LMFDB - Embedded newform 6096.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6096,2,Mod(1,6096)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6096, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6096.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6096 = 2^{4} \cdot 3 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6096.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:	$48.6768050722$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.71789.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 14x^{2} + 12x + 16 $ x^4 - x^3 - 14x^2 + 12x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3048)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.67473$ of defining polynomial
Character	$\chi$	$=$	6096.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.67473 q^{5} -4.38845 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.67473 q^{5} -4.38845 q^{7} +1.00000 q^{9} +5.47399 q^{11} +1.14872 q^{13} +1.67473 q^{15} -1.76028 q^{17} -4.14872 q^{19} -4.38845 q^{21} -5.67473 q^{23} -2.19528 q^{25} +1.00000 q^{27} -0.325271 q^{29} -5.43501 q^{31} +5.47399 q^{33} -7.34946 q^{35} +4.86244 q^{37} +1.14872 q^{39} -5.47399 q^{41} -8.87001 q^{43} +1.67473 q^{45} -1.00757 q^{47} +12.2585 q^{49} -1.76028 q^{51} -11.9722 q^{53} +9.16746 q^{55} -4.14872 q^{57} -11.0112 q^{59} +1.99243 q^{61} -4.38845 q^{63} +1.92380 q^{65} +1.34946 q^{67} -5.67473 q^{69} -10.0556 q^{71} +6.10973 q^{73} -2.19528 q^{75} -24.0223 q^{77} +11.1209 q^{79} +1.00000 q^{81} +4.71372 q^{83} -2.94799 q^{85} -0.325271 q^{87} +2.23427 q^{89} -5.04111 q^{91} -5.43501 q^{93} -6.94799 q^{95} +10.9480 q^{97} +5.47399 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + q^{5} - 5 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + q^5 - 5 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + q^{5} - 5 q^{7} + 4 q^{9} + 9 q^{11} - 14 q^{13} + q^{15} - q^{17} + 2 q^{19} - 5 q^{21} - 17 q^{23} + 9 q^{25} + 4 q^{27} - 7 q^{29} - 10 q^{31} + 9 q^{33} - 18 q^{35} - 6 q^{37} - 14 q^{39} - 9 q^{41} - 12 q^{43} + q^{45} - 6 q^{47} + 13 q^{49} - q^{51} - 5 q^{53} - 24 q^{55} + 2 q^{57} + 6 q^{61} - 5 q^{63} - q^{65} - 6 q^{67} - 17 q^{69} - 20 q^{71} + 7 q^{73} + 9 q^{75} - 8 q^{77} - 17 q^{79} + 4 q^{81} + 12 q^{83} + 14 q^{85} - 7 q^{87} - 10 q^{89} + 4 q^{91} - 10 q^{93} - 2 q^{95} + 18 q^{97} + 9 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + q^5 - 5 * q^7 + 4 * q^9 + 9 * q^11 - 14 * q^13 + q^15 - q^17 + 2 * q^19 - 5 * q^21 - 17 * q^23 + 9 * q^25 + 4 * q^27 - 7 * q^29 - 10 * q^31 + 9 * q^33 - 18 * q^35 - 6 * q^37 - 14 * q^39 - 9 * q^41 - 12 * q^43 + q^45 - 6 * q^47 + 13 * q^49 - q^51 - 5 * q^53 - 24 * q^55 + 2 * q^57 + 6 * q^61 - 5 * q^63 - q^65 - 6 * q^67 - 17 * q^69 - 20 * q^71 + 7 * q^73 + 9 * q^75 - 8 * q^77 - 17 * q^79 + 4 * q^81 + 12 * q^83 + 14 * q^85 - 7 * q^87 - 10 * q^89 + 4 * q^91 - 10 * q^93 - 2 * q^95 + 18 * q^97 + 9 * 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
$5$	1.67473	0.748962	0.374481	−	0.927235i	$-0.377821\pi$
$5$	1.67473	0.748962	0.374481	+	0.927235i	$0.377821\pi$
$6$	0	0
$7$	−4.38845	−1.65868	−0.829338	−	0.558747i	$-0.811282\pi$
$7$	−4.38845	−1.65868	−0.829338	+	0.558747i	$0.811282\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.47399	1.65047	0.825236	−	0.564789i	$-0.191042\pi$
$11$	5.47399	1.65047	0.825236	+	0.564789i	$0.191042\pi$
$12$	0	0
$13$	1.14872	0.318598	0.159299	−	0.987230i	$-0.449077\pi$
$13$	1.14872	0.318598	0.159299	+	0.987230i	$0.449077\pi$
$14$	0	0
$15$	1.67473	0.432413
$16$	0	0
$17$	−1.76028	−0.426930	−0.213465	−	0.976951i	$-0.568475\pi$
$17$	−1.76028	−0.426930	−0.213465	+	0.976951i	$0.568475\pi$
$18$	0	0
$19$	−4.14872	−0.951782	−0.475891	−	0.879504i	$-0.657874\pi$
$19$	−4.14872	−0.951782	−0.475891	+	0.879504i	$0.657874\pi$
$20$	0	0
$21$	−4.38845	−0.957637
$22$	0	0
$23$	−5.67473	−1.18326	−0.591631	−	0.806209i	$-0.701516\pi$
$23$	−5.67473	−1.18326	−0.591631	+	0.806209i	$0.701516\pi$
$24$	0	0
$25$	−2.19528	−0.439056
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.325271	−0.0604013	−0.0302006	−	0.999544i	$-0.509615\pi$
$29$	−0.325271	−0.0604013	−0.0302006	+	0.999544i	$0.509615\pi$
$30$	0	0
$31$	−5.43501	−0.976156	−0.488078	−	0.872800i	$-0.662302\pi$
$31$	−5.43501	−0.976156	−0.488078	+	0.872800i	$0.662302\pi$
$32$	0	0
$33$	5.47399	0.952900
$34$	0	0
$35$	−7.34946	−1.24229
$36$	0	0
$37$	4.86244	0.799380	0.399690	−	0.916650i	$-0.369118\pi$
$37$	4.86244	0.799380	0.399690	+	0.916650i	$0.369118\pi$
$38$	0	0
$39$	1.14872	0.183943
$40$	0	0
$41$	−5.47399	−0.854894	−0.427447	−	0.904040i	$-0.640587\pi$
$41$	−5.47399	−0.854894	−0.427447	+	0.904040i	$0.640587\pi$
$42$	0	0
$43$	−8.87001	−1.35266	−0.676332	−	0.736597i	$-0.736432\pi$
$43$	−8.87001	−1.35266	−0.676332	+	0.736597i	$0.736432\pi$
$44$	0	0
$45$	1.67473	0.249654
$46$	0	0
