LMFDB - Embedded newform 20.12.a.b.1.1

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			1.1
Root			$108.584$ 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)

$f(q)$	$=$	$q-322.338 q^{3} +3125.00 q^{5} -46319.5 q^{7} -73245.3 q^{9} +O(q^{10})$	$q-322.338 q^{3} +3125.00 q^{5} -46319.5 q^{7} -73245.3 q^{9} +317772. q^{11} +1.96937e6 q^{13} -1.00731e6 q^{15} +6.70196e6 q^{17} +3.15680e6 q^{19} +1.49305e7 q^{21} +2.51370e7 q^{23} +9.76562e6 q^{25} +8.07109e7 q^{27} +1.66550e8 q^{29} -2.25369e8 q^{31} -1.02430e8 q^{33} -1.44748e8 q^{35} -4.84476e7 q^{37} -6.34802e8 q^{39} -5.76349e8 q^{41} +1.91315e8 q^{43} -2.28892e8 q^{45} +5.49453e8 q^{47} +1.68169e8 q^{49} -2.16030e9 q^{51} +1.16938e9 q^{53} +9.93037e8 q^{55} -1.01755e9 q^{57} +5.46508e9 q^{59} +1.14271e10 q^{61} +3.39269e9 q^{63} +6.15428e9 q^{65} -5.91196e9 q^{67} -8.10261e9 q^{69} -1.78433e10 q^{71} +2.17204e10 q^{73} -3.14783e9 q^{75} -1.47190e10 q^{77} -4.51177e10 q^{79} -1.30410e10 q^{81} +1.46798e9 q^{83} +2.09436e10 q^{85} -5.36852e10 q^{87} +7.44991e10 q^{89} -9.12201e10 q^{91} +7.26450e10 q^{93} +9.86499e9 q^{95} +8.12138e10 q^{97} -2.32753e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−322.338	−0.765851	−0.382926	−	0.923779i	$-0.625083\pi$
$3$	−322.338	−0.765851	−0.382926	+	0.923779i	$0.625083\pi$
$4$	0	0
$5$	3125.00	0.447214
$5$	3125.00	0.447214
$6$	0	0
$7$	−46319.5	−1.04166	−0.520828	−	0.853661i	$-0.674377\pi$
$7$	−46319.5	−1.04166	−0.520828	+	0.853661i	$0.674377\pi$
$8$	0	0
$9$	−73245.3	−0.413472
$10$	0	0
$11$	317772.	0.594916	0.297458	−	0.954735i	$-0.403861\pi$
$11$	317772.	0.594916	0.297458	+	0.954735i	$0.403861\pi$
$12$	0	0
$13$	1.96937e6	1.47109	0.735544	−	0.677477i	$-0.236927\pi$
$13$	1.96937e6	1.47109	0.735544	+	0.677477i	$0.236927\pi$
$14$	0	0
$15$	−1.00731e6	−0.342499
$16$	0	0
$17$	6.70196e6	1.14481	0.572405	−	0.819971i	$-0.306011\pi$
$17$	6.70196e6	1.14481	0.572405	+	0.819971i	$0.306011\pi$
$18$	0	0
$19$	3.15680e6	0.292484	0.146242	−	0.989249i	$-0.453282\pi$
$19$	3.15680e6	0.292484	0.146242	+	0.989249i	$0.453282\pi$
$20$	0	0
$21$	1.49305e7	0.797754
$22$	0	0
$23$	2.51370e7	0.814349	0.407175	−	0.913350i	$-0.366514\pi$
$23$	2.51370e7	0.814349	0.407175	+	0.913350i	$0.366514\pi$
$24$	0	0
$25$	9.76562e6	0.200000
$26$	0	0
$27$	8.07109e7	1.08251
$28$	0	0
$29$	1.66550e8	1.50784	0.753919	−	0.656968i	$-0.228161\pi$
$29$	1.66550e8	1.50784	0.753919	+	0.656968i	$0.228161\pi$
$30$	0	0
$31$	−2.25369e8	−1.41386	−0.706928	−	0.707286i	$-0.749919\pi$
$31$	−2.25369e8	−1.41386	−0.706928	+	0.707286i	$0.749919\pi$
$32$	0	0
$33$	−1.02430e8	−0.455617
$34$	0	0
$35$	−1.44748e8	−0.465843
$36$	0	0
$37$	−4.84476e7	−0.114858	−0.0574292	−	0.998350i	$-0.518290\pi$
$37$	−4.84476e7	−0.114858	−0.0574292	+	0.998350i	$0.518290\pi$
$38$	0	0
$39$	−6.34802e8	−1.12663
$40$	0	0
$41$	−5.76349e8	−0.776916	−0.388458	−	0.921466i	$-0.626992\pi$
$41$	−5.76349e8	−0.776916	−0.388458	+	0.921466i	$0.626992\pi$
$42$	0	0
$43$	1.91315e8	0.198460	0.0992300	−	0.995065i	$-0.468362\pi$
$43$	1.91315e8	0.198460	0.0992300	+	0.995065i	$0.468362\pi$
$44$	0	0
$45$	−2.28892e8	−0.184910
$46$	0	0
$47$	5.49453e8	0.349456	0.174728	−	0.984617i	$-0.444095\pi$
$47$	5.49453e8	0.349456	0.174728	+	0.984617i	$0.444095\pi$
$48$	0	0
$49$	1.68169e8	0.0850488
$50$	0	0
$51$	−2.16030e9	−0.876753
$52$	0	0
$53$	1.16938e9	0.384095	0.192047	−	0.981386i	$-0.438487\pi$
$53$	1.16938e9	0.384095	0.192047	+	0.981386i	$0.438487\pi$
$54$	0	0
$55$	9.93037e8	0.266054
$56$	0	0
$57$	−1.01755e9	−0.223999
$58$	0	0
$59$	5.46508e9	0.995201	0.497600	−	0.867406i	$-0.334215\pi$
$59$	5.46508e9	0.995201	0.497600	+	0.867406i	$0.334215\pi$
$60$	0	0
$61$	1.14271e10	1.73230	0.866149	−	0.499786i	$-0.166588\pi$
$61$	1.14271e10	1.73230	0.866149	+	0.499786i	$0.166588\pi$
$62$	0	0
$63$	3.39269e9	0.430696
$64$	0	0
$65$	6.15428e9	0.657890
$66$	0	0
$67$	−5.91196e9	−0.534959	−0.267479	−	0.963564i	$-0.586191\pi$
$67$	−5.91196e9	−0.534959	−0.267479	+	0.963564i	$0.586191\pi$
$68$	0	0
$69$	−8.10261e9	−0.623670
$70$	0	0
$71$	−1.78433e10	−1.17369	−0.586846	−	0.809699i	$-0.699631\pi$
$71$	−1.78433e10	−1.17369	−0.586846	+	0.809699i	$0.699631\pi$
$72$	0	0
$73$	2.17204e10	1.22628	0.613142	−	0.789972i	$-0.289905\pi$
$73$	2.17204e10	1.22628	0.613142	+	0.789972i	$0.289905\pi$
$74$	0	0
$75$	−3.14783e9	−0.153170
$76$	0	0
$77$	−1.47190e10	−0.619698
$78$	0	0
$79$	−4.51177e10	−1.64967	−0.824837	−	0.565371i	$-0.808733\pi$
$79$	−4.51177e10	−1.64967	−0.824837	+	0.565371i	$0.808733\pi$
$80$	0	0
$81$	−1.30410e10	−0.415569
$82$	0	0
$83$	1.46798e9	0.0409062	0.0204531	−	0.999791i	$-0.493489\pi$
$83$	1.46798e9	0.0409062	0.0204531	+	0.999791i	$0.493489\pi$
$84$	0	0
$85$	2.09436e10	0.511974
$86$	0	0
$87$	−5.36852e10	−1.15478
$88$	0	0
