LMFDB - Embedded newform 768.6.d.a.385.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,6,Mod(385,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("768.385"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	768.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-480,0,-162,0,0,0,0,0,-612] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$123.174773616$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			385.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	768.385
Dual form			768.6.d.a.385.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} -34.0000i q^{5} -240.000 q^{7} -81.0000 q^{9} +124.000i q^{11} -46.0000i q^{13} -306.000 q^{15} +1954.00 q^{17} -1924.00i q^{19} +2160.00i q^{21} +2840.00 q^{23} +1969.00 q^{25} +729.000i q^{27} +8922.00i q^{29} +4648.00 q^{31} +1116.00 q^{33} +8160.00i q^{35} -4362.00i q^{37} -414.000 q^{39} +2886.00 q^{41} -11332.0i q^{43} +2754.00i q^{45} -7008.00 q^{47} +40793.0 q^{49} -17586.0i q^{51} -22594.0i q^{53} +4216.00 q^{55} -17316.0 q^{57} +28.0000i q^{59} +6386.00i q^{61} +19440.0 q^{63} -1564.00 q^{65} -39076.0i q^{67} -25560.0i q^{69} -54872.0 q^{71} -21034.0 q^{73} -17721.0i q^{75} -29760.0i q^{77} -26632.0 q^{79} +6561.00 q^{81} +56188.0i q^{83} -66436.0i q^{85} +80298.0 q^{87} -64410.0 q^{89} +11040.0i q^{91} -41832.0i q^{93} -65416.0 q^{95} -116158. q^{97} -10044.0i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 480 q^{7} - 162 q^{9} - 612 q^{15} + 3908 q^{17} + 5680 q^{23} + 3938 q^{25} + 9296 q^{31} + 2232 q^{33} - 828 q^{39} + 5772 q^{41} - 14016 q^{47} + 81586 q^{49} + 8432 q^{55} - 34632 q^{57} + 38880 q^{63}+ \cdots - 232316 q^{97}+O(q^{100}) $ 2 * q - 480 * q^7 - 162 * q^9 - 612 * q^15 + 3908 * q^17 + 5680 * q^23 + 3938 * q^25 + 9296 * q^31 + 2232 * q^33 - 828 * q^39 + 5772 * q^41 - 14016 * q^47 + 81586 * q^49 + 8432 * q^55 - 34632 * q^57 + 38880 * q^63 - 3128 * q^65 - 109744 * q^71 - 42068 * q^73 - 53264 * q^79 + 13122 * q^81 + 160596 * q^87 - 128820 * q^89 - 130832 * q^95 - 232316 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	0	0
$5$	− 34.0000i	− 0.608210i	−0.952638	−	0.304105i	$-0.901643\pi$
$5$	− 34.0000i	− 0.608210i	0.952638	−	0.304105i	$-0.0983575\pi$
$6$	0	0
$7$	−240.000	−1.85125	−0.925627	−	0.378436i	$-0.876462\pi$
$7$	−240.000	−1.85125	−0.925627	+	0.378436i	$0.876462\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	124.000i	0.308987i	0.987994	+	0.154493i	$0.0493745\pi$
$11$	124.000i	0.308987i	−0.987994	+	0.154493i	$0.950625\pi$
$12$	0	0
$13$	− 46.0000i	− 0.0754917i	−0.999287	−	0.0377459i	$-0.987982\pi$
$13$	− 46.0000i	− 0.0754917i	0.999287	−	0.0377459i	$-0.0120177\pi$
$14$	0	0
$15$	−306.000	−0.351150
$16$	0	0
$17$	1954.00	1.63984	0.819921	−	0.572476i	$-0.194017\pi$
$17$	1954.00	1.63984	0.819921	+	0.572476i	$0.194017\pi$
$18$	0	0
$19$	− 1924.00i	− 1.22270i	−0.791359	−	0.611352i	$-0.790626\pi$
$19$	− 1924.00i	− 1.22270i	0.791359	−	0.611352i	$-0.209374\pi$
$20$	0	0
$21$	2160.00i	1.06882i
$22$	0	0
$23$	2840.00	1.11943	0.559717	−	0.828684i	$-0.310910\pi$
$23$	2840.00	1.11943	0.559717	+	0.828684i	$0.310910\pi$
$24$	0	0
$25$	1969.00	0.630080
$26$	0	0
$27$	729.000i	0.192450i
$28$	0	0
$29$	8922.00i	1.97000i	0.172541	+	0.985002i	$0.444802\pi$
$29$	8922.00i	1.97000i	−0.172541	+	0.985002i	$0.555198\pi$
$30$	0	0
$31$	4648.00	0.868684	0.434342	−	0.900748i	$-0.356981\pi$
$31$	4648.00	0.868684	0.434342	+	0.900748i	$0.356981\pi$
$32$	0	0
$33$	1116.00	0.178394
$34$	0	0
$35$	8160.00i	1.12595i
$36$	0	0
$37$	− 4362.00i	− 0.523819i	−0.965092	−	0.261910i	$-0.915648\pi$
$37$	− 4362.00i	− 0.523819i	0.965092	−	0.261910i	$-0.0843522\pi$
$38$	0	0
$39$	−414.000	−0.0435852
$40$	0	0
$41$	2886.00	0.268125	0.134062	−	0.990973i	$-0.457198\pi$
$41$	2886.00	0.268125	0.134062	+	0.990973i	$0.457198\pi$
$42$	0	0
$43$	− 11332.0i	− 0.934621i	−0.884093	−	0.467310i	$-0.845223\pi$
$43$	− 11332.0i	− 0.934621i	0.884093	−	0.467310i	$-0.154777\pi$
$44$	0	0
$45$	2754.00i	0.202737i
$46$	0	0
$47$	−7008.00	−0.462753	−0.231377	−	0.972864i	$-0.574323\pi$
$47$	−7008.00	−0.462753	−0.231377	+	0.972864i	$0.574323\pi$
$48$	0	0
$49$	40793.0	2.42714
$50$	0	0
$51$	− 17586.0i	− 0.946764i
$52$	0	0
$53$	− 22594.0i	− 1.10485i	−0.833562	−	0.552425i	$-0.813703\pi$
$53$	− 22594.0i	− 1.10485i	0.833562	−	0.552425i	$-0.186297\pi$
$54$	0	0
$55$	4216.00	0.187929
$56$	0	0
$57$	−17316.0	−0.705928
$58$	0	0
$59$	28.0000i	0.00104720i	1.00000		0.000523598i	$0.000166666\pi$
$59$	28.0000i	0.00104720i	−1.00000		0.000523598i	$0.999833\pi$
$60$	0	0
$61$	6386.00i	0.219738i	0.993946	+	0.109869i	$0.0350431\pi$
$61$	6386.00i	0.219738i	−0.993946	+	0.109869i	$0.964957\pi$
$62$	0	0
$63$	19440.0	0.617085
$64$	0	0
$65$	−1564.00	−0.0459149
$66$	0	0
$67$	− 39076.0i	− 1.06346i	−0.846912	−	0.531732i	$-0.821541\pi$
$67$	− 39076.0i	− 1.06346i	0.846912	−	0.531732i	$-0.178459\pi$
$68$	0	0
$69$	− 25560.0i	− 0.646306i
$70$	0	0
$71$	−54872.0	−1.29183	−0.645914	−	0.763410i	$-0.723524\pi$
$71$	−54872.0	−1.29183	−0.645914	+	0.763410i	$0.723524\pi$
