LMFDB - Embedded newform 8040.2.a.bc.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8040,2,Mod(1,8040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8040.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:	$64.1997232251$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 37x^{8} + 132x^{7} + 358x^{6} - 1708x^{5} - 92x^{4} + 5969x^{3} - 3864x^{2} - 4752x + 3524 $ x^10 - 3x^9 - 37x^8 + 132x^7 + 358x^6 - 1708x^5 - 92x^4 + 5969x^3 - 3864x^2 - 4752*x + 3524
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.15252$ of defining polynomial
Character	$\chi$	$=$	8040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +4.74497 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +4.74497 q^{7} +1.00000 q^{9} -0.614164 q^{11} +4.83858 q^{13} -1.00000 q^{15} +7.15132 q^{17} -2.88977 q^{19} -4.74497 q^{21} -9.01284 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.36617 q^{29} -7.99531 q^{31} +0.614164 q^{33} +4.74497 q^{35} -3.03993 q^{37} -4.83858 q^{39} -2.82560 q^{41} +11.4493 q^{43} +1.00000 q^{45} +12.7326 q^{47} +15.5147 q^{49} -7.15132 q^{51} +7.25040 q^{53} -0.614164 q^{55} +2.88977 q^{57} +6.00692 q^{59} -5.77620 q^{61} +4.74497 q^{63} +4.83858 q^{65} -1.00000 q^{67} +9.01284 q^{69} +3.48689 q^{71} +11.9545 q^{73} -1.00000 q^{75} -2.91419 q^{77} -12.5216 q^{79} +1.00000 q^{81} +6.60185 q^{83} +7.15132 q^{85} -7.36617 q^{87} -3.74308 q^{89} +22.9589 q^{91} +7.99531 q^{93} -2.88977 q^{95} -9.96750 q^{97} -0.614164 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9	$ 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9} + 7 q^{11} - q^{13} - 10 q^{15} + 3 q^{17} + 4 q^{19} - q^{21} - 13 q^{23} + 10 q^{25} - 10 q^{27} + 18 q^{29} - 9 q^{31} - 7 q^{33} + q^{35} - 3 q^{37} + q^{39} + 19 q^{41} + 5 q^{43} + 10 q^{45} - 8 q^{47} + 43 q^{49} - 3 q^{51} + 17 q^{53} + 7 q^{55} - 4 q^{57} + 24 q^{59} + 21 q^{61} + q^{63} - q^{65} - 10 q^{67} + 13 q^{69} + 2 q^{71} + 25 q^{73} - 10 q^{75} + 15 q^{77} - q^{79} + 10 q^{81} - 6 q^{83} + 3 q^{85} - 18 q^{87} + 23 q^{89} + 29 q^{91} + 9 q^{93} + 4 q^{95} + 21 q^{97} + 7 q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9 + 7 * q^11 - q^13 - 10 * q^15 + 3 * q^17 + 4 * q^19 - q^21 - 13 * q^23 + 10 * q^25 - 10 * q^27 + 18 * q^29 - 9 * q^31 - 7 * q^33 + q^35 - 3 * q^37 + q^39 + 19 * q^41 + 5 * q^43 + 10 * q^45 - 8 * q^47 + 43 * q^49 - 3 * q^51 + 17 * q^53 + 7 * q^55 - 4 * q^57 + 24 * q^59 + 21 * q^61 + q^63 - q^65 - 10 * q^67 + 13 * q^69 + 2 * q^71 + 25 * q^73 - 10 * q^75 + 15 * q^77 - q^79 + 10 * q^81 - 6 * q^83 + 3 * q^85 - 18 * q^87 + 23 * q^89 + 29 * q^91 + 9 * q^93 + 4 * q^95 + 21 * q^97 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.74497	1.79343	0.896714	−	0.442610i	$-0.145947\pi$
$7$	4.74497	1.79343	0.896714	+	0.442610i	$0.145947\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.614164	−0.185177	−0.0925887	−	0.995704i	$-0.529514\pi$
$11$	−0.614164	−0.185177	−0.0925887	+	0.995704i	$0.529514\pi$
$12$	0	0
$13$	4.83858	1.34198	0.670990	−	0.741466i	$-0.265869\pi$
$13$	4.83858	1.34198	0.670990	+	0.741466i	$0.265869\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	7.15132	1.73445	0.867224	−	0.497918i	$-0.165902\pi$
$17$	7.15132	1.73445	0.867224	+	0.497918i	$0.165902\pi$
$18$	0	0
$19$	−2.88977	−0.662959	−0.331479	−	0.943462i	$-0.607548\pi$
$19$	−2.88977	−0.662959	−0.331479	+	0.943462i	$0.607548\pi$
$20$	0	0
$21$	−4.74497	−1.03544
$22$	0	0
$23$	−9.01284	−1.87931	−0.939653	−	0.342128i	$-0.888852\pi$
$23$	−9.01284	−1.87931	−0.939653	+	0.342128i	$0.888852\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.36617	1.36786	0.683931	−	0.729546i	$-0.260269\pi$
$29$	7.36617	1.36786	0.683931	+	0.729546i	$0.260269\pi$
$30$	0	0
$31$	−7.99531	−1.43600	−0.718000	−	0.696043i	$-0.754942\pi$
$31$	−7.99531	−1.43600	−0.718000	+	0.696043i	$0.754942\pi$
$32$	0	0
$33$	0.614164	0.106912
$34$	0	0
$35$	4.74497	0.802046
$36$	0	0
$37$	−3.03993	−0.499762	−0.249881	−	0.968277i	$-0.580391\pi$
$37$	−3.03993	−0.499762	−0.249881	+	0.968277i	$0.580391\pi$
$38$	0	0
$39$	−4.83858	−0.774793
$40$	0	0
$41$	−2.82560	−0.441284	−0.220642	−	0.975355i	$-0.570815\pi$
$41$	−2.82560	−0.441284	−0.220642	+	0.975355i	$0.570815\pi$
$42$	0	0
$43$	11.4493	1.74601	0.873003	−	0.487715i	$-0.162169\pi$
$43$	11.4493	1.74601	0.873003	+	0.487715i	$0.162169\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	12.7326	1.85724	0.928619	−	0.371034i	$-0.120997\pi$
$47$	12.7326	1.85724	0.928619	+	0.371034i	$0.120997\pi$
$48$	0	0
$49$	15.5147	2.21639
$50$	0	0
$51$	−7.15132	−1.00138
$52$	0	0
$53$	7.25040	0.995919	0.497960	−	0.867200i	$-0.334083\pi$
$53$	7.25040	0.995919	0.497960	+	0.867200i	$0.334083\pi$
$54$	0	0
$55$	−0.614164	−0.0828139
$56$	0	0
$57$	2.88977	0.382759
