LMFDB - Embedded newform 72.20.a.a.1.1

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			1.1
Root			$19.5591$ 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)

$f(q)$	$=$	$q-2.22342e6 q^{5} +1.60989e8 q^{7} +O(q^{10})$	$q-2.22342e6 q^{5} +1.60989e8 q^{7} +1.17593e10 q^{11} -2.44117e9 q^{13} -5.98454e11 q^{17} -2.51671e12 q^{19} +6.35737e11 q^{23} -1.41299e13 q^{25} +4.78062e13 q^{29} -2.03104e14 q^{31} -3.57946e14 q^{35} +8.95266e14 q^{37} +4.61047e14 q^{41} -2.21551e15 q^{43} +1.04653e16 q^{47} +1.45184e16 q^{49} +3.43716e16 q^{53} -2.61460e16 q^{55} -9.13126e16 q^{59} -1.19474e17 q^{61} +5.42776e15 q^{65} -1.98838e17 q^{67} +1.38863e17 q^{71} -3.63434e16 q^{73} +1.89312e18 q^{77} -4.36121e17 q^{79} -4.52389e17 q^{83} +1.33062e18 q^{85} -3.30628e18 q^{89} -3.93001e17 q^{91} +5.59572e18 q^{95} -6.03558e18 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.22342e6	−0.509105	−0.254552	−	0.967059i	$-0.581928\pi$
$5$	−2.22342e6	−0.509105	−0.254552	+	0.967059i	$0.581928\pi$
$6$	0	0
$7$	1.60989e8	1.50787	0.753934	−	0.656950i	$-0.228154\pi$
$7$	1.60989e8	1.50787	0.753934	+	0.656950i	$0.228154\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.17593e10	1.50367	0.751835	−	0.659352i	$-0.229169\pi$
$11$	1.17593e10	1.50367	0.751835	+	0.659352i	$0.229169\pi$
$12$	0	0
$13$	−2.44117e9	−0.0638464	−0.0319232	−	0.999490i	$-0.510163\pi$
$13$	−2.44117e9	−0.0638464	−0.0319232	+	0.999490i	$0.510163\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.98454e11	−1.22396	−0.611978	−	0.790874i	$-0.709626\pi$
$17$	−5.98454e11	−1.22396	−0.611978	+	0.790874i	$0.709626\pi$
$18$	0	0
$19$	−2.51671e12	−1.78927	−0.894633	−	0.446802i	$-0.852563\pi$
$19$	−2.51671e12	−1.78927	−0.894633	+	0.446802i	$0.852563\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.35737e11	0.0735975	0.0367988	−	0.999323i	$-0.488284\pi$
$23$	6.35737e11	0.0735975	0.0367988	+	0.999323i	$0.488284\pi$
$24$	0	0
$25$	−1.41299e13	−0.740812
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.78062e13	0.611932	0.305966	−	0.952043i	$-0.401021\pi$
$29$	4.78062e13	0.611932	0.305966	+	0.952043i	$0.401021\pi$
$30$	0	0
$31$	−2.03104e14	−1.37970	−0.689848	−	0.723954i	$-0.742323\pi$
$31$	−2.03104e14	−1.37970	−0.689848	+	0.723954i	$0.742323\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.57946e14	−0.767664
$36$	0	0
$37$	8.95266e14	1.13249	0.566246	−	0.824236i	$-0.308395\pi$
$37$	8.95266e14	1.13249	0.566246	+	0.824236i	$0.308395\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.61047e14	0.219937	0.109969	−	0.993935i	$-0.464925\pi$
$41$	4.61047e14	0.219937	0.109969	+	0.993935i	$0.464925\pi$
$42$	0	0
$43$	−2.21551e15	−0.672238	−0.336119	−	0.941820i	$-0.609114\pi$
$43$	−2.21551e15	−0.672238	−0.336119	+	0.941820i	$0.609114\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.04653e16	1.36403	0.682013	−	0.731340i	$-0.261105\pi$
$47$	1.04653e16	1.36403	0.682013	+	0.731340i	$0.261105\pi$
$48$	0	0
$49$	1.45184e16	1.27367
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.43716e16	1.43080	0.715400	−	0.698715i	$-0.246244\pi$
$53$	3.43716e16	1.43080	0.715400	+	0.698715i	$0.246244\pi$
$54$	0	0
$55$	−2.61460e16	−0.765525
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.13126e16	−1.37226	−0.686130	−	0.727479i	$-0.740692\pi$
$59$	−9.13126e16	−1.37226	−0.686130	+	0.727479i	$0.740692\pi$
$60$	0	0
$61$	−1.19474e17	−1.30809	−0.654047	−	0.756454i	$-0.726930\pi$
$61$	−1.19474e17	−1.30809	−0.654047	+	0.756454i	$0.726930\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.42776e15	0.0325045
$66$	0	0
$67$	−1.98838e17	−0.892872	−0.446436	−	0.894816i	$-0.647307\pi$
$67$	−1.98838e17	−0.892872	−0.446436	+	0.894816i	$0.647307\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.38863e17	0.359446	0.179723	−	0.983717i	$-0.442480\pi$
$71$	1.38863e17	0.359446	0.179723	+	0.983717i	$0.442480\pi$
$72$	0	0
$73$	−3.63434e16	−0.0722533	−0.0361267	−	0.999347i	$-0.511502\pi$
$73$	−3.63434e16	−0.0722533	−0.0361267	+	0.999347i	$0.511502\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.89312e18	2.26734
$78$	0	0
$79$	−4.36121e17	−0.409402	−0.204701	−	0.978825i	$-0.565622\pi$
$79$	−4.36121e17	−0.409402	−0.204701	+	0.978825i	$0.565622\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.52389e17	−0.265626	−0.132813	−	0.991141i	$-0.542401\pi$
$83$	−4.52389e17	−0.265626	−0.132813	+	0.991141i	$0.542401\pi$
$84$	0	0
$85$	1.33062e18	0.623123
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.30628e18	−1.00031	−0.500154	−	0.865936i	$-0.666723\pi$
$89$	−3.30628e18	−1.00031	−0.500154	+	0.865936i	$0.666723\pi$
$90$	0	0
