LMFDB - Embedded newform 288.3.e.d.161.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,3,Mod(161,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("288.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	288.e (of order $2$, degree $1$, 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:	$7.84743161358$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	288.161
Dual form			288.3.e.d.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{5} +8.00000 q^{7} +O(q^{10})$	$q+1.41421i q^{5} +8.00000 q^{7} +11.3137i q^{11} -8.00000 q^{13} -12.7279i q^{17} +32.0000 q^{19} +33.9411i q^{23} +23.0000 q^{25} +43.8406i q^{29} +40.0000 q^{31} +11.3137i q^{35} -26.0000 q^{37} -66.4680i q^{41} +16.0000 q^{43} -11.3137i q^{47} +15.0000 q^{49} +32.5269i q^{53} -16.0000 q^{55} +22.6274i q^{59} -54.0000 q^{61} -11.3137i q^{65} -80.0000 q^{67} -79.1960i q^{71} +96.0000 q^{73} +90.5097i q^{77} -104.000 q^{79} -101.823i q^{83} +18.0000 q^{85} -77.7817i q^{89} -64.0000 q^{91} +45.2548i q^{95} -80.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{7}+O(q^{10}) $ 2 * q + 16 * q^7	$ 2 q + 16 q^{7} - 16 q^{13} + 64 q^{19} + 46 q^{25} + 80 q^{31} - 52 q^{37} + 32 q^{43} + 30 q^{49} - 32 q^{55} - 108 q^{61} - 160 q^{67} + 192 q^{73} - 208 q^{79} + 36 q^{85} - 128 q^{91} - 160 q^{97}+O(q^{100}) $ 2 * q + 16 * q^7 - 16 * q^13 + 64 * q^19 + 46 * q^25 + 80 * q^31 - 52 * q^37 + 32 * q^43 + 30 * q^49 - 32 * q^55 - 108 * q^61 - 160 * q^67 + 192 * q^73 - 208 * q^79 + 36 * q^85 - 128 * q^91 - 160 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	1.41421i	0.282843i	0.989949	+	0.141421i	$0.0451672\pi$
$5$	1.41421i	0.282843i	−0.989949	+	0.141421i	$0.954833\pi$
$6$	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
$11$	11.3137i	1.02852i	0.857635	+	0.514259i	$0.171933\pi$
$11$	11.3137i	1.02852i	−0.857635	+	0.514259i	$0.828067\pi$
$12$	0	0
$13$	−8.00000	−0.615385	−0.307692	−	0.951486i	$-0.599557\pi$
$13$	−8.00000	−0.615385	−0.307692	+	0.951486i	$0.599557\pi$
$14$	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
$19$	32.0000	1.68421	0.842105	−	0.539313i	$-0.181316\pi$
$19$	32.0000	1.68421	0.842105	+	0.539313i	$0.181316\pi$
$20$	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
$29$	43.8406i	1.51175i	0.654719	+	0.755873i	$0.272787\pi$
$29$	43.8406i	1.51175i	−0.654719	+	0.755873i	$0.727213\pi$
$30$	0	0
$31$	40.0000	1.29032	0.645161	−	0.764046i	$-0.276790\pi$
$31$	40.0000	1.29032	0.645161	+	0.764046i	$0.276790\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	11.3137i	0.323249i
$36$	0	0
$37$	−26.0000	−0.702703	−0.351351	−	0.936244i	$-0.614278\pi$
$37$	−26.0000	−0.702703	−0.351351	+	0.936244i	$0.614278\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 66.4680i	− 1.62117i	−0.585620	−	0.810586i	$-0.699149\pi$
$41$	− 66.4680i	− 1.62117i	0.585620	−	0.810586i	$-0.300851\pi$
$42$	0	0
$43$	16.0000	0.372093	0.186047	−	0.982541i	$-0.440432\pi$
$43$	16.0000	0.372093	0.186047	+	0.982541i	$0.440432\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 11.3137i	− 0.240717i	−0.992730	−	0.120359i	$-0.961596\pi$
$47$	− 11.3137i	− 0.240717i	0.992730	−	0.120359i	$-0.0384044\pi$
$48$	0	0
$49$	15.0000	0.306122
$50$	0	0
$51$	0	0
$52$	0	0
$53$	32.5269i	0.613715i	0.951755	+	0.306858i	$0.0992776\pi$
$53$	32.5269i	0.613715i	−0.951755	+	0.306858i	$0.900722\pi$
$54$	0	0
$55$	−16.0000	−0.290909
$56$	0	0
$57$	0	0
$58$	0	0
$59$	22.6274i	0.383516i	0.981442	+	0.191758i	$0.0614188\pi$
$59$	22.6274i	0.383516i	−0.981442	+	0.191758i	$0.938581\pi$
$60$	0	0
$61$	−54.0000	−0.885246	−0.442623	−	0.896708i	$-0.645952\pi$
$61$	−54.0000	−0.885246	−0.442623	+	0.896708i	$0.645952\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 11.3137i	− 0.174057i
