LMFDB - Embedded newform 80.12.a.j.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{151}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 151 $ x^2 - 151
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$12.2882$ 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)

$f(q)$	$=$	$q+503.223 q^{3} -3125.00 q^{5} -15973.7 q^{7} +76086.0 q^{9} +O(q^{10})$	$q+503.223 q^{3} -3125.00 q^{5} -15973.7 q^{7} +76086.0 q^{9} -339729. q^{11} +2.02328e6 q^{13} -1.57257e6 q^{15} -2.45063e6 q^{17} +4.08504e6 q^{19} -8.03830e6 q^{21} -2.86497e7 q^{23} +9.76562e6 q^{25} -5.08562e7 q^{27} -9.41230e6 q^{29} -2.99399e8 q^{31} -1.70959e8 q^{33} +4.99177e7 q^{35} -4.57279e8 q^{37} +1.01816e9 q^{39} +1.83814e8 q^{41} -6.56811e8 q^{43} -2.37769e8 q^{45} +1.97090e8 q^{47} -1.72217e9 q^{49} -1.23321e9 q^{51} +5.15890e9 q^{53} +1.06165e9 q^{55} +2.05568e9 q^{57} +6.62200e8 q^{59} +5.58296e8 q^{61} -1.21537e9 q^{63} -6.32275e9 q^{65} -1.01206e10 q^{67} -1.44172e10 q^{69} -1.78161e10 q^{71} -2.33380e10 q^{73} +4.91428e9 q^{75} +5.42672e9 q^{77} -1.24957e10 q^{79} -3.90704e10 q^{81} -3.37037e10 q^{83} +7.65823e9 q^{85} -4.73648e9 q^{87} +2.94282e10 q^{89} -3.23192e10 q^{91} -1.50664e11 q^{93} -1.27657e10 q^{95} -1.13262e11 q^{97} -2.58486e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 220 q^{3} - 6250 q^{5} - 57900 q^{7} - 20846 q^{9}+O(q^{10}) $ 2 * q + 220 * q^3 - 6250 * q^5 - 57900 * q^7 - 20846 * q^9	$ 2 q + 220 q^{3} - 6250 q^{5} - 57900 q^{7} - 20846 q^{9} + 618176 q^{11} + 3414260 q^{13} - 687500 q^{15} + 1317940 q^{17} - 5325320 q^{19} + 3836184 q^{21} - 58943940 q^{23} + 19531250 q^{25} + 26769160 q^{27} + 94140380 q^{29} - 244543464 q^{31} - 442259840 q^{33} + 180937500 q^{35} + 21003220 q^{37} + 624203992 q^{39} - 745743316 q^{41} - 629950100 q^{43} + 65143750 q^{45} + 1402061540 q^{47} - 1941677414 q^{49} - 2300559784 q^{51} + 1138320580 q^{53} - 1931800000 q^{55} + 4720910480 q^{57} - 7317515560 q^{59} - 1516425676 q^{61} + 2848632180 q^{63} - 10669562500 q^{65} - 15734290140 q^{67} - 5837195832 q^{69} - 32938471544 q^{71} - 29982848860 q^{73} + 2148437500 q^{75} - 34734748800 q^{77} + 3302823120 q^{79} - 43884431798 q^{81} - 13299102420 q^{83} - 4118562500 q^{85} - 34064940920 q^{87} - 12674770860 q^{89} - 90637859064 q^{91} - 166200542640 q^{93} + 16641625000 q^{95} - 3080703740 q^{97} - 118700272448 q^{99}+O(q^{100}) $ 2 * q + 220 * q^3 - 6250 * q^5 - 57900 * q^7 - 20846 * q^9 + 618176 * q^11 + 3414260 * q^13 - 687500 * q^15 + 1317940 * q^17 - 5325320 * q^19 + 3836184 * q^21 - 58943940 * q^23 + 19531250 * q^25 + 26769160 * q^27 + 94140380 * q^29 - 244543464 * q^31 - 442259840 * q^33 + 180937500 * q^35 + 21003220 * q^37 + 624203992 * q^39 - 745743316 * q^41 - 629950100 * q^43 + 65143750 * q^45 + 1402061540 * q^47 - 1941677414 * q^49 - 2300559784 * q^51 + 1138320580 * q^53 - 1931800000 * q^55 + 4720910480 * q^57 - 7317515560 * q^59 - 1516425676 * q^61 + 2848632180 * q^63 - 10669562500 * q^65 - 15734290140 * q^67 - 5837195832 * q^69 - 32938471544 * q^71 - 29982848860 * q^73 + 2148437500 * q^75 - 34734748800 * q^77 + 3302823120 * q^79 - 43884431798 * q^81 - 13299102420 * q^83 - 4118562500 * q^85 - 34064940920 * q^87 - 12674770860 * q^89 - 90637859064 * q^91 - 166200542640 * q^93 + 16641625000 * q^95 - 3080703740 * q^97 - 118700272448 * 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
$3$	503.223	1.19562	0.597810	−	0.801638i	$-0.296038\pi$
$3$	503.223	1.19562	0.597810	+	0.801638i	$0.296038\pi$
$4$	0	0
$5$	−3125.00	−0.447214
$5$	−3125.00	−0.447214
$6$	0	0
$7$	−15973.7	−0.359224	−0.179612	−	0.983738i	$-0.557484\pi$
$7$	−15973.7	−0.359224	−0.179612	+	0.983738i	$0.557484\pi$
$8$	0	0
$9$	76086.0	0.429508
$10$	0	0
$11$	−339729.	−0.636024	−0.318012	−	0.948087i	$-0.603015\pi$
$11$	−339729.	−0.636024	−0.318012	+	0.948087i	$0.603015\pi$
$12$	0	0
$13$	2.02328e6	1.51136	0.755680	−	0.654941i	$-0.227307\pi$
$13$	2.02328e6	1.51136	0.755680	+	0.654941i	$0.227307\pi$
$14$	0	0
$15$	−1.57257e6	−0.534698
$16$	0	0
$17$	−2.45063e6	−0.418610	−0.209305	−	0.977850i	$-0.567120\pi$
$17$	−2.45063e6	−0.418610	−0.209305	+	0.977850i	$0.567120\pi$
$18$	0	0
$19$	4.08504e6	0.378487	0.189244	−	0.981930i	$-0.439396\pi$
$19$	4.08504e6	0.378487	0.189244	+	0.981930i	$0.439396\pi$
$20$	0	0
$21$	−8.03830e6	−0.429495
$22$	0	0
$23$	−2.86497e7	−0.928149	−0.464074	−	0.885796i	$-0.653613\pi$
$23$	−2.86497e7	−0.928149	−0.464074	+	0.885796i	$0.653613\pi$
$24$	0	0
$25$	9.76562e6	0.200000
$26$	0	0
$27$	−5.08562e7	−0.682092
$28$	0	0
$29$	−9.41230e6	−0.0852132	−0.0426066	−	0.999092i	$-0.513566\pi$
$29$	−9.41230e6	−0.0852132	−0.0426066	+	0.999092i	$0.513566\pi$
$30$	0	0
$31$	−2.99399e8	−1.87828	−0.939141	−	0.343532i	$-0.888377\pi$
$31$	−2.99399e8	−1.87828	−0.939141	+	0.343532i	$0.888377\pi$
$32$	0	0
$33$	−1.70959e8	−0.760443
$34$	0	0
$35$	4.99177e7	0.160650
$36$	0	0
$37$	−4.57279e8	−1.08411	−0.542053	−	0.840344i	$-0.682353\pi$
$37$	−4.57279e8	−1.08411	−0.542053	+	0.840344i	$0.682353\pi$
$38$	0	0
$39$	1.01816e9	1.80701
$40$	0	0
$41$	1.83814e8	0.247780	0.123890	−	0.992296i	$-0.460463\pi$
$41$	1.83814e8	0.247780	0.123890	+	0.992296i	$0.460463\pi$
$42$	0	0
$43$	−6.56811e8	−0.681340	−0.340670	−	0.940183i	$-0.610654\pi$
$43$	−6.56811e8	−0.681340	−0.340670	+	0.940183i	$0.610654\pi$
$44$	0	0
