LMFDB - Embedded newform 6975.2.a.ce.1.3

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			1.3
Root			$-0.276907$ 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)

$f(q)$	$=$	$q-0.517638 q^{2} -1.73205 q^{4} -2.17533 q^{7} +1.93185 q^{8} -3.07638 q^{11} +0.582877 q^{13} +1.12603 q^{14} +2.46410 q^{16} -1.41421 q^{17} -2.00000 q^{19} +1.59245 q^{22} +1.41421 q^{23} -0.301719 q^{26} +3.76778 q^{28} +7.27879 q^{29} +1.00000 q^{31} -5.13922 q^{32} +0.732051 q^{34} +5.36023 q^{37} +1.03528 q^{38} +0.824313 q^{41} -1.59245 q^{43} +5.32844 q^{44} -0.732051 q^{46} -3.20736 q^{47} -2.26795 q^{49} -1.00957 q^{52} -0.378937 q^{53} -4.20241 q^{56} -3.76778 q^{58} -1.12603 q^{59} +2.92820 q^{61} -0.517638 q^{62} -2.26795 q^{64} +14.4882 q^{67} +2.44949 q^{68} +8.10310 q^{71} +5.36023 q^{73} -2.77466 q^{74} +3.46410 q^{76} +6.69213 q^{77} +0.196152 q^{79} -0.426696 q^{82} +1.41421 q^{83} +0.824313 q^{86} -5.94311 q^{88} -5.85104 q^{89} -1.26795 q^{91} -2.44949 q^{92} +1.66025 q^{94} -10.2938 q^{97} +1.17398 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.517638	−0.366025	−0.183013	−	0.983111i	$-0.558585\pi$
$2$	−0.517638	−0.366025	−0.183013	+	0.983111i	$0.558585\pi$
$3$	0	0
$3$	0	0
$4$	−1.73205	−0.866025
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.17533	−0.822197	−0.411098	−	0.911591i	$-0.634855\pi$
$7$	−2.17533	−0.822197	−0.411098	+	0.911591i	$0.634855\pi$
$8$	1.93185	0.683013
$9$	0	0
$10$	0	0
$11$	−3.07638	−0.927563	−0.463781	−	0.885950i	$-0.653508\pi$
$11$	−3.07638	−0.927563	−0.463781	+	0.885950i	$0.653508\pi$
$12$	0	0
$13$	0.582877	0.161661	0.0808305	−	0.996728i	$-0.474243\pi$
$13$	0.582877	0.161661	0.0808305	+	0.996728i	$0.474243\pi$
$14$	1.12603	0.300945
$15$	0	0
$16$	2.46410	0.616025
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	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
$22$	1.59245	0.339512
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	0	0
$25$	0	0
$26$	−0.301719	−0.0591721
$27$	0	0
$28$	3.76778	0.712043
$29$	7.27879	1.35164	0.675819	−	0.737068i	$-0.263790\pi$
$29$	7.27879	1.35164	0.675819	+	0.737068i	$0.263790\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−5.13922	−0.908494
$33$	0	0
$34$	0.732051	0.125546
$35$	0	0
$36$	0	0
$37$	5.36023	0.881216	0.440608	−	0.897700i	$-0.354763\pi$
$37$	5.36023	0.881216	0.440608	+	0.897700i	$0.354763\pi$
$38$	1.03528	0.167944
$39$	0	0
$40$	0	0
$41$	0.824313	0.128736	0.0643680	−	0.997926i	$-0.479497\pi$
$41$	0.824313	0.128736	0.0643680	+	0.997926i	$0.479497\pi$
$42$	0	0
$43$	−1.59245	−0.242846	−0.121423	−	0.992601i	$-0.538746\pi$
$43$	−1.59245	−0.242846	−0.121423	+	0.992601i	$0.538746\pi$
$44$	5.32844	0.803293
$45$	0	0
$46$	−0.732051	−0.107935
$47$	−3.20736	−0.467842	−0.233921	−	0.972256i	$-0.575156\pi$
$47$	−3.20736	−0.467842	−0.233921	+	0.972256i	$0.575156\pi$
$48$	0	0
$49$	−2.26795	−0.323993
$50$	0	0
$51$	0	0
$52$	−1.00957	−0.140003
$53$	−0.378937	−0.0520511	−0.0260255	−	0.999661i	$-0.508285\pi$
$53$	−0.378937	−0.0520511	−0.0260255	+	0.999661i	$0.508285\pi$
$54$	0	0
$55$	0	0
$56$	−4.20241	−0.561571
$57$	0	0
$58$	−3.76778	−0.494734
$59$	−1.12603	−0.146597	−0.0732985	−	0.997310i	$-0.523353\pi$
$59$	−1.12603	−0.146597	−0.0732985	+	0.997310i	$0.523353\pi$
$60$	0	0
$61$	2.92820	0.374918	0.187459	−	0.982272i	$-0.439975\pi$
$61$	2.92820	0.374918	0.187459	+	0.982272i	$0.439975\pi$
$62$	−0.517638	−0.0657401
$63$	0	0
$64$	−2.26795	−0.283494
$65$	0	0
$66$	0	0
$67$	14.4882	1.77002	0.885010	−	0.465572i	$-0.154151\pi$
$67$	14.4882	1.77002	0.885010	+	0.465572i	$0.154151\pi$
$68$	2.44949	0.297044
$69$	0	0
$70$	0	0
$71$	8.10310	0.961661	0.480831	−	0.876814i	$-0.340335\pi$
$71$	8.10310	0.961661	0.480831	+	0.876814i	$0.340335\pi$
$72$	0	0
$73$	5.36023	0.627367	0.313684	−	0.949528i	$-0.398437\pi$
$73$	5.36023	0.627367	0.313684	+	0.949528i	$0.398437\pi$
$74$	−2.77466	−0.322547
$75$	0	0
$76$	3.46410	0.397360
$77$	6.69213	0.762639
$78$	0	0
$79$	0.196152	0.0220689	0.0110344	−	0.999939i	$-0.496488\pi$
$79$	0.196152	0.0220689	0.0110344	+	0.999939i	$0.496488\pi$
$80$	0	0
$81$	0	0
$82$	−0.426696	−0.0471207
$83$	1.41421	0.155230	0.0776151	−	0.996983i	$-0.475269\pi$
$83$	1.41421	0.155230	0.0776151	+	0.996983i	$0.475269\pi$
$84$	0	0
$85$	0	0
$86$	0.824313	0.0888880
$87$	0	0
$88$	−5.94311	−0.633537
$89$	−5.85104	−0.620209	−0.310104	−	0.950703i	$-0.600364\pi$
$89$	−5.85104	−0.620209	−0.310104	+	0.950703i	$0.600364\pi$
$90$	0	0
$91$	−1.26795	−0.132917
$92$	−2.44949	−0.255377
$93$	0	0
$94$	1.66025	0.171242
$95$	0	0
$96$	0	0
