LMFDB - Embedded newform 2304.3.h.g.2177.2

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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2177.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2304.2177
Dual form			2304.3.h.g.2177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{5} +8.00000 q^{7} +O(q^{10})$	$q-1.41421 q^{5} +8.00000 q^{7} +11.3137 q^{11} +8.00000i q^{13} -12.7279i q^{17} +32.0000i q^{19} +33.9411i q^{23} -23.0000 q^{25} +43.8406 q^{29} -40.0000 q^{31} -11.3137 q^{35} -26.0000i q^{37} +66.4680i q^{41} -16.0000i q^{43} +11.3137i q^{47} +15.0000 q^{49} -32.5269 q^{53} -16.0000 q^{55} +22.6274 q^{59} +54.0000i q^{61} -11.3137i q^{65} -80.0000i q^{67} -79.1960i q^{71} -96.0000 q^{73} +90.5097 q^{77} +104.000 q^{79} +101.823 q^{83} +18.0000i q^{85} +77.7817i q^{89} +64.0000i q^{91} -45.2548i q^{95} -80.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{7}+O(q^{10}) $ 4 * q + 32 * q^7	$ 4 q + 32 q^{7} - 92 q^{25} - 160 q^{31} + 60 q^{49} - 64 q^{55} - 384 q^{73} + 416 q^{79} - 320 q^{97}+O(q^{100}) $ 4 * q + 32 * q^7 - 92 * q^25 - 160 * q^31 + 60 * q^49 - 64 * q^55 - 384 * q^73 + 416 * q^79 - 320 * 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
$5$	−1.41421	−0.282843	−0.141421	−	0.989949i	$-0.545167\pi$
$5$	−1.41421	−0.282843	−0.141421	+	0.989949i	$0.545167\pi$
$6$	0	0
$7$	8.00000	1.14286	0.571429	−	0.820652i	$-0.306389\pi$
$7$	8.00000	1.14286	0.571429	+	0.820652i	$0.306389\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	11.3137	1.02852	0.514259	−	0.857635i	$-0.328067\pi$
$11$	11.3137	1.02852	0.514259	+	0.857635i	$0.328067\pi$
$12$	0	0
$13$	8.00000i	0.615385i	0.951486	+	0.307692i	$0.0995567\pi$
$13$	8.00000i	0.615385i	−0.951486	+	0.307692i	$0.900443\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 12.7279i	− 0.748701i	−0.927287	−	0.374351i	$-0.877866\pi$
$17$	− 12.7279i	− 0.748701i	0.927287	−	0.374351i	$-0.122134\pi$
$18$	0	0
$19$	32.0000i	1.68421i	0.539313	+	0.842105i	$0.318684\pi$
$19$	32.0000i	1.68421i	−0.539313	+	0.842105i	$0.681316\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	33.9411i	1.47570i	0.674964	+	0.737851i	$0.264159\pi$
$23$	33.9411i	1.47570i	−0.674964	+	0.737851i	$0.735841\pi$
$24$	0	0
$25$	−23.0000	−0.920000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	43.8406	1.51175	0.755873	−	0.654719i	$-0.227213\pi$
$29$	43.8406	1.51175	0.755873	+	0.654719i	$0.227213\pi$
$30$	0	0
$31$	−40.0000	−1.29032	−0.645161	−	0.764046i	$-0.723210\pi$
$31$	−40.0000	−1.29032	−0.645161	+	0.764046i	$0.723210\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−11.3137	−0.323249
$36$	0	0
$37$	− 26.0000i	− 0.702703i	−0.936244	−	0.351351i	$-0.885722\pi$
$37$	− 26.0000i	− 0.702703i	0.936244	−	0.351351i	$-0.114278\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	66.4680i	1.62117i	0.585620	+	0.810586i	$0.300851\pi$
$41$	66.4680i	1.62117i	−0.585620	+	0.810586i	$0.699149\pi$
$42$	0	0
$43$	− 16.0000i	− 0.372093i	−0.982541	−	0.186047i	$-0.940432\pi$
$43$	− 16.0000i	− 0.372093i	0.982541	−	0.186047i	$-0.0595675\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.3137i	0.240717i	0.992730	+	0.120359i	$0.0384044\pi$
$47$	11.3137i	0.240717i	−0.992730	+	0.120359i	$0.961596\pi$
$48$	0	0
$49$	15.0000	0.306122
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−32.5269	−0.613715	−0.306858	−	0.951755i	$-0.599278\pi$
$53$	−32.5269	−0.613715	−0.306858	+	0.951755i	$0.599278\pi$
$54$	0	0
$55$	−16.0000	−0.290909
$56$	0	0
$57$	0	0
$58$	0	0
$59$	22.6274	0.383516	0.191758	−	0.981442i	$-0.438581\pi$
$59$	22.6274	0.383516	0.191758	+	0.981442i	$0.438581\pi$
$60$	0	0
$61$	54.0000i	0.885246i	0.896708	+	0.442623i	$0.145952\pi$
$61$	54.0000i	0.885246i	−0.896708	+	0.442623i	$0.854048\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 11.3137i	− 0.174057i
$66$	0	0
$67$	− 80.0000i	− 1.19403i	−0.802230	−	0.597015i	$-0.796353\pi$
$67$	− 80.0000i	− 1.19403i	0.802230	−	0.597015i	$-0.203647\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 79.1960i	− 1.11544i	−0.830030	−	0.557718i	$-0.811677\pi$
$71$	− 79.1960i	− 1.11544i	0.830030	−	0.557718i	$-0.188323\pi$
$72$	0	0
$73$	−96.0000	−1.31507	−0.657534	−	0.753425i	$-0.728401\pi$
$73$	−96.0000	−1.31507	−0.657534	+	0.753425i	$0.728401\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	90.5097	1.17545
$78$	0	0
$79$	104.000	1.31646	0.658228	−	0.752819i	$-0.271306\pi$
$79$	104.000	1.31646	0.658228	+	0.752819i	$0.271306\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	101.823	1.22679	0.613394	−	0.789777i	$-0.289804\pi$
$83$	101.823	1.22679	0.613394	+	0.789777i	$0.289804\pi$
$84$	0	0
