LMFDB - Embedded newform 2304.3.e.f.1025.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(1025,2304)]

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([0, 0, 1]))

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

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

chi := DirichletCharacter("2304.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.e (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:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.3
Root			$-1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2304.1025
Dual form			2304.3.e.f.1025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.24264i q^{5} -12.6491 q^{7} +O(q^{10})$	$q+4.24264i q^{5} -12.6491 q^{7} -17.8885i q^{11} -10.0000 q^{13} -24.0416i q^{17} -25.2982 q^{19} +17.8885i q^{23} +7.00000 q^{25} +15.5563i q^{29} -12.6491 q^{31} -53.6656i q^{35} +64.0000 q^{37} +12.7279i q^{41} -50.5964 q^{43} +17.8885i q^{47} +111.000 q^{49} -18.3848i q^{53} +75.8947 q^{55} -42.4264i q^{65} +75.8947 q^{67} -125.220i q^{71} -96.0000 q^{73} +226.274i q^{77} +63.2456 q^{79} +125.220i q^{83} +102.000 q^{85} +55.1543i q^{89} +126.491 q^{91} -107.331i q^{95} +64.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 40 q^{13} + 28 q^{25} + 256 q^{37} + 444 q^{49} - 384 q^{73} + 408 q^{85} + 256 q^{97}+O(q^{100}) $ 4 * q - 40 * q^13 + 28 * q^25 + 256 * q^37 + 444 * q^49 - 384 * q^73 + 408 * q^85 + 256 * 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
$5$	4.24264i	0.848528i	0.905539	+	0.424264i	$0.139467\pi$
$5$	4.24264i	0.848528i	−0.905539	+	0.424264i	$0.860533\pi$
$6$	0	0
$7$	−12.6491	−1.80702	−0.903508	−	0.428571i	$-0.859017\pi$
$7$	−12.6491	−1.80702	−0.903508	+	0.428571i	$0.859017\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 17.8885i	− 1.62623i	−0.582102	−	0.813116i	$-0.697770\pi$
$11$	− 17.8885i	− 1.62623i	0.582102	−	0.813116i	$-0.302230\pi$
$12$	0	0
$13$	−10.0000	−0.769231	−0.384615	−	0.923077i	$-0.625666\pi$
$13$	−10.0000	−0.769231	−0.384615	+	0.923077i	$0.625666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 24.0416i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$17$	− 24.0416i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$18$	0	0
$19$	−25.2982	−1.33149	−0.665743	−	0.746181i	$-0.731885\pi$
$19$	−25.2982	−1.33149	−0.665743	+	0.746181i	$0.731885\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	17.8885i	0.777763i	0.921288	+	0.388881i	$0.127138\pi$
$23$	17.8885i	0.777763i	−0.921288	+	0.388881i	$0.872862\pi$
$24$	0	0
$25$	7.00000	0.280000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	15.5563i	0.536426i	0.963360	+	0.268213i	$0.0864331\pi$
$29$	15.5563i	0.536426i	−0.963360	+	0.268213i	$0.913567\pi$
$30$	0	0
$31$	−12.6491	−0.408036	−0.204018	−	0.978967i	$-0.565400\pi$
$31$	−12.6491	−0.408036	−0.204018	+	0.978967i	$0.565400\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 53.6656i	− 1.53330i
$36$	0	0
$37$	64.0000	1.72973	0.864865	−	0.502005i	$-0.167404\pi$
$37$	64.0000	1.72973	0.864865	+	0.502005i	$0.167404\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	12.7279i	0.310437i	0.987880	+	0.155219i	$0.0496082\pi$
$41$	12.7279i	0.310437i	−0.987880	+	0.155219i	$0.950392\pi$
$42$	0	0
$43$	−50.5964	−1.17666	−0.588331	−	0.808620i	$-0.700215\pi$
$43$	−50.5964	−1.17666	−0.588331	+	0.808620i	$0.700215\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	17.8885i	0.380607i	0.981725	+	0.190304i	$0.0609473\pi$
$47$	17.8885i	0.380607i	−0.981725	+	0.190304i	$0.939053\pi$
$48$	0	0
$49$	111.000	2.26531
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 18.3848i	− 0.346883i	−0.984844	−	0.173441i	$-0.944511\pi$
$53$	− 18.3848i	− 0.346883i	0.984844	−	0.173441i	$-0.0554887\pi$
$54$	0	0
$55$	75.8947	1.37990
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 42.4264i	− 0.652714i
$66$	0	0
$67$	75.8947	1.13276	0.566378	−	0.824146i	$-0.308344\pi$
$67$	75.8947	1.13276	0.566378	+	0.824146i	$0.308344\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 125.220i	− 1.76366i	−0.471568	−	0.881830i	$-0.656312\pi$
$71$	− 125.220i	− 1.76366i	0.471568	−	0.881830i	$-0.343688\pi$
$72$	0	0
$73$	−96.0000	−1.31507	−0.657534	−	0.753425i	$-0.728401\pi$
$73$	−96.0000	−1.31507	−0.657534	+	0.753425i	$0.728401\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	226.274i	2.93863i
$78$	0	0
$79$	63.2456	0.800577	0.400288	−	0.916389i	$-0.368910\pi$
$79$	63.2456	0.800577	0.400288	+	0.916389i	$0.368910\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	125.220i	1.50867i	0.656488	+	0.754336i	$0.272041\pi$
