LMFDB - Embedded newform 72.18.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [72,18,Mod(1,72)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("72.1"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	72.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-677824] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$131.919902888$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 43098290x^{2} - 18986612040x + 22664628899989 $ x^4 - 43098290x^2 - 18986612040x + 22664628899989
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{8} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6739.18$ 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)

$f(q)$	$=$	$q-484700. q^{5} -8.65967e6 q^{7} -8.03812e8 q^{11} +3.35794e9 q^{13} +5.67063e9 q^{17} +4.90853e10 q^{19} +4.86301e11 q^{23} -5.28005e11 q^{25} -1.40330e12 q^{29} +6.17478e12 q^{31} +4.19734e12 q^{35} +1.32855e13 q^{37} -1.72014e13 q^{41} -1.30978e13 q^{43} -1.21244e14 q^{47} -1.57641e14 q^{49} +6.27430e14 q^{53} +3.89608e14 q^{55} -2.36373e14 q^{59} -1.19575e15 q^{61} -1.62760e15 q^{65} -1.66133e15 q^{67} -1.72305e15 q^{71} +8.76286e15 q^{73} +6.96075e15 q^{77} +3.76431e15 q^{79} -7.79697e15 q^{83} -2.74856e15 q^{85} +5.61821e15 q^{89} -2.90787e16 q^{91} -2.37916e16 q^{95} -5.36057e16 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 677824 q^{5} + 9632880 q^{7} + 603687680 q^{11} - 2476119800 q^{13} - 7423708544 q^{17} - 19551988960 q^{19} - 458239426048 q^{23} + 1015542005996 q^{25} + 844849583040 q^{29} + 1246685956208 q^{31}+ \cdots - 17\!\cdots\!60 q^{97}+O(q^{100}) $ 4 * q - 677824 * q^5 + 9632880 * q^7 + 603687680 * q^11 - 2476119800 * q^13 - 7423708544 * q^17 - 19551988960 * q^19 - 458239426048 * q^23 + 1015542005996 * q^25 + 844849583040 * q^29 + 1246685956208 * q^31 - 3273147963648 * q^35 + 17661470383320 * q^37 + 33100198696320 * q^41 + 24133870623520 * q^43 + 11964304009728 * q^47 + 2287828660772 * q^49 - 256855748281408 * q^53 - 996530006700032 * q^55 - 905361180505600 * q^59 + 251141035416376 * q^61 + 3700013891444864 * q^65 + 2671944825444800 * q^67 - 4181227521935360 * q^71 - 1410879602368040 * q^73 - 1014782699621376 * q^77 + 7574345714454640 * q^79 - 14868236752232192 * q^83 - 788948917823488 * q^85 - 31667797088981760 * q^89 - 6872737039354656 * q^91 - 21917705446002176 * q^95 - 17206349902780360 * q^97

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
$5$	−484700.	−0.554917	−0.277459	−	0.960738i	$-0.589492\pi$
$5$	−484700.	−0.554917	−0.277459	+	0.960738i	$0.589492\pi$
$6$	0	0
$7$	−8.65967e6	−0.567764	−0.283882	−	0.958859i	$-0.591622\pi$
$7$	−8.65967e6	−0.567764	−0.283882	+	0.958859i	$0.591622\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−8.03812e8	−1.13062	−0.565310	−	0.824878i	$-0.691244\pi$
$11$	−8.03812e8	−1.13062	−0.565310	+	0.824878i	$0.691244\pi$
$12$	0	0
$13$	3.35794e9	1.14171	0.570853	−	0.821052i	$-0.306612\pi$
$13$	3.35794e9	1.14171	0.570853	+	0.821052i	$0.306612\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.67063e9	0.197159	0.0985793	−	0.995129i	$-0.468570\pi$
$17$	5.67063e9	0.197159	0.0985793	+	0.995129i	$0.468570\pi$
$18$	0	0
$19$	4.90853e10	0.663049	0.331524	−	0.943447i	$-0.392437\pi$
$19$	4.90853e10	0.663049	0.331524	+	0.943447i	$0.392437\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.86301e11	1.29485	0.647423	−	0.762131i	$-0.275847\pi$
$23$	4.86301e11	1.29485	0.647423	+	0.762131i	$0.275847\pi$
$24$	0	0
$25$	−5.28005e11	−0.692067
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.40330e12	−0.520916	−0.260458	−	0.965485i	$-0.583874\pi$
$29$	−1.40330e12	−0.520916	−0.260458	+	0.965485i	$0.583874\pi$
$30$	0	0
$31$	6.17478e12	1.30031	0.650155	−	0.759802i	$-0.274704\pi$
$31$	6.17478e12	1.30031	0.650155	+	0.759802i	$0.274704\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.19734e12	0.315062
$36$	0	0
$37$	1.32855e13	0.621820	0.310910	−	0.950439i	$-0.399366\pi$
$37$	1.32855e13	0.621820	0.310910	+	0.950439i	$0.399366\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.72014e13	−0.336435	−0.168218	−	0.985750i	$-0.553801\pi$
$41$	−1.72014e13	−0.336435	−0.168218	+	0.985750i	$0.553801\pi$
$42$	0	0
$43$	−1.30978e13	−0.170890	−0.0854449	−	0.996343i	$-0.527231\pi$
$43$	−1.30978e13	−0.170890	−0.0854449	+	0.996343i	$0.527231\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.21244e14	−0.742724	−0.371362	−	0.928488i	$-0.621109\pi$
$47$	−1.21244e14	−0.742724	−0.371362	+	0.928488i	$0.621109\pi$
$48$	0	0
$49$	−1.57641e14	−0.677644
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.27430e14	1.38427	0.692134	−	0.721769i	$-0.256671\pi$
$53$	6.27430e14	1.38427	0.692134	+	0.721769i	$0.256671\pi$
$54$	0	0
$55$	3.89608e14	0.627401
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.36373e14	−0.209583	−0.104792	−	0.994494i	$-0.533418\pi$
$59$	−2.36373e14	−0.209583	−0.104792	+	0.994494i	$0.533418\pi$
$60$	0	0