$47$	−1.00757	−0.146970	−0.0734848	−	0.997296i	$-0.523412\pi$
$47$	−1.00757	−0.146970	−0.0734848	+	0.997296i	$0.523412\pi$
$48$	0	0
$49$	12.2585	1.75121
$50$	0	0
$51$	−1.76028	−0.246488
$52$	0	0
$53$	−11.9722	−1.64450	−0.822252	−	0.569123i	$-0.807283\pi$
$53$	−11.9722	−1.64450	−0.822252	+	0.569123i	$0.807283\pi$
$54$	0	0
$55$	9.16746	1.23614
$56$	0	0
$57$	−4.14872	−0.549512
$58$	0	0
$59$	−11.0112	−1.43353	−0.716766	−	0.697314i	$-0.754378\pi$
$59$	−11.0112	−1.43353	−0.716766	+	0.697314i	$0.754378\pi$
$60$	0	0
$61$	1.99243	0.255104	0.127552	−	0.991832i	$-0.459288\pi$
$61$	1.99243	0.255104	0.127552	+	0.991832i	$0.459288\pi$
$62$	0	0
$63$	−4.38845	−0.552892
$64$	0	0
$65$	1.92380	0.238618
$66$	0	0
$67$	1.34946	0.164863	0.0824313	−	0.996597i	$-0.473731\pi$
$67$	1.34946	0.164863	0.0824313	+	0.996597i	$0.473731\pi$
$68$	0	0
$69$	−5.67473	−0.683157
$70$	0	0
$71$	−10.0556	−1.19338	−0.596690	−	0.802472i	$-0.703518\pi$
$71$	−10.0556	−1.19338	−0.596690	+	0.802472i	$0.703518\pi$
$72$	0	0
$73$	6.10973	0.715090	0.357545	−	0.933896i	$-0.383614\pi$
$73$	6.10973	0.715090	0.357545	+	0.933896i	$0.383614\pi$
$74$	0	0
$75$	−2.19528	−0.253489
$76$	0	0
$77$	−24.0223	−2.73760
$78$	0	0
$79$	11.1209	1.25120	0.625599	−	0.780145i	$-0.284855\pi$
$79$	11.1209	1.25120	0.625599	+	0.780145i	$0.284855\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.71372	0.517398	0.258699	−	0.965958i	$-0.416706\pi$
$83$	4.71372	0.517398	0.258699	+	0.965958i	$0.416706\pi$
$84$	0	0
$85$	−2.94799	−0.319754
$86$	0	0
$87$	−0.325271	−0.0348727
$88$	0	0
$89$	2.23427	0.236832	0.118416	−	0.992964i	$-0.462218\pi$
$89$	2.23427	0.236832	0.118416	+	0.992964i	$0.462218\pi$
$90$	0	0
$91$	−5.04111	−0.528452
$92$	0	0
$93$	−5.43501	−0.563584
$94$	0	0
$95$	−6.94799	−0.712848
$96$	0	0
$97$	10.9480	1.11160	0.555800	−	0.831316i	$-0.312412\pi$
$97$	10.9480	1.11160	0.555800	+	0.831316i	$0.312412\pi$
$98$	0	0
$99$	5.47399	0.550157
$100$	0	0
$101$	9.49273	0.944562	0.472281	−	0.881448i	$-0.343431\pi$
$101$	9.49273	0.944562	0.472281	+	0.881448i	$0.343431\pi$
$102$	0	0
$103$	−0.603982	−0.0595121	−0.0297560	−	0.999557i	$-0.509473\pi$
$103$	−0.603982	−0.0595121	−0.0297560	+	0.999557i	$0.509473\pi$
$104$	0	0
$105$	−7.34946	−0.717234
$106$	0	0
$107$	13.4852	1.30366	0.651830	−	0.758365i	$-0.274002\pi$
$107$	13.4852	1.30366	0.651830	+	0.758365i	$0.274002\pi$
$108$	0	0
$109$	−15.1653	−1.45258	−0.726288	−	0.687391i	$-0.758756\pi$
$109$	−15.1653	−1.45258	−0.726288	+	0.687391i	$0.758756\pi$
$110$	0	0
$111$	4.86244	0.461522
$112$	0	0
$113$	0.916571	0.0862238	0.0431119	−	0.999070i	$-0.486273\pi$
$113$	0.916571	0.0862238	0.0431119	+	0.999070i	$0.486273\pi$
$114$	0	0
$115$	−9.50363	−0.886218
$116$	0	0
$117$	1.14872	0.106199
$118$	0	0
$119$	7.72488	0.708138
$120$	0	0
$121$	18.9646	1.72405
$122$	0	0
$123$	−5.47399	−0.493573
$124$	0	0
$125$	−12.0501	−1.07780
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	0	0
$129$	−8.87001	−0.780961
$130$	0	0
$131$	−12.1264	−1.05948	−0.529742	−	0.848159i	$-0.677711\pi$
$131$	−12.1264	−1.05948	−0.529742	+	0.848159i	$0.677711\pi$
$132$	0	0
$133$	18.2064	1.57870
$134$	0	0
$135$	1.67473	0.144138
$136$	0	0
$137$	12.4386	1.06270	0.531350	−	0.847152i	$-0.321685\pi$
$137$	12.4386	1.06270	0.531350	+	0.847152i	$0.321685\pi$
$138$	0	0
$139$	8.29744	0.703780	0.351890	−	0.936041i	$-0.385539\pi$
$139$	8.29744	0.703780	0.351890	+	0.936041i	$0.385539\pi$
$140$	0	0
$141$	−1.00757	−0.0848529
$142$	0	0
$143$	6.28810	0.525837
$144$	0	0
$145$	−0.544741	−0.0452382
$146$	0	0
$147$	12.2585	1.01106
$148$	0	0
$149$	−5.20074	−0.426061	−0.213030	−	0.977046i	$-0.568333\pi$
$149$	−5.20074	−0.426061	−0.213030	+	0.977046i	$0.568333\pi$
$150$	0	0
$151$	−21.8960	−1.78187	−0.890934	−	0.454132i	$-0.849950\pi$
$151$	−21.8960	−1.78187	−0.890934	+	0.454132i	$0.849950\pi$
$152$	0	0
$153$	−1.76028	−0.142310
$154$	0	0
$155$	−9.10216	−0.731103
$156$	0	0
$157$	−20.4552	−1.63250	−0.816252	−	0.577696i	$-0.803952\pi$
$157$	−20.4552	−1.63250	−0.816252	+	0.577696i	$0.803952\pi$
$158$	0	0
$159$	−11.9722	−0.949455
$160$	0	0
$161$	24.9032	1.96265
$162$	0	0
$163$	17.2306	1.34961	0.674804	−	0.737997i	$-0.264228\pi$
$163$	17.2306	1.34961	0.674804	+	0.737997i	$0.264228\pi$
$164$	0	0
$165$	9.16746	0.713685
$166$	0	0
$167$	−12.7769	−0.988706	−0.494353	−	0.869261i	$-0.664595\pi$
$167$	−12.7769	−0.988706	−0.494353	+	0.869261i	$0.664595\pi$
$168$	0	0
$169$	−11.6804	−0.898495
$170$	0	0
$171$	−4.14872	−0.317261
$172$	0	0
$173$	18.3906	1.39821	0.699104	−	0.715020i	$-0.253582\pi$
$173$	18.3906	1.39821	0.699104	+	0.715020i	$0.253582\pi$
$174$	0	0
$175$	9.63388	0.728253
$176$	0	0
$177$	−11.0112	−0.827650
$178$	0	0
$179$	−9.49606	−0.709769	−0.354885	−	0.934910i	$-0.615480\pi$
$179$	−9.49606	−0.709769	−0.354885	+	0.934910i	$0.615480\pi$
$180$	0	0
$181$	−22.6880	−1.68639	−0.843193	−	0.537611i	$-0.819327\pi$
