LMFDB - Embedded newform 756.4.t.b.269.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,4,Mod(269,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.269"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.t (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,37] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$44.6054439643$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			269.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	756.269
Dual form			756.4.t.b.593.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(18.5000 - 0.866025i) q^{7} -62.3538i q^{13} +(-25.5000 + 14.7224i) q^{19} +(62.5000 - 108.253i) q^{25} +(-163.500 - 94.3968i) q^{31} +(55.0000 + 95.2628i) q^{37} -71.0000 q^{43} +(341.500 - 32.0429i) q^{49} +(268.500 - 155.019i) q^{61} +(440.000 - 762.102i) q^{67} +(-730.500 - 421.754i) q^{73} +(-442.000 - 765.566i) q^{79} +(-54.0000 - 1153.55i) q^{91} -465.922i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 37 q^{7} - 51 q^{19} + 125 q^{25} - 327 q^{31} + 110 q^{37} - 142 q^{43} + 683 q^{49} + 537 q^{61} + 880 q^{67} - 1461 q^{73} - 884 q^{79} - 108 q^{91}+O(q^{100}) $ 2 * q + 37 * q^7 - 51 * q^19 + 125 * q^25 - 327 * q^31 + 110 * q^37 - 142 * q^43 + 683 * q^49 + 537 * q^61 + 880 * q^67 - 1461 * q^73 - 884 * q^79 - 108 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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
$7$	18.5000	−	0.866025i	0.998906	−	0.0467610i
$7$	18.5000	−	0.866025i	0.998906	−	0.0467610i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$		−	62.3538i		−	1.33030i	−0.746712	−	0.665148i	$-0.768369\pi$
$13$		−	62.3538i		−	1.33030i	0.746712	−	0.665148i	$-0.231631\pi$
$14$		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
$19$	−25.5000	+	14.7224i	−0.307900	+	0.177766i	−0.645986	−	0.763349i	$-0.723554\pi$
$19$	−25.5000	+	14.7224i	−0.307900	+	0.177766i	0.338086	+	0.941115i	$0.390220\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	62.5000	−	108.253i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−163.500	−	94.3968i	−0.947273	−	0.546908i	−0.0550403	−	0.998484i	$-0.517529\pi$
$31$	−163.500	−	94.3968i	−0.947273	−	0.546908i	−0.892233	+	0.451576i	$0.850862\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	55.0000	+	95.2628i	0.244377	+	0.423273i	0.961956	−	0.273204i	$-0.0880833\pi$
$37$	55.0000	+	95.2628i	0.244377	+	0.423273i	−0.717579	+	0.696477i	$0.754750\pi$
$38$		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	$-0.540182\pi$
$43$	−71.0000			−0.251800			−0.125900	+	0.992043i	$0.540182\pi$
$44$		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
$49$	341.500	−	32.0429i	0.995627	−	0.0934196i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		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
$61$	268.500	−	155.019i	0.563572	−	0.325379i	−0.191006	−	0.981589i	$-0.561175\pi$
$61$	268.500	−	155.019i	0.563572	−	0.325379i	0.754578	+	0.656210i	$0.227842\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	440.000	−	762.102i	0.802307	−	1.38964i	−0.115787	−	0.993274i	$-0.536939\pi$
$67$	440.000	−	762.102i	0.802307	−	1.38964i	0.918094	−	0.396362i	$-0.129728\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−730.500	−	421.754i	−1.17121	−	0.676200i	−0.217248	−	0.976117i	$-0.569708\pi$
$73$	−730.500	−	421.754i	−1.17121	−	0.676200i	−0.953966	+	0.299916i	$0.903041\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−442.000	−	765.566i	−0.629480	−	1.09029i	−0.987656	−	0.156637i	$-0.949935\pi$
$79$	−442.000	−	765.566i	−0.629480	−	1.09029i	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
$91$	−54.0000	−	1153.55i	−0.0622059	−	1.32884i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	465.922i		−	0.487703i	−0.969813	−	0.243851i	$-0.921589\pi$
