LMFDB - Embedded newform 2394.2.f.b.2015.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(2015,2394)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2394.2015");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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			2015.1
Character	$\chi$	$=$	2394.2015
Dual form			2394.2.f.b.2015.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.90763 q^{5} +(-0.638569 - 2.56753i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +1.90763 q^{5} +(-0.638569 - 2.56753i) q^{7} +1.00000i q^{8} -1.90763i q^{10} -6.32304i q^{11} -2.98283i q^{13} +(-2.56753 + 0.638569i) q^{14} +1.00000 q^{16} -6.47753 q^{17} -1.00000i q^{19} -1.90763 q^{20} -6.32304 q^{22} -0.159186i q^{23} -1.36094 q^{25} -2.98283 q^{26} +(0.638569 + 2.56753i) q^{28} +3.91008i q^{29} +8.89827i q^{31} -1.00000i q^{32} +6.47753i q^{34} +(-1.21815 - 4.89791i) q^{35} -10.9257 q^{37} -1.00000 q^{38} +1.90763i q^{40} +8.26162 q^{41} +10.7342 q^{43} +6.32304i q^{44} -0.159186 q^{46} +4.79039 q^{47} +(-6.18446 + 3.27910i) q^{49} +1.36094i q^{50} +2.98283i q^{52} +3.69748i q^{53} -12.0620i q^{55} +(2.56753 - 0.638569i) q^{56} +3.91008 q^{58} -6.73274 q^{59} -5.53395i q^{61} +8.89827 q^{62} -1.00000 q^{64} -5.69015i q^{65} -10.8812 q^{67} +6.47753 q^{68} +(-4.89791 + 1.21815i) q^{70} +7.06392i q^{71} -8.91258i q^{73} +10.9257i q^{74} +1.00000i q^{76} +(-16.2346 + 4.03770i) q^{77} -11.5039 q^{79} +1.90763 q^{80} -8.26162i q^{82} +5.06969 q^{83} -12.3567 q^{85} -10.7342i q^{86} +6.32304 q^{88} +11.2935 q^{89} +(-7.65853 + 1.90475i) q^{91} +0.159186i q^{92} -4.79039i q^{94} -1.90763i q^{95} +4.09624i q^{97} +(3.27910 + 6.18446i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{4} + 4 q^{7}+O(q^{10}) $ 24 * q - 24 * q^4 + 4 * q^7	$ 24 q - 24 q^{4} + 4 q^{7} + 4 q^{14} + 24 q^{16} - 32 q^{17} - 8 q^{22} + 16 q^{25} - 4 q^{28} + 24 q^{35} - 24 q^{38} + 8 q^{41} + 16 q^{43} - 8 q^{46} - 16 q^{49} - 4 q^{56} + 16 q^{58} - 16 q^{59} + 16 q^{62} - 24 q^{64} - 24 q^{67} + 32 q^{68} - 16 q^{70} - 8 q^{77} - 40 q^{79} + 64 q^{83} + 40 q^{85} + 8 q^{88} + 64 q^{89} + 8 q^{91} + 16 q^{98}+O(q^{100}) $ 24 * q - 24 * q^4 + 4 * q^7 + 4 * q^14 + 24 * q^16 - 32 * q^17 - 8 * q^22 + 16 * q^25 - 4 * q^28 + 24 * q^35 - 24 * q^38 + 8 * q^41 + 16 * q^43 - 8 * q^46 - 16 * q^49 - 4 * q^56 + 16 * q^58 - 16 * q^59 + 16 * q^62 - 24 * q^64 - 24 * q^67 + 32 * q^68 - 16 * q^70 - 8 * q^77 - 40 * q^79 + 64 * q^83 + 40 * q^85 + 8 * q^88 + 64 * q^89 + 8 * q^91 + 16 * q^98

Display 100 coefficients 10 coefficients

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
$5$	1.90763			0.853119			0.426559	−	0.904460i	$-0.359726\pi$
$5$	1.90763			0.853119			0.426559	+	0.904460i	$0.359726\pi$
$6$		0			0
$7$	−0.638569	−	2.56753i	−0.241356	−	0.970437i
$7$	−0.638569	−	2.56753i	−0.241356	−	0.970437i
$8$			1.00000i			0.353553i
$9$		0			0
$10$		−	1.90763i		−	0.603246i
$11$		−	6.32304i		−	1.90647i	−0.302231	−	0.953235i	$-0.597731\pi$
$11$		−	6.32304i		−	1.90647i	0.302231	−	0.953235i	$-0.402269\pi$
$12$		0			0
$13$		−	2.98283i		−	0.827289i	−0.910438	−	0.413645i	$-0.864256\pi$
$13$		−	2.98283i		−	0.827289i	0.910438	−	0.413645i	$-0.135744\pi$
$14$	−2.56753	+	0.638569i	−0.686202	+	0.170665i
$15$		0			0
$16$	1.00000			0.250000
$17$	−6.47753			−1.57103			−0.785516	−	0.618842i	$-0.787602\pi$
$17$	−6.47753			−1.57103			−0.785516	+	0.618842i	$0.787602\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$	−1.90763			−0.426559
$21$		0			0
$22$	−6.32304			−1.34808
$23$		−	0.159186i		−	0.0331926i	−0.999862	−	0.0165963i	$-0.994717\pi$
$23$		−	0.159186i		−	0.0331926i	0.999862	−	0.0165963i	$-0.00528300\pi$
$24$		0			0
$25$	−1.36094			−0.272189
$26$	−2.98283			−0.584982
$27$		0			0
$28$	0.638569	+	2.56753i	0.120678	+	0.485218i
$29$			3.91008i			0.726083i	0.931773	+	0.363041i	$0.118262\pi$
$29$			3.91008i			0.726083i	−0.931773	+	0.363041i	$0.881738\pi$
$30$		0			0
$31$			8.89827i			1.59818i	0.601213	+	0.799088i	$0.294684\pi$
$31$			8.89827i			1.59818i	−0.601213	+	0.799088i	$0.705316\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			6.47753i			1.11089i
$35$	−1.21815	−	4.89791i	−0.205906	−	0.827897i
$36$		0			0
$37$	−10.9257			−1.79618			−0.898088	−	0.439815i	$-0.855044\pi$
$37$	−10.9257			−1.79618			−0.898088	+	0.439815i	$0.855044\pi$
$38$	−1.00000			−0.162221
$39$		0			0
$40$			1.90763i			0.301623i
$41$	8.26162			1.29025			0.645124	−	0.764078i	$-0.276806\pi$
$41$	8.26162			1.29025			0.645124	+	0.764078i	$0.276806\pi$
$42$		0			0
$43$	10.7342			1.63695			0.818474	−	0.574544i	$-0.194821\pi$
$43$	10.7342			1.63695			0.818474	+	0.574544i	$0.194821\pi$
$44$			6.32304i			0.953235i
$45$		0			0
$46$	−0.159186			−0.0234707
$47$	4.79039			0.698750			0.349375	−	0.936983i	$-0.386394\pi$
$47$	4.79039			0.698750			0.349375	+	0.936983i	$0.386394\pi$
$48$		0			0
$49$	−6.18446	+	3.27910i	−0.883494	+	0.468442i
$50$			1.36094i			0.192467i
$51$		0			0
$52$			2.98283i			0.413645i
$53$			3.69748i			0.507888i	0.967219	+	0.253944i	$0.0817279\pi$
$53$			3.69748i			0.507888i	−0.967219	+	0.253944i	$0.918272\pi$
$54$		0			0
$55$		−	12.0620i		−	1.62644i
$56$	2.56753	−	0.638569i	0.343101	−	0.0853324i
$57$		0			0
$58$	3.91008			0.513418
$59$	−6.73274			−0.876529			−0.438264	−	0.898846i	$-0.644407\pi$
$59$	−6.73274			−0.876529			−0.438264	+	0.898846i	$0.644407\pi$
$60$		0			0
$61$		−	5.53395i		−	0.708550i	−0.935141	−	0.354275i	$-0.884728\pi$
$61$		−	5.53395i		−	0.708550i	0.935141	−	0.354275i	$-0.115272\pi$
$62$	8.89827			1.13008
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	5.69015i		−	0.705776i
$66$		0			0
$67$	−10.8812			−1.32934			−0.664672	−	0.747135i	$-0.731429\pi$
$67$	−10.8812			−1.32934			−0.664672	+	0.747135i	$0.731429\pi$
$68$	6.47753			0.785516
$69$		0			0
$70$	−4.89791	+	1.21815i	−0.585412	+	0.145597i
$71$			7.06392i			0.838333i	0.907909	+	0.419167i	$0.137678\pi$
