LMFDB - Embedded newform 950.4.b.g.799.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [950,4,Mod(799,950)] 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("950.799"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	950.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-16,0,-4,0,0,-70,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$56.0518145055$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{177})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 89x^{2} + 1936 $ x^4 + 89*x^2 + 1936
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.4
Root			$7.15207i$ of defining polynomial
Character	$\chi$	$=$	950.799
Dual form			950.4.b.g.799.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +7.15207i q^{3} -4.00000 q^{4} -14.3041 q^{6} -35.1521i q^{7} -8.00000i q^{8} -24.1521 q^{9} +18.3041 q^{11} -28.6083i q^{12} -40.0645i q^{13} +70.3041 q^{14} +16.0000 q^{16} +125.281i q^{17} -48.3041i q^{18} +19.0000 q^{19} +251.410 q^{21} +36.6083i q^{22} +8.97688i q^{23} +57.2165 q^{24} +80.1289 q^{26} +20.3686i q^{27} +140.608i q^{28} -153.410 q^{29} -114.433 q^{31} +32.0000i q^{32} +130.912i q^{33} -250.562 q^{34} +96.6083 q^{36} -83.5669i q^{37} +38.0000i q^{38} +286.544 q^{39} -355.088 q^{41} +502.820i q^{42} +467.299i q^{43} -73.2165 q^{44} -17.9538 q^{46} -166.083i q^{47} +114.433i q^{48} -892.668 q^{49} -896.018 q^{51} +160.258i q^{52} +258.369i q^{53} -40.7372 q^{54} -281.217 q^{56} +135.889i q^{57} -306.820i q^{58} +371.797 q^{59} -47.3090 q^{61} -228.866i q^{62} +848.995i q^{63} -64.0000 q^{64} -261.825 q^{66} +755.539i q^{67} -501.124i q^{68} -64.2032 q^{69} +349.345 q^{71} +193.217i q^{72} +54.8479i q^{73} +167.134 q^{74} -76.0000 q^{76} -643.428i q^{77} +573.088i q^{78} -438.820 q^{79} -797.783 q^{81} -710.175i q^{82} +1073.09i q^{83} -1005.64 q^{84} -934.598 q^{86} -1097.20i q^{87} -146.433i q^{88} +501.521 q^{89} -1408.35 q^{91} -35.9075i q^{92} -818.433i q^{93} +332.165 q^{94} -228.866 q^{96} +1437.56i q^{97} -1785.34i q^{98} -442.083 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} - 4 q^{6} - 70 q^{9} + 20 q^{11} + 228 q^{14} + 64 q^{16} + 76 q^{19} + 234 q^{21} + 16 q^{24} - 52 q^{26} + 158 q^{29} - 32 q^{31} - 204 q^{34} + 280 q^{36} + 1226 q^{39} - 1580 q^{41} - 80 q^{44}+ \cdots - 704 q^{99}+O(q^{100}) $ 4 * q - 16 * q^4 - 4 * q^6 - 70 * q^9 + 20 * q^11 + 228 * q^14 + 64 * q^16 + 76 * q^19 + 234 * q^21 + 16 * q^24 - 52 * q^26 + 158 * q^29 - 32 * q^31 - 204 * q^34 + 280 * q^36 + 1226 * q^39 - 1580 * q^41 - 80 * q^44 + 620 * q^46 - 2054 * q^49 - 2706 * q^51 + 316 * q^54 - 912 * q^56 - 402 * q^59 - 1360 * q^61 - 256 * q^64 - 728 * q^66 - 2146 * q^69 + 812 * q^71 + 1520 * q^74 - 304 * q^76 - 212 * q^79 - 3404 * q^81 - 936 * q^84 - 1184 * q^86 + 1740 * q^89 - 498 * q^91 - 800 * q^94 - 64 * q^96 - 704 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	7.15207i	1.37642i	0.725513	+	0.688208i	$0.241602\pi$
$3$	7.15207i	1.37642i	−0.725513	+	0.688208i	$0.758398\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−14.3041	−0.973273
$7$	− 35.1521i	− 1.89803i	−0.315226	−	0.949017i	$-0.602080\pi$
$7$	− 35.1521i	− 1.89803i	0.315226	−	0.949017i	$-0.397920\pi$
$8$	− 8.00000i	− 0.353553i
$9$	−24.1521	−0.894521
$10$	0	0
$11$	18.3041	0.501719	0.250859	−	0.968024i	$-0.419287\pi$
$11$	18.3041	0.501719	0.250859	+	0.968024i	$0.419287\pi$
$12$	− 28.6083i	− 0.688208i
$13$	− 40.0645i	− 0.854760i	−0.904072	−	0.427380i	$-0.859437\pi$
$13$	− 40.0645i	− 0.854760i	0.904072	−	0.427380i	$-0.140563\pi$
$14$	70.3041	1.34211
$15$	0	0
$16$	16.0000	0.250000
$17$	125.281i	1.78736i	0.448706	+	0.893680i	$0.351885\pi$
$17$	125.281i	1.78736i	−0.448706	+	0.893680i	$0.648115\pi$
$18$	− 48.3041i	− 0.632522i
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	251.410	2.61248
$22$	36.6083i	0.354769i
$23$	8.97688i	0.0813830i	0.999172	+	0.0406915i	$0.0129561\pi$
$23$	8.97688i	0.0813830i	−0.999172	+	0.0406915i	$0.987044\pi$
$24$	57.2165	0.486637
$25$	0	0
$26$	80.1289	0.604407
$27$	20.3686i	0.145183i
$28$	140.608i	0.949017i
$29$	−153.410	−0.982328	−0.491164	−	0.871067i	$-0.663428\pi$
$29$	−153.410	−0.982328	−0.491164	+	0.871067i	$0.663428\pi$
$30$	0	0
$31$	−114.433	−0.662993	−0.331497	−	0.943456i	$-0.607554\pi$
$31$	−114.433	−0.662993	−0.331497	+	0.943456i	$0.607554\pi$
$32$	32.0000i	0.176777i
$33$	130.912i	0.690573i
$34$	−250.562	−1.26385
$35$	0	0
$36$	96.6083	0.447261
$37$	− 83.5669i	− 0.371306i	−0.982615	−	0.185653i	$-0.940560\pi$
$37$	− 83.5669i	− 0.371306i	0.982615	−	0.185653i	$-0.0594400\pi$
$38$	38.0000i	0.162221i
$39$	286.544	1.17651
$40$	0	0
$41$	−355.088	−1.35257	−0.676285	−	0.736640i	$-0.736411\pi$
$41$	−355.088	−1.35257	−0.676285	+	0.736640i	$0.736411\pi$
$42$	502.820i	1.84730i
$43$	467.299i	1.65727i	0.559792	+	0.828633i	$0.310881\pi$
$43$	467.299i	1.65727i	−0.559792	+	0.828633i	$0.689119\pi$
$44$	−73.2165	−0.250859
$45$	0	0
$46$	−17.9538	−0.0575464
$47$	− 166.083i	− 0.515439i	−0.966220	−	0.257720i	$-0.917029\pi$
$47$	− 166.083i	− 0.515439i	0.966220	−	0.257720i	$-0.0829711\pi$
$48$	114.433i	0.344104i
$49$	−892.668	−2.60253
$50$	0	0
$51$	−896.018	−2.46015
$52$	160.258i	0.427380i
$53$	258.369i	0.669616i	0.942286	+	0.334808i	$0.108671\pi$
$53$	258.369i	0.669616i	−0.942286	+	0.334808i	$0.891329\pi$
$54$	−40.7372	−0.102660
$55$	0	0
$56$	−281.217	−0.671056
$57$	135.889i	0.315771i
$58$	− 306.820i	− 0.694611i
$59$	371.797	0.820404	0.410202	−	0.911995i	$-0.365458\pi$
$59$	371.797	0.820404	0.410202	+	0.911995i	$0.365458\pi$
$60$	0	0
$61$	−47.3090	−0.0993000	−0.0496500	−	0.998767i	$-0.515811\pi$
