LMFDB - Embedded newform 6864.2.a.by.1.4

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.22676.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 6x + 2 $ x^4 - x^3 - 6x^2 + 6x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.4
Root			$-2.39890$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.75471 q^{5} -2.58843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.75471 q^{5} -2.58843 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.75471 q^{15} -4.34314 q^{17} +4.79780 q^{19} -2.58843 q^{21} -5.83031 q^{23} -1.92099 q^{25} +1.00000 q^{27} -9.37565 q^{29} +7.14093 q^{31} +1.00000 q^{33} -4.54194 q^{35} +2.90932 q^{37} -1.00000 q^{39} -6.92099 q^{41} -0.133767 q^{43} +1.75471 q^{45} +9.59559 q^{47} -0.300050 q^{49} -4.34314 q^{51} -2.00000 q^{53} +1.75471 q^{55} +4.79780 q^{57} +7.38622 q^{59} -12.6281 q^{61} -2.58843 q^{63} -1.75471 q^{65} -8.17345 q^{67} -5.83031 q^{69} -10.0862 q^{71} +0.0978457 q^{73} -1.92099 q^{75} -2.58843 q^{77} -6.34314 q^{79} +1.00000 q^{81} -10.7724 q^{83} -7.62094 q^{85} -9.37565 q^{87} -0.945240 q^{89} +2.58843 q^{91} +7.14093 q^{93} +8.41874 q^{95} +2.90932 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - 3 q^{15} - 2 q^{19} - 3 q^{21} - 3 q^{23} + 5 q^{25} + 4 q^{27} - 21 q^{29} - 10 q^{31} + 4 q^{33} + q^{35} + 4 q^{37} - 4 q^{39} - 15 q^{41} + 3 q^{43} - 3 q^{45} - 4 q^{47} + 5 q^{49} - 8 q^{53} - 3 q^{55} - 2 q^{57} + q^{59} - 9 q^{61} - 3 q^{63} + 3 q^{65} + 5 q^{67} - 3 q^{69} - 18 q^{71} - 27 q^{73} + 5 q^{75} - 3 q^{77} - 8 q^{79} + 4 q^{81} + 14 q^{83} - 24 q^{85} - 21 q^{87} - 20 q^{89} + 3 q^{91} - 10 q^{93} + 6 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9 + 4 * q^11 - 4 * q^13 - 3 * q^15 - 2 * q^19 - 3 * q^21 - 3 * q^23 + 5 * q^25 + 4 * q^27 - 21 * q^29 - 10 * q^31 + 4 * q^33 + q^35 + 4 * q^37 - 4 * q^39 - 15 * q^41 + 3 * q^43 - 3 * q^45 - 4 * q^47 + 5 * q^49 - 8 * q^53 - 3 * q^55 - 2 * q^57 + q^59 - 9 * q^61 - 3 * q^63 + 3 * q^65 + 5 * q^67 - 3 * q^69 - 18 * q^71 - 27 * q^73 + 5 * q^75 - 3 * q^77 - 8 * q^79 + 4 * q^81 + 14 * q^83 - 24 * q^85 - 21 * q^87 - 20 * q^89 + 3 * q^91 - 10 * q^93 + 6 * q^95 + 4 * q^97 + 4 * 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
$5$	1.75471	0.784730	0.392365	−	0.919810i	$-0.371657\pi$
$5$	1.75471	0.784730	0.392365	+	0.919810i	$0.371657\pi$
$6$	0	0
$7$	−2.58843	−0.978333	−0.489167	−	0.872190i	$-0.662699\pi$
$7$	−2.58843	−0.978333	−0.489167	+	0.872190i	$0.662699\pi$
$8$	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
$15$	1.75471	0.453064
$16$	0	0
$17$	−4.34314	−1.05337	−0.526683	−	0.850062i	$-0.676564\pi$
$17$	−4.34314	−1.05337	−0.526683	+	0.850062i	$0.676564\pi$
$18$	0	0
$19$	4.79780	1.10069	0.550345	−	0.834937i	$-0.314496\pi$
$19$	4.79780	1.10069	0.550345	+	0.834937i	$0.314496\pi$
$20$	0	0
$21$	−2.58843	−0.564841
$22$	0	0
$23$	−5.83031	−1.21570	−0.607852	−	0.794050i	$-0.707969\pi$
$23$	−5.83031	−1.21570	−0.607852	+	0.794050i	$0.707969\pi$
$24$	0	0
$25$	−1.92099	−0.384199
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.37565	−1.74102	−0.870508	−	0.492155i	$-0.836209\pi$
$29$	−9.37565	−1.74102	−0.870508	+	0.492155i	$0.836209\pi$
$30$	0	0
$31$	7.14093	1.28255	0.641275	−	0.767312i	$-0.278406\pi$
$31$	7.14093	1.28255	0.641275	+	0.767312i	$0.278406\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−4.54194	−0.767727
$36$	0	0
$37$	2.90932	0.478289	0.239145	−	0.970984i	$-0.423133\pi$
$37$	2.90932	0.478289	0.239145	+	0.970984i	$0.423133\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−6.92099	−1.08088	−0.540439	−	0.841383i	$-0.681742\pi$
$41$	−6.92099	−1.08088	−0.540439	+	0.841383i	$0.681742\pi$
$42$	0	0
$43$	−0.133767	−0.0203992	−0.0101996	−	0.999948i	$-0.503247\pi$
$43$	−0.133767	−0.0203992	−0.0101996	+	0.999948i	$0.503247\pi$
$44$	0	0
$45$	1.75471	0.261577
$46$	0	0
$47$	9.59559	1.39966	0.699830	−	0.714309i	$-0.253259\pi$
$47$	9.59559	1.39966	0.699830	+	0.714309i	$0.253259\pi$
$48$	0	0
$49$	−0.300050	−0.0428643
$50$	0	0
$51$	−4.34314	−0.608161
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	1.75471	0.236605
$56$	0	0
$57$	4.79780	0.635484
$58$	0	0
$59$	7.38622	0.961604	0.480802	−	0.876829i	$-0.340346\pi$
$59$	7.38622	0.961604	0.480802	+	0.876829i	$0.340346\pi$
$60$	0	0
$61$	−12.6281	−1.61686	−0.808432	−	0.588590i	$-0.799683\pi$
$61$	−12.6281	−1.61686	−0.808432	+	0.588590i	$0.799683\pi$
$62$	0	0
$63$	−2.58843	−0.326111
$64$	0	0
$65$	−1.75471	−0.217645
$66$	0	0
$67$	−8.17345	−0.998546	−0.499273	−	0.866445i	$-0.666399\pi$
$67$	−8.17345	−0.998546	−0.499273	+	0.866445i	$0.666399\pi$
$68$	0	0
$69$	−5.83031	−0.701887
$70$	0	0
$71$	−10.0862	−1.19701	−0.598504	−	0.801120i	$-0.704238\pi$