$91$	−3.93001e17	−0.0962720
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.59572e18	0.910924
$96$	0	0
$97$	−6.03558e18	−0.806097	−0.403049	−	0.915179i	$-0.632049\pi$
$97$	−6.03558e18	−0.806097	−0.403049	+	0.915179i	$0.632049\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.09027e18	−0.827034	−0.413517	−	0.910496i	$-0.635700\pi$
$101$	−9.09027e18	−0.827034	−0.413517	+	0.910496i	$0.635700\pi$
$102$	0	0
$103$	1.16083e19	0.876628	0.438314	−	0.898822i	$-0.355576\pi$
$103$	1.16083e19	0.876628	0.438314	+	0.898822i	$0.355576\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.99623e18	−0.525642	−0.262821	−	0.964845i	$-0.584653\pi$
$107$	−9.99623e18	−0.525642	−0.262821	+	0.964845i	$0.584653\pi$
$108$	0	0
$109$	−4.34179e18	−0.191477	−0.0957385	−	0.995407i	$-0.530521\pi$
$109$	−4.34179e18	−0.191477	−0.0957385	+	0.995407i	$0.530521\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.09237e18	0.284729	0.142365	−	0.989814i	$-0.454529\pi$
$113$	9.09237e18	0.284729	0.142365	+	0.989814i	$0.454529\pi$
$114$	0	0
$115$	−1.41351e18	−0.0374689
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.63443e19	−1.84557
$120$	0	0
$121$	7.71229e19	1.26102
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.38252e19	0.886256
$126$	0	0
$127$	9.31085e19	0.961289	0.480645	−	0.876915i	$-0.340403\pi$
$127$	9.31085e19	0.961289	0.480645	+	0.876915i	$0.340403\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.87606e19	−0.144268	−0.0721340	−	0.997395i	$-0.522981\pi$
$131$	−1.87606e19	−0.144268	−0.0721340	+	0.997395i	$0.522981\pi$
$132$	0	0
$133$	−4.05162e20	−2.69798
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.84318e20	0.926240	0.463120	−	0.886296i	$-0.346730\pi$
$137$	1.84318e20	0.926240	0.463120	+	0.886296i	$0.346730\pi$
$138$	0	0
$139$	−1.59539e20	−0.698597	−0.349299	−	0.937011i	$-0.613580\pi$
$139$	−1.59539e20	−0.698597	−0.349299	+	0.937011i	$0.613580\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.87065e19	−0.0960038
$144$	0	0
$145$	−1.06293e20	−0.311537
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.21712e20	−0.728111	−0.364055	−	0.931377i	$-0.618608\pi$
$149$	−3.21712e20	−0.728111	−0.364055	+	0.931377i	$0.618608\pi$
$150$	0	0
$151$	−1.55272e20	−0.309607	−0.154804	−	0.987945i	$-0.549474\pi$
$151$	−1.55272e20	−0.309607	−0.154804	+	0.987945i	$0.549474\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.51587e20	0.702410
$156$	0	0
$157$	−4.54202e20	−0.625464	−0.312732	−	0.949841i	$-0.601244\pi$
$157$	−4.54202e20	−0.625464	−0.312732	+	0.949841i	$0.601244\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.02346e20	0.110975
$162$	0	0
$163$	7.72395e20	0.744830	0.372415	−	0.928066i	$-0.378530\pi$
$163$	7.72395e20	0.744830	0.372415	+	0.928066i	$0.378530\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.02583e21	−0.785725	−0.392862	−	0.919597i	$-0.628515\pi$
$167$	−1.02583e21	−0.785725	−0.392862	+	0.919597i	$0.628515\pi$
$168$	0	0
$169$	−1.45596e21	−0.995924
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.92487e20	−0.379292	−0.189646	−	0.981853i	$-0.560734\pi$
$173$	−6.92487e20	−0.379292	−0.189646	+	0.981853i	$0.560734\pi$
$174$	0	0
$175$	−2.27475e21	−1.11705
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.42385e21	−0.960290	−0.480145	−	0.877189i	$-0.659416\pi$
$179$	−2.42385e21	−0.960290	−0.480145	+	0.877189i	$0.659416\pi$
$180$	0	0
$181$	−9.72457e20	−0.346676	−0.173338	−	0.984862i	$-0.555455\pi$
$181$	−9.72457e20	−0.346676	−0.173338	+	0.984862i	$0.555455\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.99056e21	−0.576558
$186$	0	0
$187$	−7.03742e21	−1.84043
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.76037e21	−1.44595	−0.722975	−	0.690874i	$-0.757226\pi$
$191$	−6.76037e21	−1.44595	−0.722975	+	0.690874i	$0.757226\pi$
$192$	0	0
$193$	1.53000e21	0.296413	0.148206	−	0.988956i	$-0.452650\pi$
$193$	1.53000e21	0.296413	0.148206	+	0.988956i	$0.452650\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.79113e21	1.08271	0.541356	−	0.840793i	$-0.317911\pi$
$197$	6.79113e21	1.08271	0.541356	+	0.840793i	$0.317911\pi$
$198$	0	0
$199$	9.45465e21	1.36943	0.684717	−	0.728809i	$-0.259926\pi$
$199$	9.45465e21	1.36943	0.684717	+	0.728809i	$0.259926\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.69624e21	0.922713
$204$	0	0
$205$	−1.02510e21	−0.111971
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.95949e22	−2.69046
$210$	0	0
$211$	−2.14995e22	−1.78544	−0.892721	−	0.450610i	$-0.851207\pi$
$211$	−2.14995e22	−1.78544	−0.892721	+	0.450610i	$0.851207\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.92601e21	0.342239
