LMFDB - Embedded newform 80.12.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,12,Mod(1,80)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	80.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:	$61.4674544448$
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			1.1
Root			$22.6867$ 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)

$f(q)$	$=$	$q-745.734 q^{3} +3125.00 q^{5} +71494.9 q^{7} +378972. q^{9} +345651. q^{11} +1.50956e6 q^{13} -2.33042e6 q^{15} -5.39291e6 q^{17} +1.11633e7 q^{19} -5.33162e7 q^{21} +5.27646e6 q^{23} +9.76562e6 q^{25} -1.50508e8 q^{27} -1.86291e7 q^{29} -7.10448e7 q^{31} -2.57763e8 q^{33} +2.23422e8 q^{35} +3.23164e8 q^{37} -1.12573e9 q^{39} -9.11277e8 q^{41} +1.16431e9 q^{43} +1.18429e9 q^{45} -2.81949e8 q^{47} +3.13420e9 q^{49} +4.02168e9 q^{51} +4.05957e9 q^{53} +1.08016e9 q^{55} -8.32485e9 q^{57} -4.89828e9 q^{59} +1.07565e10 q^{61} +2.70946e10 q^{63} +4.71739e9 q^{65} -3.70812e9 q^{67} -3.93484e9 q^{69} -3.45274e9 q^{71} -2.21136e10 q^{73} -7.28256e9 q^{75} +2.47123e10 q^{77} -7.02672e9 q^{79} +4.51052e10 q^{81} -5.55656e10 q^{83} -1.68528e10 q^{85} +1.38924e10 q^{87} -9.29706e9 q^{89} +1.07926e11 q^{91} +5.29805e10 q^{93} +3.48853e10 q^{95} +4.71888e10 q^{97} +1.30992e11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−745.734	−1.77181	−0.885905	−	0.463867i	$-0.846462\pi$
$3$	−745.734	−1.77181	−0.885905	+	0.463867i	$0.846462\pi$
$4$	0	0
$5$	3125.00	0.447214
$5$	3125.00	0.447214
$6$	0	0
$7$	71494.9	1.60782	0.803908	−	0.594754i	$-0.202751\pi$
$7$	71494.9	1.60782	0.803908	+	0.594754i	$0.202751\pi$
$8$	0	0
$9$	378972.	2.13931
$10$	0	0
$11$	345651.	0.647109	0.323555	−	0.946209i	$-0.395122\pi$
$11$	345651.	0.647109	0.323555	+	0.946209i	$0.395122\pi$
$12$	0	0
$13$	1.50956e6	1.12762	0.563810	−	0.825904i	$-0.309335\pi$
$13$	1.50956e6	1.12762	0.563810	+	0.825904i	$0.309335\pi$
$14$	0	0
$15$	−2.33042e6	−0.792377
$16$	0	0
$17$	−5.39291e6	−0.921200	−0.460600	−	0.887608i	$-0.652366\pi$
$17$	−5.39291e6	−0.921200	−0.460600	+	0.887608i	$0.652366\pi$
$18$	0	0
$19$	1.11633e7	1.03430	0.517151	−	0.855894i	$-0.326993\pi$
$19$	1.11633e7	1.03430	0.517151	+	0.855894i	$0.326993\pi$
$20$	0	0
$21$	−5.33162e7	−2.84874
$22$	0	0
$23$	5.27646e6	0.170938	0.0854692	−	0.996341i	$-0.472761\pi$
$23$	5.27646e6	0.170938	0.0854692	+	0.996341i	$0.472761\pi$
$24$	0	0
$25$	9.76562e6	0.200000
$26$	0	0
$27$	−1.50508e8	−2.01864
$28$	0	0
$29$	−1.86291e7	−0.168657	−0.0843284	−	0.996438i	$-0.526874\pi$
$29$	−1.86291e7	−0.168657	−0.0843284	+	0.996438i	$0.526874\pi$
$30$	0	0
$31$	−7.10448e7	−0.445700	−0.222850	−	0.974853i	$-0.571536\pi$
$31$	−7.10448e7	−0.445700	−0.222850	+	0.974853i	$0.571536\pi$
$32$	0	0
$33$	−2.57763e8	−1.14655
$34$	0	0
$35$	2.23422e8	0.719037
$36$	0	0
$37$	3.23164e8	0.766150	0.383075	−	0.923717i	$-0.374865\pi$
$37$	3.23164e8	0.766150	0.383075	+	0.923717i	$0.374865\pi$
$38$	0	0
$39$	−1.12573e9	−1.99793
$40$	0	0
$41$	−9.11277e8	−1.22840	−0.614199	−	0.789151i	$-0.710521\pi$
$41$	−9.11277e8	−1.22840	−0.614199	+	0.789151i	$0.710521\pi$
$42$	0	0
$43$	1.16431e9	1.20779	0.603897	−	0.797063i	$-0.293614\pi$
$43$	1.16431e9	1.20779	0.603897	+	0.797063i	$0.293614\pi$
$44$	0	0
$45$	1.18429e9	0.956729
$46$	0	0
$47$	−2.81949e8	−0.179321	−0.0896606	−	0.995972i	$-0.528578\pi$
$47$	−2.81949e8	−0.179321	−0.0896606	+	0.995972i	$0.528578\pi$
$48$	0	0
$49$	3.13420e9	1.58507
$50$	0	0
$51$	4.02168e9	1.63219
$52$	0	0
$53$	4.05957e9	1.33341	0.666704	−	0.745323i	$-0.267705\pi$
$53$	4.05957e9	1.33341	0.666704	+	0.745323i	$0.267705\pi$
$54$	0	0
$55$	1.08016e9	0.289396
$56$	0	0
$57$	−8.32485e9	−1.83259
$58$	0	0
$59$	−4.89828e9	−0.891986	−0.445993	−	0.895036i	$-0.647149\pi$
$59$	−4.89828e9	−0.891986	−0.445993	+	0.895036i	$0.647149\pi$
$60$	0	0
$61$	1.07565e10	1.63064	0.815320	−	0.579011i	$-0.196561\pi$
$61$	1.07565e10	1.63064	0.815320	+	0.579011i	$0.196561\pi$
$62$	0	0
$63$	2.70946e10	3.43962
$64$	0	0
$65$	4.71739e9	0.504287
$66$	0	0
$67$	−3.70812e9	−0.335539	−0.167769	−	0.985826i	$-0.553656\pi$
$67$	−3.70812e9	−0.335539	−0.167769	+	0.985826i	$0.553656\pi$
$68$	0	0
$69$	−3.93484e9	−0.302870
$70$	0	0
$71$	−3.45274e9	−0.227113	−0.113557	−	0.993532i	$-0.536224\pi$
$71$	−3.45274e9	−0.227113	−0.113557	+	0.993532i	$0.536224\pi$
$72$	0	0
$73$	−2.21136e10	−1.24848	−0.624242	−	0.781231i	$-0.714592\pi$
$73$	−2.21136e10	−1.24848	−0.624242	+	0.781231i	$0.714592\pi$
$74$	0	0
$75$	−7.28256e9	−0.354362
$76$	0	0
$77$	2.47123e10	1.04043
$78$	0	0
$79$	−7.02672e9	−0.256923	−0.128462	−	0.991714i	$-0.541004\pi$
$79$	−7.02672e9	−0.256923	−0.128462	+	0.991714i	$0.541004\pi$
$80$	0	0
$81$	4.51052e10	1.43734
$82$	0	0
$83$	−5.55656e10	−1.54838	−0.774188	−	0.632956i	$-0.781841\pi$
$83$	−5.55656e10	−1.54838	−0.774188	+	0.632956i	$0.781841\pi$