$61$	−47.3090	−0.0993000	−0.0496500	+	0.998767i	$0.515811\pi$
$62$	− 228.866i	− 0.468807i
$63$	848.995i	1.69783i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−261.825	−0.488309
$67$	755.539i	1.37767i	0.724919	+	0.688834i	$0.241877\pi$
$67$	755.539i	1.37767i	−0.724919	+	0.688834i	$0.758123\pi$
$68$	− 501.124i	− 0.893680i
$69$	−64.2032	−0.112017
$70$	0	0
$71$	349.345	0.583939	0.291970	−	0.956428i	$-0.405689\pi$
$71$	349.345	0.583939	0.291970	+	0.956428i	$0.405689\pi$
$72$	193.217i	0.316261i
$73$	54.8479i	0.0879379i	0.999033	+	0.0439689i	$0.0140003\pi$
$73$	54.8479i	0.0879379i	−0.999033	+	0.0439689i	$0.986000\pi$
$74$	167.134	0.262553
$75$	0	0
$76$	−76.0000	−0.114708
$77$	− 643.428i	− 0.952279i
$78$	573.088i	0.831915i
$79$	−438.820	−0.624951	−0.312475	−	0.949926i	$-0.601158\pi$
$79$	−438.820	−0.624951	−0.312475	+	0.949926i	$0.601158\pi$
$80$	0	0
$81$	−797.783	−1.09435
$82$	− 710.175i	− 0.956411i
$83$	1073.09i	1.41912i	0.704647	+	0.709558i	$0.251105\pi$
$83$	1073.09i	1.41912i	−0.704647	+	0.709558i	$0.748895\pi$
$84$	−1005.64	−1.30624
$85$	0	0
$86$	−934.598	−1.17186
$87$	− 1097.20i	− 1.35209i
$88$	− 146.433i	− 0.177384i
$89$	501.521	0.597316	0.298658	−	0.954360i	$-0.403461\pi$
$89$	501.521	0.597316	0.298658	+	0.954360i	$0.403461\pi$
$90$	0	0
$91$	−1408.35	−1.62236
$92$	− 35.9075i	− 0.0406915i
$93$	− 818.433i	− 0.912554i
$94$	332.165	0.364471
$95$	0	0
$96$	−228.866	−0.243318
$97$	1437.56i	1.50476i	0.658729	+	0.752380i	$0.271094\pi$
$97$	1437.56i	1.50476i	−0.658729	+	0.752380i	$0.728906\pi$
$98$	− 1785.34i	− 1.84027i
$99$	−442.083	−0.448798
$100$	0	0
$101$	395.124	0.389270	0.194635	−	0.980876i	$-0.437648\pi$
$101$	395.124	0.389270	0.194635	+	0.980876i	$0.437648\pi$
$102$	− 1792.04i	− 1.73959i
$103$	1285.68i	1.22992i	0.788559	+	0.614958i	$0.210827\pi$
$103$	1285.68i	1.22992i	−0.788559	+	0.614958i	$0.789173\pi$
$104$	−320.516	−0.302203
$105$	0	0
$106$	−516.737	−0.473490
$107$	− 1203.10i	− 1.08699i	−0.839413	−	0.543494i	$-0.817101\pi$
$107$	− 1203.10i	− 1.08699i	0.839413	−	0.543494i	$-0.182899\pi$
$108$	− 81.4744i	− 0.0725915i
$109$	1333.74	1.17201	0.586005	−	0.810307i	$-0.300700\pi$
$109$	1333.74	1.17201	0.586005	+	0.810307i	$0.300700\pi$
$110$	0	0
$111$	597.676	0.511071
$112$	− 562.433i	− 0.474508i
$113$	836.506i	0.696388i	0.937422	+	0.348194i	$0.113205\pi$
$113$	836.506i	0.696388i	−0.937422	+	0.348194i	$0.886795\pi$
$114$	−271.779	−0.223284
$115$	0	0
$116$	613.640	0.491164
$117$	967.640i	0.764601i
$118$	743.594i	0.580113i
$119$	4403.89	3.39247
$120$	0	0
$121$	−995.959	−0.748278
$122$	− 94.6181i	− 0.0702157i
$123$	− 2539.61i	− 1.86170i
$124$	457.732	0.331497
$125$	0	0
$126$	−1697.99	−1.20055
$127$	385.954i	0.269668i	0.990868	+	0.134834i	$0.0430501\pi$
$127$	385.954i	0.269668i	−0.990868	+	0.134834i	$0.956950\pi$
$128$	− 128.000i	− 0.0883883i
$129$	−3342.16	−2.28109
$130$	0	0
$131$	1737.90	1.15909	0.579545	−	0.814940i	$-0.303230\pi$
$131$	1737.90	1.15909	0.579545	+	0.814940i	$0.303230\pi$
$132$	− 523.650i	− 0.345287i
$133$	− 667.889i	− 0.435439i
$134$	−1511.08	−0.974159
$135$	0	0
$136$	1002.25	0.631927
$137$	− 41.7603i	− 0.0260425i	−0.999915	−	0.0130213i	$-0.995855\pi$
$137$	− 41.7603i	− 0.0260425i	0.999915	−	0.0130213i	$-0.00414491\pi$
$138$	− 128.406i	− 0.0792078i
$139$	−1536.79	−0.937763	−0.468882	−	0.883261i	$-0.655343\pi$
$139$	−1536.79	−0.937763	−0.468882	+	0.883261i	$0.655343\pi$
$140$	0	0
$141$	1187.83	0.709459
$142$	698.691i	0.412907i
$143$	− 733.345i	− 0.428849i
$144$	−386.433	−0.223630
$145$	0	0
$146$	−109.696	−0.0621815
$147$	− 6384.42i	− 3.58216i
$148$	334.268i	0.185653i
$149$	−1656.65	−0.910862	−0.455431	−	0.890271i	$-0.650515\pi$
$149$	−1656.65	−0.910862	−0.455431	+	0.890271i	$0.650515\pi$
$150$	0	0
$151$	−714.985	−0.385329	−0.192664	−	0.981265i	$-0.561713\pi$
$151$	−714.985	−0.385329	−0.192664	+	0.981265i	$0.561713\pi$
$152$	− 152.000i	− 0.0811107i
$153$	− 3025.80i	− 1.59883i
$154$	1286.86	0.673363
$155$	0	0
$156$	−1146.18	−0.588253
$157$	− 1684.23i	− 0.856153i	−0.903742	−	0.428077i	$-0.859191\pi$
$157$	− 1684.23i	− 0.856153i	0.903742	−	0.428077i	$-0.140809\pi$
$158$	− 877.640i	− 0.441907i
$159$	−1847.87	−0.921670
$160$	0	0
$161$	315.556	0.154468
$162$	− 1595.57i	− 0.773825i
$163$	702.175i	0.337415i	0.985666	+	0.168707i	$0.0539593\pi$
$163$	702.175i	0.337415i	−0.985666	+	0.168707i	$0.946041\pi$
$164$	1420.35	0.676285
$165$	0	0
$166$	−2146.18	−1.00347
$167$	282.506i	0.130904i	0.997856	+	0.0654520i	$0.0208489\pi$
$167$	282.506i	0.130904i	−0.997856	+	0.0654520i	$0.979151\pi$
$168$	− 2011.28i	− 0.923652i
$169$	591.838	0.269385
$170$	0	0
$171$	−458.889	−0.205217
$172$	− 1869.20i	− 0.828633i
$173$	2183.44i	0.959558i	0.877389	+	0.479779i	$0.159283\pi$
$173$	2183.44i	0.959558i	−0.877389	+	0.479779i	$0.840717\pi$
$174$	2194.40	0.956073
$175$	0	0
$176$	292.866	0.125430
$177$	2659.12i	1.12922i
$178$	1003.04i	0.422366i
$179$	3198.51	1.33557	0.667786	−	0.744353i	$-0.267242\pi$
$179$	3198.51	1.33557	0.667786	+	0.744353i	$0.267242\pi$
$180$	0	0
$181$	−2151.05	−0.883350	−0.441675	−	0.897175i	$-0.645616\pi$
$181$	−2151.05	−0.883350	−0.441675	+	0.897175i	$0.645616\pi$
$182$	− 2816.70i	− 1.14718i
$183$	− 338.357i	− 0.136678i
$184$	71.8150	0.0287732
$185$	0	0
$186$	1636.87	0.645273
$187$	2293.16i	0.896751i
$188$	664.331i	0.257720i
$189$	715.999	0.275562
$190$	0	0
$191$	−4435.52	−1.68033	−0.840165	−	0.542331i	$-0.817542\pi$
$191$	−4435.52	−1.68033	−0.840165	+	0.542331i	$0.817542\pi$
$192$	− 457.732i	− 0.172052i
$193$	2720.60i	1.01468i	0.861746	+	0.507339i	$0.169371\pi$
$193$	2720.60i	1.01468i	−0.861746	+	0.507339i	$0.830629\pi$
$194$	−2875.11	−1.06403
