LMFDB - Embedded newform 432.4.i.d.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,4,Mod(145,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("432.145");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	432.i (of order $3$, degree $2$, 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:	$25.4888251225$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.6831243.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 13x^{4} + 49x^{2} + 48 $ x^6 + 13x^4 + 49x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$-1.23396i$ of defining polynomial
Character	$\chi$	$=$	432.289
Dual form			432.4.i.d.145.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.92194 - 11.9892i) q^{5} +(15.3540 - 26.5939i) q^{7} +(21.9523 - 38.0225i) q^{11} +(6.11853 + 10.5976i) q^{13} -76.0286 q^{17} +44.1789 q^{19} +(39.3135 + 68.0930i) q^{23} +(-33.3265 + 57.7232i) q^{25} +(-46.3903 + 80.3504i) q^{29} +(-71.5329 - 123.899i) q^{31} -425.118 q^{35} -32.4741 q^{37} +(-167.778 - 290.599i) q^{41} +(-249.166 + 431.567i) q^{43} +(140.882 - 244.016i) q^{47} +(-299.990 - 519.599i) q^{49} +628.565 q^{53} -607.810 q^{55} +(-252.327 - 437.043i) q^{59} +(-185.951 + 322.076i) q^{61} +(84.7041 - 146.712i) q^{65} +(-81.3304 - 140.868i) q^{67} +433.512 q^{71} -629.645 q^{73} +(-674.111 - 1167.59i) q^{77} +(86.3649 - 149.588i) q^{79} +(87.4288 - 151.431i) q^{83} +(526.265 + 911.518i) q^{85} -336.716 q^{89} +375.775 q^{91} +(-305.804 - 529.668i) q^{95} +(-42.1594 + 73.0222i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{5} + 6 q^{7} + 51 q^{11} + 12 q^{13} + 222 q^{17} - 30 q^{19} + 210 q^{23} - 3 q^{25} - 456 q^{29} - 48 q^{31} - 1104 q^{35} - 96 q^{37} - 897 q^{41} - 129 q^{43} + 522 q^{47} - 225 q^{49} + 2208 q^{53}+ \cdots + 93 q^{97}+O(q^{100}) $ 6 * q - 6 * q^5 + 6 * q^7 + 51 * q^11 + 12 * q^13 + 222 * q^17 - 30 * q^19 + 210 * q^23 - 3 * q^25 - 456 * q^29 - 48 * q^31 - 1104 * q^35 - 96 * q^37 - 897 * q^41 - 129 * q^43 + 522 * q^47 - 225 * q^49 + 2208 * q^53 + 216 * q^55 + 453 * q^59 - 402 * q^61 - 1110 * q^65 + 213 * q^67 + 120 * q^71 + 750 * q^73 - 1128 * q^77 - 552 * q^79 - 612 * q^83 + 1188 * q^85 + 924 * q^89 + 264 * q^91 - 2184 * q^95 + 93 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−6.92194	−	11.9892i	−0.619117	−	1.07234i	−0.989647	−	0.143522i	$-0.954157\pi$
$5$	−6.92194	−	11.9892i	−0.619117	−	1.07234i	0.370530	−	0.928821i	$-0.379176\pi$
$6$		0			0
$7$	15.3540	−	26.5939i	0.829038	−	1.43594i	−0.0697558	−	0.997564i	$-0.522222\pi$
$7$	15.3540	−	26.5939i	0.829038	−	1.43594i	0.898794	−	0.438372i	$-0.144445\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	21.9523	−	38.0225i	0.601715	−	1.04220i	−0.390846	−	0.920456i	$-0.627818\pi$
$11$	21.9523	−	38.0225i	0.601715	−	1.04220i	0.992561	−	0.121745i	$-0.0388490\pi$
$12$		0			0
$13$	6.11853	+	10.5976i	0.130536	+	0.226096i	0.923883	−	0.382674i	$-0.124997\pi$
$13$	6.11853	+	10.5976i	0.130536	+	0.226096i	−0.793347	+	0.608770i	$0.791663\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−76.0286			−1.08468			−0.542342	−	0.840158i	$-0.682462\pi$
$17$	−76.0286			−1.08468			−0.542342	+	0.840158i	$0.682462\pi$
$18$		0			0
$19$	44.1789			0.533439			0.266720	−	0.963774i	$-0.414060\pi$
$19$	44.1789			0.533439			0.266720	+	0.963774i	$0.414060\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	39.3135	+	68.0930i	0.356410	+	0.617321i	0.987358	−	0.158504i	$-0.0506672\pi$
$23$	39.3135	+	68.0930i	0.356410	+	0.617321i	−0.630948	+	0.775825i	$0.717334\pi$
$24$		0			0
$25$	−33.3265	+	57.7232i	−0.266612	+	0.461786i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−46.3903	+	80.3504i	−0.297050	+	0.514506i	−0.975460	−	0.220178i	$-0.929336\pi$
$29$	−46.3903	+	80.3504i	−0.297050	+	0.514506i	0.678409	+	0.734684i	$0.262670\pi$
$30$		0			0
$31$	−71.5329	−	123.899i	−0.414442	−	0.717834i	0.580928	−	0.813955i	$-0.302690\pi$
$31$	−71.5329	−	123.899i	−0.414442	−	0.717834i	−0.995370	+	0.0961208i	$0.969356\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−425.118			−2.05309
$36$		0			0
$37$	−32.4741			−0.144289			−0.0721447	−	0.997394i	$-0.522984\pi$
$37$	−32.4741			−0.144289			−0.0721447	+	0.997394i	$0.522984\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−167.778	−	290.599i	−0.639084	−	1.10693i	−0.985634	−	0.168895i	$-0.945980\pi$
$41$	−167.778	−	290.599i	−0.639084	−	1.10693i	0.346550	−	0.938031i	$-0.387353\pi$
$42$		0			0
$43$	−249.166	+	431.567i	−0.883660	+	1.53054i	−0.0364186	+	0.999337i	$0.511595\pi$
$43$	−249.166	+	431.567i	−0.883660	+	1.53054i	−0.847242	+	0.531208i	$0.821738\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	140.882	−	244.016i	0.437230	−	0.757305i	−0.560244	−	0.828327i	$-0.689293\pi$
$47$	140.882	−	244.016i	0.437230	−	0.757305i	0.997475	+	0.0710223i	$0.0226261\pi$
