LMFDB - Embedded newform 2394.2.a.bc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1,2394)]

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([0, 0, 0]))

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

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

chi := DirichletCharacter("2394.1");

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.a (trivial)

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:	yes
Analytic conductor:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	2394.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.34017 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +4.34017 q^{5} -1.00000 q^{7} +1.00000 q^{8} +4.34017 q^{10} +3.07838 q^{11} +6.34017 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.07838 q^{17} -1.00000 q^{19} +4.34017 q^{20} +3.07838 q^{22} -2.34017 q^{23} +13.8371 q^{25} +6.34017 q^{26} -1.00000 q^{28} -8.83710 q^{29} -5.41855 q^{31} +1.00000 q^{32} -1.07838 q^{34} -4.34017 q^{35} -3.41855 q^{37} -1.00000 q^{38} +4.34017 q^{40} -6.68035 q^{41} -10.8371 q^{43} +3.07838 q^{44} -2.34017 q^{46} +2.73820 q^{47} +1.00000 q^{49} +13.8371 q^{50} +6.34017 q^{52} +2.00000 q^{53} +13.3607 q^{55} -1.00000 q^{56} -8.83710 q^{58} +12.8371 q^{61} -5.41855 q^{62} +1.00000 q^{64} +27.5174 q^{65} +1.07838 q^{67} -1.07838 q^{68} -4.34017 q^{70} +10.1568 q^{71} -4.15676 q^{73} -3.41855 q^{74} -1.00000 q^{76} -3.07838 q^{77} -14.8638 q^{79} +4.34017 q^{80} -6.68035 q^{82} -2.68035 q^{83} -4.68035 q^{85} -10.8371 q^{86} +3.07838 q^{88} +3.84324 q^{89} -6.34017 q^{91} -2.34017 q^{92} +2.73820 q^{94} -4.34017 q^{95} +5.60197 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8} + 2 q^{10} + 6 q^{11} + 8 q^{13} - 3 q^{14} + 3 q^{16} - 3 q^{19} + 2 q^{20} + 6 q^{22} + 4 q^{23} + 13 q^{25} + 8 q^{26} - 3 q^{28} + 2 q^{29} - 2 q^{31} + 3 q^{32} - 2 q^{35} + 4 q^{37} - 3 q^{38} + 2 q^{40} + 2 q^{41} - 4 q^{43} + 6 q^{44} + 4 q^{46} + 16 q^{47} + 3 q^{49} + 13 q^{50} + 8 q^{52} + 6 q^{53} - 4 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{61} - 2 q^{62} + 3 q^{64} + 32 q^{65} - 2 q^{70} + 24 q^{71} - 6 q^{73} + 4 q^{74} - 3 q^{76} - 6 q^{77} - 18 q^{79} + 2 q^{80} + 2 q^{82} + 14 q^{83} + 8 q^{85} - 4 q^{86} + 6 q^{88} + 18 q^{89} - 8 q^{91} + 4 q^{92} + 16 q^{94} - 2 q^{95} - 2 q^{97} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 3 * q^7 + 3 * q^8 + 2 * q^10 + 6 * q^11 + 8 * q^13 - 3 * q^14 + 3 * q^16 - 3 * q^19 + 2 * q^20 + 6 * q^22 + 4 * q^23 + 13 * q^25 + 8 * q^26 - 3 * q^28 + 2 * q^29 - 2 * q^31 + 3 * q^32 - 2 * q^35 + 4 * q^37 - 3 * q^38 + 2 * q^40 + 2 * q^41 - 4 * q^43 + 6 * q^44 + 4 * q^46 + 16 * q^47 + 3 * q^49 + 13 * q^50 + 8 * q^52 + 6 * q^53 - 4 * q^55 - 3 * q^56 + 2 * q^58 + 10 * q^61 - 2 * q^62 + 3 * q^64 + 32 * q^65 - 2 * q^70 + 24 * q^71 - 6 * q^73 + 4 * q^74 - 3 * q^76 - 6 * q^77 - 18 * q^79 + 2 * q^80 + 2 * q^82 + 14 * q^83 + 8 * q^85 - 4 * q^86 + 6 * q^88 + 18 * q^89 - 8 * q^91 + 4 * q^92 + 16 * q^94 - 2 * q^95 - 2 * q^97 + 3 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	4.34017	1.94098	0.970492	−	0.241133i	$-0.0775189\pi$
$5$	4.34017	1.94098	0.970492	+	0.241133i	$0.0775189\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	4.34017	1.37248
$11$	3.07838	0.928166	0.464083	−	0.885792i	$-0.346384\pi$
$11$	3.07838	0.928166	0.464083	+	0.885792i	$0.346384\pi$
$12$	0	0
$13$	6.34017	1.75845	0.879224	−	0.476409i	$-0.158062\pi$
$13$	6.34017	1.75845	0.879224	+	0.476409i	$0.158062\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.07838	−0.261545	−0.130773	−	0.991412i	$-0.541746\pi$
$17$	−1.07838	−0.261545	−0.130773	+	0.991412i	$0.541746\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	4.34017	0.970492
$21$	0	0
$22$	3.07838	0.656312
$23$	−2.34017	−0.487960	−0.243980	−	0.969780i	$-0.578453\pi$
$23$	−2.34017	−0.487960	−0.243980	+	0.969780i	$0.578453\pi$
$24$	0	0
$25$	13.8371	2.76742
$26$	6.34017	1.24341
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−8.83710	−1.64101	−0.820504	−	0.571640i	$-0.806307\pi$
$29$	−8.83710	−1.64101	−0.820504	+	0.571640i	$0.806307\pi$
$30$	0	0
$31$	−5.41855	−0.973200	−0.486600	−	0.873625i	$-0.661763\pi$
$31$	−5.41855	−0.973200	−0.486600	+	0.873625i	$0.661763\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.07838	−0.184940
$35$	−4.34017	−0.733623
$36$	0	0
$37$	−3.41855	−0.562006	−0.281003	−	0.959707i	$-0.590667\pi$
$37$	−3.41855	−0.562006	−0.281003	+	0.959707i	$0.590667\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	4.34017	0.686242
$41$	−6.68035	−1.04329	−0.521647	−	0.853161i	$-0.674682\pi$
$41$	−6.68035	−1.04329	−0.521647	+	0.853161i	$0.674682\pi$
$42$	0	0
$43$	−10.8371	−1.65264	−0.826321	−	0.563199i	$-0.809570\pi$
$43$	−10.8371	−1.65264	−0.826321	+	0.563199i	$0.809570\pi$
$44$	3.07838	0.464083
$45$	0	0
$46$	−2.34017	−0.345040
$47$	2.73820	0.399408	0.199704	−	0.979856i	$-0.436002\pi$
$47$	2.73820	0.399408	0.199704	+	0.979856i	$0.436002\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	13.8371	1.95686
$51$	0	0
$52$	6.34017	0.879224
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	13.3607	1.80156
