LMFDB - Embedded newform 6975.2.a.bp.1.4

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 = [5,-4,0,6,0,0,2,-9,0,0,2,0,4,-2,0,4,-19] 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:	$5$
Coefficient field:	5.5.144209.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - x^{2} + 6x - 1 $ x^5 - 6x^3 - x^2 + 6x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 775)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.177477$ 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)

$f(q)$	$=$	$q+0.264183 q^{2} -1.93021 q^{4} -2.14598 q^{7} -1.03829 q^{8} -1.13359 q^{11} +0.737342 q^{13} -0.566932 q^{14} +3.58611 q^{16} -2.91329 q^{17} +6.85889 q^{19} -0.299476 q^{22} -7.06380 q^{23} +0.194793 q^{26} +4.14218 q^{28} +7.65851 q^{29} -1.00000 q^{31} +3.02398 q^{32} -0.769643 q^{34} +5.57905 q^{37} +1.81200 q^{38} -3.02250 q^{41} +3.79962 q^{43} +2.18807 q^{44} -1.86614 q^{46} -7.55794 q^{47} -2.39477 q^{49} -1.42322 q^{52} +10.9820 q^{53} +2.22816 q^{56} +2.02325 q^{58} +9.35603 q^{59} -8.38726 q^{61} -0.264183 q^{62} -6.37334 q^{64} -13.8863 q^{67} +5.62326 q^{68} +10.3298 q^{71} +5.10463 q^{73} +1.47389 q^{74} -13.2391 q^{76} +2.43267 q^{77} +5.76964 q^{79} -0.798495 q^{82} +1.01132 q^{83} +1.00379 q^{86} +1.17700 q^{88} +6.78423 q^{89} -1.58232 q^{91} +13.6346 q^{92} -1.99668 q^{94} +15.3051 q^{97} -0.632659 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{2} + 6 q^{4} + 2 q^{7} - 9 q^{8} + 2 q^{11} + 4 q^{13} - 2 q^{14} + 4 q^{16} - 19 q^{17} + 8 q^{19} - 10 q^{22} - 12 q^{23} + 16 q^{26} + 6 q^{28} - 6 q^{29} - 5 q^{31} - 7 q^{32} + 31 q^{34}+ \cdots - 10 q^{98}+O(q^{100}) $ 5 * q - 4 * q^2 + 6 * q^4 + 2 * q^7 - 9 * q^8 + 2 * q^11 + 4 * q^13 - 2 * q^14 + 4 * q^16 - 19 * q^17 + 8 * q^19 - 10 * q^22 - 12 * q^23 + 16 * q^26 + 6 * q^28 - 6 * q^29 - 5 * q^31 - 7 * q^32 + 31 * q^34 + 16 * q^37 - 14 * q^38 + 2 * q^41 + q^43 + 4 * q^44 - 11 * q^46 - 8 * q^47 - 9 * q^49 - 26 * q^52 - 3 * q^53 + 9 * q^56 + 5 * q^58 + 4 * q^59 - 5 * q^61 + 4 * q^62 - 5 * q^64 + 13 * q^67 - 30 * q^68 - 6 * q^71 + 19 * q^73 - 4 * q^74 - 5 * q^76 - 23 * q^77 - 6 * q^79 - 27 * q^82 - 18 * q^83 + 7 * q^86 - q^88 + 31 * q^89 + 8 * q^91 + 16 * q^92 - 14 * q^94 - q^97 - 10 * q^98

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 $

$2$	0.264183	0.186806	0.0934029	−	0.995628i	$-0.470226\pi$
$2$	0.264183	0.186806	0.0934029	+	0.995628i	$0.470226\pi$
$3$	0	0
$3$	0	0
$4$	−1.93021	−0.965104
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.14598	−0.811104	−0.405552	−	0.914072i	$-0.632921\pi$
$7$	−2.14598	−0.811104	−0.405552	+	0.914072i	$0.632921\pi$
$8$	−1.03829	−0.367093
$9$	0	0
$10$	0	0
$11$	−1.13359	−0.341791	−0.170895	−	0.985289i	$-0.554666\pi$
$11$	−1.13359	−0.341791	−0.170895	+	0.985289i	$0.554666\pi$
$12$	0	0
$13$	0.737342	0.204502	0.102251	−	0.994759i	$-0.467396\pi$
$13$	0.737342	0.204502	0.102251	+	0.994759i	$0.467396\pi$
$14$	−0.566932	−0.151519
$15$	0	0
$16$	3.58611	0.896529
$17$	−2.91329	−0.706578	−0.353289	−	0.935514i	$-0.614937\pi$
$17$	−2.91329	−0.706578	−0.353289	+	0.935514i	$0.614937\pi$
$18$	0	0
$19$	6.85889	1.57354	0.786769	−	0.617248i	$-0.211753\pi$
$19$	6.85889	1.57354	0.786769	+	0.617248i	$0.211753\pi$
$20$	0	0
$21$	0	0
$22$	−0.299476	−0.0638485
$23$	−7.06380	−1.47290	−0.736452	−	0.676490i	$-0.763500\pi$
$23$	−7.06380	−1.47290	−0.736452	+	0.676490i	$0.763500\pi$
$24$	0	0
$25$	0	0
$26$	0.194793	0.0382021
$27$	0	0
$28$	4.14218	0.782799
$29$	7.65851	1.42215	0.711074	−	0.703117i	$-0.248209\pi$
$29$	7.65851	1.42215	0.711074	+	0.703117i	$0.248209\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	3.02398	0.534569
$33$	0	0
$34$	−0.769643	−0.131993
$35$	0	0
$36$	0	0
$37$	5.57905	0.917190	0.458595	−	0.888645i	$-0.348353\pi$
$37$	5.57905	0.917190	0.458595	+	0.888645i	$0.348353\pi$
$38$	1.81200	0.293946
$39$	0	0
$40$	0	0
$41$	−3.02250	−0.472036	−0.236018	−	0.971749i	$-0.575842\pi$
$41$	−3.02250	−0.472036	−0.236018	+	0.971749i	$0.575842\pi$
$42$	0	0
$43$	3.79962	0.579436	0.289718	−	0.957112i	$-0.406438\pi$
$43$	3.79962	0.579436	0.289718	+	0.957112i	$0.406438\pi$
$44$	2.18807	0.329864
$45$	0	0
$46$	−1.86614	−0.275147
$47$	−7.55794	−1.10244	−0.551219	−	0.834360i	$-0.685837\pi$
$47$	−7.55794	−1.10244	−0.551219	+	0.834360i	$0.685837\pi$
$48$	0	0
$49$	−2.39477	−0.342111
$50$	0	0
$51$	0	0
$52$	−1.42322	−0.197365
$53$	10.9820	1.50849	0.754244	−	0.656594i	$-0.228003\pi$
$53$	10.9820	1.50849	0.754244	+	0.656594i	$0.228003\pi$
$54$	0	0
$55$	0	0
$56$	2.22816	0.297750
$57$	0	0
$58$	2.02325	0.265666
$59$	9.35603	1.21805	0.609026	−	0.793151i	$-0.291561\pi$
$59$	9.35603	1.21805	0.609026	+	0.793151i	$0.291561\pi$
$60$	0	0
$61$	−8.38726	−1.07388	−0.536939	−	0.843621i	$-0.680419\pi$
$61$	−8.38726	−1.07388	−0.536939	+	0.843621i	$0.680419\pi$
