LMFDB - Embedded newform 1150.4.b.a.599.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(599,1150)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 46)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.4.b.a.599.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +9.00000i q^{3} -4.00000 q^{4} -18.0000 q^{6} +2.00000i q^{7} -8.00000i q^{8} -54.0000 q^{9} -52.0000 q^{11} -36.0000i q^{12} -43.0000i q^{13} -4.00000 q^{14} +16.0000 q^{16} -50.0000i q^{17} -108.000i q^{18} +74.0000 q^{19} -18.0000 q^{21} -104.000i q^{22} +23.0000i q^{23} +72.0000 q^{24} +86.0000 q^{26} -243.000i q^{27} -8.00000i q^{28} +7.00000 q^{29} -273.000 q^{31} +32.0000i q^{32} -468.000i q^{33} +100.000 q^{34} +216.000 q^{36} -4.00000i q^{37} +148.000i q^{38} +387.000 q^{39} +123.000 q^{41} -36.0000i q^{42} +152.000i q^{43} +208.000 q^{44} -46.0000 q^{46} +75.0000i q^{47} +144.000i q^{48} +339.000 q^{49} +450.000 q^{51} +172.000i q^{52} -86.0000i q^{53} +486.000 q^{54} +16.0000 q^{56} +666.000i q^{57} +14.0000i q^{58} +444.000 q^{59} +262.000 q^{61} -546.000i q^{62} -108.000i q^{63} -64.0000 q^{64} +936.000 q^{66} +764.000i q^{67} +200.000i q^{68} -207.000 q^{69} -21.0000 q^{71} +432.000i q^{72} -681.000i q^{73} +8.00000 q^{74} -296.000 q^{76} -104.000i q^{77} +774.000i q^{78} -426.000 q^{79} +729.000 q^{81} +246.000i q^{82} -902.000i q^{83} +72.0000 q^{84} -304.000 q^{86} +63.0000i q^{87} +416.000i q^{88} +1272.00 q^{89} +86.0000 q^{91} -92.0000i q^{92} -2457.00i q^{93} -150.000 q^{94} -288.000 q^{96} -342.000i q^{97} +678.000i q^{98} +2808.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 36 q^{6} - 108 q^{9} - 104 q^{11} - 8 q^{14} + 32 q^{16} + 148 q^{19} - 36 q^{21} + 144 q^{24} + 172 q^{26} + 14 q^{29} - 546 q^{31} + 200 q^{34} + 432 q^{36} + 774 q^{39} + 246 q^{41} + 416 q^{44}+ \cdots + 5616 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 36 * q^6 - 108 * q^9 - 104 * q^11 - 8 * q^14 + 32 * q^16 + 148 * q^19 - 36 * q^21 + 144 * q^24 + 172 * q^26 + 14 * q^29 - 546 * q^31 + 200 * q^34 + 432 * q^36 + 774 * q^39 + 246 * q^41 + 416 * q^44 - 92 * q^46 + 678 * q^49 + 900 * q^51 + 972 * q^54 + 32 * q^56 + 888 * q^59 + 524 * q^61 - 128 * q^64 + 1872 * q^66 - 414 * q^69 - 42 * q^71 + 16 * q^74 - 592 * q^76 - 852 * q^79 + 1458 * q^81 + 144 * q^84 - 608 * q^86 + 2544 * q^89 + 172 * q^91 - 300 * q^94 - 576 * q^96 + 5616 * 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
$3$	9.00000i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$3$	9.00000i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−18.0000	−1.22474
$7$	2.00000i	0.107990i	0.998541	+	0.0539949i	$0.0171955\pi$
$7$	2.00000i	0.107990i	−0.998541	+	0.0539949i	$0.982805\pi$
$8$	− 8.00000i	− 0.353553i
$9$	−54.0000	−2.00000
$10$	0	0
$11$	−52.0000	−1.42533	−0.712663	−	0.701506i	$-0.752511\pi$
$11$	−52.0000	−1.42533	−0.712663	+	0.701506i	$0.752511\pi$
$12$	− 36.0000i	− 0.866025i
$13$	− 43.0000i	− 0.917389i	−0.888594	−	0.458694i	$-0.848317\pi$
$13$	− 43.0000i	− 0.917389i	0.888594	−	0.458694i	$-0.151683\pi$
$14$	−4.00000	−0.0763604
$15$	0	0
$16$	16.0000	0.250000
$17$	− 50.0000i	− 0.713340i	−0.934230	−	0.356670i	$-0.883912\pi$
$17$	− 50.0000i	− 0.713340i	0.934230	−	0.356670i	$-0.116088\pi$
$18$	− 108.000i	− 1.41421i
$19$	74.0000	0.893514	0.446757	−	0.894655i	$-0.352579\pi$
$19$	74.0000	0.893514	0.446757	+	0.894655i	$0.352579\pi$
$20$	0	0
$21$	−18.0000	−0.187044
$22$	− 104.000i	− 1.00786i
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	72.0000	0.612372
$25$	0	0
$26$	86.0000	0.648692
$27$	− 243.000i	− 1.73205i
$28$	− 8.00000i	− 0.0539949i
$29$	7.00000	0.0448230	0.0224115	−	0.999749i	$-0.492866\pi$
$29$	7.00000	0.0448230	0.0224115	+	0.999749i	$0.492866\pi$
$30$	0	0
$31$	−273.000	−1.58169	−0.790843	−	0.612019i	$-0.790357\pi$
$31$	−273.000	−1.58169	−0.790843	+	0.612019i	$0.790357\pi$
$32$	32.0000i	0.176777i
$33$	− 468.000i	− 2.46874i
$34$	100.000	0.504408
$35$	0	0
$36$	216.000	1.00000
$37$	− 4.00000i	− 0.0177729i	−0.999961	−	0.00888643i	$-0.997171\pi$
$37$	− 4.00000i	− 0.0177729i	0.999961	−	0.00888643i	$-0.00282868\pi$
$38$	148.000i	0.631810i
$39$	387.000	1.58896
$40$	0	0
$41$	123.000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	123.000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	− 36.0000i	− 0.132260i
$43$	152.000i	0.539065i	0.962991	+	0.269532i	$0.0868691\pi$
$43$	152.000i	0.539065i	−0.962991	+	0.269532i	$0.913131\pi$
$44$	208.000	0.712663
$45$	0	0
$46$	−46.0000	−0.147442
$47$	75.0000i	0.232763i	0.993205	+	0.116382i	$0.0371296\pi$
$47$	75.0000i	0.232763i	−0.993205	+	0.116382i	$0.962870\pi$
$48$	144.000i	0.433013i
$49$	339.000	0.988338
$50$	0	0
$51$	450.000	1.23554
$52$	172.000i	0.458694i
$53$	− 86.0000i	− 0.222887i	−0.993771	−	0.111443i	$-0.964453\pi$
$53$	− 86.0000i	− 0.222887i	0.993771	−	0.111443i	$-0.0355474\pi$
$54$	486.000	1.22474
$55$	0	0
$56$	16.0000	0.0381802
$57$	666.000i	1.54761i
$58$	14.0000i	0.0316947i
$59$	444.000	0.979727	0.489863	−	0.871799i	$-0.337047\pi$
$59$	444.000	0.979727	0.489863	+	0.871799i	$0.337047\pi$
$60$	0	0
$61$	262.000	0.549929	0.274964	−	0.961454i	$-0.411334\pi$
$61$	262.000	0.549929	0.274964	+	0.961454i	$0.411334\pi$
$62$	− 546.000i	− 1.11842i
$63$	− 108.000i	− 0.215980i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	936.000	1.74566
$67$	764.000i	1.39310i	0.717510	+	0.696548i	$0.245282\pi$
$67$	764.000i	1.39310i	−0.717510	+	0.696548i	$0.754718\pi$
$68$	200.000i	0.356670i
$69$	−207.000	−0.361158
$70$	0	0
$71$	−21.0000	−0.0351020	−0.0175510	−	0.999846i	$-0.505587\pi$
