LMFDB - Embedded newform 7120.2.a.bc.1.1

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:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.737640$ 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)

$f(q)$	$=$	$q-1.19353 q^{3} -1.00000 q^{5} +0.475281 q^{7} -1.57549 q^{9} +O(q^{10})$	$q-1.19353 q^{3} -1.00000 q^{5} +0.475281 q^{7} -1.57549 q^{9} +5.19353 q^{11} -1.30490 q^{13} +1.19353 q^{15} -7.40330 q^{17} -6.83604 q^{19} -0.567260 q^{21} -2.69195 q^{23} +1.00000 q^{25} +5.46097 q^{27} -0.00689789 q^{29} +8.54230 q^{31} -6.19862 q^{33} -0.475281 q^{35} -4.56119 q^{37} +1.55743 q^{39} +3.41020 q^{41} -9.87858 q^{43} +1.57549 q^{45} +4.62493 q^{47} -6.77411 q^{49} +8.83604 q^{51} +2.86526 q^{53} -5.19353 q^{55} +8.15900 q^{57} +1.13767 q^{59} +7.30684 q^{61} -0.748801 q^{63} +1.30490 q^{65} +1.79531 q^{67} +3.21292 q^{69} -2.91492 q^{71} -2.26369 q^{73} -1.19353 q^{75} +2.46838 q^{77} +10.5104 q^{79} -1.79134 q^{81} -6.35650 q^{83} +7.40330 q^{85} +0.00823281 q^{87} -1.00000 q^{89} -0.620194 q^{91} -10.1955 q^{93} +6.83604 q^{95} +18.3113 q^{97} -8.18237 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 4 * q^9	$ 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9} + 14 q^{11} - 5 q^{13} - 2 q^{15} - 3 q^{17} + q^{19} - 4 q^{21} + 3 q^{23} + 4 q^{25} - q^{27} - 10 q^{29} + 11 q^{31} + 2 q^{35} + 3 q^{37} - 11 q^{39} - 3 q^{41} - 9 q^{43} + 4 q^{45} + 24 q^{47} + 7 q^{51} + 3 q^{53} - 14 q^{55} + 19 q^{57} + 22 q^{59} - 3 q^{61} - 6 q^{63} + 5 q^{65} + 9 q^{67} + 7 q^{69} - 16 q^{71} + 3 q^{73} + 2 q^{75} - 4 q^{77} + 27 q^{79} - 8 q^{81} - 6 q^{83} + 3 q^{85} - 4 q^{87} - 4 q^{89} + 29 q^{91} - 2 q^{93} - q^{95} + 41 q^{97} - 21 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 4 * q^9 + 14 * q^11 - 5 * q^13 - 2 * q^15 - 3 * q^17 + q^19 - 4 * q^21 + 3 * q^23 + 4 * q^25 - q^27 - 10 * q^29 + 11 * q^31 + 2 * q^35 + 3 * q^37 - 11 * q^39 - 3 * q^41 - 9 * q^43 + 4 * q^45 + 24 * q^47 + 7 * q^51 + 3 * q^53 - 14 * q^55 + 19 * q^57 + 22 * q^59 - 3 * q^61 - 6 * q^63 + 5 * q^65 + 9 * q^67 + 7 * q^69 - 16 * q^71 + 3 * q^73 + 2 * q^75 - 4 * q^77 + 27 * q^79 - 8 * q^81 - 6 * q^83 + 3 * q^85 - 4 * q^87 - 4 * q^89 + 29 * q^91 - 2 * q^93 - q^95 + 41 * q^97 - 21 * 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
$3$	−1.19353	−0.689083	−0.344542	−	0.938771i	$-0.611966\pi$
$3$	−1.19353	−0.689083	−0.344542	+	0.938771i	$0.611966\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.475281	0.179639	0.0898196	−	0.995958i	$-0.471371\pi$
$7$	0.475281	0.179639	0.0898196	+	0.995958i	$0.471371\pi$
$8$	0	0
$9$	−1.57549	−0.525164
$10$	0	0
$11$	5.19353	1.56591	0.782954	−	0.622080i	$-0.213712\pi$
$11$	5.19353	1.56591	0.782954	+	0.622080i	$0.213712\pi$
$12$	0	0
$13$	−1.30490	−0.361914	−0.180957	−	0.983491i	$-0.557920\pi$
$13$	−1.30490	−0.361914	−0.180957	+	0.983491i	$0.557920\pi$
$14$	0	0
$15$	1.19353	0.308167
$16$	0	0
$17$	−7.40330	−1.79556	−0.897782	−	0.440439i	$-0.854823\pi$
$17$	−7.40330	−1.79556	−0.897782	+	0.440439i	$0.854823\pi$
$18$	0	0
$19$	−6.83604	−1.56830	−0.784148	−	0.620574i	$-0.786900\pi$
$19$	−6.83604	−1.56830	−0.784148	+	0.620574i	$0.786900\pi$
$20$	0	0
$21$	−0.567260	−0.123786
$22$	0	0
$23$	−2.69195	−0.561311	−0.280656	−	0.959809i	$-0.590552\pi$
$23$	−2.69195	−0.561311	−0.280656	+	0.959809i	$0.590552\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.46097	1.05097
$28$	0	0
$29$	−0.00689789	−0.00128091	−0.000640453	−	1.00000i	$-0.500204\pi$
$29$	−0.00689789	−0.00128091	−0.000640453		1.00000i	$0.500204\pi$
$30$	0	0
$31$	8.54230	1.53424	0.767121	−	0.641502i	$-0.221688\pi$
$31$	8.54230	1.53424	0.767121	+	0.641502i	$0.221688\pi$
$32$	0	0
$33$	−6.19862	−1.07904
$34$	0	0
$35$	−0.475281	−0.0803371
$36$	0	0
$37$	−4.56119	−0.749855	−0.374927	−	0.927054i	$-0.622332\pi$
$37$	−4.56119	−0.749855	−0.374927	+	0.927054i	$0.622332\pi$
$38$	0	0
$39$	1.55743	0.249389
$40$	0	0
$41$	3.41020	0.532584	0.266292	−	0.963892i	$-0.414201\pi$
$41$	3.41020	0.532584	0.266292	+	0.963892i	$0.414201\pi$
$42$	0	0
$43$	−9.87858	−1.50647	−0.753235	−	0.657752i	$-0.771508\pi$
$43$	−9.87858	−1.50647	−0.753235	+	0.657752i	$0.771508\pi$
$44$	0	0
$45$	1.57549	0.234861
$46$	0	0
$47$	4.62493	0.674616	0.337308	−	0.941394i	$-0.390484\pi$
$47$	4.62493	0.674616	0.337308	+	0.941394i	$0.390484\pi$
$48$	0	0
$49$	−6.77411	−0.967730
$50$	0	0
$51$	8.83604	1.23729
$52$	0	0
$53$	2.86526	0.393574	0.196787	−	0.980446i	$-0.436949\pi$
$53$	2.86526	0.393574	0.196787	+	0.980446i	$0.436949\pi$
$54$	0	0
$55$	−5.19353	−0.700295
$56$	0	0
$57$	8.15900	1.08069
$58$	0	0
$59$	1.13767	0.148111	0.0740557	−	0.997254i	$-0.476406\pi$
$59$	1.13767	0.148111	0.0740557	+	0.997254i	$0.476406\pi$
$60$	0	0
$61$	7.30684	0.935546	0.467773	−	0.883849i	$-0.345057\pi$