$97$		−	465.922i		−	0.487703i	0.969813	−	0.243851i	$-0.0784109\pi$
$98$		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
$103$	891.000	−	514.419i	0.852358	−	0.492109i	−0.00908799	−	0.999959i	$-0.502893\pi$
$103$	891.000	−	514.419i	0.852358	−	0.492109i	0.861446	+	0.507850i	$0.169560\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$	1106.50	−	1916.51i	0.972325	−	1.68412i	0.283833	−	0.958874i	$-0.408394\pi$
$109$	1106.50	−	1916.51i	0.972325	−	1.68412i	0.688493	−	0.725243i	$-0.258273\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		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
$133$	−459.000	+	294.449i	−0.299251	+	0.191969i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		−	2026.50i		−	1.23659i	−0.785948	−	0.618293i	$-0.787825\pi$
$139$		−	2026.50i		−	1.23659i	0.785948	−	0.618293i	$-0.212175\pi$
$140$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	−1854.50	+	3212.09i	−0.999451	+	1.73110i	−0.471027	+	0.882119i	$0.656117\pi$
$151$	−1854.50	+	3212.09i	−0.999451	+	1.73110i	−0.528424	+	0.848981i	$0.677217\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−702.000	−	405.300i	−0.356852	−	0.206028i	0.310847	−	0.950460i	$-0.399387\pi$
$157$	−702.000	−	405.300i	−0.356852	−	0.206028i	−0.667699	+	0.744432i	$0.732721\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	189.500	+	328.224i	0.0910600	+	0.157721i	0.907957	−	0.419062i	$-0.137641\pi$
$163$	189.500	+	328.224i	0.0910600	+	0.157721i	−0.816897	+	0.576783i	$0.804308\pi$
$164$		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
$169$	−1691.00			−0.769686
$170$		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
$175$	1062.50	−	2056.81i	0.458957	−	0.888459i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$		−	4709.45i		−	1.93398i	−0.254814	−	0.966990i	$-0.582014\pi$
$181$		−	4709.45i		−	1.93398i	0.254814	−	0.966990i	$-0.417986\pi$
$182$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−575.000	+	995.929i	−0.214453	+	0.371443i	−0.953103	−	0.302646i	$-0.902130\pi$
$193$	−575.000	+	995.929i	−0.214453	+	0.371443i	0.738650	+	0.674089i	$0.235464\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	3049.50	+	1760.63i	1.08630	+	0.627175i	0.932588	−	0.360942i	$-0.117545\pi$
$199$	3049.50	+	1760.63i	1.08630	+	0.627175i	0.153710	+	0.988116i	$0.450878\pi$
$200$		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
$211$	2071.00			0.675704			0.337852	−	0.941199i	$-0.390300\pi$
$211$	2071.00			0.675704			0.337852	+	0.941199i	$0.390300\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3106.50	−	1604.75i	−0.971811	−	0.502015i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			5830.08i			1.75072i	0.483469	+	0.875362i	$0.339377\pi$
$223$			5830.08i			1.75072i	−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
$229$	5644.50	−	3258.85i	1.62882	−	0.940398i	0.644370	−	0.764714i	$-0.277120\pi$
$229$	5644.50	−	3258.85i	1.62882	−	0.940398i	0.984447	−	0.175684i	$-0.0562138\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−4993.50	−	2883.00i	−1.33469	−	0.770582i	−0.348673	−	0.937244i	$-0.613368\pi$
$241$	−4993.50	−	2883.00i	−1.33469	−	0.770582i	−0.986014	+	0.166662i	$0.946701\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	918.000	+	1590.02i	0.236482	+	0.409598i
$248$		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
$259$	1100.00	+	1714.73i	0.263902	+	0.411383i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		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
$271$	−3238.50	+	1869.75i	−0.725922	+	0.419111i	−0.816928	−	0.576739i	$-0.804325\pi$
$271$	−3238.50	+	1869.75i	−0.725922	+	0.419111i	0.0910064	+	0.995850i	$0.470992\pi$
