LMFDB - Embedded newform 2304.3.b.j.127.1

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			127.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2304.127
Dual form			2304.3.b.j.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.65685i q^{5} -1.65685i q^{7} -1.51472 q^{11} +0.343146i q^{13} +13.3137 q^{17} +20.8284 q^{19} -33.6569i q^{23} -33.6274 q^{25} +39.6569i q^{29} -45.2548i q^{31} -12.6863 q^{35} -29.5980i q^{37} +24.6274 q^{41} +50.0833 q^{43} -35.3137i q^{47} +46.2548 q^{49} +16.3431i q^{53} +11.5980i q^{55} +53.1127 q^{59} -34.4020i q^{61} +2.62742 q^{65} -62.4853 q^{67} +40.2843i q^{71} -55.9411 q^{73} +2.50967i q^{77} -137.941i q^{79} -114.652 q^{83} -101.941i q^{85} -2.56854 q^{89} +0.568542 q^{91} -159.480i q^{95} -138.569 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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

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
$5$	− 7.65685i	− 1.53137i	−0.643215	−	0.765685i	$-0.722400\pi$
$5$	− 7.65685i	− 1.53137i	0.643215	−	0.765685i	$-0.277600\pi$
$6$	0	0
$7$	− 1.65685i	− 0.236693i	−0.992972	−	0.118347i	$-0.962241\pi$
$7$	− 1.65685i	− 0.236693i	0.992972	−	0.118347i	$-0.0377594\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.51472	−0.137702	−0.0688508	−	0.997627i	$-0.521933\pi$
$11$	−1.51472	−0.137702	−0.0688508	+	0.997627i	$0.521933\pi$
$12$	0	0
$13$	0.343146i	0.0263958i	0.999913	+	0.0131979i	$0.00420115\pi$
$13$	0.343146i	0.0263958i	−0.999913	+	0.0131979i	$0.995799\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	13.3137	0.783159	0.391580	−	0.920144i	$-0.371929\pi$
$17$	13.3137	0.783159	0.391580	+	0.920144i	$0.371929\pi$
$18$	0	0
$19$	20.8284	1.09623	0.548117	−	0.836402i	$-0.315345\pi$
$19$	20.8284	1.09623	0.548117	+	0.836402i	$0.315345\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 33.6569i	− 1.46334i	−0.681658	−	0.731671i	$-0.738741\pi$
$23$	− 33.6569i	− 1.46334i	0.681658	−	0.731671i	$-0.261259\pi$
$24$	0	0
$25$	−33.6274	−1.34510
$26$	0	0
$27$	0	0
$28$	0	0
$29$	39.6569i	1.36748i	0.729727	+	0.683739i	$0.239647\pi$
$29$	39.6569i	1.36748i	−0.729727	+	0.683739i	$0.760353\pi$
$30$	0	0
$31$	− 45.2548i	− 1.45983i	−0.683536	−	0.729917i	$-0.739559\pi$
$31$	− 45.2548i	− 1.45983i	0.683536	−	0.729917i	$-0.260441\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−12.6863	−0.362465
$36$	0	0
$37$	− 29.5980i	− 0.799945i	−0.916527	−	0.399973i	$-0.869020\pi$
$37$	− 29.5980i	− 0.799945i	0.916527	−	0.399973i	$-0.130980\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	24.6274	0.600669	0.300334	−	0.953834i	$-0.402902\pi$
$41$	24.6274	0.600669	0.300334	+	0.953834i	$0.402902\pi$
$42$	0	0
$43$	50.0833	1.16473	0.582364	−	0.812929i	$-0.302128\pi$
$43$	50.0833	1.16473	0.582364	+	0.812929i	$0.302128\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 35.3137i	− 0.751355i	−0.926750	−	0.375678i	$-0.877410\pi$
$47$	− 35.3137i	− 0.751355i	0.926750	−	0.375678i	$-0.122590\pi$
$48$	0	0
$49$	46.2548	0.943976
$50$	0	0
$51$	0	0
$52$	0	0
$53$	16.3431i	0.308361i	0.988043	+	0.154181i	$0.0492738\pi$
$53$	16.3431i	0.308361i	−0.988043	+	0.154181i	$0.950726\pi$
$54$	0	0
$55$	11.5980i	0.210872i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	53.1127	0.900215	0.450108	−	0.892974i	$-0.351386\pi$
$59$	53.1127	0.900215	0.450108	+	0.892974i	$0.351386\pi$
$60$	0	0
$61$	− 34.4020i	− 0.563968i	−0.959419	−	0.281984i	$-0.909008\pi$
$61$	− 34.4020i	− 0.563968i	0.959419	−	0.281984i	$-0.0909924\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.62742	0.0404218
$66$	0	0
$67$	−62.4853	−0.932616	−0.466308	−	0.884622i	$-0.654416\pi$
$67$	−62.4853	−0.932616	−0.466308	+	0.884622i	$0.654416\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	40.2843i	0.567384i	0.958915	+	0.283692i	$0.0915593\pi$
$71$	40.2843i	0.567384i	−0.958915	+	0.283692i	$0.908441\pi$
$72$	0	0
$73$	−55.9411	−0.766317	−0.383158	−	0.923683i	$-0.625164\pi$
$73$	−55.9411	−0.766317	−0.383158	+	0.923683i	$0.625164\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.50967i	0.0325931i
$78$	0	0
$79$	− 137.941i	− 1.74609i	−0.487639	−	0.873045i	$-0.662142\pi$
$79$	− 137.941i	− 1.74609i	0.487639	−	0.873045i	$-0.337858\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−114.652	−1.38135	−0.690674	−	0.723167i	$-0.742686\pi$
$83$	−114.652	−1.38135	−0.690674	+	0.723167i	$0.742686\pi$
$84$	0	0
$85$	− 101.941i	− 1.19931i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.56854	−0.0288600	−0.0144300	−	0.999896i	$-0.504593\pi$