$71$			7.06392i			0.838333i	−0.907909	+	0.419167i	$0.862322\pi$
$72$		0			0
$73$		−	8.91258i		−	1.04314i	−0.853209	−	0.521569i	$-0.825347\pi$
$73$		−	8.91258i		−	1.04314i	0.853209	−	0.521569i	$-0.174653\pi$
$74$			10.9257i			1.27009i
$75$		0			0
$76$			1.00000i			0.114708i
$77$	−16.2346	+	4.03770i	−1.85011	+	0.460139i
$78$		0			0
$79$	−11.5039			−1.29429			−0.647147	−	0.762365i	$-0.724038\pi$
$79$	−11.5039			−1.29429			−0.647147	+	0.762365i	$0.724038\pi$
$80$	1.90763			0.213280
$81$		0			0
$82$		−	8.26162i		−	0.912343i
$83$	5.06969			0.556471			0.278236	−	0.960513i	$-0.410250\pi$
$83$	5.06969			0.556471			0.278236	+	0.960513i	$0.410250\pi$
$84$		0			0
$85$	−12.3567			−1.34028
$86$		−	10.7342i		−	1.15750i
$87$		0			0
$88$	6.32304			0.674039
$89$	11.2935			1.19711			0.598553	−	0.801083i	$-0.295742\pi$
$89$	11.2935			1.19711			0.598553	+	0.801083i	$0.295742\pi$
$90$		0			0
$91$	−7.65853	+	1.90475i	−0.802832	+	0.199672i
$92$			0.159186i			0.0165963i
$93$		0			0
$94$		−	4.79039i		−	0.494091i
$95$		−	1.90763i		−	0.195719i
$96$		0			0
$97$			4.09624i			0.415910i	0.978138	+	0.207955i	$0.0666807\pi$
$97$			4.09624i			0.415910i	−0.978138	+	0.207955i	$0.933319\pi$
$98$	3.27910	+	6.18446i	0.331239	+	0.624725i
$99$		0			0
$100$	1.36094			0.136094
$101$	8.52076			0.847848			0.423924	−	0.905698i	$-0.360652\pi$
$101$	8.52076			0.847848			0.423924	+	0.905698i	$0.360652\pi$
$102$		0			0
$103$		−	5.59944i		−	0.551729i	−0.961196	−	0.275865i	$-0.911036\pi$
$103$		−	5.59944i		−	0.551729i	0.961196	−	0.275865i	$-0.0889642\pi$
$104$	2.98283			0.292491
$105$		0			0
$106$	3.69748			0.359131
$107$			4.05903i			0.392401i	0.980564	+	0.196201i	$0.0628604\pi$
$107$			4.05903i			0.392401i	−0.980564	+	0.196201i	$0.937140\pi$
$108$		0			0
$109$	−18.9005			−1.81034			−0.905170	−	0.425050i	$-0.860257\pi$
$109$	−18.9005			−1.81034			−0.905170	+	0.425050i	$0.860257\pi$
$110$	−12.0620			−1.15007
$111$		0			0
$112$	−0.638569	−	2.56753i	−0.0603391	−	0.242609i
$113$		−	16.9961i		−	1.59886i	−0.600760	−	0.799430i	$-0.705135\pi$
$113$		−	16.9961i		−	1.59886i	0.600760	−	0.799430i	$-0.294865\pi$
$114$		0			0
$115$		−	0.303668i		−	0.0283172i
$116$		−	3.91008i		−	0.363041i
$117$		0			0
$118$			6.73274i			0.619799i
$119$	4.13635	+	16.6313i	0.379178	+	1.52459i
$120$		0			0
$121$	−28.9809			−2.63463
$122$	−5.53395			−0.501021
$123$		0			0
$124$		−	8.89827i		−	0.799088i
$125$	−12.1343			−1.08533
$126$		0			0
$127$	5.75581			0.510745			0.255373	−	0.966843i	$-0.417802\pi$
$127$	5.75581			0.510745			0.255373	+	0.966843i	$0.417802\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−5.69015			−0.499059
$131$	15.4764			1.35218			0.676089	−	0.736820i	$-0.263673\pi$
$131$	15.4764			1.35218			0.676089	+	0.736820i	$0.263673\pi$
$132$		0			0
$133$	−2.56753	+	0.638569i	−0.222633	+	0.0553710i
$134$			10.8812i			0.939989i
$135$		0			0
$136$		−	6.47753i		−	0.555443i
$137$		−	10.0966i		−	0.862613i	−0.902206	−	0.431306i	$-0.858053\pi$
$137$		−	10.0966i		−	0.862613i	0.902206	−	0.431306i	$-0.141947\pi$
$138$		0			0
$139$		−	1.92711i		−	0.163455i	−0.996655	−	0.0817276i	$-0.973956\pi$
$139$		−	1.92711i		−	0.163455i	0.996655	−	0.0817276i	$-0.0260437\pi$
$140$	1.21815	+	4.89791i	0.102953	+	0.413949i
$141$		0			0
$142$	7.06392			0.592791
$143$	−18.8606			−1.57720
$144$		0			0
$145$			7.45898i			0.619435i
$146$	−8.91258			−0.737610
$147$		0			0
$148$	10.9257			0.898088
$149$		−	8.47493i		−	0.694293i	−0.937811	−	0.347146i	$-0.887151\pi$
$149$		−	8.47493i		−	0.694293i	0.937811	−	0.347146i	$-0.112849\pi$
$150$		0			0
$151$	11.3265			0.921740			0.460870	−	0.887468i	$-0.347537\pi$
$151$	11.3265			0.921740			0.460870	+	0.887468i	$0.347537\pi$
$152$	1.00000			0.0811107
$153$		0			0
$154$	4.03770	+	16.2346i	0.325367	+	1.30822i
$155$			16.9746i			1.36343i
$156$		0			0
$157$		−	24.3412i		−	1.94264i	−0.237772	−	0.971321i	$-0.576417\pi$
$157$		−	24.3412i		−	1.94264i	0.237772	−	0.971321i	$-0.423583\pi$
$158$			11.5039i			0.915205i
$159$		0			0
$160$		−	1.90763i		−	0.150811i
$161$	−0.408715	+	0.101651i	−0.0322113	+	0.00801124i
$162$		0			0
$163$	−2.71446			−0.212613			−0.106306	−	0.994333i	$-0.533902\pi$
$163$	−2.71446			−0.212613			−0.106306	+	0.994333i	$0.533902\pi$
$164$	−8.26162			−0.645124
$165$		0			0
$166$		−	5.06969i		−	0.393484i
$167$	0.803303			0.0621614			0.0310807	−	0.999517i	$-0.490105\pi$
$167$	0.803303			0.0621614			0.0310807	+	0.999517i	$0.490105\pi$
$168$		0			0
$169$	4.10270			0.315592
$170$			12.3567i			0.947718i
$171$		0			0
$172$	−10.7342			−0.818474
$173$	1.01981			0.0775345			0.0387673	−	0.999248i	$-0.487657\pi$
$173$	1.01981			0.0775345			0.0387673	+	0.999248i	$0.487657\pi$
$174$		0			0
$175$	0.869057	+	3.49427i	0.0656945	+	0.264142i
$176$		−	6.32304i		−	0.476617i
$177$		0			0
$178$		−	11.2935i		−	0.846482i
$179$		−	3.69669i		−	0.276304i	−0.990411	−	0.138152i	$-0.955884\pi$
$179$		−	3.69669i		−	0.276304i	0.990411	−	0.138152i	$-0.0441162\pi$
$180$		0			0
$181$			4.90195i			0.364359i	0.983265	+	0.182180i	$0.0583153\pi$
$181$			4.90195i			0.364359i	−0.983265	+	0.182180i	$0.941685\pi$
$182$	1.90475	+	7.65853i	0.141189	+	0.567688i
$183$		0			0
$184$	0.159186			0.0117353
$185$	−20.8422			−1.53235
$186$		0			0
$187$			40.9577i			2.99512i
$188$	−4.79039			−0.349375
$189$		0			0
$190$	−1.90763			−0.138394
$191$			2.09543i			0.151620i	0.997122	+	0.0758102i	$0.0241543\pi$
$191$			2.09543i			0.151620i	−0.997122	+	0.0758102i	$0.975846\pi$
$192$		0			0
$193$	7.90248			0.568833			0.284417	−	0.958701i	$-0.408200\pi$
$193$	7.90248			0.568833			0.284417	+	0.958701i	$0.408200\pi$
$194$	4.09624			0.294093
$195$		0			0
$196$	6.18446	−	3.27910i	0.441747	−	0.234221i
$197$		−	3.61443i		−	0.257517i	−0.991676	−	0.128759i	$-0.958901\pi$
$197$		−	3.61443i		−	0.257517i	0.991676	−	0.128759i	$-0.0410993\pi$
$198$		0			0
$199$		−	21.0820i		−	1.49446i	−0.664564	−	0.747232i	$-0.731383\pi$
