LMFDB - Embedded newform 2304.3.h.l.2177.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(2177,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2304.2177");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.h (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:	$62.7794529086$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.959512576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 81 $ x^8 + 7*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2177.4
Root			$0.819051 - 1.52616i$ of defining polynomial
Character	$\chi$	$=$	2304.2177
Dual form			2304.3.h.l.2177.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.21904 q^{5} +2.00000 q^{7} +O(q^{10})$	$q-5.21904 q^{5} +2.00000 q^{7} -4.78122 q^{11} +1.38083i q^{13} -14.6807i q^{17} +26.7617i q^{19} -18.0477i q^{23} +2.23834 q^{25} +25.0180 q^{29} -39.5233 q^{31} -10.4381 q^{35} +26.0000i q^{37} +28.8228i q^{41} +9.52333i q^{43} -80.2731i q^{47} -45.0000 q^{49} -9.79874 q^{53} +24.9533 q^{55} +73.5391 q^{59} -67.5233i q^{61} -7.20661i q^{65} +102.762i q^{67} +21.9533i q^{71} +140.570 q^{73} -9.56244 q^{77} -0.476674 q^{79} -31.3142 q^{83} +76.6192i q^{85} -13.1310i q^{89} +2.76166i q^{91} -139.670i q^{95} -69.2383 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7}+O(q^{10}) $ 8 * q + 16 * q^7	$ 8 q + 16 q^{7} + 168 q^{25} - 16 q^{31} - 360 q^{49} + 800 q^{55} + 224 q^{73} - 304 q^{79} - 704 q^{97}+O(q^{100}) $ 8 * q + 16 * q^7 + 168 * q^25 - 16 * q^31 - 360 * q^49 + 800 * q^55 + 224 * q^73 - 304 * q^79 - 704 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−5.21904	−1.04381	−0.521904	−	0.853004i	$-0.674778\pi$
$5$	−5.21904	−1.04381	−0.521904	+	0.853004i	$0.674778\pi$
$6$	0	0
$7$	2.00000	0.285714	0.142857	−	0.989743i	$-0.454371\pi$
$7$	2.00000	0.285714	0.142857	+	0.989743i	$0.454371\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.78122	−0.434656	−0.217328	−	0.976099i	$-0.569734\pi$
$11$	−4.78122	−0.434656	−0.217328	+	0.976099i	$0.569734\pi$
$12$	0	0
$13$	1.38083i	0.106218i	0.998589	+	0.0531089i	$0.0169130\pi$
$13$	1.38083i	0.106218i	−0.998589	+	0.0531089i	$0.983087\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 14.6807i	− 0.863571i	−0.901976	−	0.431786i	$-0.857884\pi$
$17$	− 14.6807i	− 0.863571i	0.901976	−	0.431786i	$-0.142116\pi$
$18$	0	0
$19$	26.7617i	1.40851i	0.709948	+	0.704254i	$0.248719\pi$
$19$	26.7617i	1.40851i	−0.709948	+	0.704254i	$0.751281\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 18.0477i	− 0.784683i	−0.919820	−	0.392342i	$-0.871665\pi$
$23$	− 18.0477i	− 0.784683i	0.919820	−	0.392342i	$-0.128335\pi$
$24$	0	0
$25$	2.23834	0.0895335
$26$	0	0
$27$	0	0
$28$	0	0
$29$	25.0180	0.862691	0.431345	−	0.902187i	$-0.358039\pi$
$29$	25.0180	0.862691	0.431345	+	0.902187i	$0.358039\pi$
$30$	0	0
$31$	−39.5233	−1.27495	−0.637473	−	0.770473i	$-0.720020\pi$
$31$	−39.5233	−1.27495	−0.637473	+	0.770473i	$0.720020\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−10.4381	−0.298231
$36$	0	0
$37$	26.0000i	0.702703i	0.936244	+	0.351351i	$0.114278\pi$
$37$	26.0000i	0.702703i	−0.936244	+	0.351351i	$0.885722\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	28.8228i	0.702996i	0.936189	+	0.351498i	$0.114328\pi$
$41$	28.8228i	0.702996i	−0.936189	+	0.351498i	$0.885672\pi$
$42$	0	0
$43$	9.52333i	0.221473i	0.993850	+	0.110736i	$0.0353209\pi$
$43$	9.52333i	0.221473i	−0.993850	+	0.110736i	$0.964679\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 80.2731i	− 1.70794i	−0.520323	−	0.853969i	$-0.674189\pi$
$47$	− 80.2731i	− 1.70794i	0.520323	−	0.853969i	$-0.325811\pi$
$48$	0	0
$49$	−45.0000	−0.918367
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.79874	−0.184882	−0.0924409	−	0.995718i	$-0.529467\pi$
$53$	−9.79874	−0.184882	−0.0924409	+	0.995718i	$0.529467\pi$
$54$	0	0
$55$	24.9533	0.453697
$56$	0	0
$57$	0	0
$58$	0	0
$59$	73.5391	1.24643	0.623213	−	0.782052i	$-0.285827\pi$
$59$	73.5391	1.24643	0.623213	+	0.782052i	$0.285827\pi$
$60$	0	0
$61$	− 67.5233i	− 1.10694i	−0.832869	−	0.553470i	$-0.813303\pi$
$61$	− 67.5233i	− 1.10694i	0.832869	−	0.553470i	$-0.186697\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 7.20661i	− 0.110871i
$66$	0	0
$67$	102.762i	1.53376i	0.641793	+	0.766878i	$0.278191\pi$
$67$	102.762i	1.53376i	−0.641793	+	0.766878i	$0.721809\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	21.9533i	0.309201i	0.987977	+	0.154601i	$0.0494091\pi$
$71$	21.9533i	0.309201i	−0.987977	+	0.154601i	$0.950591\pi$
$72$	0	0
$73$	140.570	1.92562	0.962808	−	0.270186i	$-0.0870853\pi$
$73$	140.570	1.92562	0.962808	+	0.270186i	$0.0870853\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.56244	−0.124187
$78$	0	0