$195$	0	0
$196$	3570.67	1.30127
$197$	− 1254.08i	− 0.453549i	−0.973947	−	0.226775i	$-0.927182\pi$
$197$	− 1254.08i	− 0.453549i	0.973947	−	0.226775i	$-0.0728181\pi$
$198$	− 884.165i	− 0.317348i
$199$	−4155.19	−1.48017	−0.740084	−	0.672515i	$-0.765214\pi$
$199$	−4155.19	−1.48017	−0.740084	+	0.672515i	$0.765214\pi$
$200$	0	0
$201$	−5403.67	−1.89624
$202$	790.248i	0.275256i
$203$	5392.68i	1.86449i
$204$	3584.07	1.23007
$205$	0	0
$206$	−2571.35	−0.869683
$207$	− 216.810i	− 0.0727988i
$208$	− 641.032i	− 0.213690i
$209$	347.779	0.115102
$210$	0	0
$211$	−633.437	−0.206671	−0.103335	−	0.994647i	$-0.532952\pi$
$211$	−633.437	−0.206671	−0.103335	+	0.994647i	$0.532952\pi$
$212$	− 1033.47i	− 0.334808i
$213$	2498.54i	0.803743i
$214$	2406.19	0.768616
$215$	0	0
$216$	162.949	0.0513299
$217$	4022.56i	1.25838i
$218$	2667.48i	0.828737i
$219$	−392.276	−0.121039
$220$	0	0
$221$	5019.32	1.52776
$222$	1195.35i	0.361382i
$223$	− 4798.34i	− 1.44090i	−0.693507	−	0.720449i	$-0.743936\pi$
$223$	− 4798.34i	− 1.44090i	0.693507	−	0.720449i	$-0.256064\pi$
$224$	1124.87	0.335528
$225$	0	0
$226$	−1673.01	−0.492421
$227$	− 641.770i	− 0.187647i	−0.995589	−	0.0938233i	$-0.970091\pi$
$227$	− 641.770i	− 0.187647i	0.995589	−	0.0938233i	$-0.0299089\pi$
$228$	− 543.557i	− 0.157886i
$229$	1231.34	0.355325	0.177662	−	0.984091i	$-0.443146\pi$
$229$	1231.34	0.355325	0.177662	+	0.984091i	$0.443146\pi$
$230$	0	0
$231$	4601.84	1.31073
$232$	1227.28i	0.347305i
$233$	226.701i	0.0637410i	0.999492	+	0.0318705i	$0.0101464\pi$
$233$	226.701i	0.0637410i	−0.999492	+	0.0318705i	$0.989854\pi$
$234$	−1935.28	−0.540655
$235$	0	0
$236$	−1487.19	−0.410202
$237$	− 3138.47i	− 0.860192i
$238$	8807.77i	2.39884i
$239$	−2433.27	−0.658557	−0.329278	−	0.944233i	$-0.606805\pi$
$239$	−2433.27	−0.658557	−0.329278	+	0.944233i	$0.606805\pi$
$240$	0	0
$241$	−901.336	−0.240913	−0.120457	−	0.992719i	$-0.538436\pi$
$241$	−901.336	−0.240913	−0.120457	+	0.992719i	$0.538436\pi$
$242$	− 1991.92i	− 0.529113i
$243$	− 5155.85i	− 1.36110i
$244$	189.236	0.0496500
$245$	0	0
$246$	5079.22	1.31642
$247$	− 761.225i	− 0.196095i
$248$	915.465i	0.234403i
$249$	−7674.79	−1.95329
$250$	0	0
$251$	−5675.92	−1.42734	−0.713668	−	0.700484i	$-0.752967\pi$
$251$	−5675.92	−1.42734	−0.713668	+	0.700484i	$0.752967\pi$
$252$	− 3395.98i	− 0.848915i
$253$	164.314i	0.0408313i
$254$	−771.908	−0.190684
$255$	0	0
$256$	256.000	0.0625000
$257$	− 2035.88i	− 0.494143i	−0.968997	−	0.247072i	$-0.920532\pi$
$257$	− 2035.88i	− 0.494143i	0.968997	−	0.247072i	$-0.0794683\pi$
$258$	− 6684.31i	− 1.61297i
$259$	−2937.55	−0.704751
$260$	0	0
$261$	3705.17	0.878713
$262$	3475.80i	0.819601i
$263$	1924.20i	0.451147i	0.974226	+	0.225573i	$0.0724255\pi$
$263$	1924.20i	0.451147i	−0.974226	+	0.225573i	$0.927574\pi$
$264$	1047.30	0.244155
$265$	0	0
$266$	1335.78	0.307902
$267$	3586.91i	0.822155i
$268$	− 3022.16i	− 0.688834i
$269$	829.280	0.187963	0.0939815	−	0.995574i	$-0.470041\pi$
$269$	829.280	0.187963	0.0939815	+	0.995574i	$0.470041\pi$
$270$	0	0
$271$	−1223.13	−0.274169	−0.137085	−	0.990559i	$-0.543773\pi$
$271$	−1223.13	−0.274169	−0.137085	+	0.990559i	$0.543773\pi$
$272$	2004.50i	0.446840i
$273$	− 10072.6i	− 2.23305i
$274$	83.5207	0.0184148
$275$	0	0
$276$	256.813	0.0560084
$277$	− 1622.81i	− 0.352004i	−0.984390	−	0.176002i	$-0.943683\pi$
$277$	− 1622.81i	− 0.352004i	0.984390	−	0.176002i	$-0.0563166\pi$
$278$	− 3073.59i	− 0.663099i
$279$	2763.80	0.593061
$280$	0	0
$281$	3618.96	0.768288	0.384144	−	0.923273i	$-0.374497\pi$
$281$	3618.96	0.768288	0.384144	+	0.923273i	$0.374497\pi$
$282$	2375.67i	0.501663i
$283$	2102.43i	0.441613i	0.975318	+	0.220807i	$0.0708689\pi$
$283$	2102.43i	0.441613i	−0.975318	+	0.220807i	$0.929131\pi$
$284$	−1397.38	−0.291970
$285$	0	0
$286$	1466.69	0.303242
$287$	12482.1i	2.56722i
$288$	− 772.866i	− 0.158130i
$289$	−10782.3	−2.19465
$290$	0	0
$291$	−10281.5	−2.07118
$292$	− 219.392i	− 0.0439689i
$293$	− 5383.99i	− 1.07350i	−0.843741	−	0.536751i	$-0.819652\pi$
$293$	− 5383.99i	− 1.07350i	0.843741	−	0.536751i	$-0.180348\pi$
$294$	12768.8	2.53297
$295$	0	0
$296$	−668.535	−0.131276
$297$	372.830i	0.0728410i
$298$	− 3313.31i	− 0.644077i
$299$	359.654	0.0695629
$300$	0	0
$301$	16426.5	3.14555
$302$	− 1429.97i	− 0.272469i
$303$	2825.95i	0.535798i
$304$	304.000	0.0573539
$305$	0	0
$306$	6051.59	1.13054
$307$	− 3692.85i	− 0.686521i	−0.939240	−	0.343260i	$-0.888469\pi$
$307$	− 3692.85i	− 0.686521i	0.939240	−	0.343260i	$-0.111531\pi$
$308$	2573.71i	0.476139i
$309$	−9195.24	−1.69288
$310$	0	0
$311$	−4427.67	−0.807299	−0.403649	−	0.914914i	$-0.632258\pi$
$311$	−4427.67	−0.807299	−0.403649	+	0.914914i	$0.632258\pi$
$312$	− 2292.35i	− 0.415958i
$313$	− 7356.47i	− 1.32847i	−0.747523	−	0.664236i	$-0.768757\pi$
$313$	− 7356.47i	− 1.32847i	0.747523	−	0.664236i	$-0.231243\pi$
$314$	3368.46	0.605392
$315$	0	0
$316$	1755.28	0.312475
$317$	− 1612.77i	− 0.285747i	−0.989741	−	0.142874i	$-0.954366\pi$
$317$	− 1612.77i	− 0.285747i	0.989741	−	0.142874i	$-0.0456343\pi$
$318$	− 3695.74i	− 0.651719i
$319$	−2808.04	−0.492852
$320$	0	0
$321$	8604.62	1.49615
$322$	631.111i	0.109225i
$323$	2380.34i	0.410048i
$324$	3191.13	0.547177
$325$	0	0
$326$	−1404.35	−0.238588
$327$	9539.00i	1.61317i
$328$	2840.70i	0.478206i
$329$	−5838.15	−0.978321
$330$	0	0
$331$	−2262.78	−0.375752	−0.187876	−	0.982193i	$-0.560160\pi$
$331$	−2262.78	−0.375752	−0.187876	+	0.982193i	$0.560160\pi$
$332$	− 4292.35i	− 0.709558i
$333$	2018.31i	0.332141i
$334$	−565.012	−0.0925631
$335$	0	0
$336$	4022.56	0.653121
$337$	2037.63i	0.329367i	0.986346	+	0.164683i	$0.0526602\pi$
