LMFDB - Embedded newform 7120.2.a.bj.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7120,2,Mod(1,7120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7120, 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("7120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7120 = 2^{4} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7120.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:	$56.8534862392$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 8x^{5} + 6x^{4} + 19x^{3} - 10x^{2} - 12x + 6 $ x^7 - x^6 - 8x^5 + 6x^4 + 19x^3 - 10x^2 - 12*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.885013$ of defining polynomial
Character	$\chi$	$=$	7120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.76669 q^{3} +1.00000 q^{5} +3.75132 q^{7} +4.65459 q^{9} +O(q^{10})$	$q+2.76669 q^{3} +1.00000 q^{5} +3.75132 q^{7} +4.65459 q^{9} +1.54195 q^{11} +4.15218 q^{13} +2.76669 q^{15} +5.74043 q^{17} -7.03169 q^{19} +10.3788 q^{21} +6.77641 q^{23} +1.00000 q^{25} +4.57775 q^{27} -10.1743 q^{29} -0.0578180 q^{31} +4.26612 q^{33} +3.75132 q^{35} -2.18339 q^{37} +11.4878 q^{39} -8.82042 q^{41} +6.08017 q^{43} +4.65459 q^{45} -4.08434 q^{47} +7.07241 q^{49} +15.8820 q^{51} -3.49748 q^{53} +1.54195 q^{55} -19.4545 q^{57} +9.20908 q^{59} +8.32403 q^{61} +17.4609 q^{63} +4.15218 q^{65} -3.93525 q^{67} +18.7483 q^{69} +5.49996 q^{71} -13.0277 q^{73} +2.76669 q^{75} +5.78437 q^{77} -8.83960 q^{79} -1.29854 q^{81} +9.09443 q^{83} +5.74043 q^{85} -28.1493 q^{87} +1.00000 q^{89} +15.5762 q^{91} -0.159965 q^{93} -7.03169 q^{95} -2.70331 q^{97} +7.17717 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9}+O(q^{10}) $ 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9	$ 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9} + 10 q^{11} - 7 q^{13} + 8 q^{15} - 13 q^{17} + 7 q^{19} + 16 q^{21} + 13 q^{23} + 7 q^{25} + 23 q^{27} - 4 q^{29} - q^{31} - 6 q^{33} + 16 q^{35} - 5 q^{37} + 13 q^{39} + 5 q^{41} + 31 q^{43} + 11 q^{45} + 14 q^{47} + 19 q^{49} + q^{51} - 13 q^{53} + 10 q^{55} + 21 q^{57} + 14 q^{59} + 3 q^{61} + 54 q^{63} - 7 q^{65} - q^{67} + 31 q^{69} + 8 q^{71} + 9 q^{73} + 8 q^{75} + 42 q^{77} - 9 q^{79} + 35 q^{81} + 42 q^{83} - 13 q^{85} - 6 q^{87} + 7 q^{89} - 31 q^{91} + 24 q^{93} + 7 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) $ 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9 + 10 * q^11 - 7 * q^13 + 8 * q^15 - 13 * q^17 + 7 * q^19 + 16 * q^21 + 13 * q^23 + 7 * q^25 + 23 * q^27 - 4 * q^29 - q^31 - 6 * q^33 + 16 * q^35 - 5 * q^37 + 13 * q^39 + 5 * q^41 + 31 * q^43 + 11 * q^45 + 14 * q^47 + 19 * q^49 + q^51 - 13 * q^53 + 10 * q^55 + 21 * q^57 + 14 * q^59 + 3 * q^61 + 54 * q^63 - 7 * q^65 - q^67 + 31 * q^69 + 8 * q^71 + 9 * q^73 + 8 * q^75 + 42 * q^77 - 9 * q^79 + 35 * q^81 + 42 * q^83 - 13 * q^85 - 6 * q^87 + 7 * q^89 - 31 * q^91 + 24 * q^93 + 7 * q^95 - 7 * q^97 - 5 * 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
$3$	2.76669	1.59735	0.798676	−	0.601762i	$-0.205534\pi$
$3$	2.76669	1.59735	0.798676	+	0.601762i	$0.205534\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.75132	1.41787	0.708933	−	0.705276i	$-0.249177\pi$
$7$	3.75132	1.41787	0.708933	+	0.705276i	$0.249177\pi$
$8$	0	0
$9$	4.65459	1.55153
$10$	0	0
$11$	1.54195	0.464917	0.232458	−	0.972606i	$-0.425323\pi$
$11$	1.54195	0.464917	0.232458	+	0.972606i	$0.425323\pi$
$12$	0	0
$13$	4.15218	1.15161	0.575804	−	0.817588i	$-0.304689\pi$
$13$	4.15218	1.15161	0.575804	+	0.817588i	$0.304689\pi$
$14$	0	0
$15$	2.76669	0.714357
$16$	0	0
$17$	5.74043	1.39226	0.696129	−	0.717916i	$-0.254904\pi$
$17$	5.74043	1.39226	0.696129	+	0.717916i	$0.254904\pi$
$18$	0	0
$19$	−7.03169	−1.61318	−0.806590	−	0.591111i	$-0.798690\pi$
$19$	−7.03169	−1.61318	−0.806590	+	0.591111i	$0.798690\pi$
$20$	0	0
$21$	10.3788	2.26483
$22$	0	0
$23$	6.77641	1.41298	0.706490	−	0.707723i	$-0.250278\pi$
$23$	6.77641	1.41298	0.706490	+	0.707723i	$0.250278\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.57775	0.880989
$28$	0	0
$29$	−10.1743	−1.88933	−0.944664	−	0.328041i	$-0.893612\pi$
$29$	−10.1743	−1.88933	−0.944664	+	0.328041i	$0.893612\pi$
$30$	0	0
$31$	−0.0578180	−0.0103844	−0.00519221	−	0.999987i	$-0.501653\pi$
$31$	−0.0578180	−0.0103844	−0.00519221	+	0.999987i	$0.501653\pi$
$32$	0	0
$33$	4.26612	0.742636
$34$	0	0
$35$	3.75132	0.634089
$36$	0	0
$37$	−2.18339	−0.358947	−0.179473	−	0.983763i	$-0.557439\pi$
$37$	−2.18339	−0.358947	−0.179473	+	0.983763i	$0.557439\pi$
$38$	0	0
$39$	11.4878	1.83952
$40$	0	0
$41$	−8.82042	−1.37752	−0.688759	−	0.724990i	$-0.741844\pi$
$41$	−8.82042	−1.37752	−0.688759	+	0.724990i	$0.741844\pi$
$42$	0	0
$43$	6.08017	0.927218	0.463609	−	0.886040i	$-0.346554\pi$
$43$	6.08017	0.927218	0.463609	+	0.886040i	$0.346554\pi$
$44$	0	0
$45$	4.65459	0.693866
$46$	0	0
$47$	−4.08434	−0.595762	−0.297881	−	0.954603i	$-0.596280\pi$
