LMFDB - Embedded newform 1764.3.d.f.685.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(685,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1764.685");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.d (of order $2$, degree $1$, 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:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{65})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 17x^{2} + 16x + 256 $ x^4 - x^3 + 17x^2 + 16x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.3
Root			$-1.76556 + 3.05805i$ of defining polynomial
Character	$\chi$	$=$	1764.685
Dual form			1764.3.d.f.685.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.38404i q^{5} +19.5934 q^{11} -6.11609i q^{13} -8.76809i q^{17} +30.3648i q^{19} +24.0000 q^{23} +5.78016 q^{25} -13.5934 q^{29} -28.0363i q^{31} -48.5934 q^{37} -7.14387i q^{41} +53.7802 q^{43} -39.9450i q^{47} +61.5934 q^{53} +85.8983i q^{55} -77.1302i q^{59} -0.431334i q^{61} +26.8132 q^{65} -56.9669 q^{67} +123.560 q^{71} +35.8845i q^{73} -52.1868 q^{79} +136.883i q^{83} +38.4397 q^{85} +15.2650i q^{89} -133.121 q^{95} -34.1524i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 30 q^{11} + 96 q^{23} - 122 q^{25} - 6 q^{29} - 146 q^{37} + 70 q^{43} + 198 q^{53} + 204 q^{65} + 14 q^{67} + 204 q^{71} - 112 q^{79} + 444 q^{85} + 48 q^{95}+O(q^{100}) $ 4 * q + 30 * q^11 + 96 * q^23 - 122 * q^25 - 6 * q^29 - 146 * q^37 + 70 * q^43 + 198 * q^53 + 204 * q^65 + 14 * q^67 + 204 * q^71 - 112 * q^79 + 444 * q^85 + 48 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	4.38404i	0.876809i	0.898778	+	0.438404i	$0.144456\pi$
$5$	4.38404i	0.876809i	−0.898778	+	0.438404i	$0.855544\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	19.5934	1.78122	0.890608	−	0.454771i	$-0.150279\pi$
$11$	19.5934	1.78122	0.890608	+	0.454771i	$0.150279\pi$
$12$	0	0
$13$	− 6.11609i	− 0.470469i	−0.971939	−	0.235234i	$-0.924414\pi$
$13$	− 6.11609i	− 0.470469i	0.971939	−	0.235234i	$-0.0755858\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 8.76809i	− 0.515770i	−0.966176	−	0.257885i	$-0.916974\pi$
$17$	− 8.76809i	− 0.515770i	0.966176	−	0.257885i	$-0.0830255\pi$
$18$	0	0
$19$	30.3648i	1.59815i	0.601233	+	0.799074i	$0.294676\pi$
$19$	30.3648i	1.59815i	−0.601233	+	0.799074i	$0.705324\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	24.0000	1.04348	0.521739	−	0.853105i	$-0.325283\pi$
$23$	24.0000	1.04348	0.521739	+	0.853105i	$0.325283\pi$
$24$	0	0
$25$	5.78016	0.231206
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−13.5934	−0.468737	−0.234369	−	0.972148i	$-0.575302\pi$
$29$	−13.5934	−0.468737	−0.234369	+	0.972148i	$0.575302\pi$
$30$	0	0
$31$	− 28.0363i	− 0.904397i	−0.891917	−	0.452199i	$-0.850640\pi$
$31$	− 28.0363i	− 0.904397i	0.891917	−	0.452199i	$-0.149360\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−48.5934	−1.31333	−0.656667	−	0.754180i	$-0.728034\pi$
$37$	−48.5934	−1.31333	−0.656667	+	0.754180i	$0.728034\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 7.14387i	− 0.174241i	−0.996198	−	0.0871204i	$-0.972234\pi$
$41$	− 7.14387i	− 0.174241i	0.996198	−	0.0871204i	$-0.0277665\pi$
$42$	0	0
$43$	53.7802	1.25070	0.625351	−	0.780344i	$-0.284956\pi$
$43$	53.7802	1.25070	0.625351	+	0.780344i	$0.284956\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 39.9450i	− 0.849894i	−0.905218	−	0.424947i	$-0.860293\pi$
$47$	− 39.9450i	− 0.849894i	0.905218	−	0.424947i	$-0.139707\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	61.5934	1.16214	0.581070	−	0.813854i	$-0.302634\pi$
$53$	61.5934	1.16214	0.581070	+	0.813854i	$0.302634\pi$
$54$	0	0
$55$	85.8983i	1.56179i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 77.1302i	− 1.30729i	−0.756801	−	0.653646i	$-0.773239\pi$
$59$	− 77.1302i	− 1.30729i	0.756801	−	0.653646i	$-0.226761\pi$
$60$	0	0
$61$	− 0.431334i	− 0.00707105i	−0.999994	−	0.00353553i	$-0.998875\pi$
$61$	− 0.431334i	− 0.00707105i	0.999994	−	0.00353553i	$-0.00112540\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	26.8132	0.412511
$66$	0	0
$67$	−56.9669	−0.850253	−0.425126	−	0.905134i	$-0.639770\pi$
$67$	−56.9669	−0.850253	−0.425126	+	0.905134i	$0.639770\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	123.560	1.74029	0.870143	−	0.492799i	$-0.164026\pi$
$71$	123.560	1.74029	0.870143	+	0.492799i	$0.164026\pi$
$72$	0	0
$73$	35.8845i	0.491568i	0.969325	+	0.245784i	$0.0790454\pi$
$73$	35.8845i	0.491568i	−0.969325	+	0.245784i	$0.920955\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−52.1868	−0.660592	−0.330296	−	0.943877i	$-0.607149\pi$