$61$	−1.19575e15	−0.798612	−0.399306	−	0.916818i	$-0.630749\pi$
$61$	−1.19575e15	−0.798612	−0.399306	+	0.916818i	$0.630749\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.62760e15	−0.633553
$66$	0	0
$67$	−1.66133e15	−0.499827	−0.249913	−	0.968268i	$-0.580402\pi$
$67$	−1.66133e15	−0.499827	−0.249913	+	0.968268i	$0.580402\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.72305e15	−0.316665	−0.158333	−	0.987386i	$-0.550612\pi$
$71$	−1.72305e15	−0.316665	−0.158333	+	0.987386i	$0.550612\pi$
$72$	0	0
$73$	8.76286e15	1.27175	0.635875	−	0.771792i	$-0.280640\pi$
$73$	8.76286e15	1.27175	0.635875	+	0.771792i	$0.280640\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.96075e15	0.641926
$78$	0	0
$79$	3.76431e15	0.279161	0.139581	−	0.990211i	$-0.455425\pi$
$79$	3.76431e15	0.279161	0.139581	+	0.990211i	$0.455425\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.79697e15	−0.379981	−0.189991	−	0.981786i	$-0.560846\pi$
$83$	−7.79697e15	−0.379981	−0.189991	+	0.981786i	$0.560846\pi$
$84$	0	0
$85$	−2.74856e15	−0.109407
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.61821e15	0.151281	0.0756403	−	0.997135i	$-0.475900\pi$
$89$	5.61821e15	0.151281	0.0756403	+	0.997135i	$0.475900\pi$
$90$	0	0
$91$	−2.90787e16	−0.648220
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.37916e16	−0.367937
$96$	0	0
$97$	−5.36057e16	−0.694466	−0.347233	−	0.937779i	$-0.612879\pi$
$97$	−5.36057e16	−0.694466	−0.347233	+	0.937779i	$0.612879\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.91320e16	−0.543364	−0.271682	−	0.962387i	$-0.587580\pi$
$101$	−5.91320e16	−0.543364	−0.271682	+	0.962387i	$0.587580\pi$
$102$	0	0
$103$	1.10336e17	0.858223	0.429112	−	0.903251i	$-0.358827\pi$
$103$	1.10336e17	0.858223	0.429112	+	0.903251i	$0.358827\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.83279e16	0.328182	0.164091	−	0.986445i	$-0.447531\pi$
$107$	5.83279e16	0.328182	0.164091	+	0.986445i	$0.447531\pi$
$108$	0	0
$109$	4.60722e16	0.221469	0.110735	−	0.993850i	$-0.464680\pi$
$109$	4.60722e16	0.221469	0.110735	+	0.993850i	$0.464680\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.21794e17	−1.84643	−0.923215	−	0.384285i	$-0.874448\pi$
$113$	−5.21794e17	−1.84643	−0.923215	+	0.384285i	$0.874448\pi$
$114$	0	0
$115$	−2.35710e17	−0.718533
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.91058e16	−0.111940
$120$	0	0
$121$	1.40667e17	0.278303
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.25721e17	0.938957
$126$	0	0
$127$	7.24759e17	0.950302	0.475151	−	0.879904i	$-0.342393\pi$
$127$	7.24759e17	0.950302	0.475151	+	0.879904i	$0.342393\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.34265e18	−1.35256	−0.676281	−	0.736644i	$-0.736409\pi$
$131$	−1.34265e18	−1.35256	−0.676281	+	0.736644i	$0.736409\pi$
$132$	0	0
$133$	−4.25062e17	−0.376455
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.52098e18	−1.73558	−0.867789	−	0.496933i	$-0.834460\pi$
$137$	−2.52098e18	−1.73558	−0.867789	+	0.496933i	$0.834460\pi$
$138$	0	0
$139$	−6.31107e17	−0.384129	−0.192065	−	0.981382i	$-0.561518\pi$
$139$	−6.31107e17	−0.384129	−0.192065	+	0.981382i	$0.561518\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.69915e18	−1.29084
$144$	0	0
$145$	6.80180e17	0.289065
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.83545e18	−0.618955	−0.309478	−	0.950907i	$-0.600154\pi$
$149$	−1.83545e18	−0.618955	−0.309478	+	0.950907i	$0.600154\pi$
$150$	0	0
$151$	−4.25971e18	−1.28255	−0.641277	−	0.767309i	$-0.721595\pi$
$151$	−4.25971e18	−1.28255	−0.641277	+	0.767309i	$0.721595\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.99292e18	−0.721565
$156$	0	0
$157$	−8.39662e17	−0.181534	−0.0907668	−	0.995872i	$-0.528932\pi$
$157$	−8.39662e17	−0.181534	−0.0907668	+	0.995872i	$0.528932\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.21120e18	−0.735167
$162$	0	0
$163$	9.65278e18	1.51725	0.758626	−	0.651526i	$-0.225871\pi$
$163$	9.65278e18	1.51725	0.758626	+	0.651526i	$0.225871\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.02649e19	−1.31301	−0.656503	−	0.754324i	$-0.727965\pi$
$167$	−1.02649e19	−1.31301	−0.656503	+	0.754324i	$0.727965\pi$
$168$	0	0
$169$	2.62535e18	0.303494
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.76779e19	−1.67509	−0.837544	−	0.546370i	$-0.816009\pi$
$173$	−1.76779e19	−1.67509	−0.837544	+	0.546370i	$0.816009\pi$
$174$	0	0
$175$	4.57235e18	0.392931
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.58852e19	−1.12653	−0.563263	−	0.826278i	$-0.690454\pi$
$179$	−1.58852e19	−1.12653	−0.563263	+	0.826278i	$0.690454\pi$
$180$	0	0
$181$	−1.94307e19	−1.25378	−0.626890	−	0.779107i	$-0.715673\pi$
$181$	−1.94307e19	−1.25378	−0.626890	+	0.779107i	$0.715673\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.43951e18	−0.345059
$186$	0	0
$187$	−4.55812e18	−0.222912
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.89973e19	−1.18461	−0.592303	−	0.805715i	$-0.701781\pi$
