LMFDB - Embedded newform 2394.2.a.ba.1.1

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:	3.3.469.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 266)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.16425$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -4.16425 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -4.16425 q^{5} -1.00000 q^{7} -1.00000 q^{8} +4.16425 q^{10} +5.01247 q^{11} -6.32850 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.96056 q^{17} -1.00000 q^{19} -4.16425 q^{20} -5.01247 q^{22} -4.32850 q^{23} +12.3410 q^{25} +6.32850 q^{26} -1.00000 q^{28} -1.20369 q^{29} -5.69643 q^{31} -1.00000 q^{32} +4.96056 q^{34} +4.16425 q^{35} +2.79631 q^{37} +1.00000 q^{38} +4.16425 q^{40} -0.164248 q^{41} -0.796310 q^{43} +5.01247 q^{44} +4.32850 q^{46} +4.68397 q^{47} +1.00000 q^{49} -12.3410 q^{50} -6.32850 q^{52} -4.05191 q^{53} -20.8731 q^{55} +1.00000 q^{56} +1.20369 q^{58} -6.49274 q^{59} -1.64453 q^{61} +5.69643 q^{62} +1.00000 q^{64} +26.3534 q^{65} +1.03944 q^{67} -4.96056 q^{68} -4.16425 q^{70} -9.01247 q^{71} +12.0249 q^{73} -2.79631 q^{74} -1.00000 q^{76} -5.01247 q^{77} +0.467814 q^{79} -4.16425 q^{80} +0.164248 q^{82} -4.65699 q^{83} +20.6570 q^{85} +0.796310 q^{86} -5.01247 q^{88} +7.01247 q^{89} +6.32850 q^{91} -4.32850 q^{92} -4.68397 q^{94} +4.16425 q^{95} -5.64453 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 3 q^{8} + 5 q^{10} - 3 q^{11} - 4 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 3 q^{19} - 5 q^{20} + 3 q^{22} + 2 q^{23} + 4 q^{25} + 4 q^{26} - 3 q^{28} - 5 q^{29} + 4 q^{31} - 3 q^{32} + 6 q^{34} + 5 q^{35} + 7 q^{37} + 3 q^{38} + 5 q^{40} + 7 q^{41} - q^{43} - 3 q^{44} - 2 q^{46} + 11 q^{47} + 3 q^{49} - 4 q^{50} - 4 q^{52} - 3 q^{53} - 16 q^{55} + 3 q^{56} + 5 q^{58} + 3 q^{59} + 7 q^{61} - 4 q^{62} + 3 q^{64} + 28 q^{65} + 12 q^{67} - 6 q^{68} - 5 q^{70} - 9 q^{71} - 7 q^{74} - 3 q^{76} + 3 q^{77} + 15 q^{79} - 5 q^{80} - 7 q^{82} + 16 q^{83} + 32 q^{85} + q^{86} + 3 q^{88} + 3 q^{89} + 4 q^{91} + 2 q^{92} - 11 q^{94} + 5 q^{95} - 5 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 - 3 * q^8 + 5 * q^10 - 3 * q^11 - 4 * q^13 + 3 * q^14 + 3 * q^16 - 6 * q^17 - 3 * q^19 - 5 * q^20 + 3 * q^22 + 2 * q^23 + 4 * q^25 + 4 * q^26 - 3 * q^28 - 5 * q^29 + 4 * q^31 - 3 * q^32 + 6 * q^34 + 5 * q^35 + 7 * q^37 + 3 * q^38 + 5 * q^40 + 7 * q^41 - q^43 - 3 * q^44 - 2 * q^46 + 11 * q^47 + 3 * q^49 - 4 * q^50 - 4 * q^52 - 3 * q^53 - 16 * q^55 + 3 * q^56 + 5 * q^58 + 3 * q^59 + 7 * q^61 - 4 * q^62 + 3 * q^64 + 28 * q^65 + 12 * q^67 - 6 * q^68 - 5 * q^70 - 9 * q^71 - 7 * q^74 - 3 * q^76 + 3 * q^77 + 15 * q^79 - 5 * q^80 - 7 * q^82 + 16 * q^83 + 32 * q^85 + q^86 + 3 * q^88 + 3 * q^89 + 4 * q^91 + 2 * q^92 - 11 * q^94 + 5 * q^95 - 5 * 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
$5$	−4.16425	−1.86231	−0.931154	−	0.364626i	$-0.881197\pi$
$5$	−4.16425	−1.86231	−0.931154	+	0.364626i	$0.881197\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	4.16425	1.31685
$11$	5.01247	1.51132	0.755658	−	0.654967i	$-0.227317\pi$
$11$	5.01247	1.51132	0.755658	+	0.654967i	$0.227317\pi$
$12$	0	0
$13$	−6.32850	−1.75521	−0.877604	−	0.479385i	$-0.840860\pi$
$13$	−6.32850	−1.75521	−0.877604	+	0.479385i	$0.840860\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.96056	−1.20311	−0.601556	−	0.798831i	$-0.705452\pi$
$17$	−4.96056	−1.20311	−0.601556	+	0.798831i	$0.705452\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−4.16425	−0.931154
$21$	0	0
$22$	−5.01247	−1.06866
$23$	−4.32850	−0.902554	−0.451277	−	0.892384i	$-0.649031\pi$
$23$	−4.32850	−0.902554	−0.451277	+	0.892384i	$0.649031\pi$
$24$	0	0
$25$	12.3410	2.46819
$26$	6.32850	1.24112
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−1.20369	−0.223520	−0.111760	−	0.993735i	$-0.535649\pi$
$29$	−1.20369	−0.223520	−0.111760	+	0.993735i	$0.535649\pi$
$30$	0	0
$31$	−5.69643	−1.02311	−0.511555	−	0.859251i	$-0.670930\pi$
$31$	−5.69643	−1.02311	−0.511555	+	0.859251i	$0.670930\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.96056	0.850729
$35$	4.16425	0.703886
$36$	0	0
$37$	2.79631	0.459710	0.229855	−	0.973225i	$-0.426175\pi$
$37$	2.79631	0.459710	0.229855	+	0.973225i	$0.426175\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	4.16425	0.658425
$41$	−0.164248	−0.0256512	−0.0128256	−	0.999918i	$-0.504083\pi$
$41$	−0.164248	−0.0256512	−0.0128256	+	0.999918i	$0.504083\pi$
$42$	0	0
$43$	−0.796310	−0.121436	−0.0607180	−	0.998155i	$-0.519339\pi$
$43$	−0.796310	−0.121436	−0.0607180	+	0.998155i	$0.519339\pi$
$44$	5.01247	0.755658
$45$	0	0
$46$	4.32850	0.638202
$47$	4.68397	0.683227	0.341614	−	0.939841i	$-0.389027\pi$
$47$	4.68397	0.683227	0.341614	+	0.939841i	$0.389027\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−12.3410	−1.74528
$51$	0	0
$52$	−6.32850	−0.877604
$53$	−4.05191	−0.556572	−0.278286	−	0.960498i	$-0.589766\pi$
$53$	−4.05191	−0.556572	−0.278286	+	0.960498i	$0.589766\pi$
$54$	0	0
$55$	−20.8731	−2.81453
$56$	1.00000	0.133631
$57$	0	0
$58$	1.20369	0.158052
$59$	−6.49274	−0.845283	−0.422642	−	0.906297i	$-0.638897\pi$
$59$	−6.49274	−0.845283	−0.422642	+	0.906297i	$0.638897\pi$
$60$	0	0
