LMFDB - Embedded newform 2304.3.b.p.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(127,2304)]

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([1, 1, 0]))

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

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

chi := DirichletCharacter("2304.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$62.7794529086$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2304.127
Dual form			2304.3.b.p.127.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.65685i q^{5} +9.65685i q^{7} +O(q^{10})$	$q-3.65685i q^{5} +9.65685i q^{7} +18.4853 q^{11} -11.6569i q^{13} -9.31371 q^{17} -15.1716 q^{19} -22.3431i q^{23} +11.6274 q^{25} -28.3431i q^{29} +45.2548i q^{31} +35.3137 q^{35} -49.5980i q^{37} -20.6274 q^{41} +46.0833 q^{43} -12.6863i q^{47} -44.2548 q^{49} -27.6569i q^{53} -67.5980i q^{55} +9.11270 q^{59} +113.598i q^{61} -42.6274 q^{65} +45.5147 q^{67} -16.2843i q^{71} +11.9411 q^{73} +178.510i q^{77} -70.0589i q^{79} -94.6518 q^{83} +34.0589i q^{85} +110.569 q^{89} +112.569 q^{91} +55.4802i q^{95} -25.4315 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 40 q^{11} + 8 q^{17} - 72 q^{19} - 44 q^{25} + 96 q^{35} + 8 q^{41} - 8 q^{43} + 4 q^{49} - 88 q^{59} - 80 q^{65} + 216 q^{67} - 88 q^{73} + 40 q^{83} + 216 q^{89} + 224 q^{91} - 328 q^{97}+O(q^{100}) $ 4 * q + 40 * q^11 + 8 * q^17 - 72 * q^19 - 44 * q^25 + 96 * q^35 + 8 * q^41 - 8 * q^43 + 4 * q^49 - 88 * q^59 - 80 * q^65 + 216 * q^67 - 88 * q^73 + 40 * q^83 + 216 * q^89 + 224 * q^91 - 328 * 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
$5$	− 3.65685i	− 0.731371i	−0.930739	−	0.365685i	$-0.880835\pi$
$5$	− 3.65685i	− 0.731371i	0.930739	−	0.365685i	$-0.119165\pi$
$6$	0	0
$7$	9.65685i	1.37955i	0.724024	+	0.689775i	$0.242291\pi$
$7$	9.65685i	1.37955i	−0.724024	+	0.689775i	$0.757709\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	18.4853	1.68048	0.840240	−	0.542215i	$-0.182414\pi$
$11$	18.4853	1.68048	0.840240	+	0.542215i	$0.182414\pi$
$12$	0	0
$13$	− 11.6569i	− 0.896681i	−0.893863	−	0.448341i	$-0.852015\pi$
$13$	− 11.6569i	− 0.896681i	0.893863	−	0.448341i	$-0.147985\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−9.31371	−0.547865	−0.273933	−	0.961749i	$-0.588325\pi$
$17$	−9.31371	−0.547865	−0.273933	+	0.961749i	$0.588325\pi$
$18$	0	0
$19$	−15.1716	−0.798504	−0.399252	−	0.916841i	$-0.630730\pi$
$19$	−15.1716	−0.798504	−0.399252	+	0.916841i	$0.630730\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 22.3431i	− 0.971441i	−0.874114	−	0.485721i	$-0.838557\pi$
$23$	− 22.3431i	− 0.971441i	0.874114	−	0.485721i	$-0.161443\pi$
$24$	0	0
$25$	11.6274	0.465097
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 28.3431i	− 0.977350i	−0.872466	−	0.488675i	$-0.837480\pi$
$29$	− 28.3431i	− 0.977350i	0.872466	−	0.488675i	$-0.162520\pi$
$30$	0	0
$31$	45.2548i	1.45983i	0.683536	+	0.729917i	$0.260441\pi$
$31$	45.2548i	1.45983i	−0.683536	+	0.729917i	$0.739559\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	35.3137	1.00896
$36$	0	0
$37$	− 49.5980i	− 1.34049i	−0.742142	−	0.670243i	$-0.766190\pi$
$37$	− 49.5980i	− 1.34049i	0.742142	−	0.670243i	$-0.233810\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−20.6274	−0.503108	−0.251554	−	0.967843i	$-0.580942\pi$
$41$	−20.6274	−0.503108	−0.251554	+	0.967843i	$0.580942\pi$
$42$	0	0
$43$	46.0833	1.07170	0.535852	−	0.844312i	$-0.319991\pi$
$43$	46.0833	1.07170	0.535852	+	0.844312i	$0.319991\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 12.6863i	− 0.269921i	−0.990851	−	0.134961i	$-0.956909\pi$
$47$	− 12.6863i	− 0.269921i	0.990851	−	0.134961i	$-0.0430908\pi$
$48$	0	0
$49$	−44.2548	−0.903160
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 27.6569i	− 0.521827i	−0.965362	−	0.260914i	$-0.915976\pi$
$53$	− 27.6569i	− 0.521827i	0.965362	−	0.260914i	$-0.0840238\pi$
$54$	0	0
$55$	− 67.5980i	− 1.22905i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.11270	0.154453	0.0772263	−	0.997014i	$-0.475394\pi$
$59$	9.11270	0.154453	0.0772263	+	0.997014i	$0.475394\pi$
$60$	0	0
$61$	113.598i	1.86226i	0.364685	+	0.931131i	$0.381177\pi$
$61$	113.598i	1.86226i	−0.364685	+	0.931131i	$0.618823\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−42.6274	−0.655806
$66$	0	0
$67$	45.5147	0.679324	0.339662	−	0.940548i	$-0.389687\pi$
$67$	45.5147	0.679324	0.339662	+	0.940548i	$0.389687\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 16.2843i	− 0.229356i	−0.993403	−	0.114678i	$-0.963416\pi$
$71$	− 16.2843i	− 0.229356i	0.993403	−	0.114678i	$-0.0365836\pi$
$72$	0	0
$73$	11.9411	0.163577	0.0817885	−	0.996650i	$-0.473937\pi$
$73$	11.9411	0.163577	0.0817885	+	0.996650i	$0.473937\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	178.510i	2.31831i
