LMFDB - Embedded newform 8046.2.a.n.1.12

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 31 x^{10} + 82 x^{9} + 334 x^{8} - 684 x^{7} - 1561 x^{6} + 1551 x^{5} + 3573 x^{4} + \cdots - 3 $ x^12 - 3x^11 - 31x^10 + 82x^9 + 334x^8 - 684x^7 - 1561x^6 + 1551x^5 + 3573x^4 + 345x^3 - 1607x^2 - 594*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-3.46379$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.46379 q^{5} -3.72396 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.46379 q^{5} -3.72396 q^{7} +1.00000 q^{8} +3.46379 q^{10} -5.41125 q^{11} +5.45430 q^{13} -3.72396 q^{14} +1.00000 q^{16} -5.94735 q^{17} +5.41154 q^{19} +3.46379 q^{20} -5.41125 q^{22} -4.72212 q^{23} +6.99784 q^{25} +5.45430 q^{26} -3.72396 q^{28} -5.68434 q^{29} +4.89421 q^{31} +1.00000 q^{32} -5.94735 q^{34} -12.8990 q^{35} -10.8128 q^{37} +5.41154 q^{38} +3.46379 q^{40} -10.6945 q^{41} -12.8009 q^{43} -5.41125 q^{44} -4.72212 q^{46} -8.22990 q^{47} +6.86788 q^{49} +6.99784 q^{50} +5.45430 q^{52} +11.8187 q^{53} -18.7434 q^{55} -3.72396 q^{56} -5.68434 q^{58} -6.31276 q^{59} +10.9822 q^{61} +4.89421 q^{62} +1.00000 q^{64} +18.8926 q^{65} -8.30144 q^{67} -5.94735 q^{68} -12.8990 q^{70} -0.883937 q^{71} +6.87933 q^{73} -10.8128 q^{74} +5.41154 q^{76} +20.1513 q^{77} +2.27886 q^{79} +3.46379 q^{80} -10.6945 q^{82} -4.53459 q^{83} -20.6004 q^{85} -12.8009 q^{86} -5.41125 q^{88} -1.46095 q^{89} -20.3116 q^{91} -4.72212 q^{92} -8.22990 q^{94} +18.7444 q^{95} +18.1261 q^{97} +6.86788 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8} - 3 q^{10} - 14 q^{11} - 3 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} - 4 q^{19} - 3 q^{20} - 14 q^{22} - 13 q^{23} + 11 q^{25} - 3 q^{26} - 6 q^{28} - 23 q^{29} - 14 q^{31} + 12 q^{32} - 8 q^{34} - 32 q^{35} - 19 q^{37} - 4 q^{38} - 3 q^{40} - 30 q^{41} - 15 q^{43} - 14 q^{44} - 13 q^{46} + q^{47} + 14 q^{49} + 11 q^{50} - 3 q^{52} - 16 q^{53} - 7 q^{55} - 6 q^{56} - 23 q^{58} - 26 q^{59} - 16 q^{61} - 14 q^{62} + 12 q^{64} - 8 q^{65} - 39 q^{67} - 8 q^{68} - 32 q^{70} - 15 q^{71} - 2 q^{73} - 19 q^{74} - 4 q^{76} - 34 q^{77} - 13 q^{79} - 3 q^{80} - 30 q^{82} - 6 q^{83} - 11 q^{85} - 15 q^{86} - 14 q^{88} - 18 q^{89} - 35 q^{91} - 13 q^{92} + q^{94} - 51 q^{95} + 19 q^{97} + 14 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8 - 3 * q^10 - 14 * q^11 - 3 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 - 4 * q^19 - 3 * q^20 - 14 * q^22 - 13 * q^23 + 11 * q^25 - 3 * q^26 - 6 * q^28 - 23 * q^29 - 14 * q^31 + 12 * q^32 - 8 * q^34 - 32 * q^35 - 19 * q^37 - 4 * q^38 - 3 * q^40 - 30 * q^41 - 15 * q^43 - 14 * q^44 - 13 * q^46 + q^47 + 14 * q^49 + 11 * q^50 - 3 * q^52 - 16 * q^53 - 7 * q^55 - 6 * q^56 - 23 * q^58 - 26 * q^59 - 16 * q^61 - 14 * q^62 + 12 * q^64 - 8 * q^65 - 39 * q^67 - 8 * q^68 - 32 * q^70 - 15 * q^71 - 2 * q^73 - 19 * q^74 - 4 * q^76 - 34 * q^77 - 13 * q^79 - 3 * q^80 - 30 * q^82 - 6 * q^83 - 11 * q^85 - 15 * q^86 - 14 * q^88 - 18 * q^89 - 35 * q^91 - 13 * q^92 + q^94 - 51 * q^95 + 19 * q^97 + 14 * 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
$5$	3.46379	1.54905	0.774527	−	0.632541i	$-0.217988\pi$
$5$	3.46379	1.54905	0.774527	+	0.632541i	$0.217988\pi$
$6$	0	0
$7$	−3.72396	−1.40752	−0.703762	−	0.710436i	$-0.748498\pi$
$7$	−3.72396	−1.40752	−0.703762	+	0.710436i	$0.748498\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.46379	1.09535
$11$	−5.41125	−1.63155	−0.815777	−	0.578367i	$-0.803690\pi$
$11$	−5.41125	−1.63155	−0.815777	+	0.578367i	$0.803690\pi$
$12$	0	0
$13$	5.45430	1.51275	0.756375	−	0.654138i	$-0.226968\pi$
$13$	5.45430	1.51275	0.756375	+	0.654138i	$0.226968\pi$
$14$	−3.72396	−0.995270
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.94735	−1.44245	−0.721223	−	0.692703i	$-0.756420\pi$
$17$	−5.94735	−1.44245	−0.721223	+	0.692703i	$0.756420\pi$
$18$	0	0
$19$	5.41154	1.24149	0.620746	−	0.784012i	$-0.286830\pi$
$19$	5.41154	1.24149	0.620746	+	0.784012i	$0.286830\pi$
$20$	3.46379	0.774527
$21$	0	0
$22$	−5.41125	−1.15368
$23$	−4.72212	−0.984631	−0.492316	−	0.870417i	$-0.663849\pi$
$23$	−4.72212	−0.984631	−0.492316	+	0.870417i	$0.663849\pi$
$24$	0	0
$25$	6.99784	1.39957
$26$	5.45430	1.06968
$27$	0	0
$28$	−3.72396	−0.703762
$29$	−5.68434	−1.05556	−0.527778	−	0.849382i	$-0.676975\pi$
$29$	−5.68434	−1.05556	−0.527778	+	0.849382i	$0.676975\pi$
$30$	0	0
$31$	4.89421	0.879026	0.439513	−	0.898236i	$-0.355151\pi$
$31$	4.89421	0.879026	0.439513	+	0.898236i	$0.355151\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.94735	−1.01996
$35$	−12.8990	−2.18033
$36$	0	0
$37$	−10.8128	−1.77761	−0.888807	−	0.458281i	$-0.848465\pi$
$37$	−10.8128	−1.77761	−0.888807	+	0.458281i	$0.848465\pi$
$38$	5.41154	0.877868
$39$	0	0
$40$	3.46379	0.547673
$41$	−10.6945	−1.67020	−0.835101	−	0.550097i	$-0.814591\pi$
$41$	−10.6945	−1.67020	−0.835101	+	0.550097i	$0.814591\pi$
$42$	0	0
$43$	−12.8009	−1.95211	−0.976057	−	0.217514i	$-0.930205\pi$
$43$	−12.8009	−1.95211	−0.976057	+	0.217514i	$0.930205\pi$
$44$	−5.41125	−0.815777
$45$	0	0
$46$	−4.72212	−0.696239
$47$	−8.22990	−1.20045	−0.600227	−	0.799829i	$-0.704923\pi$
$47$	−8.22990	−1.20045	−0.600227	+	0.799829i	$0.704923\pi$
$48$	0	0
