LMFDB - Embedded newform 3072.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.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:	$24.5300435009$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 48)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.27133$ of defining polynomial
Character	$\chi$	$=$	3072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.79793 q^{5} +0.158942 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.79793 q^{5} +0.158942 q^{7} +1.00000 q^{9} +5.37109 q^{11} +5.95687 q^{13} +1.79793 q^{15} -3.05320 q^{17} -3.05320 q^{19} -0.158942 q^{21} +2.82843 q^{23} -1.76744 q^{25} -1.00000 q^{27} +2.96951 q^{29} -4.15894 q^{31} -5.37109 q^{33} -0.285766 q^{35} +8.46742 q^{37} -5.95687 q^{39} +2.60365 q^{41} -8.13853 q^{43} -1.79793 q^{45} -2.82843 q^{47} -6.97474 q^{49} +3.05320 q^{51} +5.03049 q^{53} -9.65685 q^{55} +3.05320 q^{57} +5.65685 q^{59} +5.18944 q^{61} +0.158942 q^{63} -10.7101 q^{65} -1.08532 q^{67} -2.82843 q^{69} -0.317883 q^{71} +1.33897 q^{73} +1.76744 q^{75} +0.853690 q^{77} -9.69382 q^{79} +1.00000 q^{81} +0.163788 q^{83} +5.48946 q^{85} -2.96951 q^{87} +14.3990 q^{89} +0.946795 q^{91} +4.15894 q^{93} +5.48946 q^{95} -0.571533 q^{97} +5.37109 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 8 q^{13} - 4 q^{15} + 4 q^{21} + 4 q^{25} - 4 q^{27} + 12 q^{29} - 12 q^{31} + 16 q^{37} - 8 q^{39} + 4 q^{45} + 4 q^{49} + 20 q^{53} - 16 q^{55} + 16 q^{61} - 4 q^{63} - 8 q^{65} + 16 q^{67} + 8 q^{71} - 8 q^{73} - 4 q^{75} + 24 q^{77} - 12 q^{79} + 4 q^{81} + 24 q^{85} - 12 q^{87} - 8 q^{89} + 16 q^{91} + 12 q^{93} + 24 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 + 8 * q^13 - 4 * q^15 + 4 * q^21 + 4 * q^25 - 4 * q^27 + 12 * q^29 - 12 * q^31 + 16 * q^37 - 8 * q^39 + 4 * q^45 + 4 * q^49 + 20 * q^53 - 16 * q^55 + 16 * q^61 - 4 * q^63 - 8 * q^65 + 16 * q^67 + 8 * q^71 - 8 * q^73 - 4 * q^75 + 24 * q^77 - 12 * q^79 + 4 * q^81 + 24 * q^85 - 12 * q^87 - 8 * q^89 + 16 * q^91 + 12 * q^93 + 24 * q^95

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
$5$	−1.79793	−0.804060	−0.402030	−	0.915627i	$-0.631695\pi$
$5$	−1.79793	−0.804060	−0.402030	+	0.915627i	$0.631695\pi$
$6$	0	0
$7$	0.158942	0.0600743	0.0300371	−	0.999549i	$-0.490437\pi$
$7$	0.158942	0.0600743	0.0300371	+	0.999549i	$0.490437\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.37109	1.61944	0.809722	−	0.586814i	$-0.199618\pi$
$11$	5.37109	1.61944	0.809722	+	0.586814i	$0.199618\pi$
$12$	0	0
$13$	5.95687	1.65214	0.826070	−	0.563568i	$-0.190572\pi$
$13$	5.95687	1.65214	0.826070	+	0.563568i	$0.190572\pi$
$14$	0	0
$15$	1.79793	0.464224
$16$	0	0
$17$	−3.05320	−0.740511	−0.370255	−	0.928930i	$-0.620730\pi$
$17$	−3.05320	−0.740511	−0.370255	+	0.928930i	$0.620730\pi$
$18$	0	0
$19$	−3.05320	−0.700453	−0.350227	−	0.936665i	$-0.613895\pi$
$19$	−3.05320	−0.700453	−0.350227	+	0.936665i	$0.613895\pi$
$20$	0	0
$21$	−0.158942	−0.0346839
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	−1.76744	−0.353488
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.96951	0.551423	0.275712	−	0.961240i	$-0.411087\pi$
$29$	2.96951	0.551423	0.275712	+	0.961240i	$0.411087\pi$
$30$	0	0
$31$	−4.15894	−0.746968	−0.373484	−	0.927637i	$-0.621837\pi$
$31$	−4.15894	−0.746968	−0.373484	+	0.927637i	$0.621837\pi$
$32$	0	0
$33$	−5.37109	−0.934986
$34$	0	0
$35$	−0.285766	−0.0483033
$36$	0	0
$37$	8.46742	1.39203	0.696017	−	0.718025i	$-0.254954\pi$
$37$	8.46742	1.39203	0.696017	+	0.718025i	$0.254954\pi$
$38$	0	0
$39$	−5.95687	−0.953863
$40$	0	0
$41$	2.60365	0.406622	0.203311	−	0.979114i	$-0.434830\pi$
$41$	2.60365	0.406622	0.203311	+	0.979114i	$0.434830\pi$
$42$	0	0
$43$	−8.13853	−1.24111	−0.620557	−	0.784162i	$-0.713093\pi$
$43$	−8.13853	−1.24111	−0.620557	+	0.784162i	$0.713093\pi$
$44$	0	0
$45$	−1.79793	−0.268020
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	−6.97474	−0.996391
$50$	0	0
$51$	3.05320	0.427534
$52$	0	0
$53$	5.03049	0.690992	0.345496	−	0.938420i	$-0.387711\pi$
$53$	5.03049	0.690992	0.345496	+	0.938420i	$0.387711\pi$
$54$	0	0
$55$	−9.65685	−1.30213
$56$	0	0
$57$	3.05320	0.404407
$58$	0	0
$59$	5.65685	0.736460	0.368230	−	0.929735i	$-0.379964\pi$
$59$	5.65685	0.736460	0.368230	+	0.929735i	$0.379964\pi$
$60$	0	0
$61$	5.18944	0.664439	0.332220	−	0.943202i	$-0.392202\pi$
$61$	5.18944	0.664439	0.332220	+	0.943202i	$0.392202\pi$
$62$	0	0
$63$	0.158942	0.0200248
$64$	0	0
$65$	−10.7101	−1.32842
$66$	0	0
$67$	−1.08532	−0.132593	−0.0662966	−	0.997800i	$-0.521118\pi$
$67$	−1.08532	−0.132593	−0.0662966	+	0.997800i	$0.521118\pi$
$68$	0	0
$69$	−2.82843	−0.340503
$70$	0	0
$71$	−0.317883	−0.0377258	−0.0188629	−	0.999822i	$-0.506005\pi$
$71$	−0.317883	−0.0377258	−0.0188629	+	0.999822i	$0.506005\pi$
$72$	0	0
$73$	1.33897	0.156715	0.0783573	−	0.996925i	$-0.475032\pi$
$73$	1.33897	0.156715	0.0783573	+	0.996925i	$0.475032\pi$