$61$	−1.64453	−0.210560	−0.105280	−	0.994443i	$-0.533574\pi$
$61$	−1.64453	−0.210560	−0.105280	+	0.994443i	$0.533574\pi$
$62$	5.69643	0.723448
$63$	0	0
$64$	1.00000	0.125000
$65$	26.3534	3.26874
$66$	0	0
$67$	1.03944	0.126988	0.0634941	−	0.997982i	$-0.479776\pi$
$67$	1.03944	0.126988	0.0634941	+	0.997982i	$0.479776\pi$
$68$	−4.96056	−0.601556
$69$	0	0
$70$	−4.16425	−0.497723
$71$	−9.01247	−1.06958	−0.534791	−	0.844984i	$-0.679610\pi$
$71$	−9.01247	−1.06958	−0.534791	+	0.844984i	$0.679610\pi$
$72$	0	0
$73$	12.0249	1.40741	0.703706	−	0.710491i	$-0.251527\pi$
$73$	12.0249	1.40741	0.703706	+	0.710491i	$0.251527\pi$
$74$	−2.79631	−0.325064
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−5.01247	−0.571223
$78$	0	0
$79$	0.467814	0.0526332	0.0263166	−	0.999654i	$-0.491622\pi$
$79$	0.467814	0.0526332	0.0263166	+	0.999654i	$0.491622\pi$
$80$	−4.16425	−0.465577
$81$	0	0
$82$	0.164248	0.0181382
$83$	−4.65699	−0.511171	−0.255586	−	0.966786i	$-0.582268\pi$
$83$	−4.65699	−0.511171	−0.255586	+	0.966786i	$0.582268\pi$
$84$	0	0
$85$	20.6570	2.24057
$86$	0.796310	0.0858683
$87$	0	0
$88$	−5.01247	−0.534331
$89$	7.01247	0.743320	0.371660	−	0.928369i	$-0.378789\pi$
$89$	7.01247	0.743320	0.371660	+	0.928369i	$0.378789\pi$
$90$	0	0
$91$	6.32850	0.663407
$92$	−4.32850	−0.451277
$93$	0	0
$94$	−4.68397	−0.483115
$95$	4.16425	0.427243
$96$	0	0
$97$	−5.64453	−0.573115	−0.286557	−	0.958063i	$-0.592511\pi$
$97$	−5.64453	−0.573115	−0.286557	+	0.958063i	$0.592511\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	12.3410	1.23410
$101$	19.3140	1.92181	0.960907	−	0.276873i	$-0.0892980\pi$
$101$	19.3140	1.92181	0.960907	+	0.276873i	$0.0892980\pi$
$102$	0	0
$103$	−7.06437	−0.696073	−0.348037	−	0.937481i	$-0.613152\pi$
$103$	−7.06437	−0.696073	−0.348037	+	0.937481i	$0.613152\pi$
$104$	6.32850	0.620560
$105$	0	0
$106$	4.05191	0.393556
$107$	19.0644	1.84302	0.921511	−	0.388352i	$-0.126955\pi$
$107$	19.0644	1.84302	0.921511	+	0.388352i	$0.126955\pi$
$108$	0	0
$109$	12.0519	1.15436	0.577182	−	0.816616i	$-0.304152\pi$
$109$	12.0519	1.15436	0.577182	+	0.816616i	$0.304152\pi$
$110$	20.8731	1.99018
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	9.61755	0.904743	0.452371	−	0.891830i	$-0.350578\pi$
$113$	9.61755	0.904743	0.452371	+	0.891830i	$0.350578\pi$
$114$	0	0
$115$	18.0249	1.68083
$116$	−1.20369	−0.111760
$117$	0	0
$118$	6.49274	0.597706
$119$	4.96056	0.454734
$120$	0	0
$121$	14.1248	1.28407
$122$	1.64453	0.148888
$123$	0	0
$124$	−5.69643	−0.511555
$125$	−30.5696	−2.73423
$126$	0	0
$127$	−7.64453	−0.678342	−0.339171	−	0.940725i	$-0.610147\pi$
$127$	−7.64453	−0.678342	−0.339171	+	0.940725i	$0.610147\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−26.3534	−2.31135
$131$	13.3679	1.16796	0.583981	−	0.811767i	$-0.301494\pi$
$131$	13.3679	1.16796	0.583981	+	0.811767i	$0.301494\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	−1.03944	−0.0897941
$135$	0	0
$136$	4.96056	0.425364
$137$	−12.2681	−1.04813	−0.524066	−	0.851678i	$-0.675585\pi$
$137$	−12.2681	−1.04813	−0.524066	+	0.851678i	$0.675585\pi$
$138$	0	0
$139$	6.73588	0.571330	0.285665	−	0.958330i	$-0.407786\pi$
$139$	6.73588	0.571330	0.285665	+	0.958330i	$0.407786\pi$
$140$	4.16425	0.351943
$141$	0	0
$142$	9.01247	0.756309
$143$	−31.7214	−2.65267
$144$	0	0
$145$	5.01247	0.416263
$146$	−12.0249	−0.995190
$147$	0	0
$148$	2.79631	0.229855
$149$	12.9606	1.06177	0.530885	−	0.847444i	$-0.321860\pi$
$149$	12.9606	1.06177	0.530885	+	0.847444i	$0.321860\pi$
$150$	0	0
$151$	19.3929	1.57817	0.789085	−	0.614285i	$-0.210555\pi$
$151$	19.3929	1.57817	0.789085	+	0.614285i	$0.210555\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	5.01247	0.403916
$155$	23.7214	1.90535
$156$	0	0
$157$	1.09135	0.0870992	0.0435496	−	0.999051i	$-0.486133\pi$
$157$	1.09135	0.0870992	0.0435496	+	0.999051i	$0.486133\pi$
$158$	−0.467814	−0.0372173
$159$	0	0
$160$	4.16425	0.329213
$161$	4.32850	0.341133
$162$	0	0
$163$	5.72341	0.448292	0.224146	−	0.974556i	$-0.428041\pi$
$163$	5.72341	0.448292	0.224146	+	0.974556i	$0.428041\pi$
$164$	−0.164248	−0.0128256
$165$	0	0
$166$	4.65699	0.361453
$167$	16.4323	1.27157	0.635785	−	0.771866i	$-0.280676\pi$
$167$	16.4323	1.27157	0.635785	+	0.771866i	$0.280676\pi$
$168$	0	0
$169$	27.0499	2.08076
$170$	−20.6570	−1.58432
$171$	0	0
$172$	−0.796310	−0.0607180
$173$	−8.02493	−0.610124	−0.305062	−	0.952332i	$-0.598677\pi$
$173$	−8.02493	−0.610124	−0.305062	+	0.952332i	$0.598677\pi$
$174$	0	0
$175$	−12.3410	−0.932889
$176$	5.01247	0.377829
$177$	0	0
$178$	−7.01247	−0.525606
$179$	−11.7214	−0.876096	−0.438048	−	0.898952i	$-0.644330\pi$
$179$	−11.7214	−0.876096	−0.438048	+	0.898952i	$0.644330\pi$
$180$	0	0
$181$	10.1038	0.751011	0.375505	−	0.926820i	$-0.377469\pi$
$181$	10.1038	0.751011	0.375505	+	0.926820i	$0.377469\pi$
$182$	−6.32850	−0.469099
$183$	0	0
$184$	4.32850	0.319101
$185$	−11.6445	−0.856123
$186$	0	0
$187$	−24.8646	−1.81828
$188$	4.68397	0.341614
$189$	0	0
$190$	−4.16425	−0.302106
$191$	−7.72136	−0.558698	−0.279349	−	0.960190i	$-0.590119\pi$
$191$	−7.72136	−0.558698	−0.279349	+	0.960190i	$0.590119\pi$
$192$	0	0
$193$	−13.2891	−0.956567	−0.478284	−	0.878205i	$-0.658741\pi$