$47$	−4.08434	−0.595762	−0.297881	+	0.954603i	$0.596280\pi$
$48$	0	0
$49$	7.07241	1.01034
$50$	0	0
$51$	15.8820	2.22393
$52$	0	0
$53$	−3.49748	−0.480415	−0.240208	−	0.970722i	$-0.577216\pi$
$53$	−3.49748	−0.480415	−0.240208	+	0.970722i	$0.577216\pi$
$54$	0	0
$55$	1.54195	0.207917
$56$	0	0
$57$	−19.4545	−2.57682
$58$	0	0
$59$	9.20908	1.19892	0.599460	−	0.800405i	$-0.295382\pi$
$59$	9.20908	1.19892	0.599460	+	0.800405i	$0.295382\pi$
$60$	0	0
$61$	8.32403	1.06578	0.532891	−	0.846184i	$-0.321105\pi$
$61$	8.32403	1.06578	0.532891	+	0.846184i	$0.321105\pi$
$62$	0	0
$63$	17.4609	2.19986
$64$	0	0
$65$	4.15218	0.515015
$66$	0	0
$67$	−3.93525	−0.480767	−0.240384	−	0.970678i	$-0.577273\pi$
$67$	−3.93525	−0.480767	−0.240384	+	0.970678i	$0.577273\pi$
$68$	0	0
$69$	18.7483	2.25702
$70$	0	0
$71$	5.49996	0.652726	0.326363	−	0.945245i	$-0.394177\pi$
$71$	5.49996	0.652726	0.326363	+	0.945245i	$0.394177\pi$
$72$	0	0
$73$	−13.0277	−1.52478	−0.762388	−	0.647120i	$-0.775973\pi$
$73$	−13.0277	−1.52478	−0.762388	+	0.647120i	$0.775973\pi$
$74$	0	0
$75$	2.76669	0.319470
$76$	0	0
$77$	5.78437	0.659190
$78$	0	0
$79$	−8.83960	−0.994533	−0.497266	−	0.867598i	$-0.665663\pi$
$79$	−8.83960	−0.994533	−0.497266	+	0.867598i	$0.665663\pi$
$80$	0	0
$81$	−1.29854	−0.144283
$82$	0	0
$83$	9.09443	0.998243	0.499122	−	0.866532i	$-0.333656\pi$
$83$	9.09443	0.998243	0.499122	+	0.866532i	$0.333656\pi$
$84$	0	0
$85$	5.74043	0.622637
$86$	0	0
$87$	−28.1493	−3.01792
$88$	0	0
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	15.5762	1.63283
$92$	0	0
$93$	−0.159965	−0.0165876
$94$	0	0
$95$	−7.03169	−0.721436
$96$	0	0
$97$	−2.70331	−0.274479	−0.137240	−	0.990538i	$-0.543823\pi$
$97$	−2.70331	−0.274479	−0.137240	+	0.990538i	$0.543823\pi$
$98$	0	0
$99$	7.17717	0.721333
$100$	0	0
$101$	−10.6908	−1.06378	−0.531888	−	0.846815i	$-0.678517\pi$
$101$	−10.6908	−1.06378	−0.531888	+	0.846815i	$0.678517\pi$
$102$	0	0
$103$	−13.1321	−1.29395	−0.646973	−	0.762513i	$-0.723966\pi$
$103$	−13.1321	−1.29395	−0.646973	+	0.762513i	$0.723966\pi$
$104$	0	0
$105$	10.3788	1.01286
$106$	0	0
$107$	−11.1134	−1.07437	−0.537187	−	0.843463i	$-0.680513\pi$
$107$	−11.1134	−1.07437	−0.537187	+	0.843463i	$0.680513\pi$
$108$	0	0
$109$	−1.18261	−0.113274	−0.0566368	−	0.998395i	$-0.518038\pi$
$109$	−1.18261	−0.113274	−0.0566368	+	0.998395i	$0.518038\pi$
$110$	0	0
$111$	−6.04077	−0.573364
$112$	0	0
$113$	−7.93908	−0.746846	−0.373423	−	0.927661i	$-0.621816\pi$
$113$	−7.93908	−0.746846	−0.373423	+	0.927661i	$0.621816\pi$
$114$	0	0
$115$	6.77641	0.631904
$116$	0	0
$117$	19.3267	1.78676
$118$	0	0
$119$	21.5342	1.97404
$120$	0	0
$121$	−8.62238	−0.783852
$122$	0	0
$123$	−24.4034	−2.20038
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.22603	0.463735	0.231868	−	0.972747i	$-0.425516\pi$
$127$	5.22603	0.463735	0.231868	+	0.972747i	$0.425516\pi$
$128$	0	0
$129$	16.8220	1.48109
$130$	0	0
$131$	3.70672	0.323857	0.161929	−	0.986802i	$-0.448229\pi$
$131$	3.70672	0.323857	0.161929	+	0.986802i	$0.448229\pi$
$132$	0	0
$133$	−26.3781	−2.28727
$134$	0	0
$135$	4.57775	0.393990
$136$	0	0
$137$	−15.4250	−1.31785	−0.658925	−	0.752209i	$-0.728989\pi$
$137$	−15.4250	−1.31785	−0.658925	+	0.752209i	$0.728989\pi$
$138$	0	0
$139$	7.00287	0.593975	0.296988	−	0.954881i	$-0.404018\pi$
$139$	7.00287	0.593975	0.296988	+	0.954881i	$0.404018\pi$
$140$	0	0
$141$	−11.3001	−0.951641
$142$	0	0
$143$	6.40248	0.535402
$144$	0	0
$145$	−10.1743	−0.844933
$146$	0	0
$147$	19.5672	1.61387
$148$	0	0
$149$	12.5398	1.02730	0.513650	−	0.858000i	$-0.328293\pi$
$149$	12.5398	1.02730	0.513650	+	0.858000i	$0.328293\pi$
$150$	0	0
$151$	−0.236140	−0.0192168	−0.00960840	−	0.999954i	$-0.503058\pi$
$151$	−0.236140	−0.0192168	−0.00960840	+	0.999954i	$0.503058\pi$
$152$	0	0
$153$	26.7194	2.16013
$154$	0	0
$155$	−0.0578180	−0.00464406
$156$	0	0
$157$	−4.98636	−0.397955	−0.198977	−	0.980004i	$-0.563762\pi$
$157$	−4.98636	−0.397955	−0.198977	+	0.980004i	$0.563762\pi$
$158$	0	0
$159$	−9.67645	−0.767392
$160$	0	0
$161$	25.4205	2.00342
$162$	0	0
$163$	10.9166	0.855055	0.427528	−	0.904002i	$-0.359385\pi$
$163$	10.9166	0.855055	0.427528	+	0.904002i	$0.359385\pi$
$164$	0	0
$165$	4.26612	0.332117
$166$	0	0
$167$	−19.6561	−1.52104	−0.760518	−	0.649317i	$-0.775055\pi$
$167$	−19.6561	−1.52104	−0.760518	+	0.649317i	$0.775055\pi$
$168$	0	0
$169$	4.24062	0.326202
$170$	0	0
$171$	−32.7297	−2.50290
$172$	0	0
$173$	−13.1762	−1.00177	−0.500883	−	0.865515i	$-0.666991\pi$
$173$	−13.1762	−1.00177	−0.500883	+	0.865515i	$0.666991\pi$
$174$	0	0
$175$	3.75132	0.283573
$176$	0	0
$177$	25.4787	1.91510
$178$	0	0
$179$	−9.74668	−0.728501	−0.364250	−	0.931301i	$-0.618675\pi$
$179$	−9.74668	−0.728501	−0.364250	+	0.931301i	$0.618675\pi$
$180$	0	0
$181$	6.93270	0.515304	0.257652	−	0.966238i	$-0.417051\pi$
$181$	6.93270	0.515304	0.257652	+	0.966238i	$0.417051\pi$