$74$	0	0
$75$	1.76744	0.204086
$76$	0	0
$77$	0.853690	0.0972870
$78$	0	0
$79$	−9.69382	−1.09064	−0.545320	−	0.838228i	$-0.683592\pi$
$79$	−9.69382	−1.09064	−0.545320	+	0.838228i	$0.683592\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.163788	0.0179781	0.00898906	−	0.999960i	$-0.497139\pi$
$83$	0.163788	0.0179781	0.00898906	+	0.999960i	$0.497139\pi$
$84$	0	0
$85$	5.48946	0.595415
$86$	0	0
$87$	−2.96951	−0.318364
$88$	0	0
$89$	14.3990	1.52629	0.763147	−	0.646225i	$-0.223653\pi$
$89$	14.3990	1.52629	0.763147	+	0.646225i	$0.223653\pi$
$90$	0	0
$91$	0.946795	0.0992511
$92$	0	0
$93$	4.15894	0.431262
$94$	0	0
$95$	5.48946	0.563206
$96$	0	0
$97$	−0.571533	−0.0580304	−0.0290152	−	0.999579i	$-0.509237\pi$
$97$	−0.571533	−0.0580304	−0.0290152	+	0.999579i	$0.509237\pi$
$98$	0	0
$99$	5.37109	0.539815
$100$	0	0
$101$	10.1158	1.00656	0.503281	−	0.864123i	$-0.332126\pi$
$101$	10.1158	1.00656	0.503281	+	0.864123i	$0.332126\pi$
$102$	0	0
$103$	11.3507	1.11841	0.559207	−	0.829028i	$-0.311106\pi$
$103$	11.3507	1.11841	0.559207	+	0.829028i	$0.311106\pi$
$104$	0	0
$105$	0.285766	0.0278879
$106$	0	0
$107$	1.02109	0.0987123	0.0493561	−	0.998781i	$-0.484283\pi$
$107$	1.02109	0.0987123	0.0493561	+	0.998781i	$0.484283\pi$
$108$	0	0
$109$	2.04313	0.195696	0.0978480	−	0.995201i	$-0.468804\pi$
$109$	2.04313	0.195696	0.0978480	+	0.995201i	$0.468804\pi$
$110$	0	0
$111$	−8.46742	−0.803692
$112$	0	0
$113$	3.53488	0.332533	0.166267	−	0.986081i	$-0.446829\pi$
$113$	3.53488	0.332533	0.166267	+	0.986081i	$0.446829\pi$
$114$	0	0
$115$	−5.08532	−0.474209
$116$	0	0
$117$	5.95687	0.550713
$118$	0	0
$119$	−0.485281	−0.0444857
$120$	0	0
$121$	17.8486	1.62260
$122$	0	0
$123$	−2.60365	−0.234763
$124$	0	0
$125$	12.1674	1.08829
$126$	0	0
$127$	1.49791	0.132918	0.0664591	−	0.997789i	$-0.478830\pi$
$127$	1.49791	0.132918	0.0664591	+	0.997789i	$0.478830\pi$
$128$	0	0
$129$	8.13853	0.716557
$130$	0	0
$131$	−14.7422	−1.28803	−0.644015	−	0.765013i	$-0.722733\pi$
$131$	−14.7422	−1.28803	−0.644015	+	0.765013i	$0.722733\pi$
$132$	0	0
$133$	−0.485281	−0.0420792
$134$	0	0
$135$	1.79793	0.154741
$136$	0	0
$137$	13.7954	1.17862	0.589309	−	0.807907i	$-0.299400\pi$
$137$	13.7954	1.17862	0.589309	+	0.807907i	$0.299400\pi$
$138$	0	0
$139$	3.42847	0.290799	0.145399	−	0.989373i	$-0.453553\pi$
$139$	3.42847	0.290799	0.145399	+	0.989373i	$0.453553\pi$
$140$	0	0
$141$	2.82843	0.238197
$142$	0	0
$143$	31.9949	2.67555
$144$	0	0
$145$	−5.33897	−0.443377
$146$	0	0
$147$	6.97474	0.575267
$148$	0	0
$149$	4.14108	0.339250	0.169625	−	0.985509i	$-0.445744\pi$
$149$	4.14108	0.339250	0.169625	+	0.985509i	$0.445744\pi$
$150$	0	0
$151$	22.6644	1.84440	0.922201	−	0.386712i	$-0.126389\pi$
$151$	22.6644	1.84440	0.922201	+	0.386712i	$0.126389\pi$
$152$	0	0
$153$	−3.05320	−0.246837
$154$	0	0
$155$	7.47750	0.600607
$156$	0	0
$157$	−3.93161	−0.313777	−0.156888	−	0.987616i	$-0.550146\pi$
$157$	−3.93161	−0.313777	−0.156888	+	0.987616i	$0.550146\pi$
$158$	0	0
$159$	−5.03049	−0.398944
$160$	0	0
$161$	0.449555	0.0354299
$162$	0	0
$163$	7.68897	0.602247	0.301123	−	0.953585i	$-0.402638\pi$
$163$	7.68897	0.602247	0.301123	+	0.953585i	$0.402638\pi$
$164$	0	0
$165$	9.65685	0.751785
$166$	0	0
$167$	3.95458	0.306015	0.153007	−	0.988225i	$-0.451104\pi$
$167$	3.95458	0.306015	0.153007	+	0.988225i	$0.451104\pi$
$168$	0	0
$169$	22.4844	1.72957
$170$	0	0
$171$	−3.05320	−0.233484
$172$	0	0
$173$	22.6011	1.71833	0.859165	−	0.511699i	$-0.170984\pi$
$173$	22.6011	1.71833	0.859165	+	0.511699i	$0.170984\pi$
$174$	0	0
$175$	−0.280920	−0.0212355
$176$	0	0
$177$	−5.65685	−0.425195
$178$	0	0
$179$	−17.2981	−1.29292	−0.646462	−	0.762946i	$-0.723752\pi$
$179$	−17.2981	−1.29292	−0.646462	+	0.762946i	$0.723752\pi$
$180$	0	0
$181$	8.14953	0.605750	0.302875	−	0.953030i	$-0.402054\pi$
$181$	8.14953	0.605750	0.302875	+	0.953030i	$0.402054\pi$
$182$	0	0
$183$	−5.18944	−0.383614
$184$	0	0
$185$	−15.2238	−1.11928
$186$	0	0
$187$	−16.3990	−1.19922
$188$	0	0
$189$	−0.158942	−0.0115613
$190$	0	0
$191$	16.1674	1.16983	0.584916	−	0.811094i	$-0.301127\pi$
$191$	16.1674	1.16983	0.584916	+	0.811094i	$0.301127\pi$
$192$	0	0
$193$	−22.1454	−1.59406	−0.797030	−	0.603940i	$-0.793597\pi$
$193$	−22.1454	−1.59406	−0.797030	+	0.603940i	$0.793597\pi$
$194$	0	0
$195$	10.7101	0.766963
$196$	0	0
$197$	20.2222	1.44077	0.720387	−	0.693572i	$-0.243964\pi$
$197$	20.2222	1.44077	0.720387	+	0.693572i	$0.243964\pi$
$198$	0	0
$199$	−25.0075	−1.77274	−0.886368	−	0.462981i	$-0.846780\pi$
$199$	−25.0075	−1.77274	−0.886368	+	0.462981i	$0.846780\pi$
$200$	0	0
$201$	1.08532	0.0765527
$202$	0	0
$203$	0.471978	0.0331264
$204$	0	0
$205$	−4.68119	−0.326948
$206$	0	0
$207$	2.82843	0.196589
$208$	0	0
$209$	−16.3990	−1.13434
$210$	0	0
$211$	26.0559	1.79376	0.896881	−	0.442273i	$-0.145828\pi$
$211$	26.0559	1.79376	0.896881	+	0.442273i	$0.145828\pi$
