LMFDB - Embedded newform 1150.4.b.s.599.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1150,4,Mod(599,1150)] 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("1150.599"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-48,0,-20,0,0,-122,0,-98] 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:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 219x^{10} + 17685x^{8} + 640366x^{6} + 10000368x^{4} + 54897345x^{2} + 95531076 $ x^12 + 219x^10 + 17685x^8 + 640366x^6 + 10000368x^4 + 54897345*x^2 + 95531076
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.12
Root			$8.71212i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.4.b.s.599.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +9.71212i q^{3} -4.00000 q^{4} -19.4242 q^{6} -33.1288i q^{7} -8.00000i q^{8} -67.3252 q^{9} +1.59705 q^{11} -38.8485i q^{12} -15.1238i q^{13} +66.2575 q^{14} +16.0000 q^{16} +63.9661i q^{17} -134.650i q^{18} +110.153 q^{19} +321.750 q^{21} +3.19409i q^{22} +23.0000i q^{23} +77.6969 q^{24} +30.2477 q^{26} -391.643i q^{27} +132.515i q^{28} -191.979 q^{29} +292.477 q^{31} +32.0000i q^{32} +15.5107i q^{33} -127.932 q^{34} +269.301 q^{36} +62.7459i q^{37} +220.306i q^{38} +146.885 q^{39} +296.314 q^{41} +643.501i q^{42} -90.3963i q^{43} -6.38819 q^{44} -46.0000 q^{46} -142.749i q^{47} +155.394i q^{48} -754.515 q^{49} -621.246 q^{51} +60.4954i q^{52} +589.530i q^{53} +783.286 q^{54} -265.030 q^{56} +1069.82i q^{57} -383.959i q^{58} +175.690 q^{59} -809.615 q^{61} +584.955i q^{62} +2230.40i q^{63} -64.0000 q^{64} -31.0214 q^{66} +431.173i q^{67} -255.864i q^{68} -223.379 q^{69} +453.657 q^{71} +538.602i q^{72} +111.788i q^{73} -125.492 q^{74} -440.612 q^{76} -52.9082i q^{77} +293.769i q^{78} +1175.57 q^{79} +1985.90 q^{81} +592.628i q^{82} -95.0748i q^{83} -1287.00 q^{84} +180.793 q^{86} -1864.52i q^{87} -12.7764i q^{88} -836.699 q^{89} -501.034 q^{91} -92.0000i q^{92} +2840.57i q^{93} +285.499 q^{94} -310.788 q^{96} -599.979i q^{97} -1509.03i q^{98} -107.522 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 48 q^{4} - 20 q^{6} - 122 q^{9} - 98 q^{11} + 168 q^{14} + 192 q^{16} + 458 q^{19} + 184 q^{21} + 80 q^{24} - 64 q^{26} + 364 q^{29} + 228 q^{31} + 700 q^{34} + 488 q^{36} + 286 q^{39} + 486 q^{41}+ \cdots + 284 q^{99}+O(q^{100}) $ 12 * q - 48 * q^4 - 20 * q^6 - 122 * q^9 - 98 * q^11 + 168 * q^14 + 192 * q^16 + 458 * q^19 + 184 * q^21 + 80 * q^24 - 64 * q^26 + 364 * q^29 + 228 * q^31 + 700 * q^34 + 488 * q^36 + 286 * q^39 + 486 * q^41 + 392 * q^44 - 552 * q^46 - 1296 * q^49 - 2062 * q^51 + 1376 * q^54 - 672 * q^56 + 1118 * q^59 - 1376 * q^61 - 768 * q^64 - 2564 * q^66 - 230 * q^69 + 1168 * q^71 + 256 * q^74 - 1832 * q^76 + 2864 * q^79 + 68 * q^81 - 736 * q^84 - 728 * q^86 + 1182 * q^89 - 5728 * q^91 + 1992 * q^94 - 320 * q^96 + 284 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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
$3$	9.71212i	1.86910i	0.355835	+	0.934549i	$0.384197\pi$
$3$	9.71212i	1.86910i	−0.355835	+	0.934549i	$0.615803\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−19.4242	−1.32165
$7$	− 33.1288i	− 1.78878i	−0.447283	−	0.894392i	$-0.647608\pi$
$7$	− 33.1288i	− 1.78878i	0.447283	−	0.894392i	$-0.352392\pi$
$8$	− 8.00000i	− 0.353553i
$9$	−67.3252	−2.49353
$10$	0	0
$11$	1.59705	0.0437753	0.0218876	−	0.999760i	$-0.493032\pi$
$11$	1.59705	0.0437753	0.0218876	+	0.999760i	$0.493032\pi$
$12$	− 38.8485i	− 0.934549i
$13$	− 15.1238i	− 0.322661i	−0.986900	−	0.161331i	$-0.948421\pi$
$13$	− 15.1238i	− 0.322661i	0.986900	−	0.161331i	$-0.0515786\pi$
$14$	66.2575	1.26486
$15$	0	0
$16$	16.0000	0.250000
$17$	63.9661i	0.912591i	0.889828	+	0.456296i	$0.150824\pi$
$17$	63.9661i	0.912591i	−0.889828	+	0.456296i	$0.849176\pi$
$18$	− 134.650i	− 1.76319i
$19$	110.153	1.33004	0.665022	−	0.746824i	$-0.268422\pi$
$19$	110.153	1.33004	0.665022	+	0.746824i	$0.268422\pi$
$20$	0	0
$21$	321.750	3.34341
$22$	3.19409i	0.0309538i
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	77.6969	0.660826
$25$	0	0
$26$	30.2477	0.228156
$27$	− 391.643i	− 2.79155i
$28$	132.515i	0.894392i
$29$	−191.979	−1.22930	−0.614649	−	0.788801i	$-0.710702\pi$
$29$	−191.979	−1.22930	−0.614649	+	0.788801i	$0.710702\pi$
$30$	0	0
$31$	292.477	1.69453	0.847266	−	0.531169i	$-0.178247\pi$
$31$	292.477	1.69453	0.847266	+	0.531169i	$0.178247\pi$
$32$	32.0000i	0.176777i
$33$	15.5107i	0.0818202i
$34$	−127.932	−0.645299
$35$	0	0
$36$	269.301	1.24676
$37$	62.7459i	0.278794i	0.990237	+	0.139397i	$0.0445163\pi$
$37$	62.7459i	0.278794i	−0.990237	+	0.139397i	$0.955484\pi$
$38$	220.306i	0.940483i
$39$	146.885	0.603086
$40$	0	0
$41$	296.314	1.12869	0.564347	−	0.825538i	$-0.309128\pi$
$41$	296.314	1.12869	0.564347	+	0.825538i	$0.309128\pi$
$42$	643.501i	2.36415i
$43$	− 90.3963i	− 0.320589i	−0.987069	−	0.160294i	$-0.948756\pi$
$43$	− 90.3963i	− 0.320589i	0.987069	−	0.160294i	$-0.0512443\pi$
$44$	−6.38819	−0.0218876
$45$	0	0
$46$	−46.0000	−0.147442
$47$	− 142.749i	− 0.443024i	−0.975158	−	0.221512i	$-0.928901\pi$
$47$	− 142.749i	− 0.443024i	0.975158	−	0.221512i	$-0.0710992\pi$
$48$	155.394i	0.467274i
$49$	−754.515	−2.19975
$50$	0	0
$51$	−621.246	−1.70572
$52$	60.4954i	0.161331i
$53$	589.530i	1.52789i	0.645281	+	0.763945i	$0.276740\pi$
$53$	589.530i	1.52789i	−0.645281	+	0.763945i	$0.723260\pi$
$54$	783.286	1.97392
$55$	0	0
$56$	−265.030	−0.632431
$57$	1069.82i	2.48598i
$58$	− 383.959i	− 0.869245i
$59$	175.690	0.387675	0.193838	−	0.981034i	$-0.437907\pi$
$59$	175.690	0.387675	0.193838	+	0.981034i	$0.437907\pi$