$66$	0	0
$67$	−80.0000	−1.19403	−0.597015	−	0.802230i	$-0.703647\pi$
$67$	−80.0000	−1.19403	−0.597015	+	0.802230i	$0.703647\pi$
$68$	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	$-0.271599\pi$
$73$	96.0000	1.31507	0.657534	+	0.753425i	$0.271599\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	90.5097i	1.17545i
$78$	0	0
$79$	−104.000	−1.31646	−0.658228	−	0.752819i	$-0.728694\pi$
$79$	−104.000	−1.31646	−0.658228	+	0.752819i	$0.728694\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 101.823i	− 1.22679i	−0.789777	−	0.613394i	$-0.789804\pi$
$83$	− 101.823i	− 1.22679i	0.789777	−	0.613394i	$-0.210196\pi$
$84$	0	0
$85$	18.0000	0.211765
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 77.7817i	− 0.873952i	−0.899473	−	0.436976i	$-0.856049\pi$
$89$	− 77.7817i	− 0.873952i	0.899473	−	0.436976i	$-0.143951\pi$
$90$	0	0
$91$	−64.0000	−0.703297
$92$	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
$101$	− 123.037i	− 1.21818i	−0.793100	−	0.609092i	$-0.791534\pi$
$101$	− 123.037i	− 1.21818i	0.793100	−	0.609092i	$-0.208466\pi$
$102$	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
$107$	− 181.019i	− 1.69177i	−0.533366	−	0.845885i	$-0.679073\pi$
$107$	− 181.019i	− 1.69177i	0.533366	−	0.845885i	$-0.320927\pi$
$108$	0	0
$109$	−88.0000	−0.807339	−0.403670	−	0.914905i	$-0.632266\pi$
$109$	−88.0000	−0.807339	−0.403670	+	0.914905i	$0.632266\pi$
$110$	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
$115$	−48.0000	−0.417391
$116$	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
$125$	67.8823i	0.543058i
$126$	0	0
$127$	−56.0000	−0.440945	−0.220472	−	0.975393i	$-0.570760\pi$
$127$	−56.0000	−0.440945	−0.220472	+	0.975393i	$0.570760\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	248.902i	1.90001i	0.312231	+	0.950006i	$0.398924\pi$
$131$	248.902i	1.90001i	−0.312231	+	0.950006i	$0.601076\pi$
$132$	0	0
$133$	256.000	1.92481
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 46.6690i	− 0.340650i	−0.985388	−	0.170325i	$-0.945518\pi$
$137$	− 46.6690i	− 0.340650i	0.985388	−	0.170325i	$-0.0544818\pi$
$138$	0	0
$139$	16.0000	0.115108	0.0575540	−	0.998342i	$-0.481670\pi$
$139$	16.0000	0.115108	0.0575540	+	0.998342i	$0.481670\pi$
$140$	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
$149$	− 182.434i	− 1.22439i	−0.790708	−	0.612193i	$-0.790288\pi$
$149$	− 182.434i	− 1.22439i	0.790708	−	0.612193i	$-0.209712\pi$
$150$	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
$155$	56.5685i	0.364958i
$156$	0	0
$157$	−10.0000	−0.0636943	−0.0318471	−	0.999493i	$-0.510139\pi$
$157$	−10.0000	−0.0636943	−0.0318471	+	0.999493i	$0.510139\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	271.529i	1.68652i
$162$	0	0
$163$	80.0000	0.490798	0.245399	−	0.969422i	$-0.421081\pi$
$163$	80.0000	0.490798	0.245399	+	0.969422i	$0.421081\pi$
$164$	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
$173$	− 55.1543i	− 0.318811i	−0.987213	−	0.159406i	$-0.949042\pi$
$173$	− 55.1543i	− 0.318811i	0.987213	−	0.159406i	$-0.0509578\pi$
$174$	0	0
$175$	184.000	1.05143
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 135.765i	− 0.758461i	−0.925302	−	0.379230i	$-0.876189\pi$
$179$	− 135.765i	− 0.758461i	0.925302	−	0.379230i	$-0.123811\pi$
$180$	0	0
$181$	8.00000	0.0441989	0.0220994	−	0.999756i	$-0.492965\pi$
$181$	8.00000	0.0441989	0.0220994	+	0.999756i	$0.492965\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 36.7696i	− 0.198754i
$186$	0	0
$187$	144.000	0.770053
$188$	0	0
$189$	0	0
$190$	0	0
$191$	67.8823i	0.355404i	0.984084	+	0.177702i	$0.0568664\pi$
$191$	67.8823i	0.355404i	−0.984084	+	0.177702i	$0.943134\pi$
$192$	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
