LMFDB - Embedded newform 2304.4.f.f.1151.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,4,Mod(1151,2304)]

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, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2304.1151");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2304.f (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:	$135.940400653$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.3
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2304.1151
Dual form			2304.4.f.f.1151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.24264 q^{5} +13.8564i q^{7} +O(q^{10})$	$q-4.24264 q^{5} +13.8564i q^{7} +58.7878i q^{11} -28.0000i q^{13} +97.5807i q^{17} -110.851 q^{19} -176.363 q^{23} -107.000 q^{25} -4.24264 q^{29} -124.708i q^{31} -58.7878i q^{35} -118.000i q^{37} -190.919i q^{41} -249.415 q^{43} -293.939 q^{47} +151.000 q^{49} +700.036 q^{53} -249.415i q^{55} -587.878i q^{59} +802.000i q^{61} +118.794i q^{65} +138.564 q^{67} +646.665 q^{71} -496.000 q^{73} -814.587 q^{77} +1011.52i q^{79} +58.7878i q^{83} -414.000i q^{85} -1124.30i q^{89} +387.979 q^{91} +470.302 q^{95} -152.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 856 q^{25} + 1208 q^{49} - 3968 q^{73} - 1216 q^{97}+O(q^{100}) $ 8 * q - 856 * q^25 + 1208 * q^49 - 3968 * q^73 - 1216 * 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
$5$	−4.24264	−0.379473	−0.189737	−	0.981835i	$-0.560763\pi$
$5$	−4.24264	−0.379473	−0.189737	+	0.981835i	$0.560763\pi$
$6$	0	0
$7$	13.8564i	0.748176i	0.927393	+	0.374088i	$0.122044\pi$
$7$	13.8564i	0.748176i	−0.927393	+	0.374088i	$0.877956\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	58.7878i	1.61138i	0.592338	+	0.805690i	$0.298205\pi$
$11$	58.7878i	1.61138i	−0.592338	+	0.805690i	$0.701795\pi$
$12$	0	0
$13$	− 28.0000i	− 0.597369i	−0.954352	−	0.298685i	$-0.903452\pi$
$13$	− 28.0000i	− 0.597369i	0.954352	−	0.298685i	$-0.0965479\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	97.5807i	1.39216i	0.717962	+	0.696082i	$0.245075\pi$
$17$	97.5807i	1.39216i	−0.717962	+	0.696082i	$0.754925\pi$
$18$	0	0
$19$	−110.851	−1.33847	−0.669237	−	0.743049i	$-0.733379\pi$
$19$	−110.851	−1.33847	−0.669237	+	0.743049i	$0.733379\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−176.363	−1.59888	−0.799441	−	0.600745i	$-0.794871\pi$
$23$	−176.363	−1.59888	−0.799441	+	0.600745i	$0.794871\pi$
$24$	0	0
$25$	−107.000	−0.856000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.24264	−0.0271668	−0.0135834	−	0.999908i	$-0.504324\pi$
$29$	−4.24264	−0.0271668	−0.0135834	+	0.999908i	$0.504324\pi$
$30$	0	0
$31$	− 124.708i	− 0.722521i	−0.932465	−	0.361261i	$-0.882346\pi$
$31$	− 124.708i	− 0.722521i	0.932465	−	0.361261i	$-0.117654\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 58.7878i	− 0.283913i
$36$	0	0
$37$	− 118.000i	− 0.524299i	−0.965027	−	0.262150i	$-0.915569\pi$
$37$	− 118.000i	− 0.524299i	0.965027	−	0.262150i	$-0.0844314\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 190.919i	− 0.727232i	−0.931549	−	0.363616i	$-0.881542\pi$
$41$	− 190.919i	− 0.727232i	0.931549	−	0.363616i	$-0.118458\pi$
$42$	0	0
$43$	−249.415	−0.884546	−0.442273	−	0.896880i	$-0.645828\pi$
$43$	−249.415	−0.884546	−0.442273	+	0.896880i	$0.645828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−293.939	−0.912242	−0.456121	−	0.889918i	$-0.650762\pi$
$47$	−293.939	−0.912242	−0.456121	+	0.889918i	$0.650762\pi$
$48$	0	0
$49$	151.000	0.440233
$50$	0	0
$51$	0	0
$52$	0	0
$53$	700.036	1.81429	0.907144	−	0.420820i	$-0.138257\pi$
$53$	700.036	1.81429	0.907144	+	0.420820i	$0.138257\pi$
$54$	0	0
$55$	− 249.415i	− 0.611476i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 587.878i	− 1.29721i	−0.761127	−	0.648603i	$-0.775354\pi$
$59$	− 587.878i	− 1.29721i	0.761127	−	0.648603i	$-0.224646\pi$
$60$	0	0
$61$	802.000i	1.68337i	0.539969	+	0.841685i	$0.318436\pi$
$61$	802.000i	1.68337i	−0.539969	+	0.841685i	$0.681564\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	118.794i	0.226686i
$66$	0	0
$67$	138.564	0.252661	0.126331	−	0.991988i	$-0.459680\pi$
$67$	138.564	0.252661	0.126331	+	0.991988i	$0.459680\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	646.665	1.08092	0.540458	−	0.841371i	$-0.318251\pi$
$71$	646.665	1.08092	0.540458	+	0.841371i	$0.318251\pi$
$72$	0	0
$73$	−496.000	−0.795238	−0.397619	−	0.917551i	$-0.630163\pi$
$73$	−496.000	−0.795238	−0.397619	+	0.917551i	$0.630163\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−814.587	−1.20559
$78$	0	0
$79$	1011.52i	1.44056i	0.693681	+	0.720282i	$0.255988\pi$
$79$	1011.52i	1.44056i	−0.693681	+	0.720282i	$0.744012\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	58.7878i	0.0777445i	0.999244	+	0.0388723i	$0.0123765\pi$