$78$	0	0
$79$	− 70.0589i	− 0.886821i	−0.896319	−	0.443411i	$-0.853768\pi$
$79$	− 70.0589i	− 0.886821i	0.896319	−	0.443411i	$-0.146232\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−94.6518	−1.14038	−0.570192	−	0.821512i	$-0.693131\pi$
$83$	−94.6518	−1.14038	−0.570192	+	0.821512i	$0.693131\pi$
$84$	0	0
$85$	34.0589i	0.400693i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	110.569	1.24234	0.621172	−	0.783675i	$-0.286657\pi$
$89$	110.569	1.24234	0.621172	+	0.783675i	$0.286657\pi$
$90$	0	0
$91$	112.569	1.23702
$92$	0	0
$93$	0	0
$94$	0	0
$95$	55.4802i	0.584002i
$96$	0	0
$97$	−25.4315	−0.262180	−0.131090	−	0.991370i	$-0.541848\pi$
$97$	−25.4315	−0.262180	−0.131090	+	0.991370i	$0.541848\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	33.5980i	0.332653i	0.986071	+	0.166327i	$0.0531906\pi$
$101$	33.5980i	0.332653i	−0.986071	+	0.166327i	$0.946809\pi$
$102$	0	0
$103$	− 99.5980i	− 0.966971i	−0.875352	−	0.483485i	$-0.839371\pi$
$103$	− 99.5980i	− 0.966971i	0.875352	−	0.483485i	$-0.160629\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	114.485	1.06996	0.534978	−	0.844866i	$-0.320320\pi$
$107$	114.485	1.06996	0.534978	+	0.844866i	$0.320320\pi$
$108$	0	0
$109$	− 70.4020i	− 0.645890i	−0.946418	−	0.322945i	$-0.895327\pi$
$109$	− 70.4020i	− 0.645890i	0.946418	−	0.322945i	$-0.104673\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.5097	−0.110705	−0.0553525	−	0.998467i	$-0.517628\pi$
$113$	−12.5097	−0.110705	−0.0553525	+	0.998467i	$0.517628\pi$
$114$	0	0
$115$	−81.7056	−0.710484
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 89.9411i	− 0.755808i
$120$	0	0
$121$	220.706	1.82401
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 133.941i	− 1.07153i
$126$	0	0
$127$	57.3726i	0.451753i	0.974156	+	0.225876i	$0.0725245\pi$
$127$	57.3726i	0.451753i	−0.974156	+	0.225876i	$0.927475\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	198.368	1.51426	0.757128	−	0.653267i	$-0.226602\pi$
$131$	198.368	1.51426	0.757128	+	0.653267i	$0.226602\pi$
$132$	0	0
$133$	− 146.510i	− 1.10158i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	53.8823	0.393301	0.196651	−	0.980474i	$-0.436994\pi$
$137$	53.8823	0.393301	0.196651	+	0.980474i	$0.436994\pi$
$138$	0	0
$139$	103.456	0.744287	0.372143	−	0.928175i	$-0.378623\pi$
$139$	103.456	0.744287	0.372143	+	0.928175i	$0.378623\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 215.480i	− 1.50685i
$144$	0	0
$145$	−103.647	−0.714805
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.0883i	0.101264i	0.998717	+	0.0506319i	$0.0161235\pi$
$149$	15.0883i	0.101264i	−0.998717	+	0.0506319i	$0.983876\pi$
$150$	0	0
$151$	− 158.676i	− 1.05084i	−0.850844	−	0.525418i	$-0.823909\pi$
$151$	− 158.676i	− 1.05084i	0.850844	−	0.525418i	$-0.176091\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	165.490	1.06768
$156$	0	0
$157$	124.108i	0.790495i	0.918575	+	0.395247i	$0.129341\pi$
$157$	124.108i	0.790495i	−0.918575	+	0.395247i	$0.870659\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	215.765	1.34015
$162$	0	0
$163$	−121.681	−0.746511	−0.373255	−	0.927729i	$-0.621758\pi$
$163$	−121.681	−0.746511	−0.373255	+	0.927729i	$0.621758\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 330.794i	− 1.98080i	−0.138224	−	0.990401i	$-0.544139\pi$
$167$	− 330.794i	− 1.98080i	0.138224	−	0.990401i	$-0.455861\pi$
$168$	0	0
$169$	33.1177	0.195963
$170$	0	0
$171$	0	0
$172$	0	0
$173$	27.6569i	0.159866i	0.996800	+	0.0799331i	$0.0254707\pi$
$173$	27.6569i	0.159866i	−0.996800	+	0.0799331i	$0.974529\pi$
$174$	0	0
$175$	112.284i	0.641624i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	91.8579	0.513172	0.256586	−	0.966521i	$-0.417402\pi$
$179$	91.8579	0.513172	0.256586	+	0.966521i	$0.417402\pi$
$180$	0	0
$181$	− 84.1076i	− 0.464683i	−0.972634	−	0.232342i	$-0.925361\pi$
$181$	− 84.1076i	− 0.464683i	0.972634	−	0.232342i	$-0.0746387\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−181.373	−0.980392
$186$	0	0
$187$	−172.167	−0.920677
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 289.137i	− 1.51381i	−0.653527	−	0.756903i	$-0.726711\pi$
$191$	− 289.137i	− 1.51381i	0.653527	−	0.756903i	$-0.273289\pi$
$192$	0	0
$193$	225.078	1.16621	0.583104	−	0.812397i	$-0.301838\pi$
$193$	225.078	1.16621	0.583104	+	0.812397i	$0.301838\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 238.167i	− 1.20897i	−0.796618	−	0.604484i	$-0.793380\pi$
$197$	− 238.167i	− 1.20897i	0.796618	−	0.604484i	$-0.206620\pi$
$198$	0	0
$199$	− 27.8335i	− 0.139867i	−0.997552	−	0.0699334i	$-0.977721\pi$
$199$	− 27.8335i	− 0.139867i	0.997552	−	0.0699334i	$-0.0222787\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	273.706	1.34830