$71$	−21.0000	−0.0351020	−0.0175510	+	0.999846i	$0.505587\pi$
$72$	432.000i	0.707107i
$73$	− 681.000i	− 1.09185i	−0.837834	−	0.545925i	$-0.816178\pi$
$73$	− 681.000i	− 1.09185i	0.837834	−	0.545925i	$-0.183822\pi$
$74$	8.00000	0.0125673
$75$	0	0
$76$	−296.000	−0.446757
$77$	− 104.000i	− 0.153921i
$78$	774.000i	1.12357i
$79$	−426.000	−0.606693	−0.303346	−	0.952880i	$-0.598104\pi$
$79$	−426.000	−0.606693	−0.303346	+	0.952880i	$0.598104\pi$
$80$	0	0
$81$	729.000	1.00000
$82$	246.000i	0.331295i
$83$	− 902.000i	− 1.19286i	−0.802665	−	0.596430i	$-0.796585\pi$
$83$	− 902.000i	− 1.19286i	0.802665	−	0.596430i	$-0.203415\pi$
$84$	72.0000	0.0935220
$85$	0	0
$86$	−304.000	−0.381176
$87$	63.0000i	0.0776357i
$88$	416.000i	0.503929i
$89$	1272.00	1.51496	0.757482	−	0.652856i	$-0.226430\pi$
$89$	1272.00	1.51496	0.757482	+	0.652856i	$0.226430\pi$
$90$	0	0
$91$	86.0000	0.0990687
$92$	− 92.0000i	− 0.104257i
$93$	− 2457.00i	− 2.73956i
$94$	−150.000	−0.164588
$95$	0	0
$96$	−288.000	−0.306186
$97$	− 342.000i	− 0.357988i	−0.983850	−	0.178994i	$-0.942716\pi$
$97$	− 342.000i	− 0.357988i	0.983850	−	0.178994i	$-0.0572843\pi$
$98$	678.000i	0.698861i
$99$	2808.00	2.85065
$100$	0	0
$101$	−1426.00	−1.40487	−0.702437	−	0.711746i	$-0.747905\pi$
$101$	−1426.00	−1.40487	−0.702437	+	0.711746i	$0.747905\pi$
$102$	900.000i	0.873660i
$103$	1190.00i	1.13839i	0.822203	+	0.569195i	$0.192745\pi$
$103$	1190.00i	1.13839i	−0.822203	+	0.569195i	$0.807255\pi$
$104$	−344.000	−0.324346
$105$	0	0
$106$	172.000	0.157605
$107$	− 1210.00i	− 1.09323i	−0.837386	−	0.546613i	$-0.815917\pi$
$107$	− 1210.00i	− 1.09323i	0.837386	−	0.546613i	$-0.184083\pi$
$108$	972.000i	0.866025i
$109$	1680.00	1.47628	0.738141	−	0.674646i	$-0.235704\pi$
$109$	1680.00	1.47628	0.738141	+	0.674646i	$0.235704\pi$
$110$	0	0
$111$	36.0000	0.0307835
$112$	32.0000i	0.0269975i
$113$	− 1030.00i	− 0.857471i	−0.903430	−	0.428736i	$-0.858959\pi$
$113$	− 1030.00i	− 0.857471i	0.903430	−	0.428736i	$-0.141041\pi$
$114$	−1332.00	−1.09433
$115$	0	0
$116$	−28.0000	−0.0224115
$117$	2322.00i	1.83478i
$118$	888.000i	0.692771i
$119$	100.000	0.0770335
$120$	0	0
$121$	1373.00	1.03156
$122$	524.000i	0.388858i
$123$	1107.00i	0.811503i
$124$	1092.00	0.790843
$125$	0	0
$126$	216.000	0.152721
$127$	− 2279.00i	− 1.59235i	−0.605066	−	0.796175i	$-0.706853\pi$
$127$	− 2279.00i	− 1.59235i	0.605066	−	0.796175i	$-0.293147\pi$
$128$	− 128.000i	− 0.0883883i
$129$	−1368.00	−0.933687
$130$	0	0
$131$	987.000	0.658279	0.329140	−	0.944281i	$-0.393241\pi$
$131$	987.000	0.658279	0.329140	+	0.944281i	$0.393241\pi$
$132$	1872.00i	1.23437i
$133$	148.000i	0.0964904i
$134$	−1528.00	−0.985068
$135$	0	0
$136$	−400.000	−0.252204
$137$	− 1644.00i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 1644.00i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	− 414.000i	− 0.255377i
$139$	−2189.00	−1.33575	−0.667873	−	0.744276i	$-0.732795\pi$
$139$	−2189.00	−1.33575	−0.667873	+	0.744276i	$0.732795\pi$
$140$	0	0
$141$	−675.000	−0.403158
$142$	− 42.0000i	− 0.0248209i
$143$	2236.00i	1.30758i
$144$	−864.000	−0.500000
$145$	0	0
$146$	1362.00	0.772054
$147$	3051.00i	1.71185i
$148$	16.0000i	0.00888643i
$149$	946.000	0.520130	0.260065	−	0.965591i	$-0.416256\pi$
$149$	946.000	0.520130	0.260065	+	0.965591i	$0.416256\pi$
$150$	0	0
$151$	−365.000	−0.196710	−0.0983552	−	0.995151i	$-0.531358\pi$
$151$	−365.000	−0.196710	−0.0983552	+	0.995151i	$0.531358\pi$
$152$	− 592.000i	− 0.315905i
$153$	2700.00i	1.42668i
$154$	208.000	0.108838
$155$	0	0
$156$	−1548.00	−0.794482
$157$	− 108.000i	− 0.0549002i	−0.999623	−	0.0274501i	$-0.991261\pi$
$157$	− 108.000i	− 0.0549002i	0.999623	−	0.0274501i	$-0.00873874\pi$
$158$	− 852.000i	− 0.428997i
$159$	774.000	0.386052
$160$	0	0
$161$	−46.0000	−0.0225174
$162$	1458.00i	0.707107i
$163$	1415.00i	0.679947i	0.940435	+	0.339973i	$0.110418\pi$
$163$	1415.00i	0.679947i	−0.940435	+	0.339973i	$0.889582\pi$
$164$	−492.000	−0.234261
$165$	0	0
$166$	1804.00	0.843479
$167$	1756.00i	0.813673i	0.913501	+	0.406836i	$0.133368\pi$
$167$	1756.00i	0.813673i	−0.913501	+	0.406836i	$0.866632\pi$
$168$	144.000i	0.0661300i
$169$	348.000	0.158398
$170$	0	0
$171$	−3996.00	−1.78703
$172$	− 608.000i	− 0.269532i
$173$	− 2358.00i	− 1.03627i	−0.855298	−	0.518137i	$-0.826626\pi$
$173$	− 2358.00i	− 1.03627i	0.855298	−	0.518137i	$-0.173374\pi$
$174$	−126.000	−0.0548968
$175$	0	0
$176$	−832.000	−0.356332
$177$	3996.00i	1.69694i
$178$	2544.00i	1.07124i
$179$	−1073.00	−0.448043	−0.224022	−	0.974584i	$-0.571919\pi$
$179$	−1073.00	−0.448043	−0.224022	+	0.974584i	$0.571919\pi$
$180$	0	0
$181$	2868.00	1.17777	0.588886	−	0.808216i	$-0.299567\pi$
$181$	2868.00	1.17777	0.588886	+	0.808216i	$0.299567\pi$
$182$	172.000i	0.0700521i
$183$	2358.00i	0.952505i
$184$	184.000	0.0737210
$185$	0	0
$186$	4914.00	1.93716
$187$	2600.00i	1.01674i
$188$	− 300.000i	− 0.116382i
$189$	486.000	0.187044
$190$	0	0
$191$	332.000	0.125773	0.0628866	−	0.998021i	$-0.479969\pi$
$191$	332.000	0.125773	0.0628866	+	0.998021i	$0.479969\pi$
$192$	− 576.000i	− 0.216506i
$193$	2143.00i	0.799257i	0.916677	+	0.399628i	$0.130861\pi$
$193$	2143.00i	0.799257i	−0.916677	+	0.399628i	$0.869139\pi$
$194$	684.000	0.253136
$195$	0	0
$196$	−1356.00	−0.494169
$197$	− 2739.00i	− 0.990587i	−0.868726	−	0.495294i	$-0.835060\pi$
$197$	− 2739.00i	− 0.990587i	0.868726	−	0.495294i	$-0.164940\pi$
$198$	5616.00i	2.01572i
$199$	−752.000	−0.267879	−0.133939	−	0.990990i	$-0.542763\pi$
$199$	−752.000	−0.267879	−0.133939	+	0.990990i	$0.542763\pi$
$200$	0	0
