LMFDB - Embedded newform 2394.2.e.d.1063.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2394,2,Mod(1063,2394)] 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("2394.1063"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2394, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,-24,0,0,-8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1063.9
Character	$\chi$	$=$	2394.1063
Dual form			2394.2.e.d.1063.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} +2.99695i q^{5} +(-0.713538 - 2.54772i) q^{7} +1.00000i q^{8} +2.99695 q^{10} -1.70021 q^{11} +1.66226 q^{13} +(-2.54772 + 0.713538i) q^{14} +1.00000 q^{16} +4.46805i q^{17} +(2.13844 - 3.79830i) q^{19} -2.99695i q^{20} +1.70021i q^{22} -0.487044 q^{23} -3.98173 q^{25} -1.66226i q^{26} +(0.713538 + 2.54772i) q^{28} +1.42708i q^{29} -3.18812 q^{31} -1.00000i q^{32} +4.46805 q^{34} +(7.63539 - 2.13844i) q^{35} +9.68312i q^{37} +(-3.79830 - 2.13844i) q^{38} -2.99695 q^{40} +3.63579 q^{41} -4.40880 q^{43} +1.70021 q^{44} +0.487044i q^{46} -2.80578i q^{47} +(-5.98173 + 3.63579i) q^{49} +3.98173i q^{50} -1.66226 q^{52} +8.53638i q^{53} -5.09543i q^{55} +(2.54772 - 0.713538i) q^{56} +1.42708 q^{58} -9.05625 q^{59} -3.96081i q^{61} +3.18812i q^{62} -1.00000 q^{64} +4.98173i q^{65} -2.67429i q^{67} -4.46805i q^{68} +(-2.13844 - 7.63539i) q^{70} +8.53638i q^{71} +3.63579i q^{73} +9.68312 q^{74} +(-2.13844 + 3.79830i) q^{76} +(1.21316 + 4.33164i) q^{77} +14.7837i q^{79} +2.99695i q^{80} -3.63579i q^{82} -1.60750i q^{83} -13.3905 q^{85} +4.40880i q^{86} -1.70021i q^{88} +7.27157 q^{89} +(-1.18609 - 4.23498i) q^{91} +0.487044 q^{92} -2.80578 q^{94} +(11.3833 + 6.40880i) q^{95} -4.57757 q^{97} +(3.63579 + 5.98173i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{4} - 8 q^{7} + 24 q^{16} + 24 q^{25} + 8 q^{28} + 32 q^{43} - 24 q^{49} + 16 q^{58} - 24 q^{64} - 64 q^{85}+O(q^{100}) $ 24 * q - 24 * q^4 - 8 * q^7 + 24 * q^16 + 24 * q^25 + 8 * q^28 + 32 * q^43 - 24 * q^49 + 16 * q^58 - 24 * q^64 - 64 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$1711$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			2.99695i			1.34028i	0.742236	+	0.670139i	$0.233766\pi$
$5$			2.99695i			1.34028i	−0.742236	+	0.670139i	$0.766234\pi$
$6$		0			0
$7$	−0.713538	−	2.54772i	−0.269692	−	0.962947i
$7$	−0.713538	−	2.54772i	−0.269692	−	0.962947i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	2.99695			0.947720
$11$	−1.70021			−0.512631			−0.256316	−	0.966593i	$-0.582509\pi$
$11$	−1.70021			−0.512631			−0.256316	+	0.966593i	$0.582509\pi$
$12$		0			0
$13$	1.66226			0.461029			0.230515	−	0.973069i	$-0.425959\pi$
$13$	1.66226			0.461029			0.230515	+	0.973069i	$0.425959\pi$
$14$	−2.54772	+	0.713538i	−0.680906	+	0.190701i
$15$		0			0
$16$	1.00000			0.250000
$17$			4.46805i			1.08366i	0.840488	+	0.541830i	$0.182269\pi$
$17$			4.46805i			1.08366i	−0.840488	+	0.541830i	$0.817731\pi$
$18$		0			0
$19$	2.13844	−	3.79830i	0.490592	−	0.871390i
$19$	2.13844	−	3.79830i	0.490592	−	0.871390i
$20$		−	2.99695i		−	0.670139i
$21$		0			0
$22$			1.70021i			0.362485i
$23$	−0.487044			−0.101556			−0.0507779	−	0.998710i	$-0.516170\pi$
$23$	−0.487044			−0.101556			−0.0507779	+	0.998710i	$0.516170\pi$
$24$		0			0
$25$	−3.98173			−0.796345
$26$		−	1.66226i		−	0.325997i
$27$		0			0
$28$	0.713538	+	2.54772i	0.134846	+	0.481473i
$29$			1.42708i			0.265001i	0.991183	+	0.132501i	$0.0423007\pi$
$29$			1.42708i			0.265001i	−0.991183	+	0.132501i	$0.957699\pi$
$30$		0			0
$31$	−3.18812			−0.572604			−0.286302	−	0.958139i	$-0.592426\pi$
$31$	−3.18812			−0.572604			−0.286302	+	0.958139i	$0.592426\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	4.46805			0.766264
$35$	7.63539	−	2.13844i	1.29062	−	0.361462i
$36$		0			0
$37$			9.68312i			1.59189i	0.605366	+	0.795947i	$0.293027\pi$
$37$			9.68312i			1.59189i	−0.605366	+	0.795947i	$0.706973\pi$
$38$	−3.79830	−	2.13844i	−0.616165	−	0.346901i
$39$		0			0
$40$	−2.99695			−0.473860
$41$	3.63579			0.567814			0.283907	−	0.958852i	$-0.408369\pi$
$41$	3.63579			0.567814			0.283907	+	0.958852i	$0.408369\pi$
$42$		0			0
$43$	−4.40880			−0.672336			−0.336168	−	0.941802i	$-0.609131\pi$
$43$	−4.40880			−0.672336			−0.336168	+	0.941802i	$0.609131\pi$
$44$	1.70021			0.256316
$45$		0			0
$46$			0.487044i			0.0718108i
$47$		−	2.80578i		−	0.409266i	−0.978839	−	0.204633i	$-0.934400\pi$
$47$		−	2.80578i		−	0.409266i	0.978839	−	0.204633i	$-0.0656000\pi$
$48$		0			0
$49$	−5.98173	+	3.63579i	−0.854532	+	0.519398i
$50$			3.98173i			0.563101i
$51$		0			0
$52$	−1.66226			−0.230515
$53$			8.53638i			1.17256i	0.810108	+	0.586281i	$0.199409\pi$
$53$			8.53638i			1.17256i	−0.810108	+	0.586281i	$0.800591\pi$
$54$		0			0
$55$		−	5.09543i		−	0.687068i
$56$	2.54772	−	0.713538i	0.340453	−	0.0953505i
$57$		0			0
$58$	1.42708			0.187384
$59$	−9.05625			−1.17902			−0.589511	−	0.807760i	$-0.700680\pi$
$59$	−9.05625			−1.17902			−0.589511	+	0.807760i	$0.700680\pi$
$60$		0			0
$61$		−	3.96081i		−	0.507130i	−0.967318	−	0.253565i	$-0.918397\pi$
$61$		−	3.96081i		−	0.507130i	0.967318	−	0.253565i	$-0.0816032\pi$
$62$			3.18812i			0.404892i
$63$		0			0
$64$	−1.00000			−0.125000
$65$			4.98173i			0.617907i
$66$		0			0
$67$		−	2.67429i		−	0.326717i	−0.986567	−	0.163358i	$-0.947767\pi$
$67$		−	2.67429i		−	0.326717i	0.986567	−	0.163358i	$-0.0522327\pi$
$68$		−	4.46805i		−	0.541830i
$69$		0			0
$70$	−2.13844	−	7.63539i	−0.255592	−	0.912604i
$71$			8.53638i			1.01308i	0.862216	+	0.506541i	$0.169076\pi$
$71$			8.53638i			1.01308i	−0.862216	+	0.506541i	$0.830924\pi$
$72$		0			0
$73$			3.63579i			0.425537i	0.977103	+	0.212768i	$0.0682479\pi$
$73$			3.63579i			0.425537i	−0.977103	+	0.212768i	$0.931752\pi$
$74$	9.68312			1.12564
$75$		0			0
$76$	−2.13844	+	3.79830i	−0.245296	+	0.435695i
$77$	1.21316	+	4.33164i	0.138253	+	0.493636i
$78$		0			0
$79$			14.7837i			1.66330i	0.555300	+	0.831650i	$0.312603\pi$
