LMFDB - Embedded newform 72.20.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,20,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 20, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	72.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:	$164.748021521$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1453}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 363 $ x^2 - x - 363
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-18.5591$ of defining polynomial
Character	$\chi$	$=$	72.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+996804. q^{5} -7.24780e7 q^{7} +O(q^{10})$	$q+996804. q^{5} -7.24780e7 q^{7} -4.59555e9 q^{11} -7.68575e9 q^{13} +6.70499e11 q^{17} -6.03766e11 q^{19} +1.41235e13 q^{23} -1.80799e13 q^{25} -1.75566e13 q^{29} +7.97149e13 q^{31} -7.22464e13 q^{35} +1.12013e15 q^{37} -3.00183e15 q^{41} -3.41815e15 q^{43} +1.14830e16 q^{47} -6.14583e15 q^{49} -2.49535e16 q^{53} -4.58086e15 q^{55} -7.22986e15 q^{59} +1.29766e17 q^{61} -7.66119e15 q^{65} +2.74592e17 q^{67} -1.56270e17 q^{71} -8.21165e17 q^{73} +3.33076e17 q^{77} +2.09830e17 q^{79} -3.15127e17 q^{83} +6.68357e17 q^{85} -2.78627e18 q^{89} +5.57048e17 q^{91} -6.01837e17 q^{95} +7.58373e18 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 1226620 q^{5} + 88510512 q^{7}+O(q^{10}) $ 2 * q - 1226620 * q^5 + 88510512 * q^7	$ 2 q - 1226620 q^{5} + 88510512 q^{7} + 7163787608 q^{11} - 10126923604 q^{13} + 72045078940 q^{17} - 3120480472232 q^{19} + 14759207090288 q^{23} - 32209737998450 q^{25} + 30249539245044 q^{29} - 123389562777920 q^{31} - 430192267170720 q^{35} + 20\!\cdots\!24 q^{37}+ \cdots + 15\!\cdots\!76 q^{97}+O(q^{100}) $ 2 * q - 1226620 * q^5 + 88510512 * q^7 + 7163787608 * q^11 - 10126923604 * q^13 + 72045078940 * q^17 - 3120480472232 * q^19 + 14759207090288 * q^23 - 32209737998450 * q^25 + 30249539245044 * q^29 - 123389562777920 * q^31 - 430192267170720 * q^35 + 2015393170174524 * q^37 - 2540784959504244 * q^41 - 5633655093389464 * q^43 + 21948339587130336 * q^47 + 8372586663064786 * q^49 + 9418125066904676 * q^53 - 30726853722003280 * q^55 - 98542449590407624 * q^59 + 10292145377839820 * q^61 - 2233432360014760 * q^65 + 75753628003984504 * q^67 - 17407052566713776 * q^71 - 857508255059832268 * q^73 + 2226194512471875648 * q^77 - 226291921444855072 * q^79 - 767515701460985048 * q^83 + 1998974022815558200 * q^85 - 6092545894435174548 * q^89 + 164047654392377376 * q^91 + 4993887710656339120 * q^95 + 1548148249522347076 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	996804.	0.228242	0.114121	−	0.993467i	$-0.463595\pi$
$5$	996804.	0.228242	0.114121	+	0.993467i	$0.463595\pi$
$6$	0	0
$7$	−7.24780e7	−0.678852	−0.339426	−	0.940633i	$-0.610233\pi$
$7$	−7.24780e7	−0.678852	−0.339426	+	0.940633i	$0.610233\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.59555e9	−0.587634	−0.293817	−	0.955862i	$-0.594926\pi$
$11$	−4.59555e9	−0.587634	−0.293817	+	0.955862i	$0.594926\pi$
$12$	0	0
$13$	−7.68575e9	−0.201013	−0.100507	−	0.994936i	$-0.532046\pi$
$13$	−7.68575e9	−0.201013	−0.100507	+	0.994936i	$0.532046\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.70499e11	1.37130	0.685652	−	0.727930i	$-0.259517\pi$
$17$	6.70499e11	1.37130	0.685652	+	0.727930i	$0.259517\pi$
$18$	0	0
$19$	−6.03766e11	−0.429249	−0.214625	−	0.976697i	$-0.568853\pi$
$19$	−6.03766e11	−0.429249	−0.214625	+	0.976697i	$0.568853\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.41235e13	1.63503	0.817517	−	0.575904i	$-0.195350\pi$
$23$	1.41235e13	1.63503	0.817517	+	0.575904i	$0.195350\pi$
$24$	0	0
$25$	−1.80799e13	−0.947906
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.75566e13	−0.224729	−0.112365	−	0.993667i	$-0.535842\pi$
$29$	−1.75566e13	−0.224729	−0.112365	+	0.993667i	$0.535842\pi$
$30$	0	0
$31$	7.97149e13	0.541506	0.270753	−	0.962649i	$-0.412727\pi$
$31$	7.97149e13	0.541506	0.270753	+	0.962649i	$0.412727\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7.22464e13	−0.154942
$36$	0	0
$37$	1.12013e15	1.41694	0.708469	−	0.705742i	$-0.249386\pi$
$37$	1.12013e15	1.41694	0.708469	+	0.705742i	$0.249386\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.00183e15	−1.43199	−0.715995	−	0.698106i	$-0.754026\pi$
$41$	−3.00183e15	−1.43199	−0.715995	+	0.698106i	$0.754026\pi$
$42$	0	0
$43$	−3.41815e15	−1.03715	−0.518574	−	0.855033i	$-0.673537\pi$
$43$	−3.41815e15	−1.03715	−0.518574	+	0.855033i	$0.673537\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.14830e16	1.49667	0.748336	−	0.663320i	$-0.230853\pi$
$47$	1.14830e16	1.49667	0.748336	+	0.663320i	$0.230853\pi$
$48$	0	0
$49$	−6.14583e15	−0.539160
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.49535e16	−1.03875	−0.519374	−	0.854547i	$-0.673835\pi$
$53$	−2.49535e16	−1.03875	−0.519374	+	0.854547i	$0.673835\pi$
$54$	0	0
$55$	−4.58086e15	−0.134122
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.22986e15	−0.108652	−0.0543258	−	0.998523i	$-0.517301\pi$
