LMFDB - Embedded newform 4020.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, 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("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.195727752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 15x^{4} - 25x^{3} - 3x^{2} + 9x + 3 $ x^6 - 15x^4 - 25x^3 - 3x^2 + 9x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.457832$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -2.59172 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -2.59172 q^{7} +1.00000 q^{9} +0.555013 q^{11} +5.55262 q^{13} -1.00000 q^{15} -0.185828 q^{17} -2.29081 q^{19} -2.59172 q^{21} -0.888313 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.39504 q^{29} -3.32155 q^{31} +0.555013 q^{33} +2.59172 q^{35} +7.98676 q^{37} +5.55262 q^{39} +5.86462 q^{41} -11.7310 q^{43} -1.00000 q^{45} +11.6969 q^{47} -0.283009 q^{49} -0.185828 q^{51} +2.91980 q^{53} -0.555013 q^{55} -2.29081 q^{57} +9.00861 q^{59} +5.74680 q^{61} -2.59172 q^{63} -5.55262 q^{65} +1.00000 q^{67} -0.888313 q^{69} -11.6850 q^{71} +3.45020 q^{73} +1.00000 q^{75} -1.43844 q^{77} +1.73304 q^{79} +1.00000 q^{81} +0.853271 q^{83} +0.185828 q^{85} -1.39504 q^{87} +14.5911 q^{89} -14.3908 q^{91} -3.32155 q^{93} +2.29081 q^{95} +5.48187 q^{97} +0.555013 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} - 6 q^{15} + 6 q^{17} - 3 q^{19} + 3 q^{21} + 6 q^{23} + 6 q^{25} + 6 q^{27} + 21 q^{29} - 3 q^{31} + 3 q^{33} - 3 q^{35} + 3 q^{39} + 9 q^{41} - 3 q^{43} - 6 q^{45} + 21 q^{47} + 3 q^{49} + 6 q^{51} + 15 q^{53} - 3 q^{55} - 3 q^{57} + 15 q^{59} + 15 q^{61} + 3 q^{63} - 3 q^{65} + 6 q^{67} + 6 q^{69} + 18 q^{71} - 6 q^{73} + 6 q^{75} + 33 q^{77} + 9 q^{79} + 6 q^{81} + 24 q^{83} - 6 q^{85} + 21 q^{87} + 24 q^{89} - 21 q^{91} - 3 q^{93} + 3 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9 + 3 * q^11 + 3 * q^13 - 6 * q^15 + 6 * q^17 - 3 * q^19 + 3 * q^21 + 6 * q^23 + 6 * q^25 + 6 * q^27 + 21 * q^29 - 3 * q^31 + 3 * q^33 - 3 * q^35 + 3 * q^39 + 9 * q^41 - 3 * q^43 - 6 * q^45 + 21 * q^47 + 3 * q^49 + 6 * q^51 + 15 * q^53 - 3 * q^55 - 3 * q^57 + 15 * q^59 + 15 * q^61 + 3 * q^63 - 3 * q^65 + 6 * q^67 + 6 * q^69 + 18 * q^71 - 6 * q^73 + 6 * q^75 + 33 * q^77 + 9 * q^79 + 6 * q^81 + 24 * q^83 - 6 * q^85 + 21 * q^87 + 24 * q^89 - 21 * q^91 - 3 * q^93 + 3 * q^95 + 9 * q^97 + 3 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.59172	−0.979577	−0.489788	−	0.871841i	$-0.662926\pi$
$7$	−2.59172	−0.979577	−0.489788	+	0.871841i	$0.662926\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.555013	0.167343	0.0836713	−	0.996493i	$-0.473335\pi$
$11$	0.555013	0.167343	0.0836713	+	0.996493i	$0.473335\pi$
$12$	0	0
$13$	5.55262	1.54002	0.770009	−	0.638033i	$-0.220252\pi$
$13$	5.55262	1.54002	0.770009	+	0.638033i	$0.220252\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−0.185828	−0.0450700	−0.0225350	−	0.999746i	$-0.507174\pi$
$17$	−0.185828	−0.0450700	−0.0225350	+	0.999746i	$0.507174\pi$
$18$	0	0
$19$	−2.29081	−0.525547	−0.262773	−	0.964858i	$-0.584637\pi$
$19$	−2.29081	−0.525547	−0.262773	+	0.964858i	$0.584637\pi$
$20$	0	0
$21$	−2.59172	−0.565559
$22$	0	0
$23$	−0.888313	−0.185226	−0.0926130	−	0.995702i	$-0.529522\pi$
$23$	−0.888313	−0.185226	−0.0926130	+	0.995702i	$0.529522\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.39504	−0.259052	−0.129526	−	0.991576i	$-0.541346\pi$
$29$	−1.39504	−0.259052	−0.129526	+	0.991576i	$0.541346\pi$
$30$	0	0
$31$	−3.32155	−0.596568	−0.298284	−	0.954477i	$-0.596414\pi$
$31$	−3.32155	−0.596568	−0.298284	+	0.954477i	$0.596414\pi$
$32$	0	0
$33$	0.555013	0.0966153
$34$	0	0
$35$	2.59172	0.438080
$36$	0	0
$37$	7.98676	1.31301	0.656507	−	0.754320i	$-0.272033\pi$
$37$	7.98676	1.31301	0.656507	+	0.754320i	$0.272033\pi$
$38$	0	0
$39$	5.55262	0.889130
$40$	0	0
$41$	5.86462	0.915900	0.457950	−	0.888978i	$-0.348584\pi$
$41$	5.86462	0.915900	0.457950	+	0.888978i	$0.348584\pi$
$42$	0	0
$43$	−11.7310	−1.78896	−0.894480	−	0.447108i	$-0.852454\pi$
$43$	−11.7310	−1.78896	−0.894480	+	0.447108i	$0.852454\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	11.6969	1.70616	0.853081	−	0.521779i	$-0.174732\pi$
$47$	11.6969	1.70616	0.853081	+	0.521779i	$0.174732\pi$
$48$	0	0
$49$	−0.283009	−0.0404298
$50$	0	0
$51$	−0.185828	−0.0260212
$52$	0	0
$53$	2.91980	0.401066	0.200533	−	0.979687i	$-0.435733\pi$
$53$	2.91980	0.401066	0.200533	+	0.979687i	$0.435733\pi$
$54$	0	0
$55$	−0.555013	−0.0748379
$56$	0	0
$57$	−2.29081	−0.303425
$58$	0	0
$59$	9.00861	1.17282	0.586410	−	0.810014i	$-0.300541\pi$
$59$	9.00861	1.17282	0.586410	+	0.810014i	$0.300541\pi$
$60$	0	0
$61$	5.74680	0.735802	0.367901	−	0.929865i	$-0.380077\pi$
$61$	5.74680	0.735802	0.367901	+	0.929865i	$0.380077\pi$