$79$			14.7837i			1.66330i	−0.555300	+	0.831650i	$0.687397\pi$
$80$			2.99695i			0.335070i
$81$		0			0
$82$		−	3.63579i		−	0.401505i
$83$		−	1.60750i		−	0.176446i	−0.996101	−	0.0882231i	$-0.971881\pi$
$83$		−	1.60750i		−	0.176446i	0.996101	−	0.0882231i	$-0.0281188\pi$
$84$		0			0
$85$	−13.3905			−1.45241
$86$			4.40880i			0.475413i
$87$		0			0
$88$		−	1.70021i		−	0.181242i
$89$	7.27157			0.770785			0.385393	−	0.922753i	$-0.374066\pi$
$89$	7.27157			0.770785			0.385393	+	0.922753i	$0.374066\pi$
$90$		0			0
$91$	−1.18609	−	4.23498i	−0.124336	−	0.443946i
$92$	0.487044			0.0507779
$93$		0			0
$94$	−2.80578			−0.289395
$95$	11.3833	+	6.40880i	1.16790	+	0.657529i
$96$		0			0
$97$	−4.57757			−0.464782			−0.232391	−	0.972622i	$-0.574655\pi$
$97$	−4.57757			−0.464782			−0.232391	+	0.972622i	$0.574655\pi$
$98$	3.63579	+	5.98173i	0.367270	+	0.604246i
$99$		0			0
$100$	3.98173			0.398173
$101$			8.22618i			0.818536i	0.912414	+	0.409268i	$0.134216\pi$
$101$			8.22618i			0.818536i	−0.912414	+	0.409268i	$0.865784\pi$
$102$		0			0
$103$	−12.6942			−1.25080			−0.625400	−	0.780304i	$-0.715064\pi$
$103$	−12.6942			−1.25080			−0.625400	+	0.780304i	$0.715064\pi$
$104$			1.66226i			0.162998i
$105$		0			0
$106$	8.53638			0.829126
$107$			9.96345i			0.963203i	0.876390	+	0.481602i	$0.159945\pi$
$107$			9.96345i			0.963203i	−0.876390	+	0.481602i	$0.840055\pi$
$108$		0			0
$109$			1.21316i			0.116200i	0.998311	+	0.0580999i	$0.0185042\pi$
$109$			1.21316i			0.116200i	−0.998311	+	0.0580999i	$0.981496\pi$
$110$	−5.09543			−0.485831
$111$		0			0
$112$	−0.713538	−	2.54772i	−0.0674230	−	0.240737i
$113$			17.7993i			1.67442i	0.546881	+	0.837210i	$0.315815\pi$
$113$			17.7993i			1.67442i	−0.546881	+	0.837210i	$0.684185\pi$
$114$		0			0
$115$		−	1.45965i		−	0.136113i
$116$		−	1.42708i		−	0.132501i
$117$		0			0
$118$			9.05625i			0.833695i
$119$	11.3833	−	3.18812i	1.04351	−	0.292255i
$120$		0			0
$121$	−8.10930			−0.737209
$122$	−3.96081			−0.358595
$123$		0			0
$124$	3.18812			0.286302
$125$			3.05172i			0.272954i
$126$		0			0
$127$		−	14.7837i		−	1.31184i	−0.754829	−	0.655922i	$-0.772280\pi$
$127$		−	14.7837i		−	1.31184i	0.754829	−	0.655922i	$-0.227720\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	4.98173			0.436926
$131$			0.764675i			0.0668100i	0.999442	+	0.0334050i	$0.0106351\pi$
$131$			0.764675i			0.0668100i	−0.999442	+	0.0334050i	$0.989365\pi$
$132$		0			0
$133$	−11.2029	−	2.73791i	−0.971410	−	0.237407i
$134$	−2.67429			−0.231024
$135$		0			0
$136$	−4.46805			−0.383132
$137$	−11.8704			−1.01415			−0.507077	−	0.861901i	$-0.669274\pi$
$137$	−11.8704			−1.01415			−0.507077	+	0.861901i	$0.669274\pi$
$138$		0			0
$139$			8.73122i			0.740572i	0.928918	+	0.370286i	$0.120740\pi$
$139$			8.73122i			0.740572i	−0.928918	+	0.370286i	$0.879260\pi$
$140$	−7.63539	+	2.13844i	−0.645308	+	0.180731i
$141$		0			0
$142$	8.53638			0.716357
$143$	−2.82619			−0.236338
$144$		0			0
$145$	−4.27688			−0.355175
$146$	3.63579			0.300900
$147$		0			0
$148$		−	9.68312i		−	0.795947i
$149$	8.70903			0.713472			0.356736	−	0.934205i	$-0.383890\pi$
$149$	8.70903			0.713472			0.356736	+	0.934205i	$0.383890\pi$
$150$		0			0
$151$			12.3574i			1.00563i	0.864393	+	0.502816i	$0.167703\pi$
$151$			12.3574i			1.00563i	−0.864393	+	0.502816i	$0.832297\pi$
$152$	3.79830	+	2.13844i	0.308083	+	0.173450i
$153$		0			0
$154$	4.33164	−	1.21316i	0.349054	−	0.0977593i
$155$		−	9.55465i		−	0.767448i
$156$		0			0
$157$			8.73122i			0.696827i	0.937341	+	0.348414i	$0.113280\pi$
$157$			8.73122i			0.696827i	−0.937341	+	0.348414i	$0.886720\pi$
$158$	14.7837			1.17613
$159$		0			0
$160$	2.99695			0.236930
$161$	0.347525	+	1.24085i	0.0273888	+	0.0977928i
$162$		0			0
$163$	9.96345			0.780398			0.390199	−	0.920731i	$-0.372406\pi$
$163$	9.96345			0.780398			0.390199	+	0.920731i	$0.372406\pi$
$164$	−3.63579			−0.283907
$165$		0			0
$166$	−1.60750			−0.124766
$167$	−15.2863			−1.18289			−0.591445	−	0.806345i	$-0.701442\pi$
$167$	−15.2863			−1.18289			−0.591445	+	0.806345i	$0.701442\pi$
$168$		0			0
$169$	−10.2369			−0.787452
$170$			13.3905i			1.02701i
$171$		0			0
$172$	4.40880			0.336168
$173$	1.04151			0.0791849			0.0395924	−	0.999216i	$-0.487394\pi$
$173$	1.04151			0.0791849			0.0395924	+	0.999216i	$0.487394\pi$
$174$		0			0
$175$	2.84111	+	10.1443i	0.214768	+	0.766838i
$176$	−1.70021			−0.128158
$177$		0			0
$178$		−	7.27157i		−	0.545027i
$179$			3.26295i			0.243885i	0.992537	+	0.121942i	$0.0389123\pi$
$179$			3.26295i			0.243885i	−0.992537	+	0.121942i	$0.961088\pi$
$180$		0			0
$181$	9.94321			0.739073			0.369536	−	0.929216i	$-0.379516\pi$
$181$	9.94321			0.739073			0.369536	+	0.929216i	$0.379516\pi$
$182$	−4.23498	+	1.18609i	−0.313918	+	0.0879187i
$183$		0			0
$184$		−	0.487044i		−	0.0359054i
$185$	−29.0199			−2.13358
$186$		0			0
$187$		−	7.59660i		−	0.555518i
$188$			2.80578i			0.204633i
$189$		0			0
$190$	6.40880	−	11.3833i	0.464943	−	0.825833i
$191$	4.86154			0.351769			0.175884	−	0.984411i	$-0.443721\pi$
$191$	4.86154			0.351769			0.175884	+	0.984411i	$0.443721\pi$
$192$		0			0
$193$			18.6712i			1.34398i	0.740560	+	0.671991i	$0.234561\pi$
$193$			18.6712i			1.34398i	−0.740560	+	0.671991i	$0.765439\pi$
$194$			4.57757i			0.328651i
$195$		0			0
$196$	5.98173	−	3.63579i	0.427266	−	0.259699i
$197$	4.61357			0.328703			0.164352	−	0.986402i	$-0.447447\pi$
$197$	4.61357			0.328703			0.164352	+	0.986402i	$0.447447\pi$
$198$		0			0
$199$		−	7.27157i		−	0.515468i	−0.966216	−	0.257734i	$-0.917024\pi$
$199$		−	7.27157i		−	0.515468i	0.966216	−	0.257734i	$-0.0829758\pi$
$200$		−	3.98173i		−	0.281551i
$201$		0			0
$202$	8.22618			0.578792
$203$	3.63579	−	1.01827i	0.255182	−	0.0714687i
$204$		0			0
$205$			10.8963i			0.761029i
$206$			12.6942i			0.884449i
$207$		0			0
$208$	1.66226			0.115257
$209$	−3.63579	+	6.45789i	−0.251493	+	0.446701i
$210$		0			0
