LMFDB - Embedded newform 9196.2.a.n.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9196,2,Mod(1,9196)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9196.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9196 = 2^{2} \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9196.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,1,0,5,0,2,0,17,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.4304296988$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.744786576.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30 $ x^6 - x^5 - 17x^4 + 13x^3 + 69x^2 - 21x - 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.27915$ of defining polynomial
Character	$\chi$	$=$	9196.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.27915 q^{3} +3.53478 q^{5} +2.34459 q^{7} +7.75283 q^{9} +5.44101 q^{13} +11.5911 q^{15} -3.37635 q^{17} +1.00000 q^{19} +7.68827 q^{21} -5.78459 q^{23} +7.49465 q^{25} +15.5853 q^{27} -9.19463 q^{29} -9.59107 q^{31} +8.28761 q^{35} -6.58617 q^{37} +17.8419 q^{39} -2.37145 q^{41} -1.18026 q^{43} +27.4045 q^{45} +1.69320 q^{47} -1.50289 q^{49} -11.0716 q^{51} +2.51125 q^{53} +3.27915 q^{57} -4.83925 q^{59} +7.62786 q^{61} +18.1772 q^{63} +19.2328 q^{65} +2.22139 q^{67} -18.9686 q^{69} +11.0379 q^{71} +14.9944 q^{73} +24.5761 q^{75} +5.18195 q^{79} +27.8479 q^{81} +4.24982 q^{83} -11.9347 q^{85} -30.1506 q^{87} +6.09139 q^{89} +12.7569 q^{91} -31.4506 q^{93} +3.53478 q^{95} -6.16698 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} - 8 q^{13} + 9 q^{15} + 2 q^{17} + 6 q^{19} - 18 q^{21} + 5 q^{23} + 13 q^{25} + 7 q^{27} - 10 q^{29} + 3 q^{31} + 4 q^{35} + 7 q^{37} + 10 q^{39} - 6 q^{41}+ \cdots + 37 q^{97}+O(q^{100}) $ 6 * q + q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9 - 8 * q^13 + 9 * q^15 + 2 * q^17 + 6 * q^19 - 18 * q^21 + 5 * q^23 + 13 * q^25 + 7 * q^27 - 10 * q^29 + 3 * q^31 + 4 * q^35 + 7 * q^37 + 10 * q^39 - 6 * q^41 - 16 * q^43 + 42 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^53 + q^57 + 15 * q^59 - 24 * q^61 + 20 * q^63 + 28 * q^65 + 25 * q^67 - 33 * q^69 - 9 * q^71 + 26 * q^73 + 28 * q^75 + 16 * q^79 + 58 * q^81 + 2 * q^83 - 12 * q^85 + 36 * q^87 + 7 * q^89 + 8 * q^91 - 55 * q^93 + 5 * q^95 + 37 * q^97

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	0
$2$	0	0
$3$	3.27915	1.89322	0.946609	−	0.322383i	$-0.104484\pi$
$3$	3.27915	1.89322	0.946609	+	0.322383i	$0.104484\pi$
$4$	0	0
$5$	3.53478	1.58080	0.790400	−	0.612591i	$-0.209873\pi$
$5$	3.53478	1.58080	0.790400	+	0.612591i	$0.209873\pi$
$6$	0	0
$7$	2.34459	0.886172	0.443086	−	0.896479i	$-0.353884\pi$
$7$	2.34459	0.886172	0.443086	+	0.896479i	$0.353884\pi$
$8$	0	0
$9$	7.75283	2.58428
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.44101	1.50906	0.754532	−	0.656263i	$-0.227864\pi$
$13$	5.44101	1.50906	0.754532	+	0.656263i	$0.227864\pi$
$14$	0	0
$15$	11.5911	2.99280
$16$	0	0
$17$	−3.37635	−0.818886	−0.409443	−	0.912336i	$-0.634277\pi$
$17$	−3.37635	−0.818886	−0.409443	+	0.912336i	$0.634277\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	7.68827	1.67772
$22$	0	0
$23$	−5.78459	−1.20617	−0.603086	−	0.797676i	$-0.706062\pi$
$23$	−5.78459	−1.20617	−0.603086	+	0.797676i	$0.706062\pi$
$24$	0	0
$25$	7.49465	1.49893
$26$	0	0
$27$	15.5853	2.99938
$28$	0	0
$29$	−9.19463	−1.70740	−0.853700	−	0.520766i	$-0.825647\pi$
$29$	−9.19463	−1.70740	−0.853700	+	0.520766i	$0.825647\pi$
$30$	0	0
$31$	−9.59107	−1.72261	−0.861304	−	0.508091i	$-0.830351\pi$
$31$	−9.59107	−1.72261	−0.861304	+	0.508091i	$0.830351\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.28761	1.40086
$36$	0	0
$37$	−6.58617	−1.08276	−0.541380	−	0.840778i	$-0.682098\pi$
$37$	−6.58617	−1.08276	−0.541380	+	0.840778i	$0.682098\pi$
$38$	0	0
$39$	17.8419	2.85699
$40$	0	0
$41$	−2.37145	−0.370359	−0.185179	−	0.982705i	$-0.559287\pi$
$41$	−2.37145	−0.370359	−0.185179	+	0.982705i	$0.559287\pi$
$42$	0	0
$43$	−1.18026	−0.179988	−0.0899941	−	0.995942i	$-0.528685\pi$
$43$	−1.18026	−0.179988	−0.0899941	+	0.995942i	$0.528685\pi$
$44$	0	0
$45$	27.4045	4.08523
$46$	0	0
$47$	1.69320	0.246979	0.123490	−	0.992346i	$-0.460591\pi$
$47$	1.69320	0.246979	0.123490	+	0.992346i	$0.460591\pi$
$48$	0	0
$49$	−1.50289	−0.214698
$50$	0	0
$51$	−11.0716	−1.55033
$52$	0	0
$53$	2.51125	0.344947	0.172474	−	0.985014i	$-0.444824\pi$
$53$	2.51125	0.344947	0.172474	+	0.985014i	$0.444824\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.27915	0.434334
$58$	0	0
$59$	−4.83925	−0.630016	−0.315008	−	0.949089i	$-0.602007\pi$
$59$	−4.83925	−0.630016	−0.315008	+	0.949089i	$0.602007\pi$
$60$	0	0
$61$	7.62786	0.976647	0.488324	−	0.872663i	$-0.337609\pi$
