LMFDB - Embedded newform 784.3.r.e.79.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,3,Mod(79,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.79"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	784.r (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,6,0,0,0,-9,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$21.3624527258$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			79.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	784.79
Dual form			784.3.r.e.655.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.00000 - 5.19615i) q^{5} +(-4.50000 + 7.79423i) q^{9} +10.0000 q^{13} +(15.0000 + 25.9808i) q^{17} +(-5.50000 - 9.52628i) q^{25} +42.0000 q^{29} +(35.0000 - 60.6218i) q^{37} +18.0000 q^{41} +(27.0000 + 46.7654i) q^{45} +(-45.0000 - 77.9423i) q^{53} +(11.0000 - 19.0526i) q^{61} +(30.0000 - 51.9615i) q^{65} +(55.0000 + 95.2628i) q^{73} +(-40.5000 - 70.1481i) q^{81} +180.000 q^{85} +(39.0000 - 67.5500i) q^{89} +130.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} - 9 q^{9} + 20 q^{13} + 30 q^{17} - 11 q^{25} + 84 q^{29} + 70 q^{37} + 36 q^{41} + 54 q^{45} - 90 q^{53} + 22 q^{61} + 60 q^{65} + 110 q^{73} - 81 q^{81} + 360 q^{85} + 78 q^{89} + 260 q^{97}+O(q^{100}) $ 2 * q + 6 * q^5 - 9 * q^9 + 20 * q^13 + 30 * q^17 - 11 * q^25 + 84 * q^29 + 70 * q^37 + 36 * q^41 + 54 * q^45 - 90 * q^53 + 22 * q^61 + 60 * q^65 + 110 * q^73 - 81 * q^81 + 360 * q^85 + 78 * q^89 + 260 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$3$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$4$		0			0
$5$	3.00000	−	5.19615i	0.600000	−	1.03923i	−0.392820	−	0.919615i	$-0.628501\pi$
$5$	3.00000	−	5.19615i	0.600000	−	1.03923i	0.992820	−	0.119615i	$-0.0381661\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	10.0000			0.769231			0.384615	−	0.923077i	$-0.374334\pi$
$13$	10.0000			0.769231			0.384615	+	0.923077i	$0.374334\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	15.0000	+	25.9808i	0.882353	+	1.52828i	0.848718	+	0.528846i	$0.177375\pi$
$17$	15.0000	+	25.9808i	0.882353	+	1.52828i	0.0336351	+	0.999434i	$0.489292\pi$
$18$		0			0
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	42.0000			1.44828			0.724138	−	0.689655i	$-0.242238\pi$
$29$	42.0000			1.44828			0.724138	+	0.689655i	$0.242238\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	35.0000	−	60.6218i	0.945946	−	1.63843i	0.192100	−	0.981375i	$-0.438470\pi$
$37$	35.0000	−	60.6218i	0.945946	−	1.63843i	0.753846	−	0.657051i	$-0.228196\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	18.0000			0.439024			0.219512	−	0.975610i	$-0.429553\pi$
$41$	18.0000			0.439024			0.219512	+	0.975610i	$0.429553\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	27.0000	+	46.7654i	0.600000	+	1.03923i
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−45.0000	−	77.9423i	−0.849057	−	1.47061i	−0.882051	−	0.471154i	$-0.843838\pi$
$53$	−45.0000	−	77.9423i	−0.849057	−	1.47061i	0.0329946	−	0.999456i	$-0.489496\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	11.0000	−	19.0526i	0.180328	−	0.312337i	−0.761664	−	0.647972i	$-0.775617\pi$
$61$	11.0000	−	19.0526i	0.180328	−	0.312337i	0.941992	+	0.335635i	$0.108951\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	30.0000	−	51.9615i	0.461538	−	0.799408i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	55.0000	+	95.2628i	0.753425	+	1.30497i	0.946154	+	0.323718i	$0.104933\pi$
$73$	55.0000	+	95.2628i	0.753425	+	1.30497i	−0.192729	+	0.981252i	$0.561734\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−40.5000	−	70.1481i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	180.000			2.11765
