LMFDB - Embedded newform 784.3.s.c.129.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,3,Mod(129,784)] 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("784.129"); S:= CuspForms(chi, 3); N := Newforms(S);

sage: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")

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:traces = [4,0,-6,0,6,0,0,0,0,0,-18] 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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			129.2
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	784.129
Dual form			784.3.s.c.705.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.621320 + 0.358719i) q^{3} +(5.74264 - 3.31552i) q^{5} +(-4.24264 - 7.34847i) q^{9} +(-2.37868 + 4.11999i) q^{11} -15.2913i q^{13} +4.75736 q^{15} +(3.25736 + 1.88064i) q^{17} +(3.62132 - 2.09077i) q^{19} +(-13.8640 - 24.0131i) q^{23} +(9.48528 - 16.4290i) q^{25} -12.5446i q^{27} +3.51472 q^{29} +(-42.3198 - 24.4334i) q^{31} +(-2.95584 + 1.70656i) q^{33} +(1.47056 + 2.54709i) q^{37} +(5.48528 - 9.50079i) q^{39} +27.9590i q^{41} +10.4853 q^{43} +(-48.7279 - 28.1331i) q^{45} +(45.6213 - 26.3395i) q^{47} +(1.34924 + 2.33696i) q^{51} +(-27.9853 + 48.4719i) q^{53} +31.5462i q^{55} +3.00000 q^{57} +(33.5330 + 19.3603i) q^{59} +(78.3823 - 45.2540i) q^{61} +(-50.6985 - 87.8124i) q^{65} +(-17.3198 + 29.9988i) q^{67} -19.8931i q^{69} -36.4264 q^{71} +(-45.5589 - 26.3034i) q^{73} +(11.7868 - 6.80511i) q^{75} +(-16.8934 - 29.2602i) q^{79} +(-33.6838 + 58.3420i) q^{81} -127.577i q^{83} +24.9411 q^{85} +(2.18377 + 1.26080i) q^{87} +(43.5883 - 25.1657i) q^{89} +(-17.5294 - 30.3619i) q^{93} +(13.8640 - 24.0131i) q^{95} -101.792i q^{97} +40.3675 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{3} + 6 q^{5} - 18 q^{11} + 36 q^{15} + 30 q^{17} + 6 q^{19} - 30 q^{23} + 4 q^{25} + 48 q^{29} - 42 q^{31} + 90 q^{33} - 62 q^{37} - 12 q^{39} + 8 q^{43} - 144 q^{45} + 174 q^{47} - 54 q^{51}+ \cdots - 144 q^{99}+O(q^{100}) $ 4 * q - 6 * q^3 + 6 * q^5 - 18 * q^11 + 36 * q^15 + 30 * q^17 + 6 * q^19 - 30 * q^23 + 4 * q^25 + 48 * q^29 - 42 * q^31 + 90 * q^33 - 62 * q^37 - 12 * q^39 + 8 * q^43 - 144 * q^45 + 174 * q^47 - 54 * q^51 - 78 * q^53 + 12 * q^57 - 78 * q^59 + 42 * q^61 - 84 * q^65 + 58 * q^67 + 24 * q^71 - 318 * q^73 + 132 * q^75 - 110 * q^79 + 18 * q^81 - 36 * q^85 - 144 * q^87 + 378 * q^89 - 138 * q^93 + 30 * q^95 - 144 * q^99

Character values

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

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

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
$3$	0.621320	+	0.358719i	0.207107	+	0.119573i	0.599966	−	0.800025i	$-0.295181\pi$
$3$	0.621320	+	0.358719i	0.207107	+	0.119573i	−0.392859	+	0.919599i	$0.628514\pi$
$4$		0			0
$5$	5.74264	−	3.31552i	1.14853	−	0.663103i	0.200000	−	0.979796i	$-0.435906\pi$
$5$	5.74264	−	3.31552i	1.14853	−	0.663103i	0.948528	+	0.316693i	$0.102572\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−4.24264	−	7.34847i	−0.471405	−	0.816497i
$10$		0			0
$11$	−2.37868	+	4.11999i	−0.216244	+	0.374545i	−0.953657	−	0.300897i	$-0.902714\pi$
$11$	−2.37868	+	4.11999i	−0.216244	+	0.374545i	0.737413	+	0.675442i	$0.236047\pi$
$12$		0			0
$13$		−	15.2913i		−	1.17625i	−0.808769	−	0.588126i	$-0.799866\pi$
$13$		−	15.2913i		−	1.17625i	0.808769	−	0.588126i	$-0.200134\pi$
$14$		0			0
$15$	4.75736			0.317157
$16$		0			0
$17$	3.25736	+	1.88064i	0.191609	+	0.110626i	0.592736	−	0.805397i	$-0.298048\pi$
$17$	3.25736	+	1.88064i	0.191609	+	0.110626i	−0.401126	+	0.916023i	$0.631381\pi$
$18$		0			0
$19$	3.62132	−	2.09077i	0.190596	−	0.110041i	−0.401666	−	0.915786i	$-0.631569\pi$
$19$	3.62132	−	2.09077i	0.190596	−	0.110041i	0.592261	+	0.805746i	$0.298235\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−13.8640	−	24.0131i	−0.602781	−	1.04405i	−0.992398	−	0.123070i	$-0.960726\pi$
$23$	−13.8640	−	24.0131i	−0.602781	−	1.04405i	0.389617	−	0.920977i	$-0.372607\pi$
$24$		0			0
$25$	9.48528	−	16.4290i	0.379411	−	0.657160i
$26$		0			0
$27$		−	12.5446i		−	0.464616i
$28$		0			0
$29$	3.51472			0.121197			0.0605986	−	0.998162i	$-0.480699\pi$
$29$	3.51472			0.121197			0.0605986	+	0.998162i	$0.480699\pi$
$30$		0			0
$31$	−42.3198	−	24.4334i	−1.36516	−	0.788173i	−0.374850	−	0.927085i	$-0.622306\pi$
$31$	−42.3198	−	24.4334i	−1.36516	−	0.788173i	−0.990305	+	0.138913i	$0.955639\pi$
$32$		0			0
$33$	−2.95584	+	1.70656i	−0.0895710	+	0.0517139i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.47056	+	2.54709i	0.0397449	+	0.0688403i	0.885214	−	0.465185i	$-0.154012\pi$
$37$	1.47056	+	2.54709i	0.0397449	+	0.0688403i	−0.845469	+	0.534025i	$0.820679\pi$
$38$		0			0
$39$	5.48528	−	9.50079i	0.140648	−	0.243610i
$40$		0			0
$41$			27.9590i			0.681927i	0.940077	+	0.340963i	$0.110753\pi$
$41$			27.9590i			0.681927i	−0.940077	+	0.340963i	$0.889247\pi$
$42$		0			0
$43$	10.4853			0.243844			0.121922	−	0.992540i	$-0.461094\pi$
$43$	10.4853			0.243844			0.121922	+	0.992540i	$0.461094\pi$
$44$		0			0
$45$	−48.7279	−	28.1331i	−1.08284	−	0.625180i
$46$		0			0
$47$	45.6213	−	26.3395i	0.970666	−	0.560415i	0.0712271	−	0.997460i	$-0.477309\pi$
$47$	45.6213	−	26.3395i	0.970666	−	0.560415i	0.899439	+	0.437046i	$0.143975\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	1.34924	+	2.33696i	0.0264557	+	0.0458227i
$52$		0			0
$53$	−27.9853	+	48.4719i	−0.528024	+	0.914565i	0.471442	+	0.881897i	$0.343734\pi$
$53$	−27.9853	+	48.4719i	−0.528024	+	0.914565i	−0.999466	+	0.0326677i	$0.989600\pi$
$54$		0			0
$55$			31.5462i			0.573567i
$56$		0			0
$57$	3.00000			0.0526316
$58$		0			0
$59$	33.5330	+	19.3603i	0.568356	+	0.328141i	0.756492	−	0.654002i	$-0.226911\pi$
$59$	33.5330	+	19.3603i	0.568356	+	0.328141i	−0.188136	+	0.982143i	$0.560245\pi$
$60$		0			0
$61$	78.3823	−	45.2540i	1.28495	−	0.741869i	0.307205	−	0.951643i	$-0.400606\pi$
$61$	78.3823	−	45.2540i	1.28495	−	0.741869i	0.977750	+	0.209774i	$0.0672729\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−50.6985	−	87.8124i	−0.779977	−	1.35096i
$66$		0			0
$67$	−17.3198	+	29.9988i	−0.258505	+	0.447743i	−0.965842	−	0.259134i	$-0.916563\pi$
$67$	−17.3198	+	29.9988i	−0.258505	+	0.447743i	0.707337	+	0.706877i	$0.249896\pi$
$68$		0			0
$69$		−	19.8931i		−	0.288306i
$70$		0			0
$71$	−36.4264			−0.513048			−0.256524	−	0.966538i	$-0.582577\pi$
$71$	−36.4264			−0.513048			−0.256524	+	0.966538i	$0.582577\pi$
$72$		0			0
