LMFDB - Embedded newform 950.4.b.g.799.3

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			799.3
Root			$-6.15207i$ of defining polynomial
Character	$\chi$	$=$	950.799
Dual form			950.4.b.g.799.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} -6.15207i q^{3} -4.00000 q^{4} +12.3041 q^{6} -21.8479i q^{7} -8.00000i q^{8} -10.8479 q^{9} -8.30413 q^{11} +24.6083i q^{12} +53.0645i q^{13} +43.6959 q^{14} +16.0000 q^{16} -74.2810i q^{17} -21.6959i q^{18} +19.0000 q^{19} -134.410 q^{21} -16.6083i q^{22} -163.977i q^{23} -49.2165 q^{24} -106.129 q^{26} -99.3686i q^{27} +87.3917i q^{28} +232.410 q^{29} +98.4331 q^{31} +32.0000i q^{32} +51.0876i q^{33} +148.562 q^{34} +43.3917 q^{36} -296.433i q^{37} +38.0000i q^{38} +326.456 q^{39} -434.912 q^{41} -268.820i q^{42} -171.299i q^{43} +33.2165 q^{44} +327.954 q^{46} +366.083i q^{47} -98.4331i q^{48} -134.332 q^{49} -456.982 q^{51} -212.258i q^{52} +138.631i q^{53} +198.737 q^{54} -174.783 q^{56} -116.889i q^{57} +464.820i q^{58} -572.797 q^{59} -632.691 q^{61} +196.866i q^{62} +237.005i q^{63} -64.0000 q^{64} -102.175 q^{66} +183.461i q^{67} +297.124i q^{68} -1008.80 q^{69} +56.6545 q^{71} +86.7835i q^{72} +68.1521i q^{73} +592.866 q^{74} -76.0000 q^{76} +181.428i q^{77} +652.912i q^{78} +332.820 q^{79} -904.217 q^{81} -869.825i q^{82} +1152.91i q^{83} +537.640 q^{84} +342.598 q^{86} -1429.80i q^{87} +66.4331i q^{88} +368.479 q^{89} +1159.35 q^{91} +655.908i q^{92} -605.567i q^{93} -732.165 q^{94} +196.866 q^{96} +426.443i q^{97} -268.664i q^{98} +90.0827 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 6.15207i	− 1.18397i	−0.805950	−	0.591983i	$-0.798345\pi$
$3$	− 6.15207i	− 1.18397i	0.805950	−	0.591983i	$-0.201655\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	12.3041	0.837190
$7$	− 21.8479i	− 1.17968i	−0.807521	−	0.589839i	$-0.799191\pi$
$7$	− 21.8479i	− 1.17968i	0.807521	−	0.589839i	$-0.200809\pi$
$8$	− 8.00000i	− 0.353553i
$9$	−10.8479	−0.401775
$10$	0	0
$11$	−8.30413	−0.227617	−0.113809	−	0.993503i	$-0.536305\pi$
$11$	−8.30413	−0.227617	−0.113809	+	0.993503i	$0.536305\pi$
$12$	24.6083i	0.591983i
$13$	53.0645i	1.13211i	0.824367	+	0.566055i	$0.191531\pi$
$13$	53.0645i	1.13211i	−0.824367	+	0.566055i	$0.808469\pi$
$14$	43.6959	0.834158
$15$	0	0
$16$	16.0000	0.250000
$17$	− 74.2810i	− 1.05975i	−0.848075	−	0.529876i	$-0.822238\pi$
$17$	− 74.2810i	− 1.05975i	0.848075	−	0.529876i	$-0.177762\pi$
$18$	− 21.6959i	− 0.284098i
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−134.410	−1.39670
$22$	− 16.6083i	− 0.160950i
$23$	− 163.977i	− 1.48659i	−0.668964	−	0.743294i	$-0.733262\pi$
$23$	− 163.977i	− 1.48659i	0.668964	−	0.743294i	$-0.266738\pi$
$24$	−49.2165	−0.418595
$25$	0	0
$26$	−106.129	−0.800523
$27$	− 99.3686i	− 0.708278i
$28$	87.3917i	0.589839i
$29$	232.410	1.48819	0.744094	−	0.668075i	$-0.232882\pi$
$29$	232.410	1.48819	0.744094	+	0.668075i	$0.232882\pi$
$30$	0	0
$31$	98.4331	0.570294	0.285147	−	0.958484i	$-0.407958\pi$
$31$	98.4331	0.570294	0.285147	+	0.958484i	$0.407958\pi$
$32$	32.0000i	0.176777i
$33$	51.0876i	0.269491i
$34$	148.562	0.749358
$35$	0	0
$36$	43.3917	0.200888
$37$	− 296.433i	− 1.31712i	−0.752530	−	0.658558i	$-0.771167\pi$
$37$	− 296.433i	− 1.31712i	0.752530	−	0.658558i	$-0.228833\pi$
$38$	38.0000i	0.162221i
$39$	326.456	1.34038
$40$	0	0
$41$	−434.912	−1.65663	−0.828316	−	0.560261i	$-0.810701\pi$
$41$	−434.912	−1.65663	−0.828316	+	0.560261i	$0.810701\pi$
$42$	− 268.820i	− 0.987615i
$43$	− 171.299i	− 0.607509i	−0.952750	−	0.303755i	$-0.901760\pi$
$43$	− 171.299i	− 0.607509i	0.952750	−	0.303755i	$-0.0982403\pi$
$44$	33.2165	0.113809
$45$	0	0
$46$	327.954	1.05118
$47$	366.083i	1.13614i	0.822980	+	0.568071i	$0.192310\pi$
$47$	366.083i	1.13614i	−0.822980	+	0.568071i	$0.807690\pi$
$48$	− 98.4331i	− 0.295991i
$49$	−134.332	−0.391639
$50$	0	0
$51$	−456.982	−1.25471
$52$	− 212.258i	− 0.566055i
$53$	138.631i	0.359292i	0.983731	+	0.179646i	$0.0574953\pi$
$53$	138.631i	0.359292i	−0.983731	+	0.179646i	$0.942505\pi$
$54$	198.737	0.500828
$55$	0	0
$56$	−174.783	−0.417079
$57$	− 116.889i	− 0.271620i
$58$	464.820i	1.05231i
$59$	−572.797	−1.26393	−0.631964	−	0.774997i	$-0.717751\pi$
$59$	−572.797	−1.26393	−0.631964	+	0.774997i	$0.717751\pi$
$60$	0	0
$61$	−632.691	−1.32800	−0.663998	−	0.747734i	$-0.731142\pi$
$61$	−632.691	−1.32800	−0.663998	+	0.747734i	$0.731142\pi$
$62$	196.866i	0.403258i
$63$	237.005i	0.473965i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−102.175	−0.190559
$67$	183.461i	0.334527i	0.985912	+	0.167264i	$0.0534931\pi$
$67$	183.461i	0.334527i	−0.985912	+	0.167264i	$0.946507\pi$
$68$	297.124i	0.529876i
$69$	−1008.80	−1.76007
$70$	0	0
$71$	56.6545	0.0946994	0.0473497	−	0.998878i	$-0.484922\pi$
$71$	56.6545	0.0946994	0.0473497	+	0.998878i	$0.484922\pi$
$72$	86.7835i	0.142049i
$73$	68.1521i	0.109268i	0.998506	+	0.0546342i	$0.0173993\pi$
$73$	68.1521i	0.109268i	−0.998506	+	0.0546342i	$0.982601\pi$
$74$	592.866	0.931342
$75$	0	0
$76$	−76.0000	−0.114708
$77$	181.428i	0.268515i
$78$	652.912i	0.947792i
$79$	332.820	0.473989	0.236995	−	0.971511i	$-0.423838\pi$
$79$	332.820	0.473989	0.236995	+	0.971511i	$0.423838\pi$
$80$	0	0
$81$	−904.217	−1.24035
$82$	− 869.825i	− 1.17142i
$83$	1152.91i	1.52468i	0.647176	+	0.762341i	$0.275950\pi$
$83$	1152.91i	1.52468i	−0.647176	+	0.762341i	$0.724050\pi$
$84$	537.640	0.698349
$85$	0	0