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−8.83710	−1.16037
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	12.8371	1.64362	0.821811	−	0.569760i	$-0.192964\pi$
$61$	12.8371	1.64362	0.821811	+	0.569760i	$0.192964\pi$
$62$	−5.41855	−0.688157
$63$	0	0
$64$	1.00000	0.125000
$65$	27.5174	3.41312
$66$	0	0
$67$	1.07838	0.131745	0.0658724	−	0.997828i	$-0.479017\pi$
$67$	1.07838	0.131745	0.0658724	+	0.997828i	$0.479017\pi$
$68$	−1.07838	−0.130773
$69$	0	0
$70$	−4.34017	−0.518750
$71$	10.1568	1.20539	0.602693	−	0.797973i	$-0.294095\pi$
$71$	10.1568	1.20539	0.602693	+	0.797973i	$0.294095\pi$
$72$	0	0
$73$	−4.15676	−0.486511	−0.243256	−	0.969962i	$-0.578215\pi$
$73$	−4.15676	−0.486511	−0.243256	+	0.969962i	$0.578215\pi$
$74$	−3.41855	−0.397398
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−3.07838	−0.350814
$78$	0	0
$79$	−14.8638	−1.67230	−0.836152	−	0.548498i	$-0.815200\pi$
$79$	−14.8638	−1.67230	−0.836152	+	0.548498i	$0.815200\pi$
$80$	4.34017	0.485246
$81$	0	0
$82$	−6.68035	−0.737721
$83$	−2.68035	−0.294206	−0.147103	−	0.989121i	$-0.546995\pi$
$83$	−2.68035	−0.294206	−0.147103	+	0.989121i	$0.546995\pi$
$84$	0	0
$85$	−4.68035	−0.507655
$86$	−10.8371	−1.16859
$87$	0	0
$88$	3.07838	0.328156
$89$	3.84324	0.407383	0.203692	−	0.979035i	$-0.434706\pi$
$89$	3.84324	0.407383	0.203692	+	0.979035i	$0.434706\pi$
$90$	0	0
$91$	−6.34017	−0.664631
$92$	−2.34017	−0.243980
$93$	0	0
$94$	2.73820	0.282424
$95$	−4.34017	−0.445292
$96$	0	0
$97$	5.60197	0.568794	0.284397	−	0.958707i	$-0.408207\pi$
$97$	5.60197	0.568794	0.284397	+	0.958707i	$0.408207\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	13.8371	1.38371
$101$	0.340173	0.0338485	0.0169242	−	0.999857i	$-0.494613\pi$
$101$	0.340173	0.0338485	0.0169242	+	0.999857i	$0.494613\pi$
$102$	0	0
$103$	−15.9421	−1.57083	−0.785413	−	0.618972i	$-0.787549\pi$
$103$	−15.9421	−1.57083	−0.785413	+	0.618972i	$0.787549\pi$
$104$	6.34017	0.621705
$105$	0	0
$106$	2.00000	0.194257
$107$	5.84324	0.564888	0.282444	−	0.959284i	$-0.408855\pi$
$107$	5.84324	0.564888	0.282444	+	0.959284i	$0.408855\pi$
$108$	0	0
$109$	−18.2557	−1.74857	−0.874287	−	0.485409i	$-0.838671\pi$
$109$	−18.2557	−1.74857	−0.874287	+	0.485409i	$0.838671\pi$
$110$	13.3607	1.27389
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	0.523590	0.0492552	0.0246276	−	0.999697i	$-0.492160\pi$
$113$	0.523590	0.0492552	0.0246276	+	0.999697i	$0.492160\pi$
$114$	0	0
$115$	−10.1568	−0.947122
$116$	−8.83710	−0.820504
$117$	0	0
$118$	0	0
$119$	1.07838	0.0988547
$120$	0	0
$121$	−1.52359	−0.138508
$122$	12.8371	1.16222
$123$	0	0
$124$	−5.41855	−0.486600
$125$	38.3545	3.43054
$126$	0	0
$127$	5.81658	0.516138	0.258069	−	0.966126i	$-0.416914\pi$
$127$	5.81658	0.516138	0.258069	+	0.966126i	$0.416914\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	27.5174	2.41344
$131$	−4.52359	−0.395228	−0.197614	−	0.980280i	$-0.563319\pi$
$131$	−4.52359	−0.395228	−0.197614	+	0.980280i	$0.563319\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	1.07838	0.0931576
$135$	0	0
$136$	−1.07838	−0.0924701
$137$	−4.68035	−0.399869	−0.199934	−	0.979809i	$-0.564073\pi$
$137$	−4.68035	−0.399869	−0.199934	+	0.979809i	$0.564073\pi$
$138$	0	0
$139$	−6.52359	−0.553324	−0.276662	−	0.960967i	$-0.589228\pi$
$139$	−6.52359	−0.553324	−0.276662	+	0.960967i	$0.589228\pi$
$140$	−4.34017	−0.366812
$141$	0	0
$142$	10.1568	0.852336
$143$	19.5174	1.63213
$144$	0	0
$145$	−38.3545	−3.18517
$146$	−4.15676	−0.344016
$147$	0	0
$148$	−3.41855	−0.281003
$149$	18.0989	1.48272	0.741360	−	0.671108i	$-0.234181\pi$
$149$	18.0989	1.48272	0.741360	+	0.671108i	$0.234181\pi$
$150$	0	0
$151$	19.8576	1.61599	0.807995	−	0.589189i	$-0.200553\pi$
$151$	19.8576	1.61599	0.807995	+	0.589189i	$0.200553\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−3.07838	−0.248063
$155$	−23.5174	−1.88897
$156$	0	0
$157$	14.3135	1.14234	0.571171	−	0.820831i	$-0.306489\pi$
$157$	14.3135	1.14234	0.571171	+	0.820831i	$0.306489\pi$
$158$	−14.8638	−1.18250
$159$	0	0
$160$	4.34017	0.343121
$161$	2.34017	0.184431
$162$	0	0
$163$	5.36069	0.419882	0.209941	−	0.977714i	$-0.432673\pi$
$163$	5.36069	0.419882	0.209941	+	0.977714i	$0.432673\pi$
$164$	−6.68035	−0.521647
$165$	0	0
$166$	−2.68035	−0.208035
$167$	−12.6803	−0.981235	−0.490617	−	0.871375i	$-0.663229\pi$
$167$	−12.6803	−0.981235	−0.490617	+	0.871375i	$0.663229\pi$
$168$	0	0
$169$	27.1978	2.09214
$170$	−4.68035	−0.358966
$171$	0	0
$172$	−10.8371	−0.826321
$173$	−1.68649	−0.128221	−0.0641107	−	0.997943i	$-0.520421\pi$
$173$	−1.68649	−0.128221	−0.0641107	+	0.997943i	$0.520421\pi$
$174$	0	0
$175$	−13.8371	−1.04599
$176$	3.07838	0.232041
$177$	0	0
$178$	3.84324	0.288063
$179$	17.0472	1.27417	0.637083	−	0.770795i	$-0.280141\pi$
$179$	17.0472	1.27417	0.637083	+	0.770795i	$0.280141\pi$
$180$	0	0
$181$	7.02052	0.521831	0.260916	−	0.965362i	$-0.415976\pi$
$181$	7.02052	0.521831	0.260916	+	0.965362i	$0.415976\pi$
$182$	−6.34017	−0.469965
$183$	0	0
$184$	−2.34017	−0.172520
$185$	−14.8371	−1.09085
$186$	0	0
$187$	−3.31965	−0.242757
$188$	2.73820	0.199704
$189$	0	0
$190$	−4.34017	−0.314869
