LMFDB - Embedded newform 8280.2.p.a.1241.26

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1241,8280)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8280.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.p (of order $2$, degree $1$, 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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.26
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +0.488657i q^{7} +O(q^{10})$	$q-1.00000 q^{5} +0.488657i q^{7} -0.477618 q^{11} -4.91681 q^{13} +0.0686460 q^{17} +2.26986i q^{19} +(-4.64530 + 1.19213i) q^{23} +1.00000 q^{25} -2.85439i q^{29} -3.06933 q^{31} -0.488657i q^{35} -5.16861i q^{37} +6.04576i q^{41} -2.17648i q^{43} -0.440529i q^{47} +6.76121 q^{49} -5.95717 q^{53} +0.477618 q^{55} -2.25315i q^{59} +3.28117i q^{61} +4.91681 q^{65} +6.36252i q^{67} -8.73638i q^{71} +13.5209 q^{73} -0.233391i q^{77} -9.13107i q^{79} +3.27923 q^{83} -0.0686460 q^{85} +10.3839 q^{89} -2.40263i q^{91} -2.26986i q^{95} +17.5807i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{5}+O(q^{10}) $ 48 * q - 48 * q^5	$ 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) $ 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times$.

$n$	$1657$	$2071$	$3961$	$4141$	$4601$
$\chi(n)$	$1$	$1$	$-1$	$1$	$-1$

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
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			0.488657i			0.184695i	0.995727	+	0.0923475i	$0.0294371\pi$
$7$			0.488657i			0.184695i	−0.995727	+	0.0923475i	$0.970563\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.477618			−0.144007			−0.0720036	−	0.997404i	$-0.522939\pi$
$11$	−0.477618			−0.144007			−0.0720036	+	0.997404i	$0.522939\pi$
$12$		0			0
$13$	−4.91681			−1.36368			−0.681838	−	0.731503i	$-0.738819\pi$
$13$	−4.91681			−1.36368			−0.681838	+	0.731503i	$0.738819\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.0686460			0.0166491			0.00832455	−	0.999965i	$-0.497350\pi$
$17$	0.0686460			0.0166491			0.00832455	+	0.999965i	$0.497350\pi$
$18$		0			0
$19$			2.26986i			0.520742i	0.965509	+	0.260371i	$0.0838449\pi$
$19$			2.26986i			0.520742i	−0.965509	+	0.260371i	$0.916155\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.64530	+	1.19213i	−0.968612	+	0.248576i
$23$	−4.64530	+	1.19213i	−0.968612	+	0.248576i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	2.85439i		−	0.530047i	−0.964242	−	0.265023i	$-0.914620\pi$
$29$		−	2.85439i		−	0.530047i	0.964242	−	0.265023i	$-0.0853797\pi$
$30$		0			0
$31$	−3.06933			−0.551268			−0.275634	−	0.961263i	$-0.588888\pi$
$31$	−3.06933			−0.551268			−0.275634	+	0.961263i	$0.588888\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	0.488657i		−	0.0825981i
$36$		0			0
$37$		−	5.16861i		−	0.849714i	−0.905261	−	0.424857i	$-0.860324\pi$
$37$		−	5.16861i		−	0.849714i	0.905261	−	0.424857i	$-0.139676\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.04576i			0.944189i	0.881548	+	0.472095i	$0.156502\pi$
$41$			6.04576i			0.944189i	−0.881548	+	0.472095i	$0.843498\pi$
$42$		0			0
$43$		−	2.17648i		−	0.331911i	−0.986133	−	0.165955i	$-0.946929\pi$
$43$		−	2.17648i		−	0.331911i	0.986133	−	0.165955i	$-0.0530708\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	0.440529i		−	0.0642577i	−0.999484	−	0.0321289i	$-0.989771\pi$
$47$		−	0.440529i		−	0.0642577i	0.999484	−	0.0321289i	$-0.0102287\pi$
$48$		0			0
$49$	6.76121			0.965888
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.95717			−0.818280			−0.409140	−	0.912472i	$-0.634171\pi$
$53$	−5.95717			−0.818280			−0.409140	+	0.912472i	$0.634171\pi$
$54$		0			0
$55$	0.477618			0.0644020
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	2.25315i		−	0.293335i	−0.989186	−	0.146668i	$-0.953145\pi$
$59$		−	2.25315i		−	0.293335i	0.989186	−	0.146668i	$-0.0468547\pi$
$60$		0			0
$61$			3.28117i			0.420111i	0.977689	+	0.210056i	$0.0673645\pi$
$61$			3.28117i			0.420111i	−0.977689	+	0.210056i	$0.932636\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.91681			0.609855
$66$		0			0
$67$			6.36252i			0.777305i	0.921384	+	0.388653i	$0.127059\pi$
$67$			6.36252i			0.777305i	−0.921384	+	0.388653i	$0.872941\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.73638i		−	1.03682i	−0.855133	−	0.518409i	$-0.826525\pi$
$71$		−	8.73638i		−	1.03682i	0.855133	−	0.518409i	$-0.173475\pi$
$72$		0			0
$73$	13.5209			1.58250			0.791249	−	0.611495i	$-0.209431\pi$
$73$	13.5209			1.58250			0.791249	+	0.611495i	$0.209431\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	0.233391i		−	0.0265974i
$78$		0			0
$79$		−	9.13107i		−	1.02733i	−0.857992	−	0.513663i	$-0.828288\pi$
$79$		−	9.13107i		−	1.02733i	0.857992	−	0.513663i	$-0.171712\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.27923			0.359943			0.179971	−	0.983672i	$-0.442399\pi$
$83$	3.27923			0.359943			0.179971	+	0.983672i	$0.442399\pi$