$86$	342.598	0.429574
$87$	− 1429.80i	− 1.76196i
$88$	66.4331i	0.0804749i
$89$	368.479	0.438862	0.219431	−	0.975628i	$-0.429580\pi$
$89$	368.479	0.438862	0.219431	+	0.975628i	$0.429580\pi$
$90$	0	0
$91$	1159.35	1.33553
$92$	655.908i	0.743294i
$93$	− 605.567i	− 0.675208i
$94$	−732.165	−0.803373
$95$	0	0
$96$	196.866	0.209298
$97$	426.443i	0.446378i	0.974775	+	0.223189i	$0.0716468\pi$
$97$	426.443i	0.446378i	−0.974775	+	0.223189i	$0.928353\pi$
$98$	− 268.664i	− 0.276931i
$99$	90.0827	0.0914510
$100$	0	0
$101$	−403.124	−0.397152	−0.198576	−	0.980086i	$-0.563632\pi$
$101$	−403.124	−0.397152	−0.198576	+	0.980086i	$0.563632\pi$
$102$	− 913.964i	− 0.887214i
$103$	− 1135.68i	− 1.08642i	−0.839596	−	0.543211i	$-0.817208\pi$
$103$	− 1135.68i	− 1.08642i	0.839596	−	0.543211i	$-0.182792\pi$
$104$	424.516	0.400262
$105$	0	0
$106$	−277.263	−0.254058
$107$	380.096i	0.343414i	0.985148	+	0.171707i	$0.0549282\pi$
$107$	380.096i	0.343414i	−0.985148	+	0.171707i	$0.945072\pi$
$108$	397.474i	0.354139i
$109$	−1180.74	−1.03756	−0.518782	−	0.854907i	$-0.673614\pi$
$109$	−1180.74	−1.03756	−0.518782	+	0.854907i	$0.673614\pi$
$110$	0	0
$111$	−1823.68	−1.55942
$112$	− 349.567i	− 0.294919i
$113$	− 1132.51i	− 0.942807i	−0.881918	−	0.471404i	$-0.843748\pi$
$113$	− 1132.51i	− 0.942807i	0.881918	−	0.471404i	$-0.156252\pi$
$114$	233.779	0.192065
$115$	0	0
$116$	−929.640	−0.744094
$117$	− 575.640i	− 0.454854i
$118$	− 1145.59i	− 0.893733i
$119$	−1622.89	−1.25017
$120$	0	0
$121$	−1262.04	−0.948190
$122$	− 1265.38i	− 0.939035i
$123$	2675.61i	1.96140i
$124$	−393.732	−0.285147
$125$	0	0
$126$	−474.010	−0.335144
$127$	40.0462i	0.0279806i	0.999902	+	0.0139903i	$0.00445339\pi$
$127$	40.0462i	0.0279806i	−0.999902	+	0.0139903i	$0.995547\pi$
$128$	− 128.000i	− 0.0883883i
$129$	−1053.84	−0.719270
$130$	0	0
$131$	−177.898	−0.118649	−0.0593244	−	0.998239i	$-0.518895\pi$
$131$	−177.898	−0.118649	−0.0593244	+	0.998239i	$0.518895\pi$
$132$	− 204.350i	− 0.134746i
$133$	− 415.111i	− 0.270637i
$134$	−366.922	−0.236547
$135$	0	0
$136$	−594.248	−0.374679
$137$	24.7603i	0.0154410i	0.999970	+	0.00772050i	$0.00245754\pi$
$137$	24.7603i	0.0154410i	−0.999970	+	0.00772050i	$0.997542\pi$
$138$	− 2017.59i	− 1.24456i
$139$	−2867.21	−1.74959	−0.874796	−	0.484491i	$-0.839005\pi$
$139$	−2867.21	−1.74959	−0.874796	+	0.484491i	$0.839005\pi$
$140$	0	0
$141$	2252.17	1.34515
$142$	113.309i	0.0669626i
$143$	− 440.655i	− 0.257688i
$144$	−173.567	−0.100444
$145$	0	0
$146$	−136.304	−0.0772645
$147$	826.421i	0.463687i
$148$	1185.73i	0.658558i
$149$	−1949.35	−1.07179	−0.535895	−	0.844285i	$-0.680026\pi$
$149$	−1949.35	−1.07179	−0.535895	+	0.844285i	$0.680026\pi$
$150$	0	0
$151$	1120.99	0.604136	0.302068	−	0.953286i	$-0.402323\pi$
$151$	1120.99	0.604136	0.302068	+	0.953286i	$0.402323\pi$
$152$	− 152.000i	− 0.0811107i
$153$	805.795i	0.425782i
$154$	−362.856	−0.189869
$155$	0	0
$156$	−1305.82	−0.670190
$157$	2360.23i	1.19979i	0.800079	+	0.599894i	$0.204791\pi$
$157$	2360.23i	1.19979i	−0.800079	+	0.599894i	$0.795209\pi$
$158$	665.640i	0.335161i
$159$	852.870	0.425390
$160$	0	0
$161$	−3582.56	−1.75370
$162$	− 1808.43i	− 0.877061i
$163$	861.825i	0.414131i	0.978327	+	0.207065i	$0.0663913\pi$
$163$	861.825i	0.414131i	−0.978327	+	0.207065i	$0.933609\pi$
$164$	1739.65	0.828316
$165$	0	0
$166$	−2305.82	−1.07811
$167$	− 1686.51i	− 0.781472i	−0.920503	−	0.390736i	$-0.872221\pi$
$167$	− 1686.51i	− 0.781472i	0.920503	−	0.390736i	$-0.127779\pi$
$168$	1075.28i	0.493807i
$169$	−618.838	−0.281674
$170$	0	0
$171$	−206.111	−0.0921736
$172$	685.197i	0.303755i
$173$	− 3191.44i	− 1.40255i	−0.712893	−	0.701273i	$-0.752616\pi$
$173$	− 3191.44i	− 1.40255i	0.712893	−	0.701273i	$-0.247384\pi$
$174$	2859.60	1.24590
$175$	0	0
$176$	−132.866	−0.0569043
$177$	3523.88i	1.49645i
$178$	736.959i	0.310322i
$179$	1229.49	0.513389	0.256695	−	0.966493i	$-0.417367\pi$
$179$	1229.49	0.513389	0.256695	+	0.966493i	$0.417367\pi$
$180$	0	0
$181$	−3108.95	−1.27672	−0.638360	−	0.769738i	$-0.720387\pi$
$181$	−3108.95	−1.27672	−0.638360	+	0.769738i	$0.720387\pi$
$182$	2318.70i	0.944359i
$183$	3892.36i	1.57230i
$184$	−1311.82	−0.525589
$185$	0	0
$186$	1211.13	0.477444
$187$	616.840i	0.241218i
$188$	− 1464.33i	− 0.568071i
$189$	−2171.00	−0.835539
$190$	0	0
$191$	−1415.48	−0.536233	−0.268117	−	0.963386i	$-0.586401\pi$
$191$	−1415.48	−0.536233	−0.268117	+	0.963386i	$0.586401\pi$
$192$	393.732i	0.147996i
$193$	1443.40i	0.538333i	0.963094	+	0.269167i	$0.0867482\pi$
$193$	1443.40i	0.538333i	−0.963094	+	0.269167i	$0.913252\pi$
$194$	−852.886	−0.315637
$195$	0	0
$196$	537.329	0.195819
$197$	− 5271.92i	− 1.90664i	−0.301954	−	0.953322i	$-0.597639\pi$
$197$	− 5271.92i	− 1.90664i	0.301954	−	0.953322i	$-0.402361\pi$
$198$	180.165i	0.0646656i
$199$	2510.19	0.894183	0.447091	−	0.894488i	$-0.352460\pi$
$199$	2510.19	0.894183	0.447091	+	0.894488i	$0.352460\pi$
$200$	0	0
$201$	1128.67	0.396069
$202$	− 806.248i	− 0.280829i
$203$	− 5077.68i	− 1.75558i
$204$	1827.93	0.627355
$205$	0	0
$206$	2271.35	0.768217
$207$	1778.81i	0.597275i
$208$	849.032i	0.283028i
$209$	−157.779	−0.0522190
$210$	0	0
$211$	1854.44	0.605046	0.302523	−	0.953142i	$-0.402171\pi$
$211$	1854.44	0.605046	0.302523	+	0.953142i	$0.402171\pi$
$212$	− 554.526i	− 0.179646i
$213$	− 348.542i	− 0.112121i
$214$	−760.192	−0.242830
$215$	0	0
$216$	−794.949	−0.250414
$217$	− 2150.56i	− 0.672763i
