LMFDB - Embedded newform 1040.2.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1040,2,Mod(1,1040)] 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("1040.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,-2,0,0,0,2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.30444181021$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 520)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1040.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{3} -1.00000 q^{5} -3.46410 q^{7} -2.46410 q^{9} -2.73205 q^{11} +1.00000 q^{13} +0.732051 q^{15} +7.46410 q^{17} +2.73205 q^{19} +2.53590 q^{21} +0.732051 q^{23} +1.00000 q^{25} +4.00000 q^{27} +9.46410 q^{29} +0.196152 q^{31} +2.00000 q^{33} +3.46410 q^{35} -0.732051 q^{39} +0.535898 q^{41} +7.26795 q^{43} +2.46410 q^{45} -4.53590 q^{47} +5.00000 q^{49} -5.46410 q^{51} -0.535898 q^{53} +2.73205 q^{55} -2.00000 q^{57} -5.66025 q^{59} -12.3923 q^{61} +8.53590 q^{63} -1.00000 q^{65} +12.9282 q^{67} -0.535898 q^{69} -9.26795 q^{71} -2.92820 q^{73} -0.732051 q^{75} +9.46410 q^{77} +6.53590 q^{79} +4.46410 q^{81} +10.3923 q^{83} -7.46410 q^{85} -6.92820 q^{87} -12.9282 q^{89} -3.46410 q^{91} -0.143594 q^{93} -2.73205 q^{95} +15.8564 q^{97} +6.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{15} + 8 q^{17} + 2 q^{19} + 12 q^{21} - 2 q^{23} + 2 q^{25} + 8 q^{27} + 12 q^{29} - 10 q^{31} + 4 q^{33} + 2 q^{39} + 8 q^{41} + 18 q^{43}+ \cdots + 10 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^15 + 8 * q^17 + 2 * q^19 + 12 * q^21 - 2 * q^23 + 2 * q^25 + 8 * q^27 + 12 * q^29 - 10 * q^31 + 4 * q^33 + 2 * q^39 + 8 * q^41 + 18 * q^43 - 2 * q^45 - 16 * q^47 + 10 * q^49 - 4 * q^51 - 8 * q^53 + 2 * q^55 - 4 * q^57 + 6 * q^59 - 4 * q^61 + 24 * q^63 - 2 * q^65 + 12 * q^67 - 8 * q^69 - 22 * q^71 + 8 * q^73 + 2 * q^75 + 12 * q^77 + 20 * q^79 + 2 * q^81 - 8 * q^85 - 12 * q^89 - 28 * q^93 - 2 * q^95 + 4 * q^97 + 10 * 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
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−2.73205	−0.823744	−0.411872	−	0.911242i	$-0.635125\pi$
$11$	−2.73205	−0.823744	−0.411872	+	0.911242i	$0.635125\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.732051	0.189015
$16$	0	0
$17$	7.46410	1.81031	0.905155	−	0.425081i	$-0.139754\pi$
$17$	7.46410	1.81031	0.905155	+	0.425081i	$0.139754\pi$
$18$	0	0
$19$	2.73205	0.626775	0.313388	−	0.949625i	$-0.398536\pi$
$19$	2.73205	0.626775	0.313388	+	0.949625i	$0.398536\pi$
$20$	0	0
$21$	2.53590	0.553378
$22$	0	0
$23$	0.732051	0.152643	0.0763216	−	0.997083i	$-0.475682\pi$
$23$	0.732051	0.152643	0.0763216	+	0.997083i	$0.475682\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	9.46410	1.75744	0.878720	−	0.477338i	$-0.158398\pi$
$29$	9.46410	1.75744	0.878720	+	0.477338i	$0.158398\pi$
$30$	0	0
$31$	0.196152	0.0352300	0.0176150	−	0.999845i	$-0.494393\pi$
$31$	0.196152	0.0352300	0.0176150	+	0.999845i	$0.494393\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	3.46410	0.585540
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−0.732051	−0.117222
$40$	0	0
$41$	0.535898	0.0836933	0.0418466	−	0.999124i	$-0.486676\pi$
$41$	0.535898	0.0836933	0.0418466	+	0.999124i	$0.486676\pi$
$42$	0	0
$43$	7.26795	1.10835	0.554176	−	0.832400i	$-0.313033\pi$
$43$	7.26795	1.10835	0.554176	+	0.832400i	$0.313033\pi$
$44$	0	0
$45$	2.46410	0.367327
$46$	0	0
$47$	−4.53590	−0.661629	−0.330814	−	0.943696i	$-0.607323\pi$
$47$	−4.53590	−0.661629	−0.330814	+	0.943696i	$0.607323\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	−5.46410	−0.765127
$52$	0	0
$53$	−0.535898	−0.0736113	−0.0368057	−	0.999322i	$-0.511718\pi$
$53$	−0.535898	−0.0736113	−0.0368057	+	0.999322i	$0.511718\pi$
$54$	0	0
$55$	2.73205	0.368390
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	−5.66025	−0.736902	−0.368451	−	0.929647i	$-0.620112\pi$
$59$	−5.66025	−0.736902	−0.368451	+	0.929647i	$0.620112\pi$
$60$	0	0
$61$	−12.3923	−1.58667	−0.793336	−	0.608784i	$-0.791658\pi$
$61$	−12.3923	−1.58667	−0.793336	+	0.608784i	$0.791658\pi$
$62$	0	0
$63$	8.53590	1.07542
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	12.9282	1.57943	0.789716	−	0.613473i	$-0.210228\pi$
$67$	12.9282	1.57943	0.789716	+	0.613473i	$0.210228\pi$
$68$	0	0
$69$	−0.535898	−0.0645146
$70$	0	0
$71$	−9.26795	−1.09990	−0.549952	−	0.835197i	$-0.685354\pi$
$71$	−9.26795	−1.09990	−0.549952	+	0.835197i	$0.685354\pi$
$72$	0	0
$73$	−2.92820	−0.342720	−0.171360	−	0.985208i	$-0.554816\pi$
$73$	−2.92820	−0.342720	−0.171360	+	0.985208i	$0.554816\pi$
$74$	0	0
$75$	−0.732051	−0.0845299