$62$	−0.264183	−0.0335513
$63$	0	0
$64$	−6.37334	−0.796668
$65$	0	0
$66$	0	0
$67$	−13.8863	−1.69648	−0.848238	−	0.529615i	$-0.822337\pi$
$67$	−13.8863	−1.69648	−0.848238	+	0.529615i	$0.822337\pi$
$68$	5.62326	0.681921
$69$	0	0
$70$	0	0
$71$	10.3298	1.22592	0.612961	−	0.790113i	$-0.289978\pi$
$71$	10.3298	1.22592	0.612961	+	0.790113i	$0.289978\pi$
$72$	0	0
$73$	5.10463	0.597452	0.298726	−	0.954339i	$-0.403438\pi$
$73$	5.10463	0.597452	0.298726	+	0.954339i	$0.403438\pi$
$74$	1.47389	0.171336
$75$	0	0
$76$	−13.2391	−1.51863
$77$	2.43267	0.277228
$78$	0	0
$79$	5.76964	0.649136	0.324568	−	0.945862i	$-0.394781\pi$
$79$	5.76964	0.649136	0.324568	+	0.945862i	$0.394781\pi$
$80$	0	0
$81$	0	0
$82$	−0.798495	−0.0881790
$83$	1.01132	0.111007	0.0555036	−	0.998458i	$-0.482324\pi$
$83$	1.01132	0.111007	0.0555036	+	0.998458i	$0.482324\pi$
$84$	0	0
$85$	0	0
$86$	1.00379	0.108242
$87$	0	0
$88$	1.17700	0.125469
$89$	6.78423	0.719127	0.359563	−	0.933121i	$-0.382926\pi$
$89$	6.78423	0.719127	0.359563	+	0.933121i	$0.382926\pi$
$90$	0	0
$91$	−1.58232	−0.165872
$92$	13.6346	1.42151
$93$	0	0
$94$	−1.99668	−0.205942
$95$	0	0
$96$	0	0
$97$	15.3051	1.55400	0.777001	−	0.629499i	$-0.216740\pi$
$97$	15.3051	1.55400	0.777001	+	0.629499i	$0.216740\pi$
$98$	−0.632659	−0.0639082
$99$	0	0
$100$	0	0
$101$	−18.4020	−1.83106	−0.915532	−	0.402245i	$-0.868230\pi$
$101$	−18.4020	−1.83106	−0.915532	+	0.402245i	$0.868230\pi$
$102$	0	0
$103$	−3.20925	−0.316216	−0.158108	−	0.987422i	$-0.550539\pi$
$103$	−3.20925	−0.316216	−0.158108	+	0.987422i	$0.550539\pi$
$104$	−0.765578	−0.0750711
$105$	0	0
$106$	2.90125	0.281794
$107$	4.18235	0.404323	0.202161	−	0.979352i	$-0.435203\pi$
$107$	4.18235	0.404323	0.202161	+	0.979352i	$0.435203\pi$
$108$	0	0
$109$	−14.8656	−1.42386	−0.711931	−	0.702250i	$-0.752179\pi$
$109$	−14.8656	−1.42386	−0.711931	+	0.702250i	$0.752179\pi$
$110$	0	0
$111$	0	0
$112$	−7.69573	−0.727178
$113$	−10.8709	−1.02265	−0.511326	−	0.859387i	$-0.670845\pi$
$113$	−10.8709	−1.02265	−0.511326	+	0.859387i	$0.670845\pi$
$114$	0	0
$115$	0	0
$116$	−14.7825	−1.37252
$117$	0	0
$118$	2.47171	0.227539
$119$	6.25187	0.573108
$120$	0	0
$121$	−9.71497	−0.883179
$122$	−2.21577	−0.200607
$123$	0	0
$124$	1.93021	0.173338
$125$	0	0
$126$	0	0
$127$	−4.22557	−0.374959	−0.187479	−	0.982269i	$-0.560032\pi$
$127$	−4.22557	−0.374959	−0.187479	+	0.982269i	$0.560032\pi$
$128$	−7.73169	−0.683391
$129$	0	0
$130$	0	0
$131$	−6.59896	−0.576554	−0.288277	−	0.957547i	$-0.593082\pi$
$131$	−6.59896	−0.576554	−0.288277	+	0.957547i	$0.593082\pi$
$132$	0	0
$133$	−14.7190	−1.27630
$134$	−3.66852	−0.316912
$135$	0	0
$136$	3.02486	0.259379
$137$	−22.0518	−1.88401	−0.942004	−	0.335601i	$-0.891061\pi$
$137$	−22.0518	−1.88401	−0.942004	+	0.335601i	$0.891061\pi$
$138$	0	0
$139$	11.2326	0.952739	0.476369	−	0.879245i	$-0.341953\pi$
$139$	11.2326	0.952739	0.476369	+	0.879245i	$0.341953\pi$
$140$	0	0
$141$	0	0
$142$	2.72896	0.229009
$143$	−0.835845	−0.0698969
$144$	0	0
$145$	0	0
$146$	1.34856	0.111607
$147$	0	0
$148$	−10.7687	−0.885183
$149$	14.2166	1.16467	0.582334	−	0.812950i	$-0.302140\pi$
$149$	14.2166	1.16467	0.582334	+	0.812950i	$0.302140\pi$
$150$	0	0
$151$	18.5380	1.50860	0.754302	−	0.656528i	$-0.227976\pi$
$151$	18.5380	1.50860	0.754302	+	0.656528i	$0.227976\pi$
$152$	−7.12155	−0.577634
$153$	0	0
$154$	0.642669	0.0517878
$155$	0	0
$156$	0	0
$157$	−14.5650	−1.16241	−0.581207	−	0.813756i	$-0.697419\pi$
$157$	−14.5650	−1.16241	−0.581207	+	0.813756i	$0.697419\pi$
$158$	1.52424	0.121262
$159$	0	0
$160$	0	0
$161$	15.1588	1.19468
$162$	0	0
$163$	3.82214	0.299373	0.149687	−	0.988733i	$-0.452173\pi$
$163$	3.82214	0.299373	0.149687	+	0.988733i	$0.452173\pi$
$164$	5.83406	0.455563
$165$	0	0
$166$	0.267175	0.0207368
$167$	11.5390	0.892916	0.446458	−	0.894805i	$-0.352685\pi$
$167$	11.5390	0.892916	0.446458	+	0.894805i	$0.352685\pi$
$168$	0	0
$169$	−12.4563	−0.958179
$170$	0	0
$171$	0	0
$172$	−7.33405	−0.559216
$173$	−5.70512	−0.433752	−0.216876	−	0.976199i	$-0.569587\pi$
$173$	−5.70512	−0.433752	−0.216876	+	0.976199i	$0.569587\pi$
$174$	0	0
$175$	0	0
$176$	−4.06519	−0.306425
$177$	0	0
$178$	1.79228	0.134337
$179$	12.0891	0.903580	0.451790	−	0.892124i	$-0.350786\pi$
$179$	12.0891	0.903580	0.451790	+	0.892124i	$0.350786\pi$
$180$	0	0
$181$	−23.0801	−1.71553	−0.857766	−	0.514040i	$-0.828148\pi$
$181$	−23.0801	−1.71553	−0.857766	+	0.514040i	$0.828148\pi$
$182$	−0.418022	−0.0309859
$183$	0	0
$184$	7.33430	0.540692
$185$	0	0
$186$	0	0
$187$	3.30249	0.241502
$188$	14.5884	1.06397
$189$	0	0
$190$	0	0
$191$	−3.50613	−0.253695	−0.126847	−	0.991922i	$-0.540486\pi$
$191$	−3.50613	−0.253695	−0.126847	+	0.991922i	$0.540486\pi$
$192$	0	0
$193$	2.19838	0.158243	0.0791216	−	0.996865i	$-0.474788\pi$
$193$	2.19838	0.158243	0.0791216	+	0.996865i	$0.474788\pi$