$85$	18.0000i	0.211765i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	77.7817i	0.873952i	0.899473	+	0.436976i	$0.143951\pi$
$89$	77.7817i	0.873952i	−0.899473	+	0.436976i	$0.856049\pi$
$90$	0	0
$91$	64.0000i	0.703297i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 45.2548i	− 0.476367i
$96$	0	0
$97$	−80.0000	−0.824742	−0.412371	−	0.911016i	$-0.635299\pi$
$97$	−80.0000	−0.824742	−0.412371	+	0.911016i	$0.635299\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	123.037	1.21818	0.609092	−	0.793100i	$-0.291534\pi$
$101$	123.037	1.21818	0.609092	+	0.793100i	$0.291534\pi$
$102$	0	0
$103$	−72.0000	−0.699029	−0.349515	−	0.936931i	$-0.613653\pi$
$103$	−72.0000	−0.699029	−0.349515	+	0.936931i	$0.613653\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−181.019	−1.69177	−0.845885	−	0.533366i	$-0.820927\pi$
$107$	−181.019	−1.69177	−0.845885	+	0.533366i	$0.820927\pi$
$108$	0	0
$109$	88.0000i	0.807339i	0.914905	+	0.403670i	$0.132266\pi$
$109$	88.0000i	0.807339i	−0.914905	+	0.403670i	$0.867734\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	137.179i	1.21397i	0.794713	+	0.606985i	$0.207621\pi$
$113$	137.179i	1.21397i	−0.794713	+	0.606985i	$0.792379\pi$
$114$	0	0
$115$	− 48.0000i	− 0.417391i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 101.823i	− 0.855659i
$120$	0	0
$121$	7.00000	0.0578512
$122$	0	0
$123$	0	0
$124$	0	0
$125$	67.8823	0.543058
$126$	0	0
$127$	56.0000	0.440945	0.220472	−	0.975393i	$-0.429240\pi$
$127$	56.0000	0.440945	0.220472	+	0.975393i	$0.429240\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−248.902	−1.90001	−0.950006	−	0.312231i	$-0.898924\pi$
$131$	−248.902	−1.90001	−0.950006	+	0.312231i	$0.898924\pi$
$132$	0	0
$133$	256.000i	1.92481i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	46.6690i	0.340650i	0.985388	+	0.170325i	$0.0544818\pi$
$137$	46.6690i	0.340650i	−0.985388	+	0.170325i	$0.945518\pi$
$138$	0	0
$139$	− 16.0000i	− 0.115108i	−0.998342	−	0.0575540i	$-0.981670\pi$
$139$	− 16.0000i	− 0.115108i	0.998342	−	0.0575540i	$-0.0183301\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	90.5097i	0.632935i
$144$	0	0
$145$	−62.0000	−0.427586
$146$	0	0
$147$	0	0
$148$	0	0
$149$	182.434	1.22439	0.612193	−	0.790708i	$-0.290288\pi$
$149$	182.434	1.22439	0.612193	+	0.790708i	$0.290288\pi$
$150$	0	0
$151$	168.000	1.11258	0.556291	−	0.830987i	$-0.312224\pi$
$151$	168.000	1.11258	0.556291	+	0.830987i	$0.312224\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	56.5685	0.364958
$156$	0	0
$157$	10.0000i	0.0636943i	0.999493	+	0.0318471i	$0.0101390\pi$
$157$	10.0000i	0.0636943i	−0.999493	+	0.0318471i	$0.989861\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	271.529i	1.68652i
$162$	0	0
$163$	80.0000i	0.490798i	0.969422	+	0.245399i	$0.0789189\pi$
$163$	80.0000i	0.490798i	−0.969422	+	0.245399i	$0.921081\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	294.156i	1.76142i	0.473660	+	0.880708i	$0.342933\pi$
$167$	294.156i	1.76142i	−0.473660	+	0.880708i	$0.657067\pi$
$168$	0	0
$169$	105.000	0.621302
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−55.1543	−0.318811	−0.159406	−	0.987213i	$-0.550958\pi$
$173$	−55.1543	−0.318811	−0.159406	+	0.987213i	$0.550958\pi$
$174$	0	0
$175$	−184.000	−1.05143
$176$	0	0
$177$	0	0
$178$	0	0
$179$	135.765	0.758461	0.379230	−	0.925302i	$-0.376189\pi$
$179$	135.765	0.758461	0.379230	+	0.925302i	$0.376189\pi$
$180$	0	0
$181$	8.00000i	0.0441989i	0.999756	+	0.0220994i	$0.00703505\pi$
$181$	8.00000i	0.0441989i	−0.999756	+	0.0220994i	$0.992965\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	36.7696i	0.198754i
$186$	0	0
$187$	− 144.000i	− 0.770053i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 67.8823i	− 0.355404i	−0.984084	−	0.177702i	$-0.943134\pi$
$191$	− 67.8823i	− 0.355404i	0.984084	−	0.177702i	$-0.0568664\pi$
$192$	0	0
$193$	−258.000	−1.33679	−0.668394	−	0.743808i	$-0.733018\pi$
$193$	−258.000	−1.33679	−0.668394	+	0.743808i	$0.733018\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	371.938	1.88801	0.944005	−	0.329930i	$-0.107025\pi$
$197$	371.938	1.88801	0.944005	+	0.329930i	$0.107025\pi$
$198$	0	0
$199$	88.0000	0.442211	0.221106	−	0.975250i	$-0.429033\pi$
$199$	88.0000	0.442211	0.221106	+	0.975250i	$0.429033\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	350.725	1.72771
$204$	0	0
$205$	− 94.0000i	− 0.458537i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	362.039i	1.73224i
$210$	0	0
$211$	368.000i	1.74408i	0.489438	+	0.872038i	$0.337202\pi$
$211$	368.000i	1.74408i	−0.489438	+	0.872038i	$0.662798\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	22.6274i	0.105244i
$216$	0	0
$217$	−320.000	−1.47465
$218$	0	0