$83$	58.7878i	0.0777445i	−0.999244	+	0.0388723i	$0.987623\pi$
$84$	0	0
$85$	− 414.000i	− 0.528289i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1124.30i	− 1.33905i	−0.742789	−	0.669525i	$-0.766497\pi$
$89$	− 1124.30i	− 1.33905i	0.742789	−	0.669525i	$-0.233503\pi$
$90$	0	0
$91$	387.979	0.446937
$92$	0	0
$93$	0	0
$94$	0	0
$95$	470.302	0.507915
$96$	0	0
$97$	−152.000	−0.159106	−0.0795529	−	0.996831i	$-0.525349\pi$
$97$	−152.000	−0.159106	−0.0795529	+	0.996831i	$0.525349\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	929.138	0.915373	0.457687	−	0.889114i	$-0.348678\pi$
$101$	929.138	0.915373	0.457687	+	0.889114i	$0.348678\pi$
$102$	0	0
$103$	1759.76i	1.68344i	0.539912	+	0.841722i	$0.318458\pi$
$103$	1759.76i	1.68344i	−0.539912	+	0.841722i	$0.681542\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	940.604i	0.849828i	0.905234	+	0.424914i	$0.139696\pi$
$107$	940.604i	0.849828i	−0.905234	+	0.424914i	$0.860304\pi$
$108$	0	0
$109$	− 628.000i	− 0.551849i	−0.961179	−	0.275924i	$-0.911016\pi$
$109$	− 628.000i	− 0.551849i	0.961179	−	0.275924i	$-0.0889839\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1243.09i	1.03487i	0.855722	+	0.517435i	$0.173113\pi$
$113$	1243.09i	1.03487i	−0.855722	+	0.517435i	$0.826887\pi$
$114$	0	0
$115$	748.246	0.606733
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1352.12	−1.04158
$120$	0	0
$121$	−2125.00	−1.59654
$122$	0	0
$123$	0	0
$124$	0	0
$125$	984.293	0.704302
$126$	0	0
$127$	− 1621.20i	− 1.13274i	−0.824151	−	0.566371i	$-0.808347\pi$
$127$	− 1621.20i	− 1.13274i	0.824151	−	0.566371i	$-0.191653\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 823.029i	− 0.548919i	−0.961599	−	0.274459i	$-0.911501\pi$
$131$	− 823.029i	− 0.548919i	0.961599	−	0.274459i	$-0.0884989\pi$
$132$	0	0
$133$	− 1536.00i	− 1.00141i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1607.96i	− 1.00275i	−0.865229	−	0.501377i	$-0.832827\pi$
$137$	− 1607.96i	− 1.00275i	0.865229	−	0.501377i	$-0.167173\pi$
$138$	0	0
$139$	2133.89	1.30211	0.651057	−	0.759029i	$-0.274326\pi$
$139$	2133.89	1.30211	0.651057	+	0.759029i	$0.274326\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1646.06	0.962589
$144$	0	0
$145$	18.0000	0.0103091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	818.830	0.450209	0.225104	−	0.974335i	$-0.427728\pi$
$149$	818.830	0.450209	0.225104	+	0.974335i	$0.427728\pi$
$150$	0	0
$151$	− 3256.26i	− 1.75490i	−0.479666	−	0.877451i	$-0.659242\pi$
$151$	− 3256.26i	− 1.75490i	0.479666	−	0.877451i	$-0.340758\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	529.090i	0.274178i
$156$	0	0
$157$	− 1730.00i	− 0.879421i	−0.898140	−	0.439710i	$-0.855081\pi$
$157$	− 1730.00i	− 0.879421i	0.898140	−	0.439710i	$-0.144919\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2443.76i	− 1.19624i
$162$	0	0
$163$	−1635.06	−0.785690	−0.392845	−	0.919605i	$-0.628509\pi$
$163$	−1635.06	−0.785690	−0.392845	+	0.919605i	$0.628509\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1998.78	0.926171	0.463085	−	0.886314i	$-0.346742\pi$
$167$	1998.78	0.926171	0.463085	+	0.886314i	$0.346742\pi$
$168$	0	0
$169$	1413.00	0.643150
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2159.50	0.949041	0.474520	−	0.880245i	$-0.342622\pi$
$173$	2159.50	0.949041	0.474520	+	0.880245i	$0.342622\pi$
$174$	0	0
$175$	− 1482.64i	− 0.640438i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2351.51i	0.981900i	0.871188	+	0.490950i	$0.163350\pi$
$179$	2351.51i	0.981900i	−0.871188	+	0.490950i	$0.836650\pi$
$180$	0	0
$181$	− 620.000i	− 0.254609i	−0.991864	−	0.127305i	$-0.959367\pi$
$181$	− 620.000i	− 0.254609i	0.991864	−	0.127305i	$-0.0406325\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	500.632i	0.198958i
$186$	0	0
$187$	−5736.55	−2.24331
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3644.84	−1.38079	−0.690396	−	0.723431i	$-0.742564\pi$
$191$	−3644.84	−1.38079	−0.690396	+	0.723431i	$0.742564\pi$
$192$	0	0
$193$	−946.000	−0.352822	−0.176411	−	0.984317i	$-0.556449\pi$
$193$	−946.000	−0.352822	−0.176411	+	0.984317i	$0.556449\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1913.43	0.692012	0.346006	−	0.938232i	$-0.387538\pi$
$197$	1913.43	0.692012	0.346006	+	0.938232i	$0.387538\pi$
$198$	0	0
$199$	− 734.390i	− 0.261605i	−0.991408	−	0.130803i	$-0.958245\pi$
$199$	− 734.390i	− 0.261605i	0.991408	−	0.130803i	$-0.0417555\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 58.7878i	− 0.0203256i
$204$	0	0
$205$	810.000i	0.275965i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 6516.70i	− 2.15679i
$210$	0	0
$211$	360.267	0.117544	0.0587720	−	0.998271i	$-0.481282\pi$