$76$	0	0
$77$	9.46410	1.07853
$78$	0	0
$79$	6.53590	0.735346	0.367673	−	0.929955i	$-0.380155\pi$
$79$	6.53590	0.735346	0.367673	+	0.929955i	$0.380155\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	10.3923	1.14070	0.570352	−	0.821401i	$-0.306807\pi$
$83$	10.3923	1.14070	0.570352	+	0.821401i	$0.306807\pi$
$84$	0	0
$85$	−7.46410	−0.809595
$86$	0	0
$87$	−6.92820	−0.742781
$88$	0	0
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	0	0
$91$	−3.46410	−0.363137
$92$	0	0
$93$	−0.143594	−0.0148900
$94$	0	0
$95$	−2.73205	−0.280302
$96$	0	0
$97$	15.8564	1.60997	0.804987	−	0.593292i	$-0.202172\pi$
$97$	15.8564	1.60997	0.804987	+	0.593292i	$0.202172\pi$
$98$	0	0
$99$	6.73205	0.676597
$100$	0	0
$101$	4.92820	0.490375	0.245187	−	0.969476i	$-0.421151\pi$
$101$	4.92820	0.490375	0.245187	+	0.969476i	$0.421151\pi$
$102$	0	0
$103$	6.19615	0.610525	0.305263	−	0.952268i	$-0.401256\pi$
$103$	6.19615	0.610525	0.305263	+	0.952268i	$0.401256\pi$
$104$	0	0
$105$	−2.53590	−0.247478
$106$	0	0
$107$	20.0526	1.93855	0.969277	−	0.245972i	$-0.0791070\pi$
$107$	20.0526	1.93855	0.969277	+	0.245972i	$0.0791070\pi$
$108$	0	0
$109$	15.8564	1.51877	0.759384	−	0.650643i	$-0.225500\pi$
$109$	15.8564	1.51877	0.759384	+	0.650643i	$0.225500\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−18.3923	−1.73020	−0.865101	−	0.501597i	$-0.832746\pi$
$113$	−18.3923	−1.73020	−0.865101	+	0.501597i	$0.832746\pi$
$114$	0	0
$115$	−0.732051	−0.0682641
$116$	0	0
$117$	−2.46410	−0.227806
$118$	0	0
$119$	−25.8564	−2.37025
$120$	0	0
$121$	−3.53590	−0.321445
$122$	0	0
$123$	−0.392305	−0.0353729
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.80385	0.160066	0.0800328	−	0.996792i	$-0.474497\pi$
$127$	1.80385	0.160066	0.0800328	+	0.996792i	$0.474497\pi$
$128$	0	0
$129$	−5.32051	−0.468445
$130$	0	0
$131$	2.92820	0.255838	0.127919	−	0.991785i	$-0.459170\pi$
$131$	2.92820	0.255838	0.127919	+	0.991785i	$0.459170\pi$
$132$	0	0
$133$	−9.46410	−0.820642
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	8.92820	0.762788	0.381394	−	0.924413i	$-0.375444\pi$
$137$	8.92820	0.762788	0.381394	+	0.924413i	$0.375444\pi$
$138$	0	0
$139$	13.4641	1.14201	0.571005	−	0.820947i	$-0.306554\pi$
$139$	13.4641	1.14201	0.571005	+	0.820947i	$0.306554\pi$
$140$	0	0
$141$	3.32051	0.279637
$142$	0	0
$143$	−2.73205	−0.228466
$144$	0	0
$145$	−9.46410	−0.785951
$146$	0	0
$147$	−3.66025	−0.301893
$148$	0	0
$149$	−7.85641	−0.643622	−0.321811	−	0.946804i	$-0.604292\pi$
$149$	−7.85641	−0.643622	−0.321811	+	0.946804i	$0.604292\pi$
$150$	0	0
$151$	−11.8038	−0.960583	−0.480292	−	0.877109i	$-0.659469\pi$
$151$	−11.8038	−0.960583	−0.480292	+	0.877109i	$0.659469\pi$
$152$	0	0
$153$	−18.3923	−1.48693
$154$	0	0
$155$	−0.196152	−0.0157553
$156$	0	0
$157$	−7.85641	−0.627009	−0.313505	−	0.949587i	$-0.601503\pi$
$157$	−7.85641	−0.627009	−0.313505	+	0.949587i	$0.601503\pi$
$158$	0	0
$159$	0.392305	0.0311118
$160$	0	0
$161$	−2.53590	−0.199857
$162$	0	0
$163$	8.92820	0.699311	0.349655	−	0.936878i	$-0.386299\pi$
$163$	8.92820	0.699311	0.349655	+	0.936878i	$0.386299\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	21.3205	1.64983	0.824915	−	0.565256i	$-0.191223\pi$
$167$	21.3205	1.64983	0.824915	+	0.565256i	$0.191223\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.73205	−0.514813
$172$	0	0
$173$	−2.39230	−0.181884	−0.0909418	−	0.995856i	$-0.528988\pi$
$173$	−2.39230	−0.181884	−0.0909418	+	0.995856i	$0.528988\pi$
$174$	0	0
$175$	−3.46410	−0.261861
$176$	0	0
$177$	4.14359	0.311452
$178$	0	0
$179$	21.8564	1.63362	0.816812	−	0.576904i	$-0.195739\pi$
$179$	21.8564	1.63362	0.816812	+	0.576904i	$0.195739\pi$
$180$	0	0
$181$	6.53590	0.485810	0.242905	−	0.970050i	$-0.421900\pi$
$181$	6.53590	0.485810	0.242905	+	0.970050i	$0.421900\pi$
$182$	0	0
$183$	9.07180	0.670607
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−20.3923	−1.49123
$188$	0	0
$189$	−13.8564	−1.00791
$190$	0	0
$191$	−18.9282	−1.36960	−0.684798	−	0.728733i	$-0.740110\pi$
$191$	−18.9282	−1.36960	−0.684798	+	0.728733i	$0.740110\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	0.732051	0.0524232
$196$	0	0
$197$	8.92820	0.636108	0.318054	−	0.948073i	$-0.396971\pi$
$197$	8.92820	0.636108	0.318054	+	0.948073i	$0.396971\pi$
$198$	0	0
$199$	6.92820	0.491127	0.245564	−	0.969380i	$-0.421027\pi$
$199$	6.92820	0.491127	0.245564	+	0.969380i	$0.421027\pi$
$200$	0	0
$201$	−9.46410	−0.667546
$202$	0	0
$203$	−32.7846	−2.30103
$204$	0	0
$205$	−0.535898	−0.0374288
$206$	0	0
$207$	−1.80385	−0.125376
$208$	0	0
$209$	−7.46410	−0.516303
