LMFDB - Embedded newform 8046.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 17x^{6} - 2x^{5} + 71x^{4} - 18x^{3} - 81x^{2} + 36x + 9 $ x^8 - 17x^6 - 2x^5 + 71x^4 - 18x^3 - 81x^2 + 36x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.58447$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.58447 q^{5} -4.00108 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.58447 q^{5} -4.00108 q^{7} +1.00000 q^{8} -3.58447 q^{10} +1.14813 q^{11} +0.738223 q^{13} -4.00108 q^{14} +1.00000 q^{16} +4.19403 q^{17} +1.58835 q^{19} -3.58447 q^{20} +1.14813 q^{22} -2.22264 q^{23} +7.84841 q^{25} +0.738223 q^{26} -4.00108 q^{28} +2.40365 q^{29} +4.76134 q^{31} +1.00000 q^{32} +4.19403 q^{34} +14.3418 q^{35} -9.04202 q^{37} +1.58835 q^{38} -3.58447 q^{40} +3.19103 q^{41} +5.30662 q^{43} +1.14813 q^{44} -2.22264 q^{46} -5.72851 q^{47} +9.00867 q^{49} +7.84841 q^{50} +0.738223 q^{52} -0.714734 q^{53} -4.11544 q^{55} -4.00108 q^{56} +2.40365 q^{58} +6.45783 q^{59} -13.7213 q^{61} +4.76134 q^{62} +1.00000 q^{64} -2.64614 q^{65} +6.34694 q^{67} +4.19403 q^{68} +14.3418 q^{70} -4.19472 q^{71} +7.45515 q^{73} -9.04202 q^{74} +1.58835 q^{76} -4.59377 q^{77} -2.56890 q^{79} -3.58447 q^{80} +3.19103 q^{82} -9.66114 q^{83} -15.0334 q^{85} +5.30662 q^{86} +1.14813 q^{88} +14.9686 q^{89} -2.95369 q^{91} -2.22264 q^{92} -5.72851 q^{94} -5.69339 q^{95} -5.33851 q^{97} +9.00867 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8	$ 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8} - 6 q^{11} - 8 q^{13} - 5 q^{14} + 8 q^{16} + 5 q^{17} - 14 q^{19} - 6 q^{22} - 21 q^{23} - 6 q^{25} - 8 q^{26} - 5 q^{28} - 3 q^{29} - 4 q^{31} + 8 q^{32} + 5 q^{34} + 2 q^{35} - 3 q^{37} - 14 q^{38} - 7 q^{41} - 12 q^{43} - 6 q^{44} - 21 q^{46} - 25 q^{47} - 7 q^{49} - 6 q^{50} - 8 q^{52} - 3 q^{53} - 9 q^{55} - 5 q^{56} - 3 q^{58} - 2 q^{59} - 17 q^{61} - 4 q^{62} + 8 q^{64} - 32 q^{65} - 14 q^{67} + 5 q^{68} + 2 q^{70} - 7 q^{71} - 10 q^{73} - 3 q^{74} - 14 q^{76} - 12 q^{77} - 33 q^{79} - 7 q^{82} - 13 q^{83} - 33 q^{85} - 12 q^{86} - 6 q^{88} + 22 q^{89} - 22 q^{91} - 21 q^{92} - 25 q^{94} + 14 q^{95} - 11 q^{97} - 7 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8 - 6 * q^11 - 8 * q^13 - 5 * q^14 + 8 * q^16 + 5 * q^17 - 14 * q^19 - 6 * q^22 - 21 * q^23 - 6 * q^25 - 8 * q^26 - 5 * q^28 - 3 * q^29 - 4 * q^31 + 8 * q^32 + 5 * q^34 + 2 * q^35 - 3 * q^37 - 14 * q^38 - 7 * q^41 - 12 * q^43 - 6 * q^44 - 21 * q^46 - 25 * q^47 - 7 * q^49 - 6 * q^50 - 8 * q^52 - 3 * q^53 - 9 * q^55 - 5 * q^56 - 3 * q^58 - 2 * q^59 - 17 * q^61 - 4 * q^62 + 8 * q^64 - 32 * q^65 - 14 * q^67 + 5 * q^68 + 2 * q^70 - 7 * q^71 - 10 * q^73 - 3 * q^74 - 14 * q^76 - 12 * q^77 - 33 * q^79 - 7 * q^82 - 13 * q^83 - 33 * q^85 - 12 * q^86 - 6 * q^88 + 22 * q^89 - 22 * q^91 - 21 * q^92 - 25 * q^94 + 14 * q^95 - 11 * q^97 - 7 * 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
$5$	−3.58447	−1.60302	−0.801511	−	0.597979i	$-0.795970\pi$
$5$	−3.58447	−1.60302	−0.801511	+	0.597979i	$0.795970\pi$
$6$	0	0
$7$	−4.00108	−1.51227	−0.756134	−	0.654417i	$-0.772914\pi$
$7$	−4.00108	−1.51227	−0.756134	+	0.654417i	$0.772914\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.58447	−1.13351
$11$	1.14813	0.346175	0.173087	−	0.984906i	$-0.444626\pi$
$11$	1.14813	0.346175	0.173087	+	0.984906i	$0.444626\pi$
$12$	0	0
$13$	0.738223	0.204746	0.102373	−	0.994746i	$-0.467356\pi$
$13$	0.738223	0.204746	0.102373	+	0.994746i	$0.467356\pi$
$14$	−4.00108	−1.06933
$15$	0	0
$16$	1.00000	0.250000
$17$	4.19403	1.01720	0.508601	−	0.861002i	$-0.330163\pi$
$17$	4.19403	1.01720	0.508601	+	0.861002i	$0.330163\pi$
$18$	0	0
$19$	1.58835	0.364393	0.182196	−	0.983262i	$-0.441679\pi$
$19$	1.58835	0.364393	0.182196	+	0.983262i	$0.441679\pi$
$20$	−3.58447	−0.801511
$21$	0	0
$22$	1.14813	0.244782
$23$	−2.22264	−0.463452	−0.231726	−	0.972781i	$-0.574437\pi$
$23$	−2.22264	−0.463452	−0.231726	+	0.972781i	$0.574437\pi$
$24$	0	0
$25$	7.84841	1.56968
$26$	0.738223	0.144777
$27$	0	0
$28$	−4.00108	−0.756134
$29$	2.40365	0.446346	0.223173	−	0.974779i	$-0.428359\pi$
$29$	2.40365	0.446346	0.223173	+	0.974779i	$0.428359\pi$
$30$	0	0
$31$	4.76134	0.855163	0.427581	−	0.903977i	$-0.359366\pi$
$31$	4.76134	0.855163	0.427581	+	0.903977i	$0.359366\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.19403	0.719270
$35$	14.3418	2.42420
$36$	0	0
$37$	−9.04202	−1.48650	−0.743249	−	0.669014i	$-0.766716\pi$
$37$	−9.04202	−1.48650	−0.743249	+	0.669014i	$0.766716\pi$
$38$	1.58835	0.257665
$39$	0	0
$40$	−3.58447	−0.566754
$41$	3.19103	0.498355	0.249177	−	0.968458i	$-0.419840\pi$
$41$	3.19103	0.498355	0.249177	+	0.968458i	$0.419840\pi$
$42$	0	0
$43$	5.30662	0.809252	0.404626	−	0.914482i	$-0.367402\pi$
$43$	5.30662	0.809252	0.404626	+	0.914482i	$0.367402\pi$
$44$	1.14813	0.173087
$45$	0	0
$46$	−2.22264	−0.327710
$47$	−5.72851	−0.835590	−0.417795	−	0.908541i	$-0.637197\pi$
$47$	−5.72851	−0.835590	−0.417795	+	0.908541i	$0.637197\pi$
$48$	0	0
$49$	9.00867	1.28695
$50$	7.84841	1.10993
$51$	0	0
$52$	0.738223	0.102373
$53$	−0.714734	−0.0981763	−0.0490881	−	0.998794i	$-0.515632\pi$
$53$	−0.714734	−0.0981763	−0.0490881	+	0.998794i	$0.515632\pi$