$212$	0	0
$213$	0.317883	0.0217810
$214$	0	0
$215$	14.6325	0.997930
$216$	0	0
$217$	−0.661029	−0.0448736
$218$	0	0
$219$	−1.33897	−0.0904793
$220$	0	0
$221$	−18.1876	−1.22343
$222$	0	0
$223$	−18.3465	−1.22857	−0.614286	−	0.789083i	$-0.710556\pi$
$223$	−18.3465	−1.22857	−0.614286	+	0.789083i	$0.710556\pi$
$224$	0	0
$225$	−1.76744	−0.117829
$226$	0	0
$227$	0.163788	0.0108710	0.00543551	−	0.999985i	$-0.498270\pi$
$227$	0.163788	0.0108710	0.00543551	+	0.999985i	$0.498270\pi$
$228$	0	0
$229$	−4.02756	−0.266148	−0.133074	−	0.991106i	$-0.542485\pi$
$229$	−4.02756	−0.266148	−0.133074	+	0.991106i	$0.542485\pi$
$230$	0	0
$231$	−0.853690	−0.0561687
$232$	0	0
$233$	−11.7211	−0.767874	−0.383937	−	0.923359i	$-0.625432\pi$
$233$	−11.7211	−0.767874	−0.383937	+	0.923359i	$0.625432\pi$
$234$	0	0
$235$	5.08532	0.331730
$236$	0	0
$237$	9.69382	0.629681
$238$	0	0
$239$	−13.6517	−0.883058	−0.441529	−	0.897247i	$-0.645564\pi$
$239$	−13.6517	−0.883058	−0.441529	+	0.897247i	$0.645564\pi$
$240$	0	0
$241$	−2.13167	−0.137313	−0.0686565	−	0.997640i	$-0.521871\pi$
$241$	−2.13167	−0.137313	−0.0686565	+	0.997640i	$0.521871\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	12.5401	0.801158
$246$	0	0
$247$	−18.1876	−1.15725
$248$	0	0
$249$	−0.163788	−0.0103797
$250$	0	0
$251$	−6.27020	−0.395771	−0.197886	−	0.980225i	$-0.563407\pi$
$251$	−6.27020	−0.395771	−0.197886	+	0.980225i	$0.563407\pi$
$252$	0	0
$253$	15.1917	0.955096
$254$	0	0
$255$	−5.48946	−0.343763
$256$	0	0
$257$	15.0853	0.940997	0.470498	−	0.882401i	$-0.344074\pi$
$257$	15.0853	0.940997	0.470498	+	0.882401i	$0.344074\pi$
$258$	0	0
$259$	1.34583	0.0836255
$260$	0	0
$261$	2.96951	0.183808
$262$	0	0
$263$	26.1706	1.61375	0.806875	−	0.590722i	$-0.201157\pi$
$263$	26.1706	1.61375	0.806875	+	0.590722i	$0.201157\pi$
$264$	0	0
$265$	−9.04449	−0.555599
$266$	0	0
$267$	−14.3990	−0.881206
$268$	0	0
$269$	−12.1580	−0.741286	−0.370643	−	0.928775i	$-0.620863\pi$
$269$	−12.1580	−0.741286	−0.370643	+	0.928775i	$0.620863\pi$
$270$	0	0
$271$	10.6644	0.647815	0.323907	−	0.946089i	$-0.395003\pi$
$271$	10.6644	0.647815	0.323907	+	0.946089i	$0.395003\pi$
$272$	0	0
$273$	−0.946795	−0.0573027
$274$	0	0
$275$	−9.49307	−0.572453
$276$	0	0
$277$	−3.76421	−0.226170	−0.113085	−	0.993585i	$-0.536073\pi$
$277$	−3.76421	−0.226170	−0.113085	+	0.993585i	$0.536073\pi$
$278$	0	0
$279$	−4.15894	−0.248989
$280$	0	0
$281$	10.4496	0.623368	0.311684	−	0.950186i	$-0.399107\pi$
$281$	10.4496	0.623368	0.311684	+	0.950186i	$0.399107\pi$
$282$	0	0
$283$	−17.6569	−1.04959	−0.524796	−	0.851228i	$-0.675858\pi$
$283$	−17.6569	−1.04959	−0.524796	+	0.851228i	$0.675858\pi$
$284$	0	0
$285$	−5.48946	−0.325167
$286$	0	0
$287$	0.413828	0.0244275
$288$	0	0
$289$	−7.67794	−0.451644
$290$	0	0
$291$	0.571533	0.0335038
$292$	0	0
$293$	−30.7465	−1.79623	−0.898114	−	0.439762i	$-0.855063\pi$
$293$	−30.7465	−1.79623	−0.898114	+	0.439762i	$0.855063\pi$
$294$	0	0
$295$	−10.1706	−0.592158
$296$	0	0
$297$	−5.37109	−0.311662
$298$	0	0
$299$	16.8486	0.974379
$300$	0	0
$301$	−1.29355	−0.0745590
$302$	0	0
$303$	−10.1158	−0.581138
$304$	0	0
$305$	−9.33026	−0.534249
$306$	0	0
$307$	21.2981	1.21555	0.607775	−	0.794110i	$-0.292062\pi$
$307$	21.2981	1.21555	0.607775	+	0.794110i	$0.292062\pi$
$308$	0	0
$309$	−11.3507	−0.645717
$310$	0	0
$311$	−1.77883	−0.100868	−0.0504342	−	0.998727i	$-0.516061\pi$
$311$	−1.77883	−0.100868	−0.0504342	+	0.998727i	$0.516061\pi$
$312$	0	0
$313$	−2.70320	−0.152794	−0.0763971	−	0.997077i	$-0.524342\pi$
$313$	−2.70320	−0.152794	−0.0763971	+	0.997077i	$0.524342\pi$
$314$	0	0
$315$	−0.285766	−0.0161011
$316$	0	0
$317$	22.0653	1.23931	0.619655	−	0.784874i	$-0.287272\pi$
$317$	22.0653	1.23931	0.619655	+	0.784874i	$0.287272\pi$
$318$	0	0
$319$	15.9495	0.892999
$320$	0	0
$321$	−1.02109	−0.0569916
$322$	0	0
$323$	9.32206	0.518693
$324$	0	0
$325$	−10.5284	−0.584011
$326$	0	0
$327$	−2.04313	−0.112985
$328$	0	0
$329$	−0.449555	−0.0247848
$330$	0	0
$331$	−21.8431	−1.20060	−0.600302	−	0.799774i	$-0.704953\pi$
$331$	−21.8431	−1.20060	−0.600302	+	0.799774i	$0.704953\pi$
$332$	0	0
$333$	8.46742	0.464012
$334$	0	0
$335$	1.95133	0.106613
$336$	0	0
$337$	18.8738	1.02812	0.514062	−	0.857753i	$-0.328140\pi$
$337$	18.8738	1.02812	0.514062	+	0.857753i	$0.328140\pi$
$338$	0	0
$339$	−3.53488	−0.191988
$340$	0	0
$341$	−22.3380	−1.20967
$342$	0	0
$343$	−2.22117	−0.119932
$344$	0	0
$345$	5.08532	0.273785
$346$	0	0
$347$	28.0490	1.50575	0.752875	−	0.658163i	$-0.228666\pi$
$347$	28.0490	1.50575	0.752875	+	0.658163i	$0.228666\pi$
$348$	0	0
$349$	16.9307	0.906279	0.453139	−	0.891440i	$-0.350304\pi$
$349$	16.9307	0.906279	0.453139	+	0.891440i	$0.350304\pi$
$350$	0	0
$351$	−5.95687	−0.317954
$352$	0	0
$353$	−12.6202	−0.671705	−0.335853	−	0.941915i	$-0.609024\pi$
$353$	−12.6202	−0.671705	−0.335853	+	0.941915i	$0.609024\pi$
$354$	0	0