$337$	2037.63i	0.329367i	−0.986346	+	0.164683i	$0.947340\pi$
$338$	1183.68i	0.190484i
$339$	−5982.75	−0.958520
$340$	0	0
$341$	−2094.60	−0.332636
$342$	− 917.779i	− 0.145110i
$343$	19322.0i	3.04166i
$344$	3738.39	0.585932
$345$	0	0
$346$	−4366.87	−0.678510
$347$	1844.33i	0.285328i	0.989771	+	0.142664i	$0.0455668\pi$
$347$	1844.33i	0.285328i	−0.989771	+	0.142664i	$0.954433\pi$
$348$	4388.79i	0.676046i
$349$	10156.9	1.55784	0.778918	−	0.627125i	$-0.215769\pi$
$349$	10156.9	1.55784	0.778918	+	0.627125i	$0.215769\pi$
$350$	0	0
$351$	816.057	0.124097
$352$	585.732i	0.0886922i
$353$	1905.43i	0.287296i	0.989629	+	0.143648i	$0.0458834\pi$
$353$	1905.43i	0.287296i	−0.989629	+	0.143648i	$0.954117\pi$
$354$	−5318.23	−0.798477
$355$	0	0
$356$	−2006.08	−0.298658
$357$	31496.9i	4.66945i
$358$	6397.01i	0.944393i
$359$	2496.50	0.367020	0.183510	−	0.983018i	$-0.441254\pi$
$359$	2496.50	0.367020	0.183510	+	0.983018i	$0.441254\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	− 4302.10i	− 0.624623i
$363$	− 7123.16i	− 1.02994i
$364$	5633.40	0.811182
$365$	0	0
$366$	676.715	0.0966460
$367$	8748.42i	1.24432i	0.782892	+	0.622158i	$0.213744\pi$
$367$	8748.42i	1.24432i	−0.782892	+	0.622158i	$0.786256\pi$
$368$	143.630i	0.0203457i
$369$	8576.10	1.20990
$370$	0	0
$371$	9082.19	1.27095
$372$	3273.73i	0.456277i
$373$	1460.62i	0.202756i	0.994848	+	0.101378i	$0.0323252\pi$
$373$	1460.62i	0.202756i	−0.994848	+	0.101378i	$0.967675\pi$
$374$	−4586.32	−0.634099
$375$	0	0
$376$	−1328.66	−0.182235
$377$	6146.29i	0.839655i
$378$	1432.00i	0.194852i
$379$	−10581.0	−1.43406	−0.717029	−	0.697043i	$-0.754499\pi$
$379$	−10581.0	−1.43406	−0.717029	+	0.697043i	$0.754499\pi$
$380$	0	0
$381$	−2760.37	−0.371176
$382$	− 8871.04i	− 1.18817i
$383$	− 2932.54i	− 0.391242i	−0.980680	−	0.195621i	$-0.937328\pi$
$383$	− 2932.54i	− 0.391242i	0.980680	−	0.195621i	$-0.0626722\pi$
$384$	915.465	0.121659
$385$	0	0
$386$	−5441.20	−0.717486
$387$	− 11286.2i	− 1.48246i
$388$	− 5750.23i	− 0.752380i
$389$	3631.19	0.473287	0.236644	−	0.971597i	$-0.423953\pi$
$389$	3631.19	0.473287	0.236644	+	0.971597i	$0.423953\pi$
$390$	0	0
$391$	−1124.63	−0.145461
$392$	7141.34i	0.920133i
$393$	12429.6i	1.59539i
$394$	2508.15	0.320708
$395$	0	0
$396$	1768.33	0.224399
$397$	2005.32i	0.253512i	0.991934	+	0.126756i	$0.0404565\pi$
$397$	2005.32i	0.253512i	−0.991934	+	0.126756i	$0.959544\pi$
$398$	− 8310.37i	− 1.04664i
$399$	4776.79	0.599345
$400$	0	0
$401$	−8187.30	−1.01959	−0.509793	−	0.860297i	$-0.670278\pi$
$401$	−8187.30	−1.01959	−0.509793	+	0.860297i	$0.670278\pi$
$402$	− 10807.3i	− 1.34085i
$403$	4584.70i	0.566700i
$404$	−1580.50	−0.194635
$405$	0	0
$406$	−10785.4	−1.31839
$407$	− 1529.62i	− 0.186291i
$408$	7168.15i	0.869794i
$409$	15565.0	1.88175	0.940877	−	0.338747i	$-0.110003\pi$
$409$	15565.0	1.88175	0.940877	+	0.338747i	$0.110003\pi$
$410$	0	0
$411$	298.673	0.0358454
$412$	− 5142.71i	− 0.614958i
$413$	− 13069.4i	− 1.55715i
$414$	433.620	0.0514765
$415$	0	0
$416$	1282.06	0.151102
$417$	− 10991.2i	− 1.29075i
$418$	695.557i	0.0813895i
$419$	1839.83	0.214514	0.107257	−	0.994231i	$-0.465793\pi$
$419$	1839.83	0.214514	0.107257	+	0.994231i	$0.465793\pi$
$420$	0	0
$421$	4658.28	0.539266	0.269633	−	0.962963i	$-0.413098\pi$
$421$	4658.28	0.539266	0.269633	+	0.962963i	$0.413098\pi$
$422$	− 1266.87i	− 0.146138i
$423$	4011.24i	0.461071i
$424$	2066.95	0.236745
$425$	0	0
$426$	−4997.08	−0.568332
$427$	1663.01i	0.188475i
$428$	4812.38i	0.543494i
$429$	5244.94	0.590275
$430$	0	0
$431$	2820.47	0.315214	0.157607	−	0.987502i	$-0.449622\pi$
$431$	2820.47	0.315214	0.157607	+	0.987502i	$0.449622\pi$
$432$	325.898i	0.0362957i
$433$	8981.39i	0.996809i	0.866945	+	0.498404i	$0.166080\pi$
$433$	8981.39i	0.996809i	−0.866945	+	0.498404i	$0.833920\pi$
$434$	−8045.12	−0.889811
$435$	0	0
$436$	−5334.96	−0.586005
$437$	170.561i	0.0186705i
$438$	− 784.552i	− 0.0855876i
$439$	−14131.1	−1.53631	−0.768157	−	0.640261i	$-0.778826\pi$
$439$	−14131.1	−1.53631	−0.768157	+	0.640261i	$0.778826\pi$
$440$	0	0
$441$	21559.8	2.32802
$442$	10038.6i	1.08029i
$443$	1659.59i	0.177990i	0.996032	+	0.0889951i	$0.0283656\pi$
$443$	1659.59i	0.177990i	−0.996032	+	0.0889951i	$0.971634\pi$
$444$	−2390.71	−0.255536
$445$	0	0
$446$	9596.68	1.01887
$447$	− 11848.5i	− 1.25372i
$448$	2249.73i	0.237254i
$449$	13732.6	1.44338	0.721692	−	0.692214i	$-0.243365\pi$
$449$	13732.6	1.44338	0.721692	+	0.692214i	$0.243365\pi$
$450$	0	0
$451$	−6499.57	−0.678609
$452$	− 3346.02i	− 0.348194i
$453$	− 5113.62i	− 0.530373i
$454$	1283.54	0.132686
$455$	0	0
$456$	1087.11	0.111642
$457$	5273.37i	0.539776i	0.962892	+	0.269888i	$0.0869867\pi$
$457$	5273.37i	0.539776i	−0.962892	+	0.269888i	$0.913013\pi$
$458$	2462.69i	0.251253i
$459$	−2551.80	−0.259494
$460$	0	0
$461$	16402.4	1.65713	0.828565	−	0.559893i	$-0.189158\pi$
$461$	16402.4	1.65713	0.828565	+	0.559893i	$0.189158\pi$
$462$	9203.68i	0.926827i
$463$	− 5296.03i	− 0.531593i	−0.964029	−	0.265796i	$-0.914365\pi$
$463$	− 5296.03i	− 0.531593i	0.964029	−	0.265796i	$-0.0856349\pi$
$464$	−2454.56	−0.245582
$465$	0	0
$466$	−453.402	−0.0450717
$467$	− 3470.94i	− 0.343931i	−0.985103	−	0.171966i	$-0.944988\pi$
$467$	− 3470.94i	− 0.343931i	0.985103	−	0.171966i	$-0.0550118\pi$
$468$	− 3870.56i	− 0.382301i
$469$	26558.8	2.61486
$470$	0	0
$471$	12045.7	1.17842
$472$	− 2974.37i	− 0.290057i
$473$	8553.51i	0.831481i
$474$	6276.94	0.608248
$475$	0	0
$476$	−17615.5	−1.69623
$477$	− 6240.14i	− 0.598986i
$478$	− 4866.54i	− 0.465670i
$479$	16294.7	1.55433	0.777163	−	0.629300i	$-0.216658\pi$
$479$	16294.7	1.55433	0.777163	+	0.629300i	$0.216658\pi$