$193$	−13.2891	−0.956567	−0.478284	+	0.878205i	$0.658741\pi$
$194$	5.64453	0.405253
$195$	0	0
$196$	1.00000	0.0714286
$197$	−11.9211	−0.849344	−0.424672	−	0.905347i	$-0.639611\pi$
$197$	−11.9211	−0.849344	−0.424672	+	0.905347i	$0.639611\pi$
$198$	0	0
$199$	20.0854	1.42381	0.711907	−	0.702274i	$-0.247832\pi$
$199$	20.0854	1.42381	0.711907	+	0.702274i	$0.247832\pi$
$200$	−12.3410	−0.872638
$201$	0	0
$202$	−19.3140	−1.35893
$203$	1.20369	0.0844825
$204$	0	0
$205$	0.683969	0.0477705
$206$	7.06437	0.492198
$207$	0	0
$208$	−6.32850	−0.438802
$209$	−5.01247	−0.346719
$210$	0	0
$211$	−23.3929	−1.61043	−0.805216	−	0.592982i	$-0.797951\pi$
$211$	−23.3929	−1.61043	−0.805216	+	0.592982i	$0.797951\pi$
$212$	−4.05191	−0.278286
$213$	0	0
$214$	−19.0644	−1.30321
$215$	3.31603	0.226151
$216$	0	0
$217$	5.69643	0.386699
$218$	−12.0519	−0.816258
$219$	0	0
$220$	−20.8731	−1.40727
$221$	31.3929	2.11171
$222$	0	0
$223$	3.89619	0.260908	0.130454	−	0.991454i	$-0.458357\pi$
$223$	3.89619	0.260908	0.130454	+	0.991454i	$0.458357\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−9.61755	−0.639750
$227$	9.92112	0.658488	0.329244	−	0.944245i	$-0.393206\pi$
$227$	9.92112	0.658488	0.329244	+	0.944245i	$0.393206\pi$
$228$	0	0
$229$	20.1642	1.33249	0.666246	−	0.745732i	$-0.267900\pi$
$229$	20.1642	1.33249	0.666246	+	0.745732i	$0.267900\pi$
$230$	−18.0249	−1.18853
$231$	0	0
$232$	1.20369	0.0790261
$233$	−13.9730	−0.915403	−0.457702	−	0.889106i	$-0.651327\pi$
$233$	−13.9730	−0.915403	−0.457702	+	0.889106i	$0.651327\pi$
$234$	0	0
$235$	−19.5052	−1.27238
$236$	−6.49274	−0.422642
$237$	0	0
$238$	−4.96056	−0.321545
$239$	−18.0249	−1.16594	−0.582968	−	0.812495i	$-0.698109\pi$
$239$	−18.0249	−1.16594	−0.582968	+	0.812495i	$0.698109\pi$
$240$	0	0
$241$	−22.9001	−1.47513	−0.737563	−	0.675278i	$-0.764024\pi$
$241$	−22.9001	−1.47513	−0.737563	+	0.675278i	$0.764024\pi$
$242$	−14.1248	−0.907977
$243$	0	0
$244$	−1.64453	−0.105280
$245$	−4.16425	−0.266044
$246$	0	0
$247$	6.32850	0.402673
$248$	5.69643	0.361724
$249$	0	0
$250$	30.5696	1.93339
$251$	22.6819	1.43167	0.715835	−	0.698269i	$-0.246046\pi$
$251$	22.6819	1.43167	0.715835	+	0.698269i	$0.246046\pi$
$252$	0	0
$253$	−21.6964	−1.36404
$254$	7.64453	0.479660
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.4927	1.02879	0.514395	−	0.857554i	$-0.328017\pi$
$257$	16.4927	1.02879	0.514395	+	0.857554i	$0.328017\pi$
$258$	0	0
$259$	−2.79631	−0.173754
$260$	26.3534	1.63437
$261$	0	0
$262$	−13.3679	−0.825874
$263$	−13.2102	−0.814574	−0.407287	−	0.913300i	$-0.633525\pi$
$263$	−13.2102	−0.814574	−0.407287	+	0.913300i	$0.633525\pi$
$264$	0	0
$265$	16.8731	1.03651
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	1.03944	0.0634941
$269$	−11.9211	−0.726843	−0.363422	−	0.931625i	$-0.618392\pi$
$269$	−11.9211	−0.726843	−0.363422	+	0.931625i	$0.618392\pi$
$270$	0	0
$271$	5.22862	0.317616	0.158808	−	0.987309i	$-0.449235\pi$
$271$	5.22862	0.317616	0.158808	+	0.987309i	$0.449235\pi$
$272$	−4.96056	−0.300778
$273$	0	0
$274$	12.2681	0.741141
$275$	61.8586	3.73022
$276$	0	0
$277$	−0.182700	−0.0109774	−0.00548868	−	0.999985i	$-0.501747\pi$
$277$	−0.182700	−0.0109774	−0.00548868	+	0.999985i	$0.501747\pi$
$278$	−6.73588	−0.403991
$279$	0	0
$280$	−4.16425	−0.248861
$281$	27.6964	1.65223	0.826115	−	0.563501i	$-0.190546\pi$
$281$	27.6964	1.65223	0.826115	+	0.563501i	$0.190546\pi$
$282$	0	0
$283$	−23.3929	−1.39056	−0.695281	−	0.718738i	$-0.744720\pi$
$283$	−23.3929	−1.39056	−0.695281	+	0.718738i	$0.744720\pi$
$284$	−9.01247	−0.534791
$285$	0	0
$286$	31.7214	1.87572
$287$	0.164248	0.00969525
$288$	0	0
$289$	7.60713	0.447478
$290$	−5.01247	−0.294342
$291$	0	0
$292$	12.0249	0.703706
$293$	11.5926	0.677248	0.338624	−	0.940922i	$-0.390039\pi$
$293$	11.5926	0.677248	0.338624	+	0.940922i	$0.390039\pi$
$294$	0	0
$295$	27.0374	1.57418
$296$	−2.79631	−0.162532
$297$	0	0
$298$	−12.9606	−0.750785
$299$	27.3929	1.58417
$300$	0	0
$301$	0.796310	0.0458985
$302$	−19.3929	−1.11593
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	6.84822	0.392128
$306$	0	0
$307$	28.1352	1.60576	0.802881	−	0.596139i	$-0.203299\pi$
$307$	28.1352	1.60576	0.802881	+	0.596139i	$0.203299\pi$
$308$	−5.01247	−0.285612
$309$	0	0
$310$	−23.7214	−1.34728
$311$	5.34096	0.302858	0.151429	−	0.988468i	$-0.451612\pi$
$311$	5.34096	0.302858	0.151429	+	0.988468i	$0.451612\pi$
$312$	0	0
$313$	1.11833	0.0632116	0.0316058	−	0.999500i	$-0.489938\pi$
$313$	1.11833	0.0632116	0.0316058	+	0.999500i	$0.489938\pi$
$314$	−1.09135	−0.0615884
$315$	0	0
$316$	0.467814	0.0263166
$317$	27.4448	1.54145	0.770726	−	0.637167i	$-0.219894\pi$
$317$	27.4448	1.54145	0.770726	+	0.637167i	$0.219894\pi$
$318$	0	0
$319$	−6.03346	−0.337809
$320$	−4.16425	−0.232789
$321$	0	0
$322$	−4.32850	−0.241218
$323$	4.96056	0.276013
$324$	0	0
$325$	−78.0997	−4.33219
$326$	−5.72341	−0.316990
$327$	0	0
$328$	0.164248	0.00906908
$329$	−4.68397	−0.258236
$330$	0	0
$331$	12.2247	0.671929	0.335965	−	0.941875i	$-0.390938\pi$
$331$	12.2247	0.671929	0.335965	+	0.941875i	$0.390938\pi$
$332$	−4.65699	−0.255586
$333$	0	0
$334$	−16.4323	−0.899136
$335$	−4.32850	−0.236491
$336$	0	0
$337$	−7.14326	−0.389118	−0.194559	−	0.980891i	$-0.562328\pi$