$355$	0.571533	0.0303338
$356$	0	0
$357$	0.485281	0.0256838
$358$	0	0
$359$	27.0867	1.42958	0.714790	−	0.699339i	$-0.246522\pi$
$359$	27.0867	1.42958	0.714790	+	0.699339i	$0.246522\pi$
$360$	0	0
$361$	−9.67794	−0.509365
$362$	0	0
$363$	−17.8486	−0.936808
$364$	0	0
$365$	−2.40738	−0.126008
$366$	0	0
$367$	−20.4937	−1.06976	−0.534882	−	0.844927i	$-0.679644\pi$
$367$	−20.4937	−1.06976	−0.534882	+	0.844927i	$0.679644\pi$
$368$	0	0
$369$	2.60365	0.135541
$370$	0	0
$371$	0.799555	0.0415108
$372$	0	0
$373$	1.46190	0.0756943	0.0378471	−	0.999284i	$-0.487950\pi$
$373$	1.46190	0.0756943	0.0378471	+	0.999284i	$0.487950\pi$
$374$	0	0
$375$	−12.1674	−0.628322
$376$	0	0
$377$	17.6890	0.911028
$378$	0	0
$379$	24.9871	1.28350	0.641751	−	0.766913i	$-0.278208\pi$
$379$	24.9871	1.28350	0.641751	+	0.766913i	$0.278208\pi$
$380$	0	0
$381$	−1.49791	−0.0767404
$382$	0	0
$383$	31.0958	1.58892	0.794460	−	0.607316i	$-0.207754\pi$
$383$	31.0958	1.58892	0.794460	+	0.607316i	$0.207754\pi$
$384$	0	0
$385$	−1.53488	−0.0782245
$386$	0	0
$387$	−8.13853	−0.413705
$388$	0	0
$389$	3.62218	0.183652	0.0918260	−	0.995775i	$-0.470730\pi$
$389$	3.62218	0.183652	0.0918260	+	0.995775i	$0.470730\pi$
$390$	0	0
$391$	−8.63577	−0.436729
$392$	0	0
$393$	14.7422	0.743644
$394$	0	0
$395$	17.4288	0.876940
$396$	0	0
$397$	7.20959	0.361839	0.180920	−	0.983498i	$-0.442093\pi$
$397$	7.20959	0.361839	0.180920	+	0.983498i	$0.442093\pi$
$398$	0	0
$399$	0.485281	0.0242945
$400$	0	0
$401$	15.2660	0.762349	0.381174	−	0.924503i	$-0.375520\pi$
$401$	15.2660	0.762349	0.381174	+	0.924503i	$0.375520\pi$
$402$	0	0
$403$	−24.7743	−1.23410
$404$	0	0
$405$	−1.79793	−0.0893400
$406$	0	0
$407$	45.4792	2.25432
$408$	0	0
$409$	−11.3779	−0.562603	−0.281302	−	0.959619i	$-0.590766\pi$
$409$	−11.3779	−0.562603	−0.281302	+	0.959619i	$0.590766\pi$
$410$	0	0
$411$	−13.7954	−0.680476
$412$	0	0
$413$	0.899110	0.0442423
$414$	0	0
$415$	−0.294481	−0.0144555
$416$	0	0
$417$	−3.42847	−0.167893
$418$	0	0
$419$	−32.9618	−1.61029	−0.805146	−	0.593077i	$-0.797913\pi$
$419$	−32.9618	−1.61029	−0.805146	+	0.593077i	$0.797913\pi$
$420$	0	0
$421$	24.9119	1.21413	0.607065	−	0.794652i	$-0.292347\pi$
$421$	24.9119	1.21413	0.607065	+	0.794652i	$0.292347\pi$
$422$	0	0
$423$	−2.82843	−0.137523
$424$	0	0
$425$	5.39635	0.261761
$426$	0	0
$427$	0.824818	0.0399157
$428$	0	0
$429$	−31.9949	−1.54473
$430$	0	0
$431$	10.3211	0.497151	0.248576	−	0.968612i	$-0.420038\pi$
$431$	10.3211	0.497151	0.248576	+	0.968612i	$0.420038\pi$
$432$	0	0
$433$	15.3137	0.735930	0.367965	−	0.929840i	$-0.380055\pi$
$433$	15.3137	0.735930	0.367965	+	0.929840i	$0.380055\pi$
$434$	0	0
$435$	5.33897	0.255984
$436$	0	0
$437$	−8.63577	−0.413105
$438$	0	0
$439$	−22.5735	−1.07738	−0.538688	−	0.842505i	$-0.681080\pi$
$439$	−22.5735	−1.07738	−0.538688	+	0.842505i	$0.681080\pi$
$440$	0	0
$441$	−6.97474	−0.332130
$442$	0	0
$443$	33.5334	1.59322	0.796610	−	0.604494i	$-0.206625\pi$
$443$	33.5334	1.59322	0.796610	+	0.604494i	$0.206625\pi$
$444$	0	0
$445$	−25.8885	−1.22723
$446$	0	0
$447$	−4.14108	−0.195866
$448$	0	0
$449$	−1.75506	−0.0828266	−0.0414133	−	0.999142i	$-0.513186\pi$
$449$	−1.75506	−0.0828266	−0.0414133	+	0.999142i	$0.513186\pi$
$450$	0	0
$451$	13.9844	0.658501
$452$	0	0
$453$	−22.6644	−1.06487
$454$	0	0
$455$	−1.70227	−0.0798039
$456$	0	0
$457$	−26.7422	−1.25095	−0.625473	−	0.780246i	$-0.715094\pi$
$457$	−26.7422	−1.25095	−0.625473	+	0.780246i	$0.715094\pi$
$458$	0	0
$459$	3.05320	0.142511
$460$	0	0
$461$	−13.0662	−0.608555	−0.304277	−	0.952584i	$-0.598415\pi$
$461$	−13.0662	−0.608555	−0.304277	+	0.952584i	$0.598415\pi$
$462$	0	0
$463$	29.4474	1.36854	0.684268	−	0.729231i	$-0.260122\pi$
$463$	29.4474	1.36854	0.684268	+	0.729231i	$0.260122\pi$
$464$	0	0
$465$	−7.47750	−0.346761
$466$	0	0
$467$	−27.7040	−1.28199	−0.640995	−	0.767545i	$-0.721478\pi$
$467$	−27.7040	−1.28199	−0.640995	+	0.767545i	$0.721478\pi$
$468$	0	0
$469$	−0.172503	−0.00796544
$470$	0	0
$471$	3.93161	0.181159
$472$	0	0
$473$	−43.7127	−2.00991
$474$	0	0
$475$	5.39635	0.247602
$476$	0	0
$477$	5.03049	0.230331
$478$	0	0
$479$	−35.5499	−1.62432	−0.812159	−	0.583436i	$-0.801708\pi$
$479$	−35.5499	−1.62432	−0.812159	+	0.583436i	$0.801708\pi$
$480$	0	0
$481$	50.4393	2.29984
$482$	0	0
$483$	−0.449555	−0.0204555
$484$	0	0
$485$	1.02758	0.0466599
$486$	0	0
$487$	−9.86632	−0.447086	−0.223543	−	0.974694i	$-0.571762\pi$
$487$	−9.86632	−0.447086	−0.223543	+	0.974694i	$0.571762\pi$
$488$	0	0
$489$	−7.68897	−0.347707
$490$	0	0
$491$	0.635767	0.0286917	0.0143459	−	0.999897i	$-0.495433\pi$
$491$	0.635767	0.0286917	0.0143459	+	0.999897i	$0.495433\pi$
$492$	0	0
$493$	−9.06651	−0.408335
$494$	0	0
$495$	−9.65685	−0.434043
$496$	0	0
$497$	−0.0505249	−0.00226635
$498$	0	0
$499$	3.82750	0.171342	0.0856712	−	0.996323i	$-0.472697\pi$
$499$	3.82750	0.171342	0.0856712	+	0.996323i	$0.472697\pi$
