LMFDB - Embedded newform 8008.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.668973.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} - x^{2} + 7x - 2 $ x^5 - x^4 - 9x^3 - x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.50189$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.830554 q^{3} +3.93077 q^{5} +1.00000 q^{7} -2.31018 q^{9} +O(q^{10})$	$q-0.830554 q^{3} +3.93077 q^{5} +1.00000 q^{7} -2.31018 q^{9} -1.00000 q^{11} -1.00000 q^{13} -3.26471 q^{15} +7.93077 q^{17} +4.36492 q^{19} -0.830554 q^{21} +1.83550 q^{23} +10.4509 q^{25} +4.41039 q^{27} -4.44186 q^{29} +1.61564 q^{31} +0.830554 q^{33} +3.93077 q^{35} +5.72080 q^{37} +0.830554 q^{39} +4.67038 q^{41} -1.06429 q^{43} -9.08078 q^{45} +2.25544 q^{47} +1.00000 q^{49} -6.58693 q^{51} -3.24095 q^{53} -3.93077 q^{55} -3.62530 q^{57} -10.4485 q^{59} +1.90473 q^{61} -2.31018 q^{63} -3.93077 q^{65} -3.90700 q^{67} -1.52448 q^{69} +8.09094 q^{71} -8.62469 q^{73} -8.68005 q^{75} -1.00000 q^{77} -10.0672 q^{79} +3.26747 q^{81} -12.3767 q^{83} +31.1740 q^{85} +3.68921 q^{87} +0.0260316 q^{89} -1.00000 q^{91} -1.34188 q^{93} +17.1575 q^{95} -1.13579 q^{97} +2.31018 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9	$ 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9} - 5 q^{11} - 5 q^{13} - 5 q^{15} + 19 q^{17} - 5 q^{19} + q^{21} + 5 q^{23} + 12 q^{25} - 11 q^{27} - 17 q^{29} + 4 q^{31} - q^{33} - q^{35} + 10 q^{37} - q^{39} + 10 q^{41} - 25 q^{43} + q^{45} + 3 q^{47} + 5 q^{49} - q^{51} + 22 q^{53} + q^{55} + 16 q^{57} + 21 q^{59} + 26 q^{61} + 6 q^{63} + q^{65} + 28 q^{67} + 16 q^{69} + 28 q^{71} - 4 q^{73} - 5 q^{77} - 11 q^{79} + 5 q^{81} + 33 q^{85} + 31 q^{87} - 37 q^{89} - 5 q^{91} - 49 q^{93} + 29 q^{95} + 18 q^{97} - 6 q^{99}+O(q^{100}) $ 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9 - 5 * q^11 - 5 * q^13 - 5 * q^15 + 19 * q^17 - 5 * q^19 + q^21 + 5 * q^23 + 12 * q^25 - 11 * q^27 - 17 * q^29 + 4 * q^31 - q^33 - q^35 + 10 * q^37 - q^39 + 10 * q^41 - 25 * q^43 + q^45 + 3 * q^47 + 5 * q^49 - q^51 + 22 * q^53 + q^55 + 16 * q^57 + 21 * q^59 + 26 * q^61 + 6 * q^63 + q^65 + 28 * q^67 + 16 * q^69 + 28 * q^71 - 4 * q^73 - 5 * q^77 - 11 * q^79 + 5 * q^81 + 33 * q^85 + 31 * q^87 - 37 * q^89 - 5 * q^91 - 49 * q^93 + 29 * q^95 + 18 * q^97 - 6 * 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
$3$	−0.830554	−0.479521	−0.239760	−	0.970832i	$-0.577069\pi$
$3$	−0.830554	−0.479521	−0.239760	+	0.970832i	$0.577069\pi$
$4$	0	0
$5$	3.93077	1.75789	0.878946	−	0.476922i	$-0.158247\pi$
$5$	3.93077	1.75789	0.878946	+	0.476922i	$0.158247\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.31018	−0.770060
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.26471	−0.842945
$16$	0	0
$17$	7.93077	1.92349	0.961747	−	0.273941i	$-0.0883273\pi$
$17$	7.93077	1.92349	0.961747	+	0.273941i	$0.0883273\pi$
$18$	0	0
$19$	4.36492	1.00138	0.500691	−	0.865626i	$-0.333079\pi$
$19$	4.36492	1.00138	0.500691	+	0.865626i	$0.333079\pi$
$20$	0	0
$21$	−0.830554	−0.181242
$22$	0	0
$23$	1.83550	0.382728	0.191364	−	0.981519i	$-0.438709\pi$
$23$	1.83550	0.382728	0.191364	+	0.981519i	$0.438709\pi$
$24$	0	0
$25$	10.4509	2.09018
$26$	0	0
$27$	4.41039	0.848780
$28$	0	0
$29$	−4.44186	−0.824833	−0.412417	−	0.910995i	$-0.635315\pi$
$29$	−4.44186	−0.824833	−0.412417	+	0.910995i	$0.635315\pi$
$30$	0	0
$31$	1.61564	0.290178	0.145089	−	0.989419i	$-0.453653\pi$
$31$	1.61564	0.290178	0.145089	+	0.989419i	$0.453653\pi$
$32$	0	0
$33$	0.830554	0.144581
$34$	0	0
$35$	3.93077	0.664421
$36$	0	0
$37$	5.72080	0.940493	0.470247	−	0.882535i	$-0.344165\pi$
$37$	5.72080	0.940493	0.470247	+	0.882535i	$0.344165\pi$
$38$	0	0
$39$	0.830554	0.132995
$40$	0	0
$41$	4.67038	0.729391	0.364696	−	0.931127i	$-0.381173\pi$
$41$	4.67038	0.729391	0.364696	+	0.931127i	$0.381173\pi$
$42$	0	0
$43$	−1.06429	−0.162303	−0.0811514	−	0.996702i	$-0.525860\pi$
$43$	−1.06429	−0.162303	−0.0811514	+	0.996702i	$0.525860\pi$
$44$	0	0
$45$	−9.08078	−1.35368
$46$	0	0
$47$	2.25544	0.328989	0.164495	−	0.986378i	$-0.447401\pi$
$47$	2.25544	0.328989	0.164495	+	0.986378i	$0.447401\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.58693	−0.922354
$52$	0	0
$53$	−3.24095	−0.445178	−0.222589	−	0.974912i	$-0.571451\pi$
$53$	−3.24095	−0.445178	−0.222589	+	0.974912i	$0.571451\pi$
$54$	0	0
$55$	−3.93077	−0.530024
$56$	0	0
$57$	−3.62530	−0.480183
$58$	0	0
$59$	−10.4485	−1.36027	−0.680137	−	0.733085i	$-0.738080\pi$
$59$	−10.4485	−1.36027	−0.680137	+	0.733085i	$0.738080\pi$
$60$	0	0
$61$	1.90473	0.243876	0.121938	−	0.992538i	$-0.461089\pi$
$61$	1.90473	0.243876	0.121938	+	0.992538i	$0.461089\pi$
$62$	0	0
$63$	−2.31018	−0.291055
$64$	0	0
$65$	−3.93077	−0.487551
$66$	0	0