$201$	−6876.00	−2.41291
$202$	− 2852.00i	− 0.993396i
$203$	14.0000i	0.00484043i
$204$	−1800.00	−0.617771
$205$	0	0
$206$	−2380.00	−0.804963
$207$	− 1242.00i	− 0.417029i
$208$	− 688.000i	− 0.229347i
$209$	−3848.00	−1.27355
$210$	0	0
$211$	−1016.00	−0.331490	−0.165745	−	0.986169i	$-0.553003\pi$
$211$	−1016.00	−0.331490	−0.165745	+	0.986169i	$0.553003\pi$
$212$	344.000i	0.111443i
$213$	− 189.000i	− 0.0607984i
$214$	2420.00	0.773027
$215$	0	0
$216$	−1944.00	−0.612372
$217$	− 546.000i	− 0.170806i
$218$	3360.00i	1.04389i
$219$	6129.00	1.89114
$220$	0	0
$221$	−2150.00	−0.654410
$222$	72.0000i	0.0217672i
$223$	1120.00i	0.336326i	0.985759	+	0.168163i	$0.0537835\pi$
$223$	1120.00i	0.336326i	−0.985759	+	0.168163i	$0.946216\pi$
$224$	−64.0000	−0.0190901
$225$	0	0
$226$	2060.00	0.606324
$227$	2706.00i	0.791205i	0.918422	+	0.395602i	$0.129464\pi$
$227$	2706.00i	0.791205i	−0.918422	+	0.395602i	$0.870536\pi$
$228$	− 2664.00i	− 0.773806i
$229$	−6140.00	−1.77180	−0.885901	−	0.463875i	$-0.846459\pi$
$229$	−6140.00	−1.77180	−0.885901	+	0.463875i	$0.846459\pi$
$230$	0	0
$231$	936.000	0.266599
$232$	− 56.0000i	− 0.0158473i
$233$	− 6567.00i	− 1.84643i	−0.384282	−	0.923216i	$-0.625551\pi$
$233$	− 6567.00i	− 1.84643i	0.384282	−	0.923216i	$-0.374449\pi$
$234$	−4644.00	−1.29738
$235$	0	0
$236$	−1776.00	−0.489863
$237$	− 3834.00i	− 1.05082i
$238$	200.000i	0.0544709i
$239$	729.000	0.197302	0.0986508	−	0.995122i	$-0.468547\pi$
$239$	729.000	0.197302	0.0986508	+	0.995122i	$0.468547\pi$
$240$	0	0
$241$	−2912.00	−0.778334	−0.389167	−	0.921167i	$-0.627237\pi$
$241$	−2912.00	−0.778334	−0.389167	+	0.921167i	$0.627237\pi$
$242$	2746.00i	0.729420i
$243$	0	0
$244$	−1048.00	−0.274964
$245$	0	0
$246$	−2214.00	−0.573819
$247$	− 3182.00i	− 0.819700i
$248$	2184.00i	0.559210i
$249$	8118.00	2.06609
$250$	0	0
$251$	398.000	0.100086	0.0500429	−	0.998747i	$-0.484064\pi$
$251$	398.000	0.100086	0.0500429	+	0.998747i	$0.484064\pi$
$252$	432.000i	0.107990i
$253$	− 1196.00i	− 0.297201i
$254$	4558.00	1.12596
$255$	0	0
$256$	256.000	0.0625000
$257$	8131.00i	1.97353i	0.162149	+	0.986766i	$0.448157\pi$
$257$	8131.00i	1.97353i	−0.162149	+	0.986766i	$0.551843\pi$
$258$	− 2736.00i	− 0.660217i
$259$	8.00000	0.00191929
$260$	0	0
$261$	−378.000	−0.0896460
$262$	1974.00i	0.465474i
$263$	1978.00i	0.463759i	0.972744	+	0.231880i	$0.0744876\pi$
$263$	1978.00i	0.463759i	−0.972744	+	0.231880i	$0.925512\pi$
$264$	−3744.00	−0.872831
$265$	0	0
$266$	−296.000	−0.0682290
$267$	11448.0i	2.62399i
$268$	− 3056.00i	− 0.696548i
$269$	8459.00	1.91730	0.958651	−	0.284584i	$-0.0918554\pi$
$269$	8459.00	1.91730	0.958651	+	0.284584i	$0.0918554\pi$
$270$	0	0
$271$	−7240.00	−1.62287	−0.811437	−	0.584440i	$-0.801314\pi$
$271$	−7240.00	−1.62287	−0.811437	+	0.584440i	$0.801314\pi$
$272$	− 800.000i	− 0.178335i
$273$	774.000i	0.171592i
$274$	3288.00	0.724947
$275$	0	0
$276$	828.000	0.180579
$277$	− 1319.00i	− 0.286105i	−0.989715	−	0.143052i	$-0.954308\pi$
$277$	− 1319.00i	− 0.286105i	0.989715	−	0.143052i	$-0.0456917\pi$
$278$	− 4378.00i	− 0.944514i
$279$	14742.0	3.16337
$280$	0	0
$281$	1770.00	0.375763	0.187881	−	0.982192i	$-0.439838\pi$
$281$	1770.00	0.375763	0.187881	+	0.982192i	$0.439838\pi$
$282$	− 1350.00i	− 0.285076i
$283$	− 4144.00i	− 0.870443i	−0.900324	−	0.435221i	$-0.856670\pi$
$283$	− 4144.00i	− 0.870443i	0.900324	−	0.435221i	$-0.143330\pi$
$284$	84.0000	0.0175510
$285$	0	0
$286$	−4472.00	−0.924598
$287$	246.000i	0.0505955i
$288$	− 1728.00i	− 0.353553i
$289$	2413.00	0.491146
$290$	0	0
$291$	3078.00	0.620053
$292$	2724.00i	0.545925i
$293$	6812.00i	1.35823i	0.734032	+	0.679115i	$0.237636\pi$
$293$	6812.00i	1.35823i	−0.734032	+	0.679115i	$0.762364\pi$
$294$	−6102.00	−1.21046
$295$	0	0
$296$	−32.0000	−0.00628366
$297$	12636.0i	2.46874i
$298$	1892.00i	0.367787i
$299$	989.000	0.191289
$300$	0	0
$301$	−304.000	−0.0582135
$302$	− 730.000i	− 0.139095i
$303$	− 12834.0i	− 2.43331i
$304$	1184.00	0.223378
$305$	0	0
$306$	−5400.00	−1.00882
$307$	− 5692.00i	− 1.05817i	−0.848567	−	0.529087i	$-0.822534\pi$
$307$	− 5692.00i	− 1.05817i	0.848567	−	0.529087i	$-0.177466\pi$
$308$	416.000i	0.0769604i
$309$	−10710.0	−1.97175
$310$	0	0
$311$	5267.00	0.960335	0.480167	−	0.877177i	$-0.340576\pi$
$311$	5267.00	0.960335	0.480167	+	0.877177i	$0.340576\pi$
$312$	− 3096.00i	− 0.561784i
$313$	− 6340.00i	− 1.14491i	−0.819935	−	0.572457i	$-0.805990\pi$
$313$	− 6340.00i	− 1.14491i	0.819935	−	0.572457i	$-0.194010\pi$
$314$	216.000	0.0388203
$315$	0	0
$316$	1704.00	0.303346
$317$	− 8794.00i	− 1.55811i	−0.626957	−	0.779054i	$-0.715700\pi$
$317$	− 8794.00i	− 1.55811i	0.626957	−	0.779054i	$-0.284300\pi$
$318$	1548.00i	0.272980i
$319$	−364.000	−0.0638874
$320$	0	0
$321$	10890.0	1.89352
$322$	− 92.0000i	− 0.0159222i
$323$	− 3700.00i	− 0.637379i
$324$	−2916.00	−0.500000
$325$	0	0
$326$	−2830.00	−0.480795
$327$	15120.0i	2.55700i
$328$	− 984.000i	− 0.165647i
$329$	−150.000	−0.0251361
$330$	0	0
$331$	8225.00	1.36582	0.682911	−	0.730502i	$-0.260714\pi$
$331$	8225.00	1.36582	0.682911	+	0.730502i	$0.260714\pi$
$332$	3608.00i	0.596430i
$333$	216.000i	0.0355457i
$334$	−3512.00	−0.575354
$335$	0	0
$336$	−288.000	−0.0467610
$337$	− 2576.00i	− 0.416391i	−0.978087	−	0.208195i	$-0.933241\pi$
$337$	− 2576.00i	− 0.416391i	0.978087	−	0.208195i	$-0.0667590\pi$
$338$	696.000i	0.112004i
$339$	9270.00	1.48518
$340$	0	0
$341$	14196.0	2.25442
$342$	− 7992.00i	− 1.26362i
$343$	1364.00i	0.214720i
$344$	1216.00	0.190588
$345$	0	0
$346$	4716.00	0.732756
$347$	596.000i	0.0922045i	0.998937	+	0.0461022i	$0.0146800\pi$
