LMFDB - Embedded newform 2304.3.b.r.127.5

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:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.5
Root			$-1.28897 - 0.581861i$ of defining polynomial
Character	$\chi$	$=$	2304.127
Dual form			2304.3.b.r.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.46308i q^{5} -5.48331i q^{7} +O(q^{10})$	$q+2.46308i q^{5} -5.48331i q^{7} -10.5830 q^{11} -8.96663i q^{13} +16.2399 q^{17} +2.96663 q^{19} -1.46141i q^{23} +18.9333 q^{25} +25.0905i q^{29} -10.5167i q^{31} +13.5058 q^{35} -16.9666i q^{37} -29.0150 q^{41} -34.9666 q^{43} +86.1254i q^{47} +18.9333 q^{49} -80.1976i q^{53} -26.0667i q^{55} -66.4208 q^{59} +0.966630i q^{61} +22.0855 q^{65} -113.800 q^{67} -90.5097i q^{71} -51.7998 q^{73} +58.0299i q^{77} -80.4499i q^{79} +79.9267 q^{83} +40.0000i q^{85} -142.694 q^{89} -49.1669 q^{91} +7.30703i q^{95} -45.8665 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 96 q^{19} - 88 q^{25} - 160 q^{43} - 88 q^{49} - 192 q^{67} + 304 q^{73} - 992 q^{91} + 112 q^{97}+O(q^{100}) $ 8 * q - 96 * q^19 - 88 * q^25 - 160 * q^43 - 88 * q^49 - 192 * q^67 + 304 * q^73 - 992 * q^91 + 112 * 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
$5$	2.46308i	0.492615i	0.969192	+	0.246308i	$0.0792173\pi$
$5$	2.46308i	0.492615i	−0.969192	+	0.246308i	$0.920783\pi$
$6$	0	0
$7$	− 5.48331i	− 0.783331i	−0.920108	−	0.391665i	$-0.871899\pi$
$7$	− 5.48331i	− 0.783331i	0.920108	−	0.391665i	$-0.128101\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−10.5830	−0.962091	−0.481046	−	0.876696i	$-0.659743\pi$
$11$	−10.5830	−0.962091	−0.481046	+	0.876696i	$0.659743\pi$
$12$	0	0
$13$	− 8.96663i	− 0.689741i	−0.938650	−	0.344870i	$-0.887923\pi$
$13$	− 8.96663i	− 0.689741i	0.938650	−	0.344870i	$-0.112077\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	16.2399	0.955286	0.477643	−	0.878554i	$-0.341491\pi$
$17$	16.2399	0.955286	0.477643	+	0.878554i	$0.341491\pi$
$18$	0	0
$19$	2.96663	0.156138	0.0780692	−	0.996948i	$-0.475124\pi$
$19$	2.96663	0.156138	0.0780692	+	0.996948i	$0.475124\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.46141i	− 0.0635394i	−0.999495	−	0.0317697i	$-0.989886\pi$
$23$	− 1.46141i	− 0.0635394i	0.999495	−	0.0317697i	$-0.0101143\pi$
$24$	0	0
$25$	18.9333	0.757330
$26$	0	0
$27$	0	0
$28$	0	0
$29$	25.0905i	0.865189i	0.901589	+	0.432595i	$0.142402\pi$
$29$	25.0905i	0.865189i	−0.901589	+	0.432595i	$0.857598\pi$
$30$	0	0
$31$	− 10.5167i	− 0.339248i	−0.985509	−	0.169624i	$-0.945745\pi$
$31$	− 10.5167i	− 0.339248i	0.985509	−	0.169624i	$-0.0542553\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	13.5058	0.385881
$36$	0	0
$37$	− 16.9666i	− 0.458558i	−0.973361	−	0.229279i	$-0.926363\pi$
$37$	− 16.9666i	− 0.458558i	0.973361	−	0.229279i	$-0.0736367\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−29.0150	−0.707682	−0.353841	−	0.935306i	$-0.615125\pi$
$41$	−29.0150	−0.707682	−0.353841	+	0.935306i	$0.615125\pi$
$42$	0	0
$43$	−34.9666	−0.813177	−0.406589	−	0.913611i	$-0.633282\pi$
$43$	−34.9666	−0.813177	−0.406589	+	0.913611i	$0.633282\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	86.1254i	1.83246i	0.400657	+	0.916228i	$0.368782\pi$
$47$	86.1254i	1.83246i	−0.400657	+	0.916228i	$0.631218\pi$
$48$	0	0
$49$	18.9333	0.386393
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 80.1976i	− 1.51316i	−0.653900	−	0.756581i	$-0.726868\pi$
$53$	− 80.1976i	− 1.51316i	0.653900	−	0.756581i	$-0.273132\pi$
$54$	0	0
$55$	− 26.0667i	− 0.473941i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−66.4208	−1.12578	−0.562889	−	0.826533i	$-0.690310\pi$
$59$	−66.4208	−1.12578	−0.562889	+	0.826533i	$0.690310\pi$
$60$	0	0
$61$	0.966630i	0.0158464i	0.999969	+	0.00792319i	$0.00252206\pi$
$61$	0.966630i	0.0158464i	−0.999969	+	0.00792319i	$0.997478\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	22.0855	0.339777
$66$	0	0
$67$	−113.800	−1.69850	−0.849252	−	0.527988i	$-0.822947\pi$
$67$	−113.800	−1.69850	−0.849252	+	0.527988i	$0.822947\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 90.5097i	− 1.27478i	−0.770540	−	0.637392i	$-0.780013\pi$
$71$	− 90.5097i	− 1.27478i	0.770540	−	0.637392i	$-0.219987\pi$
$72$	0	0
$73$	−51.7998	−0.709586	−0.354793	−	0.934945i	$-0.615449\pi$
$73$	−51.7998	−0.709586	−0.354793	+	0.934945i	$0.615449\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	58.0299i	0.753636i
$78$	0	0
$79$	− 80.4499i	− 1.01835i	−0.860662	−	0.509177i	$-0.829950\pi$
$79$	− 80.4499i	− 1.01835i	0.860662	−	0.509177i	$-0.170050\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	79.9267	0.962972	0.481486	−	0.876454i	$-0.340097\pi$
$83$	79.9267	0.962972	0.481486	+	0.876454i	$0.340097\pi$