$216$	0	0
$217$	−3.26975e22	−2.08040
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.46093e21	0.0781452
$222$	0	0
$223$	−3.89942e22	−1.91471	−0.957356	−	0.288910i	$-0.906707\pi$
$223$	−3.89942e22	−1.91471	−0.957356	+	0.288910i	$0.906707\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.69117e21	−0.0701361	−0.0350680	−	0.999385i	$-0.511165\pi$
$227$	−1.69117e21	−0.0701361	−0.0350680	+	0.999385i	$0.511165\pi$
$228$	0	0
$229$	3.34630e21	0.127681	0.0638407	−	0.997960i	$-0.479665\pi$
$229$	3.34630e21	0.127681	0.0638407	+	0.997960i	$0.479665\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.27609e22	−0.413049	−0.206524	−	0.978441i	$-0.566215\pi$
$233$	−1.27609e22	−0.413049	−0.206524	+	0.978441i	$0.566215\pi$
$234$	0	0
$235$	−2.32688e22	−0.694433
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.15938e21	0.232855	0.116428	−	0.993199i	$-0.462856\pi$
$239$	9.15938e21	0.232855	0.116428	+	0.993199i	$0.462856\pi$
$240$	0	0
$241$	−4.54596e22	−1.06774	−0.533868	−	0.845568i	$-0.679262\pi$
$241$	−4.54596e22	−1.06774	−0.533868	+	0.845568i	$0.679262\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.22806e22	−0.648431
$246$	0	0
$247$	6.14373e21	0.114238
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.68788e22	0.588677	0.294338	−	0.955701i	$-0.404901\pi$
$251$	3.68788e22	0.588677	0.294338	+	0.955701i	$0.404901\pi$
$252$	0	0
$253$	7.47585e21	0.110666
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.84420e22	−0.617813	−0.308907	−	0.951092i	$-0.599963\pi$
$257$	−4.84420e22	−0.617813	−0.308907	+	0.951092i	$0.599963\pi$
$258$	0	0
$259$	1.44128e23	1.70765
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.84645e22	−0.701271	−0.350635	−	0.936512i	$-0.614034\pi$
$263$	−6.84645e22	−0.701271	−0.350635	+	0.936512i	$0.614034\pi$
$264$	0	0
$265$	−7.64226e22	−0.728427
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.17729e23	1.79999	0.899995	−	0.435900i	$-0.143570\pi$
$269$	2.17729e23	1.79999	0.899995	+	0.435900i	$0.143570\pi$
$270$	0	0
$271$	−1.88487e23	−1.45236	−0.726180	−	0.687505i	$-0.758706\pi$
$271$	−1.88487e23	−1.45236	−0.726180	+	0.687505i	$0.758706\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.66158e23	−1.11394
$276$	0	0
$277$	2.07964e23	1.30146	0.650729	−	0.759310i	$-0.274463\pi$
$277$	2.07964e23	1.30146	0.650729	+	0.759310i	$0.274463\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3.33849e23	−1.82322	−0.911612	−	0.411052i	$-0.865162\pi$
$281$	−3.33849e23	−1.82322	−0.911612	+	0.411052i	$0.865162\pi$
$282$	0	0
$283$	−2.26611e22	−0.115694	−0.0578470	−	0.998325i	$-0.518424\pi$
$283$	−2.26611e22	−0.115694	−0.0578470	+	0.998325i	$0.518424\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.42233e22	0.331636
$288$	0	0
$289$	1.19075e23	0.498070
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.85421e23	0.680639	0.340319	−	0.940310i	$-0.389465\pi$
$293$	1.85421e23	0.680639	0.340319	+	0.940310i	$0.389465\pi$
$294$	0	0
$295$	2.03027e23	0.698625
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.55194e21	−0.00469894
$300$	0	0
$301$	−3.56671e23	−1.01365
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.65641e23	0.665958
$306$	0	0
$307$	7.93628e23	1.86983	0.934916	−	0.354870i	$-0.115475\pi$
$307$	7.93628e23	1.86983	0.934916	+	0.354870i	$0.115475\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.20440e23	1.08429	0.542147	−	0.840284i	$-0.317612\pi$
$311$	5.20440e23	1.08429	0.542147	+	0.840284i	$0.317612\pi$
$312$	0	0
$313$	−1.65345e23	−0.324130	−0.162065	−	0.986780i	$-0.551815\pi$
$313$	−1.65345e23	−0.324130	−0.162065	+	0.986780i	$0.551815\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.50226e23	−0.956044	−0.478022	−	0.878348i	$-0.658646\pi$
$317$	−5.50226e23	−0.956044	−0.478022	+	0.878348i	$0.658646\pi$
$318$	0	0
$319$	5.62169e23	0.920143
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.50614e24	2.18998
$324$	0	0
$325$	3.44934e22	0.0472982
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.68480e24	2.05677
$330$	0	0
$331$	5.52973e23	0.637291	0.318646	−	0.947874i	$-0.396772\pi$
$331$	5.52973e23	0.637291	0.318646	+	0.947874i	$0.396772\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.42102e23	0.454565
$336$	0	0
$337$	5.19323e23	0.504607	0.252303	−	0.967648i	$-0.418812\pi$
$337$	5.19323e23	0.504607	0.252303	+	0.967648i	$0.418812\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.38837e24	−2.07461
$342$	0	0
$343$	5.02207e23	0.412657
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.43821e24	−1.05850	−0.529250	−	0.848466i	$-0.677527\pi$
$347$	−1.43821e24	−1.05850	−0.529250	+	0.848466i	$0.677527\pi$