$67$	−3.90700	−0.477316	−0.238658	−	0.971104i	$-0.576707\pi$
$67$	−3.90700	−0.477316	−0.238658	+	0.971104i	$0.576707\pi$
$68$	0	0
$69$	−1.52448	−0.183526
$70$	0	0
$71$	8.09094	0.960217	0.480109	−	0.877209i	$-0.340597\pi$
$71$	8.09094	0.960217	0.480109	+	0.877209i	$0.340597\pi$
$72$	0	0
$73$	−8.62469	−1.00944	−0.504722	−	0.863282i	$-0.668405\pi$
$73$	−8.62469	−1.00944	−0.504722	+	0.863282i	$0.668405\pi$
$74$	0	0
$75$	−8.68005	−1.00229
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−10.0672	−1.13264	−0.566322	−	0.824184i	$-0.691634\pi$
$79$	−10.0672	−1.13264	−0.566322	+	0.824184i	$0.691634\pi$
$80$	0	0
$81$	3.26747	0.363052
$82$	0	0
$83$	−12.3767	−1.35852	−0.679262	−	0.733896i	$-0.737700\pi$
$83$	−12.3767	−1.35852	−0.679262	+	0.733896i	$0.737700\pi$
$84$	0	0
$85$	31.1740	3.38129
$86$	0	0
$87$	3.68921	0.395524
$88$	0	0
$89$	0.0260316	0.00275935	0.00137967	−	0.999999i	$-0.499561\pi$
$89$	0.0260316	0.00275935	0.00137967	+	0.999999i	$0.499561\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−1.34188	−0.139146
$94$	0	0
$95$	17.1575	1.76032
$96$	0	0
$97$	−1.13579	−0.115322	−0.0576610	−	0.998336i	$-0.518364\pi$
$97$	−1.13579	−0.115322	−0.0576610	+	0.998336i	$0.518364\pi$
$98$	0	0
$99$	2.31018	0.232182
$100$	0	0
$101$	9.16401	0.911853	0.455926	−	0.890018i	$-0.349308\pi$
$101$	9.16401	0.911853	0.455926	+	0.890018i	$0.349308\pi$
$102$	0	0
$103$	−17.2984	−1.70447	−0.852233	−	0.523162i	$-0.824752\pi$
$103$	−17.2984	−1.70447	−0.852233	+	0.523162i	$0.824752\pi$
$104$	0	0
$105$	−3.26471	−0.318603
$106$	0	0
$107$	−5.00928	−0.484265	−0.242132	−	0.970243i	$-0.577847\pi$
$107$	−5.00928	−0.484265	−0.242132	+	0.970243i	$0.577847\pi$
$108$	0	0
$109$	8.69427	0.832760	0.416380	−	0.909191i	$-0.363299\pi$
$109$	8.69427	0.832760	0.416380	+	0.909191i	$0.363299\pi$
$110$	0	0
$111$	−4.75143	−0.450986
$112$	0	0
$113$	17.3625	1.63332	0.816662	−	0.577117i	$-0.195822\pi$
$113$	17.3625	1.63332	0.816662	+	0.577117i	$0.195822\pi$
$114$	0	0
$115$	7.21491	0.672794
$116$	0	0
$117$	2.31018	0.213576
$118$	0	0
$119$	7.93077	0.727012
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.87901	−0.349758
$124$	0	0
$125$	21.4263	1.91642
$126$	0	0
$127$	18.8487	1.67255	0.836274	−	0.548312i	$-0.184729\pi$
$127$	18.8487	1.67255	0.836274	+	0.548312i	$0.184729\pi$
$128$	0	0
$129$	0.883951	0.0778275
$130$	0	0
$131$	1.19981	0.104828	0.0524139	−	0.998625i	$-0.483308\pi$
$131$	1.19981	0.104828	0.0524139	+	0.998625i	$0.483308\pi$
$132$	0	0
$133$	4.36492	0.378487
$134$	0	0
$135$	17.3362	1.49206
$136$	0	0
$137$	16.4439	1.40490	0.702449	−	0.711734i	$-0.252090\pi$
$137$	16.4439	1.40490	0.702449	+	0.711734i	$0.252090\pi$
$138$	0	0
$139$	−3.25701	−0.276256	−0.138128	−	0.990414i	$-0.544109\pi$
$139$	−3.25701	−0.276256	−0.138128	+	0.990414i	$0.544109\pi$
$140$	0	0
$141$	−1.87326	−0.157757
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−17.4599	−1.44997
$146$	0	0
$147$	−0.830554	−0.0685029
$148$	0	0
$149$	4.49350	0.368122	0.184061	−	0.982915i	$-0.441076\pi$
$149$	4.49350	0.368122	0.184061	+	0.982915i	$0.441076\pi$
$150$	0	0
$151$	2.14725	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$151$	2.14725	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$152$	0	0
$153$	−18.3215	−1.48121
$154$	0	0
$155$	6.35070	0.510101
$156$	0	0
$157$	−6.17417	−0.492752	−0.246376	−	0.969174i	$-0.579240\pi$
$157$	−6.17417	−0.492752	−0.246376	+	0.969174i	$0.579240\pi$
$158$	0	0
$159$	2.69178	0.213472
$160$	0	0
$161$	1.83550	0.144658
$162$	0	0
$163$	−9.83527	−0.770358	−0.385179	−	0.922842i	$-0.625860\pi$
$163$	−9.83527	−0.770358	−0.385179	+	0.922842i	$0.625860\pi$
$164$	0	0
$165$	3.26471	0.254158
$166$	0	0
$167$	15.4054	1.19211	0.596055	−	0.802944i	$-0.296734\pi$
$167$	15.4054	1.19211	0.596055	+	0.802944i	$0.296734\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−10.0838	−0.771124
$172$	0	0
$173$	3.15001	0.239491	0.119745	−	0.992805i	$-0.461792\pi$
$173$	3.15001	0.239491	0.119745	+	0.992805i	$0.461792\pi$
$174$	0	0
$175$	10.4509	0.790015
$176$	0	0
$177$	8.67801	0.652279
$178$	0	0
$179$	9.75182	0.728885	0.364443	−	0.931226i	$-0.381260\pi$
$179$	9.75182	0.728885	0.364443	+	0.931226i	$0.381260\pi$
$180$	0	0
$181$	−4.93982	−0.367174	−0.183587	−	0.983003i	$-0.558771\pi$
$181$	−4.93982	−0.367174	−0.183587	+	0.983003i	$0.558771\pi$
$182$	0	0
$183$	−1.58198	−0.116944
$184$	0	0
$185$	22.4871	1.65328
$186$	0	0
$187$	−7.93077	−0.579955
$188$	0	0
$189$	4.41039	0.320809
$190$	0	0
$191$	−17.0156	−1.23121	−0.615605	−	0.788055i	$-0.711088\pi$
$191$	−17.0156	−1.23121	−0.615605	+	0.788055i	$0.711088\pi$
$192$	0	0
$193$	21.0178	1.51290	0.756448	−	0.654053i	$-0.226933\pi$
$193$	21.0178	1.51290	0.756448	+	0.654053i	$0.226933\pi$
