LMFDB - Embedded newform 252.4.k.b.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,4,Mod(37,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, 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("252.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, degree $2$, 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:	$14.8684813214$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	252.109
Dual form			252.4.k.b.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(8.50000 - 16.4545i) q^{7} +O(q^{10})$	$q+(8.50000 - 16.4545i) q^{7} -19.0000 q^{13} +(-53.5000 - 92.6647i) q^{19} +(62.5000 - 108.253i) q^{25} +(144.500 - 250.281i) q^{31} +(-161.500 - 279.726i) q^{37} +71.0000 q^{43} +(-198.500 - 279.726i) q^{49} +(-91.0000 - 157.617i) q^{61} +(63.5000 - 109.985i) q^{67} +(135.500 - 234.693i) q^{73} +(693.500 + 1201.18i) q^{79} +(-161.500 + 312.635i) q^{91} -1330.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 17 q^{7}+O(q^{10}) $ 2 * q + 17 * q^7	$ 2 q + 17 q^{7} - 38 q^{13} - 107 q^{19} + 125 q^{25} + 289 q^{31} - 323 q^{37} + 142 q^{43} - 397 q^{49} - 182 q^{61} + 127 q^{67} + 271 q^{73} + 1387 q^{79} - 323 q^{91} - 2660 q^{97}+O(q^{100}) $ 2 * q + 17 * q^7 - 38 * q^13 - 107 * q^19 + 125 * q^25 + 289 * q^31 - 323 * q^37 + 142 * q^43 - 397 * q^49 - 182 * q^61 + 127 * q^67 + 271 * q^73 + 1387 * q^79 - 323 * q^91 - 2660 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	8.50000	−	16.4545i	0.458957	−	0.888459i
$7$	8.50000	−	16.4545i	0.458957	−	0.888459i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	−19.0000			−0.405358			−0.202679	−	0.979245i	$-0.564965\pi$
$13$	−19.0000			−0.405358			−0.202679	+	0.979245i	$0.564965\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−53.5000	−	92.6647i	−0.645986	−	1.11888i	−0.984073	−	0.177766i	$-0.943113\pi$
$19$	−53.5000	−	92.6647i	−0.645986	−	1.11888i	0.338086	−	0.941115i	$-0.390220\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	62.5000	−	108.253i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	144.500	−	250.281i	0.837192	−	1.45006i	−0.0550403	−	0.998484i	$-0.517529\pi$
$31$	144.500	−	250.281i	0.837192	−	1.45006i	0.892233	−	0.451576i	$-0.149138\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−161.500	−	279.726i	−0.717579	−	1.24288i	−0.961956	−	0.273204i	$-0.911917\pi$
$37$	−161.500	−	279.726i	−0.717579	−	1.24288i	0.244377	−	0.969680i	$-0.421417\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	71.0000			0.251800			0.125900	−	0.992043i	$-0.459818\pi$
$43$	71.0000			0.251800			0.125900	+	0.992043i	$0.459818\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	−198.500	−	279.726i	−0.578717	−	0.815528i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$53$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	−91.0000	−	157.617i	−0.191006	−	0.330832i	0.754578	−	0.656210i	$-0.227842\pi$
$61$	−91.0000	−	157.617i	−0.191006	−	0.330832i	−0.945584	+	0.325379i	$0.894508\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	63.5000	−	109.985i	0.115787	−	0.200550i	−0.802307	−	0.596912i	$-0.796394\pi$
$67$	63.5000	−	109.985i	0.115787	−	0.200550i	0.918094	+	0.396362i	$0.129728\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	135.500	−	234.693i	0.217248	−	0.376284i	−0.736718	−	0.676200i	$-0.763625\pi$
$73$	135.500	−	234.693i	0.217248	−	0.376284i	0.953966	+	0.299916i	$0.0969588\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	693.500	+	1201.18i	0.987656	+	1.71067i	0.629480	+	0.777017i	$0.283268\pi$
$79$	693.500	+	1201.18i	0.987656	+	1.71067i	0.358177	+	0.933654i	$0.383399\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	−161.500	+	312.635i	−0.186042	+	0.360144i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1330.00			−1.39218			−0.696088	−	0.717957i	$-0.745078\pi$
