LMFDB - Embedded newform 8046.2.a.m.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:	$12$
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} - 5 x^{11} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} + \cdots - 375 $ x^12 - 5x^11 - 21x^10 + 116x^9 + 106x^8 - 774x^7 - 63x^6 + 2013x^5 - 417x^4 - 2249x^3 + 761x^2 + 910*x - 375
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			1.5
Root			$1.09665$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.09665 q^{5} -2.72925 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.09665 q^{5} -2.72925 q^{7} +1.00000 q^{8} -1.09665 q^{10} +4.35836 q^{11} -3.83892 q^{13} -2.72925 q^{14} +1.00000 q^{16} +4.52192 q^{17} -1.46202 q^{19} -1.09665 q^{20} +4.35836 q^{22} +2.47990 q^{23} -3.79736 q^{25} -3.83892 q^{26} -2.72925 q^{28} -9.77560 q^{29} -0.816654 q^{31} +1.00000 q^{32} +4.52192 q^{34} +2.99302 q^{35} +3.29988 q^{37} -1.46202 q^{38} -1.09665 q^{40} +6.03409 q^{41} +2.88226 q^{43} +4.35836 q^{44} +2.47990 q^{46} +1.80668 q^{47} +0.448781 q^{49} -3.79736 q^{50} -3.83892 q^{52} -9.70841 q^{53} -4.77958 q^{55} -2.72925 q^{56} -9.77560 q^{58} +11.7716 q^{59} -12.5328 q^{61} -0.816654 q^{62} +1.00000 q^{64} +4.20994 q^{65} -9.19967 q^{67} +4.52192 q^{68} +2.99302 q^{70} +12.1052 q^{71} -9.35826 q^{73} +3.29988 q^{74} -1.46202 q^{76} -11.8950 q^{77} +6.85287 q^{79} -1.09665 q^{80} +6.03409 q^{82} -3.17103 q^{83} -4.95895 q^{85} +2.88226 q^{86} +4.35836 q^{88} -12.3920 q^{89} +10.4774 q^{91} +2.47990 q^{92} +1.80668 q^{94} +1.60332 q^{95} +6.62366 q^{97} +0.448781 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8} - 5 q^{10} - 10 q^{11} - q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} - 10 q^{19} - 5 q^{20} - 10 q^{22} - 15 q^{23} + 7 q^{25} - q^{26} - 6 q^{28} - 33 q^{29} - 6 q^{31} + 12 q^{32} - 6 q^{34} - 16 q^{35} - 13 q^{37} - 10 q^{38} - 5 q^{40} - 20 q^{41} - 11 q^{43} - 10 q^{44} - 15 q^{46} - 15 q^{47} + 2 q^{49} + 7 q^{50} - q^{52} - 4 q^{53} - 17 q^{55} - 6 q^{56} - 33 q^{58} - 10 q^{59} - 12 q^{61} - 6 q^{62} + 12 q^{64} - 40 q^{65} - 19 q^{67} - 6 q^{68} - 16 q^{70} - 47 q^{71} - 2 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{77} - 15 q^{79} - 5 q^{80} - 20 q^{82} - 18 q^{83} - 25 q^{85} - 11 q^{86} - 10 q^{88} - 24 q^{89} - 3 q^{91} - 15 q^{92} - 15 q^{94} + 3 q^{95} - 25 q^{97} + 2 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8 - 5 * q^10 - 10 * q^11 - q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 - 10 * q^19 - 5 * q^20 - 10 * q^22 - 15 * q^23 + 7 * q^25 - q^26 - 6 * q^28 - 33 * q^29 - 6 * q^31 + 12 * q^32 - 6 * q^34 - 16 * q^35 - 13 * q^37 - 10 * q^38 - 5 * q^40 - 20 * q^41 - 11 * q^43 - 10 * q^44 - 15 * q^46 - 15 * q^47 + 2 * q^49 + 7 * q^50 - q^52 - 4 * q^53 - 17 * q^55 - 6 * q^56 - 33 * q^58 - 10 * q^59 - 12 * q^61 - 6 * q^62 + 12 * q^64 - 40 * q^65 - 19 * q^67 - 6 * q^68 - 16 * q^70 - 47 * q^71 - 2 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^77 - 15 * q^79 - 5 * q^80 - 20 * q^82 - 18 * q^83 - 25 * q^85 - 11 * q^86 - 10 * q^88 - 24 * q^89 - 3 * q^91 - 15 * q^92 - 15 * q^94 + 3 * q^95 - 25 * q^97 + 2 * 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
$5$	−1.09665	−0.490436	−0.245218	−	0.969468i	$-0.578859\pi$
$5$	−1.09665	−0.490436	−0.245218	+	0.969468i	$0.578859\pi$
$6$	0	0
$7$	−2.72925	−1.03156	−0.515779	−	0.856722i	$-0.672497\pi$
$7$	−2.72925	−1.03156	−0.515779	+	0.856722i	$0.672497\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.09665	−0.346790
$11$	4.35836	1.31409	0.657047	−	0.753850i	$-0.271805\pi$
$11$	4.35836	1.31409	0.657047	+	0.753850i	$0.271805\pi$
$12$	0	0
$13$	−3.83892	−1.06472	−0.532362	−	0.846517i	$-0.678696\pi$
$13$	−3.83892	−1.06472	−0.532362	+	0.846517i	$0.678696\pi$
$14$	−2.72925	−0.729422
$15$	0	0
$16$	1.00000	0.250000
$17$	4.52192	1.09673	0.548364	−	0.836240i	$-0.315251\pi$
$17$	4.52192	1.09673	0.548364	+	0.836240i	$0.315251\pi$
$18$	0	0
$19$	−1.46202	−0.335411	−0.167706	−	0.985837i	$-0.553636\pi$
$19$	−1.46202	−0.335411	−0.167706	+	0.985837i	$0.553636\pi$
$20$	−1.09665	−0.245218
$21$	0	0
$22$	4.35836	0.929205
$23$	2.47990	0.517094	0.258547	−	0.965999i	$-0.416756\pi$
$23$	2.47990	0.517094	0.258547	+	0.965999i	$0.416756\pi$
$24$	0	0
$25$	−3.79736	−0.759473
$26$	−3.83892	−0.752874
$27$	0	0
$28$	−2.72925	−0.515779
$29$	−9.77560	−1.81528	−0.907642	−	0.419745i	$-0.862119\pi$
$29$	−9.77560	−1.81528	−0.907642	+	0.419745i	$0.862119\pi$
$30$	0	0
$31$	−0.816654	−0.146675	−0.0733377	−	0.997307i	$-0.523365\pi$
$31$	−0.816654	−0.146675	−0.0733377	+	0.997307i	$0.523365\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.52192	0.775503
$35$	2.99302	0.505913
$36$	0	0
$37$	3.29988	0.542498	0.271249	−	0.962509i	$-0.412563\pi$
$37$	3.29988	0.542498	0.271249	+	0.962509i	$0.412563\pi$
$38$	−1.46202	−0.237172
$39$	0	0
$40$	−1.09665	−0.173395
$41$	6.03409	0.942366	0.471183	−	0.882035i	$-0.343827\pi$
$41$	6.03409	0.942366	0.471183	+	0.882035i	$0.343827\pi$
$42$	0	0
$43$	2.88226	0.439540	0.219770	−	0.975552i	$-0.429469\pi$
$43$	2.88226	0.439540	0.219770	+	0.975552i	$0.429469\pi$
$44$	4.35836	0.657047
$45$	0	0
$46$	2.47990	0.365641
$47$	1.80668	0.263531	0.131765	−	0.991281i	$-0.457935\pi$
