449$	−1.74863	−0.0825230	−0.0412615	−	0.999148i	$-0.513138\pi$
$449$	−1.74863	−0.0825230	−0.0412615	+	0.999148i	$0.513138\pi$
$450$	0	0
$451$	−9.46410	−0.445647
$452$	−9.79796	−0.460857
$453$	0	0
$454$	−8.73205	−0.409815
$455$	0	0
$456$	0	0
$457$	−27.5474	−1.28861	−0.644306	−	0.764768i	$-0.722854\pi$
$457$	−27.5474	−1.28861	−0.644306	+	0.764768i	$0.722854\pi$
$458$	−11.3137	−0.528655
$459$	0	0
$460$	0	0
$461$	16.0807	0.748952	0.374476	−	0.927237i	$-0.377823\pi$
$461$	16.0807	0.748952	0.374476	+	0.927237i	$0.377823\pi$
$462$	0	0
$463$	−1.42775	−0.0663533	−0.0331766	−	0.999450i	$-0.510562\pi$
$463$	−1.42775	−0.0663533	−0.0331766	+	0.999450i	$0.510562\pi$
$464$	−4.50684	−0.209225
$465$	0	0
$466$	32.0526	1.48481
$467$	6.79367	0.314373	0.157187	−	0.987569i	$-0.449758\pi$
$467$	6.79367	0.314373	0.157187	+	0.987569i	$0.449758\pi$
$468$	0	0
$469$	13.5167	0.624141
$470$	0	0
$471$	0	0
$472$	1.12603	0.0518298
$473$	4.89898	0.225255
$474$	0	0
$475$	0	0
$476$	2.75821	0.126422
$477$	0	0
$478$	−49.6046	−2.26886
$479$	−4.62117	−0.211147	−0.105573	−	0.994412i	$-0.533668\pi$
$479$	−4.62117	−0.211147	−0.105573	+	0.994412i	$0.533668\pi$
$480$	0	0
$481$	−21.1244	−0.963188
$482$	−0.757875	−0.0345202
$483$	0	0
$484$	−14.6603	−0.666375
$485$	0	0
$486$	0	0
$487$	−24.0076	−1.08789	−0.543944	−	0.839122i	$-0.683070\pi$
$487$	−24.0076	−1.08789	−0.543944	+	0.839122i	$0.683070\pi$
$488$	−5.65685	−0.256074
$489$	0	0
$490$	0	0
$491$	−0.853392	−0.0385130	−0.0192565	−	0.999815i	$-0.506130\pi$
$491$	−0.853392	−0.0385130	−0.0192565	+	0.999815i	$0.506130\pi$
$492$	0	0
$493$	1.42775	0.0643027
$494$	−16.2369	−0.730531
$495$	0	0
$496$	−4.46410	−0.200444
$497$	7.82894	0.351176
$498$	0	0
$499$	−23.7128	−1.06153	−0.530766	−	0.847519i	$-0.678096\pi$
$499$	−23.7128	−1.06153	−0.530766	+	0.847519i	$0.678096\pi$
$500$	0	0
$501$	0	0
$502$	−30.7638	−1.37305
$503$	−11.6926	−0.521349	−0.260675	−	0.965427i	$-0.583945\pi$
$503$	−11.6926	−0.521349	−0.260675	+	0.965427i	$0.583945\pi$
$504$	0	0
$505$	0	0
$506$	−4.35066	−0.193410
$507$	0	0
$508$	−29.1152	−1.29178
$509$	36.2415	1.60638	0.803188	−	0.595725i	$-0.203135\pi$
$509$	36.2415	1.60638	0.803188	+	0.595725i	$0.203135\pi$
$510$	0	0
$511$	5.66025	0.250395
$512$	−29.2552	−1.29291
$513$	0	0
$514$	24.0526	1.06091
$515$	0	0
$516$	0	0
$517$	−12.9090	−0.567735
$518$	−10.9348	−0.480446
$519$	0	0
$520$	0	0
$521$	1.59245	0.0697665	0.0348833	−	0.999391i	$-0.488894\pi$