$83$	125.220i	1.50867i	−0.656488	+	0.754336i	$0.727959\pi$
$84$	0	0
$85$	102.000	1.20000
$86$	0	0
$87$	0	0
$88$	0	0
$89$	55.1543i	0.619712i	0.950784	+	0.309856i	$0.100281\pi$
$89$	55.1543i	0.619712i	−0.950784	+	0.309856i	$0.899719\pi$
$90$	0	0
$91$	126.491	1.39001
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 107.331i	− 1.12980i
$96$	0	0
$97$	64.0000	0.659794	0.329897	−	0.944017i	$-0.392986\pi$
$97$	64.0000	0.659794	0.329897	+	0.944017i	$0.392986\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	41.0122i	0.406061i	0.979172	+	0.203031i	$0.0650791\pi$
$101$	41.0122i	0.406061i	−0.979172	+	0.203031i	$0.934921\pi$
$102$	0	0
$103$	−139.140	−1.35088	−0.675438	−	0.737417i	$-0.736045\pi$
$103$	−139.140	−1.35088	−0.675438	+	0.737417i	$0.736045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	143.108i	1.33746i	0.743505	+	0.668731i	$0.233162\pi$
$107$	143.108i	1.33746i	−0.743505	+	0.668731i	$0.766838\pi$
$108$	0	0
$109$	−118.000	−1.08257	−0.541284	−	0.840840i	$-0.682062\pi$
$109$	−118.000	−1.08257	−0.541284	+	0.840840i	$0.682062\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 24.0416i	− 0.212758i	−0.994326	−	0.106379i	$-0.966074\pi$
$113$	− 24.0416i	− 0.212758i	0.994326	−	0.106379i	$-0.0339256\pi$
$114$	0	0
$115$	−75.8947	−0.659954
$116$	0	0
$117$	0	0
$118$	0	0
$119$	304.105i	2.55551i
$120$	0	0
$121$	−199.000	−1.64463
$122$	0	0
$123$	0	0
$124$	0	0
$125$	135.765i	1.08612i
$126$	0	0
$127$	−63.2456	−0.497996	−0.248998	−	0.968504i	$-0.580101\pi$
$127$	−63.2456	−0.497996	−0.248998	+	0.968504i	$0.580101\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	107.331i	0.819323i	0.912238	+	0.409661i	$0.134353\pi$
$131$	107.331i	0.819323i	−0.912238	+	0.409661i	$0.865647\pi$
$132$	0	0
$133$	320.000	2.40602
$134$	0	0
$135$	0	0
$136$	0	0
$137$	123.037i	0.898077i	0.893512	+	0.449039i	$0.148233\pi$
$137$	123.037i	0.898077i	−0.893512	+	0.449039i	$0.851767\pi$
$138$	0	0
$139$	126.491	0.910008	0.455004	−	0.890489i	$-0.349638\pi$
$139$	126.491	0.910008	0.455004	+	0.890489i	$0.349638\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	178.885i	1.25095i
$144$	0	0
$145$	−66.0000	−0.455172
$146$	0	0
$147$	0	0
$148$	0	0
$149$	199.404i	1.33828i	0.743135	+	0.669141i	$0.233338\pi$
$149$	199.404i	1.33828i	−0.743135	+	0.669141i	$0.766662\pi$
$150$	0	0
$151$	37.9473	0.251307	0.125653	−	0.992074i	$-0.459897\pi$
$151$	37.9473	0.251307	0.125653	+	0.992074i	$0.459897\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 53.6656i	− 0.346230i
$156$	0	0
$157$	256.000	1.63057	0.815287	−	0.579058i	$-0.196579\pi$
$157$	256.000	1.63057	0.815287	+	0.579058i	$0.196579\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 226.274i	− 1.40543i
$162$	0	0
$163$	101.193	0.620815	0.310408	−	0.950604i	$-0.399534\pi$
$163$	101.193	0.620815	0.310408	+	0.950604i	$0.399534\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 214.663i	− 1.28540i	−0.766116	−	0.642702i	$-0.777813\pi$
$167$	− 214.663i	− 1.28540i	0.766116	−	0.642702i	$-0.222187\pi$
$168$	0	0
$169$	−69.0000	−0.408284
$170$	0	0
$171$	0	0
$172$	0	0
$173$	173.948i	1.00548i	0.864437	+	0.502741i	$0.167675\pi$
$173$	173.948i	1.00548i	−0.864437	+	0.502741i	$0.832325\pi$
$174$	0	0
$175$	−88.5438	−0.505964
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 250.440i	− 1.39910i	−0.714582	−	0.699552i	$-0.753383\pi$
$179$	− 250.440i	− 1.39910i	0.714582	−	0.699552i	$-0.246617\pi$
$180$	0	0
$181$	218.000	1.20442	0.602210	−	0.798338i	$-0.294287\pi$
$181$	218.000	1.20442	0.602210	+	0.798338i	$0.294287\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	271.529i	1.46772i
$186$	0	0
$187$	−430.070	−2.29984
$188$	0	0
$189$	0	0
$190$	0	0
$191$	71.5542i	0.374629i	0.982300	+	0.187315i	$0.0599784\pi$
$191$	71.5542i	0.374629i	−0.982300	+	0.187315i	$0.940022\pi$
$192$	0	0
$193$	−30.0000	−0.155440	−0.0777202	−	0.996975i	$-0.524764\pi$
$193$	−30.0000	−0.155440	−0.0777202	+	0.996975i	$0.524764\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	131.522i	0.667624i	0.942640	+	0.333812i	$0.108335\pi$
$197$	131.522i	0.667624i	−0.942640	+	0.333812i	$0.891665\pi$
$198$	0	0
$199$	12.6491	0.0635634	0.0317817	−	0.999495i	$-0.489882\pi$
$199$	12.6491	0.0635634	0.0317817	+	0.999495i	$0.489882\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 196.774i	− 0.969330i
$204$	0	0
$205$	−54.0000	−0.263415
$206$	0	0
$207$	0	0
$208$	0	0
$209$	452.548i	2.16530i
$210$	0	0
$211$	126.491	0.599484	0.299742	−	0.954020i	$-0.403099\pi$
$211$	126.491	0.599484	0.299742	+	0.954020i	$0.403099\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 214.663i	− 0.998430i
$216$	0	0
