LMFDB - Embedded newform 784.3.s.a.705.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,3,Mod(129,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("784.129");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$21.3624527258$
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 7)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			705.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	784.705
Dual form			784.3.s.a.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 7.79423i) q^{9} +O(q^{10})$	$q+(-4.50000 + 7.79423i) q^{9} +(-3.00000 - 5.19615i) q^{11} +(9.00000 - 15.5885i) q^{23} +(-12.5000 - 21.6506i) q^{25} -54.0000 q^{29} +(19.0000 - 32.9090i) q^{37} -58.0000 q^{43} +(3.00000 + 5.19615i) q^{53} +(-59.0000 - 102.191i) q^{67} -114.000 q^{71} +(-47.0000 + 81.4064i) q^{79} +(-40.5000 - 70.1481i) q^{81} +54.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{9}+O(q^{10}) $ 2 * q - 9 * q^9	$ 2 q - 9 q^{9} - 6 q^{11} + 18 q^{23} - 25 q^{25} - 108 q^{29} + 38 q^{37} - 116 q^{43} + 6 q^{53} - 118 q^{67} - 228 q^{71} - 94 q^{79} - 81 q^{81} + 108 q^{99}+O(q^{100}) $ 2 * q - 9 * q^9 - 6 * q^11 + 18 * q^23 - 25 * q^25 - 108 * q^29 + 38 * q^37 - 116 * q^43 + 6 * q^53 - 118 * q^67 - 228 * q^71 - 94 * q^79 - 81 * q^81 + 108 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$10$		0			0
$11$	−3.00000	−	5.19615i	−0.272727	−	0.472377i	0.696832	−	0.717234i	$-0.254592\pi$
$11$	−3.00000	−	5.19615i	−0.272727	−	0.472377i	−0.969559	+	0.244857i	$0.921259\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		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
$23$	9.00000	−	15.5885i	0.391304	−	0.677759i	−0.601318	−	0.799010i	$-0.705357\pi$
$23$	9.00000	−	15.5885i	0.391304	−	0.677759i	0.992622	+	0.121251i	$0.0386906\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−54.0000			−1.86207			−0.931034	−	0.364931i	$-0.881093\pi$
$29$	−54.0000			−1.86207			−0.931034	+	0.364931i	$0.881093\pi$
$30$		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
$37$	19.0000	−	32.9090i	0.513514	−	0.889431i	−0.486364	−	0.873757i	$-0.661677\pi$
$37$	19.0000	−	32.9090i	0.513514	−	0.889431i	0.999877	−	0.0156750i	$-0.00498971\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−58.0000			−1.34884			−0.674419	−	0.738349i	$-0.735606\pi$
$43$	−58.0000			−1.34884			−0.674419	+	0.738349i	$0.735606\pi$
$44$		0			0
$45$		0			0
$46$		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
$53$	3.00000	+	5.19615i	0.0566038	+	0.0980406i	0.892939	−	0.450178i	$-0.148639\pi$
$53$	3.00000	+	5.19615i	0.0566038	+	0.0980406i	−0.836335	+	0.548219i	$0.815306\pi$
$54$		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
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−59.0000	−	102.191i	−0.880597	−	1.52524i	−0.850678	−	0.525686i	$-0.823809\pi$
$67$	−59.0000	−	102.191i	−0.880597	−	1.52524i	−0.0299186	−	0.999552i	$-0.509525\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−114.000			−1.60563			−0.802817	−	0.596226i	$-0.796666\pi$
$71$	−114.000			−1.60563			−0.802817	+	0.596226i	$0.796666\pi$
$72$		0			0
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−47.0000	+	81.4064i	−0.594937	+	1.03046i	0.398619	+	0.917117i	$0.369490\pi$
$79$	−47.0000	+	81.4064i	−0.594937	+	1.03046i	−0.993556	+	0.113344i	$0.963844\pi$
$80$		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
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$	54.0000			0.545455
$100$		0			0
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		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
$107$	93.0000	−	161.081i	0.869159	−	1.50543i	0.00630134	−	0.999980i	$-0.497994\pi$
