LMFDB - Embedded newform 6900.2.f.i.6349.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6900,2,Mod(6349,6900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6900.6349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-4,0,-4,0,0,0,0,0,0,0,4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6349.3
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	6900.6349
Dual form			6900.2.f.i.6349.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.46410i q^{7} -1.00000 q^{9} +4.19615 q^{11} +3.26795i q^{13} -7.73205i q^{17} -0.732051 q^{19} +2.46410 q^{21} +1.00000i q^{23} -1.00000i q^{27} -7.19615 q^{29} -1.00000 q^{31} +4.19615i q^{33} -11.3923i q^{37} -3.26795 q^{39} -7.73205 q^{41} -3.46410i q^{43} +0.732051i q^{47} +0.928203 q^{49} +7.73205 q^{51} -6.66025i q^{53} -0.732051i q^{57} +7.19615 q^{59} -10.7321 q^{61} +2.46410i q^{63} +5.00000i q^{67} -1.00000 q^{69} -14.1244 q^{71} +11.2679i q^{73} -10.3397i q^{77} -4.00000 q^{79} +1.00000 q^{81} +6.66025i q^{83} -7.19615i q^{87} +16.3923 q^{89} +8.05256 q^{91} -1.00000i q^{93} +4.00000i q^{97} -4.19615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} - 4 q^{11} + 4 q^{19} - 4 q^{21} - 8 q^{29} - 4 q^{31} - 20 q^{39} - 24 q^{41} - 24 q^{49} + 24 q^{51} + 8 q^{59} - 36 q^{61} - 4 q^{69} - 8 q^{71} - 16 q^{79} + 4 q^{81} + 24 q^{89} - 44 q^{91}+ \cdots + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 4 * q^11 + 4 * q^19 - 4 * q^21 - 8 * q^29 - 4 * q^31 - 20 * q^39 - 24 * q^41 - 24 * q^49 + 24 * q^51 + 8 * q^59 - 36 * q^61 - 4 * q^69 - 8 * q^71 - 16 * q^79 + 4 * q^81 + 24 * q^89 - 44 * q^91 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times$.

$n$	$277$	$1201$	$3451$	$4601$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.46410i	− 0.931343i	−0.884958	−	0.465671i	$-0.845813\pi$
$7$	− 2.46410i	− 0.931343i	0.884958	−	0.465671i	$-0.154187\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.19615	1.26519	0.632594	−	0.774484i	$-0.281990\pi$
$11$	4.19615	1.26519	0.632594	+	0.774484i	$0.281990\pi$
$12$	0	0
$13$	3.26795i	0.906366i	0.891417	+	0.453183i	$0.149712\pi$
$13$	3.26795i	0.906366i	−0.891417	+	0.453183i	$0.850288\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.73205i	− 1.87530i	−0.347584	−	0.937649i	$-0.612998\pi$
$17$	− 7.73205i	− 1.87530i	0.347584	−	0.937649i	$-0.387002\pi$
$18$	0	0
$19$	−0.732051	−0.167944	−0.0839720	−	0.996468i	$-0.526761\pi$
$19$	−0.732051	−0.167944	−0.0839720	+	0.996468i	$0.526761\pi$
$20$	0	0
$21$	2.46410	0.537711
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−7.19615	−1.33629	−0.668146	−	0.744030i	$-0.732912\pi$
$29$	−7.19615	−1.33629	−0.668146	+	0.744030i	$0.732912\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	4.19615i	0.730456i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 11.3923i	− 1.87288i	−0.350823	−	0.936442i	$-0.614098\pi$
$37$	− 11.3923i	− 1.87288i	0.350823	−	0.936442i	$-0.385902\pi$
$38$	0	0
$39$	−3.26795	−0.523291
$40$	0	0
$41$	−7.73205	−1.20754	−0.603772	−	0.797157i	$-0.706336\pi$
$41$	−7.73205	−1.20754	−0.603772	+	0.797157i	$0.706336\pi$
$42$	0	0
$43$	− 3.46410i	− 0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$	− 3.46410i	− 0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.732051i	0.106781i	0.998574	+	0.0533903i	$0.0170027\pi$
$47$	0.732051i	0.106781i	−0.998574	+	0.0533903i	$0.982997\pi$
$48$	0	0
$49$	0.928203	0.132600
$50$	0	0
$51$	7.73205	1.08270
$52$	0	0
$53$	− 6.66025i	− 0.914856i	−0.889247	−	0.457428i	$-0.848771\pi$
$53$	− 6.66025i	− 0.914856i	0.889247	−	0.457428i	$-0.151229\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 0.732051i	− 0.0969625i
$58$	0	0
$59$	7.19615	0.936859	0.468430	−	0.883501i	$-0.344820\pi$
$59$	7.19615	0.936859	0.468430	+	0.883501i	$0.344820\pi$
$60$	0	0
$61$	−10.7321	−1.37410	−0.687049	−	0.726611i	$-0.741094\pi$
$61$	−10.7321	−1.37410	−0.687049	+	0.726611i	$0.741094\pi$
$62$	0	0
$63$	2.46410i	0.310448i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.00000i	0.610847i	0.952217	+	0.305424i	$0.0987981\pi$
$67$	5.00000i	0.610847i	−0.952217	+	0.305424i	$0.901202\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−14.1244	−1.67625	−0.838126	−	0.545476i	$-0.816349\pi$
$71$	−14.1244	−1.67625	−0.838126	+	0.545476i	$0.816349\pi$
$72$	0	0
$73$	11.2679i	1.31881i	0.751786	+	0.659407i	$0.229193\pi$
$73$	11.2679i	1.31881i	−0.751786	+	0.659407i	$0.770807\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.3397i	− 1.17832i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.66025i	0.731058i	0.930800	+	0.365529i	$0.119112\pi$
