LMFDB - Embedded newform 8960.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8960,2,Mod(1,8960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8960 = 2^{8} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8960.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:	$71.5459602111$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.940.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 4 $ x^3 - 7*x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2240)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.602705$ of defining polynomial
Character	$\chi$	$=$	8960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.602705 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.63675 q^{9} +O(q^{10})$	$q+0.602705 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.63675 q^{9} -5.63675 q^{11} -4.43134 q^{13} +0.602705 q^{15} -7.46538 q^{17} +1.20541 q^{19} +0.602705 q^{21} +7.27349 q^{23} +1.00000 q^{25} -3.39730 q^{27} +6.67079 q^{29} +10.0681 q^{31} -3.39730 q^{33} +1.00000 q^{35} -4.86267 q^{37} -2.67079 q^{39} +4.79459 q^{41} -2.00000 q^{43} -2.63675 q^{45} -6.43134 q^{47} +1.00000 q^{49} -4.49942 q^{51} +2.06808 q^{53} -5.63675 q^{55} +0.726506 q^{57} +2.06808 q^{59} -1.27349 q^{61} -2.63675 q^{63} -4.43134 q^{65} +7.20541 q^{67} +4.38377 q^{69} -6.79459 q^{71} -1.27349 q^{73} +0.602705 q^{75} -5.63675 q^{77} -13.0816 q^{79} +5.86267 q^{81} +1.20541 q^{83} -7.46538 q^{85} +4.02052 q^{87} +17.2735 q^{89} -4.43134 q^{91} +6.06808 q^{93} +1.20541 q^{95} +11.4654 q^{97} +14.8627 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) $ 3 * q + 3 * q^5 + 3 * q^7 + 5 * q^9	$ 3 q + 3 q^{5} + 3 q^{7} + 5 q^{9} - 4 q^{11} - 4 q^{13} - 2 q^{17} - 4 q^{23} + 3 q^{25} - 12 q^{27} - 4 q^{29} + 8 q^{31} - 12 q^{33} + 3 q^{35} + 4 q^{37} + 16 q^{39} + 18 q^{41} - 6 q^{43} + 5 q^{45} - 10 q^{47} + 3 q^{49} + 18 q^{51} - 16 q^{53} - 4 q^{55} + 28 q^{57} - 16 q^{59} + 22 q^{61} + 5 q^{63} - 4 q^{65} + 18 q^{67} + 24 q^{69} - 24 q^{71} + 22 q^{73} - 4 q^{77} - 8 q^{79} - q^{81} - 2 q^{85} + 10 q^{87} + 26 q^{89} - 4 q^{91} - 4 q^{93} + 14 q^{97} + 26 q^{99}+O(q^{100}) $ 3 * q + 3 * q^5 + 3 * q^7 + 5 * q^9 - 4 * q^11 - 4 * q^13 - 2 * q^17 - 4 * q^23 + 3 * q^25 - 12 * q^27 - 4 * q^29 + 8 * q^31 - 12 * q^33 + 3 * q^35 + 4 * q^37 + 16 * q^39 + 18 * q^41 - 6 * q^43 + 5 * q^45 - 10 * q^47 + 3 * q^49 + 18 * q^51 - 16 * q^53 - 4 * q^55 + 28 * q^57 - 16 * q^59 + 22 * q^61 + 5 * q^63 - 4 * q^65 + 18 * q^67 + 24 * q^69 - 24 * q^71 + 22 * q^73 - 4 * q^77 - 8 * q^79 - q^81 - 2 * q^85 + 10 * q^87 + 26 * q^89 - 4 * q^91 - 4 * q^93 + 14 * q^97 + 26 * 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
$3$	0.602705	0.347972	0.173986	−	0.984748i	$-0.444335\pi$
$3$	0.602705	0.347972	0.173986	+	0.984748i	$0.444335\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.63675	−0.878916
$10$	0	0
$11$	−5.63675	−1.69954	−0.849772	−	0.527151i	$-0.823260\pi$
$11$	−5.63675	−1.69954	−0.849772	+	0.527151i	$0.823260\pi$
$12$	0	0
$13$	−4.43134	−1.22903	−0.614516	−	0.788904i	$-0.710649\pi$
$13$	−4.43134	−1.22903	−0.614516	+	0.788904i	$0.710649\pi$
$14$	0	0
$15$	0.602705	0.155618
$16$	0	0
$17$	−7.46538	−1.81062	−0.905310	−	0.424751i	$-0.860362\pi$
$17$	−7.46538	−1.81062	−0.905310	+	0.424751i	$0.860362\pi$
$18$	0	0
$19$	1.20541	0.276540	0.138270	−	0.990395i	$-0.455846\pi$
$19$	1.20541	0.276540	0.138270	+	0.990395i	$0.455846\pi$
$20$	0	0
$21$	0.602705	0.131521
$22$	0	0
$23$	7.27349	1.51663	0.758314	−	0.651889i	$-0.226023\pi$
$23$	7.27349	1.51663	0.758314	+	0.651889i	$0.226023\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.39730	−0.653810
$28$	0	0
$29$	6.67079	1.23873	0.619367	−	0.785101i	$-0.287389\pi$
$29$	6.67079	1.23873	0.619367	+	0.785101i	$0.287389\pi$
$30$	0	0
$31$	10.0681	1.80828	0.904141	−	0.427235i	$-0.140512\pi$
$31$	10.0681	1.80828	0.904141	+	0.427235i	$0.140512\pi$
$32$	0	0
$33$	−3.39730	−0.591393
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−4.86267	−0.799419	−0.399709	−	0.916642i	$-0.630889\pi$
$37$	−4.86267	−0.799419	−0.399709	+	0.916642i	$0.630889\pi$
$38$	0	0
$39$	−2.67079	−0.427668
$40$	0	0
$41$	4.79459	0.748789	0.374395	−	0.927269i	$-0.377851\pi$
$41$	4.79459	0.748789	0.374395	+	0.927269i	$0.377851\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−2.63675	−0.393063
$46$	0	0
$47$	−6.43134	−0.938107	−0.469053	−	0.883170i	$-0.655405\pi$
$47$	−6.43134	−0.938107	−0.469053	+	0.883170i	$0.655405\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.49942	−0.630045
$52$	0	0
$53$	2.06808	0.284073	0.142037	−	0.989861i	$-0.454635\pi$
$53$	2.06808	0.284073	0.142037	+	0.989861i	$0.454635\pi$
$54$	0	0
$55$	−5.63675	−0.760059
$56$	0	0
$57$	0.726506	0.0962281
$58$	0	0
$59$	2.06808	0.269242	0.134621	−	0.990897i	$-0.457018\pi$
$59$	2.06808	0.269242	0.134621	+	0.990897i	$0.457018\pi$