$107$	93.0000	−	161.081i	0.869159	−	1.50543i	0.862858	−	0.505447i	$-0.168672\pi$
$108$		0			0
$109$	−53.0000	−	91.7987i	−0.486239	−	0.842190i	0.513636	−	0.858008i	$-0.328298\pi$
$109$	−53.0000	−	91.7987i	−0.486239	−	0.842190i	−0.999875	+	0.0158181i	$0.994965\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−222.000			−1.96460			−0.982301	−	0.187310i	$-0.940023\pi$
$113$	−222.000			−1.96460			−0.982301	+	0.187310i	$0.940023\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	42.5000	−	73.6122i	0.351240	−	0.608365i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2.00000			−0.0157480			−0.00787402	−	0.999969i	$-0.502506\pi$
$127$	−2.00000			−0.0157480			−0.00787402	+	0.999969i	$0.502506\pi$
$128$		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
$137$	87.0000	+	150.688i	0.635036	+	1.09992i	0.986507	+	0.163717i	$0.0523483\pi$
$137$	87.0000	+	150.688i	0.635036	+	1.09992i	−0.351471	+	0.936199i	$0.614318\pi$
$138$		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
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−93.0000	+	161.081i	−0.624161	+	1.08108i	0.364541	+	0.931187i	$0.381226\pi$
$149$	−93.0000	+	161.081i	−0.624161	+	1.08108i	−0.988702	+	0.149892i	$0.952108\pi$
$150$		0			0
$151$	137.000	+	237.291i	0.907285	+	1.57146i	0.817821	+	0.575473i	$0.195182\pi$
$151$	137.000	+	237.291i	0.907285	+	1.57146i	0.0894642	+	0.995990i	$0.471485\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	37.0000	−	64.0859i	0.226994	−	0.393165i	−0.729922	−	0.683531i	$-0.760444\pi$
$163$	37.0000	−	64.0859i	0.226994	−	0.393165i	0.956916	+	0.290366i	$0.0937769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	169.000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−171.000	−	296.181i	−0.955307	−	1.65464i	−0.733663	−	0.679513i	$-0.762191\pi$
$179$	−171.000	−	296.181i	−0.955307	−	1.65464i	−0.221644	−	0.975128i	$-0.571142\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		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
$191$	−159.000	+	275.396i	−0.832461	+	1.44186i	0.0636205	+	0.997974i	$0.479735\pi$
$191$	−159.000	+	275.396i	−0.832461	+	1.44186i	−0.896081	+	0.443890i	$0.853598\pi$
$192$		0			0
$193$	31.0000	+	53.6936i	0.160622	+	0.278205i	0.935092	−	0.354405i	$-0.115317\pi$
$193$	31.0000	+	53.6936i	0.160622	+	0.278205i	−0.774470	+	0.632611i	$0.781983\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	282.000			1.43147			0.715736	−	0.698371i	$-0.246091\pi$
$197$	282.000			1.43147			0.715736	+	0.698371i	$0.246091\pi$
$198$		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
$205$		0			0
$206$		0			0
$207$	81.0000	+	140.296i	0.391304	+	0.677759i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	278.000			1.31754			0.658768	−	0.752346i	$-0.271078\pi$
$211$	278.000			1.31754			0.658768	+	0.752346i	$0.271078\pi$
$212$		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
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	225.000			1.00000
$226$		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
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.00000	+	15.5885i	−0.0386266	+	0.0669033i	−0.884692	−	0.466175i	$-0.845632\pi$
$233$	−9.00000	+	15.5885i	−0.0386266	+	0.0669033i	0.846066	+	0.533078i	$0.178965\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	222.000			0.928870			0.464435	−	0.885607i	$-0.346257\pi$
$239$	222.000			0.928870			0.464435	+	0.885607i	$0.346257\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$		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
$253$	−108.000			−0.426877