$59$	−7.22986e15	−0.108652	−0.0543258	+	0.998523i	$0.517301\pi$
$60$	0	0
$61$	1.29766e17	1.42078	0.710391	−	0.703808i	$-0.248518\pi$
$61$	1.29766e17	1.42078	0.710391	+	0.703808i	$0.248518\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.66119e15	−0.0458796
$66$	0	0
$67$	2.74592e17	1.23304	0.616520	−	0.787340i	$-0.288542\pi$
$67$	2.74592e17	1.23304	0.616520	+	0.787340i	$0.288542\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.56270e17	−0.404504	−0.202252	−	0.979334i	$-0.564826\pi$
$71$	−1.56270e17	−0.404504	−0.202252	+	0.979334i	$0.564826\pi$
$72$	0	0
$73$	−8.21165e17	−1.63254	−0.816269	−	0.577672i	$-0.803961\pi$
$73$	−8.21165e17	−1.63254	−0.816269	+	0.577672i	$0.803961\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.33076e17	0.398916
$78$	0	0
$79$	2.09830e17	0.196974	0.0984871	−	0.995138i	$-0.468600\pi$
$79$	2.09830e17	0.196974	0.0984871	+	0.995138i	$0.468600\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.15127e17	−0.185030	−0.0925152	−	0.995711i	$-0.529491\pi$
$83$	−3.15127e17	−0.185030	−0.0925152	+	0.995711i	$0.529491\pi$
$84$	0	0
$85$	6.68357e17	0.312989
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.78627e18	−0.842981	−0.421491	−	0.906833i	$-0.638493\pi$
$89$	−2.78627e18	−0.842981	−0.421491	+	0.906833i	$0.638493\pi$
$90$	0	0
$91$	5.57048e17	0.136458
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.01837e17	−0.0979725
$96$	0	0
$97$	7.58373e18	1.01286	0.506432	−	0.862280i	$-0.330964\pi$
$97$	7.58373e18	1.01286	0.506432	+	0.862280i	$0.330964\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.62374e18	0.329689	0.164845	−	0.986320i	$-0.447288\pi$
$101$	3.62374e18	0.329689	0.164845	+	0.986320i	$0.447288\pi$
$102$	0	0
$103$	−2.07112e19	−1.56405	−0.782026	−	0.623246i	$-0.785814\pi$
$103$	−2.07112e19	−1.56405	−0.782026	+	0.623246i	$0.785814\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.42775e19	1.27661	0.638305	−	0.769784i	$-0.279636\pi$
$107$	2.42775e19	1.27661	0.638305	+	0.769784i	$0.279636\pi$
$108$	0	0
$109$	−1.19033e19	−0.524946	−0.262473	−	0.964939i	$-0.584538\pi$
$109$	−1.19033e19	−0.524946	−0.262473	+	0.964939i	$0.584538\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.65952e18	0.0832833	0.0416417	−	0.999133i	$-0.486741\pi$
$113$	2.65952e18	0.0832833	0.0416417	+	0.999133i	$0.486741\pi$
$114$	0	0
$115$	1.40783e19	0.373183
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.85965e19	−0.930912
$120$	0	0
$121$	−4.00400e19	−0.654687
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.70346e19	−0.444593
$126$	0	0
$127$	6.57941e19	0.679284	0.339642	−	0.940555i	$-0.389694\pi$
$127$	6.57941e19	0.679284	0.339642	+	0.940555i	$0.389694\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.73472e19	0.287198	0.143599	−	0.989636i	$-0.454132\pi$
$131$	3.73472e19	0.287198	0.143599	+	0.989636i	$0.454132\pi$
$132$	0	0
$133$	4.37598e19	0.291397
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.14142e18	0.0157863	0.00789315	−	0.999969i	$-0.497488\pi$
$137$	3.14142e18	0.0157863	0.00789315	+	0.999969i	$0.497488\pi$
$138$	0	0
$139$	−3.82616e20	−1.67542	−0.837709	−	0.546117i	$-0.816105\pi$
$139$	−3.82616e20	−1.67542	−0.837709	+	0.546117i	$0.816105\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.53202e19	0.118122
$144$	0	0
$145$	−1.75005e19	−0.0512926
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.27921e20	−0.742163	−0.371081	−	0.928600i	$-0.621013\pi$
$149$	−3.27921e20	−0.742163	−0.371081	+	0.928600i	$0.621013\pi$
$150$	0	0
$151$	1.20975e20	0.241220	0.120610	−	0.992700i	$-0.461515\pi$
$151$	1.20975e20	0.241220	0.120610	+	0.992700i	$0.461515\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.94601e19	0.123594
$156$	0	0
$157$	5.68747e20	0.783199	0.391599	−	0.920136i	$-0.371922\pi$
$157$	5.68747e20	0.783199	0.391599	+	0.920136i	$0.371922\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.02364e21	−1.10995
$162$	0	0
$163$	−1.77617e21	−1.71279	−0.856393	−	0.516325i	$-0.827300\pi$
$163$	−1.77617e21	−1.71279	−0.856393	+	0.516325i	$0.827300\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.55998e21	−1.19485	−0.597424	−	0.801926i	$-0.703809\pi$
$167$	−1.55998e21	−1.19485	−0.597424	+	0.801926i	$0.703809\pi$
$168$	0	0
$169$	−1.40285e21	−0.959594
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.55034e21	0.849159	0.424580	−	0.905391i	$-0.360422\pi$
$173$	1.55034e21	0.849159	0.424580	+	0.905391i	$0.360422\pi$
$174$	0	0
$175$	1.31039e21	0.643488
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.89048e21	1.14516	0.572579	−	0.819849i	$-0.305943\pi$
$179$	2.89048e21	1.14516	0.572579	+	0.819849i	$0.305943\pi$
$180$	0	0
$181$	−5.15105e21	−1.83633	−0.918163	−	0.396204i	$-0.870327\pi$
$181$	−5.15105e21	−1.83633	−0.918163	+	0.396204i	$0.870327\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.11655e21	0.323404
$186$	0	0
$187$	−3.08131e21	−0.805824
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.17061e21	−1.53369	−0.766847	−	0.641830i	$-0.778176\pi$