$97$	−1330.00			−1.39218			−0.696088	+	0.717957i	$0.745078\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	900.500	+	1559.71i	0.861446	+	1.49207i	0.870534	+	0.492109i	$0.163774\pi$
$103$	900.500	+	1559.71i	0.861446	+	1.49207i	−0.00908799	+	0.999959i	$0.502893\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$	−1106.50	+	1916.51i	−0.972325	+	1.68412i	−0.283833	+	0.958874i	$0.591606\pi$
$109$	−1106.50	+	1916.51i	−0.972325	+	1.68412i	−0.688493	+	0.725243i	$0.741727\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	665.500	+	1152.68i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2267.00			1.58397			0.791983	−	0.610543i	$-0.209049\pi$
$127$	2267.00			1.58397			0.791983	+	0.610543i	$0.209049\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	−1979.50	+	92.6647i	−1.29056	+	0.0604139i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	467.000			0.284967			0.142484	−	0.989797i	$-0.454491\pi$
$139$	467.000			0.284967			0.142484	+	0.989797i	$0.454491\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	−874.000	+	1513.81i	−0.471027	+	0.815843i	−0.999451	−	0.0331378i	$-0.989450\pi$
$151$	−874.000	+	1513.81i	−0.471027	+	0.815843i	0.528424	+	0.848981i	$0.322783\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1925.00	−	3334.20i	0.978546	−	1.69489i	0.310847	−	0.950460i	$-0.399387\pi$
$157$	1925.00	−	3334.20i	0.978546	−	1.69489i	0.667699	−	0.744432i	$-0.267279\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1700.00	+	2944.49i	0.816897	+	1.41491i	0.907957	+	0.419062i	$0.137641\pi$
$163$	1700.00	+	2944.49i	0.816897	+	1.41491i	−0.0910600	+	0.995845i	$0.529026\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−1836.00			−0.835685
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−1250.00	−	1948.56i	−0.539949	−	0.841698i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−4699.00			−1.92969			−0.964845	−	0.262819i	$-0.915348\pi$
$181$	−4699.00			−1.92969			−0.964845	+	0.262819i	$0.915348\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	1980.50	−	3430.33i	0.738650	−	1.27938i	−0.214453	−	0.976734i	$-0.568797\pi$
$193$	1980.50	−	3430.33i	0.738650	−	1.27938i	0.953103	−	0.302646i	$-0.0978698\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	2618.00	−	4534.51i	0.932588	−	1.61529i	0.153710	−	0.988116i	$-0.450878\pi$
$199$	2618.00	−	4534.51i	0.932588	−	1.61529i	0.778879	−	0.627175i	$-0.215789\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	6032.00			1.96806			0.984028	−	0.178011i	$-0.0569664\pi$
$211$	6032.00			1.96806			0.984028	+	0.178011i	$0.0569664\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2890.00	−	4505.06i	−0.904083	−	1.40933i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−3220.00			−0.966938			−0.483469	−	0.875362i	$-0.660623\pi$
$223$	−3220.00			−0.966938			−0.483469	+	0.875362i	$0.660623\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	−1178.50	−	2041.22i	−0.340076	−	0.589029i	0.644370	−	0.764714i	$-0.277120\pi$
$229$	−1178.50	−	2041.22i	−0.340076	−	0.589029i	−0.984447	+	0.175684i	$0.943786\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	3689.00	−	6389.54i	0.986014	−	1.70783i	0.348673	−	0.937244i	$-0.386632\pi$
$241$	3689.00	−	6389.54i	0.986014	−	1.70783i	0.637341	−	0.770582i	$-0.280034\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	1016.50	+	1760.63i	0.261856	+	0.453547i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	−5975.50	+	279.726i	−1.43359	+	0.0671094i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	−406.000	−	703.213i	−0.0910064	−	0.157628i	0.816928	−	0.576739i	$-0.195675\pi$
$271$	−406.000	−	703.213i	−0.0910064	−	0.157628i	−0.907935	+	0.419111i	$0.862342\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4598.50	+	7964.84i	−0.997462	+	1.72766i	−0.437074	+	0.899425i	$0.643985\pi$
$277$	−4598.50	+	7964.84i	−0.997462	+	1.72766i	−0.560388	+	0.828230i	$0.689348\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	4734.50	−	8200.39i	0.994476	−	1.72248i	0.406340	−	0.913722i	$-0.366805\pi$