$197$	− 371.938i	− 1.88801i	−0.329930	−	0.944005i	$-0.607025\pi$
$197$	− 371.938i	− 1.88801i	0.329930	−	0.944005i	$-0.392975\pi$
$198$	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
$203$	350.725i	1.72771i
$204$	0	0
$205$	94.0000	0.458537
$206$	0	0
$207$	0	0
$208$	0	0
$209$	362.039i	1.73224i
$210$	0	0
$211$	368.000	1.74408	0.872038	−	0.489438i	$-0.162798\pi$
$211$	368.000	1.74408	0.872038	+	0.489438i	$0.162798\pi$
$212$	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
$221$	101.823i	0.460739i
$222$	0	0
$223$	104.000	0.466368	0.233184	−	0.972433i	$-0.425086\pi$
$223$	104.000	0.466368	0.233184	+	0.972433i	$0.425086\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 169.706i	− 0.747602i	−0.927509	−	0.373801i	$-0.878054\pi$
$227$	− 169.706i	− 0.747602i	0.927509	−	0.373801i	$-0.121946\pi$
$228$	0	0
$229$	344.000	1.50218	0.751092	−	0.660198i	$-0.229528\pi$
$229$	344.000	1.50218	0.751092	+	0.660198i	$0.229528\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 80.6102i	− 0.345966i	−0.984925	−	0.172983i	$-0.944659\pi$
$233$	− 80.6102i	− 0.345966i	0.984925	−	0.172983i	$-0.0553406\pi$
$234$	0	0
$235$	16.0000	0.0680851
$236$	0	0
$237$	0	0
$238$	0	0
$239$	271.529i	1.13610i	0.822992	+	0.568052i	$0.192303\pi$
$239$	271.529i	1.13610i	−0.822992	+	0.568052i	$0.807697\pi$
$240$	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
$245$	21.2132i	0.0865845i
$246$	0	0
$247$	−256.000	−1.03644
$248$	0	0
$249$	0	0
$250$	0	0
$251$	33.9411i	0.135224i	0.997712	+	0.0676118i	$0.0215379\pi$
$251$	33.9411i	0.135224i	−0.997712	+	0.0676118i	$0.978462\pi$
$252$	0	0
$253$	−384.000	−1.51779
$254$	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
$259$	−208.000	−0.803089
$260$	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
$269$	46.6690i	0.173491i	0.996231	+	0.0867454i	$0.0276467\pi$
$269$	46.6690i	0.173491i	−0.996231	+	0.0867454i	$0.972353\pi$
$270$	0	0
$271$	264.000	0.974170	0.487085	−	0.873355i	$-0.338060\pi$
$271$	264.000	0.974170	0.487085	+	0.873355i	$0.338060\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	260.215i	0.946237i
$276$	0	0
$277$	40.0000	0.144404	0.0722022	−	0.997390i	$-0.476997\pi$
$277$	40.0000	0.144404	0.0722022	+	0.997390i	$0.476997\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 190.919i	− 0.679426i	−0.940529	−	0.339713i	$-0.889670\pi$
$281$	− 190.919i	− 0.679426i	0.940529	−	0.339713i	$-0.110330\pi$
$282$	0	0
$283$	−224.000	−0.791519	−0.395760	−	0.918354i	$-0.629519\pi$
$283$	−224.000	−0.791519	−0.395760	+	0.918354i	$0.629519\pi$
$284$	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
$293$	− 352.139i	− 1.20184i	−0.799309	−	0.600920i	$-0.794801\pi$
$293$	− 352.139i	− 1.20184i	0.799309	−	0.600920i	$-0.205199\pi$
$294$	0	0
$295$	−32.0000	−0.108475
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 271.529i	− 0.908124i
$300$	0	0
$301$	128.000	0.425249
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 76.3675i	− 0.250385i
$306$	0	0
$307$	−432.000	−1.40717	−0.703583	−	0.710613i	$-0.748418\pi$
$307$	−432.000	−1.40717	−0.703583	+	0.710613i	$0.748418\pi$
$308$	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	$-0.507119\pi$
$313$	−14.0000	−0.0447284	−0.0223642	+	0.999750i	$0.507119\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	589.727i	1.86034i	0.367132	+	0.930169i	$0.380340\pi$
$317$	589.727i	1.86034i	−0.367132	+	0.930169i	$0.619660\pi$
$318$	0	0
$319$	−496.000	−1.55486
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 407.294i	− 1.26097i
$324$	0	0
$325$	−184.000	−0.566154
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 90.5097i	− 0.275105i
$330$	0	0
$331$	16.0000	0.0483384	0.0241692	−	0.999708i	$-0.492306\pi$
