LMFDB - Embedded newform 8046.2.a.c.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 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			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} +4.46410 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} +4.46410 q^{7} -1.00000 q^{8} +3.46410 q^{10} -4.73205 q^{11} -1.46410 q^{13} -4.46410 q^{14} +1.00000 q^{16} +3.26795 q^{17} +1.46410 q^{19} -3.46410 q^{20} +4.73205 q^{22} +7.73205 q^{23} +7.00000 q^{25} +1.46410 q^{26} +4.46410 q^{28} +1.73205 q^{29} +1.00000 q^{31} -1.00000 q^{32} -3.26795 q^{34} -15.4641 q^{35} -0.732051 q^{37} -1.46410 q^{38} +3.46410 q^{40} -9.19615 q^{41} -4.92820 q^{43} -4.73205 q^{44} -7.73205 q^{46} +0.535898 q^{47} +12.9282 q^{49} -7.00000 q^{50} -1.46410 q^{52} +9.73205 q^{53} +16.3923 q^{55} -4.46410 q^{56} -1.73205 q^{58} -6.73205 q^{59} -0.732051 q^{61} -1.00000 q^{62} +1.00000 q^{64} +5.07180 q^{65} +4.19615 q^{67} +3.26795 q^{68} +15.4641 q^{70} +10.1244 q^{71} -5.46410 q^{73} +0.732051 q^{74} +1.46410 q^{76} -21.1244 q^{77} -9.26795 q^{79} -3.46410 q^{80} +9.19615 q^{82} -12.1962 q^{83} -11.3205 q^{85} +4.92820 q^{86} +4.73205 q^{88} -7.73205 q^{89} -6.53590 q^{91} +7.73205 q^{92} -0.535898 q^{94} -5.07180 q^{95} +7.26795 q^{97} -12.9282 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 6 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{17} - 4 q^{19} + 6 q^{22} + 12 q^{23} + 14 q^{25} - 4 q^{26} + 2 q^{28} + 2 q^{31} - 2 q^{32} - 10 q^{34} - 24 q^{35} + 2 q^{37} + 4 q^{38} - 8 q^{41} + 4 q^{43} - 6 q^{44} - 12 q^{46} + 8 q^{47} + 12 q^{49} - 14 q^{50} + 4 q^{52} + 16 q^{53} + 12 q^{55} - 2 q^{56} - 10 q^{59} + 2 q^{61} - 2 q^{62} + 2 q^{64} + 24 q^{65} - 2 q^{67} + 10 q^{68} + 24 q^{70} - 4 q^{71} - 4 q^{73} - 2 q^{74} - 4 q^{76} - 18 q^{77} - 22 q^{79} + 8 q^{82} - 14 q^{83} + 12 q^{85} - 4 q^{86} + 6 q^{88} - 12 q^{89} - 20 q^{91} + 12 q^{92} - 8 q^{94} - 24 q^{95} + 18 q^{97} - 12 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 6 * q^11 + 4 * q^13 - 2 * q^14 + 2 * q^16 + 10 * q^17 - 4 * q^19 + 6 * q^22 + 12 * q^23 + 14 * q^25 - 4 * q^26 + 2 * q^28 + 2 * q^31 - 2 * q^32 - 10 * q^34 - 24 * q^35 + 2 * q^37 + 4 * q^38 - 8 * q^41 + 4 * q^43 - 6 * q^44 - 12 * q^46 + 8 * q^47 + 12 * q^49 - 14 * q^50 + 4 * q^52 + 16 * q^53 + 12 * q^55 - 2 * q^56 - 10 * q^59 + 2 * q^61 - 2 * q^62 + 2 * q^64 + 24 * q^65 - 2 * q^67 + 10 * q^68 + 24 * q^70 - 4 * q^71 - 4 * q^73 - 2 * q^74 - 4 * q^76 - 18 * q^77 - 22 * q^79 + 8 * q^82 - 14 * q^83 + 12 * q^85 - 4 * q^86 + 6 * q^88 - 12 * q^89 - 20 * q^91 + 12 * q^92 - 8 * q^94 - 24 * q^95 + 18 * q^97 - 12 * 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
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	4.46410	1.68727	0.843636	−	0.536916i	$-0.180411\pi$
$7$	4.46410	1.68727	0.843636	+	0.536916i	$0.180411\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.46410	1.09545
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	0	0
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	−4.46410	−1.19308
$15$	0	0
$16$	1.00000	0.250000
$17$	3.26795	0.792594	0.396297	−	0.918122i	$-0.370295\pi$
$17$	3.26795	0.792594	0.396297	+	0.918122i	$0.370295\pi$
$18$	0	0
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	−3.46410	−0.774597
$21$	0	0
$22$	4.73205	1.00888
$23$	7.73205	1.61224	0.806122	−	0.591749i	$-0.201562\pi$
$23$	7.73205	1.61224	0.806122	+	0.591749i	$0.201562\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	1.46410	0.287134
$27$	0	0
$28$	4.46410	0.843636
$29$	1.73205	0.321634	0.160817	−	0.986984i	$-0.448587\pi$
$29$	1.73205	0.321634	0.160817	+	0.986984i	$0.448587\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.26795	−0.560449
$35$	−15.4641	−2.61391
$36$	0	0
$37$	−0.732051	−0.120348	−0.0601742	−	0.998188i	$-0.519166\pi$
$37$	−0.732051	−0.120348	−0.0601742	+	0.998188i	$0.519166\pi$
$38$	−1.46410	−0.237509
$39$	0	0
$40$	3.46410	0.547723
$41$	−9.19615	−1.43620	−0.718099	−	0.695941i	$-0.754987\pi$
$41$	−9.19615	−1.43620	−0.718099	+	0.695941i	$0.754987\pi$
$42$	0	0
$43$	−4.92820	−0.751544	−0.375772	−	0.926712i	$-0.622622\pi$
$43$	−4.92820	−0.751544	−0.375772	+	0.926712i	$0.622622\pi$
$44$	−4.73205	−0.713384
$45$	0	0
$46$	−7.73205	−1.14003
$47$	0.535898	0.0781688	0.0390844	−	0.999236i	$-0.487556\pi$
$47$	0.535898	0.0781688	0.0390844	+	0.999236i	$0.487556\pi$
$48$	0	0
$49$	12.9282	1.84689
$50$	−7.00000	−0.989949
$51$	0	0
$52$	−1.46410	−0.203034
$53$	9.73205	1.33680	0.668400	−	0.743802i	$-0.266979\pi$
$53$	9.73205	1.33680	0.668400	+	0.743802i	$0.266979\pi$
$54$	0	0
$55$	16.3923	2.21034
$56$	−4.46410	−0.596541
$57$	0	0
$58$	−1.73205	−0.227429
$59$	−6.73205	−0.876438	−0.438219	−	0.898868i	$-0.644391\pi$
$59$	−6.73205	−0.876438	−0.438219	+	0.898868i	$0.644391\pi$
$60$	0	0
$61$	−0.732051	−0.0937295	−0.0468648	−	0.998901i	$-0.514923\pi$
$61$	−0.732051	−0.0937295	−0.0468648	+	0.998901i	$0.514923\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	5.07180	0.629079
$66$	0	0