$210$	0	0
$211$	−2.92820	−0.201586	−0.100793	−	0.994907i	$-0.532138\pi$
$211$	−2.92820	−0.201586	−0.100793	+	0.994907i	$0.532138\pi$
$212$	0	0
$213$	6.78461	0.464874
$214$	0	0
$215$	−7.26795	−0.495670
$216$	0	0
$217$	−0.679492	−0.0461269
$218$	0	0
$219$	2.14359	0.144851
$220$	0	0
$221$	7.46410	0.502090
$222$	0	0
$223$	−25.3205	−1.69559	−0.847793	−	0.530327i	$-0.822069\pi$
$223$	−25.3205	−1.69559	−0.847793	+	0.530327i	$0.822069\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	19.4641	1.28622	0.643112	−	0.765772i	$-0.277643\pi$
$229$	19.4641	1.28622	0.643112	+	0.765772i	$0.277643\pi$
$230$	0	0
$231$	−6.92820	−0.455842
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	4.53590	0.295889
$236$	0	0
$237$	−4.78461	−0.310794
$238$	0	0
$239$	−25.2679	−1.63445	−0.817224	−	0.576320i	$-0.804488\pi$
$239$	−25.2679	−1.63445	−0.817224	+	0.576320i	$0.804488\pi$
$240$	0	0
$241$	20.2487	1.30433	0.652167	−	0.758075i	$-0.273860\pi$
$241$	20.2487	1.30433	0.652167	+	0.758075i	$0.273860\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	0	0
$245$	−5.00000	−0.319438
$246$	0	0
$247$	2.73205	0.173836
$248$	0	0
$249$	−7.60770	−0.482118
$250$	0	0
$251$	−23.3205	−1.47198	−0.735989	−	0.676994i	$-0.763282\pi$
$251$	−23.3205	−1.47198	−0.735989	+	0.676994i	$0.763282\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	5.46410	0.342175
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−23.3205	−1.44350
$262$	0	0
$263$	11.6603	0.719002	0.359501	−	0.933145i	$-0.382947\pi$
$263$	11.6603	0.719002	0.359501	+	0.933145i	$0.382947\pi$
$264$	0	0
$265$	0.535898	0.0329200
$266$	0	0
$267$	9.46410	0.579194
$268$	0	0
$269$	−0.143594	−0.00875505	−0.00437753	−	0.999990i	$-0.501393\pi$
$269$	−0.143594	−0.00875505	−0.00437753	+	0.999990i	$0.501393\pi$
$270$	0	0
$271$	23.1244	1.40470	0.702352	−	0.711830i	$-0.252133\pi$
$271$	23.1244	1.40470	0.702352	+	0.711830i	$0.252133\pi$
$272$	0	0
$273$	2.53590	0.153480
$274$	0	0
$275$	−2.73205	−0.164749
$276$	0	0
$277$	−21.3205	−1.28103	−0.640513	−	0.767948i	$-0.721278\pi$
$277$	−21.3205	−1.28103	−0.640513	+	0.767948i	$0.721278\pi$
$278$	0	0
$279$	−0.483340	−0.0289368
$280$	0	0
$281$	−4.53590	−0.270589	−0.135295	−	0.990805i	$-0.543198\pi$
$281$	−4.53590	−0.270589	−0.135295	+	0.990805i	$0.543198\pi$
$282$	0	0
$283$	24.0526	1.42978	0.714888	−	0.699239i	$-0.246478\pi$
$283$	24.0526	1.42978	0.714888	+	0.699239i	$0.246478\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	−1.85641	−0.109580
$288$	0	0
$289$	38.7128	2.27722
$290$	0	0
$291$	−11.6077	−0.680455
$292$	0	0
$293$	6.92820	0.404750	0.202375	−	0.979308i	$-0.435134\pi$
$293$	6.92820	0.404750	0.202375	+	0.979308i	$0.435134\pi$
$294$	0	0
$295$	5.66025	0.329553
$296$	0	0
$297$	−10.9282	−0.634119
$298$	0	0
$299$	0.732051	0.0423356
$300$	0	0
$301$	−25.1769	−1.45117
$302$	0	0
$303$	−3.60770	−0.207257
$304$	0	0
$305$	12.3923	0.709581
$306$	0	0
$307$	16.5359	0.943754	0.471877	−	0.881665i	$-0.343577\pi$
$307$	16.5359	0.943754	0.471877	+	0.881665i	$0.343577\pi$
$308$	0	0
$309$	−4.53590	−0.258038
$310$	0	0
$311$	−31.3205	−1.77602	−0.888012	−	0.459821i	$-0.847914\pi$
$311$	−31.3205	−1.77602	−0.888012	+	0.459821i	$0.847914\pi$
$312$	0	0
$313$	10.3923	0.587408	0.293704	−	0.955896i	$-0.405112\pi$
$313$	10.3923	0.587408	0.293704	+	0.955896i	$0.405112\pi$
$314$	0	0
$315$	−8.53590	−0.480943
$316$	0	0
$317$	−1.85641	−0.104266	−0.0521331	−	0.998640i	$-0.516602\pi$
$317$	−1.85641	−0.104266	−0.0521331	+	0.998640i	$0.516602\pi$
$318$	0	0
$319$	−25.8564	−1.44768
$320$	0	0
$321$	−14.6795	−0.819329
$322$	0	0
$323$	20.3923	1.13466
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−11.6077	−0.641907
$328$	0	0
$329$	15.7128	0.866275
$330$	0	0
$331$	8.58846	0.472064	0.236032	−	0.971745i	$-0.424153\pi$
$331$	8.58846	0.472064	0.236032	+	0.971745i	$0.424153\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−12.9282	−0.706343
$336$	0	0
$337$	22.3923	1.21979	0.609893	−	0.792484i	$-0.291212\pi$
$337$	22.3923	1.21979	0.609893	+	0.792484i	$0.291212\pi$
$338$	0	0
$339$	13.4641	0.731270
$340$	0	0
$341$	−0.535898	−0.0290205
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0.535898	0.0288518
$346$	0	0
$347$	−2.19615	−0.117896	−0.0589478	−	0.998261i	$-0.518775\pi$
$347$	−2.19615	−0.117896	−0.0589478	+	0.998261i	$0.518775\pi$
$348$	0	0
$349$	12.5359	0.671031	0.335516	−	0.942035i	$-0.391089\pi$
$349$	12.5359	0.671031	0.335516	+	0.942035i	$0.391089\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−6.92820	−0.368751	−0.184376	−	0.982856i	$-0.559026\pi$