$79$	−52.1868	−0.660592	−0.330296	+	0.943877i	$0.607149\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	136.883i	1.64919i	0.565725	+	0.824594i	$0.308596\pi$
$83$	136.883i	1.64919i	−0.565725	+	0.824594i	$0.691404\pi$
$84$	0	0
$85$	38.4397	0.452232
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.2650i	0.171516i	0.996316	+	0.0857582i	$0.0273312\pi$
$89$	15.2650i	0.171516i	−0.996316	+	0.0857582i	$0.972669\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−133.121	−1.40127
$96$	0	0
$97$	− 34.1524i	− 0.352087i	−0.984382	−	0.176043i	$-0.943670\pi$
$97$	− 34.1524i	− 0.352087i	0.984382	−	0.176043i	$-0.0563299\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	130.874i	1.29579i	0.761732	+	0.647893i	$0.224349\pi$
$101$	130.874i	1.29579i	−0.761732	+	0.647893i	$0.775651\pi$
$102$	0	0
$103$	162.648i	1.57910i	0.613684	+	0.789552i	$0.289687\pi$
$103$	162.648i	1.57910i	−0.613684	+	0.789552i	$0.710313\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	103.967	0.971654	0.485827	−	0.874055i	$-0.338519\pi$
$107$	103.967	0.971654	0.485827	+	0.874055i	$0.338519\pi$
$108$	0	0
$109$	−15.7802	−0.144772	−0.0723861	−	0.997377i	$-0.523061\pi$
$109$	−15.7802	−0.144772	−0.0723861	+	0.997377i	$0.523061\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	59.6265	0.527668	0.263834	−	0.964568i	$-0.415013\pi$
$113$	59.6265	0.527668	0.263834	+	0.964568i	$0.415013\pi$
$114$	0	0
$115$	105.217i	0.914931i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	262.901	2.17273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	134.942i	1.07953i
$126$	0	0
$127$	−25.0000	−0.196850	−0.0984252	−	0.995144i	$-0.531381\pi$
$127$	−25.0000	−0.196850	−0.0984252	+	0.995144i	$0.531381\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	226.518i	1.72914i	0.502509	+	0.864572i	$0.332410\pi$
$131$	226.518i	1.72914i	−0.502509	+	0.864572i	$0.667590\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	247.121	1.80380	0.901900	−	0.431945i	$-0.142172\pi$
$137$	247.121	1.80380	0.901900	+	0.431945i	$0.142172\pi$
$138$	0	0
$139$	147.383i	1.06031i	0.847902	+	0.530154i	$0.177866\pi$
$139$	147.383i	1.06031i	−0.847902	+	0.530154i	$0.822134\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 119.835i	− 0.838007i
$144$	0	0
$145$	− 59.5940i	− 0.410993i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−128.813	−0.864518	−0.432259	−	0.901749i	$-0.642283\pi$
$149$	−128.813	−0.864518	−0.432259	+	0.901749i	$0.642283\pi$
$150$	0	0
$151$	62.0331	0.410815	0.205408	−	0.978677i	$-0.434148\pi$
$151$	62.0331	0.410815	0.205408	+	0.978677i	$0.434148\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	122.912	0.792983
$156$	0	0
$157$	− 250.709i	− 1.59687i	−0.602078	−	0.798437i	$-0.705660\pi$
$157$	− 250.709i	− 1.59687i	0.602078	−	0.798437i	$-0.294340\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−97.9339	−0.600821	−0.300411	−	0.953810i	$-0.597124\pi$
$163$	−97.9339	−0.600821	−0.300411	+	0.953810i	$0.597124\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 71.7689i	− 0.429754i	−0.976641	−	0.214877i	$-0.931065\pi$
$167$	− 71.7689i	− 0.429754i	0.976641	−	0.214877i	$-0.0689351\pi$
$168$	0	0
$169$	131.593	0.778659
$170$	0	0
$171$	0	0
$172$	0	0
$173$	302.024i	1.74580i	0.487897	+	0.872901i	$0.337764\pi$
$173$	302.024i	1.74580i	−0.487897	+	0.872901i	$0.662236\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−138.000	−0.770950	−0.385475	−	0.922718i	$-0.625962\pi$
$179$	−138.000	−0.770950	−0.385475	+	0.922718i	$0.625962\pi$
$180$	0	0
$181$	− 82.9733i	− 0.458416i	−0.973377	−	0.229208i	$-0.926386\pi$
$181$	− 82.9733i	− 0.458416i	0.973377	−	0.229208i	$-0.0736136\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 213.036i	− 1.15154i
$186$	0	0
$187$	− 171.797i	− 0.918698i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−172.307	−0.902133	−0.451067	−	0.892490i	$-0.648956\pi$
$191$	−172.307	−0.902133	−0.451067	+	0.892490i	$0.648956\pi$
$192$	0	0
$193$	275.307	1.42646	0.713232	−	0.700928i	$-0.247231\pi$
$193$	275.307	1.42646	0.713232	+	0.700928i	$0.247231\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	154.307	0.783286	0.391643	−	0.920117i	$-0.371907\pi$
$197$	154.307	0.783286	0.391643	+	0.920117i	$0.371907\pi$
$198$	0	0
$199$	− 187.924i	− 0.944342i	−0.881507	−	0.472171i	$-0.843470\pi$
$199$	− 187.924i	− 0.944342i	0.881507	−	0.472171i	$-0.156530\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	31.3190	0.152776
$206$	0	0
$207$	0	0
$208$	0	0
$209$	594.949i	2.84665i
$210$	0	0
$211$	−348.307	−1.65075	−0.825373	−	0.564588i	$-0.809035\pi$