$67$	4.19615	0.512642	0.256321	−	0.966592i	$-0.417490\pi$
$67$	4.19615	0.512642	0.256321	+	0.966592i	$0.417490\pi$
$68$	3.26795	0.396297
$69$	0	0
$70$	15.4641	1.84831
$71$	10.1244	1.20154	0.600770	−	0.799422i	$-0.294861\pi$
$71$	10.1244	1.20154	0.600770	+	0.799422i	$0.294861\pi$
$72$	0	0
$73$	−5.46410	−0.639525	−0.319762	−	0.947498i	$-0.603603\pi$
$73$	−5.46410	−0.639525	−0.319762	+	0.947498i	$0.603603\pi$
$74$	0.732051	0.0850992
$75$	0	0
$76$	1.46410	0.167944
$77$	−21.1244	−2.40734
$78$	0	0
$79$	−9.26795	−1.04273	−0.521363	−	0.853335i	$-0.674576\pi$
$79$	−9.26795	−1.04273	−0.521363	+	0.853335i	$0.674576\pi$
$80$	−3.46410	−0.387298
$81$	0	0
$82$	9.19615	1.01555
$83$	−12.1962	−1.33870	−0.669351	−	0.742946i	$-0.733428\pi$
$83$	−12.1962	−1.33870	−0.669351	+	0.742946i	$0.733428\pi$
$84$	0	0
$85$	−11.3205	−1.22788
$86$	4.92820	0.531422
$87$	0	0
$88$	4.73205	0.504438
$89$	−7.73205	−0.819596	−0.409798	−	0.912176i	$-0.634401\pi$
$89$	−7.73205	−0.819596	−0.409798	+	0.912176i	$0.634401\pi$
$90$	0	0
$91$	−6.53590	−0.685148
$92$	7.73205	0.806122
$93$	0	0
$94$	−0.535898	−0.0552737
$95$	−5.07180	−0.520355
$96$	0	0
$97$	7.26795	0.737948	0.368974	−	0.929440i	$-0.379709\pi$
$97$	7.26795	0.737948	0.368974	+	0.929440i	$0.379709\pi$
$98$	−12.9282	−1.30595
$99$	0	0
$100$	7.00000	0.700000
$101$	−14.3923	−1.43209	−0.716044	−	0.698055i	$-0.754049\pi$
$101$	−14.3923	−1.43209	−0.716044	+	0.698055i	$0.754049\pi$
$102$	0	0
$103$	2.07180	0.204140	0.102070	−	0.994777i	$-0.467453\pi$
$103$	2.07180	0.204140	0.102070	+	0.994777i	$0.467453\pi$
$104$	1.46410	0.143567
$105$	0	0
$106$	−9.73205	−0.945260
$107$	11.4641	1.10828	0.554138	−	0.832425i	$-0.313048\pi$
$107$	11.4641	1.10828	0.554138	+	0.832425i	$0.313048\pi$
$108$	0	0
$109$	7.92820	0.759384	0.379692	−	0.925113i	$-0.376030\pi$
$109$	7.92820	0.759384	0.379692	+	0.925113i	$0.376030\pi$
$110$	−16.3923	−1.56294
$111$	0	0
$112$	4.46410	0.421818
$113$	−5.80385	−0.545980	−0.272990	−	0.962017i	$-0.588013\pi$
$113$	−5.80385	−0.545980	−0.272990	+	0.962017i	$0.588013\pi$
$114$	0	0
$115$	−26.7846	−2.49768
$116$	1.73205	0.160817
$117$	0	0
$118$	6.73205	0.619736
$119$	14.5885	1.33732
$120$	0	0
$121$	11.3923	1.03566
$122$	0.732051	0.0662768
$123$	0	0
$124$	1.00000	0.0898027
$125$	−6.92820	−0.619677
$126$	0	0
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−5.07180	−0.444826
$131$	−2.53590	−0.221562	−0.110781	−	0.993845i	$-0.535335\pi$
$131$	−2.53590	−0.221562	−0.110781	+	0.993845i	$0.535335\pi$
$132$	0	0
$133$	6.53590	0.566734
$134$	−4.19615	−0.362492
$135$	0	0
$136$	−3.26795	−0.280224
$137$	9.19615	0.785680	0.392840	−	0.919607i	$-0.371493\pi$
$137$	9.19615	0.785680	0.392840	+	0.919607i	$0.371493\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	−15.4641	−1.30696
$141$	0	0
$142$	−10.1244	−0.849617
$143$	6.92820	0.579365
$144$	0	0
$145$	−6.00000	−0.498273
$146$	5.46410	0.452212
$147$	0	0
$148$	−0.732051	−0.0601742
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−3.12436	−0.254256	−0.127128	−	0.991886i	$-0.540576\pi$
$151$	−3.12436	−0.254256	−0.127128	+	0.991886i	$0.540576\pi$
$152$	−1.46410	−0.118754
$153$	0	0
$154$	21.1244	1.70225
$155$	−3.46410	−0.278243
$156$	0	0
$157$	10.4641	0.835126	0.417563	−	0.908648i	$-0.362884\pi$
$157$	10.4641	0.835126	0.417563	+	0.908648i	$0.362884\pi$
$158$	9.26795	0.737318
$159$	0	0
$160$	3.46410	0.273861
$161$	34.5167	2.72029
$162$	0	0
$163$	−3.92820	−0.307681	−0.153840	−	0.988096i	$-0.549164\pi$
$163$	−3.92820	−0.307681	−0.153840	+	0.988096i	$0.549164\pi$
$164$	−9.19615	−0.718099
$165$	0	0
$166$	12.1962	0.946605
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	11.3205	0.868243
$171$	0	0
$172$	−4.92820	−0.375772
$173$	17.7321	1.34814	0.674071	−	0.738666i	$-0.264544\pi$
$173$	17.7321	1.34814	0.674071	+	0.738666i	$0.264544\pi$
$174$	0	0
$175$	31.2487	2.36218
$176$	−4.73205	−0.356692
$177$	0	0
$178$	7.73205	0.579542
$179$	21.5885	1.61360	0.806799	−	0.590827i	$-0.201198\pi$
$179$	21.5885	1.61360	0.806799	+	0.590827i	$0.201198\pi$
$180$	0	0
$181$	−11.5359	−0.857457	−0.428728	−	0.903433i	$-0.641038\pi$
$181$	−11.5359	−0.857457	−0.428728	+	0.903433i	$0.641038\pi$
$182$	6.53590	0.484473
$183$	0	0
$184$	−7.73205	−0.570014
$185$	2.53590	0.186443
$186$	0	0
$187$	−15.4641	−1.13085
$188$	0.535898	0.0390844
$189$	0	0
$190$	5.07180	0.367947
$191$	−3.12436	−0.226070	−0.113035	−	0.993591i	$-0.536057\pi$
$191$	−3.12436	−0.226070	−0.113035	+	0.993591i	$0.536057\pi$
$192$	0	0
$193$	−8.58846	−0.618211	−0.309105	−	0.951028i	$-0.600030\pi$
$193$	−8.58846	−0.618211	−0.309105	+	0.951028i	$0.600030\pi$
$194$	−7.26795	−0.521808
$195$	0	0
$196$	12.9282	0.923443
$197$	21.1244	1.50505	0.752524	−	0.658565i	$-0.228836\pi$
$197$	21.1244	1.50505	0.752524	+	0.658565i	$0.228836\pi$
$198$	0	0
$199$	18.7321	1.32788	0.663940	−	0.747786i	$-0.268883\pi$
$199$	18.7321	1.32788	0.663940	+	0.747786i	$0.268883\pi$
$200$	−7.00000	−0.494975
$201$	0	0
$202$	14.3923	1.01264
$203$	7.73205	0.542684
$204$	0	0
$205$	31.8564	2.22495
$206$	−2.07180	−0.144349