$272$		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		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	865.500	+	499.697i	0.181797	+	0.104961i	0.588137	−	0.808761i	$-0.299862\pi$
$283$	865.500	+	499.697i	0.181797	+	0.104961i	−0.406340	+	0.913722i	$0.633195\pi$
$284$		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
$301$	−1313.50	+	61.4878i	−0.251525	+	0.0117744i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			10009.5i			1.86083i	0.366513	+	0.930413i	$0.380552\pi$
$307$			10009.5i			1.86083i	−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
$313$	−9559.50	+	5519.18i	−1.72631	+	0.996685i	−0.822478	+	0.568797i	$0.807409\pi$
$313$	−9559.50	+	5519.18i	−1.72631	+	0.996685i	−0.903832	+	0.427888i	$0.859258\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−6750.00	−	3897.11i	−1.15207	−	0.665148i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	4949.50	+	8572.79i	0.821901	+	1.42357i	0.904265	+	0.426971i	$0.140420\pi$
$331$	4949.50	+	8572.79i	0.821901	+	1.42357i	−0.0823644	+	0.996602i	$0.526247\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	7363.00			1.19017			0.595086	−	0.803662i	$-0.297118\pi$
$337$	7363.00			1.19017			0.595086	+	0.803662i	$0.297118\pi$
$338$		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		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$			12967.9i			1.98898i	0.104834	+	0.994490i	$0.466569\pi$
$349$			12967.9i			1.98898i	−0.104834	+	0.994490i	$0.533431\pi$
$350$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−2996.00	+	5189.22i	−0.436798	+	0.756557i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−2536.50	−	1464.45i	−0.360774	−	0.208293i	0.308646	−	0.951177i	$-0.400124\pi$
$367$	−2536.50	−	1464.45i	−0.360774	−	0.208293i	−0.669420	+	0.742884i	$0.733457\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−125.500	−	217.372i	−0.0174213	−	0.0301746i	0.857183	−	0.515011i	$-0.172212\pi$
$373$	−125.500	−	217.372i	−0.0174213	−	0.0301746i	−0.874605	+	0.484837i	$0.838879\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	8584.00			1.16340			0.581702	−	0.813402i	$-0.302387\pi$
$379$	8584.00			1.16340			0.581702	+	0.813402i	$0.302387\pi$
$380$		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		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	5938.50	−	3428.59i	0.750742	−	0.433441i	−0.0752196	−	0.997167i	$-0.523966\pi$
$397$	5938.50	−	3428.59i	0.750742	−	0.433441i	0.825962	+	0.563726i	$0.190632\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	−5886.00	+	10194.9i	−0.727550	+	1.26015i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−12420.0	−	7170.69i	−1.50154	−	0.866914i	−0.999998	−	0.00177990i	$-0.999433\pi$
$409$	−12420.0	−	7170.69i	−1.50154	−	0.866914i	−0.501541	−	0.865134i	$-0.667233\pi$
$410$		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
$421$	−10459.0			−1.21078			−0.605392	−	0.795927i	$-0.706984\pi$
$421$	−10459.0			−1.21078			−0.605392	+	0.795927i	$0.706984\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4833.00	−	3100.37i	0.547741	−	0.351376i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$		−	6673.59i		−	0.740675i	−0.928897	−	0.370338i	$-0.879242\pi$
$433$		−	6673.59i		−	0.740675i	0.928897	−	0.370338i	$-0.120758\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−9315.00	+	5378.02i	−1.01271	+	0.584690i	−0.911985	−	0.410224i	$-0.865450\pi$
$439$	−9315.00	+	5378.02i	−1.01271	+	0.584690i	−0.100728	+	0.994914i	$0.532117\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		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	0.983004	−	0.183585i	$-0.0587702\pi$
$457$	3248.50	+	5626.57i	0.332513	+	0.575929i	−0.650491	+	0.759514i	$0.725437\pi$
$458$		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
$463$	7811.00			0.784034			0.392017	−	0.919958i	$-0.371777\pi$
