LMFDB - Embedded newform 3328.2.a.m.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-4,0,0,0,-6,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{3} -3.46410 q^{5} -1.26795 q^{7} +1.00000 q^{9} -4.73205 q^{11} +1.00000 q^{13} +6.92820 q^{15} +5.46410 q^{17} -0.732051 q^{19} +2.53590 q^{21} +4.00000 q^{23} +7.00000 q^{25} +4.00000 q^{27} +2.00000 q^{29} +6.73205 q^{31} +9.46410 q^{33} +4.39230 q^{35} +8.92820 q^{37} -2.00000 q^{39} -8.92820 q^{41} -0.535898 q^{43} -3.46410 q^{45} -6.73205 q^{47} -5.39230 q^{49} -10.9282 q^{51} +2.92820 q^{53} +16.3923 q^{55} +1.46410 q^{57} -10.1962 q^{59} +2.92820 q^{61} -1.26795 q^{63} -3.46410 q^{65} -0.732051 q^{67} -8.00000 q^{69} -8.19615 q^{71} +7.46410 q^{73} -14.0000 q^{75} +6.00000 q^{77} -5.46410 q^{79} -11.0000 q^{81} -3.26795 q^{83} -18.9282 q^{85} -4.00000 q^{87} +17.3205 q^{89} -1.26795 q^{91} -13.4641 q^{93} +2.53590 q^{95} +6.39230 q^{97} -4.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} - 6 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} + 4 q^{17} + 2 q^{19} + 12 q^{21} + 8 q^{23} + 14 q^{25} + 8 q^{27} + 4 q^{29} + 10 q^{31} + 12 q^{33} - 12 q^{35} + 4 q^{37} - 4 q^{39} - 4 q^{41}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 - 6 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 + 4 * q^17 + 2 * q^19 + 12 * q^21 + 8 * q^23 + 14 * q^25 + 8 * q^27 + 4 * q^29 + 10 * q^31 + 12 * q^33 - 12 * q^35 + 4 * q^37 - 4 * q^39 - 4 * q^41 - 8 * q^43 - 10 * q^47 + 10 * q^49 - 8 * q^51 - 8 * q^53 + 12 * q^55 - 4 * q^57 - 10 * q^59 - 8 * q^61 - 6 * q^63 + 2 * q^67 - 16 * q^69 - 6 * q^71 + 8 * q^73 - 28 * q^75 + 12 * q^77 - 4 * q^79 - 22 * q^81 - 10 * q^83 - 24 * q^85 - 8 * q^87 - 6 * q^91 - 20 * q^93 + 12 * q^95 - 8 * q^97 - 6 * q^99

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	−1.26795	−0.479240	−0.239620	−	0.970867i	$-0.577023\pi$
$7$	−1.26795	−0.479240	−0.239620	+	0.970867i	$0.577023\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	6.92820	1.78885
$16$	0	0
$17$	5.46410	1.32524	0.662620	−	0.748956i	$-0.269445\pi$
$17$	5.46410	1.32524	0.662620	+	0.748956i	$0.269445\pi$
$18$	0	0
$19$	−0.732051	−0.167944	−0.0839720	−	0.996468i	$-0.526761\pi$
$19$	−0.732051	−0.167944	−0.0839720	+	0.996468i	$0.526761\pi$
$20$	0	0
$21$	2.53590	0.553378
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	6.73205	1.20911	0.604556	−	0.796563i	$-0.293351\pi$
$31$	6.73205	1.20911	0.604556	+	0.796563i	$0.293351\pi$
$32$	0	0
$33$	9.46410	1.64749
$34$	0	0
$35$	4.39230	0.742435
$36$	0	0
$37$	8.92820	1.46779	0.733894	−	0.679264i	$-0.237701\pi$
$37$	8.92820	1.46779	0.733894	+	0.679264i	$0.237701\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−8.92820	−1.39435	−0.697176	−	0.716900i	$-0.745560\pi$
$41$	−8.92820	−1.39435	−0.697176	+	0.716900i	$0.745560\pi$
$42$	0	0
$43$	−0.535898	−0.0817237	−0.0408619	−	0.999165i	$-0.513010\pi$
$43$	−0.535898	−0.0817237	−0.0408619	+	0.999165i	$0.513010\pi$
$44$	0	0
$45$	−3.46410	−0.516398
$46$	0	0
$47$	−6.73205	−0.981971	−0.490985	−	0.871168i	$-0.663363\pi$
$47$	−6.73205	−0.981971	−0.490985	+	0.871168i	$0.663363\pi$
$48$	0	0
$49$	−5.39230	−0.770329
$50$	0	0
$51$	−10.9282	−1.53025
$52$	0	0
$53$	2.92820	0.402220	0.201110	−	0.979569i	$-0.435545\pi$
$53$	2.92820	0.402220	0.201110	+	0.979569i	$0.435545\pi$
$54$	0	0
$55$	16.3923	2.21034
$56$	0	0
$57$	1.46410	0.193925
$58$	0	0
$59$	−10.1962	−1.32743	−0.663713	−	0.747987i	$-0.731020\pi$
$59$	−10.1962	−1.32743	−0.663713	+	0.747987i	$0.731020\pi$
$60$	0	0
$61$	2.92820	0.374918	0.187459	−	0.982272i	$-0.439975\pi$
$61$	2.92820	0.374918	0.187459	+	0.982272i	$0.439975\pi$
$62$	0	0
$63$	−1.26795	−0.159747
$64$	0	0
$65$	−3.46410	−0.429669
$66$	0	0
$67$	−0.732051	−0.0894342	−0.0447171	−	0.999000i	$-0.514239\pi$
$67$	−0.732051	−0.0894342	−0.0447171	+	0.999000i	$0.514239\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−8.19615	−0.972704	−0.486352	−	0.873763i	$-0.661673\pi$
$71$	−8.19615	−0.972704	−0.486352	+	0.873763i	$0.661673\pi$
$72$	0	0
$73$	7.46410	0.873607	0.436804	−	0.899557i	$-0.356111\pi$
$73$	7.46410	0.873607	0.436804	+	0.899557i	$0.356111\pi$
$74$	0	0
$75$	−14.0000	−1.61658
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	−5.46410	−0.614759	−0.307380	−	0.951587i	$-0.599452\pi$
$79$	−5.46410	−0.614759	−0.307380	+	0.951587i	$0.599452\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−3.26795	−0.358704	−0.179352	−	0.983785i	$-0.557400\pi$
$83$	−3.26795	−0.358704	−0.179352	+	0.983785i	$0.557400\pi$
$84$	0	0