$521$	1.59245	0.0697665	0.0348833	+	0.999391i	$0.488894\pi$
$522$	0	0
$523$	17.4131	0.761421	0.380710	−	0.924694i	$-0.375679\pi$
$523$	17.4131	0.761421	0.380710	+	0.924694i	$0.375679\pi$
$524$	−13.3225	−0.581995
$525$	0	0
$526$	35.8564	1.56341
$527$	1.41421	0.0616041
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	0	0
$532$	−3.90069	−0.169116
$533$	−24.9754	−1.08180
$534$	0	0
$535$	0	0
$536$	6.21362	0.268388
$537$	0	0
$538$	47.8751	2.06404
$539$	9.12801	0.393171
$540$	0	0
$541$	30.9808	1.33197	0.665983	−	0.745966i	$-0.268012\pi$
$541$	30.9808	1.33197	0.665983	+	0.745966i	$0.268012\pi$
$542$	−39.1175	−1.68024
$543$	0	0
$544$	10.7321	0.460133
$545$	0	0
$546$	0	0
$547$	−21.0121	−0.898410	−0.449205	−	0.893429i	$-0.648293\pi$
$547$	−21.0121	−0.898410	−0.449205	+	0.893429i	$0.648293\pi$
$548$	37.2228	1.59008
$549$	0	0
$550$	0	0
$551$	−2.01915	−0.0860185
$552$	0	0
$553$	−11.4812	−0.488230
$554$	−12.0424	−0.511632
$555$	0	0
$556$	−19.8564	−0.842099
$557$	−1.21114	−0.0513177	−0.0256589	−	0.999671i	$-0.508168\pi$
$557$	−1.21114	−0.0513177	−0.0256589	+	0.999671i	$0.508168\pi$
$558$	0	0
$559$	12.9282	0.546805
$560$	0	0
$561$	0	0
$562$	18.4583	0.778615
$563$	−30.8081	−1.29841	−0.649203	−	0.760615i	$-0.724897\pi$
$563$	−30.8081	−1.29841	−0.649203	+	0.760615i	$0.724897\pi$
$564$	0	0
$565$	0	0
$566$	55.2366	2.32177
$567$	0	0
$568$	3.59897	0.151009
$569$	−30.2984	−1.27018	−0.635088	−	0.772440i	$-0.719036\pi$
$569$	−30.2984	−1.27018	−0.635088	+	0.772440i	$0.719036\pi$
$570$	0	0
$571$	8.92820	0.373634	0.186817	−	0.982395i	$-0.440183\pi$
$571$	8.92820	0.373634	0.186817	+	0.982395i	$0.440183\pi$
$572$	11.5911	0.484649
$573$	0	0
$574$	−12.9282	−0.539613
$575$	0	0
$576$	0	0
$577$	−24.9936	−1.04050	−0.520248	−	0.854015i	$-0.674161\pi$
$577$	−24.9936	−1.04050	−0.520248	+	0.854015i	$0.674161\pi$
$578$	28.9778	1.20532
$579$	0	0
$580$	0	0
$581$	−1.59245	−0.0660660
$582$	0	0
$583$	−8.40482	−0.348092
$584$	2.60202	0.107673
$585$	0	0
$586$	−24.0526	−0.993602
$587$	−19.3185	−0.797361	−0.398680	−	0.917090i	$-0.630532\pi$
$587$	−19.3185	−0.797361	−0.398680	+	0.917090i	$0.630532\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	0	0
$592$	−22.4398	−0.922270
$593$	−3.03150	−0.124489	−0.0622444	−	0.998061i	$-0.519826\pi$
$593$	−3.03150	−0.124489	−0.0622444	+	0.998061i	$0.519826\pi$
$594$	0	0
$595$	0	0
$596$	19.8485	0.813025
$597$	0	0
$598$	−11.4812	−0.469501
$599$	−29.5593	−1.20776	−0.603881	−	0.797074i	$-0.706380\pi$
