LMFDB - Embedded newform 8046.2.a.g.1.5

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 9x^{7} + 25x^{6} + 29x^{5} - 58x^{4} - 43x^{3} + 34x^{2} + 25x + 2 $ x^9 - 3x^8 - 9x^7 + 25x^6 + 29x^5 - 58x^4 - 43x^3 + 34x^2 + 25x + 2
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			1.5
Root			$-0.584549$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +0.683695 q^{5} -0.639069 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +0.683695 q^{5} -0.639069 q^{7} -1.00000 q^{8} -0.683695 q^{10} +1.37847 q^{11} +1.22729 q^{13} +0.639069 q^{14} +1.00000 q^{16} +2.93824 q^{17} -1.25651 q^{19} +0.683695 q^{20} -1.37847 q^{22} -7.55461 q^{23} -4.53256 q^{25} -1.22729 q^{26} -0.639069 q^{28} +2.51033 q^{29} +5.78698 q^{31} -1.00000 q^{32} -2.93824 q^{34} -0.436928 q^{35} -2.36988 q^{37} +1.25651 q^{38} -0.683695 q^{40} +1.79132 q^{41} -10.0366 q^{43} +1.37847 q^{44} +7.55461 q^{46} +11.1030 q^{47} -6.59159 q^{49} +4.53256 q^{50} +1.22729 q^{52} -3.25589 q^{53} +0.942452 q^{55} +0.639069 q^{56} -2.51033 q^{58} -10.2114 q^{59} +1.02628 q^{61} -5.78698 q^{62} +1.00000 q^{64} +0.839089 q^{65} +8.00671 q^{67} +2.93824 q^{68} +0.436928 q^{70} -2.68750 q^{71} -15.4477 q^{73} +2.36988 q^{74} -1.25651 q^{76} -0.880936 q^{77} -7.08243 q^{79} +0.683695 q^{80} -1.79132 q^{82} -2.29043 q^{83} +2.00886 q^{85} +10.0366 q^{86} -1.37847 q^{88} +7.26133 q^{89} -0.784321 q^{91} -7.55461 q^{92} -11.1030 q^{94} -0.859067 q^{95} +0.495565 q^{97} +6.59159 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) $ 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8	$ 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8} - 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 9 q^{16} + q^{17} - 10 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 3 q^{25} + 8 q^{26} - 4 q^{28} + 4 q^{29} - 17 q^{31} - 9 q^{32} - q^{34} + 10 q^{35} - 11 q^{37} + 10 q^{38} - 4 q^{40} - 16 q^{43} + 4 q^{44} - 8 q^{46} + 7 q^{47} - 5 q^{49} + 3 q^{50} - 8 q^{52} + 12 q^{53} - 23 q^{55} + 4 q^{56} - 4 q^{58} + 6 q^{59} - 13 q^{61} + 17 q^{62} + 9 q^{64} - 24 q^{65} - 14 q^{67} + q^{68} - 10 q^{70} + 30 q^{71} - 12 q^{73} + 11 q^{74} - 10 q^{76} + 12 q^{77} - 35 q^{79} + 4 q^{80} + 5 q^{83} - 27 q^{85} + 16 q^{86} - 4 q^{88} + 23 q^{89} - 28 q^{91} + 8 q^{92} - 7 q^{94} + 32 q^{95} - 21 q^{97} + 5 q^{98}+O(q^{100}) $ 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8 - 4 * q^10 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 9 * q^16 + q^17 - 10 * q^19 + 4 * q^20 - 4 * q^22 + 8 * q^23 - 3 * q^25 + 8 * q^26 - 4 * q^28 + 4 * q^29 - 17 * q^31 - 9 * q^32 - q^34 + 10 * q^35 - 11 * q^37 + 10 * q^38 - 4 * q^40 - 16 * q^43 + 4 * q^44 - 8 * q^46 + 7 * q^47 - 5 * q^49 + 3 * q^50 - 8 * q^52 + 12 * q^53 - 23 * q^55 + 4 * q^56 - 4 * q^58 + 6 * q^59 - 13 * q^61 + 17 * q^62 + 9 * q^64 - 24 * q^65 - 14 * q^67 + q^68 - 10 * q^70 + 30 * q^71 - 12 * q^73 + 11 * q^74 - 10 * q^76 + 12 * q^77 - 35 * q^79 + 4 * q^80 + 5 * q^83 - 27 * q^85 + 16 * q^86 - 4 * q^88 + 23 * q^89 - 28 * q^91 + 8 * q^92 - 7 * q^94 + 32 * q^95 - 21 * q^97 + 5 * 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
$5$	0.683695	0.305758	0.152879	−	0.988245i	$-0.451146\pi$
$5$	0.683695	0.305758	0.152879	+	0.988245i	$0.451146\pi$
$6$	0	0
$7$	−0.639069	−0.241545	−0.120773	−	0.992680i	$-0.538537\pi$
$7$	−0.639069	−0.241545	−0.120773	+	0.992680i	$0.538537\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.683695	−0.216203
$11$	1.37847	0.415624	0.207812	−	0.978169i	$-0.433366\pi$
$11$	1.37847	0.415624	0.207812	+	0.978169i	$0.433366\pi$
$12$	0	0
$13$	1.22729	0.340388	0.170194	−	0.985411i	$-0.445561\pi$
$13$	1.22729	0.340388	0.170194	+	0.985411i	$0.445561\pi$
$14$	0.639069	0.170798
$15$	0	0
$16$	1.00000	0.250000
$17$	2.93824	0.712627	0.356313	−	0.934366i	$-0.384034\pi$
$17$	2.93824	0.712627	0.356313	+	0.934366i	$0.384034\pi$
$18$	0	0
$19$	−1.25651	−0.288262	−0.144131	−	0.989559i	$-0.546039\pi$
$19$	−1.25651	−0.288262	−0.144131	+	0.989559i	$0.546039\pi$
$20$	0.683695	0.152879
$21$	0	0
$22$	−1.37847	−0.293890
$23$	−7.55461	−1.57524	−0.787622	−	0.616158i	$-0.788688\pi$
$23$	−7.55461	−1.57524	−0.787622	+	0.616158i	$0.788688\pi$
$24$	0	0
$25$	−4.53256	−0.906512
$26$	−1.22729	−0.240691
$27$	0	0
$28$	−0.639069	−0.120773
$29$	2.51033	0.466157	0.233078	−	0.972458i	$-0.425120\pi$
$29$	2.51033	0.466157	0.233078	+	0.972458i	$0.425120\pi$
$30$	0	0
$31$	5.78698	1.03937	0.519686	−	0.854357i	$-0.326049\pi$
$31$	5.78698	1.03937	0.519686	+	0.854357i	$0.326049\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.93824	−0.503903
$35$	−0.436928	−0.0738543
$36$	0	0
$37$	−2.36988	−0.389606	−0.194803	−	0.980842i	$-0.562407\pi$
$37$	−2.36988	−0.389606	−0.194803	+	0.980842i	$0.562407\pi$
$38$	1.25651	0.203832
$39$	0	0
$40$	−0.683695	−0.108102
$41$	1.79132	0.279758	0.139879	−	0.990169i	$-0.455329\pi$
$41$	1.79132	0.279758	0.139879	+	0.990169i	$0.455329\pi$
$42$	0	0
$43$	−10.0366	−1.53056	−0.765282	−	0.643695i	$-0.777401\pi$
$43$	−10.0366	−1.53056	−0.765282	+	0.643695i	$0.777401\pi$
$44$	1.37847	0.207812
$45$	0	0
$46$	7.55461	1.11387
$47$	11.1030	1.61954	0.809770	−	0.586748i	$-0.199592\pi$
$47$	11.1030	1.61954	0.809770	+	0.586748i	$0.199592\pi$
$48$	0	0
$49$	−6.59159	−0.941656
