LMFDB - Embedded newform 8036.2.a.t.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 $ x^20 - 4x^19 - 30x^18 + 128x^17 + 348x^16 - 1644x^15 - 1934x^14 + 10948x^13 + 4748x^12 - 40524x^11 - 220x^10 + 82500x^9 - 21992x^8 - 84720x^7 + 37544x^6 + 34656x^5 - 18823x^4 - 2276x^3 + 1130x^2 + 204*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$1.06664$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.06664 q^{3} -0.715473 q^{5} -1.86229 q^{9} +O(q^{10})$	$q+1.06664 q^{3} -0.715473 q^{5} -1.86229 q^{9} -0.203291 q^{11} -4.07678 q^{13} -0.763149 q^{15} +1.49788 q^{17} -1.74419 q^{19} -4.09950 q^{23} -4.48810 q^{25} -5.18629 q^{27} -5.13540 q^{29} -3.89553 q^{31} -0.216837 q^{33} +9.19415 q^{37} -4.34844 q^{39} -1.00000 q^{41} +1.74486 q^{43} +1.33242 q^{45} +6.43307 q^{47} +1.59769 q^{51} +6.61543 q^{53} +0.145449 q^{55} -1.86042 q^{57} +12.8894 q^{59} +14.5230 q^{61} +2.91683 q^{65} -0.105967 q^{67} -4.37267 q^{69} +14.3712 q^{71} +13.2224 q^{73} -4.78716 q^{75} +6.88922 q^{79} +0.0549876 q^{81} -2.43452 q^{83} -1.07169 q^{85} -5.47760 q^{87} -4.37395 q^{89} -4.15511 q^{93} +1.24792 q^{95} +11.1054 q^{97} +0.378586 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

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
$3$	1.06664	0.615822	0.307911	−	0.951415i	$-0.400370\pi$
$3$	1.06664	0.615822	0.307911	+	0.951415i	$0.400370\pi$
$4$	0	0
$5$	−0.715473	−0.319969	−0.159985	−	0.987120i	$-0.551144\pi$
$5$	−0.715473	−0.319969	−0.159985	+	0.987120i	$0.551144\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.86229	−0.620763
$10$	0	0
$11$	−0.203291	−0.0612944	−0.0306472	−	0.999530i	$-0.509757\pi$
$11$	−0.203291	−0.0612944	−0.0306472	+	0.999530i	$0.509757\pi$
$12$	0	0
$13$	−4.07678	−1.13070	−0.565348	−	0.824853i	$-0.691258\pi$
$13$	−4.07678	−1.13070	−0.565348	+	0.824853i	$0.691258\pi$
$14$	0	0
$15$	−0.763149	−0.197044
$16$	0	0
$17$	1.49788	0.363289	0.181645	−	0.983364i	$-0.441858\pi$
$17$	1.49788	0.363289	0.181645	+	0.983364i	$0.441858\pi$
$18$	0	0
$19$	−1.74419	−0.400145	−0.200073	−	0.979781i	$-0.564118\pi$
$19$	−1.74419	−0.400145	−0.200073	+	0.979781i	$0.564118\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.09950	−0.854804	−0.427402	−	0.904062i	$-0.640571\pi$
$23$	−4.09950	−0.854804	−0.427402	+	0.904062i	$0.640571\pi$
$24$	0	0
$25$	−4.48810	−0.897620
$26$	0	0
$27$	−5.18629	−0.998102
$28$	0	0
$29$	−5.13540	−0.953620	−0.476810	−	0.879006i	$-0.658207\pi$
$29$	−5.13540	−0.953620	−0.476810	+	0.879006i	$0.658207\pi$
$30$	0	0
$31$	−3.89553	−0.699657	−0.349829	−	0.936814i	$-0.613760\pi$
$31$	−3.89553	−0.699657	−0.349829	+	0.936814i	$0.613760\pi$
$32$	0	0
$33$	−0.216837	−0.0377465
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.19415	1.51151	0.755754	−	0.654856i	$-0.227271\pi$
$37$	9.19415	1.51151	0.755754	+	0.654856i	$0.227271\pi$
$38$	0	0
$39$	−4.34844	−0.696308
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	1.74486	0.266089	0.133044	−	0.991110i	$-0.457525\pi$
$43$	1.74486	0.266089	0.133044	+	0.991110i	$0.457525\pi$
$44$	0	0
$45$	1.33242	0.198625
$46$	0	0
$47$	6.43307	0.938360	0.469180	−	0.883103i	$-0.344550\pi$
$47$	6.43307	0.938360	0.469180	+	0.883103i	$0.344550\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.59769	0.223722
$52$	0	0
$53$	6.61543	0.908699	0.454349	−	0.890824i	$-0.349872\pi$
$53$	6.61543	0.908699	0.454349	+	0.890824i	$0.349872\pi$
$54$	0	0
$55$	0.145449	0.0196123
$56$	0	0
$57$	−1.86042	−0.246418
$58$	0	0
$59$	12.8894	1.67806	0.839030	−	0.544085i	$-0.183123\pi$
$59$	12.8894	1.67806	0.839030	+	0.544085i	$0.183123\pi$
$60$	0	0
$61$	14.5230	1.85947	0.929737	−	0.368224i	$-0.120034\pi$
$61$	14.5230	1.85947	0.929737	+	0.368224i	$0.120034\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.91683	0.361788
$66$	0	0
$67$	−0.105967	−0.0129460	−0.00647299	−	0.999979i	$-0.502060\pi$
$67$	−0.105967	−0.0129460	−0.00647299	+	0.999979i	$0.502060\pi$
$68$	0	0
$69$	−4.37267	−0.526407
$70$	0	0
$71$	14.3712	1.70555	0.852775	−	0.522278i	$-0.174918\pi$
$71$	14.3712	1.70555	0.852775	+	0.522278i	$0.174918\pi$
$72$	0	0
$73$	13.2224	1.54756	0.773782	−	0.633452i	$-0.218362\pi$
$73$	13.2224	1.54756	0.773782	+	0.633452i	$0.218362\pi$
$74$	0	0
$75$	−4.78716	−0.552774
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.88922	0.775098	0.387549	−	0.921849i	$-0.373322\pi$
$79$	6.88922	0.775098	0.387549	+	0.921849i	$0.373322\pi$
$80$	0	0
$81$	0.0549876	0.00610973
$82$	0	0