$72$	0	0
$73$	−21034.0	−0.461971	−0.230986	−	0.972957i	$-0.574195\pi$
$73$	−21034.0	−0.461971	−0.230986	+	0.972957i	$0.574195\pi$
$74$	0	0
$75$	− 17721.0i	− 0.363777i
$76$	0	0
$77$	− 29760.0i	− 0.572013i
$78$	0	0
$79$	−26632.0	−0.480105	−0.240052	−	0.970760i	$-0.577165\pi$
$79$	−26632.0	−0.480105	−0.240052	+	0.970760i	$0.577165\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	56188.0i	0.895258i	0.894219	+	0.447629i	$0.147732\pi$
$83$	56188.0i	0.895258i	−0.894219	+	0.447629i	$0.852268\pi$
$84$	0	0
$85$	− 66436.0i	− 0.997370i
$86$	0	0
$87$	80298.0	1.13738
$88$	0	0
$89$	−64410.0	−0.861942	−0.430971	−	0.902366i	$-0.641829\pi$
$89$	−64410.0	−0.861942	−0.430971	+	0.902366i	$0.641829\pi$
$90$	0	0
$91$	11040.0i	0.139754i
$92$	0	0
$93$	− 41832.0i	− 0.501535i
$94$	0	0
$95$	−65416.0	−0.743661
$96$	0	0
$97$	−116158.	−1.25349	−0.626743	−	0.779226i	$-0.715613\pi$
$97$	−116158.	−1.25349	−0.626743	+	0.779226i	$0.715613\pi$
$98$	0	0
$99$	− 10044.0i	− 0.102996i
$100$	0	0
$101$	− 66834.0i	− 0.651920i	−0.945384	−	0.325960i	$-0.894313\pi$
$101$	− 66834.0i	− 0.651920i	0.945384	−	0.325960i	$-0.105687\pi$
$102$	0	0
$103$	64000.0	0.594411	0.297206	−	0.954814i	$-0.403945\pi$
$103$	64000.0	0.594411	0.297206	+	0.954814i	$0.403945\pi$
$104$	0	0
$105$	73440.0	0.650069
$106$	0	0
$107$	15084.0i	0.127367i	0.997970	+	0.0636835i	$0.0202848\pi$
$107$	15084.0i	0.127367i	−0.997970	+	0.0636835i	$0.979715\pi$
$108$	0	0
$109$	39698.0i	0.320039i	0.987114	+	0.160019i	$0.0511556\pi$
$109$	39698.0i	0.320039i	−0.987114	+	0.160019i	$0.948844\pi$
$110$	0	0
$111$	−39258.0	−0.302427
$112$	0	0
$113$	155154.	1.14305	0.571527	−	0.820583i	$-0.306351\pi$
$113$	155154.	1.14305	0.571527	+	0.820583i	$0.306351\pi$
$114$	0	0
$115$	− 96560.0i	− 0.680852i
$116$	0	0
$117$	3726.00i	0.0251639i
$118$	0	0
$119$	−468960.	−3.03577
$120$	0	0
$121$	145675.	0.904527
$122$	0	0
$123$	− 25974.0i	− 0.154802i
$124$	0	0
$125$	− 173196.i	− 0.991432i
$126$	0	0
$127$	−52072.0	−0.286480	−0.143240	−	0.989688i	$-0.545752\pi$
$127$	−52072.0	−0.286480	−0.143240	+	0.989688i	$0.545752\pi$
$128$	0	0
$129$	−101988.	−0.539604
$130$	0	0
$131$	159964.i	0.814412i	0.913336	+	0.407206i	$0.133497\pi$
$131$	159964.i	0.814412i	−0.913336	+	0.407206i	$0.866503\pi$
$132$	0	0
$133$	461760.i	2.26353i
$134$	0	0
$135$	24786.0	0.117050
$136$	0	0
$137$	262278.	1.19388	0.596940	−	0.802286i	$-0.296383\pi$
$137$	262278.	1.19388	0.596940	+	0.802286i	$0.296383\pi$
$138$	0	0
$139$	− 253524.i	− 1.11297i	−0.830859	−	0.556483i	$-0.812150\pi$
$139$	− 253524.i	− 1.11297i	0.830859	−	0.556483i	$-0.187850\pi$
$140$	0	0
$141$	63072.0i	0.267171i
$142$	0	0
$143$	5704.00	0.0233260
$144$	0	0
$145$	303348.	1.19818
$146$	0	0
$147$	− 367137.i	− 1.40131i
$148$	0	0
$149$	355630.i	1.31230i	0.754631	+	0.656149i	$0.227816\pi$
$149$	355630.i	1.31230i	−0.754631	+	0.656149i	$0.772184\pi$
$150$	0	0
$151$	−1024.00	−0.00365475	−0.00182737	−	0.999998i	$-0.500582\pi$
$151$	−1024.00	−0.00365475	−0.00182737	+	0.999998i	$0.500582\pi$
$152$	0	0
$153$	−158274.	−0.546614
$154$	0	0
$155$	− 158032.i	− 0.528343i
$156$	0	0
$157$	59954.0i	0.194119i	0.995279	+	0.0970597i	$0.0309438\pi$
$157$	59954.0i	0.194119i	−0.995279	+	0.0970597i	$0.969056\pi$
$158$	0	0
$159$	−203346.	−0.637886
$160$	0	0
$161$	−681600.	−2.07236
$162$	0	0
$163$	− 341556.i	− 1.00692i	−0.864020	−	0.503458i	$-0.832061\pi$
$163$	− 341556.i	− 1.00692i	0.864020	−	0.503458i	$-0.167939\pi$
$164$	0	0
$165$	− 37944.0i	− 0.108501i
$166$	0	0
$167$	5016.00	0.0139177	0.00695883	−	0.999976i	$-0.497785\pi$
$167$	5016.00	0.0139177	0.00695883	+	0.999976i	$0.497785\pi$
$168$	0	0
$169$	369177.	0.994301
$170$	0	0
$171$	155844.i	0.407568i
$172$	0	0
$173$	228666.i	0.580880i	0.956893	+	0.290440i	$0.0938016\pi$
$173$	228666.i	0.580880i	−0.956893	+	0.290440i	$0.906198\pi$
$174$	0	0
$175$	−472560.	−1.16644
$176$	0	0
$177$	252.000	0.000604599 0
$178$	0	0
$179$	161388.i	0.376477i	0.982123	+	0.188239i	$0.0602778\pi$
$179$	161388.i	0.376477i	−0.982123	+	0.188239i	$0.939722\pi$
$180$	0	0
$181$	− 291690.i	− 0.661797i	−0.943666	−	0.330899i	$-0.892648\pi$
$181$	− 291690.i	− 0.661797i	0.943666	−	0.330899i	$-0.107352\pi$
$182$	0	0
$183$	57474.0	0.126866
$184$	0	0
$185$	−148308.	−0.318592
$186$	0	0
$187$	242296.i	0.506690i
$188$	0	0
$189$	− 174960.i	− 0.356274i
$190$	0	0
$191$	55680.0	0.110437	0.0552187	−	0.998474i	$-0.482414\pi$
$191$	55680.0	0.110437	0.0552187	+	0.998474i	$0.482414\pi$
$192$	0	0
$193$	−176254.	−0.340601	−0.170300	−	0.985392i	$-0.554474\pi$
$193$	−176254.	−0.340601	−0.170300	+	0.985392i	$0.554474\pi$
$194$	0	0
$195$	14076.0i	0.0265090i
$196$	0	0
$197$	− 374610.i	− 0.687723i	−0.939020	−	0.343862i	$-0.888265\pi$
$197$	− 374610.i	− 0.687723i	0.939020	−	0.343862i	$-0.111735\pi$
$198$	0	0
$199$	−637760.	−1.14163	−0.570814	−	0.821079i	$-0.693372\pi$
$199$	−637760.	−1.14163	−0.570814	+	0.821079i	$0.693372\pi$
$200$	0	0
$201$	−351684.	−0.613992
$202$	0	0
$203$	− 2.14128e6i	− 3.64698i
$204$	0	0
