LMFDB - Embedded newform 1350.3.g.d.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)

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")

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);

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,-4,0,0,0,0,0,8,0,0,0,0,0,0,0,-16,12] 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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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.d.1243.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +(-1.22474 - 1.22474i) q^{7} +(2.00000 - 2.00000i) q^{8} +7.34847 q^{11} +(-7.34847 + 7.34847i) q^{13} +2.44949i q^{14} -4.00000 q^{16} +(3.00000 + 3.00000i) q^{17} -19.0000i q^{19} +(-7.34847 - 7.34847i) q^{22} +(-24.0000 + 24.0000i) q^{23} +14.6969 q^{26} +(2.44949 - 2.44949i) q^{28} +7.34847i q^{29} +13.0000 q^{31} +(4.00000 + 4.00000i) q^{32} -6.00000i q^{34} +(-13.4722 - 13.4722i) q^{37} +(-19.0000 + 19.0000i) q^{38} -7.34847 q^{41} +(15.9217 - 15.9217i) q^{43} +14.6969i q^{44} +48.0000 q^{46} +(-27.0000 - 27.0000i) q^{47} -46.0000i q^{49} +(-14.6969 - 14.6969i) q^{52} +(-21.0000 + 21.0000i) q^{53} -4.89898 q^{56} +(7.34847 - 7.34847i) q^{58} +88.1816i q^{59} -25.0000 q^{61} +(-13.0000 - 13.0000i) q^{62} -8.00000i q^{64} +(-22.0454 - 22.0454i) q^{67} +(-6.00000 + 6.00000i) q^{68} +73.4847 q^{71} +(-64.9115 + 64.9115i) q^{73} +26.9444i q^{74} +38.0000 q^{76} +(-9.00000 - 9.00000i) q^{77} -47.0000i q^{79} +(7.34847 + 7.34847i) q^{82} +(-69.0000 + 69.0000i) q^{83} -31.8434 q^{86} +(14.6969 - 14.6969i) q^{88} +146.969i q^{89} +18.0000 q^{91} +(-48.0000 - 48.0000i) q^{92} +54.0000i q^{94} +(-94.3054 - 94.3054i) q^{97} +(-46.0000 + 46.0000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 8 q^{8} - 16 q^{16} + 12 q^{17} - 96 q^{23} + 52 q^{31} + 16 q^{32} - 76 q^{38} + 192 q^{46} - 108 q^{47} - 84 q^{53} - 100 q^{61} - 52 q^{62} - 24 q^{68} + 152 q^{76} - 36 q^{77} - 276 q^{83}+ \cdots - 184 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 8 * q^8 - 16 * q^16 + 12 * q^17 - 96 * q^23 + 52 * q^31 + 16 * q^32 - 76 * q^38 + 192 * q^46 - 108 * q^47 - 84 * q^53 - 100 * q^61 - 52 * q^62 - 24 * q^68 + 152 * q^76 - 36 * q^77 - 276 * q^83 + 72 * q^91 - 192 * q^92 - 184 * 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
$7$	−1.22474	−	1.22474i	−0.174964	−	0.174964i	0.614193	−	0.789156i	$-0.289482\pi$
$7$	−1.22474	−	1.22474i	−0.174964	−	0.174964i	−0.789156	+	0.614193i	$0.789482\pi$
$8$	2.00000	−	2.00000i	0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$	7.34847			0.668043			0.334021	−	0.942566i	$-0.391594\pi$
$11$	7.34847			0.668043			0.334021	+	0.942566i	$0.391594\pi$
$12$		0			0
$13$	−7.34847	+	7.34847i	−0.565267	+	0.565267i	−0.930799	−	0.365532i	$-0.880887\pi$
$13$	−7.34847	+	7.34847i	−0.565267	+	0.565267i	0.365532	+	0.930799i	$0.380887\pi$
$14$			2.44949i			0.174964i
$15$		0			0
$16$	−4.00000			−0.250000
$17$	3.00000	+	3.00000i	0.176471	+	0.176471i	0.789815	−	0.613345i	$-0.210176\pi$
$17$	3.00000	+	3.00000i	0.176471	+	0.176471i	−0.613345	+	0.789815i	$0.710176\pi$
$18$		0			0
$19$		−	19.0000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$19$		−	19.0000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$20$		0			0
$21$		0			0
$22$	−7.34847	−	7.34847i	−0.334021	−	0.334021i
$23$	−24.0000	+	24.0000i	−1.04348	+	1.04348i	−0.0444674	+	0.999011i	$0.514159\pi$
$23$	−24.0000	+	24.0000i	−1.04348	+	1.04348i	−0.999011	+	0.0444674i	$0.985841\pi$
$24$		0			0
$25$		0			0
$26$	14.6969			0.565267
$27$		0			0
$28$	2.44949	−	2.44949i	0.0874818	−	0.0874818i
$29$			7.34847i			0.253395i	0.991941	+	0.126698i	$0.0404378\pi$
$29$			7.34847i			0.253395i	−0.991941	+	0.126698i	$0.959562\pi$
$30$		0			0
$31$	13.0000			0.419355			0.209677	−	0.977771i	$-0.432759\pi$
$31$	13.0000			0.419355			0.209677	+	0.977771i	$0.432759\pi$
$32$	4.00000	+	4.00000i	0.125000	+	0.125000i
$33$		0			0
$34$		−	6.00000i		−	0.176471i
$35$		0			0
$36$		0			0
$37$	−13.4722	−	13.4722i	−0.364113	−	0.364113i	0.501211	−	0.865325i	$-0.332888\pi$
$37$	−13.4722	−	13.4722i	−0.364113	−	0.364113i	−0.865325	+	0.501211i	$0.832888\pi$
$38$	−19.0000	+	19.0000i	−0.500000	+	0.500000i
$39$		0			0
$40$		0			0
$41$	−7.34847			−0.179231			−0.0896155	−	0.995976i	$-0.528564\pi$
$41$	−7.34847			−0.179231			−0.0896155	+	0.995976i	$0.528564\pi$
$42$		0			0
$43$	15.9217	−	15.9217i	0.370272	−	0.370272i	−0.497304	−	0.867576i	$-0.665677\pi$
$43$	15.9217	−	15.9217i	0.370272	−	0.370272i	0.867576	+	0.497304i	$0.165677\pi$
$44$			14.6969i			0.334021i
$45$		0			0
$46$	48.0000			1.04348
$47$	−27.0000	−	27.0000i	−0.574468	−	0.574468i	0.358906	−	0.933374i	$-0.383150\pi$
$47$	−27.0000	−	27.0000i	−0.574468	−	0.574468i	−0.933374	+	0.358906i	$0.883150\pi$
$48$		0			0
$49$		−	46.0000i		−	0.938776i
$50$		0			0
$51$		0			0
$52$	−14.6969	−	14.6969i	−0.282633	−	0.282633i
$53$	−21.0000	+	21.0000i	−0.396226	+	0.396226i	−0.876900	−	0.480673i	$-0.840392\pi$
$53$	−21.0000	+	21.0000i	−0.396226	+	0.396226i	0.480673	+	0.876900i	$0.340392\pi$
$54$		0			0
$55$		0			0
$56$	−4.89898			−0.0874818
$57$		0			0
$58$	7.34847	−	7.34847i	0.126698	−	0.126698i
$59$			88.1816i			1.49460i	0.664485	+	0.747302i	$0.268651\pi$
$59$			88.1816i			1.49460i	−0.664485	+	0.747302i	$0.731349\pi$
$60$		0			0
$61$	−25.0000			−0.409836			−0.204918	−	0.978779i	$-0.565693\pi$
$61$	−25.0000			−0.409836			−0.204918	+	0.978779i	$0.565693\pi$
$62$	−13.0000	−	13.0000i	−0.209677	−	0.209677i
$63$		0			0
$64$		−	8.00000i		−	0.125000i
$65$		0			0
$66$		0			0
$67$	−22.0454	−	22.0454i	−0.329036	−	0.329036i	0.523184	−	0.852220i	$-0.324744\pi$
$67$	−22.0454	−	22.0454i	−0.329036	−	0.329036i	−0.852220	+	0.523184i	$0.824744\pi$
$68$	−6.00000	+	6.00000i	−0.0882353	+	0.0882353i
$69$		0			0
$70$		0			0
$71$	73.4847			1.03500			0.517498	−	0.855685i	$-0.326864\pi$
$71$	73.4847			1.03500			0.517498	+	0.855685i	$0.326864\pi$
$72$		0			0
$73$	−64.9115	+	64.9115i	−0.889198	+	0.889198i	−0.994446	−	0.105248i	$-0.966436\pi$
$73$	−64.9115	+	64.9115i	−0.889198	+	0.889198i	0.105248	+	0.994446i	$0.466436\pi$
$74$			26.9444i			0.364113i
$75$		0			0
$76$	38.0000			0.500000
$77$	−9.00000	−	9.00000i	−0.116883	−	0.116883i
$78$		0			0