$181$	−22.6880	−1.68639	−0.843193	+	0.537611i	$0.819327\pi$
$182$	0	0
$183$	1.99243	0.147284
$184$	0	0
$185$	8.14327	0.598705
$186$	0	0
$187$	−9.63574	−0.704635
$188$	0	0
$189$	−4.38845	−0.319212
$190$	0	0
$191$	−2.78992	−0.201871	−0.100936	−	0.994893i	$-0.532184\pi$
$191$	−2.78992	−0.201871	−0.100936	+	0.994893i	$0.532184\pi$
$192$	0	0
$193$	12.7920	0.920791	0.460395	−	0.887714i	$-0.347708\pi$
$193$	12.7920	0.920791	0.460395	+	0.887714i	$0.347708\pi$
$194$	0	0
$195$	1.92380	0.137766
$196$	0	0
$197$	−4.08191	−0.290824	−0.145412	−	0.989371i	$-0.546451\pi$
$197$	−4.08191	−0.290824	−0.145412	+	0.989371i	$0.546451\pi$
$198$	0	0
$199$	−24.8108	−1.75879	−0.879394	−	0.476094i	$-0.842052\pi$
$199$	−24.8108	−1.75879	−0.879394	+	0.476094i	$0.842052\pi$
$200$	0	0
$201$	1.34946	0.0951835
$202$	0	0
$203$	1.42743	0.100186
$204$	0	0
$205$	−9.16746	−0.640283
$206$	0	0
$207$	−5.67473	−0.394421
$208$	0	0
$209$	−22.7101	−1.57089
$210$	0	0
$211$	−21.6151	−1.48805	−0.744024	−	0.668153i	$-0.767085\pi$
$211$	−21.6151	−1.48805	−0.744024	+	0.668153i	$0.767085\pi$
$212$	0	0
$213$	−10.0556	−0.688999
$214$	0	0
$215$	−14.8549	−1.01309
$216$	0	0
$217$	23.8512	1.61913
$218$	0	0
$219$	6.10973	0.412858
$220$	0	0
$221$	−2.02207	−0.136019
$222$	0	0
$223$	−0.155950	−0.0104432	−0.00522160	−	0.999986i	$-0.501662\pi$
$223$	−0.155950	−0.0104432	−0.00522160	+	0.999986i	$0.501662\pi$
$224$	0	0
$225$	−2.19528	−0.146352
$226$	0	0
$227$	−9.55378	−0.634107	−0.317053	−	0.948408i	$-0.602693\pi$
$227$	−9.55378	−0.634107	−0.317053	+	0.948408i	$0.602693\pi$
$228$	0	0
$229$	23.8533	1.57627	0.788137	−	0.615500i	$-0.211046\pi$
$229$	23.8533	1.57627	0.788137	+	0.615500i	$0.211046\pi$
$230$	0	0
$231$	−24.0223	−1.58055
$232$	0	0
$233$	13.9852	0.916201	0.458100	−	0.888900i	$-0.348530\pi$
$233$	13.9852	0.916201	0.458100	+	0.888900i	$0.348530\pi$
$234$	0	0
$235$	−1.68741	−0.110075
$236$	0	0
$237$	11.1209	0.722380
$238$	0	0
$239$	−6.79381	−0.439455	−0.219728	−	0.975561i	$-0.570517\pi$
$239$	−6.79381	−0.439455	−0.219728	+	0.975561i	$0.570517\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	20.5296	1.31159
$246$	0	0
$247$	−4.76573	−0.303236
$248$	0	0
$249$	4.71372	0.298720
$250$	0	0
$251$	−24.9237	−1.57317	−0.786587	−	0.617480i	$-0.788154\pi$
$251$	−24.9237	−1.57317	−0.786587	+	0.617480i	$0.788154\pi$
$252$	0	0
$253$	−31.0634	−1.95294
$254$	0	0
$255$	−2.94799	−0.184610
$256$	0	0
$257$	−13.0133	−0.811746	−0.405873	−	0.913929i	$-0.633032\pi$
$257$	−13.0133	−0.811746	−0.405873	+	0.913929i	$0.633032\pi$
$258$	0	0
$259$	−21.3386	−1.32591
$260$	0	0
$261$	−0.325271	−0.0201338
$262$	0	0
$263$	−6.41294	−0.395439	−0.197719	−	0.980259i	$-0.563353\pi$
$263$	−6.41294	−0.395439	−0.197719	+	0.980259i	$0.563353\pi$
$264$	0	0
$265$	−20.0501	−1.23167
$266$	0	0
$267$	2.23427	0.136735
$268$	0	0
$269$	31.0368	1.89235	0.946174	−	0.323658i	$-0.104913\pi$
$269$	31.0368	1.89235	0.946174	+	0.323658i	$0.104913\pi$
$270$	0	0
$271$	3.39961	0.206511	0.103256	−	0.994655i	$-0.467074\pi$
$271$	3.39961	0.206511	0.103256	+	0.994655i	$0.467074\pi$
$272$	0	0
$273$	−5.04111	−0.305102
$274$	0	0
$275$	−12.0170	−0.724650
$276$	0	0
$277$	6.01514	0.361415	0.180708	−	0.983537i	$-0.442161\pi$
$277$	6.01514	0.361415	0.180708	+	0.983537i	$0.442161\pi$
$278$	0	0
$279$	−5.43501	−0.325385
$280$	0	0
$281$	−26.2527	−1.56610	−0.783052	−	0.621956i	$-0.786338\pi$
$281$	−26.2527	−1.56610	−0.783052	+	0.621956i	$0.786338\pi$
$282$	0	0
$283$	−8.46854	−0.503402	−0.251701	−	0.967805i	$-0.580990\pi$
$283$	−8.46854	−0.503402	−0.251701	+	0.967805i	$0.580990\pi$
$284$	0	0
$285$	−6.94799	−0.411563
$286$	0	0
$287$	24.0223	1.41799
$288$	0	0
$289$	−13.9014	−0.817731
$290$	0	0
$291$	10.9480	0.641782
$292$	0	0
$293$	−33.1565	−1.93703	−0.968513	−	0.248962i	$-0.919910\pi$
$293$	−33.1565	−1.93703	−0.968513	+	0.248962i	$0.919910\pi$
$294$	0	0
$295$	−18.4407	−1.07366
$296$	0	0
$297$	5.47399	0.317633
$298$	0	0
$299$	−6.51869	−0.376985
$300$	0	0
$301$	38.9256	2.24363
$302$	0	0
$303$	9.49273	0.545343
$304$	0	0
$305$	3.33678	0.191063
$306$	0	0
$307$	2.87001	0.163800	0.0819001	−	0.996641i	$-0.473901\pi$
$307$	2.87001	0.163800	0.0819001	+	0.996641i	$0.473901\pi$
$308$	0	0
$309$	−0.603982	−0.0343593
$310$	0	0
$311$	18.4668	1.04715	0.523577	−	0.851978i	$-0.324597\pi$
$311$	18.4668	1.04715	0.523577	+	0.851978i	$0.324597\pi$
$312$	0	0
$313$	3.91112	0.221069	0.110535	−	0.993872i	$-0.464744\pi$
$313$	3.91112	0.221069	0.110535	+	0.993872i	$0.464744\pi$
$314$	0	0
$315$	−7.34946	−0.414095
$316$	0	0
$317$	−25.7472	−1.44611	−0.723053	−	0.690792i	$-0.757262\pi$
$317$	−25.7472	−1.44611	−0.723053	+	0.690792i	$0.757262\pi$
$318$	0	0
$319$	−1.78053	−0.0996906
$320$	0	0
$321$	13.4852	0.752668
$322$	0	0
$323$	7.30290	0.406344
$324$	0	0
$325$	−2.52177	−0.139883
$326$	0	0
$327$	−15.1653	−0.838645