$71$	−10.0862	−1.19701	−0.598504	+	0.801120i	$0.704238\pi$
$72$	0	0
$73$	0.0978457	0.0114520	0.00572599	−	0.999984i	$-0.498177\pi$
$73$	0.0978457	0.0114520	0.00572599	+	0.999984i	$0.498177\pi$
$74$	0	0
$75$	−1.92099	−0.221817
$76$	0	0
$77$	−2.58843	−0.294979
$78$	0	0
$79$	−6.34314	−0.713659	−0.356829	−	0.934170i	$-0.616142\pi$
$79$	−6.34314	−0.713659	−0.356829	+	0.934170i	$0.616142\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.7724	−1.18243	−0.591215	−	0.806514i	$-0.701351\pi$
$83$	−10.7724	−1.18243	−0.591215	+	0.806514i	$0.701351\pi$
$84$	0	0
$85$	−7.62094	−0.826607
$86$	0	0
$87$	−9.37565	−1.00518
$88$	0	0
$89$	−0.945240	−0.100195	−0.0500976	−	0.998744i	$-0.515953\pi$
$89$	−0.945240	−0.100195	−0.0500976	+	0.998744i	$0.515953\pi$
$90$	0	0
$91$	2.58843	0.271341
$92$	0	0
$93$	7.14093	0.740480
$94$	0	0
$95$	8.41874	0.863744
$96$	0	0
$97$	2.90932	0.295397	0.147698	−	0.989032i	$-0.452814\pi$
$97$	2.90932	0.295397	0.147698	+	0.989032i	$0.452814\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−14.4293	−1.43577	−0.717885	−	0.696162i	$-0.754890\pi$
$101$	−14.4293	−1.43577	−0.717885	+	0.696162i	$0.754890\pi$
$102$	0	0
$103$	−11.3862	−1.12192	−0.560959	−	0.827844i	$-0.689567\pi$
$103$	−11.3862	−1.12192	−0.560959	+	0.827844i	$0.689567\pi$
$104$	0	0
$105$	−4.54194	−0.443248
$106$	0	0
$107$	1.79063	0.173107	0.0865534	−	0.996247i	$-0.472415\pi$
$107$	1.79063	0.173107	0.0865534	+	0.996247i	$0.472415\pi$
$108$	0	0
$109$	10.0397	0.961627	0.480814	−	0.876823i	$-0.340341\pi$
$109$	10.0397	0.961627	0.480814	+	0.876823i	$0.340341\pi$
$110$	0	0
$111$	2.90932	0.276140
$112$	0	0
$113$	19.9145	1.87340	0.936698	−	0.350137i	$-0.113865\pi$
$113$	19.9145	1.87340	0.936698	+	0.350137i	$0.113865\pi$
$114$	0	0
$115$	−10.2305	−0.954000
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	11.2419	1.03054
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.92099	−0.624045
$124$	0	0
$125$	−12.1443	−1.08622
$126$	0	0
$127$	2.34314	0.207920	0.103960	−	0.994581i	$-0.466849\pi$
$127$	2.34314	0.207920	0.103960	+	0.994581i	$0.466849\pi$
$128$	0	0
$129$	−0.133767	−0.0117775
$130$	0	0
$131$	18.1375	1.58468	0.792342	−	0.610078i	$-0.208862\pi$
$131$	18.1375	1.58468	0.792342	+	0.610078i	$0.208862\pi$
$132$	0	0
$133$	−12.4187	−1.07684
$134$	0	0
$135$	1.75471	0.151021
$136$	0	0
$137$	4.23161	0.361531	0.180766	−	0.983526i	$-0.442142\pi$
$137$	4.23161	0.361531	0.180766	+	0.983526i	$0.442142\pi$
$138$	0	0
$139$	−9.85256	−0.835683	−0.417841	−	0.908520i	$-0.637213\pi$
$139$	−9.85256	−0.835683	−0.417841	+	0.908520i	$0.637213\pi$
$140$	0	0
$141$	9.59559	0.808094
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−16.4515	−1.36623
$146$	0	0
$147$	−0.300050	−0.0247477
$148$	0	0
$149$	−2.64659	−0.216817	−0.108409	−	0.994106i	$-0.534575\pi$
$149$	−2.64659	−0.216817	−0.108409	+	0.994106i	$0.534575\pi$
$150$	0	0
$151$	−3.62094	−0.294668	−0.147334	−	0.989087i	$-0.547069\pi$
$151$	−3.62094	−0.294668	−0.147334	+	0.989087i	$0.547069\pi$
$152$	0	0
$153$	−4.34314	−0.351122
$154$	0	0
$155$	12.5303	1.00645
$156$	0	0
$157$	7.97465	0.636446	0.318223	−	0.948016i	$-0.396914\pi$
$157$	7.97465	0.636446	0.318223	+	0.948016i	$0.396914\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	15.0913	1.18936
$162$	0	0
$163$	5.68968	0.445650	0.222825	−	0.974858i	$-0.428472\pi$
$163$	5.68968	0.445650	0.222825	+	0.974858i	$0.428472\pi$
$164$	0	0
$165$	1.75471	0.136604
$166$	0	0
$167$	8.29554	0.641928	0.320964	−	0.947091i	$-0.395993\pi$
$167$	8.29554	0.641928	0.320964	+	0.947091i	$0.395993\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.79780	0.366897
$172$	0	0
$173$	−19.3106	−1.46816	−0.734080	−	0.679063i	$-0.762386\pi$
$173$	−19.3106	−1.46816	−0.734080	+	0.679063i	$0.762386\pi$
$174$	0	0
$175$	4.97235	0.375874
$176$	0	0
$177$	7.38622	0.555182
$178$	0	0
$179$	−15.3261	−1.14552	−0.572762	−	0.819722i	$-0.694128\pi$
$179$	−15.3261	−1.14552	−0.572762	+	0.819722i	$0.694128\pi$
$180$	0	0
$181$	−20.8121	−1.54695	−0.773476	−	0.633825i	$-0.781484\pi$
$181$	−20.8121	−1.54695	−0.773476	+	0.633825i	$0.781484\pi$
$182$	0	0
$183$	−12.6281	−0.933497
$184$	0	0
$185$	5.10501	0.375328
$186$	0	0
$187$	−4.34314	−0.317602
$188$	0	0
$189$	−2.58843	−0.188280
$190$	0	0
$191$	−8.29554	−0.600244	−0.300122	−	0.953901i	$-0.597027\pi$
$191$	−8.29554	−0.600244	−0.300122	+	0.953901i	$0.597027\pi$
$192$	0	0
$193$	−17.5702	−1.26473	−0.632367	−	0.774669i	$-0.717916\pi$
$193$	−17.5702	−1.26473	−0.632367	+	0.774669i	$0.717916\pi$
$194$	0	0
$195$	−1.75471	−0.125657
$196$	0	0
$197$	−7.53477	−0.536830	−0.268415	−	0.963303i	$-0.586500\pi$
$197$	−7.53477	−0.536830	−0.268415	+	0.963303i	$0.586500\pi$
$198$	0	0