$61$	7.62786	0.976647	0.488324	+	0.872663i	$0.337609\pi$
$62$	0	0
$63$	18.1772	2.29012
$64$	0	0
$65$	19.2328	2.38553
$66$	0	0
$67$	2.22139	0.271386	0.135693	−	0.990751i	$-0.456674\pi$
$67$	2.22139	0.271386	0.135693	+	0.990751i	$0.456674\pi$
$68$	0	0
$69$	−18.9686	−2.28355
$70$	0	0
$71$	11.0379	1.30996	0.654978	−	0.755648i	$-0.272678\pi$
$71$	11.0379	1.30996	0.654978	+	0.755648i	$0.272678\pi$
$72$	0	0
$73$	14.9944	1.75496	0.877482	−	0.479610i	$-0.159222\pi$
$73$	14.9944	1.75496	0.877482	+	0.479610i	$0.159222\pi$
$74$	0	0
$75$	24.5761	2.83780
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.18195	0.583015	0.291507	−	0.956569i	$-0.405843\pi$
$79$	5.18195	0.583015	0.291507	+	0.956569i	$0.405843\pi$
$80$	0	0
$81$	27.8479	3.09421
$82$	0	0
$83$	4.24982	0.466478	0.233239	−	0.972419i	$-0.425068\pi$
$83$	4.24982	0.466478	0.233239	+	0.972419i	$0.425068\pi$
$84$	0	0
$85$	−11.9347	−1.29450
$86$	0	0
$87$	−30.1506	−3.23248
$88$	0	0
$89$	6.09139	0.645686	0.322843	−	0.946453i	$-0.395361\pi$
$89$	6.09139	0.645686	0.322843	+	0.946453i	$0.395361\pi$
$90$	0	0
$91$	12.7569	1.33729
$92$	0	0
$93$	−31.4506	−3.26127
$94$	0	0
$95$	3.53478	0.362661
$96$	0	0
$97$	−6.16698	−0.626162	−0.313081	−	0.949726i	$-0.601361\pi$
$97$	−6.16698	−0.626162	−0.313081	+	0.949726i	$0.601361\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.86510	−0.683103	−0.341551	−	0.939863i	$-0.610952\pi$
$101$	−6.86510	−0.683103	−0.341551	+	0.939863i	$0.610952\pi$
$102$	0	0
$103$	12.8156	1.26276	0.631380	−	0.775473i	$-0.282489\pi$
$103$	12.8156	1.26276	0.631380	+	0.775473i	$0.282489\pi$
$104$	0	0
$105$	27.1763	2.65214
$106$	0	0
$107$	−3.49076	−0.337465	−0.168732	−	0.985662i	$-0.553967\pi$
$107$	−3.49076	−0.337465	−0.168732	+	0.985662i	$0.553967\pi$
$108$	0	0
$109$	−19.9904	−1.91473	−0.957366	−	0.288877i	$-0.906718\pi$
$109$	−19.9904	−1.91473	−0.957366	+	0.288877i	$0.906718\pi$
$110$	0	0
$111$	−21.5971	−2.04990
$112$	0	0
$113$	14.0212	1.31900	0.659502	−	0.751703i	$-0.270767\pi$
$113$	14.0212	1.31900	0.659502	+	0.751703i	$0.270767\pi$
$114$	0	0
$115$	−20.4473	−1.90672
$116$	0	0
$117$	42.1832	3.89984
$118$	0	0
$119$	−7.91617	−0.725674
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−7.77636	−0.701171
$124$	0	0
$125$	8.81804	0.788710
$126$	0	0
$127$	5.54742	0.492254	0.246127	−	0.969238i	$-0.420842\pi$
$127$	5.54742	0.492254	0.246127	+	0.969238i	$0.420842\pi$
$128$	0	0
$129$	−3.87026	−0.340757
$130$	0	0
$131$	11.6672	1.01937	0.509685	−	0.860361i	$-0.329762\pi$
$131$	11.6672	1.01937	0.509685	+	0.860361i	$0.329762\pi$
$132$	0	0
$133$	2.34459	0.203302
$134$	0	0
$135$	55.0904	4.74143
$136$	0	0
$137$	−2.82797	−0.241610	−0.120805	−	0.992676i	$-0.538548\pi$
$137$	−2.82797	−0.241610	−0.120805	+	0.992676i	$0.538548\pi$
$138$	0	0
$139$	−15.4993	−1.31464	−0.657318	−	0.753614i	$-0.728309\pi$
$139$	−15.4993	−1.31464	−0.657318	+	0.753614i	$0.728309\pi$
$140$	0	0
$141$	5.55227	0.467585
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−32.5010	−2.69906
$146$	0	0
$147$	−4.92820	−0.406471
$148$	0	0
$149$	0.234586	0.0192180	0.00960900	−	0.999954i	$-0.496941\pi$
$149$	0.234586	0.0192180	0.00960900	+	0.999954i	$0.496941\pi$
$150$	0	0
$151$	−17.3110	−1.40875	−0.704375	−	0.709828i	$-0.748773\pi$
$151$	−17.3110	−1.40875	−0.704375	+	0.709828i	$0.748773\pi$
$152$	0	0
$153$	−26.1763	−2.11623
$154$	0	0
$155$	−33.9023	−2.72310
$156$	0	0
$157$	−10.8360	−0.864806	−0.432403	−	0.901680i	$-0.642334\pi$
$157$	−10.8360	−0.864806	−0.432403	+	0.901680i	$0.642334\pi$
$158$	0	0
$159$	8.23478	0.653060
$160$	0	0
$161$	−13.5625	−1.06888
$162$	0	0
$163$	−8.95716	−0.701579	−0.350789	−	0.936454i	$-0.614087\pi$
$163$	−8.95716	−0.701579	−0.350789	+	0.936454i	$0.614087\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.1193	1.09258	0.546291	−	0.837596i	$-0.316039\pi$
$167$	14.1193	1.09258	0.546291	+	0.837596i	$0.316039\pi$
$168$	0	0
$169$	16.6046	1.27728
$170$	0	0
$171$	7.75283	0.592874
$172$	0	0
$173$	−2.56411	−0.194946	−0.0974729	−	0.995238i	$-0.531076\pi$
$173$	−2.56411	−0.194946	−0.0974729	+	0.995238i	$0.531076\pi$
$174$	0	0
$175$	17.5719	1.32831
$176$	0	0
$177$	−15.8686	−1.19276
$178$	0	0
$179$	0.455979	0.0340815	0.0170407	−	0.999855i	$-0.494576\pi$
$179$	0.455979	0.0340815	0.0170407	+	0.999855i	$0.494576\pi$
$180$	0	0
$181$	−8.41804	−0.625708	−0.312854	−	0.949801i	$-0.601285\pi$
$181$	−8.41804	−0.625708	−0.312854	+	0.949801i	$0.601285\pi$
$182$	0	0
$183$	25.0129	1.84901
$184$	0	0
$185$	−23.2807	−1.71163
$186$	0	0
$187$	0	0
$188$	0	0
$189$	36.5411	2.65797
$190$	0	0
$191$	17.9694	1.30022	0.650110	−	0.759840i	$-0.274723\pi$