$191$	−2.89973e19	−1.18461	−0.592303	+	0.805715i	$0.701781\pi$
$192$	0	0
$193$	1.66625e19	0.623022	0.311511	−	0.950243i	$-0.399165\pi$
$193$	1.66625e19	0.623022	0.311511	+	0.950243i	$0.399165\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.09257e19	−1.59946	−0.799731	−	0.600358i	$-0.795025\pi$
$197$	−5.09257e19	−1.59946	−0.799731	+	0.600358i	$0.795025\pi$
$198$	0	0
$199$	−1.37945e19	−0.397608	−0.198804	−	0.980039i	$-0.563706\pi$
$199$	−1.37945e19	−0.397608	−0.198804	+	0.980039i	$0.563706\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.21521e19	0.295757
$204$	0	0
$205$	8.33753e18	0.186694
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.94553e19	−0.749657
$210$	0	0
$211$	−2.56156e18	−0.0448852	−0.0224426	−	0.999748i	$-0.507144\pi$
$211$	−2.56156e18	−0.0448852	−0.0224426	+	0.999748i	$0.507144\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.34851e18	0.0948297
$216$	0	0
$217$	−5.34715e19	−0.738269
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.90416e19	0.225097
$222$	0	0
$223$	1.00011e20	1.09510	0.547552	−	0.836772i	$-0.315560\pi$
$223$	1.00011e20	1.09510	0.547552	+	0.836772i	$0.315560\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.87714e19	−0.459140	−0.229570	−	0.973292i	$-0.573732\pi$
$227$	−4.87714e19	−0.459140	−0.229570	+	0.973292i	$0.573732\pi$
$228$	0	0
$229$	−6.62366e19	−0.578757	−0.289378	−	0.957215i	$-0.593449\pi$
$229$	−6.62366e19	−0.578757	−0.289378	+	0.957215i	$0.593449\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.21332e20	0.915059	0.457529	−	0.889195i	$-0.348734\pi$
$233$	1.21332e20	0.915059	0.457529	+	0.889195i	$0.348734\pi$
$234$	0	0
$235$	5.87669e19	0.412151
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.85803e20	−1.12894	−0.564468	−	0.825455i	$-0.690919\pi$
$239$	−1.85803e20	−1.12894	−0.564468	+	0.825455i	$0.690919\pi$
$240$	0	0
$241$	9.24458e19	0.523290	0.261645	−	0.965164i	$-0.415735\pi$
$241$	9.24458e19	0.523290	0.261645	+	0.965164i	$0.415735\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	7.64085e19	0.376036
$246$	0	0
$247$	1.64825e20	0.757008
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.40465e20	1.76476	0.882379	−	0.470540i	$-0.155941\pi$
$251$	4.40465e20	1.76476	0.882379	+	0.470540i	$0.155941\pi$
$252$	0	0
$253$	−3.90895e20	−1.46398
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.56765e20	0.841596	0.420798	−	0.907154i	$-0.361750\pi$
$257$	2.56765e20	0.841596	0.420798	+	0.907154i	$0.361750\pi$
$258$	0	0
$259$	−1.15048e20	−0.353047
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.56488e20	1.22972	0.614859	−	0.788637i	$-0.289213\pi$
$263$	4.56488e20	1.22972	0.614859	+	0.788637i	$0.289213\pi$
$264$	0	0
$265$	−3.04115e20	−0.768154
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.09953e20	1.13406	0.567030	−	0.823697i	$-0.308092\pi$
$269$	5.09953e20	1.13406	0.567030	+	0.823697i	$0.308092\pi$
$270$	0	0
$271$	1.89742e20	0.396208	0.198104	−	0.980181i	$-0.436522\pi$
$271$	1.89742e20	0.396208	0.198104	+	0.980181i	$0.436522\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.24417e20	0.782465
$276$	0	0
$277$	6.66487e19	0.115535	0.0577675	−	0.998330i	$-0.481602\pi$
$277$	6.66487e19	0.115535	0.0577675	+	0.998330i	$0.481602\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.65270e20	−0.714005	−0.357002	−	0.934104i	$-0.616201\pi$
$281$	−4.65270e20	−0.714005	−0.357002	+	0.934104i	$0.616201\pi$
$282$	0	0
$283$	1.21470e21	1.75503	0.877516	−	0.479547i	$-0.159199\pi$
$283$	1.21470e21	1.75503	0.877516	+	0.479547i	$0.159199\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.48959e20	0.191016
$288$	0	0
$289$	−7.95084e20	−0.961129
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.48758e21	1.59995	0.799974	−	0.600034i	$-0.204846\pi$
$293$	1.48758e21	1.59995	0.799974	+	0.600034i	$0.204846\pi$
$294$	0	0
$295$	1.14570e20	0.116301
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.63297e21	1.47834
$300$	0	0
$301$	1.13423e20	0.0970251
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.79579e20	0.443163
$306$	0	0
$307$	9.19688e20	0.665219	0.332609	−	0.943065i	$-0.392071\pi$
$307$	9.19688e20	0.665219	0.332609	+	0.943065i	$0.392071\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.98715e20	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$311$	−7.98715e20	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$312$	0	0
$313$	−2.67228e21	−1.63967	−0.819834	−	0.572602i	$-0.805934\pi$
$313$	−2.67228e21	−1.63967	−0.819834	+	0.572602i	$0.805934\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.22165e20	−0.177449	−0.0887247	−	0.996056i	$-0.528279\pi$
$317$	−3.22165e20	−0.177449	−0.0887247	+	0.996056i	$0.528279\pi$
$318$	0	0
$319$	1.12799e21	0.588958
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.78345e20	0.130726
$324$	0	0
$325$	−1.77301e21	−0.790137
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.04993e21	0.421692