$49$	6.86788	0.981125
$50$	6.99784	0.989645
$51$	0	0
$52$	5.45430	0.756375
$53$	11.8187	1.62343	0.811714	−	0.584055i	$-0.198535\pi$
$53$	11.8187	1.62343	0.811714	+	0.584055i	$0.198535\pi$
$54$	0	0
$55$	−18.7434	−2.52737
$56$	−3.72396	−0.497635
$57$	0	0
$58$	−5.68434	−0.746391
$59$	−6.31276	−0.821851	−0.410926	−	0.911669i	$-0.634794\pi$
$59$	−6.31276	−0.821851	−0.410926	+	0.911669i	$0.634794\pi$
$60$	0	0
$61$	10.9822	1.40613	0.703063	−	0.711127i	$-0.251815\pi$
$61$	10.9822	1.40613	0.703063	+	0.711127i	$0.251815\pi$
$62$	4.89421	0.621565
$63$	0	0
$64$	1.00000	0.125000
$65$	18.8926	2.34333
$66$	0	0
$67$	−8.30144	−1.01418	−0.507091	−	0.861892i	$-0.669279\pi$
$67$	−8.30144	−1.01418	−0.507091	+	0.861892i	$0.669279\pi$
$68$	−5.94735	−0.721223
$69$	0	0
$70$	−12.8990	−1.54173
$71$	−0.883937	−0.104904	−0.0524520	−	0.998623i	$-0.516704\pi$
$71$	−0.883937	−0.104904	−0.0524520	+	0.998623i	$0.516704\pi$
$72$	0	0
$73$	6.87933	0.805164	0.402582	−	0.915384i	$-0.368113\pi$
$73$	6.87933	0.805164	0.402582	+	0.915384i	$0.368113\pi$
$74$	−10.8128	−1.25696
$75$	0	0
$76$	5.41154	0.620746
$77$	20.1513	2.29645
$78$	0	0
$79$	2.27886	0.256392	0.128196	−	0.991749i	$-0.459081\pi$
$79$	2.27886	0.256392	0.128196	+	0.991749i	$0.459081\pi$
$80$	3.46379	0.387264
$81$	0	0
$82$	−10.6945	−1.18101
$83$	−4.53459	−0.497736	−0.248868	−	0.968537i	$-0.580059\pi$
$83$	−4.53459	−0.497736	−0.248868	+	0.968537i	$0.580059\pi$
$84$	0	0
$85$	−20.6004	−2.23443
$86$	−12.8009	−1.38035
$87$	0	0
$88$	−5.41125	−0.576841
$89$	−1.46095	−0.154860	−0.0774301	−	0.996998i	$-0.524671\pi$
$89$	−1.46095	−0.154860	−0.0774301	+	0.996998i	$0.524671\pi$
$90$	0	0
$91$	−20.3116	−2.12923
$92$	−4.72212	−0.492316
$93$	0	0
$94$	−8.22990	−0.848850
$95$	18.7444	1.92314
$96$	0	0
$97$	18.1261	1.84043	0.920213	−	0.391419i	$-0.128016\pi$
$97$	18.1261	1.84043	0.920213	+	0.391419i	$0.128016\pi$
$98$	6.86788	0.693760
$99$	0	0
$100$	6.99784	0.699784
$101$	−7.96826	−0.792871	−0.396436	−	0.918063i	$-0.629753\pi$
$101$	−7.96826	−0.792871	−0.396436	+	0.918063i	$0.629753\pi$
$102$	0	0
$103$	11.8574	1.16835	0.584174	−	0.811628i	$-0.301419\pi$
$103$	11.8574	1.16835	0.584174	+	0.811628i	$0.301419\pi$
$104$	5.45430	0.534838
$105$	0	0
$106$	11.8187	1.14794
$107$	−4.13345	−0.399596	−0.199798	−	0.979837i	$-0.564029\pi$
$107$	−4.13345	−0.399596	−0.199798	+	0.979837i	$0.564029\pi$
$108$	0	0
$109$	−16.8105	−1.61015	−0.805077	−	0.593170i	$-0.797876\pi$
$109$	−16.8105	−1.61015	−0.805077	+	0.593170i	$0.797876\pi$
$110$	−18.7434	−1.78712
$111$	0	0
$112$	−3.72396	−0.351881
$113$	9.43021	0.887120	0.443560	−	0.896245i	$-0.353715\pi$
$113$	9.43021	0.887120	0.443560	+	0.896245i	$0.353715\pi$
$114$	0	0
$115$	−16.3564	−1.52525
$116$	−5.68434	−0.527778
$117$	0	0
$118$	−6.31276	−0.581137
$119$	22.1477	2.03028
$120$	0	0
$121$	18.2816	1.66197
$122$	10.9822	0.994282
$123$	0	0
$124$	4.89421	0.439513
$125$	6.92011	0.618954
$126$	0	0
$127$	−0.238039	−0.0211225	−0.0105613	−	0.999944i	$-0.503362\pi$
$127$	−0.238039	−0.0211225	−0.0105613	+	0.999944i	$0.503362\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	18.8926	1.65699
$131$	−0.0242757	−0.00212098	−0.00106049	−	0.999999i	$-0.500338\pi$
$131$	−0.0242757	−0.00212098	−0.00106049	+	0.999999i	$0.500338\pi$
$132$	0	0
$133$	−20.1524	−1.74743
$134$	−8.30144	−0.717135
$135$	0	0
$136$	−5.94735	−0.509981
$137$	2.19280	0.187344	0.0936718	−	0.995603i	$-0.470140\pi$
$137$	2.19280	0.187344	0.0936718	+	0.995603i	$0.470140\pi$
$138$	0	0
$139$	−7.61095	−0.645552	−0.322776	−	0.946475i	$-0.604616\pi$
$139$	−7.61095	−0.645552	−0.322776	+	0.946475i	$0.604616\pi$
$140$	−12.8990	−1.09017
$141$	0	0
$142$	−0.883937	−0.0741784
$143$	−29.5146	−2.46813
$144$	0	0
$145$	−19.6894	−1.63511
$146$	6.87933	0.569337
$147$	0	0
$148$	−10.8128	−0.888807
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	2.74712	0.223558	0.111779	−	0.993733i	$-0.464345\pi$
$151$	2.74712	0.223558	0.111779	+	0.993733i	$0.464345\pi$
$152$	5.41154	0.438934
$153$	0	0
$154$	20.1513	1.62384
$155$	16.9525	1.36166
$156$	0	0
$157$	−11.6087	−0.926478	−0.463239	−	0.886233i	$-0.653313\pi$
$157$	−11.6087	−0.926478	−0.463239	+	0.886233i	$0.653313\pi$
$158$	2.27886	0.181296
$159$	0	0
$160$	3.46379	0.273837
$161$	17.5850	1.38589
$162$	0	0
$163$	−23.2573	−1.82165	−0.910826	−	0.412790i	$-0.864554\pi$
$163$	−23.2573	−1.82165	−0.910826	+	0.412790i	$0.864554\pi$
$164$	−10.6945	−0.835101
$165$	0	0
$166$	−4.53459	−0.351952
$167$	20.1860	1.56204	0.781021	−	0.624504i	$-0.214699\pi$
$167$	20.1860	1.56204	0.781021	+	0.624504i	$0.214699\pi$
$168$	0	0
$169$	16.7494	1.28841
$170$	−20.6004	−1.57998
$171$	0	0
$172$	−12.8009	−0.976057
$173$	−2.95800	−0.224893	−0.112446	−	0.993658i	$-0.535869\pi$
$173$	−2.95800	−0.224893	−0.112446	+	0.993658i	$0.535869\pi$
$174$	0	0
$175$	−26.0597	−1.96993
$176$	−5.41125	−0.407888
$177$	0	0
$178$	−1.46095	−0.109503
$179$	−11.9473	−0.892986	−0.446493	−	0.894787i	$-0.647327\pi$
$179$	−11.9473	−0.892986	−0.446493	+	0.894787i	$0.647327\pi$
$180$	0	0
$181$	−20.7182	−1.53997	−0.769986	−	0.638060i	$-0.779737\pi$
$181$	−20.7182	−1.53997	−0.769986	+	0.638060i	$0.779737\pi$
$182$	−20.3116	−1.50560
$183$	0	0
$184$	−4.72212	−0.348120