$337$	−7.14326	−0.389118	−0.194559	+	0.980891i	$0.562328\pi$
$338$	−27.0499	−1.47132
$339$	0	0
$340$	20.6570	1.12028
$341$	−28.5532	−1.54624
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0.796310	0.0429341
$345$	0	0
$346$	8.02493	0.431423
$347$	12.6570	0.679463	0.339731	−	0.940522i	$-0.389664\pi$
$347$	12.6570	0.679463	0.339731	+	0.940522i	$0.389664\pi$
$348$	0	0
$349$	−11.3140	−0.605624	−0.302812	−	0.953050i	$-0.597925\pi$
$349$	−11.3140	−0.605624	−0.302812	+	0.953050i	$0.597925\pi$
$350$	12.3410	0.659652
$351$	0	0
$352$	−5.01247	−0.267165
$353$	6.43231	0.342357	0.171179	−	0.985240i	$-0.445242\pi$
$353$	6.43231	0.342357	0.171179	+	0.985240i	$0.445242\pi$
$354$	0	0
$355$	37.5301	1.99189
$356$	7.01247	0.371660
$357$	0	0
$358$	11.7214	0.619493
$359$	−25.8002	−1.36169	−0.680843	−	0.732430i	$-0.738386\pi$
$359$	−25.8002	−1.36169	−0.680843	+	0.732430i	$0.738386\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−10.1038	−0.531045
$363$	0	0
$364$	6.32850	0.331703
$365$	−50.0748	−2.62103
$366$	0	0
$367$	3.97302	0.207390	0.103695	−	0.994609i	$-0.466933\pi$
$367$	3.97302	0.207390	0.103695	+	0.994609i	$0.466933\pi$
$368$	−4.32850	−0.225638
$369$	0	0
$370$	11.6445	0.605370
$371$	4.05191	0.210365
$372$	0	0
$373$	21.0459	1.08972	0.544858	−	0.838528i	$-0.316583\pi$
$373$	21.0459	1.08972	0.544858	+	0.838528i	$0.316583\pi$
$374$	24.8646	1.28572
$375$	0	0
$376$	−4.68397	−0.241557
$377$	7.61755	0.392324
$378$	0	0
$379$	−3.28905	−0.168947	−0.0844737	−	0.996426i	$-0.526921\pi$
$379$	−3.28905	−0.168947	−0.0844737	+	0.996426i	$0.526921\pi$
$380$	4.16425	0.213621
$381$	0	0
$382$	7.72136	0.395059
$383$	2.24961	0.114950	0.0574749	−	0.998347i	$-0.481695\pi$
$383$	2.24961	0.114950	0.0574749	+	0.998347i	$0.481695\pi$
$384$	0	0
$385$	20.8731	1.06379
$386$	13.2891	0.676395
$387$	0	0
$388$	−5.64453	−0.286557
$389$	4.07888	0.206808	0.103404	−	0.994639i	$-0.467027\pi$
$389$	4.07888	0.206808	0.103404	+	0.994639i	$0.467027\pi$
$390$	0	0
$391$	21.4718	1.08587
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	11.9211	0.600577
$395$	−1.94809	−0.0980192
$396$	0	0
$397$	−32.2226	−1.61721	−0.808604	−	0.588354i	$-0.799776\pi$
$397$	−32.2226	−1.61721	−0.808604	+	0.588354i	$0.799776\pi$
$398$	−20.0854	−1.00679
$399$	0	0
$400$	12.3410	0.617048
$401$	−19.5387	−0.975714	−0.487857	−	0.872923i	$-0.662221\pi$
$401$	−19.5387	−0.975714	−0.487857	+	0.872923i	$0.662221\pi$
$402$	0	0
$403$	36.0499	1.79577
$404$	19.3140	0.960907
$405$	0	0
$406$	−1.20369	−0.0597381
$407$	14.0164	0.694767
$408$	0	0
$409$	5.09135	0.251751	0.125876	−	0.992046i	$-0.459826\pi$
$409$	5.09135	0.251751	0.125876	+	0.992046i	$0.459826\pi$
$410$	−0.683969	−0.0337788
$411$	0	0
$412$	−7.06437	−0.348037
$413$	6.49274	0.319487
$414$	0	0
$415$	19.3929	0.951958
$416$	6.32850	0.310280
$417$	0	0
$418$	5.01247	0.245168
$419$	10.1458	0.495655	0.247827	−	0.968804i	$-0.420283\pi$
$419$	10.1458	0.495655	0.247827	+	0.968804i	$0.420283\pi$
$420$	0	0
$421$	−34.6570	−1.68908	−0.844539	−	0.535494i	$-0.820126\pi$
$421$	−34.6570	−1.68908	−0.844539	+	0.535494i	$0.820126\pi$
$422$	23.3929	1.13875
$423$	0	0
$424$	4.05191	0.196778
$425$	−61.2180	−2.96951
$426$	0	0
$427$	1.64453	0.0795842
$428$	19.0644	0.921511
$429$	0	0
$430$	−3.31603	−0.159913
$431$	−34.5880	−1.66605	−0.833023	−	0.553238i	$-0.813392\pi$
$431$	−34.5880	−1.66605	−0.833023	+	0.553238i	$0.813392\pi$
$432$	0	0
$433$	−22.9001	−1.10051	−0.550255	−	0.834997i	$-0.685469\pi$
$433$	−22.9001	−1.10051	−0.550255	+	0.834997i	$0.685469\pi$
$434$	−5.69643	−0.273438
$435$	0	0
$436$	12.0519	0.577182
$437$	4.32850	0.207060
$438$	0	0
$439$	6.35343	0.303232	0.151616	−	0.988439i	$-0.451552\pi$
$439$	6.35343	0.303232	0.151616	+	0.988439i	$0.451552\pi$
$440$	20.8731	0.995088
$441$	0	0
$442$	−31.3929	−1.49321
$443$	−28.8462	−1.37052	−0.685261	−	0.728297i	$-0.740312\pi$
$443$	−28.8462	−1.37052	−0.685261	+	0.728297i	$0.740312\pi$
$444$	0	0
$445$	−29.2016	−1.38429
$446$	−3.89619	−0.184490
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	27.6425	1.30453	0.652265	−	0.757991i	$-0.273819\pi$
$449$	27.6425	1.30453	0.652265	+	0.757991i	$0.273819\pi$
$450$	0	0
$451$	−0.823287	−0.0387671
$452$	9.61755	0.452371
$453$	0	0
$454$	−9.92112	−0.465621
$455$	−26.3534	−1.23547
$456$	0	0
$457$	−33.9191	−1.58667	−0.793334	−	0.608787i	$-0.791656\pi$
$457$	−33.9191	−1.58667	−0.793334	+	0.608787i	$0.791656\pi$
$458$	−20.1642	−0.942213
$459$	0	0
$460$	18.0249	0.840417
$461$	−3.82723	−0.178252	−0.0891259	−	0.996020i	$-0.528407\pi$
$461$	−3.82723	−0.178252	−0.0891259	+	0.996020i	$0.528407\pi$
$462$	0	0
$463$	3.44682	0.160187	0.0800937	−	0.996787i	$-0.474478\pi$
$463$	3.44682	0.160187	0.0800937	+	0.996787i	$0.474478\pi$
$464$	−1.20369	−0.0558799
$465$	0	0
$466$	13.9730	0.647288
$467$	−11.5137	−0.532792	−0.266396	−	0.963864i	$-0.585833\pi$
$467$	−11.5137	−0.532792	−0.266396	+	0.963864i	$0.585833\pi$
$468$	0	0
$469$	−1.03944	−0.0479970
$470$	19.5052	0.899708
$471$	0	0
$472$	6.49274	0.298853
$473$	−3.99147	−0.183528
$474$	0	0
$475$	−12.3410	−0.566242
$476$	4.96056	0.227367
$477$	0	0
$478$	18.0249	0.824441
$479$	−13.0210	−0.594944	−0.297472	−	0.954731i	$-0.596144\pi$