$83$	6.66025i	0.731058i	−0.930800	+	0.365529i	$0.880888\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 7.19615i	− 0.771509i
$88$	0	0
$89$	16.3923	1.73758	0.868790	−	0.495180i	$-0.164898\pi$
$89$	16.3923	1.73758	0.868790	+	0.495180i	$0.164898\pi$
$90$	0	0
$91$	8.05256	0.844138
$92$	0	0
$93$	− 1.00000i	− 0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.00000i	0.406138i	0.979164	+	0.203069i	$0.0650917\pi$
$97$	4.00000i	0.406138i	−0.979164	+	0.203069i	$0.934908\pi$
$98$	0	0
$99$	−4.19615	−0.421729
$100$	0	0
$101$	−2.80385	−0.278993	−0.139497	−	0.990223i	$-0.544548\pi$
$101$	−2.80385	−0.278993	−0.139497	+	0.990223i	$0.544548\pi$
$102$	0	0
$103$	− 16.9282i	− 1.66799i	−0.551775	−	0.833993i	$-0.686049\pi$
$103$	− 16.9282i	− 1.66799i	0.551775	−	0.833993i	$-0.313951\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.19615i	0.115636i	0.998327	+	0.0578182i	$0.0184144\pi$
$107$	1.19615i	0.115636i	−0.998327	+	0.0578182i	$0.981586\pi$
$108$	0	0
$109$	−3.66025	−0.350589	−0.175294	−	0.984516i	$-0.556088\pi$
$109$	−3.66025	−0.350589	−0.175294	+	0.984516i	$0.556088\pi$
$110$	0	0
$111$	11.3923	1.08131
$112$	0	0
$113$	15.1962i	1.42953i	0.699363	+	0.714767i	$0.253467\pi$
$113$	15.1962i	1.42953i	−0.699363	+	0.714767i	$0.746533\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.26795i	− 0.302122i
$118$	0	0
$119$	−19.0526	−1.74655
$120$	0	0
$121$	6.60770	0.600700
$122$	0	0
$123$	− 7.73205i	− 0.697176i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.1244i	0.987127i	0.869710	+	0.493563i	$0.164306\pi$
$127$	11.1244i	0.987127i	−0.869710	+	0.493563i	$0.835694\pi$
$128$	0	0
$129$	3.46410	0.304997
$130$	0	0
$131$	−21.8564	−1.90960	−0.954802	−	0.297244i	$-0.903933\pi$
$131$	−21.8564	−1.90960	−0.954802	+	0.297244i	$0.903933\pi$
$132$	0	0
$133$	1.80385i	0.156413i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0	0
$141$	−0.732051	−0.0616498
$142$	0	0
$143$	13.7128i	1.14672i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.928203i	0.0765569i
$148$	0	0
$149$	−5.80385	−0.475470	−0.237735	−	0.971330i	$-0.576405\pi$
$149$	−5.80385	−0.475470	−0.237735	+	0.971330i	$0.576405\pi$
$150$	0	0
$151$	−2.39230	−0.194683	−0.0973415	−	0.995251i	$-0.531034\pi$
$151$	−2.39230	−0.194683	−0.0973415	+	0.995251i	$0.531034\pi$
$152$	0	0
$153$	7.73205i	0.625099i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.3205i	0.983284i	0.870798	+	0.491642i	$0.163603\pi$
$157$	12.3205i	0.983284i	−0.870798	+	0.491642i	$0.836397\pi$
$158$	0	0
$159$	6.66025	0.528193
$160$	0	0
$161$	2.46410	0.194198
$162$	0	0
$163$	3.85641i	0.302057i	0.988529	+	0.151029i	$0.0482585\pi$
$163$	3.85641i	0.302057i	−0.988529	+	0.151029i	$0.951741\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.80385i	− 0.449115i	−0.974461	−	0.224558i	$-0.927906\pi$
$167$	− 5.80385i	− 0.449115i	0.974461	−	0.224558i	$-0.0720937\pi$
$168$	0	0
$169$	2.32051	0.178501
$170$	0	0
$171$	0.732051	0.0559813
$172$	0	0
$173$	− 17.8564i	− 1.35760i	−0.734324	−	0.678799i	$-0.762501\pi$
$173$	− 17.8564i	− 1.35760i	0.734324	−	0.678799i	$-0.237499\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.19615i	0.540896i
$178$	0	0
$179$	−11.4641	−0.856867	−0.428434	−	0.903573i	$-0.640934\pi$
$179$	−11.4641	−0.856867	−0.428434	+	0.903573i	$0.640934\pi$
$180$	0	0
$181$	−12.3923	−0.921113	−0.460556	−	0.887630i	$-0.652350\pi$
$181$	−12.3923	−0.921113	−0.460556	+	0.887630i	$0.652350\pi$
$182$	0	0
$183$	− 10.7321i	− 0.793336i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 32.4449i	− 2.37260i
$188$	0	0
$189$	−2.46410	−0.179237
$190$	0	0
$191$	5.80385	0.419952	0.209976	−	0.977707i	$-0.432661\pi$
$191$	5.80385	0.419952	0.209976	+	0.977707i	$0.432661\pi$
$192$	0	0
$193$	− 24.3923i	− 1.75580i	−0.478847	−	0.877898i	$-0.658945\pi$
$193$	− 24.3923i	− 1.75580i	0.478847	−	0.877898i	$-0.341055\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.46410i	− 0.531795i	−0.964001	−	0.265898i	$-0.914332\pi$
$197$	− 7.46410i	− 0.531795i	0.964001	−	0.265898i	$-0.0856683\pi$
$198$	0	0
$199$	4.53590	0.321541	0.160771	−	0.986992i	$-0.448602\pi$
$199$	4.53590	0.321541	0.160771	+	0.986992i	$0.448602\pi$
$200$	0	0
$201$	−5.00000	−0.352673
$202$	0	0
$203$	17.7321i	1.24455i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.00000i	− 0.0695048i
$208$	0	0
$209$	−3.07180	−0.212481
$210$	0	0
$211$	−17.9282	−1.23423	−0.617114	−	0.786874i	$-0.711698\pi$
$211$	−17.9282	−1.23423	−0.617114	+	0.786874i	$0.711698\pi$
$212$	0	0
$213$	− 14.1244i	− 0.967785i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.46410i	0.167274i
$218$	0	0