$84$		0			0
$85$	−0.0686460			−0.00744571
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.3839			1.10069			0.550346	−	0.834937i	$-0.314496\pi$
$89$	10.3839			1.10069			0.550346	+	0.834937i	$0.314496\pi$
$90$		0			0
$91$		−	2.40263i		−	0.251864i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	2.26986i		−	0.232883i
$96$		0			0
$97$			17.5807i			1.78505i	0.450995	+	0.892526i	$0.351069\pi$
$97$			17.5807i			1.78505i	−0.450995	+	0.892526i	$0.648931\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	18.3289i		−	1.82380i	−0.410416	−	0.911898i	$-0.634616\pi$
$101$		−	18.3289i		−	1.82380i	0.410416	−	0.911898i	$-0.365384\pi$
$102$		0			0
$103$			12.0722i			1.18951i	0.803909	+	0.594753i	$0.202750\pi$
$103$			12.0722i			1.18951i	−0.803909	+	0.594753i	$0.797250\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.830054			−0.0802443			−0.0401222	−	0.999195i	$-0.512775\pi$
$107$	−0.830054			−0.0802443			−0.0401222	+	0.999195i	$0.512775\pi$
$108$		0			0
$109$		−	3.99309i		−	0.382468i	−0.981544	−	0.191234i	$-0.938751\pi$
$109$		−	3.99309i		−	0.382468i	0.981544	−	0.191234i	$-0.0612490\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.1428			−1.33045			−0.665223	−	0.746645i	$-0.731663\pi$
$113$	−14.1428			−1.33045			−0.665223	+	0.746645i	$0.731663\pi$
$114$		0			0
$115$	4.64530	−	1.19213i	0.433177	−	0.111166i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			0.0335444i			0.00307501i
$120$		0			0
$121$	−10.7719			−0.979262
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	13.7107			1.21663			0.608313	−	0.793697i	$-0.291847\pi$
$127$	13.7107			1.21663			0.608313	+	0.793697i	$0.291847\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			8.63722i			0.754638i	0.926083	+	0.377319i	$0.123154\pi$
$131$			8.63722i			0.754638i	−0.926083	+	0.377319i	$0.876846\pi$
$132$		0			0
$133$	−1.10918			−0.0961785
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.331548			0.0283261			0.0141630	−	0.999900i	$-0.495492\pi$
$137$	0.331548			0.0283261			0.0141630	+	0.999900i	$0.495492\pi$
$138$		0			0
$139$	12.0248			1.01993			0.509967	−	0.860194i	$-0.329658\pi$
$139$	12.0248			1.01993			0.509967	+	0.860194i	$0.329658\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.34836			0.196379
$144$		0			0
$145$			2.85439i			0.237044i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	22.8306			1.87035			0.935177	−	0.354180i	$-0.115240\pi$
$149$	22.8306			1.87035			0.935177	+	0.354180i	$0.115240\pi$
$150$		0			0
$151$	19.0857			1.55318			0.776588	−	0.630009i	$-0.216949\pi$
$151$	19.0857			1.55318			0.776588	+	0.630009i	$0.216949\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.06933			0.246535
$156$		0			0
$157$		−	7.13723i		−	0.569613i	−0.958585	−	0.284807i	$-0.908071\pi$
$157$		−	7.13723i		−	0.569613i	0.958585	−	0.284807i	$-0.0919294\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.582542	−	2.26996i	−0.0459107	−	0.178898i
$162$		0			0
$163$	−1.07862			−0.0844840			−0.0422420	−	0.999107i	$-0.513450\pi$
$163$	−1.07862			−0.0844840			−0.0422420	+	0.999107i	$0.513450\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	16.2257i		−	1.25558i	−0.778382	−	0.627791i	$-0.783959\pi$
$167$		−	16.2257i		−	1.25558i	0.778382	−	0.627791i	$-0.216041\pi$
$168$		0			0
$169$	11.1750			0.859615
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	4.67906i		−	0.355742i	−0.984054	−	0.177871i	$-0.943079\pi$
$173$		−	4.67906i		−	0.355742i	0.984054	−	0.177871i	$-0.0569210\pi$
$174$		0			0
$175$			0.488657i			0.0369390i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	6.05993i		−	0.452940i	−0.974018	−	0.226470i	$-0.927281\pi$
$179$		−	6.05993i		−	0.452940i	0.974018	−	0.226470i	$-0.0727186\pi$
$180$		0			0
$181$			21.8606i			1.62489i	0.583041	+	0.812443i	$0.301863\pi$
$181$			21.8606i			1.62489i	−0.583041	+	0.812443i	$0.698137\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			5.16861i			0.380004i
$186$		0			0
$187$	−0.0327866			−0.00239759
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.6665			−1.42302			−0.711510	−	0.702676i	$-0.751988\pi$
$191$	−19.6665			−1.42302			−0.711510	+	0.702676i	$0.751988\pi$
$192$		0			0
$193$	7.08295			0.509842			0.254921	−	0.966962i	$-0.417951\pi$
$193$	7.08295			0.509842			0.254921	+	0.966962i	$0.417951\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	15.8556i		−	1.12967i	−0.825205	−	0.564834i	$-0.808940\pi$
$197$		−	15.8556i		−	1.12967i	0.825205	−	0.564834i	$-0.191060\pi$
$198$		0			0
$199$		−	5.91295i		−	0.419158i	−0.977792	−	0.209579i	$-0.932791\pi$
$199$		−	5.91295i		−	0.419158i	0.977792	−	0.209579i	$-0.0672093\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.39482			0.0978970
$204$		0			0
$205$		−	6.04576i		−	0.422254i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	1.08413i		−	0.0749907i
$210$		0			0
$211$	16.3081			1.12270			0.561348	−	0.827580i	$-0.310283\pi$