$45$	−2.37769e8	−0.192082
$46$	0	0
$47$	1.97090e8	0.125350	0.0626752	−	0.998034i	$-0.480037\pi$
$47$	1.97090e8	0.125350	0.0626752	+	0.998034i	$0.480037\pi$
$48$	0	0
$49$	−1.72217e9	−0.870958
$50$	0	0
$51$	−1.23321e9	−0.500498
$52$	0	0
$53$	5.15890e9	1.69449	0.847247	−	0.531199i	$-0.178258\pi$
$53$	5.15890e9	1.69449	0.847247	+	0.531199i	$0.178258\pi$
$54$	0	0
$55$	1.06165e9	0.284438
$56$	0	0
$57$	2.05568e9	0.452527
$58$	0	0
$59$	6.62200e8	0.120588	0.0602939	−	0.998181i	$-0.480796\pi$
$59$	6.62200e8	0.120588	0.0602939	+	0.998181i	$0.480796\pi$
$60$	0	0
$61$	5.58296e8	0.0846351	0.0423176	−	0.999104i	$-0.486526\pi$
$61$	5.58296e8	0.0846351	0.0423176	+	0.999104i	$0.486526\pi$
$62$	0	0
$63$	−1.21537e9	−0.154289
$64$	0	0
$65$	−6.32275e9	−0.675900
$66$	0	0
$67$	−1.01206e10	−0.915787	−0.457894	−	0.889007i	$-0.651396\pi$
$67$	−1.01206e10	−0.915787	−0.457894	+	0.889007i	$0.651396\pi$
$68$	0	0
$69$	−1.44172e10	−1.10971
$70$	0	0
$71$	−1.78161e10	−1.17190	−0.585952	−	0.810346i	$-0.699279\pi$
$71$	−1.78161e10	−1.17190	−0.585952	+	0.810346i	$0.699279\pi$
$72$	0	0
$73$	−2.33380e10	−1.31761	−0.658807	−	0.752312i	$-0.728939\pi$
$73$	−2.33380e10	−1.31761	−0.658807	+	0.752312i	$0.728939\pi$
$74$	0	0
$75$	4.91428e9	0.239124
$76$	0	0
$77$	5.42672e9	0.228475
$78$	0	0
$79$	−1.24957e10	−0.456888	−0.228444	−	0.973557i	$-0.573364\pi$
$79$	−1.24957e10	−0.456888	−0.228444	+	0.973557i	$0.573364\pi$
$80$	0	0
$81$	−3.90704e10	−1.24503
$82$	0	0
$83$	−3.37037e10	−0.939179	−0.469589	−	0.882885i	$-0.655598\pi$
$83$	−3.37037e10	−0.939179	−0.469589	+	0.882885i	$0.655598\pi$
$84$	0	0
$85$	7.65823e9	0.187208
$86$	0	0
$87$	−4.73648e9	−0.101883
$88$	0	0
$89$	2.94282e10	0.558623	0.279311	−	0.960201i	$-0.409894\pi$
$89$	2.94282e10	0.558623	0.279311	+	0.960201i	$0.409894\pi$
$90$	0	0
$91$	−3.23192e10	−0.542916
$92$	0	0
$93$	−1.50664e11	−2.24571
$94$	0	0
$95$	−1.27657e10	−0.169265
$96$	0	0
$97$	−1.13262e11	−1.33918	−0.669588	−	0.742732i	$-0.733529\pi$
$97$	−1.13262e11	−1.33918	−0.669588	+	0.742732i	$0.733529\pi$
$98$	0	0
$99$	−2.58486e10	−0.273177
$100$	0	0
$101$	1.19564e11	1.13196	0.565982	−	0.824417i	$-0.308497\pi$
$101$	1.19564e11	1.13196	0.565982	+	0.824417i	$0.308497\pi$
$102$	0	0
$103$	9.46874e10	0.804799	0.402399	−	0.915464i	$-0.368176\pi$
$103$	9.46874e10	0.804799	0.402399	+	0.915464i	$0.368176\pi$
$104$	0	0
$105$	2.51197e10	0.192076
$106$	0	0
$107$	−1.52769e11	−1.05299	−0.526497	−	0.850177i	$-0.676495\pi$
$107$	−1.52769e11	−1.05299	−0.526497	+	0.850177i	$0.676495\pi$
$108$	0	0
$109$	2.85078e11	1.77467	0.887337	−	0.461122i	$-0.152553\pi$
$109$	2.85078e11	1.77467	0.887337	+	0.461122i	$0.152553\pi$
$110$	0	0
$111$	−2.30113e11	−1.29618
$112$	0	0
$113$	−2.35563e11	−1.20275	−0.601374	−	0.798968i	$-0.705380\pi$
$113$	−2.35563e11	−1.20275	−0.601374	+	0.798968i	$0.705380\pi$
$114$	0	0
$115$	8.95304e10	0.415081
$116$	0	0
$117$	1.53943e11	0.649140
$118$	0	0
$119$	3.91456e10	0.150375
$120$	0	0
$121$	−1.69896e11	−0.595474
$122$	0	0
$123$	9.24991e10	0.296251
$124$	0	0
$125$	−3.05176e10	−0.0894427
$126$	0	0
$127$	1.25786e11	0.337840	0.168920	−	0.985630i	$-0.445972\pi$
$127$	1.25786e11	0.337840	0.168920	+	0.985630i	$0.445972\pi$
$128$	0	0
$129$	−3.30522e11	−0.814624
$130$	0	0
$131$	2.97347e11	0.673397	0.336698	−	0.941613i	$-0.390690\pi$
$131$	2.97347e11	0.673397	0.336698	+	0.941613i	$0.390690\pi$
$132$	0	0
$133$	−6.52530e10	−0.135962
$134$	0	0
$135$	1.58926e11	0.305041
$136$	0	0
$137$	−6.29203e10	−0.111385	−0.0556926	−	0.998448i	$-0.517737\pi$
$137$	−6.29203e10	−0.111385	−0.0556926	+	0.998448i	$0.517737\pi$
$138$	0	0
$139$	5.11416e11	0.835974	0.417987	−	0.908453i	$-0.362736\pi$
$139$	5.11416e11	0.835974	0.417987	+	0.908453i	$0.362736\pi$
$140$	0	0
$141$	9.91800e10	0.149871
$142$	0	0
$143$	−6.87368e11	−0.961260
$144$	0	0
$145$	2.94134e10	0.0381085
$146$	0	0
$147$	−8.66634e11	−1.04134
$148$	0	0
$149$	7.94154e11	0.885891	0.442945	−	0.896549i	$-0.353934\pi$
$149$	7.94154e11	0.885891	0.442945	+	0.896549i	$0.353934\pi$
$150$	0	0
$151$	−1.24629e12	−1.29195	−0.645974	−	0.763360i	$-0.723548\pi$
$151$	−1.24629e12	−1.29195	−0.645974	+	0.763360i	$0.723548\pi$
$152$	0	0
$153$	−1.86459e11	−0.179796
$154$	0	0
$155$	9.35621e11	0.839993
$156$	0	0
$157$	9.73519e11	0.814510	0.407255	−	0.913315i	$-0.366486\pi$
$157$	9.73519e11	0.814510	0.407255	+	0.913315i	$0.366486\pi$
$158$	0	0
$159$	2.59608e12	2.02597
$160$	0	0
$161$	4.57641e11	0.333413
$162$	0	0
$163$	−1.26084e12	−0.858281	−0.429140	−	0.903238i	$-0.641183\pi$
$163$	−1.26084e12	−0.858281	−0.429140	+	0.903238i	$0.641183\pi$
$164$	0	0
$165$	5.34248e11	0.340080
$166$	0	0
$167$	3.01398e12	1.79556	0.897780	−	0.440443i	$-0.145179\pi$
$167$	3.01398e12	1.79556	0.897780	+	0.440443i	$0.145179\pi$
$168$	0	0
$169$	2.30151e12	1.28421
$170$	0	0
$171$	3.10814e11	0.162563
$172$	0	0
$173$	−3.07482e11	−0.150857	−0.0754285	−	0.997151i	$-0.524032\pi$
$173$	−3.07482e11	−0.150857	−0.0754285	+	0.997151i	$0.524032\pi$
$174$	0	0
$175$	−1.55993e11	−0.0718448
$176$	0	0
$177$	3.33234e11	0.144177
$178$	0	0
$179$	1.91470e12	0.778769	0.389384	−	0.921075i	$-0.372688\pi$
$179$	1.91470e12	0.778769	0.389384	+	0.921075i	$0.372688\pi$
$180$	0	0
$181$	1.10300e11	0.0422032	0.0211016	−	0.999777i	$-0.493283\pi$
$181$	1.10300e11	0.0422032	0.0211016	+	0.999777i	$0.493283\pi$
$182$	0	0
$183$	2.80947e11	0.101191
