LMFDB - Embedded newform 8008.2.a.r.1.8

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 9x^{7} + 36x^{6} + 23x^{5} - 89x^{4} - 20x^{3} + 51x^{2} + 18x + 1 $ x^9 - 4x^8 - 9x^7 + 36x^6 + 23x^5 - 89x^4 - 20x^3 + 51x^2 + 18x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.74591$ 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)

$f(q)$	$=$	$q+2.74591 q^{3} -3.50460 q^{5} -1.00000 q^{7} +4.54004 q^{9} +O(q^{10})$	$q+2.74591 q^{3} -3.50460 q^{5} -1.00000 q^{7} +4.54004 q^{9} -1.00000 q^{11} -1.00000 q^{13} -9.62334 q^{15} -0.0357892 q^{17} +4.00697 q^{19} -2.74591 q^{21} -1.32831 q^{23} +7.28225 q^{25} +4.22882 q^{27} -4.14782 q^{29} +3.90539 q^{31} -2.74591 q^{33} +3.50460 q^{35} +3.16583 q^{37} -2.74591 q^{39} +6.72476 q^{41} -9.33451 q^{43} -15.9111 q^{45} +7.15809 q^{47} +1.00000 q^{49} -0.0982741 q^{51} -0.562036 q^{53} +3.50460 q^{55} +11.0028 q^{57} +6.24243 q^{59} -9.28739 q^{61} -4.54004 q^{63} +3.50460 q^{65} +11.9068 q^{67} -3.64743 q^{69} -8.34463 q^{71} -8.58860 q^{73} +19.9964 q^{75} +1.00000 q^{77} -6.38210 q^{79} -2.00814 q^{81} +13.9193 q^{83} +0.125427 q^{85} -11.3896 q^{87} +12.4900 q^{89} +1.00000 q^{91} +10.7239 q^{93} -14.0428 q^{95} +3.88163 q^{97} -4.54004 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) $ 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9	$ 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) $ 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9 - 9 * q^11 - 9 * q^13 - 3 * q^15 - 7 * q^17 + 13 * q^19 - 5 * q^21 + 9 * q^23 - 2 * q^25 + 5 * q^27 + q^29 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 14 * q^37 - 5 * q^39 - 2 * q^41 + 5 * q^43 - 15 * q^45 + 13 * q^47 + 9 * q^49 - 3 * q^51 + 22 * q^53 - 3 * q^55 + 16 * q^57 + 43 * q^59 - 10 * q^61 - 8 * q^63 - 3 * q^65 + 26 * q^67 - 30 * q^69 + 18 * q^71 - 8 * q^73 + 28 * q^75 + 9 * q^77 - 9 * q^79 + 33 * q^81 + 4 * q^83 + 5 * q^85 + 33 * q^87 + 7 * q^89 + 9 * q^91 + 13 * q^93 + 7 * q^95 + 2 * q^97 - 8 * 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
$3$	2.74591	1.58535	0.792677	−	0.609642i	$-0.208687\pi$
$3$	2.74591	1.58535	0.792677	+	0.609642i	$0.208687\pi$
$4$	0	0
$5$	−3.50460	−1.56731	−0.783653	−	0.621198i	$-0.786646\pi$
$5$	−3.50460	−1.56731	−0.783653	+	0.621198i	$0.786646\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.54004	1.51335
$10$	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
$15$	−9.62334	−2.48474
$16$	0	0
$17$	−0.0357892	−0.00868016	−0.00434008	−	0.999991i	$-0.501381\pi$
$17$	−0.0357892	−0.00868016	−0.00434008	+	0.999991i	$0.501381\pi$
$18$	0	0
$19$	4.00697	0.919261	0.459631	−	0.888110i	$-0.347982\pi$
$19$	4.00697	0.919261	0.459631	+	0.888110i	$0.347982\pi$
$20$	0	0
$21$	−2.74591	−0.599208
$22$	0	0
$23$	−1.32831	−0.276972	−0.138486	−	0.990364i	$-0.544224\pi$
$23$	−1.32831	−0.276972	−0.138486	+	0.990364i	$0.544224\pi$
$24$	0	0
$25$	7.28225	1.45645
$26$	0	0
$27$	4.22882	0.813838
$28$	0	0
$29$	−4.14782	−0.770231	−0.385116	−	0.922868i	$-0.625838\pi$
$29$	−4.14782	−0.770231	−0.385116	+	0.922868i	$0.625838\pi$
$30$	0	0
$31$	3.90539	0.701429	0.350714	−	0.936482i	$-0.385939\pi$
$31$	3.90539	0.701429	0.350714	+	0.936482i	$0.385939\pi$
$32$	0	0
$33$	−2.74591	−0.478002
$34$	0	0
$35$	3.50460	0.592386
$36$	0	0
$37$	3.16583	0.520459	0.260230	−	0.965547i	$-0.416202\pi$
$37$	3.16583	0.520459	0.260230	+	0.965547i	$0.416202\pi$
$38$	0	0
$39$	−2.74591	−0.439698
$40$	0	0
$41$	6.72476	1.05023	0.525116	−	0.851031i	$-0.324022\pi$
$41$	6.72476	1.05023	0.525116	+	0.851031i	$0.324022\pi$
$42$	0	0
$43$	−9.33451	−1.42350	−0.711750	−	0.702433i	$-0.752097\pi$
$43$	−9.33451	−1.42350	−0.711750	+	0.702433i	$0.752097\pi$
$44$	0	0
$45$	−15.9111	−2.37188
$46$	0	0
$47$	7.15809	1.04411	0.522057	−	0.852910i	$-0.325165\pi$
$47$	7.15809	1.04411	0.522057	+	0.852910i	$0.325165\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.0982741	−0.0137611
$52$	0	0
$53$	−0.562036	−0.0772016	−0.0386008	−	0.999255i	$-0.512290\pi$
$53$	−0.562036	−0.0772016	−0.0386008	+	0.999255i	$0.512290\pi$
$54$	0	0
$55$	3.50460	0.472561
$56$	0	0
$57$	11.0028	1.45735
$58$	0	0
$59$	6.24243	0.812695	0.406348	−	0.913719i	$-0.366802\pi$
$59$	6.24243	0.812695	0.406348	+	0.913719i	$0.366802\pi$
$60$	0	0
$61$	−9.28739	−1.18913	−0.594564	−	0.804048i	$-0.702675\pi$
$61$	−9.28739	−1.18913	−0.594564	+	0.804048i	$0.702675\pi$
$62$	0	0
$63$	−4.54004	−0.571992
$64$	0	0
$65$	3.50460	0.434693
$66$	0	0
$67$	11.9068	1.45465	0.727325	−	0.686293i	$-0.240763\pi$
$67$	11.9068	1.45465	0.727325	+	0.686293i	$0.240763\pi$
$68$	0	0
$69$	−3.64743	−0.439099
$70$	0	0
$71$	−8.34463	−0.990325	−0.495162	−	0.868800i	$-0.664891\pi$