$61$	7.30684	0.935546	0.467773	+	0.883849i	$0.345057\pi$
$62$	0	0
$63$	−0.748801	−0.0943401
$64$	0	0
$65$	1.30490	0.161853
$66$	0	0
$67$	1.79531	0.219332	0.109666	−	0.993968i	$-0.465022\pi$
$67$	1.79531	0.219332	0.109666	+	0.993968i	$0.465022\pi$
$68$	0	0
$69$	3.21292	0.386790
$70$	0	0
$71$	−2.91492	−0.345937	−0.172969	−	0.984927i	$-0.555336\pi$
$71$	−2.91492	−0.345937	−0.172969	+	0.984927i	$0.555336\pi$
$72$	0	0
$73$	−2.26369	−0.264945	−0.132473	−	0.991187i	$-0.542292\pi$
$73$	−2.26369	−0.264945	−0.132473	+	0.991187i	$0.542292\pi$
$74$	0	0
$75$	−1.19353	−0.137817
$76$	0	0
$77$	2.46838	0.281298
$78$	0	0
$79$	10.5104	1.18251	0.591257	−	0.806483i	$-0.298632\pi$
$79$	10.5104	1.18251	0.591257	+	0.806483i	$0.298632\pi$
$80$	0	0
$81$	−1.79134	−0.199038
$82$	0	0
$83$	−6.35650	−0.697716	−0.348858	−	0.937176i	$-0.613431\pi$
$83$	−6.35650	−0.697716	−0.348858	+	0.937176i	$0.613431\pi$
$84$	0	0
$85$	7.40330	0.803001
$86$	0	0
$87$	0.00823281	0.000882650 0
$88$	0	0
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−0.620194	−0.0650140
$92$	0	0
$93$	−10.1955	−1.05722
$94$	0	0
$95$	6.83604	0.701363
$96$	0	0
$97$	18.3113	1.85923	0.929617	−	0.368528i	$-0.120138\pi$
$97$	18.3113	1.85923	0.929617	+	0.368528i	$0.120138\pi$
$98$	0	0
$99$	−8.18237	−0.822359
$100$	0	0
$101$	−14.3545	−1.42832	−0.714162	−	0.699981i	$-0.753192\pi$
$101$	−14.3545	−1.42832	−0.714162	+	0.699981i	$0.753192\pi$
$102$	0	0
$103$	−5.43964	−0.535983	−0.267992	−	0.963421i	$-0.586360\pi$
$103$	−5.43964	−0.535983	−0.267992	+	0.963421i	$0.586360\pi$
$104$	0	0
$105$	0.567260	0.0553589
$106$	0	0
$107$	7.27486	0.703287	0.351643	−	0.936134i	$-0.385623\pi$
$107$	7.27486	0.703287	0.351643	+	0.936134i	$0.385623\pi$
$108$	0	0
$109$	3.92293	0.375749	0.187874	−	0.982193i	$-0.439840\pi$
$109$	3.92293	0.375749	0.187874	+	0.982193i	$0.439840\pi$
$110$	0	0
$111$	5.44390	0.516712
$112$	0	0
$113$	−15.2119	−1.43102	−0.715509	−	0.698603i	$-0.753805\pi$
$113$	−15.2119	−1.43102	−0.715509	+	0.698603i	$0.753805\pi$
$114$	0	0
$115$	2.69195	0.251026
$116$	0	0
$117$	2.05586	0.190064
$118$	0	0
$119$	−3.51865	−0.322554
$120$	0	0
$121$	15.9727	1.45207
$122$	0	0
$123$	−4.07017	−0.366995
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.3082	−1.26965	−0.634823	−	0.772658i	$-0.718927\pi$
$127$	−14.3082	−1.26965	−0.634823	+	0.772658i	$0.718927\pi$
$128$	0	0
$129$	11.7904	1.03808
$130$	0	0
$131$	5.16529	0.451294	0.225647	−	0.974209i	$-0.427550\pi$
$131$	5.16529	0.451294	0.225647	+	0.974209i	$0.427550\pi$
$132$	0	0
$133$	−3.24904	−0.281727
$134$	0	0
$135$	−5.46097	−0.470006
$136$	0	0
$137$	−10.1033	−0.863181	−0.431590	−	0.902070i	$-0.642047\pi$
$137$	−10.1033	−0.863181	−0.431590	+	0.902070i	$0.642047\pi$
$138$	0	0
$139$	21.0062	1.78172	0.890862	−	0.454275i	$-0.150102\pi$
$139$	21.0062	1.78172	0.890862	+	0.454275i	$0.150102\pi$
$140$	0	0
$141$	−5.51998	−0.464866
$142$	0	0
$143$	−6.77704	−0.566724
$144$	0	0
$145$	0.00689789	0.000572838 0
$146$	0	0
$147$	8.08508	0.666846
$148$	0	0
$149$	−4.55954	−0.373532	−0.186766	−	0.982404i	$-0.559801\pi$
$149$	−4.55954	−0.373532	−0.186766	+	0.982404i	$0.559801\pi$
$150$	0	0
$151$	−3.64071	−0.296276	−0.148138	−	0.988967i	$-0.547328\pi$
$151$	−3.64071	−0.296276	−0.148138	+	0.988967i	$0.547328\pi$
$152$	0	0
$153$	11.6639	0.942967
$154$	0	0
$155$	−8.54230	−0.686134
$156$	0	0
$157$	17.3667	1.38601	0.693006	−	0.720932i	$-0.256286\pi$
$157$	17.3667	1.38601	0.693006	+	0.720932i	$0.256286\pi$
$158$	0	0
$159$	−3.41977	−0.271205
$160$	0	0
$161$	−1.27943	−0.100834
$162$	0	0
$163$	9.06918	0.710353	0.355177	−	0.934799i	$-0.384421\pi$
$163$	9.06918	0.710353	0.355177	+	0.934799i	$0.384421\pi$
$164$	0	0
$165$	6.19862	0.482562
$166$	0	0
$167$	−4.29059	−0.332016	−0.166008	−	0.986124i	$-0.553088\pi$
$167$	−4.29059	−0.332016	−0.166008	+	0.986124i	$0.553088\pi$
$168$	0	0
$169$	−11.2972	−0.869018
$170$	0	0
$171$	10.7701	0.823613
$172$	0	0
$173$	21.1969	1.61157	0.805787	−	0.592206i	$-0.201743\pi$
$173$	21.1969	1.61157	0.805787	+	0.592206i	$0.201743\pi$
$174$	0	0
$175$	0.475281	0.0359278
$176$	0	0
$177$	−1.35783	−0.102061
$178$	0	0
$179$	−0.468991	−0.0350541	−0.0175270	−	0.999846i	$-0.505579\pi$
$179$	−0.468991	−0.0350541	−0.0175270	+	0.999846i	$0.505579\pi$
$180$	0	0
$181$	−22.2732	−1.65555	−0.827776	−	0.561058i	$-0.810394\pi$
$181$	−22.2732	−1.65555	−0.827776	+	0.561058i	$0.810394\pi$
$182$	0	0
$183$	−8.72092	−0.644669
$184$	0	0
$185$	4.56119	0.335345
$186$	0	0
$187$	−38.4493	−2.81169
$188$	0	0
$189$	2.59550	0.188795
$190$	0	0
$191$	21.3072	1.54173	0.770867	−	0.636996i	$-0.219823\pi$
$191$	21.3072	1.54173	0.770867	+	0.636996i	$0.219823\pi$
$192$	0	0
$193$	−1.33822	−0.0963275	−0.0481637	−	0.998839i	$-0.515337\pi$
$193$	−1.33822	−0.0963275	−0.0481637	+	0.998839i	$0.515337\pi$