$211$		−	5.10062i		−	0.351141i	−0.984467	−	0.175570i	$-0.943823\pi$
$211$		−	5.10062i		−	0.351141i	0.984467	−	0.175570i	$-0.0561770\pi$
$212$		−	8.53638i		−	0.586281i
$213$		0			0
$214$	9.96345			0.681088
$215$		−	13.2130i		−	0.901117i
$216$		0			0
$217$	2.27485	+	8.12243i	0.154427	+	0.551387i
$218$	1.21316			0.0821656
$219$		0			0
$220$			5.09543i			0.343534i
$221$			7.42708i			0.499599i
$222$		0			0
$223$	18.1181			1.21328			0.606640	−	0.794977i	$-0.292517\pi$
$223$	18.1181			1.21328			0.606640	+	0.794977i	$0.292517\pi$
$224$	−2.54772	+	0.713538i	−0.170227	+	0.0476753i
$225$		0			0
$226$	17.7993			1.18399
$227$	−4.67730			−0.310443			−0.155222	−	0.987880i	$-0.549609\pi$
$227$	−4.67730			−0.310443			−0.155222	+	0.987880i	$0.549609\pi$
$228$		0			0
$229$		−	26.8437i		−	1.77388i	−0.461882	−	0.886941i	$-0.652826\pi$
$229$		−	26.8437i		−	1.77388i	0.461882	−	0.886941i	$-0.347174\pi$
$230$	−1.45965			−0.0962464
$231$		0			0
$232$	−1.42708			−0.0936921
$233$	23.7407			1.55531			0.777654	−	0.628693i	$-0.216410\pi$
$233$	23.7407			1.55531			0.777654	+	0.628693i	$0.216410\pi$
$234$		0			0
$235$	8.40880			0.548530
$236$	9.05625			0.589511
$237$		0			0
$238$	−3.18812	−	11.3833i	−0.206655	−	0.737871i
$239$	−7.28786			−0.471413			−0.235706	−	0.971824i	$-0.575740\pi$
$239$	−7.28786			−0.471413			−0.235706	+	0.971824i	$0.575740\pi$
$240$		0			0
$241$	20.7327			1.33551			0.667757	−	0.744380i	$-0.267255\pi$
$241$	20.7327			1.33551			0.667757	+	0.744380i	$0.267255\pi$
$242$			8.10930i			0.521286i
$243$		0			0
$244$			3.96081i			0.253565i
$245$	−10.8963	−	17.9270i	−0.696138	−	1.14531i
$246$		0			0
$247$	3.55465	−	6.31378i	0.226177	−	0.401736i
$248$		−	3.18812i		−	0.202446i
$249$		0			0
$250$	3.05172			0.193007
$251$		−	30.6247i		−	1.93301i	−0.256643	−	0.966506i	$-0.582616\pi$
$251$		−	30.6247i		−	1.93301i	0.256643	−	0.966506i	$-0.417384\pi$
$252$		0			0
$253$	0.828075			0.0520606
$254$	−14.7837			−0.927614
$255$		0			0
$256$	1.00000			0.0625000
$257$	25.3841			1.58341			0.791707	−	0.610901i	$-0.209193\pi$
$257$	25.3841			1.58341			0.791707	+	0.610901i	$0.209193\pi$
$258$		0			0
$259$	24.6698	−	6.90927i	1.53291	−	0.429321i
$260$		−	4.98173i		−	0.308954i
$261$		0			0
$262$	0.764675			0.0472418
$263$	−19.8533			−1.22421			−0.612103	−	0.790778i	$-0.709676\pi$
$263$	−19.8533			−1.22421			−0.612103	+	0.790778i	$0.709676\pi$
$264$		0			0
$265$	−25.5831			−1.57156
$266$	−2.73791	+	11.2029i	−0.167872	+	0.686891i
$267$		0			0
$268$			2.67429i			0.163358i
$269$	−6.46198			−0.393994			−0.196997	−	0.980404i	$-0.563119\pi$
$269$	−6.46198			−0.393994			−0.196997	+	0.980404i	$0.563119\pi$
$270$		0			0
$271$			8.01473i			0.486860i	0.969918	+	0.243430i	$0.0782727\pi$
$271$			8.01473i			0.486860i	−0.969918	+	0.243430i	$0.921727\pi$
$272$			4.46805i			0.270915i
$273$		0			0
$274$			11.8704i			0.717115i
$275$	6.76975			0.408231
$276$		0			0
$277$	−3.96345			−0.238141			−0.119070	−	0.992886i	$-0.537991\pi$
$277$	−3.96345			−0.238141			−0.119070	+	0.992886i	$0.537991\pi$
$278$	8.73122			0.523664
$279$		0			0
$280$	2.13844	+	7.63539i	0.127796	+	0.456302i
$281$		−	23.5076i		−	1.40235i	−0.712990	−	0.701174i	$-0.752660\pi$
$281$		−	23.5076i		−	1.40235i	0.712990	−	0.701174i	$-0.247340\pi$
$282$		0			0
$283$		−	18.9221i		−	1.12480i	−0.826865	−	0.562401i	$-0.809878\pi$
$283$		−	18.9221i		−	1.12480i	0.826865	−	0.562401i	$-0.190122\pi$
$284$		−	8.53638i		−	0.506541i
$285$		0			0
$286$			2.82619i			0.167116i
$287$	−2.59427	−	9.26295i	−0.153135	−	0.546775i
$288$		0			0
$289$	−2.96345			−0.174321
$290$			4.27688i			0.251147i
$291$		0			0
$292$		−	3.63579i		−	0.212768i
$293$	18.9221			1.10544			0.552720	−	0.833367i	$-0.313590\pi$
$293$	18.9221			1.10544			0.552720	+	0.833367i	$0.313590\pi$
$294$		0			0
$295$		−	27.1411i		−	1.58022i
$296$	−9.68312			−0.562820
$297$		0			0
$298$		−	8.70903i		−	0.504501i
$299$	−0.809596			−0.0468202
$300$		0			0
$301$	3.14585	+	11.2324i	0.181324	+	0.647424i
$302$	12.3574			0.711089
$303$		0			0
$304$	2.13844	−	3.79830i	0.122648	−	0.217847i
$305$	11.8704			0.679695
$306$		0			0
$307$	−4.54969			−0.259665			−0.129832	−	0.991536i	$-0.541444\pi$
$307$	−4.54969			−0.259665			−0.129832	+	0.991536i	$0.541444\pi$
$308$	−1.21316	−	4.33164i	−0.0691263	−	0.246818i
$309$		0			0
$310$	−9.55465			−0.542668
$311$			18.9610i			1.07518i	0.843207	+	0.537588i	$0.180665\pi$
$311$			18.9610i			1.07518i	−0.843207	+	0.537588i	$0.819335\pi$
$312$		0			0
$313$		−	11.5574i		−	0.653263i	−0.945152	−	0.326632i	$-0.894086\pi$
$313$		−	11.5574i		−	0.653263i	0.945152	−	0.326632i	$-0.105914\pi$
$314$	8.73122			0.492731
$315$		0			0
$316$		−	14.7837i		−	0.831650i
$317$		−	23.8904i		−	1.34182i	−0.741540	−	0.670908i	$-0.765904\pi$
$317$		−	23.8904i		−	1.34182i	0.741540	−	0.670908i	$-0.234096\pi$
$318$		0			0
$319$		−	2.42632i		−	0.135848i
$320$		−	2.99695i		−	0.167535i
$321$		0			0
$322$	1.24085	−	0.347525i	0.0691499	−	0.0193668i
$323$	16.9710	+	9.55465i	0.944291	+	0.531635i
$324$		0			0
$325$	−6.61868			−0.367138
$326$		−	9.96345i		−	0.551825i
$327$		0			0
$328$			3.63579i			0.200753i
$329$	−7.14834	+	2.00203i	−0.394101	+	0.110376i
$330$		0			0
$331$			21.3455i			1.17325i	0.809857	+	0.586627i	$0.199544\pi$
$331$			21.3455i			1.17325i	−0.809857	+	0.586627i	$0.800456\pi$
$332$			1.60750i			0.0882231i
$333$		0			0
$334$			15.2863i			0.836429i
$335$	8.01473			0.437892
$336$		0			0
$337$			21.0975i			1.14925i	0.818415	+	0.574627i	$0.194853\pi$
$337$			21.0975i			1.14925i	−0.818415	+	0.574627i	$0.805147\pi$
$338$			10.2369i			0.556813i
$339$		0			0
$340$	13.3905			0.726203
$341$	5.42046			0.293534
$342$		0			0
$343$	13.5311	+	12.6455i	0.730613	+	0.682792i
$344$		−	4.40880i		−	0.237707i
$345$		0			0
$346$		−	1.04151i		−	0.0559922i
$347$	−13.2915			−0.713527			−0.356763	−	0.934195i	$-0.616120\pi$
$347$	−13.2915			−0.713527			−0.356763	+	0.934195i	$0.616120\pi$
$348$		0			0