$60$	0	0
$61$	−1.27349	−0.163054	−0.0815271	−	0.996671i	$-0.525980\pi$
$61$	−1.27349	−0.163054	−0.0815271	+	0.996671i	$0.525980\pi$
$62$	0	0
$63$	−2.63675	−0.332199
$64$	0	0
$65$	−4.43134	−0.549640
$66$	0	0
$67$	7.20541	0.880281	0.440140	−	0.897929i	$-0.354929\pi$
$67$	7.20541	0.880281	0.440140	+	0.897929i	$0.354929\pi$
$68$	0	0
$69$	4.38377	0.527744
$70$	0	0
$71$	−6.79459	−0.806369	−0.403185	−	0.915119i	$-0.632097\pi$
$71$	−6.79459	−0.806369	−0.403185	+	0.915119i	$0.632097\pi$
$72$	0	0
$73$	−1.27349	−0.149051	−0.0745256	−	0.997219i	$-0.523744\pi$
$73$	−1.27349	−0.149051	−0.0745256	+	0.997219i	$0.523744\pi$
$74$	0	0
$75$	0.602705	0.0695944
$76$	0	0
$77$	−5.63675	−0.642367
$78$	0	0
$79$	−13.0816	−1.47180	−0.735898	−	0.677092i	$-0.763240\pi$
$79$	−13.0816	−1.47180	−0.735898	+	0.677092i	$0.763240\pi$
$80$	0	0
$81$	5.86267	0.651408
$82$	0	0
$83$	1.20541	0.132311	0.0661555	−	0.997809i	$-0.478927\pi$
$83$	1.20541	0.132311	0.0661555	+	0.997809i	$0.478927\pi$
$84$	0	0
$85$	−7.46538	−0.809734
$86$	0	0
$87$	4.02052	0.431045
$88$	0	0
$89$	17.2735	1.83099	0.915493	−	0.402333i	$-0.131801\pi$
$89$	17.2735	1.83099	0.915493	+	0.402333i	$0.131801\pi$
$90$	0	0
$91$	−4.43134	−0.464530
$92$	0	0
$93$	6.06808	0.629231
$94$	0	0
$95$	1.20541	0.123672
$96$	0	0
$97$	11.4654	1.16413	0.582066	−	0.813141i	$-0.302244\pi$
$97$	11.4654	1.16413	0.582066	+	0.813141i	$0.302244\pi$
$98$	0	0
$99$	14.8627	1.49375
$100$	0	0
$101$	−4.41082	−0.438893	−0.219446	−	0.975625i	$-0.570425\pi$
$101$	−4.41082	−0.438893	−0.219446	+	0.975625i	$0.570425\pi$
$102$	0	0
$103$	5.56866	0.548697	0.274348	−	0.961630i	$-0.411538\pi$
$103$	5.56866	0.548697	0.274348	+	0.961630i	$0.411538\pi$
$104$	0	0
$105$	0.602705	0.0588180
$106$	0	0
$107$	−18.1362	−1.75329	−0.876645	−	0.481138i	$-0.840224\pi$
$107$	−18.1362	−1.75329	−0.876645	+	0.481138i	$0.840224\pi$
$108$	0	0
$109$	17.0816	1.63612	0.818061	−	0.575132i	$-0.195049\pi$
$109$	17.0816	1.63612	0.818061	+	0.575132i	$0.195049\pi$
$110$	0	0
$111$	−2.93076	−0.278175
$112$	0	0
$113$	−10.4789	−0.985772	−0.492886	−	0.870094i	$-0.664058\pi$
$113$	−10.4789	−0.985772	−0.492886	+	0.870094i	$0.664058\pi$
$114$	0	0
$115$	7.27349	0.678257
$116$	0	0
$117$	11.6843	1.08022
$118$	0	0
$119$	−7.46538	−0.684350
$120$	0	0
$121$	20.7729	1.88845
$122$	0	0
$123$	2.88972	0.260558
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.06808	0.538455	0.269228	−	0.963077i	$-0.413232\pi$
$127$	6.06808	0.538455	0.269228	+	0.963077i	$0.413232\pi$
$128$	0	0
$129$	−1.20541	−0.106130
$130$	0	0
$131$	12.8627	1.12382	0.561909	−	0.827199i	$-0.310067\pi$
$131$	12.8627	1.12382	0.561909	+	0.827199i	$0.310067\pi$
$132$	0	0
$133$	1.20541	0.104522
$134$	0	0
$135$	−3.39730	−0.292393
$136$	0	0
$137$	13.2735	1.13403	0.567016	−	0.823707i	$-0.308098\pi$
$137$	13.2735	1.13403	0.567016	+	0.823707i	$0.308098\pi$
$138$	0	0
$139$	−4.86267	−0.412447	−0.206223	−	0.978505i	$-0.566117\pi$
$139$	−4.86267	−0.412447	−0.206223	+	0.978505i	$0.566117\pi$
$140$	0	0
$141$	−3.87620	−0.326435
$142$	0	0
$143$	24.9783	2.08879
$144$	0	0
$145$	6.67079	0.553979
$146$	0	0
$147$	0.602705	0.0497103
$148$	0	0
$149$	20.1362	1.64962	0.824810	−	0.565411i	$-0.191282\pi$
$149$	20.1362	1.64962	0.824810	+	0.565411i	$0.191282\pi$
$150$	0	0
$151$	8.25997	0.672187	0.336093	−	0.941829i	$-0.390894\pi$
$151$	8.25997	0.672187	0.336093	+	0.941829i	$0.390894\pi$
$152$	0	0
$153$	19.6843	1.59138
$154$	0	0
$155$	10.0681	0.808688
$156$	0	0
$157$	−16.4108	−1.30973	−0.654863	−	0.755748i	$-0.727274\pi$
$157$	−16.4108	−1.30973	−0.654863	+	0.755748i	$0.727274\pi$
$158$	0	0
$159$	1.24644	0.0988495
$160$	0	0
$161$	7.27349	0.573232
$162$	0	0
$163$	10.4789	0.820771	0.410386	−	0.911912i	$-0.365394\pi$
$163$	10.4789	0.820771	0.410386	+	0.911912i	$0.365394\pi$
$164$	0	0
$165$	−3.39730	−0.264479
$166$	0	0
$167$	−2.39030	−0.184967	−0.0924836	−	0.995714i	$-0.529481\pi$
$167$	−2.39030	−0.184967	−0.0924836	+	0.995714i	$0.529481\pi$
$168$	0	0
$169$	6.63675	0.510519
$170$	0	0
$171$	−3.17836	−0.243055
$172$	0	0
$173$	20.5675	1.56372	0.781859	−	0.623455i	$-0.214272\pi$
$173$	20.5675	1.56372	0.781859	+	0.623455i	$0.214272\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	1.24644	0.0936885
$178$	0	0
$179$	−10.8627	−0.811914	−0.405957	−	0.913892i	$-0.633062\pi$
$179$	−10.8627	−0.811914	−0.405957	+	0.913892i	$0.633062\pi$
$180$	0	0
$181$	−21.7524	−1.61684	−0.808421	−	0.588604i	$-0.799678\pi$
$181$	−21.7524	−1.61684	−0.808421	+	0.588604i	$0.799678\pi$
$182$	0	0
$183$	−0.767541	−0.0567383
$184$	0	0
$185$	−4.86267	−0.357511
$186$	0	0
$187$	42.0804	3.07723
$188$	0	0
$189$	−3.39730	−0.247117
$190$	0	0
$191$	7.05456	0.510450	0.255225	−	0.966882i	$-0.417850\pi$
$191$	7.05456	0.510450	0.255225	+	0.966882i	$0.417850\pi$
$192$	0	0
$193$	3.54815	0.255401	0.127701	−	0.991813i	$-0.459240\pi$