$89$	7.44991e10	1.41418	0.707092	−	0.707122i	$-0.250007\pi$
$89$	7.44991e10	1.41418	0.707092	+	0.707122i	$0.250007\pi$
$90$	0	0
$91$	−9.12201e10	−1.53237
$92$	0	0
$93$	7.26450e10	1.08280
$94$	0	0
$95$	9.86499e9	0.130803
$96$	0	0
$97$	8.12138e10	0.960253	0.480126	−	0.877199i	$-0.340591\pi$
$97$	8.12138e10	0.960253	0.480126	+	0.877199i	$0.340591\pi$
$98$	0	0
$99$	−2.32753e10	−0.245981
$100$	0	0
$101$	−1.91624e11	−1.81419	−0.907095	−	0.420926i	$-0.861705\pi$
$101$	−1.91624e11	−1.81419	−0.907095	+	0.420926i	$0.861705\pi$
$102$	0	0
$103$	2.25295e11	1.91490	0.957450	−	0.288600i	$-0.0931899\pi$
$103$	2.25295e11	1.91490	0.957450	+	0.288600i	$0.0931899\pi$
$104$	0	0
$105$	4.66579e10	0.356766
$106$	0	0
$107$	2.03681e10	0.140391	0.0701955	−	0.997533i	$-0.477638\pi$
$107$	2.03681e10	0.140391	0.0701955	+	0.997533i	$0.477638\pi$
$108$	0	0
$109$	9.40097e10	0.585230	0.292615	−	0.956230i	$-0.405475\pi$
$109$	9.40097e10	0.585230	0.292615	+	0.956230i	$0.405475\pi$
$110$	0	0
$111$	1.56165e10	0.0879644
$112$	0	0
$113$	3.12193e11	1.59401	0.797006	−	0.603971i	$-0.206416\pi$
$113$	3.12193e11	1.59401	0.797006	+	0.603971i	$0.206416\pi$
$114$	0	0
$115$	7.85532e10	0.364188
$116$	0	0
$117$	−1.44247e11	−0.608253
$118$	0	0
$119$	−3.10432e11	−1.19250
$120$	0	0
$121$	−1.84333e11	−0.646075
$122$	0	0
$123$	1.85779e11	0.595002
$124$	0	0
$125$	3.05176e10	0.0894427
$126$	0	0
$127$	−5.10293e11	−1.37056	−0.685282	−	0.728278i	$-0.740321\pi$
$127$	−5.10293e11	−1.37056	−0.685282	+	0.728278i	$0.740321\pi$
$128$	0	0
$129$	−6.16682e10	−0.151991
$130$	0	0
$131$	−6.31604e11	−1.43039	−0.715193	−	0.698927i	$-0.753661\pi$
$131$	−6.31604e11	−1.43039	−0.715193	+	0.698927i	$0.753661\pi$
$132$	0	0
$133$	−1.46221e11	−0.304668
$134$	0	0
$135$	2.52222e11	0.484113
$136$	0	0
$137$	−3.82398e11	−0.676943	−0.338471	−	0.940977i	$-0.609910\pi$
$137$	−3.82398e11	−0.676943	−0.338471	+	0.940977i	$0.609910\pi$
$138$	0	0
$139$	9.74280e10	0.159258	0.0796292	−	0.996825i	$-0.474626\pi$
$139$	9.74280e10	0.159258	0.0796292	+	0.996825i	$0.474626\pi$
$140$	0	0
$141$	−1.77110e11	−0.267631
$142$	0	0
$143$	6.25810e11	0.875173
$144$	0	0
$145$	5.20467e11	0.674325
$146$	0	0
$147$	−5.42073e10	−0.0651347
$148$	0	0
$149$	−1.86847e11	−0.208431	−0.104216	−	0.994555i	$-0.533233\pi$
$149$	−1.86847e11	−0.208431	−0.104216	+	0.994555i	$0.533233\pi$
$150$	0	0
$151$	1.69395e12	1.75601	0.878005	−	0.478651i	$-0.158874\pi$
$151$	1.69395e12	1.75601	0.878005	+	0.478651i	$0.158874\pi$
$152$	0	0
$153$	−4.90888e11	−0.473346
$154$	0	0
$155$	−7.04278e11	−0.632295
$156$	0	0
$157$	4.67640e11	0.391259	0.195629	−	0.980678i	$-0.437325\pi$
$157$	4.67640e11	0.391259	0.195629	+	0.980678i	$0.437325\pi$
$158$	0	0
$159$	−3.76935e11	−0.294159
$160$	0	0
$161$	−1.16433e12	−0.848272
$162$	0	0
$163$	2.48980e12	1.69486	0.847428	−	0.530910i	$-0.178150\pi$
$163$	2.48980e12	1.69486	0.847428	+	0.530910i	$0.178150\pi$
$164$	0	0
$165$	−3.20093e11	−0.203758
$166$	0	0
$167$	2.71461e12	1.61721	0.808606	−	0.588351i	$-0.200223\pi$
$167$	2.71461e12	1.61721	0.808606	+	0.588351i	$0.200223\pi$
$168$	0	0
$169$	2.08625e12	1.16410
$170$	0	0
$171$	−2.31221e11	−0.120934
$172$	0	0
$173$	−2.77436e12	−1.36116	−0.680579	−	0.732675i	$-0.738272\pi$
$173$	−2.77436e12	−1.36116	−0.680579	+	0.732675i	$0.738272\pi$
$174$	0	0
$175$	−4.52339e11	−0.208331
$176$	0	0
$177$	−1.76160e12	−0.762175
$178$	0	0
$179$	7.59167e11	0.308778	0.154389	−	0.988010i	$-0.450659\pi$
$179$	7.59167e11	0.308778	0.154389	+	0.988010i	$0.450659\pi$
$180$	0	0
$181$	1.83029e12	0.700305	0.350152	−	0.936693i	$-0.386130\pi$
$181$	1.83029e12	0.700305	0.350152	+	0.936693i	$0.386130\pi$
$182$	0	0
$183$	−3.68339e12	−1.32668
$184$	0	0
$185$	−1.51399e11	−0.0513662
$186$	0	0
$187$	2.12970e12	0.681065
$188$	0	0
$189$	−3.73849e12	−1.12760
$190$	0	0
$191$	−7.81522e11	−0.222463	−0.111231	−	0.993795i	$-0.535479\pi$
$191$	−7.81522e11	−0.222463	−0.111231	+	0.993795i	$0.535479\pi$
$192$	0	0
$193$	−4.70627e12	−1.26506	−0.632530	−	0.774535i	$-0.717984\pi$
$193$	−4.70627e12	−1.26506	−0.632530	+	0.774535i	$0.717984\pi$
$194$	0	0
$195$	−1.98376e12	−0.503846
$196$	0	0
$197$	−8.86812e10	−0.0212945	−0.0106472	−	0.999943i	$-0.503389\pi$
$197$	−8.86812e10	−0.0212945	−0.0106472	+	0.999943i	$0.503389\pi$
$198$	0	0
$199$	2.87053e12	0.652035	0.326017	−	0.945364i	$-0.394293\pi$
$199$	2.87053e12	0.652035	0.326017	+	0.945364i	$0.394293\pi$
$200$	0	0
$201$	1.90565e12	0.409699
$202$	0	0
$203$	−7.71449e12	−1.57065
$204$	0	0
$205$	−1.80109e12	−0.347448
$206$	0	0
$207$	−1.84117e12	−0.336711
$208$	0	0
$209$	1.00314e12	0.174003
$210$	0	0
$211$	1.09016e13	1.79447	0.897237	−	0.441549i	$-0.145571\pi$
$211$	1.09016e13	1.79447	0.897237	+	0.441549i	$0.145571\pi$
$212$	0	0
$213$	5.75157e12	0.898873
$214$	0	0
$215$	5.97860e11	0.0887540
$216$	0	0