$50$	4.53256	0.641001
$51$	0	0
$52$	1.22729	0.170194
$53$	−3.25589	−0.447231	−0.223616	−	0.974677i	$-0.571786\pi$
$53$	−3.25589	−0.447231	−0.223616	+	0.974677i	$0.571786\pi$
$54$	0	0
$55$	0.942452	0.127080
$56$	0.639069	0.0853992
$57$	0	0
$58$	−2.51033	−0.329623
$59$	−10.2114	−1.32941	−0.664705	−	0.747106i	$-0.731443\pi$
$59$	−10.2114	−1.32941	−0.664705	+	0.747106i	$0.731443\pi$
$60$	0	0
$61$	1.02628	0.131402	0.0657009	−	0.997839i	$-0.479072\pi$
$61$	1.02628	0.131402	0.0657009	+	0.997839i	$0.479072\pi$
$62$	−5.78698	−0.734947
$63$	0	0
$64$	1.00000	0.125000
$65$	0.839089	0.104076
$66$	0	0
$67$	8.00671	0.978175	0.489088	−	0.872235i	$-0.337330\pi$
$67$	8.00671	0.978175	0.489088	+	0.872235i	$0.337330\pi$
$68$	2.93824	0.356313
$69$	0	0
$70$	0.436928	0.0522229
$71$	−2.68750	−0.318947	−0.159474	−	0.987202i	$-0.550980\pi$
$71$	−2.68750	−0.318947	−0.159474	+	0.987202i	$0.550980\pi$
$72$	0	0
$73$	−15.4477	−1.80802	−0.904009	−	0.427513i	$-0.859390\pi$
$73$	−15.4477	−1.80802	−0.904009	+	0.427513i	$0.859390\pi$
$74$	2.36988	0.275493
$75$	0	0
$76$	−1.25651	−0.144131
$77$	−0.880936	−0.100392
$78$	0	0
$79$	−7.08243	−0.796836	−0.398418	−	0.917204i	$-0.630441\pi$
$79$	−7.08243	−0.796836	−0.398418	+	0.917204i	$0.630441\pi$
$80$	0.683695	0.0764394
$81$	0	0
$82$	−1.79132	−0.197819
$83$	−2.29043	−0.251407	−0.125703	−	0.992068i	$-0.540119\pi$
$83$	−2.29043	−0.251407	−0.125703	+	0.992068i	$0.540119\pi$
$84$	0	0
$85$	2.00886	0.217891
$86$	10.0366	1.08227
$87$	0	0
$88$	−1.37847	−0.146945
$89$	7.26133	0.769699	0.384850	−	0.922979i	$-0.374253\pi$
$89$	7.26133	0.769699	0.384850	+	0.922979i	$0.374253\pi$
$90$	0	0
$91$	−0.784321	−0.0822191
$92$	−7.55461	−0.787622
$93$	0	0
$94$	−11.1030	−1.14519
$95$	−0.859067	−0.0881384
$96$	0	0
$97$	0.495565	0.0503170	0.0251585	−	0.999683i	$-0.491991\pi$
$97$	0.495565	0.0503170	0.0251585	+	0.999683i	$0.491991\pi$
$98$	6.59159	0.665851
$99$	0	0
$100$	−4.53256	−0.453256
$101$	−18.5785	−1.84863	−0.924316	−	0.381628i	$-0.875363\pi$
$101$	−18.5785	−1.84863	−0.924316	+	0.381628i	$0.875363\pi$
$102$	0	0
$103$	5.74954	0.566519	0.283260	−	0.959043i	$-0.408584\pi$
$103$	5.74954	0.566519	0.283260	+	0.959043i	$0.408584\pi$
$104$	−1.22729	−0.120345
$105$	0	0
$106$	3.25589	0.316240
$107$	5.00086	0.483451	0.241726	−	0.970345i	$-0.422287\pi$
$107$	5.00086	0.483451	0.241726	+	0.970345i	$0.422287\pi$
$108$	0	0
$109$	−7.81173	−0.748228	−0.374114	−	0.927383i	$-0.622053\pi$
$109$	−7.81173	−0.748228	−0.374114	+	0.927383i	$0.622053\pi$
$110$	−0.942452	−0.0898593
$111$	0	0
$112$	−0.639069	−0.0603863
$113$	18.0793	1.70076	0.850379	−	0.526171i	$-0.176373\pi$
$113$	18.0793	1.70076	0.850379	+	0.526171i	$0.176373\pi$
$114$	0	0
$115$	−5.16505	−0.481643
$116$	2.51033	0.233078
$117$	0	0
$118$	10.2114	0.940034
$119$	−1.87774	−0.172132
$120$	0	0
$121$	−9.09982	−0.827257
$122$	−1.02628	−0.0929150
$123$	0	0
$124$	5.78698	0.519686
$125$	−6.51736	−0.582931
$126$	0	0
$127$	12.3765	1.09824	0.549120	−	0.835743i	$-0.314963\pi$
$127$	12.3765	1.09824	0.549120	+	0.835743i	$0.314963\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−0.839089	−0.0735930
$131$	15.3165	1.33821	0.669104	−	0.743169i	$-0.266678\pi$
$131$	15.3165	1.33821	0.669104	+	0.743169i	$0.266678\pi$
$132$	0	0
$133$	0.802994	0.0696284
$134$	−8.00671	−0.691674
$135$	0	0
$136$	−2.93824	−0.251952
$137$	−8.12377	−0.694060	−0.347030	−	0.937854i	$-0.612810\pi$
$137$	−8.12377	−0.694060	−0.347030	+	0.937854i	$0.612810\pi$
$138$	0	0
$139$	−15.8878	−1.34759	−0.673793	−	0.738920i	$-0.735336\pi$
$139$	−15.8878	−1.34759	−0.673793	+	0.738920i	$0.735336\pi$
$140$	−0.436928	−0.0369272
$141$	0	0
$142$	2.68750	0.225530
$143$	1.69178	0.141473
$144$	0	0
$145$	1.71630	0.142531
$146$	15.4477	1.27846
$147$	0	0
$148$	−2.36988	−0.194803
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−4.92235	−0.400576	−0.200288	−	0.979737i	$-0.564188\pi$
$151$	−4.92235	−0.400576	−0.200288	+	0.979737i	$0.564188\pi$
$152$	1.25651	0.101916
$153$	0	0
$154$	0.880936	0.0709879
$155$	3.95653	0.317796
$156$	0	0
$157$	−1.05732	−0.0843835	−0.0421918	−	0.999110i	$-0.513434\pi$
$157$	−1.05732	−0.0843835	−0.0421918	+	0.999110i	$0.513434\pi$
$158$	7.08243	0.563448
$159$	0	0
$160$	−0.683695	−0.0540508
$161$	4.82791	0.380493
$162$	0	0
$163$	5.30856	0.415798	0.207899	−	0.978150i	$-0.433337\pi$
$163$	5.30856	0.415798	0.207899	+	0.978150i	$0.433337\pi$
$164$	1.79132	0.139879
$165$	0	0
$166$	2.29043	0.177771
$167$	7.75142	0.599823	0.299912	−	0.953967i	$-0.403043\pi$
$167$	7.75142	0.599823	0.299912	+	0.953967i	$0.403043\pi$
$168$	0	0
$169$	−11.4938	−0.884136
$170$	−2.00886	−0.154072
$171$	0	0
$172$	−10.0366	−0.765282
$173$	1.91298	0.145441	0.0727206	−	0.997352i	$-0.476832\pi$
$173$	1.91298	0.145441	0.0727206	+	0.997352i	$0.476832\pi$
$174$	0	0
$175$	2.89662	0.218964
$176$	1.37847	0.103906
$177$	0	0
$178$	−7.26133	−0.544260
$179$	23.0724	1.72451	0.862257	−	0.506470i	$-0.169050\pi$
$179$	23.0724	1.72451	0.862257	+	0.506470i	$0.169050\pi$
$180$	0	0
$181$	−0.420424	−0.0312499	−0.0156250	−	0.999878i	$-0.504974\pi$
$181$	−0.420424	−0.0312499	−0.0156250	+	0.999878i	$0.504974\pi$
$182$	0.784321	0.0581377
$183$	0	0
$184$	7.55461	0.556933
$185$	−1.62027	−0.119125
$186$	0	0
$187$	4.05027	0.296185