$184$	0	0
$185$	1.42900e12	0.484827
$186$	0	0
$187$	8.32552e11	0.266246
$188$	0	0
$189$	8.12359e11	0.245024
$190$	0	0
$191$	4.43991e12	1.26384	0.631918	−	0.775035i	$-0.282268\pi$
$191$	4.43991e12	1.26384	0.631918	+	0.775035i	$0.282268\pi$
$192$	0	0
$193$	3.15713e12	0.848647	0.424324	−	0.905511i	$-0.360512\pi$
$193$	3.15713e12	0.848647	0.424324	+	0.905511i	$0.360512\pi$
$194$	0	0
$195$	−3.18175e12	−0.808120
$196$	0	0
$197$	−3.58626e12	−0.861147	−0.430574	−	0.902555i	$-0.641689\pi$
$197$	−3.58626e12	−0.861147	−0.430574	+	0.902555i	$0.641689\pi$
$198$	0	0
$199$	−4.14823e12	−0.942260	−0.471130	−	0.882064i	$-0.656154\pi$
$199$	−4.14823e12	−0.942260	−0.471130	+	0.882064i	$0.656154\pi$
$200$	0	0
$201$	−5.09291e12	−1.09493
$202$	0	0
$203$	1.50349e11	0.0306106
$204$	0	0
$205$	−5.74417e11	−0.110811
$206$	0	0
$207$	−2.17984e12	−0.398647
$208$	0	0
$209$	−1.38781e12	−0.240727
$210$	0	0
$211$	−1.14275e12	−0.188104	−0.0940519	−	0.995567i	$-0.529982\pi$
$211$	−1.14275e12	−0.188104	−0.0940519	+	0.995567i	$0.529982\pi$
$212$	0	0
$213$	−8.96547e12	−1.40115
$214$	0	0
$215$	2.05253e12	0.304704
$216$	0	0
$217$	4.78249e12	0.674724
$218$	0	0
$219$	−1.17442e13	−1.57537
$220$	0	0
$221$	−4.95832e12	−0.632670
$222$	0	0
$223$	−3.54889e12	−0.430939	−0.215469	−	0.976511i	$-0.569128\pi$
$223$	−3.54889e12	−0.430939	−0.215469	+	0.976511i	$0.569128\pi$
$224$	0	0
$225$	7.43027e11	0.0859015
$226$	0	0
$227$	−3.90543e12	−0.430058	−0.215029	−	0.976608i	$-0.568985\pi$
$227$	−3.90543e12	−0.430058	−0.215029	+	0.976608i	$0.568985\pi$
$228$	0	0
$229$	−4.85126e12	−0.509048	−0.254524	−	0.967066i	$-0.581919\pi$
$229$	−4.85126e12	−0.509048	−0.254524	+	0.967066i	$0.581919\pi$
$230$	0	0
$231$	2.73085e12	0.273169
$232$	0	0
$233$	1.03473e13	0.987115	0.493557	−	0.869713i	$-0.335696\pi$
$233$	1.03473e13	0.987115	0.493557	+	0.869713i	$0.335696\pi$
$234$	0	0
$235$	−6.15905e11	−0.0560584
$236$	0	0
$237$	−6.28810e12	−0.546265
$238$	0	0
$239$	−1.85140e12	−0.153572	−0.0767859	−	0.997048i	$-0.524466\pi$
$239$	−1.85140e12	−0.153572	−0.0767859	+	0.997048i	$0.524466\pi$
$240$	0	0
$241$	2.73018e12	0.216321	0.108160	−	0.994133i	$-0.465504\pi$
$241$	2.73018e12	0.216321	0.108160	+	0.994133i	$0.465504\pi$
$242$	0	0
$243$	−1.06521e13	−0.806492
$244$	0	0
$245$	5.38178e12	0.389504
$246$	0	0
$247$	8.26518e12	0.572031
$248$	0	0
$249$	−1.69605e13	−1.12290
$250$	0	0
$251$	−1.18976e13	−0.753796	−0.376898	−	0.926255i	$-0.623009\pi$
$251$	−1.18976e13	−0.753796	−0.376898	+	0.926255i	$0.623009\pi$
$252$	0	0
$253$	9.73316e12	0.590324
$254$	0	0
$255$	3.85380e12	0.223830
$256$	0	0
$257$	3.53878e10	0.00196889	0.000984443	−	1.00000i	$-0.499687\pi$
$257$	3.53878e10	0.00196889	0.000984443		1.00000i	$0.499687\pi$
$258$	0	0
$259$	7.30442e12	0.389437
$260$	0	0
$261$	−7.16144e11	−0.0365997
$262$	0	0
$263$	1.19904e13	0.587592	0.293796	−	0.955868i	$-0.405081\pi$
$263$	1.19904e13	0.587592	0.293796	+	0.955868i	$0.405081\pi$
$264$	0	0
$265$	−1.61216e13	−0.757801
$266$	0	0
$267$	1.48089e13	0.667901
$268$	0	0
$269$	−3.81149e13	−1.64990	−0.824948	−	0.565208i	$-0.808796\pi$
$269$	−3.81149e13	−1.64990	−0.824948	+	0.565208i	$0.808796\pi$
$270$	0	0
$271$	1.13510e13	0.471740	0.235870	−	0.971785i	$-0.424206\pi$
$271$	1.13510e13	0.471740	0.235870	+	0.971785i	$0.424206\pi$
$272$	0	0
$273$	−1.62637e13	−0.649122
$274$	0	0
$275$	−3.31767e12	−0.127205
$276$	0	0
$277$	3.02533e13	1.11464	0.557320	−	0.830298i	$-0.311830\pi$
$277$	3.02533e13	1.11464	0.557320	+	0.830298i	$0.311830\pi$
$278$	0	0
$279$	−2.27801e13	−0.806736
$280$	0	0
$281$	4.72047e13	1.60731	0.803657	−	0.595093i	$-0.202885\pi$
$281$	4.72047e13	1.60731	0.803657	+	0.595093i	$0.202885\pi$
$282$	0	0
$283$	8.68805e12	0.284510	0.142255	−	0.989830i	$-0.454565\pi$
$283$	8.68805e12	0.284510	0.142255	+	0.989830i	$0.454565\pi$
$284$	0	0
$285$	−6.42401e12	−0.202376
$286$	0	0
$287$	−2.93617e12	−0.0890085
$288$	0	0
$289$	−2.82663e13	−0.824766
$290$	0	0
$291$	−5.69958e13	−1.60115
$292$	0	0
$293$	2.17869e13	0.589417	0.294709	−	0.955587i	$-0.404777\pi$
$293$	2.17869e13	0.589417	0.294709	+	0.955587i	$0.404777\pi$
$294$	0	0
$295$	−2.06938e12	−0.0539285
$296$	0	0
$297$	1.72773e13	0.433827
$298$	0	0
$299$	−5.79665e13	−1.40277
$300$	0	0
$301$	1.04917e13	0.244754
$302$	0	0
$303$	6.01673e13	1.35340
$304$	0	0
$305$	−1.74468e12	−0.0378500
$306$	0	0
$307$	−2.05380e13	−0.429829	−0.214915	−	0.976633i	$-0.568947\pi$
$307$	−2.05380e13	−0.429829	−0.214915	+	0.976633i	$0.568947\pi$
$308$	0	0
$309$	4.76488e13	0.962234
$310$	0	0
$311$	7.37156e13	1.43674	0.718369	−	0.695663i	$-0.244889\pi$
$311$	7.37156e13	1.43674	0.718369	+	0.695663i	$0.244889\pi$
$312$	0	0
$313$	−2.82026e13	−0.530634	−0.265317	−	0.964161i	$-0.585477\pi$
$313$	−2.82026e13	−0.530634	−0.265317	+	0.964161i	$0.585477\pi$
$314$	0	0
$315$	3.79803e12	0.0690003
$316$	0	0
$317$	2.56938e13	0.450819	0.225410	−	0.974264i	$-0.427628\pi$
$317$	2.56938e13	0.450819	0.225410	+	0.974264i	$0.427628\pi$
$318$	0	0
$319$	3.19763e12	0.0541976
$320$	0	0
$321$	−7.68771e13	−1.25898
$322$	0	0
$323$	−1.00109e13	−0.158439
$324$	0	0
$325$	1.97586e13	0.302272
$326$	0	0
$327$	1.43458e14	2.12184
$328$	0	0
$329$	−3.14824e12	−0.0450288
$330$	0	0
$331$	6.14309e13	0.849831	0.424916	−	0.905233i	$-0.360304\pi$
$331$	6.14309e13	0.849831	0.424916	+	0.905233i	$0.360304\pi$
$332$	0	0
