LMFDB - Embedded newform 80.12.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [80,12,Mod(1,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	80.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-604] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$61.4674544448$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1969}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 492 $ x^2 - x - 492
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-21.6867$ of defining polynomial
Character	$\chi$	$=$	80.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+141.734 q^{3} +3125.00 q^{5} -85586.9 q^{7} -157058. q^{9} -767235. q^{11} +220960. q^{13} +442919. q^{15} -930719. q^{17} +1.77341e7 q^{19} -1.21306e7 q^{21} +3.99596e7 q^{23} +9.76562e6 q^{25} -4.73683e7 q^{27} +7.68554e7 q^{29} +2.96314e7 q^{31} -1.08743e8 q^{33} -2.67459e8 q^{35} +5.40911e7 q^{37} +3.13176e7 q^{39} +1.26006e8 q^{41} +2.88676e8 q^{43} -4.90808e8 q^{45} +1.57008e9 q^{47} +5.34780e9 q^{49} -1.31915e8 q^{51} -4.09006e9 q^{53} -2.39761e9 q^{55} +2.51353e9 q^{57} -3.77882e9 q^{59} -9.64103e9 q^{61} +1.34422e10 q^{63} +6.90500e8 q^{65} +1.63819e10 q^{67} +5.66364e9 q^{69} -1.03471e10 q^{71} +4.27149e9 q^{73} +1.38412e9 q^{75} +6.56653e10 q^{77} +1.96636e10 q^{79} +2.11087e10 q^{81} +1.35791e10 q^{83} -2.90850e9 q^{85} +1.08930e10 q^{87} +2.25058e10 q^{89} -1.89113e10 q^{91} +4.19978e9 q^{93} +5.54191e10 q^{95} -1.08976e11 q^{97} +1.20501e11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 604 q^{3} + 6250 q^{5} - 14092 q^{7} + 221914 q^{9} - 421584 q^{11} + 1730524 q^{13} - 1887500 q^{15} - 6323628 q^{17} + 28897400 q^{19} - 65446816 q^{21} + 45236076 q^{23} + 19531250 q^{25} - 197876440 q^{27}+ \cdots + 251492724912 q^{99}+O(q^{100}) $ 2 * q - 604 * q^3 + 6250 * q^5 - 14092 * q^7 + 221914 * q^9 - 421584 * q^11 + 1730524 * q^13 - 1887500 * q^15 - 6323628 * q^17 + 28897400 * q^19 - 65446816 * q^21 + 45236076 * q^23 + 19531250 * q^25 - 197876440 * q^27 + 58226220 * q^29 - 41413384 * q^31 - 366506832 * q^33 - 44037500 * q^35 + 377255452 * q^37 - 1094415848 * q^39 - 785271036 * q^41 + 1452987236 * q^43 + 693481250 * q^45 + 1288127748 * q^47 + 8481998946 * q^49 + 3889762056 * q^51 - 30490836 * q^53 - 1317450000 * q^55 - 5811319600 * q^57 - 8677102440 * q^59 + 1115498764 * q^61 + 40536764356 * q^63 + 5407887500 * q^65 + 12673769708 * q^67 + 1728802848 * q^69 - 13799832984 * q^71 - 17842079516 * q^73 - 5898437500 * q^75 + 90377541264 * q^77 + 12636930320 * q^79 + 66213933922 * q^81 - 41986488924 * q^83 - 19761337500 * q^85 + 24785411160 * q^87 + 13208740020 * q^89 + 89014903096 * q^91 + 57180301568 * q^93 + 90304375000 * q^95 - 61787462828 * q^97 + 251492724912 * q^99

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
$3$	141.734	0.336750	0.168375	−	0.985723i	$-0.446148\pi$
$3$	141.734	0.336750	0.168375	+	0.985723i	$0.446148\pi$
$4$	0	0
$5$	3125.00	0.447214
$5$	3125.00	0.447214
$6$	0	0
$7$	−85586.9	−1.92472	−0.962362	−	0.271772i	$-0.912390\pi$
$7$	−85586.9	−1.92472	−0.962362	+	0.271772i	$0.912390\pi$
$8$	0	0
$9$	−157058.	−0.886599
$10$	0	0
$11$	−767235.	−1.43638	−0.718188	−	0.695849i	$-0.755028\pi$
$11$	−767235.	−1.43638	−0.718188	+	0.695849i	$0.755028\pi$
$12$	0	0
$13$	220960.	0.165054	0.0825268	−	0.996589i	$-0.473701\pi$
$13$	220960.	0.165054	0.0825268	+	0.996589i	$0.473701\pi$
$14$	0	0
$15$	442919.	0.150599
$16$	0	0
$17$	−930719.	−0.158983	−0.0794913	−	0.996836i	$-0.525330\pi$
$17$	−930719.	−0.158983	−0.0794913	+	0.996836i	$0.525330\pi$
$18$	0	0
$19$	1.77341e7	1.64310	0.821551	−	0.570135i	$-0.193109\pi$
$19$	1.77341e7	1.64310	0.821551	+	0.570135i	$0.193109\pi$
$20$	0	0
$21$	−1.21306e7	−0.648151
$22$	0	0
$23$	3.99596e7	1.29455	0.647274	−	0.762257i	$-0.275909\pi$
$23$	3.99596e7	1.29455	0.647274	+	0.762257i	$0.275909\pi$
$24$	0	0
$25$	9.76562e6	0.200000
$26$	0	0
$27$	−4.73683e7	−0.635312
$28$	0	0
$29$	7.68554e7	0.695801	0.347901	−	0.937531i	$-0.386895\pi$
$29$	7.68554e7	0.695801	0.347901	+	0.937531i	$0.386895\pi$
$30$	0	0
$31$	2.96314e7	0.185893	0.0929465	−	0.995671i	$-0.470371\pi$
$31$	2.96314e7	0.185893	0.0929465	+	0.995671i	$0.470371\pi$
$32$	0	0
$33$	−1.08743e8	−0.483700
$34$	0	0
$35$	−2.67459e8	−0.860762
$36$	0	0
$37$	5.40911e7	0.128238	0.0641189	−	0.997942i	$-0.479576\pi$
$37$	5.40911e7	0.128238	0.0641189	+	0.997942i	$0.479576\pi$
$38$	0	0
$39$	3.13176e7	0.0555818
$40$	0	0
$41$	1.26006e8	0.169856	0.0849280	−	0.996387i	$-0.472934\pi$
$41$	1.26006e8	0.169856	0.0849280	+	0.996387i	$0.472934\pi$
$42$	0	0
$43$	2.88676e8	0.299457	0.149728	−	0.988727i	$-0.452160\pi$
$43$	2.88676e8	0.299457	0.149728	+	0.988727i	$0.452160\pi$
$44$	0	0
$45$	−4.90808e8	−0.396499
$46$	0	0
$47$	1.57008e9	0.998579	0.499290	−	0.866435i	$-0.333594\pi$
$47$	1.57008e9	0.998579	0.499290	+	0.866435i	$0.333594\pi$
$48$	0	0
$49$	5.34780e9	2.70456
$50$	0	0
$51$	−1.31915e8	−0.0535374
$52$	0	0
$53$	−4.09006e9	−1.34342	−0.671711	−	0.740813i	$-0.734440\pi$
$53$	−4.09006e9	−1.34342	−0.671711	+	0.740813i	$0.734440\pi$
$54$	0	0
$55$	−2.39761e9	−0.642367
$56$	0	0