$283$	4734.50	−	8200.39i	0.994476	−	1.72248i	0.588137	−	0.808761i	$-0.299862\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2456.50	+	4254.78i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	603.500	−	1168.27i	0.115565	−	0.223714i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−3943.00			−0.733026			−0.366513	−	0.930413i	$-0.619448\pi$
$307$	−3943.00			−0.733026			−0.366513	+	0.930413i	$0.619448\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	4554.50	+	7888.63i	0.822478	+	1.42457i	0.903832	+	0.427888i	$0.140742\pi$
$313$	4554.50	+	7888.63i	0.822478	+	1.42457i	−0.0813539	+	0.996685i	$0.525924\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1187.50	+	2056.81i	−0.202679	+	0.351050i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	5445.50	+	9431.88i	0.904265	+	1.56623i	0.821901	+	0.569631i	$0.192914\pi$
$331$	5445.50	+	9431.88i	0.904265	+	1.56623i	0.0823644	+	0.996602i	$0.473753\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	12293.0			1.98707			0.993535	−	0.113529i	$-0.0362155\pi$
$337$	12293.0			1.98707			0.993535	+	0.113529i	$0.0362155\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−6290.00	+	888.542i	−0.990169	+	0.139874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	−11914.0			−1.82734			−0.913670	−	0.406456i	$-0.866764\pi$
$349$	−11914.0			−1.82734			−0.913670	+	0.406456i	$0.866764\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	−2295.00	+	3975.06i	−0.334597	+	0.579539i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−4706.50	+	8151.90i	−0.669420	+	1.15947i	0.308646	+	0.951177i	$0.400124\pi$
$367$	−4706.50	+	8151.90i	−0.669420	+	1.15947i	−0.978066	+	0.208293i	$0.933209\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6300.50	+	10912.8i	0.874605	+	1.51486i	0.857183	+	0.515011i	$0.172212\pi$
$373$	6300.50	+	10912.8i	0.874605	+	1.51486i	0.0174213	+	0.999848i	$0.494454\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−6103.00			−0.827151			−0.413575	−	0.910470i	$-0.635720\pi$
$379$	−6103.00			−0.827151			−0.413575	+	0.910470i	$0.635720\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7128.50	+	12346.9i	0.901182	+	1.56089i	0.825962	+	0.563726i	$0.190632\pi$
$397$	7128.50	+	12346.9i	0.901182	+	1.56089i	0.0752196	+	0.997167i	$0.476034\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	−2745.50	+	4755.35i	−0.339363	+	0.587793i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	8271.50	−	14326.7i	0.999998	−	1.73205i	0.498458	−	0.866914i	$-0.333900\pi$
$409$	8271.50	−	14326.7i	0.999998	−	1.73205i	0.501541	−	0.865134i	$-0.332767\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−6679.00			−0.773194			−0.386597	−	0.922249i	$-0.626350\pi$
$421$	−6679.00			−0.773194			−0.386597	+	0.922249i	$0.626350\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3367.00	+	157.617i	−0.381594	+	0.0178632i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$		0			0
$433$	16739.0			1.85779			0.928897	−	0.370338i	$-0.120758\pi$
$433$	16739.0			1.85779			0.928897	+	0.370338i	$0.120758\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−7462.00	−	12924.6i	−0.811257	−	1.40514i	−0.911985	−	0.410224i	$-0.865450\pi$
$439$	−7462.00	−	12924.6i	−0.811257	−	1.40514i	0.100728	−	0.994914i	$-0.467883\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$443$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3248.50	−	5626.57i	−0.332513	−	0.575929i	0.650491	−	0.759514i	$-0.274563\pi$
$457$	−3248.50	−	5626.57i	−0.332513	−	0.575929i	−0.983004	+	0.183585i	$0.941230\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	11969.0			1.20140			0.600698	−	0.799476i	$-0.294889\pi$
$463$	11969.0			1.20140			0.600698	+	0.799476i	$0.294889\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$	−1270.00	−	1979.73i	−0.125039	−	0.194916i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−13375.0			−1.29197