$47$	1.80668	0.263531	0.131765	+	0.991281i	$0.457935\pi$
$48$	0	0
$49$	0.448781	0.0641115
$50$	−3.79736	−0.537028
$51$	0	0
$52$	−3.83892	−0.532362
$53$	−9.70841	−1.33355	−0.666776	−	0.745258i	$-0.732326\pi$
$53$	−9.70841	−1.33355	−0.666776	+	0.745258i	$0.732326\pi$
$54$	0	0
$55$	−4.77958	−0.644479
$56$	−2.72925	−0.364711
$57$	0	0
$58$	−9.77560	−1.28360
$59$	11.7716	1.53253	0.766265	−	0.642525i	$-0.222113\pi$
$59$	11.7716	1.53253	0.766265	+	0.642525i	$0.222113\pi$
$60$	0	0
$61$	−12.5328	−1.60466	−0.802332	−	0.596878i	$-0.796408\pi$
$61$	−12.5328	−1.60466	−0.802332	+	0.596878i	$0.796408\pi$
$62$	−0.816654	−0.103715
$63$	0	0
$64$	1.00000	0.125000
$65$	4.20994	0.522179
$66$	0	0
$67$	−9.19967	−1.12392	−0.561959	−	0.827165i	$-0.689952\pi$
$67$	−9.19967	−1.12392	−0.561959	+	0.827165i	$0.689952\pi$
$68$	4.52192	0.548364
$69$	0	0
$70$	2.99302	0.357734
$71$	12.1052	1.43662	0.718310	−	0.695723i	$-0.244916\pi$
$71$	12.1052	1.43662	0.718310	+	0.695723i	$0.244916\pi$
$72$	0	0
$73$	−9.35826	−1.09530	−0.547651	−	0.836707i	$-0.684478\pi$
$73$	−9.35826	−1.09530	−0.547651	+	0.836707i	$0.684478\pi$
$74$	3.29988	0.383604
$75$	0	0
$76$	−1.46202	−0.167706
$77$	−11.8950	−1.35556
$78$	0	0
$79$	6.85287	0.771008	0.385504	−	0.922706i	$-0.374028\pi$
$79$	6.85287	0.771008	0.385504	+	0.922706i	$0.374028\pi$
$80$	−1.09665	−0.122609
$81$	0	0
$82$	6.03409	0.666353
$83$	−3.17103	−0.348066	−0.174033	−	0.984740i	$-0.555680\pi$
$83$	−3.17103	−0.348066	−0.174033	+	0.984740i	$0.555680\pi$
$84$	0	0
$85$	−4.95895	−0.537874
$86$	2.88226	0.310802
$87$	0	0
$88$	4.35836	0.464602
$89$	−12.3920	−1.31355	−0.656776	−	0.754085i	$-0.728080\pi$
$89$	−12.3920	−1.31355	−0.656776	+	0.754085i	$0.728080\pi$
$90$	0	0
$91$	10.4774	1.09833
$92$	2.47990	0.258547
$93$	0	0
$94$	1.80668	0.186344
$95$	1.60332	0.164498
$96$	0	0
$97$	6.62366	0.672531	0.336266	−	0.941767i	$-0.390836\pi$
$97$	6.62366	0.672531	0.336266	+	0.941767i	$0.390836\pi$
$98$	0.448781	0.0453337
$99$	0	0
$100$	−3.79736	−0.379736
$101$	−18.9994	−1.89051	−0.945256	−	0.326328i	$-0.894188\pi$
$101$	−18.9994	−1.89051	−0.945256	+	0.326328i	$0.894188\pi$
$102$	0	0
$103$	2.86431	0.282229	0.141114	−	0.989993i	$-0.454931\pi$
$103$	2.86431	0.282229	0.141114	+	0.989993i	$0.454931\pi$
$104$	−3.83892	−0.376437
$105$	0	0
$106$	−9.70841	−0.942964
$107$	−16.8994	−1.63372	−0.816862	−	0.576833i	$-0.804288\pi$
$107$	−16.8994	−1.63372	−0.816862	+	0.576833i	$0.804288\pi$
$108$	0	0
$109$	0.752058	0.0720341	0.0360170	−	0.999351i	$-0.488533\pi$
$109$	0.752058	0.0720341	0.0360170	+	0.999351i	$0.488533\pi$
$110$	−4.77958	−0.455715
$111$	0	0
$112$	−2.72925	−0.257889
$113$	14.1817	1.33410	0.667052	−	0.745011i	$-0.267556\pi$
$113$	14.1817	1.33410	0.667052	+	0.745011i	$0.267556\pi$
$114$	0	0
$115$	−2.71957	−0.253601
$116$	−9.77560	−0.907642
$117$	0	0
$118$	11.7716	1.08366
$119$	−12.3414	−1.13134
$120$	0	0
$121$	7.99528	0.726844
$122$	−12.5328	−1.13467
$123$	0	0
$124$	−0.816654	−0.0733377
$125$	9.64761	0.862908
$126$	0	0
$127$	4.60414	0.408551	0.204276	−	0.978913i	$-0.434516\pi$
$127$	4.60414	0.408551	0.204276	+	0.978913i	$0.434516\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.20994	0.369236
$131$	0.536620	0.0468847	0.0234423	−	0.999725i	$-0.492537\pi$
$131$	0.536620	0.0468847	0.0234423	+	0.999725i	$0.492537\pi$
$132$	0	0
$133$	3.99022	0.345996
$134$	−9.19967	−0.794730
$135$	0	0
$136$	4.52192	0.387752
$137$	−20.2425	−1.72944	−0.864718	−	0.502259i	$-0.832503\pi$
$137$	−20.2425	−1.72944	−0.864718	+	0.502259i	$0.832503\pi$
$138$	0	0
$139$	−3.15773	−0.267835	−0.133918	−	0.990992i	$-0.542756\pi$
$139$	−3.15773	−0.267835	−0.133918	+	0.990992i	$0.542756\pi$
$140$	2.99302	0.252956
$141$	0	0
$142$	12.1052	1.01584
$143$	−16.7314	−1.39915
$144$	0	0
$145$	10.7204	0.890280
$146$	−9.35826	−0.774495
$147$	0	0
$148$	3.29988	0.271249
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−24.1133	−1.96231	−0.981155	−	0.193222i	$-0.938106\pi$
$151$	−24.1133	−1.96231	−0.981155	+	0.193222i	$0.938106\pi$
$152$	−1.46202	−0.118586
$153$	0	0
$154$	−11.8950	−0.958529
$155$	0.895581	0.0719348
$156$	0	0
$157$	−6.00350	−0.479131	−0.239566	−	0.970880i	$-0.577005\pi$
$157$	−6.00350	−0.479131	−0.239566	+	0.970880i	$0.577005\pi$
$158$	6.85287	0.545185
$159$	0	0
$160$	−1.09665	−0.0866976
$161$	−6.76824	−0.533412
$162$	0	0
$163$	−19.2994	−1.51165	−0.755825	−	0.654774i	$-0.772764\pi$
$163$	−19.2994	−1.51165	−0.755825	+	0.654774i	$0.772764\pi$
$164$	6.03409	0.471183
$165$	0	0
$166$	−3.17103	−0.246120
$167$	16.5548	1.28105	0.640524	−	0.767938i	$-0.278717\pi$
$167$	16.5548	1.28105	0.640524	+	0.767938i	$0.278717\pi$
$168$	0	0
$169$	1.73730	0.133639
$170$	−4.95895	−0.380334
$171$	0	0
$172$	2.88226	0.219770
$173$	11.9508	0.908605	0.454302	−	0.890848i	$-0.349889\pi$
$173$	11.9508	0.908605	0.454302	+	0.890848i	$0.349889\pi$
$174$	0	0
$175$	10.3639	0.783440
$176$	4.35836	0.328524
$177$	0	0
$178$	−12.3920	−0.928822
$179$	−18.9136	−1.41367	−0.706833	−	0.707380i	$-0.749877\pi$
$179$	−18.9136	−1.41367	−0.706833	+	0.707380i	$0.749877\pi$
$180$	0	0
$181$	21.8386	1.62325	0.811624	−	0.584181i	$-0.198584\pi$
$181$	21.8386	1.62325	0.811624	+	0.584181i	$0.198584\pi$
$182$	10.4774	0.776633
$183$	0	0
$184$	2.47990	0.182820