$60$	0	0
$61$	−809.615	−1.69935	−0.849677	−	0.527303i	$-0.823203\pi$
$61$	−809.615	−1.69935	−0.849677	+	0.527303i	$0.823203\pi$
$62$	584.955i	1.19821i
$63$	2230.40i	4.46038i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−31.0214	−0.0578556
$67$	431.173i	0.786212i	0.919493	+	0.393106i	$0.128599\pi$
$67$	431.173i	0.786212i	−0.919493	+	0.393106i	$0.871401\pi$
$68$	− 255.864i	− 0.456296i
$69$	−223.379	−0.389734
$70$	0	0
$71$	453.657	0.758298	0.379149	−	0.925336i	$-0.376217\pi$
$71$	453.657	0.758298	0.379149	+	0.925336i	$0.376217\pi$
$72$	538.602i	0.881595i
$73$	111.788i	0.179230i	0.995976	+	0.0896149i	$0.0285636\pi$
$73$	111.788i	0.179230i	−0.995976	+	0.0896149i	$0.971436\pi$
$74$	−125.492	−0.197137
$75$	0	0
$76$	−440.612	−0.665022
$77$	− 52.9082i	− 0.0783045i
$78$	293.769i	0.426446i
$79$	1175.57	1.67420	0.837102	−	0.547047i	$-0.184248\pi$
$79$	1175.57	1.67420	0.837102	+	0.547047i	$0.184248\pi$
$80$	0	0
$81$	1985.90	2.72415
$82$	592.628i	0.798107i
$83$	− 95.0748i	− 0.125733i	−0.998022	−	0.0628663i	$-0.979976\pi$
$83$	− 95.0748i	− 0.125733i	0.998022	−	0.0628663i	$-0.0200242\pi$
$84$	−1287.00	−1.67171
$85$	0	0
$86$	180.793	0.226690
$87$	− 1864.52i	− 2.29768i
$88$	− 12.7764i	− 0.0154769i
$89$	−836.699	−0.996516	−0.498258	−	0.867029i	$-0.666027\pi$
$89$	−836.699	−0.996516	−0.498258	+	0.867029i	$0.666027\pi$
$90$	0	0
$91$	−501.034	−0.577172
$92$	− 92.0000i	− 0.104257i
$93$	2840.57i	3.16724i
$94$	285.499	0.313265
$95$	0	0
$96$	−310.788	−0.330413
$97$	− 599.979i	− 0.628027i	−0.949419	−	0.314013i	$-0.898326\pi$
$97$	− 599.979i	− 0.628027i	0.949419	−	0.314013i	$-0.101674\pi$
$98$	− 1509.03i	− 1.55546i
$99$	−107.522	−0.109155
$100$	0	0
$101$	845.212	0.832691	0.416345	−	0.909207i	$-0.363311\pi$
$101$	845.212	0.832691	0.416345	+	0.909207i	$0.363311\pi$
$102$	− 1242.49i	− 1.20613i
$103$	1103.23i	1.05538i	0.849437	+	0.527690i	$0.176942\pi$
$103$	1103.23i	1.05538i	−0.849437	+	0.527690i	$0.823058\pi$
$104$	−120.991	−0.114078
$105$	0	0
$106$	−1179.06	−1.08038
$107$	1102.86i	0.996426i	0.867055	+	0.498213i	$0.166010\pi$
$107$	1102.86i	0.996426i	−0.867055	+	0.498213i	$0.833990\pi$
$108$	1566.57i	1.39577i
$109$	1377.62	1.21057	0.605284	−	0.796010i	$-0.293060\pi$
$109$	1377.62	1.21057	0.605284	+	0.796010i	$0.293060\pi$
$110$	0	0
$111$	−609.395	−0.521092
$112$	− 530.060i	− 0.447196i
$113$	989.295i	0.823584i	0.911278	+	0.411792i	$0.135097\pi$
$113$	989.295i	0.823584i	−0.911278	+	0.411792i	$0.864903\pi$
$114$	−2139.64	−1.75785
$115$	0	0
$116$	767.917	0.614649
$117$	1018.22i	0.804565i
$118$	351.379i	0.274128i
$119$	2119.12	1.63243
$120$	0	0
$121$	−1328.45	−0.998084
$122$	− 1619.23i	− 1.20163i
$123$	2877.83i	2.10964i
$124$	−1169.91	−0.847266
$125$	0	0
$126$	−4460.80	−3.15397
$127$	− 2088.22i	− 1.45905i	−0.683954	−	0.729525i	$-0.739741\pi$
$127$	− 2088.22i	− 1.45905i	0.683954	−	0.729525i	$-0.260259\pi$
$128$	− 128.000i	− 0.0883883i
$129$	877.940	0.599211
$130$	0	0
$131$	−2153.34	−1.43617	−0.718085	−	0.695955i	$-0.754981\pi$
$131$	−2153.34	−1.43617	−0.718085	+	0.695955i	$0.754981\pi$
$132$	− 62.0428i	− 0.0409101i
$133$	− 3649.23i	− 2.37916i
$134$	−862.346	−0.555936
$135$	0	0
$136$	511.729	0.322650
$137$	1308.13i	0.815771i	0.913033	+	0.407886i	$0.133734\pi$
$137$	1308.13i	0.815771i	−0.913033	+	0.407886i	$0.866266\pi$
$138$	− 446.757i	− 0.275583i
$139$	2122.45	1.29513	0.647567	−	0.762008i	$-0.275786\pi$
$139$	2122.45	1.29513	0.647567	+	0.762008i	$0.275786\pi$
$140$	0	0
$141$	1386.40	0.828055
$142$	907.314i	0.536198i
$143$	− 24.1535i	− 0.0141246i
$144$	−1077.20	−0.623381
$145$	0	0
$146$	−223.576	−0.126735
$147$	− 7327.93i	− 4.11155i
$148$	− 250.984i	− 0.139397i
$149$	1631.01	0.896765	0.448382	−	0.893842i	$-0.352000\pi$
$149$	1631.01	0.896765	0.448382	+	0.893842i	$0.352000\pi$
$150$	0	0
$151$	1130.37	0.609192	0.304596	−	0.952482i	$-0.401479\pi$
$151$	1130.37	0.609192	0.304596	+	0.952482i	$0.401479\pi$
$152$	− 881.224i	− 0.470241i
$153$	− 4306.53i	− 2.27557i
$154$	105.816	0.0553697
$155$	0	0
$156$	−587.538	−0.301543
$157$	− 411.116i	− 0.208985i	−0.994526	−	0.104493i	$-0.966678\pi$
$157$	− 411.116i	− 0.208985i	0.994526	−	0.104493i	$-0.0333218\pi$
$158$	2351.14i	1.18384i
$159$	−5725.58	−2.85578
$160$	0	0
$161$	761.961	0.372987
$162$	3971.80i	1.92626i
$163$	− 3150.89i	− 1.51409i	−0.653362	−	0.757046i	$-0.726642\pi$
$163$	− 3150.89i	− 1.51409i	0.653362	−	0.757046i	$-0.273358\pi$
$164$	−1185.26	−0.564347
$165$	0	0
$166$	190.150	0.0889064
$167$	3685.50i	1.70774i	0.520485	+	0.853871i	$0.325751\pi$
$167$	3685.50i	1.70774i	−0.520485	+	0.853871i	$0.674249\pi$
$168$	− 2574.00i	− 1.18208i
$169$	1968.27	0.895890
$170$	0	0
$171$	−7416.07	−3.31650
$172$	361.585i	0.160294i
$173$	4516.07i	1.98468i	0.123522	+	0.992342i	$0.460581\pi$
$173$	4516.07i	1.98468i	−0.123522	+	0.992342i	$0.539419\pi$
$174$	3729.05	1.62470
$175$	0	0
$176$	25.5528	0.0109438
$177$	1706.32i	0.724603i
$178$	− 1673.40i	− 0.704643i
$179$	−3047.24	−1.27241	−0.636206	−	0.771519i	$-0.719497\pi$
$179$	−3047.24	−1.27241	−0.636206	+	0.771519i	$0.719497\pi$
$180$	0	0
$181$	−185.447	−0.0761558	−0.0380779	−	0.999275i	$-0.512124\pi$
$181$	−185.447	−0.0761558	−0.0380779	+	0.999275i	$0.512124\pi$
$182$	− 1002.07i	− 0.408122i
$183$	− 7863.08i	− 3.17626i
$184$	184.000	0.0737210
$185$	0	0
$186$	−5681.15	−2.23958
$187$	102.157i	0.0399489i
$188$	570.998i	0.221512i
$189$	−12974.6	−4.99347
$190$	0	0
$191$	3449.76	1.30689	0.653445	−	0.756974i	$-0.273323\pi$
$191$	3449.76	1.30689	0.653445	+	0.756974i	$0.273323\pi$