$188$	11.1030	0.809770
$189$	0	0
$190$	0.859067	0.0623232
$191$	−13.9592	−1.01005	−0.505026	−	0.863104i	$-0.668517\pi$
$191$	−13.9592	−1.01005	−0.505026	+	0.863104i	$0.668517\pi$
$192$	0	0
$193$	−26.1712	−1.88384	−0.941922	−	0.335831i	$-0.890983\pi$
$193$	−26.1712	−1.88384	−0.941922	+	0.335831i	$0.890983\pi$
$194$	−0.495565	−0.0355795
$195$	0	0
$196$	−6.59159	−0.470828
$197$	13.2324	0.942768	0.471384	−	0.881928i	$-0.343755\pi$
$197$	13.2324	0.942768	0.471384	+	0.881928i	$0.343755\pi$
$198$	0	0
$199$	−22.0979	−1.56648	−0.783238	−	0.621722i	$-0.786433\pi$
$199$	−22.0979	−1.56648	−0.783238	+	0.621722i	$0.786433\pi$
$200$	4.53256	0.320500
$201$	0	0
$202$	18.5785	1.30718
$203$	−1.60428	−0.112598
$204$	0	0
$205$	1.22472	0.0855380
$206$	−5.74954	−0.400589
$207$	0	0
$208$	1.22729	0.0850970
$209$	−1.73205	−0.119809
$210$	0	0
$211$	9.79151	0.674075	0.337038	−	0.941491i	$-0.390575\pi$
$211$	9.79151	0.674075	0.337038	+	0.941491i	$0.390575\pi$
$212$	−3.25589	−0.223616
$213$	0	0
$214$	−5.00086	−0.341852
$215$	−6.86196	−0.467982
$216$	0	0
$217$	−3.69828	−0.251055
$218$	7.81173	0.529077
$219$	0	0
$220$	0.942452	0.0635401
$221$	3.60606	0.242570
$222$	0	0
$223$	−24.0602	−1.61119	−0.805597	−	0.592465i	$-0.798155\pi$
$223$	−24.0602	−1.61119	−0.805597	+	0.592465i	$0.798155\pi$
$224$	0.639069	0.0426996
$225$	0	0
$226$	−18.0793	−1.20262
$227$	14.1524	0.939327	0.469664	−	0.882845i	$-0.344375\pi$
$227$	14.1524	0.939327	0.469664	+	0.882845i	$0.344375\pi$
$228$	0	0
$229$	−7.88051	−0.520759	−0.260379	−	0.965506i	$-0.583848\pi$
$229$	−7.88051	−0.520759	−0.260379	+	0.965506i	$0.583848\pi$
$230$	5.16505	0.340573
$231$	0	0
$232$	−2.51033	−0.164811
$233$	4.07402	0.266898	0.133449	−	0.991056i	$-0.457395\pi$
$233$	4.07402	0.266898	0.133449	+	0.991056i	$0.457395\pi$
$234$	0	0
$235$	7.59107	0.495187
$236$	−10.2114	−0.664705
$237$	0	0
$238$	1.87774	0.121715
$239$	−13.7267	−0.887909	−0.443954	−	0.896049i	$-0.646425\pi$
$239$	−13.7267	−0.887909	−0.443954	+	0.896049i	$0.646425\pi$
$240$	0	0
$241$	17.2144	1.10888	0.554439	−	0.832225i	$-0.312933\pi$
$241$	17.2144	1.10888	0.554439	+	0.832225i	$0.312933\pi$
$242$	9.09982	0.584959
$243$	0	0
$244$	1.02628	0.0657009
$245$	−4.50664	−0.287918
$246$	0	0
$247$	−1.54209	−0.0981210
$248$	−5.78698	−0.367473
$249$	0	0
$250$	6.51736	0.412194
$251$	11.7758	0.743282	0.371641	−	0.928377i	$-0.378795\pi$
$251$	11.7758	0.743282	0.371641	+	0.928377i	$0.378795\pi$
$252$	0	0
$253$	−10.4138	−0.654709
$254$	−12.3765	−0.776573
$255$	0	0
$256$	1.00000	0.0625000
$257$	−21.0319	−1.31193	−0.655966	−	0.754791i	$-0.727738\pi$
$257$	−21.0319	−1.31193	−0.655966	+	0.754791i	$0.727738\pi$
$258$	0	0
$259$	1.51452	0.0941074
$260$	0.839089	0.0520381
$261$	0	0
$262$	−15.3165	−0.946256
$263$	−16.8639	−1.03987	−0.519936	−	0.854205i	$-0.674044\pi$
$263$	−16.8639	−1.03987	−0.519936	+	0.854205i	$0.674044\pi$
$264$	0	0
$265$	−2.22604	−0.136744
$266$	−0.802994	−0.0492347
$267$	0	0
$268$	8.00671	0.489088
$269$	−22.4206	−1.36701	−0.683504	−	0.729947i	$-0.739545\pi$
$269$	−22.4206	−1.36701	−0.683504	+	0.729947i	$0.739545\pi$
$270$	0	0
$271$	−13.5262	−0.821661	−0.410830	−	0.911712i	$-0.634761\pi$
$271$	−13.5262	−0.821661	−0.410830	+	0.911712i	$0.634761\pi$
$272$	2.93824	0.178157
$273$	0	0
$274$	8.12377	0.490775
$275$	−6.24799	−0.376768
$276$	0	0
$277$	11.6854	0.702109	0.351054	−	0.936355i	$-0.385823\pi$
$277$	11.6854	0.702109	0.351054	+	0.936355i	$0.385823\pi$
$278$	15.8878	0.952887
$279$	0	0
$280$	0.436928	0.0261115
$281$	1.34141	0.0800215	0.0400108	−	0.999199i	$-0.487261\pi$
$281$	1.34141	0.0800215	0.0400108	+	0.999199i	$0.487261\pi$
$282$	0	0
$283$	24.3292	1.44622	0.723110	−	0.690732i	$-0.242712\pi$
$283$	24.3292	1.44622	0.723110	+	0.690732i	$0.242712\pi$
$284$	−2.68750	−0.159474
$285$	0	0
$286$	−1.69178	−0.100037
$287$	−1.14478	−0.0675742
$288$	0	0
$289$	−8.36677	−0.492163
$290$	−1.71630	−0.100785
$291$	0	0
$292$	−15.4477	−0.904009
$293$	−4.76293	−0.278253	−0.139127	−	0.990275i	$-0.544430\pi$
$293$	−4.76293	−0.278253	−0.139127	+	0.990275i	$0.544430\pi$
$294$	0	0
$295$	−6.98147	−0.406477
$296$	2.36988	0.137746
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−9.27167	−0.536194
$300$	0	0
$301$	6.41407	0.369701
$302$	4.92235	0.283250
$303$	0	0
$304$	−1.25651	−0.0720655
$305$	0.701663	0.0401771
$306$	0	0
$307$	−20.7130	−1.18215	−0.591077	−	0.806615i	$-0.701297\pi$
$307$	−20.7130	−1.18215	−0.591077	+	0.806615i	$0.701297\pi$
$308$	−0.880936	−0.0501960
$309$	0	0
$310$	−3.95653	−0.224716
$311$	17.6892	1.00306	0.501532	−	0.865139i	$-0.332770\pi$
$311$	17.6892	1.00306	0.501532	+	0.865139i	$0.332770\pi$
$312$	0	0
$313$	1.72775	0.0976583	0.0488291	−	0.998807i	$-0.484451\pi$
$313$	1.72775	0.0976583	0.0488291	+	0.998807i	$0.484451\pi$
$314$	1.05732	0.0596682
$315$	0	0
$316$	−7.08243	−0.398418
$317$	−27.4073	−1.53935	−0.769674	−	0.638437i	$-0.779581\pi$
$317$	−27.4073	−1.53935	−0.769674	+	0.638437i	$0.779581\pi$
$318$	0	0
$319$	3.46041	0.193746
$320$	0.683695	0.0382197
$321$	0	0
$322$	−4.82791	−0.269049
$323$	−3.69191	−0.205423
$324$	0	0
$325$	−5.56275	−0.308566
$326$	−5.30856	−0.294014
$327$	0	0
$328$	−1.79132	−0.0989093
$329$	−7.09558	−0.391192
$330$	0	0
$331$	−10.4910	−0.576636	−0.288318	−	0.957535i	$-0.593096\pi$