$58$	0	0
$59$	6.00692	0.782034	0.391017	−	0.920383i	$-0.372123\pi$
$59$	6.00692	0.782034	0.391017	+	0.920383i	$0.372123\pi$
$60$	0	0
$61$	−5.77620	−0.739567	−0.369783	−	0.929118i	$-0.620568\pi$
$61$	−5.77620	−0.739567	−0.369783	+	0.929118i	$0.620568\pi$
$62$	0	0
$63$	4.74497	0.597810
$64$	0	0
$65$	4.83858	0.600152
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	9.01284	1.08502
$70$	0	0
$71$	3.48689	0.413817	0.206909	−	0.978360i	$-0.433660\pi$
$71$	3.48689	0.413817	0.206909	+	0.978360i	$0.433660\pi$
$72$	0	0
$73$	11.9545	1.39917	0.699585	−	0.714550i	$-0.253368\pi$
$73$	11.9545	1.39917	0.699585	+	0.714550i	$0.253368\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−2.91419	−0.332102
$78$	0	0
$79$	−12.5216	−1.40879	−0.704396	−	0.709807i	$-0.748782\pi$
$79$	−12.5216	−1.40879	−0.704396	+	0.709807i	$0.748782\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.60185	0.724647	0.362324	−	0.932052i	$-0.381983\pi$
$83$	6.60185	0.724647	0.362324	+	0.932052i	$0.381983\pi$
$84$	0	0
$85$	7.15132	0.775669
$86$	0	0
$87$	−7.36617	−0.789736
$88$	0	0
$89$	−3.74308	−0.396765	−0.198383	−	0.980125i	$-0.563569\pi$
$89$	−3.74308	−0.396765	−0.198383	+	0.980125i	$0.563569\pi$
$90$	0	0
$91$	22.9589	2.40675
$92$	0	0
$93$	7.99531	0.829075
$94$	0	0
$95$	−2.88977	−0.296484
$96$	0	0
$97$	−9.96750	−1.01205	−0.506023	−	0.862520i	$-0.668885\pi$
$97$	−9.96750	−1.01205	−0.506023	+	0.862520i	$0.668885\pi$
$98$	0	0
$99$	−0.614164	−0.0617258
$100$	0	0
$101$	−13.8374	−1.37687	−0.688434	−	0.725299i	$-0.741702\pi$
$101$	−13.8374	−1.37687	−0.688434	+	0.725299i	$0.741702\pi$
$102$	0	0
$103$	0.915216	0.0901789	0.0450894	−	0.998983i	$-0.485643\pi$
$103$	0.915216	0.0901789	0.0450894	+	0.998983i	$0.485643\pi$
$104$	0	0
$105$	−4.74497	−0.463061
$106$	0	0
$107$	1.41601	0.136891	0.0684455	−	0.997655i	$-0.478196\pi$
$107$	1.41601	0.136891	0.0684455	+	0.997655i	$0.478196\pi$
$108$	0	0
$109$	−3.40316	−0.325964	−0.162982	−	0.986629i	$-0.552111\pi$
$109$	−3.40316	−0.325964	−0.162982	+	0.986629i	$0.552111\pi$
$110$	0	0
$111$	3.03993	0.288538
$112$	0	0
$113$	12.0015	1.12901	0.564503	−	0.825431i	$-0.309068\pi$
$113$	12.0015	1.12901	0.564503	+	0.825431i	$0.309068\pi$
$114$	0	0
$115$	−9.01284	−0.840451
$116$	0	0
$117$	4.83858	0.447327
$118$	0	0
$119$	33.9327	3.11061
$120$	0	0
$121$	−10.6228	−0.965709
$122$	0	0
$123$	2.82560	0.254775
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.0770	−1.60407	−0.802037	−	0.597275i	$-0.796250\pi$
$127$	−18.0770	−1.60407	−0.802037	+	0.597275i	$0.796250\pi$
$128$	0	0
$129$	−11.4493	−1.00806
$130$	0	0
$131$	−7.25146	−0.633563	−0.316782	−	0.948499i	$-0.602602\pi$
$131$	−7.25146	−0.633563	−0.316782	+	0.948499i	$0.602602\pi$
$132$	0	0
$133$	−13.7119	−1.18897
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	21.7191	1.85559	0.927794	−	0.373093i	$-0.121703\pi$
$137$	21.7191	1.85559	0.927794	+	0.373093i	$0.121703\pi$
$138$	0	0
$139$	−11.8993	−1.00929	−0.504643	−	0.863328i	$-0.668376\pi$
$139$	−11.8993	−1.00929	−0.504643	+	0.863328i	$0.668376\pi$
$140$	0	0
$141$	−12.7326	−1.07228
$142$	0	0
$143$	−2.97168	−0.248505
$144$	0	0
$145$	7.36617	0.611727
$146$	0	0
$147$	−15.5147	−1.27963
$148$	0	0
$149$	−9.12207	−0.747309	−0.373654	−	0.927568i	$-0.621895\pi$
$149$	−9.12207	−0.747309	−0.373654	+	0.927568i	$0.621895\pi$
$150$	0	0
$151$	5.29876	0.431207	0.215604	−	0.976481i	$-0.430828\pi$
$151$	5.29876	0.431207	0.215604	+	0.976481i	$0.430828\pi$
$152$	0	0
$153$	7.15132	0.578150
$154$	0	0
$155$	−7.99531	−0.642199
$156$	0	0
$157$	−3.94646	−0.314962	−0.157481	−	0.987522i	$-0.550337\pi$
$157$	−3.94646	−0.314962	−0.157481	+	0.987522i	$0.550337\pi$
$158$	0	0
$159$	−7.25040	−0.574994
$160$	0	0
$161$	−42.7656	−3.37040
$162$	0	0
$163$	−19.7523	−1.54712	−0.773559	−	0.633724i	$-0.781525\pi$
$163$	−19.7523	−1.54712	−0.773559	+	0.633724i	$0.781525\pi$
$164$	0	0
$165$	0.614164	0.0478126
$166$	0	0
$167$	−5.53576	−0.428370	−0.214185	−	0.976793i	$-0.568710\pi$
$167$	−5.53576	−0.428370	−0.214185	+	0.976793i	$0.568710\pi$
$168$	0	0
$169$	10.4119	0.800913
$170$	0	0
$171$	−2.88977	−0.220986
$172$	0	0
$173$	24.0517	1.82862	0.914308	−	0.405019i	$-0.132735\pi$
$173$	24.0517	1.82862	0.914308	+	0.405019i	$0.132735\pi$
$174$	0	0
$175$	4.74497	0.358686
$176$	0	0
$177$	−6.00692	−0.451508
$178$	0	0
$179$	14.8278	1.10828	0.554141	−	0.832423i	$-0.313047\pi$
$179$	14.8278	1.10828	0.554141	+	0.832423i	$0.313047\pi$
$180$	0	0
$181$	12.8119	0.952304	0.476152	−	0.879363i	$-0.342031\pi$
$181$	12.8119	0.952304	0.476152	+	0.879363i	$0.342031\pi$
$182$	0	0
$183$	5.77620	0.426989
$184$	0	0
$185$	−3.03993	−0.223500
$186$	0	0
$187$	−4.39208	−0.321181
$188$	0	0
$189$	−4.74497	−0.345145
$190$	0	0
$191$	3.86630	0.279756	0.139878	−	0.990169i	$-0.455329\pi$
