LMFDB - Embedded newform 20.12.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,12,Mod(1,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("20.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	20.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:	$15.3668636112$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46729}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 11682 $ x^2 - x - 11682
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-107.584$ of defining polynomial
Character	$\chi$	$=$	20.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+542.338 q^{3} +3125.00 q^{5} +49659.5 q^{7} +116983. q^{9} +O(q^{10})$	$q+542.338 q^{3} +3125.00 q^{5} +49659.5 q^{7} +116983. q^{9} -19451.8 q^{11} -759548. q^{13} +1.69481e6 q^{15} +2.35438e6 q^{17} +1.62826e7 q^{19} +2.69322e7 q^{21} +3.07894e7 q^{23} +9.76562e6 q^{25} -3.26290e7 q^{27} -1.24708e8 q^{29} +2.87600e8 q^{31} -1.05494e7 q^{33} +1.55186e8 q^{35} -6.80788e8 q^{37} -4.11932e8 q^{39} -3.20702e7 q^{41} -1.32076e9 q^{43} +3.65573e8 q^{45} -2.20253e9 q^{47} +4.88739e8 q^{49} +1.27687e9 q^{51} +4.55551e9 q^{53} -6.07867e7 q^{55} +8.83065e9 q^{57} -1.70865e9 q^{59} -6.50342e9 q^{61} +5.80933e9 q^{63} -2.37359e9 q^{65} +6.87165e7 q^{67} +1.66983e10 q^{69} -2.55089e10 q^{71} +1.19267e10 q^{73} +5.29627e9 q^{75} -9.65964e8 q^{77} -9.68170e9 q^{79} -3.84192e10 q^{81} -1.07710e10 q^{83} +7.35742e9 q^{85} -6.76338e10 q^{87} -7.08101e10 q^{89} -3.77188e10 q^{91} +1.55976e11 q^{93} +5.08830e10 q^{95} -1.53217e10 q^{97} -2.27553e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 220 q^{3} + 6250 q^{5} + 3340 q^{7} + 43738 q^{9}+O(q^{10}) $ 2 * q + 220 * q^3 + 6250 * q^5 + 3340 * q^7 + 43738 * q^9	$ 2 q + 220 q^{3} + 6250 q^{5} + 3340 q^{7} + 43738 q^{9} + 298320 q^{11} + 1209820 q^{13} + 687500 q^{15} + 9056340 q^{17} + 19439368 q^{19} + 41862752 q^{21} + 55926420 q^{23} + 19531250 q^{25} + 48081880 q^{27} + 41841708 q^{29} + 62230792 q^{31} - 112979280 q^{33} + 10437500 q^{35} - 729235940 q^{37} - 1046733592 q^{39} - 608419068 q^{41} - 1129440740 q^{43} + 136681250 q^{45} - 1653072900 q^{47} + 656908386 q^{49} - 883429896 q^{51} + 5724887340 q^{53} + 932250000 q^{55} + 7813100240 q^{57} + 3756433896 q^{59} + 4923703564 q^{61} + 9202019900 q^{63} + 3780687500 q^{65} - 5843244140 q^{67} + 8595645984 q^{69} - 43352162664 q^{71} + 33647099620 q^{73} + 2148437500 q^{75} - 15684992880 q^{77} - 54799425296 q^{79} - 51460199582 q^{81} - 9303032100 q^{83} + 28301062500 q^{85} - 121318983000 q^{87} + 3688968372 q^{89} - 128938931512 q^{91} + 228621221120 q^{93} + 60748025000 q^{95} + 65892157780 q^{97} - 25550825520 q^{99}+O(q^{100}) $ 2 * q + 220 * q^3 + 6250 * q^5 + 3340 * q^7 + 43738 * q^9 + 298320 * q^11 + 1209820 * q^13 + 687500 * q^15 + 9056340 * q^17 + 19439368 * q^19 + 41862752 * q^21 + 55926420 * q^23 + 19531250 * q^25 + 48081880 * q^27 + 41841708 * q^29 + 62230792 * q^31 - 112979280 * q^33 + 10437500 * q^35 - 729235940 * q^37 - 1046733592 * q^39 - 608419068 * q^41 - 1129440740 * q^43 + 136681250 * q^45 - 1653072900 * q^47 + 656908386 * q^49 - 883429896 * q^51 + 5724887340 * q^53 + 932250000 * q^55 + 7813100240 * q^57 + 3756433896 * q^59 + 4923703564 * q^61 + 9202019900 * q^63 + 3780687500 * q^65 - 5843244140 * q^67 + 8595645984 * q^69 - 43352162664 * q^71 + 33647099620 * q^73 + 2148437500 * q^75 - 15684992880 * q^77 - 54799425296 * q^79 - 51460199582 * q^81 - 9303032100 * q^83 + 28301062500 * q^85 - 121318983000 * q^87 + 3688968372 * q^89 - 128938931512 * q^91 + 228621221120 * q^93 + 60748025000 * q^95 + 65892157780 * q^97 - 25550825520 * 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
$3$	542.338	1.28856	0.644278	−	0.764792i	$-0.277158\pi$
$3$	542.338	1.28856	0.644278	+	0.764792i	$0.277158\pi$
$4$	0	0
$5$	3125.00	0.447214
$5$	3125.00	0.447214
$6$	0	0
$7$	49659.5	1.11677	0.558384	−	0.829582i	$-0.311422\pi$
$7$	49659.5	1.11677	0.558384	+	0.829582i	$0.311422\pi$
$8$	0	0
$9$	116983.	0.660374
$10$	0	0
$11$	−19451.8	−0.0364166	−0.0182083	−	0.999834i	$-0.505796\pi$
$11$	−19451.8	−0.0364166	−0.0182083	+	0.999834i	$0.505796\pi$
$12$	0	0
$13$	−759548.	−0.567371	−0.283685	−	0.958917i	$-0.591557\pi$
$13$	−759548.	−0.567371	−0.283685	+	0.958917i	$0.591557\pi$
$14$	0	0
$15$	1.69481e6	0.576259
$16$	0	0
$17$	2.35438e6	0.402167	0.201084	−	0.979574i	$-0.435554\pi$
$17$	2.35438e6	0.402167	0.201084	+	0.979574i	$0.435554\pi$
$18$	0	0
$19$	1.62826e7	1.50861	0.754307	−	0.656522i	$-0.227973\pi$
$19$	1.62826e7	1.50861	0.754307	+	0.656522i	$0.227973\pi$
$20$	0	0
$21$	2.69322e7	1.43902
$22$	0	0
$23$	3.07894e7	0.997466	0.498733	−	0.866756i	$-0.333799\pi$
$23$	3.07894e7	0.997466	0.498733	+	0.866756i	$0.333799\pi$
$24$	0	0
$25$	9.76562e6	0.200000
$26$	0	0
$27$	−3.26290e7	−0.437626
$28$	0	0
$29$	−1.24708e8	−1.12903	−0.564514	−	0.825424i	$-0.690936\pi$
$29$	−1.24708e8	−1.12903	−0.564514	+	0.825424i	$0.690936\pi$
$30$	0	0
$31$	2.87600e8	1.80426	0.902130	−	0.431464i	$-0.142003\pi$
$31$	2.87600e8	1.80426	0.902130	+	0.431464i	$0.142003\pi$
$32$	0	0
$33$	−1.05494e7	−0.0469248
$34$	0	0
$35$	1.55186e8	0.499434
$36$	0	0
$37$	−6.80788e8	−1.61400	−0.806998	−	0.590554i	$-0.798909\pi$
$37$	−6.80788e8	−1.61400	−0.806998	+	0.590554i	$0.798909\pi$
$38$	0	0
$39$	−4.11932e8	−0.731088
$40$	0	0