$191$	−20.0144	−1.44819	−0.724095	−	0.689701i	$-0.757742\pi$
$191$	−20.0144	−1.44819	−0.724095	+	0.689701i	$0.757742\pi$
$192$	0	0
$193$	−16.8371	−1.21196	−0.605981	−	0.795479i	$-0.707219\pi$
$193$	−16.8371	−1.21196	−0.605981	+	0.795479i	$0.707219\pi$
$194$	5.60197	0.402198
$195$	0	0
$196$	1.00000	0.0714286
$197$	22.0989	1.57448	0.787241	−	0.616646i	$-0.211509\pi$
$197$	22.0989	1.57448	0.787241	+	0.616646i	$0.211509\pi$
$198$	0	0
$199$	9.67420	0.685786	0.342893	−	0.939374i	$-0.388593\pi$
$199$	9.67420	0.685786	0.342893	+	0.939374i	$0.388593\pi$
$200$	13.8371	0.978431
$201$	0	0
$202$	0.340173	0.0239345
$203$	8.83710	0.620243
$204$	0	0
$205$	−28.9939	−2.02502
$206$	−15.9421	−1.11074
$207$	0	0
$208$	6.34017	0.439612
$209$	−3.07838	−0.212936
$210$	0	0
$211$	−14.0722	−0.968773	−0.484386	−	0.874854i	$-0.660957\pi$
$211$	−14.0722	−0.968773	−0.484386	+	0.874854i	$0.660957\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	5.84324	0.399436
$215$	−47.0349	−3.20775
$216$	0	0
$217$	5.41855	0.367835
$218$	−18.2557	−1.23643
$219$	0	0
$220$	13.3607	0.900778
$221$	−6.83710	−0.459913
$222$	0	0
$223$	5.78539	0.387418	0.193709	−	0.981059i	$-0.437948\pi$
$223$	5.78539	0.387418	0.193709	+	0.981059i	$0.437948\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	0.523590	0.0348287
$227$	16.1978	1.07509	0.537543	−	0.843237i	$-0.319353\pi$
$227$	16.1978	1.07509	0.537543	+	0.843237i	$0.319353\pi$
$228$	0	0
$229$	−22.6803	−1.49876	−0.749380	−	0.662140i	$-0.769648\pi$
$229$	−22.6803	−1.49876	−0.749380	+	0.662140i	$0.769648\pi$
$230$	−10.1568	−0.669717
$231$	0	0
$232$	−8.83710	−0.580184
$233$	5.84324	0.382804	0.191402	−	0.981512i	$-0.438697\pi$
$233$	5.84324	0.382804	0.191402	+	0.981512i	$0.438697\pi$
$234$	0	0
$235$	11.8843	0.775245
$236$	0	0
$237$	0	0
$238$	1.07838	0.0699008
$239$	−17.1773	−1.11111	−0.555553	−	0.831481i	$-0.687493\pi$
$239$	−17.1773	−1.11111	−0.555553	+	0.831481i	$0.687493\pi$
$240$	0	0
$241$	−1.60197	−0.103192	−0.0515959	−	0.998668i	$-0.516431\pi$
$241$	−1.60197	−0.103192	−0.0515959	+	0.998668i	$0.516431\pi$
$242$	−1.52359	−0.0979401
$243$	0	0
$244$	12.8371	0.821811
$245$	4.34017	0.277283
$246$	0	0
$247$	−6.34017	−0.403416
$248$	−5.41855	−0.344078
$249$	0	0
$250$	38.3545	2.42575
$251$	24.8371	1.56770	0.783852	−	0.620948i	$-0.213252\pi$
$251$	24.8371	1.56770	0.783852	+	0.620948i	$0.213252\pi$
$252$	0	0
$253$	−7.20394	−0.452908
$254$	5.81658	0.364965
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	3.41855	0.212418
$260$	27.5174	1.70656
$261$	0	0
$262$	−4.52359	−0.279468
$263$	11.7009	0.721506	0.360753	−	0.932661i	$-0.382520\pi$
$263$	11.7009	0.721506	0.360753	+	0.932661i	$0.382520\pi$
$264$	0	0
$265$	8.68035	0.533229
$266$	1.00000	0.0613139
$267$	0	0
$268$	1.07838	0.0658724
$269$	6.31351	0.384942	0.192471	−	0.981303i	$-0.438350\pi$
$269$	6.31351	0.384942	0.192471	+	0.981303i	$0.438350\pi$
$270$	0	0
$271$	−14.5236	−0.882245	−0.441123	−	0.897447i	$-0.645420\pi$
$271$	−14.5236	−0.882245	−0.441123	+	0.897447i	$0.645420\pi$
$272$	−1.07838	−0.0653863
$273$	0	0
$274$	−4.68035	−0.282750
$275$	42.5958	2.56862
$276$	0	0
$277$	17.8843	1.07456	0.537281	−	0.843403i	$-0.319451\pi$
$277$	17.8843	1.07456	0.537281	+	0.843403i	$0.319451\pi$
$278$	−6.52359	−0.391259
$279$	0	0
$280$	−4.34017	−0.259375
$281$	0.837101	0.0499373	0.0249686	−	0.999688i	$-0.492051\pi$
$281$	0.837101	0.0499373	0.0249686	+	0.999688i	$0.492051\pi$
$282$	0	0
$283$	−10.8371	−0.644199	−0.322099	−	0.946706i	$-0.604389\pi$
$283$	−10.8371	−0.644199	−0.322099	+	0.946706i	$0.604389\pi$
$284$	10.1568	0.602693
$285$	0	0
$286$	19.5174	1.15409
$287$	6.68035	0.394328
$288$	0	0
$289$	−15.8371	−0.931594
$290$	−38.3545	−2.25226
$291$	0	0
$292$	−4.15676	−0.243256
$293$	−19.6742	−1.14938	−0.574690	−	0.818371i	$-0.694877\pi$
$293$	−19.6742	−1.14938	−0.574690	+	0.818371i	$0.694877\pi$
$294$	0	0
$295$	0	0
$296$	−3.41855	−0.198699
$297$	0	0
$298$	18.0989	1.04844
$299$	−14.8371	−0.858052
$300$	0	0
$301$	10.8371	0.624640
$302$	19.8576	1.14268
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	55.7152	3.19025
$306$	0	0
$307$	−16.6803	−0.951998	−0.475999	−	0.879446i	$-0.657913\pi$
$307$	−16.6803	−0.951998	−0.475999	+	0.879446i	$0.657913\pi$
$308$	−3.07838	−0.175407
$309$	0	0
$310$	−23.5174	−1.33570
$311$	−29.0928	−1.64970	−0.824849	−	0.565353i	$-0.808740\pi$
$311$	−29.0928	−1.64970	−0.824849	+	0.565353i	$0.808740\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	14.3135	0.807758
$315$	0	0
$316$	−14.8638	−0.836152
$317$	31.6742	1.77900	0.889500	−	0.456935i	$-0.151053\pi$
$317$	31.6742	1.77900	0.889500	+	0.456935i	$0.151053\pi$
$318$	0	0
$319$	−27.2039	−1.52313
$320$	4.34017	0.242623
$321$	0	0
$322$	2.34017	0.130413
$323$	1.07838	0.0600025
$324$	0	0
$325$	87.7296	4.86636
$326$	5.36069	0.296901
$327$	0	0
$328$	−6.68035	−0.368860
$329$	−2.73820	−0.150962
$330$	0	0
$331$	−18.1256	−0.996271	−0.498135	−	0.867099i	$-0.665982\pi$
$331$	−18.1256	−0.996271	−0.498135	+	0.867099i	$0.665982\pi$
$332$	−2.68035	−0.147103
$333$	0	0
$334$	−12.6803	−0.693838
$335$	4.68035	0.255715
$336$	0	0