$204$	0	0
$205$	75.4315i	0.367958i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−280.451	−1.34187
$210$	0	0
$211$	42.7696	0.202699	0.101350	−	0.994851i	$-0.467684\pi$
$211$	42.7696	0.202699	0.101350	+	0.994851i	$0.467684\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 168.520i	− 0.783813i
$216$	0	0
$217$	−437.019	−2.01391
$218$	0	0
$219$	0	0
$220$	0	0
$221$	108.569i	0.491260i
$222$	0	0
$223$	− 301.255i	− 1.35092i	−0.737397	−	0.675459i	$-0.763945\pi$
$223$	− 301.255i	− 1.35092i	0.737397	−	0.675459i	$-0.236055\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	80.7797	0.355858	0.177929	−	0.984043i	$-0.443060\pi$
$227$	80.7797	0.355858	0.177929	+	0.984043i	$0.443060\pi$
$228$	0	0
$229$	195.186i	0.852340i	0.904643	+	0.426170i	$0.140137\pi$
$229$	195.186i	0.852340i	−0.904643	+	0.426170i	$0.859863\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−38.4508	−0.165025	−0.0825124	−	0.996590i	$-0.526294\pi$
$233$	−38.4508	−0.165025	−0.0825124	+	0.996590i	$0.526294\pi$
$234$	0	0
$235$	−46.3919	−0.197412
$236$	0	0
$237$	0	0
$238$	0	0
$239$	188.215i	0.787512i	0.919215	+	0.393756i	$0.128824\pi$
$239$	188.215i	0.787512i	−0.919215	+	0.393756i	$0.871176\pi$
$240$	0	0
$241$	−267.706	−1.11081	−0.555406	−	0.831579i	$-0.687437\pi$
$241$	−267.706	−1.11081	−0.555406	+	0.831579i	$0.687437\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	161.833i	0.660545i
$246$	0	0
$247$	176.853i	0.716003i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−158.652	−0.632079	−0.316039	−	0.948746i	$-0.602353\pi$
$251$	−158.652	−0.632079	−0.316039	+	0.948746i	$0.602353\pi$
$252$	0	0
$253$	− 413.019i	− 1.63249i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−338.000	−1.31518	−0.657588	−	0.753378i	$-0.728423\pi$
$257$	−338.000	−1.31518	−0.657588	+	0.753378i	$0.728423\pi$
$258$	0	0
$259$	478.960	1.84927
$260$	0	0
$261$	0	0
$262$	0	0
$263$	21.2061i	0.0806314i	0.999187	+	0.0403157i	$0.0128364\pi$
$263$	21.2061i	0.0806314i	−0.999187	+	0.0403157i	$0.987164\pi$
$264$	0	0
$265$	−101.137	−0.381649
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 403.637i	− 1.50051i	−0.661150	−	0.750254i	$-0.729931\pi$
$269$	− 403.637i	− 1.50051i	0.661150	−	0.750254i	$-0.270069\pi$
$270$	0	0
$271$	− 50.1766i	− 0.185154i	−0.995706	−	0.0925768i	$-0.970490\pi$
$271$	− 50.1766i	− 0.185154i	0.995706	−	0.0925768i	$-0.0295104\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	214.936	0.781586
$276$	0	0
$277$	229.696i	0.829226i	0.909998	+	0.414613i	$0.136083\pi$
$277$	229.696i	0.829226i	−0.909998	+	0.414613i	$0.863917\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	86.8040	0.308911	0.154456	−	0.988000i	$-0.450638\pi$
$281$	86.8040	0.308911	0.154456	+	0.988000i	$0.450638\pi$
$282$	0	0
$283$	389.848	1.37755	0.688777	−	0.724973i	$-0.258148\pi$
$283$	389.848	1.37755	0.688777	+	0.724973i	$0.258148\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 199.196i	− 0.694063i
$288$	0	0
$289$	−202.255	−0.699844
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 243.657i	− 0.831593i	−0.909458	−	0.415797i	$-0.863503\pi$
$293$	− 243.657i	− 0.831593i	0.909458	−	0.415797i	$-0.136497\pi$
$294$	0	0
$295$	− 33.3238i	− 0.112962i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−260.451	−0.871073
$300$	0	0
$301$	445.019i	1.47847i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	415.411	1.36200
$306$	0	0
$307$	276.240	0.899804	0.449902	−	0.893078i	$-0.351459\pi$
$307$	276.240	0.899804	0.449902	+	0.893078i	$0.351459\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	552.715i	1.77722i	0.458665	+	0.888609i	$0.348328\pi$
$311$	552.715i	1.77722i	−0.458665	+	0.888609i	$0.651672\pi$
$312$	0	0
$313$	372.627	1.19050	0.595251	−	0.803539i	$-0.297052\pi$
$313$	372.627	1.19050	0.595251	+	0.803539i	$0.297052\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	131.657i	0.415321i	0.978201	+	0.207661i	$0.0665850\pi$
$317$	131.657i	0.415321i	−0.978201	+	0.207661i	$0.933415\pi$
$318$	0	0
$319$	− 523.931i	− 1.64242i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	141.304	0.437472
$324$	0	0
$325$	− 135.539i	− 0.417043i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	122.510	0.372370
$330$	0	0
$331$	329.161	0.994446	0.497223	−	0.867623i	$-0.334353\pi$
$331$	329.161	0.994446	0.497223	+	0.867623i	$0.334353\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 166.441i	− 0.496838i
$336$	0	0
$337$	574.784	1.70559	0.852795	−	0.522246i	$-0.174906\pi$
$337$	574.784	1.70559	0.852795	+	0.522246i	$0.174906\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	836.548i	2.45322i
$342$	0	0
$343$	45.8234i	0.133596i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	480.406	1.38446	0.692228	−	0.721679i	$-0.256629\pi$