$191$	−7.17061e21	−1.53369	−0.766847	+	0.641830i	$0.778176\pi$
$192$	0	0
$193$	−3.25664e21	−0.630921	−0.315461	−	0.948939i	$-0.602159\pi$
$193$	−3.25664e21	−0.630921	−0.315461	+	0.948939i	$0.602159\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.19004e22	−1.89729	−0.948643	−	0.316349i	$-0.897543\pi$
$197$	−1.19004e22	−1.89729	−0.948643	+	0.316349i	$0.897543\pi$
$198$	0	0
$199$	−2.18989e21	−0.317188	−0.158594	−	0.987344i	$-0.550696\pi$
$199$	−2.18989e21	−0.317188	−0.158594	+	0.987344i	$0.550696\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.27247e21	0.152558
$204$	0	0
$205$	−2.99224e21	−0.326840
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.77464e21	0.252241
$210$	0	0
$211$	−2.32425e22	−1.93019	−0.965096	−	0.261898i	$-0.915652\pi$
$211$	−2.32425e22	−1.93019	−0.965096	+	0.261898i	$0.915652\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.40723e21	−0.236720
$216$	0	0
$217$	−5.77758e21	−0.367602
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.15329e21	−0.275650
$222$	0	0
$223$	−3.36614e21	−0.165286	−0.0826429	−	0.996579i	$-0.526336\pi$
$223$	−3.36614e21	−0.165286	−0.0826429	+	0.996579i	$0.526336\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.44209e21	0.142750	0.0713751	−	0.997450i	$-0.477261\pi$
$227$	3.44209e21	0.142750	0.0713751	+	0.997450i	$0.477261\pi$
$228$	0	0
$229$	−3.04661e22	−1.16246	−0.581232	−	0.813738i	$-0.697429\pi$
$229$	−3.04661e22	−1.16246	−0.581232	+	0.813738i	$0.697429\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.01497e22	−0.328527	−0.164264	−	0.986416i	$-0.552525\pi$
$233$	−1.01497e22	−0.328527	−0.164264	+	0.986416i	$0.552525\pi$
$234$	0	0
$235$	1.14463e22	0.341603
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.94716e22	−1.51192	−0.755961	−	0.654616i	$-0.772830\pi$
$239$	−5.94716e22	−1.51192	−0.755961	+	0.654616i	$0.772830\pi$
$240$	0	0
$241$	−6.71172e21	−0.157642	−0.0788210	−	0.996889i	$-0.525116\pi$
$241$	−6.71172e21	−0.157642	−0.0788210	+	0.996889i	$0.525116\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.12619e21	−0.123059
$246$	0	0
$247$	4.64040e21	0.0862847
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.83985e22	−1.09181	−0.545904	−	0.837848i	$-0.683814\pi$
$251$	−6.83985e22	−1.09181	−0.545904	+	0.837848i	$0.683814\pi$
$252$	0	0
$253$	−6.49051e22	−0.960802
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.76051e22	0.479603	0.239801	−	0.970822i	$-0.422918\pi$
$257$	3.76051e22	0.479603	0.239801	+	0.970822i	$0.422918\pi$
$258$	0	0
$259$	−8.11846e22	−0.961892
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.77095e23	1.81396	0.906980	−	0.421173i	$-0.138382\pi$
$263$	1.77095e23	1.81396	0.906980	+	0.421173i	$0.138382\pi$
$264$	0	0
$265$	−2.48737e22	−0.237086
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.19248e23	−0.985837	−0.492919	−	0.870075i	$-0.664070\pi$
$269$	−1.19248e23	−0.985837	−0.492919	+	0.870075i	$0.664070\pi$
$270$	0	0
$271$	−8.67879e22	−0.668730	−0.334365	−	0.942444i	$-0.608522\pi$
$271$	−8.67879e22	−0.668730	−0.334365	+	0.942444i	$0.608522\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.30869e22	0.557021
$276$	0	0
$277$	7.99312e21	0.0500217	0.0250109	−	0.999687i	$-0.492038\pi$
$277$	7.99312e21	0.0500217	0.0250109	+	0.999687i	$0.492038\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.90889e22	−0.268085	−0.134043	−	0.990976i	$-0.542796\pi$
$281$	−4.90889e22	−0.268085	−0.134043	+	0.990976i	$0.542796\pi$
$282$	0	0
$283$	2.70773e23	1.38240	0.691201	−	0.722662i	$-0.257082\pi$
$283$	2.70773e23	1.38240	0.691201	+	0.722662i	$0.257082\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.17567e23	0.972109
$288$	0	0
$289$	2.10497e23	0.880473
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.47805e23	−0.909635	−0.454818	−	0.890585i	$-0.650296\pi$
$293$	−2.47805e23	−0.909635	−0.454818	+	0.890585i	$0.650296\pi$
$294$	0	0
$295$	−7.20676e21	−0.0247988
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.08549e23	−0.328664
$300$	0	0
$301$	2.47741e23	0.704070
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.29351e23	0.324281
$306$	0	0
$307$	2.71564e23	0.639820	0.319910	−	0.947448i	$-0.396347\pi$
$307$	2.71564e23	0.639820	0.319910	+	0.947448i	$0.396347\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.17401e23	0.869620	0.434810	−	0.900522i	$-0.356816\pi$
$311$	4.17401e23	0.869620	0.434810	+	0.900522i	$0.356816\pi$
$312$	0	0
$313$	−2.21476e23	−0.434166	−0.217083	−	0.976153i	$-0.569654\pi$
$313$	−2.21476e23	−0.434166	−0.217083	+	0.976153i	$0.569654\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.33402e23	1.44808	0.724038	−	0.689760i	$-0.242284\pi$
$317$	8.33402e23	1.44808	0.724038	+	0.689760i	$0.242284\pi$
$318$	0	0
$319$	8.06823e22	0.132059
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.04825e23	−0.588631
$324$	0	0
$325$	1.38957e23	0.190542
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.32267e23	−1.01602
$330$	0	0
$331$	1.25972e24	1.45181	0.725904	−	0.687796i	$-0.241422\pi$