$599$	−29.5593	−1.20776	−0.603881	+	0.797074i	$0.706380\pi$
$600$	0	0
$601$	10.7846	0.439913	0.219957	−	0.975510i	$-0.429408\pi$
$601$	10.7846	0.439913	0.219957	+	0.975510i	$0.429408\pi$
$602$	6.69213	0.272751
$603$	0	0
$604$	−21.1244	−0.859538
$605$	0	0
$606$	0	0
$607$	−49.0820	−1.99218	−0.996089	−	0.0883571i	$-0.971838\pi$
$607$	−49.0820	−1.99218	−0.996089	+	0.0883571i	$0.971838\pi$
$608$	−15.1774	−0.615525
$609$	0	0
$610$	0	0
$611$	−34.0662	−1.37817
$612$	0	0
$613$	47.8751	1.93366	0.966829	−	0.255423i	$-0.0822148\pi$
$613$	47.8751	1.93366	0.966829	+	0.255423i	$0.0822148\pi$
$614$	38.9997	1.57390
$615$	0	0
$616$	−0.928203	−0.0373984
$617$	−30.4292	−1.22503	−0.612516	−	0.790458i	$-0.709842\pi$
$617$	−30.4292	−1.22503	−0.612516	+	0.790458i	$0.709842\pi$
$618$	0	0
$619$	41.5167	1.66870	0.834348	−	0.551238i	$-0.185845\pi$
$619$	41.5167	1.66870	0.834348	+	0.551238i	$0.185845\pi$
$620$	0	0
$621$	0	0
$622$	57.7077	2.31387
$623$	−12.7279	−0.509933
$624$	0	0
$625$	0	0
$626$	−41.7579	−1.66898
$627$	0	0
$628$	20.9312	0.835246
$629$	7.10886	0.283449
$630$	0	0
$631$	−11.6077	−0.462095	−0.231048	−	0.972942i	$-0.574215\pi$
$631$	−11.6077	−0.462095	−0.231048	+	0.972942i	$0.574215\pi$
$632$	−5.27792	−0.209944
$633$	0	0
$634$	52.4449	2.08285
$635$	0	0
$636$	0	0
$637$	24.0884	0.954418
$638$	3.10583	0.122961
$639$	0	0
$640$	0	0
$641$	46.8476	1.85037	0.925185	−	0.379516i	$-0.123909\pi$
$641$	46.8476	1.85037	0.925185	+	0.379516i	$0.123909\pi$
$642$	0	0
$643$	45.4830	1.79368	0.896838	−	0.442359i	$-0.145858\pi$
$643$	45.4830	1.79368	0.896838	+	0.442359i	$0.145858\pi$
$644$	−2.75821	−0.108689
$645$	0	0
$646$	5.46410	0.214982
$647$	−33.1833	−1.30457	−0.652284	−	0.757975i	$-0.726189\pi$
$647$	−33.1833	−1.30457	−0.652284	+	0.757975i	$0.726189\pi$
$648$	0	0
$649$	−3.46410	−0.135978
$650$	0	0
$651$	0	0
$652$	−16.8905	−0.661483
$653$	−25.0769	−0.981335	−0.490668	−	0.871347i	$-0.663247\pi$
$653$	−25.0769	−0.981335	−0.490668	+	0.871347i	$0.663247\pi$
$654$	0	0
$655$	0	0
$656$	−26.5306	−1.03585
$657$	0	0
$658$	−17.6340	−0.687443
$659$	−7.26504	−0.283006	−0.141503	−	0.989938i	$-0.545193\pi$
$659$	−7.26504	−0.283006	−0.141503	+	0.989938i	$0.545193\pi$
$660$	0	0
$661$	−41.1769	−1.60160	−0.800798	−	0.598934i	$-0.795591\pi$
$661$	−41.1769	−1.60160	−0.800798	+	0.598934i	$0.795591\pi$
$662$	−28.1827	−1.09535
$663$	0	0
$664$	−0.732051	−0.0284091
$665$	0	0
$666$	0	0
$667$	−1.42775	−0.0552828
