LMFDB - Embedded newform 8008.2.a.u.1.3

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:	$1$
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} - 2x^{9} - 16x^{8} + 26x^{7} + 93x^{6} - 113x^{5} - 230x^{4} + 197x^{3} + 201x^{2} - 115x + 8 $ x^10 - 2x^9 - 16x^8 + 26x^7 + 93x^6 - 113x^5 - 230x^4 + 197x^3 + 201x^2 - 115*x + 8
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			1.3
Root			$-1.64280$ 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)

$f(q)$	$=$	$q-1.64280 q^{3} +2.39966 q^{5} -1.00000 q^{7} -0.301198 q^{9} +O(q^{10})$	$q-1.64280 q^{3} +2.39966 q^{5} -1.00000 q^{7} -0.301198 q^{9} -1.00000 q^{11} -1.00000 q^{13} -3.94216 q^{15} -5.56268 q^{17} +1.34299 q^{19} +1.64280 q^{21} +2.38656 q^{23} +0.758347 q^{25} +5.42322 q^{27} -0.520193 q^{29} +8.43959 q^{31} +1.64280 q^{33} -2.39966 q^{35} -5.34732 q^{37} +1.64280 q^{39} +1.51389 q^{41} +8.44676 q^{43} -0.722772 q^{45} -0.408568 q^{47} +1.00000 q^{49} +9.13839 q^{51} +14.0707 q^{53} -2.39966 q^{55} -2.20628 q^{57} -14.2923 q^{59} +5.50404 q^{61} +0.301198 q^{63} -2.39966 q^{65} -5.69135 q^{67} -3.92064 q^{69} +12.2452 q^{71} +14.4010 q^{73} -1.24581 q^{75} +1.00000 q^{77} -16.5736 q^{79} -8.00568 q^{81} -5.85861 q^{83} -13.3485 q^{85} +0.854575 q^{87} -15.7968 q^{89} +1.00000 q^{91} -13.8646 q^{93} +3.22273 q^{95} -6.66235 q^{97} +0.301198 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * 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
$3$	−1.64280	−0.948473	−0.474236	−	0.880398i	$-0.657276\pi$
$3$	−1.64280	−0.948473	−0.474236	+	0.880398i	$0.657276\pi$
$4$	0	0
$5$	2.39966	1.07316	0.536579	−	0.843850i	$-0.319716\pi$
$5$	2.39966	1.07316	0.536579	+	0.843850i	$0.319716\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.301198	−0.100399
$10$	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
$15$	−3.94216	−1.01786
$16$	0	0
$17$	−5.56268	−1.34915	−0.674574	−	0.738207i	$-0.735673\pi$
$17$	−5.56268	−1.34915	−0.674574	+	0.738207i	$0.735673\pi$
$18$	0	0
$19$	1.34299	0.308104	0.154052	−	0.988063i	$-0.450768\pi$
$19$	1.34299	0.308104	0.154052	+	0.988063i	$0.450768\pi$
$20$	0	0
$21$	1.64280	0.358489
$22$	0	0
$23$	2.38656	0.497631	0.248816	−	0.968551i	$-0.419959\pi$
$23$	2.38656	0.497631	0.248816	+	0.968551i	$0.419959\pi$
$24$	0	0
$25$	0.758347	0.151669
$26$	0	0
$27$	5.42322	1.04370
$28$	0	0
$29$	−0.520193	−0.0965975	−0.0482987	−	0.998833i	$-0.515380\pi$
$29$	−0.520193	−0.0965975	−0.0482987	+	0.998833i	$0.515380\pi$
$30$	0	0
$31$	8.43959	1.51579	0.757897	−	0.652374i	$-0.226227\pi$
$31$	8.43959	1.51579	0.757897	+	0.652374i	$0.226227\pi$
$32$	0	0
$33$	1.64280	0.285975
$34$	0	0
$35$	−2.39966	−0.405616
$36$	0	0
$37$	−5.34732	−0.879094	−0.439547	−	0.898219i	$-0.644861\pi$
$37$	−5.34732	−0.879094	−0.439547	+	0.898219i	$0.644861\pi$
$38$	0	0
$39$	1.64280	0.263059
$40$	0	0
$41$	1.51389	0.236429	0.118215	−	0.992988i	$-0.462283\pi$
$41$	1.51389	0.236429	0.118215	+	0.992988i	$0.462283\pi$
$42$	0	0
$43$	8.44676	1.28812	0.644060	−	0.764975i	$-0.277249\pi$
$43$	8.44676	1.28812	0.644060	+	0.764975i	$0.277249\pi$
$44$	0	0
$45$	−0.722772	−0.107745
$46$	0	0
$47$	−0.408568	−0.0595957	−0.0297978	−	0.999556i	$-0.509486\pi$
$47$	−0.408568	−0.0595957	−0.0297978	+	0.999556i	$0.509486\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.13839	1.27963
$52$	0	0
$53$	14.0707	1.93275	0.966377	−	0.257128i	$-0.0827761\pi$
$53$	14.0707	1.93275	0.966377	+	0.257128i	$0.0827761\pi$
$54$	0	0
$55$	−2.39966	−0.323569
$56$	0	0
$57$	−2.20628	−0.292228
$58$	0	0
$59$	−14.2923	−1.86070	−0.930350	−	0.366673i	$-0.880497\pi$
$59$	−14.2923	−1.86070	−0.930350	+	0.366673i	$0.880497\pi$
$60$	0	0
$61$	5.50404	0.704720	0.352360	−	0.935864i	$-0.385379\pi$
$61$	5.50404	0.704720	0.352360	+	0.935864i	$0.385379\pi$
$62$	0	0
$63$	0.301198	0.0379474
$64$	0	0
$65$	−2.39966	−0.297641
$66$	0	0
$67$	−5.69135	−0.695309	−0.347654	−	0.937623i	$-0.613022\pi$
$67$	−5.69135	−0.695309	−0.347654	+	0.937623i	$0.613022\pi$
$68$	0	0
$69$	−3.92064	−0.471990
$70$	0	0
$71$	12.2452	1.45324	0.726619	−	0.687040i	$-0.241090\pi$
$71$	12.2452	1.45324	0.726619	+	0.687040i	$0.241090\pi$
$72$	0	0
$73$	14.4010	1.68550	0.842752	−	0.538302i	$-0.180934\pi$
$73$	14.4010	1.68550	0.842752	+	0.538302i	$0.180934\pi$
$74$	0	0
$75$	−1.24581	−0.143854
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−16.5736	−1.86467	−0.932336	−	0.361593i	$-0.882233\pi$
$79$	−16.5736	−1.86467	−0.932336	+	0.361593i	$0.882233\pi$
$80$	0	0
