LMFDB - Embedded newform 1350.3.g.b.757.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1350,3,Mod(757,1350)] 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("1350.757"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1350, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1350.g (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,-4,0,0,0,0,-8,8,0,0,8,0,-24,0,0,-16,-68] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$36.7848356886$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			757.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1350.757
Dual form			1350.3.g.b.1243.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +(-4.44949 - 4.44949i) q^{7} +(2.00000 - 2.00000i) q^{8} -10.2474 q^{11} +(-3.55051 + 3.55051i) q^{13} +8.89898i q^{14} -4.00000 q^{16} +(-15.7753 - 15.7753i) q^{17} -19.4495i q^{19} +(10.2474 + 10.2474i) q^{22} +(15.0227 - 15.0227i) q^{23} +7.10102 q^{26} +(8.89898 - 8.89898i) q^{28} +35.7980i q^{29} +27.9444 q^{31} +(4.00000 + 4.00000i) q^{32} +31.5505i q^{34} +(38.3939 + 38.3939i) q^{37} +(-19.4495 + 19.4495i) q^{38} -26.5403 q^{41} +(-30.1918 + 30.1918i) q^{43} -20.4949i q^{44} -30.0454 q^{46} +(-10.0454 - 10.0454i) q^{47} -9.40408i q^{49} +(-7.10102 - 7.10102i) q^{52} +(-47.4166 + 47.4166i) q^{53} -17.7980 q^{56} +(35.7980 - 35.7980i) q^{58} -78.0908i q^{59} +104.778 q^{61} +(-27.9444 - 27.9444i) q^{62} -8.00000i q^{64} +(85.7423 + 85.7423i) q^{67} +(31.5505 - 31.5505i) q^{68} -62.2474 q^{71} +(-52.5505 + 52.5505i) q^{73} -76.7878i q^{74} +38.8990 q^{76} +(45.5959 + 45.5959i) q^{77} +104.641i q^{79} +(26.5403 + 26.5403i) q^{82} +(-32.2145 + 32.2145i) q^{83} +60.3837 q^{86} +(-20.4949 + 20.4949i) q^{88} -69.9546i q^{89} +31.5959 q^{91} +(30.0454 + 30.0454i) q^{92} +20.0908i q^{94} +(-79.1010 - 79.1010i) q^{97} +(-9.40408 + 9.40408i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 8 q^{7} + 8 q^{8} + 8 q^{11} - 24 q^{13} - 16 q^{16} - 68 q^{17} - 8 q^{22} + 16 q^{23} + 48 q^{26} + 16 q^{28} + 4 q^{31} + 16 q^{32} + 36 q^{37} - 68 q^{38} + 80 q^{41} + 36 q^{43} - 32 q^{46}+ \cdots - 116 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 8 * q^7 + 8 * q^8 + 8 * q^11 - 24 * q^13 - 16 * q^16 - 68 * q^17 - 8 * q^22 + 16 * q^23 + 48 * q^26 + 16 * q^28 + 4 * q^31 + 16 * q^32 + 36 * q^37 - 68 * q^38 + 80 * q^41 + 36 * q^43 - 32 * q^46 + 48 * q^47 - 48 * q^52 - 28 * q^53 - 32 * q^56 + 104 * q^58 - 12 * q^61 - 4 * q^62 + 196 * q^67 + 136 * q^68 - 200 * q^71 - 220 * q^73 + 136 * q^76 + 104 * q^77 - 80 * q^82 + 72 * q^83 - 72 * q^86 + 16 * q^88 + 48 * q^91 + 32 * q^92 - 336 * q^97 - 116 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−1.00000	−	1.00000i	−0.500000	−	0.500000i
$2$	−1.00000	−	1.00000i	−0.500000	−	0.500000i
$3$		0			0
$3$		0			0
$4$			2.00000i			0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−4.44949	−	4.44949i	−0.635641	−	0.635641i	0.313836	−	0.949477i	$-0.398386\pi$
$7$	−4.44949	−	4.44949i	−0.635641	−	0.635641i	−0.949477	+	0.313836i	$0.898386\pi$
$8$	2.00000	−	2.00000i	0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$	−10.2474			−0.931586			−0.465793	−	0.884894i	$-0.654231\pi$
$11$	−10.2474			−0.931586			−0.465793	+	0.884894i	$0.654231\pi$
$12$		0			0
$13$	−3.55051	+	3.55051i	−0.273116	+	0.273116i	−0.830353	−	0.557237i	$-0.811861\pi$
$13$	−3.55051	+	3.55051i	−0.273116	+	0.273116i	0.557237	+	0.830353i	$0.311861\pi$
$14$			8.89898i			0.635641i
$15$		0			0
$16$	−4.00000			−0.250000
$17$	−15.7753	−	15.7753i	−0.927956	−	0.927956i	0.0696176	−	0.997574i	$-0.477822\pi$
$17$	−15.7753	−	15.7753i	−0.927956	−	0.927956i	−0.997574	+	0.0696176i	$0.977822\pi$
$18$		0			0
$19$		−	19.4495i		−	1.02366i	−0.859088	−	0.511829i	$-0.828968\pi$
$19$		−	19.4495i		−	1.02366i	0.859088	−	0.511829i	$-0.171032\pi$
$20$		0			0
$21$		0			0
$22$	10.2474	+	10.2474i	0.465793	+	0.465793i
$23$	15.0227	−	15.0227i	0.653161	−	0.653161i	−0.300592	−	0.953753i	$-0.597184\pi$
$23$	15.0227	−	15.0227i	0.653161	−	0.653161i	0.953753	+	0.300592i	$0.0971842\pi$
$24$		0			0
$25$		0			0
$26$	7.10102			0.273116
$27$		0			0
$28$	8.89898	−	8.89898i	0.317821	−	0.317821i
$29$			35.7980i			1.23441i	0.786801	+	0.617206i	$0.211736\pi$
$29$			35.7980i			1.23441i	−0.786801	+	0.617206i	$0.788264\pi$
$30$		0			0
$31$	27.9444			0.901432			0.450716	−	0.892667i	$-0.351169\pi$
$31$	27.9444			0.901432			0.450716	+	0.892667i	$0.351169\pi$
$32$	4.00000	+	4.00000i	0.125000	+	0.125000i
$33$		0			0
$34$			31.5505i			0.927956i
$35$		0			0
$36$		0			0
$37$	38.3939	+	38.3939i	1.03767	+	1.03767i	0.999262	+	0.0384103i	$0.0122294\pi$
$37$	38.3939	+	38.3939i	1.03767	+	1.03767i	0.0384103	+	0.999262i	$0.487771\pi$
$38$	−19.4495	+	19.4495i	−0.511829	+	0.511829i
$39$		0			0
$40$		0			0
$41$	−26.5403			−0.647325			−0.323662	−	0.946173i	$-0.604914\pi$
$41$	−26.5403			−0.647325			−0.323662	+	0.946173i	$0.604914\pi$
$42$		0			0
$43$	−30.1918	+	30.1918i	−0.702136	+	0.702136i	−0.964869	−	0.262733i	$-0.915376\pi$
$43$	−30.1918	+	30.1918i	−0.702136	+	0.702136i	0.262733	+	0.964869i	$0.415376\pi$
$44$		−	20.4949i		−	0.465793i
$45$		0			0
$46$	−30.0454			−0.653161
$47$	−10.0454	−	10.0454i	−0.213732	−	0.213732i	0.592119	−	0.805851i	$-0.298292\pi$
$47$	−10.0454	−	10.0454i	−0.213732	−	0.213732i	−0.805851	+	0.592119i	$0.798292\pi$
$48$		0			0
$49$		−	9.40408i		−	0.191920i
$50$		0			0
$51$		0			0
$52$	−7.10102	−	7.10102i	−0.136558	−	0.136558i
$53$	−47.4166	+	47.4166i	−0.894652	+	0.894652i	−0.994957	−	0.100304i	$-0.968018\pi$
$53$	−47.4166	+	47.4166i	−0.894652	+	0.894652i	0.100304	+	0.994957i	$0.468018\pi$
$54$		0			0
$55$		0			0
$56$	−17.7980			−0.317821
$57$		0			0
$58$	35.7980	−	35.7980i	0.617206	−	0.617206i
$59$		−	78.0908i		−	1.32357i	−0.749692	−	0.661787i	$-0.769799\pi$
$59$		−	78.0908i		−	1.32357i	0.749692	−	0.661787i	$-0.230201\pi$
$60$		0			0
$61$	104.778			1.71766			0.858832	−	0.512257i	$-0.171190\pi$
$61$	104.778			1.71766			0.858832	+	0.512257i	$0.171190\pi$
$62$	−27.9444	−	27.9444i	−0.450716	−	0.450716i
$63$		0			0
$64$		−	8.00000i		−	0.125000i
$65$		0			0
$66$		0			0
$67$	85.7423	+	85.7423i	1.27974	+	1.27974i	0.940814	+	0.338922i	$0.110062\pi$
$67$	85.7423	+	85.7423i	1.27974	+	1.27974i	0.338922	+	0.940814i	$0.389938\pi$
$68$	31.5505	−	31.5505i	0.463978	−	0.463978i
$69$		0			0
$70$		0			0