$194$	0	0
$195$	3.26471	0.233791
$196$	0	0
$197$	−8.39617	−0.598202	−0.299101	−	0.954221i	$-0.596687\pi$
$197$	−8.39617	−0.598202	−0.299101	+	0.954221i	$0.596687\pi$
$198$	0	0
$199$	5.20069	0.368667	0.184334	−	0.982864i	$-0.440987\pi$
$199$	5.20069	0.368667	0.184334	+	0.982864i	$0.440987\pi$
$200$	0	0
$201$	3.24497	0.228883
$202$	0	0
$203$	−4.44186	−0.311758
$204$	0	0
$205$	18.3582	1.28219
$206$	0	0
$207$	−4.24033	−0.294723
$208$	0	0
$209$	−4.36492	−0.301928
$210$	0	0
$211$	7.07667	0.487178	0.243589	−	0.969879i	$-0.421675\pi$
$211$	7.07667	0.487178	0.243589	+	0.969879i	$0.421675\pi$
$212$	0	0
$213$	−6.71996	−0.460444
$214$	0	0
$215$	−4.18348	−0.285311
$216$	0	0
$217$	1.61564	0.109677
$218$	0	0
$219$	7.16327	0.484049
$220$	0	0
$221$	−7.93077	−0.533481
$222$	0	0
$223$	8.44156	0.565289	0.282644	−	0.959225i	$-0.408788\pi$
$223$	8.44156	0.565289	0.282644	+	0.959225i	$0.408788\pi$
$224$	0	0
$225$	−24.1435	−1.60957
$226$	0	0
$227$	−12.9815	−0.861611	−0.430806	−	0.902445i	$-0.641771\pi$
$227$	−12.9815	−0.861611	−0.430806	+	0.902445i	$0.641771\pi$
$228$	0	0
$229$	2.04884	0.135391	0.0676956	−	0.997706i	$-0.478435\pi$
$229$	2.04884	0.135391	0.0676956	+	0.997706i	$0.478435\pi$
$230$	0	0
$231$	0.830554	0.0546464
$232$	0	0
$233$	−24.7160	−1.61920	−0.809601	−	0.586980i	$-0.800317\pi$
$233$	−24.7160	−1.61920	−0.809601	+	0.586980i	$0.800317\pi$
$234$	0	0
$235$	8.86559	0.578327
$236$	0	0
$237$	8.36133	0.543126
$238$	0	0
$239$	0.455859	0.0294870	0.0147435	−	0.999891i	$-0.495307\pi$
$239$	0.455859	0.0294870	0.0147435	+	0.999891i	$0.495307\pi$
$240$	0	0
$241$	20.5360	1.32284	0.661421	−	0.750015i	$-0.269954\pi$
$241$	20.5360	1.32284	0.661421	+	0.750015i	$0.269954\pi$
$242$	0	0
$243$	−15.9450	−1.02287
$244$	0	0
$245$	3.93077	0.251127
$246$	0	0
$247$	−4.36492	−0.277733
$248$	0	0
$249$	10.2795	0.651440
$250$	0	0
$251$	6.76749	0.427160	0.213580	−	0.976926i	$-0.431488\pi$
$251$	6.76749	0.427160	0.213580	+	0.976926i	$0.431488\pi$
$252$	0	0
$253$	−1.83550	−0.115397
$254$	0	0
$255$	−25.8917	−1.62140
$256$	0	0
$257$	14.2396	0.888242	0.444121	−	0.895967i	$-0.353516\pi$
$257$	14.2396	0.888242	0.444121	+	0.895967i	$0.353516\pi$
$258$	0	0
$259$	5.72080	0.355473
$260$	0	0
$261$	10.2615	0.635171
$262$	0	0
$263$	−20.7298	−1.27826	−0.639129	−	0.769100i	$-0.720705\pi$
$263$	−20.7298	−1.27826	−0.639129	+	0.769100i	$0.720705\pi$
$264$	0	0
$265$	−12.7394	−0.782575
$266$	0	0
$267$	−0.0216207	−0.00132316
$268$	0	0
$269$	−8.34116	−0.508569	−0.254285	−	0.967129i	$-0.581840\pi$
$269$	−8.34116	−0.508569	−0.254285	+	0.967129i	$0.581840\pi$
$270$	0	0
$271$	17.3516	1.05404	0.527018	−	0.849854i	$-0.323310\pi$
$271$	17.3516	1.05404	0.527018	+	0.849854i	$0.323310\pi$
$272$	0	0
$273$	0.830554	0.0502674
$274$	0	0
$275$	−10.4509	−0.630214
$276$	0	0
$277$	22.7417	1.36641	0.683207	−	0.730225i	$-0.260585\pi$
$277$	22.7417	1.36641	0.683207	+	0.730225i	$0.260585\pi$
$278$	0	0
$279$	−3.73242	−0.223454
$280$	0	0
$281$	−30.4118	−1.81421	−0.907107	−	0.420900i	$-0.861714\pi$
$281$	−30.4118	−1.81421	−0.907107	+	0.420900i	$0.861714\pi$
$282$	0	0
$283$	−12.0239	−0.714746	−0.357373	−	0.933962i	$-0.616327\pi$
$283$	−12.0239	−0.714746	−0.357373	+	0.933962i	$0.616327\pi$
$284$	0	0
$285$	−14.2502	−0.844110
$286$	0	0
$287$	4.67038	0.275684
$288$	0	0
$289$	45.8970	2.69983
$290$	0	0
$291$	0.943335	0.0552992
$292$	0	0
$293$	19.3103	1.12812	0.564059	−	0.825735i	$-0.309239\pi$
$293$	19.3103	1.12812	0.564059	+	0.825735i	$0.309239\pi$
$294$	0	0
$295$	−41.0704	−2.39121
$296$	0	0
$297$	−4.41039	−0.255917
$298$	0	0
$299$	−1.83550	−0.106150
$300$	0	0
$301$	−1.06429	−0.0613447
$302$	0	0
$303$	−7.61120	−0.437252
$304$	0	0
$305$	7.48706	0.428708
$306$	0	0
$307$	28.1514	1.60669	0.803344	−	0.595516i	$-0.203052\pi$
$307$	28.1514	1.60669	0.803344	+	0.595516i	$0.203052\pi$
$308$	0	0
$309$	14.3673	0.817327
$310$	0	0
$311$	5.69117	0.322716	0.161358	−	0.986896i	$-0.448413\pi$
$311$	5.69117	0.322716	0.161358	+	0.986896i	$0.448413\pi$
$312$	0	0
$313$	8.09286	0.457435	0.228718	−	0.973493i	$-0.426547\pi$
$313$	8.09286	0.457435	0.228718	+	0.973493i	$0.426547\pi$
$314$	0	0
$315$	−9.08078	−0.511644
$316$	0	0
$317$	−11.6538	−0.654544	−0.327272	−	0.944930i	$-0.606129\pi$
$317$	−11.6538	−0.654544	−0.327272	+	0.944930i	$0.606129\pi$
$318$	0	0
$319$	4.44186	0.248697
$320$	0	0
$321$	4.16047	0.232215
$322$	0	0
$323$	34.6172	1.92615
$324$	0	0
$325$	−10.4509	−0.579712
$326$	0	0
$327$	−7.22106	−0.399326
$328$	0	0
$329$	2.25544	0.124346
$330$	0	0
$331$	18.3359	1.00783	0.503917	−	0.863752i	$-0.331892\pi$
$331$	18.3359	1.00783	0.503917	+	0.863752i	$0.331892\pi$
$332$	0	0
$333$	−13.2161	−0.724236
$334$	0	0
$335$	−15.3575	−0.839069
$336$	0	0