$211$	360.267	0.117544	0.0587720	+	0.998271i	$0.481282\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1058.18	0.335662
$216$	0	0
$217$	1728.00	0.540573
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2732.26	0.831637
$222$	0	0
$223$	3865.94i	1.16091i	0.814293	+	0.580454i	$0.197125\pi$
$223$	3865.94i	1.16091i	−0.814293	+	0.580454i	$0.802875\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2763.02i	0.807878i	0.914786	+	0.403939i	$0.132359\pi$
$227$	2763.02i	0.807878i	−0.914786	+	0.403939i	$0.867641\pi$
$228$	0	0
$229$	700.000i	0.201997i	0.994887	+	0.100998i	$0.0322037\pi$
$229$	700.000i	0.201997i	−0.994887	+	0.100998i	$0.967796\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1964.34i	− 0.552311i	−0.961113	−	0.276155i	$-0.910940\pi$
$233$	− 1964.34i	− 0.552311i	0.961113	−	0.276155i	$-0.0890604\pi$
$234$	0	0
$235$	1247.08	0.346172
$236$	0	0
$237$	0	0
$238$	0	0
$239$	235.151	0.0636429	0.0318215	−	0.999494i	$-0.489869\pi$
$239$	235.151	0.0636429	0.0318215	+	0.999494i	$0.489869\pi$
$240$	0	0
$241$	−6488.00	−1.73414	−0.867072	−	0.498182i	$-0.834001\pi$
$241$	−6488.00	−1.73414	−0.867072	+	0.498182i	$0.834001\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−640.639	−0.167057
$246$	0	0
$247$	3103.84i	0.799564i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3586.05i	− 0.901791i	−0.892577	−	0.450896i	$-0.851105\pi$
$251$	− 3586.05i	− 0.901791i	0.892577	−	0.450896i	$-0.148895\pi$
$252$	0	0
$253$	− 10368.0i	− 2.57641i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5316.03i	− 1.29029i	−0.764060	−	0.645145i	$-0.776797\pi$
$257$	− 5316.03i	− 1.29029i	0.764060	−	0.645145i	$-0.223203\pi$
$258$	0	0
$259$	1635.06	0.392268
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5996.35	−1.40590	−0.702948	−	0.711241i	$-0.748134\pi$
$263$	−5996.35	−1.40590	−0.702948	+	0.711241i	$0.748134\pi$
$264$	0	0
$265$	−2970.00	−0.688474
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6520.94	1.47802	0.739012	−	0.673692i	$-0.235292\pi$
$269$	6520.94	1.47802	0.739012	+	0.673692i	$0.235292\pi$
$270$	0	0
$271$	− 734.390i	− 0.164616i	−0.996607	−	0.0823081i	$-0.973771\pi$
$271$	− 734.390i	− 0.164616i	0.996607	−	0.0823081i	$-0.0262292\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 6290.29i	− 1.37934i
$276$	0	0
$277$	− 4124.00i	− 0.894538i	−0.894399	−	0.447269i	$-0.852397\pi$
$277$	− 4124.00i	− 0.894538i	0.894399	−	0.447269i	$-0.147603\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 1294.01i	− 0.274711i	−0.990522	−	0.137356i	$-0.956140\pi$
$281$	− 1294.01i	− 0.274711i	0.990522	−	0.137356i	$-0.0438603\pi$
$282$	0	0
$283$	7482.46	1.57168	0.785841	−	0.618428i	$-0.212230\pi$
$283$	7482.46	1.57168	0.785841	+	0.618428i	$0.212230\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2645.45	0.544097
$288$	0	0
$289$	−4609.00	−0.938123
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3542.60	−0.706352	−0.353176	−	0.935557i	$-0.614898\pi$
$293$	−3542.60	−0.706352	−0.353176	+	0.935557i	$0.614898\pi$
$294$	0	0
$295$	2494.15i	0.492255i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4938.17i	0.955123i
$300$	0	0
$301$	− 3456.00i	− 0.661796i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 3402.60i	− 0.638794i
$306$	0	0
$307$	−249.415	−0.0463677	−0.0231839	−	0.999731i	$-0.507380\pi$
$307$	−249.415	−0.0463677	−0.0231839	+	0.999731i	$0.507380\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2233.93	0.407315	0.203657	−	0.979042i	$-0.434717\pi$
$311$	2233.93	0.407315	0.203657	+	0.979042i	$0.434717\pi$
$312$	0	0
$313$	−7946.00	−1.43493	−0.717467	−	0.696592i	$-0.754699\pi$
$313$	−7946.00	−1.43493	−0.717467	+	0.696592i	$0.754699\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8964.70	1.58835	0.794176	−	0.607688i	$-0.207903\pi$
$317$	8964.70	1.58835	0.794176	+	0.607688i	$0.207903\pi$
$318$	0	0
$319$	− 249.415i	− 0.0437761i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 10816.9i	− 1.86338i
$324$	0	0
$325$	2996.00i	0.511348i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 4072.94i	− 0.682517i
$330$	0	0
$331$	−1856.76	−0.308328	−0.154164	−	0.988045i	$-0.549268\pi$
$331$	−1856.76	−0.308328	−0.154164	+	0.988045i	$0.549268\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−587.878	−0.0958782
$336$	0	0
$337$	−1952.00	−0.315526	−0.157763	−	0.987477i	$-0.550428\pi$
$337$	−1952.00	−0.315526	−0.157763	+	0.987477i	$0.550428\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7331.28	1.16426
$342$	0	0
$343$	6845.06i	1.07755i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5819.99i	− 0.900384i	−0.892932	−	0.450192i	$-0.851356\pi$
$347$	− 5819.99i	− 0.900384i	0.892932	−	0.450192i	$-0.148644\pi$
$348$	0	0