$349$		−	4.37895i		−	0.234400i	−0.993108	−	0.117200i	$-0.962608\pi$
$349$		−	4.37895i		−	0.234400i	0.993108	−	0.117200i	$-0.0373918\pi$
$350$	10.1443	−	2.84111i	0.542236	−	0.151864i
$351$		0			0
$352$			1.70021i			0.0906212i
$353$			15.0116i			0.798989i	0.916736	+	0.399495i	$0.130814\pi$
$353$			15.0116i			0.798989i	−0.916736	+	0.399495i	$0.869186\pi$
$354$		0			0
$355$	−25.5831			−1.35781
$356$	−7.27157			−0.385393
$357$		0			0
$358$	3.26295			0.172453
$359$	−24.2278			−1.27869			−0.639347	−	0.768919i	$-0.720795\pi$
$359$	−24.2278			−1.27869			−0.639347	+	0.768919i	$0.720795\pi$
$360$		0			0
$361$	−9.85415	−	16.2449i	−0.518640	−	0.854993i
$362$		−	9.94321i		−	0.522604i
$363$		0			0
$364$	1.18609	+	4.23498i	0.0621679	+	0.221973i
$365$	−10.8963			−0.570337
$366$		0			0
$367$			15.2863i			0.797939i	0.916964	+	0.398969i	$0.130632\pi$
$367$			15.2863i			0.797939i	−0.916964	+	0.398969i	$0.869368\pi$
$368$	−0.487044			−0.0253889
$369$		0			0
$370$			29.0199i			1.50867i
$371$	21.7483	−	6.09103i	1.12911	−	0.316231i
$372$		0			0
$373$		−	33.7029i		−	1.74507i	−0.488552	−	0.872535i	$-0.662475\pi$
$373$		−	33.7029i		−	1.74507i	0.488552	−	0.872535i	$-0.337525\pi$
$374$	−7.59660			−0.392811
$375$		0			0
$376$	2.80578			0.144697
$377$			2.37218i			0.122173i
$378$		0			0
$379$			12.8755i			0.661371i	0.943741	+	0.330686i	$0.107280\pi$
$379$			12.8755i			0.661371i	−0.943741	+	0.330686i	$0.892720\pi$
$380$	−11.3833	−	6.40880i	−0.583952	−	0.328765i
$381$		0			0
$382$		−	4.86154i		−	0.248738i
$383$	32.6556			1.66863			0.834313	−	0.551292i	$-0.185865\pi$
$383$	32.6556			1.66863			0.834313	+	0.551292i	$0.185865\pi$
$384$		0			0
$385$	−12.9817	+	3.63579i	−0.661610	+	0.185297i
$386$	18.6712			0.950338
$387$		0			0
$388$	4.57757			0.232391
$389$	−32.4498			−1.64527			−0.822634	−	0.568571i	$-0.807496\pi$
$389$	−32.4498			−1.64527			−0.822634	+	0.568571i	$0.807496\pi$
$390$		0			0
$391$		−	2.17614i		−	0.110052i
$392$	−3.63579	−	5.98173i	−0.183635	−	0.302123i
$393$		0			0
$394$		−	4.61357i		−	0.232428i
$395$	−44.3062			−2.22928
$396$		0			0
$397$		−	31.6141i		−	1.58667i	−0.608787	−	0.793334i	$-0.708344\pi$
$397$		−	31.6141i		−	1.58667i	0.608787	−	0.793334i	$-0.291656\pi$
$398$	−7.27157			−0.364491
$399$		0			0
$400$	−3.98173			−0.199086
$401$		−	19.5442i		−	0.975990i	−0.872846	−	0.487995i	$-0.837728\pi$
$401$		−	19.5442i		−	0.975990i	0.872846	−	0.487995i	$-0.162272\pi$
$402$		0			0
$403$	−5.29950			−0.263987
$404$		−	8.22618i		−	0.409268i
$405$		0			0
$406$	−1.01827	−	3.63579i	−0.0505360	−	0.180441i
$407$		−	16.4633i		−	0.816055i
$408$		0			0
$409$	−2.67287			−0.132165			−0.0660825	−	0.997814i	$-0.521050\pi$
$409$	−2.67287			−0.132165			−0.0660825	+	0.997814i	$0.521050\pi$
$410$	10.8963			0.538129
$411$		0			0
$412$	12.6942			0.625400
$413$	6.46198	+	23.0728i	0.317973	+	1.13534i
$414$		0			0
$415$	4.81761			0.236487
$416$		−	1.66226i		−	0.0814992i
$417$		0			0
$418$	6.45789	+	3.63579i	0.315866	+	0.177832i
$419$			17.5994i			0.859786i	0.902880	+	0.429893i	$0.141449\pi$
$419$			17.5994i			0.859786i	−0.902880	+	0.429893i	$0.858551\pi$
$420$		0			0
$421$			18.1531i			0.884727i	0.896836	+	0.442363i	$0.145860\pi$
$421$			18.1531i			0.884727i	−0.896836	+	0.442363i	$0.854140\pi$
$422$	−5.10062			−0.248294
$423$		0			0
$424$	−8.53638			−0.414563
$425$		−	17.7905i		−	0.862968i
$426$		0			0
$427$	−10.0910	+	2.82619i	−0.488339	+	0.136769i
$428$		−	9.96345i		−	0.481602i
$429$		0			0
$430$	−13.2130			−0.637186
$431$		−	7.10930i		−	0.342443i	−0.985233	−	0.171222i	$-0.945229\pi$
$431$		−	7.10930i		−	0.342443i	0.985233	−	0.171222i	$-0.0547714\pi$
$432$		0			0
$433$	−7.02790			−0.337739			−0.168870	−	0.985638i	$-0.554012\pi$
$433$	−7.02790			−0.337739			−0.168870	+	0.985638i	$0.554012\pi$
$434$	8.12243	−	2.27485i	0.389889	−	0.109196i
$435$		0			0
$436$		−	1.21316i		−	0.0580999i
$437$	−1.04151	+	1.84994i	−0.0498224	+	0.0884946i
$438$		0			0
$439$	−4.68610			−0.223655			−0.111828	−	0.993728i	$-0.535670\pi$
$439$	−4.68610			−0.223655			−0.111828	+	0.993728i	$0.535670\pi$
$440$	5.09543			0.242915
$441$		0			0
$442$	7.42708			0.353270
$443$	−10.1702			−0.483199			−0.241600	−	0.970376i	$-0.577672\pi$
$443$	−10.1702			−0.483199			−0.241600	+	0.970376i	$0.577672\pi$
$444$		0			0
$445$			21.7926i			1.03307i
$446$		−	18.1181i		−	0.857918i
$447$		0			0
$448$	0.713538	+	2.54772i	0.0337115	+	0.120368i
$449$		−	7.40100i		−	0.349275i	−0.984633	−	0.174637i	$-0.944125\pi$
$449$		−	7.40100i		−	0.349275i	0.984633	−	0.174637i	$-0.0558753\pi$
$450$		0			0
$451$	−6.18158			−0.291079
$452$		−	17.7993i		−	0.837210i
$453$		0			0
$454$			4.67730i			0.219517i
$455$	12.6920	−	3.55465i	0.595012	−	0.166645i
$456$		0			0
$457$	0.890697			0.0416651			0.0208325	−	0.999783i	$-0.493368\pi$
$457$	0.890697			0.0416651			0.0208325	+	0.999783i	$0.493368\pi$
$458$	−26.8437			−1.25432
$459$		0			0
$460$			1.45965i			0.0680565i
$461$			12.0963i			0.563383i	0.959505	+	0.281691i	$0.0908954\pi$
$461$			12.0963i			0.563383i	−0.959505	+	0.281691i	$0.909105\pi$
$462$		0			0
$463$	32.2081			1.49684			0.748420	−	0.663226i	$-0.230813\pi$
$463$	32.2081			1.49684			0.748420	+	0.663226i	$0.230813\pi$
$464$			1.42708i			0.0662503i
$465$		0			0
$466$		−	23.7407i		−	1.09977i
$467$		−	34.9016i		−	1.61505i	−0.589832	−	0.807526i	$-0.700806\pi$
$467$		−	34.9016i		−	1.61505i	0.589832	−	0.807526i	$-0.299194\pi$
$468$		0			0
$469$	−6.81334	+	1.90821i	−0.314611	+	0.0881129i
$470$		−	8.40880i		−	0.387869i
$471$		0			0
$472$		−	9.05625i		−	0.416848i
$473$	7.49587			0.344660
$474$		0			0
$475$	−8.51468	+	15.1238i	−0.390680	+	0.693927i
$476$	−11.3833	+	3.18812i	−0.521754	+	0.146127i
$477$		0			0
$478$			7.28786i			0.333339i
$479$			25.6100i			1.17015i	0.810979	+	0.585076i	$0.198935\pi$
$479$			25.6100i			1.17015i	−0.810979	+	0.585076i	$0.801065\pi$
$480$		0			0
$481$			16.0959i			0.733910i
$482$		−	20.7327i		−	0.944350i