$79$		−	47.0000i		−	0.594937i	−0.954732	−	0.297468i	$-0.903858\pi$
$79$		−	47.0000i		−	0.594937i	0.954732	−	0.297468i	$-0.0961423\pi$
$80$		0			0
$81$		0			0
$82$	7.34847	+	7.34847i	0.0896155	+	0.0896155i
$83$	−69.0000	+	69.0000i	−0.831325	+	0.831325i	−0.987698	−	0.156373i	$-0.950020\pi$
$83$	−69.0000	+	69.0000i	−0.831325	+	0.831325i	0.156373	+	0.987698i	$0.450020\pi$
$84$		0			0
$85$		0			0
$86$	−31.8434			−0.370272
$87$		0			0
$88$	14.6969	−	14.6969i	0.167011	−	0.167011i
$89$			146.969i			1.65134i	0.564152	+	0.825671i	$0.309203\pi$
$89$			146.969i			1.65134i	−0.564152	+	0.825671i	$0.690797\pi$
$90$		0			0
$91$	18.0000			0.197802
$92$	−48.0000	−	48.0000i	−0.521739	−	0.521739i
$93$		0			0
$94$			54.0000i			0.574468i
$95$		0			0
$96$		0			0
$97$	−94.3054	−	94.3054i	−0.972220	−	0.972220i	0.0274043	−	0.999624i	$-0.491276\pi$
$97$	−94.3054	−	94.3054i	−0.972220	−	0.972220i	−0.999624	+	0.0274043i	$0.991276\pi$
$98$	−46.0000	+	46.0000i	−0.469388	+	0.469388i
$99$		0			0
$100$		0			0
$101$	7.34847			0.0727571			0.0363786	−	0.999338i	$-0.488418\pi$
$101$	7.34847			0.0727571			0.0363786	+	0.999338i	$0.488418\pi$
$102$		0			0
$103$	−37.9671	+	37.9671i	−0.368613	+	0.368613i	−0.866971	−	0.498359i	$-0.833936\pi$
$103$	−37.9671	+	37.9671i	−0.368613	+	0.368613i	0.498359	+	0.866971i	$0.333936\pi$
$104$			29.3939i			0.282633i
$105$		0			0
$106$	42.0000			0.396226
$107$	−126.000	−	126.000i	−1.17757	−	1.17757i	−0.980362	−	0.197209i	$-0.936812\pi$
$107$	−126.000	−	126.000i	−1.17757	−	1.17757i	−0.197209	−	0.980362i	$-0.563188\pi$
$108$		0			0
$109$		−	53.0000i		−	0.486239i	−0.969996	−	0.243119i	$-0.921829\pi$
$109$		−	53.0000i		−	0.486239i	0.969996	−	0.243119i	$-0.0781706\pi$
$110$		0			0
$111$		0			0
$112$	4.89898	+	4.89898i	0.0437409	+	0.0437409i
$113$	−90.0000	+	90.0000i	−0.796460	+	0.796460i	−0.982535	−	0.186075i	$-0.940423\pi$
$113$	−90.0000	+	90.0000i	−0.796460	+	0.796460i	0.186075	+	0.982535i	$0.440423\pi$
$114$		0			0
$115$		0			0
$116$	−14.6969			−0.126698
$117$		0			0
$118$	88.1816	−	88.1816i	0.747302	−	0.747302i
$119$		−	7.34847i		−	0.0617518i
$120$		0			0
$121$	−67.0000			−0.553719
$122$	25.0000	+	25.0000i	0.204918	+	0.204918i
$123$		0			0
$124$			26.0000i			0.209677i
$125$		0			0
$126$		0			0
$127$	139.621	+	139.621i	1.09938	+	1.09938i	0.994483	+	0.104894i	$0.0334503\pi$
$127$	139.621	+	139.621i	1.09938	+	1.09938i	0.104894	+	0.994483i	$0.466550\pi$
$128$	−8.00000	+	8.00000i	−0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$	−154.318			−1.17800			−0.588999	−	0.808133i	$-0.700478\pi$
$131$	−154.318			−1.17800			−0.588999	+	0.808133i	$0.700478\pi$
$132$		0			0
$133$	−23.2702	+	23.2702i	−0.174964	+	0.174964i
$134$			44.0908i			0.329036i
$135$		0			0
$136$	12.0000			0.0882353
$137$	−159.000	−	159.000i	−1.16058	−	1.16058i	−0.984348	−	0.176236i	$-0.943608\pi$
$137$	−159.000	−	159.000i	−1.16058	−	1.16058i	−0.176236	−	0.984348i	$-0.556392\pi$
$138$		0			0
$139$		−	67.0000i		−	0.482014i	−0.970523	−	0.241007i	$-0.922522\pi$
$139$		−	67.0000i		−	0.482014i	0.970523	−	0.241007i	$-0.0774777\pi$
$140$		0			0
$141$		0			0
$142$	−73.4847	−	73.4847i	−0.517498	−	0.517498i
$143$	−54.0000	+	54.0000i	−0.377622	+	0.377622i
$144$		0			0
$145$		0			0
$146$	129.823			0.889198
$147$		0			0
$148$	26.9444	−	26.9444i	0.182057	−	0.182057i
$149$			257.196i			1.72615i	0.505075	+	0.863075i	$0.331465\pi$
$149$			257.196i			1.72615i	−0.505075	+	0.863075i	$0.668535\pi$
$150$		0			0
$151$	−281.000			−1.86093			−0.930464	−	0.366384i	$-0.880596\pi$
$151$	−281.000			−1.86093			−0.930464	+	0.366384i	$0.880596\pi$
$152$	−38.0000	−	38.0000i	−0.250000	−	0.250000i
$153$		0			0
$154$			18.0000i			0.116883i
$155$		0			0
$156$		0			0
$157$	109.002	+	109.002i	0.694282	+	0.694282i	0.963171	−	0.268889i	$-0.0866565\pi$
$157$	109.002	+	109.002i	0.694282	+	0.694282i	−0.268889	+	0.963171i	$0.586656\pi$
$158$	−47.0000	+	47.0000i	−0.297468	+	0.297468i
$159$		0			0
$160$		0			0
$161$	58.7878			0.365141
$162$		0			0
$163$	−139.621	+	139.621i	−0.856570	+	0.856570i	−0.990932	−	0.134362i	$-0.957101\pi$
$163$	−139.621	+	139.621i	−0.856570	+	0.856570i	0.134362	+	0.990932i	$0.457101\pi$
$164$		−	14.6969i		−	0.0896155i
$165$		0			0
$166$	138.000			0.831325
$167$	123.000	+	123.000i	0.736527	+	0.736527i	0.971904	−	0.235377i	$-0.0756325\pi$
$167$	123.000	+	123.000i	0.736527	+	0.736527i	−0.235377	+	0.971904i	$0.575633\pi$
$168$		0			0
$169$			61.0000i			0.360947i
$170$		0			0
$171$		0			0
$172$	31.8434	+	31.8434i	0.185136	+	0.185136i
$173$	−138.000	+	138.000i	−0.797688	+	0.797688i	−0.982730	−	0.185043i	$-0.940758\pi$
$173$	−138.000	+	138.000i	−0.797688	+	0.797688i	0.185043	+	0.982730i	$0.440758\pi$
$174$		0			0
$175$		0			0
$176$	−29.3939			−0.167011
$177$		0			0
$178$	146.969	−	146.969i	0.825671	−	0.825671i
$179$		−	14.6969i		−	0.0821058i	−0.999157	−	0.0410529i	$-0.986929\pi$
$179$		−	14.6969i		−	0.0821058i	0.999157	−	0.0410529i	$-0.0130712\pi$
$180$		0			0
$181$	88.0000			0.486188			0.243094	−	0.970003i	$-0.421838\pi$
$181$	88.0000			0.486188			0.243094	+	0.970003i	$0.421838\pi$
$182$	−18.0000	−	18.0000i	−0.0989011	−	0.0989011i
$183$		0			0
$184$			96.0000i			0.521739i
$185$		0			0
$186$		0			0
$187$	22.0454	+	22.0454i	0.117890	+	0.117890i
$188$	54.0000	−	54.0000i	0.287234	−	0.287234i
$189$		0			0
$190$		0			0
$191$	286.590			1.50047			0.750236	−	0.661170i	$-0.229940\pi$
$191$	286.590			1.50047			0.750236	+	0.661170i	$0.229940\pi$
$192$		0			0
$193$	60.0125	−	60.0125i	0.310946	−	0.310946i	−0.534330	−	0.845276i	$-0.679436\pi$
$193$	60.0125	−	60.0125i	0.310946	−	0.310946i	0.845276	+	0.534330i	$0.179436\pi$
$194$			188.611i			0.972220i
$195$		0			0
$196$	92.0000			0.469388
$197$	33.0000	+	33.0000i	0.167513	+	0.167513i	0.785885	−	0.618372i	$-0.212208\pi$
$197$	33.0000	+	33.0000i	0.167513	+	0.167513i	−0.618372	+	0.785885i	$0.712208\pi$
$198$		0			0
$199$			250.000i			1.25628i	0.778100	+	0.628141i	$0.216184\pi$
$199$			250.000i			1.25628i	−0.778100	+	0.628141i	$0.783816\pi$
$200$		0			0
$201$		0			0
$202$	−7.34847	−	7.34847i	−0.0363786	−	0.0363786i
$203$	9.00000	−	9.00000i	0.0443350	−	0.0443350i
$204$		0			0
$205$		0			0
$206$	75.9342			0.368613
$207$		0			0
$208$	29.3939	−	29.3939i	0.141317	−	0.141317i
$209$		−	139.621i		−	0.668043i