$191$	3.86630	0.279756	0.139878	+	0.990169i	$0.455329\pi$
$192$	0	0
$193$	−11.8474	−0.852796	−0.426398	−	0.904536i	$-0.640218\pi$
$193$	−11.8474	−0.852796	−0.426398	+	0.904536i	$0.640218\pi$
$194$	0	0
$195$	−4.83858	−0.346498
$196$	0	0
$197$	−20.0840	−1.43093	−0.715463	−	0.698651i	$-0.753784\pi$
$197$	−20.0840	−1.43093	−0.715463	+	0.698651i	$0.753784\pi$
$198$	0	0
$199$	−10.0796	−0.714522	−0.357261	−	0.934005i	$-0.616289\pi$
$199$	−10.0796	−0.714522	−0.357261	+	0.934005i	$0.616289\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	34.9522	2.45316
$204$	0	0
$205$	−2.82560	−0.197348
$206$	0	0
$207$	−9.01284	−0.626435
$208$	0	0
$209$	1.77479	0.122765
$210$	0	0
$211$	18.5890	1.27972	0.639860	−	0.768491i	$-0.278992\pi$
$211$	18.5890	1.27972	0.639860	+	0.768491i	$0.278992\pi$
$212$	0	0
$213$	−3.48689	−0.238918
$214$	0	0
$215$	11.4493	0.780838
$216$	0	0
$217$	−37.9375	−2.57536
$218$	0	0
$219$	−11.9545	−0.807811
$220$	0	0
$221$	34.6022	2.32760
$222$	0	0
$223$	16.1421	1.08096	0.540479	−	0.841357i	$-0.318243\pi$
$223$	16.1421	1.08096	0.540479	+	0.841357i	$0.318243\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	23.6777	1.57154	0.785771	−	0.618517i	$-0.212266\pi$
$227$	23.6777	1.57154	0.785771	+	0.618517i	$0.212266\pi$
$228$	0	0
$229$	−20.1309	−1.33029	−0.665144	−	0.746715i	$-0.731630\pi$
$229$	−20.1309	−1.33029	−0.665144	+	0.746715i	$0.731630\pi$
$230$	0	0
$231$	2.91419	0.191739
$232$	0	0
$233$	−20.5003	−1.34302	−0.671509	−	0.740997i	$-0.734353\pi$
$233$	−20.5003	−1.34302	−0.671509	+	0.740997i	$0.734353\pi$
$234$	0	0
$235$	12.7326	0.830582
$236$	0	0
$237$	12.5216	0.813366
$238$	0	0
$239$	−9.12005	−0.589927	−0.294963	−	0.955509i	$-0.595307\pi$
$239$	−9.12005	−0.589927	−0.294963	+	0.955509i	$0.595307\pi$
$240$	0	0
$241$	11.0641	0.712701	0.356351	−	0.934352i	$-0.384021\pi$
$241$	11.0641	0.712701	0.356351	+	0.934352i	$0.384021\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	15.5147	0.991198
$246$	0	0
$247$	−13.9824	−0.889678
$248$	0	0
$249$	−6.60185	−0.418375
$250$	0	0
$251$	12.0112	0.758142	0.379071	−	0.925368i	$-0.376244\pi$
$251$	12.0112	0.758142	0.379071	+	0.925368i	$0.376244\pi$
$252$	0	0
$253$	5.53536	0.348005
$254$	0	0
$255$	−7.15132	−0.447833
$256$	0	0
$257$	1.53376	0.0956736	0.0478368	−	0.998855i	$-0.484767\pi$
$257$	1.53376	0.0956736	0.0478368	+	0.998855i	$0.484767\pi$
$258$	0	0
$259$	−14.4244	−0.896287
$260$	0	0
$261$	7.36617	0.455954
$262$	0	0
$263$	11.0033	0.678495	0.339247	−	0.940697i	$-0.389828\pi$
$263$	11.0033	0.678495	0.339247	+	0.940697i	$0.389828\pi$
$264$	0	0
$265$	7.25040	0.445389
$266$	0	0
$267$	3.74308	0.229072
$268$	0	0
$269$	−8.71710	−0.531491	−0.265746	−	0.964043i	$-0.585618\pi$
$269$	−8.71710	−0.531491	−0.265746	+	0.964043i	$0.585618\pi$
$270$	0	0
$271$	−11.1658	−0.678273	−0.339137	−	0.940737i	$-0.610135\pi$
$271$	−11.1658	−0.678273	−0.339137	+	0.940737i	$0.610135\pi$
$272$	0	0
$273$	−22.9589	−1.38954
$274$	0	0
$275$	−0.614164	−0.0370355
$276$	0	0
$277$	27.9408	1.67880	0.839399	−	0.543516i	$-0.182907\pi$
$277$	27.9408	1.67880	0.839399	+	0.543516i	$0.182907\pi$
$278$	0	0
$279$	−7.99531	−0.478667
$280$	0	0
$281$	−6.80061	−0.405690	−0.202845	−	0.979211i	$-0.565019\pi$
$281$	−6.80061	−0.405690	−0.202845	+	0.979211i	$0.565019\pi$
$282$	0	0
$283$	−6.71528	−0.399182	−0.199591	−	0.979879i	$-0.563961\pi$
$283$	−6.71528	−0.399182	−0.199591	+	0.979879i	$0.563961\pi$
$284$	0	0
$285$	2.88977	0.171175
$286$	0	0
$287$	−13.4074	−0.791411
$288$	0	0
$289$	34.1413	2.00831
$290$	0	0
$291$	9.96750	0.584305
$292$	0	0
$293$	23.4972	1.37272	0.686360	−	0.727262i	$-0.259207\pi$
$293$	23.4972	1.37272	0.686360	+	0.727262i	$0.259207\pi$
$294$	0	0
$295$	6.00692	0.349736
$296$	0	0
$297$	0.614164	0.0356374
$298$	0	0
$299$	−43.6093	−2.52199
$300$	0	0
$301$	54.3267	3.13134
$302$	0	0
$303$	13.8374	0.794935
$304$	0	0
$305$	−5.77620	−0.330744
$306$	0	0
$307$	8.57086	0.489165	0.244582	−	0.969629i	$-0.421349\pi$
$307$	8.57086	0.489165	0.244582	+	0.969629i	$0.421349\pi$
$308$	0	0
$309$	−0.915216	−0.0520648
$310$	0	0
$311$	−16.3429	−0.926721	−0.463361	−	0.886170i	$-0.653357\pi$
$311$	−16.3429	−0.926721	−0.463361	+	0.886170i	$0.653357\pi$
$312$	0	0
$313$	−4.54355	−0.256816	−0.128408	−	0.991721i	$-0.540987\pi$
$313$	−4.54355	−0.256816	−0.128408	+	0.991721i	$0.540987\pi$
$314$	0	0
$315$	4.74497	0.267349
$316$	0	0
$317$	4.12219	0.231525	0.115762	−	0.993277i	$-0.463069\pi$
$317$	4.12219	0.231525	0.115762	+	0.993277i	$0.463069\pi$
$318$	0	0
$319$	−4.52403	−0.253297
$320$	0	0
$321$	−1.41601	−0.0790341
$322$	0	0
$323$	−20.6657	−1.14987
$324$	0	0
$325$	4.83858	0.268396
$326$	0	0
$327$	3.40316	0.188195
$328$	0	0
$329$	60.4157	3.33083
$330$	0	0
$331$	−19.0804	−1.04876	−0.524378	−	0.851486i	$-0.675702\pi$