$199$	−12.4701	−0.883982	−0.441991	−	0.897020i	$-0.645728\pi$
$199$	−12.4701	−0.883982	−0.441991	+	0.897020i	$0.645728\pi$
$200$	0	0
$201$	−8.17345	−0.576511
$202$	0	0
$203$	24.2682	1.70329
$204$	0	0
$205$	−12.1443	−0.848197
$206$	0	0
$207$	−5.83031	−0.405235
$208$	0	0
$209$	4.79780	0.331870
$210$	0	0
$211$	−3.51999	−0.242326	−0.121163	−	0.992633i	$-0.538662\pi$
$211$	−3.51999	−0.242326	−0.121163	+	0.992633i	$0.538662\pi$
$212$	0	0
$213$	−10.0862	−0.691093
$214$	0	0
$215$	−0.234722	−0.0160079
$216$	0	0
$217$	−18.4838	−1.25476
$218$	0	0
$219$	0.0978457	0.00661180
$220$	0	0
$221$	4.34314	0.292151
$222$	0	0
$223$	20.8084	1.39343	0.696716	−	0.717348i	$-0.254644\pi$
$223$	20.8084	1.39343	0.696716	+	0.717348i	$0.254644\pi$
$224$	0	0
$225$	−1.92099	−0.128066
$226$	0	0
$227$	−5.28838	−0.351002	−0.175501	−	0.984479i	$-0.556154\pi$
$227$	−5.28838	−0.351002	−0.175501	+	0.984479i	$0.556154\pi$
$228$	0	0
$229$	−25.8261	−1.70664	−0.853318	−	0.521390i	$-0.825414\pi$
$229$	−25.8261	−1.70664	−0.853318	+	0.521390i	$0.825414\pi$
$230$	0	0
$231$	−2.58843	−0.170306
$232$	0	0
$233$	−10.1851	−0.667250	−0.333625	−	0.942706i	$-0.608272\pi$
$233$	−10.1851	−0.667250	−0.333625	+	0.942706i	$0.608272\pi$
$234$	0	0
$235$	16.8375	1.09836
$236$	0	0
$237$	−6.34314	−0.412031
$238$	0	0
$239$	22.2724	1.44068	0.720341	−	0.693620i	$-0.243985\pi$
$239$	22.2724	1.44068	0.720341	+	0.693620i	$0.243985\pi$
$240$	0	0
$241$	22.6144	1.45672	0.728362	−	0.685193i	$-0.240282\pi$
$241$	22.6144	1.45672	0.728362	+	0.685193i	$0.240282\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.526501	−0.0336369
$246$	0	0
$247$	−4.79780	−0.305276
$248$	0	0
$249$	−10.7724	−0.682676
$250$	0	0
$251$	0.339375	0.0214212	0.0107106	−	0.999943i	$-0.496591\pi$
$251$	0.339375	0.0214212	0.0107106	+	0.999943i	$0.496591\pi$
$252$	0	0
$253$	−5.83031	−0.366549
$254$	0	0
$255$	−7.62094	−0.477242
$256$	0	0
$257$	−22.0280	−1.37407	−0.687035	−	0.726625i	$-0.741088\pi$
$257$	−22.0280	−1.37407	−0.687035	+	0.726625i	$0.741088\pi$
$258$	0	0
$259$	−7.53056	−0.467926
$260$	0	0
$261$	−9.37565	−0.580338
$262$	0	0
$263$	−2.33257	−0.143832	−0.0719161	−	0.997411i	$-0.522911\pi$
$263$	−2.33257	−0.143832	−0.0719161	+	0.997411i	$0.522911\pi$
$264$	0	0
$265$	−3.50942	−0.215582
$266$	0	0
$267$	−0.945240	−0.0578477
$268$	0	0
$269$	22.1238	1.34891	0.674457	−	0.738314i	$-0.264378\pi$
$269$	22.1238	1.34891	0.674457	+	0.738314i	$0.264378\pi$
$270$	0	0
$271$	−4.37906	−0.266009	−0.133004	−	0.991115i	$-0.542462\pi$
$271$	−4.37906	−0.266009	−0.133004	+	0.991115i	$0.542462\pi$
$272$	0	0
$273$	2.58843	0.156659
$274$	0	0
$275$	−1.92099	−0.115840
$276$	0	0
$277$	25.0680	1.50619	0.753095	−	0.657912i	$-0.228560\pi$
$277$	25.0680	1.50619	0.753095	+	0.657912i	$0.228560\pi$
$278$	0	0
$279$	7.14093	0.427516
$280$	0	0
$281$	−7.16739	−0.427571	−0.213785	−	0.976881i	$-0.568579\pi$
$281$	−7.16739	−0.427571	−0.213785	+	0.976881i	$0.568579\pi$
$282$	0	0
$283$	9.53367	0.566718	0.283359	−	0.959014i	$-0.408551\pi$
$283$	9.53367	0.566718	0.283359	+	0.959014i	$0.408551\pi$
$284$	0	0
$285$	8.41874	0.498683
$286$	0	0
$287$	17.9145	1.05746
$288$	0	0
$289$	1.86283	0.109578
$290$	0	0
$291$	2.90932	0.170547
$292$	0	0
$293$	−3.11603	−0.182041	−0.0910203	−	0.995849i	$-0.529013\pi$
$293$	−3.11603	−0.182041	−0.0910203	+	0.995849i	$0.529013\pi$
$294$	0	0
$295$	12.9607	0.754600
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	5.83031	0.337176
$300$	0	0
$301$	0.346245	0.0199572
$302$	0	0
$303$	−14.4293	−0.828942
$304$	0	0
$305$	−22.1587	−1.26880
$306$	0	0
$307$	−8.13266	−0.464155	−0.232078	−	0.972697i	$-0.574552\pi$
$307$	−8.13266	−0.464155	−0.232078	+	0.972697i	$0.574552\pi$
$308$	0	0
$309$	−11.3862	−0.647739
$310$	0	0
$311$	−12.3072	−0.697878	−0.348939	−	0.937145i	$-0.613458\pi$
$311$	−12.3072	−0.697878	−0.348939	+	0.937145i	$0.613458\pi$
$312$	0	0
$313$	10.6746	0.603364	0.301682	−	0.953409i	$-0.402452\pi$
$313$	10.6746	0.603364	0.301682	+	0.953409i	$0.402452\pi$
$314$	0	0
$315$	−4.54194	−0.255909
$316$	0	0
$317$	−31.5015	−1.76930	−0.884650	−	0.466255i	$-0.845603\pi$
$317$	−31.5015	−1.76930	−0.884650	+	0.466255i	$0.845603\pi$
$318$	0	0
$319$	−9.37565	−0.524936
$320$	0	0
$321$	1.79063	0.0999433
$322$	0	0
$323$	−20.8375	−1.15943
$324$	0	0
$325$	1.92099	0.106558
$326$	0	0
$327$	10.0397	0.555196
$328$	0	0
$329$	−24.8375	−1.36933
$330$	0	0
$331$	−0.245290	−0.0134824	−0.00674118	−	0.999977i	$-0.502146\pi$
$331$	−0.245290	−0.0134824	−0.00674118	+	0.999977i	$0.502146\pi$
$332$	0	0
$333$	2.90932	0.159430
$334$	0	0
$335$	−14.3420	−0.783589
$336$	0	0
$337$	11.3514	0.618351	0.309175	−	0.951005i	$-0.399947\pi$
$337$	11.3514	0.618351	0.309175	+	0.951005i	$0.399947\pi$