$199$		−	21.0820i		−	1.49446i	0.664564	−	0.747232i	$-0.268617\pi$
$200$		−	1.36094i		−	0.0962333i
$201$		0			0
$202$		−	8.52076i		−	0.599519i
$203$	10.0393	−	2.49685i	0.704617	−	0.175245i
$204$		0			0
$205$	15.7601			1.10073
$206$	−5.59944			−0.390132
$207$		0			0
$208$		−	2.98283i		−	0.206822i
$209$	−6.32304			−0.437374
$210$		0			0
$211$	−27.8742			−1.91894			−0.959470	−	0.281811i	$-0.909065\pi$
$211$	−27.8742			−1.91894			−0.959470	+	0.281811i	$0.909065\pi$
$212$		−	3.69748i		−	0.253944i
$213$		0			0
$214$	4.05903			0.277470
$215$	20.4769			1.39651
$216$		0			0
$217$	22.8466	−	5.68216i	1.55093	−	0.385730i
$218$			18.9005i			1.28010i
$219$		0			0
$220$			12.0620i			0.813222i
$221$			19.3214i			1.29970i
$222$		0			0
$223$		−	7.38131i		−	0.494289i	−0.968979	−	0.247145i	$-0.920508\pi$
$223$		−	7.38131i		−	0.494289i	0.968979	−	0.247145i	$-0.0794923\pi$
$224$	−2.56753	+	0.638569i	−0.171551	+	0.0426662i
$225$		0			0
$226$	−16.9961			−1.13056
$227$	29.6825			1.97009			0.985047	−	0.172287i	$-0.0551156\pi$
$227$	29.6825			1.97009			0.985047	+	0.172287i	$0.0551156\pi$
$228$		0			0
$229$			23.0054i			1.52024i	0.649785	+	0.760118i	$0.274859\pi$
$229$			23.0054i			1.52024i	−0.649785	+	0.760118i	$0.725141\pi$
$230$	−0.303668			−0.0200233
$231$		0			0
$232$	−3.91008			−0.256709
$233$			8.57518i			0.561779i	0.959740	+	0.280889i	$0.0906294\pi$
$233$			8.57518i			0.561779i	−0.959740	+	0.280889i	$0.909371\pi$
$234$		0			0
$235$	9.13829			0.596116
$236$	6.73274			0.438264
$237$		0			0
$238$	16.6313	−	4.13635i	1.07805	−	0.268120i
$239$		−	1.18924i		−	0.0769259i	−0.999260	−	0.0384629i	$-0.987754\pi$
$239$		−	1.18924i		−	0.0769259i	0.999260	−	0.0384629i	$-0.0122462\pi$
$240$		0			0
$241$		−	6.83165i		−	0.440065i	−0.975492	−	0.220033i	$-0.929384\pi$
$241$		−	6.83165i		−	0.440065i	0.975492	−	0.220033i	$-0.0706164\pi$
$242$			28.9809i			1.86296i
$243$		0			0
$244$			5.53395i			0.354275i
$245$	−11.7977	+	6.25530i	−0.753725	+	0.399637i
$246$		0			0
$247$	−2.98283			−0.189793
$248$	−8.89827			−0.565041
$249$		0			0
$250$			12.1343i			0.767443i
$251$	−21.9189			−1.38351			−0.691755	−	0.722132i	$-0.743162\pi$
$251$	−21.9189			−1.38351			−0.691755	+	0.722132i	$0.743162\pi$
$252$		0			0
$253$	−1.00654			−0.0632806
$254$		−	5.75581i		−	0.361152i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−5.72995			−0.357424			−0.178712	−	0.983901i	$-0.557193\pi$
$257$	−5.72995			−0.357424			−0.178712	+	0.983901i	$0.557193\pi$
$258$		0			0
$259$	6.97683	+	28.0521i	0.433519	+	1.74308i
$260$			5.69015i			0.352888i
$261$		0			0
$262$		−	15.4764i		−	0.956135i
$263$		−	5.53123i		−	0.341070i	−0.985352	−	0.170535i	$-0.945450\pi$
$263$		−	5.53123i		−	0.341070i	0.985352	−	0.170535i	$-0.0545496\pi$
$264$		0			0
$265$			7.05342i			0.433288i
$266$	0.638569	+	2.56753i	0.0391532	+	0.157426i
$267$		0			0
$268$	10.8812			0.664672
$269$	−10.0658			−0.613722			−0.306861	−	0.951754i	$-0.599279\pi$
$269$	−10.0658			−0.613722			−0.306861	+	0.951754i	$0.599279\pi$
$270$		0			0
$271$		−	10.2081i		−	0.620099i	−0.950720	−	0.310049i	$-0.899654\pi$
$271$		−	10.2081i		−	0.620099i	0.950720	−	0.310049i	$-0.100346\pi$
$272$	−6.47753			−0.392758
$273$		0			0
$274$	−10.0966			−0.609959
$275$			8.60531i			0.518920i
$276$		0			0
$277$	−27.5935			−1.65793			−0.828966	−	0.559299i	$-0.811071\pi$
$277$	−27.5935			−1.65793			−0.828966	+	0.559299i	$0.811071\pi$
$278$	−1.92711			−0.115580
$279$		0			0
$280$	4.89791	−	1.21815i	0.292706	−	0.0727986i
$281$		−	23.1757i		−	1.38255i	−0.722593	−	0.691274i	$-0.757050\pi$
$281$		−	23.1757i		−	1.38255i	0.722593	−	0.691274i	$-0.242950\pi$
$282$		0			0
$283$			6.75569i			0.401584i	0.979634	+	0.200792i	$0.0643515\pi$
$283$			6.75569i			0.401584i	−0.979634	+	0.200792i	$0.935648\pi$
$284$		−	7.06392i		−	0.419167i
$285$		0			0
$286$			18.8606i			1.11525i
$287$	−5.27562	−	21.2120i	−0.311410	−	1.25210i
$288$		0			0
$289$	24.9584			1.46814
$290$	7.45898			0.438007
$291$		0			0
$292$			8.91258i			0.521569i
$293$	−1.28671			−0.0751707			−0.0375853	−	0.999293i	$-0.511967\pi$
$293$	−1.28671			−0.0751707			−0.0375853	+	0.999293i	$0.511967\pi$
$294$		0			0
$295$	−12.8436			−0.747783
$296$		−	10.9257i		−	0.635044i
$297$		0			0
$298$	−8.47493			−0.490939
$299$	−0.474825			−0.0274598
$300$		0			0
$301$	−6.85452	−	27.5604i	−0.395088	−	1.58855i
$302$		−	11.3265i		−	0.651769i
$303$		0			0
$304$		−	1.00000i		−	0.0573539i
$305$		−	10.5567i		−	0.604477i
$306$		0			0
$307$			6.82469i			0.389506i	0.980852	+	0.194753i	$0.0623905\pi$
$307$			6.82469i			0.389506i	−0.980852	+	0.194753i	$0.937610\pi$
$308$	16.2346	−	4.03770i	0.925054	−	0.230069i
$309$		0			0
$310$	16.9746			0.964094
$311$	6.88370			0.390339			0.195169	−	0.980770i	$-0.437474\pi$
$311$	6.88370			0.390339			0.195169	+	0.980770i	$0.437474\pi$
$312$		0			0
$313$		−	31.0256i		−	1.75367i	−0.480792	−	0.876835i	$-0.659651\pi$
$313$		−	31.0256i		−	1.75367i	0.480792	−	0.876835i	$-0.340349\pi$
$314$	−24.3412			−1.37366
$315$		0			0
$316$	11.5039			0.647147
$317$		−	30.8638i		−	1.73348i	−0.498757	−	0.866742i	$-0.666210\pi$
$317$		−	30.8638i		−	1.73348i	0.498757	−	0.866742i	$-0.333790\pi$
$318$		0			0
$319$	24.7236			1.38426
$320$	−1.90763			−0.106640
$321$		0			0
$322$	0.101651	+	0.408715i	0.00566480	+	0.0227768i
$323$			6.47753i			0.360419i
$324$		0			0
$325$			4.05947i			0.225179i
$326$			2.71446i			0.150340i
$327$		0			0
$328$			8.26162i			0.456172i
$329$	−3.05899	−	12.2995i	−0.168648	−	0.678092i
$330$		0			0
$331$	−12.7032			−0.698230			−0.349115	−	0.937080i	$-0.613518\pi$
$331$	−12.7032			−0.698230			−0.349115	+	0.937080i	$0.613518\pi$
$332$	−5.06969			−0.278236
$333$		0			0
$334$		−	0.803303i		−	0.0439548i
$335$	−20.7572			−1.13409
$336$		0			0
$337$	20.8673			1.13672			0.568358	−	0.822781i	$-0.307579\pi$
$337$	20.8673			1.13672			0.568358	+	0.822781i	$0.307579\pi$
$338$		−	4.10270i		−	0.223157i
$339$		0			0
$340$	12.3567			0.670138