$71$	−8.34463	−0.990325	−0.495162	+	0.868800i	$0.664891\pi$
$72$	0	0
$73$	−8.58860	−1.00522	−0.502610	−	0.864513i	$-0.667627\pi$
$73$	−8.58860	−1.00522	−0.502610	+	0.864513i	$0.667627\pi$
$74$	0	0
$75$	19.9964	2.30899
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−6.38210	−0.718043	−0.359021	−	0.933329i	$-0.616890\pi$
$79$	−6.38210	−0.718043	−0.359021	+	0.933329i	$0.616890\pi$
$80$	0	0
$81$	−2.00814	−0.223127
$82$	0	0
$83$	13.9193	1.52784	0.763918	−	0.645313i	$-0.223273\pi$
$83$	13.9193	1.52784	0.763918	+	0.645313i	$0.223273\pi$
$84$	0	0
$85$	0.125427	0.0136045
$86$	0	0
$87$	−11.3896	−1.22109
$88$	0	0
$89$	12.4900	1.32394	0.661969	−	0.749531i	$-0.269721\pi$
$89$	12.4900	1.32394	0.661969	+	0.749531i	$0.269721\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	10.7239	1.11201
$94$	0	0
$95$	−14.0428	−1.44076
$96$	0	0
$97$	3.88163	0.394119	0.197060	−	0.980391i	$-0.436861\pi$
$97$	3.88163	0.394119	0.197060	+	0.980391i	$0.436861\pi$
$98$	0	0
$99$	−4.54004	−0.456291
$100$	0	0
$101$	7.65446	0.761648	0.380824	−	0.924648i	$-0.375640\pi$
$101$	7.65446	0.761648	0.380824	+	0.924648i	$0.375640\pi$
$102$	0	0
$103$	−9.02043	−0.888809	−0.444404	−	0.895826i	$-0.646585\pi$
$103$	−9.02043	−0.888809	−0.444404	+	0.895826i	$0.646585\pi$
$104$	0	0
$105$	9.62334	0.939142
$106$	0	0
$107$	7.46704	0.721866	0.360933	−	0.932592i	$-0.382458\pi$
$107$	7.46704	0.721866	0.360933	+	0.932592i	$0.382458\pi$
$108$	0	0
$109$	2.92719	0.280374	0.140187	−	0.990125i	$-0.455230\pi$
$109$	2.92719	0.280374	0.140187	+	0.990125i	$0.455230\pi$
$110$	0	0
$111$	8.69310	0.825112
$112$	0	0
$113$	3.12228	0.293720	0.146860	−	0.989157i	$-0.453083\pi$
$113$	3.12228	0.293720	0.146860	+	0.989157i	$0.453083\pi$
$114$	0	0
$115$	4.65521	0.434100
$116$	0	0
$117$	−4.54004	−0.419727
$118$	0	0
$119$	0.0357892	0.00328079
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	18.4656	1.66499
$124$	0	0
$125$	−7.99839	−0.715398
$126$	0	0
$127$	10.3939	0.922309	0.461155	−	0.887320i	$-0.347435\pi$
$127$	10.3939	0.922309	0.461155	+	0.887320i	$0.347435\pi$
$128$	0	0
$129$	−25.6318	−2.25675
$130$	0	0
$131$	13.6984	1.19684	0.598418	−	0.801184i	$-0.295796\pi$
$131$	13.6984	1.19684	0.598418	+	0.801184i	$0.295796\pi$
$132$	0	0
$133$	−4.00697	−0.347448
$134$	0	0
$135$	−14.8204	−1.27553
$136$	0	0
$137$	10.1842	0.870091	0.435046	−	0.900408i	$-0.356732\pi$
$137$	10.1842	0.870091	0.435046	+	0.900408i	$0.356732\pi$
$138$	0	0
$139$	6.71920	0.569915	0.284957	−	0.958540i	$-0.408021\pi$
$139$	6.71920	0.569915	0.284957	+	0.958540i	$0.408021\pi$
$140$	0	0
$141$	19.6555	1.65529
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	14.5365	1.20719
$146$	0	0
$147$	2.74591	0.226479
$148$	0	0
$149$	23.5288	1.92755	0.963777	−	0.266708i	$-0.0859360\pi$
$149$	23.5288	1.92755	0.963777	+	0.266708i	$0.0859360\pi$
$150$	0	0
$151$	−2.76796	−0.225253	−0.112627	−	0.993637i	$-0.535926\pi$
$151$	−2.76796	−0.225253	−0.112627	+	0.993637i	$0.535926\pi$
$152$	0	0
$153$	−0.162485	−0.0131361
$154$	0	0
$155$	−13.6868	−1.09935
$156$	0	0
$157$	11.5649	0.922976	0.461488	−	0.887147i	$-0.347316\pi$
$157$	11.5649	0.922976	0.461488	+	0.887147i	$0.347316\pi$
$158$	0	0
$159$	−1.54330	−0.122392
$160$	0	0
$161$	1.32831	0.104686
$162$	0	0
$163$	−12.1780	−0.953852	−0.476926	−	0.878943i	$-0.658249\pi$
$163$	−12.1780	−0.953852	−0.476926	+	0.878943i	$0.658249\pi$
$164$	0	0
$165$	9.62334	0.749176
$166$	0	0
$167$	−0.640564	−0.0495683	−0.0247842	−	0.999693i	$-0.507890\pi$
$167$	−0.640564	−0.0495683	−0.0247842	+	0.999693i	$0.507890\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	18.1918	1.39116
$172$	0	0
$173$	22.3872	1.70207	0.851034	−	0.525110i	$-0.175976\pi$
$173$	22.3872	1.70207	0.851034	+	0.525110i	$0.175976\pi$
$174$	0	0
$175$	−7.28225	−0.550487
$176$	0	0
$177$	17.1412	1.28841
$178$	0	0
$179$	3.04087	0.227285	0.113642	−	0.993522i	$-0.463748\pi$
$179$	3.04087	0.227285	0.113642	+	0.993522i	$0.463748\pi$
$180$	0	0
$181$	13.7571	1.02256	0.511279	−	0.859415i	$-0.329172\pi$
$181$	13.7571	1.02256	0.511279	+	0.859415i	$0.329172\pi$
$182$	0	0
$183$	−25.5024	−1.88519
$184$	0	0
$185$	−11.0950	−0.815719
$186$	0	0
$187$	0.0357892	0.00261717
$188$	0	0
$189$	−4.22882	−0.307602
$190$	0	0
$191$	12.4926	0.903935	0.451967	−	0.892034i	$-0.350722\pi$
$191$	12.4926	0.903935	0.451967	+	0.892034i	$0.350722\pi$
$192$	0	0
$193$	−20.6804	−1.48861	−0.744305	−	0.667840i	$-0.767219\pi$
$193$	−20.6804	−1.48861	−0.744305	+	0.667840i	$0.767219\pi$
$194$	0	0
$195$	9.62334	0.689142
$196$	0	0
$197$	4.98929	0.355472	0.177736	−	0.984078i	$-0.443123\pi$
$197$	4.98929	0.355472	0.177736	+	0.984078i	$0.443123\pi$
