LMFDB - Embedded newform 6975.2.a.ci.1.5

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 = [11,-3,0,15,0,0,8,-9,0,0,0,0,14,14,0,27,-12] 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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 3x^{10} - 14x^{9} + 44x^{8} + 61x^{7} - 211x^{6} - 83x^{5} + 369x^{4} + 10x^{3} - 168x^{2} - 31x + 5 $ x^11 - 3x^10 - 14x^9 + 44x^8 + 61x^7 - 211x^6 - 83x^5 + 369x^4 + 10x^3 - 168x^2 - 31x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.31655$ 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)

$f(q)$	$=$	$q-1.31655 q^{2} -0.266704 q^{4} +4.78991 q^{7} +2.98422 q^{8} -0.621980 q^{11} +5.28373 q^{13} -6.30614 q^{14} -3.39546 q^{16} +1.89215 q^{17} +5.01481 q^{19} +0.818866 q^{22} +7.01697 q^{23} -6.95628 q^{26} -1.27749 q^{28} +2.99080 q^{29} +1.00000 q^{31} -1.49816 q^{32} -2.49111 q^{34} -4.38561 q^{37} -6.60224 q^{38} -6.36708 q^{41} +12.0564 q^{43} +0.165884 q^{44} -9.23817 q^{46} +5.34541 q^{47} +15.9432 q^{49} -1.40919 q^{52} +7.76340 q^{53} +14.2942 q^{56} -3.93752 q^{58} +11.9353 q^{59} +0.500639 q^{61} -1.31655 q^{62} +8.76332 q^{64} +2.30056 q^{67} -0.504644 q^{68} -5.35628 q^{71} +10.2712 q^{73} +5.77387 q^{74} -1.33747 q^{76} -2.97923 q^{77} +6.07634 q^{79} +8.38257 q^{82} -14.0387 q^{83} -15.8729 q^{86} -1.85613 q^{88} -14.4228 q^{89} +25.3086 q^{91} -1.87145 q^{92} -7.03748 q^{94} -13.7770 q^{97} -20.9900 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 3 q^{2} + 15 q^{4} + 8 q^{7} - 9 q^{8} + 14 q^{13} + 14 q^{14} + 27 q^{16} - 12 q^{17} + 12 q^{19} + 10 q^{22} - 12 q^{23} + 6 q^{26} + 22 q^{28} + 8 q^{29} + 11 q^{31} - 21 q^{32} + 2 q^{34} + 16 q^{37}+ \cdots - 17 q^{98}+O(q^{100}) $ 11 * q - 3 * q^2 + 15 * q^4 + 8 * q^7 - 9 * q^8 + 14 * q^13 + 14 * q^14 + 27 * q^16 - 12 * q^17 + 12 * q^19 + 10 * q^22 - 12 * q^23 + 6 * q^26 + 22 * q^28 + 8 * q^29 + 11 * q^31 - 21 * q^32 + 2 * q^34 + 16 * q^37 - 26 * q^38 + 8 * q^41 + 28 * q^43 + 26 * q^44 + 8 * q^46 - 14 * q^47 + 23 * q^49 + 18 * q^52 - 26 * q^53 + 42 * q^56 + 6 * q^58 + 6 * q^59 + 14 * q^61 - 3 * q^62 + 29 * q^64 + 6 * q^67 - 54 * q^68 + 18 * q^71 + 26 * q^73 + 40 * q^74 + 10 * q^76 - 2 * q^77 + 8 * q^79 - 30 * q^82 - 48 * q^83 + 12 * q^86 + 40 * q^88 + 14 * q^89 - 4 * q^91 + 30 * q^92 - 4 * q^94 + 16 * q^97 - 17 * 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 $

$2$	−1.31655	−0.930939	−0.465470	−	0.885064i	$-0.654115\pi$
$2$	−1.31655	−0.930939	−0.465470	+	0.885064i	$0.654115\pi$
$3$	0	0
$3$	0	0
$4$	−0.266704	−0.133352
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.78991	1.81042	0.905208	−	0.424969i	$-0.139715\pi$
$7$	4.78991	1.81042	0.905208	+	0.424969i	$0.139715\pi$
$8$	2.98422	1.05508
$9$	0	0
$10$	0	0
$11$	−0.621980	−0.187534	−0.0937670	−	0.995594i	$-0.529891\pi$
$11$	−0.621980	−0.187534	−0.0937670	+	0.995594i	$0.529891\pi$
$12$	0	0
$13$	5.28373	1.46544	0.732721	−	0.680529i	$-0.238250\pi$
$13$	5.28373	1.46544	0.732721	+	0.680529i	$0.238250\pi$
$14$	−6.30614	−1.68539
$15$	0	0
$16$	−3.39546	−0.848866
$17$	1.89215	0.458915	0.229457	−	0.973319i	$-0.426305\pi$
$17$	1.89215	0.458915	0.229457	+	0.973319i	$0.426305\pi$
$18$	0	0
$19$	5.01481	1.15048	0.575238	−	0.817986i	$-0.304909\pi$
$19$	5.01481	1.15048	0.575238	+	0.817986i	$0.304909\pi$
$20$	0	0
$21$	0	0
$22$	0.818866	0.174583
$23$	7.01697	1.46314	0.731570	−	0.681767i	$-0.238788\pi$
$23$	7.01697	1.46314	0.731570	+	0.681767i	$0.238788\pi$
$24$	0	0
$25$	0	0
$26$	−6.95628	−1.36424
$27$	0	0
$28$	−1.27749	−0.241422
$29$	2.99080	0.555377	0.277688	−	0.960671i	$-0.410432\pi$
$29$	2.99080	0.555377	0.277688	+	0.960671i	$0.410432\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.49816	−0.264839
$33$	0	0
$34$	−2.49111	−0.427222
$35$	0	0
$36$	0	0
$37$	−4.38561	−0.720990	−0.360495	−	0.932761i	$-0.617392\pi$
$37$	−4.38561	−0.720990	−0.360495	+	0.932761i	$0.617392\pi$
$38$	−6.60224	−1.07102
$39$	0	0
$40$	0	0
$41$	−6.36708	−0.994371	−0.497186	−	0.867644i	$-0.665633\pi$
$41$	−6.36708	−0.994371	−0.497186	+	0.867644i	$0.665633\pi$
$42$	0	0
$43$	12.0564	1.83859	0.919294	−	0.393572i	$-0.128761\pi$
$43$	12.0564	1.83859	0.919294	+	0.393572i	$0.128761\pi$
$44$	0.165884	0.0250080
$45$	0	0
$46$	−9.23817	−1.36209
$47$	5.34541	0.779708	0.389854	−	0.920877i	$-0.372526\pi$
$47$	5.34541	0.779708	0.389854	+	0.920877i	$0.372526\pi$
$48$	0	0
$49$	15.9432	2.27761
$50$	0	0
$51$	0	0
$52$	−1.40919	−0.195419
$53$	7.76340	1.06638	0.533192	−	0.845994i	$-0.320992\pi$
$53$	7.76340	1.06638	0.533192	+	0.845994i	$0.320992\pi$
$54$	0	0
$55$	0	0
$56$	14.2942	1.91014
$57$	0	0
$58$	−3.93752	−0.517022
$59$	11.9353	1.55384	0.776921	−	0.629598i	$-0.216780\pi$
$59$	11.9353	1.55384	0.776921	+	0.629598i	$0.216780\pi$
$60$	0	0
$61$	0.500639	0.0641003	0.0320501	−	0.999486i	$-0.489796\pi$
$61$	0.500639	0.0641003	0.0320501	+	0.999486i	$0.489796\pi$