$185$	−3.61881	−0.266060
$186$	0	0
$187$	19.7082	1.44120
$188$	1.80668	0.131765
$189$	0	0
$190$	1.60332	0.116317
$191$	−16.5320	−1.19621	−0.598105	−	0.801418i	$-0.704080\pi$
$191$	−16.5320	−1.19621	−0.598105	+	0.801418i	$0.704080\pi$
$192$	0	0
$193$	11.8366	0.852014	0.426007	−	0.904720i	$-0.359920\pi$
$193$	11.8366	0.852014	0.426007	+	0.904720i	$0.359920\pi$
$194$	6.62366	0.475551
$195$	0	0
$196$	0.448781	0.0320558
$197$	−17.0473	−1.21457	−0.607287	−	0.794483i	$-0.707742\pi$
$197$	−17.0473	−1.21457	−0.607287	+	0.794483i	$0.707742\pi$
$198$	0	0
$199$	4.59160	0.325490	0.162745	−	0.986668i	$-0.447965\pi$
$199$	4.59160	0.325490	0.162745	+	0.986668i	$0.447965\pi$
$200$	−3.79736	−0.268514
$201$	0	0
$202$	−18.9994	−1.33679
$203$	26.6800	1.87257
$204$	0	0
$205$	−6.61726	−0.462170
$206$	2.86431	0.199566
$207$	0	0
$208$	−3.83892	−0.266181
$209$	−6.37202	−0.440762
$210$	0	0
$211$	9.98312	0.687267	0.343633	−	0.939104i	$-0.388342\pi$
$211$	9.98312	0.687267	0.343633	+	0.939104i	$0.388342\pi$
$212$	−9.70841	−0.666776
$213$	0	0
$214$	−16.8994	−1.15522
$215$	−3.16082	−0.215566
$216$	0	0
$217$	2.22885	0.151304
$218$	0.752058	0.0509358
$219$	0	0
$220$	−4.77958	−0.322239
$221$	−17.3593	−1.16771
$222$	0	0
$223$	−12.4741	−0.835326	−0.417663	−	0.908602i	$-0.637151\pi$
$223$	−12.4741	−0.835326	−0.417663	+	0.908602i	$0.637151\pi$
$224$	−2.72925	−0.182355
$225$	0	0
$226$	14.1817	0.943354
$227$	−4.41661	−0.293141	−0.146570	−	0.989200i	$-0.546824\pi$
$227$	−4.41661	−0.293141	−0.146570	+	0.989200i	$0.546824\pi$
$228$	0	0
$229$	−21.1109	−1.39505	−0.697524	−	0.716561i	$-0.745715\pi$
$229$	−21.1109	−1.39505	−0.697524	+	0.716561i	$0.745715\pi$
$230$	−2.71957	−0.179323
$231$	0	0
$232$	−9.77560	−0.641800
$233$	3.50298	0.229488	0.114744	−	0.993395i	$-0.463395\pi$
$233$	3.50298	0.229488	0.114744	+	0.993395i	$0.463395\pi$
$234$	0	0
$235$	−1.98129	−0.129245
$236$	11.7716	0.766265
$237$	0	0
$238$	−12.3414	−0.799976
$239$	−27.2008	−1.75947	−0.879736	−	0.475463i	$-0.842281\pi$
$239$	−27.2008	−1.75947	−0.879736	+	0.475463i	$0.842281\pi$
$240$	0	0
$241$	6.93758	0.446889	0.223444	−	0.974717i	$-0.428270\pi$
$241$	6.93758	0.446889	0.223444	+	0.974717i	$0.428270\pi$
$242$	7.99528	0.513956
$243$	0	0
$244$	−12.5328	−0.802332
$245$	−0.492154	−0.0314426
$246$	0	0
$247$	5.61259	0.357121
$248$	−0.816654	−0.0518576
$249$	0	0
$250$	9.64761	0.610168
$251$	−18.5290	−1.16954	−0.584771	−	0.811199i	$-0.698815\pi$
$251$	−18.5290	−1.16954	−0.584771	+	0.811199i	$0.698815\pi$
$252$	0	0
$253$	10.8083	0.679510
$254$	4.60414	0.288889
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.3360	1.76755	0.883776	−	0.467911i	$-0.154993\pi$
$257$	28.3360	1.76755	0.883776	+	0.467911i	$0.154993\pi$
$258$	0	0
$259$	−9.00619	−0.559618
$260$	4.20994	0.261089
$261$	0	0
$262$	0.536620	0.0331525
$263$	−13.0188	−0.802773	−0.401386	−	0.915909i	$-0.631472\pi$
$263$	−13.0188	−0.802773	−0.401386	+	0.915909i	$0.631472\pi$
$264$	0	0
$265$	10.6467	0.654022
$266$	3.99022	0.244656
$267$	0	0
$268$	−9.19967	−0.561959
$269$	−27.1713	−1.65666	−0.828331	−	0.560239i	$-0.810709\pi$
$269$	−27.1713	−1.65666	−0.828331	+	0.560239i	$0.810709\pi$
$270$	0	0
$271$	0.154348	0.00937599	0.00468800	−	0.999989i	$-0.498508\pi$
$271$	0.154348	0.00937599	0.00468800	+	0.999989i	$0.498508\pi$
$272$	4.52192	0.274182
$273$	0	0
$274$	−20.2425	−1.22290
$275$	−16.5503	−0.998019
$276$	0	0
$277$	28.6038	1.71864	0.859319	−	0.511440i	$-0.170888\pi$
$277$	28.6038	1.71864	0.859319	+	0.511440i	$0.170888\pi$
$278$	−3.15773	−0.189388
$279$	0	0
$280$	2.99302	0.178867
$281$	−5.48687	−0.327319	−0.163660	−	0.986517i	$-0.552330\pi$
$281$	−5.48687	−0.327319	−0.163660	+	0.986517i	$0.552330\pi$
$282$	0	0
$283$	18.3381	1.09008	0.545042	−	0.838408i	$-0.316514\pi$
$283$	18.3381	1.09008	0.545042	+	0.838408i	$0.316514\pi$
$284$	12.1052	0.718310
$285$	0	0
$286$	−16.7314	−0.989347
$287$	−16.4685	−0.972105
$288$	0	0
$289$	3.44778	0.202811
$290$	10.7204	0.629523
$291$	0	0
$292$	−9.35826	−0.547651
$293$	−31.7560	−1.85520	−0.927602	−	0.373569i	$-0.878134\pi$
$293$	−31.7560	−1.85520	−0.927602	+	0.373569i	$0.878134\pi$
$294$	0	0
$295$	−12.9093	−0.751607
$296$	3.29988	0.191802
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−9.52012	−0.550563
$300$	0	0
$301$	−7.86639	−0.453411
$302$	−24.1133	−1.38756
$303$	0	0
$304$	−1.46202	−0.0838528
$305$	13.7441	0.786984
$306$	0	0
$307$	−34.4876	−1.96831	−0.984155	−	0.177310i	$-0.943261\pi$
$307$	−34.4876	−1.96831	−0.984155	+	0.177310i	$0.943261\pi$
$308$	−11.8950	−0.677782
$309$	0	0
$310$	0.895581	0.0508656
$311$	−7.52575	−0.426746	−0.213373	−	0.976971i	$-0.568445\pi$
$311$	−7.52575	−0.426746	−0.213373	+	0.976971i	$0.568445\pi$
$312$	0	0
$313$	−28.0840	−1.58740	−0.793700	−	0.608310i	$-0.791848\pi$
$313$	−28.0840	−1.58740	−0.793700	+	0.608310i	$0.791848\pi$
$314$	−6.00350	−0.338797
$315$	0	0
$316$	6.85287	0.385504
$317$	−25.6207	−1.43900	−0.719501	−	0.694492i	$-0.755629\pi$
$317$	−25.6207	−1.43900	−0.719501	+	0.694492i	$0.755629\pi$
$318$	0	0
$319$	−42.6056	−2.38545
$320$	−1.09665	−0.0613044
$321$	0	0
$322$	−6.76824	−0.377180
$323$	−6.61116	−0.367855
$324$	0	0
$325$	14.5778	0.808630
$326$	−19.2994	−1.06890
$327$	0	0
$328$	6.03409	0.333177