$219$	0	0
$220$	0	0
$221$	101.823	0.460739
$222$	0	0
$223$	−104.000	−0.466368	−0.233184	−	0.972433i	$-0.574914\pi$
$223$	−104.000	−0.466368	−0.233184	+	0.972433i	$0.574914\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	169.706	0.747602	0.373801	−	0.927509i	$-0.378054\pi$
$227$	169.706	0.747602	0.373801	+	0.927509i	$0.378054\pi$
$228$	0	0
$229$	344.000i	1.50218i	0.660198	+	0.751092i	$0.270472\pi$
$229$	344.000i	1.50218i	−0.660198	+	0.751092i	$0.729528\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	80.6102i	0.345966i	0.984925	+	0.172983i	$0.0553406\pi$
$233$	80.6102i	0.345966i	−0.984925	+	0.172983i	$0.944659\pi$
$234$	0	0
$235$	− 16.0000i	− 0.0680851i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 271.529i	− 1.13610i	−0.822992	−	0.568052i	$-0.807697\pi$
$239$	− 271.529i	− 1.13610i	0.822992	−	0.568052i	$-0.192303\pi$
$240$	0	0
$241$	−272.000	−1.12863	−0.564315	−	0.825559i	$-0.690860\pi$
$241$	−272.000	−1.12863	−0.564315	+	0.825559i	$0.690860\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−21.2132	−0.0865845
$246$	0	0
$247$	−256.000	−1.03644
$248$	0	0
$249$	0	0
$250$	0	0
$251$	33.9411	0.135224	0.0676118	−	0.997712i	$-0.478462\pi$
$251$	33.9411	0.135224	0.0676118	+	0.997712i	$0.478462\pi$
$252$	0	0
$253$	384.000i	1.51779i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	43.8406i	0.170586i	0.996356	+	0.0852930i	$0.0271826\pi$
$257$	43.8406i	0.170586i	−0.996356	+	0.0852930i	$0.972817\pi$
$258$	0	0
$259$	− 208.000i	− 0.803089i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 203.647i	− 0.774322i	−0.922012	−	0.387161i	$-0.873456\pi$
$263$	− 203.647i	− 0.774322i	0.922012	−	0.387161i	$-0.126544\pi$
$264$	0	0
$265$	46.0000	0.173585
$266$	0	0
$267$	0	0
$268$	0	0
$269$	46.6690	0.173491	0.0867454	−	0.996231i	$-0.472353\pi$
$269$	46.6690	0.173491	0.0867454	+	0.996231i	$0.472353\pi$
$270$	0	0
$271$	−264.000	−0.974170	−0.487085	−	0.873355i	$-0.661940\pi$
$271$	−264.000	−0.974170	−0.487085	+	0.873355i	$0.661940\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−260.215	−0.946237
$276$	0	0
$277$	40.0000i	0.144404i	0.997390	+	0.0722022i	$0.0230027\pi$
$277$	40.0000i	0.144404i	−0.997390	+	0.0722022i	$0.976997\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	190.919i	0.679426i	0.940529	+	0.339713i	$0.110330\pi$
$281$	190.919i	0.679426i	−0.940529	+	0.339713i	$0.889670\pi$
$282$	0	0
$283$	224.000i	0.791519i	0.918354	+	0.395760i	$0.129519\pi$
$283$	224.000i	0.791519i	−0.918354	+	0.395760i	$0.870481\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	531.744i	1.85277i
$288$	0	0
$289$	127.000	0.439446
$290$	0	0
$291$	0	0
$292$	0	0
$293$	352.139	1.20184	0.600920	−	0.799309i	$-0.294801\pi$
$293$	352.139	1.20184	0.600920	+	0.799309i	$0.294801\pi$
$294$	0	0
$295$	−32.0000	−0.108475
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−271.529	−0.908124
$300$	0	0
$301$	− 128.000i	− 0.425249i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 76.3675i	− 0.250385i
$306$	0	0
$307$	− 432.000i	− 1.40717i	−0.710613	−	0.703583i	$-0.751582\pi$
$307$	− 432.000i	− 1.40717i	0.710613	−	0.703583i	$-0.248418\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	203.647i	0.654813i	0.944884	+	0.327406i	$0.106175\pi$
$311$	203.647i	0.654813i	−0.944884	+	0.327406i	$0.893825\pi$
$312$	0	0
$313$	14.0000	0.0447284	0.0223642	−	0.999750i	$-0.492881\pi$
$313$	14.0000	0.0447284	0.0223642	+	0.999750i	$0.492881\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	589.727	1.86034	0.930169	−	0.367132i	$-0.119660\pi$
$317$	589.727	1.86034	0.930169	+	0.367132i	$0.119660\pi$
$318$	0	0
$319$	496.000	1.55486
$320$	0	0
$321$	0	0
$322$	0	0
$323$	407.294	1.26097
$324$	0	0
$325$	− 184.000i	− 0.566154i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	90.5097i	0.275105i
$330$	0	0
$331$	− 16.0000i	− 0.0483384i	−0.999708	−	0.0241692i	$-0.992306\pi$
$331$	− 16.0000i	− 0.0483384i	0.999708	−	0.0241692i	$-0.00769404\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	113.137i	0.337723i
$336$	0	0
$337$	128.000	0.379822	0.189911	−	0.981801i	$-0.439180\pi$
$337$	128.000	0.379822	0.189911	+	0.981801i	$0.439180\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−452.548	−1.32712
$342$	0	0
$343$	−272.000	−0.793003
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−350.725	−1.01073	−0.505367	−	0.862904i	$-0.668643\pi$
$347$	−350.725	−1.01073	−0.505367	+	0.862904i	$0.668643\pi$
$348$	0	0
$349$	− 10.0000i	− 0.0286533i	−0.999897	−	0.0143266i	$-0.995440\pi$
$349$	− 10.0000i	− 0.0286533i	0.999897	−	0.0143266i	$-0.00456047\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 247.487i	− 0.701097i	−0.936545	−	0.350549i	$-0.885995\pi$