$194$	4.04336	0.290297
$195$	0	0
$196$	4.62241	0.330172
$197$	−10.4883	−0.747259	−0.373629	−	0.927578i	$-0.621887\pi$
$197$	−10.4883	−0.747259	−0.373629	+	0.927578i	$0.621887\pi$
$198$	0	0
$199$	−13.4959	−0.956697	−0.478349	−	0.878170i	$-0.658764\pi$
$199$	−13.4959	−0.956697	−0.478349	+	0.878170i	$0.658764\pi$
$200$	0	0
$201$	0	0
$202$	−4.86149	−0.342053
$203$	−16.4350	−1.15351
$204$	0	0
$205$	0	0
$206$	−0.847829	−0.0590710
$207$	0	0
$208$	2.64419	0.183342
$209$	−7.77519	−0.537821
$210$	0	0
$211$	−13.2300	−0.910793	−0.455397	−	0.890289i	$-0.650503\pi$
$211$	−13.2300	−0.910793	−0.455397	+	0.890289i	$0.650503\pi$
$212$	−21.1975	−1.45585
$213$	0	0
$214$	1.10491	0.0755298
$215$	0	0
$216$	0	0
$217$	2.14598	0.145679
$218$	−3.92723	−0.265986
$219$	0	0
$220$	0	0
$221$	−2.14809	−0.144496
$222$	0	0
$223$	−15.2035	−1.01810	−0.509051	−	0.860736i	$-0.670004\pi$
$223$	−15.2035	−1.01810	−0.509051	+	0.860736i	$0.670004\pi$
$224$	−6.48940	−0.433591
$225$	0	0
$226$	−2.87192	−0.191037
$227$	−1.51386	−0.100479	−0.0502393	−	0.998737i	$-0.515998\pi$
$227$	−1.51386	−0.100479	−0.0502393	+	0.998737i	$0.515998\pi$
$228$	0	0
$229$	8.55623	0.565412	0.282706	−	0.959207i	$-0.408768\pi$
$229$	8.55623	0.565412	0.282706	+	0.959207i	$0.408768\pi$
$230$	0	0
$231$	0	0
$232$	−7.95179	−0.522060
$233$	−29.0101	−1.90051	−0.950257	−	0.311468i	$-0.899179\pi$
$233$	−29.0101	−1.90051	−0.950257	+	0.311468i	$0.899179\pi$
$234$	0	0
$235$	0	0
$236$	−18.0591	−1.17555
$237$	0	0
$238$	1.65164	0.107060
$239$	13.2639	0.857971	0.428986	−	0.903311i	$-0.358871\pi$
$239$	13.2639	0.857971	0.428986	+	0.903311i	$0.358871\pi$
$240$	0	0
$241$	−4.69319	−0.302315	−0.151157	−	0.988510i	$-0.548300\pi$
$241$	−4.69319	−0.302315	−0.151157	+	0.988510i	$0.548300\pi$
$242$	−2.56653	−0.164983
$243$	0	0
$244$	16.1891	1.03640
$245$	0	0
$246$	0	0
$247$	5.05735	0.321791
$248$	1.03829	0.0659318
$249$	0	0
$250$	0	0
$251$	−2.03024	−0.128148	−0.0640739	−	0.997945i	$-0.520409\pi$
$251$	−2.03024	−0.128148	−0.0640739	+	0.997945i	$0.520409\pi$
$252$	0	0
$253$	8.00747	0.503425
$254$	−1.11632	−0.0700444
$255$	0	0
$256$	10.7041	0.669007
$257$	−6.81919	−0.425369	−0.212685	−	0.977121i	$-0.568221\pi$
$257$	−6.81919	−0.425369	−0.212685	+	0.977121i	$0.568221\pi$
$258$	0	0
$259$	−11.9725	−0.743936
$260$	0	0
$261$	0	0
$262$	−1.74333	−0.107704
$263$	0.818007	0.0504404	0.0252202	−	0.999682i	$-0.491971\pi$
$263$	0.818007	0.0504404	0.0252202	+	0.999682i	$0.491971\pi$
$264$	0	0
$265$	0	0
$266$	−3.88852	−0.238421
$267$	0	0
$268$	26.8034	1.63728
$269$	−1.10427	−0.0673285	−0.0336642	−	0.999433i	$-0.510718\pi$
$269$	−1.10427	−0.0673285	−0.0336642	+	0.999433i	$0.510718\pi$
$270$	0	0
$271$	5.42896	0.329786	0.164893	−	0.986311i	$-0.447272\pi$
$271$	5.42896	0.329786	0.164893	+	0.986311i	$0.447272\pi$
$272$	−10.4474	−0.633467
$273$	0	0
$274$	−5.82570	−0.351944
$275$	0	0
$276$	0	0
$277$	−18.9283	−1.13729	−0.568646	−	0.822583i	$-0.692532\pi$
$277$	−18.9283	−1.13729	−0.568646	+	0.822583i	$0.692532\pi$
$278$	2.96747	0.177977
$279$	0	0
$280$	0	0
$281$	−9.85285	−0.587772	−0.293886	−	0.955841i	$-0.594949\pi$
$281$	−9.85285	−0.587772	−0.293886	+	0.955841i	$0.594949\pi$
$282$	0	0
$283$	−31.7526	−1.88750	−0.943748	−	0.330666i	$-0.892727\pi$
$283$	−31.7526	−1.88750	−0.943748	+	0.330666i	$0.892727\pi$
$284$	−19.9387	−1.18314
$285$	0	0
$286$	−0.220816	−0.0130571
$287$	6.48623	0.382870
$288$	0	0
$289$	−8.51272	−0.500748
$290$	0	0
$291$	0	0
$292$	−9.85300	−0.576603
$293$	19.1975	1.12153	0.560766	−	0.827974i	$-0.310507\pi$
$293$	19.1975	1.12153	0.560766	+	0.827974i	$0.310507\pi$
$294$	0	0
$295$	0	0
$296$	−5.79269	−0.336694
$297$	0	0
$298$	3.75578	0.217566
$299$	−5.20843	−0.301212
$300$	0	0
$301$	−8.15390	−0.469983
$302$	4.89744	0.281816
$303$	0	0
$304$	24.5968	1.41072
$305$	0	0
$306$	0	0
$307$	25.2908	1.44342	0.721710	−	0.692196i	$-0.243357\pi$
$307$	25.2908	1.44342	0.721710	+	0.692196i	$0.243357\pi$
$308$	−4.69555	−0.267554
$309$	0	0
$310$	0	0
$311$	9.49162	0.538220	0.269110	−	0.963109i	$-0.413270\pi$
$311$	9.49162	0.538220	0.269110	+	0.963109i	$0.413270\pi$
$312$	0	0
$313$	−19.2026	−1.08539	−0.542696	−	0.839929i	$-0.682597\pi$
$313$	−19.2026	−1.08539	−0.542696	+	0.839929i	$0.682597\pi$
$314$	−3.84783	−0.217146
$315$	0	0
$316$	−11.1366	−0.626483
$317$	6.76462	0.379939	0.189969	−	0.981790i	$-0.439161\pi$
$317$	6.76462	0.379939	0.189969	+	0.981790i	$0.439161\pi$
$318$	0	0
$319$	−8.68162	−0.486078
$320$	0	0
$321$	0	0
$322$	4.00469	0.223173
$323$	−19.9820	−1.11183
$324$	0	0
$325$	0	0
$326$	1.00975	0.0559247
$327$	0	0
$328$	3.13825	0.173281
$329$	16.2192	0.894192
$330$	0	0
$331$	−27.1257	−1.49096	−0.745481	−	0.666527i	$-0.767780\pi$
$331$	−27.1257	−1.49096	−0.745481	+	0.666527i	$0.767780\pi$
$332$	−1.95206	−0.107133
$333$	0	0
