LMFDB - Embedded newform 8016.2.a.bg.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + \cdots + 12144 $ x^13 - 2x^12 - 49x^11 + 99x^10 + 901x^9 - 1879x^8 - 7582x^7 + 16968x^6 + 26911x^5 - 72240x^4 - 14532x^3 + 112850x^2 - 72184x + 12144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.284022$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.284022 q^{5} +3.90646 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.284022 q^{5} +3.90646 q^{7} +1.00000 q^{9} -5.93953 q^{11} +5.62131 q^{13} -0.284022 q^{15} -0.211600 q^{17} +1.66151 q^{19} -3.90646 q^{21} +2.16524 q^{23} -4.91933 q^{25} -1.00000 q^{27} +4.18652 q^{29} +2.41173 q^{31} +5.93953 q^{33} +1.10952 q^{35} +0.0651212 q^{37} -5.62131 q^{39} +6.23450 q^{41} -2.49039 q^{43} +0.284022 q^{45} -5.97532 q^{47} +8.26040 q^{49} +0.211600 q^{51} +4.43542 q^{53} -1.68695 q^{55} -1.66151 q^{57} +3.03679 q^{59} -5.18295 q^{61} +3.90646 q^{63} +1.59657 q^{65} +5.08568 q^{67} -2.16524 q^{69} -6.06371 q^{71} +8.31501 q^{73} +4.91933 q^{75} -23.2025 q^{77} +2.87558 q^{79} +1.00000 q^{81} +5.50738 q^{83} -0.0600991 q^{85} -4.18652 q^{87} +0.660702 q^{89} +21.9594 q^{91} -2.41173 q^{93} +0.471905 q^{95} +17.4002 q^{97} -5.93953 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9} - 11 q^{11} + 12 q^{13} - 2 q^{15} + 15 q^{17} - 14 q^{19} + q^{21} - 9 q^{23} + 37 q^{25} - 13 q^{27} - 3 q^{29} + 17 q^{31} + 11 q^{33} - 15 q^{35} + 16 q^{37} - 12 q^{39} + 12 q^{41} - 20 q^{43} + 2 q^{45} + 6 q^{47} + 26 q^{49} - 15 q^{51} - 12 q^{53} - 7 q^{55} + 14 q^{57} - 14 q^{59} + 24 q^{61} - q^{63} + 8 q^{65} - 3 q^{67} + 9 q^{69} - 17 q^{71} + 34 q^{73} - 37 q^{75} + 30 q^{77} - 10 q^{79} + 13 q^{81} - 44 q^{83} + 25 q^{85} + 3 q^{87} + 25 q^{89} - 29 q^{91} - 17 q^{93} + 15 q^{95} + 38 q^{97} - 11 q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9 - 11 * q^11 + 12 * q^13 - 2 * q^15 + 15 * q^17 - 14 * q^19 + q^21 - 9 * q^23 + 37 * q^25 - 13 * q^27 - 3 * q^29 + 17 * q^31 + 11 * q^33 - 15 * q^35 + 16 * q^37 - 12 * q^39 + 12 * q^41 - 20 * q^43 + 2 * q^45 + 6 * q^47 + 26 * q^49 - 15 * q^51 - 12 * q^53 - 7 * q^55 + 14 * q^57 - 14 * q^59 + 24 * q^61 - q^63 + 8 * q^65 - 3 * q^67 + 9 * q^69 - 17 * q^71 + 34 * q^73 - 37 * q^75 + 30 * q^77 - 10 * q^79 + 13 * q^81 - 44 * q^83 + 25 * q^85 + 3 * q^87 + 25 * q^89 - 29 * q^91 - 17 * q^93 + 15 * q^95 + 38 * q^97 - 11 * 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
$5$	0.284022	0.127018	0.0635092	−	0.997981i	$-0.479771\pi$
$5$	0.284022	0.127018	0.0635092	+	0.997981i	$0.479771\pi$
$6$	0	0
$7$	3.90646	1.47650	0.738251	−	0.674526i	$-0.235652\pi$
$7$	3.90646	1.47650	0.738251	+	0.674526i	$0.235652\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.93953	−1.79083	−0.895417	−	0.445228i	$-0.853123\pi$
$11$	−5.93953	−1.79083	−0.895417	+	0.445228i	$0.853123\pi$
$12$	0	0
$13$	5.62131	1.55907	0.779535	−	0.626358i	$-0.215455\pi$
$13$	5.62131	1.55907	0.779535	+	0.626358i	$0.215455\pi$
$14$	0	0
$15$	−0.284022	−0.0733341
$16$	0	0
$17$	−0.211600	−0.0513206	−0.0256603	−	0.999671i	$-0.508169\pi$
$17$	−0.211600	−0.0513206	−0.0256603	+	0.999671i	$0.508169\pi$
$18$	0	0
$19$	1.66151	0.381177	0.190588	−	0.981670i	$-0.438960\pi$
$19$	1.66151	0.381177	0.190588	+	0.981670i	$0.438960\pi$
$20$	0	0
$21$	−3.90646	−0.852459
$22$	0	0
$23$	2.16524	0.451484	0.225742	−	0.974187i	$-0.427519\pi$
$23$	2.16524	0.451484	0.225742	+	0.974187i	$0.427519\pi$
$24$	0	0
$25$	−4.91933	−0.983866
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.18652	0.777416	0.388708	−	0.921361i	$-0.372921\pi$
$29$	4.18652	0.777416	0.388708	+	0.921361i	$0.372921\pi$
$30$	0	0
$31$	2.41173	0.433159	0.216580	−	0.976265i	$-0.430510\pi$
$31$	2.41173	0.433159	0.216580	+	0.976265i	$0.430510\pi$
$32$	0	0
$33$	5.93953	1.03394
$34$	0	0
$35$	1.10952	0.187543
$36$	0	0
$37$	0.0651212	0.0107059	0.00535293	−	0.999986i	$-0.498296\pi$
$37$	0.0651212	0.0107059	0.00535293	+	0.999986i	$0.498296\pi$
$38$	0	0
$39$	−5.62131	−0.900130
$40$	0	0
$41$	6.23450	0.973666	0.486833	−	0.873495i	$-0.338152\pi$
$41$	6.23450	0.973666	0.486833	+	0.873495i	$0.338152\pi$
$42$	0	0
$43$	−2.49039	−0.379781	−0.189891	−	0.981805i	$-0.560813\pi$
$43$	−2.49039	−0.379781	−0.189891	+	0.981805i	$0.560813\pi$
$44$	0	0
$45$	0.284022	0.0423395
$46$	0	0
$47$	−5.97532	−0.871591	−0.435795	−	0.900046i	$-0.643533\pi$
$47$	−5.97532	−0.871591	−0.435795	+	0.900046i	$0.643533\pi$
$48$	0	0
$49$	8.26040	1.18006
$50$	0	0
$51$	0.211600	0.0296300
$52$	0	0
$53$	4.43542	0.609252	0.304626	−	0.952472i	$-0.401468\pi$
$53$	4.43542	0.609252	0.304626	+	0.952472i	$0.401468\pi$
$54$	0	0
$55$	−1.68695	−0.227469
$56$	0	0
$57$	−1.66151	−0.220072
$58$	0	0
$59$	3.03679	0.395357	0.197679	−	0.980267i	$-0.436660\pi$