$83$	−2.43452	−0.267223	−0.133612	−	0.991034i	$-0.542657\pi$
$83$	−2.43452	−0.267223	−0.133612	+	0.991034i	$0.542657\pi$
$84$	0	0
$85$	−1.07169	−0.116241
$86$	0	0
$87$	−5.47760	−0.587261
$88$	0	0
$89$	−4.37395	−0.463638	−0.231819	−	0.972759i	$-0.574468\pi$
$89$	−4.37395	−0.463638	−0.231819	+	0.972759i	$0.574468\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.15511	−0.430864
$94$	0	0
$95$	1.24792	0.128034
$96$	0	0
$97$	11.1054	1.12758	0.563789	−	0.825919i	$-0.309343\pi$
$97$	11.1054	1.12758	0.563789	+	0.825919i	$0.309343\pi$
$98$	0	0
$99$	0.378586	0.0380493
$100$	0	0
$101$	−16.5526	−1.64705	−0.823523	−	0.567283i	$-0.807995\pi$
$101$	−16.5526	−1.64705	−0.823523	+	0.567283i	$0.807995\pi$
$102$	0	0
$103$	16.9276	1.66793	0.833964	−	0.551819i	$-0.186066\pi$
$103$	16.9276	1.66793	0.833964	+	0.551819i	$0.186066\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0473	1.16466	0.582329	−	0.812953i	$-0.302142\pi$
$107$	12.0473	1.16466	0.582329	+	0.812953i	$0.302142\pi$
$108$	0	0
$109$	−0.740391	−0.0709166	−0.0354583	−	0.999371i	$-0.511289\pi$
$109$	−0.740391	−0.0709166	−0.0354583	+	0.999371i	$0.511289\pi$
$110$	0	0
$111$	9.80680	0.930820
$112$	0	0
$113$	15.3332	1.44242	0.721212	−	0.692714i	$-0.243585\pi$
$113$	15.3332	1.44242	0.721212	+	0.692714i	$0.243585\pi$
$114$	0	0
$115$	2.93308	0.273511
$116$	0	0
$117$	7.59215	0.701894
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9587	−0.996243
$122$	0	0
$123$	−1.06664	−0.0961753
$124$	0	0
$125$	6.78848	0.607180
$126$	0	0
$127$	−17.9159	−1.58978	−0.794889	−	0.606756i	$-0.792471\pi$
$127$	−17.9159	−1.58978	−0.794889	+	0.606756i	$0.792471\pi$
$128$	0	0
$129$	1.86113	0.163863
$130$	0	0
$131$	−3.85953	−0.337208	−0.168604	−	0.985684i	$-0.553926\pi$
$131$	−3.85953	−0.337208	−0.168604	+	0.985684i	$0.553926\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.71065	0.319362
$136$	0	0
$137$	−9.19838	−0.785871	−0.392935	−	0.919566i	$-0.628540\pi$
$137$	−9.19838	−0.785871	−0.392935	+	0.919566i	$0.628540\pi$
$138$	0	0
$139$	−5.85466	−0.496586	−0.248293	−	0.968685i	$-0.579870\pi$
$139$	−5.85466	−0.496586	−0.248293	+	0.968685i	$0.579870\pi$
$140$	0	0
$141$	6.86174	0.577863
$142$	0	0
$143$	0.828772	0.0693054
$144$	0	0
$145$	3.67424	0.305129
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.8299	−1.54260	−0.771301	−	0.636470i	$-0.780394\pi$
$149$	−18.8299	−1.54260	−0.771301	+	0.636470i	$0.780394\pi$
$150$	0	0
$151$	−21.8901	−1.78139	−0.890696	−	0.454599i	$-0.849783\pi$
$151$	−21.8901	−1.78139	−0.890696	+	0.454599i	$0.849783\pi$
$152$	0	0
$153$	−2.78949	−0.225517
$154$	0	0
$155$	2.78714	0.223869
$156$	0	0
$157$	−0.114806	−0.00916251	−0.00458125	−	0.999990i	$-0.501458\pi$
$157$	−0.114806	−0.00916251	−0.00458125	+	0.999990i	$0.501458\pi$
$158$	0	0
$159$	7.05625	0.559597
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.14673	0.638101	0.319051	−	0.947738i	$-0.396636\pi$
$163$	8.14673	0.638101	0.319051	+	0.947738i	$0.396636\pi$
$164$	0	0
$165$	0.155141	0.0120777
$166$	0	0
$167$	−18.1119	−1.40154	−0.700769	−	0.713389i	$-0.747160\pi$
$167$	−18.1119	−1.40154	−0.700769	+	0.713389i	$0.747160\pi$
$168$	0	0
$169$	3.62015	0.278473
$170$	0	0
$171$	3.24819	0.248395
$172$	0	0
$173$	17.7255	1.34765	0.673824	−	0.738892i	$-0.264651\pi$
$173$	17.7255	1.34765	0.673824	+	0.738892i	$0.264651\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.7483	1.03339
$178$	0	0
$179$	23.3759	1.74720	0.873598	−	0.486648i	$-0.161781\pi$
$179$	23.3759	1.74720	0.873598	+	0.486648i	$0.161781\pi$
$180$	0	0
$181$	−18.2576	−1.35707	−0.678537	−	0.734566i	$-0.737386\pi$
$181$	−18.2576	−1.35707	−0.678537	+	0.734566i	$0.737386\pi$
$182$	0	0
$183$	15.4907	1.14511
$184$	0	0
$185$	−6.57816	−0.483636
$186$	0	0
$187$	−0.304505	−0.0222676
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.7507	0.994967	0.497484	−	0.867473i	$-0.334257\pi$
$191$	13.7507	0.994967	0.497484	+	0.867473i	$0.334257\pi$
$192$	0	0
$193$	−12.6068	−0.907458	−0.453729	−	0.891140i	$-0.649907\pi$
$193$	−12.6068	−0.907458	−0.453729	+	0.891140i	$0.649907\pi$
$194$	0	0
$195$	3.11119	0.222797
$196$	0	0
$197$	0.817144	0.0582191	0.0291095	−	0.999576i	$-0.490733\pi$
$197$	0.817144	0.0582191	0.0291095	+	0.999576i	$0.490733\pi$
$198$	0	0
$199$	25.7193	1.82320	0.911598	−	0.411083i	$-0.134849\pi$
$199$	25.7193	1.82320	0.911598	+	0.411083i	$0.134849\pi$
$200$	0	0
$201$	−0.113029	−0.00797243
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.715473	0.0499708
$206$	0	0