$41$	−3.20702e7	−0.0432305	−0.0216152	−	0.999766i	$-0.506881\pi$
$41$	−3.20702e7	−0.0432305	−0.0216152	+	0.999766i	$0.506881\pi$
$42$	0	0
$43$	−1.32076e9	−1.37008	−0.685040	−	0.728505i	$-0.740215\pi$
$43$	−1.32076e9	−1.37008	−0.685040	+	0.728505i	$0.740215\pi$
$44$	0	0
$45$	3.65573e8	0.295328
$46$	0	0
$47$	−2.20253e9	−1.40082	−0.700411	−	0.713740i	$-0.747000\pi$
$47$	−2.20253e9	−1.40082	−0.700411	+	0.713740i	$0.747000\pi$
$48$	0	0
$49$	4.88739e8	0.247172
$50$	0	0
$51$	1.27687e9	0.518215
$52$	0	0
$53$	4.55551e9	1.49630	0.748151	−	0.663528i	$-0.230942\pi$
$53$	4.55551e9	1.49630	0.748151	+	0.663528i	$0.230942\pi$
$54$	0	0
$55$	−6.07867e7	−0.0162860
$56$	0	0
$57$	8.83065e9	1.94393
$58$	0	0
$59$	−1.70865e9	−0.311148	−0.155574	−	0.987824i	$-0.549723\pi$
$59$	−1.70865e9	−0.311148	−0.155574	+	0.987824i	$0.549723\pi$
$60$	0	0
$61$	−6.50342e9	−0.985888	−0.492944	−	0.870061i	$-0.664079\pi$
$61$	−6.50342e9	−0.985888	−0.492944	+	0.870061i	$0.664079\pi$
$62$	0	0
$63$	5.80933e9	0.737485
$64$	0	0
$65$	−2.37359e9	−0.253736
$66$	0	0
$67$	6.87165e7	0.00621798	0.00310899	−	0.999995i	$-0.499010\pi$
$67$	6.87165e7	0.00621798	0.00310899	+	0.999995i	$0.499010\pi$
$68$	0	0
$69$	1.66983e10	1.28529
$70$	0	0
$71$	−2.55089e10	−1.67792	−0.838958	−	0.544196i	$-0.816835\pi$
$71$	−2.55089e10	−1.67792	−0.838958	+	0.544196i	$0.816835\pi$
$72$	0	0
$73$	1.19267e10	0.673358	0.336679	−	0.941620i	$-0.390696\pi$
$73$	1.19267e10	0.673358	0.336679	+	0.941620i	$0.390696\pi$
$74$	0	0
$75$	5.29627e9	0.257711
$76$	0	0
$77$	−9.65964e8	−0.0406689
$78$	0	0
$79$	−9.68170e9	−0.354000	−0.177000	−	0.984211i	$-0.556639\pi$
$79$	−9.68170e9	−0.354000	−0.177000	+	0.984211i	$0.556639\pi$
$80$	0	0
$81$	−3.84192e10	−1.22428
$82$	0	0
$83$	−1.07710e10	−0.300142	−0.150071	−	0.988675i	$-0.547950\pi$
$83$	−1.07710e10	−0.300142	−0.150071	+	0.988675i	$0.547950\pi$
$84$	0	0
$85$	7.35742e9	0.179855
$86$	0	0
$87$	−6.76338e10	−1.45481
$88$	0	0
$89$	−7.08101e10	−1.34416	−0.672079	−	0.740480i	$-0.734598\pi$
$89$	−7.08101e10	−1.34416	−0.672079	+	0.740480i	$0.734598\pi$
$90$	0	0
$91$	−3.77188e10	−0.633622
$92$	0	0
$93$	1.55976e11	2.32489
$94$	0	0
$95$	5.08830e10	0.674673
$96$	0	0
$97$	−1.53217e10	−0.181160	−0.0905800	−	0.995889i	$-0.528872\pi$
$97$	−1.53217e10	−0.181160	−0.0905800	+	0.995889i	$0.528872\pi$
$98$	0	0
$99$	−2.27553e9	−0.0240486
$100$	0	0
$101$	2.01154e11	1.90441	0.952207	−	0.305453i	$-0.0988078\pi$
$101$	2.01154e11	1.90441	0.952207	+	0.305453i	$0.0988078\pi$
$102$	0	0
$103$	1.24282e11	1.05634	0.528168	−	0.849140i	$-0.322879\pi$
$103$	1.24282e11	1.05634	0.528168	+	0.849140i	$0.322879\pi$
$104$	0	0
$105$	8.41632e10	0.643548
$106$	0	0
$107$	−1.78965e11	−1.23355	−0.616774	−	0.787140i	$-0.711561\pi$
$107$	−1.78965e11	−1.23355	−0.616774	+	0.787140i	$0.711561\pi$
$108$	0	0
$109$	8.15715e10	0.507800	0.253900	−	0.967230i	$-0.418287\pi$
$109$	8.15715e10	0.507800	0.253900	+	0.967230i	$0.418287\pi$
$110$	0	0
$111$	−3.69217e11	−2.07972
$112$	0	0
$113$	2.55934e11	1.30676	0.653381	−	0.757029i	$-0.273350\pi$
$113$	2.55934e11	1.30676	0.653381	+	0.757029i	$0.273350\pi$
$114$	0	0
$115$	9.62169e10	0.446080
$116$	0	0
$117$	−8.88545e10	−0.374677
$118$	0	0
$119$	1.16917e11	0.449128
$120$	0	0
$121$	−2.84933e11	−0.998674
$122$	0	0
$123$	−1.73929e10	−0.0557048
$124$	0	0
$125$	3.05176e10	0.0894427
$126$	0	0
$127$	3.21115e11	0.862461	0.431231	−	0.902242i	$-0.358080\pi$
$127$	3.21115e11	0.862461	0.431231	+	0.902242i	$0.358080\pi$
$128$	0	0
$129$	−7.16296e11	−1.76542
$130$	0	0
$131$	−6.42126e11	−1.45421	−0.727107	−	0.686524i	$-0.759136\pi$
$131$	−6.42126e11	−1.45421	−0.727107	+	0.686524i	$0.759136\pi$
$132$	0	0
$133$	8.08584e11	1.68477
$134$	0	0
$135$	−1.01966e11	−0.195713
$136$	0	0
$137$	−4.79216e11	−0.848336	−0.424168	−	0.905583i	$-0.639433\pi$
$137$	−4.79216e11	−0.848336	−0.424168	+	0.905583i	$0.639433\pi$
$138$	0	0
$139$	6.77838e11	1.10801	0.554006	−	0.832513i	$-0.313099\pi$
$139$	6.77838e11	1.10801	0.554006	+	0.832513i	$0.313099\pi$
$140$	0	0
$141$	−1.19451e12	−1.80504
$142$	0	0
$143$	1.47745e10	0.0206617
$144$	0	0
$145$	−3.89712e11	−0.504917
$146$	0	0
$147$	2.65062e11	0.318494
$148$	0	0
$149$	−7.05810e11	−0.787342	−0.393671	−	0.919251i	$-0.628795\pi$
$149$	−7.05810e11	−0.787342	−0.393671	+	0.919251i	$0.628795\pi$
$150$	0	0
$151$	6.42190e11	0.665718	0.332859	−	0.942977i	$-0.391987\pi$
$151$	6.42190e11	0.665718	0.332859	+	0.942977i	$0.391987\pi$
$152$	0	0
$153$	2.75423e11	0.265581
$154$	0	0
$155$	8.98749e11	0.806890
$156$	0	0
$157$	1.25437e12	1.04949	0.524743	−	0.851261i	$-0.324161\pi$
$157$	1.25437e12	1.04949	0.524743	+	0.851261i	$0.324161\pi$
$158$	0	0
$159$	2.47062e12	1.92807
$160$	0	0
$161$	1.52899e12	1.11394
$162$	0	0
$163$	−5.85578e10	−0.0398615	−0.0199307	−	0.999801i	$-0.506345\pi$
$163$	−5.85578e10	−0.0398615	−0.0199307	+	0.999801i	$0.506345\pi$
$164$	0	0
$165$	−3.29669e10	−0.0209854
$166$	0	0
$167$	−1.63528e12	−0.974206	−0.487103	−	0.873344i	$-0.661946\pi$
$167$	−1.63528e12	−0.974206	−0.487103	+	0.873344i	$0.661946\pi$
$168$	0	0
$169$	−1.21525e12	−0.678090
$170$	0	0
$171$	1.90479e12	0.996250
$172$	0	0
$173$	−1.99007e12	−0.976371	−0.488186	−	0.872740i	$-0.662341\pi$
$173$	−1.99007e12	−0.976371	−0.488186	+	0.872740i	$0.662341\pi$
$174$	0	0
$175$	4.84956e11	0.223354
$176$	0	0
$177$	−9.26664e11	−0.400931
$178$	0	0
$179$	−2.71716e12	−1.10515	−0.552577	−	0.833462i	$-0.686356\pi$