$48$		0			0
$49$	−299.990	−	519.599i	−0.874608	−	1.51487i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	628.565			1.62906			0.814529	−	0.580123i	$-0.196995\pi$
$53$	628.565			1.62906			0.814529	+	0.580123i	$0.196995\pi$
$54$		0			0
$55$	−607.810			−1.49013
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−252.327	−	437.043i	−0.556782	−	0.964375i	−0.997762	−	0.0668584i	$-0.978702\pi$
$59$	−252.327	−	437.043i	−0.556782	−	0.964375i	0.440980	−	0.897517i	$-0.354631\pi$
$60$		0			0
$61$	−185.951	+	322.076i	−0.390304	+	0.676026i	−0.992489	−	0.122330i	$-0.960963\pi$
$61$	−185.951	+	322.076i	−0.390304	+	0.676026i	0.602186	+	0.798356i	$0.294297\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	84.7041	−	146.712i	0.161635	−	0.279960i
$66$		0			0
$67$	−81.3304	−	140.868i	−0.148300	−	0.256863i	0.782299	−	0.622903i	$-0.214047\pi$
$67$	−81.3304	−	140.868i	−0.148300	−	0.256863i	−0.930599	+	0.366040i	$0.880713\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	433.512			0.724626			0.362313	−	0.932056i	$-0.381987\pi$
$71$	433.512			0.724626			0.362313	+	0.932056i	$0.381987\pi$
$72$		0			0
$73$	−629.645			−1.00951			−0.504756	−	0.863262i	$-0.668418\pi$
$73$	−629.645			−1.00951			−0.504756	+	0.863262i	$0.668418\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−674.111	−	1167.59i	−0.997689	−	1.72805i
$78$		0			0
$79$	86.3649	−	149.588i	0.122998	−	0.213038i	−0.797951	−	0.602723i	$-0.794083\pi$
$79$	86.3649	−	149.588i	0.122998	−	0.213038i	0.920948	+	0.389684i	$0.127416\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	87.4288	−	151.431i	0.115621	−	0.200262i	−0.802407	−	0.596778i	$-0.796447\pi$
$83$	87.4288	−	151.431i	0.115621	−	0.200262i	0.918028	+	0.396516i	$0.129781\pi$
$84$		0			0
$85$	526.265	+	911.518i	0.671547	+	1.16315i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−336.716			−0.401032			−0.200516	−	0.979690i	$-0.564262\pi$
$89$	−336.716			−0.401032			−0.200516	+	0.979690i	$0.564262\pi$
$90$		0			0
$91$	375.775			0.432879
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−305.804	−	529.668i	−0.330261	−	0.572029i
$96$		0			0
$97$	−42.1594	+	73.0222i	−0.0441303	+	0.0764358i	−0.887247	−	0.461295i	$-0.847385\pi$
$97$	−42.1594	+	73.0222i	−0.0441303	+	0.0764358i	0.843117	+	0.537731i	$0.180718\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−874.902	+	1515.38i	−0.861941	+	1.49293i	0.00811161	+	0.999967i	$0.497418\pi$
$101$	−874.902	+	1515.38i	−0.861941	+	1.49293i	−0.870053	+	0.492959i	$0.835915\pi$
$102$		0			0
$103$	55.6978	+	96.4714i	0.0532822	+	0.0922875i	0.891436	−	0.453146i	$-0.149698\pi$
$103$	55.6978	+	96.4714i	0.0532822	+	0.0922875i	−0.838154	+	0.545433i	$0.816365\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	895.520			0.809095			0.404548	−	0.914517i	$-0.367429\pi$
$107$	895.520			0.809095			0.404548	+	0.914517i	$0.367429\pi$
$108$		0			0
$109$	−716.957			−0.630019			−0.315009	−	0.949089i	$-0.602008\pi$
$109$	−716.957			−0.630019			−0.315009	+	0.949089i	$0.602008\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	115.526	+	200.096i	0.0961746	+	0.166579i	0.910098	−	0.414393i	$-0.136006\pi$
$113$	115.526	+	200.096i	0.0961746	+	0.166579i	−0.813924	+	0.580972i	$0.802673\pi$
$114$		0			0
$115$	544.252	−	942.672i	0.441320	−	0.764388i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1167.34	+	2021.90i	−0.899245	+	1.55754i
$120$		0			0
$121$	−298.306	−	516.681i	−0.224122	−	0.388190i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−807.748			−0.577978
$126$		0			0
$127$	−1715.22			−1.19843			−0.599217	−	0.800586i	$-0.704522\pi$
$127$	−1715.22			−1.19843			−0.599217	+	0.800586i	$0.704522\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−666.435	−	1154.30i	−0.444479	−	0.769859i	0.553537	−	0.832824i	$-0.313278\pi$
$131$	−666.435	−	1154.30i	−0.444479	−	0.769859i	−0.998016	+	0.0629650i	$0.979944\pi$
$132$		0			0
$133$	678.323	−	1174.89i	0.442241	−	0.765984i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1259.49	−	2181.50i	0.785443	−	1.36043i	−0.143292	−	0.989680i	$-0.545769\pi$
$137$	1259.49	−	2181.50i	0.785443	−	1.36043i	0.928734	−	0.370746i	$-0.120898\pi$
$138$		0			0
$139$	311.442	+	539.433i	0.190044	+	0.329166i	0.945265	−	0.326305i	$-0.105804\pi$
$139$	311.442	+	539.433i	0.190044	+	0.329166i	−0.755220	+	0.655471i	$0.772470\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	537.263			0.314183
$144$		0			0
$145$	1284.44			0.735636
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1204.47	+	2086.21i	0.662243	+	1.14704i	0.980025	+	0.198875i	$0.0637289\pi$
$149$	1204.47	+	2086.21i	0.662243	+	1.14704i	−0.317781	+	0.948164i	$0.602938\pi$
$150$		0			0
$151$	33.8218	−	58.5810i	0.0182277	−	0.0315712i	−0.856768	−	0.515703i	$-0.827531\pi$