$207$	0	0
$208$	−1.46410	−0.101517
$209$	−6.92820	−0.479234
$210$	0	0
$211$	−25.7846	−1.77509	−0.887543	−	0.460725i	$-0.847589\pi$
$211$	−25.7846	−1.77509	−0.887543	+	0.460725i	$0.847589\pi$
$212$	9.73205	0.668400
$213$	0	0
$214$	−11.4641	−0.783670
$215$	17.0718	1.16429
$216$	0	0
$217$	4.46410	0.303043
$218$	−7.92820	−0.536966
$219$	0	0
$220$	16.3923	1.10517
$221$	−4.78461	−0.321848
$222$	0	0
$223$	−4.73205	−0.316882	−0.158441	−	0.987368i	$-0.550647\pi$
$223$	−4.73205	−0.316882	−0.158441	+	0.987368i	$0.550647\pi$
$224$	−4.46410	−0.298270
$225$	0	0
$226$	5.80385	0.386066
$227$	−0.535898	−0.0355688	−0.0177844	−	0.999842i	$-0.505661\pi$
$227$	−0.535898	−0.0355688	−0.0177844	+	0.999842i	$0.505661\pi$
$228$	0	0
$229$	−2.19615	−0.145126	−0.0725629	−	0.997364i	$-0.523118\pi$
$229$	−2.19615	−0.145126	−0.0725629	+	0.997364i	$0.523118\pi$
$230$	26.7846	1.76612
$231$	0	0
$232$	−1.73205	−0.113715
$233$	25.9808	1.70206	0.851028	−	0.525120i	$-0.175980\pi$
$233$	25.9808	1.70206	0.851028	+	0.525120i	$0.175980\pi$
$234$	0	0
$235$	−1.85641	−0.121099
$236$	−6.73205	−0.438219
$237$	0	0
$238$	−14.5885	−0.945629
$239$	18.1244	1.17237	0.586184	−	0.810178i	$-0.300630\pi$
$239$	18.1244	1.17237	0.586184	+	0.810178i	$0.300630\pi$
$240$	0	0
$241$	15.4641	0.996130	0.498065	−	0.867140i	$-0.334044\pi$
$241$	15.4641	0.996130	0.498065	+	0.867140i	$0.334044\pi$
$242$	−11.3923	−0.732325
$243$	0	0
$244$	−0.732051	−0.0468648
$245$	−44.7846	−2.86118
$246$	0	0
$247$	−2.14359	−0.136394
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	6.92820	0.438178
$251$	−0.267949	−0.0169128	−0.00845640	−	0.999964i	$-0.502692\pi$
$251$	−0.267949	−0.0169128	−0.00845640	+	0.999964i	$0.502692\pi$
$252$	0	0
$253$	−36.5885	−2.30030
$254$	−14.0000	−0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.8038	0.673925	0.336963	−	0.941518i	$-0.390600\pi$
$257$	10.8038	0.673925	0.336963	+	0.941518i	$0.390600\pi$
$258$	0	0
$259$	−3.26795	−0.203060
$260$	5.07180	0.314539
$261$	0	0
$262$	2.53590	0.156668
$263$	26.7846	1.65161	0.825805	−	0.563956i	$-0.190721\pi$
$263$	26.7846	1.65161	0.825805	+	0.563956i	$0.190721\pi$
$264$	0	0
$265$	−33.7128	−2.07096
$266$	−6.53590	−0.400742
$267$	0	0
$268$	4.19615	0.256321
$269$	−32.6603	−1.99133	−0.995665	−	0.0930074i	$-0.970352\pi$
$269$	−32.6603	−1.99133	−0.995665	+	0.0930074i	$0.970352\pi$
$270$	0	0
$271$	−12.7846	−0.776610	−0.388305	−	0.921531i	$-0.626939\pi$
$271$	−12.7846	−0.776610	−0.388305	+	0.921531i	$0.626939\pi$
$272$	3.26795	0.198149
$273$	0	0
$274$	−9.19615	−0.555560
$275$	−33.1244	−1.99747
$276$	0	0
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	7.00000	0.419832
$279$	0	0
$280$	15.4641	0.924157
$281$	−14.0526	−0.838305	−0.419153	−	0.907916i	$-0.637673\pi$
$281$	−14.0526	−0.838305	−0.419153	+	0.907916i	$0.637673\pi$
$282$	0	0
$283$	8.46410	0.503139	0.251569	−	0.967839i	$-0.419053\pi$
$283$	8.46410	0.503139	0.251569	+	0.967839i	$0.419053\pi$
$284$	10.1244	0.600770
$285$	0	0
$286$	−6.92820	−0.409673
$287$	−41.0526	−2.42326
$288$	0	0
$289$	−6.32051	−0.371795
$290$	6.00000	0.352332
$291$	0	0
$292$	−5.46410	−0.319762
$293$	−12.1244	−0.708312	−0.354156	−	0.935186i	$-0.615232\pi$
$293$	−12.1244	−0.708312	−0.354156	+	0.935186i	$0.615232\pi$
$294$	0	0
$295$	23.3205	1.35777
$296$	0.732051	0.0425496
$297$	0	0
$298$	1.00000	0.0579284
$299$	−11.3205	−0.654682
$300$	0	0
$301$	−22.0000	−1.26806
$302$	3.12436	0.179786
$303$	0	0
$304$	1.46410	0.0839720
$305$	2.53590	0.145205
$306$	0	0
$307$	5.32051	0.303657	0.151829	−	0.988407i	$-0.451484\pi$
$307$	5.32051	0.303657	0.151829	+	0.988407i	$0.451484\pi$
$308$	−21.1244	−1.20367
$309$	0	0
$310$	3.46410	0.196748
$311$	−3.87564	−0.219768	−0.109884	−	0.993944i	$-0.535048\pi$
$311$	−3.87564	−0.219768	−0.109884	+	0.993944i	$0.535048\pi$
$312$	0	0
$313$	−29.4641	−1.66541	−0.832705	−	0.553717i	$-0.813209\pi$
$313$	−29.4641	−1.66541	−0.832705	+	0.553717i	$0.813209\pi$
$314$	−10.4641	−0.590523
$315$	0	0
$316$	−9.26795	−0.521363
$317$	23.8564	1.33991	0.669955	−	0.742402i	$-0.266314\pi$
$317$	23.8564	1.33991	0.669955	+	0.742402i	$0.266314\pi$
$318$	0	0
$319$	−8.19615	−0.458896
$320$	−3.46410	−0.193649
$321$	0	0
$322$	−34.5167	−1.92354
$323$	4.78461	0.266223
$324$	0	0
$325$	−10.2487	−0.568496
$326$	3.92820	0.217563
$327$	0	0
$328$	9.19615	0.507773
$329$	2.39230	0.131892
$330$	0	0
$331$	12.1962	0.670361	0.335181	−	0.942154i	$-0.391203\pi$
$331$	12.1962	0.670361	0.335181	+	0.942154i	$0.391203\pi$
$332$	−12.1962	−0.669351
$333$	0	0
$334$	0	0
$335$	−14.5359	−0.794181
$336$	0	0
$337$	19.0000	1.03500	0.517498	−	0.855684i	$-0.326864\pi$
$337$	19.0000	1.03500	0.517498	+	0.855684i	$0.326864\pi$
$338$	10.8564	0.590511
$339$	0	0
$340$	−11.3205	−0.613941
$341$	−4.73205	−0.256255
$342$	0	0
$343$	26.4641	1.42893
$344$	4.92820	0.265711
$345$	0	0
$346$	−17.7321	−0.953281
$347$	10.2679	0.551212	0.275606	−	0.961271i	$-0.411121\pi$
$347$	10.2679	0.551212	0.275606	+	0.961271i	$0.411121\pi$
$348$	0	0
$349$	34.3205	1.83713	0.918567	−	0.395265i	$-0.129347\pi$