$86$		0			0
$87$		0			0
$88$		0			0
$89$	39.0000	−	67.5500i	0.438202	−	0.758989i	−0.559349	−	0.828932i	$-0.688949\pi$
$89$	39.0000	−	67.5500i	0.438202	−	0.758989i	0.997551	+	0.0699439i	$0.0222820\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	130.000			1.34021			0.670103	−	0.742268i	$-0.266250\pi$
$97$	130.000			1.34021			0.670103	+	0.742268i	$0.266250\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	99.0000	+	171.473i	0.980198	+	1.69775i	0.661589	+	0.749866i	$0.269882\pi$
$101$	99.0000	+	171.473i	0.980198	+	1.69775i	0.318609	+	0.947886i	$0.396784\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	91.0000	+	157.617i	0.834862	+	1.44602i	0.894142	+	0.447783i	$0.147786\pi$
$109$	91.0000	+	157.617i	0.834862	+	1.44602i	−0.0592800	+	0.998241i	$0.518880\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−30.0000			−0.265487			−0.132743	−	0.991150i	$-0.542379\pi$
$113$	−30.0000			−0.265487			−0.132743	+	0.991150i	$0.542379\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−45.0000	+	77.9423i	−0.384615	+	0.666173i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−60.5000	+	104.789i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	84.0000			0.672000
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−105.000	−	181.865i	−0.766423	−	1.32748i	−0.939491	−	0.342574i	$-0.888701\pi$
$137$	−105.000	−	181.865i	−0.766423	−	1.32748i	0.173067	−	0.984910i	$-0.444632\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	126.000	−	218.238i	0.868966	−	1.50509i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	51.0000	−	88.3346i	0.342282	−	0.592850i	−0.642574	−	0.766223i	$-0.722134\pi$
$149$	51.0000	−	88.3346i	0.342282	−	0.592850i	0.984856	+	0.173374i	$0.0554669\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$	−270.000			−1.76471
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−85.0000	−	147.224i	−0.541401	−	0.937735i	−0.998824	−	0.0484851i	$-0.984561\pi$
$157$	−85.0000	−	147.224i	−0.541401	−	0.937735i	0.457423	−	0.889249i	$-0.348773\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−69.0000			−0.408284
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−165.000	+	285.788i	−0.953757	+	1.65196i	−0.216570	+	0.976267i	$0.569487\pi$
$173$	−165.000	+	285.788i	−0.953757	+	1.65196i	−0.737187	+	0.675689i	$0.763846\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	−38.0000			−0.209945			−0.104972	−	0.994475i	$-0.533475\pi$
$181$	−38.0000			−0.209945			−0.104972	+	0.994475i	$0.533475\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−210.000	−	363.731i	−1.13514	−	1.96611i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	95.0000	+	164.545i	0.492228	+	0.852564i	0.999960	−	0.00895123i	$-0.00284930\pi$
$193$	95.0000	+	164.545i	0.492228	+	0.852564i	−0.507732	+	0.861515i	$0.669516\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−390.000			−1.97970			−0.989848	−	0.142132i	$-0.954604\pi$
$197$	−390.000			−1.97970			−0.989848	+	0.142132i	$0.954604\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	54.0000	−	93.5307i	0.263415	−	0.456248i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	150.000	+	259.808i	0.678733	+	1.17560i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	99.0000			0.440000
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	−221.000	+	382.783i	−0.965066	+	1.67154i	−0.255627	+	0.966776i	$0.582282\pi$