$668$	−14.2165	−0.550052
$669$	0	0
$670$	0	0
$671$	17.4026	0.671821
$672$	0	0
$673$	12.1655	0.468945	0.234472	−	0.972123i	$-0.424664\pi$
$673$	12.1655	0.468945	0.234472	+	0.972123i	$0.424664\pi$
$674$	1.00957	0.0388873
$675$	0	0
$676$	8.07180	0.310454
$677$	−43.1571	−1.65866	−0.829331	−	0.558758i	$-0.811278\pi$
$677$	−43.1571	−1.65866	−0.829331	+	0.558758i	$0.811278\pi$
$678$	0	0
$679$	1.60770	0.0616977
$680$	0	0
$681$	0	0
$682$	3.07638	0.117801
$683$	6.59059	0.252182	0.126091	−	0.992019i	$-0.459757\pi$
$683$	6.59059	0.252182	0.126091	+	0.992019i	$0.459757\pi$
$684$	0	0
$685$	0	0
$686$	27.6964	1.05745
$687$	0	0
$688$	13.7333	0.523576
$689$	−22.1800	−0.844990
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−18.2832	−0.695025
$693$	0	0
$694$	−28.9282	−1.09810
$695$	0	0
$696$	0	0
$697$	8.40482	0.318355
$698$	49.5718	1.87632
$699$	0	0
$700$	0	0
$701$	38.7292	1.46278	0.731391	−	0.681958i	$-0.238872\pi$
$701$	38.7292	1.46278	0.731391	+	0.681958i	$0.238872\pi$
$702$	0	0
$703$	−10.0534	−0.379173
$704$	9.12801	0.344025
$705$	0	0
$706$	31.1244	1.17138
$707$	−5.37945	−0.202315
$708$	0	0
$709$	−32.2487	−1.21113	−0.605563	−	0.795797i	$-0.707052\pi$
$709$	−32.2487	−1.21113	−0.605563	+	0.795797i	$0.707052\pi$
$710$	0	0
$711$	0	0
$712$	−5.85104	−0.219277
$713$	−1.41421	−0.0529627
$714$	0	0
$715$	0	0
$716$	18.5684	0.693933
$717$	0	0
$718$	−54.0279	−2.01630
$719$	1.90481	0.0710376	0.0355188	−	0.999369i	$-0.488692\pi$
$719$	1.90481	0.0710376	0.0355188	+	0.999369i	$0.488692\pi$
$720$	0	0
$721$	−0.339746	−0.0126528
$722$	28.9778	1.07844
$723$	0	0
$724$	5.07180	0.188492
$725$	0	0
$726$	0	0
$727$	−4.20241	−0.155859	−0.0779294	−	0.996959i	$-0.524831\pi$
$727$	−4.20241	−0.155859	−0.0779294	+	0.996959i	$0.524831\pi$
$728$	−2.44949	−0.0907841
$729$	0	0
$730$	0	0
$731$	−4.35066	−0.160915
$732$	0	0
$733$	49.2220	1.81806	0.909029	−	0.416733i	$-0.136825\pi$
$733$	49.2220	1.81806	0.909029	+	0.416733i	$0.136825\pi$
$734$	−62.6160	−2.31120
$735$	0	0
$736$	−10.7321	−0.395589
$737$	−19.1154	−0.704126
$738$	0	0
$739$	−11.8038	−0.434212	−0.217106	−	0.976148i	$-0.569662\pi$
$739$	−11.8038	−0.434212	−0.217106	+	0.976148i	$0.569662\pi$
$740$	0	0
$741$	0	0
$742$	−11.4812	−0.421488
$743$	−42.7038	−1.56665	−0.783325	−	0.621612i	$-0.786478\pi$
$743$	−42.7038	−1.56665	−0.783325	+	0.621612i	$0.786478\pi$
$744$	0	0
$745$	0	0
$746$	35.6586	1.30556
$747$	0	0
$748$	−3.90069	−0.142623
$749$	7.10886	0.259752
$750$	0	0