$218$	− 2361.48i	− 0.733668i
$219$	419.276	0.129370
$220$	0	0
$221$	3941.68	1.19976
$222$	− 3647.35i	− 1.10268i
$223$	1880.34i	0.564649i	0.959319	+	0.282325i	$0.0911055\pi$
$223$	1880.34i	0.564649i	−0.959319	+	0.282325i	$0.908895\pi$
$224$	699.134	0.208539
$225$	0	0
$226$	2265.01	0.666665
$227$	− 1799.23i	− 0.526075i	−0.964786	−	0.263038i	$-0.915276\pi$
$227$	− 1799.23i	− 0.526075i	0.964786	−	0.263038i	$-0.0847243\pi$
$228$	467.557i	0.135810i
$229$	−4835.34	−1.39532	−0.697660	−	0.716429i	$-0.745776\pi$
$229$	−4835.34	−1.39532	−0.697660	+	0.716429i	$0.745776\pi$
$230$	0	0
$231$	1116.16	0.317913
$232$	− 1859.28i	− 0.526154i
$233$	865.299i	0.243295i	0.992573	+	0.121647i	$0.0388177\pi$
$233$	865.299i	0.243295i	−0.992573	+	0.121647i	$0.961182\pi$
$234$	1151.28	0.321630
$235$	0	0
$236$	2291.19	0.631964
$237$	− 2047.53i	− 0.561187i
$238$	− 3245.77i	− 0.884001i
$239$	4764.27	1.28943	0.644717	−	0.764421i	$-0.276975\pi$
$239$	4764.27	1.28943	0.644717	+	0.764421i	$0.276975\pi$
$240$	0	0
$241$	615.336	0.164470	0.0822350	−	0.996613i	$-0.473794\pi$
$241$	615.336	0.164470	0.0822350	+	0.996613i	$0.473794\pi$
$242$	− 2524.08i	− 0.670472i
$243$	2879.85i	0.760257i
$244$	2530.76	0.663998
$245$	0	0
$246$	−5351.22	−1.38692
$247$	1008.22i	0.259724i
$248$	− 787.465i	− 0.201629i
$249$	7092.79	1.80517
$250$	0	0
$251$	−1658.08	−0.416959	−0.208480	−	0.978027i	$-0.566852\pi$
$251$	−1658.08	−0.416959	−0.208480	+	0.978027i	$0.566852\pi$
$252$	− 948.020i	− 0.236983i
$253$	1361.69i	0.338373i
$254$	−80.0925	−0.0197852
$255$	0	0
$256$	256.000	0.0625000
$257$	− 3446.12i	− 0.836432i	−0.908348	−	0.418216i	$-0.862656\pi$
$257$	− 3446.12i	− 0.836432i	0.908348	−	0.418216i	$-0.137344\pi$
$258$	− 2107.69i	− 0.508601i
$259$	−6476.45	−1.55377
$260$	0	0
$261$	−2521.17	−0.597917
$262$	− 355.795i	− 0.0838974i
$263$	5755.80i	1.34950i	0.738048	+	0.674748i	$0.235748\pi$
$263$	5755.80i	1.34950i	−0.738048	+	0.674748i	$0.764252\pi$
$264$	408.701	0.0952795
$265$	0	0
$266$	830.221	0.191369
$267$	− 2266.91i	− 0.519598i
$268$	− 733.844i	− 0.167264i
$269$	−2257.28	−0.511631	−0.255816	−	0.966726i	$-0.582344\pi$
$269$	−2257.28	−0.511631	−0.255816	+	0.966726i	$0.582344\pi$
$270$	0	0
$271$	7012.13	1.57180	0.785898	−	0.618357i	$-0.212201\pi$
$271$	7012.13	1.57180	0.785898	+	0.618357i	$0.212201\pi$
$272$	− 1188.50i	− 0.264938i
$273$	− 7132.39i	− 1.58122i
$274$	−49.5207	−0.0109184
$275$	0	0
$276$	4035.19	0.880035
$277$	372.810i	0.0808664i	0.999182	+	0.0404332i	$0.0128738\pi$
$277$	372.810i	0.0808664i	−0.999182	+	0.0404332i	$0.987126\pi$
$278$	− 5734.41i	− 1.23715i
$279$	−1067.80	−0.229130
$280$	0	0
$281$	−1888.96	−0.401017	−0.200508	−	0.979692i	$-0.564259\pi$
$281$	−1888.96	−0.401017	−0.200508	+	0.979692i	$0.564259\pi$
$282$	4504.33i	0.951167i
$283$	− 3884.43i	− 0.815920i	−0.913000	−	0.407960i	$-0.866240\pi$
$283$	− 3884.43i	− 0.815920i	0.913000	−	0.407960i	$-0.133760\pi$
$284$	−226.618	−0.0473497
$285$	0	0
$286$	881.309	0.182213
$287$	9501.94i	1.95429i
$288$	− 347.134i	− 0.0710245i
$289$	−604.668	−0.123075
$290$	0	0
$291$	2623.51	0.528497
$292$	− 272.608i	− 0.0546342i
$293$	− 1273.01i	− 0.253823i	−0.991914	−	0.126911i	$-0.959494\pi$
$293$	− 1273.01i	− 0.253823i	0.991914	−	0.126911i	$-0.0405064\pi$
$294$	−1652.84	−0.327876
$295$	0	0
$296$	−2371.46	−0.465671
$297$	825.170i	0.161216i
$298$	− 3898.69i	− 0.757869i
$299$	8701.35	1.68298
$300$	0	0
$301$	−3742.53	−0.716665
$302$	2241.97i	0.427188i
$303$	2480.05i	0.470214i
$304$	304.000	0.0573539
$305$	0	0
$306$	−1611.59	−0.301074
$307$	− 819.153i	− 0.152285i	−0.997097	−	0.0761426i	$-0.975740\pi$
$307$	− 819.153i	− 0.152285i	0.997097	−	0.0761426i	$-0.0242604\pi$
$308$	− 725.713i	− 0.134258i
$309$	−6986.76	−1.28629
$310$	0	0
$311$	2104.67	0.383745	0.191872	−	0.981420i	$-0.438544\pi$
$311$	2104.67	0.383745	0.191872	+	0.981420i	$0.438544\pi$
$312$	− 2611.65i	− 0.473896i
$313$	2395.47i	0.432587i	0.976328	+	0.216293i	$0.0693968\pi$
$313$	2395.47i	0.432587i	−0.976328	+	0.216293i	$0.930603\pi$
$314$	−4720.46	−0.848378
$315$	0	0
$316$	−1331.28	−0.236995
$317$	− 2158.23i	− 0.382393i	−0.981552	−	0.191196i	$-0.938763\pi$
$317$	− 2158.23i	− 0.382393i	0.981552	−	0.191196i	$-0.0612368\pi$
$318$	1705.74i	0.300796i
$319$	−1929.96	−0.338737
$320$	0	0
$321$	2338.38	0.406590
$322$	− 7165.11i	− 1.24005i
$323$	− 1411.34i	− 0.243124i
$324$	3616.87	0.620176
$325$	0	0
$326$	−1723.65	−0.292835
$327$	7264.00i	1.22844i
$328$	3479.30i	0.585708i
$329$	7998.15	1.34028
$330$	0	0
$331$	517.782	0.0859815	0.0429908	−	0.999075i	$-0.486311\pi$
$331$	517.782	0.0859815	0.0429908	+	0.999075i	$0.486311\pi$
$332$	− 4611.65i	− 0.762341i
$333$	3215.69i	0.529185i
$334$	3373.01	0.552584
$335$	0	0
$336$	−2150.56	−0.349174
$337$	− 6503.63i	− 1.05126i	−0.850713	−	0.525631i	$-0.823829\pi$
$337$	− 6503.63i	− 1.05126i	0.850713	−	0.525631i	$-0.176171\pi$
$338$	− 1237.68i	− 0.199174i
$339$	−6967.25	−1.11625
$340$	0	0
$341$	−817.402	−0.129809
$342$	− 412.221i	− 0.0651766i
$343$	− 4558.96i	− 0.717670i
$344$	−1370.39	−0.214787
$345$	0	0
$346$	6382.87	0.991749
$347$	− 6058.33i	− 0.937257i	−0.883395	−	0.468628i	$-0.844748\pi$
$347$	− 6058.33i	− 0.937257i	0.883395	−	0.468628i	$-0.155252\pi$
$348$	5719.21i	0.880982i
$349$	10955.1	1.68027	0.840135	−	0.542377i	$-0.182476\pi$
$349$	10955.1	1.68027	0.840135	+	0.542377i	$0.182476\pi$
$350$	0	0
$351$	5272.94	0.801849
$352$	− 265.732i	− 0.0402374i