$479$	−13.0210	−0.594944	−0.297472	+	0.954731i	$0.596144\pi$
$480$	0	0
$481$	−17.6964	−0.806888
$482$	22.9001	1.04307
$483$	0	0
$484$	14.1248	0.642037
$485$	23.5052	1.06732
$486$	0	0
$487$	37.6240	1.70491	0.852454	−	0.522803i	$-0.175114\pi$
$487$	37.6240	1.70491	0.852454	+	0.522803i	$0.175114\pi$
$488$	1.64453	0.0744442
$489$	0	0
$490$	4.16425	0.188122
$491$	−23.8422	−1.07598	−0.537992	−	0.842950i	$-0.680817\pi$
$491$	−23.8422	−1.07598	−0.537992	+	0.842950i	$0.680817\pi$
$492$	0	0
$493$	5.97098	0.268919
$494$	−6.32850	−0.284732
$495$	0	0
$496$	−5.69643	−0.255777
$497$	9.01247	0.404264
$498$	0	0
$499$	5.40344	0.241891	0.120946	−	0.992659i	$-0.461407\pi$
$499$	5.40344	0.241891	0.120946	+	0.992659i	$0.461407\pi$
$500$	−30.5696	−1.36711
$501$	0	0
$502$	−22.6819	−1.01234
$503$	41.5571	1.85294	0.926470	−	0.376368i	$-0.122827\pi$
$503$	41.5571	1.85294	0.926470	+	0.376368i	$0.122827\pi$
$504$	0	0
$505$	−80.4282	−3.57901
$506$	21.6964	0.964524
$507$	0	0
$508$	−7.64453	−0.339171
$509$	−16.6819	−0.739413	−0.369707	−	0.929149i	$-0.620542\pi$
$509$	−16.6819	−0.739413	−0.369707	+	0.929149i	$0.620542\pi$
$510$	0	0
$511$	−12.0249	−0.531952
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−16.4927	−0.727464
$515$	29.4178	1.29630
$516$	0	0
$517$	23.4782	1.03257
$518$	2.79631	0.122863
$519$	0	0
$520$	−26.3534	−1.15567
$521$	−13.1852	−0.577656	−0.288828	−	0.957381i	$-0.593266\pi$
$521$	−13.1852	−0.577656	−0.288828	+	0.957381i	$0.593266\pi$
$522$	0	0
$523$	−22.0789	−0.965442	−0.482721	−	0.875774i	$-0.660352\pi$
$523$	−22.0789	−0.965442	−0.482721	+	0.875774i	$0.660352\pi$
$524$	13.3679	0.583981
$525$	0	0
$526$	13.2102	0.575991
$527$	28.2575	1.23092
$528$	0	0
$529$	−4.26412	−0.185397
$530$	−16.8731	−0.732923
$531$	0	0
$532$	1.00000	0.0433555
$533$	1.03944	0.0450232
$534$	0	0
$535$	−79.3888	−3.43228
$536$	−1.03944	−0.0448971
$537$	0	0
$538$	11.9211	0.513956
$539$	5.01247	0.215902
$540$	0	0
$541$	20.6321	0.887042	0.443521	−	0.896264i	$-0.353729\pi$
$541$	20.6321	0.887042	0.443521	+	0.896264i	$0.353729\pi$
$542$	−5.22862	−0.224588
$543$	0	0
$544$	4.96056	0.212682
$545$	−50.1871	−2.14978
$546$	0	0
$547$	−41.1931	−1.76129	−0.880645	−	0.473776i	$-0.842891\pi$
$547$	−41.1931	−1.76129	−0.880645	+	0.473776i	$0.842891\pi$
$548$	−12.2681	−0.524066
$549$	0	0
$550$	−61.8586	−2.63766
$551$	1.20369	0.0512789
$552$	0	0
$553$	−0.467814	−0.0198935
$554$	0.182700	0.00776216
$555$	0	0
$556$	6.73588	0.285665
$557$	27.0353	1.14552	0.572762	−	0.819722i	$-0.305872\pi$
$557$	27.0353	1.14552	0.572762	+	0.819722i	$0.305872\pi$
$558$	0	0
$559$	5.03944	0.213146
$560$	4.16425	0.175972
$561$	0	0
$562$	−27.6964	−1.16830
$563$	40.5761	1.71008	0.855039	−	0.518565i	$-0.173533\pi$
$563$	40.5761	1.71008	0.855039	+	0.518565i	$0.173533\pi$
$564$	0	0
$565$	−40.0499	−1.68491
$566$	23.3929	0.983276
$567$	0	0
$568$	9.01247	0.378155
$569$	−20.6819	−0.867031	−0.433516	−	0.901146i	$-0.642727\pi$
$569$	−20.6819	−0.867031	−0.433516	+	0.901146i	$0.642727\pi$
$570$	0	0
$571$	−5.72341	−0.239517	−0.119759	−	0.992803i	$-0.538212\pi$
$571$	−5.72341	−0.239517	−0.119759	+	0.992803i	$0.538212\pi$
$572$	−31.7214	−1.32634
$573$	0	0
$574$	−0.164248	−0.00685558
$575$	−53.4178	−2.22768
$576$	0	0
$577$	23.3638	0.972650	0.486325	−	0.873778i	$-0.338337\pi$
$577$	23.3638	0.972650	0.486325	+	0.873778i	$0.338337\pi$
$578$	−7.60713	−0.316415
$579$	0	0
$580$	5.01247	0.208131
$581$	4.65699	0.193205
$582$	0	0
$583$	−20.3100	−0.841156
$584$	−12.0249	−0.497595
$585$	0	0
$586$	−11.5926	−0.478887
$587$	7.01451	0.289520	0.144760	−	0.989467i	$-0.453759\pi$
$587$	7.01451	0.289520	0.144760	+	0.989467i	$0.453759\pi$
$588$	0	0
$589$	5.69643	0.234717
$590$	−27.0374	−1.11311
$591$	0	0
$592$	2.79631	0.114928
$593$	−8.63206	−0.354476	−0.177238	−	0.984168i	$-0.556716\pi$
$593$	−8.63206	−0.354476	−0.177238	+	0.984168i	$0.556716\pi$
$594$	0	0
$595$	−20.6570	−0.846854
$596$	12.9606	0.530885
$597$	0	0
$598$	−27.3929	−1.12018
$599$	−17.6695	−0.721954	−0.360977	−	0.932575i	$-0.617557\pi$
$599$	−17.6695	−0.721954	−0.360977	+	0.932575i	$0.617557\pi$
$600$	0	0
$601$	−11.9211	−0.486272	−0.243136	−	0.969992i	$-0.578176\pi$
$601$	−11.9211	−0.486272	−0.243136	+	0.969992i	$0.578176\pi$
$602$	−0.796310	−0.0324552
$603$	0	0
$604$	19.3929	0.789085
$605$	−58.8192	−2.39134
$606$	0	0
$607$	−39.9460	−1.62136	−0.810680	−	0.585490i	$-0.800902\pi$
$607$	−39.9460	−1.62136	−0.810680	+	0.585490i	$0.800902\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−6.84822	−0.277276
$611$	−29.6425	−1.19921
$612$	0	0
$613$	−25.7753	−1.04106	−0.520528	−	0.853845i	$-0.674265\pi$
$613$	−25.7753	−1.04106	−0.520528	+	0.853845i	$0.674265\pi$
$614$	−28.1352	−1.13545
$615$	0	0
$616$	5.01247	0.201958
$617$	5.98155	0.240808	0.120404	−	0.992725i	$-0.461581\pi$
$617$	5.98155	0.240808	0.120404	+	0.992725i	$0.461581\pi$
$618$	0	0
$619$	39.1142	1.57213	0.786067	−	0.618141i	$-0.212114\pi$
$619$	39.1142	1.57213	0.786067	+	0.618141i	$0.212114\pi$
$620$	23.7214	0.952673
$621$	0	0
$622$	−5.34096	−0.214153
$623$	−7.01247	−0.280948
$624$	0	0
$625$	65.5945	2.62378
$626$	−1.11833	−0.0446973
$627$	0	0
$628$	1.09135	0.0435496
$629$	−13.8713	−0.553083
$630$	0	0