$328$	0	0
$329$	4.42168	0.243775
$330$	0	0
$331$	−23.6469	−1.29975	−0.649876	−	0.760041i	$-0.725179\pi$
$331$	−23.6469	−1.29975	−0.649876	+	0.760041i	$0.725179\pi$
$332$	0	0
$333$	4.86244	0.266460
$334$	0	0
$335$	2.25998	0.123476
$336$	0	0
$337$	23.6953	1.29076	0.645382	−	0.763860i	$-0.276698\pi$
$337$	23.6953	1.29076	0.645382	+	0.763860i	$0.276698\pi$
$338$	0	0
$339$	0.916571	0.0497813
$340$	0	0
$341$	−29.7512	−1.61112
$342$	0	0
$343$	−23.0765	−1.24601
$344$	0	0
$345$	−9.50363	−0.511658
$346$	0	0
$347$	11.0263	0.591923	0.295962	−	0.955200i	$-0.404360\pi$
$347$	11.0263	0.591923	0.295962	+	0.955200i	$0.404360\pi$
$348$	0	0
$349$	3.07646	0.164679	0.0823394	−	0.996604i	$-0.473761\pi$
$349$	3.07646	0.164679	0.0823394	+	0.996604i	$0.473761\pi$
$350$	0	0
$351$	1.14872	0.0613143
$352$	0	0
$353$	−10.5372	−0.560837	−0.280418	−	0.959878i	$-0.590473\pi$
$353$	−10.5372	−0.560837	−0.280418	+	0.959878i	$0.590473\pi$
$354$	0	0
$355$	−16.8404	−0.893796
$356$	0	0
$357$	7.72488	0.408844
$358$	0	0
$359$	−2.97404	−0.156964	−0.0784819	−	0.996916i	$-0.525007\pi$
$359$	−2.97404	−0.156964	−0.0784819	+	0.996916i	$0.525007\pi$
$360$	0	0
$361$	−1.78810	−0.0941107
$362$	0	0
$363$	18.9646	0.995383
$364$	0	0
$365$	10.2322	0.535575
$366$	0	0
$367$	8.11878	0.423797	0.211898	−	0.977292i	$-0.432035\pi$
$367$	8.11878	0.423797	0.211898	+	0.977292i	$0.432035\pi$
$368$	0	0
$369$	−5.47399	−0.284965
$370$	0	0
$371$	52.5392	2.72770
$372$	0	0
$373$	−16.6079	−0.859925	−0.429963	−	0.902847i	$-0.641473\pi$
$373$	−16.6079	−0.859925	−0.429963	+	0.902847i	$0.641473\pi$
$374$	0	0
$375$	−12.0501	−0.622267
$376$	0	0
$377$	−0.373646	−0.0192437
$378$	0	0
$379$	34.6433	1.77950	0.889752	−	0.456443i	$-0.150877\pi$
$379$	34.6433	1.77950	0.889752	+	0.456443i	$0.150877\pi$
$380$	0	0
$381$	1.00000	0.0512316
$382$	0	0
$383$	22.0484	1.12662	0.563310	−	0.826246i	$-0.309528\pi$
$383$	22.0484	1.12662	0.563310	+	0.826246i	$0.309528\pi$
$384$	0	0
$385$	−40.2309	−2.05036
$386$	0	0
$387$	−8.87001	−0.450888
$388$	0	0
$389$	−5.34764	−0.271136	−0.135568	−	0.990768i	$-0.543286\pi$
$389$	−5.34764	−0.271136	−0.135568	+	0.990768i	$0.543286\pi$
$390$	0	0
$391$	9.98909	0.505170
$392$	0	0
$393$	−12.1264	−0.611693
$394$	0	0
$395$	18.6245	0.937100
$396$	0	0
$397$	−28.5093	−1.43084	−0.715421	−	0.698693i	$-0.753765\pi$
$397$	−28.5093	−1.43084	−0.715421	+	0.698693i	$0.753765\pi$
$398$	0	0
$399$	18.2064	0.911462
$400$	0	0
$401$	22.2249	1.10986	0.554929	−	0.831898i	$-0.312746\pi$
$401$	22.2249	1.10986	0.554929	+	0.831898i	$0.312746\pi$
$402$	0	0
$403$	−6.24331	−0.311002
$404$	0	0
$405$	1.67473	0.0832180
$406$	0	0
$407$	26.6170	1.31935
$408$	0	0
$409$	10.2974	0.509176	0.254588	−	0.967050i	$-0.418060\pi$
$409$	10.2974	0.509176	0.254588	+	0.967050i	$0.418060\pi$
$410$	0	0
$411$	12.4386	0.613551
$412$	0	0
$413$	48.3219	2.37776
$414$	0	0
$415$	7.89420	0.387511
$416$	0	0
$417$	8.29744	0.406328
$418$	0	0
$419$	39.8606	1.94732	0.973658	−	0.228012i	$-0.0732226\pi$
$419$	39.8606	1.94732	0.973658	+	0.228012i	$0.0732226\pi$
$420$	0	0
$421$	−30.0598	−1.46502	−0.732512	−	0.680754i	$-0.761652\pi$
$421$	−30.0598	−1.46502	−0.732512	+	0.680754i	$0.761652\pi$
$422$	0	0
$423$	−1.00757	−0.0489898
$424$	0	0
$425$	3.86430	0.187446
$426$	0	0
$427$	−8.74366	−0.423135
$428$	0	0
$429$	6.28810	0.303592
$430$	0	0
$431$	22.3772	1.07787	0.538937	−	0.842346i	$-0.318826\pi$
$431$	22.3772	1.07787	0.538937	+	0.842346i	$0.318826\pi$
$432$	0	0
$433$	3.88875	0.186881	0.0934406	−	0.995625i	$-0.470213\pi$
$433$	3.88875	0.186881	0.0934406	+	0.995625i	$0.470213\pi$
$434$	0	0
$435$	−0.544741	−0.0261183
$436$	0	0
$437$	23.5429	1.12621
$438$	0	0
$439$	−6.60792	−0.315379	−0.157689	−	0.987489i	$-0.550404\pi$
$439$	−6.60792	−0.315379	−0.157689	+	0.987489i	$0.550404\pi$
$440$	0	0
$441$	12.2585	0.583736
$442$	0	0
$443$	14.8123	0.703753	0.351877	−	0.936046i	$-0.385544\pi$
$443$	14.8123	0.703753	0.351877	+	0.936046i	$0.385544\pi$
$444$	0	0
$445$	3.74180	0.177378
$446$	0	0
$447$	−5.20074	−0.245986
$448$	0	0
$449$	−1.87547	−0.0885087	−0.0442543	−	0.999020i	$-0.514091\pi$
$449$	−1.87547	−0.0885087	−0.0442543	+	0.999020i	$0.514091\pi$
$450$	0	0
$451$	−29.9646	−1.41098
$452$	0	0
$453$	−21.8960	−1.02876
$454$	0	0
$455$	−8.44249	−0.395790
$456$	0	0
$457$	35.4815	1.65976	0.829878	−	0.557945i	$-0.188410\pi$
$457$	35.4815	1.65976	0.829878	+	0.557945i	$0.188410\pi$
$458$	0	0
$459$	−1.76028	−0.0821627
$460$	0	0
$461$	−8.30835	−0.386959	−0.193479	−	0.981104i	$-0.561977\pi$
$461$	−8.30835	−0.386959	−0.193479	+	0.981104i	$0.561977\pi$
$462$	0	0
$463$	21.0668	0.979056	0.489528	−	0.871988i	$-0.337169\pi$
$463$	21.0668	0.979056	0.489528	+	0.871988i	$0.337169\pi$
$464$	0	0
$465$	−9.10216	−0.422103
$466$	0	0
$467$	−16.0523	−0.742810	−0.371405	−	0.928471i	$-0.621124\pi$
$467$	−16.0523	−0.742810	−0.371405	+	0.928471i	$0.621124\pi$