$347$	480.406	1.38446	0.692228	+	0.721679i	$0.256629\pi$
$348$	0	0
$349$	− 547.186i	− 1.56787i	−0.620844	−	0.783934i	$-0.713210\pi$
$349$	− 547.186i	− 1.56787i	0.620844	−	0.783934i	$-0.286790\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	624.274	1.76848	0.884241	−	0.467031i	$-0.154676\pi$
$353$	624.274	1.76848	0.884241	+	0.467031i	$0.154676\pi$
$354$	0	0
$355$	−59.5492	−0.167744
$356$	0	0
$357$	0	0
$358$	0	0
$359$	280.617i	0.781664i	0.920462	+	0.390832i	$0.127813\pi$
$359$	280.617i	0.781664i	−0.920462	+	0.390832i	$0.872187\pi$
$360$	0	0
$361$	−130.823	−0.362392
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 43.6670i	− 0.119635i
$366$	0	0
$367$	103.196i	0.281188i	0.990067	+	0.140594i	$0.0449012\pi$
$367$	103.196i	0.281188i	−0.990067	+	0.140594i	$0.955099\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	267.078	0.719887
$372$	0	0
$373$	177.147i	0.474925i	0.971397	+	0.237463i	$0.0763158\pi$
$373$	177.147i	0.474925i	−0.971397	+	0.237463i	$0.923684\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−330.392	−0.876371
$378$	0	0
$379$	−356.759	−0.941318	−0.470659	−	0.882315i	$-0.655984\pi$
$379$	−356.759	−0.941318	−0.470659	+	0.882315i	$0.655984\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	106.039i	0.276863i	0.990372	+	0.138432i	$0.0442061\pi$
$383$	106.039i	0.276863i	−0.990372	+	0.138432i	$0.955794\pi$
$384$	0	0
$385$	652.784	1.69554
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 520.912i	− 1.33910i	−0.742765	−	0.669552i	$-0.766486\pi$
$389$	− 520.912i	− 1.33910i	0.742765	−	0.669552i	$-0.233514\pi$
$390$	0	0
$391$	208.098i	0.532219i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−256.195	−0.648595
$396$	0	0
$397$	− 80.4407i	− 0.202621i	−0.994855	−	0.101311i	$-0.967696\pi$
$397$	− 80.4407i	− 0.202621i	0.994855	−	0.101311i	$-0.0323036\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.82338	−0.00953460	−0.00476730	−	0.999989i	$-0.501517\pi$
$401$	−3.82338	−0.00953460	−0.00476730	+	0.999989i	$0.501517\pi$
$402$	0	0
$403$	527.529	1.30900
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 916.833i	− 2.25266i
$408$	0	0
$409$	−378.902	−0.926410	−0.463205	−	0.886251i	$-0.653301\pi$
$409$	−378.902	−0.926410	−0.463205	+	0.886251i	$0.653301\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	88.0000i	0.213075i
$414$	0	0
$415$	346.128i	0.834043i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−603.907	−1.44130	−0.720652	−	0.693297i	$-0.756158\pi$
$419$	−603.907	−1.44130	−0.720652	+	0.693297i	$0.756158\pi$
$420$	0	0
$421$	184.676i	0.438661i	0.975651	+	0.219330i	$0.0703873\pi$
$421$	184.676i	0.438661i	−0.975651	+	0.219330i	$0.929613\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−108.294	−0.254810
$426$	0	0
$427$	−1097.00	−2.56908
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 162.843i	− 0.377825i	−0.981994	−	0.188913i	$-0.939504\pi$
$431$	− 162.843i	− 0.377825i	0.981994	−	0.188913i	$-0.0604963\pi$
$432$	0	0
$433$	205.862	0.475432	0.237716	−	0.971335i	$-0.423601\pi$
$433$	205.862	0.475432	0.237716	+	0.971335i	$0.423601\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	338.981i	0.775699i
$438$	0	0
$439$	197.872i	0.450734i	0.974274	+	0.225367i	$0.0723581\pi$
$439$	197.872i	0.450734i	−0.974274	+	0.225367i	$0.927642\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.1522	0.0590344	0.0295172	−	0.999564i	$-0.490603\pi$
$443$	26.1522	0.0590344	0.0295172	+	0.999564i	$0.490603\pi$
$444$	0	0
$445$	− 404.333i	− 0.908614i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−321.078	−0.715096	−0.357548	−	0.933895i	$-0.616387\pi$
$449$	−321.078	−0.715096	−0.357548	+	0.933895i	$0.616387\pi$
$450$	0	0
$451$	−381.304	−0.845463
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 411.647i	− 0.904718i
$456$	0	0
$457$	−357.882	−0.783112	−0.391556	−	0.920154i	$-0.628063\pi$
$457$	−357.882	−0.783112	−0.391556	+	0.920154i	$0.628063\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	240.912i	0.522585i	0.965260	+	0.261293i	$0.0841487\pi$
$461$	240.912i	0.522585i	−0.965260	+	0.261293i	$0.915851\pi$
$462$	0	0
$463$	− 176.098i	− 0.380340i	−0.981751	−	0.190170i	$-0.939096\pi$
$463$	− 176.098i	− 0.380340i	0.981751	−	0.190170i	$-0.0609040\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	749.092	1.60405	0.802026	−	0.597289i	$-0.203755\pi$
$467$	749.092	1.60405	0.802026	+	0.597289i	$0.203755\pi$
$468$	0	0
$469$	439.529i	0.937162i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	851.862	1.80098
$474$	0	0
$475$	−176.406	−0.371381
$476$	0	0
$477$	0	0
$478$	0	0
$479$	120.235i	0.251014i	0.992093	+	0.125507i	$0.0400557\pi$
$479$	120.235i	0.251014i	−0.992093	+	0.125507i	$0.959944\pi$
$480$	0	0