$85$	−18.9282	−2.05305
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	17.3205	1.83597	0.917985	−	0.396615i	$-0.129815\pi$
$89$	17.3205	1.83597	0.917985	+	0.396615i	$0.129815\pi$
$90$	0	0
$91$	−1.26795	−0.132917
$92$	0	0
$93$	−13.4641	−1.39616
$94$	0	0
$95$	2.53590	0.260178
$96$	0	0
$97$	6.39230	0.649040	0.324520	−	0.945879i	$-0.394797\pi$
$97$	6.39230	0.649040	0.324520	+	0.945879i	$0.394797\pi$
$98$	0	0
$99$	−4.73205	−0.475589
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	6.92820	0.682656	0.341328	−	0.939944i	$-0.389123\pi$
$103$	6.92820	0.682656	0.341328	+	0.939944i	$0.389123\pi$
$104$	0	0
$105$	−8.78461	−0.857290
$106$	0	0
$107$	4.92820	0.476427	0.238214	−	0.971213i	$-0.423438\pi$
$107$	4.92820	0.476427	0.238214	+	0.971213i	$0.423438\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−17.8564	−1.69486
$112$	0	0
$113$	2.53590	0.238557	0.119279	−	0.992861i	$-0.461942\pi$
$113$	2.53590	0.238557	0.119279	+	0.992861i	$0.461942\pi$
$114$	0	0
$115$	−13.8564	−1.29212
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−6.92820	−0.635107
$120$	0	0
$121$	11.3923	1.03566
$122$	0	0
$123$	17.8564	1.61006
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	1.07180	0.0943664
$130$	0	0
$131$	−19.8564	−1.73486	−0.867431	−	0.497557i	$-0.834230\pi$
$131$	−19.8564	−1.73486	−0.867431	+	0.497557i	$0.834230\pi$
$132$	0	0
$133$	0.928203	0.0804854
$134$	0	0
$135$	−13.8564	−1.19257
$136$	0	0
$137$	−12.9282	−1.10453	−0.552265	−	0.833668i	$-0.686237\pi$
$137$	−12.9282	−1.10453	−0.552265	+	0.833668i	$0.686237\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	13.4641	1.13388
$142$	0	0
$143$	−4.73205	−0.395714
$144$	0	0
$145$	−6.92820	−0.575356
$146$	0	0
$147$	10.7846	0.889500
$148$	0	0
$149$	12.9282	1.05912	0.529560	−	0.848273i	$-0.322357\pi$
$149$	12.9282	1.05912	0.529560	+	0.848273i	$0.322357\pi$
$150$	0	0
$151$	−7.12436	−0.579772	−0.289886	−	0.957061i	$-0.593617\pi$
$151$	−7.12436	−0.579772	−0.289886	+	0.957061i	$0.593617\pi$
$152$	0	0
$153$	5.46410	0.441746
$154$	0	0
$155$	−23.3205	−1.87315
$156$	0	0
$157$	−16.9282	−1.35102	−0.675509	−	0.737352i	$-0.736076\pi$
$157$	−16.9282	−1.35102	−0.675509	+	0.737352i	$0.736076\pi$
$158$	0	0
$159$	−5.85641	−0.464443
$160$	0	0
$161$	−5.07180	−0.399714
$162$	0	0
$163$	−16.7321	−1.31056	−0.655278	−	0.755388i	$-0.727448\pi$
$163$	−16.7321	−1.31056	−0.655278	+	0.755388i	$0.727448\pi$
$164$	0	0
$165$	−32.7846	−2.55228
$166$	0	0
$167$	−5.66025	−0.438004	−0.219002	−	0.975724i	$-0.570280\pi$
$167$	−5.66025	−0.438004	−0.219002	+	0.975724i	$0.570280\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.732051	−0.0559813
$172$	0	0
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	0	0
$175$	−8.87564	−0.670936
$176$	0	0
$177$	20.3923	1.53278
$178$	0	0
$179$	10.3923	0.776757	0.388379	−	0.921500i	$-0.373035\pi$
$179$	10.3923	0.776757	0.388379	+	0.921500i	$0.373035\pi$
$180$	0	0
$181$	8.92820	0.663628	0.331814	−	0.943345i	$-0.392339\pi$
$181$	8.92820	0.663628	0.331814	+	0.943345i	$0.392339\pi$
$182$	0	0
$183$	−5.85641	−0.432918
$184$	0	0
$185$	−30.9282	−2.27389
$186$	0	0
$187$	−25.8564	−1.89081
$188$	0	0
$189$	−5.07180	−0.368919
$190$	0	0
$191$	14.5359	1.05178	0.525890	−	0.850552i	$-0.323732\pi$
$191$	14.5359	1.05178	0.525890	+	0.850552i	$0.323732\pi$
$192$	0	0
$193$	−1.60770	−0.115724	−0.0578622	−	0.998325i	$-0.518428\pi$
$193$	−1.60770	−0.115724	−0.0578622	+	0.998325i	$0.518428\pi$
$194$	0	0
$195$	6.92820	0.496139
$196$	0	0
$197$	3.07180	0.218856	0.109428	−	0.993995i	$-0.465098\pi$
$197$	3.07180	0.218856	0.109428	+	0.993995i	$0.465098\pi$
$198$	0	0
$199$	16.7846	1.18983	0.594915	−	0.803789i	$-0.297186\pi$
$199$	16.7846	1.18983	0.594915	+	0.803789i	$0.297186\pi$
$200$	0	0
$201$	1.46410	0.103270
$202$	0	0
$203$	−2.53590	−0.177985
$204$	0	0
$205$	30.9282	2.16012
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	3.46410	0.239617
$210$	0	0
$211$	7.85641	0.540857	0.270429	−	0.962740i	$-0.412835\pi$
$211$	7.85641	0.540857	0.270429	+	0.962740i	$0.412835\pi$
$212$	0	0
$213$	16.3923	1.12318
$214$	0	0
$215$	1.85641	0.126606
$216$	0	0
$217$	−8.53590	−0.579455
$218$	0	0
$219$	−14.9282	−1.00875
$220$	0	0
$221$	5.46410	0.367555
$222$	0	0
$223$	0.196152	0.0131353	0.00656767	−	0.999978i	$-0.497909\pi$
$223$	0.196152	0.0131353	0.00656767	+	0.999978i	$0.497909\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	−2.87564	−0.190863	−0.0954316	−	0.995436i	$-0.530423\pi$
$227$	−2.87564	−0.190863	−0.0954316	+	0.995436i	$0.530423\pi$
$228$	0	0