$338$	0	0
$339$	19.9145	1.08161
$340$	0	0
$341$	7.14093	0.386703
$342$	0	0
$343$	18.8956	1.02027
$344$	0	0
$345$	−10.2305	−0.550792
$346$	0	0
$347$	−6.57675	−0.353059	−0.176529	−	0.984295i	$-0.556487\pi$
$347$	−6.57675	−0.353059	−0.176529	+	0.984295i	$0.556487\pi$
$348$	0	0
$349$	22.4981	1.20430	0.602148	−	0.798385i	$-0.294312\pi$
$349$	22.4981	1.20430	0.602148	+	0.798385i	$0.294312\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	2.02911	0.107999	0.0539994	−	0.998541i	$-0.482803\pi$
$353$	2.02911	0.107999	0.0539994	+	0.998541i	$0.482803\pi$
$354$	0	0
$355$	−17.6983	−0.939328
$356$	0	0
$357$	11.2419	0.594984
$358$	0	0
$359$	−20.9421	−1.10528	−0.552642	−	0.833419i	$-0.686380\pi$
$359$	−20.9421	−1.10528	−0.552642	+	0.833419i	$0.686380\pi$
$360$	0	0
$361$	4.01884	0.211518
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	0.171691	0.00898671
$366$	0	0
$367$	1.81864	0.0949321	0.0474661	−	0.998873i	$-0.484885\pi$
$367$	1.81864	0.0949321	0.0474661	+	0.998873i	$0.484885\pi$
$368$	0	0
$369$	−6.92099	−0.360293
$370$	0	0
$371$	5.17685	0.268769
$372$	0	0
$373$	36.1587	1.87222	0.936112	−	0.351701i	$-0.114397\pi$
$373$	36.1587	1.87222	0.936112	+	0.351701i	$0.114397\pi$
$374$	0	0
$375$	−12.1443	−0.627131
$376$	0	0
$377$	9.37565	0.482871
$378$	0	0
$379$	5.71768	0.293698	0.146849	−	0.989159i	$-0.453087\pi$
$379$	5.71768	0.293698	0.146849	+	0.989159i	$0.453087\pi$
$380$	0	0
$381$	2.34314	0.120042
$382$	0	0
$383$	9.77014	0.499231	0.249616	−	0.968345i	$-0.419696\pi$
$383$	9.77014	0.499231	0.249616	+	0.968345i	$0.419696\pi$
$384$	0	0
$385$	−4.54194	−0.231479
$386$	0	0
$387$	−0.133767	−0.00679974
$388$	0	0
$389$	−12.1095	−0.613977	−0.306989	−	0.951713i	$-0.599321\pi$
$389$	−12.1095	−0.613977	−0.306989	+	0.951713i	$0.599321\pi$
$390$	0	0
$391$	25.3218	1.28058
$392$	0	0
$393$	18.1375	0.914917
$394$	0	0
$395$	−11.1304	−0.560029
$396$	0	0
$397$	28.8471	1.44780	0.723898	−	0.689907i	$-0.242348\pi$
$397$	28.8471	1.44780	0.723898	+	0.689907i	$0.242348\pi$
$398$	0	0
$399$	−12.4187	−0.621715
$400$	0	0
$401$	−25.4878	−1.27280	−0.636401	−	0.771359i	$-0.719577\pi$
$401$	−25.4878	−1.27280	−0.636401	+	0.771359i	$0.719577\pi$
$402$	0	0
$403$	−7.14093	−0.355715
$404$	0	0
$405$	1.75471	0.0871922
$406$	0	0
$407$	2.90932	0.144210
$408$	0	0
$409$	2.30656	0.114052	0.0570261	−	0.998373i	$-0.481838\pi$
$409$	2.30656	0.114052	0.0570261	+	0.998373i	$0.481838\pi$
$410$	0	0
$411$	4.23161	0.208730
$412$	0	0
$413$	−19.1187	−0.940769
$414$	0	0
$415$	−18.9025	−0.927888
$416$	0	0
$417$	−9.85256	−0.482482
$418$	0	0
$419$	19.4190	0.948682	0.474341	−	0.880341i	$-0.342686\pi$
$419$	19.4190	0.948682	0.474341	+	0.880341i	$0.342686\pi$
$420$	0	0
$421$	−30.3983	−1.48152	−0.740760	−	0.671770i	$-0.765534\pi$
$421$	−30.3983	−1.48152	−0.740760	+	0.671770i	$0.765534\pi$
$422$	0	0
$423$	9.59559	0.466554
$424$	0	0
$425$	8.34314	0.404702
$426$	0	0
$427$	32.6869	1.58183
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−27.9493	−1.34627	−0.673135	−	0.739520i	$-0.735053\pi$
$431$	−27.9493	−1.34627	−0.673135	+	0.739520i	$0.735053\pi$
$432$	0	0
$433$	3.83512	0.184304	0.0921520	−	0.995745i	$-0.470625\pi$
$433$	3.83512	0.184304	0.0921520	+	0.995745i	$0.470625\pi$
$434$	0	0
$435$	−16.4515	−0.788791
$436$	0	0
$437$	−27.9726	−1.33811
$438$	0	0
$439$	−5.23813	−0.250002	−0.125001	−	0.992157i	$-0.539893\pi$
$439$	−5.23813	−0.250002	−0.125001	+	0.992157i	$0.539893\pi$
$440$	0	0
$441$	−0.300050	−0.0142881
$442$	0	0
$443$	16.6651	0.791784	0.395892	−	0.918297i	$-0.370435\pi$
$443$	16.6651	0.791784	0.395892	+	0.918297i	$0.370435\pi$
$444$	0	0
$445$	−1.65862	−0.0786262
$446$	0	0
$447$	−2.64659	−0.125179
$448$	0	0
$449$	0.254962	0.0120324	0.00601620	−	0.999982i	$-0.498085\pi$
$449$	0.254962	0.0120324	0.00601620	+	0.999982i	$0.498085\pi$
$450$	0	0
$451$	−6.92099	−0.325897
$452$	0	0
$453$	−3.62094	−0.170127
$454$	0	0
$455$	4.54194	0.212929
$456$	0	0
$457$	22.0491	1.03142	0.515708	−	0.856765i	$-0.327529\pi$
$457$	22.0491	1.03142	0.515708	+	0.856765i	$0.327529\pi$
$458$	0	0
$459$	−4.34314	−0.202720
$460$	0	0
$461$	2.49539	0.116222	0.0581108	−	0.998310i	$-0.481492\pi$
$461$	2.49539	0.116222	0.0581108	+	0.998310i	$0.481492\pi$
$462$	0	0
$463$	3.09644	0.143904	0.0719520	−	0.997408i	$-0.477077\pi$
$463$	3.09644	0.143904	0.0719520	+	0.997408i	$0.477077\pi$
$464$	0	0
$465$	12.5303	0.581077
$466$	0	0
$467$	6.79099	0.314249	0.157125	−	0.987579i	$-0.449778\pi$
$467$	6.79099	0.314249	0.157125	+	0.987579i	$0.449778\pi$
$468$	0	0
$469$	21.1564	0.976910
$470$	0	0
$471$	7.97465	0.367452
$472$	0	0
$473$	−0.133767	−0.00615060
$474$	0	0
$475$	−9.21653	−0.422884