$185$	−37.4533	−2.75362
$186$	0	0
$187$	32.1826	2.35343
$188$	−8.22990	−0.600227
$189$	0	0
$190$	18.7444	1.35986
$191$	−17.1850	−1.24346	−0.621731	−	0.783231i	$-0.713570\pi$
$191$	−17.1850	−1.24346	−0.621731	+	0.783231i	$0.713570\pi$
$192$	0	0
$193$	0.843776	0.0607364	0.0303682	−	0.999539i	$-0.490332\pi$
$193$	0.843776	0.0607364	0.0303682	+	0.999539i	$0.490332\pi$
$194$	18.1261	1.30138
$195$	0	0
$196$	6.86788	0.490563
$197$	9.38954	0.668977	0.334489	−	0.942400i	$-0.391436\pi$
$197$	9.38954	0.668977	0.334489	+	0.942400i	$0.391436\pi$
$198$	0	0
$199$	4.08274	0.289418	0.144709	−	0.989474i	$-0.453775\pi$
$199$	4.08274	0.289418	0.144709	+	0.989474i	$0.453775\pi$
$200$	6.99784	0.494822
$201$	0	0
$202$	−7.96826	−0.560645
$203$	21.1683	1.48572
$204$	0	0
$205$	−37.0435	−2.58723
$206$	11.8574	0.826147
$207$	0	0
$208$	5.45430	0.378188
$209$	−29.2832	−2.02556
$210$	0	0
$211$	12.4094	0.854298	0.427149	−	0.904181i	$-0.359518\pi$
$211$	12.4094	0.854298	0.427149	+	0.904181i	$0.359518\pi$
$212$	11.8187	0.811714
$213$	0	0
$214$	−4.13345	−0.282557
$215$	−44.3395	−3.02393
$216$	0	0
$217$	−18.2258	−1.23725
$218$	−16.8105	−1.13855
$219$	0	0
$220$	−18.7434	−1.26368
$221$	−32.4387	−2.18206
$222$	0	0
$223$	−2.60836	−0.174669	−0.0873345	−	0.996179i	$-0.527835\pi$
$223$	−2.60836	−0.174669	−0.0873345	+	0.996179i	$0.527835\pi$
$224$	−3.72396	−0.248818
$225$	0	0
$226$	9.43021	0.627289
$227$	−1.09859	−0.0729158	−0.0364579	−	0.999335i	$-0.511607\pi$
$227$	−1.09859	−0.0729158	−0.0364579	+	0.999335i	$0.511607\pi$
$228$	0	0
$229$	19.5454	1.29160	0.645799	−	0.763508i	$-0.276524\pi$
$229$	19.5454	1.29160	0.645799	+	0.763508i	$0.276524\pi$
$230$	−16.3564	−1.07851
$231$	0	0
$232$	−5.68434	−0.373196
$233$	−22.8074	−1.49416	−0.747081	−	0.664733i	$-0.768545\pi$
$233$	−22.8074	−1.49416	−0.747081	+	0.664733i	$0.768545\pi$
$234$	0	0
$235$	−28.5067	−1.85957
$236$	−6.31276	−0.410926
$237$	0	0
$238$	22.1477	1.43562
$239$	1.17661	0.0761086	0.0380543	−	0.999276i	$-0.487884\pi$
$239$	1.17661	0.0761086	0.0380543	+	0.999276i	$0.487884\pi$
$240$	0	0
$241$	−22.0093	−1.41774	−0.708870	−	0.705339i	$-0.750795\pi$
$241$	−22.0093	−1.41774	−0.708870	+	0.705339i	$0.750795\pi$
$242$	18.2816	1.17519
$243$	0	0
$244$	10.9822	0.703063
$245$	23.7889	1.51982
$246$	0	0
$247$	29.5162	1.87807
$248$	4.89421	0.310783
$249$	0	0
$250$	6.92011	0.437666
$251$	0.340199	0.0214732	0.0107366	−	0.999942i	$-0.496582\pi$
$251$	0.340199	0.0214732	0.0107366	+	0.999942i	$0.496582\pi$
$252$	0	0
$253$	25.5526	1.60648
$254$	−0.238039	−0.0149359
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.40682	−0.337268	−0.168634	−	0.985679i	$-0.553936\pi$
$257$	−5.40682	−0.337268	−0.168634	+	0.985679i	$0.553936\pi$
$258$	0	0
$259$	40.2665	2.50204
$260$	18.8926	1.17167
$261$	0	0
$262$	−0.0242757	−0.00149976
$263$	18.4228	1.13600	0.567999	−	0.823029i	$-0.307718\pi$
$263$	18.4228	1.13600	0.567999	+	0.823029i	$0.307718\pi$
$264$	0	0
$265$	40.9376	2.51478
$266$	−20.1524	−1.23562
$267$	0	0
$268$	−8.30144	−0.507091
$269$	6.50417	0.396566	0.198283	−	0.980145i	$-0.436463\pi$
$269$	6.50417	0.396566	0.198283	+	0.980145i	$0.436463\pi$
$270$	0	0
$271$	−22.0203	−1.33764	−0.668818	−	0.743427i	$-0.733199\pi$
$271$	−22.0203	−1.33764	−0.668818	+	0.743427i	$0.733199\pi$
$272$	−5.94735	−0.360611
$273$	0	0
$274$	2.19280	0.132472
$275$	−37.8671	−2.28347
$276$	0	0
$277$	30.3999	1.82655	0.913276	−	0.407340i	$-0.133544\pi$
$277$	30.3999	1.82655	0.913276	+	0.407340i	$0.133544\pi$
$278$	−7.61095	−0.456474
$279$	0	0
$280$	−12.8990	−0.770864
$281$	−4.00120	−0.238691	−0.119346	−	0.992853i	$-0.538080\pi$
$281$	−4.00120	−0.238691	−0.119346	+	0.992853i	$0.538080\pi$
$282$	0	0
$283$	5.09021	0.302581	0.151291	−	0.988489i	$-0.451657\pi$
$283$	5.09021	0.302581	0.151291	+	0.988489i	$0.451657\pi$
$284$	−0.883937	−0.0524520
$285$	0	0
$286$	−29.5146	−1.74523
$287$	39.8259	2.35085
$288$	0	0
$289$	18.3710	1.08065
$290$	−19.6894	−1.15620
$291$	0	0
$292$	6.87933	0.402582
$293$	2.56800	0.150024	0.0750121	−	0.997183i	$-0.476100\pi$
$293$	2.56800	0.150024	0.0750121	+	0.997183i	$0.476100\pi$
$294$	0	0
$295$	−21.8661	−1.27309
$296$	−10.8128	−0.628482
$297$	0	0
$298$	1.00000	0.0579284
$299$	−25.7559	−1.48950
$300$	0	0
$301$	47.6699	2.74765
$302$	2.74712	0.158079
$303$	0	0
$304$	5.41154	0.310373
$305$	38.0400	2.17817
$306$	0	0
$307$	0.252021	0.0143836	0.00719181	−	0.999974i	$-0.497711\pi$
$307$	0.252021	0.0143836	0.00719181	+	0.999974i	$0.497711\pi$
$308$	20.1513	1.14823
$309$	0	0
$310$	16.9525	0.962838
$311$	3.21065	0.182059	0.0910295	−	0.995848i	$-0.470984\pi$
$311$	3.21065	0.182059	0.0910295	+	0.995848i	$0.470984\pi$
$312$	0	0
$313$	−17.6474	−0.997489	−0.498745	−	0.866749i	$-0.666205\pi$
$313$	−17.6474	−0.997489	−0.498745	+	0.866749i	$0.666205\pi$
$314$	−11.6087	−0.655119
$315$	0	0
$316$	2.27886	0.128196
$317$	−9.68734	−0.544095	−0.272048	−	0.962284i	$-0.587701\pi$
$317$	−9.68734	−0.544095	−0.272048	+	0.962284i	$0.587701\pi$
$318$	0	0
$319$	30.7594	1.72220
$320$	3.46379	0.193632
$321$	0	0
$322$	17.5850	0.979974
$323$	−32.1843	−1.79078
$324$	0	0
$325$	38.1683	2.11720
$326$	−23.2573	−1.28810
$327$	0	0
$328$	−10.6945	−0.590506
$329$	30.6478	1.68967