$198$	0	0
$199$	3.54355	0.251196	0.125598	−	0.992081i	$-0.459915\pi$
$199$	3.54355	0.251196	0.125598	+	0.992081i	$0.459915\pi$
$200$	0	0
$201$	32.6951	2.30614
$202$	0	0
$203$	4.14782	0.291120
$204$	0	0
$205$	−23.5676	−1.64603
$206$	0	0
$207$	−6.03059	−0.419155
$208$	0	0
$209$	−4.00697	−0.277168
$210$	0	0
$211$	7.93329	0.546150	0.273075	−	0.961993i	$-0.411959\pi$
$211$	7.93329	0.546150	0.273075	+	0.961993i	$0.411959\pi$
$212$	0	0
$213$	−22.9136	−1.57002
$214$	0	0
$215$	32.7138	2.23106
$216$	0	0
$217$	−3.90539	−0.265115
$218$	0	0
$219$	−23.5836	−1.59363
$220$	0	0
$221$	0.0357892	0.00240744
$222$	0	0
$223$	0.836208	0.0559966	0.0279983	−	0.999608i	$-0.491087\pi$
$223$	0.836208	0.0559966	0.0279983	+	0.999608i	$0.491087\pi$
$224$	0	0
$225$	33.0617	2.20412
$226$	0	0
$227$	−18.9990	−1.26101	−0.630505	−	0.776185i	$-0.717152\pi$
$227$	−18.9990	−1.26101	−0.630505	+	0.776185i	$0.717152\pi$
$228$	0	0
$229$	22.0967	1.46019	0.730096	−	0.683344i	$-0.239475\pi$
$229$	22.0967	1.46019	0.730096	+	0.683344i	$0.239475\pi$
$230$	0	0
$231$	2.74591	0.180668
$232$	0	0
$233$	28.0088	1.83492	0.917458	−	0.397833i	$-0.130238\pi$
$233$	28.0088	1.83492	0.917458	+	0.397833i	$0.130238\pi$
$234$	0	0
$235$	−25.0863	−1.63645
$236$	0	0
$237$	−17.5247	−1.13835
$238$	0	0
$239$	5.40256	0.349463	0.174731	−	0.984616i	$-0.444094\pi$
$239$	5.40256	0.349463	0.174731	+	0.984616i	$0.444094\pi$
$240$	0	0
$241$	−8.20329	−0.528420	−0.264210	−	0.964465i	$-0.585111\pi$
$241$	−8.20329	−0.528420	−0.264210	+	0.964465i	$0.585111\pi$
$242$	0	0
$243$	−18.2007	−1.16757
$244$	0	0
$245$	−3.50460	−0.223901
$246$	0	0
$247$	−4.00697	−0.254957
$248$	0	0
$249$	38.2211	2.42216
$250$	0	0
$251$	−4.88822	−0.308542	−0.154271	−	0.988029i	$-0.549303\pi$
$251$	−4.88822	−0.308542	−0.154271	+	0.988029i	$0.549303\pi$
$252$	0	0
$253$	1.32831	0.0835103
$254$	0	0
$255$	0.344412	0.0215679
$256$	0	0
$257$	−3.05914	−0.190824	−0.0954120	−	0.995438i	$-0.530417\pi$
$257$	−3.05914	−0.190824	−0.0954120	+	0.995438i	$0.530417\pi$
$258$	0	0
$259$	−3.16583	−0.196715
$260$	0	0
$261$	−18.8313	−1.16563
$262$	0	0
$263$	7.62162	0.469969	0.234984	−	0.971999i	$-0.424496\pi$
$263$	7.62162	0.469969	0.234984	+	0.971999i	$0.424496\pi$
$264$	0	0
$265$	1.96972	0.120999
$266$	0	0
$267$	34.2965	2.09891
$268$	0	0
$269$	−26.5045	−1.61601	−0.808003	−	0.589179i	$-0.799451\pi$
$269$	−26.5045	−1.61601	−0.808003	+	0.589179i	$0.799451\pi$
$270$	0	0
$271$	−7.03505	−0.427349	−0.213674	−	0.976905i	$-0.568543\pi$
$271$	−7.03505	−0.427349	−0.213674	+	0.976905i	$0.568543\pi$
$272$	0	0
$273$	2.74591	0.166190
$274$	0	0
$275$	−7.28225	−0.439136
$276$	0	0
$277$	−11.5179	−0.692043	−0.346021	−	0.938227i	$-0.612468\pi$
$277$	−11.5179	−0.692043	−0.346021	+	0.938227i	$0.612468\pi$
$278$	0	0
$279$	17.7306	1.06151
$280$	0	0
$281$	24.8614	1.48311	0.741553	−	0.670894i	$-0.234090\pi$
$281$	24.8614	1.48311	0.741553	+	0.670894i	$0.234090\pi$
$282$	0	0
$283$	−3.40575	−0.202451	−0.101225	−	0.994864i	$-0.532276\pi$
$283$	−3.40575	−0.202451	−0.101225	+	0.994864i	$0.532276\pi$
$284$	0	0
$285$	−38.5604	−2.28412
$286$	0	0
$287$	−6.72476	−0.396950
$288$	0	0
$289$	−16.9987	−0.999925
$290$	0	0
$291$	10.6586	0.624819
$292$	0	0
$293$	20.2290	1.18179	0.590896	−	0.806748i	$-0.298775\pi$
$293$	20.2290	1.18179	0.590896	+	0.806748i	$0.298775\pi$
$294$	0	0
$295$	−21.8773	−1.27374
$296$	0	0
$297$	−4.22882	−0.245381
$298$	0	0
$299$	1.32831	0.0768183
$300$	0	0
$301$	9.33451	0.538032
$302$	0	0
$303$	21.0185	1.20748
$304$	0	0
$305$	32.5486	1.86373
$306$	0	0
$307$	−26.4592	−1.51010	−0.755052	−	0.655665i	$-0.772389\pi$
$307$	−26.4592	−1.51010	−0.755052	+	0.655665i	$0.772389\pi$
$308$	0	0
$309$	−24.7693	−1.40908
$310$	0	0
$311$	−33.6688	−1.90918	−0.954592	−	0.297917i	$-0.903708\pi$
$311$	−33.6688	−1.90918	−0.954592	+	0.297917i	$0.903708\pi$
$312$	0	0
$313$	−12.2712	−0.693612	−0.346806	−	0.937937i	$-0.612734\pi$
$313$	−12.2712	−0.693612	−0.346806	+	0.937937i	$0.612734\pi$
$314$	0	0
$315$	15.9111	0.896486
$316$	0	0
$317$	14.3552	0.806270	0.403135	−	0.915141i	$-0.367921\pi$
$317$	14.3552	0.806270	0.403135	+	0.915141i	$0.367921\pi$
$318$	0	0
$319$	4.14782	0.232233
$320$	0	0
$321$	20.5039	1.14441
$322$	0	0
$323$	−0.143406	−0.00797934
$324$	0	0
$325$	−7.28225	−0.403947
$326$	0	0
$327$	8.03780	0.444491
$328$	0	0
$329$	−7.15809	−0.394638
$330$	0	0
$331$	17.1409	0.942151	0.471075	−	0.882093i	$-0.343866\pi$
$331$	17.1409	0.942151	0.471075	+	0.882093i	$0.343866\pi$
$332$	0	0
$333$	14.3730	0.787636
$334$	0	0
$335$	−41.7287	−2.27988
$336$	0	0
$337$	9.69109	0.527907	0.263954	−	0.964535i	$-0.414973\pi$
$337$	9.69109	0.527907	0.263954	+	0.964535i	$0.414973\pi$