$62$	−1.31655	−0.167202
$63$	0	0
$64$	8.76332	1.09542
$65$	0	0
$66$	0	0
$67$	2.30056	0.281058	0.140529	−	0.990077i	$-0.455120\pi$
$67$	2.30056	0.281058	0.140529	+	0.990077i	$0.455120\pi$
$68$	−0.504644	−0.0611971
$69$	0	0
$70$	0	0
$71$	−5.35628	−0.635674	−0.317837	−	0.948145i	$-0.602956\pi$
$71$	−5.35628	−0.635674	−0.317837	+	0.948145i	$0.602956\pi$
$72$	0	0
$73$	10.2712	1.20216	0.601079	−	0.799190i	$-0.294738\pi$
$73$	10.2712	1.20216	0.601079	+	0.799190i	$0.294738\pi$
$74$	5.77387	0.671198
$75$	0	0
$76$	−1.33747	−0.153418
$77$	−2.97923	−0.339515
$78$	0	0
$79$	6.07634	0.683642	0.341821	−	0.939765i	$-0.388956\pi$
$79$	6.07634	0.683642	0.341821	+	0.939765i	$0.388956\pi$
$80$	0	0
$81$	0	0
$82$	8.38257	0.925700
$83$	−14.0387	−1.54095	−0.770473	−	0.637473i	$-0.779980\pi$
$83$	−14.0387	−1.54095	−0.770473	+	0.637473i	$0.779980\pi$
$84$	0	0
$85$	0	0
$86$	−15.8729	−1.71161
$87$	0	0
$88$	−1.85613	−0.197864
$89$	−14.4228	−1.52881	−0.764405	−	0.644736i	$-0.776967\pi$
$89$	−14.4228	−1.52881	−0.764405	+	0.644736i	$0.776967\pi$
$90$	0	0
$91$	25.3086	2.65306
$92$	−1.87145	−0.195112
$93$	0	0
$94$	−7.03748	−0.725861
$95$	0	0
$96$	0	0
$97$	−13.7770	−1.39884	−0.699422	−	0.714709i	$-0.746559\pi$
$97$	−13.7770	−1.39884	−0.699422	+	0.714709i	$0.746559\pi$
$98$	−20.9900	−2.12031
$99$	0	0
$100$	0	0
$101$	5.92628	0.589687	0.294843	−	0.955546i	$-0.404733\pi$
$101$	5.92628	0.589687	0.294843	+	0.955546i	$0.404733\pi$
$102$	0	0
$103$	−16.3671	−1.61270	−0.806350	−	0.591439i	$-0.798560\pi$
$103$	−16.3671	−1.61270	−0.806350	+	0.591439i	$0.798560\pi$
$104$	15.7678	1.54616
$105$	0	0
$106$	−10.2209	−0.992740
$107$	−0.670149	−0.0647858	−0.0323929	−	0.999475i	$-0.510313\pi$
$107$	−0.670149	−0.0647858	−0.0323929	+	0.999475i	$0.510313\pi$
$108$	0	0
$109$	−2.65083	−0.253904	−0.126952	−	0.991909i	$-0.540519\pi$
$109$	−2.65083	−0.253904	−0.126952	+	0.991909i	$0.540519\pi$
$110$	0	0
$111$	0	0
$112$	−16.2640	−1.53680
$113$	−8.81055	−0.828827	−0.414413	−	0.910089i	$-0.636013\pi$
$113$	−8.81055	−0.828827	−0.414413	+	0.910089i	$0.636013\pi$
$114$	0	0
$115$	0	0
$116$	−0.797656	−0.0740605
$117$	0	0
$118$	−15.7134	−1.44653
$119$	9.06325	0.830827
$120$	0	0
$121$	−10.6131	−0.964831
$122$	−0.659115	−0.0596735
$123$	0	0
$124$	−0.266704	−0.0239507
$125$	0	0
$126$	0	0
$127$	6.01770	0.533984	0.266992	−	0.963699i	$-0.413970\pi$
$127$	6.01770	0.533984	0.266992	+	0.963699i	$0.413970\pi$
$128$	−8.54101	−0.754926
$129$	0	0
$130$	0	0
$131$	21.6015	1.88733	0.943665	−	0.330901i	$-0.107353\pi$
$131$	21.6015	1.88733	0.943665	+	0.330901i	$0.107353\pi$
$132$	0	0
$133$	24.0205	2.08284
$134$	−3.02879	−0.261648
$135$	0	0
$136$	5.64661	0.484193
$137$	−9.54736	−0.815686	−0.407843	−	0.913052i	$-0.633719\pi$
$137$	−9.54736	−0.815686	−0.407843	+	0.913052i	$0.633719\pi$
$138$	0	0
$139$	−6.63331	−0.562630	−0.281315	−	0.959616i	$-0.590771\pi$
$139$	−6.63331	−0.562630	−0.281315	+	0.959616i	$0.590771\pi$
$140$	0	0
$141$	0	0
$142$	7.05180	0.591774
$143$	−3.28637	−0.274820
$144$	0	0
$145$	0	0
$146$	−13.5226	−1.11914
$147$	0	0
$148$	1.16966	0.0961453
$149$	10.5590	0.865027	0.432514	−	0.901627i	$-0.357627\pi$
$149$	10.5590	0.865027	0.432514	+	0.901627i	$0.357627\pi$
$150$	0	0
$151$	−9.89354	−0.805125	−0.402562	−	0.915393i	$-0.631880\pi$
$151$	−9.89354	−0.805125	−0.402562	+	0.915393i	$0.631880\pi$
$152$	14.9653	1.21385
$153$	0	0
$154$	3.92229	0.316067
$155$	0	0
$156$	0	0
$157$	−16.4308	−1.31132	−0.655661	−	0.755056i	$-0.727610\pi$
$157$	−16.4308	−1.31132	−0.655661	+	0.755056i	$0.727610\pi$
$158$	−7.99979	−0.636429
$159$	0	0
$160$	0	0
$161$	33.6107	2.64889
$162$	0	0
$163$	3.54054	0.277317	0.138658	−	0.990340i	$-0.455721\pi$
$163$	3.54054	0.277317	0.138658	+	0.990340i	$0.455721\pi$
$164$	1.69812	0.132601
$165$	0	0
$166$	18.4826	1.43453
$167$	−0.0257473	−0.00199238	−0.000996192	−	1.00000i	$-0.500317\pi$
$167$	−0.0257473	−0.00199238	−0.000996192		1.00000i	$0.500317\pi$
$168$	0	0
$169$	14.9178	1.14752
$170$	0	0
$171$	0	0
$172$	−3.21549	−0.245179
$173$	−16.8380	−1.28017	−0.640083	−	0.768306i	$-0.721100\pi$
$173$	−16.8380	−1.28017	−0.640083	+	0.768306i	$0.721100\pi$
$174$	0	0
$175$	0	0
$176$	2.11191	0.159191
$177$	0	0
$178$	18.9882	1.42323
$179$	−4.18021	−0.312443	−0.156222	−	0.987722i	$-0.549931\pi$
$179$	−4.18021	−0.312443	−0.156222	+	0.987722i	$0.549931\pi$
$180$	0	0
$181$	−2.25540	−0.167643	−0.0838213	−	0.996481i	$-0.526713\pi$
$181$	−2.25540	−0.167643	−0.0838213	+	0.996481i	$0.526713\pi$
$182$	−33.3199	−2.46984
$183$	0	0
$184$	20.9402	1.54373
$185$	0	0
$186$	0	0
$187$	−1.17688	−0.0860621
$188$	−1.42564	−0.103975
$189$	0	0
$190$	0	0
$191$	−6.13086	−0.443613	−0.221807	−	0.975091i	$-0.571195\pi$
$191$	−6.13086	−0.443613	−0.221807	+	0.975091i	$0.571195\pi$
$192$	0	0
$193$	−15.1636	−1.09150	−0.545749	−	0.837948i	$-0.683755\pi$