$210$		0			0
$211$	−190.000			−0.900474			−0.450237	−	0.892909i	$-0.648660\pi$
$211$	−190.000			−0.900474			−0.450237	+	0.892909i	$0.648660\pi$
$212$	−42.0000	−	42.0000i	−0.198113	−	0.198113i
$213$		0			0
$214$			252.000i			1.17757i
$215$		0			0
$216$		0			0
$217$	−15.9217	−	15.9217i	−0.0733718	−	0.0733718i
$218$	−53.0000	+	53.0000i	−0.243119	+	0.243119i
$219$		0			0
$220$		0			0
$221$	−44.0908			−0.199506
$222$		0			0
$223$	−23.2702	+	23.2702i	−0.104350	+	0.104350i	−0.757354	−	0.653004i	$-0.773508\pi$
$223$	−23.2702	+	23.2702i	−0.104350	+	0.104350i	0.653004	+	0.757354i	$0.273508\pi$
$224$		−	9.79796i		−	0.0437409i
$225$		0			0
$226$	180.000			0.796460
$227$	−102.000	−	102.000i	−0.449339	−	0.449339i	0.445796	−	0.895135i	$-0.352921\pi$
$227$	−102.000	−	102.000i	−0.449339	−	0.449339i	−0.895135	+	0.445796i	$0.852921\pi$
$228$		0			0
$229$		−	157.000i		−	0.685590i	−0.939410	−	0.342795i	$-0.888626\pi$
$229$		−	157.000i		−	0.685590i	0.939410	−	0.342795i	$-0.111374\pi$
$230$		0			0
$231$		0			0
$232$	14.6969	+	14.6969i	0.0633489	+	0.0633489i
$233$	−81.0000	+	81.0000i	−0.347639	+	0.347639i	−0.859230	−	0.511590i	$-0.829057\pi$
$233$	−81.0000	+	81.0000i	−0.347639	+	0.347639i	0.511590	+	0.859230i	$0.329057\pi$
$234$		0			0
$235$		0			0
$236$	−176.363			−0.747302
$237$		0			0
$238$	−7.34847	+	7.34847i	−0.0308759	+	0.0308759i
$239$			389.469i			1.62958i	0.579758	+	0.814788i	$0.303147\pi$
$239$			389.469i			1.62958i	−0.579758	+	0.814788i	$0.696853\pi$
$240$		0			0
$241$	−400.000			−1.65975			−0.829876	−	0.557949i	$-0.811589\pi$
$241$	−400.000			−1.65975			−0.829876	+	0.557949i	$0.811589\pi$
$242$	67.0000	+	67.0000i	0.276860	+	0.276860i
$243$		0			0
$244$		−	50.0000i		−	0.204918i
$245$		0			0
$246$		0			0
$247$	139.621	+	139.621i	0.565267	+	0.565267i
$248$	26.0000	−	26.0000i	0.104839	−	0.104839i
$249$		0			0
$250$		0			0
$251$	−183.712			−0.731919			−0.365960	−	0.930631i	$-0.619259\pi$
$251$	−183.712			−0.731919			−0.365960	+	0.930631i	$0.619259\pi$
$252$		0			0
$253$	−176.363	+	176.363i	−0.697088	+	0.697088i
$254$		−	279.242i		−	1.09938i
$255$		0			0
$256$	16.0000			0.0625000
$257$	255.000	+	255.000i	0.992218	+	0.992218i	0.999970	−	0.00775205i	$-0.00246758\pi$
$257$	255.000	+	255.000i	0.992218	+	0.992218i	−0.00775205	+	0.999970i	$0.502468\pi$
$258$		0			0
$259$			33.0000i			0.127413i
$260$		0			0
$261$		0			0
$262$	154.318	+	154.318i	0.588999	+	0.588999i
$263$	−9.00000	+	9.00000i	−0.0342205	+	0.0342205i	−0.724010	−	0.689789i	$-0.757703\pi$
$263$	−9.00000	+	9.00000i	−0.0342205	+	0.0342205i	0.689789	+	0.724010i	$0.257703\pi$
$264$		0			0
$265$		0			0
$266$	46.5403			0.174964
$267$		0			0
$268$	44.0908	−	44.0908i	0.164518	−	0.164518i
$269$		−	323.333i		−	1.20198i	−0.799257	−	0.600990i	$-0.794773\pi$
$269$		−	323.333i		−	1.20198i	0.799257	−	0.600990i	$-0.205227\pi$
$270$		0			0
$271$	−181.000			−0.667897			−0.333948	−	0.942591i	$-0.608381\pi$
$271$	−181.000			−0.667897			−0.333948	+	0.942591i	$0.608381\pi$
$272$	−12.0000	−	12.0000i	−0.0441176	−	0.0441176i
$273$		0			0
$274$			318.000i			1.16058i
$275$		0			0
$276$		0			0
$277$	1.22474	+	1.22474i	0.00442146	+	0.00442146i	0.709314	−	0.704893i	$-0.249005\pi$
$277$	1.22474	+	1.22474i	0.00442146	+	0.00442146i	−0.704893	+	0.709314i	$0.749005\pi$
$278$	−67.0000	+	67.0000i	−0.241007	+	0.241007i
$279$		0			0
$280$		0			0
$281$	169.015			0.601476			0.300738	−	0.953707i	$-0.402767\pi$
$281$	169.015			0.601476			0.300738	+	0.953707i	$0.402767\pi$
$282$		0			0
$283$	−74.7094	+	74.7094i	−0.263991	+	0.263991i	−0.826673	−	0.562682i	$-0.809769\pi$
$283$	−74.7094	+	74.7094i	−0.263991	+	0.263991i	0.562682	+	0.826673i	$0.309769\pi$
$284$			146.969i			0.517498i
$285$		0			0
$286$	108.000			0.377622
$287$	9.00000	+	9.00000i	0.0313589	+	0.0313589i
$288$		0			0
$289$		−	271.000i		−	0.937716i
$290$		0			0
$291$		0			0
$292$	−129.823	−	129.823i	−0.444599	−	0.444599i
$293$	330.000	−	330.000i	1.12628	−	1.12628i	0.135503	−	0.990777i	$-0.456735\pi$
$293$	330.000	−	330.000i	1.12628	−	1.12628i	0.990777	−	0.135503i	$-0.0432650\pi$
$294$		0			0
$295$		0			0
$296$	−53.8888			−0.182057
$297$		0			0
$298$	257.196	−	257.196i	0.863075	−	0.863075i
$299$		−	352.727i		−	1.17969i
$300$		0			0
$301$	−39.0000			−0.129568
$302$	281.000	+	281.000i	0.930464	+	0.930464i
$303$		0			0
$304$			76.0000i			0.250000i
$305$		0			0
$306$		0			0
$307$	64.9115	+	64.9115i	0.211438	+	0.211438i	0.804878	−	0.593440i	$-0.202231\pi$
$307$	64.9115	+	64.9115i	0.211438	+	0.211438i	−0.593440	+	0.804878i	$0.702231\pi$
$308$	18.0000	−	18.0000i	0.0584416	−	0.0584416i
$309$		0			0
$310$		0			0
$311$	−139.621			−0.448942			−0.224471	−	0.974481i	$-0.572065\pi$
$311$	−139.621			−0.448942			−0.224471	+	0.974481i	$0.572065\pi$
$312$		0			0
$313$	80.8332	−	80.8332i	0.258253	−	0.258253i	−0.566090	−	0.824343i	$-0.691545\pi$
$313$	80.8332	−	80.8332i	0.258253	−	0.258253i	0.824343	+	0.566090i	$0.191545\pi$
$314$		−	218.005i		−	0.694282i
$315$		0			0
$316$	94.0000			0.297468
$317$	−219.000	−	219.000i	−0.690852	−	0.690852i	0.271568	−	0.962419i	$-0.412458\pi$
$317$	−219.000	−	219.000i	−0.690852	−	0.690852i	−0.962419	+	0.271568i	$0.912458\pi$
$318$		0			0
$319$			54.0000i			0.169279i
$320$		0			0
$321$		0			0
$322$	−58.7878	−	58.7878i	−0.182571	−	0.182571i
$323$	57.0000	−	57.0000i	0.176471	−	0.176471i
$324$		0			0
$325$		0			0
$326$	279.242			0.856570
$327$		0			0
$328$	−14.6969	+	14.6969i	−0.0448077	+	0.0448077i
$329$			66.1362i			0.201022i
$330$		0			0
$331$	−25.0000			−0.0755287			−0.0377644	−	0.999287i	$-0.512024\pi$
$331$	−25.0000			−0.0755287			−0.0377644	+	0.999287i	$0.512024\pi$
$332$	−138.000	−	138.000i	−0.415663	−	0.415663i
$333$		0			0
$334$		−	246.000i		−	0.736527i
$335$		0			0
$336$		0			0
$337$	58.7878	+	58.7878i	0.174444	+	0.174444i	0.788929	−	0.614485i	$-0.210636\pi$
$337$	58.7878	+	58.7878i	0.174444	+	0.174444i	−0.614485	+	0.788929i	$0.710636\pi$
$338$	61.0000	−	61.0000i	0.180473	−	0.180473i
$339$		0			0
$340$		0			0
$341$	95.5301			0.280147
$342$		0			0
$343$	−116.351	+	116.351i	−0.339215	+	0.339215i
$344$		−	63.6867i		−	0.185136i
$345$		0			0
$346$	276.000			0.797688
$347$	342.000	+	342.000i	0.985591	+	0.985591i	0.999898	−	0.0143069i	$-0.00455417\pi$