$79$	−0.476674	−0.00603385	−0.00301692	−	0.999995i	$-0.500960\pi$
$79$	−0.476674	−0.00603385	−0.00301692	+	0.999995i	$0.500960\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−31.3142	−0.377280	−0.188640	−	0.982046i	$-0.560408\pi$
$83$	−31.3142	−0.377280	−0.188640	+	0.982046i	$0.560408\pi$
$84$	0	0
$85$	76.6192i	0.901402i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 13.1310i	− 0.147539i	−0.997275	−	0.0737694i	$-0.976497\pi$
$89$	− 13.1310i	− 0.147539i	0.997275	−	0.0737694i	$-0.0235029\pi$
$90$	0	0
$91$	2.76166i	0.0303479i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 139.670i	− 1.47021i
$96$	0	0
$97$	−69.2383	−0.713797	−0.356899	−	0.934143i	$-0.616166\pi$
$97$	−69.2383	−0.713797	−0.356899	+	0.934143i	$0.616166\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−78.5566	−0.777788	−0.388894	−	0.921282i	$-0.627143\pi$
$101$	−78.5566	−0.777788	−0.388894	+	0.921282i	$0.627143\pi$
$102$	0	0
$103$	198.570	1.92786	0.963932	−	0.266149i	$-0.0857513\pi$
$103$	198.570	1.92786	0.963932	+	0.266149i	$0.0857513\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	177.517	1.65904	0.829518	−	0.558480i	$-0.188615\pi$
$107$	177.517	1.65904	0.829518	+	0.558480i	$0.188615\pi$
$108$	0	0
$109$	− 137.381i	− 1.26037i	−0.776443	−	0.630187i	$-0.782978\pi$
$109$	− 137.381i	− 1.26037i	0.776443	−	0.630187i	$-0.217022\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 72.5279i	− 0.641840i	−0.947106	−	0.320920i	$-0.896008\pi$
$113$	− 72.5279i	− 0.641840i	0.947106	−	0.320920i	$-0.103992\pi$
$114$	0	0
$115$	94.1917i	0.819058i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 29.3614i	− 0.246735i
$120$	0	0
$121$	−98.1400	−0.811074
$122$	0	0
$123$	0	0
$124$	0	0
$125$	118.794	0.950352
$126$	0	0
$127$	81.0467	0.638163	0.319081	−	0.947727i	$-0.396626\pi$
$127$	81.0467	0.638163	0.319081	+	0.947727i	$0.396626\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	164.452	1.25536	0.627679	−	0.778473i	$-0.284005\pi$
$131$	164.452	1.25536	0.627679	+	0.778473i	$0.284005\pi$
$132$	0	0
$133$	53.5233i	0.402431i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	224.790i	1.64081i	0.571786	+	0.820403i	$0.306251\pi$
$137$	224.790i	1.64081i	−0.571786	+	0.820403i	$0.693749\pi$
$138$	0	0
$139$	− 170.378i	− 1.22574i	−0.790183	−	0.612872i	$-0.790014\pi$
$139$	− 170.378i	− 1.22574i	0.790183	−	0.612872i	$-0.209986\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 6.60206i	− 0.0461682i
$144$	0	0
$145$	−130.570	−0.900483
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−59.5637	−0.399756	−0.199878	−	0.979821i	$-0.564055\pi$
$149$	−59.5637	−0.399756	−0.199878	+	0.979821i	$0.564055\pi$
$150$	0	0
$151$	242.570	1.60642	0.803212	−	0.595693i	$-0.203123\pi$
$151$	242.570	1.60642	0.803212	+	0.595693i	$0.203123\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	206.274	1.33080
$156$	0	0
$157$	161.617i	1.02941i	0.857369	+	0.514703i	$0.172098\pi$
$157$	161.617i	1.02941i	−0.857369	+	0.514703i	$0.827902\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 36.0954i	− 0.224195i
$162$	0	0
$163$	80.9533i	0.496646i	0.968677	+	0.248323i	$0.0798795\pi$
$163$	80.9533i	0.496646i	−0.968677	+	0.248323i	$0.920121\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 179.268i	− 1.07346i	−0.843754	−	0.536731i	$-0.819659\pi$
$167$	− 179.268i	− 1.07346i	0.843754	−	0.536731i	$-0.180341\pi$
$168$	0	0
$169$	167.093	0.988718
$170$	0	0
$171$	0	0
$172$	0	0
$173$	26.4982	0.153169	0.0765844	−	0.997063i	$-0.475599\pi$
$173$	26.4982	0.153169	0.0765844	+	0.997063i	$0.475599\pi$
$174$	0	0
$175$	4.47667	0.0255810
$176$	0	0
$177$	0	0
$178$	0	0
$179$	306.679	1.71329	0.856646	−	0.515905i	$-0.172544\pi$
$179$	306.679	1.71329	0.856646	+	0.515905i	$0.172544\pi$
$180$	0	0
$181$	57.3808i	0.317021i	0.987357	+	0.158511i	$0.0506692\pi$
$181$	57.3808i	0.317021i	−0.987357	+	0.158511i	$0.949331\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 135.695i	− 0.733486i
$186$	0	0
$187$	70.1917i	0.375357i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 216.712i	− 1.13462i	−0.823506	−	0.567308i	$-0.807985\pi$
$191$	− 216.712i	− 1.13462i	0.823506	−	0.567308i	$-0.192015\pi$
$192$	0	0
$193$	187.140	0.969637	0.484819	−	0.874615i	$-0.338886\pi$
$193$	187.140	0.969637	0.484819	+	0.874615i	$0.338886\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−65.8947	−0.334491	−0.167245	−	0.985915i	$-0.553487\pi$
$197$	−65.8947	−0.334491	−0.167245	+	0.985915i	$0.553487\pi$
$198$	0	0
$199$	114.953	0.577655	0.288828	−	0.957381i	$-0.406735\pi$
$199$	114.953	0.577655	0.288828	+	0.957381i	$0.406735\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	50.0361	0.246483
$204$	0	0
$205$	− 150.427i	− 0.733793i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 127.953i	− 0.612217i