$331$	−10.4910	−0.576636	−0.288318	+	0.957535i	$0.593096\pi$
$332$	−2.29043	−0.125703
$333$	0	0
$334$	−7.75142	−0.424139
$335$	5.47415	0.299085
$336$	0	0
$337$	33.1109	1.80366	0.901832	−	0.432086i	$-0.142222\pi$
$337$	33.1109	1.80366	0.901832	+	0.432086i	$0.142222\pi$
$338$	11.4938	0.625179
$339$	0	0
$340$	2.00886	0.108946
$341$	7.97717	0.431988
$342$	0	0
$343$	8.68596	0.468998
$344$	10.0366	0.541136
$345$	0	0
$346$	−1.91298	−0.102843
$347$	24.7393	1.32807	0.664037	−	0.747700i	$-0.268842\pi$
$347$	24.7393	1.32807	0.664037	+	0.747700i	$0.268842\pi$
$348$	0	0
$349$	−27.8523	−1.49090	−0.745450	−	0.666561i	$-0.767765\pi$
$349$	−27.8523	−1.49090	−0.745450	+	0.666561i	$0.767765\pi$
$350$	−2.89662	−0.154831
$351$	0	0
$352$	−1.37847	−0.0734726
$353$	−18.3837	−0.978466	−0.489233	−	0.872153i	$-0.662723\pi$
$353$	−18.3837	−0.978466	−0.489233	+	0.872153i	$0.662723\pi$
$354$	0	0
$355$	−1.83743	−0.0975206
$356$	7.26133	0.384850
$357$	0	0
$358$	−23.0724	−1.21942
$359$	−23.2473	−1.22695	−0.613473	−	0.789716i	$-0.710228\pi$
$359$	−23.2473	−1.22695	−0.613473	+	0.789716i	$0.710228\pi$
$360$	0	0
$361$	−17.4212	−0.916905
$362$	0.420424	0.0220970
$363$	0	0
$364$	−0.784321	−0.0411096
$365$	−10.5615	−0.552816
$366$	0	0
$367$	−2.85101	−0.148821	−0.0744106	−	0.997228i	$-0.523708\pi$
$367$	−2.85101	−0.148821	−0.0744106	+	0.997228i	$0.523708\pi$
$368$	−7.55461	−0.393811
$369$	0	0
$370$	1.62027	0.0842340
$371$	2.08074	0.108027
$372$	0	0
$373$	−4.27241	−0.221217	−0.110608	−	0.993864i	$-0.535280\pi$
$373$	−4.27241	−0.221217	−0.110608	+	0.993864i	$0.535280\pi$
$374$	−4.05027	−0.209434
$375$	0	0
$376$	−11.1030	−0.572594
$377$	3.08090	0.158674
$378$	0	0
$379$	2.12935	0.109377	0.0546887	−	0.998503i	$-0.482583\pi$
$379$	2.12935	0.109377	0.0546887	+	0.998503i	$0.482583\pi$
$380$	−0.859067	−0.0440692
$381$	0	0
$382$	13.9592	0.714214
$383$	27.7927	1.42014	0.710071	−	0.704131i	$-0.248663\pi$
$383$	27.7927	1.42014	0.710071	+	0.704131i	$0.248663\pi$
$384$	0	0
$385$	−0.602292	−0.0306956
$386$	26.1712	1.33208
$387$	0	0
$388$	0.495565	0.0251585
$389$	−2.18146	−0.110604	−0.0553021	−	0.998470i	$-0.517612\pi$
$389$	−2.18146	−0.110604	−0.0553021	+	0.998470i	$0.517612\pi$
$390$	0	0
$391$	−22.1972	−1.12256
$392$	6.59159	0.332926
$393$	0	0
$394$	−13.2324	−0.666638
$395$	−4.84222	−0.243639
$396$	0	0
$397$	36.1692	1.81528	0.907640	−	0.419750i	$-0.137882\pi$
$397$	36.1692	1.81528	0.907640	+	0.419750i	$0.137882\pi$
$398$	22.0979	1.10767
$399$	0	0
$400$	−4.53256	−0.226628
$401$	38.0499	1.90012	0.950060	−	0.312068i	$-0.101022\pi$
$401$	38.0499	1.90012	0.950060	+	0.312068i	$0.101022\pi$
$402$	0	0
$403$	7.10228	0.353790
$404$	−18.5785	−0.924316
$405$	0	0
$406$	1.60428	0.0796189
$407$	−3.26680	−0.161929
$408$	0	0
$409$	−3.27404	−0.161891	−0.0809454	−	0.996719i	$-0.525794\pi$
$409$	−3.27404	−0.161891	−0.0809454	+	0.996719i	$0.525794\pi$
$410$	−1.22472	−0.0604845
$411$	0	0
$412$	5.74954	0.283260
$413$	6.52578	0.321113
$414$	0	0
$415$	−1.56595	−0.0768696
$416$	−1.22729	−0.0601727
$417$	0	0
$418$	1.73205	0.0847175
$419$	4.93454	0.241068	0.120534	−	0.992709i	$-0.461539\pi$
$419$	4.93454	0.241068	0.120534	+	0.992709i	$0.461539\pi$
$420$	0	0
$421$	−29.3725	−1.43153	−0.715764	−	0.698342i	$-0.753922\pi$
$421$	−29.3725	−1.43153	−0.715764	+	0.698342i	$0.753922\pi$
$422$	−9.79151	−0.476643
$423$	0	0
$424$	3.25589	0.158120
$425$	−13.3177	−0.646005
$426$	0	0
$427$	−0.655864	−0.0317395
$428$	5.00086	0.241726
$429$	0	0
$430$	6.86196	0.330913
$431$	−2.39153	−0.115196	−0.0575981	−	0.998340i	$-0.518344\pi$
$431$	−2.39153	−0.115196	−0.0575981	+	0.998340i	$0.518344\pi$
$432$	0	0
$433$	−32.0988	−1.54257	−0.771285	−	0.636490i	$-0.780385\pi$
$433$	−32.0988	−1.54257	−0.771285	+	0.636490i	$0.780385\pi$
$434$	3.69828	0.177523
$435$	0	0
$436$	−7.81173	−0.374114
$437$	9.49241	0.454083
$438$	0	0
$439$	13.9582	0.666188	0.333094	−	0.942894i	$-0.391907\pi$
$439$	13.9582	0.666188	0.333094	+	0.942894i	$0.391907\pi$
$440$	−0.942452	−0.0449296
$441$	0	0
$442$	−3.60606	−0.171523
$443$	25.1491	1.19487	0.597436	−	0.801917i	$-0.296186\pi$
$443$	25.1491	1.19487	0.597436	+	0.801917i	$0.296186\pi$
$444$	0	0
$445$	4.96453	0.235341
$446$	24.0602	1.13929
$447$	0	0
$448$	−0.639069	−0.0301932
$449$	11.0525	0.521599	0.260799	−	0.965393i	$-0.416014\pi$
$449$	11.0525	0.521599	0.260799	+	0.965393i	$0.416014\pi$
$450$	0	0
$451$	2.46928	0.116274
$452$	18.0793	0.850379
$453$	0	0
$454$	−14.1524	−0.664205
$455$	−0.536236	−0.0251391
$456$	0	0
$457$	−0.164071	−0.00767492	−0.00383746	−	0.999993i	$-0.501222\pi$
$457$	−0.164071	−0.00767492	−0.00383746	+	0.999993i	$0.501222\pi$
$458$	7.88051	0.368232
$459$	0	0
$460$	−5.16505	−0.240822
$461$	−9.62526	−0.448293	−0.224147	−	0.974555i	$-0.571959\pi$
$461$	−9.62526	−0.448293	−0.224147	+	0.974555i	$0.571959\pi$
$462$	0	0
$463$	37.9510	1.76373	0.881866	−	0.471499i	$-0.156287\pi$
$463$	37.9510	1.76373	0.881866	+	0.471499i	$0.156287\pi$
$464$	2.51033	0.116539
$465$	0	0
$466$	−4.07402	−0.188725
$467$	−27.8325	−1.28793	−0.643967	−	0.765054i	$-0.722712\pi$
$467$	−27.8325	−1.28793	−0.643967	+	0.765054i	$0.722712\pi$
$468$	0	0
$469$	−5.11684	−0.236274
$470$	−7.59107	−0.350150
$471$	0	0
$472$	10.2114	0.470017
$473$	−13.8351	−0.636139