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	243.000	−	420.888i	0.931034	−	1.61260i
$262$		0			0
$263$	249.000	+	431.281i	0.946768	+	1.63985i	0.752172	+	0.658967i	$0.229006\pi$
$263$	249.000	+	431.281i	0.946768	+	1.63985i	0.194596	+	0.980883i	$0.437660\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		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
$275$	−75.0000	+	129.904i	−0.272727	+	0.472377i
$276$		0			0
$277$	227.000	+	393.176i	0.819495	+	1.41941i	0.906055	+	0.423160i	$0.139079\pi$
$277$	227.000	+	393.176i	0.819495	+	1.41941i	−0.0865605	+	0.996247i	$0.527588\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	114.000			0.405694			0.202847	−	0.979210i	$-0.434981\pi$
$281$	114.000			0.405694			0.202847	+	0.979210i	$0.434981\pi$
$282$		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
$289$	−144.500	+	250.281i	−0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		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
$305$		0			0
$306$		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
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−261.000	+	452.065i	−0.823344	+	1.42607i	0.0798346	+	0.996808i	$0.474561\pi$
$317$	−261.000	+	452.065i	−0.823344	+	1.42607i	−0.903178	+	0.429265i	$0.858773\pi$
$318$		0			0
$319$	162.000	+	280.592i	0.507837	+	0.879599i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	317.000	−	549.060i	0.957704	−	1.65879i	0.229648	−	0.973274i	$-0.426242\pi$
$331$	317.000	−	549.060i	0.957704	−	1.65879i	0.728055	−	0.685518i	$-0.240424\pi$
$332$		0			0
$333$	171.000	+	296.181i	0.513514	+	0.889431i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	226.000			0.670623			0.335312	−	0.942107i	$-0.391158\pi$
$337$	226.000			0.670623			0.335312	+	0.942107i	$0.391158\pi$
$338$		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
$347$	−339.000	−	587.165i	−0.976945	−	1.69212i	−0.673360	−	0.739314i	$-0.735150\pi$
$347$	−339.000	−	587.165i	−0.976945	−	1.69212i	−0.303585	−	0.952804i	$-0.598184\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−327.000	+	566.381i	−0.910864	+	1.57766i	−0.0980164	+	0.995185i	$0.531250\pi$
$359$	−327.000	+	566.381i	−0.910864	+	1.57766i	−0.812847	+	0.582477i	$0.802084\pi$
$360$		0			0
$361$	−180.500	−	312.635i	−0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		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
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	131.000	−	226.899i	0.351206	−	0.608307i	−0.635255	−	0.772303i	$-0.719105\pi$
$373$	131.000	−	226.899i	0.351206	−	0.608307i	0.986461	+	0.163995i	$0.0524382\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	614.000			1.62005			0.810026	−	0.586393i	$-0.199453\pi$
$379$	614.000			1.62005			0.810026	+	0.586393i	$0.199453\pi$
$380$		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
$387$	261.000	−	452.065i	0.674419	−	1.16813i
$388$		0			0
$389$	−333.000	−	576.773i	−0.856041	−	1.48271i	−0.875676	−	0.482900i	$-0.839584\pi$
$389$	−333.000	−	576.773i	−0.856041	−	1.48271i	0.0196346	−	0.999807i	$-0.493750\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−177.000	+	306.573i	−0.441397	+	0.764521i	−0.997793	−	0.0663955i	$-0.978850\pi$
$401$	−177.000	+	306.573i	−0.441397	+	0.764521i	0.556397	+	0.830917i	$0.312183\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−228.000			−0.560197
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		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
$421$	−166.000			−0.394299			−0.197150	−	0.980373i	$-0.563168\pi$
$421$	−166.000			−0.394299			−0.197150	+	0.980373i	$0.563168\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	81.0000	+	140.296i	0.187935	+	0.325513i	0.944562	−	0.328334i	$-0.106487\pi$
