LMFDB - Embedded newform 416.4.a.k.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,4,Mod(1,416)] 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("416.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	416.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.5447945624$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 110x^{4} + 2925x^{2} - 10400 $ x^6 - 110x^4 + 2925x^2 - 10400
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$5.92375$ of defining polynomial
Character	$\chi$	$=$	416.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.92375 q^{3} +15.5783 q^{5} -13.5605 q^{7} +8.09087 q^{9} -31.2241 q^{11} +13.0000 q^{13} +92.2823 q^{15} +123.842 q^{17} +122.040 q^{19} -80.3291 q^{21} +152.771 q^{23} +117.685 q^{25} -112.013 q^{27} +94.9172 q^{29} +130.569 q^{31} -184.964 q^{33} -211.250 q^{35} +17.3074 q^{37} +77.0088 q^{39} +58.9011 q^{41} -363.313 q^{43} +126.042 q^{45} -86.1168 q^{47} -159.113 q^{49} +733.608 q^{51} +701.656 q^{53} -486.420 q^{55} +722.934 q^{57} -847.635 q^{59} +201.895 q^{61} -109.716 q^{63} +202.519 q^{65} -40.8828 q^{67} +904.979 q^{69} +158.582 q^{71} -671.189 q^{73} +697.137 q^{75} +423.415 q^{77} -1062.14 q^{79} -881.991 q^{81} -940.585 q^{83} +1929.25 q^{85} +562.266 q^{87} +634.272 q^{89} -176.287 q^{91} +773.459 q^{93} +1901.18 q^{95} +393.387 q^{97} -252.630 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 16 q^{5} + 58 q^{9} + 78 q^{13} + 8 q^{17} + 340 q^{21} + 126 q^{25} + 252 q^{29} + 640 q^{33} + 888 q^{37} + 652 q^{41} + 1308 q^{45} + 642 q^{49} + 1780 q^{53} - 360 q^{57} + 2500 q^{61} + 208 q^{65}+ \cdots - 3180 q^{97}+O(q^{100}) $ 6 * q + 16 * q^5 + 58 * q^9 + 78 * q^13 + 8 * q^17 + 340 * q^21 + 126 * q^25 + 252 * q^29 + 640 * q^33 + 888 * q^37 + 652 * q^41 + 1308 * q^45 + 642 * q^49 + 1780 * q^53 - 360 * q^57 + 2500 * q^61 + 208 * q^65 + 3480 * q^69 + 916 * q^73 + 3920 * q^77 - 946 * q^81 + 2908 * q^85 - 820 * q^89 + 2600 * q^93 - 3180 * 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
$3$	5.92375	1.14003	0.570014	−	0.821635i	$-0.306938\pi$
$3$	5.92375	1.14003	0.570014	+	0.821635i	$0.306938\pi$
$4$	0	0
$5$	15.5783	1.39337	0.696685	−	0.717377i	$-0.254658\pi$
$5$	15.5783	1.39337	0.696685	+	0.717377i	$0.254658\pi$
$6$	0	0
$7$	−13.5605	−0.732199	−0.366099	−	0.930576i	$-0.619307\pi$
$7$	−13.5605	−0.732199	−0.366099	+	0.930576i	$0.619307\pi$
$8$	0	0
$9$	8.09087	0.299662
$10$	0	0
$11$	−31.2241	−0.855857	−0.427928	−	0.903813i	$-0.640756\pi$
$11$	−31.2241	−0.855857	−0.427928	+	0.903813i	$0.640756\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	92.2823	1.58848
$16$	0	0
$17$	123.842	1.76682	0.883412	−	0.468597i	$-0.155240\pi$
$17$	123.842	1.76682	0.883412	+	0.468597i	$0.155240\pi$
$18$	0	0
$19$	122.040	1.47357	0.736786	−	0.676126i	$-0.236342\pi$
$19$	122.040	1.47357	0.736786	+	0.676126i	$0.236342\pi$
$20$	0	0
$21$	−80.3291	−0.834726
$22$	0	0
$23$	152.771	1.38500	0.692499	−	0.721418i	$-0.256510\pi$
$23$	152.771	1.38500	0.692499	+	0.721418i	$0.256510\pi$
$24$	0	0
$25$	117.685	0.941480
$26$	0	0
$27$	−112.013	−0.798404
$28$	0	0
$29$	94.9172	0.607782	0.303891	−	0.952707i	$-0.401714\pi$
$29$	94.9172	0.607782	0.303891	+	0.952707i	$0.401714\pi$
$30$	0	0
$31$	130.569	0.756481	0.378240	−	0.925707i	$-0.376529\pi$
$31$	130.569	0.756481	0.378240	+	0.925707i	$0.376529\pi$
$32$	0	0
$33$	−184.964	−0.975700
$34$	0	0
$35$	−211.250	−1.02022
$36$	0	0
$37$	17.3074	0.0769004	0.0384502	−	0.999261i	$-0.487758\pi$
$37$	17.3074	0.0769004	0.0384502	+	0.999261i	$0.487758\pi$
$38$	0	0
$39$	77.0088	0.316187
$40$	0	0
$41$	58.9011	0.224361	0.112180	−	0.993688i	$-0.464217\pi$
$41$	58.9011	0.224361	0.112180	+	0.993688i	$0.464217\pi$
$42$	0	0
$43$	−363.313	−1.28848	−0.644241	−	0.764822i	$-0.722827\pi$
$43$	−363.313	−1.28848	−0.644241	+	0.764822i	$0.722827\pi$
$44$	0	0
$45$	126.042	0.417540
$46$	0	0
$47$	−86.1168	−0.267264	−0.133632	−	0.991031i	$-0.542664\pi$
$47$	−86.1168	−0.267264	−0.133632	+	0.991031i	$0.542664\pi$
$48$	0	0
$49$	−159.113	−0.463885
$50$	0	0
$51$	733.608	2.01423
$52$	0	0
$53$	701.656	1.81849	0.909244	−	0.416263i	$-0.136660\pi$
$53$	701.656	1.81849	0.909244	+	0.416263i	$0.136660\pi$
$54$	0	0
$55$	−486.420	−1.19252
$56$	0	0
$57$	722.934	1.67991
$58$	0	0
$59$	−847.635	−1.87038	−0.935192	−	0.354140i	$-0.884773\pi$
$59$	−847.635	−1.87038	−0.935192	+	0.354140i	$0.884773\pi$
$60$	0	0
$61$	201.895	0.423772	0.211886	−	0.977294i	$-0.432040\pi$
$61$	201.895	0.423772	0.211886	+	0.977294i	$0.432040\pi$
$62$	0	0
$63$	−109.716	−0.219412
$64$	0	0
$65$	202.519	0.386451
$66$	0	0
$67$	−40.8828	−0.0745467	−0.0372733	−	0.999305i	$-0.511867\pi$