$193$	−15.1636	−1.09150	−0.545749	+	0.837948i	$0.683755\pi$
$194$	18.1381	1.30224
$195$	0	0
$196$	−4.25212	−0.303723
$197$	−12.5535	−0.894402	−0.447201	−	0.894434i	$-0.647579\pi$
$197$	−12.5535	−0.894402	−0.447201	+	0.894434i	$0.647579\pi$
$198$	0	0
$199$	−26.1678	−1.85499	−0.927493	−	0.373841i	$-0.878041\pi$
$199$	−26.1678	−1.85499	−0.927493	+	0.373841i	$0.878041\pi$
$200$	0	0
$201$	0	0
$202$	−7.80222	−0.548962
$203$	14.3256	1.00546
$204$	0	0
$205$	0	0
$206$	21.5481	1.50133
$207$	0	0
$208$	−17.9407	−1.24396
$209$	−3.11911	−0.215753
$210$	0	0
$211$	2.34788	0.161635	0.0808175	−	0.996729i	$-0.474247\pi$
$211$	2.34788	0.161635	0.0808175	+	0.996729i	$0.474247\pi$
$212$	−2.07053	−0.142204
$213$	0	0
$214$	0.882283	0.0603117
$215$	0	0
$216$	0	0
$217$	4.78991	0.325160
$218$	3.48994	0.236369
$219$	0	0
$220$	0	0
$221$	9.99762	0.672513
$222$	0	0
$223$	−6.71412	−0.449610	−0.224805	−	0.974404i	$-0.572175\pi$
$223$	−6.71412	−0.449610	−0.224805	+	0.974404i	$0.572175\pi$
$224$	−7.17605	−0.479470
$225$	0	0
$226$	11.5995	0.771587
$227$	0.306205	0.0203236	0.0101618	−	0.999948i	$-0.496765\pi$
$227$	0.306205	0.0203236	0.0101618	+	0.999948i	$0.496765\pi$
$228$	0	0
$229$	−10.8908	−0.719686	−0.359843	−	0.933013i	$-0.617170\pi$
$229$	−10.8908	−0.719686	−0.359843	+	0.933013i	$0.617170\pi$
$230$	0	0
$231$	0	0
$232$	8.92520	0.585968
$233$	8.63294	0.565563	0.282781	−	0.959184i	$-0.408743\pi$
$233$	8.63294	0.565563	0.282781	+	0.959184i	$0.408743\pi$
$234$	0	0
$235$	0	0
$236$	−3.18318	−0.207208
$237$	0	0
$238$	−11.9322	−0.773449
$239$	10.3230	0.667741	0.333871	−	0.942619i	$-0.391645\pi$
$239$	10.3230	0.667741	0.333871	+	0.942619i	$0.391645\pi$
$240$	0	0
$241$	−17.9750	−1.15787	−0.578935	−	0.815374i	$-0.696532\pi$
$241$	−17.9750	−1.15787	−0.578935	+	0.815374i	$0.696532\pi$
$242$	13.9727	0.898199
$243$	0	0
$244$	−0.133522	−0.00854788
$245$	0	0
$246$	0	0
$247$	26.4969	1.68596
$248$	2.98422	0.189498
$249$	0	0
$250$	0	0
$251$	2.13566	0.134802	0.0674010	−	0.997726i	$-0.478529\pi$
$251$	2.13566	0.134802	0.0674010	+	0.997726i	$0.478529\pi$
$252$	0	0
$253$	−4.36441	−0.274388
$254$	−7.92258	−0.497107
$255$	0	0
$256$	−6.28200	−0.392625
$257$	−16.2352	−1.01272	−0.506361	−	0.862322i	$-0.669010\pi$
$257$	−16.2352	−1.01272	−0.506361	+	0.862322i	$0.669010\pi$
$258$	0	0
$259$	−21.0067	−1.30529
$260$	0	0
$261$	0	0
$262$	−28.4394	−1.75699
$263$	8.67821	0.535121	0.267561	−	0.963541i	$-0.413782\pi$
$263$	8.67821	0.535121	0.267561	+	0.963541i	$0.413782\pi$
$264$	0	0
$265$	0	0
$266$	−31.6241	−1.93900
$267$	0	0
$268$	−0.613567	−0.0374796
$269$	−4.96064	−0.302455	−0.151228	−	0.988499i	$-0.548323\pi$
$269$	−4.96064	−0.302455	−0.151228	+	0.988499i	$0.548323\pi$
$270$	0	0
$271$	−25.4292	−1.54472	−0.772358	−	0.635188i	$-0.780923\pi$
$271$	−25.4292	−1.54472	−0.772358	+	0.635188i	$0.780923\pi$
$272$	−6.42474	−0.389557
$273$	0	0
$274$	12.5696	0.759354
$275$	0	0
$276$	0	0
$277$	−5.29917	−0.318396	−0.159198	−	0.987247i	$-0.550891\pi$
$277$	−5.29917	−0.318396	−0.159198	+	0.987247i	$0.550891\pi$
$278$	8.73306	0.523774
$279$	0	0
$280$	0	0
$281$	−3.34105	−0.199310	−0.0996551	−	0.995022i	$-0.531774\pi$
$281$	−3.34105	−0.199310	−0.0996551	+	0.995022i	$0.531774\pi$
$282$	0	0
$283$	4.78809	0.284622	0.142311	−	0.989822i	$-0.454547\pi$
$283$	4.78809	0.284622	0.142311	+	0.989822i	$0.454547\pi$
$284$	1.42854	0.0847682
$285$	0	0
$286$	4.32666	0.255841
$287$	−30.4978	−1.80023
$288$	0	0
$289$	−13.4198	−0.789397
$290$	0	0
$291$	0	0
$292$	−2.73938	−0.160310
$293$	12.7321	0.743819	0.371909	−	0.928269i	$-0.378703\pi$
$293$	12.7321	0.743819	0.371909	+	0.928269i	$0.378703\pi$
$294$	0	0
$295$	0	0
$296$	−13.0876	−0.760704
$297$	0	0
$298$	−13.9014	−0.805288
$299$	37.0758	2.14415
$300$	0	0
$301$	57.7492	3.32861
$302$	13.0253	0.749522
$303$	0	0
$304$	−17.0276	−0.976600
$305$	0	0
$306$	0	0
$307$	4.30839	0.245893	0.122946	−	0.992413i	$-0.460766\pi$
$307$	4.30839	0.245893	0.122946	+	0.992413i	$0.460766\pi$
$308$	0.794571	0.0452749
$309$	0	0
$310$	0	0
$311$	5.86155	0.332378	0.166189	−	0.986094i	$-0.446854\pi$
$311$	5.86155	0.332378	0.166189	+	0.986094i	$0.446854\pi$
$312$	0	0
$313$	−8.03818	−0.454345	−0.227172	−	0.973855i	$-0.572948\pi$
$313$	−8.03818	−0.454345	−0.227172	+	0.973855i	$0.572948\pi$
$314$	21.6319	1.22076
$315$	0	0
$316$	−1.62058	−0.0911649
$317$	−7.33047	−0.411720	−0.205860	−	0.978581i	$-0.565999\pi$
$317$	−7.33047	−0.411720	−0.205860	+	0.978581i	$0.565999\pi$
$318$	0	0
$319$	−1.86021	−0.104152
$320$	0	0
$321$	0	0
$322$	−44.2500	−2.46596
$323$	9.48879	0.527971
$324$	0	0
$325$	0	0
$326$	−4.66129	−0.258165
$327$	0	0
$328$	−19.0008	−1.04914
$329$	25.6040	1.41160
$330$	0	0
$331$	−21.5538	−1.18470	−0.592352	−	0.805679i	$-0.701801\pi$
$331$	−21.5538	−1.18470	−0.592352	+	0.805679i	$0.701801\pi$
$332$	3.74417	0.205488
$333$	0	0