$219$	−11.2679	−0.761417
$220$	0	0
$221$	25.2679	1.69971
$222$	0	0
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 22.9282i	− 1.52180i	−0.648870	−	0.760899i	$-0.724758\pi$
$227$	− 22.9282i	− 1.52180i	0.648870	−	0.760899i	$-0.275242\pi$
$228$	0	0
$229$	−17.4641	−1.15406	−0.577030	−	0.816723i	$-0.695788\pi$
$229$	−17.4641	−1.15406	−0.577030	+	0.816723i	$0.695788\pi$
$230$	0	0
$231$	10.3397	0.680305
$232$	0	0
$233$	− 20.7846i	− 1.36165i	−0.732448	−	0.680823i	$-0.761622\pi$
$233$	− 20.7846i	− 1.36165i	0.732448	−	0.680823i	$-0.238378\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4.00000i	− 0.259828i
$238$	0	0
$239$	−2.12436	−0.137413	−0.0687066	−	0.997637i	$-0.521887\pi$
$239$	−2.12436	−0.137413	−0.0687066	+	0.997637i	$0.521887\pi$
$240$	0	0
$241$	−14.5885	−0.939725	−0.469863	−	0.882740i	$-0.655697\pi$
$241$	−14.5885	−0.939725	−0.469863	+	0.882740i	$0.655697\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.39230i	− 0.152219i
$248$	0	0
$249$	−6.66025	−0.422076
$250$	0	0
$251$	23.8564	1.50580	0.752902	−	0.658133i	$-0.228654\pi$
$251$	23.8564	1.50580	0.752902	+	0.658133i	$0.228654\pi$
$252$	0	0
$253$	4.19615i	0.263810i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.12436i	0.569162i	0.958652	+	0.284581i	$0.0918544\pi$
$257$	9.12436i	0.569162i	−0.958652	+	0.284581i	$0.908146\pi$
$258$	0	0
$259$	−28.0718	−1.74430
$260$	0	0
$261$	7.19615	0.445431
$262$	0	0
$263$	11.3397i	0.699239i	0.936892	+	0.349619i	$0.113689\pi$
$263$	11.3397i	0.699239i	−0.936892	+	0.349619i	$0.886311\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	16.3923i	1.00319i
$268$	0	0
$269$	1.73205	0.105605	0.0528025	−	0.998605i	$-0.483185\pi$
$269$	1.73205	0.105605	0.0528025	+	0.998605i	$0.483185\pi$
$270$	0	0
$271$	23.3923	1.42098	0.710491	−	0.703707i	$-0.248473\pi$
$271$	23.3923	1.42098	0.710491	+	0.703707i	$0.248473\pi$
$272$	0	0
$273$	8.05256i	0.487363i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 4.14359i	− 0.248964i	−0.992222	−	0.124482i	$-0.960273\pi$
$277$	− 4.14359i	− 0.248964i	0.992222	−	0.124482i	$-0.0397270\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	15.5167	0.925646	0.462823	−	0.886451i	$-0.346836\pi$
$281$	15.5167	0.925646	0.462823	+	0.886451i	$0.346836\pi$
$282$	0	0
$283$	− 30.8564i	− 1.83422i	−0.398631	−	0.917111i	$-0.630515\pi$
$283$	− 30.8564i	− 1.83422i	0.398631	−	0.917111i	$-0.369485\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	19.0526i	1.12464i
$288$	0	0
$289$	−42.7846	−2.51674
$290$	0	0
$291$	−4.00000	−0.234484
$292$	0	0
$293$	24.6603i	1.44067i	0.693628	+	0.720334i	$0.256011\pi$
$293$	24.6603i	1.44067i	−0.693628	+	0.720334i	$0.743989\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.19615i	− 0.243485i
$298$	0	0
$299$	−3.26795	−0.188990
$300$	0	0
$301$	−8.53590	−0.492001
$302$	0	0
$303$	− 2.80385i	− 0.161077i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.66025i	0.208902i	0.994530	+	0.104451i	$0.0333085\pi$
$307$	3.66025i	0.208902i	−0.994530	+	0.104451i	$0.966692\pi$
$308$	0	0
$309$	16.9282	0.963012
$310$	0	0
$311$	22.9282	1.30014	0.650070	−	0.759875i	$-0.274740\pi$
$311$	22.9282	1.30014	0.650070	+	0.759875i	$0.274740\pi$
$312$	0	0
$313$	− 22.3205i	− 1.26163i	−0.775933	−	0.630815i	$-0.782721\pi$
$313$	− 22.3205i	− 1.26163i	0.775933	−	0.630815i	$-0.217279\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 13.5167i	− 0.759171i	−0.925157	−	0.379586i	$-0.876067\pi$
$317$	− 13.5167i	− 0.759171i	0.925157	−	0.379586i	$-0.123933\pi$
$318$	0	0
$319$	−30.1962	−1.69066
$320$	0	0
$321$	−1.19615	−0.0667627
$322$	0	0
$323$	5.66025i	0.314945i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.66025i	− 0.202413i
$328$	0	0
$329$	1.80385	0.0994493
$330$	0	0
$331$	−27.2487	−1.49772	−0.748862	−	0.662726i	$-0.769400\pi$
$331$	−27.2487	−1.49772	−0.748862	+	0.662726i	$0.769400\pi$
$332$	0	0
$333$	11.3923i	0.624294i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.0718i	0.821013i	0.911858	+	0.410507i	$0.134648\pi$
$337$	15.0718i	0.821013i	−0.911858	+	0.410507i	$0.865352\pi$
$338$	0	0
$339$	−15.1962	−0.825342
$340$	0	0
$341$	−4.19615	−0.227234
$342$	0	0
$343$	− 19.5359i	− 1.05484i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.60770i	0.408402i	0.978929	+	0.204201i	$0.0654597\pi$
$347$	7.60770i	0.408402i	−0.978929	+	0.204201i	$0.934540\pi$
$348$	0	0
$349$	−0.464102	−0.0248428	−0.0124214	−	0.999923i	$-0.503954\pi$
$349$	−0.464102	−0.0248428	−0.0124214	+	0.999923i	$0.503954\pi$
$350$	0	0
$351$	3.26795	0.174430
$352$	0	0
$353$	− 1.41154i	− 0.0751288i	−0.999294	−	0.0375644i	$-0.988040\pi$