$229$	5.32051	0.351589	0.175795	−	0.984427i	$-0.443751\pi$
$229$	5.32051	0.351589	0.175795	+	0.984427i	$0.443751\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	−16.9282	−1.10900	−0.554502	−	0.832183i	$-0.687091\pi$
$233$	−16.9282	−1.10900	−0.554502	+	0.832183i	$0.687091\pi$
$234$	0	0
$235$	23.3205	1.52126
$236$	0	0
$237$	10.9282	0.709863
$238$	0	0
$239$	−18.7321	−1.21168	−0.605838	−	0.795588i	$-0.707162\pi$
$239$	−18.7321	−1.21168	−0.605838	+	0.795588i	$0.707162\pi$
$240$	0	0
$241$	−30.3923	−1.95774	−0.978870	−	0.204482i	$-0.934449\pi$
$241$	−30.3923	−1.95774	−0.978870	+	0.204482i	$0.934449\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	18.6795	1.19339
$246$	0	0
$247$	−0.732051	−0.0465793
$248$	0	0
$249$	6.53590	0.414196
$250$	0	0
$251$	6.39230	0.403479	0.201739	−	0.979439i	$-0.435341\pi$
$251$	6.39230	0.403479	0.201739	+	0.979439i	$0.435341\pi$
$252$	0	0
$253$	−18.9282	−1.19001
$254$	0	0
$255$	37.8564	2.37066
$256$	0	0
$257$	−23.8564	−1.48812	−0.744061	−	0.668112i	$-0.767103\pi$
$257$	−23.8564	−1.48812	−0.744061	+	0.668112i	$0.767103\pi$
$258$	0	0
$259$	−11.3205	−0.703422
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−27.3205	−1.68465	−0.842327	−	0.538966i	$-0.818815\pi$
$263$	−27.3205	−1.68465	−0.842327	+	0.538966i	$0.818815\pi$
$264$	0	0
$265$	−10.1436	−0.623116
$266$	0	0
$267$	−34.6410	−2.12000
$268$	0	0
$269$	−7.85641	−0.479014	−0.239507	−	0.970895i	$-0.576986\pi$
$269$	−7.85641	−0.479014	−0.239507	+	0.970895i	$0.576986\pi$
$270$	0	0
$271$	20.1962	1.22683	0.613414	−	0.789761i	$-0.289796\pi$
$271$	20.1962	1.22683	0.613414	+	0.789761i	$0.289796\pi$
$272$	0	0
$273$	2.53590	0.153480
$274$	0	0
$275$	−33.1244	−1.99747
$276$	0	0
$277$	−1.85641	−0.111541	−0.0557703	−	0.998444i	$-0.517761\pi$
$277$	−1.85641	−0.111541	−0.0557703	+	0.998444i	$0.517761\pi$
$278$	0	0
$279$	6.73205	0.403037
$280$	0	0
$281$	−9.32051	−0.556015	−0.278007	−	0.960579i	$-0.589674\pi$
$281$	−9.32051	−0.556015	−0.278007	+	0.960579i	$0.589674\pi$
$282$	0	0
$283$	19.4641	1.15702	0.578510	−	0.815675i	$-0.303634\pi$
$283$	19.4641	1.15702	0.578510	+	0.815675i	$0.303634\pi$
$284$	0	0
$285$	−5.07180	−0.300427
$286$	0	0
$287$	11.3205	0.668228
$288$	0	0
$289$	12.8564	0.756259
$290$	0	0
$291$	−12.7846	−0.749447
$292$	0	0
$293$	−32.9282	−1.92369	−0.961843	−	0.273602i	$-0.911785\pi$
$293$	−32.9282	−1.92369	−0.961843	+	0.273602i	$0.911785\pi$
$294$	0	0
$295$	35.3205	2.05644
$296$	0	0
$297$	−18.9282	−1.09833
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	0.679492	0.0391653
$302$	0	0
$303$	−24.0000	−1.37876
$304$	0	0
$305$	−10.1436	−0.580820
$306$	0	0
$307$	−0.732051	−0.0417803	−0.0208902	−	0.999782i	$-0.506650\pi$
$307$	−0.732051	−0.0417803	−0.0208902	+	0.999782i	$0.506650\pi$
$308$	0	0
$309$	−13.8564	−0.788263
$310$	0	0
$311$	1.07180	0.0607760	0.0303880	−	0.999538i	$-0.490326\pi$
$311$	1.07180	0.0607760	0.0303880	+	0.999538i	$0.490326\pi$
$312$	0	0
$313$	−0.392305	−0.0221744	−0.0110872	−	0.999939i	$-0.503529\pi$
$313$	−0.392305	−0.0221744	−0.0110872	+	0.999939i	$0.503529\pi$
$314$	0	0
$315$	4.39230	0.247478
$316$	0	0
$317$	3.46410	0.194563	0.0972817	−	0.995257i	$-0.468985\pi$
$317$	3.46410	0.194563	0.0972817	+	0.995257i	$0.468985\pi$
$318$	0	0
$319$	−9.46410	−0.529888
$320$	0	0
$321$	−9.85641	−0.550131
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	7.00000	0.388290
$326$	0	0
$327$	4.00000	0.221201
$328$	0	0
$329$	8.53590	0.470599
$330$	0	0
$331$	17.5167	0.962803	0.481401	−	0.876500i	$-0.340128\pi$
$331$	17.5167	0.962803	0.481401	+	0.876500i	$0.340128\pi$
$332$	0	0
$333$	8.92820	0.489263
$334$	0	0
$335$	2.53590	0.138551
$336$	0	0
$337$	1.46410	0.0797547	0.0398773	−	0.999205i	$-0.487303\pi$
$337$	1.46410	0.0797547	0.0398773	+	0.999205i	$0.487303\pi$
$338$	0	0
$339$	−5.07180	−0.275462
$340$	0	0
$341$	−31.8564	−1.72512
$342$	0	0
$343$	15.7128	0.848412
$344$	0	0
$345$	27.7128	1.49201
$346$	0	0
$347$	−7.07180	−0.379634	−0.189817	−	0.981819i	$-0.560789\pi$
$347$	−7.07180	−0.379634	−0.189817	+	0.981819i	$0.560789\pi$
$348$	0	0
$349$	9.60770	0.514288	0.257144	−	0.966373i	$-0.417219\pi$
$349$	9.60770	0.514288	0.257144	+	0.966373i	$0.417219\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−11.0718	−0.589292	−0.294646	−	0.955606i	$-0.595202\pi$
$353$	−11.0718	−0.589292	−0.294646	+	0.955606i	$0.595202\pi$
$354$	0	0
$355$	28.3923	1.50691
$356$	0	0
$357$	13.8564	0.733359
$358$	0	0
$359$	31.5167	1.66339	0.831693	−	0.555236i	$-0.187372\pi$
$359$	31.5167	1.66339	0.831693	+	0.555236i	$0.187372\pi$