$334$	0.0338975	0.00185479
$335$	0	0
$336$	0	0
$337$	3.39409	0.184888	0.0924440	−	0.995718i	$-0.470532\pi$
$337$	3.39409	0.184888	0.0924440	+	0.995718i	$0.470532\pi$
$338$	−19.6400	−1.06827
$339$	0	0
$340$	0	0
$341$	−0.621980	−0.0336821
$342$	0	0
$343$	42.8374	2.31300
$344$	35.9791	1.93986
$345$	0	0
$346$	22.1680	1.19176
$347$	−6.68873	−0.359070	−0.179535	−	0.983752i	$-0.557459\pi$
$347$	−6.68873	−0.359070	−0.179535	+	0.983752i	$0.557459\pi$
$348$	0	0
$349$	−7.90387	−0.423085	−0.211542	−	0.977369i	$-0.567849\pi$
$349$	−7.90387	−0.423085	−0.211542	+	0.977369i	$0.567849\pi$
$350$	0	0
$351$	0	0
$352$	0.931824	0.0496664
$353$	3.90310	0.207741	0.103871	−	0.994591i	$-0.466877\pi$
$353$	3.90310	0.207741	0.103871	+	0.994591i	$0.466877\pi$
$354$	0	0
$355$	0	0
$356$	3.84660	0.203870
$357$	0	0
$358$	5.50344	0.290866
$359$	−26.1028	−1.37766	−0.688828	−	0.724925i	$-0.741874\pi$
$359$	−26.1028	−1.37766	−0.688828	+	0.724925i	$0.741874\pi$
$360$	0	0
$361$	6.14833	0.323596
$362$	2.96934	0.156065
$363$	0	0
$364$	−6.74989	−0.353790
$365$	0	0
$366$	0	0
$367$	−1.43004	−0.0746473	−0.0373237	−	0.999303i	$-0.511883\pi$
$367$	−1.43004	−0.0746473	−0.0373237	+	0.999303i	$0.511883\pi$
$368$	−23.8259	−1.24201
$369$	0	0
$370$	0	0
$371$	37.1860	1.93060
$372$	0	0
$373$	−22.0783	−1.14317	−0.571586	−	0.820542i	$-0.693672\pi$
$373$	−22.0783	−1.14317	−0.571586	+	0.820542i	$0.693672\pi$
$374$	1.54942	0.0801186
$375$	0	0
$376$	15.9519	0.822655
$377$	15.8025	0.813873
$378$	0	0
$379$	−25.7859	−1.32453	−0.662266	−	0.749269i	$-0.730405\pi$
$379$	−25.7859	−1.32453	−0.662266	+	0.749269i	$0.730405\pi$
$380$	0	0
$381$	0	0
$382$	8.07157	0.412977
$383$	7.72696	0.394829	0.197415	−	0.980320i	$-0.436746\pi$
$383$	7.72696	0.394829	0.197415	+	0.980320i	$0.436746\pi$
$384$	0	0
$385$	0	0
$386$	19.9636	1.01612
$387$	0	0
$388$	3.67438	0.186538
$389$	−30.0977	−1.52601	−0.763007	−	0.646391i	$-0.776278\pi$
$389$	−30.0977	−1.52601	−0.763007	+	0.646391i	$0.776278\pi$
$390$	0	0
$391$	13.2772	0.671456
$392$	47.5782	2.40306
$393$	0	0
$394$	16.5273	0.832634
$395$	0	0
$396$	0	0
$397$	−2.22723	−0.111781	−0.0558907	−	0.998437i	$-0.517800\pi$
$397$	−2.22723	−0.111781	−0.0558907	+	0.998437i	$0.517800\pi$
$398$	34.4511	1.72688
$399$	0	0
$400$	0	0
$401$	1.28290	0.0640650	0.0320325	−	0.999487i	$-0.489802\pi$
$401$	1.28290	0.0640650	0.0320325	+	0.999487i	$0.489802\pi$
$402$	0	0
$403$	5.28373	0.263201
$404$	−1.58056	−0.0786358
$405$	0	0
$406$	−18.8604	−0.936025
$407$	2.72776	0.135210
$408$	0	0
$409$	34.9426	1.72780	0.863901	−	0.503661i	$-0.168014\pi$
$409$	34.9426	1.72780	0.863901	+	0.503661i	$0.168014\pi$
$410$	0	0
$411$	0	0
$412$	4.36517	0.215056
$413$	57.1690	2.81310
$414$	0	0
$415$	0	0
$416$	−7.91586	−0.388107
$417$	0	0
$418$	4.10646	0.200853
$419$	6.89813	0.336996	0.168498	−	0.985702i	$-0.446108\pi$
$419$	6.89813	0.336996	0.168498	+	0.985702i	$0.446108\pi$
$420$	0	0
$421$	25.6635	1.25076	0.625380	−	0.780320i	$-0.284944\pi$
$421$	25.6635	1.25076	0.625380	+	0.780320i	$0.284944\pi$
$422$	−3.09110	−0.150472
$423$	0	0
$424$	23.1677	1.12512
$425$	0	0
$426$	0	0
$427$	2.39802	0.116048
$428$	0.178731	0.00863930
$429$	0	0
$430$	0	0
$431$	33.4105	1.60933	0.804663	−	0.593732i	$-0.202346\pi$
$431$	33.4105	1.60933	0.804663	+	0.593732i	$0.202346\pi$
$432$	0	0
$433$	−2.84749	−0.136842	−0.0684209	−	0.997657i	$-0.521796\pi$
$433$	−2.84749	−0.136842	−0.0684209	+	0.997657i	$0.521796\pi$
$434$	−6.30614	−0.302705
$435$	0	0
$436$	0.706986	0.0338585
$437$	35.1888	1.68331
$438$	0	0
$439$	−20.9545	−1.00011	−0.500053	−	0.865995i	$-0.666686\pi$
$439$	−20.9545	−1.00011	−0.500053	+	0.865995i	$0.666686\pi$
$440$	0	0
$441$	0	0
$442$	−13.1623	−0.626069
$443$	−13.1486	−0.624709	−0.312354	−	0.949966i	$-0.601118\pi$
$443$	−13.1486	−0.624709	−0.312354	+	0.949966i	$0.601118\pi$
$444$	0	0
$445$	0	0
$446$	8.83945	0.418560
$447$	0	0
$448$	41.9755	1.98316
$449$	15.5497	0.733835	0.366918	−	0.930253i	$-0.380413\pi$
$449$	15.5497	0.733835	0.366918	+	0.930253i	$0.380413\pi$
$450$	0	0
$451$	3.96020	0.186478
$452$	2.34980	0.110526
$453$	0	0
$454$	−0.403134	−0.0189200
$455$	0	0
$456$	0	0
$457$	−6.79417	−0.317818	−0.158909	−	0.987293i	$-0.550798\pi$
$457$	−6.79417	−0.317818	−0.158909	+	0.987293i	$0.550798\pi$
$458$	14.3383	0.669984
$459$	0	0
$460$	0	0
$461$	−18.9186	−0.881129	−0.440564	−	0.897721i	$-0.645222\pi$
$461$	−18.9186	−0.881129	−0.440564	+	0.897721i	$0.645222\pi$
$462$	0	0
$463$	28.5370	1.32623	0.663113	−	0.748519i	$-0.269235\pi$
$463$	28.5370	1.32623	0.663113	+	0.748519i	$0.269235\pi$
$464$	−10.1551	−0.471440
$465$	0	0
$466$	−11.3657	−0.526505
$467$	−25.5702	−1.18325	−0.591624	−	0.806214i	$-0.701513\pi$
$467$	−25.5702	−1.18325	−0.591624	+	0.806214i	$0.701513\pi$
$468$	0	0
$469$	11.0195	0.508832
$470$	0	0
$471$	0	0
$472$	35.6176	1.63943
$473$	−7.49885	−0.344798
$474$	0	0