$333$	−3.47925e13	−0.465632
$334$	0	0
$335$	3.16269e13	0.409552
$336$	0	0
$337$	1.35566e14	1.69898	0.849488	−	0.527608i	$-0.176911\pi$
$337$	1.35566e14	1.69898	0.849488	+	0.527608i	$0.176911\pi$
$338$	0	0
$339$	−1.18540e14	−1.43803
$340$	0	0
$341$	1.01715e14	1.19463
$342$	0	0
$343$	5.90945e13	0.672093
$344$	0	0
$345$	4.50537e13	0.496279
$346$	0	0
$347$	2.88020e13	0.307334	0.153667	−	0.988123i	$-0.450892\pi$
$347$	2.88020e13	0.307334	0.153667	+	0.988123i	$0.450892\pi$
$348$	0	0
$349$	3.70742e13	0.383294	0.191647	−	0.981464i	$-0.438617\pi$
$349$	3.70742e13	0.383294	0.191647	+	0.981464i	$0.438617\pi$
$350$	0	0
$351$	−1.02896e14	−1.03089
$352$	0	0
$353$	−5.64627e13	−0.548278	−0.274139	−	0.961690i	$-0.588393\pi$
$353$	−5.64627e13	−0.548278	−0.274139	+	0.961690i	$0.588393\pi$
$354$	0	0
$355$	5.56754e13	0.524092
$356$	0	0
$357$	1.96989e13	0.179791
$358$	0	0
$359$	1.96868e14	1.74243	0.871217	−	0.490899i	$-0.163332\pi$
$359$	1.96868e14	1.74243	0.871217	+	0.490899i	$0.163332\pi$
$360$	0	0
$361$	−9.98027e13	−0.856747
$362$	0	0
$363$	−8.54954e13	−0.711961
$364$	0	0
$365$	7.29313e13	0.589255
$366$	0	0
$367$	1.00444e14	0.787514	0.393757	−	0.919214i	$-0.371175\pi$
$367$	1.00444e14	0.787514	0.393757	+	0.919214i	$0.371175\pi$
$368$	0	0
$369$	1.39856e13	0.106423
$370$	0	0
$371$	−8.24065e13	−0.608703
$372$	0	0
$373$	−7.09849e13	−0.509058	−0.254529	−	0.967065i	$-0.581921\pi$
$373$	−7.09849e13	−0.509058	−0.254529	+	0.967065i	$0.581921\pi$
$374$	0	0
$375$	−1.53571e13	−0.106940
$376$	0	0
$377$	−1.90437e13	−0.128788
$378$	0	0
$379$	−6.79976e13	−0.446661	−0.223330	−	0.974743i	$-0.571693\pi$
$379$	−6.79976e13	−0.446661	−0.223330	+	0.974743i	$0.571693\pi$
$380$	0	0
$381$	6.32982e13	0.403928
$382$	0	0
$383$	−1.89007e14	−1.17188	−0.585942	−	0.810353i	$-0.699275\pi$
$383$	−1.89007e14	−1.17188	−0.585942	+	0.810353i	$0.699275\pi$
$384$	0	0
$385$	−1.69585e13	−0.102177
$386$	0	0
$387$	−4.99741e13	−0.292641
$388$	0	0
$389$	−2.90163e14	−1.65165	−0.825826	−	0.563924i	$-0.809291\pi$
$389$	−2.90163e14	−1.65165	−0.825826	+	0.563924i	$0.809291\pi$
$390$	0	0
$391$	7.02100e13	0.388532
$392$	0	0
$393$	1.49632e14	0.805127
$394$	0	0
$395$	3.90489e13	0.204327
$396$	0	0
$397$	−5.15115e13	−0.262154	−0.131077	−	0.991372i	$-0.541843\pi$
$397$	−5.15115e13	−0.262154	−0.131077	+	0.991372i	$0.541843\pi$
$398$	0	0
$399$	−3.28368e13	−0.162559
$400$	0	0
$401$	−1.11012e14	−0.534657	−0.267329	−	0.963605i	$-0.586141\pi$
$401$	−1.11012e14	−0.534657	−0.267329	+	0.963605i	$0.586141\pi$
$402$	0	0
$403$	−6.05768e14	−2.83876
$404$	0	0
$405$	1.22095e14	0.556795
$406$	0	0
$407$	1.55351e14	0.689517
$408$	0	0
$409$	−1.92951e14	−0.833623	−0.416811	−	0.908993i	$-0.636852\pi$
$409$	−1.92951e14	−0.833623	−0.416811	+	0.908993i	$0.636852\pi$
$410$	0	0
$411$	−3.16629e13	−0.133174
$412$	0	0
$413$	−1.05778e13	−0.0433180
$414$	0	0
$415$	1.05324e14	0.420013
$416$	0	0
$417$	2.57356e14	0.999508
$418$	0	0
$419$	−2.28750e14	−0.865336	−0.432668	−	0.901553i	$-0.642428\pi$
$419$	−2.28750e14	−0.865336	−0.432668	+	0.901553i	$0.642428\pi$
$420$	0	0
$421$	−4.00332e14	−1.47526	−0.737630	−	0.675205i	$-0.764055\pi$
$421$	−4.00332e14	−1.47526	−0.737630	+	0.675205i	$0.764055\pi$
$422$	0	0
$423$	1.49958e13	0.0538389
$424$	0	0
$425$	−2.39320e13	−0.0837220
$426$	0	0
$427$	−8.91803e12	−0.0304030
$428$	0	0
$429$	−3.45899e14	−1.14930
$430$	0	0
$431$	−1.29757e14	−0.420250	−0.210125	−	0.977675i	$-0.567387\pi$
$431$	−1.29757e14	−0.420250	−0.210125	+	0.977675i	$0.567387\pi$
$432$	0	0
$433$	−3.09354e14	−0.976725	−0.488363	−	0.872641i	$-0.662406\pi$
$433$	−3.09354e14	−0.976725	−0.488363	+	0.872641i	$0.662406\pi$
$434$	0	0
$435$	1.48015e13	0.0455633
$436$	0	0
$437$	−1.17035e14	−0.351293
$438$	0	0
$439$	6.15652e13	0.180211	0.0901054	−	0.995932i	$-0.471280\pi$
$439$	6.15652e13	0.180211	0.0901054	+	0.995932i	$0.471280\pi$
$440$	0	0
$441$	−1.31033e14	−0.374083
$442$	0	0
$443$	2.42894e13	0.0676389	0.0338194	−	0.999428i	$-0.489233\pi$
$443$	2.42894e13	0.0676389	0.0338194	+	0.999428i	$0.489233\pi$
$444$	0	0
$445$	−9.19631e13	−0.249824
$446$	0	0
$447$	3.99636e14	1.05919
$448$	0	0
$449$	6.65728e14	1.72164	0.860820	−	0.508910i	$-0.169951\pi$
$449$	6.65728e14	1.72164	0.860820	+	0.508910i	$0.169951\pi$
$450$	0	0
$451$	−6.24468e13	−0.157594
$452$	0	0
$453$	−6.27160e14	−1.54468
$454$	0	0
$455$	1.00997e14	0.242800
$456$	0	0
$457$	6.99716e14	1.64204	0.821018	−	0.570902i	$-0.193406\pi$
$457$	6.99716e14	1.64204	0.821018	+	0.570902i	$0.193406\pi$
$458$	0	0
$459$	1.24630e14	0.285531
$460$	0	0
$461$	3.43320e14	0.767971	0.383985	−	0.923339i	$-0.374551\pi$
$461$	3.43320e14	0.767971	0.383985	+	0.923339i	$0.374551\pi$
$462$	0	0
$463$	−3.77708e14	−0.825012	−0.412506	−	0.910955i	$-0.635346\pi$
$463$	−3.77708e14	−0.825012	−0.412506	+	0.910955i	$0.635346\pi$
$464$	0	0
$465$	4.70826e14	1.00431
$466$	0	0
$467$	5.01733e14	1.04527	0.522637	−	0.852555i	$-0.324948\pi$
$467$	5.01733e14	1.04527	0.522637	+	0.852555i	$0.324948\pi$
$468$	0	0
$469$	1.61663e14	0.328972
$470$	0	0
$471$	4.89897e14	0.973844
$472$	0	0
$473$	2.23138e14	0.433348
$474$	0	0
$475$	3.98930e13	0.0756975
$476$	0	0
$477$	3.92520e14	0.727798
$478$	0	0
$479$	−9.56649e14	−1.73343	−0.866717	−	0.498800i	$-0.833774\pi$
$479$	−9.56649e14	−1.73343	−0.866717	+	0.498800i	$0.833774\pi$
$480$	0	0
$481$	−9.25204e14	−1.63847
$482$	0	0