$57$	2.51353e9	0.553315
$58$	0	0
$59$	−3.77882e9	−0.688129	−0.344065	−	0.938946i	$-0.611804\pi$
$59$	−3.77882e9	−0.688129	−0.344065	+	0.938946i	$0.611804\pi$
$60$	0	0
$61$	−9.64103e9	−1.46154	−0.730768	−	0.682626i	$-0.760838\pi$
$61$	−9.64103e9	−1.46154	−0.730768	+	0.682626i	$0.760838\pi$
$62$	0	0
$63$	1.34422e10	1.70646
$64$	0	0
$65$	6.90500e8	0.0738143
$66$	0	0
$67$	1.63819e10	1.48236	0.741178	−	0.671308i	$-0.234267\pi$
$67$	1.63819e10	1.48236	0.741178	+	0.671308i	$0.234267\pi$
$68$	0	0
$69$	5.66364e9	0.435939
$70$	0	0
$71$	−1.03471e10	−0.680609	−0.340304	−	0.940315i	$-0.610530\pi$
$71$	−1.03471e10	−0.680609	−0.340304	+	0.940315i	$0.610530\pi$
$72$	0	0
$73$	4.27149e9	0.241159	0.120580	−	0.992704i	$-0.461525\pi$
$73$	4.27149e9	0.241159	0.120580	+	0.992704i	$0.461525\pi$
$74$	0	0
$75$	1.38412e9	0.0673500
$76$	0	0
$77$	6.56653e10	2.76463
$78$	0	0
$79$	1.96636e10	0.718977	0.359489	−	0.933149i	$-0.382951\pi$
$79$	1.96636e10	0.718977	0.359489	+	0.933149i	$0.382951\pi$
$80$	0	0
$81$	2.11087e10	0.672658
$82$	0	0
$83$	1.35791e10	0.378391	0.189196	−	0.981939i	$-0.439412\pi$
$83$	1.35791e10	0.378391	0.189196	+	0.981939i	$0.439412\pi$
$84$	0	0
$85$	−2.90850e9	−0.0710991
$86$	0	0
$87$	1.08930e10	0.234311
$88$	0	0
$89$	2.25058e10	0.427218	0.213609	−	0.976919i	$-0.431478\pi$
$89$	2.25058e10	0.427218	0.213609	+	0.976919i	$0.431478\pi$
$90$	0	0
$91$	−1.89113e10	−0.317683
$92$	0	0
$93$	4.19978e9	0.0625994
$94$	0	0
$95$	5.54191e10	0.734818
$96$	0	0
$97$	−1.08976e11	−1.28851	−0.644255	−	0.764811i	$-0.722832\pi$
$97$	−1.08976e11	−1.28851	−0.644255	+	0.764811i	$0.722832\pi$
$98$	0	0
$99$	1.20501e11	1.27349
$100$	0	0
$101$	1.63516e11	1.54807	0.774037	−	0.633140i	$-0.218234\pi$
$101$	1.63516e11	1.54807	0.774037	+	0.633140i	$0.218234\pi$
$102$	0	0
$103$	7.69876e10	0.654359	0.327180	−	0.944962i	$-0.393902\pi$
$103$	7.69876e10	0.654359	0.327180	+	0.944962i	$0.393902\pi$
$104$	0	0
$105$	−3.79081e10	−0.289862
$106$	0	0
$107$	−1.19891e11	−0.826376	−0.413188	−	0.910646i	$-0.635585\pi$
$107$	−1.19891e11	−0.826376	−0.413188	+	0.910646i	$0.635585\pi$
$108$	0	0
$109$	−7.33426e10	−0.456573	−0.228287	−	0.973594i	$-0.573312\pi$
$109$	−7.33426e10	−0.456573	−0.228287	+	0.973594i	$0.573312\pi$
$110$	0	0
$111$	7.66655e9	0.0431841
$112$	0	0
$113$	2.18835e11	1.11734	0.558669	−	0.829391i	$-0.311312\pi$
$113$	2.18835e11	1.11734	0.558669	+	0.829391i	$0.311312\pi$
$114$	0	0
$115$	1.24874e11	0.578939
$116$	0	0
$117$	−3.47036e10	−0.146337
$118$	0	0
$119$	7.96574e10	0.305997
$120$	0	0
$121$	3.03337e11	1.06318
$122$	0	0
$123$	1.78594e10	0.0571990
$124$	0	0
$125$	3.05176e10	0.0894427
$126$	0	0
$127$	−3.93053e11	−1.05568	−0.527838	−	0.849345i	$-0.676997\pi$
$127$	−3.93053e11	−1.05568	−0.527838	+	0.849345i	$0.676997\pi$
$128$	0	0
$129$	4.09152e10	0.100842
$130$	0	0
$131$	2.63175e11	0.596008	0.298004	−	0.954565i	$-0.403679\pi$
$131$	2.63175e11	0.596008	0.298004	+	0.954565i	$0.403679\pi$
$132$	0	0
$133$	−1.51781e12	−3.16252
$134$	0	0
$135$	−1.48026e11	−0.284120
$136$	0	0
$137$	5.20521e11	0.921457	0.460728	−	0.887541i	$-0.347588\pi$
$137$	5.20521e11	0.921457	0.460728	+	0.887541i	$0.347588\pi$
$138$	0	0
$139$	−6.82675e11	−1.11592	−0.557960	−	0.829868i	$-0.688416\pi$
$139$	−6.82675e11	−1.11592	−0.557960	+	0.829868i	$0.688416\pi$
$140$	0	0
$141$	2.22533e11	0.336272
$142$	0	0
$143$	−1.69528e11	−0.237079
$144$	0	0
$145$	2.40173e11	0.311172
$146$	0	0
$147$	7.57966e11	0.910761
$148$	0	0
$149$	1.01566e12	1.13299	0.566493	−	0.824066i	$-0.308300\pi$
$149$	1.01566e12	1.13299	0.566493	+	0.824066i	$0.308300\pi$
$150$	0	0
$151$	−5.34040e11	−0.553605	−0.276803	−	0.960927i	$-0.589275\pi$
$151$	−5.34040e11	−0.553605	−0.276803	+	0.960927i	$0.589275\pi$
$152$	0	0
$153$	1.46177e11	0.140954
$154$	0	0
$155$	9.25981e10	0.0831338
$156$	0	0
$157$	−7.29675e11	−0.610494	−0.305247	−	0.952273i	$-0.598739\pi$
$157$	−7.29675e11	−0.610494	−0.305247	+	0.952273i	$0.598739\pi$
$158$	0	0
$159$	−5.79701e11	−0.452397
$160$	0	0
$161$	−3.42002e12	−2.49165
$162$	0	0
$163$	2.12171e11	0.144429	0.0722143	−	0.997389i	$-0.476993\pi$
$163$	2.12171e11	0.144429	0.0722143	+	0.997389i	$0.476993\pi$
$164$	0	0
$165$	−3.39823e11	−0.216317
$166$	0	0
$167$	2.36613e11	0.140961	0.0704804	−	0.997513i	$-0.477547\pi$
$167$	2.36613e11	0.140961	0.0704804	+	0.997513i	$0.477547\pi$
$168$	0	0
$169$	−1.74334e12	−0.972757
$170$	0	0
$171$	−2.78529e12	−1.45677
$172$	0	0
$173$	−7.36983e11	−0.361579	−0.180790	−	0.983522i	$-0.557865\pi$
$173$	−7.36983e11	−0.361579	−0.180790	+	0.983522i	$0.557865\pi$
$174$	0	0
$175$	−8.35810e11	−0.384945
$176$	0	0
$177$	−5.35588e11	−0.231727
$178$	0	0
$179$	−1.35344e11	−0.0550487	−0.0275244	−	0.999621i	$-0.508762\pi$
$179$	−1.35344e11	−0.0550487	−0.0275244	+	0.999621i	$0.508762\pi$
$180$	0	0
$181$	−1.35881e12	−0.519907	−0.259953	−	0.965621i	$-0.583707\pi$
$181$	−1.35881e12	−0.519907	−0.259953	+	0.965621i	$0.583707\pi$
$182$	0	0
$183$	−1.36646e12	−0.492172
$184$	0	0
$185$	1.69035e11	0.0573497
$186$	0	0
$187$	7.14080e11	0.228359
$188$	0	0
$189$	4.05411e12	1.22280
$190$	0	0
$191$	−1.10648e12	−0.314964	−0.157482	−	0.987522i	$-0.550338\pi$
