LMFDB - Embedded newform 624.2.c.c.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,2,Mod(337,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("624.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	624.c (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:	$4.98266508613$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	624.337
Dual form			624.2.c.c.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +2.00000i q^{11} +(3.00000 - 2.00000i) q^{13} +4.00000i q^{15} -6.00000 q^{17} +2.00000i q^{19} +2.00000i q^{21} -8.00000 q^{23} -11.0000 q^{25} -1.00000 q^{27} +6.00000 q^{29} -10.0000i q^{31} -2.00000i q^{33} -8.00000 q^{35} +4.00000i q^{37} +(-3.00000 + 2.00000i) q^{39} +4.00000 q^{43} -4.00000i q^{45} -2.00000i q^{47} +3.00000 q^{49} +6.00000 q^{51} -10.0000 q^{53} +8.00000 q^{55} -2.00000i q^{57} -14.0000i q^{59} -2.00000 q^{61} -2.00000i q^{63} +(-8.00000 - 12.0000i) q^{65} +2.00000i q^{67} +8.00000 q^{69} +6.00000i q^{71} +8.00000i q^{73} +11.0000 q^{75} +4.00000 q^{77} +1.00000 q^{81} -6.00000i q^{83} +24.0000i q^{85} -6.00000 q^{87} +(-4.00000 - 6.00000i) q^{91} +10.0000i q^{93} +8.00000 q^{95} +2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{9} + 6 q^{13} - 12 q^{17} - 16 q^{23} - 22 q^{25} - 2 q^{27} + 12 q^{29} - 16 q^{35} - 6 q^{39} + 8 q^{43} + 6 q^{49} + 12 q^{51} - 20 q^{53} + 16 q^{55} - 4 q^{61} - 16 q^{65} + 16 q^{69} + 22 q^{75} + 8 q^{77} + 2 q^{81} - 12 q^{87} - 8 q^{91} + 16 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 6 * q^13 - 12 * q^17 - 16 * q^23 - 22 * q^25 - 2 * q^27 + 12 * q^29 - 16 * q^35 - 6 * q^39 + 8 * q^43 + 6 * q^49 + 12 * q^51 - 20 * q^53 + 16 * q^55 - 4 * q^61 - 16 * q^65 + 16 * q^69 + 22 * q^75 + 8 * q^77 + 2 * q^81 - 12 * q^87 - 8 * q^91 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$		−	4.00000i		−	1.78885i	−0.447214	−	0.894427i	$-0.647584\pi$
$5$		−	4.00000i		−	1.78885i	0.447214	−	0.894427i	$-0.352416\pi$
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		0			0
$13$	3.00000	−	2.00000i	0.832050	−	0.554700i
$13$	3.00000	−	2.00000i	0.832050	−	0.554700i
$14$		0			0
$15$			4.00000i			1.03280i
$16$		0			0
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$			2.00000i			0.436436i
$22$		0			0
$23$	−8.00000			−1.66812			−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000			−1.66812			−0.834058	+	0.551677i	$0.813988\pi$
$24$		0			0
$25$	−11.0000			−2.20000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$		−	10.0000i		−	1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$		−	10.0000i		−	1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$		0			0
$33$		−	2.00000i		−	0.348155i
$34$		0			0
$35$	−8.00000			−1.35225
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$		0			0
$39$	−3.00000	+	2.00000i	−0.480384	+	0.320256i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$		−	4.00000i		−	0.596285i
$46$		0			0
$47$		−	2.00000i		−	0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$		−	2.00000i		−	0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	6.00000			0.840168
$52$		0			0
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$	8.00000			1.07872
$56$		0			0
$57$		−	2.00000i		−	0.264906i
$58$		0			0
$59$		−	14.0000i		−	1.82264i	−0.411693	−	0.911322i	$-0.635063\pi$
$59$		−	14.0000i		−	1.82264i	0.411693	−	0.911322i	$-0.364937\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		0			0
$63$		−	2.00000i		−	0.251976i
$64$		0			0
$65$	−8.00000	−	12.0000i	−0.992278	−	1.48842i
$66$		0			0
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$		0			0
$69$	8.00000			0.963087
$70$		0			0
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$		0			0
$73$			8.00000i			0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$			8.00000i			0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$		0			0
$75$	11.0000			1.27017
$76$		0			0
$77$	4.00000			0.455842
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$		0			0
$85$			24.0000i			2.60317i
$86$		0			0
$87$	−6.00000			−0.643268
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−4.00000	−	6.00000i	−0.419314	−	0.628971i
$92$		0			0
$93$			10.0000i			1.03695i
$94$		0			0
$95$	8.00000			0.820783
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$			2.00000i			0.201008i
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	−8.00000			−0.788263			−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000			−0.788263			−0.394132	+	0.919054i	$0.628955\pi$
$104$		0			0
$105$	8.00000			0.780720
$106$		0			0
$107$	4.00000			0.386695			0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000			0.386695			0.193347	+	0.981130i	$0.438066\pi$