$151$	33.8218	−	58.5810i	0.0182277	−	0.0315712i	0.874995	+	0.484131i	$0.160864\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−990.293	+	1715.24i	−0.513176	+	0.888847i
$156$		0			0
$157$	2.19676	+	3.80490i	0.00111669	+	0.00193417i	0.866583	−	0.499033i	$-0.166311\pi$
$157$	2.19676	+	3.80490i	0.00111669	+	0.00193417i	−0.865467	+	0.500967i	$0.832978\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2414.48			1.18191
$162$		0			0
$163$	2863.88			1.37617			0.688087	−	0.725628i	$-0.258451\pi$
$163$	2863.88			1.37617			0.688087	+	0.725628i	$0.258451\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−214.757	−	371.970i	−0.0995114	−	0.172359i	0.811971	−	0.583698i	$-0.198395\pi$
$167$	−214.757	−	371.970i	−0.0995114	−	0.172359i	−0.911483	+	0.411339i	$0.865061\pi$
$168$		0			0
$169$	1023.63	−	1772.97i	0.465920	−	0.806998i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1287.14	+	2229.40i	−0.565663	+	0.979756i	0.431325	+	0.902197i	$0.358046\pi$
$173$	−1287.14	+	2229.40i	−0.565663	+	0.979756i	−0.996988	+	0.0775599i	$0.975287\pi$
$174$		0			0
$175$	1023.39	+	1772.56i	0.442063	+	0.765676i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	807.448			0.337159			0.168580	−	0.985688i	$-0.446082\pi$
$179$	807.448			0.337159			0.168580	+	0.985688i	$0.446082\pi$
$180$		0			0
$181$	4296.57			1.76443			0.882215	−	0.470847i	$-0.156052\pi$
$181$	4296.57			1.76443			0.882215	+	0.470847i	$0.156052\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	224.784	+	389.337i	0.0893321	+	0.154728i
$186$		0			0
$187$	−1669.00	+	2890.79i	−0.652671	+	1.13046i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	153.147	−	265.259i	0.0580176	−	0.100489i	−0.835558	−	0.549402i	$-0.814855\pi$
$191$	153.147	−	265.259i	0.0580176	−	0.100489i	0.893575	+	0.448913i	$0.148189\pi$
$192$		0			0
$193$	856.177	+	1482.94i	0.319321	+	0.553080i	0.980347	−	0.197283i	$-0.0632119\pi$
$193$	856.177	+	1482.94i	0.319321	+	0.553080i	−0.661026	+	0.750363i	$0.729879\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	263.636			0.0953466			0.0476733	−	0.998863i	$-0.484819\pi$
$197$	263.636			0.0953466			0.0476733	+	0.998863i	$0.484819\pi$
$198$		0			0
$199$	3835.87			1.36642			0.683211	−	0.730221i	$-0.260583\pi$
$199$	3835.87			1.36642			0.683211	+	0.730221i	$0.260583\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1424.55	+	2467.40i	0.492532	+	0.853091i
$204$		0			0
$205$	−2322.69	+	4023.02i	−0.791336	+	1.37063i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	969.829	−	1679.79i	0.320978	−	0.555951i
$210$		0			0
$211$	−2034.66	−	3524.14i	−0.663848	−	1.14982i	−0.979596	−	0.200976i	$-0.935589\pi$
$211$	−2034.66	−	3524.14i	−0.663848	−	1.14982i	0.315748	−	0.948843i	$-0.397745\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6898.84			2.18836
$216$		0			0
$217$	−4393.27			−1.37435
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−465.183	−	805.720i	−0.141591	−	0.245243i
$222$		0			0
$223$	3078.07	−	5331.38i	0.924318	−	1.60097i	0.131664	−	0.991294i	$-0.457968\pi$
$223$	3078.07	−	5331.38i	0.924318	−	1.60097i	0.792654	−	0.609672i	$-0.208699\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.1307	+	21.0110i	−0.00354689	+	0.00614339i	−0.867793	−	0.496925i	$-0.834462\pi$
$227$	−12.1307	+	21.0110i	−0.00354689	+	0.00614339i	0.864247	+	0.503069i	$0.167796\pi$
$228$		0			0
$229$	316.209	+	547.691i	0.0912476	+	0.158045i	0.908036	−	0.418891i	$-0.137581\pi$
$229$	316.209	+	547.691i	0.0912476	+	0.158045i	−0.816789	+	0.576937i	$0.804248\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3163.59			0.889502			0.444751	−	0.895654i	$-0.353292\pi$
$233$	3163.59			0.889502			0.444751	+	0.895654i	$0.353292\pi$
$234$		0			0
$235$	−3900.72			−1.08279
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1110.57	−	1923.57i	−0.300573	−	0.520608i	0.675693	−	0.737183i	$-0.263845\pi$
$239$	−1110.57	−	1923.57i	−0.300573	−	0.520608i	−0.976266	+	0.216575i	$0.930511\pi$
$240$		0			0
$241$	300.730	−	520.879i	0.0803805	−	0.139223i	−0.823033	−	0.567994i	$-0.807720\pi$
$241$	300.730	−	520.879i	0.0803805	−	0.139223i	0.903413	+	0.428771i	$0.141053\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4153.03	+	7193.26i	−1.08297	+	1.87576i
$246$		0			0
$247$	270.310	+	468.191i	0.0696332	+	0.120608i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1350.71			0.339665			0.169833	−	0.985473i	$-0.445677\pi$
$251$	1350.71			0.339665			0.169833	+	0.985473i	$0.445677\pi$
$252$		0			0
$253$	3452.09			0.857830
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3521.95	−	6100.20i	−0.854838	−	1.48062i	−0.876795	−	0.480865i	$-0.840323\pi$
$257$	−3521.95	−	6100.20i	−0.854838	−	1.48062i	0.0219564	−	0.999759i	$-0.493010\pi$