$348$	0	0
$349$	−8.46494e23	−0.589905	−0.294952	−	0.955512i	$-0.595304\pi$
$349$	−8.46494e23	−0.589905	−0.294952	+	0.955512i	$0.595304\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.34329e24	−0.840063	−0.420031	−	0.907510i	$-0.637981\pi$
$353$	−1.34329e24	−0.840063	−0.420031	+	0.907510i	$0.637981\pi$
$354$	0	0
$355$	−3.08752e23	−0.182996
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.45409e24	0.774805	0.387403	−	0.921911i	$-0.373372\pi$
$359$	1.45409e24	0.774805	0.387403	+	0.921911i	$0.373372\pi$
$360$	0	0
$361$	4.35543e24	2.20147
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.08067e22	0.0367845
$366$	0	0
$367$	−1.21387e24	−0.524620	−0.262310	−	0.964984i	$-0.584484\pi$
$367$	−1.21387e24	−0.524620	−0.262310	+	0.964984i	$0.584484\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.53343e24	2.15746
$372$	0	0
$373$	3.39082e24	1.25624	0.628118	−	0.778118i	$-0.283826\pi$
$373$	3.39082e24	1.25624	0.628118	+	0.778118i	$0.283826\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.16703e23	−0.0390696
$378$	0	0
$379$	4.13450e24	1.31629	0.658144	−	0.752892i	$-0.271342\pi$
$379$	4.13450e24	1.31629	0.658144	+	0.752892i	$0.271342\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.88317e24	−1.11892	−0.559459	−	0.828858i	$-0.688991\pi$
$383$	−3.88317e24	−1.11892	−0.559459	+	0.828858i	$0.688991\pi$
$384$	0	0
$385$	−4.20921e24	−1.15431
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.58080e24	−0.641553	−0.320777	−	0.947155i	$-0.603944\pi$
$389$	−2.58080e24	−0.641553	−0.320777	+	0.947155i	$0.603944\pi$
$390$	0	0
$391$	−3.80460e23	−0.0900802
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9.69683e23	0.208429
$396$	0	0
$397$	−4.11532e24	−0.843129	−0.421564	−	0.906799i	$-0.638519\pi$
$397$	−4.11532e24	−0.843129	−0.421564	+	0.906799i	$0.638519\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.60175e24	−1.22967	−0.614833	−	0.788657i	$-0.710777\pi$
$401$	−6.60175e24	−1.22967	−0.614833	+	0.788657i	$0.710777\pi$
$402$	0	0
$403$	4.95813e23	0.0880886
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.05277e25	1.70289
$408$	0	0
$409$	−1.13278e25	−1.74893	−0.874467	−	0.485084i	$-0.838789\pi$
$409$	−1.13278e25	−1.74893	−0.874467	+	0.485084i	$0.838789\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.47003e25	−2.06919
$414$	0	0
$415$	1.00585e24	0.135231
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.26882e25	1.55729	0.778643	−	0.627468i	$-0.215908\pi$
$419$	1.26882e25	1.55729	0.778643	+	0.627468i	$0.215908\pi$
$420$	0	0
$421$	−2.67974e24	−0.314349	−0.157175	−	0.987571i	$-0.550239\pi$
$421$	−2.67974e24	−0.314349	−0.157175	+	0.987571i	$0.550239\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.45608e24	0.906722
$426$	0	0
$427$	−1.92339e25	−1.97244
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.37367e24	−0.692069	−0.346035	−	0.938222i	$-0.612472\pi$
$431$	−7.37367e24	−0.692069	−0.346035	+	0.938222i	$0.612472\pi$
$432$	0	0
$433$	−1.67233e25	−1.50206	−0.751028	−	0.660271i	$-0.770442\pi$
$433$	−1.67233e25	−1.50206	−0.751028	+	0.660271i	$0.770442\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.59997e24	−0.131686
$438$	0	0
$439$	1.48622e25	1.17130	0.585651	−	0.810563i	$-0.300839\pi$
$439$	1.48622e25	1.17130	0.585651	+	0.810563i	$0.300839\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.05393e24	−0.510030	−0.255015	−	0.966937i	$-0.582080\pi$
$443$	−7.05393e24	−0.510030	−0.255015	+	0.966937i	$0.582080\pi$
$444$	0	0
$445$	7.35126e24	0.509262
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.86136e24	−0.436586	−0.218293	−	0.975883i	$-0.570049\pi$
$449$	−6.86136e24	−0.436586	−0.218293	+	0.975883i	$0.570049\pi$
$450$	0	0
$451$	5.42161e24	0.330713
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8.73807e23	0.0490125
$456$	0	0
$457$	−1.33077e25	−0.715975	−0.357988	−	0.933726i	$-0.616537\pi$
$457$	−1.33077e25	−0.715975	−0.357988	+	0.933726i	$0.616537\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.20176e25	−1.09046	−0.545231	−	0.838286i	$-0.683558\pi$
$461$	−2.20176e25	−1.09046	−0.545231	+	0.838286i	$0.683558\pi$
$462$	0	0
$463$	−2.34544e25	−1.11482	−0.557410	−	0.830237i	$-0.688205\pi$
$463$	−2.34544e25	−1.11482	−0.557410	+	0.830237i	$0.688205\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.76282e25	−0.772143	−0.386071	−	0.922469i	$-0.626168\pi$
$467$	−1.76282e25	−0.772143	−0.386071	+	0.922469i	$0.626168\pi$
$468$	0	0
$469$	−3.20107e25	−1.34633
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.60529e25	−1.01082
$474$	0	0
$475$	3.55609e25	1.32551