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	3068.50	+	5314.80i	0.290876	+	0.503813i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	3051.50	−	5285.35i	0.283936	−	0.491791i	−0.688415	−	0.725317i	$-0.741693\pi$
$487$	3051.50	−	5285.35i	0.283936	−	0.491791i	0.972351	+	0.233526i	$0.0750265\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−10871.5	−	18830.0i	−0.975301	−	1.68927i	−0.678938	−	0.734195i	$-0.737560\pi$
$499$	−10871.5	−	18830.0i	−0.975301	−	1.68927i	−0.296363	−	0.955075i	$-0.595774\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−2710.00	−	4224.47i	−0.234605	−	0.365714i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	−11960.5	−	20716.2i	−0.999993	−	1.73204i	−0.503320	−	0.864100i	$-0.667888\pi$
$523$	−11960.5	−	20716.2i	−0.999993	−	1.73204i	−0.496673	−	0.867938i	$-0.665445\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	6083.50	−	10536.9i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−944.500	−	1635.92i	−0.0750596	−	0.130007i	0.826053	−	0.563593i	$-0.190581\pi$
$541$	−944.500	−	1635.92i	−0.0750596	−	0.130007i	−0.901112	+	0.433586i	$0.857248\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1640.00			0.128193			0.0640963	−	0.997944i	$-0.479584\pi$
$547$	1640.00			0.128193			0.0640963	+	0.997944i	$0.479584\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	25659.5	−	1201.18i	1.97315	−	0.0923675i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$	−1349.00			−0.102069
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	11970.5	−	20733.5i	0.877320	−	1.51956i	0.0230498	−	0.999734i	$-0.492662\pi$
$571$	11970.5	−	20733.5i	0.877320	−	1.51956i	0.854270	−	0.519829i	$-0.174004\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	4806.50	−	8325.10i	0.346789	−	0.600656i	−0.638888	−	0.769300i	$-0.720605\pi$
$577$	4806.50	−	8325.10i	0.346789	−	0.600656i	0.985677	+	0.168644i	$0.0539387\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−30923.0			−2.16326
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	17351.0			1.17764			0.588820	−	0.808264i	$-0.299593\pi$
$601$	17351.0			1.17764			0.588820	+	0.808264i	$0.299593\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−3068.50	−	5314.80i	−0.205184	−	0.355389i	0.745007	−	0.667056i	$-0.232446\pi$
$607$	−3068.50	−	5314.80i	−0.205184	−	0.355389i	−0.950191	+	0.311667i	$0.899113\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−8695.00	+	15060.2i	−0.572900	+	0.992292i	0.423366	+	0.905959i	$0.360848\pi$
$613$	−8695.00	+	15060.2i	−0.572900	+	0.992292i	−0.996266	+	0.0863334i	$0.972485\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	−13346.5	+	23116.8i	−0.866625	+	1.50104i	−0.00120126	+	0.999999i	$0.500382\pi$
$619$	−13346.5	+	23116.8i	−0.866625	+	1.50104i	−0.865424	+	0.501040i	$0.832951\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7812.50	−	13531.6i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1892.00			0.119365			0.0596825	−	0.998217i	$-0.480991\pi$
$631$	1892.00			0.119365			0.0596825	+	0.998217i	$0.480991\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3771.50	+	5314.80i	0.234588	+	0.330581i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	19259.0			1.18118			0.590592	−	0.806971i	$-0.298894\pi$
$643$	19259.0			1.18118			0.590592	+	0.806971i	$0.298894\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−16865.5	+	29211.9i	−0.992423	+	1.71893i	−0.389808	+	0.920896i	$0.627459\pi$
$661$	−16865.5	+	29211.9i	−0.992423	+	1.71893i	−0.602615	+	0.798032i	$0.705875\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	9899.00			0.566981			0.283491	−	0.958975i	$-0.408508\pi$
$673$	9899.00			0.566981			0.283491	+	0.958975i	$0.408508\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−11305.0	+	21884.5i	−0.638949	+	1.23689i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$683$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	10089.5	+	17475.5i	0.555460	+	0.962084i	0.997868	+	0.0652705i	$0.0207910\pi$