$353$	−6.92820	−0.368751	−0.184376	+	0.982856i	$0.559026\pi$
$354$	0	0
$355$	9.26795	0.491892
$356$	0	0
$357$	18.9282	1.00179
$358$	0	0
$359$	−11.1244	−0.587121	−0.293561	−	0.955940i	$-0.594840\pi$
$359$	−11.1244	−0.587121	−0.293561	+	0.955940i	$0.594840\pi$
$360$	0	0
$361$	−11.5359	−0.607153
$362$	0	0
$363$	2.58846	0.135859
$364$	0	0
$365$	2.92820	0.153269
$366$	0	0
$367$	1.41154	0.0736819	0.0368410	−	0.999321i	$-0.488271\pi$
$367$	1.41154	0.0736819	0.0368410	+	0.999321i	$0.488271\pi$
$368$	0	0
$369$	−1.32051	−0.0687429
$370$	0	0
$371$	1.85641	0.0963798
$372$	0	0
$373$	−7.85641	−0.406789	−0.203395	−	0.979097i	$-0.565197\pi$
$373$	−7.85641	−0.406789	−0.203395	+	0.979097i	$0.565197\pi$
$374$	0	0
$375$	0.732051	0.0378029
$376$	0	0
$377$	9.46410	0.487426
$378$	0	0
$379$	20.1962	1.03741	0.518703	−	0.854954i	$-0.326415\pi$
$379$	20.1962	1.03741	0.518703	+	0.854954i	$0.326415\pi$
$380$	0	0
$381$	−1.32051	−0.0676517
$382$	0	0
$383$	1.60770	0.0821494	0.0410747	−	0.999156i	$-0.486922\pi$
$383$	1.60770	0.0821494	0.0410747	+	0.999156i	$0.486922\pi$
$384$	0	0
$385$	−9.46410	−0.482335
$386$	0	0
$387$	−17.9090	−0.910364
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	5.46410	0.276331
$392$	0	0
$393$	−2.14359	−0.108130
$394$	0	0
$395$	−6.53590	−0.328857
$396$	0	0
$397$	−5.07180	−0.254546	−0.127273	−	0.991868i	$-0.540622\pi$
$397$	−5.07180	−0.254546	−0.127273	+	0.991868i	$0.540622\pi$
$398$	0	0
$399$	6.92820	0.346844
$400$	0	0
$401$	−10.7846	−0.538558	−0.269279	−	0.963062i	$-0.586785\pi$
$401$	−10.7846	−0.538558	−0.269279	+	0.963062i	$0.586785\pi$
$402$	0	0
$403$	0.196152	0.00977105
$404$	0	0
$405$	−4.46410	−0.221823
$406$	0	0
$407$	0	0
$408$	0	0
$409$	33.3205	1.64759	0.823797	−	0.566886i	$-0.191852\pi$
$409$	33.3205	1.64759	0.823797	+	0.566886i	$0.191852\pi$
$410$	0	0
$411$	−6.53590	−0.322392
$412$	0	0
$413$	19.6077	0.964832
$414$	0	0
$415$	−10.3923	−0.510138
$416$	0	0
$417$	−9.85641	−0.482670
$418$	0	0
$419$	25.4641	1.24400	0.622001	−	0.783016i	$-0.286320\pi$
$419$	25.4641	1.24400	0.622001	+	0.783016i	$0.286320\pi$
$420$	0	0
$421$	−19.0718	−0.929503	−0.464751	−	0.885441i	$-0.653856\pi$
$421$	−19.0718	−0.929503	−0.464751	+	0.885441i	$0.653856\pi$
$422$	0	0
$423$	11.1769	0.543440
$424$	0	0
$425$	7.46410	0.362062
$426$	0	0
$427$	42.9282	2.07744
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	22.0526	1.06223	0.531117	−	0.847298i	$-0.321772\pi$
$431$	22.0526	1.06223	0.531117	+	0.847298i	$0.321772\pi$
$432$	0	0
$433$	15.0718	0.724304	0.362152	−	0.932119i	$-0.382042\pi$
$433$	15.0718	0.724304	0.362152	+	0.932119i	$0.382042\pi$
$434$	0	0
$435$	6.92820	0.332182
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	13.8564	0.661330	0.330665	−	0.943748i	$-0.392727\pi$
$439$	13.8564	0.661330	0.330665	+	0.943748i	$0.392727\pi$
$440$	0	0
$441$	−12.3205	−0.586691
$442$	0	0
$443$	−28.4449	−1.35146	−0.675728	−	0.737151i	$-0.736171\pi$
$443$	−28.4449	−1.35146	−0.675728	+	0.737151i	$0.736171\pi$
$444$	0	0
$445$	12.9282	0.612856
$446$	0	0
$447$	5.75129	0.272027
$448$	0	0
$449$	−14.3923	−0.679215	−0.339607	−	0.940567i	$-0.610294\pi$
$449$	−14.3923	−0.679215	−0.339607	+	0.940567i	$0.610294\pi$
$450$	0	0
$451$	−1.46410	−0.0689419
$452$	0	0
$453$	8.64102	0.405990
$454$	0	0
$455$	3.46410	0.162400
$456$	0	0
$457$	−30.7846	−1.44004	−0.720022	−	0.693952i	$-0.755868\pi$
$457$	−30.7846	−1.44004	−0.720022	+	0.693952i	$0.755868\pi$
$458$	0	0
$459$	29.8564	1.39358
$460$	0	0
$461$	−12.2487	−0.570479	−0.285240	−	0.958456i	$-0.592073\pi$
$461$	−12.2487	−0.570479	−0.285240	+	0.958456i	$0.592073\pi$
$462$	0	0
$463$	−27.8564	−1.29460	−0.647298	−	0.762237i	$-0.724101\pi$
$463$	−27.8564	−1.29460	−0.647298	+	0.762237i	$0.724101\pi$
$464$	0	0
$465$	0.143594	0.00665899
$466$	0	0
$467$	−18.5885	−0.860171	−0.430086	−	0.902788i	$-0.641517\pi$
$467$	−18.5885	−0.860171	−0.430086	+	0.902788i	$0.641517\pi$
$468$	0	0
$469$	−44.7846	−2.06796
$470$	0	0
$471$	5.75129	0.265005
$472$	0	0
$473$	−19.8564	−0.912999
$474$	0	0
$475$	2.73205	0.125355
$476$	0	0
$477$	1.32051	0.0604619
$478$	0	0
$479$	−11.1244	−0.508285	−0.254142	−	0.967167i	$-0.581793\pi$
$479$	−11.1244	−0.508285	−0.254142	+	0.967167i	$0.581793\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.85641	0.0844694
$484$	0	0
$485$	−15.8564	−0.720002
$486$	0	0
$487$	39.8564	1.80607	0.903033	−	0.429571i	$-0.141335\pi$
$487$	39.8564	1.80607	0.903033	+	0.429571i	$0.141335\pi$
$488$	0	0
$489$	−6.53590	−0.295564
$490$	0	0
$491$	33.4641	1.51021	0.755107	−	0.655602i	$-0.227585\pi$
$491$	33.4641	1.51021	0.755107	+	0.655602i	$0.227585\pi$