$71$	−62.2474			−0.876725			−0.438362	−	0.898798i	$-0.644441\pi$
$71$	−62.2474			−0.876725			−0.438362	+	0.898798i	$0.644441\pi$
$72$		0			0
$73$	−52.5505	+	52.5505i	−0.719870	+	0.719870i	−0.968578	−	0.248708i	$-0.919994\pi$
$73$	−52.5505	+	52.5505i	−0.719870	+	0.719870i	0.248708	+	0.968578i	$0.419994\pi$
$74$		−	76.7878i		−	1.03767i
$75$		0			0
$76$	38.8990			0.511829
$77$	45.5959	+	45.5959i	0.592155	+	0.592155i
$78$		0			0
$79$			104.641i			1.32457i	0.749250	+	0.662287i	$0.230414\pi$
$79$			104.641i			1.32457i	−0.749250	+	0.662287i	$0.769586\pi$
$80$		0			0
$81$		0			0
$82$	26.5403	+	26.5403i	0.323662	+	0.323662i
$83$	−32.2145	+	32.2145i	−0.388127	+	0.388127i	−0.874019	−	0.485892i	$-0.838495\pi$
$83$	−32.2145	+	32.2145i	−0.388127	+	0.388127i	0.485892	+	0.874019i	$0.338495\pi$
$84$		0			0
$85$		0			0
$86$	60.3837			0.702136
$87$		0			0
$88$	−20.4949	+	20.4949i	−0.232897	+	0.232897i
$89$		−	69.9546i		−	0.786007i	−0.919537	−	0.393003i	$-0.871436\pi$
$89$		−	69.9546i		−	0.786007i	0.919537	−	0.393003i	$-0.128564\pi$
$90$		0			0
$91$	31.5959			0.347208
$92$	30.0454	+	30.0454i	0.326581	+	0.326581i
$93$		0			0
$94$			20.0908i			0.213732i
$95$		0			0
$96$		0			0
$97$	−79.1010	−	79.1010i	−0.815474	−	0.815474i	0.169974	−	0.985449i	$-0.445632\pi$
$97$	−79.1010	−	79.1010i	−0.815474	−	0.815474i	−0.985449	+	0.169974i	$0.945632\pi$
$98$	−9.40408	+	9.40408i	−0.0959600	+	0.0959600i
$99$		0			0
$100$		0			0
$101$	−84.8990			−0.840584			−0.420292	−	0.907389i	$-0.638072\pi$
$101$	−84.8990			−0.840584			−0.420292	+	0.907389i	$0.638072\pi$
$102$		0			0
$103$	−43.0806	+	43.0806i	−0.418258	+	0.418258i	−0.884603	−	0.466345i	$-0.845571\pi$
$103$	−43.0806	+	43.0806i	−0.418258	+	0.418258i	0.466345	+	0.884603i	$0.345571\pi$
$104$			14.2020i			0.136558i
$105$		0			0
$106$	94.8332			0.894652
$107$	53.8434	+	53.8434i	0.503209	+	0.503209i	0.912434	−	0.409225i	$-0.134201\pi$
$107$	53.8434	+	53.8434i	0.503209	+	0.503209i	−0.409225	+	0.912434i	$0.634201\pi$
$108$		0			0
$109$			104.576i			0.959408i	0.877430	+	0.479704i	$0.159256\pi$
$109$			104.576i			0.959408i	−0.877430	+	0.479704i	$0.840744\pi$
$110$		0			0
$111$		0			0
$112$	17.7980	+	17.7980i	0.158910	+	0.158910i
$113$	−105.283	+	105.283i	−0.931705	+	0.931705i	−0.997812	−	0.0661076i	$-0.978942\pi$
$113$	−105.283	+	105.283i	−0.931705	+	0.931705i	0.0661076	+	0.997812i	$0.478942\pi$
$114$		0			0
$115$		0			0
$116$	−71.5959			−0.617206
$117$		0			0
$118$	−78.0908	+	78.0908i	−0.661787	+	0.661787i
$119$			140.384i			1.17969i
$120$		0			0
$121$	−15.9898			−0.132147
$122$	−104.778	−	104.778i	−0.858832	−	0.858832i
$123$		0			0
$124$			55.8888i			0.450716i
$125$		0			0
$126$		0			0
$127$	170.126	+	170.126i	1.33957	+	1.33957i	0.896470	+	0.443105i	$0.146123\pi$
$127$	170.126	+	170.126i	1.33957	+	1.33957i	0.443105	+	0.896470i	$0.353877\pi$
$128$	−8.00000	+	8.00000i	−0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$	146.359			1.11724			0.558621	−	0.829423i	$-0.311331\pi$
$131$	146.359			1.11724			0.558621	+	0.829423i	$0.311331\pi$
$132$		0			0
$133$	−86.5403	+	86.5403i	−0.650679	+	0.650679i
$134$		−	171.485i		−	1.27974i
$135$		0			0
$136$	−63.1010			−0.463978
$137$	35.5732	+	35.5732i	0.259658	+	0.259658i	0.824915	−	0.565257i	$-0.191223\pi$
$137$	35.5732	+	35.5732i	0.259658	+	0.259658i	−0.565257	+	0.824915i	$0.691223\pi$
$138$		0			0
$139$		−	123.889i		−	0.891286i	−0.895211	−	0.445643i	$-0.852975\pi$
$139$		−	123.889i		−	0.891286i	0.895211	−	0.445643i	$-0.147025\pi$
$140$		0			0
$141$		0			0
$142$	62.2474	+	62.2474i	0.438362	+	0.438362i
$143$	36.3837	−	36.3837i	0.254431	−	0.254431i
$144$		0			0
$145$		0			0
$146$	105.101			0.719870
$147$		0			0
$148$	−76.7878	+	76.7878i	−0.518836	+	0.518836i
$149$			217.237i			1.45797i	0.684531	+	0.728984i	$0.260007\pi$
$149$			217.237i			1.45797i	−0.684531	+	0.728984i	$0.739993\pi$
$150$		0			0
$151$	−142.808			−0.945749			−0.472875	−	0.881130i	$-0.656784\pi$
$151$	−142.808			−0.945749			−0.472875	+	0.881130i	$0.656784\pi$
$152$	−38.8990	−	38.8990i	−0.255914	−	0.255914i
$153$		0			0
$154$		−	91.1918i		−	0.592155i
$155$		0			0
$156$		0			0
$157$	29.8332	+	29.8332i	0.190020	+	0.190020i	0.795705	−	0.605685i	$-0.207101\pi$
$157$	29.8332	+	29.8332i	0.190020	+	0.190020i	−0.605685	+	0.795705i	$0.707101\pi$
$158$	104.641	−	104.641i	0.662287	−	0.662287i
$159$		0			0
$160$		0			0
$161$	−133.687			−0.830352
$162$		0			0
$163$	61.3587	−	61.3587i	0.376434	−	0.376434i	−0.493380	−	0.869814i	$-0.664239\pi$
$163$	61.3587	−	61.3587i	0.376434	−	0.376434i	0.869814	+	0.493380i	$0.164239\pi$
$164$		−	53.0806i		−	0.323662i
$165$		0			0
$166$	64.4291			0.388127
$167$	136.240	+	136.240i	0.815806	+	0.815806i	0.985497	−	0.169692i	$-0.0542772\pi$
$167$	136.240	+	136.240i	0.815806	+	0.815806i	−0.169692	+	0.985497i	$0.554277\pi$
$168$		0			0
$169$			143.788i			0.850815i
$170$		0			0
$171$		0			0
$172$	−60.3837	−	60.3837i	−0.351068	−	0.351068i
$173$	149.800	−	149.800i	0.865897	−	0.865897i	−0.126118	−	0.992015i	$-0.540252\pi$
$173$	149.800	−	149.800i	0.865897	−	0.865897i	0.992015	+	0.126118i	$0.0402517\pi$
$174$		0			0
$175$		0			0
$176$	40.9898			0.232897
$177$		0			0
$178$	−69.9546	+	69.9546i	−0.393003	+	0.393003i
$179$			186.252i			1.04051i	0.854010	+	0.520257i	$0.174164\pi$
$179$			186.252i			1.04051i	−0.854010	+	0.520257i	$0.825836\pi$
$180$		0			0
$181$	258.373			1.42748			0.713739	−	0.700412i	$-0.247000\pi$
$181$	258.373			1.42748			0.713739	+	0.700412i	$0.247000\pi$
$182$	−31.5959	−	31.5959i	−0.173604	−	0.173604i
$183$		0			0
$184$		−	60.0908i		−	0.326581i
$185$		0			0
$186$		0			0
$187$	161.656	+	161.656i	0.864471	+	0.864471i
$188$	20.0908	−	20.0908i	0.106866	−	0.106866i
$189$		0			0
$190$		0			0
$191$	188.924			0.989131			0.494565	−	0.869140i	$-0.335327\pi$
$191$	188.924			0.989131			0.494565	+	0.869140i	$0.335327\pi$
$192$		0			0
$193$	181.237	−	181.237i	0.939053	−	0.939053i	−0.0591935	−	0.998247i	$-0.518853\pi$
$193$	181.237	−	181.237i	0.939053	−	0.939053i	0.998247	+	0.0591935i	$0.0188529\pi$
$194$			158.202i			0.815474i
$195$		0			0
$196$	18.8082			0.0959600
$197$	−181.775	−	181.775i	−0.922717	−	0.922717i	0.0745037	−	0.997221i	$-0.476263\pi$
$197$	−181.775	−	181.775i	−0.922717	−	0.922717i	−0.997221	+	0.0745037i	$0.976263\pi$
$198$		0			0
$199$			67.3235i			0.338309i	0.985590	+	0.169154i	$0.0541037\pi$