$205$	− 98124.0i	− 0.163076i
$206$	0	0
$207$	−230040.	−0.373145
$208$	0	0
$209$	238576.	0.377799
$210$	0	0
$211$	− 904628.i	− 1.39883i	−0.714717	−	0.699413i	$-0.753445\pi$
$211$	− 904628.i	− 1.39883i	0.714717	−	0.699413i	$-0.246555\pi$
$212$	0	0
$213$	493848.i	0.745838i
$214$	0	0
$215$	−385288.	−0.568446
$216$	0	0
$217$	−1.11552e6	−1.60816
$218$	0	0
$219$	189306.i	0.266719i
$220$	0	0
$221$	− 89884.0i	− 0.123795i
$222$	0	0
$223$	−619048.	−0.833609	−0.416804	−	0.908996i	$-0.636850\pi$
$223$	−619048.	−0.833609	−0.416804	+	0.908996i	$0.636850\pi$
$224$	0	0
$225$	−159489.	−0.210027
$226$	0	0
$227$	− 1.46975e6i	− 1.89312i	−0.322527	−	0.946560i	$-0.604532\pi$
$227$	− 1.46975e6i	− 1.89312i	0.322527	−	0.946560i	$-0.395468\pi$
$228$	0	0
$229$	− 3290.00i	− 0.00414579i	−0.999998	−	0.00207289i	$-0.999340\pi$
$229$	− 3290.00i	− 0.00414579i	0.999998	−	0.00207289i	$-0.000659823\pi$
$230$	0	0
$231$	−267840.	−0.330252
$232$	0	0
$233$	−935402.	−1.12878	−0.564389	−	0.825509i	$-0.690888\pi$
$233$	−935402.	−1.12878	−0.564389	+	0.825509i	$0.690888\pi$
$234$	0	0
$235$	238272.i	0.281451i
$236$	0	0
$237$	239688.i	0.277189i
$238$	0	0
$239$	875600.	0.991542	0.495771	−	0.868453i	$-0.334886\pi$
$239$	875600.	0.991542	0.495771	+	0.868453i	$0.334886\pi$
$240$	0	0
$241$	−959214.	−1.06383	−0.531916	−	0.846797i	$-0.678528\pi$
$241$	−959214.	−1.06383	−0.531916	+	0.846797i	$0.678528\pi$
$242$	0	0
$243$	− 59049.0i	− 0.0641500i
$244$	0	0
$245$	− 1.38696e6i	− 1.47621i
$246$	0	0
$247$	−88504.0	−0.0923040
$248$	0	0
$249$	505692.	0.516878
$250$	0	0
$251$	− 318868.i	− 0.319467i	−0.987160	−	0.159734i	$-0.948936\pi$
$251$	− 318868.i	− 0.319467i	0.987160	−	0.159734i	$-0.0510636\pi$
$252$	0	0
$253$	352160.i	0.345891i
$254$	0	0
$255$	−597924.	−0.575832
$256$	0	0
$257$	1.71469e6	1.61940	0.809698	−	0.586847i	$-0.199631\pi$
$257$	1.71469e6	1.61940	0.809698	+	0.586847i	$0.199631\pi$
$258$	0	0
$259$	1.04688e6i	0.969723i
$260$	0	0
$261$	− 722682.i	− 0.656668i
$262$	0	0
$263$	−1.11028e6	−0.989790	−0.494895	−	0.868953i	$-0.664794\pi$
$263$	−1.11028e6	−0.989790	−0.494895	+	0.868953i	$0.664794\pi$
$264$	0	0
$265$	−768196.	−0.671982
$266$	0	0
$267$	579690.i	0.497643i
$268$	0	0
$269$	398378.i	0.335672i	0.985815	+	0.167836i	$0.0536778\pi$
$269$	398378.i	0.335672i	−0.985815	+	0.167836i	$0.946322\pi$
$270$	0	0
$271$	−1.44198e6	−1.19271	−0.596355	−	0.802721i	$-0.703385\pi$
$271$	−1.44198e6	−1.19271	−0.596355	+	0.802721i	$0.703385\pi$
$272$	0	0
$273$	99360.0	0.0806873
$274$	0	0
$275$	244156.i	0.194686i
$276$	0	0
$277$	117238.i	0.0918056i	0.998946	+	0.0459028i	$0.0146164\pi$
$277$	117238.i	0.0918056i	−0.998946	+	0.0459028i	$0.985384\pi$
$278$	0	0
$279$	−376488.	−0.289561
$280$	0	0
$281$	1.67514e6	1.26557	0.632784	−	0.774328i	$-0.281912\pi$
$281$	1.67514e6	1.26557	0.632784	+	0.774328i	$0.281912\pi$
$282$	0	0
$283$	− 1.92468e6i	− 1.42854i	−0.699872	−	0.714269i	$-0.746760\pi$
$283$	− 1.92468e6i	− 1.42854i	0.699872	−	0.714269i	$-0.253240\pi$
$284$	0	0
$285$	588744.i	0.429353i
$286$	0	0
$287$	−692640.	−0.496367
$288$	0	0
$289$	2.39826e6	1.68908
$290$	0	0
$291$	1.04542e6i	0.723701i
$292$	0	0
$293$	1.28062e6i	0.871469i	0.900075	+	0.435734i	$0.143511\pi$
$293$	1.28062e6i	0.871469i	−0.900075	+	0.435734i	$0.856489\pi$
$294$	0	0
$295$	952.000	0.000636916 0
$296$	0	0
$297$	−90396.0	−0.0594645
$298$	0	0
$299$	− 130640.i	− 0.0845081i
$300$	0	0
$301$	2.71968e6i	1.73022i
$302$	0	0
$303$	−601506.	−0.376386
$304$	0	0
$305$	217124.	0.133647
$306$	0	0
$307$	− 2.26319e6i	− 1.37049i	−0.728314	−	0.685243i	$-0.759696\pi$
$307$	− 2.26319e6i	− 1.37049i	0.728314	−	0.685243i	$-0.240304\pi$
$308$	0	0
$309$	− 576000.i	− 0.343183i
$310$	0	0
$311$	247848.	0.145306	0.0726532	−	0.997357i	$-0.476853\pi$
$311$	247848.	0.145306	0.0726532	+	0.997357i	$0.476853\pi$
$312$	0	0
$313$	1.82391e6	1.05231	0.526154	−	0.850390i	$-0.323634\pi$
$313$	1.82391e6	1.05231	0.526154	+	0.850390i	$0.323634\pi$
$314$	0	0
$315$	− 660960.i	− 0.375317i
$316$	0	0
$317$	− 2.85629e6i	− 1.59645i	−0.602361	−	0.798224i	$-0.705773\pi$
$317$	− 2.85629e6i	− 1.59645i	0.602361	−	0.798224i	$-0.294227\pi$
$318$	0	0
$319$	−1.10633e6	−0.608705
$320$	0	0
$321$	135756.	0.0735354
$322$	0	0
$323$	− 3.75950e6i	− 2.00504i
$324$	0	0
$325$	− 90574.0i	− 0.0475658i
$326$	0	0
$327$	357282.	0.184774
$328$	0	0
$329$	1.68192e6	0.856674
$330$	0	0
$331$	147148.i	0.0738218i	0.999319	+	0.0369109i	$0.0117518\pi$
$331$	147148.i	0.0738218i	−0.999319	+	0.0369109i	$0.988248\pi$
$332$	0	0
$333$	353322.i	0.174606i
$334$	0	0
$335$	−1.32858e6	−0.646810
$336$	0	0
$337$	−3.24728e6	−1.55756	−0.778780	−	0.627297i	$-0.784161\pi$
$337$	−3.24728e6	−1.55756	−0.778780	+	0.627297i	$0.784161\pi$
$338$	0	0
$339$	− 1.39639e6i	− 0.659943i
$340$	0	0
$341$	576352.i	0.268412i
$342$	0	0
$343$	−5.75664e6	−2.64201
$344$	0	0
$345$	−869040.	−0.393090
$346$	0	0
$347$	1.55675e6i	0.694056i	0.937855	+	0.347028i	$0.112809\pi$
$347$	1.55675e6i	0.694056i	−0.937855	+	0.347028i	$0.887191\pi$
$348$	0	0