$192$	− 621.575i	− 0.233637i
$193$	1593.41i	0.594279i	0.954834	+	0.297140i	$0.0960326\pi$
$193$	1593.41i	0.594279i	−0.954834	+	0.297140i	$0.903967\pi$
$194$	1199.96	0.444082
$195$	0	0
$196$	3018.06	1.09988
$197$	91.9830i	0.0332666i	0.999862	+	0.0166333i	$0.00529479\pi$
$197$	91.9830i	0.0332666i	−0.999862	+	0.0166333i	$0.994705\pi$
$198$	− 215.043i	− 0.0771841i
$199$	3252.36	1.15856	0.579281	−	0.815128i	$-0.303333\pi$
$199$	3252.36	1.15856	0.579281	+	0.815128i	$0.303333\pi$
$200$	0	0
$201$	−4187.60	−1.46951
$202$	1690.42i	0.588801i
$203$	6360.03i	2.19895i
$204$	2484.98	0.852861
$205$	0	0
$206$	−2206.45	−0.746267
$207$	− 1548.48i	− 0.519936i
$208$	− 241.981i	− 0.0806654i
$209$	175.919	0.0582230
$210$	0	0
$211$	1591.56	0.519278	0.259639	−	0.965706i	$-0.416396\pi$
$211$	1591.56	0.519278	0.259639	+	0.965706i	$0.416396\pi$
$212$	− 2358.12i	− 0.763945i
$213$	4405.97i	1.41733i
$214$	−2205.72	−0.704580
$215$	0	0
$216$	−3133.14	−0.986960
$217$	− 9689.41i	− 3.03115i
$218$	2755.24i	0.856000i
$219$	−1085.70	−0.334998
$220$	0	0
$221$	967.413	0.294458
$222$	− 1218.79i	− 0.368468i
$223$	2299.64i	0.690561i	0.938500	+	0.345280i	$0.112216\pi$
$223$	2299.64i	0.690561i	−0.938500	+	0.345280i	$0.887784\pi$
$224$	1060.12	0.316215
$225$	0	0
$226$	−1978.59	−0.582362
$227$	− 3254.18i	− 0.951488i	−0.879584	−	0.475744i	$-0.842179\pi$
$227$	− 3254.18i	− 0.951488i	0.879584	−	0.475744i	$-0.157821\pi$
$228$	− 4279.27i	− 1.24299i
$229$	4934.80	1.42402	0.712010	−	0.702169i	$-0.247785\pi$
$229$	4934.80	1.42402	0.712010	+	0.702169i	$0.247785\pi$
$230$	0	0
$231$	513.850	0.146359
$232$	1535.83i	0.434623i
$233$	− 1194.48i	− 0.335849i	−0.985800	−	0.167925i	$-0.946294\pi$
$233$	− 1194.48i	− 0.335849i	0.985800	−	0.167925i	$-0.0537065\pi$
$234$	−2036.43	−0.568913
$235$	0	0
$236$	−702.758	−0.193838
$237$	11417.3i	3.12925i
$238$	4238.23i	1.15430i
$239$	−1552.61	−0.420208	−0.210104	−	0.977679i	$-0.567380\pi$
$239$	−1552.61	−0.420208	−0.210104	+	0.977679i	$0.567380\pi$
$240$	0	0
$241$	−4178.20	−1.11677	−0.558384	−	0.829582i	$-0.688579\pi$
$241$	−4178.20	−1.11677	−0.558384	+	0.829582i	$0.688579\pi$
$242$	− 2656.90i	− 0.705752i
$243$	8712.95i	2.30015i
$244$	3238.46	0.849677
$245$	0	0
$246$	−5755.67	−1.49174
$247$	− 1665.94i	− 0.429154i
$248$	− 2339.82i	− 0.599107i
$249$	923.377	0.235007
$250$	0	0
$251$	672.494	0.169113	0.0845567	−	0.996419i	$-0.473053\pi$
$251$	672.494	0.169113	0.0845567	+	0.996419i	$0.473053\pi$
$252$	− 8921.60i	− 2.23019i
$253$	36.7321i	0.00912777i
$254$	4176.43	1.03170
$255$	0	0
$256$	256.000	0.0625000
$257$	1986.84i	0.482239i	0.970495	+	0.241119i	$0.0775146\pi$
$257$	1986.84i	0.482239i	−0.970495	+	0.241119i	$0.922485\pi$
$258$	1755.88i	0.423707i
$259$	2078.69	0.498702
$260$	0	0
$261$	12925.0	3.06529
$262$	− 4306.68i	− 1.01553i
$263$	− 1583.85i	− 0.371347i	−0.982612	−	0.185673i	$-0.940553\pi$
$263$	− 1583.85i	− 0.371347i	0.982612	−	0.185673i	$-0.0594466\pi$
$264$	124.086	0.0289278
$265$	0	0
$266$	7298.46	1.68232
$267$	− 8126.12i	− 1.86259i
$268$	− 1724.69i	− 0.393106i
$269$	5309.66	1.20348	0.601739	−	0.798693i	$-0.294475\pi$
$269$	5309.66	1.20348	0.601739	+	0.798693i	$0.294475\pi$
$270$	0	0
$271$	−1083.04	−0.242767	−0.121383	−	0.992606i	$-0.538733\pi$
$271$	−1083.04	−0.242767	−0.121383	+	0.992606i	$0.538733\pi$
$272$	1023.46i	0.228148i
$273$	− 4866.10i	− 1.07879i
$274$	−2616.25	−0.576837
$275$	0	0
$276$	893.515	0.194867
$277$	8793.99i	1.90751i	0.300590	+	0.953753i	$0.402816\pi$
$277$	8793.99i	1.90751i	−0.300590	+	0.953753i	$0.597184\pi$
$278$	4244.90i	0.915799i
$279$	−19691.1	−4.22536
$280$	0	0
$281$	6603.86	1.40197	0.700984	−	0.713177i	$-0.252744\pi$
$281$	6603.86	1.40197	0.700984	+	0.713177i	$0.252744\pi$
$282$	2772.80i	0.585524i
$283$	7387.79i	1.55180i	0.630858	+	0.775898i	$0.282703\pi$
$283$	7387.79i	1.55180i	−0.630858	+	0.775898i	$0.717297\pi$
$284$	−1814.63	−0.379149
$285$	0	0
$286$	48.3070	0.00998759
$287$	− 9816.51i	− 2.01899i
$288$	− 2154.41i	− 0.440797i
$289$	821.341	0.167177
$290$	0	0
$291$	5827.06	1.17384
$292$	− 447.151i	− 0.0896149i
$293$	4539.89i	0.905198i	0.891714	+	0.452599i	$0.149503\pi$
$293$	4539.89i	0.905198i	−0.891714	+	0.452599i	$0.850497\pi$
$294$	14655.9	2.90730
$295$	0	0
$296$	501.967	0.0985684
$297$	− 625.472i	− 0.122201i
$298$	3262.03i	0.634108i
$299$	347.848	0.0672796
$300$	0	0
$301$	−2994.72	−0.573464
$302$	2260.73i	0.430764i
$303$	8208.80i	1.55638i
$304$	1762.45	0.332511
$305$	0	0
$306$	8613.06	1.60907
$307$	2906.64i	0.540360i	0.962810	+	0.270180i	$0.0870832\pi$
$307$	2906.64i	0.540360i	−0.962810	+	0.270180i	$0.912917\pi$
$308$	211.633i	0.0391523i
$309$	−10714.7	−1.97261
$310$	0	0
$311$	−8212.89	−1.49746	−0.748730	−	0.662875i	$-0.769336\pi$
$311$	−8212.89	−1.49746	−0.748730	+	0.662875i	$0.769336\pi$
$312$	− 1175.08i	− 0.213223i
$313$	− 201.103i	− 0.0363163i	−0.999835	−	0.0181582i	$-0.994220\pi$
$313$	− 201.103i	− 0.0363163i	0.999835	−	0.0181582i	$-0.00578024\pi$
$314$	822.233	0.147775
$315$	0	0
$316$	−4702.29	−0.837102
$317$	2545.20i	0.450954i	0.974248	+	0.225477i	$0.0723940\pi$
$317$	2545.20i	0.450954i	−0.974248	+	0.225477i	$0.927606\pi$
$318$	− 11451.2i	− 2.01934i
$319$	−306.600	−0.0538129
$320$	0	0
$321$	−10711.1	−1.86242
$322$	1523.92i	0.263742i
$323$	7046.05i	1.21379i
$324$	−7943.61	−1.36207
$325$	0	0
$326$	6301.79	1.07062
$327$	13379.6i	2.26267i
$328$	− 2370.51i	− 0.399053i
$329$	−4729.11	−0.792475
$330$	0	0
$331$	7330.61	1.21730	0.608651	−	0.793438i	$-0.291711\pi$
$331$	7330.61	1.21730	0.608651	+	0.793438i	$0.291711\pi$