$211$	−348.307	−1.65075	−0.825373	+	0.564588i	$0.809035\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	235.775i	1.09663i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−53.6265	−0.242654
$222$	0	0
$223$	− 30.3581i	− 0.136135i	−0.997681	−	0.0680675i	$-0.978317\pi$
$223$	− 30.3581i	− 0.136135i	0.997681	−	0.0680675i	$-0.0216833\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 64.4667i	− 0.283994i	−0.989867	−	0.141997i	$-0.954648\pi$
$227$	− 64.4667i	− 0.283994i	0.989867	−	0.141997i	$-0.0453523\pi$
$228$	0	0
$229$	− 10.0115i	− 0.0437185i	−0.999761	−	0.0218592i	$-0.993041\pi$
$229$	− 10.0115i	− 0.0437185i	0.999761	−	0.0218592i	$-0.00695857\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−28.5058	−0.122343	−0.0611713	−	0.998127i	$-0.519484\pi$
$233$	−28.5058	−0.122343	−0.0611713	+	0.998127i	$0.519484\pi$
$234$	0	0
$235$	175.121	0.745194
$236$	0	0
$237$	0	0
$238$	0	0
$239$	256.307	1.07242	0.536208	−	0.844086i	$-0.319856\pi$
$239$	256.307	1.07242	0.536208	+	0.844086i	$0.319856\pi$
$240$	0	0
$241$	215.694i	0.894997i	0.894285	+	0.447498i	$0.147685\pi$
$241$	215.694i	0.894997i	−0.894285	+	0.447498i	$0.852315\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	185.714	0.751879
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 371.680i	− 1.48080i	−0.672168	−	0.740398i	$-0.734637\pi$
$251$	− 371.680i	− 1.48080i	0.672168	−	0.740398i	$-0.265363\pi$
$252$	0	0
$253$	470.241	1.85866
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 74.0401i	− 0.288094i	−0.989571	−	0.144047i	$-0.953988\pi$
$257$	− 74.0401i	− 0.288094i	0.989571	−	0.144047i	$-0.0460116\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.3190	−0.0734564	−0.0367282	−	0.999325i	$-0.511694\pi$
$263$	−19.3190	−0.0734564	−0.0367282	+	0.999325i	$0.511694\pi$
$264$	0	0
$265$	270.028i	1.01897i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	208.982i	0.776884i	0.921473	+	0.388442i	$0.126987\pi$
$269$	208.982i	0.776884i	−0.921473	+	0.388442i	$0.873013\pi$
$270$	0	0
$271$	− 122.811i	− 0.453175i	−0.973991	−	0.226588i	$-0.927243\pi$
$271$	− 122.811i	− 0.453175i	0.973991	−	0.226588i	$-0.0727570\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	113.253	0.411829
$276$	0	0
$277$	27.7802	0.100289	0.0501447	−	0.998742i	$-0.484032\pi$
$277$	27.7802	0.100289	0.0501447	+	0.998742i	$0.484032\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−61.4942	−0.218841	−0.109420	−	0.993996i	$-0.534899\pi$
$281$	−61.4942	−0.218841	−0.109420	+	0.993996i	$0.534899\pi$
$282$	0	0
$283$	− 43.8910i	− 0.155092i	−0.996989	−	0.0775459i	$-0.975292\pi$
$283$	− 43.8910i	− 0.155092i	0.996989	−	0.0775459i	$-0.0247084\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	212.121	0.733981
$290$	0	0
$291$	0	0
$292$	0	0
$293$	223.600i	0.763139i	0.924340	+	0.381569i	$0.124616\pi$
$293$	223.600i	0.763139i	−0.924340	+	0.381569i	$0.875384\pi$
$294$	0	0
$295$	338.142	1.14624
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 146.786i	− 0.490924i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.89099	0.00619996
$306$	0	0
$307$	− 66.8457i	− 0.217738i	−0.994056	−	0.108869i	$-0.965277\pi$
$307$	− 66.8457i	− 0.217738i	0.994056	−	0.108869i	$-0.0347230\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	29.2225i	0.0939630i	0.998896	+	0.0469815i	$0.0149602\pi$
$311$	29.2225i	0.0939630i	−0.998896	+	0.0469815i	$0.985040\pi$
$312$	0	0
$313$	248.115i	0.792698i	0.918100	+	0.396349i	$0.129723\pi$
$313$	248.115i	0.792698i	−0.918100	+	0.396349i	$0.870277\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.6479	−0.0398987	−0.0199494	−	0.999801i	$-0.506350\pi$
$317$	−12.6479	−0.0398987	−0.0199494	+	0.999801i	$0.506350\pi$
$318$	0	0
$319$	−266.340	−0.834923
$320$	0	0
$321$	0	0
$322$	0	0
$323$	266.241	0.824276
$324$	0	0
$325$	− 35.3520i	− 0.108775i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−502.901	−1.51934	−0.759669	−	0.650310i	$-0.774639\pi$
$331$	−502.901	−1.51934	−0.759669	+	0.650310i	$0.774639\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 249.746i	− 0.745509i
$336$	0	0
$337$	−123.681	−0.367006	−0.183503	−	0.983019i	$-0.558744\pi$
$337$	−123.681	−0.367006	−0.183503	+	0.983019i	$0.558744\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 549.326i	− 1.61093i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	226.307	0.652183	0.326091	−	0.945338i	$-0.394268\pi$
$347$	226.307	0.652183	0.326091	+	0.945338i	$0.394268\pi$
$348$	0	0
$349$	− 155.669i	− 0.446043i	−0.974814	−	0.223021i	$-0.928408\pi$