$431$	81.0000	+	140.296i	0.187935	+	0.325513i	−0.756627	+	0.653847i	$0.773154\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		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
$443$	−243.000	+	420.888i	−0.548533	+	0.950087i	0.449843	+	0.893108i	$0.351480\pi$
$443$	−243.000	+	420.888i	−0.548533	+	0.950087i	−0.998375	+	0.0569787i	$0.981853\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−894.000			−1.99109			−0.995546	−	0.0942807i	$-0.969945\pi$
$449$	−894.000			−1.99109			−0.995546	+	0.0942807i	$0.969945\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	439.000	−	760.370i	0.960613	−	1.66383i	0.239646	−	0.970860i	$-0.422969\pi$
$457$	439.000	−	760.370i	0.960613	−	1.66383i	0.720967	−	0.692970i	$-0.243698\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−674.000			−1.45572			−0.727862	−	0.685724i	$-0.759486\pi$
$463$	−674.000			−1.45572			−0.727862	+	0.685724i	$0.759486\pi$
$464$		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
$473$	174.000	+	301.377i	0.367865	+	0.637160i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−54.0000			−0.113208
$478$		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
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−199.000	−	344.678i	−0.408624	−	0.707758i	0.586112	−	0.810230i	$-0.300658\pi$
$487$	−199.000	−	344.678i	−0.408624	−	0.707758i	−0.994736	+	0.102472i	$0.967325\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−954.000			−1.94297			−0.971487	−	0.237094i	$-0.923805\pi$
$491$	−954.000			−1.94297			−0.971487	+	0.237094i	$0.923805\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	149.000	−	258.076i	0.298597	−	0.517186i	−0.677218	−	0.735782i	$-0.736815\pi$
$499$	149.000	−	258.076i	0.298597	−	0.517186i	0.975815	+	0.218597i	$0.0701480\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		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
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		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
$529$	102.500	+	177.535i	0.193762	+	0.335605i
$530$		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
$541$	−37.0000	+	64.0859i	−0.0683919	+	0.118458i	−0.898194	−	0.439600i	$-0.855120\pi$
$541$	−37.0000	+	64.0859i	−0.0683919	+	0.118458i	0.829802	+	0.558058i	$0.188453\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−842.000			−1.53931			−0.769653	−	0.638463i	$-0.779571\pi$
$547$	−842.000			−1.53931			−0.769653	+	0.638463i	$0.779571\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−501.000	−	867.757i	−0.899461	−	1.55791i	−0.828184	−	0.560456i	$-0.810626\pi$
$557$	−501.000	−	867.757i	−0.899461	−	1.55791i	−0.0712774	−	0.997457i	$-0.522708\pi$
$558$		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
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	327.000	−	566.381i	0.574692	−	0.995397i	−0.421383	−	0.906883i	$-0.638455\pi$
$569$	327.000	−	566.381i	0.574692	−	0.995397i	0.996075	−	0.0885135i	$-0.0282116\pi$
$570$		0			0
$571$	−563.000	−	975.145i	−0.985989	−	1.70778i	−0.637454	−	0.770488i	$-0.720012\pi$
$571$	−563.000	−	975.145i	−0.985989	−	1.70778i	−0.348535	−	0.937296i	$-0.613321\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−450.000			−0.782609
$576$		0			0
$577$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$577$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	18.0000	−	31.1769i	0.0308748	−	0.0534767i
$584$		0			0
$585$		0			0
$586$		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