$84$	0	0
$85$	40.0000i	0.470588i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−142.694	−1.60330	−0.801652	−	0.597791i	$-0.796045\pi$
$89$	−142.694	−1.60330	−0.801652	+	0.597791i	$0.796045\pi$
$90$	0	0
$91$	−49.1669	−0.540295
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.30703i	0.0769161i
$96$	0	0
$97$	−45.8665	−0.472851	−0.236425	−	0.971650i	$-0.575976\pi$
$97$	−45.8665	−0.472851	−0.236425	+	0.971650i	$0.575976\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 83.1204i	− 0.822975i	−0.911415	−	0.411487i	$-0.865009\pi$
$101$	− 83.1204i	− 0.822975i	0.911415	−	0.411487i	$-0.134991\pi$
$102$	0	0
$103$	− 106.517i	− 1.03414i	−0.855942	−	0.517071i	$-0.827022\pi$
$103$	− 106.517i	− 1.03414i	0.855942	−	0.517071i	$-0.172978\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−114.598	−1.07101	−0.535507	−	0.844531i	$-0.679879\pi$
$107$	−114.598	−1.07101	−0.535507	+	0.844531i	$0.679879\pi$
$108$	0	0
$109$	94.7664i	0.869417i	0.900571	+	0.434708i	$0.143149\pi$
$109$	94.7664i	0.869417i	−0.900571	+	0.434708i	$0.856851\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	201.808	1.78591	0.892955	−	0.450146i	$-0.148628\pi$
$113$	201.808	1.78591	0.892955	+	0.450146i	$0.148628\pi$
$114$	0	0
$115$	3.59955	0.0313005
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 89.0483i	− 0.748305i
$120$	0	0
$121$	−9.00000	−0.0743802
$122$	0	0
$123$	0	0
$124$	0	0
$125$	108.211i	0.865687i
$126$	0	0
$127$	− 171.417i	− 1.34974i	−0.737938	−	0.674868i	$-0.764200\pi$
$127$	− 171.417i	− 1.34974i	0.737938	−	0.674868i	$-0.235800\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	169.328	1.29258	0.646290	−	0.763092i	$-0.276319\pi$
$131$	169.328	1.29258	0.646290	+	0.763092i	$0.276319\pi$
$132$	0	0
$133$	− 16.2670i	− 0.122308i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.38088	−0.0173787	−0.00868935	−	0.999962i	$-0.502766\pi$
$137$	−2.38088	−0.0173787	−0.00868935	+	0.999962i	$0.502766\pi$
$138$	0	0
$139$	−209.800	−1.50935	−0.754675	−	0.656098i	$-0.772206\pi$
$139$	−209.800	−1.50935	−0.754675	+	0.656098i	$0.772206\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	94.8939i	0.663594i
$144$	0	0
$145$	−61.7998	−0.426205
$146$	0	0
$147$	0	0
$148$	0	0
$149$	51.7246i	0.347145i	0.984821	+	0.173572i	$0.0555311\pi$
$149$	51.7246i	0.347145i	−0.984821	+	0.173572i	$0.944469\pi$
$150$	0	0
$151$	− 227.150i	− 1.50430i	−0.658991	−	0.752151i	$-0.729016\pi$
$151$	− 227.150i	− 1.50430i	0.658991	−	0.752151i	$-0.270984\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	25.9034	0.167119
$156$	0	0
$157$	− 110.766i	− 0.705519i	−0.935714	−	0.352759i	$-0.885243\pi$
$157$	− 110.766i	− 0.705519i	0.935714	−	0.352759i	$-0.114757\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.01335	−0.0497724
$162$	0	0
$163$	−276.766	−1.69795	−0.848977	−	0.528430i	$-0.822781\pi$
$163$	−276.766	−1.69795	−0.848977	+	0.528430i	$0.822781\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 80.2798i	− 0.480717i	−0.970684	−	0.240359i	$-0.922735\pi$
$167$	− 80.2798i	− 0.480717i	0.970684	−	0.240359i	$-0.0772651\pi$
$168$	0	0
$169$	88.5996	0.524258
$170$	0	0
$171$	0	0
$172$	0	0
$173$	124.533i	0.719844i	0.932982	+	0.359922i	$0.117197\pi$
$173$	124.533i	0.719844i	−0.932982	+	0.359922i	$0.882803\pi$
$174$	0	0
$175$	− 103.817i	− 0.593240i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	66.4208	0.371066	0.185533	−	0.982638i	$-0.440599\pi$
$179$	66.4208	0.371066	0.185533	+	0.982638i	$0.440599\pi$
$180$	0	0
$181$	160.967i	0.889318i	0.895700	+	0.444659i	$0.146675\pi$
$181$	160.967i	0.889318i	−0.895700	+	0.444659i	$0.853325\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	41.7901	0.225892
$186$	0	0
$187$	−171.867	−0.919072
$188$	0	0
$189$	0	0
$190$	0	0
$191$	83.2026i	0.435616i	0.975992	+	0.217808i	$0.0698906\pi$
$191$	83.2026i	0.435616i	−0.975992	+	0.217808i	$0.930109\pi$
$192$	0	0
$193$	131.666	0.682209	0.341104	−	0.940025i	$-0.389199\pi$
$193$	131.666	0.682209	0.341104	+	0.940025i	$0.389199\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 184.566i	− 0.936885i	−0.883494	−	0.468442i	$-0.844815\pi$
$197$	− 184.566i	− 0.936885i	0.883494	−	0.468442i	$-0.155185\pi$
$198$	0	0
$199$	− 168.717i	− 0.847824i	−0.905703	−	0.423912i	$-0.860657\pi$
$199$	− 168.717i	− 0.847824i	0.905703	−	0.423912i	$-0.139343\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	137.579	0.677729
$204$	0	0
$205$	− 71.4661i	− 0.348615i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−31.3959	−0.150219
$210$	0	0
$211$	−197.933	−0.938072	−0.469036	−	0.883179i	$-0.655399\pi$