$89$	−2.56854	−0.0288600	−0.0144300	+	0.999896i	$0.504593\pi$
$90$	0	0
$91$	0.568542	0.00624772
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 159.480i	− 1.67874i
$96$	0	0
$97$	−138.569	−1.42854	−0.714271	−	0.699869i	$-0.753242\pi$
$97$	−138.569	−1.42854	−0.714271	+	0.699869i	$0.753242\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	45.5980i	0.451465i	0.974189	+	0.225733i	$0.0724776\pi$
$101$	45.5980i	0.451465i	−0.974189	+	0.225733i	$0.927522\pi$
$102$	0	0
$103$	− 20.4020i	− 0.198078i	−0.995084	−	0.0990389i	$-0.968423\pi$
$103$	− 20.4020i	− 0.198078i	0.995084	−	0.0990389i	$-0.0315768\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−97.5147	−0.911353	−0.455676	−	0.890146i	$-0.650603\pi$
$107$	−97.5147	−0.911353	−0.455676	+	0.890146i	$0.650603\pi$
$108$	0	0
$109$	149.598i	1.37246i	0.727385	+	0.686229i	$0.240735\pi$
$109$	149.598i	1.37246i	−0.727385	+	0.686229i	$0.759265\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	168.510	1.49124	0.745618	−	0.666374i	$-0.232154\pi$
$113$	168.510	1.49124	0.745618	+	0.666374i	$0.232154\pi$
$114$	0	0
$115$	−257.706	−2.24092
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 22.0589i	− 0.185369i
$120$	0	0
$121$	−118.706	−0.981038
$122$	0	0
$123$	0	0
$124$	0	0
$125$	66.0589i	0.528471i
$126$	0	0
$127$	102.627i	0.808090i	0.914739	+	0.404045i	$0.132396\pi$
$127$	102.627i	0.808090i	−0.914739	+	0.404045i	$0.867604\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−45.6325	−0.348339	−0.174170	−	0.984716i	$-0.555724\pi$
$131$	−45.6325	−0.348339	−0.174170	+	0.984716i	$0.555724\pi$
$132$	0	0
$133$	− 34.5097i	− 0.259471i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−81.8823	−0.597681	−0.298840	−	0.954303i	$-0.596600\pi$
$137$	−81.8823	−0.597681	−0.298840	+	0.954303i	$0.596600\pi$
$138$	0	0
$139$	−52.5442	−0.378016	−0.189008	−	0.981976i	$-0.560527\pi$
$139$	−52.5442	−0.378016	−0.189008	+	0.981976i	$0.560527\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 0.519769i	− 0.00363475i
$144$	0	0
$145$	303.647	2.09412
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 116.912i	− 0.784642i	−0.919828	−	0.392321i	$-0.871672\pi$
$149$	− 116.912i	− 0.784642i	0.919828	−	0.392321i	$-0.128328\pi$
$150$	0	0
$151$	214.676i	1.42170i	0.703345	+	0.710848i	$0.251689\pi$
$151$	214.676i	1.42170i	−0.703345	+	0.710848i	$0.748311\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−346.510	−2.23555
$156$	0	0
$157$	136.108i	0.866928i	0.901171	+	0.433464i	$0.142709\pi$
$157$	136.108i	0.866928i	−0.901171	+	0.433464i	$0.857291\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−55.7645	−0.346363
$162$	0	0
$163$	−53.6812	−0.329333	−0.164666	−	0.986349i	$-0.552655\pi$
$163$	−53.6812	−0.329333	−0.164666	+	0.986349i	$0.552655\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 93.2061i	− 0.558120i	−0.960274	−	0.279060i	$-0.909977\pi$
$167$	− 93.2061i	− 0.558120i	0.960274	−	0.279060i	$-0.0900229\pi$
$168$	0	0
$169$	168.882	0.999303
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 16.3431i	− 0.0944691i	−0.998884	−	0.0472345i	$-0.984959\pi$
$173$	− 16.3431i	− 0.0944691i	0.998884	−	0.0472345i	$-0.0150408\pi$
$174$	0	0
$175$	55.7157i	0.318376i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−120.142	−0.671185	−0.335593	−	0.942007i	$-0.608937\pi$
$179$	−120.142	−0.671185	−0.335593	+	0.942007i	$0.608937\pi$
$180$	0	0
$181$	− 176.108i	− 0.972970i	−0.873689	−	0.486485i	$-0.838279\pi$
$181$	− 176.108i	− 0.972970i	0.873689	−	0.486485i	$-0.161721\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−226.627	−1.22501
$186$	0	0
$187$	−20.1665	−0.107842
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 62.8629i	− 0.329125i	−0.986367	−	0.164563i	$-0.947379\pi$
$191$	− 62.8629i	− 0.329125i	0.986367	−	0.164563i	$-0.0526213\pi$
$192$	0	0
$193$	−69.0782	−0.357918	−0.178959	−	0.983857i	$-0.557273\pi$
$193$	−69.0782	−0.357918	−0.178959	+	0.983857i	$0.557273\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	45.8335i	0.232657i	0.993211	+	0.116329i	$0.0371126\pi$
$197$	45.8335i	0.232657i	−0.993211	+	0.116329i	$0.962887\pi$
$198$	0	0
$199$	− 220.167i	− 1.10636i	−0.833060	−	0.553182i	$-0.813413\pi$
$199$	− 220.167i	− 1.10636i	0.833060	−	0.553182i	$-0.186587\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	65.7056	0.323673
$204$	0	0
$205$	− 188.569i	− 0.919847i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−31.5492	−0.150953