$349$	− 155.669i	− 0.446043i	0.974814	−	0.223021i	$-0.0715920\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	63.9780i	0.181241i	0.995886	+	0.0906205i	$0.0288850\pi$
$353$	63.9780i	0.181241i	−0.995886	+	0.0906205i	$0.971115\pi$
$354$	0	0
$355$	541.694i	1.52590i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	302.988	0.843979	0.421989	−	0.906601i	$-0.361332\pi$
$359$	302.988	0.843979	0.421989	+	0.906601i	$0.361332\pi$
$360$	0	0
$361$	−561.021	−1.55408
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−157.319	−0.431011
$366$	0	0
$367$	− 0.00671348i	0 1.82929e-5i	−1.00000		9.14643e-6i	$-0.999997\pi$
$367$	− 0.00671348i	0 1.82929e-5i	1.00000		9.14643e-6i	$-2.91140e-6\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−125.780	−0.337212	−0.168606	−	0.985684i	$-0.553927\pi$
$373$	−125.780	−0.337212	−0.168606	+	0.985684i	$0.553927\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	83.1384i	0.220526i
$378$	0	0
$379$	−485.076	−1.27988	−0.639942	−	0.768423i	$-0.721042\pi$
$379$	−485.076	−1.27988	−0.639942	+	0.768423i	$0.721042\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 260.138i	− 0.679211i	−0.940568	−	0.339606i	$-0.889706\pi$
$383$	− 260.138i	− 0.679211i	0.940568	−	0.339606i	$-0.110294\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−85.8677	−0.220740	−0.110370	−	0.993891i	$-0.535204\pi$
$389$	−85.8677	−0.220740	−0.110370	+	0.993891i	$0.535204\pi$
$390$	0	0
$391$	− 210.434i	− 0.538195i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 228.789i	− 0.579213i
$396$	0	0
$397$	− 463.809i	− 1.16828i	−0.811651	−	0.584142i	$-0.801431\pi$
$397$	− 463.809i	− 1.16828i	0.811651	−	0.584142i	$-0.198569\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	463.868	1.15678	0.578389	−	0.815761i	$-0.303682\pi$
$401$	463.868	1.15678	0.578389	+	0.815761i	$0.303682\pi$
$402$	0	0
$403$	−171.473	−0.425491
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−952.109	−2.33933
$408$	0	0
$409$	466.137i	1.13970i	0.821749	+	0.569850i	$0.192999\pi$
$409$	466.137i	1.13970i	−0.821749	+	0.569850i	$0.807001\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−600.099	−1.44602
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 89.9655i	− 0.214715i	−0.994220	−	0.107357i	$-0.965761\pi$
$419$	− 89.9655i	− 0.214715i	0.994220	−	0.107357i	$-0.0342389\pi$
$420$	0	0
$421$	−413.582	−0.982379	−0.491190	−	0.871053i	$-0.663438\pi$
$421$	−413.582	−0.982379	−0.491190	+	0.871053i	$0.663438\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 50.6809i	− 0.119249i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−70.1323	−0.162720	−0.0813599	−	0.996685i	$-0.525926\pi$
$431$	−70.1323	−0.162720	−0.0813599	+	0.996685i	$0.525926\pi$
$432$	0	0
$433$	431.223i	0.995897i	0.867207	+	0.497948i	$0.165913\pi$
$433$	431.223i	0.995897i	−0.867207	+	0.497948i	$0.834087\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	728.755i	1.66763i
$438$	0	0
$439$	94.3496i	0.214919i	0.994209	+	0.107460i	$0.0342716\pi$
$439$	94.3496i	0.214919i	−0.994209	+	0.107460i	$0.965728\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−412.582	−0.931336	−0.465668	−	0.884960i	$-0.654186\pi$
$443$	−412.582	−0.931336	−0.465668	+	0.884960i	$0.654186\pi$
$444$	0	0
$445$	−66.9222	−0.150387
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−617.802	−1.37595	−0.687975	−	0.725734i	$-0.741500\pi$
$449$	−617.802	−1.37595	−0.687975	+	0.725734i	$0.741500\pi$
$450$	0	0
$451$	− 139.973i	− 0.310361i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.0000	−0.0240700	−0.0120350	−	0.999928i	$-0.503831\pi$
$457$	−11.0000	−0.0240700	−0.0120350	+	0.999928i	$0.503831\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 859.960i	− 1.86542i	−0.360624	−	0.932711i	$-0.617436\pi$
$461$	− 859.960i	− 1.86542i	0.360624	−	0.932711i	$-0.382564\pi$
$462$	0	0
$463$	397.340	0.858187	0.429093	−	0.903260i	$-0.358833\pi$
$463$	397.340	0.858187	0.429093	+	0.903260i	$0.358833\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	297.151i	0.636298i	0.948041	+	0.318149i	$0.103061\pi$
$467$	297.151i	0.636298i	−0.948041	+	0.318149i	$0.896939\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1053.74	2.22777
$474$	0	0
$475$	175.513i	0.369502i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 804.420i	− 1.67937i	−0.543071	−	0.839687i	$-0.682739\pi$
$479$	− 804.420i	− 1.67937i	0.543071	−	0.839687i	$-0.317261\pi$
$480$	0	0
$481$	297.202i	0.617883i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	149.726	0.308713
$486$	0	0
$487$	60.8016	0.124849	0.0624246	−	0.998050i	$-0.480117\pi$