$97$	−10.2938	−1.04517	−0.522587	−	0.852586i	$-0.675033\pi$
$97$	−10.2938	−1.04517	−0.522587	+	0.852586i	$0.675033\pi$
$98$	1.17398	0.118590
$99$	0	0
$100$	0	0
$101$	−9.22913	−0.918333	−0.459167	−	0.888350i	$-0.651852\pi$
$101$	−9.22913	−0.918333	−0.459167	+	0.888350i	$0.651852\pi$
$102$	0	0
$103$	8.11843	0.799933	0.399967	−	0.916530i	$-0.369022\pi$
$103$	8.11843	0.799933	0.399967	+	0.916530i	$0.369022\pi$
$104$	1.12603	0.110417
$105$	0	0
$106$	0.196152	0.0190520
$107$	3.48477	0.336885	0.168443	−	0.985711i	$-0.446126\pi$
$107$	3.48477	0.336885	0.168443	+	0.985711i	$0.446126\pi$
$108$	0	0
$109$	−10.1962	−0.976614	−0.488307	−	0.872672i	$-0.662385\pi$
$109$	−10.1962	−0.976614	−0.488307	+	0.872672i	$0.662385\pi$
$110$	0	0
$111$	0	0
$112$	−5.36023	−0.506494
$113$	5.65685	0.532152	0.266076	−	0.963952i	$-0.414273\pi$
$113$	5.65685	0.532152	0.266076	+	0.963952i	$0.414273\pi$
$114$	0	0
$115$	0	0
$116$	−12.6072	−1.17055
$117$	0	0
$118$	0.582877	0.0536582
$119$	3.07638	0.282011
$120$	0	0
$121$	−1.53590	−0.139627
$122$	−1.51575	−0.137230
$123$	0	0
$124$	−1.73205	−0.155543
$125$	0	0
$126$	0	0
$127$	2.33151	0.206888	0.103444	−	0.994635i	$-0.467014\pi$
$127$	2.33151	0.206888	0.103444	+	0.994635i	$0.467014\pi$
$128$	11.4524	1.01226
$129$	0	0
$130$	0	0
$131$	11.7829	1.02948	0.514739	−	0.857347i	$-0.327889\pi$
$131$	11.7829	1.02948	0.514739	+	0.857347i	$0.327889\pi$
$132$	0	0
$133$	4.35066	0.377250
$134$	−7.49966	−0.647872
$135$	0	0
$136$	−2.73205	−0.234271
$137$	−6.79367	−0.580422	−0.290211	−	0.956963i	$-0.593726\pi$
$137$	−6.79367	−0.580422	−0.290211	+	0.956963i	$0.593726\pi$
$138$	0	0
$139$	−4.53590	−0.384730	−0.192365	−	0.981323i	$-0.561616\pi$
$139$	−4.53590	−0.384730	−0.192365	+	0.981323i	$0.561616\pi$
$140$	0	0
$141$	0	0
$142$	−4.19447	−0.351992
$143$	−1.79315	−0.149951
$144$	0	0
$145$	0	0
$146$	−2.77466	−0.229632
$147$	0	0
$148$	−9.28419	−0.763156
$149$	−9.83257	−0.805516	−0.402758	−	0.915307i	$-0.631948\pi$
$149$	−9.83257	−0.805516	−0.402758	+	0.915307i	$0.631948\pi$
$150$	0	0
$151$	−1.80385	−0.146795	−0.0733975	−	0.997303i	$-0.523384\pi$
$151$	−1.80385	−0.146795	−0.0733975	+	0.997303i	$0.523384\pi$
$152$	−3.86370	−0.313388
$153$	0	0
$154$	−3.46410	−0.279145
$155$	0	0
$156$	0	0
$157$	−15.8102	−1.26179	−0.630895	−	0.775869i	$-0.717312\pi$
$157$	−15.8102	−1.26179	−0.630895	+	0.775869i	$0.717312\pi$
$158$	−0.101536	−0.00807777
$159$	0	0
$160$	0	0
$161$	−3.07638	−0.242453
$162$	0	0
$163$	−18.8389	−1.47558	−0.737788	−	0.675033i	$-0.764130\pi$
$163$	−18.8389	−1.47558	−0.737788	+	0.675033i	$0.764130\pi$
$164$	−1.42775	−0.111489
$165$	0	0
$166$	−0.732051	−0.0568182
$167$	22.9048	1.77243	0.886214	−	0.463276i	$-0.153326\pi$
$167$	22.9048	1.77243	0.886214	+	0.463276i	$0.153326\pi$
$168$	0	0
$169$	−12.6603	−0.973866
$170$	0	0
$171$	0	0
$172$	2.75821	0.210311
$173$	0.757875	0.0576202	0.0288101	−	0.999585i	$-0.490828\pi$
$173$	0.757875	0.0576202	0.0288101	+	0.999585i	$0.490828\pi$
$174$	0	0
$175$	0	0
$176$	−7.58051	−0.571402
$177$	0	0
$178$	3.02872	0.227012
$179$	10.0534	0.751430	0.375715	−	0.926735i	$-0.377397\pi$
$179$	10.0534	0.751430	0.375715	+	0.926735i	$0.377397\pi$
$180$	0	0
$181$	−10.9282	−0.812287	−0.406143	−	0.913809i	$-0.633127\pi$
$181$	−10.9282	−0.812287	−0.406143	+	0.913809i	$0.633127\pi$
$182$	0.656339	0.0486511
$183$	0	0
$184$	2.73205	0.201409
$185$	0	0
$186$	0	0
$187$	4.35066	0.318151
$188$	5.55532	0.405163
$189$	0	0
$190$	0	0
$191$	12.3864	0.896245	0.448123	−	0.893972i	$-0.352093\pi$
$191$	12.3864	0.896245	0.448123	+	0.893972i	$0.352093\pi$
$192$	0	0
$193$	11.4595	0.824874	0.412437	−	0.910986i	$-0.364678\pi$
$193$	11.4595	0.824874	0.412437	+	0.910986i	$0.364678\pi$
$194$	5.32844	0.382560
$195$	0	0
$196$	3.92820	0.280586
$197$	1.69161	0.120523	0.0602613	−	0.998183i	$-0.480807\pi$
$197$	1.69161	0.120523	0.0602613	+	0.998183i	$0.480807\pi$
$198$	0	0
$199$	−10.1962	−0.722786	−0.361393	−	0.932414i	$-0.617699\pi$
$199$	−10.1962	−0.722786	−0.361393	+	0.932414i	$0.617699\pi$
$200$	0	0
$201$	0	0
$202$	4.77735	0.336133
$203$	−15.8338	−1.11131
$204$	0	0
$205$	0	0
$206$	−4.20241	−0.292796
$207$	0	0
$208$	1.43627	0.0995873
$209$	6.15276	0.425595
$210$	0	0
$211$	5.46410	0.376164	0.188082	−	0.982153i	$-0.439773\pi$
$211$	5.46410	0.376164	0.188082	+	0.982153i	$0.439773\pi$
$212$	0.656339	0.0450775
$213$	0	0
$214$	−1.80385	−0.123308
$215$	0	0
$216$	0	0
$217$	−2.17533	−0.147671
$218$	5.27792	0.357466
$219$	0	0
$220$	0	0
$221$	−0.824313	−0.0554493
$222$	0	0
$223$	7.96225	0.533192	0.266596	−	0.963808i	$-0.414101\pi$