$463$	7811.00			0.784034			0.392017	+	0.919958i	$0.371777\pi$
$464$		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
$469$	7480.00	−	14479.9i	0.736448	−	1.42563i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$			3680.61i			0.355532i
$476$		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
$481$	5940.00	−	3429.46i	0.563078	−	0.325093i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−7398.50	+	12814.6i	−0.688415	+	1.19237i	0.283936	+	0.958843i	$0.408360\pi$
$487$	−7398.50	+	12814.6i	−0.688415	+	1.19237i	−0.972351	+	0.233526i	$0.924974\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		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	$0.262440\pi$
$499$	10871.5	+	18830.0i	0.975301	+	1.68927i	0.296363	+	0.955075i	$0.404226\pi$
$500$		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
$511$	−13879.5	−	7169.82i	−1.20155	−	0.620693i
$512$		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
$523$	17901.0	−	10335.1i	1.49667	−	0.864100i	0.496673	−	0.867938i	$-0.334555\pi$
$523$	17901.0	−	10335.1i	1.49667	−	0.864100i	0.999993	+	0.00383755i	$0.00122153\pi$
$524$		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
$541$	−10394.5	−	18003.8i	−0.826053	−	1.43077i	−0.901112	−	0.433586i	$-0.857248\pi$
$541$	−10394.5	−	18003.8i	−0.826053	−	1.43077i	0.0750596	−	0.997179i	$-0.476085\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	21293.0			1.66439			0.832196	−	0.554481i	$-0.187083\pi$
$547$	21293.0			1.66439			0.832196	+	0.554481i	$0.187083\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−8840.00	−	13780.2i	−0.679774	−	1.05966i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$			4427.12i			0.334968i
$560$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	−314.500	+	544.730i	−0.0230498	+	0.0399234i	−0.877320	−	0.479905i	$-0.840671\pi$
$571$	−314.500	+	544.730i	−0.0230498	+	0.0399234i	0.854270	+	0.519829i	$0.174004\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−18468.0	−	10662.5i	−1.33247	−	0.769300i	−0.346789	−	0.937943i	$-0.612728\pi$
$577$	−18468.0	−	10662.5i	−1.33247	−	0.769300i	−0.985677	+	0.168644i	$0.946061\pi$
$578$		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
$589$	5559.00			0.388887
$590$		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		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$		−	23817.4i		−	1.61653i	−0.588820	−	0.808264i	$-0.700407\pi$
$601$		−	23817.4i		−	1.61653i	0.588820	−	0.808264i	$-0.299593\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	17278.5	−	9975.75i	1.15538	−	0.667056i	0.205184	−	0.978723i	$-0.434221\pi$
$607$	17278.5	−	9975.75i	1.15538	−	0.667056i	0.950191	+	0.311667i	$0.100887\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−6425.50	+	11129.3i	−0.423366	+	0.733292i	−0.996266	−	0.0863334i	$-0.972485\pi$
$613$	−6425.50	+	11129.3i	−0.423366	+	0.733292i	0.572900	+	0.819625i	$0.305818\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−13365.0	−	7716.29i	−0.867827	−	0.501040i	−0.00120126	−	0.999999i	$-0.500382\pi$
$619$	−13365.0	−	7716.29i	−0.867827	−	0.501040i	−0.866625	+	0.498959i	$0.833716\pi$
$620$		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
$631$	26459.0			1.66928			0.834640	−	0.550795i	$-0.185675\pi$
$631$	26459.0			1.66928			0.834640	+	0.550795i	$0.185675\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1998.00	−	21293.8i	−0.124276	−	1.32448i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$		−	26315.0i		−	1.61394i	−0.590592	−	0.806971i	$-0.701106\pi$
$643$		−	26315.0i		−	1.61394i	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		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−27106.5	−	15649.9i	−1.59504	−	0.920896i	−0.992423	−	0.122864i	$-0.960792\pi$