$193$	3.54815	0.255401	0.127701	+	0.991813i	$0.459240\pi$
$194$	0	0
$195$	−2.67079	−0.191259
$196$	0	0
$197$	22.8897	1.63083	0.815413	−	0.578880i	$-0.196510\pi$
$197$	22.8897	1.63083	0.815413	+	0.578880i	$0.196510\pi$
$198$	0	0
$199$	20.6151	1.46136	0.730682	−	0.682718i	$-0.239202\pi$
$199$	20.6151	1.46136	0.730682	+	0.682718i	$0.239202\pi$
$200$	0	0
$201$	4.34274	0.306313
$202$	0	0
$203$	6.67079	0.468198
$204$	0	0
$205$	4.79459	0.334869
$206$	0	0
$207$	−19.1784	−1.33299
$208$	0	0
$209$	−6.79459	−0.469992
$210$	0	0
$211$	−3.95243	−0.272097	−0.136048	−	0.990702i	$-0.543440\pi$
$211$	−3.95243	−0.272097	−0.136048	+	0.990702i	$0.543440\pi$
$212$	0	0
$213$	−4.09513	−0.280594
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	10.0681	0.683466
$218$	0	0
$219$	−0.767541	−0.0518656
$220$	0	0
$221$	33.0816	2.22531
$222$	0	0
$223$	2.39030	0.160066	0.0800332	−	0.996792i	$-0.474497\pi$
$223$	2.39030	0.160066	0.0800332	+	0.996792i	$0.474497\pi$
$224$	0	0
$225$	−2.63675	−0.175783
$226$	0	0
$227$	−0.738872	−0.0490407	−0.0245203	−	0.999699i	$-0.507806\pi$
$227$	−0.738872	−0.0490407	−0.0245203	+	0.999699i	$0.507806\pi$
$228$	0	0
$229$	12.9308	0.854489	0.427244	−	0.904136i	$-0.359484\pi$
$229$	12.9308	0.854489	0.427244	+	0.904136i	$0.359484\pi$
$230$	0	0
$231$	−3.39730	−0.223526
$232$	0	0
$233$	−18.9988	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$233$	−18.9988	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$234$	0	0
$235$	−6.43134	−0.419534
$236$	0	0
$237$	−7.88435	−0.512144
$238$	0	0
$239$	−18.6708	−1.20771	−0.603856	−	0.797093i	$-0.706370\pi$
$239$	−18.6708	−1.20771	−0.603856	+	0.797093i	$0.706370\pi$
$240$	0	0
$241$	11.5892	0.746525	0.373262	−	0.927726i	$-0.378239\pi$
$241$	11.5892	0.746525	0.373262	+	0.927726i	$0.378239\pi$
$242$	0	0
$243$	13.7253	0.880481
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−5.34158	−0.339876
$248$	0	0
$249$	0.726506	0.0460405
$250$	0	0
$251$	4.86267	0.306929	0.153465	−	0.988154i	$-0.450957\pi$
$251$	4.86267	0.306929	0.153465	+	0.988154i	$0.450957\pi$
$252$	0	0
$253$	−40.9988	−2.57758
$254$	0	0
$255$	−4.49942	−0.281765
$256$	0	0
$257$	15.6843	0.978361	0.489180	−	0.872183i	$-0.337296\pi$
$257$	15.6843	0.978361	0.489180	+	0.872183i	$0.337296\pi$
$258$	0	0
$259$	−4.86267	−0.302152
$260$	0	0
$261$	−17.5892	−1.08874
$262$	0	0
$263$	17.2054	1.06093	0.530466	−	0.847707i	$-0.322017\pi$
$263$	17.2054	1.06093	0.530466	+	0.847707i	$0.322017\pi$
$264$	0	0
$265$	2.06808	0.127041
$266$	0	0
$267$	10.4108	0.637132
$268$	0	0
$269$	−14.4789	−0.882794	−0.441397	−	0.897312i	$-0.645517\pi$
$269$	−14.4789	−0.882794	−0.441397	+	0.897312i	$0.645517\pi$
$270$	0	0
$271$	5.58918	0.339519	0.169759	−	0.985486i	$-0.445701\pi$
$271$	5.58918	0.339519	0.169759	+	0.985486i	$0.445701\pi$
$272$	0	0
$273$	−2.67079	−0.161643
$274$	0	0
$275$	−5.63675	−0.339909
$276$	0	0
$277$	3.27349	0.196685	0.0983426	−	0.995153i	$-0.468646\pi$
$277$	3.27349	0.196685	0.0983426	+	0.995153i	$0.468646\pi$
$278$	0	0
$279$	−26.5470	−1.58933
$280$	0	0
$281$	3.95243	0.235782	0.117891	−	0.993027i	$-0.462387\pi$
$281$	3.95243	0.235782	0.117891	+	0.993027i	$0.462387\pi$
$282$	0	0
$283$	−18.0804	−1.07477	−0.537386	−	0.843337i	$-0.680588\pi$
$283$	−18.0804	−1.07477	−0.537386	+	0.843337i	$0.680588\pi$
$284$	0	0
$285$	0.726506	0.0430345
$286$	0	0
$287$	4.79459	0.283016
$288$	0	0
$289$	38.7319	2.27835
$290$	0	0
$291$	6.91024	0.405085
$292$	0	0
$293$	−2.84216	−0.166041	−0.0830203	−	0.996548i	$-0.526457\pi$
$293$	−2.84216	−0.166041	−0.0830203	+	0.996548i	$0.526457\pi$
$294$	0	0
$295$	2.06808	0.120408
$296$	0	0
$297$	19.1497	1.11118
$298$	0	0
$299$	−32.2313	−1.86398
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	−2.65842	−0.152722
$304$	0	0
$305$	−1.27349	−0.0729200
$306$	0	0
$307$	8.73887	0.498754	0.249377	−	0.968407i	$-0.419774\pi$
$307$	8.73887	0.498754	0.249377	+	0.968407i	$0.419774\pi$
$308$	0	0
$309$	3.35626	0.190931
$310$	0	0
$311$	−15.1373	−0.858359	−0.429180	−	0.903219i	$-0.641197\pi$
$311$	−15.1373	−0.858359	−0.429180	+	0.903219i	$0.641197\pi$
$312$	0	0
$313$	−0.150851	−0.00852659	−0.00426330	−	0.999991i	$-0.501357\pi$
$313$	−0.150851	−0.00852659	−0.00426330	+	0.999991i	$0.501357\pi$
$314$	0	0
$315$	−2.63675	−0.148564
$316$	0	0
$317$	−22.1632	−1.24481	−0.622405	−	0.782695i	$-0.713844\pi$
$317$	−22.1632	−1.24481	−0.622405	+	0.782695i	$0.713844\pi$
$318$	0	0
$319$	−37.6015	−2.10528
$320$	0	0
$321$	−10.9308	−0.610095
$322$	0	0
$323$	−8.99884	−0.500709
$324$	0	0
$325$	−4.43134	−0.245806
$326$	0	0
$327$	10.2952	0.569324
$328$	0	0
$329$	−6.43134	−0.354571
$330$	0	0
$331$	32.5880	1.79120	0.895600	−	0.444861i	$-0.146747\pi$
$331$	32.5880	1.79120	0.895600	+	0.444861i	$0.146747\pi$
$332$	0	0
$333$	12.8216	0.702622
$334$	0	0
$335$	7.20541	0.393674
$336$	0	0
