LMFDB - Embedded newform 1368.2.s.j.505.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1368,2,Mod(505,1368)] 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("1368.505"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1368, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.s (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{18})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			505.3
Root			$-0.173648 + 0.984808i$ of defining polynomial
Character	$\chi$	$=$	1368.505
Dual form			1368.2.s.j.577.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.53209 - 2.65366i) q^{5} -2.06418 q^{7} +6.45336 q^{11} +(-0.500000 - 0.866025i) q^{13} +(0.694593 - 1.20307i) q^{17} +(-3.75877 - 2.20718i) q^{19} +(1.53209 + 2.65366i) q^{23} +(-2.19459 - 3.80115i) q^{25} +(-1.75877 - 3.04628i) q^{29} +9.45336 q^{31} +(-3.16250 + 5.47762i) q^{35} -2.38919 q^{37} +(5.06418 - 8.77141i) q^{41} +(-3.03209 + 5.25173i) q^{43} +(-3.00000 - 5.19615i) q^{47} -2.73917 q^{49} +(-5.29086 - 9.16404i) q^{53} +(9.88713 - 17.1250i) q^{55} +(-5.59627 + 9.69302i) q^{59} +(-2.56418 - 4.44129i) q^{61} -3.06418 q^{65} +(-1.72668 - 2.99070i) q^{67} +(3.36959 - 5.83629i) q^{71} +(4.56418 - 7.90539i) q^{73} -13.3209 q^{77} +(0.790859 - 1.36981i) q^{79} +17.6459 q^{83} +(-2.12836 - 3.68642i) q^{85} +(5.22668 + 9.05288i) q^{89} +(1.03209 + 1.78763i) q^{91} +(-11.6159 + 6.59289i) q^{95} +(-3.36959 + 5.83629i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{7} + 12 q^{11} - 3 q^{13} - 9 q^{25} + 12 q^{29} + 30 q^{31} - 24 q^{35} - 6 q^{37} + 12 q^{41} - 9 q^{43} - 18 q^{47} + 12 q^{49} - 6 q^{59} + 3 q^{61} + 3 q^{67} + 6 q^{71} + 9 q^{73} + 12 q^{77}+ \cdots - 6 q^{97}+O(q^{100}) $ 6 * q + 6 * q^7 + 12 * q^11 - 3 * q^13 - 9 * q^25 + 12 * q^29 + 30 * q^31 - 24 * q^35 - 6 * q^37 + 12 * q^41 - 9 * q^43 - 18 * q^47 + 12 * q^49 - 6 * q^59 + 3 * q^61 + 3 * q^67 + 6 * q^71 + 9 * q^73 + 12 * q^77 - 27 * q^79 + 24 * q^83 + 24 * q^85 + 18 * q^89 - 3 * q^91 - 48 * q^95 - 6 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.53209	−	2.65366i	0.685171	−	1.18675i	−0.288212	−	0.957567i	$-0.593061\pi$
$5$	1.53209	−	2.65366i	0.685171	−	1.18675i	0.973383	−	0.229184i	$-0.0736059\pi$
$6$		0			0
$7$	−2.06418			−0.780186			−0.390093	−	0.920775i	$-0.627557\pi$
$7$	−2.06418			−0.780186			−0.390093	+	0.920775i	$0.627557\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	6.45336			1.94576			0.972881	−	0.231306i	$-0.0742998\pi$
$11$	6.45336			1.94576			0.972881	+	0.231306i	$0.0742998\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	0.788320	−	0.615265i	$-0.210951\pi$
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	−0.926995	+	0.375073i	$0.877618\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.694593	−	1.20307i	0.168463	−	0.291787i	−0.769416	−	0.638748i	$-0.779453\pi$
$17$	0.694593	−	1.20307i	0.168463	−	0.291787i	0.937880	+	0.346960i	$0.112786\pi$
$18$		0			0
$19$	−3.75877	−	2.20718i	−0.862321	−	0.506362i
$19$	−3.75877	−	2.20718i	−0.862321	−	0.506362i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.53209	+	2.65366i	0.319463	+	0.553325i	0.980376	−	0.197137i	$-0.0631643\pi$
$23$	1.53209	+	2.65366i	0.319463	+	0.553325i	−0.660913	+	0.750462i	$0.729831\pi$
$24$		0			0
$25$	−2.19459	−	3.80115i	−0.438919	−	0.760229i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.75877	−	3.04628i	−0.326595	−	0.565680i	0.655238	−	0.755422i	$-0.272568\pi$
$29$	−1.75877	−	3.04628i	−0.326595	−	0.565680i	−0.981834	+	0.189742i	$0.939235\pi$
$30$		0			0
$31$	9.45336			1.69787			0.848937	−	0.528494i	$-0.177243\pi$
$31$	9.45336			1.69787			0.848937	+	0.528494i	$0.177243\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.16250	+	5.47762i	−0.534561	+	0.925886i
$36$		0			0
$37$	−2.38919			−0.392780			−0.196390	−	0.980526i	$-0.562922\pi$
$37$	−2.38919			−0.392780			−0.196390	+	0.980526i	$0.562922\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.06418	−	8.77141i	0.790892	−	1.36986i	−0.134524	−	0.990910i	$-0.542950\pi$
$41$	5.06418	−	8.77141i	0.790892	−	1.36986i	0.925415	−	0.378954i	$-0.123716\pi$
$42$		0			0
$43$	−3.03209	+	5.25173i	−0.462389	+	0.800882i	−0.999079	−	0.0428977i	$-0.986341\pi$
$43$	−3.03209	+	5.25173i	−0.462389	+	0.800882i	0.536690	+	0.843779i	$0.319674\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	0.559908	−	0.828554i	$-0.310836\pi$
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	−0.997503	+	0.0706177i	$0.977503\pi$
$48$		0			0
$49$	−2.73917			−0.391310
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.29086	−	9.16404i	−0.726755	−	1.25878i	−0.958247	−	0.285941i	$-0.907694\pi$
$53$	−5.29086	−	9.16404i	−0.726755	−	1.25878i	0.231492	−	0.972837i	$-0.425639\pi$
$54$		0			0
$55$	9.88713	−	17.1250i	1.33318	−	2.30914i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.59627	+	9.69302i	−0.728572	+	1.26192i	0.228915	+	0.973446i	$0.426482\pi$
$59$	−5.59627	+	9.69302i	−0.728572	+	1.26192i	−0.957487	+	0.288477i	$0.906851\pi$
$60$		0			0
$61$	−2.56418	−	4.44129i	−0.328309	−	0.568648i	0.653867	−	0.756609i	$-0.273146\pi$
$61$	−2.56418	−	4.44129i	−0.328309	−	0.568648i	−0.982177	+	0.187961i	$0.939812\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.06418			−0.380064
$66$		0			0
$67$	−1.72668	−	2.99070i	−0.210948	−	0.365372i	0.741064	−	0.671435i	$-0.234322\pi$
$67$	−1.72668	−	2.99070i	−0.210948	−	0.365372i	−0.952011	+	0.306063i	$0.900988\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.36959	−	5.83629i	0.399896	−	0.692640i	−0.593817	−	0.804600i	$-0.702380\pi$
$71$	3.36959	−	5.83629i	0.399896	−	0.692640i	0.993713	+	0.111960i	$0.0357128\pi$
$72$		0			0
$73$	4.56418	−	7.90539i	0.534197	−	0.925256i	−0.465005	−	0.885308i	$-0.653948\pi$
$73$	4.56418	−	7.90539i	0.534197	−	0.925256i	0.999202	−	0.0399477i	$-0.0127191\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−13.3209			−1.51806