$353$	− 247.487i	− 0.701097i	0.936545	−	0.350549i	$-0.114005\pi$
$354$	0	0
$355$	112.000i	0.315493i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 328.098i	− 0.913921i	−0.889487	−	0.456960i	$-0.848938\pi$
$359$	− 328.098i	− 0.913921i	0.889487	−	0.456960i	$-0.151062\pi$
$360$	0	0
$361$	−663.000	−1.83657
$362$	0	0
$363$	0	0
$364$	0	0
$365$	135.765	0.371958
$366$	0	0
$367$	696.000	1.89646	0.948229	−	0.317588i	$-0.102873\pi$
$367$	696.000	1.89646	0.948229	+	0.317588i	$0.102873\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−260.215	−0.701389
$372$	0	0
$373$	− 454.000i	− 1.21716i	−0.793493	−	0.608579i	$-0.791740\pi$
$373$	− 454.000i	− 1.21716i	0.793493	−	0.608579i	$-0.208260\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	350.725i	0.930305i
$378$	0	0
$379$	64.0000i	0.168865i	0.996429	+	0.0844327i	$0.0269078\pi$
$379$	64.0000i	0.168865i	−0.996429	+	0.0844327i	$0.973092\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 362.039i	− 0.945271i	−0.881258	−	0.472635i	$-0.843303\pi$
$383$	− 362.039i	− 0.945271i	0.881258	−	0.472635i	$-0.156697\pi$
$384$	0	0
$385$	−128.000	−0.332468
$386$	0	0
$387$	0	0
$388$	0	0
$389$	386.080	0.992494	0.496247	−	0.868181i	$-0.334711\pi$
$389$	386.080	0.992494	0.496247	+	0.868181i	$0.334711\pi$
$390$	0	0
$391$	432.000	1.10486
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−147.078	−0.372350
$396$	0	0
$397$	662.000i	1.66751i	0.552137	+	0.833753i	$0.313812\pi$
$397$	662.000i	1.66751i	−0.552137	+	0.833753i	$0.686188\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 171.120i	− 0.426733i	−0.976972	−	0.213366i	$-0.931557\pi$
$401$	− 171.120i	− 0.426733i	0.976972	−	0.213366i	$-0.0684428\pi$
$402$	0	0
$403$	− 320.000i	− 0.794045i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 294.156i	− 0.722743i
$408$	0	0
$409$	176.000	0.430318	0.215159	−	0.976579i	$-0.430973\pi$
$409$	176.000	0.430318	0.215159	+	0.976579i	$0.430973\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	181.019	0.438303
$414$	0	0
$415$	−144.000	−0.346988
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−79.1960	−0.189012	−0.0945059	−	0.995524i	$-0.530127\pi$
$419$	−79.1960	−0.189012	−0.0945059	+	0.995524i	$0.530127\pi$
$420$	0	0
$421$	488.000i	1.15914i	0.814921	+	0.579572i	$0.196780\pi$
$421$	488.000i	1.15914i	−0.814921	+	0.579572i	$0.803220\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	292.742i	0.688805i
$426$	0	0
$427$	432.000i	1.01171i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 305.470i	− 0.708747i	−0.935104	−	0.354374i	$-0.884694\pi$
$431$	− 305.470i	− 0.708747i	0.935104	−	0.354374i	$-0.115306\pi$
$432$	0	0
$433$	478.000	1.10393	0.551963	−	0.833869i	$-0.313879\pi$
$433$	478.000	1.10393	0.551963	+	0.833869i	$0.313879\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1086.12	−2.48539
$438$	0	0
$439$	392.000	0.892938	0.446469	−	0.894799i	$-0.352681\pi$
$439$	392.000	0.892938	0.446469	+	0.894799i	$0.352681\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	33.9411	0.0766165	0.0383083	−	0.999266i	$-0.487803\pi$
$443$	33.9411	0.0766165	0.0383083	+	0.999266i	$0.487803\pi$
$444$	0	0
$445$	− 110.000i	− 0.247191i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	125.865i	0.280323i	0.990129	+	0.140161i	$0.0447622\pi$
$449$	125.865i	0.280323i	−0.990129	+	0.140161i	$0.955238\pi$
$450$	0	0
$451$	752.000i	1.66741i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 90.5097i	− 0.198922i
$456$	0	0
$457$	16.0000	0.0350109	0.0175055	−	0.999847i	$-0.494428\pi$
$457$	16.0000	0.0350109	0.0175055	+	0.999847i	$0.494428\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−284.257	−0.616609	−0.308305	−	0.951288i	$-0.599762\pi$
$461$	−284.257	−0.616609	−0.308305	+	0.951288i	$0.599762\pi$
$462$	0	0
$463$	−568.000	−1.22678	−0.613391	−	0.789779i	$-0.710195\pi$
$463$	−568.000	−1.22678	−0.613391	+	0.789779i	$0.710195\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−418.607	−0.896375	−0.448188	−	0.893940i	$-0.647930\pi$
$467$	−418.607	−0.896375	−0.448188	+	0.893940i	$0.647930\pi$
$468$	0	0
$469$	− 640.000i	− 1.36461i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 181.019i	− 0.382705i
$474$	0	0
$475$	− 736.000i	− 1.54947i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 33.9411i	− 0.0708583i	−0.999372	−	0.0354291i	$-0.988720\pi$
$479$	− 33.9411i	− 0.0708583i	0.999372	−	0.0354291i	$-0.0112798\pi$
$480$	0	0
$481$	208.000	0.432432
$482$	0	0
$483$	0	0
$484$	0	0
$485$	113.137	0.233272
$486$	0	0
$487$	−424.000	−0.870637	−0.435318	−	0.900277i	$-0.643364\pi$
$487$	−424.000	−0.870637	−0.435318	+	0.900277i	$0.643364\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	724.077	1.47470	0.737350	−	0.675511i	$-0.236077\pi$