$179$	−2.71716e12	−1.10515	−0.552577	+	0.833462i	$0.686356\pi$
$180$	0	0
$181$	2.72114e12	1.04116	0.520582	−	0.853812i	$-0.325715\pi$
$181$	2.72114e12	1.04116	0.520582	+	0.853812i	$0.325715\pi$
$182$	0	0
$183$	−3.52705e12	−1.27037
$184$	0	0
$185$	−2.12746e12	−0.721801
$186$	0	0
$187$	−4.57967e10	−0.0146455
$188$	0	0
$189$	−1.62034e12	−0.488727
$190$	0	0
$191$	2.22323e12	0.632850	0.316425	−	0.948618i	$-0.397518\pi$
$191$	2.22323e12	0.632850	0.316425	+	0.948618i	$0.397518\pi$
$192$	0	0
$193$	3.75455e11	0.100924	0.0504618	−	0.998726i	$-0.483931\pi$
$193$	3.75455e11	0.100924	0.0504618	+	0.998726i	$0.483931\pi$
$194$	0	0
$195$	−1.28729e12	−0.326953
$196$	0	0
$197$	−3.51030e11	−0.0842907	−0.0421453	−	0.999111i	$-0.513419\pi$
$197$	−3.51030e11	−0.0842907	−0.0421453	+	0.999111i	$0.513419\pi$
$198$	0	0
$199$	4.02056e12	0.913260	0.456630	−	0.889657i	$-0.349056\pi$
$199$	4.02056e12	0.913260	0.456630	+	0.889657i	$0.349056\pi$
$200$	0	0
$201$	3.72675e10	0.00801221
$202$	0	0
$203$	−6.19293e12	−1.26086
$204$	0	0
$205$	−1.00219e11	−0.0193333
$206$	0	0
$207$	3.60185e12	0.658701
$208$	0	0
$209$	−3.16725e11	−0.0549386
$210$	0	0
$211$	5.85562e11	0.0963872	0.0481936	−	0.998838i	$-0.484654\pi$
$211$	5.85562e11	0.0963872	0.0481936	+	0.998838i	$0.484654\pi$
$212$	0	0
$213$	−1.38344e13	−2.16209
$214$	0	0
$215$	−4.12736e12	−0.612719
$216$	0	0
$217$	1.42821e13	2.01494
$218$	0	0
$219$	6.46832e12	0.867659
$220$	0	0
$221$	−1.78826e12	−0.228178
$222$	0	0
$223$	7.67568e12	0.932052	0.466026	−	0.884771i	$-0.345685\pi$
$223$	7.67568e12	0.932052	0.466026	+	0.884771i	$0.345685\pi$
$224$	0	0
$225$	1.14242e12	0.132075
$226$	0	0
$227$	8.03622e12	0.884931	0.442466	−	0.896785i	$-0.354104\pi$
$227$	8.03622e12	0.884931	0.442466	+	0.896785i	$0.354104\pi$
$228$	0	0
$229$	1.30248e13	1.36671	0.683354	−	0.730087i	$-0.260521\pi$
$229$	1.30248e13	1.36671	0.683354	+	0.730087i	$0.260521\pi$
$230$	0	0
$231$	−5.23879e11	−0.0524041
$232$	0	0
$233$	1.09159e13	1.04136	0.520680	−	0.853752i	$-0.325678\pi$
$233$	1.09159e13	1.04136	0.520680	+	0.853752i	$0.325678\pi$
$234$	0	0
$235$	−6.88289e12	−0.626467
$236$	0	0
$237$	−5.25075e12	−0.456148
$238$	0	0
$239$	−1.18735e13	−0.984895	−0.492447	−	0.870342i	$-0.663898\pi$
$239$	−1.18735e13	−0.984895	−0.492447	+	0.870342i	$0.663898\pi$
$240$	0	0
$241$	2.34520e13	1.85818	0.929088	−	0.369859i	$-0.120594\pi$
$241$	2.34520e13	1.85818	0.929088	+	0.369859i	$0.120594\pi$
$242$	0	0
$243$	−1.50561e13	−1.13993
$244$	0	0
$245$	1.52731e12	0.110539
$246$	0	0
$247$	−1.23674e13	−0.855944
$248$	0	0
$249$	−5.84153e12	−0.386750
$250$	0	0
$251$	−1.53634e13	−0.973378	−0.486689	−	0.873575i	$-0.661796\pi$
$251$	−1.53634e13	−0.973378	−0.486689	+	0.873575i	$0.661796\pi$
$252$	0	0
$253$	−5.98908e11	−0.0363243
$254$	0	0
$255$	3.99021e12	0.231753
$256$	0	0
$257$	−3.42146e12	−0.190362	−0.0951809	−	0.995460i	$-0.530343\pi$
$257$	−3.42146e12	−0.190362	−0.0951809	+	0.995460i	$0.530343\pi$
$258$	0	0
$259$	−3.38076e13	−1.80246
$260$	0	0
$261$	−1.45887e13	−0.745581
$262$	0	0
$263$	1.19978e13	0.587954	0.293977	−	0.955812i	$-0.405021\pi$
$263$	1.19978e13	0.587954	0.293977	+	0.955812i	$0.405021\pi$
$264$	0	0
$265$	1.42360e13	0.669167
$266$	0	0
$267$	−3.84030e13	−1.73202
$268$	0	0
$269$	1.54949e13	0.670734	0.335367	−	0.942088i	$-0.391140\pi$
$269$	1.54949e13	0.670734	0.335367	+	0.942088i	$0.391140\pi$
$270$	0	0
$271$	2.54301e13	1.05686	0.528430	−	0.848977i	$-0.322781\pi$
$271$	2.54301e13	1.05686	0.528430	+	0.848977i	$0.322781\pi$
$272$	0	0
$273$	−2.04563e13	−0.816456
$274$	0	0
$275$	−1.89959e11	−0.00728331
$276$	0	0
$277$	2.18585e12	0.0805345	0.0402672	−	0.999189i	$-0.487179\pi$
$277$	2.18585e12	0.0805345	0.0402672	+	0.999189i	$0.487179\pi$
$278$	0	0
$279$	3.36444e13	1.19149
$280$	0	0
$281$	−2.08522e13	−0.710013	−0.355007	−	0.934864i	$-0.615521\pi$
$281$	−2.08522e13	−0.710013	−0.355007	+	0.934864i	$0.615521\pi$
$282$	0	0
$283$	5.16150e13	1.69025	0.845124	−	0.534570i	$-0.179526\pi$
$283$	5.16150e13	1.69025	0.845124	+	0.534570i	$0.179526\pi$
$284$	0	0
$285$	2.75958e13	0.869353
$286$	0	0
$287$	−1.59259e12	−0.0482784
$288$	0	0
$289$	−2.87288e13	−0.838262
$290$	0	0
$291$	−8.30953e12	−0.233435
$292$	0	0
$293$	1.44670e13	0.391387	0.195693	−	0.980665i	$-0.437304\pi$
$293$	1.44670e13	0.391387	0.195693	+	0.980665i	$0.437304\pi$
$294$	0	0
$295$	−5.33952e12	−0.139149
$296$	0	0
$297$	6.34692e11	0.0159369
$298$	0	0
$299$	−2.33860e13	−0.565933
$300$	0	0
$301$	−6.55881e13	−1.53006
$302$	0	0
$303$	1.09094e14	2.45394
$304$	0	0
$305$	−2.03232e13	−0.440902
$306$	0	0
$307$	−1.67889e13	−0.351368	−0.175684	−	0.984447i	$-0.556214\pi$
$307$	−1.67889e13	−0.351368	−0.175684	+	0.984447i	$0.556214\pi$
$308$	0	0
$309$	6.74026e13	1.36115
$310$	0	0
$311$	1.98233e13	0.386361	0.193181	−	0.981163i	$-0.438120\pi$
$311$	1.98233e13	0.386361	0.193181	+	0.981163i	$0.438120\pi$
$312$	0	0
$313$	1.94059e13	0.365123	0.182561	−	0.983194i	$-0.441561\pi$
$313$	1.94059e13	0.365123	0.182561	+	0.983194i	$0.441561\pi$
$314$	0	0
$315$	1.81542e13	0.329813
$316$	0	0
$317$	9.63212e13	1.69004	0.845018	−	0.534737i	$-0.179589\pi$
$317$	9.63212e13	1.69004	0.845018	+	0.534737i	$0.179589\pi$
$318$	0	0
$319$	2.42579e12	0.0411153
$320$	0	0
$321$	−9.70592e13	−1.58949
$322$	0	0
$323$	3.83353e13	0.606715
$324$	0	0
$325$	−7.41746e12	−0.113474