$500$	0	0
$501$	−3.95458	−0.176678
$502$	0	0
$503$	−23.6719	−1.05548	−0.527739	−	0.849407i	$-0.676960\pi$
$503$	−23.6719	−1.05548	−0.527739	+	0.849407i	$0.676960\pi$
$504$	0	0
$505$	−18.1876	−0.809336
$506$	0	0
$507$	−22.4844	−0.998565
$508$	0	0
$509$	−34.7970	−1.54235	−0.771175	−	0.636623i	$-0.780331\pi$
$509$	−34.7970	−1.54235	−0.771175	+	0.636623i	$0.780331\pi$
$510$	0	0
$511$	0.212818	0.00941453
$512$	0	0
$513$	3.05320	0.134802
$514$	0	0
$515$	−20.4077	−0.899273
$516$	0	0
$517$	−15.1917	−0.668132
$518$	0	0
$519$	−22.6011	−0.992078
$520$	0	0
$521$	−14.4889	−0.634770	−0.317385	−	0.948297i	$-0.602805\pi$
$521$	−14.4889	−0.634770	−0.317385	+	0.948297i	$0.602805\pi$
$522$	0	0
$523$	27.5742	1.20574	0.602868	−	0.797841i	$-0.294025\pi$
$523$	27.5742	1.20574	0.602868	+	0.797841i	$0.294025\pi$
$524$	0	0
$525$	0.280920	0.0122603
$526$	0	0
$527$	12.6981	0.553138
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	5.65685	0.245487
$532$	0	0
$533$	15.5096	0.671796
$534$	0	0
$535$	−1.83585	−0.0793706
$536$	0	0
$537$	17.2981	0.746470
$538$	0	0
$539$	−37.4619	−1.61360
$540$	0	0
$541$	−14.1982	−0.610428	−0.305214	−	0.952284i	$-0.598728\pi$
$541$	−14.1982	−0.610428	−0.305214	+	0.952284i	$0.598728\pi$
$542$	0	0
$543$	−8.14953	−0.349730
$544$	0	0
$545$	−3.67340	−0.157351
$546$	0	0
$547$	10.1807	0.435295	0.217648	−	0.976027i	$-0.430162\pi$
$547$	10.1807	0.435295	0.217648	+	0.976027i	$0.430162\pi$
$548$	0	0
$549$	5.18944	0.221480
$550$	0	0
$551$	−9.06651	−0.386246
$552$	0	0
$553$	−1.54075	−0.0655194
$554$	0	0
$555$	15.2238	0.646216
$556$	0	0
$557$	−1.44432	−0.0611979	−0.0305990	−	0.999532i	$-0.509741\pi$
$557$	−1.44432	−0.0611979	−0.0305990	+	0.999532i	$0.509741\pi$
$558$	0	0
$559$	−48.4802	−2.05049
$560$	0	0
$561$	16.3990	0.692368
$562$	0	0
$563$	9.48585	0.399781	0.199890	−	0.979818i	$-0.435941\pi$
$563$	9.48585	0.399781	0.199890	+	0.979818i	$0.435941\pi$
$564$	0	0
$565$	−6.35547	−0.267377
$566$	0	0
$567$	0.158942	0.00667492
$568$	0	0
$569$	8.98711	0.376759	0.188380	−	0.982096i	$-0.439676\pi$
$569$	8.98711	0.376759	0.188380	+	0.982096i	$0.439676\pi$
$570$	0	0
$571$	−12.9706	−0.542801	−0.271401	−	0.962466i	$-0.587487\pi$
$571$	−12.9706	−0.542801	−0.271401	+	0.962466i	$0.587487\pi$
$572$	0	0
$573$	−16.1674	−0.675403
$574$	0	0
$575$	−4.99907	−0.208476
$576$	0	0
$577$	29.5013	1.22815	0.614077	−	0.789246i	$-0.289528\pi$
$577$	29.5013	1.22815	0.614077	+	0.789246i	$0.289528\pi$
$578$	0	0
$579$	22.1454	0.920331
$580$	0	0
$581$	0.0260328	0.00108002
$582$	0	0
$583$	27.0192	1.11902
$584$	0	0
$585$	−10.7101	−0.442806
$586$	0	0
$587$	−2.57988	−0.106483	−0.0532416	−	0.998582i	$-0.516955\pi$
$587$	−2.57988	−0.106483	−0.0532416	+	0.998582i	$0.516955\pi$
$588$	0	0
$589$	12.6981	0.523216
$590$	0	0
$591$	−20.2222	−0.831831
$592$	0	0
$593$	35.4338	1.45509	0.727546	−	0.686058i	$-0.240661\pi$
$593$	35.4338	1.45509	0.727546	+	0.686058i	$0.240661\pi$
$594$	0	0
$595$	0.872503	0.0357691
$596$	0	0
$597$	25.0075	1.02349
$598$	0	0
$599$	27.1632	1.10986	0.554930	−	0.831897i	$-0.312745\pi$
$599$	27.1632	1.10986	0.554930	+	0.831897i	$0.312745\pi$
$600$	0	0
$601$	−5.33897	−0.217781	−0.108891	−	0.994054i	$-0.534730\pi$
$601$	−5.33897	−0.217781	−0.108891	+	0.994054i	$0.534730\pi$
$602$	0	0
$603$	−1.08532	−0.0441977
$604$	0	0
$605$	−32.0906	−1.30467
$606$	0	0
$607$	−16.1084	−0.653820	−0.326910	−	0.945055i	$-0.606007\pi$
$607$	−16.1084	−0.653820	−0.326910	+	0.945055i	$0.606007\pi$
$608$	0	0
$609$	−0.471978	−0.0191255
$610$	0	0
$611$	−16.8486	−0.681621
$612$	0	0
$613$	−0.617903	−0.0249569	−0.0124784	−	0.999922i	$-0.503972\pi$
$613$	−0.617903	−0.0249569	−0.0124784	+	0.999922i	$0.503972\pi$
$614$	0	0
$615$	4.68119	0.188764
$616$	0	0
$617$	8.80641	0.354533	0.177266	−	0.984163i	$-0.443275\pi$
$617$	8.80641	0.354533	0.177266	+	0.984163i	$0.443275\pi$
$618$	0	0
$619$	−2.72847	−0.109666	−0.0548332	−	0.998496i	$-0.517463\pi$
$619$	−2.72847	−0.109666	−0.0548332	+	0.998496i	$0.517463\pi$
$620$	0	0
$621$	−2.82843	−0.113501
$622$	0	0
$623$	2.28861	0.0916910
$624$	0	0
$625$	−13.0390	−0.521559
$626$	0	0
$627$	16.3990	0.654914
$628$	0	0
$629$	−25.8528	−1.03082
$630$	0	0
$631$	−38.7864	−1.54406	−0.772030	−	0.635586i	$-0.780759\pi$
$631$	−38.7864	−1.54406	−0.772030	+	0.635586i	$0.780759\pi$
$632$	0	0
$633$	−26.0559	−1.03563
$634$	0	0
$635$	−2.69315	−0.106874
$636$	0	0
$637$	−41.5476	−1.64618
$638$	0	0
$639$	−0.317883	−0.0125753
$640$	0	0
$641$	33.1091	1.30773	0.653865	−	0.756611i	$-0.273146\pi$
$641$	33.1091	1.30773	0.653865	+	0.756611i	$0.273146\pi$
$642$	0	0
$643$	27.2797	1.07581	0.537904	−	0.843006i	$-0.319216\pi$
$643$	27.2797	1.07581	0.537904	+	0.843006i	$0.319216\pi$
$644$	0	0
$645$	−14.6325	−0.576155
$646$	0	0
$647$	−41.8477	−1.64520	−0.822601	−	0.568620i	$-0.807478\pi$
$647$	−41.8477	−1.64520	−0.822601	+	0.568620i	$0.807478\pi$
$648$	0	0
$649$	30.3835	1.19266