$210$	0	0
$211$	30.7696	0.145827	0.0729136	−	0.997338i	$-0.476770\pi$
$211$	30.7696	0.145827	0.0729136	+	0.997338i	$0.476770\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 383.480i	− 1.78363i
$216$	0	0
$217$	−74.9807	−0.345533
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.56854i	0.0206721i
$222$	0	0
$223$	− 210.745i	− 0.945046i	−0.881318	−	0.472523i	$-0.843343\pi$
$223$	− 210.745i	− 0.945046i	0.881318	−	0.472523i	$-0.156657\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−403.220	−1.77630	−0.888151	−	0.459553i	$-0.848010\pi$
$227$	−403.220	−1.77630	−0.888151	+	0.459553i	$0.848010\pi$
$228$	0	0
$229$	359.186i	1.56850i	0.620447	+	0.784249i	$0.286951\pi$
$229$	359.186i	1.56850i	−0.620447	+	0.784249i	$0.713049\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	210.451	0.903222	0.451611	−	0.892215i	$-0.350849\pi$
$233$	210.451	0.903222	0.451611	+	0.892215i	$0.350849\pi$
$234$	0	0
$235$	−270.392	−1.15060
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 332.215i	− 1.39002i	−0.718999	−	0.695011i	$-0.755399\pi$
$239$	− 332.215i	− 1.39002i	0.718999	−	0.695011i	$-0.244601\pi$
$240$	0	0
$241$	71.7056	0.297534	0.148767	−	0.988872i	$-0.452470\pi$
$241$	71.7056	0.297534	0.148767	+	0.988872i	$0.452470\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 354.167i	− 1.44558i
$246$	0	0
$247$	7.14719i	0.0289360i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−50.6518	−0.201800	−0.100900	−	0.994897i	$-0.532172\pi$
$251$	−50.6518	−0.201800	−0.100900	+	0.994897i	$0.532172\pi$
$252$	0	0
$253$	50.9807i	0.201505i
$254$	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
$259$	−49.0395	−0.189342
$260$	0	0
$261$	0	0
$262$	0	0
$263$	258.794i	0.984007i	0.870593	+	0.492004i	$0.163735\pi$
$263$	258.794i	0.984007i	−0.870593	+	0.492004i	$0.836265\pi$
$264$	0	0
$265$	125.137	0.472215
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 399.637i	− 1.48564i	−0.669492	−	0.742819i	$-0.733488\pi$
$269$	− 399.637i	− 1.48564i	0.669492	−	0.742819i	$-0.266512\pi$
$270$	0	0
$271$	− 253.823i	− 0.936618i	−0.883565	−	0.468309i	$-0.844863\pi$
$271$	− 253.823i	− 0.936618i	0.883565	−	0.468309i	$-0.155137\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	50.9361	0.185222
$276$	0	0
$277$	505.696i	1.82562i	0.408389	+	0.912808i	$0.366091\pi$
$277$	505.696i	1.82562i	−0.408389	+	0.912808i	$0.633909\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	245.196	0.872583	0.436292	−	0.899805i	$-0.356292\pi$
$281$	245.196	0.872583	0.436292	+	0.899805i	$0.356292\pi$
$282$	0	0
$283$	−22.1522	−0.0782765	−0.0391382	−	0.999234i	$-0.512461\pi$
$283$	−22.1522	−0.0782765	−0.0391382	+	0.999234i	$0.512461\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 40.8040i	− 0.142174i
$288$	0	0
$289$	−111.745	−0.386661
$290$	0	0
$291$	0	0
$292$	0	0
$293$	232.343i	0.792980i	0.918039	+	0.396490i	$0.129772\pi$
$293$	232.343i	0.792980i	−0.918039	+	0.396490i	$0.870228\pi$
$294$	0	0
$295$	− 406.676i	− 1.37856i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.5492	0.0386261
$300$	0	0
$301$	− 82.9807i	− 0.275683i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−263.411	−0.863643
$306$	0	0
$307$	408.240	1.32977	0.664885	−	0.746945i	$-0.268480\pi$
$307$	408.240	1.32977	0.664885	+	0.746945i	$0.268480\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 544.715i	− 1.75149i	−0.482770	−	0.875747i	$-0.660369\pi$
$311$	− 544.715i	− 1.75149i	0.482770	−	0.875747i	$-0.339631\pi$
$312$	0	0
$313$	327.373	1.04592	0.522959	−	0.852358i	$-0.324828\pi$
$313$	327.373	1.04592	0.522959	+	0.852358i	$0.324828\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 120.343i	− 0.379631i	−0.981820	−	0.189816i	$-0.939211\pi$
$317$	− 120.343i	− 0.379631i	0.981820	−	0.189816i	$-0.0607890\pi$
$318$	0	0
$319$	− 60.0690i	− 0.188304i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	277.304	0.858525
$324$	0	0
$325$	− 11.5391i	− 0.0355049i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−58.5097	−0.177841
$330$	0	0
$331$	61.1615	0.184778	0.0923889	−	0.995723i	$-0.470550\pi$
$331$	61.1615	0.184778	0.0923889	+	0.995723i	$0.470550\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	478.441i	1.42818i
$336$	0	0
$337$	−58.7838	−0.174433	−0.0872164	−	0.996189i	$-0.527797\pi$
$337$	−58.7838	−0.174433	−0.0872164	+	0.996189i	$0.527797\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	68.5483i	0.201022i
$342$	0	0