$81$	−8.00568	−0.889520
$82$	0	0
$83$	−5.85861	−0.643066	−0.321533	−	0.946898i	$-0.604198\pi$
$83$	−5.85861	−0.643066	−0.321533	+	0.946898i	$0.604198\pi$
$84$	0	0
$85$	−13.3485	−1.44785
$86$	0	0
$87$	0.854575	0.0916200
$88$	0	0
$89$	−15.7968	−1.67445	−0.837226	−	0.546857i	$-0.815824\pi$
$89$	−15.7968	−1.67445	−0.837226	+	0.546857i	$0.815824\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−13.8646	−1.43769
$94$	0	0
$95$	3.22273	0.330645
$96$	0	0
$97$	−6.66235	−0.676459	−0.338230	−	0.941064i	$-0.609828\pi$
$97$	−6.66235	−0.676459	−0.338230	+	0.941064i	$0.609828\pi$
$98$	0	0
$99$	0.301198	0.0302716
$100$	0	0
$101$	−12.4762	−1.24143	−0.620713	−	0.784038i	$-0.713157\pi$
$101$	−12.4762	−1.24143	−0.620713	+	0.784038i	$0.713157\pi$
$102$	0	0
$103$	5.85198	0.576613	0.288306	−	0.957538i	$-0.406908\pi$
$103$	5.85198	0.576613	0.288306	+	0.957538i	$0.406908\pi$
$104$	0	0
$105$	3.94216	0.384716
$106$	0	0
$107$	−0.437393	−0.0422843	−0.0211422	−	0.999776i	$-0.506730\pi$
$107$	−0.437393	−0.0422843	−0.0211422	+	0.999776i	$0.506730\pi$
$108$	0	0
$109$	−9.86084	−0.944498	−0.472249	−	0.881465i	$-0.656558\pi$
$109$	−9.86084	−0.944498	−0.472249	+	0.881465i	$0.656558\pi$
$110$	0	0
$111$	8.78460	0.833797
$112$	0	0
$113$	−11.3863	−1.07113	−0.535565	−	0.844494i	$-0.679901\pi$
$113$	−11.3863	−1.07113	−0.535565	+	0.844494i	$0.679901\pi$
$114$	0	0
$115$	5.72691	0.534037
$116$	0	0
$117$	0.301198	0.0278458
$118$	0	0
$119$	5.56268	0.509930
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.48702	−0.224247
$124$	0	0
$125$	−10.1785	−0.910393
$126$	0	0
$127$	17.7019	1.57079	0.785396	−	0.618993i	$-0.212459\pi$
$127$	17.7019	1.57079	0.785396	+	0.618993i	$0.212459\pi$
$128$	0	0
$129$	−13.8764	−1.22175
$130$	0	0
$131$	14.0693	1.22924	0.614622	−	0.788822i	$-0.289309\pi$
$131$	14.0693	1.22924	0.614622	+	0.788822i	$0.289309\pi$
$132$	0	0
$133$	−1.34299	−0.116452
$134$	0	0
$135$	13.0139	1.12005
$136$	0	0
$137$	−10.7925	−0.922068	−0.461034	−	0.887382i	$-0.652521\pi$
$137$	−10.7925	−0.922068	−0.461034	+	0.887382i	$0.652521\pi$
$138$	0	0
$139$	4.23174	0.358932	0.179466	−	0.983764i	$-0.442563\pi$
$139$	4.23174	0.358932	0.179466	+	0.983764i	$0.442563\pi$
$140$	0	0
$141$	0.671196	0.0565249
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−1.24828	−0.103664
$146$	0	0
$147$	−1.64280	−0.135496
$148$	0	0
$149$	−16.0923	−1.31833	−0.659165	−	0.751998i	$-0.729090\pi$
$149$	−16.0923	−1.31833	−0.659165	+	0.751998i	$0.729090\pi$
$150$	0	0
$151$	4.49449	0.365756	0.182878	−	0.983136i	$-0.441459\pi$
$151$	4.49449	0.365756	0.182878	+	0.983136i	$0.441459\pi$
$152$	0	0
$153$	1.67547	0.135454
$154$	0	0
$155$	20.2521	1.62669
$156$	0	0
$157$	−19.4728	−1.55410	−0.777050	−	0.629439i	$-0.783285\pi$
$157$	−19.4728	−1.55410	−0.777050	+	0.629439i	$0.783285\pi$
$158$	0	0
$159$	−23.1153	−1.83317
$160$	0	0
$161$	−2.38656	−0.188087
$162$	0	0
$163$	−11.5929	−0.908025	−0.454013	−	0.890995i	$-0.650008\pi$
$163$	−11.5929	−0.908025	−0.454013	+	0.890995i	$0.650008\pi$
$164$	0	0
$165$	3.94216	0.306897
$166$	0	0
$167$	3.17478	0.245672	0.122836	−	0.992427i	$-0.460801\pi$
$167$	3.17478	0.245672	0.122836	+	0.992427i	$0.460801\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.404508	−0.0309335
$172$	0	0
$173$	−5.10785	−0.388343	−0.194171	−	0.980968i	$-0.562202\pi$
$173$	−5.10785	−0.388343	−0.194171	+	0.980968i	$0.562202\pi$
$174$	0	0
$175$	−0.758347	−0.0573256
$176$	0	0
$177$	23.4794	1.76482
$178$	0	0
$179$	−18.7568	−1.40195	−0.700974	−	0.713187i	$-0.747251\pi$
$179$	−18.7568	−1.40195	−0.700974	+	0.713187i	$0.747251\pi$
$180$	0	0
$181$	3.06001	0.227449	0.113724	−	0.993512i	$-0.463722\pi$
$181$	3.06001	0.227449	0.113724	+	0.993512i	$0.463722\pi$
$182$	0	0
$183$	−9.04205	−0.668408
$184$	0	0
$185$	−12.8317	−0.943408
$186$	0	0
$187$	5.56268	0.406783
$188$	0	0
$189$	−5.42322	−0.394481
$190$	0	0
$191$	1.66617	0.120560	0.0602800	−	0.998182i	$-0.480801\pi$
$191$	1.66617	0.120560	0.0602800	+	0.998182i	$0.480801\pi$
$192$	0	0
$193$	−22.0865	−1.58982	−0.794911	−	0.606726i	$-0.792483\pi$
$193$	−22.0865	−1.58982	−0.794911	+	0.606726i	$0.792483\pi$
$194$	0	0
$195$	3.94216	0.282304
$196$	0	0
$197$	15.4822	1.10306	0.551531	−	0.834155i	$-0.314044\pi$
$197$	15.4822	1.10306	0.551531	+	0.834155i	$0.314044\pi$
$198$	0	0
$199$	3.49155	0.247510	0.123755	−	0.992313i	$-0.460506\pi$
$199$	3.49155	0.247510	0.123755	+	0.992313i	$0.460506\pi$
$200$	0	0
$201$	9.34976	0.659481
$202$	0	0
$203$	0.520193	0.0365104
$204$	0	0