$217$	160.000	0.737327
$218$	0	0
$219$	0	0
$220$	0	0
$221$	240.416i	1.08786i
$222$	0	0
$223$	−12.6491	−0.0567225	−0.0283612	−	0.999598i	$-0.509029\pi$
$223$	−12.6491	−0.0567225	−0.0283612	+	0.999598i	$0.509029\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 196.774i	− 0.866846i	−0.901191	−	0.433423i	$-0.857306\pi$
$227$	− 196.774i	− 0.866846i	0.901191	−	0.433423i	$-0.142694\pi$
$228$	0	0
$229$	−58.0000	−0.253275	−0.126638	−	0.991949i	$-0.540419\pi$
$229$	−58.0000	−0.253275	−0.126638	+	0.991949i	$0.540419\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 349.311i	− 1.49919i	−0.661898	−	0.749594i	$-0.730249\pi$
$233$	− 349.311i	− 1.49919i	0.661898	−	0.749594i	$-0.269751\pi$
$234$	0	0
$235$	−75.8947	−0.322956
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 321.994i	− 1.34725i	−0.739071	−	0.673627i	$-0.764735\pi$
$239$	− 321.994i	− 1.34725i	0.739071	−	0.673627i	$-0.235265\pi$
$240$	0	0
$241$	160.000	0.663900	0.331950	−	0.943297i	$-0.392293\pi$
$241$	160.000	0.663900	0.331950	+	0.943297i	$0.392293\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	470.933i	1.92218i
$246$	0	0
$247$	252.982	1.02422
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 53.6656i	− 0.213807i	−0.994269	−	0.106904i	$-0.965906\pi$
$251$	− 53.6656i	− 0.213807i	0.994269	−	0.106904i	$-0.0340936\pi$
$252$	0	0
$253$	320.000	1.26482
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 179.605i	− 0.698853i	−0.936964	−	0.349426i	$-0.886377\pi$
$257$	− 179.605i	− 0.698853i	0.936964	−	0.349426i	$-0.113623\pi$
$258$	0	0
$259$	−809.543	−3.12565
$260$	0	0
$261$	0	0
$262$	0	0
$263$	250.440i	0.952242i	0.879380	+	0.476121i	$0.157958\pi$
$263$	250.440i	0.952242i	−0.879380	+	0.476121i	$0.842042\pi$
$264$	0	0
$265$	78.0000	0.294340
$266$	0	0
$267$	0	0
$268$	0	0
$269$	369.110i	1.37216i	0.727528	+	0.686078i	$0.240669\pi$
$269$	369.110i	1.37216i	−0.727528	+	0.686078i	$0.759331\pi$
$270$	0	0
$271$	341.526	1.26024	0.630122	−	0.776496i	$-0.283005\pi$
$271$	341.526	1.26024	0.630122	+	0.776496i	$0.283005\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 125.220i	− 0.455345i
$276$	0	0
$277$	−230.000	−0.830325	−0.415162	−	0.909747i	$-0.636275\pi$
$277$	−230.000	−0.830325	−0.415162	+	0.909747i	$0.636275\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 352.139i	− 1.25316i	−0.779355	−	0.626582i	$-0.784453\pi$
$281$	− 352.139i	− 1.25316i	0.779355	−	0.626582i	$-0.215547\pi$
$282$	0	0
$283$	404.772	1.43029	0.715144	−	0.698977i	$-0.246361\pi$
$283$	404.772	1.43029	0.715144	+	0.698977i	$0.246361\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 160.997i	− 0.560965i
$288$	0	0
$289$	−289.000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.24264i	0.0144800i	0.999974	+	0.00724000i	$0.00230458\pi$
$293$	4.24264i	0.0144800i	−0.999974	+	0.00724000i	$0.997695\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 178.885i	− 0.598279i
$300$	0	0
$301$	640.000	2.12625
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	379.473	1.23607	0.618035	−	0.786151i	$-0.287929\pi$
$307$	379.473	1.23607	0.618035	+	0.786151i	$0.287929\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	35.7771i	0.115039i	0.998344	+	0.0575194i	$0.0183191\pi$
$311$	35.7771i	0.115039i	−0.998344	+	0.0575194i	$0.981681\pi$
$312$	0	0
$313$	430.000	1.37380	0.686901	−	0.726751i	$-0.258971\pi$
$313$	430.000	1.37380	0.686901	+	0.726751i	$0.258971\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 287.085i	− 0.905632i	−0.891604	−	0.452816i	$-0.850419\pi$
$317$	− 287.085i	− 0.905632i	0.891604	−	0.452816i	$-0.149581\pi$
$318$	0	0
$319$	278.280	0.872352
$320$	0	0
$321$	0	0
$322$	0	0
$323$	608.210i	1.88300i
$324$	0	0
$325$	−70.0000	−0.215385
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 226.274i	− 0.687763i
$330$	0	0
$331$	25.2982	0.0764297	0.0382148	−	0.999270i	$-0.487833\pi$
$331$	25.2982	0.0764297	0.0382148	+	0.999270i	$0.487833\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	321.994i	0.961175i
$336$	0	0
$337$	224.000	0.664688	0.332344	−	0.943158i	$-0.392160\pi$
$337$	224.000	0.664688	0.332344	+	0.943158i	$0.392160\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	226.274i	0.663561i
$342$	0	0
$343$	−784.245	−2.28643
$344$	0	0
$345$	0	0
$346$	0	0
$347$	375.659i	1.08259i	0.840832	+	0.541296i	$0.182066\pi$
$347$	375.659i	1.08259i	−0.840832	+	0.541296i	$0.817934\pi$
$348$	0	0
$349$	320.000	0.916905	0.458453	−	0.888719i	$-0.348404\pi$
$349$	320.000	0.916905	0.458453	+	0.888719i	$0.348404\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	179.605i	0.508796i	0.967100	+	0.254398i	$0.0818774\pi$