$331$	−19.0804	−1.04876	−0.524378	+	0.851486i	$0.675702\pi$
$332$	0	0
$333$	−3.03993	−0.166587
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−0.129063	−0.00703050	−0.00351525	−	0.999994i	$-0.501119\pi$
$337$	−0.129063	−0.00703050	−0.00351525	+	0.999994i	$0.501119\pi$
$338$	0	0
$339$	−12.0015	−0.651832
$340$	0	0
$341$	4.91043	0.265915
$342$	0	0
$343$	40.4020	2.18150
$344$	0	0
$345$	9.01284	0.485235
$346$	0	0
$347$	−1.66706	−0.0894926	−0.0447463	−	0.998998i	$-0.514248\pi$
$347$	−1.66706	−0.0894926	−0.0447463	+	0.998998i	$0.514248\pi$
$348$	0	0
$349$	13.9162	0.744918	0.372459	−	0.928049i	$-0.378515\pi$
$349$	13.9162	0.744918	0.372459	+	0.928049i	$0.378515\pi$
$350$	0	0
$351$	−4.83858	−0.258264
$352$	0	0
$353$	−21.4225	−1.14020	−0.570101	−	0.821574i	$-0.693096\pi$
$353$	−21.4225	−1.14020	−0.570101	+	0.821574i	$0.693096\pi$
$354$	0	0
$355$	3.48689	0.185065
$356$	0	0
$357$	−33.9327	−1.79591
$358$	0	0
$359$	10.5121	0.554806	0.277403	−	0.960754i	$-0.410526\pi$
$359$	10.5121	0.554806	0.277403	+	0.960754i	$0.410526\pi$
$360$	0	0
$361$	−10.6492	−0.560486
$362$	0	0
$363$	10.6228	0.557553
$364$	0	0
$365$	11.9545	0.625727
$366$	0	0
$367$	−1.54382	−0.0805866	−0.0402933	−	0.999188i	$-0.512829\pi$
$367$	−1.54382	−0.0805866	−0.0402933	+	0.999188i	$0.512829\pi$
$368$	0	0
$369$	−2.82560	−0.147095
$370$	0	0
$371$	34.4029	1.78611
$372$	0	0
$373$	−7.52041	−0.389392	−0.194696	−	0.980864i	$-0.562372\pi$
$373$	−7.52041	−0.389392	−0.194696	+	0.980864i	$0.562372\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	35.6418	1.83565
$378$	0	0
$379$	9.34126	0.479828	0.239914	−	0.970794i	$-0.422881\pi$
$379$	9.34126	0.479828	0.239914	+	0.970794i	$0.422881\pi$
$380$	0	0
$381$	18.0770	0.926112
$382$	0	0
$383$	−30.3922	−1.55297	−0.776484	−	0.630137i	$-0.782999\pi$
$383$	−30.3922	−1.55297	−0.776484	+	0.630137i	$0.782999\pi$
$384$	0	0
$385$	−2.91419	−0.148521
$386$	0	0
$387$	11.4493	0.582002
$388$	0	0
$389$	25.4782	1.29180	0.645898	−	0.763424i	$-0.276483\pi$
$389$	25.4782	1.29180	0.645898	+	0.763424i	$0.276483\pi$
$390$	0	0
$391$	−64.4536	−3.25956
$392$	0	0
$393$	7.25146	0.365788
$394$	0	0
$395$	−12.5216	−0.630031
$396$	0	0
$397$	−7.74555	−0.388738	−0.194369	−	0.980929i	$-0.562266\pi$
$397$	−7.74555	−0.388738	−0.194369	+	0.980929i	$0.562266\pi$
$398$	0	0
$399$	13.7119	0.686452
$400$	0	0
$401$	−9.85170	−0.491970	−0.245985	−	0.969274i	$-0.579111\pi$
$401$	−9.85170	−0.491970	−0.245985	+	0.969274i	$0.579111\pi$
$402$	0	0
$403$	−38.6860	−1.92708
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	1.86702	0.0925446
$408$	0	0
$409$	−6.42729	−0.317809	−0.158904	−	0.987294i	$-0.550796\pi$
$409$	−6.42729	−0.317809	−0.158904	+	0.987294i	$0.550796\pi$
$410$	0	0
$411$	−21.7191	−1.07132
$412$	0	0
$413$	28.5026	1.40252
$414$	0	0
$415$	6.60185	0.324072
$416$	0	0
$417$	11.8993	0.582711
$418$	0	0
$419$	−16.6808	−0.814912	−0.407456	−	0.913225i	$-0.633584\pi$
$419$	−16.6808	−0.814912	−0.407456	+	0.913225i	$0.633584\pi$
$420$	0	0
$421$	−2.96275	−0.144396	−0.0721979	−	0.997390i	$-0.523001\pi$
$421$	−2.96275	−0.144396	−0.0721979	+	0.997390i	$0.523001\pi$
$422$	0	0
$423$	12.7326	0.619080
$424$	0	0
$425$	7.15132	0.346890
$426$	0	0
$427$	−27.4079	−1.32636
$428$	0	0
$429$	2.97168	0.143474
$430$	0	0
$431$	−6.70484	−0.322961	−0.161480	−	0.986876i	$-0.551627\pi$
$431$	−6.70484	−0.322961	−0.161480	+	0.986876i	$0.551627\pi$
$432$	0	0
$433$	27.6260	1.32762	0.663811	−	0.747901i	$-0.268938\pi$
$433$	27.6260	1.32762	0.663811	+	0.747901i	$0.268938\pi$
$434$	0	0
$435$	−7.36617	−0.353181
$436$	0	0
$437$	26.0450	1.24590
$438$	0	0
$439$	24.6822	1.17802	0.589009	−	0.808126i	$-0.299518\pi$
$439$	24.6822	1.17802	0.589009	+	0.808126i	$0.299518\pi$
$440$	0	0
$441$	15.5147	0.738795
$442$	0	0
$443$	39.8273	1.89225	0.946126	−	0.323799i	$-0.104960\pi$
$443$	39.8273	1.89225	0.946126	+	0.323799i	$0.104960\pi$
$444$	0	0
$445$	−3.74308	−0.177439
$446$	0	0
$447$	9.12207	0.431459
$448$	0	0
$449$	−23.1344	−1.09178	−0.545889	−	0.837858i	$-0.683808\pi$
$449$	−23.1344	−1.09178	−0.545889	+	0.837858i	$0.683808\pi$
$450$	0	0
$451$	1.73538	0.0817158
$452$	0	0
$453$	−5.29876	−0.248958
$454$	0	0
$455$	22.9589	1.07633
$456$	0	0
$457$	−13.2732	−0.620894	−0.310447	−	0.950591i	$-0.600479\pi$
$457$	−13.2732	−0.620894	−0.310447	+	0.950591i	$0.600479\pi$
$458$	0	0
$459$	−7.15132	−0.333795
$460$	0	0
$461$	−17.7428	−0.826365	−0.413183	−	0.910648i	$-0.635583\pi$
$461$	−17.7428	−0.826365	−0.413183	+	0.910648i	$0.635583\pi$
$462$	0	0
$463$	1.96987	0.0915474	0.0457737	−	0.998952i	$-0.485425\pi$
$463$	1.96987	0.0915474	0.0457737	+	0.998952i	$0.485425\pi$
$464$	0	0
$465$	7.99531	0.370774
$466$	0	0
$467$	−16.2852	−0.753590	−0.376795	−	0.926297i	$-0.622974\pi$
