LMFDB - Embedded newform 2816.2.g.b.1407.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1407,2816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2816.1407");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.g (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:	$22.4858732092$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 44)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1407.4
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2816.1407
Dual form			2816.2.g.b.1407.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +1.00000i q^{5} +2.82843 q^{7} +O(q^{10})$	$q-1.73205 q^{3} +1.00000i q^{5} +2.82843 q^{7} +(-1.73205 - 2.82843i) q^{11} -4.89898 q^{13} -1.73205i q^{15} -4.89898i q^{17} +2.82843i q^{19} -4.89898 q^{21} +5.19615i q^{23} +4.00000 q^{25} +5.19615 q^{27} +1.73205i q^{31} +(3.00000 + 4.89898i) q^{33} +2.82843i q^{35} -3.00000i q^{37} +8.48528 q^{39} -4.89898i q^{41} +5.65685i q^{43} +3.46410i q^{47} +1.00000 q^{49} +8.48528i q^{51} +2.00000i q^{53} +(2.82843 - 1.73205i) q^{55} -4.89898i q^{57} +1.73205 q^{59} +9.79796 q^{61} -4.89898i q^{65} -8.66025 q^{67} -9.00000i q^{69} +12.1244i q^{71} +4.89898i q^{73} -6.92820 q^{75} +(-4.89898 - 8.00000i) q^{77} +11.3137 q^{79} -9.00000 q^{81} +4.89898 q^{85} +1.00000 q^{89} -13.8564 q^{91} -3.00000i q^{93} -2.82843 q^{95} +7.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 32 q^{25} + 24 q^{33} + 8 q^{49} - 72 q^{81} + 8 q^{89} + 56 q^{97}+O(q^{100}) $ 8 * q + 32 * q^25 + 24 * q^33 + 8 * q^49 - 72 * q^81 + 8 * q^89 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	−1.73205			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−1.73205			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$4$		0			0
$5$			1.00000i			0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$			1.00000i			0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$		0			0
$7$	2.82843			1.06904			0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843			1.06904			0.534522	+	0.845154i	$0.320491\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.73205	−	2.82843i	−0.522233	−	0.852803i
$11$	−1.73205	−	2.82843i	−0.522233	−	0.852803i
$12$		0			0
$13$	−4.89898			−1.35873			−0.679366	−	0.733799i	$-0.737745\pi$
$13$	−4.89898			−1.35873			−0.679366	+	0.733799i	$0.737745\pi$
$14$		0			0
$15$		−	1.73205i		−	0.447214i
$16$		0			0
$17$		−	4.89898i		−	1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$		−	4.89898i		−	1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$		0			0
$19$			2.82843i			0.648886i	0.945905	+	0.324443i	$0.105177\pi$
$19$			2.82843i			0.648886i	−0.945905	+	0.324443i	$0.894823\pi$
$20$		0			0
$21$	−4.89898			−1.06904
$22$		0			0
$23$			5.19615i			1.08347i	0.840548	+	0.541736i	$0.182233\pi$
$23$			5.19615i			1.08347i	−0.840548	+	0.541736i	$0.817767\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$	5.19615			1.00000
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$			1.73205i			0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$			1.73205i			0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$	3.00000	+	4.89898i	0.522233	+	0.852803i
$34$		0			0
$35$			2.82843i			0.478091i
$36$		0			0
$37$		−	3.00000i		−	0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$		−	3.00000i		−	0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$		0			0
$39$	8.48528			1.35873
$40$		0			0
$41$		−	4.89898i		−	0.765092i	−0.923936	−	0.382546i	$-0.875047\pi$
$41$		−	4.89898i		−	0.765092i	0.923936	−	0.382546i	$-0.124953\pi$
$42$		0			0
$43$			5.65685i			0.862662i	0.902194	+	0.431331i	$0.141956\pi$
$43$			5.65685i			0.862662i	−0.902194	+	0.431331i	$0.858044\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.46410i			0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$			3.46410i			0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$			8.48528i			1.18818i
$52$		0			0
$53$			2.00000i			0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$		0			0
$55$	2.82843	−	1.73205i	0.381385	−	0.233550i
$56$		0			0
$57$		−	4.89898i		−	0.648886i
$58$		0			0
$59$	1.73205			0.225494			0.112747	−	0.993624i	$-0.464035\pi$
$59$	1.73205			0.225494			0.112747	+	0.993624i	$0.464035\pi$
$60$		0			0
$61$	9.79796			1.25450			0.627250	−	0.778818i	$-0.284180\pi$
$61$	9.79796			1.25450			0.627250	+	0.778818i	$0.284180\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	4.89898i		−	0.607644i
$66$		0			0
$67$	−8.66025			−1.05802			−0.529009	−	0.848616i	$-0.677436\pi$
$67$	−8.66025			−1.05802			−0.529009	+	0.848616i	$0.677436\pi$
$68$		0			0
$69$		−	9.00000i		−	1.08347i
$70$		0			0
$71$			12.1244i			1.43890i	0.694546	+	0.719448i	$0.255605\pi$
$71$			12.1244i			1.43890i	−0.694546	+	0.719448i	$0.744395\pi$
$72$		0			0
$73$			4.89898i			0.573382i	0.958023	+	0.286691i	$0.0925553\pi$
$73$			4.89898i			0.573382i	−0.958023	+	0.286691i	$0.907445\pi$
$74$		0			0
$75$	−6.92820			−0.800000
$76$		0			0
$77$	−4.89898	−	8.00000i	−0.558291	−	0.911685i
$78$		0			0
$79$	11.3137			1.27289			0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137			1.27289			0.636446	+	0.771321i	$0.280404\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	4.89898			0.531369
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.00000			0.106000			0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000			0.106000			0.0529999	+	0.998595i	$0.483122\pi$