$223$	7.96225	0.533192	0.266596	+	0.963808i	$0.414101\pi$
$224$	11.1795	0.746960
$225$	0	0
$226$	−2.92820	−0.194781
$227$	10.1769	0.675464	0.337732	−	0.941242i	$-0.390340\pi$
$227$	10.1769	0.675464	0.337732	+	0.941242i	$0.390340\pi$
$228$	0	0
$229$	−21.8564	−1.44431	−0.722156	−	0.691730i	$-0.756849\pi$
$229$	−21.8564	−1.44431	−0.722156	+	0.691730i	$0.756849\pi$
$230$	0	0
$231$	0	0
$232$	14.0615	0.923185
$233$	11.6926	0.766011	0.383005	−	0.923746i	$-0.374889\pi$
$233$	11.6926	0.766011	0.383005	+	0.923746i	$0.374889\pi$
$234$	0	0
$235$	0	0
$236$	1.95035	0.126957
$237$	0	0
$238$	−1.59245	−0.103223
$239$	−24.9936	−1.61670	−0.808350	−	0.588702i	$-0.799639\pi$
$239$	−24.9936	−1.61670	−0.808350	+	0.588702i	$0.799639\pi$
$240$	0	0
$241$	−20.3923	−1.31358	−0.656792	−	0.754072i	$-0.728087\pi$
$241$	−20.3923	−1.31358	−0.656792	+	0.754072i	$0.728087\pi$
$242$	0.795040	0.0511071
$243$	0	0
$244$	−5.07180	−0.324689
$245$	0	0
$246$	0	0
$247$	−1.16575	−0.0741752
$248$	1.93185	0.122673
$249$	0	0
$250$	0	0
$251$	30.7638	1.94179	0.970896	−	0.239500i	$-0.0769836\pi$
$251$	30.7638	1.94179	0.970896	+	0.239500i	$0.0769836\pi$
$252$	0	0
$253$	−4.35066	−0.273523
$254$	−1.20688	−0.0757263
$255$	0	0
$256$	−1.39230	−0.0870191
$257$	27.1475	1.69341	0.846706	−	0.532061i	$-0.178582\pi$
$257$	27.1475	1.69341	0.846706	+	0.532061i	$0.178582\pi$
$258$	0	0
$259$	−11.6603	−0.724533
$260$	0	0
$261$	0	0
$262$	−6.09929	−0.376815
$263$	−15.7322	−0.970090	−0.485045	−	0.874489i	$-0.661197\pi$
$263$	−15.7322	−0.970090	−0.485045	+	0.874489i	$0.661197\pi$
$264$	0	0
$265$	0	0
$266$	−2.25207	−0.138083
$267$	0	0
$268$	−25.0944	−1.53288
$269$	−10.5760	−0.644833	−0.322416	−	0.946598i	$-0.604495\pi$
$269$	−10.5760	−0.644833	−0.322416	+	0.946598i	$0.604495\pi$
$270$	0	0
$271$	−28.2487	−1.71599	−0.857994	−	0.513661i	$-0.828289\pi$
$271$	−28.2487	−1.71599	−0.857994	+	0.513661i	$0.828289\pi$
$272$	−3.48477	−0.211295
$273$	0	0
$274$	3.51666	0.212449
$275$	0	0
$276$	0	0
$277$	−27.1135	−1.62909	−0.814546	−	0.580098i	$-0.803014\pi$
$277$	−27.1135	−1.62909	−0.814546	+	0.580098i	$0.803014\pi$
$278$	2.34795	0.140821
$279$	0	0
$280$	0	0
$281$	−18.4583	−1.10113	−0.550564	−	0.834793i	$-0.685587\pi$
$281$	−18.4583	−1.10113	−0.550564	+	0.834793i	$0.685587\pi$
$282$	0	0
$283$	10.0232	0.595820	0.297910	−	0.954594i	$-0.403710\pi$
$283$	10.0232	0.595820	0.297910	+	0.954594i	$0.403710\pi$
$284$	−14.0350	−0.832823
$285$	0	0
$286$	0.928203	0.0548858
$287$	−1.79315	−0.105846
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0	0
$291$	0	0
$292$	−9.28419	−0.543316
$293$	−27.1475	−1.58597	−0.792986	−	0.609240i	$-0.791475\pi$
$293$	−27.1475	−1.58597	−0.792986	+	0.609240i	$0.791475\pi$
$294$	0	0
$295$	0	0
$296$	10.3552	0.601882
$297$	0	0
$298$	5.08971	0.294839
$299$	0.824313	0.0476713
$300$	0	0
$301$	3.46410	0.199667
$302$	0.933740	0.0537307
$303$	0	0
$304$	−4.92820	−0.282652
$305$	0	0
$306$	0	0
$307$	8.85749	0.505524	0.252762	−	0.967529i	$-0.418661\pi$
$307$	8.85749	0.505524	0.252762	+	0.967529i	$0.418661\pi$
$308$	−11.5911	−0.660465
$309$	0	0
$310$	0	0
$311$	11.5620	0.655623	0.327812	−	0.944743i	$-0.393689\pi$
$311$	11.5620	0.655623	0.327812	+	0.944743i	$0.393689\pi$
$312$	0	0
$313$	−19.1513	−1.08249	−0.541246	−	0.840864i	$-0.682047\pi$
$313$	−19.1513	−1.08249	−0.541246	+	0.840864i	$0.682047\pi$
$314$	8.18395	0.461847
$315$	0	0
$316$	−0.339746	−0.0191122
$317$	12.4505	0.699291	0.349645	−	0.936882i	$-0.386302\pi$
$317$	12.4505	0.699291	0.349645	+	0.936882i	$0.386302\pi$
$318$	0	0
$319$	−22.3923	−1.25373
$320$	0	0
$321$	0	0
$322$	1.59245	0.0887438
$323$	2.82843	0.157378
$324$	0	0
$325$	0	0
$326$	9.75173	0.540098
$327$	0	0
$328$	1.59245	0.0879284
$329$	6.97707	0.384658
$330$	0	0
$331$	−16.5885	−0.911784	−0.455892	−	0.890035i	$-0.650680\pi$
$331$	−16.5885	−0.911784	−0.455892	+	0.890035i	$0.650680\pi$
$332$	−2.44949	−0.134433
$333$	0	0
$334$	−11.8564	−0.648754
$335$	0	0
$336$	0	0
$337$	−14.0615	−0.765981	−0.382990	−	0.923752i	$-0.625106\pi$
$337$	−14.0615	−0.765981	−0.382990	+	0.923752i	$0.625106\pi$
$338$	6.55343	0.356460
$339$	0	0
$340$	0	0
$341$	−3.07638	−0.166595
$342$	0	0
$343$	20.1608	1.08858
$344$	−3.07638	−0.165867
$345$	0	0
$346$	−0.392305	−0.0210904
$347$	29.1165	1.56305	0.781527	−	0.623871i	$-0.214441\pi$
$347$	29.1165	1.56305	0.781527	+	0.623871i	$0.214441\pi$
$348$	0	0
$349$	−8.33975	−0.446416	−0.223208	−	0.974771i	$-0.571653\pi$
$349$	−8.33975	−0.446416	−0.223208	+	0.974771i	$0.571653\pi$
$350$	0	0
$351$	0	0
$352$	15.8102	0.842685