$468$	0	0
$469$	−5.92202	−0.273454
$470$	0	0
$471$	−20.4552	−0.942526
$472$	0	0
$473$	−48.5544	−2.23253
$474$	0	0
$475$	9.10762	0.417886
$476$	0	0
$477$	−11.9722	−0.548168
$478$	0	0
$479$	7.52237	0.343706	0.171853	−	0.985123i	$-0.445025\pi$
$479$	7.52237	0.343706	0.171853	+	0.985123i	$0.445025\pi$
$480$	0	0
$481$	5.58559	0.254681
$482$	0	0
$483$	24.9032	1.13314
$484$	0	0
$485$	18.3349	0.832545
$486$	0	0
$487$	6.50389	0.294719	0.147360	−	0.989083i	$-0.452923\pi$
$487$	6.50389	0.294719	0.147360	+	0.989083i	$0.452923\pi$
$488$	0	0
$489$	17.2306	0.779196
$490$	0	0
$491$	35.1653	1.58699	0.793495	−	0.608577i	$-0.208259\pi$
$491$	35.1653	1.58699	0.793495	+	0.608577i	$0.208259\pi$
$492$	0	0
$493$	0.572567	0.0257871
$494$	0	0
$495$	9.16746	0.412047
$496$	0	0
$497$	44.1285	1.97943
$498$	0	0
$499$	22.8288	1.02196	0.510979	−	0.859593i	$-0.329283\pi$
$499$	22.8288	1.02196	0.510979	+	0.859593i	$0.329283\pi$
$500$	0	0
$501$	−12.7769	−0.570829
$502$	0	0
$503$	−7.43137	−0.331348	−0.165674	−	0.986181i	$-0.552980\pi$
$503$	−7.43137	−0.331348	−0.165674	+	0.986181i	$0.552980\pi$
$504$	0	0
$505$	15.8977	0.707440
$506$	0	0
$507$	−11.6804	−0.518746
$508$	0	0
$509$	−6.65054	−0.294780	−0.147390	−	0.989078i	$-0.547087\pi$
$509$	−6.65054	−0.294780	−0.147390	+	0.989078i	$0.547087\pi$
$510$	0	0
$511$	−26.8122	−1.18610
$512$	0	0
$513$	−4.14872	−0.183171
$514$	0	0
$515$	−1.01151	−0.0445723
$516$	0	0
$517$	−5.51544	−0.242569
$518$	0	0
$519$	18.3906	0.807256
$520$	0	0
$521$	−13.9423	−0.610822	−0.305411	−	0.952221i	$-0.598794\pi$
$521$	−13.9423	−0.610822	−0.305411	+	0.952221i	$0.598794\pi$
$522$	0	0
$523$	6.35880	0.278051	0.139026	−	0.990289i	$-0.455603\pi$
$523$	6.35880	0.278051	0.139026	+	0.990289i	$0.455603\pi$
$524$	0	0
$525$	9.63388	0.420457
$526$	0	0
$527$	9.56711	0.416750
$528$	0	0
$529$	9.20255	0.400111
$530$	0	0
$531$	−11.0112	−0.477844
$532$	0	0
$533$	−6.28810	−0.272368
$534$	0	0
$535$	22.5840	0.976391
$536$	0	0
$537$	−9.49606	−0.409785
$538$	0	0
$539$	67.1027	2.89032
$540$	0	0
$541$	0.295326	0.0126971	0.00634853	−	0.999980i	$-0.497979\pi$
$541$	0.295326	0.0126971	0.00634853	+	0.999980i	$0.497979\pi$
$542$	0	0
$543$	−22.6880	−0.973636
$544$	0	0
$545$	−25.3978	−1.08792
$546$	0	0
$547$	37.7884	1.61572	0.807858	−	0.589378i	$-0.200627\pi$
$547$	37.7884	1.61572	0.807858	+	0.589378i	$0.200627\pi$
$548$	0	0
$549$	1.99243	0.0850347
$550$	0	0
$551$	1.34946	0.0574889
$552$	0	0
$553$	−48.8035	−2.07533
$554$	0	0
$555$	8.14327	0.345663
$556$	0	0
$557$	20.6636	0.875543	0.437772	−	0.899086i	$-0.355768\pi$
$557$	20.6636	0.875543	0.437772	+	0.899086i	$0.355768\pi$
$558$	0	0
$559$	−10.1892	−0.430956
$560$	0	0
$561$	−9.63574	−0.406821
$562$	0	0
$563$	−1.13211	−0.0477126	−0.0238563	−	0.999715i	$-0.507594\pi$
$563$	−1.13211	−0.0477126	−0.0238563	+	0.999715i	$0.507594\pi$
$564$	0	0
$565$	1.53501	0.0645783
$566$	0	0
$567$	−4.38845	−0.184297
$568$	0	0
$569$	−31.8978	−1.33722	−0.668612	−	0.743611i	$-0.733111\pi$
$569$	−31.8978	−1.33722	−0.668612	+	0.743611i	$0.733111\pi$
$570$	0	0
$571$	−36.0266	−1.50766	−0.753832	−	0.657067i	$-0.771797\pi$
$571$	−36.0266	−1.50766	−0.753832	+	0.657067i	$0.771797\pi$
$572$	0	0
$573$	−2.78992	−0.116550
$574$	0	0
$575$	12.4576	0.519519
$576$	0	0
$577$	12.4849	0.519753	0.259877	−	0.965642i	$-0.416318\pi$
$577$	12.4849	0.519753	0.259877	+	0.965642i	$0.416318\pi$
$578$	0	0
$579$	12.7920	0.531619
$580$	0	0
$581$	−20.6859	−0.858195
$582$	0	0
$583$	−65.5356	−2.71421
$584$	0	0
$585$	1.92380	0.0795393
$586$	0	0
$587$	19.5631	0.807457	0.403728	−	0.914879i	$-0.367714\pi$
$587$	19.5631	0.807457	0.403728	+	0.914879i	$0.367714\pi$
$588$	0	0
$589$	22.5483	0.929088
$590$	0	0
$591$	−4.08191	−0.167907
$592$	0	0
$593$	−25.6212	−1.05214	−0.526068	−	0.850442i	$-0.676334\pi$
$593$	−25.6212	−1.05214	−0.526068	+	0.850442i	$0.676334\pi$
$594$	0	0
$595$	12.9371	0.530369
$596$	0	0
$597$	−24.8108	−1.01544
$598$	0	0
$599$	0.986720	0.0403163	0.0201581	−	0.999797i	$-0.493583\pi$
$599$	0.986720	0.0403163	0.0201581	+	0.999797i	$0.493583\pi$
$600$	0	0
$601$	−13.2787	−0.541648	−0.270824	−	0.962629i	$-0.587296\pi$
$601$	−13.2787	−0.541648	−0.270824	+	0.962629i	$0.587296\pi$
$602$	0	0
$603$	1.34946	0.0549542
$604$	0	0
$605$	31.7606	1.29125
$606$	0	0
$607$	27.6393	1.12185	0.560923	−	0.827868i	$-0.310446\pi$
$607$	27.6393	1.12185	0.560923	+	0.827868i	$0.310446\pi$
$608$	0	0
$609$	1.42743	0.0578425
$610$	0	0
$611$	−1.15742	−0.0468242
$612$	0	0
$613$	0.438250	0.0177008	0.00885038	−	0.999961i	$-0.497183\pi$
$613$	0.438250	0.0177008	0.00885038	+	0.999961i	$0.497183\pi$
$614$	0	0
$615$	−9.16746	−0.369668
$616$	0	0
$617$	35.0314	1.41031	0.705155	−	0.709053i	$-0.250877\pi$
$617$	35.0314	1.41031	0.705155	+	0.709053i	$0.250877\pi$
$618$	0	0
$619$	−30.0223	−1.20670	−0.603350	−	0.797477i	$-0.706168\pi$