$326$	0	0
$327$	4.42393e13	0.654328
$328$	0	0
$329$	−1.09376e14	−1.56439
$330$	0	0
$331$	−1.14888e14	−1.58936	−0.794679	−	0.607030i	$-0.792361\pi$
$331$	−1.14888e14	−1.58936	−0.794679	+	0.607030i	$0.792361\pi$
$332$	0	0
$333$	−7.96409e13	−1.06584
$334$	0	0
$335$	2.14739e11	0.00278077
$336$	0	0
$337$	−9.51535e13	−1.19251	−0.596253	−	0.802797i	$-0.703344\pi$
$337$	−9.51535e13	−1.19251	−0.596253	+	0.802797i	$0.703344\pi$
$338$	0	0
$339$	1.38803e14	1.68383
$340$	0	0
$341$	−5.59432e12	−0.0657050
$342$	0	0
$343$	−7.39225e13	−0.840735
$344$	0	0
$345$	5.21821e13	0.574799
$346$	0	0
$347$	−1.79048e14	−1.91054	−0.955272	−	0.295727i	$-0.904438\pi$
$347$	−1.79048e14	−1.91054	−0.955272	+	0.295727i	$0.904438\pi$
$348$	0	0
$349$	−1.42818e14	−1.47654	−0.738268	−	0.674507i	$-0.764356\pi$
$349$	−1.42818e14	−1.47654	−0.738268	+	0.674507i	$0.764356\pi$
$350$	0	0
$351$	2.47833e13	0.248296
$352$	0	0
$353$	8.34518e13	0.810354	0.405177	−	0.914238i	$-0.367210\pi$
$353$	8.34518e13	0.810354	0.405177	+	0.914238i	$0.367210\pi$
$354$	0	0
$355$	−7.97152e13	−0.750387
$356$	0	0
$357$	6.34086e13	0.578726
$358$	0	0
$359$	1.10422e14	0.977322	0.488661	−	0.872474i	$-0.337486\pi$
$359$	1.10422e14	0.977322	0.488661	+	0.872474i	$0.337486\pi$
$360$	0	0
$361$	1.48632e14	1.27592
$362$	0	0
$363$	−1.54530e14	−1.28685
$364$	0	0
$365$	3.72711e13	0.301135
$366$	0	0
$367$	6.35584e13	0.498322	0.249161	−	0.968462i	$-0.419845\pi$
$367$	6.35584e13	0.498322	0.249161	+	0.968462i	$0.419845\pi$
$368$	0	0
$369$	−3.75167e12	−0.0285483
$370$	0	0
$371$	2.26224e14	1.67102
$372$	0	0
$373$	6.93761e13	0.497521	0.248760	−	0.968565i	$-0.419977\pi$
$373$	6.93761e13	0.497521	0.248760	+	0.968565i	$0.419977\pi$
$374$	0	0
$375$	1.65508e13	0.115252
$376$	0	0
$377$	9.47216e13	0.640577
$378$	0	0
$379$	−7.64902e13	−0.502447	−0.251224	−	0.967929i	$-0.580833\pi$
$379$	−7.64902e13	−0.502447	−0.251224	+	0.967929i	$0.580833\pi$
$380$	0	0
$381$	1.74153e14	1.11133
$382$	0	0
$383$	1.21465e14	0.753110	0.376555	−	0.926394i	$-0.377109\pi$
$383$	1.21465e14	0.753110	0.376555	+	0.926394i	$0.377109\pi$
$384$	0	0
$385$	−3.01864e12	−0.0181877
$386$	0	0
$387$	−1.54506e14	−0.904766
$388$	0	0
$389$	−2.85195e14	−1.62338	−0.811688	−	0.584092i	$-0.801451\pi$
$389$	−2.85195e14	−1.62338	−0.811688	+	0.584092i	$0.801451\pi$
$390$	0	0
$391$	7.24898e13	0.401148
$392$	0	0
$393$	−3.48249e14	−1.87383
$394$	0	0
$395$	−3.02553e13	−0.158313
$396$	0	0
$397$	1.43169e14	0.728618	0.364309	−	0.931278i	$-0.381305\pi$
$397$	1.43169e14	0.728618	0.364309	+	0.931278i	$0.381305\pi$
$398$	0	0
$399$	4.38526e14	2.17092
$400$	0	0
$401$	3.83730e14	1.84812	0.924062	−	0.382242i	$-0.124848\pi$
$401$	3.83730e14	1.84812	0.924062	+	0.382242i	$0.124848\pi$
$402$	0	0
$403$	−2.18446e14	−1.02368
$404$	0	0
$405$	−1.20060e14	−0.547515
$406$	0	0
$407$	1.32425e13	0.0587762
$408$	0	0
$409$	−2.42103e13	−0.104598	−0.0522988	−	0.998631i	$-0.516655\pi$
$409$	−2.42103e13	−0.104598	−0.0522988	+	0.998631i	$0.516655\pi$
$410$	0	0
$411$	−2.59897e14	−1.09313
$412$	0	0
$413$	−8.48506e13	−0.347480
$414$	0	0
$415$	−3.36594e13	−0.134228
$416$	0	0
$417$	3.67617e14	1.42773
$418$	0	0
$419$	1.87208e14	0.708187	0.354093	−	0.935210i	$-0.384790\pi$
$419$	1.87208e14	0.708187	0.354093	+	0.935210i	$0.384790\pi$
$420$	0	0
$421$	−2.90015e14	−1.06873	−0.534366	−	0.845253i	$-0.679449\pi$
$421$	−2.90015e14	−1.06873	−0.534366	+	0.845253i	$0.679449\pi$
$422$	0	0
$423$	−2.57659e14	−0.925067
$424$	0	0
$425$	2.29919e13	0.0804334
$426$	0	0
$427$	−3.22956e14	−1.10101
$428$	0	0
$429$	8.01279e12	0.0266237
$430$	0	0
$431$	−1.21836e14	−0.394595	−0.197298	−	0.980344i	$-0.563217\pi$
$431$	−1.21836e14	−0.394595	−0.197298	+	0.980344i	$0.563217\pi$
$432$	0	0
$433$	−6.62337e13	−0.209120	−0.104560	−	0.994519i	$-0.533343\pi$
$433$	−6.62337e13	−0.209120	−0.104560	+	0.994519i	$0.533343\pi$
$434$	0	0
$435$	−2.11356e14	−0.650613
$436$	0	0
$437$	5.01331e14	1.50479
$438$	0	0
$439$	−1.40333e13	−0.0410776	−0.0205388	−	0.999789i	$-0.506538\pi$
$439$	−1.40333e13	−0.0410776	−0.0205388	+	0.999789i	$0.506538\pi$
$440$	0	0
$441$	5.71743e13	0.163226
$442$	0	0
$443$	−4.34013e14	−1.20860	−0.604299	−	0.796758i	$-0.706547\pi$
$443$	−4.34013e14	−1.20860	−0.604299	+	0.796758i	$0.706547\pi$
$444$	0	0
$445$	−2.21282e14	−0.601126
$446$	0	0
$447$	−3.82788e14	−1.01453
$448$	0	0
$449$	−3.80839e14	−0.984888	−0.492444	−	0.870344i	$-0.663896\pi$
$449$	−3.80839e14	−0.984888	−0.492444	+	0.870344i	$0.663896\pi$
$450$	0	0
$451$	6.23821e11	0.00157431
$452$	0	0
$453$	3.48284e14	0.857814
$454$	0	0
$455$	−1.17871e14	−0.283364
$456$	0	0
$457$	4.71367e14	1.10617	0.553083	−	0.833126i	$-0.313451\pi$
$457$	4.71367e14	1.10617	0.553083	+	0.833126i	$0.313451\pi$
$458$	0	0
$459$	−7.68210e13	−0.175999
$460$	0	0
$461$	−3.27142e14	−0.731782	−0.365891	−	0.930658i	$-0.619236\pi$
$461$	−3.27142e14	−0.731782	−0.365891	+	0.930658i	$0.619236\pi$
$462$	0	0
$463$	4.66836e14	1.01969	0.509846	−	0.860266i	$-0.329702\pi$
$463$	4.66836e14	1.01969	0.509846	+	0.860266i	$0.329702\pi$
$464$	0	0
$465$	4.87426e14	1.03972
$466$	0	0
$467$	−2.78595e13	−0.0580404	−0.0290202	−	0.999579i	$-0.509239\pi$
$467$	−2.78595e13	−0.0580404	−0.0290202	+	0.999579i	$0.509239\pi$
$468$	0	0
$469$	3.41243e12	0.00694404
$470$	0	0
$471$	6.80291e14	1.35232
$472$	0	0
$473$	2.56910e13	0.0498936
$474$	0	0
$475$	1.59009e14	0.301723