$229$	−221.000	+	382.783i	−0.965066	+	1.67154i	−0.709439	+	0.704767i	$0.751052\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−105.000	+	181.865i	−0.450644	+	0.780538i	−0.998426	−	0.0560830i	$-0.982139\pi$
$233$	−105.000	+	181.865i	−0.450644	+	0.780538i	0.547782	+	0.836621i	$0.315472\pi$
$234$		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
$241$	−209.000	−	361.999i	−0.867220	−	1.50207i	−0.864826	−	0.502072i	$-0.832571\pi$
$241$	−209.000	−	361.999i	−0.867220	−	1.50207i	−0.00239399	−	0.999997i	$-0.500762\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	255.000	−	441.673i	0.992218	−	1.71857i	0.388277	−	0.921543i	$-0.373070\pi$
$257$	255.000	−	441.673i	0.992218	−	1.71857i	0.603941	−	0.797029i	$-0.293596\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−189.000	+	327.358i	−0.724138	+	1.25424i
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$	−540.000			−2.03774
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−69.0000	−	119.512i	−0.256506	−	0.444281i	0.708798	−	0.705412i	$-0.249238\pi$
$269$	−69.0000	−	119.512i	−0.256506	−	0.444281i	−0.965303	+	0.261131i	$0.915905\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	115.000	+	199.186i	0.415162	+	0.719082i	0.995445	−	0.0953324i	$-0.0303914\pi$
$277$	115.000	+	199.186i	0.415162	+	0.719082i	−0.580283	+	0.814415i	$0.697058\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−462.000			−1.64413			−0.822064	−	0.569395i	$-0.807178\pi$
$281$	−462.000			−1.64413			−0.822064	+	0.569395i	$0.807178\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−305.500	+	529.142i	−1.05709	+	1.83094i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	570.000			1.94539			0.972696	−	0.232082i	$-0.0745537\pi$
$293$	570.000			1.94539			0.972696	+	0.232082i	$0.0745537\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−66.0000	−	114.315i	−0.216393	−	0.374804i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−25.0000	+	43.3013i	−0.0798722	+	0.138343i	−0.903195	−	0.429231i	$-0.858785\pi$
$313$	−25.0000	+	43.3013i	−0.0798722	+	0.138343i	0.823322	+	0.567574i	$0.192118\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	75.0000	−	129.904i	0.236593	−	0.409791i	−0.723141	−	0.690700i	$-0.757303\pi$
$317$	75.0000	−	129.904i	0.236593	−	0.409791i	0.959734	+	0.280909i	$0.0906359\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−55.0000	−	95.2628i	−0.169231	−	0.293116i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	315.000	+	545.596i	0.945946	+	1.63843i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−350.000			−1.03858			−0.519288	−	0.854599i	$-0.673803\pi$
$337$	−350.000			−1.03858			−0.519288	+	0.854599i	$0.673803\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	−598.000			−1.71347			−0.856734	−	0.515759i	$-0.827510\pi$
$349$	−598.000			−1.71347			−0.856734	+	0.515759i	$0.827510\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−225.000	−	389.711i	−0.637394	−	1.10400i	−0.986003	−	0.166730i	$-0.946679\pi$
$353$	−225.000	−	389.711i	−0.637394	−	1.10400i	0.348609	−	0.937268i	$-0.386654\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−180.500	−	312.635i	−0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	660.000			1.80822
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$	−81.0000	+	140.296i	−0.219512	+	0.380206i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	275.000	−	476.314i	0.737265	−	1.27698i	−0.216457	−	0.976292i	$-0.569450\pi$
$373$	275.000	−	476.314i	0.737265	−	1.27698i	0.953722	−	0.300689i	$-0.0972166\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	420.000			1.11406