$54$	0	0
$55$	−4.11544	−0.554926
$56$	−4.00108	−0.534667
$57$	0	0
$58$	2.40365	0.315614
$59$	6.45783	0.840738	0.420369	−	0.907353i	$-0.361901\pi$
$59$	6.45783	0.840738	0.420369	+	0.907353i	$0.361901\pi$
$60$	0	0
$61$	−13.7213	−1.75683	−0.878414	−	0.477901i	$-0.841398\pi$
$61$	−13.7213	−1.75683	−0.878414	+	0.477901i	$0.841398\pi$
$62$	4.76134	0.604691
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.64614	−0.328213
$66$	0	0
$67$	6.34694	0.775402	0.387701	−	0.921785i	$-0.373269\pi$
$67$	6.34694	0.775402	0.387701	+	0.921785i	$0.373269\pi$
$68$	4.19403	0.508601
$69$	0	0
$70$	14.3418	1.71417
$71$	−4.19472	−0.497821	−0.248911	−	0.968526i	$-0.580073\pi$
$71$	−4.19472	−0.497821	−0.248911	+	0.968526i	$0.580073\pi$
$72$	0	0
$73$	7.45515	0.872559	0.436280	−	0.899811i	$-0.356296\pi$
$73$	7.45515	0.872559	0.436280	+	0.899811i	$0.356296\pi$
$74$	−9.04202	−1.05111
$75$	0	0
$76$	1.58835	0.182196
$77$	−4.59377	−0.523509
$78$	0	0
$79$	−2.56890	−0.289024	−0.144512	−	0.989503i	$-0.546161\pi$
$79$	−2.56890	−0.289024	−0.144512	+	0.989503i	$0.546161\pi$
$80$	−3.58447	−0.400756
$81$	0	0
$82$	3.19103	0.352390
$83$	−9.66114	−1.06045	−0.530224	−	0.847858i	$-0.677892\pi$
$83$	−9.66114	−1.06045	−0.530224	+	0.847858i	$0.677892\pi$
$84$	0	0
$85$	−15.0334	−1.63060
$86$	5.30662	0.572228
$87$	0	0
$88$	1.14813	0.122391
$89$	14.9686	1.58667	0.793334	−	0.608786i	$-0.208343\pi$
$89$	14.9686	1.58667	0.793334	+	0.608786i	$0.208343\pi$
$90$	0	0
$91$	−2.95369	−0.309631
$92$	−2.22264	−0.231726
$93$	0	0
$94$	−5.72851	−0.590851
$95$	−5.69339	−0.584130
$96$	0	0
$97$	−5.33851	−0.542043	−0.271022	−	0.962573i	$-0.587361\pi$
$97$	−5.33851	−0.542043	−0.271022	+	0.962573i	$0.587361\pi$
$98$	9.00867	0.910013
$99$	0	0
$100$	7.84841	0.784841
$101$	−8.10397	−0.806375	−0.403187	−	0.915117i	$-0.632098\pi$
$101$	−8.10397	−0.806375	−0.403187	+	0.915117i	$0.632098\pi$
$102$	0	0
$103$	−8.30107	−0.817928	−0.408964	−	0.912550i	$-0.634110\pi$
$103$	−8.30107	−0.817928	−0.408964	+	0.912550i	$0.634110\pi$
$104$	0.738223	0.0723887
$105$	0	0
$106$	−0.714734	−0.0694211
$107$	5.28936	0.511342	0.255671	−	0.966764i	$-0.417704\pi$
$107$	5.28936	0.511342	0.255671	+	0.966764i	$0.417704\pi$
$108$	0	0
$109$	0.742766	0.0711441	0.0355721	−	0.999367i	$-0.488675\pi$
$109$	0.742766	0.0711441	0.0355721	+	0.999367i	$0.488675\pi$
$110$	−4.11544	−0.392392
$111$	0	0
$112$	−4.00108	−0.378067
$113$	−0.580727	−0.0546302	−0.0273151	−	0.999627i	$-0.508696\pi$
$113$	−0.580727	−0.0546302	−0.0273151	+	0.999627i	$0.508696\pi$
$114$	0	0
$115$	7.96697	0.742923
$116$	2.40365	0.223173
$117$	0	0
$118$	6.45783	0.594491
$119$	−16.7807	−1.53828
$120$	0	0
$121$	−9.68179	−0.880163
$122$	−13.7213	−1.24226
$123$	0	0
$124$	4.76134	0.427581
$125$	−10.2100	−0.913214
$126$	0	0
$127$	1.12559	0.0998804	0.0499402	−	0.998752i	$-0.484097\pi$
$127$	1.12559	0.0998804	0.0499402	+	0.998752i	$0.484097\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.64614	−0.232082
$131$	−7.91146	−0.691228	−0.345614	−	0.938377i	$-0.612329\pi$
$131$	−7.91146	−0.691228	−0.345614	+	0.938377i	$0.612329\pi$
$132$	0	0
$133$	−6.35513	−0.551059
$134$	6.34694	0.548292
$135$	0	0
$136$	4.19403	0.359635
$137$	−6.79993	−0.580958	−0.290479	−	0.956881i	$-0.593815\pi$
$137$	−6.79993	−0.580958	−0.290479	+	0.956881i	$0.593815\pi$
$138$	0	0
$139$	−20.3028	−1.72206	−0.861032	−	0.508552i	$-0.830181\pi$
$139$	−20.3028	−1.72206	−0.861032	+	0.508552i	$0.830181\pi$
$140$	14.3418	1.21210
$141$	0	0
$142$	−4.19472	−0.352013
$143$	0.847577	0.0708779
$144$	0	0
$145$	−8.61579	−0.715503
$146$	7.45515	0.616993
$147$	0	0
$148$	−9.04202	−0.743249
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−8.44910	−0.687578	−0.343789	−	0.939047i	$-0.611711\pi$
$151$	−8.44910	−0.687578	−0.343789	+	0.939047i	$0.611711\pi$
$152$	1.58835	0.128832
$153$	0	0
$154$	−4.59377	−0.370176
$155$	−17.0669	−1.37085
$156$	0	0
$157$	7.38323	0.589246	0.294623	−	0.955614i	$-0.404806\pi$
$157$	7.38323	0.589246	0.294623	+	0.955614i	$0.404806\pi$
$158$	−2.56890	−0.204371
$159$	0	0
$160$	−3.58447	−0.283377
$161$	8.89295	0.700863
$162$	0	0
$163$	−3.51001	−0.274925	−0.137463	−	0.990507i	$-0.543895\pi$
$163$	−3.51001	−0.274925	−0.137463	+	0.990507i	$0.543895\pi$
$164$	3.19103	0.249177
$165$	0	0
$166$	−9.66114	−0.749850
$167$	16.5643	1.28178	0.640892	−	0.767631i	$-0.278565\pi$
$167$	16.5643	1.28178	0.640892	+	0.767631i	$0.278565\pi$
$168$	0	0
$169$	−12.4550	−0.958079
$170$	−15.0334	−1.15301
$171$	0	0
$172$	5.30662	0.404626
$173$	−3.22617	−0.245282	−0.122641	−	0.992451i	$-0.539136\pi$
$173$	−3.22617	−0.245282	−0.122641	+	0.992451i	$0.539136\pi$
$174$	0	0
$175$	−31.4021	−2.37378
$176$	1.14813	0.0865437
$177$	0	0
$178$	14.9686	1.12194
$179$	−4.58748	−0.342884	−0.171442	−	0.985194i	$-0.554843\pi$
$179$	−4.58748	−0.342884	−0.171442	+	0.985194i	$0.554843\pi$
$180$	0	0
$181$	−19.3179	−1.43588	−0.717942	−	0.696103i	$-0.754916\pi$
$181$	−19.3179	−1.43588	−0.717942	+	0.696103i	$0.754916\pi$
$182$	−2.95369	−0.218942
$183$	0	0
$184$	−2.22264	−0.163855
$185$	32.4108	2.38289
$186$	0	0
$187$	4.81530	0.352129
$188$	−5.72851	−0.417795
$189$	0	0
$190$	−5.69339	−0.413042
$191$	14.5959	1.05612	0.528059	−	0.849207i	$-0.322920\pi$