$751$	−2.39230	−0.0872964	−0.0436482	−	0.999047i	$-0.513898\pi$
$751$	−2.39230	−0.0872964	−0.0436482	+	0.999047i	$0.513898\pi$
$752$	−36.1875	−1.31962
$753$	0	0
$754$	8.19615	0.298486
$755$	0	0
$756$	0	0
$757$	9.91342	0.360309	0.180155	−	0.983638i	$-0.442340\pi$
$757$	9.91342	0.360309	0.180155	+	0.983638i	$0.442340\pi$
$758$	−22.9048	−0.831940
$759$	0	0
$760$	0	0
$761$	18.4122	0.667442	0.333721	−	0.942672i	$-0.391696\pi$
$761$	18.4122	0.667442	0.333721	+	0.942672i	$0.391696\pi$
$762$	0	0
$763$	0.220874	0.00799618
$764$	−41.4456	−1.49945
$765$	0	0
$766$	−1.66025	−0.0599874
$767$	−9.14162	−0.330085
$768$	0	0
$769$	−24.9808	−0.900829	−0.450415	−	0.892819i	$-0.648724\pi$
$769$	−24.9808	−0.900829	−0.450415	+	0.892819i	$0.648724\pi$
$770$	0	0
$771$	0	0
$772$	−17.0305	−0.612942
$773$	38.7386	1.39333	0.696665	−	0.717397i	$-0.254667\pi$
$773$	38.7386	1.39333	0.696665	+	0.717397i	$0.254667\pi$
$774$	0	0
$775$	0	0
$776$	0.739059	0.0265307
$777$	0	0
$778$	52.6001	1.88581
$779$	−11.8862	−0.425868
$780$	0	0
$781$	−11.0718	−0.396180
$782$	3.86370	0.138166
$783$	0	0
$784$	25.5885	0.913873
$785$	0	0
$786$	0	0
$787$	−12.5264	−0.446517	−0.223259	−	0.974759i	$-0.571669\pi$
$787$	−12.5264	−0.446517	−0.223259	+	0.974759i	$0.571669\pi$
$788$	22.5259	0.802451
$789$	0	0
$790$	0	0
$791$	−6.36980	−0.226484
$792$	0	0
$793$	45.9248	1.63084
$794$	−23.7724	−0.843652
$795$	0	0
$796$	0.339746	0.0120420
$797$	−35.6327	−1.26218	−0.631088	−	0.775711i	$-0.717391\pi$
$797$	−35.6327	−1.26218	−0.631088	+	0.775711i	$0.717391\pi$
$798$	0	0
$799$	11.4641	0.405571
$800$	0	0
$801$	0	0
$802$	3.37810	0.119285
$803$	−8.00481	−0.282484
$804$	0	0
$805$	0	0
$806$	8.11843	0.285960
$807$	0	0
$808$	−2.47294	−0.0869977
$809$	−35.7005	−1.25516	−0.627581	−	0.778551i	$-0.715955\pi$
$809$	−35.7005	−1.25516	−0.627581	+	0.778551i	$0.715955\pi$
$810$	0	0
$811$	−55.0333	−1.93248	−0.966241	−	0.257641i	$-0.917055\pi$
$811$	−55.0333	−1.93248	−0.966241	+	0.257641i	$0.917055\pi$
$812$	1.96902	0.0690989
$813$	0	0
$814$	15.4641	0.542016
$815$	0	0
$816$	0	0
$817$	6.15276	0.215258
$818$	15.4548	0.540365
$819$	0	0
$820$	0	0
$821$	20.8580	0.727951	0.363975	−	0.931409i	$-0.381419\pi$
$821$	20.8580	0.727951	0.363975	+	0.931409i	$0.381419\pi$
$822$	0	0
$823$	−27.2457	−0.949724	−0.474862	−	0.880060i	$-0.657502\pi$
$823$	−27.2457	−0.949724	−0.474862	+	0.880060i	$0.657502\pi$
$824$	−0.156182	−0.00544084
$825$	0	0
$826$	−4.73205	−0.164649