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.55873e25	0.880719	0.440359	−	0.897822i	$-0.354851\pi$
$479$	2.55873e25	0.880719	0.440359	+	0.897822i	$0.354851\pi$
$480$	0	0
$481$	−2.18550e24	−0.0723056
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.34197e25	0.410388
$486$	0	0
$487$	1.39237e25	0.409478	0.204739	−	0.978817i	$-0.434365\pi$
$487$	1.39237e25	0.409478	0.204739	+	0.978817i	$0.434365\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.82993e24	0.158631	0.0793157	−	0.996850i	$-0.474726\pi$
$491$	5.82993e24	0.158631	0.0793157	+	0.996850i	$0.474726\pi$
$492$	0	0
$493$	−2.86098e25	−0.748978
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.23554e25	0.541997
$498$	0	0
$499$	1.49503e25	0.348895	0.174447	−	0.984666i	$-0.444186\pi$
$499$	1.49503e25	0.348895	0.174447	+	0.984666i	$0.444186\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.92509e25	0.416443	0.208222	−	0.978082i	$-0.433232\pi$
$503$	1.92509e25	0.416443	0.208222	+	0.978082i	$0.433232\pi$
$504$	0	0
$505$	2.02115e25	0.421047
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.27794e25	−1.02011	−0.510053	−	0.860143i	$-0.670374\pi$
$509$	−5.27794e25	−1.02011	−0.510053	+	0.860143i	$0.670374\pi$
$510$	0	0
$511$	−5.85086e24	−0.108949
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.58102e25	−0.446296
$516$	0	0
$517$	1.23065e26	2.05104
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.00961e25	−0.621074	−0.310537	−	0.950561i	$-0.600509\pi$
$521$	−4.00961e25	−0.621074	−0.310537	+	0.950561i	$0.600509\pi$
$522$	0	0
$523$	−5.30249e25	−0.791979	−0.395989	−	0.918255i	$-0.629598\pi$
$523$	−5.30249e25	−0.791979	−0.395989	+	0.918255i	$0.629598\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.21549e26	1.68869
$528$	0	0
$529$	−7.42113e25	−0.994583
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.12549e24	−0.0140422
$534$	0	0
$535$	2.22259e25	0.267607
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.70727e26	1.91518
$540$	0	0
$541$	1.31942e26	1.42892	0.714461	−	0.699676i	$-0.246672\pi$
$541$	1.31942e26	1.42892	0.714461	+	0.699676i	$0.246672\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9.65363e24	0.0974819
$546$	0	0
$547$	3.18692e25	0.310807	0.155404	−	0.987851i	$-0.450332\pi$
$547$	3.18692e25	0.310807	0.155404	+	0.987851i	$0.450332\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.20314e26	−1.09491
$552$	0	0
$553$	−7.02106e25	−0.617325
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.30222e26	1.06920	0.534600	−	0.845105i	$-0.320462\pi$
$557$	1.30222e26	1.06920	0.534600	+	0.845105i	$0.320462\pi$
$558$	0	0
$559$	5.40843e24	0.0429199
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.14771e23	0.00604235	0.00302118	−	0.999995i	$-0.499038\pi$
$563$	8.14771e23	0.00604235	0.00302118	+	0.999995i	$0.499038\pi$
$564$	0	0
$565$	−2.02162e25	−0.144957
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.13811e26	0.763161	0.381580	−	0.924336i	$-0.375380\pi$
$569$	1.13811e26	0.763161	0.381580	+	0.924336i	$0.375380\pi$
$570$	0	0
$571$	−6.33817e25	−0.411075	−0.205537	−	0.978649i	$-0.565894\pi$
$571$	−6.33817e25	−0.411075	−0.205537	+	0.978649i	$0.565894\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.98289e24	−0.0545220
$576$	0	0
$577$	1.26802e26	0.744654	0.372327	−	0.928102i	$-0.378560\pi$
$577$	1.26802e26	0.744654	0.372327	+	0.928102i	$0.378560\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7.28295e25	−0.400529
$582$	0	0
$583$	4.04187e26	2.15145
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.28792e26	−1.14125	−0.570623	−	0.821212i	$-0.693298\pi$
$587$	−2.28792e26	−1.14125	−0.570623	+	0.821212i	$0.693298\pi$
$588$	0	0
$589$	5.11156e26	2.46864
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.87324e26	0.848348	0.424174	−	0.905581i	$-0.360565\pi$
$593$	1.87324e26	0.848348	0.424174	+	0.905581i	$0.360565\pi$
$594$	0	0
$595$	2.14214e26	0.939587
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.85467e25	0.364433	0.182217	−	0.983258i	$-0.441673\pi$
$599$	8.85467e25	0.364433	0.182217	+	0.983258i	$0.441673\pi$
$600$	0	0
$601$	−3.61951e25	−0.144325	−0.0721625	−	0.997393i	$-0.522990\pi$
$601$	−3.61951e25	−0.144325	−0.0721625	+	0.997393i	$0.522990\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.71477e26	−0.641992
$606$	0	0
$607$	−2.25964e26	−0.819873	−0.409936	−	0.912114i	$-0.634449\pi$
$607$	−2.25964e26	−0.819873	−0.409936	+	0.912114i	$0.634449\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.55476e25	−0.0870882
$612$	0	0
$613$	−3.22922e26	−1.06714	−0.533572	−	0.845755i	$-0.679151\pi$