$353$	− 1806.43i	− 0.272369i	−0.990683	−	0.136185i	$-0.956516\pi$
$353$	− 1806.43i	− 0.272369i	0.990683	−	0.136185i	$-0.0434841\pi$
$354$	−7047.77	−1.05815
$355$	0	0
$356$	−1473.92	−0.219431
$357$	9984.11i	1.48015i
$358$	2458.99i	0.363021i
$359$	7964.50	1.17089	0.585446	−	0.810711i	$-0.300919\pi$
$359$	7964.50	1.17089	0.585446	+	0.810711i	$0.300919\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	− 6217.90i	− 0.902777i
$363$	7764.16i	1.12263i
$364$	−4637.40	−0.667763
$365$	0	0
$366$	−7784.71	−1.11179
$367$	7311.58i	1.03995i	0.854182	+	0.519974i	$0.174059\pi$
$367$	7311.58i	1.03995i	−0.854182	+	0.519974i	$0.825941\pi$
$368$	− 2623.63i	− 0.371647i
$369$	4717.90	0.665594
$370$	0	0
$371$	3028.81	0.423849
$372$	2422.27i	0.337604i
$373$	5518.38i	0.766035i	0.923741	+	0.383017i	$0.125115\pi$
$373$	5518.38i	0.766035i	−0.923741	+	0.383017i	$0.874885\pi$
$374$	−1233.68	−0.170567
$375$	0	0
$376$	2928.66	0.401687
$377$	12332.7i	1.68479i
$378$	− 4342.00i	− 0.590815i
$379$	1139.97	0.154502	0.0772512	−	0.997012i	$-0.475386\pi$
$379$	1139.97	0.154502	0.0772512	+	0.997012i	$0.475386\pi$
$380$	0	0
$381$	246.367	0.0331280
$382$	− 2830.96i	− 0.379174i
$383$	− 10409.5i	− 1.38877i	−0.719604	−	0.694385i	$-0.755677\pi$
$383$	− 10409.5i	− 1.38877i	0.719604	−	0.694385i	$-0.244323\pi$
$384$	−787.465	−0.104649
$385$	0	0
$386$	−2886.80	−0.380659
$387$	1858.24i	0.244082i
$388$	− 1705.77i	− 0.223189i
$389$	−10471.2	−1.36481	−0.682404	−	0.730975i	$-0.739066\pi$
$389$	−10471.2	−1.36481	−0.682404	+	0.730975i	$0.739066\pi$
$390$	0	0
$391$	−12180.4	−1.57542
$392$	1074.66i	0.138465i
$393$	1094.44i	0.140476i
$394$	10543.8	1.34820
$395$	0	0
$396$	−360.331	−0.0457255
$397$	9588.68i	1.21220i	0.795390	+	0.606098i	$0.207266\pi$
$397$	9588.68i	1.21220i	−0.795390	+	0.606098i	$0.792734\pi$
$398$	5020.37i	0.632283i
$399$	−2553.79	−0.320424
$400$	0	0
$401$	8549.30	1.06467	0.532334	−	0.846535i	$-0.321315\pi$
$401$	8549.30	1.06467	0.532334	+	0.846535i	$0.321315\pi$
$402$	2257.33i	0.280063i
$403$	5223.30i	0.645635i
$404$	1612.50	0.198576
$405$	0	0
$406$	10155.4	1.24138
$407$	2461.62i	0.299798i
$408$	3655.85i	0.443607i
$409$	−266.960	−0.0322746	−0.0161373	−	0.999870i	$-0.505137\pi$
$409$	−266.960	−0.0322746	−0.0161373	+	0.999870i	$0.505137\pi$
$410$	0	0
$411$	152.327	0.0182816
$412$	4542.71i	0.543211i
$413$	12514.4i	1.49103i
$414$	−3557.62	−0.422337
$415$	0	0
$416$	−1698.06	−0.200131
$417$	17639.2i	2.07146i
$418$	− 315.557i	− 0.0369244i
$419$	−8643.83	−1.00783	−0.503913	−	0.863755i	$-0.668107\pi$
$419$	−8643.83	−1.00783	−0.503913	+	0.863755i	$0.668107\pi$
$420$	0	0
$421$	−16801.3	−1.94500	−0.972499	−	0.232905i	$-0.925177\pi$
$421$	−16801.3	−1.94500	−0.972499	+	0.232905i	$0.925177\pi$
$422$	3708.87i	0.427832i
$423$	− 3971.24i	− 0.456474i
$424$	1109.05	0.127029
$425$	0	0
$426$	697.085	0.0792814
$427$	13823.0i	1.56661i
$428$	− 1520.38i	− 0.171707i
$429$	−2710.94	−0.305094
$430$	0	0
$431$	12053.5	1.34710	0.673548	−	0.739144i	$-0.264769\pi$
$431$	12053.5	1.34710	0.673548	+	0.739144i	$0.264769\pi$
$432$	− 1589.90i	− 0.177069i
$433$	9034.61i	1.00271i	0.865240	+	0.501357i	$0.167166\pi$
$433$	9034.61i	1.00271i	−0.865240	+	0.501357i	$0.832834\pi$
$434$	4301.12	0.475715
$435$	0	0
$436$	4722.96	0.518782
$437$	− 3115.56i	− 0.341047i
$438$	838.552i	0.0914785i
$439$	−3008.87	−0.327120	−0.163560	−	0.986533i	$-0.552298\pi$
$439$	−3008.87	−0.327120	−0.163560	+	0.986533i	$0.552298\pi$
$440$	0	0
$441$	1457.23	0.157351
$442$	7883.37i	0.848356i
$443$	− 229.594i	− 0.0246237i	−0.999924	−	0.0123119i	$-0.996081\pi$
$443$	− 229.594i	− 0.0246237i	0.999924	−	0.0123119i	$-0.00391909\pi$
$444$	7294.71	0.779710
$445$	0	0
$446$	−3760.68	−0.399267
$447$	11992.5i	1.26896i
$448$	1398.27i	0.147460i
$449$	7559.44	0.794548	0.397274	−	0.917700i	$-0.369956\pi$
$449$	7559.44	0.794548	0.397274	+	0.917700i	$0.369956\pi$
$450$	0	0
$451$	3611.57	0.377078
$452$	4530.02i	0.471404i
$453$	− 6896.38i	− 0.715276i
$454$	3598.46	0.371991
$455$	0	0
$456$	−935.114	−0.0960323
$457$	− 11556.4i	− 1.18290i	−0.806343	−	0.591449i	$-0.798556\pi$
$457$	− 11556.4i	− 1.18290i	0.806343	−	0.591449i	$-0.201444\pi$
$458$	− 9670.69i	− 0.986641i
$459$	−7381.20	−0.750599
$460$	0	0
$461$	9191.58	0.928622	0.464311	−	0.885672i	$-0.346302\pi$
$461$	9191.58	0.928622	0.464311	+	0.885672i	$0.346302\pi$
$462$	2232.32i	0.224798i
$463$	1356.03i	0.136113i	0.997681	+	0.0680564i	$0.0216798\pi$
$463$	1356.03i	0.136113i	−0.997681	+	0.0680564i	$0.978320\pi$
$464$	3718.56	0.372047
$465$	0	0
$466$	−1730.60	−0.172035
$467$	14808.9i	1.46740i	0.679473	+	0.733700i	$0.262208\pi$
$467$	14808.9i	1.46740i	−0.679473	+	0.733700i	$0.737792\pi$
$468$	2302.56i	0.227427i
$469$	4008.25	0.394635
$470$	0	0
$471$	14520.3	1.42051
$472$	4582.37i	0.446866i
$473$	1422.49i	0.138280i
$474$	4095.06	0.396819
$475$	0	0
$476$	6491.55	0.625083
$477$	− 1503.86i	− 0.144355i
$478$	9528.54i	0.911768i
$479$	−9834.66	−0.938115	−0.469058	−	0.883168i	$-0.655406\pi$
$479$	−9834.66	−0.938115	−0.469058	+	0.883168i	$0.655406\pi$
$480$	0	0
$481$	15730.1	1.49112
$482$	1230.67i	0.116298i
$483$	22040.1i	2.07632i
$484$	5048.17	0.474095
$485$	0	0
$486$	−5759.70	−0.537583
$487$	− 3687.82i	− 0.343144i	−0.985172	−	0.171572i	$-0.945115\pi$
$487$	− 3687.82i	− 0.343144i	0.985172	−	0.171572i	$-0.0548846\pi$
$488$	5061.53i	0.469518i
$489$	5302.00	0.490317
$490$	0	0