$467$	−16.2852	−0.753590	−0.376795	+	0.926297i	$0.622974\pi$
$468$	0	0
$469$	−4.74497	−0.219102
$470$	0	0
$471$	3.94646	0.181843
$472$	0	0
$473$	−7.03176	−0.323321
$474$	0	0
$475$	−2.88977	−0.132592
$476$	0	0
$477$	7.25040	0.331973
$478$	0	0
$479$	−14.9309	−0.682209	−0.341104	−	0.940025i	$-0.610801\pi$
$479$	−14.9309	−0.682209	−0.341104	+	0.940025i	$0.610801\pi$
$480$	0	0
$481$	−14.7090	−0.670671
$482$	0	0
$483$	42.7656	1.94590
$484$	0	0
$485$	−9.96750	−0.452601
$486$	0	0
$487$	22.7843	1.03246	0.516228	−	0.856451i	$-0.327336\pi$
$487$	22.7843	1.03246	0.516228	+	0.856451i	$0.327336\pi$
$488$	0	0
$489$	19.7523	0.893229
$490$	0	0
$491$	28.7686	1.29831	0.649154	−	0.760657i	$-0.275123\pi$
$491$	28.7686	1.29831	0.649154	+	0.760657i	$0.275123\pi$
$492$	0	0
$493$	52.6778	2.37249
$494$	0	0
$495$	−0.614164	−0.0276046
$496$	0	0
$497$	16.5452	0.742152
$498$	0	0
$499$	8.03990	0.359916	0.179958	−	0.983674i	$-0.442404\pi$
$499$	8.03990	0.359916	0.179958	+	0.983674i	$0.442404\pi$
$500$	0	0
$501$	5.53576	0.247319
$502$	0	0
$503$	−38.7679	−1.72858	−0.864288	−	0.502998i	$-0.832230\pi$
$503$	−38.7679	−1.72858	−0.864288	+	0.502998i	$0.832230\pi$
$504$	0	0
$505$	−13.8374	−0.615754
$506$	0	0
$507$	−10.4119	−0.462407
$508$	0	0
$509$	44.2176	1.95991	0.979955	−	0.199218i	$-0.0638402\pi$
$509$	44.2176	1.95991	0.979955	+	0.199218i	$0.0638402\pi$
$510$	0	0
$511$	56.7237	2.50931
$512$	0	0
$513$	2.88977	0.127586
$514$	0	0
$515$	0.915216	0.0403292
$516$	0	0
$517$	−7.81990	−0.343919
$518$	0	0
$519$	−24.0517	−1.05575
$520$	0	0
$521$	3.54993	0.155525	0.0777626	−	0.996972i	$-0.475222\pi$
$521$	3.54993	0.155525	0.0777626	+	0.996972i	$0.475222\pi$
$522$	0	0
$523$	−2.53646	−0.110912	−0.0554559	−	0.998461i	$-0.517661\pi$
$523$	−2.53646	−0.110912	−0.0554559	+	0.998461i	$0.517661\pi$
$524$	0	0
$525$	−4.74497	−0.207087
$526$	0	0
$527$	−57.1770	−2.49067
$528$	0	0
$529$	58.2312	2.53179
$530$	0	0
$531$	6.00692	0.260678
$532$	0	0
$533$	−13.6719	−0.592195
$534$	0	0
$535$	1.41601	0.0612196
$536$	0	0
$537$	−14.8278	−0.639867
$538$	0	0
$539$	−9.52857	−0.410425
$540$	0	0
$541$	10.8235	0.465339	0.232669	−	0.972556i	$-0.425254\pi$
$541$	10.8235	0.465339	0.232669	+	0.972556i	$0.425254\pi$
$542$	0	0
$543$	−12.8119	−0.549813
$544$	0	0
$545$	−3.40316	−0.145775
$546$	0	0
$547$	−33.6468	−1.43864	−0.719318	−	0.694681i	$-0.755545\pi$
$547$	−33.6468	−1.43864	−0.719318	+	0.694681i	$0.755545\pi$
$548$	0	0
$549$	−5.77620	−0.246522
$550$	0	0
$551$	−21.2865	−0.906837
$552$	0	0
$553$	−59.4146	−2.52657
$554$	0	0
$555$	3.03993	0.129038
$556$	0	0
$557$	27.6855	1.17307	0.586537	−	0.809923i	$-0.300491\pi$
$557$	27.6855	1.17307	0.586537	+	0.809923i	$0.300491\pi$
$558$	0	0
$559$	55.3985	2.34311
$560$	0	0
$561$	4.39208	0.185434
$562$	0	0
$563$	−30.2902	−1.27658	−0.638289	−	0.769797i	$-0.720357\pi$
$563$	−30.2902	−1.27658	−0.638289	+	0.769797i	$0.720357\pi$
$564$	0	0
$565$	12.0015	0.504907
$566$	0	0
$567$	4.74497	0.199270
$568$	0	0
$569$	−20.5056	−0.859642	−0.429821	−	0.902914i	$-0.641423\pi$
$569$	−20.5056	−0.859642	−0.429821	+	0.902914i	$0.641423\pi$
$570$	0	0
$571$	−14.8466	−0.621309	−0.310655	−	0.950523i	$-0.600548\pi$
$571$	−14.8466	−0.621309	−0.310655	+	0.950523i	$0.600548\pi$
$572$	0	0
$573$	−3.86630	−0.161517
$574$	0	0
$575$	−9.01284	−0.375861
$576$	0	0
$577$	0.0701312	0.00291960	0.00145980	−	0.999999i	$-0.499535\pi$
$577$	0.0701312	0.00291960	0.00145980	+	0.999999i	$0.499535\pi$
$578$	0	0
$579$	11.8474	0.492362
$580$	0	0
$581$	31.3256	1.29960
$582$	0	0
$583$	−4.45294	−0.184422
$584$	0	0
$585$	4.83858	0.200051
$586$	0	0
$587$	12.7643	0.526838	0.263419	−	0.964682i	$-0.415150\pi$
$587$	12.7643	0.526838	0.263419	+	0.964682i	$0.415150\pi$
$588$	0	0
$589$	23.1046	0.952009
$590$	0	0
$591$	20.0840	0.826145
$592$	0	0
$593$	−19.0784	−0.783455	−0.391727	−	0.920081i	$-0.628122\pi$
$593$	−19.0784	−0.783455	−0.391727	+	0.920081i	$0.628122\pi$
$594$	0	0
$595$	33.9327	1.39111
$596$	0	0
$597$	10.0796	0.412530
$598$	0	0
$599$	−26.4204	−1.07951	−0.539753	−	0.841823i	$-0.681482\pi$
$599$	−26.4204	−1.07951	−0.539753	+	0.841823i	$0.681482\pi$
$600$	0	0
$601$	26.0222	1.06147	0.530735	−	0.847538i	$-0.321916\pi$
$601$	26.0222	1.06147	0.530735	+	0.847538i	$0.321916\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−10.6228	−0.431878
$606$	0	0
$607$	−43.4700	−1.76439	−0.882197	−	0.470880i	$-0.843936\pi$
$607$	−43.4700	−1.76439	−0.882197	+	0.470880i	$0.843936\pi$
$608$	0	0
$609$	−34.9522	−1.41633
$610$	0	0
$611$	61.6077	2.49238
$612$	0	0
$613$	25.4337	1.02726	0.513629	−	0.858013i	$-0.328301\pi$
$613$	25.4337	1.02726	0.513629	+	0.858013i	$0.328301\pi$
$614$	0	0
$615$	2.82560	0.113939
$616$	0	0
$617$	−20.5477	−0.827220	−0.413610	−	0.910454i	$-0.635732\pi$
$617$	−20.5477	−0.827220	−0.413610	+	0.910454i	$0.635732\pi$