$78$		0			0
$79$	0.790859	−	1.36981i	0.0889786	−	0.154116i	−0.818101	−	0.575075i	$-0.804973\pi$
$79$	0.790859	−	1.36981i	0.0889786	−	0.154116i	0.907080	+	0.420959i	$0.138306\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	17.6459			1.93689			0.968444	−	0.249230i	$-0.0801774\pi$
$83$	17.6459			1.93689			0.968444	+	0.249230i	$0.0801774\pi$
$84$		0			0
$85$	−2.12836	−	3.68642i	−0.230853	−	0.399848i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.22668	+	9.05288i	0.554027	+	0.959603i	0.997979	+	0.0635523i	$0.0202430\pi$
$89$	5.22668	+	9.05288i	0.554027	+	0.959603i	−0.443951	+	0.896051i	$0.646424\pi$
$90$		0			0
$91$	1.03209	+	1.78763i	0.108192	+	0.187395i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.6159	+	6.59289i	−1.19176	+	0.676416i
$96$		0			0
$97$	−3.36959	+	5.83629i	−0.342130	+	0.592586i	−0.984828	−	0.173534i	$-0.944481\pi$
$97$	−3.36959	+	5.83629i	−0.342130	+	0.592586i	0.642698	+	0.766119i	$0.277815\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.305407	+	0.528981i	0.0303892	+	0.0526356i	0.880820	−	0.473451i	$-0.156992\pi$
$101$	0.305407	+	0.528981i	0.0303892	+	0.0526356i	−0.850431	+	0.526087i	$0.823659\pi$
$102$		0			0
$103$	0.0641778			0.00632362			0.00316181	−	0.999995i	$-0.498994\pi$
$103$	0.0641778			0.00632362			0.00316181	+	0.999995i	$0.498994\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.610815			−0.0590497			−0.0295248	−	0.999564i	$-0.509399\pi$
$107$	−0.610815			−0.0590497			−0.0295248	+	0.999564i	$0.509399\pi$
$108$		0			0
$109$	−2.30541	+	3.99308i	−0.220818	+	0.382468i	−0.955057	−	0.296424i	$-0.904206\pi$
$109$	−2.30541	+	3.99308i	−0.220818	+	0.382468i	0.734239	+	0.678891i	$0.237539\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.4884			−1.64517			−0.822587	−	0.568639i	$-0.807470\pi$
$113$	−17.4884			−1.64517			−0.822587	+	0.568639i	$0.807470\pi$
$114$		0			0
$115$	9.38919			0.875546
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.43376	+	2.48335i	−0.131433	+	0.227648i
$120$		0			0
$121$	30.6459			2.78599
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.87164			0.167405
$126$		0			0
$127$	−7.06418	−	12.2355i	−0.626844	−	1.08573i	−0.988181	−	0.153291i	$-0.951013\pi$
$127$	−7.06418	−	12.2355i	−0.626844	−	1.08573i	0.361337	−	0.932435i	$-0.382321\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.75877	−	4.77833i	0.241035	−	0.417485i	−0.719974	−	0.694001i	$-0.755847\pi$
$131$	2.75877	−	4.77833i	0.241035	−	0.417485i	0.961009	+	0.276516i	$0.0891798\pi$
$132$		0			0
$133$	7.75877	+	4.55601i	0.672771	+	0.395056i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.88713	+	15.3930i	0.759278	+	1.31511i	0.943219	+	0.332171i	$0.107781\pi$
$137$	8.88713	+	15.3930i	0.759278	+	1.31511i	−0.183941	+	0.982937i	$0.558885\pi$
$138$		0			0
$139$	5.48545	+	9.50108i	0.465270	+	0.805871i	0.999214	−	0.0396488i	$-0.0126239\pi$
$139$	5.48545	+	9.50108i	0.465270	+	0.805871i	−0.533944	+	0.845520i	$0.679291\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.22668	−	5.58878i	−0.269829	−	0.467357i
$144$		0			0
$145$	−10.7784			−0.895095
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.53209	−	11.3139i	0.535130	−	0.926872i	−0.464027	−	0.885821i	$-0.653596\pi$
$149$	6.53209	−	11.3139i	0.535130	−	0.926872i	0.999157	−	0.0410508i	$-0.0130706\pi$
$150$		0			0
$151$	4.90673			0.399304			0.199652	−	0.979867i	$-0.436019\pi$
$151$	4.90673			0.399304			0.199652	+	0.979867i	$0.436019\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	14.4834	−	25.0860i	1.16333	−	2.01495i
$156$		0			0
$157$	−5.86959	+	10.1664i	−0.468444	+	0.811369i	−0.999350	−	0.0360623i	$-0.988519\pi$
$157$	−5.86959	+	10.1664i	−0.468444	+	0.811369i	0.530906	+	0.847431i	$0.321852\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.16250	−	5.47762i	−0.249240	−	0.431697i
$162$		0			0
$163$	−13.4534			−1.05375			−0.526874	−	0.849943i	$-0.676636\pi$
$163$	−13.4534			−1.05375			−0.526874	+	0.849943i	$0.676636\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.98545	+	12.0992i	0.540551	+	0.936261i	0.998872	+	0.0474747i	$0.0151173\pi$
$167$	6.98545	+	12.0992i	0.540551	+	0.936261i	−0.458322	+	0.888786i	$0.651549\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.75877	−	3.04628i	0.133717	−	0.231604i	−0.791390	−	0.611312i	$-0.790642\pi$
$173$	1.75877	−	3.04628i	0.133717	−	0.231604i	0.925107	+	0.379708i	$0.123975\pi$
$174$		0			0
$175$	4.53003	+	7.84624i	0.342438	+	0.593120i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−11.0642			−0.826975			−0.413488	−	0.910510i	$-0.635690\pi$
$179$	−11.0642			−0.826975			−0.413488	+	0.910510i	$0.635690\pi$
$180$		0			0
$181$	−10.7588	−	18.6347i	−0.799693	−	1.38511i	−0.919816	−	0.392350i	$-0.871662\pi$
$181$	−10.7588	−	18.6347i	−0.799693	−	1.38511i	0.120123	−	0.992759i	$-0.461671\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.66044	+	6.34008i	−0.269121	+	0.466132i
$186$		0			0
$187$	4.48246	−	7.76385i	0.327790	−	0.567749i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	15.2317			1.10213			0.551065	−	0.834462i	$-0.314222\pi$
$191$	15.2317			1.10213			0.551065	+	0.834462i	$0.314222\pi$
$192$		0			0
$193$	−3.13041	+	5.42204i	−0.225332	+	0.390287i	−0.956419	−	0.291998i	$-0.905680\pi$
$193$	−3.13041	+	5.42204i	−0.225332	+	0.390287i	0.731087	+	0.682284i	$0.239013\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.71007			0.478073			0.239036	−	0.971011i	$-0.423168\pi$
$197$	6.71007			0.478073			0.239036	+	0.971011i	$0.423168\pi$
$198$		0			0
$199$	12.4659	+	21.5915i	0.883681	+	1.53058i	0.847218	+	0.531245i	$0.178276\pi$
$199$	12.4659	+	21.5915i	0.883681	+	1.53058i	0.0364626	+	0.999335i	$0.488391\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.63041	+	6.28806i	0.254805	+	0.441336i
$204$		0			0