$491$	−11197.4	−1.02919	−0.514593	−	0.857435i	$-0.672057\pi$
$491$	−11197.4	−1.02919	−0.514593	+	0.857435i	$0.672057\pi$
$492$	− 10702.4i	− 0.980698i
$493$	− 17263.6i	− 1.57711i
$494$	−2016.45	−0.183653
$495$	0	0
$496$	1574.93	0.142573
$497$	− 1237.78i	− 0.111715i
$498$	14185.6i	1.27645i
$499$	12101.6	1.08566	0.542829	−	0.839843i	$-0.317353\pi$
$499$	12101.6	1.08566	0.542829	+	0.839843i	$0.317353\pi$
$500$	0	0
$501$	−10375.5	−0.925236
$502$	− 3316.15i	− 0.294835i
$503$	− 7266.58i	− 0.644136i	−0.946716	−	0.322068i	$-0.895622\pi$
$503$	− 7266.58i	− 0.644136i	0.946716	−	0.322068i	$-0.104378\pi$
$504$	1896.04	0.167572
$505$	0	0
$506$	−2723.37	−0.239266
$507$	3807.13i	0.333493i
$508$	− 160.185i	− 0.0139903i
$509$	−2564.99	−0.223362	−0.111681	−	0.993744i	$-0.535624\pi$
$509$	−2564.99	−0.223362	−0.111681	+	0.993744i	$0.535624\pi$
$510$	0	0
$511$	1488.98	0.128902
$512$	512.000i	0.0441942i
$513$	− 1888.00i	− 0.162490i
$514$	6892.24	0.591447
$515$	0	0
$516$	4215.38	0.359635
$517$	− 3040.00i	− 0.258606i
$518$	− 12952.9i	− 1.09868i
$519$	−19633.9	−1.66057
$520$	0	0
$521$	11254.6	0.946401	0.473201	−	0.880955i	$-0.343099\pi$
$521$	11254.6	0.946401	0.473201	+	0.880955i	$0.343099\pi$
$522$	− 5042.34i	− 0.422791i
$523$	− 18357.1i	− 1.53480i	−0.641168	−	0.767401i	$-0.721550\pi$
$523$	− 18357.1i	− 1.53480i	0.641168	−	0.767401i	$-0.278450\pi$
$524$	711.591	0.0593244
$525$	0	0
$526$	−11511.6	−0.954238
$527$	− 7311.71i	− 0.604370i
$528$	817.402i	0.0673728i
$529$	−14721.4	−1.20995
$530$	0	0
$531$	6213.66	0.507815
$532$	1660.44i	0.135318i
$533$	− 23078.4i	− 1.87549i
$534$	4533.82	0.367411
$535$	0	0
$536$	1467.69	0.118273
$537$	− 7563.93i	− 0.607836i
$538$	− 4514.56i	− 0.361778i
$539$	1115.51	0.0891438
$540$	0	0
$541$	11381.8	0.904510	0.452255	−	0.891889i	$-0.350620\pi$
$541$	11381.8	0.904510	0.452255	+	0.891889i	$0.350620\pi$
$542$	14024.3i	1.11143i
$543$	19126.5i	1.51159i
$544$	2376.99	0.187340
$545$	0	0
$546$	14264.8	1.11809
$547$	8996.84i	0.703248i	0.936141	+	0.351624i	$0.114371\pi$
$547$	8996.84i	0.703248i	−0.936141	+	0.351624i	$0.885629\pi$
$548$	− 99.0413i	− 0.00772050i
$549$	6863.39	0.533556
$550$	0	0
$551$	4415.79	0.341414
$552$	8070.37i	0.622279i
$553$	− 7271.43i	− 0.559155i
$554$	−745.620	−0.0571812
$555$	0	0
$556$	11468.8	0.874796
$557$	− 10759.8i	− 0.818507i	−0.912421	−	0.409253i	$-0.865789\pi$
$557$	− 10759.8i	− 0.818507i	0.912421	−	0.409253i	$-0.134211\pi$
$558$	− 2135.59i	− 0.162019i
$559$	9089.90	0.687767
$560$	0	0
$561$	3794.84	0.285594
$562$	− 3777.91i	− 0.283562i
$563$	25381.8i	1.90003i	0.312205	+	0.950015i	$0.398932\pi$
$563$	25381.8i	1.90003i	−0.312205	+	0.950015i	$0.601068\pi$
$564$	−9008.66	−0.672576
$565$	0	0
$566$	7768.86	0.576943
$567$	19755.3i	1.46322i
$568$	− 453.236i	− 0.0334813i
$569$	5546.00	0.408612	0.204306	−	0.978907i	$-0.434506\pi$
$569$	5546.00	0.408612	0.204306	+	0.978907i	$0.434506\pi$
$570$	0	0
$571$	7714.96	0.565431	0.282715	−	0.959204i	$-0.408765\pi$
$571$	7714.96	0.565431	0.282715	+	0.959204i	$0.408765\pi$
$572$	1762.62i	0.128844i
$573$	8708.13i	0.634882i
$574$	−19003.9	−1.38189
$575$	0	0
$576$	694.268	0.0502219
$577$	− 21335.1i	− 1.53933i	−0.638448	−	0.769665i	$-0.720423\pi$
$577$	− 21335.1i	− 1.53933i	0.638448	−	0.769665i	$-0.279577\pi$
$578$	− 1209.34i	− 0.0870273i
$579$	8879.90	0.637368
$580$	0	0
$581$	25188.8	1.79863
$582$	5247.01i	0.373704i
$583$	− 1151.21i	− 0.0817811i
$584$	545.217	0.0386322
$585$	0	0
$586$	2546.02	0.179480
$587$	− 12370.9i	− 0.869846i	−0.900468	−	0.434923i	$-0.856776\pi$
$587$	− 12370.9i	− 0.869846i	0.900468	−	0.434923i	$-0.143224\pi$
$588$	− 3305.68i	− 0.231844i
$589$	1870.23	0.130834
$590$	0	0
$591$	−32433.2	−2.25740
$592$	− 4742.93i	− 0.329279i
$593$	12982.5i	0.899034i	0.893272	+	0.449517i	$0.148404\pi$
$593$	12982.5i	0.899034i	−0.893272	+	0.449517i	$0.851596\pi$
$594$	−1650.34	−0.113997
$595$	0	0
$596$	7797.38	0.535895
$597$	− 15442.8i	− 1.05868i
$598$	17402.7i	1.19005i
$599$	−13163.7	−0.897921	−0.448960	−	0.893552i	$-0.648206\pi$
$599$	−13163.7	−0.897921	−0.448960	+	0.893552i	$0.648206\pi$
$600$	0	0
$601$	−29325.4	−1.99036	−0.995180	−	0.0980640i	$-0.968735\pi$
$601$	−29325.4	−1.99036	−0.995180	+	0.0980640i	$0.968735\pi$
$602$	− 7485.07i	− 0.506758i
$603$	− 1990.17i	− 0.134405i
$604$	−4483.94	−0.302068
$605$	0	0
$606$	−4960.09	−0.332492
$607$	2170.11i	0.145110i	0.997364	+	0.0725552i	$0.0231154\pi$
$607$	2170.11i	0.145110i	−0.997364	+	0.0725552i	$0.976885\pi$
$608$	608.000i	0.0405554i
$609$	−31238.2	−2.07855
$610$	0	0
$611$	−19426.0	−1.28624
$612$	− 3223.18i	− 0.212891i
$613$	− 4917.96i	− 0.324036i	−0.986788	−	0.162018i	$-0.948200\pi$
$613$	− 4917.96i	− 0.324036i	0.986788	−	0.162018i	$-0.0518003\pi$
$614$	1638.31	0.107682
$615$	0	0
$616$	1451.43	0.0949344
$617$	− 24763.5i	− 1.61579i	−0.589328	−	0.807894i	$-0.700607\pi$
$617$	− 24763.5i	− 1.61579i	0.589328	−	0.807894i	$-0.299393\pi$
$618$	− 13973.5i	− 0.909542i
$619$	27552.0	1.78902	0.894512	−	0.447043i	$-0.147523\pi$
$619$	27552.0	1.78902	0.894512	+	0.447043i	$0.147523\pi$
$620$	0	0
$621$	−16294.2	−1.05292
$622$	4209.33i	0.271348i
$623$	− 8050.51i	− 0.517716i
$624$	5223.30	0.335095
$625$	0	0
$626$	−4790.93	−0.305885
$627$	970.664i	0.0618255i
$628$	− 9440.91i	− 0.599894i
$629$	−22019.3	−1.39582
$630$	0	0
$631$	377.839	0.0238376	0.0119188	−	0.999929i	$-0.496206\pi$
$631$	377.839	0.0238376	0.0119188	+	0.999929i	$0.496206\pi$