$353$	− 1.41154i	− 0.0751288i	0.999294	−	0.0375644i	$-0.0119599\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 19.0526i	− 1.00837i
$358$	0	0
$359$	8.73205	0.460860	0.230430	−	0.973089i	$-0.425987\pi$
$359$	8.73205	0.460860	0.230430	+	0.973089i	$0.425987\pi$
$360$	0	0
$361$	−18.4641	−0.971795
$362$	0	0
$363$	6.60770i	0.346814i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 30.4641i	− 1.59021i	−0.606470	−	0.795107i	$-0.707415\pi$
$367$	− 30.4641i	− 1.59021i	0.606470	−	0.795107i	$-0.292585\pi$
$368$	0	0
$369$	7.73205	0.402514
$370$	0	0
$371$	−16.4115	−0.852045
$372$	0	0
$373$	9.46410i	0.490033i	0.969519	+	0.245016i	$0.0787933\pi$
$373$	9.46410i	0.490033i	−0.969519	+	0.245016i	$0.921207\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 23.5167i	− 1.21117i
$378$	0	0
$379$	−9.85641	−0.506290	−0.253145	−	0.967428i	$-0.581465\pi$
$379$	−9.85641	−0.506290	−0.253145	+	0.967428i	$0.581465\pi$
$380$	0	0
$381$	−11.1244	−0.569918
$382$	0	0
$383$	− 31.0526i	− 1.58671i	−0.608758	−	0.793356i	$-0.708332\pi$
$383$	− 31.0526i	− 1.58671i	0.608758	−	0.793356i	$-0.291668\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.46410i	0.176090i
$388$	0	0
$389$	−12.5359	−0.635595	−0.317798	−	0.948159i	$-0.602943\pi$
$389$	−12.5359	−0.635595	−0.317798	+	0.948159i	$0.602943\pi$
$390$	0	0
$391$	7.73205	0.391027
$392$	0	0
$393$	− 21.8564i	− 1.10251i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	32.0000i	1.60603i	0.595956	+	0.803017i	$0.296773\pi$
$397$	32.0000i	1.60603i	−0.595956	+	0.803017i	$0.703227\pi$
$398$	0	0
$399$	−1.80385	−0.0903053
$400$	0	0
$401$	30.3923	1.51772	0.758860	−	0.651254i	$-0.225757\pi$
$401$	30.3923	1.51772	0.758860	+	0.651254i	$0.225757\pi$
$402$	0	0
$403$	− 3.26795i	− 0.162788i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 47.8038i	− 2.36955i
$408$	0	0
$409$	32.1769	1.59105	0.795523	−	0.605923i	$-0.207196\pi$
$409$	32.1769	1.59105	0.795523	+	0.605923i	$0.207196\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	− 17.7321i	− 0.872537i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 7.00000i	− 0.342791i
$418$	0	0
$419$	3.12436	0.152635	0.0763174	−	0.997084i	$-0.475684\pi$
$419$	3.12436	0.152635	0.0763174	+	0.997084i	$0.475684\pi$
$420$	0	0
$421$	13.5167	0.658762	0.329381	−	0.944197i	$-0.393160\pi$
$421$	13.5167	0.658762	0.329381	+	0.944197i	$0.393160\pi$
$422$	0	0
$423$	− 0.732051i	− 0.0355935i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	26.4449i	1.27976i
$428$	0	0
$429$	−13.7128	−0.662061
$430$	0	0
$431$	3.60770	0.173777	0.0868883	−	0.996218i	$-0.472308\pi$
$431$	3.60770	0.173777	0.0868883	+	0.996218i	$0.472308\pi$
$432$	0	0
$433$	29.3923i	1.41250i	0.707961	+	0.706252i	$0.249615\pi$
$433$	29.3923i	1.41250i	−0.707961	+	0.706252i	$0.750385\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 0.732051i	− 0.0350187i
$438$	0	0
$439$	−18.7846	−0.896541	−0.448270	−	0.893898i	$-0.647960\pi$
$439$	−18.7846	−0.896541	−0.448270	+	0.893898i	$0.647960\pi$
$440$	0	0
$441$	−0.928203	−0.0442002
$442$	0	0
$443$	1.80385i	0.0857034i	0.999081	+	0.0428517i	$0.0136443\pi$
$443$	1.80385i	0.0857034i	−0.999081	+	0.0428517i	$0.986356\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 5.80385i	− 0.274513i
$448$	0	0
$449$	−4.26795	−0.201417	−0.100708	−	0.994916i	$-0.532111\pi$
$449$	−4.26795	−0.201417	−0.100708	+	0.994916i	$0.532111\pi$
$450$	0	0
$451$	−32.4449	−1.52777
$452$	0	0
$453$	− 2.39230i	− 0.112400i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 26.4641i	− 1.23794i	−0.785415	−	0.618969i	$-0.787551\pi$
$457$	− 26.4641i	− 1.23794i	0.785415	−	0.618969i	$-0.212449\pi$
$458$	0	0
$459$	−7.73205	−0.360901
$460$	0	0
$461$	−29.3205	−1.36559	−0.682796	−	0.730609i	$-0.739236\pi$
$461$	−29.3205	−1.36559	−0.682796	+	0.730609i	$0.739236\pi$
$462$	0	0
$463$	10.5885i	0.492087i	0.969259	+	0.246044i	$0.0791307\pi$
$463$	10.5885i	0.492087i	−0.969259	+	0.246044i	$0.920869\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 29.4449i	− 1.36255i	−0.732030	−	0.681273i	$-0.761427\pi$
$467$	− 29.4449i	− 1.36255i	0.732030	−	0.681273i	$-0.238573\pi$
$468$	0	0
$469$	12.3205	0.568908
$470$	0	0
$471$	−12.3205	−0.567699
$472$	0	0
$473$	− 14.5359i	− 0.668361i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.66025i	0.304952i
$478$	0	0
$479$	34.9808	1.59831	0.799156	−	0.601124i	$-0.205280\pi$
$479$	34.9808	1.59831	0.799156	+	0.601124i	$0.205280\pi$
$480$	0	0
$481$	37.2295	1.69752
$482$	0	0
$483$	2.46410i	0.112121i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 2.73205i	− 0.123801i	−0.998082	−	0.0619005i	$-0.980284\pi$