$84$	0	0
$85$	−1.68528e10	−0.411973
$86$	0	0
$87$	1.38924e10	0.298828
$88$	0	0
$89$	−9.29706e9	−0.176482	−0.0882410	−	0.996099i	$-0.528125\pi$
$89$	−9.29706e9	−0.176482	−0.0882410	+	0.996099i	$0.528125\pi$
$90$	0	0
$91$	1.07926e11	1.81301
$92$	0	0
$93$	5.29805e10	0.789696
$94$	0	0
$95$	3.48853e10	0.462554
$96$	0	0
$97$	4.71888e10	0.557949	0.278975	−	0.960298i	$-0.410005\pi$
$97$	4.71888e10	0.557949	0.278975	+	0.960298i	$0.410005\pi$
$98$	0	0
$99$	1.30992e11	1.38437
$100$	0	0
$101$	1.61228e11	1.52642	0.763209	−	0.646151i	$-0.223622\pi$
$101$	1.61228e11	1.52642	0.763209	+	0.646151i	$0.223622\pi$
$102$	0	0
$103$	1.85579e11	1.57733	0.788666	−	0.614822i	$-0.210772\pi$
$103$	1.85579e11	1.57733	0.788666	+	0.614822i	$0.210772\pi$
$104$	0	0
$105$	−1.66613e11	−1.27400
$106$	0	0
$107$	5.45807e10	0.376208	0.188104	−	0.982149i	$-0.439766\pi$
$107$	5.45807e10	0.376208	0.188104	+	0.982149i	$0.439766\pi$
$108$	0	0
$109$	−3.54035e10	−0.220394	−0.110197	−	0.993910i	$-0.535148\pi$
$109$	−3.54035e10	−0.220394	−0.110197	+	0.993910i	$0.535148\pi$
$110$	0	0
$111$	−2.40995e11	−1.35747
$112$	0	0
$113$	1.20314e11	0.614308	0.307154	−	0.951660i	$-0.400623\pi$
$113$	1.20314e11	0.614308	0.307154	+	0.951660i	$0.400623\pi$
$114$	0	0
$115$	1.64889e10	0.0764460
$116$	0	0
$117$	5.72083e11	2.41233
$118$	0	0
$119$	−3.85566e11	−1.48112
$120$	0	0
$121$	−1.65837e11	−0.581250
$122$	0	0
$123$	6.79571e11	2.17649
$124$	0	0
$125$	3.05176e10	0.0894427
$126$	0	0
$127$	5.60687e11	1.50591	0.752957	−	0.658070i	$-0.228627\pi$
$127$	5.60687e11	1.50591	0.752957	+	0.658070i	$0.228627\pi$
$128$	0	0
$129$	−8.68267e11	−2.13998
$130$	0	0
$131$	−4.11999e11	−0.933048	−0.466524	−	0.884508i	$-0.654494\pi$
$131$	−4.11999e11	−0.933048	−0.466524	+	0.884508i	$0.654494\pi$
$132$	0	0
$133$	7.98119e11	1.66297
$134$	0	0
$135$	−4.70338e11	−0.902764
$136$	0	0
$137$	−7.86259e11	−1.39188	−0.695941	−	0.718099i	$-0.745013\pi$
$137$	−7.86259e11	−1.39188	−0.695941	+	0.718099i	$0.745013\pi$
$138$	0	0
$139$	−8.57152e11	−1.40112	−0.700562	−	0.713592i	$-0.747067\pi$
$139$	−8.57152e11	−1.40112	−0.700562	+	0.713592i	$0.747067\pi$
$140$	0	0
$141$	2.10259e11	0.317723
$142$	0	0
$143$	5.21782e11	0.729694
$144$	0	0
$145$	−5.82160e10	−0.0754256
$146$	0	0
$147$	−2.33728e12	−2.80844
$148$	0	0
$149$	−2.38606e11	−0.266168	−0.133084	−	0.991105i	$-0.542488\pi$
$149$	−2.38606e11	−0.266168	−0.133084	+	0.991105i	$0.542488\pi$
$150$	0	0
$151$	9.57309e10	0.0992382	0.0496191	−	0.998768i	$-0.484199\pi$
$151$	9.57309e10	0.0992382	0.0496191	+	0.998768i	$0.484199\pi$
$152$	0	0
$153$	−2.04376e12	−1.97073
$154$	0	0
$155$	−2.22015e11	−0.199323
$156$	0	0
$157$	−1.79935e12	−1.50545	−0.752727	−	0.658333i	$-0.771262\pi$
$157$	−1.79935e12	−1.50545	−0.752727	+	0.658333i	$0.771262\pi$
$158$	0	0
$159$	−3.02736e12	−2.36254
$160$	0	0
$161$	3.77240e11	0.274837
$162$	0	0
$163$	2.50619e12	1.70601	0.853005	−	0.521903i	$-0.174778\pi$
$163$	2.50619e12	1.70601	0.853005	+	0.521903i	$0.174778\pi$
$164$	0	0
$165$	−8.05511e11	−0.512755
$166$	0	0
$167$	2.69855e12	1.60764	0.803822	−	0.594871i	$-0.202797\pi$
$167$	2.69855e12	1.60764	0.803822	+	0.594871i	$0.202797\pi$
$168$	0	0
$169$	4.86623e11	0.271529
$170$	0	0
$171$	4.23058e12	2.21269
$172$	0	0
$173$	3.07796e12	1.51011	0.755057	−	0.655659i	$-0.227609\pi$
$173$	3.07796e12	1.51011	0.755057	+	0.655659i	$0.227609\pi$
$174$	0	0
$175$	6.98193e11	0.321563
$176$	0	0
$177$	3.65282e12	1.58043
$178$	0	0
$179$	−3.51154e11	−0.142826	−0.0714128	−	0.997447i	$-0.522751\pi$
$179$	−3.51154e11	−0.142826	−0.0714128	+	0.997447i	$0.522751\pi$
$180$	0	0
$181$	6.34106e11	0.242622	0.121311	−	0.992615i	$-0.461290\pi$
$181$	6.34106e11	0.242622	0.121311	+	0.992615i	$0.461290\pi$
$182$	0	0
$183$	−8.02151e12	−2.88918
$184$	0	0
$185$	1.00989e12	0.342633
$186$	0	0
$187$	−1.86406e12	−0.596117
$188$	0	0
$189$	−1.07606e13	−3.24560
$190$	0	0
$191$	2.61584e12	0.744609	0.372304	−	0.928111i	$-0.378568\pi$
$191$	2.61584e12	0.744609	0.372304	+	0.928111i	$0.378568\pi$
$192$	0	0
$193$	1.59819e12	0.429598	0.214799	−	0.976658i	$-0.431090\pi$
$193$	1.59819e12	0.429598	0.214799	+	0.976658i	$0.431090\pi$
$194$	0	0
$195$	−3.51792e12	−0.893501
$196$	0	0
$197$	−6.16281e10	−0.0147984	−0.00739920	−	0.999973i	$-0.502355\pi$
$197$	−6.16281e10	−0.0147984	−0.00739920	+	0.999973i	$0.502355\pi$
$198$	0	0
$199$	−2.36544e12	−0.537303	−0.268652	−	0.963237i	$-0.586578\pi$
$199$	−2.36544e12	−0.537303	−0.268652	+	0.963237i	$0.586578\pi$
$200$	0	0
$201$	2.76527e12	0.594510
$202$	0	0
$203$	−1.33189e12	−0.271169
$204$	0	0
$205$	−2.84774e12	−0.549357
$206$	0	0
$207$	1.99963e12	0.365690
$208$	0	0
$209$	3.85860e12	0.669307
$210$	0	0
$211$	−5.39734e12	−0.888436	−0.444218	−	0.895919i	$-0.646518\pi$
$211$	−5.39734e12	−0.888436	−0.444218	+	0.895919i	$0.646518\pi$