$347$	596.000i	0.0922045i	−0.998937	+	0.0461022i	$0.985320\pi$
$348$	− 252.000i	− 0.0388179i
$349$	9271.00	1.42196	0.710982	−	0.703210i	$-0.248251\pi$
$349$	9271.00	1.42196	0.710982	+	0.703210i	$0.248251\pi$
$350$	0	0
$351$	−10449.0	−1.58896
$352$	− 1664.00i	− 0.251964i
$353$	− 8141.00i	− 1.22748i	−0.789507	−	0.613742i	$-0.789664\pi$
$353$	− 8141.00i	− 1.22748i	0.789507	−	0.613742i	$-0.210336\pi$
$354$	−7992.00	−1.19992
$355$	0	0
$356$	−5088.00	−0.757482
$357$	900.000i	0.133426i
$358$	− 2146.00i	− 0.316815i
$359$	2130.00	0.313140	0.156570	−	0.987667i	$-0.449956\pi$
$359$	2130.00	0.313140	0.156570	+	0.987667i	$0.449956\pi$
$360$	0	0
$361$	−1383.00	−0.201633
$362$	5736.00i	0.832811i
$363$	12357.0i	1.78671i
$364$	−344.000	−0.0495343
$365$	0	0
$366$	−4716.00	−0.673523
$367$	− 2574.00i	− 0.366108i	−0.983103	−	0.183054i	$-0.941402\pi$
$367$	− 2574.00i	− 0.366108i	0.983103	−	0.183054i	$-0.0585984\pi$
$368$	368.000i	0.0521286i
$369$	−6642.00	−0.937043
$370$	0	0
$371$	172.000	0.0240695
$372$	9828.00i	1.36978i
$373$	4504.00i	0.625223i	0.949881	+	0.312612i	$0.101204\pi$
$373$	4504.00i	0.625223i	−0.949881	+	0.312612i	$0.898796\pi$
$374$	−5200.00	−0.718945
$375$	0	0
$376$	600.000	0.0822942
$377$	− 301.000i	− 0.0411201i
$378$	972.000i	0.132260i
$379$	−2740.00	−0.371357	−0.185679	−	0.982611i	$-0.559448\pi$
$379$	−2740.00	−0.371357	−0.185679	+	0.982611i	$0.559448\pi$
$380$	0	0
$381$	20511.0	2.75803
$382$	664.000i	0.0889351i
$383$	− 6948.00i	− 0.926961i	−0.886107	−	0.463481i	$-0.846600\pi$
$383$	− 6948.00i	− 0.926961i	0.886107	−	0.463481i	$-0.153400\pi$
$384$	1152.00	0.153093
$385$	0	0
$386$	−4286.00	−0.565160
$387$	− 8208.00i	− 1.07813i
$388$	1368.00i	0.178994i
$389$	1404.00	0.182996	0.0914982	−	0.995805i	$-0.470834\pi$
$389$	1404.00	0.182996	0.0914982	+	0.995805i	$0.470834\pi$
$390$	0	0
$391$	1150.00	0.148742
$392$	− 2712.00i	− 0.349430i
$393$	8883.00i	1.14017i
$394$	5478.00	0.700451
$395$	0	0
$396$	−11232.0	−1.42533
$397$	− 8641.00i	− 1.09239i	−0.837658	−	0.546196i	$-0.816076\pi$
$397$	− 8641.00i	− 1.09239i	0.837658	−	0.546196i	$-0.183924\pi$
$398$	− 1504.00i	− 0.189419i
$399$	−1332.00	−0.167126
$400$	0	0
$401$	−1140.00	−0.141967	−0.0709836	−	0.997477i	$-0.522614\pi$
$401$	−1140.00	−0.141967	−0.0709836	+	0.997477i	$0.522614\pi$
$402$	− 13752.0i	− 1.70619i
$403$	11739.0i	1.45102i
$404$	5704.00	0.702437
$405$	0	0
$406$	−28.0000	−0.00342270
$407$	208.000i	0.0253321i
$408$	− 3600.00i	− 0.436830i
$409$	−12529.0	−1.51472	−0.757358	−	0.652999i	$-0.773510\pi$
$409$	−12529.0	−1.51472	−0.757358	+	0.652999i	$0.773510\pi$
$410$	0	0
$411$	14796.0	1.77575
$412$	− 4760.00i	− 0.569195i
$413$	888.000i	0.105801i
$414$	2484.00	0.294884
$415$	0	0
$416$	1376.00	0.162173
$417$	− 19701.0i	− 2.31358i
$418$	− 7696.00i	− 0.900535i
$419$	−3252.00	−0.379166	−0.189583	−	0.981865i	$-0.560714\pi$
$419$	−3252.00	−0.379166	−0.189583	+	0.981865i	$0.560714\pi$
$420$	0	0
$421$	2206.00	0.255377	0.127689	−	0.991814i	$-0.459244\pi$
$421$	2206.00	0.255377	0.127689	+	0.991814i	$0.459244\pi$
$422$	− 2032.00i	− 0.234399i
$423$	− 4050.00i	− 0.465527i
$424$	−688.000	−0.0788024
$425$	0	0
$426$	378.000	0.0429910
$427$	524.000i	0.0593867i
$428$	4840.00i	0.546613i
$429$	−20124.0	−2.26479
$430$	0	0
$431$	14316.0	1.59995	0.799974	−	0.600035i	$-0.204847\pi$
$431$	14316.0	1.59995	0.799974	+	0.600035i	$0.204847\pi$
$432$	− 3888.00i	− 0.433013i
$433$	− 7828.00i	− 0.868798i	−0.900720	−	0.434399i	$-0.856961\pi$
$433$	− 7828.00i	− 0.868798i	0.900720	−	0.434399i	$-0.143039\pi$
$434$	1092.00	0.120778
$435$	0	0
$436$	−6720.00	−0.738141
$437$	1702.00i	0.186311i
$438$	12258.0i	1.33724i
$439$	16039.0	1.74374	0.871868	−	0.489742i	$-0.162909\pi$
$439$	16039.0	1.74374	0.871868	+	0.489742i	$0.162909\pi$
$440$	0	0
$441$	−18306.0	−1.97668
$442$	− 4300.00i	− 0.462738i
$443$	− 11747.0i	− 1.25986i	−0.776653	−	0.629929i	$-0.783084\pi$
$443$	− 11747.0i	− 1.25986i	0.776653	−	0.629929i	$-0.216916\pi$
$444$	−144.000	−0.0153918
$445$	0	0
$446$	−2240.00	−0.237819
$447$	8514.00i	0.900891i
$448$	− 128.000i	− 0.0134987i
$449$	2890.00	0.303758	0.151879	−	0.988399i	$-0.451468\pi$
$449$	2890.00	0.303758	0.151879	+	0.988399i	$0.451468\pi$
$450$	0	0
$451$	−6396.00	−0.667796
$452$	4120.00i	0.428736i
$453$	− 3285.00i	− 0.340713i
$454$	−5412.00	−0.559466
$455$	0	0
$456$	5328.00	0.547163
$457$	− 13126.0i	− 1.34356i	−0.740749	−	0.671782i	$-0.765529\pi$
$457$	− 13126.0i	− 1.34356i	0.740749	−	0.671782i	$-0.234471\pi$
$458$	− 12280.0i	− 1.25285i
$459$	−12150.0	−1.23554
$460$	0	0
$461$	14481.0	1.46301	0.731505	−	0.681836i	$-0.238818\pi$
$461$	14481.0	1.46301	0.731505	+	0.681836i	$0.238818\pi$
$462$	1872.00i	0.188514i
$463$	− 5272.00i	− 0.529181i	−0.964361	−	0.264590i	$-0.914763\pi$
$463$	− 5272.00i	− 0.529181i	0.964361	−	0.264590i	$-0.0852367\pi$
$464$	112.000	0.0112058
$465$	0	0
$466$	13134.0	1.30562
$467$	− 13466.0i	− 1.33433i	−0.744910	−	0.667165i	$-0.767508\pi$
$467$	− 13466.0i	− 1.33433i	0.744910	−	0.667165i	$-0.232492\pi$
$468$	− 9288.00i	− 0.917389i
$469$	−1528.00	−0.150440
$470$	0	0
$471$	972.000	0.0950900
$472$	− 3552.00i	− 0.346386i
$473$	− 7904.00i	− 0.768343i
$474$	7668.00	0.743044
$475$	0	0
$476$	−400.000	−0.0385167
$477$	4644.00i	0.445774i
$478$	1458.00i	0.139513i
$479$	4526.00	0.431729	0.215865	−	0.976423i	$-0.430743\pi$
$479$	4526.00	0.431729	0.215865	+	0.976423i	$0.430743\pi$
$480$	0	0
$481$	−172.000	−0.0163046
$482$	− 5824.00i	− 0.550365i
$483$	− 414.000i	− 0.0390014i
$484$	−5492.00	−0.515778
$485$	0	0
$486$	0	0