$330$	0	0
$331$	3.47549e21	1.32580	0.662899	−	0.748709i	$-0.269326\pi$
$331$	3.47549e21	1.32580	0.662899	+	0.748709i	$0.269326\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.05246e20	0.277362
$336$	0	0
$337$	−2.00621e21	−0.656933	−0.328467	−	0.944516i	$-0.606532\pi$
$337$	−2.00621e21	−0.656933	−0.328467	+	0.944516i	$0.606532\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.96336e21	−1.47016
$342$	0	0
$343$	3.37962e21	0.952506
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.82696e21	−0.977357	−0.488678	−	0.872464i	$-0.662521\pi$
$347$	−3.82696e21	−0.977357	−0.488678	+	0.872464i	$0.662521\pi$
$348$	0	0
$349$	−6.91280e21	−1.68127	−0.840636	−	0.541601i	$-0.817818\pi$
$349$	−6.91280e21	−1.68127	−0.840636	+	0.541601i	$0.817818\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.75477e21	0.387379	0.193689	−	0.981063i	$-0.437955\pi$
$353$	1.75477e21	0.387379	0.193689	+	0.981063i	$0.437955\pi$
$354$	0	0
$355$	8.35160e20	0.175723
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.77038e21	−1.67770	−0.838852	−	0.544359i	$-0.816773\pi$
$359$	−8.77038e21	−1.67770	−0.838852	+	0.544359i	$0.816773\pi$
$360$	0	0
$361$	−3.07102e21	−0.560366
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.24736e21	−0.705716
$366$	0	0
$367$	1.07483e22	1.70482	0.852408	−	0.522878i	$-0.175142\pi$
$367$	1.07483e22	1.70482	0.852408	+	0.522878i	$0.175142\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.43333e21	−0.785938
$372$	0	0
$373$	7.28664e21	1.00694	0.503469	−	0.864013i	$-0.332057\pi$
$373$	7.28664e21	1.00694	0.503469	+	0.864013i	$0.332057\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.71220e21	−0.594734
$378$	0	0
$379$	−4.43749e21	−0.535432	−0.267716	−	0.963498i	$-0.586269\pi$
$379$	−4.43749e21	−0.535432	−0.267716	+	0.963498i	$0.586269\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.29560e21	0.474061	0.237031	−	0.971502i	$-0.423826\pi$
$383$	4.29560e21	0.474061	0.237031	+	0.971502i	$0.423826\pi$
$384$	0	0
$385$	−3.37388e21	−0.356216
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6.37121e21	−0.616099	−0.308049	−	0.951370i	$-0.599676\pi$
$389$	−6.37121e21	−0.616099	−0.308049	+	0.951370i	$0.599676\pi$
$390$	0	0
$391$	2.75763e21	0.255290
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.82456e21	−0.154911
$396$	0	0
$397$	−2.40790e22	−1.95848	−0.979239	−	0.202707i	$-0.935026\pi$
$397$	−2.40790e22	−1.95848	−0.979239	+	0.202707i	$0.935026\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.41636e22	−1.80482	−0.902411	−	0.430876i	$-0.858205\pi$
$401$	−2.41636e22	−1.80482	−0.902411	+	0.430876i	$0.858205\pi$
$402$	0	0
$403$	2.07345e22	1.48457
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.06791e22	−0.703042
$408$	0	0
$409$	−1.99243e22	−1.25816	−0.629080	−	0.777341i	$-0.716568\pi$
$409$	−1.99243e22	−1.25816	−0.629080	+	0.777341i	$0.716568\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.04692e21	0.118994
$414$	0	0
$415$	3.77919e21	0.210858
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.06785e22	−0.549152	−0.274576	−	0.961565i	$-0.588538\pi$
$419$	−1.06785e22	−0.549152	−0.274576	+	0.961565i	$0.588538\pi$
$420$	0	0
$421$	−5.98402e21	−0.295526	−0.147763	−	0.989023i	$-0.547207\pi$
$421$	−5.98402e21	−0.295526	−0.147763	+	0.989023i	$0.547207\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.99412e21	−0.136447
$426$	0	0
$427$	1.03548e22	0.453423
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.12033e22	0.857721	0.428861	−	0.903371i	$-0.358915\pi$
$431$	2.12033e22	0.857721	0.428861	+	0.903371i	$0.358915\pi$
$432$	0	0
$433$	3.63320e22	1.41300	0.706499	−	0.707714i	$-0.250273\pi$
$433$	3.63320e22	1.41300	0.706499	+	0.707714i	$0.250273\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.38702e22	0.858547
$438$	0	0
$439$	−7.82190e21	−0.270622	−0.135311	−	0.990803i	$-0.543203\pi$
$439$	−7.82190e21	−0.270622	−0.135311	+	0.990803i	$0.543203\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.85063e20	−0.00913080	−0.00456540	−	0.999990i	$-0.501453\pi$
$443$	−2.85063e20	−0.00913080	−0.00456540	+	0.999990i	$0.501453\pi$
$444$	0	0
$445$	−2.72315e21	−0.0839482
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.51942e22	1.29119	0.645593	−	0.763682i	$-0.276610\pi$
$449$	4.51942e22	1.29119	0.645593	+	0.763682i	$0.276610\pi$
$450$	0	0
$451$	1.38267e22	0.380380
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.40944e22	0.359709
$456$	0	0
$457$	−2.18470e22	−0.537162	−0.268581	−	0.963257i	$-0.586555\pi$
$457$	−2.18470e22	−0.537162	−0.268581	+	0.963257i	$0.586555\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.50670e22	0.572326	0.286163	−	0.958181i	$-0.407620\pi$
$461$	2.50670e22	0.572326	0.286163	+	0.958181i	$0.407620\pi$
$462$	0	0
$463$	−3.46875e22	−0.763368	−0.381684	−	0.924293i	$-0.624656\pi$
$463$	−3.46875e22	−0.763368	−0.381684	+	0.924293i	$0.624656\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.78939e22	−0.570578	−0.285289	−	0.958441i	$-0.592090\pi$