$341$	56.2642			3.04688
$342$		0			0
$343$	12.3684	+	13.7849i	0.667830	+	0.744313i
$344$			10.7342i			0.578748i
$345$		0			0
$346$		−	1.01981i		−	0.0548252i
$347$		−	3.34713i		−	0.179683i	−0.995956	−	0.0898417i	$-0.971364\pi$
$347$		−	3.34713i		−	0.179683i	0.995956	−	0.0898417i	$-0.0286361\pi$
$348$		0			0
$349$			24.1080i			1.29047i	0.763984	+	0.645236i	$0.223241\pi$
$349$			24.1080i			1.29047i	−0.763984	+	0.645236i	$0.776759\pi$
$350$	3.49427	−	0.869057i	0.186777	−	0.0464530i
$351$		0			0
$352$	−6.32304			−0.337019
$353$	8.42916			0.448639			0.224319	−	0.974516i	$-0.427984\pi$
$353$	8.42916			0.448639			0.224319	+	0.974516i	$0.427984\pi$
$354$		0			0
$355$			13.4754i			0.715197i
$356$	−11.2935			−0.598553
$357$		0			0
$358$	−3.69669			−0.195376
$359$		−	11.3655i		−	0.599850i	−0.953963	−	0.299925i	$-0.903038\pi$
$359$		−	11.3655i		−	0.599850i	0.953963	−	0.299925i	$-0.0969616\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	4.90195			0.257641
$363$		0			0
$364$	7.65853	−	1.90475i	0.401416	−	0.0998358i
$365$		−	17.0019i		−	0.889920i
$366$		0			0
$367$		−	1.24677i		−	0.0650808i	−0.999470	−	0.0325404i	$-0.989640\pi$
$367$		−	1.24677i		−	0.0650808i	0.999470	−	0.0325404i	$-0.0103598\pi$
$368$		−	0.159186i		−	0.00829814i
$369$		0			0
$370$			20.8422i			1.08354i
$371$	9.49340	−	2.36110i	0.492873	−	0.122582i
$372$		0			0
$373$	3.65353			0.189172			0.0945862	−	0.995517i	$-0.469847\pi$
$373$	3.65353			0.189172			0.0945862	+	0.995517i	$0.469847\pi$
$374$	40.9577			2.11787
$375$		0			0
$376$			4.79039i			0.247045i
$377$	11.6631			0.600681
$378$		0			0
$379$	−4.28772			−0.220245			−0.110123	−	0.993918i	$-0.535124\pi$
$379$	−4.28772			−0.220245			−0.110123	+	0.993918i	$0.535124\pi$
$380$			1.90763i			0.0978594i
$381$		0			0
$382$	2.09543			0.107212
$383$	−8.21116			−0.419571			−0.209785	−	0.977747i	$-0.567277\pi$
$383$	−8.21116			−0.419571			−0.209785	+	0.977747i	$0.567277\pi$
$384$		0			0
$385$	−30.9697	+	7.70244i	−1.57836	+	0.392553i
$386$		−	7.90248i		−	0.402226i
$387$		0			0
$388$		−	4.09624i		−	0.207955i
$389$		−	17.1979i		−	0.871970i	−0.899954	−	0.435985i	$-0.856400\pi$
$389$		−	17.1979i		−	0.871970i	0.899954	−	0.435985i	$-0.143600\pi$
$390$		0			0
$391$			1.03113i			0.0521465i
$392$	−3.27910	−	6.18446i	−0.165619	−	0.312362i
$393$		0			0
$394$	−3.61443			−0.182092
$395$	−21.9453			−1.10419
$396$		0			0
$397$		−	24.6749i		−	1.23840i	−0.785235	−	0.619198i	$-0.787458\pi$
$397$		−	24.6749i		−	1.23840i	0.785235	−	0.619198i	$-0.212542\pi$
$398$	−21.0820			−1.05674
$399$		0			0
$400$	−1.36094			−0.0680472
$401$			4.96284i			0.247832i	0.992293	+	0.123916i	$0.0395454\pi$
$401$			4.96284i			0.247832i	−0.992293	+	0.123916i	$0.960455\pi$
$402$		0			0
$403$	26.5421			1.32215
$404$	−8.52076			−0.423924
$405$		0			0
$406$	−2.49685	−	10.0393i	−0.123917	−	0.498240i
$407$			69.0838i			3.42436i
$408$		0			0
$409$		−	9.50965i		−	0.470222i	−0.971969	−	0.235111i	$-0.924455\pi$
$409$		−	9.50965i		−	0.470222i	0.971969	−	0.235111i	$-0.0755454\pi$
$410$		−	15.7601i		−	0.778337i
$411$		0			0
$412$			5.59944i			0.275865i
$413$	4.29932	+	17.2865i	0.211556	+	0.850615i
$414$		0			0
$415$	9.67110			0.474736
$416$	−2.98283			−0.146245
$417$		0			0
$418$			6.32304i			0.309270i
$419$	34.5145			1.68614			0.843072	−	0.537800i	$-0.180744\pi$
$419$	34.5145			1.68614			0.843072	+	0.537800i	$0.180744\pi$
$420$		0			0
$421$	−2.26134			−0.110211			−0.0551055	−	0.998481i	$-0.517550\pi$
$421$	−2.26134			−0.110211			−0.0551055	+	0.998481i	$0.517550\pi$
$422$			27.8742i			1.35690i
$423$		0			0
$424$	−3.69748			−0.179565
$425$	8.81555			0.427617
$426$		0			0
$427$	−14.2086	+	3.53381i	−0.687603	+	0.171013i
$428$		−	4.05903i		−	0.196201i
$429$		0			0
$430$		−	20.4769i		−	0.987482i
$431$			38.4148i			1.85037i	0.379511	+	0.925187i	$0.376092\pi$
$431$			38.4148i			1.85037i	−0.379511	+	0.925187i	$0.623908\pi$
$432$		0			0
$433$			25.8100i			1.24035i	0.784464	+	0.620175i	$0.212938\pi$
$433$			25.8100i			1.24035i	−0.784464	+	0.620175i	$0.787062\pi$
$434$	−5.68216	−	22.8466i	−0.272752	−	1.09667i
$435$		0			0
$436$	18.9005			0.905170
$437$	−0.159186			−0.00761489
$438$		0			0
$439$		−	27.3489i		−	1.30529i	−0.757663	−	0.652646i	$-0.773659\pi$
$439$		−	27.3489i		−	1.30529i	0.757663	−	0.652646i	$-0.226341\pi$
$440$	12.0620			0.575035
$441$		0			0
$442$	19.3214			0.919025
$443$			24.0719i			1.14369i	0.820361	+	0.571845i	$0.193772\pi$
$443$			24.0719i			1.14369i	−0.820361	+	0.571845i	$0.806228\pi$
$444$		0			0
$445$	21.5438			1.02127
$446$	−7.38131			−0.349515
$447$		0			0
$448$	0.638569	+	2.56753i	0.0301696	+	0.121305i
$449$		−	4.71176i		−	0.222362i	−0.993800	−	0.111181i	$-0.964537\pi$
$449$		−	4.71176i		−	0.222362i	0.993800	−	0.111181i	$-0.0354633\pi$
$450$		0			0
$451$		−	52.2386i		−	2.45982i
$452$			16.9961i			0.799430i
$453$		0			0
$454$		−	29.6825i		−	1.39307i
$455$	−14.6096	+	3.63355i	−0.684911	+	0.170344i
$456$		0			0
$457$	6.23529			0.291675			0.145837	−	0.989309i	$-0.453412\pi$
$457$	6.23529			0.291675			0.145837	+	0.989309i	$0.453412\pi$
$458$	23.0054			1.07497
$459$		0			0
$460$			0.303668i			0.0141586i
$461$	33.2663			1.54937			0.774684	−	0.632348i	$-0.217909\pi$
$461$	33.2663			1.54937			0.774684	+	0.632348i	$0.217909\pi$
$462$		0			0
$463$	16.2046			0.753092			0.376546	−	0.926398i	$-0.377112\pi$
$463$	16.2046			0.753092			0.376546	+	0.926398i	$0.377112\pi$
$464$			3.91008i			0.181521i
$465$		0			0
$466$	8.57518			0.397238
$467$	−32.1363			−1.48709			−0.743545	−	0.668686i	$-0.766857\pi$
$467$	−32.1363			−1.48709			−0.743545	+	0.668686i	$0.766857\pi$
$468$		0			0
$469$	6.94837	+	27.9377i	0.320846	+	1.29004i
$470$		−	9.13829i		−	0.421518i
$471$		0			0
$472$		−	6.73274i		−	0.309900i
$473$		−	67.8727i		−	3.12079i
$474$		0			0
$475$			1.36094i			0.0624444i
$476$	−4.13635	−	16.6313i	−0.189589	−	0.762293i
$477$		0			0
$478$	−1.18924			−0.0543948