$207$	7.63445	0.530631
$208$	0	0
$209$	0.354578	0.0245267
$210$	0	0
$211$	−11.0150	−0.758306	−0.379153	−	0.925334i	$-0.623785\pi$
$211$	−11.0150	−0.758306	−0.379153	+	0.925334i	$0.623785\pi$
$212$	0	0
$213$	15.3289	1.05032
$214$	0	0
$215$	−1.24840	−0.0851401
$216$	0	0
$217$	0	0
$218$	0	0
$219$	14.1035	0.953025
$220$	0	0
$221$	−6.10653	−0.410770
$222$	0	0
$223$	25.4981	1.70748	0.853740	−	0.520700i	$-0.174329\pi$
$223$	25.4981	1.70748	0.853740	+	0.520700i	$0.174329\pi$
$224$	0	0
$225$	8.35814	0.557209
$226$	0	0
$227$	−0.861316	−0.0571676	−0.0285838	−	0.999591i	$-0.509100\pi$
$227$	−0.861316	−0.0571676	−0.0285838	+	0.999591i	$0.509100\pi$
$228$	0	0
$229$	−6.01082	−0.397206	−0.198603	−	0.980080i	$-0.563640\pi$
$229$	−6.01082	−0.397206	−0.198603	+	0.980080i	$0.563640\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.80151	−0.183533	−0.0917664	−	0.995781i	$-0.529251\pi$
$233$	−2.80151	−0.183533	−0.0917664	+	0.995781i	$0.529251\pi$
$234$	0	0
$235$	−4.60269	−0.300246
$236$	0	0
$237$	7.34828	0.477322
$238$	0	0
$239$	19.6931	1.27384	0.636919	−	0.770930i	$-0.280208\pi$
$239$	19.6931	1.27384	0.636919	+	0.770930i	$0.280208\pi$
$240$	0	0
$241$	2.80922	0.180958	0.0904789	−	0.995898i	$-0.471160\pi$
$241$	2.80922	0.180958	0.0904789	+	0.995898i	$0.471160\pi$
$242$	0	0
$243$	15.6175	1.00186
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.11069	0.452443
$248$	0	0
$249$	−2.59675	−0.164562
$250$	0	0
$251$	−1.62156	−0.102352	−0.0511760	−	0.998690i	$-0.516297\pi$
$251$	−1.62156	−0.102352	−0.0511760	+	0.998690i	$0.516297\pi$
$252$	0	0
$253$	0.833389	0.0523947
$254$	0	0
$255$	−1.14311	−0.0715840
$256$	0	0
$257$	12.7893	0.797773	0.398887	−	0.917000i	$-0.369397\pi$
$257$	12.7893	0.797773	0.398887	+	0.917000i	$0.369397\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	9.56361	0.591972
$262$	0	0
$263$	−24.4563	−1.50804	−0.754020	−	0.656852i	$-0.771888\pi$
$263$	−24.4563	−1.50804	−0.754020	+	0.656852i	$0.771888\pi$
$264$	0	0
$265$	−4.73316	−0.290756
$266$	0	0
$267$	−4.66541	−0.285518
$268$	0	0
$269$	15.5542	0.948358	0.474179	−	0.880429i	$-0.342745\pi$
$269$	15.5542	0.948358	0.474179	+	0.880429i	$0.342745\pi$
$270$	0	0
$271$	−6.47684	−0.393440	−0.196720	−	0.980460i	$-0.563029\pi$
$271$	−6.47684	−0.393440	−0.196720	+	0.980460i	$0.563029\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.912388	0.0550191
$276$	0	0
$277$	−6.79599	−0.408331	−0.204166	−	0.978936i	$-0.565448\pi$
$277$	−6.79599	−0.408331	−0.204166	+	0.978936i	$0.565448\pi$
$278$	0	0
$279$	7.25459	0.434321
$280$	0	0
$281$	−9.02577	−0.538432	−0.269216	−	0.963080i	$-0.586765\pi$
$281$	−9.02577	−0.538432	−0.269216	+	0.963080i	$0.586765\pi$
$282$	0	0
$283$	−20.9495	−1.24532	−0.622661	−	0.782492i	$-0.713948\pi$
$283$	−20.9495	−1.24532	−0.622661	+	0.782492i	$0.713948\pi$
$284$	0	0
$285$	1.33108	0.0788463
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.7564	−0.868021
$290$	0	0
$291$	11.8454	0.694388
$292$	0	0
$293$	−19.7143	−1.15172	−0.575861	−	0.817547i	$-0.695333\pi$
$293$	−19.7143	−1.15172	−0.575861	+	0.817547i	$0.695333\pi$
$294$	0	0
$295$	−9.22203	−0.536927
$296$	0	0
$297$	1.05432	0.0611781
$298$	0	0
$299$	16.7128	0.966524
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−17.6556	−1.01429
$304$	0	0
$305$	−10.3908	−0.594974
$306$	0	0
$307$	2.69256	0.153672	0.0768361	−	0.997044i	$-0.475518\pi$
$307$	2.69256	0.153672	0.0768361	+	0.997044i	$0.475518\pi$
$308$	0	0
$309$	18.0556	1.02715
$310$	0	0
$311$	14.2409	0.807527	0.403764	−	0.914863i	$-0.367702\pi$
$311$	14.2409	0.807527	0.403764	+	0.914863i	$0.367702\pi$
$312$	0	0
$313$	31.4841	1.77959	0.889794	−	0.456363i	$-0.150848\pi$
$313$	31.4841	1.77959	0.889794	+	0.456363i	$0.150848\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.8349	−1.05787	−0.528936	−	0.848662i	$-0.677409\pi$
$317$	−18.8349	−1.05787	−0.528936	+	0.848662i	$0.677409\pi$
$318$	0	0
$319$	1.04398	0.0584516
$320$	0	0
$321$	12.8501	0.717222
$322$	0	0
$323$	−2.61259	−0.145368
$324$	0	0
$325$	18.2970	1.01493
$326$	0	0
$327$	−0.789727	−0.0436720
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.16405	0.448737	0.224368	−	0.974504i	$-0.427968\pi$
$331$	8.16405	0.448737	0.224368	+	0.974504i	$0.427968\pi$
$332$	0	0
$333$	−17.1222	−0.938288
$334$	0	0
$335$	0.0758168	0.00414232
$336$	0	0
$337$	28.7590	1.56660	0.783300	−	0.621643i	$-0.213535\pi$
$337$	28.7590	1.56660	0.783300	+	0.621643i	$0.213535\pi$
$338$	0	0
$339$	16.3549	0.888277
$340$	0	0