$349$	− 4.03217e6i	− 1.77205i	−0.463639	−	0.886024i	$-0.653456\pi$
$349$	− 4.03217e6i	− 1.77205i	0.463639	−	0.886024i	$-0.346544\pi$
$350$	0	0
$351$	33534.0	0.0145284
$352$	0	0
$353$	1.79399e6	0.766271	0.383135	−	0.923692i	$-0.374844\pi$
$353$	1.79399e6	0.766271	0.383135	+	0.923692i	$0.374844\pi$
$354$	0	0
$355$	1.86565e6i	0.785704i
$356$	0	0
$357$	4.22064e6i	1.75270i
$358$	0	0
$359$	1.55278e6	0.635876	0.317938	−	0.948111i	$-0.397010\pi$
$359$	1.55278e6	0.635876	0.317938	+	0.948111i	$0.397010\pi$
$360$	0	0
$361$	−1.22568e6	−0.495003
$362$	0	0
$363$	− 1.31108e6i	− 0.522229i
$364$	0	0
$365$	715156.i	0.280976i
$366$	0	0
$367$	3.11545e6	1.20741	0.603706	−	0.797207i	$-0.293690\pi$
$367$	3.11545e6	1.20741	0.603706	+	0.797207i	$0.293690\pi$
$368$	0	0
$369$	−233766.	−0.0893749
$370$	0	0
$371$	5.42256e6i	2.04536i
$372$	0	0
$373$	− 630682.i	− 0.234714i	−0.993090	−	0.117357i	$-0.962558\pi$
$373$	− 630682.i	− 0.234714i	0.993090	−	0.117357i	$-0.0374421\pi$
$374$	0	0
$375$	−1.55876e6	−0.572403
$376$	0	0
$377$	410412.	0.148719
$378$	0	0
$379$	− 48404.0i	− 0.0173094i	−0.999963	−	0.00865472i	$-0.997245\pi$
$379$	− 48404.0i	− 0.0173094i	0.999963	−	0.00865472i	$-0.00275492\pi$
$380$	0	0
$381$	468648.i	0.165400i
$382$	0	0
$383$	−1.74182e6	−0.606747	−0.303373	−	0.952872i	$-0.598113\pi$
$383$	−1.74182e6	−0.606747	−0.303373	+	0.952872i	$0.598113\pi$
$384$	0	0
$385$	−1.01184e6	−0.347904
$386$	0	0
$387$	917892.i	0.311540i
$388$	0	0
$389$	− 3.06819e6i	− 1.02804i	−0.857779	−	0.514019i	$-0.828156\pi$
$389$	− 3.06819e6i	− 1.02804i	0.857779	−	0.514019i	$-0.171844\pi$
$390$	0	0
$391$	5.54936e6	1.83570
$392$	0	0
$393$	1.43968e6	0.470201
$394$	0	0
$395$	905488.i	0.292005i
$396$	0	0
$397$	− 5.35984e6i	− 1.70677i	−0.521280	−	0.853386i	$-0.674545\pi$
$397$	− 5.35984e6i	− 1.70677i	0.521280	−	0.853386i	$-0.325455\pi$
$398$	0	0
$399$	4.15584e6	1.30685
$400$	0	0
$401$	−2.76473e6	−0.858603	−0.429302	−	0.903161i	$-0.641240\pi$
$401$	−2.76473e6	−0.858603	−0.429302	+	0.903161i	$0.641240\pi$
$402$	0	0
$403$	− 213808.i	− 0.0655785i
$404$	0	0
$405$	− 223074.i	− 0.0675789i
$406$	0	0
$407$	540888.	0.161853
$408$	0	0
$409$	1.20893e6	0.357350	0.178675	−	0.983908i	$-0.442819\pi$
$409$	1.20893e6	0.357350	0.178675	+	0.983908i	$0.442819\pi$
$410$	0	0
$411$	− 2.36050e6i	− 0.689287i
$412$	0	0
$413$	− 6720.00i	− 0.00193863i
$414$	0	0
$415$	1.91039e6	0.544505
$416$	0	0
$417$	−2.28172e6	−0.642571
$418$	0	0
$419$	− 4.38008e6i	− 1.21884i	−0.792847	−	0.609421i	$-0.791402\pi$
$419$	− 4.38008e6i	− 1.21884i	0.792847	−	0.609421i	$-0.208598\pi$
$420$	0	0
$421$	− 922810.i	− 0.253751i	−0.991919	−	0.126875i	$-0.959505\pi$
$421$	− 922810.i	− 0.253751i	0.991919	−	0.126875i	$-0.0404948\pi$
$422$	0	0
$423$	567648.	0.154251
$424$	0	0
$425$	3.84743e6	1.03323
$426$	0	0
$427$	− 1.53264e6i	− 0.406790i
$428$	0	0
$429$	− 51336.0i	− 0.0134672i
$430$	0	0
$431$	−6.12678e6	−1.58869	−0.794345	−	0.607466i	$-0.792186\pi$
$431$	−6.12678e6	−1.58869	−0.794345	+	0.607466i	$0.792186\pi$
$432$	0	0
$433$	−1.76315e6	−0.451928	−0.225964	−	0.974136i	$-0.572553\pi$
$433$	−1.76315e6	−0.451928	−0.225964	+	0.974136i	$0.572553\pi$
$434$	0	0
$435$	− 2.73013e6i	− 0.691768i
$436$	0	0
$437$	− 5.46416e6i	− 1.36874i
$438$	0	0
$439$	3.85906e6	0.955696	0.477848	−	0.878443i	$-0.341417\pi$
$439$	3.85906e6	0.955696	0.477848	+	0.878443i	$0.341417\pi$
$440$	0	0
$441$	−3.30423e6	−0.809048
$442$	0	0
$443$	4.39396e6i	1.06377i	0.846817	+	0.531884i	$0.178516\pi$
$443$	4.39396e6i	1.06377i	−0.846817	+	0.531884i	$0.821484\pi$
$444$	0	0
$445$	2.18994e6i	0.524242i
$446$	0	0
$447$	3.20067e6	0.757656
$448$	0	0
$449$	−793390.	−0.185725	−0.0928626	−	0.995679i	$-0.529602\pi$
$449$	−793390.	−0.185725	−0.0928626	+	0.995679i	$0.529602\pi$
$450$	0	0
$451$	357864.i	0.0828470i
$452$	0	0
$453$	9216.00i	0.00211007i
$454$	0	0
$455$	375360.	0.0850001
$456$	0	0
$457$	−7.04302e6	−1.57750	−0.788748	−	0.614717i	$-0.789270\pi$
$457$	−7.04302e6	−1.57750	−0.788748	+	0.614717i	$0.789270\pi$
$458$	0	0
$459$	1.42447e6i	0.315588i
$460$	0	0
$461$	− 7.43005e6i	− 1.62832i	−0.580641	−	0.814160i	$-0.697198\pi$
$461$	− 7.43005e6i	− 1.62832i	0.580641	−	0.814160i	$-0.302802\pi$
$462$	0	0
$463$	4.10567e6	0.890086	0.445043	−	0.895509i	$-0.353188\pi$
$463$	4.10567e6	0.890086	0.445043	+	0.895509i	$0.353188\pi$
$464$	0	0
$465$	−1.42229e6	−0.305039
$466$	0	0
$467$	3.39817e6i	0.721030i	0.932753	+	0.360515i	$0.117399\pi$
$467$	3.39817e6i	0.721030i	−0.932753	+	0.360515i	$0.882601\pi$
$468$	0	0
$469$	9.37824e6i	1.96874i
$470$	0	0
$471$	539586.	0.112075
$472$	0	0
$473$	1.40517e6	0.288786
$474$	0	0
$475$	− 3.78836e6i	− 0.770401i
$476$	0	0
$477$	1.83011e6i	0.368283i
$478$	0	0
$479$	−2.78133e6	−0.553877	−0.276939	−	0.960888i	$-0.589320\pi$
$479$	−2.78133e6	−0.553877	−0.276939	+	0.960888i	$0.589320\pi$
$480$	0	0
$481$	−200652.	−0.0395440
$482$	0	0
$483$	6.13440e6i	1.19648i
$484$	0	0
$485$	3.94937e6i	0.762384i
$486$	0	0
$487$	−2.06734e6	−0.394994	−0.197497	−	0.980304i	$-0.563281\pi$
$487$	−2.06734e6	−0.394994	−0.197497	+	0.980304i	$0.563281\pi$