$337$	22.4789	1.22450	0.612252	−	0.790663i	$-0.290264\pi$
$337$	22.4789	1.22450	0.612252	+	0.790663i	$0.290264\pi$
$338$	0	0
$339$	−6.31569	−0.343021
$340$	0	0
$341$	−56.7512	−3.07325
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	4.38377	0.236014
$346$	0	0
$347$	−25.0259	−1.34346	−0.671730	−	0.740796i	$-0.734449\pi$
$347$	−25.0259	−1.34346	−0.671730	+	0.740796i	$0.734449\pi$
$348$	0	0
$349$	−13.3145	−0.712710	−0.356355	−	0.934351i	$-0.615981\pi$
$349$	−13.3145	−0.712710	−0.356355	+	0.934351i	$0.615981\pi$
$350$	0	0
$351$	15.0546	0.803553
$352$	0	0
$353$	34.8340	1.85403	0.927014	−	0.375028i	$-0.122367\pi$
$353$	34.8340	1.85403	0.927014	+	0.375028i	$0.122367\pi$
$354$	0	0
$355$	−6.79459	−0.360619
$356$	0	0
$357$	−4.49942	−0.238135
$358$	0	0
$359$	−8.51994	−0.449665	−0.224833	−	0.974397i	$-0.572184\pi$
$359$	−8.51994	−0.449665	−0.224833	+	0.974397i	$0.572184\pi$
$360$	0	0
$361$	−17.5470	−0.923526
$362$	0	0
$363$	12.5199	0.657126
$364$	0	0
$365$	−1.27349	−0.0666577
$366$	0	0
$367$	8.02052	0.418668	0.209334	−	0.977844i	$-0.432870\pi$
$367$	8.02052	0.418668	0.209334	+	0.977844i	$0.432870\pi$
$368$	0	0
$369$	−12.6421	−0.658122
$370$	0	0
$371$	2.06808	0.107370
$372$	0	0
$373$	−9.58918	−0.496509	−0.248254	−	0.968695i	$-0.579857\pi$
$373$	−9.58918	−0.496509	−0.248254	+	0.968695i	$0.579857\pi$
$374$	0	0
$375$	0.602705	0.0311235
$376$	0	0
$377$	−29.5605	−1.52244
$378$	0	0
$379$	12.4519	0.639609	0.319804	−	0.947484i	$-0.396383\pi$
$379$	12.4519	0.639609	0.319804	+	0.947484i	$0.396383\pi$
$380$	0	0
$381$	3.65726	0.187367
$382$	0	0
$383$	−21.8615	−1.11707	−0.558536	−	0.829481i	$-0.688637\pi$
$383$	−21.8615	−1.11707	−0.558536	+	0.829481i	$0.688637\pi$
$384$	0	0
$385$	−5.63675	−0.287275
$386$	0	0
$387$	5.27349	0.268067
$388$	0	0
$389$	−5.46538	−0.277106	−0.138553	−	0.990355i	$-0.544245\pi$
$389$	−5.46538	−0.277106	−0.138553	+	0.990355i	$0.544245\pi$
$390$	0	0
$391$	−54.2994	−2.74604
$392$	0	0
$393$	7.75240	0.391057
$394$	0	0
$395$	−13.0816	−0.658207
$396$	0	0
$397$	22.2518	1.11679	0.558393	−	0.829576i	$-0.311418\pi$
$397$	22.2518	1.11679	0.558393	+	0.829576i	$0.311418\pi$
$398$	0	0
$399$	0.726506	0.0363708
$400$	0	0
$401$	−0.910240	−0.0454552	−0.0227276	−	0.999742i	$-0.507235\pi$
$401$	−0.910240	−0.0454552	−0.0227276	+	0.999742i	$0.507235\pi$
$402$	0	0
$403$	−44.6151	−2.22243
$404$	0	0
$405$	5.86267	0.291319
$406$	0	0
$407$	27.4097	1.35865
$408$	0	0
$409$	6.47890	0.320361	0.160181	−	0.987088i	$-0.448792\pi$
$409$	6.47890	0.320361	0.160181	+	0.987088i	$0.448792\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	2.06808	0.101764
$414$	0	0
$415$	1.20541	0.0591712
$416$	0	0
$417$	−2.93076	−0.143520
$418$	0	0
$419$	−28.4789	−1.39129	−0.695643	−	0.718388i	$-0.744880\pi$
$419$	−28.4789	−1.39129	−0.695643	+	0.718388i	$0.744880\pi$
$420$	0	0
$421$	9.46538	0.461314	0.230657	−	0.973035i	$-0.425912\pi$
$421$	9.46538	0.461314	0.230657	+	0.973035i	$0.425912\pi$
$422$	0	0
$423$	16.9578	0.824517
$424$	0	0
$425$	−7.46538	−0.362124
$426$	0	0
$427$	−1.27349	−0.0616287
$428$	0	0
$429$	15.0546	0.726841
$430$	0	0
$431$	24.7799	1.19361	0.596803	−	0.802388i	$-0.296437\pi$
$431$	24.7799	1.19361	0.596803	+	0.802388i	$0.296437\pi$
$432$	0	0
$433$	28.4519	1.36731	0.683654	−	0.729806i	$-0.260390\pi$
$433$	28.4519	1.36731	0.683654	+	0.729806i	$0.260390\pi$
$434$	0	0
$435$	4.02052	0.192769
$436$	0	0
$437$	8.76754	0.419408
$438$	0	0
$439$	18.7946	0.897017	0.448508	−	0.893779i	$-0.351955\pi$
$439$	18.7946	0.897017	0.448508	+	0.893779i	$0.351955\pi$
$440$	0	0
$441$	−2.63675	−0.125559
$442$	0	0
$443$	−5.23246	−0.248602	−0.124301	−	0.992245i	$-0.539669\pi$
$443$	−5.23246	−0.248602	−0.124301	+	0.992245i	$0.539669\pi$
$444$	0	0
$445$	17.2735	0.818842
$446$	0	0
$447$	12.1362	0.574021
$448$	0	0
$449$	14.4584	0.682333	0.341167	−	0.940003i	$-0.389178\pi$
$449$	14.4584	0.682333	0.341167	+	0.940003i	$0.389178\pi$
$450$	0	0
$451$	−27.0259	−1.27260
$452$	0	0
$453$	4.97832	0.233902
$454$	0	0
$455$	−4.43134	−0.207744
$456$	0	0
$457$	28.9308	1.35332	0.676662	−	0.736294i	$-0.263426\pi$
$457$	28.9308	1.35332	0.676662	+	0.736294i	$0.263426\pi$
$458$	0	0
$459$	25.3621	1.18380
$460$	0	0
$461$	−35.1350	−1.63640	−0.818200	−	0.574933i	$-0.805028\pi$
$461$	−35.1350	−1.63640	−0.818200	+	0.574933i	$0.805028\pi$
$462$	0	0
$463$	24.9578	1.15989	0.579944	−	0.814656i	$-0.303074\pi$
$463$	24.9578	1.15989	0.579944	+	0.814656i	$0.303074\pi$
$464$	0	0
$465$	6.06808	0.281401
$466$	0	0
$467$	−13.8081	−0.638963	−0.319482	−	0.947592i	$-0.603509\pi$
$467$	−13.8081	−0.638963	−0.319482	+	0.947592i	$0.603509\pi$
$468$	0	0
$469$	7.20541	0.332715
$470$	0	0
$471$	−9.89088	−0.455748
$472$	0	0
$473$	11.2735	0.518356
$474$	0	0
$475$	1.20541	0.0553080
$476$	0	0
$477$	−5.45301	−0.249676
$478$	0	0
$479$	19.7524	0.902510	0.451255	−	0.892395i	$-0.350976\pi$