$487$	60.8016	0.124849	0.0624246	+	0.998050i	$0.480117\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−514.955	−1.04879	−0.524394	−	0.851475i	$-0.675708\pi$
$491$	−514.955	−1.04879	−0.524394	+	0.851475i	$0.675708\pi$
$492$	0	0
$493$	119.188i	0.241761i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	888.901	1.78136	0.890682	−	0.454627i	$-0.150227\pi$
$499$	888.901	1.78136	0.890682	+	0.454627i	$0.150227\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 822.920i	− 1.63602i	−0.575202	−	0.818012i	$-0.695076\pi$
$503$	− 822.920i	− 1.63602i	0.575202	−	0.818012i	$-0.304924\pi$
$504$	0	0
$505$	−573.759	−1.13616
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 568.157i	− 1.11622i	−0.829767	−	0.558111i	$-0.811527\pi$
$509$	− 568.157i	− 1.11622i	0.829767	−	0.558111i	$-0.188473\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−713.055	−1.38457
$516$	0	0
$517$	− 782.658i	− 1.51385i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 136.724i	− 0.262426i	−0.991354	−	0.131213i	$-0.958113\pi$
$521$	− 136.724i	− 0.262426i	0.991354	−	0.131213i	$-0.0418873\pi$
$522$	0	0
$523$	− 14.5674i	− 0.0278535i	−0.999903	−	0.0139268i	$-0.995567\pi$
$523$	− 14.5674i	− 0.0278535i	0.999903	−	0.0139268i	$-0.00443317\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−245.825	−0.466461
$528$	0	0
$529$	47.0000	0.0888469
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−43.6926	−0.0819748
$534$	0	0
$535$	455.796i	0.851954i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−924.901	−1.70961	−0.854807	−	0.518947i	$-0.826324\pi$
$541$	−924.901	−1.70961	−0.854807	+	0.518947i	$0.826324\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 69.1809i	− 0.126937i
$546$	0	0
$547$	103.626	0.189445	0.0947225	−	0.995504i	$-0.469804\pi$
$547$	103.626	0.189445	0.0947225	+	0.995504i	$0.469804\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 412.761i	− 0.749112i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−412.407	−0.740407	−0.370203	−	0.928951i	$-0.620712\pi$
$557$	−412.407	−0.740407	−0.370203	+	0.928951i	$0.620712\pi$
$558$	0	0
$559$	− 328.925i	− 0.588416i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	686.367i	1.21912i	0.792738	+	0.609562i	$0.208655\pi$
$563$	686.367i	1.21912i	−0.792738	+	0.609562i	$0.791345\pi$
$564$	0	0
$565$	261.405i	0.462664i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	146.615	0.257671	0.128836	−	0.991666i	$-0.458876\pi$
$569$	146.615	0.257671	0.128836	+	0.991666i	$0.458876\pi$
$570$	0	0
$571$	356.263	0.623928	0.311964	−	0.950094i	$-0.399013\pi$
$571$	356.263	0.623928	0.311964	+	0.950094i	$0.399013\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	138.724	0.241259
$576$	0	0
$577$	− 171.028i	− 0.296409i	−0.988957	−	0.148205i	$-0.952651\pi$
$577$	− 171.028i	− 0.296409i	0.988957	−	0.148205i	$-0.0473495\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1206.82	2.07002
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.8384i	0.0423142i	0.999776	+	0.0211571i	$0.00673502\pi$
$587$	24.8384i	0.0423142i	−0.999776	+	0.0211571i	$0.993265\pi$
$588$	0	0
$589$	851.317	1.44536
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 136.077i	− 0.229472i	−0.993396	−	0.114736i	$-0.963398\pi$
$593$	− 136.077i	− 0.229472i	0.993396	−	0.114736i	$-0.0366023\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−989.230	−1.65147	−0.825734	−	0.564059i	$-0.809239\pi$
$599$	−989.230	−1.65147	−0.825734	+	0.564059i	$0.809239\pi$
$600$	0	0
$601$	78.8184i	0.131145i	0.997848	+	0.0655727i	$0.0208874\pi$
$601$	78.8184i	0.131145i	−0.997848	+	0.0655727i	$0.979113\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1152.57i	1.90507i
$606$	0	0
$607$	− 840.449i	− 1.38459i	−0.721612	−	0.692297i	$-0.756599\pi$
$607$	− 840.449i	− 1.38459i	0.721612	−	0.692297i	$-0.243401\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−244.307	−0.399848
$612$	0	0
$613$	847.428	1.38243	0.691214	−	0.722650i	$-0.257076\pi$
$613$	847.428	1.38243	0.691214	+	0.722650i	$0.257076\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	649.669	1.05295	0.526474	−	0.850191i	$-0.323514\pi$
$617$	649.669	1.05295	0.526474	+	0.850191i	$0.323514\pi$
$618$	0	0
$619$	− 488.044i	− 0.788440i	−0.919016	−	0.394220i	$-0.871015\pi$
$619$	− 488.044i	− 0.788440i	0.919016	−	0.394220i	$-0.128985\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−447.086	−0.715337
$626$	0	0
$627$	0	0
$628$	0	0
$629$	426.071i	0.677378i
$630$	0	0
$631$	−820.473	−1.30027	−0.650137	−	0.759817i	$-0.725288\pi$
$631$	−820.473	−1.30027	−0.650137	+	0.759817i	$0.725288\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 109.601i	− 0.172600i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19.8677	0.0309949	0.0154975	−	0.999880i	$-0.495067\pi$