$199$			67.3235i			0.338309i	−0.985590	+	0.169154i	$0.945896\pi$
$200$		0			0
$201$		0			0
$202$	84.8990	+	84.8990i	0.420292	+	0.420292i
$203$	159.283	−	159.283i	0.784644	−	0.784644i
$204$		0			0
$205$		0			0
$206$	86.1612			0.418258
$207$		0			0
$208$	14.2020	−	14.2020i	0.0682790	−	0.0682790i
$209$			199.308i			0.953625i
$210$		0			0
$211$	191.520			0.907677			0.453839	−	0.891084i	$-0.350054\pi$
$211$	191.520			0.907677			0.453839	+	0.891084i	$0.350054\pi$
$212$	−94.8332	−	94.8332i	−0.447326	−	0.447326i
$213$		0			0
$214$		−	107.687i		−	0.503209i
$215$		0			0
$216$		0			0
$217$	−124.338	−	124.338i	−0.572987	−	0.572987i
$218$	104.576	−	104.576i	0.479704	−	0.479704i
$219$		0			0
$220$		0			0
$221$	112.020			0.506880
$222$		0			0
$223$	−8.36888	+	8.36888i	−0.0375286	+	0.0375286i	−0.725622	−	0.688093i	$-0.758448\pi$
$223$	−8.36888	+	8.36888i	−0.0375286	+	0.0375286i	0.688093	+	0.725622i	$0.258448\pi$
$224$		−	35.5959i		−	0.158910i
$225$		0			0
$226$	210.565			0.931705
$227$	6.91148	+	6.91148i	0.0304470	+	0.0304470i	0.722166	−	0.691719i	$-0.243146\pi$
$227$	6.91148	+	6.91148i	0.0304470	+	0.0304470i	−0.691719	+	0.722166i	$0.743146\pi$
$228$		0			0
$229$			246.242i			1.07529i	0.843171	+	0.537646i	$0.180686\pi$
$229$			246.242i			1.07529i	−0.843171	+	0.537646i	$0.819314\pi$
$230$		0			0
$231$		0			0
$232$	71.5959	+	71.5959i	0.308603	+	0.308603i
$233$	−182.586	+	182.586i	−0.783630	+	0.783630i	−0.980441	−	0.196812i	$-0.936941\pi$
$233$	−182.586	+	182.586i	−0.783630	+	0.783630i	0.196812	+	0.980441i	$0.436941\pi$
$234$		0			0
$235$		0			0
$236$	156.182			0.661787
$237$		0			0
$238$	140.384	−	140.384i	0.589847	−	0.589847i
$239$			174.409i			0.729743i	0.931058	+	0.364872i	$0.118887\pi$
$239$			174.409i			0.729743i	−0.931058	+	0.364872i	$0.881113\pi$
$240$		0			0
$241$	−190.333			−0.789762			−0.394881	−	0.918732i	$-0.629214\pi$
$241$	−190.333			−0.789762			−0.394881	+	0.918732i	$0.629214\pi$
$242$	15.9898	+	15.9898i	0.0660735	+	0.0660735i
$243$		0			0
$244$			209.555i			0.858832i
$245$		0			0
$246$		0			0
$247$	69.0556	+	69.0556i	0.279577	+	0.279577i
$248$	55.8888	−	55.8888i	0.225358	−	0.225358i
$249$		0			0
$250$		0			0
$251$	−421.707			−1.68011			−0.840054	−	0.542503i	$-0.817477\pi$
$251$	−421.707			−1.68011			−0.840054	+	0.542503i	$0.817477\pi$
$252$		0			0
$253$	−153.944	+	153.944i	−0.608476	+	0.608476i
$254$		−	340.252i		−	1.33957i
$255$		0			0
$256$	16.0000			0.0625000
$257$	88.6288	+	88.6288i	0.344859	+	0.344859i	0.858191	−	0.513331i	$-0.171589\pi$
$257$	88.6288	+	88.6288i	0.344859	+	0.344859i	−0.513331	+	0.858191i	$0.671589\pi$
$258$		0			0
$259$		−	341.666i		−	1.31917i
$260$		0			0
$261$		0			0
$262$	−146.359	−	146.359i	−0.558621	−	0.558621i
$263$	25.1214	−	25.1214i	0.0955187	−	0.0955187i	−0.657733	−	0.753251i	$-0.728484\pi$
$263$	25.1214	−	25.1214i	0.0955187	−	0.0955187i	0.753251	+	0.657733i	$0.228484\pi$
$264$		0			0
$265$		0			0
$266$	173.081			0.650679
$267$		0			0
$268$	−171.485	+	171.485i	−0.639868	+	0.639868i
$269$		−	197.621i		−	0.734650i	−0.930093	−	0.367325i	$-0.880274\pi$
$269$		−	197.621i		−	0.734650i	0.930093	−	0.367325i	$-0.119726\pi$
$270$		0			0
$271$	−158.258			−0.583977			−0.291988	−	0.956422i	$-0.594317\pi$
$271$	−158.258			−0.583977			−0.291988	+	0.956422i	$0.594317\pi$
$272$	63.1010	+	63.1010i	0.231989	+	0.231989i
$273$		0			0
$274$		−	71.1464i		−	0.259658i
$275$		0			0
$276$		0			0
$277$	−82.7321	−	82.7321i	−0.298672	−	0.298672i	0.541822	−	0.840494i	$-0.317735\pi$
$277$	−82.7321	−	82.7321i	−0.298672	−	0.298672i	−0.840494	+	0.541822i	$0.817735\pi$
$278$	−123.889	+	123.889i	−0.445643	+	0.445643i
$279$		0			0
$280$		0			0
$281$	−123.353			−0.438979			−0.219489	−	0.975615i	$-0.570439\pi$
$281$	−123.353			−0.438979			−0.219489	+	0.975615i	$0.570439\pi$
$282$		0			0
$283$	194.980	−	194.980i	0.688974	−	0.688974i	−0.273031	−	0.962005i	$-0.588026\pi$
$283$	194.980	−	194.980i	0.688974	−	0.688974i	0.962005	+	0.273031i	$0.0880263\pi$
$284$		−	124.495i		−	0.438362i
$285$		0			0
$286$	−72.7673			−0.254431
$287$	118.091	+	118.091i	0.411466	+	0.411466i
$288$		0			0
$289$			208.717i			0.722205i
$290$		0			0
$291$		0			0
$292$	−105.101	−	105.101i	−0.359935	−	0.359935i
$293$	−52.9921	+	52.9921i	−0.180860	+	0.180860i	−0.791731	−	0.610870i	$-0.790820\pi$
$293$	−52.9921	+	52.9921i	−0.180860	+	0.180860i	0.610870	+	0.791731i	$0.290820\pi$
$294$		0			0
$295$		0			0
$296$	153.576			0.518836
$297$		0			0
$298$	217.237	−	217.237i	0.728984	−	0.728984i
$299$			106.677i			0.356778i
$300$		0			0
$301$	268.677			0.892613
$302$	142.808	+	142.808i	0.472875	+	0.472875i
$303$		0			0
$304$			77.7980i			0.255914i
$305$		0			0
$306$		0			0
$307$	−186.106	−	186.106i	−0.606207	−	0.606207i	0.335746	−	0.941953i	$-0.391012\pi$
$307$	−186.106	−	186.106i	−0.606207	−	0.606207i	−0.941953	+	0.335746i	$0.891012\pi$
$308$	−91.1918	+	91.1918i	−0.296077	+	0.296077i
$309$		0			0
$310$		0			0
$311$	246.858			0.793756			0.396878	−	0.917871i	$-0.370094\pi$
$311$	246.858			0.793756			0.396878	+	0.917871i	$0.370094\pi$
$312$		0			0
$313$	210.161	−	210.161i	0.671442	−	0.671442i	−0.286607	−	0.958048i	$-0.592527\pi$
$313$	210.161	−	210.161i	0.671442	−	0.671442i	0.958048	+	0.286607i	$0.0925273\pi$
$314$		−	59.6663i		−	0.190020i
$315$		0			0
$316$	−209.283			−0.662287
$317$	244.361	+	244.361i	0.770855	+	0.770855i	0.978256	−	0.207401i	$-0.0665005\pi$
$317$	244.361	+	244.361i	0.770855	+	0.770855i	−0.207401	+	0.978256i	$0.566501\pi$
$318$		0			0
$319$		−	366.838i		−	1.14996i
$320$		0			0
$321$		0			0
$322$	133.687	+	133.687i	0.415176	+	0.415176i
$323$	−306.821	+	306.821i	−0.949909	+	0.949909i
$324$		0			0
$325$		0			0
$326$	−122.717			−0.376434
$327$		0			0
$328$	−53.0806	+	53.0806i	−0.161831	+	0.161831i
$329$			89.3939i			0.271714i
$330$		0			0
$331$	−551.464			−1.66606			−0.833028	−	0.553231i	$-0.813395\pi$
$331$	−551.464			−1.66606			−0.833028	+	0.553231i	$0.813395\pi$
$332$	−64.4291	−	64.4291i	−0.194063	−	0.194063i
$333$		0			0
$334$		−	272.479i		−	0.815806i
$335$		0			0
$336$		0			0
$337$	38.7832	+	38.7832i	0.115084	+	0.115084i	0.762303	−	0.647220i	$-0.224068\pi$
$337$	38.7832	+	38.7832i	0.115084	+	0.115084i	−0.647220	+	0.762303i	$0.724068\pi$
$338$	143.788	−	143.788i	0.425408	−	0.425408i
$339$		0			0