$205$	−15.5175	−	26.8772i	−1.08379	−	1.87718i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−24.2567	−	14.2437i	−1.67787	−	0.985260i
$210$		0			0
$211$	−4.48545	+	7.76903i	−0.308791	+	0.534842i	−0.978098	−	0.208144i	$-0.933258\pi$
$211$	−4.48545	+	7.76903i	−0.308791	+	0.534842i	0.669307	+	0.742986i	$0.266591\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	9.29086	+	16.0922i	0.633631	+	1.09748i
$216$		0			0
$217$	−19.5134			−1.32466
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.38919			−0.0934467
$222$		0			0
$223$	−12.5692	+	21.7705i	−0.841698	+	1.45786i	0.0467604	+	0.998906i	$0.485110\pi$
$223$	−12.5692	+	21.7705i	−0.841698	+	1.45786i	−0.888458	+	0.458957i	$0.848223\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	22.8384			1.51584			0.757920	−	0.652348i	$-0.226216\pi$
$227$	22.8384			1.51584			0.757920	+	0.652348i	$0.226216\pi$
$228$		0			0
$229$	−6.03508			−0.398809			−0.199405	−	0.979917i	$-0.563901\pi$
$229$	−6.03508			−0.398809			−0.199405	+	0.979917i	$0.563901\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.30541	−	3.99308i	0.151032	−	0.261596i	−0.780575	−	0.625062i	$-0.785074\pi$
$233$	2.30541	−	3.99308i	0.151032	−	0.261596i	0.931607	+	0.363467i	$0.118407\pi$
$234$		0			0
$235$	−18.3851			−1.19931
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.36009			−0.605454			−0.302727	−	0.953077i	$-0.597897\pi$
$239$	−9.36009			−0.605454			−0.302727	+	0.953077i	$0.597897\pi$
$240$		0			0
$241$	10.0817	+	17.4620i	0.649421	+	1.12483i	0.983261	+	0.182200i	$0.0583218\pi$
$241$	10.0817	+	17.4620i	0.649421	+	1.12483i	−0.333841	+	0.942629i	$0.608345\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.19665	+	7.26881i	−0.268114	+	0.464388i
$246$		0			0
$247$	−0.0320889	+	4.35878i	−0.00204177	+	0.277343i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.06418	+	10.5035i	0.382768	+	0.662973i	0.991457	−	0.130436i	$-0.0416377\pi$
$251$	6.06418	+	10.5035i	0.382768	+	0.662973i	−0.608689	+	0.793409i	$0.708304\pi$
$252$		0			0
$253$	9.88713	+	17.1250i	0.621598	+	1.07664i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	15.4192	+	26.7069i	0.961824	+	1.66593i	0.717916	+	0.696130i	$0.245096\pi$
$257$	15.4192	+	26.7069i	0.961824	+	1.66593i	0.243909	+	0.969798i	$0.421570\pi$
$258$		0			0
$259$	4.93170			0.306441
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−5.24123	+	9.07808i	−0.323188	+	0.559778i	−0.981144	−	0.193278i	$-0.938088\pi$
$263$	−5.24123	+	9.07808i	−0.323188	+	0.559778i	0.657956	+	0.753056i	$0.271421\pi$
$264$		0			0
$265$	−32.4243			−1.99181
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.901674	+	1.56175i	−0.0549760	+	0.0952213i	−0.892204	−	0.451633i	$-0.850842\pi$
$269$	−0.901674	+	1.56175i	−0.0549760	+	0.0952213i	0.837228	+	0.546854i	$0.184175\pi$
$270$		0			0
$271$	−14.1284	+	24.4710i	−0.858236	+	1.48651i	0.0153732	+	0.999882i	$0.495106\pi$
$271$	−14.1284	+	24.4710i	−0.858236	+	1.48651i	−0.873610	+	0.486627i	$0.838227\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−14.1625	−	24.5302i	−0.854031	−	1.47923i
$276$		0			0
$277$	6.48246			0.389493			0.194747	−	0.980854i	$-0.437612\pi$
$277$	6.48246			0.389493			0.194747	+	0.980854i	$0.437612\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.92127	+	5.05980i	0.174269	+	0.301842i	0.939908	−	0.341428i	$-0.110911\pi$
$281$	2.92127	+	5.05980i	0.174269	+	0.301842i	−0.765639	+	0.643270i	$0.777577\pi$
$282$		0			0
$283$	−8.12836	+	14.0787i	−0.483181	+	0.836893i	−0.999813	−	0.0193137i	$-0.993852\pi$
$283$	−8.12836	+	14.0787i	−0.483181	+	0.836893i	0.516633	+	0.856207i	$0.327185\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.4534	+	18.1058i	−0.617043	+	1.06875i
$288$		0			0
$289$	7.53508	+	13.0511i	0.443240	+	0.767714i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−26.2567			−1.53393			−0.766967	−	0.641687i	$-0.778235\pi$
$293$	−26.2567			−1.53393			−0.766967	+	0.641687i	$0.778235\pi$
$294$		0			0
$295$	17.1480	+	29.7011i	0.998393	+	1.72927i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.53209	−	2.65366i	0.0886030	−	0.153465i
$300$		0			0
$301$	6.25877	−	10.8405i	0.360750	−	0.624837i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−15.7142			−0.899792
$306$		0			0
$307$	−7.14796	+	12.3806i	−0.407955	+	0.706599i	−0.994661	−	0.103201i	$-0.967091\pi$
$307$	−7.14796	+	12.3806i	−0.407955	+	0.706599i	0.586705	+	0.809801i	$0.300425\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−22.2276			−1.26041			−0.630206	−	0.776428i	$-0.717030\pi$
$311$	−22.2276			−1.26041			−0.630206	+	0.776428i	$0.717030\pi$
$312$		0			0
$313$	−1.56624	−	2.71280i	−0.0885290	−	0.153337i	0.818361	−	0.574705i	$-0.194883\pi$
$313$	−1.56624	−	2.71280i	−0.0885290	−	0.153337i	−0.906890	+	0.421368i	$0.861550\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.04963	−	5.28211i	−0.171284	−	0.296673i	0.767585	−	0.640947i	$-0.221458\pi$
$317$	−3.04963	−	5.28211i	−0.171284	−	0.296673i	−0.938869	+	0.344274i	$0.888125\pi$
$318$		0			0
$319$	−11.3500	−	19.6588i	−0.635477	−	1.10068i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.26621	+	2.98897i	−0.293020	+	0.166311i
$324$		0			0
$325$	−2.19459	+	3.80115i	−0.121734	+	0.210850i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.19253	+	10.7258i	0.341405	+	0.591332i
$330$		0			0
$331$	−1.93582			−0.106402			−0.0532012	−	0.998584i	$-0.516942\pi$
$331$	−1.93582			−0.106402			−0.0532012	+	0.998584i	$0.516942\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.5817			−0.578141
$336$		0			0
$337$	3.58378	−	6.20729i	0.195221	−	0.338132i	−0.751752	−	0.659446i	$-0.770791\pi$
$337$	3.58378	−	6.20729i	0.195221	−	0.338132i	0.946973	+	0.321313i	$0.104124\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	61.0060			3.30366
$342$		0			0
$343$	20.1034			1.08548