$90$		0			0
$91$	−13.8564			−1.45255
$92$		0			0
$93$		−	3.00000i		−	0.311086i
$94$		0			0
$95$	−2.82843			−0.290191
$96$		0			0
$97$	7.00000			0.710742			0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000			0.710742			0.355371	+	0.934725i	$0.384354\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.89898			−0.487467			−0.243733	−	0.969842i	$-0.578372\pi$
$101$	−4.89898			−0.487467			−0.243733	+	0.969842i	$0.578372\pi$
$102$		0			0
$103$		−	3.46410i		−	0.341328i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$		−	3.46410i		−	0.341328i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$		0			0
$105$		−	4.89898i		−	0.478091i
$106$		0			0
$107$			8.48528i			0.820303i	0.912017	+	0.410152i	$0.134524\pi$
$107$			8.48528i			0.820303i	−0.912017	+	0.410152i	$0.865476\pi$
$108$		0			0
$109$	−9.79796			−0.938474			−0.469237	−	0.883072i	$-0.655471\pi$
$109$	−9.79796			−0.938474			−0.469237	+	0.883072i	$0.655471\pi$
$110$		0			0
$111$			5.19615i			0.493197i
$112$		0			0
$113$	−5.00000			−0.470360			−0.235180	−	0.971952i	$-0.575568\pi$
$113$	−5.00000			−0.470360			−0.235180	+	0.971952i	$0.575568\pi$
$114$		0			0
$115$	−5.19615			−0.484544
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	13.8564i		−	1.27021i
$120$		0			0
$121$	−5.00000	+	9.79796i	−0.454545	+	0.890724i
$122$		0			0
$123$			8.48528i			0.765092i
$124$		0			0
$125$			9.00000i			0.804984i
$126$		0			0
$127$	14.1421			1.25491			0.627456	−	0.778652i	$-0.284096\pi$
$127$	14.1421			1.25491			0.627456	+	0.778652i	$0.284096\pi$
$128$		0			0
$129$		−	9.79796i		−	0.862662i
$130$		0			0
$131$			2.82843i			0.247121i	0.992337	+	0.123560i	$0.0394313\pi$
$131$			2.82843i			0.247121i	−0.992337	+	0.123560i	$0.960569\pi$
$132$		0			0
$133$			8.00000i			0.693688i
$134$		0			0
$135$			5.19615i			0.447214i
$136$		0			0
$137$	−19.0000			−1.62328			−0.811640	−	0.584158i	$-0.801425\pi$
$137$	−19.0000			−1.62328			−0.811640	+	0.584158i	$0.801425\pi$
$138$		0			0
$139$			19.7990i			1.67933i	0.543106	+	0.839664i	$0.317248\pi$
$139$			19.7990i			1.67933i	−0.543106	+	0.839664i	$0.682752\pi$
$140$		0			0
$141$		−	6.00000i		−	0.505291i
$142$		0			0
$143$	8.48528	+	13.8564i	0.709575	+	1.15873i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1.73205			−0.142857
$148$		0			0
$149$	14.6969			1.20402			0.602010	−	0.798489i	$-0.294367\pi$
$149$	14.6969			1.20402			0.602010	+	0.798489i	$0.294367\pi$
$150$		0			0
$151$	−5.65685			−0.460348			−0.230174	−	0.973149i	$-0.573930\pi$
$151$	−5.65685			−0.460348			−0.230174	+	0.973149i	$0.573930\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.73205			−0.139122
$156$		0			0
$157$			11.0000i			0.877896i	0.898513	+	0.438948i	$0.144649\pi$
$157$			11.0000i			0.877896i	−0.898513	+	0.438948i	$0.855351\pi$
$158$		0			0
$159$		−	3.46410i		−	0.274721i
$160$		0			0
$161$			14.6969i			1.15828i
$162$		0			0
$163$	17.3205			1.35665			0.678323	−	0.734763i	$-0.262707\pi$
$163$	17.3205			1.35665			0.678323	+	0.734763i	$0.262707\pi$
$164$		0			0
$165$	−4.89898	+	3.00000i	−0.381385	+	0.233550i
$166$		0			0
$167$	19.7990			1.53209			0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990			1.53209			0.766046	+	0.642786i	$0.222221\pi$
$168$		0			0
$169$	11.0000			0.846154
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.79796			0.744925			0.372463	−	0.928047i	$-0.378514\pi$
$173$	9.79796			0.744925			0.372463	+	0.928047i	$0.378514\pi$
$174$		0			0
$175$	11.3137			0.855236
$176$		0			0
$177$	−3.00000			−0.225494
$178$		0			0
$179$	−8.66025			−0.647298			−0.323649	−	0.946177i	$-0.604910\pi$
$179$	−8.66025			−0.647298			−0.323649	+	0.946177i	$0.604910\pi$
$180$		0			0
$181$			21.0000i			1.56092i	0.625207	+	0.780459i	$0.285014\pi$
$181$			21.0000i			1.56092i	−0.625207	+	0.780459i	$0.714986\pi$
$182$		0			0
$183$	−16.9706			−1.25450
$184$		0			0
$185$	3.00000			0.220564
$186$		0			0
$187$	−13.8564	+	8.48528i	−1.01328	+	0.620505i
$188$		0			0
$189$	14.6969			1.06904
$190$		0			0
$191$		−	5.19615i		−	0.375980i	−0.982171	−	0.187990i	$-0.939803\pi$
$191$		−	5.19615i		−	0.375980i	0.982171	−	0.187990i	$-0.0601973\pi$
$192$		0			0
$193$			9.79796i			0.705273i	0.935760	+	0.352636i	$0.114715\pi$
$193$			9.79796i			0.705273i	−0.935760	+	0.352636i	$0.885285\pi$
$194$		0			0
$195$			8.48528i			0.607644i
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$			10.3923i			0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$			10.3923i			0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$		0			0
$201$	15.0000			1.05802
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.89898			0.342160
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.00000	−	4.89898i	0.553372	−	0.338869i
$210$		0			0
$211$		−	5.65685i		−	0.389434i	−0.980859	−	0.194717i	$-0.937621\pi$
$211$		−	5.65685i		−	0.389434i	0.980859	−	0.194717i	$-0.0623788\pi$
$212$		0			0
$213$		−	21.0000i		−	1.43890i
$214$		0			0
$215$	−5.65685			−0.385794
$216$		0			0
$217$			4.89898i			0.332564i
$218$		0			0
$219$		−	8.48528i		−	0.573382i
$220$		0			0
$221$			24.0000i			1.61441i
$222$		0			0
$223$			22.5167i			1.50783i	0.656974	+	0.753914i	$0.271836\pi$
$223$			22.5167i			1.50783i	−0.656974	+	0.753914i	$0.728164\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			8.48528i			0.563188i	0.959534	+	0.281594i	$0.0908631\pi$