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−87.0000	−	150.688i	−0.145242	−	0.251567i	0.784221	−	0.620481i	$-0.213063\pi$
$599$	−87.0000	−	150.688i	−0.145242	−	0.251567i	−0.929463	+	0.368915i	$0.879729\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	1062.00			1.76119
$604$		0			0
$605$		0			0
$606$		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
$613$	−109.000	−	188.794i	−0.177814	−	0.307983i	0.763318	−	0.646024i	$-0.223569\pi$
$613$	−109.000	−	188.794i	−0.177814	−	0.307983i	−0.941132	+	0.338041i	$0.890236\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−558.000			−0.904376			−0.452188	−	0.891923i	$-0.649356\pi$
$617$	−558.000			−0.904376			−0.452188	+	0.891923i	$0.649356\pi$
$618$		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
$625$	−312.500	+	541.266i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1006.00			1.59429			0.797147	−	0.603785i	$-0.206341\pi$
$631$	1006.00			1.59429			0.797147	+	0.603785i	$0.206341\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	513.000	−	888.542i	0.802817	−	1.39052i
$640$		0			0
$641$	−417.000	−	722.265i	−0.650546	−	1.12678i	−0.982991	−	0.183656i	$-0.941207\pi$
$641$	−417.000	−	722.265i	−0.650546	−	1.12678i	0.332445	−	0.943123i	$-0.392127\pi$
$642$		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
$653$	−597.000	+	1034.03i	−0.914242	+	1.58351i	−0.106235	+	0.994341i	$0.533879\pi$
$653$	−597.000	+	1034.03i	−0.914242	+	1.58351i	−0.808007	+	0.589172i	$0.799454\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−618.000			−0.937785			−0.468892	−	0.883255i	$-0.655347\pi$
$659$	−618.000			−0.937785			−0.468892	+	0.883255i	$0.655347\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−486.000	+	841.777i	−0.728636	+	1.26203i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−446.000			−0.662704			−0.331352	−	0.943507i	$-0.607505\pi$
$673$	−446.000			−0.662704			−0.331352	+	0.943507i	$0.607505\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	669.000	+	1158.74i	0.979502	+	1.69655i	0.664198	+	0.747557i	$0.268773\pi$
$683$	669.000	+	1158.74i	0.979502	+	1.69655i	0.315305	+	0.948991i	$0.397893\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		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
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1398.00			−1.99429			−0.997147	−	0.0754851i	$-0.975949\pi$
$701$	−1398.00			−1.99429			−0.997147	+	0.0754851i	$0.975949\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	691.000	−	1196.85i	0.974612	−	1.68808i	0.293403	−	0.955989i	$-0.405212\pi$
$709$	691.000	−	1196.85i	0.974612	−	1.68808i	0.681209	−	0.732089i	$-0.261454\pi$
$710$		0			0
$711$	−423.000	−	732.657i	−0.594937	−	1.03046i
$712$		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
$725$	675.000	+	1169.13i	0.931034	+	1.61260i
$726$		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
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−354.000	+	613.146i	−0.480326	+	0.831948i
$738$		0			0
$739$	613.000	+	1061.75i	0.829499	+	1.43673i	0.898432	+	0.439114i	$0.144707\pi$
$739$	613.000	+	1061.75i	0.829499	+	1.43673i	−0.0689322	+	0.997621i	$0.521959\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−114.000			−0.153432			−0.0767160	−	0.997053i	$-0.524443\pi$
$743$	−114.000			−0.153432			−0.0767160	+	0.997053i	$0.524443\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	401.000	−	694.552i	0.533955	−	0.924837i	−0.465258	−	0.885175i	$-0.654039\pi$
$751$	401.000	−	694.552i	0.533955	−	0.924837i	0.999213	−	0.0396618i	$-0.0126281\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1402.00			1.85205			0.926024	−	0.377465i	$-0.123204\pi$