$487$	− 2.73205i	− 0.123801i	0.998082	−	0.0619005i	$-0.0197162\pi$
$488$	0	0
$489$	−3.85641	−0.174393
$490$	0	0
$491$	19.0526	0.859830	0.429915	−	0.902869i	$-0.358544\pi$
$491$	19.0526	0.859830	0.429915	+	0.902869i	$0.358544\pi$
$492$	0	0
$493$	55.6410i	2.50595i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	34.8038i	1.56117i
$498$	0	0
$499$	28.4641	1.27423	0.637114	−	0.770770i	$-0.280128\pi$
$499$	28.4641	1.27423	0.637114	+	0.770770i	$0.280128\pi$
$500$	0	0
$501$	5.80385	0.259297
$502$	0	0
$503$	− 17.1962i	− 0.766739i	−0.923595	−	0.383369i	$-0.874764\pi$
$503$	− 17.1962i	− 0.766739i	0.923595	−	0.383369i	$-0.125236\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.32051i	0.103057i
$508$	0	0
$509$	32.7846	1.45315	0.726576	−	0.687086i	$-0.241110\pi$
$509$	32.7846	1.45315	0.726576	+	0.687086i	$0.241110\pi$
$510$	0	0
$511$	27.7654	1.22827
$512$	0	0
$513$	0.732051i	0.0323208i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.07180i	0.135097i
$518$	0	0
$519$	17.8564	0.783809
$520$	0	0
$521$	22.0526	0.966140	0.483070	−	0.875582i	$-0.339522\pi$
$521$	22.0526	0.966140	0.483070	+	0.875582i	$0.339522\pi$
$522$	0	0
$523$	− 36.7846i	− 1.60848i	−0.594306	−	0.804239i	$-0.702573\pi$
$523$	− 36.7846i	− 1.60848i	0.594306	−	0.804239i	$-0.297427\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.73205i	0.336813i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−7.19615	−0.312286
$532$	0	0
$533$	− 25.2679i	− 1.09448i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 11.4641i	− 0.494713i
$538$	0	0
$539$	3.89488	0.167764
$540$	0	0
$541$	−14.9282	−0.641814	−0.320907	−	0.947111i	$-0.603988\pi$
$541$	−14.9282	−0.641814	−0.320907	+	0.947111i	$0.603988\pi$
$542$	0	0
$543$	− 12.3923i	− 0.531805i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 2.92820i	− 0.125201i	−0.998039	−	0.0626005i	$-0.980061\pi$
$547$	− 2.92820i	− 0.125201i	0.998039	−	0.0626005i	$-0.0199394\pi$
$548$	0	0
$549$	10.7321	0.458033
$550$	0	0
$551$	5.26795	0.224422
$552$	0	0
$553$	9.85641i	0.419137i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 15.9808i	− 0.677127i	−0.940944	−	0.338563i	$-0.890059\pi$
$557$	− 15.9808i	− 0.677127i	0.940944	−	0.338563i	$-0.109941\pi$
$558$	0	0
$559$	11.3205	0.478806
$560$	0	0
$561$	32.4449	1.36982
$562$	0	0
$563$	− 15.5885i	− 0.656975i	−0.944508	−	0.328488i	$-0.893461\pi$
$563$	− 15.5885i	− 0.656975i	0.944508	−	0.328488i	$-0.106539\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.46410i	− 0.103483i
$568$	0	0
$569$	−0.535898	−0.0224660	−0.0112330	−	0.999937i	$-0.503576\pi$
$569$	−0.535898	−0.0224660	−0.0112330	+	0.999937i	$0.503576\pi$
$570$	0	0
$571$	−24.5885	−1.02899	−0.514497	−	0.857492i	$-0.672022\pi$
$571$	−24.5885	−1.02899	−0.514497	+	0.857492i	$0.672022\pi$
$572$	0	0
$573$	5.80385i	0.242459i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.7846i	0.698752i	0.936983	+	0.349376i	$0.113606\pi$
$577$	16.7846i	0.698752i	−0.936983	+	0.349376i	$0.886394\pi$
$578$	0	0
$579$	24.3923	1.01371
$580$	0	0
$581$	16.4115	0.680866
$582$	0	0
$583$	− 27.9474i	− 1.15746i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 24.9282i	− 1.02890i	−0.857521	−	0.514449i	$-0.827997\pi$
$587$	− 24.9282i	− 1.02890i	0.857521	−	0.514449i	$-0.172003\pi$
$588$	0	0
$589$	0.732051	0.0301636
$590$	0	0
$591$	7.46410	0.307032
$592$	0	0
$593$	31.2679i	1.28402i	0.766696	+	0.642010i	$0.221899\pi$
$593$	31.2679i	1.28402i	−0.766696	+	0.642010i	$0.778101\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.53590i	0.185642i
$598$	0	0
$599$	−3.60770	−0.147406	−0.0737032	−	0.997280i	$-0.523482\pi$
$599$	−3.60770	−0.147406	−0.0737032	+	0.997280i	$0.523482\pi$
$600$	0	0
$601$	−6.60770	−0.269534	−0.134767	−	0.990877i	$-0.543029\pi$
$601$	−6.60770	−0.269534	−0.134767	+	0.990877i	$0.543029\pi$
$602$	0	0
$603$	− 5.00000i	− 0.203616i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 28.9808i	− 1.17629i	−0.808754	−	0.588146i	$-0.799858\pi$
$607$	− 28.9808i	− 1.17629i	0.808754	−	0.588146i	$-0.200142\pi$
$608$	0	0
$609$	−17.7321	−0.718539
$610$	0	0
$611$	−2.39230	−0.0967823
$612$	0	0
$613$	41.7128i	1.68476i	0.538880	+	0.842382i	$0.318847\pi$
$613$	41.7128i	1.68476i	−0.538880	+	0.842382i	$0.681153\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 11.8756i	− 0.478095i	−0.971008	−	0.239048i	$-0.923165\pi$
$617$	− 11.8756i	− 0.478095i	0.971008	−	0.239048i	$-0.0768352\pi$
$618$	0	0
$619$	−10.9282	−0.439242	−0.219621	−	0.975585i	$-0.570482\pi$