$191$	17.9694	1.30022	0.650110	+	0.759840i	$0.274723\pi$
$192$	0	0
$193$	−17.2504	−1.24171	−0.620854	−	0.783926i	$-0.713214\pi$
$193$	−17.2504	−1.24171	−0.620854	+	0.783926i	$0.713214\pi$
$194$	0	0
$195$	63.0671	4.51633
$196$	0	0
$197$	6.88359	0.490435	0.245218	−	0.969468i	$-0.421141\pi$
$197$	6.88359	0.490435	0.245218	+	0.969468i	$0.421141\pi$
$198$	0	0
$199$	−15.0531	−1.06708	−0.533542	−	0.845773i	$-0.679140\pi$
$199$	−15.0531	−1.06708	−0.533542	+	0.845773i	$0.679140\pi$
$200$	0	0
$201$	7.28429	0.513794
$202$	0	0
$203$	−21.5576	−1.51305
$204$	0	0
$205$	−8.38256	−0.585464
$206$	0	0
$207$	−44.8470	−3.11708
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−19.2475	−1.32505	−0.662526	−	0.749039i	$-0.730516\pi$
$211$	−19.2475	−1.32505	−0.662526	+	0.749039i	$0.730516\pi$
$212$	0	0
$213$	36.1949	2.48003
$214$	0	0
$215$	−4.17196	−0.284525
$216$	0	0
$217$	−22.4871	−1.52653
$218$	0	0
$219$	49.1690	3.32253
$220$	0	0
$221$	−18.3708	−1.23575
$222$	0	0
$223$	−15.6477	−1.04785	−0.523923	−	0.851765i	$-0.675532\pi$
$223$	−15.6477	−1.04785	−0.523923	+	0.851765i	$0.675532\pi$
$224$	0	0
$225$	58.1048	3.87365
$226$	0	0
$227$	25.9837	1.72460	0.862300	−	0.506398i	$-0.169023\pi$
$227$	25.9837	1.72460	0.862300	+	0.506398i	$0.169023\pi$
$228$	0	0
$229$	−19.9887	−1.32089	−0.660445	−	0.750874i	$-0.729632\pi$
$229$	−19.9887	−1.32089	−0.660445	+	0.750874i	$0.729632\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.7578	−1.09784	−0.548920	−	0.835875i	$-0.684961\pi$
$233$	−16.7578	−1.09784	−0.548920	+	0.835875i	$0.684961\pi$
$234$	0	0
$235$	5.98510	0.390425
$236$	0	0
$237$	16.9924	1.10377
$238$	0	0
$239$	6.69553	0.433098	0.216549	−	0.976272i	$-0.430520\pi$
$239$	6.69553	0.433098	0.216549	+	0.976272i	$0.430520\pi$
$240$	0	0
$241$	−9.95052	−0.640969	−0.320485	−	0.947254i	$-0.603846\pi$
$241$	−9.95052	−0.640969	−0.320485	+	0.947254i	$0.603846\pi$
$242$	0	0
$243$	44.5618	2.85864
$244$	0	0
$245$	−5.31238	−0.339395
$246$	0	0
$247$	5.44101	0.346203
$248$	0	0
$249$	13.9358	0.883145
$250$	0	0
$251$	2.64146	0.166728	0.0833639	−	0.996519i	$-0.473434\pi$
$251$	2.64146	0.166728	0.0833639	+	0.996519i	$0.473434\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−39.1355	−2.45076
$256$	0	0
$257$	−2.24328	−0.139932	−0.0699659	−	0.997549i	$-0.522289\pi$
$257$	−2.24328	−0.139932	−0.0699659	+	0.997549i	$0.522289\pi$
$258$	0	0
$259$	−15.4419	−0.959512
$260$	0	0
$261$	−71.2844	−4.41239
$262$	0	0
$263$	8.31828	0.512927	0.256464	−	0.966554i	$-0.417443\pi$
$263$	8.31828	0.512927	0.256464	+	0.966554i	$0.417443\pi$
$264$	0	0
$265$	8.87672	0.545293
$266$	0	0
$267$	19.9746	1.22243
$268$	0	0
$269$	−31.0749	−1.89467	−0.947334	−	0.320248i	$-0.896234\pi$
$269$	−31.0749	−1.89467	−0.947334	+	0.320248i	$0.896234\pi$
$270$	0	0
$271$	23.9002	1.45184	0.725918	−	0.687781i	$-0.241415\pi$
$271$	23.9002	1.45184	0.725918	+	0.687781i	$0.241415\pi$
$272$	0	0
$273$	41.8320	2.53179
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.66668	0.100141	0.0500705	−	0.998746i	$-0.484055\pi$
$277$	1.66668	0.100141	0.0500705	+	0.998746i	$0.484055\pi$
$278$	0	0
$279$	−74.3580	−4.45170
$280$	0	0
$281$	−13.3003	−0.793432	−0.396716	−	0.917941i	$-0.629850\pi$
$281$	−13.3003	−0.793432	−0.396716	+	0.917941i	$0.629850\pi$
$282$	0	0
$283$	−13.2362	−0.786811	−0.393406	−	0.919365i	$-0.628703\pi$
$283$	−13.2362	−0.786811	−0.393406	+	0.919365i	$0.628703\pi$
$284$	0	0
$285$	11.5911	0.686596
$286$	0	0
$287$	−5.56009	−0.328202
$288$	0	0
$289$	−5.60024	−0.329426
$290$	0	0
$291$	−20.2225	−1.18546
$292$	0	0
$293$	8.08113	0.472105	0.236052	−	0.971740i	$-0.424146\pi$
$293$	8.08113	0.472105	0.236052	+	0.971740i	$0.424146\pi$
$294$	0	0
$295$	−17.1057	−0.995930
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−31.4740	−1.82019
$300$	0	0
$301$	−2.76723	−0.159501
$302$	0	0
$303$	−22.5117	−1.29326
$304$	0	0
$305$	26.9628	1.54388
$306$	0	0
$307$	4.12730	0.235557	0.117779	−	0.993040i	$-0.462423\pi$
$307$	4.12730	0.235557	0.117779	+	0.993040i	$0.462423\pi$
$308$	0	0
$309$	42.0244	2.39068
$310$	0	0
$311$	10.7342	0.608682	0.304341	−	0.952563i	$-0.401564\pi$
$311$	10.7342	0.608682	0.304341	+	0.952563i	$0.401564\pi$
$312$	0	0
$313$	7.22131	0.408172	0.204086	−	0.978953i	$-0.434578\pi$
$313$	7.22131	0.408172	0.204086	+	0.978953i	$0.434578\pi$
$314$	0	0
$315$	64.2525	3.62022
$316$	0	0
$317$	17.7601	0.997506	0.498753	−	0.866744i	$-0.333792\pi$
$317$	17.7601	0.997506	0.498753	+	0.866744i	$0.333792\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−11.4467	−0.638895
$322$	0	0
$323$	−3.37635	−0.187865
$324$	0	0
$325$	40.7785	2.26198
$326$	0	0
$327$	−65.5515	−3.62501
$328$	0	0
$329$	3.96987	0.218866
$330$	0	0