$343$	− 157.823i	− 0.460126i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	396.406	1.14238	0.571190	−	0.820818i	$-0.306482\pi$
$347$	396.406	1.14238	0.571190	+	0.820818i	$0.306482\pi$
$348$	0	0
$349$	− 7.18586i	− 0.0205899i	−0.999947	−	0.0102949i	$-0.996723\pi$
$349$	− 7.18586i	− 0.0205899i	0.999947	−	0.0102949i	$-0.00327703\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	171.726	0.486475	0.243238	−	0.969967i	$-0.421790\pi$
$353$	171.726	0.486475	0.243238	+	0.969967i	$0.421790\pi$
$354$	0	0
$355$	308.451	0.868875
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 160.617i	− 0.447402i	−0.974658	−	0.223701i	$-0.928186\pi$
$359$	− 160.617i	− 0.447402i	0.974658	−	0.223701i	$-0.0718139\pi$
$360$	0	0
$361$	72.8234	0.201727
$362$	0	0
$363$	0	0
$364$	0	0
$365$	428.333i	1.17352i
$366$	0	0
$367$	− 55.1960i	− 0.150398i	−0.997169	−	0.0751989i	$-0.976041\pi$
$367$	− 55.1960i	− 0.150398i	0.997169	−	0.0751989i	$-0.0239592\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	27.0782	0.0729871
$372$	0	0
$373$	− 346.853i	− 0.929900i	−0.885337	−	0.464950i	$-0.846072\pi$
$373$	− 346.853i	− 0.929900i	0.885337	−	0.464950i	$-0.153928\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.6081	−0.0360957
$378$	0	0
$379$	−112.759	−0.297518	−0.148759	−	0.988873i	$-0.547528\pi$
$379$	−112.759	−0.297518	−0.148759	+	0.988873i	$0.547528\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 618.039i	− 1.61368i	−0.590771	−	0.806839i	$-0.701176\pi$
$383$	− 618.039i	− 1.61368i	0.590771	−	0.806839i	$-0.298824\pi$
$384$	0	0
$385$	19.2162	0.0499121
$386$	0	0
$387$	0	0
$388$	0	0
$389$	419.088i	1.07735i	0.842514	+	0.538674i	$0.181075\pi$
$389$	419.088i	1.07735i	−0.842514	+	0.538674i	$0.818925\pi$
$390$	0	0
$391$	− 448.098i	− 1.14603i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1056.20	−2.67391
$396$	0	0
$397$	− 564.441i	− 1.42176i	−0.703311	−	0.710882i	$-0.748296\pi$
$397$	− 564.441i	− 1.42176i	0.703311	−	0.710882i	$-0.251704\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	199.823	0.498313	0.249156	−	0.968463i	$-0.419847\pi$
$401$	199.823	0.498313	0.249156	+	0.968463i	$0.419847\pi$
$402$	0	0
$403$	15.5290	0.0385335
$404$	0	0
$405$	0	0
$406$	0	0
$407$	44.8326i	0.110154i
$408$	0	0
$409$	118.902	0.290713	0.145356	−	0.989379i	$-0.453567\pi$
$409$	118.902	0.290713	0.145356	+	0.989379i	$0.453567\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 88.0000i	− 0.213075i
$414$	0	0
$415$	877.872i	2.11535i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	304.093	0.725760	0.362880	−	0.931836i	$-0.381794\pi$
$419$	304.093	0.725760	0.362880	+	0.931836i	$0.381794\pi$
$420$	0	0
$421$	188.676i	0.448162i	0.974571	+	0.224081i	$0.0719380\pi$
$421$	188.676i	0.448162i	−0.974571	+	0.224081i	$0.928062\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−447.706	−1.05343
$426$	0	0
$427$	−56.9991	−0.133487
$428$	0	0
$429$	0	0
$430$	0	0
$431$	402.843i	0.934670i	0.884080	+	0.467335i	$0.154786\pi$
$431$	402.843i	0.934670i	−0.884080	+	0.467335i	$0.845214\pi$
$432$	0	0
$433$	−721.862	−1.66712	−0.833559	−	0.552431i	$-0.813700\pi$
$433$	−721.862	−1.66712	−0.833559	+	0.552431i	$0.813700\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 701.019i	− 1.60416i
$438$	0	0
$439$	− 333.872i	− 0.760529i	−0.924878	−	0.380264i	$-0.875833\pi$
$439$	− 333.872i	− 0.760529i	0.924878	−	0.380264i	$-0.124167\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−393.848	−0.889047	−0.444523	−	0.895767i	$-0.646627\pi$
$443$	−393.848	−0.889047	−0.444523	+	0.895767i	$0.646627\pi$
$444$	0	0
$445$	19.6670i	0.0441954i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.9218	−0.0599594	−0.0299797	−	0.999551i	$-0.509544\pi$
$449$	−26.9218	−0.0599594	−0.0299797	+	0.999551i	$0.509544\pi$
$450$	0	0
$451$	−37.3036	−0.0827131
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 4.35325i	− 0.00956758i
$456$	0	0
$457$	−222.118	−0.486034	−0.243017	−	0.970022i	$-0.578137\pi$
$457$	−222.118	−0.486034	−0.243017	+	0.970022i	$0.578137\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 139.088i	− 0.301710i	−0.988556	−	0.150855i	$-0.951797\pi$
$461$	− 139.088i	− 0.301710i	0.988556	−	0.150855i	$-0.0482027\pi$
$462$	0	0
$463$	480.098i	1.03693i	0.855100	+	0.518464i	$0.173496\pi$
$463$	480.098i	1.03693i	−0.855100	+	0.518464i	$0.826504\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	105.092	0.225037	0.112519	−	0.993650i	$-0.464108\pi$
$467$	105.092	0.225037	0.112519	+	0.993650i	$0.464108\pi$