$827$	−19.4944	−0.677886	−0.338943	−	0.940807i	$-0.610069\pi$
$827$	−19.4944	−0.677886	−0.338943	+	0.940807i	$0.610069\pi$
$828$	0	0
$829$	33.3205	1.15727	0.578635	−	0.815587i	$-0.303586\pi$
$829$	33.3205	1.15727	0.578635	+	0.815587i	$0.303586\pi$
$830$	0	0
$831$	0	0
$832$	24.0884	0.835116
$833$	−8.10634	−0.280868
$834$	0	0
$835$	0	0
$836$	5.51641	0.190789
$837$	0	0
$838$	−36.3939	−1.25721
$839$	−15.5397	−0.536489	−0.268244	−	0.963351i	$-0.586443\pi$
$839$	−15.5397	−0.536489	−0.268244	+	0.963351i	$0.586443\pi$
$840$	0	0
$841$	−27.9808	−0.964854
$842$	58.3345	2.01034
$843$	0	0
$844$	−2.53590	−0.0872892
$845$	0	0
$846$	0	0
$847$	−9.53085	−0.327484
$848$	−23.5612	−0.809093
$849$	0	0
$850$	0	0
$851$	−7.10886	−0.243689
$852$	0	0
$853$	−19.8860	−0.680884	−0.340442	−	0.940265i	$-0.610577\pi$
$853$	−19.8860	−0.680884	−0.340442	+	0.940265i	$0.610577\pi$
$854$	23.7724	0.813476
$855$	0	0
$856$	3.26795	0.111696
$857$	12.8295	0.438246	0.219123	−	0.975697i	$-0.429680\pi$
$857$	12.8295	0.438246	0.219123	+	0.975697i	$0.429680\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	0	0
$862$	17.3322	0.590338
$863$	9.04008	0.307728	0.153864	−	0.988092i	$-0.450828\pi$
$863$	9.04008	0.307728	0.153864	+	0.988092i	$0.450828\pi$
$864$	0	0
$865$	0	0
$866$	−29.1327	−0.989968
$867$	0	0
$868$	1.95035	0.0661990
$869$	16.2369	0.550798
$870$	0	0
$871$	−50.4449	−1.70926
$872$	0.101536	0.00343844
$873$	0	0
$874$	−5.46410	−0.184826
$875$	0	0
$876$	0	0
$877$	28.8943	0.975691	0.487845	−	0.872930i	$-0.337783\pi$
$877$	28.8943	0.975691	0.487845	+	0.872930i	$0.337783\pi$
$878$	28.8391	0.973272
$879$	0	0
$880$	0	0
$881$	−0.582877	−0.0196376	−0.00981882	−	0.999952i	$-0.503125\pi$
$881$	−0.582877	−0.0196376	−0.00981882	+	0.999952i	$0.503125\pi$
$882$	0	0
$883$	−12.0846	−0.406680	−0.203340	−	0.979108i	$-0.565180\pi$
$883$	−12.0846	−0.406680	−0.203340	+	0.979108i	$0.565180\pi$
$884$	−10.2938	−0.346217
$885$	0	0
$886$	4.33975	0.145797
$887$	31.7690	1.06670	0.533350	−	0.845895i	$-0.320933\pi$
$887$	31.7690	1.06670	0.533350	+	0.845895i	$0.320933\pi$
$888$	0	0
$889$	−18.9282	−0.634832
$890$	0	0
$891$	0	0
$892$	26.6422	0.892047
$893$	−16.2127	−0.542537
$894$	0	0
$895$	0	0
$896$	−4.62117	−0.154382
$897$	0	0
$898$	3.37810	0.112729
$899$	1.00957	0.0336711
$900$	0	0
$901$	7.46410	0.248665
$902$	18.2832	0.608765
$903$	0	0
$904$	−2.92820	−0.0973906
$905$	0	0
$906$	0	0
$907$	56.6625	1.88145	0.940724	−	0.339173i	$-0.110147\pi$