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−189.000	−	327.358i	−0.485861	−	0.841536i	0.514007	−	0.857786i	$-0.328161\pi$
$389$	−189.000	−	327.358i	−0.485861	−	0.841536i	−0.999868	+	0.0162499i	$0.994827\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−325.000	+	562.917i	−0.818640	+	1.41793i	0.0880448	+	0.996117i	$0.471938\pi$
$397$	−325.000	+	562.917i	−0.818640	+	1.41793i	−0.906685	+	0.421809i	$0.861395\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	399.000	−	691.088i	0.995012	−	1.72341i	0.411120	−	0.911581i	$-0.365138\pi$
$401$	399.000	−	691.088i	0.995012	−	1.72341i	0.583893	−	0.811831i	$-0.301529\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−486.000			−1.20000
$406$		0			0
$407$		0			0
$408$		0			0
$409$	391.000	+	677.232i	0.955990	+	1.65582i	0.732086	+	0.681213i	$0.238547\pi$
$409$	391.000	+	677.232i	0.955990	+	1.65582i	0.223905	+	0.974611i	$0.428120\pi$
$410$		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		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	58.0000			0.137767			0.0688836	−	0.997625i	$-0.478056\pi$
$421$	58.0000			0.137767			0.0688836	+	0.997625i	$0.478056\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	165.000	−	285.788i	0.388235	−	0.672443i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$	290.000			0.669746			0.334873	−	0.942263i	$-0.391307\pi$
$433$	290.000			0.669746			0.334873	+	0.942263i	$0.391307\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$	−234.000	−	405.300i	−0.525843	−	0.910786i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−702.000			−1.56347			−0.781737	−	0.623608i	$-0.785666\pi$
$449$	−702.000			−1.56347			−0.781737	+	0.623608i	$0.785666\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−425.000	+	736.122i	−0.929978	+	1.61077i	−0.146625	+	0.989192i	$0.546841\pi$
$457$	−425.000	+	736.122i	−0.929978	+	1.61077i	−0.783353	+	0.621577i	$0.786492\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	522.000			1.13232			0.566161	−	0.824295i	$-0.308428\pi$
$461$	522.000			1.13232			0.566161	+	0.824295i	$0.308428\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	810.000			1.69811
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	350.000	−	606.218i	0.727651	−	1.26033i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	390.000	−	675.500i	0.804124	−	1.39278i
$486$		0			0
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	630.000	+	1091.19i	1.27789	+	2.21337i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	1188.00			2.35248
$506$		0			0
$507$		0			0
$508$		0			0
$509$	459.000	−	795.011i	0.901768	−	1.56191i	0.0765706	−	0.997064i	$-0.475603\pi$
$509$	459.000	−	795.011i	0.901768	−	1.56191i	0.825198	−	0.564844i	$-0.191064\pi$
$510$		0			0
$511$		0			0
$512$		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
$521$	279.000	+	483.242i	0.535509	+	0.927528i	0.999139	+	0.0414992i	$0.0132134\pi$
$521$	279.000	+	483.242i	0.535509	+	0.927528i	−0.463630	+	0.886029i	$0.653453\pi$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	−	458.127i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	180.000			0.337711
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−341.000	+	590.629i	−0.630314	+	1.09174i	0.357173	+	0.934038i	$0.383741\pi$
$541$	−341.000	+	590.629i	−0.630314	+	1.09174i	−0.987487	+	0.157698i	$0.949593\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1092.00			2.00367
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	99.0000	+	171.473i	0.180328	+	0.312337i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−165.000	−	285.788i	−0.296230	−	0.513085i	0.679040	−	0.734101i	$-0.262396\pi$