$191$	14.5959	1.05612	0.528059	+	0.849207i	$0.322920\pi$
$192$	0	0
$193$	−20.6018	−1.48295	−0.741476	−	0.670979i	$-0.765874\pi$
$193$	−20.6018	−1.48295	−0.741476	+	0.670979i	$0.765874\pi$
$194$	−5.33851	−0.383282
$195$	0	0
$196$	9.00867	0.643476
$197$	6.73218	0.479648	0.239824	−	0.970816i	$-0.422910\pi$
$197$	6.73218	0.479648	0.239824	+	0.970816i	$0.422910\pi$
$198$	0	0
$199$	−7.92167	−0.561552	−0.280776	−	0.959773i	$-0.590592\pi$
$199$	−7.92167	−0.561552	−0.280776	+	0.959773i	$0.590592\pi$
$200$	7.84841	0.554967
$201$	0	0
$202$	−8.10397	−0.570193
$203$	−9.61719	−0.674994
$204$	0	0
$205$	−11.4381	−0.798874
$206$	−8.30107	−0.578363
$207$	0	0
$208$	0.738223	0.0511866
$209$	1.82364	0.126144
$210$	0	0
$211$	−20.4048	−1.40473	−0.702363	−	0.711819i	$-0.747872\pi$
$211$	−20.4048	−1.40473	−0.702363	+	0.711819i	$0.747872\pi$
$212$	−0.714734	−0.0490881
$213$	0	0
$214$	5.28936	0.361573
$215$	−19.0214	−1.29725
$216$	0	0
$217$	−19.0505	−1.29323
$218$	0.742766	0.0503065
$219$	0	0
$220$	−4.11544	−0.277463
$221$	3.09613	0.208268
$222$	0	0
$223$	−6.23200	−0.417326	−0.208663	−	0.977988i	$-0.566911\pi$
$223$	−6.23200	−0.417326	−0.208663	+	0.977988i	$0.566911\pi$
$224$	−4.00108	−0.267334
$225$	0	0
$226$	−0.580727	−0.0386294
$227$	1.03841	0.0689220	0.0344610	−	0.999406i	$-0.489029\pi$
$227$	1.03841	0.0689220	0.0344610	+	0.999406i	$0.489029\pi$
$228$	0	0
$229$	9.32231	0.616036	0.308018	−	0.951381i	$-0.400334\pi$
$229$	9.32231	0.616036	0.308018	+	0.951381i	$0.400334\pi$
$230$	7.96697	0.525326
$231$	0	0
$232$	2.40365	0.157807
$233$	24.0216	1.57371	0.786854	−	0.617139i	$-0.211708\pi$
$233$	24.0216	1.57371	0.786854	+	0.617139i	$0.211708\pi$
$234$	0	0
$235$	20.5337	1.33947
$236$	6.45783	0.420369
$237$	0	0
$238$	−16.7807	−1.08773
$239$	−23.2098	−1.50132	−0.750658	−	0.660690i	$-0.770264\pi$
$239$	−23.2098	−1.50132	−0.750658	+	0.660690i	$0.770264\pi$
$240$	0	0
$241$	−4.67539	−0.301168	−0.150584	−	0.988597i	$-0.548115\pi$
$241$	−4.67539	−0.301168	−0.150584	+	0.988597i	$0.548115\pi$
$242$	−9.68179	−0.622369
$243$	0	0
$244$	−13.7213	−0.878414
$245$	−32.2913	−2.06301
$246$	0	0
$247$	1.17256	0.0746080
$248$	4.76134	0.302346
$249$	0	0
$250$	−10.2100	−0.645740
$251$	−1.75570	−0.110819	−0.0554094	−	0.998464i	$-0.517646\pi$
$251$	−1.75570	−0.110819	−0.0554094	+	0.998464i	$0.517646\pi$
$252$	0	0
$253$	−2.55188	−0.160435
$254$	1.12559	0.0706261
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.3857	0.710218	0.355109	−	0.934825i	$-0.384444\pi$
$257$	11.3857	0.710218	0.355109	+	0.934825i	$0.384444\pi$
$258$	0	0
$259$	36.1779	2.24798
$260$	−2.64614	−0.164106
$261$	0	0
$262$	−7.91146	−0.488772
$263$	−2.59555	−0.160048	−0.0800242	−	0.996793i	$-0.525500\pi$
$263$	−2.59555	−0.160048	−0.0800242	+	0.996793i	$0.525500\pi$
$264$	0	0
$265$	2.56194	0.157379
$266$	−6.35513	−0.389658
$267$	0	0
$268$	6.34694	0.387701
$269$	−11.4807	−0.699988	−0.349994	−	0.936752i	$-0.613816\pi$
$269$	−11.4807	−0.699988	−0.349994	+	0.936752i	$0.613816\pi$
$270$	0	0
$271$	−17.3758	−1.05551	−0.527754	−	0.849398i	$-0.676966\pi$
$271$	−17.3758	−1.05551	−0.527754	+	0.849398i	$0.676966\pi$
$272$	4.19403	0.254300
$273$	0	0
$274$	−6.79993	−0.410799
$275$	9.01101	0.543384
$276$	0	0
$277$	−6.24677	−0.375332	−0.187666	−	0.982233i	$-0.560092\pi$
$277$	−6.24677	−0.375332	−0.187666	+	0.982233i	$0.560092\pi$
$278$	−20.3028	−1.21768
$279$	0	0
$280$	14.3418	0.857084
$281$	−9.95285	−0.593737	−0.296869	−	0.954918i	$-0.595942\pi$
$281$	−9.95285	−0.593737	−0.296869	+	0.954918i	$0.595942\pi$
$282$	0	0
$283$	9.91371	0.589309	0.294654	−	0.955604i	$-0.404795\pi$
$283$	9.91371	0.589309	0.294654	+	0.955604i	$0.404795\pi$
$284$	−4.19472	−0.248911
$285$	0	0
$286$	0.847577	0.0501183
$287$	−12.7676	−0.753646
$288$	0	0
$289$	0.589891	0.0346995
$290$	−8.61579	−0.505937
$291$	0	0
$292$	7.45515	0.436280
$293$	16.0738	0.939040	0.469520	−	0.882922i	$-0.344427\pi$
$293$	16.0738	0.939040	0.469520	+	0.882922i	$0.344427\pi$
$294$	0	0
$295$	−23.1479	−1.34772
$296$	−9.04202	−0.525557
$297$	0	0
$298$	1.00000	0.0579284
$299$	−1.64080	−0.0948899
$300$	0	0
$301$	−21.2322	−1.22381
$302$	−8.44910	−0.486191
$303$	0	0
$304$	1.58835	0.0910982
$305$	49.1834	2.81624
$306$	0	0
$307$	−8.85754	−0.505526	−0.252763	−	0.967528i	$-0.581339\pi$
$307$	−8.85754	−0.505526	−0.252763	+	0.967528i	$0.581339\pi$
$308$	−4.59377	−0.261754
$309$	0	0
$310$	−17.0669	−0.969334
$311$	−25.2036	−1.42917	−0.714583	−	0.699551i	$-0.753383\pi$
$311$	−25.2036	−1.42917	−0.714583	+	0.699551i	$0.753383\pi$
$312$	0	0
$313$	25.4110	1.43631	0.718156	−	0.695882i	$-0.244986\pi$
$313$	25.4110	1.43631	0.718156	+	0.695882i	$0.244986\pi$
$314$	7.38323	0.416660
$315$	0	0
$316$	−2.56890	−0.144512
$317$	−13.2481	−0.744088	−0.372044	−	0.928215i	$-0.621343\pi$
$317$	−13.2481	−0.744088	−0.372044	+	0.928215i	$0.621343\pi$
$318$	0	0
$319$	2.75970	0.154514
$320$	−3.58447	−0.200378
$321$	0	0
$322$	8.89295	0.495585
$323$	6.66159	0.370661
$324$	0	0
$325$	5.79388	0.321387
$326$	−3.51001	−0.194402
$327$	0	0
$328$	3.19103	0.176195
$329$	22.9203	1.26363
$330$	0	0
$331$	−19.1025	−1.04997	−0.524985	−	0.851112i	$-0.675929\pi$
$331$	−19.1025	−1.04997	−0.524985	+	0.851112i	$0.675929\pi$