$211$	−197.933	−0.938072	−0.469036	+	0.883179i	$0.655399\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 86.1254i	− 0.400583i
$216$	0	0
$217$	−57.6663	−0.265743
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 145.617i	− 0.658900i
$222$	0	0
$223$	− 99.6835i	− 0.447011i	−0.974703	−	0.223506i	$-0.928250\pi$
$223$	− 99.6835i	− 0.447011i	0.974703	−	0.223506i	$-0.0717501\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−243.409	−1.07229	−0.536143	−	0.844127i	$-0.680119\pi$
$227$	−243.409	−1.07229	−0.536143	+	0.844127i	$0.680119\pi$
$228$	0	0
$229$	− 454.232i	− 1.98355i	−0.128002	−	0.991774i	$-0.540856\pi$
$229$	− 454.232i	− 1.98355i	0.128002	−	0.991774i	$-0.459144\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−77.7346	−0.333625	−0.166812	−	0.985989i	$-0.553347\pi$
$233$	−77.7346	−0.333625	−0.166812	+	0.985989i	$0.553347\pi$
$234$	0	0
$235$	−212.133	−0.902696
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 348.886i	− 1.45977i	−0.683568	−	0.729887i	$-0.739573\pi$
$239$	− 348.886i	− 1.45977i	0.683568	−	0.729887i	$-0.260427\pi$
$240$	0	0
$241$	75.7998	0.314522	0.157261	−	0.987557i	$-0.449734\pi$
$241$	75.7998	0.314522	0.157261	+	0.987557i	$0.449734\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	46.6340i	0.190343i
$246$	0	0
$247$	− 26.6007i	− 0.107695i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−131.733	−0.524834	−0.262417	−	0.964955i	$-0.584520\pi$
$251$	−131.733	−0.524834	−0.262417	+	0.964955i	$0.584520\pi$
$252$	0	0
$253$	15.4661i	0.0611307i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−305.093	−1.18713	−0.593565	−	0.804786i	$-0.702280\pi$
$257$	−305.093	−1.18713	−0.593565	+	0.804786i	$0.702280\pi$
$258$	0	0
$259$	−93.0334	−0.359202
$260$	0	0
$261$	0	0
$262$	0	0
$263$	338.656i	1.28767i	0.765166	+	0.643833i	$0.222657\pi$
$263$	338.656i	1.28767i	−0.765166	+	0.643833i	$0.777343\pi$
$264$	0	0
$265$	197.533	0.745407
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 271.234i	− 1.00830i	−0.863615	−	0.504152i	$-0.831805\pi$
$269$	− 271.234i	− 1.00830i	0.863615	−	0.504152i	$-0.168195\pi$
$270$	0	0
$271$	− 406.749i	− 1.50092i	−0.660916	−	0.750460i	$-0.729832\pi$
$271$	− 406.749i	− 1.50092i	0.660916	−	0.750460i	$-0.270168\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−200.371	−0.728621
$276$	0	0
$277$	249.234i	0.899760i	0.893089	+	0.449880i	$0.148533\pi$
$277$	249.234i	0.899760i	−0.893089	+	0.449880i	$0.851467\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	56.9461	0.202655	0.101328	−	0.994853i	$-0.467691\pi$
$281$	56.9461	0.202655	0.101328	+	0.994853i	$0.467691\pi$
$282$	0	0
$283$	−136.267	−0.481509	−0.240754	−	0.970586i	$-0.577395\pi$
$283$	−136.267	−0.481509	−0.240754	+	0.970586i	$0.577395\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	159.098i	0.554349i
$288$	0	0
$289$	−25.2670	−0.0874289
$290$	0	0
$291$	0	0
$292$	0	0
$293$	445.324i	1.51988i	0.649996	+	0.759938i	$0.274771\pi$
$293$	445.324i	1.51988i	−0.649996	+	0.759938i	$0.725229\pi$
$294$	0	0
$295$	− 163.600i	− 0.554575i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−13.1039	−0.0438257
$300$	0	0
$301$	191.733i	0.636987i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.38088	−0.00780617
$306$	0	0
$307$	−329.533	−1.07340	−0.536698	−	0.843774i	$-0.680329\pi$
$307$	−329.533	−1.07340	−0.536698	+	0.843774i	$0.680329\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	359.116i	1.15471i	0.816492	+	0.577357i	$0.195916\pi$
$311$	359.116i	1.15471i	−0.816492	+	0.577357i	$0.804084\pi$
$312$	0	0
$313$	−147.666	−0.471777	−0.235889	−	0.971780i	$-0.575800\pi$
$313$	−147.666	−0.471777	−0.235889	+	0.971780i	$0.575800\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 233.663i	− 0.737109i	−0.929606	−	0.368554i	$-0.879853\pi$
$317$	− 233.663i	− 0.737109i	0.929606	−	0.368554i	$-0.120147\pi$
$318$	0	0
$319$	− 265.533i	− 0.832391i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	48.1776	0.149157
$324$	0	0
$325$	− 169.768i	− 0.522362i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	472.253	1.43542
$330$	0	0
$331$	163.600	0.494258	0.247129	−	0.968983i	$-0.420513\pi$
$331$	163.600	0.494258	0.247129	+	0.968983i	$0.420513\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 280.297i	− 0.836709i
$336$	0	0
$337$	−35.9333	−0.106627	−0.0533134	−	0.998578i	$-0.516978\pi$
$337$	−35.9333	−0.106627	−0.0533134	+	0.998578i	$0.516978\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	111.298i	0.326387i
$342$	0	0
$343$	− 372.499i	− 1.08600i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	584.282	1.68381	0.841905	−	0.539626i	$-0.181435\pi$
$347$	584.282	1.68381	0.841905	+	0.539626i	$0.181435\pi$
$348$	0	0
$349$	− 230.766i	− 0.661222i	−0.943767	−	0.330611i	$-0.892745\pi$