$217$	1.04390e13	1.47275
$218$	0	0
$219$	−7.00129e12	−0.939152
$220$	0	0
$221$	1.31986e13	1.68411
$222$	0	0
$223$	1.61507e12	0.196117	0.0980584	−	0.995181i	$-0.468737\pi$
$223$	1.61507e12	0.196117	0.0980584	+	0.995181i	$0.468737\pi$
$224$	0	0
$225$	−7.15286e11	−0.0826944
$226$	0	0
$227$	−1.51870e13	−1.67236	−0.836182	−	0.548452i	$-0.815217\pi$
$227$	−1.51870e13	−1.67236	−0.836182	+	0.548452i	$0.815217\pi$
$228$	0	0
$229$	6.90526e12	0.724578	0.362289	−	0.932066i	$-0.381995\pi$
$229$	6.90526e12	0.724578	0.362289	+	0.932066i	$0.381995\pi$
$230$	0	0
$231$	4.74450e12	0.474597
$232$	0	0
$233$	−1.31428e13	−1.25380	−0.626902	−	0.779098i	$-0.715678\pi$
$233$	−1.31428e13	−1.25380	−0.626902	+	0.779098i	$0.715678\pi$
$234$	0	0
$235$	1.71704e12	0.156282
$236$	0	0
$237$	1.45431e13	1.26340
$238$	0	0
$239$	6.94431e12	0.576024	0.288012	−	0.957627i	$-0.407006\pi$
$239$	6.94431e12	0.576024	0.288012	+	0.957627i	$0.407006\pi$
$240$	0	0
$241$	−6.76486e12	−0.536001	−0.268000	−	0.963419i	$-0.586363\pi$
$241$	−6.76486e12	−0.536001	−0.268000	+	0.963419i	$0.586363\pi$
$242$	0	0
$243$	−1.00941e13	−0.764245
$244$	0	0
$245$	5.25529e11	0.0380350
$246$	0	0
$247$	6.21689e12	0.430269
$248$	0	0
$249$	−4.73184e11	−0.0313281
$250$	0	0
$251$	−1.68717e13	−1.06894	−0.534471	−	0.845187i	$-0.679489\pi$
$251$	−1.68717e13	−1.06894	−0.534471	+	0.845187i	$0.679489\pi$
$252$	0	0
$253$	7.98783e12	0.484469
$254$	0	0
$255$	−6.75093e12	−0.392096
$256$	0	0
$257$	−8.55471e12	−0.475963	−0.237982	−	0.971270i	$-0.576486\pi$
$257$	−8.55471e12	−0.475963	−0.237982	+	0.971270i	$0.576486\pi$
$258$	0	0
$259$	2.24407e12	0.119643
$260$	0	0
$261$	−1.21990e13	−0.623448
$262$	0	0
$263$	1.13408e13	0.555760	0.277880	−	0.960616i	$-0.410368\pi$
$263$	1.13408e13	0.555760	0.277880	+	0.960616i	$0.410368\pi$
$264$	0	0
$265$	3.65431e12	0.171772
$266$	0	0
$267$	−2.40139e13	−1.08305
$268$	0	0
$269$	−4.07033e13	−1.76194	−0.880972	−	0.473169i	$-0.843110\pi$
$269$	−4.07033e13	−1.76194	−0.880972	+	0.473169i	$0.843110\pi$
$270$	0	0
$271$	−2.46051e13	−1.02257	−0.511286	−	0.859410i	$-0.670831\pi$
$271$	−2.46051e13	−1.02257	−0.511286	+	0.859410i	$0.670831\pi$
$272$	0	0
$273$	2.94037e13	1.17357
$274$	0	0
$275$	3.10324e12	0.118983
$276$	0	0
$277$	1.40213e12	0.0516594	0.0258297	−	0.999666i	$-0.491777\pi$
$277$	1.40213e12	0.0516594	0.0258297	+	0.999666i	$0.491777\pi$
$278$	0	0
$279$	1.65072e13	0.584589
$280$	0	0
$281$	−3.18398e12	−0.108414	−0.0542071	−	0.998530i	$-0.517263\pi$
$281$	−3.18398e12	−0.108414	−0.0542071	+	0.998530i	$0.517263\pi$
$282$	0	0
$283$	−1.72628e12	−0.0565310	−0.0282655	−	0.999600i	$-0.508998\pi$
$283$	−1.72628e12	−0.0565310	−0.0282655	+	0.999600i	$0.508998\pi$
$284$	0	0
$285$	−3.17986e12	−0.100175
$286$	0	0
$287$	2.66962e13	0.809280
$288$	0	0
$289$	1.06444e13	0.310588
$290$	0	0
$291$	−2.61783e13	−0.735411
$292$	0	0
$293$	5.26240e13	1.42368	0.711839	−	0.702343i	$-0.247862\pi$
$293$	5.26240e13	1.42368	0.711839	+	0.702343i	$0.247862\pi$
$294$	0	0
$295$	1.70784e13	0.445067
$296$	0	0
$297$	2.56477e13	0.644002
$298$	0	0
$299$	4.95040e13	1.19798
$300$	0	0
$301$	−8.86163e12	−0.206727
$302$	0	0
$303$	6.17677e13	1.38940
$304$	0	0
$305$	3.57098e13	0.774707
$306$	0	0
$307$	−6.51872e13	−1.36427	−0.682137	−	0.731225i	$-0.738949\pi$
$307$	−6.51872e13	−1.36427	−0.682137	+	0.731225i	$0.738949\pi$
$308$	0	0
$309$	−7.26210e13	−1.46653
$310$	0	0
$311$	3.08407e13	0.601093	0.300546	−	0.953767i	$-0.402831\pi$
$311$	3.08407e13	0.601093	0.300546	+	0.953767i	$0.402831\pi$
$312$	0	0
$313$	−2.32028e13	−0.436563	−0.218282	−	0.975886i	$-0.570045\pi$
$313$	−2.32028e13	−0.436563	−0.218282	+	0.975886i	$0.570045\pi$
$314$	0	0
$315$	1.06021e13	0.192613
$316$	0	0
$317$	−6.76914e13	−1.18770	−0.593851	−	0.804575i	$-0.702393\pi$
$317$	−6.76914e13	−1.18770	−0.593851	+	0.804575i	$0.702393\pi$
$318$	0	0
$319$	5.29247e13	0.897036
$320$	0	0
$321$	−6.56540e12	−0.107519
$322$	0	0
$323$	2.11567e13	0.334838
$324$	0	0
$325$	1.92321e13	0.294217
$326$	0	0
$327$	−3.03029e13	−0.448199
$328$	0	0
$329$	−2.54504e13	−0.364013
$330$	0	0
$331$	−6.49600e13	−0.898653	−0.449326	−	0.893368i	$-0.648336\pi$
$331$	−6.49600e13	−0.898653	−0.449326	+	0.893368i	$0.648336\pi$
$332$	0	0
$333$	3.54856e12	0.0474907
$334$	0	0
$335$	−1.84749e13	−0.239241
$336$	0	0
$337$	3.15511e10	0.000395412 0	0.000197706	−	1.00000i	$-0.499937\pi$
$337$	3.15511e10	0.000395412 0	0.000197706		1.00000i	$0.499937\pi$
$338$	0	0
$339$	−1.00632e14	−1.22078
$340$	0	0
$341$	−7.16159e13	−0.841125
$342$	0	0
$343$	8.37993e13	0.953065
$344$	0	0
$345$	−2.53207e13	−0.278914
$346$	0	0
$347$	2.99274e13	0.319342	0.159671	−	0.987170i	$-0.448957\pi$
$347$	2.99274e13	0.319342	0.159671	+	0.987170i	$0.448957\pi$
$348$	0	0
$349$	−1.00661e14	−1.04070	−0.520348	−	0.853955i	$-0.674198\pi$