$353$	179.605i	0.508796i	−0.967100	+	0.254398i	$0.918123\pi$
$354$	0	0
$355$	531.263	1.49651
$356$	0	0
$357$	0	0
$358$	0	0
$359$	482.991i	1.34538i	0.739925	+	0.672689i	$0.234861\pi$
$359$	482.991i	1.34538i	−0.739925	+	0.672689i	$0.765139\pi$
$360$	0	0
$361$	279.000	0.772853
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 407.294i	− 1.11587i
$366$	0	0
$367$	−720.999	−1.96458	−0.982288	−	0.187378i	$-0.940001\pi$
$367$	−720.999	−1.96458	−0.982288	+	0.187378i	$0.940001\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	232.551i	0.626822i
$372$	0	0
$373$	256.000	0.686327	0.343164	−	0.939276i	$-0.388501\pi$
$373$	256.000	0.686327	0.343164	+	0.939276i	$0.388501\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 155.563i	− 0.412635i
$378$	0	0
$379$	25.2982	0.0667499	0.0333750	−	0.999443i	$-0.489374\pi$
$379$	25.2982	0.0667499	0.0333750	+	0.999443i	$0.489374\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	679.765i	1.77484i	0.460959	+	0.887421i	$0.347505\pi$
$383$	679.765i	1.77484i	−0.460959	+	0.887421i	$0.652495\pi$
$384$	0	0
$385$	−960.000	−2.49351
$386$	0	0
$387$	0	0
$388$	0	0
$389$	765.090i	1.96681i	0.181421	+	0.983406i	$0.441930\pi$
$389$	765.090i	1.96681i	−0.181421	+	0.983406i	$0.558070\pi$
$390$	0	0
$391$	430.070	1.09992
$392$	0	0
$393$	0	0
$394$	0	0
$395$	268.328i	0.679312i
$396$	0	0
$397$	−64.0000	−0.161209	−0.0806045	−	0.996746i	$-0.525685\pi$
$397$	−64.0000	−0.161209	−0.0806045	+	0.996746i	$0.525685\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	337.997i	0.842885i	0.906855	+	0.421443i	$0.138476\pi$
$401$	337.997i	0.842885i	−0.906855	+	0.421443i	$0.861524\pi$
$402$	0	0
$403$	126.491	0.313874
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1144.87i	− 2.81294i
$408$	0	0
$409$	640.000	1.56479	0.782396	−	0.622781i	$-0.213997\pi$
$409$	640.000	1.56479	0.782396	+	0.622781i	$0.213997\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−531.263	−1.28015
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 232.551i	− 0.555014i	−0.960724	−	0.277507i	$-0.910492\pi$
$419$	− 232.551i	− 0.555014i	0.960724	−	0.277507i	$-0.0895083\pi$
$420$	0	0
$421$	−518.000	−1.23040	−0.615202	−	0.788370i	$-0.710926\pi$
$421$	−518.000	−1.23040	−0.615202	+	0.788370i	$0.710926\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 168.291i	− 0.395980i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 697.653i	− 1.61868i	−0.587337	−	0.809342i	$-0.699824\pi$
$431$	− 697.653i	− 1.61868i	0.587337	−	0.809342i	$-0.300176\pi$
$432$	0	0
$433$	290.000	0.669746	0.334873	−	0.942263i	$-0.391307\pi$
$433$	290.000	0.669746	0.334873	+	0.942263i	$0.391307\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 452.548i	− 1.03558i
$438$	0	0
$439$	37.9473	0.0864404	0.0432202	−	0.999066i	$-0.486238\pi$
$439$	37.9473	0.0864404	0.0432202	+	0.999066i	$0.486238\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	697.653i	1.57484i	0.616418	+	0.787419i	$0.288583\pi$
$443$	697.653i	1.57484i	−0.616418	+	0.787419i	$0.711417\pi$
$444$	0	0
$445$	−234.000	−0.525843
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 722.663i	− 1.60949i	−0.593617	−	0.804747i	$-0.702301\pi$
$449$	− 722.663i	− 1.60949i	0.593617	−	0.804747i	$-0.297699\pi$
$450$	0	0
$451$	227.684	0.504843
$452$	0	0
$453$	0	0
$454$	0	0
$455$	536.656i	1.17946i
$456$	0	0
$457$	−64.0000	−0.140044	−0.0700219	−	0.997545i	$-0.522307\pi$
$457$	−64.0000	−0.140044	−0.0700219	+	0.997545i	$0.522307\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 581.242i	− 1.26083i	−0.776259	−	0.630414i	$-0.782885\pi$
$461$	− 581.242i	− 1.26083i	0.776259	−	0.630414i	$-0.217115\pi$
$462$	0	0
$463$	63.2456	0.136599	0.0682997	−	0.997665i	$-0.478243\pi$
$463$	63.2456	0.136599	0.0682997	+	0.997665i	$0.478243\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	160.997i	0.344747i	0.985032	+	0.172374i	$0.0551436\pi$
$467$	160.997i	0.344747i	−0.985032	+	0.172374i	$0.944856\pi$
$468$	0	0
$469$	−960.000	−2.04691
$470$	0	0
$471$	0	0
$472$	0	0
$473$	905.097i	1.91352i
$474$	0	0
$475$	−177.088	−0.372816
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 304.105i	− 0.634875i	−0.948279	−	0.317438i	$-0.897178\pi$
$479$	− 304.105i	− 0.634875i	0.948279	−	0.317438i	$-0.102822\pi$
$480$	0	0
$481$	−640.000	−1.33056
$482$	0	0
$483$	0	0
$484$	0	0
$485$	271.529i	0.559854i
$486$	0	0
$487$	771.596	1.58439	0.792193	−	0.610271i	$-0.208939\pi$
$487$	771.596	1.58439	0.792193	+	0.610271i	$0.208939\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	107.331i	0.218597i	0.994009	+	0.109299i	$0.0348605\pi$