$341$	0.791924	0.0428851
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3.12853	0.168434
$346$	0	0
$347$	−16.3427	−0.877320	−0.438660	−	0.898653i	$-0.644547\pi$
$347$	−16.3427	−0.877320	−0.438660	+	0.898653i	$0.644547\pi$
$348$	0	0
$349$	34.3702	1.83980	0.919898	−	0.392157i	$-0.128271\pi$
$349$	34.3702	1.83980	0.919898	+	0.392157i	$0.128271\pi$
$350$	0	0
$351$	21.1434	1.12855
$352$	0	0
$353$	33.6237	1.78961	0.894806	−	0.446456i	$-0.147314\pi$
$353$	33.6237	1.78961	0.894806	+	0.446456i	$0.147314\pi$
$354$	0	0
$355$	−10.2822	−0.545723
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.2344	1.49015	0.745076	−	0.666980i	$-0.232413\pi$
$359$	28.2344	1.49015	0.745076	+	0.666980i	$0.232413\pi$
$360$	0	0
$361$	−15.9578	−0.839884
$362$	0	0
$363$	−11.6889	−0.613509
$364$	0	0
$365$	−9.46027	−0.495173
$366$	0	0
$367$	−12.7369	−0.664861	−0.332431	−	0.943128i	$-0.607869\pi$
$367$	−12.7369	−0.664861	−0.332431	+	0.943128i	$0.607869\pi$
$368$	0	0
$369$	1.86229	0.0969469
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0.0901714	0.00466890	0.00233445	−	0.999997i	$-0.499257\pi$
$373$	0.0901714	0.00466890	0.00233445	+	0.999997i	$0.499257\pi$
$374$	0	0
$375$	7.24083	0.373915
$376$	0	0
$377$	20.9359	1.07825
$378$	0	0
$379$	−12.3658	−0.635186	−0.317593	−	0.948227i	$-0.602875\pi$
$379$	−12.3658	−0.635186	−0.317593	+	0.948227i	$0.602875\pi$
$380$	0	0
$381$	−19.1097	−0.979020
$382$	0	0
$383$	−14.1692	−0.724013	−0.362007	−	0.932175i	$-0.617908\pi$
$383$	−14.1692	−0.724013	−0.362007	+	0.932175i	$0.617908\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.24943	−0.165178
$388$	0	0
$389$	−18.6685	−0.946530	−0.473265	−	0.880920i	$-0.656925\pi$
$389$	−18.6685	−0.946530	−0.473265	+	0.880920i	$0.656925\pi$
$390$	0	0
$391$	−6.14055	−0.310541
$392$	0	0
$393$	−4.11671	−0.207660
$394$	0	0
$395$	−4.92905	−0.248007
$396$	0	0
$397$	−17.9221	−0.899485	−0.449743	−	0.893158i	$-0.648484\pi$
$397$	−17.9221	−0.899485	−0.449743	+	0.893158i	$0.648484\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.32147	−0.115928	−0.0579642	−	0.998319i	$-0.518461\pi$
$401$	−2.32147	−0.115928	−0.0579642	+	0.998319i	$0.518461\pi$
$402$	0	0
$403$	15.8812	0.791099
$404$	0	0
$405$	−0.0393421	−0.00195493
$406$	0	0
$407$	−1.86908	−0.0926470
$408$	0	0
$409$	2.74754	0.135857	0.0679287	−	0.997690i	$-0.478361\pi$
$409$	2.74754	0.135857	0.0679287	+	0.997690i	$0.478361\pi$
$410$	0	0
$411$	−9.81132	−0.483957
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.74183	0.0855033
$416$	0	0
$417$	−6.24479	−0.305809
$418$	0	0
$419$	−14.3395	−0.700532	−0.350266	−	0.936650i	$-0.613909\pi$
$419$	−14.3395	−0.700532	−0.350266	+	0.936650i	$0.613909\pi$
$420$	0	0
$421$	24.5005	1.19408	0.597041	−	0.802211i	$-0.296343\pi$
$421$	24.5005	1.19408	0.597041	+	0.802211i	$0.296343\pi$
$422$	0	0
$423$	−11.9802	−0.582499
$424$	0	0
$425$	−6.72263	−0.326096
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.883997	0.0426798
$430$	0	0
$431$	−33.7836	−1.62730	−0.813648	−	0.581357i	$-0.802522\pi$
$431$	−33.7836	−1.62730	−0.813648	+	0.581357i	$0.802522\pi$
$432$	0	0
$433$	14.9803	0.719907	0.359953	−	0.932970i	$-0.382793\pi$
$433$	14.9803	0.719907	0.359953	+	0.932970i	$0.382793\pi$
$434$	0	0
$435$	3.91908	0.187905
$436$	0	0
$437$	7.15031	0.342046
$438$	0	0
$439$	10.9944	0.524733	0.262367	−	0.964968i	$-0.415497\pi$
$439$	10.9944	0.524733	0.262367	+	0.964968i	$0.415497\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	32.5677	1.54734	0.773669	−	0.633590i	$-0.218419\pi$
$443$	32.5677	1.54734	0.773669	+	0.633590i	$0.218419\pi$
$444$	0	0
$445$	3.12944	0.148350
$446$	0	0
$447$	−20.0846	−0.949969
$448$	0	0
$449$	−40.8875	−1.92960	−0.964801	−	0.262982i	$-0.915294\pi$
$449$	−40.8875	−1.92960	−0.964801	+	0.262982i	$0.915294\pi$
$450$	0	0
$451$	0.203291	0.00957258
$452$	0	0
$453$	−23.3488	−1.09702
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.66835	−0.124820	−0.0624101	−	0.998051i	$-0.519879\pi$
$457$	−2.66835	−0.124820	−0.0624101	+	0.998051i	$0.519879\pi$
$458$	0	0
$459$	−7.76844	−0.362600
$460$	0	0
$461$	5.07360	0.236301	0.118151	−	0.992996i	$-0.462303\pi$
$461$	5.07360	0.236301	0.118151	+	0.992996i	$0.462303\pi$
$462$	0	0
$463$	−33.9827	−1.57931	−0.789655	−	0.613551i	$-0.789741\pi$
$463$	−33.9827	−1.57931	−0.789655	+	0.613551i	$0.789741\pi$
$464$	0	0
$465$	2.97287	0.137863
$466$	0	0
$467$	20.6133	0.953869	0.476935	−	0.878939i	$-0.341748\pi$