$481$	−578.156	−1.20199
$482$	0	0
$483$	0	0
$484$	0	0
$485$	92.9991i	0.191751i
$486$	0	0
$487$	624.284i	1.28190i	0.767584	+	0.640949i	$0.221459\pi$
$487$	624.284i	1.28190i	−0.767584	+	0.640949i	$0.778541\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	279.602	0.569455	0.284727	−	0.958609i	$-0.408097\pi$
$491$	279.602	0.569455	0.284727	+	0.958609i	$0.408097\pi$
$492$	0	0
$493$	263.980i	0.535456i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	157.255	0.316408
$498$	0	0
$499$	−260.759	−0.522564	−0.261282	−	0.965263i	$-0.584145\pi$
$499$	−260.759	−0.522564	−0.261282	+	0.965263i	$0.584145\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	735.813i	1.46285i	0.681922	+	0.731425i	$0.261144\pi$
$503$	735.813i	1.46285i	−0.681922	+	0.731425i	$0.738856\pi$
$504$	0	0
$505$	122.863	0.243293
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 508.108i	− 0.998247i	−0.866531	−	0.499123i	$-0.833655\pi$
$509$	− 508.108i	− 0.998247i	0.866531	−	0.499123i	$-0.166345\pi$
$510$	0	0
$511$	115.314i	0.225663i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−364.215	−0.707214
$516$	0	0
$517$	− 234.510i	− 0.453597i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−973.724	−1.86895	−0.934476	−	0.356026i	$-0.884131\pi$
$521$	−973.724	−1.86895	−0.934476	+	0.356026i	$0.884131\pi$
$522$	0	0
$523$	−65.8192	−0.125849	−0.0629247	−	0.998018i	$-0.520043\pi$
$523$	−65.8192	−0.125849	−0.0629247	+	0.998018i	$0.520043\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 421.490i	− 0.799792i
$528$	0	0
$529$	29.7838	0.0563022
$530$	0	0
$531$	0	0
$532$	0	0
$533$	240.451i	0.451127i
$534$	0	0
$535$	− 418.656i	− 0.782535i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−818.063	−1.51774
$540$	0	0
$541$	524.108i	0.968776i	0.874853	+	0.484388i	$0.160958\pi$
$541$	524.108i	0.968776i	−0.874853	+	0.484388i	$0.839042\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−257.450	−0.472385
$546$	0	0
$547$	−552.642	−1.01031	−0.505157	−	0.863027i	$-0.668565\pi$
$547$	−552.642	−1.01031	−0.505157	+	0.863027i	$0.668565\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	430.010i	0.780418i
$552$	0	0
$553$	676.548	1.22341
$554$	0	0
$555$	0	0
$556$	0	0
$557$	374.638i	0.672599i	0.941755	+	0.336299i	$0.109175\pi$
$557$	374.638i	0.672599i	−0.941755	+	0.336299i	$0.890825\pi$
$558$	0	0
$559$	− 537.186i	− 0.960976i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−479.220	−0.851191	−0.425595	−	0.904914i	$-0.639935\pi$
$563$	−479.220	−0.851191	−0.425595	+	0.904914i	$0.639935\pi$
$564$	0	0
$565$	45.7460i	0.0809664i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−696.705	−1.22444	−0.612219	−	0.790689i	$-0.709723\pi$
$569$	−696.705	−1.22444	−0.612219	+	0.790689i	$0.709723\pi$
$570$	0	0
$571$	307.103	0.537833	0.268916	−	0.963164i	$-0.413334\pi$
$571$	307.103	0.537833	0.268916	+	0.963164i	$0.413334\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 259.793i	− 0.451814i
$576$	0	0
$577$	−714.313	−1.23798	−0.618989	−	0.785400i	$-0.712457\pi$
$577$	−714.313	−1.23798	−0.618989	+	0.785400i	$0.712457\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 914.039i	− 1.57322i
$582$	0	0
$583$	− 511.245i	− 0.876921i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−533.279	−0.908482	−0.454241	−	0.890879i	$-0.650090\pi$
$587$	−533.279	−0.908482	−0.454241	+	0.890879i	$0.650090\pi$
$588$	0	0
$589$	− 686.587i	− 1.16568i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−29.4517	−0.0496655	−0.0248328	−	0.999692i	$-0.507905\pi$
$593$	−29.4517	−0.0496655	−0.0248328	+	0.999692i	$0.507905\pi$
$594$	0	0
$595$	−328.902	−0.552776
$596$	0	0
$597$	0	0
$598$	0	0
$599$	699.265i	1.16739i	0.811974	+	0.583694i	$0.198393\pi$
$599$	699.265i	1.16739i	−0.811974	+	0.583694i	$0.801607\pi$
$600$	0	0
$601$	−473.549	−0.787935	−0.393968	−	0.919124i	$-0.628898\pi$
$601$	−473.549	−0.787935	−0.393968	+	0.919124i	$0.628898\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 807.088i	− 1.33403i
$606$	0	0
$607$	696.431i	1.14733i	0.819089	+	0.573666i	$0.194479\pi$
$607$	696.431i	1.14733i	−0.819089	+	0.573666i	$0.805521\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−147.882	−0.242033
$612$	0	0
$613$	− 828.343i	− 1.35129i	−0.737225	−	0.675647i	$-0.763864\pi$
$613$	− 828.343i	− 1.35129i	0.737225	−	0.675647i	$-0.236136\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	885.862	1.43576	0.717878	−	0.696168i	$-0.245113\pi$
$617$	885.862	1.43576	0.717878	+	0.696168i	$0.245113\pi$
$618$	0	0
$619$	1217.40	1.96672	0.983358	−	0.181679i	$-0.0581533\pi$
$619$	1217.40	1.96672	0.983358	+	0.181679i	$0.0581533\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1067.74i	1.71388i
$624$	0	0
$625$	−199.118	−0.318588
$626$	0	0
