LMFDB - Embedded newform 7120.2.a.bc.1.4

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			1.4
Root			$-1.35567$ 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)

$f(q)$	$=$	$q+2.19353 q^{3} -1.00000 q^{5} -3.71135 q^{7} +1.81156 q^{9} +O(q^{10})$	$q+2.19353 q^{3} -1.00000 q^{5} -3.71135 q^{7} +1.81156 q^{9} +1.80647 q^{11} -6.78527 q^{13} -2.19353 q^{15} -1.92293 q^{17} +6.21801 q^{19} -8.14094 q^{21} -1.39822 q^{23} +1.00000 q^{25} -2.60687 q^{27} -4.99310 q^{29} -1.92427 q^{31} +3.96255 q^{33} +3.71135 q^{35} +2.70709 q^{37} -14.8837 q^{39} +2.91604 q^{41} -0.211586 q^{43} -1.81156 q^{45} +9.61114 q^{47} +6.77411 q^{49} -4.21801 q^{51} +10.9331 q^{53} -1.80647 q^{55} +13.6394 q^{57} +12.0984 q^{59} +10.1997 q^{61} -6.72333 q^{63} +6.78527 q^{65} -2.88548 q^{67} -3.06702 q^{69} +3.85919 q^{71} +11.5899 q^{73} +2.19353 q^{75} -6.70445 q^{77} +10.8158 q^{79} -11.1529 q^{81} +5.59257 q^{83} +1.92293 q^{85} -10.9525 q^{87} -1.00000 q^{89} +25.1825 q^{91} -4.22094 q^{93} -6.21801 q^{95} +1.07064 q^{97} +3.27254 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.19353	1.26643	0.633217	−	0.773975i	$-0.281734\pi$
$3$	2.19353	1.26643	0.633217	+	0.773975i	$0.281734\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.71135	−1.40276	−0.701379	−	0.712789i	$-0.747432\pi$
$7$	−3.71135	−1.40276	−0.701379	+	0.712789i	$0.747432\pi$
$8$	0	0
$9$	1.81156	0.603854
$10$	0	0
$11$	1.80647	0.544672	0.272336	−	0.962202i	$-0.412204\pi$
$11$	1.80647	0.544672	0.272336	+	0.962202i	$0.412204\pi$
$12$	0	0
$13$	−6.78527	−1.88190	−0.940948	−	0.338552i	$-0.890063\pi$
$13$	−6.78527	−1.88190	−0.940948	+	0.338552i	$0.890063\pi$
$14$	0	0
$15$	−2.19353	−0.566366
$16$	0	0
$17$	−1.92293	−0.466380	−0.233190	−	0.972431i	$-0.574916\pi$
$17$	−1.92293	−0.466380	−0.233190	+	0.972431i	$0.574916\pi$
$18$	0	0
$19$	6.21801	1.42651	0.713255	−	0.700905i	$-0.247220\pi$
$19$	6.21801	1.42651	0.713255	+	0.700905i	$0.247220\pi$
$20$	0	0
$21$	−8.14094	−1.77650
$22$	0	0
$23$	−1.39822	−0.291548	−0.145774	−	0.989318i	$-0.546567\pi$
$23$	−1.39822	−0.291548	−0.145774	+	0.989318i	$0.546567\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.60687	−0.501693
$28$	0	0
$29$	−4.99310	−0.927196	−0.463598	−	0.886046i	$-0.653442\pi$
$29$	−4.99310	−0.927196	−0.463598	+	0.886046i	$0.653442\pi$
$30$	0	0
$31$	−1.92427	−0.345609	−0.172805	−	0.984956i	$-0.555283\pi$
$31$	−1.92427	−0.345609	−0.172805	+	0.984956i	$0.555283\pi$
$32$	0	0
$33$	3.96255	0.689791
$34$	0	0
$35$	3.71135	0.627332
$36$	0	0
$37$	2.70709	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$37$	2.70709	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$38$	0	0
$39$	−14.8837	−2.38329
$40$	0	0
$41$	2.91604	0.455408	0.227704	−	0.973730i	$-0.426878\pi$
$41$	2.91604	0.455408	0.227704	+	0.973730i	$0.426878\pi$
$42$	0	0
$43$	−0.211586	−0.0322666	−0.0161333	−	0.999870i	$-0.505136\pi$
$43$	−0.211586	−0.0322666	−0.0161333	+	0.999870i	$0.505136\pi$
$44$	0	0
$45$	−1.81156	−0.270052
$46$	0	0
$47$	9.61114	1.40193	0.700964	−	0.713197i	$-0.252753\pi$
$47$	9.61114	1.40193	0.700964	+	0.713197i	$0.252753\pi$
$48$	0	0
$49$	6.77411	0.967730
$50$	0	0
$51$	−4.21801	−0.590639
$52$	0	0
$53$	10.9331	1.50178	0.750889	−	0.660428i	$-0.229625\pi$
$53$	10.9331	1.50178	0.750889	+	0.660428i	$0.229625\pi$
$54$	0	0
$55$	−1.80647	−0.243585
$56$	0	0
$57$	13.6394	1.80658
$58$	0	0
$59$	12.0984	1.57508	0.787539	−	0.616265i	$-0.211355\pi$
$59$	12.0984	1.57508	0.787539	+	0.616265i	$0.211355\pi$
$60$	0	0
$61$	10.1997	1.30594	0.652971	−	0.757383i	$-0.273522\pi$
$61$	10.1997	1.30594	0.652971	+	0.757383i	$0.273522\pi$
$62$	0	0
$63$	−6.72333	−0.847061
$64$	0	0
$65$	6.78527	0.841609
$66$	0	0
$67$	−2.88548	−0.352518	−0.176259	−	0.984344i	$-0.556400\pi$
$67$	−2.88548	−0.352518	−0.176259	+	0.984344i	$0.556400\pi$
$68$	0	0
$69$	−3.06702	−0.369226
$70$	0	0
$71$	3.85919	0.458002	0.229001	−	0.973426i	$-0.426454\pi$
$71$	3.85919	0.458002	0.229001	+	0.973426i	$0.426454\pi$
$72$	0	0
$73$	11.5899	1.35650	0.678250	−	0.734832i	$-0.262739\pi$
$73$	11.5899	1.35650	0.678250	+	0.734832i	$0.262739\pi$
$74$	0	0
$75$	2.19353	0.253287
$76$	0	0
$77$	−6.70445	−0.764043
$78$	0	0
$79$	10.8158	1.21688	0.608438	−	0.793602i	$-0.291797\pi$
$79$	10.8158	1.21688	0.608438	+	0.793602i	$0.291797\pi$
$80$	0	0
$81$	−11.1529	−1.23921
$82$	0	0
$83$	5.59257	0.613864	0.306932	−	0.951731i	$-0.400698\pi$
$83$	5.59257	0.613864	0.306932	+	0.951731i	$0.400698\pi$
$84$	0	0
$85$	1.92293	0.208572