$73$	−45.5589	−	26.3034i	−0.624094	−	0.360321i	0.154367	−	0.988014i	$-0.450666\pi$
$73$	−45.5589	−	26.3034i	−0.624094	−	0.360321i	−0.778461	+	0.627693i	$0.783999\pi$
$74$		0			0
$75$	11.7868	−	6.80511i	0.157157	−	0.0907348i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−16.8934	−	29.2602i	−0.213840	−	0.370383i	0.739073	−	0.673626i	$-0.235264\pi$
$79$	−16.8934	−	29.2602i	−0.213840	−	0.370383i	−0.952913	+	0.303243i	$0.901931\pi$
$80$		0			0
$81$	−33.6838	+	58.3420i	−0.415849	+	0.720272i
$82$		0			0
$83$		−	127.577i		−	1.53708i	−0.639803	−	0.768539i	$-0.720984\pi$
$83$		−	127.577i		−	1.53708i	0.639803	−	0.768539i	$-0.279016\pi$
$84$		0			0
$85$	24.9411			0.293425
$86$		0			0
$87$	2.18377	+	1.26080i	0.0251008	+	0.0144919i
$88$		0			0
$89$	43.5883	−	25.1657i	0.489756	−	0.282761i	−0.234717	−	0.972064i	$-0.575416\pi$
$89$	43.5883	−	25.1657i	0.489756	−	0.282761i	0.724473	+	0.689303i	$0.242083\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−17.5294	−	30.3619i	−0.188489	−	0.326472i
$94$		0			0
$95$	13.8640	−	24.0131i	0.145936	−	0.252769i
$96$		0			0
$97$		−	101.792i		−	1.04940i	−0.851287	−	0.524700i	$-0.824177\pi$
$97$		−	101.792i		−	1.04940i	0.851287	−	0.524700i	$-0.175823\pi$
$98$		0			0
$99$	40.3675			0.407753
$100$		0			0
$101$	51.6838	+	29.8396i	0.511720	+	0.295442i	0.733541	−	0.679646i	$-0.237866\pi$
$101$	51.6838	+	29.8396i	0.511720	+	0.295442i	−0.221820	+	0.975088i	$0.571200\pi$
$102$		0			0
$103$	104.077	−	60.0890i	1.01046	−	0.583388i	0.0991322	−	0.995074i	$-0.468393\pi$
$103$	104.077	−	60.0890i	1.01046	−	0.583388i	0.911326	+	0.411686i	$0.135060\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−56.8051	−	98.3893i	−0.530889	−	0.919526i	−0.999350	−	0.0360423i	$-0.988525\pi$
$107$	−56.8051	−	98.3893i	−0.530889	−	0.919526i	0.468462	−	0.883484i	$-0.344808\pi$
$108$		0			0
$109$	72.6543	−	125.841i	0.666553	−	1.15450i	−0.312308	−	0.949981i	$-0.601102\pi$
$109$	72.6543	−	125.841i	0.666553	−	1.15450i	0.978862	−	0.204524i	$-0.0655645\pi$
$110$		0			0
$111$			2.11008i			0.0190097i
$112$		0			0
$113$	34.5442			0.305700			0.152850	−	0.988249i	$-0.451155\pi$
$113$	34.5442			0.305700			0.152850	+	0.988249i	$0.451155\pi$
$114$		0			0
$115$	−159.231	−	91.9323i	−1.38462	−	0.799412i
$116$		0			0
$117$	−112.368	+	64.8754i	−0.960406	+	0.554491i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	49.1838	+	85.1888i	0.406477	+	0.704040i
$122$		0			0
$123$	−10.0294	+	17.3715i	−0.0815401	+	0.141232i
$124$		0			0
$125$			39.9814i			0.319851i
$126$		0			0
$127$	247.338			1.94754			0.973772	−	0.227526i	$-0.0730636\pi$
$127$	247.338			1.94754			0.973772	+	0.227526i	$0.0730636\pi$
$128$		0			0
$129$	6.51472	+	3.76127i	0.0505017	+	0.0291572i
$130$		0			0
$131$	−127.864	+	73.8223i	−0.976061	+	0.563529i	−0.901079	−	0.433656i	$-0.857223\pi$
$131$	−127.864	+	73.8223i	−0.976061	+	0.563529i	−0.0749822	+	0.997185i	$0.523890\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−41.5919	−	72.0393i	−0.308088	−	0.533624i
$136$		0			0
$137$	16.2868	−	28.2096i	0.118882	−	0.205909i	−0.800443	−	0.599409i	$-0.795402\pi$
$137$	16.2868	−	28.2096i	0.118882	−	0.205909i	0.919325	+	0.393500i	$0.128736\pi$
$138$		0			0
$139$			68.5857i			0.493422i	0.969089	+	0.246711i	$0.0793499\pi$
$139$			68.5857i			0.493422i	−0.969089	+	0.246711i	$0.920650\pi$
$140$		0			0
$141$	37.7939			0.268042
$142$		0			0
$143$	63.0000	+	36.3731i	0.440559	+	0.254357i
$144$		0			0
$145$	20.1838	−	11.6531i	0.139198	−	0.0803662i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−46.1985	−	80.0181i	−0.310057	−	0.537034i	0.668317	−	0.743876i	$-0.267015\pi$
$149$	−46.1985	−	80.0181i	−0.310057	−	0.537034i	−0.978374	+	0.206842i	$0.933681\pi$
$150$		0			0
$151$	−45.8934	+	79.4897i	−0.303930	+	0.526422i	−0.977022	−	0.213136i	$-0.931632\pi$
$151$	−45.8934	+	79.4897i	−0.303930	+	0.526422i	0.673093	+	0.739558i	$0.264965\pi$
$152$		0			0
$153$		−	31.9155i		−	0.208598i
$154$		0			0
$155$	−324.037			−2.09056
$156$		0			0
$157$	−7.32338	−	4.22815i	−0.0466457	−	0.0269309i	0.476496	−	0.879177i	$-0.341907\pi$
$157$	−7.32338	−	4.22815i	−0.0466457	−	0.0269309i	−0.523142	+	0.852246i	$0.675240\pi$
$158$		0			0
$159$	−34.7756	+	20.0777i	−0.218715	+	0.126275i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	110.989	+	192.238i	0.680913	+	1.17938i	0.974703	+	0.223506i	$0.0717503\pi$
$163$	110.989	+	192.238i	0.680913	+	1.17938i	−0.293789	+	0.955870i	$0.594916\pi$
$164$		0			0
$165$	−11.3162	+	19.6003i	−0.0685832	+	0.118790i
$166$		0			0
$167$			168.841i			1.01102i	0.862820	+	0.505511i	$0.168696\pi$
$167$			168.841i			1.01102i	−0.862820	+	0.505511i	$0.831304\pi$
$168$		0			0
$169$	−64.8234			−0.383570
$170$		0			0
$171$	−30.7279	−	17.7408i	−0.179695	−	0.103747i
$172$		0			0
$173$	142.323	−	82.1704i	0.822678	−	0.474974i	−0.0286608	−	0.999589i	$-0.509124\pi$
$173$	142.323	−	82.1704i	0.822678	−	0.474974i	0.851339	+	0.524616i	$0.175791\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	13.8898	+	24.0579i	0.0784736	+	0.135920i
$178$		0			0
$179$	92.5919	−	160.374i	0.517273	−	0.895943i	−0.482526	−	0.875882i	$-0.660280\pi$
$179$	92.5919	−	160.374i	0.517273	−	0.895943i	0.999799	−	0.0200614i	$-0.00638618\pi$
$180$		0			0
$181$			155.086i			0.856830i	0.903582	+	0.428415i	$0.140928\pi$
$181$			155.086i			0.856830i	−0.903582	+	0.428415i	$0.859072\pi$
$182$		0			0
$183$	64.9340			0.354831
$184$		0			0
$185$	16.8898	+	9.75135i	0.0912964	+	0.0527100i
$186$		0			0
$187$	−15.4964	+	8.94687i	−0.0828686	+	0.0478442i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	124.048	+	214.857i	0.649465	+	1.12491i	0.983251	+	0.182257i	$0.0583402\pi$
$191$	124.048	+	214.857i	0.649465	+	1.12491i	−0.333786	+	0.942649i	$0.608326\pi$
$192$		0			0
$193$	−77.1690	+	133.661i	−0.399840	+	0.692543i	−0.993706	−	0.112021i	$-0.964268\pi$
$193$	−77.1690	+	133.661i	−0.399840	+	0.692543i	0.593866	+	0.804564i	$0.297601\pi$
$194$		0			0
$195$		−	72.7461i		−	0.373057i
$196$		0			0
$197$	−181.103			−0.919303			−0.459651	−	0.888099i	$-0.652026\pi$
$197$	−181.103			−0.919303			−0.459651	+	0.888099i	$0.652026\pi$
$198$		0			0
$199$	301.989	+	174.353i	1.51753	+	0.876147i	0.999788	+	0.0206121i	$0.00656150\pi$
$199$	301.989	+	174.353i	1.51753	+	0.876147i	0.517744	+	0.855535i	$0.326772\pi$
$200$		0			0
$201$	−21.5223	+	12.4259i	−0.107076	+	0.0618204i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	92.6985	+	160.558i	0.452188	+	0.783212i