$491$	724.077	1.47470	0.737350	+	0.675511i	$0.236077\pi$
$492$	0	0
$493$	− 558.000i	− 1.13185i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 633.568i	− 1.27478i
$498$	0	0
$499$	− 192.000i	− 0.384770i	−0.981320	−	0.192385i	$-0.938378\pi$
$499$	− 192.000i	− 0.384770i	0.981320	−	0.192385i	$-0.0616222\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	441.235i	0.877206i	0.898681	+	0.438603i	$0.144527\pi$
$503$	441.235i	0.877206i	−0.898681	+	0.438603i	$0.855473\pi$
$504$	0	0
$505$	−174.000	−0.344554
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−250.316	−0.491780	−0.245890	−	0.969298i	$-0.579080\pi$
$509$	−250.316	−0.491780	−0.245890	+	0.969298i	$0.579080\pi$
$510$	0	0
$511$	−768.000	−1.50294
$512$	0	0
$513$	0	0
$514$	0	0
$515$	101.823	0.197715
$516$	0	0
$517$	128.000i	0.247582i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 156.978i	− 0.301301i	−0.988587	−	0.150650i	$-0.951863\pi$
$521$	− 156.978i	− 0.301301i	0.988587	−	0.150650i	$-0.0481368\pi$
$522$	0	0
$523$	− 576.000i	− 1.10134i	−0.834724	−	0.550669i	$-0.814373\pi$
$523$	− 576.000i	− 1.10134i	0.834724	−	0.550669i	$-0.185627\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	509.117i	0.966066i
$528$	0	0
$529$	−623.000	−1.17769
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−531.744	−0.997644
$534$	0	0
$535$	256.000	0.478505
$536$	0	0
$537$	0	0
$538$	0	0
$539$	169.706	0.314853
$540$	0	0
$541$	536.000i	0.990758i	0.868677	+	0.495379i	$0.164971\pi$
$541$	536.000i	0.990758i	−0.868677	+	0.495379i	$0.835029\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 124.451i	− 0.228350i
$546$	0	0
$547$	144.000i	0.263254i	0.991299	+	0.131627i	$0.0420201\pi$
$547$	144.000i	0.263254i	−0.991299	+	0.131627i	$0.957980\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1402.90i	2.54610i
$552$	0	0
$553$	832.000	1.50452
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−533.159	−0.957197	−0.478598	−	0.878034i	$-0.658855\pi$
$557$	−533.159	−0.957197	−0.478598	+	0.878034i	$0.658855\pi$
$558$	0	0
$559$	128.000	0.228980
$560$	0	0
$561$	0	0
$562$	0	0
$563$	576.999	1.02487	0.512433	−	0.858727i	$-0.328744\pi$
$563$	576.999	1.02487	0.512433	+	0.858727i	$0.328744\pi$
$564$	0	0
$565$	− 194.000i	− 0.343363i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 994.192i	− 1.74726i	−0.486589	−	0.873631i	$-0.661759\pi$
$569$	− 994.192i	− 1.74726i	0.486589	−	0.873631i	$-0.338241\pi$
$570$	0	0
$571$	− 416.000i	− 0.728546i	−0.931292	−	0.364273i	$-0.881317\pi$
$571$	− 416.000i	− 0.728546i	0.931292	−	0.364273i	$-0.118683\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 780.646i	− 1.35765i
$576$	0	0
$577$	834.000	1.44541	0.722704	−	0.691158i	$-0.242899\pi$
$577$	834.000	1.44541	0.722704	+	0.691158i	$0.242899\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	814.587	1.40204
$582$	0	0
$583$	−368.000	−0.631218
$584$	0	0
$585$	0	0
$586$	0	0
$587$	67.8823	0.115643	0.0578213	−	0.998327i	$-0.481585\pi$
$587$	67.8823	0.115643	0.0578213	+	0.998327i	$0.481585\pi$
$588$	0	0
$589$	− 1280.00i	− 2.17317i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	224.860i	0.379190i	0.981862	+	0.189595i	$0.0607176\pi$
$593$	224.860i	0.379190i	−0.981862	+	0.189595i	$0.939282\pi$
$594$	0	0
$595$	144.000i	0.242017i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 690.136i	− 1.15215i	−0.817398	−	0.576074i	$-0.804584\pi$
$599$	− 690.136i	− 1.15215i	0.817398	−	0.576074i	$-0.195416\pi$
$600$	0	0
$601$	626.000	1.04160	0.520799	−	0.853680i	$-0.325634\pi$
$601$	626.000	1.04160	0.520799	+	0.853680i	$0.325634\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.89949	−0.0163628
$606$	0	0
$607$	−232.000	−0.382208	−0.191104	−	0.981570i	$-0.561207\pi$
$607$	−232.000	−0.382208	−0.191104	+	0.981570i	$0.561207\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−90.5097	−0.148134
$612$	0	0
$613$	666.000i	1.08646i	0.839584	+	0.543230i	$0.182799\pi$
$613$	666.000i	1.08646i	−0.839584	+	0.543230i	$0.817201\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 284.257i	− 0.460708i	−0.973107	−	0.230354i	$-0.926012\pi$
$617$	− 284.257i	− 0.460708i	0.973107	−	0.230354i	$-0.0739884\pi$
$618$	0	0
$619$	768.000i	1.24071i	0.784321	+	0.620355i	$0.213012\pi$
$619$	768.000i	1.24071i	−0.784321	+	0.620355i	$0.786988\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	622.254i	0.998803i
$624$	0	0
$625$	479.000	0.766400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−330.926	−0.526114
$630$	0	0
$631$	472.000	0.748019	0.374010	−	0.927425i	$-0.377983\pi$