$349$	−1.00661e14	−1.04070	−0.520348	+	0.853955i	$0.674198\pi$
$350$	0	0
$351$	1.58950e14	1.59247
$352$	0	0
$353$	−1.54028e14	−1.49568	−0.747840	−	0.663879i	$-0.768909\pi$
$353$	−1.54028e14	−1.49568	−0.747840	+	0.663879i	$0.768909\pi$
$354$	0	0
$355$	−5.57603e13	−0.524891
$356$	0	0
$357$	1.00064e14	0.913276
$358$	0	0
$359$	1.71452e14	1.51748	0.758740	−	0.651393i	$-0.225815\pi$
$359$	1.71452e14	1.51748	0.758740	+	0.651393i	$0.225815\pi$
$360$	0	0
$361$	−1.06525e14	−0.914453
$362$	0	0
$363$	5.94174e13	0.494797
$364$	0	0
$365$	6.78761e13	0.548411
$366$	0	0
$367$	−1.71254e14	−1.34270	−0.671348	−	0.741142i	$-0.734284\pi$
$367$	−1.71254e14	−1.34270	−0.671348	+	0.741142i	$0.734284\pi$
$368$	0	0
$369$	4.22149e13	0.321233
$370$	0	0
$371$	−5.41651e13	−0.400095
$372$	0	0
$373$	−2.06032e14	−1.47753	−0.738764	−	0.673964i	$-0.764590\pi$
$373$	−2.06032e14	−1.47753	−0.738764	+	0.673964i	$0.764590\pi$
$374$	0	0
$375$	−9.83697e12	−0.0684998
$376$	0	0
$377$	3.27997e14	2.21816
$378$	0	0
$379$	2.97286e13	0.195281	0.0976403	−	0.995222i	$-0.468871\pi$
$379$	2.97286e13	0.195281	0.0976403	+	0.995222i	$0.468871\pi$
$380$	0	0
$381$	1.64487e14	1.04965
$382$	0	0
$383$	−6.83277e12	−0.0423647	−0.0211823	−	0.999776i	$-0.506743\pi$
$383$	−6.83277e12	−0.0423647	−0.0211823	+	0.999776i	$0.506743\pi$
$384$	0	0
$385$	−4.59970e13	−0.277137
$386$	0	0
$387$	−1.40129e13	−0.0820577
$388$	0	0
$389$	2.76218e14	1.57228	0.786139	−	0.618050i	$-0.212077\pi$
$389$	2.76218e14	1.57228	0.786139	+	0.618050i	$0.212077\pi$
$390$	0	0
$391$	1.68467e14	0.932274
$392$	0	0
$393$	2.03590e14	1.09546
$394$	0	0
$395$	−1.40993e14	−0.737757
$396$	0	0
$397$	2.37045e14	1.20638	0.603188	−	0.797599i	$-0.293897\pi$
$397$	2.37045e14	1.20638	0.603188	+	0.797599i	$0.293897\pi$
$398$	0	0
$399$	4.71326e13	0.233330
$400$	0	0
$401$	3.12727e13	0.150616	0.0753080	−	0.997160i	$-0.476006\pi$
$401$	3.12727e13	0.150616	0.0753080	+	0.997160i	$0.476006\pi$
$402$	0	0
$403$	−4.43835e14	−2.07990
$404$	0	0
$405$	−4.07531e13	−0.185848
$406$	0	0
$407$	−1.53953e13	−0.0683311
$408$	0	0
$409$	−2.62457e14	−1.13391	−0.566956	−	0.823748i	$-0.691879\pi$
$409$	−2.62457e14	−1.13391	−0.566956	+	0.823748i	$0.691879\pi$
$410$	0	0
$411$	1.23261e14	0.518437
$412$	0	0
$413$	−2.53140e14	−1.03666
$414$	0	0
$415$	4.58743e12	0.0182938
$416$	0	0
$417$	−3.14047e13	−0.121968
$418$	0	0
$419$	−3.02479e14	−1.14424	−0.572121	−	0.820169i	$-0.693879\pi$
$419$	−3.02479e14	−1.14424	−0.572121	+	0.820169i	$0.693879\pi$
$420$	0	0
$421$	3.11314e14	1.14722	0.573611	−	0.819128i	$-0.305542\pi$
$421$	3.11314e14	1.14722	0.573611	+	0.819128i	$0.305542\pi$
$422$	0	0
$423$	−4.02449e13	−0.144490
$424$	0	0
$425$	6.54489e13	0.228962
$426$	0	0
$427$	−5.29298e14	−1.80446
$428$	0	0
$429$	−2.01722e14	−0.670252
$430$	0	0
$431$	1.49970e14	0.485713	0.242856	−	0.970062i	$-0.421916\pi$
$431$	1.49970e14	0.485713	0.242856	+	0.970062i	$0.421916\pi$
$432$	0	0
$433$	−2.41783e14	−0.763384	−0.381692	−	0.924290i	$-0.624658\pi$
$433$	−2.41783e14	−0.763384	−0.381692	+	0.924290i	$0.624658\pi$
$434$	0	0
$435$	−1.67766e14	−0.516433
$436$	0	0
$437$	7.93524e13	0.238184
$438$	0	0
$439$	2.52346e14	0.738654	0.369327	−	0.929300i	$-0.379588\pi$
$439$	2.52346e14	0.738654	0.369327	+	0.929300i	$0.379588\pi$
$440$	0	0
$441$	−1.23176e13	−0.0351653
$442$	0	0
$443$	1.56538e14	0.435912	0.217956	−	0.975959i	$-0.430061\pi$
$443$	1.56538e14	0.435912	0.217956	+	0.975959i	$0.430061\pi$
$444$	0	0
$445$	2.32810e14	0.632442
$446$	0	0
$447$	6.02280e13	0.159627
$448$	0	0
$449$	−1.02669e14	−0.265512	−0.132756	−	0.991149i	$-0.542383\pi$
$449$	−1.02669e14	−0.265512	−0.132756	+	0.991149i	$0.542383\pi$
$450$	0	0
$451$	−1.83147e14	−0.462200
$452$	0	0
$453$	−5.46024e14	−1.34484
$454$	0	0
$455$	−2.85063e14	−0.685296
$456$	0	0
$457$	4.13279e14	0.969850	0.484925	−	0.874556i	$-0.338847\pi$
$457$	4.13279e14	0.969850	0.484925	+	0.874556i	$0.338847\pi$
$458$	0	0
$459$	5.40922e14	1.23927
$460$	0	0
$461$	2.12155e14	0.474567	0.237284	−	0.971440i	$-0.423743\pi$
$461$	2.12155e14	0.474567	0.237284	+	0.971440i	$0.423743\pi$
$462$	0	0
$463$	3.77488e14	0.824533	0.412267	−	0.911063i	$-0.364737\pi$
$463$	3.77488e14	0.824533	0.412267	+	0.911063i	$0.364737\pi$
$464$	0	0
$465$	2.27016e14	0.484244
$466$	0	0
$467$	−3.60800e14	−0.751664	−0.375832	−	0.926688i	$-0.622643\pi$
$467$	−3.60800e14	−0.751664	−0.375832	+	0.926688i	$0.622643\pi$
$468$	0	0
$469$	2.73839e14	0.557243
$470$	0	0
$471$	−1.50738e14	−0.299646
$472$	0	0
$473$	6.07946e13	0.118067
$474$	0	0
$475$	3.08281e13	0.0584967
$476$	0	0
$477$	−8.56516e13	−0.158812
$478$	0	0
$479$	−2.58776e13	−0.0468899	−0.0234449	−	0.999725i	$-0.507463\pi$
$479$	−2.58776e13	−0.0468899	−0.0234449	+	0.999725i	$0.507463\pi$
$480$	0	0
$481$	−9.54111e13	−0.168967