$487$	− 8795.00i	− 0.818356i	−0.912455	−	0.409178i	$-0.865815\pi$
$487$	− 8795.00i	− 0.818356i	0.912455	−	0.409178i	$-0.134185\pi$
$488$	− 2096.00i	− 0.194429i
$489$	−12735.0	−1.17770
$490$	0	0
$491$	−1275.00	−0.117189	−0.0585946	−	0.998282i	$-0.518662\pi$
$491$	−1275.00	−0.117189	−0.0585946	+	0.998282i	$0.518662\pi$
$492$	− 4428.00i	− 0.405751i
$493$	− 350.000i	− 0.0319741i
$494$	6364.00	0.579615
$495$	0	0
$496$	−4368.00	−0.395421
$497$	− 42.0000i	− 0.00379066i
$498$	16236.0i	1.46095i
$499$	9533.00	0.855222	0.427611	−	0.903963i	$-0.359355\pi$
$499$	9533.00	0.855222	0.427611	+	0.903963i	$0.359355\pi$
$500$	0	0
$501$	−15804.0	−1.40932
$502$	796.000i	0.0707714i
$503$	13398.0i	1.18765i	0.804595	+	0.593824i	$0.202383\pi$
$503$	13398.0i	1.18765i	−0.804595	+	0.593824i	$0.797617\pi$
$504$	−864.000	−0.0763604
$505$	0	0
$506$	2392.00	0.210153
$507$	3132.00i	0.274353i
$508$	9116.00i	0.796175i
$509$	8031.00	0.699347	0.349674	−	0.936872i	$-0.386292\pi$
$509$	8031.00	0.699347	0.349674	+	0.936872i	$0.386292\pi$
$510$	0	0
$511$	1362.00	0.117909
$512$	512.000i	0.0441942i
$513$	− 17982.0i	− 1.54761i
$514$	−16262.0	−1.39550
$515$	0	0
$516$	5472.00	0.466844
$517$	− 3900.00i	− 0.331764i
$518$	16.0000i	0.00135714i
$519$	21222.0	1.79488
$520$	0	0
$521$	−21184.0	−1.78136	−0.890679	−	0.454632i	$-0.849771\pi$
$521$	−21184.0	−1.78136	−0.890679	+	0.454632i	$0.849771\pi$
$522$	− 756.000i	− 0.0633893i
$523$	21706.0i	1.81479i	0.420275	+	0.907397i	$0.361934\pi$
$523$	21706.0i	1.81479i	−0.420275	+	0.907397i	$0.638066\pi$
$524$	−3948.00	−0.329140
$525$	0	0
$526$	−3956.00	−0.327927
$527$	13650.0i	1.12828i
$528$	− 7488.00i	− 0.617184i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	−23976.0	−1.95945
$532$	− 592.000i	− 0.0482452i
$533$	− 5289.00i	− 0.429816i
$534$	−22896.0	−1.85544
$535$	0	0
$536$	6112.00	0.492534
$537$	− 9657.00i	− 0.776034i
$538$	16918.0i	1.35574i
$539$	−17628.0	−1.40870
$540$	0	0
$541$	5781.00	0.459417	0.229709	−	0.973259i	$-0.426223\pi$
$541$	5781.00	0.459417	0.229709	+	0.973259i	$0.426223\pi$
$542$	− 14480.0i	− 1.14754i
$543$	25812.0i	2.03996i
$544$	1600.00	0.126102
$545$	0	0
$546$	−1548.00	−0.121334
$547$	7809.00i	0.610400i	0.952288	+	0.305200i	$0.0987233\pi$
$547$	7809.00i	0.610400i	−0.952288	+	0.305200i	$0.901277\pi$
$548$	6576.00i	0.512615i
$549$	−14148.0	−1.09986
$550$	0	0
$551$	518.000	0.0400500
$552$	1656.00i	0.127688i
$553$	− 852.000i	− 0.0655167i
$554$	2638.00	0.202307
$555$	0	0
$556$	8756.00	0.667873
$557$	− 20240.0i	− 1.53967i	−0.638243	−	0.769835i	$-0.720338\pi$
$557$	− 20240.0i	− 1.53967i	0.638243	−	0.769835i	$-0.279662\pi$
$558$	29484.0i	2.23684i
$559$	6536.00	0.494532
$560$	0	0
$561$	−23400.0	−1.76105
$562$	3540.00i	0.265704i
$563$	− 7612.00i	− 0.569818i	−0.958555	−	0.284909i	$-0.908037\pi$
$563$	− 7612.00i	− 0.569818i	0.958555	−	0.284909i	$-0.0919634\pi$
$564$	2700.00	0.201579
$565$	0	0
$566$	8288.00	0.615496
$567$	1458.00i	0.107990i
$568$	168.000i	0.0124104i
$569$	−19484.0	−1.43552	−0.717761	−	0.696290i	$-0.754833\pi$
$569$	−19484.0	−1.43552	−0.717761	+	0.696290i	$0.754833\pi$
$570$	0	0
$571$	6614.00	0.484741	0.242371	−	0.970184i	$-0.422075\pi$
$571$	6614.00	0.484741	0.242371	+	0.970184i	$0.422075\pi$
$572$	− 8944.00i	− 0.653789i
$573$	2988.00i	0.217846i
$574$	−492.000	−0.0357765
$575$	0	0
$576$	3456.00	0.250000
$577$	− 639.000i	− 0.0461038i	−0.999734	−	0.0230519i	$-0.992662\pi$
$577$	− 639.000i	− 0.0461038i	0.999734	−	0.0230519i	$-0.00733830\pi$
$578$	4826.00i	0.347293i
$579$	−19287.0	−1.38435
$580$	0	0
$581$	1804.00	0.128817
$582$	6156.00i	0.438444i
$583$	4472.00i	0.317687i
$584$	−5448.00	−0.386027
$585$	0	0
$586$	−13624.0	−0.960413
$587$	− 829.000i	− 0.0582904i	−0.999575	−	0.0291452i	$-0.990721\pi$
$587$	− 829.000i	− 0.0582904i	0.999575	−	0.0291452i	$-0.00927853\pi$
$588$	− 12204.0i	− 0.855926i
$589$	−20202.0	−1.41326
$590$	0	0
$591$	24651.0	1.71575
$592$	− 64.0000i	− 0.00444322i
$593$	20610.0i	1.42724i	0.700535	+	0.713618i	$0.252945\pi$
$593$	20610.0i	1.42724i	−0.700535	+	0.713618i	$0.747055\pi$
$594$	−25272.0	−1.74566
$595$	0	0
$596$	−3784.00	−0.260065
$597$	− 6768.00i	− 0.463980i
$598$	1978.00i	0.135262i
$599$	17240.0	1.17597	0.587986	−	0.808871i	$-0.299921\pi$
$599$	17240.0	1.17597	0.587986	+	0.808871i	$0.299921\pi$
$600$	0	0
$601$	−8459.00	−0.574126	−0.287063	−	0.957912i	$-0.592679\pi$
$601$	−8459.00	−0.574126	−0.287063	+	0.957912i	$0.592679\pi$
$602$	− 608.000i	− 0.0411632i
$603$	− 41256.0i	− 2.78619i
$604$	1460.00	0.0983552
$605$	0	0
$606$	25668.0	1.72061
$607$	17840.0i	1.19292i	0.802642	+	0.596461i	$0.203427\pi$
$607$	17840.0i	1.19292i	−0.802642	+	0.596461i	$0.796573\pi$
$608$	2368.00i	0.157952i
$609$	−126.000	−0.00838387
$610$	0	0
$611$	3225.00	0.213534
$612$	− 10800.0i	− 0.713340i
$613$	− 2534.00i	− 0.166961i	−0.996509	−	0.0834807i	$-0.973396\pi$
$613$	− 2534.00i	− 0.166961i	0.996509	−	0.0834807i	$-0.0266037\pi$
$614$	11384.0	0.748242
$615$	0	0
$616$	−832.000	−0.0544192
$617$	− 5610.00i	− 0.366046i	−0.983109	−	0.183023i	$-0.941412\pi$
$617$	− 5610.00i	− 0.366046i	0.983109	−	0.183023i	$-0.0585882\pi$
$618$	− 21420.0i	− 1.39424i
$619$	11948.0	0.775817	0.387908	−	0.921698i	$-0.373198\pi$
$619$	11948.0	0.775817	0.387908	+	0.921698i	$0.373198\pi$
$620$	0	0
$621$	5589.00	0.361158
$622$	10534.0i	0.679059i
$623$	2544.00i	0.163601i
$624$	6192.00	0.397241
$625$	0	0
$626$	12680.0	0.809576
$627$	− 34632.0i	− 2.20585i
$628$	432.000i	0.0274501i
$629$	−200.000	−0.0126781
$630$	0	0
$631$	−7840.00	−0.494620	−0.247310	−	0.968936i	$-0.579547\pi$