$67$	−40.8828	−0.0745467	−0.0372733	+	0.999305i	$0.511867\pi$
$68$	0	0
$69$	904.979	1.57894
$70$	0	0
$71$	158.582	0.265074	0.132537	−	0.991178i	$-0.457688\pi$
$71$	158.582	0.265074	0.132537	+	0.991178i	$0.457688\pi$
$72$	0	0
$73$	−671.189	−1.07612	−0.538060	−	0.842907i	$-0.680843\pi$
$73$	−671.189	−1.07612	−0.538060	+	0.842907i	$0.680843\pi$
$74$	0	0
$75$	697.137	1.07331
$76$	0	0
$77$	423.415	0.626657
$78$	0	0
$79$	−1062.14	−1.51266	−0.756331	−	0.654190i	$-0.773010\pi$
$79$	−1062.14	−1.51266	−0.756331	+	0.654190i	$0.773010\pi$
$80$	0	0
$81$	−881.991	−1.20986
$82$	0	0
$83$	−940.585	−1.24389	−0.621943	−	0.783062i	$-0.713657\pi$
$83$	−940.585	−1.24389	−0.621943	+	0.783062i	$0.713657\pi$
$84$	0	0
$85$	1929.25	2.46184
$86$	0	0
$87$	562.266	0.692888
$88$	0	0
$89$	634.272	0.755424	0.377712	−	0.925923i	$-0.376711\pi$
$89$	634.272	0.755424	0.377712	+	0.925923i	$0.376711\pi$
$90$	0	0
$91$	−176.287	−0.203075
$92$	0	0
$93$	773.459	0.862408
$94$	0	0
$95$	1901.18	2.05323
$96$	0	0
$97$	393.387	0.411777	0.205889	−	0.978575i	$-0.433992\pi$
$97$	393.387	0.411777	0.205889	+	0.978575i	$0.433992\pi$
$98$	0	0
$99$	−252.630	−0.256468
$100$	0	0
$101$	320.134	0.315391	0.157696	−	0.987488i	$-0.449593\pi$
$101$	320.134	0.315391	0.157696	+	0.987488i	$0.449593\pi$
$102$	0	0
$103$	−12.8240	−0.0122678	−0.00613389	−	0.999981i	$-0.501952\pi$
$103$	−12.8240	−0.0122678	−0.00613389	+	0.999981i	$0.501952\pi$
$104$	0	0
$105$	−1251.40	−1.16308
$106$	0	0
$107$	989.226	0.893758	0.446879	−	0.894594i	$-0.352535\pi$
$107$	989.226	0.893758	0.446879	+	0.894594i	$0.352535\pi$
$108$	0	0
$109$	−1437.12	−1.26285	−0.631426	−	0.775436i	$-0.717530\pi$
$109$	−1437.12	−1.26285	−0.631426	+	0.775436i	$0.717530\pi$
$110$	0	0
$111$	102.525	0.0876686
$112$	0	0
$113$	−1631.15	−1.35792	−0.678961	−	0.734174i	$-0.737570\pi$
$113$	−1631.15	−1.35792	−0.678961	+	0.734174i	$0.737570\pi$
$114$	0	0
$115$	2379.92	1.92982
$116$	0	0
$117$	105.181	0.0831113
$118$	0	0
$119$	−1679.36	−1.29367
$120$	0	0
$121$	−356.055	−0.267509
$122$	0	0
$123$	348.915	0.255778
$124$	0	0
$125$	−113.956	−0.0815405
$126$	0	0
$127$	385.344	0.269242	0.134621	−	0.990897i	$-0.457018\pi$
$127$	385.344	0.269242	0.134621	+	0.990897i	$0.457018\pi$
$128$	0	0
$129$	−2152.18	−1.46890
$130$	0	0
$131$	1985.96	1.32454	0.662269	−	0.749266i	$-0.269594\pi$
$131$	1985.96	1.32454	0.662269	+	0.749266i	$0.269594\pi$
$132$	0	0
$133$	−1654.92	−1.07895
$134$	0	0
$135$	−1744.98	−1.11247
$136$	0	0
$137$	1257.53	0.784217	0.392108	−	0.919919i	$-0.371746\pi$
$137$	1257.53	0.784217	0.392108	+	0.919919i	$0.371746\pi$
$138$	0	0
$139$	−2383.62	−1.45450	−0.727252	−	0.686370i	$-0.759203\pi$
$139$	−2383.62	−1.45450	−0.727252	+	0.686370i	$0.759203\pi$
$140$	0	0
$141$	−510.135	−0.304689
$142$	0	0
$143$	−405.913	−0.237372
$144$	0	0
$145$	1478.65	0.846865
$146$	0	0
$147$	−942.544	−0.528842
$148$	0	0
$149$	660.562	0.363190	0.181595	−	0.983373i	$-0.441874\pi$
$149$	660.562	0.363190	0.181595	+	0.983373i	$0.441874\pi$
$150$	0	0
$151$	−2139.25	−1.15291	−0.576456	−	0.817128i	$-0.695565\pi$
$151$	−2139.25	−1.15291	−0.576456	+	0.817128i	$0.695565\pi$
$152$	0	0
$153$	1001.99	0.529450
$154$	0	0
$155$	2034.05	1.05406
$156$	0	0
$157$	−273.232	−0.138894	−0.0694468	−	0.997586i	$-0.522123\pi$
$157$	−273.232	−0.138894	−0.0694468	+	0.997586i	$0.522123\pi$
$158$	0	0
$159$	4156.44	2.07313
$160$	0	0
$161$	−2071.65	−1.01409
$162$	0	0
$163$	1729.12	0.830891	0.415445	−	0.909618i	$-0.363626\pi$
$163$	1729.12	0.830891	0.415445	+	0.909618i	$0.363626\pi$
$164$	0	0
$165$	−2881.43	−1.35951
$166$	0	0
$167$	527.939	0.244630	0.122315	−	0.992491i	$-0.460968\pi$
$167$	527.939	0.244630	0.122315	+	0.992491i	$0.460968\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	987.409	0.441573
$172$	0	0
$173$	4267.36	1.87538	0.937692	−	0.347467i	$-0.112958\pi$
$173$	4267.36	1.87538	0.937692	+	0.347467i	$0.112958\pi$
$174$	0	0
$175$	−1595.87	−0.689350
$176$	0	0
$177$	−5021.18	−2.13229
$178$	0	0
$179$	−2640.88	−1.10273	−0.551366	−	0.834264i	$-0.685893\pi$
$179$	−2640.88	−1.10273	−0.551366	+	0.834264i	$0.685893\pi$
$180$	0	0
$181$	−1128.14	−0.463281	−0.231640	−	0.972801i	$-0.574409\pi$
$181$	−1128.14	−0.463281	−0.231640	+	0.972801i	$0.574409\pi$
$182$	0	0
$183$	1195.98	0.483111
$184$	0	0
$185$	269.620	0.107151
$186$	0	0
$187$	−3866.85	−1.51215
$188$	0	0
$189$	1518.95	0.584591
$190$	0	0
$191$	−3768.71	−1.42772	−0.713859	−	0.700290i	$-0.753054\pi$
$191$	−3768.71	−1.42772	−0.713859	+	0.700290i	$0.753054\pi$
$192$	0	0
$193$	−524.268	−0.195532	−0.0977659	−	0.995209i	$-0.531170\pi$
$193$	−524.268	−0.195532	−0.0977659	+	0.995209i	$0.531170\pi$
$194$	0	0
$195$	1199.67	0.440565