$468$	0	0
$469$	103.529i	0.220744i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−75.8620	−0.160385
$474$	0	0
$475$	−700.406	−1.47454
$476$	0	0
$477$	0	0
$478$	0	0
$479$	391.765i	0.817880i	0.912561	+	0.408940i	$0.134101\pi$
$479$	391.765i	0.817880i	−0.912561	+	0.408940i	$0.865899\pi$
$480$	0	0
$481$	10.1564	0.0211152
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1061.00i	2.18763i
$486$	0	0
$487$	567.716i	1.16574i	0.812565	+	0.582870i	$0.198070\pi$
$487$	567.716i	1.16574i	−0.812565	+	0.582870i	$0.801930\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	755.602	1.53890	0.769452	−	0.638704i	$-0.220529\pi$
$491$	755.602	1.53890	0.769452	+	0.638704i	$0.220529\pi$
$492$	0	0
$493$	527.980i	1.07095i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	66.7452	0.134296
$498$	0	0
$499$	−208.759	−0.418356	−0.209178	−	0.977878i	$-0.567079\pi$
$499$	−208.759	−0.418356	−0.209178	+	0.977878i	$0.567079\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	136.187i	0.270749i	0.990795	+	0.135374i	$0.0432237\pi$
$503$	136.187i	0.270749i	−0.990795	+	0.135374i	$0.956776\pi$
$504$	0	0
$505$	349.137	0.691361
$506$	0	0
$507$	0	0
$508$	0	0
$509$	247.892i	0.487018i	0.969899	+	0.243509i	$0.0782986\pi$
$509$	247.892i	0.487018i	−0.969899	+	0.243509i	$0.921701\pi$
$510$	0	0
$511$	92.6863i	0.181382i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−156.215	−0.303331
$516$	0	0
$517$	53.4903i	0.103463i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	881.724	1.69237	0.846184	−	0.532890i	$-0.178894\pi$
$521$	881.724	1.69237	0.846184	+	0.532890i	$0.178894\pi$
$522$	0	0
$523$	818.181	1.56440	0.782200	−	0.623028i	$-0.214098\pi$
$523$	818.181	1.56440	0.782200	+	0.623028i	$0.214098\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 602.510i	− 1.14328i
$528$	0	0
$529$	−603.784	−1.14137
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.45079i	0.0158551i
$534$	0	0
$535$	746.656i	1.39562i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−70.0631	−0.129987
$540$	0	0
$541$	− 263.892i	− 0.487786i	−0.969802	−	0.243893i	$-0.921575\pi$
$541$	− 263.892i	− 0.487786i	0.969802	−	0.243893i	$-0.0784246\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1145.45	2.10174
$546$	0	0
$547$	−52.6417	−0.0962371	−0.0481186	−	0.998842i	$-0.515323\pi$
$547$	−52.6417	−0.0962371	−0.0481186	+	0.998842i	$0.515323\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	825.990i	1.49907i
$552$	0	0
$553$	−228.548	−0.413288
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 725.362i	− 1.30227i	−0.758963	−	0.651133i	$-0.774294\pi$
$557$	− 725.362i	− 1.30227i	0.758963	−	0.651133i	$-0.225706\pi$
$558$	0	0
$559$	17.1859i	0.0307439i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	156.780	0.278472	0.139236	−	0.990259i	$-0.455535\pi$
$563$	156.780	0.278472	0.139236	+	0.990259i	$0.455535\pi$
$564$	0	0
$565$	− 1290.25i	− 2.28364i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	796.705	1.40018	0.700092	−	0.714053i	$-0.253142\pi$
$569$	796.705	1.40018	0.700092	+	0.714053i	$0.253142\pi$
$570$	0	0
$571$	151.103	0.264628	0.132314	−	0.991208i	$-0.457759\pi$
$571$	151.103	0.264628	0.132314	+	0.991208i	$0.457759\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1131.79i	1.96834i
$576$	0	0
$577$	462.313	0.801235	0.400618	−	0.916245i	$-0.368796\pi$
$577$	462.313	0.801235	0.400618	+	0.916245i	$0.368796\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	189.961i	0.326956i
$582$	0	0
$583$	− 24.7553i	− 0.0424619i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	278.721	0.474822	0.237411	−	0.971409i	$-0.423701\pi$
$587$	278.721	0.474822	0.237411	+	0.971409i	$0.423701\pi$
$588$	0	0
$589$	− 942.587i	− 1.60032i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−934.548	−1.57597	−0.787983	−	0.615696i	$-0.788875\pi$
$593$	−934.548	−1.57597	−0.787983	+	0.615696i	$0.788875\pi$
$594$	0	0
$595$	−168.902	−0.283868
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1004.74i	1.67735i	0.544629	+	0.838677i	$0.316670\pi$
$599$	1004.74i	1.67735i	−0.544629	+	0.838677i	$0.683330\pi$
$600$	0	0
$601$	−722.451	−1.20208	−0.601041	−	0.799218i	$-0.705247\pi$
$601$	−722.451	−1.20208	−0.601041	+	0.799218i	$0.705247\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	908.912i	1.50233i
$606$	0	0
$607$	− 344.431i	− 0.567431i	−0.958909	−	0.283715i	$-0.908433\pi$