$329$	−4.93086	−0.271847
$330$	0	0
$331$	1.31125	0.0720727	0.0360363	−	0.999350i	$-0.488527\pi$
$331$	1.31125	0.0720727	0.0360363	+	0.999350i	$0.488527\pi$
$332$	−3.17103	−0.174033
$333$	0	0
$334$	16.5548	0.905838
$335$	10.0888	0.551210
$336$	0	0
$337$	−18.4513	−1.00511	−0.502553	−	0.864547i	$-0.667606\pi$
$337$	−18.4513	−1.00511	−0.502553	+	0.864547i	$0.667606\pi$
$338$	1.73730	0.0944969
$339$	0	0
$340$	−4.95895	−0.268937
$341$	−3.55927	−0.192745
$342$	0	0
$343$	17.8799	0.965423
$344$	2.88226	0.155401
$345$	0	0
$346$	11.9508	0.642481
$347$	−7.23732	−0.388520	−0.194260	−	0.980950i	$-0.562230\pi$
$347$	−7.23732	−0.388520	−0.194260	+	0.980950i	$0.562230\pi$
$348$	0	0
$349$	−16.0951	−0.861549	−0.430774	−	0.902460i	$-0.641760\pi$
$349$	−16.0951	−0.861549	−0.430774	+	0.902460i	$0.641760\pi$
$350$	10.3639	0.553976
$351$	0	0
$352$	4.35836	0.232301
$353$	26.2451	1.39689	0.698443	−	0.715666i	$-0.253877\pi$
$353$	26.2451	1.39689	0.698443	+	0.715666i	$0.253877\pi$
$354$	0	0
$355$	−13.2751	−0.704570
$356$	−12.3920	−0.656776
$357$	0	0
$358$	−18.9136	−0.999613
$359$	35.3204	1.86414	0.932071	−	0.362277i	$-0.118001\pi$
$359$	35.3204	1.86414	0.932071	+	0.362277i	$0.118001\pi$
$360$	0	0
$361$	−16.8625	−0.887499
$362$	21.8386	1.14781
$363$	0	0
$364$	10.4774	0.549163
$365$	10.2627	0.537175
$366$	0	0
$367$	17.7864	0.928443	0.464222	−	0.885719i	$-0.346334\pi$
$367$	17.7864	0.928443	0.464222	+	0.885719i	$0.346334\pi$
$368$	2.47990	0.129274
$369$	0	0
$370$	−3.61881	−0.188133
$371$	26.4966	1.37564
$372$	0	0
$373$	5.38565	0.278858	0.139429	−	0.990232i	$-0.455473\pi$
$373$	5.38565	0.278858	0.139429	+	0.990232i	$0.455473\pi$
$374$	19.7082	1.01908
$375$	0	0
$376$	1.80668	0.0931722
$377$	37.5278	1.93278
$378$	0	0
$379$	−2.50892	−0.128875	−0.0644373	−	0.997922i	$-0.520525\pi$
$379$	−2.50892	−0.128875	−0.0644373	+	0.997922i	$0.520525\pi$
$380$	1.60332	0.0822488
$381$	0	0
$382$	−16.5320	−0.845849
$383$	13.1237	0.670591	0.335296	−	0.942113i	$-0.391164\pi$
$383$	13.1237	0.670591	0.335296	+	0.942113i	$0.391164\pi$
$384$	0	0
$385$	13.0446	0.664817
$386$	11.8366	0.602465
$387$	0	0
$388$	6.62366	0.336266
$389$	−16.8554	−0.854601	−0.427301	−	0.904110i	$-0.640535\pi$
$389$	−16.8554	−0.854601	−0.427301	+	0.904110i	$0.640535\pi$
$390$	0	0
$391$	11.2139	0.567111
$392$	0.448781	0.0226669
$393$	0	0
$394$	−17.0473	−0.858833
$395$	−7.51518	−0.378130
$396$	0	0
$397$	−28.5551	−1.43314	−0.716571	−	0.697514i	$-0.754289\pi$
$397$	−28.5551	−1.43314	−0.716571	+	0.697514i	$0.754289\pi$
$398$	4.59160	0.230156
$399$	0	0
$400$	−3.79736	−0.189868
$401$	26.0083	1.29879	0.649397	−	0.760450i	$-0.275021\pi$
$401$	26.0083	1.29879	0.649397	+	0.760450i	$0.275021\pi$
$402$	0	0
$403$	3.13507	0.156169
$404$	−18.9994	−0.945256
$405$	0	0
$406$	26.6800	1.32411
$407$	14.3821	0.712893
$408$	0	0
$409$	9.62454	0.475903	0.237951	−	0.971277i	$-0.423524\pi$
$409$	9.62454	0.475903	0.237951	+	0.971277i	$0.423524\pi$
$410$	−6.61726	−0.326803
$411$	0	0
$412$	2.86431	0.141114
$413$	−32.1275	−1.58089
$414$	0	0
$415$	3.47750	0.170704
$416$	−3.83892	−0.188219
$417$	0	0
$418$	−6.37202	−0.311666
$419$	−17.5251	−0.856157	−0.428078	−	0.903742i	$-0.640809\pi$
$419$	−17.5251	−0.856157	−0.428078	+	0.903742i	$0.640809\pi$
$420$	0	0
$421$	31.8934	1.55439	0.777195	−	0.629260i	$-0.216642\pi$
$421$	31.8934	1.55439	0.777195	+	0.629260i	$0.216642\pi$
$422$	9.98312	0.485971
$423$	0	0
$424$	−9.70841	−0.471482
$425$	−17.1714	−0.832935
$426$	0	0
$427$	34.2052	1.65530
$428$	−16.8994	−0.816862
$429$	0	0
$430$	−3.16082	−0.152428
$431$	−6.40574	−0.308553	−0.154277	−	0.988028i	$-0.549305\pi$
$431$	−6.40574	−0.308553	−0.154277	+	0.988028i	$0.549305\pi$
$432$	0	0
$433$	12.4910	0.600281	0.300141	−	0.953895i	$-0.402966\pi$
$433$	12.4910	0.600281	0.300141	+	0.953895i	$0.402966\pi$
$434$	2.22885	0.106988
$435$	0	0
$436$	0.752058	0.0360170
$437$	−3.62567	−0.173439
$438$	0	0
$439$	31.2958	1.49367	0.746834	−	0.665010i	$-0.231573\pi$
$439$	31.2958	1.49367	0.746834	+	0.665010i	$0.231573\pi$
$440$	−4.77958	−0.227858
$441$	0	0
$442$	−17.3593	−0.825698
$443$	39.4216	1.87298	0.936488	−	0.350700i	$-0.114056\pi$
$443$	39.4216	1.87298	0.936488	+	0.350700i	$0.114056\pi$
$444$	0	0
$445$	13.5897	0.644213
$446$	−12.4741	−0.590665
$447$	0	0
$448$	−2.72925	−0.128945
$449$	−3.51295	−0.165787	−0.0828933	−	0.996558i	$-0.526416\pi$
$449$	−3.51295	−0.165787	−0.0828933	+	0.996558i	$0.526416\pi$
$450$	0	0
$451$	26.2987	1.23836
$452$	14.1817	0.667052
$453$	0	0
$454$	−4.41661	−0.207282
$455$	−11.4900	−0.538658
$456$	0	0
$457$	13.0035	0.608278	0.304139	−	0.952628i	$-0.401631\pi$
$457$	13.0035	0.608278	0.304139	+	0.952628i	$0.401631\pi$
$458$	−21.1109	−0.986448
$459$	0	0
$460$	−2.71957	−0.126801
$461$	−7.06541	−0.329069	−0.164534	−	0.986371i	$-0.552612\pi$
$461$	−7.06541	−0.329069	−0.164534	+	0.986371i	$0.552612\pi$
$462$	0	0
$463$	−20.9332	−0.972847	−0.486423	−	0.873723i	$-0.661699\pi$
$463$	−20.9332	−0.972847	−0.486423	+	0.873723i	$0.661699\pi$
$464$	−9.77560	−0.453821
$465$	0	0
$466$	3.50298	0.162272
$467$	28.2094	1.30538	0.652688	−	0.757627i	$-0.273641\pi$
$467$	28.2094	1.30538	0.652688	+	0.757627i	$0.273641\pi$
$468$	0	0
$469$	25.1082	1.15939
$470$	−1.98129	−0.0913899
