LMFDB - Embedded newform 6864.2.a.cf.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2172244.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 16x^{2} + 5x - 8 $ x^5 - 2x^4 - 8x^3 + 16x^2 + 5x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.71661$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.169303 q^{5} +1.46187 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.169303 q^{5} +1.46187 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +0.169303 q^{15} +6.72578 q^{17} +6.80398 q^{19} +1.46187 q^{21} -5.29802 q^{23} -4.97134 q^{25} +1.00000 q^{27} -4.23175 q^{29} +7.17280 q^{31} +1.00000 q^{33} +0.247499 q^{35} +5.03028 q^{37} +1.00000 q^{39} +0.720338 q^{41} -12.1322 q^{43} +0.169303 q^{45} +10.4213 q^{47} -4.86292 q^{49} +6.72578 q^{51} +12.9346 q^{53} +0.169303 q^{55} +6.80398 q^{57} -3.92725 q^{59} +1.23168 q^{61} +1.46187 q^{63} +0.169303 q^{65} -0.504017 q^{67} -5.29802 q^{69} +12.0184 q^{71} -3.12327 q^{73} -4.97134 q^{75} +1.46187 q^{77} -6.05246 q^{79} +1.00000 q^{81} +16.0109 q^{83} +1.13869 q^{85} -4.23175 q^{87} -1.80392 q^{89} +1.46187 q^{91} +7.17280 q^{93} +1.15193 q^{95} -9.83614 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + q^5 + 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} + q^{5} + 5 q^{7} + 5 q^{9} + 5 q^{11} + 5 q^{13} + q^{15} + 4 q^{19} + 5 q^{21} + 5 q^{23} + 4 q^{25} + 5 q^{27} + 11 q^{29} + 8 q^{31} + 5 q^{33} + 5 q^{35} - 8 q^{37} + 5 q^{39} - q^{41} - q^{43} + q^{45} + 18 q^{47} + 10 q^{49} + 2 q^{53} + q^{55} + 4 q^{57} + 13 q^{59} + 9 q^{61} + 5 q^{63} + q^{65} - 5 q^{67} + 5 q^{69} + 24 q^{71} - 13 q^{73} + 4 q^{75} + 5 q^{77} + 6 q^{79} + 5 q^{81} + 22 q^{83} - 22 q^{85} + 11 q^{87} - 14 q^{89} + 5 q^{91} + 8 q^{93} + 32 q^{95} - 20 q^{97} + 5 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + q^5 + 5 * q^7 + 5 * q^9 + 5 * q^11 + 5 * q^13 + q^15 + 4 * q^19 + 5 * q^21 + 5 * q^23 + 4 * q^25 + 5 * q^27 + 11 * q^29 + 8 * q^31 + 5 * q^33 + 5 * q^35 - 8 * q^37 + 5 * q^39 - q^41 - q^43 + q^45 + 18 * q^47 + 10 * q^49 + 2 * q^53 + q^55 + 4 * q^57 + 13 * q^59 + 9 * q^61 + 5 * q^63 + q^65 - 5 * q^67 + 5 * q^69 + 24 * q^71 - 13 * q^73 + 4 * q^75 + 5 * q^77 + 6 * q^79 + 5 * q^81 + 22 * q^83 - 22 * q^85 + 11 * q^87 - 14 * q^89 + 5 * q^91 + 8 * q^93 + 32 * q^95 - 20 * q^97 + 5 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.169303	0.0757145	0.0378572	−	0.999283i	$-0.487947\pi$
$5$	0.169303	0.0757145	0.0378572	+	0.999283i	$0.487947\pi$
$6$	0	0
$7$	1.46187	0.552537	0.276268	−	0.961081i	$-0.410902\pi$
$7$	1.46187	0.552537	0.276268	+	0.961081i	$0.410902\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.169303	0.0437138
$16$	0	0
$17$	6.72578	1.63124	0.815621	−	0.578587i	$-0.196396\pi$
$17$	6.72578	1.63124	0.815621	+	0.578587i	$0.196396\pi$
$18$	0	0
$19$	6.80398	1.56094	0.780470	−	0.625193i	$-0.214980\pi$
$19$	6.80398	1.56094	0.780470	+	0.625193i	$0.214980\pi$
$20$	0	0
$21$	1.46187	0.319007
$22$	0	0
$23$	−5.29802	−1.10471	−0.552356	−	0.833608i	$-0.686271\pi$
$23$	−5.29802	−1.10471	−0.552356	+	0.833608i	$0.686271\pi$
$24$	0	0
$25$	−4.97134	−0.994267
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.23175	−0.785815	−0.392908	−	0.919578i	$-0.628531\pi$
$29$	−4.23175	−0.785815	−0.392908	+	0.919578i	$0.628531\pi$
$30$	0	0
$31$	7.17280	1.28827	0.644137	−	0.764910i	$-0.277217\pi$
$31$	7.17280	1.28827	0.644137	+	0.764910i	$0.277217\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	0.247499	0.0418350
$36$	0	0
$37$	5.03028	0.826973	0.413486	−	0.910510i	$-0.364311\pi$
$37$	5.03028	0.826973	0.413486	+	0.910510i	$0.364311\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	0.720338	0.112498	0.0562489	−	0.998417i	$-0.482086\pi$
$41$	0.720338	0.112498	0.0562489	+	0.998417i	$0.482086\pi$
$42$	0	0
$43$	−12.1322	−1.85015	−0.925073	−	0.379790i	$-0.875996\pi$
$43$	−12.1322	−1.85015	−0.925073	+	0.379790i	$0.875996\pi$
$44$	0	0
$45$	0.169303	0.0252382
$46$	0	0
$47$	10.4213	1.52010	0.760050	−	0.649864i	$-0.225174\pi$
$47$	10.4213	1.52010	0.760050	+	0.649864i	$0.225174\pi$
$48$	0	0
$49$	−4.86292	−0.694703
$50$	0	0
$51$	6.72578	0.941798
$52$	0	0
$53$	12.9346	1.77671	0.888355	−	0.459158i	$-0.151849\pi$
$53$	12.9346	1.77671	0.888355	+	0.459158i	$0.151849\pi$
$54$	0	0
$55$	0.169303	0.0228288
$56$	0	0
$57$	6.80398	0.901209
$58$	0	0
$59$	−3.92725	−0.511284	−0.255642	−	0.966771i	$-0.582287\pi$
$59$	−3.92725	−0.511284	−0.255642	+	0.966771i	$0.582287\pi$
$60$	0	0
$61$	1.23168	0.157701	0.0788504	−	0.996886i	$-0.474875\pi$
$61$	1.23168	0.157701	0.0788504	+	0.996886i	$0.474875\pi$
$62$	0	0
$63$	1.46187	0.184179
$64$	0	0
$65$	0.169303	0.0209994
$66$	0	0
$67$	−0.504017	−0.0615754	−0.0307877	−	0.999526i	$-0.509802\pi$