$210$	0	0
$211$	103.332i	0.489723i	0.969558	+	0.244862i	$0.0787426\pi$
$211$	103.332i	0.489723i	−0.969558	+	0.244862i	$0.921257\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 49.7026i	− 0.231175i
$216$	0	0
$217$	−79.0467	−0.364270
$218$	0	0
$219$	0	0
$220$	0	0
$221$	20.2716	0.0917267
$222$	0	0
$223$	241.047	1.08093	0.540463	−	0.841368i	$-0.318249\pi$
$223$	241.047	1.08093	0.540463	+	0.841368i	$0.318249\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.5579	−0.0993738	−0.0496869	−	0.998765i	$-0.515822\pi$
$227$	−22.5579	−0.0993738	−0.0496869	+	0.998765i	$0.515822\pi$
$228$	0	0
$229$	− 212.997i	− 0.930120i	−0.885279	−	0.465060i	$-0.846033\pi$
$229$	− 212.997i	− 0.930120i	0.885279	−	0.465060i	$-0.153967\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 63.9107i	− 0.274295i	−0.990551	−	0.137147i	$-0.956207\pi$
$233$	− 63.9107i	− 0.274295i	0.990551	−	0.137147i	$-0.0437934\pi$
$234$	0	0
$235$	418.948i	1.78276i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 174.417i	− 0.729779i	−0.931051	−	0.364890i	$-0.881107\pi$
$239$	− 174.417i	− 0.729779i	0.931051	−	0.364890i	$-0.118893\pi$
$240$	0	0
$241$	−47.3316	−0.196397	−0.0981984	−	0.995167i	$-0.531308\pi$
$241$	−47.3316	−0.196397	−0.0981984	+	0.995167i	$0.531308\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	234.857	0.958598
$246$	0	0
$247$	−36.9533	−0.149609
$248$	0	0
$249$	0	0
$250$	0	0
$251$	205.731	0.819647	0.409824	−	0.912165i	$-0.365590\pi$
$251$	205.731	0.819647	0.409824	+	0.912165i	$0.365590\pi$
$252$	0	0
$253$	86.2901i	0.341067i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	251.796i	0.979751i	0.871792	+	0.489875i	$0.162958\pi$
$257$	251.796i	0.979751i	−0.871792	+	0.489875i	$0.837042\pi$
$258$	0	0
$259$	52.0000i	0.200772i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 206.607i	− 0.785578i	−0.919629	−	0.392789i	$-0.871510\pi$
$263$	− 206.607i	− 0.785578i	0.919629	−	0.392789i	$-0.128490\pi$
$264$	0	0
$265$	51.1400	0.192981
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−430.963	−1.60209	−0.801047	−	0.598601i	$-0.795723\pi$
$269$	−430.963	−1.60209	−0.801047	+	0.598601i	$0.795723\pi$
$270$	0	0
$271$	290.953	1.07363	0.536814	−	0.843700i	$-0.319628\pi$
$271$	290.953	1.07363	0.536814	+	0.843700i	$0.319628\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.7020	−0.0389163
$276$	0	0
$277$	57.9508i	0.209209i	0.994514	+	0.104604i	$0.0333576\pi$
$277$	57.9508i	0.209209i	−0.994514	+	0.104604i	$0.966642\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 124.517i	− 0.443120i	−0.975147	−	0.221560i	$-0.928885\pi$
$281$	− 124.517i	− 0.443120i	0.975147	−	0.221560i	$-0.0711149\pi$
$282$	0	0
$283$	254.093i	0.897856i	0.893568	+	0.448928i	$0.148194\pi$
$283$	254.093i	0.897856i	−0.893568	+	0.448928i	$0.851806\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	57.6457i	0.200856i
$288$	0	0
$289$	73.4767	0.254245
$290$	0	0
$291$	0	0
$292$	0	0
$293$	239.909	0.818802	0.409401	−	0.912355i	$-0.365738\pi$
$293$	239.909	0.818802	0.409401	+	0.912355i	$0.365738\pi$
$294$	0	0
$295$	−383.803	−1.30103
$296$	0	0
$297$	0	0
$298$	0	0
$299$	24.9209	0.0833473
$300$	0	0
$301$	19.0467i	0.0632779i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	352.407i	1.15543i
$306$	0	0
$307$	3.71501i	0.0121010i	0.999982	+	0.00605051i	$0.00192595\pi$
$307$	3.71501i	0.0121010i	−0.999982	+	0.00605051i	$0.998074\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 473.828i	− 1.52356i	−0.647835	−	0.761781i	$-0.724325\pi$
$311$	− 473.828i	− 1.52356i	0.647835	−	0.761781i	$-0.275675\pi$
$312$	0	0
$313$	−310.000	−0.990415	−0.495208	−	0.868775i	$-0.664908\pi$
$313$	−310.000	−0.990415	−0.495208	+	0.868775i	$0.664908\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	475.342	1.49950	0.749752	−	0.661719i	$-0.230173\pi$
$317$	475.342	1.49950	0.749752	+	0.661719i	$0.230173\pi$
$318$	0	0
$319$	−119.617	−0.374974
$320$	0	0
$321$	0	0
$322$	0	0
$323$	392.880	1.21635
$324$	0	0
$325$	3.09077i	0.00951005i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 160.546i	− 0.487982i
$330$	0	0
$331$	390.762i	1.18055i	0.807203	+	0.590274i	$0.200980\pi$
$331$	390.762i	1.18055i	−0.807203	+	0.590274i	$0.799020\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 536.317i	− 1.60095i
$336$	0	0
$337$	316.000	0.937685	0.468843	−	0.883282i	$-0.344671\pi$
$337$	316.000	0.937685	0.468843	+	0.883282i	$0.344671\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	188.970	0.554163
$342$	0	0
$343$	−188.000	−0.548105
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−485.475	−1.39906	−0.699531	−	0.714602i	$-0.746608\pi$
$347$	−485.475	−1.39906	−0.699531	+	0.714602i	$0.746608\pi$
$348$	0	0