$467$	20.6133	0.953869	0.476935	+	0.878939i	$0.341748\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−0.122456	−0.00564248
$472$	0	0
$473$	−0.354714	−0.0163097
$474$	0	0
$475$	7.82811	0.359178
$476$	0	0
$477$	−12.3198	−0.564086
$478$	0	0
$479$	−0.766787	−0.0350354	−0.0175177	−	0.999847i	$-0.505576\pi$
$479$	−0.766787	−0.0350354	−0.0175177	+	0.999847i	$0.505576\pi$
$480$	0	0
$481$	−37.4825	−1.70906
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.94558	−0.360790
$486$	0	0
$487$	−5.65740	−0.256361	−0.128181	−	0.991751i	$-0.540914\pi$
$487$	−5.65740	−0.256361	−0.128181	+	0.991751i	$0.540914\pi$
$488$	0	0
$489$	8.68960	0.392957
$490$	0	0
$491$	5.55831	0.250843	0.125422	−	0.992104i	$-0.459972\pi$
$491$	5.55831	0.250843	0.125422	+	0.992104i	$0.459972\pi$
$492$	0	0
$493$	−7.69222	−0.346440
$494$	0	0
$495$	−0.270868	−0.0121746
$496$	0	0
$497$	0	0
$498$	0	0
$499$	35.1207	1.57222	0.786108	−	0.618089i	$-0.212093\pi$
$499$	35.1207	1.57222	0.786108	+	0.618089i	$0.212093\pi$
$500$	0	0
$501$	−19.3187	−0.863098
$502$	0	0
$503$	−8.33640	−0.371702	−0.185851	−	0.982578i	$-0.559504\pi$
$503$	−8.33640	−0.371702	−0.185851	+	0.982578i	$0.559504\pi$
$504$	0	0
$505$	11.8429	0.527004
$506$	0	0
$507$	3.86138	0.171490
$508$	0	0
$509$	29.4727	1.30635	0.653177	−	0.757205i	$-0.273436\pi$
$509$	29.4727	1.30635	0.653177	+	0.757205i	$0.273436\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	9.04589	0.399386
$514$	0	0
$515$	−12.1112	−0.533685
$516$	0	0
$517$	−1.30778	−0.0575163
$518$	0	0
$519$	18.9067	0.829911
$520$	0	0
$521$	11.8169	0.517707	0.258853	−	0.965917i	$-0.416655\pi$
$521$	11.8169	0.517707	0.258853	+	0.965917i	$0.416655\pi$
$522$	0	0
$523$	36.4835	1.59531	0.797656	−	0.603113i	$-0.206073\pi$
$523$	36.4835	1.59531	0.797656	+	0.603113i	$0.206073\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.83503	−0.254178
$528$	0	0
$529$	−6.19413	−0.269310
$530$	0	0
$531$	−24.0038	−1.04168
$532$	0	0
$533$	4.07678	0.176585
$534$	0	0
$535$	−8.61953	−0.372655
$536$	0	0
$537$	24.9335	1.07596
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.4703	0.751107	0.375553	−	0.926801i	$-0.377453\pi$
$541$	17.4703	0.751107	0.375553	+	0.926801i	$0.377453\pi$
$542$	0	0
$543$	−19.4742	−0.835717
$544$	0	0
$545$	0.529730	0.0226911
$546$	0	0
$547$	22.4215	0.958673	0.479337	−	0.877631i	$-0.340877\pi$
$547$	22.4215	0.958673	0.479337	+	0.877631i	$0.340877\pi$
$548$	0	0
$549$	−27.0459	−1.15429
$550$	0	0
$551$	8.95713	0.381587
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7.01650	−0.297834
$556$	0	0
$557$	−5.30562	−0.224806	−0.112403	−	0.993663i	$-0.535855\pi$
$557$	−5.30562	−0.224806	−0.112403	+	0.993663i	$0.535855\pi$
$558$	0	0
$559$	−7.11341	−0.300865
$560$	0	0
$561$	−0.324796	−0.0137129
$562$	0	0
$563$	−8.36390	−0.352496	−0.176248	−	0.984346i	$-0.556396\pi$
$563$	−8.36390	−0.352496	−0.176248	+	0.984346i	$0.556396\pi$
$564$	0	0
$565$	−10.9705	−0.461531
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.7650	1.28974	0.644868	−	0.764294i	$-0.276912\pi$
$569$	30.7650	1.28974	0.644868	+	0.764294i	$0.276912\pi$
$570$	0	0
$571$	−10.4697	−0.438145	−0.219072	−	0.975709i	$-0.570303\pi$
$571$	−10.4697	−0.438145	−0.219072	+	0.975709i	$0.570303\pi$
$572$	0	0
$573$	14.6670	0.612723
$574$	0	0
$575$	18.3989	0.767289
$576$	0	0
$577$	19.1722	0.798150	0.399075	−	0.916918i	$-0.369331\pi$
$577$	19.1722	0.798150	0.399075	+	0.916918i	$0.369331\pi$
$578$	0	0
$579$	−13.4469	−0.558833
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.34485	−0.0556982
$584$	0	0
$585$	−5.43197	−0.224585
$586$	0	0
$587$	23.2967	0.961559	0.480779	−	0.876842i	$-0.340354\pi$
$587$	23.2967	0.961559	0.480779	+	0.876842i	$0.340354\pi$
$588$	0	0
$589$	6.79455	0.279964
$590$	0	0
$591$	0.871595	0.0358526
$592$	0	0
$593$	−3.59717	−0.147718	−0.0738590	−	0.997269i	$-0.523531\pi$
$593$	−3.59717	−0.147718	−0.0738590	+	0.997269i	$0.523531\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	27.4332	1.12276
$598$	0	0
$599$	−42.0452	−1.71792	−0.858961	−	0.512041i	$-0.828889\pi$
$599$	−42.0452	−1.71792	−0.858961	+	0.512041i	$0.828889\pi$
$600$	0	0
$601$	19.9085	0.812084	0.406042	−	0.913854i	$-0.366909\pi$
$601$	19.9085	0.812084	0.406042	+	0.913854i	$0.366909\pi$
$602$	0	0
$603$	0.197342	0.00803639
$604$	0	0
$605$	7.84063	0.318767
$606$	0	0
$607$	37.6134	1.52668	0.763341	−	0.645996i	$-0.223558\pi$
$607$	37.6134	1.52668	0.763341	+	0.645996i	$0.223558\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−26.2262	−1.06100
$612$	0	0