$191$	−1.10648e12	−0.314964	−0.157482	+	0.987522i	$0.550338\pi$
$192$	0	0
$193$	6.17025e12	1.65858	0.829292	−	0.558816i	$-0.188744\pi$
$193$	6.17025e12	1.65858	0.829292	+	0.558816i	$0.188744\pi$
$194$	0	0
$195$	9.78674e10	0.0248570
$196$	0	0
$197$	2.00333e12	0.481047	0.240523	−	0.970643i	$-0.422681\pi$
$197$	2.00333e12	0.481047	0.240523	+	0.970643i	$0.422681\pi$
$198$	0	0
$199$	7.73897e12	1.75789	0.878944	−	0.476925i	$-0.158249\pi$
$199$	7.73897e12	1.75789	0.878944	+	0.476925i	$0.158249\pi$
$200$	0	0
$201$	2.32187e12	0.499183
$202$	0	0
$203$	−6.57782e12	−1.33922
$204$	0	0
$205$	3.93770e11	0.0759619
$206$	0	0
$207$	−6.27599e12	−1.14775
$208$	0	0
$209$	−1.36062e13	−2.36011
$210$	0	0
$211$	9.01879e12	1.48455	0.742275	−	0.670096i	$-0.233747\pi$
$211$	9.01879e12	1.48455	0.742275	+	0.670096i	$0.233747\pi$
$212$	0	0
$213$	−1.46654e12	−0.229195
$214$	0	0
$215$	9.02112e11	0.133921
$216$	0	0
$217$	−2.53606e12	−0.357792
$218$	0	0
$219$	6.05416e11	0.0812103
$220$	0	0
$221$	−2.05652e11	−0.0262407
$222$	0	0
$223$	−2.06620e12	−0.250897	−0.125448	−	0.992100i	$-0.540037\pi$
$223$	−2.06620e12	−0.250897	−0.125448	+	0.992100i	$0.540037\pi$
$224$	0	0
$225$	−1.53377e12	−0.177320
$226$	0	0
$227$	1.56806e13	1.72672	0.863359	−	0.504591i	$-0.168357\pi$
$227$	1.56806e13	1.72672	0.863359	+	0.504591i	$0.168357\pi$
$228$	0	0
$229$	6.02309e12	0.632010	0.316005	−	0.948758i	$-0.397658\pi$
$229$	6.02309e12	0.632010	0.316005	+	0.948758i	$0.397658\pi$
$230$	0	0
$231$	9.30701e12	0.930989
$232$	0	0
$233$	1.43129e13	1.36544	0.682718	−	0.730682i	$-0.260798\pi$
$233$	1.43129e13	1.36544	0.682718	+	0.730682i	$0.260798\pi$
$234$	0	0
$235$	4.90649e12	0.446578
$236$	0	0
$237$	2.78701e12	0.242116
$238$	0	0
$239$	−2.22886e13	−1.84881	−0.924407	−	0.381407i	$-0.875440\pi$
$239$	−2.22886e13	−1.84881	−0.924407	+	0.381407i	$0.875440\pi$
$240$	0	0
$241$	2.06638e13	1.63726	0.818629	−	0.574323i	$-0.194735\pi$
$241$	2.06638e13	1.63726	0.818629	+	0.574323i	$0.194735\pi$
$242$	0	0
$243$	1.13830e13	0.861830
$244$	0	0
$245$	1.67119e13	1.20952
$246$	0	0
$247$	3.91853e12	0.271200
$248$	0	0
$249$	1.92462e12	0.127423
$250$	0	0
$251$	1.23384e13	0.781721	0.390861	−	0.920450i	$-0.372177\pi$
$251$	1.23384e13	0.781721	0.390861	+	0.920450i	$0.372177\pi$
$252$	0	0
$253$	−3.06584e13	−1.85946
$254$	0	0
$255$	−4.12233e11	−0.0239426
$256$	0	0
$257$	1.19160e13	0.662975	0.331487	−	0.943460i	$-0.392450\pi$
$257$	1.19160e13	0.662975	0.331487	+	0.943460i	$0.392450\pi$
$258$	0	0
$259$	−4.62949e12	−0.246822
$260$	0	0
$261$	−1.20708e13	−0.616897
$262$	0	0
$263$	−1.54645e12	−0.0757845	−0.0378923	−	0.999282i	$-0.512064\pi$
$263$	−1.54645e12	−0.0757845	−0.0378923	+	0.999282i	$0.512064\pi$
$264$	0	0
$265$	−1.27814e13	−0.600797
$266$	0	0
$267$	3.18984e12	0.143866
$268$	0	0
$269$	1.43195e13	0.619855	0.309927	−	0.950760i	$-0.399695\pi$
$269$	1.43195e13	0.619855	0.309927	+	0.950760i	$0.399695\pi$
$270$	0	0
$271$	−2.26776e13	−0.942468	−0.471234	−	0.882008i	$-0.656191\pi$
$271$	−2.26776e13	−0.942468	−0.471234	+	0.882008i	$0.656191\pi$
$272$	0	0
$273$	−2.68038e12	−0.106980
$274$	0	0
$275$	−7.49253e12	−0.287275
$276$	0	0
$277$	9.25283e12	0.340907	0.170454	−	0.985366i	$-0.445477\pi$
$277$	9.25283e12	0.340907	0.170454	+	0.985366i	$0.445477\pi$
$278$	0	0
$279$	−4.65386e12	−0.164813
$280$	0	0
$281$	−2.26511e12	−0.0771265	−0.0385633	−	0.999256i	$-0.512278\pi$
$281$	−2.26511e12	−0.0771265	−0.0385633	+	0.999256i	$0.512278\pi$
$282$	0	0
$283$	3.60409e13	1.18024	0.590121	−	0.807315i	$-0.299080\pi$
$283$	3.60409e13	1.18024	0.590121	+	0.807315i	$0.299080\pi$
$284$	0	0
$285$	7.85478e12	0.247450
$286$	0	0
$287$	−1.07845e13	−0.326926
$288$	0	0
$289$	−3.34057e13	−0.974725
$290$	0	0
$291$	−1.54457e13	−0.433905
$292$	0	0
$293$	1.63348e13	0.441918	0.220959	−	0.975283i	$-0.429081\pi$
$293$	1.63348e13	0.441918	0.220959	+	0.975283i	$0.429081\pi$
$294$	0	0
$295$	−1.18088e13	−0.307741
$296$	0	0
$297$	3.63426e13	0.912548
$298$	0	0
$299$	8.82948e12	0.213670
$300$	0	0
$301$	−2.47069e13	−0.576371
$302$	0	0
$303$	2.31758e13	0.521314
$304$	0	0
$305$	−3.01282e13	−0.653618
$306$	0	0
$307$	5.61322e13	1.17476	0.587382	−	0.809310i	$-0.300158\pi$
$307$	5.61322e13	1.17476	0.587382	+	0.809310i	$0.300158\pi$
$308$	0	0
$309$	1.09118e13	0.220355
$310$	0	0
$311$	−6.92340e13	−1.34939	−0.674694	−	0.738097i	$-0.735725\pi$
$311$	−6.92340e13	−1.34939	−0.674694	+	0.738097i	$0.735725\pi$
$312$	0	0
$313$	−3.23822e13	−0.609273	−0.304637	−	0.952469i	$-0.598535\pi$
$313$	−3.23822e13	−0.609273	−0.304637	+	0.952469i	$0.598535\pi$
$314$	0	0
$315$	4.20067e13	0.763151
$316$	0	0
$317$	−1.84023e13	−0.322885	−0.161442	−	0.986882i	$-0.551615\pi$
$317$	−1.84023e13	−0.322885	−0.161442	+	0.986882i	$0.551615\pi$
$318$	0	0
$319$	−5.89661e13	−0.999433
$320$	0	0
$321$	−1.69927e13	−0.278282
$322$	0	0
$323$	−1.65055e13	−0.261224
$324$	0	0
$325$	2.15781e12	0.0330107
$326$	0	0
$327$	−1.03951e13	−0.153751
$328$	0	0
$329$	−1.34378e14	−1.92199
$330$	0	0
$331$	−6.74629e13	−0.933277	−0.466639	−	0.884448i	$-0.654535\pi$