$227$			8.48528i			0.563188i	−0.959534	+	0.281594i	$0.909137\pi$
$228$		0			0
$229$			1.00000i			0.0660819i	0.999454	+	0.0330409i	$0.0105192\pi$
$229$			1.00000i			0.0660819i	−0.999454	+	0.0330409i	$0.989481\pi$
$230$		0			0
$231$	8.48528	+	13.8564i	0.558291	+	0.911685i
$232$		0			0
$233$			19.5959i			1.28377i	0.766800	+	0.641886i	$0.221848\pi$
$233$			19.5959i			1.28377i	−0.766800	+	0.641886i	$0.778152\pi$
$234$		0			0
$235$	−3.46410			−0.225973
$236$		0			0
$237$	−19.5959			−1.27289
$238$		0			0
$239$	−5.65685			−0.365911			−0.182956	−	0.983121i	$-0.558567\pi$
$239$	−5.65685			−0.365911			−0.182956	+	0.983121i	$0.558567\pi$
$240$		0			0
$241$			29.3939i			1.89343i	0.322078	+	0.946713i	$0.395619\pi$
$241$			29.3939i			1.89343i	−0.322078	+	0.946713i	$0.604381\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			1.00000i			0.0638877i
$246$		0			0
$247$		−	13.8564i		−	0.881662i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	22.5167			1.42124			0.710620	−	0.703577i	$-0.248415\pi$
$251$	22.5167			1.42124			0.710620	+	0.703577i	$0.248415\pi$
$252$		0			0
$253$	14.6969	−	9.00000i	0.923989	−	0.565825i
$254$		0			0
$255$	−8.48528			−0.531369
$256$		0			0
$257$	−22.0000			−1.37232			−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000			−1.37232			−0.686161	+	0.727450i	$0.740706\pi$
$258$		0			0
$259$		−	8.48528i		−	0.527250i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	14.1421			0.872041			0.436021	−	0.899937i	$-0.356387\pi$
$263$	14.1421			0.872041			0.436021	+	0.899937i	$0.356387\pi$
$264$		0			0
$265$	−2.00000			−0.122859
$266$		0			0
$267$	−1.73205			−0.106000
$268$		0			0
$269$		−	10.0000i		−	0.609711i	−0.952399	−	0.304855i	$-0.901392\pi$
$269$		−	10.0000i		−	0.609711i	0.952399	−	0.304855i	$-0.0986081\pi$
$270$		0			0
$271$	2.82843			0.171815			0.0859074	−	0.996303i	$-0.472621\pi$
$271$	2.82843			0.171815			0.0859074	+	0.996303i	$0.472621\pi$
$272$		0			0
$273$	24.0000			1.45255
$274$		0			0
$275$	−6.92820	−	11.3137i	−0.417786	−	0.682242i
$276$		0			0
$277$	9.79796			0.588702			0.294351	−	0.955697i	$-0.404896\pi$
$277$	9.79796			0.588702			0.294351	+	0.955697i	$0.404896\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$			11.3137i			0.672530i	0.941767	+	0.336265i	$0.109164\pi$
$283$			11.3137i			0.672530i	−0.941767	+	0.336265i	$0.890836\pi$
$284$		0			0
$285$	4.89898			0.290191
$286$		0			0
$287$		−	13.8564i		−	0.817918i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$	−12.1244			−0.710742
$292$		0			0
$293$	4.89898			0.286201			0.143101	−	0.989708i	$-0.454293\pi$
$293$	4.89898			0.286201			0.143101	+	0.989708i	$0.454293\pi$
$294$		0			0
$295$			1.73205i			0.100844i
$296$		0			0
$297$	−9.00000	−	14.6969i	−0.522233	−	0.852803i
$298$		0			0
$299$		−	25.4558i		−	1.47215i
$300$		0			0
$301$			16.0000i			0.922225i
$302$		0			0
$303$	8.48528			0.487467
$304$		0			0
$305$			9.79796i			0.561029i
$306$		0			0
$307$			19.7990i			1.12999i	0.825095	+	0.564994i	$0.191122\pi$
$307$			19.7990i			1.12999i	−0.825095	+	0.564994i	$0.808878\pi$
$308$		0			0
$309$			6.00000i			0.341328i
$310$		0			0
$311$			10.3923i			0.589294i	0.955606	+	0.294647i	$0.0952020\pi$
$311$			10.3923i			0.589294i	−0.955606	+	0.294647i	$0.904798\pi$
$312$		0			0
$313$	−27.0000			−1.52613			−0.763065	−	0.646322i	$-0.776306\pi$
$313$	−27.0000			−1.52613			−0.763065	+	0.646322i	$0.776306\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	25.0000i		−	1.40414i	−0.712108	−	0.702070i	$-0.752259\pi$
$317$		−	25.0000i		−	1.40414i	0.712108	−	0.702070i	$-0.247741\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		−	14.6969i		−	0.820303i
$322$		0			0
$323$	13.8564			0.770991
$324$		0			0
$325$	−19.5959			−1.08699
$326$		0			0
$327$	16.9706			0.938474
$328$		0			0
$329$			9.79796i			0.540179i
$330$		0			0
$331$	−5.19615			−0.285606			−0.142803	−	0.989751i	$-0.545612\pi$
$331$	−5.19615			−0.285606			−0.142803	+	0.989751i	$0.545612\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	8.66025i		−	0.473160i
$336$		0			0
$337$		−	34.2929i		−	1.86805i	−0.357206	−	0.934025i	$-0.616271\pi$
$337$		−	34.2929i		−	1.86805i	0.357206	−	0.934025i	$-0.383729\pi$
$338$		0			0
$339$	8.66025			0.470360
$340$		0			0
$341$	4.89898	−	3.00000i	0.265295	−	0.162459i
$342$		0			0
$343$	−16.9706			−0.916324
$344$		0			0
$345$	9.00000			0.484544
$346$		0			0
$347$		−	25.4558i		−	1.36654i	−0.730165	−	0.683271i	$-0.760557\pi$
$347$		−	25.4558i		−	1.36654i	0.730165	−	0.683271i	$-0.239443\pi$
$348$		0			0
$349$	−4.89898			−0.262236			−0.131118	−	0.991367i	$-0.541857\pi$
$349$	−4.89898			−0.262236			−0.131118	+	0.991367i	$0.541857\pi$
$350$		0			0
$351$	−25.4558			−1.35873
$352$		0			0
$353$	−17.0000			−0.904819			−0.452409	−	0.891810i	$-0.649435\pi$
$353$	−17.0000			−0.904819			−0.452409	+	0.891810i	$0.649435\pi$
$354$		0			0
$355$	−12.1244			−0.643494
$356$		0			0
$357$			24.0000i			1.27021i
$358$		0			0
$359$	8.48528			0.447836			0.223918	−	0.974608i	$-0.428115\pi$
$359$	8.48528			0.447836			0.223918	+	0.974608i	$0.428115\pi$
$360$		0			0
$361$	11.0000			0.578947
$362$		0			0
$363$	8.66025	−	16.9706i	0.454545	−	0.890724i
$364$		0			0
$365$	−4.89898			−0.256424
$366$		0			0
$367$			1.73205i			0.0904123i	0.998978	+	0.0452062i	$0.0143945\pi$