$331$	1.25972e24	1.45181	0.725904	+	0.687796i	$0.241422\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.73714e23	0.281431
$336$	0	0
$337$	8.60548e23	0.836163	0.418082	−	0.908409i	$-0.362703\pi$
$337$	8.60548e23	0.836163	0.418082	+	0.908409i	$0.362703\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.66333e23	−0.318207
$342$	0	0
$343$	1.27161e24	1.04486
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.55690e24	1.88184	0.940922	−	0.338623i	$-0.109961\pi$
$347$	2.55690e24	1.88184	0.940922	+	0.338623i	$0.109961\pi$
$348$	0	0
$349$	1.73272e24	1.20750	0.603751	−	0.797173i	$-0.293672\pi$
$349$	1.73272e24	1.20750	0.603751	+	0.797173i	$0.293672\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.44110e23	−0.465348	−0.232674	−	0.972555i	$-0.574747\pi$
$353$	−7.44110e23	−0.465348	−0.232674	+	0.972555i	$0.574747\pi$
$354$	0	0
$355$	−1.55771e23	−0.0923246
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.67777e24	0.893995	0.446997	−	0.894535i	$-0.352493\pi$
$359$	1.67777e24	0.893995	0.446997	+	0.894535i	$0.352493\pi$
$360$	0	0
$361$	−1.61389e24	−0.815745
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.18541e23	−0.372613
$366$	0	0
$367$	1.06458e24	0.460096	0.230048	−	0.973179i	$-0.426112\pi$
$367$	1.06458e24	0.460096	0.230048	+	0.973179i	$0.426112\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.80858e24	0.705156
$372$	0	0
$373$	−3.55316e24	−1.31638	−0.658189	−	0.752853i	$-0.728677\pi$
$373$	−3.55316e24	−1.31638	−0.658189	+	0.752853i	$0.728677\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.34936e23	0.0451736
$378$	0	0
$379$	−2.96331e24	−0.943418	−0.471709	−	0.881754i	$-0.656363\pi$
$379$	−2.96331e24	−0.943418	−0.471709	+	0.881754i	$0.656363\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.14872e22	−0.0119544	−0.00597718	−	0.999982i	$-0.501903\pi$
$383$	−4.14872e22	−0.0119544	−0.00597718	+	0.999982i	$0.501903\pi$
$384$	0	0
$385$	3.32012e23	0.0910493
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.32009e24	−0.328158	−0.164079	−	0.986447i	$-0.552465\pi$
$389$	−1.32009e24	−0.328158	−0.164079	+	0.986447i	$0.552465\pi$
$390$	0	0
$391$	9.46978e24	2.24213
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.09159e23	0.0449577
$396$	0	0
$397$	−2.49987e24	−0.512162	−0.256081	−	0.966655i	$-0.582431\pi$
$397$	−2.49987e24	−0.512162	−0.256081	+	0.966655i	$0.582431\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.22882e24	−0.973939	−0.486970	−	0.873419i	$-0.661898\pi$
$401$	−5.22882e24	−0.973939	−0.486970	+	0.873419i	$0.661898\pi$
$402$	0	0
$403$	−6.12669e23	−0.108850
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.14760e24	−0.832641
$408$	0	0
$409$	−3.59253e24	−0.554663	−0.277331	−	0.960774i	$-0.589450\pi$
$409$	−3.59253e24	−0.554663	−0.277331	+	0.960774i	$0.589450\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.24006e23	0.0737583
$414$	0	0
$415$	−3.14120e23	−0.0422317
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.64100e24	0.937815	0.468907	−	0.883247i	$-0.344648\pi$
$419$	7.64100e24	0.937815	0.468907	+	0.883247i	$0.344648\pi$
$420$	0	0
$421$	−6.63037e24	−0.777782	−0.388891	−	0.921284i	$-0.627142\pi$
$421$	−6.63037e24	−0.777782	−0.388891	+	0.921284i	$0.627142\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.21225e25	−1.29987
$426$	0	0
$427$	−9.40518e24	−0.964500
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.83021e24	0.828776	0.414388	−	0.910100i	$-0.363996\pi$
$431$	8.83021e24	0.828776	0.414388	+	0.910100i	$0.363996\pi$
$432$	0	0
$433$	−1.89602e24	−0.170298	−0.0851488	−	0.996368i	$-0.527137\pi$
$433$	−1.89602e24	−0.170298	−0.0851488	+	0.996368i	$0.527137\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.52727e24	−0.701837
$438$	0	0
$439$	−2.03577e25	−1.60441	−0.802206	−	0.597047i	$-0.796340\pi$
$439$	−2.03577e25	−1.60441	−0.802206	+	0.597047i	$0.796340\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.63050e24	0.696327	0.348164	−	0.937434i	$-0.386805\pi$
$443$	9.63050e24	0.696327	0.348164	+	0.937434i	$0.386805\pi$
$444$	0	0
$445$	−2.77736e24	−0.192403
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.24366e25	−1.42764	−0.713818	−	0.700331i	$-0.753036\pi$
$449$	−2.24366e25	−1.42764	−0.713818	+	0.700331i	$0.753036\pi$
$450$	0	0
$451$	1.37951e25	0.841485
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.55268e23	0.0311454
$456$	0	0
$457$	−1.43601e25	−0.772597	−0.386299	−	0.922374i	$-0.626247\pi$
$457$	−1.43601e25	−0.772597	−0.386299	+	0.922374i	$0.626247\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.20527e25	−1.58747	−0.793736	−	0.608262i	$-0.791867\pi$
$461$	−3.20527e25	−1.58747	−0.793736	+	0.608262i	$0.791867\pi$
$462$	0	0
$463$	4.99392e24	0.237368	0.118684	−	0.992932i	$-0.462132\pi$
$463$	4.99392e24	0.237368	0.118684	+	0.992932i	$0.462132\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.24714e24	0.361238	0.180619	−	0.983553i	$-0.442190\pi$
$467$	8.24714e24	0.361238	0.180619	+	0.983553i	$0.442190\pi$
$468$	0	0