$194$	0	0
$195$	−1.55743	−0.111530
$196$	0	0
$197$	−9.65959	−0.688217	−0.344109	−	0.938930i	$-0.611819\pi$
$197$	−9.65959	−0.688217	−0.344109	+	0.938930i	$0.611819\pi$
$198$	0	0
$199$	−9.85738	−0.698771	−0.349386	−	0.936979i	$-0.613610\pi$
$199$	−9.85738	−0.698771	−0.349386	+	0.936979i	$0.613610\pi$
$200$	0	0
$201$	−2.14275	−0.151138
$202$	0	0
$203$	−0.00327843	−0.000230101 0
$204$	0	0
$205$	−3.41020	−0.238179
$206$	0	0
$207$	4.24116	0.294781
$208$	0	0
$209$	−35.5032	−2.45581
$210$	0	0
$211$	12.2510	0.843396	0.421698	−	0.906736i	$-0.361434\pi$
$211$	12.2510	0.843396	0.421698	+	0.906736i	$0.361434\pi$
$212$	0	0
$213$	3.47903	0.238379
$214$	0	0
$215$	9.87858	0.673714
$216$	0	0
$217$	4.05999	0.275610
$218$	0	0
$219$	2.70178	0.182569
$220$	0	0
$221$	9.66057	0.649841
$222$	0	0
$223$	22.4242	1.50163	0.750817	−	0.660511i	$-0.229660\pi$
$223$	22.4242	1.50163	0.750817	+	0.660511i	$0.229660\pi$
$224$	0	0
$225$	−1.57549	−0.105033
$226$	0	0
$227$	23.1665	1.53762	0.768809	−	0.639479i	$-0.220850\pi$
$227$	23.1665	1.53762	0.768809	+	0.639479i	$0.220850\pi$
$228$	0	0
$229$	4.15788	0.274761	0.137380	−	0.990518i	$-0.456132\pi$
$229$	4.15788	0.274761	0.137380	+	0.990518i	$0.456132\pi$
$230$	0	0
$231$	−2.94608	−0.193838
$232$	0	0
$233$	−11.7721	−0.771215	−0.385607	−	0.922663i	$-0.626008\pi$
$233$	−11.7721	−0.771215	−0.385607	+	0.922663i	$0.626008\pi$
$234$	0	0
$235$	−4.62493	−0.301697
$236$	0	0
$237$	−12.5445	−0.814850
$238$	0	0
$239$	20.3834	1.31849	0.659246	−	0.751927i	$-0.270875\pi$
$239$	20.3834	1.31849	0.659246	+	0.751927i	$0.270875\pi$
$240$	0	0
$241$	−11.3468	−0.730908	−0.365454	−	0.930829i	$-0.619086\pi$
$241$	−11.3468	−0.730908	−0.365454	+	0.930829i	$0.619086\pi$
$242$	0	0
$243$	−14.2449	−0.913811
$244$	0	0
$245$	6.77411	0.432782
$246$	0	0
$247$	8.92036	0.567589
$248$	0	0
$249$	7.58665	0.480785
$250$	0	0
$251$	16.0015	1.01000	0.505002	−	0.863118i	$-0.331492\pi$
$251$	16.0015	1.01000	0.505002	+	0.863118i	$0.331492\pi$
$252$	0	0
$253$	−13.9807	−0.878962
$254$	0	0
$255$	−8.83604	−0.553334
$256$	0	0
$257$	9.71532	0.606025	0.303012	−	0.952987i	$-0.402008\pi$
$257$	9.71532	0.606025	0.303012	+	0.952987i	$0.402008\pi$
$258$	0	0
$259$	−2.16784	−0.134703
$260$	0	0
$261$	0.0108676	0.000672686 0
$262$	0	0
$263$	−23.1738	−1.42896	−0.714479	−	0.699657i	$-0.753336\pi$
$263$	−23.1738	−1.42896	−0.714479	+	0.699657i	$0.753336\pi$
$264$	0	0
$265$	−2.86526	−0.176012
$266$	0	0
$267$	1.19353	0.0730427
$268$	0	0
$269$	−4.92771	−0.300448	−0.150224	−	0.988652i	$-0.547999\pi$
$269$	−4.92771	−0.300448	−0.150224	+	0.988652i	$0.547999\pi$
$270$	0	0
$271$	−16.5952	−1.00809	−0.504044	−	0.863678i	$-0.668155\pi$
$271$	−16.5952	−1.00809	−0.504044	+	0.863678i	$0.668155\pi$
$272$	0	0
$273$	0.740218	0.0448000
$274$	0	0
$275$	5.19353	0.313181
$276$	0	0
$277$	−12.9860	−0.780256	−0.390128	−	0.920761i	$-0.627569\pi$
$277$	−12.9860	−0.780256	−0.390128	+	0.920761i	$0.627569\pi$
$278$	0	0
$279$	−13.4583	−0.805730
$280$	0	0
$281$	29.4497	1.75682	0.878412	−	0.477904i	$-0.158603\pi$
$281$	29.4497	1.75682	0.878412	+	0.477904i	$0.158603\pi$
$282$	0	0
$283$	14.8999	0.885708	0.442854	−	0.896594i	$-0.353966\pi$
$283$	14.8999	0.885708	0.442854	+	0.896594i	$0.353966\pi$
$284$	0	0
$285$	−8.15900	−0.483298
$286$	0	0
$287$	1.62080	0.0956729
$288$	0	0
$289$	37.8089	2.22405
$290$	0	0
$291$	−21.8551	−1.28117
$292$	0	0
$293$	−2.70575	−0.158072	−0.0790358	−	0.996872i	$-0.525184\pi$
$293$	−2.70575	−0.158072	−0.0790358	+	0.996872i	$0.525184\pi$
$294$	0	0
$295$	−1.13767	−0.0662374
$296$	0	0
$297$	28.3617	1.64571
$298$	0	0
$299$	3.51273	0.203147
$300$	0	0
$301$	−4.69510	−0.270621
$302$	0	0
$303$	17.1325	0.984234
$304$	0	0
$305$	−7.30684	−0.418389
$306$	0	0
$307$	15.4273	0.880481	0.440241	−	0.897880i	$-0.354893\pi$
$307$	15.4273	0.880481	0.440241	+	0.897880i	$0.354893\pi$
$308$	0	0
$309$	6.49235	0.369337
$310$	0	0
$311$	9.28894	0.526728	0.263364	−	0.964697i	$-0.415168\pi$
$311$	9.28894	0.526728	0.263364	+	0.964697i	$0.415168\pi$
$312$	0	0
$313$	−5.56675	−0.314651	−0.157326	−	0.987547i	$-0.550287\pi$
$313$	−5.56675	−0.314651	−0.157326	+	0.987547i	$0.550287\pi$
$314$	0	0
$315$	0.748801	0.0421902
$316$	0	0
$317$	30.9404	1.73778	0.868892	−	0.495002i	$-0.164833\pi$
$317$	30.9404	1.73778	0.868892	+	0.495002i	$0.164833\pi$
$318$	0	0
$319$	−0.0358244	−0.00200578
$320$	0	0
$321$	−8.68274	−0.484623
$322$	0	0
$323$	50.6093	2.81598
$324$	0	0
$325$	−1.30490	−0.0723829
$326$	0	0
$327$	−4.68213	−0.258922
$328$	0	0
$329$	2.19814	0.121187
$330$	0	0
$331$	−0.0734460	−0.00403696	−0.00201848	−	0.999998i	$-0.500643\pi$
$331$	−0.0734460	−0.00403696	−0.00201848	+	0.999998i	$0.500643\pi$
$332$	0	0
$333$	7.18612	0.393797
$334$	0	0
$335$	−1.79531	−0.0980884
$336$	0	0