$86$	0	0
$87$	−10.9525	−1.17423
$88$	0	0
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	25.1825	2.63984
$92$	0	0
$93$	−4.22094	−0.437691
$94$	0	0
$95$	−6.21801	−0.637954
$96$	0	0
$97$	1.07064	0.108707	0.0543536	−	0.998522i	$-0.482690\pi$
$97$	1.07064	0.108707	0.0543536	+	0.998522i	$0.482690\pi$
$98$	0	0
$99$	3.27254	0.328902
$100$	0	0
$101$	−13.8603	−1.37915	−0.689576	−	0.724213i	$-0.742203\pi$
$101$	−13.8603	−1.37915	−0.689576	+	0.724213i	$0.742203\pi$
$102$	0	0
$103$	−2.85216	−0.281032	−0.140516	−	0.990078i	$-0.544876\pi$
$103$	−2.85216	−0.281032	−0.140516	+	0.990078i	$0.544876\pi$
$104$	0	0
$105$	8.14094	0.794475
$106$	0	0
$107$	1.48908	0.143954	0.0719772	−	0.997406i	$-0.477069\pi$
$107$	1.48908	0.143954	0.0719772	+	0.997406i	$0.477069\pi$
$108$	0	0
$109$	9.40330	0.900673	0.450337	−	0.892859i	$-0.351304\pi$
$109$	9.40330	0.900673	0.450337	+	0.892859i	$0.351304\pi$
$110$	0	0
$111$	5.93807	0.563616
$112$	0	0
$113$	12.6841	1.19322	0.596609	−	0.802532i	$-0.296515\pi$
$113$	12.6841	1.19322	0.596609	+	0.802532i	$0.296515\pi$
$114$	0	0
$115$	1.39822	0.130384
$116$	0	0
$117$	−12.2919	−1.13639
$118$	0	0
$119$	7.13668	0.654218
$120$	0	0
$121$	−7.73666	−0.703332
$122$	0	0
$123$	6.39641	0.576744
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.25413	−0.111286	−0.0556429	−	0.998451i	$-0.517721\pi$
$127$	−1.25413	−0.111286	−0.0556429	+	0.998451i	$0.517721\pi$
$128$	0	0
$129$	−0.464120	−0.0408635
$130$	0	0
$131$	2.27240	0.198541	0.0992704	−	0.995060i	$-0.468349\pi$
$131$	2.27240	0.198541	0.0992704	+	0.995060i	$0.468349\pi$
$132$	0	0
$133$	−23.0772	−2.00105
$134$	0	0
$135$	2.60687	0.224364
$136$	0	0
$137$	−15.5836	−1.33140	−0.665700	−	0.746219i	$-0.731867\pi$
$137$	−15.5836	−1.33140	−0.665700	+	0.746219i	$0.731867\pi$
$138$	0	0
$139$	21.5004	1.82364	0.911819	−	0.410593i	$-0.134678\pi$
$139$	21.5004	1.82364	0.911819	+	0.410593i	$0.134678\pi$
$140$	0	0
$141$	21.0823	1.77545
$142$	0	0
$143$	−12.2574	−1.02502
$144$	0	0
$145$	4.99310	0.414655
$146$	0	0
$147$	14.8592	1.22557
$148$	0	0
$149$	−17.0028	−1.39292	−0.696461	−	0.717595i	$-0.745243\pi$
$149$	−17.0028	−1.39292	−0.696461	+	0.717595i	$0.745243\pi$
$150$	0	0
$151$	17.7866	1.44745	0.723727	−	0.690087i	$-0.242428\pi$
$151$	17.7866	1.44745	0.723727	+	0.690087i	$0.242428\pi$
$152$	0	0
$153$	−3.48351	−0.281625
$154$	0	0
$155$	1.92427	0.154561
$156$	0	0
$157$	−8.13061	−0.648893	−0.324447	−	0.945904i	$-0.605178\pi$
$157$	−8.13061	−0.648893	−0.324447	+	0.945904i	$0.605178\pi$
$158$	0	0
$159$	23.9821	1.90190
$160$	0	0
$161$	5.18926	0.408971
$162$	0	0
$163$	−23.0135	−1.80255	−0.901276	−	0.433244i	$-0.857369\pi$
$163$	−23.0135	−1.80255	−0.901276	+	0.433244i	$0.857369\pi$
$164$	0	0
$165$	−3.96255	−0.308484
$166$	0	0
$167$	−5.88974	−0.455762	−0.227881	−	0.973689i	$-0.573180\pi$
$167$	−5.88974	−0.455762	−0.227881	+	0.973689i	$0.573180\pi$
$168$	0	0
$169$	33.0399	2.54153
$170$	0	0
$171$	11.2643	0.861403
$172$	0	0
$173$	−15.0723	−1.14593	−0.572964	−	0.819581i	$-0.694206\pi$
$173$	−15.0723	−1.14593	−0.572964	+	0.819581i	$0.694206\pi$
$174$	0	0
$175$	−3.71135	−0.280552
$176$	0	0
$177$	26.5382	1.99473
$178$	0	0
$179$	−4.65562	−0.347977	−0.173989	−	0.984748i	$-0.555666\pi$
$179$	−4.65562	−0.347977	−0.173989	+	0.984748i	$0.555666\pi$
$180$	0	0
$181$	9.62070	0.715101	0.357551	−	0.933894i	$-0.383612\pi$
$181$	9.62070	0.715101	0.357551	+	0.933894i	$0.383612\pi$
$182$	0	0
$183$	22.3734	1.65389
$184$	0	0
$185$	−2.70709	−0.199029
$186$	0	0
$187$	−3.47373	−0.254024
$188$	0	0
$189$	9.67501	0.703754
$190$	0	0
$191$	−13.3629	−0.966907	−0.483454	−	0.875370i	$-0.660618\pi$
$191$	−13.3629	−0.966907	−0.483454	+	0.875370i	$0.660618\pi$
$192$	0	0
$193$	0.449680	0.0323687	0.0161843	−	0.999869i	$-0.494848\pi$
$193$	0.449680	0.0323687	0.0161843	+	0.999869i	$0.494848\pi$
$194$	0	0
$195$	14.8837	1.06584
$196$	0	0
$197$	8.56942	0.610546	0.305273	−	0.952265i	$-0.401252\pi$
$197$	8.56942	0.610546	0.305273	+	0.952265i	$0.401252\pi$
$198$	0	0
$199$	8.67704	0.615099	0.307550	−	0.951532i	$-0.400491\pi$
$199$	8.67704	0.615099	0.307550	+	0.951532i	$0.400491\pi$
$200$	0	0
$201$	−6.32938	−0.446440
$202$	0	0
$203$	18.5311	1.30063
$204$	0	0
$205$	−2.91604	−0.203665
$206$	0	0
$207$	−2.53295	−0.176052
$208$	0	0
$209$	11.2327	0.776980
$210$	0	0
$211$	−25.1953	−1.73452	−0.867258	−	0.497858i	$-0.834120\pi$
$211$	−25.1953	−1.73452	−0.867258	+	0.497858i	$0.834120\pi$
$212$	0	0
$213$	8.46524	0.580029
$214$	0	0