$349$	34.3205	1.83713	0.918567	+	0.395265i	$0.129347\pi$
$350$	−31.2487	−1.67031
$351$	0	0
$352$	4.73205	0.252219
$353$	6.92820	0.368751	0.184376	−	0.982856i	$-0.440974\pi$
$353$	6.92820	0.368751	0.184376	+	0.982856i	$0.440974\pi$
$354$	0	0
$355$	−35.0718	−1.86142
$356$	−7.73205	−0.409798
$357$	0	0
$358$	−21.5885	−1.14099
$359$	−24.3923	−1.28738	−0.643688	−	0.765288i	$-0.722597\pi$
$359$	−24.3923	−1.28738	−0.643688	+	0.765288i	$0.722597\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	11.5359	0.606313
$363$	0	0
$364$	−6.53590	−0.342574
$365$	18.9282	0.990747
$366$	0	0
$367$	34.8564	1.81949	0.909745	−	0.415168i	$-0.136277\pi$
$367$	34.8564	1.81949	0.909745	+	0.415168i	$0.136277\pi$
$368$	7.73205	0.403061
$369$	0	0
$370$	−2.53590	−0.131835
$371$	43.4449	2.25554
$372$	0	0
$373$	13.9282	0.721175	0.360588	−	0.932725i	$-0.382576\pi$
$373$	13.9282	0.721175	0.360588	+	0.932725i	$0.382576\pi$
$374$	15.4641	0.799630
$375$	0	0
$376$	−0.535898	−0.0276368
$377$	−2.53590	−0.130605
$378$	0	0
$379$	35.8564	1.84182	0.920910	−	0.389775i	$-0.127447\pi$
$379$	35.8564	1.84182	0.920910	+	0.389775i	$0.127447\pi$
$380$	−5.07180	−0.260178
$381$	0	0
$382$	3.12436	0.159856
$383$	19.6603	1.00459	0.502296	−	0.864696i	$-0.332489\pi$
$383$	19.6603	1.00459	0.502296	+	0.864696i	$0.332489\pi$
$384$	0	0
$385$	73.1769	3.72944
$386$	8.58846	0.437141
$387$	0	0
$388$	7.26795	0.368974
$389$	−23.4641	−1.18968	−0.594839	−	0.803845i	$-0.702784\pi$
$389$	−23.4641	−1.18968	−0.594839	+	0.803845i	$0.702784\pi$
$390$	0	0
$391$	25.2679	1.27786
$392$	−12.9282	−0.652973
$393$	0	0
$394$	−21.1244	−1.06423
$395$	32.1051	1.61538
$396$	0	0
$397$	−34.7846	−1.74579	−0.872895	−	0.487909i	$-0.837760\pi$
$397$	−34.7846	−1.74579	−0.872895	+	0.487909i	$0.837760\pi$
$398$	−18.7321	−0.938953
$399$	0	0
$400$	7.00000	0.350000
$401$	−5.12436	−0.255898	−0.127949	−	0.991781i	$-0.540839\pi$
$401$	−5.12436	−0.255898	−0.127949	+	0.991781i	$0.540839\pi$
$402$	0	0
$403$	−1.46410	−0.0729321
$404$	−14.3923	−0.716044
$405$	0	0
$406$	−7.73205	−0.383735
$407$	3.46410	0.171709
$408$	0	0
$409$	−2.39230	−0.118292	−0.0591459	−	0.998249i	$-0.518838\pi$
$409$	−2.39230	−0.118292	−0.0591459	+	0.998249i	$0.518838\pi$
$410$	−31.8564	−1.57328
$411$	0	0
$412$	2.07180	0.102070
$413$	−30.0526	−1.47879
$414$	0	0
$415$	42.2487	2.07391
$416$	1.46410	0.0717835
$417$	0	0
$418$	6.92820	0.338869
$419$	−3.58846	−0.175308	−0.0876538	−	0.996151i	$-0.527937\pi$
$419$	−3.58846	−0.175308	−0.0876538	+	0.996151i	$0.527937\pi$
$420$	0	0
$421$	1.60770	0.0783543	0.0391771	−	0.999232i	$-0.487526\pi$
$421$	1.60770	0.0783543	0.0391771	+	0.999232i	$0.487526\pi$
$422$	25.7846	1.25518
$423$	0	0
$424$	−9.73205	−0.472630
$425$	22.8756	1.10963
$426$	0	0
$427$	−3.26795	−0.158147
$428$	11.4641	0.554138
$429$	0	0
$430$	−17.0718	−0.823275
$431$	12.2487	0.590000	0.295000	−	0.955497i	$-0.404680\pi$
$431$	12.2487	0.590000	0.295000	+	0.955497i	$0.404680\pi$
$432$	0	0
$433$	−14.9808	−0.719929	−0.359965	−	0.932966i	$-0.617211\pi$
$433$	−14.9808	−0.719929	−0.359965	+	0.932966i	$0.617211\pi$
$434$	−4.46410	−0.214284
$435$	0	0
$436$	7.92820	0.379692
$437$	11.3205	0.541533
$438$	0	0
$439$	26.1962	1.25027	0.625137	−	0.780515i	$-0.285043\pi$
$439$	26.1962	1.25027	0.625137	+	0.780515i	$0.285043\pi$
$440$	−16.3923	−0.781472
$441$	0	0
$442$	4.78461	0.227581
$443$	14.6603	0.696530	0.348265	−	0.937396i	$-0.386771\pi$
$443$	14.6603	0.696530	0.348265	+	0.937396i	$0.386771\pi$
$444$	0	0
$445$	26.7846	1.26971
$446$	4.73205	0.224069
$447$	0	0
$448$	4.46410	0.210909
$449$	−17.8756	−0.843604	−0.421802	−	0.906688i	$-0.638602\pi$
$449$	−17.8756	−0.843604	−0.421802	+	0.906688i	$0.638602\pi$
$450$	0	0
$451$	43.5167	2.04912
$452$	−5.80385	−0.272990
$453$	0	0
$454$	0.535898	0.0251510
$455$	22.6410	1.06143
$456$	0	0
$457$	−23.0718	−1.07925	−0.539627	−	0.841904i	$-0.681435\pi$
$457$	−23.0718	−1.07925	−0.539627	+	0.841904i	$0.681435\pi$
$458$	2.19615	0.102619
$459$	0	0
$460$	−26.7846	−1.24884
$461$	−5.32051	−0.247801	−0.123900	−	0.992295i	$-0.539540\pi$
$461$	−5.32051	−0.247801	−0.123900	+	0.992295i	$0.539540\pi$
$462$	0	0
$463$	31.6410	1.47048	0.735241	−	0.677805i	$-0.237069\pi$
$463$	31.6410	1.47048	0.735241	+	0.677805i	$0.237069\pi$
$464$	1.73205	0.0804084
$465$	0	0
$466$	−25.9808	−1.20354
$467$	36.1051	1.67075	0.835373	−	0.549684i	$-0.185252\pi$
$467$	36.1051	1.67075	0.835373	+	0.549684i	$0.185252\pi$
$468$	0	0
$469$	18.7321	0.864966
$470$	1.85641	0.0856296
$471$	0	0
$472$	6.73205	0.309868
$473$	23.3205	1.07228
$474$	0	0
$475$	10.2487	0.470243
$476$	14.5885	0.668661
$477$	0	0
$478$	−18.1244	−0.828989
$479$	8.26795	0.377772	0.188886	−	0.981999i	$-0.439512\pi$
$479$	8.26795	0.377772	0.188886	+	0.981999i	$0.439512\pi$
$480$	0	0
$481$	1.07180	0.0488697
$482$	−15.4641	−0.704371
$483$	0	0
$484$	11.3923	0.517832
$485$	−25.1769	−1.14322
$486$	0	0
$487$	20.3923	0.924064	0.462032	−	0.886863i	$-0.347121\pi$
$487$	20.3923	0.924064	0.462032	+	0.886863i	$0.347121\pi$