$337$	16.9619	0.923971	0.461986	−	0.886887i	$-0.347137\pi$
$337$	16.9619	0.923971	0.461986	+	0.886887i	$0.347137\pi$
$338$	0	0
$339$	18.1559	0.986091
$340$	0	0
$341$	44.3647	2.40248
$342$	0	0
$343$	−6.54657	−0.353481
$344$	0	0
$345$	−3.21292	−0.172978
$346$	0	0
$347$	21.2221	1.13926	0.569630	−	0.821901i	$-0.307087\pi$
$347$	21.2221	1.13926	0.569630	+	0.821901i	$0.307087\pi$
$348$	0	0
$349$	7.48298	0.400555	0.200277	−	0.979739i	$-0.435816\pi$
$349$	7.48298	0.400555	0.200277	+	0.979739i	$0.435816\pi$
$350$	0	0
$351$	−7.12603	−0.380359
$352$	0	0
$353$	2.09475	0.111492	0.0557461	−	0.998445i	$-0.482246\pi$
$353$	2.09475	0.111492	0.0557461	+	0.998445i	$0.482246\pi$
$354$	0	0
$355$	2.91492	0.154708
$356$	0	0
$357$	4.19960	0.222266
$358$	0	0
$359$	1.57842	0.0833059	0.0416529	−	0.999132i	$-0.486738\pi$
$359$	1.57842	0.0833059	0.0416529	+	0.999132i	$0.486738\pi$
$360$	0	0
$361$	27.7315	1.45955
$362$	0	0
$363$	−19.0639	−1.00059
$364$	0	0
$365$	2.26369	0.118487
$366$	0	0
$367$	−34.6898	−1.81079	−0.905397	−	0.424567i	$-0.860426\pi$
$367$	−34.6898	−1.81079	−0.905397	+	0.424567i	$0.860426\pi$
$368$	0	0
$369$	−5.37275	−0.279694
$370$	0	0
$371$	1.36180	0.0707013
$372$	0	0
$373$	28.3861	1.46977	0.734887	−	0.678189i	$-0.237235\pi$
$373$	28.3861	1.46977	0.734887	+	0.678189i	$0.237235\pi$
$374$	0	0
$375$	1.19353	0.0616335
$376$	0	0
$377$	0.00900106	0.000463578 0
$378$	0	0
$379$	15.3669	0.789344	0.394672	−	0.918822i	$-0.370858\pi$
$379$	15.3669	0.789344	0.394672	+	0.918822i	$0.370858\pi$
$380$	0	0
$381$	17.0772	0.874891
$382$	0	0
$383$	23.8489	1.21862	0.609311	−	0.792932i	$-0.291446\pi$
$383$	23.8489	1.21862	0.609311	+	0.792932i	$0.291446\pi$
$384$	0	0
$385$	−2.46838	−0.125800
$386$	0	0
$387$	15.5636	0.791144
$388$	0	0
$389$	11.7669	0.596603	0.298302	−	0.954472i	$-0.403580\pi$
$389$	11.7669	0.596603	0.298302	+	0.954472i	$0.403580\pi$
$390$	0	0
$391$	19.9294	1.00787
$392$	0	0
$393$	−6.16492	−0.310979
$394$	0	0
$395$	−10.5104	−0.528836
$396$	0	0
$397$	−11.2778	−0.566019	−0.283009	−	0.959117i	$-0.591333\pi$
$397$	−11.2778	−0.566019	−0.283009	+	0.959117i	$0.591333\pi$
$398$	0	0
$399$	3.87782	0.194134
$400$	0	0
$401$	12.9916	0.648770	0.324385	−	0.945925i	$-0.394843\pi$
$401$	12.9916	0.648770	0.324385	+	0.945925i	$0.394843\pi$
$402$	0	0
$403$	−11.1469	−0.555264
$404$	0	0
$405$	1.79134	0.0890125
$406$	0	0
$407$	−23.6887	−1.17420
$408$	0	0
$409$	−14.2198	−0.703124	−0.351562	−	0.936165i	$-0.614349\pi$
$409$	−14.2198	−0.703124	−0.351562	+	0.936165i	$0.614349\pi$
$410$	0	0
$411$	12.0585	0.594804
$412$	0	0
$413$	0.540710	0.0266066
$414$	0	0
$415$	6.35650	0.312028
$416$	0	0
$417$	−25.0715	−1.22776
$418$	0	0
$419$	32.8278	1.60374	0.801871	−	0.597497i	$-0.203838\pi$
$419$	32.8278	1.60374	0.801871	+	0.597497i	$0.203838\pi$
$420$	0	0
$421$	35.9329	1.75126	0.875630	−	0.482982i	$-0.160447\pi$
$421$	35.9329	1.75126	0.875630	+	0.482982i	$0.160447\pi$
$422$	0	0
$423$	−7.28655	−0.354284
$424$	0	0
$425$	−7.40330	−0.359113
$426$	0	0
$427$	3.47280	0.168061
$428$	0	0
$429$	8.08858	0.390520
$430$	0	0
$431$	24.7729	1.19327	0.596634	−	0.802513i	$-0.296504\pi$
$431$	24.7729	1.19327	0.596634	+	0.802513i	$0.296504\pi$
$432$	0	0
$433$	28.1739	1.35395	0.676976	−	0.736005i	$-0.263290\pi$
$433$	28.1739	1.35395	0.676976	+	0.736005i	$0.263290\pi$
$434$	0	0
$435$	−0.00823281	−0.000394733 0
$436$	0	0
$437$	18.4023	0.880302
$438$	0	0
$439$	0.128231	0.00612014	0.00306007	−	0.999995i	$-0.499026\pi$
$439$	0.128231	0.00612014	0.00306007	+	0.999995i	$0.499026\pi$
$440$	0	0
$441$	10.6726	0.508217
$442$	0	0
$443$	−3.96327	−0.188301	−0.0941504	−	0.995558i	$-0.530013\pi$
$443$	−3.96327	−0.188301	−0.0941504	+	0.995558i	$0.530013\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	0	0
$447$	5.44193	0.257395
$448$	0	0
$449$	−5.39593	−0.254650	−0.127325	−	0.991861i	$-0.540639\pi$
$449$	−5.39593	−0.254650	−0.127325	+	0.991861i	$0.540639\pi$
$450$	0	0
$451$	17.7110	0.833977
$452$	0	0
$453$	4.34528	0.204159
$454$	0	0
$455$	0.620194	0.0290751
$456$	0	0
$457$	0.665823	0.0311459	0.0155729	−	0.999879i	$-0.495043\pi$
$457$	0.665823	0.0311459	0.0155729	+	0.999879i	$0.495043\pi$
$458$	0	0
$459$	−40.4293	−1.88708
$460$	0	0
$461$	42.6998	1.98873	0.994365	−	0.106010i	$-0.0338076\pi$
$461$	42.6998	1.98873	0.994365	+	0.106010i	$0.0338076\pi$
$462$	0	0
$463$	−7.05789	−0.328008	−0.164004	−	0.986460i	$-0.552441\pi$
$463$	−7.05789	−0.328008	−0.164004	+	0.986460i	$0.552441\pi$
$464$	0	0
$465$	10.1955	0.472804
$466$	0	0
$467$	30.6233	1.41708	0.708540	−	0.705671i	$-0.249354\pi$
$467$	30.6233	1.41708	0.708540	+	0.705671i	$0.249354\pi$
$468$	0	0
$469$	0.853277	0.0394007
$470$	0	0
$471$	−20.7276	−0.955077
$472$	0	0
$473$	−51.3047	−2.35899
$474$	0	0
$475$	−6.83604	−0.313659
$476$	0	0
$477$	−4.51420	−0.206691