$211$	16.3081			1.12270			0.561348	+	0.827580i	$0.310283\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			2.17648i			0.148435i
$216$		0			0
$217$		−	1.49985i		−	0.101817i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.337519			−0.0227040
$222$		0			0
$223$	3.19461			0.213927			0.106964	−	0.994263i	$-0.465887\pi$
$223$	3.19461			0.213927			0.106964	+	0.994263i	$0.465887\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−28.3014			−1.87843			−0.939216	−	0.343326i	$-0.888447\pi$
$227$	−28.3014			−1.87843			−0.939216	+	0.343326i	$0.888447\pi$
$228$		0			0
$229$			7.96687i			0.526465i	0.964732	+	0.263233i	$0.0847887\pi$
$229$			7.96687i			0.526465i	−0.964732	+	0.263233i	$0.915211\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	20.5271i		−	1.34478i	−0.740198	−	0.672389i	$-0.765268\pi$
$233$		−	20.5271i		−	1.34478i	0.740198	−	0.672389i	$-0.234732\pi$
$234$		0			0
$235$			0.440529i			0.0287369i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			25.8843i			1.67431i	0.546962	+	0.837157i	$0.315784\pi$
$239$			25.8843i			1.67431i	−0.546962	+	0.837157i	$0.684216\pi$
$240$		0			0
$241$		−	10.4182i		−	0.671097i	−0.942023	−	0.335549i	$-0.891078\pi$
$241$		−	10.4182i		−	0.671097i	0.942023	−	0.335549i	$-0.108922\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.76121			−0.431958
$246$		0			0
$247$		−	11.1605i		−	0.710124i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.46023			0.407766			0.203883	−	0.978995i	$-0.434644\pi$
$251$	6.46023			0.407766			0.203883	+	0.978995i	$0.434644\pi$
$252$		0			0
$253$	2.21868	−	0.569382i	0.139487	−	0.0357967i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	10.2754i		−	0.640961i	−0.947255	−	0.320480i	$-0.896156\pi$
$257$		−	10.2754i		−	0.640961i	0.947255	−	0.320480i	$-0.103844\pi$
$258$		0			0
$259$	2.52568			0.156938
$260$		0			0
$261$		0			0
$262$		0			0
$263$	28.2722			1.74334			0.871670	−	0.490094i	$-0.163038\pi$
$263$	28.2722			1.74334			0.871670	+	0.490094i	$0.163038\pi$
$264$		0			0
$265$	5.95717			0.365946
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	16.8772i		−	1.02902i	−0.857484	−	0.514511i	$-0.827973\pi$
$269$		−	16.8772i		−	1.02902i	0.857484	−	0.514511i	$-0.172027\pi$
$270$		0			0
$271$	17.2612			1.04855			0.524273	−	0.851550i	$-0.324337\pi$
$271$	17.2612			1.04855			0.524273	+	0.851550i	$0.324337\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.477618			−0.0288015
$276$		0			0
$277$	−12.0237			−0.722436			−0.361218	−	0.932481i	$-0.617639\pi$
$277$	−12.0237			−0.722436			−0.361218	+	0.932481i	$0.617639\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.0597			0.958043			0.479021	−	0.877803i	$-0.340992\pi$
$281$	16.0597			0.958043			0.479021	+	0.877803i	$0.340992\pi$
$282$		0			0
$283$			21.9028i			1.30199i	0.759084	+	0.650993i	$0.225647\pi$
$283$			21.9028i			1.30199i	−0.759084	+	0.650993i	$0.774353\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.95430			−0.174387
$288$		0			0
$289$	−16.9953			−0.999723
$290$		0			0
$291$		0			0
$292$		0			0
$293$	22.2825			1.30176			0.650880	−	0.759181i	$-0.274400\pi$
$293$	22.2825			1.30176			0.650880	+	0.759181i	$0.274400\pi$
$294$		0			0
$295$			2.25315i			0.131183i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	22.8401	−	5.86146i	1.32087	−	0.338977i
$300$		0			0
$301$	1.06355			0.0613023
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	3.28117i		−	0.187879i
$306$		0			0
$307$	14.9419			0.852777			0.426388	−	0.904540i	$-0.359786\pi$
$307$	14.9419			0.852777			0.426388	+	0.904540i	$0.359786\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			20.9265i			1.18663i	0.804969	+	0.593317i	$0.202182\pi$
$311$			20.9265i			1.18663i	−0.804969	+	0.593317i	$0.797818\pi$
$312$		0			0
$313$		−	21.9377i		−	1.23999i	−0.784604	−	0.619997i	$-0.787134\pi$
$313$		−	21.9377i		−	1.23999i	0.784604	−	0.619997i	$-0.212866\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	28.1562i		−	1.58141i	−0.612199	−	0.790704i	$-0.709715\pi$
$317$		−	28.1562i		−	1.58141i	0.612199	−	0.790704i	$-0.290285\pi$
$318$		0			0
$319$			1.36331i			0.0763306i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			0.155817i			0.00866989i
$324$		0			0
$325$	−4.91681			−0.272735
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.215267			0.0118681
$330$		0			0
$331$	−5.62411			−0.309129			−0.154565	−	0.987983i	$-0.549397\pi$
$331$	−5.62411			−0.309129			−0.154565	+	0.987983i	$0.549397\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	6.36252i		−	0.347621i
$336$		0			0
$337$			18.9823i			1.03403i	0.855975	+	0.517017i	$0.172958\pi$
$337$			18.9823i			1.03403i	−0.855975	+	0.517017i	$0.827042\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.46597			0.0793866
$342$		0			0
$343$			6.72452i			0.363090i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			29.2227i			1.56876i	0.620284	+	0.784378i	$0.287017\pi$