$483$		0			0
$484$	8.10930			0.368605
$485$		−	13.7188i		−	0.622937i
$486$		0			0
$487$		−	39.0026i		−	1.76738i	−0.468075	−	0.883689i	$-0.655052\pi$
$487$		−	39.0026i		−	1.76738i	0.468075	−	0.883689i	$-0.344948\pi$
$488$	3.96081			0.179298
$489$		0			0
$490$	−17.9270	+	10.8963i	−0.809857	+	0.492244i
$491$	−19.3973			−0.875388			−0.437694	−	0.899124i	$-0.644205\pi$
$491$	−19.3973			−0.875388			−0.437694	+	0.899124i	$0.644205\pi$
$492$		0			0
$493$	−6.37624			−0.287172
$494$	−6.31378	−	3.55465i	−0.284070	−	0.159931i
$495$		0			0
$496$	−3.18812			−0.143151
$497$	21.7483	−	6.09103i	0.975544	−	0.273220i
$498$		0			0
$499$	−18.1171			−0.811033			−0.405517	−	0.914088i	$-0.632908\pi$
$499$	−18.1171			−0.811033			−0.405517	+	0.914088i	$0.632908\pi$
$500$		−	3.05172i		−	0.136477i
$501$		0			0
$502$	−30.6247			−1.36685
$503$			25.7439i			1.14786i	0.818903	+	0.573932i	$0.194583\pi$
$503$			25.7439i			1.14786i	−0.818903	+	0.573932i	$0.805417\pi$
$504$		0			0
$505$	−24.6535			−1.09707
$506$		−	0.828075i		−	0.0368124i
$507$		0			0
$508$			14.7837i			0.655922i
$509$	−24.1106			−1.06868			−0.534342	−	0.845268i	$-0.679441\pi$
$509$	−24.1106			−1.06868			−0.534342	+	0.845268i	$0.679441\pi$
$510$		0			0
$511$	9.26295	−	2.59427i	0.409769	−	0.114764i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	25.3841i		−	1.11964i
$515$		−	38.0440i		−	1.67642i
$516$		0			0
$517$			4.77041i			0.209802i
$518$	−6.90927	−	24.6698i	−0.303576	−	1.08393i
$519$		0			0
$520$	−4.98173			−0.218463
$521$	−2.08303			−0.0912592			−0.0456296	−	0.998958i	$-0.514529\pi$
$521$	−2.08303			−0.0912592			−0.0456296	+	0.998958i	$0.514529\pi$
$522$		0			0
$523$	19.2069			0.839858			0.419929	−	0.907557i	$-0.362055\pi$
$523$	19.2069			0.839858			0.419929	+	0.907557i	$0.362055\pi$
$524$		−	0.764675i		−	0.0334050i
$525$		0			0
$526$			19.8533i			0.865644i
$527$		−	14.2447i		−	0.620508i
$528$		0			0
$529$	−22.7628			−0.989686
$530$			25.5831i			1.11126i
$531$		0			0
$532$	11.2029	+	2.73791i	0.485705	+	0.118703i
$533$	6.04364			0.261779
$534$		0			0
$535$	−29.8600			−1.29096
$536$	2.67429			0.115512
$537$		0			0
$538$			6.46198i			0.278596i
$539$	10.1702	−	6.18158i	0.438060	−	0.266260i
$540$		0			0
$541$	6.57292			0.282592			0.141296	−	0.989967i	$-0.454873\pi$
$541$	6.57292			0.282592			0.141296	+	0.989967i	$0.454873\pi$
$542$	8.01473			0.344262
$543$		0			0
$544$	4.46805			0.191566
$545$	−3.63579			−0.155740
$546$		0			0
$547$			32.9368i			1.40828i	0.710063	+	0.704138i	$0.248666\pi$
$547$			32.9368i			1.40828i	−0.710063	+	0.704138i	$0.751334\pi$
$548$	11.8704			0.507077
$549$		0			0
$550$		−	6.76975i		−	0.288663i
$551$	5.42046	+	3.05172i	0.230919	+	0.130007i
$552$		0			0
$553$	37.6648	−	10.5488i	1.60167	−	0.448579i
$554$			3.96345i			0.168391i
$555$		0			0
$556$		−	8.73122i		−	0.370286i
$557$	−26.6852			−1.13069			−0.565343	−	0.824856i	$-0.691256\pi$
$557$	−26.6852			−1.13069			−0.565343	+	0.824856i	$0.691256\pi$
$558$		0			0
$559$	−7.32859			−0.309967
$560$	7.63539	−	2.13844i	0.322654	−	0.0903656i
$561$		0			0
$562$	−23.5076			−0.991610
$563$	−11.1393			−0.469465			−0.234732	−	0.972060i	$-0.575421\pi$
$563$	−11.1393			−0.469465			−0.234732	+	0.972060i	$0.575421\pi$
$564$		0			0
$565$	−53.3438			−2.24419
$566$	−18.9221			−0.795355
$567$		0			0
$568$	−8.53638			−0.358178
$569$		−	7.92691i		−	0.332313i	−0.986099	−	0.166157i	$-0.946864\pi$
$569$		−	7.92691i		−	0.332313i	0.986099	−	0.166157i	$-0.0531357\pi$
$570$		0			0
$571$	−8.99220			−0.376312			−0.188156	−	0.982139i	$-0.560251\pi$
$571$	−8.99220			−0.376312			−0.188156	+	0.982139i	$0.560251\pi$
$572$	2.82619			0.118169
$573$		0			0
$574$	−9.26295	+	2.59427i	−0.386628	+	0.108283i
$575$	1.93928			0.0808735
$576$		0			0
$577$		−	13.1102i		−	0.545783i	−0.962045	−	0.272892i	$-0.912020\pi$
$577$		−	13.1102i		−	0.545783i	0.962045	−	0.272892i	$-0.0879801\pi$
$578$			2.96345i			0.123263i
$579$		0			0
$580$	4.27688			0.177588
$581$	−4.09546	+	1.14701i	−0.169908	+	0.0475861i
$582$		0			0
$583$		−	14.5136i		−	0.601092i
$584$	−3.63579			−0.150450
$585$		0			0
$586$		−	18.9221i		−	0.781664i
$587$		−	17.5212i		−	0.723179i	−0.932337	−	0.361589i	$-0.882234\pi$
$587$		−	17.5212i		−	0.723179i	0.932337	−	0.361589i	$-0.117766\pi$
$588$		0			0
$589$	−6.81761	+	12.1094i	−0.280915	+	0.498961i
$590$	−27.1411			−1.11738
$591$		0			0
$592$			9.68312i			0.397974i
$593$			26.6485i			1.09432i	0.837027	+	0.547161i	$0.184292\pi$
$593$			26.6485i			1.09432i	−0.837027	+	0.547161i	$0.815708\pi$
$594$		0			0
$595$	9.55465	+	34.1153i	0.391703	+	1.39859i
$596$	−8.70903			−0.356736
$597$		0			0
$598$			0.809596i			0.0331068i
$599$			24.8176i			1.01402i	0.861940	+	0.507010i	$0.169249\pi$
$599$			24.8176i			1.01402i	−0.861940	+	0.507010i	$0.830751\pi$
$600$		0			0
$601$	17.0015			0.693505			0.346753	−	0.937957i	$-0.387284\pi$
$601$	17.0015			0.693505			0.346753	+	0.937957i	$0.387284\pi$
$602$	11.2324	−	3.14585i	0.457798	−	0.128215i
$603$		0			0
$604$		−	12.3574i		−	0.502816i
$605$		−	24.3032i		−	0.988066i
$606$		0			0
$607$	38.4720			1.56153			0.780766	−	0.624824i	$-0.214829\pi$
$607$	38.4720			1.56153			0.780766	+	0.624824i	$0.214829\pi$
$608$	−3.79830	−	2.13844i	−0.154041	−	0.0867252i
$609$		0			0
$610$		−	11.8704i		−	0.480617i
$611$		−	4.66395i		−	0.188683i
$612$		0			0
$613$	−9.67176			−0.390639			−0.195319	−	0.980740i	$-0.562574\pi$
$613$	−9.67176			−0.390639			−0.195319	+	0.980740i	$0.562574\pi$
$614$			4.54969i			0.183611i
$615$		0			0
$616$	−4.33164	+	1.21316i	−0.174527	+	0.0488796i
$617$	−14.9917			−0.603545			−0.301772	−	0.953380i	$-0.597578\pi$
$617$	−14.9917			−0.603545			−0.301772	+	0.953380i	$0.597578\pi$
$618$		0			0
$619$			17.4624i			0.701875i	0.936399	+	0.350937i	$0.114137\pi$
$619$			17.4624i			0.701875i	−0.936399	+	0.350937i	$0.885863\pi$
$620$			9.55465i			0.383724i
$621$		0			0
$622$	18.9610			0.760265