$480$	0	0
$481$	−3348.06	−0.317378
$482$	− 1802.67i	− 0.170352i
$483$	2256.88i	0.212612i
$484$	3983.83	0.374139
$485$	0	0
$486$	10311.7	0.962445
$487$	2245.82i	0.208969i	0.994527	+	0.104485i	$0.0333193\pi$
$487$	2245.82i	0.208969i	−0.994527	+	0.104485i	$0.966681\pi$
$488$	378.472i	0.0351079i
$489$	−5022.00	−0.464423
$490$	0	0
$491$	−7578.64	−0.696577	−0.348288	−	0.937387i	$-0.613237\pi$
$491$	−7578.64	−0.696577	−0.348288	+	0.937387i	$0.613237\pi$
$492$	10158.4i	0.930849i
$493$	− 19219.4i	− 1.75577i
$494$	1522.45	0.138660
$495$	0	0
$496$	−1830.93	−0.165748
$497$	− 12280.2i	− 1.10834i
$498$	− 15349.6i	− 1.38119i
$499$	10558.4	0.947209	0.473604	−	0.880738i	$-0.342953\pi$
$499$	10558.4	0.947209	0.473604	+	0.880738i	$0.342953\pi$
$500$	0	0
$501$	−2020.50	−0.180178
$502$	− 11351.8i	− 1.00928i
$503$	− 5816.42i	− 0.515590i	−0.966200	−	0.257795i	$-0.917004\pi$
$503$	− 5816.42i	− 0.515590i	0.966200	−	0.257795i	$-0.0829958\pi$
$504$	6791.96	0.600274
$505$	0	0
$506$	−328.628	−0.0288721
$507$	4232.87i	0.370785i
$508$	− 1543.82i	− 0.134834i
$509$	−18051.0	−1.57190	−0.785950	−	0.618290i	$-0.787826\pi$
$509$	−18051.0	−1.57190	−0.785950	+	0.618290i	$0.787826\pi$
$510$	0	0
$511$	1928.02	0.166909
$512$	512.000i	0.0441942i
$513$	387.004i	0.0333073i
$514$	4071.76	0.349412
$515$	0	0
$516$	13368.6	1.14054
$517$	− 3040.00i	− 0.258606i
$518$	− 5875.10i	− 0.498334i
$519$	−15616.1	−1.32075
$520$	0	0
$521$	−0.648976	−5.45723e−5 0	−2.72861e−5	−	1.00000i	$-0.500009\pi$
$521$	−0.648976	−5.45723e−5 0	−2.72861e−5		1.00000i	$0.500009\pi$
$522$	7410.34i	0.621344i
$523$	16912.1i	1.41399i	0.707219	+	0.706994i	$0.249949\pi$
$523$	16912.1i	1.41399i	−0.707219	+	0.706994i	$0.750051\pi$
$524$	−6951.59	−0.579545
$525$	0	0
$526$	−3848.41	−0.319009
$527$	− 14336.3i	− 1.18501i
$528$	2094.60i	0.172643i
$529$	12086.4	0.993377
$530$	0	0
$531$	−8979.66	−0.733868
$532$	2671.56i	0.217719i
$533$	14226.4i	1.15612i
$534$	−7173.82	−0.581351
$535$	0	0
$536$	6044.31	0.487079
$537$	22875.9i	1.83830i
$538$	1658.56i	0.132910i
$539$	−16339.5	−1.30574
$540$	0	0
$541$	−8893.75	−0.706788	−0.353394	−	0.935475i	$-0.614972\pi$
$541$	−8893.75	−0.706788	−0.353394	+	0.935475i	$0.614972\pi$
$542$	− 2446.26i	− 0.193867i
$543$	− 15384.5i	− 1.21586i
$544$	−4008.99	−0.315963
$545$	0	0
$546$	20145.2	1.57900
$547$	− 16972.8i	− 1.32670i	−0.748308	−	0.663351i	$-0.769134\pi$
$547$	− 16972.8i	− 1.32670i	0.748308	−	0.663351i	$-0.230866\pi$
$548$	167.041i	0.0130213i
$549$	1142.61	0.0888260
$550$	0	0
$551$	−2914.79	−0.225362
$552$	513.626i	0.0396039i
$553$	15425.4i	1.18618i
$554$	3245.62	0.248905
$555$	0	0
$556$	6147.17	0.468882
$557$	17045.8i	1.29669i	0.761348	+	0.648343i	$0.224538\pi$
$557$	17045.8i	1.29669i	−0.761348	+	0.648343i	$0.775462\pi$
$558$	5527.59i	0.419358i
$559$	18722.1	1.41657
$560$	0	0
$561$	−16400.8	−1.23430
$562$	7237.91i	0.543261i
$563$	− 6973.83i	− 0.522046i	−0.965333	−	0.261023i	$-0.915940\pi$
$563$	− 6973.83i	− 0.522046i	0.965333	−	0.261023i	$-0.0840598\pi$
$564$	−4751.34	−0.354730
$565$	0	0
$566$	−4204.86	−0.312268
$567$	28043.7i	2.07712i
$568$	− 2794.76i	− 0.206454i
$569$	21644.0	1.59466	0.797332	−	0.603541i	$-0.206244\pi$
$569$	21644.0	1.59466	0.797332	+	0.603541i	$0.206244\pi$
$570$	0	0
$571$	−13891.0	−1.01807	−0.509035	−	0.860746i	$-0.669998\pi$
$571$	−13891.0	−1.01807	−0.509035	+	0.860746i	$0.669998\pi$
$572$	2933.38i	0.214425i
$573$	− 31723.1i	− 2.31283i
$574$	−24964.1	−1.81530
$575$	0	0
$576$	1545.73	0.111815
$577$	2386.14i	0.172160i	0.996288	+	0.0860798i	$0.0274340\pi$
$577$	2386.14i	0.172160i	−0.996288	+	0.0860798i	$0.972566\pi$
$578$	− 21564.7i	− 1.55185i
$579$	−19457.9	−1.39662
$580$	0	0
$581$	37721.2	2.69353
$582$	− 20563.0i	− 1.46454i
$583$	4729.21i	0.335959i
$584$	438.783	0.0310907
$585$	0	0
$586$	10768.0	0.759080
$587$	826.851i	0.0581393i	0.999577	+	0.0290697i	$0.00925447\pi$
$587$	826.851i	0.0581393i	−0.999577	+	0.0290697i	$0.990746\pi$
$588$	25537.7i	1.79108i
$589$	−2174.23	−0.152101
$590$	0	0
$591$	8969.23	0.624272
$592$	− 1337.07i	− 0.0928265i
$593$	9789.50i	0.677920i	0.940801	+	0.338960i	$0.110075\pi$
$593$	9789.50i	0.677920i	−0.940801	+	0.338960i	$0.889925\pi$
$594$	−745.659	−0.0515064
$595$	0	0
$596$	6626.62	0.455431
$597$	− 29718.2i	− 2.03733i
$598$	719.308i	0.0491884i
$599$	23555.7	1.60678	0.803389	−	0.595455i	$-0.203028\pi$
$599$	23555.7	1.60678	0.803389	+	0.595455i	$0.203028\pi$
$600$	0	0
$601$	12263.4	0.832334	0.416167	−	0.909288i	$-0.363373\pi$
$601$	12263.4	0.832334	0.416167	+	0.909288i	$0.363373\pi$
$602$	32853.1i	2.22424i
$603$	− 18247.8i	− 1.23235i
$604$	2859.94	0.192664
$605$	0	0
$606$	−5651.91	−0.378866
$607$	− 464.109i	− 0.0310340i	−0.999880	−	0.0155170i	$-0.995061\pi$
$607$	− 464.109i	− 0.0310340i	0.999880	−	0.0155170i	$-0.00493941\pi$
$608$	608.000i	0.0405554i
$609$	−38568.8	−2.56632
$610$	0	0
$611$	−6654.02	−0.440577
$612$	12103.2i	0.799415i
$613$	11526.0i	0.759427i	0.925104	+	0.379714i	$0.123977\pi$
$613$	11526.0i	0.759427i	−0.925104	+	0.379714i	$0.876023\pi$
$614$	7385.69	0.485444
$615$	0	0
$616$	−5147.43	−0.336681
$617$	− 17632.5i	− 1.15050i	−0.817978	−	0.575249i	$-0.804905\pi$
$617$	− 17632.5i	− 1.15050i	0.817978	−	0.575249i	$-0.195095\pi$
$618$	− 18390.5i	− 1.19705i
$619$	−16538.0	−1.07386	−0.536928	−	0.843628i	$-0.680415\pi$
$619$	−16538.0	−1.07386	−0.536928	+	0.843628i	$0.680415\pi$
$620$	0	0
$621$	−182.846	−0.0118154
$622$	− 8855.33i	− 0.570846i
$623$	− 17629.5i	− 1.13372i
$624$	4584.70	0.294126
$625$	0	0
$626$	14712.9	0.939372
$627$	2487.34i	0.158428i
$628$	6736.91i	0.428077i
$629$	10469.3	0.663657
$630$	0	0