$469$	−1.99019e25	−0.837051
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.57083e25	0.609463
$474$	0	0
$475$	1.09160e25	0.406888
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.71323e25	0.933899	0.466950	−	0.884284i	$-0.345353\pi$
$479$	2.71323e25	0.933899	0.466950	+	0.884284i	$0.345353\pi$
$480$	0	0
$481$	−8.60902e24	−0.284823
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.55949e24	0.231178
$486$	0	0
$487$	−5.00346e25	−1.47145	−0.735725	−	0.677281i	$-0.763158\pi$
$487$	−5.00346e25	−1.47145	−0.735725	+	0.677281i	$0.763158\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.80743e25	−1.03600	−0.517998	−	0.855382i	$-0.673323\pi$
$491$	−3.80743e25	−1.03600	−0.517998	+	0.855382i	$0.673323\pi$
$492$	0	0
$493$	−1.17717e25	−0.308172
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.13262e25	0.274598
$498$	0	0
$499$	1.94991e25	0.455049	0.227525	−	0.973772i	$-0.426937\pi$
$499$	1.94991e25	0.455049	0.227525	+	0.973772i	$0.426937\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.11673e25	0.890546	0.445273	−	0.895395i	$-0.353107\pi$
$503$	4.11673e25	0.890546	0.445273	+	0.895395i	$0.353107\pi$
$504$	0	0
$505$	3.61216e24	0.0752488
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6.45581e24	−0.124776	−0.0623881	−	0.998052i	$-0.519872\pi$
$509$	−6.45581e24	−0.124776	−0.0623881	+	0.998052i	$0.519872\pi$
$510$	0	0
$511$	5.95164e25	1.10825
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.06450e25	−0.356982
$516$	0	0
$517$	−5.27708e25	−0.879495
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.61781e25	−1.17997	−0.589986	−	0.807413i	$-0.700867\pi$
$521$	−7.61781e25	−1.17997	−0.589986	+	0.807413i	$0.700867\pi$
$522$	0	0
$523$	7.54461e25	1.12686	0.563431	−	0.826163i	$-0.309481\pi$
$523$	7.54461e25	1.12686	0.563431	+	0.826163i	$0.309481\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.34488e25	0.742569
$528$	0	0
$529$	1.24857e26	1.67334
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.30713e25	0.287849
$534$	0	0
$535$	2.41999e25	0.291376
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.82435e25	0.316829
$540$	0	0
$541$	1.27935e26	1.38553	0.692763	−	0.721165i	$-0.256393\pi$
$541$	1.27935e26	1.38553	0.692763	+	0.721165i	$0.256393\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.18652e25	−0.119814
$546$	0	0
$547$	−1.03216e26	−1.00662	−0.503310	−	0.864106i	$-0.667885\pi$
$547$	−1.03216e26	−1.00662	−0.503310	+	0.864106i	$0.667885\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.06001e25	0.0964649
$552$	0	0
$553$	−1.52080e25	−0.133716
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.15924e26	0.951811	0.475905	−	0.879497i	$-0.342120\pi$
$557$	1.15924e26	0.951811	0.475905	+	0.879497i	$0.342120\pi$
$558$	0	0
$559$	2.62710e25	0.208480
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.69128e25	0.496226	0.248113	−	0.968731i	$-0.420190\pi$
$563$	6.69128e25	0.496226	0.248113	+	0.968731i	$0.420190\pi$
$564$	0	0
$565$	2.65102e24	0.0190087
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.98116e26	1.32847	0.664236	−	0.747523i	$-0.268757\pi$
$569$	1.98116e26	1.32847	0.664236	+	0.747523i	$0.268757\pi$
$570$	0	0
$571$	2.67644e26	1.73586	0.867930	−	0.496687i	$-0.165450\pi$
$571$	2.67644e26	1.73586	0.867930	+	0.496687i	$0.165450\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.55350e26	−1.54986
$576$	0	0
$577$	−1.86979e25	−0.109805	−0.0549027	−	0.998492i	$-0.517485\pi$
$577$	−1.86979e25	−0.109805	−0.0549027	+	0.998492i	$0.517485\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.28398e25	0.125608
$582$	0	0
$583$	1.14675e26	0.610403
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.83068e26	0.913167	0.456583	−	0.889681i	$-0.349073\pi$
$587$	1.83068e26	0.913167	0.456583	+	0.889681i	$0.349073\pi$
$588$	0	0
$589$	−4.81291e25	−0.232441
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.12500e25	−0.413250	−0.206625	−	0.978420i	$-0.566248\pi$
$593$	−9.12500e25	−0.413250	−0.206625	+	0.978420i	$0.566248\pi$
$594$	0	0
$595$	−4.84412e25	−0.212473
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.37311e26	−0.976705	−0.488353	−	0.872646i	$-0.662402\pi$
$599$	−2.37311e26	−0.976705	−0.488353	+	0.872646i	$0.662402\pi$
$600$	0	0
$601$	2.29560e26	0.915355	0.457677	−	0.889118i	$-0.348681\pi$
$601$	2.29560e26	0.915355	0.457677	+	0.889118i	$0.348681\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.99121e25	−0.149427
$606$	0	0
$607$	−1.10962e26	−0.402608	−0.201304	−	0.979529i	$-0.564518\pi$
$607$	−1.10962e26	−0.402608	−0.201304	+	0.979529i	$0.564518\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.82556e25	−0.300851
$612$	0	0
$613$	−1.27027e25	−0.0419779	−0.0209890	−	0.999780i	$-0.506681\pi$
$613$	−1.27027e25	−0.0419779	−0.0209890	+	0.999780i	$0.506681\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.22914e26	1.31384	0.656921	−	0.753960i	$-0.271859\pi$
$617$	4.22914e26	1.31384	0.656921	+	0.753960i	$0.271859\pi$
$618$	0	0
$619$	−5.49720e26	−1.65608	−0.828039	−	0.560671i	$-0.810543\pi$