$108$		0			0
$109$		−	4.00000i		−	0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$		−	4.00000i		−	0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$		0			0
$111$		−	4.00000i		−	0.379663i
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$			32.0000i			2.98402i
$116$		0			0
$117$	3.00000	−	2.00000i	0.277350	−	0.184900i
$118$		0			0
$119$			12.0000i			1.10004i
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$			24.0000i			2.14663i
$126$		0			0
$127$	16.0000			1.41977			0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000			1.41977			0.709885	+	0.704317i	$0.248747\pi$
$128$		0			0
$129$	−4.00000			−0.352180
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	4.00000			0.346844
$134$		0			0
$135$			4.00000i			0.344265i
$136$		0			0
$137$		−	16.0000i		−	1.36697i	−0.729964	−	0.683486i	$-0.760463\pi$
$137$		−	16.0000i		−	1.36697i	0.729964	−	0.683486i	$-0.239537\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$			2.00000i			0.168430i
$142$		0			0
$143$	4.00000	+	6.00000i	0.334497	+	0.501745i
$144$		0			0
$145$		−	24.0000i		−	1.99309i
$146$		0			0
$147$	−3.00000			−0.247436
$148$		0			0
$149$		−	20.0000i		−	1.63846i	−0.573462	−	0.819232i	$-0.694400\pi$
$149$		−	20.0000i		−	1.63846i	0.573462	−	0.819232i	$-0.305600\pi$
$150$		0			0
$151$		−	2.00000i		−	0.162758i	−0.996683	−	0.0813788i	$-0.974068\pi$
$151$		−	2.00000i		−	0.162758i	0.996683	−	0.0813788i	$-0.0259324\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$	−40.0000			−3.21288
$156$		0			0
$157$	22.0000			1.75579			0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000			1.75579			0.877896	+	0.478852i	$0.158947\pi$
$158$		0			0
$159$	10.0000			0.793052
$160$		0			0
$161$			16.0000i			1.26098i
$162$		0			0
$163$			2.00000i			0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$			2.00000i			0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$		0			0
$165$	−8.00000			−0.622799
$166$		0			0
$167$		−	10.0000i		−	0.773823i	−0.922117	−	0.386912i	$-0.873542\pi$
$167$		−	10.0000i		−	0.773823i	0.922117	−	0.386912i	$-0.126458\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$			2.00000i			0.152944i
$172$		0			0
$173$	14.0000			1.06440			0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000			1.06440			0.532200	+	0.846619i	$0.321365\pi$
$174$		0			0
$175$			22.0000i			1.66304i
$176$		0			0
$177$			14.0000i			1.05230i
$178$		0			0
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		0			0
$183$	2.00000			0.147844
$184$		0			0
$185$	16.0000			1.17634
$186$		0			0
$187$		−	12.0000i		−	0.877527i
$188$		0			0
$189$			2.00000i			0.145479i
$190$		0			0
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$		0			0
$193$		−	16.0000i		−	1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$		0			0
$195$	8.00000	+	12.0000i	0.572892	+	0.859338i
$196$		0			0
$197$			12.0000i			0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$			12.0000i			0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$		0			0
$199$	−24.0000			−1.70131			−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000			−1.70131			−0.850657	+	0.525720i	$0.823796\pi$
$200$		0			0
$201$		−	2.00000i		−	0.141069i
$202$		0			0
$203$		−	12.0000i		−	0.842235i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−8.00000			−0.556038
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		0			0
$213$		−	6.00000i		−	0.411113i
$214$		0			0
$215$		−	16.0000i		−	1.09119i
$216$		0			0
$217$	−20.0000			−1.35769
$218$		0			0
$219$		−	8.00000i		−	0.540590i
$220$		0			0
$221$	−18.0000	+	12.0000i	−1.21081	+	0.807207i
$222$		0			0
$223$		−	26.0000i		−	1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$		−	26.0000i		−	1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$		0			0
$225$	−11.0000			−0.733333
$226$		0			0
$227$			10.0000i			0.663723i	0.943328	+	0.331862i	$0.107677\pi$
$227$			10.0000i			0.663723i	−0.943328	+	0.331862i	$0.892323\pi$
$228$		0			0
$229$		−	28.0000i		−	1.85029i	−0.379611	−	0.925146i	$-0.623942\pi$
$229$		−	28.0000i		−	1.85029i	0.379611	−	0.925146i	$-0.376058\pi$
$230$		0			0
$231$	−4.00000			−0.263181
$232$		0			0
$233$	−22.0000			−1.44127			−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000			−1.44127			−0.720634	+	0.693316i	$0.756149\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$			30.0000i			1.94054i	0.242028	+	0.970269i	$0.422188\pi$
$239$			30.0000i			1.94054i	−0.242028	+	0.970269i	$0.577812\pi$
$240$		0			0
$241$			16.0000i			1.03065i	0.856995	+	0.515325i	$0.172329\pi$
$241$			16.0000i			1.03065i	−0.856995	+	0.515325i	$0.827671\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	12.0000i		−	0.766652i