$632$	− 2662.56i	− 0.167581i
$633$	− 11408.6i	− 0.716354i
$634$	4316.47	0.270393
$635$	0	0
$636$	−3411.48	−0.212695
$637$	− 7128.27i	− 0.443379i
$638$	− 3859.93i	− 0.239523i
$639$	−614.584	−0.0380479
$640$	0	0
$641$	−14047.3	−0.865576	−0.432788	−	0.901496i	$-0.642470\pi$
$641$	−14047.3	−0.865576	−0.432788	+	0.901496i	$0.642470\pi$
$642$	4676.75i	0.287503i
$643$	5768.94i	0.353818i	0.984227	+	0.176909i	$0.0566098\pi$
$643$	5768.94i	0.353818i	−0.984227	+	0.176909i	$0.943390\pi$
$644$	14330.2	0.876848
$645$	0	0
$646$	2822.68	0.171915
$647$	− 4568.59i	− 0.277604i	−0.990320	−	0.138802i	$-0.955675\pi$
$647$	− 4568.59i	− 0.277604i	0.990320	−	0.138802i	$-0.0443251\pi$
$648$	7233.73i	0.438531i
$649$	4756.58	0.287692
$650$	0	0
$651$	−13230.4	−0.796528
$652$	− 3447.30i	− 0.207065i
$653$	− 16532.7i	− 0.990774i	−0.868672	−	0.495387i	$-0.835026\pi$
$653$	− 16532.7i	− 0.990774i	0.868672	−	0.495387i	$-0.164974\pi$
$654$	−14528.0	−0.868638
$655$	0	0
$656$	−6958.60	−0.414158
$657$	− 739.309i	− 0.0439014i
$658$	15996.3i	0.947721i
$659$	18630.0	1.10125	0.550624	−	0.834753i	$-0.314390\pi$
$659$	18630.0	1.10125	0.550624	+	0.834753i	$0.314390\pi$
$660$	0	0
$661$	28851.3	1.69771	0.848854	−	0.528627i	$-0.177293\pi$
$661$	28851.3	1.69771	0.848854	+	0.528627i	$0.177293\pi$
$662$	1035.56i	0.0607981i
$663$	− 24249.5i	− 1.42047i
$664$	9223.30	0.539056
$665$	0	0
$666$	−6431.37	−0.374190
$667$	− 38109.9i	− 2.21232i
$668$	6746.02i	0.390736i
$669$	11568.0	0.668525
$670$	0	0
$671$	5253.95	0.302275
$672$	− 4301.12i	− 0.246904i
$673$	− 3873.41i	− 0.221856i	−0.993828	−	0.110928i	$-0.964618\pi$
$673$	− 3873.41i	− 0.221856i	0.993828	−	0.110928i	$-0.0353822\pi$
$674$	13007.3	0.743354
$675$	0	0
$676$	2475.35	0.140837
$677$	− 5025.24i	− 0.285282i	−0.989775	−	0.142641i	$-0.954441\pi$
$677$	− 5025.24i	− 0.285282i	0.989775	−	0.142641i	$-0.0455594\pi$
$678$	− 13934.5i	− 0.789309i
$679$	9316.90	0.526583
$680$	0	0
$681$	−11069.0	−0.622855
$682$	− 1634.80i	− 0.0917886i
$683$	− 14157.0i	− 0.793122i	−0.918008	−	0.396561i	$-0.870204\pi$
$683$	− 14157.0i	− 0.793122i	0.918008	−	0.396561i	$-0.129796\pi$
$684$	824.443	0.0460868
$685$	0	0
$686$	9117.92	0.507469
$687$	29747.4i	1.65201i
$688$	− 2740.79i	− 0.151877i
$689$	−7356.40	−0.406758
$690$	0	0
$691$	−2437.65	−0.134200	−0.0671002	−	0.997746i	$-0.521375\pi$
$691$	−2437.65	−0.134200	−0.0671002	+	0.997746i	$0.521375\pi$
$692$	12765.7i	0.701273i
$693$	− 1968.12i	− 0.107883i
$694$	12116.7	0.662741
$695$	0	0
$696$	−11438.4	−0.622948
$697$	32305.7i	1.75562i
$698$	21910.2i	1.18813i
$699$	5323.38	0.288052
$700$	0	0
$701$	9496.96	0.511691	0.255845	−	0.966718i	$-0.417646\pi$
$701$	9496.96	0.511691	0.255845	+	0.966718i	$0.417646\pi$
$702$	10545.9i	0.566993i
$703$	− 5632.23i	− 0.302167i
$704$	531.465	0.0284522
$705$	0	0
$706$	3612.85	0.192594
$707$	8807.43i	0.468511i
$708$	− 14095.5i	− 0.748224i
$709$	−6350.33	−0.336377	−0.168189	−	0.985755i	$-0.553792\pi$
$709$	−6350.33	−0.336377	−0.168189	+	0.985755i	$0.553792\pi$
$710$	0	0
$711$	−3610.41	−0.190437
$712$	− 2947.83i	− 0.155161i
$713$	− 16140.7i	− 0.847792i
$714$	−19968.2	−1.04663
$715$	0	0
$716$	−4917.98	−0.256695
$717$	− 29310.1i	− 1.52665i
$718$	15929.0i	0.827946i
$719$	19722.6	1.02299	0.511496	−	0.859286i	$-0.329092\pi$
$719$	19722.6	1.02299	0.511496	+	0.859286i	$0.329092\pi$
$720$	0	0
$721$	−24812.2	−1.28163
$722$	722.000i	0.0372161i
$723$	− 3785.59i	− 0.194727i
$724$	12435.8	0.638360
$725$	0	0
$726$	−15528.3	−0.793816
$727$	− 28325.8i	− 1.44504i	−0.691350	−	0.722520i	$-0.742984\pi$
$727$	− 28325.8i	− 1.44504i	0.691350	−	0.722520i	$-0.257016\pi$
$728$	− 9274.79i	− 0.472179i
$729$	−6696.82	−0.340234
$730$	0	0
$731$	−12724.3	−0.643809
$732$	− 15569.4i	− 0.786151i
$733$	33981.8i	1.71234i	0.516694	+	0.856170i	$0.327163\pi$
$733$	33981.8i	1.71234i	−0.516694	+	0.856170i	$0.672837\pi$
$734$	−14623.2	−0.735355
$735$	0	0
$736$	5247.26	0.262794
$737$	− 1523.49i	− 0.0761443i
$738$	9435.80i	0.470646i
$739$	−9147.10	−0.455320	−0.227660	−	0.973741i	$-0.573107\pi$
$739$	−9147.10	−0.455320	−0.227660	+	0.973741i	$0.573107\pi$
$740$	0	0
$741$	6202.67	0.307504
$742$	6057.62i	0.299706i
$743$	− 34159.2i	− 1.68665i	−0.537405	−	0.843324i	$-0.680596\pi$
$743$	− 34159.2i	− 1.68665i	0.537405	−	0.843324i	$-0.319404\pi$
$744$	−4844.54	−0.238722
$745$	0	0
$746$	−11036.8	−0.541668
$747$	− 12506.7i	− 0.612579i
$748$	− 2467.36i	− 0.120609i
$749$	8304.31	0.405117
$750$	0	0
$751$	28361.8	1.37808	0.689038	−	0.724725i	$-0.258033\pi$
$751$	28361.8	1.37808	0.689038	+	0.724725i	$0.258033\pi$
$752$	5857.32i	0.284035i
$753$	10200.6i	0.493666i
$754$	−24665.4	−1.19133
$755$	0	0
$756$	8683.99	0.417770
$757$	− 11464.9i	− 0.550462i	−0.961378	−	0.275231i	$-0.911246\pi$
$757$	− 11464.9i	− 0.550462i	0.961378	−	0.275231i	$-0.0887543\pi$
$758$	2279.94i	0.109250i
$759$	8377.18	0.400623
$760$	0	0
$761$	14289.3	0.680666	0.340333	−	0.940305i	$-0.389460\pi$
$761$	14289.3	0.680666	0.340333	+	0.940305i	$0.389460\pi$
$762$	492.734i	0.0234250i
$763$	25796.7i	1.22399i
$764$	5661.92	0.268117
$765$	0	0
$766$	20818.9	0.982008
$767$	− 30395.2i	− 1.43091i
$768$	− 1574.93i	− 0.0739979i
$769$	−100.811	−0.00472734	−0.00236367	−	0.999997i	$-0.500752\pi$
$769$	−100.811	−0.00472734	−0.00236367	+	0.999997i	$0.500752\pi$
$770$	0	0
$771$	−21200.8	−0.990307
$772$	− 5773.61i	− 0.269167i
$773$	52.1073i	0.00242454i	0.999999	+	0.00121227i	$0.000385878\pi$