$212$	0	0
$213$	2.57482e12	0.402402
$214$	0	0
$215$	3.63847e12	0.540142
$216$	0	0
$217$	−5.07934e12	−0.716604
$218$	0	0
$219$	1.64908e13	2.21208
$220$	0	0
$221$	−8.14094e12	−1.03876
$222$	0	0
$223$	9.37124e12	1.13794	0.568971	−	0.822357i	$-0.307342\pi$
$223$	9.37124e12	1.13794	0.568971	+	0.822357i	$0.307342\pi$
$224$	0	0
$225$	3.70090e12	0.427862
$226$	0	0
$227$	6.51561e12	0.717485	0.358742	−	0.933437i	$-0.383206\pi$
$227$	6.51561e12	0.717485	0.358742	+	0.933437i	$0.383206\pi$
$228$	0	0
$229$	−1.74808e12	−0.183429	−0.0917144	−	0.995785i	$-0.529235\pi$
$229$	−1.74808e12	−0.183429	−0.0917144	+	0.995785i	$0.529235\pi$
$230$	0	0
$231$	−1.84288e13	−1.84345
$232$	0	0
$233$	8.64493e12	0.824715	0.412358	−	0.911022i	$-0.364705\pi$
$233$	8.64493e12	0.824715	0.412358	+	0.911022i	$0.364705\pi$
$234$	0	0
$235$	−8.81089e11	−0.0801949
$236$	0	0
$237$	5.24006e12	0.455219
$238$	0	0
$239$	−6.71025e12	−0.556609	−0.278304	−	0.960493i	$-0.589772\pi$
$239$	−6.71025e12	−0.556609	−0.278304	+	0.960493i	$0.589772\pi$
$240$	0	0
$241$	−1.20329e13	−0.953404	−0.476702	−	0.879065i	$-0.658168\pi$
$241$	−1.20329e13	−0.953404	−0.476702	+	0.879065i	$0.658168\pi$
$242$	0	0
$243$	−6.97444e12	−0.528050
$244$	0	0
$245$	9.79438e12	0.708865
$246$	0	0
$247$	1.68517e13	1.16630
$248$	0	0
$249$	4.14371e13	2.74343
$250$	0	0
$251$	2.23212e13	1.41420	0.707101	−	0.707113i	$-0.250003\pi$
$251$	2.23212e13	1.41420	0.707101	+	0.707113i	$0.250003\pi$
$252$	0	0
$253$	1.82381e12	0.110616
$254$	0	0
$255$	1.25677e13	0.729938
$256$	0	0
$257$	1.80055e13	1.00178	0.500890	−	0.865511i	$-0.333006\pi$
$257$	1.80055e13	1.00178	0.500890	+	0.865511i	$0.333006\pi$
$258$	0	0
$259$	2.31046e13	1.23183
$260$	0	0
$261$	−7.05993e12	−0.360809
$262$	0	0
$263$	−1.38894e13	−0.680653	−0.340326	−	0.940307i	$-0.610538\pi$
$263$	−1.38894e13	−0.680653	−0.340326	+	0.940307i	$0.610538\pi$
$264$	0	0
$265$	1.26862e13	0.596318
$266$	0	0
$267$	6.93313e12	0.312693
$268$	0	0
$269$	2.71611e13	1.17574	0.587868	−	0.808957i	$-0.299967\pi$
$269$	2.71611e13	1.17574	0.587868	+	0.808957i	$0.299967\pi$
$270$	0	0
$271$	7.24542e12	0.301115	0.150558	−	0.988601i	$-0.451893\pi$
$271$	7.24542e12	0.301115	0.150558	+	0.988601i	$0.451893\pi$
$272$	0	0
$273$	−8.04842e13	−3.21230
$274$	0	0
$275$	3.37549e12	0.129422
$276$	0	0
$277$	3.97300e13	1.46380	0.731898	−	0.681415i	$-0.238635\pi$
$277$	3.97300e13	1.46380	0.731898	+	0.681415i	$0.238635\pi$
$278$	0	0
$279$	−2.69240e13	−0.953491
$280$	0	0
$281$	5.95255e12	0.202683	0.101342	−	0.994852i	$-0.467686\pi$
$281$	5.95255e12	0.202683	0.101342	+	0.994852i	$0.467686\pi$
$282$	0	0
$283$	3.85782e13	1.26333	0.631665	−	0.775241i	$-0.282372\pi$
$283$	3.85782e13	1.26333	0.631665	+	0.775241i	$0.282372\pi$
$284$	0	0
$285$	−2.60152e13	−0.819558
$286$	0	0
$287$	−6.51517e13	−1.97504
$288$	0	0
$289$	−5.18843e12	−0.151390
$290$	0	0
$291$	−3.51903e13	−0.988580
$292$	0	0
$293$	−5.32529e13	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$293$	−5.32529e13	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$294$	0	0
$295$	−1.53071e13	−0.398908
$296$	0	0
$297$	−5.20232e13	−1.30628
$298$	0	0
$299$	7.96516e12	0.192754
$300$	0	0
$301$	8.32424e13	1.94191
$302$	0	0
$303$	−1.20233e14	−2.70452
$304$	0	0
$305$	3.36142e13	0.729244
$306$	0	0
$307$	5.74336e13	1.20200	0.601001	−	0.799249i	$-0.294769\pi$
$307$	5.74336e13	1.20200	0.601001	+	0.799249i	$0.294769\pi$
$308$	0	0
$309$	−1.38392e14	−2.79473
$310$	0	0
$311$	2.58616e13	0.504050	0.252025	−	0.967721i	$-0.418903\pi$
$311$	2.58616e13	0.504050	0.252025	+	0.967721i	$0.418903\pi$
$312$	0	0
$313$	−3.57307e13	−0.672277	−0.336138	−	0.941813i	$-0.609121\pi$
$313$	−3.57307e13	−0.672277	−0.336138	+	0.941813i	$0.609121\pi$
$314$	0	0
$315$	8.46707e13	1.53824
$316$	0	0
$317$	−1.78631e13	−0.313422	−0.156711	−	0.987644i	$-0.550089\pi$
$317$	−1.78631e13	−0.313422	−0.156711	+	0.987644i	$0.550089\pi$
$318$	0	0
$319$	−6.43917e12	−0.109139
$320$	0	0
$321$	−4.07027e13	−0.666570
$322$	0	0
$323$	−6.02026e13	−0.952800
$324$	0	0
$325$	1.47418e13	0.225524
$326$	0	0
$327$	2.64016e13	0.390496
$328$	0	0
$329$	−2.01579e13	−0.288315
$330$	0	0
$331$	−7.78278e13	−1.07667	−0.538333	−	0.842732i	$-0.680946\pi$
$331$	−7.78278e13	−1.07667	−0.538333	+	0.842732i	$0.680946\pi$
$332$	0	0
$333$	1.22470e14	1.63903
$334$	0	0
$335$	−1.15879e13	−0.150057
$336$	0	0
$337$	4.95598e12	0.0621105	0.0310552	−	0.999518i	$-0.490113\pi$
$337$	4.95598e12	0.0621105	0.0310552	+	0.999518i	$0.490113\pi$
$338$	0	0
$339$	−8.97226e13	−1.08844
$340$	0	0
$341$	−2.45567e13	−0.288417
$342$	0	0
$343$	8.27106e13	0.940684
$344$	0	0
$345$	−1.22964e13	−0.135448
$346$	0	0
$347$	6.55706e13	0.699676	0.349838	−	0.936810i	$-0.386237\pi$
$347$	6.55706e13	0.699676	0.349838	+	0.936810i	$0.386237\pi$
$348$	0	0