$215$	0.211586	0.0144300
$216$	0	0
$217$	7.14163	0.484806
$218$	0	0
$219$	25.4228	1.71792
$220$	0	0
$221$	13.0476	0.877679
$222$	0	0
$223$	−16.2438	−1.08777	−0.543884	−	0.839161i	$-0.683047\pi$
$223$	−16.2438	−1.08777	−0.543884	+	0.839161i	$0.683047\pi$
$224$	0	0
$225$	1.81156	0.120771
$226$	0	0
$227$	13.1941	0.875726	0.437863	−	0.899042i	$-0.355736\pi$
$227$	13.1941	0.875726	0.437863	+	0.899042i	$0.355736\pi$
$228$	0	0
$229$	2.36998	0.156613	0.0783063	−	0.996929i	$-0.475049\pi$
$229$	2.36998	0.156613	0.0783063	+	0.996929i	$0.475049\pi$
$230$	0	0
$231$	−14.7064	−0.967610
$232$	0	0
$233$	−9.67877	−0.634077	−0.317039	−	0.948413i	$-0.602688\pi$
$233$	−9.67877	−0.634077	−0.317039	+	0.948413i	$0.602688\pi$
$234$	0	0
$235$	−9.61114	−0.626961
$236$	0	0
$237$	23.7248	1.54109
$238$	0	0
$239$	23.8871	1.54513	0.772564	−	0.634936i	$-0.218974\pi$
$239$	23.8871	1.54513	0.772564	+	0.634936i	$0.218974\pi$
$240$	0	0
$241$	−15.7221	−1.01275	−0.506376	−	0.862313i	$-0.669015\pi$
$241$	−15.7221	−1.01275	−0.506376	+	0.862313i	$0.669015\pi$
$242$	0	0
$243$	−16.6436	−1.06769
$244$	0	0
$245$	−6.77411	−0.432782
$246$	0	0
$247$	−42.1909	−2.68454
$248$	0	0
$249$	12.2674	0.777418
$250$	0	0
$251$	−14.2933	−0.902183	−0.451091	−	0.892478i	$-0.648965\pi$
$251$	−14.2933	−0.902183	−0.451091	+	0.892478i	$0.648965\pi$
$252$	0	0
$253$	−2.52584	−0.158798
$254$	0	0
$255$	4.21801	0.264142
$256$	0	0
$257$	−8.51369	−0.531070	−0.265535	−	0.964101i	$-0.585549\pi$
$257$	−8.51369	−0.531070	−0.265535	+	0.964101i	$0.585549\pi$
$258$	0	0
$259$	−10.0469	−0.624286
$260$	0	0
$261$	−9.04531	−0.559891
$262$	0	0
$263$	19.3754	1.19474	0.597370	−	0.801966i	$-0.296212\pi$
$263$	19.3754	1.19474	0.597370	+	0.801966i	$0.296212\pi$
$264$	0	0
$265$	−10.9331	−0.671616
$266$	0	0
$267$	−2.19353	−0.134242
$268$	0	0
$269$	−7.39853	−0.451096	−0.225548	−	0.974232i	$-0.572417\pi$
$269$	−7.39853	−0.451096	−0.225548	+	0.974232i	$0.572417\pi$
$270$	0	0
$271$	27.2477	1.65518	0.827590	−	0.561333i	$-0.189711\pi$
$271$	27.2477	1.65518	0.827590	+	0.561333i	$0.189711\pi$
$272$	0	0
$273$	55.2385	3.34319
$274$	0	0
$275$	1.80647	0.108934
$276$	0	0
$277$	28.4582	1.70989	0.854943	−	0.518722i	$-0.173592\pi$
$277$	28.4582	1.70989	0.854943	+	0.518722i	$0.173592\pi$
$278$	0	0
$279$	−3.48593	−0.208697
$280$	0	0
$281$	22.1815	1.32323	0.661617	−	0.749842i	$-0.269870\pi$
$281$	22.1815	1.32323	0.661617	+	0.749842i	$0.269870\pi$
$282$	0	0
$283$	−0.247445	−0.0147091	−0.00735455	−	0.999973i	$-0.502341\pi$
$283$	−0.247445	−0.0147091	−0.00735455	+	0.999973i	$0.502341\pi$
$284$	0	0
$285$	−13.6394	−0.807927
$286$	0	0
$287$	−10.8224	−0.638828
$288$	0	0
$289$	−13.3023	−0.782490
$290$	0	0
$291$	2.34848	0.137671
$292$	0	0
$293$	−11.3844	−0.665085	−0.332542	−	0.943088i	$-0.607906\pi$
$293$	−11.3844	−0.665085	−0.332542	+	0.943088i	$0.607906\pi$
$294$	0	0
$295$	−12.0984	−0.704396
$296$	0	0
$297$	−4.70925	−0.273258
$298$	0	0
$299$	9.48727	0.548663
$300$	0	0
$301$	0.785269	0.0452622
$302$	0	0
$303$	−30.4030	−1.74660
$304$	0	0
$305$	−10.1997	−0.584035
$306$	0	0
$307$	6.37110	0.363618	0.181809	−	0.983334i	$-0.441805\pi$
$307$	6.37110	0.363618	0.181809	+	0.983334i	$0.441805\pi$
$308$	0	0
$309$	−6.25629	−0.355908
$310$	0	0
$311$	30.5996	1.73514	0.867572	−	0.497312i	$-0.165679\pi$
$311$	30.5996	1.73514	0.867572	+	0.497312i	$0.165679\pi$
$312$	0	0
$313$	−19.2316	−1.08704	−0.543518	−	0.839398i	$-0.682908\pi$
$313$	−19.2316	−1.08704	−0.543518	+	0.839398i	$0.682908\pi$
$314$	0	0
$315$	6.72333	0.378817
$316$	0	0
$317$	−9.70429	−0.545047	−0.272524	−	0.962149i	$-0.587858\pi$
$317$	−9.70429	−0.545047	−0.272524	+	0.962149i	$0.587858\pi$
$318$	0	0
$319$	−9.01990	−0.505018
$320$	0	0
$321$	3.26633	0.182309
$322$	0	0
$323$	−11.9568	−0.665296
$324$	0	0
$325$	−6.78527	−0.376379
$326$	0	0
$327$	20.6264	1.14064
$328$	0	0
$329$	−35.6703	−1.96657
$330$	0	0
$331$	28.9275	1.59000	0.795001	−	0.606608i	$-0.207470\pi$
$331$	28.9275	1.59000	0.795001	+	0.606608i	$0.207470\pi$
$332$	0	0
$333$	4.90405	0.268740
$334$	0	0
$335$	2.88548	0.157651
$336$	0	0
$337$	2.30865	0.125760	0.0628802	−	0.998021i	$-0.479971\pi$
$337$	2.30865	0.125760	0.0628802	+	0.998021i	$0.479971\pi$
$338$	0	0
$339$	27.8229	1.51113
$340$	0	0
$341$	−3.47614	−0.188244
$342$	0	0
$343$	0.838363	0.0452673
$344$	0	0
$345$	3.06702	0.165123
$346$	0	0
$347$	34.4649	1.85017	0.925085	−	0.379759i	$-0.123993\pi$
$347$	34.4649	1.85017	0.925085	+	0.379759i	$0.123993\pi$
$348$	0	0