$182$	0	0
$183$	23.0300	1.70243
$184$	0	0
$185$	−2.18339	−0.160526
$186$	0	0
$187$	8.85148	0.647285
$188$	0	0
$189$	17.1726	1.24912
$190$	0	0
$191$	−4.37217	−0.316359	−0.158180	−	0.987410i	$-0.550563\pi$
$191$	−4.37217	−0.316359	−0.158180	+	0.987410i	$0.550563\pi$
$192$	0	0
$193$	2.05206	0.147711	0.0738554	−	0.997269i	$-0.476470\pi$
$193$	2.05206	0.147711	0.0738554	+	0.997269i	$0.476470\pi$
$194$	0	0
$195$	11.4878	0.822660
$196$	0	0
$197$	−6.48648	−0.462143	−0.231071	−	0.972937i	$-0.574223\pi$
$197$	−6.48648	−0.462143	−0.231071	+	0.972937i	$0.574223\pi$
$198$	0	0
$199$	−13.4260	−0.951745	−0.475873	−	0.879514i	$-0.657868\pi$
$199$	−13.4260	−0.951745	−0.475873	+	0.879514i	$0.657868\pi$
$200$	0	0
$201$	−10.8876	−0.767954
$202$	0	0
$203$	−38.1672	−2.67881
$204$	0	0
$205$	−8.82042	−0.616045
$206$	0	0
$207$	31.5414	2.19228
$208$	0	0
$209$	−10.8426	−0.749995
$210$	0	0
$211$	20.1294	1.38576	0.692882	−	0.721051i	$-0.256341\pi$
$211$	20.1294	1.38576	0.692882	+	0.721051i	$0.256341\pi$
$212$	0	0
$213$	15.2167	1.04263
$214$	0	0
$215$	6.08017	0.414664
$216$	0	0
$217$	−0.216894	−0.0147237
$218$	0	0
$219$	−36.0436	−2.43560
$220$	0	0
$221$	23.8353	1.60334
$222$	0	0
$223$	8.26176	0.553248	0.276624	−	0.960978i	$-0.410784\pi$
$223$	8.26176	0.553248	0.276624	+	0.960978i	$0.410784\pi$
$224$	0	0
$225$	4.65459	0.310306
$226$	0	0
$227$	21.1024	1.40061	0.700306	−	0.713842i	$-0.253047\pi$
$227$	21.1024	1.40061	0.700306	+	0.713842i	$0.253047\pi$
$228$	0	0
$229$	−13.3006	−0.878930	−0.439465	−	0.898260i	$-0.644832\pi$
$229$	−13.3006	−0.878930	−0.439465	+	0.898260i	$0.644832\pi$
$230$	0	0
$231$	16.0036	1.05296
$232$	0	0
$233$	5.42111	0.355148	0.177574	−	0.984107i	$-0.443175\pi$
$233$	5.42111	0.355148	0.177574	+	0.984107i	$0.443175\pi$
$234$	0	0
$235$	−4.08434	−0.266433
$236$	0	0
$237$	−24.4565	−1.58862
$238$	0	0
$239$	−10.4763	−0.677657	−0.338828	−	0.940848i	$-0.610031\pi$
$239$	−10.4763	−0.677657	−0.338828	+	0.940848i	$0.610031\pi$
$240$	0	0
$241$	−13.1333	−0.845991	−0.422995	−	0.906132i	$-0.639021\pi$
$241$	−13.1333	−0.845991	−0.422995	+	0.906132i	$0.639021\pi$
$242$	0	0
$243$	−17.3259	−1.11146
$244$	0	0
$245$	7.07241	0.451839
$246$	0	0
$247$	−29.1969	−1.85775
$248$	0	0
$249$	25.1615	1.59455
$250$	0	0
$251$	27.1469	1.71350	0.856748	−	0.515735i	$-0.172481\pi$
$251$	27.1469	1.71350	0.856748	+	0.515735i	$0.172481\pi$
$252$	0	0
$253$	10.4489	0.656918
$254$	0	0
$255$	15.8820	0.994570
$256$	0	0
$257$	−18.1588	−1.13272	−0.566358	−	0.824159i	$-0.691648\pi$
$257$	−18.1588	−1.13272	−0.566358	+	0.824159i	$0.691648\pi$
$258$	0	0
$259$	−8.19059	−0.508938
$260$	0	0
$261$	−47.3574	−2.93135
$262$	0	0
$263$	23.7946	1.46724	0.733619	−	0.679561i	$-0.237830\pi$
$263$	23.7946	1.46724	0.733619	+	0.679561i	$0.237830\pi$
$264$	0	0
$265$	−3.49748	−0.214848
$266$	0	0
$267$	2.76669	0.169319
$268$	0	0
$269$	0.159138	0.00970280	0.00485140	−	0.999988i	$-0.498456\pi$
$269$	0.159138	0.00970280	0.00485140	+	0.999988i	$0.498456\pi$
$270$	0	0
$271$	1.15143	0.0699447	0.0349723	−	0.999388i	$-0.488866\pi$
$271$	1.15143	0.0699447	0.0349723	+	0.999388i	$0.488866\pi$
$272$	0	0
$273$	43.0945	2.60820
$274$	0	0
$275$	1.54195	0.0929834
$276$	0	0
$277$	−1.65637	−0.0995216	−0.0497608	−	0.998761i	$-0.515846\pi$
$277$	−1.65637	−0.0995216	−0.0497608	+	0.998761i	$0.515846\pi$
$278$	0	0
$279$	−0.269119	−0.0161118
$280$	0	0
$281$	4.08763	0.243848	0.121924	−	0.992539i	$-0.461094\pi$
$281$	4.08763	0.243848	0.121924	+	0.992539i	$0.461094\pi$
$282$	0	0
$283$	14.7662	0.877757	0.438879	−	0.898546i	$-0.355376\pi$
$283$	14.7662	0.877757	0.438879	+	0.898546i	$0.355376\pi$
$284$	0	0
$285$	−19.4545	−1.15239
$286$	0	0
$287$	−33.0882	−1.95314
$288$	0	0
$289$	15.9525	0.938384
$290$	0	0
$291$	−7.47922	−0.438440
$292$	0	0
$293$	3.43348	0.200586	0.100293	−	0.994958i	$-0.468022\pi$
$293$	3.43348	0.200586	0.100293	+	0.994958i	$0.468022\pi$
$294$	0	0
$295$	9.20908	0.536173
$296$	0	0
$297$	7.05869	0.409586
$298$	0	0
$299$	28.1369	1.62720
$300$	0	0
$301$	22.8087	1.31467
$302$	0	0
$303$	−29.5782	−1.69922
$304$	0	0
$305$	8.32403	0.476633
$306$	0	0
$307$	22.5178	1.28516	0.642581	−	0.766218i	$-0.277864\pi$
$307$	22.5178	1.28516	0.642581	+	0.766218i	$0.277864\pi$
$308$	0	0
$309$	−36.3326	−2.06689
$310$	0	0
$311$	−11.4828	−0.651129	−0.325564	−	0.945520i	$-0.605554\pi$
$311$	−11.4828	−0.651129	−0.325564	+	0.945520i	$0.605554\pi$
$312$	0	0
$313$	−2.22350	−0.125679	−0.0628397	−	0.998024i	$-0.520016\pi$
$313$	−2.22350	−0.125679	−0.0628397	+	0.998024i	$0.520016\pi$
$314$	0	0
$315$	17.4609	0.983809
$316$	0	0
$317$	32.5253	1.82680	0.913402	−	0.407059i	$-0.133446\pi$
$317$	32.5253	1.82680	0.913402	+	0.407059i	$0.133446\pi$
$318$	0	0
$319$	−15.6884	−0.878380
$320$	0	0
$321$	−30.7474	−1.71615
$322$	0	0
$323$	−40.3649	−2.24597
$324$	0	0
$325$	4.15218	0.230322
$326$	0	0