$631$	−7840.00	−0.494620	−0.247310	+	0.968936i	$0.579547\pi$
$632$	3408.00i	0.214498i
$633$	− 9144.00i	− 0.574157i
$634$	17588.0	1.10175
$635$	0	0
$636$	−3096.00	−0.193026
$637$	− 14577.0i	− 0.906690i
$638$	− 728.000i	− 0.0451752i
$639$	1134.00	0.0702040
$640$	0	0
$641$	−2320.00	−0.142956	−0.0714778	−	0.997442i	$-0.522771\pi$
$641$	−2320.00	−0.142956	−0.0714778	+	0.997442i	$0.522771\pi$
$642$	21780.0i	1.33892i
$643$	− 1864.00i	− 0.114322i	−0.998365	−	0.0571610i	$-0.981795\pi$
$643$	− 1864.00i	− 0.114322i	0.998365	−	0.0571610i	$-0.0182048\pi$
$644$	184.000	0.0112587
$645$	0	0
$646$	7400.00	0.450695
$647$	11939.0i	0.725457i	0.931895	+	0.362728i	$0.118155\pi$
$647$	11939.0i	0.725457i	−0.931895	+	0.362728i	$0.881845\pi$
$648$	− 5832.00i	− 0.353553i
$649$	−23088.0	−1.39643
$650$	0	0
$651$	4914.00	0.295845
$652$	− 5660.00i	− 0.339973i
$653$	− 10503.0i	− 0.629424i	−0.949187	−	0.314712i	$-0.898092\pi$
$653$	− 10503.0i	− 0.629424i	0.949187	−	0.314712i	$-0.101908\pi$
$654$	−30240.0	−1.80807
$655$	0	0
$656$	1968.00	0.117130
$657$	36774.0i	2.18370i
$658$	− 300.000i	− 0.0177739i
$659$	−10950.0	−0.647271	−0.323635	−	0.946182i	$-0.604905\pi$
$659$	−10950.0	−0.647271	−0.323635	+	0.946182i	$0.604905\pi$
$660$	0	0
$661$	−3210.00	−0.188887	−0.0944437	−	0.995530i	$-0.530107\pi$
$661$	−3210.00	−0.188887	−0.0944437	+	0.995530i	$0.530107\pi$
$662$	16450.0i	0.965782i
$663$	− 19350.0i	− 1.13347i
$664$	−7216.00	−0.421740
$665$	0	0
$666$	−432.000	−0.0251346
$667$	161.000i	0.00934624i
$668$	− 7024.00i	− 0.406836i
$669$	−10080.0	−0.582534
$670$	0	0
$671$	−13624.0	−0.783828
$672$	− 576.000i	− 0.0330650i
$673$	13517.0i	0.774208i	0.922036	+	0.387104i	$0.126525\pi$
$673$	13517.0i	0.774208i	−0.922036	+	0.387104i	$0.873475\pi$
$674$	5152.00	0.294433
$675$	0	0
$676$	−1392.00	−0.0791989
$677$	− 7494.00i	− 0.425433i	−0.977114	−	0.212716i	$-0.931769\pi$
$677$	− 7494.00i	− 0.425433i	0.977114	−	0.212716i	$-0.0682310\pi$
$678$	18540.0i	1.05018i
$679$	684.000	0.0386591
$680$	0	0
$681$	−24354.0	−1.37041
$682$	28392.0i	1.59411i
$683$	− 17865.0i	− 1.00086i	−0.865778	−	0.500428i	$-0.833176\pi$
$683$	− 17865.0i	− 1.00086i	0.865778	−	0.500428i	$-0.166824\pi$
$684$	15984.0	0.893514
$685$	0	0
$686$	−2728.00	−0.151830
$687$	− 55260.0i	− 3.06885i
$688$	2432.00i	0.134766i
$689$	−3698.00	−0.204474
$690$	0	0
$691$	22364.0	1.23121	0.615605	−	0.788055i	$-0.288912\pi$
$691$	22364.0	1.23121	0.615605	+	0.788055i	$0.288912\pi$
$692$	9432.00i	0.518137i
$693$	5616.00i	0.307842i
$694$	−1192.00	−0.0651984
$695$	0	0
$696$	504.000	0.0274484
$697$	− 6150.00i	− 0.334215i
$698$	18542.0i	1.00548i
$699$	59103.0	3.19811
$700$	0	0
$701$	7842.00	0.422522	0.211261	−	0.977430i	$-0.432243\pi$
$701$	7842.00	0.422522	0.211261	+	0.977430i	$0.432243\pi$
$702$	− 20898.0i	− 1.12357i
$703$	− 296.000i	− 0.0158803i
$704$	3328.00	0.178166
$705$	0	0
$706$	16282.0	0.867962
$707$	− 2852.00i	− 0.151712i
$708$	− 15984.0i	− 0.848468i
$709$	−11234.0	−0.595066	−0.297533	−	0.954712i	$-0.596164\pi$
$709$	−11234.0	−0.595066	−0.297533	+	0.954712i	$0.596164\pi$
$710$	0	0
$711$	23004.0	1.21339
$712$	− 10176.0i	− 0.535620i
$713$	− 6279.00i	− 0.329804i
$714$	−1800.00	−0.0943464
$715$	0	0
$716$	4292.00	0.224022
$717$	6561.00i	0.341736i
$718$	4260.00i	0.221423i
$719$	17568.0	0.911232	0.455616	−	0.890176i	$-0.349419\pi$
$719$	17568.0	0.911232	0.455616	+	0.890176i	$0.349419\pi$
$720$	0	0
$721$	−2380.00	−0.122935
$722$	− 2766.00i	− 0.142576i
$723$	− 26208.0i	− 1.34811i
$724$	−11472.0	−0.588886
$725$	0	0
$726$	−24714.0	−1.26339
$727$	− 35664.0i	− 1.81940i	−0.415265	−	0.909701i	$-0.636311\pi$
$727$	− 35664.0i	− 1.81940i	0.415265	−	0.909701i	$-0.363689\pi$
$728$	− 688.000i	− 0.0350261i
$729$	19683.0	1.00000
$730$	0	0
$731$	7600.00	0.384536
$732$	− 9432.00i	− 0.476252i
$733$	27914.0i	1.40659i	0.710900	+	0.703293i	$0.248288\pi$
$733$	27914.0i	1.40659i	−0.710900	+	0.703293i	$0.751712\pi$
$734$	5148.00	0.258878
$735$	0	0
$736$	−736.000	−0.0368605
$737$	− 39728.0i	− 1.98562i
$738$	− 13284.0i	− 0.662589i
$739$	−39529.0	−1.96766	−0.983828	−	0.179116i	$-0.942676\pi$
$739$	−39529.0	−1.96766	−0.983828	+	0.179116i	$0.942676\pi$
$740$	0	0
$741$	28638.0	1.41976
$742$	344.000i	0.0170197i
$743$	− 10062.0i	− 0.496822i	−0.968655	−	0.248411i	$-0.920092\pi$
$743$	− 10062.0i	− 0.496822i	0.968655	−	0.248411i	$-0.0799084\pi$
$744$	−19656.0	−0.968581
$745$	0	0
$746$	−9008.00	−0.442100
$747$	48708.0i	2.38572i
$748$	− 10400.0i	− 0.508371i
$749$	2420.00	0.118057
$750$	0	0
$751$	25644.0	1.24602	0.623011	−	0.782213i	$-0.285909\pi$
$751$	25644.0	1.24602	0.623011	+	0.782213i	$0.285909\pi$
$752$	1200.00i	0.0581908i
$753$	3582.00i	0.173354i
$754$	602.000	0.0290763
$755$	0	0
$756$	−1944.00	−0.0935220
$757$	37368.0i	1.79414i	0.441890	+	0.897069i	$0.354308\pi$
$757$	37368.0i	1.79414i	−0.441890	+	0.897069i	$0.645692\pi$
$758$	− 5480.00i	− 0.262589i
$759$	10764.0	0.514767
$760$	0	0
$761$	105.000	0.00500164	0.00250082	−	0.999997i	$-0.499204\pi$
$761$	105.000	0.00500164	0.00250082	+	0.999997i	$0.499204\pi$
$762$	41022.0i	1.95022i
$763$	3360.00i	0.159424i
$764$	−1328.00	−0.0628866
$765$	0	0
$766$	13896.0	0.655461
$767$	− 19092.0i	− 0.898790i
$768$	2304.00i	0.108253i
$769$	15464.0	0.725157	0.362579	−	0.931953i	$-0.381896\pi$
$769$	15464.0	0.725157	0.362579	+	0.931953i	$0.381896\pi$
$770$	0	0
$771$	−73179.0	−3.41826
$772$	− 8572.00i	− 0.399628i
$773$	− 35168.0i	− 1.63636i	−0.574963	−	0.818179i	$-0.694984\pi$
$773$	− 35168.0i	− 1.63636i	0.574963	−	0.818179i	$-0.305016\pi$
$774$	16416.0	0.762353
$775$	0	0
$776$	−2736.00	−0.126568