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.80840	+	8.32839i	−0.258128	+	0.447092i	−0.965741	−	0.259510i	$-0.916439\pi$
$347$	−4.80840	+	8.32839i	−0.258128	+	0.447092i	0.707612	+	0.706601i	$0.249772\pi$
$348$		0			0
$349$	31.5134			1.68687			0.843437	−	0.537227i	$-0.180528\pi$
$349$	31.5134			1.68687			0.843437	+	0.537227i	$0.180528\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	25.8425			1.37546			0.687730	−	0.725967i	$-0.258607\pi$
$353$	25.8425			1.37546			0.687730	+	0.725967i	$0.258607\pi$
$354$		0			0
$355$	−10.3250	−	17.8834i	−0.547995	−	0.949154i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.1284	−	27.9351i	0.851222	−	1.47436i	−0.0288840	−	0.999583i	$-0.509195\pi$
$359$	16.1284	−	27.9351i	0.851222	−	1.47436i	0.880106	−	0.474777i	$-0.157471\pi$
$360$		0			0
$361$	9.25671	+	16.5926i	0.487195	+	0.873293i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−13.9855	−	24.2235i	−0.732032	−	1.26792i
$366$		0			0
$367$	−14.6334	−	25.3458i	−0.763858	−	1.32304i	−0.940848	−	0.338828i	$-0.889970\pi$
$367$	−14.6334	−	25.3458i	−0.763858	−	1.32304i	0.176991	−	0.984213i	$-0.443364\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.9213	+	18.9162i	0.567004	+	0.982080i
$372$		0			0
$373$	8.03920			0.416254			0.208127	−	0.978102i	$-0.433263\pi$
$373$	8.03920			0.416254			0.208127	+	0.978102i	$0.433263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.75877	+	3.04628i	−0.0905813	+	0.156891i
$378$		0			0
$379$	8.19253			0.420822			0.210411	−	0.977613i	$-0.432520\pi$
$379$	8.19253			0.420822			0.210411	+	0.977613i	$0.432520\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.74422	+	4.75313i	−0.140223	+	0.242874i	−0.927581	−	0.373623i	$-0.878115\pi$
$383$	−2.74422	+	4.75313i	−0.140223	+	0.242874i	0.787357	+	0.616497i	$0.211449\pi$
$384$		0			0
$385$	−20.4088	+	35.3491i	−1.04013	+	1.80155i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4.22668	+	7.32083i	0.214301	+	0.371181i	0.953056	−	0.302793i	$-0.0979192\pi$
$389$	4.22668	+	7.32083i	0.214301	+	0.371181i	−0.738755	+	0.673974i	$0.764586\pi$
$390$		0			0
$391$	4.25671			0.215271
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.42333	−	4.19734i	−0.121931	−	0.211191i
$396$		0			0
$397$	−12.5351	+	21.7114i	−0.629118	+	1.08966i	0.358611	+	0.933487i	$0.383250\pi$
$397$	−12.5351	+	21.7114i	−0.629118	+	1.08966i	−0.987729	+	0.156177i	$0.950083\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.8726	+	22.2960i	−0.642826	+	1.11341i	0.341973	+	0.939710i	$0.388905\pi$
$401$	−12.8726	+	22.2960i	−0.642826	+	1.11341i	−0.984799	+	0.173697i	$0.944429\pi$
$402$		0			0
$403$	−4.72668	−	8.18685i	−0.235453	−	0.407816i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−15.4183			−0.764256
$408$		0			0
$409$	3.21213	+	5.56358i	0.158830	+	0.275101i	0.934447	−	0.356102i	$-0.115895\pi$
$409$	3.21213	+	5.56358i	0.158830	+	0.275101i	−0.775617	+	0.631204i	$0.782561\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	11.5517	−	20.0081i	0.568421	−	0.984535i
$414$		0			0
$415$	27.0351	−	46.8261i	1.32710	−	2.29860i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16.2668			−0.794686			−0.397343	−	0.917670i	$-0.630068\pi$
$419$	−16.2668			−0.794686			−0.397343	+	0.917670i	$0.630068\pi$
$420$		0			0
$421$	14.7297	−	25.5125i	0.717880	−	1.24341i	−0.243958	−	0.969786i	$-0.578446\pi$
$421$	14.7297	−	25.5125i	0.717880	−	1.24341i	0.961838	−	0.273619i	$-0.0882209\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.09739			−0.295767
$426$		0			0
$427$	5.29292	+	9.16760i	0.256142	+	0.443651i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.51754	−	14.7528i	−0.410276	−	0.710618i	0.584644	−	0.811290i	$-0.301234\pi$
$431$	−8.51754	−	14.7528i	−0.410276	−	0.710618i	−0.994920	+	0.100672i	$0.967901\pi$
$432$		0			0
$433$	6.84049	+	11.8481i	0.328733	+	0.569382i	0.982261	−	0.187520i	$-0.0600451\pi$
$433$	6.84049	+	11.8481i	0.328733	+	0.569382i	−0.653528	+	0.756903i	$0.726712\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.0983261	−	13.3561i	0.00470357	−	0.638908i
$438$		0			0
$439$	−0.707081	+	1.22470i	−0.0337471	+	0.0584518i	−0.882406	−	0.470489i	$-0.844077\pi$
$439$	−0.707081	+	1.22470i	−0.0337471	+	0.0584518i	0.848659	+	0.528941i	$0.177411\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.369585	+	0.640140i	0.0175595	+	0.0304140i	0.874672	−	0.484716i	$-0.161077\pi$
$443$	0.369585	+	0.640140i	0.0175595	+	0.0304140i	−0.857112	+	0.515130i	$0.827744\pi$
$444$		0			0
$445$	32.0310			1.51841
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−7.51754			−0.354775			−0.177387	−	0.984141i	$-0.556765\pi$
$449$	−7.51754			−0.354775			−0.177387	+	0.984141i	$0.556765\pi$
$450$		0			0
$451$	32.6810	−	56.6051i	1.53889	−	2.66543i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.32501			0.296521
$456$		0			0
$457$	−15.6810			−0.733525			−0.366763	−	0.930315i	$-0.619534\pi$
$457$	−15.6810			−0.733525			−0.366763	+	0.930315i	$0.619534\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.7442	−	18.6095i	0.500408	−	0.866733i	−0.499592	−	0.866261i	$-0.666517\pi$
$461$	10.7442	−	18.6095i	0.500408	−	0.866733i	1.00000		0.000471567i	$-0.000150104\pi$
$462$		0			0
$463$	−9.62092			−0.447122			−0.223561	−	0.974690i	$-0.571768\pi$
$463$	−9.62092			−0.447122			−0.223561	+	0.974690i	$0.571768\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.87164			−0.0866094			−0.0433047	−	0.999062i	$-0.513789\pi$
$467$	−1.87164			−0.0866094			−0.0433047	+	0.999062i	$0.513789\pi$
$468$		0			0
$469$	3.56418	+	6.17334i	0.164578	+	0.285058i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−19.5672	+	33.8913i	−0.899699	+	1.55833i
$474$		0			0
$475$	−0.140844	+	19.1315i	−0.00646237	+	0.877813i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.36959	+	5.83629i	0.153960	+	0.266667i	0.932680	−	0.360705i	$-0.117464\pi$