$349$	1.14750e14	1.18635	0.593177	−	0.805072i	$-0.297873\pi$
$349$	1.14750e14	1.18635	0.593177	+	0.805072i	$0.297873\pi$
$350$	0	0
$351$	−2.27202e14	−2.27626
$352$	0	0
$353$	−5.10272e13	−0.495497	−0.247749	−	0.968824i	$-0.579691\pi$
$353$	−5.10272e13	−0.495497	−0.247749	+	0.968824i	$0.579691\pi$
$354$	0	0
$355$	−1.07898e13	−0.101568
$356$	0	0
$357$	2.87530e14	2.62426
$358$	0	0
$359$	1.16657e14	1.03251	0.516253	−	0.856436i	$-0.327326\pi$
$359$	1.16657e14	1.03251	0.516253	+	0.856436i	$0.327326\pi$
$360$	0	0
$361$	8.12884e12	0.0697813
$362$	0	0
$363$	1.23671e14	1.02986
$364$	0	0
$365$	−6.91049e13	−0.558339
$366$	0	0
$367$	−1.44058e14	−1.12947	−0.564733	−	0.825274i	$-0.691021\pi$
$367$	−1.44058e14	−1.12947	−0.564733	+	0.825274i	$0.691021\pi$
$368$	0	0
$369$	−3.45349e14	−2.62793
$370$	0	0
$371$	2.90239e14	2.14387
$372$	0	0
$373$	6.90594e13	0.495250	0.247625	−	0.968856i	$-0.420350\pi$
$373$	6.90594e13	0.495250	0.247625	+	0.968856i	$0.420350\pi$
$374$	0	0
$375$	−2.27580e13	−0.158475
$376$	0	0
$377$	−2.81219e13	−0.190181
$378$	0	0
$379$	−2.28579e14	−1.50149	−0.750743	−	0.660595i	$-0.770304\pi$
$379$	−2.28579e14	−1.50149	−0.750743	+	0.660595i	$0.770304\pi$
$380$	0	0
$381$	−4.18123e14	−2.66819
$382$	0	0
$383$	7.16725e13	0.444385	0.222192	−	0.975003i	$-0.428679\pi$
$383$	7.16725e13	0.444385	0.222192	+	0.975003i	$0.428679\pi$
$384$	0	0
$385$	7.72259e13	0.465295
$386$	0	0
$387$	4.41242e14	2.58385
$388$	0	0
$389$	5.31179e13	0.302356	0.151178	−	0.988507i	$-0.451693\pi$
$389$	5.31179e13	0.302356	0.151178	+	0.988507i	$0.451693\pi$
$390$	0	0
$391$	−2.84555e13	−0.157469
$392$	0	0
$393$	3.07242e14	1.65318
$394$	0	0
$395$	−2.19585e13	−0.114900
$396$	0	0
$397$	9.39721e12	0.0478246	0.0239123	−	0.999714i	$-0.492388\pi$
$397$	9.39721e12	0.0478246	0.0239123	+	0.999714i	$0.492388\pi$
$398$	0	0
$399$	−5.95185e14	−2.94646
$400$	0	0
$401$	2.20216e14	1.06061	0.530303	−	0.847808i	$-0.322078\pi$
$401$	2.20216e14	1.06061	0.530303	+	0.847808i	$0.322078\pi$
$402$	0	0
$403$	−1.07247e14	−0.502581
$404$	0	0
$405$	1.40954e14	0.642797
$406$	0	0
$407$	1.11702e14	0.495783
$408$	0	0
$409$	−3.05103e14	−1.31816	−0.659080	−	0.752072i	$-0.729054\pi$
$409$	−3.05103e14	−1.31816	−0.659080	+	0.752072i	$0.729054\pi$
$410$	0	0
$411$	5.86340e14	2.46615
$412$	0	0
$413$	−3.50203e14	−1.43415
$414$	0	0
$415$	−1.73642e14	−0.692455
$416$	0	0
$417$	6.39208e14	2.48253
$418$	0	0
$419$	−4.21440e14	−1.59426	−0.797130	−	0.603808i	$-0.793649\pi$
$419$	−4.21440e14	−1.59426	−0.797130	+	0.603808i	$0.793649\pi$
$420$	0	0
$421$	1.11120e12	0.00409487	0.00204744	−	0.999998i	$-0.499348\pi$
$421$	1.11120e12	0.00409487	0.00204744	+	0.999998i	$0.499348\pi$
$422$	0	0
$423$	−1.06851e14	−0.383624
$424$	0	0
$425$	−5.26651e13	−0.184240
$426$	0	0
$427$	7.69037e14	2.62177
$428$	0	0
$429$	−3.89110e14	−1.29288
$430$	0	0
$431$	−2.29204e14	−0.742332	−0.371166	−	0.928567i	$-0.621042\pi$
$431$	−2.29204e14	−0.742332	−0.371166	+	0.928567i	$0.621042\pi$
$432$	0	0
$433$	1.54032e14	0.486325	0.243162	−	0.969986i	$-0.421815\pi$
$433$	1.54032e14	0.486325	0.243162	+	0.969986i	$0.421815\pi$
$434$	0	0
$435$	4.34137e13	0.133640
$436$	0	0
$437$	5.89027e13	0.176802
$438$	0	0
$439$	1.25797e14	0.368227	0.184114	−	0.982905i	$-0.441059\pi$
$439$	1.25797e14	0.368227	0.184114	+	0.982905i	$0.441059\pi$
$440$	0	0
$441$	1.18778e15	3.39096
$442$	0	0
$443$	1.34623e14	0.374885	0.187443	−	0.982276i	$-0.439980\pi$
$443$	1.34623e14	0.374885	0.187443	+	0.982276i	$0.439980\pi$
$444$	0	0
$445$	−2.90533e13	−0.0789252
$446$	0	0
$447$	1.77937e14	0.471600
$448$	0	0
$449$	4.28972e14	1.10937	0.554683	−	0.832062i	$-0.312840\pi$
$449$	4.28972e14	1.10937	0.554683	+	0.832062i	$0.312840\pi$
$450$	0	0
$451$	−3.14984e14	−0.794908
$452$	0	0
$453$	−7.13898e13	−0.175831
$454$	0	0
$455$	3.37269e14	0.810801
$456$	0	0
$457$	−7.89196e14	−1.85202	−0.926011	−	0.377497i	$-0.876785\pi$
$457$	−7.89196e14	−1.85202	−0.926011	+	0.377497i	$0.876785\pi$
$458$	0	0
$459$	8.11677e14	1.85957
$460$	0	0
$461$	−6.01739e14	−1.34602	−0.673012	−	0.739632i	$-0.735000\pi$
$461$	−6.01739e14	−1.34602	−0.673012	+	0.739632i	$0.735000\pi$
$462$	0	0
$463$	−2.36873e14	−0.517393	−0.258696	−	0.965959i	$-0.583293\pi$
$463$	−2.36873e14	−0.517393	−0.258696	+	0.965959i	$0.583293\pi$
$464$	0	0
$465$	1.65564e14	0.353163
$466$	0	0
$467$	−8.17060e14	−1.70220	−0.851101	−	0.525002i	$-0.824064\pi$
$467$	−8.17060e14	−1.70220	−0.851101	+	0.525002i	$0.824064\pi$
$468$	0	0
$469$	−2.65112e14	−0.539484
$470$	0	0
$471$	1.34184e15	2.66738
$472$	0	0
$473$	4.02445e14	0.781574
$474$	0	0
$475$	1.09017e14	0.206860
$476$	0	0
$477$	1.53847e15	2.85257
$478$	0	0
$479$	−2.80917e14	−0.509018	−0.254509	−	0.967070i	$-0.581914\pi$
$479$	−2.80917e14	−0.509018	−0.254509	+	0.967070i	$0.581914\pi$
$480$	0	0
$481$	4.87837e14	0.863927