$557$	−165.000	−	285.788i	−0.296230	−	0.513085i	−0.975270	+	0.221016i	$0.929063\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$	−90.0000	+	155.885i	−0.159292	+	0.275902i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	231.000	−	400.104i	0.405975	−	0.703170i	−0.588459	−	0.808527i	$-0.700265\pi$
$569$	231.000	−	400.104i	0.405975	−	0.703170i	0.994434	+	0.105357i	$0.0335985\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	575.000	+	995.929i	0.996534	+	1.72605i	0.570311	+	0.821429i	$0.306823\pi$
$577$	575.000	+	995.929i	0.996534	+	1.72605i	0.426223	+	0.904618i	$0.359844\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	270.000	+	467.654i	0.461538	+	0.799408i
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−465.000	+	805.404i	−0.784148	+	1.35818i	0.145358	+	0.989379i	$0.453567\pi$
$593$	−465.000	+	805.404i	−0.784148	+	1.35818i	−0.929506	+	0.368806i	$0.879767\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	−1102.00			−1.83361			−0.916805	−	0.399334i	$-0.869241\pi$
$601$	−1102.00			−1.83361			−0.916805	+	0.399334i	$0.869241\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	363.000	+	628.734i	0.600000	+	1.03923i
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	35.0000	+	60.6218i	0.0570962	+	0.0988936i	0.893161	−	0.449738i	$-0.148482\pi$
$613$	35.0000	+	60.6218i	0.0570962	+	0.0988936i	−0.836065	+	0.548631i	$0.815149\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	210.000			0.340357			0.170178	−	0.985413i	$-0.445566\pi$
$617$	210.000			0.340357			0.170178	+	0.985413i	$0.445566\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	389.500	−	674.634i	0.623200	−	1.07941i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2100.00			3.33863
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−609.000	−	1054.82i	−0.950078	−	1.64558i	−0.745250	−	0.666785i	$-0.767670\pi$
$641$	−609.000	−	1054.82i	−0.950078	−	1.64558i	−0.204828	−	0.978798i	$-0.565664\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	315.000	−	545.596i	0.482389	−	0.835522i	−0.517407	−	0.855740i	$-0.673103\pi$
$653$	315.000	−	545.596i	0.482389	−	0.835522i	0.999796	+	0.0202175i	$0.00643585\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−990.000			−1.50685
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−589.000	−	1020.18i	−0.891074	−	1.54339i	−0.838589	−	0.544764i	$-0.816619\pi$
$661$	−589.000	−	1020.18i	−0.891074	−	1.54339i	−0.0524847	−	0.998622i	$-0.516714\pi$
$662$		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
$673$	770.000			1.14413			0.572065	−	0.820208i	$-0.306142\pi$
$673$	770.000			1.14413			0.572065	+	0.820208i	$0.306142\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	675.000	−	1169.13i	0.997046	−	1.72693i	0.432004	−	0.901872i	$-0.357807\pi$
$677$	675.000	−	1169.13i	0.997046	−	1.72693i	0.565042	−	0.825062i	$-0.308860\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$		0			0
$685$	−1260.00			−1.83942
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−450.000	−	779.423i	−0.653120	−	1.13124i
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	270.000	+	467.654i	0.387374	+	0.670952i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1302.00			−1.85735			−0.928673	−	0.370899i	$-0.879050\pi$
$701$	−1302.00			−1.85735			−0.928673	+	0.370899i	$0.879050\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	259.000	−	448.601i	0.365303	−	0.632724i	−0.623522	−	0.781806i	$-0.714299\pi$