$59$	3.03679	0.395357	0.197679	+	0.980267i	$0.436660\pi$
$60$	0	0
$61$	−5.18295	−0.663609	−0.331804	−	0.943348i	$-0.607657\pi$
$61$	−5.18295	−0.663609	−0.331804	+	0.943348i	$0.607657\pi$
$62$	0	0
$63$	3.90646	0.492167
$64$	0	0
$65$	1.59657	0.198031
$66$	0	0
$67$	5.08568	0.621315	0.310658	−	0.950522i	$-0.399451\pi$
$67$	5.08568	0.621315	0.310658	+	0.950522i	$0.399451\pi$
$68$	0	0
$69$	−2.16524	−0.260664
$70$	0	0
$71$	−6.06371	−0.719630	−0.359815	−	0.933024i	$-0.617160\pi$
$71$	−6.06371	−0.719630	−0.359815	+	0.933024i	$0.617160\pi$
$72$	0	0
$73$	8.31501	0.973199	0.486599	−	0.873625i	$-0.338237\pi$
$73$	8.31501	0.973199	0.486599	+	0.873625i	$0.338237\pi$
$74$	0	0
$75$	4.91933	0.568035
$76$	0	0
$77$	−23.2025	−2.64417
$78$	0	0
$79$	2.87558	0.323528	0.161764	−	0.986830i	$-0.448282\pi$
$79$	2.87558	0.323528	0.161764	+	0.986830i	$0.448282\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.50738	0.604514	0.302257	−	0.953227i	$-0.402260\pi$
$83$	5.50738	0.604514	0.302257	+	0.953227i	$0.402260\pi$
$84$	0	0
$85$	−0.0600991	−0.00651866
$86$	0	0
$87$	−4.18652	−0.448842
$88$	0	0
$89$	0.660702	0.0700343	0.0350171	−	0.999387i	$-0.488851\pi$
$89$	0.660702	0.0700343	0.0350171	+	0.999387i	$0.488851\pi$
$90$	0	0
$91$	21.9594	2.30197
$92$	0	0
$93$	−2.41173	−0.250085
$94$	0	0
$95$	0.471905	0.0484164
$96$	0	0
$97$	17.4002	1.76672	0.883360	−	0.468694i	$-0.155275\pi$
$97$	17.4002	1.76672	0.883360	+	0.468694i	$0.155275\pi$
$98$	0	0
$99$	−5.93953	−0.596945
$100$	0	0
$101$	0.510196	0.0507664	0.0253832	−	0.999678i	$-0.491919\pi$
$101$	0.510196	0.0507664	0.0253832	+	0.999678i	$0.491919\pi$
$102$	0	0
$103$	−6.73893	−0.664007	−0.332003	−	0.943278i	$-0.607725\pi$
$103$	−6.73893	−0.664007	−0.332003	+	0.943278i	$0.607725\pi$
$104$	0	0
$105$	−1.10952	−0.108278
$106$	0	0
$107$	−6.94201	−0.671109	−0.335555	−	0.942021i	$-0.608924\pi$
$107$	−6.94201	−0.671109	−0.335555	+	0.942021i	$0.608924\pi$
$108$	0	0
$109$	−2.47553	−0.237112	−0.118556	−	0.992947i	$-0.537827\pi$
$109$	−2.47553	−0.237112	−0.118556	+	0.992947i	$0.537827\pi$
$110$	0	0
$111$	−0.0651212	−0.00618104
$112$	0	0
$113$	5.81283	0.546825	0.273413	−	0.961897i	$-0.411848\pi$
$113$	5.81283	0.546825	0.273413	+	0.961897i	$0.411848\pi$
$114$	0	0
$115$	0.614976	0.0573468
$116$	0	0
$117$	5.62131	0.519690
$118$	0	0
$119$	−0.826607	−0.0757750
$120$	0	0
$121$	24.2780	2.20709
$122$	0	0
$123$	−6.23450	−0.562146
$124$	0	0
$125$	−2.81731	−0.251987
$126$	0	0
$127$	9.44159	0.837806	0.418903	−	0.908031i	$-0.362415\pi$
$127$	9.44159	0.837806	0.418903	+	0.908031i	$0.362415\pi$
$128$	0	0
$129$	2.49039	0.219267
$130$	0	0
$131$	−10.0575	−0.878731	−0.439365	−	0.898308i	$-0.644797\pi$
$131$	−10.0575	−0.878731	−0.439365	+	0.898308i	$0.644797\pi$
$132$	0	0
$133$	6.49062	0.562808
$134$	0	0
$135$	−0.284022	−0.0244447
$136$	0	0
$137$	7.06922	0.603964	0.301982	−	0.953314i	$-0.402352\pi$
$137$	7.06922	0.603964	0.301982	+	0.953314i	$0.402352\pi$
$138$	0	0
$139$	2.77510	0.235380	0.117690	−	0.993050i	$-0.462451\pi$
$139$	2.77510	0.235380	0.117690	+	0.993050i	$0.462451\pi$
$140$	0	0
$141$	5.97532	0.503213
$142$	0	0
$143$	−33.3879	−2.79204
$144$	0	0
$145$	1.18906	0.0987462
$146$	0	0
$147$	−8.26040	−0.681306
$148$	0	0
$149$	−4.33933	−0.355492	−0.177746	−	0.984076i	$-0.556881\pi$
$149$	−4.33933	−0.355492	−0.177746	+	0.984076i	$0.556881\pi$
$150$	0	0
$151$	10.7758	0.876921	0.438460	−	0.898751i	$-0.355524\pi$
$151$	10.7758	0.876921	0.438460	+	0.898751i	$0.355524\pi$
$152$	0	0
$153$	−0.211600	−0.0171069
$154$	0	0
$155$	0.684984	0.0550192
$156$	0	0
$157$	2.05881	0.164311	0.0821554	−	0.996620i	$-0.473820\pi$
$157$	2.05881	0.164311	0.0821554	+	0.996620i	$0.473820\pi$
$158$	0	0
$159$	−4.43542	−0.351752
$160$	0	0
$161$	8.45842	0.666617
$162$	0	0
$163$	−16.7791	−1.31424	−0.657120	−	0.753786i	$-0.728225\pi$
$163$	−16.7791	−1.31424	−0.657120	+	0.753786i	$0.728225\pi$
$164$	0	0
$165$	1.68695	0.131329
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	18.5991	1.43070
$170$	0	0
$171$	1.66151	0.127059
$172$	0	0
$173$	−5.76336	−0.438180	−0.219090	−	0.975705i	$-0.570309\pi$
$173$	−5.76336	−0.438180	−0.219090	+	0.975705i	$0.570309\pi$
$174$	0	0
$175$	−19.2172	−1.45268
$176$	0	0
$177$	−3.03679	−0.228259
$178$	0	0
$179$	−24.6981	−1.84602	−0.923011	−	0.384774i	$-0.874279\pi$
$179$	−24.6981	−1.84602	−0.923011	+	0.384774i	$0.874279\pi$
$180$	0	0
$181$	3.85640	0.286644	0.143322	−	0.989676i	$-0.454222\pi$
$181$	3.85640	0.286644	0.143322	+	0.989676i	$0.454222\pi$
$182$	0	0
$183$	5.18295	0.383135
$184$	0	0
$185$	0.0184959	0.00135984
$186$	0	0
$187$	1.25681	0.0919067
$188$	0	0
$189$	−3.90646	−0.284153
$190$	0	0
$191$	10.3499	0.748891	0.374445	−	0.927249i	$-0.377833\pi$
$191$	10.3499	0.748891	0.374445	+	0.927249i	$0.377833\pi$
$192$	0	0