$205$	3.63281	0.253726
$206$	0	0
$207$	−0.718827	−0.0499619
$208$	0	0
$209$	−1.34299	−0.0928969
$210$	0	0
$211$	−7.41014	−0.510135	−0.255068	−	0.966923i	$-0.582098\pi$
$211$	−7.41014	−0.510135	−0.255068	+	0.966923i	$0.582098\pi$
$212$	0	0
$213$	−20.1165	−1.37836
$214$	0	0
$215$	20.2693	1.38236
$216$	0	0
$217$	−8.43959	−0.572916
$218$	0	0
$219$	−23.6579	−1.59865
$220$	0	0
$221$	5.56268	0.374186
$222$	0	0
$223$	−21.0017	−1.40638	−0.703189	−	0.711003i	$-0.748241\pi$
$223$	−21.0017	−1.40638	−0.703189	+	0.711003i	$0.748241\pi$
$224$	0	0
$225$	−0.228413	−0.0152275
$226$	0	0
$227$	−2.50544	−0.166292	−0.0831459	−	0.996537i	$-0.526497\pi$
$227$	−2.50544	−0.166292	−0.0831459	+	0.996537i	$0.526497\pi$
$228$	0	0
$229$	11.2660	0.744479	0.372239	−	0.928137i	$-0.378590\pi$
$229$	11.2660	0.744479	0.372239	+	0.928137i	$0.378590\pi$
$230$	0	0
$231$	−1.64280	−0.108089
$232$	0	0
$233$	−8.20880	−0.537776	−0.268888	−	0.963171i	$-0.586656\pi$
$233$	−8.20880	−0.537776	−0.268888	+	0.963171i	$0.586656\pi$
$234$	0	0
$235$	−0.980421	−0.0639556
$236$	0	0
$237$	27.2271	1.76859
$238$	0	0
$239$	14.6157	0.945411	0.472705	−	0.881220i	$-0.343277\pi$
$239$	14.6157	0.945411	0.472705	+	0.881220i	$0.343277\pi$
$240$	0	0
$241$	−3.28925	−0.211879	−0.105940	−	0.994373i	$-0.533785\pi$
$241$	−3.28925	−0.211879	−0.105940	+	0.994373i	$0.533785\pi$
$242$	0	0
$243$	−3.11789	−0.200013
$244$	0	0
$245$	2.39966	0.153308
$246$	0	0
$247$	−1.34299	−0.0854527
$248$	0	0
$249$	9.62454	0.609930
$250$	0	0
$251$	3.50916	0.221496	0.110748	−	0.993849i	$-0.464675\pi$
$251$	3.50916	0.221496	0.110748	+	0.993849i	$0.464675\pi$
$252$	0	0
$253$	−2.38656	−0.150041
$254$	0	0
$255$	21.9290	1.37325
$256$	0	0
$257$	−19.4294	−1.21197	−0.605985	−	0.795476i	$-0.707221\pi$
$257$	−19.4294	−1.21197	−0.605985	+	0.795476i	$0.707221\pi$
$258$	0	0
$259$	5.34732	0.332266
$260$	0	0
$261$	0.156681	0.00969833
$262$	0	0
$263$	5.84088	0.360164	0.180082	−	0.983652i	$-0.442364\pi$
$263$	5.84088	0.360164	0.180082	+	0.983652i	$0.442364\pi$
$264$	0	0
$265$	33.7648	2.07415
$266$	0	0
$267$	25.9510	1.58817
$268$	0	0
$269$	−1.75819	−0.107199	−0.0535993	−	0.998563i	$-0.517069\pi$
$269$	−1.75819	−0.107199	−0.0535993	+	0.998563i	$0.517069\pi$
$270$	0	0
$271$	4.64648	0.282254	0.141127	−	0.989992i	$-0.454927\pi$
$271$	4.64648	0.282254	0.141127	+	0.989992i	$0.454927\pi$
$272$	0	0
$273$	−1.64280	−0.0994270
$274$	0	0
$275$	−0.758347	−0.0457300
$276$	0	0
$277$	10.2026	0.613012	0.306506	−	0.951869i	$-0.400840\pi$
$277$	10.2026	0.613012	0.306506	+	0.951869i	$0.400840\pi$
$278$	0	0
$279$	−2.54199	−0.152185
$280$	0	0
$281$	21.9924	1.31196	0.655978	−	0.754780i	$-0.272256\pi$
$281$	21.9924	1.31196	0.655978	+	0.754780i	$0.272256\pi$
$282$	0	0
$283$	3.24165	0.192696	0.0963479	−	0.995348i	$-0.469284\pi$
$283$	3.24165	0.192696	0.0963479	+	0.995348i	$0.469284\pi$
$284$	0	0
$285$	−5.29430	−0.313607
$286$	0	0
$287$	−1.51389	−0.0893619
$288$	0	0
$289$	13.9434	0.820201
$290$	0	0
$291$	10.9449	0.641603
$292$	0	0
$293$	−24.1111	−1.40859	−0.704293	−	0.709909i	$-0.748736\pi$
$293$	−24.1111	−1.40859	−0.704293	+	0.709909i	$0.748736\pi$
$294$	0	0
$295$	−34.2966	−1.99683
$296$	0	0
$297$	−5.42322	−0.314687
$298$	0	0
$299$	−2.38656	−0.138018
$300$	0	0
$301$	−8.44676	−0.486863
$302$	0	0
$303$	20.4959	1.17746
$304$	0	0
$305$	13.2078	0.756277
$306$	0	0
$307$	1.81645	0.103670	0.0518350	−	0.998656i	$-0.483493\pi$
$307$	1.81645	0.103670	0.0518350	+	0.998656i	$0.483493\pi$
$308$	0	0
$309$	−9.61365	−0.546901
$310$	0	0
$311$	−5.76410	−0.326852	−0.163426	−	0.986556i	$-0.552255\pi$
$311$	−5.76410	−0.326852	−0.163426	+	0.986556i	$0.552255\pi$
$312$	0	0
$313$	−24.4540	−1.38222	−0.691111	−	0.722749i	$-0.742878\pi$
$313$	−24.4540	−1.38222	−0.691111	+	0.722749i	$0.742878\pi$
$314$	0	0
$315$	0.722772	0.0407236
$316$	0	0
$317$	−29.6759	−1.66677	−0.833383	−	0.552695i	$-0.813599\pi$
$317$	−29.6759	−1.66677	−0.833383	+	0.552695i	$0.813599\pi$
$318$	0	0
$319$	0.520193	0.0291252
$320$	0	0
$321$	0.718550	0.0401055
$322$	0	0
$323$	−7.47065	−0.415678
$324$	0	0
$325$	−0.758347	−0.0420655
$326$	0	0
$327$	16.1994	0.895830
$328$	0	0
$329$	0.408568	0.0225251
$330$	0	0
$331$	20.7437	1.14017	0.570087	−	0.821584i	$-0.306909\pi$
$331$	20.7437	1.14017	0.570087	+	0.821584i	$0.306909\pi$
$332$	0	0
$333$	1.61060	0.0882606
$334$	0	0
$335$	−13.6573	−0.746176
$336$	0	0
$337$	−2.30768	−0.125707	−0.0628536	−	0.998023i	$-0.520020\pi$
$337$	−2.30768	−0.125707	−0.0628536	+	0.998023i	$0.520020\pi$
$338$	0	0