$492$	0	0
$493$	70.6410	3.18151
$494$	0	0
$495$	−6.73205	−0.302583
$496$	0	0
$497$	32.1051	1.44011
$498$	0	0
$499$	−24.1962	−1.08317	−0.541584	−	0.840646i	$-0.682175\pi$
$499$	−24.1962	−1.08317	−0.541584	+	0.840646i	$0.682175\pi$
$500$	0	0
$501$	−15.6077	−0.697300
$502$	0	0
$503$	41.1244	1.83364	0.916822	−	0.399296i	$-0.130745\pi$
$503$	41.1244	1.83364	0.916822	+	0.399296i	$0.130745\pi$
$504$	0	0
$505$	−4.92820	−0.219302
$506$	0	0
$507$	−0.732051	−0.0325115
$508$	0	0
$509$	−3.46410	−0.153544	−0.0767718	−	0.997049i	$-0.524461\pi$
$509$	−3.46410	−0.153544	−0.0767718	+	0.997049i	$0.524461\pi$
$510$	0	0
$511$	10.1436	0.448726
$512$	0	0
$513$	10.9282	0.482492
$514$	0	0
$515$	−6.19615	−0.273035
$516$	0	0
$517$	12.3923	0.545013
$518$	0	0
$519$	1.75129	0.0768730
$520$	0	0
$521$	8.39230	0.367674	0.183837	−	0.982957i	$-0.441148\pi$
$521$	8.39230	0.367674	0.183837	+	0.982957i	$0.441148\pi$
$522$	0	0
$523$	−17.5167	−0.765950	−0.382975	−	0.923759i	$-0.625100\pi$
$523$	−17.5167	−0.765950	−0.382975	+	0.923759i	$0.625100\pi$
$524$	0	0
$525$	2.53590	0.110676
$526$	0	0
$527$	1.46410	0.0637773
$528$	0	0
$529$	−22.4641	−0.976700
$530$	0	0
$531$	13.9474	0.605267
$532$	0	0
$533$	0.535898	0.0232123
$534$	0	0
$535$	−20.0526	−0.866948
$536$	0	0
$537$	−16.0000	−0.690451
$538$	0	0
$539$	−13.6603	−0.588389
$540$	0	0
$541$	38.3923	1.65061	0.825307	−	0.564684i	$-0.191002\pi$
$541$	38.3923	1.65061	0.825307	+	0.564684i	$0.191002\pi$
$542$	0	0
$543$	−4.78461	−0.205327
$544$	0	0
$545$	−15.8564	−0.679214
$546$	0	0
$547$	8.33975	0.356582	0.178291	−	0.983978i	$-0.442943\pi$
$547$	8.33975	0.356582	0.178291	+	0.983978i	$0.442943\pi$
$548$	0	0
$549$	30.5359	1.30324
$550$	0	0
$551$	25.8564	1.10152
$552$	0	0
$553$	−22.6410	−0.962794
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.8564	0.926086	0.463043	−	0.886336i	$-0.346758\pi$
$557$	21.8564	0.926086	0.463043	+	0.886336i	$0.346758\pi$
$558$	0	0
$559$	7.26795	0.307401
$560$	0	0
$561$	14.9282	0.630269
$562$	0	0
$563$	−40.7321	−1.71665	−0.858326	−	0.513105i	$-0.828495\pi$
$563$	−40.7321	−1.71665	−0.858326	+	0.513105i	$0.828495\pi$
$564$	0	0
$565$	18.3923	0.773770
$566$	0	0
$567$	−15.4641	−0.649431
$568$	0	0
$569$	−17.4641	−0.732133	−0.366067	−	0.930589i	$-0.619296\pi$
$569$	−17.4641	−0.732133	−0.366067	+	0.930589i	$0.619296\pi$
$570$	0	0
$571$	27.3205	1.14333	0.571664	−	0.820488i	$-0.306298\pi$
$571$	27.3205	1.14333	0.571664	+	0.820488i	$0.306298\pi$
$572$	0	0
$573$	13.8564	0.578860
$574$	0	0
$575$	0.732051	0.0305286
$576$	0	0
$577$	−30.6410	−1.27560	−0.637801	−	0.770201i	$-0.720156\pi$
$577$	−30.6410	−1.27560	−0.637801	+	0.770201i	$0.720156\pi$
$578$	0	0
$579$	1.46410	0.0608460
$580$	0	0
$581$	−36.0000	−1.49353
$582$	0	0
$583$	1.46410	0.0606369
$584$	0	0
$585$	2.46410	0.101878
$586$	0	0
$587$	−1.21539	−0.0501645	−0.0250823	−	0.999685i	$-0.507985\pi$
$587$	−1.21539	−0.0501645	−0.0250823	+	0.999685i	$0.507985\pi$
$588$	0	0
$589$	0.535898	0.0220813
$590$	0	0
$591$	−6.53590	−0.268851
$592$	0	0
$593$	40.6410	1.66893	0.834463	−	0.551064i	$-0.185778\pi$
$593$	40.6410	1.66893	0.834463	+	0.551064i	$0.185778\pi$
$594$	0	0
$595$	25.8564	1.06001
$596$	0	0
$597$	−5.07180	−0.207575
$598$	0	0
$599$	33.4641	1.36731	0.683653	−	0.729807i	$-0.260390\pi$
$599$	33.4641	1.36731	0.683653	+	0.729807i	$0.260390\pi$
$600$	0	0
$601$	−3.85641	−0.157306	−0.0786531	−	0.996902i	$-0.525062\pi$
$601$	−3.85641	−0.157306	−0.0786531	+	0.996902i	$0.525062\pi$
$602$	0	0
$603$	−31.8564	−1.29729
$604$	0	0
$605$	3.53590	0.143755
$606$	0	0
$607$	13.5167	0.548624	0.274312	−	0.961641i	$-0.411550\pi$
$607$	13.5167	0.548624	0.274312	+	0.961641i	$0.411550\pi$
$608$	0	0
$609$	24.0000	0.972529
$610$	0	0
$611$	−4.53590	−0.183503
$612$	0	0
$613$	−0.143594	−0.00579969	−0.00289984	−	0.999996i	$-0.500923\pi$
$613$	−0.143594	−0.00579969	−0.00289984	+	0.999996i	$0.500923\pi$
$614$	0	0
$615$	0.392305	0.0158193
$616$	0	0
$617$	−45.7128	−1.84033	−0.920164	−	0.391533i	$-0.871945\pi$
$617$	−45.7128	−1.84033	−0.920164	+	0.391533i	$0.871945\pi$
$618$	0	0
$619$	18.3397	0.737137	0.368568	−	0.929601i	$-0.379848\pi$
$619$	18.3397	0.737137	0.368568	+	0.929601i	$0.379848\pi$
$620$	0	0
$621$	2.92820	0.117505
$622$	0	0
$623$	44.7846	1.79426
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.46410	0.218215
$628$	0	0
$629$	0	0
$630$	0	0
$631$	7.80385	0.310666	0.155333	−	0.987862i	$-0.450355\pi$
$631$	7.80385	0.310666	0.155333	+	0.987862i	$0.450355\pi$
$632$	0	0
$633$	2.14359	0.0852002
$634$	0	0
$635$	−1.80385	−0.0715835