$332$	−9.66114	−0.530224
$333$	0	0
$334$	16.5643	0.906358
$335$	−22.7504	−1.24299
$336$	0	0
$337$	−10.4574	−0.569650	−0.284825	−	0.958580i	$-0.591936\pi$
$337$	−10.4574	−0.569650	−0.284825	+	0.958580i	$0.591936\pi$
$338$	−12.4550	−0.677464
$339$	0	0
$340$	−15.0334	−0.815299
$341$	5.46665	0.296036
$342$	0	0
$343$	−8.03685	−0.433949
$344$	5.30662	0.286114
$345$	0	0
$346$	−3.22617	−0.173440
$347$	−8.52699	−0.457753	−0.228877	−	0.973455i	$-0.573505\pi$
$347$	−8.52699	−0.457753	−0.228877	+	0.973455i	$0.573505\pi$
$348$	0	0
$349$	−34.3510	−1.83877	−0.919384	−	0.393360i	$-0.871313\pi$
$349$	−34.3510	−1.83877	−0.919384	+	0.393360i	$0.871313\pi$
$350$	−31.4021	−1.67852
$351$	0	0
$352$	1.14813	0.0611956
$353$	−8.96344	−0.477076	−0.238538	−	0.971133i	$-0.576668\pi$
$353$	−8.96344	−0.477076	−0.238538	+	0.971133i	$0.576668\pi$
$354$	0	0
$355$	15.0358	0.798019
$356$	14.9686	0.793334
$357$	0	0
$358$	−4.58748	−0.242456
$359$	25.1520	1.32747	0.663736	−	0.747967i	$-0.268970\pi$
$359$	25.1520	1.32747	0.663736	+	0.747967i	$0.268970\pi$
$360$	0	0
$361$	−16.4771	−0.867218
$362$	−19.3179	−1.01532
$363$	0	0
$364$	−2.95369	−0.154816
$365$	−26.7227	−1.39873
$366$	0	0
$367$	−10.3047	−0.537903	−0.268952	−	0.963154i	$-0.586677\pi$
$367$	−10.3047	−0.537903	−0.268952	+	0.963154i	$0.586677\pi$
$368$	−2.22264	−0.115863
$369$	0	0
$370$	32.4108	1.68496
$371$	2.85971	0.148469
$372$	0	0
$373$	26.0716	1.34993	0.674967	−	0.737848i	$-0.264158\pi$
$373$	26.0716	1.34993	0.674967	+	0.737848i	$0.264158\pi$
$374$	4.81530	0.248993
$375$	0	0
$376$	−5.72851	−0.295426
$377$	1.77443	0.0913876
$378$	0	0
$379$	−12.2436	−0.628910	−0.314455	−	0.949272i	$-0.601822\pi$
$379$	−12.2436	−0.628910	−0.314455	+	0.949272i	$0.601822\pi$
$380$	−5.69339	−0.292065
$381$	0	0
$382$	14.5959	0.746789
$383$	−16.6460	−0.850569	−0.425284	−	0.905060i	$-0.639826\pi$
$383$	−16.6460	−0.850569	−0.425284	+	0.905060i	$0.639826\pi$
$384$	0	0
$385$	16.4662	0.839196
$386$	−20.6018	−1.04861
$387$	0	0
$388$	−5.33851	−0.271022
$389$	−24.1756	−1.22575	−0.612875	−	0.790180i	$-0.709987\pi$
$389$	−24.1756	−1.22575	−0.612875	+	0.790180i	$0.709987\pi$
$390$	0	0
$391$	−9.32180	−0.471424
$392$	9.00867	0.455006
$393$	0	0
$394$	6.73218	0.339162
$395$	9.20815	0.463312
$396$	0	0
$397$	26.5827	1.33415	0.667073	−	0.744992i	$-0.267547\pi$
$397$	26.5827	1.33415	0.667073	+	0.744992i	$0.267547\pi$
$398$	−7.92167	−0.397078
$399$	0	0
$400$	7.84841	0.392421
$401$	19.3434	0.965966	0.482983	−	0.875630i	$-0.339553\pi$
$401$	19.3434	0.965966	0.482983	+	0.875630i	$0.339553\pi$
$402$	0	0
$403$	3.51493	0.175091
$404$	−8.10397	−0.403187
$405$	0	0
$406$	−9.61719	−0.477293
$407$	−10.3814	−0.514588
$408$	0	0
$409$	−9.33365	−0.461519	−0.230760	−	0.973011i	$-0.574121\pi$
$409$	−9.33365	−0.461519	−0.230760	+	0.973011i	$0.574121\pi$
$410$	−11.4381	−0.564889
$411$	0	0
$412$	−8.30107	−0.408964
$413$	−25.8383	−1.27142
$414$	0	0
$415$	34.6300	1.69992
$416$	0.738223	0.0361944
$417$	0	0
$418$	1.82364	0.0891969
$419$	−19.2809	−0.941932	−0.470966	−	0.882151i	$-0.656094\pi$
$419$	−19.2809	−0.941932	−0.470966	+	0.882151i	$0.656094\pi$
$420$	0	0
$421$	−16.6428	−0.811119	−0.405559	−	0.914069i	$-0.632923\pi$
$421$	−16.6428	−0.811119	−0.405559	+	0.914069i	$0.632923\pi$
$422$	−20.4048	−0.993291
$423$	0	0
$424$	−0.714734	−0.0347106
$425$	32.9165	1.59668
$426$	0	0
$427$	54.8999	2.65679
$428$	5.28936	0.255671
$429$	0	0
$430$	−19.0214	−0.917294
$431$	28.6125	1.37821	0.689107	−	0.724660i	$-0.258003\pi$
$431$	28.6125	1.37821	0.689107	+	0.724660i	$0.258003\pi$
$432$	0	0
$433$	29.7265	1.42857	0.714283	−	0.699857i	$-0.246753\pi$
$433$	29.7265	1.42857	0.714283	+	0.699857i	$0.246753\pi$
$434$	−19.0505	−0.914455
$435$	0	0
$436$	0.742766	0.0355721
$437$	−3.53033	−0.168878
$438$	0	0
$439$	0.359854	0.0171749	0.00858745	−	0.999963i	$-0.497266\pi$
$439$	0.359854	0.0171749	0.00858745	+	0.999963i	$0.497266\pi$
$440$	−4.11544	−0.196196
$441$	0	0
$442$	3.09613	0.147268
$443$	−25.6758	−1.21989	−0.609947	−	0.792442i	$-0.708809\pi$
$443$	−25.6758	−1.21989	−0.609947	+	0.792442i	$0.708809\pi$
$444$	0	0
$445$	−53.6545	−2.54347
$446$	−6.23200	−0.295094
$447$	0	0
$448$	−4.00108	−0.189033
$449$	−32.8395	−1.54979	−0.774896	−	0.632088i	$-0.782198\pi$
$449$	−32.8395	−1.54979	−0.774896	+	0.632088i	$0.782198\pi$
$450$	0	0
$451$	3.66372	0.172518
$452$	−0.580727	−0.0273151
$453$	0	0
$454$	1.03841	0.0487352
$455$	10.5874	0.496346
$456$	0	0
$457$	18.6648	0.873101	0.436551	−	0.899680i	$-0.356200\pi$
$457$	18.6648	0.873101	0.436551	+	0.899680i	$0.356200\pi$
$458$	9.32231	0.435603
$459$	0	0
$460$	7.96697	0.371462
$461$	36.1601	1.68414	0.842072	−	0.539365i	$-0.181336\pi$
$461$	36.1601	1.68414	0.842072	+	0.539365i	$0.181336\pi$
$462$	0	0
$463$	3.07692	0.142996	0.0714982	−	0.997441i	$-0.477222\pi$
$463$	3.07692	0.142996	0.0714982	+	0.997441i	$0.477222\pi$
$464$	2.40365	0.111586
$465$	0	0
$466$	24.0216	1.11278
$467$	32.1937	1.48975	0.744874	−	0.667205i	$-0.232510\pi$
$467$	32.1937	1.48975	0.744874	+	0.667205i	$0.232510\pi$
$468$	0	0
$469$	−25.3946	−1.17261
$470$	20.5337	0.947148
$471$	0	0
$472$	6.45783	0.297246
$473$	6.09270	0.280142
$474$	0	0