$196$	0	0
$197$	−1171.41	−0.423653	−0.211826	−	0.977307i	$-0.567941\pi$
$197$	−1171.41	−0.423653	−0.211826	+	0.977307i	$0.567941\pi$
$198$	0	0
$199$	2087.39	0.743574	0.371787	−	0.928318i	$-0.378745\pi$
$199$	2087.39	0.743574	0.371787	+	0.928318i	$0.378745\pi$
$200$	0	0
$201$	−242.180	−0.0849852
$202$	0	0
$203$	−1287.13	−0.445017
$204$	0	0
$205$	917.581	0.312618
$206$	0	0
$207$	1236.05	0.415031
$208$	0	0
$209$	−3810.59	−1.26117
$210$	0	0
$211$	2702.18	0.881639	0.440819	−	0.897596i	$-0.354688\pi$
$211$	2702.18	0.881639	0.440819	+	0.897596i	$0.354688\pi$
$212$	0	0
$213$	939.403	0.302192
$214$	0	0
$215$	−5659.82	−1.79533
$216$	0	0
$217$	−1770.58	−0.553894
$218$	0	0
$219$	−3975.96	−1.22681
$220$	0	0
$221$	1609.94	0.490029
$222$	0	0
$223$	−2997.54	−0.900134	−0.450067	−	0.892995i	$-0.648600\pi$
$223$	−2997.54	−0.900134	−0.450067	+	0.892995i	$0.648600\pi$
$224$	0	0
$225$	952.174	0.282126
$226$	0	0
$227$	−1157.59	−0.338466	−0.169233	−	0.985576i	$-0.554129\pi$
$227$	−1157.59	−0.338466	−0.169233	+	0.985576i	$0.554129\pi$
$228$	0	0
$229$	−2913.12	−0.840630	−0.420315	−	0.907378i	$-0.638080\pi$
$229$	−2913.12	−0.840630	−0.420315	+	0.907378i	$0.638080\pi$
$230$	0	0
$231$	2508.21	0.714406
$232$	0	0
$233$	−5676.59	−1.59608	−0.798038	−	0.602608i	$-0.794128\pi$
$233$	−5676.59	−1.59608	−0.798038	+	0.602608i	$0.794128\pi$
$234$	0	0
$235$	−1341.56	−0.372398
$236$	0	0
$237$	−6291.86	−1.72447
$238$	0	0
$239$	180.549	0.0488651	0.0244325	−	0.999701i	$-0.492222\pi$
$239$	180.549	0.0488651	0.0244325	+	0.999701i	$0.492222\pi$
$240$	0	0
$241$	3912.01	1.04562	0.522811	−	0.852448i	$-0.324883\pi$
$241$	3912.01	1.04562	0.522811	+	0.852448i	$0.324883\pi$
$242$	0	0
$243$	−2200.35	−0.580874
$244$	0	0
$245$	−2478.71	−0.646364
$246$	0	0
$247$	1586.52	0.408695
$248$	0	0
$249$	−5571.79	−1.41806
$250$	0	0
$251$	−5748.27	−1.44553	−0.722764	−	0.691094i	$-0.757129\pi$
$251$	−5748.27	−1.44553	−0.722764	+	0.691094i	$0.757129\pi$
$252$	0	0
$253$	−4770.14	−1.18536
$254$	0	0
$255$	11428.4	2.80656
$256$	0	0
$257$	−4387.90	−1.06502	−0.532509	−	0.846425i	$-0.678751\pi$
$257$	−4387.90	−1.06502	−0.532509	+	0.846425i	$0.678751\pi$
$258$	0	0
$259$	−234.697	−0.0563064
$260$	0	0
$261$	767.963	0.182129
$262$	0	0
$263$	4413.18	1.03471	0.517354	−	0.855772i	$-0.326917\pi$
$263$	4413.18	1.03471	0.517354	+	0.855772i	$0.326917\pi$
$264$	0	0
$265$	10930.6	2.53383
$266$	0	0
$267$	3757.27	0.861203
$268$	0	0
$269$	5793.95	1.31325	0.656624	−	0.754218i	$-0.271984\pi$
$269$	5793.95	1.31325	0.656624	+	0.754218i	$0.271984\pi$
$270$	0	0
$271$	8188.99	1.83559	0.917796	−	0.397051i	$-0.129967\pi$
$271$	8188.99	1.83559	0.917796	+	0.397051i	$0.129967\pi$
$272$	0	0
$273$	−1044.28	−0.231511
$274$	0	0
$275$	−3674.61	−0.805772
$276$	0	0
$277$	54.4386	0.0118083	0.00590415	−	0.999983i	$-0.498121\pi$
$277$	54.4386	0.0118083	0.00590415	+	0.999983i	$0.498121\pi$
$278$	0	0
$279$	1056.42	0.226688
$280$	0	0
$281$	−3841.77	−0.815590	−0.407795	−	0.913074i	$-0.633702\pi$
$281$	−3841.77	−0.815590	−0.407795	+	0.913074i	$0.633702\pi$
$282$	0	0
$283$	7935.53	1.66685	0.833425	−	0.552632i	$-0.186377\pi$
$283$	7935.53	1.66685	0.833425	+	0.552632i	$0.186377\pi$
$284$	0	0
$285$	11262.1	2.34074
$286$	0	0
$287$	−798.728	−0.164277
$288$	0	0
$289$	10423.8	2.12167
$290$	0	0
$291$	2330.33	0.469437
$292$	0	0
$293$	−7776.81	−1.55060	−0.775301	−	0.631593i	$-0.782402\pi$
$293$	−7776.81	−1.55060	−0.775301	+	0.631593i	$0.782402\pi$
$294$	0	0
$295$	−13204.8	−2.60614
$296$	0	0
$297$	3497.51	0.683320
$298$	0	0
$299$	1986.02	0.384130
$300$	0	0
$301$	4926.71	0.943425
$302$	0	0
$303$	1896.40	0.359555
$304$	0	0
$305$	3145.20	0.590470
$306$	0	0
$307$	−3258.04	−0.605688	−0.302844	−	0.953040i	$-0.597936\pi$
$307$	−3258.04	−0.605688	−0.302844	+	0.953040i	$0.597936\pi$
$308$	0	0
$309$	−75.9660	−0.0139856
$310$	0	0
$311$	5109.12	0.931549	0.465775	−	0.884903i	$-0.345776\pi$
$311$	5109.12	0.931549	0.465775	+	0.884903i	$0.345776\pi$
$312$	0	0
$313$	767.796	0.138653	0.0693265	−	0.997594i	$-0.477915\pi$
$313$	767.796	0.138653	0.0693265	+	0.997594i	$0.477915\pi$
$314$	0	0
$315$	−1709.20	−0.305722
$316$	0	0
$317$	−4814.85	−0.853088	−0.426544	−	0.904467i	$-0.640269\pi$
$317$	−4814.85	−0.853088	−0.426544	+	0.904467i	$0.640269\pi$
$318$	0	0
$319$	−2963.71	−0.520174
$320$	0	0
$321$	5859.93	1.01891
$322$	0	0
$323$	15113.6	2.60354
$324$	0	0
$325$	1529.90	0.261119
$326$	0	0
$327$	−8513.13	−1.43969
$328$	0	0
$329$	1167.79	0.195691
$330$	0	0
$331$	3866.07	0.641990	0.320995	−	0.947081i	$-0.395983\pi$
$331$	3866.07	0.641990	0.320995	+	0.947081i	$0.395983\pi$
$332$	0	0
$333$	140.032	0.0230441
$334$	0	0
$335$	−636.886	−0.103871