$338$	0	0
$339$	8.57352	0.465650
$340$	0	0
$341$	−3.90539	−0.211489
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	12.7828	0.688203
$346$	0	0
$347$	15.9643	0.857008	0.428504	−	0.903540i	$-0.359041\pi$
$347$	15.9643	0.857008	0.428504	+	0.903540i	$0.359041\pi$
$348$	0	0
$349$	−20.3023	−1.08676	−0.543379	−	0.839487i	$-0.682855\pi$
$349$	−20.3023	−1.08676	−0.543379	+	0.839487i	$0.682855\pi$
$350$	0	0
$351$	−4.22882	−0.225718
$352$	0	0
$353$	22.0121	1.17159	0.585794	−	0.810460i	$-0.300783\pi$
$353$	22.0121	1.17159	0.585794	+	0.810460i	$0.300783\pi$
$354$	0	0
$355$	29.2446	1.55214
$356$	0	0
$357$	0.0982741	0.00520122
$358$	0	0
$359$	−28.4790	−1.50307	−0.751533	−	0.659695i	$-0.770685\pi$
$359$	−28.4790	−1.50307	−0.751533	+	0.659695i	$0.770685\pi$
$360$	0	0
$361$	−2.94422	−0.154959
$362$	0	0
$363$	2.74591	0.144123
$364$	0	0
$365$	30.0996	1.57549
$366$	0	0
$367$	8.43315	0.440207	0.220103	−	0.975477i	$-0.429361\pi$
$367$	8.43315	0.440207	0.220103	+	0.975477i	$0.429361\pi$
$368$	0	0
$369$	30.5307	1.58936
$370$	0	0
$371$	0.562036	0.0291795
$372$	0	0
$373$	−0.181268	−0.00938571	−0.00469285	−	0.999989i	$-0.501494\pi$
$373$	−0.181268	−0.00938571	−0.00469285	+	0.999989i	$0.501494\pi$
$374$	0	0
$375$	−21.9629	−1.13416
$376$	0	0
$377$	4.14782	0.213624
$378$	0	0
$379$	−6.69113	−0.343700	−0.171850	−	0.985123i	$-0.554974\pi$
$379$	−6.69113	−0.343700	−0.171850	+	0.985123i	$0.554974\pi$
$380$	0	0
$381$	28.5407	1.46219
$382$	0	0
$383$	12.6428	0.646017	0.323009	−	0.946396i	$-0.395306\pi$
$383$	12.6428	0.646017	0.323009	+	0.946396i	$0.395306\pi$
$384$	0	0
$385$	−3.50460	−0.178611
$386$	0	0
$387$	−42.3791	−2.15425
$388$	0	0
$389$	22.7035	1.15111	0.575556	−	0.817762i	$-0.304786\pi$
$389$	22.7035	1.15111	0.575556	+	0.817762i	$0.304786\pi$
$390$	0	0
$391$	0.0475393	0.00240416
$392$	0	0
$393$	37.6146	1.89741
$394$	0	0
$395$	22.3668	1.12539
$396$	0	0
$397$	12.3905	0.621861	0.310930	−	0.950433i	$-0.399359\pi$
$397$	12.3905	0.621861	0.310930	+	0.950433i	$0.399359\pi$
$398$	0	0
$399$	−11.0028	−0.550828
$400$	0	0
$401$	−28.6846	−1.43244	−0.716221	−	0.697874i	$-0.754130\pi$
$401$	−28.6846	−1.43244	−0.716221	+	0.697874i	$0.754130\pi$
$402$	0	0
$403$	−3.90539	−0.194541
$404$	0	0
$405$	7.03774	0.349708
$406$	0	0
$407$	−3.16583	−0.156924
$408$	0	0
$409$	28.4847	1.40848	0.704239	−	0.709963i	$-0.251288\pi$
$409$	28.4847	1.40848	0.704239	+	0.709963i	$0.251288\pi$
$410$	0	0
$411$	27.9648	1.37940
$412$	0	0
$413$	−6.24243	−0.307170
$414$	0	0
$415$	−48.7815	−2.39459
$416$	0	0
$417$	18.4503	0.903517
$418$	0	0
$419$	1.53373	0.0749275	0.0374638	−	0.999298i	$-0.488072\pi$
$419$	1.53373	0.0749275	0.0374638	+	0.999298i	$0.488072\pi$
$420$	0	0
$421$	22.8659	1.11442	0.557208	−	0.830373i	$-0.311873\pi$
$421$	22.8659	1.11442	0.557208	+	0.830373i	$0.311873\pi$
$422$	0	0
$423$	32.4980	1.58011
$424$	0	0
$425$	−0.260626	−0.0126422
$426$	0	0
$427$	9.28739	0.449448
$428$	0	0
$429$	2.74591	0.132574
$430$	0	0
$431$	−16.1416	−0.777511	−0.388756	−	0.921341i	$-0.627095\pi$
$431$	−16.1416	−0.777511	−0.388756	+	0.921341i	$0.627095\pi$
$432$	0	0
$433$	−14.9813	−0.719957	−0.359978	−	0.932961i	$-0.617216\pi$
$433$	−14.9813	−0.719957	−0.359978	+	0.932961i	$0.617216\pi$
$434$	0	0
$435$	39.9159	1.91382
$436$	0	0
$437$	−5.32250	−0.254610
$438$	0	0
$439$	−2.53714	−0.121091	−0.0605455	−	0.998165i	$-0.519284\pi$
$439$	−2.53714	−0.121091	−0.0605455	+	0.998165i	$0.519284\pi$
$440$	0	0
$441$	4.54004	0.216193
$442$	0	0
$443$	23.9612	1.13843	0.569216	−	0.822188i	$-0.307247\pi$
$443$	23.9612	1.13843	0.569216	+	0.822188i	$0.307247\pi$
$444$	0	0
$445$	−43.7726	−2.07502
$446$	0	0
$447$	64.6081	3.05586
$448$	0	0
$449$	12.5696	0.593195	0.296598	−	0.955003i	$-0.404148\pi$
$449$	12.5696	0.593195	0.296598	+	0.955003i	$0.404148\pi$
$450$	0	0
$451$	−6.72476	−0.316657
$452$	0	0
$453$	−7.60058	−0.357106
$454$	0	0
$455$	−3.50460	−0.164298
$456$	0	0
$457$	20.6714	0.966968	0.483484	−	0.875353i	$-0.339371\pi$
$457$	20.6714	0.966968	0.483484	+	0.875353i	$0.339371\pi$
$458$	0	0
$459$	−0.151346	−0.00706424
$460$	0	0
$461$	−37.1682	−1.73110	−0.865548	−	0.500825i	$-0.833030\pi$
$461$	−37.1682	−1.73110	−0.865548	+	0.500825i	$0.833030\pi$
$462$	0	0
$463$	−8.08516	−0.375749	−0.187875	−	0.982193i	$-0.560160\pi$
$463$	−8.08516	−0.375749	−0.187875	+	0.982193i	$0.560160\pi$
$464$	0	0
$465$	−37.5829	−1.74287
$466$	0	0
$467$	13.1550	0.608740	0.304370	−	0.952554i	$-0.401554\pi$
$467$	13.1550	0.608740	0.304370	+	0.952554i	$0.401554\pi$
$468$	0	0
$469$	−11.9068	−0.549806
$470$	0	0
$471$	31.7561	1.46324
$472$	0	0
$473$	9.33451	0.429201
$474$	0	0