$67$	−0.504017	−0.0615754	−0.0307877	+	0.999526i	$0.509802\pi$
$68$	0	0
$69$	−5.29802	−0.637806
$70$	0	0
$71$	12.0184	1.42632	0.713158	−	0.701003i	$-0.247264\pi$
$71$	12.0184	1.42632	0.713158	+	0.701003i	$0.247264\pi$
$72$	0	0
$73$	−3.12327	−0.365551	−0.182775	−	0.983155i	$-0.558508\pi$
$73$	−3.12327	−0.365551	−0.182775	+	0.983155i	$0.558508\pi$
$74$	0	0
$75$	−4.97134	−0.574041
$76$	0	0
$77$	1.46187	0.166596
$78$	0	0
$79$	−6.05246	−0.680955	−0.340478	−	0.940253i	$-0.610589\pi$
$79$	−6.05246	−0.680955	−0.340478	+	0.940253i	$0.610589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.0109	1.75742	0.878712	−	0.477353i	$-0.158404\pi$
$83$	16.0109	1.75742	0.878712	+	0.477353i	$0.158404\pi$
$84$	0	0
$85$	1.13869	0.123509
$86$	0	0
$87$	−4.23175	−0.453691
$88$	0	0
$89$	−1.80392	−0.191215	−0.0956074	−	0.995419i	$-0.530479\pi$
$89$	−1.80392	−0.191215	−0.0956074	+	0.995419i	$0.530479\pi$
$90$	0	0
$91$	1.46187	0.153246
$92$	0	0
$93$	7.17280	0.743785
$94$	0	0
$95$	1.15193	0.118186
$96$	0	0
$97$	−9.83614	−0.998709	−0.499354	−	0.866398i	$-0.666429\pi$
$97$	−9.83614	−0.998709	−0.499354	+	0.866398i	$0.666429\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−17.3218	−1.72359	−0.861793	−	0.507261i	$-0.830658\pi$
$101$	−17.3218	−1.72359	−0.861793	+	0.507261i	$0.830658\pi$
$102$	0	0
$103$	−4.26585	−0.420327	−0.210164	−	0.977666i	$-0.567400\pi$
$103$	−4.26585	−0.420327	−0.210164	+	0.977666i	$0.567400\pi$
$104$	0	0
$105$	0.247499	0.0241535
$106$	0	0
$107$	−1.59253	−0.153956	−0.0769781	−	0.997033i	$-0.524527\pi$
$107$	−1.59253	−0.153956	−0.0769781	+	0.997033i	$0.524527\pi$
$108$	0	0
$109$	−1.32480	−0.126893	−0.0634463	−	0.997985i	$-0.520209\pi$
$109$	−1.32480	−0.126893	−0.0634463	+	0.997985i	$0.520209\pi$
$110$	0	0
$111$	5.03028	0.477453
$112$	0	0
$113$	−19.8056	−1.86315	−0.931577	−	0.363544i	$-0.881566\pi$
$113$	−19.8056	−1.86315	−0.931577	+	0.363544i	$0.881566\pi$
$114$	0	0
$115$	−0.896969	−0.0836428
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	9.83225	0.901321
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.720338	0.0649507
$124$	0	0
$125$	−1.68817	−0.150995
$126$	0	0
$127$	15.8243	1.40418	0.702089	−	0.712089i	$-0.252251\pi$
$127$	15.8243	1.40418	0.702089	+	0.712089i	$0.252251\pi$
$128$	0	0
$129$	−12.1322	−1.06818
$130$	0	0
$131$	10.3302	0.902552	0.451276	−	0.892384i	$-0.350969\pi$
$131$	10.3302	0.902552	0.451276	+	0.892384i	$0.350969\pi$
$132$	0	0
$133$	9.94656	0.862477
$134$	0	0
$135$	0.169303	0.0145713
$136$	0	0
$137$	5.00894	0.427943	0.213972	−	0.976840i	$-0.431360\pi$
$137$	5.00894	0.427943	0.213972	+	0.976840i	$0.431360\pi$
$138$	0	0
$139$	−18.3521	−1.55660	−0.778302	−	0.627890i	$-0.783919\pi$
$139$	−18.3521	−1.55660	−0.778302	+	0.627890i	$0.783919\pi$
$140$	0	0
$141$	10.4213	0.877631
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−0.716446	−0.0594976
$146$	0	0
$147$	−4.86292	−0.401087
$148$	0	0
$149$	−10.0624	−0.824347	−0.412174	−	0.911105i	$-0.635230\pi$
$149$	−10.0624	−0.824347	−0.412174	+	0.911105i	$0.635230\pi$
$150$	0	0
$151$	10.9862	0.894044	0.447022	−	0.894523i	$-0.352485\pi$
$151$	10.9862	0.894044	0.447022	+	0.894523i	$0.352485\pi$
$152$	0	0
$153$	6.72578	0.543747
$154$	0	0
$155$	1.21438	0.0975410
$156$	0	0
$157$	9.43133	0.752702	0.376351	−	0.926477i	$-0.377179\pi$
$157$	9.43133	0.752702	0.376351	+	0.926477i	$0.377179\pi$
$158$	0	0
$159$	12.9346	1.02578
$160$	0	0
$161$	−7.74504	−0.610394
$162$	0	0
$163$	5.75502	0.450768	0.225384	−	0.974270i	$-0.427636\pi$
$163$	5.75502	0.450768	0.225384	+	0.974270i	$0.427636\pi$
$164$	0	0
$165$	0.169303	0.0131802
$166$	0	0
$167$	−1.52432	−0.117955	−0.0589776	−	0.998259i	$-0.518784\pi$
$167$	−1.52432	−0.117955	−0.0589776	+	0.998259i	$0.518784\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.80398	0.520313
$172$	0	0
$173$	−3.99656	−0.303853	−0.151927	−	0.988392i	$-0.548548\pi$
$173$	−3.99656	−0.303853	−0.151927	+	0.988392i	$0.548548\pi$
$174$	0	0
$175$	−7.26747	−0.549369
$176$	0	0
$177$	−3.92725	−0.295190
$178$	0	0
$179$	−4.23719	−0.316703	−0.158351	−	0.987383i	$-0.550618\pi$
$179$	−4.23719	−0.316703	−0.158351	+	0.987383i	$0.550618\pi$
$180$	0	0
$181$	13.4718	1.00135	0.500676	−	0.865635i	$-0.333085\pi$
$181$	13.4718	1.00135	0.500676	+	0.865635i	$0.333085\pi$
$182$	0	0
$183$	1.23168	0.0910486
$184$	0	0
$185$	0.851640	0.0626138
$186$	0	0
$187$	6.72578	0.491838
$188$	0	0
$189$	1.46187	0.106336
$190$	0	0
$191$	13.2739	0.960465	0.480232	−	0.877141i	$-0.340552\pi$
$191$	13.2739	0.960465	0.480232	+	0.877141i	$0.340552\pi$
$192$	0	0
$193$	−5.14259	−0.370171	−0.185086	−	0.982722i	$-0.559256\pi$
$193$	−5.14259	−0.370171	−0.185086	+	0.982722i	$0.559256\pi$
$194$	0	0
$195$	0.169303	0.0121240
$196$	0	0
$197$	1.86934	0.133185	0.0665925	−	0.997780i	$-0.478787\pi$