$773$	52.1073i	0.00242454i	−0.999999	+	0.00121227i	$0.999614\pi$
$774$	−3716.49	−0.172592
$775$	0	0
$776$	3411.54	0.157819
$777$	39843.6i	1.83961i
$778$	− 20942.4i	− 0.965065i
$779$	−8263.34	−0.380057
$780$	0	0
$781$	−470.467	−0.0215552
$782$	− 24360.7i	− 1.11399i
$783$	− 23094.3i	− 1.05405i
$784$	−2149.31	−0.0979097
$785$	0	0
$786$	−2188.88	−0.0993316
$787$	− 15261.8i	− 0.691266i	−0.938370	−	0.345633i	$-0.887664\pi$
$787$	− 15261.8i	− 0.691266i	0.938370	−	0.345633i	$-0.112336\pi$
$788$	21087.7i	0.953322i
$789$	35410.0	1.59776
$790$	0	0
$791$	−24742.9	−1.11221
$792$	− 720.662i	− 0.0323328i
$793$	− 33573.4i	− 1.50344i
$794$	−19177.4	−0.857152
$795$	0	0
$796$	−10040.7	−0.447091
$797$	− 24432.3i	− 1.08587i	−0.839775	−	0.542934i	$-0.817313\pi$
$797$	− 24432.3i	− 1.08587i	0.839775	−	0.542934i	$-0.182687\pi$
$798$	− 5107.58i	− 0.226574i
$799$	27193.0	1.20403
$800$	0	0
$801$	−3997.24	−0.176324
$802$	17098.6i	0.752834i
$803$	− 565.944i	− 0.0248714i
$804$	−4514.66	−0.198035
$805$	0	0
$806$	−10446.6	−0.456533
$807$	13886.9i	0.605754i
$808$	3224.99i	0.140414i
$809$	−3635.86	−0.158010	−0.0790050	−	0.996874i	$-0.525174\pi$
$809$	−3635.86	−0.158010	−0.0790050	+	0.996874i	$0.525174\pi$
$810$	0	0
$811$	1087.93	0.0471051	0.0235526	−	0.999723i	$-0.492502\pi$
$811$	1087.93	0.0471051	0.0235526	+	0.999723i	$0.492502\pi$
$812$	20310.7i	0.877791i
$813$	− 43139.1i	− 1.86095i
$814$	−4923.24	−0.211990
$815$	0	0
$816$	−7311.71	−0.313678
$817$	− 3254.69i	− 0.139372i
$818$	− 533.920i	− 0.0228216i
$819$	−12576.5	−0.536581
$820$	0	0
$821$	10953.7	0.465636	0.232818	−	0.972520i	$-0.425205\pi$
$821$	10953.7	0.465636	0.232818	+	0.972520i	$0.425205\pi$
$822$	304.655i	0.0129271i
$823$	13502.5i	0.571893i	0.958246	+	0.285947i	$0.0923080\pi$
$823$	13502.5i	0.571893i	−0.958246	+	0.285947i	$0.907692\pi$
$824$	−9085.41	−0.384108
$825$	0	0
$826$	−25028.9	−1.05432
$827$	− 26812.4i	− 1.12740i	−0.825980	−	0.563699i	$-0.809378\pi$
$827$	− 26812.4i	− 1.12740i	0.825980	−	0.563699i	$-0.190622\pi$
$828$	− 7115.24i	− 0.298637i
$829$	4497.94	0.188444	0.0942218	−	0.995551i	$-0.469964\pi$
$829$	4497.94	0.188444	0.0942218	+	0.995551i	$0.469964\pi$
$830$	0	0
$831$	2293.55	0.0957430
$832$	− 3396.13i	− 0.141514i
$833$	9978.33i	0.415040i
$834$	−35278.5	−1.46474
$835$	0	0
$836$	631.114	0.0261095
$837$	− 9781.16i	− 0.403926i
$838$	− 17287.7i	− 0.712640i
$839$	−30324.9	−1.24783	−0.623917	−	0.781491i	$-0.714460\pi$
$839$	−30324.9	−1.24783	−0.623917	+	0.781491i	$0.714460\pi$
$840$	0	0
$841$	29625.4	1.21470
$842$	− 33602.6i	− 1.37532i
$843$	11621.0i	0.474790i
$844$	−7417.75	−0.302523
$845$	0	0
$846$	7942.48	0.322776
$847$	27573.0i	1.11856i
$848$	2218.10i	0.0898230i
$849$	−23897.3	−0.966022
$850$	0	0
$851$	−48608.2	−1.95801
$852$	1394.17i	0.0560604i
$853$	17230.8i	0.691642i	0.938300	+	0.345821i	$0.112400\pi$
$853$	17230.8i	0.691642i	−0.938300	+	0.345821i	$0.887600\pi$
$854$	−27646.0	−1.10776
$855$	0	0
$856$	3040.77	0.121415
$857$	6724.12i	0.268018i	0.990980	+	0.134009i	$0.0427851\pi$
$857$	6724.12i	0.268018i	−0.990980	+	0.134009i	$0.957215\pi$
$858$	− 5421.87i	− 0.215734i
$859$	36183.5	1.43721	0.718606	−	0.695417i	$-0.244781\pi$
$859$	36183.5	1.43721	0.718606	+	0.695417i	$0.244781\pi$
$860$	0	0
$861$	58456.6	2.31381
$862$	24107.1i	0.952541i
$863$	14350.8i	0.566056i	0.959112	+	0.283028i	$0.0913390\pi$
$863$	14350.8i	0.566056i	−0.959112	+	0.283028i	$0.908661\pi$
$864$	3179.80	0.125207
$865$	0	0
$866$	−18069.2	−0.709027
$867$	3719.96i	0.145717i
$868$	8602.24i	0.336381i
$869$	−2763.78	−0.107888
$870$	0	0
$871$	−9735.27	−0.378722
$872$	9445.93i	0.366834i
$873$	− 4626.02i	− 0.179344i
$874$	6231.12	0.241157
$875$	0	0
$876$	−1677.10	−0.0646851
$877$	42978.6i	1.65483i	0.561592	+	0.827414i	$0.310189\pi$
$877$	42978.6i	1.65483i	−0.561592	+	0.827414i	$0.689811\pi$
$878$	− 6017.74i	− 0.231309i
$879$	−7831.65	−0.300518
$880$	0	0
$881$	−7298.75	−0.279116	−0.139558	−	0.990214i	$-0.544568\pi$
$881$	−7298.75	−0.279116	−0.139558	+	0.990214i	$0.544568\pi$
$882$	2914.45i	0.111264i
$883$	46722.1i	1.78066i	0.455314	+	0.890331i	$0.349527\pi$
$883$	46722.1i	1.78066i	−0.455314	+	0.890331i	$0.650473\pi$
$884$	−15766.7	−0.599878
$885$	0	0
$886$	459.187	0.0174116
$887$	− 35268.7i	− 1.33507i	−0.744578	−	0.667535i	$-0.767349\pi$
$887$	− 35268.7i	− 1.33507i	0.744578	−	0.667535i	$-0.232651\pi$
$888$	14589.4i	0.551338i
$889$	874.928	0.0330080
$890$	0	0
$891$	7508.74	0.282326
$892$	− 7521.35i	− 0.282325i
$893$	6955.57i	0.260649i
$894$	−23985.0	−0.897292
$895$	0	0
$896$	−2796.54	−0.104270
$897$	− 53531.3i	− 1.99259i
$898$	15118.9i	0.561830i
$899$	22876.8	0.848704
$900$	0	0
$901$	10297.7	0.380761
$902$	7223.14i	0.266635i
$903$	23024.3i	0.848507i
$904$	−9060.05	−0.333333
$905$	0	0
$906$	13792.8	0.505777
$907$	46292.7i	1.69474i	0.531007	+	0.847368i	$0.321814\pi$
$907$	46292.7i	1.69474i	−0.531007	+	0.847368i	$0.678186\pi$
$908$	7196.92i	0.263038i
$909$	4373.06	0.159566
$910$	0	0
$911$	−42085.5	−1.53058	−0.765288	−	0.643688i	$-0.777403\pi$
$911$	−42085.5	−1.53058	−0.765288	+	0.643688i	$0.777403\pi$
$912$	− 1870.23i	− 0.0679051i
$913$	− 9573.94i	− 0.347044i
$914$	23112.7	0.836435
$915$	0	0
$916$	19341.4	0.697660
$917$	3886.70i	0.139967i
$918$	− 14762.4i	− 0.530754i
$919$	−21331.8	−0.765691	−0.382845	−	0.923812i	$-0.625056\pi$
$919$	−21331.8	−0.765691	−0.382845	+	0.923812i	$0.625056\pi$
$920$	0	0