$479$	19.7524	0.902510	0.451255	+	0.892395i	$0.350976\pi$
$480$	0	0
$481$	21.5481	0.982511
$482$	0	0
$483$	4.38377	0.199468
$484$	0	0
$485$	11.4654	0.520616
$486$	0	0
$487$	33.6843	1.52638	0.763191	−	0.646173i	$-0.223632\pi$
$487$	33.6843	1.52638	0.763191	+	0.646173i	$0.223632\pi$
$488$	0	0
$489$	6.31569	0.285605
$490$	0	0
$491$	−21.7729	−0.982598	−0.491299	−	0.870991i	$-0.663478\pi$
$491$	−21.7729	−0.982598	−0.491299	+	0.870991i	$0.663478\pi$
$492$	0	0
$493$	−49.8000	−2.24288
$494$	0	0
$495$	14.8627	0.668028
$496$	0	0
$497$	−6.79459	−0.304779
$498$	0	0
$499$	−20.8692	−0.934234	−0.467117	−	0.884196i	$-0.654707\pi$
$499$	−20.8692	−0.934234	−0.467117	+	0.884196i	$0.654707\pi$
$500$	0	0
$501$	−1.44065	−0.0643634
$502$	0	0
$503$	−15.1578	−0.675855	−0.337927	−	0.941172i	$-0.609726\pi$
$503$	−15.1578	−0.675855	−0.337927	+	0.941172i	$0.609726\pi$
$504$	0	0
$505$	−4.41082	−0.196279
$506$	0	0
$507$	4.00000	0.177646
$508$	0	0
$509$	10.4789	0.464469	0.232235	−	0.972660i	$-0.425396\pi$
$509$	10.4789	0.464469	0.232235	+	0.972660i	$0.425396\pi$
$510$	0	0
$511$	−1.27349	−0.0563360
$512$	0	0
$513$	−4.09513	−0.180805
$514$	0	0
$515$	5.56866	0.245385
$516$	0	0
$517$	36.2518	1.59435
$518$	0	0
$519$	12.3961	0.544130
$520$	0	0
$521$	21.7934	0.954788	0.477394	−	0.878689i	$-0.341582\pi$
$521$	21.7934	0.954788	0.477394	+	0.878689i	$0.341582\pi$
$522$	0	0
$523$	2.79459	0.122199	0.0610994	−	0.998132i	$-0.480539\pi$
$523$	2.79459	0.122199	0.0610994	+	0.998132i	$0.480539\pi$
$524$	0	0
$525$	0.602705	0.0263042
$526$	0	0
$527$	−75.1621	−3.27411
$528$	0	0
$529$	29.9037	1.30016
$530$	0	0
$531$	−5.45301	−0.236641
$532$	0	0
$533$	−21.2464	−0.920286
$534$	0	0
$535$	−18.1362	−0.784095
$536$	0	0
$537$	−6.54699	−0.282523
$538$	0	0
$539$	−5.63675	−0.242792
$540$	0	0
$541$	23.0546	0.991193	0.495596	−	0.868553i	$-0.334950\pi$
$541$	23.0546	0.991193	0.495596	+	0.868553i	$0.334950\pi$
$542$	0	0
$543$	−13.1103	−0.562616
$544$	0	0
$545$	17.0816	0.731696
$546$	0	0
$547$	−14.7265	−0.629660	−0.314830	−	0.949148i	$-0.601947\pi$
$547$	−14.7265	−0.629660	−0.314830	+	0.949148i	$0.601947\pi$
$548$	0	0
$549$	3.35788	0.143311
$550$	0	0
$551$	8.04103	0.342560
$552$	0	0
$553$	−13.0816	−0.556287
$554$	0	0
$555$	−2.93076	−0.124404
$556$	0	0
$557$	28.9578	1.22698	0.613491	−	0.789702i	$-0.289765\pi$
$557$	28.9578	1.22698	0.613491	+	0.789702i	$0.289765\pi$
$558$	0	0
$559$	8.86267	0.374851
$560$	0	0
$561$	25.3621	1.07079
$562$	0	0
$563$	−23.6162	−0.995305	−0.497653	−	0.867376i	$-0.665805\pi$
$563$	−23.6162	−0.995305	−0.497653	+	0.867376i	$0.665805\pi$
$564$	0	0
$565$	−10.4789	−0.440851
$566$	0	0
$567$	5.86267	0.246209
$568$	0	0
$569$	−13.2325	−0.554734	−0.277367	−	0.960764i	$-0.589462\pi$
$569$	−13.2325	−0.554734	−0.277367	+	0.960764i	$0.589462\pi$
$570$	0	0
$571$	5.13733	0.214990	0.107495	−	0.994206i	$-0.465717\pi$
$571$	5.13733	0.214990	0.107495	+	0.994206i	$0.465717\pi$
$572$	0	0
$573$	4.25182	0.177622
$574$	0	0
$575$	7.27349	0.303326
$576$	0	0
$577$	21.7400	0.905049	0.452525	−	0.891752i	$-0.350523\pi$
$577$	21.7400	0.905049	0.452525	+	0.891752i	$0.350523\pi$
$578$	0	0
$579$	2.13849	0.0888724
$580$	0	0
$581$	1.20541	0.0500088
$582$	0	0
$583$	−11.6573	−0.482795
$584$	0	0
$585$	11.6843	0.483087
$586$	0	0
$587$	−18.6584	−0.770116	−0.385058	−	0.922892i	$-0.625818\pi$
$587$	−18.6584	−0.770116	−0.385058	+	0.922892i	$0.625818\pi$
$588$	0	0
$589$	12.1362	0.500062
$590$	0	0
$591$	13.7957	0.567481
$592$	0	0
$593$	8.67079	0.356067	0.178033	−	0.984024i	$-0.443027\pi$
$593$	8.67079	0.356067	0.178033	+	0.984024i	$0.443027\pi$
$594$	0	0
$595$	−7.46538	−0.306051
$596$	0	0
$597$	12.4248	0.508513
$598$	0	0
$599$	29.3292	1.19836	0.599180	−	0.800615i	$-0.295494\pi$
$599$	29.3292	1.19836	0.599180	+	0.800615i	$0.295494\pi$
$600$	0	0
$601$	−39.5188	−1.61200	−0.806002	−	0.591912i	$-0.798373\pi$
$601$	−39.5188	−1.61200	−0.806002	+	0.591912i	$0.798373\pi$
$602$	0	0
$603$	−18.9988	−0.773693
$604$	0	0
$605$	20.7729	0.844539
$606$	0	0
$607$	−35.3891	−1.43640	−0.718201	−	0.695836i	$-0.755034\pi$
$607$	−35.3891	−1.43640	−0.718201	+	0.695836i	$0.755034\pi$
$608$	0	0
$609$	4.02052	0.162920
$610$	0	0
$611$	28.4994	1.15296
$612$	0	0
$613$	−44.6561	−1.80364	−0.901822	−	0.432109i	$-0.857770\pi$
$613$	−44.6561	−1.80364	−0.901822	+	0.432109i	$0.857770\pi$
$614$	0	0
$615$	2.88972	0.116525
$616$	0	0
$617$	3.16438	0.127393	0.0636965	−	0.997969i	$-0.479711\pi$
$617$	3.16438	0.127393	0.0636965	+	0.997969i	$0.479711\pi$
$618$	0	0
$619$	44.3134	1.78110	0.890552	−	0.454881i	$-0.150318\pi$
$619$	44.3134	1.78110	0.890552	+	0.454881i	$0.150318\pi$
$620$	0	0
$621$	−24.7102	−0.991586
$622$	0	0
$623$	17.2735	0.692048
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.09513	−0.163544
$628$	0	0
$629$	36.3017	1.44744
$630$	0	0
$631$	−42.6708	−1.69870	−0.849349	−	0.527832i	$-0.823005\pi$