$631$	−8881.84	−0.560349	−0.280175	−	0.959949i	$-0.590392\pi$
$631$	−8881.84	−0.560349	−0.280175	+	0.959949i	$0.590392\pi$
$632$	3510.56i	0.220953i
$633$	− 4530.38i	− 0.284465i
$634$	3225.53	0.202054
$635$	0	0
$636$	7391.48	0.460835
$637$	35764.3i	2.22454i
$638$	− 5616.07i	− 0.348499i
$639$	−8437.42	−0.522346
$640$	0	0
$641$	22459.3	1.38391	0.691956	−	0.721939i	$-0.256749\pi$
$641$	22459.3	1.38391	0.691956	+	0.721939i	$0.256749\pi$
$642$	17209.2i	1.05794i
$643$	19685.1i	1.20731i	0.797244	+	0.603657i	$0.206290\pi$
$643$	19685.1i	1.20731i	−0.797244	+	0.603657i	$0.793710\pi$
$644$	−1262.22	−0.0772338
$645$	0	0
$646$	−4760.68	−0.289948
$647$	11755.6i	0.714312i	0.934045	+	0.357156i	$0.116254\pi$
$647$	11755.6i	0.714312i	−0.934045	+	0.357156i	$0.883746\pi$
$648$	6382.27i	0.386912i
$649$	6805.42	0.411612
$650$	0	0
$651$	−28769.6	−1.73206
$652$	− 2808.70i	− 0.168707i
$653$	18350.7i	1.09972i	0.835256	+	0.549862i	$0.185320\pi$
$653$	18350.7i	1.09972i	−0.835256	+	0.549862i	$0.814680\pi$
$654$	−19078.0	−1.14069
$655$	0	0
$656$	−5681.40	−0.338142
$657$	− 1324.69i	− 0.0786623i
$658$	− 11676.3i	− 0.691777i
$659$	−14617.0	−0.864034	−0.432017	−	0.901866i	$-0.642198\pi$
$659$	−14617.0	−0.864034	−0.432017	+	0.901866i	$0.642198\pi$
$660$	0	0
$661$	−2932.29	−0.172546	−0.0862729	−	0.996272i	$-0.527496\pi$
$661$	−2932.29	−0.172546	−0.0862729	+	0.996272i	$0.527496\pi$
$662$	− 4525.56i	− 0.265696i
$663$	35898.5i	2.10284i
$664$	8584.70	0.501733
$665$	0	0
$666$	−4036.63	−0.234859
$667$	− 1377.14i	− 0.0799448i
$668$	− 1130.02i	− 0.0654520i
$669$	34318.0	1.98328
$670$	0	0
$671$	−865.951	−0.0498207
$672$	8045.12i	0.461826i
$673$	21271.4i	1.21835i	0.793034	+	0.609177i	$0.208500\pi$
$673$	21271.4i	1.21835i	−0.793034	+	0.609177i	$0.791500\pi$
$674$	−4075.25	−0.232898
$675$	0	0
$676$	−2367.35	−0.134692
$677$	32878.2i	1.86649i	0.359241	+	0.933245i	$0.383036\pi$
$677$	32878.2i	1.86649i	−0.359241	+	0.933245i	$0.616964\pi$
$678$	− 11965.5i	− 0.677776i
$679$	50533.1	2.85609
$680$	0	0
$681$	4589.98	0.258280
$682$	− 4189.20i	− 0.235209i
$683$	− 7771.01i	− 0.435358i	−0.976020	−	0.217679i	$-0.930151\pi$
$683$	− 7771.01i	− 0.435358i	0.976020	−	0.217679i	$-0.0698485\pi$
$684$	1835.56	0.102609
$685$	0	0
$686$	−38643.9	−2.15078
$687$	8806.65i	0.489075i
$688$	7476.79i	0.414317i
$689$	10351.4	0.572361
$690$	0	0
$691$	3655.65	0.201255	0.100628	−	0.994924i	$-0.467915\pi$
$691$	3655.65	0.201255	0.100628	+	0.994924i	$0.467915\pi$
$692$	− 8733.74i	− 0.479779i
$693$	15540.1i	0.851833i
$694$	−3688.66	−0.201757
$695$	0	0
$696$	−8777.59	−0.478037
$697$	− 44485.7i	− 2.41753i
$698$	20313.8i	1.10156i
$699$	−1621.38	−0.0877342
$700$	0	0
$701$	−28207.0	−1.51977	−0.759887	−	0.650055i	$-0.774746\pi$
$701$	−28207.0	−1.51977	−0.759887	+	0.650055i	$0.774746\pi$
$702$	1632.11i	0.0877496i
$703$	− 1587.77i	− 0.0851834i
$704$	−1171.46	−0.0627148
$705$	0	0
$706$	−3810.85	−0.203149
$707$	− 13889.4i	− 0.738848i
$708$	− 10636.5i	− 0.564608i
$709$	−36417.7	−1.92905	−0.964524	−	0.263997i	$-0.914959\pi$
$709$	−36417.7	−1.92905	−0.964524	+	0.263997i	$0.914959\pi$
$710$	0	0
$711$	10598.4	0.559031
$712$	− 4012.17i	− 0.211183i
$713$	− 1027.25i	− 0.0539563i
$714$	−62993.8	−3.30180
$715$	0	0
$716$	−12794.0	−0.667786
$717$	− 17402.9i	− 0.906448i
$718$	4993.00i	0.259522i
$719$	−27227.6	−1.41227	−0.706133	−	0.708079i	$-0.749562\pi$
$719$	−27227.6	−1.41227	−0.706133	+	0.708079i	$0.749562\pi$
$720$	0	0
$721$	45194.2	2.33442
$722$	722.000i	0.0372161i
$723$	− 6446.41i	− 0.331597i
$724$	8604.20	0.441675
$725$	0	0
$726$	14246.3	0.728279
$727$	− 27647.2i	− 1.41043i	−0.708996	−	0.705213i	$-0.750851\pi$
$727$	− 27647.2i	− 1.41043i	0.708996	−	0.705213i	$-0.249149\pi$
$728$	11266.8i	0.573592i
$729$	15334.8	0.779090
$730$	0	0
$731$	−58543.7	−2.96213
$732$	1353.43i	0.0683391i
$733$	− 32139.8i	− 1.61952i	−0.586760	−	0.809761i	$-0.699597\pi$
$733$	− 32139.8i	− 1.61952i	0.586760	−	0.809761i	$-0.300403\pi$
$734$	−17496.8	−0.879864
$735$	0	0
$736$	−287.260	−0.0143866
$737$	13829.5i	0.691202i
$738$	17152.2i	0.855530i
$739$	−16224.9	−0.807635	−0.403818	−	0.914839i	$-0.632317\pi$
$739$	−16224.9	−0.807635	−0.403818	+	0.914839i	$0.632317\pi$
$740$	0	0
$741$	5444.33	0.269909
$742$	18164.4i	0.898700i
$743$	22463.2i	1.10915i	0.832135	+	0.554573i	$0.187118\pi$
$743$	22463.2i	1.10915i	−0.832135	+	0.554573i	$0.812882\pi$
$744$	−6547.46	−0.322637
$745$	0	0
$746$	−2921.24	−0.143370
$747$	− 25917.3i	− 1.26943i
$748$	− 9172.64i	− 0.448376i
$749$	−42291.3	−2.06314
$750$	0	0
$751$	9310.24	0.452377	0.226189	−	0.974083i	$-0.427373\pi$
$751$	9310.24	0.452377	0.226189	+	0.974083i	$0.427373\pi$
$752$	− 2657.32i	− 0.128860i
$753$	− 40594.6i	− 1.96461i
$754$	−12292.6	−0.593726
$755$	0	0
$756$	−2863.99	−0.137781
$757$	− 28707.1i	− 1.37830i	−0.724617	−	0.689152i	$-0.757983\pi$
$757$	− 28707.1i	− 1.37830i	0.724617	−	0.689152i	$-0.242017\pi$
$758$	− 21161.9i	− 1.01403i
$759$	−1175.18	−0.0562009
$760$	0	0
$761$	−20820.3	−0.991768	−0.495884	−	0.868389i	$-0.665156\pi$
$761$	−20820.3	−0.991768	−0.495884	+	0.868389i	$0.665156\pi$
$762$	− 5520.73i	− 0.262461i
$763$	− 46883.7i	− 2.22452i
$764$	17742.1	0.840165
$765$	0	0
$766$	5865.08	0.276650
$767$	− 14895.8i	− 0.701249i
$768$	1830.93i	0.0860260i
$769$	−28026.2	−1.31424	−0.657120	−	0.753786i	$-0.728225\pi$
$769$	−28026.2	−1.31424	−0.657120	+	0.753786i	$0.728225\pi$
$770$	0	0
$771$	14560.8	0.680146
$772$	− 10882.4i	− 0.507339i
$773$	− 16405.1i	− 0.763326i	−0.924302	−	0.381663i	$-0.875352\pi$
$773$	− 16405.1i	− 0.763326i	0.924302	−	0.381663i	$-0.124648\pi$
$774$	22572.5	1.04826
$775$	0	0
$776$	11500.5	0.532013