$349$	− 230.766i	− 0.661222i	0.943767	−	0.330611i	$-0.107255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−50.0166	−0.141690	−0.0708450	−	0.997487i	$-0.522570\pi$
$353$	−50.0166	−0.141690	−0.0708450	+	0.997487i	$0.522570\pi$
$354$	0	0
$355$	222.932	0.627978
$356$	0	0
$357$	0	0
$358$	0	0
$359$	253.992i	0.707499i	0.935340	+	0.353749i	$0.115093\pi$
$359$	253.992i	0.707499i	−0.935340	+	0.353749i	$0.884907\pi$
$360$	0	0
$361$	−352.199	−0.975621
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 127.587i	− 0.349553i
$366$	0	0
$367$	348.682i	0.950088i	0.879962	+	0.475044i	$0.157568\pi$
$367$	348.682i	0.950088i	−0.879962	+	0.475044i	$0.842432\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−439.749	−1.18531
$372$	0	0
$373$	646.499i	1.73324i	0.498967	+	0.866621i	$0.333713\pi$
$373$	646.499i	1.73324i	−0.498967	+	0.866621i	$0.666287\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	224.977	0.596756
$378$	0	0
$379$	−274.433	−0.724097	−0.362048	−	0.932159i	$-0.617922\pi$
$379$	−274.433	−0.724097	−0.362048	+	0.932159i	$0.617922\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 272.990i	− 0.712769i	−0.934339	−	0.356384i	$-0.884009\pi$
$383$	− 272.990i	− 0.712769i	0.934339	−	0.356384i	$-0.115991\pi$
$384$	0	0
$385$	−142.932	−0.371252
$386$	0	0
$387$	0	0
$388$	0	0
$389$	412.679i	1.06087i	0.847725	+	0.530436i	$0.177972\pi$
$389$	412.679i	1.06087i	−0.847725	+	0.530436i	$0.822028\pi$
$390$	0	0
$391$	− 23.7330i	− 0.0606983i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	198.154	0.501656
$396$	0	0
$397$	384.700i	0.969017i	0.874787	+	0.484508i	$0.161001\pi$
$397$	384.700i	0.969017i	−0.874787	+	0.484508i	$0.838999\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−561.466	−1.40016	−0.700082	−	0.714063i	$-0.746853\pi$
$401$	−561.466	−1.40016	−0.700082	+	0.714063i	$0.746853\pi$
$402$	0	0
$403$	−94.2992	−0.233993
$404$	0	0
$405$	0	0
$406$	0	0
$407$	179.558i	0.441174i
$408$	0	0
$409$	253.333	0.619395	0.309698	−	0.950835i	$-0.399772\pi$
$409$	253.333	0.619395	0.309698	+	0.950835i	$0.399772\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	364.206i	0.881856i
$414$	0	0
$415$	196.865i	0.474374i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−622.985	−1.48684	−0.743418	−	0.668827i	$-0.766797\pi$
$419$	−622.985	−1.48684	−0.743418	+	0.668827i	$0.766797\pi$
$420$	0	0
$421$	− 487.300i	− 1.15748i	−0.815511	−	0.578741i	$-0.803544\pi$
$421$	− 487.300i	− 1.15748i	0.815511	−	0.578741i	$-0.196456\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	307.473	0.723467
$426$	0	0
$427$	5.30033	0.0124130
$428$	0	0
$429$	0	0
$430$	0	0
$431$	176.635i	0.409826i	0.978780	+	0.204913i	$0.0656912\pi$
$431$	176.635i	0.409826i	−0.978780	+	0.204913i	$0.934309\pi$
$432$	0	0
$433$	−257.066	−0.593685	−0.296843	−	0.954926i	$-0.595934\pi$
$433$	−257.066	−0.593685	−0.296843	+	0.954926i	$0.595934\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.33545i	− 0.00992094i
$438$	0	0
$439$	638.482i	1.45440i	0.686425	+	0.727201i	$0.259179\pi$
$439$	638.482i	1.45440i	−0.686425	+	0.727201i	$0.740821\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−104.722	−0.236392	−0.118196	−	0.992990i	$-0.537711\pi$
$443$	−104.722	−0.236392	−0.118196	+	0.992990i	$0.537711\pi$
$444$	0	0
$445$	− 351.466i	− 0.789811i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	219.132	0.488043	0.244022	−	0.969770i	$-0.421533\pi$
$449$	219.132	0.488043	0.244022	+	0.969770i	$0.421533\pi$
$450$	0	0
$451$	307.066	0.680855
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 121.102i	− 0.266158i
$456$	0	0
$457$	46.4004	0.101533	0.0507664	−	0.998711i	$-0.483834\pi$
$457$	46.4004	0.101533	0.0507664	+	0.998711i	$0.483834\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 517.377i	− 1.12229i	−0.827717	−	0.561146i	$-0.810361\pi$
$461$	− 517.377i	− 1.12229i	0.827717	−	0.561146i	$-0.189639\pi$
$462$	0	0
$463$	− 111.016i	− 0.239776i	−0.992787	−	0.119888i	$-0.961747\pi$
$463$	− 111.016i	− 0.239776i	0.992787	−	0.119888i	$-0.0382535\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	233.934	0.500930	0.250465	−	0.968126i	$-0.419416\pi$
$467$	233.934	0.500930	0.250465	+	0.968126i	$0.419416\pi$
$468$	0	0
$469$	624.000i	1.33049i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	370.052	0.782351
$474$	0	0
$475$	56.1680	0.118248
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 610.185i	− 1.27387i	−0.770916	−	0.636937i	$-0.780201\pi$
$479$	− 610.185i	− 1.27387i	0.770916	−	0.636937i	$-0.219799\pi$
$480$	0	0
$481$	−152.133	−0.316286
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 112.973i	− 0.232933i
$486$	0	0