$62$	0	0
$63$	−2.59172	−0.326526
$64$	0	0
$65$	−5.55262	−0.688717
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−0.888313	−0.106940
$70$	0	0
$71$	−11.6850	−1.38675	−0.693377	−	0.720575i	$-0.743878\pi$
$71$	−11.6850	−1.38675	−0.693377	+	0.720575i	$0.743878\pi$
$72$	0	0
$73$	3.45020	0.403815	0.201908	−	0.979405i	$-0.435286\pi$
$73$	3.45020	0.403815	0.201908	+	0.979405i	$0.435286\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−1.43844	−0.163925
$78$	0	0
$79$	1.73304	0.194983	0.0974913	−	0.995236i	$-0.468918\pi$
$79$	1.73304	0.194983	0.0974913	+	0.995236i	$0.468918\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.853271	0.0936587	0.0468293	−	0.998903i	$-0.485088\pi$
$83$	0.853271	0.0936587	0.0468293	+	0.998903i	$0.485088\pi$
$84$	0	0
$85$	0.185828	0.0201559
$86$	0	0
$87$	−1.39504	−0.149564
$88$	0	0
$89$	14.5911	1.54666	0.773329	−	0.634005i	$-0.218590\pi$
$89$	14.5911	1.54666	0.773329	+	0.634005i	$0.218590\pi$
$90$	0	0
$91$	−14.3908	−1.50857
$92$	0	0
$93$	−3.32155	−0.344429
$94$	0	0
$95$	2.29081	0.235032
$96$	0	0
$97$	5.48187	0.556599	0.278300	−	0.960494i	$-0.410229\pi$
$97$	5.48187	0.556599	0.278300	+	0.960494i	$0.410229\pi$
$98$	0	0
$99$	0.555013	0.0557809
$100$	0	0
$101$	−10.9657	−1.09113	−0.545565	−	0.838069i	$-0.683685\pi$
$101$	−10.9657	−1.09113	−0.545565	+	0.838069i	$0.683685\pi$
$102$	0	0
$103$	−11.1870	−1.10229	−0.551143	−	0.834411i	$-0.685808\pi$
$103$	−11.1870	−1.10229	−0.551143	+	0.834411i	$0.685808\pi$
$104$	0	0
$105$	2.59172	0.252926
$106$	0	0
$107$	15.6031	1.50841	0.754203	−	0.656642i	$-0.228024\pi$
$107$	15.6031	1.50841	0.754203	+	0.656642i	$0.228024\pi$
$108$	0	0
$109$	17.3247	1.65941	0.829703	−	0.558205i	$-0.188510\pi$
$109$	17.3247	1.65941	0.829703	+	0.558205i	$0.188510\pi$
$110$	0	0
$111$	7.98676	0.758069
$112$	0	0
$113$	11.2366	1.05705	0.528526	−	0.848917i	$-0.322745\pi$
$113$	11.2366	1.05705	0.528526	+	0.848917i	$0.322745\pi$
$114$	0	0
$115$	0.888313	0.0828356
$116$	0	0
$117$	5.55262	0.513340
$118$	0	0
$119$	0.481614	0.0441495
$120$	0	0
$121$	−10.6920	−0.971996
$122$	0	0
$123$	5.86462	0.528795
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.11359	0.808700	0.404350	−	0.914604i	$-0.367498\pi$
$127$	9.11359	0.808700	0.404350	+	0.914604i	$0.367498\pi$
$128$	0	0
$129$	−11.7310	−1.03286
$130$	0	0
$131$	−4.31390	−0.376907	−0.188454	−	0.982082i	$-0.560348\pi$
$131$	−4.31390	−0.376907	−0.188454	+	0.982082i	$0.560348\pi$
$132$	0	0
$133$	5.93712	0.514813
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−6.22792	−0.532087	−0.266044	−	0.963961i	$-0.585717\pi$
$137$	−6.22792	−0.532087	−0.266044	+	0.963961i	$0.585717\pi$
$138$	0	0
$139$	20.1346	1.70780	0.853899	−	0.520439i	$-0.174232\pi$
$139$	20.1346	1.70780	0.853899	+	0.520439i	$0.174232\pi$
$140$	0	0
$141$	11.6969	0.985053
$142$	0	0
$143$	3.08177	0.257711
$144$	0	0
$145$	1.39504	0.115852
$146$	0	0
$147$	−0.283009	−0.0233422
$148$	0	0
$149$	15.0754	1.23503	0.617513	−	0.786561i	$-0.288140\pi$
$149$	15.0754	1.23503	0.617513	+	0.786561i	$0.288140\pi$
$150$	0	0
$151$	23.6332	1.92324	0.961622	−	0.274378i	$-0.0884719\pi$
$151$	23.6332	1.92324	0.961622	+	0.274378i	$0.0884719\pi$
$152$	0	0
$153$	−0.185828	−0.0150233
$154$	0	0
$155$	3.32155	0.266793
$156$	0	0
$157$	7.16618	0.571924	0.285962	−	0.958241i	$-0.407687\pi$
$157$	7.16618	0.571924	0.285962	+	0.958241i	$0.407687\pi$
$158$	0	0
$159$	2.91980	0.231555
$160$	0	0
$161$	2.30225	0.181443
$162$	0	0
$163$	−19.4483	−1.52331	−0.761655	−	0.647982i	$-0.775613\pi$
$163$	−19.4483	−1.52331	−0.761655	+	0.647982i	$0.775613\pi$
$164$	0	0
$165$	−0.555013	−0.0432077
$166$	0	0
$167$	−5.64067	−0.436488	−0.218244	−	0.975894i	$-0.570033\pi$
$167$	−5.64067	−0.436488	−0.218244	+	0.975894i	$0.570033\pi$
$168$	0	0
$169$	17.8315	1.37166
$170$	0	0
$171$	−2.29081	−0.175182
$172$	0	0
$173$	−13.4733	−1.02436	−0.512178	−	0.858879i	$-0.671161\pi$
$173$	−13.4733	−1.02436	−0.512178	+	0.858879i	$0.671161\pi$
$174$	0	0
$175$	−2.59172	−0.195915
$176$	0	0
$177$	9.00861	0.677128
$178$	0	0
$179$	17.6586	1.31987	0.659933	−	0.751325i	$-0.270585\pi$
$179$	17.6586	1.31987	0.659933	+	0.751325i	$0.270585\pi$
$180$	0	0
$181$	10.4214	0.774615	0.387308	−	0.921951i	$-0.373405\pi$
$181$	10.4214	0.774615	0.387308	+	0.921951i	$0.373405\pi$
$182$	0	0
$183$	5.74680	0.424816
$184$	0	0
$185$	−7.98676	−0.587198
$186$	0	0
$187$	−0.103137	−0.00754214
$188$	0	0
$189$	−2.59172	−0.188520
$190$	0	0
$191$	19.8649	1.43738	0.718688	−	0.695332i	$-0.244743\pi$
$191$	19.8649	1.43738	0.718688	+	0.695332i	$0.244743\pi$
$192$	0	0
$193$	−4.23272	−0.304678	−0.152339	−	0.988328i	$-0.548680\pi$
$193$	−4.23272	−0.304678	−0.152339	+	0.988328i	$0.548680\pi$
$194$	0	0