$193$	25.4995	1.83549	0.917746	−	0.397169i	$-0.130007\pi$
$193$	25.4995	1.83549	0.917746	+	0.397169i	$0.130007\pi$
$194$	0	0
$195$	−1.59657	−0.114333
$196$	0	0
$197$	4.63585	0.330291	0.165145	−	0.986269i	$-0.447191\pi$
$197$	4.63585	0.330291	0.165145	+	0.986269i	$0.447191\pi$
$198$	0	0
$199$	−3.77710	−0.267752	−0.133876	−	0.990998i	$-0.542742\pi$
$199$	−3.77710	−0.267752	−0.133876	+	0.990998i	$0.542742\pi$
$200$	0	0
$201$	−5.08568	−0.358716
$202$	0	0
$203$	16.3544	1.14786
$204$	0	0
$205$	1.77073	0.123673
$206$	0	0
$207$	2.16524	0.150495
$208$	0	0
$209$	−9.86858	−0.682624
$210$	0	0
$211$	1.10023	0.0757431	0.0378716	−	0.999283i	$-0.487942\pi$
$211$	1.10023	0.0757431	0.0378716	+	0.999283i	$0.487942\pi$
$212$	0	0
$213$	6.06371	0.415479
$214$	0	0
$215$	−0.707326	−0.0482392
$216$	0	0
$217$	9.42132	0.639561
$218$	0	0
$219$	−8.31501	−0.561876
$220$	0	0
$221$	−1.18947	−0.0800125
$222$	0	0
$223$	−2.73735	−0.183306	−0.0916531	−	0.995791i	$-0.529215\pi$
$223$	−2.73735	−0.183306	−0.0916531	+	0.995791i	$0.529215\pi$
$224$	0	0
$225$	−4.91933	−0.327955
$226$	0	0
$227$	2.50676	0.166380	0.0831898	−	0.996534i	$-0.473489\pi$
$227$	2.50676	0.166380	0.0831898	+	0.996534i	$0.473489\pi$
$228$	0	0
$229$	−8.95306	−0.591635	−0.295818	−	0.955244i	$-0.595592\pi$
$229$	−8.95306	−0.591635	−0.295818	+	0.955244i	$0.595592\pi$
$230$	0	0
$231$	23.2025	1.52661
$232$	0	0
$233$	−5.73726	−0.375860	−0.187930	−	0.982182i	$-0.560178\pi$
$233$	−5.73726	−0.375860	−0.187930	+	0.982182i	$0.560178\pi$
$234$	0	0
$235$	−1.69712	−0.110708
$236$	0	0
$237$	−2.87558	−0.186789
$238$	0	0
$239$	−27.2677	−1.76380	−0.881901	−	0.471434i	$-0.843736\pi$
$239$	−27.2677	−1.76380	−0.881901	+	0.471434i	$0.843736\pi$
$240$	0	0
$241$	−6.56847	−0.423112	−0.211556	−	0.977366i	$-0.567853\pi$
$241$	−6.56847	−0.423112	−0.211556	+	0.977366i	$0.567853\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.34613	0.149889
$246$	0	0
$247$	9.33986	0.594281
$248$	0	0
$249$	−5.50738	−0.349016
$250$	0	0
$251$	−9.28770	−0.586235	−0.293117	−	0.956076i	$-0.594693\pi$
$251$	−9.28770	−0.586235	−0.293117	+	0.956076i	$0.594693\pi$
$252$	0	0
$253$	−12.8605	−0.808533
$254$	0	0
$255$	0.0600991	0.00376355
$256$	0	0
$257$	5.88100	0.366847	0.183423	−	0.983034i	$-0.441282\pi$
$257$	5.88100	0.366847	0.183423	+	0.983034i	$0.441282\pi$
$258$	0	0
$259$	0.254393	0.0158072
$260$	0	0
$261$	4.18652	0.259139
$262$	0	0
$263$	25.2830	1.55902	0.779509	−	0.626391i	$-0.215469\pi$
$263$	25.2830	1.55902	0.779509	+	0.626391i	$0.215469\pi$
$264$	0	0
$265$	1.25976	0.0773862
$266$	0	0
$267$	−0.660702	−0.0404343
$268$	0	0
$269$	−28.0769	−1.71188	−0.855940	−	0.517076i	$-0.827021\pi$
$269$	−28.0769	−1.71188	−0.855940	+	0.517076i	$0.827021\pi$
$270$	0	0
$271$	13.1988	0.801767	0.400884	−	0.916129i	$-0.368703\pi$
$271$	13.1988	0.801767	0.400884	+	0.916129i	$0.368703\pi$
$272$	0	0
$273$	−21.9594	−1.32904
$274$	0	0
$275$	29.2185	1.76194
$276$	0	0
$277$	24.7372	1.48631	0.743156	−	0.669119i	$-0.233328\pi$
$277$	24.7372	1.48631	0.743156	+	0.669119i	$0.233328\pi$
$278$	0	0
$279$	2.41173	0.144386
$280$	0	0
$281$	10.9315	0.652119	0.326059	−	0.945349i	$-0.394279\pi$
$281$	10.9315	0.652119	0.326059	+	0.945349i	$0.394279\pi$
$282$	0	0
$283$	−28.3015	−1.68235	−0.841174	−	0.540765i	$-0.818135\pi$
$283$	−28.3015	−1.68235	−0.841174	+	0.540765i	$0.818135\pi$
$284$	0	0
$285$	−0.471905	−0.0279532
$286$	0	0
$287$	24.3548	1.43762
$288$	0	0
$289$	−16.9552	−0.997366
$290$	0	0
$291$	−17.4002	−1.02002
$292$	0	0
$293$	8.06024	0.470884	0.235442	−	0.971888i	$-0.424346\pi$
$293$	8.06024	0.470884	0.235442	+	0.971888i	$0.424346\pi$
$294$	0	0
$295$	0.862516	0.0502176
$296$	0	0
$297$	5.93953	0.344646
$298$	0	0
$299$	12.1715	0.703895
$300$	0	0
$301$	−9.72861	−0.560748
$302$	0	0
$303$	−0.510196	−0.0293100
$304$	0	0
$305$	−1.47207	−0.0842905
$306$	0	0
$307$	28.5806	1.63118	0.815592	−	0.578628i	$-0.196412\pi$
$307$	28.5806	1.63118	0.815592	+	0.578628i	$0.196412\pi$
$308$	0	0
$309$	6.73893	0.383365
$310$	0	0
$311$	28.5057	1.61641	0.808205	−	0.588901i	$-0.200439\pi$
$311$	28.5057	1.61641	0.808205	+	0.588901i	$0.200439\pi$
$312$	0	0
$313$	30.2771	1.71136	0.855682	−	0.517501i	$-0.173138\pi$
$313$	30.2771	1.71136	0.855682	+	0.517501i	$0.173138\pi$
$314$	0	0
$315$	1.10952	0.0625143
$316$	0	0
$317$	−27.6851	−1.55495	−0.777475	−	0.628914i	$-0.783500\pi$
$317$	−27.6851	−1.55495	−0.777475	+	0.628914i	$0.783500\pi$
$318$	0	0
$319$	−24.8659	−1.39222
$320$	0	0
$321$	6.94201	0.387465
$322$	0	0
$323$	−0.351576	−0.0195622
$324$	0	0
$325$	−27.6531	−1.53392
$326$	0	0
$327$	2.47553	0.136897
$328$	0	0
$329$	−23.3423	−1.28690
$330$	0	0
$331$	19.4216	1.06751	0.533754	−	0.845640i	$-0.320781\pi$
$331$	19.4216	1.06751	0.533754	+	0.845640i	$0.320781\pi$
$332$	0	0