$331$	5.86554	0.322399	0.161200	−	0.986922i	$-0.448464\pi$
$331$	5.86554	0.322399	0.161200	+	0.986922i	$0.448464\pi$
$332$	0	0
$333$	−51.0615	−2.79815
$334$	0	0
$335$	7.85213	0.429008
$336$	0	0
$337$	6.27598	0.341874	0.170937	−	0.985282i	$-0.445320\pi$
$337$	6.27598	0.341874	0.170937	+	0.985282i	$0.445320\pi$
$338$	0	0
$339$	45.9776	2.49716
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−19.9358	−1.07643
$344$	0	0
$345$	−67.0496	−3.60983
$346$	0	0
$347$	12.8165	0.688025	0.344012	−	0.938965i	$-0.388214\pi$
$347$	12.8165	0.688025	0.344012	+	0.938965i	$0.388214\pi$
$348$	0	0
$349$	−8.94535	−0.478834	−0.239417	−	0.970917i	$-0.576956\pi$
$349$	−8.94535	−0.478834	−0.239417	+	0.970917i	$0.576956\pi$
$350$	0	0
$351$	84.7996	4.52627
$352$	0	0
$353$	33.8419	1.80122	0.900611	−	0.434626i	$-0.143119\pi$
$353$	33.8419	1.80122	0.900611	+	0.434626i	$0.143119\pi$
$354$	0	0
$355$	39.0165	2.07078
$356$	0	0
$357$	−25.9583	−1.37386
$358$	0	0
$359$	2.70804	0.142925	0.0714625	−	0.997443i	$-0.477233\pi$
$359$	2.70804	0.142925	0.0714625	+	0.997443i	$0.477233\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	53.0019	2.77425
$366$	0	0
$367$	22.5295	1.17603	0.588016	−	0.808850i	$-0.299909\pi$
$367$	22.5295	1.17603	0.588016	+	0.808850i	$0.299909\pi$
$368$	0	0
$369$	−18.3855	−0.957111
$370$	0	0
$371$	5.88786	0.305683
$372$	0	0
$373$	−25.5197	−1.32136	−0.660680	−	0.750668i	$-0.729732\pi$
$373$	−25.5197	−1.32136	−0.660680	+	0.750668i	$0.729732\pi$
$374$	0	0
$375$	28.9157	1.49320
$376$	0	0
$377$	−50.0281	−2.57658
$378$	0	0
$379$	−12.0860	−0.620818	−0.310409	−	0.950603i	$-0.600466\pi$
$379$	−12.0860	−0.620818	−0.310409	+	0.950603i	$0.600466\pi$
$380$	0	0
$381$	18.1908	0.931944
$382$	0	0
$383$	19.4116	0.991885	0.495943	−	0.868355i	$-0.334823\pi$
$383$	19.4116	0.991885	0.495943	+	0.868355i	$0.334823\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.15037	−0.465139
$388$	0	0
$389$	11.2668	0.571250	0.285625	−	0.958342i	$-0.407799\pi$
$389$	11.2668	0.571250	0.285625	+	0.958342i	$0.407799\pi$
$390$	0	0
$391$	19.5308	0.987716
$392$	0	0
$393$	38.2586	1.92989
$394$	0	0
$395$	18.3170	0.921630
$396$	0	0
$397$	11.9391	0.599204	0.299602	−	0.954064i	$-0.403146\pi$
$397$	11.9391	0.599204	0.299602	+	0.954064i	$0.403146\pi$
$398$	0	0
$399$	7.68827	0.384895
$400$	0	0
$401$	9.90453	0.494608	0.247304	−	0.968938i	$-0.420455\pi$
$401$	9.90453	0.494608	0.247304	+	0.968938i	$0.420455\pi$
$402$	0	0
$403$	−52.1851	−2.59953
$404$	0	0
$405$	98.4362	4.89134
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−11.1425	−0.550960	−0.275480	−	0.961307i	$-0.588837\pi$
$409$	−11.1425	−0.550960	−0.275480	+	0.961307i	$0.588837\pi$
$410$	0	0
$411$	−9.27336	−0.457421
$412$	0	0
$413$	−11.3461	−0.558303
$414$	0	0
$415$	15.0222	0.737409
$416$	0	0
$417$	−50.8246	−2.48889
$418$	0	0
$419$	−7.88093	−0.385009	−0.192504	−	0.981296i	$-0.561661\pi$
$419$	−7.88093	−0.385009	−0.192504	+	0.981296i	$0.561661\pi$
$420$	0	0
$421$	37.8040	1.84245	0.921226	−	0.389027i	$-0.127189\pi$
$421$	37.8040	1.84245	0.921226	+	0.389027i	$0.127189\pi$
$422$	0	0
$423$	13.1271	0.638262
$424$	0	0
$425$	−25.3046	−1.22745
$426$	0	0
$427$	17.8842	0.865478
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.80267	0.472178	0.236089	−	0.971731i	$-0.424134\pi$
$431$	9.80267	0.472178	0.236089	+	0.971731i	$0.424134\pi$
$432$	0	0
$433$	−22.1855	−1.06617	−0.533083	−	0.846063i	$-0.678967\pi$
$433$	−22.1855	−1.06617	−0.533083	+	0.846063i	$0.678967\pi$
$434$	0	0
$435$	−106.576	−5.10991
$436$	0	0
$437$	−5.78459	−0.276715
$438$	0	0
$439$	−18.9073	−0.902396	−0.451198	−	0.892424i	$-0.649003\pi$
$439$	−18.9073	−0.902396	−0.451198	+	0.892424i	$0.649003\pi$
$440$	0	0
$441$	−11.6516	−0.554840
$442$	0	0
$443$	4.46822	0.212292	0.106146	−	0.994351i	$-0.466149\pi$
$443$	4.46822	0.212292	0.106146	+	0.994351i	$0.466149\pi$
$444$	0	0
$445$	21.5317	1.02070
$446$	0	0
$447$	0.769242	0.0363839
$448$	0	0
$449$	−28.6131	−1.35033	−0.675167	−	0.737665i	$-0.735929\pi$
$449$	−28.6131	−1.35033	−0.675167	+	0.737665i	$0.735929\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−56.7654	−2.66707
$454$	0	0
$455$	45.0930	2.11399
$456$	0	0
$457$	14.8196	0.693232	0.346616	−	0.938007i	$-0.387331\pi$
$457$	14.8196	0.693232	0.346616	+	0.938007i	$0.387331\pi$
$458$	0	0
$459$	−52.6213	−2.45615
$460$	0	0
$461$	−36.4495	−1.69762	−0.848811	−	0.528697i	$-0.822681\pi$
$461$	−36.4495	−1.69762	−0.848811	+	0.528697i	$0.822681\pi$
$462$	0	0
$463$	−32.9591	−1.53174	−0.765871	−	0.642995i	$-0.777692\pi$
$463$	−32.9591	−1.53174	−0.765871	+	0.642995i	$0.777692\pi$
$464$	0	0
$465$	−111.171	−5.15542
$466$	0	0