$339$	18.7054	1.01594
$340$	0	0
$341$	−8.43959	−0.457029
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−9.40819	−0.506520
$346$	0	0
$347$	−18.5379	−0.995167	−0.497583	−	0.867416i	$-0.665779\pi$
$347$	−18.5379	−0.995167	−0.497583	+	0.867416i	$0.665779\pi$
$348$	0	0
$349$	27.6518	1.48017	0.740084	−	0.672514i	$-0.234786\pi$
$349$	27.6518	1.48017	0.740084	+	0.672514i	$0.234786\pi$
$350$	0	0
$351$	−5.42322	−0.289470
$352$	0	0
$353$	−13.6831	−0.728277	−0.364139	−	0.931345i	$-0.618637\pi$
$353$	−13.6831	−0.728277	−0.364139	+	0.931345i	$0.618637\pi$
$354$	0	0
$355$	29.3843	1.55956
$356$	0	0
$357$	−9.13839	−0.483655
$358$	0	0
$359$	−19.1398	−1.01016	−0.505081	−	0.863072i	$-0.668537\pi$
$359$	−19.1398	−1.01016	−0.505081	+	0.863072i	$0.668537\pi$
$360$	0	0
$361$	−17.1964	−0.905072
$362$	0	0
$363$	−1.64280	−0.0862248
$364$	0	0
$365$	34.5573	1.80881
$366$	0	0
$367$	−4.48344	−0.234033	−0.117017	−	0.993130i	$-0.537333\pi$
$367$	−4.48344	−0.234033	−0.117017	+	0.993130i	$0.537333\pi$
$368$	0	0
$369$	−0.455980	−0.0237374
$370$	0	0
$371$	−14.0707	−0.730513
$372$	0	0
$373$	30.6354	1.58624	0.793120	−	0.609065i	$-0.208455\pi$
$373$	30.6354	1.58624	0.793120	+	0.609065i	$0.208455\pi$
$374$	0	0
$375$	16.7213	0.863483
$376$	0	0
$377$	0.520193	0.0267913
$378$	0	0
$379$	10.0887	0.518222	0.259111	−	0.965848i	$-0.416570\pi$
$379$	10.0887	0.518222	0.259111	+	0.965848i	$0.416570\pi$
$380$	0	0
$381$	−29.0808	−1.48985
$382$	0	0
$383$	27.9318	1.42725	0.713625	−	0.700528i	$-0.247052\pi$
$383$	27.9318	1.42725	0.713625	+	0.700528i	$0.247052\pi$
$384$	0	0
$385$	2.39966	0.122298
$386$	0	0
$387$	−2.54415	−0.129326
$388$	0	0
$389$	34.1525	1.73160	0.865801	−	0.500388i	$-0.166809\pi$
$389$	34.1525	1.73160	0.865801	+	0.500388i	$0.166809\pi$
$390$	0	0
$391$	−13.2756	−0.671378
$392$	0	0
$393$	−23.1132	−1.16590
$394$	0	0
$395$	−39.7709	−2.00109
$396$	0	0
$397$	−10.2140	−0.512626	−0.256313	−	0.966594i	$-0.582508\pi$
$397$	−10.2140	−0.512626	−0.256313	+	0.966594i	$0.582508\pi$
$398$	0	0
$399$	2.20628	0.110452
$400$	0	0
$401$	−24.1600	−1.20649	−0.603247	−	0.797554i	$-0.706127\pi$
$401$	−24.1600	−1.20649	−0.603247	+	0.797554i	$0.706127\pi$
$402$	0	0
$403$	−8.43959	−0.420406
$404$	0	0
$405$	−19.2109	−0.954597
$406$	0	0
$407$	5.34732	0.265057
$408$	0	0
$409$	−17.1025	−0.845666	−0.422833	−	0.906208i	$-0.638964\pi$
$409$	−17.1025	−0.845666	−0.422833	+	0.906208i	$0.638964\pi$
$410$	0	0
$411$	17.7300	0.874556
$412$	0	0
$413$	14.2923	0.703278
$414$	0	0
$415$	−14.0586	−0.690111
$416$	0	0
$417$	−6.95192	−0.340437
$418$	0	0
$419$	16.9863	0.829834	0.414917	−	0.909859i	$-0.363811\pi$
$419$	16.9863	0.829834	0.414917	+	0.909859i	$0.363811\pi$
$420$	0	0
$421$	−26.8970	−1.31088	−0.655439	−	0.755248i	$-0.727516\pi$
$421$	−26.8970	−1.31088	−0.655439	+	0.755248i	$0.727516\pi$
$422$	0	0
$423$	0.123060	0.00598338
$424$	0	0
$425$	−4.21844	−0.204624
$426$	0	0
$427$	−5.50404	−0.266359
$428$	0	0
$429$	−1.64280	−0.0793153
$430$	0	0
$431$	35.9479	1.73155	0.865775	−	0.500433i	$-0.166826\pi$
$431$	35.9479	1.73155	0.865775	+	0.500433i	$0.166826\pi$
$432$	0	0
$433$	−0.938349	−0.0450942	−0.0225471	−	0.999746i	$-0.507178\pi$
$433$	−0.938349	−0.0450942	−0.0225471	+	0.999746i	$0.507178\pi$
$434$	0	0
$435$	2.05069	0.0983228
$436$	0	0
$437$	3.20513	0.153322
$438$	0	0
$439$	−13.0850	−0.624515	−0.312257	−	0.949998i	$-0.601085\pi$
$439$	−13.0850	−0.624515	−0.312257	+	0.949998i	$0.601085\pi$
$440$	0	0
$441$	−0.301198	−0.0143428
$442$	0	0
$443$	−39.1274	−1.85900	−0.929499	−	0.368824i	$-0.879761\pi$
$443$	−39.1274	−1.85900	−0.929499	+	0.368824i	$0.879761\pi$
$444$	0	0
$445$	−37.9068	−1.79695
$446$	0	0
$447$	26.4364	1.25040
$448$	0	0
$449$	9.62089	0.454038	0.227019	−	0.973890i	$-0.427102\pi$
$449$	9.62089	0.454038	0.227019	+	0.973890i	$0.427102\pi$
$450$	0	0
$451$	−1.51389	−0.0712862
$452$	0	0
$453$	−7.38356	−0.346910
$454$	0	0
$455$	2.39966	0.112498
$456$	0	0
$457$	−13.8035	−0.645702	−0.322851	−	0.946450i	$-0.604641\pi$
$457$	−13.8035	−0.645702	−0.322851	+	0.946450i	$0.604641\pi$
$458$	0	0
$459$	−30.1676	−1.40810
$460$	0	0
$461$	32.3567	1.50700	0.753501	−	0.657447i	$-0.228364\pi$
$461$	32.3567	1.50700	0.753501	+	0.657447i	$0.228364\pi$
$462$	0	0
$463$	−27.5056	−1.27829	−0.639147	−	0.769084i	$-0.720713\pi$
$463$	−27.5056	−1.27829	−0.639147	+	0.769084i	$0.720713\pi$
$464$	0	0
$465$	−33.2702	−1.54287
$466$	0	0
$467$	−1.52725	−0.0706727	−0.0353363	−	0.999375i	$-0.511250\pi$