$488$	0.732051	0.0331384
$489$	0	0
$490$	44.7846	2.02316
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	0	0
$493$	5.66025	0.254925
$494$	2.14359	0.0964448
$495$	0	0
$496$	1.00000	0.0449013
$497$	45.1962	2.02732
$498$	0	0
$499$	7.39230	0.330925	0.165463	−	0.986216i	$-0.447088\pi$
$499$	7.39230	0.330925	0.165463	+	0.986216i	$0.447088\pi$
$500$	−6.92820	−0.309839
$501$	0	0
$502$	0.267949	0.0119592
$503$	21.4641	0.957037	0.478518	−	0.878077i	$-0.341174\pi$
$503$	21.4641	0.957037	0.478518	+	0.878077i	$0.341174\pi$
$504$	0	0
$505$	49.8564	2.21858
$506$	36.5885	1.62656
$507$	0	0
$508$	14.0000	0.621150
$509$	36.9808	1.63914	0.819572	−	0.572977i	$-0.194211\pi$
$509$	36.9808	1.63914	0.819572	+	0.572977i	$0.194211\pi$
$510$	0	0
$511$	−24.3923	−1.07905
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−10.8038	−0.476537
$515$	−7.17691	−0.316253
$516$	0	0
$517$	−2.53590	−0.111529
$518$	3.26795	0.143585
$519$	0	0
$520$	−5.07180	−0.222413
$521$	−21.3205	−0.934068	−0.467034	−	0.884239i	$-0.654678\pi$
$521$	−21.3205	−0.934068	−0.467034	+	0.884239i	$0.654678\pi$
$522$	0	0
$523$	12.9282	0.565311	0.282655	−	0.959222i	$-0.408785\pi$
$523$	12.9282	0.565311	0.282655	+	0.959222i	$0.408785\pi$
$524$	−2.53590	−0.110781
$525$	0	0
$526$	−26.7846	−1.16786
$527$	3.26795	0.142354
$528$	0	0
$529$	36.7846	1.59933
$530$	33.7128	1.46439
$531$	0	0
$532$	6.53590	0.283367
$533$	13.4641	0.583195
$534$	0	0
$535$	−39.7128	−1.71693
$536$	−4.19615	−0.181246
$537$	0	0
$538$	32.6603	1.40808
$539$	−61.1769	−2.63508
$540$	0	0
$541$	22.5359	0.968894	0.484447	−	0.874821i	$-0.339021\pi$
$541$	22.5359	0.968894	0.484447	+	0.874821i	$0.339021\pi$
$542$	12.7846	0.549146
$543$	0	0
$544$	−3.26795	−0.140112
$545$	−27.4641	−1.17643
$546$	0	0
$547$	−4.53590	−0.193941	−0.0969705	−	0.995287i	$-0.530915\pi$
$547$	−4.53590	−0.193941	−0.0969705	+	0.995287i	$0.530915\pi$
$548$	9.19615	0.392840
$549$	0	0
$550$	33.1244	1.41243
$551$	2.53590	0.108033
$552$	0	0
$553$	−41.3731	−1.75936
$554$	−5.00000	−0.212430
$555$	0	0
$556$	−7.00000	−0.296866
$557$	3.87564	0.164216	0.0821082	−	0.996623i	$-0.473835\pi$
$557$	3.87564	0.164216	0.0821082	+	0.996623i	$0.473835\pi$
$558$	0	0
$559$	7.21539	0.305178
$560$	−15.4641	−0.653478
$561$	0	0
$562$	14.0526	0.592771
$563$	13.3397	0.562203	0.281102	−	0.959678i	$-0.409300\pi$
$563$	13.3397	0.562203	0.281102	+	0.959678i	$0.409300\pi$
$564$	0	0
$565$	20.1051	0.845829
$566$	−8.46410	−0.355773
$567$	0	0
$568$	−10.1244	−0.424809
$569$	−12.1244	−0.508279	−0.254140	−	0.967168i	$-0.581792\pi$
$569$	−12.1244	−0.508279	−0.254140	+	0.967168i	$0.581792\pi$
$570$	0	0
$571$	−14.9282	−0.624726	−0.312363	−	0.949963i	$-0.601120\pi$
$571$	−14.9282	−0.624726	−0.312363	+	0.949963i	$0.601120\pi$
$572$	6.92820	0.289683
$573$	0	0
$574$	41.0526	1.71350
$575$	54.1244	2.25714
$576$	0	0
$577$	−2.21539	−0.0922279	−0.0461140	−	0.998936i	$-0.514684\pi$
$577$	−2.21539	−0.0922279	−0.0461140	+	0.998936i	$0.514684\pi$
$578$	6.32051	0.262898
$579$	0	0
$580$	−6.00000	−0.249136
$581$	−54.4449	−2.25875
$582$	0	0
$583$	−46.0526	−1.90730
$584$	5.46410	0.226106
$585$	0	0
$586$	12.1244	0.500853
$587$	46.3923	1.91482	0.957408	−	0.288740i	$-0.0932362\pi$
$587$	46.3923	1.91482	0.957408	+	0.288740i	$0.0932362\pi$
$588$	0	0
$589$	1.46410	0.0603273
$590$	−23.3205	−0.960090
$591$	0	0
$592$	−0.732051	−0.0300871
$593$	45.5885	1.87209	0.936047	−	0.351876i	$-0.114456\pi$
$593$	45.5885	1.87209	0.936047	+	0.351876i	$0.114456\pi$
$594$	0	0
$595$	−50.5359	−2.07177
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	11.3205	0.462930
$599$	5.85641	0.239286	0.119643	−	0.992817i	$-0.461825\pi$
$599$	5.85641	0.239286	0.119643	+	0.992817i	$0.461825\pi$
$600$	0	0
$601$	29.3923	1.19894	0.599469	−	0.800398i	$-0.295379\pi$
$601$	29.3923	1.19894	0.599469	+	0.800398i	$0.295379\pi$
$602$	22.0000	0.896653
$603$	0	0
$604$	−3.12436	−0.127128
$605$	−39.4641	−1.60444
$606$	0	0
$607$	18.1962	0.738559	0.369280	−	0.929318i	$-0.379604\pi$
$607$	18.1962	0.738559	0.369280	+	0.929318i	$0.379604\pi$
$608$	−1.46410	−0.0593772
$609$	0	0
$610$	−2.53590	−0.102676
$611$	−0.784610	−0.0317419
$612$	0	0
$613$	22.2487	0.898617	0.449308	−	0.893377i	$-0.351670\pi$
$613$	22.2487	0.898617	0.449308	+	0.893377i	$0.351670\pi$
$614$	−5.32051	−0.214718
$615$	0	0
$616$	21.1244	0.851125
$617$	−28.2487	−1.13725	−0.568625	−	0.822597i	$-0.692524\pi$
$617$	−28.2487	−1.13725	−0.568625	+	0.822597i	$0.692524\pi$
$618$	0	0
$619$	−19.4641	−0.782328	−0.391164	−	0.920321i	$-0.627928\pi$
$619$	−19.4641	−0.782328	−0.391164	+	0.920321i	$0.627928\pi$
$620$	−3.46410	−0.139122
$621$	0	0
$622$	3.87564	0.155399
$623$	−34.5167	−1.38288
$624$	0	0
$625$	−11.0000	−0.440000
$626$	29.4641	1.17762
$627$	0	0
$628$	10.4641	0.417563
$629$	−2.39230	−0.0953874
$630$	0	0
$631$	37.8564	1.50704	0.753520	−	0.657425i	$-0.228354\pi$
$631$	37.8564	1.50704	0.753520	+	0.657425i	$0.228354\pi$
$632$	9.26795	0.368659
$633$	0	0
$634$	−23.8564	−0.947459
$635$	−48.4974	−1.92456
$636$	0	0