$475$	0	0
$476$	−2.41720	−0.110792
$477$	0	0
$478$	−13.5908	−0.621627
$479$	−16.4499	−0.751616	−0.375808	−	0.926698i	$-0.622635\pi$
$479$	−16.4499	−0.751616	−0.375808	+	0.926698i	$0.622635\pi$
$480$	0	0
$481$	−23.1724	−1.05657
$482$	23.6649	1.07791
$483$	0	0
$484$	2.83056	0.128662
$485$	0	0
$486$	0	0
$487$	19.8000	0.897222	0.448611	−	0.893727i	$-0.351919\pi$
$487$	19.8000	0.897222	0.448611	+	0.893727i	$0.351919\pi$
$488$	1.49402	0.0676310
$489$	0	0
$490$	0	0
$491$	3.30945	0.149354	0.0746768	−	0.997208i	$-0.476207\pi$
$491$	3.30945	0.149354	0.0746768	+	0.997208i	$0.476207\pi$
$492$	0	0
$493$	5.65905	0.254871
$494$	−34.8844	−1.56952
$495$	0	0
$496$	−3.39546	−0.152461
$497$	−25.6561	−1.15083
$498$	0	0
$499$	14.8241	0.663620	0.331810	−	0.943346i	$-0.392341\pi$
$499$	14.8241	0.663620	0.331810	+	0.943346i	$0.392341\pi$
$500$	0	0
$501$	0	0
$502$	−2.81170	−0.125492
$503$	20.8025	0.927539	0.463769	−	0.885956i	$-0.346497\pi$
$503$	20.8025	0.927539	0.463769	+	0.885956i	$0.346497\pi$
$504$	0	0
$505$	0	0
$506$	5.74596	0.255439
$507$	0	0
$508$	−1.60494	−0.0712078
$509$	23.8685	1.05795	0.528976	−	0.848637i	$-0.322576\pi$
$509$	23.8685	1.05795	0.528976	+	0.848637i	$0.322576\pi$
$510$	0	0
$511$	49.1983	2.17641
$512$	25.3526	1.12044
$513$	0	0
$514$	21.3744	0.942783
$515$	0	0
$516$	0	0
$517$	−3.32473	−0.146222
$518$	27.6563	1.21515
$519$	0	0
$520$	0	0
$521$	−24.8541	−1.08888	−0.544439	−	0.838800i	$-0.683257\pi$
$521$	−24.8541	−1.08888	−0.544439	+	0.838800i	$0.683257\pi$
$522$	0	0
$523$	−36.1177	−1.57932	−0.789658	−	0.613548i	$-0.789742\pi$
$523$	−36.1177	−1.57932	−0.789658	+	0.613548i	$0.789742\pi$
$524$	−5.76119	−0.251679
$525$	0	0
$526$	−11.4253	−0.498166
$527$	1.89215	0.0824235
$528$	0	0
$529$	26.2379	1.14078
$530$	0	0
$531$	0	0
$532$	−6.40635	−0.277751
$533$	−33.6419	−1.45719
$534$	0	0
$535$	0	0
$536$	6.86538	0.296539
$537$	0	0
$538$	6.53091	0.281568
$539$	−9.91638	−0.427129
$540$	0	0
$541$	5.49151	0.236098	0.118049	−	0.993008i	$-0.462336\pi$
$541$	5.49151	0.236098	0.118049	+	0.993008i	$0.462336\pi$
$542$	33.4788	1.43804
$543$	0	0
$544$	−2.83475	−0.121539
$545$	0	0
$546$	0	0
$547$	−25.5042	−1.09048	−0.545240	−	0.838280i	$-0.683561\pi$
$547$	−25.5042	−1.09048	−0.545240	+	0.838280i	$0.683561\pi$
$548$	2.54632	0.108773
$549$	0	0
$550$	0	0
$551$	14.9983	0.638948
$552$	0	0
$553$	29.1052	1.23768
$554$	6.97660	0.296407
$555$	0	0
$556$	1.76913	0.0750277
$557$	26.2473	1.11213	0.556067	−	0.831137i	$-0.312310\pi$
$557$	26.2473	1.11213	0.556067	+	0.831137i	$0.312310\pi$
$558$	0	0
$559$	63.7029	2.69434
$560$	0	0
$561$	0	0
$562$	4.39865	0.185546
$563$	−8.39380	−0.353756	−0.176878	−	0.984233i	$-0.556600\pi$
$563$	−8.39380	−0.353756	−0.176878	+	0.984233i	$0.556600\pi$
$564$	0	0
$565$	0	0
$566$	−6.30375	−0.264966
$567$	0	0
$568$	−15.9843	−0.670688
$569$	−0.746059	−0.0312764	−0.0156382	−	0.999878i	$-0.504978\pi$
$569$	−0.746059	−0.0312764	−0.0156382	+	0.999878i	$0.504978\pi$
$570$	0	0
$571$	−9.61849	−0.402521	−0.201261	−	0.979538i	$-0.564504\pi$
$571$	−9.61849	−0.402521	−0.201261	+	0.979538i	$0.564504\pi$
$572$	0.876487	0.0366478
$573$	0	0
$574$	40.1517	1.67590
$575$	0	0
$576$	0	0
$577$	29.7986	1.24053	0.620265	−	0.784392i	$-0.287025\pi$
$577$	29.7986	1.24053	0.620265	+	0.784392i	$0.287025\pi$
$578$	17.6677	0.734881
$579$	0	0
$580$	0	0
$581$	−67.2441	−2.78975
$582$	0	0
$583$	−4.82868	−0.199983
$584$	30.6517	1.26838
$585$	0	0
$586$	−16.7624	−0.692450
$587$	−4.90068	−0.202273	−0.101136	−	0.994873i	$-0.532248\pi$
$587$	−4.90068	−0.202273	−0.101136	+	0.994873i	$0.532248\pi$
$588$	0	0
$589$	5.01481	0.206632
$590$	0	0
$591$	0	0
$592$	14.8912	0.612024
$593$	−26.4295	−1.08533	−0.542666	−	0.839949i	$-0.682585\pi$
$593$	−26.4295	−1.08533	−0.542666	+	0.839949i	$0.682585\pi$
$594$	0	0
$595$	0	0
$596$	−2.81612	−0.115353
$597$	0	0
$598$	−48.8120	−1.99607
$599$	3.65198	0.149216	0.0746079	−	0.997213i	$-0.476229\pi$
$599$	3.65198	0.149216	0.0746079	+	0.997213i	$0.476229\pi$
$600$	0	0
$601$	18.9524	0.773085	0.386543	−	0.922272i	$-0.373669\pi$
$601$	18.9524	0.773085	0.386543	+	0.922272i	$0.373669\pi$
$602$	−76.0296	−3.09873
$603$	0	0
$604$	2.63864	0.107365
$605$	0	0
$606$	0	0
$607$	21.2031	0.860608	0.430304	−	0.902684i	$-0.358406\pi$
$607$	21.2031	0.860608	0.430304	+	0.902684i	$0.358406\pi$
$608$	−7.51298	−0.304692
$609$	0	0
$610$	0	0
$611$	28.2437	1.14262
$612$	0	0
$613$	−19.5022	−0.787688	−0.393844	−	0.919177i	$-0.628855\pi$
$613$	−19.5022	−0.787688	−0.393844	+	0.919177i	$0.628855\pi$
$614$	−5.67220	−0.228911
$615$	0	0
$616$	−8.89068	−0.358216
$617$	49.5357	1.99423	0.997117	−	0.0758823i	$-0.0241773\pi$
$617$	49.5357	1.99423	0.997117	+	0.0758823i	$0.0241773\pi$
$618$	0	0
$619$	−11.2865	−0.453641	−0.226821	−	0.973937i	$-0.572833\pi$
$619$	−11.2865	−0.453641	−0.226821	+	0.973937i	$0.572833\pi$
$620$	0	0