$340$		0			0
$341$	−286.359			−0.839762
$342$		0			0
$343$	−259.868	+	259.868i	−0.757634	+	0.757634i
$344$			120.767i			0.351068i
$345$		0			0
$346$	−299.601			−0.865897
$347$	402.722	+	402.722i	1.16058	+	1.16058i	0.984348	+	0.176234i	$0.0563914\pi$
$347$	402.722	+	402.722i	1.16058	+	1.16058i	0.176234	+	0.984348i	$0.443609\pi$
$348$		0			0
$349$		−	491.272i		−	1.40766i	−0.710370	−	0.703829i	$-0.751472\pi$
$349$		−	491.272i		−	1.40766i	0.710370	−	0.703829i	$-0.248528\pi$
$350$		0			0
$351$		0			0
$352$	−40.9898	−	40.9898i	−0.116448	−	0.116448i
$353$	−68.5857	+	68.5857i	−0.194294	+	0.194294i	−0.797549	−	0.603255i	$-0.793870\pi$
$353$	−68.5857	+	68.5857i	−0.194294	+	0.194294i	0.603255	+	0.797549i	$0.293870\pi$
$354$		0			0
$355$		0			0
$356$	139.909			0.393003
$357$		0			0
$358$	186.252	−	186.252i	0.520257	−	0.520257i
$359$		−	353.773i		−	0.985440i	−0.870188	−	0.492720i	$-0.836003\pi$
$359$		−	353.773i		−	0.985440i	0.870188	−	0.492720i	$-0.163997\pi$
$360$		0			0
$361$	−17.2827			−0.0478744
$362$	−258.373	−	258.373i	−0.713739	−	0.713739i
$363$		0			0
$364$			63.1918i			0.173604i
$365$		0			0
$366$		0			0
$367$	−240.702	−	240.702i	−0.655862	−	0.655862i	0.298536	−	0.954398i	$-0.403502\pi$
$367$	−240.702	−	240.702i	−0.655862	−	0.655862i	−0.954398	+	0.298536i	$0.903502\pi$
$368$	−60.0908	+	60.0908i	−0.163290	+	0.163290i
$369$		0			0
$370$		0			0
$371$	421.959			1.13736
$372$		0			0
$373$	−423.060	+	423.060i	−1.13421	+	1.13421i	−0.144740	+	0.989470i	$0.546235\pi$
$373$	−423.060	+	423.060i	−1.13421	+	1.13421i	−0.989470	+	0.144740i	$0.953765\pi$
$374$		−	323.312i		−	0.864471i
$375$		0			0
$376$	−40.1816			−0.106866
$377$	−127.101	−	127.101i	−0.337138	−	0.337138i
$378$		0			0
$379$		−	57.1066i		−	0.150677i	−0.997158	−	0.0753386i	$-0.975996\pi$
$379$		−	57.1066i		−	0.150677i	0.997158	−	0.0753386i	$-0.0240038\pi$
$380$		0			0
$381$		0			0
$382$	−188.924	−	188.924i	−0.494565	−	0.494565i
$383$	362.487	−	362.487i	0.946441	−	0.946441i	−0.0521957	−	0.998637i	$-0.516622\pi$
$383$	362.487	−	362.487i	0.946441	−	0.946441i	0.998637	+	0.0521957i	$0.0166219\pi$
$384$		0			0
$385$		0			0
$386$	−362.474			−0.939053
$387$		0			0
$388$	158.202	−	158.202i	0.407737	−	0.407737i
$389$		−	477.106i		−	1.22649i	−0.789892	−	0.613246i	$-0.789863\pi$
$389$		−	477.106i		−	1.22649i	0.789892	−	0.613246i	$-0.210137\pi$
$390$		0			0
$391$	−473.974			−1.21221
$392$	−18.8082	−	18.8082i	−0.0479800	−	0.0479800i
$393$		0			0
$394$			363.551i			0.922717i
$395$		0			0
$396$		0			0
$397$	−130.297	−	130.297i	−0.328205	−	0.328205i	0.523699	−	0.851904i	$-0.324552\pi$
$397$	−130.297	−	130.297i	−0.328205	−	0.328205i	−0.851904	+	0.523699i	$0.824552\pi$
$398$	67.3235	−	67.3235i	0.169154	−	0.169154i
$399$		0			0
$400$		0			0
$401$	−91.7821			−0.228883			−0.114442	−	0.993430i	$-0.536508\pi$
$401$	−91.7821			−0.228883			−0.114442	+	0.993430i	$0.536508\pi$
$402$		0			0
$403$	−99.2168	+	99.2168i	−0.246196	+	0.246196i
$404$		−	169.798i		−	0.420292i
$405$		0			0
$406$	−318.565			−0.784644
$407$	−393.439	−	393.439i	−0.966681	−	0.966681i
$408$		0			0
$409$		−	23.2520i		−	0.0568509i	−0.999596	−	0.0284255i	$-0.990951\pi$
$409$		−	23.2520i		−	0.0568509i	0.999596	−	0.0284255i	$-0.00904933\pi$
$410$		0			0
$411$		0			0
$412$	−86.1612	−	86.1612i	−0.209129	−	0.209129i
$413$	−347.464	+	347.464i	−0.841318	+	0.841318i
$414$		0			0
$415$		0			0
$416$	−28.4041			−0.0682790
$417$		0			0
$418$	199.308	−	199.308i	0.476813	−	0.476813i
$419$			675.551i			1.61229i	0.591716	+	0.806146i	$0.298451\pi$
$419$			675.551i			1.61229i	−0.591716	+	0.806146i	$0.701549\pi$
$420$		0			0
$421$	307.252			0.729815			0.364907	−	0.931044i	$-0.381101\pi$
$421$	307.252			0.729815			0.364907	+	0.931044i	$0.381101\pi$
$422$	−191.520	−	191.520i	−0.453839	−	0.453839i
$423$		0			0
$424$			189.666i			0.447326i
$425$		0			0
$426$		0			0
$427$	−466.207	−	466.207i	−1.09182	−	1.09182i
$428$	−107.687	+	107.687i	−0.251605	+	0.251605i
$429$		0			0
$430$		0			0
$431$	−680.120			−1.57801			−0.789003	−	0.614390i	$-0.789402\pi$
$431$	−680.120			−1.57801			−0.789003	+	0.614390i	$0.789402\pi$
$432$		0			0
$433$	416.479	−	416.479i	0.961845	−	0.961845i	−0.0374530	−	0.999298i	$-0.511924\pi$
$433$	416.479	−	416.479i	0.961845	−	0.961845i	0.999298	+	0.0374530i	$0.0119244\pi$
$434$			248.677i			0.572987i
$435$		0			0
$436$	−209.151			−0.479704
$437$	−292.184	−	292.184i	−0.668613	−	0.668613i
$438$		0			0
$439$			325.277i			0.740950i	0.928842	+	0.370475i	$0.120805\pi$
$439$			325.277i			0.740950i	−0.928842	+	0.370475i	$0.879195\pi$
$440$		0			0
$441$		0			0
$442$	−112.020	−	112.020i	−0.253440	−	0.253440i
$443$	−310.330	+	310.330i	−0.700520	+	0.700520i	−0.964522	−	0.264002i	$-0.914957\pi$
$443$	−310.330	+	310.330i	−0.700520	+	0.700520i	0.264002	+	0.964522i	$0.414957\pi$
$444$		0			0
$445$		0			0
$446$	16.7378			0.0375286
$447$		0			0
$448$	−35.5959	+	35.5959i	−0.0794552	+	0.0794552i
$449$			184.697i			0.411352i	0.978620	+	0.205676i	$0.0659393\pi$
$449$			184.697i			0.411352i	−0.978620	+	0.205676i	$0.934061\pi$
$450$		0			0
$451$	271.970			0.603039
$452$	−210.565	−	210.565i	−0.465852	−	0.465852i
$453$		0			0
$454$		−	13.8230i		−	0.0304470i
$455$		0			0
$456$		0			0
$457$	−520.182	−	520.182i	−1.13825	−	1.13825i	−0.988763	−	0.149490i	$-0.952237\pi$
$457$	−520.182	−	520.182i	−1.13825	−	1.13825i	−0.149490	−	0.988763i	$-0.547763\pi$
$458$	246.242	−	246.242i	0.537646	−	0.537646i
$459$		0			0
$460$		0			0
$461$	−808.286			−1.75333			−0.876666	−	0.481099i	$-0.840238\pi$
$461$	−808.286			−1.75333			−0.876666	+	0.481099i	$0.840238\pi$
$462$		0			0
$463$	38.8934	−	38.8934i	0.0840029	−	0.0840029i	−0.663857	−	0.747860i	$-0.731082\pi$
$463$	38.8934	−	38.8934i	0.0840029	−	0.0840029i	0.747860	+	0.663857i	$0.231082\pi$
$464$		−	143.192i		−	0.308603i
$465$		0			0
$466$	365.171			0.783630
$467$	−80.8411	−	80.8411i	−0.173107	−	0.173107i	0.615236	−	0.788343i	$-0.289061\pi$
$467$	−80.8411	−	80.8411i	−0.173107	−	0.173107i	−0.788343	+	0.615236i	$0.789061\pi$
$468$		0			0
$469$		−	763.019i		−	1.62691i
$470$		0			0
$471$		0			0
$472$	−156.182	−	156.182i	−0.330893	−	0.330893i
$473$	309.389	−	309.389i	0.654100	−	0.654100i
$474$		0			0
$475$		0			0
$476$	−280.767			−0.589847
$477$		0			0
$478$	174.409	−	174.409i	0.364872	−	0.364872i