$347$	342.000	+	342.000i	0.985591	+	0.985591i	−0.0143069	+	0.999898i	$0.504554\pi$
$348$		0			0
$349$		−	403.000i		−	1.15473i	−0.816487	−	0.577364i	$-0.804081\pi$
$349$		−	403.000i		−	1.15473i	0.816487	−	0.577364i	$-0.195919\pi$
$350$		0			0
$351$		0			0
$352$	29.3939	+	29.3939i	0.0835053	+	0.0835053i
$353$	306.000	−	306.000i	0.866856	−	0.866856i	−0.125267	−	0.992123i	$-0.539979\pi$
$353$	306.000	−	306.000i	0.866856	−	0.866856i	0.992123	+	0.125267i	$0.0399789\pi$
$354$		0			0
$355$		0			0
$356$	−293.939			−0.825671
$357$		0			0
$358$	−14.6969	+	14.6969i	−0.0410529	+	0.0410529i
$359$		−	66.1362i		−	0.184223i	−0.995749	−	0.0921117i	$-0.970638\pi$
$359$		−	66.1362i		−	0.184223i	0.995749	−	0.0921117i	$-0.0293617\pi$
$360$		0			0
$361$		0			0
$362$	−88.0000	−	88.0000i	−0.243094	−	0.243094i
$363$		0			0
$364$			36.0000i			0.0989011i
$365$		0			0
$366$		0			0
$367$	110.227	+	110.227i	0.300346	+	0.300346i	0.841149	−	0.540803i	$-0.181880\pi$
$367$	110.227	+	110.227i	0.300346	+	0.300346i	−0.540803	+	0.841149i	$0.681880\pi$
$368$	96.0000	−	96.0000i	0.260870	−	0.260870i
$369$		0			0
$370$		0			0
$371$	51.4393			0.138650
$372$		0			0
$373$	−111.452	+	111.452i	−0.298798	+	0.298798i	−0.840543	−	0.541745i	$-0.817764\pi$
$373$	−111.452	+	111.452i	−0.298798	+	0.298798i	0.541745	+	0.840543i	$0.317764\pi$
$374$		−	44.0908i		−	0.117890i
$375$		0			0
$376$	−108.000			−0.287234
$377$	−54.0000	−	54.0000i	−0.143236	−	0.143236i
$378$		0			0
$379$			416.000i			1.09763i	0.835945	+	0.548813i	$0.184920\pi$
$379$			416.000i			1.09763i	−0.835945	+	0.548813i	$0.815080\pi$
$380$		0			0
$381$		0			0
$382$	−286.590	−	286.590i	−0.750236	−	0.750236i
$383$	321.000	−	321.000i	0.838120	−	0.838120i	−0.150491	−	0.988611i	$-0.548086\pi$
$383$	321.000	−	321.000i	0.838120	−	0.838120i	0.988611	+	0.150491i	$0.0480855\pi$
$384$		0			0
$385$		0			0
$386$	−120.025			−0.310946
$387$		0			0
$388$	188.611	−	188.611i	0.486110	−	0.486110i
$389$		−	756.892i		−	1.94574i	−0.231355	−	0.972869i	$-0.574316\pi$
$389$		−	756.892i		−	1.94574i	0.231355	−	0.972869i	$-0.425684\pi$
$390$		0			0
$391$	−144.000			−0.368286
$392$	−92.0000	−	92.0000i	−0.234694	−	0.234694i
$393$		0			0
$394$		−	66.0000i		−	0.167513i
$395$		0			0
$396$		0			0
$397$	−336.805	−	336.805i	−0.848375	−	0.848375i	0.141555	−	0.989930i	$-0.454790\pi$
$397$	−336.805	−	336.805i	−0.848375	−	0.848375i	−0.989930	+	0.141555i	$0.954790\pi$
$398$	250.000	−	250.000i	0.628141	−	0.628141i
$399$		0			0
$400$		0			0
$401$	−558.484			−1.39273			−0.696364	−	0.717689i	$-0.745200\pi$
$401$	−558.484			−1.39273			−0.696364	+	0.717689i	$0.745200\pi$
$402$		0			0
$403$	−95.5301	+	95.5301i	−0.237047	+	0.237047i
$404$			14.6969i			0.0363786i
$405$		0			0
$406$	−18.0000			−0.0443350
$407$	−99.0000	−	99.0000i	−0.243243	−	0.243243i
$408$		0			0
$409$		−	550.000i		−	1.34474i	−0.740214	−	0.672372i	$-0.765276\pi$
$409$		−	550.000i		−	1.34474i	0.740214	−	0.672372i	$-0.234724\pi$
$410$		0			0
$411$		0			0
$412$	−75.9342	−	75.9342i	−0.184306	−	0.184306i
$413$	108.000	−	108.000i	0.261501	−	0.261501i
$414$		0			0
$415$		0			0
$416$	−58.7878			−0.141317
$417$		0			0
$418$	−139.621	+	139.621i	−0.334021	+	0.334021i
$419$			58.7878i			0.140305i	0.997536	+	0.0701525i	$0.0223486\pi$
$419$			58.7878i			0.140305i	−0.997536	+	0.0701525i	$0.977651\pi$
$420$		0			0
$421$	583.000			1.38480			0.692399	−	0.721515i	$-0.256554\pi$
$421$	583.000			1.38480			0.692399	+	0.721515i	$0.256554\pi$
$422$	190.000	+	190.000i	0.450237	+	0.450237i
$423$		0			0
$424$			84.0000i			0.198113i
$425$		0			0
$426$		0			0
$427$	30.6186	+	30.6186i	0.0717064	+	0.0717064i
$428$	252.000	−	252.000i	0.588785	−	0.588785i
$429$		0			0
$430$		0			0
$431$	−183.712			−0.426245			−0.213123	−	0.977025i	$-0.568363\pi$
$431$	−183.712			−0.426245			−0.213123	+	0.977025i	$0.568363\pi$
$432$		0			0
$433$	−515.618	+	515.618i	−1.19080	+	1.19080i	−0.213960	+	0.976842i	$0.568636\pi$
$433$	−515.618	+	515.618i	−1.19080	+	1.19080i	−0.976842	+	0.213960i	$0.931364\pi$
$434$			31.8434i			0.0733718i
$435$		0			0
$436$	106.000			0.243119
$437$	456.000	+	456.000i	1.04348	+	1.04348i
$438$		0			0
$439$		−	29.0000i		−	0.0660592i	−0.999454	−	0.0330296i	$-0.989484\pi$
$439$		−	29.0000i		−	0.0660592i	0.999454	−	0.0330296i	$-0.0105156\pi$
$440$		0			0
$441$		0			0
$442$	44.0908	+	44.0908i	0.0997530	+	0.0997530i
$443$	60.0000	−	60.0000i	0.135440	−	0.135440i	−0.636136	−	0.771577i	$-0.719468\pi$
$443$	60.0000	−	60.0000i	0.135440	−	0.135440i	0.771577	+	0.636136i	$0.219468\pi$
$444$		0			0
$445$		0			0
$446$	46.5403			0.104350
$447$		0			0
$448$	−9.79796	+	9.79796i	−0.0218704	+	0.0218704i
$449$		−	521.741i		−	1.16201i	−0.813901	−	0.581004i	$-0.802660\pi$
$449$		−	521.741i		−	1.16201i	0.813901	−	0.581004i	$-0.197340\pi$
$450$		0			0
$451$	−54.0000			−0.119734
$452$	−180.000	−	180.000i	−0.398230	−	0.398230i
$453$		0			0
$454$			204.000i			0.449339i
$455$		0			0
$456$		0			0
$457$	−448.257	−	448.257i	−0.980868	−	0.980868i	0.0189525	−	0.999820i	$-0.493967\pi$
$457$	−448.257	−	448.257i	−0.980868	−	0.980868i	−0.999820	+	0.0189525i	$0.993967\pi$
$458$	−157.000	+	157.000i	−0.342795	+	0.342795i
$459$		0			0
$460$		0			0
$461$	−146.969			−0.318806			−0.159403	−	0.987214i	$-0.550957\pi$
$461$	−146.969			−0.318806			−0.159403	+	0.987214i	$0.550957\pi$
$462$		0			0
$463$	603.799	−	603.799i	1.30410	−	1.30410i	0.378501	−	0.925601i	$-0.376440\pi$
$463$	603.799	−	603.799i	1.30410	−	1.30410i	0.925601	−	0.378501i	$-0.123560\pi$
$464$		−	29.3939i		−	0.0633489i
$465$		0			0
$466$	162.000			0.347639
$467$	330.000	+	330.000i	0.706638	+	0.706638i	0.965827	−	0.259189i	$-0.0834551\pi$
$467$	330.000	+	330.000i	0.706638	+	0.706638i	−0.259189	+	0.965827i	$0.583455\pi$
$468$		0			0
$469$			54.0000i			0.115139i
$470$		0			0
$471$		0			0
$472$	176.363	+	176.363i	0.373651	+	0.373651i
$473$	117.000	−	117.000i	0.247357	−	0.247357i
$474$		0			0
$475$		0			0
$476$	14.6969			0.0308759
$477$		0			0
$478$	389.469	−	389.469i	0.814788	−	0.814788i
$479$		−	881.816i		−	1.84095i	−0.390798	−	0.920476i	$-0.627801\pi$
$479$		−	881.816i		−	1.84095i	0.390798	−	0.920476i	$-0.372199\pi$
$480$		0			0
$481$	198.000			0.411642
$482$	400.000	+	400.000i	0.829876	+	0.829876i