$641$	19.8677	0.0309949	0.0154975	+	0.999880i	$0.495067\pi$
$642$	0	0
$643$	− 1013.95i	− 1.57691i	−0.615092	−	0.788456i	$-0.710881\pi$
$643$	− 1013.95i	− 1.57691i	0.615092	−	0.788456i	$-0.289119\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 133.462i	− 0.206279i	−0.994667	−	0.103139i	$-0.967111\pi$
$647$	− 133.462i	− 0.206279i	0.994667	−	0.103139i	$-0.0328888\pi$
$648$	0	0
$649$	− 1511.24i	− 2.32857i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−604.757	−0.926121	−0.463060	−	0.886327i	$-0.653249\pi$
$653$	−604.757	−0.926121	−0.463060	+	0.886327i	$0.653249\pi$
$654$	0	0
$655$	−993.064	−1.51613
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−335.451	−0.509031	−0.254515	−	0.967069i	$-0.581916\pi$
$659$	−335.451	−0.509031	−0.254515	+	0.967069i	$0.581916\pi$
$660$	0	0
$661$	702.273i	1.06244i	0.847234	+	0.531220i	$0.178266\pi$
$661$	702.273i	1.06244i	−0.847234	+	0.531220i	$0.821734\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−326.241	−0.489117
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 8.45130i	− 0.0125951i
$672$	0	0
$673$	−1119.48	−1.66342	−0.831711	−	0.555209i	$-0.812638\pi$
$673$	−1119.48	−1.66342	−0.831711	+	0.555209i	$0.812638\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	260.626i	0.384973i	0.981300	+	0.192486i	$0.0616551\pi$
$677$	260.626i	0.384973i	−0.981300	+	0.192486i	$0.938345\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	200.516	0.293581	0.146790	−	0.989168i	$-0.453106\pi$
$683$	200.516	0.293581	0.146790	+	0.989168i	$0.453106\pi$
$684$	0	0
$685$	1083.39i	1.58159i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 376.711i	− 0.546750i
$690$	0	0
$691$	− 994.120i	− 1.43867i	−0.694664	−	0.719334i	$-0.744447\pi$
$691$	− 994.120i	− 1.43867i	0.694664	−	0.719334i	$-0.255553\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−646.132	−0.929687
$696$	0	0
$697$	−62.6381	−0.0898681
$698$	0	0
$699$	0	0
$700$	0	0
$701$	148.231	0.211457	0.105729	−	0.994395i	$-0.466283\pi$
$701$	148.231	0.211457	0.105729	+	0.994395i	$0.466283\pi$
$702$	0	0
$703$	− 1475.53i	− 2.09890i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	711.977	1.00420	0.502099	−	0.864810i	$-0.332561\pi$
$709$	711.977	1.00420	0.502099	+	0.864810i	$0.332561\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 672.872i	− 0.943719i
$714$	0	0
$715$	525.362	0.734772
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 749.527i	− 1.04246i	−0.853417	−	0.521228i	$-0.825474\pi$
$719$	− 749.527i	− 1.04246i	0.853417	−	0.521228i	$-0.174526\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−78.5719	−0.108375
$726$	0	0
$727$	− 865.702i	− 1.19079i	−0.803434	−	0.595393i	$-0.796996\pi$
$727$	− 865.702i	− 1.19079i	0.803434	−	0.595393i	$-0.203004\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 471.549i	− 0.645074i
$732$	0	0
$733$	− 96.4995i	− 0.131650i	−0.997831	−	0.0658251i	$-0.979032\pi$
$733$	− 96.4995i	− 0.131650i	0.997831	−	0.0658251i	$-0.0209679\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1116.18	−1.51448
$738$	0	0
$739$	−420.901	−0.569555	−0.284777	−	0.958594i	$-0.591920\pi$
$739$	−420.901	−0.569555	−0.284777	+	0.958594i	$0.591920\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−920.043	−1.23828	−0.619141	−	0.785280i	$-0.712519\pi$
$743$	−920.043	−1.23828	−0.619141	+	0.785280i	$0.712519\pi$
$744$	0	0
$745$	− 564.723i	− 0.758017i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−320.362	−0.426580	−0.213290	−	0.976989i	$-0.568418\pi$
$751$	−320.362	−0.426580	−0.213290	+	0.976989i	$0.568418\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	271.956i	0.360206i
$756$	0	0
$757$	330.813	0.437006	0.218503	−	0.975836i	$-0.429883\pi$
$757$	330.813	0.437006	0.218503	+	0.975836i	$0.429883\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	266.291i	0.349923i	0.984575	+	0.174961i	$0.0559800\pi$
$761$	266.291i	0.349923i	−0.984575	+	0.174961i	$0.944020\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−471.735	−0.615040
$768$	0	0
$769$	− 241.301i	− 0.313785i	−0.987616	−	0.156893i	$-0.949852\pi$
$769$	− 241.301i	− 0.313785i	0.987616	−	0.156893i	$-0.0501477\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	929.774i	1.20281i	0.798943	+	0.601406i	$0.205393\pi$
$773$	929.774i	1.20281i	−0.798943	+	0.601406i	$0.794607\pi$
$774$	0	0
$775$	− 162.054i	− 0.209102i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	216.922	0.278462
$780$	0	0
$781$	2420.97	3.09983