$479$			311.773i			0.650883i	0.945562	+	0.325442i	$0.105513\pi$
$479$			311.773i			0.650883i	−0.945562	+	0.325442i	$0.894487\pi$
$480$		0			0
$481$	−272.636			−0.566810
$482$	190.333	+	190.333i	0.394881	+	0.394881i
$483$		0			0
$484$		−	31.9796i		−	0.0660735i
$485$		0			0
$486$		0			0
$487$	323.424	+	323.424i	0.664116	+	0.664116i	0.956348	−	0.292232i	$-0.0943978\pi$
$487$	323.424	+	323.424i	0.664116	+	0.664116i	−0.292232	+	0.956348i	$0.594398\pi$
$488$	209.555	−	209.555i	0.429416	−	0.429416i
$489$		0			0
$490$		0			0
$491$	753.914			1.53547			0.767733	−	0.640770i	$-0.221385\pi$
$491$	753.914			1.53547			0.767733	+	0.640770i	$0.221385\pi$
$492$		0			0
$493$	564.722	−	564.722i	1.14548	−	1.14548i
$494$		−	138.111i		−	0.279577i
$495$		0			0
$496$	−111.778			−0.225358
$497$	276.969	+	276.969i	0.557282	+	0.557282i
$498$		0			0
$499$			535.388i			1.07292i	0.843925	+	0.536461i	$0.180239\pi$
$499$			535.388i			1.07292i	−0.843925	+	0.536461i	$0.819761\pi$
$500$		0			0
$501$		0			0
$502$	421.707	+	421.707i	0.840054	+	0.840054i
$503$	209.704	−	209.704i	0.416906	−	0.416906i	−0.467230	−	0.884136i	$-0.654748\pi$
$503$	209.704	−	209.704i	0.416906	−	0.416906i	0.884136	+	0.467230i	$0.154748\pi$
$504$		0			0
$505$		0			0
$506$	307.889			0.608476
$507$		0			0
$508$	−340.252	+	340.252i	−0.669787	+	0.669787i
$509$			57.2122i			0.112401i	0.998419	+	0.0562006i	$0.0178986\pi$
$509$			57.2122i			0.112401i	−0.998419	+	0.0562006i	$0.982101\pi$
$510$		0			0
$511$	467.646			0.915158
$512$	−16.0000	−	16.0000i	−0.0312500	−	0.0312500i
$513$		0			0
$514$		−	177.258i		−	0.344859i
$515$		0			0
$516$		0			0
$517$	102.940	+	102.940i	0.199110	+	0.199110i
$518$	−341.666	+	341.666i	−0.659587	+	0.659587i
$519$		0			0
$520$		0			0
$521$	−344.611			−0.661441			−0.330720	−	0.943729i	$-0.607292\pi$
$521$	−344.611			−0.661441			−0.330720	+	0.943729i	$0.607292\pi$
$522$		0			0
$523$	66.7923	−	66.7923i	0.127710	−	0.127710i	−0.640363	−	0.768073i	$-0.721216\pi$
$523$	66.7923	−	66.7923i	0.127710	−	0.127710i	0.768073	+	0.640363i	$0.221216\pi$
$524$			292.717i			0.558621i
$525$		0			0
$526$	−50.2429			−0.0955187
$527$	−440.830	−	440.830i	−0.836489	−	0.836489i
$528$		0			0
$529$			77.6367i			0.146761i
$530$		0			0
$531$		0			0
$532$	−173.081	−	173.081i	−0.325339	−	0.325339i
$533$	94.2316	−	94.2316i	0.176795	−	0.176795i
$534$		0			0
$535$		0			0
$536$	342.969			0.639868
$537$		0			0
$538$	−197.621	+	197.621i	−0.367325	+	0.367325i
$539$			96.3678i			0.178790i
$540$		0			0
$541$	−943.383			−1.74378			−0.871888	−	0.489706i	$-0.837104\pi$
$541$	−943.383			−1.74378			−0.871888	+	0.489706i	$0.837104\pi$
$542$	158.258	+	158.258i	0.291988	+	0.291988i
$543$		0			0
$544$		−	126.202i		−	0.231989i
$545$		0			0
$546$		0			0
$547$	−669.929	−	669.929i	−1.22473	−	1.22473i	−0.965930	−	0.258802i	$-0.916672\pi$
$547$	−669.929	−	669.929i	−1.22473	−	1.22473i	−0.258802	−	0.965930i	$-0.583328\pi$
$548$	−71.1464	+	71.1464i	−0.129829	+	0.129829i
$549$		0			0
$550$		0			0
$551$	696.252			1.26362
$552$		0			0
$553$	465.601	−	465.601i	0.841954	−	0.841954i
$554$			165.464i			0.298672i
$555$		0			0
$556$	247.778			0.445643
$557$	128.586	+	128.586i	0.230854	+	0.230854i	0.813049	−	0.582195i	$-0.197806\pi$
$557$	128.586	+	128.586i	0.230854	+	0.230854i	−0.582195	+	0.813049i	$0.697806\pi$
$558$		0			0
$559$		−	214.393i		−	0.383529i
$560$		0			0
$561$		0			0
$562$	123.353	+	123.353i	0.219489	+	0.219489i
$563$	53.5301	−	53.5301i	0.0950801	−	0.0950801i	−0.657967	−	0.753047i	$-0.728583\pi$
$563$	53.5301	−	53.5301i	0.0950801	−	0.0950801i	0.753047	+	0.657967i	$0.228583\pi$
$564$		0			0
$565$		0			0
$566$	−389.959			−0.688974
$567$		0			0
$568$	−124.495	+	124.495i	−0.219181	+	0.219181i
$569$		−	449.959i		−	0.790789i	−0.918511	−	0.395395i	$-0.870608\pi$
$569$		−	449.959i		−	0.790789i	0.918511	−	0.395395i	$-0.129392\pi$
$570$		0			0
$571$	28.7526			0.0503547			0.0251774	−	0.999683i	$-0.491985\pi$
$571$	28.7526			0.0503547			0.0251774	+	0.999683i	$0.491985\pi$
$572$	72.7673	+	72.7673i	0.127216	+	0.127216i
$573$		0			0
$574$		−	236.182i		−	0.411466i
$575$		0			0
$576$		0			0
$577$	664.135	+	664.135i	1.15101	+	1.15101i	0.986350	+	0.164665i	$0.0526542\pi$
$577$	664.135	+	664.135i	1.15101	+	1.15101i	0.164665	+	0.986350i	$0.447346\pi$
$578$	208.717	−	208.717i	0.361103	−	0.361103i
$579$		0			0
$580$		0			0
$581$	286.677			0.493419
$582$		0			0
$583$	485.899	−	485.899i	0.833446	−	0.833446i
$584$			210.202i			0.359935i
$585$		0			0
$586$	105.984			0.180860
$587$	304.083	+	304.083i	0.518029	+	0.518029i	0.916975	−	0.398946i	$-0.130624\pi$
$587$	304.083	+	304.083i	0.518029	+	0.518029i	−0.398946	+	0.916975i	$0.630624\pi$
$588$		0			0
$589$		−	543.504i		−	0.922757i
$590$		0			0
$591$		0			0
$592$	−153.576	−	153.576i	−0.259418	−	0.259418i
$593$	−588.102	+	588.102i	−0.991741	+	0.991741i	−0.999966	−	0.00822539i	$-0.997382\pi$
$593$	−588.102	+	588.102i	−0.991741	+	0.991741i	0.00822539	+	0.999966i	$0.497382\pi$
$594$		0			0
$595$		0			0
$596$	−434.474			−0.728984
$597$		0			0
$598$	106.677	−	106.677i	0.178389	−	0.178389i
$599$			160.252i			0.267533i	0.991013	+	0.133766i	$0.0427072\pi$
$599$			160.252i			0.267533i	−0.991013	+	0.133766i	$0.957293\pi$
$600$		0			0
$601$	−1157.79			−1.92644			−0.963218	−	0.268722i	$-0.913399\pi$
$601$	−1157.79			−1.92644			−0.963218	+	0.268722i	$0.913399\pi$
$602$	−268.677	−	268.677i	−0.446307	−	0.446307i
$603$		0			0
$604$		−	285.616i		−	0.472875i
$605$		0			0
$606$		0			0
$607$	−312.303	−	312.303i	−0.514503	−	0.514503i	0.401400	−	0.915903i	$-0.368524\pi$
$607$	−312.303	−	312.303i	−0.514503	−	0.514503i	−0.915903	+	0.401400i	$0.868524\pi$
$608$	77.7980	−	77.7980i	0.127957	−	0.127957i
$609$		0			0
$610$		0			0
$611$	71.3326			0.116747
$612$		0			0
$613$	−687.959	+	687.959i	−1.12228	+	1.12228i	−0.130885	+	0.991398i	$0.541782\pi$
$613$	−687.959	+	687.959i	−1.12228	+	1.12228i	−0.991398	+	0.130885i	$0.958218\pi$
$614$			372.211i			0.606207i
$615$		0			0
$616$	182.384			0.296077
$617$	−213.421	−	213.421i	−0.345901	−	0.345901i	0.512679	−	0.858580i	$-0.328653\pi$
$617$	−213.421	−	213.421i	−0.345901	−	0.345901i	−0.858580	+	0.512679i	$0.828653\pi$
$618$		0			0
$619$		−	86.9082i		−	0.140401i	−0.997533	−	0.0702004i	$-0.977636\pi$
$619$		−	86.9082i		−	0.140401i	0.997533	−	0.0702004i	$-0.0223639\pi$
$620$		0			0