$483$		0			0
$484$		−	134.000i		−	0.276860i
$485$		0			0
$486$		0			0
$487$	−205.757	−	205.757i	−0.422499	−	0.422499i	0.463564	−	0.886063i	$-0.346570\pi$
$487$	−205.757	−	205.757i	−0.422499	−	0.422499i	−0.886063	+	0.463564i	$0.846570\pi$
$488$	−50.0000	+	50.0000i	−0.102459	+	0.102459i
$489$		0			0
$490$		0			0
$491$	−169.015			−0.344226			−0.172113	−	0.985077i	$-0.555059\pi$
$491$	−169.015			−0.344226			−0.172113	+	0.985077i	$0.555059\pi$
$492$		0			0
$493$	−22.0454	+	22.0454i	−0.0447169	+	0.0447169i
$494$		−	279.242i		−	0.565267i
$495$		0			0
$496$	−52.0000			−0.104839
$497$	−90.0000	−	90.0000i	−0.181087	−	0.181087i
$498$		0			0
$499$		−	293.000i		−	0.587174i	−0.955932	−	0.293587i	$-0.905151\pi$
$499$		−	293.000i		−	0.587174i	0.955932	−	0.293587i	$-0.0948491\pi$
$500$		0			0
$501$		0			0
$502$	183.712	+	183.712i	0.365960	+	0.365960i
$503$	453.000	−	453.000i	0.900596	−	0.900596i	−0.0948912	−	0.995488i	$-0.530250\pi$
$503$	453.000	−	453.000i	0.900596	−	0.900596i	0.995488	+	0.0948912i	$0.0302503\pi$
$504$		0			0
$505$		0			0
$506$	352.727			0.697088
$507$		0			0
$508$	−279.242	+	279.242i	−0.549689	+	0.549689i
$509$			764.241i			1.50146i	0.660612	+	0.750728i	$0.270297\pi$
$509$			764.241i			1.50146i	−0.660612	+	0.750728i	$0.729703\pi$
$510$		0			0
$511$	159.000			0.311155
$512$	−16.0000	−	16.0000i	−0.0312500	−	0.0312500i
$513$		0			0
$514$		−	510.000i		−	0.992218i
$515$		0			0
$516$		0			0
$517$	−198.409	−	198.409i	−0.383769	−	0.383769i
$518$	33.0000	−	33.0000i	0.0637066	−	0.0637066i
$519$		0			0
$520$		0			0
$521$	73.4847			0.141045			0.0705227	−	0.997510i	$-0.477533\pi$
$521$	73.4847			0.141045			0.0705227	+	0.997510i	$0.477533\pi$
$522$		0			0
$523$	42.8661	−	42.8661i	0.0819619	−	0.0819619i	−0.664937	−	0.746899i	$-0.731542\pi$
$523$	42.8661	−	42.8661i	0.0819619	−	0.0819619i	0.746899	+	0.664937i	$0.231542\pi$
$524$		−	308.636i		−	0.588999i
$525$		0			0
$526$	18.0000			0.0342205
$527$	39.0000	+	39.0000i	0.0740038	+	0.0740038i
$528$		0			0
$529$		−	623.000i		−	1.17769i
$530$		0			0
$531$		0			0
$532$	−46.5403	−	46.5403i	−0.0874818	−	0.0874818i
$533$	54.0000	−	54.0000i	0.101313	−	0.101313i
$534$		0			0
$535$		0			0
$536$	−88.1816			−0.164518
$537$		0			0
$538$	−323.333	+	323.333i	−0.600990	+	0.600990i
$539$		−	338.030i		−	0.627142i
$540$		0			0
$541$	143.000			0.264325			0.132163	−	0.991228i	$-0.457808\pi$
$541$	143.000			0.264325			0.132163	+	0.991228i	$0.457808\pi$
$542$	181.000	+	181.000i	0.333948	+	0.333948i
$543$		0			0
$544$			24.0000i			0.0441176i
$545$		0			0
$546$		0			0
$547$	498.471	+	498.471i	0.911282	+	0.911282i	0.996373	−	0.0850913i	$-0.0271182\pi$
$547$	498.471	+	498.471i	0.911282	+	0.911282i	−0.0850913	+	0.996373i	$0.527118\pi$
$548$	318.000	−	318.000i	0.580292	−	0.580292i
$549$		0			0
$550$		0			0
$551$	139.621			0.253395
$552$		0			0
$553$	−57.5630	+	57.5630i	−0.104092	+	0.104092i
$554$		−	2.44949i		−	0.00442146i
$555$		0			0
$556$	134.000			0.241007
$557$	−45.0000	−	45.0000i	−0.0807899	−	0.0807899i	0.665557	−	0.746347i	$-0.268194\pi$
$557$	−45.0000	−	45.0000i	−0.0807899	−	0.0807899i	−0.746347	+	0.665557i	$0.768194\pi$
$558$		0			0
$559$			234.000i			0.418605i
$560$		0			0
$561$		0			0
$562$	−169.015	−	169.015i	−0.300738	−	0.300738i
$563$	36.0000	−	36.0000i	0.0639432	−	0.0639432i	−0.674412	−	0.738355i	$-0.735603\pi$
$563$	36.0000	−	36.0000i	0.0639432	−	0.0639432i	0.738355	+	0.674412i	$0.235603\pi$
$564$		0			0
$565$		0			0
$566$	149.419			0.263991
$567$		0			0
$568$	146.969	−	146.969i	0.258749	−	0.258749i
$569$			176.363i			0.309953i	0.987918	+	0.154977i	$0.0495302\pi$
$569$			176.363i			0.309953i	−0.987918	+	0.154977i	$0.950470\pi$
$570$		0			0
$571$	−769.000			−1.34676			−0.673380	−	0.739297i	$-0.735158\pi$
$571$	−769.000			−1.34676			−0.673380	+	0.739297i	$0.735158\pi$
$572$	−108.000	−	108.000i	−0.188811	−	0.188811i
$573$		0			0
$574$		−	18.0000i		−	0.0313589i
$575$		0			0
$576$		0			0
$577$	−461.729	−	461.729i	−0.800223	−	0.800223i	0.182907	−	0.983130i	$-0.441449\pi$
$577$	−461.729	−	461.729i	−0.800223	−	0.800223i	−0.983130	+	0.182907i	$0.941449\pi$
$578$	−271.000	+	271.000i	−0.468858	+	0.468858i
$579$		0			0
$580$		0			0
$581$	169.015			0.290903
$582$		0			0
$583$	−154.318	+	154.318i	−0.264696	+	0.264696i
$584$			259.646i			0.444599i
$585$		0			0
$586$	−660.000			−1.12628
$587$	111.000	+	111.000i	0.189097	+	0.189097i	0.795306	−	0.606209i	$-0.207310\pi$
$587$	111.000	+	111.000i	0.189097	+	0.189097i	−0.606209	+	0.795306i	$0.707310\pi$
$588$		0			0
$589$		−	247.000i		−	0.419355i
$590$		0			0
$591$		0			0
$592$	53.8888	+	53.8888i	0.0910283	+	0.0910283i
$593$	102.000	−	102.000i	0.172007	−	0.172007i	−0.615854	−	0.787860i	$-0.711189\pi$
$593$	102.000	−	102.000i	0.172007	−	0.172007i	0.787860	+	0.615854i	$0.211189\pi$
$594$		0			0
$595$		0			0
$596$	−514.393			−0.863075
$597$		0			0
$598$	−352.727	+	352.727i	−0.589844	+	0.589844i
$599$		−	176.363i		−	0.294429i	−0.989105	−	0.147215i	$-0.952969\pi$
$599$		−	176.363i		−	0.294429i	0.989105	−	0.147215i	$-0.0470308\pi$
$600$		0			0
$601$	439.000			0.730449			0.365225	−	0.930919i	$-0.380992\pi$
$601$	439.000			0.730449			0.365225	+	0.930919i	$0.380992\pi$
$602$	39.0000	+	39.0000i	0.0647841	+	0.0647841i
$603$		0			0
$604$		−	562.000i		−	0.930464i
$605$		0			0
$606$		0			0
$607$	221.679	+	221.679i	0.365204	+	0.365204i	0.865725	−	0.500521i	$-0.166858\pi$
$607$	221.679	+	221.679i	0.365204	+	0.365204i	−0.500521	+	0.865725i	$0.666858\pi$
$608$	76.0000	−	76.0000i	0.125000	−	0.125000i
$609$		0			0
$610$		0			0
$611$	396.817			0.649456
$612$		0			0
$613$	263.320	−	263.320i	0.429560	−	0.429560i	−0.458919	−	0.888478i	$-0.651763\pi$
$613$	263.320	−	263.320i	0.429560	−	0.429560i	0.888478	+	0.458919i	$0.151763\pi$
$614$		−	129.823i		−	0.211438i
$615$		0			0
$616$	−36.0000			−0.0584416
$617$	336.000	+	336.000i	0.544571	+	0.544571i	0.924865	−	0.380295i	$-0.124177\pi$
$617$	336.000	+	336.000i	0.544571	+	0.544571i	−0.380295	+	0.924865i	$0.624177\pi$
$618$		0			0
$619$			835.000i			1.34895i	0.738298	+	0.674475i	$0.235630\pi$
$619$			835.000i			1.34895i	−0.738298	+	0.674475i	$0.764370\pi$
$620$		0			0
$621$		0			0
$622$	139.621	+	139.621i	0.224471	+	0.224471i