$349$	19.3154	1.03393	0.516965	−	0.856007i	$-0.327062\pi$
$349$	19.3154	1.03393	0.516965	+	0.856007i	$0.327062\pi$
$350$	0	0
$351$	17.6883	0.944133
$352$	0	0
$353$	1.41183	0.0751441	0.0375721	−	0.999294i	$-0.488038\pi$
$353$	1.41183	0.0751441	0.0375721	+	0.999294i	$0.488038\pi$
$354$	0	0
$355$	−3.85919	−0.204825
$356$	0	0
$357$	15.6545	0.828524
$358$	0	0
$359$	17.2200	0.908834	0.454417	−	0.890789i	$-0.349848\pi$
$359$	17.2200	0.908834	0.454417	+	0.890789i	$0.349848\pi$
$360$	0	0
$361$	19.6636	1.03493
$362$	0	0
$363$	−16.9706	−0.890724
$364$	0	0
$365$	−11.5899	−0.606645
$366$	0	0
$367$	20.4193	1.06588	0.532939	−	0.846154i	$-0.321087\pi$
$367$	20.4193	1.06588	0.532939	+	0.846154i	$0.321087\pi$
$368$	0	0
$369$	5.28258	0.275000
$370$	0	0
$371$	−40.5766	−2.10663
$372$	0	0
$373$	−0.00410346	−0.000212469 0	−0.000106235	−	1.00000i	$-0.500034\pi$
$373$	−0.00410346	−0.000212469 0	−0.000106235		1.00000i	$0.500034\pi$
$374$	0	0
$375$	−2.19353	−0.113273
$376$	0	0
$377$	33.8795	1.74489
$378$	0	0
$379$	−33.3456	−1.71285	−0.856424	−	0.516273i	$-0.827319\pi$
$379$	−33.3456	−1.71285	−0.856424	+	0.516273i	$0.827319\pi$
$380$	0	0
$381$	−2.75096	−0.140936
$382$	0	0
$383$	−34.6473	−1.77039	−0.885196	−	0.465218i	$-0.845976\pi$
$383$	−34.6473	−1.77039	−0.885196	+	0.465218i	$0.845976\pi$
$384$	0	0
$385$	6.70445	0.341690
$386$	0	0
$387$	−0.383301	−0.0194843
$388$	0	0
$389$	30.7954	1.56139	0.780696	−	0.624912i	$-0.214865\pi$
$389$	30.7954	1.56139	0.780696	+	0.624912i	$0.214865\pi$
$390$	0	0
$391$	2.68868	0.135972
$392$	0	0
$393$	4.98458	0.251439
$394$	0	0
$395$	−10.8158	−0.544203
$396$	0	0
$397$	30.1664	1.51401	0.757004	−	0.653411i	$-0.226662\pi$
$397$	30.1664	1.51401	0.757004	+	0.653411i	$0.226662\pi$
$398$	0	0
$399$	−50.6205	−2.53419
$400$	0	0
$401$	−7.51947	−0.375504	−0.187752	−	0.982216i	$-0.560120\pi$
$401$	−7.51947	−0.375504	−0.187752	+	0.982216i	$0.560120\pi$
$402$	0	0
$403$	13.0567	0.650400
$404$	0	0
$405$	11.1529	0.554194
$406$	0	0
$407$	4.89028	0.242402
$408$	0	0
$409$	32.8936	1.62648	0.813241	−	0.581927i	$-0.197701\pi$
$409$	32.8936	1.62648	0.813241	+	0.581927i	$0.197701\pi$
$410$	0	0
$411$	−34.1831	−1.68613
$412$	0	0
$413$	−44.9014	−2.20945
$414$	0	0
$415$	−5.59257	−0.274528
$416$	0	0
$417$	47.1616	2.30952
$418$	0	0
$419$	−15.0851	−0.736957	−0.368479	−	0.929636i	$-0.620121\pi$
$419$	−15.0851	−0.736957	−0.368479	+	0.929636i	$0.620121\pi$
$420$	0	0
$421$	−23.0575	−1.12375	−0.561876	−	0.827221i	$-0.689920\pi$
$421$	−23.0575	−1.12375	−0.561876	+	0.827221i	$0.689920\pi$
$422$	0	0
$423$	17.4112	0.846560
$424$	0	0
$425$	−1.92293	−0.0932760
$426$	0	0
$427$	−37.8548	−1.83192
$428$	0	0
$429$	−26.8870	−1.29811
$430$	0	0
$431$	12.8238	0.617703	0.308851	−	0.951110i	$-0.400055\pi$
$431$	12.8238	0.617703	0.308851	+	0.951110i	$0.400055\pi$
$432$	0	0
$433$	−28.7231	−1.38034	−0.690171	−	0.723646i	$-0.742465\pi$
$433$	−28.7231	−1.38034	−0.690171	+	0.723646i	$0.742465\pi$
$434$	0	0
$435$	10.9525	0.525132
$436$	0	0
$437$	−8.69411	−0.415896
$438$	0	0
$439$	23.6489	1.12870	0.564349	−	0.825536i	$-0.309127\pi$
$439$	23.6489	1.12870	0.564349	+	0.825536i	$0.309127\pi$
$440$	0	0
$441$	12.2717	0.584367
$442$	0	0
$443$	35.5043	1.68686	0.843430	−	0.537239i	$-0.180533\pi$
$443$	35.5043	1.68686	0.843430	+	0.537239i	$0.180533\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	0	0
$447$	−37.2960	−1.76404
$448$	0	0
$449$	32.7779	1.54688	0.773442	−	0.633867i	$-0.218533\pi$
$449$	32.7779	1.54688	0.773442	+	0.633867i	$0.218533\pi$
$450$	0	0
$451$	5.26774	0.248048
$452$	0	0
$453$	39.0154	1.83310
$454$	0	0
$455$	−25.1825	−1.18057
$456$	0	0
$457$	28.7506	1.34490	0.672448	−	0.740145i	$-0.265243\pi$
$457$	28.7506	1.34490	0.672448	+	0.740145i	$0.265243\pi$
$458$	0	0
$459$	5.01285	0.233980
$460$	0	0
$461$	−11.6097	−0.540717	−0.270358	−	0.962760i	$-0.587142\pi$
$461$	−11.6097	−0.540717	−0.270358	+	0.962760i	$0.587142\pi$
$462$	0	0
$463$	18.7448	0.871145	0.435573	−	0.900154i	$-0.356546\pi$
$463$	18.7448	0.871145	0.435573	+	0.900154i	$0.356546\pi$
$464$	0	0
$465$	4.22094	0.195741
$466$	0	0
$467$	0.634016	0.0293387	0.0146694	−	0.999892i	$-0.495330\pi$
$467$	0.634016	0.0293387	0.0146694	+	0.999892i	$0.495330\pi$
$468$	0	0
$469$	10.7090	0.494497
$470$	0	0
$471$	−17.8347	−0.821780
$472$	0	0
$473$	−0.382224	−0.0175747
$474$	0	0
$475$	6.21801	0.285302
$476$	0	0
$477$	19.8060	0.906854
$478$	0	0
$479$	−29.2698	−1.33737	−0.668686	−	0.743545i	$-0.733143\pi$
$479$	−29.2698	−1.33737	−0.668686	+	0.743545i	$0.733143\pi$