$487$	104.183i	0.213928i	0.994263	+	0.106964i	$0.0341130\pi$
$487$	104.183i	0.213928i	−0.994263	+	0.106964i	$0.965887\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	423.320	0.862159	0.431080	−	0.902314i	$-0.358133\pi$
$491$	423.320	0.862159	0.431080	+	0.902314i	$0.358133\pi$
$492$	0	0
$493$	407.466i	0.826503i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−496.293	−0.998577
$498$	0	0
$499$	−126.932	−0.254373	−0.127187	−	0.991879i	$-0.540595\pi$
$499$	−126.932	−0.254373	−0.127187	+	0.991879i	$0.540595\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 274.452i	− 0.545630i	−0.962067	−	0.272815i	$-0.912045\pi$
$503$	− 274.452i	− 0.545630i	0.962067	−	0.272815i	$-0.0879547\pi$
$504$	0	0
$505$	204.732	0.405410
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 750.416i	− 1.47429i	−0.675732	−	0.737147i	$-0.736172\pi$
$509$	− 750.416i	− 1.47429i	0.675732	−	0.737147i	$-0.263828\pi$
$510$	0	0
$511$	284.034i	0.555840i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	262.359	0.509434
$516$	0	0
$517$	− 911.466i	− 1.76299i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−822.387	−1.57848	−0.789239	−	0.614086i	$-0.789525\pi$
$521$	−822.387	−1.57848	−0.789239	+	0.614086i	$0.789525\pi$
$522$	0	0
$523$	184.366	0.352516	0.176258	−	0.984344i	$-0.443601\pi$
$523$	184.366	0.352516	0.176258	+	0.984344i	$0.443601\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 170.789i	− 0.324079i
$528$	0	0
$529$	526.864	0.995963
$530$	0	0
$531$	0	0
$532$	0	0
$533$	260.167i	0.488117i
$534$	0	0
$535$	− 282.265i	− 0.527598i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−200.371	−0.371745
$540$	0	0
$541$	− 432.967i	− 0.800308i	−0.916448	−	0.400154i	$-0.868957\pi$
$541$	− 432.967i	− 0.800308i	0.916448	−	0.400154i	$-0.131043\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−233.417	−0.428288
$546$	0	0
$547$	557.567	1.01932	0.509659	−	0.860376i	$-0.329771\pi$
$547$	557.567	1.01932	0.509659	+	0.860376i	$0.329771\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	74.4342i	0.135089i
$552$	0	0
$553$	−441.132	−0.797708
$554$	0	0
$555$	0	0
$556$	0	0
$557$	656.984i	1.17950i	0.807584	+	0.589752i	$0.200774\pi$
$557$	656.984i	1.17950i	−0.807584	+	0.589752i	$0.799226\pi$
$558$	0	0
$559$	313.533i	0.560882i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	204.706	0.363599	0.181799	−	0.983336i	$-0.441808\pi$
$563$	204.706	0.363599	0.181799	+	0.983336i	$0.441808\pi$
$564$	0	0
$565$	497.068i	0.879766i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−251.611	−0.442199	−0.221100	−	0.975251i	$-0.570965\pi$
$569$	−251.611	−0.442199	−0.221100	+	0.975251i	$0.570965\pi$
$570$	0	0
$571$	832.198	1.45744	0.728720	−	0.684812i	$-0.240116\pi$
$571$	832.198	1.45744	0.728720	+	0.684812i	$0.240116\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 27.6692i	− 0.0481203i
$576$	0	0
$577$	539.132	0.934372	0.467186	−	0.884159i	$-0.345268\pi$
$577$	539.132	0.934372	0.467186	+	0.884159i	$0.345268\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 438.263i	− 0.754325i
$582$	0	0
$583$	848.732i	1.45580i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−84.6640	−0.144232	−0.0721159	−	0.997396i	$-0.522975\pi$
$587$	−84.6640	−0.144232	−0.0721159	+	0.997396i	$0.522975\pi$
$588$	0	0
$589$	− 31.1991i	− 0.0529696i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	923.717	1.55770	0.778851	−	0.627209i	$-0.215803\pi$
$593$	923.717	1.55770	0.778851	+	0.627209i	$0.215803\pi$
$594$	0	0
$595$	219.333	0.368626
$596$	0	0
$597$	0	0
$598$	0	0
$599$	538.674i	0.899288i	0.893208	+	0.449644i	$0.148449\pi$
$599$	538.674i	0.899288i	−0.893208	+	0.449644i	$0.851551\pi$
$600$	0	0
$601$	164.334	0.273434	0.136717	−	0.990610i	$-0.456345\pi$
$601$	164.334	0.273434	0.136717	+	0.990610i	$0.456345\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 22.1677i	− 0.0366408i
$606$	0	0
$607$	368.984i	0.607881i	0.952691	+	0.303941i	$0.0983024\pi$
$607$	368.984i	0.607881i	−0.952691	+	0.303941i	$0.901698\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	772.255	1.26392
$612$	0	0
$613$	− 968.433i	− 1.57982i	−0.613220	−	0.789912i	$-0.710126\pi$
$613$	− 968.433i	− 1.57982i	0.613220	−	0.789912i	$-0.289874\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−53.2682	−0.0863342	−0.0431671	−	0.999068i	$-0.513745\pi$
$617$	−53.2682	−0.0863342	−0.0431671	+	0.999068i	$0.513745\pi$
$618$	0	0
$619$	1066.80	1.72342	0.861711	−	0.507399i	$-0.169393\pi$
$619$	1066.80	1.72342	0.861711	+	0.507399i	$0.169393\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	782.436i	1.25592i
$624$	0	0
$625$	206.800	0.330880
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 275.536i	− 0.438054i