$631$	−42.6708	−1.69870	−0.849349	+	0.527832i	$0.823005\pi$
$632$	0	0
$633$	−2.38215	−0.0946820
$634$	0	0
$635$	6.06808	0.240805
$636$	0	0
$637$	−4.43134	−0.175576
$638$	0	0
$639$	17.9156	0.708731
$640$	0	0
$641$	−13.1784	−0.520514	−0.260257	−	0.965539i	$-0.583807\pi$
$641$	−13.1784	−0.520514	−0.260257	+	0.965539i	$0.583807\pi$
$642$	0	0
$643$	6.71182	0.264688	0.132344	−	0.991204i	$-0.457750\pi$
$643$	6.71182	0.264688	0.132344	+	0.991204i	$0.457750\pi$
$644$	0	0
$645$	−1.20541	−0.0474630
$646$	0	0
$647$	−17.8615	−0.702209	−0.351104	−	0.936336i	$-0.614194\pi$
$647$	−17.8615	−0.702209	−0.351104	+	0.936336i	$0.614194\pi$
$648$	0	0
$649$	−11.6573	−0.457588
$650$	0	0
$651$	6.06808	0.237827
$652$	0	0
$653$	−26.1632	−1.02385	−0.511923	−	0.859031i	$-0.671067\pi$
$653$	−26.1632	−1.02385	−0.511923	+	0.859031i	$0.671067\pi$
$654$	0	0
$655$	12.8627	0.502586
$656$	0	0
$657$	3.35788	0.131003
$658$	0	0
$659$	30.6766	1.19499	0.597496	−	0.801872i	$-0.296162\pi$
$659$	30.6766	1.19499	0.597496	+	0.801872i	$0.296162\pi$
$660$	0	0
$661$	14.9988	0.583387	0.291694	−	0.956512i	$-0.405781\pi$
$661$	14.9988	0.583387	0.291694	+	0.956512i	$0.405781\pi$
$662$	0	0
$663$	19.9384	0.774345
$664$	0	0
$665$	1.20541	0.0467438
$666$	0	0
$667$	48.5199	1.87870
$668$	0	0
$669$	1.44065	0.0556986
$670$	0	0
$671$	7.17836	0.277118
$672$	0	0
$673$	16.3157	0.628924	0.314462	−	0.949270i	$-0.398176\pi$
$673$	16.3157	0.628924	0.314462	+	0.949270i	$0.398176\pi$
$674$	0	0
$675$	−3.39730	−0.130762
$676$	0	0
$677$	2.88319	0.110810	0.0554050	−	0.998464i	$-0.482355\pi$
$677$	2.88319	0.110810	0.0554050	+	0.998464i	$0.482355\pi$
$678$	0	0
$679$	11.4654	0.440001
$680$	0	0
$681$	−0.445322	−0.0170648
$682$	0	0
$683$	−2.86267	−0.109537	−0.0547686	−	0.998499i	$-0.517442\pi$
$683$	−2.86267	−0.109537	−0.0547686	+	0.998499i	$0.517442\pi$
$684$	0	0
$685$	13.2735	0.507154
$686$	0	0
$687$	7.79343	0.297338
$688$	0	0
$689$	−9.16438	−0.349135
$690$	0	0
$691$	−24.9578	−0.949440	−0.474720	−	0.880137i	$-0.657451\pi$
$691$	−24.9578	−0.949440	−0.474720	+	0.880137i	$0.657451\pi$
$692$	0	0
$693$	14.8627	0.564586
$694$	0	0
$695$	−4.86267	−0.184452
$696$	0	0
$697$	−35.7934	−1.35577
$698$	0	0
$699$	−11.4507	−0.433105
$700$	0	0
$701$	34.0394	1.28565	0.642825	−	0.766013i	$-0.277762\pi$
$701$	34.0394	1.28565	0.642825	+	0.766013i	$0.277762\pi$
$702$	0	0
$703$	−5.86151	−0.221071
$704$	0	0
$705$	−3.87620	−0.145986
$706$	0	0
$707$	−4.41082	−0.165886
$708$	0	0
$709$	−35.7648	−1.34317	−0.671587	−	0.740926i	$-0.734387\pi$
$709$	−35.7648	−1.34317	−0.671587	+	0.740926i	$0.734387\pi$
$710$	0	0
$711$	34.4929	1.29358
$712$	0	0
$713$	73.2301	2.74249
$714$	0	0
$715$	24.9783	0.934136
$716$	0	0
$717$	−11.2530	−0.420250
$718$	0	0
$719$	−5.97295	−0.222753	−0.111377	−	0.993778i	$-0.535526\pi$
$719$	−5.97295	−0.222753	−0.111377	+	0.993778i	$0.535526\pi$
$720$	0	0
$721$	5.56866	0.207388
$722$	0	0
$723$	6.98486	0.259770
$724$	0	0
$725$	6.67079	0.247747
$726$	0	0
$727$	35.5048	1.31680	0.658400	−	0.752668i	$-0.271234\pi$
$727$	35.5048	1.31680	0.658400	+	0.752668i	$0.271234\pi$
$728$	0	0
$729$	−9.31569	−0.345025
$730$	0	0
$731$	14.9308	0.552234
$732$	0	0
$733$	−22.2518	−0.821890	−0.410945	−	0.911660i	$-0.634801\pi$
$733$	−22.2518	−0.821890	−0.410945	+	0.911660i	$0.634801\pi$
$734$	0	0
$735$	0.602705	0.0222311
$736$	0	0
$737$	−40.6151	−1.49608
$738$	0	0
$739$	34.0452	1.25237	0.626187	−	0.779673i	$-0.284615\pi$
$739$	34.0452	1.25237	0.626187	+	0.779673i	$0.284615\pi$
$740$	0	0
$741$	−3.21939	−0.118267
$742$	0	0
$743$	−33.4367	−1.22667	−0.613337	−	0.789821i	$-0.710173\pi$
$743$	−33.4367	−1.22667	−0.613337	+	0.789821i	$0.710173\pi$
$744$	0	0
$745$	20.1362	0.737732
$746$	0	0
$747$	−3.17836	−0.116290
$748$	0	0
$749$	−18.1362	−0.662681
$750$	0	0
$751$	−36.8340	−1.34409	−0.672046	−	0.740509i	$-0.734584\pi$
$751$	−36.8340	−1.34409	−0.672046	+	0.740509i	$0.734584\pi$
$752$	0	0
$753$	2.93076	0.106803
$754$	0	0
$755$	8.25997	0.300611
$756$	0	0
$757$	8.76754	0.318662	0.159331	−	0.987225i	$-0.449066\pi$
$757$	8.76754	0.318662	0.159331	+	0.987225i	$0.449066\pi$
$758$	0	0
$759$	−24.7102	−0.896924
$760$	0	0
$761$	14.2723	0.517372	0.258686	−	0.965962i	$-0.416711\pi$
$761$	14.2723	0.517372	0.258686	+	0.965962i	$0.416711\pi$
$762$	0	0
$763$	17.0816	0.618396
$764$	0	0
$765$	19.6843	0.711688
$766$	0	0
$767$	−9.16438	−0.330906
$768$	0	0
$769$	−28.9718	−1.04475	−0.522375	−	0.852716i	$-0.674954\pi$
$769$	−28.9718	−1.04475	−0.522375	+	0.852716i	$0.674954\pi$
$770$	0	0
$771$	9.45301	0.340442
$772$	0	0
$773$	−9.88435	−0.355515	−0.177758	−	0.984074i	$-0.556884\pi$
$773$	−9.88435	−0.355515	−0.177758	+	0.984074i	$0.556884\pi$
$774$	0	0
$775$	10.0681	0.361656
$776$	0	0
$777$	−2.93076	−0.105140
$778$	0	0
$779$	5.77945	0.207070
$780$	0	0
$781$	38.2994	1.37046
$782$	0	0