$482$	0	0
$483$	3.75309e14	0.649650
$484$	0	0
$485$	2.53793e14	0.429438
$486$	0	0
$487$	3.54633e14	0.586637	0.293318	−	0.956015i	$-0.405240\pi$
$487$	3.54633e14	0.586637	0.293318	+	0.956015i	$0.405240\pi$
$488$	0	0
$489$	−8.02557e14	−1.29801
$490$	0	0
$491$	1.72416e14	0.272664	0.136332	−	0.990663i	$-0.456469\pi$
$491$	1.72416e14	0.272664	0.136332	+	0.990663i	$0.456469\pi$
$492$	0	0
$493$	1.11621e15	1.72619
$494$	0	0
$495$	−7.27353e13	−0.110006
$496$	0	0
$497$	8.26492e14	1.22258
$498$	0	0
$499$	7.01194e14	1.01458	0.507289	−	0.861776i	$-0.330648\pi$
$499$	7.01194e14	1.01458	0.507289	+	0.861776i	$0.330648\pi$
$500$	0	0
$501$	−8.75021e14	−1.23854
$502$	0	0
$503$	7.18922e14	0.995538	0.497769	−	0.867310i	$-0.334153\pi$
$503$	7.18922e14	0.995538	0.497769	+	0.867310i	$0.334153\pi$
$504$	0	0
$505$	−5.98825e14	−0.811330
$506$	0	0
$507$	−6.72478e14	−0.891526
$508$	0	0
$509$	−1.13548e15	−1.47309	−0.736547	−	0.676386i	$-0.763545\pi$
$509$	−1.13548e15	−1.47309	−0.736547	+	0.676386i	$0.763545\pi$
$510$	0	0
$511$	−1.00608e15	−1.27737
$512$	0	0
$513$	2.54788e14	0.316616
$514$	0	0
$515$	7.04045e14	0.856369
$516$	0	0
$517$	1.74601e14	0.207897
$518$	0	0
$519$	8.94280e14	1.04244
$520$	0	0
$521$	−1.52530e15	−1.74079	−0.870396	−	0.492352i	$-0.836137\pi$
$521$	−1.52530e15	−1.74079	−0.870396	+	0.492352i	$0.836137\pi$
$522$	0	0
$523$	3.67077e14	0.410202	0.205101	−	0.978741i	$-0.434248\pi$
$523$	3.67077e14	0.410202	0.205101	+	0.978741i	$0.434248\pi$
$524$	0	0
$525$	1.45806e14	0.159551
$526$	0	0
$527$	−1.51042e15	−1.61859
$528$	0	0
$529$	−3.20940e14	−0.336835
$530$	0	0
$531$	−4.00292e14	−0.411488
$532$	0	0
$533$	−1.13504e15	−1.14291
$534$	0	0
$535$	6.36502e13	0.0627847
$536$	0	0
$537$	−2.44708e14	−0.236478
$538$	0	0
$539$	5.34394e13	0.0505969
$540$	0	0
$541$	5.85922e14	0.543569	0.271785	−	0.962358i	$-0.412386\pi$
$541$	5.85922e14	0.543569	0.271785	+	0.962358i	$0.412386\pi$
$542$	0	0
$543$	−5.89971e14	−0.536329
$544$	0	0
$545$	2.93780e14	0.261723
$546$	0	0
$547$	−6.35737e14	−0.555069	−0.277535	−	0.960716i	$-0.589517\pi$
$547$	−6.35737e14	−0.555069	−0.277535	+	0.960716i	$0.589517\pi$
$548$	0	0
$549$	−8.36983e14	−0.716257
$550$	0	0
$551$	5.25763e14	0.441018
$552$	0	0
$553$	2.08983e15	1.71839
$554$	0	0
$555$	4.88015e13	0.0393389
$556$	0	0
$557$	−1.06572e15	−0.842246	−0.421123	−	0.907004i	$-0.638364\pi$
$557$	−1.06572e15	−0.842246	−0.421123	+	0.907004i	$0.638364\pi$
$558$	0	0
$559$	3.76770e14	0.291952
$560$	0	0
$561$	−6.86481e14	−0.521594
$562$	0	0
$563$	1.14286e15	0.851522	0.425761	−	0.904836i	$-0.360006\pi$
$563$	1.14286e15	0.851522	0.425761	+	0.904836i	$0.360006\pi$
$564$	0	0
$565$	9.75603e14	0.712864
$566$	0	0
$567$	6.04052e14	0.432880
$568$	0	0
$569$	−4.03257e14	−0.283442	−0.141721	−	0.989907i	$-0.545264\pi$
$569$	−4.03257e14	−0.283442	−0.141721	+	0.989907i	$0.545264\pi$
$570$	0	0
$571$	1.35092e15	0.931386	0.465693	−	0.884946i	$-0.345805\pi$
$571$	1.35092e15	0.931386	0.465693	+	0.884946i	$0.345805\pi$
$572$	0	0
$573$	2.51914e14	0.170373
$574$	0	0
$575$	2.45479e14	0.162870
$576$	0	0
$577$	1.33874e15	0.871421	0.435711	−	0.900087i	$-0.356497\pi$
$577$	1.33874e15	0.871421	0.435711	+	0.900087i	$0.356497\pi$
$578$	0	0
$579$	1.51701e15	0.968848
$580$	0	0
$581$	−6.79959e13	−0.0426103
$582$	0	0
$583$	3.71596e14	0.228504
$584$	0	0
$585$	−4.50772e14	−0.272019
$586$	0	0
$587$	1.41689e15	0.839123	0.419562	−	0.907727i	$-0.362184\pi$
$587$	1.41689e15	0.839123	0.419562	+	0.907727i	$0.362184\pi$
$588$	0	0
$589$	−7.11444e14	−0.413530
$590$	0	0
$591$	2.85853e13	0.0163084
$592$	0	0
$593$	−9.36699e14	−0.524565	−0.262282	−	0.964991i	$-0.584475\pi$
$593$	−9.36699e14	−0.524565	−0.262282	+	0.964991i	$0.584475\pi$
$594$	0	0
$595$	−9.70099e14	−0.533301
$596$	0	0
$597$	−9.25281e14	−0.499361
$598$	0	0
$599$	−2.38262e15	−1.26243	−0.631214	−	0.775609i	$-0.717443\pi$
$599$	−2.38262e15	−1.26243	−0.631214	+	0.775609i	$0.717443\pi$
$600$	0	0
$601$	1.79176e15	0.932114	0.466057	−	0.884755i	$-0.345674\pi$
$601$	1.79176e15	0.932114	0.466057	+	0.884755i	$0.345674\pi$
$602$	0	0
$603$	4.33023e14	0.221190
$604$	0	0
$605$	−5.76040e14	−0.288934
$606$	0	0
$607$	−2.96072e15	−1.45834	−0.729170	−	0.684332i	$-0.760094\pi$
$607$	−2.96072e15	−1.45834	−0.729170	+	0.684332i	$0.760094\pi$
$608$	0	0
$609$	2.48667e15	1.20288
$610$	0	0
$611$	1.08208e15	0.514081
$612$	0	0
$613$	−3.04029e14	−0.141867	−0.0709336	−	0.997481i	$-0.522598\pi$
$613$	−3.04029e14	−0.141867	−0.0709336	+	0.997481i	$0.522598\pi$
$614$	0	0
$615$	5.80560e14	0.266093
$616$	0	0
$617$	3.93397e15	1.77118	0.885591	−	0.464467i	$-0.153754\pi$
$617$	3.93397e15	1.77118	0.885591	+	0.464467i	$0.153754\pi$
$618$	0	0
$619$	4.65120e14	0.205715	0.102858	−	0.994696i	$-0.467201\pi$
$619$	4.65120e14	0.205715	0.102858	+	0.994696i	$0.467201\pi$