$479$	3.36959	+	5.83629i	0.153960	+	0.266667i	−0.778720	+	0.627372i	$0.784131\pi$
$480$		0			0
$481$	1.19459	+	2.06910i	0.0544687	+	0.0943426i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.3250	+	17.8834i	0.468834	+	0.812045i
$486$		0			0
$487$	−2.38507			−0.108078			−0.0540388	−	0.998539i	$-0.517209\pi$
$487$	−2.38507			−0.108078			−0.0540388	+	0.998539i	$0.517209\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.36959	+	9.30039i	−0.242326	+	0.419721i	−0.961376	−	0.275237i	$-0.911244\pi$
$491$	−5.36959	+	9.30039i	−0.242326	+	0.419721i	0.719050	+	0.694958i	$0.244577\pi$
$492$		0			0
$493$	−4.88652			−0.220078
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.95542	+	12.0471i	−0.311993	+	0.540388i
$498$		0			0
$499$	−14.2733	+	24.7221i	−0.638961	+	1.10671i	0.346700	+	0.937976i	$0.387302\pi$
$499$	−14.2733	+	24.7221i	−0.638961	+	1.10671i	−0.985661	+	0.168738i	$0.946031\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.75877	+	3.04628i	0.0784197	+	0.135827i	0.902568	−	0.430547i	$-0.141679\pi$
$503$	1.75877	+	3.04628i	0.0784197	+	0.135827i	−0.824149	+	0.566374i	$0.808346\pi$
$504$		0			0
$505$	1.87164			0.0832871
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.61081	+	6.25411i	0.160047	+	0.277209i	0.934885	−	0.354950i	$-0.115502\pi$
$509$	3.61081	+	6.25411i	0.160047	+	0.277209i	−0.774839	+	0.632159i	$0.782169\pi$
$510$		0			0
$511$	−9.42127	+	16.3181i	−0.416773	+	0.721871i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.0983261	−	0.170306i	0.00433276	−	0.00750457i
$516$		0			0
$517$	−19.3601	−	33.5327i	−0.851456	−	1.47476i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−16.8075			−0.736348			−0.368174	−	0.929757i	$-0.620017\pi$
$521$	−16.8075			−0.736348			−0.368174	+	0.929757i	$0.620017\pi$
$522$		0			0
$523$	3.88413	+	6.72752i	0.169841	+	0.294174i	0.938364	−	0.345649i	$-0.112341\pi$
$523$	3.88413	+	6.72752i	0.169841	+	0.294174i	−0.768523	+	0.639823i	$0.779008\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.56624	−	11.3731i	0.286030	−	0.495418i
$528$		0			0
$529$	6.80541	−	11.7873i	0.295887	−	0.512492i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−10.1284			−0.438708
$534$		0			0
$535$	−0.935822	+	1.62089i	−0.0404591	+	0.0700773i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−17.6769			−0.761396
$540$		0			0
$541$	18.1655	+	31.4636i	0.780996	+	1.35272i	0.931362	+	0.364094i	$0.118621\pi$
$541$	18.1655	+	31.4636i	0.780996	+	1.35272i	−0.150367	+	0.988630i	$0.548045\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.06418	+	12.2355i	0.302596	+	0.524112i
$546$		0			0
$547$	−15.8105	−	27.3845i	−0.676006	−	1.17088i	−0.976174	−	0.216991i	$-0.930376\pi$
$547$	−15.8105	−	27.3845i	−0.676006	−	1.17088i	0.300167	−	0.953887i	$-0.402958\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.112874	+	15.3322i	−0.00480859	+	0.653173i
$552$		0			0
$553$	−1.63247	+	2.82753i	−0.0694199	+	0.120239i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.01960	+	3.49805i	0.0855732	+	0.148217i	0.905635	−	0.424057i	$-0.139395\pi$
$557$	2.01960	+	3.49805i	0.0855732	+	0.148217i	−0.820062	+	0.572275i	$0.806061\pi$
$558$		0			0
$559$	6.06418			0.256487
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.61081			0.362903			0.181451	−	0.983400i	$-0.441921\pi$
$563$	8.61081			0.362903			0.181451	+	0.983400i	$0.441921\pi$
$564$		0			0
$565$	−26.7939	+	46.4083i	−1.12723	+	1.95241i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	27.6851			1.16062			0.580310	−	0.814396i	$-0.302931\pi$
$569$	27.6851			1.16062			0.580310	+	0.814396i	$0.302931\pi$
$570$		0			0
$571$	24.1533			1.01079			0.505393	−	0.862889i	$-0.331348\pi$
$571$	24.1533			1.01079			0.505393	+	0.862889i	$0.331348\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.72462	−	11.6474i	0.280436	−	0.485730i
$576$		0			0
$577$	28.8675			1.20177			0.600885	−	0.799335i	$-0.294815\pi$
$577$	28.8675			1.20177			0.600885	+	0.799335i	$0.294815\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−36.4243			−1.51113
$582$		0			0
$583$	−34.1438	−	59.1389i	−1.41409	−	2.44928i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.14290	−	8.90777i	0.212270	−	0.367663i	−0.740154	−	0.672437i	$-0.765248\pi$
$587$	5.14290	−	8.90777i	0.212270	−	0.367663i	0.952425	+	0.304774i	$0.0985809\pi$
$588$		0			0
$589$	−35.5330	−	20.8653i	−1.46411	−	0.859739i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7.96585	−	13.7973i	−0.327118	−	0.566586i	0.654820	−	0.755784i	$-0.272744\pi$
$593$	−7.96585	−	13.7973i	−0.327118	−	0.566586i	−0.981939	+	0.189199i	$0.939411\pi$
$594$		0			0
$595$	4.39330	+	7.60943i	0.180108	+	0.311956i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−9.03003	−	15.6405i	−0.368957	−	0.639052i	0.620446	−	0.784249i	$-0.286952\pi$
$599$	−9.03003	−	15.6405i	−0.368957	−	0.639052i	−0.989403	+	0.145197i	$0.953618\pi$
$600$		0			0
$601$	−7.86753			−0.320923			−0.160462	−	0.987042i	$-0.551298\pi$
$601$	−7.86753			−0.320923			−0.160462	+	0.987042i	$0.551298\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	46.9522	−	81.3237i	1.90888	−	3.30628i
$606$		0			0
$607$	−17.2668			−0.700838			−0.350419	−	0.936593i	$-0.613961\pi$
$607$	−17.2668			−0.700838			−0.350419	+	0.936593i	$0.613961\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.00000	+	5.19615i	−0.121367	+	0.210214i
$612$		0			0
$613$	13.7939	−	23.8917i	0.557128	−	0.964975i	−0.440606	−	0.897701i	$-0.645236\pi$
$613$	13.7939	−	23.8917i	0.557128	−	0.964975i	0.997735	−	0.0672742i	$-0.0214302\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.24628	+	14.2830i	0.331983	+	0.575011i	0.982901	−	0.184137i	$-0.0589491\pi$
$617$	8.24628	+	14.2830i	0.331983	+	0.575011i	−0.650918	+	0.759148i	$0.725616\pi$
$618$		0			0
$619$	24.4884			0.984274			0.492137	−	0.870518i	$-0.336216\pi$