$331$	16.0000	0.0483384	0.0241692	+	0.999708i	$0.492306\pi$
$332$	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
$341$	452.548i	1.32712i
$342$	0	0
$343$	−272.000	−0.793003
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 350.725i	− 1.01073i	−0.862904	−	0.505367i	$-0.831357\pi$
$347$	− 350.725i	− 1.01073i	0.862904	−	0.505367i	$-0.168643\pi$
$348$	0	0
$349$	10.0000	0.0286533	0.0143266	−	0.999897i	$-0.495440\pi$
$349$	10.0000	0.0286533	0.0143266	+	0.999897i	$0.495440\pi$
$350$	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
$355$	112.000	0.315493
$356$	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
$365$	135.765i	0.371958i
$366$	0	0
$367$	−696.000	−1.89646	−0.948229	−	0.317588i	$-0.897127\pi$
$367$	−696.000	−1.89646	−0.948229	+	0.317588i	$0.897127\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	260.215i	0.701389i
$372$	0	0
$373$	−454.000	−1.21716	−0.608579	−	0.793493i	$-0.708260\pi$
$373$	−454.000	−1.21716	−0.608579	+	0.793493i	$0.708260\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 350.725i	− 0.930305i
$378$	0	0
$379$	−64.0000	−0.168865	−0.0844327	−	0.996429i	$-0.526908\pi$
$379$	−64.0000	−0.168865	−0.0844327	+	0.996429i	$0.526908\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	362.039i	0.945271i	0.881258	+	0.472635i	$0.156697\pi$
$383$	362.039i	0.945271i	−0.881258	+	0.472635i	$0.843303\pi$
$384$	0	0
$385$	−128.000	−0.332468
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 386.080i	− 0.992494i	−0.868181	−	0.496247i	$-0.834711\pi$
$389$	− 386.080i	− 0.992494i	0.868181	−	0.496247i	$-0.165289\pi$
$390$	0	0
$391$	432.000	1.10486
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 147.078i	− 0.372350i
$396$	0	0
$397$	−662.000	−1.66751	−0.833753	−	0.552137i	$-0.813812\pi$
$397$	−662.000	−1.66751	−0.833753	+	0.552137i	$0.813812\pi$
$398$	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
$403$	−320.000	−0.794045
$404$	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	$-0.569027\pi$
$409$	−176.000	−0.430318	−0.215159	+	0.976579i	$0.569027\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	181.019i	0.438303i
$414$	0	0
$415$	144.000	0.346988
$416$	0	0
$417$	0	0
$418$	0	0
$419$	79.1960i	0.189012i	0.995524	+	0.0945059i	$0.0301271\pi$
$419$	79.1960i	0.189012i	−0.995524	+	0.0945059i	$0.969873\pi$
$420$	0	0
$421$	488.000	1.15914	0.579572	−	0.814921i	$-0.303220\pi$
$421$	488.000	1.15914	0.579572	+	0.814921i	$0.303220\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 292.742i	− 0.688805i
$426$	0	0
$427$	−432.000	−1.01171
$428$	0	0
$429$	0	0
$430$	0	0
$431$	305.470i	0.708747i	0.935104	+	0.354374i	$0.115306\pi$
$431$	305.470i	0.708747i	−0.935104	+	0.354374i	$0.884694\pi$
$432$	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
$437$	1086.12i	2.48539i
$438$	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
$443$	33.9411i	0.0766165i	0.999266	+	0.0383083i	$0.0121969\pi$
$443$	33.9411i	0.0766165i	−0.999266	+	0.0383083i	$0.987803\pi$
$444$	0	0
$445$	110.000	0.247191
$446$	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
$451$	752.000	1.66741
$452$	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	$-0.505572\pi$
$457$	−16.0000	−0.0350109	−0.0175055	+	0.999847i	$0.505572\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 284.257i	− 0.616609i	−0.951288	−	0.308305i	$-0.900238\pi$
$461$	− 284.257i	− 0.616609i	0.951288	−	0.308305i	$-0.0997616\pi$
$462$	0	0
$463$	568.000	1.22678	0.613391	−	0.789779i	$-0.289805\pi$
$463$	568.000	1.22678	0.613391	+	0.789779i	$0.289805\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	418.607i	0.896375i	0.893940	+	0.448188i	$0.147930\pi$
$467$	418.607i	0.896375i	−0.893940	+	0.448188i	$0.852070\pi$
$468$	0	0