$621$		0			0
$622$	−246.858	−	246.858i	−0.396878	−	0.396878i
$623$	−311.262	+	311.262i	−0.499618	+	0.499618i
$624$		0			0
$625$		0			0
$626$	−420.322			−0.671442
$627$		0			0
$628$	−59.6663	+	59.6663i	−0.0950101	+	0.0950101i
$629$		−	1211.35i		−	1.92583i
$630$		0			0
$631$	302.298			0.479078			0.239539	−	0.970887i	$-0.423004\pi$
$631$	302.298			0.479078			0.239539	+	0.970887i	$0.423004\pi$
$632$	209.283	+	209.283i	0.331143	+	0.331143i
$633$		0			0
$634$		−	488.722i		−	0.770855i
$635$		0			0
$636$		0			0
$637$	33.3893	+	33.3893i	0.0524165	+	0.0524165i
$638$	−366.838	+	366.838i	−0.574981	+	0.574981i
$639$		0			0
$640$		0			0
$641$	−667.060			−1.04066			−0.520328	−	0.853967i	$-0.674190\pi$
$641$	−667.060			−1.04066			−0.520328	+	0.853967i	$0.674190\pi$
$642$		0			0
$643$	−246.581	+	246.581i	−0.383485	+	0.383485i	−0.872356	−	0.488871i	$-0.837409\pi$
$643$	−246.581	+	246.581i	−0.383485	+	0.383485i	0.488871	+	0.872356i	$0.337409\pi$
$644$		−	267.373i		−	0.415176i
$645$		0			0
$646$	613.641			0.949909
$647$	43.7900	+	43.7900i	0.0676817	+	0.0676817i	0.740137	−	0.672456i	$-0.234760\pi$
$647$	43.7900	+	43.7900i	0.0676817	+	0.0676817i	−0.672456	+	0.740137i	$0.734760\pi$
$648$		0			0
$649$			800.232i			1.23302i
$650$		0			0
$651$		0			0
$652$	122.717	+	122.717i	0.188217	+	0.188217i
$653$	72.4268	−	72.4268i	0.110914	−	0.110914i	−0.649472	−	0.760386i	$-0.725010\pi$
$653$	72.4268	−	72.4268i	0.110914	−	0.110914i	0.760386	+	0.649472i	$0.225010\pi$
$654$		0			0
$655$		0			0
$656$	106.161			0.161831
$657$		0			0
$658$	89.3939	−	89.3939i	0.135857	−	0.135857i
$659$		−	346.636i		−	0.526003i	−0.964795	−	0.263001i	$-0.915288\pi$
$659$		−	346.636i		−	0.526003i	0.964795	−	0.263001i	$-0.0847124\pi$
$660$		0			0
$661$	−474.152			−0.717325			−0.358663	−	0.933467i	$-0.616767\pi$
$661$	−474.152			−0.717325			−0.358663	+	0.933467i	$0.616767\pi$
$662$	551.464	+	551.464i	0.833028	+	0.833028i
$663$		0			0
$664$			128.858i			0.194063i
$665$		0			0
$666$		0			0
$667$	537.782	+	537.782i	0.806270	+	0.806270i
$668$	−272.479	+	272.479i	−0.407903	+	0.407903i
$669$		0			0
$670$		0			0
$671$	−1073.70			−1.60015
$672$		0			0
$673$	70.7332	−	70.7332i	0.105101	−	0.105101i	−0.652601	−	0.757702i	$-0.726322\pi$
$673$	70.7332	−	70.7332i	0.105101	−	0.105101i	0.757702	+	0.652601i	$0.226322\pi$
$674$		−	77.5663i		−	0.115084i
$675$		0			0
$676$	−287.576			−0.425408
$677$	−162.009	−	162.009i	−0.239305	−	0.239305i	0.577258	−	0.816562i	$-0.304123\pi$
$677$	−162.009	−	162.009i	−0.239305	−	0.239305i	−0.816562	+	0.577258i	$0.804123\pi$
$678$		0			0
$679$			703.918i			1.03670i
$680$		0			0
$681$		0			0
$682$	286.359	+	286.359i	0.419881	+	0.419881i
$683$	855.304	−	855.304i	1.25228	−	1.25228i	0.297578	−	0.954697i	$-0.403821\pi$
$683$	855.304	−	855.304i	1.25228	−	1.25228i	0.954697	−	0.297578i	$-0.0961789\pi$
$684$		0			0
$685$		0			0
$686$	519.737			0.757634
$687$		0			0
$688$	120.767	−	120.767i	0.175534	−	0.175534i
$689$		−	336.706i		−	0.488688i
$690$		0			0
$691$	122.430			0.177178			0.0885891	−	0.996068i	$-0.471764\pi$
$691$	122.430			0.177178			0.0885891	+	0.996068i	$0.471764\pi$
$692$	299.601	+	299.601i	0.432949	+	0.432949i
$693$		0			0
$694$		−	805.444i		−	1.16058i
$695$		0			0
$696$		0			0
$697$	418.680	+	418.680i	0.600689	+	0.600689i
$698$	−491.272	+	491.272i	−0.703829	+	0.703829i
$699$		0			0
$700$		0			0
$701$	1322.17			1.88613			0.943063	−	0.332613i	$-0.107930\pi$
$701$	1322.17			1.88613			0.943063	+	0.332613i	$0.107930\pi$
$702$		0			0
$703$	746.741	−	746.741i	1.06222	−	1.06222i
$704$			81.9796i			0.116448i
$705$		0			0
$706$	137.171			0.194294
$707$	377.757	+	377.757i	0.534310	+	0.534310i
$708$		0			0
$709$		−	779.909i		−	1.10001i	−0.835160	−	0.550006i	$-0.814625\pi$
$709$		−	779.909i		−	1.10001i	0.835160	−	0.550006i	$-0.185375\pi$
$710$		0			0
$711$		0			0
$712$	−139.909	−	139.909i	−0.196502	−	0.196502i
$713$	419.800	−	419.800i	0.588780	−	0.588780i
$714$		0			0
$715$		0			0
$716$	−372.504			−0.520257
$717$		0			0
$718$	−353.773	+	353.773i	−0.492720	+	0.492720i
$719$		−	572.652i		−	0.796456i	−0.917287	−	0.398228i	$-0.869625\pi$
$719$		−	572.652i		−	0.796456i	0.917287	−	0.398228i	$-0.130375\pi$
$720$		0			0
$721$	383.373			0.531725
$722$	17.2827	+	17.2827i	0.0239372	+	0.0239372i
$723$		0			0
$724$			516.747i			0.713739i
$725$		0			0
$726$		0			0
$727$	−231.748	−	231.748i	−0.318773	−	0.318773i	0.529523	−	0.848296i	$-0.322371\pi$
$727$	−231.748	−	231.748i	−0.318773	−	0.318773i	−0.848296	+	0.529523i	$0.822371\pi$
$728$	63.1918	−	63.1918i	0.0868020	−	0.0868020i
$729$		0			0
$730$		0			0
$731$	952.568			1.30310
$732$		0			0
$733$	−739.626	+	739.626i	−1.00904	+	1.00904i	−0.00908011	+	0.999959i	$0.502890\pi$
$733$	−739.626	+	739.626i	−1.00904	+	1.00904i	−0.999959	+	0.00908011i	$0.997110\pi$
$734$			481.403i			0.655862i
$735$		0			0
$736$	120.182			0.163290
$737$	−878.640	−	878.640i	−1.19218	−	1.19218i
$738$		0			0
$739$			506.539i			0.685439i	0.939438	+	0.342719i	$0.111348\pi$
$739$			506.539i			0.685439i	−0.939438	+	0.342719i	$0.888652\pi$
$740$		0			0
$741$		0			0
$742$	−421.959	−	421.959i	−0.568678	−	0.568678i
$743$	10.7219	−	10.7219i	0.0144306	−	0.0144306i	−0.699855	−	0.714285i	$-0.746752\pi$
$743$	10.7219	−	10.7219i	0.0144306	−	0.0144306i	0.714285	+	0.699855i	$0.246752\pi$
$744$		0			0
$745$		0			0
$746$	846.120			1.13421
$747$		0			0
$748$	−323.312	+	323.312i	−0.432236	+	0.432236i
$749$		−	479.151i		−	0.639721i
$750$		0			0
$751$	813.075			1.08266			0.541328	−	0.840811i	$-0.317922\pi$
$751$	813.075			1.08266			0.541328	+	0.840811i	$0.317922\pi$
$752$	40.1816	+	40.1816i	0.0534330	+	0.0534330i
$753$		0			0
$754$			254.202i			0.337138i
$755$		0			0
$756$		0			0
$757$	−132.514	−	132.514i	−0.175052	−	0.175052i	0.614143	−	0.789195i	$-0.289502\pi$
$757$	−132.514	−	132.514i	−0.175052	−	0.175052i	−0.789195	+	0.614143i	$0.789502\pi$
$758$	−57.1066	+	57.1066i	−0.0753386	+	0.0753386i
$759$		0			0
$760$		0			0
$761$	412.252			0.541724			0.270862	−	0.962618i	$-0.412691\pi$
$761$	412.252			0.541724			0.270862	+	0.962618i	$0.412691\pi$
$762$		0			0
$763$	465.308	−	465.308i	0.609840	−	0.609840i
$764$			377.848i			0.494565i
$765$		0			0
$766$	−724.974			−0.946441
$767$	277.262	+	277.262i	0.361489	+	0.361489i
$768$		0			0
$769$			867.393i			1.12795i	0.825792	+	0.563975i	$0.190728\pi$