$330$	0	0
$331$	23.3524	1.28356	0.641781	−	0.766888i	$-0.278196\pi$
$331$	23.3524	1.28356	0.641781	+	0.766888i	$0.278196\pi$
$332$	−4.53459	−0.248868
$333$	0	0
$334$	20.1860	1.10453
$335$	−28.7544	−1.57102
$336$	0	0
$337$	11.1102	0.605213	0.302607	−	0.953116i	$-0.402143\pi$
$337$	11.1102	0.605213	0.302607	+	0.953116i	$0.402143\pi$
$338$	16.7494	0.911047
$339$	0	0
$340$	−20.6004	−1.11721
$341$	−26.4838	−1.43418
$342$	0	0
$343$	0.492021	0.0265666
$344$	−12.8009	−0.690177
$345$	0	0
$346$	−2.95800	−0.159023
$347$	−13.1429	−0.705549	−0.352774	−	0.935708i	$-0.614762\pi$
$347$	−13.1429	−0.705549	−0.352774	+	0.935708i	$0.614762\pi$
$348$	0	0
$349$	−10.0265	−0.536708	−0.268354	−	0.963320i	$-0.586480\pi$
$349$	−10.0265	−0.536708	−0.268354	+	0.963320i	$0.586480\pi$
$350$	−26.0597	−1.39295
$351$	0	0
$352$	−5.41125	−0.288421
$353$	−22.1184	−1.17724	−0.588621	−	0.808409i	$-0.700329\pi$
$353$	−22.1184	−1.17724	−0.588621	+	0.808409i	$0.700329\pi$
$354$	0	0
$355$	−3.06177	−0.162502
$356$	−1.46095	−0.0774301
$357$	0	0
$358$	−11.9473	−0.631437
$359$	19.5158	1.03000	0.515002	−	0.857189i	$-0.327791\pi$
$359$	19.5158	1.03000	0.515002	+	0.857189i	$0.327791\pi$
$360$	0	0
$361$	10.2848	0.541303
$362$	−20.7182	−1.08893
$363$	0	0
$364$	−20.3116	−1.06462
$365$	23.8285	1.24724
$366$	0	0
$367$	3.06129	0.159798	0.0798991	−	0.996803i	$-0.474540\pi$
$367$	3.06129	0.159798	0.0798991	+	0.996803i	$0.474540\pi$
$368$	−4.72212	−0.246158
$369$	0	0
$370$	−37.4533	−1.94710
$371$	−44.0125	−2.28501
$372$	0	0
$373$	−16.5213	−0.855442	−0.427721	−	0.903911i	$-0.640683\pi$
$373$	−16.5213	−0.855442	−0.427721	+	0.903911i	$0.640683\pi$
$374$	32.1826	1.66412
$375$	0	0
$376$	−8.22990	−0.424425
$377$	−31.0041	−1.59679
$378$	0	0
$379$	−24.7081	−1.26917	−0.634584	−	0.772854i	$-0.718829\pi$
$379$	−24.7081	−1.26917	−0.634584	+	0.772854i	$0.718829\pi$
$380$	18.7444	0.961569
$381$	0	0
$382$	−17.1850	−0.879260
$383$	4.53673	0.231816	0.115908	−	0.993260i	$-0.463022\pi$
$383$	4.53673	0.231816	0.115908	+	0.993260i	$0.463022\pi$
$384$	0	0
$385$	69.7998	3.55733
$386$	0.843776	0.0429471
$387$	0	0
$388$	18.1261	0.920213
$389$	31.1647	1.58011	0.790056	−	0.613035i	$-0.210052\pi$
$389$	31.1647	1.58011	0.790056	+	0.613035i	$0.210052\pi$
$390$	0	0
$391$	28.0841	1.42028
$392$	6.86788	0.346880
$393$	0	0
$394$	9.38954	0.473038
$395$	7.89350	0.397165
$396$	0	0
$397$	14.0089	0.703088	0.351544	−	0.936171i	$-0.385657\pi$
$397$	14.0089	0.703088	0.351544	+	0.936171i	$0.385657\pi$
$398$	4.08274	0.204649
$399$	0	0
$400$	6.99784	0.349892
$401$	−19.7883	−0.988178	−0.494089	−	0.869411i	$-0.664498\pi$
$401$	−19.7883	−0.988178	−0.494089	+	0.869411i	$0.664498\pi$
$402$	0	0
$403$	26.6945	1.32975
$404$	−7.96826	−0.396436
$405$	0	0
$406$	21.1683	1.05056
$407$	58.5108	2.90027
$408$	0	0
$409$	−12.8739	−0.636575	−0.318288	−	0.947994i	$-0.603108\pi$
$409$	−12.8739	−0.636575	−0.318288	+	0.947994i	$0.603108\pi$
$410$	−37.0435	−1.82945
$411$	0	0
$412$	11.8574	0.584174
$413$	23.5085	1.15678
$414$	0	0
$415$	−15.7069	−0.771020
$416$	5.45430	0.267419
$417$	0	0
$418$	−29.2832	−1.43229
$419$	−17.4280	−0.851412	−0.425706	−	0.904862i	$-0.639974\pi$
$419$	−17.4280	−0.851412	−0.425706	+	0.904862i	$0.639974\pi$
$420$	0	0
$421$	26.0347	1.26885	0.634427	−	0.772983i	$-0.281236\pi$
$421$	26.0347	1.26885	0.634427	+	0.772983i	$0.281236\pi$
$422$	12.4094	0.604080
$423$	0	0
$424$	11.8187	0.573968
$425$	−41.6187	−2.01880
$426$	0	0
$427$	−40.8973	−1.97916
$428$	−4.13345	−0.199798
$429$	0	0
$430$	−44.3395	−2.13824
$431$	−4.09204	−0.197107	−0.0985533	−	0.995132i	$-0.531421\pi$
$431$	−4.09204	−0.197107	−0.0985533	+	0.995132i	$0.531421\pi$
$432$	0	0
$433$	21.9423	1.05448	0.527241	−	0.849716i	$-0.323227\pi$
$433$	21.9423	1.05448	0.527241	+	0.849716i	$0.323227\pi$
$434$	−18.2258	−0.874868
$435$	0	0
$436$	−16.8105	−0.805077
$437$	−25.5540	−1.22241
$438$	0	0
$439$	−26.8522	−1.28158	−0.640792	−	0.767715i	$-0.721394\pi$
$439$	−26.8522	−1.28158	−0.640792	+	0.767715i	$0.721394\pi$
$440$	−18.7434	−0.893559
$441$	0	0
$442$	−32.4387	−1.54295
$443$	−1.31682	−0.0625639	−0.0312819	−	0.999511i	$-0.509959\pi$
$443$	−1.31682	−0.0625639	−0.0312819	+	0.999511i	$0.509959\pi$
$444$	0	0
$445$	−5.06042	−0.239887
$446$	−2.60836	−0.123510
$447$	0	0
$448$	−3.72396	−0.175941
$449$	9.00217	0.424838	0.212419	−	0.977179i	$-0.431866\pi$
$449$	9.00217	0.424838	0.212419	+	0.977179i	$0.431866\pi$
$450$	0	0
$451$	57.8707	2.72502
$452$	9.43021	0.443560
$453$	0	0
$454$	−1.09859	−0.0515592
$455$	−70.3551	−3.29830
$456$	0	0
$457$	0.899724	0.0420873	0.0210437	−	0.999779i	$-0.493301\pi$
$457$	0.899724	0.0420873	0.0210437	+	0.999779i	$0.493301\pi$
$458$	19.5454	0.913298
$459$	0	0
$460$	−16.3564	−0.762623
$461$	2.41367	0.112416	0.0562079	−	0.998419i	$-0.482099\pi$
$461$	2.41367	0.112416	0.0562079	+	0.998419i	$0.482099\pi$
$462$	0	0
$463$	−10.3255	−0.479869	−0.239934	−	0.970789i	$-0.577126\pi$
$463$	−10.3255	−0.479869	−0.239934	+	0.970789i	$0.577126\pi$
$464$	−5.68434	−0.263889
$465$	0	0
$466$	−22.8074	−1.05653
$467$	−11.2771	−0.521844	−0.260922	−	0.965360i	$-0.584027\pi$
$467$	−11.2771	−0.521844	−0.260922	+	0.965360i	$0.584027\pi$
$468$	0	0
$469$	30.9142	1.42749