$619$	24.4884			0.984274			0.492137	+	0.870518i	$0.336216\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.7888	−	18.6867i	−0.432244	−	0.748669i
$624$		0			0
$625$	13.8405	−	23.9724i	0.553620	−	0.958897i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.65951	+	2.87436i	−0.0661690	+	0.114608i
$630$		0			0
$631$	−9.83544	−	17.0355i	−0.391543	−	0.678172i	0.601111	−	0.799166i	$-0.294725\pi$
$631$	−9.83544	−	17.0355i	−0.391543	−	0.678172i	−0.992653	+	0.120994i	$0.961392\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−43.2918			−1.71798
$636$		0			0
$637$	1.36959	+	2.37219i	0.0542649	+	0.0939896i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.06418	−	7.03936i	0.160525	−	0.278038i	−0.774532	−	0.632535i	$-0.782014\pi$
$641$	4.06418	−	7.03936i	0.160525	−	0.278038i	0.935057	+	0.354497i	$0.115348\pi$
$642$		0			0
$643$	−3.91921	+	6.78828i	−0.154559	+	0.267704i	−0.932898	−	0.360140i	$-0.882729\pi$
$643$	−3.91921	+	6.78828i	−0.154559	+	0.267704i	0.778340	+	0.627844i	$0.216062\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	39.7743			1.56369			0.781844	−	0.623475i	$-0.214280\pi$
$647$	39.7743			1.56369			0.781844	+	0.623475i	$0.214280\pi$
$648$		0			0
$649$	−36.1147	+	62.5526i	−1.41763	+	2.45540i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.94593			−0.350081			−0.175041	−	0.984561i	$-0.556006\pi$
$653$	−8.94593			−0.350081			−0.175041	+	0.984561i	$0.556006\pi$
$654$		0			0
$655$	−8.45336	−	14.6417i	−0.330300	−	0.572097i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5.93676	−	10.2828i	−0.231263	−	0.400560i	0.726917	−	0.686725i	$-0.240953\pi$
$659$	−5.93676	−	10.2828i	−0.231263	−	0.400560i	−0.958180	+	0.286166i	$0.907619\pi$
$660$		0			0
$661$	−21.9067	−	37.9436i	−0.852073	−	1.47583i	−0.879334	−	0.476205i	$-0.842012\pi$
$661$	−21.9067	−	37.9436i	−0.852073	−	1.47583i	0.0272613	−	0.999628i	$-0.491321\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	23.9772	−	13.6089i	0.929796	−	0.527730i
$666$		0			0
$667$	5.38919	−	9.33434i	0.208670	−	0.361427i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−16.5476	−	28.6612i	−0.638812	−	1.10645i
$672$		0			0
$673$	−37.6810			−1.45249			−0.726247	−	0.687433i	$-0.758737\pi$
$673$	−37.6810			−1.45249			−0.726247	+	0.687433i	$0.758737\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	35.1052			1.34920			0.674602	−	0.738182i	$-0.264315\pi$
$677$	35.1052			1.34920			0.674602	+	0.738182i	$0.264315\pi$
$678$		0			0
$679$	6.95542	−	12.0471i	0.266925	−	0.462327i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−37.6168			−1.43937			−0.719683	−	0.694302i	$-0.755713\pi$
$683$	−37.6168			−1.43937			−0.719683	+	0.694302i	$0.755713\pi$
$684$		0			0
$685$	54.4635			2.08094
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.29086	+	9.16404i	−0.201566	+	0.349122i
$690$		0			0
$691$	−2.69997			−0.102712			−0.0513558	−	0.998680i	$-0.516354\pi$
$691$	−2.69997			−0.102712			−0.0513558	+	0.998680i	$0.516354\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	33.6168			1.27516
$696$		0			0
$697$	−7.03508	−	12.1851i	−0.266473	−	0.461544i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10.8821	+	18.8483i	−0.411010	+	0.711891i	−0.995000	−	0.0998700i	$-0.968157\pi$
$701$	−10.8821	+	18.8483i	−0.411010	+	0.711891i	0.583990	+	0.811761i	$0.301491\pi$
$702$		0			0
$703$	8.98040	+	5.27336i	0.338702	+	0.198889i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.630415	−	1.09191i	−0.0237092	−	0.0410655i
$708$		0			0
$709$	13.3033	+	23.0421i	0.499618	+	0.865363i	1.00000		0.000441366i	$-0.000140491\pi$
$709$	13.3033	+	23.0421i	0.499618	+	0.865363i	−0.500382	+	0.865805i	$0.666807\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	14.4834	+	25.0860i	0.542407	+	0.939477i
$714$		0			0
$715$	−19.7743			−0.739515
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.81790	+	6.61279i	−0.142383	+	0.246615i	−0.928394	−	0.371598i	$-0.878810\pi$
$719$	−3.81790	+	6.61279i	−0.142383	+	0.246615i	0.786010	+	0.618214i	$0.212143\pi$
$720$		0			0
$721$	−0.132474			−0.00493360
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.71957	+	13.3707i	−0.286698	+	0.496575i
$726$		0			0
$727$	−20.3726	+	35.2863i	−0.755577	+	1.30870i	0.189510	+	0.981879i	$0.439310\pi$
$727$	−20.3726	+	35.2863i	−0.755577	+	1.30870i	−0.945087	+	0.326819i	$0.894023\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.21213	+	7.29563i	0.155791	+	0.269839i
$732$		0			0
$733$	48.4742			1.79044			0.895218	−	0.445628i	$-0.147020\pi$
$733$	48.4742			1.79044			0.895218	+	0.445628i	$0.147020\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.1429	−	19.3001i	−0.410454	−	0.710927i
$738$		0			0
$739$	5.30840	−	9.19442i	0.195273	−	0.338222i	−0.751717	−	0.659486i	$-0.770774\pi$
$739$	5.30840	−	9.19442i	0.195273	−	0.338222i	0.946990	+	0.321263i	$0.104107\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	23.7793	−	41.1870i	0.872378	−	1.51100i	0.0128483	−	0.999917i	$-0.495910\pi$
$743$	23.7793	−	41.1870i	0.872378	−	1.51100i	0.859530	−	0.511086i	$-0.170757\pi$
$744$		0			0
$745$	−20.0155	−	34.6678i	−0.733311	−	1.27013i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.26083			0.0460697
$750$		0			0
$751$	7.96791	+	13.8008i	0.290753	+	0.503599i	0.973988	−	0.226599i	$-0.0727608\pi$
$751$	7.96791	+	13.8008i	0.290753	+	0.503599i	−0.683235	+	0.730199i	$0.739427\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.51754	−	13.0208i	0.273591	−	0.473874i
$756$		0			0
$757$	9.17499	−	15.8916i	0.333471	−	0.577588i	−0.649719	−	0.760174i	$-0.725113\pi$
$757$	9.17499	−	15.8916i	0.333471	−	0.577588i	0.983190	+	0.182586i	$0.0584468\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	36.7802			1.33328			0.666641	−	0.745379i	$-0.267731\pi$
$761$	36.7802			1.33328			0.666641	+	0.745379i	$0.267731\pi$
$762$		0			0