$332$	380.299i	0.0628663i
$333$	− 4224.38i	− 0.695179i
$334$	−7371.01	−1.20756
$335$	0	0
$336$	5148.01	0.835853
$337$	− 7988.83i	− 1.29133i	−0.763620	−	0.645666i	$-0.776580\pi$
$337$	− 7988.83i	− 1.29133i	0.763620	−	0.645666i	$-0.223420\pi$
$338$	3936.54i	0.633490i
$339$	−9608.15	−1.53936
$340$	0	0
$341$	467.100	0.0741786
$342$	− 14832.1i	− 2.34512i
$343$	13633.0i	2.14610i
$344$	−723.171	−0.113345
$345$	0	0
$346$	−9032.14	−1.40338
$347$	9300.59i	1.43885i	0.694569	+	0.719427i	$0.255595\pi$
$347$	9300.59i	1.43885i	−0.694569	+	0.719427i	$0.744405\pi$
$348$	7458.10i	1.14884i
$349$	627.258	0.0962073	0.0481037	−	0.998842i	$-0.484682\pi$
$349$	627.258	0.0962073	0.0481037	+	0.998842i	$0.484682\pi$
$350$	0	0
$351$	−5923.15	−0.900724
$352$	51.1055i	0.00773845i
$353$	− 10275.9i	− 1.54938i	−0.632339	−	0.774691i	$-0.717905\pi$
$353$	− 10275.9i	− 1.54938i	0.632339	−	0.774691i	$-0.282095\pi$
$354$	−3412.63	−0.512371
$355$	0	0
$356$	3346.80	0.498258
$357$	20581.1i	3.05117i
$358$	− 6094.49i	− 0.899731i
$359$	−9083.80	−1.33544	−0.667722	−	0.744411i	$-0.732730\pi$
$359$	−9083.80	−1.33544	−0.667722	+	0.744411i	$0.732730\pi$
$360$	0	0
$361$	5274.67	0.769015
$362$	− 370.895i	− 0.0538503i
$363$	− 12902.1i	− 1.86552i
$364$	2004.14	0.288586
$365$	0	0
$366$	15726.2	2.24595
$367$	8578.97i	1.22021i	0.792319	+	0.610107i	$0.208873\pi$
$367$	8578.97i	1.22021i	−0.792319	+	0.610107i	$0.791127\pi$
$368$	368.000i	0.0521286i
$369$	−19949.4	−2.81443
$370$	0	0
$371$	19530.4	2.73307
$372$	− 11362.3i	− 1.58362i
$373$	− 10462.6i	− 1.45236i	−0.687503	−	0.726181i	$-0.741293\pi$
$373$	− 10462.6i	− 1.45236i	0.687503	−	0.726181i	$-0.258707\pi$
$374$	−204.314	−0.0282482
$375$	0	0
$376$	−1142.00	−0.156633
$377$	2903.46i	0.396647i
$378$	− 25949.3i	− 3.53092i
$379$	7320.67	0.992184	0.496092	−	0.868270i	$-0.334768\pi$
$379$	7320.67	0.992184	0.496092	+	0.868270i	$0.334768\pi$
$380$	0	0
$381$	20281.0	2.72711
$382$	6899.52i	0.924111i
$383$	2226.12i	0.296995i	0.988913	+	0.148498i	$0.0474437\pi$
$383$	2226.12i	0.296995i	−0.988913	+	0.148498i	$0.952556\pi$
$384$	1243.15	0.165206
$385$	0	0
$386$	−3186.81	−0.420219
$387$	6085.95i	0.799396i
$388$	2399.92i	0.314013i
$389$	−1358.18	−0.177024	−0.0885121	−	0.996075i	$-0.528211\pi$
$389$	−1358.18	−0.177024	−0.0885121	+	0.996075i	$0.528211\pi$
$390$	0	0
$391$	−1471.22	−0.190288
$392$	6036.12i	0.777729i
$393$	− 20913.5i	− 2.68434i
$394$	−183.966	−0.0235230
$395$	0	0
$396$	430.086	0.0545774
$397$	− 9938.92i	− 1.25647i	−0.778022	−	0.628237i	$-0.783777\pi$
$397$	− 9938.92i	− 1.25647i	0.778022	−	0.628237i	$-0.216223\pi$
$398$	6504.73i	0.819227i
$399$	35441.7	4.44688
$400$	0	0
$401$	−13384.5	−1.66680	−0.833401	−	0.552669i	$-0.813609\pi$
$401$	−13384.5	−1.66680	−0.833401	+	0.552669i	$0.813609\pi$
$402$	− 8375.21i	− 1.03910i
$403$	− 4423.38i	− 0.546760i
$404$	−3380.85	−0.416345
$405$	0	0
$406$	−12720.1	−1.55489
$407$	100.208i	0.0122043i
$408$	4969.97i	0.603064i
$409$	−2957.56	−0.357560	−0.178780	−	0.983889i	$-0.557215\pi$
$409$	−2957.56	−0.357560	−0.178780	+	0.983889i	$0.557215\pi$
$410$	0	0
$411$	−12704.7	−1.52476
$412$	− 4412.91i	− 0.527690i
$413$	− 5820.38i	− 0.693467i
$414$	3096.96	0.367650
$415$	0	0
$416$	483.963	0.0570390
$417$	20613.5i	2.42073i
$418$	351.839i	0.0411699i
$419$	13178.4	1.53654	0.768268	−	0.640128i	$-0.221119\pi$
$419$	13178.4	1.53654	0.768268	+	0.640128i	$0.221119\pi$
$420$	0	0
$421$	−1367.55	−0.158314	−0.0791569	−	0.996862i	$-0.525223\pi$
$421$	−1367.55	−0.158314	−0.0791569	+	0.996862i	$0.525223\pi$
$422$	3183.12i	0.367185i
$423$	9610.63i	1.10469i
$424$	4716.24	0.540191
$425$	0	0
$426$	−8811.94	−1.00221
$427$	26821.6i	3.03978i
$428$	− 4411.44i	− 0.498213i
$429$	234.581	0.0264002
$430$	0	0
$431$	34.6193	0.00386903	0.00193451	−	0.999998i	$-0.499384\pi$
$431$	34.6193	0.00386903	0.00193451	+	0.999998i	$0.499384\pi$
$432$	− 6266.29i	− 0.697886i
$433$	3270.08i	0.362933i	0.983397	+	0.181466i	$0.0580843\pi$
$433$	3270.08i	0.362933i	−0.983397	+	0.181466i	$0.941916\pi$
$434$	19378.8	2.14335
$435$	0	0
$436$	−5510.47	−0.605284
$437$	2533.52i	0.277333i
$438$	− 2171.39i	− 0.236879i
$439$	−6565.41	−0.713781	−0.356891	−	0.934146i	$-0.616163\pi$
$439$	−6565.41	−0.713781	−0.356891	+	0.934146i	$0.616163\pi$
$440$	0	0
$441$	50797.8	5.48514
$442$	1934.83i	0.208213i
$443$	8809.74i	0.944838i	0.881374	+	0.472419i	$0.156619\pi$
$443$	8809.74i	0.944838i	−0.881374	+	0.472419i	$0.843381\pi$
$444$	2437.58	0.260546
$445$	0	0
$446$	−4599.27	−0.488300
$447$	15840.6i	1.67614i
$448$	2120.24i	0.223598i
$449$	8977.39	0.943584	0.471792	−	0.881710i	$-0.343607\pi$
$449$	8977.39	0.943584	0.471792	+	0.881710i	$0.343607\pi$
$450$	0	0
$451$	473.227	0.0494089
$452$	− 3957.18i	− 0.411792i
$453$	10978.3i	1.13864i
$454$	6508.37	0.672803
$455$	0	0
$456$	8558.55	0.878927
$457$	856.195i	0.0876392i	0.999039	+	0.0438196i	$0.0139527\pi$
$457$	856.195i	0.0876392i	−0.999039	+	0.0438196i	$0.986047\pi$
$458$	9869.60i	1.00693i
$459$	25051.9	2.54754
$460$	0	0
$461$	367.756	0.0371543	0.0185771	−	0.999827i	$-0.494086\pi$
$461$	367.756	0.0371543	0.0185771	+	0.999827i	$0.494086\pi$
$462$	1027.70i	0.103491i
$463$	7330.47i	0.735801i	0.929865	+	0.367901i	$0.119923\pi$
$463$	7330.47i	0.735801i	−0.929865	+	0.367901i	$0.880077\pi$
$464$	−3071.67	−0.307325
$465$	0	0
$466$	2388.96	0.237481
$467$	11503.1i	1.13983i	0.821705	+	0.569914i	$0.193023\pi$
$467$	11503.1i	1.13983i	−0.821705	+	0.569914i	$0.806977\pi$
$468$	− 4072.86i	− 0.402282i
$469$	14284.2	1.40636
$470$	0	0
$471$	3992.81	0.390614
$472$	− 1405.52i	− 0.137064i