$475$	29.1797	1.33886
$476$	0	0
$477$	−2.55167	−0.116833
$478$	0	0
$479$	−12.6244	−0.576825	−0.288412	−	0.957506i	$-0.593127\pi$
$479$	−12.6244	−0.576825	−0.288412	+	0.957506i	$0.593127\pi$
$480$	0	0
$481$	−3.16583	−0.144349
$482$	0	0
$483$	3.64743	0.165964
$484$	0	0
$485$	−13.6036	−0.617706
$486$	0	0
$487$	24.2427	1.09854	0.549271	−	0.835644i	$-0.314905\pi$
$487$	24.2427	1.09854	0.549271	+	0.835644i	$0.314905\pi$
$488$	0	0
$489$	−33.4396	−1.51219
$490$	0	0
$491$	−14.1285	−0.637610	−0.318805	−	0.947820i	$-0.603282\pi$
$491$	−14.1285	−0.637610	−0.318805	+	0.947820i	$0.603282\pi$
$492$	0	0
$493$	0.148447	0.00668573
$494$	0	0
$495$	15.9111	0.715149
$496$	0	0
$497$	8.34463	0.374308
$498$	0	0
$499$	23.8730	1.06870	0.534351	−	0.845263i	$-0.320556\pi$
$499$	23.8730	1.06870	0.534351	+	0.845263i	$0.320556\pi$
$500$	0	0
$501$	−1.75893	−0.0785833
$502$	0	0
$503$	−32.8704	−1.46562	−0.732810	−	0.680433i	$-0.761792\pi$
$503$	−32.8704	−1.46562	−0.732810	+	0.680433i	$0.761792\pi$
$504$	0	0
$505$	−26.8259	−1.19374
$506$	0	0
$507$	2.74591	0.121950
$508$	0	0
$509$	3.32047	0.147177	0.0735885	−	0.997289i	$-0.476555\pi$
$509$	3.32047	0.147177	0.0735885	+	0.997289i	$0.476555\pi$
$510$	0	0
$511$	8.58860	0.379937
$512$	0	0
$513$	16.9448	0.748129
$514$	0	0
$515$	31.6130	1.39304
$516$	0	0
$517$	−7.15809	−0.314812
$518$	0	0
$519$	61.4734	2.69838
$520$	0	0
$521$	−13.3534	−0.585024	−0.292512	−	0.956262i	$-0.594491\pi$
$521$	−13.3534	−0.585024	−0.292512	+	0.956262i	$0.594491\pi$
$522$	0	0
$523$	39.1402	1.71148	0.855741	−	0.517404i	$-0.173102\pi$
$523$	39.1402	1.71148	0.855741	+	0.517404i	$0.173102\pi$
$524$	0	0
$525$	−19.9964	−0.872716
$526$	0	0
$527$	−0.139771	−0.00608851
$528$	0	0
$529$	−21.2356	−0.923286
$530$	0	0
$531$	28.3409	1.22989
$532$	0	0
$533$	−6.72476	−0.291282
$534$	0	0
$535$	−26.1690	−1.13139
$536$	0	0
$537$	8.34995	0.360327
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−27.0513	−1.16303	−0.581514	−	0.813537i	$-0.697539\pi$
$541$	−27.0513	−1.16303	−0.581514	+	0.813537i	$0.697539\pi$
$542$	0	0
$543$	37.7758	1.62112
$544$	0	0
$545$	−10.2586	−0.439431
$546$	0	0
$547$	3.69772	0.158103	0.0790516	−	0.996871i	$-0.474811\pi$
$547$	3.69772	0.158103	0.0790516	+	0.996871i	$0.474811\pi$
$548$	0	0
$549$	−42.1651	−1.79956
$550$	0	0
$551$	−16.6202	−0.708044
$552$	0	0
$553$	6.38210	0.271395
$554$	0	0
$555$	−30.4659	−1.29320
$556$	0	0
$557$	−2.44610	−0.103645	−0.0518223	−	0.998656i	$-0.516503\pi$
$557$	−2.44610	−0.103645	−0.0518223	+	0.998656i	$0.516503\pi$
$558$	0	0
$559$	9.33451	0.394808
$560$	0	0
$561$	0.0982741	0.00414914
$562$	0	0
$563$	−30.2372	−1.27434	−0.637172	−	0.770722i	$-0.719896\pi$
$563$	−30.2372	−1.27434	−0.637172	+	0.770722i	$0.719896\pi$
$564$	0	0
$565$	−10.9424	−0.460349
$566$	0	0
$567$	2.00814	0.0843340
$568$	0	0
$569$	−17.4107	−0.729893	−0.364946	−	0.931029i	$-0.618913\pi$
$569$	−17.4107	−0.729893	−0.364946	+	0.931029i	$0.618913\pi$
$570$	0	0
$571$	10.8212	0.452854	0.226427	−	0.974028i	$-0.427296\pi$
$571$	10.8212	0.452854	0.226427	+	0.974028i	$0.427296\pi$
$572$	0	0
$573$	34.3037	1.43306
$574$	0	0
$575$	−9.67310	−0.403396
$576$	0	0
$577$	−20.9833	−0.873544	−0.436772	−	0.899572i	$-0.643878\pi$
$577$	−20.9833	−0.873544	−0.436772	+	0.899572i	$0.643878\pi$
$578$	0	0
$579$	−56.7867	−2.35997
$580$	0	0
$581$	−13.9193	−0.577468
$582$	0	0
$583$	0.562036	0.0232772
$584$	0	0
$585$	15.9111	0.657841
$586$	0	0
$587$	13.2428	0.546590	0.273295	−	0.961930i	$-0.411887\pi$
$587$	13.2428	0.546590	0.273295	+	0.961930i	$0.411887\pi$
$588$	0	0
$589$	15.6488	0.644796
$590$	0	0
$591$	13.7002	0.563550
$592$	0	0
$593$	−16.1402	−0.662798	−0.331399	−	0.943491i	$-0.607521\pi$
$593$	−16.1402	−0.662798	−0.331399	+	0.943491i	$0.607521\pi$
$594$	0	0
$595$	−0.125427	−0.00514201
$596$	0	0
$597$	9.73029	0.398234
$598$	0	0
$599$	48.3677	1.97625	0.988126	−	0.153644i	$-0.0491010\pi$
$599$	48.3677	1.97625	0.988126	+	0.153644i	$0.0491010\pi$
$600$	0	0
$601$	12.3247	0.502734	0.251367	−	0.967892i	$-0.419120\pi$
$601$	12.3247	0.502734	0.251367	+	0.967892i	$0.419120\pi$
$602$	0	0
$603$	54.0575	2.20139
$604$	0	0
$605$	−3.50460	−0.142482
$606$	0	0
$607$	−23.5260	−0.954890	−0.477445	−	0.878662i	$-0.658437\pi$
$607$	−23.5260	−0.954890	−0.477445	+	0.878662i	$0.658437\pi$
$608$	0	0
$609$	11.3896	0.461528
$610$	0	0
$611$	−7.15809	−0.289585
$612$	0	0
$613$	0.497515	0.0200944	0.0100472	−	0.999950i	$-0.496802\pi$
$613$	0.497515	0.0200944	0.0100472	+	0.999950i	$0.496802\pi$
$614$	0	0
$615$	−64.7147	−2.60955
$616$	0	0
$617$	22.2593	0.896125	0.448063	−	0.894002i	$-0.352114\pi$
$617$	22.2593	0.896125	0.448063	+	0.894002i	$0.352114\pi$