$206$		0			0
$207$	−117.640	+	203.758i	−0.568307	+	0.984337i
$208$		0			0
$209$			19.8931i			0.0951823i
$210$		0			0
$211$	−364.073			−1.72547			−0.862733	−	0.505660i	$-0.831249\pi$
$211$	−364.073			−1.72547			−0.862733	+	0.505660i	$0.831249\pi$
$212$		0			0
$213$	−22.6325	−	13.0669i	−0.106256	−	0.0613468i
$214$		0			0
$215$	60.2132	−	34.7641i	0.280061	−	0.161694i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−18.8711	−	32.6857i	−0.0861694	−	0.149250i
$220$		0			0
$221$	28.7574	−	49.8092i	0.130124	−	0.225381i
$222$		0			0
$223$			123.231i			0.552603i	0.961071	+	0.276302i	$0.0891089\pi$
$223$			123.231i			0.552603i	−0.961071	+	0.276302i	$0.910891\pi$
$224$		0			0
$225$	−160.971			−0.715425
$226$		0			0
$227$	−66.1432	−	38.1878i	−0.291380	−	0.168228i	0.347184	−	0.937797i	$-0.387138\pi$
$227$	−66.1432	−	38.1878i	−0.291380	−	0.168228i	−0.638564	+	0.769569i	$0.720471\pi$
$228$		0			0
$229$	−309.419	+	178.643i	−1.35117	+	0.780101i	−0.988414	−	0.151782i	$-0.951499\pi$
$229$	−309.419	+	178.643i	−1.35117	+	0.780101i	−0.362760	+	0.931883i	$0.618166\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	136.537	+	236.488i	0.585994	+	1.01497i	0.994751	+	0.102328i	$0.0326291\pi$
$233$	136.537	+	236.488i	0.585994	+	1.01497i	−0.408757	+	0.912643i	$0.634038\pi$
$234$		0			0
$235$	174.658	−	302.516i	0.743225	−	1.28730i
$236$		0			0
$237$		−	24.2400i		−	0.102278i
$238$		0			0
$239$	265.103			1.10922			0.554608	−	0.832112i	$-0.312868\pi$
$239$	265.103			1.10922			0.554608	+	0.832112i	$0.312868\pi$
$240$		0			0
$241$	75.8970	+	43.8191i	0.314925	+	0.181822i	0.649128	−	0.760679i	$-0.275134\pi$
$241$	75.8970	+	43.8191i	0.314925	+	0.181822i	−0.334203	+	0.942501i	$0.608467\pi$
$242$		0			0
$243$	−139.632	+	80.6168i	−0.574619	+	0.331757i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−31.9706	−	55.3746i	−0.129435	−	0.224189i
$248$		0			0
$249$	45.7645	−	79.2664i	0.183793	−	0.318339i
$250$		0			0
$251$		−	495.655i		−	1.97472i	−0.158491	−	0.987360i	$-0.550663\pi$
$251$		−	495.655i		−	1.97472i	0.158491	−	0.987360i	$-0.449337\pi$
$252$		0			0
$253$	131.912			0.521390
$254$		0			0
$255$	15.4964	+	8.94687i	0.0607703	+	0.0350858i
$256$		0			0
$257$	−346.875	+	200.268i	−1.34971	+	0.779254i	−0.988208	−	0.153119i	$-0.951068\pi$
$257$	−346.875	+	200.268i	−1.34971	+	0.779254i	−0.361499	+	0.932372i	$0.617735\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−14.9117	−	25.8278i	−0.0571329	−	0.0989571i
$262$		0			0
$263$	−16.1726	+	28.0118i	−0.0614928	+	0.106509i	−0.895133	−	0.445799i	$-0.852919\pi$
$263$	−16.1726	+	28.0118i	−0.0614928	+	0.106509i	0.833640	+	0.552308i	$0.186253\pi$
$264$		0			0
$265$			371.142i			1.40054i
$266$		0			0
$267$	36.1097			0.135242
$268$		0			0
$269$	265.838	+	153.482i	0.988246	+	0.570564i	0.904749	−	0.425944i	$-0.140058\pi$
$269$	265.838	+	153.482i	0.988246	+	0.570564i	0.0834963	+	0.996508i	$0.473391\pi$
$270$		0			0
$271$	−65.8051	+	37.9926i	−0.242823	+	0.140194i	−0.616474	−	0.787376i	$-0.711439\pi$
$271$	−65.8051	+	37.9926i	−0.242823	+	0.140194i	0.373650	+	0.927570i	$0.378106\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	45.1249	+	78.1586i	0.164091	+	0.284213i
$276$		0			0
$277$	−139.206	+	241.111i	−0.502547	+	0.870438i	0.497448	+	0.867494i	$0.334270\pi$
$277$	−139.206	+	241.111i	−0.502547	+	0.870438i	−0.999996	+	0.00294398i	$0.999063\pi$
$278$		0			0
$279$			414.648i			1.48619i
$280$		0			0
$281$	394.690			1.40459			0.702296	−	0.711885i	$-0.252158\pi$
$281$	394.690			1.40459			0.702296	+	0.711885i	$0.252158\pi$
$282$		0			0
$283$	126.783	+	73.1981i	0.447996	+	0.258650i	0.706983	−	0.707230i	$-0.250056\pi$
$283$	126.783	+	73.1981i	0.447996	+	0.258650i	−0.258988	+	0.965881i	$0.583389\pi$
$284$		0			0
$285$	17.2279	−	9.94655i	0.0604488	−	0.0349002i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−137.426	−	238.030i	−0.475524	−	0.823632i
$290$		0			0
$291$	36.5147	−	63.2453i	0.125480	−	0.217338i
$292$		0			0
$293$		−	299.678i		−	1.02279i	−0.859345	−	0.511396i	$-0.829128\pi$
$293$		−	299.678i		−	1.02279i	0.859345	−	0.511396i	$-0.170872\pi$
$294$		0			0
$295$	256.757			0.870364
$296$		0			0
$297$	51.6838	+	29.8396i	0.174019	+	0.100470i
$298$		0			0
$299$	−367.191	+	211.998i	−1.22806	+	0.709023i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	21.4081	+	37.0799i	0.0706539	+	0.122376i
$304$		0			0
$305$	300.081	−	519.755i	0.983871	−	1.70412i
$306$		0			0
$307$		−	20.9886i		−	0.0683666i	−0.999416	−	0.0341833i	$-0.989117\pi$
$307$		−	20.9886i		−	0.0683666i	0.999416	−	0.0341833i	$-0.0108830\pi$
$308$		0			0
$309$	86.2203			0.279030
$310$		0			0
$311$	157.651	+	91.0197i	0.506916	+	0.292668i	0.731565	−	0.681772i	$-0.238790\pi$
$311$	157.651	+	91.0197i	0.506916	+	0.292668i	−0.224649	+	0.974440i	$0.572124\pi$
$312$		0			0
$313$	84.8087	−	48.9643i	0.270954	−	0.156435i	−0.358367	−	0.933581i	$-0.616666\pi$
$313$	84.8087	−	48.9643i	0.270954	−	0.156435i	0.629321	+	0.777145i	$0.283333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	240.985	+	417.399i	0.760206	+	1.31672i	0.942744	+	0.333517i	$0.108235\pi$
$317$	240.985	+	417.399i	0.760206	+	1.31672i	−0.182538	+	0.983199i	$0.558431\pi$
$318$		0			0
$319$	−8.36039	+	14.4806i	−0.0262081	+	0.0453938i
$320$		0			0
$321$		−	81.5084i		−	0.253920i
$322$		0			0
$323$	15.7279			0.0486933
$324$		0			0
$325$	−251.220	−	145.042i	−0.772986	−	0.446283i
$326$		0			0
$327$	90.2832	−	52.1250i	0.276095	−	0.159404i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	112.504	+	194.862i	0.339890	+	0.588707i	0.984412	−	0.175879i	$-0.0562768\pi$
$331$	112.504	+	194.862i	0.339890	+	0.588707i	−0.644522	+	0.764586i	$0.722944\pi$
$332$		0			0
$333$	12.4781	−	21.6128i	0.0374719	−	0.0649032i
$334$		0			0
$335$			229.696i			0.685661i
$336$		0			0
$337$	−264.368			−0.784473			−0.392237	−	0.919864i	$-0.628299\pi$
$337$	−264.368			−0.784473			−0.392237	+	0.919864i	$0.628299\pi$
$338$		0			0
$339$	21.4630	+	12.3917i	0.0633126	+	0.0365536i
$340$		0			0
$341$	201.331	−	116.238i	0.590412	−	0.340875i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−65.9558	−	114.239i	−0.191176	−	0.331127i
$346$		0			0
$347$	−95.6285	+	165.633i	−0.275586	+	0.477330i	−0.970283	−	0.241973i	$-0.922205\pi$
$347$	−95.6285	+	165.633i	−0.275586	+	0.477330i	0.694697	+	0.719303i	$0.255539\pi$
$348$		0			0
$349$		−	135.448i		−	0.388104i	−0.980991	−	0.194052i	$-0.937837\pi$