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1099.12	1.40015
$786$	0	0
$787$	1109.78i	1.41014i	0.709138	+	0.705070i	$0.249084\pi$
$787$	1109.78i	1.41014i	−0.709138	+	0.705070i	$0.750916\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.63808	−0.00332671
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 231.047i	− 0.289896i	−0.989439	−	0.144948i	$-0.953699\pi$
$797$	− 231.047i	− 0.289896i	0.989439	−	0.144948i	$-0.0463014\pi$
$798$	0	0
$799$	−350.241	−0.438350
$800$	0	0
$801$	0	0
$802$	0	0
$803$	703.098i	0.875589i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	318.945	0.394247	0.197123	−	0.980379i	$-0.436840\pi$
$809$	318.945	0.394247	0.197123	+	0.980379i	$0.436840\pi$
$810$	0	0
$811$	196.807i	0.242672i	0.992612	+	0.121336i	$0.0387178\pi$
$811$	196.807i	0.242672i	−0.992612	+	0.121336i	$0.961282\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 429.346i	− 0.526805i
$816$	0	0
$817$	1633.02i	1.99881i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1384.01	1.68576	0.842881	−	0.538101i	$-0.180858\pi$
$821$	1384.01	1.68576	0.842881	+	0.538101i	$0.180858\pi$
$822$	0	0
$823$	276.506	0.335973	0.167987	−	0.985789i	$-0.446273\pi$
$823$	276.506	0.335973	0.167987	+	0.985789i	$0.446273\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1467.81	−1.77486	−0.887431	−	0.460940i	$-0.847512\pi$
$827$	−1467.81	−1.77486	−0.887431	+	0.460940i	$0.847512\pi$
$828$	0	0
$829$	78.7343i	0.0949750i	0.998872	+	0.0474875i	$0.0151214\pi$
$829$	78.7343i	0.0949750i	−0.998872	+	0.0474875i	$0.984879\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	314.638	0.376812
$836$	0	0
$837$	0	0
$838$	0	0
$839$	591.398i	0.704884i	0.935834	+	0.352442i	$0.114649\pi$
$839$	591.398i	0.704884i	−0.935834	+	0.352442i	$0.885351\pi$
$840$	0	0
$841$	−656.220	−0.780285
$842$	0	0
$843$	0	0
$844$	0	0
$845$	576.911i	0.682735i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1166.24	−1.37044
$852$	0	0
$853$	1048.80i	1.22954i	0.788707	+	0.614770i	$0.210751\pi$
$853$	1048.80i	1.22954i	−0.788707	+	0.614770i	$0.789249\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 107.171i	− 0.125054i	−0.998043	−	0.0625271i	$-0.980084\pi$
$857$	− 107.171i	− 0.125054i	0.998043	−	0.0625271i	$-0.0199160\pi$
$858$	0	0
$859$	1506.32i	1.75358i	0.480874	+	0.876790i	$0.340320\pi$
$859$	1506.32i	1.75358i	−0.480874	+	0.876790i	$0.659680\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−375.362	−0.434950	−0.217475	−	0.976066i	$-0.569782\pi$
$863$	−375.362	−0.434950	−0.217475	+	0.976066i	$0.569782\pi$
$864$	0	0
$865$	−1324.09	−1.53073
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1022.52	−1.17666
$870$	0	0
$871$	348.415i	0.400017i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1408.79	1.60637	0.803187	−	0.595727i	$-0.203136\pi$
$877$	1408.79	1.60637	0.803187	+	0.595727i	$0.203136\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1236.04i	− 1.40299i	−0.712673	−	0.701497i	$-0.752515\pi$
$881$	− 1236.04i	− 1.40299i	0.712673	−	0.701497i	$-0.247485\pi$
$882$	0	0
$883$	−72.9437	−0.0826089	−0.0413045	−	0.999147i	$-0.513151\pi$
$883$	−72.9437	−0.0826089	−0.0413045	+	0.999147i	$0.513151\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1442.23i	1.62597i	0.582287	+	0.812983i	$0.302158\pi$
$887$	1442.23i	1.62597i	−0.582287	+	0.812983i	$0.697842\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1212.92	1.35826
$894$	0	0
$895$	− 604.998i	− 0.675975i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	381.108i	0.423925i
$900$	0	0
$901$	− 540.056i	− 0.599396i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	363.759	0.401943
$906$	0	0
$907$	−1209.76	−1.33380	−0.666900	−	0.745147i	$-0.732379\pi$
$907$	−1209.76	−1.33380	−0.666900	+	0.745147i	$0.732379\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1328.22	1.45798	0.728989	−	0.684525i	$-0.239991\pi$
$911$	1328.22	1.45798	0.728989	+	0.684525i	$0.239991\pi$
$912$	0	0
$913$	2681.99i	2.93756i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−488.858	−0.531946	−0.265973	−	0.963981i	$-0.585693\pi$
$919$	−488.858	−0.531946	−0.265973	+	0.963981i	$0.585693\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 755.707i	− 0.818750i
$924$	0	0
$925$	−280.878	−0.303651
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 951.536i	− 1.02426i	−0.858908	−	0.512129i	$-0.828857\pi$
$929$	− 951.536i	− 1.02426i	0.858908	−	0.512129i	$-0.171143\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	753.164	0.805522
$936$	0	0
$937$	− 309.707i	− 0.330530i	−0.986249	−	0.165265i	$-0.947152\pi$