$347$			29.2227i			1.56876i	−0.620284	+	0.784378i	$0.712983\pi$
$348$		0			0
$349$	−7.19871			−0.385338			−0.192669	−	0.981264i	$-0.561714\pi$
$349$	−7.19871			−0.385338			−0.192669	+	0.981264i	$0.561714\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			0.256413i			0.0136475i	0.999977	+	0.00682374i	$0.00217208\pi$
$353$			0.256413i			0.0136475i	−0.999977	+	0.00682374i	$0.997828\pi$
$354$		0			0
$355$			8.73638i			0.463679i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5.98763			0.316015			0.158008	−	0.987438i	$-0.449493\pi$
$359$	5.98763			0.316015			0.158008	+	0.987438i	$0.449493\pi$
$360$		0			0
$361$	13.8477			0.728828
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−13.5209			−0.707714
$366$		0			0
$367$			24.3065i			1.26879i	0.773010	+	0.634394i	$0.218750\pi$
$367$			24.3065i			1.26879i	−0.773010	+	0.634394i	$0.781250\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	2.91101i		−	0.151132i
$372$		0			0
$373$		−	13.5468i		−	0.701426i	−0.936483	−	0.350713i	$-0.885939\pi$
$373$		−	13.5468i		−	0.701426i	0.936483	−	0.350713i	$-0.114061\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			14.0345i			0.722813i
$378$		0			0
$379$		−	36.3096i		−	1.86510i	−0.361041	−	0.932550i	$-0.617578\pi$
$379$		−	36.3096i		−	1.86510i	0.361041	−	0.932550i	$-0.382422\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−14.7485			−0.753613			−0.376807	−	0.926292i	$-0.622978\pi$
$383$	−14.7485			−0.753613			−0.376807	+	0.926292i	$0.622978\pi$
$384$		0			0
$385$			0.233391i			0.0118947i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.45233			0.377848			0.188924	−	0.981992i	$-0.439500\pi$
$389$	7.45233			0.377848			0.188924	+	0.981992i	$0.439500\pi$
$390$		0			0
$391$	−0.318882	+	0.0818348i	−0.0161265	+	0.00413856i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			9.13107i			0.459434i
$396$		0			0
$397$	29.8237			1.49681			0.748405	−	0.663242i	$-0.230820\pi$
$397$	29.8237			1.49681			0.748405	+	0.663242i	$0.230820\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	11.8531			0.591913			0.295957	−	0.955201i	$-0.404362\pi$
$401$	11.8531			0.591913			0.295957	+	0.955201i	$0.404362\pi$
$402$		0			0
$403$	15.0913			0.751752
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.46862i			0.122365i
$408$		0			0
$409$	−1.78774			−0.0883980			−0.0441990	−	0.999023i	$-0.514074\pi$
$409$	−1.78774			−0.0883980			−0.0441990	+	0.999023i	$0.514074\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.10102			0.0541775
$414$		0			0
$415$	−3.27923			−0.160971
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.8972			−0.581214			−0.290607	−	0.956842i	$-0.593857\pi$
$419$	−11.8972			−0.581214			−0.290607	+	0.956842i	$0.593857\pi$
$420$		0			0
$421$		−	32.3121i		−	1.57480i	−0.616445	−	0.787398i	$-0.711428\pi$
$421$		−	32.3121i		−	1.57480i	0.616445	−	0.787398i	$-0.288572\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.0686460			0.00332982
$426$		0			0
$427$	−1.60337			−0.0775924
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9.18357			0.442357			0.221178	−	0.975233i	$-0.429010\pi$
$431$	9.18357			0.442357			0.221178	+	0.975233i	$0.429010\pi$
$432$		0			0
$433$			1.75229i			0.0842096i	0.999113	+	0.0421048i	$0.0134063\pi$
$433$			1.75229i			0.0842096i	−0.999113	+	0.0421048i	$0.986594\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.70597	−	10.5442i	−0.129444	−	0.504397i
$438$		0			0
$439$	25.6757			1.22543			0.612717	−	0.790303i	$-0.290077\pi$
$439$	25.6757			1.22543			0.612717	+	0.790303i	$0.290077\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			0.614786i			0.0292093i	0.999893	+	0.0146047i	$0.00464898\pi$
$443$			0.614786i			0.0292093i	−0.999893	+	0.0146047i	$0.995351\pi$
$444$		0			0
$445$	−10.3839			−0.492244
$446$		0			0
$447$		0			0
$448$		0			0
$449$			4.49807i			0.212277i	0.994351	+	0.106138i	$0.0338487\pi$
$449$			4.49807i			0.212277i	−0.994351	+	0.106138i	$0.966151\pi$
$450$		0			0
$451$		−	2.88756i		−	0.135970i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			2.40263i			0.112637i
$456$		0			0
$457$			19.5296i			0.913557i	0.889580	+	0.456779i	$0.150997\pi$
$457$			19.5296i			0.913557i	−0.889580	+	0.456779i	$0.849003\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	27.5125i		−	1.28138i	−0.767798	−	0.640692i	$-0.778647\pi$
$461$		−	27.5125i		−	1.28138i	0.767798	−	0.640692i	$-0.221353\pi$
$462$		0			0
$463$	28.4632			1.32279			0.661397	−	0.750036i	$-0.269964\pi$
$463$	28.4632			1.32279			0.661397	+	0.750036i	$0.269964\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.63927			−0.168406			−0.0842028	−	0.996449i	$-0.526834\pi$
$467$	−3.63927			−0.168406			−0.0842028	+	0.996449i	$0.526834\pi$
$468$		0			0
$469$	−3.10909			−0.143564
$470$		0			0
$471$		0			0
$472$		0			0
$473$			1.03953i			0.0477976i