$619$	−10.9282	−0.439242	−0.219621	+	0.975585i	$0.570482\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	− 40.3923i	− 1.61828i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.07180i	− 0.122676i
$628$	0	0
$629$	−88.0859	−3.51221
$630$	0	0
$631$	34.8372	1.38685	0.693423	−	0.720531i	$-0.256102\pi$
$631$	34.8372	1.38685	0.693423	+	0.720531i	$0.256102\pi$
$632$	0	0
$633$	− 17.9282i	− 0.712582i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.03332i	0.120185i
$638$	0	0
$639$	14.1244	0.558751
$640$	0	0
$641$	10.0526	0.397052	0.198526	−	0.980096i	$-0.436385\pi$
$641$	10.0526	0.397052	0.198526	+	0.980096i	$0.436385\pi$
$642$	0	0
$643$	1.53590i	0.0605699i	0.999541	+	0.0302850i	$0.00964148\pi$
$643$	1.53590i	0.0605699i	−0.999541	+	0.0302850i	$0.990359\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 26.7321i	− 1.05095i	−0.850810	−	0.525473i	$-0.823889\pi$
$647$	− 26.7321i	− 1.05095i	0.850810	−	0.525473i	$-0.176111\pi$
$648$	0	0
$649$	30.1962	1.18530
$650$	0	0
$651$	−2.46410	−0.0965758
$652$	0	0
$653$	33.3731i	1.30599i	0.757363	+	0.652995i	$0.226488\pi$
$653$	33.3731i	1.30599i	−0.757363	+	0.652995i	$0.773512\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 11.2679i	− 0.439605i
$658$	0	0
$659$	50.4449	1.96505	0.982526	−	0.186123i	$-0.0595923\pi$
$659$	50.4449	1.96505	0.982526	+	0.186123i	$0.0595923\pi$
$660$	0	0
$661$	28.5359	1.10992	0.554959	−	0.831878i	$-0.312734\pi$
$661$	28.5359	1.10992	0.554959	+	0.831878i	$0.312734\pi$
$662$	0	0
$663$	25.2679i	0.981326i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 7.19615i	− 0.278636i
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	−45.0333	−1.73849
$672$	0	0
$673$	27.9090i	1.07581i	0.843005	+	0.537906i	$0.180784\pi$
$673$	27.9090i	1.07581i	−0.843005	+	0.537906i	$0.819216\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 20.8038i	− 0.799557i	−0.916612	−	0.399778i	$-0.869087\pi$
$677$	− 20.8038i	− 0.799557i	0.916612	−	0.399778i	$-0.130913\pi$
$678$	0	0
$679$	9.85641	0.378254
$680$	0	0
$681$	22.9282	0.878611
$682$	0	0
$683$	7.26795i	0.278100i	0.990285	+	0.139050i	$0.0444049\pi$
$683$	7.26795i	0.278100i	−0.990285	+	0.139050i	$0.955595\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 17.4641i	− 0.666297i
$688$	0	0
$689$	21.7654	0.829195
$690$	0	0
$691$	−0.535898	−0.0203865	−0.0101933	−	0.999948i	$-0.503245\pi$
$691$	−0.535898	−0.0203865	−0.0101933	+	0.999948i	$0.503245\pi$
$692$	0	0
$693$	10.3397i	0.392774i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	59.7846i	2.26450i
$698$	0	0
$699$	20.7846	0.786146
$700$	0	0
$701$	−18.5885	−0.702076	−0.351038	−	0.936361i	$-0.614171\pi$
$701$	−18.5885	−0.702076	−0.351038	+	0.936361i	$0.614171\pi$
$702$	0	0
$703$	8.33975i	0.314539i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.90897i	0.259838i
$708$	0	0
$709$	−31.1244	−1.16890	−0.584450	−	0.811430i	$-0.698690\pi$
$709$	−31.1244	−1.16890	−0.584450	+	0.811430i	$0.698690\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	− 1.00000i	− 0.0374503i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 2.12436i	− 0.0793355i
$718$	0	0
$719$	−14.2679	−0.532105	−0.266052	−	0.963959i	$-0.585719\pi$
$719$	−14.2679	−0.532105	−0.266052	+	0.963959i	$0.585719\pi$
$720$	0	0
$721$	−41.7128	−1.55347
$722$	0	0
$723$	− 14.5885i	− 0.542551i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	29.0000i	1.07555i	0.843088	+	0.537775i	$0.180735\pi$
$727$	29.0000i	1.07555i	−0.843088	+	0.537775i	$0.819265\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−26.7846	−0.990665
$732$	0	0
$733$	− 15.6410i	− 0.577714i	−0.957372	−	0.288857i	$-0.906725\pi$
$733$	− 15.6410i	− 0.577714i	0.957372	−	0.288857i	$-0.0932752\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20.9808i	0.772836i
$738$	0	0
$739$	1.67949	0.0617811	0.0308906	−	0.999523i	$-0.490166\pi$
$739$	1.67949	0.0617811	0.0308906	+	0.999523i	$0.490166\pi$
$740$	0	0
$741$	2.39230	0.0878835
$742$	0	0
$743$	17.8564i	0.655088i	0.944836	+	0.327544i	$0.106221\pi$
$743$	17.8564i	0.655088i	−0.944836	+	0.327544i	$0.893779\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 6.66025i	− 0.243686i
$748$	0	0
$749$	2.94744	0.107697
$750$	0	0
$751$	−49.1244	−1.79257	−0.896287	−	0.443475i	$-0.853745\pi$
$751$	−49.1244	−1.79257	−0.896287	+	0.443475i	$0.853745\pi$
$752$	0	0
$753$	23.8564i	0.869376i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	23.1436i	0.841168i	0.907254	+	0.420584i	$0.138175\pi$
$757$	23.1436i	0.841168i	−0.907254	+	0.420584i	$0.861825\pi$
$758$	0	0
$759$	−4.19615	−0.152311
$760$	0	0
$761$	5.19615	0.188360	0.0941802	−	0.995555i	$-0.469977\pi$
$761$	5.19615	0.188360	0.0941802	+	0.995555i	$0.469977\pi$