$467$	−2.78939e22	−0.570578	−0.285289	+	0.958441i	$0.592090\pi$
$468$	0	0
$469$	1.43865e22	0.283784
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.05282e22	0.193212
$474$	0	0
$475$	−2.59173e22	−0.458874
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9.33721e22	1.53945	0.769724	−	0.638377i	$-0.220394\pi$
$479$	9.33721e22	1.53945	0.769724	+	0.638377i	$0.220394\pi$
$480$	0	0
$481$	4.46121e22	0.709936
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.59827e22	0.385371
$486$	0	0
$487$	3.39236e22	0.485854	0.242927	−	0.970045i	$-0.421893\pi$
$487$	3.39236e22	0.485854	0.242927	+	0.970045i	$0.421893\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.36955e23	1.82973	0.914864	−	0.403763i	$-0.132298\pi$
$491$	1.36955e23	1.82973	0.914864	+	0.403763i	$0.132298\pi$
$492$	0	0
$493$	−7.95760e21	−0.102703
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.49210e22	0.179791
$498$	0	0
$499$	−9.60929e22	−1.11902	−0.559508	−	0.828825i	$-0.689010\pi$
$499$	−9.60929e22	−1.11902	−0.559508	+	0.828825i	$0.689010\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.78161e19	0.000955534 0	0.000477767	−	1.00000i	$-0.499848\pi$
$503$	8.78161e19	0.000955534 0	0.000477767		1.00000i	$0.499848\pi$
$504$	0	0
$505$	2.86613e22	0.301522
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.48440e22	0.441167	0.220583	−	0.975368i	$-0.429204\pi$
$509$	4.48440e22	0.441167	0.220583	+	0.975368i	$0.429204\pi$
$510$	0	0
$511$	−7.58834e22	−0.722053
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.34798e22	−0.476243
$516$	0	0
$517$	9.74572e22	0.839739
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.40072e23	1.13039	0.565196	−	0.824956i	$-0.308800\pi$
$521$	1.40072e23	1.13039	0.565196	+	0.824956i	$0.308800\pi$
$522$	0	0
$523$	−3.87014e22	−0.302317	−0.151159	−	0.988510i	$-0.548300\pi$
$523$	−3.87014e22	−0.302317	−0.151159	+	0.988510i	$0.548300\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.50149e22	0.256367
$528$	0	0
$529$	9.54385e22	0.676628
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.77613e22	−0.384110
$534$	0	0
$535$	−2.82716e22	−0.182114
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.26714e23	0.766158
$540$	0	0
$541$	1.71513e23	1.00489	0.502447	−	0.864608i	$-0.332433\pi$
$541$	1.71513e23	1.00489	0.502447	+	0.864608i	$0.332433\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.23312e22	−0.122897
$546$	0	0
$547$	−4.18991e22	−0.223518	−0.111759	−	0.993735i	$-0.535648\pi$
$547$	−4.18991e22	−0.223518	−0.111759	+	0.993735i	$0.535648\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.88814e22	−0.345393
$552$	0	0
$553$	−3.25977e22	−0.158498
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.03398e23	−1.38753	−0.693766	−	0.720200i	$-0.744050\pi$
$557$	−3.03398e23	−1.38753	−0.693766	+	0.720200i	$0.744050\pi$
$558$	0	0
$559$	−4.39816e22	−0.195106
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.79700e23	−1.16781	−0.583903	−	0.811823i	$-0.698475\pi$
$563$	−2.79700e23	−1.16781	−0.583903	+	0.811823i	$0.698475\pi$
$564$	0	0
$565$	2.52914e23	1.02462
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.97640e23	−0.754086	−0.377043	−	0.926196i	$-0.623059\pi$
$569$	−1.97640e23	−0.754086	−0.377043	+	0.926196i	$0.623059\pi$
$570$	0	0
$571$	−4.38294e23	−1.62315	−0.811574	−	0.584249i	$-0.801389\pi$
$571$	−4.38294e23	−1.62315	−0.811574	+	0.584249i	$0.801389\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.56769e23	−0.896121
$576$	0	0
$577$	−1.05986e23	−0.359133	−0.179566	−	0.983746i	$-0.557469\pi$
$577$	−1.05986e23	−0.359133	−0.179566	+	0.983746i	$0.557469\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.75192e22	0.215740
$582$	0	0
$583$	−5.04336e23	−1.56508
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.65195e23	0.483696	0.241848	−	0.970314i	$-0.422247\pi$
$587$	1.65195e23	0.483696	0.241848	+	0.970314i	$0.422247\pi$
$588$	0	0
$589$	3.03091e23	0.862169
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.85479e22	0.0498116	0.0249058	−	0.999690i	$-0.492071\pi$
$593$	1.85479e22	0.0498116	0.0249058	+	0.999690i	$0.492071\pi$
$594$	0	0
$595$	2.38016e22	0.0621172
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.11584e23	−1.50775	−0.753873	−	0.657020i	$-0.771817\pi$
$599$	−6.11584e23	−1.50775	−0.753873	+	0.657020i	$0.771817\pi$
$600$	0	0
$601$	6.16474e23	1.47734	0.738672	−	0.674065i	$-0.235453\pi$
$601$	6.16474e23	1.47734	0.738672	+	0.674065i	$0.235453\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.81814e22	−0.154435
$606$	0	0
$607$	1.17867e23	0.259590	0.129795	−	0.991541i	$-0.458568\pi$
$607$	1.17867e23	0.259590	0.129795	+	0.991541i	$0.458568\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.07129e23	−0.847973
$612$	0	0
$613$	−1.09211e23	−0.221234	−0.110617	−	0.993863i	$-0.535283\pi$