$327$	−3.27192	−0.180938
$328$	0	0
$329$	−15.3217	−0.844710
$330$	0	0
$331$	27.2476	1.49766	0.748831	−	0.662761i	$-0.230615\pi$
$331$	27.2476	1.49766	0.748831	+	0.662761i	$0.230615\pi$
$332$	0	0
$333$	−10.1628	−0.556917
$334$	0	0
$335$	−3.93525	−0.215006
$336$	0	0
$337$	16.5317	0.900536	0.450268	−	0.892893i	$-0.351328\pi$
$337$	16.5317	0.900536	0.450268	+	0.892893i	$0.351328\pi$
$338$	0	0
$339$	−21.9650	−1.19297
$340$	0	0
$341$	−0.0891528	−0.00482789
$342$	0	0
$343$	0.271619	0.0146661
$344$	0	0
$345$	18.7483	1.00937
$346$	0	0
$347$	33.2231	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$347$	33.2231	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$348$	0	0
$349$	−5.87046	−0.314239	−0.157119	−	0.987580i	$-0.550221\pi$
$349$	−5.87046	−0.314239	−0.157119	+	0.987580i	$0.550221\pi$
$350$	0	0
$351$	19.0077	1.01455
$352$	0	0
$353$	31.2340	1.66242	0.831210	−	0.555958i	$-0.187649\pi$
$353$	31.2340	1.66242	0.831210	+	0.555958i	$0.187649\pi$
$354$	0	0
$355$	5.49996	0.291908
$356$	0	0
$357$	59.5785	3.15323
$358$	0	0
$359$	4.01656	0.211986	0.105993	−	0.994367i	$-0.466198\pi$
$359$	4.01656	0.211986	0.105993	+	0.994367i	$0.466198\pi$
$360$	0	0
$361$	30.4447	1.60235
$362$	0	0
$363$	−23.8555	−1.25209
$364$	0	0
$365$	−13.0277	−0.681901
$366$	0	0
$367$	−13.7718	−0.718882	−0.359441	−	0.933168i	$-0.617033\pi$
$367$	−13.7718	−0.718882	−0.359441	+	0.933168i	$0.617033\pi$
$368$	0	0
$369$	−41.0555	−2.13726
$370$	0	0
$371$	−13.1202	−0.681165
$372$	0	0
$373$	−24.2574	−1.25600	−0.628000	−	0.778214i	$-0.716126\pi$
$373$	−24.2574	−1.25600	−0.628000	+	0.778214i	$0.716126\pi$
$374$	0	0
$375$	2.76669	0.142871
$376$	0	0
$377$	−42.2457	−2.17577
$378$	0	0
$379$	−29.7093	−1.52606	−0.763032	−	0.646361i	$-0.776290\pi$
$379$	−29.7093	−1.52606	−0.763032	+	0.646361i	$0.776290\pi$
$380$	0	0
$381$	14.4588	0.740748
$382$	0	0
$383$	−31.1003	−1.58915	−0.794575	−	0.607166i	$-0.792306\pi$
$383$	−31.1003	−1.58915	−0.794575	+	0.607166i	$0.792306\pi$
$384$	0	0
$385$	5.78437	0.294799
$386$	0	0
$387$	28.3007	1.43861
$388$	0	0
$389$	−2.70571	−0.137185	−0.0685924	−	0.997645i	$-0.521851\pi$
$389$	−2.70571	−0.137185	−0.0685924	+	0.997645i	$0.521851\pi$
$390$	0	0
$391$	38.8995	1.96723
$392$	0	0
$393$	10.2553	0.517314
$394$	0	0
$395$	−8.83960	−0.444768
$396$	0	0
$397$	−22.9150	−1.15007	−0.575036	−	0.818128i	$-0.695012\pi$
$397$	−22.9150	−1.15007	−0.575036	+	0.818128i	$0.695012\pi$
$398$	0	0
$399$	−72.9802	−3.65358
$400$	0	0
$401$	37.1991	1.85764	0.928818	−	0.370535i	$-0.120826\pi$
$401$	37.1991	1.85764	0.928818	+	0.370535i	$0.120826\pi$
$402$	0	0
$403$	−0.240071	−0.0119588
$404$	0	0
$405$	−1.29854	−0.0645251
$406$	0	0
$407$	−3.36669	−0.166880
$408$	0	0
$409$	25.6275	1.26720	0.633598	−	0.773663i	$-0.281577\pi$
$409$	25.6275	1.26720	0.633598	+	0.773663i	$0.281577\pi$
$410$	0	0
$411$	−42.6764	−2.10507
$412$	0	0
$413$	34.5462	1.69991
$414$	0	0
$415$	9.09443	0.446428
$416$	0	0
$417$	19.3748	0.948787
$418$	0	0
$419$	28.1678	1.37609	0.688043	−	0.725670i	$-0.258470\pi$
$419$	28.1678	1.37609	0.688043	+	0.725670i	$0.258470\pi$
$420$	0	0
$421$	−20.4045	−0.994453	−0.497227	−	0.867621i	$-0.665648\pi$
$421$	−20.4045	−0.994453	−0.497227	+	0.867621i	$0.665648\pi$
$422$	0	0
$423$	−19.0109	−0.924343
$424$	0	0
$425$	5.74043	0.278452
$426$	0	0
$427$	31.2261	1.51114
$428$	0	0
$429$	17.7137	0.855225
$430$	0	0
$431$	−20.4146	−0.983338	−0.491669	−	0.870782i	$-0.663613\pi$
$431$	−20.4146	−0.983338	−0.491669	+	0.870782i	$0.663613\pi$
$432$	0	0
$433$	16.1740	0.777275	0.388637	−	0.921391i	$-0.372946\pi$
$433$	16.1740	0.777275	0.388637	+	0.921391i	$0.372946\pi$
$434$	0	0
$435$	−28.1493	−1.34965
$436$	0	0
$437$	−47.6496	−2.27939
$438$	0	0
$439$	−9.85761	−0.470478	−0.235239	−	0.971938i	$-0.575587\pi$
$439$	−9.85761	−0.470478	−0.235239	+	0.971938i	$0.575587\pi$
$440$	0	0
$441$	32.9192	1.56758
$442$	0	0
$443$	−7.14208	−0.339331	−0.169665	−	0.985502i	$-0.554269\pi$
$443$	−7.14208	−0.339331	−0.169665	+	0.985502i	$0.554269\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	0	0
$447$	34.6938	1.64096
$448$	0	0
$449$	−34.4416	−1.62540	−0.812701	−	0.582681i	$-0.802004\pi$
$449$	−34.4416	−1.62540	−0.812701	+	0.582681i	$0.802004\pi$
$450$	0	0
$451$	−13.6007	−0.640432
$452$	0	0
$453$	−0.653327	−0.0306960
$454$	0	0
$455$	15.5762	0.730222
$456$	0	0
$457$	28.5729	1.33658	0.668292	−	0.743899i	$-0.267026\pi$
$457$	28.5729	1.33658	0.668292	+	0.743899i	$0.267026\pi$
$458$	0	0
$459$	26.2783	1.22656
$460$	0	0
$461$	24.3926	1.13608	0.568038	−	0.823002i	$-0.307703\pi$
$461$	24.3926	1.13608	0.568038	+	0.823002i	$0.307703\pi$
$462$	0	0
$463$	9.51283	0.442099	0.221049	−	0.975263i	$-0.429052\pi$
$463$	9.51283	0.442099	0.221049	+	0.975263i	$0.429052\pi$
$464$	0	0
$465$	−0.159965	−0.00741819
$466$	0	0
$467$	11.5000	0.532157	0.266079	−	0.963951i	$-0.414272\pi$
$467$	11.5000	0.532157	0.266079	+	0.963951i	$0.414272\pi$