$349$		−	135.448i		−	0.388104i	0.980991	−	0.194052i	$-0.0621630\pi$
$350$		0			0
$351$	−191.823			−0.546505
$352$		0			0
$353$	301.802	+	174.245i	0.854962	+	0.493612i	0.862322	−	0.506360i	$-0.169009\pi$
$353$	301.802	+	174.245i	0.854962	+	0.493612i	−0.00736010	+	0.999973i	$0.502343\pi$
$354$		0			0
$355$	−209.184	+	120.772i	−0.589250	+	0.340204i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	152.415	+	263.991i	0.424555	+	0.735351i	0.996379	−	0.0850256i	$-0.0270972\pi$
$359$	152.415	+	263.991i	0.424555	+	0.735351i	−0.571824	+	0.820377i	$0.693764\pi$
$360$		0			0
$361$	−171.757	+	297.492i	−0.475782	+	0.824079i
$362$		0			0
$363$			70.5727i			0.194415i
$364$		0			0
$365$	−348.838			−0.955720
$366$		0			0
$367$	−82.2761	−	47.5021i	−0.224186	−	0.129434i	0.383701	−	0.923457i	$-0.374649\pi$
$367$	−82.2761	−	47.5021i	−0.224186	−	0.129434i	−0.607887	+	0.794024i	$0.707983\pi$
$368$		0			0
$369$	205.456	−	118.620i	0.556791	−	0.321463i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−126.779	−	219.588i	−0.339891	−	0.588708i	0.644521	−	0.764586i	$-0.277057\pi$
$373$	−126.779	−	219.588i	−0.339891	−	0.588708i	−0.984412	+	0.175879i	$0.943723\pi$
$374$		0			0
$375$	−14.3421	+	24.8412i	−0.0382456	+	0.0662433i
$376$		0			0
$377$		−	53.7446i		−	0.142559i
$378$		0			0
$379$	−508.250			−1.34103			−0.670514	−	0.741897i	$-0.733926\pi$
$379$	−508.250			−1.34103			−0.670514	+	0.741897i	$0.733926\pi$
$380$		0			0
$381$	153.676	+	88.7250i	0.403350	+	0.232874i
$382$		0			0
$383$	413.753	−	238.881i	1.08030	−	0.623709i	0.149320	−	0.988789i	$-0.452292\pi$
$383$	413.753	−	238.881i	1.08030	−	0.623709i	0.930976	+	0.365080i	$0.118958\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−44.4853	−	77.0508i	−0.114949	−	0.199098i
$388$		0			0
$389$	−85.1102	+	147.415i	−0.218792	+	0.378959i	−0.954439	−	0.298406i	$-0.903545\pi$
$389$	−85.1102	+	147.415i	−0.218792	+	0.378959i	0.735647	+	0.677365i	$0.236878\pi$
$390$		0			0
$391$		−	104.292i		−	0.266732i
$392$		0			0
$393$	−105.926			−0.269532
$394$		0			0
$395$	−194.025	−	112.021i	−0.491204	−	0.283597i
$396$		0			0
$397$	−211.786	+	122.275i	−0.533467	+	0.307997i	−0.742427	−	0.669927i	$-0.766325\pi$
$397$	−211.786	+	122.275i	−0.533467	+	0.307997i	0.208960	+	0.977924i	$0.432992\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	208.786	+	361.629i	0.520664	+	0.901817i	0.999711	+	0.0240277i	$0.00764899\pi$
$401$	208.786	+	361.629i	0.520664	+	0.901817i	−0.479047	+	0.877789i	$0.659018\pi$
$402$		0			0
$403$	−373.617	+	647.124i	−0.927090	+	1.60577i
$404$		0			0
$405$			446.716i			1.10300i
$406$		0			0
$407$	−13.9920			−0.0343784
$408$		0			0
$409$	−266.919	−	154.106i	−0.652614	−	0.376787i	0.136843	−	0.990593i	$-0.456304\pi$
$409$	−266.919	−	154.106i	−0.652614	−	0.376787i	−0.789457	+	0.613806i	$0.789638\pi$
$410$		0			0
$411$	20.2386	−	11.6848i	0.0492424	−	0.0284301i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−422.985	−	732.631i	−1.01924	−	1.76538i
$416$		0			0
$417$	−24.6030	+	42.6137i	−0.0590001	+	0.102191i
$418$		0			0
$419$		−	103.142i		−	0.246163i	−0.992397	−	0.123081i	$-0.960722\pi$
$419$		−	103.142i		−	0.246163i	0.992397	−	0.123081i	$-0.0392776\pi$
$420$		0			0
$421$	−165.220			−0.392447			−0.196224	−	0.980559i	$-0.562868\pi$
$421$	−165.220			−0.392447			−0.196224	+	0.980559i	$0.562868\pi$
$422$		0			0
$423$	−387.110	−	223.498i	−0.915153	−	0.528364i
$424$		0			0
$425$	61.7939	−	35.6767i	0.145398	−	0.0839453i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	26.0955	+	45.1987i	0.0608286	+	0.105358i
$430$		0			0
$431$	297.268	−	514.883i	0.689717	−	1.19463i	−0.282212	−	0.959352i	$-0.591068\pi$
$431$	297.268	−	514.883i	0.689717	−	1.19463i	0.971929	−	0.235273i	$-0.0755984\pi$
$432$		0			0
$433$			40.6267i			0.0938261i	0.998899	+	0.0469131i	$0.0149384\pi$
$433$			40.6267i			0.0938261i	−0.998899	+	0.0469131i	$0.985062\pi$
$434$		0			0
$435$	16.7208			0.0384386
$436$		0			0
$437$	−100.412	−	57.9727i	−0.229775	−	0.132661i
$438$		0			0
$439$	126.959	−	73.3001i	0.289201	−	0.166971i	−0.348380	−	0.937353i	$-0.613268\pi$
$439$	126.959	−	73.3001i	0.289201	−	0.166971i	0.637582	+	0.770383i	$0.279935\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	53.6802	+	92.9768i	0.121174	+	0.209880i	0.920231	−	0.391376i	$-0.128001\pi$
$443$	53.6802	+	92.9768i	0.121174	+	0.209880i	−0.799057	+	0.601256i	$0.794667\pi$
$444$		0			0
$445$	166.875	−	289.035i	0.374999	−	0.649518i
$446$		0			0
$447$		−	66.2892i		−	0.148298i
$448$		0			0
$449$	135.161			0.301028			0.150514	−	0.988608i	$-0.451907\pi$
$449$	135.161			0.301028			0.150514	+	0.988608i	$0.451907\pi$
$450$		0			0
$451$	−115.191	−	66.5055i	−0.255412	−	0.147462i
$452$		0			0
$453$	−57.0290	+	32.9257i	−0.125892	+	0.0726837i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−79.8675	−	138.335i	−0.174765	−	0.302702i	0.765315	−	0.643656i	$-0.222583\pi$
$457$	−79.8675	−	138.335i	−0.174765	−	0.302702i	−0.940080	+	0.340954i	$0.889250\pi$
$458$		0			0
$459$	23.5919	−	40.8623i	0.0513984	−	0.0890247i
$460$		0			0
$461$		−	310.250i		−	0.672993i	−0.941685	−	0.336497i	$-0.890758\pi$
$461$		−	310.250i		−	0.672993i	0.941685	−	0.336497i	$-0.109242\pi$
$462$		0			0
$463$	326.014			0.704135			0.352067	−	0.935975i	$-0.385479\pi$
$463$	326.014			0.704135			0.352067	+	0.935975i	$0.385479\pi$
$464$		0			0
$465$	−201.331	−	116.238i	−0.432969	−	0.249975i
$466$		0			0
$467$	515.769	−	297.779i	1.10443	−	0.637643i	0.167048	−	0.985949i	$-0.446576\pi$
$467$	515.769	−	297.779i	1.10443	−	0.637643i	0.937381	+	0.348306i	$0.113243\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−3.03344	−	5.25408i	−0.00644043	−	0.0111551i
$472$		0			0
$473$	−24.9411	+	43.1993i	−0.0527297	+	0.0913304i
$474$		0			0
$475$		−	79.3262i		−	0.167002i
$476$		0			0
$477$	474.926			0.995652
$478$		0			0
$479$	438.798	+	253.340i	0.916071	+	0.528894i	0.882379	−	0.470539i	$-0.155940\pi$
$479$	438.798	+	253.340i	0.916071	+	0.528894i	0.0336914	+	0.999432i	$0.489274\pi$
$480$		0			0
$481$	38.9483	−	22.4868i	0.0809735	−	0.0467501i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−337.492	−	584.554i	−0.695861	−	1.20527i
$486$		0			0
$487$	105.651	−	182.992i	0.216942	−	0.375755i	−0.736930	−	0.675970i	$-0.763725\pi$
$487$	105.651	−	182.992i	0.216942	−	0.375755i	0.953872	+	0.300215i	$0.0970584\pi$
$488$		0			0
$489$			159.255i			0.325676i
$490$		0			0
$491$	784.161			1.59707			0.798534	−	0.601949i	$-0.205609\pi$