$661$	−27106.5	−	15649.9i	−1.59504	−	0.920896i	−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
$673$	33949.0			1.94448			0.972242	−	0.233977i	$-0.0751742\pi$
$673$	33949.0			1.94448			0.972242	+	0.233977i	$0.0751742\pi$
$674$		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
$679$	−403.500	−	8619.55i	−0.0228055	−	0.487169i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	26161.5	−	15104.3i	1.44028	−	0.831543i	0.442408	−	0.896814i	$-0.354124\pi$
$691$	26161.5	−	15104.3i	1.44028	−	0.831543i	0.997868	+	0.0652705i	$0.0207910\pi$
$692$		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		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−2805.00	−	1619.47i	−0.150487	−	0.0868839i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−4311.50	−	7467.74i	−0.228381	−	0.395567i	0.728948	−	0.684569i	$-0.240010\pi$
$709$	−4311.50	−	7467.74i	−0.228381	−	0.395567i	−0.957328	+	0.289003i	$0.906676\pi$
$710$		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
$721$	16038.0	−	10288.4i	0.828414	−	0.531428i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	9510.69i		−	0.485188i	−0.970128	−	0.242594i	$-0.922002\pi$
$727$		−	9510.69i		−	0.485188i	0.970128	−	0.242594i	$-0.0779984\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	4615.50	−	2664.76i	0.232575	−	0.134277i	−0.379184	−	0.925321i	$-0.623795\pi$
$733$	4615.50	−	2664.76i	0.232575	−	0.134277i	0.611759	+	0.791044i	$0.290462\pi$
$734$		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		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	20510.5	+	35525.2i	0.996590	+	1.72614i	0.569757	+	0.821813i	$0.307037\pi$
$751$	20510.5	+	35525.2i	0.996590	+	1.72614i	0.426832	+	0.904331i	$0.359630\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−24137.0			−1.15888			−0.579441	−	0.815014i	$-0.696729\pi$
$757$	−24137.0			−1.15888			−0.579441	+	0.815014i	$0.696729\pi$
$758$		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
$763$	18810.5	−	36413.8i	0.892511	−	1.72774i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			17211.4i			0.807098i	0.914958	+	0.403549i	$0.132224\pi$
$769$			17211.4i			0.807098i	−0.914958	+	0.403549i	$0.867776\pi$
$770$		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
$775$	−20437.5	+	11799.6i	−0.947273	+	0.546908i
$776$		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
$787$	−36073.5	−	20827.0i	−1.63390	−	0.943334i	−0.982874	−	0.184281i	$-0.941004\pi$
$787$	−36073.5	−	20827.0i	−1.63390	−	0.943334i	−0.651029	−	0.759053i	$-0.725662\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−9666.00	−	16742.0i	−0.432850	−	0.749717i
$794$		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		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$			24162.1i			1.04617i	0.852280	+	0.523087i	$0.175220\pi$
$811$			24162.1i			1.04617i	−0.852280	+	0.523087i	$0.824780\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1810.50	−	1045.29i	0.0775292	−	0.0447615i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	−22805.5	+	39500.3i	−0.965917	+	1.67302i	−0.258786	+	0.965935i	$0.583323\pi$
$823$	−22805.5	+	39500.3i	−0.965917	+	1.67302i	−0.707131	+	0.707083i	$0.750011\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−6505.50	−	3755.95i	−0.272552	−	0.157358i	0.357495	−	0.933915i	$-0.383631\pi$
$829$	−6505.50	−	3755.95i	−0.272552	−	0.157358i	−0.630047	+	0.776557i	$0.716964\pi$
$830$		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
$847$	−13310.0	−	20748.2i	−0.539949	−	0.841698i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			31736.4i			1.27390i	0.770907	+	0.636948i	$0.219803\pi$
$853$			31736.4i			1.27390i	−0.770907	+	0.636948i	$0.780197\pi$
$854$		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