$336$	0	0
$337$	8280.96	1.33855	0.669277	−	0.743013i	$-0.266604\pi$
$337$	8280.96	1.33855	0.669277	+	0.743013i	$0.266604\pi$
$338$	0	0
$339$	−9662.50	−1.54807
$340$	0	0
$341$	−4076.90	−0.647439
$342$	0	0
$343$	6808.90	1.07185
$344$	0	0
$345$	14098.1	2.20004
$346$	0	0
$347$	11445.0	1.77061	0.885304	−	0.465012i	$-0.153950\pi$
$347$	11445.0	1.77061	0.885304	+	0.465012i	$0.153950\pi$
$348$	0	0
$349$	6637.37	1.01802	0.509012	−	0.860759i	$-0.330011\pi$
$349$	6637.37	1.01802	0.509012	+	0.860759i	$0.330011\pi$
$350$	0	0
$351$	−1456.17	−0.221438
$352$	0	0
$353$	−2346.97	−0.353872	−0.176936	−	0.984222i	$-0.556619\pi$
$353$	−2346.97	−0.353872	−0.176936	+	0.984222i	$0.556619\pi$
$354$	0	0
$355$	2470.45	0.369346
$356$	0	0
$357$	−9948.09	−1.47481
$358$	0	0
$359$	−10364.6	−1.52374	−0.761872	−	0.647728i	$-0.775719\pi$
$359$	−10364.6	−1.52374	−0.761872	+	0.647728i	$0.775719\pi$
$360$	0	0
$361$	8034.73	1.17141
$362$	0	0
$363$	−2109.18	−0.304968
$364$	0	0
$365$	−10456.0	−1.49943
$366$	0	0
$367$	−10430.2	−1.48352	−0.741762	−	0.670663i	$-0.766010\pi$
$367$	−10430.2	−1.48352	−0.741762	+	0.670663i	$0.766010\pi$
$368$	0	0
$369$	476.561	0.0672324
$370$	0	0
$371$	−9514.81	−1.33149
$372$	0	0
$373$	9098.80	1.26305	0.631525	−	0.775355i	$-0.282429\pi$
$373$	9098.80	1.26305	0.631525	+	0.775355i	$0.282429\pi$
$374$	0	0
$375$	−675.050	−0.0929584
$376$	0	0
$377$	1233.92	0.168568
$378$	0	0
$379$	−5770.72	−0.782116	−0.391058	−	0.920366i	$-0.627891\pi$
$379$	−5770.72	−0.782116	−0.391058	+	0.920366i	$0.627891\pi$
$380$	0	0
$381$	2282.68	0.306943
$382$	0	0
$383$	−14.7089	−0.00196238	−0.000981191	−	1.00000i	$-0.500312\pi$
$383$	−14.7089	−0.00196238	−0.000981191		1.00000i	$0.500312\pi$
$384$	0	0
$385$	6596.10	0.873165
$386$	0	0
$387$	−2939.52	−0.386109
$388$	0	0
$389$	−11102.4	−1.44708	−0.723540	−	0.690283i	$-0.757486\pi$
$389$	−11102.4	−1.44708	−0.723540	+	0.690283i	$0.757486\pi$
$390$	0	0
$391$	18919.4	2.44705
$392$	0	0
$393$	11764.4	1.51001
$394$	0	0
$395$	−16546.4	−2.10770
$396$	0	0
$397$	8594.09	1.08646	0.543231	−	0.839583i	$-0.317201\pi$
$397$	8594.09	1.08646	0.543231	+	0.839583i	$0.317201\pi$
$398$	0	0
$399$	−9803.36	−1.23003
$400$	0	0
$401$	−3092.92	−0.385170	−0.192585	−	0.981280i	$-0.561687\pi$
$401$	−3092.92	−0.385170	−0.192585	+	0.981280i	$0.561687\pi$
$402$	0	0
$403$	1697.40	0.209810
$404$	0	0
$405$	−13740.0	−1.68579
$406$	0	0
$407$	−540.408	−0.0658158
$408$	0	0
$409$	−515.273	−0.0622949	−0.0311474	−	0.999515i	$-0.509916\pi$
$409$	−515.273	−0.0622949	−0.0311474	+	0.999515i	$0.509916\pi$
$410$	0	0
$411$	7449.27	0.894028
$412$	0	0
$413$	11494.4	1.36949
$414$	0	0
$415$	−14652.8	−1.73319
$416$	0	0
$417$	−14120.0	−1.65818
$418$	0	0
$419$	8422.50	0.982019	0.491009	−	0.871154i	$-0.336628\pi$
$419$	8422.50	0.982019	0.491009	+	0.871154i	$0.336628\pi$
$420$	0	0
$421$	−12950.7	−1.49924	−0.749620	−	0.661869i	$-0.769764\pi$
$421$	−12950.7	−1.49924	−0.749620	+	0.661869i	$0.769764\pi$
$422$	0	0
$423$	−696.760	−0.0800890
$424$	0	0
$425$	14574.3	1.66343
$426$	0	0
$427$	−2737.80	−0.310285
$428$	0	0
$429$	−2404.53	−0.270610
$430$	0	0
$431$	−2351.34	−0.262785	−0.131392	−	0.991330i	$-0.541945\pi$
$431$	−2351.34	−0.262785	−0.131392	+	0.991330i	$0.541945\pi$
$432$	0	0
$433$	−5506.04	−0.611093	−0.305547	−	0.952177i	$-0.598839\pi$
$433$	−5506.04	−0.611093	−0.305547	+	0.952177i	$0.598839\pi$
$434$	0	0
$435$	8759.18	0.965449
$436$	0	0
$437$	18644.2	2.04090
$438$	0	0
$439$	16403.4	1.78335	0.891676	−	0.452674i	$-0.149530\pi$
$439$	16403.4	1.78335	0.891676	+	0.452674i	$0.149530\pi$
$440$	0	0
$441$	−1287.36	−0.139009
$442$	0	0
$443$	13759.5	1.47570	0.737848	−	0.674967i	$-0.235842\pi$
$443$	13759.5	1.47570	0.737848	+	0.674967i	$0.235842\pi$
$444$	0	0
$445$	9880.91	1.05258
$446$	0	0
$447$	3913.01	0.414047
$448$	0	0
$449$	−15882.1	−1.66932	−0.834660	−	0.550766i	$-0.814336\pi$
$449$	−15882.1	−1.66932	−0.834660	+	0.550766i	$0.814336\pi$
$450$	0	0
$451$	−1839.13	−0.192021
$452$	0	0
$453$	−12672.4	−1.31435
$454$	0	0
$455$	−2746.25	−0.282959
$456$	0	0
$457$	−9542.58	−0.976768	−0.488384	−	0.872629i	$-0.662414\pi$
$457$	−9542.58	−0.976768	−0.488384	+	0.872629i	$0.662414\pi$
$458$	0	0
$459$	−13871.9	−1.41064
$460$	0	0
$461$	−12978.8	−1.31124	−0.655619	−	0.755092i	$-0.727592\pi$
$461$	−12978.8	−1.31124	−0.655619	+	0.755092i	$0.727592\pi$
$462$	0	0
$463$	−9196.97	−0.923152	−0.461576	−	0.887101i	$-0.652716\pi$
$463$	−9196.97	−0.923152	−0.461576	+	0.887101i	$0.652716\pi$
$464$	0	0
$465$	12049.2	1.20165
$466$	0	0
$467$	8635.66	0.855698	0.427849	−	0.903850i	$-0.359272\pi$
$467$	8635.66	0.855698	0.427849	+	0.903850i	$0.359272\pi$
$468$	0	0