$195$	−5.55262	−0.397631
$196$	0	0
$197$	16.9353	1.20659	0.603294	−	0.797519i	$-0.293855\pi$
$197$	16.9353	1.20659	0.603294	+	0.797519i	$0.293855\pi$
$198$	0	0
$199$	−3.84324	−0.272440	−0.136220	−	0.990679i	$-0.543495\pi$
$199$	−3.84324	−0.272440	−0.136220	+	0.990679i	$0.543495\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	3.61555	0.253762
$204$	0	0
$205$	−5.86462	−0.409603
$206$	0	0
$207$	−0.888313	−0.0617420
$208$	0	0
$209$	−1.27143	−0.0879464
$210$	0	0
$211$	27.2584	1.87655	0.938274	−	0.345893i	$-0.112424\pi$
$211$	27.2584	1.87655	0.938274	+	0.345893i	$0.112424\pi$
$212$	0	0
$213$	−11.6850	−0.800643
$214$	0	0
$215$	11.7310	0.800047
$216$	0	0
$217$	8.60852	0.584384
$218$	0	0
$219$	3.45020	0.233143
$220$	0	0
$221$	−1.03183	−0.0694087
$222$	0	0
$223$	−2.84326	−0.190399	−0.0951995	−	0.995458i	$-0.530349\pi$
$223$	−2.84326	−0.190399	−0.0951995	+	0.995458i	$0.530349\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−25.7298	−1.70774	−0.853872	−	0.520483i	$-0.825752\pi$
$227$	−25.7298	−1.70774	−0.853872	+	0.520483i	$0.825752\pi$
$228$	0	0
$229$	−20.5116	−1.35544	−0.677722	−	0.735318i	$-0.737033\pi$
$229$	−20.5116	−1.35544	−0.677722	+	0.735318i	$0.737033\pi$
$230$	0	0
$231$	−1.43844	−0.0946421
$232$	0	0
$233$	6.93481	0.454315	0.227157	−	0.973858i	$-0.427057\pi$
$233$	6.93481	0.454315	0.227157	+	0.973858i	$0.427057\pi$
$234$	0	0
$235$	−11.6969	−0.763018
$236$	0	0
$237$	1.73304	0.112573
$238$	0	0
$239$	−3.64142	−0.235544	−0.117772	−	0.993041i	$-0.537575\pi$
$239$	−3.64142	−0.235544	−0.117772	+	0.993041i	$0.537575\pi$
$240$	0	0
$241$	−22.5267	−1.45108	−0.725538	−	0.688183i	$-0.758409\pi$
$241$	−22.5267	−1.45108	−0.725538	+	0.688183i	$0.758409\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.283009	0.0180808
$246$	0	0
$247$	−12.7200	−0.809352
$248$	0	0
$249$	0.853271	0.0540739
$250$	0	0
$251$	19.6475	1.24014	0.620068	−	0.784548i	$-0.287105\pi$
$251$	19.6475	1.24014	0.620068	+	0.784548i	$0.287105\pi$
$252$	0	0
$253$	−0.493025	−0.0309962
$254$	0	0
$255$	0.185828	0.0116370
$256$	0	0
$257$	9.60737	0.599291	0.299646	−	0.954051i	$-0.403132\pi$
$257$	9.60737	0.599291	0.299646	+	0.954051i	$0.403132\pi$
$258$	0	0
$259$	−20.6994	−1.28620
$260$	0	0
$261$	−1.39504	−0.0863508
$262$	0	0
$263$	16.6492	1.02663	0.513317	−	0.858199i	$-0.328416\pi$
$263$	16.6492	1.02663	0.513317	+	0.858199i	$0.328416\pi$
$264$	0	0
$265$	−2.91980	−0.179362
$266$	0	0
$267$	14.5911	0.892963
$268$	0	0
$269$	12.9576	0.790038	0.395019	−	0.918673i	$-0.370738\pi$
$269$	12.9576	0.790038	0.395019	+	0.918673i	$0.370738\pi$
$270$	0	0
$271$	−29.4319	−1.78786	−0.893930	−	0.448207i	$-0.852063\pi$
$271$	−29.4319	−1.78786	−0.893930	+	0.448207i	$0.852063\pi$
$272$	0	0
$273$	−14.3908	−0.870971
$274$	0	0
$275$	0.555013	0.0334685
$276$	0	0
$277$	13.6288	0.818873	0.409436	−	0.912339i	$-0.365725\pi$
$277$	13.6288	0.818873	0.409436	+	0.912339i	$0.365725\pi$
$278$	0	0
$279$	−3.32155	−0.198856
$280$	0	0
$281$	17.7784	1.06057	0.530285	−	0.847819i	$-0.322085\pi$
$281$	17.7784	1.06057	0.530285	+	0.847819i	$0.322085\pi$
$282$	0	0
$283$	−10.4991	−0.624109	−0.312055	−	0.950064i	$-0.601017\pi$
$283$	−10.4991	−0.624109	−0.312055	+	0.950064i	$0.601017\pi$
$284$	0	0
$285$	2.29081	0.135696
$286$	0	0
$287$	−15.1994	−0.897194
$288$	0	0
$289$	−16.9655	−0.997969
$290$	0	0
$291$	5.48187	0.321353
$292$	0	0
$293$	14.7883	0.863943	0.431972	−	0.901887i	$-0.357818\pi$
$293$	14.7883	0.863943	0.431972	+	0.901887i	$0.357818\pi$
$294$	0	0
$295$	−9.00861	−0.524501
$296$	0	0
$297$	0.555013	0.0322051
$298$	0	0
$299$	−4.93246	−0.285252
$300$	0	0
$301$	30.4034	1.75242
$302$	0	0
$303$	−10.9657	−0.629964
$304$	0	0
$305$	−5.74680	−0.329061
$306$	0	0
$307$	−25.8573	−1.47575	−0.737876	−	0.674936i	$-0.764171\pi$
$307$	−25.8573	−1.47575	−0.737876	+	0.674936i	$0.764171\pi$
$308$	0	0
$309$	−11.1870	−0.636405
$310$	0	0
$311$	−22.7273	−1.28875	−0.644375	−	0.764710i	$-0.722882\pi$
$311$	−22.7273	−1.28875	−0.644375	+	0.764710i	$0.722882\pi$
$312$	0	0
$313$	−25.3090	−1.43055	−0.715275	−	0.698844i	$-0.753698\pi$
$313$	−25.3090	−1.43055	−0.715275	+	0.698844i	$0.753698\pi$
$314$	0	0
$315$	2.59172	0.146027
$316$	0	0
$317$	−25.1284	−1.41135	−0.705677	−	0.708534i	$-0.749357\pi$
$317$	−25.1284	−1.41135	−0.705677	+	0.708534i	$0.749357\pi$
$318$	0	0
$319$	−0.774265	−0.0433505
$320$	0	0
$321$	15.6031	0.870878
$322$	0	0
$323$	0.425697	0.0236864
$324$	0	0
$325$	5.55262	0.308004
$326$	0	0
$327$	17.3247	0.958059
$328$	0	0
$329$	−30.3149	−1.67132
$330$	0	0
$331$	18.7348	1.02976	0.514879	−	0.857263i	$-0.327837\pi$
$331$	18.7348	1.02976	0.514879	+	0.857263i	$0.327837\pi$
$332$	0	0
$333$	7.98676	0.437672
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−28.5676	−1.55618	−0.778089	−	0.628155i	$-0.783811\pi$