$631$	472.000	0.748019	0.374010	+	0.927425i	$0.377983\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−79.1960	−0.124718
$636$	0	0
$637$	120.000i	0.188383i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	371.938i	0.580247i	0.956989	+	0.290123i	$0.0936963\pi$
$641$	371.938i	0.580247i	−0.956989	+	0.290123i	$0.906304\pi$
$642$	0	0
$643$	− 240.000i	− 0.373250i	−0.982431	−	0.186625i	$-0.940245\pi$
$643$	− 240.000i	− 0.373250i	0.982431	−	0.186625i	$-0.0597550\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 712.764i	− 1.10164i	−0.834623	−	0.550822i	$-0.814314\pi$
$647$	− 712.764i	− 1.10164i	0.834623	−	0.550822i	$-0.185686\pi$
$648$	0	0
$649$	256.000	0.394453
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1163.90	−1.78239	−0.891193	−	0.453625i	$-0.850131\pi$
$653$	−1163.90	−1.78239	−0.891193	+	0.453625i	$0.850131\pi$
$654$	0	0
$655$	352.000	0.537405
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−203.647	−0.309024	−0.154512	−	0.987991i	$-0.549381\pi$
$659$	−203.647	−0.309024	−0.154512	+	0.987991i	$0.549381\pi$
$660$	0	0
$661$	− 1018.00i	− 1.54009i	−0.637989	−	0.770045i	$-0.720234\pi$
$661$	− 1018.00i	− 1.54009i	0.637989	−	0.770045i	$-0.279766\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 362.039i	− 0.544419i
$666$	0	0
$667$	1488.00i	2.23088i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	610.940i	0.910492i
$672$	0	0
$673$	−382.000	−0.567608	−0.283804	−	0.958882i	$-0.591596\pi$
$673$	−382.000	−0.567608	−0.283804	+	0.958882i	$0.591596\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	247.487	0.365565	0.182782	−	0.983153i	$-0.441490\pi$
$677$	247.487	0.365565	0.182782	+	0.983153i	$0.441490\pi$
$678$	0	0
$679$	−640.000	−0.942563
$680$	0	0
$681$	0	0
$682$	0	0
$683$	169.706	0.248471	0.124235	−	0.992253i	$-0.460352\pi$
$683$	169.706	0.248471	0.124235	+	0.992253i	$0.460352\pi$
$684$	0	0
$685$	− 66.0000i	− 0.0963504i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 260.215i	− 0.377671i
$690$	0	0
$691$	1040.00i	1.50507i	0.658555	+	0.752533i	$0.271168\pi$
$691$	1040.00i	1.50507i	−0.658555	+	0.752533i	$0.728832\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22.6274i	0.0325574i
$696$	0	0
$697$	846.000	1.21377
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−657.609	−0.938102	−0.469051	−	0.883171i	$-0.655404\pi$
$701$	−657.609	−0.938102	−0.469051	+	0.883171i	$0.655404\pi$
$702$	0	0
$703$	832.000	1.18350
$704$	0	0
$705$	0	0
$706$	0	0
$707$	984.293	1.39221
$708$	0	0
$709$	952.000i	1.34274i	0.741124	+	0.671368i	$0.234293\pi$
$709$	952.000i	1.34274i	−0.741124	+	0.671368i	$0.765707\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1357.65i	− 1.90413i
$714$	0	0
$715$	− 128.000i	− 0.179021i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 192.333i	− 0.267501i	−0.991015	−	0.133750i	$-0.957298\pi$
$719$	− 192.333i	− 0.267501i	0.991015	−	0.133750i	$-0.0427020\pi$
$720$	0	0
$721$	−576.000	−0.798890
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1008.33	−1.39081
$726$	0	0
$727$	−648.000	−0.891334	−0.445667	−	0.895199i	$-0.647033\pi$
$727$	−648.000	−0.891334	−0.445667	+	0.895199i	$0.647033\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−203.647	−0.278587
$732$	0	0
$733$	− 1208.00i	− 1.64802i	−0.566574	−	0.824011i	$-0.691731\pi$
$733$	− 1208.00i	− 1.64802i	0.566574	−	0.824011i	$-0.308269\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 905.097i	− 1.22808i
$738$	0	0
$739$	− 1312.00i	− 1.77537i	−0.460449	−	0.887686i	$-0.652312\pi$
$739$	− 1312.00i	− 1.77537i	0.460449	−	0.887686i	$-0.347688\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 610.940i	− 0.822261i	−0.911576	−	0.411131i	$-0.865134\pi$
$743$	− 610.940i	− 0.822261i	0.911576	−	0.411131i	$-0.134866\pi$
$744$	0	0
$745$	−258.000	−0.346309
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1448.15	−1.93345
$750$	0	0
$751$	632.000	0.841545	0.420772	−	0.907166i	$-0.361759\pi$
$751$	632.000	0.841545	0.420772	+	0.907166i	$0.361759\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−237.588	−0.314686
$756$	0	0
$757$	− 840.000i	− 1.10964i	−0.831969	−	0.554822i	$-0.812786\pi$
$757$	− 840.000i	− 1.10964i	0.831969	−	0.554822i	$-0.187214\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 521.845i	− 0.685736i	−0.939384	−	0.342868i	$-0.888602\pi$
$761$	− 521.845i	− 0.685736i	0.939384	−	0.342868i	$-0.111398\pi$
$762$	0	0
$763$	704.000i	0.922674i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	181.019i	0.236010i
$768$	0	0
$769$	130.000	0.169051	0.0845254	−	0.996421i	$-0.473063\pi$
$769$	130.000	0.169051	0.0845254	+	0.996421i	$0.473063\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	284.257	0.367732	0.183866	−	0.982951i	$-0.441139\pi$