$777$	72.0000i	0.00332431i
$778$	2808.00i	0.129398i
$779$	9102.00	0.418630
$780$	0	0
$781$	1092.00	0.0500318
$782$	2300.00i	0.105176i
$783$	− 1701.00i	− 0.0776357i
$784$	5424.00	0.247085
$785$	0	0
$786$	−17766.0	−0.806224
$787$	− 21216.0i	− 0.960951i	−0.877008	−	0.480476i	$-0.840464\pi$
$787$	− 21216.0i	− 0.960951i	0.877008	−	0.480476i	$-0.159536\pi$
$788$	10956.0i	0.495294i
$789$	−17802.0	−0.803255
$790$	0	0
$791$	2060.00	0.0925982
$792$	− 22464.0i	− 1.00786i
$793$	− 11266.0i	− 0.504499i
$794$	17282.0	0.772437
$795$	0	0
$796$	3008.00	0.133939
$797$	9506.00i	0.422484i	0.977434	+	0.211242i	$0.0677508\pi$
$797$	9506.00i	0.422484i	−0.977434	+	0.211242i	$0.932249\pi$
$798$	− 2664.00i	− 0.118176i
$799$	3750.00	0.166039
$800$	0	0
$801$	−68688.0	−3.02993
$802$	− 2280.00i	− 0.100386i
$803$	35412.0i	1.55624i
$804$	27504.0	1.20646
$805$	0	0
$806$	−23478.0	−1.02603
$807$	76131.0i	3.32087i
$808$	11408.0i	0.496698i
$809$	20550.0	0.893077	0.446539	−	0.894764i	$-0.352657\pi$
$809$	20550.0	0.893077	0.446539	+	0.894764i	$0.352657\pi$
$810$	0	0
$811$	−5161.00	−0.223461	−0.111731	−	0.993739i	$-0.535639\pi$
$811$	−5161.00	−0.223461	−0.111731	+	0.993739i	$0.535639\pi$
$812$	− 56.0000i	− 0.00242022i
$813$	− 65160.0i	− 2.81090i
$814$	−416.000	−0.0179125
$815$	0	0
$816$	7200.00	0.308885
$817$	11248.0i	0.481662i
$818$	− 25058.0i	− 1.07107i
$819$	−4644.00	−0.198137
$820$	0	0
$821$	7866.00	0.334379	0.167190	−	0.985925i	$-0.446531\pi$
$821$	7866.00	0.334379	0.167190	+	0.985925i	$0.446531\pi$
$822$	29592.0i	1.25564i
$823$	22317.0i	0.945227i	0.881270	+	0.472613i	$0.156689\pi$
$823$	22317.0i	0.945227i	−0.881270	+	0.472613i	$0.843311\pi$
$824$	9520.00	0.402482
$825$	0	0
$826$	−1776.00	−0.0748123
$827$	− 26196.0i	− 1.10148i	−0.834677	−	0.550740i	$-0.814346\pi$
$827$	− 26196.0i	− 1.10148i	0.834677	−	0.550740i	$-0.185654\pi$
$828$	4968.00i	0.208514i
$829$	−5886.00	−0.246597	−0.123299	−	0.992370i	$-0.539347\pi$
$829$	−5886.00	−0.246597	−0.123299	+	0.992370i	$0.539347\pi$
$830$	0	0
$831$	11871.0	0.495548
$832$	2752.00i	0.114674i
$833$	− 16950.0i	− 0.705021i
$834$	39402.0	1.63595
$835$	0	0
$836$	15392.0	0.636774
$837$	66339.0i	2.73956i
$838$	− 6504.00i	− 0.268111i
$839$	−32394.0	−1.33297	−0.666487	−	0.745517i	$-0.732203\pi$
$839$	−32394.0	−1.33297	−0.666487	+	0.745517i	$0.732203\pi$
$840$	0	0
$841$	−24340.0	−0.997991
$842$	4412.00i	0.180579i
$843$	15930.0i	0.650840i
$844$	4064.00	0.165745
$845$	0	0
$846$	8100.00	0.329177
$847$	2746.00i	0.111397i
$848$	− 1376.00i	− 0.0557217i
$849$	37296.0	1.50765
$850$	0	0
$851$	92.0000	0.00370590
$852$	756.000i	0.0303992i
$853$	31286.0i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$853$	31286.0i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$854$	−1048.00	−0.0419928
$855$	0	0
$856$	−9680.00	−0.386514
$857$	− 2913.00i	− 0.116110i	−0.998313	−	0.0580550i	$-0.981510\pi$
$857$	− 2913.00i	− 0.116110i	0.998313	−	0.0580550i	$-0.0184899\pi$
$858$	− 40248.0i	− 1.60145i
$859$	−15451.0	−0.613715	−0.306858	−	0.951755i	$-0.599278\pi$
$859$	−15451.0	−0.613715	−0.306858	+	0.951755i	$0.599278\pi$
$860$	0	0
$861$	−2214.00	−0.0876341
$862$	28632.0i	1.13133i
$863$	20627.0i	0.813617i	0.913514	+	0.406808i	$0.133358\pi$
$863$	20627.0i	0.813617i	−0.913514	+	0.406808i	$0.866642\pi$
$864$	7776.00	0.306186
$865$	0	0
$866$	15656.0	0.614333
$867$	21717.0i	0.850690i
$868$	2184.00i	0.0854030i
$869$	22152.0	0.864735
$870$	0	0
$871$	32852.0	1.27801
$872$	− 13440.0i	− 0.521945i
$873$	18468.0i	0.715976i
$874$	−3404.00	−0.131741
$875$	0	0
$876$	−24516.0	−0.945569
$877$	− 6966.00i	− 0.268216i	−0.990967	−	0.134108i	$-0.957183\pi$
$877$	− 6966.00i	− 0.268216i	0.990967	−	0.134108i	$-0.0428168\pi$
$878$	32078.0i	1.23301i
$879$	−61308.0	−2.35252
$880$	0	0
$881$	−37590.0	−1.43750	−0.718751	−	0.695268i	$-0.755286\pi$
$881$	−37590.0	−1.43750	−0.718751	+	0.695268i	$0.755286\pi$
$882$	− 36612.0i	− 1.39772i
$883$	− 27876.0i	− 1.06240i	−0.847245	−	0.531202i	$-0.821741\pi$
$883$	− 27876.0i	− 1.06240i	0.847245	−	0.531202i	$-0.178259\pi$
$884$	8600.00	0.327205
$885$	0	0
$886$	23494.0	0.890854
$887$	9471.00i	0.358518i	0.983802	+	0.179259i	$0.0573699\pi$
$887$	9471.00i	0.358518i	−0.983802	+	0.179259i	$0.942630\pi$
$888$	− 288.000i	− 0.0108836i
$889$	4558.00	0.171958
$890$	0	0
$891$	−37908.0	−1.42533
$892$	− 4480.00i	− 0.168163i
$893$	5550.00i	0.207977i
$894$	−17028.0	−0.637026
$895$	0	0
$896$	256.000	0.00954504
$897$	8901.00i	0.331322i
$898$	5780.00i	0.214790i
$899$	−1911.00	−0.0708959
$900$	0	0
$901$	−4300.00	−0.158994
$902$	− 12792.0i	− 0.472203i
$903$	− 2736.00i	− 0.100829i
$904$	−8240.00	−0.303162
$905$	0	0
$906$	6570.00	0.240920
$907$	28366.0i	1.03845i	0.854636	+	0.519227i	$0.173780\pi$
$907$	28366.0i	1.03845i	−0.854636	+	0.519227i	$0.826220\pi$
$908$	− 10824.0i	− 0.395602i
$909$	77004.0	2.80975
$910$	0	0
$911$	−7210.00	−0.262215	−0.131108	−	0.991368i	$-0.541853\pi$
$911$	−7210.00	−0.262215	−0.131108	+	0.991368i	$0.541853\pi$
$912$	10656.0i	0.386903i
$913$	46904.0i	1.70021i
$914$	26252.0	0.950043
$915$	0	0
$916$	24560.0	0.885901
$917$	1974.00i	0.0710875i
$918$	− 24300.0i	− 0.873660i
$919$	−17198.0	−0.617312	−0.308656	−	0.951174i	$-0.599879\pi$
$919$	−17198.0	−0.617312	−0.308656	+	0.951174i	$0.599879\pi$
$920$	0	0
$921$	51228.0	1.83281
$922$	28962.0i	1.03450i
$923$	903.000i	0.0322022i
$924$	−3744.00	−0.133299
$925$	0	0
$926$	10544.0	0.374187
$927$	− 64260.0i	− 2.27678i
$928$	224.000i	0.00792366i
$929$	51033.0	1.80230	0.901151	−	0.433505i	$-0.142724\pi$
$929$	51033.0	1.80230	0.901151	+	0.433505i	$0.142724\pi$