$631$	−15.2351	−0.606500	−0.303250	−	0.952911i	$-0.598072\pi$
$631$	−15.2351	−0.606500	−0.303250	+	0.952911i	$0.598072\pi$
$632$	−0.467814	−0.0186086
$633$	0	0
$634$	−27.4448	−1.08997
$635$	31.8337	1.26328
$636$	0	0
$637$	−6.32850	−0.250744
$638$	6.03346	0.238867
$639$	0	0
$640$	4.16425	0.164606
$641$	−10.9186	−0.431258	−0.215629	−	0.976475i	$-0.569180\pi$
$641$	−10.9186	−0.431258	−0.215629	+	0.976475i	$0.569180\pi$
$642$	0	0
$643$	0.607132	0.0239429	0.0119715	−	0.999928i	$-0.496189\pi$
$643$	0.607132	0.0239429	0.0119715	+	0.999928i	$0.496189\pi$
$644$	4.32850	0.170567
$645$	0	0
$646$	−4.96056	−0.195171
$647$	16.4139	0.645295	0.322648	−	0.946519i	$-0.395427\pi$
$647$	16.4139	0.645295	0.322648	+	0.946519i	$0.395427\pi$
$648$	0	0
$649$	−32.5447	−1.27749
$650$	78.0997	3.06332
$651$	0	0
$652$	5.72341	0.224146
$653$	35.6923	1.39675	0.698375	−	0.715732i	$-0.253907\pi$
$653$	35.6923	1.39675	0.698375	+	0.715732i	$0.253907\pi$
$654$	0	0
$655$	−55.6674	−2.17511
$656$	−0.164248	−0.00641280
$657$	0	0
$658$	4.68397	0.182600
$659$	−3.55064	−0.138313	−0.0691566	−	0.997606i	$-0.522031\pi$
$659$	−3.55064	−0.138313	−0.0691566	+	0.997606i	$0.522031\pi$
$660$	0	0
$661$	45.8921	1.78500	0.892498	−	0.451052i	$-0.148951\pi$
$661$	45.8921	1.78500	0.892498	+	0.451052i	$0.148951\pi$
$662$	−12.2247	−0.475126
$663$	0	0
$664$	4.65699	0.180726
$665$	−4.16425	−0.161483
$666$	0	0
$667$	5.21017	0.201739
$668$	16.4323	0.635785
$669$	0	0
$670$	4.32850	0.167224
$671$	−8.24313	−0.318223
$672$	0	0
$673$	11.0353	0.425381	0.212691	−	0.977120i	$-0.431777\pi$
$673$	11.0353	0.425381	0.212691	+	0.977120i	$0.431777\pi$
$674$	7.14326	0.275148
$675$	0	0
$676$	27.0499	1.04038
$677$	0.128745	0.00494807	0.00247403	−	0.999997i	$-0.499212\pi$
$677$	0.128745	0.00494807	0.00247403	+	0.999997i	$0.499212\pi$
$678$	0	0
$679$	5.64453	0.216617
$680$	−20.6570	−0.792159
$681$	0	0
$682$	28.5532	1.09336
$683$	−20.8817	−0.799015	−0.399508	−	0.916730i	$-0.630819\pi$
$683$	−20.8817	−0.799015	−0.399508	+	0.916730i	$0.630819\pi$
$684$	0	0
$685$	51.0873	1.95194
$686$	1.00000	0.0381802
$687$	0	0
$688$	−0.796310	−0.0303590
$689$	25.6425	0.976901
$690$	0	0
$691$	23.8791	0.908405	0.454203	−	0.890898i	$-0.349924\pi$
$691$	23.8791	0.908405	0.454203	+	0.890898i	$0.349924\pi$
$692$	−8.02493	−0.305062
$693$	0	0
$694$	−12.6570	−0.480453
$695$	−28.0499	−1.06399
$696$	0	0
$697$	0.814761	0.0308613
$698$	11.3140	0.428241
$699$	0	0
$700$	−12.3410	−0.466444
$701$	0.839691	0.0317147	0.0158574	−	0.999874i	$-0.494952\pi$
$701$	0.839691	0.0317147	0.0158574	+	0.999874i	$0.494952\pi$
$702$	0	0
$703$	−2.79631	−0.105465
$704$	5.01247	0.188914
$705$	0	0
$706$	−6.43231	−0.242083
$707$	−19.3140	−0.726377
$708$	0	0
$709$	32.4572	1.21896	0.609479	−	0.792802i	$-0.291379\pi$
$709$	32.4572	1.21896	0.609479	+	0.792802i	$0.291379\pi$
$710$	−37.5301	−1.40848
$711$	0	0
$712$	−7.01247	−0.262803
$713$	24.6570	0.923412
$714$	0	0
$715$	132.096	4.94010
$716$	−11.7214	−0.438048
$717$	0	0
$718$	25.8002	0.962857
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	7.06437	0.263091
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	10.1038	0.375505
$725$	−14.8547	−0.551690
$726$	0	0
$727$	13.2870	0.492788	0.246394	−	0.969170i	$-0.420754\pi$
$727$	13.2870	0.492788	0.246394	+	0.969170i	$0.420754\pi$
$728$	−6.32850	−0.234550
$729$	0	0
$730$	50.0748	1.85335
$731$	3.95014	0.146101
$732$	0	0
$733$	19.3993	0.716531	0.358266	−	0.933620i	$-0.383368\pi$
$733$	19.3993	0.716531	0.358266	+	0.933620i	$0.383368\pi$
$734$	−3.97302	−0.146647
$735$	0	0
$736$	4.32850	0.159550
$737$	5.21017	0.191919
$738$	0	0
$739$	27.2536	1.00254	0.501269	−	0.865291i	$-0.332867\pi$
$739$	27.2536	1.00254	0.501269	+	0.865291i	$0.332867\pi$
$740$	−11.6445	−0.428061
$741$	0	0
$742$	−4.05191	−0.148750
$743$	−39.5322	−1.45030	−0.725148	−	0.688593i	$-0.758229\pi$
$743$	−39.5322	−1.45030	−0.725148	+	0.688593i	$0.758229\pi$
$744$	0	0
$745$	−53.9710	−1.97734
$746$	−21.0459	−0.770546
$747$	0	0
$748$	−24.8646	−0.909141
$749$	−19.0644	−0.696597
$750$	0	0
$751$	4.13932	0.151046	0.0755229	−	0.997144i	$-0.475937\pi$
$751$	4.13932	0.151046	0.0755229	+	0.997144i	$0.475937\pi$
$752$	4.68397	0.170807
$753$	0	0
$754$	−7.61755	−0.277415
$755$	−80.7567	−2.93904
$756$	0	0
$757$	−14.2745	−0.518817	−0.259409	−	0.965768i	$-0.583528\pi$
$757$	−14.2745	−0.518817	−0.259409	+	0.965768i	$0.583528\pi$
$758$	3.28905	0.119464
$759$	0	0
$760$	−4.16425	−0.151053
$761$	20.5781	0.745956	0.372978	−	0.927840i	$-0.378337\pi$
$761$	20.5781	0.745956	0.372978	+	0.927840i	$0.378337\pi$
$762$	0	0
$763$	−12.0519	−0.436308
$764$	−7.72136	−0.279349
$765$	0	0
$766$	−2.24961	−0.0812818
$767$	41.0893	1.48365
$768$	0	0
$769$	34.7109	1.25171	0.625854	−	0.779940i	$-0.284750\pi$
$769$	34.7109	1.25171	0.625854	+	0.779940i	$0.284750\pi$
$770$	−20.8731	−0.752216
$771$	0	0
$772$	−13.2891	−0.478284
$773$	−4.18270	−0.150441	−0.0752206	−	0.997167i	$-0.523966\pi$
$773$	−4.18270	−0.150441	−0.0752206	+	0.997167i	$0.523966\pi$
$774$	0	0
$775$	−70.2995	−2.52523
$776$	5.64453	0.202627
$777$	0	0
$778$	−4.07888	−0.146235
$779$	0.164248	0.00588479
$780$	0	0
$781$	−45.1747	−1.61648
$782$	−21.4718	−0.767828
$783$	0	0