$619$	−5.49720e26	−1.65608	−0.828039	+	0.560671i	$0.810543\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.01943e26	0.572259
$624$	0	0
$625$	3.07930e26	0.846431
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.51045e26	1.94305
$630$	0	0
$631$	2.83709e25	0.0712187	0.0356093	−	0.999366i	$-0.488663\pi$
$631$	2.83709e25	0.0712187	0.0356093	+	0.999366i	$0.488663\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.55838e25	0.155041
$636$	0	0
$637$	4.72353e25	0.108378
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.77075e26	−1.89620	−0.948102	−	0.317965i	$-0.897001\pi$
$641$	−8.77075e26	−1.89620	−0.948102	+	0.317965i	$0.897001\pi$
$642$	0	0
$643$	−4.19589e26	−0.880683	−0.440341	−	0.897830i	$-0.645143\pi$
$643$	−4.19589e26	−0.880683	−0.440341	+	0.897830i	$0.645143\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.89932e26	−0.771611	−0.385805	−	0.922580i	$-0.626076\pi$
$647$	−3.89932e26	−0.771611	−0.385805	+	0.922580i	$0.626076\pi$
$648$	0	0
$649$	3.32252e25	0.0638473
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.24023e26	−0.949895	−0.474947	−	0.880014i	$-0.657533\pi$
$653$	−5.24023e26	−0.949895	−0.474947	+	0.880014i	$0.657533\pi$
$654$	0	0
$655$	3.72279e25	0.0655505
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7.67459e26	−1.27539	−0.637696	−	0.770288i	$-0.720112\pi$
$659$	−7.67459e26	−1.27539	−0.637696	+	0.770288i	$0.720112\pi$
$660$	0	0
$661$	−1.32790e26	−0.214413	−0.107206	−	0.994237i	$-0.534191\pi$
$661$	−1.32790e26	−0.214413	−0.107206	+	0.994237i	$0.534191\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.36199e25	0.0665088
$666$	0	0
$667$	−2.47960e26	−0.367440
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.96346e26	−0.834899
$672$	0	0
$673$	−3.37509e26	−0.459349	−0.229674	−	0.973268i	$-0.573766\pi$
$673$	−3.37509e26	−0.459349	−0.229674	+	0.973268i	$0.573766\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.12451e27	1.44667	0.723336	−	0.690496i	$-0.242608\pi$
$677$	1.12451e27	1.44667	0.723336	+	0.690496i	$0.242608\pi$
$678$	0	0
$679$	−5.49654e26	−0.687585
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7.05871e26	−0.835081	−0.417540	−	0.908658i	$-0.637108\pi$
$683$	−7.05871e26	−0.835081	−0.417540	+	0.908658i	$0.637108\pi$
$684$	0	0
$685$	3.13138e24	0.00360309
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.91786e26	0.208802
$690$	0	0
$691$	−4.70572e25	−0.0498408	−0.0249204	−	0.999689i	$-0.507933\pi$
$691$	−4.70572e25	−0.0498408	−0.0249204	+	0.999689i	$0.507933\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.81394e26	−0.382400
$696$	0	0
$697$	−2.01273e27	−1.96369
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.38593e25	−0.0497668	−0.0248834	−	0.999690i	$-0.507921\pi$
$701$	−5.38593e25	−0.0497668	−0.0248834	+	0.999690i	$0.507921\pi$
$702$	0	0
$703$	−6.76295e26	−0.608220
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.62642e26	−0.223810
$708$	0	0
$709$	−2.80682e26	−0.232849	−0.116425	−	0.993200i	$-0.537143\pi$
$709$	−2.80682e26	−0.232849	−0.116425	+	0.993200i	$0.537143\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.12585e27	0.885381
$714$	0	0
$715$	3.52074e25	0.0269604
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.82032e27	1.32198	0.660988	−	0.750397i	$-0.270138\pi$
$719$	1.82032e27	1.32198	0.660988	+	0.750397i	$0.270138\pi$
$720$	0	0
$721$	1.50111e27	1.06176
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.17421e26	0.213022
$726$	0	0
$727$	−1.47227e27	−0.962522	−0.481261	−	0.876577i	$-0.659821\pi$
$727$	−1.47227e27	−0.962522	−0.481261	+	0.876577i	$0.659821\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.29187e27	−1.42224
$732$	0	0
$733$	2.23845e27	1.35350	0.676751	−	0.736212i	$-0.263387\pi$
$733$	2.23845e27	1.35350	0.676751	+	0.736212i	$0.263387\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.26190e27	−0.724576
$738$	0	0
$739$	4.64717e26	0.260056	0.130028	−	0.991510i	$-0.458493\pi$
$739$	4.64717e26	0.260056	0.130028	+	0.991510i	$0.458493\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.63164e26	−0.299393	−0.149696	−	0.988732i	$-0.547830\pi$
$743$	−5.63164e26	−0.299393	−0.149696	+	0.988732i	$0.547830\pi$
$744$	0	0
$745$	−3.26873e26	−0.169392
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.75959e27	−0.866629
$750$	0	0
$751$	−6.04104e26	−0.290090	−0.145045	−	0.989425i	$-0.546333\pi$
$751$	−6.04104e26	−0.290090	−0.145045	+	0.989425i	$0.546333\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.20588e26	0.0550564
$756$	0	0
$757$	−3.91726e27	−1.74410	−0.872051	−	0.489415i	$-0.837211\pi$
$757$	−3.91726e27	−1.74410	−0.872051	+	0.489415i	$0.837211\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.21509e27	1.36157	0.680783	−	0.732486i	$-0.261640\pi$
$761$	3.21509e27	1.36157	0.680783	+	0.732486i	$0.261640\pi$
$762$	0	0
$763$	8.62725e26	0.356360
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.55669e25	0.0218404
$768$	0	0
$769$	3.87198e27	1.48468	0.742339	−	0.670024i	$-0.233716\pi$