$337$	24.1568	1.31590	0.657951	−	0.753061i	$-0.271423\pi$
$337$	24.1568	1.31590	0.657951	+	0.753061i	$0.271423\pi$
$338$	27.1978	1.47936
$339$	0	0
$340$	−4.68035	−0.253827
$341$	−16.6803	−0.903291
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−10.8371	−0.584297
$345$	0	0
$346$	−1.68649	−0.0906662
$347$	23.9565	1.28605	0.643027	−	0.765844i	$-0.277678\pi$
$347$	23.9565	1.28605	0.643027	+	0.765844i	$0.277678\pi$
$348$	0	0
$349$	−10.6803	−0.571706	−0.285853	−	0.958274i	$-0.592277\pi$
$349$	−10.6803	−0.571706	−0.285853	+	0.958274i	$0.592277\pi$
$350$	−13.8371	−0.739624
$351$	0	0
$352$	3.07838	0.164078
$353$	−35.9155	−1.91159	−0.955794	−	0.294037i	$-0.905001\pi$
$353$	−35.9155	−1.91159	−0.955794	+	0.294037i	$0.905001\pi$
$354$	0	0
$355$	44.0821	2.33963
$356$	3.84324	0.203692
$357$	0	0
$358$	17.0472	0.900972
$359$	−1.85762	−0.0980415	−0.0490207	−	0.998798i	$-0.515610\pi$
$359$	−1.85762	−0.0980415	−0.0490207	+	0.998798i	$0.515610\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	7.02052	0.368990
$363$	0	0
$364$	−6.34017	−0.332315
$365$	−18.0410	−0.944311
$366$	0	0
$367$	−12.1978	−0.636720	−0.318360	−	0.947970i	$-0.603132\pi$
$367$	−12.1978	−0.636720	−0.318360	+	0.947970i	$0.603132\pi$
$368$	−2.34017	−0.121990
$369$	0	0
$370$	−14.8371	−0.771344
$371$	−2.00000	−0.103835
$372$	0	0
$373$	−22.9360	−1.18758	−0.593790	−	0.804620i	$-0.702369\pi$
$373$	−22.9360	−1.18758	−0.593790	+	0.804620i	$0.702369\pi$
$374$	−3.31965	−0.171655
$375$	0	0
$376$	2.73820	0.141212
$377$	−56.0288	−2.88563
$378$	0	0
$379$	28.9093	1.48497	0.742486	−	0.669861i	$-0.233646\pi$
$379$	28.9093	1.48497	0.742486	+	0.669861i	$0.233646\pi$
$380$	−4.34017	−0.222646
$381$	0	0
$382$	−20.0144	−1.02402
$383$	−26.5236	−1.35529	−0.677646	−	0.735388i	$-0.737000\pi$
$383$	−26.5236	−1.35529	−0.677646	+	0.735388i	$0.737000\pi$
$384$	0	0
$385$	−13.3607	−0.680924
$386$	−16.8371	−0.856986
$387$	0	0
$388$	5.60197	0.284397
$389$	4.05786	0.205742	0.102871	−	0.994695i	$-0.467197\pi$
$389$	4.05786	0.205742	0.102871	+	0.994695i	$0.467197\pi$
$390$	0	0
$391$	2.52359	0.127623
$392$	1.00000	0.0505076
$393$	0	0
$394$	22.0989	1.11333
$395$	−64.5113	−3.24592
$396$	0	0
$397$	−12.8371	−0.644276	−0.322138	−	0.946693i	$-0.604401\pi$
$397$	−12.8371	−0.644276	−0.322138	+	0.946693i	$0.604401\pi$
$398$	9.67420	0.484924
$399$	0	0
$400$	13.8371	0.691855
$401$	29.8310	1.48969	0.744843	−	0.667239i	$-0.232524\pi$
$401$	29.8310	1.48969	0.744843	+	0.667239i	$0.232524\pi$
$402$	0	0
$403$	−34.3545	−1.71132
$404$	0.340173	0.0169242
$405$	0	0
$406$	8.83710	0.438578
$407$	−10.5236	−0.521635
$408$	0	0
$409$	−17.4329	−0.862003	−0.431001	−	0.902351i	$-0.641840\pi$
$409$	−17.4329	−0.862003	−0.431001	+	0.902351i	$0.641840\pi$
$410$	−28.9939	−1.43190
$411$	0	0
$412$	−15.9421	−0.785413
$413$	0	0
$414$	0	0
$415$	−11.6332	−0.571050
$416$	6.34017	0.310853
$417$	0	0
$418$	−3.07838	−0.150568
$419$	35.9877	1.75811	0.879057	−	0.476716i	$-0.158173\pi$
$419$	35.9877	1.75811	0.879057	+	0.476716i	$0.158173\pi$
$420$	0	0
$421$	−1.57531	−0.0767757	−0.0383879	−	0.999263i	$-0.512222\pi$
$421$	−1.57531	−0.0767757	−0.0383879	+	0.999263i	$0.512222\pi$
$422$	−14.0722	−0.685026
$423$	0	0
$424$	2.00000	0.0971286
$425$	−14.9216	−0.723805
$426$	0	0
$427$	−12.8371	−0.621231
$428$	5.84324	0.282444
$429$	0	0
$430$	−47.0349	−2.26822
$431$	−25.3607	−1.22158	−0.610791	−	0.791792i	$-0.709148\pi$
$431$	−25.3607	−1.22158	−0.610791	+	0.791792i	$0.709148\pi$
$432$	0	0
$433$	−7.13170	−0.342728	−0.171364	−	0.985208i	$-0.554817\pi$
$433$	−7.13170	−0.342728	−0.171364	+	0.985208i	$0.554817\pi$
$434$	5.41855	0.260099
$435$	0	0
$436$	−18.2557	−0.874287
$437$	2.34017	0.111946
$438$	0	0
$439$	−13.7321	−0.655396	−0.327698	−	0.944783i	$-0.606273\pi$
$439$	−13.7321	−0.655396	−0.327698	+	0.944783i	$0.606273\pi$
$440$	13.3607	0.636946
$441$	0	0
$442$	−6.83710	−0.325208
$443$	−22.2823	−1.05866	−0.529332	−	0.848415i	$-0.677558\pi$
$443$	−22.2823	−1.05866	−0.529332	+	0.848415i	$0.677558\pi$
$444$	0	0
$445$	16.6803	0.790724
$446$	5.78539	0.273946
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	36.7214	1.73299	0.866495	−	0.499186i	$-0.166368\pi$
$449$	36.7214	1.73299	0.866495	+	0.499186i	$0.166368\pi$
$450$	0	0
$451$	−20.5646	−0.968351
$452$	0.523590	0.0246276
$453$	0	0
$454$	16.1978	0.760200
$455$	−27.5174	−1.29004
$456$	0	0
$457$	22.5113	1.05303	0.526517	−	0.850164i	$-0.323498\pi$
$457$	22.5113	1.05303	0.526517	+	0.850164i	$0.323498\pi$
$458$	−22.6803	−1.05978
$459$	0	0
$460$	−10.1568	−0.473561
$461$	24.8515	1.15745	0.578724	−	0.815523i	$-0.303551\pi$
$461$	24.8515	1.15745	0.578724	+	0.815523i	$0.303551\pi$
$462$	0	0
$463$	24.6803	1.14699	0.573496	−	0.819208i	$-0.305587\pi$
$463$	24.6803	1.14699	0.573496	+	0.819208i	$0.305587\pi$
$464$	−8.83710	−0.410252
$465$	0	0
$466$	5.84324	0.270683
$467$	−30.9939	−1.43422	−0.717112	−	0.696958i	$-0.754537\pi$
$467$	−30.9939	−1.43422	−0.717112	+	0.696958i	$0.754537\pi$
$468$	0	0
$469$	−1.07838	−0.0497949
$470$	11.8843	0.548181
$471$	0	0
$472$	0	0
$473$	−33.3607	−1.53393
$474$	0	0
$475$	−13.8371	−0.634890
$476$	1.07838	0.0494274
$477$	0	0
$478$	−17.1773	−0.785670