$360$	0	0
$361$	−18.4641	−0.971795
$362$	0	0
$363$	−22.7846	−1.19588
$364$	0	0
$365$	−25.8564	−1.35339
$366$	0	0
$367$	22.2487	1.16137	0.580687	−	0.814127i	$-0.302784\pi$
$367$	22.2487	1.16137	0.580687	+	0.814127i	$0.302784\pi$
$368$	0	0
$369$	−8.92820	−0.464784
$370$	0	0
$371$	−3.71281	−0.192760
$372$	0	0
$373$	14.7846	0.765518	0.382759	−	0.923848i	$-0.374974\pi$
$373$	14.7846	0.765518	0.382759	+	0.923848i	$0.374974\pi$
$374$	0	0
$375$	13.8564	0.715542
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	14.5885	0.749359	0.374679	−	0.927154i	$-0.377753\pi$
$379$	14.5885	0.749359	0.374679	+	0.927154i	$0.377753\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	10.3397	0.528336	0.264168	−	0.964477i	$-0.414903\pi$
$383$	10.3397	0.528336	0.264168	+	0.964477i	$0.414903\pi$
$384$	0	0
$385$	−20.7846	−1.05928
$386$	0	0
$387$	−0.535898	−0.0272412
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	21.8564	1.10533
$392$	0	0
$393$	39.7128	2.00325
$394$	0	0
$395$	18.9282	0.952381
$396$	0	0
$397$	−4.53590	−0.227650	−0.113825	−	0.993501i	$-0.536310\pi$
$397$	−4.53590	−0.227650	−0.113825	+	0.993501i	$0.536310\pi$
$398$	0	0
$399$	−1.85641	−0.0929366
$400$	0	0
$401$	4.53590	0.226512	0.113256	−	0.993566i	$-0.463872\pi$
$401$	4.53590	0.226512	0.113256	+	0.993566i	$0.463872\pi$
$402$	0	0
$403$	6.73205	0.335347
$404$	0	0
$405$	38.1051	1.89346
$406$	0	0
$407$	−42.2487	−2.09419
$408$	0	0
$409$	−12.9282	−0.639259	−0.319629	−	0.947543i	$-0.603558\pi$
$409$	−12.9282	−0.639259	−0.319629	+	0.947543i	$0.603558\pi$
$410$	0	0
$411$	25.8564	1.27540
$412$	0	0
$413$	12.9282	0.636155
$414$	0	0
$415$	11.3205	0.555702
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	30.7846	1.50393	0.751963	−	0.659205i	$-0.229107\pi$
$419$	30.7846	1.50393	0.751963	+	0.659205i	$0.229107\pi$
$420$	0	0
$421$	−24.2487	−1.18181	−0.590905	−	0.806741i	$-0.701229\pi$
$421$	−24.2487	−1.18181	−0.590905	+	0.806741i	$0.701229\pi$
$422$	0	0
$423$	−6.73205	−0.327324
$424$	0	0
$425$	38.2487	1.85534
$426$	0	0
$427$	−3.71281	−0.179676
$428$	0	0
$429$	9.46410	0.456931
$430$	0	0
$431$	−39.1244	−1.88455	−0.942277	−	0.334835i	$-0.891320\pi$
$431$	−39.1244	−1.88455	−0.942277	+	0.334835i	$0.891320\pi$
$432$	0	0
$433$	−6.78461	−0.326048	−0.163024	−	0.986622i	$-0.552125\pi$
$433$	−6.78461	−0.326048	−0.163024	+	0.986622i	$0.552125\pi$
$434$	0	0
$435$	13.8564	0.664364
$436$	0	0
$437$	−2.92820	−0.140075
$438$	0	0
$439$	2.92820	0.139756	0.0698778	−	0.997556i	$-0.477739\pi$
$439$	2.92820	0.139756	0.0698778	+	0.997556i	$0.477739\pi$
$440$	0	0
$441$	−5.39230	−0.256776
$442$	0	0
$443$	−30.0000	−1.42534	−0.712672	−	0.701498i	$-0.752515\pi$
$443$	−30.0000	−1.42534	−0.712672	+	0.701498i	$0.752515\pi$
$444$	0	0
$445$	−60.0000	−2.84427
$446$	0	0
$447$	−25.8564	−1.22297
$448$	0	0
$449$	17.6077	0.830959	0.415479	−	0.909603i	$-0.363614\pi$
$449$	17.6077	0.830959	0.415479	+	0.909603i	$0.363614\pi$
$450$	0	0
$451$	42.2487	1.98941
$452$	0	0
$453$	14.2487	0.669463
$454$	0	0
$455$	4.39230	0.205914
$456$	0	0
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	0	0
$459$	21.8564	1.02017
$460$	0	0
$461$	20.9282	0.974724	0.487362	−	0.873200i	$-0.337959\pi$
$461$	20.9282	0.974724	0.487362	+	0.873200i	$0.337959\pi$
$462$	0	0
$463$	−1.66025	−0.0771585	−0.0385793	−	0.999256i	$-0.512283\pi$
$463$	−1.66025	−0.0771585	−0.0385793	+	0.999256i	$0.512283\pi$
$464$	0	0
$465$	46.6410	2.16293
$466$	0	0
$467$	−36.2487	−1.67739	−0.838695	−	0.544601i	$-0.816681\pi$
$467$	−36.2487	−1.67739	−0.838695	+	0.544601i	$0.816681\pi$
$468$	0	0
$469$	0.928203	0.0428604
$470$	0	0
$471$	33.8564	1.56002
$472$	0	0
$473$	2.53590	0.116601
$474$	0	0
$475$	−5.12436	−0.235122
$476$	0	0
$477$	2.92820	0.134073
$478$	0	0
$479$	−21.2679	−0.971757	−0.485879	−	0.874026i	$-0.661500\pi$
$479$	−21.2679	−0.971757	−0.485879	+	0.874026i	$0.661500\pi$
$480$	0	0
$481$	8.92820	0.407091
$482$	0	0
$483$	10.1436	0.461549
$484$	0	0
$485$	−22.1436	−1.00549
$486$	0	0
$487$	−42.4449	−1.92336	−0.961680	−	0.274174i	$-0.911596\pi$
$487$	−42.4449	−1.92336	−0.961680	+	0.274174i	$0.911596\pi$
$488$	0	0
$489$	33.4641	1.51330
$490$	0	0
$491$	−24.2487	−1.09433	−0.547165	−	0.837025i	$-0.684293\pi$
$491$	−24.2487	−1.09433	−0.547165	+	0.837025i	$0.684293\pi$
$492$	0	0
$493$	10.9282	0.492182
$494$	0	0
$495$	16.3923	0.736779
$496$	0	0
$497$	10.3923	0.466159
$498$	0	0
$499$	7.26795	0.325358	0.162679	−	0.986679i	$-0.447986\pi$
$499$	7.26795	0.325358	0.162679	+	0.986679i	$0.447986\pi$