$613$	16.3767	0.661448	0.330724	−	0.943728i	$-0.392707\pi$
$613$	16.3767	0.661448	0.330724	+	0.943728i	$0.392707\pi$
$614$	0	0
$615$	0.763149	0.0307731
$616$	0	0
$617$	22.8650	0.920509	0.460255	−	0.887787i	$-0.347758\pi$
$617$	22.8650	0.920509	0.460255	+	0.887787i	$0.347758\pi$
$618$	0	0
$619$	−14.3938	−0.578536	−0.289268	−	0.957248i	$-0.593412\pi$
$619$	−14.3938	−0.578536	−0.289268	+	0.957248i	$0.593412\pi$
$620$	0	0
$621$	21.2612	0.853182
$622$	0	0
$623$	0	0
$624$	0	0
$625$	17.5835	0.703341
$626$	0	0
$627$	0.378205	0.0151041
$628$	0	0
$629$	13.7717	0.549115
$630$	0	0
$631$	−34.5118	−1.37389	−0.686946	−	0.726709i	$-0.741049\pi$
$631$	−34.5118	−1.37389	−0.686946	+	0.726709i	$0.741049\pi$
$632$	0	0
$633$	−11.7490	−0.466981
$634$	0	0
$635$	12.8183	0.508680
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−26.7634	−1.05874
$640$	0	0
$641$	27.1524	1.07245	0.536227	−	0.844074i	$-0.319849\pi$
$641$	27.1524	1.07245	0.536227	+	0.844074i	$0.319849\pi$
$642$	0	0
$643$	29.5059	1.16360	0.581800	−	0.813332i	$-0.302349\pi$
$643$	29.5059	1.16360	0.581800	+	0.813332i	$0.302349\pi$
$644$	0	0
$645$	−1.33159	−0.0524312
$646$	0	0
$647$	−22.5174	−0.885251	−0.442625	−	0.896707i	$-0.645953\pi$
$647$	−22.5174	−0.885251	−0.442625	+	0.896707i	$0.645953\pi$
$648$	0	0
$649$	−2.62030	−0.102856
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14.7254	−0.576248	−0.288124	−	0.957593i	$-0.593032\pi$
$653$	−14.7254	−0.576248	−0.288124	+	0.957593i	$0.593032\pi$
$654$	0	0
$655$	2.76139	0.107896
$656$	0	0
$657$	−24.6239	−0.960671
$658$	0	0
$659$	−19.4025	−0.755816	−0.377908	−	0.925843i	$-0.623356\pi$
$659$	−19.4025	−0.755816	−0.377908	+	0.925843i	$0.623356\pi$
$660$	0	0
$661$	4.44756	0.172990	0.0864950	−	0.996252i	$-0.472433\pi$
$661$	4.44756	0.172990	0.0864950	+	0.996252i	$0.472433\pi$
$662$	0	0
$663$	−6.51344	−0.252961
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.0526	0.815159
$668$	0	0
$669$	27.1972	1.05150
$670$	0	0
$671$	−2.95238	−0.113975
$672$	0	0
$673$	−34.2080	−1.31862	−0.659310	−	0.751871i	$-0.729152\pi$
$673$	−34.2080	−1.31862	−0.659310	+	0.751871i	$0.729152\pi$
$674$	0	0
$675$	23.2766	0.895916
$676$	0	0
$677$	−7.76820	−0.298556	−0.149278	−	0.988795i	$-0.547695\pi$
$677$	−7.76820	−0.298556	−0.149278	+	0.988795i	$0.547695\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−0.918711	−0.0352051
$682$	0	0
$683$	37.2431	1.42507	0.712534	−	0.701637i	$-0.247547\pi$
$683$	37.2431	1.42507	0.712534	+	0.701637i	$0.247547\pi$
$684$	0	0
$685$	6.58119	0.251454
$686$	0	0
$687$	−6.41135	−0.244608
$688$	0	0
$689$	−26.9696	−1.02746
$690$	0	0
$691$	−1.92228	−0.0731271	−0.0365635	−	0.999331i	$-0.511641\pi$
$691$	−1.92228	−0.0731271	−0.0365635	+	0.999331i	$0.511641\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.18885	0.158892
$696$	0	0
$697$	−1.49788	−0.0567362
$698$	0	0
$699$	−2.98818	−0.113024
$700$	0	0
$701$	50.1648	1.89470	0.947350	−	0.320201i	$-0.103750\pi$
$701$	50.1648	1.89470	0.947350	+	0.320201i	$0.103750\pi$
$702$	0	0
$703$	−16.0364	−0.604823
$704$	0	0
$705$	−4.90939	−0.184898
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−41.8140	−1.57036	−0.785180	−	0.619268i	$-0.787429\pi$
$709$	−41.8140	−1.57036	−0.785180	+	0.619268i	$0.787429\pi$
$710$	0	0
$711$	−12.8297	−0.481152
$712$	0	0
$713$	15.9697	0.598070
$714$	0	0
$715$	−0.592964	−0.0221756
$716$	0	0
$717$	21.0053	0.784458
$718$	0	0
$719$	−22.9341	−0.855298	−0.427649	−	0.903945i	$-0.640658\pi$
$719$	−22.9341	−0.855298	−0.427649	+	0.903945i	$0.640658\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	2.99641	0.111438
$724$	0	0
$725$	23.0482	0.855989
$726$	0	0
$727$	−18.6789	−0.692762	−0.346381	−	0.938094i	$-0.612589\pi$
$727$	−18.6789	−0.692762	−0.346381	+	0.938094i	$0.612589\pi$
$728$	0	0
$729$	16.4932	0.610861
$730$	0	0
$731$	2.61359	0.0966671
$732$	0	0
$733$	52.7201	1.94726	0.973631	−	0.228127i	$-0.0732600\pi$
$733$	52.7201	1.94726	0.973631	+	0.228127i	$0.0732600\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.0215422	0.000793517 0
$738$	0	0
$739$	−10.8600	−0.399490	−0.199745	−	0.979848i	$-0.564011\pi$
$739$	−10.8600	−0.399490	−0.199745	+	0.979848i	$0.564011\pi$
$740$	0	0
$741$	7.58452	0.278624
$742$	0	0
$743$	−32.6159	−1.19656	−0.598281	−	0.801286i	$-0.704149\pi$
$743$	−32.6159	−1.19656	−0.598281	+	0.801286i	$0.704149\pi$
$744$	0	0
$745$	13.4723	0.493585
$746$	0	0
$747$	4.53378	0.165882
$748$	0	0
$749$	0	0
$750$	0	0
$751$	12.5099	0.456491	0.228246	−	0.973604i	$-0.426701\pi$