$367$			1.73205i			0.0904123i	−0.998978	+	0.0452062i	$0.985606\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			5.65685i			0.293689i
$372$		0			0
$373$	14.6969			0.760979			0.380489	−	0.924785i	$-0.375756\pi$
$373$	14.6969			0.760979			0.380489	+	0.924785i	$0.375756\pi$
$374$		0			0
$375$		−	15.5885i		−	0.804984i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−12.1244			−0.622786			−0.311393	−	0.950281i	$-0.600796\pi$
$379$	−12.1244			−0.622786			−0.311393	+	0.950281i	$0.600796\pi$
$380$		0			0
$381$	−24.4949			−1.25491
$382$		0			0
$383$		−	32.9090i		−	1.68157i	−0.541370	−	0.840785i	$-0.682094\pi$
$383$		−	32.9090i		−	1.68157i	0.541370	−	0.840785i	$-0.317906\pi$
$384$		0			0
$385$	8.00000	−	4.89898i	0.407718	−	0.249675i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	19.0000i		−	0.963338i	−0.876353	−	0.481669i	$-0.840031\pi$
$389$		−	19.0000i		−	0.963338i	0.876353	−	0.481669i	$-0.159969\pi$
$390$		0			0
$391$	25.4558			1.28736
$392$		0			0
$393$		−	4.89898i		−	0.247121i
$394$		0			0
$395$			11.3137i			0.569254i
$396$		0			0
$397$			6.00000i			0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$			6.00000i			0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$		0			0
$399$		−	13.8564i		−	0.693688i
$400$		0			0
$401$	10.0000			0.499376			0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000			0.499376			0.249688	+	0.968326i	$0.419672\pi$
$402$		0			0
$403$		−	8.48528i		−	0.422682i
$404$		0			0
$405$		−	9.00000i		−	0.447214i
$406$		0			0
$407$	−8.48528	+	5.19615i	−0.420600	+	0.257564i
$408$		0			0
$409$		−	29.3939i		−	1.45343i	−0.686937	−	0.726717i	$-0.741045\pi$
$409$		−	29.3939i		−	1.45343i	0.686937	−	0.726717i	$-0.258955\pi$
$410$		0			0
$411$	32.9090			1.62328
$412$		0			0
$413$	4.89898			0.241063
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	34.2929i		−	1.67933i
$418$		0			0
$419$	−10.3923			−0.507697			−0.253849	−	0.967244i	$-0.581697\pi$
$419$	−10.3923			−0.507697			−0.253849	+	0.967244i	$0.581697\pi$
$420$		0			0
$421$		−	30.0000i		−	1.46211i	−0.682318	−	0.731055i	$-0.739028\pi$
$421$		−	30.0000i		−	1.46211i	0.682318	−	0.731055i	$-0.260972\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	19.5959i		−	0.950542i
$426$		0			0
$427$	27.7128			1.34112
$428$		0			0
$429$	−14.6969	−	24.0000i	−0.709575	−	1.15873i
$430$		0			0
$431$	−14.1421			−0.681203			−0.340601	−	0.940208i	$-0.610631\pi$
$431$	−14.1421			−0.681203			−0.340601	+	0.940208i	$0.610631\pi$
$432$		0			0
$433$	−21.0000			−1.00920			−0.504598	−	0.863355i	$-0.668359\pi$
$433$	−21.0000			−1.00920			−0.504598	+	0.863355i	$0.668359\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.6969			−0.703050
$438$		0			0
$439$	−22.6274			−1.07995			−0.539974	−	0.841682i	$-0.681566\pi$
$439$	−22.6274			−1.07995			−0.539974	+	0.841682i	$0.681566\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.19615			−0.246877			−0.123438	−	0.992352i	$-0.539392\pi$
$443$	−5.19615			−0.246877			−0.123438	+	0.992352i	$0.539392\pi$
$444$		0			0
$445$			1.00000i			0.0474045i
$446$		0			0
$447$	−25.4558			−1.20402
$448$		0			0
$449$	23.0000			1.08544			0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000			1.08544			0.542719	+	0.839915i	$0.317395\pi$
$450$		0			0
$451$	−13.8564	+	8.48528i	−0.652473	+	0.399556i
$452$		0			0
$453$	9.79796			0.460348
$454$		0			0
$455$		−	13.8564i		−	0.649598i
$456$		0			0
$457$			19.5959i			0.916658i	0.888783	+	0.458329i	$0.151552\pi$
$457$			19.5959i			0.916658i	−0.888783	+	0.458329i	$0.848448\pi$
$458$		0			0
$459$		−	25.4558i		−	1.18818i
$460$		0			0
$461$	29.3939			1.36901			0.684505	−	0.729008i	$-0.260019\pi$
$461$	29.3939			1.36901			0.684505	+	0.729008i	$0.260019\pi$
$462$		0			0
$463$			36.3731i			1.69040i	0.534450	+	0.845200i	$0.320519\pi$
$463$			36.3731i			1.69040i	−0.534450	+	0.845200i	$0.679481\pi$
$464$		0			0
$465$	3.00000			0.139122
$466$		0			0
$467$	32.9090			1.52285			0.761423	−	0.648256i	$-0.224501\pi$
$467$	32.9090			1.52285			0.761423	+	0.648256i	$0.224501\pi$
$468$		0			0
$469$	−24.4949			−1.13107
$470$		0			0
$471$		−	19.0526i		−	0.877896i
$472$		0			0
$473$	16.0000	−	9.79796i	0.735681	−	0.450511i
$474$		0			0
$475$			11.3137i			0.519109i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−31.1127			−1.42158			−0.710788	−	0.703407i	$-0.751661\pi$
$479$	−31.1127			−1.42158			−0.710788	+	0.703407i	$0.751661\pi$
$480$		0			0
$481$			14.6969i			0.670123i
$482$		0			0
$483$		−	25.4558i		−	1.15828i
$484$		0			0
$485$			7.00000i			0.317854i
$486$		0			0
$487$			39.8372i			1.80519i	0.430486	+	0.902597i	$0.358342\pi$
$487$			39.8372i			1.80519i	−0.430486	+	0.902597i	$0.641658\pi$
$488$		0			0
$489$	−30.0000			−1.35665
$490$		0			0
$491$		−	33.9411i		−	1.53174i	−0.642995	−	0.765871i	$-0.722308\pi$
$491$		−	33.9411i		−	1.53174i	0.642995	−	0.765871i	$-0.277692\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			34.2929i			1.53824i
$498$		0			0
$499$	17.3205			0.775372			0.387686	−	0.921791i	$-0.373274\pi$
$499$	17.3205			0.775372			0.387686	+	0.921791i	$0.373274\pi$
$500$		0			0
$501$	−34.2929			−1.53209
$502$		0			0
$503$	−39.5980			−1.76559			−0.882793	−	0.469762i	$-0.844340\pi$
$503$	−39.5980			−1.76559			−0.882793	+	0.469762i	$0.844340\pi$
$504$		0			0
$505$		−	4.89898i		−	0.218002i
$506$		0			0
$507$	−19.0526			−0.846154
$508$		0			0
$509$		−	1.00000i		−	0.0443242i	−0.999754	−	0.0221621i	$-0.992945\pi$