$769$	3.87198e27	1.48468	0.742339	+	0.670024i	$0.233716\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.22257e27	0.811240	0.405620	−	0.914042i	$-0.367056\pi$
$773$	2.22257e27	0.811240	0.405620	+	0.914042i	$0.367056\pi$
$774$	0	0
$775$	−1.44123e27	−0.513297
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.81240e27	0.614680
$780$	0	0
$781$	7.18148e26	0.237700
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.66929e26	0.178759
$786$	0	0
$787$	3.32013e27	1.02187	0.510934	−	0.859620i	$-0.329300\pi$
$787$	3.32013e27	1.02187	0.510934	+	0.859620i	$0.329300\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.92757e26	−0.0565370
$792$	0	0
$793$	−9.97349e26	−0.285596
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.80480e27	−1.03867	−0.519335	−	0.854571i	$-0.673820\pi$
$797$	−3.80480e27	−1.03867	−0.519335	+	0.854571i	$0.673820\pi$
$798$	0	0
$799$	7.69936e27	2.05239
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.77370e27	0.959334
$804$	0	0
$805$	−1.02037e27	−0.253336
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.75938e26	0.160102	0.0800508	−	0.996791i	$-0.474492\pi$
$809$	6.75938e26	0.160102	0.0800508	+	0.996791i	$0.474492\pi$
$810$	0	0
$811$	1.86888e27	0.432398	0.216199	−	0.976349i	$-0.430634\pi$
$811$	1.86888e27	0.432398	0.216199	+	0.976349i	$0.430634\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.77050e27	−0.390929
$816$	0	0
$817$	2.06376e27	0.445195
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.78616e27	−0.367841	−0.183921	−	0.982941i	$-0.558879\pi$
$821$	−1.78616e27	−0.367841	−0.183921	+	0.982941i	$0.558879\pi$
$822$	0	0
$823$	4.84426e27	0.974831	0.487415	−	0.873170i	$-0.337940\pi$
$823$	4.84426e27	0.974831	0.487415	+	0.873170i	$0.337940\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.46068e26	−0.143376	−0.0716879	−	0.997427i	$-0.522839\pi$
$827$	−7.46068e26	−0.143376	−0.0716879	+	0.997427i	$0.522839\pi$
$828$	0	0
$829$	9.45101e27	1.77505	0.887524	−	0.460761i	$-0.152424\pi$
$829$	9.45101e27	1.77505	0.887524	+	0.460761i	$0.152424\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.12077e27	−0.739352
$834$	0	0
$835$	−1.55499e27	−0.272714
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.87983e27	1.48822	0.744108	−	0.668059i	$-0.232875\pi$
$839$	8.87983e27	1.48822	0.744108	+	0.668059i	$0.232875\pi$
$840$	0	0
$841$	−5.79503e27	−0.949497
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.39837e27	−0.219019
$846$	0	0
$847$	2.90202e27	0.444435
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.58201e28	2.31674
$852$	0	0
$853$	−8.81716e26	−0.126274	−0.0631369	−	0.998005i	$-0.520110\pi$
$853$	−8.81716e26	−0.126274	−0.0631369	+	0.998005i	$0.520110\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.68989e26	0.0231495	0.0115747	−	0.999933i	$-0.496316\pi$
$857$	1.68989e26	0.0231495	0.0115747	+	0.999933i	$0.496316\pi$
$858$	0	0
$859$	−1.45204e28	−1.94556	−0.972779	−	0.231735i	$-0.925560\pi$
$859$	−1.45204e28	−1.94556	−0.972779	+	0.231735i	$0.925560\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.45753e27	−0.186860	−0.0934299	−	0.995626i	$-0.529783\pi$
$863$	−1.45753e27	−0.186860	−0.0934299	+	0.995626i	$0.529783\pi$
$864$	0	0
$865$	1.54539e27	0.193813
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9.64282e26	−0.115749
$870$	0	0
$871$	−2.11044e27	−0.247857
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.68420e27	0.301813
$876$	0	0
$877$	8.44834e27	0.929555	0.464777	−	0.885428i	$-0.346134\pi$
$877$	8.44834e27	0.929555	0.464777	+	0.885428i	$0.346134\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−3.79803e26	−0.0400209	−0.0200105	−	0.999800i	$-0.506370\pi$
$881$	−3.79803e26	−0.0400209	−0.0200105	+	0.999800i	$0.506370\pi$
$882$	0	0
$883$	1.64147e27	0.169280	0.0846400	−	0.996412i	$-0.473026\pi$
$883$	1.64147e27	0.169280	0.0846400	+	0.996412i	$0.473026\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.53135e28	−1.51286	−0.756432	−	0.654072i	$-0.773059\pi$
$887$	−1.53135e28	−1.51286	−0.756432	+	0.654072i	$0.773059\pi$
$888$	0	0
$889$	−4.76863e27	−0.461133
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−6.93306e27	−0.642445
$894$	0	0
$895$	2.88124e27	0.261373
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.39952e27	−0.121692
$900$	0	0
$901$	−1.67313e28	−1.42444
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.13459e27	−0.419126
$906$	0	0
$907$	2.27504e28	1.81853	0.909265	−	0.416218i	$-0.136645\pi$
$907$	2.27504e28	1.81853	0.909265	+	0.416218i	$0.136645\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.90023e28	1.45674	0.728369	−	0.685185i	$-0.240279\pi$
$911$	1.90023e28	1.45674	0.728369	+	0.685185i	$0.240279\pi$
$912$	0	0
$913$	1.44818e27	0.108730
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.70685e27	−0.194965
$918$	0	0
$919$	7.01793e27	0.495122	0.247561	−	0.968872i	$-0.420371\pi$
$919$	7.01793e27	0.495122	0.247561	+	0.968872i	$0.420371\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.20106e27	0.0813106