$480$	0	0
$481$	−18.3683	−0.837523
$482$	0	0
$483$	11.3828	0.517935
$484$	0	0
$485$	−1.07064	−0.0486154
$486$	0	0
$487$	−8.96526	−0.406255	−0.203127	−	0.979152i	$-0.565111\pi$
$487$	−8.96526	−0.406255	−0.203127	+	0.979152i	$0.565111\pi$
$488$	0	0
$489$	−50.4806	−2.28281
$490$	0	0
$491$	−21.4601	−0.968479	−0.484240	−	0.874935i	$-0.660904\pi$
$491$	−21.4601	−0.968479	−0.484240	+	0.874935i	$0.660904\pi$
$492$	0	0
$493$	9.60141	0.432426
$494$	0	0
$495$	−3.27254	−0.147090
$496$	0	0
$497$	−14.3228	−0.642465
$498$	0	0
$499$	−21.8821	−0.979576	−0.489788	−	0.871841i	$-0.662926\pi$
$499$	−21.8821	−0.979576	−0.489788	+	0.871841i	$0.662926\pi$
$500$	0	0
$501$	−12.9193	−0.577192
$502$	0	0
$503$	8.91940	0.397696	0.198848	−	0.980030i	$-0.436280\pi$
$503$	8.91940	0.397696	0.198848	+	0.980030i	$0.436280\pi$
$504$	0	0
$505$	13.8603	0.616776
$506$	0	0
$507$	72.4739	3.21868
$508$	0	0
$509$	−5.76572	−0.255561	−0.127780	−	0.991802i	$-0.540785\pi$
$509$	−5.76572	−0.255561	−0.127780	+	0.991802i	$0.540785\pi$
$510$	0	0
$511$	−43.0143	−1.90284
$512$	0	0
$513$	−16.2096	−0.715670
$514$	0	0
$515$	2.85216	0.125681
$516$	0	0
$517$	17.3623	0.763591
$518$	0	0
$519$	−33.0616	−1.45124
$520$	0	0
$521$	14.8063	0.648676	0.324338	−	0.945941i	$-0.394858\pi$
$521$	14.8063	0.648676	0.324338	+	0.945941i	$0.394858\pi$
$522$	0	0
$523$	−34.3182	−1.50063	−0.750315	−	0.661081i	$-0.770098\pi$
$523$	−34.3182	−1.50063	−0.750315	+	0.661081i	$0.770098\pi$
$524$	0	0
$525$	−8.14094	−0.355300
$526$	0	0
$527$	3.70024	0.161185
$528$	0	0
$529$	−21.0450	−0.915000
$530$	0	0
$531$	21.9170	0.951117
$532$	0	0
$533$	−19.7861	−0.857031
$534$	0	0
$535$	−1.48908	−0.0643784
$536$	0	0
$537$	−10.2122	−0.440690
$538$	0	0
$539$	12.2372	0.527095
$540$	0	0
$541$	−19.0816	−0.820384	−0.410192	−	0.911999i	$-0.634538\pi$
$541$	−19.0816	−0.820384	−0.410192	+	0.911999i	$0.634538\pi$
$542$	0	0
$543$	21.1033	0.905628
$544$	0	0
$545$	−9.40330	−0.402793
$546$	0	0
$547$	16.2497	0.694789	0.347394	−	0.937719i	$-0.387067\pi$
$547$	16.2497	0.694789	0.347394	+	0.937719i	$0.387067\pi$
$548$	0	0
$549$	18.4774	0.788598
$550$	0	0
$551$	−31.0472	−1.32265
$552$	0	0
$553$	−40.1413	−1.70698
$554$	0	0
$555$	−5.93807	−0.252057
$556$	0	0
$557$	34.7023	1.47038	0.735191	−	0.677860i	$-0.237092\pi$
$557$	34.7023	1.47038	0.735191	+	0.677860i	$0.237092\pi$
$558$	0	0
$559$	1.43567	0.0607223
$560$	0	0
$561$	−7.61972	−0.321705
$562$	0	0
$563$	−11.6228	−0.489842	−0.244921	−	0.969543i	$-0.578762\pi$
$563$	−11.6228	−0.489842	−0.244921	+	0.969543i	$0.578762\pi$
$564$	0	0
$565$	−12.6841	−0.533623
$566$	0	0
$567$	41.3924	1.73832
$568$	0	0
$569$	42.6106	1.78633	0.893164	−	0.449732i	$-0.148480\pi$
$569$	42.6106	1.78633	0.893164	+	0.449732i	$0.148480\pi$
$570$	0	0
$571$	3.41985	0.143116	0.0715582	−	0.997436i	$-0.477203\pi$
$571$	3.41985	0.143116	0.0715582	+	0.997436i	$0.477203\pi$
$572$	0	0
$573$	−29.3119	−1.22452
$574$	0	0
$575$	−1.39822	−0.0583096
$576$	0	0
$577$	6.58744	0.274239	0.137119	−	0.990555i	$-0.456216\pi$
$577$	6.58744	0.274239	0.137119	+	0.990555i	$0.456216\pi$
$578$	0	0
$579$	0.986386	0.0409928
$580$	0	0
$581$	−20.7560	−0.861103
$582$	0	0
$583$	19.7504	0.817977
$584$	0	0
$585$	12.2919	0.508209
$586$	0	0
$587$	3.93893	0.162577	0.0812884	−	0.996691i	$-0.474097\pi$
$587$	3.93893	0.162577	0.0812884	+	0.996691i	$0.474097\pi$
$588$	0	0
$589$	−11.9651	−0.493014
$590$	0	0
$591$	18.7973	0.773216
$592$	0	0
$593$	−20.9067	−0.858537	−0.429269	−	0.903177i	$-0.641229\pi$
$593$	−20.9067	−0.858537	−0.429269	+	0.903177i	$0.641229\pi$
$594$	0	0
$595$	−7.13668	−0.292575
$596$	0	0
$597$	19.0333	0.778982
$598$	0	0
$599$	−8.80091	−0.359595	−0.179798	−	0.983704i	$-0.557544\pi$
$599$	−8.80091	−0.359595	−0.179798	+	0.983704i	$0.557544\pi$
$600$	0	0
$601$	3.04731	0.124303	0.0621513	−	0.998067i	$-0.480204\pi$
$601$	3.04731	0.124303	0.0621513	+	0.998067i	$0.480204\pi$
$602$	0	0
$603$	−5.22723	−0.212869
$604$	0	0
$605$	7.73666	0.314540
$606$	0	0
$607$	37.2207	1.51074	0.755371	−	0.655298i	$-0.227457\pi$
$607$	37.2207	1.51074	0.755371	+	0.655298i	$0.227457\pi$
$608$	0	0
$609$	40.6486	1.64716
$610$	0	0
$611$	−65.2141	−2.63828
$612$	0	0
$613$	7.03831	0.284275	0.142137	−	0.989847i	$-0.454603\pi$
$613$	7.03831	0.284275	0.142137	+	0.989847i	$0.454603\pi$
$614$	0	0
$615$	−6.39641	−0.257928
$616$	0	0
$617$	−41.9256	−1.68786	−0.843931	−	0.536452i	$-0.819764\pi$
$617$	−41.9256	−1.68786	−0.843931	+	0.536452i	$0.819764\pi$