$650$	0	0
$651$	0.661029	0.0259078
$652$	0	0
$653$	−20.8937	−0.817634	−0.408817	−	0.912616i	$-0.634059\pi$
$653$	−20.8937	−0.817634	−0.408817	+	0.912616i	$0.634059\pi$
$654$	0	0
$655$	26.5054	1.03565
$656$	0	0
$657$	1.33897	0.0522382
$658$	0	0
$659$	−3.15142	−0.122762	−0.0613809	−	0.998114i	$-0.519550\pi$
$659$	−3.15142	−0.122762	−0.0613809	+	0.998114i	$0.519550\pi$
$660$	0	0
$661$	−25.5527	−0.993886	−0.496943	−	0.867783i	$-0.665544\pi$
$661$	−25.5527	−0.993886	−0.496943	+	0.867783i	$0.665544\pi$
$662$	0	0
$663$	18.1876	0.706346
$664$	0	0
$665$	0.872503	0.0338342
$666$	0	0
$667$	8.39903	0.325212
$668$	0	0
$669$	18.3465	0.709317
$670$	0	0
$671$	27.8729	1.07602
$672$	0	0
$673$	20.7981	0.801706	0.400853	−	0.916142i	$-0.368714\pi$
$673$	20.7981	0.801706	0.400853	+	0.916142i	$0.368714\pi$
$674$	0	0
$675$	1.76744	0.0680287
$676$	0	0
$677$	−41.0423	−1.57738	−0.788692	−	0.614789i	$-0.789241\pi$
$677$	−41.0423	−1.57738	−0.788692	+	0.614789i	$0.789241\pi$
$678$	0	0
$679$	−0.0908404	−0.00348613
$680$	0	0
$681$	−0.163788	−0.00627639
$682$	0	0
$683$	26.5784	1.01699	0.508497	−	0.861064i	$-0.330201\pi$
$683$	26.5784	1.01699	0.508497	+	0.861064i	$0.330201\pi$
$684$	0	0
$685$	−24.8032	−0.947680
$686$	0	0
$687$	4.02756	0.153661
$688$	0	0
$689$	29.9660	1.14161
$690$	0	0
$691$	14.7899	0.562633	0.281316	−	0.959615i	$-0.409229\pi$
$691$	14.7899	0.562633	0.281316	+	0.959615i	$0.409229\pi$
$692$	0	0
$693$	0.853690	0.0324290
$694$	0	0
$695$	−6.16415	−0.233820
$696$	0	0
$697$	−7.94948	−0.301108
$698$	0	0
$699$	11.7211	0.443332
$700$	0	0
$701$	25.9245	0.979155	0.489577	−	0.871960i	$-0.337151\pi$
$701$	25.9245	0.979155	0.489577	+	0.871960i	$0.337151\pi$
$702$	0	0
$703$	−25.8528	−0.975055
$704$	0	0
$705$	−5.08532	−0.191524
$706$	0	0
$707$	1.60782	0.0604685
$708$	0	0
$709$	−20.6082	−0.773958	−0.386979	−	0.922089i	$-0.626481\pi$
$709$	−20.6082	−0.773958	−0.386979	+	0.922089i	$0.626481\pi$
$710$	0	0
$711$	−9.69382	−0.363547
$712$	0	0
$713$	−11.7633	−0.440538
$714$	0	0
$715$	−57.5247	−2.15130
$716$	0	0
$717$	13.6517	0.509834
$718$	0	0
$719$	44.0949	1.64446	0.822230	−	0.569155i	$-0.192730\pi$
$719$	44.0949	1.64446	0.822230	+	0.569155i	$0.192730\pi$
$720$	0	0
$721$	1.80409	0.0671880
$722$	0	0
$723$	2.13167	0.0792777
$724$	0	0
$725$	−5.24842	−0.194921
$726$	0	0
$727$	9.23457	0.342491	0.171246	−	0.985228i	$-0.445221\pi$
$727$	9.23457	0.342491	0.171246	+	0.985228i	$0.445221\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.8486	0.919058
$732$	0	0
$733$	25.8467	0.954670	0.477335	−	0.878721i	$-0.341603\pi$
$733$	25.8467	0.954670	0.477335	+	0.878721i	$0.341603\pi$
$734$	0	0
$735$	−12.5401	−0.462549
$736$	0	0
$737$	−5.82936	−0.214727
$738$	0	0
$739$	−24.0403	−0.884337	−0.442169	−	0.896932i	$-0.645791\pi$
$739$	−24.0403	−0.884337	−0.442169	+	0.896932i	$0.645791\pi$
$740$	0	0
$741$	18.1876	0.668137
$742$	0	0
$743$	17.8748	0.655762	0.327881	−	0.944719i	$-0.393665\pi$
$743$	17.8748	0.655762	0.327881	+	0.944719i	$0.393665\pi$
$744$	0	0
$745$	−7.44538	−0.272778
$746$	0	0
$747$	0.163788	0.00599271
$748$	0	0
$749$	0.162293	0.00593007
$750$	0	0
$751$	−35.0731	−1.27984	−0.639918	−	0.768443i	$-0.721032\pi$
$751$	−35.0731	−1.27984	−0.639918	+	0.768443i	$0.721032\pi$
$752$	0	0
$753$	6.27020	0.228499
$754$	0	0
$755$	−40.7490	−1.48301
$756$	0	0
$757$	−46.3962	−1.68630	−0.843150	−	0.537679i	$-0.819301\pi$
$757$	−46.3962	−1.68630	−0.843150	+	0.537679i	$0.819301\pi$
$758$	0	0
$759$	−15.1917	−0.551425
$760$	0	0
$761$	−10.5531	−0.382550	−0.191275	−	0.981536i	$-0.561262\pi$
$761$	−10.5531	−0.382550	−0.191275	+	0.981536i	$0.561262\pi$
$762$	0	0
$763$	0.324738	0.0117563
$764$	0	0
$765$	5.48946	0.198472
$766$	0	0
$767$	33.6972	1.21673
$768$	0	0
$769$	−35.2068	−1.26959	−0.634795	−	0.772681i	$-0.718915\pi$
$769$	−35.2068	−1.26959	−0.634795	+	0.772681i	$0.718915\pi$
$770$	0	0
$771$	−15.0853	−0.543285
$772$	0	0
$773$	27.4212	0.986271	0.493136	−	0.869952i	$-0.335851\pi$
$773$	27.4212	0.986271	0.493136	+	0.869952i	$0.335851\pi$
$774$	0	0
$775$	7.35067	0.264044
$776$	0	0
$777$	−1.34583	−0.0482812
$778$	0	0
$779$	−7.94948	−0.284820
$780$	0	0
$781$	−1.70738	−0.0610948
$782$	0	0
$783$	−2.96951	−0.106121
$784$	0	0
$785$	7.06877	0.252295
$786$	0	0
$787$	−9.46058	−0.337233	−0.168617	−	0.985682i	$-0.553930\pi$
$787$	−9.46058	−0.337233	−0.168617	+	0.985682i	$0.553930\pi$
$788$	0	0
$789$	−26.1706	−0.931700
$790$	0	0
$791$	0.561839	0.0199767
$792$	0	0
$793$	30.9128	1.09775
$794$	0	0
$795$	9.04449	0.320775
$796$	0	0
$797$	19.1791	0.679359	0.339680	−	0.940541i	$-0.389681\pi$
$797$	19.1791	0.679359	0.339680	+	0.940541i	$0.389681\pi$
$798$	0	0
$799$	8.63577	0.305511
$800$	0	0
$801$	14.3990	0.508765
$802$	0	0
$803$	7.19173	0.253791
$804$	0	0
$805$	−0.808269	−0.0284878
$806$	0	0
$807$	12.1580	0.427982
$808$	0	0
$809$	−43.1578	−1.51735	−0.758673	−	0.651472i	$-0.774152\pi$