$762$	0	0
$763$	9.01924i	0.326518i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	23.5167i	0.849137i
$768$	0	0
$769$	9.66025	0.348358	0.174179	−	0.984714i	$-0.444273\pi$
$769$	9.66025	0.348358	0.174179	+	0.984714i	$0.444273\pi$
$770$	0	0
$771$	−9.12436	−0.328606
$772$	0	0
$773$	− 45.0333i	− 1.61974i	−0.586612	−	0.809868i	$-0.699539\pi$
$773$	− 45.0333i	− 1.61974i	0.586612	−	0.809868i	$-0.300461\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 28.0718i	− 1.00707i
$778$	0	0
$779$	5.66025	0.202800
$780$	0	0
$781$	−59.2679	−2.12077
$782$	0	0
$783$	7.19615i	0.257170i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	45.3923i	1.61806i	0.587767	+	0.809030i	$0.300007\pi$
$787$	45.3923i	1.61806i	−0.587767	+	0.809030i	$0.699993\pi$
$788$	0	0
$789$	−11.3397	−0.403706
$790$	0	0
$791$	37.4449	1.33139
$792$	0	0
$793$	− 35.0718i	− 1.24544i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.26795i	0.151179i	0.997139	+	0.0755893i	$0.0240838\pi$
$797$	4.26795i	0.151179i	−0.997139	+	0.0755893i	$0.975916\pi$
$798$	0	0
$799$	5.66025	0.200245
$800$	0	0
$801$	−16.3923	−0.579194
$802$	0	0
$803$	47.2820i	1.66855i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.73205i	0.0609711i
$808$	0	0
$809$	23.1962	0.815533	0.407767	−	0.913086i	$-0.366308\pi$
$809$	23.1962	0.815533	0.407767	+	0.913086i	$0.366308\pi$
$810$	0	0
$811$	−33.9282	−1.19138	−0.595690	−	0.803214i	$-0.703121\pi$
$811$	−33.9282	−1.19138	−0.595690	+	0.803214i	$0.703121\pi$
$812$	0	0
$813$	23.3923i	0.820404i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.53590i	0.0887199i
$818$	0	0
$819$	−8.05256	−0.281379
$820$	0	0
$821$	−1.85641	−0.0647890	−0.0323945	−	0.999475i	$-0.510313\pi$
$821$	−1.85641	−0.0647890	−0.0323945	+	0.999475i	$0.510313\pi$
$822$	0	0
$823$	− 20.7846i	− 0.724506i	−0.932080	−	0.362253i	$-0.882008\pi$
$823$	− 20.7846i	− 0.724506i	0.932080	−	0.362253i	$-0.117992\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 40.5167i	− 1.40890i	−0.709752	−	0.704451i	$-0.751193\pi$
$827$	− 40.5167i	− 1.40890i	0.709752	−	0.704451i	$-0.248807\pi$
$828$	0	0
$829$	−55.4974	−1.92751	−0.963753	−	0.266798i	$-0.914034\pi$
$829$	−55.4974	−1.92751	−0.963753	+	0.266798i	$0.914034\pi$
$830$	0	0
$831$	4.14359	0.143740
$832$	0	0
$833$	− 7.17691i	− 0.248665i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.00000i	0.0345651i
$838$	0	0
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	22.7846	0.785676
$842$	0	0
$843$	15.5167i	0.534422i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 16.2820i	− 0.559457i
$848$	0	0
$849$	30.8564	1.05899
$850$	0	0
$851$	11.3923	0.390523
$852$	0	0
$853$	31.4641i	1.07731i	0.842526	+	0.538655i	$0.181067\pi$
$853$	31.4641i	1.07731i	−0.842526	+	0.538655i	$0.818933\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.7128i	1.15161i	0.817588	+	0.575804i	$0.195311\pi$
$857$	33.7128i	1.15161i	−0.817588	+	0.575804i	$0.804689\pi$
$858$	0	0
$859$	19.0000	0.648272	0.324136	−	0.946011i	$-0.394927\pi$
$859$	19.0000	0.648272	0.324136	+	0.946011i	$0.394927\pi$
$860$	0	0
$861$	−19.0526	−0.649309
$862$	0	0
$863$	− 51.9615i	− 1.76879i	−0.466738	−	0.884395i	$-0.654571\pi$
$863$	− 51.9615i	− 1.76879i	0.466738	−	0.884395i	$-0.345429\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 42.7846i	− 1.45304i
$868$	0	0
$869$	−16.7846	−0.569379
$870$	0	0
$871$	−16.3397	−0.553651
$872$	0	0
$873$	− 4.00000i	− 0.135379i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7.46410i	− 0.252045i	−0.992027	−	0.126022i	$-0.959779\pi$
$877$	− 7.46410i	− 0.252045i	0.992027	−	0.126022i	$-0.0402211\pi$
$878$	0	0
$879$	−24.6603	−0.831770
$880$	0	0
$881$	7.94744	0.267756	0.133878	−	0.990998i	$-0.457257\pi$
$881$	7.94744	0.267756	0.133878	+	0.990998i	$0.457257\pi$
$882$	0	0
$883$	45.6603i	1.53659i	0.640096	+	0.768295i	$0.278895\pi$
$883$	45.6603i	1.53659i	−0.640096	+	0.768295i	$0.721105\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 17.3205i	− 0.581566i	−0.956789	−	0.290783i	$-0.906084\pi$
$887$	− 17.3205i	− 0.581566i	0.956789	−	0.290783i	$-0.0939157\pi$
$888$	0	0
$889$	27.4115	0.919354
$890$	0	0
$891$	4.19615	0.140576
$892$	0	0
$893$	− 0.535898i	− 0.0179332i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3.26795i	− 0.109114i
$898$	0	0
$899$	7.19615	0.240005
$900$	0	0
$901$	−51.4974	−1.71563
$902$	0	0
$903$	− 8.53590i	− 0.284057i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19.9282i	0.661705i	0.943682	+	0.330853i	$0.107336\pi$
$907$	19.9282i	0.661705i	−0.943682	+	0.330853i	$0.892664\pi$
$908$	0	0
$909$	2.80385	0.0929978