$479$	−4.14290			−0.189294			−0.0946470	−	0.995511i	$-0.530172\pi$
$479$	−4.14290			−0.189294			−0.0946470	+	0.995511i	$0.530172\pi$
$480$		0			0
$481$			32.5896i			1.48596i
$482$	−6.83165			−0.311173
$483$		0			0
$484$	28.9809			1.31731
$485$			7.81411i			0.354820i
$486$		0			0
$487$	6.75885			0.306273			0.153136	−	0.988205i	$-0.451063\pi$
$487$	6.75885			0.306273			0.153136	+	0.988205i	$0.451063\pi$
$488$	5.53395			0.250510
$489$		0			0
$490$	6.25530	+	11.7977i	0.282586	+	0.532964i
$491$		−	34.3605i		−	1.55067i	−0.631550	−	0.775335i	$-0.717581\pi$
$491$		−	34.3605i		−	1.55067i	0.631550	−	0.775335i	$-0.282419\pi$
$492$		0			0
$493$		−	25.3276i		−	1.14070i
$494$			2.98283i			0.134204i
$495$		0			0
$496$			8.89827i			0.399544i
$497$	18.1369	−	4.51080i	0.813549	−	0.202337i
$498$		0			0
$499$	29.3178			1.31244			0.656222	−	0.754568i	$-0.272153\pi$
$499$	29.3178			1.31244			0.656222	+	0.754568i	$0.272153\pi$
$500$	12.1343			0.542664
$501$		0			0
$502$			21.9189i			0.978289i
$503$	−15.6061			−0.695843			−0.347922	−	0.937524i	$-0.613112\pi$
$503$	−15.6061			−0.695843			−0.347922	+	0.937524i	$0.613112\pi$
$504$		0			0
$505$	16.2545			0.723315
$506$			1.00654i			0.0447461i
$507$		0			0
$508$	−5.75581			−0.255373
$509$	20.0212			0.887423			0.443712	−	0.896170i	$-0.353661\pi$
$509$	20.0212			0.887423			0.443712	+	0.896170i	$0.353661\pi$
$510$		0			0
$511$	−22.8833	+	5.69130i	−1.01230	+	0.251768i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			5.72995i			0.252737i
$515$		−	10.6817i		−	0.470690i
$516$		0			0
$517$		−	30.2898i		−	1.33215i
$518$	28.0521	−	6.97683i	1.23254	−	0.306544i
$519$		0			0
$520$	5.69015			0.249529
$521$	−36.8008			−1.61227			−0.806136	−	0.591730i	$-0.798445\pi$
$521$	−36.8008			−1.61227			−0.806136	+	0.591730i	$0.798445\pi$
$522$		0			0
$523$		−	11.0988i		−	0.485318i	−0.970112	−	0.242659i	$-0.921980\pi$
$523$		−	11.0988i		−	0.485318i	0.970112	−	0.242659i	$-0.0780196\pi$
$524$	−15.4764			−0.676089
$525$		0			0
$526$	−5.53123			−0.241173
$527$		−	57.6388i		−	2.51079i
$528$		0			0
$529$	22.9747			0.998898
$530$	7.05342			0.306381
$531$		0			0
$532$	2.56753	−	0.638569i	0.111317	−	0.0276855i
$533$		−	24.6430i		−	1.06741i
$534$		0			0
$535$			7.74313i			0.334765i
$536$		−	10.8812i		−	0.469994i
$537$		0			0
$538$			10.0658i			0.433967i
$539$	20.7339	+	39.1046i	0.893071	+	1.68435i
$540$		0			0
$541$	−24.5005			−1.05336			−0.526679	−	0.850064i	$-0.676563\pi$
$541$	−24.5005			−1.05336			−0.526679	+	0.850064i	$0.676563\pi$
$542$	−10.2081			−0.438476
$543$		0			0
$544$			6.47753i			0.277722i
$545$	−36.0552			−1.54443
$546$		0			0
$547$	0.185711			0.00794041			0.00397021	−	0.999992i	$-0.498736\pi$
$547$	0.185711			0.00794041			0.00397021	+	0.999992i	$0.498736\pi$
$548$			10.0966i			0.431306i
$549$		0			0
$550$	8.60531			0.366932
$551$	3.91008			0.166575
$552$		0			0
$553$	7.34606	+	29.5368i	0.312386	+	1.25603i
$554$			27.5935i			1.17234i
$555$		0			0
$556$			1.92711i			0.0817276i
$557$			7.90378i			0.334894i	0.985881	+	0.167447i	$0.0535523\pi$
$557$			7.90378i			0.334894i	−0.985881	+	0.167447i	$0.946448\pi$
$558$		0			0
$559$		−	32.0183i		−	1.35423i
$560$	−1.21815	−	4.89791i	−0.0514764	−	0.206974i
$561$		0			0
$562$	−23.1757			−0.977609
$563$	22.8094			0.961300			0.480650	−	0.876913i	$-0.340401\pi$
$563$	22.8094			0.961300			0.480650	+	0.876913i	$0.340401\pi$
$564$		0			0
$565$		−	32.4223i		−	1.36402i
$566$	6.75569			0.283963
$567$		0			0
$568$	−7.06392			−0.296396
$569$			7.65839i			0.321057i	0.987031	+	0.160528i	$0.0513198\pi$
$569$			7.65839i			0.321057i	−0.987031	+	0.160528i	$0.948680\pi$
$570$		0			0
$571$	−18.6620			−0.780982			−0.390491	−	0.920607i	$-0.627695\pi$
$571$	−18.6620			−0.780982			−0.390491	+	0.920607i	$0.627695\pi$
$572$	18.8606			0.788601
$573$		0			0
$574$	−21.2120	+	5.27562i	−0.885371	+	0.220200i
$575$			0.216643i			0.00903464i
$576$		0			0
$577$			5.52255i			0.229907i	0.993371	+	0.114953i	$0.0366719\pi$
$577$			5.52255i			0.229907i	−0.993371	+	0.114953i	$0.963328\pi$
$578$		−	24.9584i		−	1.03813i
$579$		0			0
$580$		−	7.45898i		−	0.309717i
$581$	−3.23735	−	13.0166i	−0.134308	−	0.540020i
$582$		0			0
$583$	23.3793			0.968273
$584$	8.91258			0.368805
$585$		0			0
$586$			1.28671i			0.0531537i
$587$	−36.7462			−1.51668			−0.758338	−	0.651862i	$-0.773988\pi$
$587$	−36.7462			−1.51668			−0.758338	+	0.651862i	$0.773988\pi$
$588$		0			0
$589$	8.89827			0.366647
$590$			12.8436i			0.528762i
$591$		0			0
$592$	−10.9257			−0.449044
$593$	−2.49672			−0.102528			−0.0512639	−	0.998685i	$-0.516325\pi$
$593$	−2.49672			−0.102528			−0.0512639	+	0.998685i	$0.516325\pi$
$594$		0			0
$595$	7.89063	+	31.7263i	0.323484	+	1.30065i
$596$			8.47493i			0.347146i
$597$		0			0
$598$			0.474825i			0.0194170i
$599$			3.05572i			0.124854i	0.998050	+	0.0624268i	$0.0198840\pi$
$599$			3.05572i			0.124854i	−0.998050	+	0.0624268i	$0.980116\pi$
$600$		0			0
$601$		−	18.4891i		−	0.754186i	−0.926175	−	0.377093i	$-0.876924\pi$
$601$		−	18.4891i		−	0.754186i	0.926175	−	0.377093i	$-0.123076\pi$
$602$	−27.5604	+	6.85452i	−1.12328	+	0.279369i
$603$		0			0
$604$	−11.3265			−0.460870
$605$	−55.2848			−2.24765
$606$		0			0
$607$			11.5125i			0.467280i	0.972323	+	0.233640i	$0.0750637\pi$
$607$			11.5125i			0.467280i	−0.972323	+	0.233640i	$0.924936\pi$
$608$	−1.00000			−0.0405554
$609$		0			0
$610$	−10.5567			−0.427430
$611$		−	14.2889i		−	0.578068i
$612$		0			0
$613$	−24.5737			−0.992524			−0.496262	−	0.868173i	$-0.665294\pi$
$613$	−24.5737			−0.992524			−0.496262	+	0.868173i	$0.665294\pi$
$614$	6.82469			0.275422
$615$		0			0
$616$	−4.03770	−	16.2346i	−0.162684	−	0.654112i
$617$		−	27.2006i		−	1.09505i	−0.836788	−	0.547527i	$-0.815569\pi$
$617$		−	27.2006i		−	1.09505i	0.836788	−	0.547527i	$-0.184431\pi$
$618$		0			0
$619$		−	13.6144i		−	0.547207i	−0.961843	−	0.273604i	$-0.911784\pi$
$619$		−	13.6144i		−	0.547207i	0.961843	−	0.273604i	$-0.0882157\pi$