$334$	3.04841	0.166802
$335$	0	0
$336$	0	0
$337$	−28.0407	−1.52747	−0.763737	−	0.645527i	$-0.776638\pi$
$337$	−28.0407	−1.52747	−0.763737	+	0.645527i	$0.776638\pi$
$338$	−3.29075	−0.178993
$339$	0	0
$340$	0	0
$341$	1.13359	0.0613875
$342$	0	0
$343$	20.1610	1.08859
$344$	−3.94512	−0.212707
$345$	0	0
$346$	−1.50720	−0.0810274
$347$	−19.2914	−1.03561	−0.517807	−	0.855497i	$-0.673252\pi$
$347$	−19.2914	−1.03561	−0.517807	+	0.855497i	$0.673252\pi$
$348$	0	0
$349$	20.8611	1.11667	0.558336	−	0.829615i	$-0.311440\pi$
$349$	20.8611	1.11667	0.558336	+	0.829615i	$0.311440\pi$
$350$	0	0
$351$	0	0
$352$	−3.42796	−0.182711
$353$	14.3753	0.765123	0.382561	−	0.923930i	$-0.375042\pi$
$353$	14.3753	0.765123	0.382561	+	0.923930i	$0.375042\pi$
$354$	0	0
$355$	0	0
$356$	−13.0950	−0.694032
$357$	0	0
$358$	3.19373	0.168794
$359$	−21.2523	−1.12165	−0.560827	−	0.827933i	$-0.689517\pi$
$359$	−21.2523	−1.12165	−0.560827	+	0.827933i	$0.689517\pi$
$360$	0	0
$361$	28.0444	1.47602
$362$	−6.09738	−0.320471
$363$	0	0
$364$	3.05421	0.160084
$365$	0	0
$366$	0	0
$367$	0.665621	0.0347451	0.0173726	−	0.999849i	$-0.494470\pi$
$367$	0.665621	0.0347451	0.0173726	+	0.999849i	$0.494470\pi$
$368$	−25.3316	−1.32050
$369$	0	0
$370$	0	0
$371$	−23.5671	−1.22354
$372$	0	0
$373$	−18.6381	−0.965045	−0.482522	−	0.875884i	$-0.660279\pi$
$373$	−18.6381	−0.965045	−0.482522	+	0.875884i	$0.660279\pi$
$374$	0.872462	0.0451139
$375$	0	0
$376$	7.84736	0.404697
$377$	5.64694	0.290832
$378$	0	0
$379$	−14.5982	−0.749861	−0.374931	−	0.927053i	$-0.622333\pi$
$379$	−14.5982	−0.749861	−0.374931	+	0.927053i	$0.622333\pi$
$380$	0	0
$381$	0	0
$382$	−0.926261	−0.0473917
$383$	2.77909	0.142005	0.0710024	−	0.997476i	$-0.477380\pi$
$383$	2.77909	0.142005	0.0710024	+	0.997476i	$0.477380\pi$
$384$	0	0
$385$	0	0
$386$	0.580776	0.0295607
$387$	0	0
$388$	−29.5421	−1.49977
$389$	−37.9321	−1.92323	−0.961616	−	0.274398i	$-0.911522\pi$
$389$	−37.9321	−1.92323	−0.961616	+	0.274398i	$0.911522\pi$
$390$	0	0
$391$	20.5789	1.04072
$392$	2.48648	0.125586
$393$	0	0
$394$	−2.77083	−0.139592
$395$	0	0
$396$	0	0
$397$	10.3664	0.520277	0.260138	−	0.965571i	$-0.416232\pi$
$397$	10.3664	0.520277	0.260138	+	0.965571i	$0.416232\pi$
$398$	−3.56538	−0.178717
$399$	0	0
$400$	0	0
$401$	12.3610	0.617279	0.308640	−	0.951179i	$-0.400126\pi$
$401$	12.3610	0.617279	0.308640	+	0.951179i	$0.400126\pi$
$402$	0	0
$403$	−0.737342	−0.0367296
$404$	35.5196	1.76717
$405$	0	0
$406$	−4.34185	−0.215482
$407$	−6.32437	−0.313487
$408$	0	0
$409$	33.8499	1.67377	0.836885	−	0.547379i	$-0.184375\pi$
$409$	33.8499	1.67377	0.836885	+	0.547379i	$0.184375\pi$
$410$	0	0
$411$	0	0
$412$	6.19451	0.305182
$413$	−20.0778	−0.987966
$414$	0	0
$415$	0	0
$416$	2.22971	0.109320
$417$	0	0
$418$	−2.05407	−0.100468
$419$	−24.9078	−1.21682	−0.608412	−	0.793621i	$-0.708193\pi$
$419$	−24.9078	−1.21682	−0.608412	+	0.793621i	$0.708193\pi$
$420$	0	0
$421$	7.03930	0.343074	0.171537	−	0.985178i	$-0.445127\pi$
$421$	7.03930	0.343074	0.171537	+	0.985178i	$0.445127\pi$
$422$	−3.49515	−0.170141
$423$	0	0
$424$	−11.4025	−0.553755
$425$	0	0
$426$	0	0
$427$	17.9989	0.871027
$428$	−8.07279	−0.390213
$429$	0	0
$430$	0	0
$431$	6.94074	0.334324	0.167162	−	0.985929i	$-0.446540\pi$
$431$	6.94074	0.334324	0.167162	+	0.985929i	$0.446540\pi$
$432$	0	0
$433$	16.2700	0.781885	0.390943	−	0.920415i	$-0.372149\pi$
$433$	16.2700	0.781885	0.390943	+	0.920415i	$0.372149\pi$
$434$	0.566932	0.0272136
$435$	0	0
$436$	28.6936	1.37417
$437$	−48.4498	−2.31767
$438$	0	0
$439$	13.1144	0.625916	0.312958	−	0.949767i	$-0.398680\pi$
$439$	13.1144	0.625916	0.312958	+	0.949767i	$0.398680\pi$
$440$	0	0
$441$	0	0
$442$	−0.567490	−0.0269927
$443$	26.3432	1.25160	0.625801	−	0.779983i	$-0.284772\pi$
$443$	26.3432	1.25160	0.625801	+	0.779983i	$0.284772\pi$
$444$	0	0
$445$	0	0
$446$	−4.01651	−0.190187
$447$	0	0
$448$	13.6771	0.646181
$449$	12.8845	0.608058	0.304029	−	0.952663i	$-0.401668\pi$
$449$	12.8845	0.608058	0.304029	+	0.952663i	$0.401668\pi$
$450$	0	0
$451$	3.42629	0.161338
$452$	20.9832	0.986965
$453$	0	0
$454$	−0.399937	−0.0187700
$455$	0	0
$456$	0	0
$457$	−0.122280	−0.00572001	−0.00286000	−	0.999996i	$-0.500910\pi$
$457$	−0.122280	−0.00572001	−0.00286000	+	0.999996i	$0.500910\pi$
$458$	2.26041	0.105622
$459$	0	0
$460$	0	0
$461$	23.2014	1.08060	0.540298	−	0.841474i	$-0.318311\pi$
$461$	23.2014	1.08060	0.540298	+	0.841474i	$0.318311\pi$
$462$	0	0
$463$	−3.99507	−0.185667	−0.0928333	−	0.995682i	$-0.529592\pi$
$463$	−3.99507	−0.185667	−0.0928333	+	0.995682i	$0.529592\pi$
$464$	27.4643	1.27500
$465$	0	0
$466$	−7.66397	−0.355027
$467$	−38.5295	−1.78293	−0.891467	−	0.453085i	$-0.850323\pi$
$467$	−38.5295	−1.78293	−0.891467	+	0.453085i	$0.850323\pi$
$468$	0	0
$469$	29.7996	1.37602
$470$	0	0
$471$	0	0
$472$	−9.71431	−0.447138