$331$	−6.74629e13	−0.933277	−0.466639	+	0.884448i	$0.654535\pi$
$332$	0	0
$333$	−8.49546e12	−0.113696
$334$	0	0
$335$	5.11934e13	0.662930
$336$	0	0
$337$	1.07897e14	1.35222	0.676109	−	0.736802i	$-0.263665\pi$
$337$	1.07897e14	1.35222	0.676109	+	0.736802i	$0.263665\pi$
$338$	0	0
$339$	3.10163e13	0.376264
$340$	0	0
$341$	−2.27342e13	−0.267012
$342$	0	0
$343$	−2.88468e14	−3.28081
$344$	0	0
$345$	1.76989e13	0.194958
$346$	0	0
$347$	1.29756e13	0.138458	0.0692288	−	0.997601i	$-0.477946\pi$
$347$	1.29756e13	0.138458	0.0692288	+	0.997601i	$0.477946\pi$
$348$	0	0
$349$	−7.76358e13	−0.802643	−0.401321	−	0.915937i	$-0.631449\pi$
$349$	−7.76358e13	−0.802643	−0.401321	+	0.915937i	$0.631449\pi$
$350$	0	0
$351$	−1.04665e13	−0.104861
$352$	0	0
$353$	−4.21031e13	−0.408839	−0.204420	−	0.978883i	$-0.565531\pi$
$353$	−4.21031e13	−0.408839	−0.204420	+	0.978883i	$0.565531\pi$
$354$	0	0
$355$	−3.23347e13	−0.304377
$356$	0	0
$357$	1.12902e13	0.103045
$358$	0	0
$359$	−1.84963e14	−1.63707	−0.818533	−	0.574460i	$-0.805212\pi$
$359$	−1.84963e14	−1.63707	−0.818533	+	0.574460i	$0.805212\pi$
$360$	0	0
$361$	1.98008e14	1.69978
$362$	0	0
$363$	4.29933e13	0.358025
$364$	0	0
$365$	1.33484e13	0.107850
$366$	0	0
$367$	1.68959e14	1.32470	0.662351	−	0.749194i	$-0.269559\pi$
$367$	1.68959e14	1.32470	0.662351	+	0.749194i	$0.269559\pi$
$368$	0	0
$369$	−1.97903e13	−0.150594
$370$	0	0
$371$	3.50056e14	2.58572
$372$	0	0
$373$	−1.87187e14	−1.34238	−0.671191	−	0.741284i	$-0.734217\pi$
$373$	−1.87187e14	−1.34238	−0.671191	+	0.741284i	$0.734217\pi$
$374$	0	0
$375$	4.32538e12	0.0301198
$376$	0	0
$377$	1.69820e13	0.114845
$378$	0	0
$379$	7.47613e13	0.491090	0.245545	−	0.969385i	$-0.421033\pi$
$379$	7.47613e13	0.491090	0.245545	+	0.969385i	$0.421033\pi$
$380$	0	0
$381$	−5.57090e13	−0.355499
$382$	0	0
$383$	1.55521e14	0.964264	0.482132	−	0.876099i	$-0.339863\pi$
$383$	1.55521e14	0.964264	0.482132	+	0.876099i	$0.339863\pi$
$384$	0	0
$385$	2.05204e14	1.23638
$386$	0	0
$387$	−4.53390e13	−0.265498
$388$	0	0
$389$	−8.76448e13	−0.498889	−0.249444	−	0.968389i	$-0.580248\pi$
$389$	−8.76448e13	−0.498889	−0.249444	+	0.968389i	$0.580248\pi$
$390$	0	0
$391$	−3.71912e13	−0.205811
$392$	0	0
$393$	3.73009e13	0.200706
$394$	0	0
$395$	6.14489e13	0.321536
$396$	0	0
$397$	5.19767e13	0.264521	0.132261	−	0.991215i	$-0.457776\pi$
$397$	5.19767e13	0.264521	0.132261	+	0.991215i	$0.457776\pi$
$398$	0	0
$399$	−2.15125e14	−1.06498
$400$	0	0
$401$	−3.40753e14	−1.64114	−0.820570	−	0.571546i	$-0.806344\pi$
$401$	−3.40753e14	−1.64114	−0.820570	+	0.571546i	$0.806344\pi$
$402$	0	0
$403$	6.54735e12	0.0306823
$404$	0	0
$405$	6.59648e13	0.300822
$406$	0	0
$407$	−4.15005e13	−0.184198
$408$	0	0
$409$	3.34635e14	1.44575	0.722875	−	0.690979i	$-0.242820\pi$
$409$	3.34635e14	1.44575	0.722875	+	0.690979i	$0.242820\pi$
$410$	0	0
$411$	7.37756e13	0.310301
$412$	0	0
$413$	3.23417e14	1.32446
$414$	0	0
$415$	4.24347e13	0.169222
$416$	0	0
$417$	−9.67584e13	−0.375786
$418$	0	0
$419$	−1.37830e14	−0.521396	−0.260698	−	0.965420i	$-0.583953\pi$
$419$	−1.37830e14	−0.521396	−0.260698	+	0.965420i	$0.583953\pi$
$420$	0	0
$421$	4.35453e14	1.60468	0.802342	−	0.596864i	$-0.203587\pi$
$421$	4.35453e14	1.60468	0.802342	+	0.596864i	$0.203587\pi$
$422$	0	0
$423$	−2.46594e14	−0.885340
$424$	0	0
$425$	−9.08905e12	−0.0317965
$426$	0	0
$427$	8.25146e14	2.81305
$428$	0	0
$429$	−2.40279e13	−0.0798365
$430$	0	0
$431$	−1.51185e14	−0.489647	−0.244824	−	0.969568i	$-0.578730\pi$
$431$	−1.51185e14	−0.489647	−0.244824	+	0.969568i	$0.578730\pi$
$432$	0	0
$433$	−3.75044e13	−0.118413	−0.0592065	−	0.998246i	$-0.518857\pi$
$433$	−3.75044e13	−0.118413	−0.0592065	+	0.998246i	$0.518857\pi$
$434$	0	0
$435$	3.40407e13	0.104787
$436$	0	0
$437$	7.08648e14	2.12707
$438$	0	0
$439$	1.61934e14	0.474004	0.237002	−	0.971509i	$-0.423835\pi$
$439$	1.61934e14	0.474004	0.237002	+	0.971509i	$0.423835\pi$
$440$	0	0
$441$	−8.39917e14	−2.39786
$442$	0	0
$443$	1.29342e14	0.360179	0.180090	−	0.983650i	$-0.442361\pi$
$443$	1.29342e14	0.360179	0.180090	+	0.983650i	$0.442361\pi$
$444$	0	0
$445$	7.03306e13	0.191058
$446$	0	0
$447$	1.43954e14	0.381533
$448$	0	0
$449$	1.30255e13	0.0336852	0.0168426	−	0.999858i	$-0.494639\pi$
$449$	1.30255e13	0.0336852	0.0168426	+	0.999858i	$0.494639\pi$
$450$	0	0
$451$	−9.66764e13	−0.243977
$452$	0	0
$453$	−7.56916e13	−0.186427
$454$	0	0
$455$	−5.90978e13	−0.142072
$456$	0	0
$457$	3.45724e14	0.811318	0.405659	−	0.914025i	$-0.367042\pi$
$457$	3.45724e14	0.811318	0.405659	+	0.914025i	$0.367042\pi$
$458$	0	0
$459$	4.40866e13	0.101004
$460$	0	0
$461$	2.96945e14	0.664234	0.332117	−	0.943238i	$-0.392237\pi$
$461$	2.96945e14	0.664234	0.332117	+	0.943238i	$0.392237\pi$
$462$	0	0
$463$	−4.55258e14	−0.994402	−0.497201	−	0.867635i	$-0.665639\pi$
$463$	−4.55258e14	−0.994402	−0.497201	+	0.867635i	$0.665639\pi$
$464$	0	0
$465$	1.31243e13	0.0279953
$466$	0	0
$467$	3.66499e14	0.763537	0.381768	−	0.924258i	$-0.375315\pi$
$467$	3.66499e14	0.763537	0.381768	+	0.924258i	$0.375315\pi$
$468$	0	0
$469$	−1.40208e15	−2.85313
$470$	0	0
$471$	−1.03420e14	−0.205584
$472$	0	0
$473$	−2.21482e14	−0.430132