$349$	561.233i	1.60812i	0.594549	+	0.804059i	$0.297330\pi$
$349$	561.233i	1.60812i	−0.594549	+	0.804059i	$0.702670\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 114.280i	− 0.323740i	−0.986812	−	0.161870i	$-0.948247\pi$
$353$	− 114.280i	− 0.323740i	0.986812	−	0.161870i	$-0.0517525\pi$
$354$	0	0
$355$	− 114.575i	− 0.322747i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 217.789i	− 0.606654i	−0.952886	−	0.303327i	$-0.901903\pi$
$359$	− 217.789i	− 0.606654i	0.952886	−	0.303327i	$-0.0980975\pi$
$360$	0	0
$361$	−355.187	−0.983896
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−733.640	−2.00997
$366$	0	0
$367$	143.907	0.392116	0.196058	−	0.980592i	$-0.437186\pi$
$367$	143.907	0.392116	0.196058	+	0.980592i	$0.437186\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−19.5975	−0.0528234
$372$	0	0
$373$	− 400.663i	− 1.07416i	−0.843530	−	0.537082i	$-0.819527\pi$
$373$	− 400.663i	− 1.07416i	0.843530	−	0.537082i	$-0.180473\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	34.5457i	0.0916331i
$378$	0	0
$379$	156.668i	0.413373i	0.978407	+	0.206686i	$0.0662680\pi$
$379$	156.668i	0.413373i	−0.978407	+	0.206686i	$0.933732\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	483.390i	1.26211i	0.775736	+	0.631057i	$0.217379\pi$
$383$	483.390i	1.26211i	−0.775736	+	0.631057i	$0.782621\pi$
$384$	0	0
$385$	49.9067	0.129628
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−726.933	−1.86872	−0.934362	−	0.356326i	$-0.884029\pi$
$389$	−726.933	−1.86872	−0.934362	+	0.356326i	$0.884029\pi$
$390$	0	0
$391$	−264.953	−0.677630
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.48778	0.00629817
$396$	0	0
$397$	94.1866i	0.237246i	0.992939	+	0.118623i	$0.0378480\pi$
$397$	94.1866i	0.237246i	−0.992939	+	0.118623i	$0.962152\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.42898i	0.00605731i	0.999995	+	0.00302865i	$0.000964052\pi$
$401$	2.42898i	0.00605731i	−0.999995	+	0.00302865i	$0.999036\pi$
$402$	0	0
$403$	− 54.5751i	− 0.135422i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 124.312i	− 0.305434i
$408$	0	0
$409$	66.7617	0.163231	0.0816157	−	0.996664i	$-0.473992\pi$
$409$	66.7617	0.163231	0.0816157	+	0.996664i	$0.473992\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	147.078	0.356122
$414$	0	0
$415$	163.430	0.393807
$416$	0	0
$417$	0	0
$418$	0	0
$419$	241.021	0.575229	0.287614	−	0.957746i	$-0.407138\pi$
$419$	241.021	0.575229	0.287614	+	0.957746i	$0.407138\pi$
$420$	0	0
$421$	− 698.806i	− 1.65987i	−0.557859	−	0.829936i	$-0.688377\pi$
$421$	− 698.806i	− 1.65987i	0.557859	−	0.829936i	$-0.311623\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 32.8604i	− 0.0773185i
$426$	0	0
$427$	− 135.047i	− 0.316269i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	783.599i	1.81810i	0.416692	+	0.909048i	$0.363189\pi$
$431$	783.599i	1.81810i	−0.416692	+	0.909048i	$0.636811\pi$
$432$	0	0
$433$	317.233	0.732640	0.366320	−	0.930489i	$-0.380617\pi$
$433$	317.233	0.732640	0.366320	+	0.930489i	$0.380617\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	482.987	1.10523
$438$	0	0
$439$	219.140	0.499180	0.249590	−	0.968352i	$-0.419704\pi$
$439$	219.140	0.499180	0.249590	+	0.968352i	$0.419704\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−229.707	−0.518526	−0.259263	−	0.965807i	$-0.583480\pi$
$443$	−229.707	−0.518526	−0.259263	+	0.965807i	$0.583480\pi$
$444$	0	0
$445$	68.5309i	0.154002i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 107.074i	− 0.238471i	−0.992866	−	0.119236i	$-0.961956\pi$
$449$	− 107.074i	− 0.238471i	0.992866	−	0.119236i	$-0.0380444\pi$
$450$	0	0
$451$	− 137.808i	− 0.305562i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 14.4132i	− 0.0316774i
$456$	0	0
$457$	−605.042	−1.32394	−0.661971	−	0.749529i	$-0.730280\pi$
$457$	−605.042	−1.32394	−0.661971	+	0.749529i	$0.730280\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−305.435	−0.662550	−0.331275	−	0.943534i	$-0.607479\pi$
$461$	−305.435	−0.662550	−0.331275	+	0.943534i	$0.607479\pi$
$462$	0	0
$463$	59.1400	0.127732	0.0638660	−	0.997958i	$-0.479657\pi$
$463$	59.1400	0.127732	0.0638660	+	0.997958i	$0.479657\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	565.477	1.21087	0.605435	−	0.795894i	$-0.292999\pi$
$467$	565.477	1.21087	0.605435	+	0.795894i	$0.292999\pi$
$468$	0	0
$469$	205.523i	0.438216i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 45.5331i	− 0.0962645i
$474$	0	0
$475$	59.9016i	0.126109i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 249.576i	− 0.521035i	−0.965469	−	0.260517i	$-0.916107\pi$
$479$	− 249.576i	− 0.521035i	0.965469	−	0.260517i	$-0.0838932\pi$
$480$	0	0
$481$	−35.9016	−0.0746395
$482$	0	0
$483$	0	0
$484$	0	0