$468$	0	0
$469$	−14.7624	−0.681664
$470$	0	0
$471$	−13.7957	−0.635673
$472$	0	0
$473$	9.37535	0.431079
$474$	0	0
$475$	−7.03169	−0.322636
$476$	0	0
$477$	−16.2793	−0.745379
$478$	0	0
$479$	37.9454	1.73377	0.866884	−	0.498510i	$-0.166119\pi$
$479$	37.9454	1.73377	0.866884	+	0.498510i	$0.166119\pi$
$480$	0	0
$481$	−9.06583	−0.413366
$482$	0	0
$483$	70.3307	3.20016
$484$	0	0
$485$	−2.70331	−0.122751
$486$	0	0
$487$	−24.8040	−1.12398	−0.561988	−	0.827145i	$-0.689963\pi$
$487$	−24.8040	−1.12398	−0.561988	+	0.827145i	$0.689963\pi$
$488$	0	0
$489$	30.2029	1.36582
$490$	0	0
$491$	27.1395	1.22479	0.612395	−	0.790552i	$-0.290206\pi$
$491$	27.1395	1.22479	0.612395	+	0.790552i	$0.290206\pi$
$492$	0	0
$493$	−58.4051	−2.63043
$494$	0	0
$495$	7.17717	0.322590
$496$	0	0
$497$	20.6321	0.925478
$498$	0	0
$499$	33.3081	1.49108	0.745538	−	0.666463i	$-0.232192\pi$
$499$	33.3081	1.49108	0.745538	+	0.666463i	$0.232192\pi$
$500$	0	0
$501$	−54.3824	−2.42963
$502$	0	0
$503$	26.6651	1.18894	0.594469	−	0.804119i	$-0.297362\pi$
$503$	26.6651	1.18894	0.594469	+	0.804119i	$0.297362\pi$
$504$	0	0
$505$	−10.6908	−0.475735
$506$	0	0
$507$	11.7325	0.521059
$508$	0	0
$509$	28.5259	1.26439	0.632193	−	0.774811i	$-0.282155\pi$
$509$	28.5259	1.26439	0.632193	+	0.774811i	$0.282155\pi$
$510$	0	0
$511$	−48.8710	−2.16193
$512$	0	0
$513$	−32.1893	−1.42119
$514$	0	0
$515$	−13.1321	−0.578671
$516$	0	0
$517$	−6.29786	−0.276980
$518$	0	0
$519$	−36.4545	−1.60017
$520$	0	0
$521$	−3.47227	−0.152123	−0.0760614	−	0.997103i	$-0.524235\pi$
$521$	−3.47227	−0.152123	−0.0760614	+	0.997103i	$0.524235\pi$
$522$	0	0
$523$	16.5706	0.724584	0.362292	−	0.932065i	$-0.381994\pi$
$523$	16.5706	0.724584	0.362292	+	0.932065i	$0.381994\pi$
$524$	0	0
$525$	10.3788	0.452966
$526$	0	0
$527$	−0.331900	−0.0144578
$528$	0	0
$529$	22.9197	0.996511
$530$	0	0
$531$	42.8645	1.86016
$532$	0	0
$533$	−36.6240	−1.58636
$534$	0	0
$535$	−11.1134	−0.480475
$536$	0	0
$537$	−26.9661	−1.16367
$538$	0	0
$539$	10.9053	0.469726
$540$	0	0
$541$	−29.0138	−1.24740	−0.623700	−	0.781664i	$-0.714371\pi$
$541$	−29.0138	−1.24740	−0.623700	+	0.781664i	$0.714371\pi$
$542$	0	0
$543$	19.1807	0.823121
$544$	0	0
$545$	−1.18261	−0.0506575
$546$	0	0
$547$	−32.6731	−1.39700	−0.698500	−	0.715610i	$-0.746149\pi$
$547$	−32.6731	−1.39700	−0.698500	+	0.715610i	$0.746149\pi$
$548$	0	0
$549$	38.7450	1.65359
$550$	0	0
$551$	71.5428	3.04783
$552$	0	0
$553$	−33.1602	−1.41011
$554$	0	0
$555$	−6.04077	−0.256416
$556$	0	0
$557$	−10.9818	−0.465315	−0.232658	−	0.972559i	$-0.574742\pi$
$557$	−10.9818	−0.465315	−0.232658	+	0.972559i	$0.574742\pi$
$558$	0	0
$559$	25.2460	1.06779
$560$	0	0
$561$	24.4893	1.03394
$562$	0	0
$563$	33.8545	1.42680	0.713399	−	0.700758i	$-0.247155\pi$
$563$	33.8545	1.42680	0.713399	+	0.700758i	$0.247155\pi$
$564$	0	0
$565$	−7.93908	−0.333999
$566$	0	0
$567$	−4.87125	−0.204573
$568$	0	0
$569$	9.27784	0.388947	0.194474	−	0.980908i	$-0.437700\pi$
$569$	9.27784	0.388947	0.194474	+	0.980908i	$0.437700\pi$
$570$	0	0
$571$	16.6644	0.697383	0.348692	−	0.937238i	$-0.386626\pi$
$571$	16.6644	0.697383	0.348692	+	0.937238i	$0.386626\pi$
$572$	0	0
$573$	−12.0965	−0.505337
$574$	0	0
$575$	6.77641	0.282596
$576$	0	0
$577$	9.60454	0.399842	0.199921	−	0.979812i	$-0.435931\pi$
$577$	9.60454	0.399842	0.199921	+	0.979812i	$0.435931\pi$
$578$	0	0
$579$	5.67743	0.235946
$580$	0	0
$581$	34.1161	1.41538
$582$	0	0
$583$	−5.39295	−0.223353
$584$	0	0
$585$	19.3267	0.799062
$586$	0	0
$587$	−42.5202	−1.75500	−0.877498	−	0.479580i	$-0.840789\pi$
$587$	−42.5202	−1.75500	−0.877498	+	0.479580i	$0.840789\pi$
$588$	0	0
$589$	0.406559	0.0167520
$590$	0	0
$591$	−17.9461	−0.738204
$592$	0	0
$593$	−16.5448	−0.679412	−0.339706	−	0.940532i	$-0.610328\pi$
$593$	−16.5448	−0.679412	−0.339706	+	0.940532i	$0.610328\pi$
$594$	0	0
$595$	21.5342	0.882816
$596$	0	0
$597$	−37.1457	−1.52027
$598$	0	0
$599$	−3.61576	−0.147736	−0.0738680	−	0.997268i	$-0.523534\pi$
$599$	−3.61576	−0.147736	−0.0738680	+	0.997268i	$0.523534\pi$
$600$	0	0
$601$	−6.04796	−0.246701	−0.123351	−	0.992363i	$-0.539364\pi$
$601$	−6.04796	−0.246701	−0.123351	+	0.992363i	$0.539364\pi$
$602$	0	0
$603$	−18.3170	−0.745925
$604$	0	0
$605$	−8.62238	−0.350549
$606$	0	0
$607$	13.4200	0.544701	0.272350	−	0.962198i	$-0.412199\pi$
$607$	13.4200	0.544701	0.272350	+	0.962198i	$0.412199\pi$
$608$	0	0
$609$	−105.597	−4.27900
$610$	0	0
$611$	−16.9589	−0.686084
$612$	0	0
$613$	11.6728	0.471461	0.235731	−	0.971818i	$-0.424252\pi$
$613$	11.6728	0.471461	0.235731	+	0.971818i	$0.424252\pi$
$614$	0	0
$615$	−24.4034	−0.984040
$616$	0	0
$617$	−28.0621	−1.12974	−0.564869	−	0.825180i	$-0.691073\pi$
$617$	−28.0621	−1.12974	−0.564869	+	0.825180i	$0.691073\pi$
$618$	0	0
$619$	−20.9899	−0.843655	−0.421827	−	0.906676i	$-0.638611\pi$
$619$	−20.9899	−0.843655	−0.421827	+	0.906676i	$0.638611\pi$