$353$	−13.2827	−0.706968	−0.353484	−	0.935441i	$-0.615003\pi$
$353$	−13.2827	−0.706968	−0.353484	+	0.935441i	$0.615003\pi$
$354$	0	0
$355$	0	0
$356$	10.1343	0.537116
$357$	0	0
$358$	−5.20405	−0.275042
$359$	16.7288	0.882912	0.441456	−	0.897283i	$-0.354462\pi$
$359$	16.7288	0.882912	0.441456	+	0.897283i	$0.354462\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	5.65685	0.297318
$363$	0	0
$364$	2.19615	0.115110
$365$	0	0
$366$	0	0
$367$	27.8107	1.45171	0.725854	−	0.687849i	$-0.241445\pi$
$367$	27.8107	1.45171	0.725854	+	0.687849i	$0.241445\pi$
$368$	3.48477	0.181656
$369$	0	0
$370$	0	0
$371$	0.824313	0.0427962
$372$	0	0
$373$	−9.55470	−0.494724	−0.247362	−	0.968923i	$-0.579564\pi$
$373$	−9.55470	−0.494724	−0.247362	+	0.968923i	$0.579564\pi$
$374$	−2.25207	−0.116452
$375$	0	0
$376$	−6.19615	−0.319542
$377$	4.24264	0.218507
$378$	0	0
$379$	−15.8564	−0.814489	−0.407244	−	0.913319i	$-0.633510\pi$
$379$	−15.8564	−0.814489	−0.407244	+	0.913319i	$0.633510\pi$
$380$	0	0
$381$	0	0
$382$	−6.41165	−0.328049
$383$	−30.2533	−1.54587	−0.772935	−	0.634485i	$-0.781212\pi$
$383$	−30.2533	−1.54587	−0.772935	+	0.634485i	$0.781212\pi$
$384$	0	0
$385$	0	0
$386$	−5.93188	−0.301925
$387$	0	0
$388$	17.8293	0.905146
$389$	−36.6148	−1.85644	−0.928222	−	0.372026i	$-0.878663\pi$
$389$	−36.6148	−1.85644	−0.928222	+	0.372026i	$0.878663\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	−4.38134	−0.221291
$393$	0	0
$394$	−0.875644	−0.0441143
$395$	0	0
$396$	0	0
$397$	6.36980	0.319691	0.159846	−	0.987142i	$-0.448900\pi$
$397$	6.36980	0.319691	0.159846	+	0.987142i	$0.448900\pi$
$398$	5.27792	0.264558
$399$	0	0
$400$	0	0
$401$	12.6072	0.629575	0.314788	−	0.949162i	$-0.398067\pi$
$401$	12.6072	0.629575	0.314788	+	0.949162i	$0.398067\pi$
$402$	0	0
$403$	0.582877	0.0290352
$404$	15.9853	0.795300
$405$	0	0
$406$	8.19615	0.406768
$407$	−16.4901	−0.817383
$408$	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
$412$	−14.0615	−0.692762
$413$	2.44949	0.120532
$414$	0	0
$415$	0	0
$416$	−2.99553	−0.146868
$417$	0	0
$418$	−3.18490	−0.155779
$419$	9.75173	0.476403	0.238202	−	0.971216i	$-0.423442\pi$
$419$	9.75173	0.476403	0.238202	+	0.971216i	$0.423442\pi$
$420$	0	0
$421$	−19.8038	−0.965180	−0.482590	−	0.875846i	$-0.660304\pi$
$421$	−19.8038	−0.965180	−0.482590	+	0.875846i	$0.660304\pi$
$422$	−2.82843	−0.137686
$423$	0	0
$424$	−0.732051	−0.0355515
$425$	0	0
$426$	0	0
$427$	−6.36980	−0.308256
$428$	−6.03579	−0.291751
$429$	0	0
$430$	0	0
$431$	−22.6607	−1.09153	−0.545763	−	0.837939i	$-0.683760\pi$
$431$	−22.6607	−1.09153	−0.545763	+	0.837939i	$0.683760\pi$
$432$	0	0
$433$	16.0807	0.772788	0.386394	−	0.922334i	$-0.373720\pi$
$433$	16.0807	0.772788	0.386394	+	0.922334i	$0.373720\pi$
$434$	1.12603	0.0540513
$435$	0	0
$436$	17.6603	0.845773
$437$	−2.82843	−0.135302
$438$	0	0
$439$	−1.07180	−0.0511541	−0.0255770	−	0.999673i	$-0.508142\pi$
$439$	−1.07180	−0.0511541	−0.0255770	+	0.999673i	$0.508142\pi$
$440$	0	0
$441$	0	0
$442$	0.426696	0.0202958
$443$	−41.8444	−1.98809	−0.994044	−	0.108983i	$-0.965241\pi$
$443$	−41.8444	−1.98809	−0.994044	+	0.108983i	$0.965241\pi$
$444$	0	0
$445$	0	0
$446$	−4.12157	−0.195162
$447$	0	0
$448$	4.93353	0.233088
$449$	12.6072	0.594972	0.297486	−	0.954726i	$-0.403852\pi$
$449$	12.6072	0.594972	0.297486	+	0.954726i	$0.403852\pi$
$450$	0	0
$451$	−2.53590	−0.119411
$452$	−9.79796	−0.460857
$453$	0	0
$454$	−5.26795	−0.247237
$455$	0	0
$456$	0	0
$457$	38.1463	1.78441	0.892205	−	0.451631i	$-0.149158\pi$
$457$	38.1463	1.78441	0.892205	+	0.451631i	$0.149158\pi$
$458$	11.3137	0.528655
$459$	0	0
$460$	0	0
$461$	15.0802	0.702354	0.351177	−	0.936309i	$-0.385782\pi$
$461$	15.0802	0.702354	0.351177	+	0.936309i	$0.385782\pi$
$462$	0	0
$463$	10.2938	0.478392	0.239196	−	0.970971i	$-0.423116\pi$
$463$	10.2938	0.478392	0.239196	+	0.970971i	$0.423116\pi$
$464$	17.9357	0.832643
$465$	0	0
$466$	−6.05256	−0.280379
$467$	−21.4906	−0.994467	−0.497233	−	0.867617i	$-0.665651\pi$
$467$	−21.4906	−0.994467	−0.497233	+	0.867617i	$0.665651\pi$
$468$	0	0
$469$	−31.5167	−1.45530
$470$	0	0
$471$	0	0
$472$	−2.17533	−0.100128
$473$	4.89898	0.225255
$474$	0	0
$475$	0	0
$476$	−5.32844	−0.244229
$477$	0	0
$478$	12.9376	0.591754
$479$	−24.9127	−1.13829	−0.569146	−	0.822236i	$-0.692726\pi$
$479$	−24.9127	−1.13829	−0.569146	+	0.822236i	$0.692726\pi$
$480$	0	0
$481$	3.12436	0.142458
$482$	10.5558	0.480805
$483$	0	0
$484$	2.66025	0.120921
$485$	0	0
$486$	0	0
$487$	−28.9765	−1.31305	−0.656525	−	0.754305i	$-0.727974\pi$