$924$	0	0
$925$	−2.02518e28	−1.34312
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.04079e28	−0.662545	−0.331273	−	0.943535i	$-0.607478\pi$
$929$	−1.04079e28	−0.662545	−0.331273	+	0.943535i	$0.607478\pi$
$930$	0	0
$931$	3.71064e27	0.231434
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.07146e27	−0.183923
$936$	0	0
$937$	−2.36290e28	−1.38650	−0.693249	−	0.720698i	$-0.743821\pi$
$937$	−2.36290e28	−1.38650	−0.693249	+	0.720698i	$0.743821\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.44433e27	0.250441	0.125220	−	0.992129i	$-0.460036\pi$
$941$	4.44433e27	0.250441	0.125220	+	0.992129i	$0.460036\pi$
$942$	0	0
$943$	−4.23963e28	−2.34135
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.77081e28	−0.939390	−0.469695	−	0.882829i	$-0.655636\pi$
$947$	−1.77081e28	−0.939390	−0.469695	+	0.882829i	$0.655636\pi$
$948$	0	0
$949$	6.31127e27	0.328162
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.35882e27	−0.217764	−0.108882	−	0.994055i	$-0.534727\pi$
$953$	−4.35882e27	−0.217764	−0.108882	+	0.994055i	$0.534727\pi$
$954$	0	0
$955$	−7.14769e27	−0.350053
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.27684e26	−0.0107166
$960$	0	0
$961$	−1.53162e28	−0.706771
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.24623e27	−0.144003
$966$	0	0
$967$	−7.37579e27	−0.320817	−0.160408	−	0.987051i	$-0.551281\pi$
$967$	−7.37579e27	−0.320817	−0.160408	+	0.987051i	$0.551281\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.32649e28	0.554780	0.277390	−	0.960757i	$-0.410531\pi$
$971$	1.32649e28	0.554780	0.277390	+	0.960757i	$0.410531\pi$
$972$	0	0
$973$	2.77313e28	1.13736
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.57429e28	−0.620991	−0.310496	−	0.950575i	$-0.600495\pi$
$977$	−1.57429e28	−0.620991	−0.310496	+	0.950575i	$0.600495\pi$
$978$	0	0
$979$	1.28044e28	0.495364
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.23838e27	0.0833059	0.0416529	−	0.999132i	$-0.486738\pi$
$983$	2.23838e27	0.0833059	0.0416529	+	0.999132i	$0.486738\pi$
$984$	0	0
$985$	−1.18624e28	−0.433040
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.82761e28	−1.69577
$990$	0	0
$991$	−2.72873e27	−0.0940288	−0.0470144	−	0.998894i	$-0.514971\pi$
$991$	−2.72873e27	−0.0940288	−0.0470144	+	0.998894i	$0.514971\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.18289e27	−0.0723956
$996$	0	0
$997$	1.56740e28	0.510008	0.255004	−	0.966940i	$-0.417923\pi$
$997$	1.56740e28	0.510008	0.255004	+	0.966940i	$0.417923\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.20.a.a.1.2		2
3.2	odd	2		8.20.a.a.1.2	✓	2
12.11	even	2		16.20.a.e.1.1		2
24.5	odd	2		64.20.a.k.1.1		2
24.11	even	2		64.20.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.20.a.a.1.2	✓	2	3.2	odd	2
16.20.a.e.1.1		2	12.11	even	2
64.20.a.j.1.2		2	24.11	even	2
64.20.a.k.1.1		2	24.5	odd	2
72.20.a.a.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

\(f(q)\)	\(=\)	\(q+996804. q^{5} -7.24780e7 q^{7} +O(q^{10})\)	\(q+996804. q^{5} -7.24780e7 q^{7} -4.59555e9 q^{11} -7.68575e9 q^{13} +6.70499e11 q^{17} -6.03766e11 q^{19} +1.41235e13 q^{23} -1.80799e13 q^{25} -1.75566e13 q^{29} +7.97149e13 q^{31} -7.22464e13 q^{35} +1.12013e15 q^{37} -3.00183e15 q^{41} -3.41815e15 q^{43} +1.14830e16 q^{47} -6.14583e15 q^{49} -2.49535e16 q^{53} -4.58086e15 q^{55} -7.22986e15 q^{59} +1.29766e17 q^{61} -7.66119e15 q^{65} +2.74592e17 q^{67} -1.56270e17 q^{71} -8.21165e17 q^{73} +3.33076e17 q^{77} +2.09830e17 q^{79} -3.15127e17 q^{83} +6.68357e17 q^{85} -2.78627e18 q^{89} +5.57048e17 q^{91} -6.01837e17 q^{95} +7.58373e18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 1226620 q^{5} + 88510512 q^{7}+O(q^{10}) \) 2 * q - 1226620 * q^5 + 88510512 * q^7	\( 2 q - 1226620 q^{5} + 88510512 q^{7} + 7163787608 q^{11} - 10126923604 q^{13} + 72045078940 q^{17} - 3120480472232 q^{19} + 14759207090288 q^{23} - 32209737998450 q^{25} + 30249539245044 q^{29} - 123389562777920 q^{31} - 430192267170720 q^{35} + 20\!\cdots\!24 q^{37}+ \cdots + 15\!\cdots\!76 q^{97}+O(q^{100}) \) 2 * q - 1226620 * q^5 + 88510512 * q^7 + 7163787608 * q^11 - 10126923604 * q^13 + 72045078940 * q^17 - 3120480472232 * q^19 + 14759207090288 * q^23 - 32209737998450 * q^25 + 30249539245044 * q^29 - 123389562777920 * q^31 - 430192267170720 * q^35 + 2015393170174524 * q^37 - 2540784959504244 * q^41 - 5633655093389464 * q^43 + 21948339587130336 * q^47 + 8372586663064786 * q^49 + 9418125066904676 * q^53 - 30726853722003280 * q^55 - 98542449590407624 * q^59 + 10292145377839820 * q^61 - 2233432360014760 * q^65 + 75753628003984504 * q^67 - 17407052566713776 * q^71 - 857508255059832268 * q^73 + 2226194512471875648 * q^77 - 226291921444855072 * q^79 - 767515701460985048 * q^83 + 1998974022815558200 * q^85 - 6092545894435174548 * q^89 + 164047654392377376 * q^91 + 4993887710656339120 * q^95 + 1548148249522347076 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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