$471$	0	0
$472$	11.7716	0.541831
$473$	12.5619	0.577597
$474$	0	0
$475$	5.55184	0.254736
$476$	−12.3414	−0.565669
$477$	0	0
$478$	−27.2008	−1.24413
$479$	−19.7500	−0.902399	−0.451200	−	0.892423i	$-0.649004\pi$
$479$	−19.7500	−0.902399	−0.451200	+	0.892423i	$0.649004\pi$
$480$	0	0
$481$	−12.6680	−0.577611
$482$	6.93758	0.315998
$483$	0	0
$484$	7.99528	0.363422
$485$	−7.26382	−0.329833
$486$	0	0
$487$	−22.8585	−1.03582	−0.517908	−	0.855437i	$-0.673289\pi$
$487$	−22.8585	−1.03582	−0.517908	+	0.855437i	$0.673289\pi$
$488$	−12.5328	−0.567334
$489$	0	0
$490$	−0.492154	−0.0222333
$491$	9.36712	0.422732	0.211366	−	0.977407i	$-0.432209\pi$
$491$	9.36712	0.422732	0.211366	+	0.977407i	$0.432209\pi$
$492$	0	0
$493$	−44.2045	−1.99087
$494$	5.61259	0.252522
$495$	0	0
$496$	−0.816654	−0.0366689
$497$	−33.0380	−1.48196
$498$	0	0
$499$	−10.1322	−0.453580	−0.226790	−	0.973944i	$-0.572823\pi$
$499$	−10.1322	−0.453580	−0.226790	+	0.973944i	$0.572823\pi$
$500$	9.64761	0.431454
$501$	0	0
$502$	−18.5290	−0.826991
$503$	15.1255	0.674413	0.337206	−	0.941431i	$-0.390518\pi$
$503$	15.1255	0.674413	0.337206	+	0.941431i	$0.390518\pi$
$504$	0	0
$505$	20.8357	0.927175
$506$	10.8083	0.480486
$507$	0	0
$508$	4.60414	0.204276
$509$	−6.98576	−0.309638	−0.154819	−	0.987943i	$-0.549479\pi$
$509$	−6.98576	−0.309638	−0.154819	+	0.987943i	$0.549479\pi$
$510$	0	0
$511$	25.5410	1.12987
$512$	1.00000	0.0441942
$513$	0	0
$514$	28.3360	1.24985
$515$	−3.14114	−0.138415
$516$	0	0
$517$	7.87414	0.346304
$518$	−9.00619	−0.395709
$519$	0	0
$520$	4.20994	0.184618
$521$	40.9358	1.79343	0.896714	−	0.442611i	$-0.145948\pi$
$521$	40.9358	1.79343	0.896714	+	0.442611i	$0.145948\pi$
$522$	0	0
$523$	−19.9444	−0.872109	−0.436055	−	0.899920i	$-0.643625\pi$
$523$	−19.9444	−0.872109	−0.436055	+	0.899920i	$0.643625\pi$
$524$	0.536620	0.0234423
$525$	0	0
$526$	−13.0188	−0.567646
$527$	−3.69285	−0.160863
$528$	0	0
$529$	−16.8501	−0.732614
$530$	10.6467	0.462463
$531$	0	0
$532$	3.99022	0.172998
$533$	−23.1644	−1.00336
$534$	0	0
$535$	18.5327	0.801237
$536$	−9.19967	−0.397365
$537$	0	0
$538$	−27.1713	−1.17144
$539$	1.95595	0.0842486
$540$	0	0
$541$	27.8738	1.19839	0.599195	−	0.800603i	$-0.295487\pi$
$541$	27.8738	1.19839	0.599195	+	0.800603i	$0.295487\pi$
$542$	0.154348	0.00662983
$543$	0	0
$544$	4.52192	0.193876
$545$	−0.824742	−0.0353281
$546$	0	0
$547$	8.04603	0.344023	0.172012	−	0.985095i	$-0.444973\pi$
$547$	8.04603	0.344023	0.172012	+	0.985095i	$0.444973\pi$
$548$	−20.2425	−0.864718
$549$	0	0
$550$	−16.5503	−0.705706
$551$	14.2922	0.608867
$552$	0	0
$553$	−18.7032	−0.795339
$554$	28.6038	1.21526
$555$	0	0
$556$	−3.15773	−0.133918
$557$	−18.9214	−0.801726	−0.400863	−	0.916138i	$-0.631290\pi$
$557$	−18.9214	−0.801726	−0.400863	+	0.916138i	$0.631290\pi$
$558$	0	0
$559$	−11.0648	−0.467989
$560$	2.99302	0.126478
$561$	0	0
$562$	−5.48687	−0.231450
$563$	28.7214	1.21046	0.605232	−	0.796049i	$-0.293080\pi$
$563$	28.7214	1.21046	0.605232	+	0.796049i	$0.293080\pi$
$564$	0	0
$565$	−15.5523	−0.654292
$566$	18.3381	0.770806
$567$	0	0
$568$	12.1052	0.507922
$569$	−27.6719	−1.16007	−0.580033	−	0.814593i	$-0.696960\pi$
$569$	−27.6719	−1.16007	−0.580033	+	0.814593i	$0.696960\pi$
$570$	0	0
$571$	8.38536	0.350916	0.175458	−	0.984487i	$-0.443859\pi$
$571$	8.38536	0.350916	0.175458	+	0.984487i	$0.443859\pi$
$572$	−16.7314	−0.699574
$573$	0	0
$574$	−16.4685	−0.687382
$575$	−9.41707	−0.392719
$576$	0	0
$577$	34.4251	1.43314	0.716568	−	0.697517i	$-0.245712\pi$
$577$	34.4251	1.43314	0.716568	+	0.697517i	$0.245712\pi$
$578$	3.44778	0.143409
$579$	0	0
$580$	10.7204	0.445140
$581$	8.65452	0.359050
$582$	0	0
$583$	−42.3127	−1.75241
$584$	−9.35826	−0.387248
$585$	0	0
$586$	−31.7560	−1.31183
$587$	38.9122	1.60608	0.803039	−	0.595927i	$-0.203215\pi$
$587$	38.9122	1.60608	0.803039	+	0.595927i	$0.203215\pi$
$588$	0	0
$589$	1.19397	0.0491966
$590$	−12.9093	−0.531467
$591$	0	0
$592$	3.29988	0.135624
$593$	−9.78088	−0.401653	−0.200826	−	0.979627i	$-0.564363\pi$
$593$	−9.78088	−0.401653	−0.200826	+	0.979627i	$0.564363\pi$
$594$	0	0
$595$	13.5342	0.554848
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−9.52012	−0.389307
$599$	−37.3040	−1.52420	−0.762100	−	0.647459i	$-0.775832\pi$
$599$	−37.3040	−1.52420	−0.762100	+	0.647459i	$0.775832\pi$
$600$	0	0
$601$	24.1603	0.985520	0.492760	−	0.870165i	$-0.335988\pi$
$601$	24.1603	0.985520	0.492760	+	0.870165i	$0.335988\pi$
$602$	−7.86639	−0.320610
$603$	0	0
$604$	−24.1133	−0.981155
$605$	−8.76800	−0.356470
$606$	0	0
$607$	−42.8379	−1.73874	−0.869369	−	0.494164i	$-0.835474\pi$
$607$	−42.8379	−1.73874	−0.869369	+	0.494164i	$0.835474\pi$
$608$	−1.46202	−0.0592929
$609$	0	0
$610$	13.7441	0.556482
$611$	−6.93569	−0.280588
$612$	0	0
$613$	30.7697	1.24278	0.621389	−	0.783502i	$-0.286569\pi$
$613$	30.7697	1.24278	0.621389	+	0.783502i	$0.286569\pi$
$614$	−34.4876	−1.39181
$615$	0	0
$616$	−11.8950	−0.479264
$617$	−35.1120	−1.41356	−0.706778	−	0.707435i	$-0.749852\pi$
$617$	−35.1120	−1.41356	−0.706778	+	0.707435i	$0.749852\pi$
$618$	0	0
$619$	21.8156	0.876842	0.438421	−	0.898770i	$-0.355538\pi$
$619$	21.8156	0.876842	0.438421	+	0.898770i	$0.355538\pi$
$620$	0.895581	0.0359674
$621$	0	0