$467$	6.28650	0.290905	0.145452	−	0.989365i	$-0.453536\pi$
$467$	6.28650	0.290905	0.145452	+	0.989365i	$0.453536\pi$
$468$	0	0
$469$	5.20826	0.240495
$470$	0	0
$471$	−35.5329	−1.63727
$472$	0	0
$473$	0	0
$474$	0	0
$475$	7.49465	0.343878
$476$	0	0
$477$	19.4693	0.891439
$478$	0	0
$479$	18.8832	0.862797	0.431399	−	0.902161i	$-0.358020\pi$
$479$	18.8832	0.862797	0.431399	+	0.902161i	$0.358020\pi$
$480$	0	0
$481$	−35.8354	−1.63396
$482$	0	0
$483$	−44.4735	−2.02362
$484$	0	0
$485$	−21.7989	−0.989837
$486$	0	0
$487$	24.5268	1.11142	0.555708	−	0.831377i	$-0.312447\pi$
$487$	24.5268	1.11142	0.555708	+	0.831377i	$0.312447\pi$
$488$	0	0
$489$	−29.3719	−1.32824
$490$	0	0
$491$	34.8056	1.57075	0.785377	−	0.619017i	$-0.212469\pi$
$491$	34.8056	1.57075	0.785377	+	0.619017i	$0.212469\pi$
$492$	0	0
$493$	31.0443	1.39816
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.8794	1.16085
$498$	0	0
$499$	−9.63700	−0.431411	−0.215706	−	0.976458i	$-0.569205\pi$
$499$	−9.63700	−0.431411	−0.215706	+	0.976458i	$0.569205\pi$
$500$	0	0
$501$	46.2992	2.06850
$502$	0	0
$503$	−32.1848	−1.43505	−0.717524	−	0.696534i	$-0.754724\pi$
$503$	−32.1848	−1.43505	−0.717524	+	0.696534i	$0.754724\pi$
$504$	0	0
$505$	−24.2666	−1.07985
$506$	0	0
$507$	54.4490	2.41816
$508$	0	0
$509$	13.3645	0.592371	0.296185	−	0.955131i	$-0.404285\pi$
$509$	13.3645	0.592371	0.296185	+	0.955131i	$0.404285\pi$
$510$	0	0
$511$	35.1558	1.55520
$512$	0	0
$513$	15.5853	0.688106
$514$	0	0
$515$	45.3004	1.99617
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−8.40811	−0.369075
$520$	0	0
$521$	15.8705	0.695298	0.347649	−	0.937625i	$-0.386980\pi$
$521$	15.8705	0.695298	0.347649	+	0.937625i	$0.386980\pi$
$522$	0	0
$523$	17.2073	0.752423	0.376211	−	0.926534i	$-0.377227\pi$
$523$	17.2073	0.752423	0.376211	+	0.926534i	$0.377227\pi$
$524$	0	0
$525$	57.6209	2.51478
$526$	0	0
$527$	32.3828	1.41062
$528$	0	0
$529$	10.4615	0.454849
$530$	0	0
$531$	−37.5179	−1.62814
$532$	0	0
$533$	−12.9031	−0.558896
$534$	0	0
$535$	−12.3391	−0.533465
$536$	0	0
$537$	1.49522	0.0645237
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−35.0905	−1.50866	−0.754330	−	0.656496i	$-0.772038\pi$
$541$	−35.0905	−1.50866	−0.754330	+	0.656496i	$0.772038\pi$
$542$	0	0
$543$	−27.6040	−1.18460
$544$	0	0
$545$	−70.6616	−3.02681
$546$	0	0
$547$	2.78485	0.119072	0.0595358	−	0.998226i	$-0.481038\pi$
$547$	2.78485	0.119072	0.0595358	+	0.998226i	$0.481038\pi$
$548$	0	0
$549$	59.1375	2.52393
$550$	0	0
$551$	−9.19463	−0.391704
$552$	0	0
$553$	12.1496	0.516652
$554$	0	0
$555$	−76.3408	−3.24049
$556$	0	0
$557$	−2.49963	−0.105913	−0.0529564	−	0.998597i	$-0.516864\pi$
$557$	−2.49963	−0.105913	−0.0529564	+	0.998597i	$0.516864\pi$
$558$	0	0
$559$	−6.42181	−0.271614
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.06488	−0.255604	−0.127802	−	0.991800i	$-0.540792\pi$
$563$	−6.06488	−0.255604	−0.127802	+	0.991800i	$0.540792\pi$
$564$	0	0
$565$	49.5618	2.08508
$566$	0	0
$567$	65.2920	2.74201
$568$	0	0
$569$	14.9447	0.626513	0.313256	−	0.949669i	$-0.398580\pi$
$569$	14.9447	0.626513	0.313256	+	0.949669i	$0.398580\pi$
$570$	0	0
$571$	−27.3521	−1.14465	−0.572325	−	0.820027i	$-0.693958\pi$
$571$	−27.3521	−1.14465	−0.572325	+	0.820027i	$0.693958\pi$
$572$	0	0
$573$	58.9244	2.46160
$574$	0	0
$575$	−43.3535	−1.80797
$576$	0	0
$577$	−1.83560	−0.0764170	−0.0382085	−	0.999270i	$-0.512165\pi$
$577$	−1.83560	−0.0764170	−0.0382085	+	0.999270i	$0.512165\pi$
$578$	0	0
$579$	−56.5666	−2.35083
$580$	0	0
$581$	9.96409	0.413380
$582$	0	0
$583$	0	0
$584$	0	0
$585$	149.108	6.16487
$586$	0	0
$587$	31.1835	1.28708	0.643540	−	0.765412i	$-0.277465\pi$
$587$	31.1835	1.28708	0.643540	+	0.765412i	$0.277465\pi$
$588$	0	0
$589$	−9.59107	−0.395193
$590$	0	0
$591$	22.5723	0.928501
$592$	0	0
$593$	−20.4998	−0.841827	−0.420913	−	0.907101i	$-0.638290\pi$
$593$	−20.4998	−0.841827	−0.420913	+	0.907101i	$0.638290\pi$
$594$	0	0
$595$	−27.9819	−1.14715
$596$	0	0
$597$	−49.3613	−2.02023
$598$	0	0
$599$	−21.8573	−0.893065	−0.446532	−	0.894768i	$-0.647341\pi$
$599$	−21.8573	−0.893065	−0.446532	+	0.894768i	$0.647341\pi$
$600$	0	0
$601$	−2.12818	−0.0868102	−0.0434051	−	0.999058i	$-0.513821\pi$
$601$	−2.12818	−0.0868102	−0.0434051	+	0.999058i	$0.513821\pi$
$602$	0	0
$603$	17.2221	0.701338
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11.3933	0.462439	0.231219	−	0.972902i	$-0.425729\pi$
$607$	11.3933	0.462439	0.231219	+	0.972902i	$0.425729\pi$
$608$	0	0
$609$	−70.6908	−2.86454
$610$	0	0
$611$	9.21273	0.372707
$612$	0	0
$613$	44.3930	1.79302	0.896508	−	0.443028i	$-0.146096\pi$
$613$	44.3930	1.79302	0.896508	+	0.443028i	$0.146096\pi$
$614$	0	0