$474$	0	0
$475$	1.73185e14	0.328620
$476$	0	0
$477$	6.42379e14	1.19108
$478$	0	0
$479$	−5.78377e14	−1.04801	−0.524005	−	0.851715i	$-0.675563\pi$
$479$	−5.78377e14	−1.04801	−0.524005	+	0.851715i	$0.675563\pi$
$480$	0	0
$481$	1.19520e13	0.0211661
$482$	0	0
$483$	−4.84734e14	−0.839062
$484$	0	0
$485$	−3.40551e14	−0.576239
$486$	0	0
$487$	4.75203e14	0.786086	0.393043	−	0.919520i	$-0.371422\pi$
$487$	4.75203e14	0.786086	0.393043	+	0.919520i	$0.371422\pi$
$488$	0	0
$489$	3.00718e13	0.0486363
$490$	0	0
$491$	−7.59641e14	−1.20132	−0.600662	−	0.799503i	$-0.705096\pi$
$491$	−7.59641e14	−1.20132	−0.600662	+	0.799503i	$0.705096\pi$
$492$	0	0
$493$	−7.15307e13	−0.110620
$494$	0	0
$495$	3.76565e14	0.569522
$496$	0	0
$497$	8.85576e14	1.30998
$498$	0	0
$499$	4.91039e14	0.710499	0.355249	−	0.934772i	$-0.384396\pi$
$499$	4.91039e14	0.710499	0.355249	+	0.934772i	$0.384396\pi$
$500$	0	0
$501$	3.35362e13	0.0474686
$502$	0	0
$503$	4.25544e14	0.589278	0.294639	−	0.955609i	$-0.404801\pi$
$503$	4.25544e14	0.589278	0.294639	+	0.955609i	$0.404801\pi$
$504$	0	0
$505$	5.10987e14	0.692320
$506$	0	0
$507$	−2.47090e14	−0.327576
$508$	0	0
$509$	−1.41466e15	−1.83529	−0.917644	−	0.397404i	$-0.869911\pi$
$509$	−1.41466e15	−1.83529	−0.917644	+	0.397404i	$0.869911\pi$
$510$	0	0
$511$	−3.65584e14	−0.464165
$512$	0	0
$513$	−8.40035e14	−1.04388
$514$	0	0
$515$	2.40586e14	0.292638
$516$	0	0
$517$	−1.20462e15	−1.43434
$518$	0	0
$519$	−1.04456e14	−0.121762
$520$	0	0
$521$	6.27267e14	0.715888	0.357944	−	0.933743i	$-0.383478\pi$
$521$	6.27267e14	0.715888	0.357944	+	0.933743i	$0.383478\pi$
$522$	0	0
$523$	−2.23774e14	−0.250063	−0.125032	−	0.992153i	$-0.539903\pi$
$523$	−2.23774e14	−0.250063	−0.125032	+	0.992153i	$0.539903\pi$
$524$	0	0
$525$	−1.18463e14	−0.129630
$526$	0	0
$527$	−2.75785e13	−0.0295537
$528$	0	0
$529$	6.43961e14	0.675855
$530$	0	0
$531$	5.93495e14	0.610095
$532$	0	0
$533$	2.78424e13	0.0280354
$534$	0	0
$535$	−3.74661e14	−0.369566
$536$	0	0
$537$	−1.91829e13	−0.0185377
$538$	0	0
$539$	−4.10302e15	−3.88477
$540$	0	0
$541$	5.47756e14	0.508162	0.254081	−	0.967183i	$-0.418227\pi$
$541$	5.47756e14	0.508162	0.254081	+	0.967183i	$0.418227\pi$
$542$	0	0
$543$	−1.92589e14	−0.175079
$544$	0	0
$545$	−2.29196e14	−0.204186
$546$	0	0
$547$	−4.33518e14	−0.378509	−0.189255	−	0.981928i	$-0.560607\pi$
$547$	−4.33518e14	−0.378509	−0.189255	+	0.981928i	$0.560607\pi$
$548$	0	0
$549$	1.51420e15	1.29580
$550$	0	0
$551$	1.36296e15	1.14327
$552$	0	0
$553$	−1.68295e15	−1.38383
$554$	0	0
$555$	2.39580e13	0.0193125
$556$	0	0
$557$	−7.36061e14	−0.581715	−0.290857	−	0.956766i	$-0.593941\pi$
$557$	−7.36061e14	−0.581715	−0.290857	+	0.956766i	$0.593941\pi$
$558$	0	0
$559$	6.37858e13	0.0494264
$560$	0	0
$561$	1.01209e14	0.0768998
$562$	0	0
$563$	9.03199e14	0.672957	0.336478	−	0.941691i	$-0.390764\pi$
$563$	9.03199e14	0.672957	0.336478	+	0.941691i	$0.390764\pi$
$564$	0	0
$565$	6.83858e14	0.499689
$566$	0	0
$567$	−1.80663e15	−1.29468
$568$	0	0
$569$	−2.25786e15	−1.58701	−0.793503	−	0.608566i	$-0.791745\pi$
$569$	−2.25786e15	−1.58701	−0.793503	+	0.608566i	$0.791745\pi$
$570$	0	0
$571$	−9.15100e14	−0.630913	−0.315457	−	0.948940i	$-0.602158\pi$
$571$	−9.15100e14	−0.630913	−0.315457	+	0.948940i	$0.602158\pi$
$572$	0	0
$573$	−1.56826e14	−0.106064
$574$	0	0
$575$	3.90231e14	0.258910
$576$	0	0
$577$	2.84484e14	0.185179	0.0925893	−	0.995704i	$-0.470486\pi$
$577$	2.84484e14	0.185179	0.0925893	+	0.995704i	$0.470486\pi$
$578$	0	0
$579$	8.74535e14	0.558528
$580$	0	0
$581$	−1.16219e15	−0.728299
$582$	0	0
$583$	3.13804e15	1.92966
$584$	0	0
$585$	−1.08449e14	−0.0654437
$586$	0	0
$587$	1.99879e15	1.18374	0.591871	−	0.806032i	$-0.298389\pi$
$587$	1.99879e15	1.18374	0.591871	+	0.806032i	$0.298389\pi$
$588$	0	0
$589$	5.25486e14	0.305441
$590$	0	0
$591$	2.83940e14	0.161992
$592$	0	0
$593$	8.52303e14	0.477302	0.238651	−	0.971105i	$-0.423295\pi$
$593$	8.52303e14	0.477302	0.238651	+	0.971105i	$0.423295\pi$
$594$	0	0
$595$	2.48929e14	0.136846
$596$	0	0
$597$	1.09688e15	0.591969
$598$	0	0
$599$	−1.08984e14	−0.0577449	−0.0288725	−	0.999583i	$-0.509192\pi$
$599$	−1.08984e14	−0.0577449	−0.0288725	+	0.999583i	$0.509192\pi$
$600$	0	0
$601$	3.21315e14	0.167156	0.0835778	−	0.996501i	$-0.473365\pi$
$601$	3.21315e14	0.167156	0.0835778	+	0.996501i	$0.473365\pi$
$602$	0	0
$603$	−2.57291e15	−1.31426
$604$	0	0
$605$	9.47929e14	0.475468
$606$	0	0
$607$	3.08680e15	1.52044	0.760222	−	0.649664i	$-0.225090\pi$
$607$	3.08680e15	1.52044	0.760222	+	0.649664i	$0.225090\pi$
$608$	0	0
$609$	−9.32301e14	−0.450984
$610$	0	0
$611$	3.46924e14	0.164819
$612$	0	0
$613$	8.70867e14	0.406368	0.203184	−	0.979141i	$-0.434871\pi$
$613$	8.70867e14	0.406368	0.203184	+	0.979141i	$0.434871\pi$
$614$	0	0
$615$	5.58106e13	0.0255802
$616$	0	0
$617$	3.06108e15	1.37818	0.689091	−	0.724675i	$-0.258010\pi$
$617$	3.06108e15	1.37818	0.689091	+	0.724675i	$0.258010\pi$
$618$	0	0
$619$	1.21317e15	0.536566	0.268283	−	0.963340i	$-0.413544\pi$
$619$	1.21317e15	0.536566	0.268283	+	0.963340i	$0.413544\pi$
$620$	0	0
$621$	−1.89282e15	−0.822442