$619$	−30.0223	−1.20670	−0.603350	+	0.797477i	$0.706168\pi$
$620$	0	0
$621$	−5.67473	−0.227719
$622$	0	0
$623$	−9.80497	−0.392828
$624$	0	0
$625$	−9.20433	−0.368173
$626$	0	0
$627$	−22.7101	−0.906953
$628$	0	0
$629$	−8.55924	−0.341279
$630$	0	0
$631$	−18.8721	−0.751288	−0.375644	−	0.926764i	$-0.622578\pi$
$631$	−18.8721	−0.751288	−0.375644	+	0.926764i	$0.622578\pi$
$632$	0	0
$633$	−21.6151	−0.859125
$634$	0	0
$635$	1.67473	0.0664596
$636$	0	0
$637$	14.0816	0.557932
$638$	0	0
$639$	−10.0556	−0.397793
$640$	0	0
$641$	−25.5112	−1.00763	−0.503815	−	0.863812i	$-0.668071\pi$
$641$	−25.5112	−1.00763	−0.503815	+	0.863812i	$0.668071\pi$
$642$	0	0
$643$	−26.6321	−1.05027	−0.525133	−	0.851020i	$-0.675984\pi$
$643$	−26.6321	−1.05027	−0.525133	+	0.851020i	$0.675984\pi$
$644$	0	0
$645$	−14.8549	−0.584910
$646$	0	0
$647$	31.8833	1.25346	0.626731	−	0.779236i	$-0.284393\pi$
$647$	31.8833	1.25346	0.626731	+	0.779236i	$0.284393\pi$
$648$	0	0
$649$	−60.2750	−2.36600
$650$	0	0
$651$	23.8512	0.934803
$652$	0	0
$653$	11.4610	0.448502	0.224251	−	0.974531i	$-0.428006\pi$
$653$	11.4610	0.448502	0.224251	+	0.974531i	$0.428006\pi$
$654$	0	0
$655$	−20.3084	−0.793513
$656$	0	0
$657$	6.10973	0.238363
$658$	0	0
$659$	−17.5837	−0.684965	−0.342482	−	0.939524i	$-0.611268\pi$
$659$	−17.5837	−0.684965	−0.342482	+	0.939524i	$0.611268\pi$
$660$	0	0
$661$	33.4353	1.30048	0.650240	−	0.759728i	$-0.274668\pi$
$661$	33.4353	1.30048	0.650240	+	0.759728i	$0.274668\pi$
$662$	0	0
$663$	−2.02207	−0.0785307
$664$	0	0
$665$	30.4909	1.18238
$666$	0	0
$667$	1.84582	0.0714706
$668$	0	0
$669$	−0.155950	−0.00602938
$670$	0	0
$671$	10.9065	0.421042
$672$	0	0
$673$	−18.1989	−0.701515	−0.350757	−	0.936466i	$-0.614076\pi$
$673$	−18.1989	−0.701515	−0.350757	+	0.936466i	$0.614076\pi$
$674$	0	0
$675$	−2.19528	−0.0844965
$676$	0	0
$677$	45.8231	1.76112	0.880562	−	0.473932i	$-0.157166\pi$
$677$	45.8231	1.76112	0.880562	+	0.473932i	$0.157166\pi$
$678$	0	0
$679$	−48.0446	−1.84378
$680$	0	0
$681$	−9.55378	−0.366102
$682$	0	0
$683$	−29.5147	−1.12935	−0.564674	−	0.825314i	$-0.690998\pi$
$683$	−29.5147	−1.12935	−0.564674	+	0.825314i	$0.690998\pi$
$684$	0	0
$685$	20.8313	0.795922
$686$	0	0
$687$	23.8533	0.910062
$688$	0	0
$689$	−13.7527	−0.523936
$690$	0	0
$691$	10.0042	0.380579	0.190290	−	0.981728i	$-0.439057\pi$
$691$	10.0042	0.380579	0.190290	+	0.981728i	$0.439057\pi$
$692$	0	0
$693$	−24.0223	−0.912533
$694$	0	0
$695$	13.8960	0.527104
$696$	0	0
$697$	9.63574	0.364980
$698$	0	0
$699$	13.9852	0.528969
$700$	0	0
$701$	25.3700	0.958212	0.479106	−	0.877757i	$-0.340961\pi$
$701$	25.3700	0.958212	0.479106	+	0.877757i	$0.340961\pi$
$702$	0	0
$703$	−20.1729	−0.760836
$704$	0	0
$705$	−1.68741	−0.0635516
$706$	0	0
$707$	−41.6583	−1.56672
$708$	0	0
$709$	−16.9123	−0.635156	−0.317578	−	0.948232i	$-0.602870\pi$
$709$	−16.9123	−0.635156	−0.317578	+	0.948232i	$0.602870\pi$
$710$	0	0
$711$	11.1209	0.417066
$712$	0	0
$713$	30.8422	1.15505
$714$	0	0
$715$	10.5309	0.393832
$716$	0	0
$717$	−6.79381	−0.253719
$718$	0	0
$719$	−18.6454	−0.695357	−0.347679	−	0.937614i	$-0.613030\pi$
$719$	−18.6454	−0.695357	−0.347679	+	0.937614i	$0.613030\pi$
$720$	0	0
$721$	2.65054	0.0987113
$722$	0	0
$723$	8.00000	0.297523
$724$	0	0
$725$	0.714061	0.0265196
$726$	0	0
$727$	−36.5834	−1.35680	−0.678401	−	0.734692i	$-0.737327\pi$
$727$	−36.5834	−1.35680	−0.678401	+	0.734692i	$0.737327\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	15.6137	0.577493
$732$	0	0
$733$	−18.7049	−0.690882	−0.345441	−	0.938440i	$-0.612271\pi$
$733$	−18.7049	−0.690882	−0.345441	+	0.938440i	$0.612271\pi$
$734$	0	0
$735$	20.5296	0.757246
$736$	0	0
$737$	7.38693	0.272101
$738$	0	0
$739$	12.9978	0.478133	0.239066	−	0.971003i	$-0.423159\pi$
$739$	12.9978	0.478133	0.239066	+	0.971003i	$0.423159\pi$
$740$	0	0
$741$	−4.76573	−0.175073
$742$	0	0
$743$	27.6233	1.01340	0.506701	−	0.862122i	$-0.330865\pi$
$743$	27.6233	1.01340	0.506701	+	0.862122i	$0.330865\pi$
$744$	0	0
$745$	−8.70982	−0.319103
$746$	0	0
$747$	4.71372	0.172466
$748$	0	0
$749$	−59.1789	−2.16235
$750$	0	0
$751$	−26.9386	−0.983003	−0.491502	−	0.870877i	$-0.663552\pi$
$751$	−26.9386	−0.983003	−0.491502	+	0.870877i	$0.663552\pi$
$752$	0	0
$753$	−24.9237	−0.908272
$754$	0	0
$755$	−36.6698	−1.33455
$756$	0	0
$757$	12.1303	0.440883	0.220441	−	0.975400i	$-0.429250\pi$
$757$	12.1303	0.440883	0.220441	+	0.975400i	$0.429250\pi$
$758$	0	0
$759$	−31.0634	−1.12753
$760$	0	0
$761$	−1.44011	−0.0522041	−0.0261021	−	0.999659i	$-0.508309\pi$
$761$	−1.44011	−0.0522041	−0.0261021	+	0.999659i	$0.508309\pi$
$762$	0	0
$763$	66.5523	2.40935
$764$	0	0
$765$	−2.94799	−0.106585
$766$	0	0
$767$	−12.6488	−0.456721
$768$	0	0
$769$	−29.8706	−1.07716	−0.538581	−	0.842574i	$-0.681039\pi$
$769$	−29.8706	−1.07716	−0.538581	+	0.842574i	$0.681039\pi$
$770$	0	0
$771$	−13.0133	−0.468662