$618$	0	0
$619$	−20.3289	−0.817086	−0.408543	−	0.912739i	$-0.633963\pi$
$619$	−20.3289	−0.817086	−0.408543	+	0.912739i	$0.633963\pi$
$620$	0	0
$621$	−5.61720	−0.225410
$622$	0	0
$623$	−12.4900	−0.500402
$624$	0	0
$625$	−8.38005	−0.335202
$626$	0	0
$627$	−11.0028	−0.439409
$628$	0	0
$629$	−0.113303	−0.00451767
$630$	0	0
$631$	4.46703	0.177830	0.0889148	−	0.996039i	$-0.471660\pi$
$631$	4.46703	0.177830	0.0889148	+	0.996039i	$0.471660\pi$
$632$	0	0
$633$	21.7841	0.865842
$634$	0	0
$635$	−36.4265	−1.44554
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−37.8850	−1.49871
$640$	0	0
$641$	−8.44709	−0.333640	−0.166820	−	0.985987i	$-0.553350\pi$
$641$	−8.44709	−0.333640	−0.166820	+	0.985987i	$0.553350\pi$
$642$	0	0
$643$	−20.7503	−0.818313	−0.409157	−	0.912464i	$-0.634177\pi$
$643$	−20.7503	−0.818313	−0.409157	+	0.912464i	$0.634177\pi$
$644$	0	0
$645$	89.8292	3.53702
$646$	0	0
$647$	−21.8278	−0.858139	−0.429070	−	0.903271i	$-0.641158\pi$
$647$	−21.8278	−0.858139	−0.429070	+	0.903271i	$0.641158\pi$
$648$	0	0
$649$	−6.24243	−0.245037
$650$	0	0
$651$	−10.7239	−0.420301
$652$	0	0
$653$	−49.6344	−1.94235	−0.971173	−	0.238377i	$-0.923384\pi$
$653$	−49.6344	−1.94235	−0.971173	+	0.238377i	$0.923384\pi$
$654$	0	0
$655$	−48.0075	−1.87581
$656$	0	0
$657$	−38.9926	−1.52125
$658$	0	0
$659$	8.67552	0.337950	0.168975	−	0.985620i	$-0.445954\pi$
$659$	8.67552	0.337950	0.168975	+	0.985620i	$0.445954\pi$
$660$	0	0
$661$	−8.59309	−0.334232	−0.167116	−	0.985937i	$-0.553446\pi$
$661$	−8.59309	−0.334232	−0.167116	+	0.985937i	$0.553446\pi$
$662$	0	0
$663$	0.0982741	0.00381665
$664$	0	0
$665$	14.0428	0.544558
$666$	0	0
$667$	5.50960	0.213333
$668$	0	0
$669$	2.29615	0.0887745
$670$	0	0
$671$	9.28739	0.358536
$672$	0	0
$673$	−3.07972	−0.118714	−0.0593572	−	0.998237i	$-0.518905\pi$
$673$	−3.07972	−0.118714	−0.0593572	+	0.998237i	$0.518905\pi$
$674$	0	0
$675$	30.7954	1.18531
$676$	0	0
$677$	−29.0505	−1.11650	−0.558251	−	0.829672i	$-0.688527\pi$
$677$	−29.0505	−1.11650	−0.558251	+	0.829672i	$0.688527\pi$
$678$	0	0
$679$	−3.88163	−0.148963
$680$	0	0
$681$	−52.1697	−1.99915
$682$	0	0
$683$	39.5813	1.51454	0.757268	−	0.653104i	$-0.226534\pi$
$683$	39.5813	1.51454	0.757268	+	0.653104i	$0.226534\pi$
$684$	0	0
$685$	−35.6914	−1.36370
$686$	0	0
$687$	60.6757	2.31492
$688$	0	0
$689$	0.562036	0.0214119
$690$	0	0
$691$	28.0474	1.06697	0.533486	−	0.845809i	$-0.320882\pi$
$691$	28.0474	1.06697	0.533486	+	0.845809i	$0.320882\pi$
$692$	0	0
$693$	4.54004	0.172462
$694$	0	0
$695$	−23.5481	−0.893232
$696$	0	0
$697$	−0.240674	−0.00911618
$698$	0	0
$699$	76.9097	2.90899
$700$	0	0
$701$	15.6272	0.590232	0.295116	−	0.955461i	$-0.404642\pi$
$701$	15.6272	0.590232	0.295116	+	0.955461i	$0.404642\pi$
$702$	0	0
$703$	12.6854	0.478438
$704$	0	0
$705$	−68.8847	−2.59435
$706$	0	0
$707$	−7.65446	−0.287876
$708$	0	0
$709$	11.7004	0.439419	0.219709	−	0.975565i	$-0.429489\pi$
$709$	11.7004	0.439419	0.219709	+	0.975565i	$0.429489\pi$
$710$	0	0
$711$	−28.9750	−1.08665
$712$	0	0
$713$	−5.18758	−0.194276
$714$	0	0
$715$	−3.50460	−0.131065
$716$	0	0
$717$	14.8350	0.554022
$718$	0	0
$719$	19.6585	0.733137	0.366569	−	0.930391i	$-0.380532\pi$
$719$	19.6585	0.733137	0.366569	+	0.930391i	$0.380532\pi$
$720$	0	0
$721$	9.02043	0.335938
$722$	0	0
$723$	−22.5255	−0.837733
$724$	0	0
$725$	−30.2055	−1.12180
$726$	0	0
$727$	10.5410	0.390943	0.195471	−	0.980709i	$-0.437376\pi$
$727$	10.5410	0.390943	0.195471	+	0.980709i	$0.437376\pi$
$728$	0	0
$729$	−43.9530	−1.62789
$730$	0	0
$731$	0.334075	0.0123562
$732$	0	0
$733$	−23.5004	−0.868006	−0.434003	−	0.900911i	$-0.642899\pi$
$733$	−23.5004	−0.868006	−0.434003	+	0.900911i	$0.642899\pi$
$734$	0	0
$735$	−9.62334	−0.354962
$736$	0	0
$737$	−11.9068	−0.438594
$738$	0	0
$739$	−13.0019	−0.478282	−0.239141	−	0.970985i	$-0.576866\pi$
$739$	−13.0019	−0.478282	−0.239141	+	0.970985i	$0.576866\pi$
$740$	0	0
$741$	−11.0028	−0.404197
$742$	0	0
$743$	28.8486	1.05835	0.529175	−	0.848512i	$-0.322501\pi$
$743$	28.8486	1.05835	0.529175	+	0.848512i	$0.322501\pi$
$744$	0	0
$745$	−82.4592	−3.02107
$746$	0	0
$747$	63.1940	2.31215
$748$	0	0
$749$	−7.46704	−0.272840
$750$	0	0
$751$	−32.0306	−1.16881	−0.584407	−	0.811460i	$-0.698673\pi$
$751$	−32.0306	−1.16881	−0.584407	+	0.811460i	$0.698673\pi$
$752$	0	0
$753$	−13.4226	−0.489148
$754$	0	0
$755$	9.70060	0.353041
$756$	0	0
$757$	45.3189	1.64714	0.823572	−	0.567211i	$-0.191978\pi$
$757$	45.3189	1.64714	0.823572	+	0.567211i	$0.191978\pi$
$758$	0	0
$759$	3.64743	0.132393
$760$	0	0
$761$	−46.6952	−1.69270	−0.846350	−	0.532628i	$-0.821205\pi$
$761$	−46.6952	−1.69270	−0.846350	+	0.532628i	$0.821205\pi$