$482$	0	0
$483$	−2.81321e14	−0.486960
$484$	0	0
$485$	1.47465e14	0.249522
$486$	0	0
$487$	4.06414e14	0.672294	0.336147	−	0.941810i	$-0.390876\pi$
$487$	4.06414e14	0.672294	0.336147	+	0.941810i	$0.390876\pi$
$488$	0	0
$489$	−1.86895e15	−3.02273
$490$	0	0
$491$	−1.02479e15	−1.62065	−0.810324	−	0.585982i	$-0.800709\pi$
$491$	−1.02479e15	−1.62065	−0.810324	+	0.585982i	$0.800709\pi$
$492$	0	0
$493$	1.00465e14	0.155367
$494$	0	0
$495$	4.09350e14	0.619108
$496$	0	0
$497$	−2.46853e14	−0.365156
$498$	0	0
$499$	−1.01593e14	−0.146997	−0.0734987	−	0.997295i	$-0.523416\pi$
$499$	−1.01593e14	−0.146997	−0.0734987	+	0.997295i	$0.523416\pi$
$500$	0	0
$501$	−2.01240e15	−2.84844
$502$	0	0
$503$	−9.67924e14	−1.34035	−0.670174	−	0.742204i	$-0.733780\pi$
$503$	−9.67924e14	−1.34035	−0.670174	+	0.742204i	$0.733780\pi$
$504$	0	0
$505$	5.03838e14	0.682635
$506$	0	0
$507$	−3.62891e14	−0.481097
$508$	0	0
$509$	9.40088e14	1.21961	0.609805	−	0.792552i	$-0.291248\pi$
$509$	9.40088e14	1.21961	0.609805	+	0.792552i	$0.291248\pi$
$510$	0	0
$511$	−1.58101e15	−2.00733
$512$	0	0
$513$	−1.68017e15	−2.08789
$514$	0	0
$515$	5.79933e14	0.705404
$516$	0	0
$517$	−9.74557e13	−0.116040
$518$	0	0
$519$	−2.29534e15	−2.67564
$520$	0	0
$521$	−1.00574e15	−1.14783	−0.573916	−	0.818914i	$-0.694576\pi$
$521$	−1.00574e15	−1.14783	−0.573916	+	0.818914i	$0.694576\pi$
$522$	0	0
$523$	1.24911e15	1.39586	0.697931	−	0.716165i	$-0.254104\pi$
$523$	1.24911e15	1.39586	0.697931	+	0.716165i	$0.254104\pi$
$524$	0	0
$525$	−5.20666e14	−0.569749
$526$	0	0
$527$	3.83138e14	0.410579
$528$	0	0
$529$	−9.24969e14	−0.970780
$530$	0	0
$531$	−1.85631e15	−1.90823
$532$	0	0
$533$	−1.37563e15	−1.38517
$534$	0	0
$535$	1.70565e14	0.168246
$536$	0	0
$537$	2.61868e14	0.253060
$538$	0	0
$539$	1.08334e15	1.02571
$540$	0	0
$541$	1.47684e15	1.37009	0.685046	−	0.728500i	$-0.259782\pi$
$541$	1.47684e15	1.37009	0.685046	+	0.728500i	$0.259782\pi$
$542$	0	0
$543$	−4.72874e14	−0.429880
$544$	0	0
$545$	−1.10636e14	−0.0985632
$546$	0	0
$547$	2.32000e14	0.202562	0.101281	−	0.994858i	$-0.467706\pi$
$547$	2.32000e14	0.202562	0.101281	+	0.994858i	$0.467706\pi$
$548$	0	0
$549$	4.07643e15	3.48844
$550$	0	0
$551$	−2.07962e14	−0.174442
$552$	0	0
$553$	−5.02375e14	−0.413085
$554$	0	0
$555$	−7.53108e14	−0.607080
$556$	0	0
$557$	2.34238e15	1.85120	0.925602	−	0.378498i	$-0.123559\pi$
$557$	2.34238e15	1.85120	0.925602	+	0.378498i	$0.123559\pi$
$558$	0	0
$559$	1.75760e15	1.36193
$560$	0	0
$561$	1.39010e15	1.05621
$562$	0	0
$563$	−2.32309e15	−1.73089	−0.865446	−	0.501002i	$-0.832965\pi$
$563$	−2.32309e15	−1.73089	−0.865446	+	0.501002i	$0.832965\pi$
$564$	0	0
$565$	3.75983e14	0.274727
$566$	0	0
$567$	3.22479e15	2.31098
$568$	0	0
$569$	−2.48598e14	−0.174735	−0.0873676	−	0.996176i	$-0.527845\pi$
$569$	−2.48598e14	−0.174735	−0.0873676	+	0.996176i	$0.527845\pi$
$570$	0	0
$571$	2.44700e15	1.68708	0.843540	−	0.537066i	$-0.180467\pi$
$571$	2.44700e15	1.68708	0.843540	+	0.537066i	$0.180467\pi$
$572$	0	0
$573$	−1.95072e15	−1.31931
$574$	0	0
$575$	5.15280e13	0.0341877
$576$	0	0
$577$	−2.05338e15	−1.33660	−0.668302	−	0.743890i	$-0.732979\pi$
$577$	−2.05338e15	−1.33660	−0.668302	+	0.743890i	$0.732979\pi$
$578$	0	0
$579$	−1.19182e15	−0.761166
$580$	0	0
$581$	−3.97266e15	−2.48950
$582$	0	0
$583$	1.40319e15	0.862860
$584$	0	0
$585$	1.78776e15	1.07883
$586$	0	0
$587$	−3.00291e14	−0.177841	−0.0889207	−	0.996039i	$-0.528342\pi$
$587$	−3.00291e14	−0.177841	−0.0889207	+	0.996039i	$0.528342\pi$
$588$	0	0
$589$	−7.93094e14	−0.460989
$590$	0	0
$591$	4.59582e13	0.0262200
$592$	0	0
$593$	−2.71460e15	−1.52022	−0.760108	−	0.649796i	$-0.774854\pi$
$593$	−2.71460e15	−1.52022	−0.760108	+	0.649796i	$0.774854\pi$
$594$	0	0
$595$	−1.20489e15	−0.662377
$596$	0	0
$597$	1.76399e15	0.951999
$598$	0	0
$599$	2.10006e15	1.11272	0.556358	−	0.830943i	$-0.312198\pi$
$599$	2.10006e15	1.11272	0.556358	+	0.830943i	$0.312198\pi$
$600$	0	0
$601$	2.84187e15	1.47841	0.739204	−	0.673481i	$-0.235202\pi$
$601$	2.84187e15	1.47841	0.739204	+	0.673481i	$0.235202\pi$
$602$	0	0
$603$	−1.40528e15	−0.717821
$604$	0	0
$605$	−5.18242e14	−0.259943
$606$	0	0
$607$	3.78069e14	0.186223	0.0931114	−	0.995656i	$-0.470319\pi$
$607$	3.78069e14	0.186223	0.0931114	+	0.995656i	$0.470319\pi$
$608$	0	0
$609$	9.93235e14	0.480460
$610$	0	0
$611$	−4.25619e14	−0.202206
$612$	0	0
$613$	2.97141e15	1.38653	0.693267	−	0.720681i	$-0.256171\pi$
$613$	2.97141e15	1.38653	0.693267	+	0.720681i	$0.256171\pi$
$614$	0	0
$615$	2.12366e15	0.973355
$616$	0	0
$617$	−1.19545e15	−0.538222	−0.269111	−	0.963109i	$-0.586730\pi$
$617$	−1.19545e15	−0.538222	−0.269111	+	0.963109i	$0.586730\pi$
$618$	0	0
$619$	−3.17735e14	−0.140529	−0.0702645	−	0.997528i	$-0.522384\pi$
$619$	−3.17735e14	−0.140529	−0.0702645	+	0.997528i	$0.522384\pi$