$469$	−640.000	−1.36461
$470$	0	0
$471$	0	0
$472$	0	0
$473$	181.019i	0.382705i
$474$	0	0
$475$	736.000	1.54947
$476$	0	0
$477$	0	0
$478$	0	0
$479$	33.9411i	0.0708583i	0.999372	+	0.0354291i	$0.0112798\pi$
$479$	33.9411i	0.0708583i	−0.999372	+	0.0354291i	$0.988720\pi$
$480$	0	0
$481$	208.000	0.432432
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 113.137i	− 0.233272i
$486$	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
$491$	724.077i	1.47470i	0.675511	+	0.737350i	$0.263923\pi$
$491$	724.077i	1.47470i	−0.675511	+	0.737350i	$0.736077\pi$
$492$	0	0
$493$	558.000	1.13185
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 633.568i	− 1.27478i
$498$	0	0
$499$	−192.000	−0.384770	−0.192385	−	0.981320i	$-0.561622\pi$
$499$	−192.000	−0.384770	−0.192385	+	0.981320i	$0.561622\pi$
$500$	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
$509$	− 250.316i	− 0.491780i	−0.969298	−	0.245890i	$-0.920920\pi$
$509$	− 250.316i	− 0.491780i	0.969298	−	0.245890i	$-0.0790801\pi$
$510$	0	0
$511$	768.000	1.50294
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 101.823i	− 0.197715i
$516$	0	0
$517$	128.000	0.247582
$518$	0	0
$519$	0	0
$520$	0	0
$521$	156.978i	0.301301i	0.988587	+	0.150650i	$0.0481368\pi$
$521$	156.978i	0.301301i	−0.988587	+	0.150650i	$0.951863\pi$
$522$	0	0
$523$	576.000	1.10134	0.550669	−	0.834724i	$-0.314373\pi$
$523$	576.000	1.10134	0.550669	+	0.834724i	$0.314373\pi$
$524$	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
$533$	531.744i	0.997644i
$534$	0	0
$535$	256.000	0.478505
$536$	0	0
$537$	0	0
$538$	0	0
$539$	169.706i	0.314853i
$540$	0	0
$541$	−536.000	−0.990758	−0.495379	−	0.868677i	$-0.664971\pi$
$541$	−536.000	−0.990758	−0.495379	+	0.868677i	$0.664971\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 124.451i	− 0.228350i
$546$	0	0
$547$	144.000	0.263254	0.131627	−	0.991299i	$-0.457980\pi$
$547$	144.000	0.263254	0.131627	+	0.991299i	$0.457980\pi$
$548$	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
$557$	− 533.159i	− 0.957197i	−0.878034	−	0.478598i	$-0.841145\pi$
$557$	− 533.159i	− 0.957197i	0.878034	−	0.478598i	$-0.158855\pi$
$558$	0	0
$559$	−128.000	−0.228980
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 576.999i	− 1.02487i	−0.858727	−	0.512433i	$-0.828744\pi$
$563$	− 576.999i	− 1.02487i	0.858727	−	0.512433i	$-0.171256\pi$
$564$	0	0
$565$	−194.000	−0.343363
$566$	0	0
$567$	0	0
$568$	0	0
$569$	994.192i	1.74726i	0.486589	+	0.873631i	$0.338241\pi$
$569$	994.192i	1.74726i	−0.486589	+	0.873631i	$0.661759\pi$
$570$	0	0
$571$	416.000	0.728546	0.364273	−	0.931292i	$-0.381317\pi$
$571$	416.000	0.728546	0.364273	+	0.931292i	$0.381317\pi$
$572$	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
$581$	− 814.587i	− 1.40204i
$582$	0	0
$583$	−368.000	−0.631218
$584$	0	0
$585$	0	0
$586$	0	0
$587$	67.8823i	0.115643i	0.998327	+	0.0578213i	$0.0184154\pi$
$587$	67.8823i	0.115643i	−0.998327	+	0.0578213i	$0.981585\pi$
$588$	0	0
$589$	1280.00	2.17317
$590$	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
$595$	144.000	0.242017
$596$	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	$-0.674366\pi$
$601$	−626.000	−1.04160	−0.520799	+	0.853680i	$0.674366\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 9.89949i	− 0.0163628i
$606$	0	0
$607$	232.000	0.382208	0.191104	−	0.981570i	$-0.438793\pi$
$607$	232.000	0.382208	0.191104	+	0.981570i	$0.438793\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	90.5097i	0.148134i
$612$	0	0
$613$	666.000	1.08646	0.543230	−	0.839584i	$-0.317201\pi$
$613$	666.000	1.08646	0.543230	+	0.839584i	$0.317201\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	284.257i	0.460708i	0.973107	+	0.230354i	$0.0739884\pi$