$751$	12.5099	0.456491	0.228246	+	0.973604i	$0.426701\pi$
$752$	0	0
$753$	−1.72961	−0.0630307
$754$	0	0
$755$	15.6618	0.569991
$756$	0	0
$757$	−45.9033	−1.66838	−0.834192	−	0.551474i	$-0.814066\pi$
$757$	−45.9033	−1.66838	−0.834192	+	0.551474i	$0.814066\pi$
$758$	0	0
$759$	0.888922	0.0322658
$760$	0	0
$761$	16.5517	0.599998	0.299999	−	0.953940i	$-0.403014\pi$
$761$	16.5517	0.599998	0.299999	+	0.953940i	$0.403014\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.99580	0.0721583
$766$	0	0
$767$	−52.5474	−1.89738
$768$	0	0
$769$	13.3183	0.480269	0.240134	−	0.970740i	$-0.422809\pi$
$769$	13.3183	0.480269	0.240134	+	0.970740i	$0.422809\pi$
$770$	0	0
$771$	13.6415	0.491286
$772$	0	0
$773$	16.1812	0.581998	0.290999	−	0.956723i	$-0.406012\pi$
$773$	16.1812	0.581998	0.290999	+	0.956723i	$0.406012\pi$
$774$	0	0
$775$	17.4835	0.628026
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.74419	0.0624922
$780$	0	0
$781$	−2.92153	−0.104541
$782$	0	0
$783$	26.6337	0.951810
$784$	0	0
$785$	0.0821405	0.00293172
$786$	0	0
$787$	−17.8464	−0.636156	−0.318078	−	0.948065i	$-0.603037\pi$
$787$	−17.8464	−0.636156	−0.318078	+	0.948065i	$0.603037\pi$
$788$	0	0
$789$	−26.0859	−0.928684
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−59.2069	−2.10250
$794$	0	0
$795$	−5.04855	−0.179054
$796$	0	0
$797$	−7.13372	−0.252689	−0.126345	−	0.991986i	$-0.540325\pi$
$797$	−7.13372	−0.252689	−0.126345	+	0.991986i	$0.540325\pi$
$798$	0	0
$799$	9.63597	0.340896
$800$	0	0
$801$	8.14556	0.287809
$802$	0	0
$803$	−2.68799	−0.0948571
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.5907	0.584020
$808$	0	0
$809$	25.3449	0.891081	0.445540	−	0.895262i	$-0.353012\pi$
$809$	25.3449	0.891081	0.445540	+	0.895262i	$0.353012\pi$
$810$	0	0
$811$	1.80874	0.0635136	0.0317568	−	0.999496i	$-0.489890\pi$
$811$	1.80874	0.0635136	0.0317568	+	0.999496i	$0.489890\pi$
$812$	0	0
$813$	−6.90842	−0.242289
$814$	0	0
$815$	−5.82877	−0.204173
$816$	0	0
$817$	−3.04337	−0.106474
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−48.5388	−1.69402	−0.847008	−	0.531581i	$-0.821598\pi$
$821$	−48.5388	−1.69402	−0.847008	+	0.531581i	$0.821598\pi$
$822$	0	0
$823$	29.2821	1.02071	0.510355	−	0.859964i	$-0.329514\pi$
$823$	29.2821	1.02071	0.510355	+	0.859964i	$0.329514\pi$
$824$	0	0
$825$	0.973186	0.0338820
$826$	0	0
$827$	−17.6267	−0.612940	−0.306470	−	0.951880i	$-0.599148\pi$
$827$	−17.6267	−0.612940	−0.306470	+	0.951880i	$0.599148\pi$
$828$	0	0
$829$	45.7256	1.58812	0.794058	−	0.607843i	$-0.207965\pi$
$829$	45.7256	1.58812	0.794058	+	0.607843i	$0.207965\pi$
$830$	0	0
$831$	−7.24884	−0.251459
$832$	0	0
$833$	0	0
$834$	0	0
$835$	12.9585	0.448449
$836$	0	0
$837$	20.2033	0.698329
$838$	0	0
$839$	24.4240	0.843210	0.421605	−	0.906780i	$-0.361467\pi$
$839$	24.4240	0.843210	0.421605	+	0.906780i	$0.361467\pi$
$840$	0	0
$841$	−2.62763	−0.0906081
$842$	0	0
$843$	−9.62720	−0.331578
$844$	0	0
$845$	−2.59012	−0.0891029
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−22.3455	−0.766896
$850$	0	0
$851$	−37.6914	−1.29204
$852$	0	0
$853$	−1.48129	−0.0507185	−0.0253592	−	0.999678i	$-0.508073\pi$
$853$	−1.48129	−0.0507185	−0.0253592	+	0.999678i	$0.508073\pi$
$854$	0	0
$855$	−2.32399	−0.0794789
$856$	0	0
$857$	0.323927	0.0110652	0.00553258	−	0.999985i	$-0.498239\pi$
$857$	0.323927	0.0110652	0.00553258	+	0.999985i	$0.498239\pi$
$858$	0	0
$859$	−28.8939	−0.985847	−0.492924	−	0.870073i	$-0.664072\pi$
$859$	−28.8939	−0.985847	−0.492924	+	0.870073i	$0.664072\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	30.8493	1.05012	0.525062	−	0.851064i	$-0.324042\pi$
$863$	30.8493	1.05012	0.525062	+	0.851064i	$0.324042\pi$
$864$	0	0
$865$	−12.6821	−0.431206
$866$	0	0
$867$	−15.7397	−0.534547
$868$	0	0
$869$	−1.40051	−0.0475092
$870$	0	0
$871$	0.432006	0.0146380
$872$	0	0
$873$	−20.6814	−0.699959
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−30.8540	−1.04187	−0.520933	−	0.853597i	$-0.674416\pi$
$877$	−30.8540	−1.04187	−0.520933	+	0.853597i	$0.674416\pi$
$878$	0	0
$879$	−21.0280	−0.709257
$880$	0	0
$881$	−53.8585	−1.81454	−0.907270	−	0.420550i	$-0.861837\pi$
$881$	−53.8585	−1.81454	−0.907270	+	0.420550i	$0.861837\pi$
$882$	0	0
$883$	14.1687	0.476814	0.238407	−	0.971165i	$-0.423375\pi$
$883$	14.1687	0.476814	0.238407	+	0.971165i	$0.423375\pi$
$884$	0	0
$885$	−9.83654	−0.330652
$886$	0	0
$887$	−32.6309	−1.09564	−0.547820	−	0.836596i	$-0.684542\pi$