$636$	0	0
$637$	5.00000	0.198107
$638$	0	0
$639$	22.8372	0.903424
$640$	0	0
$641$	−0.928203	−0.0366618	−0.0183309	−	0.999832i	$-0.505835\pi$
$641$	−0.928203	−0.0366618	−0.0183309	+	0.999832i	$0.505835\pi$
$642$	0	0
$643$	9.60770	0.378891	0.189445	−	0.981891i	$-0.439331\pi$
$643$	9.60770	0.378891	0.189445	+	0.981891i	$0.439331\pi$
$644$	0	0
$645$	5.32051	0.209495
$646$	0	0
$647$	−0.339746	−0.0133568	−0.00667840	−	0.999978i	$-0.502126\pi$
$647$	−0.339746	−0.0133568	−0.00667840	+	0.999978i	$0.502126\pi$
$648$	0	0
$649$	15.4641	0.607019
$650$	0	0
$651$	0.497423	0.0194955
$652$	0	0
$653$	−33.7128	−1.31928	−0.659642	−	0.751580i	$-0.729292\pi$
$653$	−33.7128	−1.31928	−0.659642	+	0.751580i	$0.729292\pi$
$654$	0	0
$655$	−2.92820	−0.114414
$656$	0	0
$657$	7.21539	0.281499
$658$	0	0
$659$	−39.3205	−1.53171	−0.765855	−	0.643014i	$-0.777684\pi$
$659$	−39.3205	−1.53171	−0.765855	+	0.643014i	$0.777684\pi$
$660$	0	0
$661$	20.9282	0.814013	0.407006	−	0.913425i	$-0.366573\pi$
$661$	20.9282	0.814013	0.407006	+	0.913425i	$0.366573\pi$
$662$	0	0
$663$	−5.46410	−0.212208
$664$	0	0
$665$	9.46410	0.367002
$666$	0	0
$667$	6.92820	0.268261
$668$	0	0
$669$	18.5359	0.716639
$670$	0	0
$671$	33.8564	1.30701
$672$	0	0
$673$	28.5359	1.09998	0.549989	−	0.835172i	$-0.314632\pi$
$673$	28.5359	1.09998	0.549989	+	0.835172i	$0.314632\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	9.60770	0.369254	0.184627	−	0.982809i	$-0.440892\pi$
$677$	9.60770	0.369254	0.184627	+	0.982809i	$0.440892\pi$
$678$	0	0
$679$	−54.9282	−2.10795
$680$	0	0
$681$	13.1769	0.504940
$682$	0	0
$683$	−24.9282	−0.953851	−0.476926	−	0.878944i	$-0.658249\pi$
$683$	−24.9282	−0.953851	−0.476926	+	0.878944i	$0.658249\pi$
$684$	0	0
$685$	−8.92820	−0.341129
$686$	0	0
$687$	−14.2487	−0.543622
$688$	0	0
$689$	−0.535898	−0.0204161
$690$	0	0
$691$	−43.9090	−1.67038	−0.835188	−	0.549965i	$-0.814641\pi$
$691$	−43.9090	−1.67038	−0.835188	+	0.549965i	$0.814641\pi$
$692$	0	0
$693$	−23.3205	−0.885873
$694$	0	0
$695$	−13.4641	−0.510722
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	−13.1769	−0.498397
$700$	0	0
$701$	29.7128	1.12224	0.561119	−	0.827735i	$-0.310371\pi$
$701$	29.7128	1.12224	0.561119	+	0.827735i	$0.310371\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−3.32051	−0.125058
$706$	0	0
$707$	−17.0718	−0.642051
$708$	0	0
$709$	5.60770	0.210601	0.105301	−	0.994440i	$-0.466419\pi$
$709$	5.60770	0.210601	0.105301	+	0.994440i	$0.466419\pi$
$710$	0	0
$711$	−16.1051	−0.603989
$712$	0	0
$713$	0.143594	0.00537762
$714$	0	0
$715$	2.73205	0.102173
$716$	0	0
$717$	18.4974	0.690799
$718$	0	0
$719$	34.6410	1.29189	0.645946	−	0.763383i	$-0.276463\pi$
$719$	34.6410	1.29189	0.645946	+	0.763383i	$0.276463\pi$
$720$	0	0
$721$	−21.4641	−0.799365
$722$	0	0
$723$	−14.8231	−0.551276
$724$	0	0
$725$	9.46410	0.351488
$726$	0	0
$727$	34.9808	1.29736	0.648682	−	0.761059i	$-0.275320\pi$
$727$	34.9808	1.29736	0.648682	+	0.761059i	$0.275320\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	54.2487	2.00646
$732$	0	0
$733$	−15.8564	−0.585670	−0.292835	−	0.956163i	$-0.594599\pi$
$733$	−15.8564	−0.585670	−0.292835	+	0.956163i	$0.594599\pi$
$734$	0	0
$735$	3.66025	0.135011
$736$	0	0
$737$	−35.3205	−1.30105
$738$	0	0
$739$	−1.26795	−0.0466423	−0.0233211	−	0.999728i	$-0.507424\pi$
$739$	−1.26795	−0.0466423	−0.0233211	+	0.999728i	$0.507424\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	−25.3205	−0.928919	−0.464460	−	0.885594i	$-0.653751\pi$
$743$	−25.3205	−0.928919	−0.464460	+	0.885594i	$0.653751\pi$
$744$	0	0
$745$	7.85641	0.287836
$746$	0	0
$747$	−25.6077	−0.936937
$748$	0	0
$749$	−69.4641	−2.53816
$750$	0	0
$751$	−45.4641	−1.65901	−0.829504	−	0.558500i	$-0.811377\pi$
$751$	−45.4641	−1.65901	−0.829504	+	0.558500i	$0.811377\pi$
$752$	0	0
$753$	17.0718	0.622131
$754$	0	0
$755$	11.8038	0.429586
$756$	0	0
$757$	−23.4641	−0.852817	−0.426409	−	0.904531i	$-0.640221\pi$
$757$	−23.4641	−0.852817	−0.426409	+	0.904531i	$0.640221\pi$
$758$	0	0
$759$	1.46410	0.0531435
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	0	0
$763$	−54.9282	−1.98853
$764$	0	0
$765$	18.3923	0.664975
$766$	0	0
$767$	−5.66025	−0.204380
$768$	0	0
$769$	15.0718	0.543503	0.271751	−	0.962367i	$-0.412397\pi$
$769$	15.0718	0.543503	0.271751	+	0.962367i	$0.412397\pi$
$770$	0	0
$771$	−4.39230	−0.158185
$772$	0	0
$773$	5.07180	0.182420	0.0912099	−	0.995832i	$-0.470927\pi$
$773$	5.07180	0.182420	0.0912099	+	0.995832i	$0.470927\pi$
$774$	0	0
$775$	0.196152	0.00704600
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.46410	0.0524569