$473$	−4.30722	−0.198046
$474$	0	0
$475$	0	0
$476$	−12.0674	−0.553108
$477$	0	0
$478$	3.50410	0.160274
$479$	4.15155	0.189689	0.0948445	−	0.995492i	$-0.469765\pi$
$479$	4.15155	0.189689	0.0948445	+	0.995492i	$0.469765\pi$
$480$	0	0
$481$	4.11366	0.187567
$482$	−1.23986	−0.0564741
$483$	0	0
$484$	18.7519	0.852359
$485$	0	0
$486$	0	0
$487$	−1.26418	−0.0572856	−0.0286428	−	0.999590i	$-0.509119\pi$
$487$	−1.26418	−0.0572856	−0.0286428	+	0.999590i	$0.509119\pi$
$488$	8.70844	0.394213
$489$	0	0
$490$	0	0
$491$	36.5423	1.64913	0.824565	−	0.565767i	$-0.191420\pi$
$491$	36.5423	1.64913	0.824565	+	0.565767i	$0.191420\pi$
$492$	0	0
$493$	−22.3115	−1.00486
$494$	1.33607	0.0601124
$495$	0	0
$496$	−3.58611	−0.161021
$497$	−22.1675	−0.994350
$498$	0	0
$499$	−15.7894	−0.706831	−0.353416	−	0.935466i	$-0.614980\pi$
$499$	−15.7894	−0.706831	−0.353416	+	0.935466i	$0.614980\pi$
$500$	0	0
$501$	0	0
$502$	−0.536356	−0.0239388
$503$	−18.5564	−0.827390	−0.413695	−	0.910416i	$-0.635762\pi$
$503$	−18.5564	−0.827390	−0.413695	+	0.910416i	$0.635762\pi$
$504$	0	0
$505$	0	0
$506$	2.11544	0.0940427
$507$	0	0
$508$	8.15623	0.361874
$509$	−23.0935	−1.02360	−0.511801	−	0.859104i	$-0.671022\pi$
$509$	−23.0935	−1.02360	−0.511801	+	0.859104i	$0.671022\pi$
$510$	0	0
$511$	−10.9544	−0.484596
$512$	18.2912	0.808366
$513$	0	0
$514$	−1.80151	−0.0794614
$515$	0	0
$516$	0	0
$517$	8.56762	0.376803
$518$	−3.16294	−0.138972
$519$	0	0
$520$	0	0
$521$	−36.7618	−1.61056	−0.805281	−	0.592893i	$-0.797986\pi$
$521$	−36.7618	−1.61056	−0.805281	+	0.592893i	$0.797986\pi$
$522$	0	0
$523$	−31.7538	−1.38850	−0.694249	−	0.719735i	$-0.744263\pi$
$523$	−31.7538	−1.38850	−0.694249	+	0.719735i	$0.744263\pi$
$524$	12.7374	0.556434
$525$	0	0
$526$	0.216104	0.00942256
$527$	2.91329	0.126905
$528$	0	0
$529$	26.8973	1.16945
$530$	0	0
$531$	0	0
$532$	28.4108	1.23176
$533$	−2.22862	−0.0965322
$534$	0	0
$535$	0	0
$536$	14.4180	0.622764
$537$	0	0
$538$	−0.291729	−0.0125773
$539$	2.71470	0.116930
$540$	0	0
$541$	19.0037	0.817034	0.408517	−	0.912751i	$-0.366046\pi$
$541$	19.0037	0.817034	0.408517	+	0.912751i	$0.366046\pi$
$542$	1.43424	0.0616058
$543$	0	0
$544$	−8.80974	−0.377715
$545$	0	0
$546$	0	0
$547$	−3.98428	−0.170356	−0.0851778	−	0.996366i	$-0.527146\pi$
$547$	−3.98428	−0.170356	−0.0851778	+	0.996366i	$0.527146\pi$
$548$	42.5645	1.81826
$549$	0	0
$550$	0	0
$551$	52.5288	2.23780
$552$	0	0
$553$	−12.3815	−0.526516
$554$	−5.00054	−0.212453
$555$	0	0
$556$	−21.6813	−0.919492
$557$	−41.8896	−1.77492	−0.887460	−	0.460885i	$-0.847532\pi$
$557$	−41.8896	−1.77492	−0.887460	+	0.460885i	$0.847532\pi$
$558$	0	0
$559$	2.80162	0.118496
$560$	0	0
$561$	0	0
$562$	−2.60296	−0.109799
$563$	−13.4064	−0.565010	−0.282505	−	0.959266i	$-0.591165\pi$
$563$	−13.4064	−0.565010	−0.282505	+	0.959266i	$0.591165\pi$
$564$	0	0
$565$	0	0
$566$	−8.38850	−0.352595
$567$	0	0
$568$	−10.7254	−0.450027
$569$	3.75926	0.157597	0.0787983	−	0.996891i	$-0.474892\pi$
$569$	3.75926	0.157597	0.0787983	+	0.996891i	$0.474892\pi$
$570$	0	0
$571$	−10.0725	−0.421521	−0.210761	−	0.977538i	$-0.567594\pi$
$571$	−10.0725	−0.421521	−0.210761	+	0.977538i	$0.567594\pi$
$572$	1.61335	0.0674577
$573$	0	0
$574$	1.71355	0.0715223
$575$	0	0
$576$	0	0
$577$	15.5629	0.647894	0.323947	−	0.946075i	$-0.394990\pi$
$577$	15.5629	0.647894	0.323947	+	0.946075i	$0.394990\pi$
$578$	−2.24892	−0.0935426
$579$	0	0
$580$	0	0
$581$	−2.17028	−0.0900383
$582$	0	0
$583$	−12.4491	−0.515588
$584$	−5.30011	−0.219320
$585$	0	0
$586$	5.07167	0.209509
$587$	−20.9217	−0.863531	−0.431765	−	0.901986i	$-0.642109\pi$
$587$	−20.9217	−0.863531	−0.431765	+	0.901986i	$0.642109\pi$
$588$	0	0
$589$	−6.85889	−0.282616
$590$	0	0
$591$	0	0
$592$	20.0071	0.822287
$593$	−13.9196	−0.571610	−0.285805	−	0.958288i	$-0.592261\pi$
$593$	−13.9196	−0.571610	−0.285805	+	0.958288i	$0.592261\pi$
$594$	0	0
$595$	0	0
$596$	−27.4409	−1.12402
$597$	0	0
$598$	−1.37598	−0.0562680
$599$	20.1940	0.825104	0.412552	−	0.910934i	$-0.364638\pi$
$599$	20.1940	0.825104	0.412552	+	0.910934i	$0.364638\pi$
$600$	0	0
$601$	−14.5143	−0.592049	−0.296025	−	0.955180i	$-0.595661\pi$
$601$	−14.5143	−0.592049	−0.296025	+	0.955180i	$0.595661\pi$
$602$	−2.15412	−0.0877955
$603$	0	0
$604$	−35.7822	−1.45596
$605$	0	0
$606$	0	0
$607$	−36.6572	−1.48787	−0.743934	−	0.668253i	$-0.767042\pi$
$607$	−36.6572	−1.48787	−0.743934	+	0.668253i	$0.767042\pi$
$608$	20.7411	0.841165
$609$	0	0
$610$	0	0
$611$	−5.57278	−0.225451
$612$	0	0
$613$	10.3229	0.416938	0.208469	−	0.978029i	$-0.433152\pi$
$613$	10.3229	0.416938	0.208469	+	0.978029i	$0.433152\pi$
$614$	6.68139	0.269639
$615$	0	0
$616$	−2.52582	−0.101768
$617$	−23.3921	−0.941729	−0.470865	−	0.882206i	$-0.656058\pi$
$617$	−23.3921	−0.941729	−0.470865	+	0.882206i	$0.656058\pi$
$618$	0	0