$907$	56.6625	1.88145	0.940724	+	0.339173i	$0.110147\pi$
$908$	7.82894	0.259813
$909$	0	0
$910$	0	0
$911$	−7.42122	−0.245876	−0.122938	−	0.992414i	$-0.539232\pi$
$911$	−7.42122	−0.245876	−0.122938	+	0.992414i	$0.539232\pi$
$912$	0	0
$913$	2.25207	0.0745325
$914$	53.2174	1.76028
$915$	0	0
$916$	10.1436	0.335154
$917$	−8.66115	−0.286016
$918$	0	0
$919$	2.39230	0.0789149	0.0394574	−	0.999221i	$-0.487437\pi$
$919$	2.39230	0.0789149	0.0394574	+	0.999221i	$0.487437\pi$
$920$	0	0
$921$	0	0
$922$	−31.0655	−1.02309
$923$	−29.2180	−0.961723
$924$	0	0
$925$	0	0
$926$	2.75821	0.0906402
$927$	0	0
$928$	7.66135	0.251496
$929$	−28.5916	−0.938061	−0.469030	−	0.883182i	$-0.655397\pi$
$929$	−28.5916	−0.938061	−0.469030	+	0.883182i	$0.655397\pi$
$930$	0	0
$931$	11.4641	0.375721
$932$	−28.7375	−0.941329
$933$	0	0
$934$	−13.1244	−0.429442
$935$	0	0
$936$	0	0
$937$	14.7785	0.482791	0.241395	−	0.970427i	$-0.422395\pi$
$937$	14.7785	0.482791	0.241395	+	0.970427i	$0.422395\pi$
$938$	−26.1122	−0.852593
$939$	0	0
$940$	0	0
$941$	−43.2360	−1.40945	−0.704727	−	0.709478i	$-0.748931\pi$
$941$	−43.2360	−1.40945	−0.704727	+	0.709478i	$0.748931\pi$
$942$	0	0
$943$	−8.40482	−0.273699
$944$	−9.71088	−0.316062
$945$	0	0
$946$	−9.46410	−0.307704
$947$	−47.2239	−1.53457	−0.767284	−	0.641307i	$-0.778393\pi$
$947$	−47.2239	−1.53457	−0.767284	+	0.641307i	$0.778393\pi$
$948$	0	0
$949$	−21.1244	−0.685726
$950$	0	0
$951$	0	0
$952$	0.824313	0.0267161
$953$	21.2875	0.689571	0.344785	−	0.938682i	$-0.387952\pi$
$953$	21.2875	0.689571	0.344785	+	0.938682i	$0.387952\pi$
$954$	0	0
$955$	0	0
$956$	44.4743	1.43840
$957$	0	0
$958$	8.92741	0.288432
$959$	24.1991	0.781430
$960$	0	0
$961$	1.00000	0.0322581
$962$	40.8091	1.31574
$963$	0	0
$964$	0.679492	0.0218850
$965$	0	0
$966$	0	0
$967$	−33.8402	−1.08823	−0.544113	−	0.839012i	$-0.683134\pi$
$967$	−33.8402	−1.08823	−0.544113	+	0.839012i	$0.683134\pi$
$968$	−4.38134	−0.140822
$969$	0	0
$970$	0	0
$971$	17.9855	0.577182	0.288591	−	0.957452i	$-0.406813\pi$
$971$	17.9855	0.577182	0.288591	+	0.957452i	$0.406813\pi$
$972$	0	0
$973$	−12.9090	−0.413842
$974$	46.3791	1.48608
$975$	0	0
$976$	48.7846	1.56156
$977$	−42.5007	−1.35972	−0.679860	−	0.733342i	$-0.737959\pi$
$977$	−42.5007	−1.35972	−0.679860	+	0.733342i	$0.737959\pi$
$978$	0	0
$979$	18.0000	0.575282
$980$	0	0
$981$	0	0
$982$	1.64863	0.0526098
$983$	−41.2624	−1.31607	−0.658033	−	0.752989i	$-0.728611\pi$