$487$	−28.9765	−1.31305	−0.656525	+	0.754305i	$0.727974\pi$
$488$	5.65685	0.256074
$489$	0	0
$490$	0	0
$491$	−22.9624	−1.03628	−0.518139	−	0.855296i	$-0.673375\pi$
$491$	−22.9624	−1.03628	−0.518139	+	0.855296i	$0.673375\pi$
$492$	0	0
$493$	−10.2938	−0.463608
$494$	0.603439	0.0271500
$495$	0	0
$496$	2.46410	0.110641
$497$	−17.6269	−0.790675
$498$	0	0
$499$	31.7128	1.41966	0.709830	−	0.704373i	$-0.248772\pi$
$499$	31.7128	1.41966	0.709830	+	0.704373i	$0.248772\pi$
$500$	0	0
$501$	0	0
$502$	−15.9245	−0.710745
$503$	16.5916	0.739784	0.369892	−	0.929075i	$-0.379395\pi$
$503$	16.5916	0.739784	0.369892	+	0.929075i	$0.379395\pi$
$504$	0	0
$505$	0	0
$506$	2.25207	0.100117
$507$	0	0
$508$	−4.03829	−0.179170
$509$	0.743468	0.0329536	0.0164768	−	0.999864i	$-0.494755\pi$
$509$	0.743468	0.0329536	0.0164768	+	0.999864i	$0.494755\pi$
$510$	0	0
$511$	−11.6603	−0.515819
$512$	−22.1841	−0.980408
$513$	0	0
$514$	−14.0526	−0.619832
$515$	0	0
$516$	0	0
$517$	9.86707	0.433953
$518$	6.03579	0.265197
$519$	0	0
$520$	0	0
$521$	3.07638	0.134779	0.0673893	−	0.997727i	$-0.478533\pi$
$521$	3.07638	0.134779	0.0673893	+	0.997727i	$0.478533\pi$
$522$	0	0
$523$	−18.5684	−0.811938	−0.405969	−	0.913887i	$-0.633066\pi$
$523$	−18.5684	−0.811938	−0.405969	+	0.913887i	$0.633066\pi$
$524$	−20.4086	−0.891554
$525$	0	0
$526$	8.14359	0.355078
$527$	−1.41421	−0.0616041
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	0	0
$532$	−7.53556	−0.326708
$533$	0.480473	0.0208116
$534$	0	0
$535$	0	0
$536$	27.9891	1.20895
$537$	0	0
$538$	5.47456	0.236025
$539$	6.97707	0.300524
$540$	0	0
$541$	−20.9808	−0.902033	−0.451017	−	0.892516i	$-0.648939\pi$
$541$	−20.9808	−0.902033	−0.451017	+	0.892516i	$0.648939\pi$
$542$	14.6226	0.628095
$543$	0	0
$544$	7.26795	0.311611
$545$	0	0
$546$	0	0
$547$	2.91439	0.124610	0.0623051	−	0.998057i	$-0.480155\pi$
$547$	2.91439	0.124610	0.0623051	+	0.998057i	$0.480155\pi$
$548$	11.7670	0.502660
$549$	0	0
$550$	0	0
$551$	−14.5576	−0.620174
$552$	0	0
$553$	−0.426696	−0.0181450
$554$	14.0350	0.596289
$555$	0	0
$556$	7.85641	0.333186
$557$	−37.9807	−1.60929	−0.804647	−	0.593754i	$-0.797645\pi$
$557$	−37.9807	−1.60929	−0.804647	+	0.593754i	$0.797645\pi$
$558$	0	0
$559$	−0.928203	−0.0392588
$560$	0	0
$561$	0	0
$562$	9.55470	0.403041
$563$	−27.9797	−1.17920	−0.589601	−	0.807695i	$-0.700715\pi$
$563$	−27.9797	−1.17920	−0.589601	+	0.807695i	$0.700715\pi$
$564$	0	0
$565$	0	0
$566$	−5.18841	−0.218085
$567$	0	0
$568$	15.6540	0.656827
$569$	0.0808455	0.00338922	0.00169461	−	0.999999i	$-0.499461\pi$
$569$	0.0808455	0.00338922	0.00169461	+	0.999999i	$0.499461\pi$
$570$	0	0
$571$	−4.92820	−0.206239	−0.103119	−	0.994669i	$-0.532882\pi$
$571$	−4.92820	−0.206239	−0.103119	+	0.994669i	$0.532882\pi$
$572$	3.10583	0.129861
$573$	0	0
$574$	0.928203	0.0387425
$575$	0	0
$576$	0	0
$577$	25.6772	1.06896	0.534479	−	0.845182i	$-0.320508\pi$
$577$	25.6772	1.06896	0.534479	+	0.845182i	$0.320508\pi$
$578$	7.76457	0.322964
$579$	0	0
$580$	0	0
$581$	−3.07638	−0.127630
$582$	0	0
$583$	1.16575	0.0482806
$584$	10.3552	0.428500
$585$	0	0
$586$	14.0526	0.580506
$587$	−5.17638	−0.213652	−0.106826	−	0.994278i	$-0.534069\pi$
$587$	−5.17638	−0.213652	−0.106826	+	0.994278i	$0.534069\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	0	0
$592$	13.2081	0.542852
$593$	42.2233	1.73390	0.866952	−	0.498391i	$-0.166076\pi$
$593$	42.2233	1.73390	0.866952	+	0.498391i	$0.166076\pi$
$594$	0	0
$595$	0	0
$596$	17.0305	0.697597
$597$	0	0
$598$	−0.426696	−0.0174489
$599$	−19.8052	−0.809217	−0.404609	−	0.914490i	$-0.632592\pi$
$599$	−19.8052	−0.809217	−0.404609	+	0.914490i	$0.632592\pi$
$600$	0	0
$601$	−30.7846	−1.25573	−0.627865	−	0.778322i	$-0.716071\pi$
$601$	−30.7846	−1.25573	−0.627865	+	0.778322i	$0.716071\pi$
$602$	−1.79315	−0.0730834
$603$	0	0
$604$	3.12436	0.127128
$605$	0	0
$606$	0	0
$607$	26.9992	1.09586	0.547931	−	0.836523i	$-0.315416\pi$
$607$	26.9992	1.09586	0.547931	+	0.836523i	$0.315416\pi$
$608$	10.2784	0.416845
$609$	0	0
$610$	0	0
$611$	−1.86950	−0.0756319
$612$	0	0
$613$	5.47456	0.221115	0.110558	−	0.993870i	$-0.464736\pi$
$613$	5.47456	0.221115	0.110558	+	0.993870i	$0.464736\pi$
$614$	−4.58498	−0.185035
$615$	0	0
$616$	12.9282	0.520892
$617$	−33.2576	−1.33890	−0.669450	−	0.742857i	$-0.733470\pi$
$617$	−33.2576	−1.33890	−0.669450	+	0.742857i	$0.733470\pi$
$618$	0	0
$619$	−3.51666	−0.141347	−0.0706733	−	0.997500i	$-0.522515\pi$
$619$	−3.51666	−0.141347	−0.0706733	+	0.997500i	$0.522515\pi$
$620$	0	0
$621$	0	0
$622$	−5.98495	−0.239975
$623$	12.7279	0.509933