$469$	554.391	0.0545830
$470$	0	0
$471$	−1618.56	−0.158343
$472$	0	0
$473$	11344.1	1.10276
$474$	0	0
$475$	14362.3	1.38734
$476$	0	0
$477$	5677.01	0.544932
$478$	0	0
$479$	−9568.41	−0.912717	−0.456359	−	0.889796i	$-0.650847\pi$
$479$	−9568.41	−0.912717	−0.456359	+	0.889796i	$0.650847\pi$
$480$	0	0
$481$	224.996	0.0213283
$482$	0	0
$483$	−12272.0	−1.15610
$484$	0	0
$485$	6128.32	0.573758
$486$	0	0
$487$	−12663.1	−1.17827	−0.589135	−	0.808034i	$-0.700532\pi$
$487$	−12663.1	−1.17827	−0.589135	+	0.808034i	$0.700532\pi$
$488$	0	0
$489$	10242.9	0.947238
$490$	0	0
$491$	−1316.49	−0.121002	−0.0605012	−	0.998168i	$-0.519270\pi$
$491$	−1316.49	−0.121002	−0.0605012	+	0.998168i	$0.519270\pi$
$492$	0	0
$493$	11754.7	1.07384
$494$	0	0
$495$	−3935.56	−0.357354
$496$	0	0
$497$	−2150.46	−0.194087
$498$	0	0
$499$	−12781.2	−1.14663	−0.573314	−	0.819336i	$-0.694342\pi$
$499$	−12781.2	−1.14663	−0.573314	+	0.819336i	$0.694342\pi$
$500$	0	0
$501$	3127.38	0.278884
$502$	0	0
$503$	8283.67	0.734295	0.367148	−	0.930163i	$-0.380334\pi$
$503$	8283.67	0.734295	0.367148	+	0.930163i	$0.380334\pi$
$504$	0	0
$505$	4987.16	0.439457
$506$	0	0
$507$	1001.11	0.0876944
$508$	0	0
$509$	−856.974	−0.0746261	−0.0373131	−	0.999304i	$-0.511880\pi$
$509$	−856.974	−0.0746261	−0.0373131	+	0.999304i	$0.511880\pi$
$510$	0	0
$511$	9101.67	0.787934
$512$	0	0
$513$	−13670.1	−1.17651
$514$	0	0
$515$	−199.776	−0.0170936
$516$	0	0
$517$	2688.92	0.228740
$518$	0	0
$519$	25278.8	2.13799
$520$	0	0
$521$	−8750.87	−0.735859	−0.367930	−	0.929854i	$-0.619933\pi$
$521$	−8750.87	−0.735859	−0.367930	+	0.929854i	$0.619933\pi$
$522$	0	0
$523$	9938.29	0.830920	0.415460	−	0.909612i	$-0.363621\pi$
$523$	9938.29	0.830920	0.415460	+	0.909612i	$0.363621\pi$
$524$	0	0
$525$	−9453.53	−0.785878
$526$	0	0
$527$	16169.9	1.33657
$528$	0	0
$529$	11172.0	0.918222
$530$	0	0
$531$	−6858.11	−0.560483
$532$	0	0
$533$	765.714	0.0622265
$534$	0	0
$535$	15410.5	1.24534
$536$	0	0
$537$	−15644.0	−1.25714
$538$	0	0
$539$	4968.15	0.397019
$540$	0	0
$541$	3470.03	0.275764	0.137882	−	0.990449i	$-0.455971\pi$
$541$	3470.03	0.275764	0.137882	+	0.990449i	$0.455971\pi$
$542$	0	0
$543$	−6682.81	−0.528153
$544$	0	0
$545$	−22387.9	−1.75962
$546$	0	0
$547$	−2147.82	−0.167887	−0.0839435	−	0.996471i	$-0.526752\pi$
$547$	−2147.82	−0.167887	−0.0839435	+	0.996471i	$0.526752\pi$
$548$	0	0
$549$	1633.51	0.126988
$550$	0	0
$551$	11583.7	0.895611
$552$	0	0
$553$	14403.2	1.10757
$554$	0	0
$555$	1597.17	0.122155
$556$	0	0
$557$	−13751.3	−1.04607	−0.523037	−	0.852310i	$-0.675201\pi$
$557$	−13751.3	−1.04607	−0.523037	+	0.852310i	$0.675201\pi$
$558$	0	0
$559$	−4723.07	−0.357361
$560$	0	0
$561$	−22906.2	−1.72389
$562$	0	0
$563$	1077.51	0.0806598	0.0403299	−	0.999186i	$-0.487159\pi$
$563$	1077.51	0.0806598	0.0403299	+	0.999186i	$0.487159\pi$
$564$	0	0
$565$	−25410.5	−1.89209
$566$	0	0
$567$	11960.3	0.885861
$568$	0	0
$569$	11538.5	0.850120	0.425060	−	0.905165i	$-0.360253\pi$
$569$	11538.5	0.850120	0.425060	+	0.905165i	$0.360253\pi$
$570$	0	0
$571$	−6799.09	−0.498306	−0.249153	−	0.968464i	$-0.580152\pi$
$571$	−6799.09	−0.498306	−0.249153	+	0.968464i	$0.580152\pi$
$572$	0	0
$573$	−22324.9	−1.62764
$574$	0	0
$575$	17978.9	1.30395
$576$	0	0
$577$	5900.94	0.425753	0.212876	−	0.977079i	$-0.431717\pi$
$577$	5900.94	0.425753	0.212876	+	0.977079i	$0.431717\pi$
$578$	0	0
$579$	−3105.64	−0.222912
$580$	0	0
$581$	12754.8	0.910772
$582$	0	0
$583$	−21908.6	−1.55637
$584$	0	0
$585$	1638.55	0.115805
$586$	0	0
$587$	20703.8	1.45577	0.727884	−	0.685700i	$-0.240504\pi$
$587$	20703.8	1.45577	0.727884	+	0.685700i	$0.240504\pi$
$588$	0	0
$589$	15934.6	1.11473
$590$	0	0
$591$	−6939.16	−0.482976
$592$	0	0
$593$	−19600.4	−1.35732	−0.678662	−	0.734451i	$-0.737440\pi$
$593$	−19600.4	−1.35732	−0.678662	+	0.734451i	$0.737440\pi$
$594$	0	0
$595$	−26161.6	−1.80256
$596$	0	0
$597$	12365.2	0.847695
$598$	0	0
$599$	−27794.2	−1.89589	−0.947946	−	0.318431i	$-0.896844\pi$
$599$	−27794.2	−1.89589	−0.947946	+	0.318431i	$0.896844\pi$
$600$	0	0
$601$	−14969.0	−1.01597	−0.507985	−	0.861366i	$-0.669609\pi$
$601$	−14969.0	−1.01597	−0.507985	+	0.861366i	$0.669609\pi$
$602$	0	0
$603$	−330.777	−0.0223388
$604$	0	0
$605$	−5546.75	−0.372739
$606$	0	0
$607$	11670.7	0.780394	0.390197	−	0.920731i	$-0.372407\pi$
$607$	11670.7	0.780394	0.390197	+	0.920731i	$0.372407\pi$
$608$	0	0
$609$	−7624.62	−0.507332
$610$	0	0
$611$	−1119.52	−0.0741258
$612$	0	0
$613$	−10167.0	−0.669889	−0.334944	−	0.942238i	$-0.608717\pi$
$613$	−10167.0	−0.669889	−0.334944	+	0.942238i	$0.608717\pi$
$614$	0	0
$615$	5435.53	0.356393
$616$	0	0
$617$	−29642.7	−1.93415	−0.967075	−	0.254493i	$-0.918091\pi$