$197$	1.86934	0.133185	0.0665925	+	0.997780i	$0.478787\pi$
$198$	0	0
$199$	13.4134	0.950853	0.475426	−	0.879755i	$-0.342294\pi$
$199$	13.4134	0.950853	0.475426	+	0.879755i	$0.342294\pi$
$200$	0	0
$201$	−0.504017	−0.0355506
$202$	0	0
$203$	−6.18628	−0.434192
$204$	0	0
$205$	0.121955	0.00851772
$206$	0	0
$207$	−5.29802	−0.368238
$208$	0	0
$209$	6.80398	0.470641
$210$	0	0
$211$	8.21586	0.565603	0.282801	−	0.959178i	$-0.408736\pi$
$211$	8.21586	0.565603	0.282801	+	0.959178i	$0.408736\pi$
$212$	0	0
$213$	12.0184	0.823484
$214$	0	0
$215$	−2.05402	−0.140083
$216$	0	0
$217$	10.4857	0.711818
$218$	0	0
$219$	−3.12327	−0.211051
$220$	0	0
$221$	6.72578	0.452425
$222$	0	0
$223$	−1.32530	−0.0887489	−0.0443745	−	0.999015i	$-0.514129\pi$
$223$	−1.32530	−0.0887489	−0.0443745	+	0.999015i	$0.514129\pi$
$224$	0	0
$225$	−4.97134	−0.331422
$226$	0	0
$227$	20.2447	1.34368	0.671842	−	0.740694i	$-0.265503\pi$
$227$	20.2447	1.34368	0.671842	+	0.740694i	$0.265503\pi$
$228$	0	0
$229$	−13.8516	−0.915337	−0.457669	−	0.889123i	$-0.651315\pi$
$229$	−13.8516	−0.915337	−0.457669	+	0.889123i	$0.651315\pi$
$230$	0	0
$231$	1.46187	0.0961843
$232$	0	0
$233$	22.2878	1.46012	0.730060	−	0.683383i	$-0.239492\pi$
$233$	22.2878	1.46012	0.730060	+	0.683383i	$0.239492\pi$
$234$	0	0
$235$	1.76435	0.115094
$236$	0	0
$237$	−6.05246	−0.393150
$238$	0	0
$239$	−6.30445	−0.407801	−0.203900	−	0.978992i	$-0.565362\pi$
$239$	−6.30445	−0.407801	−0.203900	+	0.978992i	$0.565362\pi$
$240$	0	0
$241$	−18.7821	−1.20986	−0.604932	−	0.796277i	$-0.706800\pi$
$241$	−18.7821	−1.20986	−0.604932	+	0.796277i	$0.706800\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.823306	−0.0525991
$246$	0	0
$247$	6.80398	0.432927
$248$	0	0
$249$	16.0109	1.01465
$250$	0	0
$251$	15.6981	0.990856	0.495428	−	0.868649i	$-0.335011\pi$
$251$	15.6981	0.990856	0.495428	+	0.868649i	$0.335011\pi$
$252$	0	0
$253$	−5.29802	−0.333083
$254$	0	0
$255$	1.13869	0.0713078
$256$	0	0
$257$	5.50047	0.343110	0.171555	−	0.985175i	$-0.445121\pi$
$257$	5.50047	0.343110	0.171555	+	0.985175i	$0.445121\pi$
$258$	0	0
$259$	7.35364	0.456933
$260$	0	0
$261$	−4.23175	−0.261938
$262$	0	0
$263$	16.8625	1.03979	0.519894	−	0.854231i	$-0.325971\pi$
$263$	16.8625	1.03979	0.519894	+	0.854231i	$0.325971\pi$
$264$	0	0
$265$	2.18987	0.134523
$266$	0	0
$267$	−1.80392	−0.110398
$268$	0	0
$269$	−13.9227	−0.848883	−0.424441	−	0.905455i	$-0.639530\pi$
$269$	−13.9227	−0.848883	−0.424441	+	0.905455i	$0.639530\pi$
$270$	0	0
$271$	26.7189	1.62306	0.811529	−	0.584312i	$-0.198636\pi$
$271$	26.7189	1.62306	0.811529	+	0.584312i	$0.198636\pi$
$272$	0	0
$273$	1.46187	0.0884767
$274$	0	0
$275$	−4.97134	−0.299783
$276$	0	0
$277$	−12.0560	−0.724376	−0.362188	−	0.932105i	$-0.617970\pi$
$277$	−12.0560	−0.724376	−0.362188	+	0.932105i	$0.617970\pi$
$278$	0	0
$279$	7.17280	0.429424
$280$	0	0
$281$	14.5560	0.868339	0.434170	−	0.900831i	$-0.357042\pi$
$281$	14.5560	0.868339	0.434170	+	0.900831i	$0.357042\pi$
$282$	0	0
$283$	20.0219	1.19018	0.595090	−	0.803659i	$-0.297117\pi$
$283$	20.0219	1.19018	0.595090	+	0.803659i	$0.297117\pi$
$284$	0	0
$285$	1.15193	0.0682346
$286$	0	0
$287$	1.05304	0.0621592
$288$	0	0
$289$	28.2362	1.66095
$290$	0	0
$291$	−9.83614	−0.576605
$292$	0	0
$293$	−29.9852	−1.75175	−0.875876	−	0.482537i	$-0.839716\pi$
$293$	−29.9852	−1.75175	−0.875876	+	0.482537i	$0.839716\pi$
$294$	0	0
$295$	−0.664894	−0.0387116
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−5.29802	−0.306392
$300$	0	0
$301$	−17.7358	−1.02227
$302$	0	0
$303$	−17.3218	−0.995112
$304$	0	0
$305$	0.208527	0.0119402
$306$	0	0
$307$	−31.9883	−1.82567	−0.912833	−	0.408332i	$-0.866110\pi$
$307$	−31.9883	−1.82567	−0.912833	+	0.408332i	$0.866110\pi$
$308$	0	0
$309$	−4.26585	−0.242676
$310$	0	0
$311$	−4.84618	−0.274802	−0.137401	−	0.990516i	$-0.543875\pi$
$311$	−4.84618	−0.274802	−0.137401	+	0.990516i	$0.543875\pi$
$312$	0	0
$313$	−18.2931	−1.03399	−0.516993	−	0.855989i	$-0.672949\pi$
$313$	−18.2931	−1.03399	−0.516993	+	0.855989i	$0.672949\pi$
$314$	0	0
$315$	0.247499	0.0139450
$316$	0	0
$317$	8.11944	0.456033	0.228017	−	0.973657i	$-0.426776\pi$
$317$	8.11944	0.456033	0.228017	+	0.973657i	$0.426776\pi$
$318$	0	0
$319$	−4.23175	−0.236932
$320$	0	0
$321$	−1.59253	−0.0888866
$322$	0	0
$323$	45.7621	2.54627
$324$	0	0
$325$	−4.97134	−0.275760
$326$	0	0
$327$	−1.32480	−0.0732615
$328$	0	0
$329$	15.2346	0.839911
$330$	0	0
$331$	−6.05901	−0.333033	−0.166517	−	0.986039i	$-0.553252\pi$
$331$	−6.05901	−0.333033	−0.166517	+	0.986039i	$0.553252\pi$
$332$	0	0
$333$	5.03028	0.275658
$334$	0	0
$335$	−0.0853314	−0.00466215
$336$	0	0
$337$	−3.19803	−0.174208	−0.0871039	−	0.996199i	$-0.527761\pi$
$337$	−3.19803	−0.174208	−0.0871039	+	0.996199i	$0.527761\pi$