$620$	0	0
$621$	31.0207	1.24482
$622$	0	0
$623$	3.75132	0.150293
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−29.9980	−1.19801
$628$	0	0
$629$	−12.5336	−0.499747
$630$	0	0
$631$	−29.7837	−1.18567	−0.592836	−	0.805323i	$-0.701992\pi$
$631$	−29.7837	−1.18567	−0.592836	+	0.805323i	$0.701992\pi$
$632$	0	0
$633$	55.6919	2.21355
$634$	0	0
$635$	5.22603	0.207389
$636$	0	0
$637$	29.3659	1.16352
$638$	0	0
$639$	25.6001	1.01272
$640$	0	0
$641$	18.9560	0.748715	0.374358	−	0.927284i	$-0.377863\pi$
$641$	18.9560	0.748715	0.374358	+	0.927284i	$0.377863\pi$
$642$	0	0
$643$	−48.3147	−1.90535	−0.952673	−	0.303996i	$-0.901679\pi$
$643$	−48.3147	−1.90535	−0.952673	+	0.303996i	$0.901679\pi$
$644$	0	0
$645$	16.8220	0.662365
$646$	0	0
$647$	10.0752	0.396096	0.198048	−	0.980192i	$-0.436540\pi$
$647$	10.0752	0.396096	0.198048	+	0.980192i	$0.436540\pi$
$648$	0	0
$649$	14.2000	0.557398
$650$	0	0
$651$	−0.600079	−0.0235190
$652$	0	0
$653$	10.2900	0.402678	0.201339	−	0.979522i	$-0.435471\pi$
$653$	10.2900	0.402678	0.201339	+	0.979522i	$0.435471\pi$
$654$	0	0
$655$	3.70672	0.144833
$656$	0	0
$657$	−60.6386	−2.36574
$658$	0	0
$659$	−32.9627	−1.28405	−0.642023	−	0.766686i	$-0.721905\pi$
$659$	−32.9627	−1.28405	−0.642023	+	0.766686i	$0.721905\pi$
$660$	0	0
$661$	40.9469	1.59265	0.796325	−	0.604869i	$-0.206775\pi$
$661$	40.9469	1.59265	0.796325	+	0.604869i	$0.206775\pi$
$662$	0	0
$663$	65.9450	2.56109
$664$	0	0
$665$	−26.3781	−1.02290
$666$	0	0
$667$	−68.9455	−2.66958
$668$	0	0
$669$	22.8578	0.883732
$670$	0	0
$671$	12.8353	0.495500
$672$	0	0
$673$	−19.3257	−0.744952	−0.372476	−	0.928042i	$-0.621491\pi$
$673$	−19.3257	−0.744952	−0.372476	+	0.928042i	$0.621491\pi$
$674$	0	0
$675$	4.57775	0.176198
$676$	0	0
$677$	13.6476	0.524518	0.262259	−	0.964998i	$-0.415533\pi$
$677$	13.6476	0.524518	0.262259	+	0.964998i	$0.415533\pi$
$678$	0	0
$679$	−10.1410	−0.389175
$680$	0	0
$681$	58.3838	2.23727
$682$	0	0
$683$	34.1585	1.30704	0.653520	−	0.756909i	$-0.273291\pi$
$683$	34.1585	1.30704	0.653520	+	0.756909i	$0.273291\pi$
$684$	0	0
$685$	−15.4250	−0.589360
$686$	0	0
$687$	−36.7987	−1.40396
$688$	0	0
$689$	−14.5222	−0.553250
$690$	0	0
$691$	4.07591	0.155055	0.0775275	−	0.996990i	$-0.475297\pi$
$691$	4.07591	0.155055	0.0775275	+	0.996990i	$0.475297\pi$
$692$	0	0
$693$	26.9239	1.02275
$694$	0	0
$695$	7.00287	0.265634
$696$	0	0
$697$	−50.6330	−1.91786
$698$	0	0
$699$	14.9985	0.567297
$700$	0	0
$701$	−4.59497	−0.173550	−0.0867749	−	0.996228i	$-0.527656\pi$
$701$	−4.59497	−0.173550	−0.0867749	+	0.996228i	$0.527656\pi$
$702$	0	0
$703$	15.3529	0.579046
$704$	0	0
$705$	−11.3001	−0.425587
$706$	0	0
$707$	−40.1047	−1.50829
$708$	0	0
$709$	−1.21881	−0.0457735	−0.0228867	−	0.999738i	$-0.507286\pi$
$709$	−1.21881	−0.0457735	−0.0228867	+	0.999738i	$0.507286\pi$
$710$	0	0
$711$	−41.1447	−1.54305
$712$	0	0
$713$	−0.391799	−0.0146730
$714$	0	0
$715$	6.40248	0.239439
$716$	0	0
$717$	−28.9848	−1.08246
$718$	0	0
$719$	−19.6350	−0.732263	−0.366131	−	0.930563i	$-0.619318\pi$
$719$	−19.6350	−0.732263	−0.366131	+	0.930563i	$0.619318\pi$
$720$	0	0
$721$	−49.2628	−1.83464
$722$	0	0
$723$	−36.3358	−1.35134
$724$	0	0
$725$	−10.1743	−0.377865
$726$	0	0
$727$	−21.2231	−0.787121	−0.393561	−	0.919299i	$-0.628757\pi$
$727$	−21.2231	−0.787121	−0.393561	+	0.919299i	$0.628757\pi$
$728$	0	0
$729$	−44.0399	−1.63111
$730$	0	0
$731$	34.9028	1.29093
$732$	0	0
$733$	18.4194	0.680338	0.340169	−	0.940364i	$-0.389516\pi$
$733$	18.4194	0.680338	0.340169	+	0.940364i	$0.389516\pi$
$734$	0	0
$735$	19.5672	0.721746
$736$	0	0
$737$	−6.06798	−0.223517
$738$	0	0
$739$	9.08147	0.334067	0.167034	−	0.985951i	$-0.446581\pi$
$739$	9.08147	0.334067	0.167034	+	0.985951i	$0.446581\pi$
$740$	0	0
$741$	−80.7788	−2.96748
$742$	0	0
$743$	0.0809652	0.00297033	0.00148516	−	0.999999i	$-0.499527\pi$
$743$	0.0809652	0.00297033	0.00148516	+	0.999999i	$0.499527\pi$
$744$	0	0
$745$	12.5398	0.459422
$746$	0	0
$747$	42.3309	1.54881
$748$	0	0
$749$	−41.6900	−1.52332
$750$	0	0
$751$	6.49039	0.236838	0.118419	−	0.992964i	$-0.462217\pi$
$751$	6.49039	0.236838	0.118419	+	0.992964i	$0.462217\pi$
$752$	0	0
$753$	75.1071	2.73706
$754$	0	0
$755$	−0.236140	−0.00859402
$756$	0	0
$757$	−28.1185	−1.02199	−0.510993	−	0.859585i	$-0.670722\pi$
$757$	−28.1185	−1.02199	−0.510993	+	0.859585i	$0.670722\pi$
$758$	0	0
$759$	28.9090	1.04933
$760$	0	0
$761$	2.54060	0.0920966	0.0460483	−	0.998939i	$-0.485337\pi$
$761$	2.54060	0.0920966	0.0460483	+	0.998939i	$0.485337\pi$
$762$	0	0
$763$	−4.43635	−0.160607
$764$	0	0
$765$	26.7194	0.966041
$766$	0	0
$767$	38.2378	1.38069
$768$	0	0
$769$	0.0116365	0.000419624 0	0.000209812	−	1.00000i	$-0.499933\pi$
$769$	0.0116365	0.000419624 0	0.000209812		1.00000i	$0.499933\pi$
$770$	0	0
$771$	−50.2399	−1.80935
$772$	0	0
$773$	12.9008	0.464010	0.232005	−	0.972715i	$-0.425471\pi$