$337$	−24.8600	−1.35421	−0.677104	−	0.735887i	$-0.736765\pi$
$337$	−24.8600	−1.35421	−0.677104	+	0.735887i	$0.736765\pi$
$338$	0	0
$339$	−14.4205	−0.783212
$340$	0	0
$341$	−1.61564	−0.0874918
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−5.99238	−0.322619
$346$	0	0
$347$	13.8423	0.743095	0.371547	−	0.928414i	$-0.378827\pi$
$347$	13.8423	0.743095	0.371547	+	0.928414i	$0.378827\pi$
$348$	0	0
$349$	0.573934	0.0307220	0.0153610	−	0.999882i	$-0.495110\pi$
$349$	0.573934	0.0307220	0.0153610	+	0.999882i	$0.495110\pi$
$350$	0	0
$351$	−4.41039	−0.235409
$352$	0	0
$353$	−4.20042	−0.223566	−0.111783	−	0.993733i	$-0.535656\pi$
$353$	−4.20042	−0.223566	−0.111783	+	0.993733i	$0.535656\pi$
$354$	0	0
$355$	31.8036	1.68796
$356$	0	0
$357$	−6.58693	−0.348617
$358$	0	0
$359$	−24.7380	−1.30562	−0.652810	−	0.757522i	$-0.726410\pi$
$359$	−24.7380	−1.30562	−0.652810	+	0.757522i	$0.726410\pi$
$360$	0	0
$361$	0.0525588	0.00276625
$362$	0	0
$363$	−0.830554	−0.0435928
$364$	0	0
$365$	−33.9016	−1.77449
$366$	0	0
$367$	−19.1792	−1.00115	−0.500573	−	0.865694i	$-0.666877\pi$
$367$	−19.1792	−1.00115	−0.500573	+	0.865694i	$0.666877\pi$
$368$	0	0
$369$	−10.7894	−0.561675
$370$	0	0
$371$	−3.24095	−0.168261
$372$	0	0
$373$	9.59935	0.497036	0.248518	−	0.968627i	$-0.420057\pi$
$373$	9.59935	0.497036	0.248518	+	0.968627i	$0.420057\pi$
$374$	0	0
$375$	−17.7957	−0.918964
$376$	0	0
$377$	4.44186	0.228768
$378$	0	0
$379$	3.41932	0.175639	0.0878195	−	0.996136i	$-0.472010\pi$
$379$	3.41932	0.175639	0.0878195	+	0.996136i	$0.472010\pi$
$380$	0	0
$381$	−15.6548	−0.802021
$382$	0	0
$383$	−23.7390	−1.21301	−0.606503	−	0.795082i	$-0.707428\pi$
$383$	−23.7390	−1.21301	−0.606503	+	0.795082i	$0.707428\pi$
$384$	0	0
$385$	−3.93077	−0.200330
$386$	0	0
$387$	2.45870	0.124983
$388$	0	0
$389$	30.2210	1.53227	0.766133	−	0.642682i	$-0.222178\pi$
$389$	30.2210	1.53227	0.766133	+	0.642682i	$0.222178\pi$
$390$	0	0
$391$	14.5569	0.736174
$392$	0	0
$393$	−0.996506	−0.0502671
$394$	0	0
$395$	−39.5717	−1.99107
$396$	0	0
$397$	−1.79536	−0.0901067	−0.0450534	−	0.998985i	$-0.514346\pi$
$397$	−1.79536	−0.0901067	−0.0450534	+	0.998985i	$0.514346\pi$
$398$	0	0
$399$	−3.62530	−0.181492
$400$	0	0
$401$	36.3314	1.81430	0.907151	−	0.420804i	$-0.138252\pi$
$401$	36.3314	1.81430	0.907151	+	0.420804i	$0.138252\pi$
$402$	0	0
$403$	−1.61564	−0.0804808
$404$	0	0
$405$	12.8437	0.638207
$406$	0	0
$407$	−5.72080	−0.283569
$408$	0	0
$409$	−34.1855	−1.69036	−0.845182	−	0.534478i	$-0.820508\pi$
$409$	−34.1855	−1.69036	−0.845182	+	0.534478i	$0.820508\pi$
$410$	0	0
$411$	−13.6575	−0.673677
$412$	0	0
$413$	−10.4485	−0.514135
$414$	0	0
$415$	−48.6500	−2.38814
$416$	0	0
$417$	2.70512	0.132470
$418$	0	0
$419$	−35.0201	−1.71084	−0.855422	−	0.517932i	$-0.826702\pi$
$419$	−35.0201	−1.71084	−0.855422	+	0.517932i	$0.826702\pi$
$420$	0	0
$421$	−0.581067	−0.0283195	−0.0141597	−	0.999900i	$-0.504507\pi$
$421$	−0.581067	−0.0283195	−0.0141597	+	0.999900i	$0.504507\pi$
$422$	0	0
$423$	−5.21046	−0.253341
$424$	0	0
$425$	82.8837	4.02045
$426$	0	0
$427$	1.90473	0.0921765
$428$	0	0
$429$	−0.830554	−0.0400995
$430$	0	0
$431$	5.84827	0.281701	0.140851	−	0.990031i	$-0.455016\pi$
$431$	5.84827	0.281701	0.140851	+	0.990031i	$0.455016\pi$
$432$	0	0
$433$	21.1995	1.01878	0.509391	−	0.860535i	$-0.329871\pi$
$433$	21.1995	1.01878	0.509391	+	0.860535i	$0.329871\pi$
$434$	0	0
$435$	14.5014	0.695289
$436$	0	0
$437$	8.01181	0.383257
$438$	0	0
$439$	4.94511	0.236017	0.118009	−	0.993013i	$-0.462349\pi$
$439$	4.94511	0.236017	0.118009	+	0.993013i	$0.462349\pi$
$440$	0	0
$441$	−2.31018	−0.110009
$442$	0	0
$443$	−3.00276	−0.142665	−0.0713327	−	0.997453i	$-0.522725\pi$
$443$	−3.00276	−0.142665	−0.0713327	+	0.997453i	$0.522725\pi$
$444$	0	0
$445$	0.102324	0.00485064
$446$	0	0
$447$	−3.73210	−0.176522
$448$	0	0
$449$	1.48142	0.0699127	0.0349563	−	0.999389i	$-0.488871\pi$
$449$	1.48142	0.0699127	0.0349563	+	0.999389i	$0.488871\pi$
$450$	0	0
$451$	−4.67038	−0.219920
$452$	0	0
$453$	−1.78341	−0.0837918
$454$	0	0
$455$	−3.93077	−0.184277
$456$	0	0
$457$	−17.6396	−0.825146	−0.412573	−	0.910925i	$-0.635370\pi$
$457$	−17.6396	−0.825146	−0.412573	+	0.910925i	$0.635370\pi$
$458$	0	0
$459$	34.9778	1.63262
$460$	0	0
$461$	36.3214	1.69166	0.845828	−	0.533456i	$-0.179107\pi$
$461$	36.3214	1.69166	0.845828	+	0.533456i	$0.179107\pi$
$462$	0	0
$463$	−0.723252	−0.0336124	−0.0168062	−	0.999859i	$-0.505350\pi$
$463$	−0.723252	−0.0336124	−0.0168062	+	0.999859i	$0.505350\pi$
$464$	0	0
$465$	−5.27460	−0.244604
$466$	0	0
$467$	32.3836	1.49853	0.749266	−	0.662269i	$-0.230406\pi$
$467$	32.3836	1.49853	0.749266	+	0.662269i	$0.230406\pi$
$468$	0	0
$469$	−3.90700	−0.180408
$470$	0	0
$471$	5.12798	0.236285
$472$	0	0