$349$	226.000i	0.0346633i	0.999850	+	0.0173317i	$0.00551712\pi$
$349$	226.000i	0.0346633i	−0.999850	+	0.0173317i	$0.994483\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1845.55i	− 0.278268i	−0.990274	−	0.139134i	$-0.955568\pi$
$353$	− 1845.55i	− 0.278268i	0.990274	−	0.139134i	$-0.0444319\pi$
$354$	0	0
$355$	−2743.57	−0.410179
$356$	0	0
$357$	0	0
$358$	0	0
$359$	58.7878	0.00864262	0.00432131	−	0.999991i	$-0.498624\pi$
$359$	58.7878	0.00864262	0.00432131	+	0.999991i	$0.498624\pi$
$360$	0	0
$361$	5429.00	0.791515
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2104.35	0.301772
$366$	0	0
$367$	2480.30i	0.352780i	0.984320	+	0.176390i	$0.0564421\pi$
$367$	2480.30i	0.352780i	−0.984320	+	0.176390i	$0.943558\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9699.98i	1.35741i
$372$	0	0
$373$	− 5338.00i	− 0.740995i	−0.928833	−	0.370498i	$-0.879187\pi$
$373$	− 5338.00i	− 0.740995i	0.928833	−	0.370498i	$-0.120813\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	118.794i	0.0162286i
$378$	0	0
$379$	−5376.29	−0.728658	−0.364329	−	0.931270i	$-0.618702\pi$
$379$	−5376.29	−0.728658	−0.364329	+	0.931270i	$0.618702\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6349.08	0.847057	0.423528	−	0.905883i	$-0.360791\pi$
$383$	6349.08	0.847057	0.423528	+	0.905883i	$0.360791\pi$
$384$	0	0
$385$	3456.00	0.457491
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11238.8	1.46485	0.732426	−	0.680847i	$-0.238388\pi$
$389$	11238.8	1.46485	0.732426	+	0.680847i	$0.238388\pi$
$390$	0	0
$391$	− 17209.7i	− 2.22591i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 4291.51i	− 0.546656i
$396$	0	0
$397$	466.000i	0.0589115i	0.999566	+	0.0294558i	$0.00937741\pi$
$397$	466.000i	0.0589115i	−0.999566	+	0.0294558i	$0.990623\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 7403.41i	− 0.921966i	−0.887409	−	0.460983i	$-0.847497\pi$
$401$	− 7403.41i	− 0.921966i	0.887409	−	0.460983i	$-0.152503\pi$
$402$	0	0
$403$	−3491.81	−0.431612
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6936.95	0.844845
$408$	0	0
$409$	−4024.00	−0.486489	−0.243244	−	0.969965i	$-0.578212\pi$
$409$	−4024.00	−0.486489	−0.243244	+	0.969965i	$0.578212\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8145.87	0.970538
$414$	0	0
$415$	− 249.415i	− 0.0295020i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15108.5i	1.76157i	0.473520	+	0.880783i	$0.342983\pi$
$419$	15108.5i	1.76157i	−0.473520	+	0.880783i	$0.657017\pi$
$420$	0	0
$421$	− 8540.00i	− 0.988632i	−0.869282	−	0.494316i	$-0.835419\pi$
$421$	− 8540.00i	− 0.988632i	0.869282	−	0.494316i	$-0.164581\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 10441.1i	− 1.19169i
$426$	0	0
$427$	−11112.8	−1.25946
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16049.1	−1.79363	−0.896817	−	0.442403i	$-0.854126\pi$
$431$	−16049.1	−1.79363	−0.896817	+	0.442403i	$0.854126\pi$
$432$	0	0
$433$	−11266.0	−1.25037	−0.625184	−	0.780477i	$-0.714976\pi$
$433$	−11266.0	−1.25037	−0.625184	+	0.780477i	$0.714976\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19550.1	2.14006
$438$	0	0
$439$	6110.68i	0.664343i	0.943219	+	0.332172i	$0.107781\pi$
$439$	6110.68i	0.664343i	−0.943219	+	0.332172i	$0.892219\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13580.0i	1.45644i	0.685342	+	0.728221i	$0.259653\pi$
$443$	13580.0i	1.45644i	−0.685342	+	0.728221i	$0.740347\pi$
$444$	0	0
$445$	4770.00i	0.508134i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	547.301i	0.0575250i	0.999586	+	0.0287625i	$0.00915665\pi$
$449$	547.301i	0.0575250i	−0.999586	+	0.0287625i	$0.990843\pi$
$450$	0	0
$451$	11223.7	1.17185
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1646.06	−0.169601
$456$	0	0
$457$	−15784.0	−1.61563	−0.807817	−	0.589434i	$-0.799351\pi$
$457$	−15784.0	−1.61563	−0.807817	+	0.589434i	$0.799351\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18620.9	−1.88127	−0.940634	−	0.339424i	$-0.889768\pi$
$461$	−18620.9	−1.88127	−0.940634	+	0.339424i	$0.889768\pi$
$462$	0	0
$463$	− 8244.56i	− 0.827554i	−0.910378	−	0.413777i	$-0.864209\pi$
$463$	− 8244.56i	− 0.827554i	0.910378	−	0.413777i	$-0.135791\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2057.57i	0.203882i	0.994790	+	0.101941i	$0.0325053\pi$
$467$	2057.57i	0.203882i	−0.994790	+	0.101941i	$0.967495\pi$
$468$	0	0
$469$	1920.00i	0.189035i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 14662.6i	− 1.42534i
$474$	0	0
$475$	11861.1	1.14573
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2175.15	−0.207484	−0.103742	−	0.994604i	$-0.533082\pi$
$479$	−2175.15	−0.207484	−0.103742	+	0.994604i	$0.533082\pi$
$480$	0	0
$481$	−3304.00	−0.313200
$482$	0	0
$483$	0	0
$484$	0	0
$485$	644.881	0.0603764