$485$	361.357	0.745067
$486$	0	0
$487$	435.897	0.895065	0.447532	−	0.894268i	$-0.352303\pi$
$487$	435.897	0.895065	0.447532	+	0.894268i	$0.352303\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−335.506	−0.683311	−0.341655	−	0.939825i	$-0.610988\pi$
$491$	−335.506	−0.683311	−0.341655	+	0.939825i	$0.610988\pi$
$492$	0	0
$493$	− 367.282i	− 0.744995i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	43.9066i	0.0883433i
$498$	0	0
$499$	− 712.373i	− 1.42760i	−0.700349	−	0.713801i	$-0.746972\pi$
$499$	− 712.373i	− 1.42760i	0.700349	−	0.713801i	$-0.253028\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	527.832i	1.04937i	0.851297	+	0.524683i	$0.175816\pi$
$503$	527.832i	1.04937i	−0.851297	+	0.524683i	$0.824184\pi$
$504$	0	0
$505$	409.990	0.811861
$506$	0	0
$507$	0	0
$508$	0	0
$509$	736.294	1.44655	0.723275	−	0.690560i	$-0.242636\pi$
$509$	736.294	1.44655	0.723275	+	0.690560i	$0.242636\pi$
$510$	0	0
$511$	281.140	0.550176
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1036.34	−2.01232
$516$	0	0
$517$	383.803i	0.742366i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 372.001i	− 0.714013i	−0.934102	−	0.357006i	$-0.883797\pi$
$521$	− 372.001i	− 0.714013i	0.934102	−	0.357006i	$-0.116203\pi$
$522$	0	0
$523$	− 251.135i	− 0.480181i	−0.970750	−	0.240091i	$-0.922823\pi$
$523$	− 251.135i	− 0.480181i	0.970750	−	0.240091i	$-0.0771772\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	580.231i	1.10101i
$528$	0	0
$529$	203.280	0.384272
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−39.7995	−0.0746707
$534$	0	0
$535$	−926.467	−1.73171
$536$	0	0
$537$	0	0
$538$	0	0
$539$	215.155	0.399174
$540$	0	0
$541$	966.992i	1.78742i	0.448648	+	0.893708i	$0.351906\pi$
$541$	966.992i	1.78742i	−0.448648	+	0.893708i	$0.648094\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	716.996i	1.31559i
$546$	0	0
$547$	− 293.710i	− 0.536947i	−0.963287	−	0.268473i	$-0.913481\pi$
$547$	− 293.710i	− 0.536947i	0.963287	−	0.268473i	$-0.0865192\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	669.524i	1.21511i
$552$	0	0
$553$	−0.953348	−0.00172396
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−566.193	−1.01650	−0.508252	−	0.861208i	$-0.669708\pi$
$557$	−566.193	−1.01650	−0.508252	+	0.861208i	$0.669708\pi$
$558$	0	0
$559$	−13.1501	−0.0235243
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−150.108	−0.266622	−0.133311	−	0.991074i	$-0.542561\pi$
$563$	−150.108	−0.266622	−0.133311	+	0.991074i	$0.542561\pi$
$564$	0	0
$565$	378.526i	0.669957i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	899.707i	1.58121i	0.612328	+	0.790604i	$0.290233\pi$
$569$	899.707i	1.58121i	−0.612328	+	0.790604i	$0.709767\pi$
$570$	0	0
$571$	− 680.570i	− 1.19189i	−0.803025	−	0.595946i	$-0.796777\pi$
$571$	− 680.570i	− 1.19189i	0.803025	−	0.595946i	$-0.203223\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 40.3969i	− 0.0702554i
$576$	0	0
$577$	−249.233	−0.431947	−0.215973	−	0.976399i	$-0.569292\pi$
$577$	−249.233	−0.431947	−0.215973	+	0.976399i	$0.569292\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−62.6284	−0.107794
$582$	0	0
$583$	46.8499	0.0803600
$584$	0	0
$585$	0	0
$586$	0	0
$587$	602.587	1.02655	0.513277	−	0.858223i	$-0.328431\pi$
$587$	602.587	1.02655	0.513277	+	0.858223i	$0.328431\pi$
$588$	0	0
$589$	− 1057.71i	− 1.79577i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	154.142i	0.259936i	0.991518	+	0.129968i	$0.0414875\pi$
$593$	154.142i	0.259936i	−0.991518	+	0.129968i	$0.958513\pi$
$594$	0	0
$595$	153.238i	0.257543i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	309.244i	0.516267i	0.966109	+	0.258133i	$0.0831074\pi$
$599$	309.244i	0.516267i	−0.966109	+	0.258133i	$0.916893\pi$
$600$	0	0
$601$	811.327	1.34996	0.674981	−	0.737836i	$-0.264152\pi$
$601$	811.327	1.34996	0.674981	+	0.737836i	$0.264152\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	512.196	0.846605
$606$	0	0
$607$	−202.197	−0.333108	−0.166554	−	0.986032i	$-0.553264\pi$
$607$	−202.197	−0.333108	−0.166554	+	0.986032i	$0.553264\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	110.844	0.181413
$612$	0	0
$613$	− 694.560i	− 1.13305i	−0.824044	−	0.566525i	$-0.808287\pi$
$613$	− 694.560i	− 1.13305i	0.824044	−	0.566525i	$-0.191713\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 896.267i	− 1.45262i	−0.687367	−	0.726311i	$-0.741233\pi$
$617$	− 896.267i	− 1.45262i	0.687367	−	0.726311i	$-0.258767\pi$
$618$	0	0
$619$	− 672.570i	− 1.08654i	−0.839557	−	0.543271i	$-0.817185\pi$
$619$	− 672.570i	− 1.08654i	0.839557	−	0.543271i	$-0.182815\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 26.2619i	− 0.0421540i
$624$	0	0
$625$	−675.948	−1.08152
$626$	0	0
$627$	0	0
$628$	0	0