$509$		−	1.00000i		−	0.0443242i	0.999754	−	0.0221621i	$-0.00705500\pi$
$510$		0			0
$511$			13.8564i			0.612971i
$512$		0			0
$513$			14.6969i			0.648886i
$514$		0			0
$515$	3.46410			0.152647
$516$		0			0
$517$	9.79796	−	6.00000i	0.430914	−	0.263880i
$518$		0			0
$519$	−16.9706			−0.744925
$520$		0			0
$521$	17.0000			0.744784			0.372392	−	0.928076i	$-0.378538\pi$
$521$	17.0000			0.744784			0.372392	+	0.928076i	$0.378538\pi$
$522$		0			0
$523$			36.7696i			1.60782i	0.594751	+	0.803910i	$0.297251\pi$
$523$			36.7696i			1.60782i	−0.594751	+	0.803910i	$0.702749\pi$
$524$		0			0
$525$	−19.5959			−0.855236
$526$		0			0
$527$	8.48528			0.369625
$528$		0			0
$529$	−4.00000			−0.173913
$530$		0			0
$531$		0			0
$532$		0			0
$533$			24.0000i			1.03956i
$534$		0			0
$535$	−8.48528			−0.366851
$536$		0			0
$537$	15.0000			0.647298
$538$		0			0
$539$	−1.73205	−	2.82843i	−0.0746047	−	0.121829i
$540$		0			0
$541$	−14.6969			−0.631871			−0.315935	−	0.948781i	$-0.602318\pi$
$541$	−14.6969			−0.631871			−0.315935	+	0.948781i	$0.602318\pi$
$542$		0			0
$543$		−	36.3731i		−	1.56092i
$544$		0			0
$545$		−	9.79796i		−	0.419698i
$546$		0			0
$547$		−	5.65685i		−	0.241870i	−0.992660	−	0.120935i	$-0.961411\pi$
$547$		−	5.65685i		−	0.241870i	0.992660	−	0.120935i	$-0.0385892\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	32.0000			1.36078
$554$		0			0
$555$	−5.19615			−0.220564
$556$		0			0
$557$	24.4949			1.03788			0.518941	−	0.854810i	$-0.326326\pi$
$557$	24.4949			1.03788			0.518941	+	0.854810i	$0.326326\pi$
$558$		0			0
$559$		−	27.7128i		−	1.17213i
$560$		0			0
$561$	24.0000	−	14.6969i	1.01328	−	0.620505i
$562$		0			0
$563$		−	11.3137i		−	0.476816i	−0.971165	−	0.238408i	$-0.923374\pi$
$563$		−	11.3137i		−	0.476816i	0.971165	−	0.238408i	$-0.0766255\pi$
$564$		0			0
$565$		−	5.00000i		−	0.210352i
$566$		0			0
$567$	−25.4558			−1.06904
$568$		0			0
$569$			19.5959i			0.821504i	0.911747	+	0.410752i	$0.134734\pi$
$569$			19.5959i			0.821504i	−0.911747	+	0.410752i	$0.865266\pi$
$570$		0			0
$571$		−	5.65685i		−	0.236732i	−0.992970	−	0.118366i	$-0.962234\pi$
$571$		−	5.65685i		−	0.236732i	0.992970	−	0.118366i	$-0.0377656\pi$
$572$		0			0
$573$			9.00000i			0.375980i
$574$		0			0
$575$			20.7846i			0.866778i
$576$		0			0
$577$	−17.0000			−0.707719			−0.353860	−	0.935299i	$-0.615131\pi$
$577$	−17.0000			−0.707719			−0.353860	+	0.935299i	$0.615131\pi$
$578$		0			0
$579$		−	16.9706i		−	0.705273i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	5.65685	−	3.46410i	0.234283	−	0.143468i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.3923			0.428936			0.214468	−	0.976731i	$-0.431198\pi$
$587$	10.3923			0.428936			0.214468	+	0.976731i	$0.431198\pi$
$588$		0			0
$589$	−4.89898			−0.201859
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	9.79796i		−	0.402354i	−0.979555	−	0.201177i	$-0.935523\pi$
$593$		−	9.79796i		−	0.402354i	0.979555	−	0.201177i	$-0.0644766\pi$
$594$		0			0
$595$	13.8564			0.568057
$596$		0			0
$597$		−	18.0000i		−	0.736691i
$598$		0			0
$599$		−	45.0333i		−	1.84001i	−0.391905	−	0.920006i	$-0.628184\pi$
$599$		−	45.0333i		−	1.84001i	0.391905	−	0.920006i	$-0.371816\pi$
$600$		0			0
$601$		−	4.89898i		−	0.199834i	−0.994996	−	0.0999168i	$-0.968142\pi$
$601$		−	4.89898i		−	0.199834i	0.994996	−	0.0999168i	$-0.0318577\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.79796	−	5.00000i	−0.398344	−	0.203279i
$606$		0			0
$607$	22.6274			0.918419			0.459209	−	0.888328i	$-0.348133\pi$
$607$	22.6274			0.918419			0.459209	+	0.888328i	$0.348133\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	16.9706i		−	0.686555i
$612$		0			0
$613$	9.79796			0.395736			0.197868	−	0.980229i	$-0.436598\pi$
$613$	9.79796			0.395736			0.197868	+	0.980229i	$0.436598\pi$
$614$		0			0
$615$	−8.48528			−0.342160
$616$		0			0
$617$	−2.00000			−0.0805170			−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000			−0.0805170			−0.0402585	+	0.999189i	$0.512818\pi$
$618$		0			0
$619$	1.73205			0.0696170			0.0348085	−	0.999394i	$-0.488918\pi$
$619$	1.73205			0.0696170			0.0348085	+	0.999394i	$0.488918\pi$
$620$		0			0
$621$			27.0000i			1.08347i
$622$		0			0
$623$	2.82843			0.113319
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$	−13.8564	+	8.48528i	−0.553372	+	0.338869i
$628$		0			0
$629$	−14.6969			−0.586005
$630$		0			0
$631$		−	36.3731i		−	1.44799i	−0.689806	−	0.723994i	$-0.742304\pi$
$631$		−	36.3731i		−	1.44799i	0.689806	−	0.723994i	$-0.257696\pi$
$632$		0			0
$633$			9.79796i			0.389434i
$634$		0			0
$635$			14.1421i			0.561214i
$636$		0			0
$637$	−4.89898			−0.194105
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.0000			0.750455			0.375227	−	0.926933i	$-0.377565\pi$
$641$	19.0000			0.750455			0.375227	+	0.926933i	$0.377565\pi$
$642$		0			0
$643$	25.9808			1.02458			0.512291	−	0.858812i	$-0.328797\pi$
$643$	25.9808			1.02458			0.512291	+	0.858812i	$0.328797\pi$
$644$		0			0
$645$	9.79796			0.385794
$646$		0			0
$647$		−	1.73205i		−	0.0680939i	−0.999420	−	0.0340470i	$-0.989160\pi$
$647$		−	1.73205i		−	0.0680939i	0.999420	−	0.0340470i	$-0.0108396\pi$
$648$		0			0
$649$	−3.00000	−	4.89898i	−0.117760	−	0.192302i
$650$		0			0
$651$		−	8.48528i		−	0.332564i
$652$		0			0
$653$			31.0000i			1.21312i	0.795036	+	0.606562i	$0.207452\pi$