$483$	2.30295e14	0.398635
$484$	0	0
$485$	3.53942e14	0.598898
$486$	0	0
$487$	−1.81856e14	−0.300828	−0.150414	−	0.988623i	$-0.548061\pi$
$487$	−1.81856e14	−0.300828	−0.150414	+	0.988623i	$0.548061\pi$
$488$	0	0
$489$	−6.34485e14	−1.02618
$490$	0	0
$491$	−9.28536e14	−1.46842	−0.734210	−	0.678922i	$-0.762447\pi$
$491$	−9.28536e14	−1.46842	−0.734210	+	0.678922i	$0.762447\pi$
$492$	0	0
$493$	2.30661e13	0.0356711
$494$	0	0
$495$	8.07770e13	0.122168
$496$	0	0
$497$	2.84589e14	0.420976
$498$	0	0
$499$	−1.10604e15	−1.60036	−0.800178	−	0.599763i	$-0.795261\pi$
$499$	−1.10604e15	−1.60036	−0.800178	+	0.599763i	$0.795261\pi$
$500$	0	0
$501$	1.51670e15	2.14681
$502$	0	0
$503$	1.07467e15	1.48816	0.744081	−	0.668089i	$-0.232888\pi$
$503$	1.07467e15	1.48816	0.744081	+	0.668089i	$0.232888\pi$
$504$	0	0
$505$	−3.73637e14	−0.506230
$506$	0	0
$507$	1.15817e15	1.53542
$508$	0	0
$509$	−1.50116e15	−1.94750	−0.973752	−	0.227610i	$-0.926909\pi$
$509$	−1.50116e15	−1.94750	−0.973752	+	0.227610i	$0.926909\pi$
$510$	0	0
$511$	3.72793e14	0.473318
$512$	0	0
$513$	−2.07750e14	−0.258163
$514$	0	0
$515$	−2.95898e14	−0.359917
$516$	0	0
$517$	−6.69571e13	−0.0797258
$518$	0	0
$519$	−1.54732e14	−0.180368
$520$	0	0
$521$	−8.31060e13	−0.0948473	−0.0474237	−	0.998875i	$-0.515101\pi$
$521$	−8.31060e13	−0.0948473	−0.0474237	+	0.998875i	$0.515101\pi$
$522$	0	0
$523$	9.42223e14	1.05292	0.526459	−	0.850201i	$-0.323519\pi$
$523$	9.42223e14	1.05292	0.526459	+	0.850201i	$0.323519\pi$
$524$	0	0
$525$	−7.84991e13	−0.0858990
$526$	0	0
$527$	7.33717e14	0.786267
$528$	0	0
$529$	−1.32002e14	−0.138540
$530$	0	0
$531$	5.03841e13	0.0517933
$532$	0	0
$533$	3.71906e14	0.374485
$534$	0	0
$535$	4.77405e14	0.470913
$536$	0	0
$537$	9.63519e14	0.931112
$538$	0	0
$539$	5.85071e14	0.553950
$540$	0	0
$541$	1.18229e15	1.09683	0.548416	−	0.836206i	$-0.315231\pi$
$541$	1.18229e15	1.09683	0.548416	+	0.836206i	$0.315231\pi$
$542$	0	0
$543$	5.55056e13	0.0504589
$544$	0	0
$545$	−8.90870e14	−0.793658
$546$	0	0
$547$	−1.57701e15	−1.37691	−0.688453	−	0.725281i	$-0.741710\pi$
$547$	−1.57701e15	−1.37691	−0.688453	+	0.725281i	$0.741710\pi$
$548$	0	0
$549$	4.24785e13	0.0363514
$550$	0	0
$551$	−3.84496e13	−0.0322521
$552$	0	0
$553$	1.99601e14	0.164125
$554$	0	0
$555$	7.19104e14	0.579669
$556$	0	0
$557$	4.53070e14	0.358065	0.179033	−	0.983843i	$-0.442703\pi$
$557$	4.53070e14	0.358065	0.179033	+	0.983843i	$0.442703\pi$
$558$	0	0
$559$	−1.32891e15	−1.02975
$560$	0	0
$561$	4.18959e14	0.318329
$562$	0	0
$563$	−4.23450e13	−0.0315505	−0.0157752	−	0.999876i	$-0.505022\pi$
$563$	−4.23450e13	−0.0315505	−0.0157752	+	0.999876i	$0.505022\pi$
$564$	0	0
$565$	7.36133e14	0.537885
$566$	0	0
$567$	6.24097e14	0.447245
$568$	0	0
$569$	−9.72594e14	−0.683619	−0.341810	−	0.939769i	$-0.611040\pi$
$569$	−9.72594e14	−0.683619	−0.341810	+	0.939769i	$0.611040\pi$
$570$	0	0
$571$	2.07663e15	1.43173	0.715863	−	0.698241i	$-0.246034\pi$
$571$	2.07663e15	1.43173	0.715863	+	0.698241i	$0.246034\pi$
$572$	0	0
$573$	2.23426e15	1.51107
$574$	0	0
$575$	−2.79783e14	−0.185630
$576$	0	0
$577$	2.62999e15	1.71194	0.855968	−	0.517028i	$-0.172962\pi$
$577$	2.62999e15	1.71194	0.855968	+	0.517028i	$0.172962\pi$
$578$	0	0
$579$	1.58874e15	1.01466
$580$	0	0
$581$	5.38371e14	0.337375
$582$	0	0
$583$	−1.75263e15	−1.07774
$584$	0	0
$585$	−4.81073e14	−0.290304
$586$	0	0
$587$	2.89275e15	1.71317	0.856587	−	0.516003i	$-0.172581\pi$
$587$	2.89275e15	1.71317	0.856587	+	0.516003i	$0.172581\pi$
$588$	0	0
$589$	−1.22306e15	−0.710906
$590$	0	0
$591$	−1.80469e15	−1.02960
$592$	0	0
$593$	−1.15734e15	−0.648125	−0.324063	−	0.946036i	$-0.605049\pi$
$593$	−1.15734e15	−0.648125	−0.324063	+	0.946036i	$0.605049\pi$
$594$	0	0
$595$	−1.22330e14	−0.0672496
$596$	0	0
$597$	−2.08748e15	−1.12659
$598$	0	0
$599$	−2.42056e14	−0.128253	−0.0641265	−	0.997942i	$-0.520426\pi$
$599$	−2.42056e14	−0.128253	−0.0641265	+	0.997942i	$0.520426\pi$
$600$	0	0
$601$	−3.08465e15	−1.60471	−0.802354	−	0.596848i	$-0.796419\pi$
$601$	−3.08465e15	−1.60471	−0.802354	+	0.596848i	$0.796419\pi$
$602$	0	0
$603$	−7.70035e14	−0.393337
$604$	0	0
$605$	5.30924e14	0.266304
$606$	0	0
$607$	−2.71223e15	−1.33595	−0.667973	−	0.744186i	$-0.732838\pi$
$607$	−2.71223e15	−1.33595	−0.667973	+	0.744186i	$0.732838\pi$
$608$	0	0
$609$	7.56589e13	0.0365987
$610$	0	0
$611$	3.98768e14	0.189449
$612$	0	0
$613$	−1.80466e15	−0.842097	−0.421049	−	0.907038i	$-0.638338\pi$
$613$	−1.80466e15	−0.842097	−0.421049	+	0.907038i	$0.638338\pi$
$614$	0	0
$615$	−2.89060e14	−0.132487
$616$	0	0
$617$	4.28183e14	0.192780	0.0963898	−	0.995344i	$-0.469270\pi$
$617$	4.28183e14	0.192780	0.0963898	+	0.995344i	$0.469270\pi$
$618$	0	0
$619$	−2.39110e15	−1.05754	−0.528772	−	0.848764i	$-0.677347\pi$
$619$	−2.39110e15	−1.05754	−0.528772	+	0.848764i	$0.677347\pi$
$620$	0	0
$621$	1.45702e15	0.633083
$622$	0	0
$623$	−4.70076e14	−0.200671
$624$	0	0
$625$	9.53674e13	0.0400000
$626$	0	0
$627$	−6.98376e14	−0.287818
$628$	0	0
$629$	1.12062e15	0.453818
$630$	0	0
$631$	−3.26028e15	−1.29746	−0.648730	−	0.761019i	$-0.724699\pi$
$631$	−3.26028e15	−1.29746	−0.648730	+	0.761019i	$0.724699\pi$
$632$	0	0
$633$	−5.75057e14	−0.224901
$634$	0	0
$635$	−3.93080e14	−0.151087
$636$	0	0
$637$	−3.48443e15	−1.31633
$638$	0	0
$639$	−1.35556e15	−0.503342
$640$	0	0
$641$	−2.32104e15	−0.847157	−0.423579	−	0.905859i	$-0.639226\pi$