$474$		0			0
$475$			2.26986i			0.104148i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.59859			0.118732			0.0593662	−	0.998236i	$-0.481092\pi$
$479$	2.59859			0.118732			0.0593662	+	0.998236i	$0.481092\pi$
$480$		0			0
$481$			25.4130i			1.15874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	17.5807i		−	0.798300i
$486$		0			0
$487$	−34.4106			−1.55929			−0.779647	−	0.626220i	$-0.784601\pi$
$487$	−34.4106			−1.55929			−0.779647	+	0.626220i	$0.784601\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	13.4169i		−	0.605498i	−0.953070	−	0.302749i	$-0.902096\pi$
$491$		−	13.4169i		−	0.605498i	0.953070	−	0.302749i	$-0.0979043\pi$
$492$		0			0
$493$		−	0.195943i		−	0.00882481i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.26910			0.191495
$498$		0			0
$499$	26.6443			1.19276			0.596381	−	0.802701i	$-0.296605\pi$
$499$	26.6443			1.19276			0.596381	+	0.802701i	$0.296605\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5.45861			−0.243387			−0.121694	−	0.992568i	$-0.538833\pi$
$503$	−5.45861			−0.243387			−0.121694	+	0.992568i	$0.538833\pi$
$504$		0			0
$505$			18.3289i			0.815627i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	13.2896i		−	0.589050i	−0.955644	−	0.294525i	$-0.904839\pi$
$509$		−	13.2896i		−	0.589050i	0.955644	−	0.294525i	$-0.0951613\pi$
$510$		0			0
$511$			6.60707i			0.292279i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	12.0722i		−	0.531963i
$516$		0			0
$517$			0.210404i			0.00925358i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11.2119			−0.491201			−0.245601	−	0.969371i	$-0.578985\pi$
$521$	−11.2119			−0.491201			−0.245601	+	0.969371i	$0.578985\pi$
$522$		0			0
$523$			21.8051i			0.953471i	0.879047	+	0.476736i	$0.158180\pi$
$523$			21.8051i			0.953471i	−0.879047	+	0.476736i	$0.841820\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.210697			−0.00917813
$528$		0			0
$529$	20.1577	−	11.0756i	0.876420	−	0.481547i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	29.7258i		−	1.28757i
$534$		0			0
$535$	0.830054			0.0358863
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.22928			−0.139095
$540$		0			0
$541$	11.8423			0.509139			0.254570	−	0.967054i	$-0.418066\pi$
$541$	11.8423			0.509139			0.254570	+	0.967054i	$0.418066\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			3.99309i			0.171045i
$546$		0			0
$547$	27.7019			1.18445			0.592224	−	0.805773i	$-0.298250\pi$
$547$	27.7019			1.18445			0.592224	+	0.805773i	$0.298250\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.47907			0.276018
$552$		0			0
$553$	4.46196			0.189742
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.79968			0.203369			0.101685	−	0.994817i	$-0.467577\pi$
$557$	4.79968			0.203369			0.101685	+	0.994817i	$0.467577\pi$
$558$		0			0
$559$			10.7014i			0.452619i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	33.9764			1.43193			0.715967	−	0.698134i	$-0.245986\pi$
$563$	33.9764			1.43193			0.715967	+	0.698134i	$0.245986\pi$
$564$		0			0
$565$	14.1428			0.594993
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10.5365			−0.441712			−0.220856	−	0.975306i	$-0.570885\pi$
$569$	−10.5365			−0.441712			−0.220856	+	0.975306i	$0.570885\pi$
$570$		0			0
$571$			16.1162i			0.674442i	0.941425	+	0.337221i	$0.109487\pi$
$571$			16.1162i			0.674442i	−0.941425	+	0.337221i	$0.890513\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.64530	+	1.19213i	−0.193722	+	0.0497152i
$576$		0			0
$577$	1.93736			0.0806534			0.0403267	−	0.999187i	$-0.487160\pi$
$577$	1.93736			0.0806534			0.0403267	+	0.999187i	$0.487160\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			1.60242i			0.0664796i
$582$		0			0
$583$	2.84525			0.117838
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	33.6865i		−	1.39039i	−0.718821	−	0.695195i	$-0.755318\pi$
$587$		−	33.6865i		−	1.39039i	0.718821	−	0.695195i	$-0.244682\pi$
$588$		0			0
$589$		−	6.96696i		−	0.287069i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	5.23620i		−	0.215025i	−0.994204	−	0.107512i	$-0.965711\pi$
$593$		−	5.23620i		−	0.215025i	0.994204	−	0.107512i	$-0.0342885\pi$
$594$		0			0
$595$		−	0.0335444i		−	0.00137519i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	24.4592i		−	0.999376i	−0.866206	−	0.499688i	$-0.833448\pi$
$599$		−	24.4592i		−	0.999376i	0.866206	−	0.499688i	$-0.166552\pi$
$600$		0			0
$601$	9.64776			0.393540			0.196770	−	0.980450i	$-0.436955\pi$
$601$	9.64776			0.393540			0.196770	+	0.980450i	$0.436955\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.7719			0.437939
$606$		0			0
$607$	39.5325			1.60458			0.802288	−	0.596937i	$-0.203616\pi$
$607$	39.5325			1.60458			0.802288	+	0.596937i	$0.203616\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			2.16599i			0.0876268i
$612$		0			0
$613$		−	20.9202i		−	0.844959i	−0.906372	−	0.422480i	$-0.861160\pi$