$470$	−28.5067	−1.31491
$471$	0	0
$472$	−6.31276	−0.290568
$473$	69.2687	3.18498
$474$	0	0
$475$	37.8691	1.73755
$476$	22.1477	1.01514
$477$	0	0
$478$	1.17661	0.0538169
$479$	14.2371	0.650509	0.325254	−	0.945627i	$-0.394550\pi$
$479$	14.2371	0.650509	0.325254	+	0.945627i	$0.394550\pi$
$480$	0	0
$481$	−58.9763	−2.68909
$482$	−22.0093	−1.00249
$483$	0	0
$484$	18.2816	0.830984
$485$	62.7850	2.85092
$486$	0	0
$487$	7.25573	0.328789	0.164394	−	0.986395i	$-0.447433\pi$
$487$	7.25573	0.328789	0.164394	+	0.986395i	$0.447433\pi$
$488$	10.9822	0.497141
$489$	0	0
$490$	23.7889	1.07467
$491$	−9.21094	−0.415684	−0.207842	−	0.978162i	$-0.566644\pi$
$491$	−9.21094	−0.415684	−0.207842	+	0.978162i	$0.566644\pi$
$492$	0	0
$493$	33.8068	1.52258
$494$	29.5162	1.32799
$495$	0	0
$496$	4.89421	0.219756
$497$	3.29175	0.147655
$498$	0	0
$499$	−34.1216	−1.52749	−0.763747	−	0.645516i	$-0.776642\pi$
$499$	−34.1216	−1.52749	−0.763747	+	0.645516i	$0.776642\pi$
$500$	6.92011	0.309477
$501$	0	0
$502$	0.340199	0.0151838
$503$	−2.34544	−0.104578	−0.0522890	−	0.998632i	$-0.516652\pi$
$503$	−2.34544	−0.104578	−0.0522890	+	0.998632i	$0.516652\pi$
$504$	0	0
$505$	−27.6004	−1.22820
$506$	25.5526	1.13595
$507$	0	0
$508$	−0.238039	−0.0105613
$509$	−1.10156	−0.0488258	−0.0244129	−	0.999702i	$-0.507772\pi$
$509$	−1.10156	−0.0488258	−0.0244129	+	0.999702i	$0.507772\pi$
$510$	0	0
$511$	−25.6183	−1.13329
$512$	1.00000	0.0441942
$513$	0	0
$514$	−5.40682	−0.238485
$515$	41.0717	1.80983
$516$	0	0
$517$	44.5341	1.95861
$518$	40.2665	1.76921
$519$	0	0
$520$	18.8926	0.828493
$521$	8.55087	0.374620	0.187310	−	0.982301i	$-0.440023\pi$
$521$	8.55087	0.374620	0.187310	+	0.982301i	$0.440023\pi$
$522$	0	0
$523$	−18.3580	−0.802740	−0.401370	−	0.915916i	$-0.631466\pi$
$523$	−18.3580	−0.802740	−0.401370	+	0.915916i	$0.631466\pi$
$524$	−0.0242757	−0.00106049
$525$	0	0
$526$	18.4228	0.803272
$527$	−29.1076	−1.26795
$528$	0	0
$529$	−0.701541	−0.0305018
$530$	40.9376	1.77822
$531$	0	0
$532$	−20.1524	−0.873715
$533$	−58.3311	−2.52660
$534$	0	0
$535$	−14.3174	−0.618995
$536$	−8.30144	−0.358568
$537$	0	0
$538$	6.50417	0.280415
$539$	−37.1638	−1.60076
$540$	0	0
$541$	−37.9301	−1.63074	−0.815372	−	0.578937i	$-0.803468\pi$
$541$	−37.9301	−1.63074	−0.815372	+	0.578937i	$0.803468\pi$
$542$	−22.0203	−0.945851
$543$	0	0
$544$	−5.94735	−0.254991
$545$	−58.2281	−2.49422
$546$	0	0
$547$	28.1125	1.20200	0.601001	−	0.799248i	$-0.294769\pi$
$547$	28.1125	1.20200	0.601001	+	0.799248i	$0.294769\pi$
$548$	2.19280	0.0936718
$549$	0	0
$550$	−37.8671	−1.61466
$551$	−30.7611	−1.31046
$552$	0	0
$553$	−8.48639	−0.360878
$554$	30.3999	1.29157
$555$	0	0
$556$	−7.61095	−0.322776
$557$	30.8310	1.30635	0.653175	−	0.757207i	$-0.273436\pi$
$557$	30.8310	1.30635	0.653175	+	0.757207i	$0.273436\pi$
$558$	0	0
$559$	−69.8198	−2.95306
$560$	−12.8990	−0.545083
$561$	0	0
$562$	−4.00120	−0.168780
$563$	−6.47841	−0.273033	−0.136516	−	0.990638i	$-0.543591\pi$
$563$	−6.47841	−0.273033	−0.136516	+	0.990638i	$0.543591\pi$
$564$	0	0
$565$	32.6643	1.37420
$566$	5.09021	0.213957
$567$	0	0
$568$	−0.883937	−0.0370892
$569$	10.7510	0.450707	0.225354	−	0.974277i	$-0.427646\pi$
$569$	10.7510	0.450707	0.225354	+	0.974277i	$0.427646\pi$
$570$	0	0
$571$	−32.2527	−1.34973	−0.674867	−	0.737940i	$-0.735799\pi$
$571$	−32.2527	−1.34973	−0.674867	+	0.737940i	$0.735799\pi$
$572$	−29.5146	−1.23407
$573$	0	0
$574$	39.8259	1.66230
$575$	−33.0447	−1.37806
$576$	0	0
$577$	29.0518	1.20944	0.604721	−	0.796438i	$-0.293285\pi$
$577$	29.0518	1.20944	0.604721	+	0.796438i	$0.293285\pi$
$578$	18.3710	0.764134
$579$	0	0
$580$	−19.6894	−0.817557
$581$	16.8866	0.700576
$582$	0	0
$583$	−63.9541	−2.64871
$584$	6.87933	0.284669
$585$	0	0
$586$	2.56800	0.106083
$587$	37.6130	1.55245	0.776227	−	0.630454i	$-0.217131\pi$
$587$	37.6130	1.55245	0.776227	+	0.630454i	$0.217131\pi$
$588$	0	0
$589$	26.4852	1.09130
$590$	−21.8661	−0.900212
$591$	0	0
$592$	−10.8128	−0.444404
$593$	−25.5257	−1.04821	−0.524106	−	0.851653i	$-0.675601\pi$
$593$	−25.5257	−1.04821	−0.524106	+	0.851653i	$0.675601\pi$
$594$	0	0
$595$	76.7150	3.14501
$596$	1.00000	0.0409616
$597$	0	0
$598$	−25.7559	−1.05324
$599$	−21.1821	−0.865478	−0.432739	−	0.901519i	$-0.642453\pi$
$599$	−21.1821	−0.865478	−0.432739	+	0.901519i	$0.642453\pi$
$600$	0	0
$601$	29.6931	1.21120	0.605602	−	0.795767i	$-0.292932\pi$
$601$	29.6931	1.21120	0.605602	+	0.795767i	$0.292932\pi$
$602$	47.6699	1.94288
$603$	0	0
$604$	2.74712	0.111779
$605$	63.3238	2.57448
$606$	0	0
$607$	6.32992	0.256923	0.128462	−	0.991714i	$-0.458996\pi$
$607$	6.32992	0.256923	0.128462	+	0.991714i	$0.458996\pi$
$608$	5.41154	0.219467
$609$	0	0
$610$	38.0400	1.54020
$611$	−44.8884	−1.81599
$612$	0	0
$613$	8.36343	0.337796	0.168898	−	0.985634i	$-0.445979\pi$
$613$	8.36343	0.337796	0.168898	+	0.985634i	$0.445979\pi$
$614$	0.252021	0.0101708
$615$	0	0
$616$	20.1513	0.811918
$617$	47.7212	1.92118	0.960592	−	0.277963i	$-0.0896593\pi$
$617$	47.7212	1.92118	0.960592	+	0.277963i	$0.0896593\pi$
$618$	0	0
$619$	−29.7786	−1.19690	−0.598451	−	0.801159i	$-0.704217\pi$
$619$	−29.7786	−1.19690	−0.598451	+	0.801159i	$0.704217\pi$
$620$	16.9525	0.680829
$621$	0	0
$622$	3.21065	0.128735