$607$	− 344.431i	− 0.567431i	0.958909	−	0.283715i	$-0.0915671\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.1177	0.0198326
$612$	0	0
$613$	839.657i	1.36975i	0.728660	+	0.684875i	$0.240143\pi$
$613$	839.657i	1.36975i	−0.728660	+	0.684875i	$0.759857\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−41.8620	−0.0678477	−0.0339239	−	0.999424i	$-0.510800\pi$
$617$	−41.8620	−0.0678477	−0.0339239	+	0.999424i	$0.510800\pi$
$618$	0	0
$619$	−1098.60	−1.77480	−0.887401	−	0.460997i	$-0.847492\pi$
$619$	−1098.60	−1.77480	−0.887401	+	0.460997i	$0.847492\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.25570i	0.00683098i
$624$	0	0
$625$	−334.882	−0.535812
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 394.059i	− 0.626485i
$630$	0	0
$631$	1042.32i	1.65186i	0.563774	+	0.825929i	$0.309349\pi$
$631$	1042.32i	1.65186i	−0.563774	+	0.825929i	$0.690651\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	785.803	1.23749
$636$	0	0
$637$	15.8721i	0.0249170i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−448.412	−0.699551	−0.349775	−	0.936834i	$-0.613742\pi$
$641$	−448.412	−0.699551	−0.349775	+	0.936834i	$0.613742\pi$
$642$	0	0
$643$	−523.740	−0.814526	−0.407263	−	0.913311i	$-0.633517\pi$
$643$	−523.740	−0.814526	−0.407263	+	0.913311i	$0.633517\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 465.283i	− 0.719140i	−0.933118	−	0.359570i	$-0.882923\pi$
$647$	− 465.283i	− 0.719140i	0.933118	−	0.359570i	$-0.117077\pi$
$648$	0	0
$649$	−80.4508	−0.123961
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 283.559i	− 0.434241i	−0.976145	−	0.217120i	$-0.930334\pi$
$653$	− 283.559i	− 0.434241i	0.976145	−	0.217120i	$-0.0696664\pi$
$654$	0	0
$655$	349.401i	0.533437i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−209.710	−0.318224	−0.159112	−	0.987261i	$-0.550863\pi$
$659$	−209.710	−0.318224	−0.159112	+	0.987261i	$0.550863\pi$
$660$	0	0
$661$	31.4214i	0.0475361i	0.999718	+	0.0237680i	$0.00756632\pi$
$661$	31.4214i	0.0475361i	−0.999718	+	0.0237680i	$0.992434\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−264.235	−0.397347
$666$	0	0
$667$	1334.72	2.00109
$668$	0	0
$669$	0	0
$670$	0	0
$671$	52.1094i	0.0776593i
$672$	0	0
$673$	1327.00	1.97177	0.985883	−	0.167433i	$-0.0535478\pi$
$673$	1327.00	1.97177	0.985883	+	0.167433i	$0.0535478\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	211.559i	0.312495i	0.987718	+	0.156248i	$0.0499398\pi$
$677$	211.559i	0.312495i	−0.987718	+	0.156248i	$0.950060\pi$
$678$	0	0
$679$	229.588i	0.338126i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	801.799	1.17394	0.586969	−	0.809610i	$-0.300321\pi$
$683$	801.799	1.17394	0.586969	+	0.809610i	$0.300321\pi$
$684$	0	0
$685$	626.960i	0.915271i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.60808	−0.00813945
$690$	0	0
$691$	173.161	0.250595	0.125298	−	0.992119i	$-0.460011\pi$
$691$	173.161	0.250595	0.125298	+	0.992119i	$0.460011\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	402.323i	0.578882i
$696$	0	0
$697$	327.882	0.470419
$698$	0	0
$699$	0	0
$700$	0	0
$701$	764.912i	1.09117i	0.838055	+	0.545586i	$0.183693\pi$
$701$	764.912i	1.09117i	−0.838055	+	0.545586i	$0.816307\pi$
$702$	0	0
$703$	− 616.479i	− 0.876927i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	75.5492	0.106859
$708$	0	0
$709$	− 252.656i	− 0.356355i	−0.983998	−	0.178178i	$-0.942980\pi$
$709$	− 252.656i	− 0.356355i	0.983998	−	0.178178i	$-0.0570202\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1523.14	−2.13623
$714$	0	0
$715$	−3.97980	−0.00556615
$716$	0	0
$717$	0	0
$718$	0	0
$719$	806.725i	1.12201i	0.827813	+	0.561005i	$0.189585\pi$
$719$	806.725i	1.12201i	−0.827813	+	0.561005i	$0.810415\pi$
$720$	0	0
$721$	−33.8032	−0.0468837
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1333.56i	− 1.83939i
$726$	0	0
$727$	− 1185.75i	− 1.63102i	−0.578740	−	0.815512i	$-0.696455\pi$
$727$	− 1185.75i	− 1.63102i	0.578740	−	0.815512i	$-0.303545\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	666.794	0.912167
$732$	0	0
$733$	785.911i	1.07218i	0.844160	+	0.536092i	$0.180100\pi$
$733$	785.911i	1.07218i	−0.844160	+	0.536092i	$0.819900\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	94.6476	0.128423
$738$	0	0
$739$	1074.28	1.45369	0.726846	−	0.686800i	$-0.240985\pi$
$739$	1074.28	1.45369	0.726846	+	0.686800i	$0.240985\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	385.852i	0.519316i	0.965701	+	0.259658i	$0.0836099\pi$
$743$	385.852i	0.519316i	−0.965701	+	0.259658i	$0.916390\pi$