$478$	0	0
$479$	−24.7778	−1.13213	−0.566063	−	0.824362i	$-0.691534\pi$
$479$	−24.7778	−1.13213	−0.566063	+	0.824362i	$0.691534\pi$
$480$	0	0
$481$	5.95190	0.271383
$482$	0	0
$483$	1.52704	0.0694827
$484$	0	0
$485$	−18.3113	−0.831474
$486$	0	0
$487$	−41.0479	−1.86006	−0.930029	−	0.367486i	$-0.880218\pi$
$487$	−41.0479	−1.86006	−0.930029	+	0.367486i	$0.880218\pi$
$488$	0	0
$489$	−10.8243	−0.489492
$490$	0	0
$491$	−32.1154	−1.44935	−0.724674	−	0.689092i	$-0.758010\pi$
$491$	−32.1154	−1.44935	−0.724674	+	0.689092i	$0.758010\pi$
$492$	0	0
$493$	0.0510671	0.00229995
$494$	0	0
$495$	8.18237	0.367770
$496$	0	0
$497$	−1.38540	−0.0621439
$498$	0	0
$499$	29.5346	1.32215	0.661074	−	0.750321i	$-0.270101\pi$
$499$	29.5346	1.32215	0.661074	+	0.750321i	$0.270101\pi$
$500$	0	0
$501$	5.12094	0.228787
$502$	0	0
$503$	−17.9883	−0.802058	−0.401029	−	0.916065i	$-0.631347\pi$
$503$	−17.9883	−0.802058	−0.401029	+	0.916065i	$0.631347\pi$
$504$	0	0
$505$	14.3545	0.638766
$506$	0	0
$507$	13.4836	0.598826
$508$	0	0
$509$	28.2936	1.25409	0.627045	−	0.778983i	$-0.284264\pi$
$509$	28.2936	1.25409	0.627045	+	0.778983i	$0.284264\pi$
$510$	0	0
$511$	−1.07589	−0.0475946
$512$	0	0
$513$	−37.3315	−1.64822
$514$	0	0
$515$	5.43964	0.239699
$516$	0	0
$517$	24.0197	1.05639
$518$	0	0
$519$	−25.2991	−1.11051
$520$	0	0
$521$	−13.2784	−0.581739	−0.290870	−	0.956763i	$-0.593945\pi$
$521$	−13.2784	−0.581739	−0.290870	+	0.956763i	$0.593945\pi$
$522$	0	0
$523$	23.7559	1.03877	0.519386	−	0.854540i	$-0.326161\pi$
$523$	23.7559	1.03877	0.519386	+	0.854540i	$0.326161\pi$
$524$	0	0
$525$	−0.567260	−0.0247573
$526$	0	0
$527$	−63.2413	−2.75483
$528$	0	0
$529$	−15.7534	−0.684930
$530$	0	0
$531$	−1.79238	−0.0777828
$532$	0	0
$533$	−4.44997	−0.192750
$534$	0	0
$535$	−7.27486	−0.314519
$536$	0	0
$537$	0.559754	0.0241552
$538$	0	0
$539$	−35.1815	−1.51538
$540$	0	0
$541$	−12.9184	−0.555404	−0.277702	−	0.960667i	$-0.589573\pi$
$541$	−12.9184	−0.555404	−0.277702	+	0.960667i	$0.589573\pi$
$542$	0	0
$543$	26.5836	1.14081
$544$	0	0
$545$	−3.92293	−0.168040
$546$	0	0
$547$	−39.0481	−1.66958	−0.834788	−	0.550571i	$-0.814410\pi$
$547$	−39.0481	−1.66958	−0.834788	+	0.550571i	$0.814410\pi$
$548$	0	0
$549$	−11.5119	−0.491315
$550$	0	0
$551$	0.0471543	0.00200884
$552$	0	0
$553$	4.99540	0.212426
$554$	0	0
$555$	−5.44390	−0.231081
$556$	0	0
$557$	5.51252	0.233573	0.116786	−	0.993157i	$-0.462741\pi$
$557$	5.51252	0.233573	0.116786	+	0.993157i	$0.462741\pi$
$558$	0	0
$559$	12.8906	0.545213
$560$	0	0
$561$	45.8902	1.93749
$562$	0	0
$563$	37.2064	1.56806	0.784031	−	0.620722i	$-0.213160\pi$
$563$	37.2064	1.56806	0.784031	+	0.620722i	$0.213160\pi$
$564$	0	0
$565$	15.2119	0.639971
$566$	0	0
$567$	−0.851390	−0.0357550
$568$	0	0
$569$	−27.0351	−1.13337	−0.566685	−	0.823935i	$-0.691774\pi$
$569$	−27.0351	−1.13337	−0.566685	+	0.823935i	$0.691774\pi$
$570$	0	0
$571$	−32.8494	−1.37471	−0.687353	−	0.726324i	$-0.741227\pi$
$571$	−32.8494	−1.37471	−0.687353	+	0.726324i	$0.741227\pi$
$572$	0	0
$573$	−25.4307	−1.06238
$574$	0	0
$575$	−2.69195	−0.112262
$576$	0	0
$577$	35.2109	1.46585	0.732925	−	0.680309i	$-0.238155\pi$
$577$	35.2109	1.46585	0.732925	+	0.680309i	$0.238155\pi$
$578$	0	0
$579$	1.59721	0.0663776
$580$	0	0
$581$	−3.02112	−0.125337
$582$	0	0
$583$	14.8808	0.616301
$584$	0	0
$585$	−2.05586	−0.0849994
$586$	0	0
$587$	−40.2094	−1.65962	−0.829811	−	0.558045i	$-0.811552\pi$
$587$	−40.2094	−1.65962	−0.829811	+	0.558045i	$0.811552\pi$
$588$	0	0
$589$	−58.3956	−2.40615
$590$	0	0
$591$	11.5290	0.474239
$592$	0	0
$593$	29.5937	1.21527	0.607633	−	0.794218i	$-0.292119\pi$
$593$	29.5937	1.21527	0.607633	+	0.794218i	$0.292119\pi$
$594$	0	0
$595$	3.51865	0.144250
$596$	0	0
$597$	11.7651	0.481511
$598$	0	0
$599$	8.74518	0.357318	0.178659	−	0.983911i	$-0.442824\pi$
$599$	8.74518	0.357318	0.178659	+	0.983911i	$0.442824\pi$
$600$	0	0
$601$	3.42482	0.139701	0.0698507	−	0.997557i	$-0.477748\pi$
$601$	3.42482	0.139701	0.0698507	+	0.997557i	$0.477748\pi$
$602$	0	0
$603$	−2.82850	−0.115185
$604$	0	0
$605$	−15.9727	−0.649384
$606$	0	0
$607$	20.5908	0.835756	0.417878	−	0.908503i	$-0.362774\pi$
$607$	20.5908	0.835756	0.417878	+	0.908503i	$0.362774\pi$
$608$	0	0
$609$	0.00391290	0.000158559 0
$610$	0	0
$611$	−6.03508	−0.244153
$612$	0	0
$613$	−26.4547	−1.06850	−0.534248	−	0.845328i	$-0.679405\pi$
$613$	−26.4547	−1.06850	−0.534248	+	0.845328i	$0.679405\pi$
$614$	0	0
$615$	4.07017	0.164125
$616$	0	0
$617$	−43.8302	−1.76454	−0.882268	−	0.470747i	$-0.843985\pi$
$617$	−43.8302	−1.76454	−0.882268	+	0.470747i	$0.843985\pi$
$618$	0	0
$619$	0.0432321	0.00173765	0.000868823	−	1.00000i	$-0.499723\pi$
$619$	0.0432321	0.00173765	0.000868823		1.00000i	$0.499723\pi$
$620$	0	0
$621$	−14.7007	−0.589919
$622$	0	0