$887$	−32.6309	−1.09564	−0.547820	+	0.836596i	$0.684542\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.0111785	−0.000374493 0
$892$	0	0
$893$	−11.2205	−0.375480
$894$	0	0
$895$	−16.7248	−0.559049
$896$	0	0
$897$	17.8264	0.595207
$898$	0	0
$899$	20.0051	0.667207
$900$	0	0
$901$	9.90911	0.330120
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.0628	0.434222
$906$	0	0
$907$	48.1270	1.59803	0.799016	−	0.601310i	$-0.205354\pi$
$907$	48.1270	1.59803	0.799016	+	0.601310i	$0.205354\pi$
$908$	0	0
$909$	30.8257	1.02243
$910$	0	0
$911$	−20.3867	−0.675441	−0.337720	−	0.941247i	$-0.609656\pi$
$911$	−20.3867	−0.675441	−0.337720	+	0.941247i	$0.609656\pi$
$912$	0	0
$913$	0.494915	0.0163793
$914$	0	0
$915$	−11.0832	−0.366398
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−2.66405	−0.0878789	−0.0439395	−	0.999034i	$-0.513991\pi$
$919$	−2.66405	−0.0878789	−0.0439395	+	0.999034i	$0.513991\pi$
$920$	0	0
$921$	2.87198	0.0946348
$922$	0	0
$923$	−58.5883	−1.92846
$924$	0	0
$925$	−41.2642	−1.35676
$926$	0	0
$927$	−31.5241	−1.03539
$928$	0	0
$929$	−31.5871	−1.03634	−0.518170	−	0.855278i	$-0.673387\pi$
$929$	−31.5871	−1.03634	−0.518170	+	0.855278i	$0.673387\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	15.1898	0.497293
$934$	0	0
$935$	0.217865	0.00712495
$936$	0	0
$937$	−7.30876	−0.238767	−0.119383	−	0.992848i	$-0.538092\pi$
$937$	−7.30876	−0.238767	−0.119383	+	0.992848i	$0.538092\pi$
$938$	0	0
$939$	33.5821	1.09591
$940$	0	0
$941$	2.93375	0.0956374	0.0478187	−	0.998856i	$-0.484773\pi$
$941$	2.93375	0.0956374	0.0478187	+	0.998856i	$0.484773\pi$
$942$	0	0
$943$	4.09950	0.133498
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.7181	−0.933212	−0.466606	−	0.884465i	$-0.654523\pi$
$947$	−28.7181	−0.933212	−0.466606	+	0.884465i	$0.654523\pi$
$948$	0	0
$949$	−53.9048	−1.74982
$950$	0	0
$951$	−20.0899	−0.651461
$952$	0	0
$953$	39.0891	1.26622	0.633110	−	0.774062i	$-0.281778\pi$
$953$	39.0891	1.26622	0.633110	+	0.774062i	$0.281778\pi$
$954$	0	0
$955$	−9.83827	−0.318359
$956$	0	0
$957$	1.11355	0.0359958
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.8249	−0.510480
$962$	0	0
$963$	−22.4356	−0.722976
$964$	0	0
$965$	9.01983	0.290359
$966$	0	0
$967$	4.10133	0.131890	0.0659450	−	0.997823i	$-0.478994\pi$
$967$	4.10133	0.131890	0.0659450	+	0.997823i	$0.478994\pi$
$968$	0	0
$969$	−2.78668	−0.0895211
$970$	0	0
$971$	32.9460	1.05729	0.528644	−	0.848844i	$-0.322701\pi$
$971$	32.9460	1.05729	0.528644	+	0.848844i	$0.322701\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	19.5162	0.625019
$976$	0	0
$977$	−28.9811	−0.927186	−0.463593	−	0.886048i	$-0.653440\pi$
$977$	−28.9811	−0.927186	−0.463593	+	0.886048i	$0.653440\pi$
$978$	0	0
$979$	0.889183	0.0284184
$980$	0	0
$981$	1.37882	0.0440224
$982$	0	0
$983$	−35.2495	−1.12428	−0.562142	−	0.827040i	$-0.690023\pi$
$983$	−35.2495	−1.12428	−0.562142	+	0.827040i	$0.690023\pi$
$984$	0	0
$985$	−0.584644	−0.0186283
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.15305	−0.227454
$990$	0	0
$991$	48.6066	1.54404	0.772021	−	0.635598i	$-0.219246\pi$
$991$	48.6066	1.54404	0.772021	+	0.635598i	$0.219246\pi$
$992$	0	0
$993$	8.70807	0.276342
$994$	0	0
$995$	−18.4015	−0.583366
$996$	0	0
$997$	31.6793	1.00329	0.501647	−	0.865072i	$-0.332728\pi$
$997$	31.6793	1.00329	0.501647	+	0.865072i	$0.332728\pi$
$998$	0	0
$999$	−47.6835	−1.50864

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.t.1.13	yes	20
7.6	odd	2		8036.2.a.s.1.8	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.8	✓	20	7.6	odd	2
8036.2.a.t.1.13	yes	20	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.06664 q^{3} -0.715473 q^{5} -1.86229 q^{9} +O(q^{10})\)	\(q+1.06664 q^{3} -0.715473 q^{5} -1.86229 q^{9} -0.203291 q^{11} -4.07678 q^{13} -0.763149 q^{15} +1.49788 q^{17} -1.74419 q^{19} -4.09950 q^{23} -4.48810 q^{25} -5.18629 q^{27} -5.13540 q^{29} -3.89553 q^{31} -0.216837 q^{33} +9.19415 q^{37} -4.34844 q^{39} -1.00000 q^{41} +1.74486 q^{43} +1.33242 q^{45} +6.43307 q^{47} +1.59769 q^{51} +6.61543 q^{53} +0.145449 q^{55} -1.86042 q^{57} +12.8894 q^{59} +14.5230 q^{61} +2.91683 q^{65} -0.105967 q^{67} -4.37267 q^{69} +14.3712 q^{71} +13.2224 q^{73} -4.78716 q^{75} +6.88922 q^{79} +0.0549876 q^{81} -2.43452 q^{83} -1.07169 q^{85} -5.47760 q^{87} -4.37395 q^{89} -4.15511 q^{93} +1.24792 q^{95} +11.1054 q^{97} +0.378586 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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