$620$		−	16.9746i		−	0.681717i
$621$		0			0
$622$		−	6.88370i		−	0.276011i
$623$	−7.21167	−	28.9964i	−0.288929	−	1.16172i
$624$		0			0
$625$	−16.3431			−0.653724
$626$	−31.0256			−1.24003
$627$		0			0
$628$			24.3412i			0.971321i
$629$	70.7716			2.82185
$630$		0			0
$631$	20.0849			0.799569			0.399784	−	0.916609i	$-0.369085\pi$
$631$	20.0849			0.799569			0.399784	+	0.916609i	$0.369085\pi$
$632$		−	11.5039i		−	0.457602i
$633$		0			0
$634$	−30.8638			−1.22576
$635$	10.9800			0.435726
$636$		0			0
$637$	9.78100	+	18.4472i	0.387537	+	0.730905i
$638$		−	24.7236i		−	0.978816i
$639$		0			0
$640$			1.90763i			0.0754057i
$641$		−	18.0809i		−	0.714155i	−0.934075	−	0.357077i	$-0.883773\pi$
$641$		−	18.0809i		−	0.714155i	0.934075	−	0.357077i	$-0.116227\pi$
$642$		0			0
$643$		−	36.9488i		−	1.45712i	−0.684984	−	0.728558i	$-0.740191\pi$
$643$		−	36.9488i		−	1.45712i	0.684984	−	0.728558i	$-0.259809\pi$
$644$	0.408715	−	0.101651i	0.0161056	−	0.00400562i
$645$		0			0
$646$	6.47753			0.254855
$647$	12.3791			0.486672			0.243336	−	0.969942i	$-0.421758\pi$
$647$	12.3791			0.486672			0.243336	+	0.969942i	$0.421758\pi$
$648$		0			0
$649$			42.5714i			1.67108i
$650$	4.05947			0.159226
$651$		0			0
$652$	2.71446			0.106306
$653$		−	13.8824i		−	0.543259i	−0.962402	−	0.271629i	$-0.912438\pi$
$653$		−	13.8824i		−	0.543259i	0.962402	−	0.271629i	$-0.0875625\pi$
$654$		0			0
$655$	29.5232			1.15357
$656$	8.26162			0.322562
$657$		0			0
$658$	−12.2995	+	3.05899i	−0.479484	+	0.119252i
$659$		−	18.0260i		−	0.702195i	−0.936339	−	0.351097i	$-0.885809\pi$
$659$		−	18.0260i		−	0.702195i	0.936339	−	0.351097i	$-0.114191\pi$
$660$		0			0
$661$			3.82462i			0.148761i	0.997230	+	0.0743803i	$0.0236979\pi$
$661$			3.82462i			0.148761i	−0.997230	+	0.0743803i	$0.976302\pi$
$662$			12.7032i			0.493723i
$663$		0			0
$664$			5.06969i			0.196742i
$665$	−4.89791	+	1.21815i	−0.189933	+	0.0472380i
$666$		0			0
$667$	0.622429			0.0241005
$668$	−0.803303			−0.0310807
$669$		0			0
$670$			20.7572i			0.801922i
$671$	−34.9914			−1.35083
$672$		0			0
$673$	3.75819			0.144868			0.0724338	−	0.997373i	$-0.476923\pi$
$673$	3.75819			0.144868			0.0724338	+	0.997373i	$0.476923\pi$
$674$		−	20.8673i		−	0.803780i
$675$		0			0
$676$	−4.10270			−0.157796
$677$	47.7211			1.83407			0.917036	−	0.398805i	$-0.130575\pi$
$677$	47.7211			1.83407			0.917036	+	0.398805i	$0.130575\pi$
$678$		0			0
$679$	10.5172	−	2.61573i	0.403614	−	0.100383i
$680$		−	12.3567i		−	0.473859i
$681$		0			0
$682$		−	56.2642i		−	2.15447i
$683$		−	9.90959i		−	0.379180i	−0.981863	−	0.189590i	$-0.939284\pi$
$683$		−	9.90959i		−	0.379180i	0.981863	−	0.189590i	$-0.0607158\pi$
$684$		0			0
$685$		−	19.2606i		−	0.735911i
$686$	13.7849	−	12.3684i	0.526309	−	0.472227i
$687$		0			0
$688$	10.7342			0.409237
$689$	11.0290			0.420170
$690$		0			0
$691$			34.2788i			1.30403i	0.758207	+	0.652014i	$0.226076\pi$
$691$			34.2788i			1.30403i	−0.758207	+	0.652014i	$0.773924\pi$
$692$	−1.01981			−0.0387673
$693$		0			0
$694$	−3.34713			−0.127055
$695$		−	3.67621i		−	0.139447i
$696$		0			0
$697$	−53.5149			−2.02702
$698$	24.1080			0.912501
$699$		0			0
$700$	−0.869057	−	3.49427i	−0.0328473	−	0.132071i
$701$		−	1.95271i		−	0.0737530i	−0.999320	−	0.0368765i	$-0.988259\pi$
$701$		−	1.95271i		−	0.0737530i	0.999320	−	0.0368765i	$-0.0117408\pi$
$702$		0			0
$703$			10.9257i			0.412071i
$704$			6.32304i			0.238309i
$705$		0			0
$706$		−	8.42916i		−	0.317235i
$707$	−5.44110	−	21.8774i	−0.204634	−	0.822782i
$708$		0			0
$709$	28.4547			1.06864			0.534320	−	0.845282i	$-0.320568\pi$
$709$	28.4547			1.06864			0.534320	+	0.845282i	$0.320568\pi$
$710$	13.4754			0.505721
$711$		0			0
$712$			11.2935i			0.423241i
$713$	1.41648			0.0530476
$714$		0			0
$715$	−35.9791			−1.34554
$716$			3.69669i			0.138152i
$717$		0			0
$718$	−11.3655			−0.424158
$719$	14.5553			0.542821			0.271411	−	0.962464i	$-0.412510\pi$
$719$	14.5553			0.542821			0.271411	+	0.962464i	$0.412510\pi$
$720$		0			0
$721$	−14.3768	+	3.57563i	−0.535418	+	0.133163i
$722$			1.00000i			0.0372161i
$723$		0			0
$724$		−	4.90195i		−	0.182180i
$725$		−	5.32139i		−	0.197632i
$726$		0			0
$727$		−	5.13492i		−	0.190444i	−0.995456	−	0.0952219i	$-0.969644\pi$
$727$		−	5.13492i		−	0.190444i	0.995456	−	0.0952219i	$-0.0303561\pi$
$728$	−1.90475	−	7.65853i	−0.0705946	−	0.283844i
$729$		0			0
$730$	−17.0019			−0.629269
$731$	−69.5309			−2.57169
$732$		0			0
$733$			33.1838i			1.22567i	0.790211	+	0.612835i	$0.209971\pi$
$733$			33.1838i			1.22567i	−0.790211	+	0.612835i	$0.790029\pi$
$734$	−1.24677			−0.0460191
$735$		0			0
$736$	−0.159186			−0.00586767
$737$			68.8020i			2.53436i
$738$		0			0
$739$	24.5975			0.904834			0.452417	−	0.891807i	$-0.350562\pi$
$739$	24.5975			0.904834			0.452417	+	0.891807i	$0.350562\pi$
$740$	20.8422			0.766176
$741$		0			0
$742$	−2.36110	−	9.49340i	−0.0866785	−	0.348514i
$743$			4.46587i			0.163837i	0.996639	+	0.0819184i	$0.0261047\pi$
$743$			4.46587i			0.163837i	−0.996639	+	0.0819184i	$0.973895\pi$
$744$		0			0
$745$		−	16.1670i		−	0.592314i
$746$		−	3.65353i		−	0.133765i
$747$		0			0
$748$		−	40.9577i		−	1.49756i
$749$	10.4217	−	2.59197i	0.380801	−	0.0947086i
$750$		0			0
$751$	21.7595			0.794014			0.397007	−	0.917816i	$-0.370049\pi$
$751$	21.7595			0.794014			0.397007	+	0.917816i	$0.370049\pi$
$752$	4.79039			0.174687
$753$		0			0
$754$		−	11.6631i		−	0.424745i
$755$	21.6068			0.786354
$756$		0			0
$757$	−28.9136			−1.05088			−0.525441	−	0.850830i	$-0.676100\pi$
$757$	−28.9136			−1.05088			−0.525441	+	0.850830i	$0.676100\pi$
$758$			4.28772i			0.155737i
$759$		0			0
$760$	1.90763			0.0691971
$761$	4.72706			0.171356			0.0856780	−	0.996323i	$-0.472694\pi$
$761$	4.72706			0.171356			0.0856780	+	0.996323i	$0.472694\pi$
$762$		0			0
$763$	12.0693	+	48.5277i	0.436937	+	1.75682i
$764$		−	2.09543i		−	0.0758102i
$765$		0			0
$766$			8.21116i			0.296681i