$617$	−29642.7	−1.93415	−0.967075	+	0.254493i	$0.918091\pi$
$618$	0	0
$619$	−7708.29	−0.500521	−0.250260	−	0.968179i	$-0.580516\pi$
$619$	−7708.29	−0.500521	−0.250260	+	0.968179i	$0.580516\pi$
$620$	0	0
$621$	−17112.4	−1.10579
$622$	0	0
$623$	−8601.05	−0.553120
$624$	0	0
$625$	−16485.9	−1.05510
$626$	0	0
$627$	−22573.0	−1.43776
$628$	0	0
$629$	2143.37	0.135870
$630$	0	0
$631$	−11781.1	−0.743261	−0.371631	−	0.928381i	$-0.621201\pi$
$631$	−11781.1	−0.743261	−0.371631	+	0.928381i	$0.621201\pi$
$632$	0	0
$633$	16007.1	1.00509
$634$	0	0
$635$	6003.02	0.375154
$636$	0	0
$637$	−2068.46	−0.128659
$638$	0	0
$639$	1283.07	0.0794326
$640$	0	0
$641$	5136.98	0.316534	0.158267	−	0.987396i	$-0.449409\pi$
$641$	5136.98	0.316534	0.158267	+	0.987396i	$0.449409\pi$
$642$	0	0
$643$	28455.1	1.74520	0.872598	−	0.488440i	$-0.162434\pi$
$643$	28455.1	1.74520	0.872598	+	0.488440i	$0.162434\pi$
$644$	0	0
$645$	−33527.4	−2.04673
$646$	0	0
$647$	2529.14	0.153680	0.0768398	−	0.997043i	$-0.475517\pi$
$647$	2529.14	0.153680	0.0768398	+	0.997043i	$0.475517\pi$
$648$	0	0
$649$	26466.7	1.60078
$650$	0	0
$651$	−10488.5	−0.631454
$652$	0	0
$653$	23415.6	1.40325	0.701625	−	0.712546i	$-0.252458\pi$
$653$	23415.6	1.40325	0.701625	+	0.712546i	$0.252458\pi$
$654$	0	0
$655$	30938.0	1.84557
$656$	0	0
$657$	−5430.51	−0.322472
$658$	0	0
$659$	3586.47	0.212002	0.106001	−	0.994366i	$-0.466195\pi$
$659$	3586.47	0.212002	0.106001	+	0.994366i	$0.466195\pi$
$660$	0	0
$661$	2012.65	0.118431	0.0592155	−	0.998245i	$-0.481140\pi$
$661$	2012.65	0.118431	0.0592155	+	0.998245i	$0.481140\pi$
$662$	0	0
$663$	9536.90	0.558646
$664$	0	0
$665$	−25781.0	−1.50337
$666$	0	0
$667$	14500.6	0.841778
$668$	0	0
$669$	−17756.7	−1.02618
$670$	0	0
$671$	−6304.01	−0.362688
$672$	0	0
$673$	−6742.06	−0.386163	−0.193081	−	0.981183i	$-0.561848\pi$
$673$	−6742.06	−0.386163	−0.193081	+	0.981183i	$0.561848\pi$
$674$	0	0
$675$	−13182.2	−0.751681
$676$	0	0
$677$	−28466.9	−1.61606	−0.808030	−	0.589141i	$-0.799466\pi$
$677$	−28466.9	−1.61606	−0.808030	+	0.589141i	$0.799466\pi$
$678$	0	0
$679$	−5334.53	−0.301503
$680$	0	0
$681$	−6857.27	−0.385861
$682$	0	0
$683$	−435.701	−0.0244094	−0.0122047	−	0.999926i	$-0.503885\pi$
$683$	−435.701	−0.0244094	−0.0122047	+	0.999926i	$0.503885\pi$
$684$	0	0
$685$	19590.2	1.09270
$686$	0	0
$687$	−17256.6	−0.958341
$688$	0	0
$689$	9121.53	0.504358
$690$	0	0
$691$	9418.40	0.518513	0.259257	−	0.965808i	$-0.416522\pi$
$691$	9418.40	0.518513	0.259257	+	0.965808i	$0.416522\pi$
$692$	0	0
$693$	3425.80	0.187785
$694$	0	0
$695$	−37132.9	−2.02666
$696$	0	0
$697$	7294.40	0.396406
$698$	0	0
$699$	−33626.7	−1.81957
$700$	0	0
$701$	31304.2	1.68665	0.843325	−	0.537403i	$-0.180595\pi$
$701$	31304.2	1.68665	0.843325	+	0.537403i	$0.180595\pi$
$702$	0	0
$703$	2112.19	0.113318
$704$	0	0
$705$	−7947.06	−0.424544
$706$	0	0
$707$	−4341.18	−0.230929
$708$	0	0
$709$	14314.5	0.758243	0.379121	−	0.925347i	$-0.376226\pi$
$709$	14314.5	0.758243	0.379121	+	0.925347i	$0.376226\pi$
$710$	0	0
$711$	−8593.65	−0.453287
$712$	0	0
$713$	19947.2	1.04772
$714$	0	0
$715$	−6323.46	−0.330747
$716$	0	0
$717$	1069.53	0.0557075
$718$	0	0
$719$	17747.3	0.920531	0.460265	−	0.887781i	$-0.347754\pi$
$719$	17747.3	0.920531	0.460265	+	0.887781i	$0.347754\pi$
$720$	0	0
$721$	173.899	0.00898246
$722$	0	0
$723$	23173.8	1.19204
$724$	0	0
$725$	11170.3	0.572214
$726$	0	0
$727$	−17528.1	−0.894195	−0.447097	−	0.894485i	$-0.647542\pi$
$727$	−17528.1	−0.894195	−0.447097	+	0.894485i	$0.647542\pi$
$728$	0	0
$729$	10779.4	0.547652
$730$	0	0
$731$	−44993.3	−2.27652
$732$	0	0
$733$	26782.4	1.34956	0.674781	−	0.738018i	$-0.264238\pi$
$733$	26782.4	1.34956	0.674781	+	0.738018i	$0.264238\pi$
$734$	0	0
$735$	−14683.3	−0.736872
$736$	0	0
$737$	1276.53	0.0638013
$738$	0	0
$739$	38218.8	1.90244	0.951219	−	0.308515i	$-0.0998321\pi$
$739$	38218.8	1.90244	0.951219	+	0.308515i	$0.0998321\pi$
$740$	0	0
$741$	9398.15	0.465924
$742$	0	0
$743$	26630.3	1.31490	0.657450	−	0.753498i	$-0.271635\pi$
$743$	26630.3	1.31490	0.657450	+	0.753498i	$0.271635\pi$
$744$	0	0
$745$	10290.5	0.506058
$746$	0	0
$747$	−7610.15	−0.372746
$748$	0	0
$749$	−13414.4	−0.654408
$750$	0	0
$751$	14713.7	0.714928	0.357464	−	0.933927i	$-0.383641\pi$
$751$	14713.7	0.714928	0.357464	+	0.933927i	$0.383641\pi$
$752$	0	0
$753$	−34051.4	−1.64794
$754$	0	0
$755$	−33326.0	−1.60643
$756$	0	0
$757$	23212.4	1.11449	0.557245	−	0.830348i	$-0.311859\pi$
$757$	23212.4	1.11449	0.557245	+	0.830348i	$0.311859\pi$
$758$	0	0
$759$	−28257.2	−1.35134
$760$	0	0
$761$	23159.6	1.10320	0.551600	−	0.834109i	$-0.314018\pi$