$491$	784.161			1.59707			0.798534	+	0.601949i	$0.205609\pi$
$492$		0			0
$493$	11.4487	+	6.60991i	0.0232225	+	0.0134075i
$494$		0			0
$495$	231.816	−	133.839i	0.468316	−	0.270382i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−85.7462	−	148.517i	−0.171836	−	0.297629i	0.767226	−	0.641377i	$-0.221637\pi$
$499$	−85.7462	−	148.517i	−0.171836	−	0.297629i	−0.939062	+	0.343748i	$0.888303\pi$
$500$		0			0
$501$	−60.5665	+	104.904i	−0.120891	+	0.209390i
$502$		0			0
$503$		−	20.0883i		−	0.0399370i	−0.999801	−	0.0199685i	$-0.993643\pi$
$503$		−	20.0883i		−	0.0399370i	0.999801	−	0.0199685i	$-0.00635659\pi$
$504$		0			0
$505$	395.735			0.783634
$506$		0			0
$507$	−40.2761	−	23.2534i	−0.0794400	−	0.0458647i
$508$		0			0
$509$	412.890	−	238.382i	0.811178	−	0.468334i	−0.0361865	−	0.999345i	$-0.511521\pi$
$509$	412.890	−	238.382i	0.811178	−	0.468334i	0.847365	+	0.531011i	$0.178188\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−26.2279	−	45.4281i	−0.0511266	−	0.0885538i
$514$		0			0
$515$	398.452	−	690.139i	0.773693	−	1.34008i
$516$		0			0
$517$			250.613i			0.484744i
$518$		0			0
$519$	117.905			0.227176
$520$		0			0
$521$	−739.823	−	427.137i	−1.42001	−	0.819841i	−0.423707	−	0.905799i	$-0.639271\pi$
$521$	−739.823	−	427.137i	−1.42001	−	0.819841i	−0.996299	+	0.0859587i	$0.972605\pi$
$522$		0			0
$523$	−513.554	+	296.501i	−0.981940	+	0.566923i	−0.902855	−	0.429945i	$-0.858533\pi$
$523$	−513.554	+	296.501i	−0.981940	+	0.566923i	−0.0790845	+	0.996868i	$0.525200\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−91.9005	−	159.176i	−0.174384	−	0.302043i
$528$		0			0
$529$	−119.919	+	207.706i	−0.226690	+	0.392638i
$530$		0			0
$531$		−	328.555i		−	0.618748i
$532$		0			0
$533$	427.529			0.802118
$534$		0			0
$535$	−652.422	−	376.676i	−1.21948	−	0.704068i
$536$		0			0
$537$	115.058	−	66.4290i	0.214262	−	0.123704i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−427.595	−	740.617i	−0.790380	−	1.36898i	−0.925732	−	0.378180i	$-0.876550\pi$
$541$	−427.595	−	740.617i	−0.790380	−	1.36898i	0.135352	−	0.990798i	$-0.456783\pi$
$542$		0			0
$543$	−55.6325	+	96.3583i	−0.102454	+	0.177455i
$544$		0			0
$545$		−	963.546i		−	1.76797i
$546$		0			0
$547$	−415.897			−0.760323			−0.380161	−	0.924920i	$-0.624132\pi$
$547$	−415.897			−0.760323			−0.380161	+	0.924920i	$0.624132\pi$
$548$		0			0
$549$	−665.095	−	383.993i	−1.21147	−	0.699441i
$550$		0			0
$551$	12.7279	−	7.34847i	0.0230997	−	0.0133366i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	6.99600	+	12.1174i	0.0126054	+	0.0218332i
$556$		0			0
$557$	292.110	−	505.950i	0.524435	−	0.908348i	−0.475160	−	0.879899i	$-0.657610\pi$
$557$	292.110	−	505.950i	0.524435	−	0.908348i	0.999595	−	0.0284485i	$-0.00905667\pi$
$558$		0			0
$559$		−	160.333i		−	0.286822i
$560$		0			0
$561$	−12.8377			−0.0228835
$562$		0			0
$563$	789.076	+	455.573i	1.40156	+	0.809189i	0.994552	−	0.104237i	$-0.0332402\pi$
$563$	789.076	+	455.573i	1.40156	+	0.809189i	0.407004	+	0.913426i	$0.366573\pi$
$564$		0			0
$565$	198.375	−	114.532i	0.351106	−	0.202711i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−350.000	−	606.217i	−0.615113	−	1.06541i	−0.990365	−	0.138485i	$-0.955777\pi$
$569$	−350.000	−	606.217i	−0.615113	−	1.06541i	0.375251	−	0.926923i	$-0.377556\pi$
$570$		0			0
$571$	−281.231	+	487.107i	−0.492525	+	0.853077i	−0.999963	−	0.00861055i	$-0.997259\pi$
$571$	−281.231	+	487.107i	−0.492525	+	0.853077i	0.507438	+	0.861688i	$0.330592\pi$
$572$		0			0
$573$			177.993i			0.310634i
$574$		0			0
$575$	−526.014			−0.914807
$576$		0			0
$577$	573.014	+	330.830i	0.993092	+	0.573362i	0.906197	−	0.422856i	$-0.138972\pi$
$577$	573.014	+	330.830i	0.993092	+	0.573362i	0.0868946	+	0.996218i	$0.472306\pi$
$578$		0			0
$579$	−95.8934	+	55.3641i	−0.165619	+	0.0956202i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−133.136	−	230.598i	−0.228364	−	0.395538i
$584$		0			0
$585$	−430.191	+	745.113i	−0.735369	+	1.27370i
$586$		0			0
$587$		−	823.029i		−	1.40209i	−0.713116	−	0.701046i	$-0.752717\pi$
$587$		−	823.029i		−	1.40209i	0.713116	−	0.701046i	$-0.247283\pi$
$588$		0			0
$589$	−204.338			−0.346924
$590$		0			0
$591$	−112.523	−	64.9650i	−0.190394	−	0.109924i
$592$		0			0
$593$	538.890	−	311.128i	0.908752	−	0.524668i	0.0287225	−	0.999587i	$-0.490856\pi$
$593$	538.890	−	311.128i	0.908752	−	0.524668i	0.880029	+	0.474919i	$0.157523\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	125.088	+	216.659i	0.209527	+	0.362912i
$598$		0			0
$599$	−256.422	+	444.137i	−0.428084	+	0.741463i	−0.996703	−	0.0811377i	$-0.974145\pi$
$599$	−256.422	+	444.137i	−0.428084	+	0.741463i	0.568619	+	0.822601i	$0.307478\pi$
$600$		0			0
$601$		−	680.160i		−	1.13171i	−0.824504	−	0.565857i	$-0.808546\pi$
$601$		−	680.160i		−	1.13171i	0.824504	−	0.565857i	$-0.191454\pi$
$602$		0			0
$603$	293.927			0.487441
$604$		0			0
$605$	564.889	+	326.139i	0.933701	+	0.539073i
$606$		0			0
$607$	33.5482	−	19.3690i	0.0552688	−	0.0319095i	−0.472111	−	0.881539i	$-0.656508\pi$
$607$	33.5482	−	19.3690i	0.0552688	−	0.0319095i	0.527380	+	0.849630i	$0.323175\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−402.765	−	697.609i	−0.659189	−	1.14175i
$612$		0			0
$613$	−200.552	+	347.366i	−0.327164	+	0.566665i	−0.981948	−	0.189151i	$-0.939426\pi$
$613$	−200.552	+	347.366i	−0.327164	+	0.566665i	0.654784	+	0.755816i	$0.272760\pi$
$614$		0			0
$615$			133.011i			0.216278i
$616$		0			0
$617$	−959.044			−1.55437			−0.777183	−	0.629275i	$-0.783352\pi$
$617$	−959.044			−1.55437			−0.777183	+	0.629275i	$0.783352\pi$
$618$		0			0
$619$	−869.951	−	502.267i	−1.40541	−	0.811416i	−0.410473	−	0.911873i	$-0.634636\pi$
$619$	−869.951	−	502.267i	−1.40541	−	0.811416i	−0.994941	+	0.100457i	$0.967970\pi$
$620$		0			0
$621$	−301.235	+	173.918i	−0.485081	+	0.280061i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	369.691	+	640.323i	0.591505	+	1.02452i
$626$		0			0
$627$	−7.13604	+	12.3600i	−0.0113812	+	0.0197129i
$628$		0			0
$629$			11.0624i			0.0175873i
$630$		0			0
$631$	386.514			0.612542			0.306271	−	0.951944i	$-0.400919\pi$
$631$	386.514			0.612542			0.306271	+	0.951944i	$0.400919\pi$
$632$		0			0
$633$	−226.206	−	130.600i	−0.357356	−	0.206319i
$634$		0			0
$635$	1420.37	−	820.053i	2.23681	−	1.29142i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	154.544	+	267.678i	0.241853	+	0.418902i
$640$		0			0
$641$	496.074	−	859.225i	0.773906	−	1.34044i	−0.161502	−	0.986872i	$-0.551634\pi$