$630$	0	0
$631$	468.049i	0.741758i	0.928681	+	0.370879i	$0.120944\pi$
$631$	468.049i	0.741758i	−0.928681	+	0.370879i	$0.879056\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	422.212	0.664901
$636$	0	0
$637$	− 169.768i	− 0.266511i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	603.043	0.940784	0.470392	−	0.882458i	$-0.344113\pi$
$641$	603.043	0.940784	0.470392	+	0.882458i	$0.344113\pi$
$642$	0	0
$643$	352.633	0.548418	0.274209	−	0.961670i	$-0.411584\pi$
$643$	352.633	0.548418	0.274209	+	0.961670i	$0.411584\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 897.790i	− 1.38762i	−0.720158	−	0.693810i	$-0.755931\pi$
$647$	− 897.790i	− 1.38762i	0.720158	−	0.693810i	$-0.244069\pi$
$648$	0	0
$649$	702.932	1.08310
$650$	0	0
$651$	0	0
$652$	0	0
$653$	169.459i	0.259508i	0.991546	+	0.129754i	$0.0414188\pi$
$653$	169.459i	0.259508i	−0.991546	+	0.129754i	$0.958581\pi$
$654$	0	0
$655$	417.068i	0.636745i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−36.4864	−0.0553663	−0.0276832	−	0.999617i	$-0.508813\pi$
$659$	−36.4864	−0.0553663	−0.0276832	+	0.999617i	$0.508813\pi$
$660$	0	0
$661$	1038.77i	1.57151i	0.618539	+	0.785754i	$0.287725\pi$
$661$	1038.77i	1.57151i	−0.618539	+	0.785754i	$0.712275\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	40.0668	0.0602508
$666$	0	0
$667$	36.6674	0.0549736
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 10.2298i	− 0.0152457i
$672$	0	0
$673$	−593.600	−0.882020	−0.441010	−	0.897502i	$-0.645380\pi$
$673$	−593.600	−0.882020	−0.441010	+	0.897502i	$0.645380\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 296.455i	− 0.437895i	−0.975737	−	0.218948i	$-0.929738\pi$
$677$	− 296.455i	− 0.437895i	0.975737	−	0.218948i	$-0.0702624\pi$
$678$	0	0
$679$	251.501i	0.370398i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1169.67	−1.71255	−0.856275	−	0.516520i	$-0.827227\pi$
$683$	−1169.67	−1.71255	−0.856275	+	0.516520i	$0.827227\pi$
$684$	0	0
$685$	− 5.86429i	− 0.00856101i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−719.102	−1.04369
$690$	0	0
$691$	1242.70	1.79841	0.899204	−	0.437530i	$-0.144147\pi$
$691$	1242.70	1.79841	0.899204	+	0.437530i	$0.144147\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 516.753i	− 0.743529i
$696$	0	0
$697$	−471.199	−0.676039
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1362.28i	1.94333i	0.236358	+	0.971666i	$0.424046\pi$
$701$	1362.28i	1.94333i	−0.236358	+	0.971666i	$0.575954\pi$
$702$	0	0
$703$	− 50.3337i	− 0.0715984i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−455.776	−0.644661
$708$	0	0
$709$	− 1261.97i	− 1.77992i	−0.456036	−	0.889962i	$-0.650731\pi$
$709$	− 1261.97i	− 1.77992i	0.456036	−	0.889962i	$-0.349269\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−15.3692	−0.0215556
$714$	0	0
$715$	−233.731	−0.326896
$716$	0	0
$717$	0	0
$718$	0	0
$719$	951.764i	1.32373i	0.749622	+	0.661867i	$0.230235\pi$
$719$	951.764i	1.32373i	−0.749622	+	0.661867i	$0.769765\pi$
$720$	0	0
$721$	−584.065	−0.810076
$722$	0	0
$723$	0	0
$724$	0	0
$725$	475.045i	0.655234i
$726$	0	0
$727$	982.383i	1.35128i	0.737230	+	0.675642i	$0.236133\pi$
$727$	982.383i	1.35128i	−0.737230	+	0.675642i	$0.763867\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−567.853	−0.776817
$732$	0	0
$733$	165.699i	0.226055i	0.993592	+	0.113028i	$0.0360549\pi$
$733$	165.699i	0.226055i	−0.993592	+	0.113028i	$0.963945\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1204.34	1.63412
$738$	0	0
$739$	−248.999	−0.336940	−0.168470	−	0.985707i	$-0.553883\pi$
$739$	−248.999	−0.336940	−0.168470	+	0.985707i	$0.553883\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 538.674i	− 0.724998i	−0.931984	−	0.362499i	$-0.881924\pi$
$743$	− 538.674i	− 0.724998i	0.931984	−	0.362499i	$-0.118076\pi$
$744$	0	0
$745$	−127.402	−0.171009
$746$	0	0
$747$	0	0
$748$	0	0
$749$	628.380i	0.838958i
$750$	0	0
$751$	979.882i	1.30477i	0.757888	+	0.652385i	$0.226231\pi$
$751$	979.882i	1.30477i	−0.757888	+	0.652385i	$0.773769\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	559.487	0.741042
$756$	0	0
$757$	1151.90i	1.52166i	0.648950	+	0.760831i	$0.275209\pi$
$757$	1151.90i	1.52166i	−0.648950	+	0.760831i	$0.724791\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−268.064	−0.352253	−0.176126	−	0.984368i	$-0.556357\pi$
$761$	−268.064	−0.352253	−0.176126	+	0.984368i	$0.556357\pi$
$762$	0	0
$763$	519.634	0.681041
$764$	0	0
$765$	0	0
$766$	0	0
$767$	595.571i	0.776494i
$768$	0	0
$769$	428.467	0.557174	0.278587	−	0.960411i	$-0.410134\pi$
$769$	428.467	0.557174	0.278587	+	0.960411i	$0.410134\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1118.14i	− 1.44649i	−0.690592	−	0.723245i	$-0.742650\pi$