$613$	−3.22922e26	−1.06714	−0.533572	+	0.845755i	$0.679151\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.57559e26	1.73213	0.866067	−	0.499927i	$-0.166640\pi$
$617$	5.57559e26	1.73213	0.866067	+	0.499927i	$0.166640\pi$
$618$	0	0
$619$	−3.69876e26	−1.11428	−0.557141	−	0.830418i	$-0.688102\pi$
$619$	−3.69876e26	−1.11428	−0.557141	+	0.830418i	$0.688102\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.32273e26	−1.50833
$624$	0	0
$625$	1.05361e26	0.289615
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.35775e26	−1.38612
$630$	0	0
$631$	−1.48829e26	−0.373603	−0.186801	−	0.982398i	$-0.559812\pi$
$631$	−1.48829e26	−0.373603	−0.186801	+	0.982398i	$0.559812\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.07020e26	−0.489397
$636$	0	0
$637$	−3.54419e25	−0.0813192
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.74129e26	0.592657	0.296329	−	0.955086i	$-0.404238\pi$
$641$	2.74129e26	0.592657	0.296329	+	0.955086i	$0.404238\pi$
$642$	0	0
$643$	−2.47762e26	−0.520032	−0.260016	−	0.965604i	$-0.583728\pi$
$643$	−2.47762e26	−0.520032	−0.260016	+	0.965604i	$0.583728\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.36261e26	0.269638	0.134819	−	0.990870i	$-0.456955\pi$
$647$	1.36261e26	0.269638	0.134819	+	0.990870i	$0.456955\pi$
$648$	0	0
$649$	−1.07378e27	−2.06343
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.13257e25	0.111165	0.0555824	−	0.998454i	$-0.482298\pi$
$653$	6.13257e25	0.111165	0.0555824	+	0.998454i	$0.482298\pi$
$654$	0	0
$655$	4.17128e25	0.0734475
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.67926e26	0.445250	0.222625	−	0.974904i	$-0.428537\pi$
$659$	2.67926e26	0.445250	0.222625	+	0.974904i	$0.428537\pi$
$660$	0	0
$661$	3.12853e26	0.505157	0.252578	−	0.967576i	$-0.418721\pi$
$661$	3.12853e26	0.505157	0.252578	+	0.967576i	$0.418721\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.00848e26	1.37355
$666$	0	0
$667$	3.03922e25	0.0450367
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.40493e27	−1.96694
$672$	0	0
$673$	7.60450e26	1.03497	0.517485	−	0.855692i	$-0.326868\pi$
$673$	7.60450e26	1.03497	0.517485	+	0.855692i	$0.326868\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.44198e27	1.85510	0.927550	−	0.373700i	$-0.121911\pi$
$677$	1.44198e27	1.85510	0.927550	+	0.373700i	$0.121911\pi$
$678$	0	0
$679$	−9.71659e26	−1.21549
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.31240e26	−0.983399	−0.491700	−	0.870765i	$-0.663624\pi$
$683$	−8.31240e26	−0.983399	−0.491700	+	0.870765i	$0.663624\pi$
$684$	0	0
$685$	−4.09818e26	−0.471553
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.39069e25	−0.0913514
$690$	0	0
$691$	−2.10819e26	−0.223290	−0.111645	−	0.993748i	$-0.535612\pi$
$691$	−2.10819e26	−0.223290	−0.111645	+	0.993748i	$0.535612\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.54723e26	0.355659
$696$	0	0
$697$	−2.75915e26	−0.269194
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9.41456e26	−0.869919	−0.434959	−	0.900450i	$-0.643237\pi$
$701$	−9.41456e26	−0.869919	−0.434959	+	0.900450i	$0.643237\pi$
$702$	0	0
$703$	−2.25313e27	−2.02633
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.46343e27	−1.24706
$708$	0	0
$709$	−1.83074e27	−1.51875	−0.759376	−	0.650652i	$-0.774495\pi$
$709$	−1.83074e27	−1.51875	−0.759376	+	0.650652i	$0.774495\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.29121e26	−0.101542
$714$	0	0
$715$	6.38268e25	0.0488760
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.73532e27	−1.26025	−0.630124	−	0.776495i	$-0.716996\pi$
$719$	−1.73532e27	−1.26025	−0.630124	+	0.776495i	$0.716996\pi$
$720$	0	0
$721$	1.86880e27	1.32184
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.75495e26	−0.453326
$726$	0	0
$727$	5.33232e26	0.348610	0.174305	−	0.984692i	$-0.444232\pi$
$727$	5.33232e26	0.348610	0.174305	+	0.984692i	$0.444232\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.32588e27	0.822790
$732$	0	0
$733$	3.90931e26	0.236381	0.118190	−	0.992991i	$-0.462291\pi$
$733$	3.90931e26	0.236381	0.118190	+	0.992991i	$0.462291\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.33820e27	−1.34258
$738$	0	0
$739$	7.87032e26	0.440423	0.220212	−	0.975452i	$-0.429325\pi$
$739$	7.87032e26	0.440423	0.220212	+	0.975452i	$0.429325\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.68492e27	1.95900	0.979500	−	0.201445i	$-0.0645637\pi$
$743$	3.68492e27	1.95900	0.979500	+	0.201445i	$0.0645637\pi$
$744$	0	0
$745$	7.15302e26	0.370685
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.60928e27	−0.792600
$750$	0	0
$751$	3.86208e26	0.185457	0.0927283	−	0.995691i	$-0.470441\pi$
$751$	3.86208e26	0.185457	0.0927283	+	0.995691i	$0.470441\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.45235e26	0.157623