$623$	−0.475281	−0.0190417
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	42.3740	1.69225
$628$	0	0
$629$	33.7679	1.34641
$630$	0	0
$631$	29.9042	1.19047	0.595233	−	0.803553i	$-0.297060\pi$
$631$	29.9042	1.19047	0.595233	+	0.803553i	$0.297060\pi$
$632$	0	0
$633$	−14.6219	−0.581170
$634$	0	0
$635$	14.3082	0.567803
$636$	0	0
$637$	8.83954	0.350235
$638$	0	0
$639$	4.59243	0.181674
$640$	0	0
$641$	−16.3131	−0.644330	−0.322165	−	0.946684i	$-0.604411\pi$
$641$	−16.3131	−0.644330	−0.322165	+	0.946684i	$0.604411\pi$
$642$	0	0
$643$	−17.2134	−0.678832	−0.339416	−	0.940636i	$-0.610229\pi$
$643$	−17.2134	−0.678832	−0.339416	+	0.940636i	$0.610229\pi$
$644$	0	0
$645$	−11.7904	−0.464245
$646$	0	0
$647$	−28.3178	−1.11329	−0.556645	−	0.830751i	$-0.687912\pi$
$647$	−28.3178	−1.11329	−0.556645	+	0.830751i	$0.687912\pi$
$648$	0	0
$649$	5.90850	0.231929
$650$	0	0
$651$	−4.84571	−0.189918
$652$	0	0
$653$	27.2771	1.06743	0.533717	−	0.845663i	$-0.320795\pi$
$653$	27.2771	1.06743	0.533717	+	0.845663i	$0.320795\pi$
$654$	0	0
$655$	−5.16529	−0.201825
$656$	0	0
$657$	3.56644	0.139140
$658$	0	0
$659$	36.6207	1.42654	0.713270	−	0.700889i	$-0.247213\pi$
$659$	36.6207	1.42654	0.713270	+	0.700889i	$0.247213\pi$
$660$	0	0
$661$	−39.6430	−1.54193	−0.770967	−	0.636875i	$-0.780227\pi$
$661$	−39.6430	−1.54193	−0.770967	+	0.636875i	$0.780227\pi$
$662$	0	0
$663$	−11.5302	−0.447794
$664$	0	0
$665$	3.24904	0.125992
$666$	0	0
$667$	0.0185688	0.000718987 0
$668$	0	0
$669$	−26.7638	−1.03475
$670$	0	0
$671$	37.9483	1.46498
$672$	0	0
$673$	7.70607	0.297047	0.148524	−	0.988909i	$-0.452548\pi$
$673$	7.70607	0.297047	0.148524	+	0.988909i	$0.452548\pi$
$674$	0	0
$675$	5.46097	0.210193
$676$	0	0
$677$	−16.9048	−0.649706	−0.324853	−	0.945765i	$-0.605315\pi$
$677$	−16.9048	−0.649706	−0.324853	+	0.945765i	$0.605315\pi$
$678$	0	0
$679$	8.70302	0.333991
$680$	0	0
$681$	−27.6499	−1.05955
$682$	0	0
$683$	7.11167	0.272120	0.136060	−	0.990701i	$-0.456556\pi$
$683$	7.11167	0.272120	0.136060	+	0.990701i	$0.456556\pi$
$684$	0	0
$685$	10.1033	0.386026
$686$	0	0
$687$	−4.96255	−0.189333
$688$	0	0
$689$	−3.73888	−0.142440
$690$	0	0
$691$	−18.0632	−0.687156	−0.343578	−	0.939124i	$-0.611639\pi$
$691$	−18.0632	−0.687156	−0.343578	+	0.939124i	$0.611639\pi$
$692$	0	0
$693$	−3.88892	−0.147728
$694$	0	0
$695$	−21.0062	−0.796811
$696$	0	0
$697$	−25.2468	−0.956289
$698$	0	0
$699$	14.0503	0.531431
$700$	0	0
$701$	14.5894	0.551034	0.275517	−	0.961296i	$-0.411151\pi$
$701$	14.5894	0.551034	0.275517	+	0.961296i	$0.411151\pi$
$702$	0	0
$703$	31.1805	1.17599
$704$	0	0
$705$	5.51998	0.207895
$706$	0	0
$707$	−6.82240	−0.256583
$708$	0	0
$709$	19.9577	0.749527	0.374763	−	0.927120i	$-0.377724\pi$
$709$	19.9577	0.749527	0.374763	+	0.927120i	$0.377724\pi$
$710$	0	0
$711$	−16.5591	−0.621014
$712$	0	0
$713$	−22.9955	−0.861188
$714$	0	0
$715$	6.77704	0.253447
$716$	0	0
$717$	−24.3281	−0.908551
$718$	0	0
$719$	1.62038	0.0604298	0.0302149	−	0.999543i	$-0.490381\pi$
$719$	1.62038	0.0604298	0.0302149	+	0.999543i	$0.490381\pi$
$720$	0	0
$721$	−2.58535	−0.0962836
$722$	0	0
$723$	13.5427	0.503657
$724$	0	0
$725$	−0.00689789	−0.000256181 0
$726$	0	0
$727$	−5.61596	−0.208284	−0.104142	−	0.994562i	$-0.533210\pi$
$727$	−5.61596	−0.208284	−0.104142	+	0.994562i	$0.533210\pi$
$728$	0	0
$729$	22.3757	0.828730
$730$	0	0
$731$	73.1342	2.70496
$732$	0	0
$733$	48.1350	1.77791	0.888953	−	0.457998i	$-0.151433\pi$
$733$	48.1350	1.77791	0.888953	+	0.457998i	$0.151433\pi$
$734$	0	0
$735$	−8.08508	−0.298223
$736$	0	0
$737$	9.32400	0.343454
$738$	0	0
$739$	8.98910	0.330669	0.165335	−	0.986238i	$-0.447130\pi$
$739$	8.98910	0.330669	0.165335	+	0.986238i	$0.447130\pi$
$740$	0	0
$741$	−10.6467	−0.391116
$742$	0	0
$743$	−2.20072	−0.0807365	−0.0403683	−	0.999185i	$-0.512853\pi$
$743$	−2.20072	−0.0807365	−0.0403683	+	0.999185i	$0.512853\pi$
$744$	0	0
$745$	4.55954	0.167049
$746$	0	0
$747$	10.0146	0.366416
$748$	0	0
$749$	3.45760	0.126338
$750$	0	0
$751$	22.7671	0.830781	0.415391	−	0.909643i	$-0.363645\pi$
$751$	22.7671	0.830781	0.415391	+	0.909643i	$0.363645\pi$
$752$	0	0
$753$	−19.0982	−0.695977
$754$	0	0
$755$	3.64071	0.132499
$756$	0	0
$757$	−30.2022	−1.09772	−0.548859	−	0.835915i	$-0.684937\pi$
$757$	−30.2022	−1.09772	−0.548859	+	0.835915i	$0.684937\pi$
$758$	0	0
$759$	16.6864	0.605678
$760$	0	0
$761$	17.4802	0.633656	0.316828	−	0.948483i	$-0.397382\pi$
$761$	17.4802	0.633656	0.316828	+	0.948483i	$0.397382\pi$
$762$	0	0
$763$	1.86449	0.0674992
$764$	0	0
$765$	−11.6639	−0.421708
$766$	0	0
$767$	−1.48454	−0.0536036
$768$	0	0
$769$	−28.0588	−1.01183	−0.505913	−	0.862585i	$-0.668844\pi$
$769$	−28.0588	−1.01183	−0.505913	+	0.862585i	$0.668844\pi$
$770$	0	0
$771$	−11.5955	−0.417602
$772$	0	0