$773$	− 1118.14i	− 1.44649i	0.690592	−	0.723245i	$-0.257350\pi$
$774$	0	0
$775$	− 199.115i	− 0.256923i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−86.0767	−0.110496
$780$	0	0
$781$	957.864i	1.22646i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	272.826	0.347549
$786$	0	0
$787$	711.832	0.904488	0.452244	−	0.891894i	$-0.350624\pi$
$787$	711.832	0.904488	0.452244	+	0.891894i	$0.350624\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1106.58i	− 1.39896i
$792$	0	0
$793$	8.66741	0.0109299
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 209.033i	− 0.262274i	−0.991364	−	0.131137i	$-0.958137\pi$
$797$	− 209.033i	− 0.262274i	0.991364	−	0.131137i	$-0.0418628\pi$
$798$	0	0
$799$	1398.67i	1.75052i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	548.197	0.682687
$804$	0	0
$805$	− 19.7375i	− 0.0245186i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−581.170	−0.718381	−0.359190	−	0.933264i	$-0.616947\pi$
$809$	−581.170	−0.718381	−0.359190	+	0.933264i	$0.616947\pi$
$810$	0	0
$811$	−171.432	−0.211383	−0.105691	−	0.994399i	$-0.533706\pi$
$811$	−171.432	−0.211383	−0.105691	+	0.994399i	$0.533706\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 681.697i	− 0.836437i
$816$	0	0
$817$	−103.733	−0.126968
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1134.49i	− 1.38184i	−0.722931	−	0.690921i	$-0.757205\pi$
$821$	− 1134.49i	− 1.38184i	0.722931	−	0.690921i	$-0.242795\pi$
$822$	0	0
$823$	379.249i	0.460812i	0.973095	+	0.230406i	$0.0740055\pi$
$823$	379.249i	0.460812i	−0.973095	+	0.230406i	$0.925995\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1177.33	−1.42362	−0.711809	−	0.702373i	$-0.752124\pi$
$827$	−1177.33	−1.42362	−0.711809	+	0.702373i	$0.752124\pi$
$828$	0	0
$829$	− 1278.56i	− 1.54230i	−0.636656	−	0.771148i	$-0.719683\pi$
$829$	− 1278.56i	− 1.54230i	0.636656	−	0.771148i	$-0.280317\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	307.473	0.369116
$834$	0	0
$835$	197.735	0.236809
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1015.97i	1.21093i	0.795873	+	0.605464i	$0.207012\pi$
$839$	1015.97i	1.21093i	−0.795873	+	0.605464i	$0.792988\pi$
$840$	0	0
$841$	211.467	0.251447
$842$	0	0
$843$	0	0
$844$	0	0
$845$	218.227i	0.258257i
$846$	0	0
$847$	49.3498i	0.0582643i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24.7951	−0.0291365
$852$	0	0
$853$	1509.43i	1.76956i	0.466012	+	0.884778i	$0.345690\pi$
$853$	1509.43i	1.76956i	−0.466012	+	0.884778i	$0.654310\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−844.260	−0.985134	−0.492567	−	0.870275i	$-0.663941\pi$
$857$	−844.260	−0.985134	−0.492567	+	0.870275i	$0.663941\pi$
$858$	0	0
$859$	989.231	1.15161	0.575804	−	0.817588i	$-0.304689\pi$
$859$	989.231	1.15161	0.575804	+	0.817588i	$0.304689\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1680.23i	1.94696i	0.228774	+	0.973480i	$0.426528\pi$
$863$	1680.23i	1.94696i	−0.228774	+	0.973480i	$0.573472\pi$
$864$	0	0
$865$	−306.734	−0.354606
$866$	0	0
$867$	0	0
$868$	0	0
$869$	851.402i	0.979749i
$870$	0	0
$871$	1020.40i	1.17153i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	593.355	0.678120
$876$	0	0
$877$	591.899i	0.674913i	0.941341	+	0.337457i	$0.109567\pi$
$877$	591.899i	0.674913i	−0.941341	+	0.337457i	$0.890433\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	797.050	0.904711	0.452355	−	0.891838i	$-0.350584\pi$
$881$	797.050	0.904711	0.452355	+	0.891838i	$0.350584\pi$
$882$	0	0
$883$	276.964	0.313663	0.156831	−	0.987625i	$-0.449872\pi$
$883$	276.964	0.313663	0.156831	+	0.987625i	$0.449872\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	910.845i	1.02688i	0.858125	+	0.513441i	$0.171630\pi$
$887$	910.845i	1.02688i	−0.858125	+	0.513441i	$0.828370\pi$
$888$	0	0
$889$	−939.931	−1.05729
$890$	0	0
$891$	0	0
$892$	0	0
$893$	255.502i	0.286117i
$894$	0	0
$895$	163.600i	0.182793i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	263.869	0.293514
$900$	0	0
$901$	− 1302.40i	− 1.44550i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−396.473	−0.438092
$906$	0	0
$907$	1472.83	1.62385	0.811924	−	0.583763i	$-0.198420\pi$
$907$	1472.83	1.62385	0.811924	+	0.583763i	$0.198420\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	36.5352i	0.0401045i	0.999799	+	0.0200522i	$0.00638325\pi$
$911$	36.5352i	0.0401045i	−0.999799	+	0.0200522i	$0.993617\pi$
$912$	0	0
$913$	−845.864	−0.926467
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 928.479i	− 1.01252i
$918$	0	0
$919$	− 467.150i	− 0.508324i	−0.967162	−	0.254162i	$-0.918200\pi$
$919$	− 467.150i	− 0.508324i	0.967162	−	0.254162i	$-0.0817996\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−811.567	−0.879270