$486$	0	0
$487$	− 11708.7i	− 1.08947i	−0.838609	−	0.544733i	$-0.816631\pi$
$487$	− 11708.7i	− 1.08947i	0.838609	−	0.544733i	$-0.183369\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1646.06i	0.151294i	0.997135	+	0.0756472i	$0.0241023\pi$
$491$	1646.06i	0.151294i	−0.997135	+	0.0756472i	$0.975898\pi$
$492$	0	0
$493$	− 414.000i	− 0.0378207i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8960.46i	0.808715i
$498$	0	0
$499$	2992.98	0.268506	0.134253	−	0.990947i	$-0.457137\pi$
$499$	2992.98	0.268506	0.134253	+	0.990947i	$0.457137\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9112.10	−0.807731	−0.403865	−	0.914818i	$-0.632334\pi$
$503$	−9112.10	−0.807731	−0.403865	+	0.914818i	$0.632334\pi$
$504$	0	0
$505$	−3942.00	−0.347360
$506$	0	0
$507$	0	0
$508$	0	0
$509$	12384.3	1.07843	0.539217	−	0.842167i	$-0.318720\pi$
$509$	12384.3	1.07843	0.539217	+	0.842167i	$0.318720\pi$
$510$	0	0
$511$	− 6872.78i	− 0.594978i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 7466.04i	− 0.638822i
$516$	0	0
$517$	− 17280.0i	− 1.46997i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 3882.02i	− 0.326438i	−0.986590	−	0.163219i	$-0.947812\pi$
$521$	− 3882.02i	− 0.326438i	0.986590	−	0.163219i	$-0.0521877\pi$
$522$	0	0
$523$	15574.6	1.30216	0.651080	−	0.759009i	$-0.274316\pi$
$523$	15574.6	1.30216	0.651080	+	0.759009i	$0.274316\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12169.1	1.00587
$528$	0	0
$529$	18937.0	1.55642
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5345.73	−0.434426
$534$	0	0
$535$	− 3990.65i	− 0.322487i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8876.95i	0.709383i
$540$	0	0
$541$	− 11188.0i	− 0.889112i	−0.895751	−	0.444556i	$-0.853361\pi$
$541$	− 11188.0i	− 0.889112i	0.895751	−	0.444556i	$-0.146639\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2664.38i	0.209412i
$546$	0	0
$547$	−21588.3	−1.68747	−0.843737	−	0.536757i	$-0.819649\pi$
$547$	−21588.3	−1.68747	−0.843737	+	0.536757i	$0.819649\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	470.302	0.0363621
$552$	0	0
$553$	−14016.0	−1.07780
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−16346.9	−1.24352	−0.621760	−	0.783208i	$-0.713582\pi$
$557$	−16346.9	−1.24352	−0.621760	+	0.783208i	$0.713582\pi$
$558$	0	0
$559$	6983.63i	0.528401i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 21692.7i	− 1.62387i	−0.583749	−	0.811934i	$-0.698415\pi$
$563$	− 21692.7i	− 1.62387i	0.583749	−	0.811934i	$-0.301585\pi$
$564$	0	0
$565$	− 5274.00i	− 0.392706i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 9414.42i	− 0.693626i	−0.937934	−	0.346813i	$-0.887264\pi$
$569$	− 9414.42i	− 0.693626i	0.937934	−	0.346813i	$-0.112736\pi$
$570$	0	0
$571$	−10364.6	−0.759623	−0.379811	−	0.925064i	$-0.624011\pi$
$571$	−10364.6	−0.759623	−0.379811	+	0.925064i	$0.624011\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18870.9	1.36864
$576$	0	0
$577$	−9406.00	−0.678643	−0.339321	−	0.940670i	$-0.610197\pi$
$577$	−9406.00	−0.678643	−0.339321	+	0.940670i	$0.610197\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−814.587	−0.0581665
$582$	0	0
$583$	41153.5i	2.92351i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 21986.6i	− 1.54597i	−0.634424	−	0.772985i	$-0.718763\pi$
$587$	− 21986.6i	− 1.54597i	0.634424	−	0.772985i	$-0.281237\pi$
$588$	0	0
$589$	13824.0i	0.967076i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	13419.5i	0.929295i	0.885496	+	0.464647i	$0.153819\pi$
$593$	13419.5i	0.929295i	−0.885496	+	0.464647i	$0.846181\pi$
$594$	0	0
$595$	5736.55	0.395253
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6878.17	0.469172	0.234586	−	0.972095i	$-0.424626\pi$
$599$	6878.17	0.469172	0.234586	+	0.972095i	$0.424626\pi$
$600$	0	0
$601$	−6086.00	−0.413067	−0.206533	−	0.978440i	$-0.566218\pi$
$601$	−6086.00	−0.413067	−0.206533	+	0.978440i	$0.566218\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9015.61	0.605846
$606$	0	0
$607$	7967.43i	0.532765i	0.963867	+	0.266382i	$0.0858284\pi$
$607$	7967.43i	0.532765i	−0.963867	+	0.266382i	$0.914172\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8230.29i	0.544946i
$612$	0	0
$613$	20438.0i	1.34663i	0.739357	+	0.673314i	$0.235130\pi$
$613$	20438.0i	1.34663i	−0.739357	+	0.673314i	$0.764870\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3568.06i	− 0.232812i	−0.993202	−	0.116406i	$-0.962863\pi$
$617$	− 3568.06i	− 0.232812i	0.993202	−	0.116406i	$-0.0371373\pi$
$618$	0	0
$619$	−1884.47	−0.122364	−0.0611820	−	0.998127i	$-0.519487\pi$
$619$	−1884.47	−0.122364	−0.0611820	+	0.998127i	$0.519487\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15578.8	1.00185
$624$	0	0
$625$	9199.00	0.588736