$338$	0	0
$339$	−19.8056	−1.07569
$340$	0	0
$341$	7.17280	0.388429
$342$	0	0
$343$	−17.3421	−0.936386
$344$	0	0
$345$	−0.896969	−0.0482912
$346$	0	0
$347$	3.58261	0.192324	0.0961622	−	0.995366i	$-0.469343\pi$
$347$	3.58261	0.192324	0.0961622	+	0.995366i	$0.469343\pi$
$348$	0	0
$349$	4.75243	0.254392	0.127196	−	0.991878i	$-0.459402\pi$
$349$	4.75243	0.254392	0.127196	+	0.991878i	$0.459402\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	5.89049	0.313519	0.156760	−	0.987637i	$-0.449895\pi$
$353$	5.89049	0.313519	0.156760	+	0.987637i	$0.449895\pi$
$354$	0	0
$355$	2.03474	0.107993
$356$	0	0
$357$	9.83225	0.520378
$358$	0	0
$359$	17.1342	0.904306	0.452153	−	0.891940i	$-0.350656\pi$
$359$	17.1342	0.904306	0.452153	+	0.891940i	$0.350656\pi$
$360$	0	0
$361$	27.2941	1.43653
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−0.528778	−0.0276775
$366$	0	0
$367$	−23.2841	−1.21542	−0.607711	−	0.794158i	$-0.707912\pi$
$367$	−23.2841	−1.21542	−0.607711	+	0.794158i	$0.707912\pi$
$368$	0	0
$369$	0.720338	0.0374993
$370$	0	0
$371$	18.9088	0.981697
$372$	0	0
$373$	−10.8175	−0.560110	−0.280055	−	0.959984i	$-0.590353\pi$
$373$	−10.8175	−0.560110	−0.280055	+	0.959984i	$0.590353\pi$
$374$	0	0
$375$	−1.68817	−0.0871770
$376$	0	0
$377$	−4.23175	−0.217946
$378$	0	0
$379$	−25.4050	−1.30497	−0.652484	−	0.757803i	$-0.726273\pi$
$379$	−25.4050	−1.30497	−0.652484	+	0.757803i	$0.726273\pi$
$380$	0	0
$381$	15.8243	0.810702
$382$	0	0
$383$	22.4029	1.14474	0.572368	−	0.819997i	$-0.306025\pi$
$383$	22.4029	1.14474	0.572368	+	0.819997i	$0.306025\pi$
$384$	0	0
$385$	0.247499	0.0126137
$386$	0	0
$387$	−12.1322	−0.616715
$388$	0	0
$389$	−8.84646	−0.448534	−0.224267	−	0.974528i	$-0.571999\pi$
$389$	−8.84646	−0.448534	−0.224267	+	0.974528i	$0.571999\pi$
$390$	0	0
$391$	−35.6333	−1.80205
$392$	0	0
$393$	10.3302	0.521089
$394$	0	0
$395$	−1.02470	−0.0515582
$396$	0	0
$397$	−5.44164	−0.273108	−0.136554	−	0.990633i	$-0.543603\pi$
$397$	−5.44164	−0.273108	−0.136554	+	0.990633i	$0.543603\pi$
$398$	0	0
$399$	9.94656	0.497951
$400$	0	0
$401$	4.80646	0.240023	0.120011	−	0.992773i	$-0.461707\pi$
$401$	4.80646	0.240023	0.120011	+	0.992773i	$0.461707\pi$
$402$	0	0
$403$	7.17280	0.357303
$404$	0	0
$405$	0.169303	0.00841272
$406$	0	0
$407$	5.03028	0.249342
$408$	0	0
$409$	−10.3006	−0.509330	−0.254665	−	0.967029i	$-0.581965\pi$
$409$	−10.3006	−0.509330	−0.254665	+	0.967029i	$0.581965\pi$
$410$	0	0
$411$	5.00894	0.247073
$412$	0	0
$413$	−5.74114	−0.282503
$414$	0	0
$415$	2.71069	0.133062
$416$	0	0
$417$	−18.3521	−0.898706
$418$	0	0
$419$	−6.71741	−0.328167	−0.164083	−	0.986446i	$-0.552467\pi$
$419$	−6.71741	−0.328167	−0.164083	+	0.986446i	$0.552467\pi$
$420$	0	0
$421$	26.2699	1.28032	0.640158	−	0.768244i	$-0.278869\pi$
$421$	26.2699	1.28032	0.640158	+	0.768244i	$0.278869\pi$
$422$	0	0
$423$	10.4213	0.506700
$424$	0	0
$425$	−33.4361	−1.62189
$426$	0	0
$427$	1.80057	0.0871355
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	5.97238	0.287680	0.143840	−	0.989601i	$-0.454055\pi$
$431$	5.97238	0.287680	0.143840	+	0.989601i	$0.454055\pi$
$432$	0	0
$433$	19.1175	0.918728	0.459364	−	0.888248i	$-0.348077\pi$
$433$	19.1175	0.918728	0.459364	+	0.888248i	$0.348077\pi$
$434$	0	0
$435$	−0.716446	−0.0343510
$436$	0	0
$437$	−36.0476	−1.72439
$438$	0	0
$439$	25.0129	1.19380	0.596901	−	0.802315i	$-0.296399\pi$
$439$	25.0129	1.19380	0.596901	+	0.802315i	$0.296399\pi$
$440$	0	0
$441$	−4.86292	−0.231568
$442$	0	0
$443$	37.8615	1.79885	0.899427	−	0.437071i	$-0.143984\pi$
$443$	37.8615	1.79885	0.899427	+	0.437071i	$0.143984\pi$
$444$	0	0
$445$	−0.305408	−0.0144777
$446$	0	0
$447$	−10.0624	−0.475937
$448$	0	0
$449$	−36.1295	−1.70506	−0.852528	−	0.522681i	$-0.824932\pi$
$449$	−36.1295	−1.70506	−0.852528	+	0.522681i	$0.824932\pi$
$450$	0	0
$451$	0.720338	0.0339194
$452$	0	0
$453$	10.9862	0.516176
$454$	0	0
$455$	0.247499	0.0116030
$456$	0	0
$457$	−9.63104	−0.450521	−0.225261	−	0.974299i	$-0.572323\pi$
$457$	−9.63104	−0.450521	−0.225261	+	0.974299i	$0.572323\pi$
$458$	0	0
$459$	6.72578	0.313933
$460$	0	0
$461$	39.0377	1.81817	0.909083	−	0.416615i	$-0.136784\pi$
$461$	39.0377	1.81817	0.909083	+	0.416615i	$0.136784\pi$
$462$	0	0
$463$	−10.4168	−0.484110	−0.242055	−	0.970263i	$-0.577821\pi$
$463$	−10.4168	−0.484110	−0.242055	+	0.970263i	$0.577821\pi$
$464$	0	0
$465$	1.21438	0.0563153
$466$	0	0
$467$	−6.82610	−0.315874	−0.157937	−	0.987449i	$-0.550484\pi$
$467$	−6.82610	−0.315874	−0.157937	+	0.987449i	$0.550484\pi$
$468$	0	0
$469$	−0.736809	−0.0340227
$470$	0	0
$471$	9.43133	0.434573
$472$	0	0
$473$	−12.1322	−0.557840
$474$	0	0
$475$	−33.8249	−1.55199
$476$	0	0
$477$	12.9346	0.592236
$478$	0	0
$479$	−26.2244	−1.19822	−0.599112	−	0.800665i	$-0.704480\pi$