$479$	11.7321	0.536052	0.268026	−	0.963412i	$-0.413629\pi$
$479$	11.7321	0.536052	0.268026	+	0.963412i	$0.413629\pi$
$480$	0	0
$481$	−21.6742	−0.988259
$482$	−1.60197	−0.0729677
$483$	0	0
$484$	−1.52359	−0.0692541
$485$	24.3135	1.10402
$486$	0	0
$487$	−34.0677	−1.54375	−0.771877	−	0.635771i	$-0.780682\pi$
$487$	−34.0677	−1.54375	−0.771877	+	0.635771i	$0.780682\pi$
$488$	12.8371	0.581108
$489$	0	0
$490$	4.34017	0.196069
$491$	41.4329	1.86984	0.934921	−	0.354856i	$-0.115470\pi$
$491$	41.4329	1.86984	0.934921	+	0.354856i	$0.115470\pi$
$492$	0	0
$493$	9.52973	0.429198
$494$	−6.34017	−0.285258
$495$	0	0
$496$	−5.41855	−0.243300
$497$	−10.1568	−0.455593
$498$	0	0
$499$	18.0410	0.807628	0.403814	−	0.914841i	$-0.367684\pi$
$499$	18.0410	0.807628	0.403814	+	0.914841i	$0.367684\pi$
$500$	38.3545	1.71527
$501$	0	0
$502$	24.8371	1.10853
$503$	17.6286	0.786022	0.393011	−	0.919534i	$-0.371433\pi$
$503$	17.6286	0.786022	0.393011	+	0.919534i	$0.371433\pi$
$504$	0	0
$505$	1.47641	0.0656994
$506$	−7.20394	−0.320254
$507$	0	0
$508$	5.81658	0.258069
$509$	−18.3668	−0.814096	−0.407048	−	0.913407i	$-0.633442\pi$
$509$	−18.3668	−0.814096	−0.407048	+	0.913407i	$0.633442\pi$
$510$	0	0
$511$	4.15676	0.183884
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	−69.1917	−3.04895
$516$	0	0
$517$	8.42923	0.370717
$518$	3.41855	0.150202
$519$	0	0
$520$	27.5174	1.20672
$521$	−10.7337	−0.470251	−0.235125	−	0.971965i	$-0.575550\pi$
$521$	−10.7337	−0.470251	−0.235125	+	0.971965i	$0.575550\pi$
$522$	0	0
$523$	10.3545	0.452773	0.226386	−	0.974038i	$-0.427309\pi$
$523$	10.3545	0.452773	0.226386	+	0.974038i	$0.427309\pi$
$524$	−4.52359	−0.197614
$525$	0	0
$526$	11.7009	0.510182
$527$	5.84324	0.254536
$528$	0	0
$529$	−17.5236	−0.761895
$530$	8.68035	0.377050
$531$	0	0
$532$	1.00000	0.0433555
$533$	−42.3545	−1.83458
$534$	0	0
$535$	25.3607	1.09644
$536$	1.07838	0.0465788
$537$	0	0
$538$	6.31351	0.272195
$539$	3.07838	0.132595
$540$	0	0
$541$	13.5174	0.581160	0.290580	−	0.956851i	$-0.406152\pi$
$541$	13.5174	0.581160	0.290580	+	0.956851i	$0.406152\pi$
$542$	−14.5236	−0.623842
$543$	0	0
$544$	−1.07838	−0.0462351
$545$	−79.2327	−3.39396
$546$	0	0
$547$	−23.2351	−0.993463	−0.496731	−	0.867904i	$-0.665467\pi$
$547$	−23.2351	−0.993463	−0.496731	+	0.867904i	$0.665467\pi$
$548$	−4.68035	−0.199934
$549$	0	0
$550$	42.5958	1.81629
$551$	8.83710	0.376473
$552$	0	0
$553$	14.8638	0.632072
$554$	17.8843	0.759830
$555$	0	0
$556$	−6.52359	−0.276662
$557$	−13.9011	−0.589009	−0.294504	−	0.955650i	$-0.595155\pi$
$557$	−13.9011	−0.589009	−0.294504	+	0.955650i	$0.595155\pi$
$558$	0	0
$559$	−68.7091	−2.90609
$560$	−4.34017	−0.183406
$561$	0	0
$562$	0.837101	0.0353110
$563$	22.7214	0.957592	0.478796	−	0.877926i	$-0.341073\pi$
$563$	22.7214	0.957592	0.478796	+	0.877926i	$0.341073\pi$
$564$	0	0
$565$	2.27247	0.0956037
$566$	−10.8371	−0.455517
$567$	0	0
$568$	10.1568	0.426168
$569$	−27.3074	−1.14478	−0.572392	−	0.819980i	$-0.693985\pi$
$569$	−27.3074	−1.14478	−0.572392	+	0.819980i	$0.693985\pi$
$570$	0	0
$571$	−43.4017	−1.81631	−0.908153	−	0.418639i	$-0.862507\pi$
$571$	−43.4017	−1.81631	−0.908153	+	0.418639i	$0.862507\pi$
$572$	19.5174	0.816065
$573$	0	0
$574$	6.68035	0.278832
$575$	−32.3812	−1.35039
$576$	0	0
$577$	−30.1445	−1.25493	−0.627465	−	0.778644i	$-0.715908\pi$
$577$	−30.1445	−1.25493	−0.627465	+	0.778644i	$0.715908\pi$
$578$	−15.8371	−0.658737
$579$	0	0
$580$	−38.3545	−1.59259
$581$	2.68035	0.111199
$582$	0	0
$583$	6.15676	0.254987
$584$	−4.15676	−0.172008
$585$	0	0
$586$	−19.6742	−0.812734
$587$	−8.47027	−0.349605	−0.174803	−	0.984603i	$-0.555929\pi$
$587$	−8.47027	−0.349605	−0.174803	+	0.984603i	$0.555929\pi$
$588$	0	0
$589$	5.41855	0.223267
$590$	0	0
$591$	0	0
$592$	−3.41855	−0.140502
$593$	−21.7587	−0.893524	−0.446762	−	0.894653i	$-0.647423\pi$
$593$	−21.7587	−0.893524	−0.446762	+	0.894653i	$0.647423\pi$
$594$	0	0
$595$	4.68035	0.191875
$596$	18.0989	0.741360
$597$	0	0
$598$	−14.8371	−0.606734
$599$	−24.9939	−1.02122	−0.510611	−	0.859812i	$-0.670581\pi$
$599$	−24.9939	−1.02122	−0.510611	+	0.859812i	$0.670581\pi$
$600$	0	0
$601$	15.3919	0.627848	0.313924	−	0.949448i	$-0.398356\pi$
$601$	15.3919	0.627848	0.313924	+	0.949448i	$0.398356\pi$
$602$	10.8371	0.441687
$603$	0	0
$604$	19.8576	0.807995
$605$	−6.61265	−0.268842
$606$	0	0
$607$	41.6697	1.69132	0.845660	−	0.533722i	$-0.179207\pi$
$607$	41.6697	1.69132	0.845660	+	0.533722i	$0.179207\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	55.7152	2.25584
$611$	17.3607	0.702338
$612$	0	0
$613$	0.639308	0.0258214	0.0129107	−	0.999917i	$-0.495890\pi$
$613$	0.639308	0.0258214	0.0129107	+	0.999917i	$0.495890\pi$
$614$	−16.6803	−0.673164
$615$	0	0
$616$	−3.07838	−0.124031
$617$	12.7961	0.515150	0.257575	−	0.966258i	$-0.417077\pi$
$617$	12.7961	0.515150	0.257575	+	0.966258i	$0.417077\pi$
$618$	0	0
$619$	14.3545	0.576958	0.288479	−	0.957486i	$-0.406850\pi$
$619$	14.3545	0.576958	0.288479	+	0.957486i	$0.406850\pi$
$620$	−23.5174	−0.944483
$621$	0	0
$622$	−29.0928	−1.16651
$623$	−3.84324	−0.153976
$624$	0	0
$625$	97.2799	3.89119
$626$	−14.0000	−0.559553