$623$	−5.18854	−	18.5259i	−0.207875	−	0.742225i
$624$		0			0
$625$	−29.0545			−1.16218
$626$	−11.5574			−0.461927
$627$		0			0
$628$		−	8.73122i		−	0.348414i
$629$	−43.2646			−1.72507
$630$		0			0
$631$	17.0988			0.680694			0.340347	−	0.940300i	$-0.389455\pi$
$631$	17.0988			0.680694			0.340347	+	0.940300i	$0.389455\pi$
$632$	−14.7837			−0.588065
$633$		0			0
$634$	−23.8904			−0.948808
$635$	44.3062			1.75824
$636$		0			0
$637$	−9.94321	+	6.04364i	−0.393964	+	0.239458i
$638$	−2.42632			−0.0960590
$639$		0			0
$640$	−2.99695			−0.118465
$641$		−	16.7811i		−	0.662812i	−0.943488	−	0.331406i	$-0.892477\pi$
$641$		−	16.7811i		−	0.662812i	0.943488	−	0.331406i	$-0.107523\pi$
$642$		0			0
$643$		−	23.5994i		−	0.930669i	−0.885135	−	0.465335i	$-0.845934\pi$
$643$		−	23.5994i		−	0.930669i	0.885135	−	0.465335i	$-0.154066\pi$
$644$	−0.347525	−	1.24085i	−0.0136944	−	0.0488964i
$645$		0			0
$646$	9.55465	−	16.9710i	0.375923	−	0.667714i
$647$			21.1385i			0.831039i	0.909584	+	0.415519i	$0.136400\pi$
$647$			21.1385i			0.831039i	−0.909584	+	0.415519i	$0.863600\pi$
$648$		0			0
$649$	15.3975			0.604404
$650$			6.61868i			0.259606i
$651$		0			0
$652$	−9.96345			−0.390199
$653$	35.1551			1.37573			0.687863	−	0.725840i	$-0.258549\pi$
$653$	35.1551			1.37573			0.687863	+	0.725840i	$0.258549\pi$
$654$		0			0
$655$	−2.29170			−0.0895440
$656$	3.63579			0.141954
$657$		0			0
$658$	2.00203	+	7.14834i	0.0780474	+	0.278671i
$659$		−	8.15365i		−	0.317621i	−0.987309	−	0.158811i	$-0.949234\pi$
$659$		−	8.15365i		−	0.317621i	0.987309	−	0.158811i	$-0.0507659\pi$
$660$		0			0
$661$	25.4188			0.988678			0.494339	−	0.869269i	$-0.335410\pi$
$661$	25.4188			0.988678			0.494339	+	0.869269i	$0.335410\pi$
$662$	21.3455			0.829616
$663$		0			0
$664$	1.60750			0.0623832
$665$	8.20539	−	33.5744i	0.318191	−	1.30196i
$666$		0			0
$667$		−	0.695049i		−	0.0269124i
$668$	15.2863			0.591445
$669$		0			0
$670$		−	8.01473i		−	0.309636i
$671$			6.73419i			0.259971i
$672$		0			0
$673$			18.6712i			0.719721i	0.933006	+	0.359861i	$0.117176\pi$
$673$			18.6712i			0.719721i	−0.933006	+	0.359861i	$0.882824\pi$
$674$	21.0975			0.812646
$675$		0			0
$676$	10.2369			0.393726
$677$	42.4550			1.63168			0.815840	−	0.578278i	$-0.196275\pi$
$677$	42.4550			1.63168			0.815840	+	0.578278i	$0.196275\pi$
$678$		0			0
$679$	3.26627	+	11.6624i	0.125348	+	0.447560i
$680$		−	13.3905i		−	0.513503i
$681$		0			0
$682$		−	5.42046i		−	0.207560i
$683$			17.4816i			0.668913i	0.942411	+	0.334457i	$0.108553\pi$
$683$			17.4816i			0.668913i	−0.942411	+	0.334457i	$0.891447\pi$
$684$		0			0
$685$		−	35.5749i		−	1.35925i
$686$	12.6455	−	13.5311i	0.482807	−	0.516621i
$687$		0			0
$688$	−4.40880			−0.168084
$689$			14.1897i			0.540585i
$690$		0			0
$691$		−	20.7068i		−	0.787722i	−0.919170	−	0.393861i	$-0.871139\pi$
$691$		−	20.7068i		−	0.787722i	0.919170	−	0.393861i	$-0.128861\pi$
$692$	−1.04151			−0.0395924
$693$		0			0
$694$			13.2915i			0.504540i
$695$	−26.1671			−0.992573
$696$		0			0
$697$			16.2449i			0.615318i
$698$	−4.37895			−0.165746
$699$		0			0
$700$	−2.84111	−	10.1443i	−0.107384	−	0.383419i
$701$	−10.6572			−0.402517			−0.201259	−	0.979538i	$-0.564503\pi$
$701$	−10.6572			−0.402517			−0.201259	+	0.979538i	$0.564503\pi$
$702$		0			0
$703$	36.7794	+	20.7068i	1.38716	+	0.780970i
$704$	1.70021			0.0640789
$705$		0			0
$706$	15.0116			0.564971
$707$	20.9580	−	5.86969i	0.788206	−	0.220753i
$708$		0			0
$709$	−18.8071			−0.706317			−0.353158	−	0.935564i	$-0.614892\pi$
$709$	−18.8071			−0.706317			−0.353158	+	0.935564i	$0.614892\pi$
$710$			25.5831i			0.960118i
$711$		0			0
$712$			7.27157i			0.272514i
$713$	1.55276			0.0581512
$714$		0			0
$715$		−	8.46996i		−	0.316758i
$716$		−	3.26295i		−	0.121942i
$717$		0			0
$718$			24.2278i			0.904173i
$719$			16.5106i			0.615742i	0.951428	+	0.307871i	$0.0996166\pi$
$719$			16.5106i			0.615742i	−0.951428	+	0.307871i	$0.900383\pi$
$720$		0			0
$721$	9.05782	+	32.3413i	0.337331	+	1.20445i
$722$	−16.2449	+	9.85415i	−0.604571	+	0.366734i
$723$		0			0
$724$	−9.94321			−0.369536
$725$		−	5.68223i		−	0.211033i
$726$		0			0
$727$			0.743160i			0.0275623i	0.999905	+	0.0137811i	$0.00438681\pi$
$727$			0.743160i			0.0275623i	−0.999905	+	0.0137811i	$0.995613\pi$
$728$	4.23498	−	1.18609i	0.156959	−	0.0439594i
$729$		0			0
$730$			10.8963i			0.403289i
$731$		−	19.6987i		−	0.728584i
$732$		0			0
$733$		−	21.0051i		−	0.775842i	−0.921693	−	0.387921i	$-0.873193\pi$
$733$		−	21.0051i		−	0.775842i	0.921693	−	0.387921i	$-0.126807\pi$
$734$	15.2863			0.564228
$735$		0			0
$736$			0.487044i			0.0179527i
$737$			4.54685i			0.167485i
$738$		0			0
$739$	17.7083			0.651410			0.325705	−	0.945471i	$-0.394398\pi$
$739$	17.7083			0.651410			0.325705	+	0.945471i	$0.394398\pi$
$740$	29.0199			1.06679
$741$		0			0
$742$	−6.09103	−	21.7483i	−0.223609	−	0.798404i
$743$			36.7915i			1.34975i	0.737932	+	0.674875i	$0.235803\pi$
$743$			36.7915i			1.34975i	−0.737932	+	0.674875i	$0.764197\pi$
$744$		0			0
$745$			26.1006i			0.956250i
$746$	−33.7029			−1.23395
$747$		0			0
$748$			7.59660i			0.277759i
$749$	25.3841	−	7.10930i	0.927514	−	0.259768i
$750$		0			0
$751$		−	28.8014i		−	1.05098i	−0.850801	−	0.525489i	$-0.823883\pi$
$751$		−	28.8014i		−	1.05098i	0.850801	−	0.525489i	$-0.176117\pi$
$752$		−	2.80578i		−	0.102316i
$753$		0			0
$754$	2.37218			0.0863896
$755$	−37.0346			−1.34783
$756$		0			0
$757$	−0.291697			−0.0106019			−0.00530095	−	0.999986i	$-0.501687\pi$
$757$	−0.291697			−0.0106019			−0.00530095	+	0.999986i	$0.501687\pi$
$758$	12.8755			0.467660
$759$		0			0
$760$	−6.40880	+	11.3833i	−0.232472	+	0.412917i
$761$		−	45.9335i		−	1.66509i	−0.553958	−	0.832545i	$-0.686883\pi$
$761$		−	45.9335i		−	1.66509i	0.553958	−	0.832545i	$-0.313117\pi$
$762$		0			0
$763$	3.09079	−	0.865636i	0.111894	−	0.0313381i
$764$	−4.86154			−0.175884
$765$		0			0
$766$		−	32.6556i		−	1.17990i
$767$	−15.0539			−0.543564