$474$	0	0
$475$	5.69519	0.261313
$476$	−1.87774	−0.0860659
$477$	0	0
$478$	13.7267	0.627846
$479$	−14.8934	−0.680495	−0.340248	−	0.940336i	$-0.610511\pi$
$479$	−14.8934	−0.680495	−0.340248	+	0.940336i	$0.610511\pi$
$480$	0	0
$481$	−2.90852	−0.132617
$482$	−17.2144	−0.784095
$483$	0	0
$484$	−9.09982	−0.413628
$485$	0.338815	0.0153848
$486$	0	0
$487$	18.5223	0.839326	0.419663	−	0.907680i	$-0.362148\pi$
$487$	18.5223	0.839326	0.419663	+	0.907680i	$0.362148\pi$
$488$	−1.02628	−0.0464575
$489$	0	0
$490$	4.50664	0.203589
$491$	−33.3676	−1.50586	−0.752929	−	0.658102i	$-0.771359\pi$
$491$	−33.3676	−1.50586	−0.752929	+	0.658102i	$0.771359\pi$
$492$	0	0
$493$	7.37595	0.332196
$494$	1.54209	0.0693820
$495$	0	0
$496$	5.78698	0.259843
$497$	1.71750	0.0770403
$498$	0	0
$499$	−27.6463	−1.23762	−0.618808	−	0.785542i	$-0.712384\pi$
$499$	−27.6463	−1.23762	−0.618808	+	0.785542i	$0.712384\pi$
$500$	−6.51736	−0.291465
$501$	0	0
$502$	−11.7758	−0.525580
$503$	20.7047	0.923176	0.461588	−	0.887094i	$-0.347280\pi$
$503$	20.7047	0.923176	0.461588	+	0.887094i	$0.347280\pi$
$504$	0	0
$505$	−12.7020	−0.565233
$506$	10.4138	0.462949
$507$	0	0
$508$	12.3765	0.549120
$509$	−17.9958	−0.797648	−0.398824	−	0.917028i	$-0.630582\pi$
$509$	−17.9958	−0.797648	−0.398824	+	0.917028i	$0.630582\pi$
$510$	0	0
$511$	9.87216	0.436719
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	21.0319	0.927675
$515$	3.93093	0.173218
$516$	0	0
$517$	15.3051	0.673119
$518$	−1.51452	−0.0665440
$519$	0	0
$520$	−0.839089	−0.0367965
$521$	−26.7908	−1.17373	−0.586864	−	0.809686i	$-0.699637\pi$
$521$	−26.7908	−1.17373	−0.586864	+	0.809686i	$0.699637\pi$
$522$	0	0
$523$	−33.3656	−1.45897	−0.729487	−	0.683995i	$-0.760241\pi$
$523$	−33.3656	−1.45897	−0.729487	+	0.683995i	$0.760241\pi$
$524$	15.3165	0.669104
$525$	0	0
$526$	16.8639	0.735300
$527$	17.0035	0.740684
$528$	0	0
$529$	34.0721	1.48140
$530$	2.22604	0.0966928
$531$	0	0
$532$	0.802994	0.0348142
$533$	2.19847	0.0952261
$534$	0	0
$535$	3.41906	0.147819
$536$	−8.00671	−0.345837
$537$	0	0
$538$	22.4206	0.966621
$539$	−9.08630	−0.391375
$540$	0	0
$541$	−14.4014	−0.619165	−0.309582	−	0.950873i	$-0.600189\pi$
$541$	−14.4014	−0.619165	−0.309582	+	0.950873i	$0.600189\pi$
$542$	13.5262	0.581002
$543$	0	0
$544$	−2.93824	−0.125976
$545$	−5.34084	−0.228776
$546$	0	0
$547$	−24.5526	−1.04979	−0.524897	−	0.851166i	$-0.675896\pi$
$547$	−24.5526	−1.04979	−0.524897	+	0.851166i	$0.675896\pi$
$548$	−8.12377	−0.347030
$549$	0	0
$550$	6.24799	0.266415
$551$	−3.15425	−0.134375
$552$	0	0
$553$	4.52616	0.192472
$554$	−11.6854	−0.496466
$555$	0	0
$556$	−15.8878	−0.673793
$557$	17.0938	0.724290	0.362145	−	0.932122i	$-0.382045\pi$
$557$	17.0938	0.724290	0.362145	+	0.932122i	$0.382045\pi$
$558$	0	0
$559$	−12.3178	−0.520986
$560$	−0.436928	−0.0184636
$561$	0	0
$562$	−1.34141	−0.0565838
$563$	29.1527	1.22864	0.614321	−	0.789056i	$-0.289430\pi$
$563$	29.1527	1.22864	0.614321	+	0.789056i	$0.289430\pi$
$564$	0	0
$565$	12.3607	0.520020
$566$	−24.3292	−1.02263
$567$	0	0
$568$	2.68750	0.112765
$569$	−17.0923	−0.716545	−0.358273	−	0.933617i	$-0.616634\pi$
$569$	−17.0923	−0.716545	−0.358273	+	0.933617i	$0.616634\pi$
$570$	0	0
$571$	−6.80880	−0.284940	−0.142470	−	0.989799i	$-0.545504\pi$
$571$	−6.80880	−0.284940	−0.142470	+	0.989799i	$0.545504\pi$
$572$	1.69178	0.0707367
$573$	0	0
$574$	1.14478	0.0477821
$575$	34.2417	1.42798
$576$	0	0
$577$	42.9978	1.79002	0.895012	−	0.446043i	$-0.147167\pi$
$577$	42.9978	1.79002	0.895012	+	0.446043i	$0.147167\pi$
$578$	8.36677	0.348012
$579$	0	0
$580$	1.71630	0.0712655
$581$	1.46374	0.0607262
$582$	0	0
$583$	−4.48814	−0.185880
$584$	15.4477	0.639231
$585$	0	0
$586$	4.76293	0.196755
$587$	−40.7800	−1.68317	−0.841585	−	0.540125i	$-0.818377\pi$
$587$	−40.7800	−1.68317	−0.841585	+	0.540125i	$0.818377\pi$
$588$	0	0
$589$	−7.27137	−0.299612
$590$	6.98147	0.287423
$591$	0	0
$592$	−2.36988	−0.0974014
$593$	−13.2489	−0.544066	−0.272033	−	0.962288i	$-0.587696\pi$
$593$	−13.2489	−0.544066	−0.272033	+	0.962288i	$0.587696\pi$
$594$	0	0
$595$	−1.28380	−0.0526306
$596$	1.00000	0.0409616
$597$	0	0
$598$	9.27167	0.379147
$599$	42.3211	1.72919	0.864597	−	0.502467i	$-0.167574\pi$
$599$	42.3211	1.72919	0.864597	+	0.502467i	$0.167574\pi$
$600$	0	0
$601$	44.3599	1.80948	0.904739	−	0.425966i	$-0.140066\pi$
$601$	44.3599	1.80948	0.904739	+	0.425966i	$0.140066\pi$
$602$	−6.41407	−0.261418
$603$	0	0
$604$	−4.92235	−0.200288
$605$	−6.22150	−0.252940
$606$	0	0
$607$	−8.49059	−0.344622	−0.172311	−	0.985043i	$-0.555123\pi$
$607$	−8.49059	−0.344622	−0.172311	+	0.985043i	$0.555123\pi$
$608$	1.25651	0.0509580
$609$	0	0
$610$	−0.701663	−0.0284095
$611$	13.6266	0.551272
$612$	0	0
$613$	−40.3767	−1.63080	−0.815399	−	0.578899i	$-0.803482\pi$
$613$	−40.3767	−1.63080	−0.815399	+	0.578899i	$0.803482\pi$
$614$	20.7130	0.835910
$615$	0	0
$616$	0.880936	0.0354939
$617$	−41.2562	−1.66091	−0.830455	−	0.557085i	$-0.811920\pi$
$617$	−41.2562	−1.66091	−0.830455	+	0.557085i	$0.811920\pi$
$618$	0	0
$619$	25.1365	1.01032	0.505160	−	0.863025i	$-0.331433\pi$
$619$	25.1365	1.01032	0.505160	+	0.863025i	$0.331433\pi$
$620$	3.95653	0.158898
$621$	0	0
$622$	−17.6892	−0.709274
$623$	−4.64049	−0.185917
$624$	0	0
$625$	18.2069	0.728277