$641$	496.074	−	859.225i	0.773906	−	1.34044i	0.935407	−	0.353572i	$-0.115033\pi$
$642$		0			0
$643$			944.986i			1.46965i	0.678256	+	0.734826i	$0.262736\pi$
$643$			944.986i			1.46965i	−0.678256	+	0.734826i	$0.737264\pi$
$644$		0			0
$645$	49.8823			0.0773368
$646$		0			0
$647$	2.50357	+	1.44544i	0.00386951	+	0.00223406i	0.501934	−	0.864906i	$-0.332622\pi$
$647$	2.50357	+	1.44544i	0.00386951	+	0.00223406i	−0.498064	+	0.867140i	$0.665956\pi$
$648$		0			0
$649$	−159.529	+	92.1039i	−0.245807	+	0.141917i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	161.529	+	279.777i	0.247365	+	0.428449i	0.962794	−	0.270237i	$-0.0871019\pi$
$653$	161.529	+	279.777i	0.247365	+	0.428449i	−0.715429	+	0.698686i	$0.753769\pi$
$654$		0			0
$655$	−489.518	+	847.870i	−0.747356	+	1.29446i
$656$		0			0
$657$			446.384i			0.679428i
$658$		0			0
$659$	−295.955			−0.449098			−0.224549	−	0.974463i	$-0.572091\pi$
$659$	−295.955			−0.449098			−0.224549	+	0.974463i	$0.572091\pi$
$660$		0			0
$661$	−17.9710	−	10.3756i	−0.0271876	−	0.0156968i	0.486345	−	0.873767i	$-0.338330\pi$
$661$	−17.9710	−	10.3756i	−0.0271876	−	0.0156968i	−0.513532	+	0.858070i	$0.671663\pi$
$662$		0			0
$663$	35.7351	−	20.6316i	0.0538990	−	0.0311186i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−48.7279	−	84.3992i	−0.0730554	−	0.126536i
$668$		0			0
$669$	−44.2052	+	76.5656i	−0.0660765	+	0.114448i
$670$		0			0
$671$			430.579i			0.641698i
$672$		0			0
$673$	627.044			0.931714			0.465857	−	0.884860i	$-0.345746\pi$
$673$	627.044			0.931714			0.465857	+	0.884860i	$0.345746\pi$
$674$		0			0
$675$	−206.095	−	118.989i	−0.305327	−	0.176280i
$676$		0			0
$677$	94.6097	−	54.6230i	0.139749	−	0.0806838i	−0.428496	−	0.903544i	$-0.640956\pi$
$677$	94.6097	−	54.6230i	0.139749	−	0.0806838i	0.568244	+	0.822860i	$0.307623\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−27.3974	−	47.4537i	−0.0402311	−	0.0696824i
$682$		0			0
$683$	396.783	−	687.248i	0.580941	−	1.00622i	−0.414427	−	0.910083i	$-0.636018\pi$
$683$	396.783	−	687.248i	0.580941	−	1.00622i	0.995368	−	0.0961370i	$-0.0306487\pi$
$684$		0			0
$685$		−	215.996i		−	0.315323i
$686$		0			0
$687$	−256.331			−0.373116
$688$		0			0
$689$	741.198	+	427.931i	1.07576	+	0.621090i
$690$		0			0
$691$	159.253	−	91.9447i	0.230467	−	0.133060i	−0.380320	−	0.924855i	$-0.624186\pi$
$691$	159.253	−	91.9447i	0.230467	−	0.133060i	0.610788	+	0.791794i	$0.290853\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	227.397	+	393.863i	0.327190	+	0.566710i
$696$		0			0
$697$	−52.5807	+	91.0725i	−0.0754386	+	0.130664i
$698$		0			0
$699$			195.913i			0.280277i
$700$		0			0
$701$	−1043.82			−1.48905			−0.744525	−	0.667595i	$-0.767324\pi$
$701$	−1043.82			−1.48905			−0.744525	+	0.667595i	$0.767324\pi$
$702$		0			0
$703$	10.6508	+	6.14922i	0.0151504	+	0.00874711i
$704$		0			0
$705$	217.037	−	125.306i	0.307854	−	0.177740i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	490.279	+	849.188i	0.691507	+	1.19773i	0.971344	+	0.237678i	$0.0763864\pi$
$709$	490.279	+	849.188i	0.691507	+	1.19773i	−0.279836	+	0.960048i	$0.590280\pi$
$710$		0			0
$711$	−143.345	+	248.281i	−0.201611	+	0.349200i
$712$		0			0
$713$			1354.97i			1.90038i
$714$		0			0
$715$	482.382			0.674660
$716$		0			0
$717$	164.714	+	95.0975i	0.229726	+	0.132632i
$718$		0			0
$719$	−674.187	+	389.242i	−0.937673	+	0.541366i	−0.889230	−	0.457460i	$-0.848759\pi$
$719$	−674.187	+	389.242i	−0.937673	+	0.541366i	−0.0484429	+	0.998826i	$0.515426\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	31.4376	+	54.4514i	0.0434821	+	0.0753132i
$724$		0			0
$725$	33.3381	−	57.7433i	0.0459836	−	0.0796459i
$726$		0			0
$727$		−	735.255i		−	1.01135i	−0.862723	−	0.505677i	$-0.831243\pi$
$727$		−	735.255i		−	1.01135i	0.862723	−	0.505677i	$-0.168757\pi$
$728$		0			0
$729$	490.632			0.673021
$730$		0			0
$731$	34.1543	+	19.7190i	0.0467227	+	0.0269754i
$732$		0			0
$733$	414.705	−	239.430i	0.565764	−	0.326644i	−0.189692	−	0.981844i	$-0.560749\pi$
$733$	414.705	−	239.430i	0.565764	−	0.326644i	0.755456	+	0.655200i	$0.227415\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−82.3965	−	142.715i	−0.111800	−	0.193643i
$738$		0			0
$739$	−9.95227	+	17.2378i	−0.0134672	+	0.0233259i	−0.872680	−	0.488292i	$-0.837620\pi$
$739$	−9.95227	+	17.2378i	−0.0134672	+	0.0233259i	0.859213	+	0.511618i	$0.170954\pi$
$740$		0			0
$741$		−	45.8739i		−	0.0619080i
$742$		0			0
$743$	−43.3095			−0.0582901			−0.0291450	−	0.999575i	$-0.509278\pi$
$743$	−43.3095			−0.0582901			−0.0291450	+	0.999575i	$0.509278\pi$
$744$		0			0
$745$	−530.603	−	306.344i	−0.712218	−	0.411199i
$746$		0			0
$747$	−937.499	+	541.265i	−1.25502	+	0.724585i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	112.665	+	195.142i	0.150020	+	0.259842i	0.931235	−	0.364420i	$-0.118733\pi$
$751$	112.665	+	195.142i	0.150020	+	0.259842i	−0.781215	+	0.624263i	$0.785400\pi$
$752$		0			0
$753$	177.801	−	307.961i	0.236124	−	0.408978i
$754$		0			0
$755$			608.641i			0.806147i
$756$		0			0
$757$	935.779			1.23617			0.618084	−	0.786112i	$-0.287909\pi$
$757$	935.779			1.23617			0.618084	+	0.786112i	$0.287909\pi$
$758$		0			0
$759$	81.9594	+	47.3193i	0.107983	+	0.0623443i
$760$		0			0
$761$	−1214.79	+	701.357i	−1.59630	+	0.921625i	−0.604110	+	0.796901i	$0.706471\pi$
$761$	−1214.79	+	701.357i	−1.59630	+	0.921625i	−0.992191	+	0.124724i	$0.960195\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−105.816	−	183.279i	−0.138322	−	0.239581i
$766$		0			0
$767$	296.044	−	512.763i	0.385976	−	0.668530i
$768$		0			0
$769$		−	1.72330i		−	0.00224097i	−0.999999	−	0.00112048i	$-0.999643\pi$
$769$		−	1.72330i		−	0.00224097i	0.999999	−	0.00112048i	$-0.000356661\pi$
$770$		0			0
$771$	−287.360			−0.372711
$772$		0			0
$773$	−194.213	−	112.129i	−0.251245	−	0.145057i	0.369089	−	0.929394i	$-0.379670\pi$
$773$	−194.213	−	112.129i	−0.251245	−	0.145057i	−0.620334	+	0.784337i	$0.713003\pi$
$774$		0			0
$775$	−802.831	+	463.514i	−1.03591	+	0.598083i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	58.4558	+	101.248i	0.0750396	+	0.129972i
$780$		0			0
$781$	86.6468	−	150.077i	0.110943	−	0.192160i
$782$		0			0
$783$		−	44.0908i		−	0.0563101i
$784$		0			0
$785$	−56.0740			−0.0714319
$786$		0			0
$787$	60.7979	+	35.1017i	0.0772528	+	0.0446019i	0.538129	−	0.842863i	$-0.319131\pi$
$787$	60.7979	+	35.1017i	0.0772528	+	0.0446019i	−0.460876	+	0.887465i	$0.652465\pi$
$788$		0			0