$488$	0	0
$489$	−3.07400e6	−0.581343
$490$	0	0
$491$	7.65976e6i	1.43387i	0.697138	+	0.716937i	$0.254457\pi$
$491$	7.65976e6i	1.43387i	−0.697138	+	0.716937i	$0.745543\pi$
$492$	0	0
$493$	1.74336e7i	3.23050i
$494$	0	0
$495$	−341496.	−0.0626430
$496$	0	0
$497$	1.31693e7	2.39150
$498$	0	0
$499$	− 386580.i	− 0.0695005i	−0.999396	−	0.0347503i	$-0.988936\pi$
$499$	− 386580.i	− 0.0695005i	0.999396	−	0.0347503i	$-0.0110636\pi$
$500$	0	0
$501$	− 45144.0i	− 0.00803537i
$502$	0	0
$503$	−2.57326e6	−0.453485	−0.226743	−	0.973955i	$-0.572808\pi$
$503$	−2.57326e6	−0.453485	−0.226743	+	0.973955i	$0.572808\pi$
$504$	0	0
$505$	−2.27236e6	−0.396504
$506$	0	0
$507$	− 3.32259e6i	− 0.574060i
$508$	0	0
$509$	− 360678.i	− 0.0617057i	−0.999524	−	0.0308528i	$-0.990178\pi$
$509$	− 360678.i	− 0.0617057i	0.999524	−	0.0308528i	$-0.00982232\pi$
$510$	0	0
$511$	5.04816e6	0.855226
$512$	0	0
$513$	1.40260e6	0.235309
$514$	0	0
$515$	− 2.17600e6i	− 0.361527i
$516$	0	0
$517$	− 868992.i	− 0.142985i
$518$	0	0
$519$	2.05799e6	0.335371
$520$	0	0
$521$	1.55908e6	0.251636	0.125818	−	0.992053i	$-0.459844\pi$
$521$	1.55908e6	0.251636	0.125818	+	0.992053i	$0.459844\pi$
$522$	0	0
$523$	9.18220e6i	1.46789i	0.679210	+	0.733944i	$0.262322\pi$
$523$	9.18220e6i	1.46789i	−0.679210	+	0.733944i	$0.737678\pi$
$524$	0	0
$525$	4.25304e6i	0.673444i
$526$	0	0
$527$	9.08219e6	1.42451
$528$	0	0
$529$	1.62926e6	0.253134
$530$	0	0
$531$	− 2268.00i	0 0.000349065i
$532$	0	0
$533$	− 132756.i	− 0.0202412i
$534$	0	0
$535$	512856.	0.0774660
$536$	0	0
$537$	1.45249e6	0.217359
$538$	0	0
$539$	5.05833e6i	0.749955i
$540$	0	0
$541$	6.67773e6i	0.980925i	0.871462	+	0.490462i	$0.163172\pi$
$541$	6.67773e6i	0.980925i	−0.871462	+	0.490462i	$0.836828\pi$
$542$	0	0
$543$	−2.62521e6	−0.382089
$544$	0	0
$545$	1.34973e6	0.194651
$546$	0	0
$547$	8.89656e6i	1.27132i	0.771971	+	0.635658i	$0.219271\pi$
$547$	8.89656e6i	1.27132i	−0.771971	+	0.635658i	$0.780729\pi$
$548$	0	0
$549$	− 517266.i	− 0.0732459i
$550$	0	0
$551$	1.71659e7	2.40873
$552$	0	0
$553$	6.39168e6	0.888796
$554$	0	0
$555$	1.33477e6i	0.183939i
$556$	0	0
$557$	4.46070e6i	0.609207i	0.952479	+	0.304603i	$0.0985239\pi$
$557$	4.46070e6i	0.609207i	−0.952479	+	0.304603i	$0.901476\pi$
$558$	0	0
$559$	−521272.	−0.0705562
$560$	0	0
$561$	2.18066e6	0.292538
$562$	0	0
$563$	6.37660e6i	0.847849i	0.905698	+	0.423924i	$0.139348\pi$
$563$	6.37660e6i	0.847849i	−0.905698	+	0.423924i	$0.860652\pi$
$564$	0	0
$565$	− 5.27524e6i	− 0.695218i
$566$	0	0
$567$	−1.57464e6	−0.205695
$568$	0	0
$569$	−5.51143e6	−0.713648	−0.356824	−	0.934172i	$-0.616140\pi$
$569$	−5.51143e6	−0.713648	−0.356824	+	0.934172i	$0.616140\pi$
$570$	0	0
$571$	− 1.35431e6i	− 0.173831i	−0.996216	−	0.0869155i	$-0.972299\pi$
$571$	− 1.35431e6i	− 0.173831i	0.996216	−	0.0869155i	$-0.0277010\pi$
$572$	0	0
$573$	− 501120.i	− 0.0637610i
$574$	0	0
$575$	5.59196e6	0.705333
$576$	0	0
$577$	−5.00736e6	−0.626137	−0.313068	−	0.949731i	$-0.601357\pi$
$577$	−5.00736e6	−0.626137	−0.313068	+	0.949731i	$0.601357\pi$
$578$	0	0
$579$	1.58629e6i	0.196646i
$580$	0	0
$581$	− 1.34851e7i	− 1.65735i
$582$	0	0
$583$	2.80166e6	0.341384
$584$	0	0
$585$	126684.	0.0153050
$586$	0	0
$587$	− 2.69964e6i	− 0.323378i	−0.986842	−	0.161689i	$-0.948306\pi$
$587$	− 2.69964e6i	− 0.323378i	0.986842	−	0.161689i	$-0.0516941\pi$
$588$	0	0
$589$	− 8.94275e6i	− 1.06214i
$590$	0	0
$591$	−3.37149e6	−0.397057
$592$	0	0
$593$	1.31035e7	1.53021	0.765103	−	0.643908i	$-0.222688\pi$
$593$	1.31035e7	1.53021	0.765103	+	0.643908i	$0.222688\pi$
$594$	0	0
$595$	1.59446e7i	1.84639i
$596$	0	0
$597$	5.73984e6i	0.659119i
$598$	0	0
$599$	−5.22804e6	−0.595349	−0.297675	−	0.954667i	$-0.596211\pi$
$599$	−5.22804e6	−0.595349	−0.297675	+	0.954667i	$0.596211\pi$
$600$	0	0
$601$	−1.02248e7	−1.15470	−0.577351	−	0.816496i	$-0.695913\pi$
$601$	−1.02248e7	−1.15470	−0.577351	+	0.816496i	$0.695913\pi$
$602$	0	0
$603$	3.16516e6i	0.354488i
$604$	0	0
$605$	− 4.95295e6i	− 0.550143i
$606$	0	0
$607$	−8.81684e6	−0.971273	−0.485636	−	0.874161i	$-0.661412\pi$
$607$	−8.81684e6	−0.971273	−0.485636	+	0.874161i	$0.661412\pi$
$608$	0	0
$609$	−1.92715e7	−2.10558
$610$	0	0
$611$	322368.i	0.0349340i
$612$	0	0
$613$	1.13600e7i	1.22103i	0.792006	+	0.610514i	$0.209037\pi$
$613$	1.13600e7i	1.22103i	−0.792006	+	0.610514i	$0.790963\pi$
$614$	0	0
$615$	−883116.	−0.0941521
$616$	0	0
$617$	4.77356e6	0.504812	0.252406	−	0.967621i	$-0.418778\pi$
$617$	4.77356e6	0.504812	0.252406	+	0.967621i	$0.418778\pi$
$618$	0	0
$619$	2.55931e6i	0.268470i	0.990950	+	0.134235i	$0.0428577\pi$
$619$	2.55931e6i	0.268470i	−0.990950	+	0.134235i	$0.957142\pi$
$620$	0	0
$621$	2.07036e6i	0.215435i
$622$	0	0
$623$	1.54584e7	1.59567
$624$	0	0
$625$	264461.	0.0270808
$626$	0	0
$627$	− 2.14718e6i	− 0.218122i
$628$	0	0
$629$	− 8.52335e6i	− 0.858981i
$630$	0	0
$631$	−8.41981e6	−0.841839	−0.420919	−	0.907098i	$-0.638292\pi$
$631$	−8.41981e6	−0.841839	−0.420919	+	0.907098i	$0.638292\pi$
$632$	0	0
$633$	−8.14165e6	−0.807613