$246$		0			0
$247$	4.00000	+	6.00000i	0.254514	+	0.381771i
$248$		0			0
$249$			6.00000i			0.380235i
$250$		0			0
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$		0			0
$253$		−	16.0000i		−	1.00591i
$254$		0			0
$255$		−	24.0000i		−	1.50294i
$256$		0			0
$257$	2.00000			0.124757			0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000			0.124757			0.0623783	+	0.998053i	$0.480131\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$			40.0000i			2.45718i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$		0			0
$271$			22.0000i			1.33640i	0.743980	+	0.668202i	$0.232936\pi$
$271$			22.0000i			1.33640i	−0.743980	+	0.668202i	$0.767064\pi$
$272$		0			0
$273$	4.00000	+	6.00000i	0.242091	+	0.363137i
$274$		0			0
$275$		−	22.0000i		−	1.32665i
$276$		0			0
$277$	−10.0000			−0.600842			−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000			−0.600842			−0.300421	+	0.953807i	$0.597127\pi$
$278$		0			0
$279$		−	10.0000i		−	0.598684i
$280$		0			0
$281$			16.0000i			0.954480i	0.878773	+	0.477240i	$0.158363\pi$
$281$			16.0000i			0.954480i	−0.878773	+	0.477240i	$0.841637\pi$
$282$		0			0
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$		0			0
$285$	−8.00000			−0.473879
$286$		0			0
$287$		0			0
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$			12.0000i			0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$			12.0000i			0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$		0			0
$295$	−56.0000			−3.26045
$296$		0			0
$297$		−	2.00000i		−	0.116052i
$298$		0			0
$299$	−24.0000	+	16.0000i	−1.38796	+	0.925304i
$300$		0			0
$301$		−	8.00000i		−	0.461112i
$302$		0			0
$303$	−6.00000			−0.344691
$304$		0			0
$305$			8.00000i			0.458079i
$306$		0			0
$307$		−	30.0000i		−	1.71219i	−0.516818	−	0.856095i	$-0.672884\pi$
$307$		−	30.0000i		−	1.71219i	0.516818	−	0.856095i	$-0.327116\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$	−14.0000			−0.791327			−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000			−0.791327			−0.395663	+	0.918396i	$0.629485\pi$
$314$		0			0
$315$	−8.00000			−0.450749
$316$		0			0
$317$		−	12.0000i		−	0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$		−	12.0000i		−	0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$		0			0
$319$			12.0000i			0.671871i
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$	−33.0000	+	22.0000i	−1.83051	+	1.22034i
$326$		0			0
$327$			4.00000i			0.221201i
$328$		0			0
$329$	−4.00000			−0.220527
$330$		0			0
$331$		−	6.00000i		−	0.329790i	−0.986311	−	0.164895i	$-0.947272\pi$
$331$		−	6.00000i		−	0.329790i	0.986311	−	0.164895i	$-0.0527285\pi$
$332$		0			0
$333$			4.00000i			0.219199i
$334$		0			0
$335$	8.00000			0.437087
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$	20.0000			1.08306
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$		−	32.0000i		−	1.72282i
$346$		0			0
$347$	4.00000			0.214731			0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000			0.214731			0.107366	+	0.994220i	$0.465758\pi$
$348$		0			0
$349$		−	20.0000i		−	1.07058i	−0.844670	−	0.535288i	$-0.820203\pi$
$349$		−	20.0000i		−	1.07058i	0.844670	−	0.535288i	$-0.179797\pi$
$350$		0			0
$351$	−3.00000	+	2.00000i	−0.160128	+	0.106752i
$352$		0			0
$353$		−	8.00000i		−	0.425797i	−0.977074	−	0.212899i	$-0.931710\pi$
$353$		−	8.00000i		−	0.425797i	0.977074	−	0.212899i	$-0.0682904\pi$
$354$		0			0
$355$	24.0000			1.27379
$356$		0			0
$357$		−	12.0000i		−	0.635107i
$358$		0			0
$359$		−	26.0000i		−	1.37223i	−0.727494	−	0.686114i	$-0.759315\pi$
$359$		−	26.0000i		−	1.37223i	0.727494	−	0.686114i	$-0.240685\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	−7.00000			−0.367405
$364$		0			0
$365$	32.0000			1.67496
$366$		0			0
$367$	−16.0000			−0.835193			−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000			−0.835193			−0.417597	+	0.908633i	$0.637127\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			20.0000i			1.03835i
$372$		0			0
$373$	−18.0000			−0.932005			−0.466002	−	0.884783i	$-0.654306\pi$
$373$	−18.0000			−0.932005			−0.466002	+	0.884783i	$0.654306\pi$
$374$		0			0
$375$		−	24.0000i		−	1.23935i
$376$		0			0
$377$	18.0000	−	12.0000i	0.927047	−	0.618031i
$378$		0			0
$379$			26.0000i			1.33553i	0.744372	+	0.667765i	$0.232749\pi$
$379$			26.0000i			1.33553i	−0.744372	+	0.667765i	$0.767251\pi$
$380$		0			0
$381$	−16.0000			−0.819705
$382$		0			0
$383$		−	2.00000i		−	0.102195i	−0.998694	−	0.0510976i	$-0.983728\pi$
$383$		−	2.00000i		−	0.102195i	0.998694	−	0.0510976i	$-0.0162720\pi$
$384$		0			0
$385$		−	16.0000i		−	0.815436i