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−7.25356	−0.331424	−0.165712	−	0.986174i	$-0.552992\pi$
$479$	−7.25356	−0.331424	−0.165712	+	0.986174i	$0.552992\pi$
$480$	0	0
$481$	−2.90932	−0.132654
$482$	0	0
$483$	15.0913	0.686679
$484$	0	0
$485$	5.10501	0.231807
$486$	0	0
$487$	4.14544	0.187848	0.0939239	−	0.995579i	$-0.470059\pi$
$487$	4.14544	0.187848	0.0939239	+	0.995579i	$0.470059\pi$
$488$	0	0
$489$	5.68968	0.257296
$490$	0	0
$491$	28.3983	1.28160	0.640798	−	0.767710i	$-0.278604\pi$
$491$	28.3983	1.28160	0.640798	+	0.767710i	$0.278604\pi$
$492$	0	0
$493$	40.7197	1.83392
$494$	0	0
$495$	1.75471	0.0788683
$496$	0	0
$497$	26.1073	1.17107
$498$	0	0
$499$	−24.4621	−1.09507	−0.547537	−	0.836781i	$-0.684435\pi$
$499$	−24.4621	−1.09507	−0.547537	+	0.836781i	$0.684435\pi$
$500$	0	0
$501$	8.29554	0.370617
$502$	0	0
$503$	36.0075	1.60550	0.802748	−	0.596318i	$-0.203370\pi$
$503$	36.0075	1.60550	0.802748	+	0.596318i	$0.203370\pi$
$504$	0	0
$505$	−25.3192	−1.12669
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−35.3805	−1.56821	−0.784107	−	0.620626i	$-0.786879\pi$
$509$	−35.3805	−1.56821	−0.784107	+	0.620626i	$0.786879\pi$
$510$	0	0
$511$	−0.253266	−0.0112038
$512$	0	0
$513$	4.79780	0.211828
$514$	0	0
$515$	−19.9795	−0.880403
$516$	0	0
$517$	9.59559	0.422014
$518$	0	0
$519$	−19.3106	−0.847642
$520$	0	0
$521$	39.5751	1.73382	0.866908	−	0.498467i	$-0.166104\pi$
$521$	39.5751	1.73382	0.866908	+	0.498467i	$0.166104\pi$
$522$	0	0
$523$	27.2758	1.19269	0.596344	−	0.802729i	$-0.296620\pi$
$523$	27.2758	1.19269	0.596344	+	0.802729i	$0.296620\pi$
$524$	0	0
$525$	4.97235	0.217011
$526$	0	0
$527$	−31.0140	−1.35099
$528$	0	0
$529$	10.9925	0.477937
$530$	0	0
$531$	7.38622	0.320535
$532$	0	0
$533$	6.92099	0.299782
$534$	0	0
$535$	3.14204	0.135842
$536$	0	0
$537$	−15.3261	−0.661368
$538$	0	0
$539$	−0.300050	−0.0129241
$540$	0	0
$541$	4.84650	0.208367	0.104184	−	0.994558i	$-0.466777\pi$
$541$	4.84650	0.208367	0.104184	+	0.994558i	$0.466777\pi$
$542$	0	0
$543$	−20.8121	−0.893134
$544$	0	0
$545$	17.6167	0.754618
$546$	0	0
$547$	−43.8675	−1.87564	−0.937820	−	0.347121i	$-0.887159\pi$
$547$	−43.8675	−1.87564	−0.937820	+	0.347121i	$0.887159\pi$
$548$	0	0
$549$	−12.6281	−0.538954
$550$	0	0
$551$	−44.9825	−1.91632
$552$	0	0
$553$	16.4187	0.698196
$554$	0	0
$555$	5.10501	0.216696
$556$	0	0
$557$	−39.9028	−1.69074	−0.845368	−	0.534184i	$-0.820619\pi$
$557$	−39.9028	−1.69074	−0.845368	+	0.534184i	$0.820619\pi$
$558$	0	0
$559$	0.133767	0.00565773
$560$	0	0
$561$	−4.34314	−0.183367
$562$	0	0
$563$	18.5768	0.782917	0.391458	−	0.920196i	$-0.371971\pi$
$563$	18.5768	0.782917	0.391458	+	0.920196i	$0.371971\pi$
$564$	0	0
$565$	34.9441	1.47011
$566$	0	0
$567$	−2.58843	−0.108704
$568$	0	0
$569$	3.46869	0.145415	0.0727076	−	0.997353i	$-0.476836\pi$
$569$	3.46869	0.145415	0.0727076	+	0.997353i	$0.476836\pi$
$570$	0	0
$571$	−35.0808	−1.46808	−0.734042	−	0.679104i	$-0.762369\pi$
$571$	−35.0808	−1.46808	−0.734042	+	0.679104i	$0.762369\pi$
$572$	0	0
$573$	−8.29554	−0.346551
$574$	0	0
$575$	11.2000	0.467072
$576$	0	0
$577$	23.2562	0.968169	0.484084	−	0.875021i	$-0.339153\pi$
$577$	23.2562	0.968169	0.484084	+	0.875021i	$0.339153\pi$
$578$	0	0
$579$	−17.5702	−0.730194
$580$	0	0
$581$	27.8837	1.15681
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	−1.75471	−0.0725483
$586$	0	0
$587$	−38.4844	−1.58842	−0.794211	−	0.607642i	$-0.792116\pi$
$587$	−38.4844	−1.58842	−0.794211	+	0.607642i	$0.792116\pi$
$588$	0	0
$589$	34.2607	1.41169
$590$	0	0
$591$	−7.53477	−0.309939
$592$	0	0
$593$	4.56794	0.187583	0.0937914	−	0.995592i	$-0.470101\pi$
$593$	4.56794	0.187583	0.0937914	+	0.995592i	$0.470101\pi$
$594$	0	0
$595$	19.7262	0.808697
$596$	0	0
$597$	−12.4701	−0.510367
$598$	0	0
$599$	−23.0680	−0.942532	−0.471266	−	0.881991i	$-0.656203\pi$
$599$	−23.0680	−0.942532	−0.471266	+	0.881991i	$0.656203\pi$
$600$	0	0
$601$	3.84199	0.156718	0.0783590	−	0.996925i	$-0.475032\pi$
$601$	3.84199	0.156718	0.0783590	+	0.996925i	$0.475032\pi$
$602$	0	0
$603$	−8.17345	−0.332849
$604$	0	0
$605$	1.75471	0.0713391
$606$	0	0
$607$	−34.7134	−1.40897	−0.704486	−	0.709718i	$-0.748823\pi$
$607$	−34.7134	−1.40897	−0.704486	+	0.709718i	$0.748823\pi$
$608$	0	0
$609$	24.2682	0.983396
$610$	0	0
$611$	−9.59559	−0.388196
$612$	0	0
$613$	4.31402	0.174242	0.0871209	−	0.996198i	$-0.472233\pi$
$613$	4.31402	0.174242	0.0871209	+	0.996198i	$0.472233\pi$
$614$	0	0
$615$	−12.1443	−0.489707
$616$	0	0
$617$	35.9156	1.44591	0.722954	−	0.690897i	$-0.242784\pi$
$617$	35.9156	1.44591	0.722954	+	0.690897i	$0.242784\pi$
$618$	0	0
$619$	20.6640	0.830557	0.415279	−	0.909694i	$-0.363684\pi$
$619$	20.6640	0.830557	0.415279	+	0.909694i	$0.363684\pi$