$622$	0	0
$623$	−1.92620e15	−0.822276
$624$	0	0
$625$	9.53674e13	0.0400000
$626$	0	0
$627$	−1.92847e15	−0.794768
$628$	0	0
$629$	−5.03436e13	−0.0203876
$630$	0	0
$631$	−2.95354e15	−1.17539	−0.587693	−	0.809084i	$-0.699964\pi$
$631$	−2.95354e15	−1.17539	−0.587693	+	0.809084i	$0.699964\pi$
$632$	0	0
$633$	1.27827e15	0.499922
$634$	0	0
$635$	−1.22829e15	−0.472112
$636$	0	0
$637$	1.18165e15	0.446398
$638$	0	0
$639$	1.62510e15	0.603427
$640$	0	0
$641$	−1.24519e15	−0.454483	−0.227241	−	0.973838i	$-0.572971\pi$
$641$	−1.24519e15	−0.454483	−0.227241	+	0.973838i	$0.572971\pi$
$642$	0	0
$643$	−2.82493e14	−0.101356	−0.0506778	−	0.998715i	$-0.516138\pi$
$643$	−2.82493e14	−0.101356	−0.0506778	+	0.998715i	$0.516138\pi$
$644$	0	0
$645$	1.27860e14	0.0450979
$646$	0	0
$647$	3.45042e15	1.19646	0.598230	−	0.801324i	$-0.295871\pi$
$647$	3.45042e15	1.19646	0.598230	+	0.801324i	$0.295871\pi$
$648$	0	0
$649$	2.89924e15	0.988413
$650$	0	0
$651$	−3.59446e14	−0.120487
$652$	0	0
$653$	3.55303e15	1.17105	0.585527	−	0.810653i	$-0.300888\pi$
$653$	3.55303e15	1.17105	0.585527	+	0.810653i	$0.300888\pi$
$654$	0	0
$655$	8.22421e14	0.266543
$656$	0	0
$657$	−6.70874e14	−0.213812
$658$	0	0
$659$	−1.76602e14	−0.0553511	−0.0276756	−	0.999617i	$-0.508811\pi$
$659$	−1.76602e14	−0.0553511	−0.0276756	+	0.999617i	$0.508811\pi$
$660$	0	0
$661$	3.46137e15	1.06694	0.533470	−	0.845819i	$-0.320888\pi$
$661$	3.46137e15	1.06694	0.533470	+	0.845819i	$0.320888\pi$
$662$	0	0
$663$	−2.91479e13	−0.00883654
$664$	0	0
$665$	−4.74315e15	−1.41432
$666$	0	0
$667$	3.07111e15	0.900748
$668$	0	0
$669$	−2.92851e14	−0.0844895
$670$	0	0
$671$	7.39693e15	2.09932
$672$	0	0
$673$	6.92295e15	1.93290	0.966448	−	0.256864i	$-0.0826893\pi$
$673$	6.92295e15	1.93290	0.966448	+	0.256864i	$0.0826893\pi$
$674$	0	0
$675$	−4.62581e14	−0.127062
$676$	0	0
$677$	−3.89173e15	−1.05173	−0.525865	−	0.850568i	$-0.676258\pi$
$677$	−3.89173e15	−1.05173	−0.525865	+	0.850568i	$0.676258\pi$
$678$	0	0
$679$	9.32695e15	2.48002
$680$	0	0
$681$	2.22248e15	0.581472
$682$	0	0
$683$	−4.79007e15	−1.23318	−0.616592	−	0.787283i	$-0.711487\pi$
$683$	−4.79007e15	−1.23318	−0.616592	+	0.787283i	$0.711487\pi$
$684$	0	0
$685$	1.62663e15	0.412088
$686$	0	0
$687$	8.53677e14	0.212829
$688$	0	0
$689$	−9.03740e14	−0.221737
$690$	0	0
$691$	−1.60990e15	−0.388749	−0.194374	−	0.980927i	$-0.562268\pi$
$691$	−1.60990e15	−0.388749	−0.194374	+	0.980927i	$0.562268\pi$
$692$	0	0
$693$	−1.03133e16	−2.45112
$694$	0	0
$695$	−2.13336e15	−0.499054
$696$	0	0
$697$	−1.17276e14	−0.0270041
$698$	0	0
$699$	2.02863e15	0.459810
$700$	0	0
$701$	−2.46356e15	−0.549686	−0.274843	−	0.961489i	$-0.588626\pi$
$701$	−2.46356e15	−0.549686	−0.274843	+	0.961489i	$0.588626\pi$
$702$	0	0
$703$	9.59257e14	0.210708
$704$	0	0
$705$	6.95417e14	0.150385
$706$	0	0
$707$	−1.39948e16	−2.97962
$708$	0	0
$709$	6.21323e15	1.30246	0.651228	−	0.758882i	$-0.274254\pi$
$709$	6.21323e15	1.30246	0.651228	+	0.758882i	$0.274254\pi$
$710$	0	0
$711$	−3.08834e15	−0.637445
$712$	0	0
$713$	1.18406e15	0.240647
$714$	0	0
$715$	−5.29776e14	−0.106025
$716$	0	0
$717$	−3.15905e15	−0.622588
$718$	0	0
$719$	−7.12825e15	−1.38348	−0.691742	−	0.722145i	$-0.743156\pi$
$719$	−7.12825e15	−1.38348	−0.691742	+	0.722145i	$0.743156\pi$
$720$	0	0
$721$	−6.58914e15	−1.25946
$722$	0	0
$723$	2.92877e15	0.551346
$724$	0	0
$725$	7.50541e14	0.139160
$726$	0	0
$727$	−5.59273e15	−1.02137	−0.510686	−	0.859767i	$-0.670609\pi$
$727$	−5.59273e15	−1.02137	−0.510686	+	0.859767i	$0.670609\pi$
$728$	0	0
$729$	−2.12599e15	−0.382437
$730$	0	0
$731$	−2.68676e14	−0.0476084
$732$	0	0
$733$	−2.75558e15	−0.480995	−0.240498	−	0.970650i	$-0.577311\pi$
$733$	−2.75558e15	−0.480995	−0.240498	+	0.970650i	$0.577311\pi$
$734$	0	0
$735$	2.36864e15	0.407304
$736$	0	0
$737$	−1.25688e16	−2.12922
$738$	0	0
$739$	−1.99004e15	−0.332138	−0.166069	−	0.986114i	$-0.553107\pi$
$739$	−1.99004e15	−0.332138	−0.166069	+	0.986114i	$0.553107\pi$
$740$	0	0
$741$	5.55389e14	0.0913266
$742$	0	0
$743$	7.01045e15	1.13581	0.567907	−	0.823093i	$-0.307753\pi$
$743$	7.01045e15	1.13581	0.567907	+	0.823093i	$0.307753\pi$
$744$	0	0
$745$	3.17394e15	0.506687
$746$	0	0
$747$	−2.13271e15	−0.335481
$748$	0	0
$749$	1.02611e16	1.59054
$750$	0	0
$751$	9.37589e15	1.43216	0.716082	−	0.698016i	$-0.245934\pi$
$751$	9.37589e15	1.43216	0.716082	+	0.698016i	$0.245934\pi$
$752$	0	0
$753$	1.74877e15	0.263245
$754$	0	0
$755$	−1.66887e15	−0.247580
$756$	0	0
$757$	1.88289e15	0.275294	0.137647	−	0.990481i	$-0.456046\pi$
$757$	1.88289e15	0.275294	0.137647	+	0.990481i	$0.456046\pi$
$758$	0	0
$759$	−4.34534e15	−0.626173
$760$	0	0
$761$	6.71695e15	0.954018	0.477009	−	0.878898i	$-0.341721\pi$
$761$	6.71695e15	0.954018	0.477009	+	0.878898i	$0.341721\pi$
$762$	0	0
$763$	6.27717e15	0.878777
$764$	0	0
$765$	4.56804e14	0.0630365
$766$	0	0
$767$	−8.34968e14	−0.113578
$768$	0	0
$769$	−5.72271e15	−0.767374	−0.383687	−	0.923463i	$-0.625346\pi$
$769$	−5.72271e15	−0.767374	−0.383687	+	0.923463i	$0.625346\pi$
$770$	0	0
$771$	1.68890e15	0.223257
$772$	0	0
$773$	−3.24936e15	−0.423458	−0.211729	−	0.977328i	$-0.567909\pi$