$626$	−1.72775	−0.0690548
$627$	0	0
$628$	−1.05732	−0.0421918
$629$	−6.96326	−0.277643
$630$	0	0
$631$	−25.1878	−1.00271	−0.501355	−	0.865242i	$-0.667165\pi$
$631$	−25.1878	−1.00271	−0.501355	+	0.865242i	$0.667165\pi$
$632$	7.08243	0.281724
$633$	0	0
$634$	27.4073	1.08848
$635$	8.46178	0.335795
$636$	0	0
$637$	−8.08977	−0.320528
$638$	−3.46041	−0.136999
$639$	0	0
$640$	−0.683695	−0.0270254
$641$	22.2446	0.878609	0.439304	−	0.898338i	$-0.355225\pi$
$641$	22.2446	0.878609	0.439304	+	0.898338i	$0.355225\pi$
$642$	0	0
$643$	−35.5526	−1.40206	−0.701028	−	0.713133i	$-0.747275\pi$
$643$	−35.5526	−1.40206	−0.701028	+	0.713133i	$0.747275\pi$
$644$	4.82791	0.190247
$645$	0	0
$646$	3.69191	0.145256
$647$	−36.5412	−1.43658	−0.718291	−	0.695742i	$-0.755076\pi$
$647$	−36.5412	−1.43658	−0.718291	+	0.695742i	$0.755076\pi$
$648$	0	0
$649$	−14.0761	−0.552534
$650$	5.56275	0.218189
$651$	0	0
$652$	5.30856	0.207899
$653$	19.4662	0.761771	0.380885	−	0.924622i	$-0.375619\pi$
$653$	19.4662	0.761771	0.380885	+	0.924622i	$0.375619\pi$
$654$	0	0
$655$	10.4718	0.409167
$656$	1.79132	0.0699394
$657$	0	0
$658$	7.09558	0.276615
$659$	1.02847	0.0400637	0.0200318	−	0.999799i	$-0.493623\pi$
$659$	1.02847	0.0400637	0.0200318	+	0.999799i	$0.493623\pi$
$660$	0	0
$661$	−22.9972	−0.894488	−0.447244	−	0.894412i	$-0.647595\pi$
$661$	−22.9972	−0.894488	−0.447244	+	0.894412i	$0.647595\pi$
$662$	10.4910	0.407743
$663$	0	0
$664$	2.29043	0.0888857
$665$	0.549003	0.0212894
$666$	0	0
$667$	−18.9646	−0.734311
$668$	7.75142	0.299912
$669$	0	0
$670$	−5.47415	−0.211485
$671$	1.41469	0.0546137
$672$	0	0
$673$	−13.8495	−0.533858	−0.266929	−	0.963716i	$-0.586009\pi$
$673$	−13.8495	−0.533858	−0.266929	+	0.963716i	$0.586009\pi$
$674$	−33.1109	−1.27538
$675$	0	0
$676$	−11.4938	−0.442068
$677$	10.2057	0.392235	0.196118	−	0.980580i	$-0.437167\pi$
$677$	10.2057	0.392235	0.196118	+	0.980580i	$0.437167\pi$
$678$	0	0
$679$	−0.316700	−0.0121538
$680$	−2.00886	−0.0770361
$681$	0	0
$682$	−7.97717	−0.305461
$683$	−17.0019	−0.650561	−0.325280	−	0.945618i	$-0.605459\pi$
$683$	−17.0019	−0.650561	−0.325280	+	0.945618i	$0.605459\pi$
$684$	0	0
$685$	−5.55418	−0.212214
$686$	−8.68596	−0.331632
$687$	0	0
$688$	−10.0366	−0.382641
$689$	−3.99591	−0.152232
$690$	0	0
$691$	−15.9421	−0.606466	−0.303233	−	0.952916i	$-0.598066\pi$
$691$	−15.9421	−0.606466	−0.303233	+	0.952916i	$0.598066\pi$
$692$	1.91298	0.0727206
$693$	0	0
$694$	−24.7393	−0.939090
$695$	−10.8624	−0.412035
$696$	0	0
$697$	5.26333	0.199363
$698$	27.8523	1.05423
$699$	0	0
$700$	2.89662	0.109482
$701$	−29.6601	−1.12024	−0.560122	−	0.828410i	$-0.689246\pi$
$701$	−29.6601	−1.12024	−0.560122	+	0.828410i	$0.689246\pi$
$702$	0	0
$703$	2.97777	0.112309
$704$	1.37847	0.0519530
$705$	0	0
$706$	18.3837	0.691880
$707$	11.8730	0.446528
$708$	0	0
$709$	28.9912	1.08879	0.544394	−	0.838829i	$-0.316760\pi$
$709$	28.9912	1.08879	0.544394	+	0.838829i	$0.316760\pi$
$710$	1.83743	0.0689575
$711$	0	0
$712$	−7.26133	−0.272130
$713$	−43.7183	−1.63726
$714$	0	0
$715$	1.15666	0.0432566
$716$	23.0724	0.862257
$717$	0	0
$718$	23.2473	0.867582
$719$	−28.5349	−1.06417	−0.532086	−	0.846690i	$-0.678592\pi$
$719$	−28.5349	−1.06417	−0.532086	+	0.846690i	$0.678592\pi$
$720$	0	0
$721$	−3.67435	−0.136840
$722$	17.4212	0.648350
$723$	0	0
$724$	−0.420424	−0.0156250
$725$	−11.3782	−0.422577
$726$	0	0
$727$	−26.4341	−0.980388	−0.490194	−	0.871613i	$-0.663074\pi$
$727$	−26.4341	−0.980388	−0.490194	+	0.871613i	$0.663074\pi$
$728$	0.784321	0.0290689
$729$	0	0
$730$	10.5615	0.390900
$731$	−29.4898	−1.09072
$732$	0	0
$733$	−31.2748	−1.15516	−0.577581	−	0.816334i	$-0.696003\pi$
$733$	−31.2748	−1.15516	−0.577581	+	0.816334i	$0.696003\pi$
$734$	2.85101	0.105233
$735$	0	0
$736$	7.55461	0.278467
$737$	11.0370	0.406553
$738$	0	0
$739$	−27.9612	−1.02857	−0.514284	−	0.857620i	$-0.671942\pi$
$739$	−27.9612	−1.02857	−0.514284	+	0.857620i	$0.671942\pi$
$740$	−1.62027	−0.0595625
$741$	0	0
$742$	−2.08074	−0.0763863
$743$	−20.2955	−0.744570	−0.372285	−	0.928118i	$-0.621426\pi$
$743$	−20.2955	−0.744570	−0.372285	+	0.928118i	$0.621426\pi$
$744$	0	0
$745$	0.683695	0.0250486
$746$	4.27241	0.156424
$747$	0	0
$748$	4.05027	0.148092
$749$	−3.19589	−0.116775
$750$	0	0
$751$	39.7018	1.44874	0.724369	−	0.689412i	$-0.242131\pi$
$751$	39.7018	1.44874	0.724369	+	0.689412i	$0.242131\pi$
$752$	11.1030	0.404885
$753$	0	0
$754$	−3.08090	−0.112200
$755$	−3.36539	−0.122479
$756$	0	0
$757$	−1.35049	−0.0490843	−0.0245421	−	0.999699i	$-0.507813\pi$
$757$	−1.35049	−0.0490843	−0.0245421	+	0.999699i	$0.507813\pi$
$758$	−2.12935	−0.0773416
$759$	0	0
$760$	0.859067	0.0311616
$761$	−32.4735	−1.17716	−0.588582	−	0.808438i	$-0.700313\pi$
$761$	−32.4735	−1.17716	−0.588582	+	0.808438i	$0.700313\pi$
$762$	0	0
$763$	4.99223	0.180731
$764$	−13.9592	−0.505026
$765$	0	0
$766$	−27.7927	−1.00419
$767$	−12.5323	−0.452515
$768$	0	0
$769$	42.3996	1.52897	0.764485	−	0.644641i	$-0.222993\pi$
$769$	42.3996	1.52897	0.764485	+	0.644641i	$0.222993\pi$
$770$	0.602292	0.0217051
$771$	0	0
$772$	−26.1712	−0.941922
$773$	−47.7751	−1.71835	−0.859175	−	0.511682i	$-0.829022\pi$
$773$	−47.7751	−1.71835	−0.859175	+	0.511682i	$0.829022\pi$
$774$	0	0
$775$	−26.2298	−0.942203
$776$	−0.495565	−0.0177897