$772$	0	0
$773$	36.7154	1.32056	0.660281	−	0.751018i	$-0.270437\pi$
$773$	36.7154	1.32056	0.660281	+	0.751018i	$0.270437\pi$
$774$	0	0
$775$	11.9314	0.428588
$776$	0	0
$777$	−21.3386	−0.765516
$778$	0	0
$779$	22.7101	0.813673
$780$	0	0
$781$	−55.0443	−1.96964
$782$	0	0
$783$	−0.325271	−0.0116242
$784$	0	0
$785$	−34.2569	−1.22268
$786$	0	0
$787$	28.9667	1.03255	0.516276	−	0.856422i	$-0.327318\pi$
$787$	28.9667	1.03255	0.516276	+	0.856422i	$0.327318\pi$
$788$	0	0
$789$	−6.41294	−0.228307
$790$	0	0
$791$	−4.02232	−0.143017
$792$	0	0
$793$	2.28875	0.0812758
$794$	0	0
$795$	−20.0501	−0.711106
$796$	0	0
$797$	26.4482	0.936845	0.468422	−	0.883505i	$-0.344823\pi$
$797$	26.4482	0.936845	0.468422	+	0.883505i	$0.344823\pi$
$798$	0	0
$799$	1.77361	0.0627457
$800$	0	0
$801$	2.23427	0.0789440
$802$	0	0
$803$	33.4446	1.18024
$804$	0	0
$805$	41.7062	1.46995
$806$	0	0
$807$	31.0368	1.09255
$808$	0	0
$809$	−32.7457	−1.15128	−0.575639	−	0.817704i	$-0.695247\pi$
$809$	−32.7457	−1.15128	−0.575639	+	0.817704i	$0.695247\pi$
$810$	0	0
$811$	39.5761	1.38971	0.694853	−	0.719152i	$-0.255469\pi$
$811$	39.5761	1.38971	0.694853	+	0.719152i	$0.255469\pi$
$812$	0	0
$813$	3.39961	0.119229
$814$	0	0
$815$	28.8566	1.01080
$816$	0	0
$817$	36.7992	1.28744
$818$	0	0
$819$	−5.04111	−0.176151
$820$	0	0
$821$	−45.7974	−1.59834	−0.799169	−	0.601106i	$-0.794727\pi$
$821$	−45.7974	−1.59834	−0.799169	+	0.601106i	$0.794727\pi$
$822$	0	0
$823$	14.4035	0.502076	0.251038	−	0.967977i	$-0.419228\pi$
$823$	14.4035	0.502076	0.251038	+	0.967977i	$0.419228\pi$
$824$	0	0
$825$	−12.0170	−0.418377
$826$	0	0
$827$	−2.33830	−0.0813105	−0.0406553	−	0.999173i	$-0.512945\pi$
$827$	−2.33830	−0.0813105	−0.0406553	+	0.999173i	$0.512945\pi$
$828$	0	0
$829$	−17.4556	−0.606258	−0.303129	−	0.952950i	$-0.598031\pi$
$829$	−17.4556	−0.606258	−0.303129	+	0.952950i	$0.598031\pi$
$830$	0	0
$831$	6.01514	0.208663
$832$	0	0
$833$	−21.5783	−0.747643
$834$	0	0
$835$	−21.3978	−0.740503
$836$	0	0
$837$	−5.43501	−0.187861
$838$	0	0
$839$	30.2140	1.04310	0.521551	−	0.853220i	$-0.325354\pi$
$839$	30.2140	1.04310	0.521551	+	0.853220i	$0.325354\pi$
$840$	0	0
$841$	−28.8942	−0.996352
$842$	0	0
$843$	−26.2527	−0.904191
$844$	0	0
$845$	−19.5616	−0.672938
$846$	0	0
$847$	−83.2251	−2.85965
$848$	0	0
$849$	−8.46854	−0.290640
$850$	0	0
$851$	−27.5930	−0.945877
$852$	0	0
$853$	6.09312	0.208624	0.104312	−	0.994545i	$-0.466736\pi$
$853$	6.09312	0.208624	0.104312	+	0.994545i	$0.466736\pi$
$854$	0	0
$855$	−6.94799	−0.237616
$856$	0	0
$857$	12.0928	0.413081	0.206541	−	0.978438i	$-0.433779\pi$
$857$	12.0928	0.413081	0.206541	+	0.978438i	$0.433779\pi$
$858$	0	0
$859$	4.27512	0.145865	0.0729326	−	0.997337i	$-0.476764\pi$
$859$	4.27512	0.145865	0.0729326	+	0.997337i	$0.476764\pi$
$860$	0	0
$861$	24.0223	0.818679
$862$	0	0
$863$	−26.6270	−0.906392	−0.453196	−	0.891411i	$-0.649716\pi$
$863$	−26.6270	−0.906392	−0.453196	+	0.891411i	$0.649716\pi$
$864$	0	0
$865$	30.7992	1.04720
$866$	0	0
$867$	−13.9014	−0.472117
$868$	0	0
$869$	60.8757	2.06507
$870$	0	0
$871$	1.55015	0.0525249
$872$	0	0
$873$	10.9480	0.370533
$874$	0	0
$875$	52.8814	1.78772
$876$	0	0
$877$	−41.8345	−1.41265	−0.706326	−	0.707887i	$-0.749649\pi$
$877$	−41.8345	−1.41265	−0.706326	+	0.707887i	$0.749649\pi$
$878$	0	0
$879$	−33.1565	−1.11834
$880$	0	0
$881$	−29.6617	−0.999328	−0.499664	−	0.866219i	$-0.666543\pi$
$881$	−29.6617	−0.999328	−0.499664	+	0.866219i	$0.666543\pi$
$882$	0	0
$883$	11.8104	0.397452	0.198726	−	0.980055i	$-0.436320\pi$
$883$	11.8104	0.397452	0.198726	+	0.980055i	$0.436320\pi$
$884$	0	0
$885$	−18.4407	−0.619878
$886$	0	0
$887$	−26.2436	−0.881173	−0.440586	−	0.897710i	$-0.645229\pi$
$887$	−26.2436	−0.881173	−0.440586	+	0.897710i	$0.645229\pi$
$888$	0	0
$889$	−4.38845	−0.147184
$890$	0	0
$891$	5.47399	0.183386
$892$	0	0
$893$	4.18014	0.139883
$894$	0	0
$895$	−15.9033	−0.531590
$896$	0	0
$897$	−6.51869	−0.217653
$898$	0	0
$899$	1.76785	0.0589611
$900$	0	0
$901$	21.0743	0.702088
$902$	0	0
$903$	38.9256	1.29536
$904$	0	0
$905$	−37.9963	−1.26304
$906$	0	0
$907$	41.4664	1.37687	0.688434	−	0.725299i	$-0.258298\pi$
$907$	41.4664	1.37687	0.688434	+	0.725299i	$0.258298\pi$
$908$	0	0
$909$	9.49273	0.314854
$910$	0	0
$911$	13.8050	0.457379	0.228690	−	0.973499i	$-0.426556\pi$
$911$	13.8050	0.457379	0.228690	+	0.973499i	$0.426556\pi$
$912$	0	0
$913$	25.8029	0.853950
$914$	0	0
$915$	3.33678	0.110310
$916$	0	0
$917$	53.2158	1.75734
$918$	0	0
$919$	42.2379	1.39330	0.696649	−	0.717412i	$-0.254673\pi$
$919$	42.2379	1.39330	0.696649	+	0.717412i	$0.254673\pi$
$920$	0	0
$921$	2.87001	0.0945701
$922$	0	0
$923$	−11.5511	−0.380209
$924$	0	0
$925$	−10.6744	−0.350973
$926$	0	0
$927$	−0.603982	−0.0198374
$928$	0	0
$929$	38.0526	1.24847	0.624233	−	0.781238i	$-0.285412\pi$
$929$	38.0526	1.24847	0.624233	+	0.781238i	$0.285412\pi$