$629$	381.699	0.606834
$630$	0	0
$631$	218.953	0.346994	0.173497	−	0.984834i	$-0.444493\pi$
$631$	218.953	0.346994	0.173497	+	0.984834i	$0.444493\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−422.985	−0.666119
$636$	0	0
$637$	− 62.1374i	− 0.0975470i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1193.18i	− 1.86144i	−0.365734	−	0.930720i	$-0.619182\pi$
$641$	− 1193.18i	− 1.86144i	0.365734	−	0.930720i	$-0.380818\pi$
$642$	0	0
$643$	− 1009.33i	− 1.56971i	−0.619676	−	0.784857i	$-0.712736\pi$
$643$	− 1009.33i	− 1.56971i	0.619676	−	0.784857i	$-0.287264\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	637.605i	0.985479i	0.870177	+	0.492740i	$0.164005\pi$
$647$	637.605i	0.985479i	−0.870177	+	0.492740i	$0.835995\pi$
$648$	0	0
$649$	−351.606	−0.541767
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−939.073	−1.43809	−0.719045	−	0.694964i	$-0.755420\pi$
$653$	−939.073	−1.43809	−0.719045	+	0.694964i	$0.755420\pi$
$654$	0	0
$655$	−858.280	−1.31035
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1224.84	−1.85864	−0.929318	−	0.369281i	$-0.879604\pi$
$659$	−1224.84	−1.85864	−0.929318	+	0.369281i	$0.879604\pi$
$660$	0	0
$661$	1034.18i	1.56456i	0.622925	+	0.782282i	$0.285944\pi$
$661$	1034.18i	1.56456i	−0.622925	+	0.782282i	$0.714056\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 279.340i	− 0.420060i
$666$	0	0
$667$	− 451.518i	− 0.676939i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	322.844i	0.481138i
$672$	0	0
$673$	−18.9533	−0.0281625	−0.0140812	−	0.999901i	$-0.504482\pi$
$673$	−18.9533	−0.0281625	−0.0140812	+	0.999901i	$0.504482\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1024.66	−1.51353	−0.756767	−	0.653685i	$-0.773222\pi$
$677$	−1024.66	−1.51353	−0.756767	+	0.653685i	$0.773222\pi$
$678$	0	0
$679$	−138.477	−0.203942
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−965.904	−1.41421	−0.707104	−	0.707109i	$-0.749999\pi$
$683$	−965.904	−1.41421	−0.707104	+	0.707109i	$0.749999\pi$
$684$	0	0
$685$	− 1173.19i	− 1.71268i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 13.5304i	− 0.0196377i
$690$	0	0
$691$	419.233i	0.606705i	0.952878	+	0.303353i	$0.0981060\pi$
$691$	419.233i	0.606705i	−0.952878	+	0.303353i	$0.901894\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	889.210i	1.27944i
$696$	0	0
$697$	423.140	0.607087
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1085.27	1.54817	0.774086	−	0.633081i	$-0.218210\pi$
$701$	1085.27	1.54817	0.774086	+	0.633081i	$0.218210\pi$
$702$	0	0
$703$	−695.803	−0.989763
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−157.113	−0.222225
$708$	0	0
$709$	369.469i	0.521113i	0.965459	+	0.260556i	$0.0839060\pi$
$709$	369.469i	0.521113i	−0.965459	+	0.260556i	$0.916094\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	713.306i	1.00043i
$714$	0	0
$715$	34.4564i	0.0481907i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	468.567i	0.651692i	0.945423	+	0.325846i	$0.105649\pi$
$719$	468.567i	0.651692i	−0.945423	+	0.325846i	$0.894351\pi$
$720$	0	0
$721$	397.140	0.550818
$722$	0	0
$723$	0	0
$724$	0	0
$725$	55.9988	0.0772397
$726$	0	0
$727$	502.570	0.691293	0.345646	−	0.938365i	$-0.387660\pi$
$727$	502.570	0.691293	0.345646	+	0.938365i	$0.387660\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	139.809	0.191257
$732$	0	0
$733$	1189.37i	1.62261i	0.584625	+	0.811303i	$0.301241\pi$
$733$	1189.37i	1.62261i	−0.584625	+	0.811303i	$0.698759\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 491.326i	− 0.666657i
$738$	0	0
$739$	1431.04i	1.93645i	0.250081	+	0.968225i	$0.419543\pi$
$739$	1431.04i	1.93645i	−0.250081	+	0.968225i	$0.580457\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 203.508i	− 0.273900i	−0.990578	−	0.136950i	$-0.956270\pi$
$743$	− 203.508i	− 0.273900i	0.990578	−	0.136950i	$-0.0437300\pi$
$744$	0	0
$745$	310.865	0.417269
$746$	0	0
$747$	0	0
$748$	0	0
$749$	355.034	0.474010
$750$	0	0
$751$	−930.570	−1.23911	−0.619554	−	0.784954i	$-0.712686\pi$
$751$	−930.570	−1.23911	−0.619554	+	0.784954i	$0.712686\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1265.98	−1.67680
$756$	0	0
$757$	− 306.894i	− 0.405408i	−0.979240	−	0.202704i	$-0.935027\pi$
$757$	− 306.894i	− 0.405408i	0.979240	−	0.202704i	$-0.0649730\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	939.701i	1.23482i	0.786640	+	0.617412i	$0.211819\pi$
$761$	939.701i	1.23482i	−0.786640	+	0.617412i	$0.788181\pi$
$762$	0	0
$763$	− 274.762i	− 0.360107i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	101.545i	0.132393i
$768$	0	0
$769$	−1238.19	−1.61013	−0.805063	−	0.593190i	$-0.797868\pi$
$769$	−1238.19	−1.61013	−0.805063	+	0.593190i	$0.797868\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−697.502	−0.902332	−0.451166	−	0.892440i	$-0.648992\pi$