$617$	284.257i	0.460708i	−0.973107	+	0.230354i	$0.926012\pi$
$618$	0	0
$619$	−768.000	−1.24071	−0.620355	−	0.784321i	$-0.713012\pi$
$619$	−768.000	−1.24071	−0.620355	+	0.784321i	$0.713012\pi$
$620$	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
$629$	330.926i	0.526114i
$630$	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
$635$	− 79.1960i	− 0.124718i
$636$	0	0
$637$	−120.000	−0.188383
$638$	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
$643$	−240.000	−0.373250	−0.186625	−	0.982431i	$-0.559755\pi$
$643$	−240.000	−0.373250	−0.186625	+	0.982431i	$0.559755\pi$
$644$	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
$653$	− 1163.90i	− 1.78239i	−0.453625	−	0.891193i	$-0.649869\pi$
$653$	− 1163.90i	− 1.78239i	0.453625	−	0.891193i	$-0.350131\pi$
$654$	0	0
$655$	−352.000	−0.537405
$656$	0	0
$657$	0	0
$658$	0	0
$659$	203.647i	0.309024i	0.987991	+	0.154512i	$0.0493805\pi$
$659$	203.647i	0.309024i	−0.987991	+	0.154512i	$0.950619\pi$
$660$	0	0
$661$	−1018.00	−1.54009	−0.770045	−	0.637989i	$-0.779766\pi$
$661$	−1018.00	−1.54009	−0.770045	+	0.637989i	$0.779766\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	362.039i	0.544419i
$666$	0	0
$667$	−1488.00	−2.23088
$668$	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
$677$	− 247.487i	− 0.365565i	−0.983153	−	0.182782i	$-0.941490\pi$
$677$	− 247.487i	− 0.365565i	0.983153	−	0.182782i	$-0.0585104\pi$
$678$	0	0
$679$	−640.000	−0.942563
$680$	0	0
$681$	0	0
$682$	0	0
$683$	169.706i	0.248471i	0.992253	+	0.124235i	$0.0396478\pi$
$683$	169.706i	0.248471i	−0.992253	+	0.124235i	$0.960352\pi$
$684$	0	0
$685$	66.0000	0.0963504
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 260.215i	− 0.377671i
$690$	0	0
$691$	1040.00	1.50507	0.752533	−	0.658555i	$-0.228832\pi$
$691$	1040.00	1.50507	0.752533	+	0.658555i	$0.228832\pi$
$692$	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
$701$	− 657.609i	− 0.938102i	−0.883171	−	0.469051i	$-0.844596\pi$
$701$	− 657.609i	− 0.938102i	0.883171	−	0.469051i	$-0.155404\pi$
$702$	0	0
$703$	−832.000	−1.18350
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 984.293i	− 1.39221i
$708$	0	0
$709$	952.000	1.34274	0.671368	−	0.741124i	$-0.265707\pi$
$709$	952.000	1.34274	0.671368	+	0.741124i	$0.265707\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1357.65i	1.90413i
$714$	0	0
$715$	128.000	0.179021
$716$	0	0
$717$	0	0
$718$	0	0
$719$	192.333i	0.267501i	0.991015	+	0.133750i	$0.0427020\pi$
$719$	192.333i	0.267501i	−0.991015	+	0.133750i	$0.957298\pi$
$720$	0	0
$721$	−576.000	−0.798890
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1008.33i	1.39081i
$726$	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
$731$	− 203.647i	− 0.278587i
$732$	0	0
$733$	1208.00	1.64802	0.824011	−	0.566574i	$-0.191731\pi$
$733$	1208.00	1.64802	0.824011	+	0.566574i	$0.191731\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 905.097i	− 1.22808i
$738$	0	0
$739$	−1312.00	−1.77537	−0.887686	−	0.460449i	$-0.847688\pi$
$739$	−1312.00	−1.77537	−0.887686	+	0.460449i	$0.847688\pi$
$740$	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
$749$	− 1448.15i	− 1.93345i
$750$	0	0
$751$	−632.000	−0.841545	−0.420772	−	0.907166i	$-0.638241\pi$
$751$	−632.000	−0.841545	−0.420772	+	0.907166i	$0.638241\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	237.588i	0.314686i
$756$	0	0
$757$	−840.000	−1.10964	−0.554822	−	0.831969i	$-0.687214\pi$
$757$	−840.000	−1.10964	−0.554822	+	0.831969i	$0.687214\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	521.845i	0.685736i	0.939384	+	0.342868i	$0.111398\pi$
$761$	521.845i	0.685736i	−0.939384	+	0.342868i	$0.888602\pi$
$762$	0	0
$763$	−704.000	−0.922674