$627$	0	0
$628$	0	0
$629$	461.941i	0.734406i
$630$	0	0
$631$	261.677i	0.414702i	0.978267	+	0.207351i	$0.0664842\pi$
$631$	261.677i	0.414702i	−0.978267	+	0.207351i	$0.933516\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	209.803	0.330399
$636$	0	0
$637$	515.872i	0.809846i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−923.588	−1.44085	−0.720427	−	0.693530i	$-0.756054\pi$
$641$	−923.588	−1.44085	−0.720427	+	0.693530i	$0.756054\pi$
$642$	0	0
$643$	416.260	0.647372	0.323686	−	0.946165i	$-0.395078\pi$
$643$	416.260	0.647372	0.323686	+	0.946165i	$0.395078\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	745.283i	1.15191i	0.817483	+	0.575953i	$0.195369\pi$
$647$	745.283i	1.15191i	−0.817483	+	0.575953i	$0.804631\pi$
$648$	0	0
$649$	168.451	0.259554
$650$	0	0
$651$	0	0
$652$	0	0
$653$	928.441i	1.42181i	0.703289	+	0.710904i	$0.251714\pi$
$653$	928.441i	1.42181i	−0.703289	+	0.710904i	$0.748286\pi$
$654$	0	0
$655$	− 725.401i	− 1.10748i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1085.71	−1.64751	−0.823756	−	0.566945i	$-0.808125\pi$
$659$	−1085.71	−1.64751	−0.823756	+	0.566945i	$0.808125\pi$
$660$	0	0
$661$	251.421i	0.380365i	0.981749	+	0.190183i	$0.0609080\pi$
$661$	251.421i	0.380365i	−0.981749	+	0.190183i	$0.939092\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−535.765	−0.805661
$666$	0	0
$667$	−633.275	−0.949438
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2099.89i	3.12949i
$672$	0	0
$673$	173.001	0.257059	0.128530	−	0.991706i	$-0.458974\pi$
$673$	173.001	0.257059	0.128530	+	0.991706i	$0.458974\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 856.441i	− 1.26505i	−0.774539	−	0.632526i	$-0.782018\pi$
$677$	− 856.441i	− 1.26505i	0.774539	−	0.632526i	$-0.217982\pi$
$678$	0	0
$679$	− 245.588i	− 0.361691i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−762.201	−1.11596	−0.557980	−	0.829854i	$-0.688424\pi$
$683$	−762.201	−1.11596	−0.557980	+	0.829854i	$0.688424\pi$
$684$	0	0
$685$	− 197.040i	− 0.287649i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−322.392	−0.467913
$690$	0	0
$691$	217.161	0.314271	0.157136	−	0.987577i	$-0.449774\pi$
$691$	217.161	0.314271	0.157136	+	0.987577i	$0.449774\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 378.323i	− 0.544350i
$696$	0	0
$697$	192.118	0.275635
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 663.088i	− 0.945918i	−0.881085	−	0.472959i	$-0.843186\pi$
$701$	− 663.088i	− 0.945918i	0.881085	−	0.472959i	$-0.156814\pi$
$702$	0	0
$703$	752.479i	1.07038i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−324.451	−0.458912
$708$	0	0
$709$	− 912.656i	− 1.28724i	−0.765344	−	0.643622i	$-0.777431\pi$
$709$	− 912.656i	− 1.28724i	0.765344	−	0.643622i	$-0.222569\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1011.14	1.41814
$714$	0	0
$715$	−787.980	−1.10207
$716$	0	0
$717$	0	0
$718$	0	0
$719$	105.275i	0.146419i	0.997317	+	0.0732093i	$0.0233241\pi$
$719$	105.275i	0.146419i	−0.997317	+	0.0732093i	$0.976676\pi$
$720$	0	0
$721$	961.803	1.33398
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 329.558i	− 0.454562i
$726$	0	0
$727$	− 518.246i	− 0.712855i	−0.934323	−	0.356428i	$-0.883995\pi$
$727$	− 518.246i	− 0.712855i	0.934323	−	0.356428i	$-0.116005\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−429.206	−0.587149
$732$	0	0
$733$	469.911i	0.641079i	0.947235	+	0.320539i	$0.103864\pi$
$733$	469.911i	0.641079i	−0.947235	+	0.320539i	$0.896136\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	841.352	1.14159
$738$	0	0
$739$	334.278	0.452339	0.226169	−	0.974088i	$-0.427380\pi$
$739$	334.278	0.452339	0.226169	+	0.974088i	$0.427380\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 937.852i	− 1.26225i	−0.775681	−	0.631125i	$-0.782593\pi$
$743$	− 937.852i	− 1.26225i	0.775681	−	0.631125i	$-0.217407\pi$
$744$	0	0
$745$	55.1758	0.0740614
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1105.57i	1.47606i
$750$	0	0
$751$	1193.67i	1.58943i	0.606980	+	0.794717i	$0.292381\pi$
$751$	1193.67i	1.58943i	−0.606980	+	0.794717i	$0.707619\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−580.256	−0.768551
$756$	0	0
$757$	129.383i	0.170915i	0.996342	+	0.0854575i	$0.0272352\pi$
$757$	129.383i	0.170915i	−0.996342	+	0.0854575i	$0.972765\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1055.45	1.38693	0.693463	−	0.720493i	$-0.256084\pi$
$761$	1055.45	1.38693	0.693463	+	0.720493i	$0.256084\pi$
$762$	0	0
$763$	679.862	0.891038
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 106.225i	− 0.138495i
$768$	0	0
$769$	−285.980	−0.371885	−0.185943	−	0.982561i	$-0.559534\pi$
$769$	−285.980	−0.371885	−0.185943	+	0.982561i	$0.559534\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	688.656i	0.890887i	0.895310	+	0.445444i	$0.146954\pi$