$615$	−27.4877	−1.10841
$616$	0	0
$617$	37.3951	1.50547	0.752736	−	0.658323i	$-0.228734\pi$
$617$	37.3951	1.50547	0.752736	+	0.658323i	$0.228734\pi$
$618$	0	0
$619$	−15.6910	−0.630673	−0.315337	−	0.948980i	$-0.602117\pi$
$619$	−15.6910	−0.630673	−0.315337	+	0.948980i	$0.602117\pi$
$620$	0	0
$621$	−90.1544	−3.61777
$622$	0	0
$623$	14.2818	0.572189
$624$	0	0
$625$	−6.30344	−0.252138
$626$	0	0
$627$	0	0
$628$	0	0
$629$	22.2372	0.886657
$630$	0	0
$631$	−7.72618	−0.307574	−0.153787	−	0.988104i	$-0.549147\pi$
$631$	−7.72618	−0.307574	−0.153787	+	0.988104i	$0.549147\pi$
$632$	0	0
$633$	−63.1154	−2.50861
$634$	0	0
$635$	19.6089	0.778155
$636$	0	0
$637$	−8.17723	−0.323994
$638$	0	0
$639$	85.5749	3.38529
$640$	0	0
$641$	−7.26960	−0.287132	−0.143566	−	0.989641i	$-0.545857\pi$
$641$	−7.26960	−0.287132	−0.143566	+	0.989641i	$0.545857\pi$
$642$	0	0
$643$	−6.00629	−0.236865	−0.118432	−	0.992962i	$-0.537787\pi$
$643$	−6.00629	−0.236865	−0.118432	+	0.992962i	$0.537787\pi$
$644$	0	0
$645$	−13.6805	−0.538669
$646$	0	0
$647$	46.0802	1.81160	0.905801	−	0.423704i	$-0.139270\pi$
$647$	46.0802	1.81160	0.905801	+	0.423704i	$0.139270\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−73.7388	−2.89005
$652$	0	0
$653$	−37.2226	−1.45663	−0.728317	−	0.685240i	$-0.759697\pi$
$653$	−37.2226	−1.45663	−0.728317	+	0.685240i	$0.759697\pi$
$654$	0	0
$655$	41.2410	1.61142
$656$	0	0
$657$	116.249	4.53531
$658$	0	0
$659$	−34.9204	−1.36030	−0.680152	−	0.733071i	$-0.738086\pi$
$659$	−34.9204	−1.36030	−0.680152	+	0.733071i	$0.738086\pi$
$660$	0	0
$661$	−18.6194	−0.724211	−0.362105	−	0.932137i	$-0.617942\pi$
$661$	−18.6194	−0.724211	−0.362105	+	0.932137i	$0.617942\pi$
$662$	0	0
$663$	−60.2405	−2.33955
$664$	0	0
$665$	8.28761	0.321380
$666$	0	0
$667$	53.1872	2.05942
$668$	0	0
$669$	−51.3111	−1.98380
$670$	0	0
$671$	0	0
$672$	0	0
$673$	39.6581	1.52871	0.764355	−	0.644796i	$-0.223058\pi$
$673$	39.6581	1.52871	0.764355	+	0.644796i	$0.223058\pi$
$674$	0	0
$675$	116.806	4.49587
$676$	0	0
$677$	43.8777	1.68636	0.843179	−	0.537632i	$-0.180681\pi$
$677$	43.8777	1.68636	0.843179	+	0.537632i	$0.180681\pi$
$678$	0	0
$679$	−14.4591	−0.554888
$680$	0	0
$681$	85.2046	3.26505
$682$	0	0
$683$	−5.57257	−0.213228	−0.106614	−	0.994300i	$-0.534001\pi$
$683$	−5.57257	−0.213228	−0.106614	+	0.994300i	$0.534001\pi$
$684$	0	0
$685$	−9.99626	−0.381937
$686$	0	0
$687$	−65.5460	−2.50073
$688$	0	0
$689$	13.6638	0.520547
$690$	0	0
$691$	−3.03739	−0.115548	−0.0577739	−	0.998330i	$-0.518400\pi$
$691$	−3.03739	−0.115548	−0.0577739	+	0.998330i	$0.518400\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−54.7866	−2.07818
$696$	0	0
$697$	8.00687	0.303282
$698$	0	0
$699$	−54.9514	−2.07845
$700$	0	0
$701$	−6.85374	−0.258862	−0.129431	−	0.991588i	$-0.541315\pi$
$701$	−6.85374	−0.258862	−0.129431	+	0.991588i	$0.541315\pi$
$702$	0	0
$703$	−6.58617	−0.248402
$704$	0	0
$705$	19.6260	0.739159
$706$	0	0
$707$	−16.0959	−0.605347
$708$	0	0
$709$	22.4947	0.844804	0.422402	−	0.906409i	$-0.361187\pi$
$709$	22.4947	0.844804	0.422402	+	0.906409i	$0.361187\pi$
$710$	0	0
$711$	40.1748	1.50667
$712$	0	0
$713$	55.4805	2.07776
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.9557	0.819949
$718$	0	0
$719$	16.5930	0.618816	0.309408	−	0.950929i	$-0.399869\pi$
$719$	16.5930	0.618816	0.309408	+	0.950929i	$0.399869\pi$
$720$	0	0
$721$	30.0474	1.11902
$722$	0	0
$723$	−32.6293	−1.21350
$724$	0	0
$725$	−68.9105	−2.55927
$726$	0	0
$727$	12.7104	0.471402	0.235701	−	0.971826i	$-0.424261\pi$
$727$	12.7104	0.471402	0.235701	+	0.971826i	$0.424261\pi$
$728$	0	0
$729$	62.5811	2.31782
$730$	0	0
$731$	3.98498	0.147390
$732$	0	0
$733$	19.8202	0.732077	0.366039	−	0.930600i	$-0.380714\pi$
$733$	19.8202	0.732077	0.366039	+	0.930600i	$0.380714\pi$
$734$	0	0
$735$	−17.4201	−0.642550
$736$	0	0
$737$	0	0
$738$	0	0
$739$	48.1166	1.77000	0.885000	−	0.465592i	$-0.154158\pi$
$739$	48.1166	1.77000	0.885000	+	0.465592i	$0.154158\pi$
$740$	0	0
$741$	17.8419	0.655438
$742$	0	0
$743$	−35.5155	−1.30294	−0.651468	−	0.758676i	$-0.725847\pi$
$743$	−35.5155	−1.30294	−0.651468	+	0.758676i	$0.725847\pi$
$744$	0	0
$745$	0.829208	0.0303798
$746$	0	0
$747$	32.9481	1.20551
$748$	0	0
$749$	−8.18441	−0.299052
$750$	0	0
$751$	−8.43033	−0.307627	−0.153814	−	0.988100i	$-0.549156\pi$
$751$	−8.43033	−0.307627	−0.153814	+	0.988100i	$0.549156\pi$
$752$	0	0
$753$	8.66176	0.315652
$754$	0	0
$755$	−61.1906	−2.22695
$756$	0	0
$757$	−39.6118	−1.43972	−0.719858	−	0.694121i	$-0.755793\pi$
$757$	−39.6118	−1.43972	−0.719858	+	0.694121i	$0.755793\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.8269	1.47998	0.739988	−	0.672621i	$-0.234831\pi$