$757$	1402.00			1.85205			0.926024	+	0.377465i	$0.123204\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	342.000	+	592.361i	0.437900	+	0.758465i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		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
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		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
$809$	87.0000	+	150.688i	0.107540	+	0.186265i	0.914773	−	0.403968i	$-0.132369\pi$
$809$	87.0000	+	150.688i	0.107540	+	0.186265i	−0.807233	+	0.590233i	$0.799036\pi$
$810$		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
$821$	579.000	−	1002.86i	0.705238	−	1.22151i	−0.261368	−	0.965239i	$-0.584174\pi$
$821$	579.000	−	1002.86i	0.705238	−	1.22151i	0.966606	−	0.256268i	$-0.0824930\pi$
$822$		0			0
$823$	−311.000	−	538.668i	−0.377886	−	0.654517i	0.612869	−	0.790185i	$-0.290015\pi$
$823$	−311.000	−	538.668i	−0.377886	−	0.654517i	−0.990754	+	0.135667i	$0.956682\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−282.000			−0.340992			−0.170496	−	0.985358i	$-0.554537\pi$
$827$	−282.000			−0.340992			−0.170496	+	0.985358i	$0.554537\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		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
$841$	2075.00			2.46730
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−342.000	−	592.361i	−0.401880	−	0.696077i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		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
$863$	−831.000	+	1439.33i	−0.962920	+	1.66783i	−0.247818	+	0.968807i	$0.579713\pi$
$863$	−831.000	+	1439.33i	−0.962920	+	1.66783i	−0.715102	+	0.699020i	$0.753620\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	564.000			0.649022
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−373.000	+	646.055i	−0.425314	+	0.736665i	−0.996450	−	0.0841907i	$-0.973170\pi$
$877$	−373.000	+	646.055i	−0.425314	+	0.736665i	0.571136	+	0.820855i	$0.306503\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	1622.00			1.83692			0.918460	−	0.395514i	$-0.129434\pi$
$883$	1622.00			1.83692			0.918460	+	0.395514i	$0.129434\pi$
$884$		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
$891$	−243.000	+	420.888i	−0.272727	+	0.472377i
$892$		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
$907$	893.000	+	1546.72i	0.984564	+	1.70532i	0.643856	+	0.765147i	$0.277334\pi$
$907$	893.000	+	1546.72i	0.984564	+	1.70532i	0.340709	+	0.940169i	$0.389333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1566.00			1.71899			0.859495	−	0.511144i	$-0.170778\pi$
$911$	1566.00			1.71899			0.859495	+	0.511144i	$0.170778\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	233.000	−	403.568i	0.253536	−	0.439138i	−0.710961	−	0.703232i	$-0.751740\pi$
$919$	233.000	−	403.568i	0.253536	−	0.439138i	0.964497	+	0.264094i	$0.0850729\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−950.000			−1.02703
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−747.000	+	1293.84i	−0.788807	+	1.36625i	0.137892	+	0.990447i	$0.455967\pi$
$947$	−747.000	+	1293.84i	−0.788807	+	1.36625i	−0.926698	+	0.375806i	$0.877366\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1458.00			1.52991			0.764953	−	0.644086i	$-0.222762\pi$
$953$	1458.00			1.52991			0.764953	+	0.644086i	$0.222762\pi$
$954$		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
$963$	837.000	+	1449.73i	0.869159	+	1.50543i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	334.000			0.345398			0.172699	−	0.984975i	$-0.444751\pi$
$967$	334.000			0.345398			0.172699	+	0.984975i	$0.444751\pi$
$968$		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