$921$	−5039.49	−0.180300
$922$	18383.2i	0.656635i
$923$	3006.34i	0.107210i
$924$	−4464.63	−0.158956
$925$	0	0
$926$	−2712.07	−0.0962463
$927$	12319.7i	0.436498i
$928$	7437.12i	0.263077i
$929$	34331.5	1.21247	0.606233	−	0.795287i	$-0.292680\pi$
$929$	34331.5	1.21247	0.606233	+	0.795287i	$0.292680\pi$
$930$	0	0
$931$	−2552.31	−0.0898481
$932$	− 3461.20i	− 0.121647i
$933$	− 12948.0i	− 0.454341i
$934$	−29617.9	−1.03761
$935$	0	0
$936$	−4605.12	−0.160815
$937$	− 13625.1i	− 0.475041i	−0.971383	−	0.237521i	$-0.923665\pi$
$937$	− 13625.1i	− 0.475041i	0.971383	−	0.237521i	$-0.0763347\pi$
$938$	8016.49i	0.279049i
$939$	14737.1	0.512168
$940$	0	0
$941$	7086.48	0.245497	0.122748	−	0.992438i	$-0.460829\pi$
$941$	7086.48	0.245497	0.122748	+	0.992438i	$0.460829\pi$
$942$	29040.6i	1.00445i
$943$	71315.6i	2.46273i
$944$	−9164.75	−0.315982
$945$	0	0
$946$	−2844.98	−0.0977784
$947$	− 38735.9i	− 1.32920i	−0.747202	−	0.664598i	$-0.768603\pi$
$947$	− 38735.9i	− 1.32920i	0.747202	−	0.664598i	$-0.231397\pi$
$948$	8190.12i	0.280594i
$949$	−3616.45	−0.123704
$950$	0	0
$951$	−13277.6	−0.452740
$952$	12983.1i	0.442000i
$953$	30964.3i	1.05250i	0.850330	+	0.526249i	$0.176402\pi$
$953$	30964.3i	1.05250i	−0.850330	+	0.526249i	$0.823598\pi$
$954$	3007.73	0.102074
$955$	0	0
$956$	−19057.1	−0.644717
$957$	11873.3i	0.401053i
$958$	− 19669.3i	− 0.663347i
$959$	540.962	0.0182154
$960$	0	0
$961$	−20101.9	−0.674765
$962$	31460.1i	1.05438i
$963$	− 4123.26i	− 0.137975i
$964$	−2461.34	−0.0822350
$965$	0	0
$966$	−44080.2	−1.46818
$967$	− 10968.0i	− 0.364743i	−0.983230	−	0.182371i	$-0.941623\pi$
$967$	− 10968.0i	− 0.364743i	0.983230	−	0.182371i	$-0.0583773\pi$
$968$	10096.3i	0.335236i
$969$	−8682.65	−0.287850
$970$	0	0
$971$	−30081.2	−0.994183	−0.497091	−	0.867698i	$-0.665599\pi$
$971$	−30081.2	−0.994183	−0.497091	+	0.867698i	$0.665599\pi$
$972$	− 11519.4i	− 0.380128i
$973$	62642.5i	2.06395i
$974$	7375.64	0.242640
$975$	0	0
$976$	−10123.1	−0.331999
$977$	− 26628.1i	− 0.871962i	−0.899956	−	0.435981i	$-0.856401\pi$
$977$	− 26628.1i	− 0.871962i	0.899956	−	0.435981i	$-0.143599\pi$
$978$	10604.0i	0.346706i
$979$	−3059.90	−0.0998926
$980$	0	0
$981$	12808.6	0.416867
$982$	− 22394.7i	− 0.727744i
$983$	25495.3i	0.827236i	0.910450	+	0.413618i	$0.135735\pi$
$983$	25495.3i	0.827236i	−0.910450	+	0.413618i	$0.864265\pi$
$984$	21404.9	0.693458
$985$	0	0
$986$	34527.3	1.11519
$987$	− 49205.2i	− 1.58685i
$988$	− 4032.90i	− 0.129862i
$989$	−28089.1	−0.903116
$990$	0	0
$991$	−54385.4	−1.74330	−0.871649	−	0.490130i	$-0.836949\pi$
$991$	−54385.4	−1.74330	−0.871649	+	0.490130i	$0.836949\pi$
$992$	3149.86i	0.100815i
$993$	− 3185.43i	− 0.101799i
$994$	2475.57	0.0789942
$995$	0	0
$996$	−28371.2	−0.902586
$997$	36846.5i	1.17045i	0.810870	+	0.585227i	$0.198994\pi$
$997$	36846.5i	1.17045i	−0.810870	+	0.585227i	$0.801006\pi$
$998$	24203.3i	0.767677i
$999$	−29456.1	−0.932884

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.4.b.g.799.3		4
5.2	odd	4		38.4.a.b.1.1	✓	2
5.3	odd	4		950.4.a.h.1.2		2
5.4	even	2	inner	950.4.b.g.799.2		4
15.2	even	4		342.4.a.k.1.1		2
20.7	even	4		304.4.a.d.1.2		2
35.27	even	4		1862.4.a.b.1.2		2
40.27	even	4		1216.4.a.l.1.1		2
40.37	odd	4		1216.4.a.j.1.2		2
95.37	even	4		722.4.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.b.1.1	✓	2	5.2	odd	4
304.4.a.d.1.2		2	20.7	even	4
342.4.a.k.1.1		2	15.2	even	4
722.4.a.i.1.2		2	95.37	even	4
950.4.a.h.1.2		2	5.3	odd	4
950.4.b.g.799.2		4	5.4	even	2	inner
950.4.b.g.799.3		4	1.1	even	1	trivial
1216.4.a.j.1.2		2	40.37	odd	4
1216.4.a.l.1.1		2	40.27	even	4
1862.4.a.b.1.2		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)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -6.15207i q^{3} -4.00000 q^{4} +12.3041 q^{6} -21.8479i q^{7} -8.00000i q^{8} -10.8479 q^{9} -8.30413 q^{11} +24.6083i q^{12} +53.0645i q^{13} +43.6959 q^{14} +16.0000 q^{16} -74.2810i q^{17} -21.6959i q^{18} +19.0000 q^{19} -134.410 q^{21} -16.6083i q^{22} -163.977i q^{23} -49.2165 q^{24} -106.129 q^{26} -99.3686i q^{27} +87.3917i q^{28} +232.410 q^{29} +98.4331 q^{31} +32.0000i q^{32} +51.0876i q^{33} +148.562 q^{34} +43.3917 q^{36} -296.433i q^{37} +38.0000i q^{38} +326.456 q^{39} -434.912 q^{41} -268.820i q^{42} -171.299i q^{43} +33.2165 q^{44} +327.954 q^{46} +366.083i q^{47} -98.4331i q^{48} -134.332 q^{49} -456.982 q^{51} -212.258i q^{52} +138.631i q^{53} +198.737 q^{54} -174.783 q^{56} -116.889i q^{57} +464.820i q^{58} -572.797 q^{59} -632.691 q^{61} +196.866i q^{62} +237.005i q^{63} -64.0000 q^{64} -102.175 q^{66} +183.461i q^{67} +297.124i q^{68} -1008.80 q^{69} +56.6545 q^{71} +86.7835i q^{72} +68.1521i q^{73} +592.866 q^{74} -76.0000 q^{76} +181.428i q^{77} +652.912i q^{78} +332.820 q^{79} -904.217 q^{81} -869.825i q^{82} +1152.91i q^{83} +537.640 q^{84} +342.598 q^{86} -1429.80i q^{87} +66.4331i q^{88} +368.479 q^{89} +1159.35 q^{91} +655.908i q^{92} -605.567i q^{93} -732.165 q^{94} +196.866 q^{96} +426.443i q^{97} -268.664i q^{98} +90.0827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} - 4 q^{6} - 70 q^{9} + 20 q^{11} + 228 q^{14} + 64 q^{16} + 76 q^{19} + 234 q^{21} + 16 q^{24} - 52 q^{26} + 158 q^{29} - 32 q^{31} - 204 q^{34} + 280 q^{36} + 1226 q^{39} - 1580 q^{41} - 80 q^{44}+ \cdots - 704 q^{99}+O(q^{100}) \) 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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