$761$	40.8269	1.47998	0.739988	+	0.672621i	$0.234831\pi$
$762$	0	0
$763$	−46.8693	−1.69678
$764$	0	0
$765$	−92.5274	−3.34534
$766$	0	0
$767$	−26.3304	−0.950735
$768$	0	0
$769$	7.64570	0.275711	0.137856	−	0.990452i	$-0.455979\pi$
$769$	7.64570	0.275711	0.137856	+	0.990452i	$0.455979\pi$
$770$	0	0
$771$	−7.35604	−0.264921
$772$	0	0
$773$	−12.7578	−0.458867	−0.229433	−	0.973324i	$-0.573687\pi$
$773$	−12.7578	−0.458867	−0.229433	+	0.973324i	$0.573687\pi$
$774$	0	0
$775$	−71.8817	−2.58207
$776$	0	0
$777$	−50.6363	−1.81657
$778$	0	0
$779$	−2.37145	−0.0849662
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−143.301	−5.12115
$784$	0	0
$785$	−38.3028	−1.36709
$786$	0	0
$787$	−27.0124	−0.962889	−0.481445	−	0.876476i	$-0.659888\pi$
$787$	−27.0124	−0.962889	−0.481445	+	0.876476i	$0.659888\pi$
$788$	0	0
$789$	27.2769	0.971084
$790$	0	0
$791$	32.8740	1.16886
$792$	0	0
$793$	41.5033	1.47382
$794$	0	0
$795$	29.1081	1.03236
$796$	0	0
$797$	−17.2729	−0.611837	−0.305919	−	0.952058i	$-0.598964\pi$
$797$	−17.2729	−0.611837	−0.305919	+	0.952058i	$0.598964\pi$
$798$	0	0
$799$	−5.71685	−0.202248
$800$	0	0
$801$	47.2255	1.66863
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−47.9405	−1.68968
$806$	0	0
$807$	−101.899	−3.58702
$808$	0	0
$809$	38.7686	1.36303	0.681515	−	0.731804i	$-0.261321\pi$
$809$	38.7686	1.36303	0.681515	+	0.731804i	$0.261321\pi$
$810$	0	0
$811$	−15.4408	−0.542199	−0.271100	−	0.962551i	$-0.587387\pi$
$811$	−15.4408	−0.542199	−0.271100	+	0.962551i	$0.587387\pi$
$812$	0	0
$813$	78.3725	2.74864
$814$	0	0
$815$	−31.6616	−1.10906
$816$	0	0
$817$	−1.18026	−0.0412921
$818$	0	0
$819$	98.9025	3.45593
$820$	0	0
$821$	−34.9424	−1.21950	−0.609750	−	0.792594i	$-0.708730\pi$
$821$	−34.9424	−1.21950	−0.609750	+	0.792594i	$0.708730\pi$
$822$	0	0
$823$	−20.9332	−0.729684	−0.364842	−	0.931069i	$-0.618877\pi$
$823$	−20.9332	−0.729684	−0.364842	+	0.931069i	$0.618877\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31.8329	−1.10694	−0.553470	−	0.832869i	$-0.686697\pi$
$827$	−31.8329	−1.10694	−0.553470	+	0.832869i	$0.686697\pi$
$828$	0	0
$829$	−15.1128	−0.524891	−0.262445	−	0.964947i	$-0.584529\pi$
$829$	−15.1128	−0.524891	−0.262445	+	0.964947i	$0.584529\pi$
$830$	0	0
$831$	5.46529	0.189589
$832$	0	0
$833$	5.07428	0.175813
$834$	0	0
$835$	49.9085	1.72715
$836$	0	0
$837$	−149.479	−5.16676
$838$	0	0
$839$	39.0512	1.34820	0.674099	−	0.738641i	$-0.264532\pi$
$839$	39.0512	1.34820	0.674099	+	0.738641i	$0.264532\pi$
$840$	0	0
$841$	55.5412	1.91521
$842$	0	0
$843$	−43.6139	−1.50214
$844$	0	0
$845$	58.6935	2.01912
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−43.4036	−1.48961
$850$	0	0
$851$	38.0983	1.30599
$852$	0	0
$853$	14.8573	0.508703	0.254351	−	0.967112i	$-0.418138\pi$
$853$	14.8573	0.508703	0.254351	+	0.967112i	$0.418138\pi$
$854$	0	0
$855$	27.4045	0.937216
$856$	0	0
$857$	−42.3513	−1.44669	−0.723347	−	0.690485i	$-0.757397\pi$
$857$	−42.3513	−1.44669	−0.723347	+	0.690485i	$0.757397\pi$
$858$	0	0
$859$	23.9371	0.816723	0.408361	−	0.912820i	$-0.366100\pi$
$859$	23.9371	0.816723	0.408361	+	0.912820i	$0.366100\pi$
$860$	0	0
$861$	−18.2324	−0.621358
$862$	0	0
$863$	−41.0021	−1.39573	−0.697865	−	0.716230i	$-0.745866\pi$
$863$	−41.0021	−1.39573	−0.697865	+	0.716230i	$0.745866\pi$
$864$	0	0
$865$	−9.06357	−0.308171
$866$	0	0
$867$	−18.3640	−0.623676
$868$	0	0
$869$	0	0
$870$	0	0
$871$	12.0866	0.409540
$872$	0	0
$873$	−47.8116	−1.61818
$874$	0	0
$875$	20.6747	0.698933
$876$	0	0
$877$	1.31855	0.0445242	0.0222621	−	0.999752i	$-0.492913\pi$
$877$	1.31855	0.0445242	0.0222621	+	0.999752i	$0.492913\pi$
$878$	0	0
$879$	26.4992	0.893797
$880$	0	0
$881$	−25.5816	−0.861865	−0.430933	−	0.902384i	$-0.641815\pi$
$881$	−25.5816	−0.861865	−0.430933	+	0.902384i	$0.641815\pi$
$882$	0	0
$883$	−23.2089	−0.781040	−0.390520	−	0.920594i	$-0.627705\pi$
$883$	−23.2089	−0.781040	−0.390520	+	0.920594i	$0.627705\pi$
$884$	0	0
$885$	−56.0920	−1.88551
$886$	0	0
$887$	33.6865	1.13108	0.565540	−	0.824721i	$-0.308668\pi$
$887$	33.6865	1.13108	0.565540	+	0.824721i	$0.308668\pi$
$888$	0	0
$889$	13.0064	0.436222
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.69320	0.0566609
$894$	0	0
$895$	1.61179	0.0538760
$896$	0	0
$897$	−103.208	−3.44602
$898$	0	0
$899$	88.1863	2.94118
$900$	0	0
$901$	−8.47887	−0.282472
$902$	0	0
$903$	−9.07417	−0.301969
$904$	0	0
$905$	−29.7559	−0.989119
$906$	0	0
$907$	40.5536	1.34656	0.673280	−	0.739388i	$-0.264885\pi$
$907$	40.5536	1.34656	0.673280	+	0.739388i	$0.264885\pi$
$908$	0	0
$909$	−53.2240	−1.76533
$910$	0	0
$911$	−14.4247	−0.477911	−0.238956	−	0.971031i	$-0.576805\pi$