$386$		0			0
$387$	4.00000			0.203331
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$	48.0000			2.42746
$392$		0			0
$393$	−12.0000			−0.605320
$394$		0			0
$395$		0			0
$396$		0			0
$397$			28.0000i			1.40528i	0.711546	+	0.702640i	$0.247995\pi$
$397$			28.0000i			1.40528i	−0.711546	+	0.702640i	$0.752005\pi$
$398$		0			0
$399$	−4.00000			−0.200250
$400$		0			0
$401$		−	8.00000i		−	0.399501i	−0.979847	−	0.199750i	$-0.935987\pi$
$401$		−	8.00000i		−	0.399501i	0.979847	−	0.199750i	$-0.0640132\pi$
$402$		0			0
$403$	−20.0000	−	30.0000i	−0.996271	−	1.49441i
$404$		0			0
$405$		−	4.00000i		−	0.198762i
$406$		0			0
$407$	−8.00000			−0.396545
$408$		0			0
$409$			8.00000i			0.395575i	0.980245	+	0.197787i	$0.0633755\pi$
$409$			8.00000i			0.395575i	−0.980245	+	0.197787i	$0.936624\pi$
$410$		0			0
$411$			16.0000i			0.789222i
$412$		0			0
$413$	−28.0000			−1.37779
$414$		0			0
$415$	−24.0000			−1.17811
$416$		0			0
$417$	−4.00000			−0.195881
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$			4.00000i			0.194948i	0.995238	+	0.0974740i	$0.0310763\pi$
$421$			4.00000i			0.194948i	−0.995238	+	0.0974740i	$0.968924\pi$
$422$		0			0
$423$		−	2.00000i		−	0.0972433i
$424$		0			0
$425$	66.0000			3.20147
$426$		0			0
$427$			4.00000i			0.193574i
$428$		0			0
$429$	−4.00000	−	6.00000i	−0.193122	−	0.289683i
$430$		0			0
$431$		−	2.00000i		−	0.0963366i	−0.998839	−	0.0481683i	$-0.984662\pi$
$431$		−	2.00000i		−	0.0963366i	0.998839	−	0.0481683i	$-0.0153384\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$			24.0000i			1.15071i
$436$		0			0
$437$		−	16.0000i		−	0.765384i
$438$		0			0
$439$	−24.0000			−1.14546			−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000			−1.14546			−0.572729	+	0.819745i	$0.694115\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$	−12.0000			−0.570137			−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000			−0.570137			−0.285069	+	0.958507i	$0.592016\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			20.0000i			0.945968i
$448$		0			0
$449$			8.00000i			0.377543i	0.982021	+	0.188772i	$0.0604506\pi$
$449$			8.00000i			0.377543i	−0.982021	+	0.188772i	$0.939549\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			2.00000i			0.0939682i
$454$		0			0
$455$	−24.0000	+	16.0000i	−1.12514	+	0.750092i
$456$		0			0
$457$		−	8.00000i		−	0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$		−	8.00000i		−	0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$		0			0
$459$	6.00000			0.280056
$460$		0			0
$461$		−	12.0000i		−	0.558896i	−0.960161	−	0.279448i	$-0.909849\pi$
$461$		−	12.0000i		−	0.558896i	0.960161	−	0.279448i	$-0.0901514\pi$
$462$		0			0
$463$			22.0000i			1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$			22.0000i			1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$		0			0
$465$	40.0000			1.85496
$466$		0			0
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$		0			0
$471$	−22.0000			−1.01371
$472$		0			0
$473$			8.00000i			0.367840i
$474$		0			0
$475$		−	22.0000i		−	1.00943i
$476$		0			0
$477$	−10.0000			−0.457869
$478$		0			0
$479$			14.0000i			0.639676i	0.947472	+	0.319838i	$0.103629\pi$
$479$			14.0000i			0.639676i	−0.947472	+	0.319838i	$0.896371\pi$
$480$		0			0
$481$	8.00000	+	12.0000i	0.364769	+	0.547153i
$482$		0			0
$483$		−	16.0000i		−	0.728025i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	18.0000i		−	0.815658i	−0.913058	−	0.407829i	$-0.866286\pi$
$487$		−	18.0000i		−	0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$		0			0
$489$		−	2.00000i		−	0.0904431i
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$	−36.0000			−1.62136
$494$		0			0
$495$	8.00000			0.359573
$496$		0			0
$497$	12.0000			0.538274
$498$		0			0
$499$			18.0000i			0.805791i	0.915246	+	0.402895i	$0.131996\pi$
$499$			18.0000i			0.805791i	−0.915246	+	0.402895i	$0.868004\pi$
$500$		0			0
$501$			10.0000i			0.446767i
$502$		0			0
$503$	8.00000			0.356702			0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000			0.356702			0.178351	+	0.983967i	$0.442924\pi$
$504$		0			0
$505$		−	24.0000i		−	1.06799i
$506$		0			0
$507$	−5.00000	+	12.0000i	−0.222058	+	0.532939i
$508$		0			0
$509$			20.0000i			0.886484i	0.896402	+	0.443242i	$0.146172\pi$
$509$			20.0000i			0.886484i	−0.896402	+	0.443242i	$0.853828\pi$
$510$		0			0
$511$	16.0000			0.707798
$512$		0			0
$513$		−	2.00000i		−	0.0883022i
$514$		0			0
$515$			32.0000i			1.41009i
$516$		0			0
$517$	4.00000			0.175920
$518$		0			0
$519$	−14.0000			−0.614532
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$		0			0
$525$		−	22.0000i		−	0.960159i
$526$		0			0