$634$	0	0
$635$	1.77045e6i	0.174240i
$636$	0	0
$637$	− 1.87648e6i	− 0.183229i
$638$	0	0
$639$	4.44463e6	0.430610
$640$	0	0
$641$	−1.21494e7	−1.16791	−0.583957	−	0.811785i	$-0.698496\pi$
$641$	−1.21494e7	−1.16791	−0.583957	+	0.811785i	$0.698496\pi$
$642$	0	0
$643$	− 1.08968e7i	− 1.03937i	−0.854358	−	0.519685i	$-0.826049\pi$
$643$	− 1.08968e7i	− 1.03937i	0.854358	−	0.519685i	$-0.173951\pi$
$644$	0	0
$645$	3.46759e6i	0.328193i
$646$	0	0
$647$	1.32166e7	1.24124	0.620622	−	0.784110i	$-0.286880\pi$
$647$	1.32166e7	1.24124	0.620622	+	0.784110i	$0.286880\pi$
$648$	0	0
$649$	−3472.00	−0.000323570 0
$650$	0	0
$651$	1.00397e7i	0.928469i
$652$	0	0
$653$	− 1.65915e7i	− 1.52266i	−0.648365	−	0.761329i	$-0.724547\pi$
$653$	− 1.65915e7i	− 1.52266i	0.648365	−	0.761329i	$-0.275453\pi$
$654$	0	0
$655$	5.43878e6	0.495334
$656$	0	0
$657$	1.70375e6	0.153990
$658$	0	0
$659$	− 2.29372e6i	− 0.205743i	−0.994695	−	0.102872i	$-0.967197\pi$
$659$	− 2.29372e6i	− 0.205743i	0.994695	−	0.102872i	$-0.0328031\pi$
$660$	0	0
$661$	− 719194.i	− 0.0640239i	−0.999487	−	0.0320120i	$-0.989809\pi$
$661$	− 719194.i	− 0.0640239i	0.999487	−	0.0320120i	$-0.0101915\pi$
$662$	0	0
$663$	−808956.	−0.0714728
$664$	0	0
$665$	1.56998e7	1.37671
$666$	0	0
$667$	2.53385e7i	2.20529i
$668$	0	0
$669$	5.57143e6i	0.481284i
$670$	0	0
$671$	−791864.	−0.0678960
$672$	0	0
$673$	8.64695e6	0.735911	0.367955	−	0.929843i	$-0.380058\pi$
$673$	8.64695e6	0.735911	0.367955	+	0.929843i	$0.380058\pi$
$674$	0	0
$675$	1.43540e6i	0.121259i
$676$	0	0
$677$	− 1.69592e7i	− 1.42211i	−0.703135	−	0.711056i	$-0.748217\pi$
$677$	− 1.69592e7i	− 1.42211i	0.703135	−	0.711056i	$-0.251783\pi$
$678$	0	0
$679$	2.78779e7	2.32052
$680$	0	0
$681$	−1.32277e7	−1.09299
$682$	0	0
$683$	− 1.87105e7i	− 1.53473i	−0.641209	−	0.767367i	$-0.721567\pi$
$683$	− 1.87105e7i	− 1.53473i	0.641209	−	0.767367i	$-0.278433\pi$
$684$	0	0
$685$	− 8.91745e6i	− 0.726130i
$686$	0	0
$687$	−29610.0	−0.00239357
$688$	0	0
$689$	−1.03932e6	−0.0834071
$690$	0	0
$691$	− 1.16204e7i	− 0.925820i	−0.886405	−	0.462910i	$-0.846805\pi$
$691$	− 1.16204e7i	− 0.925820i	0.886405	−	0.462910i	$-0.153195\pi$
$692$	0	0
$693$	2.41056e6i	0.190671i
$694$	0	0
$695$	−8.61982e6	−0.676918
$696$	0	0
$697$	5.63924e6	0.439682
$698$	0	0
$699$	8.41862e6i	0.651700i
$700$	0	0
$701$	− 2.23497e7i	− 1.71781i	−0.512132	−	0.858907i	$-0.671144\pi$
$701$	− 2.23497e7i	− 1.71781i	0.512132	−	0.858907i	$-0.328856\pi$
$702$	0	0
$703$	−8.39249e6	−0.640475
$704$	0	0
$705$	2.14445e6	0.162496
$706$	0	0
$707$	1.60402e7i	1.20687i
$708$	0	0
$709$	1.02353e7i	0.764687i	0.924020	+	0.382344i	$0.124883\pi$
$709$	1.02353e7i	0.764687i	−0.924020	+	0.382344i	$0.875117\pi$
$710$	0	0
$711$	2.15719e6	0.160035
$712$	0	0
$713$	1.32003e7	0.972435
$714$	0	0
$715$	− 193936.i	− 0.0141871i
$716$	0	0
$717$	− 7.88040e6i	− 0.572467i
$718$	0	0
$719$	1.70339e7	1.22883	0.614416	−	0.788982i	$-0.289392\pi$
$719$	1.70339e7	1.22883	0.614416	+	0.788982i	$0.289392\pi$
$720$	0	0
$721$	−1.53600e7	−1.10041
$722$	0	0
$723$	8.63293e6i	0.614203i
$724$	0	0
$725$	1.75674e7i	1.24126i
$726$	0	0
$727$	−1.62280e7	−1.13875	−0.569377	−	0.822077i	$-0.692815\pi$
$727$	−1.62280e7	−1.13875	−0.569377	+	0.822077i	$0.692815\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	− 2.21427e7i	− 1.53263i
$732$	0	0
$733$	2.17495e7i	1.49517i	0.664168	+	0.747583i	$0.268786\pi$
$733$	2.17495e7i	1.49517i	−0.664168	+	0.747583i	$0.731214\pi$
$734$	0	0
$735$	−1.24827e7	−0.852293
$736$	0	0
$737$	4.84542e6	0.328597
$738$	0	0
$739$	1.96200e7i	1.32156i	0.750578	+	0.660781i	$0.229775\pi$
$739$	1.96200e7i	1.32156i	−0.750578	+	0.660781i	$0.770225\pi$
$740$	0	0
$741$	796536.i	0.0532917i
$742$	0	0
$743$	1.74018e7	1.15644	0.578218	−	0.815882i	$-0.303748\pi$
$743$	1.74018e7	1.15644	0.578218	+	0.815882i	$0.303748\pi$
$744$	0	0
$745$	1.20914e7	0.798154
$746$	0	0
$747$	− 4.55123e6i	− 0.298419i
$748$	0	0
$749$	− 3.62016e6i	− 0.235789i
$750$	0	0
$751$	2.62693e7	1.69961	0.849803	−	0.527101i	$-0.176721\pi$
$751$	2.62693e7	1.69961	0.849803	+	0.527101i	$0.176721\pi$
$752$	0	0
$753$	−2.86981e6	−0.184445
$754$	0	0
$755$	34816.0i	0.00222286i
$756$	0	0
$757$	− 5.70356e6i	− 0.361748i	−0.983506	−	0.180874i	$-0.942107\pi$
$757$	− 5.70356e6i	− 0.361748i	0.983506	−	0.180874i	$-0.0578927\pi$
$758$	0	0
$759$	3.16944e6	0.199700
$760$	0	0
$761$	2.13762e7	1.33804	0.669020	−	0.743244i	$-0.266714\pi$
$761$	2.13762e7	1.33804	0.669020	+	0.743244i	$0.266714\pi$
$762$	0	0
$763$	− 9.52752e6i	− 0.592473i
$764$	0	0
$765$	5.38132e6i	0.332457i
$766$	0	0
$767$	1288.00	7.90547e−5 0
$768$	0	0
$769$	−2.01523e6	−0.122888	−0.0614439	−	0.998111i	$-0.519571\pi$
$769$	−2.01523e6	−0.122888	−0.0614439	+	0.998111i	$0.519571\pi$
$770$	0	0
$771$	− 1.54322e7i	− 0.934958i
$772$	0	0
$773$	− 1.27674e7i	− 0.768520i	−0.923225	−	0.384260i	$-0.874457\pi$
$773$	− 1.27674e7i	− 0.768520i	0.923225	−	0.384260i	$-0.125543\pi$
$774$	0	0
$775$	9.15191e6	0.547340
$776$	0	0
$777$	9.42192e6	0.559870
$778$	0	0
$779$	− 5.55266e6i	− 0.327837i