$773$	284.257	0.367732	0.183866	+	0.982951i	$0.441139\pi$
$774$	0	0
$775$	920.000	1.18710
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2126.98	−2.73039
$780$	0	0
$781$	− 896.000i	− 1.14725i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 14.1421i	− 0.0180155i
$786$	0	0
$787$	− 864.000i	− 1.09784i	−0.835875	−	0.548920i	$-0.815039\pi$
$787$	− 864.000i	− 1.09784i	0.835875	−	0.548920i	$-0.184961\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1097.43i	1.38740i
$792$	0	0
$793$	−432.000	−0.544767
$794$	0	0
$795$	0	0
$796$	0	0
$797$	759.433	0.952864	0.476432	−	0.879211i	$-0.341930\pi$
$797$	759.433	0.952864	0.476432	+	0.879211i	$0.341930\pi$
$798$	0	0
$799$	144.000	0.180225
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1086.12	−1.35257
$804$	0	0
$805$	− 384.000i	− 0.477019i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 906.511i	− 1.12053i	−0.828313	−	0.560266i	$-0.810699\pi$
$809$	− 906.511i	− 1.12053i	0.828313	−	0.560266i	$-0.189301\pi$
$810$	0	0
$811$	− 1472.00i	− 1.81504i	−0.420005	−	0.907522i	$-0.637972\pi$
$811$	− 1472.00i	− 1.81504i	0.420005	−	0.907522i	$-0.362028\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 113.137i	− 0.138819i
$816$	0	0
$817$	512.000	0.626683
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1189.35	−1.44866	−0.724332	−	0.689451i	$-0.757852\pi$
$821$	−1189.35	−1.44866	−0.724332	+	0.689451i	$0.757852\pi$
$822$	0	0
$823$	−664.000	−0.806804	−0.403402	−	0.915023i	$-0.632172\pi$
$823$	−664.000	−0.806804	−0.403402	+	0.915023i	$0.632172\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	248.902	0.300969	0.150485	−	0.988612i	$-0.451917\pi$
$827$	248.902	0.300969	0.150485	+	0.988612i	$0.451917\pi$
$828$	0	0
$829$	280.000i	0.337756i	0.985637	+	0.168878i	$0.0540144\pi$
$829$	280.000i	0.337756i	−0.985637	+	0.168878i	$0.945986\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 190.919i	− 0.229194i
$834$	0	0
$835$	− 416.000i	− 0.498204i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 554.372i	− 0.660753i	−0.943849	−	0.330376i	$-0.892824\pi$
$839$	− 554.372i	− 0.660753i	0.943849	−	0.330376i	$-0.107176\pi$
$840$	0	0
$841$	1081.00	1.28537
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−148.492	−0.175731
$846$	0	0
$847$	56.0000	0.0661157
$848$	0	0
$849$	0	0
$850$	0	0
$851$	882.469	1.03698
$852$	0	0
$853$	− 762.000i	− 0.893318i	−0.894704	−	0.446659i	$-0.852614\pi$
$853$	− 762.000i	− 0.893318i	0.894704	−	0.446659i	$-0.147386\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1537.25i	1.79376i	0.442277	+	0.896879i	$0.354171\pi$
$857$	1537.25i	1.79376i	−0.442277	+	0.896879i	$0.645829\pi$
$858$	0	0
$859$	− 1552.00i	− 1.80675i	−0.428849	−	0.903376i	$-0.641081\pi$
$859$	− 1552.00i	− 1.80675i	0.428849	−	0.903376i	$-0.358919\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 22.6274i	− 0.0262195i	−0.999914	−	0.0131097i	$-0.995827\pi$
$863$	− 22.6274i	− 0.0262195i	0.999914	−	0.0131097i	$-0.00417308\pi$
$864$	0	0
$865$	78.0000	0.0901734
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1176.63	1.35400
$870$	0	0
$871$	640.000	0.734788
$872$	0	0
$873$	0	0
$874$	0	0
$875$	543.058	0.620638
$876$	0	0
$877$	86.0000i	0.0980616i	0.998797	+	0.0490308i	$0.0156132\pi$
$877$	86.0000i	0.0980616i	−0.998797	+	0.0490308i	$0.984387\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 657.609i	− 0.746435i	−0.927744	−	0.373218i	$-0.878255\pi$
$881$	− 657.609i	− 0.746435i	0.927744	−	0.373218i	$-0.121745\pi$
$882$	0	0
$883$	496.000i	0.561721i	0.959749	+	0.280861i	$0.0906199\pi$
$883$	496.000i	0.561721i	−0.959749	+	0.280861i	$0.909380\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1018.23i	− 1.14795i	−0.818872	−	0.573976i	$-0.805400\pi$
$887$	− 1018.23i	− 1.14795i	0.818872	−	0.573976i	$-0.194600\pi$
$888$	0	0
$889$	448.000	0.503937
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−362.039	−0.405418
$894$	0	0
$895$	−192.000	−0.214525
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1753.62	−1.95064
$900$	0	0
$901$	414.000i	0.459489i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 11.3137i	− 0.0125013i
$906$	0	0
$907$	880.000i	0.970232i	0.874450	+	0.485116i	$0.161223\pi$
$907$	880.000i	0.970232i	−0.874450	+	0.485116i	$0.838777\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1561.29i	− 1.71382i	−0.515464	−	0.856911i	$-0.672381\pi$
$911$	− 1561.29i	− 1.71382i	0.515464	−	0.856911i	$-0.327619\pi$
$912$	0	0
$913$	1152.00	1.26177
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1991.21	−2.17144
$918$	0	0
$919$	264.000	0.287269	0.143634	−	0.989631i	$-0.454121\pi$