$773$	−3.24936e15	−0.423458	−0.211729	+	0.977328i	$0.567909\pi$
$774$	0	0
$775$	2.89369e14	0.0371786
$776$	0	0
$777$	−6.56157e14	−0.0831174
$778$	0	0
$779$	2.23461e15	0.279091
$780$	0	0
$781$	7.93865e15	0.977611
$782$	0	0
$783$	−3.64051e15	−0.442051
$784$	0	0
$785$	−2.28023e15	−0.273021
$786$	0	0
$787$	−1.29754e16	−1.53201	−0.766003	−	0.642838i	$-0.777757\pi$
$787$	−1.29754e16	−1.53201	−0.766003	+	0.642838i	$0.777757\pi$
$788$	0	0
$789$	−2.19185e14	−0.0255204
$790$	0	0
$791$	−1.87294e16	−2.15057
$792$	0	0
$793$	−2.13028e15	−0.241232
$794$	0	0
$795$	−1.81157e15	−0.202318
$796$	0	0
$797$	−1.21467e16	−1.33795	−0.668974	−	0.743286i	$-0.733266\pi$
$797$	−1.21467e16	−1.33795	−0.668974	+	0.743286i	$0.733266\pi$
$798$	0	0
$799$	−1.46130e15	−0.158757
$800$	0	0
$801$	−3.53473e15	−0.378771
$802$	0	0
$803$	−3.27724e15	−0.346395
$804$	0	0
$805$	−1.06876e16	−1.11430
$806$	0	0
$807$	2.02956e15	0.208736
$808$	0	0
$809$	3.06008e15	0.310468	0.155234	−	0.987878i	$-0.450387\pi$
$809$	3.06008e15	0.310468	0.155234	+	0.987878i	$0.450387\pi$
$810$	0	0
$811$	−1.63765e15	−0.163910	−0.0819552	−	0.996636i	$-0.526116\pi$
$811$	−1.63765e15	−0.163910	−0.0819552	+	0.996636i	$0.526116\pi$
$812$	0	0
$813$	−3.21419e15	−0.317376
$814$	0	0
$815$	6.63033e14	0.0645904
$816$	0	0
$817$	5.11941e15	0.492038
$818$	0	0
$819$	2.97018e15	0.281657
$820$	0	0
$821$	1.07044e15	0.100156	0.0500778	−	0.998745i	$-0.484053\pi$
$821$	1.07044e15	0.100156	0.0500778	+	0.998745i	$0.484053\pi$
$822$	0	0
$823$	3.24575e15	0.299651	0.149825	−	0.988712i	$-0.452129\pi$
$823$	3.24575e15	0.299651	0.149825	+	0.988712i	$0.452129\pi$
$824$	0	0
$825$	−1.06195e15	−0.0967400
$826$	0	0
$827$	2.05440e16	1.84674	0.923369	−	0.383913i	$-0.125424\pi$
$827$	2.05440e16	1.84674	0.923369	+	0.383913i	$0.125424\pi$
$828$	0	0
$829$	5.85872e15	0.519700	0.259850	−	0.965649i	$-0.416327\pi$
$829$	5.85872e15	0.519700	0.259850	+	0.965649i	$0.416327\pi$
$830$	0	0
$831$	1.31144e15	0.114800
$832$	0	0
$833$	−4.97730e15	−0.429978
$834$	0	0
$835$	7.39417e14	0.0630396
$836$	0	0
$837$	−1.40359e15	−0.118100
$838$	0	0
$839$	2.25383e16	1.87168	0.935838	−	0.352430i	$-0.114645\pi$
$839$	2.25383e16	1.87168	0.935838	+	0.352430i	$0.114645\pi$
$840$	0	0
$841$	−6.29376e15	−0.515861
$842$	0	0
$843$	−3.21043e14	−0.0259724
$844$	0	0
$845$	−5.44793e15	−0.435030
$846$	0	0
$847$	−2.59617e16	−2.04632
$848$	0	0
$849$	5.10823e15	0.397446
$850$	0	0
$851$	2.16146e15	0.166010
$852$	0	0
$853$	1.84471e16	1.39865	0.699326	−	0.714803i	$-0.253484\pi$
$853$	1.84471e16	1.39865	0.699326	+	0.714803i	$0.253484\pi$
$854$	0	0
$855$	−8.70403e15	−0.651489
$856$	0	0
$857$	−1.53957e15	−0.113764	−0.0568820	−	0.998381i	$-0.518116\pi$
$857$	−1.53957e15	−0.113764	−0.0568820	+	0.998381i	$0.518116\pi$
$858$	0	0
$859$	2.14973e16	1.56827	0.784136	−	0.620589i	$-0.213107\pi$
$859$	2.14973e16	1.56827	0.784136	+	0.620589i	$0.213107\pi$
$860$	0	0
$861$	−1.52853e15	−0.110092
$862$	0	0
$863$	1.16923e16	0.831462	0.415731	−	0.909488i	$-0.363526\pi$
$863$	1.16923e16	0.831462	0.415731	+	0.909488i	$0.363526\pi$
$864$	0	0
$865$	−2.30307e15	−0.161703
$866$	0	0
$867$	−4.73472e15	−0.328239
$868$	0	0
$869$	−1.50866e16	−1.03272
$870$	0	0
$871$	3.61974e15	0.244668
$872$	0	0
$873$	1.71156e16	1.14239
$874$	0	0
$875$	−2.61191e15	−0.172152
$876$	0	0
$877$	1.13573e16	0.739225	0.369613	−	0.929186i	$-0.379490\pi$
$877$	1.13573e16	0.739225	0.369613	+	0.929186i	$0.379490\pi$
$878$	0	0
$879$	2.31520e15	0.148816
$880$	0	0
$881$	1.12847e16	0.716346	0.358173	−	0.933655i	$-0.383400\pi$
$881$	1.12847e16	0.716346	0.358173	+	0.933655i	$0.383400\pi$
$882$	0	0
$883$	5.36286e15	0.336211	0.168106	−	0.985769i	$-0.446235\pi$
$883$	5.36286e15	0.336211	0.168106	+	0.985769i	$0.446235\pi$
$884$	0	0
$885$	−1.67371e15	−0.103632
$886$	0	0
$887$	−8.12685e15	−0.496984	−0.248492	−	0.968634i	$-0.579935\pi$
$887$	−8.12685e15	−0.496984	−0.248492	+	0.968634i	$0.579935\pi$
$888$	0	0
$889$	3.36402e16	2.03188
$890$	0	0
$891$	−1.61953e16	−0.966190
$892$	0	0
$893$	2.78439e16	1.64077
$894$	0	0
$895$	−4.22950e14	−0.0246185
$896$	0	0
$897$	1.25144e15	0.0719534
$898$	0	0
$899$	2.27733e15	0.129345
$900$	0	0
$901$	3.80670e15	0.213581
$902$	0	0
$903$	−3.50181e15	−0.194093
$904$	0	0
$905$	−4.24627e15	−0.232509
$906$	0	0
$907$	1.18191e16	0.639361	0.319680	−	0.947525i	$-0.396424\pi$
$907$	1.18191e16	0.639361	0.319680	+	0.947525i	$0.396424\pi$
$908$	0	0
$909$	−2.56815e16	−1.37252
$910$	0	0
$911$	−1.50443e16	−0.794366	−0.397183	−	0.917739i	$-0.630012\pi$
$911$	−1.50443e16	−0.794366	−0.397183	+	0.917739i	$0.630012\pi$
$912$	0	0
$913$	−1.04183e16	−0.543513
$914$	0	0
$915$	−4.27020e15	−0.220106
$916$	0	0
$917$	−2.25243e16	−1.14715
$918$	0	0
$919$	3.98208e15	0.200389	0.100195	−	0.994968i	$-0.468053\pi$
$919$	3.98208e15	0.200389	0.100195	+	0.994968i	$0.468053\pi$
$920$	0	0
$921$	7.95585e15	0.395602
$922$	0	0
$923$	−2.28629e15	−0.112337
$924$	0	0
$925$	5.28233e14	0.0256476
$926$	0	0
$927$	−1.20916e16	−0.580154
$928$	0	0
$929$	4.06546e16	1.92763	0.963814	−	0.266575i	$-0.0858919\pi$