$777$	0	0
$778$	2.18146	0.0782090
$779$	−2.25081	−0.0806435
$780$	0	0
$781$	−3.70463	−0.132562
$782$	22.1972	0.793771
$783$	0	0
$784$	−6.59159	−0.235414
$785$	−0.722886	−0.0258009
$786$	0	0
$787$	−9.34661	−0.333171	−0.166585	−	0.986027i	$-0.553274\pi$
$787$	−9.34661	−0.333171	−0.166585	+	0.986027i	$0.553274\pi$
$788$	13.2324	0.471384
$789$	0	0
$790$	4.84222	0.172279
$791$	−11.5539	−0.410810
$792$	0	0
$793$	1.25954	0.0447276
$794$	−36.1692	−1.28360
$795$	0	0
$796$	−22.0979	−0.783238
$797$	−50.5937	−1.79212	−0.896061	−	0.443931i	$-0.853584\pi$
$797$	−50.5937	−1.79212	−0.896061	+	0.443931i	$0.853584\pi$
$798$	0	0
$799$	32.6232	1.15413
$800$	4.53256	0.160250
$801$	0	0
$802$	−38.0499	−1.34359
$803$	−21.2942	−0.751456
$804$	0	0
$805$	3.30082	0.116339
$806$	−7.10228	−0.250167
$807$	0	0
$808$	18.5785	0.653590
$809$	−3.65250	−0.128415	−0.0642075	−	0.997937i	$-0.520452\pi$
$809$	−3.65250	−0.128415	−0.0642075	+	0.997937i	$0.520452\pi$
$810$	0	0
$811$	−40.0359	−1.40585	−0.702925	−	0.711264i	$-0.748123\pi$
$811$	−40.0359	−1.40585	−0.702925	+	0.711264i	$0.748123\pi$
$812$	−1.60428	−0.0562990
$813$	0	0
$814$	3.26680	0.114501
$815$	3.62943	0.127134
$816$	0	0
$817$	12.6110	0.441204
$818$	3.27404	0.114474
$819$	0	0
$820$	1.22472	0.0427690
$821$	−19.9066	−0.694744	−0.347372	−	0.937727i	$-0.612926\pi$
$821$	−19.9066	−0.694744	−0.347372	+	0.937727i	$0.612926\pi$
$822$	0	0
$823$	−2.99252	−0.104313	−0.0521564	−	0.998639i	$-0.516609\pi$
$823$	−2.99252	−0.104313	−0.0521564	+	0.998639i	$0.516609\pi$
$824$	−5.74954	−0.200295
$825$	0	0
$826$	−6.52578	−0.227061
$827$	29.0535	1.01029	0.505145	−	0.863034i	$-0.331439\pi$
$827$	29.0535	1.01029	0.505145	+	0.863034i	$0.331439\pi$
$828$	0	0
$829$	50.9263	1.76874	0.884371	−	0.466785i	$-0.154588\pi$
$829$	50.9263	1.76874	0.884371	+	0.466785i	$0.154588\pi$
$830$	1.56595	0.0543550
$831$	0	0
$832$	1.22729	0.0425485
$833$	−19.3676	−0.671049
$834$	0	0
$835$	5.29961	0.183401
$836$	−1.73205	−0.0599043
$837$	0	0
$838$	−4.93454	−0.170461
$839$	−31.3519	−1.08239	−0.541194	−	0.840898i	$-0.682028\pi$
$839$	−31.3519	−1.08239	−0.541194	+	0.840898i	$0.682028\pi$
$840$	0	0
$841$	−22.6982	−0.782698
$842$	29.3725	1.01224
$843$	0	0
$844$	9.79151	0.337038
$845$	−7.85823	−0.270331
$846$	0	0
$847$	5.81542	0.199820
$848$	−3.25589	−0.111808
$849$	0	0
$850$	13.3177	0.456794
$851$	17.9035	0.613724
$852$	0	0
$853$	−29.7165	−1.01747	−0.508736	−	0.860923i	$-0.669887\pi$
$853$	−29.7165	−1.01747	−0.508736	+	0.860923i	$0.669887\pi$
$854$	0.655864	0.0224432
$855$	0	0
$856$	−5.00086	−0.170926
$857$	54.8894	1.87499	0.937493	−	0.348005i	$-0.113141\pi$
$857$	54.8894	1.87499	0.937493	+	0.348005i	$0.113141\pi$
$858$	0	0
$859$	21.5798	0.736295	0.368147	−	0.929767i	$-0.379992\pi$
$859$	21.5798	0.736295	0.368147	+	0.929767i	$0.379992\pi$
$860$	−6.86196	−0.233991
$861$	0	0
$862$	2.39153	0.0814560
$863$	20.8297	0.709051	0.354526	−	0.935046i	$-0.384642\pi$
$863$	20.8297	0.709051	0.354526	+	0.935046i	$0.384642\pi$
$864$	0	0
$865$	1.30790	0.0444698
$866$	32.0988	1.09076
$867$	0	0
$868$	−3.69828	−0.125528
$869$	−9.76291	−0.331184
$870$	0	0
$871$	9.82652	0.332959
$872$	7.81173	0.264538
$873$	0	0
$874$	−9.49241	−0.321085
$875$	4.16504	0.140804
$876$	0	0
$877$	19.7031	0.665328	0.332664	−	0.943045i	$-0.392053\pi$
$877$	19.7031	0.665328	0.332664	+	0.943045i	$0.392053\pi$
$878$	−13.9582	−0.471066
$879$	0	0
$880$	0.942452	0.0317700
$881$	−30.5817	−1.03032	−0.515162	−	0.857093i	$-0.672268\pi$
$881$	−30.5817	−1.03032	−0.515162	+	0.857093i	$0.672268\pi$
$882$	0	0
$883$	−12.4083	−0.417573	−0.208786	−	0.977961i	$-0.566951\pi$
$883$	−12.4083	−0.417573	−0.208786	+	0.977961i	$0.566951\pi$
$884$	3.60606	0.121285
$885$	0	0
$886$	−25.1491	−0.844902
$887$	−8.80165	−0.295530	−0.147765	−	0.989022i	$-0.547208\pi$
$887$	−8.80165	−0.295530	−0.147765	+	0.989022i	$0.547208\pi$
$888$	0	0
$889$	−7.90946	−0.265275
$890$	−4.96453	−0.166412
$891$	0	0
$892$	−24.0602	−0.805597
$893$	−13.9510	−0.466852
$894$	0	0
$895$	15.7745	0.527284
$896$	0.639069	0.0213498
$897$	0	0
$898$	−11.0525	−0.368826
$899$	14.5272	0.484510
$900$	0	0
$901$	−9.56657	−0.318709
$902$	−2.46928	−0.0822181
$903$	0	0
$904$	−18.0793	−0.601309
$905$	−0.287442	−0.00955490
$906$	0	0
$907$	−40.8567	−1.35662	−0.678312	−	0.734774i	$-0.737288\pi$
$907$	−40.8567	−1.35662	−0.678312	+	0.734774i	$0.737288\pi$
$908$	14.1524	0.469664
$909$	0	0
$910$	0.536236	0.0177760
$911$	−50.3927	−1.66958	−0.834792	−	0.550566i	$-0.814412\pi$
$911$	−50.3927	−1.66958	−0.834792	+	0.550566i	$0.814412\pi$
$912$	0	0
$913$	−3.15728	−0.104491
$914$	0.164071	0.00542699
$915$	0	0
$916$	−7.88051	−0.260379
$917$	−9.78829	−0.323238
$918$	0	0
$919$	4.21729	0.139116	0.0695578	−	0.997578i	$-0.477841\pi$
$919$	4.21729	0.139116	0.0695578	+	0.997578i	$0.477841\pi$
$920$	5.16505	0.170287
$921$	0	0
$922$	9.62526	0.316991
$923$	−3.29833	−0.108566
$924$	0	0
$925$	10.7416	0.353182
$926$	−37.9510	−1.24715
$927$	0	0
$928$	−2.51033	−0.0824057
$929$	−31.9962	−1.04976	−0.524881	−	0.851176i	$-0.675890\pi$
$929$	−31.9962	−1.04976	−0.524881	+	0.851176i	$0.675890\pi$
$930$	0	0
$931$	8.28237	0.271444
$932$	4.07402	0.133449
$933$	0	0
$934$	27.8325	0.910706
$935$	2.76915	0.0905607
$936$	0	0