$258$		0			0
$259$	−498.607	+	863.613i	−0.119621	+	0.207190i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1162.13	+	2012.87i	−0.272472	+	0.471935i	−0.969494	−	0.245114i	$-0.921175\pi$
$263$	−1162.13	+	2012.87i	−0.272472	+	0.471935i	0.697022	+	0.717050i	$0.254508\pi$
$264$		0			0
$265$	−4350.89	−	7535.97i	−1.00858	−	1.74691i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3675.87			0.833167			0.416584	−	0.909097i	$-0.363227\pi$
$269$	3675.87			0.833167			0.416584	+	0.909097i	$0.363227\pi$
$270$		0			0
$271$	1881.48			0.421741			0.210871	−	0.977514i	$-0.432370\pi$
$271$	1881.48			0.421741			0.210871	+	0.977514i	$0.432370\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1463.19	+	2534.31i	0.320849	+	0.555727i
$276$		0			0
$277$	−732.008	+	1267.87i	−0.158780	+	0.275015i	−0.934429	−	0.356149i	$-0.884089\pi$
$277$	−732.008	+	1267.87i	−0.158780	+	0.275015i	0.775649	+	0.631165i	$0.217423\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4116.28	+	7129.61i	−0.873867	+	1.51358i	−0.0159023	+	0.999874i	$0.505062\pi$
$281$	−4116.28	+	7129.61i	−0.873867	+	1.51358i	−0.857965	+	0.513709i	$0.828271\pi$
$282$		0			0
$283$	844.364	+	1462.48i	0.177358	+	0.307192i	0.940975	−	0.338477i	$-0.109912\pi$
$283$	844.364	+	1462.48i	0.177358	+	0.307192i	−0.763617	+	0.645669i	$0.776578\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10304.2			−2.11930
$288$		0			0
$289$	867.343			0.176540
$290$		0			0
$291$		0			0
$292$		0			0
$293$	255.866	+	443.173i	0.0510166	+	0.0883633i	0.890406	−	0.455167i	$-0.150421\pi$
$293$	255.866	+	443.173i	0.0510166	+	0.0883633i	−0.839389	+	0.543531i	$0.817087\pi$
$294$		0			0
$295$	−3493.18	+	6050.37i	−0.689427	+	1.19412i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−481.082	+	833.258i	−0.0930491	+	0.161166i
$300$		0			0
$301$	7651.37	+	13252.6i	1.46518	+	2.53776i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5148.55			0.966575
$306$		0			0
$307$	3171.98			0.589690			0.294845	−	0.955545i	$-0.404732\pi$
$307$	3171.98			0.589690			0.294845	+	0.955545i	$0.404732\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2530.39	−	4382.76i	−0.461367	−	0.799112i	0.537662	−	0.843160i	$-0.319308\pi$
$311$	−2530.39	−	4382.76i	−0.461367	−	0.799112i	−0.999029	+	0.0440487i	$0.985974\pi$
$312$		0			0
$313$	4732.92	−	8197.66i	0.854698	−	1.48038i	−0.0222263	−	0.999753i	$-0.507075\pi$
$313$	4732.92	−	8197.66i	0.854698	−	1.48038i	0.876925	−	0.480628i	$-0.159591\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3208.57	−	5557.40i	0.568489	−	0.984652i	−0.428226	−	0.903671i	$-0.640861\pi$
$317$	3208.57	−	5557.40i	0.568489	−	0.984652i	0.996716	−	0.0809808i	$-0.0258052\pi$
$318$		0			0
$319$	2036.75	+	3527.75i	0.357479	+	0.619172i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3358.86			−0.578613
$324$		0			0
$325$	−815.637			−0.139210
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4326.22	−	7493.23i	−0.724961	−	1.25567i
$330$		0			0
$331$	3332.37	−	5771.83i	0.553364	−	0.958455i	−0.444665	−	0.895697i	$-0.646677\pi$
$331$	3332.37	−	5771.83i	0.553364	−	0.958455i	0.998029	−	0.0627575i	$-0.0199895\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1125.93	+	1950.17i	−0.183630	+	0.318056i
$336$		0			0
$337$	−5244.05	−	9082.95i	−0.847660	−	1.46819i	−0.883291	−	0.468825i	$-0.844678\pi$
$337$	−5244.05	−	9082.95i	−0.847660	−	1.46819i	0.0356314	−	0.999365i	$-0.488656\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6281.25			−0.997503
$342$		0			0
$343$	−7891.37			−1.24226
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1074.48	−	1861.05i	−0.166228	−	0.287915i	0.770863	−	0.637001i	$-0.219825\pi$
$347$	−1074.48	−	1861.05i	−0.166228	−	0.287915i	−0.937091	+	0.349086i	$0.886492\pi$
$348$		0			0
$349$	1388.53	−	2405.00i	0.212969	−	0.368874i	−0.739673	−	0.672966i	$-0.765020\pi$
$349$	1388.53	−	2405.00i	0.212969	−	0.368874i	0.952642	+	0.304093i	$0.0983532\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3898.72	−	6752.78i	0.587841	−	1.01817i	−0.406673	−	0.913574i	$-0.633311\pi$
$353$	3898.72	−	6752.78i	0.587841	−	1.01817i	0.994515	−	0.104597i	$-0.0333554\pi$
$354$		0			0
$355$	−3000.75	−	5197.45i	−0.448629	−	0.777047i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2309.31			0.339500			0.169750	−	0.985487i	$-0.445704\pi$
$359$	2309.31			0.339500			0.169750	+	0.985487i	$0.445704\pi$
$360$		0			0
$361$	−4907.22			−0.715443
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4358.36	+	7548.91i	0.625006	+	1.08254i
$366$		0			0
$367$	−3105.66	+	5379.16i	−0.441728	+	0.765095i	−0.997818	−	0.0660271i	$-0.978968\pi$
$367$	−3105.66	+	5379.16i	−0.441728	+	0.765095i	0.556090	+	0.831122i	$0.312301\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9650.99	−	16716.0i	1.35055	−	2.33922i