$500$	0	0
$501$	11.3205	0.505763
$502$	0	0
$503$	30.5359	1.36153	0.680764	−	0.732503i	$-0.261648\pi$
$503$	30.5359	1.36153	0.680764	+	0.732503i	$0.261648\pi$
$504$	0	0
$505$	−41.5692	−1.84981
$506$	0	0
$507$	−2.00000	−0.0888231
$508$	0	0
$509$	14.3923	0.637928	0.318964	−	0.947767i	$-0.396665\pi$
$509$	14.3923	0.637928	0.318964	+	0.947767i	$0.396665\pi$
$510$	0	0
$511$	−9.46410	−0.418667
$512$	0	0
$513$	−2.92820	−0.129283
$514$	0	0
$515$	−24.0000	−1.05757
$516$	0	0
$517$	31.8564	1.40104
$518$	0	0
$519$	−13.8564	−0.608229
$520$	0	0
$521$	25.1769	1.10302	0.551510	−	0.834168i	$-0.314052\pi$
$521$	25.1769	1.10302	0.551510	+	0.834168i	$0.314052\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	0	0
$525$	17.7513	0.774730
$526$	0	0
$527$	36.7846	1.60236
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−10.1962	−0.442475
$532$	0	0
$533$	−8.92820	−0.386723
$534$	0	0
$535$	−17.0718	−0.738078
$536$	0	0
$537$	−20.7846	−0.896922
$538$	0	0
$539$	25.5167	1.09908
$540$	0	0
$541$	−3.07180	−0.132067	−0.0660334	−	0.997817i	$-0.521034\pi$
$541$	−3.07180	−0.132067	−0.0660334	+	0.997817i	$0.521034\pi$
$542$	0	0
$543$	−17.8564	−0.766292
$544$	0	0
$545$	6.92820	0.296772
$546$	0	0
$547$	11.8564	0.506943	0.253472	−	0.967343i	$-0.418428\pi$
$547$	11.8564	0.506943	0.253472	+	0.967343i	$0.418428\pi$
$548$	0	0
$549$	2.92820	0.124973
$550$	0	0
$551$	−1.46410	−0.0623728
$552$	0	0
$553$	6.92820	0.294617
$554$	0	0
$555$	61.8564	2.62566
$556$	0	0
$557$	34.7846	1.47387	0.736936	−	0.675963i	$-0.236272\pi$
$557$	34.7846	1.47387	0.736936	+	0.675963i	$0.236272\pi$
$558$	0	0
$559$	−0.535898	−0.0226661
$560$	0	0
$561$	51.7128	2.18332
$562$	0	0
$563$	7.46410	0.314574	0.157287	−	0.987553i	$-0.449725\pi$
$563$	7.46410	0.314574	0.157287	+	0.987553i	$0.449725\pi$
$564$	0	0
$565$	−8.78461	−0.369571
$566$	0	0
$567$	13.9474	0.585737
$568$	0	0
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	0	0
$571$	2.67949	0.112133	0.0560666	−	0.998427i	$-0.482144\pi$
$571$	2.67949	0.112133	0.0560666	+	0.998427i	$0.482144\pi$
$572$	0	0
$573$	−29.0718	−1.21449
$574$	0	0
$575$	28.0000	1.16768
$576$	0	0
$577$	−7.07180	−0.294403	−0.147201	−	0.989107i	$-0.547027\pi$
$577$	−7.07180	−0.294403	−0.147201	+	0.989107i	$0.547027\pi$
$578$	0	0
$579$	3.21539	0.133627
$580$	0	0
$581$	4.14359	0.171905
$582$	0	0
$583$	−13.8564	−0.573874
$584$	0	0
$585$	−3.46410	−0.143223
$586$	0	0
$587$	−4.33975	−0.179120	−0.0895602	−	0.995981i	$-0.528546\pi$
$587$	−4.33975	−0.179120	−0.0895602	+	0.995981i	$0.528546\pi$
$588$	0	0
$589$	−4.92820	−0.203063
$590$	0	0
$591$	−6.14359	−0.252714
$592$	0	0
$593$	−7.85641	−0.322624	−0.161312	−	0.986903i	$-0.551573\pi$
$593$	−7.85641	−0.322624	−0.161312	+	0.986903i	$0.551573\pi$
$594$	0	0
$595$	24.0000	0.983904
$596$	0	0
$597$	−33.5692	−1.37390
$598$	0	0
$599$	−45.1769	−1.84588	−0.922939	−	0.384945i	$-0.874220\pi$
$599$	−45.1769	−1.84588	−0.922939	+	0.384945i	$0.874220\pi$
$600$	0	0
$601$	4.39230	0.179166	0.0895829	−	0.995979i	$-0.471447\pi$
$601$	4.39230	0.179166	0.0895829	+	0.995979i	$0.471447\pi$
$602$	0	0
$603$	−0.732051	−0.0298114
$604$	0	0
$605$	−39.4641	−1.60444
$606$	0	0
$607$	−7.32051	−0.297130	−0.148565	−	0.988903i	$-0.547465\pi$
$607$	−7.32051	−0.297130	−0.148565	+	0.988903i	$0.547465\pi$
$608$	0	0
$609$	5.07180	0.205520
$610$	0	0
$611$	−6.73205	−0.272350
$612$	0	0
$613$	−24.6410	−0.995241	−0.497621	−	0.867395i	$-0.665793\pi$
$613$	−24.6410	−0.995241	−0.497621	+	0.867395i	$0.665793\pi$
$614$	0	0
$615$	−61.8564	−2.49429
$616$	0	0
$617$	−41.3205	−1.66350	−0.831751	−	0.555150i	$-0.812661\pi$
$617$	−41.3205	−1.66350	−0.831751	+	0.555150i	$0.812661\pi$
$618$	0	0
$619$	−18.5885	−0.747133	−0.373567	−	0.927603i	$-0.621865\pi$
$619$	−18.5885	−0.747133	−0.373567	+	0.927603i	$0.621865\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	0	0
$623$	−21.9615	−0.879870
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−6.92820	−0.276686
$628$	0	0
$629$	48.7846	1.94517
$630$	0	0
$631$	−42.0526	−1.67409	−0.837043	−	0.547137i	$-0.815718\pi$
$631$	−42.0526	−1.67409	−0.837043	+	0.547137i	$0.815718\pi$
$632$	0	0
$633$	−15.7128	−0.624528
$634$	0	0
$635$	−13.8564	−0.549875
$636$	0	0
$637$	−5.39230	−0.213651
$638$	0	0
$639$	−8.19615	−0.324235
$640$	0	0
$641$	−22.2487	−0.878771	−0.439386	−	0.898299i	$-0.644804\pi$
$641$	−22.2487	−0.878771	−0.439386	+	0.898299i	$0.644804\pi$
$642$	0	0
$643$	12.7321	0.502103	0.251052	−	0.967974i	$-0.419224\pi$
$643$	12.7321	0.502103	0.251052	+	0.967974i	$0.419224\pi$
$644$	0	0