$780$	0	0
$781$	25.3205	0.906039
$782$	0	0
$783$	37.8564	1.35288
$784$	0	0
$785$	7.85641	0.280407
$786$	0	0
$787$	−18.3923	−0.655615	−0.327807	−	0.944745i	$-0.606310\pi$
$787$	−18.3923	−0.655615	−0.327807	+	0.944745i	$0.606310\pi$
$788$	0	0
$789$	−8.53590	−0.303886
$790$	0	0
$791$	63.7128	2.26537
$792$	0	0
$793$	−12.3923	−0.440064
$794$	0	0
$795$	−0.392305	−0.0139136
$796$	0	0
$797$	46.7846	1.65720	0.828598	−	0.559844i	$-0.189139\pi$
$797$	46.7846	1.65720	0.828598	+	0.559844i	$0.189139\pi$
$798$	0	0
$799$	−33.8564	−1.19775
$800$	0	0
$801$	31.8564	1.12559
$802$	0	0
$803$	8.00000	0.282314
$804$	0	0
$805$	2.53590	0.0893787
$806$	0	0
$807$	0.105118	0.00370032
$808$	0	0
$809$	8.67949	0.305155	0.152577	−	0.988292i	$-0.451243\pi$
$809$	8.67949	0.305155	0.152577	+	0.988292i	$0.451243\pi$
$810$	0	0
$811$	−35.9090	−1.26093	−0.630467	−	0.776216i	$-0.717137\pi$
$811$	−35.9090	−1.26093	−0.630467	+	0.776216i	$0.717137\pi$
$812$	0	0
$813$	−16.9282	−0.593698
$814$	0	0
$815$	−8.92820	−0.312741
$816$	0	0
$817$	19.8564	0.694688
$818$	0	0
$819$	8.53590	0.298268
$820$	0	0
$821$	−12.9282	−0.451197	−0.225599	−	0.974220i	$-0.572434\pi$
$821$	−12.9282	−0.451197	−0.225599	+	0.974220i	$0.572434\pi$
$822$	0	0
$823$	−26.5885	−0.926815	−0.463408	−	0.886145i	$-0.653373\pi$
$823$	−26.5885	−0.926815	−0.463408	+	0.886145i	$0.653373\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	−3.46410	−0.120459	−0.0602293	−	0.998185i	$-0.519183\pi$
$827$	−3.46410	−0.120459	−0.0602293	+	0.998185i	$0.519183\pi$
$828$	0	0
$829$	44.3923	1.54181	0.770904	−	0.636951i	$-0.219805\pi$
$829$	44.3923	1.54181	0.770904	+	0.636951i	$0.219805\pi$
$830$	0	0
$831$	15.6077	0.541425
$832$	0	0
$833$	37.3205	1.29308
$834$	0	0
$835$	−21.3205	−0.737827
$836$	0	0
$837$	0.784610	0.0271201
$838$	0	0
$839$	−34.4449	−1.18917	−0.594584	−	0.804033i	$-0.702683\pi$
$839$	−34.4449	−1.18917	−0.594584	+	0.804033i	$0.702683\pi$
$840$	0	0
$841$	60.5692	2.08859
$842$	0	0
$843$	3.32051	0.114364
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	12.2487	0.420871
$848$	0	0
$849$	−17.6077	−0.604295
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−7.71281	−0.264082	−0.132041	−	0.991244i	$-0.542153\pi$
$853$	−7.71281	−0.264082	−0.132041	+	0.991244i	$0.542153\pi$
$854$	0	0
$855$	6.73205	0.230231
$856$	0	0
$857$	−4.14359	−0.141542	−0.0707712	−	0.997493i	$-0.522546\pi$
$857$	−4.14359	−0.141542	−0.0707712	+	0.997493i	$0.522546\pi$
$858$	0	0
$859$	12.3923	0.422820	0.211410	−	0.977397i	$-0.432194\pi$
$859$	12.3923	0.422820	0.211410	+	0.977397i	$0.432194\pi$
$860$	0	0
$861$	1.35898	0.0463140
$862$	0	0
$863$	33.6077	1.14402	0.572010	−	0.820247i	$-0.306164\pi$
$863$	33.6077	1.14402	0.572010	+	0.820247i	$0.306164\pi$
$864$	0	0
$865$	2.39230	0.0813408
$866$	0	0
$867$	−28.3397	−0.962468
$868$	0	0
$869$	−17.8564	−0.605737
$870$	0	0
$871$	12.9282	0.438055
$872$	0	0
$873$	−39.0718	−1.32238
$874$	0	0
$875$	3.46410	0.117108
$876$	0	0
$877$	−41.7128	−1.40854	−0.704271	−	0.709931i	$-0.748726\pi$
$877$	−41.7128	−1.40854	−0.704271	+	0.709931i	$0.748726\pi$
$878$	0	0
$879$	−5.07180	−0.171067
$880$	0	0
$881$	5.46410	0.184090	0.0920451	−	0.995755i	$-0.470660\pi$
$881$	5.46410	0.184090	0.0920451	+	0.995755i	$0.470660\pi$
$882$	0	0
$883$	−22.5885	−0.760162	−0.380081	−	0.924953i	$-0.624104\pi$
$883$	−22.5885	−0.760162	−0.380081	+	0.924953i	$0.624104\pi$
$884$	0	0
$885$	−4.14359	−0.139285
$886$	0	0
$887$	−12.4449	−0.417858	−0.208929	−	0.977931i	$-0.566998\pi$
$887$	−12.4449	−0.417858	−0.208929	+	0.977931i	$0.566998\pi$
$888$	0	0
$889$	−6.24871	−0.209575
$890$	0	0
$891$	−12.1962	−0.408586
$892$	0	0
$893$	−12.3923	−0.414693
$894$	0	0
$895$	−21.8564	−0.730579
$896$	0	0
$897$	−0.535898	−0.0178931
$898$	0	0
$899$	1.85641	0.0619146
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	18.4308	0.613338
$904$	0	0
$905$	−6.53590	−0.217261
$906$	0	0
$907$	36.8372	1.22316	0.611579	−	0.791183i	$-0.290535\pi$
$907$	36.8372	1.22316	0.611579	+	0.791183i	$0.290535\pi$
$908$	0	0
$909$	−12.1436	−0.402778
$910$	0	0
$911$	20.7846	0.688625	0.344312	−	0.938855i	$-0.388112\pi$
$911$	20.7846	0.688625	0.344312	+	0.938855i	$0.388112\pi$
$912$	0	0
$913$	−28.3923	−0.939648
$914$	0	0
$915$	−9.07180	−0.299904
$916$	0	0
$917$	−10.1436	−0.334971
$918$	0	0
$919$	−39.3205	−1.29706	−0.648532	−	0.761187i	$-0.724617\pi$
$919$	−39.3205	−1.29706	−0.648532	+	0.761187i	$0.724617\pi$
$920$	0	0
$921$	−12.1051	−0.398877
$922$	0	0
$923$	−9.26795	−0.305058
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−15.2679	−0.501465