$744$	0	0
$745$	−895.176	−1.20158
$746$	0	0
$747$	0	0
$748$	0	0
$749$	161.568i	0.215711i
$750$	0	0
$751$	− 729.665i	− 0.971592i	−0.874072	−	0.485796i	$-0.838530\pi$
$751$	− 729.665i	− 0.971592i	0.874072	−	0.485796i	$-0.161470\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1643.74	2.17714
$756$	0	0
$757$	− 570.617i	− 0.753788i	−0.926256	−	0.376894i	$-0.876992\pi$
$757$	− 570.617i	− 0.753788i	0.926256	−	0.376894i	$-0.123008\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−347.450	−0.456570	−0.228285	−	0.973594i	$-0.573312\pi$
$761$	−347.450	−0.456570	−0.228285	+	0.973594i	$0.573312\pi$
$762$	0	0
$763$	247.862	0.324852
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.2254i	0.0237619i
$768$	0	0
$769$	505.980	0.657971	0.328986	−	0.944335i	$-0.393293\pi$
$769$	505.980	0.657971	0.328986	+	0.944335i	$0.393293\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	476.656i	0.616631i	0.951284	+	0.308316i	$0.0997653\pi$
$773$	476.656i	0.616631i	−0.951284	+	0.308316i	$0.900235\pi$
$774$	0	0
$775$	1521.80i	1.96362i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	512.950	0.658473
$780$	0	0
$781$	− 61.0193i	− 0.0781298i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1042.16	1.32759
$786$	0	0
$787$	884.731	1.12418	0.562091	−	0.827076i	$-0.309997\pi$
$787$	884.731	1.12418	0.562091	+	0.827076i	$0.309997\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 279.196i	− 0.352966i
$792$	0	0
$793$	11.8049	0.0148864
$794$	0	0
$795$	0	0
$796$	0	0
$797$	956.441i	1.20005i	0.799981	+	0.600026i	$0.204843\pi$
$797$	956.441i	1.20005i	−0.799981	+	0.600026i	$0.795157\pi$
$798$	0	0
$799$	− 470.156i	− 0.588431i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	84.7351	0.105523
$804$	0	0
$805$	426.981i	0.530411i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1185.72	1.46567	0.732833	−	0.680408i	$-0.238198\pi$
$809$	1185.72	1.46567	0.732833	+	0.680408i	$0.238198\pi$
$810$	0	0
$811$	1017.95	1.25517	0.627587	−	0.778547i	$-0.284043\pi$
$811$	1017.95	1.25517	0.627587	+	0.778547i	$0.284043\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	411.029i	0.504331i
$816$	0	0
$817$	1043.16	1.27681
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1359.19i	− 1.65552i	−0.561079	−	0.827762i	$-0.689614\pi$
$821$	− 1359.19i	− 1.65552i	0.561079	−	0.827762i	$-0.310386\pi$
$822$	0	0
$823$	− 360.382i	− 0.437888i	−0.975737	−	0.218944i	$-0.929739\pi$
$823$	− 360.382i	− 0.437888i	0.975737	−	0.218944i	$-0.0702612\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−464.475	−0.561639	−0.280819	−	0.959761i	$-0.590606\pi$
$827$	−464.475	−0.561639	−0.280819	+	0.959761i	$0.590606\pi$
$828$	0	0
$829$	103.166i	0.124446i	0.998062	+	0.0622230i	$0.0198190\pi$
$829$	103.166i	0.124446i	−0.998062	+	0.0622230i	$0.980181\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	615.823	0.739284
$834$	0	0
$835$	−713.665	−0.854689
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 687.204i	− 0.819075i	−0.912293	−	0.409538i	$-0.865690\pi$
$839$	− 687.204i	− 0.819075i	0.912293	−	0.409538i	$-0.134310\pi$
$840$	0	0
$841$	−731.666	−0.869995
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1293.11i	− 1.53030i
$846$	0	0
$847$	196.678i	0.232205i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−996.175	−1.17059
$852$	0	0
$853$	− 693.833i	− 0.813404i	−0.913561	−	0.406702i	$-0.866679\pi$
$853$	− 693.833i	− 0.813404i	0.913561	−	0.406702i	$-0.133321\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.35325	−0.00274591	−0.00137296	−	0.999999i	$-0.500437\pi$
$857$	−2.35325	−0.00274591	−0.00137296	+	0.999999i	$0.500437\pi$
$858$	0	0
$859$	1422.49	1.65599	0.827994	−	0.560737i	$-0.189482\pi$
$859$	1422.49	1.65599	0.827994	+	0.560737i	$0.189482\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 501.961i	− 0.581647i	−0.956777	−	0.290823i	$-0.906071\pi$
$863$	− 501.961i	− 0.581647i	0.956777	−	0.290823i	$-0.0939292\pi$
$864$	0	0
$865$	−125.137	−0.144667
$866$	0	0
$867$	0	0
$868$	0	0
$869$	208.942i	0.240440i
$870$	0	0
$871$	− 21.4416i	− 0.0246172i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	109.450	0.125086
$876$	0	0
$877$	1560.73i	1.77963i	0.456324	+	0.889814i	$0.349166\pi$
$877$	1560.73i	1.77963i	−0.456324	+	0.889814i	$0.650834\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−665.294	−0.755157	−0.377579	−	0.925978i	$-0.623243\pi$
$881$	−665.294	−0.755157	−0.377579	+	0.925978i	$0.623243\pi$
$882$	0	0