$491$	107.331i	0.218597i	−0.994009	+	0.109299i	$0.965140\pi$
$492$	0	0
$493$	374.000	0.758621
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1583.92i	3.18696i
$498$	0	0
$499$	−708.350	−1.41954	−0.709770	−	0.704434i	$-0.751201\pi$
$499$	−708.350	−1.41954	−0.709770	+	0.704434i	$0.751201\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	482.991i	0.960220i	0.877208	+	0.480110i	$0.159403\pi$
$503$	482.991i	0.960220i	−0.877208	+	0.480110i	$0.840597\pi$
$504$	0	0
$505$	−174.000	−0.344554
$506$	0	0
$507$	0	0
$508$	0	0
$509$	241.831i	0.475109i	0.971374	+	0.237555i	$0.0763458\pi$
$509$	241.831i	0.475109i	−0.971374	+	0.237555i	$0.923654\pi$
$510$	0	0
$511$	1214.31	2.37635
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 590.322i	− 1.14626i
$516$	0	0
$517$	320.000	0.618956
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 284.257i	− 0.545599i	−0.962071	−	0.272799i	$-0.912050\pi$
$521$	− 284.257i	− 0.545599i	0.962071	−	0.272799i	$-0.0879495\pi$
$522$	0	0
$523$	278.280	0.532085	0.266042	−	0.963961i	$-0.414284\pi$
$523$	278.280	0.532085	0.266042	+	0.963961i	$0.414284\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	304.105i	0.577050i
$528$	0	0
$529$	209.000	0.395085
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 127.279i	− 0.238798i
$534$	0	0
$535$	−607.157	−1.13487
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1985.63i	− 3.68391i
$540$	0	0
$541$	−662.000	−1.22366	−0.611830	−	0.790989i	$-0.709566\pi$
$541$	−662.000	−1.22366	−0.611830	+	0.790989i	$0.709566\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 500.632i	− 0.918590i
$546$	0	0
$547$	151.789	0.277494	0.138747	−	0.990328i	$-0.455692\pi$
$547$	151.789	0.277494	0.138747	+	0.990328i	$0.455692\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 393.548i	− 0.714243i
$552$	0	0
$553$	−800.000	−1.44665
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 400.222i	− 0.718532i	−0.933235	−	0.359266i	$-0.883027\pi$
$557$	− 400.222i	− 0.718532i	0.933235	−	0.359266i	$-0.116973\pi$
$558$	0	0
$559$	505.964	0.905124
$560$	0	0
$561$	0	0
$562$	0	0
$563$	268.328i	0.476604i	0.971191	+	0.238302i	$0.0765908\pi$
$563$	268.328i	0.476604i	−0.971191	+	0.238302i	$0.923409\pi$
$564$	0	0
$565$	102.000	0.180531
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 578.413i	− 1.01654i	−0.861197	−	0.508272i	$-0.830285\pi$
$569$	− 578.413i	− 1.01654i	0.861197	−	0.508272i	$-0.169715\pi$
$570$	0	0
$571$	−50.5964	−0.0886102	−0.0443051	−	0.999018i	$-0.514107\pi$
$571$	−50.5964	−0.0886102	−0.0443051	+	0.999018i	$0.514107\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	125.220i	0.217774i
$576$	0	0
$577$	−30.0000	−0.0519931	−0.0259965	−	0.999662i	$-0.508276\pi$
$577$	−30.0000	−0.0519931	−0.0259965	+	0.999662i	$0.508276\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1583.92i	− 2.72619i
$582$	0	0
$583$	−328.877	−0.564111
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 286.217i	− 0.487592i	−0.969826	−	0.243796i	$-0.921607\pi$
$587$	− 286.217i	− 0.487592i	0.969826	−	0.243796i	$-0.0783928\pi$
$588$	0	0
$589$	320.000	0.543294
$590$	0	0
$591$	0	0
$592$	0	0
$593$	250.316i	0.422118i	0.977473	+	0.211059i	$0.0676912\pi$
$593$	250.316i	0.422118i	−0.977473	+	0.211059i	$0.932309\pi$
$594$	0	0
$595$	−1290.21	−2.16842
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 482.991i	− 0.806328i	−0.915128	−	0.403164i	$-0.867910\pi$
$599$	− 482.991i	− 0.806328i	0.915128	−	0.403164i	$-0.132090\pi$
$600$	0	0
$601$	−658.000	−1.09484	−0.547421	−	0.836857i	$-0.684390\pi$
$601$	−658.000	−1.09484	−0.547421	+	0.836857i	$0.684390\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 844.285i	− 1.39551i
$606$	0	0
$607$	−164.438	−0.270904	−0.135452	−	0.990784i	$-0.543249\pi$
$607$	−164.438	−0.270904	−0.135452	+	0.990784i	$0.543249\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 178.885i	− 0.292775i
$612$	0	0
$613$	−576.000	−0.939641	−0.469821	−	0.882762i	$-0.655681\pi$
$613$	−576.000	−0.939641	−0.469821	+	0.882762i	$0.655681\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 193.747i	− 0.314015i	−0.987597	−	0.157008i	$-0.949815\pi$
$617$	− 193.747i	− 0.314015i	0.987597	−	0.157008i	$-0.0501847\pi$
$618$	0	0
$619$	−556.561	−0.899129	−0.449565	−	0.893248i	$-0.648421\pi$
$619$	−556.561	−0.899129	−0.449565	+	0.893248i	$0.648421\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 697.653i	− 1.11983i
$624$	0	0
$625$	−401.000	−0.641600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1538.66i	− 2.44621i
$630$	0	0
$631$	518.614	0.821891	0.410946	−	0.911660i	$-0.365199\pi$