$618$	0	0
$619$	7.92233	0.318425	0.159213	−	0.987244i	$-0.449104\pi$
$619$	7.92233	0.318425	0.159213	+	0.987244i	$0.449104\pi$
$620$	0	0
$621$	3.64497	0.146268
$622$	0	0
$623$	3.71135	0.148692
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	24.6392	0.983993
$628$	0	0
$629$	−5.20555	−0.207559
$630$	0	0
$631$	−27.9812	−1.11391	−0.556956	−	0.830542i	$-0.688031\pi$
$631$	−27.9812	−1.11391	−0.556956	+	0.830542i	$0.688031\pi$
$632$	0	0
$633$	−55.2666	−2.19665
$634$	0	0
$635$	1.25413	0.0497685
$636$	0	0
$637$	−45.9641	−1.82117
$638$	0	0
$639$	6.99116	0.276566
$640$	0	0
$641$	−10.8328	−0.427868	−0.213934	−	0.976848i	$-0.568628\pi$
$641$	−10.8328	−0.427868	−0.213934	+	0.976848i	$0.568628\pi$
$642$	0	0
$643$	−33.1604	−1.30772	−0.653859	−	0.756616i	$-0.726851\pi$
$643$	−33.1604	−1.30772	−0.653859	+	0.756616i	$0.726851\pi$
$644$	0	0
$645$	0.464120	0.0182747
$646$	0	0
$647$	−7.80677	−0.306916	−0.153458	−	0.988155i	$-0.549041\pi$
$647$	−7.80677	−0.306916	−0.153458	+	0.988155i	$0.549041\pi$
$648$	0	0
$649$	21.8554	0.857901
$650$	0	0
$651$	15.6654	0.613974
$652$	0	0
$653$	−9.79178	−0.383182	−0.191591	−	0.981475i	$-0.561365\pi$
$653$	−9.79178	−0.383182	−0.191591	+	0.981475i	$0.561365\pi$
$654$	0	0
$655$	−2.27240	−0.0887901
$656$	0	0
$657$	20.9959	0.819127
$658$	0	0
$659$	33.0449	1.28725	0.643623	−	0.765342i	$-0.277430\pi$
$659$	33.0449	1.28725	0.643623	+	0.765342i	$0.277430\pi$
$660$	0	0
$661$	21.9348	0.853165	0.426582	−	0.904449i	$-0.359717\pi$
$661$	21.9348	0.853165	0.426582	+	0.904449i	$0.359717\pi$
$662$	0	0
$663$	28.6203	1.11152
$664$	0	0
$665$	23.0772	0.894895
$666$	0	0
$667$	6.98143	0.270322
$668$	0	0
$669$	−35.6313	−1.37758
$670$	0	0
$671$	18.4255	0.711310
$672$	0	0
$673$	12.6202	0.486472	0.243236	−	0.969967i	$-0.421791\pi$
$673$	12.6202	0.486472	0.243236	+	0.969967i	$0.421791\pi$
$674$	0	0
$675$	−2.60687	−0.100339
$676$	0	0
$677$	−43.1296	−1.65761	−0.828803	−	0.559541i	$-0.810977\pi$
$677$	−43.1296	−1.65761	−0.828803	+	0.559541i	$0.810977\pi$
$678$	0	0
$679$	−3.97353	−0.152490
$680$	0	0
$681$	28.9417	1.10905
$682$	0	0
$683$	33.1031	1.26666	0.633328	−	0.773884i	$-0.281689\pi$
$683$	33.1031	1.26666	0.633328	+	0.773884i	$0.281689\pi$
$684$	0	0
$685$	15.5836	0.595420
$686$	0	0
$687$	5.19862	0.198340
$688$	0	0
$689$	−74.1841	−2.82619
$690$	0	0
$691$	−11.4778	−0.436637	−0.218319	−	0.975878i	$-0.570057\pi$
$691$	−11.4778	−0.436637	−0.218319	+	0.975878i	$0.570057\pi$
$692$	0	0
$693$	−12.1455	−0.461370
$694$	0	0
$695$	−21.5004	−0.815556
$696$	0	0
$697$	−5.60735	−0.212393
$698$	0	0
$699$	−21.2306	−0.803017
$700$	0	0
$701$	27.7156	1.04680	0.523401	−	0.852087i	$-0.324663\pi$
$701$	27.7156	1.04680	0.523401	+	0.852087i	$0.324663\pi$
$702$	0	0
$703$	16.8327	0.634857
$704$	0	0
$705$	−21.0823	−0.794005
$706$	0	0
$707$	51.4404	1.93462
$708$	0	0
$709$	21.6735	0.813965	0.406983	−	0.913436i	$-0.366581\pi$
$709$	21.6735	0.813965	0.406983	+	0.913436i	$0.366581\pi$
$710$	0	0
$711$	19.5935	0.734815
$712$	0	0
$713$	2.69054	0.100762
$714$	0	0
$715$	12.2574	0.458401
$716$	0	0
$717$	52.3970	1.95680
$718$	0	0
$719$	−13.5991	−0.507160	−0.253580	−	0.967314i	$-0.581608\pi$
$719$	−13.5991	−0.507160	−0.253580	+	0.967314i	$0.581608\pi$
$720$	0	0
$721$	10.5854	0.394219
$722$	0	0
$723$	−34.4869	−1.28258
$724$	0	0
$725$	−4.99310	−0.185439
$726$	0	0
$727$	−22.5512	−0.836379	−0.418189	−	0.908360i	$-0.637335\pi$
$727$	−22.5512	−0.836379	−0.418189	+	0.908360i	$0.637335\pi$
$728$	0	0
$729$	−3.04947	−0.112943
$730$	0	0
$731$	0.406866	0.0150485
$732$	0	0
$733$	11.3716	0.420019	0.210009	−	0.977699i	$-0.432651\pi$
$733$	11.3716	0.420019	0.210009	+	0.977699i	$0.432651\pi$
$734$	0	0
$735$	−14.8592	−0.548089
$736$	0	0
$737$	−5.21254	−0.192007
$738$	0	0
$739$	51.8437	1.90710	0.953551	−	0.301232i	$-0.0973977\pi$
$739$	51.8437	1.90710	0.953551	+	0.301232i	$0.0973977\pi$
$740$	0	0
$741$	−92.5468	−3.39979
$742$	0	0
$743$	−20.9239	−0.767623	−0.383812	−	0.923411i	$-0.625389\pi$
$743$	−20.9239	−0.767623	−0.383812	+	0.923411i	$0.625389\pi$
$744$	0	0
$745$	17.0028	0.622933
$746$	0	0
$747$	10.1313	0.370684
$748$	0	0
$749$	−5.52648	−0.201933
$750$	0	0
$751$	−27.2392	−0.993972	−0.496986	−	0.867759i	$-0.665560\pi$
$751$	−27.2392	−0.993972	−0.496986	+	0.867759i	$0.665560\pi$
$752$	0	0
$753$	−31.3527	−1.14255
$754$	0	0
$755$	−17.7866	−0.647321
$756$	0	0
$757$	−18.6306	−0.677142	−0.338571	−	0.940941i	$-0.609943\pi$
$757$	−18.6306	−0.677142	−0.338571	+	0.940941i	$0.609943\pi$