$613$		−	20.9202i		−	0.844959i	0.906372	−	0.422480i	$-0.138840\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−34.5230			−1.38985			−0.694923	−	0.719084i	$-0.744561\pi$
$617$	−34.5230			−1.38985			−0.694923	+	0.719084i	$0.744561\pi$
$618$		0			0
$619$			4.97391i			0.199918i	0.994992	+	0.0999591i	$0.0318712\pi$
$619$			4.97391i			0.199918i	−0.994992	+	0.0999591i	$0.968129\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			5.07417i			0.203292i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	0.354804i		−	0.0141470i
$630$		0			0
$631$		−	21.8330i		−	0.869157i	−0.900634	−	0.434579i	$-0.856897\pi$
$631$		−	21.8330i		−	0.869157i	0.900634	−	0.434579i	$-0.143103\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.7107			−0.544091
$636$		0			0
$637$	−33.2436			−1.31716
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.18190			−0.0466822			−0.0233411	−	0.999728i	$-0.507430\pi$
$641$	−1.18190			−0.0466822			−0.0233411	+	0.999728i	$0.507430\pi$
$642$		0			0
$643$		−	44.4715i		−	1.75378i	−0.480688	−	0.876892i	$-0.659613\pi$
$643$		−	44.4715i		−	1.75378i	0.480688	−	0.876892i	$-0.340387\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.14925i		−	0.0451817i	−0.999745	−	0.0225909i	$-0.992808\pi$
$647$		−	1.14925i		−	0.0451817i	0.999745	−	0.0225909i	$-0.00719151\pi$
$648$		0			0
$649$			1.07614i			0.0422424i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	3.67210i		−	0.143701i	−0.997415	−	0.0718503i	$-0.977110\pi$
$653$		−	3.67210i		−	0.143701i	0.997415	−	0.0718503i	$-0.0228904\pi$
$654$		0			0
$655$		−	8.63722i		−	0.337484i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.6641			−0.493325			−0.246663	−	0.969101i	$-0.579334\pi$
$659$	−12.6641			−0.493325			−0.246663	+	0.969101i	$0.579334\pi$
$660$		0			0
$661$		−	23.7090i		−	0.922173i	−0.887355	−	0.461086i	$-0.847460\pi$
$661$		−	23.7090i		−	0.922173i	0.887355	−	0.461086i	$-0.152540\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.10918			0.0430123
$666$		0			0
$667$	3.40280	+	13.2595i	0.131757	+	0.513410i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1.56715i		−	0.0604991i
$672$		0			0
$673$	−5.96633			−0.229985			−0.114993	−	0.993366i	$-0.536684\pi$
$673$	−5.96633			−0.229985			−0.114993	+	0.993366i	$0.536684\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−8.25186			−0.317145			−0.158572	−	0.987347i	$-0.550689\pi$
$677$	−8.25186			−0.317145			−0.158572	+	0.987347i	$0.550689\pi$
$678$		0			0
$679$	−8.59095			−0.329690
$680$		0			0
$681$		0			0
$682$		0			0
$683$			4.06441i			0.155520i	0.996972	+	0.0777601i	$0.0247768\pi$
$683$			4.06441i			0.155520i	−0.996972	+	0.0777601i	$0.975223\pi$
$684$		0			0
$685$	−0.331548			−0.0126678
$686$		0			0
$687$		0			0
$688$		0			0
$689$	29.2903			1.11587
$690$		0			0
$691$	17.8924			0.680659			0.340330	−	0.940306i	$-0.389461\pi$
$691$	17.8924			0.680659			0.340330	+	0.940306i	$0.389461\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12.0248			−0.456128
$696$		0			0
$697$			0.415018i			0.0157199i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22.9103			0.865310			0.432655	−	0.901559i	$-0.357577\pi$
$701$	22.9103			0.865310			0.432655	+	0.901559i	$0.357577\pi$
$702$		0			0
$703$	11.7320			0.442482
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8.95656			0.336846
$708$		0			0
$709$			48.7002i			1.82898i	0.404614	+	0.914488i	$0.367406\pi$
$709$			48.7002i			1.82898i	−0.404614	+	0.914488i	$0.632594\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	14.2580	−	3.65903i	0.533965	−	0.137032i
$714$		0			0
$715$	−2.34836			−0.0878235
$716$		0			0
$717$		0			0
$718$		0			0
$719$			34.9310i			1.30270i	0.758775	+	0.651352i	$0.225798\pi$
$719$			34.9310i			1.30270i	−0.758775	+	0.651352i	$0.774202\pi$
$720$		0			0
$721$	−5.89915			−0.219696
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	2.85439i		−	0.106009i
$726$		0			0
$727$			38.9966i			1.44630i	0.690690	+	0.723151i	$0.257307\pi$
$727$			38.9966i			1.44630i	−0.690690	+	0.723151i	$0.742693\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	0.149407i		−	0.00552602i
$732$		0			0
$733$		−	10.2487i		−	0.378545i	−0.981925	−	0.189273i	$-0.939387\pi$
$733$		−	10.2487i		−	0.378545i	0.981925	−	0.189273i	$-0.0606130\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	3.03885i		−	0.111938i
$738$		0			0
$739$	−23.3080			−0.857398			−0.428699	−	0.903447i	$-0.641028\pi$
$739$	−23.3080			−0.857398			−0.428699	+	0.903447i	$0.641028\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	37.0143			1.35792			0.678961	−	0.734174i	$-0.262431\pi$
$743$	37.0143			1.35792			0.678961	+	0.734174i	$0.262431\pi$
$744$		0			0
$745$	−22.8306			−0.836448
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	0.405612i		−	0.0148207i
$750$		0			0
$751$			30.2486i			1.10379i	0.833915	+	0.551893i	$0.186094\pi$