$479$	−26.2244	−1.19822	−0.599112	+	0.800665i	$0.704480\pi$
$480$	0	0
$481$	5.03028	0.229361
$482$	0	0
$483$	−7.74504	−0.352411
$484$	0	0
$485$	−1.66529	−0.0756167
$486$	0	0
$487$	−22.1940	−1.00571	−0.502853	−	0.864372i	$-0.667716\pi$
$487$	−22.1940	−1.00571	−0.502853	+	0.864372i	$0.667716\pi$
$488$	0	0
$489$	5.75502	0.260251
$490$	0	0
$491$	−31.9338	−1.44115	−0.720576	−	0.693376i	$-0.756122\pi$
$491$	−31.9338	−1.44115	−0.720576	+	0.693376i	$0.756122\pi$
$492$	0	0
$493$	−28.4618	−1.28186
$494$	0	0
$495$	0.169303	0.00760959
$496$	0	0
$497$	17.5693	0.788092
$498$	0	0
$499$	−39.3209	−1.76025	−0.880123	−	0.474747i	$-0.842540\pi$
$499$	−39.3209	−1.76025	−0.880123	+	0.474747i	$0.842540\pi$
$500$	0	0
$501$	−1.52432	−0.0681015
$502$	0	0
$503$	16.9623	0.756310	0.378155	−	0.925742i	$-0.376559\pi$
$503$	16.9623	0.756310	0.378155	+	0.925742i	$0.376559\pi$
$504$	0	0
$505$	−2.93263	−0.130500
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	30.2437	1.34053	0.670264	−	0.742123i	$-0.266181\pi$
$509$	30.2437	1.34053	0.670264	+	0.742123i	$0.266181\pi$
$510$	0	0
$511$	−4.56583	−0.201980
$512$	0	0
$513$	6.80398	0.300403
$514$	0	0
$515$	−0.722221	−0.0318249
$516$	0	0
$517$	10.4213	0.458328
$518$	0	0
$519$	−3.99656	−0.175430
$520$	0	0
$521$	23.2926	1.02047	0.510233	−	0.860036i	$-0.329559\pi$
$521$	23.2926	1.02047	0.510233	+	0.860036i	$0.329559\pi$
$522$	0	0
$523$	−10.2861	−0.449782	−0.224891	−	0.974384i	$-0.572203\pi$
$523$	−10.2861	−0.449782	−0.224891	+	0.974384i	$0.572203\pi$
$524$	0	0
$525$	−7.26747	−0.317178
$526$	0	0
$527$	48.2427	2.10149
$528$	0	0
$529$	5.06899	0.220391
$530$	0	0
$531$	−3.92725	−0.170428
$532$	0	0
$533$	0.720338	0.0312013
$534$	0	0
$535$	−0.269621	−0.0116567
$536$	0	0
$537$	−4.23719	−0.182848
$538$	0	0
$539$	−4.86292	−0.209461
$540$	0	0
$541$	31.4278	1.35119	0.675594	−	0.737274i	$-0.263887\pi$
$541$	31.4278	1.35119	0.675594	+	0.737274i	$0.263887\pi$
$542$	0	0
$543$	13.4718	0.578130
$544$	0	0
$545$	−0.224292	−0.00960761
$546$	0	0
$547$	6.41396	0.274241	0.137121	−	0.990554i	$-0.456215\pi$
$547$	6.41396	0.274241	0.137121	+	0.990554i	$0.456215\pi$
$548$	0	0
$549$	1.23168	0.0525669
$550$	0	0
$551$	−28.7927	−1.22661
$552$	0	0
$553$	−8.84794	−0.376253
$554$	0	0
$555$	0.851640	0.0361501
$556$	0	0
$557$	40.6435	1.72212	0.861061	−	0.508501i	$-0.169800\pi$
$557$	40.6435	1.72212	0.861061	+	0.508501i	$0.169800\pi$
$558$	0	0
$559$	−12.1322	−0.513138
$560$	0	0
$561$	6.72578	0.283963
$562$	0	0
$563$	−35.7972	−1.50867	−0.754335	−	0.656489i	$-0.772041\pi$
$563$	−35.7972	−1.50867	−0.754335	+	0.656489i	$0.772041\pi$
$564$	0	0
$565$	−3.35314	−0.141068
$566$	0	0
$567$	1.46187	0.0613930
$568$	0	0
$569$	−0.718784	−0.0301330	−0.0150665	−	0.999886i	$-0.504796\pi$
$569$	−0.718784	−0.0301330	−0.0150665	+	0.999886i	$0.504796\pi$
$570$	0	0
$571$	41.9442	1.75531	0.877654	−	0.479294i	$-0.159107\pi$
$571$	41.9442	1.75531	0.877654	+	0.479294i	$0.159107\pi$
$572$	0	0
$573$	13.2739	0.554525
$574$	0	0
$575$	26.3382	1.09838
$576$	0	0
$577$	27.2971	1.13639	0.568196	−	0.822893i	$-0.307641\pi$
$577$	27.2971	1.13639	0.568196	+	0.822893i	$0.307641\pi$
$578$	0	0
$579$	−5.14259	−0.213719
$580$	0	0
$581$	23.4059	0.971041
$582$	0	0
$583$	12.9346	0.535698
$584$	0	0
$585$	0.169303	0.00699981
$586$	0	0
$587$	1.53119	0.0631990	0.0315995	−	0.999501i	$-0.489940\pi$
$587$	1.53119	0.0631990	0.0315995	+	0.999501i	$0.489940\pi$
$588$	0	0
$589$	48.8036	2.01092
$590$	0	0
$591$	1.86934	0.0768944
$592$	0	0
$593$	15.2913	0.627939	0.313970	−	0.949433i	$-0.398341\pi$
$593$	15.2913	0.627939	0.313970	+	0.949433i	$0.398341\pi$
$594$	0	0
$595$	1.66463	0.0682431
$596$	0	0
$597$	13.4134	0.548975
$598$	0	0
$599$	32.3443	1.32155	0.660776	−	0.750583i	$-0.270227\pi$
$599$	32.3443	1.32155	0.660776	+	0.750583i	$0.270227\pi$
$600$	0	0
$601$	15.8783	0.647691	0.323846	−	0.946110i	$-0.395024\pi$
$601$	15.8783	0.647691	0.323846	+	0.946110i	$0.395024\pi$
$602$	0	0
$603$	−0.504017	−0.0205251
$604$	0	0
$605$	0.169303	0.00688314
$606$	0	0
$607$	15.8243	0.642288	0.321144	−	0.947030i	$-0.395933\pi$
$607$	15.8243	0.642288	0.321144	+	0.947030i	$0.395933\pi$
$608$	0	0
$609$	−6.18628	−0.250681
$610$	0	0
$611$	10.4213	0.421600
$612$	0	0
$613$	−25.0853	−1.01318	−0.506592	−	0.862186i	$-0.669095\pi$
$613$	−25.0853	−1.01318	−0.506592	+	0.862186i	$0.669095\pi$
$614$	0	0
$615$	0.121955	0.00491771
$616$	0	0
$617$	−42.3649	−1.70555	−0.852773	−	0.522282i	$-0.825081\pi$
$617$	−42.3649	−1.70555	−0.852773	+	0.522282i	$0.825081\pi$
$618$	0	0
$619$	8.87965	0.356903	0.178452	−	0.983949i	$-0.442891\pi$
$619$	8.87965	0.356903	0.178452	+	0.983949i	$0.442891\pi$
$620$	0	0
$621$	−5.29802	−0.212602
$622$	0	0
$623$	−2.63710	−0.105653
$624$	0	0