$911$	−14.4247	−0.477911	−0.238956	+	0.971031i	$0.576805\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	88.4150	2.92291
$916$	0	0
$917$	27.3549	0.903337
$918$	0	0
$919$	−39.7419	−1.31096	−0.655482	−	0.755211i	$-0.727534\pi$
$919$	−39.7419	−1.31096	−0.655482	+	0.755211i	$0.727534\pi$
$920$	0	0
$921$	13.5340	0.445962
$922$	0	0
$923$	60.0573	1.97681
$924$	0	0
$925$	−49.3611	−1.62298
$926$	0	0
$927$	99.3574	3.26332
$928$	0	0
$929$	−20.4506	−0.670964	−0.335482	−	0.942047i	$-0.608899\pi$
$929$	−20.4506	−0.670964	−0.335482	+	0.942047i	$0.608899\pi$
$930$	0	0
$931$	−1.50289	−0.0492552
$932$	0	0
$933$	35.1991	1.15237
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.32869	0.272086	0.136043	−	0.990703i	$-0.456561\pi$
$937$	8.32869	0.272086	0.136043	+	0.990703i	$0.456561\pi$
$938$	0	0
$939$	23.6798	0.772760
$940$	0	0
$941$	−51.0299	−1.66353	−0.831763	−	0.555131i	$-0.812668\pi$
$941$	−51.0299	−1.66353	−0.831763	+	0.555131i	$0.812668\pi$
$942$	0	0
$943$	13.7179	0.446716
$944$	0	0
$945$	129.165	4.20172
$946$	0	0
$947$	−22.5652	−0.733270	−0.366635	−	0.930365i	$-0.619490\pi$
$947$	−22.5652	−0.733270	−0.366635	+	0.930365i	$0.619490\pi$
$948$	0	0
$949$	81.5848	2.64835
$950$	0	0
$951$	58.2380	1.88850
$952$	0	0
$953$	−34.1812	−1.10724	−0.553619	−	0.832770i	$-0.686754\pi$
$953$	−34.1812	−1.10724	−0.553619	+	0.832770i	$0.686754\pi$
$954$	0	0
$955$	63.5178	2.05539
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.63045	−0.214108
$960$	0	0
$961$	60.9886	1.96738
$962$	0	0
$963$	−27.0633	−0.872103
$964$	0	0
$965$	−60.9762	−1.96289
$966$	0	0
$967$	−22.0673	−0.709636	−0.354818	−	0.934935i	$-0.615457\pi$
$967$	−22.0673	−0.709636	−0.354818	+	0.934935i	$0.615457\pi$
$968$	0	0
$969$	−11.0716	−0.355670
$970$	0	0
$971$	13.2403	0.424902	0.212451	−	0.977172i	$-0.431855\pi$
$971$	13.2403	0.424902	0.212451	+	0.977172i	$0.431855\pi$
$972$	0	0
$973$	−36.3396	−1.16499
$974$	0	0
$975$	133.719	4.28243
$976$	0	0
$977$	6.65904	0.213041	0.106521	−	0.994310i	$-0.466029\pi$
$977$	6.65904	0.213041	0.106521	+	0.994310i	$0.466029\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−154.982	−4.94820
$982$	0	0
$983$	30.9907	0.988449	0.494225	−	0.869334i	$-0.335452\pi$
$983$	30.9907	0.988449	0.494225	+	0.869334i	$0.335452\pi$
$984$	0	0
$985$	24.3319	0.775280
$986$	0	0
$987$	13.0178	0.414361
$988$	0	0
$989$	6.82733	0.217097
$990$	0	0
$991$	59.2229	1.88128	0.940639	−	0.339408i	$-0.110227\pi$
$991$	59.2229	1.88128	0.940639	+	0.339408i	$0.110227\pi$
$992$	0	0
$993$	19.2340	0.610373
$994$	0	0
$995$	−53.2093	−1.68685
$996$	0	0
$997$	28.5785	0.905092	0.452546	−	0.891741i	$-0.350516\pi$
$997$	28.5785	0.905092	0.452546	+	0.891741i	$0.350516\pi$
$998$	0	0
$999$	−102.647	−3.24761

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.n.1.6		6
11.10	odd	2		836.2.a.e.1.6	✓	6
33.32	even	2		7524.2.a.q.1.1		6
44.43	even	2		3344.2.a.w.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.e.1.6	✓	6	11.10	odd	2
3344.2.a.w.1.1		6	44.43	even	2
7524.2.a.q.1.1		6	33.32	even	2
9196.2.a.n.1.6		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\(q+3.27915 q^{3} +3.53478 q^{5} +2.34459 q^{7} +7.75283 q^{9} +5.44101 q^{13} +11.5911 q^{15} -3.37635 q^{17} +1.00000 q^{19} +7.68827 q^{21} -5.78459 q^{23} +7.49465 q^{25} +15.5853 q^{27} -9.19463 q^{29} -9.59107 q^{31} +8.28761 q^{35} -6.58617 q^{37} +17.8419 q^{39} -2.37145 q^{41} -1.18026 q^{43} +27.4045 q^{45} +1.69320 q^{47} -1.50289 q^{49} -11.0716 q^{51} +2.51125 q^{53} +3.27915 q^{57} -4.83925 q^{59} +7.62786 q^{61} +18.1772 q^{63} +19.2328 q^{65} +2.22139 q^{67} -18.9686 q^{69} +11.0379 q^{71} +14.9944 q^{73} +24.5761 q^{75} +5.18195 q^{79} +27.8479 q^{81} +4.24982 q^{83} -11.9347 q^{85} -30.1506 q^{87} +6.09139 q^{89} +12.7569 q^{91} -31.4506 q^{93} +3.53478 q^{95} -6.16698 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} - 8 q^{13} + 9 q^{15} + 2 q^{17} + 6 q^{19} - 18 q^{21} + 5 q^{23} + 13 q^{25} + 7 q^{27} - 10 q^{29} + 3 q^{31} + 4 q^{35} + 7 q^{37} + 10 q^{39} - 6 q^{41}+ \cdots + 37 q^{97}+O(q^{100}) \) 6 * q + q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9 - 8 * q^13 + 9 * q^15 + 2 * q^17 + 6 * q^19 - 18 * q^21 + 5 * q^23 + 13 * q^25 + 7 * q^27 - 10 * q^29 + 3 * q^31 + 4 * q^35 + 7 * q^37 + 10 * q^39 - 6 * q^41 - 16 * q^43 + 42 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^53 + q^57 + 15 * q^59 - 24 * q^61 + 20 * q^63 + 28 * q^65 + 25 * q^67 - 33 * q^69 - 9 * q^71 + 26 * q^73 + 28 * q^75 + 16 * q^79 + 58 * q^81 + 2 * q^83 - 12 * q^85 + 36 * q^87 + 7 * q^89 + 8 * q^91 - 55 * q^93 + 5 * q^95 + 37 * q^97

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