$691$	10089.5	+	17475.5i	0.555460	+	0.962084i	−0.442408	+	0.896814i	$0.645876\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−17280.5	+	29930.7i	−0.927093	+	1.60577i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−18073.0	−	31303.4i	−0.957328	−	1.65814i	−0.728948	−	0.684569i	$-0.759990\pi$
$709$	−18073.0	−	31303.4i	−0.957328	−	1.65814i	−0.228381	−	0.973572i	$-0.573343\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	33318.5	−	1559.71i	1.72101	−	0.0805641i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	38033.0			1.94026			0.970128	−	0.242594i	$-0.0779984\pi$
$727$	38033.0			1.94026			0.970128	+	0.242594i	$0.0779984\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	19665.5	+	34061.6i	0.990944	+	1.71637i	0.611759	+	0.791044i	$0.290462\pi$
$733$	19665.5	+	34061.6i	0.990944	+	1.71637i	0.379184	+	0.925321i	$0.376205\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−3023.50	+	5236.86i	−0.150502	+	0.260678i	−0.931412	−	0.363966i	$-0.881422\pi$
$739$	−3023.50	+	5236.86i	−0.150502	+	0.260678i	0.780910	+	0.624644i	$0.214756\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	8784.50	+	15215.2i	0.426832	+	0.739295i	0.996590	−	0.0825179i	$-0.0262962\pi$
$751$	8784.50	+	15215.2i	0.426832	+	0.739295i	−0.569757	+	0.821813i	$0.692963\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−41470.0			−1.99109			−0.995543	−	0.0943039i	$-0.969937\pi$
$757$	−41470.0			−1.99109			−0.995543	+	0.0943039i	$0.969937\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	22130.0	+	34497.3i	1.05001	+	1.63681i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−34417.0			−1.61393			−0.806963	−	0.590602i	$-0.798890\pi$
$769$	−34417.0			−1.61393			−0.806963	+	0.590602i	$0.798890\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	−18062.5	−	31285.2i	−0.837192	−	1.45006i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−21700.0	+	37585.5i	−0.982874	+	1.70239i	−0.331844	+	0.943334i	$0.607671\pi$
$787$	−21700.0	+	37585.5i	−0.982874	+	1.70239i	−0.651029	+	0.759053i	$0.725662\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1729.00	+	2994.72i	0.0774257	+	0.134105i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$809$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$810$		0			0
$811$	39368.0			1.70456			0.852280	−	0.523087i	$-0.175220\pi$
$811$	39368.0			1.70456			0.852280	+	0.523087i	$0.175220\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−3798.50	−	6579.19i	−0.162659	−	0.281734i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$		0			0
$823$	6110.00	−	10582.8i	0.258786	−	0.448231i	−0.707131	−	0.707083i	$-0.750011\pi$
$823$	6110.00	−	10582.8i	0.258786	−	0.448231i	0.965917	+	0.258852i	$0.0833441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−15038.5	+	26047.4i	−0.630047	+	1.09127i	0.357495	+	0.933915i	$0.383631\pi$
$829$	−15038.5	+	26047.4i	−0.630047	+	1.09127i	−0.987542	+	0.157358i	$0.949702\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−24389.0			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	24623.5	−	1152.68i	0.998906	−	0.0467610i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	8279.00			0.332318			0.166159	−	0.986099i	$-0.446863\pi$
$853$	8279.00			0.332318			0.166159	+	0.986099i	$0.446863\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	−15652.0	−	27110.1i	−0.621699	−	1.07681i	−0.989169	−	0.146779i	$-0.953109\pi$
$859$	−15652.0	−	27110.1i	−0.621699	−	1.07681i	0.367470	−	0.930035i	$-0.380224\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−1206.50	+	2089.72i	−0.0469354	+	0.0812944i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−25075.0	−	43431.2i	−0.965476	−	1.67225i	−0.708330	−	0.705881i	$-0.750551\pi$