$620$	0	0
$621$	2.02883e15	0.881540
$622$	0	0
$623$	−3.45076e15	−1.47309
$624$	0	0
$625$	9.53674e13	0.0400000
$626$	0	0
$627$	−3.23350e14	−0.133261
$628$	0	0
$629$	−3.24694e14	−0.131491
$630$	0	0
$631$	−3.11552e15	−1.23985	−0.619925	−	0.784661i	$-0.712837\pi$
$631$	−3.11552e15	−1.23985	−0.619925	+	0.784661i	$0.712837\pi$
$632$	0	0
$633$	−3.51400e15	−1.37430
$634$	0	0
$635$	−1.59467e15	−0.612935
$636$	0	0
$637$	3.31187e14	0.125114
$638$	0	0
$639$	1.30694e15	0.485289
$640$	0	0
$641$	−2.68239e15	−0.979047	−0.489523	−	0.871990i	$-0.662829\pi$
$641$	−2.68239e15	−0.979047	−0.489523	+	0.871990i	$0.662829\pi$
$642$	0	0
$643$	−3.24755e15	−1.16519	−0.582594	−	0.812763i	$-0.697962\pi$
$643$	−3.24755e15	−1.16519	−0.582594	+	0.812763i	$0.697962\pi$
$644$	0	0
$645$	−1.92713e14	−0.0679724
$646$	0	0
$647$	2.20147e15	0.763376	0.381688	−	0.924291i	$-0.375343\pi$
$647$	2.20147e15	0.763376	0.381688	+	0.924291i	$0.375343\pi$
$648$	0	0
$649$	1.73665e15	0.592061
$650$	0	0
$651$	−3.36488e15	−1.12791
$652$	0	0
$653$	1.76341e15	0.581208	0.290604	−	0.956843i	$-0.406144\pi$
$653$	1.76341e15	0.581208	0.290604	+	0.956843i	$0.406144\pi$
$654$	0	0
$655$	−1.97376e15	−0.639688
$656$	0	0
$657$	−1.59091e15	−0.507034
$658$	0	0
$659$	−2.25728e15	−0.707483	−0.353741	−	0.935343i	$-0.615091\pi$
$659$	−2.25728e15	−0.707483	−0.353741	+	0.935343i	$0.615091\pi$
$660$	0	0
$661$	5.06357e15	1.56081	0.780403	−	0.625277i	$-0.215014\pi$
$661$	5.06357e15	1.56081	0.780403	+	0.625277i	$0.215014\pi$
$662$	0	0
$663$	−4.25442e15	−1.28978
$664$	0	0
$665$	−4.56941e14	−0.136252
$666$	0	0
$667$	4.18656e15	1.22791
$668$	0	0
$669$	−5.20598e14	−0.150196
$670$	0	0
$671$	3.63122e15	1.03057
$672$	0	0
$673$	−2.34714e15	−0.655323	−0.327662	−	0.944795i	$-0.606261\pi$
$673$	−2.34714e15	−0.655323	−0.327662	+	0.944795i	$0.606261\pi$
$674$	0	0
$675$	7.88193e14	0.216502
$676$	0	0
$677$	1.22198e15	0.330237	0.165118	−	0.986274i	$-0.447199\pi$
$677$	1.22198e15	0.330237	0.165118	+	0.986274i	$0.447199\pi$
$678$	0	0
$679$	−3.76178e15	−1.00025
$680$	0	0
$681$	4.89536e15	1.28078
$682$	0	0
$683$	5.01192e15	1.29030	0.645149	−	0.764057i	$-0.276795\pi$
$683$	5.01192e15	1.29030	0.645149	+	0.764057i	$0.276795\pi$
$684$	0	0
$685$	−1.19499e15	−0.302738
$686$	0	0
$687$	−2.22583e15	−0.554919
$688$	0	0
$689$	2.30294e15	0.565037
$690$	0	0
$691$	4.25993e15	1.02866	0.514332	−	0.857591i	$-0.328040\pi$
$691$	4.25993e15	1.02866	0.514332	+	0.857591i	$0.328040\pi$
$692$	0	0
$693$	1.07810e15	0.256228
$694$	0	0
$695$	3.04462e14	0.0712225
$696$	0	0
$697$	−3.86267e15	−0.889421
$698$	0	0
$699$	4.23642e15	0.960228
$700$	0	0
$701$	3.06037e15	0.682849	0.341425	−	0.939909i	$-0.389091\pi$
$701$	3.06037e15	0.682849	0.341425	+	0.939909i	$0.389091\pi$
$702$	0	0
$703$	−1.52939e14	−0.0335942
$704$	0	0
$705$	−5.53468e14	−0.119688
$706$	0	0
$707$	8.87594e15	1.88976
$708$	0	0
$709$	3.38201e15	0.708959	0.354479	−	0.935064i	$-0.384658\pi$
$709$	3.38201e15	0.708959	0.354479	+	0.935064i	$0.384658\pi$
$710$	0	0
$711$	3.30466e15	0.682094
$712$	0	0
$713$	−5.66511e15	−1.15137
$714$	0	0
$715$	1.95565e15	0.391389
$716$	0	0
$717$	−2.23841e15	−0.441149
$718$	0	0
$719$	−5.50086e15	−1.06763	−0.533816	−	0.845601i	$-0.679243\pi$
$719$	−5.50086e15	−1.06763	−0.533816	+	0.845601i	$0.679243\pi$
$720$	0	0
$721$	−1.04355e16	−1.99467
$722$	0	0
$723$	2.18057e15	0.410497
$724$	0	0
$725$	1.62646e15	0.301567
$726$	0	0
$727$	−5.91333e15	−1.07992	−0.539961	−	0.841690i	$-0.681561\pi$
$727$	−5.91333e15	−1.07992	−0.539961	+	0.841690i	$0.681561\pi$
$728$	0	0
$729$	5.56388e15	1.00087
$730$	0	0
$731$	1.28219e15	0.227199
$732$	0	0
$733$	6.66898e14	0.116409	0.0582046	−	0.998305i	$-0.481462\pi$
$733$	6.66898e14	0.116409	0.0582046	+	0.998305i	$0.481462\pi$
$734$	0	0
$735$	−1.69398e14	−0.0291291
$736$	0	0
$737$	−1.87865e15	−0.318255
$738$	0	0
$739$	−7.79002e15	−1.30015	−0.650076	−	0.759869i	$-0.725263\pi$
$739$	−7.79002e15	−1.30015	−0.650076	+	0.759869i	$0.725263\pi$
$740$	0	0
$741$	−2.00394e15	−0.329522
$742$	0	0
$743$	4.40631e14	0.0713898	0.0356949	−	0.999363i	$-0.488636\pi$
$743$	4.40631e14	0.0713898	0.0356949	+	0.999363i	$0.488636\pi$
$744$	0	0
$745$	−5.83898e14	−0.0932132
$746$	0	0
$747$	−1.07522e14	−0.0169136
$748$	0	0
$749$	−9.43439e14	−0.146239
$750$	0	0
$751$	2.68780e14	0.0410560	0.0205280	−	0.999789i	$-0.493465\pi$
$751$	2.68780e14	0.0410560	0.0205280	+	0.999789i	$0.493465\pi$
$752$	0	0
$753$	5.43840e15	0.818651
$754$	0	0
$755$	5.29359e15	0.785312
$756$	0	0
$757$	−1.09457e16	−1.60035	−0.800176	−	0.599766i	$-0.795261\pi$
$757$	−1.09457e16	−1.60035	−0.800176	+	0.599766i	$0.795261\pi$
$758$	0	0
$759$	−2.57478e15	−0.371031
$760$	0	0
$761$	−8.77462e15	−1.24627	−0.623136	−	0.782114i	$-0.714142\pi$
$761$	−8.77462e15	−1.24627	−0.623136	+	0.782114i	$0.714142\pi$