$769$			867.393i			1.12795i	−0.825792	+	0.563975i	$0.809272\pi$
$770$		0			0
$771$		0			0
$772$	362.474	+	362.474i	0.469527	+	0.469527i
$773$	−295.578	+	295.578i	−0.382377	+	0.382377i	−0.871958	−	0.489581i	$-0.837150\pi$
$773$	−295.578	+	295.578i	−0.382377	+	0.382377i	0.489581	+	0.871958i	$0.337150\pi$
$774$		0			0
$775$		0			0
$776$	−316.404			−0.407737
$777$		0			0
$778$	−477.106	+	477.106i	−0.613246	+	0.613246i
$779$			516.195i			0.662639i
$780$		0			0
$781$	637.878			0.816745
$782$	473.974	+	473.974i	0.606105	+	0.606105i
$783$		0			0
$784$			37.6163i			0.0479800i
$785$		0			0
$786$		0			0
$787$	−179.899	−	179.899i	−0.228588	−	0.228588i	0.583514	−	0.812103i	$-0.301677\pi$
$787$	−179.899	−	179.899i	−0.228588	−	0.228588i	−0.812103	+	0.583514i	$0.801677\pi$
$788$	363.551	−	363.551i	0.461359	−	0.461359i
$789$		0			0
$790$		0			0
$791$	936.908			1.18446
$792$		0			0
$793$	−372.014	+	372.014i	−0.469122	+	0.469122i
$794$			260.595i			0.328205i
$795$		0			0
$796$	−134.647			−0.169154
$797$	40.0023	+	40.0023i	0.0501911	+	0.0501911i	0.731757	−	0.681566i	$-0.238701\pi$
$797$	40.0023	+	40.0023i	0.0501911	+	0.0501911i	−0.681566	+	0.731757i	$0.738701\pi$
$798$		0			0
$799$			316.938i			0.396668i
$800$		0			0
$801$		0			0
$802$	91.7821	+	91.7821i	0.114442	+	0.114442i
$803$	538.509	−	538.509i	0.670621	−	0.670621i
$804$		0			0
$805$		0			0
$806$	198.434			0.246196
$807$		0			0
$808$	−169.798	+	169.798i	−0.210146	+	0.210146i
$809$			135.875i			0.167954i	0.996468	+	0.0839771i	$0.0267623\pi$
$809$			135.875i			0.167954i	−0.996468	+	0.0839771i	$0.973238\pi$
$810$		0			0
$811$	1361.17			1.67838			0.839192	−	0.543835i	$-0.183028\pi$
$811$	1361.17			1.67838			0.839192	+	0.543835i	$0.183028\pi$
$812$	318.565	+	318.565i	0.392322	+	0.392322i
$813$		0			0
$814$			786.879i			0.966681i
$815$		0			0
$816$		0			0
$817$	587.216	+	587.216i	0.718746	+	0.718746i
$818$	−23.2520	+	23.2520i	−0.0284255	+	0.0284255i
$819$		0			0
$820$		0			0
$821$	−369.630			−0.450219			−0.225110	−	0.974333i	$-0.572274\pi$
$821$	−369.630			−0.450219			−0.225110	+	0.974333i	$0.572274\pi$
$822$		0			0
$823$	−752.514	+	752.514i	−0.914355	+	0.914355i	−0.996611	−	0.0822561i	$-0.973787\pi$
$823$	−752.514	+	752.514i	−0.914355	+	0.914355i	0.0822561	+	0.996611i	$0.473787\pi$
$824$			172.322i			0.209129i
$825$		0			0
$826$	694.929			0.841318
$827$	263.579	+	263.579i	0.318717	+	0.318717i	0.848274	−	0.529557i	$-0.177642\pi$
$827$	263.579	+	263.579i	0.318717	+	0.318717i	−0.529557	+	0.848274i	$0.677642\pi$
$828$		0			0
$829$			367.171i			0.442909i	0.975171	+	0.221454i	$0.0710804\pi$
$829$			367.171i			0.442909i	−0.975171	+	0.221454i	$0.928920\pi$
$830$		0			0
$831$		0			0
$832$	28.4041	+	28.4041i	0.0341395	+	0.0341395i
$833$	−148.352	+	148.352i	−0.178093	+	0.178093i
$834$		0			0
$835$		0			0
$836$	−398.615			−0.476813
$837$		0			0
$838$	675.551	−	675.551i	0.806146	−	0.806146i
$839$			601.970i			0.717486i	0.933436	+	0.358743i	$0.116794\pi$
$839$			601.970i			0.717486i	−0.933436	+	0.358743i	$0.883206\pi$
$840$		0			0
$841$	−440.494			−0.523774
$842$	−307.252	−	307.252i	−0.364907	−	0.364907i
$843$		0			0
$844$			383.040i			0.453839i
$845$		0			0
$846$		0			0
$847$	71.1464	+	71.1464i	0.0839981	+	0.0839981i
$848$	189.666	−	189.666i	0.223663	−	0.223663i
$849$		0			0
$850$		0			0
$851$	1153.56			1.35553
$852$		0			0
$853$	−370.767	+	370.767i	−0.434663	+	0.434663i	−0.890211	−	0.455548i	$-0.849443\pi$
$853$	−370.767	+	370.767i	−0.434663	+	0.434663i	0.455548	+	0.890211i	$0.349443\pi$
$854$			932.413i			1.09182i
$855$		0			0
$856$	215.373			0.251605
$857$	128.073	+	128.073i	0.149443	+	0.149443i	0.777869	−	0.628426i	$-0.216301\pi$
$857$	128.073	+	128.073i	0.149443	+	0.149443i	−0.628426	+	0.777869i	$0.716301\pi$
$858$		0			0
$859$		−	518.610i		−	0.603737i	−0.953350	−	0.301868i	$-0.902390\pi$
$859$		−	518.610i		−	0.603737i	0.953350	−	0.301868i	$-0.0976103\pi$
$860$		0			0
$861$		0			0
$862$	680.120	+	680.120i	0.789003	+	0.789003i
$863$	519.729	−	519.729i	0.602235	−	0.602235i	−0.338670	−	0.940905i	$-0.609977\pi$
$863$	519.729	−	519.729i	0.602235	−	0.602235i	0.940905	+	0.338670i	$0.109977\pi$
$864$		0			0
$865$		0			0
$866$	−832.958			−0.961845
$867$		0			0
$868$	248.677	−	248.677i	0.286494	−	0.286494i
$869$		−	1072.31i		−	1.23395i
$870$		0			0
$871$	−608.858			−0.699033
$872$	209.151	+	209.151i	0.239852	+	0.239852i
$873$		0			0
$874$			584.368i			0.668613i
$875$		0			0
$876$		0			0
$877$	339.915	+	339.915i	0.387588	+	0.387588i	0.873826	−	0.486238i	$-0.161631\pi$
$877$	339.915	+	339.915i	0.387588	+	0.387588i	−0.486238	+	0.873826i	$0.661631\pi$
$878$	325.277	−	325.277i	0.370475	−	0.370475i
$879$		0			0
$880$		0			0
$881$	−1325.62			−1.50468			−0.752340	−	0.658775i	$-0.771075\pi$
$881$	−1325.62			−1.50468			−0.752340	+	0.658775i	$0.771075\pi$
$882$		0			0
$883$	−393.358	+	393.358i	−0.445479	+	0.445479i	−0.893848	−	0.448370i	$-0.852005\pi$
$883$	−393.358	+	393.358i	−0.445479	+	0.445479i	0.448370	+	0.893848i	$0.352005\pi$
$884$			224.041i			0.253440i
$885$		0			0
$886$	620.661			0.700520
$887$	−430.936	−	430.936i	−0.485836	−	0.485836i	0.421153	−	0.906989i	$-0.361625\pi$
$887$	−430.936	−	430.936i	−0.485836	−	0.485836i	−0.906989	+	0.421153i	$0.861625\pi$
$888$		0			0
$889$		−	1513.95i		−	1.70298i
$890$		0			0
$891$		0			0
$892$	−16.7378	−	16.7378i	−0.0187643	−	0.0187643i
$893$	−195.378	+	195.378i	−0.218788	+	0.218788i
$894$		0			0
$895$		0			0
$896$	71.1918			0.0794552
$897$		0			0
$898$	184.697	−	184.697i	0.205676	−	0.205676i
$899$			1000.35i			1.11274i
$900$		0			0
$901$	1496.02			1.66040
$902$	−271.970	−	271.970i	−0.301519	−	0.301519i
$903$		0			0
$904$			421.131i			0.465852i
$905$		0			0
$906$		0			0
$907$	1199.45	+	1199.45i	1.32244	+	1.32244i	0.911796	+	0.410644i	$0.134696\pi$
$907$	1199.45	+	1199.45i	1.32244	+	1.32244i	0.410644	+	0.911796i	$0.365304\pi$
$908$	−13.8230	+	13.8230i	−0.0152235	+	0.0152235i
$909$		0			0
$910$		0			0
$911$	1159.45			1.27273			0.636363	−	0.771390i	$-0.280438\pi$
$911$	1159.45			1.27273			0.636363	+	0.771390i	$0.280438\pi$
$912$		0			0
$913$	330.117	−	330.117i	0.361574	−	0.361574i
$914$			1040.36i			1.13825i
$915$		0			0
$916$	−492.484			−0.537646
$917$	−651.221	−	651.221i	−0.710165	−	0.710165i
$918$		0			0