$877$	−25075.0	−	43431.2i	−0.965476	−	1.67225i	−0.257146	−	0.966373i	$-0.582782\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	52109.0			1.98597			0.992983	−	0.118260i	$-0.0377315\pi$
$883$	52109.0			1.98597			0.992983	+	0.118260i	$0.0377315\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	19269.5	−	37302.3i	0.726972	−	1.40729i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	24723.5	−	42822.4i	0.905105	−	1.56769i	0.0843291	−	0.996438i	$-0.473125\pi$
$907$	24723.5	−	42822.4i	0.905105	−	1.56769i	0.820776	−	0.571250i	$-0.193541\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−23408.5	−	40544.7i	−0.840234	−	1.45533i	−0.889697	−	0.456552i	$-0.849084\pi$
$919$	−23408.5	−	40544.7i	−0.840234	−	1.45533i	0.0494625	−	0.998776i	$-0.484249\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−40375.0			−1.43516
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	−15301.0	+	33359.3i	−0.538636	+	1.17434i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	40283.0			1.40447			0.702235	−	0.711945i	$-0.252186\pi$
$937$	40283.0			1.40447			0.702235	+	0.711945i	$0.252186\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$	−2574.50	+	4459.16i	−0.0880630	+	0.152530i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−26865.0	−	46531.5i	−0.901782	−	1.56193i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−3907.00			−0.129928			−0.0649641	−	0.997888i	$-0.520693\pi$
$967$	−3907.00			−0.129928			−0.0649641	+	0.997888i	$0.520693\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$	3969.50	−	7684.24i	0.130788	−	0.253181i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	7020.50	−	12159.9i	0.225039	−	0.389779i	−0.731292	−	0.682064i	$-0.761082\pi$
$991$	7020.50	−	12159.9i	0.225039	−	0.389779i	0.956331	+	0.292286i	$0.0944158\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	31446.5	−	54466.9i	0.998917	−	1.73018i	0.459172	−	0.888347i	$-0.348146\pi$
$997$	31446.5	−	54466.9i	0.998917	−	1.73018i	0.539745	−	0.841828i	$-0.318520\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.4.k.b.109.1	yes	2
3.2	odd	2	CM	252.4.k.b.109.1	yes	2
7.2	even	3	inner	252.4.k.b.37.1	✓	2
7.3	odd	6		1764.4.a.g.1.1		1
7.4	even	3		1764.4.a.f.1.1		1
7.5	odd	6		1764.4.k.i.1549.1		2
7.6	odd	2		1764.4.k.i.361.1		2
21.2	odd	6	inner	252.4.k.b.37.1	✓	2
21.5	even	6		1764.4.k.i.1549.1		2
21.11	odd	6		1764.4.a.f.1.1		1
21.17	even	6		1764.4.a.g.1.1		1
21.20	even	2		1764.4.k.i.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.k.b.37.1	✓	2	7.2	even	3	inner
252.4.k.b.37.1	✓	2	21.2	odd	6	inner
252.4.k.b.109.1	yes	2	1.1	even	1	trivial
252.4.k.b.109.1	yes	2	3.2	odd	2	CM
1764.4.a.f.1.1		1	7.4	even	3
1764.4.a.f.1.1		1	21.11	odd	6
1764.4.a.g.1.1		1	7.3	odd	6
1764.4.a.g.1.1		1	21.17	even	6
1764.4.k.i.361.1		2	7.6	odd	2
1764.4.k.i.361.1		2	21.20	even	2
1764.4.k.i.1549.1		2	7.5	odd	6
1764.4.k.i.1549.1		2	21.5	even	6

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(8.50000 - 16.4545i) q^{7} +O(q^{10})\)	\(q+(8.50000 - 16.4545i) q^{7} -19.0000 q^{13} +(-53.5000 - 92.6647i) q^{19} +(62.5000 - 108.253i) q^{25} +(144.500 - 250.281i) q^{31} +(-161.500 - 279.726i) q^{37} +71.0000 q^{43} +(-198.500 - 279.726i) q^{49} +(-91.0000 - 157.617i) q^{61} +(63.5000 - 109.985i) q^{67} +(135.500 - 234.693i) q^{73} +(693.500 + 1201.18i) q^{79} +(-161.500 + 312.635i) q^{91} -1330.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 17 q^{7}+O(q^{10}) \) 2 * q + 17 * q^7	\( 2 q + 17 q^{7} - 38 q^{13} - 107 q^{19} + 125 q^{25} + 289 q^{31} - 323 q^{37} + 142 q^{43} - 397 q^{49} - 182 q^{61} + 127 q^{67} + 271 q^{73} + 1387 q^{79} - 323 q^{91} - 2660 q^{97}+O(q^{100}) \) 2 * q + 17 * q^7 - 38 * q^13 - 107 * q^19 + 125 * q^25 + 289 * q^31 - 323 * q^37 + 142 * q^43 - 397 * q^49 - 182 * q^61 + 127 * q^67 + 271 * q^73 + 1387 * q^79 - 323 * q^91 - 2660 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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