$919$	264.000	0.287269	0.143634	+	0.989631i	$0.454121\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	633.568	0.686422
$924$	0	0
$925$	598.000i	0.646486i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 284.257i	− 0.305982i	−0.988228	−	0.152991i	$-0.951110\pi$
$929$	− 284.257i	− 0.305982i	0.988228	−	0.152991i	$-0.0488905\pi$
$930$	0	0
$931$	480.000i	0.515575i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	203.647i	0.217804i
$936$	0	0
$937$	−1070.00	−1.14194	−0.570971	−	0.820970i	$-0.693433\pi$
$937$	−1070.00	−1.14194	−0.570971	+	0.820970i	$0.693433\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	575.585	0.611674	0.305837	−	0.952084i	$-0.401064\pi$
$941$	575.585	0.611674	0.305837	+	0.952084i	$0.401064\pi$
$942$	0	0
$943$	−2256.00	−2.39236
$944$	0	0
$945$	0	0
$946$	0	0
$947$	610.940	0.645132	0.322566	−	0.946547i	$-0.395455\pi$
$947$	610.940	0.645132	0.322566	+	0.946547i	$0.395455\pi$
$948$	0	0
$949$	− 768.000i	− 0.809273i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1039.45i	− 1.09071i	−0.838205	−	0.545355i	$-0.816395\pi$
$953$	− 1039.45i	− 1.09071i	0.838205	−	0.545355i	$-0.183605\pi$
$954$	0	0
$955$	96.0000i	0.100524i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	373.352i	0.389314i
$960$	0	0
$961$	639.000	0.664932
$962$	0	0
$963$	0	0
$964$	0	0
$965$	364.867	0.378101
$966$	0	0
$967$	696.000	0.719752	0.359876	−	0.933000i	$-0.382819\pi$
$967$	696.000	0.719752	0.359876	+	0.933000i	$0.382819\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1572.61	1.61957	0.809787	−	0.586725i	$-0.199583\pi$
$971$	1572.61	1.61957	0.809787	+	0.586725i	$0.199583\pi$
$972$	0	0
$973$	− 128.000i	− 0.131552i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 507.703i	− 0.519655i	−0.965655	−	0.259827i	$-0.916334\pi$
$977$	− 507.703i	− 0.519655i	0.965655	−	0.259827i	$-0.0836657\pi$
$978$	0	0
$979$	880.000i	0.898876i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	475.176i	0.483393i	0.970352	+	0.241697i	$0.0777039\pi$
$983$	475.176i	0.483393i	−0.970352	+	0.241697i	$0.922296\pi$
$984$	0	0
$985$	−526.000	−0.534010
$986$	0	0
$987$	0	0
$988$	0	0
$989$	543.058	0.549098
$990$	0	0
$991$	−520.000	−0.524723	−0.262361	−	0.964970i	$-0.584501\pi$
$991$	−520.000	−0.524723	−0.262361	+	0.964970i	$0.584501\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−124.451	−0.125076
$996$	0	0
$997$	− 486.000i	− 0.487462i	−0.969843	−	0.243731i	$-0.921629\pi$
$997$	− 486.000i	− 0.487462i	0.969843	−	0.243731i	$-0.0783715\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.h.g.2177.2		4
3.2	odd	2	inner	2304.3.h.g.2177.4		4
4.3	odd	2		2304.3.h.b.2177.2		4
8.3	odd	2		2304.3.h.b.2177.3		4
8.5	even	2	inner	2304.3.h.g.2177.3		4
12.11	even	2		2304.3.h.b.2177.4		4
16.3	odd	4		288.3.e.d.161.2	yes	2
16.5	even	4		576.3.e.b.449.1		2
16.11	odd	4		576.3.e.g.449.1		2
16.13	even	4		288.3.e.a.161.2	yes	2
24.5	odd	2	inner	2304.3.h.g.2177.1		4
24.11	even	2		2304.3.h.b.2177.1		4
48.5	odd	4		576.3.e.b.449.2		2
48.11	even	4		576.3.e.g.449.2		2
48.29	odd	4		288.3.e.a.161.1	✓	2
48.35	even	4		288.3.e.d.161.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.e.a.161.1	✓	2	48.29	odd	4
288.3.e.a.161.2	yes	2	16.13	even	4
288.3.e.d.161.1	yes	2	48.35	even	4
288.3.e.d.161.2	yes	2	16.3	odd	4
576.3.e.b.449.1		2	16.5	even	4
576.3.e.b.449.2		2	48.5	odd	4
576.3.e.g.449.1		2	16.11	odd	4
576.3.e.g.449.2		2	48.11	even	4
2304.3.h.b.2177.1		4	24.11	even	2
2304.3.h.b.2177.2		4	4.3	odd	2
2304.3.h.b.2177.3		4	8.3	odd	2
2304.3.h.b.2177.4		4	12.11	even	2
2304.3.h.g.2177.1		4	24.5	odd	2	inner
2304.3.h.g.2177.2		4	1.1	even	1	trivial
2304.3.h.g.2177.3		4	8.5	even	2	inner
2304.3.h.g.2177.4		4	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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{5} +8.00000 q^{7} +O(q^{10})\)	\(q-1.41421 q^{5} +8.00000 q^{7} +11.3137 q^{11} +8.00000i q^{13} -12.7279i q^{17} +32.0000i q^{19} +33.9411i q^{23} -23.0000 q^{25} +43.8406 q^{29} -40.0000 q^{31} -11.3137 q^{35} -26.0000i q^{37} +66.4680i q^{41} -16.0000i q^{43} +11.3137i q^{47} +15.0000 q^{49} -32.5269 q^{53} -16.0000 q^{55} +22.6274 q^{59} +54.0000i q^{61} -11.3137i q^{65} -80.0000i q^{67} -79.1960i q^{71} -96.0000 q^{73} +90.5097 q^{77} +104.000 q^{79} +101.823 q^{83} +18.0000i q^{85} +77.7817i q^{89} +64.0000i q^{91} -45.2548i q^{95} -80.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{7}+O(q^{10}) \) 4 * q + 32 * q^7	\( 4 q + 32 q^{7} - 92 q^{25} - 160 q^{31} + 60 q^{49} - 64 q^{55} - 384 q^{73} + 416 q^{79} - 320 q^{97}+O(q^{100}) \) 4 * q + 32 * q^7 - 92 * q^25 - 160 * q^31 + 60 * q^49 - 64 * q^55 - 384 * q^73 + 416 * q^79 - 320 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more