$613$	−1.09211e23	−0.221234	−0.110617	+	0.993863i	$0.535283\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.33478e23	−1.21425	−0.607124	−	0.794607i	$-0.707677\pi$
$617$	−6.33478e23	−1.21425	−0.607124	+	0.794607i	$0.707677\pi$
$618$	0	0
$619$	4.19708e23	0.782666	0.391333	−	0.920249i	$-0.372014\pi$
$619$	4.19708e23	0.782666	0.391333	+	0.920249i	$0.372014\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.86518e22	−0.0858916
$624$	0	0
$625$	9.95487e22	0.171023
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.53374e22	0.122597
$630$	0	0
$631$	−8.84513e23	−1.40105	−0.700527	−	0.713626i	$-0.747052\pi$
$631$	−8.84513e23	−1.40105	−0.700527	+	0.713626i	$0.747052\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.51291e23	−0.527339
$636$	0	0
$637$	−5.29348e23	−0.773671
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.03257e24	−1.43096	−0.715478	−	0.698636i	$-0.753791\pi$
$641$	−1.03257e24	−1.43096	−0.715478	+	0.698636i	$0.753791\pi$
$642$	0	0
$643$	5.84752e23	0.789184	0.394592	−	0.918856i	$-0.370886\pi$
$643$	5.84752e23	0.789184	0.394592	+	0.918856i	$0.370886\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.89705e23	−0.370911	−0.185456	−	0.982653i	$-0.559376\pi$
$647$	−2.89705e23	−0.370911	−0.185456	+	0.982653i	$0.559376\pi$
$648$	0	0
$649$	1.90000e23	0.236959
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.54002e23	0.655767	0.327883	−	0.944718i	$-0.393665\pi$
$653$	5.54002e23	0.655767	0.327883	+	0.944718i	$0.393665\pi$
$654$	0	0
$655$	6.50783e23	0.750560
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.67095e24	1.82994	0.914969	−	0.403524i	$-0.132215\pi$
$659$	1.67095e24	1.82994	0.914969	+	0.403524i	$0.132215\pi$
$660$	0	0
$661$	−3.59471e23	−0.383664	−0.191832	−	0.981428i	$-0.561443\pi$
$661$	−3.59471e23	−0.383664	−0.191832	+	0.981428i	$0.561443\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.06028e23	0.208902
$666$	0	0
$667$	−6.82426e23	−0.674507
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.61156e23	0.902927
$672$	0	0
$673$	−7.73956e23	−0.708905	−0.354452	−	0.935074i	$-0.615333\pi$
$673$	−7.73956e23	−0.708905	−0.354452	+	0.935074i	$0.615333\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.87302e23	−0.685706	−0.342853	−	0.939389i	$-0.611393\pi$
$677$	−7.87302e23	−0.685706	−0.342853	+	0.939389i	$0.611393\pi$
$678$	0	0
$679$	4.64207e23	0.394293
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−4.91474e23	−0.397122	−0.198561	−	0.980088i	$-0.563627\pi$
$683$	−4.91474e23	−0.397122	−0.198561	+	0.980088i	$0.563627\pi$
$684$	0	0
$685$	1.22192e24	0.963102
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.10687e24	1.58043
$690$	0	0
$691$	−9.36732e23	−0.685570	−0.342785	−	0.939414i	$-0.611370\pi$
$691$	−9.36732e23	−0.685570	−0.342785	+	0.939414i	$0.611370\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.05898e23	0.213160
$696$	0	0
$697$	−9.75429e22	−0.0663311
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.64436e23	0.236058	0.118029	−	0.993010i	$-0.462342\pi$
$701$	3.64436e23	0.236058	0.118029	+	0.993010i	$0.462342\pi$
$702$	0	0
$703$	6.52125e23	0.412297
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.12063e23	0.308503
$708$	0	0
$709$	−2.02149e24	−1.18899	−0.594494	−	0.804100i	$-0.702648\pi$
$709$	−2.02149e24	−1.18899	−0.594494	+	0.804100i	$0.702648\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.00280e24	1.68370
$714$	0	0
$715$	1.30828e24	0.716308
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.18677e23	0.218617	0.109308	−	0.994008i	$-0.465136\pi$
$719$	4.18677e23	0.218617	0.109308	+	0.994008i	$0.465136\pi$
$720$	0	0
$721$	−9.55472e23	−0.487268
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.40950e23	0.360509
$726$	0	0
$727$	2.25868e24	1.07353	0.536763	−	0.843733i	$-0.319647\pi$
$727$	2.25868e24	1.07353	0.536763	+	0.843733i	$0.319647\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.42728e22	−0.0336924
$732$	0	0
$733$	−1.59589e24	−0.707325	−0.353663	−	0.935373i	$-0.615064\pi$
$733$	−1.59589e24	−0.707325	−0.353663	+	0.935373i	$0.615064\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.33540e24	0.565114
$738$	0	0
$739$	3.93421e24	1.62697	0.813485	−	0.581585i	$-0.197567\pi$
$739$	3.93421e24	1.62697	0.813485	+	0.581585i	$0.197567\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.09794e24	1.22368	0.611841	−	0.790981i	$-0.290429\pi$
$743$	3.09794e24	1.22368	0.611841	+	0.790981i	$0.290429\pi$
$744$	0	0
$745$	8.89643e23	0.343469
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.05101e23	−0.186330
$750$	0	0
$751$	1.49050e24	0.537518	0.268759	−	0.963207i	$-0.413386\pi$
$751$	1.49050e24	0.537518	0.268759	+	0.963207i	$0.413386\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.06468e24	0.711712
$756$	0	0
$757$	−4.26149e24	−1.43631	−0.718153	−	0.695885i	$-0.755012\pi$
$757$	−4.26149e24	−1.43631	−0.718153	+	0.695885i	$0.755012\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.46465e24	−0.794303	−0.397152	−	0.917753i	$-0.630001\pi$