$809$	−43.1578	−1.51735	−0.758673	+	0.651472i	$0.774152\pi$
$810$	0	0
$811$	−3.87518	−0.136076	−0.0680380	−	0.997683i	$-0.521674\pi$
$811$	−3.87518	−0.136076	−0.0680380	+	0.997683i	$0.521674\pi$
$812$	0	0
$813$	−10.6644	−0.374016
$814$	0	0
$815$	−13.8243	−0.484242
$816$	0	0
$817$	24.8486	0.869342
$818$	0	0
$819$	0.946795	0.0330837
$820$	0	0
$821$	5.62084	0.196169	0.0980843	−	0.995178i	$-0.468729\pi$
$821$	5.62084	0.196169	0.0980843	+	0.995178i	$0.468729\pi$
$822$	0	0
$823$	38.5255	1.34291	0.671457	−	0.741043i	$-0.265669\pi$
$823$	38.5255	1.34291	0.671457	+	0.741043i	$0.265669\pi$
$824$	0	0
$825$	9.49307	0.330506
$826$	0	0
$827$	4.23674	0.147326	0.0736629	−	0.997283i	$-0.476531\pi$
$827$	4.23674	0.147326	0.0736629	+	0.997283i	$0.476531\pi$
$828$	0	0
$829$	−35.3128	−1.22646	−0.613231	−	0.789903i	$-0.710131\pi$
$829$	−35.3128	−1.22646	−0.613231	+	0.789903i	$0.710131\pi$
$830$	0	0
$831$	3.76421	0.130579
$832$	0	0
$833$	21.2953	0.737838
$834$	0	0
$835$	−7.11007	−0.246054
$836$	0	0
$837$	4.15894	0.143754
$838$	0	0
$839$	39.6005	1.36716	0.683580	−	0.729876i	$-0.260422\pi$
$839$	39.6005	1.36716	0.683580	+	0.729876i	$0.260422\pi$
$840$	0	0
$841$	−20.1820	−0.695932
$842$	0	0
$843$	−10.4496	−0.359902
$844$	0	0
$845$	−40.4253	−1.39067
$846$	0	0
$847$	2.83688	0.0974765
$848$	0	0
$849$	17.6569	0.605982
$850$	0	0
$851$	23.9495	0.820977
$852$	0	0
$853$	10.8683	0.372124	0.186062	−	0.982538i	$-0.440428\pi$
$853$	10.8683	0.372124	0.186062	+	0.982538i	$0.440428\pi$
$854$	0	0
$855$	5.48946	0.187735
$856$	0	0
$857$	−34.0082	−1.16170	−0.580849	−	0.814011i	$-0.697279\pi$
$857$	−34.0082	−1.16170	−0.580849	+	0.814011i	$0.697279\pi$
$858$	0	0
$859$	12.9973	0.443463	0.221731	−	0.975108i	$-0.428829\pi$
$859$	12.9973	0.443463	0.221731	+	0.975108i	$0.428829\pi$
$860$	0	0
$861$	−0.413828	−0.0141032
$862$	0	0
$863$	25.8307	0.879289	0.439644	−	0.898172i	$-0.355104\pi$
$863$	25.8307	0.879289	0.439644	+	0.898172i	$0.355104\pi$
$864$	0	0
$865$	−40.6353	−1.38164
$866$	0	0
$867$	7.67794	0.260757
$868$	0	0
$869$	−52.0663	−1.76623
$870$	0	0
$871$	−6.46512	−0.219062
$872$	0	0
$873$	−0.571533	−0.0193435
$874$	0	0
$875$	1.93391	0.0653780
$876$	0	0
$877$	−45.2150	−1.52680	−0.763400	−	0.645926i	$-0.776472\pi$
$877$	−45.2150	−1.52680	−0.763400	+	0.645926i	$0.776472\pi$
$878$	0	0
$879$	30.7465	1.03705
$880$	0	0
$881$	−34.7403	−1.17043	−0.585215	−	0.810878i	$-0.698990\pi$
$881$	−34.7403	−1.17043	−0.585215	+	0.810878i	$0.698990\pi$
$882$	0	0
$883$	−48.9366	−1.64685	−0.823424	−	0.567427i	$-0.807939\pi$
$883$	−48.9366	−1.64685	−0.823424	+	0.567427i	$0.807939\pi$
$884$	0	0
$885$	10.1706	0.341882
$886$	0	0
$887$	−22.9284	−0.769860	−0.384930	−	0.922946i	$-0.625774\pi$
$887$	−22.9284	−0.769860	−0.384930	+	0.922946i	$0.625774\pi$
$888$	0	0
$889$	0.238081	0.00798497
$890$	0	0
$891$	5.37109	0.179938
$892$	0	0
$893$	8.63577	0.288985
$894$	0	0
$895$	31.1009	1.03959
$896$	0	0
$897$	−16.8486	−0.562558
$898$	0	0
$899$	−12.3500	−0.411896
$900$	0	0
$901$	−15.3591	−0.511687
$902$	0	0
$903$	1.29355	0.0430467
$904$	0	0
$905$	−14.6523	−0.487059
$906$	0	0
$907$	23.1680	0.769280	0.384640	−	0.923067i	$-0.374326\pi$
$907$	23.1680	0.769280	0.384640	+	0.923067i	$0.374326\pi$
$908$	0	0
$909$	10.1158	0.335520
$910$	0	0
$911$	−29.4078	−0.974324	−0.487162	−	0.873312i	$-0.661968\pi$
$911$	−29.4078	−0.974324	−0.487162	+	0.873312i	$0.661968\pi$
$912$	0	0
$913$	0.879722	0.0291146
$914$	0	0
$915$	9.33026	0.308449
$916$	0	0
$917$	−2.34315	−0.0773775
$918$	0	0
$919$	6.86029	0.226300	0.113150	−	0.993578i	$-0.463906\pi$
$919$	6.86029	0.226300	0.113150	+	0.993578i	$0.463906\pi$
$920$	0	0
$921$	−21.2981	−0.701798
$922$	0	0
$923$	−1.89359	−0.0623283
$924$	0	0
$925$	−14.9656	−0.492067
$926$	0	0
$927$	11.3507	0.372805
$928$	0	0
$929$	16.1385	0.529488	0.264744	−	0.964319i	$-0.414713\pi$
$929$	16.1385	0.529488	0.264744	+	0.964319i	$0.414713\pi$
$930$	0	0
$931$	21.2953	0.697925
$932$	0	0
$933$	1.77883	0.0582363
$934$	0	0
$935$	29.4844	0.964241
$936$	0	0
$937$	−34.7669	−1.13579	−0.567893	−	0.823102i	$-0.692241\pi$
$937$	−34.7669	−1.13579	−0.567893	+	0.823102i	$0.692241\pi$
$938$	0	0
$939$	2.70320	0.0882157
$940$	0	0
$941$	52.7024	1.71805	0.859025	−	0.511934i	$-0.171071\pi$
$941$	52.7024	1.71805	0.859025	+	0.511934i	$0.171071\pi$
$942$	0	0
$943$	7.36423	0.239812
$944$	0	0
$945$	0.285766	0.00929598
$946$	0	0
$947$	26.1972	0.851296	0.425648	−	0.904889i	$-0.360046\pi$
$947$	26.1972	0.851296	0.425648	+	0.904889i	$0.360046\pi$
$948$	0	0
$949$	7.97608	0.258915
$950$	0	0
$951$	−22.0653	−0.715516
$952$	0	0
$953$	−11.1752	−0.362000	−0.181000	−	0.983483i	$-0.557933\pi$
$953$	−11.1752	−0.362000	−0.181000	+	0.983483i	$0.557933\pi$
$954$	0	0
$955$	−29.0679	−0.940615
$956$	0	0
$957$	−15.9495	−0.515573
$958$	0	0
$959$	2.19266	0.0708047
$960$	0	0
$961$	−13.7032	−0.442039
$962$	0	0
$963$	1.02109	0.0329041
$964$	0	0