$780$	0	0
$781$	− 6.80413e6i	− 0.399158i
$782$	0	0
$783$	−6.50414e6	−0.379128
$784$	0	0
$785$	2.03844e6	0.118065
$786$	0	0
$787$	− 2.72384e7i	− 1.56764i	−0.620990	−	0.783818i	$-0.713269\pi$
$787$	− 2.72384e7i	− 1.56764i	0.620990	−	0.783818i	$-0.286731\pi$
$788$	0	0
$789$	9.99252e6i	0.571456i
$790$	0	0
$791$	−3.72370e7	−2.11608
$792$	0	0
$793$	293756.	0.0165884
$794$	0	0
$795$	6.91376e6i	0.387969i
$796$	0	0
$797$	7.66724e6i	0.427556i	0.976882	+	0.213778i	$0.0685770\pi$
$797$	7.66724e6i	0.427556i	−0.976882	+	0.213778i	$0.931423\pi$
$798$	0	0
$799$	−1.36936e7	−0.758843
$800$	0	0
$801$	5.21721e6	0.287314
$802$	0	0
$803$	− 2.60822e6i	− 0.142743i
$804$	0	0
$805$	2.31744e7i	1.26043i
$806$	0	0
$807$	3.58540e6	0.193800
$808$	0	0
$809$	1.05541e7	0.566956	0.283478	−	0.958979i	$-0.408512\pi$
$809$	1.05541e7	0.566956	0.283478	+	0.958979i	$0.408512\pi$
$810$	0	0
$811$	1.32883e6i	0.0709442i	0.999371	+	0.0354721i	$0.0112935\pi$
$811$	1.32883e6i	0.0709442i	−0.999371	+	0.0354721i	$0.988707\pi$
$812$	0	0
$813$	1.29778e7i	0.688611i
$814$	0	0
$815$	−1.16129e7	−0.612416
$816$	0	0
$817$	−2.18028e7	−1.14276
$818$	0	0
$819$	− 894240.i	− 0.0465848i
$820$	0	0
$821$	− 6.15933e6i	− 0.318915i	−0.987205	−	0.159458i	$-0.949025\pi$
$821$	− 6.15933e6i	− 0.318915i	0.987205	−	0.159458i	$-0.0509746\pi$
$822$	0	0
$823$	1.00734e7	0.518414	0.259207	−	0.965822i	$-0.416539\pi$
$823$	1.00734e7	0.518414	0.259207	+	0.965822i	$0.416539\pi$
$824$	0	0
$825$	2.19740e6	0.112402
$826$	0	0
$827$	6.49152e6i	0.330052i	0.986289	+	0.165026i	$0.0527708\pi$
$827$	6.49152e6i	0.330052i	−0.986289	+	0.165026i	$0.947229\pi$
$828$	0	0
$829$	1.93536e7i	0.978082i	0.872261	+	0.489041i	$0.162653\pi$
$829$	1.93536e7i	0.978082i	−0.872261	+	0.489041i	$0.837347\pi$
$830$	0	0
$831$	1.05514e6	0.0530040
$832$	0	0
$833$	7.97095e7	3.98013
$834$	0	0
$835$	− 170544.i	− 0.00846487i
$836$	0	0
$837$	3.38839e6i	0.167178i
$838$	0	0
$839$	−2.78622e7	−1.36650	−0.683251	−	0.730183i	$-0.739435\pi$
$839$	−2.78622e7	−1.36650	−0.683251	+	0.730183i	$0.739435\pi$
$840$	0	0
$841$	−5.90909e7	−2.88092
$842$	0	0
$843$	− 1.50763e7i	− 0.730677i
$844$	0	0
$845$	− 1.25520e7i	− 0.604744i
$846$	0	0
$847$	−3.49620e7	−1.67451
$848$	0	0
$849$	−1.73221e7	−0.824766
$850$	0	0
$851$	− 1.23881e7i	− 0.586381i
$852$	0	0
$853$	1.07651e7i	0.506577i	0.967391	+	0.253288i	$0.0815121\pi$
$853$	1.07651e7i	0.506577i	−0.967391	+	0.253288i	$0.918488\pi$
$854$	0	0
$855$	5.29870e6	0.247887
$856$	0	0
$857$	−1.22439e7	−0.569465	−0.284733	−	0.958607i	$-0.591905\pi$
$857$	−1.22439e7	−0.569465	−0.284733	+	0.958607i	$0.591905\pi$
$858$	0	0
$859$	1.38664e6i	0.0641179i	0.999486	+	0.0320590i	$0.0102064\pi$
$859$	1.38664e6i	0.0641179i	−0.999486	+	0.0320590i	$0.989794\pi$
$860$	0	0
$861$	6.23376e6i	0.286578i
$862$	0	0
$863$	1.09856e7	0.502109	0.251055	−	0.967973i	$-0.419223\pi$
$863$	1.09856e7	0.502109	0.251055	+	0.967973i	$0.419223\pi$
$864$	0	0
$865$	7.77464e6	0.353297
$866$	0	0
$867$	− 2.15843e7i	− 0.975194i
$868$	0	0
$869$	− 3.30237e6i	− 0.148346i
$870$	0	0
$871$	−1.79750e6	−0.0802828
$872$	0	0
$873$	9.40880e6	0.417829
$874$	0	0
$875$	4.15670e7i	1.83539i
$876$	0	0
$877$	− 8.17798e6i	− 0.359044i	−0.983754	−	0.179522i	$-0.942545\pi$
$877$	− 8.17798e6i	− 0.359044i	0.983754	−	0.179522i	$-0.0574550\pi$
$878$	0	0
$879$	1.15256e7	0.503143
$880$	0	0
$881$	4.66520e6	0.202503	0.101251	−	0.994861i	$-0.467715\pi$
$881$	4.66520e6	0.202503	0.101251	+	0.994861i	$0.467715\pi$
$882$	0	0
$883$	3.82201e7i	1.64964i	0.565393	+	0.824822i	$0.308724\pi$
$883$	3.82201e7i	1.64964i	−0.565393	+	0.824822i	$0.691276\pi$
$884$	0	0
$885$	− 8568.00i	0 0.000367723i
$886$	0	0
$887$	−7.72172e6	−0.329538	−0.164769	−	0.986332i	$-0.552688\pi$
$887$	−7.72172e6	−0.329538	−0.164769	+	0.986332i	$0.552688\pi$
$888$	0	0
$889$	1.24973e7	0.530348
$890$	0	0
$891$	813564.i	0.0343319i
$892$	0	0
$893$	1.34834e7i	0.565810i
$894$	0	0
$895$	5.48719e6	0.228977
$896$	0	0
$897$	−1.17576e6	−0.0487908
$898$	0	0
$899$	4.14695e7i	1.71131i
$900$	0	0
$901$	− 4.41487e7i	− 1.81178i
$902$	0	0
$903$	2.44771e7	0.998944
$904$	0	0
$905$	−9.91746e6	−0.402512
$906$	0	0
$907$	− 4.33137e7i	− 1.74826i	−0.485689	−	0.874131i	$-0.661431\pi$
$907$	− 4.33137e7i	− 1.74826i	0.485689	−	0.874131i	$-0.338569\pi$
$908$	0	0
$909$	5.41355e6i	0.217307i
$910$	0	0
$911$	−3.44456e6	−0.137511	−0.0687556	−	0.997634i	$-0.521903\pi$
$911$	−3.44456e6	−0.137511	−0.0687556	+	0.997634i	$0.521903\pi$
$912$	0	0
$913$	−6.96731e6	−0.276623
$914$	0	0
$915$	− 1.95412e6i	− 0.0771610i
$916$	0	0
$917$	− 3.83914e7i	− 1.50768i
$918$	0	0
$919$	−4.37073e7	−1.70712	−0.853562	−	0.520991i	$-0.825563\pi$
$919$	−4.37073e7	−1.70712	−0.853562	+	0.520991i	$0.825563\pi$
$920$	0	0
$921$	−2.03687e7	−0.791251
$922$	0	0
$923$	2.52411e6i	0.0975224i
$924$	0	0
$925$	− 8.58878e6i	− 0.330048i
$926$	0	0
$927$	−5.18400e6	−0.198137
$928$	0	0
$929$	−4.13022e7	−1.57012	−0.785062	−	0.619418i	$-0.787369\pi$
$929$	−4.13022e7	−1.57012	−0.785062	+	0.619418i	$0.787369\pi$
$930$	0	0