$763$	4.75877	−	8.24243i	0.172279	−	0.298396i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	11.1925			0.404139
$768$		0			0
$769$	−21.0175	−	36.4034i	−0.757912	−	1.31274i	−0.943914	−	0.330193i	$-0.892886\pi$
$769$	−21.0175	−	36.4034i	−0.757912	−	1.31274i	0.186002	−	0.982549i	$-0.440447\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.9709	+	39.7868i	0.826206	+	1.43103i	0.900994	+	0.433831i	$0.142839\pi$
$773$	22.9709	+	39.7868i	0.826206	+	1.43103i	−0.0747881	+	0.997199i	$0.523828\pi$
$774$		0			0
$775$	−20.7463	−	35.9336i	−0.745228	−	1.29077i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−38.3952	+	21.7922i	−1.37565	+	0.780786i
$780$		0			0
$781$	21.7452	−	37.6637i	0.778103	−	1.34771i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	17.9855	+	31.1517i	0.641928	+	1.11185i
$786$		0			0
$787$	−17.3250			−0.617570			−0.308785	−	0.951132i	$-0.599922\pi$
$787$	−17.3250			−0.617570			−0.308785	+	0.951132i	$0.599922\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	36.0993			1.28354
$792$		0			0
$793$	−2.56418	+	4.44129i	−0.0910566	+	0.157715i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21.8324			0.773345			0.386672	−	0.922217i	$-0.373624\pi$
$797$	21.8324			0.773345			0.386672	+	0.922217i	$0.373624\pi$
$798$		0			0
$799$	−8.33511			−0.294875
$800$		0			0
$801$		0			0
$802$		0			0
$803$	29.4543	−	51.0163i	1.03942	−	1.80033i
$804$		0			0
$805$	−19.3809			−0.683089
$806$		0			0
$807$		0			0
$808$		0			0
$809$	34.6709			1.21896			0.609482	−	0.792800i	$-0.291378\pi$
$809$	34.6709			1.21896			0.609482	+	0.792800i	$0.291378\pi$
$810$		0			0
$811$	14.4534	+	25.0340i	0.507526	+	0.879061i	0.999962	+	0.00871245i	$0.00277329\pi$
$811$	14.4534	+	25.0340i	0.507526	+	0.879061i	−0.492436	+	0.870349i	$0.663893\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−20.6117	+	35.7006i	−0.721998	+	1.25054i
$816$		0			0
$817$	22.9884	−	13.0477i	0.804264	−	0.456481i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.7588	−	23.8309i	−0.480184	−	0.831704i	0.519557	−	0.854436i	$-0.326097\pi$
$821$	−13.7588	−	23.8309i	−0.480184	−	0.831704i	−0.999742	+	0.0227319i	$0.992764\pi$
$822$		0			0
$823$	21.9709	+	38.0547i	0.765858	+	1.32650i	0.939792	+	0.341747i	$0.111019\pi$
$823$	21.9709	+	38.0547i	0.765858	+	1.32650i	−0.173934	+	0.984757i	$0.555648\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.980400	+	1.69810i	0.0340918	+	0.0590488i	0.882568	−	0.470185i	$-0.155813\pi$
$827$	0.980400	+	1.69810i	0.0340918	+	0.0590488i	−0.848476	+	0.529234i	$0.822479\pi$
$828$		0			0
$829$	−10.4284			−0.362193			−0.181096	−	0.983465i	$-0.557965\pi$
$829$	−10.4284			−0.362193			−0.181096	+	0.983465i	$0.557965\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.90261	+	3.29541i	−0.0659214	+	0.114179i
$834$		0			0
$835$	42.8093			1.48148
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16.1976	−	28.0550i	0.559203	−	0.968568i	−0.438360	−	0.898799i	$-0.644441\pi$
$839$	16.1976	−	28.0550i	0.559203	−	0.968568i	0.997563	−	0.0697683i	$-0.0222260\pi$
$840$		0			0
$841$	8.31345	−	14.3993i	0.286671	−	0.496528i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−18.3851	−	31.8439i	−0.632466	−	1.09546i
$846$		0			0
$847$	−63.2586			−2.17359
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.66044	−	6.34008i	−0.125478	−	0.217335i
$852$		0			0
$853$	14.5838	−	25.2598i	0.499339	−	0.864881i	−0.500661	−	0.865644i	$-0.666910\pi$
$853$	14.5838	−	25.2598i	0.499339	−	0.864881i	1.00000		0.000763028i	$0.000242879\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.77837	−	13.4725i	0.265704	−	0.460213i	−0.702044	−	0.712134i	$-0.747729\pi$
$857$	7.77837	−	13.4725i	0.265704	−	0.460213i	0.967748	+	0.251921i	$0.0810623\pi$
$858$		0			0
$859$	−15.0767	−	26.1136i	−0.514409	−	0.890983i	−0.999860	−	0.0167190i	$-0.994678\pi$
$859$	−15.0767	−	26.1136i	−0.514409	−	0.890983i	0.485451	−	0.874264i	$-0.338655\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	19.7351			0.671789			0.335894	−	0.941900i	$-0.390961\pi$
$863$	19.7351			0.671789			0.335894	+	0.941900i	$0.390961\pi$
$864$		0			0
$865$	−5.38919	−	9.33434i	−0.183238	−	0.317377i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	5.10370	−	8.83987i	0.173131	−	0.299872i
$870$		0			0
$871$	−1.72668	+	2.99070i	−0.0585064	+	0.101336i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.86341			−0.130607
$876$		0			0
$877$	5.32295	−	9.21962i	0.179743	−	0.311324i	−0.762049	−	0.647519i	$-0.775807\pi$
$877$	5.32295	−	9.21962i	0.179743	−	0.311324i	0.941793	+	0.336195i	$0.109140\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−39.9709			−1.34665			−0.673327	−	0.739345i	$-0.735135\pi$
$881$	−39.9709			−1.34665			−0.673327	+	0.739345i	$0.735135\pi$
$882$		0			0
$883$	−14.5496	−	25.2007i	−0.489634	−	0.848071i	0.510295	−	0.860000i	$-0.329536\pi$
$883$	−14.5496	−	25.2007i	−0.489634	−	0.848071i	−0.999929	+	0.0119285i	$0.996203\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.72462	+	16.8435i	0.326521	+	0.565551i	0.981819	−	0.189820i	$-0.0607903\pi$
$887$	9.72462	+	16.8435i	0.326521	+	0.565551i	−0.655298	+	0.755370i	$0.727457\pi$
$888$		0			0
$889$	14.5817	+	25.2563i	0.489055	+	0.847068i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.192533	+	26.1527i	−0.00644288	+	0.875166i
$894$		0			0
$895$	−16.9513	+	29.3605i	−0.566620	+	0.981414i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−16.6263	−	28.7976i	−0.554518	−	0.960453i
$900$		0			0
$901$	−14.7000			−0.489727
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−65.9336			−2.19171
$906$		0			0
$907$	−0.778371	+	1.34818i	−0.0258454	+	0.0447655i	−0.878659	−	0.477450i	$-0.841561\pi$
$907$	−0.778371	+	1.34818i	−0.0258454	+	0.0447655i	0.852813	+	0.522216i	$0.174894\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−43.9418			−1.45586			−0.727929	−	0.685653i	$-0.759517\pi$