$623$	180.000	−	180.000i	0.288925	−	0.288925i
$624$		0			0
$625$		0			0
$626$	−161.666			−0.258253
$627$		0			0
$628$	−218.005	+	218.005i	−0.347141	+	0.347141i
$629$		−	80.8332i		−	0.128511i
$630$		0			0
$631$	−52.0000			−0.0824089			−0.0412044	−	0.999151i	$-0.513119\pi$
$631$	−52.0000			−0.0824089			−0.0412044	+	0.999151i	$0.513119\pi$
$632$	−94.0000	−	94.0000i	−0.148734	−	0.148734i
$633$		0			0
$634$			438.000i			0.690852i
$635$		0			0
$636$		0			0
$637$	338.030	+	338.030i	0.530659	+	0.530659i
$638$	54.0000	−	54.0000i	0.0846395	−	0.0846395i
$639$		0			0
$640$		0			0
$641$	764.241			1.19226			0.596132	−	0.802887i	$-0.296704\pi$
$641$	764.241			1.19226			0.596132	+	0.802887i	$0.296704\pi$
$642$		0			0
$643$	595.226	−	595.226i	0.925701	−	0.925701i	−0.0717232	−	0.997425i	$-0.522850\pi$
$643$	595.226	−	595.226i	0.925701	−	0.925701i	0.997425	+	0.0717232i	$0.0228498\pi$
$644$			117.576i			0.182571i
$645$		0			0
$646$	−114.000			−0.176471
$647$	−138.000	−	138.000i	−0.213292	−	0.213292i	0.592372	−	0.805664i	$-0.298191\pi$
$647$	−138.000	−	138.000i	−0.213292	−	0.213292i	−0.805664	+	0.592372i	$0.798191\pi$
$648$		0			0
$649$			648.000i			0.998459i
$650$		0			0
$651$		0			0
$652$	−279.242	−	279.242i	−0.428285	−	0.428285i
$653$	−444.000	+	444.000i	−0.679939	+	0.679939i	−0.959986	−	0.280047i	$-0.909650\pi$
$653$	−444.000	+	444.000i	−0.679939	+	0.679939i	0.280047	+	0.959986i	$0.409650\pi$
$654$		0			0
$655$		0			0
$656$	29.3939			0.0448077
$657$		0			0
$658$	66.1362	−	66.1362i	0.100511	−	0.100511i
$659$		−	36.7423i		−	0.0557547i	−0.999611	−	0.0278773i	$-0.991125\pi$
$659$		−	36.7423i		−	0.0557547i	0.999611	−	0.0278773i	$-0.00887479\pi$
$660$		0			0
$661$	1039.00			1.57186			0.785930	−	0.618315i	$-0.212184\pi$
$661$	1039.00			1.57186			0.785930	+	0.618315i	$0.212184\pi$
$662$	25.0000	+	25.0000i	0.0377644	+	0.0377644i
$663$		0			0
$664$			276.000i			0.415663i
$665$		0			0
$666$		0			0
$667$	−176.363	−	176.363i	−0.264413	−	0.264413i
$668$	−246.000	+	246.000i	−0.368263	+	0.368263i
$669$		0			0
$670$		0			0
$671$	−183.712			−0.273788
$672$		0			0
$673$	111.452	−	111.452i	0.165604	−	0.165604i	−0.619440	−	0.785044i	$-0.712640\pi$
$673$	111.452	−	111.452i	0.165604	−	0.165604i	0.785044	+	0.619440i	$0.212640\pi$
$674$		−	117.576i		−	0.174444i
$675$		0			0
$676$	−122.000			−0.180473
$677$	171.000	+	171.000i	0.252585	+	0.252585i	0.822030	−	0.569445i	$-0.192842\pi$
$677$	171.000	+	171.000i	0.252585	+	0.252585i	−0.569445	+	0.822030i	$0.692842\pi$
$678$		0			0
$679$			231.000i			0.340206i
$680$		0			0
$681$		0			0
$682$	−95.5301	−	95.5301i	−0.140073	−	0.140073i
$683$	−417.000	+	417.000i	−0.610542	+	0.610542i	−0.943087	−	0.332545i	$-0.892093\pi$
$683$	−417.000	+	417.000i	−0.610542	+	0.610542i	0.332545	+	0.943087i	$0.392093\pi$
$684$		0			0
$685$		0			0
$686$	232.702			0.339215
$687$		0			0
$688$	−63.6867	+	63.6867i	−0.0925679	+	0.0925679i
$689$		−	308.636i		−	0.447947i
$690$		0			0
$691$	40.0000			0.0578871			0.0289436	−	0.999581i	$-0.490786\pi$
$691$	40.0000			0.0578871			0.0289436	+	0.999581i	$0.490786\pi$
$692$	−276.000	−	276.000i	−0.398844	−	0.398844i
$693$		0			0
$694$		−	684.000i		−	0.985591i
$695$		0			0
$696$		0			0
$697$	−22.0454	−	22.0454i	−0.0316290	−	0.0316290i
$698$	−403.000	+	403.000i	−0.577364	+	0.577364i
$699$		0			0
$700$		0			0
$701$	−102.879			−0.146760			−0.0733799	−	0.997304i	$-0.523379\pi$
$701$	−102.879			−0.146760			−0.0733799	+	0.997304i	$0.523379\pi$
$702$		0			0
$703$	−255.972	+	255.972i	−0.364113	+	0.364113i
$704$		−	58.7878i		−	0.0835053i
$705$		0			0
$706$	−612.000			−0.866856
$707$	−9.00000	−	9.00000i	−0.0127298	−	0.0127298i
$708$		0			0
$709$			227.000i			0.320169i	0.987103	+	0.160085i	$0.0511767\pi$
$709$			227.000i			0.320169i	−0.987103	+	0.160085i	$0.948823\pi$
$710$		0			0
$711$		0			0
$712$	293.939	+	293.939i	0.412835	+	0.412835i
$713$	−312.000	+	312.000i	−0.437588	+	0.437588i
$714$		0			0
$715$		0			0
$716$	29.3939			0.0410529
$717$		0			0
$718$	−66.1362	+	66.1362i	−0.0921117	+	0.0921117i
$719$			609.923i			0.848293i	0.905593	+	0.424147i	$0.139426\pi$
$719$			609.923i			0.848293i	−0.905593	+	0.424147i	$0.860574\pi$
$720$		0			0
$721$	93.0000			0.128988
$722$		0			0
$723$		0			0
$724$			176.000i			0.243094i
$725$		0			0
$726$		0			0
$727$	983.470	+	983.470i	1.35278	+	1.35278i	0.882538	+	0.470240i	$0.155833\pi$
$727$	983.470	+	983.470i	1.35278	+	1.35278i	0.470240	+	0.882538i	$0.344167\pi$
$728$	36.0000	−	36.0000i	0.0494505	−	0.0494505i
$729$		0			0
$730$		0			0
$731$	95.5301			0.130684
$732$		0			0
$733$	360.075	−	360.075i	0.491235	−	0.491235i	−0.417460	−	0.908695i	$-0.637080\pi$
$733$	360.075	−	360.075i	0.491235	−	0.491235i	0.908695	+	0.417460i	$0.137080\pi$
$734$		−	220.454i		−	0.300346i
$735$		0			0
$736$	−192.000			−0.260870
$737$	−162.000	−	162.000i	−0.219810	−	0.219810i
$738$		0			0
$739$			1298.00i			1.75643i	0.478268	+	0.878214i	$0.341265\pi$
$739$			1298.00i			1.75643i	−0.478268	+	0.878214i	$0.658735\pi$
$740$		0			0
$741$		0			0
$742$	−51.4393	−	51.4393i	−0.0693252	−	0.0693252i
$743$	−234.000	+	234.000i	−0.314939	+	0.314939i	−0.846820	−	0.531880i	$-0.821486\pi$
$743$	−234.000	+	234.000i	−0.314939	+	0.314939i	0.531880	+	0.846820i	$0.321486\pi$
$744$		0			0
$745$		0			0
$746$	222.904			0.298798
$747$		0			0
$748$	−44.0908	+	44.0908i	−0.0589449	+	0.0589449i
$749$			308.636i			0.412064i
$750$		0			0
$751$	−383.000			−0.509987			−0.254993	−	0.966943i	$-0.582073\pi$
$751$	−383.000			−0.509987			−0.254993	+	0.966943i	$0.582073\pi$
$752$	108.000	+	108.000i	0.143617	+	0.143617i
$753$		0			0
$754$			108.000i			0.143236i
$755$		0			0
$756$		0			0
$757$	−905.086	−	905.086i	−1.19562	−	1.19562i	−0.975464	−	0.220159i	$-0.929342\pi$
$757$	−905.086	−	905.086i	−1.19562	−	1.19562i	−0.220159	−	0.975464i	$-0.570658\pi$
$758$	416.000	−	416.000i	0.548813	−	0.548813i
$759$		0			0
$760$		0			0
$761$	−1036.13			−1.36154			−0.680771	−	0.732496i	$-0.738355\pi$
$761$	−1036.13			−1.36154			−0.680771	+	0.732496i	$0.738355\pi$
$762$		0			0
$763$	−64.9115	+	64.9115i	−0.0850740	+	0.0850740i
$764$			573.181i			0.750236i
$765$		0			0
$766$	−642.000			−0.838120
$767$	−648.000	−	648.000i	−0.844850	−	0.844850i
$768$		0			0