$527$			60.0000i			2.61364i
$528$		0			0
$529$	41.0000			1.78261
$530$		0			0
$531$		−	14.0000i		−	0.607548i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	16.0000i		−	0.691740i
$536$		0			0
$537$	4.00000			0.172613
$538$		0			0
$539$			6.00000i			0.258438i
$540$		0			0
$541$		−	20.0000i		−	0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$		0			0
$543$	−14.0000			−0.600798
$544$		0			0
$545$	−16.0000			−0.685365
$546$		0			0
$547$	44.0000			1.88130			0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000			1.88130			0.940652	+	0.339372i	$0.110215\pi$
$548$		0			0
$549$	−2.00000			−0.0853579
$550$		0			0
$551$			12.0000i			0.511217i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−16.0000			−0.679162
$556$		0			0
$557$			20.0000i			0.847427i	0.905796	+	0.423714i	$0.139274\pi$
$557$			20.0000i			0.847427i	−0.905796	+	0.423714i	$0.860726\pi$
$558$		0			0
$559$	12.0000	−	8.00000i	0.507546	−	0.338364i
$560$		0			0
$561$			12.0000i			0.506640i
$562$		0			0
$563$	−4.00000			−0.168580			−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000			−0.168580			−0.0842900	+	0.996441i	$0.526862\pi$
$564$		0			0
$565$			24.0000i			1.00969i
$566$		0			0
$567$		−	2.00000i		−	0.0839921i
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$	−16.0000			−0.668410
$574$		0			0
$575$	88.0000			3.66985
$576$		0			0
$577$			32.0000i			1.33218i	0.745873	+	0.666089i	$0.232033\pi$
$577$			32.0000i			1.33218i	−0.745873	+	0.666089i	$0.767967\pi$
$578$		0			0
$579$			16.0000i			0.664937i
$580$		0			0
$581$	−12.0000			−0.497844
$582$		0			0
$583$		−	20.0000i		−	0.828315i
$584$		0			0
$585$	−8.00000	−	12.0000i	−0.330759	−	0.496139i
$586$		0			0
$587$			18.0000i			0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$			18.0000i			0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$		0			0
$589$	20.0000			0.824086
$590$		0			0
$591$		−	12.0000i		−	0.493614i
$592$		0			0
$593$			8.00000i			0.328521i	0.986417	+	0.164260i	$0.0525237\pi$
$593$			8.00000i			0.328521i	−0.986417	+	0.164260i	$0.947476\pi$
$594$		0			0
$595$	48.0000			1.96781
$596$		0			0
$597$	24.0000			0.982255
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	−14.0000			−0.571072			−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000			−0.571072			−0.285536	+	0.958368i	$0.592172\pi$
$602$		0			0
$603$			2.00000i			0.0814463i
$604$		0			0
$605$		−	28.0000i		−	1.13836i
$606$		0			0
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$		0			0
$609$			12.0000i			0.486265i
$610$		0			0
$611$	−4.00000	−	6.00000i	−0.161823	−	0.242734i
$612$		0			0
$613$			4.00000i			0.161558i	0.996732	+	0.0807792i	$0.0257409\pi$
$613$			4.00000i			0.161558i	−0.996732	+	0.0807792i	$0.974259\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			48.0000i			1.93241i	0.257780	+	0.966204i	$0.417009\pi$
$617$			48.0000i			1.93241i	−0.257780	+	0.966204i	$0.582991\pi$
$618$		0			0
$619$			26.0000i			1.04503i	0.852631	+	0.522514i	$0.175006\pi$
$619$			26.0000i			1.04503i	−0.852631	+	0.522514i	$0.824994\pi$
$620$		0			0
$621$	8.00000			0.321029
$622$		0			0
$623$		0			0
$624$		0			0
$625$	41.0000			1.64000
$626$		0			0
$627$	4.00000			0.159745
$628$		0			0
$629$		−	24.0000i		−	0.956943i
$630$		0			0
$631$		−	34.0000i		−	1.35352i	−0.736204	−	0.676759i	$-0.763384\pi$
$631$		−	34.0000i		−	1.35352i	0.736204	−	0.676759i	$-0.236616\pi$
$632$		0			0
$633$	−12.0000			−0.476957
$634$		0			0
$635$		−	64.0000i		−	2.53976i
$636$		0			0
$637$	9.00000	−	6.00000i	0.356593	−	0.237729i
$638$		0			0
$639$			6.00000i			0.237356i
$640$		0			0
$641$	10.0000			0.394976			0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000			0.394976			0.197488	+	0.980305i	$0.436722\pi$
$642$		0			0
$643$		−	14.0000i		−	0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$		−	14.0000i		−	0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$		0			0
$645$			16.0000i			0.629999i
$646$		0			0
$647$	24.0000			0.943537			0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		0			0
$649$	28.0000			1.09910
$650$		0			0
$651$	20.0000			0.783862
$652$		0			0
$653$	22.0000			0.860927			0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000			0.860927			0.430463	+	0.902608i	$0.358350\pi$
$654$		0			0
$655$		−	48.0000i		−	1.87552i
$656$		0			0
$657$			8.00000i			0.312110i
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$			4.00000i			0.155582i	0.996970	+	0.0777910i	$0.0247867\pi$
$661$			4.00000i			0.155582i	−0.996970	+	0.0777910i	$0.975213\pi$
$662$		0			0
$663$	18.0000	−	12.0000i	0.699062	−	0.466041i
$664$		0			0
$665$		−	16.0000i		−	0.620453i