$625$	24.5709	0.982835
$626$	0	0
$627$	6.80398	0.271725
$628$	0	0
$629$	33.8326	1.34899
$630$	0	0
$631$	−23.0862	−0.919048	−0.459524	−	0.888165i	$-0.651980\pi$
$631$	−23.0862	−0.919048	−0.459524	+	0.888165i	$0.651980\pi$
$632$	0	0
$633$	8.21586	0.326551
$634$	0	0
$635$	2.67909	0.106317
$636$	0	0
$637$	−4.86292	−0.192676
$638$	0	0
$639$	12.0184	0.475439
$640$	0	0
$641$	−49.0640	−1.93791	−0.968955	−	0.247237i	$-0.920477\pi$
$641$	−49.0640	−1.93791	−0.968955	+	0.247237i	$0.920477\pi$
$642$	0	0
$643$	−12.7668	−0.503472	−0.251736	−	0.967796i	$-0.581001\pi$
$643$	−12.7668	−0.503472	−0.251736	+	0.967796i	$0.581001\pi$
$644$	0	0
$645$	−2.05402	−0.0808768
$646$	0	0
$647$	−10.3683	−0.407621	−0.203810	−	0.979010i	$-0.565333\pi$
$647$	−10.3683	−0.407621	−0.203810	+	0.979010i	$0.565333\pi$
$648$	0	0
$649$	−3.92725	−0.154158
$650$	0	0
$651$	10.4857	0.410968
$652$	0	0
$653$	18.9307	0.740817	0.370409	−	0.928869i	$-0.379218\pi$
$653$	18.9307	0.740817	0.370409	+	0.928869i	$0.379218\pi$
$654$	0	0
$655$	1.74893	0.0683363
$656$	0	0
$657$	−3.12327	−0.121850
$658$	0	0
$659$	9.33917	0.363803	0.181901	−	0.983317i	$-0.441775\pi$
$659$	9.33917	0.363803	0.181901	+	0.983317i	$0.441775\pi$
$660$	0	0
$661$	3.10985	0.120959	0.0604796	−	0.998169i	$-0.480737\pi$
$661$	3.10985	0.120959	0.0604796	+	0.998169i	$0.480737\pi$
$662$	0	0
$663$	6.72578	0.261208
$664$	0	0
$665$	1.68398	0.0653020
$666$	0	0
$667$	22.4199	0.868100
$668$	0	0
$669$	−1.32530	−0.0512392
$670$	0	0
$671$	1.23168	0.0475486
$672$	0	0
$673$	15.3905	0.593262	0.296631	−	0.954992i	$-0.404137\pi$
$673$	15.3905	0.593262	0.296631	+	0.954992i	$0.404137\pi$
$674$	0	0
$675$	−4.97134	−0.191347
$676$	0	0
$677$	−26.3887	−1.01420	−0.507100	−	0.861887i	$-0.669282\pi$
$677$	−26.3887	−1.01420	−0.507100	+	0.861887i	$0.669282\pi$
$678$	0	0
$679$	−14.3792	−0.551823
$680$	0	0
$681$	20.2447	0.775777
$682$	0	0
$683$	−26.3448	−1.00805	−0.504027	−	0.863688i	$-0.668149\pi$
$683$	−26.3448	−1.00805	−0.504027	+	0.863688i	$0.668149\pi$
$684$	0	0
$685$	0.848028	0.0324015
$686$	0	0
$687$	−13.8516	−0.528470
$688$	0	0
$689$	12.9346	0.492771
$690$	0	0
$691$	−18.3557	−0.698284	−0.349142	−	0.937070i	$-0.613527\pi$
$691$	−18.3557	−0.698284	−0.349142	+	0.937070i	$0.613527\pi$
$692$	0	0
$693$	1.46187	0.0555320
$694$	0	0
$695$	−3.10706	−0.117858
$696$	0	0
$697$	4.84483	0.183511
$698$	0	0
$699$	22.2878	0.843001
$700$	0	0
$701$	−11.9392	−0.450939	−0.225469	−	0.974250i	$-0.572392\pi$
$701$	−11.9392	−0.450939	−0.225469	+	0.974250i	$0.572392\pi$
$702$	0	0
$703$	34.2259	1.29086
$704$	0	0
$705$	1.76435	0.0664494
$706$	0	0
$707$	−25.3223	−0.952344
$708$	0	0
$709$	35.4427	1.33108	0.665539	−	0.746363i	$-0.268202\pi$
$709$	35.4427	1.33108	0.665539	+	0.746363i	$0.268202\pi$
$710$	0	0
$711$	−6.05246	−0.226985
$712$	0	0
$713$	−38.0016	−1.42317
$714$	0	0
$715$	0.169303	0.00633156
$716$	0	0
$717$	−6.30445	−0.235444
$718$	0	0
$719$	−21.5206	−0.802581	−0.401291	−	0.915951i	$-0.631438\pi$
$719$	−21.5206	−0.802581	−0.401291	+	0.915951i	$0.631438\pi$
$720$	0	0
$721$	−6.23614	−0.232246
$722$	0	0
$723$	−18.7821	−0.698515
$724$	0	0
$725$	21.0374	0.781311
$726$	0	0
$727$	−19.9536	−0.740037	−0.370018	−	0.929024i	$-0.620649\pi$
$727$	−19.9536	−0.740037	−0.370018	+	0.929024i	$0.620649\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−81.5986	−3.01803
$732$	0	0
$733$	−9.02391	−0.333306	−0.166653	−	0.986016i	$-0.553296\pi$
$733$	−9.02391	−0.333306	−0.166653	+	0.986016i	$0.553296\pi$
$734$	0	0
$735$	−0.823306	−0.0303681
$736$	0	0
$737$	−0.504017	−0.0185657
$738$	0	0
$739$	−52.6309	−1.93606	−0.968029	−	0.250839i	$-0.919293\pi$
$739$	−52.6309	−1.93606	−0.968029	+	0.250839i	$0.919293\pi$
$740$	0	0
$741$	6.80398	0.249950
$742$	0	0
$743$	29.0711	1.06651	0.533257	−	0.845953i	$-0.320968\pi$
$743$	29.0711	1.06651	0.533257	+	0.845953i	$0.320968\pi$
$744$	0	0
$745$	−1.70360	−0.0624150
$746$	0	0
$747$	16.0109	0.585808
$748$	0	0
$749$	−2.32809	−0.0850664
$750$	0	0
$751$	−6.44794	−0.235289	−0.117644	−	0.993056i	$-0.537534\pi$
$751$	−6.44794	−0.235289	−0.117644	+	0.993056i	$0.537534\pi$
$752$	0	0
$753$	15.6981	0.572071
$754$	0	0
$755$	1.85999	0.0676921
$756$	0	0
$757$	−36.6510	−1.33210	−0.666052	−	0.745905i	$-0.732017\pi$
$757$	−36.6510	−1.33210	−0.666052	+	0.745905i	$0.732017\pi$
$758$	0	0
$759$	−5.29802	−0.192306
$760$	0	0
$761$	−31.6839	−1.14854	−0.574270	−	0.818666i	$-0.694714\pi$
$761$	−31.6839	−1.14854	−0.574270	+	0.818666i	$0.694714\pi$
$762$	0	0
$763$	−1.93669	−0.0701128
$764$	0	0
$765$	1.13869	0.0411696
$766$	0	0
$767$	−3.92725	−0.141805
$768$	0	0
$769$	5.40643	0.194961	0.0974804	−	0.995237i	$-0.468922\pi$
$769$	5.40643	0.194961	0.0974804	+	0.995237i	$0.468922\pi$
$770$	0	0
$771$	5.50047	0.198094
$772$	0	0