$773$	−2.19223	−0.0788489	−0.0394245	−	0.999223i	$-0.512552\pi$
$773$	−2.19223	−0.0788489	−0.0394245	+	0.999223i	$0.512552\pi$
$774$	0	0
$775$	8.54230	0.306849
$776$	0	0
$777$	2.58738	0.0928218
$778$	0	0
$779$	−23.3123	−0.835249
$780$	0	0
$781$	−15.1387	−0.541706
$782$	0	0
$783$	−0.0376692	−0.00134619
$784$	0	0
$785$	−17.3667	−0.619843
$786$	0	0
$787$	36.7899	1.31142	0.655709	−	0.755013i	$-0.272370\pi$
$787$	36.7899	1.31142	0.655709	+	0.755013i	$0.272370\pi$
$788$	0	0
$789$	27.6585	0.984670
$790$	0	0
$791$	−7.22994	−0.257067
$792$	0	0
$793$	−9.53471	−0.338587
$794$	0	0
$795$	3.41977	0.121287
$796$	0	0
$797$	−42.4563	−1.50388	−0.751941	−	0.659231i	$-0.770882\pi$
$797$	−42.4563	−1.50388	−0.751941	+	0.659231i	$0.770882\pi$
$798$	0	0
$799$	−34.2398	−1.21132
$800$	0	0
$801$	1.57549	0.0556673
$802$	0	0
$803$	−11.7566	−0.414880
$804$	0	0
$805$	1.27943	0.0450941
$806$	0	0
$807$	5.88135	0.207033
$808$	0	0
$809$	28.4853	1.00149	0.500744	−	0.865595i	$-0.333060\pi$
$809$	28.4853	1.00149	0.500744	+	0.865595i	$0.333060\pi$
$810$	0	0
$811$	17.7783	0.624282	0.312141	−	0.950036i	$-0.398954\pi$
$811$	17.7783	0.624282	0.312141	+	0.950036i	$0.398954\pi$
$812$	0	0
$813$	19.8069	0.694657
$814$	0	0
$815$	−9.06918	−0.317680
$816$	0	0
$817$	67.5304	2.36259
$818$	0	0
$819$	0.977111	0.0341430
$820$	0	0
$821$	31.1566	1.08737	0.543687	−	0.839288i	$-0.317028\pi$
$821$	31.1566	1.08737	0.543687	+	0.839288i	$0.317028\pi$
$822$	0	0
$823$	−20.4644	−0.713346	−0.356673	−	0.934229i	$-0.616089\pi$
$823$	−20.4644	−0.713346	−0.356673	+	0.934229i	$0.616089\pi$
$824$	0	0
$825$	−6.19862	−0.215808
$826$	0	0
$827$	16.0701	0.558812	0.279406	−	0.960173i	$-0.409862\pi$
$827$	16.0701	0.558812	0.279406	+	0.960173i	$0.409862\pi$
$828$	0	0
$829$	−34.1508	−1.18611	−0.593053	−	0.805163i	$-0.702078\pi$
$829$	−34.1508	−1.18611	−0.593053	+	0.805163i	$0.702078\pi$
$830$	0	0
$831$	15.4992	0.537661
$832$	0	0
$833$	50.1508	1.73762
$834$	0	0
$835$	4.29059	0.148482
$836$	0	0
$837$	46.6493	1.61244
$838$	0	0
$839$	2.87004	0.0990846	0.0495423	−	0.998772i	$-0.484224\pi$
$839$	2.87004	0.0990846	0.0495423	+	0.998772i	$0.484224\pi$
$840$	0	0
$841$	−29.0000	−0.999998
$842$	0	0
$843$	−35.1491	−1.21060
$844$	0	0
$845$	11.2972	0.388637
$846$	0	0
$847$	7.59153	0.260848
$848$	0	0
$849$	−17.7835	−0.610327
$850$	0	0
$851$	12.2785	0.420902
$852$	0	0
$853$	22.6448	0.775344	0.387672	−	0.921797i	$-0.373279\pi$
$853$	22.6448	0.775344	0.387672	+	0.921797i	$0.373279\pi$
$854$	0	0
$855$	−10.7701	−0.368331
$856$	0	0
$857$	22.5002	0.768594	0.384297	−	0.923210i	$-0.374444\pi$
$857$	22.5002	0.768594	0.384297	+	0.923210i	$0.374444\pi$
$858$	0	0
$859$	38.7537	1.32226	0.661129	−	0.750272i	$-0.270077\pi$
$859$	38.7537	1.32226	0.661129	+	0.750272i	$0.270077\pi$
$860$	0	0
$861$	−1.93447	−0.0659266
$862$	0	0
$863$	−29.5580	−1.00617	−0.503083	−	0.864238i	$-0.667801\pi$
$863$	−29.5580	−1.00617	−0.503083	+	0.864238i	$0.667801\pi$
$864$	0	0
$865$	−21.1969	−0.720718
$866$	0	0
$867$	−45.1259	−1.53256
$868$	0	0
$869$	54.5861	1.85171
$870$	0	0
$871$	−2.34270	−0.0793795
$872$	0	0
$873$	−28.8494	−0.976403
$874$	0	0
$875$	−0.475281	−0.0160674
$876$	0	0
$877$	−0.306586	−0.0103527	−0.00517633	−	0.999987i	$-0.501648\pi$
$877$	−0.306586	−0.0103527	−0.00517633	+	0.999987i	$0.501648\pi$
$878$	0	0
$879$	3.22939	0.108925
$880$	0	0
$881$	4.86851	0.164024	0.0820121	−	0.996631i	$-0.473865\pi$
$881$	4.86851	0.164024	0.0820121	+	0.996631i	$0.473865\pi$
$882$	0	0
$883$	−54.1155	−1.82113	−0.910565	−	0.413366i	$-0.864353\pi$
$883$	−54.1155	−1.82113	−0.910565	+	0.413366i	$0.864353\pi$
$884$	0	0
$885$	1.35783	0.0456431
$886$	0	0
$887$	42.0954	1.41343	0.706713	−	0.707500i	$-0.250177\pi$
$887$	42.0954	1.41343	0.706713	+	0.707500i	$0.250177\pi$
$888$	0	0
$889$	−6.80040	−0.228078
$890$	0	0
$891$	−9.30338	−0.311675
$892$	0	0
$893$	−31.6162	−1.05800
$894$	0	0
$895$	0.468991	0.0156767
$896$	0	0
$897$	−4.19254	−0.139985
$898$	0	0
$899$	−0.0589238	−0.00196522
$900$	0	0
$901$	−21.2124	−0.706688
$902$	0	0
$903$	5.60373	0.186480
$904$	0	0
$905$	22.2732	0.740386
$906$	0	0
$907$	3.59534	0.119381	0.0596907	−	0.998217i	$-0.480989\pi$
$907$	3.59534	0.119381	0.0596907	+	0.998217i	$0.480989\pi$
$908$	0	0
$909$	22.6154	0.750105
$910$	0	0
$911$	−51.3001	−1.69965	−0.849823	−	0.527068i	$-0.823291\pi$
$911$	−51.3001	−1.69965	−0.849823	+	0.527068i	$0.823291\pi$
$912$	0	0
$913$	−33.0127	−1.09256
$914$	0	0
$915$	8.72092	0.288305
$916$	0	0
$917$	2.45496	0.0810700
$918$	0	0
$919$	24.6752	0.813959	0.406979	−	0.913437i	$-0.366582\pi$
$919$	24.6752	0.813959	0.406979	+	0.913437i	$0.366582\pi$
$920$	0	0
$921$	−18.4129	−0.606725
$922$	0	0
$923$	3.80368	0.125200
$924$	0	0
$925$	−4.56119	−0.149971
$926$	0	0
$927$	8.57011	0.281479
$928$	0	0
$929$	−6.23375	−0.204523	−0.102261	−	0.994758i	$-0.532608\pi$