$475$	12.4660	0.571981
$476$	−16.7807	−0.769141
$477$	0	0
$478$	−23.2098	−1.06159
$479$	−3.55290	−0.162336	−0.0811681	−	0.996700i	$-0.525865\pi$
$479$	−3.55290	−0.162336	−0.0811681	+	0.996700i	$0.525865\pi$
$480$	0	0
$481$	−6.67503	−0.304355
$482$	−4.67539	−0.212958
$483$	0	0
$484$	−9.68179	−0.440082
$485$	19.1357	0.868908
$486$	0	0
$487$	−5.94156	−0.269238	−0.134619	−	0.990897i	$-0.542981\pi$
$487$	−5.94156	−0.269238	−0.134619	+	0.990897i	$0.542981\pi$
$488$	−13.7213	−0.621132
$489$	0	0
$490$	−32.2913	−1.45877
$491$	−0.576627	−0.0260228	−0.0130114	−	0.999915i	$-0.504142\pi$
$491$	−0.576627	−0.0260228	−0.0130114	+	0.999915i	$0.504142\pi$
$492$	0	0
$493$	10.0810	0.454024
$494$	1.17256	0.0527558
$495$	0	0
$496$	4.76134	0.213791
$497$	16.7834	0.752839
$498$	0	0
$499$	−1.31313	−0.0587840	−0.0293920	−	0.999568i	$-0.509357\pi$
$499$	−1.31313	−0.0587840	−0.0293920	+	0.999568i	$0.509357\pi$
$500$	−10.2100	−0.456607
$501$	0	0
$502$	−1.75570	−0.0783608
$503$	−8.98574	−0.400654	−0.200327	−	0.979729i	$-0.564200\pi$
$503$	−8.98574	−0.400654	−0.200327	+	0.979729i	$0.564200\pi$
$504$	0	0
$505$	29.0484	1.29264
$506$	−2.55188	−0.113445
$507$	0	0
$508$	1.12559	0.0499402
$509$	43.8069	1.94171	0.970854	−	0.239671i	$-0.0770396\pi$
$509$	43.8069	1.94171	0.970854	+	0.239671i	$0.0770396\pi$
$510$	0	0
$511$	−29.8287	−1.31954
$512$	1.00000	0.0441942
$513$	0	0
$514$	11.3857	0.502200
$515$	29.7549	1.31116
$516$	0	0
$517$	−6.57709	−0.289260
$518$	36.1779	1.58956
$519$	0	0
$520$	−2.64614	−0.116041
$521$	32.6400	1.42999	0.714993	−	0.699132i	$-0.246430\pi$
$521$	32.6400	1.42999	0.714993	+	0.699132i	$0.246430\pi$
$522$	0	0
$523$	7.41479	0.324226	0.162113	−	0.986772i	$-0.448169\pi$
$523$	7.41479	0.324226	0.162113	+	0.986772i	$0.448169\pi$
$524$	−7.91146	−0.345614
$525$	0	0
$526$	−2.59555	−0.113171
$527$	19.9692	0.869873
$528$	0	0
$529$	−18.0599	−0.785213
$530$	2.56194	0.111284
$531$	0	0
$532$	−6.35513	−0.275530
$533$	2.35569	0.102036
$534$	0	0
$535$	−18.9596	−0.819693
$536$	6.34694	0.274146
$537$	0	0
$538$	−11.4807	−0.494966
$539$	10.3431	0.445510
$540$	0	0
$541$	−9.31819	−0.400620	−0.200310	−	0.979733i	$-0.564195\pi$
$541$	−9.31819	−0.400620	−0.200310	+	0.979733i	$0.564195\pi$
$542$	−17.3758	−0.746356
$543$	0	0
$544$	4.19403	0.179818
$545$	−2.66242	−0.114046
$546$	0	0
$547$	24.6160	1.05251	0.526253	−	0.850328i	$-0.323597\pi$
$547$	24.6160	1.05251	0.526253	+	0.850328i	$0.323597\pi$
$548$	−6.79993	−0.290479
$549$	0	0
$550$	9.01101	0.384231
$551$	3.81783	0.162645
$552$	0	0
$553$	10.2784	0.437082
$554$	−6.24677	−0.265400
$555$	0	0
$556$	−20.3028	−0.861032
$557$	−1.09903	−0.0465675	−0.0232838	−	0.999729i	$-0.507412\pi$
$557$	−1.09903	−0.0465675	−0.0232838	+	0.999729i	$0.507412\pi$
$558$	0	0
$559$	3.91747	0.165691
$560$	14.3418	0.606050
$561$	0	0
$562$	−9.95285	−0.419836
$563$	−28.4348	−1.19838	−0.599191	−	0.800606i	$-0.704511\pi$
$563$	−28.4348	−1.19838	−0.599191	+	0.800606i	$0.704511\pi$
$564$	0	0
$565$	2.08160	0.0875735
$566$	9.91371	0.416704
$567$	0	0
$568$	−4.19472	−0.176006
$569$	−33.0449	−1.38531	−0.692657	−	0.721267i	$-0.743560\pi$
$569$	−33.0449	−1.38531	−0.692657	+	0.721267i	$0.743560\pi$
$570$	0	0
$571$	−19.2456	−0.805402	−0.402701	−	0.915332i	$-0.631928\pi$
$571$	−19.2456	−0.805402	−0.402701	+	0.915332i	$0.631928\pi$
$572$	0.847577	0.0354390
$573$	0	0
$574$	−12.7676	−0.532908
$575$	−17.4442	−0.727472
$576$	0	0
$577$	11.4884	0.478268	0.239134	−	0.970987i	$-0.423136\pi$
$577$	11.4884	0.478268	0.239134	+	0.970987i	$0.423136\pi$
$578$	0.589891	0.0245362
$579$	0	0
$580$	−8.61579	−0.357751
$581$	38.6550	1.60368
$582$	0	0
$583$	−0.820608	−0.0339861
$584$	7.45515	0.308496
$585$	0	0
$586$	16.0738	0.664002
$587$	12.1525	0.501589	0.250794	−	0.968040i	$-0.419308\pi$
$587$	12.1525	0.501589	0.250794	+	0.968040i	$0.419308\pi$
$588$	0	0
$589$	7.56269	0.311615
$590$	−23.1479	−0.952983
$591$	0	0
$592$	−9.04202	−0.371625
$593$	−14.2186	−0.583886	−0.291943	−	0.956436i	$-0.594302\pi$
$593$	−14.2186	−0.583886	−0.291943	+	0.956436i	$0.594302\pi$
$594$	0	0
$595$	60.1498	2.46590
$596$	1.00000	0.0409616
$597$	0	0
$598$	−1.64080	−0.0670973
$599$	−26.1745	−1.06946	−0.534731	−	0.845022i	$-0.679587\pi$
$599$	−26.1745	−1.06946	−0.534731	+	0.845022i	$0.679587\pi$
$600$	0	0
$601$	−33.4872	−1.36597	−0.682986	−	0.730432i	$-0.739319\pi$
$601$	−33.4872	−1.36597	−0.682986	+	0.730432i	$0.739319\pi$
$602$	−21.2322	−0.865361
$603$	0	0
$604$	−8.44910	−0.343789
$605$	34.7041	1.41092
$606$	0	0
$607$	29.6994	1.20546	0.602731	−	0.797944i	$-0.294079\pi$
$607$	29.6994	1.20546	0.602731	+	0.797944i	$0.294079\pi$
$608$	1.58835	0.0644162
$609$	0	0
$610$	49.1834	1.99138
$611$	−4.22892	−0.171084
$612$	0	0
$613$	−0.157755	−0.00637168	−0.00318584	−	0.999995i	$-0.501014\pi$
$613$	−0.157755	−0.00637168	−0.00318584	+	0.999995i	$0.501014\pi$
$614$	−8.85754	−0.357461
$615$	0	0
$616$	−4.59377	−0.185088
$617$	23.7253	0.955143	0.477572	−	0.878593i	$-0.341517\pi$
$617$	23.7253	0.955143	0.477572	+	0.878593i	$0.341517\pi$
$618$	0	0
$619$	5.82368	0.234073	0.117037	−	0.993128i	$-0.462660\pi$
$619$	5.82368	0.234073	0.117037	+	0.993128i	$0.462660\pi$
$620$	−17.0669	−0.685423
$621$	0	0
$622$	−25.2036	−1.01057