$773$	−25.1099	−0.903139	−0.451569	−	0.892236i	$-0.649136\pi$
$773$	−25.1099	−0.903139	−0.451569	+	0.892236i	$0.649136\pi$
$774$	0	0
$775$	−35.6584	−1.28089
$776$	0	0
$777$	7.35364	0.263810
$778$	0	0
$779$	4.90116	0.175602
$780$	0	0
$781$	12.0184	0.430051
$782$	0	0
$783$	−4.23175	−0.151230
$784$	0	0
$785$	1.59675	0.0569904
$786$	0	0
$787$	−46.6625	−1.66334	−0.831669	−	0.555272i	$-0.812614\pi$
$787$	−46.6625	−1.66334	−0.831669	+	0.555272i	$0.812614\pi$
$788$	0	0
$789$	16.8625	0.600322
$790$	0	0
$791$	−28.9533	−1.02946
$792$	0	0
$793$	1.23168	0.0437383
$794$	0	0
$795$	2.18987	0.0776667
$796$	0	0
$797$	42.8734	1.51865	0.759327	−	0.650709i	$-0.225528\pi$
$797$	42.8734	1.51865	0.759327	+	0.650709i	$0.225528\pi$
$798$	0	0
$799$	70.0913	2.47965
$800$	0	0
$801$	−1.80392	−0.0637383
$802$	0	0
$803$	−3.12327	−0.110218
$804$	0	0
$805$	−1.31126	−0.0462157
$806$	0	0
$807$	−13.9227	−0.490103
$808$	0	0
$809$	−15.5992	−0.548439	−0.274219	−	0.961667i	$-0.588419\pi$
$809$	−15.5992	−0.548439	−0.274219	+	0.961667i	$0.588419\pi$
$810$	0	0
$811$	1.72773	0.0606688	0.0303344	−	0.999540i	$-0.490343\pi$
$811$	1.72773	0.0606688	0.0303344	+	0.999540i	$0.490343\pi$
$812$	0	0
$813$	26.7189	0.937073
$814$	0	0
$815$	0.974340	0.0341296
$816$	0	0
$817$	−82.5473	−2.88797
$818$	0	0
$819$	1.46187	0.0510820
$820$	0	0
$821$	22.9823	0.802088	0.401044	−	0.916059i	$-0.368647\pi$
$821$	22.9823	0.802088	0.401044	+	0.916059i	$0.368647\pi$
$822$	0	0
$823$	−24.7293	−0.862011	−0.431005	−	0.902349i	$-0.641841\pi$
$823$	−24.7293	−0.862011	−0.431005	+	0.902349i	$0.641841\pi$
$824$	0	0
$825$	−4.97134	−0.173080
$826$	0	0
$827$	−5.12470	−0.178203	−0.0891016	−	0.996023i	$-0.528400\pi$
$827$	−5.12470	−0.178203	−0.0891016	+	0.996023i	$0.528400\pi$
$828$	0	0
$829$	−47.5783	−1.65246	−0.826231	−	0.563332i	$-0.809519\pi$
$829$	−47.5783	−1.65246	−0.826231	+	0.563332i	$0.809519\pi$
$830$	0	0
$831$	−12.0560	−0.418219
$832$	0	0
$833$	−32.7070	−1.13323
$834$	0	0
$835$	−0.258071	−0.00893092
$836$	0	0
$837$	7.17280	0.247928
$838$	0	0
$839$	−42.4415	−1.46524	−0.732621	−	0.680637i	$-0.761703\pi$
$839$	−42.4415	−1.46524	−0.732621	+	0.680637i	$0.761703\pi$
$840$	0	0
$841$	−11.0923	−0.382494
$842$	0	0
$843$	14.5560	0.501336
$844$	0	0
$845$	0.169303	0.00582419
$846$	0	0
$847$	1.46187	0.0502306
$848$	0	0
$849$	20.0219	0.687150
$850$	0	0
$851$	−26.6505	−0.913568
$852$	0	0
$853$	−28.4447	−0.973927	−0.486963	−	0.873422i	$-0.661896\pi$
$853$	−28.4447	−0.973927	−0.486963	+	0.873422i	$0.661896\pi$
$854$	0	0
$855$	1.15193	0.0393953
$856$	0	0
$857$	38.5420	1.31657	0.658284	−	0.752769i	$-0.271282\pi$
$857$	38.5420	1.31657	0.658284	+	0.752769i	$0.271282\pi$
$858$	0	0
$859$	−52.5831	−1.79411	−0.897056	−	0.441918i	$-0.854298\pi$
$859$	−52.5831	−1.79411	−0.897056	+	0.441918i	$0.854298\pi$
$860$	0	0
$861$	1.05304	0.0358876
$862$	0	0
$863$	16.1893	0.551091	0.275546	−	0.961288i	$-0.411141\pi$
$863$	16.1893	0.551091	0.275546	+	0.961288i	$0.411141\pi$
$864$	0	0
$865$	−0.676629	−0.0230061
$866$	0	0
$867$	28.2362	0.958950
$868$	0	0
$869$	−6.05246	−0.205316
$870$	0	0
$871$	−0.504017	−0.0170780
$872$	0	0
$873$	−9.83614	−0.332903
$874$	0	0
$875$	−2.46790	−0.0834302
$876$	0	0
$877$	−42.1952	−1.42483	−0.712415	−	0.701758i	$-0.752399\pi$
$877$	−42.1952	−1.42483	−0.712415	+	0.701758i	$0.752399\pi$
$878$	0	0
$879$	−29.9852	−1.01137
$880$	0	0
$881$	−44.8413	−1.51074	−0.755370	−	0.655298i	$-0.772543\pi$
$881$	−44.8413	−1.51074	−0.755370	+	0.655298i	$0.772543\pi$
$882$	0	0
$883$	0.137720	0.00463464	0.00231732	−	0.999997i	$-0.499262\pi$
$883$	0.137720	0.00463464	0.00231732	+	0.999997i	$0.499262\pi$
$884$	0	0
$885$	−0.664894	−0.0223502
$886$	0	0
$887$	−16.0809	−0.539943	−0.269972	−	0.962868i	$-0.587014\pi$
$887$	−16.0809	−0.539943	−0.269972	+	0.962868i	$0.587014\pi$
$888$	0	0
$889$	23.1331	0.775860
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	70.9062	2.37279
$894$	0	0
$895$	−0.717368	−0.0239790
$896$	0	0
$897$	−5.29802	−0.176896
$898$	0	0
$899$	−30.3535	−1.01235
$900$	0	0
$901$	86.9956	2.89824
$902$	0	0
$903$	−17.7358	−0.590210
$904$	0	0
$905$	2.28081	0.0758168
$906$	0	0
$907$	−45.3518	−1.50588	−0.752942	−	0.658087i	$-0.771366\pi$
$907$	−45.3518	−1.50588	−0.752942	+	0.658087i	$0.771366\pi$
$908$	0	0
$909$	−17.3218	−0.574528
$910$	0	0
$911$	−1.90417	−0.0630878	−0.0315439	−	0.999502i	$-0.510042\pi$
$911$	−1.90417	−0.0630878	−0.0315439	+	0.999502i	$0.510042\pi$
$912$	0	0
$913$	16.0109	0.529883
$914$	0	0
$915$	0.208527	0.00689370
$916$	0	0
$917$	15.1014	0.498693
$918$	0	0
$919$	60.0572	1.98110	0.990552	−	0.137137i	$-0.0437900\pi$
$919$	60.0572	1.98110	0.990552	+	0.137137i	$0.0437900\pi$
$920$	0	0
$921$	−31.9883	−1.05405
$922$	0	0
$923$	12.0184	0.395589
$924$	0	0
$925$	−25.0072	−0.822232