$762$	0	0
$763$	−2.92719	−0.105971
$764$	0	0
$765$	0.569444	0.0205883
$766$	0	0
$767$	−6.24243	−0.225401
$768$	0	0
$769$	−16.6527	−0.600511	−0.300255	−	0.953859i	$-0.597072\pi$
$769$	−16.6527	−0.600511	−0.300255	+	0.953859i	$0.597072\pi$
$770$	0	0
$771$	−8.40014	−0.302524
$772$	0	0
$773$	9.91368	0.356570	0.178285	−	0.983979i	$-0.442945\pi$
$773$	9.91368	0.356570	0.178285	+	0.983979i	$0.442945\pi$
$774$	0	0
$775$	28.4400	1.02160
$776$	0	0
$777$	−8.69310	−0.311863
$778$	0	0
$779$	26.9459	0.965437
$780$	0	0
$781$	8.34463	0.298594
$782$	0	0
$783$	−17.5404	−0.626843
$784$	0	0
$785$	−40.5302	−1.44659
$786$	0	0
$787$	−14.1895	−0.505800	−0.252900	−	0.967492i	$-0.581384\pi$
$787$	−14.1895	−0.505800	−0.252900	+	0.967492i	$0.581384\pi$
$788$	0	0
$789$	20.9283	0.745067
$790$	0	0
$791$	−3.12228	−0.111016
$792$	0	0
$793$	9.28739	0.329805
$794$	0	0
$795$	5.40867	0.191826
$796$	0	0
$797$	−49.4472	−1.75151	−0.875754	−	0.482757i	$-0.839636\pi$
$797$	−49.4472	−1.75151	−0.875754	+	0.482757i	$0.839636\pi$
$798$	0	0
$799$	−0.256182	−0.00906308
$800$	0	0
$801$	56.7052	2.00358
$802$	0	0
$803$	8.58860	0.303085
$804$	0	0
$805$	−4.65521	−0.164075
$806$	0	0
$807$	−72.7790	−2.56194
$808$	0	0
$809$	−8.43721	−0.296636	−0.148318	−	0.988940i	$-0.547386\pi$
$809$	−8.43721	−0.296636	−0.148318	+	0.988940i	$0.547386\pi$
$810$	0	0
$811$	15.8260	0.555725	0.277862	−	0.960621i	$-0.410374\pi$
$811$	15.8260	0.555725	0.277862	+	0.960621i	$0.410374\pi$
$812$	0	0
$813$	−19.3176	−0.677499
$814$	0	0
$815$	42.6790	1.49498
$816$	0	0
$817$	−37.4031	−1.30857
$818$	0	0
$819$	4.54004	0.158642
$820$	0	0
$821$	8.17054	0.285154	0.142577	−	0.989784i	$-0.454461\pi$
$821$	8.17054	0.285154	0.142577	+	0.989784i	$0.454461\pi$
$822$	0	0
$823$	17.3209	0.603769	0.301885	−	0.953344i	$-0.402384\pi$
$823$	17.3209	0.603769	0.301885	+	0.953344i	$0.402384\pi$
$824$	0	0
$825$	−19.9964	−0.696187
$826$	0	0
$827$	−47.6994	−1.65867	−0.829336	−	0.558750i	$-0.811281\pi$
$827$	−47.6994	−1.65867	−0.829336	+	0.558750i	$0.811281\pi$
$828$	0	0
$829$	−45.5678	−1.58263	−0.791317	−	0.611406i	$-0.790604\pi$
$829$	−45.5678	−1.58263	−0.791317	+	0.611406i	$0.790604\pi$
$830$	0	0
$831$	−31.6271	−1.09713
$832$	0	0
$833$	−0.0357892	−0.00124002
$834$	0	0
$835$	2.24492	0.0776887
$836$	0	0
$837$	16.5152	0.570849
$838$	0	0
$839$	−26.1846	−0.903994	−0.451997	−	0.892019i	$-0.649288\pi$
$839$	−26.1846	−0.903994	−0.451997	+	0.892019i	$0.649288\pi$
$840$	0	0
$841$	−11.7956	−0.406744
$842$	0	0
$843$	68.2672	2.35125
$844$	0	0
$845$	−3.50460	−0.120562
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−9.35189	−0.320956
$850$	0	0
$851$	−4.20521	−0.144153
$852$	0	0
$853$	9.42470	0.322696	0.161348	−	0.986898i	$-0.448416\pi$
$853$	9.42470	0.322696	0.161348	+	0.986898i	$0.448416\pi$
$854$	0	0
$855$	−63.7551	−2.18038
$856$	0	0
$857$	−56.4899	−1.92966	−0.964829	−	0.262878i	$-0.915328\pi$
$857$	−56.4899	−1.92966	−0.964829	+	0.262878i	$0.915328\pi$
$858$	0	0
$859$	6.41818	0.218986	0.109493	−	0.993988i	$-0.465077\pi$
$859$	6.41818	0.218986	0.109493	+	0.993988i	$0.465077\pi$
$860$	0	0
$861$	−18.4656	−0.629306
$862$	0	0
$863$	−20.6372	−0.702497	−0.351248	−	0.936282i	$-0.614243\pi$
$863$	−20.6372	−0.702497	−0.351248	+	0.936282i	$0.614243\pi$
$864$	0	0
$865$	−78.4583	−2.66766
$866$	0	0
$867$	−46.6770	−1.58523
$868$	0	0
$869$	6.38210	0.216498
$870$	0	0
$871$	−11.9068	−0.403447
$872$	0	0
$873$	17.6227	0.596440
$874$	0	0
$875$	7.99839	0.270395
$876$	0	0
$877$	−5.73201	−0.193556	−0.0967780	−	0.995306i	$-0.530854\pi$
$877$	−5.73201	−0.193556	−0.0967780	+	0.995306i	$0.530854\pi$
$878$	0	0
$879$	55.5471	1.87356
$880$	0	0
$881$	19.5525	0.658742	0.329371	−	0.944201i	$-0.393163\pi$
$881$	19.5525	0.658742	0.329371	+	0.944201i	$0.393163\pi$
$882$	0	0
$883$	−24.8212	−0.835300	−0.417650	−	0.908608i	$-0.637146\pi$
$883$	−24.8212	−0.835300	−0.417650	+	0.908608i	$0.637146\pi$
$884$	0	0
$885$	−60.0731	−2.01933
$886$	0	0
$887$	−26.7271	−0.897407	−0.448704	−	0.893681i	$-0.648114\pi$
$887$	−26.7271	−0.897407	−0.448704	+	0.893681i	$0.648114\pi$
$888$	0	0
$889$	−10.3939	−0.348600
$890$	0	0
$891$	2.00814	0.0672753
$892$	0	0
$893$	28.6822	0.959814
$894$	0	0
$895$	−10.6570	−0.356225
$896$	0	0
$897$	3.64743	0.121784
$898$	0	0
$899$	−16.1989	−0.540262
$900$	0	0
$901$	0.0201148	0.000670123 0
$902$	0	0
$903$	25.6318	0.852972
$904$	0	0
$905$	−48.2132	−1.60266
$906$	0	0
$907$	2.29657	0.0762562	0.0381281	−	0.999273i	$-0.487860\pi$
$907$	2.29657	0.0762562	0.0381281	+	0.999273i	$0.487860\pi$
$908$	0	0
$909$	34.7516	1.15264
$910$	0	0
$911$	12.6109	0.417819	0.208910	−	0.977935i	$-0.433009\pi$