$623$	5.44051	0.217969
$624$	0	0
$625$	−11.0194	−0.440776
$626$	−17.6474	−0.705332
$627$	0	0
$628$	−11.6087	−0.463239
$629$	64.3076	2.56411
$630$	0	0
$631$	27.3209	1.08763	0.543815	−	0.839205i	$-0.316979\pi$
$631$	27.3209	1.08763	0.543815	+	0.839205i	$0.316979\pi$
$632$	2.27886	0.0906482
$633$	0	0
$634$	−9.68734	−0.384733
$635$	−0.824516	−0.0327199
$636$	0	0
$637$	37.4595	1.48420
$638$	30.7594	1.21778
$639$	0	0
$640$	3.46379	0.136918
$641$	26.6147	1.05122	0.525610	−	0.850726i	$-0.323837\pi$
$641$	26.6147	1.05122	0.525610	+	0.850726i	$0.323837\pi$
$642$	0	0
$643$	25.3467	0.999577	0.499789	−	0.866147i	$-0.333411\pi$
$643$	25.3467	0.999577	0.499789	+	0.866147i	$0.333411\pi$
$644$	17.5850	0.692946
$645$	0	0
$646$	−32.1843	−1.26628
$647$	4.22827	0.166230	0.0831151	−	0.996540i	$-0.473513\pi$
$647$	4.22827	0.166230	0.0831151	+	0.996540i	$0.473513\pi$
$648$	0	0
$649$	34.1599	1.34089
$650$	38.1683	1.49709
$651$	0	0
$652$	−23.2573	−0.910826
$653$	−11.3973	−0.446010	−0.223005	−	0.974817i	$-0.571587\pi$
$653$	−11.3973	−0.446010	−0.223005	+	0.974817i	$0.571587\pi$
$654$	0	0
$655$	−0.0840858	−0.00328551
$656$	−10.6945	−0.417551
$657$	0	0
$658$	30.6478	1.19478
$659$	−22.0121	−0.857471	−0.428735	−	0.903430i	$-0.641041\pi$
$659$	−22.0121	−0.857471	−0.428735	+	0.903430i	$0.641041\pi$
$660$	0	0
$661$	−35.6587	−1.38696	−0.693481	−	0.720475i	$-0.743924\pi$
$661$	−35.6587	−1.38696	−0.693481	+	0.720475i	$0.743924\pi$
$662$	23.3524	0.907616
$663$	0	0
$664$	−4.53459	−0.175976
$665$	−69.8035	−2.70686
$666$	0	0
$667$	26.8422	1.03933
$668$	20.1860	0.781021
$669$	0	0
$670$	−28.7544	−1.11088
$671$	−59.4275	−2.29417
$672$	0	0
$673$	−36.5338	−1.40827	−0.704137	−	0.710064i	$-0.748666\pi$
$673$	−36.5338	−1.40827	−0.704137	+	0.710064i	$0.748666\pi$
$674$	11.1102	0.427950
$675$	0	0
$676$	16.7494	0.644207
$677$	20.0748	0.771539	0.385770	−	0.922595i	$-0.373936\pi$
$677$	20.0748	0.771539	0.385770	+	0.922595i	$0.373936\pi$
$678$	0	0
$679$	−67.5008	−2.59044
$680$	−20.6004	−0.789989
$681$	0	0
$682$	−26.4838	−1.01412
$683$	21.0937	0.807129	0.403565	−	0.914951i	$-0.367771\pi$
$683$	21.0937	0.807129	0.403565	+	0.914951i	$0.367771\pi$
$684$	0	0
$685$	7.59541	0.290206
$686$	0.492021	0.0187855
$687$	0	0
$688$	−12.8009	−0.488029
$689$	64.4629	2.45584
$690$	0	0
$691$	22.4326	0.853377	0.426689	−	0.904399i	$-0.359680\pi$
$691$	22.4326	0.853377	0.426689	+	0.904399i	$0.359680\pi$
$692$	−2.95800	−0.112446
$693$	0	0
$694$	−13.1429	−0.498898
$695$	−26.3627	−0.999996
$696$	0	0
$697$	63.6041	2.40918
$698$	−10.0265	−0.379510
$699$	0	0
$700$	−26.0597	−0.984964
$701$	31.3649	1.18464	0.592318	−	0.805704i	$-0.298213\pi$
$701$	31.3649	1.18464	0.592318	+	0.805704i	$0.298213\pi$
$702$	0	0
$703$	−58.5139	−2.20689
$704$	−5.41125	−0.203944
$705$	0	0
$706$	−22.1184	−0.832436
$707$	29.6735	1.11599
$708$	0	0
$709$	19.4658	0.731055	0.365528	−	0.930801i	$-0.380889\pi$
$709$	19.4658	0.731055	0.365528	+	0.930801i	$0.380889\pi$
$710$	−3.06177	−0.114906
$711$	0	0
$712$	−1.46095	−0.0547513
$713$	−23.1111	−0.865516
$714$	0	0
$715$	−102.232	−3.82327
$716$	−11.9473	−0.446493
$717$	0	0
$718$	19.5158	0.728323
$719$	8.77430	0.327226	0.163613	−	0.986525i	$-0.447685\pi$
$719$	8.77430	0.327226	0.163613	+	0.986525i	$0.447685\pi$
$720$	0	0
$721$	−44.1566	−1.64448
$722$	10.2848	0.382759
$723$	0	0
$724$	−20.7182	−0.769986
$725$	−39.7782	−1.47732
$726$	0	0
$727$	4.57286	0.169598	0.0847990	−	0.996398i	$-0.472975\pi$
$727$	4.57286	0.169598	0.0847990	+	0.996398i	$0.472975\pi$
$728$	−20.3116	−0.752798
$729$	0	0
$730$	23.8285	0.881934
$731$	76.1313	2.81582
$732$	0	0
$733$	22.0022	0.812670	0.406335	−	0.913724i	$-0.366807\pi$
$733$	22.0022	0.812670	0.406335	+	0.913724i	$0.366807\pi$
$734$	3.06129	0.112994
$735$	0	0
$736$	−4.72212	−0.174060
$737$	44.9212	1.65469
$738$	0	0
$739$	−31.4305	−1.15619	−0.578095	−	0.815969i	$-0.696204\pi$
$739$	−31.4305	−1.15619	−0.578095	+	0.815969i	$0.696204\pi$
$740$	−37.4533	−1.37681
$741$	0	0
$742$	−44.0125	−1.61575
$743$	37.2913	1.36809	0.684043	−	0.729441i	$-0.260220\pi$
$743$	37.2913	1.36809	0.684043	+	0.729441i	$0.260220\pi$
$744$	0	0
$745$	3.46379	0.126903
$746$	−16.5213	−0.604889
$747$	0	0
$748$	32.1826	1.17671
$749$	15.3928	0.562441
$750$	0	0
$751$	20.7149	0.755898	0.377949	−	0.925826i	$-0.376630\pi$
$751$	20.7149	0.755898	0.377949	+	0.925826i	$0.376630\pi$
$752$	−8.22990	−0.300114
$753$	0	0
$754$	−31.0041	−1.12910
$755$	9.51546	0.346303
$756$	0	0
$757$	1.38254	0.0502492	0.0251246	−	0.999684i	$-0.492002\pi$
$757$	1.38254	0.0502492	0.0251246	+	0.999684i	$0.492002\pi$
$758$	−24.7081	−0.897438
$759$	0	0
$760$	18.7444	0.679932
$761$	−40.2126	−1.45771	−0.728853	−	0.684670i	$-0.759946\pi$
$761$	−40.2126	−1.45771	−0.728853	+	0.684670i	$0.759946\pi$
$762$	0	0
$763$	62.6017	2.26633
$764$	−17.1850	−0.621731
$765$	0	0
$766$	4.53673	0.163919
$767$	−34.4317	−1.24326
$768$	0	0
$769$	4.72751	0.170478	0.0852391	−	0.996361i	$-0.472835\pi$
$769$	4.72751	0.170478	0.0852391	+	0.996361i	$0.472835\pi$
$770$	69.7998	2.51541
$771$	0	0
$772$	0.843776	0.0303682
$773$	−0.478036	−0.0171937	−0.00859687	−	0.999963i	$-0.502737\pi$
$773$	−0.478036	−0.0171937	−0.00859687	+	0.999963i	$0.502737\pi$