$476$	0	0
$477$	5.32918e14	0.988120
$478$	0	0
$479$	6.05624e14	1.09738	0.548691	−	0.836025i	$-0.315126\pi$
$479$	6.05624e14	1.09738	0.548691	+	0.836025i	$0.315126\pi$
$480$	0	0
$481$	5.17092e14	0.915734
$482$	0	0
$483$	8.29227e14	1.43537
$484$	0	0
$485$	−4.78803e13	−0.0810172
$486$	0	0
$487$	−7.89431e14	−1.30588	−0.652942	−	0.757408i	$-0.726466\pi$
$487$	−7.89431e14	−1.30588	−0.652942	+	0.757408i	$0.726466\pi$
$488$	0	0
$489$	−3.17581e13	−0.0513637
$490$	0	0
$491$	−1.81410e14	−0.286889	−0.143444	−	0.989658i	$-0.545818\pi$
$491$	−1.81410e14	−0.286889	−0.143444	+	0.989658i	$0.545818\pi$
$492$	0	0
$493$	−2.93609e14	−0.454058
$494$	0	0
$495$	−7.11103e12	−0.0107548
$496$	0	0
$497$	−1.26676e15	−1.87384
$498$	0	0
$499$	4.89614e14	0.708436	0.354218	−	0.935163i	$-0.384747\pi$
$499$	4.89614e14	0.708436	0.354218	+	0.935163i	$0.384747\pi$
$500$	0	0
$501$	−8.86873e14	−1.25532
$502$	0	0
$503$	2.37938e14	0.329488	0.164744	−	0.986336i	$-0.447320\pi$
$503$	2.37938e14	0.329488	0.164744	+	0.986336i	$0.447320\pi$
$504$	0	0
$505$	6.28607e14	0.851680
$506$	0	0
$507$	−6.59074e14	−0.873757
$508$	0	0
$509$	−8.23454e14	−1.06830	−0.534148	−	0.845391i	$-0.679368\pi$
$509$	−8.23454e14	−1.06830	−0.534148	+	0.845391i	$0.679368\pi$
$510$	0	0
$511$	5.92276e14	0.751985
$512$	0	0
$513$	−5.31285e14	−0.660210
$514$	0	0
$515$	3.88380e14	0.472408
$516$	0	0
$517$	4.28430e13	0.0510131
$518$	0	0
$519$	−1.07929e15	−1.25811
$520$	0	0
$521$	−9.50125e14	−1.08436	−0.542180	−	0.840263i	$-0.682401\pi$
$521$	−9.50125e14	−1.08436	−0.542180	+	0.840263i	$0.682401\pi$
$522$	0	0
$523$	1.61137e15	1.80067	0.900337	−	0.435194i	$-0.143320\pi$
$523$	1.61137e15	1.80067	0.900337	+	0.435194i	$0.143320\pi$
$524$	0	0
$525$	2.63010e14	0.287804
$526$	0	0
$527$	6.77118e14	0.725614
$528$	0	0
$529$	−4.82246e12	−0.00506130
$530$	0	0
$531$	−1.99883e14	−0.205474
$532$	0	0
$533$	2.43588e13	0.0245277
$534$	0	0
$535$	−5.59264e14	−0.551660
$536$	0	0
$537$	−1.47362e15	−1.42405
$538$	0	0
$539$	−9.50683e12	−0.00900114
$540$	0	0
$541$	−1.03512e15	−0.960294	−0.480147	−	0.877188i	$-0.659417\pi$
$541$	−1.03512e15	−0.960294	−0.480147	+	0.877188i	$0.659417\pi$
$542$	0	0
$543$	1.47578e15	1.34160
$544$	0	0
$545$	2.54911e14	0.227095
$546$	0	0
$547$	5.24788e14	0.458198	0.229099	−	0.973403i	$-0.426422\pi$
$547$	5.24788e14	0.458198	0.229099	+	0.973403i	$0.426422\pi$
$548$	0	0
$549$	−7.60791e14	−0.651055
$550$	0	0
$551$	−2.03056e15	−1.70327
$552$	0	0
$553$	−4.80789e14	−0.395336
$554$	0	0
$555$	−1.15380e15	−0.930081
$556$	0	0
$557$	2.26339e15	1.78878	0.894388	−	0.447292i	$-0.147612\pi$
$557$	2.26339e15	1.78878	0.894388	+	0.447292i	$0.147612\pi$
$558$	0	0
$559$	1.00318e15	0.777344
$560$	0	0
$561$	−2.48373e13	−0.0188716
$562$	0	0
$563$	6.47720e14	0.482604	0.241302	−	0.970450i	$-0.422426\pi$
$563$	6.47720e14	0.482604	0.241302	+	0.970450i	$0.422426\pi$
$564$	0	0
$565$	7.99794e14	0.584402
$566$	0	0
$567$	−1.90788e15	−1.36724
$568$	0	0
$569$	2.63615e14	0.185290	0.0926451	−	0.995699i	$-0.470468\pi$
$569$	2.63615e14	0.185290	0.0926451	+	0.995699i	$0.470468\pi$
$570$	0	0
$571$	−1.94491e14	−0.134091	−0.0670457	−	0.997750i	$-0.521357\pi$
$571$	−1.94491e14	−0.134091	−0.0670457	+	0.997750i	$0.521357\pi$
$572$	0	0
$573$	1.20574e15	0.815461
$574$	0	0
$575$	3.00678e14	0.199493
$576$	0	0
$577$	7.85826e14	0.511516	0.255758	−	0.966741i	$-0.417675\pi$
$577$	7.85826e14	0.511516	0.255758	+	0.966741i	$0.417675\pi$
$578$	0	0
$579$	2.03624e14	0.130046
$580$	0	0
$581$	−5.34883e14	−0.335189
$582$	0	0
$583$	−8.86126e13	−0.0544902
$584$	0	0
$585$	−2.77670e14	−0.167561
$586$	0	0
$587$	1.69825e15	1.00576	0.502878	−	0.864357i	$-0.332274\pi$
$587$	1.69825e15	1.00576	0.502878	+	0.864357i	$0.332274\pi$
$588$	0	0
$589$	4.68286e15	2.72193
$590$	0	0
$591$	−1.90377e14	−0.108613
$592$	0	0
$593$	−1.57747e15	−0.883406	−0.441703	−	0.897161i	$-0.645625\pi$
$593$	−1.57747e15	−0.883406	−0.441703	+	0.897161i	$0.645625\pi$
$594$	0	0
$595$	3.65366e14	0.200856
$596$	0	0
$597$	2.18050e15	1.17679
$598$	0	0
$599$	2.36133e14	0.125115	0.0625573	−	0.998041i	$-0.480074\pi$
$599$	2.36133e14	0.125115	0.0625573	+	0.998041i	$0.480074\pi$
$600$	0	0
$601$	2.87143e15	1.49379	0.746893	−	0.664944i	$-0.231545\pi$
$601$	2.87143e15	1.49379	0.746893	+	0.664944i	$0.231545\pi$
$602$	0	0
$603$	8.03868e12	0.00410619
$604$	0	0
$605$	−8.90417e14	−0.446621
$606$	0	0
$607$	−9.22001e14	−0.454144	−0.227072	−	0.973878i	$-0.572915\pi$
$607$	−9.22001e14	−0.454144	−0.227072	+	0.973878i	$0.572915\pi$
$608$	0	0
$609$	−3.35866e15	−1.62469
$610$	0	0
$611$	1.67292e15	0.794785
$612$	0	0
$613$	−4.25894e14	−0.198732	−0.0993662	−	0.995051i	$-0.531682\pi$
$613$	−4.25894e14	−0.198732	−0.0993662	+	0.995051i	$0.531682\pi$
$614$	0	0
$615$	−5.43527e13	−0.0249120
$616$	0	0
$617$	−2.66401e15	−1.19941	−0.599705	−	0.800221i	$-0.704715\pi$
$617$	−2.66401e15	−1.19941	−0.599705	+	0.800221i	$0.704715\pi$
$618$	0	0
$619$	−1.71903e15	−0.760301	−0.380151	−	0.924925i	$-0.624128\pi$
$619$	−1.71903e15	−0.760301	−0.380151	+	0.924925i	$0.624128\pi$
$620$	0	0
$621$	−1.00463e15	−0.436518
$622$	0	0
$623$	−3.51639e15	−1.50111
$624$	0	0
$625$	9.53674e13	0.0400000
$626$	0	0
$627$	−1.71772e14	−0.0707914
$628$	0	0
$629$	−1.60283e15	−0.649097
$630$	0	0
$631$	−4.92062e13	−0.0195821	−0.00979103	−	0.999952i	$-0.503117\pi$