$618$	0	0
$619$	−23.6041	−0.948729	−0.474365	−	0.880329i	$-0.657322\pi$
$619$	−23.6041	−0.948729	−0.474365	+	0.880329i	$0.657322\pi$
$620$	0	0
$621$	9.01284	0.361673
$622$	0	0
$623$	−17.7608	−0.711570
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.77479	−0.0708784
$628$	0	0
$629$	−21.7395	−0.866811
$630$	0	0
$631$	25.4483	1.01308	0.506541	−	0.862216i	$-0.330924\pi$
$631$	25.4483	1.01308	0.506541	+	0.862216i	$0.330924\pi$
$632$	0	0
$633$	−18.5890	−0.738847
$634$	0	0
$635$	−18.0770	−0.717364
$636$	0	0
$637$	75.0691	2.97435
$638$	0	0
$639$	3.48689	0.137939
$640$	0	0
$641$	−12.3985	−0.489713	−0.244856	−	0.969559i	$-0.578741\pi$
$641$	−12.3985	−0.489713	−0.244856	+	0.969559i	$0.578741\pi$
$642$	0	0
$643$	27.9845	1.10360	0.551800	−	0.833977i	$-0.313941\pi$
$643$	27.9845	1.10360	0.551800	+	0.833977i	$0.313941\pi$
$644$	0	0
$645$	−11.4493	−0.450817
$646$	0	0
$647$	29.7259	1.16865	0.584323	−	0.811521i	$-0.301360\pi$
$647$	29.7259	1.16865	0.584323	+	0.811521i	$0.301360\pi$
$648$	0	0
$649$	−3.68923	−0.144815
$650$	0	0
$651$	37.9375	1.48689
$652$	0	0
$653$	47.6651	1.86528	0.932640	−	0.360809i	$-0.117499\pi$
$653$	47.6651	1.86528	0.932640	+	0.360809i	$0.117499\pi$
$654$	0	0
$655$	−7.25146	−0.283338
$656$	0	0
$657$	11.9545	0.466390
$658$	0	0
$659$	27.9924	1.09043	0.545215	−	0.838296i	$-0.316448\pi$
$659$	27.9924	1.09043	0.545215	+	0.838296i	$0.316448\pi$
$660$	0	0
$661$	−11.8624	−0.461396	−0.230698	−	0.973025i	$-0.574101\pi$
$661$	−11.8624	−0.461396	−0.230698	+	0.973025i	$0.574101\pi$
$662$	0	0
$663$	−34.6022	−1.34384
$664$	0	0
$665$	−13.7119	−0.531723
$666$	0	0
$667$	−66.3901	−2.57063
$668$	0	0
$669$	−16.1421	−0.624091
$670$	0	0
$671$	3.54753	0.136951
$672$	0	0
$673$	−37.3771	−1.44078	−0.720391	−	0.693568i	$-0.756038\pi$
$673$	−37.3771	−1.44078	−0.720391	+	0.693568i	$0.756038\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	8.45656	0.325012	0.162506	−	0.986708i	$-0.448042\pi$
$677$	8.45656	0.325012	0.162506	+	0.986708i	$0.448042\pi$
$678$	0	0
$679$	−47.2955	−1.81503
$680$	0	0
$681$	−23.6777	−0.907330
$682$	0	0
$683$	−5.97377	−0.228580	−0.114290	−	0.993447i	$-0.536459\pi$
$683$	−5.97377	−0.228580	−0.114290	+	0.993447i	$0.536459\pi$
$684$	0	0
$685$	21.7191	0.829844
$686$	0	0
$687$	20.1309	0.768043
$688$	0	0
$689$	35.0817	1.33650
$690$	0	0
$691$	8.51722	0.324010	0.162005	−	0.986790i	$-0.448204\pi$
$691$	8.51722	0.324010	0.162005	+	0.986790i	$0.448204\pi$
$692$	0	0
$693$	−2.91419	−0.110701
$694$	0	0
$695$	−11.8993	−0.451366
$696$	0	0
$697$	−20.2067	−0.765384
$698$	0	0
$699$	20.5003	0.775392
$700$	0	0
$701$	29.7777	1.12469	0.562345	−	0.826903i	$-0.309899\pi$
$701$	29.7777	1.12469	0.562345	+	0.826903i	$0.309899\pi$
$702$	0	0
$703$	8.78470	0.331321
$704$	0	0
$705$	−12.7326	−0.479537
$706$	0	0
$707$	−65.6578	−2.46932
$708$	0	0
$709$	26.1419	0.981779	0.490890	−	0.871222i	$-0.336672\pi$
$709$	26.1419	0.981779	0.490890	+	0.871222i	$0.336672\pi$
$710$	0	0
$711$	−12.5216	−0.469597
$712$	0	0
$713$	72.0604	2.69868
$714$	0	0
$715$	−2.97168	−0.111135
$716$	0	0
$717$	9.12005	0.340594
$718$	0	0
$719$	21.0197	0.783902	0.391951	−	0.919986i	$-0.371800\pi$
$719$	21.0197	0.783902	0.391951	+	0.919986i	$0.371800\pi$
$720$	0	0
$721$	4.34267	0.161729
$722$	0	0
$723$	−11.0641	−0.411478
$724$	0	0
$725$	7.36617	0.273573
$726$	0	0
$727$	4.51628	0.167499	0.0837497	−	0.996487i	$-0.473310\pi$
$727$	4.51628	0.167499	0.0837497	+	0.996487i	$0.473310\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	81.8777	3.02836
$732$	0	0
$733$	43.3217	1.60012	0.800062	−	0.599918i	$-0.204800\pi$
$733$	43.3217	1.60012	0.800062	+	0.599918i	$0.204800\pi$
$734$	0	0
$735$	−15.5147	−0.572268
$736$	0	0
$737$	0.614164	0.0226230
$738$	0	0
$739$	6.52213	0.239920	0.119960	−	0.992779i	$-0.461723\pi$
$739$	6.52213	0.239920	0.119960	+	0.992779i	$0.461723\pi$
$740$	0	0
$741$	13.9824	0.513656
$742$	0	0
$743$	34.0574	1.24944	0.624722	−	0.780847i	$-0.285212\pi$
$743$	34.0574	1.24944	0.624722	+	0.780847i	$0.285212\pi$
$744$	0	0
$745$	−9.12207	−0.334207
$746$	0	0
$747$	6.60185	0.241549
$748$	0	0
$749$	6.71893	0.245504
$750$	0	0
$751$	−23.7400	−0.866284	−0.433142	−	0.901326i	$-0.642595\pi$
$751$	−23.7400	−0.866284	−0.433142	+	0.901326i	$0.642595\pi$
$752$	0	0
$753$	−12.0112	−0.437713
$754$	0	0
$755$	5.29876	0.192842
$756$	0	0
$757$	−21.1597	−0.769062	−0.384531	−	0.923112i	$-0.625637\pi$
$757$	−21.1597	−0.769062	−0.384531	+	0.923112i	$0.625637\pi$
$758$	0	0
$759$	−5.53536	−0.200921
$760$	0	0
$761$	−5.28827	−0.191699	−0.0958497	−	0.995396i	$-0.530557\pi$
$761$	−5.28827	−0.191699	−0.0958497	+	0.995396i	$0.530557\pi$
$762$	0	0
$763$	−16.1479	−0.584593
$764$	0	0
$765$	7.15132	0.258556
$766$	0	0
$767$	29.0650	1.04948
$768$	0	0