$372$		0			0
$373$	−4349.61	−	7533.75i	−0.603792	−	1.04580i	−0.992241	−	0.124328i	$-0.960323\pi$
$373$	−4349.61	−	7533.75i	−0.603792	−	1.04580i	0.388450	−	0.921470i	$-0.373011\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1135.36			−0.155104
$378$		0			0
$379$	−5954.77			−0.807061			−0.403531	−	0.914966i	$-0.632217\pi$
$379$	−5954.77			−0.807061			−0.403531	+	0.914966i	$0.632217\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2128.35	−	3686.41i	−0.283952	−	0.491819i	0.688403	−	0.725329i	$-0.258312\pi$
$383$	−2128.35	−	3686.41i	−0.283952	−	0.491819i	−0.972354	+	0.233510i	$0.924979\pi$
$384$		0			0
$385$	−9332.31	+	16164.0i	−1.23537	+	2.13973i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	801.019	−	1387.41i	0.104404	−	0.180834i	−0.809090	−	0.587684i	$-0.800040\pi$
$389$	801.019	−	1387.41i	0.104404	−	0.180834i	0.913495	+	0.406851i	$0.133373\pi$
$390$		0			0
$391$	−2988.95	−	5177.02i	−0.386593	−	0.669598i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2391.25			−0.304600
$396$		0			0
$397$	−7987.87			−1.00982			−0.504912	−	0.863171i	$-0.668475\pi$
$397$	−7987.87			−1.00982			−0.504912	+	0.863171i	$0.668475\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3534.75	−	6122.36i	−0.440191	−	0.762434i	0.557512	−	0.830169i	$-0.311756\pi$
$401$	−3534.75	−	6122.36i	−0.440191	−	0.762434i	−0.997703	+	0.0677350i	$0.978423\pi$
$402$		0			0
$403$	875.352	−	1516.15i	0.108200	−	0.187407i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−712.881	+	1234.75i	−0.0868211	+	0.150379i
$408$		0			0
$409$	5932.38	+	10275.2i	0.717207	+	1.24224i	0.962102	+	0.272689i	$0.0879129\pi$
$409$	5932.38	+	10275.2i	0.717207	+	1.24224i	−0.244896	+	0.969549i	$0.578754\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−15496.9			−1.84637
$414$		0			0
$415$	−2420.71			−0.286332
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6171.30	+	10689.0i	0.719541	+	1.24628i	0.961182	+	0.275916i	$0.0889810\pi$
$419$	6171.30	+	10689.0i	0.719541	+	1.24628i	−0.241641	+	0.970366i	$0.577686\pi$
$420$		0			0
$421$	−2732.93	+	4733.58i	−0.316377	+	0.547982i	−0.979729	−	0.200326i	$-0.935800\pi$
$421$	−2732.93	+	4733.58i	−0.316377	+	0.547982i	0.663352	+	0.748308i	$0.269133\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2533.77	−	4388.61i	0.289190	−	0.500892i
$426$		0			0
$427$	5710.17	+	9890.30i	0.647153	+	1.12090i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8997.67			1.00557			0.502787	−	0.864410i	$-0.332308\pi$
$431$	8997.67			1.00557			0.502787	+	0.864410i	$0.332308\pi$
$432$		0			0
$433$	−10967.4			−1.21723			−0.608615	−	0.793466i	$-0.708275\pi$
$433$	−10967.4			−1.21723			−0.608615	+	0.793466i	$0.708275\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1736.83	+	3008.28i	0.190123	+	0.329303i
$438$		0			0
$439$	3563.51	−	6172.18i	0.387419	−	0.671029i	−0.604683	−	0.796467i	$-0.706700\pi$
$439$	3563.51	−	6172.18i	0.387419	−	0.671029i	0.992102	+	0.125437i	$0.0400333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4434.28	+	7680.39i	−0.475573	+	0.823716i	−0.999608	−	0.0279799i	$-0.991093\pi$
$443$	−4434.28	+	7680.39i	−0.475573	+	0.823716i	0.524036	+	0.851696i	$0.324426\pi$
$444$		0			0
$445$	2330.73	+	4036.94i	0.248286	+	0.430043i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9835.05			−1.03373			−0.516865	−	0.856067i	$-0.672901\pi$
$449$	−9835.05			−1.03373			−0.516865	+	0.856067i	$0.672901\pi$
$450$		0			0
$451$	−14732.4			−1.53819
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2601.09	−	4505.23i	−0.268003	−	0.464194i
$456$		0			0
$457$	4443.25	−	7695.93i	0.454806	−	0.787747i	−0.543871	−	0.839169i	$-0.683042\pi$
$457$	4443.25	−	7695.93i	0.454806	−	0.787747i	0.998677	+	0.0514218i	$0.0163753\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	932.894	−	1615.82i	0.0942499	−	0.163246i	−0.815045	−	0.579397i	$-0.803288\pi$
$461$	932.894	−	1615.82i	0.0942499	−	0.163246i	0.909295	+	0.416151i	$0.136621\pi$
$462$		0			0
$463$	−6811.70	−	11798.2i	−0.683729	−	1.18425i	−0.973834	−	0.227259i	$-0.927024\pi$
$463$	−6811.70	−	11798.2i	−0.683729	−	1.18425i	0.290105	−	0.956995i	$-0.406310\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9328.97			−0.924397			−0.462199	−	0.886776i	$-0.652939\pi$
$467$	−9328.97			−0.924397			−0.462199	+	0.886776i	$0.652939\pi$
$468$		0			0
$469$	−4994.99			−0.491785
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10939.5	+	18947.8i	1.06342	+	1.84190i
$474$		0			0
$475$	−1472.33	+	2550.15i	−0.142221	+	0.246335i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1330.15	+	2303.89i	−0.126881	+	0.219765i	−0.922467	−	0.386077i	$-0.873830\pi$
$479$	−1330.15	+	2303.89i	−0.126881	+	0.219765i	0.795586	+	0.605841i	$0.207163\pi$
$480$		0			0