$624$	0	0
$625$	0	0
$626$	9.91342	0.396220
$627$	0	0
$628$	27.3840	1.09274
$629$	−7.58051	−0.302255
$630$	0	0
$631$	−32.3923	−1.28952	−0.644759	−	0.764386i	$-0.723042\pi$
$631$	−32.3923	−1.28952	−0.644759	+	0.764386i	$0.723042\pi$
$632$	0.378937	0.0150733
$633$	0	0
$634$	−6.44486	−0.255958
$635$	0	0
$636$	0	0
$637$	−1.32194	−0.0523770
$638$	11.5911	0.458896
$639$	0	0
$640$	0	0
$641$	−32.0515	−1.26596	−0.632979	−	0.774169i	$-0.718168\pi$
$641$	−32.0515	−1.26596	−0.632979	+	0.774169i	$0.718168\pi$
$642$	0	0
$643$	−42.6532	−1.68208	−0.841038	−	0.540976i	$-0.818055\pi$
$643$	−42.6532	−1.68208	−0.841038	+	0.540976i	$0.818055\pi$
$644$	5.32844	0.209970
$645$	0	0
$646$	−1.46410	−0.0576043
$647$	23.3853	0.919371	0.459685	−	0.888082i	$-0.347962\pi$
$647$	23.3853	0.919371	0.459685	+	0.888082i	$0.347962\pi$
$648$	0	0
$649$	3.46410	0.135978
$650$	0	0
$651$	0	0
$652$	32.6299	1.27789
$653$	20.1779	0.789623	0.394812	−	0.918762i	$-0.370810\pi$
$653$	20.1779	0.789623	0.394812	+	0.918762i	$0.370810\pi$
$654$	0	0
$655$	0	0
$656$	2.03119	0.0793047
$657$	0	0
$658$	−3.61160	−0.140795
$659$	23.2641	0.906241	0.453121	−	0.891449i	$-0.350311\pi$
$659$	23.2641	0.906241	0.453121	+	0.891449i	$0.350311\pi$
$660$	0	0
$661$	21.1769	0.823687	0.411843	−	0.911255i	$-0.364885\pi$
$661$	21.1769	0.823687	0.411843	+	0.911255i	$0.364885\pi$
$662$	8.58682	0.333736
$663$	0	0
$664$	2.73205	0.106024
$665$	0	0
$666$	0	0
$667$	10.2938	0.398576
$668$	−39.6723	−1.53497
$669$	0	0
$670$	0	0
$671$	−9.00826	−0.347760
$672$	0	0
$673$	−46.1086	−1.77736	−0.888678	−	0.458533i	$-0.848375\pi$
$673$	−46.1086	−1.77736	−0.888678	+	0.458533i	$0.848375\pi$
$674$	7.27879	0.280368
$675$	0	0
$676$	21.9282	0.843392
$677$	−20.5297	−0.789019	−0.394509	−	0.918892i	$-0.629085\pi$
$677$	−20.5297	−0.789019	−0.394509	+	0.918892i	$0.629085\pi$
$678$	0	0
$679$	22.3923	0.859338
$680$	0	0
$681$	0	0
$682$	1.59245	0.0609781
$683$	17.9043	0.685089	0.342545	−	0.939502i	$-0.388711\pi$
$683$	17.9043	0.685089	0.342545	+	0.939502i	$0.388711\pi$
$684$	0	0
$685$	0	0
$686$	−10.4360	−0.398449
$687$	0	0
$688$	−3.92396	−0.149600
$689$	−0.220874	−0.00841463
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−1.31268	−0.0499005
$693$	0	0
$694$	−15.0718	−0.572118
$695$	0	0
$696$	0	0
$697$	−1.16575	−0.0441561
$698$	4.31697	0.163400
$699$	0	0
$700$	0	0
$701$	−31.7498	−1.19917	−0.599586	−	0.800310i	$-0.704668\pi$
$701$	−31.7498	−1.19917	−0.599586	+	0.800310i	$0.704668\pi$
$702$	0	0
$703$	−10.7205	−0.404330
$704$	6.97707	0.262958
$705$	0	0
$706$	6.87564	0.258768
$707$	20.0764	0.755050
$708$	0	0
$709$	16.2487	0.610233	0.305117	−	0.952315i	$-0.401305\pi$
$709$	16.2487	0.610233	0.305117	+	0.952315i	$0.401305\pi$
$710$	0	0
$711$	0	0
$712$	−11.3033	−0.423610
$713$	1.41421	0.0529627
$714$	0	0
$715$	0	0
$716$	−17.4131	−0.650757
$717$	0	0
$718$	−8.65946	−0.323168
$719$	−28.2908	−1.05507	−0.527535	−	0.849533i	$-0.676884\pi$
$719$	−28.2908	−1.05507	−0.527535	+	0.849533i	$0.676884\pi$
$720$	0	0
$721$	−17.6603	−0.657702
$722$	7.76457	0.288967
$723$	0	0
$724$	18.9282	0.703461
$725$	0	0
$726$	0	0
$727$	0.582877	0.0216177	0.0108089	−	0.999942i	$-0.496559\pi$
$727$	0.582877	0.0216177	0.0108089	+	0.999942i	$0.496559\pi$
$728$	−2.44949	−0.0907841
$729$	0	0
$730$	0	0
$731$	2.25207	0.0832956
$732$	0	0
$733$	25.4792	0.941096	0.470548	−	0.882374i	$-0.344056\pi$
$733$	25.4792	0.941096	0.470548	+	0.882374i	$0.344056\pi$
$734$	−14.3959	−0.531362
$735$	0	0
$736$	−7.26795	−0.267900
$737$	−44.5713	−1.64180
$738$	0	0
$739$	−22.1962	−0.816499	−0.408249	−	0.912870i	$-0.633861\pi$
$739$	−22.1962	−0.816499	−0.408249	+	0.912870i	$0.633861\pi$
$740$	0	0
$741$	0	0
$742$	−0.426696	−0.0156645
$743$	28.0069	1.02747	0.513736	−	0.857948i	$-0.328261\pi$
$743$	28.0069	1.02747	0.513736	+	0.857948i	$0.328261\pi$
$744$	0	0
$745$	0	0
$746$	4.94588	0.181081
$747$	0	0
$748$	−7.53556	−0.275527
$749$	−7.58051	−0.276986
$750$	0	0
$751$	18.3923	0.671145	0.335572	−	0.942014i	$-0.391070\pi$
$751$	18.3923	0.671145	0.335572	+	0.942014i	$0.391070\pi$
$752$	−7.90327	−0.288203
$753$	0	0
$754$	−2.19615	−0.0799792
$755$	0	0
$756$	0	0
$757$	−41.7579	−1.51772	−0.758859	−	0.651255i	$-0.774243\pi$
$757$	−41.7579	−1.51772	−0.758859	+	0.651255i	$0.774243\pi$
$758$	8.20788	0.298124
$759$	0	0
$760$	0	0
$761$	−1.72947	−0.0626933	−0.0313466	−	0.999509i	$-0.509980\pi$
$761$	−1.72947	−0.0626933	−0.0313466	+	0.999509i	$0.509980\pi$
$762$	0	0
$763$	22.1800	0.802969
$764$	−21.4538	−0.776171
$765$	0	0
$766$	15.6603	0.565828
$767$	−0.656339	−0.0236990