$761$	23159.6	1.10320	0.551600	+	0.834109i	$0.314018\pi$
$762$	0	0
$763$	19488.0	0.924658
$764$	0	0
$765$	15609.3	0.737720
$766$	0	0
$767$	−11019.3	−0.518751
$768$	0	0
$769$	−25390.7	−1.19065	−0.595327	−	0.803483i	$-0.702978\pi$
$769$	−25390.7	−1.19065	−0.595327	+	0.803483i	$0.702978\pi$
$770$	0	0
$771$	−25992.8	−1.21415
$772$	0	0
$773$	40500.2	1.88446	0.942232	−	0.334961i	$-0.108723\pi$
$773$	40500.2	1.88446	0.942232	+	0.334961i	$0.108723\pi$
$774$	0	0
$775$	15366.0	0.712211
$776$	0	0
$777$	−1390.29	−0.0641908
$778$	0	0
$779$	7188.28	0.330612
$780$	0	0
$781$	−4951.59	−0.226865
$782$	0	0
$783$	−10632.0	−0.485256
$784$	0	0
$785$	−4256.51	−0.193530
$786$	0	0
$787$	2207.54	0.0999875	0.0499937	−	0.998750i	$-0.484080\pi$
$787$	2207.54	0.0999875	0.0499937	+	0.998750i	$0.484080\pi$
$788$	0	0
$789$	26142.6	1.17960
$790$	0	0
$791$	22119.2	0.994269
$792$	0	0
$793$	2624.64	0.117533
$794$	0	0
$795$	64750.5	2.88863
$796$	0	0
$797$	−3762.68	−0.167228	−0.0836141	−	0.996498i	$-0.526646\pi$
$797$	−3762.68	−0.167228	−0.0836141	+	0.996498i	$0.526646\pi$
$798$	0	0
$799$	−10664.8	−0.472209
$800$	0	0
$801$	5131.81	0.226372
$802$	0	0
$803$	20957.3	0.921004
$804$	0	0
$805$	−32272.9	−1.41301
$806$	0	0
$807$	34322.0	1.49714
$808$	0	0
$809$	−33126.0	−1.43961	−0.719806	−	0.694175i	$-0.755769\pi$
$809$	−33126.0	−1.43961	−0.719806	+	0.694175i	$0.755769\pi$
$810$	0	0
$811$	32188.4	1.39370	0.696849	−	0.717218i	$-0.254585\pi$
$811$	32188.4	1.39370	0.696849	+	0.717218i	$0.254585\pi$
$812$	0	0
$813$	48509.6	2.09263
$814$	0	0
$815$	26936.9	1.15774
$816$	0	0
$817$	−44338.7	−1.89867
$818$	0	0
$819$	−1426.31	−0.0608540
$820$	0	0
$821$	32238.4	1.37044	0.685218	−	0.728338i	$-0.259707\pi$
$821$	32238.4	1.37044	0.685218	+	0.728338i	$0.259707\pi$
$822$	0	0
$823$	−16481.8	−0.698079	−0.349040	−	0.937108i	$-0.613492\pi$
$823$	−16481.8	−0.698079	−0.349040	+	0.937108i	$0.613492\pi$
$824$	0	0
$825$	−21767.5	−0.918602
$826$	0	0
$827$	−3735.08	−0.157052	−0.0785258	−	0.996912i	$-0.525021\pi$
$827$	−3735.08	−0.157052	−0.0785258	+	0.996912i	$0.525021\pi$
$828$	0	0
$829$	36792.9	1.54146	0.770729	−	0.637163i	$-0.219892\pi$
$829$	36792.9	1.54146	0.770729	+	0.637163i	$0.219892\pi$
$830$	0	0
$831$	322.481	0.0134618
$832$	0	0
$833$	−19704.8	−0.819604
$834$	0	0
$835$	8224.42	0.340860
$836$	0	0
$837$	−14625.4	−0.603977
$838$	0	0
$839$	−19594.5	−0.806291	−0.403145	−	0.915136i	$-0.632083\pi$
$839$	−19594.5	−0.806291	−0.403145	+	0.915136i	$0.632083\pi$
$840$	0	0
$841$	−15379.7	−0.630601
$842$	0	0
$843$	−22757.7	−0.929794
$844$	0	0
$845$	2632.74	0.107182
$846$	0	0
$847$	4828.28	0.195870
$848$	0	0
$849$	47008.2	1.90025
$850$	0	0
$851$	2644.07	0.106507
$852$	0	0
$853$	−44842.5	−1.79997	−0.899987	−	0.435917i	$-0.856424\pi$
$853$	−44842.5	−1.79997	−0.899987	+	0.435917i	$0.856424\pi$
$854$	0	0
$855$	15382.2	0.615275
$856$	0	0
$857$	−14503.6	−0.578102	−0.289051	−	0.957314i	$-0.593340\pi$
$857$	−14503.6	−0.578102	−0.289051	+	0.957314i	$0.593340\pi$
$858$	0	0
$859$	15305.4	0.607932	0.303966	−	0.952683i	$-0.401689\pi$
$859$	15305.4	0.607932	0.303966	+	0.952683i	$0.401689\pi$
$860$	0	0
$861$	−4731.47	−0.187280
$862$	0	0
$863$	13818.6	0.545066	0.272533	−	0.962146i	$-0.412139\pi$
$863$	13818.6	0.545066	0.272533	+	0.962146i	$0.412139\pi$
$864$	0	0
$865$	66478.4	2.61310
$866$	0	0
$867$	61747.8	2.41876
$868$	0	0
$869$	33164.4	1.29462
$870$	0	0
$871$	−531.476	−0.0206755
$872$	0	0
$873$	3182.84	0.123394
$874$	0	0
$875$	1545.31	0.0597039
$876$	0	0
$877$	−6929.88	−0.266825	−0.133413	−	0.991061i	$-0.542594\pi$
$877$	−6929.88	−0.266825	−0.133413	+	0.991061i	$0.542594\pi$
$878$	0	0
$879$	−46067.9	−1.76773
$880$	0	0
$881$	49020.5	1.87462	0.937312	−	0.348491i	$-0.113306\pi$
$881$	49020.5	1.87462	0.937312	+	0.348491i	$0.113306\pi$
$882$	0	0
$883$	−10612.9	−0.404474	−0.202237	−	0.979337i	$-0.564821\pi$
$883$	−10612.9	−0.404474	−0.202237	+	0.979337i	$0.564821\pi$
$884$	0	0
$885$	−78221.7	−2.97107
$886$	0	0
$887$	−9294.89	−0.351851	−0.175926	−	0.984403i	$-0.556292\pi$
$887$	−9294.89	−0.351851	−0.175926	+	0.984403i	$0.556292\pi$
$888$	0	0
$889$	−5225.46	−0.197139
$890$	0	0
$891$	27539.4	1.03547
$892$	0	0
$893$	−10509.7	−0.393833
$894$	0	0
$895$	−41140.6	−1.53651
$896$	0	0
$897$	11764.7	0.437918
$898$	0	0
$899$	12393.2	0.459775
$900$	0	0
$901$	86894.3	3.21295
$902$	0	0
$903$	29184.6	1.07553
$904$	0	0
$905$	−17574.5	−0.645522
$906$	0	0
$907$	−39128.3	−1.43245	−0.716226	−	0.697868i	$-0.754132\pi$
$907$	−39128.3	−1.43245	−0.716226	+	0.697868i	$0.754132\pi$
$908$	0	0
$909$	2590.16	0.0945108
$910$	0	0
$911$	2556.70	0.0929828	0.0464914	−	0.998919i	$-0.485196\pi$
$911$	2556.70	0.0929828	0.0464914	+	0.998919i	$0.485196\pi$