$769$	3.54443	0.127815	0.0639076	−	0.997956i	$-0.479644\pi$
$769$	3.54443	0.127815	0.0639076	+	0.997956i	$0.479644\pi$
$770$	0	0
$771$	−1.53376	−0.0552372
$772$	0	0
$773$	−9.03449	−0.324948	−0.162474	−	0.986713i	$-0.551947\pi$
$773$	−9.03449	−0.324948	−0.162474	+	0.986713i	$0.551947\pi$
$774$	0	0
$775$	−7.99531	−0.287200
$776$	0	0
$777$	14.4244	0.517471
$778$	0	0
$779$	8.16532	0.292553
$780$	0	0
$781$	−2.14152	−0.0766296
$782$	0	0
$783$	−7.36617	−0.263245
$784$	0	0
$785$	−3.94646	−0.140855
$786$	0	0
$787$	−11.9177	−0.424819	−0.212409	−	0.977181i	$-0.568131\pi$
$787$	−11.9177	−0.424819	−0.212409	+	0.977181i	$0.568131\pi$
$788$	0	0
$789$	−11.0033	−0.391729
$790$	0	0
$791$	56.9467	2.02479
$792$	0	0
$793$	−27.9486	−0.992485
$794$	0	0
$795$	−7.25040	−0.257145
$796$	0	0
$797$	−39.6381	−1.40405	−0.702027	−	0.712151i	$-0.747721\pi$
$797$	−39.6381	−1.40405	−0.702027	+	0.712151i	$0.747721\pi$
$798$	0	0
$799$	91.0548	3.22129
$800$	0	0
$801$	−3.74308	−0.132255
$802$	0	0
$803$	−7.34203	−0.259095
$804$	0	0
$805$	−42.7656	−1.50729
$806$	0	0
$807$	8.71710	0.306857
$808$	0	0
$809$	46.1689	1.62321	0.811606	−	0.584205i	$-0.198594\pi$
$809$	46.1689	1.62321	0.811606	+	0.584205i	$0.198594\pi$
$810$	0	0
$811$	20.5441	0.721401	0.360701	−	0.932682i	$-0.382538\pi$
$811$	20.5441	0.721401	0.360701	+	0.932682i	$0.382538\pi$
$812$	0	0
$813$	11.1658	0.391601
$814$	0	0
$815$	−19.7523	−0.691892
$816$	0	0
$817$	−33.0859	−1.15753
$818$	0	0
$819$	22.9589	0.802249
$820$	0	0
$821$	29.9196	1.04420	0.522101	−	0.852884i	$-0.325149\pi$
$821$	29.9196	1.04420	0.522101	+	0.852884i	$0.325149\pi$
$822$	0	0
$823$	47.8236	1.66703	0.833513	−	0.552500i	$-0.186326\pi$
$823$	47.8236	1.66703	0.833513	+	0.552500i	$0.186326\pi$
$824$	0	0
$825$	0.614164	0.0213824
$826$	0	0
$827$	7.22573	0.251263	0.125632	−	0.992077i	$-0.459904\pi$
$827$	7.22573	0.251263	0.125632	+	0.992077i	$0.459904\pi$
$828$	0	0
$829$	−2.16550	−0.0752108	−0.0376054	−	0.999293i	$-0.511973\pi$
$829$	−2.16550	−0.0752108	−0.0376054	+	0.999293i	$0.511973\pi$
$830$	0	0
$831$	−27.9408	−0.969255
$832$	0	0
$833$	110.951	3.84421
$834$	0	0
$835$	−5.53576	−0.191573
$836$	0	0
$837$	7.99531	0.276358
$838$	0	0
$839$	−27.6455	−0.954428	−0.477214	−	0.878787i	$-0.658353\pi$
$839$	−27.6455	−0.954428	−0.477214	+	0.878787i	$0.658353\pi$
$840$	0	0
$841$	25.2604	0.871048
$842$	0	0
$843$	6.80061	0.234225
$844$	0	0
$845$	10.4119	0.358179
$846$	0	0
$847$	−50.4048	−1.73193
$848$	0	0
$849$	6.71528	0.230468
$850$	0	0
$851$	27.3984	0.939205
$852$	0	0
$853$	−28.3748	−0.971536	−0.485768	−	0.874088i	$-0.661460\pi$
$853$	−28.3748	−0.971536	−0.485768	+	0.874088i	$0.661460\pi$
$854$	0	0
$855$	−2.88977	−0.0988281
$856$	0	0
$857$	25.6600	0.876529	0.438264	−	0.898846i	$-0.355593\pi$
$857$	25.6600	0.876529	0.438264	+	0.898846i	$0.355593\pi$
$858$	0	0
$859$	31.0029	1.05780	0.528902	−	0.848683i	$-0.322604\pi$
$859$	31.0029	1.05780	0.528902	+	0.848683i	$0.322604\pi$
$860$	0	0
$861$	13.4074	0.456921
$862$	0	0
$863$	6.39761	0.217777	0.108889	−	0.994054i	$-0.465271\pi$
$863$	6.39761	0.217777	0.108889	+	0.994054i	$0.465271\pi$
$864$	0	0
$865$	24.0517	0.817782
$866$	0	0
$867$	−34.1413	−1.15950
$868$	0	0
$869$	7.69032	0.260876
$870$	0	0
$871$	−4.83858	−0.163949
$872$	0	0
$873$	−9.96750	−0.337349
$874$	0	0
$875$	4.74497	0.160409
$876$	0	0
$877$	12.5786	0.424748	0.212374	−	0.977189i	$-0.431881\pi$
$877$	12.5786	0.424748	0.212374	+	0.977189i	$0.431881\pi$
$878$	0	0
$879$	−23.4972	−0.792540
$880$	0	0
$881$	−11.7135	−0.394637	−0.197318	−	0.980339i	$-0.563223\pi$
$881$	−11.7135	−0.394637	−0.197318	+	0.980339i	$0.563223\pi$
$882$	0	0
$883$	4.58763	0.154386	0.0771929	−	0.997016i	$-0.475404\pi$
$883$	4.58763	0.154386	0.0771929	+	0.997016i	$0.475404\pi$
$884$	0	0
$885$	−6.00692	−0.201920
$886$	0	0
$887$	−56.6136	−1.90090	−0.950449	−	0.310880i	$-0.899376\pi$
$887$	−56.6136	−1.90090	−0.950449	+	0.310880i	$0.899376\pi$
$888$	0	0
$889$	−85.7747	−2.87679
$890$	0	0
$891$	−0.614164	−0.0205753
$892$	0	0
$893$	−36.7943	−1.23127
$894$	0	0
$895$	14.8278	0.495639
$896$	0	0
$897$	43.6093	1.45607
$898$	0	0
$899$	−58.8948	−1.96425
$900$	0	0
$901$	51.8499	1.72737
$902$	0	0
$903$	−54.3267	−1.80788
$904$	0	0
$905$	12.8119	0.425883
$906$	0	0
$907$	36.0827	1.19811	0.599053	−	0.800710i	$-0.295544\pi$
$907$	36.0827	1.19811	0.599053	+	0.800710i	$0.295544\pi$
$908$	0	0
$909$	−13.8374	−0.458956
$910$	0	0
$911$	23.7589	0.787167	0.393583	−	0.919289i	$-0.371235\pi$
$911$	23.7589	0.787167	0.393583	+	0.919289i	$0.371235\pi$
$912$	0	0
$913$	−4.05462	−0.134188
$914$	0	0
$915$	5.77620	0.190955
$916$	0	0
$917$	−34.4079	−1.13625
$918$	0	0
$919$	2.46380	0.0812731	0.0406366	−	0.999174i	$-0.487061\pi$
$919$	2.46380	0.0812731	0.0406366	+	0.999174i	$0.487061\pi$
$920$	0	0