$623$	−59.8906	−2.39947
$624$	0	0
$625$	−2.64449	−0.105780
$626$	25.4110	1.01563
$627$	0	0
$628$	7.38323	0.294623
$629$	−37.9225	−1.51207
$630$	0	0
$631$	38.9996	1.55255	0.776275	−	0.630394i	$-0.217107\pi$
$631$	38.9996	1.55255	0.776275	+	0.630394i	$0.217107\pi$
$632$	−2.56890	−0.102185
$633$	0	0
$634$	−13.2481	−0.526149
$635$	−4.03466	−0.160111
$636$	0	0
$637$	6.65041	0.263499
$638$	2.75970	0.109258
$639$	0	0
$640$	−3.58447	−0.141689
$641$	15.5069	0.612487	0.306243	−	0.951953i	$-0.400928\pi$
$641$	15.5069	0.612487	0.306243	+	0.951953i	$0.400928\pi$
$642$	0	0
$643$	−4.61504	−0.181999	−0.0909997	−	0.995851i	$-0.529006\pi$
$643$	−4.61504	−0.181999	−0.0909997	+	0.995851i	$0.529006\pi$
$644$	8.89295	0.350431
$645$	0	0
$646$	6.66159	0.262097
$647$	4.58826	0.180383	0.0901916	−	0.995924i	$-0.471252\pi$
$647$	4.58826	0.180383	0.0901916	+	0.995924i	$0.471252\pi$
$648$	0	0
$649$	7.41444	0.291042
$650$	5.79388	0.227255
$651$	0	0
$652$	−3.51001	−0.137463
$653$	33.0317	1.29263	0.646316	−	0.763070i	$-0.276309\pi$
$653$	33.0317	1.29263	0.646316	+	0.763070i	$0.276309\pi$
$654$	0	0
$655$	28.3584	1.10805
$656$	3.19103	0.124589
$657$	0	0
$658$	22.9203	0.893525
$659$	29.9577	1.16699	0.583493	−	0.812118i	$-0.301686\pi$
$659$	29.9577	1.16699	0.583493	+	0.812118i	$0.301686\pi$
$660$	0	0
$661$	48.0296	1.86813	0.934067	−	0.357097i	$-0.116233\pi$
$661$	48.0296	1.86813	0.934067	+	0.357097i	$0.116233\pi$
$662$	−19.1025	−0.742440
$663$	0	0
$664$	−9.66114	−0.374925
$665$	22.7797	0.883361
$666$	0	0
$667$	−5.34243	−0.206860
$668$	16.5643	0.640892
$669$	0	0
$670$	−22.7504	−0.878924
$671$	−15.7538	−0.608169
$672$	0	0
$673$	−43.3732	−1.67192	−0.835958	−	0.548794i	$-0.815087\pi$
$673$	−43.3732	−1.67192	−0.835958	+	0.548794i	$0.815087\pi$
$674$	−10.4574	−0.402803
$675$	0	0
$676$	−12.4550	−0.479039
$677$	13.9188	0.534943	0.267471	−	0.963566i	$-0.413812\pi$
$677$	13.9188	0.534943	0.267471	+	0.963566i	$0.413812\pi$
$678$	0	0
$679$	21.3598	0.819714
$680$	−15.0334	−0.576503
$681$	0	0
$682$	5.46665	0.209329
$683$	−23.7919	−0.910371	−0.455186	−	0.890397i	$-0.650427\pi$
$683$	−23.7919	−0.910371	−0.455186	+	0.890397i	$0.650427\pi$
$684$	0	0
$685$	24.3741	0.931288
$686$	−8.03685	−0.306848
$687$	0	0
$688$	5.30662	0.202313
$689$	−0.527633	−0.0201012
$690$	0	0
$691$	32.5022	1.23644	0.618221	−	0.786004i	$-0.287854\pi$
$691$	32.5022	1.23644	0.618221	+	0.786004i	$0.287854\pi$
$692$	−3.22617	−0.122641
$693$	0	0
$694$	−8.52699	−0.323680
$695$	72.7748	2.76051
$696$	0	0
$697$	13.3833	0.506927
$698$	−34.3510	−1.30021
$699$	0	0
$700$	−31.4021	−1.18689
$701$	−13.9219	−0.525823	−0.262912	−	0.964820i	$-0.584683\pi$
$701$	−13.9219	−0.525823	−0.262912	+	0.964820i	$0.584683\pi$
$702$	0	0
$703$	−14.3619	−0.541669
$704$	1.14813	0.0432718
$705$	0	0
$706$	−8.96344	−0.337344
$707$	32.4246	1.21945
$708$	0	0
$709$	−48.1002	−1.80644	−0.903220	−	0.429179i	$-0.858803\pi$
$709$	−48.1002	−1.80644	−0.903220	+	0.429179i	$0.858803\pi$
$710$	15.0358	0.564285
$711$	0	0
$712$	14.9686	0.560972
$713$	−10.5827	−0.396326
$714$	0	0
$715$	−3.03811	−0.113619
$716$	−4.58748	−0.171442
$717$	0	0
$718$	25.1520	0.938664
$719$	−5.67089	−0.211489	−0.105744	−	0.994393i	$-0.533723\pi$
$719$	−5.67089	−0.211489	−0.105744	+	0.994393i	$0.533723\pi$
$720$	0	0
$721$	33.2133	1.23693
$722$	−16.4771	−0.613216
$723$	0	0
$724$	−19.3179	−0.717942
$725$	18.8648	0.700621
$726$	0	0
$727$	24.4806	0.907934	0.453967	−	0.891019i	$-0.350008\pi$
$727$	24.4806	0.907934	0.453967	+	0.891019i	$0.350008\pi$
$728$	−2.95369	−0.109471
$729$	0	0
$730$	−26.7227	−0.989053
$731$	22.2561	0.823173
$732$	0	0
$733$	−32.8568	−1.21359	−0.606796	−	0.794858i	$-0.707546\pi$
$733$	−32.8568	−1.21359	−0.606796	+	0.794858i	$0.707546\pi$
$734$	−10.3047	−0.380355
$735$	0	0
$736$	−2.22264	−0.0819274
$737$	7.28712	0.268424
$738$	0	0
$739$	21.5953	0.794398	0.397199	−	0.917733i	$-0.369982\pi$
$739$	21.5953	0.794398	0.397199	+	0.917733i	$0.369982\pi$
$740$	32.4108	1.19145
$741$	0	0
$742$	2.85971	0.104983
$743$	−4.67353	−0.171455	−0.0857275	−	0.996319i	$-0.527321\pi$
$743$	−4.67353	−0.171455	−0.0857275	+	0.996319i	$0.527321\pi$
$744$	0	0
$745$	−3.58447	−0.131325
$746$	26.0716	0.954548
$747$	0	0
$748$	4.81530	0.176065
$749$	−21.1632	−0.773286
$750$	0	0
$751$	20.2100	0.737472	0.368736	−	0.929534i	$-0.379791\pi$
$751$	20.2100	0.737472	0.368736	+	0.929534i	$0.379791\pi$
$752$	−5.72851	−0.208897
$753$	0	0
$754$	1.77443	0.0646208
$755$	30.2855	1.10220
$756$	0	0
$757$	−6.18270	−0.224714	−0.112357	−	0.993668i	$-0.535840\pi$
$757$	−6.18270	−0.224714	−0.112357	+	0.993668i	$0.535840\pi$
$758$	−12.2436	−0.444707
$759$	0	0
$760$	−5.69339	−0.206521
$761$	−31.5581	−1.14398	−0.571990	−	0.820260i	$-0.693829\pi$
$761$	−31.5581	−1.14398	−0.571990	+	0.820260i	$0.693829\pi$
$762$	0	0
$763$	−2.97187	−0.107589
$764$	14.5959	0.528059
$765$	0	0
$766$	−16.6460	−0.601443
$767$	4.76732	0.172138
$768$	0	0
$769$	−19.3756	−0.698703	−0.349352	−	0.936992i	$-0.613598\pi$
$769$	−19.3756	−0.698703	−0.349352	+	0.936992i	$0.613598\pi$
$770$	16.4662	0.593401
$771$	0	0
$772$	−20.6018	−0.741476
$773$	50.9935	1.83411	0.917054	−	0.398763i	$-0.130560\pi$