$911$	12.6109	0.417819	0.208910	+	0.977935i	$0.433009\pi$
$912$	0	0
$913$	−13.9193	−0.460660
$914$	0	0
$915$	89.3757	2.95467
$916$	0	0
$917$	−13.6984	−0.452361
$918$	0	0
$919$	−9.14429	−0.301642	−0.150821	−	0.988561i	$-0.548192\pi$
$919$	−9.14429	−0.301642	−0.150821	+	0.988561i	$0.548192\pi$
$920$	0	0
$921$	−72.6546	−2.39405
$922$	0	0
$923$	8.34463	0.274667
$924$	0	0
$925$	23.0544	0.758023
$926$	0	0
$927$	−40.9531	−1.34508
$928$	0	0
$929$	0.117904	0.00386831	0.00193415	−	0.999998i	$-0.499384\pi$
$929$	0.117904	0.00386831	0.00193415	+	0.999998i	$0.499384\pi$
$930$	0	0
$931$	4.00697	0.131323
$932$	0	0
$933$	−92.4517	−3.02673
$934$	0	0
$935$	−0.125427	−0.00410190
$936$	0	0
$937$	−4.89007	−0.159752	−0.0798759	−	0.996805i	$-0.525452\pi$
$937$	−4.89007	−0.159752	−0.0798759	+	0.996805i	$0.525452\pi$
$938$	0	0
$939$	−33.6958	−1.09962
$940$	0	0
$941$	−39.6052	−1.29109	−0.645547	−	0.763721i	$-0.723370\pi$
$941$	−39.6052	−1.29109	−0.645547	+	0.763721i	$0.723370\pi$
$942$	0	0
$943$	−8.93258	−0.290885
$944$	0	0
$945$	14.8204	0.482106
$946$	0	0
$947$	12.1981	0.396384	0.198192	−	0.980163i	$-0.436493\pi$
$947$	12.1981	0.396384	0.198192	+	0.980163i	$0.436493\pi$
$948$	0	0
$949$	8.58860	0.278798
$950$	0	0
$951$	39.4182	1.27822
$952$	0	0
$953$	59.6865	1.93344	0.966718	−	0.255846i	$-0.0823539\pi$
$953$	59.6865	1.93344	0.966718	+	0.255846i	$0.0823539\pi$
$954$	0	0
$955$	−43.7817	−1.41674
$956$	0	0
$957$	11.3896	0.368172
$958$	0	0
$959$	−10.1842	−0.328864
$960$	0	0
$961$	−15.7479	−0.507998
$962$	0	0
$963$	33.9007	1.09243
$964$	0	0
$965$	72.4768	2.33311
$966$	0	0
$967$	43.4378	1.39687	0.698433	−	0.715676i	$-0.253881\pi$
$967$	43.4378	1.39687	0.698433	+	0.715676i	$0.253881\pi$
$968$	0	0
$969$	−0.393781	−0.0126501
$970$	0	0
$971$	−22.3615	−0.717616	−0.358808	−	0.933411i	$-0.616817\pi$
$971$	−22.3615	−0.717616	−0.358808	+	0.933411i	$0.616817\pi$
$972$	0	0
$973$	−6.71920	−0.215408
$974$	0	0
$975$	−19.9964	−0.640399
$976$	0	0
$977$	16.6628	0.533090	0.266545	−	0.963822i	$-0.414118\pi$
$977$	16.6628	0.533090	0.266545	+	0.963822i	$0.414118\pi$
$978$	0	0
$979$	−12.4900	−0.399183
$980$	0	0
$981$	13.2895	0.424303
$982$	0	0
$983$	41.8458	1.33467	0.667337	−	0.744756i	$-0.267434\pi$
$983$	41.8458	1.33467	0.667337	+	0.744756i	$0.267434\pi$
$984$	0	0
$985$	−17.4855	−0.557134
$986$	0	0
$987$	−19.6555	−0.625641
$988$	0	0
$989$	12.3991	0.394270
$990$	0	0
$991$	41.5262	1.31912	0.659562	−	0.751651i	$-0.270742\pi$
$991$	41.5262	1.31912	0.659562	+	0.751651i	$0.270742\pi$
$992$	0	0
$993$	47.0675	1.49364
$994$	0	0
$995$	−12.4188	−0.393701
$996$	0	0
$997$	−12.2981	−0.389485	−0.194743	−	0.980854i	$-0.562387\pi$
$997$	−12.2981	−0.389485	−0.194743	+	0.980854i	$0.562387\pi$
$998$	0	0
$999$	13.3877	0.423569

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.r.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.r.1.8	✓	9	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)

\(f(q)\)	\(=\)	\(q+2.74591 q^{3} -3.50460 q^{5} -1.00000 q^{7} +4.54004 q^{9} +O(q^{10})\)	\(q+2.74591 q^{3} -3.50460 q^{5} -1.00000 q^{7} +4.54004 q^{9} -1.00000 q^{11} -1.00000 q^{13} -9.62334 q^{15} -0.0357892 q^{17} +4.00697 q^{19} -2.74591 q^{21} -1.32831 q^{23} +7.28225 q^{25} +4.22882 q^{27} -4.14782 q^{29} +3.90539 q^{31} -2.74591 q^{33} +3.50460 q^{35} +3.16583 q^{37} -2.74591 q^{39} +6.72476 q^{41} -9.33451 q^{43} -15.9111 q^{45} +7.15809 q^{47} +1.00000 q^{49} -0.0982741 q^{51} -0.562036 q^{53} +3.50460 q^{55} +11.0028 q^{57} +6.24243 q^{59} -9.28739 q^{61} -4.54004 q^{63} +3.50460 q^{65} +11.9068 q^{67} -3.64743 q^{69} -8.34463 q^{71} -8.58860 q^{73} +19.9964 q^{75} +1.00000 q^{77} -6.38210 q^{79} -2.00814 q^{81} +13.9193 q^{83} +0.125427 q^{85} -11.3896 q^{87} +12.4900 q^{89} +1.00000 q^{91} +10.7239 q^{93} -14.0428 q^{95} +3.88163 q^{97} -4.54004 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) \) 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9	\( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) \) 9 * q + 5 * q^3 + 3 * q^5 - 9 * q^7 + 8 * q^9 - 9 * q^11 - 9 * q^13 - 3 * q^15 - 7 * q^17 + 13 * q^19 - 5 * q^21 + 9 * q^23 - 2 * q^25 + 5 * q^27 + q^29 + 10 * q^31 - 5 * q^33 - 3 * q^35 + 14 * q^37 - 5 * q^39 - 2 * q^41 + 5 * q^43 - 15 * q^45 + 13 * q^47 + 9 * q^49 - 3 * q^51 + 22 * q^53 - 3 * q^55 + 16 * q^57 + 43 * q^59 - 10 * q^61 - 8 * q^63 - 3 * q^65 + 26 * q^67 - 30 * q^69 + 18 * q^71 - 8 * q^73 + 28 * q^75 + 9 * q^77 - 9 * q^79 + 33 * q^81 + 4 * q^83 + 5 * q^85 + 33 * q^87 + 7 * q^89 + 9 * q^91 + 13 * q^93 + 7 * q^95 + 2 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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