$756$	0	0
$757$	3.93310e27	1.75116	0.875578	−	0.483078i	$-0.160481\pi$
$757$	3.93310e27	1.75116	0.875578	+	0.483078i	$0.160481\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.44215e27	0.610742	0.305371	−	0.952234i	$-0.401220\pi$
$761$	1.44215e27	0.610742	0.305371	+	0.952234i	$0.401220\pi$
$762$	0	0
$763$	−6.98978e26	−0.288722
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.22910e26	0.0876139
$768$	0	0
$769$	−4.77621e27	−1.83140	−0.915701	−	0.401861i	$-0.868364\pi$
$769$	−4.77621e27	−1.83140	−0.915701	+	0.401861i	$0.868364\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.06095e27	1.11725	0.558625	−	0.829420i	$-0.311329\pi$
$773$	3.06095e27	1.11725	0.558625	+	0.829420i	$0.311329\pi$
$774$	0	0
$775$	2.86984e27	1.02210
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.16032e27	−0.393526
$780$	0	0
$781$	1.63294e27	0.540488
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.00988e27	0.318427
$786$	0	0
$787$	2.46755e27	0.759461	0.379731	−	0.925097i	$-0.376017\pi$
$787$	2.46755e27	0.759461	0.379731	+	0.925097i	$0.376017\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.46377e27	0.429334
$792$	0	0
$793$	2.91656e26	0.0835171
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.72784e27	1.29065	0.645326	−	0.763907i	$-0.276722\pi$
$797$	4.72784e27	1.29065	0.645326	+	0.763907i	$0.276722\pi$
$798$	0	0
$799$	−6.26301e27	−1.66951
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.27374e26	−0.108645
$804$	0	0
$805$	−2.27560e26	−0.0564981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4.73872e27	−1.12241	−0.561203	−	0.827678i	$-0.689661\pi$
$809$	−4.73872e27	−1.12241	−0.561203	+	0.827678i	$0.689661\pi$
$810$	0	0
$811$	−1.55266e27	−0.359234	−0.179617	−	0.983737i	$-0.557486\pi$
$811$	−1.55266e27	−0.359234	−0.179617	+	0.983737i	$0.557486\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.71736e27	−0.379197
$816$	0	0
$817$	5.57580e27	1.20281
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.90129e27	1.62719	0.813595	−	0.581432i	$-0.197507\pi$
$821$	7.90129e27	1.62719	0.813595	+	0.581432i	$0.197507\pi$
$822$	0	0
$823$	7.55379e27	1.52008	0.760040	−	0.649877i	$-0.225180\pi$
$823$	7.55379e27	1.52008	0.760040	+	0.649877i	$0.225180\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.52722e27	1.83090	0.915448	−	0.402436i	$-0.131836\pi$
$827$	9.52722e27	1.83090	0.915448	+	0.402436i	$0.131836\pi$
$828$	0	0
$829$	1.29069e27	0.242412	0.121206	−	0.992627i	$-0.461324\pi$
$829$	1.29069e27	0.242412	0.121206	+	0.992627i	$0.461324\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.68861e27	−1.55892
$834$	0	0
$835$	2.28086e27	0.400016
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.22538e27	−1.54613	−0.773065	−	0.634327i	$-0.781277\pi$
$839$	−9.22538e27	−1.54613	−0.773065	+	0.634327i	$0.781277\pi$
$840$	0	0
$841$	−3.81783e27	−0.625540
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.23722e27	0.507030
$846$	0	0
$847$	1.24159e28	1.90145
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.69154e26	0.0833487
$852$	0	0
$853$	−6.63470e27	−0.950180	−0.475090	−	0.879937i	$-0.657584\pi$
$853$	−6.63470e27	−0.950180	−0.475090	+	0.879937i	$0.657584\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.92147e27	−0.400206	−0.200103	−	0.979775i	$-0.564128\pi$
$857$	−2.92147e27	−0.400206	−0.200103	+	0.979775i	$0.564128\pi$
$858$	0	0
$859$	8.47684e27	1.13579	0.567896	−	0.823100i	$-0.307757\pi$
$859$	8.47684e27	1.13579	0.567896	+	0.823100i	$0.307757\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.99055e27	1.28082	0.640408	−	0.768035i	$-0.278765\pi$
$863$	9.99055e27	1.28082	0.640408	+	0.768035i	$0.278765\pi$
$864$	0	0
$865$	1.53969e27	0.193099
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5.12850e27	−0.615605
$870$	0	0
$871$	4.85398e26	0.0570066
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.18850e28	1.33636
$876$	0	0
$877$	−3.66387e27	−0.403129	−0.201564	−	0.979475i	$-0.564603\pi$
$877$	−3.66387e27	−0.403129	−0.201564	+	0.979475i	$0.564603\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−3.90178e27	−0.411141	−0.205571	−	0.978642i	$-0.565905\pi$
$881$	−3.90178e27	−0.411141	−0.205571	+	0.978642i	$0.565905\pi$
$882$	0	0
$883$	−6.02824e27	−0.621676	−0.310838	−	0.950463i	$-0.600610\pi$
$883$	−6.02824e27	−0.621676	−0.310838	+	0.950463i	$0.600610\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.43600e28	1.41867	0.709333	−	0.704874i	$-0.248996\pi$
$887$	1.43600e28	1.41867	0.709333	+	0.704874i	$0.248996\pi$
$888$	0	0
$889$	1.49894e28	1.44950