$983$	−41.2624	−1.31607	−0.658033	+	0.752989i	$0.728611\pi$
$984$	0	0
$985$	0	0
$986$	−2.75821	−0.0878391
$987$	0	0
$988$	14.5576	0.463138
$989$	4.35066	0.138343
$990$	0	0
$991$	−25.7128	−0.816794	−0.408397	−	0.912804i	$-0.633912\pi$
$991$	−25.7128	−0.816794	−0.408397	+	0.912804i	$0.633912\pi$
$992$	7.58871	0.240942
$993$	0	0
$994$	−15.1244	−0.479715
$995$	0	0
$996$	0	0
$997$	1.20688	0.0382222	0.0191111	−	0.999817i	$-0.493916\pi$
$997$	1.20688	0.0382222	0.0191111	+	0.999817i	$0.493916\pi$
$998$	45.8096	1.45008
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.ce.1.2		8
3.2	odd	2	inner	6975.2.a.ce.1.8		8
5.2	odd	4		1395.2.c.d.559.1	✓	8
5.3	odd	4		1395.2.c.d.559.7	yes	8
5.4	even	2	inner	6975.2.a.ce.1.7		8
15.2	even	4		1395.2.c.d.559.8	yes	8
15.8	even	4		1395.2.c.d.559.2	yes	8
15.14	odd	2	inner	6975.2.a.ce.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1395.2.c.d.559.1	✓	8	5.2	odd	4
1395.2.c.d.559.2	yes	8	15.8	even	4
1395.2.c.d.559.7	yes	8	5.3	odd	4
1395.2.c.d.559.8	yes	8	15.2	even	4
6975.2.a.ce.1.1		8	15.14	odd	2	inner
6975.2.a.ce.1.2		8	1.1	even	1	trivial
6975.2.a.ce.1.7		8	5.4	even	2	inner
6975.2.a.ce.1.8		8	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q-1.93185 q^{2} +1.73205 q^{4} +1.12603 q^{7} +0.517638 q^{8} -1.59245 q^{11} -4.20241 q^{13} -2.17533 q^{14} -4.46410 q^{16} +1.41421 q^{17} -2.00000 q^{19} +3.07638 q^{22} -1.41421 q^{23} +8.11843 q^{26} +1.95035 q^{28} +1.00957 q^{29} +1.00000 q^{31} +7.58871 q^{32} -2.73205 q^{34} +5.02672 q^{37} +3.86370 q^{38} +5.94311 q^{41} -3.07638 q^{43} -2.75821 q^{44} +2.73205 q^{46} +8.10634 q^{47} -5.73205 q^{49} -7.27879 q^{52} +5.27792 q^{53} +0.582877 q^{56} -1.95035 q^{58} +2.17533 q^{59} -10.9282 q^{61} -1.93185 q^{62} -5.73205 q^{64} +12.0038 q^{67} +2.44949 q^{68} +6.95268 q^{71} +5.02672 q^{73} -9.71088 q^{74} -3.46410 q^{76} -1.79315 q^{77} -10.1962 q^{79} -11.4812 q^{82} -1.41421 q^{83} +5.94311 q^{86} -0.824313 q^{88} -11.3033 q^{89} -4.73205 q^{91} -2.44949 q^{92} -15.6603 q^{94} +1.42775 q^{97} +11.0735 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{16} - 16 q^{19} + 8 q^{31} - 8 q^{34} + 8 q^{46} - 32 q^{49} - 32 q^{61} - 32 q^{64} - 40 q^{79} - 24 q^{91} - 56 q^{94}+O(q^{100}) \) 8 * q - 8 * q^16 - 16 * q^19 + 8 * q^31 - 8 * q^34 + 8 * q^46 - 32 * q^49 - 32 * q^61 - 32 * q^64 - 40 * q^79 - 24 * q^91 - 56 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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