$773$	−697.502	−0.902332	−0.451166	+	0.892440i	$0.648992\pi$
$774$	0	0
$775$	−88.4665	−0.114150
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−771.347	−0.990176
$780$	0	0
$781$	− 104.964i	− 0.134396i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 843.483i	− 1.07450i
$786$	0	0
$787$	− 206.575i	− 0.262484i	−0.991350	−	0.131242i	$-0.958103\pi$
$787$	− 206.575i	− 0.262484i	0.991350	−	0.131242i	$-0.0418965\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 145.056i	− 0.183383i
$792$	0	0
$793$	93.2383	0.117577
$794$	0	0
$795$	0	0
$796$	0	0
$797$	566.596	0.710911	0.355455	−	0.934693i	$-0.384326\pi$
$797$	566.596	0.710911	0.355455	+	0.934693i	$0.384326\pi$
$798$	0	0
$799$	−1178.47	−1.47493
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−672.096	−0.836981
$804$	0	0
$805$	188.383i	0.234017i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	806.091i	0.996404i	0.867061	+	0.498202i	$0.166006\pi$
$809$	806.091i	0.996404i	−0.867061	+	0.498202i	$0.833994\pi$
$810$	0	0
$811$	− 420.472i	− 0.518461i	−0.965816	−	0.259230i	$-0.916531\pi$
$811$	− 420.472i	− 0.518461i	0.965816	−	0.259230i	$-0.0834689\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 422.498i	− 0.518403i
$816$	0	0
$817$	−254.860	−0.311946
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−386.042	−0.470209	−0.235105	−	0.971970i	$-0.575543\pi$
$821$	−386.042	−0.470209	−0.235105	+	0.971970i	$0.575543\pi$
$822$	0	0
$823$	−998.570	−1.21333	−0.606665	−	0.794958i	$-0.707493\pi$
$823$	−998.570	−1.21333	−0.606665	+	0.794958i	$0.707493\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−684.604	−0.827816	−0.413908	−	0.910319i	$-0.635837\pi$
$827$	−684.604	−0.827816	−0.413908	+	0.910319i	$0.635837\pi$
$828$	0	0
$829$	− 533.567i	− 0.643628i	−0.946803	−	0.321814i	$-0.895707\pi$
$829$	− 533.567i	− 0.643628i	0.946803	−	0.321814i	$-0.104293\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	660.632i	0.793076i
$834$	0	0
$835$	935.606i	1.12049i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1441.02i	1.71754i	0.512360	+	0.858771i	$0.328771\pi$
$839$	1441.02i	1.71754i	−0.512360	+	0.858771i	$0.671229\pi$
$840$	0	0
$841$	−215.098	−0.255765
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−872.066	−1.03203
$846$	0	0
$847$	−196.280	−0.231735
$848$	0	0
$849$	0	0
$850$	0	0
$851$	469.241	0.551399
$852$	0	0
$853$	670.383i	0.785913i	0.919557	+	0.392956i	$0.128548\pi$
$853$	670.383i	0.785913i	−0.919557	+	0.392956i	$0.871452\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	223.046i	0.260264i	0.991497	+	0.130132i	$0.0415401\pi$
$857$	223.046i	0.260264i	−0.991497	+	0.130132i	$0.958460\pi$
$858$	0	0
$859$	− 1174.48i	− 1.36726i	−0.729829	−	0.683630i	$-0.760400\pi$
$859$	− 1174.48i	− 1.36726i	0.729829	−	0.683630i	$-0.239600\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	780.785i	0.904734i	0.891832	+	0.452367i	$0.149420\pi$
$863$	780.785i	0.904734i	−0.891832	+	0.452367i	$0.850580\pi$
$864$	0	0
$865$	−138.295	−0.159879
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.27908	0.00262265
$870$	0	0
$871$	−141.897	−0.162912
$872$	0	0
$873$	0	0
$874$	0	0
$875$	237.588	0.271529
$876$	0	0
$877$	− 1096.28i	− 1.25003i	−0.780611	−	0.625017i	$-0.785092\pi$
$877$	− 1096.28i	− 1.25003i	0.780611	−	0.625017i	$-0.214908\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1164.43i	1.32171i	0.750514	+	0.660854i	$0.229806\pi$
$881$	1164.43i	1.32171i	−0.750514	+	0.660854i	$0.770194\pi$
$882$	0	0
$883$	11.8032i	0.0133672i	0.999978	+	0.00668360i	$0.00212747\pi$
$883$	11.8032i	0.0133672i	−0.999978	+	0.00668360i	$0.997873\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 295.505i	− 0.333151i	−0.986029	−	0.166575i	$-0.946729\pi$
$887$	− 295.505i	− 0.333151i	0.986029	−	0.166575i	$-0.0532709\pi$
$888$	0	0
$889$	162.093	0.182332
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2148.24	2.40565
$894$	0	0
$895$	−1600.57	−1.78835
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−988.796	−1.09988
$900$	0	0
$901$	143.852i	0.159659i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 299.473i	− 0.330909i
$906$	0	0
$907$	− 423.617i	− 0.467053i	−0.972350	−	0.233526i	$-0.924974\pi$
$907$	− 423.617i	− 0.467053i	0.972350	−	0.233526i	$-0.0750265\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1008.39i	− 1.10691i	−0.832880	−	0.553454i	$-0.813309\pi$
$911$	− 1008.39i	− 1.10691i	0.832880	−	0.553454i	$-0.186691\pi$
$912$	0	0
$913$	149.720	0.163987
$914$	0	0
$915$	0	0
$916$	0	0
$917$	328.904	0.358674
$918$	0	0
$919$	1067.51	1.16160	0.580802	−	0.814045i	$-0.302739\pi$