$619$	31.4862	1.26554	0.632769	−	0.774340i	$-0.281918\pi$
$619$	31.4862	1.26554	0.632769	+	0.774340i	$0.281918\pi$
$620$	0	0
$621$	0	0
$622$	2.50753	0.100543
$623$	−14.5588	−0.583286
$624$	0	0
$625$	0	0
$626$	−5.07299	−0.202758
$627$	0	0
$628$	28.1135	1.12185
$629$	−16.2534	−0.648066
$630$	0	0
$631$	−16.6711	−0.663668	−0.331834	−	0.943338i	$-0.607667\pi$
$631$	−16.6711	−0.663668	−0.331834	+	0.943338i	$0.607667\pi$
$632$	−5.99059	−0.238293
$633$	0	0
$634$	1.78710	0.0709747
$635$	0	0
$636$	0	0
$637$	−1.76577	−0.0699622
$638$	−2.29354	−0.0908021
$639$	0	0
$640$	0	0
$641$	42.2037	1.66695	0.833473	−	0.552560i	$-0.186349\pi$
$641$	42.2037	1.66695	0.833473	+	0.552560i	$0.186349\pi$
$642$	0	0
$643$	−30.9789	−1.22169	−0.610845	−	0.791751i	$-0.709170\pi$
$643$	−30.9789	−1.22169	−0.610845	+	0.791751i	$0.709170\pi$
$644$	−29.2596	−1.15299
$645$	0	0
$646$	−5.27890	−0.207695
$647$	−8.32177	−0.327163	−0.163581	−	0.986530i	$-0.552305\pi$
$647$	−8.32177	−0.327163	−0.163581	+	0.986530i	$0.552305\pi$
$648$	0	0
$649$	−10.6059	−0.416319
$650$	0	0
$651$	0	0
$652$	−7.37753	−0.288926
$653$	18.7569	0.734012	0.367006	−	0.930218i	$-0.380383\pi$
$653$	18.7569	0.734012	0.367006	+	0.930218i	$0.380383\pi$
$654$	0	0
$655$	0	0
$656$	−10.8390	−0.423194
$657$	0	0
$658$	4.28483	0.167040
$659$	25.3896	0.989037	0.494518	−	0.869167i	$-0.335344\pi$
$659$	25.3896	0.989037	0.494518	+	0.869167i	$0.335344\pi$
$660$	0	0
$661$	−3.43376	−0.133558	−0.0667789	−	0.997768i	$-0.521272\pi$
$661$	−3.43376	−0.133558	−0.0667789	+	0.997768i	$0.521272\pi$
$662$	−7.16615	−0.278520
$663$	0	0
$664$	−1.05005	−0.0407499
$665$	0	0
$666$	0	0
$667$	−54.0982	−2.09469
$668$	−22.2727	−0.861756
$669$	0	0
$670$	0	0
$671$	9.50773	0.367042
$672$	0	0
$673$	−8.30298	−0.320056	−0.160028	−	0.987112i	$-0.551158\pi$
$673$	−8.30298	−0.320056	−0.160028	+	0.987112i	$0.551158\pi$
$674$	−7.40788	−0.285341
$675$	0	0
$676$	24.0433	0.924742
$677$	6.44471	0.247690	0.123845	−	0.992302i	$-0.460477\pi$
$677$	6.44471	0.247690	0.123845	+	0.992302i	$0.460477\pi$
$678$	0	0
$679$	−32.8445	−1.26046
$680$	0	0
$681$	0	0
$682$	0.299476	0.0114675
$683$	44.0122	1.68408	0.842040	−	0.539416i	$-0.181355\pi$
$683$	44.0122	1.68408	0.842040	+	0.539416i	$0.181355\pi$
$684$	0	0
$685$	0	0
$686$	5.32619	0.203355
$687$	0	0
$688$	13.6259	0.519481
$689$	8.09746	0.308489
$690$	0	0
$691$	23.7146	0.902146	0.451073	−	0.892487i	$-0.351041\pi$
$691$	23.7146	0.902146	0.451073	+	0.892487i	$0.351041\pi$
$692$	11.0121	0.418616
$693$	0	0
$694$	−5.09645	−0.193459
$695$	0	0
$696$	0	0
$697$	8.80544	0.333530
$698$	5.51116	0.208601
$699$	0	0
$700$	0	0
$701$	−47.7474	−1.80340	−0.901698	−	0.432366i	$-0.857679\pi$
$701$	−47.7474	−1.80340	−0.901698	+	0.432366i	$0.857679\pi$
$702$	0	0
$703$	38.2661	1.44323
$704$	7.22478	0.272294
$705$	0	0
$706$	3.79773	0.142929
$707$	39.4902	1.48518
$708$	0	0
$709$	18.8673	0.708576	0.354288	−	0.935136i	$-0.384723\pi$
$709$	18.8673	0.708576	0.354288	+	0.935136i	$0.384723\pi$
$710$	0	0
$711$	0	0
$712$	−7.04403	−0.263986
$713$	7.06380	0.264541
$714$	0	0
$715$	0	0
$716$	−23.3344	−0.872049
$717$	0	0
$718$	−5.61450	−0.209531
$719$	−2.09269	−0.0780440	−0.0390220	−	0.999238i	$-0.512424\pi$
$719$	−2.09269	−0.0780440	−0.0390220	+	0.999238i	$0.512424\pi$
$720$	0	0
$721$	6.88697	0.256484
$722$	7.40885	0.275729
$723$	0	0
$724$	44.5494	1.65567
$725$	0	0
$726$	0	0
$727$	−11.5637	−0.428872	−0.214436	−	0.976738i	$-0.568791\pi$
$727$	−11.5637	−0.428872	−0.214436	+	0.976738i	$0.568791\pi$
$728$	1.64291	0.0608904
$729$	0	0
$730$	0	0
$731$	−11.0694	−0.409417
$732$	0	0
$733$	19.6851	0.727087	0.363544	−	0.931577i	$-0.381567\pi$
$733$	19.6851	0.727087	0.363544	+	0.931577i	$0.381567\pi$
$734$	0.175846	0.00649059
$735$	0	0
$736$	−21.3608	−0.787369
$737$	15.7414	0.579840
$738$	0	0
$739$	−1.35449	−0.0498258	−0.0249129	−	0.999690i	$-0.507931\pi$
$739$	−1.35449	−0.0498258	−0.0249129	+	0.999690i	$0.507931\pi$
$740$	0	0
$741$	0	0
$742$	−6.22602	−0.228564
$743$	28.3972	1.04179	0.520895	−	0.853621i	$-0.325598\pi$
$743$	28.3972	1.04179	0.520895	+	0.853621i	$0.325598\pi$
$744$	0	0
$745$	0	0
$746$	−4.92387	−0.180276
$747$	0	0
$748$	−6.37449	−0.233074
$749$	−8.97523	−0.327948
$750$	0	0
$751$	16.4155	0.599010	0.299505	−	0.954095i	$-0.403178\pi$
$751$	16.4155	0.599010	0.299505	+	0.954095i	$0.403178\pi$
$752$	−27.1036	−0.988368
$753$	0	0
$754$	1.49183	0.0543291
$755$	0	0
$756$	0	0
$757$	−19.3374	−0.702829	−0.351415	−	0.936220i	$-0.614299\pi$
$757$	−19.3374	−0.702829	−0.351415	+	0.936220i	$0.614299\pi$
$758$	−3.85661	−0.140078
$759$	0	0
$760$	0	0
$761$	−21.6923	−0.786346	−0.393173	−	0.919464i	$-0.628623\pi$
$761$	−21.6923	−0.786346	−0.393173	+	0.919464i	$0.628623\pi$
$762$	0	0
$763$	31.9012	1.15490
$764$	6.76756	0.244842
$765$	0	0
$766$	0.734188	0.0265273
$767$	6.89859	0.249094