$653$			31.0000i			1.21312i	−0.795036	+	0.606562i	$0.792548\pi$
$654$		0			0
$655$	−2.82843			−0.110516
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	45.2548i		−	1.76288i	−0.472298	−	0.881439i	$-0.656575\pi$
$659$		−	45.2548i		−	1.76288i	0.472298	−	0.881439i	$-0.343425\pi$
$660$		0			0
$661$			17.0000i			0.661223i	0.943767	+	0.330612i	$0.107255\pi$
$661$			17.0000i			0.661223i	−0.943767	+	0.330612i	$0.892745\pi$
$662$		0			0
$663$		−	41.5692i		−	1.61441i
$664$		0			0
$665$	−8.00000			−0.310227
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	39.0000i		−	1.50783i
$670$		0			0
$671$	−16.9706	−	27.7128i	−0.655141	−	1.06984i
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	20.7846			0.800000
$676$		0			0
$677$	24.4949			0.941415			0.470708	−	0.882289i	$-0.343999\pi$
$677$	24.4949			0.941415			0.470708	+	0.882289i	$0.343999\pi$
$678$		0			0
$679$	19.7990			0.759815
$680$		0			0
$681$		−	14.6969i		−	0.563188i
$682$		0			0
$683$	−3.46410			−0.132550			−0.0662751	−	0.997801i	$-0.521111\pi$
$683$	−3.46410			−0.132550			−0.0662751	+	0.997801i	$0.521111\pi$
$684$		0			0
$685$		−	19.0000i		−	0.725953i
$686$		0			0
$687$		−	1.73205i		−	0.0660819i
$688$		0			0
$689$		−	9.79796i		−	0.373273i
$690$		0			0
$691$	−15.5885			−0.593013			−0.296506	−	0.955031i	$-0.595822\pi$
$691$	−15.5885			−0.593013			−0.296506	+	0.955031i	$0.595822\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−19.7990			−0.751018
$696$		0			0
$697$	−24.0000			−0.909065
$698$		0			0
$699$		−	33.9411i		−	1.28377i
$700$		0			0
$701$	−34.2929			−1.29522			−0.647612	−	0.761971i	$-0.724232\pi$
$701$	−34.2929			−1.29522			−0.647612	+	0.761971i	$0.724232\pi$
$702$		0			0
$703$	8.48528			0.320028
$704$		0			0
$705$	6.00000			0.225973
$706$		0			0
$707$	−13.8564			−0.521124
$708$		0			0
$709$			25.0000i			0.938895i	0.882960	+	0.469447i	$0.155547\pi$
$709$			25.0000i			0.938895i	−0.882960	+	0.469447i	$0.844453\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.00000			−0.337053
$714$		0			0
$715$	−13.8564	+	8.48528i	−0.518200	+	0.317332i
$716$		0			0
$717$	9.79796			0.365911
$718$		0			0
$719$			15.5885i			0.581351i	0.956822	+	0.290676i	$0.0938801\pi$
$719$			15.5885i			0.581351i	−0.956822	+	0.290676i	$0.906120\pi$
$720$		0			0
$721$		−	9.79796i		−	0.364895i
$722$		0			0
$723$		−	50.9117i		−	1.89343i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			25.9808i			0.963573i	0.876289	+	0.481787i	$0.160012\pi$
$727$			25.9808i			0.963573i	−0.876289	+	0.481787i	$0.839988\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	27.7128			1.02500
$732$		0			0
$733$	9.79796			0.361896			0.180948	−	0.983493i	$-0.442083\pi$
$733$	9.79796			0.361896			0.180948	+	0.983493i	$0.442083\pi$
$734$		0			0
$735$		−	1.73205i		−	0.0638877i
$736$		0			0
$737$	15.0000	+	24.4949i	0.552532	+	0.902281i
$738$		0			0
$739$			5.65685i			0.208091i	0.994573	+	0.104045i	$0.0331787\pi$
$739$			5.65685i			0.208091i	−0.994573	+	0.104045i	$0.966821\pi$
$740$		0			0
$741$			24.0000i			0.881662i
$742$		0			0
$743$	−33.9411			−1.24518			−0.622590	−	0.782549i	$-0.713919\pi$
$743$	−33.9411			−1.24518			−0.622590	+	0.782549i	$0.713919\pi$
$744$		0			0
$745$			14.6969i			0.538454i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			24.0000i			0.876941i
$750$		0			0
$751$		−	5.19615i		−	0.189610i	−0.995496	−	0.0948051i	$-0.969777\pi$
$751$		−	5.19615i		−	0.189610i	0.995496	−	0.0948051i	$-0.0302228\pi$
$752$		0			0
$753$	−39.0000			−1.42124
$754$		0			0
$755$		−	5.65685i		−	0.205874i
$756$		0			0
$757$		−	30.0000i		−	1.09037i	−0.838316	−	0.545184i	$-0.816460\pi$
$757$		−	30.0000i		−	1.09037i	0.838316	−	0.545184i	$-0.183540\pi$
$758$		0			0
$759$	−25.4558	+	15.5885i	−0.923989	+	0.565825i
$760$		0			0
$761$			44.0908i			1.59829i	0.601138	+	0.799145i	$0.294714\pi$
$761$			44.0908i			1.59829i	−0.601138	+	0.799145i	$0.705286\pi$
$762$		0			0
$763$	−27.7128			−1.00327
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.48528			−0.306386
$768$		0			0
$769$			4.89898i			0.176662i	0.996091	+	0.0883309i	$0.0281533\pi$
$769$			4.89898i			0.176662i	−0.996091	+	0.0883309i	$0.971847\pi$
$770$		0			0
$771$	38.1051			1.37232
$772$		0			0
$773$			10.0000i			0.359675i	0.983696	+	0.179838i	$0.0575572\pi$
$773$			10.0000i			0.359675i	−0.983696	+	0.179838i	$0.942443\pi$
$774$		0			0
$775$			6.92820i			0.248868i
$776$		0			0
$777$			14.6969i			0.527250i
$778$		0			0
$779$	13.8564			0.496457
$780$		0			0
$781$	34.2929	−	21.0000i	1.22709	−	0.751439i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11.0000			−0.392607
$786$		0			0
$787$			28.2843i			1.00823i	0.863638	+	0.504113i	$0.168180\pi$
$787$			28.2843i			1.00823i	−0.863638	+	0.504113i	$0.831820\pi$
$788$		0			0
$789$	−24.4949			−0.872041
$790$		0			0
$791$	−14.1421			−0.502836
$792$		0			0
$793$	−48.0000			−1.70453
$794$		0			0
$795$	3.46410			0.122859
$796$		0			0
$797$			7.00000i			0.247953i	0.992285	+	0.123976i	$0.0395647\pi$
$797$			7.00000i			0.247953i	−0.992285	+	0.123976i	$0.960435\pi$
$798$		0			0
$799$	16.9706			0.600375
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13.8564	−	8.48528i	0.488982	−	0.299439i
$804$		0			0
$805$	−14.6969			−0.517999
$806$		0			0
$807$			17.3205i			0.609711i
$808$		0			0
$809$		−	44.0908i		−	1.55015i	−0.631869	−	0.775075i	$-0.717712\pi$