$777$	− 21009.6i	− 0.970030i
$778$	7262.38i	0.334665i
$779$	−6746.66	−0.310301
$780$	0	0
$781$	6394.47	0.292973
$782$	− 2249.26i	− 0.102856i
$783$	− 3124.75i	− 0.142617i
$784$	−14282.7	−0.650633
$785$	0	0
$786$	−24859.1	−1.12811
$787$	28468.8i	1.28946i	0.764411	+	0.644730i	$0.223030\pi$
$787$	28468.8i	1.28946i	−0.764411	+	0.644730i	$0.776970\pi$
$788$	5016.30i	0.226775i
$789$	−13762.0	−0.620965
$790$	0	0
$791$	29404.9	1.32177
$792$	3536.66i	0.158674i
$793$	1895.41i	0.0848777i
$794$	−4010.64	−0.179260
$795$	0	0
$796$	16620.7	0.740084
$797$	− 7256.68i	− 0.322515i	−0.986912	−	0.161258i	$-0.948445\pi$
$797$	− 7256.68i	− 0.322515i	0.986912	−	0.161258i	$-0.0515550\pi$
$798$	9553.58i	0.423801i
$799$	20807.0	0.921275
$800$	0	0
$801$	−12112.8	−0.534311
$802$	− 16374.6i	− 0.720957i
$803$	1003.94i	0.0441201i
$804$	21614.7	0.948122
$805$	0	0
$806$	−9169.40	−0.400718
$807$	5931.06i	0.258715i
$808$	− 3160.99i	− 0.137628i
$809$	−10647.1	−0.462711	−0.231356	−	0.972869i	$-0.574316\pi$
$809$	−10647.1	−0.462711	−0.231356	+	0.972869i	$0.574316\pi$
$810$	0	0
$811$	−27076.9	−1.17238	−0.586190	−	0.810174i	$-0.699373\pi$
$811$	−27076.9	−1.17238	−0.586190	+	0.810174i	$0.699373\pi$
$812$	− 21570.7i	− 0.932246i
$813$	− 8747.91i	− 0.377371i
$814$	3059.24	0.131728
$815$	0	0
$816$	−14336.3	−0.615037
$817$	8878.69i	0.380203i
$818$	31129.9i	1.33060i
$819$	34014.5	1.45124
$820$	0	0
$821$	1880.29	0.0799301	0.0399650	−	0.999201i	$-0.487275\pi$
$821$	1880.29	0.0799301	0.0399650	+	0.999201i	$0.487275\pi$
$822$	597.345i	0.0253465i
$823$	36904.5i	1.56307i	0.623859	+	0.781537i	$0.285564\pi$
$823$	36904.5i	1.56307i	−0.623859	+	0.781537i	$0.714436\pi$
$824$	10285.4	0.434841
$825$	0	0
$826$	26138.9	1.10107
$827$	23091.4i	0.970939i	0.874254	+	0.485470i	$0.161351\pi$
$827$	23091.4i	0.970939i	−0.874254	+	0.485470i	$0.838649\pi$
$828$	867.240i	0.0363994i
$829$	15527.1	0.650515	0.325258	−	0.945625i	$-0.394549\pi$
$829$	15527.1	0.650515	0.325258	+	0.945625i	$0.394549\pi$
$830$	0	0
$831$	11606.4	0.484504
$832$	2564.13i	0.106845i
$833$	− 111834.i	− 4.65166i
$834$	21982.5	0.912700
$835$	0	0
$836$	−1391.11	−0.0575511
$837$	− 2330.84i	− 0.0962553i
$838$	3679.66i	0.151685i
$839$	−25695.1	−1.05732	−0.528661	−	0.848833i	$-0.677306\pi$
$839$	−25695.1	−1.05732	−0.528661	+	0.848833i	$0.677306\pi$
$840$	0	0
$841$	−854.386	−0.0350316
$842$	9316.57i	0.381319i
$843$	25883.0i	1.05748i
$844$	2533.75	0.103335
$845$	0	0
$846$	−8022.48	−0.326027
$847$	35010.0i	1.42026i
$848$	4133.90i	0.167404i
$849$	−15036.7	−0.607843
$850$	0	0
$851$	750.170	0.0302180
$852$	− 9994.17i	− 0.401872i
$853$	− 4534.78i	− 0.182026i	−0.995850	−	0.0910129i	$-0.970990\pi$
$853$	− 4534.78i	− 0.182026i	0.995850	−	0.0910129i	$-0.0290104\pi$
$854$	−3326.02	−0.133272
$855$	0	0
$856$	−9624.77	−0.384308
$857$	− 48754.1i	− 1.94330i	−0.236422	−	0.971651i	$-0.575975\pi$
$857$	− 48754.1i	− 1.94330i	0.236422	−	0.971651i	$-0.424025\pi$
$858$	10489.9i	0.417387i
$859$	11730.5	0.465937	0.232968	−	0.972484i	$-0.425156\pi$
$859$	11730.5	0.465937	0.232968	+	0.972484i	$0.425156\pi$
$860$	0	0
$861$	−89272.6	−3.53357
$862$	5640.93i	0.222890i
$863$	9907.21i	0.390783i	0.980725	+	0.195391i	$0.0625977\pi$
$863$	9907.21i	0.390783i	−0.980725	+	0.195391i	$0.937402\pi$
$864$	−651.795	−0.0256650
$865$	0	0
$866$	−17962.8	−0.704850
$867$	− 77116.0i	− 3.02076i
$868$	− 16090.2i	− 0.629192i
$869$	−8032.22	−0.313549
$870$	0	0
$871$	30270.3	1.17758
$872$	− 10669.9i	− 0.414368i
$873$	− 34720.0i	− 1.34604i
$874$	−341.121	−0.0132021
$875$	0	0
$876$	1569.10	0.0605195
$877$	− 23741.6i	− 0.914136i	−0.889432	−	0.457068i	$-0.848900\pi$
$877$	− 23741.6i	− 0.914136i	0.889432	−	0.457068i	$-0.151100\pi$
$878$	− 28262.3i	− 1.08634i
$879$	38506.7	1.47758
$880$	0	0
$881$	−24381.3	−0.932378	−0.466189	−	0.884685i	$-0.654373\pi$
$881$	−24381.3	−0.932378	−0.466189	+	0.884685i	$0.654373\pi$
$882$	43119.5i	1.64616i
$883$	343.893i	0.0131064i	0.999979	+	0.00655319i	$0.00208596\pi$
$883$	343.893i	0.0131064i	−0.999979	+	0.00655319i	$0.997914\pi$
$884$	−20077.3	−0.763882
$885$	0	0
$886$	−3319.19	−0.125858
$887$	− 33805.3i	− 1.27967i	−0.768511	−	0.639837i	$-0.779002\pi$
$887$	− 33805.3i	− 1.27967i	0.768511	−	0.639837i	$-0.220998\pi$
$888$	− 4781.41i	− 0.180691i
$889$	13567.1	0.511839
$890$	0	0
$891$	−14602.7	−0.549057
$892$	19193.4i	0.720449i
$893$	− 3155.57i	− 0.118250i
$894$	23697.0	0.886517
$895$	0	0
$896$	−4499.46	−0.167764
$897$	2572.27i	0.0957475i
$898$	27465.1i	1.02063i
$899$	17555.2	0.651277
$900$	0	0
$901$	−32368.7	−1.19684
$902$	− 12999.1i	− 0.479849i
$903$	117484.i	4.32958i
$904$	6692.05	0.246210
$905$	0	0
$906$	10227.2	0.375030
$907$	26456.3i	0.968540i	0.874919	+	0.484270i	$0.160915\pi$
$907$	26456.3i	0.968540i	−0.874919	+	0.484270i	$0.839085\pi$
$908$	2567.08i	0.0938233i
$909$	−9543.06	−0.348211
$910$	0	0
$911$	−310.509	−0.0112927	−0.00564633	−	0.999984i	$-0.501797\pi$
$911$	−310.509	−0.0112927	−0.00564633	+	0.999984i	$0.501797\pi$
$912$	2174.23i	0.0789429i
$913$	19641.9i	0.711997i
$914$	−10546.7	−0.381679
$915$	0	0
$916$	−4925.37	−0.177662
$917$	− 61090.7i	− 2.19999i
$918$	− 5103.60i	− 0.183490i
$919$	17316.8	0.621575	0.310787	−	0.950479i	$-0.399407\pi$
$919$	17316.8	0.621575	0.310787	+	0.950479i	$0.399407\pi$
$920$	0	0
$921$	26411.5	0.944938
$922$	32804.8i	1.17177i
$923$	− 13996.3i	− 0.499128i
$924$	−18407.4	−0.655366
$925$	0	0
$926$	10592.1	0.375893
$927$	− 31051.7i	− 1.10019i
$928$	− 4909.12i	− 0.173653i
$929$	36473.5	1.28811	0.644056	−	0.764978i	$-0.277250\pi$
$929$	36473.5	1.28811	0.644056	+	0.764978i	$0.277250\pi$
$930$	0	0
$931$	−16960.7	−0.597061