$645$	−3.71281	−0.146192
$646$	0	0
$647$	37.8564	1.48829	0.744144	−	0.668019i	$-0.232857\pi$
$647$	37.8564	1.48829	0.744144	+	0.668019i	$0.232857\pi$
$648$	0	0
$649$	48.2487	1.89393
$650$	0	0
$651$	17.0718	0.669096
$652$	0	0
$653$	−3.07180	−0.120209	−0.0601043	−	0.998192i	$-0.519143\pi$
$653$	−3.07180	−0.120209	−0.0601043	+	0.998192i	$0.519143\pi$
$654$	0	0
$655$	68.7846	2.68764
$656$	0	0
$657$	7.46410	0.291202
$658$	0	0
$659$	−14.0000	−0.545363	−0.272681	−	0.962104i	$-0.587910\pi$
$659$	−14.0000	−0.545363	−0.272681	+	0.962104i	$0.587910\pi$
$660$	0	0
$661$	−17.3205	−0.673690	−0.336845	−	0.941560i	$-0.609360\pi$
$661$	−17.3205	−0.673690	−0.336845	+	0.941560i	$0.609360\pi$
$662$	0	0
$663$	−10.9282	−0.424416
$664$	0	0
$665$	−3.21539	−0.124687
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	−0.392305	−0.0151674
$670$	0	0
$671$	−13.8564	−0.534921
$672$	0	0
$673$	33.1769	1.27888	0.639438	−	0.768843i	$-0.279167\pi$
$673$	33.1769	1.27888	0.639438	+	0.768843i	$0.279167\pi$
$674$	0	0
$675$	28.0000	1.07772
$676$	0	0
$677$	−21.0718	−0.809855	−0.404927	−	0.914349i	$-0.632703\pi$
$677$	−21.0718	−0.809855	−0.404927	+	0.914349i	$0.632703\pi$
$678$	0	0
$679$	−8.10512	−0.311046
$680$	0	0
$681$	5.75129	0.220390
$682$	0	0
$683$	−17.8038	−0.681245	−0.340623	−	0.940200i	$-0.610638\pi$
$683$	−17.8038	−0.681245	−0.340623	+	0.940200i	$0.610638\pi$
$684$	0	0
$685$	44.7846	1.71113
$686$	0	0
$687$	−10.6410	−0.405980
$688$	0	0
$689$	2.92820	0.111556
$690$	0	0
$691$	32.8372	1.24918	0.624592	−	0.780951i	$-0.285265\pi$
$691$	32.8372	1.24918	0.624592	+	0.780951i	$0.285265\pi$
$692$	0	0
$693$	6.00000	0.227921
$694$	0	0
$695$	34.6410	1.31401
$696$	0	0
$697$	−48.7846	−1.84785
$698$	0	0
$699$	33.8564	1.28057
$700$	0	0
$701$	−40.6410	−1.53499	−0.767495	−	0.641055i	$-0.778497\pi$
$701$	−40.6410	−1.53499	−0.767495	+	0.641055i	$0.778497\pi$
$702$	0	0
$703$	−6.53590	−0.246506
$704$	0	0
$705$	−46.6410	−1.75660
$706$	0	0
$707$	−15.2154	−0.572234
$708$	0	0
$709$	−29.3205	−1.10115	−0.550577	−	0.834784i	$-0.685592\pi$
$709$	−29.3205	−1.10115	−0.550577	+	0.834784i	$0.685592\pi$
$710$	0	0
$711$	−5.46410	−0.204920
$712$	0	0
$713$	26.9282	1.00847
$714$	0	0
$715$	16.3923	0.613037
$716$	0	0
$717$	37.4641	1.39912
$718$	0	0
$719$	−30.9282	−1.15343	−0.576714	−	0.816946i	$-0.695665\pi$
$719$	−30.9282	−1.15343	−0.576714	+	0.816946i	$0.695665\pi$
$720$	0	0
$721$	−8.78461	−0.327156
$722$	0	0
$723$	60.7846	2.26060
$724$	0	0
$725$	14.0000	0.519947
$726$	0	0
$727$	−22.9282	−0.850360	−0.425180	−	0.905109i	$-0.639789\pi$
$727$	−22.9282	−0.850360	−0.425180	+	0.905109i	$0.639789\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−2.92820	−0.108304
$732$	0	0
$733$	37.3205	1.37846	0.689232	−	0.724541i	$-0.257948\pi$
$733$	37.3205	1.37846	0.689232	+	0.724541i	$0.257948\pi$
$734$	0	0
$735$	−37.3590	−1.37801
$736$	0	0
$737$	3.46410	0.127602
$738$	0	0
$739$	−39.7654	−1.46279	−0.731396	−	0.681953i	$-0.761131\pi$
$739$	−39.7654	−1.46279	−0.731396	+	0.681953i	$0.761131\pi$
$740$	0	0
$741$	1.46410	0.0537851
$742$	0	0
$743$	−32.1962	−1.18116	−0.590581	−	0.806978i	$-0.701101\pi$
$743$	−32.1962	−1.18116	−0.590581	+	0.806978i	$0.701101\pi$
$744$	0	0
$745$	−44.7846	−1.64078
$746$	0	0
$747$	−3.26795	−0.119568
$748$	0	0
$749$	−6.24871	−0.228323
$750$	0	0
$751$	−1.07180	−0.0391104	−0.0195552	−	0.999809i	$-0.506225\pi$
$751$	−1.07180	−0.0391104	−0.0195552	+	0.999809i	$0.506225\pi$
$752$	0	0
$753$	−12.7846	−0.465897
$754$	0	0
$755$	24.6795	0.898179
$756$	0	0
$757$	−40.7846	−1.48234	−0.741171	−	0.671316i	$-0.765729\pi$
$757$	−40.7846	−1.48234	−0.741171	+	0.671316i	$0.765729\pi$
$758$	0	0
$759$	37.8564	1.37410
$760$	0	0
$761$	−18.7846	−0.680942	−0.340471	−	0.940255i	$-0.610586\pi$
$761$	−18.7846	−0.680942	−0.340471	+	0.940255i	$0.610586\pi$
$762$	0	0
$763$	2.53590	0.0918057
$764$	0	0
$765$	−18.9282	−0.684351
$766$	0	0
$767$	−10.1962	−0.368162
$768$	0	0
$769$	31.4641	1.13462	0.567312	−	0.823503i	$-0.307983\pi$
$769$	31.4641	1.13462	0.567312	+	0.823503i	$0.307983\pi$
$770$	0	0
$771$	47.7128	1.71833
$772$	0	0
$773$	−27.4641	−0.987815	−0.493908	−	0.869514i	$-0.664432\pi$
$773$	−27.4641	−0.987815	−0.493908	+	0.869514i	$0.664432\pi$
$774$	0	0
$775$	47.1244	1.69276
$776$	0	0
$777$	22.6410	0.812242
$778$	0	0
$779$	6.53590	0.234173
$780$	0	0
$781$	38.7846	1.38782
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	58.6410	2.09299
$786$	0	0
$787$	29.1244	1.03817	0.519086	−	0.854722i	$-0.326273\pi$
$787$	29.1244	1.03817	0.519086	+	0.854722i	$0.326273\pi$