$637$	−18.9282	−0.749963
$638$	8.19615	0.324489
$639$	0	0
$640$	3.46410	0.136931
$641$	0.875644	0.0345859	0.0172929	−	0.999850i	$-0.494495\pi$
$641$	0.875644	0.0345859	0.0172929	+	0.999850i	$0.494495\pi$
$642$	0	0
$643$	−44.0526	−1.73726	−0.868632	−	0.495458i	$-0.835000\pi$
$643$	−44.0526	−1.73726	−0.868632	+	0.495458i	$0.835000\pi$
$644$	34.5167	1.36015
$645$	0	0
$646$	−4.78461	−0.188248
$647$	−8.51666	−0.334824	−0.167412	−	0.985887i	$-0.553541\pi$
$647$	−8.51666	−0.334824	−0.167412	+	0.985887i	$0.553541\pi$
$648$	0	0
$649$	31.8564	1.25047
$650$	10.2487	0.401988
$651$	0	0
$652$	−3.92820	−0.153840
$653$	1.85641	0.0726468	0.0363234	−	0.999340i	$-0.488435\pi$
$653$	1.85641	0.0726468	0.0363234	+	0.999340i	$0.488435\pi$
$654$	0	0
$655$	8.78461	0.343243
$656$	−9.19615	−0.359049
$657$	0	0
$658$	−2.39230	−0.0932618
$659$	46.5167	1.81203	0.906016	−	0.423244i	$-0.139109\pi$
$659$	46.5167	1.81203	0.906016	+	0.423244i	$0.139109\pi$
$660$	0	0
$661$	−37.2487	−1.44881	−0.724403	−	0.689376i	$-0.757885\pi$
$661$	−37.2487	−1.44881	−0.724403	+	0.689376i	$0.757885\pi$
$662$	−12.1962	−0.474017
$663$	0	0
$664$	12.1962	0.473303
$665$	−22.6410	−0.877981
$666$	0	0
$667$	13.3923	0.518552
$668$	0	0
$669$	0	0
$670$	14.5359	0.561571
$671$	3.46410	0.133730
$672$	0	0
$673$	42.5885	1.64166	0.820832	−	0.571169i	$-0.193510\pi$
$673$	42.5885	1.64166	0.820832	+	0.571169i	$0.193510\pi$
$674$	−19.0000	−0.731853
$675$	0	0
$676$	−10.8564	−0.417554
$677$	2.39230	0.0919437	0.0459719	−	0.998943i	$-0.485362\pi$
$677$	2.39230	0.0919437	0.0459719	+	0.998943i	$0.485362\pi$
$678$	0	0
$679$	32.4449	1.24512
$680$	11.3205	0.434122
$681$	0	0
$682$	4.73205	0.181200
$683$	−15.6077	−0.597212	−0.298606	−	0.954376i	$-0.596522\pi$
$683$	−15.6077	−0.597212	−0.298606	+	0.954376i	$0.596522\pi$
$684$	0	0
$685$	−31.8564	−1.21717
$686$	−26.4641	−1.01040
$687$	0	0
$688$	−4.92820	−0.187886
$689$	−14.2487	−0.542833
$690$	0	0
$691$	−6.48334	−0.246638	−0.123319	−	0.992367i	$-0.539354\pi$
$691$	−6.48334	−0.246638	−0.123319	+	0.992367i	$0.539354\pi$
$692$	17.7321	0.674071
$693$	0	0
$694$	−10.2679	−0.389766
$695$	24.2487	0.919806
$696$	0	0
$697$	−30.0526	−1.13832
$698$	−34.3205	−1.29905
$699$	0	0
$700$	31.2487	1.18109
$701$	21.6603	0.818097	0.409048	−	0.912513i	$-0.365861\pi$
$701$	21.6603	0.818097	0.409048	+	0.912513i	$0.365861\pi$
$702$	0	0
$703$	−1.07180	−0.0404236
$704$	−4.73205	−0.178346
$705$	0	0
$706$	−6.92820	−0.260746
$707$	−64.2487	−2.41632
$708$	0	0
$709$	−10.3923	−0.390291	−0.195146	−	0.980774i	$-0.562518\pi$
$709$	−10.3923	−0.390291	−0.195146	+	0.980774i	$0.562518\pi$
$710$	35.0718	1.31622
$711$	0	0
$712$	7.73205	0.289771
$713$	7.73205	0.289568
$714$	0	0
$715$	−24.0000	−0.897549
$716$	21.5885	0.806799
$717$	0	0
$718$	24.3923	0.910313
$719$	33.8038	1.26067	0.630335	−	0.776323i	$-0.282917\pi$
$719$	33.8038	1.26067	0.630335	+	0.776323i	$0.282917\pi$
$720$	0	0
$721$	9.24871	0.344440
$722$	16.8564	0.627330
$723$	0	0
$724$	−11.5359	−0.428728
$725$	12.1244	0.450287
$726$	0	0
$727$	21.5167	0.798009	0.399004	−	0.916949i	$-0.369356\pi$
$727$	21.5167	0.798009	0.399004	+	0.916949i	$0.369356\pi$
$728$	6.53590	0.242237
$729$	0	0
$730$	−18.9282	−0.700564
$731$	−16.1051	−0.595669
$732$	0	0
$733$	27.5359	1.01706	0.508531	−	0.861044i	$-0.330189\pi$
$733$	27.5359	1.01706	0.508531	+	0.861044i	$0.330189\pi$
$734$	−34.8564	−1.28657
$735$	0	0
$736$	−7.73205	−0.285007
$737$	−19.8564	−0.731420
$738$	0	0
$739$	−44.5885	−1.64021	−0.820106	−	0.572211i	$-0.806086\pi$
$739$	−44.5885	−1.64021	−0.820106	+	0.572211i	$0.806086\pi$
$740$	2.53590	0.0932215
$741$	0	0
$742$	−43.4449	−1.59491
$743$	−30.3731	−1.11428	−0.557140	−	0.830419i	$-0.688101\pi$
$743$	−30.3731	−1.11428	−0.557140	+	0.830419i	$0.688101\pi$
$744$	0	0
$745$	3.46410	0.126915
$746$	−13.9282	−0.509948
$747$	0	0
$748$	−15.4641	−0.565424
$749$	51.1769	1.86996
$750$	0	0
$751$	16.0718	0.586468	0.293234	−	0.956041i	$-0.405268\pi$
$751$	16.0718	0.586468	0.293234	+	0.956041i	$0.405268\pi$
$752$	0.535898	0.0195422
$753$	0	0
$754$	2.53590	0.0923520
$755$	10.8231	0.393892
$756$	0	0
$757$	39.0000	1.41748	0.708740	−	0.705470i	$-0.249264\pi$
$757$	39.0000	1.41748	0.708740	+	0.705470i	$0.249264\pi$
$758$	−35.8564	−1.30236
$759$	0	0
$760$	5.07180	0.183973
$761$	−5.60770	−0.203279	−0.101639	−	0.994821i	$-0.532409\pi$
$761$	−5.60770	−0.203279	−0.101639	+	0.994821i	$0.532409\pi$
$762$	0	0
$763$	35.3923	1.28129
$764$	−3.12436	−0.113035
$765$	0	0
$766$	−19.6603	−0.710354
$767$	9.85641	0.355894
$768$	0	0
$769$	−37.0718	−1.33684	−0.668422	−	0.743783i	$-0.733030\pi$
$769$	−37.0718	−1.33684	−0.668422	+	0.743783i	$0.733030\pi$
$770$	−73.1769	−2.63711
$771$	0	0
$772$	−8.58846	−0.309105
$773$	−30.2679	−1.08866	−0.544331	−	0.838870i	$-0.683217\pi$
$773$	−30.2679	−1.08866	−0.544331	+	0.838870i	$0.683217\pi$
$774$	0	0
$775$	7.00000	0.251447
$776$	−7.26795	−0.260904
$777$	0	0
$778$	23.4641	0.841229
$779$	−13.4641	−0.482402
$780$	0	0
$781$	−47.9090	−1.71432
$782$	−25.2679	−0.903580
$783$	0	0