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11514.5	0.729911
$630$	0	0
$631$	− 9491.64i	− 0.598821i	−0.954124	−	0.299411i	$-0.903210\pi$
$631$	− 9491.64i	− 0.598821i	0.954124	−	0.299411i	$-0.0967900\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6878.17i	0.429845i
$636$	0	0
$637$	− 4228.00i	− 0.262982i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 16363.9i	− 1.00832i	−0.863610	−	0.504161i	$-0.831802\pi$
$641$	− 16363.9i	− 1.00832i	0.863610	−	0.504161i	$-0.168198\pi$
$642$	0	0
$643$	24193.3	1.48381	0.741905	−	0.670505i	$-0.233922\pi$
$643$	24193.3	1.48381	0.741905	+	0.670505i	$0.233922\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14520.6	0.882323	0.441161	−	0.897428i	$-0.354567\pi$
$647$	14520.6	0.882323	0.441161	+	0.897428i	$0.354567\pi$
$648$	0	0
$649$	34560.0	2.09029
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4187.49	−0.250948	−0.125474	−	0.992097i	$-0.540045\pi$
$653$	−4187.49	−0.250948	−0.125474	+	0.992097i	$0.540045\pi$
$654$	0	0
$655$	3491.81i	0.208300i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 4115.14i	− 0.243252i	−0.992576	−	0.121626i	$-0.961189\pi$
$659$	− 4115.14i	− 0.243252i	0.992576	−	0.121626i	$-0.0388109\pi$
$660$	0	0
$661$	− 22438.0i	− 1.32033i	−0.751121	−	0.660164i	$-0.770487\pi$
$661$	− 22438.0i	− 1.32033i	0.751121	−	0.660164i	$-0.229513\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6516.70i	0.380010i
$666$	0	0
$667$	748.246	0.0434366
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−47147.8	−2.71255
$672$	0	0
$673$	18962.0	1.08608	0.543040	−	0.839707i	$-0.317273\pi$
$673$	18962.0	1.08608	0.543040	+	0.839707i	$0.317273\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2278.30	−0.129338	−0.0646692	−	0.997907i	$-0.520599\pi$
$677$	−2278.30	−0.129338	−0.0646692	+	0.997907i	$0.520599\pi$
$678$	0	0
$679$	− 2106.17i	− 0.119039i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 11581.2i	− 0.648817i	−0.945917	−	0.324408i	$-0.894835\pi$
$683$	− 11581.2i	− 0.648817i	0.945917	−	0.324408i	$-0.105165\pi$
$684$	0	0
$685$	6822.00i	0.380519i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 19601.0i	− 1.08380i
$690$	0	0
$691$	−11722.5	−0.645363	−0.322681	−	0.946508i	$-0.604584\pi$
$691$	−11722.5	−0.645363	−0.322681	+	0.946508i	$0.604584\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9053.31	−0.494118
$696$	0	0
$697$	18630.0	1.01243
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9601.10	0.517302	0.258651	−	0.965971i	$-0.416722\pi$
$701$	9601.10	0.517302	0.258651	+	0.965971i	$0.416722\pi$
$702$	0	0
$703$	13080.4i	0.701762i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12874.5i	0.684860i
$708$	0	0
$709$	14572.0i	0.771880i	0.922524	+	0.385940i	$0.126123\pi$
$709$	14572.0i	0.771880i	−0.922524	+	0.385940i	$0.873877\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	21993.8i	1.15523i
$714$	0	0
$715$	−6983.63	−0.365277
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9817.55	−0.509225	−0.254613	−	0.967043i	$-0.581948\pi$
$719$	−9817.55	−0.509225	−0.254613	+	0.967043i	$0.581948\pi$
$720$	0	0
$721$	−24384.0	−1.25951
$722$	0	0
$723$	0	0
$724$	0	0
$725$	453.963	0.0232548
$726$	0	0
$727$	− 33047.5i	− 1.68592i	−0.537975	−	0.842961i	$-0.680811\pi$
$727$	− 33047.5i	− 1.68592i	0.537975	−	0.842961i	$-0.319189\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 24338.1i	− 1.23143i
$732$	0	0
$733$	− 28636.0i	− 1.44297i	−0.692432	−	0.721483i	$-0.743461\pi$
$733$	− 28636.0i	− 1.44297i	0.692432	−	0.721483i	$-0.256539\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8145.87i	0.407133i
$738$	0	0
$739$	−1496.49	−0.0744917	−0.0372458	−	0.999306i	$-0.511858\pi$
$739$	−1496.49	−0.0744917	−0.0372458	+	0.999306i	$0.511858\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13050.9	−0.644402	−0.322201	−	0.946671i	$-0.604423\pi$
$743$	−13050.9	−0.644402	−0.322201	+	0.946671i	$0.604423\pi$
$744$	0	0
$745$	−3474.00	−0.170842
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−13033.4	−0.635821
$750$	0	0
$751$	8244.56i	0.400597i	0.979735	+	0.200298i	$0.0641912\pi$
$751$	8244.56i	0.400597i	−0.979735	+	0.200298i	$0.935809\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	13815.1i	0.665939i
$756$	0	0
$757$	17740.0i	0.851745i	0.904783	+	0.425873i	$0.140033\pi$
$757$	17740.0i	0.851745i	−0.904783	+	0.425873i	$0.859967\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17254.8i	0.821927i	0.911652	+	0.410964i	$0.134808\pi$
$761$	17254.8i	0.821927i	−0.911652	+	0.410964i	$0.865192\pi$
$762$	0	0
$763$	8701.82	0.412880
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16460.6	−0.774911
$768$	0	0
$769$	−17038.0	−0.798967	−0.399484	−	0.916740i	$-0.630811\pi$