$473$	1.06429	0.0489361
$474$	0	0
$475$	45.6174	2.09307
$476$	0	0
$477$	7.48717	0.342814
$478$	0	0
$479$	1.14341	0.0522439	0.0261220	−	0.999659i	$-0.491684\pi$
$479$	1.14341	0.0522439	0.0261220	+	0.999659i	$0.491684\pi$
$480$	0	0
$481$	−5.72080	−0.260846
$482$	0	0
$483$	−1.52448	−0.0693663
$484$	0	0
$485$	−4.46452	−0.202723
$486$	0	0
$487$	−8.62251	−0.390723	−0.195362	−	0.980731i	$-0.562588\pi$
$487$	−8.62251	−0.390723	−0.195362	+	0.980731i	$0.562588\pi$
$488$	0	0
$489$	8.16873	0.369403
$490$	0	0
$491$	2.45206	0.110660	0.0553299	−	0.998468i	$-0.482379\pi$
$491$	2.45206	0.110660	0.0553299	+	0.998468i	$0.482379\pi$
$492$	0	0
$493$	−35.2274	−1.58656
$494$	0	0
$495$	9.08078	0.408150
$496$	0	0
$497$	8.09094	0.362928
$498$	0	0
$499$	19.0621	0.853336	0.426668	−	0.904408i	$-0.359687\pi$
$499$	19.0621	0.853336	0.426668	+	0.904408i	$0.359687\pi$
$500$	0	0
$501$	−12.7951	−0.571641
$502$	0	0
$503$	−16.3290	−0.728073	−0.364037	−	0.931385i	$-0.618602\pi$
$503$	−16.3290	−0.728073	−0.364037	+	0.931385i	$0.618602\pi$
$504$	0	0
$505$	36.0216	1.60294
$506$	0	0
$507$	−0.830554	−0.0368862
$508$	0	0
$509$	−24.0299	−1.06511	−0.532554	−	0.846396i	$-0.678768\pi$
$509$	−24.0299	−1.06511	−0.532554	+	0.846396i	$0.678768\pi$
$510$	0	0
$511$	−8.62469	−0.381534
$512$	0	0
$513$	19.2510	0.849953
$514$	0	0
$515$	−67.9961	−2.99627
$516$	0	0
$517$	−2.25544	−0.0991940
$518$	0	0
$519$	−2.61625	−0.114841
$520$	0	0
$521$	15.2769	0.669292	0.334646	−	0.942344i	$-0.391383\pi$
$521$	15.2769	0.669292	0.334646	+	0.942344i	$0.391383\pi$
$522$	0	0
$523$	4.76481	0.208351	0.104175	−	0.994559i	$-0.466780\pi$
$523$	4.76481	0.208351	0.104175	+	0.994559i	$0.466780\pi$
$524$	0	0
$525$	−8.68005	−0.378828
$526$	0	0
$527$	12.8133	0.558155
$528$	0	0
$529$	−19.6309	−0.853519
$530$	0	0
$531$	24.1378	1.04749
$532$	0	0
$533$	−4.67038	−0.202297
$534$	0	0
$535$	−19.6903	−0.851285
$536$	0	0
$537$	−8.09941	−0.349515
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	5.30174	0.227940	0.113970	−	0.993484i	$-0.463643\pi$
$541$	5.30174	0.227940	0.113970	+	0.993484i	$0.463643\pi$
$542$	0	0
$543$	4.10278	0.176067
$544$	0	0
$545$	34.1751	1.46390
$546$	0	0
$547$	25.9714	1.11046	0.555229	−	0.831697i	$-0.312631\pi$
$547$	25.9714	1.11046	0.555229	+	0.831697i	$0.312631\pi$
$548$	0	0
$549$	−4.40028	−0.187799
$550$	0	0
$551$	−19.3884	−0.825973
$552$	0	0
$553$	−10.0672	−0.428100
$554$	0	0
$555$	−18.6768	−0.792784
$556$	0	0
$557$	−12.3002	−0.521178	−0.260589	−	0.965450i	$-0.583917\pi$
$557$	−12.3002	−0.521178	−0.260589	+	0.965450i	$0.583917\pi$
$558$	0	0
$559$	1.06429	0.0450147
$560$	0	0
$561$	6.58693	0.278100
$562$	0	0
$563$	7.73805	0.326120	0.163060	−	0.986616i	$-0.447864\pi$
$563$	7.73805	0.326120	0.163060	+	0.986616i	$0.447864\pi$
$564$	0	0
$565$	68.2478	2.87121
$566$	0	0
$567$	3.26747	0.137221
$568$	0	0
$569$	11.4204	0.478768	0.239384	−	0.970925i	$-0.423054\pi$
$569$	11.4204	0.478768	0.239384	+	0.970925i	$0.423054\pi$
$570$	0	0
$571$	40.6088	1.69943	0.849713	−	0.527245i	$-0.176775\pi$
$571$	40.6088	1.69943	0.849713	+	0.527245i	$0.176775\pi$
$572$	0	0
$573$	14.1324	0.590390
$574$	0	0
$575$	19.1826	0.799971
$576$	0	0
$577$	−19.8823	−0.827710	−0.413855	−	0.910343i	$-0.635818\pi$
$577$	−19.8823	−0.827710	−0.413855	+	0.910343i	$0.635818\pi$
$578$	0	0
$579$	−17.4564	−0.725465
$580$	0	0
$581$	−12.3767	−0.513473
$582$	0	0
$583$	3.24095	0.134226
$584$	0	0
$585$	9.08078	0.375444
$586$	0	0
$587$	−43.8388	−1.80942	−0.904711	−	0.426025i	$-0.859913\pi$
$587$	−43.8388	−1.80942	−0.904711	+	0.426025i	$0.859913\pi$
$588$	0	0
$589$	7.05215	0.290579
$590$	0	0
$591$	6.97347	0.286850
$592$	0	0
$593$	14.9247	0.612884	0.306442	−	0.951889i	$-0.400861\pi$
$593$	14.9247	0.612884	0.306442	+	0.951889i	$0.400861\pi$
$594$	0	0
$595$	31.1740	1.27801
$596$	0	0
$597$	−4.31946	−0.176784
$598$	0	0
$599$	17.6022	0.719206	0.359603	−	0.933105i	$-0.382912\pi$
$599$	17.6022	0.719206	0.359603	+	0.933105i	$0.382912\pi$
$600$	0	0
$601$	6.88188	0.280718	0.140359	−	0.990101i	$-0.455174\pi$
$601$	6.88188	0.280718	0.140359	+	0.990101i	$0.455174\pi$
$602$	0	0
$603$	9.02587	0.367562
$604$	0	0
$605$	3.93077	0.159808
$606$	0	0
$607$	25.2321	1.02414	0.512070	−	0.858944i	$-0.328879\pi$
$607$	25.2321	1.02414	0.512070	+	0.858944i	$0.328879\pi$
$608$	0	0
$609$	3.68921	0.149494
$610$	0	0
$611$	−2.25544	−0.0912452
$612$	0	0
$613$	−46.7285	−1.88735	−0.943673	−	0.330878i	$-0.892655\pi$
$613$	−46.7285	−1.88735	−0.943673	+	0.330878i	$0.892655\pi$
$614$	0	0
$615$	−15.2475	−0.614837
$616$	0	0
$617$	31.6513	1.27423	0.637117	−	0.770767i	$-0.280127\pi$
$617$	31.6513	1.27423	0.637117	+	0.770767i	$0.280127\pi$
$618$	0	0
$619$	−36.7582	−1.47744	−0.738719	−	0.674014i	$-0.764569\pi$