$965$	39.8159	1.28172
$966$	0	0
$967$	12.8452	0.413075	0.206537	−	0.978439i	$-0.433780\pi$
$967$	12.8452	0.413075	0.206537	+	0.978439i	$0.433780\pi$
$968$	0	0
$969$	−9.32206	−0.299468
$970$	0	0
$971$	−8.40774	−0.269817	−0.134909	−	0.990858i	$-0.543074\pi$
$971$	−8.40774	−0.269817	−0.134909	+	0.990858i	$0.543074\pi$
$972$	0	0
$973$	0.544926	0.0174695
$974$	0	0
$975$	10.5284	0.337179
$976$	0	0
$977$	−40.4156	−1.29301	−0.646504	−	0.762910i	$-0.723770\pi$
$977$	−40.4156	−1.29301	−0.646504	+	0.762910i	$0.723770\pi$
$978$	0	0
$979$	77.3385	2.47175
$980$	0	0
$981$	2.04313	0.0652320
$982$	0	0
$983$	13.9202	0.443985	0.221993	−	0.975048i	$-0.428744\pi$
$983$	13.9202	0.443985	0.221993	+	0.975048i	$0.428744\pi$
$984$	0	0
$985$	−36.3582	−1.15847
$986$	0	0
$987$	0.449555	0.0143095
$988$	0	0
$989$	−23.0192	−0.731969
$990$	0	0
$991$	−41.0309	−1.30339	−0.651695	−	0.758481i	$-0.725942\pi$
$991$	−41.0309	−1.30339	−0.651695	+	0.758481i	$0.725942\pi$
$992$	0	0
$993$	21.8431	0.693169
$994$	0	0
$995$	44.9618	1.42539
$996$	0	0
$997$	−38.6427	−1.22383	−0.611913	−	0.790925i	$-0.709600\pi$
$997$	−38.6427	−1.22383	−0.611913	+	0.790925i	$0.709600\pi$
$998$	0	0
$999$	−8.46742	−0.267897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.a.n.1.1		4
3.2	odd	2		9216.2.a.x.1.4		4
4.3	odd	2		3072.2.a.t.1.1		4
8.3	odd	2		3072.2.a.i.1.4		4
8.5	even	2		3072.2.a.o.1.4		4
12.11	even	2		9216.2.a.y.1.4		4
16.3	odd	4		3072.2.d.f.1537.8		8
16.5	even	4		3072.2.d.i.1537.5		8
16.11	odd	4		3072.2.d.f.1537.1		8
16.13	even	4		3072.2.d.i.1537.4		8
24.5	odd	2		9216.2.a.bn.1.1		4
24.11	even	2		9216.2.a.bo.1.1		4
32.3	odd	8		48.2.j.a.13.1	✓	8
32.5	even	8		384.2.j.a.97.3		8
32.11	odd	8		48.2.j.a.37.1	yes	8
32.13	even	8		384.2.j.a.289.3		8
32.19	odd	8		384.2.j.b.289.1		8
32.21	even	8		192.2.j.a.49.2		8
32.27	odd	8		384.2.j.b.97.1		8
32.29	even	8		192.2.j.a.145.2		8
96.5	odd	8		1152.2.k.f.865.3		8
96.11	even	8		144.2.k.b.37.4		8
96.29	odd	8		576.2.k.b.145.2		8
96.35	even	8		144.2.k.b.109.4		8
96.53	odd	8		576.2.k.b.433.2		8
96.59	even	8		1152.2.k.c.865.3		8
96.77	odd	8		1152.2.k.f.289.3		8
96.83	even	8		1152.2.k.c.289.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.1	✓	8	32.3	odd	8
48.2.j.a.37.1	yes	8	32.11	odd	8
144.2.k.b.37.4		8	96.11	even	8
144.2.k.b.109.4		8	96.35	even	8
192.2.j.a.49.2		8	32.21	even	8
192.2.j.a.145.2		8	32.29	even	8
384.2.j.a.97.3		8	32.5	even	8
384.2.j.a.289.3		8	32.13	even	8
384.2.j.b.97.1		8	32.27	odd	8
384.2.j.b.289.1		8	32.19	odd	8
576.2.k.b.145.2		8	96.29	odd	8
576.2.k.b.433.2		8	96.53	odd	8
1152.2.k.c.289.3		8	96.83	even	8
1152.2.k.c.865.3		8	96.59	even	8
1152.2.k.f.289.3		8	96.77	odd	8
1152.2.k.f.865.3		8	96.5	odd	8
3072.2.a.i.1.4		4	8.3	odd	2
3072.2.a.n.1.1		4	1.1	even	1	trivial
3072.2.a.o.1.4		4	8.5	even	2
3072.2.a.t.1.1		4	4.3	odd	2
3072.2.d.f.1537.1		8	16.11	odd	4
3072.2.d.f.1537.8		8	16.3	odd	4
3072.2.d.i.1537.4		8	16.13	even	4
3072.2.d.i.1537.5		8	16.5	even	4
9216.2.a.x.1.4		4	3.2	odd	2
9216.2.a.y.1.4		4	12.11	even	2
9216.2.a.bn.1.1		4	24.5	odd	2
9216.2.a.bo.1.1		4	24.11	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:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.79793 q^{5} +0.158942 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.79793 q^{5} +0.158942 q^{7} +1.00000 q^{9} +5.37109 q^{11} +5.95687 q^{13} +1.79793 q^{15} -3.05320 q^{17} -3.05320 q^{19} -0.158942 q^{21} +2.82843 q^{23} -1.76744 q^{25} -1.00000 q^{27} +2.96951 q^{29} -4.15894 q^{31} -5.37109 q^{33} -0.285766 q^{35} +8.46742 q^{37} -5.95687 q^{39} +2.60365 q^{41} -8.13853 q^{43} -1.79793 q^{45} -2.82843 q^{47} -6.97474 q^{49} +3.05320 q^{51} +5.03049 q^{53} -9.65685 q^{55} +3.05320 q^{57} +5.65685 q^{59} +5.18944 q^{61} +0.158942 q^{63} -10.7101 q^{65} -1.08532 q^{67} -2.82843 q^{69} -0.317883 q^{71} +1.33897 q^{73} +1.76744 q^{75} +0.853690 q^{77} -9.69382 q^{79} +1.00000 q^{81} +0.163788 q^{83} +5.48946 q^{85} -2.96951 q^{87} +14.3990 q^{89} +0.946795 q^{91} +4.15894 q^{93} +5.48946 q^{95} -0.571533 q^{97} +5.37109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 8 q^{13} - 4 q^{15} + 4 q^{21} + 4 q^{25} - 4 q^{27} + 12 q^{29} - 12 q^{31} + 16 q^{37} - 8 q^{39} + 4 q^{45} + 4 q^{49} + 20 q^{53} - 16 q^{55} + 16 q^{61} - 4 q^{63} - 8 q^{65} + 16 q^{67} + 8 q^{71} - 8 q^{73} - 4 q^{75} + 24 q^{77} - 12 q^{79} + 4 q^{81} + 24 q^{85} - 12 q^{87} - 8 q^{89} + 16 q^{91} + 12 q^{93} + 24 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 + 8 * q^13 - 4 * q^15 + 4 * q^21 + 4 * q^25 - 4 * q^27 + 12 * q^29 - 12 * q^31 + 16 * q^37 - 8 * q^39 + 4 * q^45 + 4 * q^49 + 20 * q^53 - 16 * q^55 + 16 * q^61 - 4 * q^63 - 8 * q^65 + 16 * q^67 + 8 * q^71 - 8 * q^73 - 4 * q^75 + 24 * q^77 - 12 * q^79 + 4 * q^81 + 24 * q^85 - 12 * q^87 - 8 * q^89 + 16 * q^91 + 12 * q^93 + 24 * q^95

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