$764$	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
$773$	− 284.257i	− 0.367732i	−0.982951	−	0.183866i	$-0.941139\pi$
$773$	− 284.257i	− 0.367732i	0.982951	−	0.183866i	$-0.0588613\pi$
$774$	0	0
$775$	920.000	1.18710
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2126.98i	− 2.73039i
$780$	0	0
$781$	896.000	1.14725
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 14.1421i	− 0.0180155i
$786$	0	0
$787$	−864.000	−1.09784	−0.548920	−	0.835875i	$-0.684961\pi$
$787$	−864.000	−1.09784	−0.548920	+	0.835875i	$0.684961\pi$
$788$	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
$797$	759.433i	0.952864i	0.879211	+	0.476432i	$0.158070\pi$
$797$	759.433i	0.952864i	−0.879211	+	0.476432i	$0.841930\pi$
$798$	0	0
$799$	−144.000	−0.180225
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1086.12i	1.35257i
$804$	0	0
$805$	−384.000	−0.477019
$806$	0	0
$807$	0	0
$808$	0	0
$809$	906.511i	1.12053i	0.828313	+	0.560266i	$0.189301\pi$
$809$	906.511i	1.12053i	−0.828313	+	0.560266i	$0.810699\pi$
$810$	0	0
$811$	1472.00	1.81504	0.907522	−	0.420005i	$-0.137972\pi$
$811$	1472.00	1.81504	0.907522	+	0.420005i	$0.137972\pi$
$812$	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
$821$	1189.35i	1.44866i	0.689451	+	0.724332i	$0.257852\pi$
$821$	1189.35i	1.44866i	−0.689451	+	0.724332i	$0.742148\pi$
$822$	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
$827$	248.902i	0.300969i	0.988612	+	0.150485i	$0.0480834\pi$
$827$	248.902i	0.300969i	−0.988612	+	0.150485i	$0.951917\pi$
$828$	0	0
$829$	−280.000	−0.337756	−0.168878	−	0.985637i	$-0.554014\pi$
$829$	−280.000	−0.337756	−0.168878	+	0.985637i	$0.554014\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 190.919i	− 0.229194i
$834$	0	0
$835$	−416.000	−0.498204
$836$	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
$845$	− 148.492i	− 0.175731i
$846$	0	0
$847$	−56.0000	−0.0661157
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 882.469i	− 1.03698i
$852$	0	0
$853$	−762.000	−0.893318	−0.446659	−	0.894704i	$-0.647386\pi$
$853$	−762.000	−0.893318	−0.446659	+	0.894704i	$0.647386\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1537.25i	− 1.79376i	−0.442277	−	0.896879i	$-0.645829\pi$
$857$	− 1537.25i	− 1.79376i	0.442277	−	0.896879i	$-0.354171\pi$
$858$	0	0
$859$	1552.00	1.80675	0.903376	−	0.428849i	$-0.141081\pi$
$859$	1552.00	1.80675	0.903376	+	0.428849i	$0.141081\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22.6274i	0.0262195i	0.999914	+	0.0131097i	$0.00417308\pi$
$863$	22.6274i	0.0262195i	−0.999914	+	0.0131097i	$0.995827\pi$
$864$	0	0
$865$	78.0000	0.0901734
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1176.63i	− 1.35400i
$870$	0	0
$871$	640.000	0.734788
$872$	0	0
$873$	0	0
$874$	0	0
$875$	543.058i	0.620638i
$876$	0	0
$877$	−86.0000	−0.0980616	−0.0490308	−	0.998797i	$-0.515613\pi$
$877$	−86.0000	−0.0980616	−0.0490308	+	0.998797i	$0.515613\pi$
$878$	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
$883$	496.000	0.561721	0.280861	−	0.959749i	$-0.409380\pi$
$883$	496.000	0.561721	0.280861	+	0.959749i	$0.409380\pi$
$884$	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
$893$	− 362.039i	− 0.405418i
$894$	0	0
$895$	192.000	0.214525
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1753.62i	1.95064i
$900$	0	0
$901$	414.000	0.459489
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.3137i	0.0125013i
$906$	0	0
$907$	−880.000	−0.970232	−0.485116	−	0.874450i	$-0.661223\pi$
$907$	−880.000	−0.970232	−0.485116	+	0.874450i	$0.661223\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1561.29i	1.71382i	0.515464	+	0.856911i	$0.327619\pi$
$911$	1561.29i	1.71382i	−0.515464	+	0.856911i	$0.672381\pi$
$912$	0	0
$913$	1152.00	1.26177