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	27.9474i	0.924925i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	53.8564i	1.77850i
$918$	0	0
$919$	22.9282	0.756332	0.378166	−	0.925738i	$-0.376555\pi$
$919$	22.9282	0.756332	0.378166	+	0.925738i	$0.376555\pi$
$920$	0	0
$921$	−3.66025	−0.120609
$922$	0	0
$923$	− 46.1577i	− 1.51930i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.9282i	0.555995i
$928$	0	0
$929$	−10.5167	−0.345040	−0.172520	−	0.985006i	$-0.555191\pi$
$929$	−10.5167	−0.345040	−0.172520	+	0.985006i	$0.555191\pi$
$930$	0	0
$931$	−0.679492	−0.0222694
$932$	0	0
$933$	22.9282i	0.750636i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.07180i	0.165688i	0.996563	+	0.0828442i	$0.0264004\pi$
$937$	5.07180i	0.165688i	−0.996563	+	0.0828442i	$0.973600\pi$
$938$	0	0
$939$	22.3205	0.728402
$940$	0	0
$941$	22.1962	0.723574	0.361787	−	0.932261i	$-0.382167\pi$
$941$	22.1962	0.723574	0.361787	+	0.932261i	$0.382167\pi$
$942$	0	0
$943$	− 7.73205i	− 0.251790i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.67949i	− 0.282046i	−0.990006	−	0.141023i	$-0.954961\pi$
$947$	− 8.67949i	− 0.282046i	0.990006	−	0.141023i	$-0.0450391\pi$
$948$	0	0
$949$	−36.8231	−1.19533
$950$	0	0
$951$	13.5167	0.438308
$952$	0	0
$953$	25.7128i	0.832920i	0.909154	+	0.416460i	$0.136729\pi$
$953$	25.7128i	0.832920i	−0.909154	+	0.416460i	$0.863271\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 30.1962i	− 0.976103i
$958$	0	0
$959$	−29.5692	−0.954840
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	− 1.19615i	− 0.0385455i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 16.8756i	− 0.542684i	−0.962483	−	0.271342i	$-0.912533\pi$
$967$	− 16.8756i	− 0.542684i	0.962483	−	0.271342i	$-0.0874675\pi$
$968$	0	0
$969$	−5.66025	−0.181834
$970$	0	0
$971$	−47.3205	−1.51859	−0.759294	−	0.650748i	$-0.774455\pi$
$971$	−47.3205	−1.51859	−0.759294	+	0.650748i	$0.774455\pi$
$972$	0	0
$973$	17.2487i	0.552968i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.87564i	0.315950i	0.987443	+	0.157975i	$0.0504965\pi$
$977$	9.87564i	0.315950i	−0.987443	+	0.157975i	$0.949503\pi$
$978$	0	0
$979$	68.7846	2.19837
$980$	0	0
$981$	3.66025	0.116863
$982$	0	0
$983$	− 11.1962i	− 0.357102i	−0.983931	−	0.178551i	$-0.942859\pi$
$983$	− 11.1962i	− 0.357102i	0.983931	−	0.178551i	$-0.0571409\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.80385i	0.0574171i
$988$	0	0
$989$	3.46410	0.110152
$990$	0	0
$991$	28.7128	0.912093	0.456046	−	0.889956i	$-0.349265\pi$
$991$	28.7128	0.912093	0.456046	+	0.889956i	$0.349265\pi$
$992$	0	0
$993$	− 27.2487i	− 0.864712i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 53.4641i	− 1.69323i	−0.532210	−	0.846613i	$-0.678638\pi$
$997$	− 53.4641i	− 1.69323i	0.532210	−	0.846613i	$-0.321362\pi$
$998$	0	0
$999$	−11.3923	−0.360437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.f.i.6349.3		4
5.2	odd	4		6900.2.a.s.1.2		2
5.3	odd	4		1380.2.a.h.1.1	✓	2
5.4	even	2	inner	6900.2.f.i.6349.2		4
15.8	even	4		4140.2.a.o.1.1		2
20.3	even	4		5520.2.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.h.1.1	✓	2	5.3	odd	4
4140.2.a.o.1.1		2	15.8	even	4
5520.2.a.bp.1.2		2	20.3	even	4
6900.2.a.s.1.2		2	5.2	odd	4
6900.2.f.i.6349.2		4	5.4	even	2	inner
6900.2.f.i.6349.3		4	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.46410i q^{7} -1.00000 q^{9} +4.19615 q^{11} +3.26795i q^{13} -7.73205i q^{17} -0.732051 q^{19} +2.46410 q^{21} +1.00000i q^{23} -1.00000i q^{27} -7.19615 q^{29} -1.00000 q^{31} +4.19615i q^{33} -11.3923i q^{37} -3.26795 q^{39} -7.73205 q^{41} -3.46410i q^{43} +0.732051i q^{47} +0.928203 q^{49} +7.73205 q^{51} -6.66025i q^{53} -0.732051i q^{57} +7.19615 q^{59} -10.7321 q^{61} +2.46410i q^{63} +5.00000i q^{67} -1.00000 q^{69} -14.1244 q^{71} +11.2679i q^{73} -10.3397i q^{77} -4.00000 q^{79} +1.00000 q^{81} +6.66025i q^{83} -7.19615i q^{87} +16.3923 q^{89} +8.05256 q^{91} -1.00000i q^{93} +4.00000i q^{97} -4.19615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} - 4 q^{11} + 4 q^{19} - 4 q^{21} - 8 q^{29} - 4 q^{31} - 20 q^{39} - 24 q^{41} - 24 q^{49} + 24 q^{51} + 8 q^{59} - 36 q^{61} - 4 q^{69} - 8 q^{71} - 16 q^{79} + 4 q^{81} + 24 q^{89} - 44 q^{91}+ \cdots + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 4 * q^11 + 4 * q^19 - 4 * q^21 - 8 * q^29 - 4 * q^31 - 20 * q^39 - 24 * q^41 - 24 * q^49 + 24 * q^51 + 8 * q^59 - 36 * q^61 - 4 * q^69 - 8 * q^71 - 16 * q^79 + 4 * q^81 + 24 * q^89 - 44 * q^91 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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