$621$	0	0
$622$	−7.71700	−0.309424
$623$	−69.0838	−2.76778
$624$	0	0
$625$	0	0
$626$	10.5826	0.422968
$627$	0	0
$628$	4.38216	0.174867
$629$	−8.29825	−0.330873
$630$	0	0
$631$	38.7681	1.54333	0.771667	−	0.636027i	$-0.219423\pi$
$631$	38.7681	1.54333	0.771667	+	0.636027i	$0.219423\pi$
$632$	18.1332	0.721298
$633$	0	0
$634$	9.65091	0.383287
$635$	0	0
$636$	0	0
$637$	84.2398	3.33770
$638$	2.44906	0.0969592
$639$	0	0
$640$	0	0
$641$	−33.9507	−1.34097	−0.670485	−	0.741923i	$-0.733914\pi$
$641$	−33.9507	−1.34097	−0.670485	+	0.741923i	$0.733914\pi$
$642$	0	0
$643$	−8.50487	−0.335399	−0.167700	−	0.985838i	$-0.553634\pi$
$643$	−8.50487	−0.335399	−0.167700	+	0.985838i	$0.553634\pi$
$644$	−8.96409	−0.353234
$645$	0	0
$646$	−12.4924	−0.491509
$647$	29.2340	1.14931	0.574654	−	0.818396i	$-0.305136\pi$
$647$	29.2340	1.14931	0.574654	+	0.818396i	$0.305136\pi$
$648$	0	0
$649$	−7.42351	−0.291398
$650$	0	0
$651$	0	0
$652$	−0.944276	−0.0369807
$653$	21.5608	0.843741	0.421870	−	0.906656i	$-0.361374\pi$
$653$	21.5608	0.843741	0.421870	+	0.906656i	$0.361374\pi$
$654$	0	0
$655$	0	0
$656$	21.6192	0.844088
$657$	0	0
$658$	−33.7089	−1.31411
$659$	40.4708	1.57652	0.788259	−	0.615343i	$-0.210983\pi$
$659$	40.4708	1.57652	0.788259	+	0.615343i	$0.210983\pi$
$660$	0	0
$661$	−3.28789	−0.127884	−0.0639420	−	0.997954i	$-0.520367\pi$
$661$	−3.28789	−0.127884	−0.0639420	+	0.997954i	$0.520367\pi$
$662$	28.3766	1.10289
$663$	0	0
$664$	−41.8946	−1.62582
$665$	0	0
$666$	0	0
$667$	20.9863	0.812594
$668$	0.00686689	0.000265688 0
$669$	0	0
$670$	0	0
$671$	−0.311387	−0.0120210
$672$	0	0
$673$	6.62171	0.255248	0.127624	−	0.991823i	$-0.459265\pi$
$673$	6.62171	0.255248	0.127624	+	0.991823i	$0.459265\pi$
$674$	−4.46848	−0.172120
$675$	0	0
$676$	−3.97862	−0.153024
$677$	−22.2862	−0.856528	−0.428264	−	0.903654i	$-0.640875\pi$
$677$	−22.2862	−0.856528	−0.428264	+	0.903654i	$0.640875\pi$
$678$	0	0
$679$	−65.9907	−2.53249
$680$	0	0
$681$	0	0
$682$	0.818866	0.0313560
$683$	17.4636	0.668225	0.334113	−	0.942533i	$-0.391563\pi$
$683$	17.4636	0.668225	0.334113	+	0.942533i	$0.391563\pi$
$684$	0	0
$685$	0	0
$686$	−56.3974	−2.15326
$687$	0	0
$688$	−40.9371	−1.56071
$689$	41.0197	1.56273
$690$	0	0
$691$	−12.0521	−0.458482	−0.229241	−	0.973370i	$-0.573624\pi$
$691$	−12.0521	−0.458482	−0.229241	+	0.973370i	$0.573624\pi$
$692$	4.49074	0.170713
$693$	0	0
$694$	8.80603	0.334272
$695$	0	0
$696$	0	0
$697$	−12.0475	−0.456332
$698$	10.4058	0.393866
$699$	0	0
$700$	0	0
$701$	10.2896	0.388632	0.194316	−	0.980939i	$-0.437751\pi$
$701$	10.2896	0.388632	0.194316	+	0.980939i	$0.437751\pi$
$702$	0	0
$703$	−21.9930	−0.829482
$704$	−5.45061	−0.205428
$705$	0	0
$706$	−5.13862	−0.193394
$707$	28.3863	1.06758
$708$	0	0
$709$	19.8908	0.747016	0.373508	−	0.927627i	$-0.378155\pi$
$709$	19.8908	0.747016	0.373508	+	0.927627i	$0.378155\pi$
$710$	0	0
$711$	0	0
$712$	−43.0407	−1.61302
$713$	7.01697	0.262788
$714$	0	0
$715$	0	0
$716$	1.11488	0.0416649
$717$	0	0
$718$	34.3656	1.28251
$719$	−4.49572	−0.167662	−0.0838310	−	0.996480i	$-0.526716\pi$
$719$	−4.49572	−0.167662	−0.0838310	+	0.996480i	$0.526716\pi$
$720$	0	0
$721$	−78.3970	−2.91966
$722$	−8.09457	−0.301249
$723$	0	0
$724$	0.601524	0.0223554
$725$	0	0
$726$	0	0
$727$	45.4891	1.68710	0.843549	−	0.537051i	$-0.180462\pi$
$727$	45.4891	1.68710	0.843549	+	0.537051i	$0.180462\pi$
$728$	75.5264	2.79920
$729$	0	0
$730$	0	0
$731$	22.8126	0.843755
$732$	0	0
$733$	−9.12610	−0.337080	−0.168540	−	0.985695i	$-0.553905\pi$
$733$	−9.12610	−0.337080	−0.168540	+	0.985695i	$0.553905\pi$
$734$	1.88271	0.0694921
$735$	0	0
$736$	−10.5125	−0.387497
$737$	−1.43090	−0.0527079
$738$	0	0
$739$	40.3532	1.48442	0.742208	−	0.670170i	$-0.233779\pi$
$739$	40.3532	1.48442	0.742208	+	0.670170i	$0.233779\pi$
$740$	0	0
$741$	0	0
$742$	−48.9571	−1.79727
$743$	−5.62475	−0.206352	−0.103176	−	0.994663i	$-0.532900\pi$
$743$	−5.62475	−0.206352	−0.103176	+	0.994663i	$0.532900\pi$
$744$	0	0
$745$	0	0
$746$	29.0672	1.06422
$747$	0	0
$748$	0.313878	0.0114765
$749$	−3.20996	−0.117289
$750$	0	0
$751$	−28.0500	−1.02356	−0.511779	−	0.859117i	$-0.671013\pi$
$751$	−28.0500	−1.02356	−0.511779	+	0.859117i	$0.671013\pi$
$752$	−18.1501	−0.661867
$753$	0	0
$754$	−20.8048	−0.757666
$755$	0	0
$756$	0	0
$757$	0.869840	0.0316149	0.0158074	−	0.999875i	$-0.494968\pi$
$757$	0.869840	0.0316149	0.0158074	+	0.999875i	$0.494968\pi$
$758$	33.9483	1.23306
$759$	0	0
$760$	0	0
$761$	23.6066	0.855739	0.427870	−	0.903840i	$-0.359264\pi$
$761$	23.6066	0.855739	0.427870	+	0.903840i	$0.359264\pi$
$762$	0	0
$763$	−12.6972	−0.459671
$764$	1.63512	0.0591567
$765$	0	0
$766$	−10.1729	−0.367562
$767$	63.0628	2.27707
$768$	0	0
$769$	9.17036	0.330692	0.165346	−	0.986236i	$-0.447126\pi$
$769$	9.17036	0.330692	0.165346	+	0.986236i	$0.447126\pi$
$770$	0	0
$771$	0	0