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1991.21i	2.17144i
$918$	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
$923$	633.568i	0.686422i
$924$	0	0
$925$	−598.000	−0.646486
$926$	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
$931$	480.000	0.515575
$932$	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	$-0.306567\pi$
$937$	1070.00	1.14194	0.570971	+	0.820970i	$0.306567\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	575.585i	0.611674i	0.952084	+	0.305837i	$0.0989362\pi$
$941$	575.585i	0.611674i	−0.952084	+	0.305837i	$0.901064\pi$
$942$	0	0
$943$	2256.00	2.39236
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 610.940i	− 0.645132i	−0.946547	−	0.322566i	$-0.895455\pi$
$947$	− 610.940i	− 0.645132i	0.946547	−	0.322566i	$-0.104545\pi$
$948$	0	0
$949$	−768.000	−0.809273
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1039.45i	1.09071i	0.838205	+	0.545355i	$0.183605\pi$
$953$	1039.45i	1.09071i	−0.838205	+	0.545355i	$0.816395\pi$
$954$	0	0
$955$	−96.0000	−0.100524
$956$	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
$965$	− 364.867i	− 0.378101i
$966$	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
$971$	1572.61i	1.61957i	0.586725	+	0.809787i	$0.300417\pi$
$971$	1572.61i	1.61957i	−0.586725	+	0.809787i	$0.699583\pi$
$972$	0	0
$973$	128.000	0.131552
$974$	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
$979$	880.000	0.898876
$980$	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
$989$	543.058i	0.549098i
$990$	0	0
$991$	520.000	0.524723	0.262361	−	0.964970i	$-0.415499\pi$
$991$	520.000	0.524723	0.262361	+	0.964970i	$0.415499\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	124.451i	0.125076i
$996$	0	0
$997$	−486.000	−0.487462	−0.243731	−	0.969843i	$-0.578371\pi$
$997$	−486.000	−0.487462	−0.243731	+	0.969843i	$0.578371\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.e.a.161.1	✓	2	12.11	even	2
288.3.e.a.161.2	yes	2	4.3	odd	2
288.3.e.d.161.1	yes	2	3.2	odd	2	inner
288.3.e.d.161.2	yes	2	1.1	even	1	trivial
576.3.e.b.449.1		2	8.3	odd	2
576.3.e.b.449.2		2	24.11	even	2
576.3.e.g.449.1		2	8.5	even	2
576.3.e.g.449.2		2	24.5	odd	2
2304.3.h.b.2177.1		4	48.29	odd	4
2304.3.h.b.2177.2		4	16.5	even	4
2304.3.h.b.2177.3		4	16.13	even	4
2304.3.h.b.2177.4		4	48.5	odd	4
2304.3.h.g.2177.1		4	48.35	even	4
2304.3.h.g.2177.2		4	16.11	odd	4
2304.3.h.g.2177.3		4	16.3	odd	4
2304.3.h.g.2177.4		4	48.11	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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	288.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{5} +8.00000 q^{7} +O(q^{10})\)	\(q+1.41421i q^{5} +8.00000 q^{7} +11.3137i q^{11} -8.00000 q^{13} -12.7279i q^{17} +32.0000 q^{19} +33.9411i q^{23} +23.0000 q^{25} +43.8406i q^{29} +40.0000 q^{31} +11.3137i q^{35} -26.0000 q^{37} -66.4680i q^{41} +16.0000 q^{43} -11.3137i q^{47} +15.0000 q^{49} +32.5269i q^{53} -16.0000 q^{55} +22.6274i q^{59} -54.0000 q^{61} -11.3137i q^{65} -80.0000 q^{67} -79.1960i q^{71} +96.0000 q^{73} +90.5097i q^{77} -104.000 q^{79} -101.823i q^{83} +18.0000 q^{85} -77.7817i q^{89} -64.0000 q^{91} +45.2548i q^{95} -80.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{7}+O(q^{10}) \) 2 * q + 16 * q^7	\( 2 q + 16 q^{7} - 16 q^{13} + 64 q^{19} + 46 q^{25} + 80 q^{31} - 52 q^{37} + 32 q^{43} + 30 q^{49} - 32 q^{55} - 108 q^{61} - 160 q^{67} + 192 q^{73} - 208 q^{79} + 36 q^{85} - 128 q^{91} - 160 q^{97}+O(q^{100}) \) 2 * q + 16 * q^7 - 16 * q^13 + 64 * q^19 + 46 * q^25 + 80 * q^31 - 52 * q^37 + 32 * q^43 + 30 * q^49 - 32 * q^55 - 108 * q^61 - 160 * q^67 + 192 * q^73 - 208 * q^79 + 36 * q^85 - 128 * q^91 - 160 * q^97

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