$337$	−28.5676	−1.55618	−0.778089	+	0.628155i	$0.783811\pi$
$338$	0	0
$339$	11.2366	0.610290
$340$	0	0
$341$	−1.84350	−0.0998313
$342$	0	0
$343$	18.8755	1.01918
$344$	0	0
$345$	0.888313	0.0478252
$346$	0	0
$347$	7.33485	0.393755	0.196878	−	0.980428i	$-0.436920\pi$
$347$	7.33485	0.393755	0.196878	+	0.980428i	$0.436920\pi$
$348$	0	0
$349$	−0.490240	−0.0262420	−0.0131210	−	0.999914i	$-0.504177\pi$
$349$	−0.490240	−0.0262420	−0.0131210	+	0.999914i	$0.504177\pi$
$350$	0	0
$351$	5.55262	0.296377
$352$	0	0
$353$	−27.9530	−1.48779	−0.743895	−	0.668297i	$-0.767024\pi$
$353$	−27.9530	−1.48779	−0.743895	+	0.668297i	$0.767024\pi$
$354$	0	0
$355$	11.6850	0.620176
$356$	0	0
$357$	0.481614	0.0254897
$358$	0	0
$359$	21.9750	1.15979	0.579897	−	0.814690i	$-0.303093\pi$
$359$	21.9750	1.15979	0.579897	+	0.814690i	$0.303093\pi$
$360$	0	0
$361$	−13.7522	−0.723800
$362$	0	0
$363$	−10.6920	−0.561182
$364$	0	0
$365$	−3.45020	−0.180592
$366$	0	0
$367$	14.4768	0.755684	0.377842	−	0.925870i	$-0.376666\pi$
$367$	14.4768	0.755684	0.377842	+	0.925870i	$0.376666\pi$
$368$	0	0
$369$	5.86462	0.305300
$370$	0	0
$371$	−7.56730	−0.392875
$372$	0	0
$373$	10.7751	0.557915	0.278958	−	0.960303i	$-0.410011\pi$
$373$	10.7751	0.557915	0.278958	+	0.960303i	$0.410011\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−7.74612	−0.398946
$378$	0	0
$379$	−22.9005	−1.17632	−0.588160	−	0.808745i	$-0.700147\pi$
$379$	−22.9005	−1.17632	−0.588160	+	0.808745i	$0.700147\pi$
$380$	0	0
$381$	9.11359	0.466903
$382$	0	0
$383$	35.6968	1.82402	0.912010	−	0.410167i	$-0.134530\pi$
$383$	35.6968	1.82402	0.912010	+	0.410167i	$0.134530\pi$
$384$	0	0
$385$	1.43844	0.0733095
$386$	0	0
$387$	−11.7310	−0.596320
$388$	0	0
$389$	31.6775	1.60611	0.803056	−	0.595904i	$-0.203206\pi$
$389$	31.6775	1.60611	0.803056	+	0.595904i	$0.203206\pi$
$390$	0	0
$391$	0.165074	0.00834814
$392$	0	0
$393$	−4.31390	−0.217608
$394$	0	0
$395$	−1.73304	−0.0871989
$396$	0	0
$397$	18.3673	0.921829	0.460914	−	0.887445i	$-0.347522\pi$
$397$	18.3673	0.921829	0.460914	+	0.887445i	$0.347522\pi$
$398$	0	0
$399$	5.93712	0.297228
$400$	0	0
$401$	−25.5580	−1.27631	−0.638153	−	0.769909i	$-0.720301\pi$
$401$	−25.5580	−1.27631	−0.638153	+	0.769909i	$0.720301\pi$
$402$	0	0
$403$	−18.4433	−0.918726
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	4.43275	0.219723
$408$	0	0
$409$	−1.02869	−0.0508657	−0.0254328	−	0.999677i	$-0.508096\pi$
$409$	−1.02869	−0.0508657	−0.0254328	+	0.999677i	$0.508096\pi$
$410$	0	0
$411$	−6.22792	−0.307201
$412$	0	0
$413$	−23.3478	−1.14887
$414$	0	0
$415$	−0.853271	−0.0418854
$416$	0	0
$417$	20.1346	0.985998
$418$	0	0
$419$	−13.1306	−0.641470	−0.320735	−	0.947169i	$-0.603930\pi$
$419$	−13.1306	−0.641470	−0.320735	+	0.947169i	$0.603930\pi$
$420$	0	0
$421$	−24.3744	−1.18794	−0.593968	−	0.804488i	$-0.702440\pi$
$421$	−24.3744	−1.18794	−0.593968	+	0.804488i	$0.702440\pi$
$422$	0	0
$423$	11.6969	0.568720
$424$	0	0
$425$	−0.185828	−0.00901400
$426$	0	0
$427$	−14.8941	−0.720774
$428$	0	0
$429$	3.08177	0.148789
$430$	0	0
$431$	−7.81841	−0.376600	−0.188300	−	0.982112i	$-0.560298\pi$
$431$	−7.81841	−0.376600	−0.188300	+	0.982112i	$0.560298\pi$
$432$	0	0
$433$	−30.5024	−1.46585	−0.732927	−	0.680308i	$-0.761846\pi$
$433$	−30.5024	−1.46585	−0.732927	+	0.680308i	$0.761846\pi$
$434$	0	0
$435$	1.39504	0.0668871
$436$	0	0
$437$	2.03495	0.0973450
$438$	0	0
$439$	−25.0100	−1.19366	−0.596830	−	0.802368i	$-0.703573\pi$
$439$	−25.0100	−1.19366	−0.596830	+	0.802368i	$0.703573\pi$
$440$	0	0
$441$	−0.283009	−0.0134766
$442$	0	0
$443$	−29.0228	−1.37892	−0.689458	−	0.724326i	$-0.742151\pi$
$443$	−29.0228	−1.37892	−0.689458	+	0.724326i	$0.742151\pi$
$444$	0	0
$445$	−14.5911	−0.691686
$446$	0	0
$447$	15.0754	0.713042
$448$	0	0
$449$	18.1375	0.855962	0.427981	−	0.903788i	$-0.359225\pi$
$449$	18.1375	0.855962	0.427981	+	0.903788i	$0.359225\pi$
$450$	0	0
$451$	3.25494	0.153269
$452$	0	0
$453$	23.6332	1.11039
$454$	0	0
$455$	14.3908	0.674651
$456$	0	0
$457$	15.2384	0.712821	0.356411	−	0.934329i	$-0.384000\pi$
$457$	15.2384	0.712821	0.356411	+	0.934329i	$0.384000\pi$
$458$	0	0
$459$	−0.185828	−0.00867373
$460$	0	0
$461$	3.56516	0.166046	0.0830230	−	0.996548i	$-0.473542\pi$
$461$	3.56516	0.166046	0.0830230	+	0.996548i	$0.473542\pi$
$462$	0	0
$463$	−29.2852	−1.36100	−0.680499	−	0.732749i	$-0.738237\pi$
$463$	−29.2852	−1.36100	−0.680499	+	0.732749i	$0.738237\pi$
$464$	0	0
$465$	3.32155	0.154033
$466$	0	0
$467$	20.3770	0.942934	0.471467	−	0.881884i	$-0.343725\pi$
$467$	20.3770	0.942934	0.471467	+	0.881884i	$0.343725\pi$
$468$	0	0
$469$	−2.59172	−0.119674
$470$	0	0
$471$	7.16618	0.330200
$472$	0	0
$473$	−6.51085	−0.299369