$333$	0.0651212	0.00356862
$334$	0	0
$335$	1.44444	0.0789184
$336$	0	0
$337$	14.5405	0.792071	0.396036	−	0.918235i	$-0.370386\pi$
$337$	14.5405	0.792071	0.396036	+	0.918235i	$0.370386\pi$
$338$	0	0
$339$	−5.81283	−0.315710
$340$	0	0
$341$	−14.3245	−0.775717
$342$	0	0
$343$	4.92369	0.265854
$344$	0	0
$345$	−0.614976	−0.0331092
$346$	0	0
$347$	16.7308	0.898157	0.449079	−	0.893492i	$-0.351752\pi$
$347$	16.7308	0.898157	0.449079	+	0.893492i	$0.351752\pi$
$348$	0	0
$349$	3.21327	0.172002	0.0860011	−	0.996295i	$-0.472591\pi$
$349$	3.21327	0.172002	0.0860011	+	0.996295i	$0.472591\pi$
$350$	0	0
$351$	−5.62131	−0.300043
$352$	0	0
$353$	−3.58189	−0.190645	−0.0953224	−	0.995446i	$-0.530388\pi$
$353$	−3.58189	−0.190645	−0.0953224	+	0.995446i	$0.530388\pi$
$354$	0	0
$355$	−1.72223	−0.0914063
$356$	0	0
$357$	0.826607	0.0437487
$358$	0	0
$359$	36.0308	1.90163	0.950815	−	0.309758i	$-0.100248\pi$
$359$	36.0308	1.90163	0.950815	+	0.309758i	$0.100248\pi$
$360$	0	0
$361$	−16.2394	−0.854704
$362$	0	0
$363$	−24.2780	−1.27426
$364$	0	0
$365$	2.36164	0.123614
$366$	0	0
$367$	9.14426	0.477326	0.238663	−	0.971102i	$-0.423291\pi$
$367$	9.14426	0.477326	0.238663	+	0.971102i	$0.423291\pi$
$368$	0	0
$369$	6.23450	0.324555
$370$	0	0
$371$	17.3268	0.899562
$372$	0	0
$373$	9.07464	0.469867	0.234934	−	0.972011i	$-0.424513\pi$
$373$	9.07464	0.469867	0.234934	+	0.972011i	$0.424513\pi$
$374$	0	0
$375$	2.81731	0.145485
$376$	0	0
$377$	23.5337	1.21205
$378$	0	0
$379$	−24.4853	−1.25773	−0.628863	−	0.777516i	$-0.716480\pi$
$379$	−24.4853	−1.25773	−0.628863	+	0.777516i	$0.716480\pi$
$380$	0	0
$381$	−9.44159	−0.483707
$382$	0	0
$383$	−2.13348	−0.109016	−0.0545078	−	0.998513i	$-0.517359\pi$
$383$	−2.13348	−0.109016	−0.0545078	+	0.998513i	$0.517359\pi$
$384$	0	0
$385$	−6.59001	−0.335858
$386$	0	0
$387$	−2.49039	−0.126594
$388$	0	0
$389$	27.5348	1.39607	0.698034	−	0.716064i	$-0.254058\pi$
$389$	27.5348	1.39607	0.698034	+	0.716064i	$0.254058\pi$
$390$	0	0
$391$	−0.458166	−0.0231704
$392$	0	0
$393$	10.0575	0.507336
$394$	0	0
$395$	0.816726	0.0410939
$396$	0	0
$397$	1.61900	0.0812553	0.0406277	−	0.999174i	$-0.487064\pi$
$397$	1.61900	0.0812553	0.0406277	+	0.999174i	$0.487064\pi$
$398$	0	0
$399$	−6.49062	−0.324937
$400$	0	0
$401$	−15.0026	−0.749193	−0.374597	−	0.927188i	$-0.622219\pi$
$401$	−15.0026	−0.749193	−0.374597	+	0.927188i	$0.622219\pi$
$402$	0	0
$403$	13.5571	0.675326
$404$	0	0
$405$	0.284022	0.0141132
$406$	0	0
$407$	−0.386789	−0.0191724
$408$	0	0
$409$	38.1509	1.88644	0.943220	−	0.332168i	$-0.107780\pi$
$409$	38.1509	1.88644	0.943220	+	0.332168i	$0.107780\pi$
$410$	0	0
$411$	−7.06922	−0.348699
$412$	0	0
$413$	11.8631	0.583745
$414$	0	0
$415$	1.56422	0.0767844
$416$	0	0
$417$	−2.77510	−0.135897
$418$	0	0
$419$	18.3176	0.894871	0.447436	−	0.894316i	$-0.352337\pi$
$419$	18.3176	0.894871	0.447436	+	0.894316i	$0.352337\pi$
$420$	0	0
$421$	6.47766	0.315702	0.157851	−	0.987463i	$-0.449543\pi$
$421$	6.47766	0.315702	0.157851	+	0.987463i	$0.449543\pi$
$422$	0	0
$423$	−5.97532	−0.290530
$424$	0	0
$425$	1.04093	0.0504926
$426$	0	0
$427$	−20.2470	−0.979820
$428$	0	0
$429$	33.3879	1.61198
$430$	0	0
$431$	−2.77403	−0.133620	−0.0668102	−	0.997766i	$-0.521282\pi$
$431$	−2.77403	−0.133620	−0.0668102	+	0.997766i	$0.521282\pi$
$432$	0	0
$433$	11.2036	0.538411	0.269205	−	0.963083i	$-0.413239\pi$
$433$	11.2036	0.538411	0.269205	+	0.963083i	$0.413239\pi$
$434$	0	0
$435$	−1.18906	−0.0570111
$436$	0	0
$437$	3.59757	0.172095
$438$	0	0
$439$	−18.7525	−0.895007	−0.447503	−	0.894282i	$-0.647687\pi$
$439$	−18.7525	−0.895007	−0.447503	+	0.894282i	$0.647687\pi$
$440$	0	0
$441$	8.26040	0.393352
$442$	0	0
$443$	27.0953	1.28734	0.643668	−	0.765305i	$-0.277412\pi$
$443$	27.0953	1.28734	0.643668	+	0.765305i	$0.277412\pi$
$444$	0	0
$445$	0.187654	0.00889564
$446$	0	0
$447$	4.33933	0.205243
$448$	0	0
$449$	6.21568	0.293336	0.146668	−	0.989186i	$-0.453145\pi$
$449$	6.21568	0.293336	0.146668	+	0.989186i	$0.453145\pi$
$450$	0	0
$451$	−37.0300	−1.74367
$452$	0	0
$453$	−10.7758	−0.506290
$454$	0	0
$455$	6.23694	0.292392
$456$	0	0
$457$	−6.77365	−0.316858	−0.158429	−	0.987370i	$-0.550643\pi$
$457$	−6.77365	−0.316858	−0.158429	+	0.987370i	$0.550643\pi$
$458$	0	0
$459$	0.211600	0.00987666
$460$	0	0
$461$	−24.4579	−1.13912	−0.569559	−	0.821950i	$-0.692886\pi$
$461$	−24.4579	−1.13912	−0.569559	+	0.821950i	$0.692886\pi$
$462$	0	0
$463$	27.1130	1.26005	0.630023	−	0.776576i	$-0.283045\pi$
$463$	27.1130	1.26005	0.630023	+	0.776576i	$0.283045\pi$
$464$	0	0
$465$	−0.684984	−0.0317654
$466$	0	0
$467$	11.3996	0.527509	0.263755	−	0.964590i	$-0.415039\pi$
$467$	11.3996	0.527509	0.263755	+	0.964590i	$0.415039\pi$
$468$	0	0
$469$	19.8670	0.917373
$470$	0	0
$471$	−2.05881	−0.0948649
$472$	0	0