$631$	518.614	0.821891	0.410946	+	0.911660i	$0.365199\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 268.328i	− 0.422564i
$636$	0	0
$637$	−1110.00	−1.74254
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 861.256i	− 1.34361i	−0.740727	−	0.671807i	$-0.765519\pi$
$641$	− 861.256i	− 1.34361i	0.740727	−	0.671807i	$-0.234481\pi$
$642$	0	0
$643$	303.579	0.472129	0.236064	−	0.971737i	$-0.424142\pi$
$643$	303.579	0.472129	0.236064	+	0.971737i	$0.424142\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	268.328i	0.414727i	0.978264	+	0.207363i	$0.0664882\pi$
$647$	268.328i	0.414727i	−0.978264	+	0.207363i	$0.933512\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	490.732i	0.751504i	0.926720	+	0.375752i	$0.122616\pi$
$653$	490.732i	0.751504i	−0.926720	+	0.375752i	$0.877384\pi$
$654$	0	0
$655$	−455.368	−0.695218
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 643.988i	− 0.977219i	−0.872502	−	0.488610i	$-0.837504\pi$
$659$	− 643.988i	− 0.977219i	0.872502	−	0.488610i	$-0.162496\pi$
$660$	0	0
$661$	640.000	0.968230	0.484115	−	0.875004i	$-0.339142\pi$
$661$	640.000	0.968230	0.484115	+	0.875004i	$0.339142\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1357.65i	2.04157i
$666$	0	0
$667$	−278.280	−0.417212
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	830.000	1.23328	0.616642	−	0.787244i	$-0.288493\pi$
$673$	830.000	1.23328	0.616642	+	0.787244i	$0.288493\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	230.517i	0.340498i	0.985401	+	0.170249i	$0.0544571\pi$
$677$	230.517i	0.340498i	−0.985401	+	0.170249i	$0.945543\pi$
$678$	0	0
$679$	−809.543	−1.19226
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1055.42i	1.54528i	0.634846	+	0.772638i	$0.281063\pi$
$683$	1055.42i	1.54528i	−0.634846	+	0.772638i	$0.718937\pi$
$684$	0	0
$685$	−522.000	−0.762044
$686$	0	0
$687$	0	0
$688$	0	0
$689$	183.848i	0.266833i
$690$	0	0
$691$	809.543	1.17155	0.585776	−	0.810473i	$-0.300790\pi$
$691$	809.543	1.17155	0.585776	+	0.810473i	$0.300790\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	536.656i	0.772167i
$696$	0	0
$697$	306.000	0.439024
$698$	0	0
$699$	0	0
$700$	0	0
$701$	196.576i	0.280422i	0.990122	+	0.140211i	$0.0447781\pi$
$701$	196.576i	0.280422i	−0.990122	+	0.140211i	$0.955222\pi$
$702$	0	0
$703$	−1619.09	−2.30311
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 518.768i	− 0.733759i
$708$	0	0
$709$	−698.000	−0.984485	−0.492243	−	0.870458i	$-0.663823\pi$
$709$	−698.000	−0.984485	−0.492243	+	0.870458i	$0.663823\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 226.274i	− 0.317355i
$714$	0	0
$715$	−758.947	−1.06146
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1234.31i	1.71670i	0.513062	+	0.858352i	$0.328511\pi$
$719$	1234.31i	1.71670i	−0.513062	+	0.858352i	$0.671489\pi$
$720$	0	0
$721$	1760.00	2.44105
$722$	0	0
$723$	0	0
$724$	0	0
$725$	108.894i	0.150199i
$726$	0	0
$727$	468.017	0.643765	0.321882	−	0.946780i	$-0.395684\pi$
$727$	468.017	0.643765	0.321882	+	0.946780i	$0.395684\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1216.42i	1.66405i
$732$	0	0
$733$	−1130.00	−1.54161	−0.770805	−	0.637071i	$-0.780146\pi$
$733$	−1130.00	−1.54161	−0.770805	+	0.637071i	$0.780146\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1357.65i	− 1.84212i
$738$	0	0
$739$	−50.5964	−0.0684661	−0.0342330	−	0.999414i	$-0.510899\pi$
$739$	−50.5964	−0.0684661	−0.0342330	+	0.999414i	$0.510899\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 143.108i	− 0.192609i	−0.995352	−	0.0963044i	$-0.969298\pi$
$743$	− 143.108i	− 0.192609i	0.995352	−	0.0963044i	$-0.0307022\pi$
$744$	0	0
$745$	−846.000	−1.13557
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1810.19i	− 2.41681i
$750$	0	0
$751$	−771.596	−1.02742	−0.513712	−	0.857963i	$-0.671730\pi$
$751$	−771.596	−1.02742	−0.513712	+	0.857963i	$0.671730\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	160.997i	0.213241i
$756$	0	0
$757$	−90.0000	−0.118890	−0.0594452	−	0.998232i	$-0.518933\pi$
$757$	−90.0000	−0.118890	−0.0594452	+	0.998232i	$0.518933\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1076.22i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$761$	1076.22i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$762$	0	0
$763$	1492.60	1.95622
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−482.000	−0.626788	−0.313394	−	0.949623i	$-0.601466\pi$
$769$	−482.000	−0.626788	−0.313394	+	0.949623i	$0.601466\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1262.89i	− 1.63376i	−0.576811	−	0.816878i	$-0.695703\pi$
$773$	− 1262.89i	− 1.63376i	0.576811	−	0.816878i	$-0.304297\pi$