$474$	0	0
$475$	−2.29081	−0.105109
$476$	0	0
$477$	2.91980	0.133689
$478$	0	0
$479$	25.0674	1.14536	0.572680	−	0.819779i	$-0.305904\pi$
$479$	25.0674	1.14536	0.572680	+	0.819779i	$0.305904\pi$
$480$	0	0
$481$	44.3474	2.02207
$482$	0	0
$483$	2.30225	0.104756
$484$	0	0
$485$	−5.48187	−0.248919
$486$	0	0
$487$	22.7106	1.02911	0.514557	−	0.857456i	$-0.327956\pi$
$487$	22.7106	1.02911	0.514557	+	0.857456i	$0.327956\pi$
$488$	0	0
$489$	−19.4483	−0.879484
$490$	0	0
$491$	−7.14000	−0.322224	−0.161112	−	0.986936i	$-0.551508\pi$
$491$	−7.14000	−0.322224	−0.161112	+	0.986936i	$0.551508\pi$
$492$	0	0
$493$	0.259238	0.0116755
$494$	0	0
$495$	−0.555013	−0.0249460
$496$	0	0
$497$	30.2842	1.35843
$498$	0	0
$499$	−3.97471	−0.177932	−0.0889662	−	0.996035i	$-0.528356\pi$
$499$	−3.97471	−0.177932	−0.0889662	+	0.996035i	$0.528356\pi$
$500$	0	0
$501$	−5.64067	−0.252007
$502$	0	0
$503$	6.21243	0.276999	0.138499	−	0.990363i	$-0.455772\pi$
$503$	6.21243	0.276999	0.138499	+	0.990363i	$0.455772\pi$
$504$	0	0
$505$	10.9657	0.487968
$506$	0	0
$507$	17.8315	0.791927
$508$	0	0
$509$	40.8268	1.80961	0.904807	−	0.425822i	$-0.140015\pi$
$509$	40.8268	1.80961	0.904807	+	0.425822i	$0.140015\pi$
$510$	0	0
$511$	−8.94194	−0.395568
$512$	0	0
$513$	−2.29081	−0.101142
$514$	0	0
$515$	11.1870	0.492957
$516$	0	0
$517$	6.49190	0.285514
$518$	0	0
$519$	−13.4733	−0.591412
$520$	0	0
$521$	33.6283	1.47328	0.736642	−	0.676283i	$-0.236410\pi$
$521$	33.6283	1.47328	0.736642	+	0.676283i	$0.236410\pi$
$522$	0	0
$523$	23.6068	1.03225	0.516127	−	0.856512i	$-0.327373\pi$
$523$	23.6068	1.03225	0.516127	+	0.856512i	$0.327373\pi$
$524$	0	0
$525$	−2.59172	−0.113112
$526$	0	0
$527$	0.617239	0.0268873
$528$	0	0
$529$	−22.2109	−0.965691
$530$	0	0
$531$	9.00861	0.390940
$532$	0	0
$533$	32.5640	1.41050
$534$	0	0
$535$	−15.6031	−0.674579
$536$	0	0
$537$	17.6586	0.762024
$538$	0	0
$539$	−0.157073	−0.00676563
$540$	0	0
$541$	−15.2952	−0.657594	−0.328797	−	0.944401i	$-0.606643\pi$
$541$	−15.2952	−0.657594	−0.328797	+	0.944401i	$0.606643\pi$
$542$	0	0
$543$	10.4214	0.447224
$544$	0	0
$545$	−17.3247	−0.742109
$546$	0	0
$547$	−34.3464	−1.46855	−0.734273	−	0.678854i	$-0.762477\pi$
$547$	−34.3464	−1.46855	−0.734273	+	0.678854i	$0.762477\pi$
$548$	0	0
$549$	5.74680	0.245267
$550$	0	0
$551$	3.19577	0.136144
$552$	0	0
$553$	−4.49156	−0.191000
$554$	0	0
$555$	−7.98676	−0.339019
$556$	0	0
$557$	1.83573	0.0777825	0.0388912	−	0.999243i	$-0.487617\pi$
$557$	1.83573	0.0777825	0.0388912	+	0.999243i	$0.487617\pi$
$558$	0	0
$559$	−65.1377	−2.75503
$560$	0	0
$561$	−0.103137	−0.00435445
$562$	0	0
$563$	−9.97854	−0.420545	−0.210273	−	0.977643i	$-0.567435\pi$
$563$	−9.97854	−0.420545	−0.210273	+	0.977643i	$0.567435\pi$
$564$	0	0
$565$	−11.2366	−0.472728
$566$	0	0
$567$	−2.59172	−0.108842
$568$	0	0
$569$	13.1749	0.552321	0.276160	−	0.961112i	$-0.410938\pi$
$569$	13.1749	0.552321	0.276160	+	0.961112i	$0.410938\pi$
$570$	0	0
$571$	−12.5420	−0.524868	−0.262434	−	0.964950i	$-0.584525\pi$
$571$	−12.5420	−0.524868	−0.262434	+	0.964950i	$0.584525\pi$
$572$	0	0
$573$	19.8649	0.829870
$574$	0	0
$575$	−0.888313	−0.0370452
$576$	0	0
$577$	1.54978	0.0645182	0.0322591	−	0.999480i	$-0.489730\pi$
$577$	1.54978	0.0645182	0.0322591	+	0.999480i	$0.489730\pi$
$578$	0	0
$579$	−4.23272	−0.175906
$580$	0	0
$581$	−2.21144	−0.0917459
$582$	0	0
$583$	1.62053	0.0671154
$584$	0	0
$585$	−5.55262	−0.229572
$586$	0	0
$587$	−44.3396	−1.83009	−0.915045	−	0.403351i	$-0.867845\pi$
$587$	−44.3396	−1.83009	−0.915045	+	0.403351i	$0.867845\pi$
$588$	0	0
$589$	7.60903	0.313525
$590$	0	0
$591$	16.9353	0.696624
$592$	0	0
$593$	−42.8725	−1.76056	−0.880281	−	0.474453i	$-0.842646\pi$
$593$	−42.8725	−1.76056	−0.880281	+	0.474453i	$0.842646\pi$
$594$	0	0
$595$	−0.481614	−0.0197443
$596$	0	0
$597$	−3.84324	−0.157293
$598$	0	0
$599$	9.26522	0.378567	0.189283	−	0.981923i	$-0.439384\pi$
$599$	9.26522	0.378567	0.189283	+	0.981923i	$0.439384\pi$
$600$	0	0
$601$	28.9859	1.18236	0.591180	−	0.806539i	$-0.298662\pi$
$601$	28.9859	1.18236	0.591180	+	0.806539i	$0.298662\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	10.6920	0.434690
$606$	0	0
$607$	−1.04351	−0.0423549	−0.0211774	−	0.999776i	$-0.506741\pi$
$607$	−1.04351	−0.0423549	−0.0211774	+	0.999776i	$0.506741\pi$
$608$	0	0
$609$	3.61555	0.146509
$610$	0	0
$611$	64.9481	2.62752
$612$	0	0
$613$	−7.41274	−0.299398	−0.149699	−	0.988732i	$-0.547830\pi$
$613$	−7.41274	−0.299398	−0.149699	+	0.988732i	$0.547830\pi$
$614$	0	0
$615$	−5.86462	−0.236484
$616$	0	0
$617$	2.44106	0.0982735	0.0491367	−	0.998792i	$-0.484353\pi$
$617$	2.44106	0.0982735	0.0491367	+	0.998792i	$0.484353\pi$
$618$	0	0
$619$	4.07274	0.163697	0.0818487	−	0.996645i	$-0.473918\pi$