$473$	− 144.367i	− 0.0140339i
$474$	−22834.6	−2.21271
$475$	0	0
$476$	−8476.47	−0.816215
$477$	− 39690.2i	− 3.80983i
$478$	− 3105.21i	− 0.297132i
$479$	−8580.47	−0.818479	−0.409240	−	0.912427i	$-0.634206\pi$
$479$	−8580.47	−0.818479	−0.409240	+	0.912427i	$0.634206\pi$
$480$	0	0
$481$	948.959	0.0899559
$482$	− 8356.39i	− 0.789675i
$483$	7400.26i	0.697150i
$484$	5313.80	0.499042
$485$	0	0
$486$	−17425.9	−1.62645
$487$	− 15826.8i	− 1.47265i	−0.676628	−	0.736325i	$-0.736559\pi$
$487$	− 15826.8i	− 1.47265i	0.676628	−	0.736325i	$-0.263441\pi$
$488$	6476.92i	0.600813i
$489$	30601.8	2.82999
$490$	0	0
$491$	−6219.11	−0.571618	−0.285809	−	0.958287i	$-0.592262\pi$
$491$	−6219.11	−0.571618	−0.285809	+	0.958287i	$0.592262\pi$
$492$	− 11511.3i	− 1.05482i
$493$	− 12280.2i	− 1.12185i
$494$	3331.87	0.303457
$495$	0	0
$496$	4679.64	0.423633
$497$	− 15029.1i	− 1.35643i
$498$	1846.75i	0.166175i
$499$	−12820.1	−1.15011	−0.575056	−	0.818114i	$-0.695020\pi$
$499$	−12820.1	−1.15011	−0.575056	+	0.818114i	$0.695020\pi$
$500$	0	0
$501$	−35794.0	−3.19194
$502$	1344.99i	0.119581i
$503$	− 423.446i	− 0.0375358i	−0.999824	−	0.0187679i	$-0.994026\pi$
$503$	− 423.446i	− 0.0375358i	0.999824	−	0.0187679i	$-0.00597437\pi$
$504$	17843.2	1.57698
$505$	0	0
$506$	−73.4642	−0.00645431
$507$	19116.1i	1.67451i
$508$	8352.87i	0.729525i
$509$	13039.2	1.13546	0.567731	−	0.823214i	$-0.307821\pi$
$509$	13039.2	1.13546	0.567731	+	0.823214i	$0.307821\pi$
$510$	0	0
$511$	3703.39	0.320603
$512$	512.000i	0.0441942i
$513$	− 43140.6i	− 3.71288i
$514$	−3973.67	−0.340994
$515$	0	0
$516$	−3511.76	−0.299606
$517$	− 227.978i	− 0.0193935i
$518$	4157.39i	0.352635i
$519$	−43860.6	−3.70957
$520$	0	0
$521$	13783.7	1.15907	0.579535	−	0.814948i	$-0.303234\pi$
$521$	13783.7	1.15907	0.579535	+	0.814948i	$0.303234\pi$
$522$	25850.1i	2.16749i
$523$	4223.40i	0.353109i	0.984291	+	0.176555i	$0.0564953\pi$
$523$	4223.40i	0.353109i	−0.984291	+	0.176555i	$0.943505\pi$
$524$	8613.36	0.718085
$525$	0	0
$526$	3167.69	0.262582
$527$	18708.6i	1.54641i
$528$	248.171i	0.0204551i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	−11828.3	−0.966678
$532$	14596.9i	1.18958i
$533$	− 4481.40i	− 0.364186i
$534$	16252.2	1.31705
$535$	0	0
$536$	3449.39	0.277968
$537$	− 29595.2i	− 2.37826i
$538$	10619.3i	0.850987i
$539$	−1205.00	−0.0962947
$540$	0	0
$541$	−8572.42	−0.681252	−0.340626	−	0.940199i	$-0.610639\pi$
$541$	−8572.42	−0.681252	−0.340626	+	0.940199i	$0.610639\pi$
$542$	− 2166.07i	− 0.171662i
$543$	− 1801.09i	− 0.142343i
$544$	−2046.91	−0.161325
$545$	0	0
$546$	9732.20	0.762820
$547$	− 17318.1i	− 1.35369i	−0.736125	−	0.676845i	$-0.763347\pi$
$547$	− 17318.1i	− 1.35369i	0.736125	−	0.676845i	$-0.236653\pi$
$548$	− 5232.50i	− 0.407886i
$549$	54507.5	4.23738
$550$	0	0
$551$	−21147.1	−1.63502
$552$	1787.03i	0.137792i
$553$	− 38945.2i	− 2.99479i
$554$	−17588.0	−1.34881
$555$	0	0
$556$	−8489.79	−0.647567
$557$	17447.9i	1.32727i	0.748057	+	0.663635i	$0.230987\pi$
$557$	17447.9i	1.32727i	−0.748057	+	0.663635i	$0.769013\pi$
$558$	− 39382.2i	− 2.98778i
$559$	−1367.14	−0.103442
$560$	0	0
$561$	−992.159	−0.0746684
$562$	13207.7i	0.991341i
$563$	− 22775.6i	− 1.70493i	−0.522784	−	0.852465i	$-0.675107\pi$
$563$	− 22775.6i	− 1.70493i	0.522784	−	0.852465i	$-0.324893\pi$
$564$	−5545.59	−0.414028
$565$	0	0
$566$	−14775.6	−1.09729
$567$	− 65790.5i	− 4.87291i
$568$	− 3629.26i	− 0.268099i
$569$	14993.4	1.10467	0.552333	−	0.833624i	$-0.313738\pi$
$569$	14993.4	1.10467	0.552333	+	0.833624i	$0.313738\pi$
$570$	0	0
$571$	5161.10	0.378258	0.189129	−	0.981952i	$-0.439434\pi$
$571$	5161.10	0.378258	0.189129	+	0.981952i	$0.439434\pi$
$572$	96.6140i	0.00706229i
$573$	33504.5i	2.44270i
$574$	19633.0	1.42764
$575$	0	0
$576$	4308.81	0.311691
$577$	− 7474.63i	− 0.539294i	−0.962959	−	0.269647i	$-0.913093\pi$
$577$	− 7474.63i	− 0.539294i	0.962959	−	0.269647i	$-0.0869070\pi$
$578$	1642.68i	0.118212i
$579$	−15475.3	−1.11077
$580$	0	0
$581$	−3149.71	−0.224909
$582$	11654.1i	0.830033i
$583$	941.507i	0.0668838i
$584$	894.302	0.0633673
$585$	0	0
$586$	−9079.78	−0.640072
$587$	− 10806.8i	− 0.759869i	−0.925013	−	0.379934i	$-0.875947\pi$
$587$	− 10806.8i	− 0.759869i	0.925013	−	0.379934i	$-0.124053\pi$
$588$	29311.7i	2.05577i
$589$	32217.2	2.25380
$590$	0	0
$591$	−893.349	−0.0621785
$592$	1003.93i	0.0696984i
$593$	2031.95i	0.140712i	0.997522	+	0.0703559i	$0.0224135\pi$
$593$	2031.95i	0.140712i	−0.997522	+	0.0703559i	$0.977587\pi$
$594$	1250.94	0.0864089
$595$	0	0
$596$	−6524.06	−0.448382
$597$	31587.3i	2.16547i
$598$	695.697i	0.0475738i
$599$	17544.8	1.19677	0.598383	−	0.801210i	$-0.295810\pi$
$599$	17544.8	1.19677	0.598383	+	0.801210i	$0.295810\pi$
$600$	0	0
$601$	38.6142	0.00262081	0.00131041	−	0.999999i	$-0.499583\pi$
$601$	38.6142	0.00262081	0.00131041	+	0.999999i	$0.499583\pi$
$602$	− 5989.44i	− 0.405500i
$603$	− 29028.8i	− 1.96044i
$604$	−4521.47	−0.304596
$605$	0	0
$606$	−16417.6	−1.10053
$607$	10452.9i	0.698961i	0.936944	+	0.349480i	$0.113642\pi$
$607$	10452.9i	0.698961i	−0.936944	+	0.349480i	$0.886358\pi$
$608$	3524.89i	0.235121i
$609$	−61769.4	−4.11005
$610$	0	0
$611$	−2158.92	−0.142947
$612$	17226.1i	1.13778i
$613$	− 27674.5i	− 1.82343i	−0.410823	−	0.911715i	$-0.634759\pi$
$613$	− 27674.5i	− 1.82343i	0.410823	−	0.911715i	$-0.365241\pi$
$614$	−5813.27	−0.382092
$615$	0	0
$616$	−423.266	−0.0276848
$617$	− 24533.5i	− 1.60078i	−0.599480	−	0.800390i	$-0.704626\pi$
$617$	− 24533.5i	− 1.60078i	0.599480	−	0.800390i	$-0.295374\pi$
$618$	− 21429.3i	− 1.39485i