$622$	−7.52575	−0.301755
$623$	33.8209	1.35501
$624$	0	0
$625$	8.40680	0.336272
$626$	−28.0840	−1.12246
$627$	0	0
$628$	−6.00350	−0.239566
$629$	14.9218	0.594972
$630$	0	0
$631$	−12.1031	−0.481815	−0.240907	−	0.970548i	$-0.577445\pi$
$631$	−12.1031	−0.481815	−0.240907	+	0.970548i	$0.577445\pi$
$632$	6.85287	0.272592
$633$	0	0
$634$	−25.6207	−1.01753
$635$	−5.04912	−0.200368
$636$	0	0
$637$	−1.72283	−0.0682611
$638$	−42.6056	−1.68677
$639$	0	0
$640$	−1.09665	−0.0433488
$641$	13.1368	0.518873	0.259437	−	0.965760i	$-0.416463\pi$
$641$	13.1368	0.518873	0.259437	+	0.965760i	$0.416463\pi$
$642$	0	0
$643$	−11.8722	−0.468195	−0.234097	−	0.972213i	$-0.575213\pi$
$643$	−11.8722	−0.468195	−0.234097	+	0.972213i	$0.575213\pi$
$644$	−6.76824	−0.266706
$645$	0	0
$646$	−6.61116	−0.260113
$647$	11.6314	0.457279	0.228639	−	0.973511i	$-0.426572\pi$
$647$	11.6314	0.457279	0.228639	+	0.973511i	$0.426572\pi$
$648$	0	0
$649$	51.3048	2.01389
$650$	14.5778	0.571788
$651$	0	0
$652$	−19.2994	−0.755825
$653$	47.2362	1.84850	0.924248	−	0.381793i	$-0.124693\pi$
$653$	47.2362	1.84850	0.924248	+	0.381793i	$0.124693\pi$
$654$	0	0
$655$	−0.588483	−0.0229939
$656$	6.03409	0.235591
$657$	0	0
$658$	−4.93086	−0.192225
$659$	−43.1846	−1.68223	−0.841117	−	0.540853i	$-0.818101\pi$
$659$	−43.1846	−1.68223	−0.841117	+	0.540853i	$0.818101\pi$
$660$	0	0
$661$	40.8820	1.59013	0.795063	−	0.606526i	$-0.207438\pi$
$661$	40.8820	1.59013	0.795063	+	0.606526i	$0.207438\pi$
$662$	1.31125	0.0509631
$663$	0	0
$664$	−3.17103	−0.123060
$665$	−4.37587	−0.169689
$666$	0	0
$667$	−24.2425	−0.938673
$668$	16.5548	0.640524
$669$	0	0
$670$	10.0888	0.389764
$671$	−54.6225	−2.10868
$672$	0	0
$673$	−5.77879	−0.222756	−0.111378	−	0.993778i	$-0.535526\pi$
$673$	−5.77879	−0.222756	−0.111378	+	0.993778i	$0.535526\pi$
$674$	−18.4513	−0.710717
$675$	0	0
$676$	1.73730	0.0668194
$677$	34.1968	1.31429	0.657146	−	0.753764i	$-0.271764\pi$
$677$	34.1968	1.31429	0.657146	+	0.753764i	$0.271764\pi$
$678$	0	0
$679$	−18.0776	−0.693755
$680$	−4.95895	−0.190167
$681$	0	0
$682$	−3.55927	−0.136292
$683$	6.45987	0.247180	0.123590	−	0.992333i	$-0.460559\pi$
$683$	6.45987	0.247180	0.123590	+	0.992333i	$0.460559\pi$
$684$	0	0
$685$	22.1989	0.848176
$686$	17.8799	0.682657
$687$	0	0
$688$	2.88226	0.109885
$689$	37.2698	1.41987
$690$	0	0
$691$	−13.7970	−0.524863	−0.262432	−	0.964951i	$-0.584524\pi$
$691$	−13.7970	−0.524863	−0.262432	+	0.964951i	$0.584524\pi$
$692$	11.9508	0.454302
$693$	0	0
$694$	−7.23732	−0.274725
$695$	3.46292	0.131356
$696$	0	0
$697$	27.2857	1.03352
$698$	−16.0951	−0.609207
$699$	0	0
$700$	10.3639	0.391720
$701$	−28.5634	−1.07882	−0.539412	−	0.842042i	$-0.681354\pi$
$701$	−28.5634	−1.07882	−0.539412	+	0.842042i	$0.681354\pi$
$702$	0	0
$703$	−4.82451	−0.181960
$704$	4.35836	0.164262
$705$	0	0
$706$	26.2451	0.987748
$707$	51.8541	1.95017
$708$	0	0
$709$	−28.6004	−1.07411	−0.537055	−	0.843547i	$-0.680463\pi$
$709$	−28.6004	−1.07411	−0.537055	+	0.843547i	$0.680463\pi$
$710$	−13.2751	−0.498206
$711$	0	0
$712$	−12.3920	−0.464411
$713$	−2.02522	−0.0758450
$714$	0	0
$715$	18.3484	0.686192
$716$	−18.9136	−0.706833
$717$	0	0
$718$	35.3204	1.31815
$719$	−18.5267	−0.690931	−0.345465	−	0.938432i	$-0.612279\pi$
$719$	−18.5267	−0.690931	−0.345465	+	0.938432i	$0.612279\pi$
$720$	0	0
$721$	−7.81741	−0.291135
$722$	−16.8625	−0.627557
$723$	0	0
$724$	21.8386	0.811624
$725$	37.1215	1.37866
$726$	0	0
$727$	−23.9428	−0.887989	−0.443995	−	0.896029i	$-0.646439\pi$
$727$	−23.9428	−0.887989	−0.443995	+	0.896029i	$0.646439\pi$
$728$	10.4774	0.388317
$729$	0	0
$730$	10.2627	0.379840
$731$	13.0333	0.482056
$732$	0	0
$733$	−25.7865	−0.952444	−0.476222	−	0.879325i	$-0.657994\pi$
$733$	−25.7865	−0.952444	−0.476222	+	0.879325i	$0.657994\pi$
$734$	17.7864	0.656508
$735$	0	0
$736$	2.47990	0.0914102
$737$	−40.0955	−1.47693
$738$	0	0
$739$	24.5497	0.903077	0.451538	−	0.892252i	$-0.350875\pi$
$739$	24.5497	0.903077	0.451538	+	0.892252i	$0.350875\pi$
$740$	−3.61881	−0.133030
$741$	0	0
$742$	26.4966	0.972722
$743$	−9.44491	−0.346500	−0.173250	−	0.984878i	$-0.555427\pi$
$743$	−9.44491	−0.346500	−0.173250	+	0.984878i	$0.555427\pi$
$744$	0	0
$745$	1.09665	0.0401780
$746$	5.38565	0.197183
$747$	0	0
$748$	19.7082	0.720601
$749$	46.1226	1.68528
$750$	0	0
$751$	24.0856	0.878895	0.439447	−	0.898268i	$-0.355174\pi$
$751$	24.0856	0.878895	0.439447	+	0.898268i	$0.355174\pi$
$752$	1.80668	0.0658827
$753$	0	0
$754$	37.5278	1.36668
$755$	26.4438	0.962387
$756$	0	0
$757$	40.5109	1.47240	0.736198	−	0.676767i	$-0.236619\pi$
$757$	40.5109	1.47240	0.736198	+	0.676767i	$0.236619\pi$
$758$	−2.50892	−0.0911281
$759$	0	0
$760$	1.60332	0.0581587
$761$	20.8544	0.755972	0.377986	−	0.925811i	$-0.376617\pi$
$761$	20.8544	0.755972	0.377986	+	0.925811i	$0.376617\pi$
$762$	0	0
$763$	−2.05255	−0.0743073
$764$	−16.5320	−0.598105
$765$	0	0
$766$	13.1237	0.474180
$767$	−45.1902	−1.63172
$768$	0	0
$769$	45.6339	1.64560	0.822800	−	0.568331i	$-0.192411\pi$
$769$	45.6339	1.64560	0.822800	+	0.568331i	$0.192411\pi$
$770$	13.0446	0.470097
$771$	0	0
$772$	11.8366	0.426007
$773$	−12.8727	−0.462997	−0.231499	−	0.972835i	$-0.574363\pi$
$773$	−12.8727	−0.462997	−0.231499	+	0.972835i	$0.574363\pi$