$937$	−55.0823	−1.79946	−0.899730	−	0.436447i	$-0.856237\pi$
$937$	−55.0823	−1.79946	−0.899730	+	0.436447i	$0.856237\pi$
$938$	5.11684	0.167071
$939$	0	0
$940$	7.59107	0.247593
$941$	−3.82358	−0.124645	−0.0623225	−	0.998056i	$-0.519851\pi$
$941$	−3.82358	−0.124645	−0.0623225	+	0.998056i	$0.519851\pi$
$942$	0	0
$943$	−13.5327	−0.440687
$944$	−10.2114	−0.332352
$945$	0	0
$946$	13.8351	0.449818
$947$	−20.1008	−0.653188	−0.326594	−	0.945165i	$-0.605901\pi$
$947$	−20.1008	−0.653188	−0.326594	+	0.945165i	$0.605901\pi$
$948$	0	0
$949$	−18.9588	−0.615428
$950$	−5.69519	−0.184776
$951$	0	0
$952$	1.87774	0.0608577
$953$	−33.0458	−1.07046	−0.535229	−	0.844707i	$-0.679775\pi$
$953$	−33.0458	−1.07046	−0.535229	+	0.844707i	$0.679775\pi$
$954$	0	0
$955$	−9.54383	−0.308831
$956$	−13.7267	−0.443954
$957$	0	0
$958$	14.8934	0.481183
$959$	5.19165	0.167647
$960$	0	0
$961$	2.48910	0.0802937
$962$	2.90852	0.0937744
$963$	0	0
$964$	17.2144	0.554439
$965$	−17.8931	−0.576000
$966$	0	0
$967$	17.4137	0.559986	0.279993	−	0.960002i	$-0.409668\pi$
$967$	17.4137	0.559986	0.279993	+	0.960002i	$0.409668\pi$
$968$	9.09982	0.292479
$969$	0	0
$970$	−0.338815	−0.0108787
$971$	28.5581	0.916474	0.458237	−	0.888830i	$-0.348481\pi$
$971$	28.5581	0.916474	0.458237	+	0.888830i	$0.348481\pi$
$972$	0	0
$973$	10.1534	0.325503
$974$	−18.5223	−0.593493
$975$	0	0
$976$	1.02628	0.0328504
$977$	33.7204	1.07881	0.539406	−	0.842046i	$-0.318649\pi$
$977$	33.7204	1.07881	0.539406	+	0.842046i	$0.318649\pi$
$978$	0	0
$979$	10.0095	0.319905
$980$	−4.50664	−0.143959
$981$	0	0
$982$	33.3676	1.06480
$983$	26.3171	0.839384	0.419692	−	0.907667i	$-0.362138\pi$
$983$	26.3171	0.839384	0.419692	+	0.907667i	$0.362138\pi$
$984$	0	0
$985$	9.04691	0.288258
$986$	−7.37595	−0.234898
$987$	0	0
$988$	−1.54209	−0.0490605
$989$	75.8224	2.41101
$990$	0	0
$991$	47.9413	1.52291	0.761453	−	0.648221i	$-0.224487\pi$
$991$	47.9413	1.52291	0.761453	+	0.648221i	$0.224487\pi$
$992$	−5.78698	−0.183737
$993$	0	0
$994$	−1.71750	−0.0544757
$995$	−15.1082	−0.478962
$996$	0	0
$997$	40.7657	1.29106	0.645531	−	0.763734i	$-0.276636\pi$
$997$	40.7657	1.29106	0.645531	+	0.763734i	$0.276636\pi$
$998$	27.6463	0.875127
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.g.1.5	✓	9
3.2	odd	2		8046.2.a.h.1.5	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.g.1.5	✓	9	1.1	even	1	trivial
8046.2.a.h.1.5	yes	9	3.2	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.683695 q^{5} -0.639069 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.683695 q^{5} -0.639069 q^{7} -1.00000 q^{8} -0.683695 q^{10} +1.37847 q^{11} +1.22729 q^{13} +0.639069 q^{14} +1.00000 q^{16} +2.93824 q^{17} -1.25651 q^{19} +0.683695 q^{20} -1.37847 q^{22} -7.55461 q^{23} -4.53256 q^{25} -1.22729 q^{26} -0.639069 q^{28} +2.51033 q^{29} +5.78698 q^{31} -1.00000 q^{32} -2.93824 q^{34} -0.436928 q^{35} -2.36988 q^{37} +1.25651 q^{38} -0.683695 q^{40} +1.79132 q^{41} -10.0366 q^{43} +1.37847 q^{44} +7.55461 q^{46} +11.1030 q^{47} -6.59159 q^{49} +4.53256 q^{50} +1.22729 q^{52} -3.25589 q^{53} +0.942452 q^{55} +0.639069 q^{56} -2.51033 q^{58} -10.2114 q^{59} +1.02628 q^{61} -5.78698 q^{62} +1.00000 q^{64} +0.839089 q^{65} +8.00671 q^{67} +2.93824 q^{68} +0.436928 q^{70} -2.68750 q^{71} -15.4477 q^{73} +2.36988 q^{74} -1.25651 q^{76} -0.880936 q^{77} -7.08243 q^{79} +0.683695 q^{80} -1.79132 q^{82} -2.29043 q^{83} +2.00886 q^{85} +10.0366 q^{86} -1.37847 q^{88} +7.26133 q^{89} -0.784321 q^{91} -7.55461 q^{92} -11.1030 q^{94} -0.859067 q^{95} +0.495565 q^{97} +6.59159 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) \) 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8	\( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8} - 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 9 q^{16} + q^{17} - 10 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 3 q^{25} + 8 q^{26} - 4 q^{28} + 4 q^{29} - 17 q^{31} - 9 q^{32} - q^{34} + 10 q^{35} - 11 q^{37} + 10 q^{38} - 4 q^{40} - 16 q^{43} + 4 q^{44} - 8 q^{46} + 7 q^{47} - 5 q^{49} + 3 q^{50} - 8 q^{52} + 12 q^{53} - 23 q^{55} + 4 q^{56} - 4 q^{58} + 6 q^{59} - 13 q^{61} + 17 q^{62} + 9 q^{64} - 24 q^{65} - 14 q^{67} + q^{68} - 10 q^{70} + 30 q^{71} - 12 q^{73} + 11 q^{74} - 10 q^{76} + 12 q^{77} - 35 q^{79} + 4 q^{80} + 5 q^{83} - 27 q^{85} + 16 q^{86} - 4 q^{88} + 23 q^{89} - 28 q^{91} + 8 q^{92} - 7 q^{94} + 32 q^{95} - 21 q^{97} + 5 q^{98}+O(q^{100}) \) 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8 - 4 * q^10 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 9 * q^16 + q^17 - 10 * q^19 + 4 * q^20 - 4 * q^22 + 8 * q^23 - 3 * q^25 + 8 * q^26 - 4 * q^28 + 4 * q^29 - 17 * q^31 - 9 * q^32 - q^34 + 10 * q^35 - 11 * q^37 + 10 * q^38 - 4 * q^40 - 16 * q^43 + 4 * q^44 - 8 * q^46 + 7 * q^47 - 5 * q^49 + 3 * q^50 - 8 * q^52 + 12 * q^53 - 23 * q^55 + 4 * q^56 - 4 * q^58 + 6 * q^59 - 13 * q^61 + 17 * q^62 + 9 * q^64 - 24 * q^65 - 14 * q^67 + q^68 - 10 * q^70 + 30 * q^71 - 12 * q^73 + 11 * q^74 - 10 * q^76 + 12 * q^77 - 35 * q^79 + 4 * q^80 + 5 * q^83 - 27 * q^85 + 16 * q^86 - 4 * q^88 + 23 * q^89 - 28 * q^91 + 8 * q^92 - 7 * q^94 + 32 * q^95 - 21 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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