$784$	1.00000	0.0357143
$785$	−4.54465	−0.162206
$786$	0	0
$787$	6.77138	0.241374	0.120687	−	0.992691i	$-0.461490\pi$
$787$	6.77138	0.241374	0.120687	+	0.992691i	$0.461490\pi$
$788$	−11.9211	−0.424672
$789$	0	0
$790$	1.94809	0.0693100
$791$	−9.61755	−0.341961
$792$	0	0
$793$	10.4074	0.369577
$794$	32.2226	1.14354
$795$	0	0
$796$	20.0854	0.711907
$797$	34.8107	1.23306	0.616529	−	0.787333i	$-0.288538\pi$
$797$	34.8107	1.23306	0.616529	+	0.787333i	$0.288538\pi$
$798$	0	0
$799$	−23.2351	−0.821999
$800$	−12.3410	−0.436319
$801$	0	0
$802$	19.5387	0.689934
$803$	60.2745	2.12704
$804$	0	0
$805$	−18.0249	−0.635295
$806$	−36.0499	−1.26980
$807$	0	0
$808$	−19.3140	−0.679464
$809$	23.3324	0.820325	0.410162	−	0.912013i	$-0.365472\pi$
$809$	23.3324	0.820325	0.410162	+	0.912013i	$0.365472\pi$
$810$	0	0
$811$	−7.91907	−0.278076	−0.139038	−	0.990287i	$-0.544401\pi$
$811$	−7.91907	−0.278076	−0.139038	+	0.990287i	$0.544401\pi$
$812$	1.20369	0.0422412
$813$	0	0
$814$	−14.0164	−0.491275
$815$	−23.8337	−0.834858
$816$	0	0
$817$	0.796310	0.0278593
$818$	−5.09135	−0.178015
$819$	0	0
$820$	0.683969	0.0238852
$821$	29.1852	1.01857	0.509286	−	0.860597i	$-0.329910\pi$
$821$	29.1852	1.01857	0.509286	+	0.860597i	$0.329910\pi$
$822$	0	0
$823$	15.0104	0.523230	0.261615	−	0.965172i	$-0.415745\pi$
$823$	15.0104	0.523230	0.261615	+	0.965172i	$0.415745\pi$
$824$	7.06437	0.246099
$825$	0	0
$826$	−6.49274	−0.225911
$827$	−12.3784	−0.430438	−0.215219	−	0.976566i	$-0.569046\pi$
$827$	−12.3784	−0.430438	−0.215219	+	0.976566i	$0.569046\pi$
$828$	0	0
$829$	−32.5611	−1.13089	−0.565446	−	0.824785i	$-0.691296\pi$
$829$	−32.5611	−1.13089	−0.565446	+	0.824785i	$0.691296\pi$
$830$	−19.3929	−0.673136
$831$	0	0
$832$	−6.32850	−0.219401
$833$	−4.96056	−0.171873
$834$	0	0
$835$	−68.4282	−2.36806
$836$	−5.01247	−0.173360
$837$	0	0
$838$	−10.1458	−0.350481
$839$	−6.35343	−0.219345	−0.109672	−	0.993968i	$-0.534980\pi$
$839$	−6.35343	−0.219345	−0.109672	+	0.993968i	$0.534980\pi$
$840$	0	0
$841$	−27.5511	−0.950039
$842$	34.6570	1.19436
$843$	0	0
$844$	−23.3929	−0.805216
$845$	−112.642	−3.87501
$846$	0	0
$847$	−14.1248	−0.485334
$848$	−4.05191	−0.139143
$849$	0	0
$850$	61.2180	2.09976
$851$	−12.1038	−0.414913
$852$	0	0
$853$	−38.2930	−1.31113	−0.655564	−	0.755140i	$-0.727569\pi$
$853$	−38.2930	−1.31113	−0.655564	+	0.755140i	$0.727569\pi$
$854$	−1.64453	−0.0562745
$855$	0	0
$856$	−19.0644	−0.651607
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	−13.8002	−0.470858	−0.235429	−	0.971891i	$-0.575650\pi$
$859$	−13.8002	−0.470858	−0.235429	+	0.971891i	$0.575650\pi$
$860$	3.31603	0.113076
$861$	0	0
$862$	34.5880	1.17807
$863$	15.0459	0.512169	0.256085	−	0.966654i	$-0.417567\pi$
$863$	15.0459	0.512169	0.256085	+	0.966654i	$0.417567\pi$
$864$	0	0
$865$	33.4178	1.13624
$866$	22.9001	0.778178
$867$	0	0
$868$	5.69643	0.193350
$869$	2.34490	0.0795453
$870$	0	0
$871$	−6.57811	−0.222891
$872$	−12.0519	−0.408129
$873$	0	0
$874$	−4.32850	−0.146414
$875$	30.5696	1.03344
$876$	0	0
$877$	0.659039	0.0222542	0.0111271	−	0.999938i	$-0.496458\pi$
$877$	0.659039	0.0222542	0.0111271	+	0.999938i	$0.496458\pi$
$878$	−6.35343	−0.214418
$879$	0	0
$880$	−20.8731	−0.703634
$881$	−39.8501	−1.34258	−0.671292	−	0.741193i	$-0.734260\pi$
$881$	−39.8501	−1.34258	−0.671292	+	0.741193i	$0.734260\pi$
$882$	0	0
$883$	13.5158	0.454842	0.227421	−	0.973797i	$-0.426971\pi$
$883$	13.5158	0.454842	0.227421	+	0.973797i	$0.426971\pi$
$884$	31.3929	1.05586
$885$	0	0
$886$	28.8462	0.969106
$887$	−52.4822	−1.76218	−0.881089	−	0.472950i	$-0.843189\pi$
$887$	−52.4822	−1.76218	−0.881089	+	0.472950i	$0.843189\pi$
$888$	0	0
$889$	7.64453	0.256389
$890$	29.2016	0.978841
$891$	0	0
$892$	3.89619	0.130454
$893$	−4.68397	−0.156743
$894$	0	0
$895$	48.8107	1.63156
$896$	1.00000	0.0334077
$897$	0	0
$898$	−27.6425	−0.922441
$899$	6.85674	0.228685
$900$	0	0
$901$	20.0997	0.669619
$902$	0.823287	0.0274125
$903$	0	0
$904$	−9.61755	−0.319875
$905$	−42.0748	−1.39861
$906$	0	0
$907$	27.2891	0.906118	0.453059	−	0.891480i	$-0.350333\pi$
$907$	27.2891	0.906118	0.453059	+	0.891480i	$0.350333\pi$
$908$	9.92112	0.329244
$909$	0	0
$910$	26.3534	0.873608
$911$	−32.3968	−1.07335	−0.536677	−	0.843788i	$-0.680321\pi$
$911$	−32.3968	−1.07335	−0.536677	+	0.843788i	$0.680321\pi$
$912$	0	0
$913$	−23.3430	−0.772541
$914$	33.9191	1.12194
$915$	0	0
$916$	20.1642	0.666246
$917$	−13.3679	−0.441448
$918$	0	0
$919$	59.0063	1.94644	0.973219	−	0.229878i	$-0.0738327\pi$
$919$	59.0063	1.94644	0.973219	+	0.229878i	$0.0738327\pi$
$920$	−18.0249	−0.594264
$921$	0	0
$922$	3.82723	0.126043
$923$	57.0353	1.87734
$924$	0	0
$925$	34.5091	1.13465
$926$	−3.44682	−0.113270
$927$	0	0
$928$	1.20369	0.0395131
$929$	55.0353	1.80565	0.902826	−	0.430007i	$-0.141489\pi$
$929$	55.0353	1.80565	0.902826	+	0.430007i	$0.141489\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	−13.9730	−0.457702
$933$	0	0
$934$	11.5137	0.376741
$935$	103.542	3.38620
$936$	0	0
$937$	17.9829	0.587477	0.293739	−	0.955886i	$-0.405100\pi$
$937$	17.9829	0.587477	0.293739	+	0.955886i	$0.405100\pi$
$938$	1.03944	0.0339390
$939$	0	0
$940$	−19.5052	−0.636190
$941$	−15.1433	−0.493656	−0.246828	−	0.969059i	$-0.579388\pi$