$666$		0			0
$667$	−48.0000			−1.85857
$668$		0			0
$669$			26.0000i			1.00522i
$670$		0			0
$671$		−	4.00000i		−	0.154418i
$672$		0			0
$673$	26.0000			1.00223			0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000			1.00223			0.501113	+	0.865382i	$0.332924\pi$
$674$		0			0
$675$	11.0000			0.423390
$676$		0			0
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	10.0000i		−	0.383201i
$682$		0			0
$683$		−	30.0000i		−	1.14792i	−0.818884	−	0.573959i	$-0.805407\pi$
$683$		−	30.0000i		−	1.14792i	0.818884	−	0.573959i	$-0.194593\pi$
$684$		0			0
$685$	−64.0000			−2.44531
$686$		0			0
$687$			28.0000i			1.06827i
$688$		0			0
$689$	−30.0000	+	20.0000i	−1.14291	+	0.761939i
$690$		0			0
$691$			50.0000i			1.90209i	0.309053	+	0.951045i	$0.399988\pi$
$691$			50.0000i			1.90209i	−0.309053	+	0.951045i	$0.600012\pi$
$692$		0			0
$693$	4.00000			0.151947
$694$		0			0
$695$		−	16.0000i		−	0.606915i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	22.0000			0.832116
$700$		0			0
$701$	38.0000			1.43524			0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000			1.43524			0.717620	+	0.696435i	$0.245231\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$		0			0
$705$	8.00000			0.301297
$706$		0			0
$707$		−	12.0000i		−	0.451306i
$708$		0			0
$709$		−	28.0000i		−	1.05156i	−0.850620	−	0.525781i	$-0.823773\pi$
$709$		−	28.0000i		−	1.05156i	0.850620	−	0.525781i	$-0.176227\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			80.0000i			2.99602i
$714$		0			0
$715$	24.0000	−	16.0000i	0.897549	−	0.598366i
$716$		0			0
$717$		−	30.0000i		−	1.12037i
$718$		0			0
$719$	−32.0000			−1.19340			−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000			−1.19340			−0.596699	+	0.802465i	$0.703521\pi$
$720$		0			0
$721$			16.0000i			0.595871i
$722$		0			0
$723$		−	16.0000i		−	0.595046i
$724$		0			0
$725$	−66.0000			−2.45118
$726$		0			0
$727$	24.0000			0.890111			0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000			0.890111			0.445055	+	0.895503i	$0.353184\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−24.0000			−0.887672
$732$		0			0
$733$		−	4.00000i		−	0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$		−	4.00000i		−	0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$		0			0
$735$			12.0000i			0.442627i
$736$		0			0
$737$	−4.00000			−0.147342
$738$		0			0
$739$			34.0000i			1.25071i	0.780340	+	0.625355i	$0.215046\pi$
$739$			34.0000i			1.25071i	−0.780340	+	0.625355i	$0.784954\pi$
$740$		0			0
$741$	−4.00000	−	6.00000i	−0.146944	−	0.220416i
$742$		0			0
$743$			38.0000i			1.39408i	0.717030	+	0.697042i	$0.245501\pi$
$743$			38.0000i			1.39408i	−0.717030	+	0.697042i	$0.754499\pi$
$744$		0			0
$745$	−80.0000			−2.93097
$746$		0			0
$747$		−	6.00000i		−	0.219529i
$748$		0			0
$749$		−	8.00000i		−	0.292314i
$750$		0			0
$751$	−16.0000			−0.583848			−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000			−0.583848			−0.291924	+	0.956441i	$0.594295\pi$
$752$		0			0
$753$	−4.00000			−0.145768
$754$		0			0
$755$	−8.00000			−0.291150
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$		0			0
$759$			16.0000i			0.580763i
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−8.00000			−0.289619
$764$		0			0
$765$			24.0000i			0.867722i
$766$		0			0
$767$	−28.0000	−	42.0000i	−1.01102	−	1.51653i
$768$		0			0
$769$			48.0000i			1.73092i	0.500974	+	0.865462i	$0.332975\pi$
$769$			48.0000i			1.73092i	−0.500974	+	0.865462i	$0.667025\pi$
$770$		0			0
$771$	−2.00000			−0.0720282
$772$		0			0
$773$		−	20.0000i		−	0.719350i	−0.933078	−	0.359675i	$-0.882888\pi$
$773$		−	20.0000i		−	0.719350i	0.933078	−	0.359675i	$-0.117112\pi$
$774$		0			0
$775$			110.000i			3.95132i
$776$		0			0
$777$	−8.00000			−0.286998
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−12.0000			−0.429394
$782$		0			0
$783$	−6.00000			−0.214423
$784$		0			0
$785$		−	88.0000i		−	3.14085i
$786$		0			0
$787$			2.00000i			0.0712923i	0.999364	+	0.0356462i	$0.0113489\pi$
$787$			2.00000i			0.0712923i	−0.999364	+	0.0356462i	$0.988651\pi$
$788$		0			0
$789$	−24.0000			−0.854423
$790$		0			0
$791$			12.0000i			0.426671i
$792$		0			0
$793$	−6.00000	+	4.00000i	−0.213066	+	0.142044i
$794$		0			0
$795$		−	40.0000i		−	1.41865i
$796$		0			0
$797$	38.0000			1.34603			0.673015	−	0.739629i	$-0.264999\pi$
$797$	38.0000			1.34603			0.673015	+	0.739629i	$0.264999\pi$
$798$		0			0
$799$			12.0000i			0.424529i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−16.0000			−0.564628
$804$		0			0
$805$	64.0000			2.25570
$806$		0			0
$807$	10.0000			0.352017
$808$		0			0