$767$			20.0827i			0.725143i
$768$		0			0
$769$			49.5720i			1.78761i	0.448455	+	0.893806i	$0.351975\pi$
$769$			49.5720i			1.78761i	−0.448455	+	0.893806i	$0.648025\pi$
$770$	7.70244	+	30.9697i	0.277577	+	1.11607i
$771$		0			0
$772$	−7.90248			−0.284417
$773$	−30.1504			−1.08443			−0.542217	−	0.840239i	$-0.682415\pi$
$773$	−30.1504			−1.08443			−0.542217	+	0.840239i	$0.682415\pi$
$774$		0			0
$775$		−	12.1100i		−	0.435006i
$776$	−4.09624			−0.147046
$777$		0			0
$778$	−17.1979			−0.616576
$779$		−	8.26162i		−	0.296003i
$780$		0			0
$781$	44.6655			1.59826
$782$	1.03113			0.0368732
$783$		0			0
$784$	−6.18446	+	3.27910i	−0.220874	+	0.117111i
$785$		−	46.4341i		−	1.65730i
$786$		0			0
$787$			6.41706i			0.228744i	0.993438	+	0.114372i	$0.0364855\pi$
$787$			6.41706i			0.228744i	−0.993438	+	0.114372i	$0.963514\pi$
$788$			3.61443i			0.128759i
$789$		0			0
$790$			21.9453i			0.780778i
$791$	−43.6381	+	10.8532i	−1.55159	+	0.385895i
$792$		0			0
$793$	−16.5069			−0.586176
$794$	−24.6749			−0.875678
$795$		0			0
$796$			21.0820i			0.747232i
$797$	35.4664			1.25628			0.628142	−	0.778099i	$-0.283816\pi$
$797$	35.4664			1.25628			0.628142	+	0.778099i	$0.283816\pi$
$798$		0			0
$799$	−31.0299			−1.09776
$800$			1.36094i			0.0481166i
$801$		0			0
$802$	4.96284			0.175244
$803$	−56.3546			−1.98871
$804$		0			0
$805$	−0.779678	+	0.193913i	−0.0274800	+	0.00683453i
$806$		−	26.5421i		−	0.934905i
$807$		0			0
$808$			8.52076i			0.299759i
$809$			35.9140i			1.26267i	0.775511	+	0.631335i	$0.217493\pi$
$809$			35.9140i			1.26267i	−0.775511	+	0.631335i	$0.782507\pi$
$810$		0			0
$811$			25.8890i			0.909085i	0.890725	+	0.454542i	$0.150197\pi$
$811$			25.8890i			0.909085i	−0.890725	+	0.454542i	$0.849803\pi$
$812$	−10.0393	+	2.49685i	−0.352309	+	0.0876224i
$813$		0			0
$814$	69.0838			2.42139
$815$	−5.17818			−0.181384
$816$		0			0
$817$		−	10.7342i		−	0.375541i
$818$	−9.50965			−0.332497
$819$		0			0
$820$	−15.7601			−0.550367
$821$			50.2666i			1.75432i	0.480202	+	0.877158i	$0.340563\pi$
$821$			50.2666i			1.75432i	−0.480202	+	0.877158i	$0.659437\pi$
$822$		0			0
$823$	−6.47280			−0.225628			−0.112814	−	0.993616i	$-0.535986\pi$
$823$	−6.47280			−0.225628			−0.112814	+	0.993616i	$0.535986\pi$
$824$	5.59944			0.195066
$825$		0			0
$826$	17.2865	−	4.29932i	0.601476	−	0.149593i
$827$		−	2.73934i		−	0.0952562i	−0.998865	−	0.0476281i	$-0.984834\pi$
$827$		−	2.73934i		−	0.0952562i	0.998865	−	0.0476281i	$-0.0151662\pi$
$828$		0			0
$829$			20.5113i			0.712386i	0.934412	+	0.356193i	$0.115925\pi$
$829$			20.5113i			0.712386i	−0.934412	+	0.356193i	$0.884075\pi$
$830$		−	9.67110i		−	0.335689i
$831$		0			0
$832$			2.98283i			0.103411i
$833$	40.0600	−	21.2404i	1.38800	−	0.735937i
$834$		0			0
$835$	1.53241			0.0530311
$836$	6.32304			0.218687
$837$		0			0
$838$		−	34.5145i		−	1.19228i
$839$	9.90706			0.342030			0.171015	−	0.985268i	$-0.445295\pi$
$839$	9.90706			0.342030			0.171015	+	0.985268i	$0.445295\pi$
$840$		0			0
$841$	13.7113			0.472804
$842$			2.26134i			0.0779309i
$843$		0			0
$844$	27.8742			0.959470
$845$	7.82644			0.269238
$846$		0			0
$847$	18.5063	+	74.4094i	0.635884	+	2.55674i
$848$			3.69748i			0.126972i
$849$		0			0
$850$		−	8.81555i		−	0.302371i
$851$			1.73922i			0.0596197i
$852$		0			0
$853$		−	30.0888i		−	1.03022i	−0.857124	−	0.515110i	$-0.827751\pi$
$853$		−	30.0888i		−	1.03022i	0.857124	−	0.515110i	$-0.172249\pi$
$854$	3.53381	+	14.2086i	0.120925	+	0.486209i
$855$		0			0
$856$	−4.05903			−0.138735
$857$	1.78579			0.0610013			0.0305006	−	0.999535i	$-0.490290\pi$
$857$	1.78579			0.0610013			0.0305006	+	0.999535i	$0.490290\pi$
$858$		0			0
$859$		−	35.0468i		−	1.19578i	−0.801578	−	0.597890i	$-0.796006\pi$
$859$		−	35.0468i		−	1.19578i	0.801578	−	0.597890i	$-0.203994\pi$
$860$	−20.4769			−0.698255
$861$		0			0
$862$	38.4148			1.30841
$863$			4.89042i			0.166472i	0.996530	+	0.0832360i	$0.0265255\pi$
$863$			4.89042i			0.166472i	−0.996530	+	0.0832360i	$0.973474\pi$
$864$		0			0
$865$	1.94542			0.0661461
$866$	25.8100			0.877060
$867$		0			0
$868$	−22.8466	+	5.68216i	−0.775465	+	0.192865i
$869$			72.7400i			2.46753i
$870$		0			0
$871$			32.4567i			1.09975i
$872$		−	18.9005i		−	0.640052i
$873$		0			0
$874$			0.159186i			0.00538454i
$875$	7.74861	+	31.1553i	0.261951	+	1.05324i
$876$		0			0
$877$	−30.4127			−1.02696			−0.513482	−	0.858100i	$-0.671645\pi$
$877$	−30.4127			−1.02696			−0.513482	+	0.858100i	$0.671645\pi$
$878$	−27.3489			−0.922981
$879$		0			0
$880$		−	12.0620i		−	0.406611i
$881$	46.9569			1.58202			0.791009	−	0.611804i	$-0.209556\pi$
$881$	46.9569			1.58202			0.791009	+	0.611804i	$0.209556\pi$
$882$		0			0
$883$	46.9566			1.58022			0.790108	−	0.612967i	$-0.210024\pi$
$883$	46.9566			1.58022			0.790108	+	0.612967i	$0.210024\pi$
$884$		−	19.3214i		−	0.649849i
$885$		0			0
$886$	24.0719			0.808712
$887$	32.0401			1.07580			0.537901	−	0.843008i	$-0.319217\pi$
$887$	32.0401			1.07580			0.537901	+	0.843008i	$0.319217\pi$
$888$		0			0
$889$	−3.67548	−	14.7782i	−0.123272	−	0.495646i
$890$		−	21.5438i		−	0.722150i
$891$		0			0
$892$			7.38131i			0.247145i
$893$		−	4.79039i		−	0.160304i
$894$		0			0
$895$		−	7.05192i		−	0.235720i
$896$	2.56753	−	0.638569i	0.0857753	−	0.0213331i
$897$		0			0
$898$	−4.71176			−0.157233
$899$	−34.7929			−1.16041
$900$		0			0
$901$		−	23.9505i		−	0.797907i
$902$	−52.2386			−1.73935
$903$		0			0
$904$	16.9961			0.565282
$905$			9.35112i			0.310842i
$906$		0			0
$907$	31.2608			1.03800			0.519000	−	0.854774i	$-0.326305\pi$
$907$	31.2608			1.03800			0.519000	+	0.854774i	$0.326305\pi$
$908$	−29.6825			−0.985047
$909$		0			0
$910$	3.63355	+	14.6096i	0.120451	+	0.484305i
$911$			44.7328i			1.48206i	0.671470	+	0.741032i	$0.265663\pi$
$911$			44.7328i			1.48206i	−0.671470	+	0.741032i	$0.734337\pi$
$912$		0			0
$913$		−	32.0559i		−	1.06090i
$914$		−	6.23529i		−	0.206245i
$915$		0			0
$916$		−	23.0054i		−	0.760118i