$921$	−8.57086	−0.282419
$922$	0	0
$923$	16.8716	0.555335
$924$	0	0
$925$	−3.03993	−0.0999523
$926$	0	0
$927$	0.915216	0.0300596
$928$	0	0
$929$	16.3029	0.534881	0.267440	−	0.963574i	$-0.413822\pi$
$929$	16.3029	0.534881	0.267440	+	0.963574i	$0.413822\pi$
$930$	0	0
$931$	−44.8339	−1.46937
$932$	0	0
$933$	16.3429	0.535043
$934$	0	0
$935$	−4.39208	−0.143636
$936$	0	0
$937$	7.74356	0.252971	0.126485	−	0.991968i	$-0.459630\pi$
$937$	7.74356	0.252971	0.126485	+	0.991968i	$0.459630\pi$
$938$	0	0
$939$	4.54355	0.148273
$940$	0	0
$941$	−2.55569	−0.0833132	−0.0416566	−	0.999132i	$-0.513264\pi$
$941$	−2.55569	−0.0833132	−0.0416566	+	0.999132i	$0.513264\pi$
$942$	0	0
$943$	25.4666	0.829308
$944$	0	0
$945$	−4.74497	−0.154354
$946$	0	0
$947$	−32.4961	−1.05598	−0.527991	−	0.849250i	$-0.677055\pi$
$947$	−32.4961	−1.05598	−0.527991	+	0.849250i	$0.677055\pi$
$948$	0	0
$949$	57.8429	1.87766
$950$	0	0
$951$	−4.12219	−0.133671
$952$	0	0
$953$	−24.3340	−0.788256	−0.394128	−	0.919056i	$-0.628953\pi$
$953$	−24.3340	−0.788256	−0.394128	+	0.919056i	$0.628953\pi$
$954$	0	0
$955$	3.86630	0.125111
$956$	0	0
$957$	4.52403	0.146241
$958$	0	0
$959$	103.056	3.32786
$960$	0	0
$961$	32.9250	1.06210
$962$	0	0
$963$	1.41601	0.0456304
$964$	0	0
$965$	−11.8474	−0.381382
$966$	0	0
$967$	22.4275	0.721219	0.360610	−	0.932717i	$-0.382569\pi$
$967$	22.4275	0.721219	0.360610	+	0.932717i	$0.382569\pi$
$968$	0	0
$969$	20.6657	0.663877
$970$	0	0
$971$	0.120931	0.00388087	0.00194044	−	0.999998i	$-0.499382\pi$
$971$	0.120931	0.00388087	0.00194044	+	0.999998i	$0.499382\pi$
$972$	0	0
$973$	−56.4618	−1.81008
$974$	0	0
$975$	−4.83858	−0.154959
$976$	0	0
$977$	−4.81414	−0.154018	−0.0770090	−	0.997030i	$-0.524537\pi$
$977$	−4.81414	−0.154018	−0.0770090	+	0.997030i	$0.524537\pi$
$978$	0	0
$979$	2.29886	0.0734719
$980$	0	0
$981$	−3.40316	−0.108655
$982$	0	0
$983$	−7.55259	−0.240890	−0.120445	−	0.992720i	$-0.538432\pi$
$983$	−7.55259	−0.240890	−0.120445	+	0.992720i	$0.538432\pi$
$984$	0	0
$985$	−20.0840	−0.639929
$986$	0	0
$987$	−60.4157	−1.92305
$988$	0	0
$989$	−103.191	−3.28128
$990$	0	0
$991$	−41.9815	−1.33359	−0.666793	−	0.745243i	$-0.732333\pi$
$991$	−41.9815	−1.33359	−0.666793	+	0.745243i	$0.732333\pi$
$992$	0	0
$993$	19.0804	0.605500
$994$	0	0
$995$	−10.0796	−0.319544
$996$	0	0
$997$	−10.2951	−0.326048	−0.163024	−	0.986622i	$-0.552125\pi$
$997$	−10.2951	−0.326048	−0.163024	+	0.986622i	$0.552125\pi$
$998$	0	0
$999$	3.03993	0.0961792

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.bc.1.10	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.bc.1.10	✓	10	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.74497 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.74497 q^{7} +1.00000 q^{9} -0.614164 q^{11} +4.83858 q^{13} -1.00000 q^{15} +7.15132 q^{17} -2.88977 q^{19} -4.74497 q^{21} -9.01284 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.36617 q^{29} -7.99531 q^{31} +0.614164 q^{33} +4.74497 q^{35} -3.03993 q^{37} -4.83858 q^{39} -2.82560 q^{41} +11.4493 q^{43} +1.00000 q^{45} +12.7326 q^{47} +15.5147 q^{49} -7.15132 q^{51} +7.25040 q^{53} -0.614164 q^{55} +2.88977 q^{57} +6.00692 q^{59} -5.77620 q^{61} +4.74497 q^{63} +4.83858 q^{65} -1.00000 q^{67} +9.01284 q^{69} +3.48689 q^{71} +11.9545 q^{73} -1.00000 q^{75} -2.91419 q^{77} -12.5216 q^{79} +1.00000 q^{81} +6.60185 q^{83} +7.15132 q^{85} -7.36617 q^{87} -3.74308 q^{89} +22.9589 q^{91} +7.99531 q^{93} -2.88977 q^{95} -9.96750 q^{97} -0.614164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9	\( 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9} + 7 q^{11} - q^{13} - 10 q^{15} + 3 q^{17} + 4 q^{19} - q^{21} - 13 q^{23} + 10 q^{25} - 10 q^{27} + 18 q^{29} - 9 q^{31} - 7 q^{33} + q^{35} - 3 q^{37} + q^{39} + 19 q^{41} + 5 q^{43} + 10 q^{45} - 8 q^{47} + 43 q^{49} - 3 q^{51} + 17 q^{53} + 7 q^{55} - 4 q^{57} + 24 q^{59} + 21 q^{61} + q^{63} - q^{65} - 10 q^{67} + 13 q^{69} + 2 q^{71} + 25 q^{73} - 10 q^{75} + 15 q^{77} - q^{79} + 10 q^{81} - 6 q^{83} + 3 q^{85} - 18 q^{87} + 23 q^{89} + 29 q^{91} + 9 q^{93} + 4 q^{95} + 21 q^{97} + 7 q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9 + 7 * q^11 - q^13 - 10 * q^15 + 3 * q^17 + 4 * q^19 - q^21 - 13 * q^23 + 10 * q^25 - 10 * q^27 + 18 * q^29 - 9 * q^31 - 7 * q^33 + q^35 - 3 * q^37 + q^39 + 19 * q^41 + 5 * q^43 + 10 * q^45 - 8 * q^47 + 43 * q^49 - 3 * q^51 + 17 * q^53 + 7 * q^55 - 4 * q^57 + 24 * q^59 + 21 * q^61 + q^63 - q^65 - 10 * q^67 + 13 * q^69 + 2 * q^71 + 25 * q^73 - 10 * q^75 + 15 * q^77 - q^79 + 10 * q^81 - 6 * q^83 + 3 * q^85 - 18 * q^87 + 23 * q^89 + 29 * q^91 + 9 * q^93 + 4 * q^95 + 21 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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