$481$	−198.694	−	344.147i	−0.0188350	−	0.0326232i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1167.30			0.109287
$486$		0			0
$487$	20450.7			1.90289			0.951447	−	0.307813i	$-0.0995971\pi$
$487$	20450.7			1.90289			0.951447	+	0.307813i	$0.0995971\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−10327.4	−	17887.7i	−0.949228	−	1.64411i	−0.747055	−	0.664762i	$-0.768533\pi$
$491$	−10327.4	−	17887.7i	−0.949228	−	1.64411i	−0.202173	−	0.979350i	$-0.564800\pi$
$492$		0			0
$493$	3526.99	−	6108.92i	0.322206	−	0.558077i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6656.15	−	11528.8i	0.600743	−	1.04052i
$498$		0			0
$499$	8733.31	+	15126.5i	0.783480	+	1.35703i	0.929903	+	0.367805i	$0.119891\pi$
$499$	8733.31	+	15126.5i	0.783480	+	1.35703i	−0.146423	+	0.989222i	$0.546776\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	19951.2			1.76855			0.884275	−	0.466966i	$-0.154653\pi$
$503$	19951.2			1.76855			0.884275	+	0.466966i	$0.154653\pi$
$504$		0			0
$505$	24224.1			2.13457
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3638.88	−	6302.73i	−0.316878	−	0.548848i	0.662957	−	0.748657i	$-0.269301\pi$
$509$	−3638.88	−	6302.73i	−0.316878	−	0.548848i	−0.979835	+	0.199809i	$0.935968\pi$
$510$		0			0
$511$	−9667.57	+	16744.7i	−0.836924	+	1.44959i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	771.074	−	1335.54i	0.0659759	−	0.114274i
$516$		0			0
$517$	−6185.39	−	10713.4i	−0.526176	−	0.911364i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11664.5			0.980866			0.490433	−	0.871479i	$-0.336839\pi$
$521$	11664.5			0.980866			0.490433	+	0.871479i	$0.336839\pi$
$522$		0			0
$523$	8925.51			0.746243			0.373122	−	0.927782i	$-0.378287\pi$
$523$	8925.51			0.746243			0.373122	+	0.927782i	$0.378287\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5438.55	+	9419.84i	0.449539	+	0.778624i
$528$		0			0
$529$	2992.39	−	5182.98i	0.245943	−	0.425986i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2053.10	−	3556.08i	0.166848	−	0.288988i
$534$		0			0
$535$	−6198.74	−	10736.5i	−0.500925	−	0.867627i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−26341.9			−2.10506
$540$		0			0
$541$	12334.1			0.980193			0.490097	−	0.871668i	$-0.336962\pi$
$541$	12334.1			0.980193			0.490097	+	0.871668i	$0.336962\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4962.74	+	8595.71i	0.390056	+	0.675596i
$546$		0			0
$547$	2284.81	−	3957.41i	0.178595	−	0.309336i	−0.762804	−	0.646629i	$-0.776178\pi$
$547$	2284.81	−	3957.41i	0.178595	−	0.309336i	0.941400	+	0.337293i	$0.109511\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2049.47	+	3549.79i	−0.158458	+	0.274458i
$552$		0			0
$553$	−2652.09	−	4593.56i	−0.203939	−	0.353233i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8154.40			0.620310			0.310155	−	0.950686i	$-0.399619\pi$
$557$	8154.40			0.620310			0.310155	+	0.950686i	$0.399619\pi$
$558$		0			0
$559$	−6098.10			−0.461399
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8086.82	+	14006.8i	0.605362	+	1.04852i	0.991994	+	0.126284i	$0.0403050\pi$
$563$	8086.82	+	14006.8i	0.605362	+	1.04852i	−0.386632	+	0.922234i	$0.626362\pi$
$564$		0			0
$565$	1599.32	−	2770.11i	0.119087	−	0.206264i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4548.76	+	7878.69i	−0.335139	+	0.580478i	−0.983512	−	0.180845i	$-0.942117\pi$
$569$	−4548.76	+	7878.69i	−0.335139	+	0.580478i	0.648372	+	0.761323i	$0.275450\pi$
$570$		0			0
$571$	−1647.54	−	2853.63i	−0.120749	−	0.209143i	0.799314	−	0.600913i	$-0.205196\pi$
$571$	−1647.54	−	2853.63i	−0.120749	−	0.209143i	−0.920063	+	0.391770i	$0.871863\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5240.73			−0.380093
$576$		0			0
$577$	15544.7			1.12155			0.560775	−	0.827968i	$-0.310503\pi$
$577$	15544.7			1.12155			0.560775	+	0.827968i	$0.310503\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2684.76	−	4650.14i	−0.191709	−	0.332049i
$582$		0			0
$583$	13798.4	−	23899.6i	0.980229	−	1.69781i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2344.33	−	4060.50i	0.164840	−	0.285511i	−0.771759	−	0.635916i	$-0.780623\pi$
$587$	2344.33	−	4060.50i	0.164840	−	0.285511i	0.936598	+	0.350405i	$0.113956\pi$
$588$		0			0
$589$	−3160.25	−	5473.71i	−0.221079	−	0.382921i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16637.6			−1.15215			−0.576075	−	0.817397i	$-0.695416\pi$
$593$	−16637.6			−1.15215			−0.576075	+	0.817397i	$0.695416\pi$
$594$		0			0
$595$	32321.1			2.22695
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2933.73	+	5081.37i	0.200115	+	0.346610i	0.948565	−	0.316581i	$-0.102535\pi$
$599$	2933.73	+	5081.37i	0.200115	+	0.346610i	−0.748450	+	0.663191i	$0.769202\pi$
$600$		0			0
$601$	−10337.9	+	17905.8i	−0.701649	+	1.21529i	0.266238	+	0.963907i	$0.414219\pi$