$773$	50.9935	1.83411	0.917054	+	0.398763i	$0.130560\pi$
$774$	0	0
$775$	37.3690	1.34233
$776$	−5.33851	−0.191641
$777$	0	0
$778$	−24.1756	−0.866736
$779$	5.06847	0.181597
$780$	0	0
$781$	−4.81609	−0.172333
$782$	−9.32180	−0.333347
$783$	0	0
$784$	9.00867	0.321738
$785$	−26.4650	−0.944575
$786$	0	0
$787$	−43.0364	−1.53408	−0.767041	−	0.641599i	$-0.778272\pi$
$787$	−43.0364	−1.53408	−0.767041	+	0.641599i	$0.778272\pi$
$788$	6.73218	0.239824
$789$	0	0
$790$	9.20815	0.327611
$791$	2.32354	0.0826155
$792$	0	0
$793$	−10.1294	−0.359704
$794$	26.5827	0.943384
$795$	0	0
$796$	−7.92167	−0.280776
$797$	16.5275	0.585433	0.292716	−	0.956199i	$-0.405441\pi$
$797$	16.5275	0.585433	0.292716	+	0.956199i	$0.405441\pi$
$798$	0	0
$799$	−24.0256	−0.849963
$800$	7.84841	0.277483
$801$	0	0
$802$	19.3434	0.683041
$803$	8.55949	0.302058
$804$	0	0
$805$	−31.8765	−1.12350
$806$	3.51493	0.123808
$807$	0	0
$808$	−8.10397	−0.285097
$809$	−15.8718	−0.558024	−0.279012	−	0.960288i	$-0.590007\pi$
$809$	−15.8718	−0.558024	−0.279012	+	0.960288i	$0.590007\pi$
$810$	0	0
$811$	−21.3824	−0.750836	−0.375418	−	0.926856i	$-0.622501\pi$
$811$	−21.3824	−0.750836	−0.375418	+	0.926856i	$0.622501\pi$
$812$	−9.61719	−0.337497
$813$	0	0
$814$	−10.3814	−0.363869
$815$	12.5815	0.440712
$816$	0	0
$817$	8.42878	0.294886
$818$	−9.33365	−0.326344
$819$	0	0
$820$	−11.4381	−0.399437
$821$	10.8762	0.379581	0.189790	−	0.981825i	$-0.439219\pi$
$821$	10.8762	0.379581	0.189790	+	0.981825i	$0.439219\pi$
$822$	0	0
$823$	15.4775	0.539512	0.269756	−	0.962929i	$-0.413057\pi$
$823$	15.4775	0.539512	0.269756	+	0.962929i	$0.413057\pi$
$824$	−8.30107	−0.289181
$825$	0	0
$826$	−25.8383	−0.899030
$827$	−32.4509	−1.12843	−0.564213	−	0.825629i	$-0.690820\pi$
$827$	−32.4509	−1.12843	−0.564213	+	0.825629i	$0.690820\pi$
$828$	0	0
$829$	−25.3518	−0.880505	−0.440252	−	0.897874i	$-0.645111\pi$
$829$	−25.3518	−0.880505	−0.440252	+	0.897874i	$0.645111\pi$
$830$	34.6300	1.20203
$831$	0	0
$832$	0.738223	0.0255933
$833$	37.7826	1.30909
$834$	0	0
$835$	−59.3742	−2.05473
$836$	1.82364	0.0630718
$837$	0	0
$838$	−19.2809	−0.666046
$839$	−5.53227	−0.190995	−0.0954976	−	0.995430i	$-0.530444\pi$
$839$	−5.53227	−0.190995	−0.0954976	+	0.995430i	$0.530444\pi$
$840$	0	0
$841$	−23.2225	−0.800775
$842$	−16.6428	−0.573547
$843$	0	0
$844$	−20.4048	−0.702363
$845$	44.6446	1.53582
$846$	0	0
$847$	38.7377	1.33104
$848$	−0.714734	−0.0245441
$849$	0	0
$850$	32.9165	1.12903
$851$	20.0971	0.688920
$852$	0	0
$853$	−38.3426	−1.31283	−0.656413	−	0.754401i	$-0.727927\pi$
$853$	−38.3426	−1.31283	−0.656413	+	0.754401i	$0.727927\pi$
$854$	54.8999	1.87864
$855$	0	0
$856$	5.28936	0.180787
$857$	30.5593	1.04388	0.521942	−	0.852981i	$-0.325208\pi$
$857$	30.5593	1.04388	0.521942	+	0.852981i	$0.325208\pi$
$858$	0	0
$859$	22.8135	0.778388	0.389194	−	0.921156i	$-0.372754\pi$
$859$	22.8135	0.778388	0.389194	+	0.921156i	$0.372754\pi$
$860$	−19.0214	−0.648625
$861$	0	0
$862$	28.6125	0.974544
$863$	33.1507	1.12846	0.564232	−	0.825616i	$-0.309172\pi$
$863$	33.1507	1.12846	0.564232	+	0.825616i	$0.309172\pi$
$864$	0	0
$865$	11.5641	0.393192
$866$	29.7265	1.01015
$867$	0	0
$868$	−19.0505	−0.646617
$869$	−2.94944	−0.100053
$870$	0	0
$871$	4.68545	0.158761
$872$	0.742766	0.0251532
$873$	0	0
$874$	−3.53033	−0.119415
$875$	40.8512	1.38102
$876$	0	0
$877$	10.8999	0.368064	0.184032	−	0.982920i	$-0.441085\pi$
$877$	10.8999	0.368064	0.184032	+	0.982920i	$0.441085\pi$
$878$	0.359854	0.0121445
$879$	0	0
$880$	−4.11544	−0.138731
$881$	−18.6624	−0.628751	−0.314375	−	0.949299i	$-0.601795\pi$
$881$	−18.6624	−0.628751	−0.314375	+	0.949299i	$0.601795\pi$
$882$	0	0
$883$	−25.5953	−0.861350	−0.430675	−	0.902507i	$-0.641724\pi$
$883$	−25.5953	−0.861350	−0.430675	+	0.902507i	$0.641724\pi$
$884$	3.09613	0.104134
$885$	0	0
$886$	−25.6758	−0.862595
$887$	32.4241	1.08869	0.544347	−	0.838860i	$-0.316777\pi$
$887$	32.4241	1.08869	0.544347	+	0.838860i	$0.316777\pi$
$888$	0	0
$889$	−4.50360	−0.151046
$890$	−53.6545	−1.79850
$891$	0	0
$892$	−6.23200	−0.208663
$893$	−9.09889	−0.304483
$894$	0	0
$895$	16.4437	0.549651
$896$	−4.00108	−0.133667
$897$	0	0
$898$	−32.8395	−1.09587
$899$	11.4446	0.381698
$900$	0	0
$901$	−2.99762	−0.0998651
$902$	3.66372	0.121988
$903$	0	0
$904$	−0.580727	−0.0193147
$905$	69.2442	2.30176
$906$	0	0
$907$	−8.53057	−0.283253	−0.141626	−	0.989920i	$-0.545233\pi$
$907$	−8.53057	−0.283253	−0.141626	+	0.989920i	$0.545233\pi$
$908$	1.03841	0.0344610
$909$	0	0
$910$	10.5874	0.350969
$911$	−18.3960	−0.609487	−0.304743	−	0.952435i	$-0.598571\pi$
$911$	−18.3960	−0.609487	−0.304743	+	0.952435i	$0.598571\pi$
$912$	0	0
$913$	−11.0923	−0.367100
$914$	18.6648	0.617376
$915$	0	0
$916$	9.32231	0.308018
$917$	31.6544	1.04532
$918$	0	0
$919$	10.2944	0.339583	0.169791	−	0.985480i	$-0.445691\pi$
$919$	10.2944	0.339583	0.169791	+	0.985480i	$0.445691\pi$
$920$	7.96697	0.262663
$921$	0	0
$922$	36.1601	1.19087
$923$	−3.09664	−0.101927
$924$	0	0
$925$	−70.9655	−2.33333
$926$	3.07692	0.101114
$927$	0	0
$928$	2.40365	0.0789035
$929$	−34.3854	−1.12815	−0.564075	−	0.825724i	$-0.690767\pi$
$929$	−34.3854	−1.12815	−0.564075	+	0.825724i	$0.690767\pi$
$930$	0	0
$931$	14.3089	0.468956