$772$	4.04418	0.145553
$773$	−18.7108	−0.672982	−0.336491	−	0.941687i	$-0.609240\pi$
$773$	−18.7108	−0.672982	−0.336491	+	0.941687i	$0.609240\pi$
$774$	0	0
$775$	0	0
$776$	−41.1137	−1.47589
$777$	0	0
$778$	39.6250	1.42063
$779$	−31.9297	−1.14400
$780$	0	0
$781$	3.33150	0.119210
$782$	−17.4800	−0.625085
$783$	0	0
$784$	−54.1347	−1.93338
$785$	0	0
$786$	0	0
$787$	6.09367	0.217216	0.108608	−	0.994085i	$-0.465361\pi$
$787$	6.09367	0.217216	0.108608	+	0.994085i	$0.465361\pi$
$788$	3.34807	0.119270
$789$	0	0
$790$	0	0
$791$	−42.2017	−1.50052
$792$	0	0
$793$	2.64524	0.0939352
$794$	2.93225	0.104062
$795$	0	0
$796$	6.97904	0.247366
$797$	−49.8813	−1.76689	−0.883443	−	0.468539i	$-0.844781\pi$
$797$	−49.8813	−1.76689	−0.883443	+	0.468539i	$0.844781\pi$
$798$	0	0
$799$	10.1143	0.357819
$800$	0	0
$801$	0	0
$802$	−1.68900	−0.0596406
$803$	−6.38851	−0.225446
$804$	0	0
$805$	0	0
$806$	−6.95628	−0.245024
$807$	0	0
$808$	17.6853	0.622168
$809$	12.2723	0.431470	0.215735	−	0.976452i	$-0.430785\pi$
$809$	12.2723	0.431470	0.215735	+	0.976452i	$0.430785\pi$
$810$	0	0
$811$	−20.6374	−0.724677	−0.362338	−	0.932047i	$-0.618022\pi$
$811$	−20.6374	−0.724677	−0.362338	+	0.932047i	$0.618022\pi$
$812$	−3.82070	−0.134080
$813$	0	0
$814$	−3.59123	−0.125872
$815$	0	0
$816$	0	0
$817$	60.4607	2.11525
$818$	−46.0036	−1.60848
$819$	0	0
$820$	0	0
$821$	42.2359	1.47404	0.737022	−	0.675868i	$-0.236231\pi$
$821$	42.2359	1.47404	0.737022	+	0.675868i	$0.236231\pi$
$822$	0	0
$823$	41.6944	1.45337	0.726687	−	0.686968i	$-0.241059\pi$
$823$	41.6944	1.45337	0.726687	+	0.686968i	$0.241059\pi$
$824$	−48.8431	−1.70153
$825$	0	0
$826$	−75.2657	−2.61883
$827$	−45.3564	−1.57720	−0.788598	−	0.614909i	$-0.789193\pi$
$827$	−45.3564	−1.57720	−0.788598	+	0.614909i	$0.789193\pi$
$828$	0	0
$829$	−10.9376	−0.379878	−0.189939	−	0.981796i	$-0.560829\pi$
$829$	−10.9376	−0.379878	−0.189939	+	0.981796i	$0.560829\pi$
$830$	0	0
$831$	0	0
$832$	46.3030	1.60527
$833$	30.1671	1.04523
$834$	0	0
$835$	0	0
$836$	0.831878	0.0287711
$837$	0	0
$838$	−9.08172	−0.313723
$839$	−19.5053	−0.673396	−0.336698	−	0.941613i	$-0.609310\pi$
$839$	−19.5053	−0.673396	−0.336698	+	0.941613i	$0.609310\pi$
$840$	0	0
$841$	−20.0551	−0.691557
$842$	−33.7872	−1.16438
$843$	0	0
$844$	−0.626189	−0.0215543
$845$	0	0
$846$	0	0
$847$	−50.8360	−1.74675
$848$	−26.3603	−0.905217
$849$	0	0
$850$	0	0
$851$	−30.7737	−1.05491
$852$	0	0
$853$	8.51435	0.291526	0.145763	−	0.989320i	$-0.453436\pi$
$853$	8.51435	0.291526	0.145763	+	0.989320i	$0.453436\pi$
$854$	−3.15710	−0.108034
$855$	0	0
$856$	−1.99987	−0.0683543
$857$	−37.0174	−1.26449	−0.632245	−	0.774769i	$-0.717866\pi$
$857$	−37.0174	−1.26449	−0.632245	+	0.774769i	$0.717866\pi$
$858$	0	0
$859$	−13.5056	−0.460804	−0.230402	−	0.973096i	$-0.574004\pi$
$859$	−13.5056	−0.460804	−0.230402	+	0.973096i	$0.574004\pi$
$860$	0	0
$861$	0	0
$862$	−43.9865	−1.49818
$863$	0.465096	0.0158320	0.00791602	−	0.999969i	$-0.497480\pi$
$863$	0.465096	0.0158320	0.00791602	+	0.999969i	$0.497480\pi$
$864$	0	0
$865$	0	0
$866$	3.74886	0.127391
$867$	0	0
$868$	−1.27749	−0.0433607
$869$	−3.77936	−0.128206
$870$	0	0
$871$	12.1555	0.411874
$872$	−7.91067	−0.267889
$873$	0	0
$874$	−46.3277	−1.56706
$875$	0	0
$876$	0	0
$877$	−57.7330	−1.94951	−0.974753	−	0.223287i	$-0.928321\pi$
$877$	−57.7330	−1.94951	−0.974753	+	0.223287i	$0.928321\pi$
$878$	27.5876	0.931038
$879$	0	0
$880$	0	0
$881$	39.2778	1.32330	0.661652	−	0.749811i	$-0.269856\pi$
$881$	39.2778	1.32330	0.661652	+	0.749811i	$0.269856\pi$
$882$	0	0
$883$	50.7061	1.70640	0.853199	−	0.521586i	$-0.174659\pi$
$883$	50.7061	1.70640	0.853199	+	0.521586i	$0.174659\pi$
$884$	−2.66640	−0.0896808
$885$	0	0
$886$	17.3108	0.581566
$887$	−22.4218	−0.752850	−0.376425	−	0.926447i	$-0.622847\pi$
$887$	−22.4218	−0.752850	−0.376425	+	0.926447i	$0.622847\pi$
$888$	0	0
$889$	28.8242	0.966734
$890$	0	0
$891$	0	0
$892$	1.79068	0.0599564
$893$	26.8062	0.897035
$894$	0	0
$895$	0	0
$896$	−40.9107	−1.36673
$897$	0	0
$898$	−20.4719	−0.683156
$899$	2.99080	0.0997486
$900$	0	0
$901$	14.6895	0.489380
$902$	−5.21379	−0.173600
$903$	0	0
$904$	−26.2926	−0.874480
$905$	0	0
$906$	0	0
$907$	3.31143	0.109954	0.0549771	−	0.998488i	$-0.482491\pi$
$907$	3.31143	0.109954	0.0549771	+	0.998488i	$0.482491\pi$
$908$	−0.0816661	−0.00271018
$909$	0	0
$910$	0	0
$911$	24.8537	0.823440	0.411720	−	0.911310i	$-0.364928\pi$
$911$	24.8537	0.823440	0.411720	+	0.911310i	$0.364928\pi$
$912$	0	0
$913$	8.73178	0.288980
$914$	8.94485	0.295869
$915$	0	0
$916$	2.90462	0.0959714
$917$	103.469	3.41685
$918$	0	0
$919$	42.1133	1.38919	0.694595	−	0.719401i	$-0.255584\pi$
$919$	42.1133	1.38919	0.694595	+	0.719401i	$0.255584\pi$
$920$	0	0
$921$	0	0
$922$	24.9073	0.820278
$923$	−28.3011	−0.931543
$924$	0	0
$925$	0	0
$926$	−37.5703	−1.23464
$927$	0	0
$928$	−4.48069	−0.147086