$758$	0	0
$759$	−5.54049	−0.201107
$760$	0	0
$761$	−4.06378	−0.147312	−0.0736559	−	0.997284i	$-0.523467\pi$
$761$	−4.06378	−0.147312	−0.0736559	+	0.997284i	$0.523467\pi$
$762$	0	0
$763$	−34.8989	−1.26343
$764$	0	0
$765$	3.48351	0.125947
$766$	0	0
$767$	−82.0909	−2.96413
$768$	0	0
$769$	−32.7396	−1.18062	−0.590310	−	0.807177i	$-0.700994\pi$
$769$	−32.7396	−1.18062	−0.590310	+	0.807177i	$0.700994\pi$
$770$	0	0
$771$	−18.6750	−0.672564
$772$	0	0
$773$	19.0463	0.685049	0.342524	−	0.939509i	$-0.388718\pi$
$773$	19.0463	0.685049	0.342524	+	0.939509i	$0.388718\pi$
$774$	0	0
$775$	−1.92427	−0.0691218
$776$	0	0
$777$	−22.0382	−0.790617
$778$	0	0
$779$	18.1319	0.649644
$780$	0	0
$781$	6.97152	0.249461
$782$	0	0
$783$	13.0164	0.465168
$784$	0	0
$785$	8.13061	0.290194
$786$	0	0
$787$	−36.9703	−1.31785	−0.658924	−	0.752210i	$-0.728988\pi$
$787$	−36.9703	−1.31785	−0.658924	+	0.752210i	$0.728988\pi$
$788$	0	0
$789$	42.5005	1.51306
$790$	0	0
$791$	−47.0750	−1.67379
$792$	0	0
$793$	−69.2079	−2.45765
$794$	0	0
$795$	−23.9821	−0.850557
$796$	0	0
$797$	−10.1404	−0.359191	−0.179596	−	0.983741i	$-0.557479\pi$
$797$	−10.1404	−0.359191	−0.179596	+	0.983741i	$0.557479\pi$
$798$	0	0
$799$	−18.4816	−0.653832
$800$	0	0
$801$	−1.81156	−0.0640084
$802$	0	0
$803$	20.9369	0.738847
$804$	0	0
$805$	−5.18926	−0.182898
$806$	0	0
$807$	−16.2289	−0.571284
$808$	0	0
$809$	0.167200	0.00587845	0.00293922	−	0.999996i	$-0.499064\pi$
$809$	0.167200	0.00587845	0.00293922	+	0.999996i	$0.499064\pi$
$810$	0	0
$811$	41.1102	1.44357	0.721787	−	0.692115i	$-0.243321\pi$
$811$	41.1102	1.44357	0.721787	+	0.692115i	$0.243321\pi$
$812$	0	0
$813$	59.7686	2.09618
$814$	0	0
$815$	23.0135	0.806126
$816$	0	0
$817$	−1.31564	−0.0460285
$818$	0	0
$819$	45.6196	1.59408
$820$	0	0
$821$	−43.1698	−1.50664	−0.753318	−	0.657656i	$-0.771548\pi$
$821$	−43.1698	−1.50664	−0.753318	+	0.657656i	$0.771548\pi$
$822$	0	0
$823$	8.46445	0.295052	0.147526	−	0.989058i	$-0.452869\pi$
$823$	8.46445	0.295052	0.147526	+	0.989058i	$0.452869\pi$
$824$	0	0
$825$	3.96255	0.137958
$826$	0	0
$827$	−6.41763	−0.223163	−0.111581	−	0.993755i	$-0.535592\pi$
$827$	−6.41763	−0.223163	−0.111581	+	0.993755i	$0.535592\pi$
$828$	0	0
$829$	−16.7935	−0.583261	−0.291631	−	0.956531i	$-0.594198\pi$
$829$	−16.7935	−0.583261	−0.291631	+	0.956531i	$0.594198\pi$
$830$	0	0
$831$	62.4238	2.16546
$832$	0	0
$833$	−13.0262	−0.451330
$834$	0	0
$835$	5.88974	0.203823
$836$	0	0
$837$	5.01633	0.173390
$838$	0	0
$839$	7.92834	0.273717	0.136858	−	0.990591i	$-0.456299\pi$
$839$	7.92834	0.273717	0.136858	+	0.990591i	$0.456299\pi$
$840$	0	0
$841$	−4.06893	−0.140308
$842$	0	0
$843$	48.6556	1.67579
$844$	0	0
$845$	−33.0399	−1.13661
$846$	0	0
$847$	28.7134	0.986605
$848$	0	0
$849$	−0.542778	−0.0186281
$850$	0	0
$851$	−3.78509	−0.129751
$852$	0	0
$853$	−27.1727	−0.930375	−0.465187	−	0.885212i	$-0.654013\pi$
$853$	−27.1727	−0.930375	−0.465187	+	0.885212i	$0.654013\pi$
$854$	0	0
$855$	−11.2643	−0.385231
$856$	0	0
$857$	−43.2642	−1.47788	−0.738938	−	0.673773i	$-0.764672\pi$
$857$	−43.2642	−1.47788	−0.738938	+	0.673773i	$0.764672\pi$
$858$	0	0
$859$	−18.4487	−0.629463	−0.314731	−	0.949181i	$-0.601914\pi$
$859$	−18.4487	−0.629463	−0.314731	+	0.949181i	$0.601914\pi$
$860$	0	0
$861$	−23.7393	−0.809033
$862$	0	0
$863$	−1.54533	−0.0526037	−0.0263018	−	0.999654i	$-0.508373\pi$
$863$	−1.54533	−0.0526037	−0.0263018	+	0.999654i	$0.508373\pi$
$864$	0	0
$865$	15.0723	0.512475
$866$	0	0
$867$	−29.1790	−0.990971
$868$	0	0
$869$	19.5385	0.662798
$870$	0	0
$871$	19.5788	0.663401
$872$	0	0
$873$	1.93953	0.0656433
$874$	0	0
$875$	3.71135	0.125466
$876$	0	0
$877$	47.7230	1.61149	0.805745	−	0.592262i	$-0.201765\pi$
$877$	47.7230	1.61149	0.805745	+	0.592262i	$0.201765\pi$
$878$	0	0
$879$	−24.9720	−0.842286
$880$	0	0
$881$	28.2004	0.950095	0.475047	−	0.879960i	$-0.342431\pi$
$881$	28.2004	0.950095	0.475047	+	0.879960i	$0.342431\pi$
$882$	0	0
$883$	11.2269	0.377816	0.188908	−	0.981995i	$-0.439505\pi$
$883$	11.2269	0.377816	0.188908	+	0.981995i	$0.439505\pi$
$884$	0	0
$885$	−26.5382	−0.892071
$886$	0	0
$887$	−12.8249	−0.430619	−0.215309	−	0.976546i	$-0.569076\pi$
$887$	−12.8249	−0.430619	−0.215309	+	0.976546i	$0.569076\pi$
$888$	0	0
$889$	4.65450	0.156107
$890$	0	0
$891$	−20.1475	−0.674965
$892$	0	0
$893$	59.7621	1.99986
$894$	0	0
$895$	4.65562	0.155620
$896$	0	0
$897$	20.8106	0.694845
$898$	0	0
$899$	9.60807	0.320447
$900$	0	0
$901$	−21.0237	−0.700400
$902$	0	0