$467$	−1.52725	−0.0706727	−0.0353363	+	0.999375i	$0.511250\pi$
$468$	0	0
$469$	5.69135	0.262802
$470$	0	0
$471$	31.9900	1.47402
$472$	0	0
$473$	−8.44676	−0.388383
$474$	0	0
$475$	1.01846	0.0467300
$476$	0	0
$477$	−4.23806	−0.194048
$478$	0	0
$479$	7.53266	0.344176	0.172088	−	0.985082i	$-0.444949\pi$
$479$	7.53266	0.344176	0.172088	+	0.985082i	$0.444949\pi$
$480$	0	0
$481$	5.34732	0.243817
$482$	0	0
$483$	3.92064	0.178395
$484$	0	0
$485$	−15.9874	−0.725948
$486$	0	0
$487$	−24.1988	−1.09655	−0.548276	−	0.836297i	$-0.684716\pi$
$487$	−24.1988	−1.09655	−0.548276	+	0.836297i	$0.684716\pi$
$488$	0	0
$489$	19.0448	0.861237
$490$	0	0
$491$	−3.32992	−0.150277	−0.0751385	−	0.997173i	$-0.523940\pi$
$491$	−3.32992	−0.150277	−0.0751385	+	0.997173i	$0.523940\pi$
$492$	0	0
$493$	2.89367	0.130324
$494$	0	0
$495$	0.722772	0.0324862
$496$	0	0
$497$	−12.2452	−0.549273
$498$	0	0
$499$	−31.8711	−1.42675	−0.713373	−	0.700785i	$-0.752833\pi$
$499$	−31.8711	−1.42675	−0.713373	+	0.700785i	$0.752833\pi$
$500$	0	0
$501$	−5.21554	−0.233013
$502$	0	0
$503$	2.79722	0.124722	0.0623609	−	0.998054i	$-0.480137\pi$
$503$	2.79722	0.124722	0.0623609	+	0.998054i	$0.480137\pi$
$504$	0	0
$505$	−29.9385	−1.33225
$506$	0	0
$507$	−1.64280	−0.0729594
$508$	0	0
$509$	−9.97182	−0.441993	−0.220997	−	0.975275i	$-0.570931\pi$
$509$	−9.97182	−0.441993	−0.220997	+	0.975275i	$0.570931\pi$
$510$	0	0
$511$	−14.4010	−0.637061
$512$	0	0
$513$	7.28335	0.321568
$514$	0	0
$515$	14.0427	0.618797
$516$	0	0
$517$	0.408568	0.0179688
$518$	0	0
$519$	8.39119	0.368333
$520$	0	0
$521$	−7.63123	−0.334330	−0.167165	−	0.985929i	$-0.553461\pi$
$521$	−7.63123	−0.334330	−0.167165	+	0.985929i	$0.553461\pi$
$522$	0	0
$523$	−6.93305	−0.303161	−0.151581	−	0.988445i	$-0.548436\pi$
$523$	−6.93305	−0.303161	−0.151581	+	0.988445i	$0.548436\pi$
$524$	0	0
$525$	1.24581	0.0543718
$526$	0	0
$527$	−46.9467	−2.04503
$528$	0	0
$529$	−17.3044	−0.752363
$530$	0	0
$531$	4.30482	0.186813
$532$	0	0
$533$	−1.51389	−0.0655737
$534$	0	0
$535$	−1.04959	−0.0453778
$536$	0	0
$537$	30.8137	1.32971
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	7.13757	0.306868	0.153434	−	0.988159i	$-0.450967\pi$
$541$	7.13757	0.306868	0.153434	+	0.988159i	$0.450967\pi$
$542$	0	0
$543$	−5.02700	−0.215729
$544$	0	0
$545$	−23.6626	−1.01360
$546$	0	0
$547$	−38.6511	−1.65260	−0.826301	−	0.563228i	$-0.809559\pi$
$547$	−38.6511	−1.65260	−0.826301	+	0.563228i	$0.809559\pi$
$548$	0	0
$549$	−1.65781	−0.0707535
$550$	0	0
$551$	−0.698617	−0.0297621
$552$	0	0
$553$	16.5736	0.704780
$554$	0	0
$555$	21.0800	0.894796
$556$	0	0
$557$	−29.4362	−1.24725	−0.623626	−	0.781723i	$-0.714341\pi$
$557$	−29.4362	−1.24725	−0.623626	+	0.781723i	$0.714341\pi$
$558$	0	0
$559$	−8.44676	−0.357260
$560$	0	0
$561$	−9.13839	−0.385823
$562$	0	0
$563$	−30.9492	−1.30435	−0.652177	−	0.758066i	$-0.726144\pi$
$563$	−30.9492	−1.30435	−0.652177	+	0.758066i	$0.726144\pi$
$564$	0	0
$565$	−27.3231	−1.14949
$566$	0	0
$567$	8.00568	0.336207
$568$	0	0
$569$	30.8226	1.29215	0.646076	−	0.763273i	$-0.276409\pi$
$569$	30.8226	1.29215	0.646076	+	0.763273i	$0.276409\pi$
$570$	0	0
$571$	−44.3529	−1.85611	−0.928056	−	0.372442i	$-0.878521\pi$
$571$	−44.3529	−1.85611	−0.928056	+	0.372442i	$0.878521\pi$
$572$	0	0
$573$	−2.73719	−0.114348
$574$	0	0
$575$	1.80984	0.0754754
$576$	0	0
$577$	20.8723	0.868925	0.434462	−	0.900690i	$-0.356938\pi$
$577$	20.8723	0.868925	0.434462	+	0.900690i	$0.356938\pi$
$578$	0	0
$579$	36.2838	1.50790
$580$	0	0
$581$	5.85861	0.243056
$582$	0	0
$583$	−14.0707	−0.582748
$584$	0	0
$585$	0.722772	0.0298830
$586$	0	0
$587$	−15.6679	−0.646682	−0.323341	−	0.946283i	$-0.604806\pi$
$587$	−15.6679	−0.646682	−0.323341	+	0.946283i	$0.604806\pi$
$588$	0	0
$589$	11.3343	0.467023
$590$	0	0
$591$	−25.4342	−1.04622
$592$	0	0
$593$	24.6491	1.01222	0.506108	−	0.862470i	$-0.331084\pi$
$593$	24.6491	1.01222	0.506108	+	0.862470i	$0.331084\pi$
$594$	0	0
$595$	13.3485	0.547236
$596$	0	0
$597$	−5.73593	−0.234756
$598$	0	0
$599$	24.7032	1.00934	0.504672	−	0.863311i	$-0.331614\pi$
$599$	24.7032	1.00934	0.504672	+	0.863311i	$0.331614\pi$
$600$	0	0
$601$	−4.85788	−0.198157	−0.0990785	−	0.995080i	$-0.531590\pi$
$601$	−4.85788	−0.198157	−0.0990785	+	0.995080i	$0.531590\pi$
$602$	0	0
$603$	1.71422	0.0698086
$604$	0	0
$605$	2.39966	0.0975599
$606$	0	0
$607$	−16.3748	−0.664633	−0.332316	−	0.943168i	$-0.607830\pi$
$607$	−16.3748	−0.664633	−0.332316	+	0.943168i	$0.607830\pi$
$608$	0	0
$609$	−0.854575	−0.0346291
$610$	0	0