$773$	12.9008	0.464010	0.232005	+	0.972715i	$0.425471\pi$
$774$	0	0
$775$	−0.0578180	−0.00207688
$776$	0	0
$777$	−22.6608	−0.812953
$778$	0	0
$779$	62.0225	2.22219
$780$	0	0
$781$	8.48070	0.303463
$782$	0	0
$783$	−46.5756	−1.66448
$784$	0	0
$785$	−4.98636	−0.177971
$786$	0	0
$787$	−55.1951	−1.96749	−0.983746	−	0.179566i	$-0.942531\pi$
$787$	−55.1951	−1.96749	−0.983746	+	0.179566i	$0.942531\pi$
$788$	0	0
$789$	65.8323	2.34369
$790$	0	0
$791$	−29.7820	−1.05893
$792$	0	0
$793$	34.5629	1.22736
$794$	0	0
$795$	−9.67645	−0.343188
$796$	0	0
$797$	31.5728	1.11837	0.559183	−	0.829044i	$-0.311115\pi$
$797$	31.5728	1.11837	0.559183	+	0.829044i	$0.311115\pi$
$798$	0	0
$799$	−23.4458	−0.829455
$800$	0	0
$801$	4.65459	0.164462
$802$	0	0
$803$	−20.0881	−0.708894
$804$	0	0
$805$	25.4205	0.895955
$806$	0	0
$807$	0.440285	0.0154988
$808$	0	0
$809$	37.2161	1.30845	0.654225	−	0.756300i	$-0.272995\pi$
$809$	37.2161	1.30845	0.654225	+	0.756300i	$0.272995\pi$
$810$	0	0
$811$	−45.2306	−1.58826	−0.794131	−	0.607746i	$-0.792074\pi$
$811$	−45.2306	−1.58826	−0.794131	+	0.607746i	$0.792074\pi$
$812$	0	0
$813$	3.18567	0.111726
$814$	0	0
$815$	10.9166	0.382392
$816$	0	0
$817$	−42.7539	−1.49577
$818$	0	0
$819$	72.5007	2.53338
$820$	0	0
$821$	19.6062	0.684261	0.342131	−	0.939652i	$-0.388851\pi$
$821$	19.6062	0.684261	0.342131	+	0.939652i	$0.388851\pi$
$822$	0	0
$823$	19.4397	0.677625	0.338812	−	0.940854i	$-0.389975\pi$
$823$	19.4397	0.677625	0.338812	+	0.940854i	$0.389975\pi$
$824$	0	0
$825$	4.26612	0.148527
$826$	0	0
$827$	15.8859	0.552407	0.276203	−	0.961099i	$-0.410924\pi$
$827$	15.8859	0.552407	0.276203	+	0.961099i	$0.410924\pi$
$828$	0	0
$829$	9.64772	0.335079	0.167539	−	0.985865i	$-0.446418\pi$
$829$	9.64772	0.335079	0.167539	+	0.985865i	$0.446418\pi$
$830$	0	0
$831$	−4.58267	−0.158971
$832$	0	0
$833$	40.5987	1.40666
$834$	0	0
$835$	−19.6561	−0.680228
$836$	0	0
$837$	−0.264677	−0.00914856
$838$	0	0
$839$	−38.4916	−1.32888	−0.664439	−	0.747342i	$-0.731330\pi$
$839$	−38.4916	−1.32888	−0.664439	+	0.747342i	$0.731330\pi$
$840$	0	0
$841$	74.5172	2.56956
$842$	0	0
$843$	11.3092	0.389510
$844$	0	0
$845$	4.24062	0.145882
$846$	0	0
$847$	−32.3453	−1.11140
$848$	0	0
$849$	40.8534	1.40209
$850$	0	0
$851$	−14.7955	−0.507184
$852$	0	0
$853$	7.63504	0.261419	0.130709	−	0.991421i	$-0.458275\pi$
$853$	7.63504	0.261419	0.130709	+	0.991421i	$0.458275\pi$
$854$	0	0
$855$	−32.7297	−1.11933
$856$	0	0
$857$	−4.22874	−0.144451	−0.0722255	−	0.997388i	$-0.523010\pi$
$857$	−4.22874	−0.144451	−0.0722255	+	0.997388i	$0.523010\pi$
$858$	0	0
$859$	44.5369	1.51958	0.759789	−	0.650170i	$-0.225302\pi$
$859$	44.5369	1.51958	0.759789	+	0.650170i	$0.225302\pi$
$860$	0	0
$861$	−91.5450	−3.11985
$862$	0	0
$863$	−3.64429	−0.124053	−0.0620266	−	0.998074i	$-0.519756\pi$
$863$	−3.64429	−0.124053	−0.0620266	+	0.998074i	$0.519756\pi$
$864$	0	0
$865$	−13.1762	−0.448004
$866$	0	0
$867$	44.1358	1.49893
$868$	0	0
$869$	−13.6303	−0.462375
$870$	0	0
$871$	−16.3399	−0.553656
$872$	0	0
$873$	−12.5828	−0.425863
$874$	0	0
$875$	3.75132	0.126818
$876$	0	0
$877$	−40.7502	−1.37604	−0.688018	−	0.725694i	$-0.741519\pi$
$877$	−40.7502	−1.37604	−0.688018	+	0.725694i	$0.741519\pi$
$878$	0	0
$879$	9.49939	0.320407
$880$	0	0
$881$	−11.4553	−0.385940	−0.192970	−	0.981205i	$-0.561812\pi$
$881$	−11.4553	−0.385940	−0.192970	+	0.981205i	$0.561812\pi$
$882$	0	0
$883$	5.39868	0.181680	0.0908401	−	0.995865i	$-0.471045\pi$
$883$	5.39868	0.181680	0.0908401	+	0.995865i	$0.471045\pi$
$884$	0	0
$885$	25.4787	0.856457
$886$	0	0
$887$	25.7317	0.863987	0.431994	−	0.901877i	$-0.357810\pi$
$887$	25.7317	0.863987	0.431994	+	0.901877i	$0.357810\pi$
$888$	0	0
$889$	19.6045	0.657515
$890$	0	0
$891$	−2.00230	−0.0670794
$892$	0	0
$893$	28.7198	0.961071
$894$	0	0
$895$	−9.74668	−0.325796
$896$	0	0
$897$	77.8462	2.59921
$898$	0	0
$899$	0.588260	0.0196196
$900$	0	0
$901$	−20.0770	−0.668862
$902$	0	0
$903$	63.1046	2.09999
$904$	0	0
$905$	6.93270	0.230451
$906$	0	0
$907$	−6.59646	−0.219032	−0.109516	−	0.993985i	$-0.534930\pi$
$907$	−6.59646	−0.219032	−0.109516	+	0.993985i	$0.534930\pi$
$908$	0	0
$909$	−49.7614	−1.65048
$910$	0	0
$911$	−22.1874	−0.735102	−0.367551	−	0.930003i	$-0.619804\pi$
$911$	−22.1874	−0.735102	−0.367551	+	0.930003i	$0.619804\pi$
$912$	0	0
$913$	14.0232	0.464100
$914$	0	0
$915$	23.0300	0.761350
$916$	0	0
$917$	13.9051	0.459186
$918$	0	0
$919$	−15.0284	−0.495742	−0.247871	−	0.968793i	$-0.579731\pi$
$919$	−15.0284	−0.495742	−0.247871	+	0.968793i	$0.579731\pi$
$920$	0	0
$921$	62.3000	2.05285
$922$	0	0
$923$	22.8369	0.751684
$924$	0	0
$925$	−2.18339	−0.0717894
$926$	0	0
$927$	−61.1247	−2.00760
$928$	0	0
$929$	−21.3719	−0.701189	−0.350594	−	0.936527i	$-0.614020\pi$
$929$	−21.3719	−0.701189	−0.350594	+	0.936527i	$0.614020\pi$
$930$	0	0
$931$	−49.7310	−1.62987
$932$	0	0