$930$	0	0
$931$	25086.0	0.883094
$932$	26268.0i	0.923216i
$933$	47403.0i	1.66335i
$934$	26932.0	0.943514
$935$	0	0
$936$	18576.0	0.648692
$937$	33328.0i	1.16198i	0.813910	+	0.580992i	$0.197335\pi$
$937$	33328.0i	1.16198i	−0.813910	+	0.580992i	$0.802665\pi$
$938$	− 3056.00i	− 0.106377i
$939$	57060.0	1.98305
$940$	0	0
$941$	−20166.0	−0.698611	−0.349305	−	0.937009i	$-0.613582\pi$
$941$	−20166.0	−0.698611	−0.349305	+	0.937009i	$0.613582\pi$
$942$	1944.00i	0.0672388i
$943$	2829.00i	0.0976934i
$944$	7104.00	0.244932
$945$	0	0
$946$	15808.0	0.543301
$947$	− 28629.0i	− 0.982384i	−0.871051	−	0.491192i	$-0.836561\pi$
$947$	− 28629.0i	− 0.982384i	0.871051	−	0.491192i	$-0.163439\pi$
$948$	15336.0i	0.525412i
$949$	−29283.0	−1.00165
$950$	0	0
$951$	79146.0	2.69872
$952$	− 800.000i	− 0.0272355i
$953$	− 38146.0i	− 1.29661i	−0.761380	−	0.648305i	$-0.775478\pi$
$953$	− 38146.0i	− 1.29661i	0.761380	−	0.648305i	$-0.224522\pi$
$954$	−9288.00	−0.315210
$955$	0	0
$956$	−2916.00	−0.0986508
$957$	− 3276.00i	− 0.110656i
$958$	9052.00i	0.305279i
$959$	3288.00	0.110714
$960$	0	0
$961$	44738.0	1.50173
$962$	− 344.000i	− 0.0115291i
$963$	65340.0i	2.18645i
$964$	11648.0	0.389167
$965$	0	0
$966$	828.000	0.0275781
$967$	44621.0i	1.48388i	0.670465	+	0.741941i	$0.266095\pi$
$967$	44621.0i	1.48388i	−0.670465	+	0.741941i	$0.733905\pi$
$968$	− 10984.0i	− 0.364710i
$969$	33300.0	1.10397
$970$	0	0
$971$	5950.00	0.196647	0.0983237	−	0.995154i	$-0.468652\pi$
$971$	5950.00	0.196647	0.0983237	+	0.995154i	$0.468652\pi$
$972$	0	0
$973$	− 4378.00i	− 0.144247i
$974$	17590.0	0.578665
$975$	0	0
$976$	4192.00	0.137482
$977$	40836.0i	1.33722i	0.743615	+	0.668608i	$0.233109\pi$
$977$	40836.0i	1.33722i	−0.743615	+	0.668608i	$0.766891\pi$
$978$	− 25470.0i	− 0.832762i
$979$	−66144.0	−2.15932
$980$	0	0
$981$	−90720.0	−2.95257
$982$	− 2550.00i	− 0.0828653i
$983$	− 26874.0i	− 0.871971i	−0.899954	−	0.435985i	$-0.856400\pi$
$983$	− 26874.0i	− 0.871971i	0.899954	−	0.435985i	$-0.143600\pi$
$984$	8856.00	0.286910
$985$	0	0
$986$	700.000	0.0226091
$987$	− 1350.00i	− 0.0435370i
$988$	12728.0i	0.409850i
$989$	−3496.00	−0.112403
$990$	0	0
$991$	21472.0	0.688275	0.344138	−	0.938919i	$-0.388171\pi$
$991$	21472.0	0.688275	0.344138	+	0.938919i	$0.388171\pi$
$992$	− 8736.00i	− 0.279605i
$993$	74025.0i	2.36567i
$994$	84.0000	0.00268040
$995$	0	0
$996$	−32472.0	−1.03305
$997$	6286.00i	0.199679i	0.995004	+	0.0998393i	$0.0318329\pi$
$997$	6286.00i	0.199679i	−0.995004	+	0.0998393i	$0.968167\pi$
$998$	19066.0i	0.604733i
$999$	−972.000	−0.0307835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.b.a.599.2		2
5.2	odd	4		1150.4.a.d.1.1		1
5.3	odd	4		46.4.a.b.1.1	✓	1
5.4	even	2	inner	1150.4.b.a.599.1		2
15.8	even	4		414.4.a.b.1.1		1
20.3	even	4		368.4.a.e.1.1		1
35.13	even	4		2254.4.a.b.1.1		1
40.3	even	4		1472.4.a.a.1.1		1
40.13	odd	4		1472.4.a.j.1.1		1
115.68	even	4		1058.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.4.a.b.1.1	✓	1	5.3	odd	4
368.4.a.e.1.1		1	20.3	even	4
414.4.a.b.1.1		1	15.8	even	4
1058.4.a.b.1.1		1	115.68	even	4
1150.4.a.d.1.1		1	5.2	odd	4
1150.4.b.a.599.1		2	5.4	even	2	inner
1150.4.b.a.599.2		2	1.1	even	1	trivial
1472.4.a.a.1.1		1	40.3	even	4
1472.4.a.j.1.1		1	40.13	odd	4
2254.4.a.b.1.1		1	35.13	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +9.00000i q^{3} -4.00000 q^{4} -18.0000 q^{6} +2.00000i q^{7} -8.00000i q^{8} -54.0000 q^{9} -52.0000 q^{11} -36.0000i q^{12} -43.0000i q^{13} -4.00000 q^{14} +16.0000 q^{16} -50.0000i q^{17} -108.000i q^{18} +74.0000 q^{19} -18.0000 q^{21} -104.000i q^{22} +23.0000i q^{23} +72.0000 q^{24} +86.0000 q^{26} -243.000i q^{27} -8.00000i q^{28} +7.00000 q^{29} -273.000 q^{31} +32.0000i q^{32} -468.000i q^{33} +100.000 q^{34} +216.000 q^{36} -4.00000i q^{37} +148.000i q^{38} +387.000 q^{39} +123.000 q^{41} -36.0000i q^{42} +152.000i q^{43} +208.000 q^{44} -46.0000 q^{46} +75.0000i q^{47} +144.000i q^{48} +339.000 q^{49} +450.000 q^{51} +172.000i q^{52} -86.0000i q^{53} +486.000 q^{54} +16.0000 q^{56} +666.000i q^{57} +14.0000i q^{58} +444.000 q^{59} +262.000 q^{61} -546.000i q^{62} -108.000i q^{63} -64.0000 q^{64} +936.000 q^{66} +764.000i q^{67} +200.000i q^{68} -207.000 q^{69} -21.0000 q^{71} +432.000i q^{72} -681.000i q^{73} +8.00000 q^{74} -296.000 q^{76} -104.000i q^{77} +774.000i q^{78} -426.000 q^{79} +729.000 q^{81} +246.000i q^{82} -902.000i q^{83} +72.0000 q^{84} -304.000 q^{86} +63.0000i q^{87} +416.000i q^{88} +1272.00 q^{89} +86.0000 q^{91} -92.0000i q^{92} -2457.00i q^{93} -150.000 q^{94} -288.000 q^{96} -342.000i q^{97} +678.000i q^{98} +2808.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 36 q^{6} - 108 q^{9} - 104 q^{11} - 8 q^{14} + 32 q^{16} + 148 q^{19} - 36 q^{21} + 144 q^{24} + 172 q^{26} + 14 q^{29} - 546 q^{31} + 200 q^{34} + 432 q^{36} + 774 q^{39} + 246 q^{41} + 416 q^{44}+ \cdots + 5616 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 36 * q^6 - 108 * q^9 - 104 * q^11 - 8 * q^14 + 32 * q^16 + 148 * q^19 - 36 * q^21 + 144 * q^24 + 172 * q^26 + 14 * q^29 - 546 * q^31 + 200 * q^34 + 432 * q^36 + 774 * q^39 + 246 * q^41 + 416 * q^44 - 92 * q^46 + 678 * q^49 + 900 * q^51 + 972 * q^54 + 32 * q^56 + 888 * q^59 + 524 * q^61 - 128 * q^64 + 1872 * q^66 - 414 * q^69 - 42 * q^71 + 16 * q^74 - 592 * q^76 - 852 * q^79 + 1458 * q^81 + 144 * q^84 - 608 * q^86 + 2544 * q^89 + 172 * q^91 - 300 * q^94 - 576 * q^96 + 5616 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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