$784$	12.9282	0.461722
$785$	−36.2487	−1.29377
$786$	0	0
$787$	−9.12436	−0.325248	−0.162624	−	0.986688i	$-0.551996\pi$
$787$	−9.12436	−0.325248	−0.162624	+	0.986688i	$0.551996\pi$
$788$	21.1244	0.752524
$789$	0	0
$790$	−32.1051	−1.14225
$791$	−25.9090	−0.921217
$792$	0	0
$793$	1.07180	0.0380606
$794$	34.7846	1.23446
$795$	0	0
$796$	18.7321	0.663940
$797$	41.3205	1.46365	0.731824	−	0.681494i	$-0.238669\pi$
$797$	41.3205	1.46365	0.731824	+	0.681494i	$0.238669\pi$
$798$	0	0
$799$	1.75129	0.0619561
$800$	−7.00000	−0.247487
$801$	0	0
$802$	5.12436	0.180947
$803$	25.8564	0.912453
$804$	0	0
$805$	−119.569	−4.21426
$806$	1.46410	0.0515708
$807$	0	0
$808$	14.3923	0.506320
$809$	45.0333	1.58329	0.791644	−	0.610983i	$-0.209226\pi$
$809$	45.0333	1.58329	0.791644	+	0.610983i	$0.209226\pi$
$810$	0	0
$811$	38.3923	1.34814	0.674068	−	0.738669i	$-0.264545\pi$
$811$	38.3923	1.34814	0.674068	+	0.738669i	$0.264545\pi$
$812$	7.73205	0.271342
$813$	0	0
$814$	−3.46410	−0.121417
$815$	13.6077	0.476657
$816$	0	0
$817$	−7.21539	−0.252435
$818$	2.39230	0.0836450
$819$	0	0
$820$	31.8564	1.11247
$821$	43.0526	1.50254	0.751272	−	0.659992i	$-0.229440\pi$
$821$	43.0526	1.50254	0.751272	+	0.659992i	$0.229440\pi$
$822$	0	0
$823$	37.0718	1.29224	0.646121	−	0.763235i	$-0.276390\pi$
$823$	37.0718	1.29224	0.646121	+	0.763235i	$0.276390\pi$
$824$	−2.07180	−0.0721745
$825$	0	0
$826$	30.0526	1.04566
$827$	23.6077	0.820920	0.410460	−	0.911879i	$-0.365368\pi$
$827$	23.6077	0.820920	0.410460	+	0.911879i	$0.365368\pi$
$828$	0	0
$829$	6.17691	0.214533	0.107267	−	0.994230i	$-0.465790\pi$
$829$	6.17691	0.214533	0.107267	+	0.994230i	$0.465790\pi$
$830$	−42.2487	−1.46647
$831$	0	0
$832$	−1.46410	−0.0507586
$833$	42.2487	1.46383
$834$	0	0
$835$	0	0
$836$	−6.92820	−0.239617
$837$	0	0
$838$	3.58846	0.123961
$839$	−16.5167	−0.570218	−0.285109	−	0.958495i	$-0.592030\pi$
$839$	−16.5167	−0.570218	−0.285109	+	0.958495i	$0.592030\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	−1.60770	−0.0554048
$843$	0	0
$844$	−25.7846	−0.887543
$845$	37.6077	1.29374
$846$	0	0
$847$	50.8564	1.74745
$848$	9.73205	0.334200
$849$	0	0
$850$	−22.8756	−0.784628
$851$	−5.66025	−0.194031
$852$	0	0
$853$	−56.6410	−1.93935	−0.969676	−	0.244395i	$-0.921411\pi$
$853$	−56.6410	−1.93935	−0.969676	+	0.244395i	$0.921411\pi$
$854$	3.26795	0.111827
$855$	0	0
$856$	−11.4641	−0.391835
$857$	13.8038	0.471530	0.235765	−	0.971810i	$-0.424240\pi$
$857$	13.8038	0.471530	0.235765	+	0.971810i	$0.424240\pi$
$858$	0	0
$859$	56.0526	1.91249	0.956244	−	0.292569i	$-0.0945102\pi$
$859$	56.0526	1.91249	0.956244	+	0.292569i	$0.0945102\pi$
$860$	17.0718	0.582143
$861$	0	0
$862$	−12.2487	−0.417193
$863$	−47.2295	−1.60771	−0.803855	−	0.594825i	$-0.797221\pi$
$863$	−47.2295	−1.60771	−0.803855	+	0.594825i	$0.797221\pi$
$864$	0	0
$865$	−61.4256	−2.08853
$866$	14.9808	0.509067
$867$	0	0
$868$	4.46410	0.151521
$869$	43.8564	1.48773
$870$	0	0
$871$	−6.14359	−0.208168
$872$	−7.92820	−0.268483
$873$	0	0
$874$	−11.3205	−0.382922
$875$	−30.9282	−1.04556
$876$	0	0
$877$	31.0718	1.04922	0.524610	−	0.851343i	$-0.324211\pi$
$877$	31.0718	1.04922	0.524610	+	0.851343i	$0.324211\pi$
$878$	−26.1962	−0.884077
$879$	0	0
$880$	16.3923	0.552584
$881$	44.9090	1.51302	0.756511	−	0.653981i	$-0.226902\pi$
$881$	44.9090	1.51302	0.756511	+	0.653981i	$0.226902\pi$
$882$	0	0
$883$	48.2487	1.62370	0.811849	−	0.583867i	$-0.198461\pi$
$883$	48.2487	1.62370	0.811849	+	0.583867i	$0.198461\pi$
$884$	−4.78461	−0.160924
$885$	0	0
$886$	−14.6603	−0.492521
$887$	−35.9090	−1.20571	−0.602853	−	0.797853i	$-0.705969\pi$
$887$	−35.9090	−1.20571	−0.602853	+	0.797853i	$0.705969\pi$
$888$	0	0
$889$	62.4974	2.09610
$890$	−26.7846	−0.897822
$891$	0	0
$892$	−4.73205	−0.158441
$893$	0.784610	0.0262560
$894$	0	0
$895$	−74.7846	−2.49977
$896$	−4.46410	−0.149135
$897$	0	0
$898$	17.8756	0.596518
$899$	1.73205	0.0577671
$900$	0	0
$901$	31.8038	1.05954
$902$	−43.5167	−1.44895
$903$	0	0
$904$	5.80385	0.193033
$905$	39.9615	1.32837
$906$	0	0
$907$	−0.248711	−0.00825832	−0.00412916	−	0.999991i	$-0.501314\pi$
$907$	−0.248711	−0.00825832	−0.00412916	+	0.999991i	$0.501314\pi$
$908$	−0.535898	−0.0177844
$909$	0	0
$910$	−22.6410	−0.750542
$911$	−43.7128	−1.44827	−0.724135	−	0.689658i	$-0.757761\pi$
$911$	−43.7128	−1.44827	−0.724135	+	0.689658i	$0.757761\pi$
$912$	0	0
$913$	57.7128	1.91002
$914$	23.0718	0.763147
$915$	0	0
$916$	−2.19615	−0.0725629
$917$	−11.3205	−0.373836
$918$	0	0
$919$	−7.78461	−0.256791	−0.128395	−	0.991723i	$-0.540983\pi$
$919$	−7.78461	−0.256791	−0.128395	+	0.991723i	$0.540983\pi$
$920$	26.7846	0.883062
$921$	0	0
$922$	5.32051	0.175222
$923$	−14.8231	−0.487908
$924$	0	0
$925$	−5.12436	−0.168488
$926$	−31.6410	−1.03979
$927$	0	0
$928$	−1.73205	−0.0568574
$929$	22.3923	0.734668	0.367334	−	0.930089i	$-0.380271\pi$
$929$	22.3923	0.734668	0.367334	+	0.930089i	$0.380271\pi$
$930$	0	0
$931$	18.9282	0.620347
$932$	25.9808	0.851028
$933$	0	0
$934$	−36.1051	−1.18140
$935$	53.5692	1.75190