$473$	14.7918	0.680126
$474$	0	0
$475$	−8.17352	−0.375027
$476$	0	0
$477$	4.43542	0.203084
$478$	0	0
$479$	−14.3408	−0.655249	−0.327624	−	0.944808i	$-0.606248\pi$
$479$	−14.3408	−0.655249	−0.327624	+	0.944808i	$0.606248\pi$
$480$	0	0
$481$	0.366067	0.0166912
$482$	0	0
$483$	−8.45842	−0.384871
$484$	0	0
$485$	4.94203	0.224406
$486$	0	0
$487$	18.0212	0.816618	0.408309	−	0.912844i	$-0.366119\pi$
$487$	18.0212	0.816618	0.408309	+	0.912844i	$0.366119\pi$
$488$	0	0
$489$	16.7791	0.758776
$490$	0	0
$491$	6.85437	0.309333	0.154667	−	0.987967i	$-0.450570\pi$
$491$	6.85437	0.309333	0.154667	+	0.987967i	$0.450570\pi$
$492$	0	0
$493$	−0.885868	−0.0398975
$494$	0	0
$495$	−1.68695	−0.0758230
$496$	0	0
$497$	−23.6876	−1.06254
$498$	0	0
$499$	−22.7551	−1.01866	−0.509329	−	0.860572i	$-0.670106\pi$
$499$	−22.7551	−1.01866	−0.509329	+	0.860572i	$0.670106\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	25.9313	1.15622	0.578109	−	0.815960i	$-0.303791\pi$
$503$	25.9313	1.15622	0.578109	+	0.815960i	$0.303791\pi$
$504$	0	0
$505$	0.144907	0.00644827
$506$	0	0
$507$	−18.5991	−0.826015
$508$	0	0
$509$	−16.4159	−0.727622	−0.363811	−	0.931473i	$-0.618525\pi$
$509$	−16.4159	−0.727622	−0.363811	+	0.931473i	$0.618525\pi$
$510$	0	0
$511$	32.4822	1.43693
$512$	0	0
$513$	−1.66151	−0.0733575
$514$	0	0
$515$	−1.91400	−0.0843411
$516$	0	0
$517$	35.4906	1.56087
$518$	0	0
$519$	5.76336	0.252983
$520$	0	0
$521$	−25.2661	−1.10693	−0.553464	−	0.832873i	$-0.686695\pi$
$521$	−25.2661	−1.10693	−0.553464	+	0.832873i	$0.686695\pi$
$522$	0	0
$523$	−37.9788	−1.66070	−0.830349	−	0.557244i	$-0.811859\pi$
$523$	−37.9788	−1.66070	−0.830349	+	0.557244i	$0.811859\pi$
$524$	0	0
$525$	19.2172	0.838705
$526$	0	0
$527$	−0.510323	−0.0222300
$528$	0	0
$529$	−18.3117	−0.796162
$530$	0	0
$531$	3.03679	0.131786
$532$	0	0
$533$	35.0461	1.51801
$534$	0	0
$535$	−1.97168	−0.0852432
$536$	0	0
$537$	24.6981	1.06580
$538$	0	0
$539$	−49.0628	−2.11329
$540$	0	0
$541$	−30.2348	−1.29989	−0.649947	−	0.759979i	$-0.725209\pi$
$541$	−30.2348	−1.29989	−0.649947	+	0.759979i	$0.725209\pi$
$542$	0	0
$543$	−3.85640	−0.165494
$544$	0	0
$545$	−0.703103	−0.0301176
$546$	0	0
$547$	31.9660	1.36677	0.683384	−	0.730059i	$-0.260508\pi$
$547$	31.9660	1.36677	0.683384	+	0.730059i	$0.260508\pi$
$548$	0	0
$549$	−5.18295	−0.221203
$550$	0	0
$551$	6.95594	0.296333
$552$	0	0
$553$	11.2333	0.477689
$554$	0	0
$555$	−0.0184959	−0.000785105 0
$556$	0	0
$557$	9.42023	0.399148	0.199574	−	0.979883i	$-0.436044\pi$
$557$	9.42023	0.399148	0.199574	+	0.979883i	$0.436044\pi$
$558$	0	0
$559$	−13.9993	−0.592106
$560$	0	0
$561$	−1.25681	−0.0530624
$562$	0	0
$563$	20.7864	0.876041	0.438020	−	0.898965i	$-0.355680\pi$
$563$	20.7864	0.876041	0.438020	+	0.898965i	$0.355680\pi$
$564$	0	0
$565$	1.65097	0.0694569
$566$	0	0
$567$	3.90646	0.164056
$568$	0	0
$569$	−12.6387	−0.529844	−0.264922	−	0.964270i	$-0.585346\pi$
$569$	−12.6387	−0.529844	−0.264922	+	0.964270i	$0.585346\pi$
$570$	0	0
$571$	25.9073	1.08419	0.542094	−	0.840318i	$-0.317632\pi$
$571$	25.9073	1.08419	0.542094	+	0.840318i	$0.317632\pi$
$572$	0	0
$573$	−10.3499	−0.432372
$574$	0	0
$575$	−10.6515	−0.444200
$576$	0	0
$577$	−4.33510	−0.180472	−0.0902362	−	0.995920i	$-0.528762\pi$
$577$	−4.33510	−0.180472	−0.0902362	+	0.995920i	$0.528762\pi$
$578$	0	0
$579$	−25.4995	−1.05972
$580$	0	0
$581$	21.5143	0.892565
$582$	0	0
$583$	−26.3443	−1.09107
$584$	0	0
$585$	1.59657	0.0660102
$586$	0	0
$587$	11.7232	0.483869	0.241934	−	0.970293i	$-0.422218\pi$
$587$	11.7232	0.483869	0.241934	+	0.970293i	$0.422218\pi$
$588$	0	0
$589$	4.00711	0.165110
$590$	0	0
$591$	−4.63585	−0.190693
$592$	0	0
$593$	6.54590	0.268808	0.134404	−	0.990927i	$-0.457088\pi$
$593$	6.54590	0.268808	0.134404	+	0.990927i	$0.457088\pi$
$594$	0	0
$595$	−0.234774	−0.00962481
$596$	0	0
$597$	3.77710	0.154587
$598$	0	0
$599$	39.9982	1.63428	0.817141	−	0.576438i	$-0.195558\pi$
$599$	39.9982	1.63428	0.817141	+	0.576438i	$0.195558\pi$
$600$	0	0
$601$	−24.2386	−0.988714	−0.494357	−	0.869259i	$-0.664596\pi$
$601$	−24.2386	−0.988714	−0.494357	+	0.869259i	$0.664596\pi$
$602$	0	0
$603$	5.08568	0.207105
$604$	0	0
$605$	6.89547	0.280341
$606$	0	0
$607$	−14.4936	−0.588279	−0.294139	−	0.955762i	$-0.595033\pi$
$607$	−14.4936	−0.588279	−0.294139	+	0.955762i	$0.595033\pi$
$608$	0	0
$609$	−16.3544	−0.662715
$610$	0	0
$611$	−33.5891	−1.35887
$612$	0	0
$613$	2.55905	0.103359	0.0516795	−	0.998664i	$-0.483543\pi$
$613$	2.55905	0.103359	0.0516795	+	0.998664i	$0.483543\pi$
$614$	0	0
$615$	−1.77073	−0.0714029
$616$	0	0
$617$	27.2707	1.09788	0.548938	−	0.835863i	$-0.315032\pi$
$617$	27.2707	1.09788	0.548938	+	0.835863i	$0.315032\pi$
$618$	0	0
$619$	16.4994	0.663166	0.331583	−	0.943426i	$-0.392417\pi$