$620$	0	0
$621$	−7.94151e14	−0.345064
$622$	0	0
$623$	−6.64693e14	−0.283750
$624$	0	0
$625$	9.53674e13	0.0400000
$626$	0	0
$627$	−2.87749e15	−1.18588
$628$	0	0
$629$	−1.74280e15	−0.705778
$630$	0	0
$631$	7.14141e14	0.284199	0.142099	−	0.989852i	$-0.454615\pi$
$631$	7.14141e14	0.284199	0.142099	+	0.989852i	$0.454615\pi$
$632$	0	0
$633$	4.02498e15	1.57414
$634$	0	0
$635$	1.75215e15	0.673465
$636$	0	0
$637$	4.73128e15	1.78736
$638$	0	0
$639$	−1.30849e15	−0.485866
$640$	0	0
$641$	3.74739e15	1.36776	0.683880	−	0.729594i	$-0.260291\pi$
$641$	3.74739e15	1.36776	0.683880	+	0.729594i	$0.260291\pi$
$642$	0	0
$643$	−3.72691e14	−0.133718	−0.0668588	−	0.997762i	$-0.521298\pi$
$643$	−3.72691e14	−0.133718	−0.0668588	+	0.997762i	$0.521298\pi$
$644$	0	0
$645$	−2.71333e15	−0.957028
$646$	0	0
$647$	−3.55565e15	−1.23295	−0.616475	−	0.787374i	$-0.711440\pi$
$647$	−3.55565e15	−1.23295	−0.616475	+	0.787374i	$0.711440\pi$
$648$	0	0
$649$	−1.69310e15	−0.577212
$650$	0	0
$651$	3.78784e15	1.26969
$652$	0	0
$653$	5.07282e15	1.67196	0.835982	−	0.548757i	$-0.184899\pi$
$653$	5.07282e15	1.67196	0.835982	+	0.548757i	$0.184899\pi$
$654$	0	0
$655$	−1.28750e15	−0.417272
$656$	0	0
$657$	−8.38043e15	−2.67090
$658$	0	0
$659$	−1.66566e15	−0.522054	−0.261027	−	0.965331i	$-0.584061\pi$
$659$	−1.66566e15	−0.522054	−0.261027	+	0.965331i	$0.584061\pi$
$660$	0	0
$661$	3.06978e14	0.0946235	0.0473118	−	0.998880i	$-0.484935\pi$
$661$	3.06978e14	0.0946235	0.0473118	+	0.998880i	$0.484935\pi$
$662$	0	0
$663$	6.07098e15	1.84049
$664$	0	0
$665$	2.49412e15	0.743701
$666$	0	0
$667$	−9.82960e13	−0.0288299
$668$	0	0
$669$	−6.98845e15	−2.01622
$670$	0	0
$671$	3.71800e15	1.05520
$672$	0	0
$673$	3.20107e15	0.893742	0.446871	−	0.894599i	$-0.352538\pi$
$673$	3.20107e15	0.893742	0.446871	+	0.894599i	$0.352538\pi$
$674$	0	0
$675$	−1.46981e15	−0.403728
$676$	0	0
$677$	−4.88443e15	−1.32001	−0.660004	−	0.751262i	$-0.729445\pi$
$677$	−4.88443e15	−1.32001	−0.660004	+	0.751262i	$0.729445\pi$
$678$	0	0
$679$	3.37376e15	0.897079
$680$	0	0
$681$	−4.85891e15	−1.27125
$682$	0	0
$683$	−3.43048e15	−0.883163	−0.441582	−	0.897221i	$-0.645582\pi$
$683$	−3.43048e15	−0.883163	−0.441582	+	0.897221i	$0.645582\pi$
$684$	0	0
$685$	−2.45706e15	−0.622469
$686$	0	0
$687$	1.30361e15	0.325001
$688$	0	0
$689$	6.12818e15	1.50358
$690$	0	0
$691$	−1.26067e15	−0.304420	−0.152210	−	0.988348i	$-0.548639\pi$
$691$	−1.26067e15	−0.304420	−0.152210	+	0.988348i	$0.548639\pi$
$692$	0	0
$693$	9.36527e15	2.22581
$694$	0	0
$695$	−2.67860e15	−0.626602
$696$	0	0
$697$	4.91444e15	1.13160
$698$	0	0
$699$	−6.44682e15	−1.46124
$700$	0	0
$701$	−1.46416e15	−0.326693	−0.163347	−	0.986569i	$-0.552229\pi$
$701$	−1.46416e15	−0.326693	−0.163347	+	0.986569i	$0.552229\pi$
$702$	0	0
$703$	3.60758e15	0.792431
$704$	0	0
$705$	6.57058e14	0.142090
$706$	0	0
$707$	1.15270e16	2.45420
$708$	0	0
$709$	−2.00407e15	−0.420105	−0.210053	−	0.977690i	$-0.567364\pi$
$709$	−2.00407e15	−0.420105	−0.210053	+	0.977690i	$0.567364\pi$
$710$	0	0
$711$	−2.66293e15	−0.549639
$712$	0	0
$713$	−3.74865e14	−0.0761873
$714$	0	0
$715$	1.63057e15	0.326329
$716$	0	0
$717$	5.00406e15	0.986205
$718$	0	0
$719$	6.94101e15	1.34714	0.673571	−	0.739122i	$-0.264759\pi$
$719$	6.94101e15	1.34714	0.673571	+	0.739122i	$0.264759\pi$
$720$	0	0
$721$	1.32679e16	2.53606
$722$	0	0
$723$	8.97336e15	1.68925
$724$	0	0
$725$	−1.81925e14	−0.0337313
$726$	0	0
$727$	8.96895e14	0.163796	0.0818978	−	0.996641i	$-0.473902\pi$
$727$	8.96895e14	0.163796	0.0818978	+	0.996641i	$0.473902\pi$
$728$	0	0
$729$	−2.78918e15	−0.501735
$730$	0	0
$731$	−6.27903e15	−1.11262
$732$	0	0
$733$	−1.11078e16	−1.93890	−0.969451	−	0.245286i	$-0.921118\pi$
$733$	−1.11078e16	−1.93890	−0.969451	+	0.245286i	$0.921118\pi$
$734$	0	0
$735$	−7.30400e15	−1.25597
$736$	0	0
$737$	−1.28171e15	−0.217130
$738$	0	0
$739$	−5.03625e15	−0.840548	−0.420274	−	0.907397i	$-0.638066\pi$
$739$	−5.03625e15	−0.840548	−0.420274	+	0.907397i	$0.638066\pi$
$740$	0	0
$741$	−1.25669e16	−2.06646
$742$	0	0
$743$	−1.36645e15	−0.221389	−0.110695	−	0.993854i	$-0.535308\pi$
$743$	−1.36645e15	−0.221389	−0.110695	+	0.993854i	$0.535308\pi$
$744$	0	0
$745$	−7.45643e14	−0.119034
$746$	0	0
$747$	−2.10578e16	−3.31246
$748$	0	0
$749$	3.90225e15	0.604874
$750$	0	0
$751$	9.78301e15	1.49435	0.747176	−	0.664627i	$-0.231409\pi$
$751$	9.78301e15	1.49435	0.747176	+	0.664627i	$0.231409\pi$
$752$	0	0
$753$	−1.66456e16	−2.50570
$754$	0	0
$755$	2.99159e14	0.0443807
$756$	0	0
$757$	−1.39502e15	−0.203964	−0.101982	−	0.994786i	$-0.532518\pi$
$757$	−1.39502e15	−0.203964	−0.101982	+	0.994786i	$0.532518\pi$
$758$	0	0
$759$	−1.36008e15	−0.195990
$760$	0	0
$761$	6.09995e15	0.866385	0.433192	−	0.901301i	$-0.357387\pi$
$761$	6.09995e15	0.866385	0.433192	+	0.901301i	$0.357387\pi$