$762$	0	0
$763$	−4.35448e15	−0.609609
$764$	0	0
$765$	−1.53402e15	−0.211687
$766$	0	0
$767$	1.07628e16	1.46403
$768$	0	0
$769$	3.16898e15	0.424936	0.212468	−	0.977168i	$-0.431850\pi$
$769$	3.16898e15	0.424936	0.212468	+	0.977168i	$0.431850\pi$
$770$	0	0
$771$	2.75751e15	0.364517
$772$	0	0
$773$	−6.13394e15	−0.799378	−0.399689	−	0.916651i	$-0.630882\pi$
$773$	−6.13394e15	−0.799378	−0.399689	+	0.916651i	$0.630882\pi$
$774$	0	0
$775$	−2.20087e15	−0.282771
$776$	0	0
$777$	−7.23348e14	−0.0916287
$778$	0	0
$779$	−1.81942e15	−0.227235
$780$	0	0
$781$	−5.67009e15	−0.698248
$782$	0	0
$783$	1.34424e16	1.63225
$784$	0	0
$785$	1.46138e15	0.174976
$786$	0	0
$787$	−1.04791e16	−1.23727	−0.618633	−	0.785680i	$-0.712313\pi$
$787$	−1.04791e16	−1.23727	−0.618633	+	0.785680i	$0.712313\pi$
$788$	0	0
$789$	−3.65557e15	−0.425629
$790$	0	0
$791$	−1.44606e16	−1.66041
$792$	0	0
$793$	2.25042e16	2.54836
$794$	0	0
$795$	−1.17792e15	−0.131552
$796$	0	0
$797$	1.44584e15	0.159257	0.0796286	−	0.996825i	$-0.474627\pi$
$797$	1.44584e15	0.159257	0.0796286	+	0.996825i	$0.474627\pi$
$798$	0	0
$799$	3.68242e15	0.400061
$800$	0	0
$801$	−5.45671e15	−0.584725
$802$	0	0
$803$	6.90212e15	0.729536
$804$	0	0
$805$	−3.63854e15	−0.379359
$806$	0	0
$807$	1.31202e16	1.34939
$808$	0	0
$809$	−7.03288e15	−0.713537	−0.356768	−	0.934193i	$-0.616121\pi$
$809$	−7.03288e15	−0.713537	−0.356768	+	0.934193i	$0.616121\pi$
$810$	0	0
$811$	−1.18983e16	−1.19088	−0.595441	−	0.803399i	$-0.703023\pi$
$811$	−1.18983e16	−1.19088	−0.595441	+	0.803399i	$0.703023\pi$
$812$	0	0
$813$	7.93116e15	0.783138
$814$	0	0
$815$	7.78063e15	0.757963
$816$	0	0
$817$	6.03943e14	0.0580463
$818$	0	0
$819$	6.68145e15	0.633591
$820$	0	0
$821$	7.65247e15	0.716002	0.358001	−	0.933721i	$-0.383458\pi$
$821$	7.65247e15	0.716002	0.358001	+	0.933721i	$0.383458\pi$
$822$	0	0
$823$	−7.71911e15	−0.712637	−0.356318	−	0.934365i	$-0.615968\pi$
$823$	−7.71911e15	−0.712637	−0.356318	+	0.934365i	$0.615968\pi$
$824$	0	0
$825$	−1.00029e15	−0.0911234
$826$	0	0
$827$	9.37712e15	0.842926	0.421463	−	0.906846i	$-0.361517\pi$
$827$	9.37712e15	0.842926	0.421463	+	0.906846i	$0.361517\pi$
$828$	0	0
$829$	1.44345e16	1.28042	0.640209	−	0.768201i	$-0.278848\pi$
$829$	1.44345e16	1.28042	0.640209	+	0.768201i	$0.278848\pi$
$830$	0	0
$831$	−4.51959e14	−0.0395634
$832$	0	0
$833$	1.12706e15	0.0973646
$834$	0	0
$835$	8.48316e15	0.723239
$836$	0	0
$837$	−1.81897e16	−1.53051
$838$	0	0
$839$	5.83144e15	0.484268	0.242134	−	0.970243i	$-0.422153\pi$
$839$	5.83144e15	0.484268	0.242134	+	0.970243i	$0.422153\pi$
$840$	0	0
$841$	1.55382e16	1.27357
$842$	0	0
$843$	1.02632e15	0.0830292
$844$	0	0
$845$	6.51953e15	0.520601
$846$	0	0
$847$	8.53820e15	0.672989
$848$	0	0
$849$	5.56446e14	0.0432943
$850$	0	0
$851$	−1.21783e15	−0.0935348
$852$	0	0
$853$	8.78821e15	0.666317	0.333158	−	0.942871i	$-0.391886\pi$
$853$	8.78821e15	0.666317	0.333158	+	0.942871i	$0.391886\pi$
$854$	0	0
$855$	−7.22564e14	−0.0540832
$856$	0	0
$857$	−1.69999e16	−1.25618	−0.628090	−	0.778141i	$-0.716163\pi$
$857$	−1.69999e16	−1.25618	−0.628090	+	0.778141i	$0.716163\pi$
$858$	0	0
$859$	−8.82248e15	−0.643618	−0.321809	−	0.946805i	$-0.604291\pi$
$859$	−8.82248e15	−0.643618	−0.321809	+	0.946805i	$0.604291\pi$
$860$	0	0
$861$	−8.60519e15	−0.619788
$862$	0	0
$863$	−1.35285e16	−0.962035	−0.481017	−	0.876711i	$-0.659733\pi$
$863$	−1.35285e16	−0.962035	−0.481017	+	0.876711i	$0.659733\pi$
$864$	0	0
$865$	−8.66986e15	−0.608728
$866$	0	0
$867$	−3.43110e15	−0.237864
$868$	0	0
$869$	−1.43371e16	−0.981417
$870$	0	0
$871$	−1.16428e16	−0.786971
$872$	0	0
$873$	−5.94853e15	−0.397038
$874$	0	0
$875$	−1.41356e15	−0.0931686
$876$	0	0
$877$	−6.09366e13	−0.00396625	−0.00198313	−	0.999998i	$-0.500631\pi$
$877$	−6.09366e13	−0.00396625	−0.00198313	+	0.999998i	$0.500631\pi$
$878$	0	0
$879$	−1.69627e16	−1.09033
$880$	0	0
$881$	1.47034e16	0.933362	0.466681	−	0.884426i	$-0.345450\pi$
$881$	1.47034e16	0.933362	0.466681	+	0.884426i	$0.345450\pi$
$882$	0	0
$883$	−7.54689e15	−0.473134	−0.236567	−	0.971615i	$-0.576022\pi$
$883$	−7.54689e15	−0.473134	−0.236567	+	0.971615i	$0.576022\pi$
$884$	0	0
$885$	−5.50501e15	−0.340855
$886$	0	0
$887$	1.09156e16	0.667525	0.333762	−	0.942657i	$-0.391682\pi$
$887$	1.09156e16	0.667525	0.333762	+	0.942657i	$0.391682\pi$
$888$	0	0
$889$	2.36365e16	1.42766
$890$	0	0
$891$	−4.14406e15	−0.247229
$892$	0	0
$893$	1.73451e15	0.102210
$894$	0	0
$895$	2.37240e15	0.138090
$896$	0	0
$897$	−1.59570e16	−0.917473
$898$	0	0
$899$	−3.75351e16	−2.13186
$900$	0	0
$901$	7.83714e15	0.439715
$902$	0	0
$903$	2.85644e15	0.158322
$904$	0	0
$905$	5.71965e15	0.313186
$906$	0	0
$907$	−2.62328e16	−1.41907	−0.709535	−	0.704670i	$-0.751095\pi$