$930$	0	0
$931$	−50.8569	−1.66677
$932$	0	0
$933$	18.4668	0.604575
$934$	0	0
$935$	−16.1373	−0.527745
$936$	0	0
$937$	18.6396	0.608930	0.304465	−	0.952523i	$-0.401522\pi$
$937$	18.6396	0.608930	0.304465	+	0.952523i	$0.401522\pi$
$938$	0	0
$939$	3.91112	0.127634
$940$	0	0
$941$	−14.1267	−0.460516	−0.230258	−	0.973130i	$-0.573957\pi$
$941$	−14.1267	−0.460516	−0.230258	+	0.973130i	$0.573957\pi$
$942$	0	0
$943$	31.0634	1.01156
$944$	0	0
$945$	−7.34946	−0.239078
$946$	0	0
$947$	24.2258	0.787234	0.393617	−	0.919275i	$-0.371224\pi$
$947$	24.2258	0.787234	0.393617	+	0.919275i	$0.371224\pi$
$948$	0	0
$949$	7.01839	0.227827
$950$	0	0
$951$	−25.7472	−0.834910
$952$	0	0
$953$	−48.4038	−1.56795	−0.783976	−	0.620791i	$-0.786812\pi$
$953$	−48.4038	−1.56795	−0.783976	+	0.620791i	$0.786812\pi$
$954$	0	0
$955$	−4.67236	−0.151194
$956$	0	0
$957$	−1.78053	−0.0575564
$958$	0	0
$959$	−54.5861	−1.76268
$960$	0	0
$961$	−1.46071	−0.0471198
$962$	0	0
$963$	13.4852	0.434553
$964$	0	0
$965$	21.4232	0.689637
$966$	0	0
$967$	0.477328	0.0153498	0.00767492	−	0.999971i	$-0.497557\pi$
$967$	0.477328	0.0153498	0.00767492	+	0.999971i	$0.497557\pi$
$968$	0	0
$969$	7.30290	0.234603
$970$	0	0
$971$	−41.9201	−1.34528	−0.672640	−	0.739970i	$-0.734840\pi$
$971$	−41.9201	−1.34528	−0.672640	+	0.739970i	$0.734840\pi$
$972$	0	0
$973$	−36.4129	−1.16734
$974$	0	0
$975$	−2.52177	−0.0807613
$976$	0	0
$977$	−8.54409	−0.273350	−0.136675	−	0.990616i	$-0.543642\pi$
$977$	−8.54409	−0.273350	−0.136675	+	0.990616i	$0.543642\pi$
$978$	0	0
$979$	12.2304	0.390885
$980$	0	0
$981$	−15.1653	−0.484192
$982$	0	0
$983$	10.6617	0.340054	0.170027	−	0.985439i	$-0.445615\pi$
$983$	10.6617	0.340054	0.170027	+	0.985439i	$0.445615\pi$
$984$	0	0
$985$	−6.83609	−0.217816
$986$	0	0
$987$	4.42168	0.140744
$988$	0	0
$989$	50.3349	1.60056
$990$	0	0
$991$	−5.47218	−0.173829	−0.0869147	−	0.996216i	$-0.527701\pi$
$991$	−5.47218	−0.173829	−0.0869147	+	0.996216i	$0.527701\pi$
$992$	0	0
$993$	−23.6469	−0.750412
$994$	0	0
$995$	−41.5513	−1.31727
$996$	0	0
$997$	49.9725	1.58265	0.791323	−	0.611399i	$-0.209393\pi$
$997$	49.9725	1.58265	0.791323	+	0.611399i	$0.209393\pi$
$998$	0	0
$999$	4.86244	0.153841

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6096.2.a.ba.1.3		4
4.3	odd	2		3048.2.a.h.1.3	✓	4
12.11	even	2		9144.2.a.s.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3048.2.a.h.1.3	✓	4	4.3	odd	2
6096.2.a.ba.1.3		4	1.1	even	1	trivial
9144.2.a.s.1.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6096 = 2^{4} \cdot 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6096.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.67473 q^{5} -4.38845 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.67473 q^{5} -4.38845 q^{7} +1.00000 q^{9} +5.47399 q^{11} +1.14872 q^{13} +1.67473 q^{15} -1.76028 q^{17} -4.14872 q^{19} -4.38845 q^{21} -5.67473 q^{23} -2.19528 q^{25} +1.00000 q^{27} -0.325271 q^{29} -5.43501 q^{31} +5.47399 q^{33} -7.34946 q^{35} +4.86244 q^{37} +1.14872 q^{39} -5.47399 q^{41} -8.87001 q^{43} +1.67473 q^{45} -1.00757 q^{47} +12.2585 q^{49} -1.76028 q^{51} -11.9722 q^{53} +9.16746 q^{55} -4.14872 q^{57} -11.0112 q^{59} +1.99243 q^{61} -4.38845 q^{63} +1.92380 q^{65} +1.34946 q^{67} -5.67473 q^{69} -10.0556 q^{71} +6.10973 q^{73} -2.19528 q^{75} -24.0223 q^{77} +11.1209 q^{79} +1.00000 q^{81} +4.71372 q^{83} -2.94799 q^{85} -0.325271 q^{87} +2.23427 q^{89} -5.04111 q^{91} -5.43501 q^{93} -6.94799 q^{95} +10.9480 q^{97} +5.47399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + q^{5} - 5 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + q^5 - 5 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + q^{5} - 5 q^{7} + 4 q^{9} + 9 q^{11} - 14 q^{13} + q^{15} - q^{17} + 2 q^{19} - 5 q^{21} - 17 q^{23} + 9 q^{25} + 4 q^{27} - 7 q^{29} - 10 q^{31} + 9 q^{33} - 18 q^{35} - 6 q^{37} - 14 q^{39} - 9 q^{41} - 12 q^{43} + q^{45} - 6 q^{47} + 13 q^{49} - q^{51} - 5 q^{53} - 24 q^{55} + 2 q^{57} + 6 q^{61} - 5 q^{63} - q^{65} - 6 q^{67} - 17 q^{69} - 20 q^{71} + 7 q^{73} + 9 q^{75} - 8 q^{77} - 17 q^{79} + 4 q^{81} + 12 q^{83} + 14 q^{85} - 7 q^{87} - 10 q^{89} + 4 q^{91} - 10 q^{93} - 2 q^{95} + 18 q^{97} + 9 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + q^5 - 5 * q^7 + 4 * q^9 + 9 * q^11 - 14 * q^13 + q^15 - q^17 + 2 * q^19 - 5 * q^21 - 17 * q^23 + 9 * q^25 + 4 * q^27 - 7 * q^29 - 10 * q^31 + 9 * q^33 - 18 * q^35 - 6 * q^37 - 14 * q^39 - 9 * q^41 - 12 * q^43 + q^45 - 6 * q^47 + 13 * q^49 - q^51 - 5 * q^53 - 24 * q^55 + 2 * q^57 + 6 * q^61 - 5 * q^63 - q^65 - 6 * q^67 - 17 * q^69 - 20 * q^71 + 7 * q^73 + 9 * q^75 - 8 * q^77 - 17 * q^79 + 4 * q^81 + 12 * q^83 + 14 * q^85 - 7 * q^87 - 10 * q^89 + 4 * q^91 - 10 * q^93 - 2 * q^95 + 18 * q^97 + 9 * q^99

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