$751$			30.2486i			1.10379i	−0.833915	+	0.551893i	$0.813906\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−19.0857			−0.694601
$756$		0			0
$757$			1.53142i			0.0556603i	0.999613	+	0.0278302i	$0.00885976\pi$
$757$			1.53142i			0.0556603i	−0.999613	+	0.0278302i	$0.991140\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	6.10735i		−	0.221391i	−0.993854	−	0.110696i	$-0.964692\pi$
$761$		−	6.10735i		−	0.221391i	0.993854	−	0.110696i	$-0.0353079\pi$
$762$		0			0
$763$	1.95125			0.0706400
$764$		0			0
$765$		0			0
$766$		0			0
$767$			11.0783i			0.400014i
$768$		0			0
$769$		−	23.2168i		−	0.837221i	−0.908166	−	0.418610i	$-0.862517\pi$
$769$		−	23.2168i		−	0.837221i	0.908166	−	0.418610i	$-0.137483\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	17.0419			0.612955			0.306477	−	0.951878i	$-0.400850\pi$
$773$	17.0419			0.612955			0.306477	+	0.951878i	$0.400850\pi$
$774$		0			0
$775$	−3.06933			−0.110254
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−13.7230			−0.491679
$780$		0			0
$781$			4.17265i			0.149309i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			7.13723i			0.254739i
$786$		0			0
$787$		−	6.41481i		−	0.228663i	−0.993443	−	0.114332i	$-0.963527\pi$
$787$		−	6.41481i		−	0.228663i	0.993443	−	0.114332i	$-0.0364726\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	6.91100i		−	0.245727i
$792$		0			0
$793$		−	16.1329i		−	0.572896i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−54.2270			−1.92082			−0.960410	−	0.278592i	$-0.910132\pi$
$797$	−54.2270			−1.92082			−0.960410	+	0.278592i	$0.910132\pi$
$798$		0			0
$799$		−	0.0302405i		−	0.00106983i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.45781			−0.227891
$804$		0			0
$805$	0.582542	+	2.26996i	0.0205319	+	0.0800056i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1.11751i		−	0.0392897i	−0.999807	−	0.0196448i	$-0.993746\pi$
$809$		−	1.11751i		−	0.0392897i	0.999807	−	0.0196448i	$-0.00625355\pi$
$810$		0			0
$811$	−6.32571			−0.222126			−0.111063	−	0.993813i	$-0.535426\pi$
$811$	−6.32571			−0.222126			−0.111063	+	0.993813i	$0.535426\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.07862			0.0377824
$816$		0			0
$817$	4.94032			0.172840
$818$		0			0
$819$		0			0
$820$		0			0
$821$			4.54360i			0.158573i	0.996852	+	0.0792864i	$0.0252642\pi$
$821$			4.54360i			0.158573i	−0.996852	+	0.0792864i	$0.974736\pi$
$822$		0			0
$823$	11.3482			0.395572			0.197786	−	0.980245i	$-0.436625\pi$
$823$	11.3482			0.395572			0.197786	+	0.980245i	$0.436625\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−40.7718			−1.41777			−0.708887	−	0.705322i	$-0.750803\pi$
$827$	−40.7718			−1.41777			−0.708887	+	0.705322i	$0.750803\pi$
$828$		0			0
$829$	44.7971			1.55587			0.777933	−	0.628347i	$-0.216268\pi$
$829$	44.7971			1.55587			0.777933	+	0.628347i	$0.216268\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.464131			0.0160812
$834$		0			0
$835$			16.2257i			0.561513i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.99750			0.207057			0.103528	−	0.994626i	$-0.466987\pi$
$839$	5.99750			0.207057			0.103528	+	0.994626i	$0.466987\pi$
$840$		0			0
$841$	20.8525			0.719050
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−11.1750			−0.384431
$846$		0			0
$847$		−	5.26376i		−	0.180865i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.16164	+	24.0097i	0.211218	+	0.823043i
$852$		0			0
$853$	7.70592			0.263846			0.131923	−	0.991260i	$-0.457885\pi$
$853$	7.70592			0.263846			0.131923	+	0.991260i	$0.457885\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	5.06687i		−	0.173081i	−0.996248	−	0.0865406i	$-0.972419\pi$
$857$		−	5.06687i		−	0.173081i	0.996248	−	0.0865406i	$-0.0275812\pi$
$858$		0			0
$859$	8.86980			0.302634			0.151317	−	0.988485i	$-0.451649\pi$
$859$	8.86980			0.302634			0.151317	+	0.988485i	$0.451649\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			46.9300i			1.59752i	0.601652	+	0.798758i	$0.294509\pi$
$863$			46.9300i			1.59752i	−0.601652	+	0.798758i	$0.705491\pi$
$864$		0			0
$865$			4.67906i			0.159093i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			4.36116i			0.147942i
$870$		0			0
$871$		−	31.2833i		−	1.05999i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	0.488657i		−	0.0165196i
$876$		0			0
$877$	−21.8943			−0.739317			−0.369658	−	0.929168i	$-0.620525\pi$
$877$	−21.8943			−0.739317			−0.369658	+	0.929168i	$0.620525\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−5.98139			−0.201518			−0.100759	−	0.994911i	$-0.532127\pi$
$881$	−5.98139			−0.201518			−0.100759	+	0.994911i	$0.532127\pi$
$882$		0			0
$883$	−24.0137			−0.808124			−0.404062	−	0.914732i	$-0.632402\pi$
$883$	−24.0137			−0.808124			−0.404062	+	0.914732i	$0.632402\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	11.5538i		−	0.387938i	−0.981008	−	0.193969i	$-0.937864\pi$
$887$		−	11.5538i		−	0.387938i	0.981008	−	0.193969i	$-0.0621362\pi$