$932$	24.0216	0.786854
$933$	0	0
$934$	32.1937	1.05341
$935$	−17.2603	−0.564472
$936$	0	0
$937$	−7.80306	−0.254915	−0.127457	−	0.991844i	$-0.540682\pi$
$937$	−7.80306	−0.254915	−0.127457	+	0.991844i	$0.540682\pi$
$938$	−25.3946	−0.829164
$939$	0	0
$940$	20.5337	0.669735
$941$	11.0486	0.360173	0.180087	−	0.983651i	$-0.442362\pi$
$941$	11.0486	0.360173	0.180087	+	0.983651i	$0.442362\pi$
$942$	0	0
$943$	−7.09249	−0.230963
$944$	6.45783	0.210184
$945$	0	0
$946$	6.09270	0.198091
$947$	−35.1164	−1.14113	−0.570565	−	0.821253i	$-0.693276\pi$
$947$	−35.1164	−1.14113	−0.570565	+	0.821253i	$0.693276\pi$
$948$	0	0
$949$	5.50356	0.178653
$950$	12.4660	0.404452
$951$	0	0
$952$	−16.7807	−0.543864
$953$	22.4676	0.727797	0.363899	−	0.931439i	$-0.381445\pi$
$953$	22.4676	0.727797	0.363899	+	0.931439i	$0.381445\pi$
$954$	0	0
$955$	−52.3184	−1.69298
$956$	−23.2098	−0.750658
$957$	0	0
$958$	−3.55290	−0.114789
$959$	27.2071	0.878563
$960$	0	0
$961$	−8.32960	−0.268697
$962$	−6.67503	−0.215211
$963$	0	0
$964$	−4.67539	−0.150584
$965$	73.8466	2.37721
$966$	0	0
$967$	9.62397	0.309486	0.154743	−	0.987955i	$-0.450545\pi$
$967$	9.62397	0.309486	0.154743	+	0.987955i	$0.450545\pi$
$968$	−9.68179	−0.311185
$969$	0	0
$970$	19.1357	0.614410
$971$	−15.3226	−0.491725	−0.245863	−	0.969305i	$-0.579071\pi$
$971$	−15.3226	−0.491725	−0.245863	+	0.969305i	$0.579071\pi$
$972$	0	0
$973$	81.2333	2.60422
$974$	−5.94156	−0.190380
$975$	0	0
$976$	−13.7213	−0.439207
$977$	−21.1806	−0.677629	−0.338814	−	0.940853i	$-0.610026\pi$
$977$	−21.1806	−0.677629	−0.338814	+	0.940853i	$0.610026\pi$
$978$	0	0
$979$	17.1859	0.549264
$980$	−32.2913	−1.03151
$981$	0	0
$982$	−0.576627	−0.0184009
$983$	41.8433	1.33459	0.667297	−	0.744792i	$-0.267451\pi$
$983$	41.8433	1.33459	0.667297	+	0.744792i	$0.267451\pi$
$984$	0	0
$985$	−24.1313	−0.768886
$986$	10.0810	0.321043
$987$	0	0
$988$	1.17256	0.0373040
$989$	−11.7947	−0.375049
$990$	0	0
$991$	−48.9591	−1.55524	−0.777618	−	0.628736i	$-0.783572\pi$
$991$	−48.9591	−1.55524	−0.777618	+	0.628736i	$0.783572\pi$
$992$	4.76134	0.151173
$993$	0	0
$994$	16.7834	0.532338
$995$	28.3950	0.900181
$996$	0	0
$997$	−20.1156	−0.637067	−0.318533	−	0.947912i	$-0.603190\pi$
$997$	−20.1156	−0.637067	−0.318533	+	0.947912i	$0.603190\pi$
$998$	−1.31313	−0.0415665
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.f.1.1	yes	8
3.2	odd	2		8046.2.a.e.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.e.1.8	✓	8	3.2	odd	2
8046.2.a.f.1.1	yes	8	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.58447 q^{5} -4.00108 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.58447 q^{5} -4.00108 q^{7} +1.00000 q^{8} -3.58447 q^{10} +1.14813 q^{11} +0.738223 q^{13} -4.00108 q^{14} +1.00000 q^{16} +4.19403 q^{17} +1.58835 q^{19} -3.58447 q^{20} +1.14813 q^{22} -2.22264 q^{23} +7.84841 q^{25} +0.738223 q^{26} -4.00108 q^{28} +2.40365 q^{29} +4.76134 q^{31} +1.00000 q^{32} +4.19403 q^{34} +14.3418 q^{35} -9.04202 q^{37} +1.58835 q^{38} -3.58447 q^{40} +3.19103 q^{41} +5.30662 q^{43} +1.14813 q^{44} -2.22264 q^{46} -5.72851 q^{47} +9.00867 q^{49} +7.84841 q^{50} +0.738223 q^{52} -0.714734 q^{53} -4.11544 q^{55} -4.00108 q^{56} +2.40365 q^{58} +6.45783 q^{59} -13.7213 q^{61} +4.76134 q^{62} +1.00000 q^{64} -2.64614 q^{65} +6.34694 q^{67} +4.19403 q^{68} +14.3418 q^{70} -4.19472 q^{71} +7.45515 q^{73} -9.04202 q^{74} +1.58835 q^{76} -4.59377 q^{77} -2.56890 q^{79} -3.58447 q^{80} +3.19103 q^{82} -9.66114 q^{83} -15.0334 q^{85} +5.30662 q^{86} +1.14813 q^{88} +14.9686 q^{89} -2.95369 q^{91} -2.22264 q^{92} -5.72851 q^{94} -5.69339 q^{95} -5.33851 q^{97} +9.00867 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8	\( 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8} - 6 q^{11} - 8 q^{13} - 5 q^{14} + 8 q^{16} + 5 q^{17} - 14 q^{19} - 6 q^{22} - 21 q^{23} - 6 q^{25} - 8 q^{26} - 5 q^{28} - 3 q^{29} - 4 q^{31} + 8 q^{32} + 5 q^{34} + 2 q^{35} - 3 q^{37} - 14 q^{38} - 7 q^{41} - 12 q^{43} - 6 q^{44} - 21 q^{46} - 25 q^{47} - 7 q^{49} - 6 q^{50} - 8 q^{52} - 3 q^{53} - 9 q^{55} - 5 q^{56} - 3 q^{58} - 2 q^{59} - 17 q^{61} - 4 q^{62} + 8 q^{64} - 32 q^{65} - 14 q^{67} + 5 q^{68} + 2 q^{70} - 7 q^{71} - 10 q^{73} - 3 q^{74} - 14 q^{76} - 12 q^{77} - 33 q^{79} - 7 q^{82} - 13 q^{83} - 33 q^{85} - 12 q^{86} - 6 q^{88} + 22 q^{89} - 22 q^{91} - 21 q^{92} - 25 q^{94} + 14 q^{95} - 11 q^{97} - 7 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8 - 6 * q^11 - 8 * q^13 - 5 * q^14 + 8 * q^16 + 5 * q^17 - 14 * q^19 - 6 * q^22 - 21 * q^23 - 6 * q^25 - 8 * q^26 - 5 * q^28 - 3 * q^29 - 4 * q^31 + 8 * q^32 + 5 * q^34 + 2 * q^35 - 3 * q^37 - 14 * q^38 - 7 * q^41 - 12 * q^43 - 6 * q^44 - 21 * q^46 - 25 * q^47 - 7 * q^49 - 6 * q^50 - 8 * q^52 - 3 * q^53 - 9 * q^55 - 5 * q^56 - 3 * q^58 - 2 * q^59 - 17 * q^61 - 4 * q^62 + 8 * q^64 - 32 * q^65 - 14 * q^67 + 5 * q^68 + 2 * q^70 - 7 * q^71 - 10 * q^73 - 3 * q^74 - 14 * q^76 - 12 * q^77 - 33 * q^79 - 7 * q^82 - 13 * q^83 - 33 * q^85 - 12 * q^86 - 6 * q^88 + 22 * q^89 - 22 * q^91 - 21 * q^92 - 25 * q^94 + 14 * q^95 - 11 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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