$774$	0	0
$775$	3.10113	0.111396
$776$	6.62366	0.237776
$777$	0	0
$778$	−16.8554	−0.604294
$779$	−8.82198	−0.316080
$780$	0	0
$781$	52.7587	1.88785
$782$	11.2139	0.401008
$783$	0	0
$784$	0.448781	0.0160279
$785$	6.58372	0.234983
$786$	0	0
$787$	49.3025	1.75745	0.878723	−	0.477332i	$-0.158396\pi$
$787$	49.3025	1.75745	0.878723	+	0.477332i	$0.158396\pi$
$788$	−17.0473	−0.607287
$789$	0	0
$790$	−7.51518	−0.267378
$791$	−38.7054	−1.37621
$792$	0	0
$793$	48.1125	1.70853
$794$	−28.5551	−1.01338
$795$	0	0
$796$	4.59160	0.162745
$797$	−4.83675	−0.171326	−0.0856632	−	0.996324i	$-0.527301\pi$
$797$	−4.83675	−0.171326	−0.0856632	+	0.996324i	$0.527301\pi$
$798$	0	0
$799$	8.16965	0.289021
$800$	−3.79736	−0.134257
$801$	0	0
$802$	26.0083	0.918386
$803$	−40.7866	−1.43933
$804$	0	0
$805$	7.42238	0.261604
$806$	3.13507	0.110428
$807$	0	0
$808$	−18.9994	−0.668397
$809$	16.3692	0.575510	0.287755	−	0.957704i	$-0.407091\pi$
$809$	16.3692	0.575510	0.287755	+	0.957704i	$0.407091\pi$
$810$	0	0
$811$	−38.7722	−1.36147	−0.680737	−	0.732528i	$-0.738340\pi$
$811$	−38.7722	−1.36147	−0.680737	+	0.732528i	$0.738340\pi$
$812$	26.6800	0.936285
$813$	0	0
$814$	14.3821	0.504091
$815$	21.1647	0.741366
$816$	0	0
$817$	−4.21393	−0.147427
$818$	9.62454	0.336514
$819$	0	0
$820$	−6.61726	−0.231085
$821$	−29.6576	−1.03506	−0.517529	−	0.855666i	$-0.673148\pi$
$821$	−29.6576	−1.03506	−0.517529	+	0.855666i	$0.673148\pi$
$822$	0	0
$823$	29.6906	1.03495	0.517474	−	0.855699i	$-0.326872\pi$
$823$	29.6906	1.03495	0.517474	+	0.855699i	$0.326872\pi$
$824$	2.86431	0.0997830
$825$	0	0
$826$	−32.1275	−1.11786
$827$	13.8789	0.482618	0.241309	−	0.970448i	$-0.422423\pi$
$827$	13.8789	0.482618	0.241309	+	0.970448i	$0.422423\pi$
$828$	0	0
$829$	39.1836	1.36090	0.680451	−	0.732793i	$-0.261784\pi$
$829$	39.1836	1.36090	0.680451	+	0.732793i	$0.261784\pi$
$830$	3.47750	0.120706
$831$	0	0
$832$	−3.83892	−0.133091
$833$	2.02935	0.0703129
$834$	0	0
$835$	−18.1548	−0.628271
$836$	−6.37202	−0.220381
$837$	0	0
$838$	−17.5251	−0.605394
$839$	4.43681	0.153176	0.0765879	−	0.997063i	$-0.475597\pi$
$839$	4.43681	0.153176	0.0765879	+	0.997063i	$0.475597\pi$
$840$	0	0
$841$	66.5624	2.29526
$842$	31.8934	1.09912
$843$	0	0
$844$	9.98312	0.343633
$845$	−1.90521	−0.0655412
$846$	0	0
$847$	−21.8211	−0.749781
$848$	−9.70841	−0.333388
$849$	0	0
$850$	−17.1714	−0.588974
$851$	8.18337	0.280522
$852$	0	0
$853$	−53.3902	−1.82804	−0.914022	−	0.405664i	$-0.867040\pi$
$853$	−53.3902	−1.82804	−0.914022	+	0.405664i	$0.867040\pi$
$854$	34.2052	1.17048
$855$	0	0
$856$	−16.8994	−0.577609
$857$	−34.9541	−1.19401	−0.597004	−	0.802238i	$-0.703642\pi$
$857$	−34.9541	−1.19401	−0.597004	+	0.802238i	$0.703642\pi$
$858$	0	0
$859$	8.50093	0.290048	0.145024	−	0.989428i	$-0.453674\pi$
$859$	8.50093	0.290048	0.145024	+	0.989428i	$0.453674\pi$
$860$	−3.16082	−0.107783
$861$	0	0
$862$	−6.40574	−0.218180
$863$	20.3417	0.692440	0.346220	−	0.938153i	$-0.387465\pi$
$863$	20.3417	0.692440	0.346220	+	0.938153i	$0.387465\pi$
$864$	0	0
$865$	−13.1058	−0.445612
$866$	12.4910	0.424463
$867$	0	0
$868$	2.22885	0.0756521
$869$	29.8672	1.01318
$870$	0	0
$871$	35.3168	1.19666
$872$	0.752058	0.0254679
$873$	0	0
$874$	−3.62567	−0.122640
$875$	−26.3307	−0.890140
$876$	0	0
$877$	−11.1783	−0.377466	−0.188733	−	0.982028i	$-0.560438\pi$
$877$	−11.1783	−0.377466	−0.188733	+	0.982028i	$0.560438\pi$
$878$	31.2958	1.05618
$879$	0	0
$880$	−4.77958	−0.161120
$881$	−13.5491	−0.456480	−0.228240	−	0.973605i	$-0.573297\pi$
$881$	−13.5491	−0.456480	−0.228240	+	0.973605i	$0.573297\pi$
$882$	0	0
$883$	31.6753	1.06596	0.532979	−	0.846129i	$-0.321073\pi$
$883$	31.6753	1.06596	0.532979	+	0.846129i	$0.321073\pi$
$884$	−17.3593	−0.583856
$885$	0	0
$886$	39.4216	1.32439
$887$	−2.92676	−0.0982710	−0.0491355	−	0.998792i	$-0.515647\pi$
$887$	−2.92676	−0.0982710	−0.0491355	+	0.998792i	$0.515647\pi$
$888$	0	0
$889$	−12.5658	−0.421444
$890$	13.5897	0.455527
$891$	0	0
$892$	−12.4741	−0.417663
$893$	−2.64140	−0.0883912
$894$	0	0
$895$	20.7415	0.693312
$896$	−2.72925	−0.0911777
$897$	0	0
$898$	−3.51295	−0.117229
$899$	7.98329	0.266258
$900$	0	0
$901$	−43.9007	−1.46254
$902$	26.2987	0.875651
$903$	0	0
$904$	14.1817	0.471677
$905$	−23.9492	−0.796098
$906$	0	0
$907$	−8.72412	−0.289680	−0.144840	−	0.989455i	$-0.546267\pi$
$907$	−8.72412	−0.289680	−0.144840	+	0.989455i	$0.546267\pi$
$908$	−4.41661	−0.146570
$909$	0	0
$910$	−11.4900	−0.380889
$911$	18.4481	0.611214	0.305607	−	0.952158i	$-0.401141\pi$
$911$	18.4481	0.611214	0.305607	+	0.952158i	$0.401141\pi$
$912$	0	0
$913$	−13.8205	−0.457391
$914$	13.0035	0.430118
$915$	0	0
$916$	−21.1109	−0.697524
$917$	−1.46457	−0.0483643
$918$	0	0
$919$	−33.8809	−1.11763	−0.558814	−	0.829293i	$-0.688743\pi$
$919$	−33.8809	−1.11763	−0.558814	+	0.829293i	$0.688743\pi$
$920$	−2.71957	−0.0896616
$921$	0	0
$922$	−7.06541	−0.232687
$923$	−46.4708	−1.52960
$924$	0	0
$925$	−12.5309	−0.412012
$926$	−20.9332	−0.687906
$927$	0	0
$928$	−9.77560	−0.320900
$929$	6.69462	0.219643	0.109822	−	0.993951i	$-0.464972\pi$
$929$	6.69462	0.219643	0.109822	+	0.993951i	$0.464972\pi$
$930$	0	0
$931$	−0.656128	−0.0215037
$932$	3.50298	0.114744
$933$	0	0
$934$	28.2094	0.923040