$924$	0	0
$925$	− 321.234i	− 0.347280i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−762.190	−0.820441	−0.410220	−	0.911986i	$-0.634548\pi$
$929$	−762.190	−0.820441	−0.410220	+	0.911986i	$0.634548\pi$
$930$	0	0
$931$	56.1680	0.0603308
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 423.320i	− 0.452749i
$936$	0	0
$937$	−123.335	−0.131627	−0.0658137	−	0.997832i	$-0.520964\pi$
$937$	−123.335	−0.131627	−0.0658137	+	0.997832i	$0.520964\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 735.638i	− 0.781762i	−0.920441	−	0.390881i	$-0.872170\pi$
$941$	− 735.638i	− 0.781762i	0.920441	−	0.390881i	$-0.127830\pi$
$942$	0	0
$943$	42.4027i	0.0449657i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	549.610	0.580370	0.290185	−	0.956971i	$-0.406283\pi$
$947$	549.610	0.580370	0.290185	+	0.956971i	$0.406283\pi$
$948$	0	0
$949$	464.469i	0.489430i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1217.33	−1.27737	−0.638684	−	0.769469i	$-0.720521\pi$
$953$	−1217.33	−1.27737	−0.638684	+	0.769469i	$0.720521\pi$
$954$	0	0
$955$	−204.934	−0.214591
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.0551i	0.0136133i
$960$	0	0
$961$	850.399	0.884911
$962$	0	0
$963$	0	0
$964$	0	0
$965$	324.304i	0.336066i
$966$	0	0
$967$	514.616i	0.532178i	0.963949	+	0.266089i	$0.0857314\pi$
$967$	514.616i	0.532178i	−0.963949	+	0.266089i	$0.914269\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1248.49	−1.28578	−0.642889	−	0.765959i	$-0.722264\pi$
$971$	−1248.49	−1.28578	−0.642889	+	0.765959i	$0.722264\pi$
$972$	0	0
$973$	1150.40i	1.18232i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	126.028	0.128995	0.0644973	−	0.997918i	$-0.479456\pi$
$977$	126.028	0.128995	0.0644973	+	0.997918i	$0.479456\pi$
$978$	0	0
$979$	1510.13	1.54252
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1018.89i	− 1.03651i	−0.855226	−	0.518256i	$-0.826581\pi$
$983$	− 1018.89i	− 1.03651i	0.855226	−	0.518256i	$-0.173419\pi$
$984$	0	0
$985$	454.601	0.461524
$986$	0	0
$987$	0	0
$988$	0	0
$989$	51.1005i	0.0516688i
$990$	0	0
$991$	− 859.981i	− 0.867791i	−0.900963	−	0.433895i	$-0.857139\pi$
$991$	− 859.981i	− 0.867791i	0.900963	−	0.433895i	$-0.142861\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	415.562	0.417651
$996$	0	0
$997$	− 1104.43i	− 1.10776i	−0.832598	−	0.553878i	$-0.813147\pi$
$997$	− 1104.43i	− 1.10776i	0.832598	−	0.553878i	$-0.186853\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.b.r.127.5		8
3.2	odd	2	inner	2304.3.b.r.127.3		8
4.3	odd	2		2304.3.b.s.127.6		8
8.3	odd	2	inner	2304.3.b.r.127.4		8
8.5	even	2		2304.3.b.s.127.3		8
12.11	even	2		2304.3.b.s.127.4		8
16.3	odd	4		1152.3.g.d.127.5	yes	8
16.5	even	4		1152.3.g.e.127.4	yes	8
16.11	odd	4		1152.3.g.e.127.3	yes	8
16.13	even	4		1152.3.g.d.127.6	yes	8
24.5	odd	2		2304.3.b.s.127.5		8
24.11	even	2	inner	2304.3.b.r.127.6		8
48.5	odd	4		1152.3.g.e.127.6	yes	8
48.11	even	4		1152.3.g.e.127.5	yes	8
48.29	odd	4		1152.3.g.d.127.4	yes	8
48.35	even	4		1152.3.g.d.127.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.g.d.127.3	✓	8	48.35	even	4
1152.3.g.d.127.4	yes	8	48.29	odd	4
1152.3.g.d.127.5	yes	8	16.3	odd	4
1152.3.g.d.127.6	yes	8	16.13	even	4
1152.3.g.e.127.3	yes	8	16.11	odd	4
1152.3.g.e.127.4	yes	8	16.5	even	4
1152.3.g.e.127.5	yes	8	48.11	even	4
1152.3.g.e.127.6	yes	8	48.5	odd	4
2304.3.b.r.127.3		8	3.2	odd	2	inner
2304.3.b.r.127.4		8	8.3	odd	2	inner
2304.3.b.r.127.5		8	1.1	even	1	trivial
2304.3.b.r.127.6		8	24.11	even	2	inner
2304.3.b.s.127.3		8	8.5	even	2
2304.3.b.s.127.4		8	12.11	even	2
2304.3.b.s.127.5		8	24.5	odd	2
2304.3.b.s.127.6		8	4.3	odd	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)

\(f(q)\)	\(=\)	\(q+2.46308i q^{5} -5.48331i q^{7} +O(q^{10})\)	\(q+2.46308i q^{5} -5.48331i q^{7} -10.5830 q^{11} -8.96663i q^{13} +16.2399 q^{17} +2.96663 q^{19} -1.46141i q^{23} +18.9333 q^{25} +25.0905i q^{29} -10.5167i q^{31} +13.5058 q^{35} -16.9666i q^{37} -29.0150 q^{41} -34.9666 q^{43} +86.1254i q^{47} +18.9333 q^{49} -80.1976i q^{53} -26.0667i q^{55} -66.4208 q^{59} +0.966630i q^{61} +22.0855 q^{65} -113.800 q^{67} -90.5097i q^{71} -51.7998 q^{73} +58.0299i q^{77} -80.4499i q^{79} +79.9267 q^{83} +40.0000i q^{85} -142.694 q^{89} -49.1669 q^{91} +7.30703i q^{95} -45.8665 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 96 q^{19} - 88 q^{25} - 160 q^{43} - 88 q^{49} - 192 q^{67} + 304 q^{73} - 992 q^{91} + 112 q^{97}+O(q^{100}) \) 8 * q - 96 * q^19 - 88 * q^25 - 160 * q^43 - 88 * q^49 - 192 * q^67 + 304 * q^73 - 992 * q^91 + 112 * q^97

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