$631$	−4.92062e13	−0.0195821	−0.00979103	+	0.999952i	$0.503117\pi$
$632$	0	0
$633$	3.17572e14	0.124200
$634$	0	0
$635$	1.00348e15	0.385704
$636$	0	0
$637$	−3.71221e14	−0.140238
$638$	0	0
$639$	−2.98411e15	−1.10805
$640$	0	0
$641$	4.72099e15	1.72311	0.861557	−	0.507661i	$-0.169490\pi$
$641$	4.72099e15	1.72311	0.861557	+	0.507661i	$0.169490\pi$
$642$	0	0
$643$	−3.18915e15	−1.14423	−0.572117	−	0.820172i	$-0.693878\pi$
$643$	−3.18915e15	−1.14423	−0.572117	+	0.820172i	$0.693878\pi$
$644$	0	0
$645$	−2.23842e15	−0.789522
$646$	0	0
$647$	−2.51792e15	−0.873110	−0.436555	−	0.899677i	$-0.643802\pi$
$647$	−2.51792e15	−0.873110	−0.436555	+	0.899677i	$0.643802\pi$
$648$	0	0
$649$	3.32362e13	0.0113309
$650$	0	0
$651$	7.74570e15	2.59636
$652$	0	0
$653$	−3.80457e15	−1.25396	−0.626979	−	0.779036i	$-0.715709\pi$
$653$	−3.80457e15	−1.25396	−0.626979	+	0.779036i	$0.715709\pi$
$654$	0	0
$655$	−2.00664e15	−0.650344
$656$	0	0
$657$	1.39523e15	0.444668
$658$	0	0
$659$	3.23977e15	1.01542	0.507709	−	0.861529i	$-0.330493\pi$
$659$	3.23977e15	1.01542	0.507709	+	0.861529i	$0.330493\pi$
$660$	0	0
$661$	−2.71166e15	−0.835848	−0.417924	−	0.908482i	$-0.637242\pi$
$661$	−2.71166e15	−0.835848	−0.417924	+	0.908482i	$0.637242\pi$
$662$	0	0
$663$	−9.69842e14	−0.294020
$664$	0	0
$665$	2.52683e15	0.753453
$666$	0	0
$667$	−3.83968e15	−1.12617
$668$	0	0
$669$	4.16281e15	1.20100
$670$	0	0
$671$	1.26503e14	0.0359026
$672$	0	0
$673$	3.62755e15	1.01282	0.506409	−	0.862294i	$-0.330973\pi$
$673$	3.62755e15	1.01282	0.506409	+	0.862294i	$0.330973\pi$
$674$	0	0
$675$	−3.18643e14	−0.0875253
$676$	0	0
$677$	4.05118e15	1.09482	0.547412	−	0.836863i	$-0.315613\pi$
$677$	4.05118e15	1.09482	0.547412	+	0.836863i	$0.315613\pi$
$678$	0	0
$679$	−7.60868e14	−0.202314
$680$	0	0
$681$	4.35835e15	1.14028
$682$	0	0
$683$	−4.56724e15	−1.17582	−0.587908	−	0.808928i	$-0.700048\pi$
$683$	−4.56724e15	−1.17582	−0.587908	+	0.808928i	$0.700048\pi$
$684$	0	0
$685$	−1.49755e15	−0.379388
$686$	0	0
$687$	7.06384e15	1.76108
$688$	0	0
$689$	−3.46013e15	−0.848958
$690$	0	0
$691$	−6.26218e15	−1.51215	−0.756077	−	0.654482i	$-0.772887\pi$
$691$	−6.26218e15	−1.51215	−0.756077	+	0.654482i	$0.772887\pi$
$692$	0	0
$693$	−1.13002e14	−0.0268567
$694$	0	0
$695$	2.11824e15	0.495518
$696$	0	0
$697$	−7.55052e13	−0.0173859
$698$	0	0
$699$	5.92009e15	1.34185
$700$	0	0
$701$	−1.12469e15	−0.250949	−0.125474	−	0.992097i	$-0.540045\pi$
$701$	−1.12469e15	−0.250949	−0.125474	+	0.992097i	$0.540045\pi$
$702$	0	0
$703$	−1.10850e16	−2.43490
$704$	0	0
$705$	−3.73285e15	−0.807237
$706$	0	0
$707$	9.98922e15	2.12679
$708$	0	0
$709$	4.16576e15	0.873253	0.436626	−	0.899643i	$-0.356173\pi$
$709$	4.16576e15	0.873253	0.436626	+	0.899643i	$0.356173\pi$
$710$	0	0
$711$	−1.13260e15	−0.233772
$712$	0	0
$713$	8.85503e15	1.79969
$714$	0	0
$715$	4.61705e13	0.00924019
$716$	0	0
$717$	−6.43944e15	−1.26909
$718$	0	0
$719$	−8.31711e14	−0.161422	−0.0807111	−	0.996738i	$-0.525719\pi$
$719$	−8.31711e14	−0.161422	−0.0807111	+	0.996738i	$0.525719\pi$
$720$	0	0
$721$	6.17176e15	1.17968
$722$	0	0
$723$	1.27189e16	2.39436
$724$	0	0
$725$	−1.21785e15	−0.225806
$726$	0	0
$727$	1.54083e15	0.281395	0.140697	−	0.990053i	$-0.455066\pi$
$727$	1.54083e15	0.281395	0.140697	+	0.990053i	$0.455066\pi$
$728$	0	0
$729$	−1.35962e15	−0.244577
$730$	0	0
$731$	−3.10956e15	−0.551001
$732$	0	0
$733$	4.81955e14	0.0841268	0.0420634	−	0.999115i	$-0.486607\pi$
$733$	4.81955e14	0.0841268	0.0420634	+	0.999115i	$0.486607\pi$
$734$	0	0
$735$	8.28318e14	0.142435
$736$	0	0
$737$	−1.33666e12	−0.000226438 0
$738$	0	0
$739$	−1.56414e15	−0.261055	−0.130528	−	0.991445i	$-0.541667\pi$
$739$	−1.56414e15	−0.261055	−0.130528	+	0.991445i	$0.541667\pi$
$740$	0	0
$741$	−6.70731e15	−1.10293
$742$	0	0
$743$	−5.48466e14	−0.0888610	−0.0444305	−	0.999012i	$-0.514147\pi$
$743$	−5.48466e14	−0.0888610	−0.0444305	+	0.999012i	$0.514147\pi$
$744$	0	0
$745$	−2.20566e15	−0.352110
$746$	0	0
$747$	−1.26003e15	−0.198206
$748$	0	0
$749$	−8.88729e15	−1.37759
$750$	0	0
$751$	−9.51331e15	−1.45315	−0.726577	−	0.687085i	$-0.758890\pi$
$751$	−9.51331e15	−1.45315	−0.726577	+	0.687085i	$0.758890\pi$
$752$	0	0
$753$	−8.33215e15	−1.25425
$754$	0	0
$755$	2.00684e15	0.297718
$756$	0	0
$757$	−6.27757e15	−0.917834	−0.458917	−	0.888479i	$-0.651762\pi$
$757$	−6.27757e15	−0.917834	−0.458917	+	0.888479i	$0.651762\pi$
$758$	0	0
$759$	−3.24810e14	−0.0468059
$760$	0	0
$761$	−2.56418e15	−0.364193	−0.182097	−	0.983281i	$-0.558288\pi$
$761$	−2.56418e15	−0.364193	−0.182097	+	0.983281i	$0.558288\pi$
$762$	0	0
$763$	4.05080e15	0.567095
$764$	0	0
$765$	8.60696e14	0.118771
$766$	0	0
$767$	1.29780e15	0.176536
$768$	0	0
$769$	5.83938e14	0.0783018	0.0391509	−	0.999233i	$-0.487535\pi$
$769$	5.83938e14	0.0783018	0.0391509	+	0.999233i	$0.487535\pi$
$770$	0	0
$771$	−1.85559e15	−0.245292
$772$	0	0
$773$	2.35645e15	0.307094	0.153547	−	0.988141i	$-0.450930\pi$
$773$	2.35645e15	0.307094	0.153547	+	0.988141i	$0.450930\pi$
$774$	0	0
$775$	2.80859e15	0.360852
$776$	0	0
$777$	−1.83351e16	−2.32257
$778$	0	0
$779$	−5.22185e14	−0.0652181
$780$	0	0
$781$	4.96192e14	0.0611040
$782$	0	0
$783$	4.06910e15	0.494093
$784$	0	0
$785$	3.91990e15	0.469344
$786$	0	0
$787$	−3.54576e15	−0.418647	−0.209324	−	0.977846i	$-0.567126\pi$
$787$	−3.54576e15	−0.418647	−0.209324	+	0.977846i	$0.567126\pi$
$788$	0	0
$789$	6.50684e15	0.757612