$620$	0	0
$621$	−5.83031	−0.233962
$622$	0	0
$623$	2.44668	0.0980243
$624$	0	0
$625$	−11.7048	−0.468193
$626$	0	0
$627$	4.79780	0.191605
$628$	0	0
$629$	−12.6356	−0.503813
$630$	0	0
$631$	−43.4667	−1.73038	−0.865191	−	0.501443i	$-0.832803\pi$
$631$	−43.4667	−1.73038	−0.865191	+	0.501443i	$0.832803\pi$
$632$	0	0
$633$	−3.51999	−0.139907
$634$	0	0
$635$	4.11152	0.163161
$636$	0	0
$637$	0.300050	0.0118884
$638$	0	0
$639$	−10.0862	−0.399003
$640$	0	0
$641$	−12.2305	−0.483076	−0.241538	−	0.970391i	$-0.577652\pi$
$641$	−12.2305	−0.483076	−0.241538	+	0.970391i	$0.577652\pi$
$642$	0	0
$643$	16.7577	0.660858	0.330429	−	0.943831i	$-0.392807\pi$
$643$	16.7577	0.660858	0.330429	+	0.943831i	$0.392807\pi$
$644$	0	0
$645$	−0.234722	−0.00924216
$646$	0	0
$647$	−22.7958	−0.896195	−0.448098	−	0.893985i	$-0.647898\pi$
$647$	−22.7958	−0.896195	−0.448098	+	0.893985i	$0.647898\pi$
$648$	0	0
$649$	7.38622	0.289935
$650$	0	0
$651$	−18.4838	−0.724436
$652$	0	0
$653$	−35.4308	−1.38651	−0.693257	−	0.720691i	$-0.743825\pi$
$653$	−35.4308	−1.38651	−0.693257	+	0.720691i	$0.743825\pi$
$654$	0	0
$655$	31.8261	1.24355
$656$	0	0
$657$	0.0978457	0.00381732
$658$	0	0
$659$	15.2123	0.592588	0.296294	−	0.955097i	$-0.404249\pi$
$659$	15.2123	0.592588	0.296294	+	0.955097i	$0.404249\pi$
$660$	0	0
$661$	12.8676	0.500493	0.250246	−	0.968182i	$-0.419488\pi$
$661$	12.8676	0.500493	0.250246	+	0.968182i	$0.419488\pi$
$662$	0	0
$663$	4.34314	0.168673
$664$	0	0
$665$	−21.7913	−0.845030
$666$	0	0
$667$	54.6630	2.11656
$668$	0	0
$669$	20.8084	0.804498
$670$	0	0
$671$	−12.6281	−0.487503
$672$	0	0
$673$	−28.7217	−1.10714	−0.553571	−	0.832802i	$-0.686735\pi$
$673$	−28.7217	−1.10714	−0.553571	+	0.832802i	$0.686735\pi$
$674$	0	0
$675$	−1.92099	−0.0739391
$676$	0	0
$677$	−24.7112	−0.949727	−0.474864	−	0.880059i	$-0.657503\pi$
$677$	−24.7112	−0.949727	−0.474864	+	0.880059i	$0.657503\pi$
$678$	0	0
$679$	−7.53056	−0.288996
$680$	0	0
$681$	−5.28838	−0.202651
$682$	0	0
$683$	26.1353	1.00004	0.500020	−	0.866014i	$-0.333326\pi$
$683$	26.1353	1.00004	0.500020	+	0.866014i	$0.333326\pi$
$684$	0	0
$685$	7.42525	0.283704
$686$	0	0
$687$	−25.8261	−0.985327
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	13.4735	0.512556	0.256278	−	0.966603i	$-0.417504\pi$
$691$	13.4735	0.512556	0.256278	+	0.966603i	$0.417504\pi$
$692$	0	0
$693$	−2.58843	−0.0983262
$694$	0	0
$695$	−17.2884	−0.655785
$696$	0	0
$697$	30.0588	1.13856
$698$	0	0
$699$	−10.1851	−0.385237
$700$	0	0
$701$	−39.8087	−1.50355	−0.751777	−	0.659417i	$-0.770803\pi$
$701$	−39.8087	−1.50355	−0.751777	+	0.659417i	$0.770803\pi$
$702$	0	0
$703$	13.9583	0.526448
$704$	0	0
$705$	16.8375	0.634136
$706$	0	0
$707$	37.3492	1.40466
$708$	0	0
$709$	−25.6824	−0.964523	−0.482262	−	0.876027i	$-0.660185\pi$
$709$	−25.6824	−0.964523	−0.482262	+	0.876027i	$0.660185\pi$
$710$	0	0
$711$	−6.34314	−0.237886
$712$	0	0
$713$	−41.6339	−1.55920
$714$	0	0
$715$	−1.75471	−0.0656224
$716$	0	0
$717$	22.2724	0.831778
$718$	0	0
$719$	41.9938	1.56611	0.783053	−	0.621955i	$-0.213661\pi$
$719$	41.9938	1.56611	0.783053	+	0.621955i	$0.213661\pi$
$720$	0	0
$721$	29.4724	1.09761
$722$	0	0
$723$	22.6144	0.841040
$724$	0	0
$725$	18.0106	0.668896
$726$	0	0
$727$	8.32355	0.308703	0.154352	−	0.988016i	$-0.450671\pi$
$727$	8.32355	0.308703	0.154352	+	0.988016i	$0.450671\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.580967	0.0214878
$732$	0	0
$733$	−18.6021	−0.687084	−0.343542	−	0.939137i	$-0.611627\pi$
$733$	−18.6021	−0.687084	−0.343542	+	0.939137i	$0.611627\pi$
$734$	0	0
$735$	−0.526501	−0.0194203
$736$	0	0
$737$	−8.17345	−0.301073
$738$	0	0
$739$	−30.4379	−1.11968	−0.559838	−	0.828602i	$-0.689136\pi$
$739$	−30.4379	−1.11968	−0.559838	+	0.828602i	$0.689136\pi$
$740$	0	0
$741$	−4.79780	−0.176251
$742$	0	0
$743$	30.7793	1.12918	0.564592	−	0.825370i	$-0.309034\pi$
$743$	30.7793	1.12918	0.564592	+	0.825370i	$0.309034\pi$
$744$	0	0
$745$	−4.64400	−0.170143
$746$	0	0
$747$	−10.7724	−0.394143
$748$	0	0
$749$	−4.63492	−0.169356
$750$	0	0
$751$	−6.50025	−0.237198	−0.118599	−	0.992942i	$-0.537840\pi$
$751$	−6.50025	−0.237198	−0.118599	+	0.992942i	$0.537840\pi$
$752$	0	0
$753$	0.339375	0.0123675
$754$	0	0
$755$	−6.35370	−0.231235
$756$	0	0
$757$	32.8121	1.19258	0.596288	−	0.802770i	$-0.296641\pi$
$757$	32.8121	1.19258	0.596288	+	0.802770i	$0.296641\pi$
$758$	0	0
$759$	−5.83031	−0.211627
$760$	0	0
$761$	37.5982	1.36294	0.681468	−	0.731848i	$-0.261342\pi$
$761$	37.5982	1.36294	0.681468	+	0.731848i	$0.261342\pi$
$762$	0	0
$763$	−25.9870	−0.940792
$764$	0	0
$765$	−7.62094	−0.275536
$766$	0	0
$767$	−7.38622	−0.266701
$768$	0	0
$769$	−47.3287	−1.70672	−0.853358	−	0.521325i	$-0.825438\pi$