$932$	− 906.803i	− 0.0318705i
$933$	− 31667.0i	− 1.11118i
$934$	6941.88	0.243196
$935$	0	0
$936$	7741.12	0.270327
$937$	− 4777.88i	− 0.166581i	−0.996525	−	0.0832905i	$-0.973457\pi$
$937$	− 4777.88i	− 0.166581i	0.996525	−	0.0832905i	$-0.0265429\pi$
$938$	53117.5i	1.84899i
$939$	52613.9	1.82853
$940$	0	0
$941$	31366.5	1.08663	0.543315	−	0.839529i	$-0.317169\pi$
$941$	31366.5	1.08663	0.543315	+	0.839529i	$0.317169\pi$
$942$	24091.4i	0.833271i
$943$	− 3187.58i	− 0.110076i
$944$	5948.75	0.205101
$945$	0	0
$946$	−17107.0	−0.587946
$947$	− 26496.1i	− 0.909195i	−0.890697	−	0.454597i	$-0.849783\pi$
$947$	− 26496.1i	− 0.909195i	0.890697	−	0.454597i	$-0.150217\pi$
$948$	12553.9i	0.430096i
$949$	2197.45	0.0751658
$950$	0	0
$951$	11534.6	0.393307
$952$	− 35231.1i	− 1.19942i
$953$	− 11928.3i	− 0.405451i	−0.979236	−	0.202725i	$-0.935020\pi$
$953$	− 11928.3i	− 0.405451i	0.979236	−	0.202725i	$-0.0649798\pi$
$954$	12480.3	0.423547
$955$	0	0
$956$	9733.07	0.329278
$957$	− 20083.3i	− 0.678370i
$958$	32589.3i	1.09907i
$959$	−1467.96	−0.0494296
$960$	0	0
$961$	−16696.1	−0.560440
$962$	− 6696.13i	− 0.224420i
$963$	29057.3i	0.972333i
$964$	3605.34	0.120457
$965$	0	0
$966$	−4513.75	−0.150339
$967$	58320.0i	1.93945i	0.244209	+	0.969723i	$0.421472\pi$
$967$	58320.0i	1.93945i	−0.244209	+	0.969723i	$0.578528\pi$
$968$	7967.67i	0.264556i
$969$	−17024.3	−0.564397
$970$	0	0
$971$	−27526.8	−0.909760	−0.454880	−	0.890553i	$-0.650318\pi$
$971$	−27526.8	−0.909760	−0.454880	+	0.890553i	$0.650318\pi$
$972$	20623.4i	0.680551i
$973$	54021.5i	1.77991i
$974$	−4491.64	−0.147763
$975$	0	0
$976$	−756.945	−0.0248250
$977$	36194.1i	1.18521i	0.805493	+	0.592605i	$0.201901\pi$
$977$	36194.1i	1.18521i	−0.805493	+	0.592605i	$0.798099\pi$
$978$	− 10044.0i	− 0.328397i
$979$	9179.90	0.299684
$980$	0	0
$981$	−32212.6	−1.04839
$982$	− 15157.3i	− 0.492554i
$983$	− 58081.3i	− 1.88454i	−0.334850	−	0.942271i	$-0.608686\pi$
$983$	− 58081.3i	− 1.88454i	0.334850	−	0.942271i	$-0.391314\pi$
$984$	−20316.9	−0.658210
$985$	0	0
$986$	38438.7	1.24152
$987$	− 41754.8i	− 1.34658i
$988$	3044.90i	0.0980477i
$989$	−4194.89	−0.134873
$990$	0	0
$991$	54921.4	1.76048	0.880240	−	0.474529i	$-0.157382\pi$
$991$	54921.4	1.76048	0.880240	+	0.474529i	$0.157382\pi$
$992$	− 3661.86i	− 0.117202i
$993$	− 16183.6i	− 0.517190i
$994$	24560.4	0.783712
$995$	0	0
$996$	30699.2	0.976647
$997$	33999.5i	1.08001i	0.841661	+	0.540007i	$0.181578\pi$
$997$	33999.5i	1.08001i	−0.841661	+	0.540007i	$0.818422\pi$
$998$	21116.7i	0.669778i
$999$	1702.14	0.0539073

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.4.b.g.799.4		4
5.2	odd	4		38.4.a.b.1.2	✓	2
5.3	odd	4		950.4.a.h.1.1		2
5.4	even	2	inner	950.4.b.g.799.1		4
15.2	even	4		342.4.a.k.1.2		2
20.7	even	4		304.4.a.d.1.1		2
35.27	even	4		1862.4.a.b.1.1		2
40.27	even	4		1216.4.a.l.1.2		2
40.37	odd	4		1216.4.a.j.1.1		2
95.37	even	4		722.4.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.b.1.2	✓	2	5.2	odd	4
304.4.a.d.1.1		2	20.7	even	4
342.4.a.k.1.2		2	15.2	even	4
722.4.a.i.1.1		2	95.37	even	4
950.4.a.h.1.1		2	5.3	odd	4
950.4.b.g.799.1		4	5.4	even	2	inner
950.4.b.g.799.4		4	1.1	even	1	trivial
1216.4.a.j.1.1		2	40.37	odd	4
1216.4.a.l.1.2		2	40.27	even	4
1862.4.a.b.1.1		2	35.27	even	4

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:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	950.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +7.15207i q^{3} -4.00000 q^{4} -14.3041 q^{6} -35.1521i q^{7} -8.00000i q^{8} -24.1521 q^{9} +18.3041 q^{11} -28.6083i q^{12} -40.0645i q^{13} +70.3041 q^{14} +16.0000 q^{16} +125.281i q^{17} -48.3041i q^{18} +19.0000 q^{19} +251.410 q^{21} +36.6083i q^{22} +8.97688i q^{23} +57.2165 q^{24} +80.1289 q^{26} +20.3686i q^{27} +140.608i q^{28} -153.410 q^{29} -114.433 q^{31} +32.0000i q^{32} +130.912i q^{33} -250.562 q^{34} +96.6083 q^{36} -83.5669i q^{37} +38.0000i q^{38} +286.544 q^{39} -355.088 q^{41} +502.820i q^{42} +467.299i q^{43} -73.2165 q^{44} -17.9538 q^{46} -166.083i q^{47} +114.433i q^{48} -892.668 q^{49} -896.018 q^{51} +160.258i q^{52} +258.369i q^{53} -40.7372 q^{54} -281.217 q^{56} +135.889i q^{57} -306.820i q^{58} +371.797 q^{59} -47.3090 q^{61} -228.866i q^{62} +848.995i q^{63} -64.0000 q^{64} -261.825 q^{66} +755.539i q^{67} -501.124i q^{68} -64.2032 q^{69} +349.345 q^{71} +193.217i q^{72} +54.8479i q^{73} +167.134 q^{74} -76.0000 q^{76} -643.428i q^{77} +573.088i q^{78} -438.820 q^{79} -797.783 q^{81} -710.175i q^{82} +1073.09i q^{83} -1005.64 q^{84} -934.598 q^{86} -1097.20i q^{87} -146.433i q^{88} +501.521 q^{89} -1408.35 q^{91} -35.9075i q^{92} -818.433i q^{93} +332.165 q^{94} -228.866 q^{96} +1437.56i q^{97} -1785.34i q^{98} -442.083 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} - 4 q^{6} - 70 q^{9} + 20 q^{11} + 228 q^{14} + 64 q^{16} + 76 q^{19} + 234 q^{21} + 16 q^{24} - 52 q^{26} + 158 q^{29} - 32 q^{31} - 204 q^{34} + 280 q^{36} + 1226 q^{39} - 1580 q^{41} - 80 q^{44}+ \cdots - 704 q^{99}+O(q^{100}) \) 4 * q - 16 * q^4 - 4 * q^6 - 70 * q^9 + 20 * q^11 + 228 * q^14 + 64 * q^16 + 76 * q^19 + 234 * q^21 + 16 * q^24 - 52 * q^26 + 158 * q^29 - 32 * q^31 - 204 * q^34 + 280 * q^36 + 1226 * q^39 - 1580 * q^41 - 80 * q^44 + 620 * q^46 - 2054 * q^49 - 2706 * q^51 + 316 * q^54 - 912 * q^56 - 402 * q^59 - 1360 * q^61 - 256 * q^64 - 728 * q^66 - 2146 * q^69 + 812 * q^71 + 1520 * q^74 - 304 * q^76 - 212 * q^79 - 3404 * q^81 - 936 * q^84 - 1184 * q^86 + 1740 * q^89 - 498 * q^91 - 800 * q^94 - 64 * q^96 - 704 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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