$769$		−	872.000i		−	1.13394i	−0.823738	−	0.566970i	$-0.808115\pi$
$769$		−	872.000i		−	1.13394i	0.823738	−	0.566970i	$-0.191885\pi$
$770$		0			0
$771$		0			0
$772$	120.025	+	120.025i	0.155473	+	0.155473i
$773$	237.000	−	237.000i	0.306598	−	0.306598i	−0.536991	−	0.843588i	$-0.680439\pi$
$773$	237.000	−	237.000i	0.306598	−	0.306598i	0.843588	+	0.536991i	$0.180439\pi$
$774$		0			0
$775$		0			0
$776$	−377.221			−0.486110
$777$		0			0
$778$	−756.892	+	756.892i	−0.972869	+	0.972869i
$779$			139.621i			0.179231i
$780$		0			0
$781$	540.000			0.691421
$782$	144.000	+	144.000i	0.184143	+	0.184143i
$783$		0			0
$784$			184.000i			0.234694i
$785$		0			0
$786$		0			0
$787$	909.985	+	909.985i	1.15627	+	1.15627i	0.985271	+	0.171000i	$0.0546999\pi$
$787$	909.985	+	909.985i	1.15627	+	1.15627i	0.171000	+	0.985271i	$0.445300\pi$
$788$	−66.0000	+	66.0000i	−0.0837563	+	0.0837563i
$789$		0			0
$790$		0			0
$791$	220.454			0.278703
$792$		0			0
$793$	183.712	−	183.712i	0.231667	−	0.231667i
$794$			673.610i			0.848375i
$795$		0			0
$796$	−500.000			−0.628141
$797$	−750.000	−	750.000i	−0.941029	−	0.941029i	0.0573266	−	0.998355i	$-0.481742\pi$
$797$	−750.000	−	750.000i	−0.941029	−	0.941029i	−0.998355	+	0.0573266i	$0.981742\pi$
$798$		0			0
$799$		−	162.000i		−	0.202753i
$800$		0			0
$801$		0			0
$802$	558.484	+	558.484i	0.696364	+	0.696364i
$803$	−477.000	+	477.000i	−0.594022	+	0.594022i
$804$		0			0
$805$		0			0
$806$	191.060			0.237047
$807$		0			0
$808$	14.6969	−	14.6969i	0.0181893	−	0.0181893i
$809$			1543.18i			1.90751i	0.300579	+	0.953757i	$0.402820\pi$
$809$			1543.18i			1.90751i	−0.300579	+	0.953757i	$0.597180\pi$
$810$		0			0
$811$	−35.0000			−0.0431566			−0.0215783	−	0.999767i	$-0.506869\pi$
$811$	−35.0000			−0.0431566			−0.0215783	+	0.999767i	$0.506869\pi$
$812$	18.0000	+	18.0000i	0.0221675	+	0.0221675i
$813$		0			0
$814$			198.000i			0.243243i
$815$		0			0
$816$		0			0
$817$	−302.512	−	302.512i	−0.370272	−	0.370272i
$818$	−550.000	+	550.000i	−0.672372	+	0.672372i
$819$		0			0
$820$		0			0
$821$	−698.105			−0.850310			−0.425155	−	0.905121i	$-0.639780\pi$
$821$	−698.105			−0.850310			−0.425155	+	0.905121i	$0.639780\pi$
$822$		0			0
$823$	1021.44	−	1021.44i	1.24111	−	1.24111i	0.281575	−	0.959539i	$-0.409143\pi$
$823$	1021.44	−	1021.44i	1.24111	−	1.24111i	0.959539	−	0.281575i	$-0.0908569\pi$
$824$			151.868i			0.184306i
$825$		0			0
$826$	−216.000			−0.261501
$827$	−663.000	−	663.000i	−0.801693	−	0.801693i	0.181667	−	0.983360i	$-0.441851\pi$
$827$	−663.000	−	663.000i	−0.801693	−	0.801693i	−0.983360	+	0.181667i	$0.941851\pi$
$828$		0			0
$829$			395.000i			0.476478i	0.971207	+	0.238239i	$0.0765701\pi$
$829$			395.000i			0.476478i	−0.971207	+	0.238239i	$0.923430\pi$
$830$		0			0
$831$		0			0
$832$	58.7878	+	58.7878i	0.0706584	+	0.0706584i
$833$	138.000	−	138.000i	0.165666	−	0.165666i
$834$		0			0
$835$		0			0
$836$	279.242			0.334021
$837$		0			0
$838$	58.7878	−	58.7878i	0.0701525	−	0.0701525i
$839$			1499.09i			1.78676i	0.449307	+	0.893378i	$0.351671\pi$
$839$			1499.09i			1.78676i	−0.449307	+	0.893378i	$0.648329\pi$
$840$		0			0
$841$	787.000			0.935791
$842$	−583.000	−	583.000i	−0.692399	−	0.692399i
$843$		0			0
$844$		−	380.000i		−	0.450237i
$845$		0			0
$846$		0			0
$847$	82.0579	+	82.0579i	0.0968806	+	0.0968806i
$848$	84.0000	−	84.0000i	0.0990566	−	0.0990566i
$849$		0			0
$850$		0			0
$851$	646.665			0.759889
$852$		0			0
$853$	617.271	−	617.271i	0.723648	−	0.723648i	−0.245699	−	0.969346i	$-0.579017\pi$
$853$	617.271	−	617.271i	0.723648	−	0.723648i	0.969346	+	0.245699i	$0.0790174\pi$
$854$		−	61.2372i		−	0.0717064i
$855$		0			0
$856$	−504.000			−0.588785
$857$	300.000	+	300.000i	0.350058	+	0.350058i	0.860131	−	0.510073i	$-0.170382\pi$
$857$	300.000	+	300.000i	0.350058	+	0.350058i	−0.510073	+	0.860131i	$0.670382\pi$
$858$		0			0
$859$			1615.00i			1.88009i	0.341046	+	0.940047i	$0.389219\pi$
$859$			1615.00i			1.88009i	−0.341046	+	0.940047i	$0.610781\pi$
$860$		0			0
$861$		0			0
$862$	183.712	+	183.712i	0.213123	+	0.213123i
$863$	−579.000	+	579.000i	−0.670915	+	0.670915i	−0.957927	−	0.287012i	$-0.907338\pi$
$863$	−579.000	+	579.000i	−0.670915	+	0.670915i	0.287012	+	0.957927i	$0.407338\pi$
$864$		0			0
$865$		0			0
$866$	1031.24			1.19080
$867$		0			0
$868$	31.8434	−	31.8434i	0.0366859	−	0.0366859i
$869$		−	345.378i		−	0.397443i
$870$		0			0
$871$	324.000			0.371986
$872$	−106.000	−	106.000i	−0.121560	−	0.121560i
$873$		0			0
$874$		−	912.000i		−	1.04348i
$875$		0			0
$876$		0			0
$877$	−96.7548	−	96.7548i	−0.110325	−	0.110325i	0.649789	−	0.760114i	$-0.274857\pi$
$877$	−96.7548	−	96.7548i	−0.110325	−	0.110325i	−0.760114	+	0.649789i	$0.774857\pi$
$878$	−29.0000	+	29.0000i	−0.0330296	+	0.0330296i
$879$		0			0
$880$		0			0
$881$	286.590			0.325301			0.162651	−	0.986684i	$-0.447996\pi$
$881$	286.590			0.325301			0.162651	+	0.986684i	$0.447996\pi$
$882$		0			0
$883$	278.017	−	278.017i	0.314855	−	0.314855i	−0.531932	−	0.846787i	$-0.678534\pi$
$883$	278.017	−	278.017i	0.314855	−	0.314855i	0.846787	+	0.531932i	$0.178534\pi$
$884$		−	88.1816i		−	0.0997530i
$885$		0			0
$886$	−120.000			−0.135440
$887$	−285.000	−	285.000i	−0.321308	−	0.321308i	0.527961	−	0.849269i	$-0.322957\pi$
$887$	−285.000	−	285.000i	−0.321308	−	0.321308i	−0.849269	+	0.527961i	$0.822957\pi$
$888$		0			0
$889$		−	342.000i		−	0.384702i
$890$		0			0
$891$		0			0
$892$	−46.5403	−	46.5403i	−0.0521752	−	0.0521752i
$893$	−513.000	+	513.000i	−0.574468	+	0.574468i
$894$		0			0
$895$		0			0
$896$	19.5959			0.0218704
$897$		0			0
$898$	−521.741	+	521.741i	−0.581004	+	0.581004i
$899$			95.5301i			0.106263i
$900$		0			0
$901$	−126.000			−0.139845
$902$	54.0000	+	54.0000i	0.0598670	+	0.0598670i
$903$		0			0
$904$			360.000i			0.398230i
$905$		0			0
$906$		0			0
$907$	331.906	+	331.906i	0.365938	+	0.365938i	0.865993	−	0.500055i	$-0.166687\pi$
$907$	331.906	+	331.906i	0.365938	+	0.365938i	−0.500055	+	0.865993i	$0.666687\pi$
$908$	204.000	−	204.000i	0.224670	−	0.224670i
$909$		0			0
$910$		0			0
$911$	−1080.22			−1.18576			−0.592879	−	0.805292i	$-0.702009\pi$
$911$	−1080.22			−1.18576			−0.592879	+	0.805292i	$0.702009\pi$
$912$		0			0
$913$	−507.044	+	507.044i	−0.555361	+	0.555361i