$788$	0	0
$789$	54.6410	1.94527
$790$	0	0
$791$	−3.21539	−0.114326
$792$	0	0
$793$	2.92820	0.103984
$794$	0	0
$795$	20.2872	0.719512
$796$	0	0
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	−36.7846	−1.30135
$800$	0	0
$801$	17.3205	0.611990
$802$	0	0
$803$	−35.3205	−1.24643
$804$	0	0
$805$	17.5692	0.619234
$806$	0	0
$807$	15.7128	0.553117
$808$	0	0
$809$	5.46410	0.192108	0.0960538	−	0.995376i	$-0.469378\pi$
$809$	5.46410	0.192108	0.0960538	+	0.995376i	$0.469378\pi$
$810$	0	0
$811$	21.4115	0.751861	0.375930	−	0.926648i	$-0.377323\pi$
$811$	21.4115	0.751861	0.375930	+	0.926648i	$0.377323\pi$
$812$	0	0
$813$	−40.3923	−1.41662
$814$	0	0
$815$	57.9615	2.03030
$816$	0	0
$817$	0.392305	0.0137250
$818$	0	0
$819$	−1.26795	−0.0443057
$820$	0	0
$821$	48.2487	1.68389	0.841946	−	0.539562i	$-0.181410\pi$
$821$	48.2487	1.68389	0.841946	+	0.539562i	$0.181410\pi$
$822$	0	0
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	0	0
$825$	66.2487	2.30648
$826$	0	0
$827$	8.73205	0.303643	0.151822	−	0.988408i	$-0.451486\pi$
$827$	8.73205	0.303643	0.151822	+	0.988408i	$0.451486\pi$
$828$	0	0
$829$	−28.7846	−0.999731	−0.499865	−	0.866103i	$-0.666617\pi$
$829$	−28.7846	−0.999731	−0.499865	+	0.866103i	$0.666617\pi$
$830$	0	0
$831$	3.71281	0.128796
$832$	0	0
$833$	−29.4641	−1.02087
$834$	0	0
$835$	19.6077	0.678552
$836$	0	0
$837$	26.9282	0.930775
$838$	0	0
$839$	−20.9808	−0.724336	−0.362168	−	0.932113i	$-0.617963\pi$
$839$	−20.9808	−0.724336	−0.362168	+	0.932113i	$0.617963\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	18.6410	0.642031
$844$	0	0
$845$	−3.46410	−0.119169
$846$	0	0
$847$	−14.4449	−0.496331
$848$	0	0
$849$	−38.9282	−1.33601
$850$	0	0
$851$	35.7128	1.22422
$852$	0	0
$853$	55.1769	1.88922	0.944611	−	0.328193i	$-0.106440\pi$
$853$	55.1769	1.88922	0.944611	+	0.328193i	$0.106440\pi$
$854$	0	0
$855$	2.53590	0.0867259
$856$	0	0
$857$	7.85641	0.268370	0.134185	−	0.990956i	$-0.457158\pi$
$857$	7.85641	0.268370	0.134185	+	0.990956i	$0.457158\pi$
$858$	0	0
$859$	18.0000	0.614152	0.307076	−	0.951685i	$-0.400649\pi$
$859$	18.0000	0.614152	0.307076	+	0.951685i	$0.400649\pi$
$860$	0	0
$861$	−22.6410	−0.771604
$862$	0	0
$863$	−1.26795	−0.0431615	−0.0215807	−	0.999767i	$-0.506870\pi$
$863$	−1.26795	−0.0431615	−0.0215807	+	0.999767i	$0.506870\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	0	0
$867$	−25.7128	−0.873253
$868$	0	0
$869$	25.8564	0.877119
$870$	0	0
$871$	−0.732051	−0.0248046
$872$	0	0
$873$	6.39230	0.216347
$874$	0	0
$875$	8.78461	0.296974
$876$	0	0
$877$	−23.4641	−0.792326	−0.396163	−	0.918180i	$-0.629659\pi$
$877$	−23.4641	−0.792326	−0.396163	+	0.918180i	$0.629659\pi$
$878$	0	0
$879$	65.8564	2.22128
$880$	0	0
$881$	−33.7128	−1.13581	−0.567907	−	0.823093i	$-0.692247\pi$
$881$	−33.7128	−1.13581	−0.567907	+	0.823093i	$0.692247\pi$
$882$	0	0
$883$	−31.4641	−1.05885	−0.529426	−	0.848356i	$-0.677593\pi$
$883$	−31.4641	−1.05885	−0.529426	+	0.848356i	$0.677593\pi$
$884$	0	0
$885$	−70.6410	−2.37457
$886$	0	0
$887$	30.2487	1.01565	0.507826	−	0.861460i	$-0.330449\pi$
$887$	30.2487	1.01565	0.507826	+	0.861460i	$0.330449\pi$
$888$	0	0
$889$	−5.07180	−0.170103
$890$	0	0
$891$	52.0526	1.74383
$892$	0	0
$893$	4.92820	0.164916
$894$	0	0
$895$	−36.0000	−1.20335
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	13.4641	0.449053
$900$	0	0
$901$	16.0000	0.533037
$902$	0	0
$903$	−1.35898	−0.0452242
$904$	0	0
$905$	−30.9282	−1.02809
$906$	0	0
$907$	22.1051	0.733988	0.366994	−	0.930223i	$-0.380387\pi$
$907$	22.1051	0.733988	0.366994	+	0.930223i	$0.380387\pi$
$908$	0	0
$909$	12.0000	0.398015
$910$	0	0
$911$	−3.32051	−0.110013	−0.0550067	−	0.998486i	$-0.517518\pi$
$911$	−3.32051	−0.110013	−0.0550067	+	0.998486i	$0.517518\pi$
$912$	0	0
$913$	15.4641	0.511787
$914$	0	0
$915$	20.2872	0.670674
$916$	0	0
$917$	25.1769	0.831415
$918$	0	0
$919$	−35.3205	−1.16512	−0.582558	−	0.812789i	$-0.697948\pi$
$919$	−35.3205	−1.16512	−0.582558	+	0.812789i	$0.697948\pi$
$920$	0	0
$921$	1.46410	0.0482438
$922$	0	0
$923$	−8.19615	−0.269780
$924$	0	0
$925$	62.4974	2.05490
$926$	0	0
$927$	6.92820	0.227552
$928$	0	0
$929$	20.5359	0.673761	0.336880	−	0.941547i	$-0.390628\pi$
$929$	20.5359	0.673761	0.336880	+	0.941547i	$0.390628\pi$
$930$	0	0
$931$	3.94744	0.129372
$932$	0	0
$933$	−2.14359	−0.0701781
$934$	0	0
$935$	89.5692	2.92923
$936$	0	0
$937$	13.7128	0.447978	0.223989	−	0.974592i	$-0.428092\pi$
$937$	13.7128	0.447978	0.223989	+	0.974592i	$0.428092\pi$
$938$	0	0