$773$	688.656i	0.890887i	−0.895310	+	0.445444i	$0.853046\pi$
$774$	0	0
$775$	526.197i	0.678964i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	312.950	0.401733
$780$	0	0
$781$	− 301.019i	− 0.385428i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	453.844	0.578145
$786$	0	0
$787$	−1535.27	−1.95079	−0.975393	−	0.220472i	$-0.929240\pi$
$787$	−1535.27	−1.95079	−0.975393	+	0.220472i	$0.929240\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 120.804i	− 0.152723i
$792$	0	0
$793$	1324.20	1.66986
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 311.559i	− 0.390915i	−0.980712	−	0.195458i	$-0.937381\pi$
$797$	− 311.559i	− 0.390915i	0.980712	−	0.195458i	$-0.0626192\pi$
$798$	0	0
$799$	118.156i	0.147880i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	220.735	0.274888
$804$	0	0
$805$	− 789.019i	− 0.980148i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−669.724	−0.827842	−0.413921	−	0.910313i	$-0.635841\pi$
$809$	−669.724	−0.827842	−0.413921	+	0.910313i	$0.635841\pi$
$810$	0	0
$811$	5.94531	0.00733084	0.00366542	−	0.999993i	$-0.498833\pi$
$811$	5.94531	0.00733084	0.00366542	+	0.999993i	$0.498833\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	444.971i	0.545976i
$816$	0	0
$817$	−699.156	−0.855760
$818$	0	0
$819$	0	0
$820$	0	0
$821$	804.814i	0.980285i	0.871642	+	0.490143i	$0.163055\pi$
$821$	804.814i	0.980285i	−0.871642	+	0.490143i	$0.836945\pi$
$822$	0	0
$823$	352.382i	0.428167i	0.976815	+	0.214084i	$0.0686765\pi$
$823$	352.382i	0.428167i	−0.976815	+	0.214084i	$0.931323\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	51.5248	0.0623033	0.0311516	−	0.999515i	$-0.490083\pi$
$827$	51.5248	0.0623033	0.0311516	+	0.999515i	$0.490083\pi$
$828$	0	0
$829$	1243.17i	1.49960i	0.661666	+	0.749798i	$0.269850\pi$
$829$	1243.17i	1.49960i	−0.661666	+	0.749798i	$0.730150\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	412.177	0.494810
$834$	0	0
$835$	−1209.67	−1.44870
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1383.20i	1.64863i	0.566128	+	0.824317i	$0.308441\pi$
$839$	1383.20i	1.64863i	−0.566128	+	0.824317i	$0.691559\pi$
$840$	0	0
$841$	37.6661	0.0447873
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 121.107i	− 0.143322i
$846$	0	0
$847$	2131.32i	2.51632i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1108.17	−1.30220
$852$	0	0
$853$	886.167i	1.03888i	0.854506	+	0.519441i	$0.173860\pi$
$853$	886.167i	1.03888i	−0.854506	+	0.519441i	$0.826140\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−409.647	−0.478001	−0.239000	−	0.971019i	$-0.576820\pi$
$857$	−409.647	−0.478001	−0.239000	+	0.971019i	$0.576820\pi$
$858$	0	0
$859$	506.494	0.589632	0.294816	−	0.955554i	$-0.404742\pi$
$859$	506.494	0.589632	0.294816	+	0.955554i	$0.404742\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1226.04i	− 1.42067i	−0.703863	−	0.710335i	$-0.748543\pi$
$863$	− 1226.04i	− 1.42067i	0.703863	−	0.710335i	$-0.251457\pi$
$864$	0	0
$865$	101.137	0.116921
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1295.06i	− 1.49029i
$870$	0	0
$871$	− 530.558i	− 0.609137i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1293.45	1.47823
$876$	0	0
$877$	1052.73i	1.20038i	0.799857	+	0.600190i	$0.204908\pi$
$877$	1052.73i	1.20038i	−0.799857	+	0.600190i	$0.795092\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	149.294	0.169459	0.0847296	−	0.996404i	$-0.472997\pi$
$881$	149.294	0.169459	0.0847296	+	0.996404i	$0.472997\pi$
$882$	0	0
$883$	−1621.05	−1.83585	−0.917923	−	0.396758i	$-0.870135\pi$
$883$	−1621.05	−1.83585	−0.917923	+	0.396758i	$0.870135\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 266.420i	− 0.300361i	−0.988659	−	0.150181i	$-0.952014\pi$
$887$	− 266.420i	− 0.300361i	0.988659	−	0.150181i	$-0.0479855\pi$
$888$	0	0
$889$	−554.039	−0.623216
$890$	0	0
$891$	0	0
$892$	0	0
$893$	192.471i	0.215533i
$894$	0	0
$895$	− 335.911i	− 0.375319i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1282.66	1.42677
$900$	0	0
$901$	257.588i	0.285891i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−307.569	−0.339856
$906$	0	0
$907$	1060.43	1.16917	0.584584	−	0.811333i	$-0.301258\pi$
$907$	1060.43	1.16917	0.584584	+	0.811333i	$0.301258\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	558.018i	0.612534i	0.951946	+	0.306267i	$0.0990800\pi$
$911$	558.018i	0.612534i	−0.951946	+	0.306267i	$0.900920\pi$
$912$	0	0
$913$	−1749.67	−1.91639
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1915.61i	2.08899i
$918$	0	0
$919$	1271.68i	1.38376i	0.722013	+	0.691880i	$0.243217\pi$
$919$	1271.68i	1.38376i	−0.722013	+	0.691880i	$0.756783\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−189.823	−0.205659