$911$	−43.9418			−1.45586			−0.727929	+	0.685653i	$0.759517\pi$
$912$		0			0
$913$	113.875			3.76872
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−5.69459	+	9.86332i	−0.188052	+	0.325716i
$918$		0			0
$919$	43.4843			1.43442			0.717208	−	0.696859i	$-0.245420\pi$
$919$	43.4843			1.43442			0.717208	+	0.696859i	$0.245420\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−6.73917			−0.221822
$924$		0			0
$925$	5.24329	+	9.08164i	0.172398	+	0.298603i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.63547	−	14.9571i	0.283320	−	0.490725i	−0.688880	−	0.724875i	$-0.741897\pi$
$929$	8.63547	−	14.9571i	0.283320	−	0.490725i	0.972201	+	0.234150i	$0.0752307\pi$
$930$		0			0
$931$	10.2959	+	6.04584i	0.337435	+	0.198144i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−13.7351	−	23.7898i	−0.449184	−	0.778010i
$936$		0			0
$937$	−14.6088	−	25.3031i	−0.477247	−	0.826616i	0.522413	−	0.852693i	$-0.325032\pi$
$937$	−14.6088	−	25.3031i	−0.477247	−	0.826616i	−0.999660	+	0.0260763i	$0.991699\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.6459	−	23.6354i	−0.444844	−	0.770492i	0.553198	−	0.833050i	$-0.313407\pi$
$941$	−13.6459	−	23.6354i	−0.444844	−	0.770492i	−0.998041	+	0.0625584i	$0.980074\pi$
$942$		0			0
$943$	31.0351			1.01064
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−25.6655	+	44.4539i	−0.834017	+	1.44456i	0.0608120	+	0.998149i	$0.480631\pi$
$947$	−25.6655	+	44.4539i	−0.834017	+	1.44456i	−0.894829	+	0.446410i	$0.852702\pi$
$948$		0			0
$949$	−9.12836			−0.296319
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−27.5672	+	47.7477i	−0.892988	+	1.54670i	−0.0567134	+	0.998390i	$0.518062\pi$
$953$	−27.5672	+	47.7477i	−0.892988	+	1.54670i	−0.836275	+	0.548311i	$0.815271\pi$
$954$		0			0
$955$	23.3364	−	40.4198i	0.755147	−	1.30795i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.3446	−	31.7738i	−0.592378	−	1.02603i
$960$		0			0
$961$	58.3661			1.88278
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.59215	+	16.6141i	0.308782	+	0.534826i
$966$		0			0
$967$	−25.1759	+	43.6060i	−0.809603	+	1.40227i	0.103536	+	0.994626i	$0.466984\pi$
$967$	−25.1759	+	43.6060i	−0.809603	+	1.40227i	−0.913139	+	0.407648i	$0.866349\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	27.9959	−	48.4903i	0.898431	−	1.55613i	0.0689300	−	0.997621i	$-0.478041\pi$
$971$	27.9959	−	48.4903i	0.898431	−	1.55613i	0.829501	−	0.558506i	$-0.188625\pi$
$972$		0			0
$973$	−11.3229	−	19.6119i	−0.362997	−	0.628729i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.8283			0.474400			0.237200	−	0.971461i	$-0.423770\pi$
$977$	14.8283			0.474400			0.237200	+	0.971461i	$0.423770\pi$
$978$		0			0
$979$	33.7297	+	58.4215i	1.07801	+	1.86716i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8.41921	−	14.5825i	0.268531	−	0.465110i	−0.699952	−	0.714190i	$-0.746795\pi$
$983$	8.41921	−	14.5825i	0.268531	−	0.465110i	0.968483	+	0.249081i	$0.0801284\pi$
$984$		0			0
$985$	10.2804	−	17.8062i	0.327562	−	0.567354i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18.5817			−0.590864
$990$		0			0
$991$	−27.2050	+	47.1205i	−0.864196	+	1.49683i	0.00364729	+	0.999993i	$0.498839\pi$
$991$	−27.2050	+	47.1205i	−0.864196	+	1.49683i	−0.867843	+	0.496838i	$0.834494\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	76.3952			2.42189
$996$		0			0
$997$	−27.1614	−	47.0449i	−0.860209	−	1.48993i	−0.871726	−	0.489993i	$-0.836999\pi$
$997$	−27.1614	−	47.0449i	−0.860209	−	1.48993i	0.0115170	−	0.999934i	$-0.496334\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.s.j.505.3		6
3.2	odd	2		456.2.q.f.49.1	✓	6
4.3	odd	2		2736.2.s.x.1873.3		6
12.11	even	2		912.2.q.k.49.1		6
19.7	even	3	inner	1368.2.s.j.577.3		6
57.8	even	6		8664.2.a.z.1.3		3
57.11	odd	6		8664.2.a.x.1.3		3
57.26	odd	6		456.2.q.f.121.1	yes	6
76.7	odd	6		2736.2.s.x.577.3		6
228.83	even	6		912.2.q.k.577.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.q.f.49.1	✓	6	3.2	odd	2
456.2.q.f.121.1	yes	6	57.26	odd	6
912.2.q.k.49.1		6	12.11	even	2
912.2.q.k.577.1		6	228.83	even	6
1368.2.s.j.505.3		6	1.1	even	1	trivial
1368.2.s.j.577.3		6	19.7	even	3	inner
2736.2.s.x.577.3		6	76.7	odd	6
2736.2.s.x.1873.3		6	4.3	odd	2
8664.2.a.x.1.3		3	57.11	odd	6
8664.2.a.z.1.3		3	57.8	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.53209 - 2.65366i) q^{5} -2.06418 q^{7} +6.45336 q^{11} +(-0.500000 - 0.866025i) q^{13} +(0.694593 - 1.20307i) q^{17} +(-3.75877 - 2.20718i) q^{19} +(1.53209 + 2.65366i) q^{23} +(-2.19459 - 3.80115i) q^{25} +(-1.75877 - 3.04628i) q^{29} +9.45336 q^{31} +(-3.16250 + 5.47762i) q^{35} -2.38919 q^{37} +(5.06418 - 8.77141i) q^{41} +(-3.03209 + 5.25173i) q^{43} +(-3.00000 - 5.19615i) q^{47} -2.73917 q^{49} +(-5.29086 - 9.16404i) q^{53} +(9.88713 - 17.1250i) q^{55} +(-5.59627 + 9.69302i) q^{59} +(-2.56418 - 4.44129i) q^{61} -3.06418 q^{65} +(-1.72668 - 2.99070i) q^{67} +(3.36959 - 5.83629i) q^{71} +(4.56418 - 7.90539i) q^{73} -13.3209 q^{77} +(0.790859 - 1.36981i) q^{79} +17.6459 q^{83} +(-2.12836 - 3.68642i) q^{85} +(5.22668 + 9.05288i) q^{89} +(1.03209 + 1.78763i) q^{91} +(-11.6159 + 6.59289i) q^{95} +(-3.36959 + 5.83629i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{7} + 12 q^{11} - 3 q^{13} - 9 q^{25} + 12 q^{29} + 30 q^{31} - 24 q^{35} - 6 q^{37} + 12 q^{41} - 9 q^{43} - 18 q^{47} + 12 q^{49} - 6 q^{59} + 3 q^{61} + 3 q^{67} + 6 q^{71} + 9 q^{73} + 12 q^{77}+ \cdots - 6 q^{97}+O(q^{100}) \) 6 * q + 6 * q^7 + 12 * q^11 - 3 * q^13 - 9 * q^25 + 12 * q^29 + 30 * q^31 - 24 * q^35 - 6 * q^37 + 12 * q^41 - 9 * q^43 - 18 * q^47 + 12 * q^49 - 6 * q^59 + 3 * q^61 + 3 * q^67 + 6 * q^71 + 9 * q^73 + 12 * q^77 - 27 * q^79 + 24 * q^83 + 24 * q^85 + 18 * q^89 - 3 * q^91 - 48 * q^95 - 6 * q^97

Introduction

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