$619$	16.4994	0.663166	0.331583	+	0.943426i	$0.392417\pi$
$620$	0	0
$621$	−2.16524	−0.0868881
$622$	0	0
$623$	2.58100	0.103406
$624$	0	0
$625$	23.7965	0.951859
$626$	0	0
$627$	9.86858	0.394113
$628$	0	0
$629$	−0.0137797	−0.000549432 0
$630$	0	0
$631$	28.3397	1.12818	0.564092	−	0.825712i	$-0.309226\pi$
$631$	28.3397	1.12818	0.564092	+	0.825712i	$0.309226\pi$
$632$	0	0
$633$	−1.10023	−0.0437303
$634$	0	0
$635$	2.68162	0.106417
$636$	0	0
$637$	46.4342	1.83979
$638$	0	0
$639$	−6.06371	−0.239877
$640$	0	0
$641$	33.4715	1.32204	0.661022	−	0.750366i	$-0.270123\pi$
$641$	33.4715	1.32204	0.661022	+	0.750366i	$0.270123\pi$
$642$	0	0
$643$	−1.51010	−0.0595524	−0.0297762	−	0.999557i	$-0.509479\pi$
$643$	−1.51010	−0.0595524	−0.0297762	+	0.999557i	$0.509479\pi$
$644$	0	0
$645$	0.707326	0.0278509
$646$	0	0
$647$	21.6453	0.850964	0.425482	−	0.904967i	$-0.360105\pi$
$647$	21.6453	0.850964	0.425482	+	0.904967i	$0.360105\pi$
$648$	0	0
$649$	−18.0371	−0.708019
$650$	0	0
$651$	−9.42132	−0.369251
$652$	0	0
$653$	20.2062	0.790731	0.395365	−	0.918524i	$-0.370618\pi$
$653$	20.2062	0.790731	0.395365	+	0.918524i	$0.370618\pi$
$654$	0	0
$655$	−2.85656	−0.111615
$656$	0	0
$657$	8.31501	0.324400
$658$	0	0
$659$	11.5989	0.451828	0.225914	−	0.974147i	$-0.427463\pi$
$659$	11.5989	0.451828	0.225914	+	0.974147i	$0.427463\pi$
$660$	0	0
$661$	−44.1475	−1.71714	−0.858570	−	0.512697i	$-0.828646\pi$
$661$	−44.1475	−1.71714	−0.858570	+	0.512697i	$0.828646\pi$
$662$	0	0
$663$	1.18947	0.0461952
$664$	0	0
$665$	1.84348	0.0714869
$666$	0	0
$667$	9.06482	0.350991
$668$	0	0
$669$	2.73735	0.105832
$670$	0	0
$671$	30.7843	1.18841
$672$	0	0
$673$	−21.8201	−0.841104	−0.420552	−	0.907269i	$-0.638164\pi$
$673$	−21.8201	−0.841104	−0.420552	+	0.907269i	$0.638164\pi$
$674$	0	0
$675$	4.91933	0.189345
$676$	0	0
$677$	−31.8137	−1.22270	−0.611351	−	0.791360i	$-0.709373\pi$
$677$	−31.8137	−1.22270	−0.611351	+	0.791360i	$0.709373\pi$
$678$	0	0
$679$	67.9730	2.60857
$680$	0	0
$681$	−2.50676	−0.0960593
$682$	0	0
$683$	27.4526	1.05045	0.525223	−	0.850965i	$-0.323982\pi$
$683$	27.4526	1.05045	0.525223	+	0.850965i	$0.323982\pi$
$684$	0	0
$685$	2.00781	0.0767145
$686$	0	0
$687$	8.95306	0.341581
$688$	0	0
$689$	24.9329	0.949867
$690$	0	0
$691$	23.1220	0.879603	0.439802	−	0.898095i	$-0.355049\pi$
$691$	23.1220	0.879603	0.439802	+	0.898095i	$0.355049\pi$
$692$	0	0
$693$	−23.2025	−0.881390
$694$	0	0
$695$	0.788187	0.0298976
$696$	0	0
$697$	−1.31922	−0.0499691
$698$	0	0
$699$	5.73726	0.217003
$700$	0	0
$701$	48.6779	1.83854	0.919270	−	0.393628i	$-0.128780\pi$
$701$	48.6779	1.83854	0.919270	+	0.393628i	$0.128780\pi$
$702$	0	0
$703$	0.108200	0.00408083
$704$	0	0
$705$	1.69712	0.0639173
$706$	0	0
$707$	1.99306	0.0749567
$708$	0	0
$709$	51.3682	1.92917	0.964587	−	0.263766i	$-0.0849647\pi$
$709$	51.3682	1.92917	0.964587	+	0.263766i	$0.0849647\pi$
$710$	0	0
$711$	2.87558	0.107843
$712$	0	0
$713$	5.22198	0.195565
$714$	0	0
$715$	−9.48289	−0.354640
$716$	0	0
$717$	27.2677	1.01833
$718$	0	0
$719$	17.1498	0.639580	0.319790	−	0.947489i	$-0.396388\pi$
$719$	17.1498	0.639580	0.319790	+	0.947489i	$0.396388\pi$
$720$	0	0
$721$	−26.3253	−0.980407
$722$	0	0
$723$	6.56847	0.244284
$724$	0	0
$725$	−20.5949	−0.764874
$726$	0	0
$727$	−12.0019	−0.445126	−0.222563	−	0.974918i	$-0.571442\pi$
$727$	−12.0019	−0.445126	−0.222563	+	0.974918i	$0.571442\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.526968	0.0194906
$732$	0	0
$733$	4.00307	0.147857	0.0739285	−	0.997264i	$-0.476446\pi$
$733$	4.00307	0.147857	0.0739285	+	0.997264i	$0.476446\pi$
$734$	0	0
$735$	−2.34613	−0.0865384
$736$	0	0
$737$	−30.2066	−1.11267
$738$	0	0
$739$	34.4777	1.26828	0.634140	−	0.773218i	$-0.281354\pi$
$739$	34.4777	1.26828	0.634140	+	0.773218i	$0.281354\pi$
$740$	0	0
$741$	−9.33986	−0.343108
$742$	0	0
$743$	−26.9128	−0.987334	−0.493667	−	0.869651i	$-0.664344\pi$
$743$	−26.9128	−0.987334	−0.493667	+	0.869651i	$0.664344\pi$
$744$	0	0
$745$	−1.23246	−0.0451540
$746$	0	0
$747$	5.50738	0.201505
$748$	0	0
$749$	−27.1186	−0.990894
$750$	0	0
$751$	41.9616	1.53120	0.765600	−	0.643317i	$-0.222442\pi$
$751$	41.9616	1.53120	0.765600	+	0.643317i	$0.222442\pi$
$752$	0	0
$753$	9.28770	0.338463
$754$	0	0
$755$	3.06056	0.111385
$756$	0	0
$757$	10.7842	0.391957	0.195979	−	0.980608i	$-0.437212\pi$
$757$	10.7842	0.391957	0.195979	+	0.980608i	$0.437212\pi$
$758$	0	0
$759$	12.8605	0.466807
$760$	0	0
$761$	−53.5138	−1.93987	−0.969936	−	0.243359i	$-0.921751\pi$
$761$	−53.5138	−1.93987	−0.969936	+	0.243359i	$0.921751\pi$
$762$	0	0
$763$	−9.67053	−0.350097
$764$	0	0
$765$	−0.0600991	−0.00217289
$766$	0	0
$767$	17.0708	0.616389
$768$	0	0
$769$	36.2997	1.30900	0.654501	−	0.756061i	$-0.272879\pi$
$769$	36.2997	1.30900	0.654501	+	0.756061i	$0.272879\pi$