$783$	−22.6626	−0.809897
$784$	0	0
$785$	−16.4108	−0.585727
$786$	0	0
$787$	3.91723	0.139634	0.0698171	−	0.997560i	$-0.477758\pi$
$787$	3.91723	0.139634	0.0698171	+	0.997560i	$0.477758\pi$
$788$	0	0
$789$	10.3698	0.369174
$790$	0	0
$791$	−10.4789	−0.372587
$792$	0	0
$793$	5.64328	0.200399
$794$	0	0
$795$	1.24644	0.0442068
$796$	0	0
$797$	−26.0205	−0.921694	−0.460847	−	0.887480i	$-0.652454\pi$
$797$	−26.0205	−0.921694	−0.460847	+	0.887480i	$0.652454\pi$
$798$	0	0
$799$	48.0124	1.69856
$800$	0	0
$801$	−45.5458	−1.60928
$802$	0	0
$803$	7.17836	0.253319
$804$	0	0
$805$	7.27349	0.256357
$806$	0	0
$807$	−8.72651	−0.307188
$808$	0	0
$809$	45.7729	1.60929	0.804645	−	0.593756i	$-0.202356\pi$
$809$	45.7729	1.60929	0.804645	+	0.593756i	$0.202356\pi$
$810$	0	0
$811$	−28.2313	−0.991335	−0.495668	−	0.868512i	$-0.665077\pi$
$811$	−28.2313	−0.991335	−0.495668	+	0.868512i	$0.665077\pi$
$812$	0	0
$813$	3.36863	0.118143
$814$	0	0
$815$	10.4789	0.367060
$816$	0	0
$817$	−2.41082	−0.0843439
$818$	0	0
$819$	11.6843	0.408283
$820$	0	0
$821$	−4.56167	−0.159203	−0.0796017	−	0.996827i	$-0.525365\pi$
$821$	−4.56167	−0.159203	−0.0796017	+	0.996827i	$0.525365\pi$
$822$	0	0
$823$	24.4789	0.853281	0.426640	−	0.904421i	$-0.359697\pi$
$823$	24.4789	0.853281	0.426640	+	0.904421i	$0.359697\pi$
$824$	0	0
$825$	−3.39730	−0.118279
$826$	0	0
$827$	23.3826	0.813093	0.406547	−	0.913630i	$-0.366733\pi$
$827$	23.3826	0.813093	0.406547	+	0.913630i	$0.366733\pi$
$828$	0	0
$829$	9.69830	0.336836	0.168418	−	0.985716i	$-0.446134\pi$
$829$	9.69830	0.336836	0.168418	+	0.985716i	$0.446134\pi$
$830$	0	0
$831$	1.97295	0.0684409
$832$	0	0
$833$	−7.46538	−0.258660
$834$	0	0
$835$	−2.39030	−0.0827198
$836$	0	0
$837$	−34.2043	−1.18227
$838$	0	0
$839$	35.2031	1.21535	0.607673	−	0.794187i	$-0.292103\pi$
$839$	35.2031	1.21535	0.607673	+	0.794187i	$0.292103\pi$
$840$	0	0
$841$	15.4994	0.534463
$842$	0	0
$843$	2.38215	0.0820456
$844$	0	0
$845$	6.63675	0.228311
$846$	0	0
$847$	20.7729	0.713766
$848$	0	0
$849$	−10.8972	−0.373990
$850$	0	0
$851$	−35.3686	−1.21242
$852$	0	0
$853$	19.6433	0.672573	0.336287	−	0.941760i	$-0.390829\pi$
$853$	19.6433	0.672573	0.336287	+	0.941760i	$0.390829\pi$
$854$	0	0
$855$	−3.17836	−0.108698
$856$	0	0
$857$	−14.0410	−0.479633	−0.239816	−	0.970818i	$-0.577087\pi$
$857$	−14.0410	−0.479633	−0.239816	+	0.970818i	$0.577087\pi$
$858$	0	0
$859$	−53.3663	−1.82083	−0.910417	−	0.413691i	$-0.864239\pi$
$859$	−53.3663	−1.82083	−0.910417	+	0.413691i	$0.864239\pi$
$860$	0	0
$861$	2.88972	0.0984815
$862$	0	0
$863$	−33.5188	−1.14099	−0.570496	−	0.821300i	$-0.693249\pi$
$863$	−33.5188	−1.14099	−0.570496	+	0.821300i	$0.693249\pi$
$864$	0	0
$865$	20.5675	0.699316
$866$	0	0
$867$	23.3439	0.792800
$868$	0	0
$869$	73.7377	2.50138
$870$	0	0
$871$	−31.9296	−1.08189
$872$	0	0
$873$	−30.2313	−1.02317
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−1.97295	−0.0666218	−0.0333109	−	0.999445i	$-0.510605\pi$
$877$	−1.97295	−0.0666218	−0.0333109	+	0.999445i	$0.510605\pi$
$878$	0	0
$879$	−1.71298	−0.0577774
$880$	0	0
$881$	−13.9590	−0.470290	−0.235145	−	0.971960i	$-0.575556\pi$
$881$	−13.9590	−0.470290	−0.235145	+	0.971960i	$0.575556\pi$
$882$	0	0
$883$	−25.5458	−0.859686	−0.429843	−	0.902904i	$-0.641431\pi$
$883$	−25.5458	−0.859686	−0.429843	+	0.902904i	$0.641431\pi$
$884$	0	0
$885$	1.24644	0.0418988
$886$	0	0
$887$	−17.8615	−0.599731	−0.299865	−	0.953981i	$-0.596942\pi$
$887$	−17.8615	−0.599731	−0.299865	+	0.953981i	$0.596942\pi$
$888$	0	0
$889$	6.06808	0.203517
$890$	0	0
$891$	−33.0464	−1.10710
$892$	0	0
$893$	−7.75240	−0.259424
$894$	0	0
$895$	−10.8627	−0.363099
$896$	0	0
$897$	−19.4260	−0.648614
$898$	0	0
$899$	67.1621	2.23998
$900$	0	0
$901$	−15.4390	−0.514349
$902$	0	0
$903$	−1.20541	−0.0401135
$904$	0	0
$905$	−21.7524	−0.723074
$906$	0	0
$907$	−20.0681	−0.666350	−0.333175	−	0.942865i	$-0.608120\pi$
$907$	−20.0681	−0.666350	−0.333175	+	0.942865i	$0.608120\pi$
$908$	0	0
$909$	11.6302	0.385750
$910$	0	0
$911$	−6.10912	−0.202404	−0.101202	−	0.994866i	$-0.532269\pi$
$911$	−6.10912	−0.202404	−0.101202	+	0.994866i	$0.532269\pi$
$912$	0	0
$913$	−6.79459	−0.224868
$914$	0	0
$915$	−0.767541	−0.0253741
$916$	0	0
$917$	12.8627	0.424763
$918$	0	0
$919$	0.779907	0.0257267	0.0128634	−	0.999917i	$-0.495905\pi$
$919$	0.779907	0.0257267	0.0128634	+	0.999917i	$0.495905\pi$
$920$	0	0
$921$	5.26696	0.173552
$922$	0	0
$923$	30.1091	0.991054
$924$	0	0
$925$	−4.86267	−0.159884
$926$	0	0
$927$	−14.6832	−0.482258
$928$	0	0
$929$	16.2453	0.532991	0.266495	−	0.963836i	$-0.414134\pi$
$929$	16.2453	0.532991	0.266495	+	0.963836i	$0.414134\pi$
$930$	0	0
$931$	1.20541	0.0395057
$932$	0	0
$933$	−9.12334	−0.298685
$934$	0	0
$935$	42.0804	1.37618
$936$	0	0
$937$	−40.2870	−1.31612	−0.658060	−	0.752966i	$-0.728623\pi$
$937$	−40.2870	−1.31612	−0.658060	+	0.752966i	$0.728623\pi$