$601$	−10337.9	+	17905.8i	−0.701649	+	1.21529i	−0.967887	+	0.251385i	$0.919114\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4129.71	+	7152.87i	−0.277515	+	0.480671i
$606$		0			0
$607$	6303.86	+	10918.6i	0.421525	+	0.730103i	0.996089	−	0.0883567i	$-0.0281615\pi$
$607$	6303.86	+	10918.6i	0.421525	+	0.730103i	−0.574564	+	0.818460i	$0.694828\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3447.97			0.228298
$612$		0			0
$613$	12469.7			0.821608			0.410804	−	0.911724i	$-0.365248\pi$
$613$	12469.7			0.821608			0.410804	+	0.911724i	$0.365248\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9293.48	+	16096.8i	0.606388	+	1.05030i	0.991830	+	0.127563i	$0.0407156\pi$
$617$	9293.48	+	16096.8i	0.606388	+	1.05030i	−0.385442	+	0.922732i	$0.625951\pi$
$618$		0			0
$619$	−5972.96	+	10345.5i	−0.387841	+	0.671761i	−0.992159	−	0.124983i	$-0.960112\pi$
$619$	−5972.96	+	10345.5i	−0.387841	+	0.671761i	0.604318	+	0.796743i	$0.293446\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5169.94	+	8954.60i	−0.332471	+	0.575856i
$624$		0			0
$625$	9757.00	+	16899.6i	0.624448	+	1.08158i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2468.96			0.156509
$630$		0			0
$631$	−4285.35			−0.270360			−0.135180	−	0.990821i	$-0.543161\pi$
$631$	−4285.35			−0.270360			−0.135180	+	0.990821i	$0.543161\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11872.7	+	20564.0i	0.741972	+	1.28513i
$636$		0			0
$637$	3671.00	−	6358.36i	0.228336	−	0.395490i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10879.1	+	18843.2i	−0.670357	+	1.16109i	0.307446	+	0.951566i	$0.400526\pi$
$641$	−10879.1	+	18843.2i	−0.670357	+	1.16109i	−0.977803	+	0.209527i	$0.932808\pi$
$642$		0			0
$643$	−3671.54	−	6359.30i	−0.225181	−	0.390025i	0.731193	−	0.682171i	$-0.238964\pi$
$643$	−3671.54	−	6359.30i	−0.225181	−	0.390025i	−0.956374	+	0.292146i	$0.905631\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9483.24			0.576236			0.288118	−	0.957595i	$-0.406970\pi$
$647$	9483.24			0.576236			0.288118	+	0.957595i	$0.406970\pi$
$648$		0			0
$649$	−22156.6			−1.34010
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2392.09	+	4143.23i	0.143353	+	0.248295i	0.928757	−	0.370688i	$-0.120878\pi$
$653$	2392.09	+	4143.23i	0.143353	+	0.248295i	−0.785404	+	0.618984i	$0.787545\pi$
$654$		0			0
$655$	−9226.05	+	15980.0i	−0.550369	+	0.953266i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	16034.0	−	27771.6i	0.947791	−	1.64162i	0.197727	−	0.980257i	$-0.436644\pi$
$659$	16034.0	−	27771.6i	0.947791	−	1.64162i	0.750064	−	0.661365i	$-0.230023\pi$
$660$		0			0
$661$	3762.64	+	6517.09i	0.221407	+	0.383488i	0.955235	−	0.295847i	$-0.0956019\pi$
$661$	3762.64	+	6517.09i	0.221407	+	0.383488i	−0.733829	+	0.679335i	$0.762269\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−18781.3			−1.09520
$666$		0			0
$667$	−7295.07			−0.423487
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8164.08	+	14140.6i	0.469703	+	0.813550i
$672$		0			0
$673$	3670.98	−	6358.32i	0.210261	−	0.364183i	−0.741535	−	0.670914i	$-0.765902\pi$
$673$	3670.98	−	6358.32i	0.210261	−	0.364183i	0.951796	+	0.306731i	$0.0992352\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4760.52	−	8245.45i	0.270253	−	0.468092i	−0.698673	−	0.715441i	$-0.746226\pi$
$677$	4760.52	−	8245.45i	0.270253	−	0.468092i	0.968927	+	0.247348i	$0.0795592\pi$
$678$		0			0
$679$	1294.63	+	2242.36i	0.0731713	+	0.126736i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1475.06			−0.0826377			−0.0413188	−	0.999146i	$-0.513156\pi$
$683$	−1475.06			−0.0826377			−0.0413188	+	0.999146i	$0.513156\pi$
$684$		0			0
$685$	−34872.5			−1.94512
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3845.89	+	6661.28i	0.212651	+	0.368323i
$690$		0			0
$691$	17080.4	−	29584.0i	0.940329	−	1.62870i	0.175484	−	0.984482i	$-0.443851\pi$
$691$	17080.4	−	29584.0i	0.940329	−	1.62870i	0.764844	−	0.644215i	$-0.222816\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4311.56	−	7467.84i	0.235319	−	0.407585i
$696$		0			0
$697$	12755.9	+	22093.8i	0.693205	+	1.20067i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	29427.2			1.58552			0.792759	−	0.609535i	$-0.208644\pi$
$701$	29427.2			1.58552			0.792759	+	0.609535i	$0.208644\pi$
$702$		0			0
$703$	−1434.67			−0.0769696
$704$		0			0
$705$		0			0
$706$		0			0
$707$	26866.5	+	46534.1i	1.42916	+	2.47538i
$708$		0			0
$709$	−9846.43	+	17054.5i	−0.521566	+	0.903379i	0.478119	+	0.878295i	$0.341319\pi$
$709$	−9846.43	+	17054.5i	−0.521566	+	0.903379i	−0.999685	+	0.0250840i	$0.992015\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5624.42	−	9741.79i	0.295423	−	0.511687i
$714$		0			0
$715$	−3718.90	−	6441.32i	−0.194516	−	0.336912i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−33746.9			−1.75041			−0.875206	−	0.483751i	$-0.839274\pi$