$619$	4.07274	0.163697	0.0818487	+	0.996645i	$0.473918\pi$
$620$	0	0
$621$	−0.888313	−0.0356468
$622$	0	0
$623$	−37.8161	−1.51507
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.27143	−0.0507759
$628$	0	0
$629$	−1.48417	−0.0591776
$630$	0	0
$631$	−30.7734	−1.22507	−0.612535	−	0.790444i	$-0.709850\pi$
$631$	−30.7734	−1.22507	−0.612535	+	0.790444i	$0.709850\pi$
$632$	0	0
$633$	27.2584	1.08343
$634$	0	0
$635$	−9.11359	−0.361662
$636$	0	0
$637$	−1.57144	−0.0622627
$638$	0	0
$639$	−11.6850	−0.462252
$640$	0	0
$641$	−12.7423	−0.503290	−0.251645	−	0.967820i	$-0.580972\pi$
$641$	−12.7423	−0.503290	−0.251645	+	0.967820i	$0.580972\pi$
$642$	0	0
$643$	6.93195	0.273369	0.136685	−	0.990615i	$-0.456355\pi$
$643$	6.93195	0.273369	0.136685	+	0.990615i	$0.456355\pi$
$644$	0	0
$645$	11.7310	0.461907
$646$	0	0
$647$	14.7455	0.579704	0.289852	−	0.957071i	$-0.406394\pi$
$647$	14.7455	0.579704	0.289852	+	0.957071i	$0.406394\pi$
$648$	0	0
$649$	4.99989	0.196263
$650$	0	0
$651$	8.60852	0.337394
$652$	0	0
$653$	−23.7761	−0.930432	−0.465216	−	0.885197i	$-0.654023\pi$
$653$	−23.7761	−0.930432	−0.465216	+	0.885197i	$0.654023\pi$
$654$	0	0
$655$	4.31390	0.168558
$656$	0	0
$657$	3.45020	0.134605
$658$	0	0
$659$	−36.8307	−1.43472	−0.717360	−	0.696703i	$-0.754650\pi$
$659$	−36.8307	−1.43472	−0.717360	+	0.696703i	$0.754650\pi$
$660$	0	0
$661$	−10.1029	−0.392957	−0.196478	−	0.980508i	$-0.562951\pi$
$661$	−10.1029	−0.392957	−0.196478	+	0.980508i	$0.562951\pi$
$662$	0	0
$663$	−1.03183	−0.0400731
$664$	0	0
$665$	−5.93712	−0.230232
$666$	0	0
$667$	1.23923	0.0479833
$668$	0	0
$669$	−2.84326	−0.109927
$670$	0	0
$671$	3.18955	0.123131
$672$	0	0
$673$	−25.7292	−0.991788	−0.495894	−	0.868383i	$-0.665160\pi$
$673$	−25.7292	−0.991788	−0.495894	+	0.868383i	$0.665160\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−18.4420	−0.708785	−0.354392	−	0.935097i	$-0.615312\pi$
$677$	−18.4420	−0.708785	−0.354392	+	0.935097i	$0.615312\pi$
$678$	0	0
$679$	−14.2074	−0.545232
$680$	0	0
$681$	−25.7298	−0.985966
$682$	0	0
$683$	−31.2539	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$683$	−31.2539	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$684$	0	0
$685$	6.22792	0.237957
$686$	0	0
$687$	−20.5116	−0.782566
$688$	0	0
$689$	16.2125	0.617649
$690$	0	0
$691$	−6.56495	−0.249742	−0.124871	−	0.992173i	$-0.539852\pi$
$691$	−6.56495	−0.249742	−0.124871	+	0.992173i	$0.539852\pi$
$692$	0	0
$693$	−1.43844	−0.0546416
$694$	0	0
$695$	−20.1346	−0.763750
$696$	0	0
$697$	−1.08981	−0.0412796
$698$	0	0
$699$	6.93481	0.262299
$700$	0	0
$701$	29.4783	1.11338	0.556691	−	0.830720i	$-0.312071\pi$
$701$	29.4783	1.11338	0.556691	+	0.830720i	$0.312071\pi$
$702$	0	0
$703$	−18.2961	−0.690051
$704$	0	0
$705$	−11.6969	−0.440529
$706$	0	0
$707$	28.4200	1.06884
$708$	0	0
$709$	38.6530	1.45164	0.725822	−	0.687883i	$-0.241460\pi$
$709$	38.6530	1.45164	0.725822	+	0.687883i	$0.241460\pi$
$710$	0	0
$711$	1.73304	0.0649942
$712$	0	0
$713$	2.95058	0.110500
$714$	0	0
$715$	−3.08177	−0.115252
$716$	0	0
$717$	−3.64142	−0.135991
$718$	0	0
$719$	−27.6110	−1.02972	−0.514858	−	0.857275i	$-0.672155\pi$
$719$	−27.6110	−1.02972	−0.514858	+	0.857275i	$0.672155\pi$
$720$	0	0
$721$	28.9935	1.07977
$722$	0	0
$723$	−22.5267	−0.837779
$724$	0	0
$725$	−1.39504	−0.0518105
$726$	0	0
$727$	30.2682	1.12259	0.561293	−	0.827617i	$-0.310304\pi$
$727$	30.2682	1.12259	0.561293	+	0.827617i	$0.310304\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.17995	0.0806284
$732$	0	0
$733$	47.0158	1.73657	0.868284	−	0.496068i	$-0.165223\pi$
$733$	47.0158	1.73657	0.868284	+	0.496068i	$0.165223\pi$
$734$	0	0
$735$	0.283009	0.0104389
$736$	0	0
$737$	0.555013	0.0204442
$738$	0	0
$739$	−18.2046	−0.669669	−0.334834	−	0.942277i	$-0.608680\pi$
$739$	−18.2046	−0.669669	−0.334834	+	0.942277i	$0.608680\pi$
$740$	0	0
$741$	−12.7200	−0.467280
$742$	0	0
$743$	−26.7706	−0.982118	−0.491059	−	0.871126i	$-0.663390\pi$
$743$	−26.7706	−0.982118	−0.491059	+	0.871126i	$0.663390\pi$
$744$	0	0
$745$	−15.0754	−0.552320
$746$	0	0
$747$	0.853271	0.0312196
$748$	0	0
$749$	−40.4387	−1.47760
$750$	0	0
$751$	−14.7310	−0.537543	−0.268771	−	0.963204i	$-0.586618\pi$
$751$	−14.7310	−0.537543	−0.268771	+	0.963204i	$0.586618\pi$
$752$	0	0
$753$	19.6475	0.715993
$754$	0	0
$755$	−23.6332	−0.860101
$756$	0	0
$757$	12.9109	0.469254	0.234627	−	0.972085i	$-0.424613\pi$
$757$	12.9109	0.469254	0.234627	+	0.972085i	$0.424613\pi$
$758$	0	0
$759$	−0.493025	−0.0178957
$760$	0	0
$761$	31.2743	1.13369	0.566846	−	0.823824i	$-0.308164\pi$
$761$	31.2743	1.13369	0.566846	+	0.823824i	$0.308164\pi$
$762$	0	0
$763$	−44.9007	−1.62552
$764$	0	0
$765$	0.185828	0.00671864
$766$	0	0
$767$	50.0213	1.80617
$768$	0	0