$917$	−9.88275	−	39.7362i	−0.326357	−	1.31220i
$918$		0			0
$919$	25.7901			0.850737			0.425369	−	0.905020i	$-0.360144\pi$
$919$	25.7901			0.850737			0.425369	+	0.905020i	$0.360144\pi$
$920$	0.303668			0.0100116
$921$		0			0
$922$		−	33.2663i		−	1.09557i
$923$	21.0705			0.693544
$924$		0			0
$925$	14.8693			0.488899
$926$		−	16.2046i		−	0.532517i
$927$		0			0
$928$	3.91008			0.128355
$929$	−47.1698			−1.54759			−0.773796	−	0.633435i	$-0.781644\pi$
$929$	−47.1698			−1.54759			−0.773796	+	0.633435i	$0.781644\pi$
$930$		0			0
$931$	3.27910	+	6.18446i	0.107468	+	0.202687i
$932$		−	8.57518i		−	0.280889i
$933$		0			0
$934$			32.1363i			1.05153i
$935$			78.1322i			2.55519i
$936$		0			0
$937$		−	33.2473i		−	1.08614i	−0.839687	−	0.543070i	$-0.817262\pi$
$937$		−	33.2473i		−	1.08614i	0.839687	−	0.543070i	$-0.182738\pi$
$938$	27.9377	−	6.94837i	0.912199	−	0.226872i
$939$		0			0
$940$	−9.13829			−0.298058
$941$	−50.5931			−1.64929			−0.824644	−	0.565651i	$-0.808625\pi$
$941$	−50.5931			−1.64929			−0.824644	+	0.565651i	$0.808625\pi$
$942$		0			0
$943$		−	1.31513i		−	0.0428266i
$944$	−6.73274			−0.219132
$945$		0			0
$946$	−67.8727			−2.20673
$947$			57.4609i			1.86723i	0.358281	+	0.933614i	$0.383363\pi$
$947$			57.4609i			1.86723i	−0.358281	+	0.933614i	$0.616637\pi$
$948$		0			0
$949$	−26.5847			−0.862977
$950$	1.36094			0.0441548
$951$		0			0
$952$	−16.6313	+	4.13635i	−0.539023	+	0.134060i
$953$		−	34.9482i		−	1.13208i	−0.824376	−	0.566042i	$-0.808474\pi$
$953$		−	34.9482i		−	1.13208i	0.824376	−	0.566042i	$-0.191526\pi$
$954$		0			0
$955$			3.99732i			0.129350i
$956$			1.18924i			0.0384629i
$957$		0			0
$958$			4.14290i			0.133851i
$959$	−25.9234	+	6.44739i	−0.837111	+	0.208197i
$960$		0			0
$961$	−48.1793			−1.55417
$962$	32.5896			1.05073
$963$		0			0
$964$			6.83165i			0.220033i
$965$	15.0750			0.485282
$966$		0			0
$967$	−13.3941			−0.430725			−0.215362	−	0.976534i	$-0.569093\pi$
$967$	−13.3941			−0.430725			−0.215362	+	0.976534i	$0.569093\pi$
$968$		−	28.9809i		−	0.931481i
$969$		0			0
$970$	7.81411			0.250896
$971$	61.2110			1.96436			0.982178	−	0.187955i	$-0.0601860\pi$
$971$	61.2110			1.96436			0.982178	+	0.187955i	$0.0601860\pi$
$972$		0			0
$973$	−4.94792	+	1.23059i	−0.158623	+	0.0394510i
$974$		−	6.75885i		−	0.216567i
$975$		0			0
$976$		−	5.53395i		−	0.177138i
$977$			16.8158i			0.537986i	0.963142	+	0.268993i	$0.0866908\pi$
$977$			16.8158i			0.537986i	−0.963142	+	0.268993i	$0.913309\pi$
$978$		0			0
$979$		−	71.4092i		−	2.28225i
$980$	11.7977	−	6.25530i	0.376863	−	0.199818i
$981$		0			0
$982$	−34.3605			−1.09649
$983$	48.9398			1.56094			0.780469	−	0.625194i	$-0.214980\pi$
$983$	48.9398			1.56094			0.780469	+	0.625194i	$0.214980\pi$
$984$		0			0
$985$		−	6.89500i		−	0.219693i
$986$	−25.3276			−0.806596
$987$		0			0
$988$	2.98283			0.0948966
$989$		−	1.70873i		−	0.0543345i
$990$		0			0
$991$	38.2440			1.21486			0.607430	−	0.794373i	$-0.292200\pi$
$991$	38.2440			1.21486			0.607430	+	0.794373i	$0.292200\pi$
$992$	8.89827			0.282520
$993$		0			0
$994$	−4.51080	−	18.1369i	−0.143074	−	0.575266i
$995$		−	40.2167i		−	1.27495i
$996$		0			0
$997$		−	30.5427i		−	0.967297i	−0.875262	−	0.483648i	$-0.839311\pi$
$997$		−	30.5427i		−	0.967297i	0.875262	−	0.483648i	$-0.160689\pi$
$998$		−	29.3178i		−	0.928038i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.f.b.2015.1	yes	24
3.2	odd	2		2394.2.f.a.2015.2	yes	24
7.6	odd	2		2394.2.f.a.2015.1	✓	24
21.20	even	2	inner	2394.2.f.b.2015.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.f.a.2015.1	✓	24	7.6	odd	2
2394.2.f.a.2015.2	yes	24	3.2	odd	2
2394.2.f.b.2015.1	yes	24	1.1	even	1	trivial
2394.2.f.b.2015.2	yes	24	21.20	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.90763 q^{5} +(-0.638569 - 2.56753i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.90763 q^{5} +(-0.638569 - 2.56753i) q^{7} +1.00000i q^{8} -1.90763i q^{10} -6.32304i q^{11} -2.98283i q^{13} +(-2.56753 + 0.638569i) q^{14} +1.00000 q^{16} -6.47753 q^{17} -1.00000i q^{19} -1.90763 q^{20} -6.32304 q^{22} -0.159186i q^{23} -1.36094 q^{25} -2.98283 q^{26} +(0.638569 + 2.56753i) q^{28} +3.91008i q^{29} +8.89827i q^{31} -1.00000i q^{32} +6.47753i q^{34} +(-1.21815 - 4.89791i) q^{35} -10.9257 q^{37} -1.00000 q^{38} +1.90763i q^{40} +8.26162 q^{41} +10.7342 q^{43} +6.32304i q^{44} -0.159186 q^{46} +4.79039 q^{47} +(-6.18446 + 3.27910i) q^{49} +1.36094i q^{50} +2.98283i q^{52} +3.69748i q^{53} -12.0620i q^{55} +(2.56753 - 0.638569i) q^{56} +3.91008 q^{58} -6.73274 q^{59} -5.53395i q^{61} +8.89827 q^{62} -1.00000 q^{64} -5.69015i q^{65} -10.8812 q^{67} +6.47753 q^{68} +(-4.89791 + 1.21815i) q^{70} +7.06392i q^{71} -8.91258i q^{73} +10.9257i q^{74} +1.00000i q^{76} +(-16.2346 + 4.03770i) q^{77} -11.5039 q^{79} +1.90763 q^{80} -8.26162i q^{82} +5.06969 q^{83} -12.3567 q^{85} -10.7342i q^{86} +6.32304 q^{88} +11.2935 q^{89} +(-7.65853 + 1.90475i) q^{91} +0.159186i q^{92} -4.79039i q^{94} -1.90763i q^{95} +4.09624i q^{97} +(3.27910 + 6.18446i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{4} + 4 q^{7}+O(q^{10}) \) 24 * q - 24 * q^4 + 4 * q^7	\( 24 q - 24 q^{4} + 4 q^{7} + 4 q^{14} + 24 q^{16} - 32 q^{17} - 8 q^{22} + 16 q^{25} - 4 q^{28} + 24 q^{35} - 24 q^{38} + 8 q^{41} + 16 q^{43} - 8 q^{46} - 16 q^{49} - 4 q^{56} + 16 q^{58} - 16 q^{59} + 16 q^{62} - 24 q^{64} - 24 q^{67} + 32 q^{68} - 16 q^{70} - 8 q^{77} - 40 q^{79} + 64 q^{83} + 40 q^{85} + 8 q^{88} + 64 q^{89} + 8 q^{91} + 16 q^{98}+O(q^{100}) \) 24 * q - 24 * q^4 + 4 * q^7 + 4 * q^14 + 24 * q^16 - 32 * q^17 - 8 * q^22 + 16 * q^25 - 4 * q^28 + 24 * q^35 - 24 * q^38 + 8 * q^41 + 16 * q^43 - 8 * q^46 - 16 * q^49 - 4 * q^56 + 16 * q^58 - 16 * q^59 + 16 * q^62 - 24 * q^64 - 24 * q^67 + 32 * q^68 - 16 * q^70 - 8 * q^77 - 40 * q^79 + 64 * q^83 + 40 * q^85 + 8 * q^88 + 64 * q^89 + 8 * q^91 + 16 * q^98

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