$883$	−817.052	−0.925314	−0.462657	−	0.886537i	$-0.653104\pi$
$883$	−817.052	−0.925314	−0.462657	+	0.886537i	$0.653104\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1170.42i	1.31953i	0.751473	+	0.659764i	$0.229344\pi$
$887$	1170.42i	1.31953i	−0.751473	+	0.659764i	$0.770656\pi$
$888$	0	0
$889$	170.039	0.191270
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 735.529i	− 0.823661i
$894$	0	0
$895$	919.911i	1.02783i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1794.66	1.99629
$900$	0	0
$901$	217.588i	0.241496i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1348.43	−1.48998
$906$	0	0
$907$	936.435	1.03245	0.516226	−	0.856452i	$-0.327336\pi$
$907$	936.435	1.03245	0.516226	+	0.856452i	$0.327336\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 958.018i	− 1.05161i	−0.850605	−	0.525806i	$-0.823764\pi$
$911$	− 958.018i	− 1.05161i	0.850605	−	0.525806i	$-0.176236\pi$
$912$	0	0
$913$	173.665	0.190214
$914$	0	0
$915$	0	0
$916$	0	0
$917$	75.6063i	0.0824497i
$918$	0	0
$919$	− 255.675i	− 0.278210i	−0.990278	−	0.139105i	$-0.955577\pi$
$919$	− 255.675i	− 0.278210i	0.990278	−	0.139105i	$-0.0444226\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−13.8234	−0.0149766
$924$	0	0
$925$	995.304i	1.07600i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−868.960	−0.935372	−0.467686	−	0.883895i	$-0.654912\pi$
$929$	−868.960	−0.935372	−0.467686	+	0.883895i	$0.654912\pi$
$930$	0	0
$931$	963.415	1.03482
$932$	0	0
$933$	0	0
$934$	0	0
$935$	154.412i	0.165147i
$936$	0	0
$937$	1122.57	1.19805	0.599023	−	0.800732i	$-0.295556\pi$
$937$	1122.57	1.19805	0.599023	+	0.800732i	$0.295556\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	765.537i	0.813536i	0.913531	+	0.406768i	$0.133344\pi$
$941$	765.537i	0.813536i	−0.913531	+	0.406768i	$0.866656\pi$
$942$	0	0
$943$	− 828.881i	− 0.878983i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	249.563	0.263531	0.131765	−	0.991281i	$-0.457935\pi$
$947$	249.563	0.263531	0.131765	+	0.991281i	$0.457935\pi$
$948$	0	0
$949$	− 19.1960i	− 0.0202276i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−709.960	−0.744973	−0.372487	−	0.928038i	$-0.621495\pi$
$953$	−709.960	−0.744973	−0.372487	+	0.928038i	$0.621495\pi$
$954$	0	0
$955$	−481.332	−0.504013
$956$	0	0
$957$	0	0
$958$	0	0
$959$	135.667i	0.141467i
$960$	0	0
$961$	−1087.00	−1.13111
$962$	0	0
$963$	0	0
$964$	0	0
$965$	528.922i	0.548105i
$966$	0	0
$967$	1525.64i	1.57770i	0.614585	+	0.788850i	$0.289323\pi$
$967$	1525.64i	1.57770i	−0.614585	+	0.788850i	$0.710677\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	288.662	0.297283	0.148642	−	0.988891i	$-0.452510\pi$
$971$	288.662	0.297283	0.148642	+	0.988891i	$0.452510\pi$
$972$	0	0
$973$	87.0580i	0.0894738i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−541.667	−0.554419	−0.277209	−	0.960810i	$-0.589410\pi$
$977$	−541.667	−0.554419	−0.277209	+	0.960810i	$0.589410\pi$
$978$	0	0
$979$	3.89062	0.00397407
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1007.01i	1.02442i	0.858859	+	0.512212i	$0.171174\pi$
$983$	1007.01i	1.02442i	−0.858859	+	0.512212i	$0.828826\pi$
$984$	0	0
$985$	350.940	0.356285
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1685.65i	− 1.70439i
$990$	0	0
$991$	− 183.098i	− 0.184761i	−0.995724	−	0.0923806i	$-0.970552\pi$
$991$	− 183.098i	− 0.184761i	0.995724	−	0.0923806i	$-0.0294477\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1685.78	−1.69425
$996$	0	0
$997$	678.950i	0.680993i	0.940246	+	0.340497i	$0.110595\pi$
$997$	678.950i	0.680993i	−0.940246	+	0.340497i	$0.889405\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q-7.65685i q^{5} -1.65685i q^{7} -1.51472 q^{11} +0.343146i q^{13} +13.3137 q^{17} +20.8284 q^{19} -33.6569i q^{23} -33.6274 q^{25} +39.6569i q^{29} -45.2548i q^{31} -12.6863 q^{35} -29.5980i q^{37} +24.6274 q^{41} +50.0833 q^{43} -35.3137i q^{47} +46.2548 q^{49} +16.3431i q^{53} +11.5980i q^{55} +53.1127 q^{59} -34.4020i q^{61} +2.62742 q^{65} -62.4853 q^{67} +40.2843i q^{71} -55.9411 q^{73} +2.50967i q^{77} -137.941i q^{79} -114.652 q^{83} -101.941i q^{85} -2.56854 q^{89} +0.568542 q^{91} -159.480i q^{95} -138.569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 40 q^{11} + 8 q^{17} + 72 q^{19} - 44 q^{25} - 96 q^{35} + 8 q^{41} + 8 q^{43} + 4 q^{49} + 88 q^{59} - 80 q^{65} - 216 q^{67} - 88 q^{73} - 40 q^{83} + 216 q^{89} - 224 q^{91} - 328 q^{97}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more