$627$	0	0
$628$	14.3135	0.571171
$629$	3.68649	0.146990
$630$	0	0
$631$	3.00614	0.119673	0.0598363	−	0.998208i	$-0.480942\pi$
$631$	3.00614	0.119673	0.0598363	+	0.998208i	$0.480942\pi$
$632$	−14.8638	−0.591249
$633$	0	0
$634$	31.6742	1.25794
$635$	25.2450	1.00182
$636$	0	0
$637$	6.34017	0.251207
$638$	−27.2039	−1.07701
$639$	0	0
$640$	4.34017	0.171560
$641$	19.6742	0.777084	0.388542	−	0.921431i	$-0.372979\pi$
$641$	19.6742	0.777084	0.388542	+	0.921431i	$0.372979\pi$
$642$	0	0
$643$	40.1445	1.58314	0.791572	−	0.611076i	$-0.209263\pi$
$643$	40.1445	1.58314	0.791572	+	0.611076i	$0.209263\pi$
$644$	2.34017	0.0922157
$645$	0	0
$646$	1.07838	0.0424282
$647$	−25.6286	−1.00757	−0.503783	−	0.863830i	$-0.668059\pi$
$647$	−25.6286	−1.00757	−0.503783	+	0.863830i	$0.668059\pi$
$648$	0	0
$649$	0	0
$650$	87.7296	3.44104
$651$	0	0
$652$	5.36069	0.209941
$653$	−5.73206	−0.224313	−0.112156	−	0.993691i	$-0.535776\pi$
$653$	−5.73206	−0.224313	−0.112156	+	0.993691i	$0.535776\pi$
$654$	0	0
$655$	−19.6332	−0.767131
$656$	−6.68035	−0.260824
$657$	0	0
$658$	−2.73820	−0.106746
$659$	−10.8904	−0.424231	−0.212115	−	0.977245i	$-0.568035\pi$
$659$	−10.8904	−0.424231	−0.212115	+	0.977245i	$0.568035\pi$
$660$	0	0
$661$	6.82273	0.265373	0.132687	−	0.991158i	$-0.457640\pi$
$661$	6.82273	0.265373	0.132687	+	0.991158i	$0.457640\pi$
$662$	−18.1256	−0.704470
$663$	0	0
$664$	−2.68035	−0.104018
$665$	4.34017	0.168305
$666$	0	0
$667$	20.6803	0.800746
$668$	−12.6803	−0.490617
$669$	0	0
$670$	4.68035	0.180818
$671$	39.5174	1.52555
$672$	0	0
$673$	−12.3545	−0.476233	−0.238116	−	0.971237i	$-0.576530\pi$
$673$	−12.3545	−0.476233	−0.238116	+	0.971237i	$0.576530\pi$
$674$	24.1568	0.930483
$675$	0	0
$676$	27.1978	1.04607
$677$	−48.3545	−1.85842	−0.929208	−	0.369557i	$-0.879510\pi$
$677$	−48.3545	−1.85842	−0.929208	+	0.369557i	$0.879510\pi$
$678$	0	0
$679$	−5.60197	−0.214984
$680$	−4.68035	−0.179483
$681$	0	0
$682$	−16.6803	−0.638723
$683$	40.0821	1.53370	0.766849	−	0.641828i	$-0.221824\pi$
$683$	40.0821	1.53370	0.766849	+	0.641828i	$0.221824\pi$
$684$	0	0
$685$	−20.3135	−0.776139
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−10.8371	−0.413161
$689$	12.6803	0.483083
$690$	0	0
$691$	42.3012	1.60921	0.804607	−	0.593807i	$-0.202376\pi$
$691$	42.3012	1.60921	0.804607	+	0.593807i	$0.202376\pi$
$692$	−1.68649	−0.0641107
$693$	0	0
$694$	23.9565	0.909377
$695$	−28.3135	−1.07399
$696$	0	0
$697$	7.20394	0.272869
$698$	−10.6803	−0.404257
$699$	0	0
$700$	−13.8371	−0.522993
$701$	50.7792	1.91791	0.958953	−	0.283566i	$-0.0915175\pi$
$701$	50.7792	1.91791	0.958953	+	0.283566i	$0.0915175\pi$
$702$	0	0
$703$	3.41855	0.128933
$704$	3.07838	0.116021
$705$	0	0
$706$	−35.9155	−1.35170
$707$	−0.340173	−0.0127935
$708$	0	0
$709$	−5.94668	−0.223332	−0.111666	−	0.993746i	$-0.535619\pi$
$709$	−5.94668	−0.223332	−0.111666	+	0.993746i	$0.535619\pi$
$710$	44.0821	1.65437
$711$	0	0
$712$	3.84324	0.144032
$713$	12.6803	0.474883
$714$	0	0
$715$	84.7091	3.16794
$716$	17.0472	0.637083
$717$	0	0
$718$	−1.85762	−0.0693258
$719$	30.2557	1.12835	0.564173	−	0.825657i	$-0.309195\pi$
$719$	30.2557	1.12835	0.564173	+	0.825657i	$0.309195\pi$
$720$	0	0
$721$	15.9421	0.593716
$722$	1.00000	0.0372161
$723$	0	0
$724$	7.02052	0.260916
$725$	−122.280	−4.54136
$726$	0	0
$727$	2.95282	0.109514	0.0547570	−	0.998500i	$-0.482562\pi$
$727$	2.95282	0.109514	0.0547570	+	0.998500i	$0.482562\pi$
$728$	−6.34017	−0.234982
$729$	0	0
$730$	−18.0410	−0.667729
$731$	11.6865	0.432240
$732$	0	0
$733$	44.3545	1.63827	0.819136	−	0.573599i	$-0.194453\pi$
$733$	44.3545	1.63827	0.819136	+	0.573599i	$0.194453\pi$
$734$	−12.1978	−0.450229
$735$	0	0
$736$	−2.34017	−0.0862599
$737$	3.31965	0.122281
$738$	0	0
$739$	11.8310	0.435209	0.217604	−	0.976037i	$-0.430176\pi$
$739$	11.8310	0.435209	0.217604	+	0.976037i	$0.430176\pi$
$740$	−14.8371	−0.545423
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	5.84324	0.214368	0.107184	−	0.994239i	$-0.465817\pi$
$743$	5.84324	0.214368	0.107184	+	0.994239i	$0.465817\pi$
$744$	0	0
$745$	78.5523	2.87794
$746$	−22.9360	−0.839747
$747$	0	0
$748$	−3.31965	−0.121379
$749$	−5.84324	−0.213508
$750$	0	0
$751$	−32.0554	−1.16972	−0.584859	−	0.811135i	$-0.698850\pi$
$751$	−32.0554	−1.16972	−0.584859	+	0.811135i	$0.698850\pi$
$752$	2.73820	0.0998521
$753$	0	0
$754$	−56.0288	−2.04045
$755$	86.1855	3.13661
$756$	0	0
$757$	2.73367	0.0993569	0.0496785	−	0.998765i	$-0.484180\pi$
$757$	2.73367	0.0993569	0.0496785	+	0.998765i	$0.484180\pi$
$758$	28.9093	1.05003
$759$	0	0
$760$	−4.34017	−0.157435
$761$	−24.9093	−0.902963	−0.451481	−	0.892281i	$-0.649104\pi$
$761$	−24.9093	−0.902963	−0.451481	+	0.892281i	$0.649104\pi$
$762$	0	0
$763$	18.2557	0.660899
$764$	−20.0144	−0.724095
$765$	0	0
$766$	−26.5236	−0.958336
$767$	0	0
$768$	0	0
$769$	−49.3484	−1.77955	−0.889775	−	0.456400i	$-0.849138\pi$
$769$	−49.3484	−1.77955	−0.889775	+	0.456400i	$0.849138\pi$
$770$	−13.3607	−0.481486
$771$	0	0
$772$	−16.8371	−0.605981
$773$	−44.3545	−1.59532	−0.797661	−	0.603106i	$-0.793930\pi$
$773$	−44.3545	−1.59532	−0.797661	+	0.603106i	$0.793930\pi$