$641$	−2.32104e15	−0.847157	−0.423579	+	0.905859i	$0.639226\pi$
$642$	0	0
$643$	2.55397e15	0.916336	0.458168	−	0.888866i	$-0.348506\pi$
$643$	2.55397e15	0.916336	0.458168	+	0.888866i	$0.348506\pi$
$644$	0	0
$645$	1.03288e15	0.364311
$646$	0	0
$647$	1.70408e15	0.590902	0.295451	−	0.955358i	$-0.404530\pi$
$647$	1.70408e15	0.590902	0.295451	+	0.955358i	$0.404530\pi$
$648$	0	0
$649$	−2.24969e14	−0.0766966
$650$	0	0
$651$	2.40666e15	0.806713
$652$	0	0
$653$	−2.20439e15	−0.726552	−0.363276	−	0.931682i	$-0.618342\pi$
$653$	−2.20439e15	−0.726552	−0.363276	+	0.931682i	$0.618342\pi$
$654$	0	0
$655$	−9.29208e14	−0.301152
$656$	0	0
$657$	−1.77570e15	−0.565925
$658$	0	0
$659$	−1.97759e14	−0.0619821	−0.0309910	−	0.999520i	$-0.509866\pi$
$659$	−1.97759e14	−0.0619821	−0.0309910	+	0.999520i	$0.509866\pi$
$660$	0	0
$661$	−3.43414e15	−1.05855	−0.529274	−	0.848451i	$-0.677536\pi$
$661$	−3.43414e15	−1.05855	−0.529274	+	0.848451i	$0.677536\pi$
$662$	0	0
$663$	−2.49514e15	−0.756433
$664$	0	0
$665$	2.03916e14	0.0608039
$666$	0	0
$667$	2.69660e14	0.0790905
$668$	0	0
$669$	−1.78588e15	−0.515239
$670$	0	0
$671$	−1.89670e14	−0.0538299
$672$	0	0
$673$	2.01250e15	0.561892	0.280946	−	0.959724i	$-0.409352\pi$
$673$	2.01250e15	0.561892	0.280946	+	0.959724i	$0.409352\pi$
$674$	0	0
$675$	−4.96643e14	−0.136418
$676$	0	0
$677$	−1.01903e15	−0.275392	−0.137696	−	0.990475i	$-0.543970\pi$
$677$	−1.01903e15	−0.275392	−0.137696	+	0.990475i	$0.543970\pi$
$678$	0	0
$679$	1.80920e15	0.481064
$680$	0	0
$681$	−1.96530e15	−0.514186
$682$	0	0
$683$	−2.86885e15	−0.738575	−0.369287	−	0.929315i	$-0.620398\pi$
$683$	−2.86885e15	−0.738575	−0.369287	+	0.929315i	$0.620398\pi$
$684$	0	0
$685$	1.96626e14	0.0498130
$686$	0	0
$687$	−2.44126e15	−0.608629
$688$	0	0
$689$	1.04379e16	2.56099
$690$	0	0
$691$	1.79931e15	0.434487	0.217243	−	0.976117i	$-0.430293\pi$
$691$	1.79931e15	0.434487	0.217243	+	0.976117i	$0.430293\pi$
$692$	0	0
$693$	4.12897e14	0.0981316
$694$	0	0
$695$	−1.59817e15	−0.373859
$696$	0	0
$697$	−4.50460e14	−0.103723
$698$	0	0
$699$	5.20697e15	1.18021
$700$	0	0
$701$	2.99337e15	0.667899	0.333949	−	0.942591i	$-0.391619\pi$
$701$	2.99337e15	0.667899	0.333949	+	0.942591i	$0.391619\pi$
$702$	0	0
$703$	−1.86800e15	−0.410321
$704$	0	0
$705$	−3.09937e14	−0.0670245
$706$	0	0
$707$	−1.90987e15	−0.406629
$708$	0	0
$709$	2.74187e15	0.574769	0.287384	−	0.957815i	$-0.407214\pi$
$709$	2.74187e15	0.574769	0.287384	+	0.957815i	$0.407214\pi$
$710$	0	0
$711$	−9.50744e14	−0.196237
$712$	0	0
$713$	8.57770e15	1.74333
$714$	0	0
$715$	2.14802e15	0.429889
$716$	0	0
$717$	−9.31666e14	−0.183614
$718$	0	0
$719$	4.38021e15	0.850131	0.425065	−	0.905163i	$-0.360251\pi$
$719$	4.38021e15	0.850131	0.425065	+	0.905163i	$0.360251\pi$
$720$	0	0
$721$	−1.51250e15	−0.289103
$722$	0	0
$723$	1.37389e15	0.258637
$724$	0	0
$725$	−9.19170e13	−0.0170426
$726$	0	0
$727$	−6.60205e15	−1.20570	−0.602851	−	0.797854i	$-0.705969\pi$
$727$	−6.60205e15	−1.20570	−0.602851	+	0.797854i	$0.705969\pi$
$728$	0	0
$729$	1.56084e15	0.280773
$730$	0	0
$731$	1.60960e15	0.285216
$732$	0	0
$733$	−8.22795e15	−1.43622	−0.718108	−	0.695932i	$-0.754992\pi$
$733$	−8.22795e15	−1.43622	−0.718108	+	0.695932i	$0.754992\pi$
$734$	0	0
$735$	2.70823e15	0.465699
$736$	0	0
$737$	3.43826e15	0.582462
$738$	0	0
$739$	1.10770e16	1.84875	0.924376	−	0.381483i	$-0.124586\pi$
$739$	1.10770e16	1.84875	0.924376	+	0.381483i	$0.124586\pi$
$740$	0	0
$741$	4.15923e15	0.683931
$742$	0	0
$743$	3.86160e15	0.625646	0.312823	−	0.949811i	$-0.398725\pi$
$743$	3.86160e15	0.625646	0.312823	+	0.949811i	$0.398725\pi$
$744$	0	0
$745$	−2.48173e15	−0.396182
$746$	0	0
$747$	−2.56438e15	−0.403384
$748$	0	0
$749$	2.44029e15	0.378260
$750$	0	0
$751$	−1.05947e15	−0.161834	−0.0809168	−	0.996721i	$-0.525785\pi$
$751$	−1.05947e15	−0.161834	−0.0809168	+	0.996721i	$0.525785\pi$
$752$	0	0
$753$	−5.98714e15	−0.901254
$754$	0	0
$755$	3.89465e15	0.577776
$756$	0	0
$757$	−7.04375e15	−1.02986	−0.514928	−	0.857233i	$-0.672182\pi$
$757$	−7.04375e15	−1.02986	−0.514928	+	0.857233i	$0.672182\pi$
$758$	0	0
$759$	4.89794e15	0.705804
$760$	0	0
$761$	3.63598e15	0.516424	0.258212	−	0.966088i	$-0.416867\pi$
$761$	3.63598e15	0.516424	0.258212	+	0.966088i	$0.416867\pi$
$762$	0	0
$763$	−4.55374e15	−0.637505
$764$	0	0
$765$	5.82684e14	0.0804072
$766$	0	0
$767$	1.33982e15	0.182251
$768$	0	0
$769$	−5.34829e15	−0.717167	−0.358583	−	0.933498i	$-0.616740\pi$
$769$	−5.34829e15	−0.717167	−0.358583	+	0.933498i	$0.616740\pi$
$770$	0	0
$771$	1.78079e13	0.00235404
$772$	0	0
$773$	8.06669e15	1.05126	0.525628	−	0.850715i	$-0.323831\pi$
$773$	8.06669e15	1.05126	0.525628	+	0.850715i	$0.323831\pi$
$774$	0	0
$775$	−2.92382e15	−0.375656
$776$	0	0
$777$	3.67575e15	0.465618
$778$	0	0
$779$	7.50886e14	0.0937816
$780$	0	0
$781$	6.05266e15	0.745359
$782$	0	0
$783$	4.78674e14	0.0581233
$784$	0	0
$785$	−3.04225e15	−0.364260
$786$	0	0
$787$	8.56068e15	1.01076	0.505379	−	0.862897i	$-0.331353\pi$
$787$	8.56068e15	1.01076	0.505379	+	0.862897i	$0.331353\pi$
$788$	0	0
$789$	6.03383e15	0.702537
$790$	0	0
$791$	3.76279e15	0.432056
$792$	0	0
$793$	1.12959e15	0.127914
$794$	0	0
$795$	−8.11274e15	−0.906042
$796$	0	0
$797$	−1.47001e16	−1.61919	−0.809596	−	0.586987i	$-0.800314\pi$
$797$	−1.47001e16	−1.61919	−0.809596	+	0.586987i	$0.800314\pi$