$769$	−17038.0	−0.798967	−0.399484	+	0.916740i	$0.630811\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−38739.6	−1.80254	−0.901271	−	0.433256i	$-0.857365\pi$
$773$	−38739.6	−1.80254	−0.901271	+	0.433256i	$0.857365\pi$
$774$	0	0
$775$	13343.7i	0.618478i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21163.6i	0.973382i
$780$	0	0
$781$	38016.0i	1.74177i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7339.77i	0.333717i
$786$	0	0
$787$	−3103.84	−0.140584	−0.0702921	−	0.997526i	$-0.522393\pi$
$787$	−3103.84	−0.140584	−0.0702921	+	0.997526i	$0.522393\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−17224.8	−0.774265
$792$	0	0
$793$	22456.0	1.00559
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1098.84	−0.0488370	−0.0244185	−	0.999702i	$-0.507773\pi$
$797$	−1098.84	−0.0488370	−0.0244185	+	0.999702i	$0.507773\pi$
$798$	0	0
$799$	− 28682.8i	− 1.26999i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 29158.7i	− 1.28143i
$804$	0	0
$805$	10368.0i	0.453943i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38739.6i	1.68357i	0.539811	+	0.841786i	$0.318496\pi$
$809$	38739.6i	1.68357i	−0.539811	+	0.841786i	$0.681504\pi$
$810$	0	0
$811$	−34419.3	−1.49029	−0.745145	−	0.666902i	$-0.767620\pi$
$811$	−34419.3	−1.49029	−0.745145	+	0.666902i	$0.767620\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6936.95	0.298148
$816$	0	0
$817$	27648.0	1.18394
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−35387.9	−1.50432	−0.752159	−	0.658982i	$-0.770987\pi$
$821$	−35387.9	−1.50432	−0.752159	+	0.658982i	$0.770987\pi$
$822$	0	0
$823$	11708.7i	0.495915i	0.968771	+	0.247958i	$0.0797594\pi$
$823$	11708.7i	0.495915i	−0.968771	+	0.247958i	$0.920241\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31392.7i	1.31999i	0.751271	+	0.659994i	$0.229441\pi$
$827$	31392.7i	1.31999i	−0.751271	+	0.659994i	$0.770559\pi$
$828$	0	0
$829$	− 43444.0i	− 1.82011i	−0.414486	−	0.910056i	$-0.636039\pi$
$829$	− 43444.0i	− 1.82011i	0.414486	−	0.910056i	$-0.363961\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	14734.7i	0.612877i
$834$	0	0
$835$	−8480.12	−0.351457
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15343.6	0.631371	0.315685	−	0.948864i	$-0.397766\pi$
$839$	15343.6	0.631371	0.315685	+	0.948864i	$0.397766\pi$
$840$	0	0
$841$	−24371.0	−0.999262
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−5994.85	−0.244058
$846$	0	0
$847$	− 29444.9i	− 1.19450i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	20810.9i	0.838293i
$852$	0	0
$853$	− 4870.00i	− 0.195481i	−0.995212	−	0.0977407i	$-0.968838\pi$
$853$	− 4870.00i	− 0.195481i	0.995212	−	0.0977407i	$-0.0311616\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9414.42i	0.375251i	0.982241	+	0.187626i	$0.0600792\pi$
$857$	9414.42i	0.375251i	−0.982241	+	0.187626i	$0.939921\pi$
$858$	0	0
$859$	16600.0	0.659353	0.329676	−	0.944094i	$-0.393060\pi$
$859$	16600.0	0.659353	0.329676	+	0.944094i	$0.393060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36095.7	1.42377	0.711884	−	0.702297i	$-0.247842\pi$
$863$	36095.7	1.42377	0.711884	+	0.702297i	$0.247842\pi$
$864$	0	0
$865$	−9162.00	−0.360136
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−59464.9	−2.32130
$870$	0	0
$871$	− 3879.79i	− 0.150932i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13638.8i	0.526942i
$876$	0	0
$877$	19954.0i	0.768300i	0.923271	+	0.384150i	$0.125505\pi$
$877$	19954.0i	0.768300i	−0.923271	+	0.384150i	$0.874495\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 8744.08i	− 0.334388i	−0.985924	−	0.167194i	$-0.946529\pi$
$881$	− 8744.08i	− 0.334388i	0.985924	−	0.167194i	$-0.0534706\pi$
$882$	0	0
$883$	−30068.4	−1.14596	−0.572980	−	0.819570i	$-0.694213\pi$
$883$	−30068.4	−1.14596	−0.572980	+	0.819570i	$0.694213\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28100.5	−1.06372	−0.531862	−	0.846831i	$-0.678508\pi$
$887$	−28100.5	−1.06372	−0.531862	+	0.846831i	$0.678508\pi$
$888$	0	0
$889$	22464.0	0.847490
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32583.5	1.22101
$894$	0	0
$895$	− 9976.61i	− 0.372605i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	529.090i	0.0196286i
$900$	0	0
$901$	68310.0i	2.52579i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2630.44i	0.0966173i
$906$	0	0
$907$	−1136.23	−0.0415962	−0.0207981	−	0.999784i	$-0.506621\pi$
$907$	−1136.23	−0.0415962	−0.0207981	+	0.999784i	$0.506621\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39387.8	1.43247	0.716233	−	0.697862i	$-0.245865\pi$
$911$	39387.8	1.43247	0.716233	+	0.697862i	$0.245865\pi$
$912$	0	0
$913$	−3456.00	−0.125276
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11404.2	0.410688
$918$	0	0