$931$	− 7.84857e7i	− 2.96768i
$932$	0	0
$933$	− 2.23063e6i	− 0.0838926i
$934$	0	0
$935$	8.23806e6	0.308174
$936$	0	0
$937$	−9.57460e6	−0.356264	−0.178132	−	0.984007i	$-0.557005\pi$
$937$	−9.57460e6	−0.356264	−0.178132	+	0.984007i	$0.557005\pi$
$938$	0	0
$939$	− 1.64152e7i	− 0.607550i
$940$	0	0
$941$	− 8.71623e6i	− 0.320889i	−0.987045	−	0.160444i	$-0.948707\pi$
$941$	− 8.71623e6i	− 0.320889i	0.987045	−	0.160444i	$-0.0512927\pi$
$942$	0	0
$943$	8.19624e6	0.300148
$944$	0	0
$945$	−5.94864e6	−0.216690
$946$	0	0
$947$	− 1.30605e7i	− 0.473244i	−0.971602	−	0.236622i	$-0.923960\pi$
$947$	− 1.30605e7i	− 0.473244i	0.971602	−	0.236622i	$-0.0760403\pi$
$948$	0	0
$949$	967564.i	0.0348750i
$950$	0	0
$951$	−2.57066e7	−0.921710
$952$	0	0
$953$	−1.13875e7	−0.406158	−0.203079	−	0.979162i	$-0.565095\pi$
$953$	−1.13875e7	−0.406158	−0.203079	+	0.979162i	$0.565095\pi$
$954$	0	0
$955$	− 1.89312e6i	− 0.0671691i
$956$	0	0
$957$	9.95695e6i	0.351436i
$958$	0	0
$959$	−6.29467e7	−2.21017
$960$	0	0
$961$	−7.02525e6	−0.245388
$962$	0	0
$963$	− 1.22180e6i	− 0.0424557i
$964$	0	0
$965$	5.99264e6i	0.207157i
$966$	0	0
$967$	4.62711e7	1.59127	0.795634	−	0.605778i	$-0.207138\pi$
$967$	4.62711e7	1.59127	0.795634	+	0.605778i	$0.207138\pi$
$968$	0	0
$969$	−3.38355e7	−1.15761
$970$	0	0
$971$	− 1.63206e7i	− 0.555506i	−0.960653	−	0.277753i	$-0.910410\pi$
$971$	− 1.63206e7i	− 0.555506i	0.960653	−	0.277753i	$-0.0895896\pi$
$972$	0	0
$973$	6.08458e7i	2.06038i
$974$	0	0
$975$	−815166.	−0.0274621
$976$	0	0
$977$	−1.95213e7	−0.654294	−0.327147	−	0.944973i	$-0.606087\pi$
$977$	−1.95213e7	−0.654294	−0.327147	+	0.944973i	$0.606087\pi$
$978$	0	0
$979$	− 7.98684e6i	− 0.266329i
$980$	0	0
$981$	− 3.21554e6i	− 0.106680i
$982$	0	0
$983$	−4.33962e7	−1.43241	−0.716207	−	0.697888i	$-0.754123\pi$
$983$	−4.33962e7	−1.43241	−0.716207	+	0.697888i	$0.754123\pi$
$984$	0	0
$985$	−1.27367e7	−0.418281
$986$	0	0
$987$	− 1.51373e7i	− 0.494601i
$988$	0	0
$989$	− 3.21829e7i	− 1.04625i
$990$	0	0
$991$	−3.83518e7	−1.24051	−0.620257	−	0.784399i	$-0.712972\pi$
$991$	−3.83518e7	−1.24051	−0.620257	+	0.784399i	$0.712972\pi$
$992$	0	0
$993$	1.32433e6	0.0426210
$994$	0	0
$995$	2.16838e7i	0.694350i
$996$	0	0
$997$	− 7.82206e6i	− 0.249220i	−0.992206	−	0.124610i	$-0.960232\pi$
$997$	− 7.82206e6i	− 0.249220i	0.992206	−	0.124610i	$-0.0397680\pi$
$998$	0	0
$999$	3.17990e6	0.100809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.6.d.a.385.1		2
4.3	odd	2		768.6.d.r.385.2		2
8.3	odd	2		768.6.d.r.385.1		2
8.5	even	2	inner	768.6.d.a.385.2		2
16.3	odd	4		24.6.a.a.1.1	✓	1
16.5	even	4		192.6.a.f.1.1		1
16.11	odd	4		192.6.a.n.1.1		1
16.13	even	4		48.6.a.d.1.1		1
48.5	odd	4		576.6.a.l.1.1		1
48.11	even	4		576.6.a.k.1.1		1
48.29	odd	4		144.6.a.i.1.1		1
48.35	even	4		72.6.a.e.1.1		1
80.3	even	4		600.6.f.f.49.1		2
80.19	odd	4		600.6.a.i.1.1		1
80.67	even	4		600.6.f.f.49.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.6.a.a.1.1	✓	1	16.3	odd	4
48.6.a.d.1.1		1	16.13	even	4
72.6.a.e.1.1		1	48.35	even	4
144.6.a.i.1.1		1	48.29	odd	4
192.6.a.f.1.1		1	16.5	even	4
192.6.a.n.1.1		1	16.11	odd	4
576.6.a.k.1.1		1	48.11	even	4
576.6.a.l.1.1		1	48.5	odd	4
600.6.a.i.1.1		1	80.19	odd	4
600.6.f.f.49.1		2	80.3	even	4
600.6.f.f.49.2		2	80.67	even	4
768.6.d.a.385.1		2	1.1	even	1	trivial
768.6.d.a.385.2		2	8.5	even	2	inner
768.6.d.r.385.1		2	8.3	odd	2
768.6.d.r.385.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.00000i q^{3} -34.0000i q^{5} -240.000 q^{7} -81.0000 q^{9} +124.000i q^{11} -46.0000i q^{13} -306.000 q^{15} +1954.00 q^{17} -1924.00i q^{19} +2160.00i q^{21} +2840.00 q^{23} +1969.00 q^{25} +729.000i q^{27} +8922.00i q^{29} +4648.00 q^{31} +1116.00 q^{33} +8160.00i q^{35} -4362.00i q^{37} -414.000 q^{39} +2886.00 q^{41} -11332.0i q^{43} +2754.00i q^{45} -7008.00 q^{47} +40793.0 q^{49} -17586.0i q^{51} -22594.0i q^{53} +4216.00 q^{55} -17316.0 q^{57} +28.0000i q^{59} +6386.00i q^{61} +19440.0 q^{63} -1564.00 q^{65} -39076.0i q^{67} -25560.0i q^{69} -54872.0 q^{71} -21034.0 q^{73} -17721.0i q^{75} -29760.0i q^{77} -26632.0 q^{79} +6561.00 q^{81} +56188.0i q^{83} -66436.0i q^{85} +80298.0 q^{87} -64410.0 q^{89} +11040.0i q^{91} -41832.0i q^{93} -65416.0 q^{95} -116158. q^{97} -10044.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 480 q^{7} - 162 q^{9} - 612 q^{15} + 3908 q^{17} + 5680 q^{23} + 3938 q^{25} + 9296 q^{31} + 2232 q^{33} - 828 q^{39} + 5772 q^{41} - 14016 q^{47} + 81586 q^{49} + 8432 q^{55} - 34632 q^{57} + 38880 q^{63}+ \cdots - 232316 q^{97}+O(q^{100}) \) 2 * q - 480 * q^7 - 162 * q^9 - 612 * q^15 + 3908 * q^17 + 5680 * q^23 + 3938 * q^25 + 9296 * q^31 + 2232 * q^33 - 828 * q^39 + 5772 * q^41 - 14016 * q^47 + 81586 * q^49 + 8432 * q^55 - 34632 * q^57 + 38880 * q^63 - 3128 * q^65 - 109744 * q^71 - 42068 * q^73 - 53264 * q^79 + 13122 * q^81 + 160596 * q^87 - 128820 * q^89 - 130832 * q^95 - 232316 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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