$768$	0	0
$769$	26.9808	0.972951	0.486476	−	0.873694i	$-0.338282\pi$
$769$	26.9808	0.972951	0.486476	+	0.873694i	$0.338282\pi$
$770$	0	0
$771$	0	0
$772$	−19.8485	−0.714362
$773$	−9.34469	−0.336105	−0.168053	−	0.985778i	$-0.553748\pi$
$773$	−9.34469	−0.336105	−0.168053	+	0.985778i	$0.553748\pi$
$774$	0	0
$775$	0	0
$776$	−19.8860	−0.713867
$777$	0	0
$778$	18.9532	0.679506
$779$	−1.64863	−0.0590682
$780$	0	0
$781$	−24.9282	−0.892001
$782$	1.03528	0.0370214
$783$	0	0
$784$	−5.58846	−0.199588
$785$	0	0
$786$	0	0
$787$	−28.5498	−1.01769	−0.508845	−	0.860858i	$-0.669927\pi$
$787$	−28.5498	−1.01769	−0.508845	+	0.860858i	$0.669927\pi$
$788$	−2.92996	−0.104376
$789$	0	0
$790$	0	0
$791$	−12.3055	−0.437534
$792$	0	0
$793$	1.70678	0.0606096
$794$	−3.29725	−0.117015
$795$	0	0
$796$	17.6603	0.625951
$797$	20.9358	0.741584	0.370792	−	0.928716i	$-0.379086\pi$
$797$	20.9358	0.741584	0.370792	+	0.928716i	$0.379086\pi$
$798$	0	0
$799$	4.53590	0.160469
$800$	0	0
$801$	0	0
$802$	−6.52598	−0.230440
$803$	−16.4901	−0.581923
$804$	0	0
$805$	0	0
$806$	−0.301719	−0.0106276
$807$	0	0
$808$	−17.8293	−0.627233
$809$	53.5861	1.88399	0.941994	−	0.335629i	$-0.108949\pi$
$809$	53.5861	1.88399	0.941994	+	0.335629i	$0.108949\pi$
$810$	0	0
$811$	35.0333	1.23019	0.615093	−	0.788454i	$-0.289118\pi$
$811$	35.0333	1.23019	0.615093	+	0.788454i	$0.289118\pi$
$812$	27.4249	0.962424
$813$	0	0
$814$	8.53590	0.299183
$815$	0	0
$816$	0	0
$817$	3.18490	0.111426
$818$	4.14110	0.144790
$819$	0	0
$820$	0	0
$821$	24.3093	0.848401	0.424200	−	0.905568i	$-0.360555\pi$
$821$	24.3093	0.848401	0.424200	+	0.905568i	$0.360555\pi$
$822$	0	0
$823$	30.0279	1.04671	0.523353	−	0.852116i	$-0.324681\pi$
$823$	30.0279	1.04671	0.523353	+	0.852116i	$0.324681\pi$
$824$	15.6836	0.546364
$825$	0	0
$826$	−1.26795	−0.0441176
$827$	−39.2934	−1.36636	−0.683182	−	0.730248i	$-0.739405\pi$
$827$	−39.2934	−1.36636	−0.683182	+	0.730248i	$0.739405\pi$
$828$	0	0
$829$	−1.32051	−0.0458631	−0.0229316	−	0.999737i	$-0.507300\pi$
$829$	−1.32051	−0.0458631	−0.0229316	+	0.999737i	$0.507300\pi$
$830$	0	0
$831$	0	0
$832$	−1.32194	−0.0458299
$833$	3.20736	0.111129
$834$	0	0
$835$	0	0
$836$	−10.6569	−0.368576
$837$	0	0
$838$	−5.04787	−0.174376
$839$	39.2494	1.35504	0.677521	−	0.735504i	$-0.263054\pi$
$839$	39.2494	1.35504	0.677521	+	0.735504i	$0.263054\pi$
$840$	0	0
$841$	23.9808	0.826923
$842$	10.2512	0.353281
$843$	0	0
$844$	−9.46410	−0.325768
$845$	0	0
$846$	0	0
$847$	3.34108	0.114801
$848$	−0.933740	−0.0320648
$849$	0	0
$850$	0	0
$851$	7.58051	0.259856
$852$	0	0
$853$	0.739059	0.0253049	0.0126524	−	0.999920i	$-0.495972\pi$
$853$	0.739059	0.0253049	0.0126524	+	0.999920i	$0.495972\pi$
$854$	3.29725	0.112830
$855$	0	0
$856$	6.73205	0.230097
$857$	−32.4254	−1.10763	−0.553815	−	0.832640i	$-0.686828\pi$
$857$	−32.4254	−1.10763	−0.553815	+	0.832640i	$0.686828\pi$
$858$	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
$862$	11.7300	0.399526
$863$	20.3538	0.692851	0.346426	−	0.938077i	$-0.387395\pi$
$863$	20.3538	0.692851	0.346426	+	0.938077i	$0.387395\pi$
$864$	0	0
$865$	0	0
$866$	−8.32398	−0.282860
$867$	0	0
$868$	3.76778	0.127887
$869$	−0.603439	−0.0204703
$870$	0	0
$871$	8.44486	0.286143
$872$	−19.6975	−0.667040
$873$	0	0
$874$	1.46410	0.0495240
$875$	0	0
$876$	0	0
$877$	−18.1417	−0.612601	−0.306301	−	0.951935i	$-0.599091\pi$
$877$	−18.1417	−0.612601	−0.306301	+	0.951935i	$0.599091\pi$
$878$	0.554803	0.0187237
$879$	0	0
$880$	0	0
$881$	4.20241	0.141583	0.0707914	−	0.997491i	$-0.477448\pi$
$881$	4.20241	0.141583	0.0707914	+	0.997491i	$0.477448\pi$
$882$	0	0
$883$	15.8102	0.532055	0.266027	−	0.963965i	$-0.414289\pi$
$883$	15.8102	0.532055	0.266027	+	0.963965i	$0.414289\pi$
$884$	1.42775	0.0480205
$885$	0	0
$886$	21.6603	0.727690
$887$	−21.9711	−0.737717	−0.368858	−	0.929486i	$-0.620251\pi$
$887$	−21.9711	−0.737717	−0.368858	+	0.929486i	$0.620251\pi$
$888$	0	0
$889$	−5.07180	−0.170103
$890$	0	0
$891$	0	0
$892$	−13.7910	−0.461758
$893$	6.41473	0.214661
$894$	0	0
$895$	0	0
$896$	−24.9127	−0.832276
$897$	0	0
$898$	−6.52598	−0.217775
$899$	7.27879	0.242761
$900$	0	0
$901$	0.535898	0.0178534
$902$	1.31268	0.0437074
$903$	0	0
$904$	10.9282	0.363467
$905$	0	0
$906$	0	0
$907$	−34.1080	−1.13254	−0.566269	−	0.824220i	$-0.691614\pi$
$907$	−34.1080	−1.13254	−0.566269	+	0.824220i	$0.691614\pi$
$908$	−17.6269	−0.584969
$909$	0	0
$910$	0	0
$911$	38.9477	1.29040	0.645198	−	0.764016i	$-0.276775\pi$
$911$	38.9477	1.29040	0.645198	+	0.764016i	$0.276775\pi$
$912$	0	0
$913$	−4.35066	−0.143986