$798$	0	0
$799$	−4.82995e14	−0.0524729
$800$	0	0
$801$	2.23907e15	0.239933
$802$	0	0
$803$	7.92861e15	0.838033
$804$	0	0
$805$	−1.43013e15	−0.149107
$806$	0	0
$807$	−1.91803e16	−1.97265
$808$	0	0
$809$	5.20792e15	0.528382	0.264191	−	0.964470i	$-0.414895\pi$
$809$	5.20792e15	0.528382	0.264191	+	0.964470i	$0.414895\pi$
$810$	0	0
$811$	−1.95571e16	−1.95745	−0.978724	−	0.205183i	$-0.934221\pi$
$811$	−1.95571e16	−1.95745	−0.978724	+	0.205183i	$0.934221\pi$
$812$	0	0
$813$	5.71207e15	0.564022
$814$	0	0
$815$	3.94014e15	0.383835
$816$	0	0
$817$	−2.68310e15	−0.257879
$818$	0	0
$819$	−2.45904e15	−0.233187
$820$	0	0
$821$	1.02440e16	0.958476	0.479238	−	0.877685i	$-0.340913\pi$
$821$	1.02440e16	0.958476	0.479238	+	0.877685i	$0.340913\pi$
$822$	0	0
$823$	1.40372e16	1.29593	0.647964	−	0.761671i	$-0.275621\pi$
$823$	1.40372e16	1.29593	0.647964	+	0.761671i	$0.275621\pi$
$824$	0	0
$825$	−1.66953e15	−0.152089
$826$	0	0
$827$	−8.80160e14	−0.0791191	−0.0395596	−	0.999217i	$-0.512595\pi$
$827$	−8.80160e14	−0.0791191	−0.0395596	+	0.999217i	$0.512595\pi$
$828$	0	0
$829$	4.54642e14	0.0403292	0.0201646	−	0.999797i	$-0.493581\pi$
$829$	4.54642e14	0.0403292	0.0201646	+	0.999797i	$0.493581\pi$
$830$	0	0
$831$	1.52242e16	1.33269
$832$	0	0
$833$	4.22041e15	0.364592
$834$	0	0
$835$	−9.41870e15	−0.802999
$836$	0	0
$837$	1.52263e16	1.28116
$838$	0	0
$839$	−1.32346e16	−1.09906	−0.549529	−	0.835474i	$-0.685193\pi$
$839$	−1.32346e16	−1.09906	−0.549529	+	0.835474i	$0.685193\pi$
$840$	0	0
$841$	−1.21119e16	−0.992739
$842$	0	0
$843$	2.37545e16	1.92174
$844$	0	0
$845$	−7.19220e15	−0.574315
$846$	0	0
$847$	2.71386e15	0.213908
$848$	0	0
$849$	4.37202e15	0.340166
$850$	0	0
$851$	1.31009e16	1.00621
$852$	0	0
$853$	−2.60074e15	−0.197186	−0.0985932	−	0.995128i	$-0.531434\pi$
$853$	−2.60074e15	−0.197186	−0.0985932	+	0.995128i	$0.531434\pi$
$854$	0	0
$855$	−9.71294e14	−0.0727005
$856$	0	0
$857$	−1.68895e16	−1.24802	−0.624011	−	0.781415i	$-0.714498\pi$
$857$	−1.68895e16	−1.24802	−0.624011	+	0.781415i	$0.714498\pi$
$858$	0	0
$859$	−8.98241e15	−0.655285	−0.327643	−	0.944802i	$-0.606254\pi$
$859$	−8.98241e15	−0.655285	−0.327643	+	0.944802i	$0.606254\pi$
$860$	0	0
$861$	−1.47755e15	−0.106420
$862$	0	0
$863$	1.16178e16	0.826164	0.413082	−	0.910694i	$-0.364452\pi$
$863$	1.16178e16	0.826164	0.413082	+	0.910694i	$0.364452\pi$
$864$	0	0
$865$	9.60880e14	0.0674653
$866$	0	0
$867$	−1.42242e16	−0.986107
$868$	0	0
$869$	4.24514e15	0.290592
$870$	0	0
$871$	−2.04768e16	−1.38408
$872$	0	0
$873$	−8.61761e15	−0.575186
$874$	0	0
$875$	4.87477e14	0.0321300
$876$	0	0
$877$	4.73202e15	0.307999	0.153999	−	0.988071i	$-0.450785\pi$
$877$	4.73202e15	0.307999	0.153999	+	0.988071i	$0.450785\pi$
$878$	0	0
$879$	1.09636e16	0.704719
$880$	0	0
$881$	2.81982e15	0.179000	0.0895002	−	0.995987i	$-0.471473\pi$
$881$	2.81982e15	0.179000	0.0895002	+	0.995987i	$0.471473\pi$
$882$	0	0
$883$	−1.91741e16	−1.20208	−0.601038	−	0.799221i	$-0.705246\pi$
$883$	−1.91741e16	−1.20208	−0.601038	+	0.799221i	$0.705246\pi$
$884$	0	0
$885$	−1.04136e15	−0.0644780
$886$	0	0
$887$	−2.08389e15	−0.127437	−0.0637183	−	0.997968i	$-0.520296\pi$
$887$	−2.08389e15	−0.127437	−0.0637183	+	0.997968i	$0.520296\pi$
$888$	0	0
$889$	−2.00926e15	−0.121360
$890$	0	0
$891$	1.32734e16	0.791869
$892$	0	0
$893$	8.05119e14	0.0474435
$894$	0	0
$895$	−5.98343e15	−0.348276
$896$	0	0
$897$	−2.91700e16	−1.67718
$898$	0	0
$899$	2.81803e15	0.160054
$900$	0	0
$901$	−1.26426e16	−0.709332
$902$	0	0
$903$	5.27965e15	0.292632
$904$	0	0
$905$	−3.44689e14	−0.0188738
$906$	0	0
$907$	−3.20443e16	−1.73345	−0.866723	−	0.498791i	$-0.833778\pi$
$907$	−3.20443e16	−1.73345	−0.866723	+	0.498791i	$0.833778\pi$
$908$	0	0
$909$	9.09714e15	0.486187
$910$	0	0
$911$	8.44282e15	0.445797	0.222898	−	0.974842i	$-0.428448\pi$
$911$	8.44282e15	0.445797	0.222898	+	0.974842i	$0.428448\pi$
$912$	0	0
$913$	1.14501e16	0.597340
$914$	0	0
$915$	−8.77960e14	−0.0452542
$916$	0	0
$917$	−4.74971e15	−0.241900
$918$	0	0
$919$	1.08277e16	0.544880	0.272440	−	0.962173i	$-0.412169\pi$
$919$	1.08277e16	0.544880	0.272440	+	0.962173i	$0.412169\pi$
$920$	0	0
$921$	−1.03352e16	−0.513913
$922$	0	0
$923$	−3.60470e16	−1.77117
$924$	0	0
$925$	−4.46562e15	−0.216821
$926$	0	0
$927$	7.20438e15	0.345667
$928$	0	0
$929$	−1.30338e16	−0.617994	−0.308997	−	0.951063i	$-0.599993\pi$
$929$	−1.30338e16	−0.617994	−0.308997	+	0.951063i	$0.599993\pi$
$930$	0	0
$931$	−7.03513e15	−0.329647
$932$	0	0
$933$	3.70954e16	1.71779
$934$	0	0
$935$	−2.60173e15	−0.119069
$936$	0	0
$937$	−2.06679e16	−0.934819	−0.467410	−	0.884041i	$-0.654813\pi$
$937$	−2.06679e16	−0.934819	−0.467410	+	0.884041i	$0.654813\pi$
$938$	0	0
$939$	−1.41922e16	−0.634436
$940$	0	0
$941$	−3.48692e16	−1.54063	−0.770317	−	0.637661i	$-0.779902\pi$
$941$	−3.48692e16	−1.54063	−0.770317	+	0.637661i	$0.779902\pi$
$942$	0	0
$943$	−5.26621e15	−0.229977
$944$	0	0
$945$	−2.53862e15	−0.109578
$946$	0	0
$947$	2.53080e15	0.107977	0.0539887	−	0.998542i	$-0.482807\pi$
$947$	2.53080e15	0.107977	0.0539887	+	0.998542i	$0.482807\pi$
$948$	0	0
$949$	−4.72194e16	−1.99139
$950$	0	0
$951$	1.29297e16	0.539008
$952$	0	0
$953$	−1.68411e16	−0.693999	−0.347000	−	0.937865i	$-0.612799\pi$
$953$	−1.68411e16	−0.693999	−0.347000	+	0.937865i	$0.612799\pi$
$954$	0	0
$955$	−1.38747e16	−0.565205