$709$	259.000	−	448.601i	0.365303	−	0.632724i	0.988825	+	0.149082i	$0.0476320\pi$
$710$		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		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−231.000	−	400.104i	−0.318621	−	0.551867i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	729.000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−725.000	+	1255.74i	−0.989086	+	1.71315i	−0.366943	+	0.930243i	$0.619596\pi$
$733$	−725.000	+	1255.74i	−0.989086	+	1.71315i	−0.622143	+	0.782904i	$0.713738\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−306.000	−	530.008i	−0.410738	−	0.711420i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1190.00			−1.57199			−0.785997	−	0.618230i	$-0.787850\pi$
$757$	−1190.00			−1.57199			−0.785997	+	0.618230i	$0.787850\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	39.0000	−	67.5500i	0.0512484	−	0.0887648i	−0.839263	−	0.543725i	$-0.817013\pi$
$761$	39.0000	−	67.5500i	0.0512484	−	0.0887648i	0.890512	+	0.454961i	$0.150347\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−810.000	+	1402.96i	−1.05882	+	1.83394i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	962.000			1.25098			0.625488	−	0.780234i	$-0.284900\pi$
$769$	962.000			1.25098			0.625488	+	0.780234i	$0.284900\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	195.000	+	337.750i	0.252264	+	0.436934i	0.964149	−	0.265362i	$-0.0854916\pi$
$773$	195.000	+	337.750i	0.252264	+	0.436934i	−0.711885	+	0.702296i	$0.752158\pi$
$774$		0			0
$775$		0			0
$776$		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
$785$	−1020.00			−1.29936
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	110.000	−	190.526i	0.138714	−	0.240259i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1110.00			−1.39272			−0.696361	−	0.717691i	$-0.745199\pi$
$797$	−1110.00			−1.39272			−0.696361	+	0.717691i	$0.745199\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	351.000	+	607.950i	0.438202	+	0.758989i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	759.000	+	1314.63i	0.938195	+	1.62500i	0.768835	+	0.639448i	$0.220837\pi$
$809$	759.000	+	1314.63i	0.938195	+	1.62500i	0.169361	+	0.985554i	$0.445830\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−429.000	+	743.050i	−0.522533	+	0.905055i	0.477123	+	0.878837i	$0.341680\pi$
$821$	−429.000	+	743.050i	−0.522533	+	0.905055i	−0.999656	+	0.0262179i	$0.991654\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−629.000	−	1089.46i	−0.758745	−	1.31419i	−0.943491	−	0.331399i	$-0.892479\pi$
$829$	−629.000	−	1089.46i	−0.758745	−	1.31419i	0.184745	−	0.982786i	$-0.440854\pi$
$830$		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		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	923.000			1.09750
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−207.000	+	358.535i	−0.244970	+	0.424301i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	410.000			0.480657			0.240328	−	0.970692i	$-0.422745\pi$
$853$	410.000			0.480657			0.240328	+	0.970692i	$0.422745\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−825.000	−	1428.94i	−0.962660	−	1.66738i	−0.715774	−	0.698333i	$-0.753926\pi$
$857$	−825.000	−	1428.94i	−0.962660	−	1.66738i	−0.246887	−	0.969044i	$-0.579408\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$	990.000	+	1714.73i	1.14451	+	1.98235i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−585.000	+	1013.25i	−0.670103	+	1.16065i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−805.000	+	1394.30i	−0.917902	+	1.58985i	−0.115306	+	0.993330i	$0.536785\pi$