$774$	0	0
$775$	34.2489	1.23026
$776$	18.1261	0.650689
$777$	0	0
$778$	31.1647	1.11731
$779$	−57.8738	−2.07354
$780$	0	0
$781$	4.78321	0.171157
$782$	28.0841	1.00429
$783$	0	0
$784$	6.86788	0.245281
$785$	−40.2103	−1.43517
$786$	0	0
$787$	8.16119	0.290915	0.145457	−	0.989365i	$-0.453535\pi$
$787$	8.16119	0.290915	0.145457	+	0.989365i	$0.453535\pi$
$788$	9.38954	0.334489
$789$	0	0
$790$	7.89350	0.280838
$791$	−35.1177	−1.24864
$792$	0	0
$793$	59.9002	2.12712
$794$	14.0089	0.497159
$795$	0	0
$796$	4.08274	0.144709
$797$	−53.1778	−1.88365	−0.941827	−	0.336098i	$-0.890893\pi$
$797$	−53.1778	−1.88365	−0.941827	+	0.336098i	$0.890893\pi$
$798$	0	0
$799$	48.9461	1.73159
$800$	6.99784	0.247411
$801$	0	0
$802$	−19.7883	−0.698747
$803$	−37.2258	−1.31367
$804$	0	0
$805$	60.9108	2.14682
$806$	26.6945	0.940273
$807$	0	0
$808$	−7.96826	−0.280322
$809$	−5.56478	−0.195647	−0.0978236	−	0.995204i	$-0.531188\pi$
$809$	−5.56478	−0.195647	−0.0978236	+	0.995204i	$0.531188\pi$
$810$	0	0
$811$	−18.6131	−0.653594	−0.326797	−	0.945094i	$-0.605969\pi$
$811$	−18.6131	−0.653594	−0.326797	+	0.945094i	$0.605969\pi$
$812$	21.1683	0.742861
$813$	0	0
$814$	58.5108	2.05080
$815$	−80.5584	−2.82184
$816$	0	0
$817$	−69.2724	−2.42353
$818$	−12.8739	−0.450127
$819$	0	0
$820$	−37.0435	−1.29362
$821$	−52.9148	−1.84674	−0.923370	−	0.383912i	$-0.874577\pi$
$821$	−52.9148	−1.84674	−0.923370	+	0.383912i	$0.874577\pi$
$822$	0	0
$823$	37.9567	1.32309	0.661543	−	0.749907i	$-0.269902\pi$
$823$	37.9567	1.32309	0.661543	+	0.749907i	$0.269902\pi$
$824$	11.8574	0.413073
$825$	0	0
$826$	23.5085	0.817964
$827$	38.1529	1.32670	0.663352	−	0.748307i	$-0.269133\pi$
$827$	38.1529	1.32670	0.663352	+	0.748307i	$0.269133\pi$
$828$	0	0
$829$	−13.0479	−0.453173	−0.226587	−	0.973991i	$-0.572757\pi$
$829$	−13.0479	−0.453173	−0.226587	+	0.973991i	$0.572757\pi$
$830$	−15.7069	−0.545193
$831$	0	0
$832$	5.45430	0.189094
$833$	−40.8457	−1.41522
$834$	0	0
$835$	69.9202	2.41969
$836$	−29.2832	−1.01278
$837$	0	0
$838$	−17.4280	−0.602039
$839$	31.5390	1.08885	0.544424	−	0.838810i	$-0.316748\pi$
$839$	31.5390	1.08885	0.544424	+	0.838810i	$0.316748\pi$
$840$	0	0
$841$	3.31178	0.114199
$842$	26.0347	0.897215
$843$	0	0
$844$	12.4094	0.427149
$845$	58.0164	1.99582
$846$	0	0
$847$	−68.0801	−2.33926
$848$	11.8187	0.405857
$849$	0	0
$850$	−41.6187	−1.42751
$851$	51.0594	1.75029
$852$	0	0
$853$	−30.8429	−1.05604	−0.528021	−	0.849232i	$-0.677066\pi$
$853$	−30.8429	−1.05604	−0.528021	+	0.849232i	$0.677066\pi$
$854$	−40.8973	−1.39948
$855$	0	0
$856$	−4.13345	−0.141278
$857$	−15.4634	−0.528219	−0.264110	−	0.964493i	$-0.585078\pi$
$857$	−15.4634	−0.528219	−0.264110	+	0.964493i	$0.585078\pi$
$858$	0	0
$859$	31.4288	1.07234	0.536168	−	0.844112i	$-0.319871\pi$
$859$	31.4288	1.07234	0.536168	+	0.844112i	$0.319871\pi$
$860$	−44.3395	−1.51197
$861$	0	0
$862$	−4.09204	−0.139375
$863$	31.7402	1.08045	0.540224	−	0.841521i	$-0.318340\pi$
$863$	31.7402	1.08045	0.540224	+	0.841521i	$0.318340\pi$
$864$	0	0
$865$	−10.2459	−0.348371
$866$	21.9423	0.745631
$867$	0	0
$868$	−18.2258	−0.618625
$869$	−12.3315	−0.418317
$870$	0	0
$871$	−45.2785	−1.53420
$872$	−16.8105	−0.569276
$873$	0	0
$874$	−25.5540	−0.864376
$875$	−25.7702	−0.871193
$876$	0	0
$877$	−29.5509	−0.997863	−0.498931	−	0.866642i	$-0.666274\pi$
$877$	−29.5509	−0.997863	−0.498931	+	0.866642i	$0.666274\pi$
$878$	−26.8522	−0.906216
$879$	0	0
$880$	−18.7434	−0.631841
$881$	−16.4745	−0.555040	−0.277520	−	0.960720i	$-0.589512\pi$
$881$	−16.4745	−0.555040	−0.277520	+	0.960720i	$0.589512\pi$
$882$	0	0
$883$	5.97164	0.200962	0.100481	−	0.994939i	$-0.467962\pi$
$883$	5.97164	0.200962	0.100481	+	0.994939i	$0.467962\pi$
$884$	−32.4387	−1.09103
$885$	0	0
$886$	−1.31682	−0.0442393
$887$	21.7725	0.731048	0.365524	−	0.930802i	$-0.380890\pi$
$887$	21.7725	0.731048	0.365524	+	0.930802i	$0.380890\pi$
$888$	0	0
$889$	0.886447	0.0297305
$890$	−5.06042	−0.169626
$891$	0	0
$892$	−2.60836	−0.0873345
$893$	−44.5364	−1.49036
$894$	0	0
$895$	−41.3831	−1.38328
$896$	−3.72396	−0.124409
$897$	0	0
$898$	9.00217	0.300406
$899$	−27.8204	−0.927861
$900$	0	0
$901$	−70.2902	−2.34171
$902$	57.8707	1.92688
$903$	0	0
$904$	9.43021	0.313644
$905$	−71.7635	−2.38550
$906$	0	0
$907$	33.1641	1.10120	0.550598	−	0.834771i	$-0.314400\pi$
$907$	33.1641	1.10120	0.550598	+	0.834771i	$0.314400\pi$
$908$	−1.09859	−0.0364579
$909$	0	0
$910$	−70.3551	−2.33225
$911$	−18.7236	−0.620342	−0.310171	−	0.950681i	$-0.600386\pi$
$911$	−18.7236	−0.620342	−0.310171	+	0.950681i	$0.600386\pi$
$912$	0	0
$913$	24.5378	0.812083
$914$	0.899724	0.0297602
$915$	0	0
$916$	19.5454	0.645799
$917$	0.0904016	0.00298532
$918$	0	0
$919$	−5.10012	−0.168237	−0.0841187	−	0.996456i	$-0.526807\pi$
$919$	−5.10012	−0.168237	−0.0841187	+	0.996456i	$0.526807\pi$
$920$	−16.3564	−0.539256
$921$	0	0
$922$	2.41367	0.0794900
$923$	−4.82126	−0.158694
$924$	0	0
$925$	−75.6664	−2.48789
$926$	−10.3255	−0.339318
$927$	0	0
$928$	−5.68434	−0.186598
$929$	−22.0558	−0.723627	−0.361813	−	0.932251i	$-0.617842\pi$
$929$	−22.0558	−0.723627	−0.361813	+	0.932251i	$0.617842\pi$
$930$	0	0
$931$	37.1658	1.21806
$932$	−22.8074	−0.747081
$933$	0	0