$789$	−20.0968	+	11.6029i	−0.0254712	+	0.0147058i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−691.992	−	1198.57i	−0.872625	−	1.51143i
$794$		0			0
$795$	−133.136	+	230.598i	−0.167467	+	0.290061i
$796$		0			0
$797$		−	1305.38i		−	1.63787i	−0.573889	−	0.818933i	$-0.694566\pi$
$797$		−	1305.38i		−	1.63787i	0.573889	−	0.818933i	$-0.305434\pi$
$798$		0			0
$799$	198.140			0.247985
$800$		0			0
$801$	−369.859	−	213.538i	−0.461747	−	0.266590i
$802$		0			0
$803$	216.740	−	125.135i	0.269913	−	0.155834i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	110.114	+	190.723i	0.136448	+	0.236335i
$808$		0			0
$809$	−381.382	+	660.573i	−0.471424	+	0.816531i	−0.999466	−	0.0326879i	$-0.989593\pi$
$809$	−381.382	+	660.573i	−0.471424	+	0.816531i	0.528041	+	0.849219i	$0.322927\pi$
$810$		0			0
$811$			1214.98i			1.49813i	0.662498	+	0.749064i	$0.269496\pi$
$811$			1214.98i			1.49813i	−0.662498	+	0.749064i	$0.730504\pi$
$812$		0			0
$813$	−54.5147			−0.0670538
$814$		0			0
$815$	1274.74	+	735.970i	1.56410	+	0.903031i
$816$		0			0
$817$	37.9706	−	21.9223i	0.0464756	−	0.0268327i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	291.684	+	505.211i	0.355279	+	0.615361i	0.987166	−	0.159700i	$-0.0510527\pi$
$821$	291.684	+	505.211i	0.355279	+	0.615361i	−0.631887	+	0.775061i	$0.717719\pi$
$822$		0			0
$823$	515.371	−	892.648i	0.626210	−	1.08463i	−0.362096	−	0.932141i	$-0.617939\pi$
$823$	515.371	−	892.648i	0.626210	−	1.08463i	0.988306	−	0.152486i	$-0.0487280\pi$
$824$		0			0
$825$			64.7487i			0.0784833i
$826$		0			0
$827$	−152.102			−0.183920			−0.0919599	−	0.995763i	$-0.529313\pi$
$827$	−152.102			−0.183920			−0.0919599	+	0.995763i	$0.529313\pi$
$828$		0			0
$829$	−532.095	−	307.205i	−0.641852	−	0.370573i	0.143476	−	0.989654i	$-0.454172\pi$
$829$	−532.095	−	307.205i	−0.641852	−	0.370573i	−0.785327	+	0.619081i	$0.787505\pi$
$830$		0			0
$831$	−172.983	+	99.8715i	−0.208162	+	0.120182i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	559.794	+	969.592i	0.670412	+	1.16119i
$836$		0			0
$837$	−306.507	+	530.886i	−0.366197	+	0.634272i
$838$		0			0
$839$		−	1546.14i		−	1.84284i	−0.388568	−	0.921420i	$-0.627030\pi$
$839$		−	1546.14i		−	1.84284i	0.388568	−	0.921420i	$-0.372970\pi$
$840$		0			0
$841$	−828.647			−0.985311
$842$		0			0
$843$	245.229	+	141.583i	0.290901	+	0.167952i
$844$		0			0
$845$	−372.257	+	214.923i	−0.440541	+	0.254347i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	52.5152	+	90.9589i	0.0618553	+	0.107137i
$850$		0			0
$851$	40.7756	−	70.6255i	0.0479150	−	0.0829912i
$852$		0			0
$853$			1235.15i			1.44800i	0.689798	+	0.724002i	$0.257699\pi$
$853$			1235.15i			1.44800i	−0.689798	+	0.724002i	$0.742301\pi$
$854$		0			0
$855$	−235.279			−0.275180
$856$		0			0
$857$	953.219	+	550.341i	1.11227	+	0.642172i	0.939417	−	0.342775i	$-0.111367\pi$
$857$	953.219	+	550.341i	1.11227	+	0.642172i	0.172857	+	0.984947i	$0.444700\pi$
$858$		0			0
$859$	512.488	−	295.885i	0.596610	−	0.344453i	−0.171096	−	0.985254i	$-0.554731\pi$
$859$	512.488	−	295.885i	0.596610	−	0.344453i	0.767707	+	0.640801i	$0.221398\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−32.3635	−	56.0553i	−0.0375012	−	0.0649540i	0.846666	−	0.532125i	$-0.178606\pi$
$863$	−32.3635	−	56.0553i	−0.0375012	−	0.0649540i	−0.884167	+	0.467171i	$0.845273\pi$
$864$		0			0
$865$	544.875	−	943.751i	0.629913	−	1.09104i
$866$		0			0
$867$		−	197.190i		−	0.227440i
$868$		0			0
$869$	160.736			0.184967
$870$		0			0
$871$	458.720	+	264.842i	0.526659	+	0.304067i
$872$		0			0
$873$	−748.014	+	431.866i	−0.856832	+	0.494692i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	152.096	+	263.438i	0.173427	+	0.300385i	0.939616	−	0.342231i	$-0.111182\pi$
$877$	152.096	+	263.438i	0.173427	+	0.300385i	−0.766188	+	0.642616i	$0.777849\pi$
$878$		0			0
$879$	107.500	−	186.196i	0.122299	−	0.211827i
$880$		0			0
$881$		−	863.732i		−	0.980400i	−0.871610	−	0.490200i	$-0.836924\pi$
$881$		−	863.732i		−	0.980400i	0.871610	−	0.490200i	$-0.163076\pi$
$882$		0			0
$883$	567.456			0.642645			0.321323	−	0.946970i	$-0.395873\pi$
$883$	567.456			0.642645			0.321323	+	0.946970i	$0.395873\pi$
$884$		0			0
$885$	159.529	+	92.1039i	0.180258	+	0.104072i
$886$		0			0
$887$	−770.555	+	444.880i	−0.868721	+	0.501556i	−0.866923	−	0.498442i	$-0.833906\pi$
$887$	−770.555	+	444.880i	−0.868721	+	0.501556i	−0.00179783	+	0.999998i	$0.500572\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−160.246	−	277.554i	−0.179849	−	0.311508i
$892$		0			0
$893$	110.140	−	190.767i	0.123337	−	0.213625i
$894$		0			0
$895$		−	1227.96i		−	1.37202i
$896$		0			0
$897$	−304.191			−0.339120
$898$		0			0
$899$	−148.742	−	85.8764i	−0.165453	−	0.0955243i
$900$		0			0
$901$	−182.316	+	105.260i	−0.202349	+	0.116826i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	514.191	+	890.605i	0.568167	+	0.984094i
$906$		0			0
$907$	186.989	−	323.874i	0.206162	−	0.357083i	−0.744340	−	0.667800i	$-0.767236\pi$
$907$	186.989	−	323.874i	0.206162	−	0.357083i	0.950502	+	0.310717i	$0.100569\pi$
$908$		0			0
$909$		−	506.395i		−	0.557091i
$910$		0			0
$911$	−1133.75			−1.24451			−0.622256	−	0.782814i	$-0.713784\pi$
$911$	−1133.75			−1.24451			−0.622256	+	0.782814i	$0.713784\pi$
$912$		0			0
$913$	525.618	+	303.466i	0.575704	+	0.332383i
$914$		0			0
$915$	372.893	−	215.290i	0.407533	−	0.235289i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	228.151	+	395.169i	0.248260	+	0.429999i	0.963043	−	0.269347i	$-0.0868078\pi$
$919$	228.151	+	395.169i	0.248260	+	0.429999i	−0.714783	+	0.699346i	$0.753475\pi$
$920$		0			0
$921$	7.52900	−	13.0406i	0.00817481	−	0.0141592i
$922$		0			0
$923$			557.007i			0.603474i
$924$		0			0
$925$	55.7948			0.0603187
$926$		0			0
$927$	−883.124	−	509.872i	−0.952669	−	0.550024i
$928$		0			0
$929$	824.058	−	475.770i	0.887037	−	0.512131i	0.0140650	−	0.999901i	$-0.495523\pi$
$929$	824.058	−	475.770i	0.887037	−	0.512131i	0.872972	+	0.487770i	$0.162189\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	65.3011	+	113.105i	0.0699904	+	0.121227i
$934$		0			0
$935$	−59.3269	+	102.757i	−0.0634513	+	0.109901i
$936$		0			0
$937$			1295.71i			1.38283i	0.722460	+	0.691413i	$0.243011\pi$
$937$			1295.71i			1.38283i	−0.722460	+	0.691413i	$0.756989\pi$
$938$		0			0
$939$	70.2578			0.0748219
$940$		0			0
$941$	1175.10	+	678.446i	1.24878	+	0.720984i	0.970866	−	0.239622i	$-0.0770234\pi$