$938$	0	0
$939$	−0.0909185	−0.00296701
$940$	0	0
$941$	3.63021	0.118342	0.0591708	−	0.998248i	$-0.481154\pi$
$941$	3.63021	0.118342	0.0591708	+	0.998248i	$0.481154\pi$
$942$	0	0
$943$	34.8734	1.13563
$944$	0	0
$945$	−3.39730	−0.110514
$946$	0	0
$947$	−46.0247	−1.49560	−0.747801	−	0.663922i	$-0.768890\pi$
$947$	−46.0247	−1.49560	−0.747801	+	0.663922i	$0.768890\pi$
$948$	0	0
$949$	5.64328	0.183189
$950$	0	0
$951$	−13.3579	−0.433159
$952$	0	0
$953$	−1.75240	−0.0567657	−0.0283829	−	0.999597i	$-0.509036\pi$
$953$	−1.75240	−0.0567657	−0.0283829	+	0.999597i	$0.509036\pi$
$954$	0	0
$955$	7.05456	0.228280
$956$	0	0
$957$	−22.6626	−0.732579
$958$	0	0
$959$	13.2735	0.428623
$960$	0	0
$961$	70.3663	2.26988
$962$	0	0
$963$	47.8205	1.54099
$964$	0	0
$965$	3.54815	0.114219
$966$	0	0
$967$	−27.6162	−0.888078	−0.444039	−	0.896007i	$-0.646455\pi$
$967$	−27.6162	−0.888078	−0.444039	+	0.896007i	$0.646455\pi$
$968$	0	0
$969$	−5.42365	−0.174233
$970$	0	0
$971$	0.615071	0.0197386	0.00986928	−	0.999951i	$-0.496858\pi$
$971$	0.615071	0.0197386	0.00986928	+	0.999951i	$0.496858\pi$
$972$	0	0
$973$	−4.86267	−0.155890
$974$	0	0
$975$	−2.67079	−0.0855337
$976$	0	0
$977$	31.0940	0.994784	0.497392	−	0.867526i	$-0.334291\pi$
$977$	31.0940	0.994784	0.497392	+	0.867526i	$0.334291\pi$
$978$	0	0
$979$	−97.3663	−3.11184
$980$	0	0
$981$	−45.0399	−1.43801
$982$	0	0
$983$	−10.4313	−0.332708	−0.166354	−	0.986066i	$-0.553199\pi$
$983$	−10.4313	−0.332708	−0.166354	+	0.986066i	$0.553199\pi$
$984$	0	0
$985$	22.8897	0.729327
$986$	0	0
$987$	−3.87620	−0.123381
$988$	0	0
$989$	−14.5470	−0.462567
$990$	0	0
$991$	−1.89088	−0.0600658	−0.0300329	−	0.999549i	$-0.509561\pi$
$991$	−1.89088	−0.0600658	−0.0300329	+	0.999549i	$0.509561\pi$
$992$	0	0
$993$	19.6410	0.623287
$994$	0	0
$995$	20.6151	0.653542
$996$	0	0
$997$	21.1578	0.670076	0.335038	−	0.942205i	$-0.391251\pi$
$997$	21.1578	0.670076	0.335038	+	0.942205i	$0.391251\pi$
$998$	0	0
$999$	16.5199	0.522668

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8960.2.a.bq.1.2		3
4.3	odd	2		8960.2.a.bn.1.2		3
8.3	odd	2		8960.2.a.bh.1.2		3
8.5	even	2		8960.2.a.bk.1.2		3
16.3	odd	4		2240.2.b.f.1121.3	yes	6
16.5	even	4		2240.2.b.e.1121.3	✓	6
16.11	odd	4		2240.2.b.f.1121.4	yes	6
16.13	even	4		2240.2.b.e.1121.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.b.e.1121.3	✓	6	16.5	even	4
2240.2.b.e.1121.4	yes	6	16.13	even	4
2240.2.b.f.1121.3	yes	6	16.3	odd	4
2240.2.b.f.1121.4	yes	6	16.11	odd	4
8960.2.a.bh.1.2		3	8.3	odd	2
8960.2.a.bk.1.2		3	8.5	even	2
8960.2.a.bn.1.2		3	4.3	odd	2
8960.2.a.bq.1.2		3	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:	\( N \)	\(=\)	\( 8960 = 2^{8} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8960.a (trivial)

\(f(q)\)	\(=\)	\(q+0.602705 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.63675 q^{9} +O(q^{10})\)	\(q+0.602705 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.63675 q^{9} -5.63675 q^{11} -4.43134 q^{13} +0.602705 q^{15} -7.46538 q^{17} +1.20541 q^{19} +0.602705 q^{21} +7.27349 q^{23} +1.00000 q^{25} -3.39730 q^{27} +6.67079 q^{29} +10.0681 q^{31} -3.39730 q^{33} +1.00000 q^{35} -4.86267 q^{37} -2.67079 q^{39} +4.79459 q^{41} -2.00000 q^{43} -2.63675 q^{45} -6.43134 q^{47} +1.00000 q^{49} -4.49942 q^{51} +2.06808 q^{53} -5.63675 q^{55} +0.726506 q^{57} +2.06808 q^{59} -1.27349 q^{61} -2.63675 q^{63} -4.43134 q^{65} +7.20541 q^{67} +4.38377 q^{69} -6.79459 q^{71} -1.27349 q^{73} +0.602705 q^{75} -5.63675 q^{77} -13.0816 q^{79} +5.86267 q^{81} +1.20541 q^{83} -7.46538 q^{85} +4.02052 q^{87} +17.2735 q^{89} -4.43134 q^{91} +6.06808 q^{93} +1.20541 q^{95} +11.4654 q^{97} +14.8627 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) \) 3 * q + 3 * q^5 + 3 * q^7 + 5 * q^9	\( 3 q + 3 q^{5} + 3 q^{7} + 5 q^{9} - 4 q^{11} - 4 q^{13} - 2 q^{17} - 4 q^{23} + 3 q^{25} - 12 q^{27} - 4 q^{29} + 8 q^{31} - 12 q^{33} + 3 q^{35} + 4 q^{37} + 16 q^{39} + 18 q^{41} - 6 q^{43} + 5 q^{45} - 10 q^{47} + 3 q^{49} + 18 q^{51} - 16 q^{53} - 4 q^{55} + 28 q^{57} - 16 q^{59} + 22 q^{61} + 5 q^{63} - 4 q^{65} + 18 q^{67} + 24 q^{69} - 24 q^{71} + 22 q^{73} - 4 q^{77} - 8 q^{79} - q^{81} - 2 q^{85} + 10 q^{87} + 26 q^{89} - 4 q^{91} - 4 q^{93} + 14 q^{97} + 26 q^{99}+O(q^{100}) \) 3 * q + 3 * q^5 + 3 * q^7 + 5 * q^9 - 4 * q^11 - 4 * q^13 - 2 * q^17 - 4 * q^23 + 3 * q^25 - 12 * q^27 - 4 * q^29 + 8 * q^31 - 12 * q^33 + 3 * q^35 + 4 * q^37 + 16 * q^39 + 18 * q^41 - 6 * q^43 + 5 * q^45 - 10 * q^47 + 3 * q^49 + 18 * q^51 - 16 * q^53 - 4 * q^55 + 28 * q^57 - 16 * q^59 + 22 * q^61 + 5 * q^63 - 4 * q^65 + 18 * q^67 + 24 * q^69 - 24 * q^71 + 22 * q^73 - 4 * q^77 - 8 * q^79 - q^81 - 2 * q^85 + 10 * q^87 + 26 * q^89 - 4 * q^91 - 4 * q^93 + 14 * q^97 + 26 * q^99

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