$941$	−15.1433	−0.493656	−0.246828	+	0.969059i	$0.579388\pi$
$942$	0	0
$943$	0.710947	0.0231516
$944$	−6.49274	−0.211321
$945$	0	0
$946$	3.99147	0.129774
$947$	26.9875	0.876977	0.438489	−	0.898737i	$-0.355514\pi$
$947$	26.9875	0.876977	0.438489	+	0.898737i	$0.355514\pi$
$948$	0	0
$949$	−76.0997	−2.47030
$950$	12.3410	0.400394
$951$	0	0
$952$	−4.96056	−0.160773
$953$	−56.5781	−1.83274	−0.916372	−	0.400327i	$-0.868897\pi$
$953$	−56.5781	−1.83274	−0.916372	+	0.400327i	$0.868897\pi$
$954$	0	0
$955$	32.1537	1.04047
$956$	−18.0249	−0.582968
$957$	0	0
$958$	13.0210	0.420689
$959$	12.2681	0.396156
$960$	0	0
$961$	1.44936	0.0467536
$962$	17.6964	0.570556
$963$	0	0
$964$	−22.9001	−0.737563
$965$	55.3389	1.78142
$966$	0	0
$967$	−26.1786	−0.841847	−0.420924	−	0.907096i	$-0.638294\pi$
$967$	−26.1786	−0.841847	−0.420924	+	0.907096i	$0.638294\pi$
$968$	−14.1248	−0.453988
$969$	0	0
$970$	−23.5052	−0.754707
$971$	53.3369	1.71166	0.855831	−	0.517256i	$-0.173046\pi$
$971$	53.3369	1.71166	0.855831	+	0.517256i	$0.173046\pi$
$972$	0	0
$973$	−6.73588	−0.215942
$974$	−37.6240	−1.20555
$975$	0	0
$976$	−1.64453	−0.0526400
$977$	−25.9460	−0.830088	−0.415044	−	0.909801i	$-0.636234\pi$
$977$	−25.9460	−0.830088	−0.415044	+	0.909801i	$0.636234\pi$
$978$	0	0
$979$	35.1497	1.12339
$980$	−4.16425	−0.133022
$981$	0	0
$982$	23.8422	0.760836
$983$	16.6570	0.531276	0.265638	−	0.964073i	$-0.414417\pi$
$983$	16.6570	0.531276	0.265638	+	0.964073i	$0.414417\pi$
$984$	0	0
$985$	49.6425	1.58174
$986$	−5.97098	−0.190155
$987$	0	0
$988$	6.32850	0.201336
$989$	3.44682	0.109603
$990$	0	0
$991$	24.2476	0.770249	0.385125	−	0.922865i	$-0.374158\pi$
$991$	24.2476	0.770249	0.385125	+	0.922865i	$0.374158\pi$
$992$	5.69643	0.180862
$993$	0	0
$994$	−9.01247	−0.285858
$995$	−83.6404	−2.65158
$996$	0	0
$997$	11.9935	0.379839	0.189919	−	0.981800i	$-0.439177\pi$
$997$	11.9935	0.379839	0.189919	+	0.981800i	$0.439177\pi$
$998$	−5.40344	−0.171043
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.ba.1.1		3
3.2	odd	2		266.2.a.d.1.2	✓	3
12.11	even	2		2128.2.a.s.1.2		3
15.14	odd	2		6650.2.a.cd.1.2		3
21.20	even	2		1862.2.a.r.1.2		3
24.5	odd	2		8512.2.a.bm.1.2		3
24.11	even	2		8512.2.a.bj.1.2		3
57.56	even	2		5054.2.a.r.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
266.2.a.d.1.2	✓	3	3.2	odd	2
1862.2.a.r.1.2		3	21.20	even	2
2128.2.a.s.1.2		3	12.11	even	2
2394.2.a.ba.1.1		3	1.1	even	1	trivial
5054.2.a.r.1.2		3	57.56	even	2
6650.2.a.cd.1.2		3	15.14	odd	2
8512.2.a.bj.1.2		3	24.11	even	2
8512.2.a.bm.1.2		3	24.5	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.16425 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.16425 q^{5} -1.00000 q^{7} -1.00000 q^{8} +4.16425 q^{10} +5.01247 q^{11} -6.32850 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.96056 q^{17} -1.00000 q^{19} -4.16425 q^{20} -5.01247 q^{22} -4.32850 q^{23} +12.3410 q^{25} +6.32850 q^{26} -1.00000 q^{28} -1.20369 q^{29} -5.69643 q^{31} -1.00000 q^{32} +4.96056 q^{34} +4.16425 q^{35} +2.79631 q^{37} +1.00000 q^{38} +4.16425 q^{40} -0.164248 q^{41} -0.796310 q^{43} +5.01247 q^{44} +4.32850 q^{46} +4.68397 q^{47} +1.00000 q^{49} -12.3410 q^{50} -6.32850 q^{52} -4.05191 q^{53} -20.8731 q^{55} +1.00000 q^{56} +1.20369 q^{58} -6.49274 q^{59} -1.64453 q^{61} +5.69643 q^{62} +1.00000 q^{64} +26.3534 q^{65} +1.03944 q^{67} -4.96056 q^{68} -4.16425 q^{70} -9.01247 q^{71} +12.0249 q^{73} -2.79631 q^{74} -1.00000 q^{76} -5.01247 q^{77} +0.467814 q^{79} -4.16425 q^{80} +0.164248 q^{82} -4.65699 q^{83} +20.6570 q^{85} +0.796310 q^{86} -5.01247 q^{88} +7.01247 q^{89} +6.32850 q^{91} -4.32850 q^{92} -4.68397 q^{94} +4.16425 q^{95} -5.64453 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 3 q^{8} + 5 q^{10} - 3 q^{11} - 4 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 3 q^{19} - 5 q^{20} + 3 q^{22} + 2 q^{23} + 4 q^{25} + 4 q^{26} - 3 q^{28} - 5 q^{29} + 4 q^{31} - 3 q^{32} + 6 q^{34} + 5 q^{35} + 7 q^{37} + 3 q^{38} + 5 q^{40} + 7 q^{41} - q^{43} - 3 q^{44} - 2 q^{46} + 11 q^{47} + 3 q^{49} - 4 q^{50} - 4 q^{52} - 3 q^{53} - 16 q^{55} + 3 q^{56} + 5 q^{58} + 3 q^{59} + 7 q^{61} - 4 q^{62} + 3 q^{64} + 28 q^{65} + 12 q^{67} - 6 q^{68} - 5 q^{70} - 9 q^{71} - 7 q^{74} - 3 q^{76} + 3 q^{77} + 15 q^{79} - 5 q^{80} - 7 q^{82} + 16 q^{83} + 32 q^{85} + q^{86} + 3 q^{88} + 3 q^{89} + 4 q^{91} + 2 q^{92} - 11 q^{94} + 5 q^{95} - 5 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 - 3 * q^8 + 5 * q^10 - 3 * q^11 - 4 * q^13 + 3 * q^14 + 3 * q^16 - 6 * q^17 - 3 * q^19 - 5 * q^20 + 3 * q^22 + 2 * q^23 + 4 * q^25 + 4 * q^26 - 3 * q^28 - 5 * q^29 + 4 * q^31 - 3 * q^32 + 6 * q^34 + 5 * q^35 + 7 * q^37 + 3 * q^38 + 5 * q^40 + 7 * q^41 - q^43 - 3 * q^44 - 2 * q^46 + 11 * q^47 + 3 * q^49 - 4 * q^50 - 4 * q^52 - 3 * q^53 - 16 * q^55 + 3 * q^56 + 5 * q^58 + 3 * q^59 + 7 * q^61 - 4 * q^62 + 3 * q^64 + 28 * q^65 + 12 * q^67 - 6 * q^68 - 5 * q^70 - 9 * q^71 - 7 * q^74 - 3 * q^76 + 3 * q^77 + 15 * q^79 - 5 * q^80 - 7 * q^82 + 16 * q^83 + 32 * q^85 + q^86 + 3 * q^88 + 3 * q^89 + 4 * q^91 + 2 * q^92 - 11 * q^94 + 5 * q^95 - 5 * q^97 - 3 * q^98

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