$719$	−33746.9			−1.75041			−0.875206	+	0.483751i	$0.839274\pi$
$720$		0			0
$721$	3420.74			0.176692
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3092.05	−	5355.60i	−0.158395	−	0.274347i
$726$		0			0
$727$	10380.4	−	17979.4i	0.529558	−	0.917221i	−0.469848	−	0.882748i	$-0.655691\pi$
$727$	10380.4	−	17979.4i	0.529558	−	0.917221i	0.999406	−	0.0344737i	$-0.0109755\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	18943.7	−	32811.4i	0.958492	−	1.66016i
$732$		0			0
$733$	−7122.02	−	12335.7i	−0.358878	−	0.621596i	0.628895	−	0.777490i	$-0.283507\pi$
$733$	−7122.02	−	12335.7i	−0.358878	−	0.621596i	−0.987774	+	0.155894i	$0.950174\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7141.55			−0.356937
$738$		0			0
$739$	−27490.2			−1.36839			−0.684196	−	0.729298i	$-0.739847\pi$
$739$	−27490.2			−1.36839			−0.684196	+	0.729298i	$0.739847\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−18234.0	−	31582.3i	−0.900326	−	1.55941i	−0.827072	−	0.562096i	$-0.809995\pi$
$743$	−18234.0	−	31582.3i	−0.900326	−	1.55941i	−0.0732537	−	0.997313i	$-0.523338\pi$
$744$		0			0
$745$	16674.6	−	28881.2i	0.820013	−	1.42030i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	13749.8	−	23815.4i	0.670771	−	1.16181i
$750$		0			0
$751$	19341.5	+	33500.5i	0.939791	+	1.62777i	0.765860	+	0.643007i	$0.222314\pi$
$751$	19341.5	+	33500.5i	0.939791	+	1.62777i	0.173931	+	0.984758i	$0.444353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−936.449			−0.0451402
$756$		0			0
$757$	37768.4			1.81337			0.906683	−	0.421813i	$-0.138606\pi$
$757$	37768.4			1.81337			0.906683	+	0.421813i	$0.138606\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10406.6	−	18024.7i	−0.495714	−	0.858603i	0.504273	−	0.863544i	$-0.331760\pi$
$761$	−10406.6	−	18024.7i	−0.495714	−	0.858603i	−0.999988	+	0.00494154i	$0.998427\pi$
$762$		0			0
$763$	−11008.2	+	19066.7i	−0.522310	+	0.904667i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3087.74	−	5348.12i	0.145361	−	0.251772i
$768$		0			0
$769$	−3060.85	−	5301.55i	−0.143533	−	0.248607i	0.785291	−	0.619126i	$-0.212513\pi$
$769$	−3060.85	−	5301.55i	−0.143533	−	0.248607i	−0.928825	+	0.370519i	$0.879180\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	34251.2			1.59370			0.796850	−	0.604177i	$-0.206498\pi$
$773$	34251.2			1.59370			0.796850	+	0.604177i	$0.206498\pi$
$774$		0			0
$775$	9535.78			0.441981
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7412.23	−	12838.4i	−0.340912	−	0.590478i
$780$		0			0
$781$	9516.59	−	16483.2i	0.436018	−	0.755206i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	30.4117	−	52.6746i	0.00138273	−	0.00239495i
$786$		0			0
$787$	−4561.34	−	7900.47i	−0.206600	−	0.357842i	0.744041	−	0.668134i	$-0.232907\pi$
$787$	−4561.34	−	7900.47i	−0.206600	−	0.357842i	−0.950641	+	0.310292i	$0.899573\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7095.12			0.318930
$792$		0			0
$793$	−4550.97			−0.203795
$794$		0			0
$795$		0			0
$796$		0

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	432.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-6.92194 - 11.9892i) q^{5} +(15.3540 - 26.5939i) q^{7} +(21.9523 - 38.0225i) q^{11} +(6.11853 + 10.5976i) q^{13} -76.0286 q^{17} +44.1789 q^{19} +(39.3135 + 68.0930i) q^{23} +(-33.3265 + 57.7232i) q^{25} +(-46.3903 + 80.3504i) q^{29} +(-71.5329 - 123.899i) q^{31} -425.118 q^{35} -32.4741 q^{37} +(-167.778 - 290.599i) q^{41} +(-249.166 + 431.567i) q^{43} +(140.882 - 244.016i) q^{47} +(-299.990 - 519.599i) q^{49} +628.565 q^{53} -607.810 q^{55} +(-252.327 - 437.043i) q^{59} +(-185.951 + 322.076i) q^{61} +(84.7041 - 146.712i) q^{65} +(-81.3304 - 140.868i) q^{67} +433.512 q^{71} -629.645 q^{73} +(-674.111 - 1167.59i) q^{77} +(86.3649 - 149.588i) q^{79} +(87.4288 - 151.431i) q^{83} +(526.265 + 911.518i) q^{85} -336.716 q^{89} +375.775 q^{91} +(-305.804 - 529.668i) q^{95} +(-42.1594 + 73.0222i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{5} + 6 q^{7} + 51 q^{11} + 12 q^{13} + 222 q^{17} - 30 q^{19} + 210 q^{23} - 3 q^{25} - 456 q^{29} - 48 q^{31} - 1104 q^{35} - 96 q^{37} - 897 q^{41} - 129 q^{43} + 522 q^{47} - 225 q^{49} + 2208 q^{53}+ \cdots + 93 q^{97}+O(q^{100}) \) 6 * q - 6 * q^5 + 6 * q^7 + 51 * q^11 + 12 * q^13 + 222 * q^17 - 30 * q^19 + 210 * q^23 - 3 * q^25 - 456 * q^29 - 48 * q^31 - 1104 * q^35 - 96 * q^37 - 897 * q^41 - 129 * q^43 + 522 * q^47 - 225 * q^49 + 2208 * q^53 + 216 * q^55 + 453 * q^59 - 402 * q^61 - 1110 * q^65 + 213 * q^67 + 120 * q^71 + 750 * q^73 - 1128 * q^77 - 552 * q^79 - 612 * q^83 + 1188 * q^85 + 924 * q^89 + 264 * q^91 - 2184 * q^95 + 93 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more