$768$		0			0
$769$		−	31.9391i		−	1.15176i	−0.817536	−	0.575878i	$-0.804660\pi$
$769$		−	31.9391i		−	1.15176i	0.817536	−	0.575878i	$-0.195340\pi$
$770$	3.63579	+	12.9817i	0.131025	+	0.467829i
$771$		0			0
$772$		−	18.6712i		−	0.671991i
$773$	−24.5745			−0.883882			−0.441941	−	0.897044i	$-0.645710\pi$
$773$	−24.5745			−0.883882			−0.441941	+	0.897044i	$0.645710\pi$
$774$		0			0
$775$	12.6942			0.455990
$776$		−	4.57757i		−	0.164325i
$777$		0			0
$778$			32.4498i			1.16338i
$779$	7.77491	−	13.8098i	0.278565	−	0.494788i
$780$		0			0
$781$		−	14.5136i		−	0.519337i
$782$	−2.17614			−0.0778185
$783$		0			0
$784$	−5.98173	+	3.63579i	−0.213633	+	0.129849i
$785$	−26.1671			−0.933942
$786$		0			0
$787$	43.9716			1.56742			0.783709	−	0.621129i	$-0.213326\pi$
$787$	43.9716			1.56742			0.783709	+	0.621129i	$0.213326\pi$
$788$	−4.61357			−0.164352
$789$		0			0
$790$			44.3062i			1.57634i
$791$	45.3477	−	12.7005i	1.61238	−	0.451578i
$792$		0			0
$793$		−	6.58392i		−	0.233802i
$794$	−31.6141			−1.12194
$795$		0			0
$796$			7.27157i			0.257734i
$797$	6.23006			0.220680			0.110340	−	0.993894i	$-0.464806\pi$
$797$	6.23006			0.220680			0.110340	+	0.993894i	$0.464806\pi$
$798$		0			0
$799$	12.5364			0.443505
$800$			3.98173i			0.140775i
$801$		0			0
$802$	−19.5442			−0.690129
$803$		−	6.18158i		−	0.218143i
$804$		0			0
$805$	−3.71877	+	1.04151i	−0.131070	+	0.0367086i
$806$			5.29950i			0.186667i
$807$		0			0
$808$	−8.22618			−0.289396
$809$	49.2127			1.73023			0.865114	−	0.501575i	$-0.167246\pi$
$809$	49.2127			1.73023			0.865114	+	0.501575i	$0.167246\pi$
$810$		0			0
$811$	6.37624			0.223900			0.111950	−	0.993714i	$-0.464290\pi$
$811$	6.37624			0.223900			0.111950	+	0.993714i	$0.464290\pi$
$812$	−3.63579	+	1.01827i	−0.127591	+	0.0357344i
$813$		0			0
$814$	−16.4633			−0.577038
$815$			29.8600i			1.04595i
$816$		0			0
$817$	−9.42796	+	16.7460i	−0.329843	+	0.585867i
$818$			2.67287i			0.0934547i
$819$		0			0
$820$		−	10.8963i		−	0.380515i
$821$	30.3025			1.05756			0.528782	−	0.848758i	$-0.322649\pi$
$821$	30.3025			1.05756			0.528782	+	0.848758i	$0.322649\pi$
$822$		0			0
$823$	53.2548			1.85635			0.928173	−	0.372149i	$-0.121379\pi$
$823$	53.2548			1.85635			0.928173	+	0.372149i	$0.121379\pi$
$824$		−	12.6942i		−	0.442225i
$825$		0			0
$826$	23.0728	−	6.46198i	0.802804	−	0.224841i
$827$		−	16.6640i		−	0.579462i	−0.957108	−	0.289731i	$-0.906434\pi$
$827$		−	16.6640i		−	0.579462i	0.957108	−	0.289731i	$-0.0935659\pi$
$828$		0			0
$829$	−39.0455			−1.35611			−0.678054	−	0.735012i	$-0.737176\pi$
$829$	−39.0455			−1.35611			−0.678054	+	0.735012i	$0.737176\pi$
$830$		−	4.81761i		−	0.167222i
$831$		0			0
$832$	−1.66226			−0.0576286
$833$	−16.2449	−	26.7266i	−0.562851	−	0.926023i
$834$		0			0
$835$		−	45.8123i		−	1.58540i
$836$	3.63579	−	6.45789i	0.125746	−	0.223351i
$837$		0			0
$838$	17.5994			0.607961
$839$	13.2033			0.455828			0.227914	−	0.973681i	$-0.426810\pi$
$839$	13.2033			0.455828			0.227914	+	0.973681i	$0.426810\pi$
$840$		0			0
$841$	26.9635			0.929774
$842$	18.1531			0.625596
$843$		0			0
$844$			5.10062i			0.175570i
$845$		−	30.6794i		−	1.05540i
$846$		0			0
$847$	5.78630	+	20.6602i	0.198819	+	0.709893i
$848$			8.53638i			0.293140i
$849$		0			0
$850$	−17.7905			−0.610211
$851$		−	4.71611i		−	0.161666i
$852$		0			0
$853$		−	38.8857i		−	1.33142i	−0.746210	−	0.665710i	$-0.768129\pi$
$853$		−	38.8857i		−	1.33142i	0.746210	−	0.665710i	$-0.231871\pi$
$854$	2.82619	+	10.0910i	0.0967102	+	0.345308i
$855$		0			0
$856$	−9.96345			−0.340544
$857$	−45.1158			−1.54112			−0.770562	−	0.637365i	$-0.780025\pi$
$857$	−45.1158			−1.54112			−0.770562	+	0.637365i	$0.780025\pi$
$858$		0			0
$859$			2.91930i			0.0996051i	0.998759	+	0.0498026i	$0.0158592\pi$
$859$			2.91930i			0.0996051i	−0.998759	+	0.0498026i	$0.984141\pi$
$860$			13.2130i			0.450559i
$861$		0			0
$862$	−7.10930			−0.242144
$863$			52.6975i			1.79384i	0.442189	+	0.896922i	$0.354202\pi$
$863$			52.6975i			1.79384i	−0.442189	+	0.896922i	$0.645798\pi$
$864$		0			0
$865$			3.12137i			0.106130i
$866$			7.02790i			0.238818i
$867$		0			0
$868$	−2.27485	−	8.12243i	−0.0772133	−	0.275693i
$869$		−	25.1354i		−	0.852659i
$870$		0			0
$871$		−	4.44538i		−	0.150626i
$872$	−1.21316			−0.0410828
$873$		0			0
$874$	1.84994	+	1.04151i	0.0625751	+	0.0352298i
$875$	7.77491	−	2.17751i	0.262840	−	0.0736134i
$876$		0			0
$877$			8.29302i			0.280035i	0.990149	+	0.140018i	$0.0447159\pi$
$877$			8.29302i			0.280035i	−0.990149	+	0.140018i	$0.955284\pi$
$878$			4.68610i			0.158148i
$879$		0			0
$880$		−	5.09543i		−	0.171767i
$881$			16.9163i			0.569926i	0.958538	+	0.284963i	$0.0919814\pi$
$881$			16.9163i			0.569926i	−0.958538	+	0.284963i	$0.908019\pi$
$882$		0			0
$883$	11.1899			0.376569			0.188284	−	0.982115i	$-0.439707\pi$
$883$	11.1899			0.376569			0.188284	+	0.982115i	$0.439707\pi$
$884$		−	7.42708i		−	0.249800i
$885$		0			0
$886$			10.1702i			0.341673i
$887$	12.9240			0.433944			0.216972	−	0.976178i	$-0.430382\pi$
$887$	12.9240			0.433944			0.216972	+	0.976178i	$0.430382\pi$
$888$		0			0
$889$	−37.6648	+	10.5488i	−1.26324	+	0.353794i
$890$	21.7926			0.730488
$891$		0			0
$892$	−18.1181			−0.606640
$893$	−10.6572	−	6.00000i	−0.356630	−	0.200782i
$894$		0			0
$895$	−9.77892			−0.326873
$896$	2.54772	−	0.713538i	0.0851133	−	0.0238376i
$897$		0			0
$898$	−7.40100			−0.246975
$899$		−	4.54969i		−	0.151741i
$900$		0			0
$901$	−38.1410			−1.27066
$902$			6.18158i			0.205824i
$903$		0			0
$904$	−17.7993			−0.591997
$905$			29.7993i			0.990563i
$906$		0			0
$907$		−	12.8755i		−	0.427525i	−0.976886	−	0.213762i	$-0.931428\pi$
$907$		−	12.8755i		−	0.427525i	0.976886	−	0.213762i	$-0.0685718\pi$
$908$	4.67730			0.155222
$909$		0			0
$910$	−3.55465	−	12.6920i	−0.117836	−	0.420737i
$911$		−	27.0362i		−	0.895750i	−0.894096	−	0.447875i	$-0.852181\pi$
$911$		−	27.0362i		−	0.895750i	0.894096	−	0.447875i	$-0.147819\pi$