$914$	−19.7460	−0.653139
$915$	0	0
$916$	37.8564	1.25081
$917$	−25.6317	−0.846434
$918$	0	0
$919$	−18.3923	−0.606706	−0.303353	−	0.952878i	$-0.598106\pi$
$919$	−18.3923	−0.606706	−0.303353	+	0.952878i	$0.598106\pi$
$920$	0	0
$921$	0	0
$922$	−7.80607	−0.257079
$923$	4.72311	0.155463
$924$	0	0
$925$	0	0
$926$	−5.32844	−0.175104
$927$	0	0
$928$	−37.4073	−1.22795
$929$	46.0056	1.50940	0.754698	−	0.656072i	$-0.227783\pi$
$929$	46.0056	1.50940	0.754698	+	0.656072i	$0.227783\pi$
$930$	0	0
$931$	4.53590	0.148658
$932$	−20.2523	−0.663385
$933$	0	0
$934$	11.1244	0.364000
$935$	0	0
$936$	0	0
$937$	24.1991	0.790551	0.395275	−	0.918563i	$-0.370649\pi$
$937$	24.1991	0.790551	0.395275	+	0.918563i	$0.370649\pi$
$938$	16.3142	0.532678
$939$	0	0
$940$	0	0
$941$	49.6855	1.61970	0.809850	−	0.586637i	$-0.199549\pi$
$941$	49.6855	1.61970	0.809850	+	0.586637i	$0.199549\pi$
$942$	0	0
$943$	1.16575	0.0379622
$944$	−2.77466	−0.0903074
$945$	0	0
$946$	−2.53590	−0.0824492
$947$	17.8300	0.579396	0.289698	−	0.957118i	$-0.406445\pi$
$947$	17.8300	0.579396	0.289698	+	0.957118i	$0.406445\pi$
$948$	0	0
$949$	3.12436	0.101421
$950$	0	0
$951$	0	0
$952$	5.94311	0.192617
$953$	32.6012	1.05606	0.528029	−	0.849226i	$-0.322931\pi$
$953$	32.6012	1.05606	0.528029	+	0.849226i	$0.322931\pi$
$954$	0	0
$955$	0	0
$956$	43.2902	1.40010
$957$	0	0
$958$	12.8958	0.416644
$959$	14.7785	0.477221
$960$	0	0
$961$	1.00000	0.0322581
$962$	−1.61729	−0.0521434
$963$	0	0
$964$	35.3205	1.13760
$965$	0	0
$966$	0	0
$967$	−17.5170	−0.563307	−0.281654	−	0.959516i	$-0.590883\pi$
$967$	−17.5170	−0.563307	−0.281654	+	0.959516i	$0.590883\pi$
$968$	−2.96713	−0.0953671
$969$	0	0
$970$	0	0
$971$	−13.2107	−0.423951	−0.211975	−	0.977275i	$-0.567990\pi$
$971$	−13.2107	−0.423951	−0.211975	+	0.977275i	$0.567990\pi$
$972$	0	0
$973$	9.86707	0.316324
$974$	14.9993	0.480609
$975$	0	0
$976$	7.21539	0.230959
$977$	−11.3880	−0.364336	−0.182168	−	0.983267i	$-0.558311\pi$
$977$	−11.3880	−0.364336	−0.182168	+	0.983267i	$0.558311\pi$
$978$	0	0
$979$	18.0000	0.575282
$980$	0	0
$981$	0	0
$982$	11.8862	0.379304
$983$	−46.9192	−1.49649	−0.748246	−	0.663422i	$-0.769104\pi$
$983$	−46.9192	−1.49649	−0.748246	+	0.663422i	$0.769104\pi$
$984$	0	0
$985$	0	0
$986$	5.32844	0.169692
$987$	0	0
$988$	2.01915	0.0642376
$989$	−2.25207	−0.0716115
$990$	0	0
$991$	29.7128	0.943859	0.471929	−	0.881636i	$-0.343558\pi$
$991$	29.7128	0.943859	0.471929	+	0.881636i	$0.343558\pi$
$992$	−5.13922	−0.163170
$993$	0	0
$994$	9.12436	0.289407
$995$	0	0
$996$	0	0
$997$	−32.4737	−1.02845	−0.514227	−	0.857654i	$-0.671921\pi$
$997$	−32.4737	−1.02845	−0.514227	+	0.857654i	$0.671921\pi$
$998$	−16.4158	−0.519632
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.ce.1.3		8
3.2	odd	2	inner	6975.2.a.ce.1.5		8
5.2	odd	4		1395.2.c.d.559.4	yes	8
5.3	odd	4		1395.2.c.d.559.6	yes	8
5.4	even	2	inner	6975.2.a.ce.1.6		8
15.2	even	4		1395.2.c.d.559.5	yes	8
15.8	even	4		1395.2.c.d.559.3	✓	8
15.14	odd	2	inner	6975.2.a.ce.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1395.2.c.d.559.3	✓	8	15.8	even	4
1395.2.c.d.559.4	yes	8	5.2	odd	4
1395.2.c.d.559.5	yes	8	15.2	even	4
1395.2.c.d.559.6	yes	8	5.3	odd	4
6975.2.a.ce.1.3		8	1.1	even	1	trivial
6975.2.a.ce.1.4		8	15.14	odd	2	inner
6975.2.a.ce.1.5		8	3.2	odd	2	inner
6975.2.a.ce.1.6		8	5.4	even	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)

\(f(q)\)	\(=\)	\(q-0.517638 q^{2} -1.73205 q^{4} -2.17533 q^{7} +1.93185 q^{8} -3.07638 q^{11} +0.582877 q^{13} +1.12603 q^{14} +2.46410 q^{16} -1.41421 q^{17} -2.00000 q^{19} +1.59245 q^{22} +1.41421 q^{23} -0.301719 q^{26} +3.76778 q^{28} +7.27879 q^{29} +1.00000 q^{31} -5.13922 q^{32} +0.732051 q^{34} +5.36023 q^{37} +1.03528 q^{38} +0.824313 q^{41} -1.59245 q^{43} +5.32844 q^{44} -0.732051 q^{46} -3.20736 q^{47} -2.26795 q^{49} -1.00957 q^{52} -0.378937 q^{53} -4.20241 q^{56} -3.76778 q^{58} -1.12603 q^{59} +2.92820 q^{61} -0.517638 q^{62} -2.26795 q^{64} +14.4882 q^{67} +2.44949 q^{68} +8.10310 q^{71} +5.36023 q^{73} -2.77466 q^{74} +3.46410 q^{76} +6.69213 q^{77} +0.196152 q^{79} -0.426696 q^{82} +1.41421 q^{83} +0.824313 q^{86} -5.94311 q^{88} -5.85104 q^{89} -1.26795 q^{91} -2.44949 q^{92} +1.66025 q^{94} -10.2938 q^{97} +1.17398 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{16} - 16 q^{19} + 8 q^{31} - 8 q^{34} + 8 q^{46} - 32 q^{49} - 32 q^{61} - 32 q^{64} - 40 q^{79} - 24 q^{91} - 56 q^{94}+O(q^{100}) \) 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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