$809$		−	44.0908i		−	1.55015i	0.631869	−	0.775075i	$-0.282288\pi$
$810$		0			0
$811$		−	39.5980i		−	1.39047i	−0.718781	−	0.695237i	$-0.755300\pi$
$811$		−	39.5980i		−	1.39047i	0.718781	−	0.695237i	$-0.244700\pi$
$812$		0			0
$813$	−4.89898			−0.171815
$814$		0			0
$815$			17.3205i			0.606711i
$816$		0			0
$817$	−16.0000			−0.559769
$818$		0			0
$819$		0			0
$820$		0			0
$821$	48.9898			1.70976			0.854878	−	0.518829i	$-0.173632\pi$
$821$	48.9898			1.70976			0.854878	+	0.518829i	$0.173632\pi$
$822$		0			0
$823$		−	8.66025i		−	0.301877i	−0.988543	−	0.150939i	$-0.951770\pi$
$823$		−	8.66025i		−	0.301877i	0.988543	−	0.150939i	$-0.0482296\pi$
$824$		0			0
$825$	12.0000	+	19.5959i	0.417786	+	0.682242i
$826$		0			0
$827$			25.4558i			0.885186i	0.896723	+	0.442593i	$0.145941\pi$
$827$			25.4558i			0.885186i	−0.896723	+	0.442593i	$0.854059\pi$
$828$		0			0
$829$		−	17.0000i		−	0.590434i	−0.955430	−	0.295217i	$-0.904608\pi$
$829$		−	17.0000i		−	0.590434i	0.955430	−	0.295217i	$-0.0953920\pi$
$830$		0			0
$831$	−16.9706			−0.588702
$832$		0			0
$833$		−	4.89898i		−	0.169740i
$834$		0			0
$835$			19.7990i			0.685172i
$836$		0			0
$837$			9.00000i			0.311086i
$838$		0			0
$839$			25.9808i			0.896956i	0.893794	+	0.448478i	$0.148034\pi$
$839$			25.9808i			0.896956i	−0.893794	+	0.448478i	$0.851966\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$			11.0000i			0.378412i
$846$		0			0
$847$	−14.1421	+	27.7128i	−0.485930	+	0.952224i
$848$		0			0
$849$		−	19.5959i		−	0.672530i
$850$		0			0
$851$	15.5885			0.534365
$852$		0			0
$853$	39.1918			1.34190			0.670951	−	0.741501i	$-0.265886\pi$
$853$	39.1918			1.34190			0.670951	+	0.741501i	$0.265886\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	29.3939i		−	1.00408i	−0.864846	−	0.502038i	$-0.832584\pi$
$857$		−	29.3939i		−	1.00408i	0.864846	−	0.502038i	$-0.167416\pi$
$858$		0			0
$859$	15.5885			0.531871			0.265936	−	0.963991i	$-0.414319\pi$
$859$	15.5885			0.531871			0.265936	+	0.963991i	$0.414319\pi$
$860$		0			0
$861$			24.0000i			0.817918i
$862$		0			0
$863$			31.1769i			1.06127i	0.847599	+	0.530637i	$0.178047\pi$
$863$			31.1769i			1.06127i	−0.847599	+	0.530637i	$0.821953\pi$
$864$		0			0
$865$			9.79796i			0.333141i
$866$		0			0
$867$	12.1244			0.411765
$868$		0			0
$869$	−19.5959	−	32.0000i	−0.664746	−	1.08553i
$870$		0			0
$871$	42.4264			1.43756
$872$		0			0
$873$		0			0
$874$		0			0
$875$			25.4558i			0.860565i
$876$		0			0
$877$	53.8888			1.81969			0.909847	−	0.414943i	$-0.136199\pi$
$877$	53.8888			1.81969			0.909847	+	0.414943i	$0.136199\pi$
$878$		0			0
$879$	−8.48528			−0.286201
$880$		0			0
$881$	−49.0000			−1.65085			−0.825426	−	0.564510i	$-0.809065\pi$
$881$	−49.0000			−1.65085			−0.825426	+	0.564510i	$0.809065\pi$
$882$		0			0
$883$	−24.2487			−0.816034			−0.408017	−	0.912974i	$-0.633780\pi$
$883$	−24.2487			−0.816034			−0.408017	+	0.912974i	$0.633780\pi$
$884$		0			0
$885$		−	3.00000i		−	0.100844i
$886$		0			0
$887$	16.9706			0.569816			0.284908	−	0.958555i	$-0.408037\pi$
$887$	16.9706			0.569816			0.284908	+	0.958555i	$0.408037\pi$
$888$		0			0
$889$	40.0000			1.34156
$890$		0			0
$891$	15.5885	+	25.4558i	0.522233	+	0.852803i
$892$		0			0
$893$	−9.79796			−0.327876
$894$		0			0
$895$		−	8.66025i		−	0.289480i
$896$		0			0
$897$			44.0908i			1.47215i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	9.79796			0.326417
$902$		0			0
$903$		−	27.7128i		−	0.922225i
$904$		0			0
$905$	−21.0000			−0.698064
$906$		0			0
$907$	−3.46410			−0.115024			−0.0575118	−	0.998345i	$-0.518317\pi$
$907$	−3.46410			−0.115024			−0.0575118	+	0.998345i	$0.518317\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	24.2487i		−	0.803396i	−0.915772	−	0.401698i	$-0.868420\pi$
$911$		−	24.2487i		−	0.803396i	0.915772	−	0.401698i	$-0.131580\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		−	16.9706i		−	0.561029i
$916$		0			0
$917$			8.00000i			0.264183i
$918$		0			0
$919$	−11.3137			−0.373205			−0.186602	−	0.982436i	$-0.559748\pi$
$919$	−11.3137			−0.373205			−0.186602	+	0.982436i	$0.559748\pi$
$920$		0			0
$921$		−	34.2929i		−	1.12999i
$922$		0			0
$923$		−	59.3970i		−	1.95508i
$924$		0			0
$925$		−	12.0000i		−	0.394558i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−22.0000			−0.721797			−0.360898	−	0.932605i	$-0.617530\pi$
$929$	−22.0000			−0.721797			−0.360898	+	0.932605i	$0.617530\pi$
$930$		0			0
$931$			2.82843i			0.0926980i
$932$		0			0
$933$		−	18.0000i		−	0.589294i
$934$		0			0
$935$	−8.48528	−	13.8564i	−0.277498	−	0.453153i
$936$		0			0
$937$		−	4.89898i		−	0.160043i	−0.996793	−	0.0800213i	$-0.974501\pi$
$937$		−	4.89898i		−	0.160043i	0.996793	−	0.0800213i	$-0.0254988\pi$
$938$		0			0
$939$	46.7654			1.52613
$940$		0			0
$941$	39.1918			1.27762			0.638809	−	0.769366i	$-0.279428\pi$
$941$	39.1918			1.27762			0.638809	+	0.769366i	$0.279428\pi$
$942$		0			0
$943$	25.4558			0.828956
$944$		0			0
$945$			14.6969i			0.478091i
$946$		0			0
$947$	−43.3013			−1.40710			−0.703551	−	0.710645i	$-0.748403\pi$
$947$	−43.3013			−1.40710			−0.703551	+	0.710645i	$0.748403\pi$
$948$		0			0
$949$		−	24.0000i		−	0.779073i
$950$		0			0
$951$			43.3013i			1.40414i
$952$		0			0
$953$			34.2929i			1.11085i	0.831565	+	0.555427i	$0.187445\pi$
$953$			34.2929i			1.11085i	−0.831565	+	0.555427i	$0.812555\pi$
$954$		0			0
$955$	5.19615			0.168144