$941$	1175.10	+	678.446i	1.24878	+	0.720984i	0.277915	+	0.960606i	$0.410357\pi$
$942$		0			0
$943$	671.382	−	387.622i	0.711964	−	0.411052i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−354.731	−	614.412i	−0.374584	−	0.648799i	0.615681	−	0.787996i	$-0.288881\pi$
$947$	−354.731	−	614.412i	−0.374584	−	0.648799i	−0.990265	+	0.139197i	$0.955548\pi$
$948$		0			0
$949$	−402.213	+	696.654i	−0.423828	+	0.734092i
$950$		0			0
$951$			345.784i			0.363601i
$952$		0			0
$953$	936.603			0.982794			0.491397	−	0.870936i	$-0.336486\pi$
$953$	936.603			0.982794			0.491397	+	0.870936i	$0.336486\pi$
$954$		0			0
$955$	1424.72	+	822.564i	1.49186	+	0.861324i
$956$		0			0
$957$	−10.3890	+	5.99807i	−0.0108558	+	0.00626757i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	713.477	+	1235.78i	0.742432	+	1.28593i
$962$		0			0
$963$	−482.007	+	834.861i	−0.500527	+	0.866938i
$964$		0			0
$965$			1023.42i			1.06054i
$966$		0			0
$967$	−1374.37			−1.42127			−0.710635	−	0.703561i	$-0.751592\pi$
$967$	−1374.37			−1.42127			−0.710635	+	0.703561i	$0.751592\pi$
$968$		0			0
$969$	9.77208	+	5.64191i	0.0100847	+	0.00582241i
$970$		0			0
$971$	−27.2466	+	15.7309i	−0.0280604	+	0.0162007i	−0.513965	−	0.857811i	$-0.671824\pi$
$971$	−27.2466	+	15.7309i	−0.0280604	+	0.0162007i	0.485904	+	0.874012i	$0.338490\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−104.059	−	180.235i	−0.106727	−	0.184857i
$976$		0			0
$977$	270.949	−	469.297i	0.277327	−	0.480345i	−0.693392	−	0.720560i	$-0.743885\pi$
$977$	270.949	−	469.297i	0.277327	−	0.480345i	0.970720	+	0.240215i	$0.0772181\pi$
$978$		0			0
$979$			239.445i			0.244581i
$980$		0			0
$981$	−1232.98			−1.25687
$982$		0			0
$983$	19.4161	+	11.2099i	0.0197519	+	0.0114038i	0.509843	−	0.860267i	$-0.329703\pi$
$983$	19.4161	+	11.2099i	0.0197519	+	0.0114038i	−0.490092	+	0.871671i	$0.663037\pi$
$984$		0			0
$985$	−1040.01	+	600.448i	−1.05584	+	0.609592i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−145.368	−	251.784i	−0.146984	−	0.254584i
$990$		0			0
$991$	339.017	−	587.195i	0.342096	−	0.592528i	−0.642726	−	0.766097i	$-0.722196\pi$
$991$	339.017	−	587.195i	0.342096	−	0.592528i	0.984822	+	0.173568i	$0.0555298\pi$
$992$		0			0
$993$			161.429i			0.162567i
$994$		0			0
$995$	2312.28			2.32390
$996$		0			0
$997$	−758.779	−	438.081i	−0.761062	−	0.439400i	0.0686147	−	0.997643i	$-0.478142\pi$
$997$	−758.779	−	438.081i	−0.761062	−	0.439400i	−0.829677	+	0.558244i	$0.811475\pi$
$998$		0			0
$999$	31.9523	−	18.4477i	0.0319843	−	0.0184661i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.3.s.c.129.2		4
4.3	odd	2		98.3.d.a.31.1		4
7.2	even	3		112.3.s.b.33.1		4
7.3	odd	6		784.3.c.e.97.3		4
7.4	even	3		784.3.c.e.97.2		4
7.5	odd	6	inner	784.3.s.c.705.2		4
7.6	odd	2		112.3.s.b.17.1		4
12.11	even	2		882.3.n.b.325.2		4
21.2	odd	6		1008.3.cg.l.145.2		4
21.20	even	2		1008.3.cg.l.577.2		4
28.3	even	6		98.3.b.b.97.3		4
28.11	odd	6		98.3.b.b.97.4		4
28.19	even	6		98.3.d.a.19.1		4
28.23	odd	6		14.3.d.a.5.1	yes	4
28.27	even	2		14.3.d.a.3.1	✓	4
56.13	odd	2		448.3.s.c.129.2		4
56.27	even	2		448.3.s.d.129.1		4
56.37	even	6		448.3.s.c.257.2		4
56.51	odd	6		448.3.s.d.257.1		4
84.11	even	6		882.3.c.f.685.1		4
84.23	even	6		126.3.n.c.19.2		4
84.47	odd	6		882.3.n.b.19.2		4
84.59	odd	6		882.3.c.f.685.2		4
84.83	odd	2		126.3.n.c.73.2		4
140.23	even	12		350.3.i.a.299.3		8
140.27	odd	4		350.3.i.a.199.3		8
140.79	odd	6		350.3.k.a.201.2		4
140.83	odd	4		350.3.i.a.199.2		8
140.107	even	12		350.3.i.a.299.2		8
140.139	even	2		350.3.k.a.101.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.3.d.a.3.1	✓	4	28.27	even	2
14.3.d.a.5.1	yes	4	28.23	odd	6
98.3.b.b.97.3		4	28.3	even	6
98.3.b.b.97.4		4	28.11	odd	6
98.3.d.a.19.1		4	28.19	even	6
98.3.d.a.31.1		4	4.3	odd	2
112.3.s.b.17.1		4	7.6	odd	2
112.3.s.b.33.1		4	7.2	even	3
126.3.n.c.19.2		4	84.23	even	6
126.3.n.c.73.2		4	84.83	odd	2
350.3.i.a.199.2		8	140.83	odd	4
350.3.i.a.199.3		8	140.27	odd	4
350.3.i.a.299.2		8	140.107	even	12
350.3.i.a.299.3		8	140.23	even	12
350.3.k.a.101.2		4	140.139	even	2
350.3.k.a.201.2		4	140.79	odd	6
448.3.s.c.129.2		4	56.13	odd	2
448.3.s.c.257.2		4	56.37	even	6
448.3.s.d.129.1		4	56.27	even	2
448.3.s.d.257.1		4	56.51	odd	6
784.3.c.e.97.2		4	7.4	even	3
784.3.c.e.97.3		4	7.3	odd	6
784.3.s.c.129.2		4	1.1	even	1	trivial
784.3.s.c.705.2		4	7.5	odd	6	inner
882.3.c.f.685.1		4	84.11	even	6
882.3.c.f.685.2		4	84.59	odd	6
882.3.n.b.19.2		4	84.47	odd	6
882.3.n.b.325.2		4	12.11	even	2
1008.3.cg.l.145.2		4	21.2	odd	6
1008.3.cg.l.577.2		4	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.621320 + 0.358719i) q^{3} +(5.74264 - 3.31552i) q^{5} +(-4.24264 - 7.34847i) q^{9} +(-2.37868 + 4.11999i) q^{11} -15.2913i q^{13} +4.75736 q^{15} +(3.25736 + 1.88064i) q^{17} +(3.62132 - 2.09077i) q^{19} +(-13.8640 - 24.0131i) q^{23} +(9.48528 - 16.4290i) q^{25} -12.5446i q^{27} +3.51472 q^{29} +(-42.3198 - 24.4334i) q^{31} +(-2.95584 + 1.70656i) q^{33} +(1.47056 + 2.54709i) q^{37} +(5.48528 - 9.50079i) q^{39} +27.9590i q^{41} +10.4853 q^{43} +(-48.7279 - 28.1331i) q^{45} +(45.6213 - 26.3395i) q^{47} +(1.34924 + 2.33696i) q^{51} +(-27.9853 + 48.4719i) q^{53} +31.5462i q^{55} +3.00000 q^{57} +(33.5330 + 19.3603i) q^{59} +(78.3823 - 45.2540i) q^{61} +(-50.6985 - 87.8124i) q^{65} +(-17.3198 + 29.9988i) q^{67} -19.8931i q^{69} -36.4264 q^{71} +(-45.5589 - 26.3034i) q^{73} +(11.7868 - 6.80511i) q^{75} +(-16.8934 - 29.2602i) q^{79} +(-33.6838 + 58.3420i) q^{81} -127.577i q^{83} +24.9411 q^{85} +(2.18377 + 1.26080i) q^{87} +(43.5883 - 25.1657i) q^{89} +(-17.5294 - 30.3619i) q^{93} +(13.8640 - 24.0131i) q^{95} -101.792i q^{97} +40.3675 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{3} + 6 q^{5} - 18 q^{11} + 36 q^{15} + 30 q^{17} + 6 q^{19} - 30 q^{23} + 4 q^{25} + 48 q^{29} - 42 q^{31} + 90 q^{33} - 62 q^{37} - 12 q^{39} + 8 q^{43} - 144 q^{45} + 174 q^{47} - 54 q^{51}+ \cdots - 144 q^{99}+O(q^{100}) \) 4 * q - 6 * q^3 + 6 * q^5 - 18 * q^11 + 36 * q^15 + 30 * q^17 + 6 * q^19 - 30 * q^23 + 4 * q^25 + 48 * q^29 - 42 * q^31 + 90 * q^33 - 62 * q^37 - 12 * q^39 + 8 * q^43 - 144 * q^45 + 174 * q^47 - 54 * q^51 - 78 * q^53 + 12 * q^57 - 78 * q^59 + 42 * q^61 - 84 * q^65 + 58 * q^67 + 24 * q^71 - 318 * q^73 + 132 * q^75 - 110 * q^79 + 18 * q^81 - 36 * q^85 - 144 * q^87 + 378 * q^89 - 138 * q^93 + 30 * q^95 - 144 * q^99

Introduction

L-functions

Modular forms

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