$774$	0	0
$775$	−74.9770	−2.69325
$776$	5.60197	0.201099
$777$	0	0
$778$	4.05786	0.145481
$779$	6.68035	0.239348
$780$	0	0
$781$	31.2663	1.11880
$782$	2.52359	0.0902434
$783$	0	0
$784$	1.00000	0.0357143
$785$	62.1231	2.21727
$786$	0	0
$787$	28.5646	1.01822	0.509110	−	0.860702i	$-0.329975\pi$
$787$	28.5646	1.01822	0.509110	+	0.860702i	$0.329975\pi$
$788$	22.0989	0.787241
$789$	0	0
$790$	−64.5113	−2.29521
$791$	−0.523590	−0.0186167
$792$	0	0
$793$	81.3894	2.89022
$794$	−12.8371	−0.455572
$795$	0	0
$796$	9.67420	0.342893
$797$	−37.0882	−1.31373	−0.656866	−	0.754007i	$-0.728118\pi$
$797$	−37.0882	−1.31373	−0.656866	+	0.754007i	$0.728118\pi$
$798$	0	0
$799$	−2.95282	−0.104463
$800$	13.8371	0.489215
$801$	0	0
$802$	29.8310	1.05337
$803$	−12.7961	−0.451563
$804$	0	0
$805$	10.1568	0.357979
$806$	−34.3545	−1.21009
$807$	0	0
$808$	0.340173	0.0119672
$809$	19.4641	0.684322	0.342161	−	0.939641i	$-0.388841\pi$
$809$	19.4641	0.684322	0.342161	+	0.939641i	$0.388841\pi$
$810$	0	0
$811$	−46.9816	−1.64975	−0.824873	−	0.565318i	$-0.808753\pi$
$811$	−46.9816	−1.64975	−0.824873	+	0.565318i	$0.808753\pi$
$812$	8.83710	0.310121
$813$	0	0
$814$	−10.5236	−0.368852
$815$	23.2663	0.814984
$816$	0	0
$817$	10.8371	0.379142
$818$	−17.4329	−0.609528
$819$	0	0
$820$	−28.9939	−1.01251
$821$	26.9483	0.940502	0.470251	−	0.882533i	$-0.344163\pi$
$821$	26.9483	0.940502	0.470251	+	0.882533i	$0.344163\pi$
$822$	0	0
$823$	30.3545	1.05809	0.529047	−	0.848593i	$-0.322550\pi$
$823$	30.3545	1.05809	0.529047	+	0.848593i	$0.322550\pi$
$824$	−15.9421	−0.555371
$825$	0	0
$826$	0	0
$827$	42.4079	1.47467	0.737333	−	0.675529i	$-0.236085\pi$
$827$	42.4079	1.47467	0.737333	+	0.675529i	$0.236085\pi$
$828$	0	0
$829$	−9.54411	−0.331481	−0.165740	−	0.986169i	$-0.553001\pi$
$829$	−9.54411	−0.331481	−0.165740	+	0.986169i	$0.553001\pi$
$830$	−11.6332	−0.403793
$831$	0	0
$832$	6.34017	0.219806
$833$	−1.07838	−0.0373636
$834$	0	0
$835$	−55.0349	−1.90456
$836$	−3.07838	−0.106468
$837$	0	0
$838$	35.9877	1.24317
$839$	16.5646	0.571874	0.285937	−	0.958248i	$-0.407695\pi$
$839$	16.5646	0.571874	0.285937	+	0.958248i	$0.407695\pi$
$840$	0	0
$841$	49.0944	1.69291
$842$	−1.57531	−0.0542886
$843$	0	0
$844$	−14.0722	−0.484386
$845$	118.043	4.06081
$846$	0	0
$847$	1.52359	0.0523512
$848$	2.00000	0.0686803
$849$	0	0
$850$	−14.9216	−0.511807
$851$	8.00000	0.274236
$852$	0	0
$853$	−14.1445	−0.484297	−0.242149	−	0.970239i	$-0.577852\pi$
$853$	−14.1445	−0.484297	−0.242149	+	0.970239i	$0.577852\pi$
$854$	−12.8371	−0.439277
$855$	0	0
$856$	5.84324	0.199718
$857$	−19.5297	−0.667123	−0.333561	−	0.942728i	$-0.608250\pi$
$857$	−19.5297	−0.667123	−0.333561	+	0.942728i	$0.608250\pi$
$858$	0	0
$859$	−11.6332	−0.396918	−0.198459	−	0.980109i	$-0.563594\pi$
$859$	−11.6332	−0.396918	−0.198459	+	0.980109i	$0.563594\pi$
$860$	−47.0349	−1.60388
$861$	0	0
$862$	−25.3607	−0.863789
$863$	5.10957	0.173932	0.0869660	−	0.996211i	$-0.472283\pi$
$863$	5.10957	0.173932	0.0869660	+	0.996211i	$0.472283\pi$
$864$	0	0
$865$	−7.31965	−0.248876
$866$	−7.13170	−0.242345
$867$	0	0
$868$	5.41855	0.183918
$869$	−45.7563	−1.55218
$870$	0	0
$871$	6.83710	0.231666
$872$	−18.2557	−0.618214
$873$	0	0
$874$	2.34017	0.0791575
$875$	−38.3545	−1.29662
$876$	0	0
$877$	−4.15222	−0.140211	−0.0701053	−	0.997540i	$-0.522334\pi$
$877$	−4.15222	−0.140211	−0.0701053	+	0.997540i	$0.522334\pi$
$878$	−13.7321	−0.463435
$879$	0	0
$880$	13.3607	0.450389
$881$	20.0845	0.676665	0.338332	−	0.941027i	$-0.390137\pi$
$881$	20.0845	0.676665	0.338332	+	0.941027i	$0.390137\pi$
$882$	0	0
$883$	−16.6803	−0.561338	−0.280669	−	0.959805i	$-0.590556\pi$
$883$	−16.6803	−0.561338	−0.280669	+	0.959805i	$0.590556\pi$
$884$	−6.83710	−0.229957
$885$	0	0
$886$	−22.2823	−0.748589
$887$	−45.6742	−1.53359	−0.766795	−	0.641892i	$-0.778150\pi$
$887$	−45.6742	−1.53359	−0.766795	+	0.641892i	$0.778150\pi$
$888$	0	0
$889$	−5.81658	−0.195082
$890$	16.6803	0.559126
$891$	0	0
$892$	5.78539	0.193709
$893$	−2.73820	−0.0916305
$894$	0	0
$895$	73.9877	2.47314
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	36.7214	1.22541
$899$	47.8843	1.59703
$900$	0	0
$901$	−2.15676	−0.0718519
$902$	−20.5646	−0.684727
$903$	0	0
$904$	0.523590	0.0174144
$905$	30.4703	1.01287
$906$	0	0
$907$	−34.4391	−1.14353	−0.571765	−	0.820417i	$-0.693741\pi$
$907$	−34.4391	−1.14353	−0.571765	+	0.820417i	$0.693741\pi$
$908$	16.1978	0.537543
$909$	0	0
$910$	−27.5174	−0.912194
$911$	−16.3668	−0.542257	−0.271129	−	0.962543i	$-0.587397\pi$
$911$	−16.3668	−0.542257	−0.271129	+	0.962543i	$0.587397\pi$
$912$	0	0
$913$	−8.25112	−0.273072
$914$	22.5113	0.744608
$915$	0	0
$916$	−22.6803	−0.749380
$917$	4.52359	0.149382
$918$	0	0
$919$	−14.3545	−0.473513	−0.236756	−	0.971569i	$-0.576084\pi$
$919$	−14.3545	−0.473513	−0.236756	+	0.971569i	$0.576084\pi$
$920$	−10.1568	−0.334858
$921$	0	0
$922$	24.8515	0.818440
$923$	64.3956	2.11961
$924$	0	0
$925$	−47.3028	−1.55531
$926$	24.6803	0.811046
$927$	0	0
$928$	−8.83710	−0.290092
$929$	−1.39189	−0.0456664	−0.0228332	−	0.999739i	$-0.507269\pi$