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.63382e28	−2.44061
$894$	0	0
$895$	5.38925e27	0.488888
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9.70964e27	−0.844279
$900$	0	0
$901$	−2.05698e28	−1.75124
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.16218e27	0.176495
$906$	0	0
$907$	8.84058e27	0.706662	0.353331	−	0.935498i	$-0.385049\pi$
$907$	8.84058e27	0.706662	0.353331	+	0.935498i	$0.385049\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.77289e28	−1.35912	−0.679561	−	0.733619i	$-0.737830\pi$
$911$	−1.77289e28	−1.35912	−0.679561	+	0.733619i	$0.737830\pi$
$912$	0	0
$913$	−5.31979e27	−0.399413
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.02024e27	−0.217537
$918$	0	0
$919$	−1.67320e28	−1.18046	−0.590230	−	0.807235i	$-0.700963\pi$
$919$	−1.67320e28	−1.18046	−0.590230	+	0.807235i	$0.700963\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.38989e26	−0.0229493
$924$	0	0
$925$	−1.26500e28	−0.838965
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.04557e28	1.30216	0.651082	−	0.759008i	$-0.274315\pi$
$929$	2.04557e28	1.30216	0.651082	+	0.759008i	$0.274315\pi$
$930$	0	0
$931$	−3.65387e28	−2.27893
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.56472e28	0.936970
$936$	0	0
$937$	−2.17132e27	−0.127408	−0.0637042	−	0.997969i	$-0.520291\pi$
$937$	−2.17132e27	−0.127408	−0.0637042	+	0.997969i	$0.520291\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8.37859e27	−0.472138	−0.236069	−	0.971736i	$-0.575859\pi$
$941$	−8.37859e27	−0.472138	−0.236069	+	0.971736i	$0.575859\pi$
$942$	0	0
$943$	2.93105e26	0.0161868
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.69589e28	−0.899648	−0.449824	−	0.893117i	$-0.648513\pi$
$947$	−1.69589e28	−0.899648	−0.449824	+	0.893117i	$0.648513\pi$
$948$	0	0
$949$	8.87204e25	0.00461311
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.53298e27	−0.176506	−0.0882529	−	0.996098i	$-0.528128\pi$
$953$	−3.53298e27	−0.176506	−0.0882529	+	0.996098i	$0.528128\pi$
$954$	0	0
$955$	1.50312e28	0.736140
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.96732e28	1.39665
$960$	0	0
$961$	1.95807e28	0.903560
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.40184e27	−0.150905
$966$	0	0
$967$	−9.62859e27	−0.418805	−0.209402	−	0.977830i	$-0.567152\pi$
$967$	−9.62859e27	−0.418805	−0.209402	+	0.977830i	$0.567152\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.59014e28	1.08328	0.541640	−	0.840611i	$-0.317804\pi$
$971$	2.59014e28	1.08328	0.541640	+	0.840611i	$0.317804\pi$
$972$	0	0
$973$	−2.56840e28	−1.05339
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.99288e28	−1.57503	−0.787514	−	0.616297i	$-0.788632\pi$
$977$	−3.99288e28	−1.57503	−0.787514	+	0.616297i	$0.788632\pi$
$978$	0	0
$979$	−3.88796e28	−1.50413
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.74542e27	−0.139393	−0.0696966	−	0.997568i	$-0.522203\pi$
$983$	−3.74542e27	−0.139393	−0.0696966	+	0.997568i	$0.522203\pi$
$984$	0	0
$985$	−1.50996e28	−0.551214
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.40848e27	−0.0494750
$990$	0	0
$991$	−3.40811e28	−1.17440	−0.587198	−	0.809443i	$-0.699769\pi$
$991$	−3.40811e28	−1.17440	−0.587198	+	0.809443i	$0.699769\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.10217e28	−0.697186
$996$	0	0
$997$	−4.93862e27	−0.160695	−0.0803473	−	0.996767i	$-0.525603\pi$
$997$	−4.93862e27	−0.160695	−0.0803473	+	0.996767i	$0.525603\pi$
$998$	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.1.1		2
3.2	odd	2		8.20.a.a.1.1	✓	2
12.11	even	2		16.20.a.e.1.2		2
24.5	odd	2		64.20.a.k.1.2		2
24.11	even	2		64.20.a.j.1.1		2

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

\(f(q)\)	\(=\)	\(q-2.22342e6 q^{5} +1.60989e8 q^{7} +O(q^{10})\)	\(q-2.22342e6 q^{5} +1.60989e8 q^{7} +1.17593e10 q^{11} -2.44117e9 q^{13} -5.98454e11 q^{17} -2.51671e12 q^{19} +6.35737e11 q^{23} -1.41299e13 q^{25} +4.78062e13 q^{29} -2.03104e14 q^{31} -3.57946e14 q^{35} +8.95266e14 q^{37} +4.61047e14 q^{41} -2.21551e15 q^{43} +1.04653e16 q^{47} +1.45184e16 q^{49} +3.43716e16 q^{53} -2.61460e16 q^{55} -9.13126e16 q^{59} -1.19474e17 q^{61} +5.42776e15 q^{65} -1.98838e17 q^{67} +1.38863e17 q^{71} -3.63434e16 q^{73} +1.89312e18 q^{77} -4.36121e17 q^{79} -4.52389e17 q^{83} +1.33062e18 q^{85} -3.30628e18 q^{89} -3.93001e17 q^{91} +5.59572e18 q^{95} -6.03558e18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 1226620 q^{5} + 88510512 q^{7}+O(q^{10}) \) 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

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Groups

Database

Properties

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