$769$	10.6774	0.385036	0.192518	−	0.981293i	$-0.438335\pi$
$769$	10.6774	0.385036	0.192518	+	0.981293i	$0.438335\pi$
$770$	0	0
$771$	9.60737	0.346001
$772$	0	0
$773$	17.7619	0.638851	0.319426	−	0.947611i	$-0.396510\pi$
$773$	17.7619	0.638851	0.319426	+	0.947611i	$0.396510\pi$
$774$	0	0
$775$	−3.32155	−0.119314
$776$	0	0
$777$	−20.6994	−0.742587
$778$	0	0
$779$	−13.4347	−0.481348
$780$	0	0
$781$	−6.48533	−0.232063
$782$	0	0
$783$	−1.39504	−0.0498547
$784$	0	0
$785$	−7.16618	−0.255772
$786$	0	0
$787$	−6.76761	−0.241239	−0.120620	−	0.992699i	$-0.538488\pi$
$787$	−6.76761	−0.241239	−0.120620	+	0.992699i	$0.538488\pi$
$788$	0	0
$789$	16.6492	0.592728
$790$	0	0
$791$	−29.1221	−1.03546
$792$	0	0
$793$	31.9098	1.13315
$794$	0	0
$795$	−2.91980	−0.103555
$796$	0	0
$797$	−43.8700	−1.55396	−0.776978	−	0.629528i	$-0.783248\pi$
$797$	−43.8700	−1.55396	−0.776978	+	0.629528i	$0.783248\pi$
$798$	0	0
$799$	−2.17361	−0.0768967
$800$	0	0
$801$	14.5911	0.515553
$802$	0	0
$803$	1.91491	0.0675756
$804$	0	0
$805$	−2.30225	−0.0811438
$806$	0	0
$807$	12.9576	0.456128
$808$	0	0
$809$	33.2929	1.17052	0.585258	−	0.810847i	$-0.300993\pi$
$809$	33.2929	1.17052	0.585258	+	0.810847i	$0.300993\pi$
$810$	0	0
$811$	−14.8674	−0.522064	−0.261032	−	0.965330i	$-0.584063\pi$
$811$	−14.8674	−0.522064	−0.261032	+	0.965330i	$0.584063\pi$
$812$	0	0
$813$	−29.4319	−1.03222
$814$	0	0
$815$	19.4483	0.681245
$816$	0	0
$817$	26.8734	0.940182
$818$	0	0
$819$	−14.3908	−0.502855
$820$	0	0
$821$	0.469447	0.0163838	0.00819190	−	0.999966i	$-0.497392\pi$
$821$	0.469447	0.0163838	0.00819190	+	0.999966i	$0.497392\pi$
$822$	0	0
$823$	16.2132	0.565155	0.282578	−	0.959244i	$-0.408811\pi$
$823$	16.2132	0.565155	0.282578	+	0.959244i	$0.408811\pi$
$824$	0	0
$825$	0.555013	0.0193231
$826$	0	0
$827$	50.4593	1.75464	0.877321	−	0.479904i	$-0.159328\pi$
$827$	50.4593	1.75464	0.877321	+	0.479904i	$0.159328\pi$
$828$	0	0
$829$	−3.90119	−0.135494	−0.0677470	−	0.997703i	$-0.521581\pi$
$829$	−3.90119	−0.135494	−0.0677470	+	0.997703i	$0.521581\pi$
$830$	0	0
$831$	13.6288	0.472776
$832$	0	0
$833$	0.0525911	0.00182217
$834$	0	0
$835$	5.64067	0.195203
$836$	0	0
$837$	−3.32155	−0.114810
$838$	0	0
$839$	−10.5448	−0.364048	−0.182024	−	0.983294i	$-0.558265\pi$
$839$	−10.5448	−0.364048	−0.182024	+	0.983294i	$0.558265\pi$
$840$	0	0
$841$	−27.0539	−0.932892
$842$	0	0
$843$	17.7784	0.612320
$844$	0	0
$845$	−17.8315	−0.613424
$846$	0	0
$847$	27.7105	0.952145
$848$	0	0
$849$	−10.4991	−0.360330
$850$	0	0
$851$	−7.09474	−0.243205
$852$	0	0
$853$	21.4265	0.733630	0.366815	−	0.930294i	$-0.380448\pi$
$853$	21.4265	0.733630	0.366815	+	0.930294i	$0.380448\pi$
$854$	0	0
$855$	2.29081	0.0783439
$856$	0	0
$857$	−20.8336	−0.711663	−0.355832	−	0.934550i	$-0.615802\pi$
$857$	−20.8336	−0.711663	−0.355832	+	0.934550i	$0.615802\pi$
$858$	0	0
$859$	15.3190	0.522677	0.261339	−	0.965247i	$-0.415836\pi$
$859$	15.3190	0.522677	0.261339	+	0.965247i	$0.415836\pi$
$860$	0	0
$861$	−15.1994	−0.517995
$862$	0	0
$863$	−31.5886	−1.07529	−0.537645	−	0.843171i	$-0.680686\pi$
$863$	−31.5886	−1.07529	−0.537645	+	0.843171i	$0.680686\pi$
$864$	0	0
$865$	13.4733	0.458106
$866$	0	0
$867$	−16.9655	−0.576177
$868$	0	0
$869$	0.961862	0.0326289
$870$	0	0
$871$	5.55262	0.188143
$872$	0	0
$873$	5.48187	0.185533
$874$	0	0
$875$	2.59172	0.0876160
$876$	0	0
$877$	−1.82660	−0.0616800	−0.0308400	−	0.999524i	$-0.509818\pi$
$877$	−1.82660	−0.0616800	−0.0308400	+	0.999524i	$0.509818\pi$
$878$	0	0
$879$	14.7883	0.498798
$880$	0	0
$881$	3.30232	0.111258	0.0556289	−	0.998452i	$-0.482284\pi$
$881$	3.30232	0.111258	0.0556289	+	0.998452i	$0.482284\pi$
$882$	0	0
$883$	−40.5403	−1.36429	−0.682146	−	0.731216i	$-0.738953\pi$
$883$	−40.5403	−1.36429	−0.682146	+	0.731216i	$0.738953\pi$
$884$	0	0
$885$	−9.00861	−0.302821
$886$	0	0
$887$	−10.3321	−0.346917	−0.173458	−	0.984841i	$-0.555494\pi$
$887$	−10.3321	−0.346917	−0.173458	+	0.984841i	$0.555494\pi$
$888$	0	0
$889$	−23.6198	−0.792183
$890$	0	0
$891$	0.555013	0.0185936
$892$	0	0
$893$	−26.7952	−0.896668
$894$	0	0
$895$	−17.6586	−0.590262
$896$	0	0
$897$	−4.93246	−0.164690
$898$	0	0
$899$	4.63370	0.154543
$900$	0	0
$901$	−0.542582	−0.0180760
$902$	0	0
$903$	30.4034	1.01176
$904$	0	0
$905$	−10.4214	−0.346419
$906$	0	0
$907$	−1.11151	−0.0369071	−0.0184536	−	0.999830i	$-0.505874\pi$
$907$	−1.11151	−0.0369071	−0.0184536	+	0.999830i	$0.505874\pi$
$908$	0	0
$909$	−10.9657	−0.363710
$910$	0	0
$911$	38.8583	1.28743	0.643716	−	0.765265i	$-0.277392\pi$
$911$	38.8583	1.28743	0.643716	+	0.765265i	$0.277392\pi$
$912$	0	0
$913$	0.473576	0.0156731
$914$	0	0
$915$	−5.74680	−0.189983
$916$	0	0
$917$	11.1804	0.369210
$918$	0	0
$919$	−4.43530	−0.146307	−0.0731536	−	0.997321i	$-0.523306\pi$