$769$	−47.3287	−1.70672	−0.853358	+	0.521325i	$0.825438\pi$
$770$	0	0
$771$	−22.0280	−0.793319
$772$	0	0
$773$	11.7388	0.422216	0.211108	−	0.977463i	$-0.432293\pi$
$773$	11.7388	0.422216	0.211108	+	0.977463i	$0.432293\pi$
$774$	0	0
$775$	−13.7177	−0.492754
$776$	0	0
$777$	−7.53056	−0.270157
$778$	0	0
$779$	−33.2055	−1.18971
$780$	0	0
$781$	−10.0862	−0.360912
$782$	0	0
$783$	−9.37565	−0.335058
$784$	0	0
$785$	13.9932	0.499438
$786$	0	0
$787$	32.3359	1.15265	0.576325	−	0.817221i	$-0.304486\pi$
$787$	32.3359	1.15265	0.576325	+	0.817221i	$0.304486\pi$
$788$	0	0
$789$	−2.33257	−0.0830416
$790$	0	0
$791$	−51.5472	−1.83281
$792$	0	0
$793$	12.6281	0.448437
$794$	0	0
$795$	−3.50942	−0.124466
$796$	0	0
$797$	37.0960	1.31401	0.657004	−	0.753887i	$-0.271824\pi$
$797$	37.0960	1.31401	0.657004	+	0.753887i	$0.271824\pi$
$798$	0	0
$799$	−41.6750	−1.47435
$800$	0	0
$801$	−0.945240	−0.0333984
$802$	0	0
$803$	0.0978457	0.00345290
$804$	0	0
$805$	26.4809	0.933329
$806$	0	0
$807$	22.1238	0.778796
$808$	0	0
$809$	−36.1972	−1.27263	−0.636314	−	0.771431i	$-0.719542\pi$
$809$	−36.1972	−1.27263	−0.636314	+	0.771431i	$0.719542\pi$
$810$	0	0
$811$	34.8588	1.22406	0.612029	−	0.790835i	$-0.290353\pi$
$811$	34.8588	1.22406	0.612029	+	0.790835i	$0.290353\pi$
$812$	0	0
$813$	−4.37906	−0.153580
$814$	0	0
$815$	9.98373	0.349715
$816$	0	0
$817$	−0.641785	−0.0224532
$818$	0	0
$819$	2.58843	0.0904469
$820$	0	0
$821$	−48.0500	−1.67696	−0.838478	−	0.544935i	$-0.816554\pi$
$821$	−48.0500	−1.67696	−0.838478	+	0.544935i	$0.816554\pi$
$822$	0	0
$823$	43.7353	1.52452	0.762259	−	0.647272i	$-0.224090\pi$
$823$	43.7353	1.52452	0.762259	+	0.647272i	$0.224090\pi$
$824$	0	0
$825$	−1.92099	−0.0668804
$826$	0	0
$827$	−24.0882	−0.837628	−0.418814	−	0.908072i	$-0.637554\pi$
$827$	−24.0882	−0.837628	−0.418814	+	0.908072i	$0.637554\pi$
$828$	0	0
$829$	26.5215	0.921128	0.460564	−	0.887626i	$-0.347647\pi$
$829$	26.5215	0.921128	0.460564	+	0.887626i	$0.347647\pi$
$830$	0	0
$831$	25.0680	0.869599
$832$	0	0
$833$	1.30316	0.0451518
$834$	0	0
$835$	14.5563	0.503740
$836$	0	0
$837$	7.14093	0.246827
$838$	0	0
$839$	−30.1073	−1.03942	−0.519710	−	0.854343i	$-0.673960\pi$
$839$	−30.1073	−1.03942	−0.519710	+	0.854343i	$0.673960\pi$
$840$	0	0
$841$	58.9029	2.03113
$842$	0	0
$843$	−7.16739	−0.246858
$844$	0	0
$845$	1.75471	0.0603639
$846$	0	0
$847$	−2.58843	−0.0889394
$848$	0	0
$849$	9.53367	0.327195
$850$	0	0
$851$	−16.9622	−0.581458
$852$	0	0
$853$	39.8706	1.36514	0.682572	−	0.730818i	$-0.260861\pi$
$853$	39.8706	1.36514	0.682572	+	0.730818i	$0.260861\pi$
$854$	0	0
$855$	8.41874	0.287915
$856$	0	0
$857$	17.0876	0.583702	0.291851	−	0.956464i	$-0.405729\pi$
$857$	17.0876	0.583702	0.291851	+	0.956464i	$0.405729\pi$
$858$	0	0
$859$	−2.95100	−0.100687	−0.0503435	−	0.998732i	$-0.516032\pi$
$859$	−2.95100	−0.100687	−0.0503435	+	0.998732i	$0.516032\pi$
$860$	0	0
$861$	17.9145	0.610524
$862$	0	0
$863$	32.1935	1.09588	0.547939	−	0.836518i	$-0.315412\pi$
$863$	32.1935	1.09588	0.547939	+	0.836518i	$0.315412\pi$
$864$	0	0
$865$	−33.8845	−1.15211
$866$	0	0
$867$	1.86283	0.0632650
$868$	0	0
$869$	−6.34314	−0.215176
$870$	0	0
$871$	8.17345	0.276947
$872$	0	0
$873$	2.90932	0.0984655
$874$	0	0
$875$	31.4347	1.06269
$876$	0	0
$877$	3.00451	0.101455	0.0507276	−	0.998713i	$-0.483846\pi$
$877$	3.00451	0.101455	0.0507276	+	0.998713i	$0.483846\pi$
$878$	0	0
$879$	−3.11603	−0.105101
$880$	0	0
$881$	44.0361	1.48361	0.741807	−	0.670613i	$-0.233969\pi$
$881$	44.0361	1.48361	0.741807	+	0.670613i	$0.233969\pi$
$882$	0	0
$883$	22.4776	0.756430	0.378215	−	0.925718i	$-0.376538\pi$
$883$	22.4776	0.756430	0.378215	+	0.925718i	$0.376538\pi$
$884$	0	0
$885$	12.9607	0.435668
$886$	0	0
$887$	−46.0452	−1.54605	−0.773023	−	0.634378i	$-0.781256\pi$
$887$	−46.0452	−1.54605	−0.773023	+	0.634378i	$0.781256\pi$
$888$	0	0
$889$	−6.06503	−0.203415
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	46.0377	1.54059
$894$	0	0
$895$	−26.8928	−0.898926
$896$	0	0
$897$	5.83031	0.194668
$898$	0	0
$899$	−66.9509	−2.23294
$900$	0	0
$901$	8.68627	0.289382
$902$	0	0
$903$	0.346245	0.0115223
$904$	0	0
$905$	−36.5192	−1.21394
$906$	0	0
$907$	−33.5449	−1.11384	−0.556920	−	0.830566i	$-0.688017\pi$
$907$	−33.5449	−1.11384	−0.556920	+	0.830566i	$0.688017\pi$
$908$	0	0
$909$	−14.4293	−0.478590
$910$	0	0
$911$	7.09820	0.235174	0.117587	−	0.993063i	$-0.462484\pi$
$911$	7.09820	0.235174	0.117587	+	0.993063i	$0.462484\pi$
$912$	0	0
$913$	−10.7724	−0.356516
$914$	0	0
$915$	−22.1587	−0.732543
$916$	0	0
$917$	−46.9476	−1.55035
$918$	0	0
$919$	−25.3175	−0.835147	−0.417573	−	0.908643i	$-0.637119\pi$
$919$	−25.3175	−0.835147	−0.417573	+	0.908643i	$0.637119\pi$