$790$	0	0
$791$	1.27096e16	1.45935
$792$	0	0
$793$	4.93966e15	0.559364
$794$	0	0
$795$	7.72070e15	0.862259
$796$	0	0
$797$	4.80052e15	0.528771	0.264385	−	0.964417i	$-0.414831\pi$
$797$	4.80052e15	0.528771	0.264385	+	0.964417i	$0.414831\pi$
$798$	0	0
$799$	−5.18557e15	−0.563365
$800$	0	0
$801$	−8.28360e15	−0.887647
$802$	0	0
$803$	−2.31996e14	−0.0245214
$804$	0	0
$805$	4.77808e15	0.498169
$806$	0	0
$807$	8.40345e15	0.864278
$808$	0	0
$809$	−3.91609e14	−0.0397316	−0.0198658	−	0.999803i	$-0.506324\pi$
$809$	−3.91609e14	−0.0397316	−0.0198658	+	0.999803i	$0.506324\pi$
$810$	0	0
$811$	5.86491e15	0.587012	0.293506	−	0.955957i	$-0.405178\pi$
$811$	5.86491e15	0.587012	0.293506	+	0.955957i	$0.405178\pi$
$812$	0	0
$813$	1.37917e16	1.36182
$814$	0	0
$815$	−1.82993e14	−0.0178266
$816$	0	0
$817$	−2.15053e16	−2.06692
$818$	0	0
$819$	−4.41247e15	−0.418427
$820$	0	0
$821$	−1.66455e15	−0.155743	−0.0778715	−	0.996963i	$-0.524812\pi$
$821$	−1.66455e15	−0.155743	−0.0778715	+	0.996963i	$0.524812\pi$
$822$	0	0
$823$	1.67289e16	1.54443	0.772216	−	0.635360i	$-0.219148\pi$
$823$	1.67289e16	1.54443	0.772216	+	0.635360i	$0.219148\pi$
$824$	0	0
$825$	−1.03022e14	−0.00938495
$826$	0	0
$827$	1.91438e16	1.72087	0.860435	−	0.509560i	$-0.170192\pi$
$827$	1.91438e16	1.72087	0.860435	+	0.509560i	$0.170192\pi$
$828$	0	0
$829$	1.02576e16	0.909906	0.454953	−	0.890515i	$-0.349656\pi$
$829$	1.02576e16	0.909906	0.454953	+	0.890515i	$0.349656\pi$
$830$	0	0
$831$	1.18547e15	0.103773
$832$	0	0
$833$	1.15068e15	0.0994043
$834$	0	0
$835$	−5.11024e15	−0.435678
$836$	0	0
$837$	−9.38411e15	−0.789592
$838$	0	0
$839$	−1.89422e16	−1.57304	−0.786519	−	0.617566i	$-0.788119\pi$
$839$	−1.89422e16	−1.57304	−0.786519	+	0.617566i	$0.788119\pi$
$840$	0	0
$841$	3.35153e15	0.274704
$842$	0	0
$843$	−1.13089e16	−0.914891
$844$	0	0
$845$	−3.79765e15	−0.303251
$846$	0	0
$847$	−1.41496e16	−1.11529
$848$	0	0
$849$	2.79928e16	2.17798
$850$	0	0
$851$	−2.09611e16	−1.60991
$852$	0	0
$853$	1.33491e16	1.01212	0.506059	−	0.862499i	$-0.331102\pi$
$853$	1.33491e16	1.01212	0.506059	+	0.862499i	$0.331102\pi$
$854$	0	0
$855$	5.95247e15	0.445537
$856$	0	0
$857$	2.46363e16	1.82046	0.910231	−	0.414101i	$-0.135904\pi$
$857$	2.46363e16	1.82046	0.910231	+	0.414101i	$0.135904\pi$
$858$	0	0
$859$	−2.13423e16	−1.55696	−0.778482	−	0.627668i	$-0.784010\pi$
$859$	−2.13423e16	−1.55696	−0.778482	+	0.627668i	$0.784010\pi$
$860$	0	0
$861$	−8.63721e14	−0.0622094
$862$	0	0
$863$	9.98852e15	0.710300	0.355150	−	0.934809i	$-0.384430\pi$
$863$	9.98852e15	0.710300	0.355150	+	0.934809i	$0.384430\pi$
$864$	0	0
$865$	−6.21897e15	−0.436646
$866$	0	0
$867$	−1.55807e16	−1.08015
$868$	0	0
$869$	1.88326e14	0.0128915
$870$	0	0
$871$	−5.21935e13	−0.00352790
$872$	0	0
$873$	−1.79238e15	−0.119633
$874$	0	0
$875$	1.51549e15	0.0998868
$876$	0	0
$877$	2.31195e16	1.50480	0.752402	−	0.658705i	$-0.228895\pi$
$877$	2.31195e16	1.50480	0.752402	+	0.658705i	$0.228895\pi$
$878$	0	0
$879$	7.84599e15	0.504323
$880$	0	0
$881$	−2.73216e16	−1.73436	−0.867180	−	0.497995i	$-0.834070\pi$
$881$	−2.73216e16	−1.73436	−0.867180	+	0.497995i	$0.834070\pi$
$882$	0	0
$883$	−1.06010e16	−0.664604	−0.332302	−	0.943173i	$-0.607825\pi$
$883$	−1.06010e16	−0.664604	−0.332302	+	0.943173i	$0.607825\pi$
$884$	0	0
$885$	−2.89583e15	−0.179302
$886$	0	0
$887$	2.11039e16	1.29058	0.645288	−	0.763939i	$-0.276737\pi$
$887$	2.11039e16	1.29058	0.645288	+	0.763939i	$0.276737\pi$
$888$	0	0
$889$	1.59464e16	0.963170
$890$	0	0
$891$	7.47321e14	0.0445841
$892$	0	0
$893$	−3.58628e16	−2.11330
$894$	0	0
$895$	−8.49112e15	−0.494240
$896$	0	0
$897$	−1.26831e16	−0.729236
$898$	0	0
$899$	−3.58659e16	−2.03706
$900$	0	0
$901$	1.07254e16	0.601764
$902$	0	0
$903$	−3.55709e16	−1.97157
$904$	0	0
$905$	8.50357e15	0.465623
$906$	0	0
$907$	6.86803e15	0.371528	0.185764	−	0.982594i	$-0.440524\pi$
$907$	6.86803e15	0.371528	0.185764	+	0.982594i	$0.440524\pi$
$908$	0	0
$909$	2.35317e16	1.25763
$910$	0	0
$911$	−4.56797e15	−0.241197	−0.120599	−	0.992701i	$-0.538481\pi$
$911$	−4.56797e15	−0.241197	−0.120599	+	0.992701i	$0.538481\pi$
$912$	0	0
$913$	2.09515e14	0.0109301
$914$	0	0
$915$	−1.10220e16	−0.568127
$916$	0	0
$917$	−3.18877e16	−1.62402
$918$	0	0
$919$	2.45493e16	1.23539	0.617695	−	0.786418i	$-0.288067\pi$
$919$	2.45493e16	1.23539	0.617695	+	0.786418i	$0.288067\pi$
$920$	0	0
$921$	−9.10527e15	−0.452757
$922$	0	0
$923$	1.93752e16	0.952001
$924$	0	0
$925$	−6.64832e15	−0.322799
$926$	0	0
$927$	1.45389e16	0.697577
$928$	0	0
$929$	−1.63221e16	−0.773909	−0.386954	−	0.922099i	$-0.626473\pi$
$929$	−1.63221e16	−0.773909	−0.386954	+	0.922099i	$0.626473\pi$
$930$	0	0
$931$	7.95793e15	0.372887
$932$	0	0
$933$	1.07509e16	0.497848
$934$	0	0
$935$	−1.43115e14	−0.00654969
$936$	0	0
$937$	9.76542e15	0.441696	0.220848	−	0.975308i	$-0.429118\pi$
$937$	9.76542e15	0.441696	0.220848	+	0.975308i	$0.429118\pi$
$938$	0	0
$939$	1.05245e16	0.470481
$940$	0	0
$941$	4.42567e16	1.95540	0.977702	−	0.209999i	$-0.0673460\pi$
$941$	4.42567e16	1.95540	0.977702	+	0.209999i	$0.0673460\pi$
$942$	0	0
$943$	−9.87421e14	−0.0431209
$944$	0	0
$945$	−5.06357e15	−0.218566
$946$	0	0
$947$	2.15494e16	0.919412	0.459706	−	0.888071i	$-0.347955\pi$
$947$	2.15494e16	0.919412	0.459706	+	0.888071i	$0.347955\pi$
$948$	0	0
$949$	−9.05893e15	−0.382044
$950$	0	0
$951$	5.22386e16	2.17771