$929$	−1.39189	−0.0456664	−0.0228332	+	0.999739i	$0.507269\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	5.84324	0.191402
$933$	0	0
$934$	−30.9939	−1.01415
$935$	−14.4079	−0.471188
$936$	0	0
$937$	−20.7214	−0.676938	−0.338469	−	0.940978i	$-0.609909\pi$
$937$	−20.7214	−0.676938	−0.338469	+	0.940978i	$0.609909\pi$
$938$	−1.07838	−0.0352103
$939$	0	0
$940$	11.8843	0.387623
$941$	27.4140	0.893671	0.446836	−	0.894616i	$-0.352551\pi$
$941$	27.4140	0.893671	0.446836	+	0.894616i	$0.352551\pi$
$942$	0	0
$943$	15.6332	0.509086
$944$	0	0
$945$	0	0
$946$	−33.3607	−1.08465
$947$	57.4329	1.86632	0.933160	−	0.359462i	$-0.117040\pi$
$947$	57.4329	1.86632	0.933160	+	0.359462i	$0.117040\pi$
$948$	0	0
$949$	−26.3545	−0.855505
$950$	−13.8371	−0.448935
$951$	0	0
$952$	1.07838	0.0349504
$953$	−28.8371	−0.934125	−0.467063	−	0.884224i	$-0.654688\pi$
$953$	−28.8371	−0.934125	−0.467063	+	0.884224i	$0.654688\pi$
$954$	0	0
$955$	−86.8659	−2.81091
$956$	−17.1773	−0.555553
$957$	0	0
$958$	11.7321	0.379046
$959$	4.68035	0.151136
$960$	0	0
$961$	−1.63931	−0.0528809
$962$	−21.6742	−0.698804
$963$	0	0
$964$	−1.60197	−0.0515959
$965$	−73.0759	−2.35240
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	−1.52359	−0.0489701
$969$	0	0
$970$	24.3135	0.780660
$971$	32.9315	1.05682	0.528410	−	0.848989i	$-0.322788\pi$
$971$	32.9315	1.05682	0.528410	+	0.848989i	$0.322788\pi$
$972$	0	0
$973$	6.52359	0.209137
$974$	−34.0677	−1.09160
$975$	0	0
$976$	12.8371	0.410906
$977$	16.1568	0.516900	0.258450	−	0.966025i	$-0.416788\pi$
$977$	16.1568	0.516900	0.258450	+	0.966025i	$0.416788\pi$
$978$	0	0
$979$	11.8310	0.378119
$980$	4.34017	0.138642
$981$	0	0
$982$	41.4329	1.32218
$983$	−3.31965	−0.105881	−0.0529403	−	0.998598i	$-0.516859\pi$
$983$	−3.31965	−0.105881	−0.0529403	+	0.998598i	$0.516859\pi$
$984$	0	0
$985$	95.9130	3.05604
$986$	9.52973	0.303489
$987$	0	0
$988$	−6.34017	−0.201708
$989$	25.3607	0.806423
$990$	0	0
$991$	39.7419	1.26244	0.631222	−	0.775603i	$-0.282554\pi$
$991$	39.7419	1.26244	0.631222	+	0.775603i	$0.282554\pi$
$992$	−5.41855	−0.172039
$993$	0	0
$994$	−10.1568	−0.322153
$995$	41.9877	1.33110
$996$	0	0
$997$	−19.0472	−0.603230	−0.301615	−	0.953430i	$-0.597526\pi$
$997$	−19.0472	−0.603230	−0.301615	+	0.953430i	$0.597526\pi$
$998$	18.0410	0.571079
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.bc.1.3	yes	3
3.2	odd	2		2394.2.a.bb.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.a.bb.1.1	✓	3	3.2	odd	2
2394.2.a.bc.1.3	yes	3	1.1	even	1	trivial

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.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.34017 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.34017 q^{5} -1.00000 q^{7} +1.00000 q^{8} +4.34017 q^{10} +3.07838 q^{11} +6.34017 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.07838 q^{17} -1.00000 q^{19} +4.34017 q^{20} +3.07838 q^{22} -2.34017 q^{23} +13.8371 q^{25} +6.34017 q^{26} -1.00000 q^{28} -8.83710 q^{29} -5.41855 q^{31} +1.00000 q^{32} -1.07838 q^{34} -4.34017 q^{35} -3.41855 q^{37} -1.00000 q^{38} +4.34017 q^{40} -6.68035 q^{41} -10.8371 q^{43} +3.07838 q^{44} -2.34017 q^{46} +2.73820 q^{47} +1.00000 q^{49} +13.8371 q^{50} +6.34017 q^{52} +2.00000 q^{53} +13.3607 q^{55} -1.00000 q^{56} -8.83710 q^{58} +12.8371 q^{61} -5.41855 q^{62} +1.00000 q^{64} +27.5174 q^{65} +1.07838 q^{67} -1.07838 q^{68} -4.34017 q^{70} +10.1568 q^{71} -4.15676 q^{73} -3.41855 q^{74} -1.00000 q^{76} -3.07838 q^{77} -14.8638 q^{79} +4.34017 q^{80} -6.68035 q^{82} -2.68035 q^{83} -4.68035 q^{85} -10.8371 q^{86} +3.07838 q^{88} +3.84324 q^{89} -6.34017 q^{91} -2.34017 q^{92} +2.73820 q^{94} -4.34017 q^{95} +5.60197 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 2 q^{5} - 3 q^{7} + 3 q^{8} + 2 q^{10} + 6 q^{11} + 8 q^{13} - 3 q^{14} + 3 q^{16} - 3 q^{19} + 2 q^{20} + 6 q^{22} + 4 q^{23} + 13 q^{25} + 8 q^{26} - 3 q^{28} + 2 q^{29} - 2 q^{31} + 3 q^{32} - 2 q^{35} + 4 q^{37} - 3 q^{38} + 2 q^{40} + 2 q^{41} - 4 q^{43} + 6 q^{44} + 4 q^{46} + 16 q^{47} + 3 q^{49} + 13 q^{50} + 8 q^{52} + 6 q^{53} - 4 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{61} - 2 q^{62} + 3 q^{64} + 32 q^{65} - 2 q^{70} + 24 q^{71} - 6 q^{73} + 4 q^{74} - 3 q^{76} - 6 q^{77} - 18 q^{79} + 2 q^{80} + 2 q^{82} + 14 q^{83} + 8 q^{85} - 4 q^{86} + 6 q^{88} + 18 q^{89} - 8 q^{91} + 4 q^{92} + 16 q^{94} - 2 q^{95} - 2 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^5 - 3 * q^7 + 3 * q^8 + 2 * q^10 + 6 * q^11 + 8 * q^13 - 3 * q^14 + 3 * q^16 - 3 * q^19 + 2 * q^20 + 6 * q^22 + 4 * q^23 + 13 * q^25 + 8 * q^26 - 3 * q^28 + 2 * q^29 - 2 * q^31 + 3 * q^32 - 2 * q^35 + 4 * q^37 - 3 * q^38 + 2 * q^40 + 2 * q^41 - 4 * q^43 + 6 * q^44 + 4 * q^46 + 16 * q^47 + 3 * q^49 + 13 * q^50 + 8 * q^52 + 6 * q^53 - 4 * q^55 - 3 * q^56 + 2 * q^58 + 10 * q^61 - 2 * q^62 + 3 * q^64 + 32 * q^65 - 2 * q^70 + 24 * q^71 - 6 * q^73 + 4 * q^74 - 3 * q^76 - 6 * q^77 - 18 * q^79 + 2 * q^80 + 2 * q^82 + 14 * q^83 + 8 * q^85 - 4 * q^86 + 6 * q^88 + 18 * q^89 - 8 * q^91 + 4 * q^92 + 16 * q^94 - 2 * q^95 - 2 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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