$877$	−805.000	+	1394.30i	−0.917902	+	1.58985i	−0.802596	+	0.596523i	$0.796549\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	738.000			0.837684			0.418842	−	0.908059i	$-0.362436\pi$
$881$	738.000			0.837684			0.418842	+	0.908059i	$0.362436\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$		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
$901$	1350.00	−	2338.27i	1.49834	−	2.59519i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−114.000	+	197.454i	−0.125967	+	0.218181i
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		0			0
$909$	−1782.00			−1.96040
$910$		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
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−770.000			−0.832432
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−129.000	+	223.435i	−0.138859	+	0.240511i	−0.927065	−	0.374901i	$-0.877677\pi$
$929$	−129.000	+	223.435i	−0.138859	+	0.240511i	0.788206	+	0.615411i	$0.211010\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−430.000			−0.458911			−0.229456	−	0.973319i	$-0.573695\pi$
$937$	−430.000			−0.458911			−0.229456	+	0.973319i	$0.573695\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−741.000	−	1283.45i	−0.787460	−	1.36392i	−0.927518	−	0.373778i	$-0.878062\pi$
$941$	−741.000	−	1283.45i	−0.787460	−	1.36392i	0.140058	−	0.990143i	$-0.455271\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$	550.000	+	952.628i	0.579557	+	1.00382i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1230.00			−1.29066			−0.645331	−	0.763903i	$-0.723280\pi$
$953$	−1230.00			−1.29066			−0.645331	+	0.763903i	$0.723280\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−480.500	+	832.250i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1140.00			1.18135
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−945.000	−	1636.79i	−0.967247	−	1.67532i	−0.703454	−	0.710741i	$-0.748360\pi$
$977$	−945.000	−	1636.79i	−0.967247	−	1.67532i	−0.263793	−	0.964579i	$-0.584974\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1638.00			−1.66972
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$	−1170.00	+	2026.50i	−1.18782	+	2.05736i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−925.000	−	1602.15i	−0.927783	−	1.60697i	−0.787023	−	0.616924i	$-0.788378\pi$
$997$	−925.000	−	1602.15i	−0.927783	−	1.60697i	−0.140761	−	0.990044i	$-0.544955\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.3.r.e.79.1		2
4.3	odd	2	CM	784.3.r.e.79.1		2
7.2	even	3		16.3.c.a.15.1	✓	1
7.3	odd	6		784.3.r.d.655.1		2
7.4	even	3	inner	784.3.r.e.655.1		2
7.5	odd	6		784.3.d.b.687.1		1
7.6	odd	2		784.3.r.d.79.1		2
21.2	odd	6		144.3.g.a.127.1		1
28.3	even	6		784.3.r.d.655.1		2
28.11	odd	6	inner	784.3.r.e.655.1		2
28.19	even	6		784.3.d.b.687.1		1
28.23	odd	6		16.3.c.a.15.1	✓	1
28.27	even	2		784.3.r.d.79.1		2
35.2	odd	12		400.3.h.a.399.1		2
35.9	even	6		400.3.b.a.351.1		1
35.23	odd	12		400.3.h.a.399.2		2
56.37	even	6		64.3.c.a.63.1		1
56.51	odd	6		64.3.c.a.63.1		1
63.2	odd	6		1296.3.o.b.271.1		2
63.16	even	3		1296.3.o.o.271.1		2
63.23	odd	6		1296.3.o.b.703.1		2
63.58	even	3		1296.3.o.o.703.1		2
84.23	even	6		144.3.g.a.127.1		1
105.2	even	12		3600.3.j.a.1999.1		2
105.23	even	12		3600.3.j.a.1999.2		2
105.44	odd	6		3600.3.e.c.3151.1		1
112.37	even	12		256.3.d.b.127.1		2
112.51	odd	12		256.3.d.b.127.2		2
112.93	even	12		256.3.d.b.127.2		2
112.107	odd	12		256.3.d.b.127.1		2
140.23	even	12		400.3.h.a.399.2		2
140.79	odd	6		400.3.b.a.351.1		1
140.107	even	12		400.3.h.a.399.1		2
168.107	even	6		576.3.g.b.127.1		1
168.149	odd	6		576.3.g.b.127.1		1