$611$	0.408568	0.0165289
$612$	0	0
$613$	−21.0395	−0.849779	−0.424890	−	0.905245i	$-0.639687\pi$
$613$	−21.0395	−0.849779	−0.424890	+	0.905245i	$0.639687\pi$
$614$	0	0
$615$	−5.96799	−0.240652
$616$	0	0
$617$	−30.6569	−1.23420	−0.617101	−	0.786884i	$-0.711693\pi$
$617$	−30.6569	−1.23420	−0.617101	+	0.786884i	$0.711693\pi$
$618$	0	0
$619$	19.5794	0.786964	0.393482	−	0.919332i	$-0.371270\pi$
$619$	19.5794	0.786964	0.393482	+	0.919332i	$0.371270\pi$
$620$	0	0
$621$	12.9428	0.519377
$622$	0	0
$623$	15.7968	0.632883
$624$	0	0
$625$	−28.2166	−1.12867
$626$	0	0
$627$	2.20628	0.0881102
$628$	0	0
$629$	29.7454	1.18603
$630$	0	0
$631$	4.73796	0.188615	0.0943076	−	0.995543i	$-0.469936\pi$
$631$	4.73796	0.188615	0.0943076	+	0.995543i	$0.469936\pi$
$632$	0	0
$633$	12.1734	0.483850
$634$	0	0
$635$	42.4785	1.68571
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−3.68824	−0.145904
$640$	0	0
$641$	−13.3647	−0.527873	−0.263936	−	0.964540i	$-0.585021\pi$
$641$	−13.3647	−0.527873	−0.263936	+	0.964540i	$0.585021\pi$
$642$	0	0
$643$	29.2377	1.15302	0.576511	−	0.817090i	$-0.304414\pi$
$643$	29.2377	1.15302	0.576511	+	0.817090i	$0.304414\pi$
$644$	0	0
$645$	−33.2985	−1.31113
$646$	0	0
$647$	−35.9618	−1.41381	−0.706903	−	0.707311i	$-0.749908\pi$
$647$	−35.9618	−1.41381	−0.706903	+	0.707311i	$0.749908\pi$
$648$	0	0
$649$	14.2923	0.561022
$650$	0	0
$651$	13.8646	0.543396
$652$	0	0
$653$	6.89836	0.269954	0.134977	−	0.990849i	$-0.456904\pi$
$653$	6.89836	0.269954	0.134977	+	0.990849i	$0.456904\pi$
$654$	0	0
$655$	33.7616	1.31917
$656$	0	0
$657$	−4.33754	−0.169224
$658$	0	0
$659$	−24.0182	−0.935617	−0.467808	−	0.883830i	$-0.654956\pi$
$659$	−24.0182	−0.935617	−0.467808	+	0.883830i	$0.654956\pi$
$660$	0	0
$661$	31.7016	1.23305	0.616525	−	0.787335i	$-0.288540\pi$
$661$	31.7016	1.23305	0.616525	+	0.787335i	$0.288540\pi$
$662$	0	0
$663$	−9.13839	−0.354906
$664$	0	0
$665$	−3.22273	−0.124972
$666$	0	0
$667$	−1.24147	−0.0480699
$668$	0	0
$669$	34.5017	1.33391
$670$	0	0
$671$	−5.50404	−0.212481
$672$	0	0
$673$	34.6161	1.33435	0.667177	−	0.744899i	$-0.267502\pi$
$673$	34.6161	1.33435	0.667177	+	0.744899i	$0.267502\pi$
$674$	0	0
$675$	4.11268	0.158297
$676$	0	0
$677$	25.8616	0.993941	0.496970	−	0.867768i	$-0.334446\pi$
$677$	25.8616	0.993941	0.496970	+	0.867768i	$0.334446\pi$
$678$	0	0
$679$	6.66235	0.255678
$680$	0	0
$681$	4.11594	0.157723
$682$	0	0
$683$	2.70904	0.103659	0.0518293	−	0.998656i	$-0.483495\pi$
$683$	2.70904	0.103659	0.0518293	+	0.998656i	$0.483495\pi$
$684$	0	0
$685$	−25.8984	−0.989525
$686$	0	0
$687$	−18.5078	−0.706118
$688$	0	0
$689$	−14.0707	−0.536050
$690$	0	0
$691$	28.0188	1.06589	0.532944	−	0.846151i	$-0.321086\pi$
$691$	28.0188	1.06589	0.532944	+	0.846151i	$0.321086\pi$
$692$	0	0
$693$	−0.301198	−0.0114416
$694$	0	0
$695$	10.1547	0.385191
$696$	0	0
$697$	−8.42127	−0.318978
$698$	0	0
$699$	13.4854	0.510066
$700$	0	0
$701$	9.95128	0.375855	0.187927	−	0.982183i	$-0.439823\pi$
$701$	9.95128	0.375855	0.187927	+	0.982183i	$0.439823\pi$
$702$	0	0
$703$	−7.18143	−0.270853
$704$	0	0
$705$	1.61064	0.0606602
$706$	0	0
$707$	12.4762	0.469215
$708$	0	0
$709$	−42.7166	−1.60426	−0.802129	−	0.597151i	$-0.796299\pi$
$709$	−42.7166	−1.60426	−0.802129	+	0.597151i	$0.796299\pi$
$710$	0	0
$711$	4.99193	0.187212
$712$	0	0
$713$	20.1415	0.754307
$714$	0	0
$715$	2.39966	0.0897420
$716$	0	0
$717$	−24.0107	−0.896696
$718$	0	0
$719$	−33.4226	−1.24645	−0.623225	−	0.782042i	$-0.714178\pi$
$719$	−33.4226	−1.24645	−0.623225	+	0.782042i	$0.714178\pi$
$720$	0	0
$721$	−5.85198	−0.217939
$722$	0	0
$723$	5.40358	0.200961
$724$	0	0
$725$	−0.394487	−0.0146509
$726$	0	0
$727$	29.5522	1.09603	0.548016	−	0.836468i	$-0.315383\pi$
$727$	29.5522	1.09603	0.548016	+	0.836468i	$0.315383\pi$
$728$	0	0
$729$	29.1391	1.07923
$730$	0	0
$731$	−46.9866	−1.73786
$732$	0	0
$733$	6.64126	0.245300	0.122650	−	0.992450i	$-0.460861\pi$
$733$	6.64126	0.245300	0.122650	+	0.992450i	$0.460861\pi$
$734$	0	0
$735$	−3.94216	−0.145409
$736$	0	0
$737$	5.69135	0.209643
$738$	0	0
$739$	20.9747	0.771568	0.385784	−	0.922589i	$-0.373931\pi$
$739$	20.9747	0.771568	0.385784	+	0.922589i	$0.373931\pi$
$740$	0	0
$741$	2.20628	0.0810496
$742$	0	0
$743$	19.0260	0.697995	0.348998	−	0.937124i	$-0.386522\pi$
$743$	19.0260	0.697995	0.348998	+	0.937124i	$0.386522\pi$
$744$	0	0
$745$	−38.6159	−1.41478
$746$	0	0
$747$	1.76460	0.0645634
$748$	0	0
$749$	0.437393	0.0159820
$750$	0	0
$751$	−3.82055	−0.139414	−0.0697070	−	0.997568i	$-0.522206\pi$