$935$	−21.6129	−0.706817
$936$	0	0
$937$	11.4433	0.373836	0.186918	−	0.982375i	$-0.440150\pi$
$937$	11.4433	0.373836	0.186918	+	0.982375i	$0.440150\pi$
$938$	25.1082	0.819810
$939$	0	0
$940$	−1.98129	−0.0646225
$941$	30.0674	0.980171	0.490085	−	0.871674i	$-0.336966\pi$
$941$	30.0674	0.980171	0.490085	+	0.871674i	$0.336966\pi$
$942$	0	0
$943$	14.9639	0.487292
$944$	11.7716	0.383132
$945$	0	0
$946$	12.5619	0.408423
$947$	−18.9500	−0.615793	−0.307896	−	0.951420i	$-0.599625\pi$
$947$	−18.9500	−0.615793	−0.307896	+	0.951420i	$0.599625\pi$
$948$	0	0
$949$	35.9256	1.16619
$950$	5.55184	0.180125
$951$	0	0
$952$	−12.3414	−0.399988
$953$	−43.1005	−1.39616	−0.698081	−	0.716019i	$-0.745962\pi$
$953$	−43.1005	−1.39616	−0.698081	+	0.716019i	$0.745962\pi$
$954$	0	0
$955$	18.1297	0.586664
$956$	−27.2008	−0.879736
$957$	0	0
$958$	−19.7500	−0.638093
$959$	55.2468	1.78401
$960$	0	0
$961$	−30.3331	−0.978486
$962$	−12.6680	−0.408432
$963$	0	0
$964$	6.93758	0.223444
$965$	−12.9805	−0.417858
$966$	0	0
$967$	59.9741	1.92864	0.964319	−	0.264743i	$-0.0852870\pi$
$967$	59.9741	1.92864	0.964319	+	0.264743i	$0.0852870\pi$
$968$	7.99528	0.256978
$969$	0	0
$970$	−7.26382	−0.233227
$971$	22.6541	0.727006	0.363503	−	0.931593i	$-0.381581\pi$
$971$	22.6541	0.727006	0.363503	+	0.931593i	$0.381581\pi$
$972$	0	0
$973$	8.61822	0.276287
$974$	−22.8585	−0.732432
$975$	0	0
$976$	−12.5328	−0.401166
$977$	23.1749	0.741430	0.370715	−	0.928747i	$-0.379113\pi$
$977$	23.1749	0.741430	0.370715	+	0.928747i	$0.379113\pi$
$978$	0	0
$979$	−54.0089	−1.72613
$980$	−0.492154	−0.0157213
$981$	0	0
$982$	9.36712	0.298917
$983$	−24.3734	−0.777390	−0.388695	−	0.921367i	$-0.627074\pi$
$983$	−24.3734	−0.777390	−0.388695	+	0.921367i	$0.627074\pi$
$984$	0	0
$985$	18.6949	0.595670
$986$	−44.2045	−1.40776
$987$	0	0
$988$	5.61259	0.178560
$989$	7.14770	0.227284
$990$	0	0
$991$	7.73172	0.245606	0.122803	−	0.992431i	$-0.460812\pi$
$991$	7.73172	0.245606	0.122803	+	0.992431i	$0.460812\pi$
$992$	−0.816654	−0.0259288
$993$	0	0
$994$	−33.0380	−1.04790
$995$	−5.03537	−0.159632
$996$	0	0
$997$	5.46445	0.173061	0.0865305	−	0.996249i	$-0.472422\pi$
$997$	5.46445	0.173061	0.0865305	+	0.996249i	$0.472422\pi$
$998$	−10.1322	−0.320729
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.m.1.5	yes	12
3.2	odd	2		8046.2.a.l.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.l.1.8	✓	12	3.2	odd	2
8046.2.a.m.1.5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.09665 q^{5} -2.72925 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.09665 q^{5} -2.72925 q^{7} +1.00000 q^{8} -1.09665 q^{10} +4.35836 q^{11} -3.83892 q^{13} -2.72925 q^{14} +1.00000 q^{16} +4.52192 q^{17} -1.46202 q^{19} -1.09665 q^{20} +4.35836 q^{22} +2.47990 q^{23} -3.79736 q^{25} -3.83892 q^{26} -2.72925 q^{28} -9.77560 q^{29} -0.816654 q^{31} +1.00000 q^{32} +4.52192 q^{34} +2.99302 q^{35} +3.29988 q^{37} -1.46202 q^{38} -1.09665 q^{40} +6.03409 q^{41} +2.88226 q^{43} +4.35836 q^{44} +2.47990 q^{46} +1.80668 q^{47} +0.448781 q^{49} -3.79736 q^{50} -3.83892 q^{52} -9.70841 q^{53} -4.77958 q^{55} -2.72925 q^{56} -9.77560 q^{58} +11.7716 q^{59} -12.5328 q^{61} -0.816654 q^{62} +1.00000 q^{64} +4.20994 q^{65} -9.19967 q^{67} +4.52192 q^{68} +2.99302 q^{70} +12.1052 q^{71} -9.35826 q^{73} +3.29988 q^{74} -1.46202 q^{76} -11.8950 q^{77} +6.85287 q^{79} -1.09665 q^{80} +6.03409 q^{82} -3.17103 q^{83} -4.95895 q^{85} +2.88226 q^{86} +4.35836 q^{88} -12.3920 q^{89} +10.4774 q^{91} +2.47990 q^{92} +1.80668 q^{94} +1.60332 q^{95} +6.62366 q^{97} +0.448781 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8} - 5 q^{10} - 10 q^{11} - q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} - 10 q^{19} - 5 q^{20} - 10 q^{22} - 15 q^{23} + 7 q^{25} - q^{26} - 6 q^{28} - 33 q^{29} - 6 q^{31} + 12 q^{32} - 6 q^{34} - 16 q^{35} - 13 q^{37} - 10 q^{38} - 5 q^{40} - 20 q^{41} - 11 q^{43} - 10 q^{44} - 15 q^{46} - 15 q^{47} + 2 q^{49} + 7 q^{50} - q^{52} - 4 q^{53} - 17 q^{55} - 6 q^{56} - 33 q^{58} - 10 q^{59} - 12 q^{61} - 6 q^{62} + 12 q^{64} - 40 q^{65} - 19 q^{67} - 6 q^{68} - 16 q^{70} - 47 q^{71} - 2 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{77} - 15 q^{79} - 5 q^{80} - 20 q^{82} - 18 q^{83} - 25 q^{85} - 11 q^{86} - 10 q^{88} - 24 q^{89} - 3 q^{91} - 15 q^{92} - 15 q^{94} + 3 q^{95} - 25 q^{97} + 2 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8 - 5 * q^10 - 10 * q^11 - q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 - 10 * q^19 - 5 * q^20 - 10 * q^22 - 15 * q^23 + 7 * q^25 - q^26 - 6 * q^28 - 33 * q^29 - 6 * q^31 + 12 * q^32 - 6 * q^34 - 16 * q^35 - 13 * q^37 - 10 * q^38 - 5 * q^40 - 20 * q^41 - 11 * q^43 - 10 * q^44 - 15 * q^46 - 15 * q^47 + 2 * q^49 + 7 * q^50 - q^52 - 4 * q^53 - 17 * q^55 - 6 * q^56 - 33 * q^58 - 10 * q^59 - 12 * q^61 - 6 * q^62 + 12 * q^64 - 40 * q^65 - 19 * q^67 - 6 * q^68 - 16 * q^70 - 47 * q^71 - 2 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^77 - 15 * q^79 - 5 * q^80 - 20 * q^82 - 18 * q^83 - 25 * q^85 - 11 * q^86 - 10 * q^88 - 24 * q^89 - 3 * q^91 - 15 * q^92 - 15 * q^94 + 3 * q^95 - 25 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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