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−53.7401			−1.73536
$960$		0			0
$961$	28.0000			0.903226
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.79796			−0.315407
$966$		0			0
$967$	45.2548			1.45530			0.727649	−	0.685950i	$-0.240613\pi$
$967$	45.2548			1.45530			0.727649	+	0.685950i	$0.240613\pi$
$968$		0			0
$969$	−24.0000			−0.770991
$970$		0			0
$971$	−32.9090			−1.05610			−0.528049	−	0.849214i	$-0.677076\pi$
$971$	−32.9090			−1.05610			−0.528049	+	0.849214i	$0.677076\pi$
$972$		0			0
$973$			56.0000i			1.79528i
$974$		0			0
$975$	33.9411			1.08699
$976$		0			0
$977$	19.0000			0.607864			0.303932	−	0.952694i	$-0.401700\pi$
$977$	19.0000			0.607864			0.303932	+	0.952694i	$0.401700\pi$
$978$		0			0
$979$	−1.73205	−	2.82843i	−0.0553566	−	0.0903969i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1.73205i		−	0.0552438i	−0.999618	−	0.0276219i	$-0.991207\pi$
$983$		−	1.73205i		−	0.0552438i	0.999618	−	0.0276219i	$-0.00879345\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	16.9706i		−	0.540179i
$988$		0			0
$989$	−29.3939			−0.934671
$990$		0			0
$991$		−	51.9615i		−	1.65061i	−0.564686	−	0.825306i	$-0.691003\pi$
$991$		−	51.9615i		−	1.65061i	0.564686	−	0.825306i	$-0.308997\pi$
$992$		0			0
$993$	9.00000			0.285606
$994$		0			0
$995$	−10.3923			−0.329458
$996$		0			0
$997$	−4.89898			−0.155152			−0.0775761	−	0.996986i	$-0.524718\pi$
$997$	−4.89898			−0.155152			−0.0775761	+	0.996986i	$0.524718\pi$
$998$		0			0
$999$		−	15.5885i		−	0.493197i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.g.b.1407.4		8
4.3	odd	2	inner	2816.2.g.b.1407.7		8
8.3	odd	2	inner	2816.2.g.b.1407.1		8
8.5	even	2	inner	2816.2.g.b.1407.6		8
11.10	odd	2	inner	2816.2.g.b.1407.3		8
16.3	odd	4		704.2.e.b.703.4		4
16.5	even	4		44.2.c.a.43.3	yes	4
16.11	odd	4		44.2.c.a.43.1	✓	4
16.13	even	4		704.2.e.b.703.1		4
44.43	even	2	inner	2816.2.g.b.1407.8		8
48.5	odd	4		396.2.h.b.307.2		4
48.11	even	4		396.2.h.b.307.4		4
88.21	odd	2	inner	2816.2.g.b.1407.5		8
88.43	even	2	inner	2816.2.g.b.1407.2		8
176.5	even	20		484.2.g.g.239.3		16
176.21	odd	4		44.2.c.a.43.2	yes	4
176.27	odd	20		484.2.g.g.239.4		16
176.37	even	20		484.2.g.g.215.4		16
176.43	even	4		44.2.c.a.43.4	yes	4
176.53	even	20		484.2.g.g.403.1		16
176.59	odd	20		484.2.g.g.215.3		16
176.69	even	20		484.2.g.g.475.2		16
176.75	odd	20		484.2.g.g.403.2		16
176.85	odd	20		484.2.g.g.475.3		16
176.91	odd	20		484.2.g.g.475.1		16
176.101	odd	20		484.2.g.g.403.4		16
176.107	even	20		484.2.g.g.475.4		16
176.109	odd	4		704.2.e.b.703.2		4
176.117	odd	20		484.2.g.g.215.1		16
176.123	even	20		484.2.g.g.403.3		16
176.131	even	4		704.2.e.b.703.3		4
176.139	even	20		484.2.g.g.215.2		16
176.149	odd	20		484.2.g.g.239.2		16
176.171	even	20		484.2.g.g.239.1		16
528.197	even	4		396.2.h.b.307.3		4
528.395	odd	4		396.2.h.b.307.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.2.c.a.43.1	✓	4	16.11	odd	4
44.2.c.a.43.2	yes	4	176.21	odd	4
44.2.c.a.43.3	yes	4	16.5	even	4
44.2.c.a.43.4	yes	4	176.43	even	4
396.2.h.b.307.1		4	528.395	odd	4
396.2.h.b.307.2		4	48.5	odd	4
396.2.h.b.307.3		4	528.197	even	4
396.2.h.b.307.4		4	48.11	even	4
484.2.g.g.215.1		16	176.117	odd	20
484.2.g.g.215.2		16	176.139	even	20
484.2.g.g.215.3		16	176.59	odd	20
484.2.g.g.215.4		16	176.37	even	20
484.2.g.g.239.1		16	176.171	even	20
484.2.g.g.239.2		16	176.149	odd	20
484.2.g.g.239.3		16	176.5	even	20
484.2.g.g.239.4		16	176.27	odd	20
484.2.g.g.403.1		16	176.53	even	20
484.2.g.g.403.2		16	176.75	odd	20
484.2.g.g.403.3		16	176.123	even	20
484.2.g.g.403.4		16	176.101	odd	20
484.2.g.g.475.1		16	176.91	odd	20
484.2.g.g.475.2		16	176.69	even	20
484.2.g.g.475.3		16	176.85	odd	20
484.2.g.g.475.4		16	176.107	even	20
704.2.e.b.703.1		4	16.13	even	4
704.2.e.b.703.2		4	176.109	odd	4
704.2.e.b.703.3		4	176.131	even	4
704.2.e.b.703.4		4	16.3	odd	4
2816.2.g.b.1407.1		8	8.3	odd	2	inner
2816.2.g.b.1407.2		8	88.43	even	2	inner
2816.2.g.b.1407.3		8	11.10	odd	2	inner
2816.2.g.b.1407.4		8	1.1	even	1	trivial
2816.2.g.b.1407.5		8	88.21	odd	2	inner
2816.2.g.b.1407.6		8	8.5	even	2	inner
2816.2.g.b.1407.7		8	4.3	odd	2	inner
2816.2.g.b.1407.8		8	44.43	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:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +1.00000i q^{5} +2.82843 q^{7} +O(q^{10})\)	\(q-1.73205 q^{3} +1.00000i q^{5} +2.82843 q^{7} +(-1.73205 - 2.82843i) q^{11} -4.89898 q^{13} -1.73205i q^{15} -4.89898i q^{17} +2.82843i q^{19} -4.89898 q^{21} +5.19615i q^{23} +4.00000 q^{25} +5.19615 q^{27} +1.73205i q^{31} +(3.00000 + 4.89898i) q^{33} +2.82843i q^{35} -3.00000i q^{37} +8.48528 q^{39} -4.89898i q^{41} +5.65685i q^{43} +3.46410i q^{47} +1.00000 q^{49} +8.48528i q^{51} +2.00000i q^{53} +(2.82843 - 1.73205i) q^{55} -4.89898i q^{57} +1.73205 q^{59} +9.79796 q^{61} -4.89898i q^{65} -8.66025 q^{67} -9.00000i q^{69} +12.1244i q^{71} +4.89898i q^{73} -6.92820 q^{75} +(-4.89898 - 8.00000i) q^{77} +11.3137 q^{79} -9.00000 q^{81} +4.89898 q^{85} +1.00000 q^{89} -13.8564 q^{91} -3.00000i q^{93} -2.82843 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 32 q^{25} + 24 q^{33} + 8 q^{49} - 72 q^{81} + 8 q^{89} + 56 q^{97}+O(q^{100}) \) 8 * q + 32 * q^25 + 24 * q^33 + 8 * q^49 - 72 * q^81 + 8 * q^89 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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