252.23	even	6		1296.3.o.b.703.1		2
252.79	odd	6		1296.3.o.o.271.1		2
252.191	even	6		1296.3.o.b.271.1		2
252.247	odd	6		1296.3.o.o.703.1		2
280.37	odd	12		1600.3.h.b.1599.2		2
280.93	odd	12		1600.3.h.b.1599.1		2
280.107	even	12		1600.3.h.b.1599.2		2
280.149	even	6		1600.3.b.b.1151.1		1
280.163	even	12		1600.3.h.b.1599.1		2
280.219	odd	6		1600.3.b.b.1151.1		1
336.107	even	12		2304.3.b.f.127.2		2
336.149	odd	12		2304.3.b.f.127.2		2
336.275	even	12		2304.3.b.f.127.1		2
336.317	odd	12		2304.3.b.f.127.1		2
420.23	odd	12		3600.3.j.a.1999.2		2
420.107	odd	12		3600.3.j.a.1999.1		2
420.359	even	6		3600.3.e.c.3151.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.3.c.a.15.1	✓	1	7.2	even	3
16.3.c.a.15.1	✓	1	28.23	odd	6
64.3.c.a.63.1		1	56.37	even	6
64.3.c.a.63.1		1	56.51	odd	6
144.3.g.a.127.1		1	21.2	odd	6
144.3.g.a.127.1		1	84.23	even	6
256.3.d.b.127.1		2	112.37	even	12
256.3.d.b.127.1		2	112.107	odd	12
256.3.d.b.127.2		2	112.51	odd	12
256.3.d.b.127.2		2	112.93	even	12
400.3.b.a.351.1		1	35.9	even	6
400.3.b.a.351.1		1	140.79	odd	6
400.3.h.a.399.1		2	35.2	odd	12
400.3.h.a.399.1		2	140.107	even	12
400.3.h.a.399.2		2	35.23	odd	12
400.3.h.a.399.2		2	140.23	even	12
576.3.g.b.127.1		1	168.107	even	6
576.3.g.b.127.1		1	168.149	odd	6
784.3.d.b.687.1		1	7.5	odd	6
784.3.d.b.687.1		1	28.19	even	6
784.3.r.d.79.1		2	7.6	odd	2
784.3.r.d.79.1		2	28.27	even	2
784.3.r.d.655.1		2	7.3	odd	6
784.3.r.d.655.1		2	28.3	even	6
784.3.r.e.79.1		2	1.1	even	1	trivial
784.3.r.e.79.1		2	4.3	odd	2	CM
784.3.r.e.655.1		2	7.4	even	3	inner
784.3.r.e.655.1		2	28.11	odd	6	inner
1296.3.o.b.271.1		2	63.2	odd	6
1296.3.o.b.271.1		2	252.191	even	6
1296.3.o.b.703.1		2	63.23	odd	6
1296.3.o.b.703.1		2	252.23	even	6
1296.3.o.o.271.1		2	63.16	even	3
1296.3.o.o.271.1		2	252.79	odd	6
1296.3.o.o.703.1		2	63.58	even	3
1296.3.o.o.703.1		2	252.247	odd	6
1600.3.b.b.1151.1		1	280.149	even	6
1600.3.b.b.1151.1		1	280.219	odd	6
1600.3.h.b.1599.1		2	280.93	odd	12
1600.3.h.b.1599.1		2	280.163	even	12
1600.3.h.b.1599.2		2	280.37	odd	12
1600.3.h.b.1599.2		2	280.107	even	12
2304.3.b.f.127.1		2	336.275	even	12
2304.3.b.f.127.1		2	336.317	odd	12
2304.3.b.f.127.2		2	336.107	even	12
2304.3.b.f.127.2		2	336.149	odd	12
3600.3.e.c.3151.1		1	105.44	odd	6
3600.3.e.c.3151.1		1	420.359	even	6
3600.3.j.a.1999.1		2	105.2	even	12
3600.3.j.a.1999.1		2	420.107	odd	12
3600.3.j.a.1999.2		2	105.23	even	12
3600.3.j.a.1999.2		2	420.23	odd	12

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	784.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.00000 - 5.19615i) q^{5} +(-4.50000 + 7.79423i) q^{9} +10.0000 q^{13} +(15.0000 + 25.9808i) q^{17} +(-5.50000 - 9.52628i) q^{25} +42.0000 q^{29} +(35.0000 - 60.6218i) q^{37} +18.0000 q^{41} +(27.0000 + 46.7654i) q^{45} +(-45.0000 - 77.9423i) q^{53} +(11.0000 - 19.0526i) q^{61} +(30.0000 - 51.9615i) q^{65} +(55.0000 + 95.2628i) q^{73} +(-40.5000 - 70.1481i) q^{81} +180.000 q^{85} +(39.0000 - 67.5500i) q^{89} +130.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} - 9 q^{9} + 20 q^{13} + 30 q^{17} - 11 q^{25} + 84 q^{29} + 70 q^{37} + 36 q^{41} + 54 q^{45} - 90 q^{53} + 22 q^{61} + 60 q^{65} + 110 q^{73} - 81 q^{81} + 360 q^{85} + 78 q^{89} + 260 q^{97}+O(q^{100}) \) 2 * q + 6 * q^5 - 9 * q^9 + 20 * q^13 + 30 * q^17 - 11 * q^25 + 84 * q^29 + 70 * q^37 + 36 * q^41 + 54 * q^45 - 90 * q^53 + 22 * q^61 + 60 * q^65 + 110 * q^73 - 81 * q^81 + 360 * q^85 + 78 * q^89 + 260 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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