$762$	0	0
$763$	−2.53117e15	−0.354353
$764$	0	0
$765$	−6.38676e15	−0.881339
$766$	0	0
$767$	−7.39427e15	−1.00582
$768$	0	0
$769$	6.60626e15	0.885851	0.442925	−	0.896558i	$-0.353941\pi$
$769$	6.60626e15	0.885851	0.442925	+	0.896558i	$0.353941\pi$
$770$	0	0
$771$	−1.34273e16	−1.77496
$772$	0	0
$773$	3.74520e15	0.488076	0.244038	−	0.969766i	$-0.421528\pi$
$773$	3.74520e15	0.488076	0.244038	+	0.969766i	$0.421528\pi$
$774$	0	0
$775$	−6.93797e14	−0.0891400
$776$	0	0
$777$	−1.72299e16	−2.18257
$778$	0	0
$779$	−1.01729e16	−1.27054
$780$	0	0
$781$	−1.19344e15	−0.146967
$782$	0	0
$783$	2.80384e15	0.340458
$784$	0	0
$785$	−5.62297e15	−0.673259
$786$	0	0
$787$	1.01107e16	1.19377	0.596885	−	0.802326i	$-0.296405\pi$
$787$	1.01107e16	1.19377	0.596885	+	0.802326i	$0.296405\pi$
$788$	0	0
$789$	1.03578e16	1.20599
$790$	0	0
$791$	8.60188e15	0.987694
$792$	0	0
$793$	1.62377e16	1.83874
$794$	0	0
$795$	−9.46050e15	−1.05656
$796$	0	0
$797$	−1.40032e16	−1.54244	−0.771219	−	0.636570i	$-0.780353\pi$
$797$	−1.40032e16	−1.54244	−0.771219	+	0.636570i	$0.780353\pi$
$798$	0	0
$799$	1.52052e15	0.165191
$800$	0	0
$801$	−3.52333e15	−0.377550
$802$	0	0
$803$	−7.64357e15	−0.807906
$804$	0	0
$805$	1.17888e15	0.122911
$806$	0	0
$807$	−2.02550e16	−2.08318
$808$	0	0
$809$	2.05016e15	0.208004	0.104002	−	0.994577i	$-0.466835\pi$
$809$	2.05016e15	0.208004	0.104002	+	0.994577i	$0.466835\pi$
$810$	0	0
$811$	1.45631e16	1.45760	0.728801	−	0.684725i	$-0.240078\pi$
$811$	1.45631e16	1.45760	0.728801	+	0.684725i	$0.240078\pi$
$812$	0	0
$813$	−5.40316e15	−0.533519
$814$	0	0
$815$	7.83183e15	0.762951
$816$	0	0
$817$	1.29975e16	1.24922
$818$	0	0
$819$	4.09011e16	3.87858
$820$	0	0
$821$	−5.13429e15	−0.480389	−0.240195	−	0.970725i	$-0.577211\pi$
$821$	−5.13429e15	−0.480389	−0.240195	+	0.970725i	$0.577211\pi$
$822$	0	0
$823$	6.16638e15	0.569287	0.284643	−	0.958633i	$-0.408125\pi$
$823$	6.16638e15	0.569287	0.284643	+	0.958633i	$0.408125\pi$
$824$	0	0
$825$	−2.51722e15	−0.229311
$826$	0	0
$827$	2.77673e15	0.249605	0.124803	−	0.992182i	$-0.460170\pi$
$827$	2.77673e15	0.249605	0.124803	+	0.992182i	$0.460170\pi$
$828$	0	0
$829$	1.33865e16	1.18745	0.593726	−	0.804667i	$-0.297656\pi$
$829$	1.33865e16	1.18745	0.593726	+	0.804667i	$0.297656\pi$
$830$	0	0
$831$	−2.96281e16	−2.59357
$832$	0	0
$833$	−1.69025e16	−1.46017
$834$	0	0
$835$	8.43296e15	0.718960
$836$	0	0
$837$	1.06928e16	0.899709
$838$	0	0
$839$	−1.47887e16	−1.22812	−0.614059	−	0.789260i	$-0.710464\pi$
$839$	−1.47887e16	−1.22812	−0.614059	+	0.789260i	$0.710464\pi$
$840$	0	0
$841$	−1.18535e16	−0.971555
$842$	0	0
$843$	−4.43902e15	−0.359116
$844$	0	0
$845$	1.52070e15	0.121431
$846$	0	0
$847$	−1.18565e16	−0.934542
$848$	0	0
$849$	−2.87691e16	−2.23838
$850$	0	0
$851$	1.70517e15	0.130965
$852$	0	0
$853$	1.97554e16	1.49784	0.748922	−	0.662658i	$-0.230572\pi$
$853$	1.97554e16	1.49784	0.748922	+	0.662658i	$0.230572\pi$
$854$	0	0
$855$	1.32206e16	0.989547
$856$	0	0
$857$	−2.32233e16	−1.71605	−0.858023	−	0.513612i	$-0.828307\pi$
$857$	−2.32233e16	−1.71605	−0.858023	+	0.513612i	$0.828307\pi$
$858$	0	0
$859$	1.67344e16	1.22081	0.610405	−	0.792089i	$-0.291007\pi$
$859$	1.67344e16	1.22081	0.610405	+	0.792089i	$0.291007\pi$
$860$	0	0
$861$	4.85859e16	3.49939
$862$	0	0
$863$	2.46024e16	1.74951	0.874757	−	0.484563i	$-0.161021\pi$
$863$	2.46024e16	1.74951	0.874757	+	0.484563i	$0.161021\pi$
$864$	0	0
$865$	9.61864e15	0.675344
$866$	0	0
$867$	3.86919e15	0.268234
$868$	0	0
$869$	−2.42879e15	−0.166257
$870$	0	0
$871$	−5.59764e15	−0.378360
$872$	0	0
$873$	1.78833e16	1.19363
$874$	0	0
$875$	2.18185e15	0.143807
$876$	0	0
$877$	−4.55865e15	−0.296714	−0.148357	−	0.988934i	$-0.547399\pi$
$877$	−4.55865e15	−0.296714	−0.148357	+	0.988934i	$0.547399\pi$
$878$	0	0
$879$	3.97125e16	2.55263
$880$	0	0
$881$	1.67834e16	1.06540	0.532701	−	0.846304i	$-0.321177\pi$
$881$	1.67834e16	1.06540	0.532701	+	0.846304i	$0.321177\pi$
$882$	0	0
$883$	1.13628e16	0.712362	0.356181	−	0.934417i	$-0.384079\pi$
$883$	1.13628e16	0.712362	0.356181	+	0.934417i	$0.384079\pi$
$884$	0	0
$885$	1.14151e16	0.706789
$886$	0	0
$887$	−1.55603e16	−0.951563	−0.475782	−	0.879563i	$-0.657835\pi$
$887$	−1.55603e16	−0.951563	−0.475782	+	0.879563i	$0.657835\pi$
$888$	0	0
$889$	4.00863e16	2.42123
$890$	0	0
$891$	1.55906e16	0.930115
$892$	0	0
$893$	−3.14747e15	−0.185472
$894$	0	0
$895$	−1.09736e15	−0.0638736
$896$	0	0
$897$	−5.93989e15	−0.341523
$898$	0	0
$899$	1.32350e15	0.0751704
$900$	0	0
$901$	−2.18929e16	−1.22834
$902$	0	0
$903$	−6.20767e16	−3.44069
$904$	0	0
$905$	1.98158e15	0.108504
$906$	0	0
$907$	−3.40764e16	−1.84338	−0.921688	−	0.387932i	$-0.873189\pi$
$907$	−3.40764e16	−1.84338	−0.921688	+	0.387932i	$0.873189\pi$
$908$	0	0
$909$	6.11011e16	3.26548
$910$	0	0