$933$	−31.7693	−1.04008
$934$	0	0
$935$	8.85148	0.289474
$936$	0	0
$937$	−9.16254	−0.299327	−0.149664	−	0.988737i	$-0.547819\pi$
$937$	−9.16254	−0.299327	−0.149664	+	0.988737i	$0.547819\pi$
$938$	0	0
$939$	−6.15173	−0.200754
$940$	0	0
$941$	−49.0960	−1.60048	−0.800241	−	0.599678i	$-0.795295\pi$
$941$	−49.0960	−1.60048	−0.800241	+	0.599678i	$0.795295\pi$
$942$	0	0
$943$	−59.7708	−1.94641
$944$	0	0
$945$	17.1726	0.558625
$946$	0	0
$947$	−15.2650	−0.496045	−0.248023	−	0.968754i	$-0.579781\pi$
$947$	−15.2650	−0.496045	−0.248023	+	0.968754i	$0.579781\pi$
$948$	0	0
$949$	−54.0934	−1.75594
$950$	0	0
$951$	89.9876	2.91805
$952$	0	0
$953$	−4.11784	−0.133390	−0.0666949	−	0.997773i	$-0.521245\pi$
$953$	−4.11784	−0.133390	−0.0666949	+	0.997773i	$0.521245\pi$
$954$	0	0
$955$	−4.37217	−0.141480
$956$	0	0
$957$	−43.4049	−1.40308
$958$	0	0
$959$	−57.8643	−1.86853
$960$	0	0
$961$	−30.9967	−0.999892
$962$	0	0
$963$	−51.7284	−1.66693
$964$	0	0
$965$	2.05206	0.0660583
$966$	0	0
$967$	34.1373	1.09778	0.548891	−	0.835894i	$-0.315050\pi$
$967$	34.1373	1.09778	0.548891	+	0.835894i	$0.315050\pi$
$968$	0	0
$969$	−111.677	−3.58760
$970$	0	0
$971$	16.7233	0.536675	0.268338	−	0.963325i	$-0.413526\pi$
$971$	16.7233	0.536675	0.268338	+	0.963325i	$0.413526\pi$
$972$	0	0
$973$	26.2700	0.842177
$974$	0	0
$975$	11.4878	0.367905
$976$	0	0
$977$	39.4703	1.26277	0.631384	−	0.775470i	$-0.282487\pi$
$977$	39.4703	1.26277	0.631384	+	0.775470i	$0.282487\pi$
$978$	0	0
$979$	1.54195	0.0492811
$980$	0	0
$981$	−5.50457	−0.175747
$982$	0	0
$983$	24.8433	0.792380	0.396190	−	0.918169i	$-0.370332\pi$
$983$	24.8433	0.792380	0.396190	+	0.918169i	$0.370332\pi$
$984$	0	0
$985$	−6.48648	−0.206677
$986$	0	0
$987$	−42.3903	−1.34930
$988$	0	0
$989$	41.2018	1.31014
$990$	0	0
$991$	0.276840	0.00879411	0.00439706	−	0.999990i	$-0.498600\pi$
$991$	0.276840	0.00879411	0.00439706	+	0.999990i	$0.498600\pi$
$992$	0	0
$993$	75.3857	2.39229
$994$	0	0
$995$	−13.4260	−0.425633
$996$	0	0
$997$	−61.1434	−1.93643	−0.968216	−	0.250117i	$-0.919531\pi$
$997$	−61.1434	−1.93643	−0.968216	+	0.250117i	$0.919531\pi$
$998$	0	0
$999$	−9.99501	−0.316228

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7120.2.a.bj.1.5		7
4.3	odd	2		445.2.a.f.1.2	✓	7
12.11	even	2		4005.2.a.o.1.6		7
20.19	odd	2		2225.2.a.k.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.f.1.2	✓	7	4.3	odd	2
2225.2.a.k.1.6		7	20.19	odd	2
4005.2.a.o.1.6		7	12.11	even	2
7120.2.a.bj.1.5		7	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:	\( N \)	\(=\)	\( 7120 = 2^{4} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7120.a (trivial)

\(f(q)\)	\(=\)	\(q+2.76669 q^{3} +1.00000 q^{5} +3.75132 q^{7} +4.65459 q^{9} +O(q^{10})\)	\(q+2.76669 q^{3} +1.00000 q^{5} +3.75132 q^{7} +4.65459 q^{9} +1.54195 q^{11} +4.15218 q^{13} +2.76669 q^{15} +5.74043 q^{17} -7.03169 q^{19} +10.3788 q^{21} +6.77641 q^{23} +1.00000 q^{25} +4.57775 q^{27} -10.1743 q^{29} -0.0578180 q^{31} +4.26612 q^{33} +3.75132 q^{35} -2.18339 q^{37} +11.4878 q^{39} -8.82042 q^{41} +6.08017 q^{43} +4.65459 q^{45} -4.08434 q^{47} +7.07241 q^{49} +15.8820 q^{51} -3.49748 q^{53} +1.54195 q^{55} -19.4545 q^{57} +9.20908 q^{59} +8.32403 q^{61} +17.4609 q^{63} +4.15218 q^{65} -3.93525 q^{67} +18.7483 q^{69} +5.49996 q^{71} -13.0277 q^{73} +2.76669 q^{75} +5.78437 q^{77} -8.83960 q^{79} -1.29854 q^{81} +9.09443 q^{83} +5.74043 q^{85} -28.1493 q^{87} +1.00000 q^{89} +15.5762 q^{91} -0.159965 q^{93} -7.03169 q^{95} -2.70331 q^{97} +7.17717 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9}+O(q^{10}) \) 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9	\( 7 q + 8 q^{3} + 7 q^{5} + 16 q^{7} + 11 q^{9} + 10 q^{11} - 7 q^{13} + 8 q^{15} - 13 q^{17} + 7 q^{19} + 16 q^{21} + 13 q^{23} + 7 q^{25} + 23 q^{27} - 4 q^{29} - q^{31} - 6 q^{33} + 16 q^{35} - 5 q^{37} + 13 q^{39} + 5 q^{41} + 31 q^{43} + 11 q^{45} + 14 q^{47} + 19 q^{49} + q^{51} - 13 q^{53} + 10 q^{55} + 21 q^{57} + 14 q^{59} + 3 q^{61} + 54 q^{63} - 7 q^{65} - q^{67} + 31 q^{69} + 8 q^{71} + 9 q^{73} + 8 q^{75} + 42 q^{77} - 9 q^{79} + 35 q^{81} + 42 q^{83} - 13 q^{85} - 6 q^{87} + 7 q^{89} - 31 q^{91} + 24 q^{93} + 7 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) \) 7 * q + 8 * q^3 + 7 * q^5 + 16 * q^7 + 11 * q^9 + 10 * q^11 - 7 * q^13 + 8 * q^15 - 13 * q^17 + 7 * q^19 + 16 * q^21 + 13 * q^23 + 7 * q^25 + 23 * q^27 - 4 * q^29 - q^31 - 6 * q^33 + 16 * q^35 - 5 * q^37 + 13 * q^39 + 5 * q^41 + 31 * q^43 + 11 * q^45 + 14 * q^47 + 19 * q^49 + q^51 - 13 * q^53 + 10 * q^55 + 21 * q^57 + 14 * q^59 + 3 * q^61 + 54 * q^63 - 7 * q^65 - q^67 + 31 * q^69 + 8 * q^71 + 9 * q^73 + 8 * q^75 + 42 * q^77 - 9 * q^79 + 35 * q^81 + 42 * q^83 - 13 * q^85 - 6 * q^87 + 7 * q^89 - 31 * q^91 + 24 * q^93 + 7 * q^95 - 7 * q^97 - 5 * q^99

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