$907$	−2.62328e16	−1.41907	−0.709535	+	0.704670i	$0.751095\pi$
$908$	0	0
$909$	1.40356e16	0.750117
$910$	0	0
$911$	2.73010e16	1.44154	0.720772	−	0.693173i	$-0.243788\pi$
$911$	2.73010e16	1.44154	0.720772	+	0.693173i	$0.243788\pi$
$912$	0	0
$913$	4.66481e14	0.0243358
$914$	0	0
$915$	−1.15106e16	−0.593311
$916$	0	0
$917$	2.92556e16	1.48997
$918$	0	0
$919$	2.05271e16	1.03298	0.516490	−	0.856293i	$-0.327238\pi$
$919$	2.05271e16	1.03298	0.516490	+	0.856293i	$0.327238\pi$
$920$	0	0
$921$	2.10123e16	1.04483
$922$	0	0
$923$	−3.51400e16	−1.72660
$924$	0	0
$925$	−4.73121e14	−0.0229717
$926$	0	0
$927$	−1.65018e16	−0.791757
$928$	0	0
$929$	−2.80839e16	−1.33159	−0.665796	−	0.746134i	$-0.731908\pi$
$929$	−2.80839e16	−1.33159	−0.665796	+	0.746134i	$0.731908\pi$
$930$	0	0
$931$	5.30876e14	0.0248754
$932$	0	0
$933$	−9.94111e15	−0.460348
$934$	0	0
$935$	6.65530e15	0.304582
$936$	0	0
$937$	3.72912e16	1.68670	0.843352	−	0.537362i	$-0.180579\pi$
$937$	3.72912e16	1.68670	0.843352	+	0.537362i	$0.180579\pi$
$938$	0	0
$939$	7.47915e15	0.334342
$940$	0	0
$941$	−1.72223e16	−0.760937	−0.380469	−	0.924794i	$-0.624237\pi$
$941$	−1.72223e16	−0.760937	−0.380469	+	0.924794i	$0.624237\pi$
$942$	0	0
$943$	−1.44877e16	−0.632681
$944$	0	0
$945$	−1.16828e16	−0.504279
$946$	0	0
$947$	4.57432e16	1.95165	0.975824	−	0.218557i	$-0.0701349\pi$
$947$	4.57432e16	1.95165	0.975824	+	0.218557i	$0.0701349\pi$
$948$	0	0
$949$	4.27754e16	1.80397
$950$	0	0
$951$	2.18195e16	0.909603
$952$	0	0
$953$	−4.13428e15	−0.170368	−0.0851841	−	0.996365i	$-0.527148\pi$
$953$	−4.13428e15	−0.170368	−0.0851841	+	0.996365i	$0.527148\pi$
$954$	0	0
$955$	−2.44226e15	−0.0994884
$956$	0	0
$957$	−1.70596e16	−0.686996
$958$	0	0
$959$	1.77125e16	0.705142
$960$	0	0
$961$	2.53827e16	0.998986
$962$	0	0
$963$	−1.49187e15	−0.0580477
$964$	0	0
$965$	−1.47071e16	−0.565752
$966$	0	0
$967$	2.48419e16	0.944799	0.472400	−	0.881385i	$-0.343388\pi$
$967$	2.48419e16	0.944799	0.472400	+	0.881385i	$0.343388\pi$
$968$	0	0
$969$	−6.81962e15	−0.256436
$970$	0	0
$971$	−3.83204e16	−1.42470	−0.712351	−	0.701824i	$-0.752369\pi$
$971$	−3.83204e16	−1.42470	−0.712351	+	0.701824i	$0.752369\pi$
$972$	0	0
$973$	−4.51281e15	−0.165893
$974$	0	0
$975$	−6.19924e15	−0.225327
$976$	0	0
$977$	4.89629e15	0.175973	0.0879866	−	0.996122i	$-0.471957\pi$
$977$	4.89629e15	0.175973	0.0879866	+	0.996122i	$0.471957\pi$
$978$	0	0
$979$	2.36737e16	0.841320
$980$	0	0
$981$	−6.88577e15	−0.241976
$982$	0	0
$983$	−4.17672e16	−1.45141	−0.725707	−	0.688004i	$-0.758487\pi$
$983$	−4.17672e16	−1.45141	−0.725707	+	0.688004i	$0.758487\pi$
$984$	0	0
$985$	−2.77129e14	−0.00952319
$986$	0	0
$987$	8.20363e15	0.278780
$988$	0	0
$989$	4.80910e15	0.161616
$990$	0	0
$991$	5.64170e16	1.87502	0.937508	−	0.347963i	$-0.113127\pi$
$991$	5.64170e16	1.87502	0.937508	+	0.347963i	$0.113127\pi$
$992$	0	0
$993$	2.09391e16	0.688234
$994$	0	0
$995$	8.97041e15	0.291599
$996$	0	0
$997$	−6.38236e15	−0.205191	−0.102595	−	0.994723i	$-0.532715\pi$
$997$	−6.38236e15	−0.205191	−0.102595	+	0.994723i	$0.532715\pi$
$998$	0	0
$999$	−3.91025e15	−0.124335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.12.a.b.1.1	✓	2
3.2	odd	2		180.12.a.c.1.1		2
4.3	odd	2		80.12.a.i.1.2		2
5.2	odd	4		100.12.c.c.49.3		4
5.3	odd	4		100.12.c.c.49.2		4
5.4	even	2		100.12.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.12.a.b.1.1	✓	2	1.1	even	1	trivial
80.12.a.i.1.2		2	4.3	odd	2
100.12.a.c.1.2		2	5.4	even	2
100.12.c.c.49.2		4	5.3	odd	4
100.12.c.c.49.3		4	5.2	odd	4
180.12.a.c.1.1		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)

\(f(q)\)	\(=\)	\(q-322.338 q^{3} +3125.00 q^{5} -46319.5 q^{7} -73245.3 q^{9} +O(q^{10})\)	\(q-322.338 q^{3} +3125.00 q^{5} -46319.5 q^{7} -73245.3 q^{9} +317772. q^{11} +1.96937e6 q^{13} -1.00731e6 q^{15} +6.70196e6 q^{17} +3.15680e6 q^{19} +1.49305e7 q^{21} +2.51370e7 q^{23} +9.76562e6 q^{25} +8.07109e7 q^{27} +1.66550e8 q^{29} -2.25369e8 q^{31} -1.02430e8 q^{33} -1.44748e8 q^{35} -4.84476e7 q^{37} -6.34802e8 q^{39} -5.76349e8 q^{41} +1.91315e8 q^{43} -2.28892e8 q^{45} +5.49453e8 q^{47} +1.68169e8 q^{49} -2.16030e9 q^{51} +1.16938e9 q^{53} +9.93037e8 q^{55} -1.01755e9 q^{57} +5.46508e9 q^{59} +1.14271e10 q^{61} +3.39269e9 q^{63} +6.15428e9 q^{65} -5.91196e9 q^{67} -8.10261e9 q^{69} -1.78433e10 q^{71} +2.17204e10 q^{73} -3.14783e9 q^{75} -1.47190e10 q^{77} -4.51177e10 q^{79} -1.30410e10 q^{81} +1.46798e9 q^{83} +2.09436e10 q^{85} -5.36852e10 q^{87} +7.44991e10 q^{89} -9.12201e10 q^{91} +7.26450e10 q^{93} +9.86499e9 q^{95} +8.12138e10 q^{97} -2.32753e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 220 q^{3} + 6250 q^{5} + 3340 q^{7} + 43738 q^{9}+O(q^{10}) \) 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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