$751$	−3.82055	−0.139414	−0.0697070	+	0.997568i	$0.522206\pi$
$752$	0	0
$753$	−5.76486	−0.210083
$754$	0	0
$755$	10.7852	0.392514
$756$	0	0
$757$	−9.96730	−0.362268	−0.181134	−	0.983458i	$-0.557977\pi$
$757$	−9.96730	−0.362268	−0.181134	+	0.983458i	$0.557977\pi$
$758$	0	0
$759$	3.92064	0.142310
$760$	0	0
$761$	−30.1817	−1.09409	−0.547043	−	0.837105i	$-0.684246\pi$
$761$	−30.1817	−1.09409	−0.547043	+	0.837105i	$0.684246\pi$
$762$	0	0
$763$	9.86084	0.356987
$764$	0	0
$765$	4.02055	0.145363
$766$	0	0
$767$	14.2923	0.516065
$768$	0	0
$769$	−30.6444	−1.10507	−0.552533	−	0.833491i	$-0.686339\pi$
$769$	−30.6444	−1.10507	−0.552533	+	0.833491i	$0.686339\pi$
$770$	0	0
$771$	31.9186	1.14952
$772$	0	0
$773$	8.77540	0.315629	0.157815	−	0.987469i	$-0.449555\pi$
$773$	8.77540	0.315629	0.157815	+	0.987469i	$0.449555\pi$
$774$	0	0
$775$	6.40013	0.229900
$776$	0	0
$777$	−8.78460	−0.315146
$778$	0	0
$779$	2.03314	0.0728449
$780$	0	0
$781$	−12.2452	−0.438168
$782$	0	0
$783$	−2.82112	−0.100819
$784$	0	0
$785$	−46.7281	−1.66780
$786$	0	0
$787$	−34.6006	−1.23338	−0.616690	−	0.787206i	$-0.711527\pi$
$787$	−34.6006	−1.23338	−0.616690	+	0.787206i	$0.711527\pi$
$788$	0	0
$789$	−9.59541	−0.341606
$790$	0	0
$791$	11.3863	0.404849
$792$	0	0
$793$	−5.50404	−0.195454
$794$	0	0
$795$	−55.4688	−1.96728
$796$	0	0
$797$	37.1131	1.31461	0.657307	−	0.753623i	$-0.271696\pi$
$797$	37.1131	1.31461	0.657307	+	0.753623i	$0.271696\pi$
$798$	0	0
$799$	2.27273	0.0804034
$800$	0	0
$801$	4.75796	0.168114
$802$	0	0
$803$	−14.4010	−0.508199
$804$	0	0
$805$	−5.72691	−0.201847
$806$	0	0
$807$	2.88836	0.101675
$808$	0	0
$809$	−19.0435	−0.669532	−0.334766	−	0.942301i	$-0.608657\pi$
$809$	−19.0435	−0.669532	−0.334766	+	0.942301i	$0.608657\pi$
$810$	0	0
$811$	−19.8623	−0.697460	−0.348730	−	0.937223i	$-0.613387\pi$
$811$	−19.8623	−0.697460	−0.348730	+	0.937223i	$0.613387\pi$
$812$	0	0
$813$	−7.63325	−0.267710
$814$	0	0
$815$	−27.8189	−0.974455
$816$	0	0
$817$	11.3440	0.396875
$818$	0	0
$819$	−0.301198	−0.0105247
$820$	0	0
$821$	21.5743	0.752947	0.376474	−	0.926427i	$-0.377137\pi$
$821$	21.5743	0.752947	0.376474	+	0.926427i	$0.377137\pi$
$822$	0	0
$823$	−15.9308	−0.555312	−0.277656	−	0.960681i	$-0.589558\pi$
$823$	−15.9308	−0.555312	−0.277656	+	0.960681i	$0.589558\pi$
$824$	0	0
$825$	1.24581	0.0433737
$826$	0	0
$827$	−52.1864	−1.81470	−0.907349	−	0.420377i	$-0.861898\pi$
$827$	−52.1864	−1.81470	−0.907349	+	0.420377i	$0.861898\pi$
$828$	0	0
$829$	−9.68440	−0.336353	−0.168177	−	0.985757i	$-0.553788\pi$
$829$	−9.68440	−0.336353	−0.168177	+	0.985757i	$0.553788\pi$
$830$	0	0
$831$	−16.7608	−0.581425
$832$	0	0
$833$	−5.56268	−0.192735
$834$	0	0
$835$	7.61839	0.263645
$836$	0	0
$837$	45.7697	1.58203
$838$	0	0
$839$	−4.47194	−0.154388	−0.0771942	−	0.997016i	$-0.524596\pi$
$839$	−4.47194	−0.154388	−0.0771942	+	0.997016i	$0.524596\pi$
$840$	0	0
$841$	−28.7294	−0.990669
$842$	0	0
$843$	−36.1292	−1.24436
$844$	0	0
$845$	2.39966	0.0825507
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−5.32539	−0.182767
$850$	0	0
$851$	−12.7617	−0.437465
$852$	0	0
$853$	−32.6193	−1.11686	−0.558432	−	0.829550i	$-0.688597\pi$
$853$	−32.6193	−1.11686	−0.558432	+	0.829550i	$0.688597\pi$
$854$	0	0
$855$	−0.970680	−0.0331965
$856$	0	0
$857$	26.4158	0.902347	0.451174	−	0.892436i	$-0.351006\pi$
$857$	26.4158	0.902347	0.451174	+	0.892436i	$0.351006\pi$
$858$	0	0
$859$	31.5661	1.07702	0.538511	−	0.842618i	$-0.318987\pi$
$859$	31.5661	1.07702	0.538511	+	0.842618i	$0.318987\pi$
$860$	0	0
$861$	2.48702	0.0847574
$862$	0	0
$863$	−26.3330	−0.896385	−0.448193	−	0.893937i	$-0.647932\pi$
$863$	−26.3330	−0.896385	−0.448193	+	0.893937i	$0.647932\pi$
$864$	0	0
$865$	−12.2571	−0.416753
$866$	0	0
$867$	−22.9063	−0.777938
$868$	0	0
$869$	16.5736	0.562220
$870$	0	0
$871$	5.69135	0.192844
$872$	0	0
$873$	2.00669	0.0679162
$874$	0	0
$875$	10.1785	0.344096
$876$	0	0
$877$	49.7176	1.67884	0.839422	−	0.543480i	$-0.182894\pi$
$877$	49.7176	1.67884	0.839422	+	0.543480i	$0.182894\pi$
$878$	0	0
$879$	39.6098	1.33601
$880$	0	0
$881$	−20.8053	−0.700950	−0.350475	−	0.936572i	$-0.613980\pi$
$881$	−20.8053	−0.700950	−0.350475	+	0.936572i	$0.613980\pi$
$882$	0	0
$883$	2.46311	0.0828903	0.0414452	−	0.999141i	$-0.486804\pi$
$883$	2.46311	0.0828903	0.0414452	+	0.999141i	$0.486804\pi$
$884$	0	0
$885$	56.3426	1.89393
$886$	0	0
$887$	−42.6209	−1.43107	−0.715535	−	0.698577i	$-0.753817\pi$
$887$	−42.6209	−1.43107	−0.715535	+	0.698577i	$0.753817\pi$