$977$	−81.0000	−	140.296i	−0.0829069	−	0.143599i	0.821591	−	0.570078i	$-0.193087\pi$
$977$	−81.0000	−	140.296i	−0.0829069	−	0.143599i	−0.904497	+	0.426479i	$0.859754\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	954.000			0.972477
$982$		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
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−522.000	+	904.131i	−0.527806	+	0.914187i
$990$		0			0
$991$	−703.000	−	1217.63i	−0.709384	−	1.22869i	−0.965086	−	0.261934i	$-0.915640\pi$
$991$	−703.000	−	1217.63i	−0.709384	−	1.22869i	0.255701	−	0.966756i	$-0.417694\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.3.s.a.705.1		2
4.3	odd	2		49.3.d.a.19.1		2
7.2	even	3		112.3.c.a.97.1		1
7.3	odd	6	inner	784.3.s.a.129.1		2
7.4	even	3	inner	784.3.s.a.129.1		2
7.5	odd	6		112.3.c.a.97.1		1
7.6	odd	2	CM	784.3.s.a.705.1		2
12.11	even	2		441.3.m.a.19.1		2
21.2	odd	6		1008.3.f.a.433.1		1
21.5	even	6		1008.3.f.a.433.1		1
28.3	even	6		49.3.d.a.31.1		2
28.11	odd	6		49.3.d.a.31.1		2
28.19	even	6		7.3.b.a.6.1	✓	1
28.23	odd	6		7.3.b.a.6.1	✓	1
28.27	even	2		49.3.d.a.19.1		2
56.5	odd	6		448.3.c.b.321.1		1
56.19	even	6		448.3.c.a.321.1		1
56.37	even	6		448.3.c.b.321.1		1
56.51	odd	6		448.3.c.a.321.1		1
84.11	even	6		441.3.m.a.325.1		2
84.23	even	6		63.3.d.a.55.1		1
84.47	odd	6		63.3.d.a.55.1		1
84.59	odd	6		441.3.m.a.325.1		2
84.83	odd	2		441.3.m.a.19.1		2
140.19	even	6		175.3.d.a.76.1		1
140.23	even	12		175.3.c.a.174.2		2
140.47	odd	12		175.3.c.a.174.1		2
140.79	odd	6		175.3.d.a.76.1		1
140.103	odd	12		175.3.c.a.174.2		2
140.107	even	12		175.3.c.a.174.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.3.b.a.6.1	✓	1	28.19	even	6
7.3.b.a.6.1	✓	1	28.23	odd	6
49.3.d.a.19.1		2	4.3	odd	2
49.3.d.a.19.1		2	28.27	even	2
49.3.d.a.31.1		2	28.3	even	6
49.3.d.a.31.1		2	28.11	odd	6
63.3.d.a.55.1		1	84.23	even	6
63.3.d.a.55.1		1	84.47	odd	6
112.3.c.a.97.1		1	7.2	even	3
112.3.c.a.97.1		1	7.5	odd	6
175.3.c.a.174.1		2	140.47	odd	12
175.3.c.a.174.1		2	140.107	even	12
175.3.c.a.174.2		2	140.23	even	12
175.3.c.a.174.2		2	140.103	odd	12
175.3.d.a.76.1		1	140.19	even	6
175.3.d.a.76.1		1	140.79	odd	6
441.3.m.a.19.1		2	12.11	even	2
441.3.m.a.19.1		2	84.83	odd	2
441.3.m.a.325.1		2	84.11	even	6
441.3.m.a.325.1		2	84.59	odd	6
448.3.c.a.321.1		1	56.19	even	6
448.3.c.a.321.1		1	56.51	odd	6
448.3.c.b.321.1		1	56.5	odd	6
448.3.c.b.321.1		1	56.37	even	6
784.3.s.a.129.1		2	7.3	odd	6	inner
784.3.s.a.129.1		2	7.4	even	3	inner
784.3.s.a.705.1		2	1.1	even	1	trivial
784.3.s.a.705.1		2	7.6	odd	2	CM
1008.3.f.a.433.1		1	21.2	odd	6
1008.3.f.a.433.1		1	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	784.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)	\(q+(-4.50000 + 7.79423i) q^{9} +(-3.00000 - 5.19615i) q^{11} +(9.00000 - 15.5885i) q^{23} +(-12.5000 - 21.6506i) q^{25} -54.0000 q^{29} +(19.0000 - 32.9090i) q^{37} -58.0000 q^{43} +(3.00000 + 5.19615i) q^{53} +(-59.0000 - 102.191i) q^{67} -114.000 q^{71} +(-47.0000 + 81.4064i) q^{79} +(-40.5000 - 70.1481i) q^{81} +54.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{9}+O(q^{10}) \) 2 * q - 9 * q^9	\( 2 q - 9 q^{9} - 6 q^{11} + 18 q^{23} - 25 q^{25} - 108 q^{29} + 38 q^{37} - 116 q^{43} + 6 q^{53} - 118 q^{67} - 228 q^{71} - 94 q^{79} - 81 q^{81} + 108 q^{99}+O(q^{100}) \) 2 * q - 9 * q^9 - 6 * q^11 + 18 * q^23 - 25 * q^25 - 108 * q^29 + 38 * q^37 - 116 * q^43 + 6 * q^53 - 118 * q^67 - 228 * q^71 - 94 * q^79 - 81 * q^81 + 108 * q^99

Introduction

L-functions

Modular forms

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Database

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