$956$	0	0
$957$	1.60912e15	0.0647997
$958$	0	0
$959$	1.00507e15	0.0400122
$960$	0	0
$961$	6.42312e16	2.52794
$962$	0	0
$963$	−1.16236e16	−0.452269
$964$	0	0
$965$	−9.86603e15	−0.379527
$966$	0	0
$967$	2.82761e16	1.07541	0.537705	−	0.843133i	$-0.319292\pi$
$967$	2.82761e16	1.07541	0.537705	+	0.843133i	$0.319292\pi$
$968$	0	0
$969$	−5.03773e15	−0.189432
$970$	0	0
$971$	−2.55089e16	−0.948386	−0.474193	−	0.880421i	$-0.657260\pi$
$971$	−2.55089e16	−0.948386	−0.474193	+	0.880421i	$0.657260\pi$
$972$	0	0
$973$	−8.16918e15	−0.300302
$974$	0	0
$975$	9.94298e15	0.361402
$976$	0	0
$977$	5.16659e16	1.85688	0.928441	−	0.371480i	$-0.121150\pi$
$977$	5.16659e16	1.85688	0.928441	+	0.371480i	$0.121150\pi$
$978$	0	0
$979$	−9.99762e15	−0.355297
$980$	0	0
$981$	2.16905e16	0.762236
$982$	0	0
$983$	−1.06975e15	−0.0371740	−0.0185870	−	0.999827i	$-0.505917\pi$
$983$	−1.06975e15	−0.0371740	−0.0185870	+	0.999827i	$0.505917\pi$
$984$	0	0
$985$	1.12071e16	0.385117
$986$	0	0
$987$	−1.58427e15	−0.0538374
$988$	0	0
$989$	1.88175e16	0.632385
$990$	0	0
$991$	−2.05333e16	−0.682423	−0.341211	−	0.939987i	$-0.610837\pi$
$991$	−2.05333e16	−0.682423	−0.341211	+	0.939987i	$0.610837\pi$
$992$	0	0
$993$	3.09134e16	1.01608
$994$	0	0
$995$	1.29632e16	0.421391
$996$	0	0
$997$	−1.61931e16	−0.520603	−0.260301	−	0.965527i	$-0.583822\pi$
$997$	−1.61931e16	−0.520603	−0.260301	+	0.965527i	$0.583822\pi$
$998$	0	0
$999$	2.32555e16	0.739461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.12.a.j.1.2		2
4.3	odd	2		5.12.a.b.1.1	✓	2
12.11	even	2		45.12.a.d.1.2		2
20.3	even	4		25.12.b.c.24.4		4
20.7	even	4		25.12.b.c.24.1		4
20.19	odd	2		25.12.a.c.1.2		2
28.27	even	2		245.12.a.b.1.1		2
60.23	odd	4		225.12.b.f.199.1		4
60.47	odd	4		225.12.b.f.199.4		4
60.59	even	2		225.12.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.12.a.b.1.1	✓	2	4.3	odd	2
25.12.a.c.1.2		2	20.19	odd	2
25.12.b.c.24.1		4	20.7	even	4
25.12.b.c.24.4		4	20.3	even	4
45.12.a.d.1.2		2	12.11	even	2
80.12.a.j.1.2		2	1.1	even	1	trivial
225.12.a.h.1.1		2	60.59	even	2
225.12.b.f.199.1		4	60.23	odd	4
225.12.b.f.199.4		4	60.47	odd	4
245.12.a.b.1.1		2	28.27	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)

\(f(q)\)	\(=\)	\(q+503.223 q^{3} -3125.00 q^{5} -15973.7 q^{7} +76086.0 q^{9} +O(q^{10})\)	\(q+503.223 q^{3} -3125.00 q^{5} -15973.7 q^{7} +76086.0 q^{9} -339729. q^{11} +2.02328e6 q^{13} -1.57257e6 q^{15} -2.45063e6 q^{17} +4.08504e6 q^{19} -8.03830e6 q^{21} -2.86497e7 q^{23} +9.76562e6 q^{25} -5.08562e7 q^{27} -9.41230e6 q^{29} -2.99399e8 q^{31} -1.70959e8 q^{33} +4.99177e7 q^{35} -4.57279e8 q^{37} +1.01816e9 q^{39} +1.83814e8 q^{41} -6.56811e8 q^{43} -2.37769e8 q^{45} +1.97090e8 q^{47} -1.72217e9 q^{49} -1.23321e9 q^{51} +5.15890e9 q^{53} +1.06165e9 q^{55} +2.05568e9 q^{57} +6.62200e8 q^{59} +5.58296e8 q^{61} -1.21537e9 q^{63} -6.32275e9 q^{65} -1.01206e10 q^{67} -1.44172e10 q^{69} -1.78161e10 q^{71} -2.33380e10 q^{73} +4.91428e9 q^{75} +5.42672e9 q^{77} -1.24957e10 q^{79} -3.90704e10 q^{81} -3.37037e10 q^{83} +7.65823e9 q^{85} -4.73648e9 q^{87} +2.94282e10 q^{89} -3.23192e10 q^{91} -1.50664e11 q^{93} -1.27657e10 q^{95} -1.13262e11 q^{97} -2.58486e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 220 q^{3} - 6250 q^{5} - 57900 q^{7} - 20846 q^{9}+O(q^{10}) \) 2 * q + 220 * q^3 - 6250 * q^5 - 57900 * q^7 - 20846 * q^9	\( 2 q + 220 q^{3} - 6250 q^{5} - 57900 q^{7} - 20846 q^{9} + 618176 q^{11} + 3414260 q^{13} - 687500 q^{15} + 1317940 q^{17} - 5325320 q^{19} + 3836184 q^{21} - 58943940 q^{23} + 19531250 q^{25} + 26769160 q^{27} + 94140380 q^{29} - 244543464 q^{31} - 442259840 q^{33} + 180937500 q^{35} + 21003220 q^{37} + 624203992 q^{39} - 745743316 q^{41} - 629950100 q^{43} + 65143750 q^{45} + 1402061540 q^{47} - 1941677414 q^{49} - 2300559784 q^{51} + 1138320580 q^{53} - 1931800000 q^{55} + 4720910480 q^{57} - 7317515560 q^{59} - 1516425676 q^{61} + 2848632180 q^{63} - 10669562500 q^{65} - 15734290140 q^{67} - 5837195832 q^{69} - 32938471544 q^{71} - 29982848860 q^{73} + 2148437500 q^{75} - 34734748800 q^{77} + 3302823120 q^{79} - 43884431798 q^{81} - 13299102420 q^{83} - 4118562500 q^{85} - 34064940920 q^{87} - 12674770860 q^{89} - 90637859064 q^{91} - 166200542640 q^{93} + 16641625000 q^{95} - 3080703740 q^{97} - 118700272448 q^{99}+O(q^{100}) \) 2 * q + 220 * q^3 - 6250 * q^5 - 57900 * q^7 - 20846 * q^9 + 618176 * q^11 + 3414260 * q^13 - 687500 * q^15 + 1317940 * q^17 - 5325320 * q^19 + 3836184 * q^21 - 58943940 * q^23 + 19531250 * q^25 + 26769160 * q^27 + 94140380 * q^29 - 244543464 * q^31 - 442259840 * q^33 + 180937500 * q^35 + 21003220 * q^37 + 624203992 * q^39 - 745743316 * q^41 - 629950100 * q^43 + 65143750 * q^45 + 1402061540 * q^47 - 1941677414 * q^49 - 2300559784 * q^51 + 1138320580 * q^53 - 1931800000 * q^55 + 4720910480 * q^57 - 7317515560 * q^59 - 1516425676 * q^61 + 2848632180 * q^63 - 10669562500 * q^65 - 15734290140 * q^67 - 5837195832 * q^69 - 32938471544 * q^71 - 29982848860 * q^73 + 2148437500 * q^75 - 34734748800 * q^77 + 3302823120 * q^79 - 43884431798 * q^81 - 13299102420 * q^83 - 4118562500 * q^85 - 34064940920 * q^87 - 12674770860 * q^89 - 90637859064 * q^91 - 166200542640 * q^93 + 16641625000 * q^95 - 3080703740 * q^97 - 118700272448 * q^99

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