$619$	−36.7582	−1.47744	−0.738719	+	0.674014i	$0.764569\pi$
$620$	0	0
$621$	8.09527	0.324852
$622$	0	0
$623$	0.0260316	0.00104294
$624$	0	0
$625$	31.9670	1.27868
$626$	0	0
$627$	3.62530	0.144781
$628$	0	0
$629$	45.3703	1.80903
$630$	0	0
$631$	43.7174	1.74036	0.870180	−	0.492734i	$-0.164002\pi$
$631$	43.7174	1.74036	0.870180	+	0.492734i	$0.164002\pi$
$632$	0	0
$633$	−5.87756	−0.233612
$634$	0	0
$635$	74.0896	2.94016
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−18.6915	−0.739425
$640$	0	0
$641$	15.5013	0.612264	0.306132	−	0.951989i	$-0.400965\pi$
$641$	15.5013	0.612264	0.306132	+	0.951989i	$0.400965\pi$
$642$	0	0
$643$	−11.4587	−0.451885	−0.225943	−	0.974141i	$-0.572546\pi$
$643$	−11.4587	−0.451885	−0.225943	+	0.974141i	$0.572546\pi$
$644$	0	0
$645$	3.47460	0.136812
$646$	0	0
$647$	−27.3797	−1.07641	−0.538204	−	0.842815i	$-0.680897\pi$
$647$	−27.3797	−1.07641	−0.538204	+	0.842815i	$0.680897\pi$
$648$	0	0
$649$	10.4485	0.410138
$650$	0	0
$651$	−1.34188	−0.0525923
$652$	0	0
$653$	22.2087	0.869095	0.434548	−	0.900649i	$-0.356908\pi$
$653$	22.2087	0.869095	0.434548	+	0.900649i	$0.356908\pi$
$654$	0	0
$655$	4.71617	0.184276
$656$	0	0
$657$	19.9246	0.777332
$658$	0	0
$659$	38.8773	1.51444	0.757221	−	0.653158i	$-0.226556\pi$
$659$	38.8773	1.51444	0.757221	+	0.653158i	$0.226556\pi$
$660$	0	0
$661$	−10.1744	−0.395736	−0.197868	−	0.980229i	$-0.563402\pi$
$661$	−10.1744	−0.395736	−0.197868	+	0.980229i	$0.563402\pi$
$662$	0	0
$663$	6.58693	0.255815
$664$	0	0
$665$	17.1575	0.665339
$666$	0	0
$667$	−8.15303	−0.315687
$668$	0	0
$669$	−7.01117	−0.271067
$670$	0	0
$671$	−1.90473	−0.0735314
$672$	0	0
$673$	−21.2566	−0.819380	−0.409690	−	0.912225i	$-0.634363\pi$
$673$	−21.2566	−0.819380	−0.409690	+	0.912225i	$0.634363\pi$
$674$	0	0
$675$	46.0926	1.77411
$676$	0	0
$677$	−29.8385	−1.14679	−0.573393	−	0.819280i	$-0.694373\pi$
$677$	−29.8385	−1.14679	−0.573393	+	0.819280i	$0.694373\pi$
$678$	0	0
$679$	−1.13579	−0.0435876
$680$	0	0
$681$	10.7818	0.413160
$682$	0	0
$683$	−36.0336	−1.37879	−0.689394	−	0.724386i	$-0.742123\pi$
$683$	−36.0336	−1.37879	−0.689394	+	0.724386i	$0.742123\pi$
$684$	0	0
$685$	64.6371	2.46966
$686$	0	0
$687$	−1.70167	−0.0649229
$688$	0	0
$689$	3.24095	0.123470
$690$	0	0
$691$	42.2705	1.60804	0.804022	−	0.594599i	$-0.202689\pi$
$691$	42.2705	1.60804	0.804022	+	0.594599i	$0.202689\pi$
$692$	0	0
$693$	2.31018	0.0877565
$694$	0	0
$695$	−12.8025	−0.485628
$696$	0	0
$697$	37.0397	1.40298
$698$	0	0
$699$	20.5280	0.776441
$700$	0	0
$701$	−17.9241	−0.676985	−0.338493	−	0.940969i	$-0.609917\pi$
$701$	−17.9241	−0.676985	−0.338493	+	0.940969i	$0.609917\pi$
$702$	0	0
$703$	24.9708	0.941793
$704$	0	0
$705$	−7.36335	−0.277320
$706$	0	0
$707$	9.16401	0.344648
$708$	0	0
$709$	−5.92651	−0.222575	−0.111287	−	0.993788i	$-0.535497\pi$
$709$	−5.92651	−0.222575	−0.111287	+	0.993788i	$0.535497\pi$
$710$	0	0
$711$	23.2570	0.872205
$712$	0	0
$713$	2.96551	0.111059
$714$	0	0
$715$	3.93077	0.147002
$716$	0	0
$717$	−0.378615	−0.0141396
$718$	0	0
$719$	−28.0414	−1.04577	−0.522884	−	0.852404i	$-0.675144\pi$
$719$	−28.0414	−1.04577	−0.522884	+	0.852404i	$0.675144\pi$
$720$	0	0
$721$	−17.2984	−0.644228
$722$	0	0
$723$	−17.0563	−0.634330
$724$	0	0
$725$	−46.4215	−1.72405
$726$	0	0
$727$	−33.6093	−1.24650	−0.623250	−	0.782022i	$-0.714188\pi$
$727$	−33.6093	−1.24650	−0.623250	+	0.782022i	$0.714188\pi$
$728$	0	0
$729$	3.44075	0.127435
$730$	0	0
$731$	−8.44064	−0.312188
$732$	0	0
$733$	12.5522	0.463626	0.231813	−	0.972760i	$-0.425534\pi$
$733$	12.5522	0.463626	0.231813	+	0.972760i	$0.425534\pi$
$734$	0	0
$735$	−3.26471	−0.120421
$736$	0	0
$737$	3.90700	0.143916
$738$	0	0
$739$	−19.5696	−0.719881	−0.359940	−	0.932975i	$-0.617203\pi$
$739$	−19.5696	−0.719881	−0.359940	+	0.932975i	$0.617203\pi$
$740$	0	0
$741$	3.62530	0.133179
$742$	0	0
$743$	19.1048	0.700889	0.350444	−	0.936584i	$-0.386031\pi$
$743$	19.1048	0.700889	0.350444	+	0.936584i	$0.386031\pi$
$744$	0	0
$745$	17.6629	0.647119
$746$	0	0
$747$	28.5925	1.04614
$748$	0	0
$749$	−5.00928	−0.183035
$750$	0	0
$751$	42.8140	1.56231	0.781153	−	0.624340i	$-0.214632\pi$
$751$	42.8140	1.56231	0.781153	+	0.624340i	$0.214632\pi$
$752$	0	0
$753$	−5.62077	−0.204832
$754$	0	0
$755$	8.44034	0.307175
$756$	0	0
$757$	−5.92751	−0.215439	−0.107720	−	0.994181i	$-0.534355\pi$
$757$	−5.92751	−0.215439	−0.107720	+	0.994181i	$0.534355\pi$
$758$	0	0
$759$	1.52448	0.0553351
$760$	0	0
$761$	−17.0440	−0.617843	−0.308922	−	0.951087i	$-0.599968\pi$
$761$	−17.0440	−0.617843	−0.308922	+	0.951087i	$0.599968\pi$
$762$	0	0
$763$	8.69427	0.314754
$764$	0	0
$765$	−72.0175	−2.60380
$766$	0	0
$767$	10.4485	0.377272
$768$	0	0
$769$	−27.2498	−0.982653	−0.491327	−	0.870975i	$-0.663488\pi$