$888$		0			0
$889$			6.69982i			0.224705i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.999940			0.0334617
$894$		0			0
$895$			6.05993i			0.202561i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			8.76107i			0.292198i
$900$		0			0
$901$	−0.408936			−0.0136236
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	21.8606i		−	0.726671i
$906$		0			0
$907$		−	0.430914i		−	0.0143083i	−0.999974	−	0.00715413i	$-0.997723\pi$
$907$		−	0.430914i		−	0.0143083i	0.999974	−	0.00715413i	$-0.00227725\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	56.4174			1.86919			0.934597	−	0.355710i	$-0.115761\pi$
$911$	56.4174			1.86919			0.934597	+	0.355710i	$0.115761\pi$
$912$		0			0
$913$	−1.56622			−0.0518344
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.22064			−0.139378
$918$		0			0
$919$		−	19.9844i		−	0.659226i	−0.944116	−	0.329613i	$-0.893082\pi$
$919$		−	19.9844i		−	0.659226i	0.944116	−	0.329613i	$-0.106918\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			42.9551i			1.41388i
$924$		0			0
$925$		−	5.16861i		−	0.169943i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	43.9053i		−	1.44049i	−0.693721	−	0.720244i	$-0.744030\pi$
$929$		−	43.9053i		−	1.44049i	0.693721	−	0.720244i	$-0.255970\pi$
$930$		0			0
$931$			15.3470i			0.502979i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.0327866			0.00107224
$936$		0			0
$937$		−	9.04753i		−	0.295570i	−0.989020	−	0.147785i	$-0.952786\pi$
$937$		−	9.04753i		−	0.295570i	0.989020	−	0.147785i	$-0.0472144\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−30.7043			−1.00093			−0.500465	−	0.865757i	$-0.666838\pi$
$941$	−30.7043			−1.00093			−0.500465	+	0.865757i	$0.666838\pi$
$942$		0			0
$943$	−7.20732	−	28.0844i	−0.234703	−	0.914554i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			31.6819i			1.02952i	0.857333	+	0.514762i	$0.172120\pi$
$947$			31.6819i			1.02952i	−0.857333	+	0.514762i	$0.827880\pi$
$948$		0			0
$949$	−66.4795			−2.15801
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9.94470			0.322140			0.161070	−	0.986943i	$-0.448505\pi$
$953$	9.94470			0.322140			0.161070	+	0.986943i	$0.448505\pi$
$954$		0			0
$955$	19.6665			0.636394
$956$		0			0
$957$		0			0
$958$		0			0
$959$			0.162014i			0.00523169i
$960$		0			0
$961$	−21.5792			−0.696103
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.08295			−0.228008
$966$		0			0
$967$	−35.9474			−1.15599			−0.577995	−	0.816040i	$-0.696165\pi$
$967$	−35.9474			−1.15599			−0.577995	+	0.816040i	$0.696165\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−34.4577			−1.10580			−0.552900	−	0.833248i	$-0.686479\pi$
$971$	−34.4577			−1.10580			−0.552900	+	0.833248i	$0.686479\pi$
$972$		0			0
$973$			5.87602i			0.188377i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.31973			0.170193			0.0850967	−	0.996373i	$-0.472880\pi$
$977$	5.31973			0.170193			0.0850967	+	0.996373i	$0.472880\pi$
$978$		0			0
$979$	−4.95954			−0.158508
$980$		0			0
$981$		0			0
$982$		0			0
$983$	21.4607			0.684489			0.342245	−	0.939611i	$-0.388813\pi$
$983$	21.4607			0.684489			0.342245	+	0.939611i	$0.388813\pi$
$984$		0			0
$985$			15.8556i			0.505203i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.59465	+	10.1104i	0.0825050	+	0.321493i
$990$		0			0
$991$	−31.7807			−1.00955			−0.504773	−	0.863252i	$-0.668424\pi$
$991$	−31.7807			−1.00955			−0.504773	+	0.863252i	$0.668424\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			5.91295i			0.187453i
$996$		0			0
$997$	−26.7049			−0.845753			−0.422877	−	0.906187i	$-0.638980\pi$
$997$	−26.7049			−0.845753			−0.422877	+	0.906187i	$0.638980\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.p.a.1241.26	yes	48
3.2	odd	2		8280.2.p.b.1241.26	yes	48
23.22	odd	2		8280.2.p.b.1241.23	yes	48
69.68	even	2	inner	8280.2.p.a.1241.23	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.23	✓	48	69.68	even	2	inner
8280.2.p.a.1241.26	yes	48	1.1	even	1	trivial
8280.2.p.b.1241.23	yes	48	23.22	odd	2
8280.2.p.b.1241.26	yes	48	3.2	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +0.488657i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +0.488657i q^{7} -0.477618 q^{11} -4.91681 q^{13} +0.0686460 q^{17} +2.26986i q^{19} +(-4.64530 + 1.19213i) q^{23} +1.00000 q^{25} -2.85439i q^{29} -3.06933 q^{31} -0.488657i q^{35} -5.16861i q^{37} +6.04576i q^{41} -2.17648i q^{43} -0.440529i q^{47} +6.76121 q^{49} -5.95717 q^{53} +0.477618 q^{55} -2.25315i q^{59} +3.28117i q^{61} +4.91681 q^{65} +6.36252i q^{67} -8.73638i q^{71} +13.5209 q^{73} -0.233391i q^{77} -9.13107i q^{79} +3.27923 q^{83} -0.0686460 q^{85} +10.3839 q^{89} -2.40263i q^{91} -2.26986i q^{95} +17.5807i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 48 * q - 48 * q^5	\( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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