$929$	38.1059	1.25021	0.625107	−	0.780539i	$-0.285055\pi$
$929$	38.1059	1.25021	0.625107	+	0.780539i	$0.285055\pi$
$930$	0	0
$931$	79.9524	2.62033
$932$	−2.30244	−0.0754188
$933$	0	0
$934$	33.6644	1.10153
$935$	0	0
$936$	0	0
$937$	−13.0559	−0.426517	−0.213259	−	0.976996i	$-0.568408\pi$
$937$	−13.0559	−0.426517	−0.213259	+	0.976996i	$0.568408\pi$
$938$	−14.5076	−0.473692
$939$	0	0
$940$	0	0
$941$	25.9851	0.847092	0.423546	−	0.905875i	$-0.360785\pi$
$941$	25.9851	0.847092	0.423546	+	0.905875i	$0.360785\pi$
$942$	0	0
$943$	−44.6776	−1.45490
$944$	−40.5258	−1.31900
$945$	0	0
$946$	9.87259	0.320986
$947$	−54.9507	−1.78566	−0.892830	−	0.450395i	$-0.851283\pi$
$947$	−54.9507	−1.78566	−0.892830	+	0.450395i	$0.851283\pi$
$948$	0	0
$949$	54.2705	1.76169
$950$	0	0
$951$	0	0
$952$	27.0467	0.876590
$953$	−27.6241	−0.894833	−0.447416	−	0.894326i	$-0.647656\pi$
$953$	−27.6241	−0.894833	−0.447416	+	0.894326i	$0.647656\pi$
$954$	0	0
$955$	0	0
$956$	−2.75319	−0.0890445
$957$	0	0
$958$	21.6571	0.699709
$959$	−45.7310	−1.47673
$960$	0	0
$961$	1.00000	0.0322581
$962$	30.5075	0.983602
$963$	0	0
$964$	4.79399	0.154404
$965$	0	0
$966$	0	0
$967$	19.1622	0.616216	0.308108	−	0.951351i	$-0.400304\pi$
$967$	19.1622	0.616216	0.308108	+	0.951351i	$0.400304\pi$
$968$	−31.6720	−1.01798
$969$	0	0
$970$	0	0
$971$	8.68260	0.278638	0.139319	−	0.990248i	$-0.455509\pi$
$971$	8.68260	0.278638	0.139319	+	0.990248i	$0.455509\pi$
$972$	0	0
$973$	−31.7730	−1.01859
$974$	−26.0676	−0.835259
$975$	0	0
$976$	−1.69990	−0.0544125
$977$	−12.3964	−0.396594	−0.198297	−	0.980142i	$-0.563541\pi$
$977$	−12.3964	−0.396594	−0.198297	+	0.980142i	$0.563541\pi$
$978$	0	0
$979$	8.97067	0.286704
$980$	0	0
$981$	0	0
$982$	−4.35705	−0.139039
$983$	−39.2608	−1.25223	−0.626113	−	0.779732i	$-0.715355\pi$
$983$	−39.2608	−1.25223	−0.626113	+	0.779732i	$0.715355\pi$
$984$	0	0
$985$	0	0
$986$	−7.45040	−0.237269
$987$	0	0
$988$	−7.06682	−0.224825
$989$	84.5996	2.69011
$990$	0	0
$991$	41.1045	1.30573	0.652863	−	0.757476i	$-0.273568\pi$
$991$	41.1045	1.30573	0.652863	+	0.757476i	$0.273568\pi$
$992$	−1.49816	−0.0475666
$993$	0	0
$994$	33.7775	1.07136
$995$	0	0
$996$	0	0
$997$	−40.6680	−1.28797	−0.643984	−	0.765039i	$-0.722720\pi$
$997$	−40.6680	−1.28797	−0.643984	+	0.765039i	$0.722720\pi$
$998$	−19.5167	−0.617790
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.ci.1.5		11
3.2	odd	2		2325.2.a.bd.1.7		11
5.2	odd	4		1395.2.c.h.559.8		22
5.3	odd	4		1395.2.c.h.559.15		22
5.4	even	2		6975.2.a.cj.1.7		11
15.2	even	4		465.2.c.b.94.15	yes	22
15.8	even	4		465.2.c.b.94.8	✓	22
15.14	odd	2		2325.2.a.bc.1.5		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.c.b.94.8	✓	22	15.8	even	4
465.2.c.b.94.15	yes	22	15.2	even	4
1395.2.c.h.559.8		22	5.2	odd	4
1395.2.c.h.559.15		22	5.3	odd	4
2325.2.a.bc.1.5		11	15.14	odd	2
2325.2.a.bd.1.7		11	3.2	odd	2
6975.2.a.ci.1.5		11	1.1	even	1	trivial
6975.2.a.cj.1.7		11	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)

\(f(q)\)	\(=\)	\(q-1.31655 q^{2} -0.266704 q^{4} +4.78991 q^{7} +2.98422 q^{8} -0.621980 q^{11} +5.28373 q^{13} -6.30614 q^{14} -3.39546 q^{16} +1.89215 q^{17} +5.01481 q^{19} +0.818866 q^{22} +7.01697 q^{23} -6.95628 q^{26} -1.27749 q^{28} +2.99080 q^{29} +1.00000 q^{31} -1.49816 q^{32} -2.49111 q^{34} -4.38561 q^{37} -6.60224 q^{38} -6.36708 q^{41} +12.0564 q^{43} +0.165884 q^{44} -9.23817 q^{46} +5.34541 q^{47} +15.9432 q^{49} -1.40919 q^{52} +7.76340 q^{53} +14.2942 q^{56} -3.93752 q^{58} +11.9353 q^{59} +0.500639 q^{61} -1.31655 q^{62} +8.76332 q^{64} +2.30056 q^{67} -0.504644 q^{68} -5.35628 q^{71} +10.2712 q^{73} +5.77387 q^{74} -1.33747 q^{76} -2.97923 q^{77} +6.07634 q^{79} +8.38257 q^{82} -14.0387 q^{83} -15.8729 q^{86} -1.85613 q^{88} -14.4228 q^{89} +25.3086 q^{91} -1.87145 q^{92} -7.03748 q^{94} -13.7770 q^{97} -20.9900 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 3 q^{2} + 15 q^{4} + 8 q^{7} - 9 q^{8} + 14 q^{13} + 14 q^{14} + 27 q^{16} - 12 q^{17} + 12 q^{19} + 10 q^{22} - 12 q^{23} + 6 q^{26} + 22 q^{28} + 8 q^{29} + 11 q^{31} - 21 q^{32} + 2 q^{34} + 16 q^{37}+ \cdots - 17 q^{98}+O(q^{100}) \) 11 * q - 3 * q^2 + 15 * q^4 + 8 * q^7 - 9 * q^8 + 14 * q^13 + 14 * q^14 + 27 * q^16 - 12 * q^17 + 12 * q^19 + 10 * q^22 - 12 * q^23 + 6 * q^26 + 22 * q^28 + 8 * q^29 + 11 * q^31 - 21 * q^32 + 2 * q^34 + 16 * q^37 - 26 * q^38 + 8 * q^41 + 28 * q^43 + 26 * q^44 + 8 * q^46 - 14 * q^47 + 23 * q^49 + 18 * q^52 - 26 * q^53 + 42 * q^56 + 6 * q^58 + 6 * q^59 + 14 * q^61 - 3 * q^62 + 29 * q^64 + 6 * q^67 - 54 * q^68 + 18 * q^71 + 26 * q^73 + 40 * q^74 + 10 * q^76 - 2 * q^77 + 8 * q^79 - 30 * q^82 - 48 * q^83 + 12 * q^86 + 40 * q^88 + 14 * q^89 - 4 * q^91 + 30 * q^92 - 4 * q^94 + 16 * q^97 - 17 * q^98

Introduction

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