$761$	−2.46465e24	−0.794303	−0.397152	+	0.917753i	$0.630001\pi$
$762$	0	0
$763$	−3.98970e23	−0.125742
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.93728e23	−0.239283
$768$	0	0
$769$	−3.52425e24	−1.03918	−0.519592	−	0.854414i	$-0.673916\pi$
$769$	−3.52425e24	−1.03918	−0.519592	+	0.854414i	$0.673916\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5.09437e24	1.43736	0.718678	−	0.695343i	$-0.244747\pi$
$773$	5.09437e24	1.43736	0.718678	+	0.695343i	$0.244747\pi$
$774$	0	0
$775$	−3.26031e24	−0.899902
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.44336e23	−0.223073
$780$	0	0
$781$	1.38500e24	0.358028
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.06984e23	0.100736
$786$	0	0
$787$	3.97377e24	0.962538	0.481269	−	0.876573i	$-0.340176\pi$
$787$	3.97377e24	0.962538	0.481269	+	0.876573i	$0.340176\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.51857e24	1.04834
$792$	0	0
$793$	−4.01525e24	−0.911780
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.02040e24	−0.657157	−0.328579	−	0.944477i	$-0.606570\pi$
$797$	−3.02040e24	−0.657157	−0.328579	+	0.944477i	$0.606570\pi$
$798$	0	0
$799$	−6.87529e23	−0.146434
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.04369e24	−1.43787
$804$	0	0
$805$	2.04117e24	0.407957
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7.06671e24	−1.35411	−0.677056	−	0.735931i	$-0.736745\pi$
$809$	−7.06671e24	−1.35411	−0.677056	+	0.735931i	$0.736745\pi$
$810$	0	0
$811$	7.01786e24	1.31682	0.658412	−	0.752658i	$-0.271228\pi$
$811$	7.01786e24	1.31682	0.658412	+	0.752658i	$0.271228\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.67870e24	−0.841949
$816$	0	0
$817$	−6.42909e23	−0.113308
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.08902e24	−0.691357	−0.345679	−	0.938353i	$-0.612351\pi$
$821$	−4.08902e24	−0.691357	−0.345679	+	0.938353i	$0.612351\pi$
$822$	0	0
$823$	1.18447e24	0.196167	0.0980835	−	0.995178i	$-0.468729\pi$
$823$	1.18447e24	0.196167	0.0980835	+	0.995178i	$0.468729\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.42303e24	−0.702947	−0.351474	−	0.936198i	$-0.614319\pi$
$827$	−4.42303e24	−0.702947	−0.351474	+	0.936198i	$0.614319\pi$
$828$	0	0
$829$	−1.70464e24	−0.265411	−0.132706	−	0.991156i	$-0.542366\pi$
$829$	−1.70464e24	−0.265411	−0.132706	+	0.991156i	$0.542366\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.93922e23	−0.133603
$834$	0	0
$835$	4.97542e24	0.728609
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.65201e23	−0.0513517	−0.0256759	−	0.999670i	$-0.508174\pi$
$839$	−3.65201e23	−0.0513517	−0.0256759	+	0.999670i	$0.508174\pi$
$840$	0	0
$841$	−5.28789e24	−0.728646
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.27251e24	−0.168414
$846$	0	0
$847$	−1.21813e24	−0.158010
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.46077e24	0.805162
$852$	0	0
$853$	6.64517e24	0.811782	0.405891	−	0.913921i	$-0.366961\pi$
$853$	6.64517e24	0.811782	0.405891	+	0.913921i	$0.366961\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.34537e24	−0.392742	−0.196371	−	0.980530i	$-0.562916\pi$
$857$	−3.34537e24	−0.392742	−0.196371	+	0.980530i	$0.562916\pi$
$858$	0	0
$859$	−1.52661e25	−1.75706	−0.878529	−	0.477688i	$-0.841475\pi$
$859$	−1.52661e25	−1.75706	−0.878529	+	0.477688i	$0.841475\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.29506e25	−1.43284	−0.716419	−	0.697670i	$-0.754220\pi$
$863$	−1.29506e25	−1.43284	−0.716419	+	0.697670i	$0.754220\pi$
$864$	0	0
$865$	8.56846e24	0.929535
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.02580e24	−0.315625
$870$	0	0
$871$	−5.57864e24	−0.570656
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.41854e24	−0.533106
$876$	0	0
$877$	−3.39683e24	−0.327776	−0.163888	−	0.986479i	$-0.552404\pi$
$877$	−3.39683e24	−0.327776	−0.163888	+	0.986479i	$0.552404\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.16744e25	−1.08377	−0.541887	−	0.840451i	$-0.682290\pi$
$881$	−1.16744e25	−1.08377	−0.541887	+	0.840451i	$0.682290\pi$
$882$	0	0
$883$	1.03857e25	0.945733	0.472866	−	0.881134i	$-0.343219\pi$
$883$	1.03857e25	0.945733	0.472866	+	0.881134i	$0.343219\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.20725e23	−0.0631566	−0.0315783	−	0.999501i	$-0.510053\pi$
$887$	−7.20725e23	−0.0631566	−0.0315783	+	0.999501i	$0.510053\pi$
$888$	0	0
$889$	−6.27617e24	−0.539548
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.95128e24	−0.492463
$894$	0	0
$895$	7.69954e24	0.625128
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.66507e24	−0.677353
$900$	0	0
$901$	3.55792e24	0.272920
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.41809e24	0.695745
$906$	0	0
$907$	−6.80337e24	−0.493244	−0.246622	−	0.969112i	$-0.579321\pi$
$907$	−6.80337e24	−0.493244	−0.246622	+	0.969112i	$0.579321\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.70926e24	−0.328887	−0.164444	−	0.986386i	$-0.552583\pi$