$774$	0	0
$775$	−88.5438	−0.114250
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 321.994i	− 0.413342i
$780$	0	0
$781$	−2240.00	−2.86812
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1086.12i	1.38359i
$786$	0	0
$787$	531.263	0.675048	0.337524	−	0.941317i	$-0.390411\pi$
$787$	531.263	0.675048	0.337524	+	0.941317i	$0.390411\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	304.105i	0.384457i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	603.869i	0.757678i	0.925463	+	0.378839i	$0.123677\pi$
$797$	603.869i	0.757678i	−0.925463	+	0.378839i	$0.876323\pi$
$798$	0	0
$799$	430.070	0.538260
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1717.30i	2.13861i
$804$	0	0
$805$	960.000	1.19255
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 168.291i	− 0.208024i	−0.994576	−	0.104012i	$-0.966832\pi$
$809$	− 168.291i	− 0.208024i	0.994576	−	0.104012i	$-0.0331680\pi$
$810$	0	0
$811$	632.456	0.779847	0.389923	−	0.920847i	$-0.372502\pi$
$811$	632.456	0.779847	0.389923	+	0.920847i	$0.372502\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	429.325i	0.526779i
$816$	0	0
$817$	1280.00	1.56671
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 479.418i	− 0.583944i	−0.956427	−	0.291972i	$-0.905689\pi$
$821$	− 479.418i	− 0.583944i	0.956427	−	0.291972i	$-0.0943115\pi$
$822$	0	0
$823$	746.298	0.906801	0.453401	−	0.891307i	$-0.350211\pi$
$823$	746.298	0.906801	0.453401	+	0.891307i	$0.350211\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	679.765i	0.821965i	0.911643	+	0.410982i	$0.134814\pi$
$827$	679.765i	0.821965i	−0.911643	+	0.410982i	$0.865186\pi$
$828$	0	0
$829$	−598.000	−0.721351	−0.360676	−	0.932691i	$-0.617454\pi$
$829$	−598.000	−0.721351	−0.360676	+	0.932691i	$0.617454\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2668.62i	− 3.20363i
$834$	0	0
$835$	910.736	1.09070
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1234.31i	− 1.47117i	−0.677434	−	0.735584i	$-0.736908\pi$
$839$	− 1234.31i	− 1.47117i	0.677434	−	0.735584i	$-0.263092\pi$
$840$	0	0
$841$	599.000	0.712247
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 292.742i	− 0.346440i
$846$	0	0
$847$	2517.17	2.97187
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1144.87i	1.34532i
$852$	0	0
$853$	256.000	0.300117	0.150059	−	0.988677i	$-0.452054\pi$
$853$	256.000	0.300117	0.150059	+	0.988677i	$0.452054\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 371.938i	− 0.434000i	−0.976172	−	0.217000i	$-0.930373\pi$
$857$	− 371.938i	− 0.434000i	0.976172	−	0.217000i	$-0.0696272\pi$
$858$	0	0
$859$	−860.140	−1.00133	−0.500663	−	0.865642i	$-0.666911\pi$
$859$	−860.140	−1.00133	−0.500663	+	0.865642i	$0.666911\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 572.433i	− 0.663306i	−0.943401	−	0.331653i	$-0.892394\pi$
$863$	− 572.433i	− 0.663306i	0.943401	−	0.331653i	$-0.107606\pi$
$864$	0	0
$865$	−738.000	−0.853179
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1131.37i	− 1.30192i
$870$	0	0
$871$	−758.947	−0.871351
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1717.30i	− 1.96263i
$876$	0	0
$877$	704.000	0.802737	0.401368	−	0.915917i	$-0.368535\pi$
$877$	704.000	0.802737	0.401368	+	0.915917i	$0.368535\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	69.2965i	0.0786566i	0.999226	+	0.0393283i	$0.0125218\pi$
$881$	69.2965i	0.0786566i	−0.999226	+	0.0393283i	$0.987478\pi$
$882$	0	0
$883$	556.561	0.630307	0.315153	−	0.949041i	$-0.397944\pi$
$883$	556.561	0.630307	0.315153	+	0.949041i	$0.397944\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1609.97i	1.81507i	0.419974	+	0.907536i	$0.362039\pi$
$887$	1609.97i	1.81507i	−0.419974	+	0.907536i	$0.637961\pi$
$888$	0	0
$889$	800.000	0.899888
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 452.548i	− 0.506773i
$894$	0	0
$895$	1062.53	1.18718
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 196.774i	− 0.218881i
$900$	0	0
$901$	−442.000	−0.490566
$902$	0	0
$903$	0	0
$904$	0	0
$905$	924.896i	1.02198i
$906$	0	0
$907$	1770.88	1.95245	0.976227	−	0.216751i	$-0.0695461\pi$
$907$	1770.88	1.95245	0.976227	+	0.216751i	$0.0695461\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 858.650i	− 0.942536i	−0.881990	−	0.471268i	$-0.843797\pi$
$911$	− 858.650i	− 0.942536i	0.881990	−	0.471268i	$-0.156203\pi$
$912$	0	0
$913$	2240.00	2.45345
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1357.65i	− 1.48053i
$918$	0	0
$919$	37.9473	0.0412920	0.0206460	−	0.999787i	$-0.493428\pi$
$919$	37.9473	0.0412920	0.0206460	+	0.999787i	$0.493428\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1252.20i	1.35666i