$768$	0	0
$769$	−23.6351	−0.852305	−0.426153	−	0.904651i	$-0.640131\pi$
$769$	−23.6351	−0.852305	−0.426153	+	0.904651i	$0.640131\pi$
$770$	0	0
$771$	0	0
$772$	−4.24334	−0.152721
$773$	−3.67852	−0.132307	−0.0661536	−	0.997809i	$-0.521073\pi$
$773$	−3.67852	−0.132307	−0.0661536	+	0.997809i	$0.521073\pi$
$774$	0	0
$775$	0	0
$776$	−15.8913	−0.570463
$777$	0	0
$778$	−10.0210	−0.359271
$779$	−20.7310	−0.742766
$780$	0	0
$781$	−11.7098	−0.419009
$782$	5.43661	0.194413
$783$	0	0
$784$	−8.58793	−0.306712
$785$	0	0
$786$	0	0
$787$	3.62197	0.129109	0.0645547	−	0.997914i	$-0.479437\pi$
$787$	3.62197	0.129109	0.0645547	+	0.997914i	$0.479437\pi$
$788$	20.2445	0.721182
$789$	0	0
$790$	0	0
$791$	23.3288	0.829477
$792$	0	0
$793$	−6.18427	−0.219610
$794$	2.73864	0.0971907
$795$	0	0
$796$	26.0498	0.923312
$797$	4.34495	0.153906	0.0769529	−	0.997035i	$-0.475481\pi$
$797$	4.34495	0.153906	0.0769529	+	0.997035i	$0.475481\pi$
$798$	0	0
$799$	22.0185	0.778958
$800$	0	0
$801$	0	0
$802$	3.26557	0.115311
$803$	−5.78658	−0.204204
$804$	0	0
$805$	0	0
$806$	−0.194793	−0.00686130
$807$	0	0
$808$	19.1067	0.672170
$809$	8.74314	0.307392	0.153696	−	0.988118i	$-0.450882\pi$
$809$	8.74314	0.307392	0.153696	+	0.988118i	$0.450882\pi$
$810$	0	0
$811$	28.7426	1.00929	0.504644	−	0.863328i	$-0.331624\pi$
$811$	28.7426	1.00929	0.504644	+	0.863328i	$0.331624\pi$
$812$	31.7229	1.11326
$813$	0	0
$814$	−1.67079	−0.0585612
$815$	0	0
$816$	0	0
$817$	26.0612	0.911764
$818$	8.94257	0.312670
$819$	0	0
$820$	0	0
$821$	−53.2555	−1.85863	−0.929315	−	0.369288i	$-0.879602\pi$
$821$	−53.2555	−1.85863	−0.929315	+	0.369288i	$0.879602\pi$
$822$	0	0
$823$	−22.7844	−0.794213	−0.397106	−	0.917773i	$-0.629986\pi$
$823$	−22.7844	−0.794213	−0.397106	+	0.917773i	$0.629986\pi$
$824$	3.33214	0.116081
$825$	0	0
$826$	−5.30423	−0.184558
$827$	12.7870	0.444648	0.222324	−	0.974973i	$-0.428636\pi$
$827$	12.7870	0.444648	0.222324	+	0.974973i	$0.428636\pi$
$828$	0	0
$829$	5.19645	0.180480	0.0902400	−	0.995920i	$-0.471237\pi$
$829$	5.19645	0.180480	0.0902400	+	0.995920i	$0.471237\pi$
$830$	0	0
$831$	0	0
$832$	−4.69933	−0.162920
$833$	6.97668	0.241728
$834$	0	0
$835$	0	0
$836$	15.0077	0.519053
$837$	0	0
$838$	−6.58021	−0.227310
$839$	6.70333	0.231425	0.115712	−	0.993283i	$-0.463085\pi$
$839$	6.70333	0.231425	0.115712	+	0.993283i	$0.463085\pi$
$840$	0	0
$841$	29.6527	1.02251
$842$	1.85966	0.0640882
$843$	0	0
$844$	25.5367	0.879010
$845$	0	0
$846$	0	0
$847$	20.8481	0.716350
$848$	39.3826	1.35240
$849$	0	0
$850$	0	0
$851$	−39.4093	−1.35093
$852$	0	0
$853$	7.30984	0.250284	0.125142	−	0.992139i	$-0.460061\pi$
$853$	7.30984	0.250284	0.125142	+	0.992139i	$0.460061\pi$
$854$	4.75500	0.162713
$855$	0	0
$856$	−4.34251	−0.148424
$857$	24.4667	0.835768	0.417884	−	0.908500i	$-0.362772\pi$
$857$	24.4667	0.835768	0.417884	+	0.908500i	$0.362772\pi$
$858$	0	0
$859$	−35.9327	−1.22601	−0.613005	−	0.790079i	$-0.710039\pi$
$859$	−35.9327	−1.22601	−0.613005	+	0.790079i	$0.710039\pi$
$860$	0	0
$861$	0	0
$862$	1.83363	0.0624535
$863$	−14.7380	−0.501686	−0.250843	−	0.968028i	$-0.580708\pi$
$863$	−14.7380	−0.501686	−0.250843	+	0.968028i	$0.580708\pi$
$864$	0	0
$865$	0	0
$866$	4.29826	0.146061
$867$	0	0
$868$	−4.14218	−0.140595
$869$	−6.54042	−0.221869
$870$	0	0
$871$	−10.2389	−0.346933
$872$	15.4348	0.522689
$873$	0	0
$874$	−12.7996	−0.432954
$875$	0	0
$876$	0	0
$877$	−1.10555	−0.0373317	−0.0186658	−	0.999826i	$-0.505942\pi$
$877$	−1.10555	−0.0373317	−0.0186658	+	0.999826i	$0.505942\pi$
$878$	3.46460	0.116925
$879$	0	0
$880$	0	0
$881$	4.50629	0.151821	0.0759104	−	0.997115i	$-0.475814\pi$
$881$	4.50629	0.151821	0.0759104	+	0.997115i	$0.475814\pi$
$882$	0	0
$883$	22.1741	0.746218	0.373109	−	0.927787i	$-0.378292\pi$
$883$	22.1741	0.746218	0.373109	+	0.927787i	$0.378292\pi$
$884$	4.14627	0.139454
$885$	0	0
$886$	6.95942	0.233806
$887$	27.5123	0.923773	0.461887	−	0.886939i	$-0.347173\pi$
$887$	27.5123	0.923773	0.461887	+	0.886939i	$0.347173\pi$
$888$	0	0
$889$	9.06798	0.304130
$890$	0	0
$891$	0	0
$892$	29.3459	0.982575
$893$	−51.8391	−1.73473
$894$	0	0
$895$	0	0
$896$	16.5920	0.554301
$897$	0	0
$898$	3.40387	0.113589
$899$	−7.65851	−0.255425
$900$	0	0
$901$	−31.9937	−1.06586
$902$	0.905167	0.0301388
$903$	0	0
$904$	11.2872	0.375408
$905$	0	0
$906$	0	0
$907$	5.13775	0.170596	0.0852982	−	0.996355i	$-0.472816\pi$
$907$	5.13775	0.170596	0.0852982	+	0.996355i	$0.472816\pi$
$908$	2.92207	0.0969722
$909$	0	0
$910$	0	0
$911$	24.8391	0.822957	0.411478	−	0.911420i	$-0.365013\pi$
$911$	24.8391	0.822957	0.411478	+	0.911420i	$0.365013\pi$
$912$	0	0
$913$	−1.14643	−0.0379412
$914$	−0.0323043	−0.00106853
$915$	0	0
$916$	−16.5153	−0.545681
$917$	14.1612	0.467645
$918$	0	0
$919$	−24.2553	−0.800108	−0.400054	−	0.916492i	$-0.631009\pi$