$859$	6400.50	−	3695.33i	0.254228	−	0.146779i	−0.367470	−	0.930035i	$-0.619776\pi$
$859$	6400.50	−	3695.33i	0.254228	−	0.146779i	0.621699	+	0.783256i	$0.286443\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−47520.0	−	27435.7i	−1.84863	−	1.06731i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	18396.5	+	31863.7i	0.708330	+	1.22686i	0.965476	+	0.260491i	$0.0838846\pi$
$877$	18396.5	+	31863.7i	0.708330	+	1.22686i	−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
$883$	31429.0			1.19781			0.598907	−	0.800818i	$-0.295602\pi$
$883$	31429.0			1.19781			0.598907	+	0.800818i	$0.295602\pi$
$884$		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
$889$	41939.5	−	1963.28i	1.58223	−	0.0740678i
$890$		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
$907$	22420.0	−	38832.6i	0.820776	−	1.42163i	−0.0843291	−	0.996438i	$-0.526875\pi$
$907$	22420.0	−	38832.6i	0.820776	−	1.42163i	0.905105	−	0.425188i	$-0.139792\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		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	$0.150916\pi$
$919$	23408.5	+	40544.7i	0.840234	+	1.45533i	−0.0494625	+	0.998776i	$0.515751\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	13750.0			0.488754
$926$		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
$931$	−8236.50	+	5844.81i	−0.289947	+	0.205753i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	14466.1i		−	0.504361i	−0.967680	−	0.252181i	$-0.918852\pi$
$937$		−	14466.1i		−	0.504361i	0.967680	−	0.252181i	$-0.0811477\pi$
$938$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	−26298.0	+	45549.5i	−0.899546	+	1.55806i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2926.00	+	5067.98i	0.0982176	+	0.170118i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−50020.0			−1.66343			−0.831714	−	0.555204i	$-0.812640\pi$
$967$	−50020.0			−1.66343			−0.831714	+	0.555204i	$0.812640\pi$
$968$		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
$973$	−1755.00	−	37490.2i	−0.0578240	−	1.23523i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		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
$991$	−22814.0	+	39515.0i	−0.731292	+	1.26664i	0.225039	+	0.974350i	$0.427749\pi$
$991$	−22814.0	+	39515.0i	−0.731292	+	1.26664i	−0.956331	+	0.292286i	$0.905584\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2536.50	−	1464.45i	−0.0805735	−	0.0465191i	0.459172	−	0.888347i	$-0.348146\pi$
$997$	−2536.50	−	1464.45i	−0.0805735	−	0.0465191i	−0.539745	+	0.841828i	$0.681480\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.t.b.269.1	✓	2
3.2	odd	2	CM	756.4.t.b.269.1	✓	2
7.5	odd	6	inner	756.4.t.b.593.1	yes	2
21.5	even	6	inner	756.4.t.b.593.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.4.t.b.269.1	✓	2	1.1	even	1	trivial
756.4.t.b.269.1	✓	2	3.2	odd	2	CM
756.4.t.b.593.1	yes	2	7.5	odd	6	inner
756.4.t.b.593.1	yes	2	21.5	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(18.5000 - 0.866025i) q^{7} -62.3538i q^{13} +(-25.5000 + 14.7224i) q^{19} +(62.5000 - 108.253i) q^{25} +(-163.500 - 94.3968i) q^{31} +(55.0000 + 95.2628i) q^{37} -71.0000 q^{43} +(341.500 - 32.0429i) q^{49} +(268.500 - 155.019i) q^{61} +(440.000 - 762.102i) q^{67} +(-730.500 - 421.754i) q^{73} +(-442.000 - 765.566i) q^{79} +(-54.0000 - 1153.55i) q^{91} -465.922i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 37 q^{7} - 51 q^{19} + 125 q^{25} - 327 q^{31} + 110 q^{37} - 142 q^{43} + 683 q^{49} + 537 q^{61} + 880 q^{67} - 1461 q^{73} - 884 q^{79} - 108 q^{91}+O(q^{100}) \) 2 * q + 37 * q^7 - 51 * q^19 + 125 * q^25 - 327 * q^31 + 110 * q^37 - 142 * q^43 + 683 * q^49 + 537 * q^61 + 880 * q^67 - 1461 * q^73 - 884 * q^79 - 108 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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