$903$	1.72251	0.0573215
$904$	0	0
$905$	−9.62070	−0.319803
$906$	0	0
$907$	45.9112	1.52446	0.762229	−	0.647308i	$-0.224105\pi$
$907$	45.9112	1.52446	0.762229	+	0.647308i	$0.224105\pi$
$908$	0	0
$909$	−25.1088	−0.832806
$910$	0	0
$911$	−55.7475	−1.84700	−0.923499	−	0.383600i	$-0.874684\pi$
$911$	−55.7475	−1.84700	−0.923499	+	0.383600i	$0.874684\pi$
$912$	0	0
$913$	10.1028	0.334355
$914$	0	0
$915$	−22.3734	−0.739642
$916$	0	0
$917$	−8.43368	−0.278505
$918$	0	0
$919$	49.0675	1.61859	0.809293	−	0.587405i	$-0.199850\pi$
$919$	49.0675	1.61859	0.809293	+	0.587405i	$0.199850\pi$
$920$	0	0
$921$	13.9752	0.460498
$922$	0	0
$923$	−26.1856	−0.861911
$924$	0	0
$925$	2.70709	0.0890084
$926$	0	0
$927$	−5.16686	−0.169702
$928$	0	0
$929$	−0.564626	−0.0185248	−0.00926238	−	0.999957i	$-0.502948\pi$
$929$	−0.564626	−0.0185248	−0.00926238	+	0.999957i	$0.502948\pi$
$930$	0	0
$931$	42.1215	1.38048
$932$	0	0
$933$	67.1210	2.19744
$934$	0	0
$935$	3.47373	0.113603
$936$	0	0
$937$	21.4710	0.701426	0.350713	−	0.936483i	$-0.385939\pi$
$937$	21.4710	0.701426	0.350713	+	0.936483i	$0.385939\pi$
$938$	0	0
$939$	−42.1851	−1.37666
$940$	0	0
$941$	32.9258	1.07335	0.536676	−	0.843789i	$-0.319680\pi$
$941$	32.9258	1.07335	0.536676	+	0.843789i	$0.319680\pi$
$942$	0	0
$943$	−4.07725	−0.132773
$944$	0	0
$945$	−9.67501	−0.314728
$946$	0	0
$947$	46.4200	1.50845	0.754224	−	0.656617i	$-0.228013\pi$
$947$	46.4200	1.50845	0.754224	+	0.656617i	$0.228013\pi$
$948$	0	0
$949$	−78.6408	−2.55279
$950$	0	0
$951$	−21.2866	−0.690266
$952$	0	0
$953$	40.6651	1.31727	0.658636	−	0.752462i	$-0.271134\pi$
$953$	40.6651	1.31727	0.658636	+	0.752462i	$0.271134\pi$
$954$	0	0
$955$	13.3629	0.432414
$956$	0	0
$957$	−19.7854	−0.639571
$958$	0	0
$959$	57.8363	1.86763
$960$	0	0
$961$	−27.2972	−0.880554
$962$	0	0
$963$	2.69755	0.0869274
$964$	0	0
$965$	−0.449680	−0.0144757
$966$	0	0
$967$	−13.7463	−0.442051	−0.221026	−	0.975268i	$-0.570940\pi$
$967$	−13.7463	−0.442051	−0.221026	+	0.975268i	$0.570940\pi$
$968$	0	0
$969$	−26.2276	−0.842553
$970$	0	0
$971$	10.4871	0.336548	0.168274	−	0.985740i	$-0.446181\pi$
$971$	10.4871	0.336548	0.168274	+	0.985740i	$0.446181\pi$
$972$	0	0
$973$	−79.7954	−2.55812
$974$	0	0
$975$	−14.8837	−0.476659
$976$	0	0
$977$	26.8721	0.859716	0.429858	−	0.902897i	$-0.358564\pi$
$977$	26.8721	0.859716	0.429858	+	0.902897i	$0.358564\pi$
$978$	0	0
$979$	−1.80647	−0.0577351
$980$	0	0
$981$	17.0347	0.543875
$982$	0	0
$983$	−5.47408	−0.174596	−0.0872980	−	0.996182i	$-0.527823\pi$
$983$	−5.47408	−0.174596	−0.0872980	+	0.996182i	$0.527823\pi$
$984$	0	0
$985$	−8.56942	−0.273044
$986$	0	0
$987$	−78.2437	−2.49052
$988$	0	0
$989$	0.295843	0.00940725
$990$	0	0
$991$	−25.1620	−0.799296	−0.399648	−	0.916669i	$-0.630868\pi$
$991$	−25.1620	−0.799296	−0.399648	+	0.916669i	$0.630868\pi$
$992$	0	0
$993$	63.4534	2.01363
$994$	0	0
$995$	−8.67704	−0.275081
$996$	0	0
$997$	11.3414	0.359186	0.179593	−	0.983741i	$-0.442522\pi$
$997$	11.3414	0.359186	0.179593	+	0.983741i	$0.442522\pi$
$998$	0	0
$999$	−7.05703	−0.223274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7120.2.a.bc.1.4		4
4.3	odd	2		445.2.a.d.1.3	✓	4
12.11	even	2		4005.2.a.l.1.2		4
20.19	odd	2		2225.2.a.i.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.d.1.3	✓	4	4.3	odd	2
2225.2.a.i.1.2		4	20.19	odd	2
4005.2.a.l.1.2		4	12.11	even	2
7120.2.a.bc.1.4		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)

\(f(q)\)	\(=\)	\(q+2.19353 q^{3} -1.00000 q^{5} -3.71135 q^{7} +1.81156 q^{9} +O(q^{10})\)	\(q+2.19353 q^{3} -1.00000 q^{5} -3.71135 q^{7} +1.81156 q^{9} +1.80647 q^{11} -6.78527 q^{13} -2.19353 q^{15} -1.92293 q^{17} +6.21801 q^{19} -8.14094 q^{21} -1.39822 q^{23} +1.00000 q^{25} -2.60687 q^{27} -4.99310 q^{29} -1.92427 q^{31} +3.96255 q^{33} +3.71135 q^{35} +2.70709 q^{37} -14.8837 q^{39} +2.91604 q^{41} -0.211586 q^{43} -1.81156 q^{45} +9.61114 q^{47} +6.77411 q^{49} -4.21801 q^{51} +10.9331 q^{53} -1.80647 q^{55} +13.6394 q^{57} +12.0984 q^{59} +10.1997 q^{61} -6.72333 q^{63} +6.78527 q^{65} -2.88548 q^{67} -3.06702 q^{69} +3.85919 q^{71} +11.5899 q^{73} +2.19353 q^{75} -6.70445 q^{77} +10.8158 q^{79} -11.1529 q^{81} +5.59257 q^{83} +1.92293 q^{85} -10.9525 q^{87} -1.00000 q^{89} +25.1825 q^{91} -4.22094 q^{93} -6.21801 q^{95} +1.07064 q^{97} +3.27254 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) 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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