$924$	0	0
$925$	448.000	0.484324
$926$	0	0
$927$	0	0
$928$	0	0
$929$	948.937i	1.02146i	0.859741	+	0.510731i	$0.170625\pi$
$929$	948.937i	1.02146i	−0.859741	+	0.510731i	$0.829375\pi$
$930$	0	0
$931$	−2808.10	−3.01622
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1824.63i	− 1.95148i
$936$	0	0
$937$	−1810.00	−1.93170	−0.965848	−	0.259108i	$-0.916572\pi$
$937$	−1810.00	−1.93170	−0.965848	+	0.259108i	$0.916572\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	550.129i	0.584622i	0.956323	+	0.292311i	$0.0944242\pi$
$941$	550.129i	0.584622i	−0.956323	+	0.292311i	$0.905576\pi$
$942$	0	0
$943$	−227.684	−0.241446
$944$	0	0
$945$	0	0
$946$	0	0
$947$	214.663i	0.226676i	0.993556	+	0.113338i	$0.0361544\pi$
$947$	214.663i	0.226676i	−0.993556	+	0.113338i	$0.963846\pi$
$948$	0	0
$949$	960.000	1.01159
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 80.6102i	− 0.0845857i	−0.999105	−	0.0422929i	$-0.986534\pi$
$953$	− 80.6102i	− 0.0845857i	0.999105	−	0.0422929i	$-0.0134662\pi$
$954$	0	0
$955$	−303.579	−0.317883
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1556.30i	− 1.62284i
$960$	0	0
$961$	−801.000	−0.833507
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 127.279i	− 0.131896i
$966$	0	0
$967$	−341.526	−0.353181	−0.176590	−	0.984284i	$-0.556507\pi$
$967$	−341.526	−0.353181	−0.176590	+	0.984284i	$0.556507\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1198.53i	1.23433i	0.786835	+	0.617164i	$0.211718\pi$
$971$	1198.53i	1.23433i	−0.786835	+	0.617164i	$0.788282\pi$
$972$	0	0
$973$	−1600.00	−1.64440
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 428.507i	− 0.438594i	−0.975658	−	0.219297i	$-0.929624\pi$
$977$	− 428.507i	− 0.438594i	0.975658	−	0.219297i	$-0.0703764\pi$
$978$	0	0
$979$	986.631	1.00779
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1466.86i	1.49223i	0.665818	+	0.746114i	$0.268083\pi$
$983$	1466.86i	1.49223i	−0.665818	+	0.746114i	$0.731917\pi$
$984$	0	0
$985$	−558.000	−0.566497
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 905.097i	− 0.915163i
$990$	0	0
$991$	−973.982	−0.982827	−0.491413	−	0.870926i	$-0.663520\pi$
$991$	−973.982	−0.982827	−0.491413	+	0.870926i	$0.663520\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	53.6656i	0.0539353i
$996$	0	0
$997$	−896.000	−0.898696	−0.449348	−	0.893357i	$-0.648344\pi$
$997$	−896.000	−0.898696	−0.449348	+	0.893357i	$0.648344\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.e.f.1025.3		4
3.2	odd	2	inner	2304.3.e.f.1025.1		4
4.3	odd	2	inner	2304.3.e.f.1025.4		4
8.3	odd	2		2304.3.e.k.1025.2		4
8.5	even	2		2304.3.e.k.1025.1		4
12.11	even	2	inner	2304.3.e.f.1025.2		4
16.3	odd	4		1152.3.h.e.449.5	yes	8
16.5	even	4		1152.3.h.e.449.4	yes	8
16.11	odd	4		1152.3.h.e.449.2	yes	8
16.13	even	4		1152.3.h.e.449.7	yes	8
24.5	odd	2		2304.3.e.k.1025.3		4
24.11	even	2		2304.3.e.k.1025.4		4
48.5	odd	4		1152.3.h.e.449.8	yes	8
48.11	even	4		1152.3.h.e.449.6	yes	8
48.29	odd	4		1152.3.h.e.449.3	yes	8
48.35	even	4		1152.3.h.e.449.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.h.e.449.1	✓	8	48.35	even	4
1152.3.h.e.449.2	yes	8	16.11	odd	4
1152.3.h.e.449.3	yes	8	48.29	odd	4
1152.3.h.e.449.4	yes	8	16.5	even	4
1152.3.h.e.449.5	yes	8	16.3	odd	4
1152.3.h.e.449.6	yes	8	48.11	even	4
1152.3.h.e.449.7	yes	8	16.13	even	4
1152.3.h.e.449.8	yes	8	48.5	odd	4
2304.3.e.f.1025.1		4	3.2	odd	2	inner
2304.3.e.f.1025.2		4	12.11	even	2	inner
2304.3.e.f.1025.3		4	1.1	even	1	trivial
2304.3.e.f.1025.4		4	4.3	odd	2	inner
2304.3.e.k.1025.1		4	8.5	even	2
2304.3.e.k.1025.2		4	8.3	odd	2
2304.3.e.k.1025.3		4	24.5	odd	2
2304.3.e.k.1025.4		4	24.11	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.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.24264i q^{5} -12.6491 q^{7} +O(q^{10})\)	\(q+4.24264i q^{5} -12.6491 q^{7} -17.8885i q^{11} -10.0000 q^{13} -24.0416i q^{17} -25.2982 q^{19} +17.8885i q^{23} +7.00000 q^{25} +15.5563i q^{29} -12.6491 q^{31} -53.6656i q^{35} +64.0000 q^{37} +12.7279i q^{41} -50.5964 q^{43} +17.8885i q^{47} +111.000 q^{49} -18.3848i q^{53} +75.8947 q^{55} -42.4264i q^{65} +75.8947 q^{67} -125.220i q^{71} -96.0000 q^{73} +226.274i q^{77} +63.2456 q^{79} +125.220i q^{83} +102.000 q^{85} +55.1543i q^{89} +126.491 q^{91} -107.331i q^{95} +64.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 40 q^{13} + 28 q^{25} + 256 q^{37} + 444 q^{49} - 384 q^{73} + 408 q^{85} + 256 q^{97}+O(q^{100}) \) 4 * q - 40 * q^13 + 28 * q^25 + 256 * q^37 + 444 * q^49 - 384 * q^73 + 408 * q^85 + 256 * q^97

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