$928$	0	0
$929$	43.1769	1.41659	0.708294	−	0.705917i	$-0.249465\pi$
$929$	43.1769	1.41659	0.708294	+	0.705917i	$0.249465\pi$
$930$	0	0
$931$	13.6603	0.447697
$932$	0	0
$933$	22.9282	0.750636
$934$	0	0
$935$	20.3923	0.666900
$936$	0	0
$937$	−27.0718	−0.884397	−0.442199	−	0.896917i	$-0.645801\pi$
$937$	−27.0718	−0.884397	−0.442199	+	0.896917i	$0.645801\pi$
$938$	0	0
$939$	−7.60770	−0.248268
$940$	0	0
$941$	22.3923	0.729968	0.364984	−	0.931014i	$-0.381074\pi$
$941$	22.3923	0.729968	0.364984	+	0.931014i	$0.381074\pi$
$942$	0	0
$943$	0.392305	0.0127752
$944$	0	0
$945$	13.8564	0.450749
$946$	0	0
$947$	−22.3923	−0.727652	−0.363826	−	0.931467i	$-0.618530\pi$
$947$	−22.3923	−0.727652	−0.363826	+	0.931467i	$0.618530\pi$
$948$	0	0
$949$	−2.92820	−0.0950535
$950$	0	0
$951$	1.35898	0.0440681
$952$	0	0
$953$	−1.21539	−0.0393704	−0.0196852	−	0.999806i	$-0.506266\pi$
$953$	−1.21539	−0.0393704	−0.0196852	+	0.999806i	$0.506266\pi$
$954$	0	0
$955$	18.9282	0.612502
$956$	0	0
$957$	18.9282	0.611862
$958$	0	0
$959$	−30.9282	−0.998724
$960$	0	0
$961$	−30.9615	−0.998759
$962$	0	0
$963$	−49.4115	−1.59226
$964$	0	0
$965$	2.00000	0.0643823
$966$	0	0
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	0	0
$969$	−14.9282	−0.479563
$970$	0	0
$971$	−43.7128	−1.40281	−0.701405	−	0.712762i	$-0.747444\pi$
$971$	−43.7128	−1.40281	−0.701405	+	0.712762i	$0.747444\pi$
$972$	0	0
$973$	−46.6410	−1.49524
$974$	0	0
$975$	−0.732051	−0.0234444
$976$	0	0
$977$	−2.92820	−0.0936815	−0.0468408	−	0.998902i	$-0.514915\pi$
$977$	−2.92820	−0.0936815	−0.0468408	+	0.998902i	$0.514915\pi$
$978$	0	0
$979$	35.3205	1.12885
$980$	0	0
$981$	−39.0718	−1.24747
$982$	0	0
$983$	15.8564	0.505741	0.252870	−	0.967500i	$-0.418625\pi$
$983$	15.8564	0.505741	0.252870	+	0.967500i	$0.418625\pi$
$984$	0	0
$985$	−8.92820	−0.284476
$986$	0	0
$987$	−11.5026	−0.366131
$988$	0	0
$989$	5.32051	0.169182
$990$	0	0
$991$	53.8564	1.71081	0.855403	−	0.517964i	$-0.173310\pi$
$991$	53.8564	1.71081	0.855403	+	0.517964i	$0.173310\pi$
$992$	0	0
$993$	−6.28719	−0.199518
$994$	0	0
$995$	−6.92820	−0.219639
$996$	0	0
$997$	−9.32051	−0.295183	−0.147592	−	0.989048i	$-0.547152\pi$
$997$	−9.32051	−0.295183	−0.147592	+	0.989048i	$0.547152\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.a.l.1.1		2
3.2	odd	2		9360.2.a.cr.1.1		2
4.3	odd	2		520.2.a.e.1.2	✓	2
5.4	even	2		5200.2.a.bn.1.2		2
8.3	odd	2		4160.2.a.bk.1.1		2
8.5	even	2		4160.2.a.w.1.2		2
12.11	even	2		4680.2.a.bd.1.2		2
20.3	even	4		2600.2.d.m.1249.3		4
20.7	even	4		2600.2.d.m.1249.2		4
20.19	odd	2		2600.2.a.v.1.1		2
52.51	odd	2		6760.2.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.e.1.2	✓	2	4.3	odd	2
1040.2.a.l.1.1		2	1.1	even	1	trivial
2600.2.a.v.1.1		2	20.19	odd	2
2600.2.d.m.1249.2		4	20.7	even	4
2600.2.d.m.1249.3		4	20.3	even	4
4160.2.a.w.1.2		2	8.5	even	2
4160.2.a.bk.1.1		2	8.3	odd	2
4680.2.a.bd.1.2		2	12.11	even	2
5200.2.a.bn.1.2		2	5.4	even	2
6760.2.a.o.1.2		2	52.51	odd	2
9360.2.a.cr.1.1		2	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{3} -1.00000 q^{5} -3.46410 q^{7} -2.46410 q^{9} -2.73205 q^{11} +1.00000 q^{13} +0.732051 q^{15} +7.46410 q^{17} +2.73205 q^{19} +2.53590 q^{21} +0.732051 q^{23} +1.00000 q^{25} +4.00000 q^{27} +9.46410 q^{29} +0.196152 q^{31} +2.00000 q^{33} +3.46410 q^{35} -0.732051 q^{39} +0.535898 q^{41} +7.26795 q^{43} +2.46410 q^{45} -4.53590 q^{47} +5.00000 q^{49} -5.46410 q^{51} -0.535898 q^{53} +2.73205 q^{55} -2.00000 q^{57} -5.66025 q^{59} -12.3923 q^{61} +8.53590 q^{63} -1.00000 q^{65} +12.9282 q^{67} -0.535898 q^{69} -9.26795 q^{71} -2.92820 q^{73} -0.732051 q^{75} +9.46410 q^{77} +6.53590 q^{79} +4.46410 q^{81} +10.3923 q^{83} -7.46410 q^{85} -6.92820 q^{87} -12.9282 q^{89} -3.46410 q^{91} -0.143594 q^{93} -2.73205 q^{95} +15.8564 q^{97} +6.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{15} + 8 q^{17} + 2 q^{19} + 12 q^{21} - 2 q^{23} + 2 q^{25} + 8 q^{27} + 12 q^{29} - 10 q^{31} + 4 q^{33} + 2 q^{39} + 8 q^{41} + 18 q^{43}+ \cdots + 10 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^15 + 8 * q^17 + 2 * q^19 + 12 * q^21 - 2 * q^23 + 2 * q^25 + 8 * q^27 + 12 * q^29 - 10 * q^31 + 4 * q^33 + 2 * q^39 + 8 * q^41 + 18 * q^43 - 2 * q^45 - 16 * q^47 + 10 * q^49 - 4 * q^51 - 8 * q^53 + 2 * q^55 - 4 * q^57 + 6 * q^59 - 4 * q^61 + 24 * q^63 - 2 * q^65 + 12 * q^67 - 8 * q^69 - 22 * q^71 + 8 * q^73 + 2 * q^75 + 12 * q^77 + 20 * q^79 + 2 * q^81 - 8 * q^85 - 12 * q^89 - 28 * q^93 - 2 * q^95 + 4 * q^97 + 10 * q^99

Introduction

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