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$			26.0000i			0.912983i	0.889728	+	0.456492i	$0.150894\pi$
$811$			26.0000i			0.912983i	−0.889728	+	0.456492i	$0.849106\pi$
$812$		0			0
$813$		−	22.0000i		−	0.771574i
$814$		0			0
$815$	8.00000			0.280228
$816$		0			0
$817$			8.00000i			0.279885i
$818$		0			0
$819$	−4.00000	−	6.00000i	−0.139771	−	0.209657i
$820$		0			0
$821$			28.0000i			0.977207i	0.872506	+	0.488603i	$0.162493\pi$
$821$			28.0000i			0.977207i	−0.872506	+	0.488603i	$0.837507\pi$
$822$		0			0
$823$	−24.0000			−0.836587			−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000			−0.836587			−0.418294	+	0.908312i	$0.637372\pi$
$824$		0			0
$825$			22.0000i			0.765942i
$826$		0			0
$827$			2.00000i			0.0695468i	0.999395	+	0.0347734i	$0.0110710\pi$
$827$			2.00000i			0.0695468i	−0.999395	+	0.0347734i	$0.988929\pi$
$828$		0			0
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$	10.0000			0.346896
$832$		0			0
$833$	−18.0000			−0.623663
$834$		0			0
$835$	−40.0000			−1.38426
$836$		0			0
$837$			10.0000i			0.345651i
$838$		0			0
$839$		−	42.0000i		−	1.45000i	−0.688748	−	0.725001i	$-0.741839\pi$
$839$		−	42.0000i		−	1.45000i	0.688748	−	0.725001i	$-0.258161\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		−	16.0000i		−	0.551069i
$844$		0			0
$845$	−48.0000	−	20.0000i	−1.65125	−	0.688021i
$846$		0			0
$847$		−	14.0000i		−	0.481046i
$848$		0			0
$849$	−4.00000			−0.137280
$850$		0			0
$851$		−	32.0000i		−	1.09695i
$852$		0			0
$853$		−	28.0000i		−	0.958702i	−0.877623	−	0.479351i	$-0.840872\pi$
$853$		−	28.0000i		−	0.958702i	0.877623	−	0.479351i	$-0.159128\pi$
$854$		0			0
$855$	8.00000			0.273594
$856$		0			0
$857$	−38.0000			−1.29806			−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000			−1.29806			−0.649028	+	0.760765i	$0.724824\pi$
$858$		0			0
$859$	−28.0000			−0.955348			−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000			−0.955348			−0.477674	+	0.878537i	$0.658520\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	18.0000i		−	0.612727i	−0.951915	−	0.306364i	$-0.900888\pi$
$863$		−	18.0000i		−	0.612727i	0.951915	−	0.306364i	$-0.0991123\pi$
$864$		0			0
$865$		−	56.0000i		−	1.90406i
$866$		0			0
$867$	−19.0000			−0.645274
$868$		0			0
$869$		0			0
$870$		0			0
$871$	4.00000	+	6.00000i	0.135535	+	0.203302i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	48.0000			1.62270
$876$		0			0
$877$			28.0000i			0.945493i	0.881199	+	0.472746i	$0.156737\pi$
$877$			28.0000i			0.945493i	−0.881199	+	0.472746i	$0.843263\pi$
$878$		0			0
$879$		−	12.0000i		−	0.404750i
$880$		0			0
$881$	−14.0000			−0.471672			−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000			−0.471672			−0.235836	+	0.971793i	$0.575783\pi$
$882$		0			0
$883$	−52.0000			−1.74994			−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000			−1.74994			−0.874970	+	0.484178i	$0.839119\pi$
$884$		0			0
$885$	56.0000			1.88242
$886$		0			0
$887$	−40.0000			−1.34307			−0.671534	−	0.740973i	$-0.734364\pi$
$887$	−40.0000			−1.34307			−0.671534	+	0.740973i	$0.734364\pi$
$888$		0			0
$889$		−	32.0000i		−	1.07325i
$890$		0			0
$891$			2.00000i			0.0670025i
$892$		0			0
$893$	4.00000			0.133855
$894$		0			0
$895$			16.0000i			0.534821i
$896$		0			0
$897$	24.0000	−	16.0000i	0.801337	−	0.534224i
$898$		0			0
$899$		−	60.0000i		−	2.00111i
$900$		0			0
$901$	60.0000			1.99889
$902$		0			0
$903$			8.00000i			0.266223i
$904$		0			0
$905$		−	56.0000i		−	1.86150i
$906$		0			0
$907$	−12.0000			−0.398453			−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000			−0.398453			−0.199227	+	0.979953i	$0.563843\pi$
$908$		0			0
$909$	6.00000			0.199007
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	12.0000			0.397142
$914$		0			0
$915$		−	8.00000i		−	0.264472i
$916$		0			0
$917$		−	24.0000i		−	0.792550i
$918$		0			0
$919$	24.0000			0.791687			0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000			0.791687			0.395843	+	0.918318i	$0.370452\pi$
$920$		0			0
$921$			30.0000i			0.988534i
$922$		0			0
$923$	12.0000	+	18.0000i	0.394985	+	0.592477i
$924$		0			0
$925$		−	44.0000i		−	1.44671i
$926$		0			0
$927$	−8.00000			−0.262754
$928$		0			0
$929$		−	24.0000i		−	0.787414i	−0.919236	−	0.393707i	$-0.871192\pi$
$929$		−	24.0000i		−	0.787414i	0.919236	−	0.393707i	$-0.128808\pi$
$930$		0			0
$931$			6.00000i			0.196642i
$932$		0			0
$933$	−24.0000			−0.785725
$934$		0			0
$935$	−48.0000			−1.56977
$936$		0			0
$937$	10.0000			0.326686			0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000			0.326686			0.163343	+	0.986569i	$0.447772\pi$
$938$		0			0
$939$	14.0000			0.456873
$940$		0			0