$939$	0.784610	0.0256048
$940$	0	0
$941$	32.2487	1.05128	0.525639	−	0.850708i	$-0.323826\pi$
$941$	32.2487	1.05128	0.525639	+	0.850708i	$0.323826\pi$
$942$	0	0
$943$	−35.7128	−1.16297
$944$	0	0
$945$	17.5692	0.571527
$946$	0	0
$947$	48.4449	1.57425	0.787123	−	0.616796i	$-0.211570\pi$
$947$	48.4449	1.57425	0.787123	+	0.616796i	$0.211570\pi$
$948$	0	0
$949$	7.46410	0.242295
$950$	0	0
$951$	−6.92820	−0.224662
$952$	0	0
$953$	31.8564	1.03193	0.515965	−	0.856610i	$-0.327433\pi$
$953$	31.8564	1.03193	0.515965	+	0.856610i	$0.327433\pi$
$954$	0	0
$955$	−50.3538	−1.62941
$956$	0	0
$957$	18.9282	0.611862
$958$	0	0
$959$	16.3923	0.529335
$960$	0	0
$961$	14.3205	0.461952
$962$	0	0
$963$	4.92820	0.158809
$964$	0	0
$965$	5.56922	0.179280
$966$	0	0
$967$	−8.98076	−0.288802	−0.144401	−	0.989519i	$-0.546125\pi$
$967$	−8.98076	−0.288802	−0.144401	+	0.989519i	$0.546125\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	−16.1436	−0.518073	−0.259036	−	0.965868i	$-0.583405\pi$
$971$	−16.1436	−0.518073	−0.259036	+	0.965868i	$0.583405\pi$
$972$	0	0
$973$	12.6795	0.406486
$974$	0	0
$975$	−14.0000	−0.448359
$976$	0	0
$977$	−58.3923	−1.86814	−0.934068	−	0.357096i	$-0.883767\pi$
$977$	−58.3923	−1.86814	−0.934068	+	0.357096i	$0.883767\pi$
$978$	0	0
$979$	−81.9615	−2.61950
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	16.1962	0.516577	0.258289	−	0.966068i	$-0.416841\pi$
$983$	16.1962	0.516577	0.258289	+	0.966068i	$0.416841\pi$
$984$	0	0
$985$	−10.6410	−0.339051
$986$	0	0
$987$	−17.0718	−0.543401
$988$	0	0
$989$	−2.14359	−0.0681623
$990$	0	0
$991$	8.67949	0.275713	0.137857	−	0.990452i	$-0.455979\pi$
$991$	8.67949	0.275713	0.137857	+	0.990452i	$0.455979\pi$
$992$	0	0
$993$	−35.0333	−1.11175
$994$	0	0
$995$	−58.1436	−1.84328
$996$	0	0
$997$	−33.5692	−1.06315	−0.531574	−	0.847012i	$-0.678399\pi$
$997$	−33.5692	−1.06315	−0.531574	+	0.847012i	$0.678399\pi$
$998$	0	0
$999$	35.7128	1.12990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.m.1.1		2
4.3	odd	2		3328.2.a.bd.1.1		2
8.3	odd	2		3328.2.a.n.1.2		2
8.5	even	2		3328.2.a.bc.1.2		2
16.3	odd	4		104.2.b.b.53.1	✓	4
16.5	even	4		416.2.b.b.209.3		4
16.11	odd	4		104.2.b.b.53.2	yes	4
16.13	even	4		416.2.b.b.209.2		4
48.5	odd	4		3744.2.g.b.1873.3		4
48.11	even	4		936.2.g.b.469.3		4
48.29	odd	4		3744.2.g.b.1873.1		4
48.35	even	4		936.2.g.b.469.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.b.b.53.1	✓	4	16.3	odd	4
104.2.b.b.53.2	yes	4	16.11	odd	4
416.2.b.b.209.2		4	16.13	even	4
416.2.b.b.209.3		4	16.5	even	4
936.2.g.b.469.3		4	48.11	even	4
936.2.g.b.469.4		4	48.35	even	4
3328.2.a.m.1.1		2	1.1	even	1	trivial
3328.2.a.n.1.2		2	8.3	odd	2
3328.2.a.bc.1.2		2	8.5	even	2
3328.2.a.bd.1.1		2	4.3	odd	2
3744.2.g.b.1873.1		4	48.29	odd	4
3744.2.g.b.1873.3		4	48.5	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} -3.46410 q^{5} -1.26795 q^{7} +1.00000 q^{9} -4.73205 q^{11} +1.00000 q^{13} +6.92820 q^{15} +5.46410 q^{17} -0.732051 q^{19} +2.53590 q^{21} +4.00000 q^{23} +7.00000 q^{25} +4.00000 q^{27} +2.00000 q^{29} +6.73205 q^{31} +9.46410 q^{33} +4.39230 q^{35} +8.92820 q^{37} -2.00000 q^{39} -8.92820 q^{41} -0.535898 q^{43} -3.46410 q^{45} -6.73205 q^{47} -5.39230 q^{49} -10.9282 q^{51} +2.92820 q^{53} +16.3923 q^{55} +1.46410 q^{57} -10.1962 q^{59} +2.92820 q^{61} -1.26795 q^{63} -3.46410 q^{65} -0.732051 q^{67} -8.00000 q^{69} -8.19615 q^{71} +7.46410 q^{73} -14.0000 q^{75} +6.00000 q^{77} -5.46410 q^{79} -11.0000 q^{81} -3.26795 q^{83} -18.9282 q^{85} -4.00000 q^{87} +17.3205 q^{89} -1.26795 q^{91} -13.4641 q^{93} +2.53590 q^{95} +6.39230 q^{97} -4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 6 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} + 4 q^{17} + 2 q^{19} + 12 q^{21} + 8 q^{23} + 14 q^{25} + 8 q^{27} + 4 q^{29} + 10 q^{31} + 12 q^{33} - 12 q^{35} + 4 q^{37} - 4 q^{39} - 4 q^{41}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 - 6 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 + 4 * q^17 + 2 * q^19 + 12 * q^21 + 8 * q^23 + 14 * q^25 + 8 * q^27 + 4 * q^29 + 10 * q^31 + 12 * q^33 - 12 * q^35 + 4 * q^37 - 4 * q^39 - 4 * q^41 - 8 * q^43 - 10 * q^47 + 10 * q^49 - 8 * q^51 - 8 * q^53 + 12 * q^55 - 4 * q^57 - 10 * q^59 - 8 * q^61 - 6 * q^63 + 2 * q^67 - 16 * q^69 - 6 * q^71 + 8 * q^73 - 28 * q^75 + 12 * q^77 - 4 * q^79 - 22 * q^81 - 10 * q^83 - 24 * q^85 - 8 * q^87 - 6 * q^91 - 20 * q^93 + 12 * q^95 - 8 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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