$914$			896.513i			0.980868i
$915$		0			0
$916$	314.000			0.342795
$917$	189.000	+	189.000i	0.206107	+	0.206107i
$918$		0			0
$919$			1457.00i			1.58542i	0.609600	+	0.792709i	$0.291330\pi$
$919$			1457.00i			1.58542i	−0.609600	+	0.792709i	$0.708670\pi$
$920$		0			0
$921$		0			0
$922$	146.969	+	146.969i	0.159403	+	0.159403i
$923$	−540.000	+	540.000i	−0.585049	+	0.585049i
$924$		0			0
$925$		0			0
$926$	−1207.60			−1.30410
$927$		0			0
$928$	−29.3939	+	29.3939i	−0.0316744	+	0.0316744i
$929$		−	73.4847i		−	0.0791009i	−0.999218	−	0.0395504i	$-0.987407\pi$
$929$		−	73.4847i		−	0.0791009i	0.999218	−	0.0395504i	$-0.0125926\pi$
$930$		0			0
$931$	−874.000			−0.938776
$932$	−162.000	−	162.000i	−0.173820	−	0.173820i
$933$		0			0
$934$		−	660.000i		−	0.706638i
$935$		0			0
$936$		0			0
$937$	491.123	+	491.123i	0.524144	+	0.524144i	0.918820	−	0.394676i	$-0.129143\pi$
$937$	491.123	+	491.123i	0.524144	+	0.524144i	−0.394676	+	0.918820i	$0.629143\pi$
$938$	54.0000	−	54.0000i	0.0575693	−	0.0575693i
$939$		0			0
$940$		0			0
$941$	1756.28			1.86640			0.933201	−	0.359355i	$-0.117003\pi$
$941$	1756.28			1.86640			0.933201	+	0.359355i	$0.117003\pi$
$942$		0			0
$943$	176.363	−	176.363i	0.187024	−	0.187024i
$944$		−	352.727i		−	0.373651i
$945$		0			0
$946$	−234.000			−0.247357
$947$	−87.0000	−	87.0000i	−0.0918691	−	0.0918691i	0.659679	−	0.751548i	$-0.270692\pi$
$947$	−87.0000	−	87.0000i	−0.0918691	−	0.0918691i	−0.751548	+	0.659679i	$0.770692\pi$
$948$		0			0
$949$		−	954.000i		−	1.00527i
$950$		0			0
$951$		0			0
$952$	−14.6969	−	14.6969i	−0.0154380	−	0.0154380i
$953$	−639.000	+	639.000i	−0.670514	+	0.670514i	−0.957835	−	0.287320i	$-0.907236\pi$
$953$	−639.000	+	639.000i	−0.670514	+	0.670514i	0.287320	+	0.957835i	$0.407236\pi$
$954$		0			0
$955$		0			0
$956$	−778.938			−0.814788
$957$		0			0
$958$	−881.816	+	881.816i	−0.920476	+	0.920476i
$959$			389.469i			0.406120i
$960$		0			0
$961$	−792.000			−0.824142
$962$	−198.000	−	198.000i	−0.205821	−	0.205821i
$963$		0			0
$964$		−	800.000i		−	0.829876i
$965$		0			0
$966$		0			0
$967$	−821.804	−	821.804i	−0.849849	−	0.849849i	0.140265	−	0.990114i	$-0.455205\pi$
$967$	−821.804	−	821.804i	−0.849849	−	0.849849i	−0.990114	+	0.140265i	$0.955205\pi$
$968$	−134.000	+	134.000i	−0.138430	+	0.138430i
$969$		0			0
$970$		0			0
$971$	176.363			0.181631			0.0908153	−	0.995868i	$-0.471053\pi$
$971$	176.363			0.181631			0.0908153	+	0.995868i	$0.471053\pi$
$972$		0			0
$973$	−82.0579	+	82.0579i	−0.0843350	+	0.0843350i
$974$			411.514i			0.422499i
$975$		0			0
$976$	100.000			0.102459
$977$	−828.000	−	828.000i	−0.847492	−	0.847492i	0.142327	−	0.989820i	$-0.454541\pi$
$977$	−828.000	−	828.000i	−0.847492	−	0.847492i	−0.989820	+	0.142327i	$0.954541\pi$
$978$		0			0
$979$			1080.00i			1.10317i
$980$		0			0
$981$		0			0
$982$	169.015	+	169.015i	0.172113	+	0.172113i
$983$	−555.000	+	555.000i	−0.564598	+	0.564598i	−0.930610	−	0.366012i	$-0.880723\pi$
$983$	−555.000	+	555.000i	−0.564598	+	0.564598i	0.366012	+	0.930610i	$0.380723\pi$
$984$		0			0
$985$		0			0
$986$	44.0908			0.0447169
$987$		0			0
$988$	−279.242	+	279.242i	−0.282633	+	0.282633i
$989$			764.241i			0.772741i
$990$		0			0
$991$	503.000			0.507568			0.253784	−	0.967261i	$-0.418325\pi$
$991$	503.000			0.507568			0.253784	+	0.967261i	$0.418325\pi$
$992$	52.0000	+	52.0000i	0.0524194	+	0.0524194i
$993$		0			0
$994$			180.000i			0.181087i
$995$		0			0
$996$		0			0
$997$	−1256.59	−	1256.59i	−1.26037	−	1.26037i	−0.950914	−	0.309455i	$-0.899853\pi$
$997$	−1256.59	−	1256.59i	−1.26037	−	1.26037i	−0.309455	−	0.950914i	$-0.600147\pi$
$998$	−293.000	+	293.000i	−0.293587	+	0.293587i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.3.g.d.757.1	✓	4
3.2	odd	2		1350.3.g.i.757.1	yes	4
5.2	odd	4		1350.3.g.i.1243.2	yes	4
5.3	odd	4	inner	1350.3.g.d.1243.1	yes	4
5.4	even	2		1350.3.g.i.757.2	yes	4
15.2	even	4	inner	1350.3.g.d.1243.2	yes	4
15.8	even	4		1350.3.g.i.1243.1	yes	4
15.14	odd	2	inner	1350.3.g.d.757.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.3.g.d.757.1	✓	4	1.1	even	1	trivial
1350.3.g.d.757.2	yes	4	15.14	odd	2	inner
1350.3.g.d.1243.1	yes	4	5.3	odd	4	inner
1350.3.g.d.1243.2	yes	4	15.2	even	4	inner
1350.3.g.i.757.1	yes	4	3.2	odd	2
1350.3.g.i.757.2	yes	4	5.4	even	2
1350.3.g.i.1243.1	yes	4	15.8	even	4
1350.3.g.i.1243.2	yes	4	5.2	odd	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)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +(-1.22474 - 1.22474i) q^{7} +(2.00000 - 2.00000i) q^{8} +7.34847 q^{11} +(-7.34847 + 7.34847i) q^{13} +2.44949i q^{14} -4.00000 q^{16} +(3.00000 + 3.00000i) q^{17} -19.0000i q^{19} +(-7.34847 - 7.34847i) q^{22} +(-24.0000 + 24.0000i) q^{23} +14.6969 q^{26} +(2.44949 - 2.44949i) q^{28} +7.34847i q^{29} +13.0000 q^{31} +(4.00000 + 4.00000i) q^{32} -6.00000i q^{34} +(-13.4722 - 13.4722i) q^{37} +(-19.0000 + 19.0000i) q^{38} -7.34847 q^{41} +(15.9217 - 15.9217i) q^{43} +14.6969i q^{44} +48.0000 q^{46} +(-27.0000 - 27.0000i) q^{47} -46.0000i q^{49} +(-14.6969 - 14.6969i) q^{52} +(-21.0000 + 21.0000i) q^{53} -4.89898 q^{56} +(7.34847 - 7.34847i) q^{58} +88.1816i q^{59} -25.0000 q^{61} +(-13.0000 - 13.0000i) q^{62} -8.00000i q^{64} +(-22.0454 - 22.0454i) q^{67} +(-6.00000 + 6.00000i) q^{68} +73.4847 q^{71} +(-64.9115 + 64.9115i) q^{73} +26.9444i q^{74} +38.0000 q^{76} +(-9.00000 - 9.00000i) q^{77} -47.0000i q^{79} +(7.34847 + 7.34847i) q^{82} +(-69.0000 + 69.0000i) q^{83} -31.8434 q^{86} +(14.6969 - 14.6969i) q^{88} +146.969i q^{89} +18.0000 q^{91} +(-48.0000 - 48.0000i) q^{92} +54.0000i q^{94} +(-94.3054 - 94.3054i) q^{97} +(-46.0000 + 46.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 8 q^{8} - 16 q^{16} + 12 q^{17} - 96 q^{23} + 52 q^{31} + 16 q^{32} - 76 q^{38} + 192 q^{46} - 108 q^{47} - 84 q^{53} - 100 q^{61} - 52 q^{62} - 24 q^{68} + 152 q^{76} - 36 q^{77} - 276 q^{83}+ \cdots - 184 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 8 * q^8 - 16 * q^16 + 12 * q^17 - 96 * q^23 + 52 * q^31 + 16 * q^32 - 76 * q^38 + 192 * q^46 - 108 * q^47 - 84 * q^53 - 100 * q^61 - 52 * q^62 - 24 * q^68 + 152 * q^76 - 36 * q^77 - 276 * q^83 + 72 * q^91 - 192 * q^92 - 184 * q^98

Introduction

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