$912$	0	0
$913$	29368.9	1.06459
$914$	0	0
$915$	18631.4	0.673152
$916$	0	0
$917$	−26930.7	−0.969824
$918$	0	0
$919$	5774.65	0.207278	0.103639	−	0.994615i	$-0.466951\pi$
$919$	5774.65	0.207278	0.103639	+	0.994615i	$0.466951\pi$
$920$	0	0
$921$	−19299.8	−0.690501
$922$	0	0
$923$	2061.57	0.0735183
$924$	0	0
$925$	2036.82	0.0724002
$926$	0	0
$927$	−103.757	−0.00367619
$928$	0	0
$929$	−21935.8	−0.774694	−0.387347	−	0.921934i	$-0.626608\pi$
$929$	−21935.8	−0.774694	−0.387347	+	0.921934i	$0.626608\pi$
$930$	0	0
$931$	−19418.1	−0.683568
$932$	0	0
$933$	30265.2	1.06199
$934$	0	0
$935$	−60239.1	−2.10698
$936$	0	0
$937$	16455.4	0.573718	0.286859	−	0.957973i	$-0.407389\pi$
$937$	16455.4	0.573718	0.286859	+	0.957973i	$0.407389\pi$
$938$	0	0
$939$	4548.23	0.158068
$940$	0	0
$941$	37160.8	1.28736	0.643681	−	0.765294i	$-0.277406\pi$
$941$	37160.8	1.28736	0.643681	+	0.765294i	$0.277406\pi$
$942$	0	0
$943$	8998.38	0.310740
$944$	0	0
$945$	23662.8	0.814551
$946$	0	0
$947$	−39560.5	−1.35749	−0.678746	−	0.734373i	$-0.737476\pi$
$947$	−39560.5	−1.35749	−0.678746	+	0.734373i	$0.737476\pi$
$948$	0	0
$949$	−8725.46	−0.298462
$950$	0	0
$951$	−28522.0	−0.972544
$952$	0	0
$953$	−16999.1	−0.577813	−0.288907	−	0.957357i	$-0.593292\pi$
$953$	−16999.1	−0.577813	−0.288907	+	0.957357i	$0.593292\pi$
$954$	0	0
$955$	−58710.2	−1.98934
$956$	0	0
$957$	−17556.3	−0.593013
$958$	0	0
$959$	−17052.7	−0.574202
$960$	0	0
$961$	−12742.7	−0.427737
$962$	0	0
$963$	8003.70	0.267825
$964$	0	0
$965$	−8167.23	−0.272448
$966$	0	0
$967$	38502.5	1.28041	0.640206	−	0.768204i	$-0.278849\pi$
$967$	38502.5	1.28041	0.640206	+	0.768204i	$0.278849\pi$
$968$	0	0
$969$	89529.4	2.96811
$970$	0	0
$971$	12104.1	0.400041	0.200020	−	0.979792i	$-0.435899\pi$
$971$	12104.1	0.400041	0.200020	+	0.979792i	$0.435899\pi$
$972$	0	0
$973$	32323.1	1.06499
$974$	0	0
$975$	9062.78	0.297683
$976$	0	0
$977$	−24134.7	−0.790315	−0.395157	−	0.918613i	$-0.629310\pi$
$977$	−24134.7	−0.790315	−0.395157	+	0.918613i	$0.629310\pi$
$978$	0	0
$979$	−19804.6	−0.646534
$980$	0	0
$981$	−11627.5	−0.378429
$982$	0	0
$983$	16208.1	0.525897	0.262949	−	0.964810i	$-0.415305\pi$
$983$	16208.1	0.525897	0.262949	+	0.964810i	$0.415305\pi$
$984$	0	0
$985$	−18248.7	−0.590305
$986$	0	0
$987$	6917.69	0.223093
$988$	0	0
$989$	−55503.8	−1.78455
$990$	0	0
$991$	7719.43	0.247443	0.123721	−	0.992317i	$-0.460517\pi$
$991$	7719.43	0.247443	0.123721	+	0.992317i	$0.460517\pi$
$992$	0	0
$993$	22901.7	0.731886
$994$	0	0
$995$	32518.1	1.03607
$996$	0	0
$997$	791.172	0.0251321	0.0125660	−	0.999921i	$-0.496000\pi$
$997$	791.172	0.0251321	0.0125660	+	0.999921i	$0.496000\pi$
$998$	0	0
$999$	−1938.65	−0.0613977

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.4.a.k.1.5	yes	6
4.3	odd	2	inner	416.4.a.k.1.2	✓	6
8.3	odd	2		832.4.a.bh.1.5		6
8.5	even	2		832.4.a.bh.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.4.a.k.1.2	✓	6	4.3	odd	2	inner
416.4.a.k.1.5	yes	6	1.1	even	1	trivial
832.4.a.bh.1.2		6	8.5	even	2
832.4.a.bh.1.5		6	8.3	odd	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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	416.a (trivial)

\(f(q)\)	\(=\)	\(q+5.92375 q^{3} +15.5783 q^{5} -13.5605 q^{7} +8.09087 q^{9} -31.2241 q^{11} +13.0000 q^{13} +92.2823 q^{15} +123.842 q^{17} +122.040 q^{19} -80.3291 q^{21} +152.771 q^{23} +117.685 q^{25} -112.013 q^{27} +94.9172 q^{29} +130.569 q^{31} -184.964 q^{33} -211.250 q^{35} +17.3074 q^{37} +77.0088 q^{39} +58.9011 q^{41} -363.313 q^{43} +126.042 q^{45} -86.1168 q^{47} -159.113 q^{49} +733.608 q^{51} +701.656 q^{53} -486.420 q^{55} +722.934 q^{57} -847.635 q^{59} +201.895 q^{61} -109.716 q^{63} +202.519 q^{65} -40.8828 q^{67} +904.979 q^{69} +158.582 q^{71} -671.189 q^{73} +697.137 q^{75} +423.415 q^{77} -1062.14 q^{79} -881.991 q^{81} -940.585 q^{83} +1929.25 q^{85} +562.266 q^{87} +634.272 q^{89} -176.287 q^{91} +773.459 q^{93} +1901.18 q^{95} +393.387 q^{97} -252.630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 16 q^{5} + 58 q^{9} + 78 q^{13} + 8 q^{17} + 340 q^{21} + 126 q^{25} + 252 q^{29} + 640 q^{33} + 888 q^{37} + 652 q^{41} + 1308 q^{45} + 642 q^{49} + 1780 q^{53} - 360 q^{57} + 2500 q^{61} + 208 q^{65}+ \cdots - 3180 q^{97}+O(q^{100}) \) 6 * q + 16 * q^5 + 58 * q^9 + 78 * q^13 + 8 * q^17 + 340 * q^21 + 126 * q^25 + 252 * q^29 + 640 * q^33 + 888 * q^37 + 652 * q^41 + 1308 * q^45 + 642 * q^49 + 1780 * q^53 - 360 * q^57 + 2500 * q^61 + 208 * q^65 + 3480 * q^69 + 916 * q^73 + 3920 * q^77 - 946 * q^81 + 2908 * q^85 - 820 * q^89 + 2600 * q^93 - 3180 * q^97

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