$919$	−24.2553	−0.800108	−0.400054	+	0.916492i	$0.631009\pi$
$920$	0	0
$921$	0	0
$922$	6.12942	0.201862
$923$	7.61659	0.250703
$924$	0	0
$925$	0	0
$926$	−1.05543	−0.0346836
$927$	0	0
$928$	23.1592	0.760237
$929$	−20.3390	−0.667301	−0.333650	−	0.942697i	$-0.608281\pi$
$929$	−20.3390	−0.667301	−0.333650	+	0.942697i	$0.608281\pi$
$930$	0	0
$931$	−16.4255	−0.538324
$932$	55.9955	1.83419
$933$	0	0
$934$	−10.1789	−0.333062
$935$	0	0
$936$	0	0
$937$	56.2552	1.83778	0.918888	−	0.394518i	$-0.129088\pi$
$937$	56.2552	1.83778	0.918888	+	0.394518i	$0.129088\pi$
$938$	7.87256	0.257048
$939$	0	0
$940$	0	0
$941$	−34.4730	−1.12379	−0.561894	−	0.827210i	$-0.689927\pi$
$941$	−34.4730	−1.12379	−0.561894	+	0.827210i	$0.689927\pi$
$942$	0	0
$943$	21.3504	0.695263
$944$	33.5518	1.09202
$945$	0	0
$946$	−1.13789	−0.0369961
$947$	−39.4986	−1.28353	−0.641766	−	0.766901i	$-0.721798\pi$
$947$	−39.4986	−1.28353	−0.641766	+	0.766901i	$0.721798\pi$
$948$	0	0
$949$	3.76386	0.122180
$950$	0	0
$951$	0	0
$952$	−6.49128	−0.210384
$953$	21.9633	0.711462	0.355731	−	0.934588i	$-0.384232\pi$
$953$	21.9633	0.711462	0.355731	+	0.934588i	$0.384232\pi$
$954$	0	0
$955$	0	0
$956$	−25.6021	−0.828031
$957$	0	0
$958$	1.09677	0.0354350
$959$	47.3226	1.52813
$960$	0	0
$961$	1.00000	0.0322581
$962$	1.08676	0.0350386
$963$	0	0
$964$	9.05882	0.291765
$965$	0	0
$966$	0	0
$967$	40.1621	1.29153	0.645763	−	0.763538i	$-0.276539\pi$
$967$	40.1621	1.29153	0.645763	+	0.763538i	$0.276539\pi$
$968$	10.0870	0.324208
$969$	0	0
$970$	0	0
$971$	57.3568	1.84067	0.920333	−	0.391136i	$-0.127918\pi$
$971$	57.3568	1.84067	0.920333	+	0.391136i	$0.127918\pi$
$972$	0	0
$973$	−24.1050	−0.772770
$974$	−0.333976	−0.0107013
$975$	0	0
$976$	−30.0777	−0.962762
$977$	−13.1468	−0.420605	−0.210302	−	0.977636i	$-0.567445\pi$
$977$	−13.1468	−0.420605	−0.210302	+	0.977636i	$0.567445\pi$
$978$	0	0
$979$	−7.69055	−0.245791
$980$	0	0
$981$	0	0
$982$	9.65385	0.308067
$983$	1.42361	0.0454062	0.0227031	−	0.999742i	$-0.492773\pi$
$983$	1.42361	0.0454062	0.0227031	+	0.999742i	$0.492773\pi$
$984$	0	0
$985$	0	0
$986$	−5.89432	−0.187713
$987$	0	0
$988$	−9.76172	−0.310562
$989$	−26.8397	−0.853454
$990$	0	0
$991$	2.78624	0.0885077	0.0442539	−	0.999020i	$-0.485909\pi$
$991$	2.78624	0.0885077	0.0442539	+	0.999020i	$0.485909\pi$
$992$	−3.02398	−0.0960115
$993$	0	0
$994$	−5.85629	−0.185750
$995$	0	0
$996$	0	0
$997$	−22.0927	−0.699684	−0.349842	−	0.936809i	$-0.613765\pi$
$997$	−22.0927	−0.699684	−0.349842	+	0.936809i	$0.613765\pi$
$998$	−4.17130	−0.132040
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bp.1.4		5
3.2	odd	2		775.2.a.k.1.2	yes	5
5.4	even	2		6975.2.a.by.1.2		5
15.2	even	4		775.2.b.g.249.5		10
15.8	even	4		775.2.b.g.249.6		10
15.14	odd	2		775.2.a.h.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
775.2.a.h.1.4	✓	5	15.14	odd	2
775.2.a.k.1.2	yes	5	3.2	odd	2
775.2.b.g.249.5		10	15.2	even	4
775.2.b.g.249.6		10	15.8	even	4
6975.2.a.bp.1.4		5	1.1	even	1	trivial
6975.2.a.by.1.2		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+0.264183 q^{2} -1.93021 q^{4} -2.14598 q^{7} -1.03829 q^{8} -1.13359 q^{11} +0.737342 q^{13} -0.566932 q^{14} +3.58611 q^{16} -2.91329 q^{17} +6.85889 q^{19} -0.299476 q^{22} -7.06380 q^{23} +0.194793 q^{26} +4.14218 q^{28} +7.65851 q^{29} -1.00000 q^{31} +3.02398 q^{32} -0.769643 q^{34} +5.57905 q^{37} +1.81200 q^{38} -3.02250 q^{41} +3.79962 q^{43} +2.18807 q^{44} -1.86614 q^{46} -7.55794 q^{47} -2.39477 q^{49} -1.42322 q^{52} +10.9820 q^{53} +2.22816 q^{56} +2.02325 q^{58} +9.35603 q^{59} -8.38726 q^{61} -0.264183 q^{62} -6.37334 q^{64} -13.8863 q^{67} +5.62326 q^{68} +10.3298 q^{71} +5.10463 q^{73} +1.47389 q^{74} -13.2391 q^{76} +2.43267 q^{77} +5.76964 q^{79} -0.798495 q^{82} +1.01132 q^{83} +1.00379 q^{86} +1.17700 q^{88} +6.78423 q^{89} -1.58232 q^{91} +13.6346 q^{92} -1.99668 q^{94} +15.3051 q^{97} -0.632659 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} + 6 q^{4} + 2 q^{7} - 9 q^{8} + 2 q^{11} + 4 q^{13} - 2 q^{14} + 4 q^{16} - 19 q^{17} + 8 q^{19} - 10 q^{22} - 12 q^{23} + 16 q^{26} + 6 q^{28} - 6 q^{29} - 5 q^{31} - 7 q^{32} + 31 q^{34}+ \cdots - 10 q^{98}+O(q^{100}) \) 5 * q - 4 * q^2 + 6 * q^4 + 2 * q^7 - 9 * q^8 + 2 * q^11 + 4 * q^13 - 2 * q^14 + 4 * q^16 - 19 * q^17 + 8 * q^19 - 10 * q^22 - 12 * q^23 + 16 * q^26 + 6 * q^28 - 6 * q^29 - 5 * q^31 - 7 * q^32 + 31 * q^34 + 16 * q^37 - 14 * q^38 + 2 * q^41 + q^43 + 4 * q^44 - 11 * q^46 - 8 * q^47 - 9 * q^49 - 26 * q^52 - 3 * q^53 + 9 * q^56 + 5 * q^58 + 4 * q^59 - 5 * q^61 + 4 * q^62 - 5 * q^64 + 13 * q^67 - 30 * q^68 - 6 * q^71 + 19 * q^73 - 4 * q^74 - 5 * q^76 - 23 * q^77 - 6 * q^79 - 27 * q^82 - 18 * q^83 + 7 * q^86 - q^88 + 31 * q^89 + 8 * q^91 + 16 * q^92 - 14 * q^94 - q^97 - 10 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more