$919$	1067.51	1.16160	0.580802	+	0.814045i	$0.302739\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−30.3138	−0.0328427
$924$	0	0
$925$	58.1968i	0.0629154i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 720.842i	− 0.775934i	−0.921673	−	0.387967i	$-0.873178\pi$
$929$	− 720.842i	− 0.775934i	0.921673	−	0.387967i	$-0.126822\pi$
$930$	0	0
$931$	− 1204.27i	− 1.29353i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 366.333i	− 0.391800i
$936$	0	0
$937$	1032.47	1.10189	0.550943	−	0.834543i	$-0.314268\pi$
$937$	1032.47	1.10189	0.550943	+	0.834543i	$0.314268\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1597.62	−1.69779	−0.848893	−	0.528565i	$-0.822730\pi$
$941$	−1597.62	−1.69779	−0.848893	+	0.528565i	$0.822730\pi$
$942$	0	0
$943$	520.187	0.551629
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−625.214	−0.660205	−0.330103	−	0.943945i	$-0.607083\pi$
$947$	−625.214	−0.660205	−0.330103	+	0.943945i	$0.607083\pi$
$948$	0	0
$949$	194.103i	0.204535i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 807.036i	− 0.846838i	−0.905934	−	0.423419i	$-0.860830\pi$
$953$	− 807.036i	− 0.846838i	0.905934	−	0.423419i	$-0.139170\pi$
$954$	0	0
$955$	1131.03i	1.18432i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	449.581i	0.468802i
$960$	0	0
$961$	601.093	0.625487
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−976.690	−1.01211
$966$	0	0
$967$	−1296.47	−1.34071	−0.670355	−	0.742041i	$-0.733858\pi$
$967$	−1296.47	−1.34071	−0.670355	+	0.742041i	$0.733858\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−200.075	−0.206050	−0.103025	−	0.994679i	$-0.532852\pi$
$971$	−200.075	−0.206050	−0.103025	+	0.994679i	$0.532852\pi$
$972$	0	0
$973$	− 340.757i	− 0.350212i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1309.95i	− 1.34079i	−0.742003	−	0.670396i	$-0.766124\pi$
$977$	− 1309.95i	− 1.34079i	0.742003	−	0.670396i	$-0.233876\pi$
$978$	0	0
$979$	62.7820i	0.0641287i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1553.47i	1.58033i	0.612893	+	0.790166i	$0.290006\pi$
$983$	1553.47i	1.58033i	−0.612893	+	0.790166i	$0.709994\pi$
$984$	0	0
$985$	343.907	0.349144
$986$	0	0
$987$	0	0
$988$	0	0
$989$	171.874	0.173786
$990$	0	0
$991$	922.757	0.931137	0.465568	−	0.885012i	$-0.345850\pi$
$991$	922.757	0.931137	0.465568	+	0.885012i	$0.345850\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−599.946	−0.602960
$996$	0	0
$997$	− 787.710i	− 0.790080i	−0.918664	−	0.395040i	$-0.870731\pi$
$997$	− 787.710i	− 0.790080i	0.918664	−	0.395040i	$-0.129269\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.h.l.2177.4		8
3.2	odd	2	inner	2304.3.h.l.2177.6		8
4.3	odd	2		2304.3.h.j.2177.4		8
8.3	odd	2		2304.3.h.j.2177.5		8
8.5	even	2	inner	2304.3.h.l.2177.5		8
12.11	even	2		2304.3.h.j.2177.6		8
16.3	odd	4		1152.3.e.g.1025.3	yes	4
16.5	even	4		1152.3.e.a.1025.2	✓	4
16.11	odd	4		1152.3.e.e.1025.2	yes	4
16.13	even	4		1152.3.e.c.1025.3	yes	4
24.5	odd	2	inner	2304.3.h.l.2177.3		8
24.11	even	2		2304.3.h.j.2177.3		8
48.5	odd	4		1152.3.e.a.1025.3	yes	4
48.11	even	4		1152.3.e.e.1025.3	yes	4
48.29	odd	4		1152.3.e.c.1025.2	yes	4
48.35	even	4		1152.3.e.g.1025.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.e.a.1025.2	✓	4	16.5	even	4
1152.3.e.a.1025.3	yes	4	48.5	odd	4
1152.3.e.c.1025.2	yes	4	48.29	odd	4
1152.3.e.c.1025.3	yes	4	16.13	even	4
1152.3.e.e.1025.2	yes	4	16.11	odd	4
1152.3.e.e.1025.3	yes	4	48.11	even	4
1152.3.e.g.1025.2	yes	4	48.35	even	4
1152.3.e.g.1025.3	yes	4	16.3	odd	4
2304.3.h.j.2177.3		8	24.11	even	2
2304.3.h.j.2177.4		8	4.3	odd	2
2304.3.h.j.2177.5		8	8.3	odd	2
2304.3.h.j.2177.6		8	12.11	even	2
2304.3.h.l.2177.3		8	24.5	odd	2	inner
2304.3.h.l.2177.4		8	1.1	even	1	trivial
2304.3.h.l.2177.5		8	8.5	even	2	inner
2304.3.h.l.2177.6		8	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.21904 q^{5} +2.00000 q^{7} +O(q^{10})\)	\(q-5.21904 q^{5} +2.00000 q^{7} -4.78122 q^{11} +1.38083i q^{13} -14.6807i q^{17} +26.7617i q^{19} -18.0477i q^{23} +2.23834 q^{25} +25.0180 q^{29} -39.5233 q^{31} -10.4381 q^{35} +26.0000i q^{37} +28.8228i q^{41} +9.52333i q^{43} -80.2731i q^{47} -45.0000 q^{49} -9.79874 q^{53} +24.9533 q^{55} +73.5391 q^{59} -67.5233i q^{61} -7.20661i q^{65} +102.762i q^{67} +21.9533i q^{71} +140.570 q^{73} -9.56244 q^{77} -0.476674 q^{79} -31.3142 q^{83} +76.6192i q^{85} -13.1310i q^{89} +2.76166i q^{91} -139.670i q^{95} -69.2383 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7}+O(q^{10}) \) 8 * q + 16 * q^7	\( 8 q + 16 q^{7} + 168 q^{25} - 16 q^{31} - 360 q^{49} + 800 q^{55} + 224 q^{73} - 304 q^{79} - 704 q^{97}+O(q^{100}) \) 8 * q + 16 * q^7 + 168 * q^25 - 16 * q^31 - 360 * q^49 + 800 * q^55 + 224 * q^73 - 304 * q^79 - 704 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more