$929$	4.06546e16	1.92763	0.963814	+	0.266575i	$0.0858919\pi$
$930$	0	0
$931$	9.48384e16	4.44387
$932$	0	0
$933$	−9.81282e15	−0.454407
$934$	0	0
$935$	2.23150e15	0.102125
$936$	0	0
$937$	−1.18458e15	−0.0535792	−0.0267896	−	0.999641i	$-0.508528\pi$
$937$	−1.18458e15	−0.0535792	−0.0267896	+	0.999641i	$0.508528\pi$
$938$	0	0
$939$	−4.58966e15	−0.205173
$940$	0	0
$941$	−1.02223e16	−0.451652	−0.225826	−	0.974168i	$-0.572508\pi$
$941$	−1.02223e16	−0.451652	−0.225826	+	0.974168i	$0.572508\pi$
$942$	0	0
$943$	5.03516e15	0.219887
$944$	0	0
$945$	1.26691e16	0.546853
$946$	0	0
$947$	1.68932e16	0.720755	0.360378	−	0.932807i	$-0.382648\pi$
$947$	1.68932e16	0.720755	0.360378	+	0.932807i	$0.382648\pi$
$948$	0	0
$949$	9.43829e14	0.0398042
$950$	0	0
$951$	−2.60824e15	−0.108731
$952$	0	0
$953$	−3.76078e16	−1.54977	−0.774884	−	0.632103i	$-0.782192\pi$
$953$	−3.76078e16	−1.54977	−0.774884	+	0.632103i	$0.782192\pi$
$954$	0	0
$955$	−3.45775e15	−0.140856
$956$	0	0
$957$	−8.35751e15	−0.336559
$958$	0	0
$959$	−4.45498e16	−1.77355
$960$	0	0
$961$	−2.45305e16	−0.965444
$962$	0	0
$963$	1.88300e16	0.732664
$964$	0	0
$965$	1.92820e16	0.741741
$966$	0	0
$967$	3.52262e16	1.33974	0.669869	−	0.742479i	$-0.266350\pi$
$967$	3.52262e16	1.33974	0.669869	+	0.742479i	$0.266350\pi$
$968$	0	0
$969$	−2.33939e15	−0.0879674
$970$	0	0
$971$	−3.56726e16	−1.32626	−0.663131	−	0.748504i	$-0.730773\pi$
$971$	−3.56726e16	−1.32626	−0.663131	+	0.748504i	$0.730773\pi$
$972$	0	0
$973$	5.84281e16	2.14784
$974$	0	0
$975$	3.05836e14	0.0111164
$976$	0	0
$977$	−4.79343e16	−1.72277	−0.861383	−	0.507957i	$-0.830401\pi$
$977$	−4.79343e16	−1.72277	−0.861383	+	0.507957i	$0.830401\pi$
$978$	0	0
$979$	−1.72672e16	−0.613646
$980$	0	0
$981$	1.15191e16	0.404798
$982$	0	0
$983$	−2.05131e16	−0.712833	−0.356416	−	0.934327i	$-0.616002\pi$
$983$	−2.05131e16	−0.712833	−0.356416	+	0.934327i	$0.616002\pi$
$984$	0	0
$985$	6.26039e15	0.215131
$986$	0	0
$987$	−1.90460e16	−0.647230
$988$	0	0
$989$	1.15354e16	0.387661
$990$	0	0
$991$	2.11314e16	0.702301	0.351150	−	0.936319i	$-0.385791\pi$
$991$	2.11314e16	0.702301	0.351150	+	0.936319i	$0.385791\pi$
$992$	0	0
$993$	−9.56179e15	−0.314281
$994$	0	0
$995$	2.41843e16	0.786151
$996$	0	0
$997$	4.57566e16	1.47106	0.735531	−	0.677491i	$-0.236933\pi$
$997$	4.57566e16	1.47106	0.735531	+	0.677491i	$0.236933\pi$
$998$	0	0
$999$	−2.56220e15	−0.0814711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.12.a.g.1.2		2
4.3	odd	2		10.12.a.d.1.1	✓	2
12.11	even	2		90.12.a.l.1.2		2
20.3	even	4		50.12.b.f.49.1		4
20.7	even	4		50.12.b.f.49.4		4
20.19	odd	2		50.12.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.12.a.d.1.1	✓	2	4.3	odd	2
50.12.a.f.1.2		2	20.19	odd	2
50.12.b.f.49.1		4	20.3	even	4
50.12.b.f.49.4		4	20.7	even	4
80.12.a.g.1.2		2	1.1	even	1	trivial
90.12.a.l.1.2		2	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(f(q)\)	\(=\)	\(q+141.734 q^{3} +3125.00 q^{5} -85586.9 q^{7} -157058. q^{9} -767235. q^{11} +220960. q^{13} +442919. q^{15} -930719. q^{17} +1.77341e7 q^{19} -1.21306e7 q^{21} +3.99596e7 q^{23} +9.76562e6 q^{25} -4.73683e7 q^{27} +7.68554e7 q^{29} +2.96314e7 q^{31} -1.08743e8 q^{33} -2.67459e8 q^{35} +5.40911e7 q^{37} +3.13176e7 q^{39} +1.26006e8 q^{41} +2.88676e8 q^{43} -4.90808e8 q^{45} +1.57008e9 q^{47} +5.34780e9 q^{49} -1.31915e8 q^{51} -4.09006e9 q^{53} -2.39761e9 q^{55} +2.51353e9 q^{57} -3.77882e9 q^{59} -9.64103e9 q^{61} +1.34422e10 q^{63} +6.90500e8 q^{65} +1.63819e10 q^{67} +5.66364e9 q^{69} -1.03471e10 q^{71} +4.27149e9 q^{73} +1.38412e9 q^{75} +6.56653e10 q^{77} +1.96636e10 q^{79} +2.11087e10 q^{81} +1.35791e10 q^{83} -2.90850e9 q^{85} +1.08930e10 q^{87} +2.25058e10 q^{89} -1.89113e10 q^{91} +4.19978e9 q^{93} +5.54191e10 q^{95} -1.08976e11 q^{97} +1.20501e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 604 q^{3} + 6250 q^{5} - 14092 q^{7} + 221914 q^{9} - 421584 q^{11} + 1730524 q^{13} - 1887500 q^{15} - 6323628 q^{17} + 28897400 q^{19} - 65446816 q^{21} + 45236076 q^{23} + 19531250 q^{25} - 197876440 q^{27}+ \cdots + 251492724912 q^{99}+O(q^{100}) \) 2 * q - 604 * q^3 + 6250 * q^5 - 14092 * q^7 + 221914 * q^9 - 421584 * q^11 + 1730524 * q^13 - 1887500 * q^15 - 6323628 * q^17 + 28897400 * q^19 - 65446816 * q^21 + 45236076 * q^23 + 19531250 * q^25 - 197876440 * q^27 + 58226220 * q^29 - 41413384 * q^31 - 366506832 * q^33 - 44037500 * q^35 + 377255452 * q^37 - 1094415848 * q^39 - 785271036 * q^41 + 1452987236 * q^43 + 693481250 * q^45 + 1288127748 * q^47 + 8481998946 * q^49 + 3889762056 * q^51 - 30490836 * q^53 - 1317450000 * q^55 - 5811319600 * q^57 - 8677102440 * q^59 + 1115498764 * q^61 + 40536764356 * q^63 + 5407887500 * q^65 + 12673769708 * q^67 + 1728802848 * q^69 - 13799832984 * q^71 - 17842079516 * q^73 - 5898437500 * q^75 + 90377541264 * q^77 + 12636930320 * q^79 + 66213933922 * q^81 - 41986488924 * q^83 - 19761337500 * q^85 + 24785411160 * q^87 + 13208740020 * q^89 + 89014903096 * q^91 + 57180301568 * q^93 + 90304375000 * q^95 - 61787462828 * q^97 + 251492724912 * q^99

Introduction

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