$937$	− 309.707i	− 0.330530i	0.986249	−	0.165265i	$-0.0528480\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1172.51i	− 1.24602i	−0.782213	−	0.623011i	$-0.785909\pi$
$941$	− 1172.51i	− 1.24602i	0.782213	−	0.623011i	$-0.214091\pi$
$942$	0	0
$943$	− 171.453i	− 0.181816i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	724.856	0.765424	0.382712	−	0.923868i	$-0.374990\pi$
$947$	724.856	0.765424	0.382712	+	0.923868i	$0.374990\pi$
$948$	0	0
$949$	219.473	0.231267
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−27.5371	−0.0288951	−0.0144476	−	0.999896i	$-0.504599\pi$
$953$	−27.5371	−0.0288951	−0.0144476	+	0.999896i	$0.504599\pi$
$954$	0	0
$955$	− 755.403i	− 0.790998i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	174.965	0.182066
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1206.96i	1.25074i
$966$	0	0
$967$	1279.81	1.32349	0.661744	−	0.749730i	$-0.269817\pi$
$967$	1279.81	1.32349	0.661744	+	0.749730i	$0.269817\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1112.77i	1.14600i	0.819554	+	0.573001i	$0.194221\pi$
$971$	1112.77i	1.14600i	−0.819554	+	0.573001i	$0.805779\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	435.759	0.446017	0.223009	−	0.974816i	$-0.428412\pi$
$977$	435.759	0.446017	0.223009	+	0.974816i	$0.428412\pi$
$978$	0	0
$979$	299.092i	0.305508i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	281.556i	0.286425i	0.989692	+	0.143213i	$0.0457433\pi$
$983$	281.556i	0.286425i	−0.989692	+	0.143213i	$0.954257\pi$
$984$	0	0
$985$	676.490i	0.686792i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1290.72	1.30508
$990$	0	0
$991$	−617.572	−0.623181	−0.311590	−	0.950217i	$-0.600862\pi$
$991$	−617.572	−0.623181	−0.311590	+	0.950217i	$0.600862\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	823.868	0.828008
$996$	0	0
$997$	− 1505.63i	− 1.51016i	−0.655634	−	0.755079i	$-0.727599\pi$
$997$	− 1505.63i	− 1.51016i	0.655634	−	0.755079i	$-0.272401\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.d.f.685.3		4
3.2	odd	2		588.3.d.b.97.1		4
7.2	even	3		1764.3.z.h.325.2		4
7.3	odd	6		1764.3.z.h.901.2		4
7.4	even	3		252.3.z.e.145.1		4
7.5	odd	6		252.3.z.e.73.1		4
7.6	odd	2	inner	1764.3.d.f.685.2		4
12.11	even	2		2352.3.f.f.97.3		4
21.2	odd	6		588.3.m.d.325.1		4
21.5	even	6		84.3.m.b.73.2	yes	4
21.11	odd	6		84.3.m.b.61.2	✓	4
21.17	even	6		588.3.m.d.313.1		4
21.20	even	2		588.3.d.b.97.4		4
28.11	odd	6		1008.3.cg.m.145.1		4
28.19	even	6		1008.3.cg.m.577.1		4
84.11	even	6		336.3.bh.f.145.2		4
84.47	odd	6		336.3.bh.f.241.2		4
84.83	odd	2		2352.3.f.f.97.2		4
105.32	even	12		2100.3.be.d.649.3		8
105.47	odd	12		2100.3.be.d.1249.2		8
105.53	even	12		2100.3.be.d.649.2		8
105.68	odd	12		2100.3.be.d.1249.3		8
105.74	odd	6		2100.3.bd.f.901.1		4
105.89	even	6		2100.3.bd.f.1501.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.m.b.61.2	✓	4	21.11	odd	6
84.3.m.b.73.2	yes	4	21.5	even	6
252.3.z.e.73.1		4	7.5	odd	6
252.3.z.e.145.1		4	7.4	even	3
336.3.bh.f.145.2		4	84.11	even	6
336.3.bh.f.241.2		4	84.47	odd	6
588.3.d.b.97.1		4	3.2	odd	2
588.3.d.b.97.4		4	21.20	even	2
588.3.m.d.313.1		4	21.17	even	6
588.3.m.d.325.1		4	21.2	odd	6
1008.3.cg.m.145.1		4	28.11	odd	6
1008.3.cg.m.577.1		4	28.19	even	6
1764.3.d.f.685.2		4	7.6	odd	2	inner
1764.3.d.f.685.3		4	1.1	even	1	trivial
1764.3.z.h.325.2		4	7.2	even	3
1764.3.z.h.901.2		4	7.3	odd	6
2100.3.bd.f.901.1		4	105.74	odd	6
2100.3.bd.f.1501.2		4	105.89	even	6
2100.3.be.d.649.2		8	105.53	even	12
2100.3.be.d.649.3		8	105.32	even	12
2100.3.be.d.1249.2		8	105.47	odd	12
2100.3.be.d.1249.3		8	105.68	odd	12
2352.3.f.f.97.2		4	84.83	odd	2
2352.3.f.f.97.3		4	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.38404i q^{5} +19.5934 q^{11} -6.11609i q^{13} -8.76809i q^{17} +30.3648i q^{19} +24.0000 q^{23} +5.78016 q^{25} -13.5934 q^{29} -28.0363i q^{31} -48.5934 q^{37} -7.14387i q^{41} +53.7802 q^{43} -39.9450i q^{47} +61.5934 q^{53} +85.8983i q^{55} -77.1302i q^{59} -0.431334i q^{61} +26.8132 q^{65} -56.9669 q^{67} +123.560 q^{71} +35.8845i q^{73} -52.1868 q^{79} +136.883i q^{83} +38.4397 q^{85} +15.2650i q^{89} -133.121 q^{95} -34.1524i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 30 q^{11} + 96 q^{23} - 122 q^{25} - 6 q^{29} - 146 q^{37} + 70 q^{43} + 198 q^{53} + 204 q^{65} + 14 q^{67} + 204 q^{71} - 112 q^{79} + 444 q^{85} + 48 q^{95}+O(q^{100}) \) 4 * q + 30 * q^11 + 96 * q^23 - 122 * q^25 - 6 * q^29 - 146 * q^37 + 70 * q^43 + 198 * q^53 + 204 * q^65 + 14 * q^67 + 204 * q^71 - 112 * q^79 + 444 * q^85 + 48 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more