$911$	−4.70926e24	−0.328887	−0.164444	+	0.986386i	$0.552583\pi$
$912$	0	0
$913$	6.26730e24	0.429615
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.16269e25	0.767936
$918$	0	0
$919$	−2.09220e25	−1.35651	−0.678253	−	0.734828i	$-0.737263\pi$
$919$	−2.09220e25	−1.35651	−0.678253	+	0.734828i	$0.737263\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.78588e24	−0.361539
$924$	0	0
$925$	−7.01483e24	−0.430341
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.73931e25	1.02859	0.514296	−	0.857613i	$-0.328053\pi$
$929$	1.73931e25	1.02859	0.514296	+	0.857613i	$0.328053\pi$
$930$	0	0
$931$	−7.73784e24	−0.449311
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.20932e24	0.123697
$936$	0	0
$937$	2.24671e25	1.23527	0.617634	−	0.786466i	$-0.288091\pi$
$937$	2.24671e25	1.23527	0.617634	+	0.786466i	$0.288091\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.98079e24	−0.105033	−0.0525166	−	0.998620i	$-0.516724\pi$
$941$	−1.98079e24	−0.105033	−0.0525166	+	0.998620i	$0.516724\pi$
$942$	0	0
$943$	−8.36506e24	−0.435632
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.99450e25	1.50435	0.752177	−	0.658962i	$-0.229004\pi$
$947$	2.99450e25	1.50435	0.752177	+	0.658962i	$0.229004\pi$
$948$	0	0
$949$	2.94252e25	1.45196
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.23958e25	−0.590181	−0.295091	−	0.955469i	$-0.595350\pi$
$953$	−1.23958e25	−0.590181	−0.295091	+	0.955469i	$0.595350\pi$
$954$	0	0
$955$	1.40550e25	0.657359
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.18308e25	0.985399
$960$	0	0
$961$	1.55778e25	0.690807
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−8.07632e24	−0.345726
$966$	0	0
$967$	−3.16830e25	−1.33261	−0.666303	−	0.745681i	$-0.732124\pi$
$967$	−3.16830e25	−1.33261	−0.666303	+	0.745681i	$0.732124\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.08361e25	1.65837	0.829183	−	0.558977i	$-0.188806\pi$
$971$	4.08361e25	1.65837	0.829183	+	0.558977i	$0.188806\pi$
$972$	0	0
$973$	5.46518e24	0.218095
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.32294e25	1.28061	0.640307	−	0.768119i	$-0.278807\pi$
$977$	3.32294e25	1.28061	0.640307	+	0.768119i	$0.278807\pi$
$978$	0	0
$979$	−4.51599e24	−0.171041
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−9.07807e24	−0.332115	−0.166058	−	0.986116i	$-0.553104\pi$
$983$	−9.07807e24	−0.332115	−0.166058	+	0.986116i	$0.553104\pi$
$984$	0	0
$985$	2.46837e25	0.887569
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.36947e24	−0.221276
$990$	0	0
$991$	−5.16244e25	−1.76290	−0.881452	−	0.472273i	$-0.843434\pi$
$991$	−5.16244e25	−1.76290	−0.881452	+	0.472273i	$0.843434\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.68621e24	0.220640
$996$	0	0
$997$	−7.25042e24	−0.235209	−0.117605	−	0.993061i	$-0.537522\pi$
$997$	−7.25042e24	−0.235209	−0.117605	+	0.993061i	$0.537522\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.18.a.g.1.2	✓	4
3.2	odd	2		72.18.a.h.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.18.a.g.1.2	✓	4	1.1	even	1	trivial
72.18.a.h.1.3	yes	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

\(f(q)\)	\(=\)	\(q-484700. q^{5} -8.65967e6 q^{7} -8.03812e8 q^{11} +3.35794e9 q^{13} +5.67063e9 q^{17} +4.90853e10 q^{19} +4.86301e11 q^{23} -5.28005e11 q^{25} -1.40330e12 q^{29} +6.17478e12 q^{31} +4.19734e12 q^{35} +1.32855e13 q^{37} -1.72014e13 q^{41} -1.30978e13 q^{43} -1.21244e14 q^{47} -1.57641e14 q^{49} +6.27430e14 q^{53} +3.89608e14 q^{55} -2.36373e14 q^{59} -1.19575e15 q^{61} -1.62760e15 q^{65} -1.66133e15 q^{67} -1.72305e15 q^{71} +8.76286e15 q^{73} +6.96075e15 q^{77} +3.76431e15 q^{79} -7.79697e15 q^{83} -2.74856e15 q^{85} +5.61821e15 q^{89} -2.90787e16 q^{91} -2.37916e16 q^{95} -5.36057e16 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 677824 q^{5} + 9632880 q^{7} + 603687680 q^{11} - 2476119800 q^{13} - 7423708544 q^{17} - 19551988960 q^{19} - 458239426048 q^{23} + 1015542005996 q^{25} + 844849583040 q^{29} + 1246685956208 q^{31}+ \cdots - 17\!\cdots\!60 q^{97}+O(q^{100}) \) 4 * q - 677824 * q^5 + 9632880 * q^7 + 603687680 * q^11 - 2476119800 * q^13 - 7423708544 * q^17 - 19551988960 * q^19 - 458239426048 * q^23 + 1015542005996 * q^25 + 844849583040 * q^29 + 1246685956208 * q^31 - 3273147963648 * q^35 + 17661470383320 * q^37 + 33100198696320 * q^41 + 24133870623520 * q^43 + 11964304009728 * q^47 + 2287828660772 * q^49 - 256855748281408 * q^53 - 996530006700032 * q^55 - 905361180505600 * q^59 + 251141035416376 * q^61 + 3700013891444864 * q^65 + 2671944825444800 * q^67 - 4181227521935360 * q^71 - 1410879602368040 * q^73 - 1014782699621376 * q^77 + 7574345714454640 * q^79 - 14868236752232192 * q^83 - 788948917823488 * q^85 - 31667797088981760 * q^89 - 6872737039354656 * q^91 - 21917705446002176 * q^95 - 17206349902780360 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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