$770$	0	0
$771$	−5.88100	−0.211799
$772$	0	0
$773$	−7.72086	−0.277700	−0.138850	−	0.990313i	$-0.544341\pi$
$773$	−7.72086	−0.277700	−0.138850	+	0.990313i	$0.544341\pi$
$774$	0	0
$775$	−11.8641	−0.426171
$776$	0	0
$777$	−0.254393	−0.00912631
$778$	0	0
$779$	10.3587	0.371139
$780$	0	0
$781$	36.0156	1.28874
$782$	0	0
$783$	−4.18652	−0.149614
$784$	0	0
$785$	0.584747	0.0208705
$786$	0	0
$787$	−37.9982	−1.35449	−0.677244	−	0.735758i	$-0.736826\pi$
$787$	−37.9982	−1.35449	−0.677244	+	0.735758i	$0.736826\pi$
$788$	0	0
$789$	−25.2830	−0.900099
$790$	0	0
$791$	22.7076	0.807388
$792$	0	0
$793$	−29.1350	−1.03461
$794$	0	0
$795$	−1.25976	−0.0446790
$796$	0	0
$797$	−3.80242	−0.134689	−0.0673443	−	0.997730i	$-0.521453\pi$
$797$	−3.80242	−0.134689	−0.0673443	+	0.997730i	$0.521453\pi$
$798$	0	0
$799$	1.26438	0.0447306
$800$	0	0
$801$	0.660702	0.0233448
$802$	0	0
$803$	−49.3872	−1.74284
$804$	0	0
$805$	2.40238	0.0846726
$806$	0	0
$807$	28.0769	0.988354
$808$	0	0
$809$	−33.4039	−1.17442	−0.587208	−	0.809436i	$-0.699773\pi$
$809$	−33.4039	−1.17442	−0.587208	+	0.809436i	$0.699773\pi$
$810$	0	0
$811$	36.5294	1.28272	0.641360	−	0.767240i	$-0.278370\pi$
$811$	36.5294	1.28272	0.641360	+	0.767240i	$0.278370\pi$
$812$	0	0
$813$	−13.1988	−0.462901
$814$	0	0
$815$	−4.76562	−0.166933
$816$	0	0
$817$	−4.13781	−0.144764
$818$	0	0
$819$	21.9594	0.767323
$820$	0	0
$821$	−13.8920	−0.484833	−0.242416	−	0.970172i	$-0.577940\pi$
$821$	−13.8920	−0.484833	−0.242416	+	0.970172i	$0.577940\pi$
$822$	0	0
$823$	−22.8120	−0.795178	−0.397589	−	0.917564i	$-0.630153\pi$
$823$	−22.8120	−0.795178	−0.397589	+	0.917564i	$0.630153\pi$
$824$	0	0
$825$	−29.2185	−1.01726
$826$	0	0
$827$	31.7370	1.10361	0.551803	−	0.833975i	$-0.313940\pi$
$827$	31.7370	1.10361	0.551803	+	0.833975i	$0.313940\pi$
$828$	0	0
$829$	−16.3832	−0.569012	−0.284506	−	0.958674i	$-0.591830\pi$
$829$	−16.3832	−0.569012	−0.284506	+	0.958674i	$0.591830\pi$
$830$	0	0
$831$	−24.7372	−0.858122
$832$	0	0
$833$	−1.74790	−0.0605612
$834$	0	0
$835$	0.284022	0.00982898
$836$	0	0
$837$	−2.41173	−0.0833616
$838$	0	0
$839$	−36.5169	−1.26070	−0.630352	−	0.776310i	$-0.717089\pi$
$839$	−36.5169	−1.26070	−0.630352	+	0.776310i	$0.717089\pi$
$840$	0	0
$841$	−11.4731	−0.395624
$842$	0	0
$843$	−10.9315	−0.376501
$844$	0	0
$845$	5.28255	0.181725
$846$	0	0
$847$	94.8408	3.25877
$848$	0	0
$849$	28.3015	0.971304
$850$	0	0
$851$	0.141003	0.00483353
$852$	0	0
$853$	31.9108	1.09261	0.546303	−	0.837588i	$-0.316035\pi$
$853$	31.9108	1.09261	0.546303	+	0.837588i	$0.316035\pi$
$854$	0	0
$855$	0.471905	0.0161388
$856$	0	0
$857$	28.9696	0.989583	0.494791	−	0.869012i	$-0.335244\pi$
$857$	28.9696	0.989583	0.494791	+	0.869012i	$0.335244\pi$
$858$	0	0
$859$	45.7885	1.56228	0.781142	−	0.624354i	$-0.214638\pi$
$859$	45.7885	1.56228	0.781142	+	0.624354i	$0.214638\pi$
$860$	0	0
$861$	−24.3548	−0.830010
$862$	0	0
$863$	−32.1532	−1.09451	−0.547254	−	0.836966i	$-0.684327\pi$
$863$	−32.1532	−1.09451	−0.547254	+	0.836966i	$0.684327\pi$
$864$	0	0
$865$	−1.63692	−0.0556569
$866$	0	0
$867$	16.9552	0.575830
$868$	0	0
$869$	−17.0796	−0.579384
$870$	0	0
$871$	28.5882	0.968674
$872$	0	0
$873$	17.4002	0.588907
$874$	0	0
$875$	−11.0057	−0.372060
$876$	0	0
$877$	−28.5974	−0.965664	−0.482832	−	0.875713i	$-0.660392\pi$
$877$	−28.5974	−0.965664	−0.482832	+	0.875713i	$0.660392\pi$
$878$	0	0
$879$	−8.06024	−0.271865
$880$	0	0
$881$	9.33502	0.314505	0.157252	−	0.987558i	$-0.449736\pi$
$881$	9.33502	0.314505	0.157252	+	0.987558i	$0.449736\pi$
$882$	0	0
$883$	50.0530	1.68442	0.842209	−	0.539151i	$-0.181255\pi$
$883$	50.0530	1.68442	0.842209	+	0.539151i	$0.181255\pi$
$884$	0	0
$885$	−0.862516	−0.0289931
$886$	0	0
$887$	35.2474	1.18349	0.591746	−	0.806124i	$-0.298439\pi$
$887$	35.2474	1.18349	0.591746	+	0.806124i	$0.298439\pi$
$888$	0	0
$889$	36.8832	1.23702
$890$	0	0
$891$	−5.93953	−0.198982
$892$	0	0
$893$	−9.92806	−0.332230
$894$	0	0
$895$	−7.01479	−0.234479
$896$	0	0
$897$	−12.1715	−0.406394
$898$	0	0
$899$	10.0967	0.336745
$900$	0	0
$901$	−0.938537	−0.0312672
$902$	0	0
$903$	9.72861	0.323748
$904$	0	0
$905$	1.09530	0.0364090
$906$	0	0
$907$	−3.29975	−0.109566	−0.0547832	−	0.998498i	$-0.517447\pi$
$907$	−3.29975	−0.109566	−0.0547832	+	0.998498i	$0.517447\pi$
$908$	0	0
$909$	0.510196	0.0169221
$910$	0	0
$911$	−9.05582	−0.300033	−0.150016	−	0.988684i	$-0.547933\pi$
$911$	−9.05582	−0.300033	−0.150016	+	0.988684i	$0.547933\pi$
$912$	0	0
$913$	−32.7112	−1.08258
$914$	0	0
$915$	1.47207	0.0486652
$916$	0	0
$917$	−39.2893	−1.29745
$918$	0	0
$919$	−33.3598	−1.10044	−0.550220	−	0.835020i	$-0.685456\pi$
$919$	−33.3598	−1.10044	−0.550220	+	0.835020i	$0.685456\pi$
$920$	0	0
$921$	−28.5806	−0.941764
$922$	0	0
$923$	−34.0860	−1.12195