$912$		0			0
$913$			2.73308i			0.0904518i
$914$		−	0.890697i		−	0.0294616i
$915$		0			0
$916$			26.8437i			0.886941i
$917$	1.94818	−	0.545625i	0.0643345	−	0.0180181i
$918$		0			0
$919$	33.5621			1.10711			0.553556	−	0.832812i	$-0.313270\pi$
$919$	33.5621			1.10711			0.553556	+	0.832812i	$0.313270\pi$
$920$	1.45965			0.0481232
$921$		0			0
$922$	12.0963			0.398372
$923$			14.1897i			0.467060i
$924$		0			0
$925$		−	38.5555i		−	1.26770i
$926$		−	32.2081i		−	1.05843i
$927$		0			0
$928$	1.42708			0.0468461
$929$		−	3.67898i		−	0.120704i	−0.998177	−	0.0603518i	$-0.980778\pi$
$929$		−	3.67898i		−	0.120704i	0.998177	−	0.0603518i	$-0.0192222\pi$
$930$		0			0
$931$	1.01824	+	30.4953i	0.0333715	+	0.999443i
$932$	−23.7407			−0.777654
$933$		0			0
$934$	−34.9016			−1.14201
$935$	22.7666			0.744549
$936$		0			0
$937$			39.3969i			1.28704i	0.765429	+	0.643521i	$0.222527\pi$
$937$			39.3969i			1.28704i	−0.765429	+	0.643521i	$0.777473\pi$
$938$	1.90821	+	6.81334i	0.0623053	+	0.222464i
$939$		0			0
$940$	−8.40880			−0.274265
$941$	−21.0051			−0.684747			−0.342374	−	0.939564i	$-0.611231\pi$
$941$	−21.0051			−0.684747			−0.342374	+	0.939564i	$0.611231\pi$
$942$		0			0
$943$	−1.77079			−0.0576648
$944$	−9.05625			−0.294756
$945$		0			0
$946$		−	7.49587i		−	0.243712i
$947$	−53.0602			−1.72423			−0.862113	−	0.506716i	$-0.830859\pi$
$947$	−53.0602			−1.72423			−0.862113	+	0.506716i	$0.830859\pi$
$948$		0			0
$949$			6.04364i			0.196185i
$950$	15.1238	+	8.51468i	0.490681	+	0.276253i
$951$		0			0
$952$	3.18812	+	11.3833i	0.103328	+	0.368936i
$953$			37.1428i			1.20317i	0.798807	+	0.601587i	$0.205465\pi$
$953$			37.1428i			1.20317i	−0.798807	+	0.601587i	$0.794535\pi$
$954$		0			0
$955$			14.5698i			0.471468i
$956$	7.28786			0.235706
$957$		0			0
$958$	25.6100			0.827422
$959$	8.46996	+	30.2423i	0.273509	+	0.976576i
$960$		0			0
$961$	−20.8359			−0.672125
$962$	16.0959			0.518953
$963$		0			0
$964$	−20.7327			−0.667757
$965$	−55.9567			−1.80131
$966$		0			0
$967$	53.4890			1.72009			0.860046	−	0.510217i	$-0.170435\pi$
$967$	53.4890			1.72009			0.860046	+	0.510217i	$0.170435\pi$
$968$		−	8.10930i		−	0.260643i
$969$		0			0
$970$	−13.7188			−0.440483
$971$	−51.2794			−1.64563			−0.822817	−	0.568307i	$-0.807599\pi$
$971$	−51.2794			−1.64563			−0.822817	+	0.568307i	$0.807599\pi$
$972$		0			0
$973$	22.2447	−	6.23006i	0.713132	−	0.199726i
$974$	−39.0026			−1.24972
$975$		0			0
$976$		−	3.96081i		−	0.126783i
$977$		−	24.3827i		−	0.780073i	−0.920799	−	0.390036i	$-0.872462\pi$
$977$		−	24.3827i		−	0.780073i	0.920799	−	0.390036i	$-0.127538\pi$
$978$		0			0
$979$	−12.3632			−0.395128
$980$	10.8963	+	17.9270i	0.348069	+	0.572656i
$981$		0			0
$982$			19.3973i			0.618993i
$983$	−26.1272			−0.833329			−0.416665	−	0.909060i	$-0.636801\pi$
$983$	−26.1272			−0.833329			−0.416665	+	0.909060i	$0.636801\pi$
$984$		0			0
$985$			13.8267i			0.440554i
$986$			6.37624i			0.203061i
$987$		0			0
$988$	−3.55465	+	6.31378i	−0.113089	+	0.200868i
$989$	2.14728			0.0682796
$990$		0			0
$991$		−	26.3751i		−	0.837831i	−0.908025	−	0.418916i	$-0.862410\pi$
$991$		−	26.3751i		−	0.837831i	0.908025	−	0.418916i	$-0.137590\pi$
$992$			3.18812i			0.101223i
$993$		0			0
$994$	−6.09103	−	21.7483i	−0.193196	−	0.689814i
$995$	21.7926			0.690871
$996$		0			0
$997$			18.7359i			0.593371i	0.954975	+	0.296686i	$0.0958813\pi$
$997$			18.7359i			0.593371i	−0.954975	+	0.296686i	$0.904119\pi$
$998$			18.1171i			0.573487i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.e.d.1063.9	✓	24
3.2	odd	2	inner	2394.2.e.d.1063.20	yes	24
7.6	odd	2	inner	2394.2.e.d.1063.17	yes	24
19.18	odd	2	inner	2394.2.e.d.1063.19	yes	24
21.20	even	2	inner	2394.2.e.d.1063.12	yes	24
57.56	even	2	inner	2394.2.e.d.1063.10	yes	24
133.132	even	2	inner	2394.2.e.d.1063.11	yes	24
399.398	odd	2	inner	2394.2.e.d.1063.18	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.e.d.1063.9	✓	24	1.1	even	1	trivial
2394.2.e.d.1063.10	yes	24	57.56	even	2	inner
2394.2.e.d.1063.11	yes	24	133.132	even	2	inner
2394.2.e.d.1063.12	yes	24	21.20	even	2	inner
2394.2.e.d.1063.17	yes	24	7.6	odd	2	inner
2394.2.e.d.1063.18	yes	24	399.398	odd	2	inner
2394.2.e.d.1063.19	yes	24	19.18	odd	2	inner
2394.2.e.d.1063.20	yes	24	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.99695i q^{5} +(-0.713538 - 2.54772i) q^{7} +1.00000i q^{8} +2.99695 q^{10} -1.70021 q^{11} +1.66226 q^{13} +(-2.54772 + 0.713538i) q^{14} +1.00000 q^{16} +4.46805i q^{17} +(2.13844 - 3.79830i) q^{19} -2.99695i q^{20} +1.70021i q^{22} -0.487044 q^{23} -3.98173 q^{25} -1.66226i q^{26} +(0.713538 + 2.54772i) q^{28} +1.42708i q^{29} -3.18812 q^{31} -1.00000i q^{32} +4.46805 q^{34} +(7.63539 - 2.13844i) q^{35} +9.68312i q^{37} +(-3.79830 - 2.13844i) q^{38} -2.99695 q^{40} +3.63579 q^{41} -4.40880 q^{43} +1.70021 q^{44} +0.487044i q^{46} -2.80578i q^{47} +(-5.98173 + 3.63579i) q^{49} +3.98173i q^{50} -1.66226 q^{52} +8.53638i q^{53} -5.09543i q^{55} +(2.54772 - 0.713538i) q^{56} +1.42708 q^{58} -9.05625 q^{59} -3.96081i q^{61} +3.18812i q^{62} -1.00000 q^{64} +4.98173i q^{65} -2.67429i q^{67} -4.46805i q^{68} +(-2.13844 - 7.63539i) q^{70} +8.53638i q^{71} +3.63579i q^{73} +9.68312 q^{74} +(-2.13844 + 3.79830i) q^{76} +(1.21316 + 4.33164i) q^{77} +14.7837i q^{79} +2.99695i q^{80} -3.63579i q^{82} -1.60750i q^{83} -13.3905 q^{85} +4.40880i q^{86} -1.70021i q^{88} +7.27157 q^{89} +(-1.18609 - 4.23498i) q^{91} +0.487044 q^{92} -2.80578 q^{94} +(11.3833 + 6.40880i) q^{95} -4.57757 q^{97} +(3.63579 + 5.98173i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{4} - 8 q^{7} + 24 q^{16} + 24 q^{25} + 8 q^{28} + 32 q^{43} - 24 q^{49} + 16 q^{58} - 24 q^{64} - 64 q^{85}+O(q^{100}) \) 24 * q - 24 * q^4 - 8 * q^7 + 24 * q^16 + 24 * q^25 + 8 * q^28 + 32 * q^43 - 24 * q^49 + 16 * q^58 - 24 * q^64 - 64 * q^85

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