$888$	0	0
$889$	−17.7019	−0.593704
$890$	0	0
$891$	8.00568	0.268201
$892$	0	0
$893$	−0.548704	−0.0183617
$894$	0	0
$895$	−45.0098	−1.50451
$896$	0	0
$897$	3.92064	0.130906
$898$	0	0
$899$	−4.39022	−0.146422
$900$	0	0
$901$	−78.2706	−2.60757
$902$	0	0
$903$	13.8764	0.461777
$904$	0	0
$905$	7.34298	0.244089
$906$	0	0
$907$	51.3137	1.70384	0.851922	−	0.523669i	$-0.175437\pi$
$907$	51.3137	1.70384	0.851922	+	0.523669i	$0.175437\pi$
$908$	0	0
$909$	3.75780	0.124638
$910$	0	0
$911$	8.21512	0.272179	0.136089	−	0.990697i	$-0.456547\pi$
$911$	8.21512	0.272179	0.136089	+	0.990697i	$0.456547\pi$
$912$	0	0
$913$	5.85861	0.193892
$914$	0	0
$915$	−21.6978	−0.717308
$916$	0	0
$917$	−14.0693	−0.464611
$918$	0	0
$919$	3.84321	0.126776	0.0633879	−	0.997989i	$-0.479809\pi$
$919$	3.84321	0.126776	0.0633879	+	0.997989i	$0.479809\pi$
$920$	0	0
$921$	−2.98406	−0.0983282
$922$	0	0
$923$	−12.2452	−0.403056
$924$	0	0
$925$	−4.05513	−0.133332
$926$	0	0
$927$	−1.76261	−0.0578916
$928$	0	0
$929$	36.6152	1.20130	0.600652	−	0.799511i	$-0.294908\pi$
$929$	36.6152	1.20130	0.600652	+	0.799511i	$0.294908\pi$
$930$	0	0
$931$	1.34299	0.0440149
$932$	0	0
$933$	9.46928	0.310010
$934$	0	0
$935$	13.3485	0.436543
$936$	0	0
$937$	28.9713	0.946451	0.473225	−	0.880941i	$-0.343090\pi$
$937$	28.9713	0.946451	0.473225	+	0.880941i	$0.343090\pi$
$938$	0	0
$939$	40.1731	1.31100
$940$	0	0
$941$	35.6723	1.16288	0.581441	−	0.813588i	$-0.302489\pi$
$941$	35.6723	1.16288	0.581441	+	0.813588i	$0.302489\pi$
$942$	0	0
$943$	3.61298	0.117655
$944$	0	0
$945$	−13.0139	−0.423341
$946$	0	0
$947$	7.52494	0.244528	0.122264	−	0.992498i	$-0.460985\pi$
$947$	7.52494	0.244528	0.122264	+	0.992498i	$0.460985\pi$
$948$	0	0
$949$	−14.4010	−0.467475
$950$	0	0
$951$	48.7517	1.58088
$952$	0	0
$953$	−30.3891	−0.984401	−0.492200	−	0.870482i	$-0.663807\pi$
$953$	−30.3891	−0.984401	−0.492200	+	0.870482i	$0.663807\pi$
$954$	0	0
$955$	3.99824	0.129380
$956$	0	0
$957$	−0.854575	−0.0276245
$958$	0	0
$959$	10.7925	0.348509
$960$	0	0
$961$	40.2266	1.29763
$962$	0	0
$963$	0.131742	0.00424532
$964$	0	0
$965$	−53.0000	−1.70613
$966$	0	0
$967$	−23.8860	−0.768121	−0.384060	−	0.923308i	$-0.625475\pi$
$967$	−23.8860	−0.768121	−0.384060	+	0.923308i	$0.625475\pi$
$968$	0	0
$969$	12.2728	0.394259
$970$	0	0
$971$	41.6845	1.33772	0.668860	−	0.743388i	$-0.266783\pi$
$971$	41.6845	1.33772	0.668860	+	0.743388i	$0.266783\pi$
$972$	0	0
$973$	−4.23174	−0.135663
$974$	0	0
$975$	1.24581	0.0398980
$976$	0	0
$977$	−39.8330	−1.27437	−0.637185	−	0.770711i	$-0.719901\pi$
$977$	−39.8330	−1.27437	−0.637185	+	0.770711i	$0.719901\pi$
$978$	0	0
$979$	15.7968	0.504866
$980$	0	0
$981$	2.97007	0.0948271
$982$	0	0
$983$	−15.2512	−0.486438	−0.243219	−	0.969971i	$-0.578203\pi$
$983$	−15.2512	−0.486438	−0.243219	+	0.969971i	$0.578203\pi$
$984$	0	0
$985$	37.1519	1.18376
$986$	0	0
$987$	−0.671196	−0.0213644
$988$	0	0
$989$	20.1587	0.641008
$990$	0	0
$991$	−25.3637	−0.805705	−0.402853	−	0.915265i	$-0.631981\pi$
$991$	−25.3637	−0.805705	−0.402853	+	0.915265i	$0.631981\pi$
$992$	0	0
$993$	−34.0777	−1.08142
$994$	0	0
$995$	8.37852	0.265617
$996$	0	0
$997$	−31.1588	−0.986809	−0.493404	−	0.869800i	$-0.664248\pi$
$997$	−31.1588	−0.986809	−0.493404	+	0.869800i	$0.664248\pi$
$998$	0	0
$999$	−28.9997	−0.917510

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.u.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.u.1.3	✓	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.64280 q^{3} +2.39966 q^{5} -1.00000 q^{7} -0.301198 q^{9} +O(q^{10})\)	\(q-1.64280 q^{3} +2.39966 q^{5} -1.00000 q^{7} -0.301198 q^{9} -1.00000 q^{11} -1.00000 q^{13} -3.94216 q^{15} -5.56268 q^{17} +1.34299 q^{19} +1.64280 q^{21} +2.38656 q^{23} +0.758347 q^{25} +5.42322 q^{27} -0.520193 q^{29} +8.43959 q^{31} +1.64280 q^{33} -2.39966 q^{35} -5.34732 q^{37} +1.64280 q^{39} +1.51389 q^{41} +8.44676 q^{43} -0.722772 q^{45} -0.408568 q^{47} +1.00000 q^{49} +9.13839 q^{51} +14.0707 q^{53} -2.39966 q^{55} -2.20628 q^{57} -14.2923 q^{59} +5.50404 q^{61} +0.301198 q^{63} -2.39966 q^{65} -5.69135 q^{67} -3.92064 q^{69} +12.2452 q^{71} +14.4010 q^{73} -1.24581 q^{75} +1.00000 q^{77} -16.5736 q^{79} -8.00568 q^{81} -5.85861 q^{83} -13.3485 q^{85} +0.854575 q^{87} -15.7968 q^{89} +1.00000 q^{91} -13.8646 q^{93} +3.22273 q^{95} -6.66235 q^{97} +0.301198 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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