$924$	0	0
$925$	−0.320353	−0.0105331
$926$	0	0
$927$	−6.73893	−0.221336
$928$	0	0
$929$	15.4418	0.506628	0.253314	−	0.967384i	$-0.418479\pi$
$929$	15.4418	0.506628	0.253314	+	0.967384i	$0.418479\pi$
$930$	0	0
$931$	13.7247	0.449810
$932$	0	0
$933$	−28.5057	−0.933235
$934$	0	0
$935$	0.356960	0.0116738
$936$	0	0
$937$	−6.31490	−0.206299	−0.103149	−	0.994666i	$-0.532892\pi$
$937$	−6.31490	−0.206299	−0.103149	+	0.994666i	$0.532892\pi$
$938$	0	0
$939$	−30.2771	−0.988057
$940$	0	0
$941$	−29.4186	−0.959020	−0.479510	−	0.877536i	$-0.659186\pi$
$941$	−29.4186	−0.959020	−0.479510	+	0.877536i	$0.659186\pi$
$942$	0	0
$943$	13.4992	0.439595
$944$	0	0
$945$	−1.10952	−0.0360926
$946$	0	0
$947$	−38.6029	−1.25443	−0.627213	−	0.778848i	$-0.715804\pi$
$947$	−38.6029	−1.25443	−0.627213	+	0.778848i	$0.715804\pi$
$948$	0	0
$949$	46.7412	1.51729
$950$	0	0
$951$	27.6851	0.897751
$952$	0	0
$953$	33.7981	1.09483	0.547414	−	0.836862i	$-0.315612\pi$
$953$	33.7981	1.09483	0.547414	+	0.836862i	$0.315612\pi$
$954$	0	0
$955$	2.93959	0.0951229
$956$	0	0
$957$	24.8659	0.803801
$958$	0	0
$959$	27.6156	0.891754
$960$	0	0
$961$	−25.1836	−0.812373
$962$	0	0
$963$	−6.94201	−0.223703
$964$	0	0
$965$	7.24240	0.233141
$966$	0	0
$967$	−27.6619	−0.889546	−0.444773	−	0.895643i	$-0.646716\pi$
$967$	−27.6619	−0.889546	−0.444773	+	0.895643i	$0.646716\pi$
$968$	0	0
$969$	0.351576	0.0112943
$970$	0	0
$971$	−39.0471	−1.25308	−0.626540	−	0.779389i	$-0.715530\pi$
$971$	−39.0471	−1.25308	−0.626540	+	0.779389i	$0.715530\pi$
$972$	0	0
$973$	10.8408	0.347540
$974$	0	0
$975$	27.6531	0.885607
$976$	0	0
$977$	−47.4175	−1.51702	−0.758509	−	0.651662i	$-0.774072\pi$
$977$	−47.4175	−1.51702	−0.758509	+	0.651662i	$0.774072\pi$
$978$	0	0
$979$	−3.92426	−0.125420
$980$	0	0
$981$	−2.47553	−0.0790375
$982$	0	0
$983$	−27.5610	−0.879058	−0.439529	−	0.898228i	$-0.644855\pi$
$983$	−27.5610	−0.879058	−0.439529	+	0.898228i	$0.644855\pi$
$984$	0	0
$985$	1.31668	0.0419530
$986$	0	0
$987$	23.3423	0.742995
$988$	0	0
$989$	−5.39230	−0.171465
$990$	0	0
$991$	−32.5362	−1.03355	−0.516773	−	0.856122i	$-0.672867\pi$
$991$	−32.5362	−1.03355	−0.516773	+	0.856122i	$0.672867\pi$
$992$	0	0
$993$	−19.4216	−0.616326
$994$	0	0
$995$	−1.07278	−0.0340094
$996$	0	0
$997$	8.74378	0.276918	0.138459	−	0.990368i	$-0.455785\pi$
$997$	8.74378	0.276918	0.138459	+	0.990368i	$0.455785\pi$
$998$	0	0
$999$	−0.0651212	−0.00206035

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bg.1.6		13
4.3	odd	2		4008.2.a.m.1.6	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.m.1.6	✓	13	4.3	odd	2
8016.2.a.bg.1.6		13	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.284022 q^{5} +3.90646 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.284022 q^{5} +3.90646 q^{7} +1.00000 q^{9} -5.93953 q^{11} +5.62131 q^{13} -0.284022 q^{15} -0.211600 q^{17} +1.66151 q^{19} -3.90646 q^{21} +2.16524 q^{23} -4.91933 q^{25} -1.00000 q^{27} +4.18652 q^{29} +2.41173 q^{31} +5.93953 q^{33} +1.10952 q^{35} +0.0651212 q^{37} -5.62131 q^{39} +6.23450 q^{41} -2.49039 q^{43} +0.284022 q^{45} -5.97532 q^{47} +8.26040 q^{49} +0.211600 q^{51} +4.43542 q^{53} -1.68695 q^{55} -1.66151 q^{57} +3.03679 q^{59} -5.18295 q^{61} +3.90646 q^{63} +1.59657 q^{65} +5.08568 q^{67} -2.16524 q^{69} -6.06371 q^{71} +8.31501 q^{73} +4.91933 q^{75} -23.2025 q^{77} +2.87558 q^{79} +1.00000 q^{81} +5.50738 q^{83} -0.0600991 q^{85} -4.18652 q^{87} +0.660702 q^{89} +21.9594 q^{91} -2.41173 q^{93} +0.471905 q^{95} +17.4002 q^{97} -5.93953 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9} - 11 q^{11} + 12 q^{13} - 2 q^{15} + 15 q^{17} - 14 q^{19} + q^{21} - 9 q^{23} + 37 q^{25} - 13 q^{27} - 3 q^{29} + 17 q^{31} + 11 q^{33} - 15 q^{35} + 16 q^{37} - 12 q^{39} + 12 q^{41} - 20 q^{43} + 2 q^{45} + 6 q^{47} + 26 q^{49} - 15 q^{51} - 12 q^{53} - 7 q^{55} + 14 q^{57} - 14 q^{59} + 24 q^{61} - q^{63} + 8 q^{65} - 3 q^{67} + 9 q^{69} - 17 q^{71} + 34 q^{73} - 37 q^{75} + 30 q^{77} - 10 q^{79} + 13 q^{81} - 44 q^{83} + 25 q^{85} + 3 q^{87} + 25 q^{89} - 29 q^{91} - 17 q^{93} + 15 q^{95} + 38 q^{97} - 11 q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9 - 11 * q^11 + 12 * q^13 - 2 * q^15 + 15 * q^17 - 14 * q^19 + q^21 - 9 * q^23 + 37 * q^25 - 13 * q^27 - 3 * q^29 + 17 * q^31 + 11 * q^33 - 15 * q^35 + 16 * q^37 - 12 * q^39 + 12 * q^41 - 20 * q^43 + 2 * q^45 + 6 * q^47 + 26 * q^49 - 15 * q^51 - 12 * q^53 - 7 * q^55 + 14 * q^57 - 14 * q^59 + 24 * q^61 - q^63 + 8 * q^65 - 3 * q^67 + 9 * q^69 - 17 * q^71 + 34 * q^73 - 37 * q^75 + 30 * q^77 - 10 * q^79 + 13 * q^81 - 44 * q^83 + 25 * q^85 + 3 * q^87 + 25 * q^89 - 29 * q^91 - 17 * q^93 + 15 * q^95 + 38 * q^97 - 11 * q^99

Introduction

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