$924$	0	0
$925$	− 576.696i	− 0.623456i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−439.040	−0.472594	−0.236297	−	0.971681i	$-0.575934\pi$
$929$	−439.040	−0.472594	−0.236297	+	0.971681i	$0.575934\pi$
$930$	0	0
$931$	671.415	0.721177
$932$	0	0
$933$	0	0
$934$	0	0
$935$	629.588i	0.673356i
$936$	0	0
$937$	1009.43	1.07730	0.538651	−	0.842529i	$-0.318934\pi$
$937$	1009.43	1.07730	0.538651	+	0.842529i	$0.318934\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1689.54i	1.79547i	0.440536	+	0.897735i	$0.354788\pi$
$941$	1689.54i	1.79547i	−0.440536	+	0.897735i	$0.645212\pi$
$942$	0	0
$943$	460.881i	0.488740i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	61.5635	0.0650090	0.0325045	−	0.999472i	$-0.489652\pi$
$947$	61.5635	0.0650090	0.0325045	+	0.999472i	$0.489652\pi$
$948$	0	0
$949$	− 139.196i	− 0.146676i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	873.960	0.917061	0.458531	−	0.888679i	$-0.348376\pi$
$953$	873.960	0.917061	0.458531	+	0.888679i	$0.348376\pi$
$954$	0	0
$955$	−1057.33	−1.10715
$956$	0	0
$957$	0	0
$958$	0	0
$959$	520.333i	0.542579i
$960$	0	0
$961$	−1087.00	−1.13111
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 823.078i	− 0.852931i
$966$	0	0
$967$	722.363i	0.747015i	0.927627	+	0.373507i	$0.121845\pi$
$967$	722.363i	0.747015i	−0.927627	+	0.373507i	$0.878155\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−475.338	−0.489535	−0.244767	−	0.969582i	$-0.578712\pi$
$971$	−475.338	−0.489535	−0.244767	+	0.969582i	$0.578712\pi$
$972$	0	0
$973$	999.058i	1.02678i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−926.333	−0.948140	−0.474070	−	0.880487i	$-0.657216\pi$
$977$	−926.333	−0.948140	−0.474070	+	0.880487i	$0.657216\pi$
$978$	0	0
$979$	2043.89	2.08773
$980$	0	0
$981$	0	0
$982$	0	0
$983$	248.991i	0.253297i	0.991948	+	0.126648i	$0.0404220\pi$
$983$	248.991i	0.253297i	−0.991948	+	0.126648i	$0.959578\pi$
$984$	0	0
$985$	−870.940	−0.884203
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1029.65i	− 1.04110i
$990$	0	0
$991$	− 680.902i	− 0.687085i	−0.939137	−	0.343543i	$-0.888373\pi$
$991$	− 680.902i	− 0.687085i	0.939137	−	0.343543i	$-0.111627\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−101.783	−0.102294
$996$	0	0
$997$	146.950i	0.147393i	0.997281	+	0.0736963i	$0.0234795\pi$
$997$	146.950i	0.147393i	−0.997281	+	0.0736963i	$0.976520\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.b.p.127.2		4
3.2	odd	2		256.3.d.d.127.4		4
4.3	odd	2		2304.3.b.j.127.2		4
8.3	odd	2	inner	2304.3.b.p.127.3		4
8.5	even	2		2304.3.b.j.127.3		4
12.11	even	2		256.3.d.e.127.2		4
16.3	odd	4		1152.3.g.b.127.2		4
16.5	even	4		1152.3.g.a.127.3		4
16.11	odd	4		1152.3.g.a.127.4		4
16.13	even	4		1152.3.g.b.127.1		4
24.5	odd	2		256.3.d.e.127.1		4
24.11	even	2		256.3.d.d.127.3		4
48.5	odd	4		128.3.c.b.127.2	yes	4
48.11	even	4		128.3.c.b.127.3	yes	4
48.29	odd	4		128.3.c.a.127.3	yes	4
48.35	even	4		128.3.c.a.127.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.c.a.127.2	✓	4	48.35	even	4
128.3.c.a.127.3	yes	4	48.29	odd	4
128.3.c.b.127.2	yes	4	48.5	odd	4
128.3.c.b.127.3	yes	4	48.11	even	4
256.3.d.d.127.3		4	24.11	even	2
256.3.d.d.127.4		4	3.2	odd	2
256.3.d.e.127.1		4	24.5	odd	2
256.3.d.e.127.2		4	12.11	even	2
1152.3.g.a.127.3		4	16.5	even	4
1152.3.g.a.127.4		4	16.11	odd	4
1152.3.g.b.127.1		4	16.13	even	4
1152.3.g.b.127.2		4	16.3	odd	4
2304.3.b.j.127.2		4	4.3	odd	2
2304.3.b.j.127.3		4	8.5	even	2
2304.3.b.p.127.2		4	1.1	even	1	trivial
2304.3.b.p.127.3		4	8.3	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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.65685i q^{5} +9.65685i q^{7} +O(q^{10})\)	\(q-3.65685i q^{5} +9.65685i q^{7} +18.4853 q^{11} -11.6569i q^{13} -9.31371 q^{17} -15.1716 q^{19} -22.3431i q^{23} +11.6274 q^{25} -28.3431i q^{29} +45.2548i q^{31} +35.3137 q^{35} -49.5980i q^{37} -20.6274 q^{41} +46.0833 q^{43} -12.6863i q^{47} -44.2548 q^{49} -27.6569i q^{53} -67.5980i q^{55} +9.11270 q^{59} +113.598i q^{61} -42.6274 q^{65} +45.5147 q^{67} -16.2843i q^{71} +11.9411 q^{73} +178.510i q^{77} -70.0589i q^{79} -94.6518 q^{83} +34.0589i q^{85} +110.569 q^{89} +112.569 q^{91} +55.4802i q^{95} -25.4315 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 40 q^{11} + 8 q^{17} - 72 q^{19} - 44 q^{25} + 96 q^{35} + 8 q^{41} - 8 q^{43} + 4 q^{49} - 88 q^{59} - 80 q^{65} + 216 q^{67} - 88 q^{73} + 40 q^{83} + 216 q^{89} + 224 q^{91} - 328 q^{97}+O(q^{100}) \) 4 * q + 40 * q^11 + 8 * q^17 - 72 * q^19 - 44 * q^25 + 96 * q^35 + 8 * q^41 - 8 * q^43 + 4 * q^49 - 88 * q^59 - 80 * q^65 + 216 * q^67 - 88 * q^73 + 40 * q^83 + 216 * q^89 + 224 * q^91 - 328 * q^97

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