$619$	1794.30	0.116509	0.0582545	−	0.998302i	$-0.481446\pi$
$619$	1794.30	0.116509	0.0582545	+	0.998302i	$0.481446\pi$
$620$	0	0
$621$	9007.79	0.582077
$622$	− 16425.8i	− 1.05886i
$623$	27718.8i	1.78255i
$624$	2350.15	0.150771
$625$	0	0
$626$	402.206	0.0256795
$627$	1708.55i	0.108824i
$628$	1644.47i	0.104493i
$629$	−4013.61	−0.254425
$630$	0	0
$631$	−12771.3	−0.805735	−0.402868	−	0.915258i	$-0.631986\pi$
$631$	−12771.3	−0.805735	−0.402868	+	0.915258i	$0.631986\pi$
$632$	− 9404.57i	− 0.591921i
$633$	15457.4i	0.970581i
$634$	−5090.39	−0.318873
$635$	0	0
$636$	22902.3	1.42789
$637$	11411.2i	0.709775i
$638$	− 613.200i	− 0.0380514i
$639$	−30542.5	−1.89084
$640$	0	0
$641$	15626.1	0.962863	0.481432	−	0.876484i	$-0.340117\pi$
$641$	15626.1	0.962863	0.481432	+	0.876484i	$0.340117\pi$
$642$	− 21422.2i	− 1.31693i
$643$	12756.7i	0.782384i	0.920309	+	0.391192i	$0.127937\pi$
$643$	12756.7i	0.782384i	−0.920309	+	0.391192i	$0.872063\pi$
$644$	−3047.85	−0.186494
$645$	0	0
$646$	−14092.1	−0.858276
$647$	− 22.8807i	− 0.00139031i	−1.00000		0.000695157i	$-0.999779\pi$
$647$	− 22.8807i	− 0.00139031i	1.00000		0.000695157i	$-0.000221275\pi$
$648$	− 15887.2i	− 0.963131i
$649$	280.584	0.0169706
$650$	0	0
$651$	94104.7	5.66552
$652$	12603.6i	0.757046i
$653$	5985.07i	0.358674i	0.983788	+	0.179337i	$0.0573952\pi$
$653$	5985.07i	0.358674i	−0.983788	+	0.179337i	$0.942605\pi$
$654$	−26759.2	−1.59995
$655$	0	0
$656$	4741.02	0.282173
$657$	− 7526.13i	− 0.446914i
$658$	− 9458.22i	− 0.560364i
$659$	12224.7	0.722622	0.361311	−	0.932445i	$-0.382329\pi$
$659$	12224.7	0.722622	0.361311	+	0.932445i	$0.382329\pi$
$660$	0	0
$661$	−24940.4	−1.46758	−0.733790	−	0.679376i	$-0.762251\pi$
$661$	−24940.4	−1.46758	−0.733790	+	0.679376i	$0.762251\pi$
$662$	14661.2i	0.860762i
$663$	9395.62i	0.550371i
$664$	−760.598	−0.0444532
$665$	0	0
$666$	8448.76	0.491566
$667$	− 4415.52i	− 0.256326i
$668$	− 14742.0i	− 0.853871i
$669$	−22334.3	−1.29073
$670$	0	0
$671$	−1292.99	−0.0743897
$672$	10296.0i	0.591038i
$673$	15461.4i	0.885576i	0.896626	+	0.442788i	$0.146011\pi$
$673$	15461.4i	0.885576i	−0.896626	+	0.442788i	$0.853989\pi$
$674$	15977.7	0.913110
$675$	0	0
$676$	−7873.08	−0.447945
$677$	− 6904.24i	− 0.391952i	−0.980609	−	0.195976i	$-0.937212\pi$
$677$	− 6904.24i	− 0.391952i	0.980609	−	0.195976i	$-0.0627875\pi$
$678$	− 19216.3i	− 1.08849i
$679$	−19876.6	−1.12341
$680$	0	0
$681$	31605.0	1.77842
$682$	934.200i	0.0524522i
$683$	9644.73i	0.540330i	0.962814	+	0.270165i	$0.0870782\pi$
$683$	9644.73i	0.540330i	−0.962814	+	0.270165i	$0.912922\pi$
$684$	29664.3	1.65825
$685$	0	0
$686$	−27265.9	−1.51752
$687$	47927.4i	2.66163i
$688$	− 1446.34i	− 0.0801472i
$689$	8915.96	0.492991
$690$	0	0
$691$	−704.001	−0.0387575	−0.0193788	−	0.999812i	$-0.506169\pi$
$691$	−704.001	−0.0387575	−0.0193788	+	0.999812i	$0.506169\pi$
$692$	− 18064.3i	− 0.992342i
$693$	3562.05i	0.195254i
$694$	−18601.2	−1.01742
$695$	0	0
$696$	−14916.2	−0.812352
$697$	18954.0i	1.03004i
$698$	1254.52i	0.0680289i
$699$	11600.9	0.627735
$700$	0	0
$701$	2557.38	0.137790	0.0688952	−	0.997624i	$-0.478053\pi$
$701$	2557.38	0.137790	0.0688952	+	0.997624i	$0.478053\pi$
$702$	− 11846.3i	− 0.636908i
$703$	6911.64i	0.370807i
$704$	−102.211	−0.00547191
$705$	0	0
$706$	20551.8	1.09558
$707$	− 28000.8i	− 1.48950i
$708$	− 6825.27i	− 0.362301i
$709$	17999.7	0.953448	0.476724	−	0.879053i	$-0.341824\pi$
$709$	17999.7	0.953448	0.476724	+	0.879053i	$0.341824\pi$
$710$	0	0
$711$	−79145.6	−4.17467
$712$	6693.59i	0.352322i
$713$	6726.98i	0.353334i
$714$	−41162.2	−2.15750
$715$	0	0
$716$	12189.0	0.636206
$717$	− 15079.1i	− 0.785410i
$718$	− 18167.6i	− 0.944301i
$719$	24565.1	1.27416	0.637082	−	0.770796i	$-0.280141\pi$
$719$	24565.1	1.27416	0.637082	+	0.770796i	$0.280141\pi$
$720$	0	0
$721$	36548.5	1.88785
$722$	10549.3i	0.543775i
$723$	− 40579.1i	− 2.08735i
$724$	741.790	0.0380779
$725$	0	0
$726$	25804.1	1.31912
$727$	− 16301.8i	− 0.831638i	−0.909447	−	0.415819i	$-0.863495\pi$
$727$	− 16301.8i	− 0.831638i	0.909447	−	0.415819i	$-0.136505\pi$
$728$	4008.27i	0.204061i
$729$	−31001.8	−1.57506
$730$	0	0
$731$	5782.30	0.292566
$732$	31452.3i	1.58813i
$733$	6884.70i	0.346920i	0.984841	+	0.173460i	$0.0554947\pi$
$733$	6884.70i	0.346920i	−0.984841	+	0.173460i	$0.944505\pi$
$734$	−17157.9	−0.862821
$735$	0	0
$736$	−736.000	−0.0368605
$737$	688.604i	0.0344166i
$738$	− 39898.8i	− 1.99010i
$739$	32833.4	1.63436	0.817182	−	0.576380i	$-0.195535\pi$
$739$	32833.4	1.63436	0.817182	+	0.576380i	$0.195535\pi$
$740$	0	0
$741$	16179.8	0.802130
$742$	39060.8i	1.93257i
$743$	32228.8i	1.59133i	0.605736	+	0.795666i	$0.292879\pi$
$743$	32228.8i	1.59133i	−0.605736	+	0.795666i	$0.707121\pi$
$744$	22724.6	1.11979
$745$	0	0
$746$	20925.1	1.02698
$747$	6400.93i	0.313518i
$748$	− 408.627i	− 0.0199745i
$749$	36536.4	1.78239
$750$	0	0
$751$	−10938.3	−0.531481	−0.265741	−	0.964045i	$-0.585616\pi$
$751$	−10938.3	−0.531481	−0.265741	+	0.964045i	$0.585616\pi$
$752$	− 2283.99i	− 0.110756i
$753$	6531.34i	0.316090i
$754$	−5806.93	−0.280472
$755$	0	0
$756$	51898.6	2.49674
$757$	13881.7i	0.666496i	0.942839	+	0.333248i	$0.108145\pi$
$757$	13881.7i	0.666496i	−0.942839	+	0.333248i	$0.891855\pi$
$758$	14641.3i	0.701580i
$759$	−356.746	−0.0170607
$760$	0	0
$761$	−32811.6	−1.56297	−0.781485	−	0.623924i	$-0.785538\pi$
$761$	−32811.6	−1.56297	−0.781485	+	0.623924i	$0.785538\pi$
$762$	40562.0i	1.92836i
$763$	− 45638.8i	− 2.16544i
$764$	−13799.0	−0.653445
$765$	0	0
$766$	−4452.23	−0.210007
$767$	− 2657.10i	− 0.125088i