$941$			20.0000i			0.651981i	0.945373	+	0.325991i	$0.105698\pi$
$941$			20.0000i			0.651981i	−0.945373	+	0.325991i	$0.894302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	8.00000			0.260240
$946$		0			0
$947$			42.0000i			1.36482i	0.730971	+	0.682408i	$0.239067\pi$
$947$			42.0000i			1.36482i	−0.730971	+	0.682408i	$0.760933\pi$
$948$		0			0
$949$	16.0000	+	24.0000i	0.519382	+	0.779073i
$950$		0			0
$951$			12.0000i			0.389127i
$952$		0			0
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$		−	64.0000i		−	2.07099i
$956$		0			0
$957$		−	12.0000i		−	0.387905i
$958$		0			0
$959$	−32.0000			−1.03333
$960$		0			0
$961$	−69.0000			−2.22581
$962$		0			0
$963$	4.00000			0.128898
$964$		0			0
$965$	−64.0000			−2.06023
$966$		0			0
$967$		−	34.0000i		−	1.09337i	−0.837340	−	0.546683i	$-0.815890\pi$
$967$		−	34.0000i		−	1.09337i	0.837340	−	0.546683i	$-0.184110\pi$
$968$		0			0
$969$			12.0000i			0.385496i
$970$		0			0
$971$	52.0000			1.66876			0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000			1.66876			0.834380	+	0.551190i	$0.185826\pi$
$972$		0			0
$973$		−	8.00000i		−	0.256468i
$974$		0			0
$975$	33.0000	−	22.0000i	1.05685	−	0.704564i
$976$		0			0
$977$		−	24.0000i		−	0.767828i	−0.923369	−	0.383914i	$-0.874576\pi$
$977$		−	24.0000i		−	0.767828i	0.923369	−	0.383914i	$-0.125424\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	4.00000i		−	0.127710i
$982$		0			0
$983$			22.0000i			0.701691i	0.936433	+	0.350846i	$0.114106\pi$
$983$			22.0000i			0.701691i	−0.936433	+	0.350846i	$0.885894\pi$
$984$		0			0
$985$	48.0000			1.52941
$986$		0			0
$987$	4.00000			0.127321
$988$		0			0
$989$	−32.0000			−1.01754
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$		0			0
$993$			6.00000i			0.190404i
$994$		0			0
$995$			96.0000i			3.04340i
$996$		0			0
$997$	6.00000			0.190022			0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000			0.190022			0.0950110	+	0.995476i	$0.469711\pi$
$998$		0			0
$999$		−	4.00000i		−	0.126554i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.2.c.c.337.1		2
3.2	odd	2		1872.2.c.i.1585.2		2
4.3	odd	2		312.2.c.c.25.1	✓	2
8.3	odd	2		2496.2.c.a.961.2		2
8.5	even	2		2496.2.c.h.961.2		2
12.11	even	2		936.2.c.c.649.2		2
13.5	odd	4		8112.2.a.a.1.1		1
13.8	odd	4		8112.2.a.p.1.1		1
13.12	even	2	inner	624.2.c.c.337.2		2
39.38	odd	2		1872.2.c.i.1585.1		2
52.31	even	4		4056.2.a.k.1.1		1
52.47	even	4		4056.2.a.r.1.1		1
52.51	odd	2		312.2.c.c.25.2	yes	2
104.51	odd	2		2496.2.c.a.961.1		2
104.77	even	2		2496.2.c.h.961.1		2
156.155	even	2		936.2.c.c.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.c.c.25.1	✓	2	4.3	odd	2
312.2.c.c.25.2	yes	2	52.51	odd	2
624.2.c.c.337.1		2	1.1	even	1	trivial
624.2.c.c.337.2		2	13.12	even	2	inner
936.2.c.c.649.1		2	156.155	even	2
936.2.c.c.649.2		2	12.11	even	2
1872.2.c.i.1585.1		2	39.38	odd	2
1872.2.c.i.1585.2		2	3.2	odd	2
2496.2.c.a.961.1		2	104.51	odd	2
2496.2.c.a.961.2		2	8.3	odd	2
2496.2.c.h.961.1		2	104.77	even	2
2496.2.c.h.961.2		2	8.5	even	2
4056.2.a.k.1.1		1	52.31	even	4
4056.2.a.r.1.1		1	52.47	even	4
8112.2.a.a.1.1		1	13.5	odd	4
8112.2.a.p.1.1		1	13.8	odd	4

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +2.00000i q^{11} +(3.00000 - 2.00000i) q^{13} +4.00000i q^{15} -6.00000 q^{17} +2.00000i q^{19} +2.00000i q^{21} -8.00000 q^{23} -11.0000 q^{25} -1.00000 q^{27} +6.00000 q^{29} -10.0000i q^{31} -2.00000i q^{33} -8.00000 q^{35} +4.00000i q^{37} +(-3.00000 + 2.00000i) q^{39} +4.00000 q^{43} -4.00000i q^{45} -2.00000i q^{47} +3.00000 q^{49} +6.00000 q^{51} -10.0000 q^{53} +8.00000 q^{55} -2.00000i q^{57} -14.0000i q^{59} -2.00000 q^{61} -2.00000i q^{63} +(-8.00000 - 12.0000i) q^{65} +2.00000i q^{67} +8.00000 q^{69} +6.00000i q^{71} +8.00000i q^{73} +11.0000 q^{75} +4.00000 q^{77} +1.00000 q^{81} -6.00000i q^{83} +24.0000i q^{85} -6.00000 q^{87} +(-4.00000 - 6.00000i) q^{91} +10.0000i q^{93} +8.00000 q^{95} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} + 6 q^{13} - 12 q^{17} - 16 q^{23} - 22 q^{25} - 2 q^{27} + 12 q^{29} - 16 q^{35} - 6 q^{39} + 8 q^{43} + 6 q^{49} + 12 q^{51} - 20 q^{53} + 16 q^{55} - 4 q^{61} - 16 q^{65} + 16 q^{69} + 22 q^{75} + 8 q^{77} + 2 q^{81} - 12 q^{87} - 8 q^{91} + 16 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 6 * q^13 - 12 * q^17 - 16 * q^23 - 22 * q^25 - 2 * q^27 + 12 * q^29 - 16 * q^35 - 6 * q^39 + 8 * q^43 + 6 * q^49 + 12 * q^51 - 20 * q^53 + 16 * q^55 - 4 * q^61 - 16 * q^65 + 16 * q^69 + 22 * q^75 + 8 * q^77 + 2 * q^81 - 12 * q^87 - 8 * q^91 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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