$919$			279.362i			0.303985i	0.988382	+	0.151993i	$0.0485690\pi$
$919$			279.362i			0.303985i	−0.988382	+	0.151993i	$0.951431\pi$
$920$		0			0
$921$		0			0
$922$	808.286	+	808.286i	0.876666	+	0.876666i
$923$	221.010	−	221.010i	0.239448	−	0.239448i
$924$		0			0
$925$		0			0
$926$	−77.7867			−0.0840029
$927$		0			0
$928$	−143.192	+	143.192i	−0.154302	+	0.154302i
$929$			1140.90i			1.22810i	0.789269	+	0.614048i	$0.210460\pi$
$929$			1140.90i			1.22810i	−0.789269	+	0.614048i	$0.789540\pi$
$930$		0			0
$931$	−182.905			−0.196460
$932$	−365.171	−	365.171i	−0.391815	−	0.391815i
$933$		0			0
$934$			161.682i			0.173107i
$935$		0			0
$936$		0			0
$937$	267.535	+	267.535i	0.285523	+	0.285523i	0.835307	−	0.549784i	$-0.185290\pi$
$937$	267.535	+	267.535i	0.285523	+	0.285523i	−0.549784	+	0.835307i	$0.685290\pi$
$938$	−763.019	+	763.019i	−0.813453	+	0.813453i
$939$		0			0
$940$		0			0
$941$	−859.065			−0.912928			−0.456464	−	0.889742i	$-0.650884\pi$
$941$	−859.065			−0.912928			−0.456464	+	0.889742i	$0.650884\pi$
$942$		0			0
$943$	−398.707	+	398.707i	−0.422807	+	0.422807i
$944$			312.363i			0.330893i
$945$		0			0
$946$	−618.779			−0.654100
$947$	−675.159	−	675.159i	−0.712945	−	0.712945i	0.254205	−	0.967150i	$-0.418186\pi$
$947$	−675.159	−	675.159i	−0.712945	−	0.712945i	−0.967150	+	0.254205i	$0.918186\pi$
$948$		0			0
$949$		−	373.162i		−	0.393216i
$950$		0			0
$951$		0			0
$952$	280.767	+	280.767i	0.294924	+	0.294924i
$953$	297.873	−	297.873i	0.312563	−	0.312563i	−0.533338	−	0.845902i	$-0.679063\pi$
$953$	297.873	−	297.873i	0.312563	−	0.312563i	0.845902	+	0.533338i	$0.179063\pi$
$954$		0			0
$955$		0			0
$956$	−348.817			−0.364872
$957$		0			0
$958$	311.773	−	311.773i	0.325442	−	0.325442i
$959$		−	316.565i		−	0.330099i
$960$		0			0
$961$	−180.111			−0.187421
$962$	272.636	+	272.636i	0.283405	+	0.283405i
$963$		0			0
$964$		−	380.665i		−	0.394881i
$965$		0			0
$966$		0			0
$967$	−140.955	−	140.955i	−0.145765	−	0.145765i	0.630458	−	0.776223i	$-0.282867\pi$
$967$	−140.955	−	140.955i	−0.145765	−	0.145765i	−0.776223	+	0.630458i	$0.782867\pi$
$968$	−31.9796	+	31.9796i	−0.0330368	+	0.0330368i
$969$		0			0
$970$		0			0
$971$	1252.45			1.28985			0.644927	−	0.764245i	$-0.276888\pi$
$971$	1252.45			1.28985			0.644927	+	0.764245i	$0.276888\pi$
$972$		0			0
$973$	−551.242	+	551.242i	−0.566538	+	0.566538i
$974$		−	646.849i		−	0.664116i
$975$		0			0
$976$	−419.110			−0.429416
$977$	−192.172	−	192.172i	−0.196696	−	0.196696i	0.601886	−	0.798582i	$-0.294416\pi$
$977$	−192.172	−	192.172i	−0.196696	−	0.196696i	−0.798582	+	0.601886i	$0.794416\pi$
$978$		0			0
$979$			716.856i			0.732233i
$980$		0			0
$981$		0			0
$982$	−753.914	−	753.914i	−0.767733	−	0.767733i
$983$	994.178	−	994.178i	1.01137	−	1.01137i	0.0114370	−	0.999935i	$-0.496359\pi$
$983$	994.178	−	994.178i	1.01137	−	1.01137i	0.999935	−	0.0114370i	$-0.00364060\pi$
$984$		0			0
$985$		0			0
$986$	−1129.44			−1.14548
$987$		0			0
$988$	−138.111	+	138.111i	−0.139789	+	0.139789i
$989$			907.126i			0.917215i
$990$		0			0
$991$	651.166			0.657080			0.328540	−	0.944490i	$-0.393444\pi$
$991$	651.166			0.657080			0.328540	+	0.944490i	$0.393444\pi$
$992$	111.778	+	111.778i	0.112679	+	0.112679i
$993$		0			0
$994$		−	553.939i		−	0.557282i
$995$		0			0
$996$		0			0
$997$	430.853	+	430.853i	0.432149	+	0.432149i	0.889359	−	0.457210i	$-0.151151\pi$
$997$	430.853	+	430.853i	0.432149	+	0.432149i	−0.457210	+	0.889359i	$0.651151\pi$
$998$	535.388	−	535.388i	0.536461	−	0.536461i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.3.g.b.757.1		4
3.2	odd	2		1350.3.g.h.757.1		4
5.2	odd	4		270.3.g.d.163.2	yes	4
5.3	odd	4	inner	1350.3.g.b.1243.1		4
5.4	even	2		270.3.g.d.217.2	yes	4
15.2	even	4		270.3.g.a.163.1	✓	4
15.8	even	4		1350.3.g.h.1243.1		4
15.14	odd	2		270.3.g.a.217.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.g.a.163.1	✓	4	15.2	even	4
270.3.g.a.217.1	yes	4	15.14	odd	2
270.3.g.d.163.2	yes	4	5.2	odd	4
270.3.g.d.217.2	yes	4	5.4	even	2
1350.3.g.b.757.1		4	1.1	even	1	trivial
1350.3.g.b.1243.1		4	5.3	odd	4	inner
1350.3.g.h.757.1		4	3.2	odd	2
1350.3.g.h.1243.1		4	15.8	even	4

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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1350.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +(-4.44949 - 4.44949i) q^{7} +(2.00000 - 2.00000i) q^{8} -10.2474 q^{11} +(-3.55051 + 3.55051i) q^{13} +8.89898i q^{14} -4.00000 q^{16} +(-15.7753 - 15.7753i) q^{17} -19.4495i q^{19} +(10.2474 + 10.2474i) q^{22} +(15.0227 - 15.0227i) q^{23} +7.10102 q^{26} +(8.89898 - 8.89898i) q^{28} +35.7980i q^{29} +27.9444 q^{31} +(4.00000 + 4.00000i) q^{32} +31.5505i q^{34} +(38.3939 + 38.3939i) q^{37} +(-19.4495 + 19.4495i) q^{38} -26.5403 q^{41} +(-30.1918 + 30.1918i) q^{43} -20.4949i q^{44} -30.0454 q^{46} +(-10.0454 - 10.0454i) q^{47} -9.40408i q^{49} +(-7.10102 - 7.10102i) q^{52} +(-47.4166 + 47.4166i) q^{53} -17.7980 q^{56} +(35.7980 - 35.7980i) q^{58} -78.0908i q^{59} +104.778 q^{61} +(-27.9444 - 27.9444i) q^{62} -8.00000i q^{64} +(85.7423 + 85.7423i) q^{67} +(31.5505 - 31.5505i) q^{68} -62.2474 q^{71} +(-52.5505 + 52.5505i) q^{73} -76.7878i q^{74} +38.8990 q^{76} +(45.5959 + 45.5959i) q^{77} +104.641i q^{79} +(26.5403 + 26.5403i) q^{82} +(-32.2145 + 32.2145i) q^{83} +60.3837 q^{86} +(-20.4949 + 20.4949i) q^{88} -69.9546i q^{89} +31.5959 q^{91} +(30.0454 + 30.0454i) q^{92} +20.0908i q^{94} +(-79.1010 - 79.1010i) q^{97} +(-9.40408 + 9.40408i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 8 q^{7} + 8 q^{8} + 8 q^{11} - 24 q^{13} - 16 q^{16} - 68 q^{17} - 8 q^{22} + 16 q^{23} + 48 q^{26} + 16 q^{28} + 4 q^{31} + 16 q^{32} + 36 q^{37} - 68 q^{38} + 80 q^{41} + 36 q^{43} - 32 q^{46}+ \cdots - 116 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 8 * q^7 + 8 * q^8 + 8 * q^11 - 24 * q^13 - 16 * q^16 - 68 * q^17 - 8 * q^22 + 16 * q^23 + 48 * q^26 + 16 * q^28 + 4 * q^31 + 16 * q^32 + 36 * q^37 - 68 * q^38 + 80 * q^41 + 36 * q^43 - 32 * q^46 + 48 * q^47 - 48 * q^52 - 28 * q^53 - 32 * q^56 + 104 * q^58 - 12 * q^61 - 4 * q^62 + 196 * q^67 + 136 * q^68 - 200 * q^71 - 220 * q^73 + 136 * q^76 + 104 * q^77 - 80 * q^82 + 72 * q^83 - 72 * q^86 + 16 * q^88 + 48 * q^91 + 32 * q^92 - 336 * q^97 - 116 * q^98

Introduction

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