$920$	0	0
$921$	−8.13266	−0.267980
$922$	0	0
$923$	10.0862	0.331990
$924$	0	0
$925$	−5.58878	−0.183758
$926$	0	0
$927$	−11.3862	−0.373973
$928$	0	0
$929$	21.7616	0.713974	0.356987	−	0.934109i	$-0.383804\pi$
$929$	21.7616	0.713974	0.356987	+	0.934109i	$0.383804\pi$
$930$	0	0
$931$	−1.43958	−0.0471803
$932$	0	0
$933$	−12.3072	−0.402920
$934$	0	0
$935$	−7.62094	−0.249231
$936$	0	0
$937$	12.0610	0.394017	0.197008	−	0.980402i	$-0.436877\pi$
$937$	12.0610	0.394017	0.197008	+	0.980402i	$0.436877\pi$
$938$	0	0
$939$	10.6746	0.348352
$940$	0	0
$941$	−7.51364	−0.244938	−0.122469	−	0.992472i	$-0.539081\pi$
$941$	−7.51364	−0.244938	−0.122469	+	0.992472i	$0.539081\pi$
$942$	0	0
$943$	40.3516	1.31403
$944$	0	0
$945$	−4.54194	−0.147749
$946$	0	0
$947$	43.5403	1.41487	0.707435	−	0.706779i	$-0.249852\pi$
$947$	43.5403	1.41487	0.707435	+	0.706779i	$0.249852\pi$
$948$	0	0
$949$	−0.0978457	−0.00317621
$950$	0	0
$951$	−31.5015	−1.02151
$952$	0	0
$953$	−49.5138	−1.60391	−0.801954	−	0.597386i	$-0.796206\pi$
$953$	−49.5138	−1.60391	−0.801954	+	0.597386i	$0.796206\pi$
$954$	0	0
$955$	−14.5563	−0.471030
$956$	0	0
$957$	−9.37565	−0.303072
$958$	0	0
$959$	−10.9532	−0.353698
$960$	0	0
$961$	19.9929	0.644932
$962$	0	0
$963$	1.79063	0.0577023
$964$	0	0
$965$	−30.8307	−0.992474
$966$	0	0
$967$	56.4824	1.81635	0.908176	−	0.418588i	$-0.137475\pi$
$967$	56.4824	1.81635	0.908176	+	0.418588i	$0.137475\pi$
$968$	0	0
$969$	−20.8375	−0.669396
$970$	0	0
$971$	−14.0985	−0.452442	−0.226221	−	0.974076i	$-0.572637\pi$
$971$	−14.0985	−0.452442	−0.226221	+	0.974076i	$0.572637\pi$
$972$	0	0
$973$	25.5026	0.817576
$974$	0	0
$975$	1.92099	0.0615210
$976$	0	0
$977$	21.5597	0.689755	0.344877	−	0.938648i	$-0.387920\pi$
$977$	21.5597	0.689755	0.344877	+	0.938648i	$0.387920\pi$
$978$	0	0
$979$	−0.945240	−0.0302100
$980$	0	0
$981$	10.0397	0.320542
$982$	0	0
$983$	9.70231	0.309456	0.154728	−	0.987957i	$-0.450550\pi$
$983$	9.70231	0.309456	0.154728	+	0.987957i	$0.450550\pi$
$984$	0	0
$985$	−13.2213	−0.421267
$986$	0	0
$987$	−24.8375	−0.790586
$988$	0	0
$989$	0.779902	0.0247994
$990$	0	0
$991$	18.4284	0.585398	0.292699	−	0.956205i	$-0.405447\pi$
$991$	18.4284	0.585398	0.292699	+	0.956205i	$0.405447\pi$
$992$	0	0
$993$	−0.245290	−0.00778405
$994$	0	0
$995$	−21.8814	−0.693687
$996$	0	0
$997$	−59.4292	−1.88214	−0.941071	−	0.338209i	$-0.890179\pi$
$997$	−59.4292	−1.88214	−0.941071	+	0.338209i	$0.890179\pi$
$998$	0	0
$999$	2.90932	0.0920468

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.by.1.4		4
4.3	odd	2		3432.2.a.q.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.q.1.4	✓	4	4.3	odd	2
6864.2.a.by.1.4		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.75471 q^{5} -2.58843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.75471 q^{5} -2.58843 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.75471 q^{15} -4.34314 q^{17} +4.79780 q^{19} -2.58843 q^{21} -5.83031 q^{23} -1.92099 q^{25} +1.00000 q^{27} -9.37565 q^{29} +7.14093 q^{31} +1.00000 q^{33} -4.54194 q^{35} +2.90932 q^{37} -1.00000 q^{39} -6.92099 q^{41} -0.133767 q^{43} +1.75471 q^{45} +9.59559 q^{47} -0.300050 q^{49} -4.34314 q^{51} -2.00000 q^{53} +1.75471 q^{55} +4.79780 q^{57} +7.38622 q^{59} -12.6281 q^{61} -2.58843 q^{63} -1.75471 q^{65} -8.17345 q^{67} -5.83031 q^{69} -10.0862 q^{71} +0.0978457 q^{73} -1.92099 q^{75} -2.58843 q^{77} -6.34314 q^{79} +1.00000 q^{81} -10.7724 q^{83} -7.62094 q^{85} -9.37565 q^{87} -0.945240 q^{89} +2.58843 q^{91} +7.14093 q^{93} +8.41874 q^{95} +2.90932 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - 3 q^{15} - 2 q^{19} - 3 q^{21} - 3 q^{23} + 5 q^{25} + 4 q^{27} - 21 q^{29} - 10 q^{31} + 4 q^{33} + q^{35} + 4 q^{37} - 4 q^{39} - 15 q^{41} + 3 q^{43} - 3 q^{45} - 4 q^{47} + 5 q^{49} - 8 q^{53} - 3 q^{55} - 2 q^{57} + q^{59} - 9 q^{61} - 3 q^{63} + 3 q^{65} + 5 q^{67} - 3 q^{69} - 18 q^{71} - 27 q^{73} + 5 q^{75} - 3 q^{77} - 8 q^{79} + 4 q^{81} + 14 q^{83} - 24 q^{85} - 21 q^{87} - 20 q^{89} + 3 q^{91} - 10 q^{93} + 6 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9 + 4 * q^11 - 4 * q^13 - 3 * q^15 - 2 * q^19 - 3 * q^21 - 3 * q^23 + 5 * q^25 + 4 * q^27 - 21 * q^29 - 10 * q^31 + 4 * q^33 + q^35 + 4 * q^37 - 4 * q^39 - 15 * q^41 + 3 * q^43 - 3 * q^45 - 4 * q^47 + 5 * q^49 - 8 * q^53 - 3 * q^55 - 2 * q^57 + q^59 - 9 * q^61 - 3 * q^63 + 3 * q^65 + 5 * q^67 - 3 * q^69 - 18 * q^71 - 27 * q^73 + 5 * q^75 - 3 * q^77 - 8 * q^79 + 4 * q^81 + 14 * q^83 - 24 * q^85 - 21 * q^87 - 20 * q^89 + 3 * q^91 - 10 * q^93 + 6 * q^95 + 4 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

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