$769$	−27.2498	−0.982653	−0.491327	+	0.870975i	$0.663488\pi$
$770$	0	0
$771$	−11.8268	−0.425930
$772$	0	0
$773$	−28.3900	−1.02112	−0.510559	−	0.859842i	$-0.670562\pi$
$773$	−28.3900	−1.02112	−0.510559	+	0.859842i	$0.670562\pi$
$774$	0	0
$775$	16.8849	0.606524
$776$	0	0
$777$	−4.75143	−0.170457
$778$	0	0
$779$	20.3859	0.730400
$780$	0	0
$781$	−8.09094	−0.289516
$782$	0	0
$783$	−19.5904	−0.700102
$784$	0	0
$785$	−24.2692	−0.866205
$786$	0	0
$787$	−23.1949	−0.826811	−0.413405	−	0.910547i	$-0.635661\pi$
$787$	−23.1949	−0.826811	−0.413405	+	0.910547i	$0.635661\pi$
$788$	0	0
$789$	17.2173	0.612951
$790$	0	0
$791$	17.3625	0.617338
$792$	0	0
$793$	−1.90473	−0.0676391
$794$	0	0
$795$	10.5808	0.375261
$796$	0	0
$797$	10.2174	0.361917	0.180959	−	0.983491i	$-0.442080\pi$
$797$	10.2174	0.361917	0.180959	+	0.983491i	$0.442080\pi$
$798$	0	0
$799$	17.8873	0.632808
$800$	0	0
$801$	−0.0601378	−0.00212486
$802$	0	0
$803$	8.62469	0.304359
$804$	0	0
$805$	7.21491	0.254292
$806$	0	0
$807$	6.92778	0.243869
$808$	0	0
$809$	−31.9180	−1.12218	−0.561089	−	0.827756i	$-0.689617\pi$
$809$	−31.9180	−1.12218	−0.561089	+	0.827756i	$0.689617\pi$
$810$	0	0
$811$	−16.3797	−0.575170	−0.287585	−	0.957755i	$-0.592852\pi$
$811$	−16.3797	−0.575170	−0.287585	+	0.957755i	$0.592852\pi$
$812$	0	0
$813$	−14.4115	−0.505432
$814$	0	0
$815$	−38.6602	−1.35421
$816$	0	0
$817$	−4.64555	−0.162527
$818$	0	0
$819$	2.31018	0.0807242
$820$	0	0
$821$	13.8567	0.483604	0.241802	−	0.970326i	$-0.422262\pi$
$821$	13.8567	0.483604	0.241802	+	0.970326i	$0.422262\pi$
$822$	0	0
$823$	−1.49284	−0.0520372	−0.0260186	−	0.999661i	$-0.508283\pi$
$823$	−1.49284	−0.0520372	−0.0260186	+	0.999661i	$0.508283\pi$
$824$	0	0
$825$	8.68005	0.302200
$826$	0	0
$827$	−29.8158	−1.03680	−0.518398	−	0.855140i	$-0.673471\pi$
$827$	−29.8158	−1.03680	−0.518398	+	0.855140i	$0.673471\pi$
$828$	0	0
$829$	24.1800	0.839805	0.419903	−	0.907569i	$-0.362064\pi$
$829$	24.1800	0.839805	0.419903	+	0.907569i	$0.362064\pi$
$830$	0	0
$831$	−18.8882	−0.655224
$832$	0	0
$833$	7.93077	0.274785
$834$	0	0
$835$	60.5552	2.09560
$836$	0	0
$837$	7.12561	0.246297
$838$	0	0
$839$	10.5044	0.362651	0.181325	−	0.983423i	$-0.441961\pi$
$839$	10.5044	0.362651	0.181325	+	0.983423i	$0.441961\pi$
$840$	0	0
$841$	−9.26985	−0.319650
$842$	0	0
$843$	25.2586	0.869953
$844$	0	0
$845$	3.93077	0.135222
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	9.98649	0.342735
$850$	0	0
$851$	10.5005	0.359953
$852$	0	0
$853$	−9.45198	−0.323630	−0.161815	−	0.986821i	$-0.551735\pi$
$853$	−9.45198	−0.323630	−0.161815	+	0.986821i	$0.551735\pi$
$854$	0	0
$855$	−39.6369	−1.35555
$856$	0	0
$857$	34.6464	1.18350	0.591748	−	0.806123i	$-0.298438\pi$
$857$	34.6464	1.18350	0.591748	+	0.806123i	$0.298438\pi$
$858$	0	0
$859$	22.6704	0.773503	0.386751	−	0.922184i	$-0.373597\pi$
$859$	22.6704	0.773503	0.386751	+	0.922184i	$0.373597\pi$
$860$	0	0
$861$	−3.87901	−0.132196
$862$	0	0
$863$	19.9863	0.680343	0.340172	−	0.940363i	$-0.389515\pi$
$863$	19.9863	0.680343	0.340172	+	0.940363i	$0.389515\pi$
$864$	0	0
$865$	12.3819	0.420999
$866$	0	0
$867$	−38.1200	−1.29462
$868$	0	0
$869$	10.0672	0.341505
$870$	0	0
$871$	3.90700	0.132384
$872$	0	0
$873$	2.62388	0.0888048
$874$	0	0
$875$	21.4263	0.724340
$876$	0	0
$877$	−22.4779	−0.759024	−0.379512	−	0.925187i	$-0.623908\pi$
$877$	−22.4779	−0.759024	−0.379512	+	0.925187i	$0.623908\pi$
$878$	0	0
$879$	−16.0382	−0.540955
$880$	0	0
$881$	7.77915	0.262086	0.131043	−	0.991377i	$-0.458167\pi$
$881$	7.77915	0.262086	0.131043	+	0.991377i	$0.458167\pi$
$882$	0	0
$883$	−28.7808	−0.968550	−0.484275	−	0.874916i	$-0.660916\pi$
$883$	−28.7808	−0.968550	−0.484275	+	0.874916i	$0.660916\pi$
$884$	0	0
$885$	34.1112	1.14664
$886$	0	0
$887$	30.5794	1.02676	0.513379	−	0.858162i	$-0.328394\pi$
$887$	30.5794	1.02676	0.513379	+	0.858162i	$0.328394\pi$
$888$	0	0
$889$	18.8487	0.632164
$890$	0	0
$891$	−3.26747	−0.109464
$892$	0	0
$893$	9.84481	0.329444
$894$	0	0
$895$	38.3321	1.28130
$896$	0	0
$897$	1.52448	0.0509009
$898$	0	0
$899$	−7.17645	−0.239348
$900$	0	0
$901$	−25.7032	−0.856297
$902$	0	0
$903$	0.883951	0.0294160
$904$	0	0
$905$	−19.4173	−0.645451
$906$	0	0
$907$	24.7371	0.821382	0.410691	−	0.911775i	$-0.365288\pi$
$907$	24.7371	0.821382	0.410691	+	0.911775i	$0.365288\pi$
$908$	0	0
$909$	−21.1705	−0.702181
$910$	0	0
$911$	14.5877	0.483311	0.241655	−	0.970362i	$-0.422310\pi$
$911$	14.5877	0.483311	0.241655	+	0.970362i	$0.422310\pi$
$912$	0	0
$913$	12.3767	0.409610
$914$	0	0
$915$	−6.21841	−0.205574
$916$	0	0
$917$	1.19981	0.0396212
$918$	0	0
$919$	−57.3669	−1.89236	−0.946180	−	0.323641i	$-0.895093\pi$
$919$	−57.3669	−1.89236	−0.946180	+	0.323641i	$0.895093\pi$
$920$	0	0