$911$	−1.60732e16	−0.848697	−0.424348	−	0.905499i	$-0.639497\pi$
$911$	−1.60732e16	−0.848697	−0.424348	+	0.905499i	$0.639497\pi$
$912$	0	0
$913$	−1.92063e16	−1.00197
$914$	0	0
$915$	−2.50672e16	−1.29208
$916$	0	0
$917$	−2.94559e16	−1.50017
$918$	0	0
$919$	7.32080e15	0.368403	0.184201	−	0.982889i	$-0.441030\pi$
$919$	7.32080e15	0.368403	0.184201	+	0.982889i	$0.441030\pi$
$920$	0	0
$921$	−4.28302e16	−2.12972
$922$	0	0
$923$	−5.21213e15	−0.256098
$924$	0	0
$925$	3.15590e15	0.153230
$926$	0	0
$927$	7.03292e16	3.37440
$928$	0	0
$929$	−1.41658e16	−0.671666	−0.335833	−	0.941922i	$-0.609018\pi$
$929$	−1.41658e16	−0.671666	−0.335833	+	0.941922i	$0.609018\pi$
$930$	0	0
$931$	3.49880e16	1.63944
$932$	0	0
$933$	−1.92859e16	−0.893081
$934$	0	0
$935$	−5.82520e15	−0.266592
$936$	0	0
$937$	1.51385e16	0.684721	0.342361	−	0.939569i	$-0.388774\pi$
$937$	1.51385e16	0.684721	0.342361	+	0.939569i	$0.388774\pi$
$938$	0	0
$939$	2.66456e16	1.19115
$940$	0	0
$941$	8.81297e15	0.389385	0.194693	−	0.980864i	$-0.437629\pi$
$941$	8.81297e15	0.389385	0.194693	+	0.980864i	$0.437629\pi$
$942$	0	0
$943$	−4.80832e15	−0.209981
$944$	0	0
$945$	−3.36268e16	−1.45148
$946$	0	0
$947$	−6.85225e15	−0.292353	−0.146177	−	0.989258i	$-0.546697\pi$
$947$	−6.85225e15	−0.292353	−0.146177	+	0.989258i	$0.546697\pi$
$948$	0	0
$949$	−3.33818e16	−1.40782
$950$	0	0
$951$	1.33211e16	0.555325
$952$	0	0
$953$	3.21185e16	1.32356	0.661781	−	0.749697i	$-0.269801\pi$
$953$	3.21185e16	1.32356	0.661781	+	0.749697i	$0.269801\pi$
$954$	0	0
$955$	8.17451e15	0.332999
$956$	0	0
$957$	4.80191e15	0.193374
$958$	0	0
$959$	−5.62135e16	−2.23789
$960$	0	0
$961$	−2.03611e16	−0.801351
$962$	0	0
$963$	2.06846e16	0.804827
$964$	0	0
$965$	4.99433e15	0.192122
$966$	0	0
$967$	−2.01801e16	−0.767500	−0.383750	−	0.923437i	$-0.625367\pi$
$967$	−2.01801e16	−0.767500	−0.383750	+	0.923437i	$0.625367\pi$
$968$	0	0
$969$	4.48952e16	1.68818
$970$	0	0
$971$	−7.19924e15	−0.267658	−0.133829	−	0.991004i	$-0.542727\pi$
$971$	−7.19924e15	−0.267658	−0.133829	+	0.991004i	$0.542727\pi$
$972$	0	0
$973$	−6.12821e16	−2.25275
$974$	0	0
$975$	−1.09935e16	−0.399586
$976$	0	0
$977$	3.14860e16	1.13161	0.565806	−	0.824538i	$-0.308565\pi$
$977$	3.14860e16	1.13161	0.565806	+	0.824538i	$0.308565\pi$
$978$	0	0
$979$	−3.21353e15	−0.114203
$980$	0	0
$981$	−1.34169e16	−0.471491
$982$	0	0
$983$	1.68059e15	0.0584006	0.0292003	−	0.999574i	$-0.490704\pi$
$983$	1.68059e15	0.0584006	0.0292003	+	0.999574i	$0.490704\pi$
$984$	0	0
$985$	−1.92588e14	−0.00661805
$986$	0	0
$987$	1.50324e16	0.510840
$988$	0	0
$989$	6.14345e15	0.206458
$990$	0	0
$991$	2.21678e15	0.0736746	0.0368373	−	0.999321i	$-0.488272\pi$
$991$	2.21678e15	0.0736746	0.0368373	+	0.999321i	$0.488272\pi$
$992$	0	0
$993$	5.80389e16	1.90765
$994$	0	0
$995$	−7.39199e15	−0.240289
$996$	0	0
$997$	2.59411e14	0.00833999	0.00416999	−	0.999991i	$-0.498673\pi$
$997$	2.59411e14	0.00833999	0.00416999	+	0.999991i	$0.498673\pi$
$998$	0	0
$999$	−4.86389e16	−1.54658

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.12.a.g.1.1		2
4.3	odd	2		10.12.a.d.1.2	✓	2
12.11	even	2		90.12.a.l.1.1		2
20.3	even	4		50.12.b.f.49.2		4
20.7	even	4		50.12.b.f.49.3		4
20.19	odd	2		50.12.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.12.a.d.1.2	✓	2	4.3	odd	2
50.12.a.f.1.1		2	20.19	odd	2
50.12.b.f.49.2		4	20.3	even	4
50.12.b.f.49.3		4	20.7	even	4
80.12.a.g.1.1		2	1.1	even	1	trivial
90.12.a.l.1.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-745.734 q^{3} +3125.00 q^{5} +71494.9 q^{7} +378972. q^{9} +345651. q^{11} +1.50956e6 q^{13} -2.33042e6 q^{15} -5.39291e6 q^{17} +1.11633e7 q^{19} -5.33162e7 q^{21} +5.27646e6 q^{23} +9.76562e6 q^{25} -1.50508e8 q^{27} -1.86291e7 q^{29} -7.10448e7 q^{31} -2.57763e8 q^{33} +2.23422e8 q^{35} +3.23164e8 q^{37} -1.12573e9 q^{39} -9.11277e8 q^{41} +1.16431e9 q^{43} +1.18429e9 q^{45} -2.81949e8 q^{47} +3.13420e9 q^{49} +4.02168e9 q^{51} +4.05957e9 q^{53} +1.08016e9 q^{55} -8.32485e9 q^{57} -4.89828e9 q^{59} +1.07565e10 q^{61} +2.70946e10 q^{63} +4.71739e9 q^{65} -3.70812e9 q^{67} -3.93484e9 q^{69} -3.45274e9 q^{71} -2.21136e10 q^{73} -7.28256e9 q^{75} +2.47123e10 q^{77} -7.02672e9 q^{79} +4.51052e10 q^{81} -5.55656e10 q^{83} -1.68528e10 q^{85} +1.38924e10 q^{87} -9.29706e9 q^{89} +1.07926e11 q^{91} +5.29805e10 q^{93} +3.48853e10 q^{95} +4.71888e10 q^{97} +1.30992e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 604 q^{3} + 6250 q^{5} - 14092 q^{7} + 221914 q^{9} - 421584 q^{11} + 1730524 q^{13} - 1887500 q^{15} - 6323628 q^{17} + 28897400 q^{19} - 65446816 q^{21} + 45236076 q^{23} + 19531250 q^{25} - 197876440 q^{27}+ \cdots + 251492724912 q^{99}+O(q^{100}) \) 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

L-functions

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