$768$	2486.30i	0.116819i
$769$	−9987.66	−0.468354	−0.234177	−	0.972194i	$-0.575240\pi$
$769$	−9987.66	−0.468354	−0.234177	+	0.972194i	$0.575240\pi$
$770$	0	0
$771$	−19296.4	−0.901352
$772$	− 6373.62i	− 0.297140i
$773$	− 3484.56i	− 0.162135i	−0.996709	−	0.0810677i	$-0.974167\pi$
$773$	− 3484.56i	− 0.162135i	0.996709	−	0.0810677i	$-0.0258330\pi$
$774$	−12171.9	−0.565258
$775$	0	0
$776$	−4799.83	−0.222041
$777$	20188.5i	0.932122i
$778$	− 2716.36i	− 0.125175i
$779$	32639.8	1.50121
$780$	0	0
$781$	724.512	0.0331947
$782$	− 2942.44i	− 0.134554i
$783$	75187.3i	3.43164i
$784$	−12072.2	−0.549938
$785$	0	0
$786$	41827.0	1.89812
$787$	− 12421.1i	− 0.562597i	−0.959620	−	0.281298i	$-0.909235\pi$
$787$	− 12421.1i	− 0.562597i	0.959620	−	0.281298i	$-0.0907650\pi$
$788$	− 367.932i	− 0.0166333i
$789$	15382.5	0.694083
$790$	0	0
$791$	32774.1	1.47322
$792$	860.172i	0.0385920i
$793$	12244.5i	0.548316i
$794$	19877.8	0.888461
$795$	0	0
$796$	−13009.5	−0.579281
$797$	− 24083.7i	− 1.07037i	−0.844734	−	0.535187i	$-0.820241\pi$
$797$	− 24083.7i	− 1.07037i	0.844734	−	0.535187i	$-0.179759\pi$
$798$	70883.5i	3.14442i
$799$	9131.12	0.404300
$800$	0	0
$801$	56330.9	2.48484
$802$	− 26768.9i	− 1.17861i
$803$	178.530i	0.00784583i
$804$	16750.4	0.734753
$805$	0	0
$806$	8846.76	0.386618
$807$	51568.0i	2.24942i
$808$	− 6761.70i	− 0.294401i
$809$	−3687.22	−0.160242	−0.0801209	−	0.996785i	$-0.525531\pi$
$809$	−3687.22	−0.160242	−0.0801209	+	0.996785i	$0.525531\pi$
$810$	0	0
$811$	19516.5	0.845028	0.422514	−	0.906356i	$-0.361148\pi$
$811$	19516.5	0.845028	0.422514	+	0.906356i	$0.361148\pi$
$812$	− 25440.1i	− 1.09948i
$813$	− 10518.6i	− 0.453755i
$814$	−200.416	−0.00862971
$815$	0	0
$816$	−9939.93	−0.426431
$817$	− 9957.42i	− 0.426397i
$818$	− 5915.12i	− 0.252833i
$819$	33732.2	1.43919
$820$	0	0
$821$	−25386.0	−1.07914	−0.539571	−	0.841940i	$-0.681414\pi$
$821$	−25386.0	−1.07914	−0.539571	+	0.841940i	$0.681414\pi$
$822$	− 25409.3i	− 1.07817i
$823$	− 34018.2i	− 1.44083i	−0.693545	−	0.720413i	$-0.743952\pi$
$823$	− 34018.2i	− 1.44083i	0.693545	−	0.720413i	$-0.256048\pi$
$824$	8825.82	0.373133
$825$	0	0
$826$	11640.8	0.490355
$827$	− 23934.0i	− 1.00637i	−0.864180	−	0.503183i	$-0.832162\pi$
$827$	− 23934.0i	− 1.00637i	0.864180	−	0.503183i	$-0.167838\pi$
$828$	6193.92i	0.259968i
$829$	26149.2	1.09554	0.547768	−	0.836630i	$-0.315478\pi$
$829$	26149.2	1.09554	0.547768	+	0.836630i	$0.315478\pi$
$830$	0	0
$831$	−85408.2	−3.56532
$832$	967.926i	0.0403327i
$833$	− 48263.3i	− 2.00747i
$834$	−41226.9	−1.71172
$835$	0	0
$836$	−703.678	−0.0291115
$837$	− 114547.i	− 4.73036i
$838$	26356.9i	1.08649i
$839$	−13611.0	−0.560074	−0.280037	−	0.959989i	$-0.590347\pi$
$839$	−13611.0	−0.560074	−0.280037	+	0.959989i	$0.590347\pi$
$840$	0	0
$841$	12467.0	0.511175
$842$	− 2735.09i	− 0.111945i
$843$	64137.4i	2.62042i
$844$	−6366.24	−0.259639
$845$	0	0
$846$	−19221.3	−0.781135
$847$	44009.9i	1.78536i
$848$	9432.48i	0.381973i
$849$	−71751.0	−2.90046
$850$	0	0
$851$	−1443.16	−0.0581325
$852$	− 17623.9i	− 0.708667i
$853$	27108.9i	1.08815i	0.839038	+	0.544073i	$0.183119\pi$
$853$	27108.9i	1.08815i	−0.839038	+	0.544073i	$0.816881\pi$
$854$	−53643.1	−2.14945
$855$	0	0
$856$	8822.89	0.352290
$857$	− 11549.4i	− 0.460349i	−0.973149	−	0.230175i	$-0.926070\pi$
$857$	− 11549.4i	− 0.460349i	0.973149	−	0.230175i	$-0.0739297\pi$
$858$	469.163i	0.0186678i
$859$	15491.8	0.615335	0.307668	−	0.951494i	$-0.400451\pi$
$859$	15491.8	0.615335	0.307668	+	0.951494i	$0.400451\pi$
$860$	0	0
$861$	95339.1	3.77369
$862$	69.2386i	0.00273582i
$863$	4709.17i	0.185750i	0.995678	+	0.0928749i	$0.0296057\pi$
$863$	4709.17i	0.185750i	−0.995678	+	0.0928749i	$0.970394\pi$
$864$	12532.6	0.493480
$865$	0	0
$866$	−6540.15	−0.256632
$867$	7976.96i	0.312470i
$868$	38757.6i	1.51558i
$869$	1877.44	0.0732887
$870$	0	0
$871$	6520.99	0.253680
$872$	− 11020.9i	− 0.428000i
$873$	40393.7i	1.56600i
$874$	−5067.04	−0.196104
$875$	0	0
$876$	4342.78	0.167499
$877$	22566.1i	0.868875i	0.900702	+	0.434438i	$0.143053\pi$
$877$	22566.1i	0.868875i	−0.900702	+	0.434438i	$0.856947\pi$
$878$	− 13130.8i	− 0.504720i
$879$	−44091.9	−1.69190
$880$	0	0
$881$	15121.2	0.578261	0.289130	−	0.957290i	$-0.406634\pi$
$881$	15121.2	0.578261	0.289130	+	0.957290i	$0.406634\pi$
$882$	101596.i	3.87858i
$883$	48306.7i	1.84105i	0.390678	+	0.920527i	$0.372240\pi$
$883$	48306.7i	1.84105i	−0.390678	+	0.920527i	$0.627760\pi$
$884$	−3869.65	−0.147229
$885$	0	0
$886$	−17619.5	−0.668101
$887$	15632.8i	0.591767i	0.955224	+	0.295883i	$0.0956140\pi$
$887$	15632.8i	0.591767i	−0.955224	+	0.295883i	$0.904386\pi$
$888$	4875.16i	0.184234i
$889$	−69180.0	−2.60993
$890$	0	0
$891$	3171.58	0.119250
$892$	− 9198.55i	− 0.345280i
$893$	− 15724.3i	− 0.589241i
$894$	−31681.2	−1.18521
$895$	0	0
$896$	−4240.48	−0.158108
$897$	3378.34i	0.125752i
$898$	17954.8i	0.667215i
$899$	−56149.6	−2.08309
$900$	0	0
$901$	−37709.9	−1.39434
$902$	946.454i	0.0349373i
$903$	− 29085.1i	− 1.07186i
$904$	7914.36	0.291181
$905$	0	0
$906$	−21956.5	−0.805139
$907$	− 15321.2i	− 0.560897i	−0.959869	−	0.280449i	$-0.909517\pi$
$907$	− 15321.2i	− 0.560897i	0.959869	−	0.280449i	$-0.0904832\pi$
$908$	13016.7i	0.475744i
$909$	−56904.1	−2.07634
$910$	0	0
$911$	27566.7	1.00255	0.501275	−	0.865288i	$-0.332864\pi$
$911$	27566.7	1.00255	0.501275	+	0.865288i	$0.332864\pi$
$912$	17117.1i	0.621495i
$913$	− 151.839i	− 0.00550398i
$914$	−1712.39	−0.0619702
$915$	0	0
$916$	−19739.2	−0.712010
$917$	71337.5i	2.56900i
$918$	50103.7i	1.80138i