$919$	−4.43530	−0.146307	−0.0731536	+	0.997321i	$0.523306\pi$
$920$	0	0
$921$	−25.8573	−0.852026
$922$	0	0
$923$	−64.8823	−2.13563
$924$	0	0
$925$	7.98676	0.262603
$926$	0	0
$927$	−11.1870	−0.367429
$928$	0	0
$929$	42.3768	1.39034	0.695168	−	0.718847i	$-0.255330\pi$
$929$	42.3768	1.39034	0.695168	+	0.718847i	$0.255330\pi$
$930$	0	0
$931$	0.648318	0.0212478
$932$	0	0
$933$	−22.7273	−0.744060
$934$	0	0
$935$	0.103137	0.00337295
$936$	0	0
$937$	−19.3496	−0.632122	−0.316061	−	0.948739i	$-0.602360\pi$
$937$	−19.3496	−0.632122	−0.316061	+	0.948739i	$0.602360\pi$
$938$	0	0
$939$	−25.3090	−0.825928
$940$	0	0
$941$	45.5229	1.48400	0.742002	−	0.670397i	$-0.233876\pi$
$941$	45.5229	1.48400	0.742002	+	0.670397i	$0.233876\pi$
$942$	0	0
$943$	−5.20962	−0.169649
$944$	0	0
$945$	2.59172	0.0843085
$946$	0	0
$947$	−60.2821	−1.95891	−0.979453	−	0.201671i	$-0.935363\pi$
$947$	−60.2821	−1.95891	−0.979453	+	0.201671i	$0.935363\pi$
$948$	0	0
$949$	19.1576	0.621883
$950$	0	0
$951$	−25.1284	−0.814845
$952$	0	0
$953$	−5.20417	−0.168579	−0.0842897	−	0.996441i	$-0.526862\pi$
$953$	−5.20417	−0.168579	−0.0842897	+	0.996441i	$0.526862\pi$
$954$	0	0
$955$	−19.8649	−0.642814
$956$	0	0
$957$	−0.774265	−0.0250284
$958$	0	0
$959$	16.1410	0.521220
$960$	0	0
$961$	−19.9673	−0.644106
$962$	0	0
$963$	15.6031	0.502802
$964$	0	0
$965$	4.23272	0.136256
$966$	0	0
$967$	−36.1822	−1.16354	−0.581770	−	0.813353i	$-0.697640\pi$
$967$	−36.1822	−1.16354	−0.581770	+	0.813353i	$0.697640\pi$
$968$	0	0
$969$	0.425697	0.0136754
$970$	0	0
$971$	−15.7045	−0.503983	−0.251991	−	0.967729i	$-0.581085\pi$
$971$	−15.7045	−0.503983	−0.251991	+	0.967729i	$0.581085\pi$
$972$	0	0
$973$	−52.1833	−1.67292
$974$	0	0
$975$	5.55262	0.177826
$976$	0	0
$977$	35.3486	1.13090	0.565452	−	0.824781i	$-0.308702\pi$
$977$	35.3486	1.13090	0.565452	+	0.824781i	$0.308702\pi$
$978$	0	0
$979$	8.09827	0.258822
$980$	0	0
$981$	17.3247	0.553136
$982$	0	0
$983$	0.788593	0.0251522	0.0125761	−	0.999921i	$-0.495997\pi$
$983$	0.788593	0.0251522	0.0125761	+	0.999921i	$0.495997\pi$
$984$	0	0
$985$	−16.9353	−0.539602
$986$	0	0
$987$	−30.3149	−0.964934
$988$	0	0
$989$	10.4208	0.331362
$990$	0	0
$991$	51.0610	1.62201	0.811003	−	0.585042i	$-0.198922\pi$
$991$	51.0610	1.62201	0.811003	+	0.585042i	$0.198922\pi$
$992$	0	0
$993$	18.7348	0.594531
$994$	0	0
$995$	3.84324	0.121839
$996$	0	0
$997$	29.1766	0.924034	0.462017	−	0.886871i	$-0.347126\pi$
$997$	29.1766	0.924034	0.462017	+	0.886871i	$0.347126\pi$
$998$	0	0
$999$	7.98676	0.252690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.g.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.g.1.2	✓	6	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.59172 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.59172 q^{7} +1.00000 q^{9} +0.555013 q^{11} +5.55262 q^{13} -1.00000 q^{15} -0.185828 q^{17} -2.29081 q^{19} -2.59172 q^{21} -0.888313 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.39504 q^{29} -3.32155 q^{31} +0.555013 q^{33} +2.59172 q^{35} +7.98676 q^{37} +5.55262 q^{39} +5.86462 q^{41} -11.7310 q^{43} -1.00000 q^{45} +11.6969 q^{47} -0.283009 q^{49} -0.185828 q^{51} +2.91980 q^{53} -0.555013 q^{55} -2.29081 q^{57} +9.00861 q^{59} +5.74680 q^{61} -2.59172 q^{63} -5.55262 q^{65} +1.00000 q^{67} -0.888313 q^{69} -11.6850 q^{71} +3.45020 q^{73} +1.00000 q^{75} -1.43844 q^{77} +1.73304 q^{79} +1.00000 q^{81} +0.853271 q^{83} +0.185828 q^{85} -1.39504 q^{87} +14.5911 q^{89} -14.3908 q^{91} -3.32155 q^{93} +2.29081 q^{95} +5.48187 q^{97} +0.555013 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} - 6 q^{15} + 6 q^{17} - 3 q^{19} + 3 q^{21} + 6 q^{23} + 6 q^{25} + 6 q^{27} + 21 q^{29} - 3 q^{31} + 3 q^{33} - 3 q^{35} + 3 q^{39} + 9 q^{41} - 3 q^{43} - 6 q^{45} + 21 q^{47} + 3 q^{49} + 6 q^{51} + 15 q^{53} - 3 q^{55} - 3 q^{57} + 15 q^{59} + 15 q^{61} + 3 q^{63} - 3 q^{65} + 6 q^{67} + 6 q^{69} + 18 q^{71} - 6 q^{73} + 6 q^{75} + 33 q^{77} + 9 q^{79} + 6 q^{81} + 24 q^{83} - 6 q^{85} + 21 q^{87} + 24 q^{89} - 21 q^{91} - 3 q^{93} + 3 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9 + 3 * q^11 + 3 * q^13 - 6 * q^15 + 6 * q^17 - 3 * q^19 + 3 * q^21 + 6 * q^23 + 6 * q^25 + 6 * q^27 + 21 * q^29 - 3 * q^31 + 3 * q^33 - 3 * q^35 + 3 * q^39 + 9 * q^41 - 3 * q^43 - 6 * q^45 + 21 * q^47 + 3 * q^49 + 6 * q^51 + 15 * q^53 - 3 * q^55 - 3 * q^57 + 15 * q^59 + 15 * q^61 + 3 * q^63 - 3 * q^65 + 6 * q^67 + 6 * q^69 + 18 * q^71 - 6 * q^73 + 6 * q^75 + 33 * q^77 + 9 * q^79 + 6 * q^81 + 24 * q^83 - 6 * q^85 + 21 * q^87 + 24 * q^89 - 21 * q^91 - 3 * q^93 + 3 * q^95 + 9 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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