$936$	0	0
$937$	59.5167	1.94432	0.972162	−	0.234309i	$-0.0752826\pi$
$937$	59.5167	1.94432	0.972162	+	0.234309i	$0.0752826\pi$
$938$	−18.7321	−0.611623
$939$	0	0
$940$	−1.85641	−0.0605493
$941$	39.9808	1.30334	0.651668	−	0.758505i	$-0.274070\pi$
$941$	39.9808	1.30334	0.651668	+	0.758505i	$0.274070\pi$
$942$	0	0
$943$	−71.1051	−2.31550
$944$	−6.73205	−0.219110
$945$	0	0
$946$	−23.3205	−0.758215
$947$	−17.9808	−0.584296	−0.292148	−	0.956373i	$-0.594370\pi$
$947$	−17.9808	−0.584296	−0.292148	+	0.956373i	$0.594370\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	−10.2487	−0.332512
$951$	0	0
$952$	−14.5885	−0.472815
$953$	24.1244	0.781465	0.390732	−	0.920504i	$-0.372222\pi$
$953$	24.1244	0.781465	0.390732	+	0.920504i	$0.372222\pi$
$954$	0	0
$955$	10.8231	0.350227
$956$	18.1244	0.586184
$957$	0	0
$958$	−8.26795	−0.267125
$959$	41.0526	1.32566
$960$	0	0
$961$	−30.0000	−0.967742
$962$	−1.07180	−0.0345561
$963$	0	0
$964$	15.4641	0.498065
$965$	29.7513	0.957728
$966$	0	0
$967$	−24.9282	−0.801637	−0.400818	−	0.916157i	$-0.631274\pi$
$967$	−24.9282	−0.801637	−0.400818	+	0.916157i	$0.631274\pi$
$968$	−11.3923	−0.366163
$969$	0	0
$970$	25.1769	0.808382
$971$	−7.94744	−0.255046	−0.127523	−	0.991836i	$-0.540703\pi$
$971$	−7.94744	−0.255046	−0.127523	+	0.991836i	$0.540703\pi$
$972$	0	0
$973$	−31.2487	−1.00179
$974$	−20.3923	−0.653412
$975$	0	0
$976$	−0.732051	−0.0234324
$977$	36.0000	1.15174	0.575871	−	0.817541i	$-0.304663\pi$
$977$	36.0000	1.15174	0.575871	+	0.817541i	$0.304663\pi$
$978$	0	0
$979$	36.5885	1.16937
$980$	−44.7846	−1.43059
$981$	0	0
$982$	24.0000	0.765871
$983$	21.0526	0.671472	0.335736	−	0.941956i	$-0.391015\pi$
$983$	21.0526	0.671472	0.335736	+	0.941956i	$0.391015\pi$
$984$	0	0
$985$	−73.1769	−2.33161
$986$	−5.66025	−0.180259
$987$	0	0
$988$	−2.14359	−0.0681968
$989$	−38.1051	−1.21167
$990$	0	0
$991$	−51.5692	−1.63815	−0.819075	−	0.573686i	$-0.805513\pi$
$991$	−51.5692	−1.63815	−0.819075	+	0.573686i	$0.805513\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	−45.1962	−1.43353
$995$	−64.8897	−2.05714
$996$	0	0
$997$	−12.0526	−0.381708	−0.190854	−	0.981618i	$-0.561126\pi$
$997$	−12.0526	−0.381708	−0.190854	+	0.981618i	$0.561126\pi$
$998$	−7.39230	−0.233999
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.c.1.1	✓	2
3.2	odd	2		8046.2.a.d.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.c.1.1	✓	2	1.1	even	1	trivial
8046.2.a.d.1.2	yes	2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} +4.46410 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} +4.46410 q^{7} -1.00000 q^{8} +3.46410 q^{10} -4.73205 q^{11} -1.46410 q^{13} -4.46410 q^{14} +1.00000 q^{16} +3.26795 q^{17} +1.46410 q^{19} -3.46410 q^{20} +4.73205 q^{22} +7.73205 q^{23} +7.00000 q^{25} +1.46410 q^{26} +4.46410 q^{28} +1.73205 q^{29} +1.00000 q^{31} -1.00000 q^{32} -3.26795 q^{34} -15.4641 q^{35} -0.732051 q^{37} -1.46410 q^{38} +3.46410 q^{40} -9.19615 q^{41} -4.92820 q^{43} -4.73205 q^{44} -7.73205 q^{46} +0.535898 q^{47} +12.9282 q^{49} -7.00000 q^{50} -1.46410 q^{52} +9.73205 q^{53} +16.3923 q^{55} -4.46410 q^{56} -1.73205 q^{58} -6.73205 q^{59} -0.732051 q^{61} -1.00000 q^{62} +1.00000 q^{64} +5.07180 q^{65} +4.19615 q^{67} +3.26795 q^{68} +15.4641 q^{70} +10.1244 q^{71} -5.46410 q^{73} +0.732051 q^{74} +1.46410 q^{76} -21.1244 q^{77} -9.26795 q^{79} -3.46410 q^{80} +9.19615 q^{82} -12.1962 q^{83} -11.3205 q^{85} +4.92820 q^{86} +4.73205 q^{88} -7.73205 q^{89} -6.53590 q^{91} +7.73205 q^{92} -0.535898 q^{94} -5.07180 q^{95} +7.26795 q^{97} -12.9282 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 6 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{17} - 4 q^{19} + 6 q^{22} + 12 q^{23} + 14 q^{25} - 4 q^{26} + 2 q^{28} + 2 q^{31} - 2 q^{32} - 10 q^{34} - 24 q^{35} + 2 q^{37} + 4 q^{38} - 8 q^{41} + 4 q^{43} - 6 q^{44} - 12 q^{46} + 8 q^{47} + 12 q^{49} - 14 q^{50} + 4 q^{52} + 16 q^{53} + 12 q^{55} - 2 q^{56} - 10 q^{59} + 2 q^{61} - 2 q^{62} + 2 q^{64} + 24 q^{65} - 2 q^{67} + 10 q^{68} + 24 q^{70} - 4 q^{71} - 4 q^{73} - 2 q^{74} - 4 q^{76} - 18 q^{77} - 22 q^{79} + 8 q^{82} - 14 q^{83} + 12 q^{85} - 4 q^{86} + 6 q^{88} - 12 q^{89} - 20 q^{91} + 12 q^{92} - 8 q^{94} - 24 q^{95} + 18 q^{97} - 12 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 6 * q^11 + 4 * q^13 - 2 * q^14 + 2 * q^16 + 10 * q^17 - 4 * q^19 + 6 * q^22 + 12 * q^23 + 14 * q^25 - 4 * q^26 + 2 * q^28 + 2 * q^31 - 2 * q^32 - 10 * q^34 - 24 * q^35 + 2 * q^37 + 4 * q^38 - 8 * q^41 + 4 * q^43 - 6 * q^44 - 12 * q^46 + 8 * q^47 + 12 * q^49 - 14 * q^50 + 4 * q^52 + 16 * q^53 + 12 * q^55 - 2 * q^56 - 10 * q^59 + 2 * q^61 - 2 * q^62 + 2 * q^64 + 24 * q^65 - 2 * q^67 + 10 * q^68 + 24 * q^70 - 4 * q^71 - 4 * q^73 - 2 * q^74 - 4 * q^76 - 18 * q^77 - 22 * q^79 + 8 * q^82 - 14 * q^83 + 12 * q^85 - 4 * q^86 + 6 * q^88 - 12 * q^89 - 20 * q^91 + 12 * q^92 - 8 * q^94 - 24 * q^95 + 18 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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