$919$	−6372.54	−0.228738	−0.114369	−	0.993438i	$-0.536485\pi$
$919$	−6372.54	−0.228738	−0.114369	+	0.993438i	$0.536485\pi$
$920$	0	0
$921$	−28229.6	−1.00999
$922$	735.512i	0.0262720i
$923$	− 6861.04i	− 0.244674i
$924$	−2055.40	−0.0731794
$925$	0	0
$926$	−14660.9	−0.520290
$927$	− 74275.0i	− 2.63162i
$928$	− 6143.34i	− 0.217311i
$929$	−39918.6	−1.40978	−0.704890	−	0.709316i	$-0.749004\pi$
$929$	−39918.6	−1.40978	−0.704890	+	0.709316i	$0.749004\pi$
$930$	0	0
$931$	−83112.0	−2.92576
$932$	4777.91i	0.167925i
$933$	− 79764.5i	− 2.79890i
$934$	−23006.2	−0.805980
$935$	0	0
$936$	8145.72	0.284457
$937$	11758.4i	0.409957i	0.978766	+	0.204979i	$0.0657125\pi$
$937$	11758.4i	0.409957i	−0.978766	+	0.204979i	$0.934288\pi$
$938$	28568.5i	0.994449i
$939$	1953.14	0.0678788
$940$	0	0
$941$	−26312.9	−0.911559	−0.455779	−	0.890093i	$-0.650639\pi$
$941$	−26312.9	−0.911559	−0.455779	+	0.890093i	$0.650639\pi$
$942$	7985.62i	0.276206i
$943$	6815.22i	0.235349i
$944$	2811.03	0.0969188
$945$	0	0
$946$	288.734	0.00992343
$947$	8785.42i	0.301466i	0.988575	+	0.150733i	$0.0481633\pi$
$947$	8785.42i	0.301466i	−0.988575	+	0.150733i	$0.951837\pi$
$948$	− 45669.1i	− 1.56463i
$949$	1690.66	0.0578305
$950$	0	0
$951$	−24719.2	−0.842877
$952$	− 16952.9i	− 0.577151i
$953$	11133.8i	0.378447i	0.981934	+	0.189223i	$0.0605970\pi$
$953$	11133.8i	0.378447i	−0.981934	+	0.189223i	$0.939403\pi$
$954$	79380.5	2.69396
$955$	0	0
$956$	6210.42	0.210104
$957$	− 2977.73i	− 0.100581i
$958$	− 17160.9i	− 0.578752i
$959$	43336.6	1.45924
$960$	0	0
$961$	55752.0	1.87144
$962$	1897.92i	0.0636084i
$963$	− 74250.3i	− 2.48461i
$964$	16712.8	0.558384
$965$	0	0
$966$	−14800.5	−0.492959
$967$	30546.1i	1.01582i	0.861410	+	0.507910i	$0.169582\pi$
$967$	30546.1i	1.01582i	−0.861410	+	0.507910i	$0.830418\pi$
$968$	10627.6i	0.352876i
$969$	−68432.1	−2.26868
$970$	0	0
$971$	−26757.1	−0.884322	−0.442161	−	0.896936i	$-0.645788\pi$
$971$	−26757.1	−0.884322	−0.442161	+	0.896936i	$0.645788\pi$
$972$	− 34851.8i	− 1.15007i
$973$	− 70314.1i	− 2.31672i
$974$	31653.6	1.04132
$975$	0	0
$976$	−12953.8	−0.424839
$977$	36550.8i	1.19689i	0.801163	+	0.598447i	$0.204215\pi$
$977$	36550.8i	1.19689i	−0.801163	+	0.598447i	$0.795785\pi$
$978$	61203.7i	2.00110i
$979$	−1336.25	−0.0436228
$980$	0	0
$981$	−92748.4	−3.01858
$982$	− 12438.2i	− 0.404195i
$983$	7459.39i	0.242032i	0.992651	+	0.121016i	$0.0386152\pi$
$983$	7459.39i	0.242032i	−0.992651	+	0.121016i	$0.961385\pi$
$984$	23022.7	0.745870
$985$	0	0
$986$	24560.3	0.793266
$987$	− 45929.7i	− 1.48121i
$988$	6663.74i	0.214577i
$989$	2079.12	0.0668474
$990$	0	0
$991$	−56932.3	−1.82494	−0.912470	−	0.409145i	$-0.865827\pi$
$991$	−56932.3	−1.82494	−0.912470	+	0.409145i	$0.865827\pi$
$992$	9359.27i	0.299554i
$993$	71195.7i	2.27526i
$994$	30058.2	0.959142
$995$	0	0
$996$	−3693.51	−0.117503
$997$	34243.6i	1.08777i	0.839160	+	0.543885i	$0.183047\pi$
$997$	34243.6i	1.08777i	−0.839160	+	0.543885i	$0.816953\pi$
$998$	− 25640.2i	− 0.813252i
$999$	24574.0	0.778265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.b.s.599.12		12
5.2	odd	4		1150.4.a.w.1.6	✓	6
5.3	odd	4		1150.4.a.x.1.1	yes	6
5.4	even	2	inner	1150.4.b.s.599.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.w.1.6	✓	6	5.2	odd	4
1150.4.a.x.1.1	yes	6	5.3	odd	4
1150.4.b.s.599.1		12	5.4	even	2	inner
1150.4.b.s.599.12		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +9.71212i q^{3} -4.00000 q^{4} -19.4242 q^{6} -33.1288i q^{7} -8.00000i q^{8} -67.3252 q^{9} +1.59705 q^{11} -38.8485i q^{12} -15.1238i q^{13} +66.2575 q^{14} +16.0000 q^{16} +63.9661i q^{17} -134.650i q^{18} +110.153 q^{19} +321.750 q^{21} +3.19409i q^{22} +23.0000i q^{23} +77.6969 q^{24} +30.2477 q^{26} -391.643i q^{27} +132.515i q^{28} -191.979 q^{29} +292.477 q^{31} +32.0000i q^{32} +15.5107i q^{33} -127.932 q^{34} +269.301 q^{36} +62.7459i q^{37} +220.306i q^{38} +146.885 q^{39} +296.314 q^{41} +643.501i q^{42} -90.3963i q^{43} -6.38819 q^{44} -46.0000 q^{46} -142.749i q^{47} +155.394i q^{48} -754.515 q^{49} -621.246 q^{51} +60.4954i q^{52} +589.530i q^{53} +783.286 q^{54} -265.030 q^{56} +1069.82i q^{57} -383.959i q^{58} +175.690 q^{59} -809.615 q^{61} +584.955i q^{62} +2230.40i q^{63} -64.0000 q^{64} -31.0214 q^{66} +431.173i q^{67} -255.864i q^{68} -223.379 q^{69} +453.657 q^{71} +538.602i q^{72} +111.788i q^{73} -125.492 q^{74} -440.612 q^{76} -52.9082i q^{77} +293.769i q^{78} +1175.57 q^{79} +1985.90 q^{81} +592.628i q^{82} -95.0748i q^{83} -1287.00 q^{84} +180.793 q^{86} -1864.52i q^{87} -12.7764i q^{88} -836.699 q^{89} -501.034 q^{91} -92.0000i q^{92} +2840.57i q^{93} +285.499 q^{94} -310.788 q^{96} -599.979i q^{97} -1509.03i q^{98} -107.522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 48 q^{4} - 20 q^{6} - 122 q^{9} - 98 q^{11} + 168 q^{14} + 192 q^{16} + 458 q^{19} + 184 q^{21} + 80 q^{24} - 64 q^{26} + 364 q^{29} + 228 q^{31} + 700 q^{34} + 488 q^{36} + 286 q^{39} + 486 q^{41}+ \cdots + 284 q^{99}+O(q^{100}) \) 12 * q - 48 * q^4 - 20 * q^6 - 122 * q^9 - 98 * q^11 + 168 * q^14 + 192 * q^16 + 458 * q^19 + 184 * q^21 + 80 * q^24 - 64 * q^26 + 364 * q^29 + 228 * q^31 + 700 * q^34 + 488 * q^36 + 286 * q^39 + 486 * q^41 + 392 * q^44 - 552 * q^46 - 1296 * q^49 - 2062 * q^51 + 1376 * q^54 - 672 * q^56 + 1118 * q^59 - 1376 * q^61 - 768 * q^64 - 2564 * q^66 - 230 * q^69 + 1168 * q^71 + 256 * q^74 - 1832 * q^76 + 2864 * q^79 + 68 * q^81 - 736 * q^84 - 728 * q^86 + 1182 * q^89 - 5728 * q^91 + 1992 * q^94 - 320 * q^96 + 284 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more