$919$	− 5861.26i	− 0.210386i	−0.994452	−	0.105193i	$-0.966454\pi$
$919$	− 5861.26i	− 0.210386i	0.994452	−	0.105193i	$-0.0335461\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 18106.6i	− 0.645706i
$924$	0	0
$925$	12626.0i	0.448800i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6614.28i	0.233592i	0.993156	+	0.116796i	$0.0372624\pi$
$929$	6614.28i	0.233592i	−0.993156	+	0.116796i	$0.962738\pi$
$930$	0	0
$931$	−16738.5	−0.589241
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24338.1	0.851275
$936$	0	0
$937$	21610.0	0.753434	0.376717	−	0.926328i	$-0.377053\pi$
$937$	21610.0	0.753434	0.376717	+	0.926328i	$0.377053\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−50865.0	−1.76212	−0.881059	−	0.473007i	$-0.843168\pi$
$941$	−50865.0	−1.76212	−0.881059	+	0.473007i	$0.843168\pi$
$942$	0	0
$943$	33671.1i	1.16276i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 22927.2i	− 0.786731i	−0.919382	−	0.393366i	$-0.871311\pi$
$947$	− 22927.2i	− 0.786731i	0.919382	−	0.393366i	$-0.128689\pi$
$948$	0	0
$949$	13888.0i	0.475051i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8981.67i	0.305294i	0.988281	+	0.152647i	$0.0487797\pi$
$953$	8981.67i	0.305294i	−0.988281	+	0.152647i	$0.951220\pi$
$954$	0	0
$955$	15463.7	0.523974
$956$	0	0
$957$	0	0
$958$	0	0
$959$	22280.6	0.750236
$960$	0	0
$961$	14239.0	0.477963
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4013.54	0.133886
$966$	0	0
$967$	983.805i	0.0327167i	0.999866	+	0.0163583i	$0.00520725\pi$
$967$	983.805i	0.0327167i	−0.999866	+	0.0163583i	$0.994793\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12639.4i	0.417731i	0.977944	+	0.208865i	$0.0669771\pi$
$971$	12639.4i	0.417731i	−0.977944	+	0.208865i	$0.933023\pi$
$972$	0	0
$973$	29568.0i	0.974210i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31501.6i	1.03155i	0.856724	+	0.515776i	$0.172496\pi$
$977$	31501.6i	1.03155i	−0.856724	+	0.515776i	$0.827504\pi$
$978$	0	0
$979$	66095.1	2.15772
$980$	0	0
$981$	0	0
$982$	0	0
$983$	56318.7	1.82735	0.913676	−	0.406444i	$-0.133231\pi$
$983$	56318.7	1.82735	0.913676	+	0.406444i	$0.133231\pi$
$984$	0	0
$985$	−8118.00	−0.262600
$986$	0	0
$987$	0	0
$988$	0	0
$989$	43987.7	1.41428
$990$	0	0
$991$	5473.28i	0.175443i	0.996145	+	0.0877217i	$0.0279586\pi$
$991$	5473.28i	0.175443i	−0.996145	+	0.0877217i	$0.972041\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3115.75i	0.0992723i
$996$	0	0
$997$	− 22858.0i	− 0.726098i	−0.931770	−	0.363049i	$-0.881736\pi$
$997$	− 22858.0i	− 0.726098i	0.931770	−	0.363049i	$-0.118264\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.4.f.f.1151.3		8
3.2	odd	2	inner	2304.4.f.f.1151.7		8
4.3	odd	2	inner	2304.4.f.f.1151.1		8
8.3	odd	2	inner	2304.4.f.f.1151.6		8
8.5	even	2	inner	2304.4.f.f.1151.8		8
12.11	even	2	inner	2304.4.f.f.1151.5		8
16.3	odd	4		144.4.c.b.143.4	yes	4
16.5	even	4		576.4.c.c.575.1		4
16.11	odd	4		576.4.c.c.575.2		4
16.13	even	4		144.4.c.b.143.3	yes	4
24.5	odd	2	inner	2304.4.f.f.1151.4		8
24.11	even	2	inner	2304.4.f.f.1151.2		8
48.5	odd	4		576.4.c.c.575.3		4
48.11	even	4		576.4.c.c.575.4		4
48.29	odd	4		144.4.c.b.143.1	✓	4
48.35	even	4		144.4.c.b.143.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.c.b.143.1	✓	4	48.29	odd	4
144.4.c.b.143.2	yes	4	48.35	even	4
144.4.c.b.143.3	yes	4	16.13	even	4
144.4.c.b.143.4	yes	4	16.3	odd	4
576.4.c.c.575.1		4	16.5	even	4
576.4.c.c.575.2		4	16.11	odd	4
576.4.c.c.575.3		4	48.5	odd	4
576.4.c.c.575.4		4	48.11	even	4
2304.4.f.f.1151.1		8	4.3	odd	2	inner
2304.4.f.f.1151.2		8	24.11	even	2	inner
2304.4.f.f.1151.3		8	1.1	even	1	trivial
2304.4.f.f.1151.4		8	24.5	odd	2	inner
2304.4.f.f.1151.5		8	12.11	even	2	inner
2304.4.f.f.1151.6		8	8.3	odd	2	inner
2304.4.f.f.1151.7		8	3.2	odd	2	inner
2304.4.f.f.1151.8		8	8.5	even	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2304.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.24264 q^{5} +13.8564i q^{7} +O(q^{10})\)	\(q-4.24264 q^{5} +13.8564i q^{7} +58.7878i q^{11} -28.0000i q^{13} +97.5807i q^{17} -110.851 q^{19} -176.363 q^{23} -107.000 q^{25} -4.24264 q^{29} -124.708i q^{31} -58.7878i q^{35} -118.000i q^{37} -190.919i q^{41} -249.415 q^{43} -293.939 q^{47} +151.000 q^{49} +700.036 q^{53} -249.415i q^{55} -587.878i q^{59} +802.000i q^{61} +118.794i q^{65} +138.564 q^{67} +646.665 q^{71} -496.000 q^{73} -814.587 q^{77} +1011.52i q^{79} +58.7878i q^{83} -414.000i q^{85} -1124.30i q^{89} +387.979 q^{91} +470.302 q^{95} -152.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 856 q^{25} + 1208 q^{49} - 3968 q^{73} - 1216 q^{97}+O(q^{100}) \) 8 * q - 856 * q^25 + 1208 * q^49 - 3968 * q^73 - 1216 * q^97

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