$934$	−11.2771	−0.369000
$935$	111.474	3.64559
$936$	0	0
$937$	43.4294	1.41878	0.709388	−	0.704818i	$-0.248971\pi$
$937$	43.4294	1.41878	0.709388	+	0.704818i	$0.248971\pi$
$938$	30.9142	1.00939
$939$	0	0
$940$	−28.5067	−0.929785
$941$	4.85491	0.158265	0.0791327	−	0.996864i	$-0.474785\pi$
$941$	4.85491	0.158265	0.0791327	+	0.996864i	$0.474785\pi$
$942$	0	0
$943$	50.5008	1.64453
$944$	−6.31276	−0.205463
$945$	0	0
$946$	69.2687	2.25212
$947$	−3.91485	−0.127215	−0.0636077	−	0.997975i	$-0.520261\pi$
$947$	−3.91485	−0.127215	−0.0636077	+	0.997975i	$0.520261\pi$
$948$	0	0
$949$	37.5219	1.21801
$950$	37.8691	1.22864
$951$	0	0
$952$	22.1477	0.717811
$953$	14.1503	0.458375	0.229187	−	0.973382i	$-0.426393\pi$
$953$	14.1503	0.458375	0.229187	+	0.973382i	$0.426393\pi$
$954$	0	0
$955$	−59.5252	−1.92619
$956$	1.17661	0.0380543
$957$	0	0
$958$	14.2371	0.459979
$959$	−8.16591	−0.263691
$960$	0	0
$961$	−7.04673	−0.227314
$962$	−58.9763	−1.90147
$963$	0	0
$964$	−22.0093	−0.708870
$965$	2.92266	0.0940839
$966$	0	0
$967$	57.0322	1.83403	0.917016	−	0.398850i	$-0.130591\pi$
$967$	57.0322	1.83403	0.917016	+	0.398850i	$0.130591\pi$
$968$	18.2816	0.587594
$969$	0	0
$970$	62.7850	2.01590
$971$	−13.2501	−0.425216	−0.212608	−	0.977138i	$-0.568196\pi$
$971$	−13.2501	−0.425216	−0.212608	+	0.977138i	$0.568196\pi$
$972$	0	0
$973$	28.3429	0.908631
$974$	7.25573	0.232489
$975$	0	0
$976$	10.9822	0.351532
$977$	25.1158	0.803526	0.401763	−	0.915744i	$-0.368398\pi$
$977$	25.1158	0.803526	0.401763	+	0.915744i	$0.368398\pi$
$978$	0	0
$979$	7.90556	0.252663
$980$	23.7889	0.759908
$981$	0	0
$982$	−9.21094	−0.293933
$983$	−30.8442	−0.983778	−0.491889	−	0.870658i	$-0.663694\pi$
$983$	−30.8442	−0.983778	−0.491889	+	0.870658i	$0.663694\pi$
$984$	0	0
$985$	32.5234	1.03628
$986$	33.8068	1.07663
$987$	0	0
$988$	29.5162	0.939034
$989$	60.4473	1.92211
$990$	0	0
$991$	−11.8688	−0.377025	−0.188512	−	0.982071i	$-0.560367\pi$
$991$	−11.8688	−0.377025	−0.188512	+	0.982071i	$0.560367\pi$
$992$	4.89421	0.155391
$993$	0	0
$994$	3.29175	0.104408
$995$	14.1418	0.448324
$996$	0	0
$997$	−42.8911	−1.35838	−0.679188	−	0.733964i	$-0.737668\pi$
$997$	−42.8911	−1.35838	−0.679188	+	0.733964i	$0.737668\pi$
$998$	−34.1216	−1.08010
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.n.1.12	yes	12
3.2	odd	2		8046.2.a.k.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.k.1.1	✓	12	3.2	odd	2
8046.2.a.n.1.12	yes	12	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.46379 q^{5} -3.72396 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.46379 q^{5} -3.72396 q^{7} +1.00000 q^{8} +3.46379 q^{10} -5.41125 q^{11} +5.45430 q^{13} -3.72396 q^{14} +1.00000 q^{16} -5.94735 q^{17} +5.41154 q^{19} +3.46379 q^{20} -5.41125 q^{22} -4.72212 q^{23} +6.99784 q^{25} +5.45430 q^{26} -3.72396 q^{28} -5.68434 q^{29} +4.89421 q^{31} +1.00000 q^{32} -5.94735 q^{34} -12.8990 q^{35} -10.8128 q^{37} +5.41154 q^{38} +3.46379 q^{40} -10.6945 q^{41} -12.8009 q^{43} -5.41125 q^{44} -4.72212 q^{46} -8.22990 q^{47} +6.86788 q^{49} +6.99784 q^{50} +5.45430 q^{52} +11.8187 q^{53} -18.7434 q^{55} -3.72396 q^{56} -5.68434 q^{58} -6.31276 q^{59} +10.9822 q^{61} +4.89421 q^{62} +1.00000 q^{64} +18.8926 q^{65} -8.30144 q^{67} -5.94735 q^{68} -12.8990 q^{70} -0.883937 q^{71} +6.87933 q^{73} -10.8128 q^{74} +5.41154 q^{76} +20.1513 q^{77} +2.27886 q^{79} +3.46379 q^{80} -10.6945 q^{82} -4.53459 q^{83} -20.6004 q^{85} -12.8009 q^{86} -5.41125 q^{88} -1.46095 q^{89} -20.3116 q^{91} -4.72212 q^{92} -8.22990 q^{94} +18.7444 q^{95} +18.1261 q^{97} +6.86788 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8} - 3 q^{10} - 14 q^{11} - 3 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} - 4 q^{19} - 3 q^{20} - 14 q^{22} - 13 q^{23} + 11 q^{25} - 3 q^{26} - 6 q^{28} - 23 q^{29} - 14 q^{31} + 12 q^{32} - 8 q^{34} - 32 q^{35} - 19 q^{37} - 4 q^{38} - 3 q^{40} - 30 q^{41} - 15 q^{43} - 14 q^{44} - 13 q^{46} + q^{47} + 14 q^{49} + 11 q^{50} - 3 q^{52} - 16 q^{53} - 7 q^{55} - 6 q^{56} - 23 q^{58} - 26 q^{59} - 16 q^{61} - 14 q^{62} + 12 q^{64} - 8 q^{65} - 39 q^{67} - 8 q^{68} - 32 q^{70} - 15 q^{71} - 2 q^{73} - 19 q^{74} - 4 q^{76} - 34 q^{77} - 13 q^{79} - 3 q^{80} - 30 q^{82} - 6 q^{83} - 11 q^{85} - 15 q^{86} - 14 q^{88} - 18 q^{89} - 35 q^{91} - 13 q^{92} + q^{94} - 51 q^{95} + 19 q^{97} + 14 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8 - 3 * q^10 - 14 * q^11 - 3 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 - 4 * q^19 - 3 * q^20 - 14 * q^22 - 13 * q^23 + 11 * q^25 - 3 * q^26 - 6 * q^28 - 23 * q^29 - 14 * q^31 + 12 * q^32 - 8 * q^34 - 32 * q^35 - 19 * q^37 - 4 * q^38 - 3 * q^40 - 30 * q^41 - 15 * q^43 - 14 * q^44 - 13 * q^46 + q^47 + 14 * q^49 + 11 * q^50 - 3 * q^52 - 16 * q^53 - 7 * q^55 - 6 * q^56 - 23 * q^58 - 26 * q^59 - 16 * q^61 - 14 * q^62 + 12 * q^64 - 8 * q^65 - 39 * q^67 - 8 * q^68 - 32 * q^70 - 15 * q^71 - 2 * q^73 - 19 * q^74 - 4 * q^76 - 34 * q^77 - 13 * q^79 - 3 * q^80 - 30 * q^82 - 6 * q^83 - 11 * q^85 - 15 * q^86 - 14 * q^88 - 18 * q^89 - 35 * q^91 - 13 * q^92 + q^94 - 51 * q^95 + 19 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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