$926$	0	0
$927$	−4.26585	−0.140109
$928$	0	0
$929$	15.4685	0.507506	0.253753	−	0.967269i	$-0.418335\pi$
$929$	15.4685	0.507506	0.253753	+	0.967269i	$0.418335\pi$
$930$	0	0
$931$	−33.0872	−1.08439
$932$	0	0
$933$	−4.84618	−0.158657
$934$	0	0
$935$	1.13869	0.0372393
$936$	0	0
$937$	−47.9243	−1.56562	−0.782810	−	0.622261i	$-0.786214\pi$
$937$	−47.9243	−1.56562	−0.782810	+	0.622261i	$0.786214\pi$
$938$	0	0
$939$	−18.2931	−0.596972
$940$	0	0
$941$	−3.31714	−0.108136	−0.0540678	−	0.998537i	$-0.517219\pi$
$941$	−3.31714	−0.108136	−0.0540678	+	0.998537i	$0.517219\pi$
$942$	0	0
$943$	−3.81636	−0.124278
$944$	0	0
$945$	0.247499	0.00805116
$946$	0	0
$947$	20.7581	0.674547	0.337274	−	0.941407i	$-0.390495\pi$
$947$	20.7581	0.674547	0.337274	+	0.941407i	$0.390495\pi$
$948$	0	0
$949$	−3.12327	−0.101386
$950$	0	0
$951$	8.11944	0.263291
$952$	0	0
$953$	14.2806	0.462596	0.231298	−	0.972883i	$-0.425703\pi$
$953$	14.2806	0.462596	0.231298	+	0.972883i	$0.425703\pi$
$954$	0	0
$955$	2.24731	0.0727211
$956$	0	0
$957$	−4.23175	−0.136793
$958$	0	0
$959$	7.32245	0.236454
$960$	0	0
$961$	20.4491	0.659648
$962$	0	0
$963$	−1.59253	−0.0513187
$964$	0	0
$965$	−0.870654	−0.0280273
$966$	0	0
$967$	34.9044	1.12245	0.561224	−	0.827664i	$-0.310330\pi$
$967$	34.9044	1.12245	0.561224	+	0.827664i	$0.310330\pi$
$968$	0	0
$969$	45.7621	1.47009
$970$	0	0
$971$	45.4184	1.45754	0.728772	−	0.684756i	$-0.240091\pi$
$971$	45.4184	1.45754	0.728772	+	0.684756i	$0.240091\pi$
$972$	0	0
$973$	−26.8285	−0.860081
$974$	0	0
$975$	−4.97134	−0.159210
$976$	0	0
$977$	57.9835	1.85506	0.927529	−	0.373752i	$-0.121929\pi$
$977$	57.9835	1.85506	0.927529	+	0.373752i	$0.121929\pi$
$978$	0	0
$979$	−1.80392	−0.0576534
$980$	0	0
$981$	−1.32480	−0.0422975
$982$	0	0
$983$	−15.4019	−0.491244	−0.245622	−	0.969366i	$-0.578992\pi$
$983$	−15.4019	−0.491244	−0.245622	+	0.969366i	$0.578992\pi$
$984$	0	0
$985$	0.316484	0.0100840
$986$	0	0
$987$	15.2346	0.484923
$988$	0	0
$989$	64.2767	2.04388
$990$	0	0
$991$	10.1165	0.321360	0.160680	−	0.987007i	$-0.448631\pi$
$991$	10.1165	0.321360	0.160680	+	0.987007i	$0.448631\pi$
$992$	0	0
$993$	−6.05901	−0.192277
$994$	0	0
$995$	2.27093	0.0719933
$996$	0	0
$997$	−44.1473	−1.39816	−0.699080	−	0.715044i	$-0.746407\pi$
$997$	−44.1473	−1.39816	−0.699080	+	0.715044i	$0.746407\pi$
$998$	0	0
$999$	5.03028	0.159151

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.cf.1.3		5
4.3	odd	2		3432.2.a.w.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.w.1.3	✓	5	4.3	odd	2
6864.2.a.cf.1.3		5	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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.169303 q^{5} +1.46187 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.169303 q^{5} +1.46187 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +0.169303 q^{15} +6.72578 q^{17} +6.80398 q^{19} +1.46187 q^{21} -5.29802 q^{23} -4.97134 q^{25} +1.00000 q^{27} -4.23175 q^{29} +7.17280 q^{31} +1.00000 q^{33} +0.247499 q^{35} +5.03028 q^{37} +1.00000 q^{39} +0.720338 q^{41} -12.1322 q^{43} +0.169303 q^{45} +10.4213 q^{47} -4.86292 q^{49} +6.72578 q^{51} +12.9346 q^{53} +0.169303 q^{55} +6.80398 q^{57} -3.92725 q^{59} +1.23168 q^{61} +1.46187 q^{63} +0.169303 q^{65} -0.504017 q^{67} -5.29802 q^{69} +12.0184 q^{71} -3.12327 q^{73} -4.97134 q^{75} +1.46187 q^{77} -6.05246 q^{79} +1.00000 q^{81} +16.0109 q^{83} +1.13869 q^{85} -4.23175 q^{87} -1.80392 q^{89} +1.46187 q^{91} +7.17280 q^{93} +1.15193 q^{95} -9.83614 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + q^5 + 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} + q^{5} + 5 q^{7} + 5 q^{9} + 5 q^{11} + 5 q^{13} + q^{15} + 4 q^{19} + 5 q^{21} + 5 q^{23} + 4 q^{25} + 5 q^{27} + 11 q^{29} + 8 q^{31} + 5 q^{33} + 5 q^{35} - 8 q^{37} + 5 q^{39} - q^{41} - q^{43} + q^{45} + 18 q^{47} + 10 q^{49} + 2 q^{53} + q^{55} + 4 q^{57} + 13 q^{59} + 9 q^{61} + 5 q^{63} + q^{65} - 5 q^{67} + 5 q^{69} + 24 q^{71} - 13 q^{73} + 4 q^{75} + 5 q^{77} + 6 q^{79} + 5 q^{81} + 22 q^{83} - 22 q^{85} + 11 q^{87} - 14 q^{89} + 5 q^{91} + 8 q^{93} + 32 q^{95} - 20 q^{97} + 5 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + q^5 + 5 * q^7 + 5 * q^9 + 5 * q^11 + 5 * q^13 + q^15 + 4 * q^19 + 5 * q^21 + 5 * q^23 + 4 * q^25 + 5 * q^27 + 11 * q^29 + 8 * q^31 + 5 * q^33 + 5 * q^35 - 8 * q^37 + 5 * q^39 - q^41 - q^43 + q^45 + 18 * q^47 + 10 * q^49 + 2 * q^53 + q^55 + 4 * q^57 + 13 * q^59 + 9 * q^61 + 5 * q^63 + q^65 - 5 * q^67 + 5 * q^69 + 24 * q^71 - 13 * q^73 + 4 * q^75 + 5 * q^77 + 6 * q^79 + 5 * q^81 + 22 * q^83 - 22 * q^85 + 11 * q^87 - 14 * q^89 + 5 * q^91 + 8 * q^93 + 32 * q^95 - 20 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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