$921$	−23.3813	−0.770440
$922$	0	0
$923$	−8.09094	−0.266316
$924$	0	0
$925$	59.7875	1.96580
$926$	0	0
$927$	39.9625	1.31254
$928$	0	0
$929$	14.0393	0.460614	0.230307	−	0.973118i	$-0.426027\pi$
$929$	14.0393	0.460614	0.230307	+	0.973118i	$0.426027\pi$
$930$	0	0
$931$	4.36492	0.143055
$932$	0	0
$933$	−4.72682	−0.154749
$934$	0	0
$935$	−31.1740	−1.01950
$936$	0	0
$937$	9.76400	0.318976	0.159488	−	0.987200i	$-0.449016\pi$
$937$	9.76400	0.318976	0.159488	+	0.987200i	$0.449016\pi$
$938$	0	0
$939$	−6.72155	−0.219350
$940$	0	0
$941$	−25.9364	−0.845501	−0.422751	−	0.906246i	$-0.638935\pi$
$941$	−25.9364	−0.845501	−0.422751	+	0.906246i	$0.638935\pi$
$942$	0	0
$943$	8.57248	0.279158
$944$	0	0
$945$	17.3362	0.563947
$946$	0	0
$947$	7.04401	0.228900	0.114450	−	0.993429i	$-0.463490\pi$
$947$	7.04401	0.228900	0.114450	+	0.993429i	$0.463490\pi$
$948$	0	0
$949$	8.62469	0.279969
$950$	0	0
$951$	9.67913	0.313867
$952$	0	0
$953$	−49.5112	−1.60383	−0.801913	−	0.597441i	$-0.796184\pi$
$953$	−49.5112	−1.60383	−0.801913	+	0.597441i	$0.796184\pi$
$954$	0	0
$955$	−66.8845	−2.16433
$956$	0	0
$957$	−3.68921	−0.119255
$958$	0	0
$959$	16.4439	0.531001
$960$	0	0
$961$	−28.3897	−0.915797
$962$	0	0
$963$	11.5723	0.372913
$964$	0	0
$965$	82.6162	2.65951
$966$	0	0
$967$	−6.79115	−0.218389	−0.109194	−	0.994020i	$-0.534827\pi$
$967$	−6.79115	−0.218389	−0.109194	+	0.994020i	$0.534827\pi$
$968$	0	0
$969$	−28.7514	−0.923629
$970$	0	0
$971$	44.0252	1.41283	0.706417	−	0.707796i	$-0.250310\pi$
$971$	44.0252	1.41283	0.706417	+	0.707796i	$0.250310\pi$
$972$	0	0
$973$	−3.25701	−0.104415
$974$	0	0
$975$	8.68005	0.277984
$976$	0	0
$977$	7.69910	0.246316	0.123158	−	0.992387i	$-0.460698\pi$
$977$	7.69910	0.246316	0.123158	+	0.992387i	$0.460698\pi$
$978$	0	0
$979$	−0.0260316	−0.000831975 0
$980$	0	0
$981$	−20.0853	−0.641275
$982$	0	0
$983$	3.70465	0.118160	0.0590800	−	0.998253i	$-0.481183\pi$
$983$	3.70465	0.118160	0.0590800	+	0.998253i	$0.481183\pi$
$984$	0	0
$985$	−33.0034	−1.05157
$986$	0	0
$987$	−1.87326	−0.0596266
$988$	0	0
$989$	−1.95350	−0.0621178
$990$	0	0
$991$	54.0033	1.71547	0.857736	−	0.514091i	$-0.171871\pi$
$991$	54.0033	1.71547	0.857736	+	0.514091i	$0.171871\pi$
$992$	0	0
$993$	−15.2290	−0.483277
$994$	0	0
$995$	20.4427	0.648077
$996$	0	0
$997$	−15.5895	−0.493725	−0.246863	−	0.969050i	$-0.579400\pi$
$997$	−15.5895	−0.493725	−0.246863	+	0.969050i	$0.579400\pi$
$998$	0	0
$999$	25.2309	0.798272

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.j.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.j.1.2	✓	5	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-0.830554 q^{3} +3.93077 q^{5} +1.00000 q^{7} -2.31018 q^{9} +O(q^{10})\)	\(q-0.830554 q^{3} +3.93077 q^{5} +1.00000 q^{7} -2.31018 q^{9} -1.00000 q^{11} -1.00000 q^{13} -3.26471 q^{15} +7.93077 q^{17} +4.36492 q^{19} -0.830554 q^{21} +1.83550 q^{23} +10.4509 q^{25} +4.41039 q^{27} -4.44186 q^{29} +1.61564 q^{31} +0.830554 q^{33} +3.93077 q^{35} +5.72080 q^{37} +0.830554 q^{39} +4.67038 q^{41} -1.06429 q^{43} -9.08078 q^{45} +2.25544 q^{47} +1.00000 q^{49} -6.58693 q^{51} -3.24095 q^{53} -3.93077 q^{55} -3.62530 q^{57} -10.4485 q^{59} +1.90473 q^{61} -2.31018 q^{63} -3.93077 q^{65} -3.90700 q^{67} -1.52448 q^{69} +8.09094 q^{71} -8.62469 q^{73} -8.68005 q^{75} -1.00000 q^{77} -10.0672 q^{79} +3.26747 q^{81} -12.3767 q^{83} +31.1740 q^{85} +3.68921 q^{87} +0.0260316 q^{89} -1.00000 q^{91} -1.34188 q^{93} +17.1575 q^{95} -1.13579 q^{97} +2.31018 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9	\( 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9} - 5 q^{11} - 5 q^{13} - 5 q^{15} + 19 q^{17} - 5 q^{19} + q^{21} + 5 q^{23} + 12 q^{25} - 11 q^{27} - 17 q^{29} + 4 q^{31} - q^{33} - q^{35} + 10 q^{37} - q^{39} + 10 q^{41} - 25 q^{43} + q^{45} + 3 q^{47} + 5 q^{49} - q^{51} + 22 q^{53} + q^{55} + 16 q^{57} + 21 q^{59} + 26 q^{61} + 6 q^{63} + q^{65} + 28 q^{67} + 16 q^{69} + 28 q^{71} - 4 q^{73} - 5 q^{77} - 11 q^{79} + 5 q^{81} + 33 q^{85} + 31 q^{87} - 37 q^{89} - 5 q^{91} - 49 q^{93} + 29 q^{95} + 18 q^{97} - 6 q^{99}+O(q^{100}) \) 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9 - 5 * q^11 - 5 * q^13 - 5 * q^15 + 19 * q^17 - 5 * q^19 + q^21 + 5 * q^23 + 12 * q^25 - 11 * q^27 - 17 * q^29 + 4 * q^31 - q^33 - q^35 + 10 * q^37 - q^39 + 10 * q^41 - 25 * q^43 + q^45 + 3 * q^47 + 5 * q^49 - q^51 + 22 * q^53 + q^55 + 16 * q^57 + 21 * q^59 + 26 * q^61 + 6 * q^63 + q^65 + 28 * q^67 + 16 * q^69 + 28 * q^71 - 4 * q^73 - 5 * q^77 - 11 * q^79 + 5 * q^81 + 33 * q^85 + 31 * q^87 - 37 * q^89 - 5 * q^91 - 49 * q^93 + 29 * q^95 + 18 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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