$929$	−6.23375	−0.204523	−0.102261	+	0.994758i	$0.532608\pi$
$930$	0	0
$931$	46.3081	1.51769
$932$	0	0
$933$	−11.0866	−0.362959
$934$	0	0
$935$	38.4493	1.25743
$936$	0	0
$937$	−10.7283	−0.350479	−0.175240	−	0.984526i	$-0.556070\pi$
$937$	−10.7283	−0.350479	−0.175240	+	0.984526i	$0.556070\pi$
$938$	0	0
$939$	6.64407	0.216821
$940$	0	0
$941$	11.6152	0.378644	0.189322	−	0.981915i	$-0.439371\pi$
$941$	11.6152	0.378644	0.189322	+	0.981915i	$0.439371\pi$
$942$	0	0
$943$	−9.18011	−0.298945
$944$	0	0
$945$	−2.59550	−0.0844315
$946$	0	0
$947$	−9.67739	−0.314473	−0.157236	−	0.987561i	$-0.550258\pi$
$947$	−9.67739	−0.314473	−0.157236	+	0.987561i	$0.550258\pi$
$948$	0	0
$949$	2.95390	0.0958875
$950$	0	0
$951$	−36.9282	−1.19748
$952$	0	0
$953$	18.4382	0.597273	0.298636	−	0.954367i	$-0.403468\pi$
$953$	18.4382	0.597273	0.298636	+	0.954367i	$0.403468\pi$
$954$	0	0
$955$	−21.3072	−0.689485
$956$	0	0
$957$	0.0427573	0.00138215
$958$	0	0
$959$	−4.80189	−0.155061
$960$	0	0
$961$	41.9709	1.35390
$962$	0	0
$963$	−11.4615	−0.369341
$964$	0	0
$965$	1.33822	0.0430790
$966$	0	0
$967$	45.7382	1.47084	0.735420	−	0.677611i	$-0.236985\pi$
$967$	45.7382	1.47084	0.735420	+	0.677611i	$0.236985\pi$
$968$	0	0
$969$	−60.4036	−1.94044
$970$	0	0
$971$	18.8604	0.605259	0.302629	−	0.953108i	$-0.402136\pi$
$971$	18.8604	0.605259	0.302629	+	0.953108i	$0.402136\pi$
$972$	0	0
$973$	9.98384	0.320067
$974$	0	0
$975$	1.55743	0.0498778
$976$	0	0
$977$	−5.89342	−0.188547	−0.0942736	−	0.995546i	$-0.530053\pi$
$977$	−5.89342	−0.188547	−0.0942736	+	0.995546i	$0.530053\pi$
$978$	0	0
$979$	−5.19353	−0.165986
$980$	0	0
$981$	−6.18056	−0.197330
$982$	0	0
$983$	−2.88660	−0.0920683	−0.0460341	−	0.998940i	$-0.514658\pi$
$983$	−2.88660	−0.0920683	−0.0460341	+	0.998940i	$0.514658\pi$
$984$	0	0
$985$	9.65959	0.307780
$986$	0	0
$987$	−2.62354	−0.0835082
$988$	0	0
$989$	26.5927	0.845599
$990$	0	0
$991$	−15.8725	−0.504206	−0.252103	−	0.967700i	$-0.581122\pi$
$991$	−15.8725	−0.504206	−0.252103	+	0.967700i	$0.581122\pi$
$992$	0	0
$993$	0.0876598	0.00278180
$994$	0	0
$995$	9.85738	0.312500
$996$	0	0
$997$	−28.4316	−0.900437	−0.450218	−	0.892918i	$-0.648654\pi$
$997$	−28.4316	−0.900437	−0.450218	+	0.892918i	$0.648654\pi$
$998$	0	0
$999$	−24.9085	−0.788071

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7120.2.a.bc.1.1		4
4.3	odd	2		445.2.a.d.1.2	✓	4
12.11	even	2		4005.2.a.l.1.3		4
20.19	odd	2		2225.2.a.i.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.d.1.2	✓	4	4.3	odd	2
2225.2.a.i.1.3		4	20.19	odd	2
4005.2.a.l.1.3		4	12.11	even	2
7120.2.a.bc.1.1		4	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)

\(f(q)\)	\(=\)	\(q-1.19353 q^{3} -1.00000 q^{5} +0.475281 q^{7} -1.57549 q^{9} +O(q^{10})\)	\(q-1.19353 q^{3} -1.00000 q^{5} +0.475281 q^{7} -1.57549 q^{9} +5.19353 q^{11} -1.30490 q^{13} +1.19353 q^{15} -7.40330 q^{17} -6.83604 q^{19} -0.567260 q^{21} -2.69195 q^{23} +1.00000 q^{25} +5.46097 q^{27} -0.00689789 q^{29} +8.54230 q^{31} -6.19862 q^{33} -0.475281 q^{35} -4.56119 q^{37} +1.55743 q^{39} +3.41020 q^{41} -9.87858 q^{43} +1.57549 q^{45} +4.62493 q^{47} -6.77411 q^{49} +8.83604 q^{51} +2.86526 q^{53} -5.19353 q^{55} +8.15900 q^{57} +1.13767 q^{59} +7.30684 q^{61} -0.748801 q^{63} +1.30490 q^{65} +1.79531 q^{67} +3.21292 q^{69} -2.91492 q^{71} -2.26369 q^{73} -1.19353 q^{75} +2.46838 q^{77} +10.5104 q^{79} -1.79134 q^{81} -6.35650 q^{83} +7.40330 q^{85} +0.00823281 q^{87} -1.00000 q^{89} -0.620194 q^{91} -10.1955 q^{93} +6.83604 q^{95} +18.3113 q^{97} -8.18237 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 4 * q^9	\( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9} + 14 q^{11} - 5 q^{13} - 2 q^{15} - 3 q^{17} + q^{19} - 4 q^{21} + 3 q^{23} + 4 q^{25} - q^{27} - 10 q^{29} + 11 q^{31} + 2 q^{35} + 3 q^{37} - 11 q^{39} - 3 q^{41} - 9 q^{43} + 4 q^{45} + 24 q^{47} + 7 q^{51} + 3 q^{53} - 14 q^{55} + 19 q^{57} + 22 q^{59} - 3 q^{61} - 6 q^{63} + 5 q^{65} + 9 q^{67} + 7 q^{69} - 16 q^{71} + 3 q^{73} + 2 q^{75} - 4 q^{77} + 27 q^{79} - 8 q^{81} - 6 q^{83} + 3 q^{85} - 4 q^{87} - 4 q^{89} + 29 q^{91} - 2 q^{93} - q^{95} + 41 q^{97} - 21 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 - 4 * q^9 + 14 * q^11 - 5 * q^13 - 2 * q^15 - 3 * q^17 + q^19 - 4 * q^21 + 3 * q^23 + 4 * q^25 - q^27 - 10 * q^29 + 11 * q^31 + 2 * q^35 + 3 * q^37 - 11 * q^39 - 3 * q^41 - 9 * q^43 + 4 * q^45 + 24 * q^47 + 7 * q^51 + 3 * q^53 - 14 * q^55 + 19 * q^57 + 22 * q^59 - 3 * q^61 - 6 * q^63 + 5 * q^65 + 9 * q^67 + 7 * q^69 - 16 * q^71 + 3 * q^73 + 2 * q^75 - 4 * q^77 + 27 * q^79 - 8 * q^81 - 6 * q^83 + 3 * q^85 - 4 * q^87 - 4 * q^89 + 29 * q^91 - 2 * q^93 - q^95 + 41 * q^97 - 21 * q^99

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