$952$	0	0
$953$	−3.05469e16	−1.25880	−0.629399	−	0.777083i	$-0.716699\pi$
$953$	−3.05469e16	−1.25880	−0.629399	+	0.777083i	$0.716699\pi$
$954$	0	0
$955$	6.94759e15	0.283019
$956$	0	0
$957$	1.31560e15	0.0529794
$958$	0	0
$959$	−2.37976e16	−0.947395
$960$	0	0
$961$	5.73052e16	2.25536
$962$	0	0
$963$	−2.09359e16	−0.814604
$964$	0	0
$965$	1.17330e15	0.0451344
$966$	0	0
$967$	−2.67248e15	−0.101641	−0.0508204	−	0.998708i	$-0.516184\pi$
$967$	−2.67248e15	−0.101641	−0.0508204	+	0.998708i	$0.516184\pi$
$968$	0	0
$969$	2.07907e16	0.781786
$970$	0	0
$971$	1.05715e16	0.393036	0.196518	−	0.980500i	$-0.437037\pi$
$971$	1.05715e16	0.393036	0.196518	+	0.980500i	$0.437037\pi$
$972$	0	0
$973$	3.36611e16	1.23739
$974$	0	0
$975$	−4.02277e15	−0.146218
$976$	0	0
$977$	2.22544e16	0.799826	0.399913	−	0.916553i	$-0.369040\pi$
$977$	2.22544e16	0.799826	0.399913	+	0.916553i	$0.369040\pi$
$978$	0	0
$979$	1.37738e15	0.0489496
$980$	0	0
$981$	9.54251e15	0.335338
$982$	0	0
$983$	−5.53819e15	−0.192453	−0.0962263	−	0.995359i	$-0.530677\pi$
$983$	−5.53819e15	−0.192453	−0.0962263	+	0.995359i	$0.530677\pi$
$984$	0	0
$985$	−1.09697e15	−0.0376959
$986$	0	0
$987$	−5.93189e16	−2.01581
$988$	0	0
$989$	−4.06653e16	−1.36661
$990$	0	0
$991$	−2.34700e16	−0.780024	−0.390012	−	0.920810i	$-0.627529\pi$
$991$	−2.34700e16	−0.780024	−0.390012	+	0.920810i	$0.627529\pi$
$992$	0	0
$993$	−6.23082e16	−2.04798
$994$	0	0
$995$	1.25642e16	0.408422
$996$	0	0
$997$	−5.95725e16	−1.91524	−0.957618	−	0.288041i	$-0.906996\pi$
$997$	−5.95725e16	−1.91524	−0.957618	+	0.288041i	$0.906996\pi$
$998$	0	0
$999$	2.22135e16	0.706328

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.12.a.b.1.2	✓	2
3.2	odd	2		180.12.a.c.1.2		2
4.3	odd	2		80.12.a.i.1.1		2
5.2	odd	4		100.12.c.c.49.1		4
5.3	odd	4		100.12.c.c.49.4		4
5.4	even	2		100.12.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.12.a.b.1.2	✓	2	1.1	even	1	trivial
80.12.a.i.1.1		2	4.3	odd	2
100.12.a.c.1.1		2	5.4	even	2
100.12.c.c.49.1		4	5.2	odd	4
100.12.c.c.49.4		4	5.3	odd	4
180.12.a.c.1.2		2	3.2	odd	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:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	20.a (trivial)

\(f(q)\)	\(=\)	\(q+542.338 q^{3} +3125.00 q^{5} +49659.5 q^{7} +116983. q^{9} +O(q^{10})\)	\(q+542.338 q^{3} +3125.00 q^{5} +49659.5 q^{7} +116983. q^{9} -19451.8 q^{11} -759548. q^{13} +1.69481e6 q^{15} +2.35438e6 q^{17} +1.62826e7 q^{19} +2.69322e7 q^{21} +3.07894e7 q^{23} +9.76562e6 q^{25} -3.26290e7 q^{27} -1.24708e8 q^{29} +2.87600e8 q^{31} -1.05494e7 q^{33} +1.55186e8 q^{35} -6.80788e8 q^{37} -4.11932e8 q^{39} -3.20702e7 q^{41} -1.32076e9 q^{43} +3.65573e8 q^{45} -2.20253e9 q^{47} +4.88739e8 q^{49} +1.27687e9 q^{51} +4.55551e9 q^{53} -6.07867e7 q^{55} +8.83065e9 q^{57} -1.70865e9 q^{59} -6.50342e9 q^{61} +5.80933e9 q^{63} -2.37359e9 q^{65} +6.87165e7 q^{67} +1.66983e10 q^{69} -2.55089e10 q^{71} +1.19267e10 q^{73} +5.29627e9 q^{75} -9.65964e8 q^{77} -9.68170e9 q^{79} -3.84192e10 q^{81} -1.07710e10 q^{83} +7.35742e9 q^{85} -6.76338e10 q^{87} -7.08101e10 q^{89} -3.77188e10 q^{91} +1.55976e11 q^{93} +5.08830e10 q^{95} -1.53217e10 q^{97} -2.27553e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 220 q^{3} + 6250 q^{5} + 3340 q^{7} + 43738 q^{9}+O(q^{10}) \) 2 * q + 220 * q^3 + 6250 * q^5 + 3340 * q^7 + 43738 * q^9	\( 2 q + 220 q^{3} + 6250 q^{5} + 3340 q^{7} + 43738 q^{9} + 298320 q^{11} + 1209820 q^{13} + 687500 q^{15} + 9056340 q^{17} + 19439368 q^{19} + 41862752 q^{21} + 55926420 q^{23} + 19531250 q^{25} + 48081880 q^{27} + 41841708 q^{29} + 62230792 q^{31} - 112979280 q^{33} + 10437500 q^{35} - 729235940 q^{37} - 1046733592 q^{39} - 608419068 q^{41} - 1129440740 q^{43} + 136681250 q^{45} - 1653072900 q^{47} + 656908386 q^{49} - 883429896 q^{51} + 5724887340 q^{53} + 932250000 q^{55} + 7813100240 q^{57} + 3756433896 q^{59} + 4923703564 q^{61} + 9202019900 q^{63} + 3780687500 q^{65} - 5843244140 q^{67} + 8595645984 q^{69} - 43352162664 q^{71} + 33647099620 q^{73} + 2148437500 q^{75} - 15684992880 q^{77} - 54799425296 q^{79} - 51460199582 q^{81} - 9303032100 q^{83} + 28301062500 q^{85} - 121318983000 q^{87} + 3688968372 q^{89} - 128938931512 q^{91} + 228621221120 q^{93} + 60748025000 q^{95} + 65892157780 q^{97} - 25550825520 q^{99}+O(q^{100}) \) 2 * q + 220 * q^3 + 6250 * q^5 + 3340 * q^7 + 43738 * q^9 + 298320 * q^11 + 1209820 * q^13 + 687500 * q^15 + 9056340 * q^17 + 19439368 * q^19 + 41862752 * q^21 + 55926420 * q^23 + 19531250 * q^25 + 48081880 * q^27 + 41841708 * q^29 + 62230792 * q^31 - 112979280 * q^33 + 10437500 * q^35 - 729235940 * q^37 - 1046733592 * q^39 - 608419068 * q^41 - 1129440740 * q^43 + 136681250 * q^45 - 1653072900 * q^47 + 656908386 * q^49 - 883429896 * q^51 + 5724887340 * q^53 + 932250000 * q^55 + 7813100240 * q^57 + 3756433896 * q^59 + 4923703564 * q^61 + 9202019900 * q^63 + 3780687500 * q^65 - 5843244140 * q^67 + 8595645984 * q^69 - 43352162664 * q^71 + 33647099620 * q^73 + 2148437500 * q^75 - 15684992880 * q^77 - 54799425296 * q^79 - 51460199582 * q^81 - 9303032100 * q^83 + 28301062500 * q^85 - 121318983000 * q^87 + 3688968372 * q^89 - 128938931512 * q^91 + 228621221120 * q^93 + 60748025000 * q^95 + 65892157780 * q^97 - 25550825520 * q^99

Introduction

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