LMFDB - Embedded newform 72.22.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,22,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, 22, names="a")

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

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 22 $
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:	$201.223687887$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{537541}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 134385 $ x^2 - x - 134385
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2}\cdot 7 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-366.086$ 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)

$f(q)$	$=$	$q+3.80644e6 q^{5} -1.01851e9 q^{7} +O(q^{10})$	$q+3.80644e6 q^{5} -1.01851e9 q^{7} -1.06383e11 q^{11} +3.96922e11 q^{13} +3.67718e12 q^{17} -7.39341e12 q^{19} -2.08605e14 q^{23} -4.62348e14 q^{25} +1.10405e15 q^{29} -7.87819e15 q^{31} -3.87689e15 q^{35} +1.14668e16 q^{37} -9.57394e15 q^{41} -1.54549e17 q^{43} -2.63961e17 q^{47} +4.78814e17 q^{49} +1.43334e18 q^{53} -4.04939e17 q^{55} +5.89337e18 q^{59} +2.96513e18 q^{61} +1.51086e18 q^{65} -1.75878e19 q^{67} -3.35158e19 q^{71} -6.12183e19 q^{73} +1.08352e20 q^{77} -5.83561e19 q^{79} +1.25379e20 q^{83} +1.39970e19 q^{85} +4.73528e20 q^{89} -4.04268e20 q^{91} -2.81426e19 q^{95} -2.83797e20 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 21948620 q^{5} - 659451408 q^{7}+O(q^{10}) $ 2 * q - 21948620 * q^5 - 659451408 * q^7	$ 2 q - 21948620 q^{5} - 659451408 q^{7} + 29910896872 q^{11} + 4468866812 q^{13} + 17665404721820 q^{17} - 22467979297496 q^{19} - 120995273049584 q^{23} - 275862509554850 q^{25} + 33\!\cdots\!84 q^{29}+ \cdots - 12\!\cdots\!20 q^{97}+O(q^{100}) $ 2 * q - 21948620 * q^5 - 659451408 * q^7 + 29910896872 * q^11 + 4468866812 * q^13 + 17665404721820 * q^17 - 22467979297496 * q^19 - 120995273049584 * q^23 - 275862509554850 * q^25 + 3318133806711684 * q^29 - 6983846005801568 * q^31 - 13124419284174240 * q^35 - 23423034370792788 * q^37 - 152179406345334996 * q^41 + 35137855888657880 * q^43 + 84350534118008928 * q^47 + 49189617082449970 * q^49 - 389728341256520684 * q^53 - 3915186255016270960 * q^55 + 3799253986972943944 * q^59 + 1500415393700389916 * q^61 + 11618501316836358040 * q^65 - 15727134260135911736 * q^67 + 15483319715102645936 * q^71 - 75370240511639281708 * q^73 + 157288770168613331136 * q^77 - 53554722400730974336 * q^79 + 299101821652772652632 * q^83 - 346270420314818644520 * q^85 + 614802488682208004364 * q^89 - 545181052826294967264 * q^91 + 360103706555542618640 * q^95 - 1201555028804369106620 * 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
$5$	3.80644e6	0.174314	0.0871572	−	0.996195i	$-0.472222\pi$
$5$	3.80644e6	0.174314	0.0871572	+	0.996195i	$0.472222\pi$
$6$	0	0
$7$	−1.01851e9	−1.36281	−0.681405	−	0.731907i	$-0.738631\pi$
$7$	−1.01851e9	−1.36281	−0.681405	+	0.731907i	$0.738631\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.06383e11	−1.23665	−0.618326	−	0.785922i	$-0.712189\pi$
$11$	−1.06383e11	−1.23665	−0.618326	+	0.785922i	$0.712189\pi$
$12$	0	0
$13$	3.96922e11	0.798545	0.399273	−	0.916832i	$-0.369263\pi$
$13$	3.96922e11	0.798545	0.399273	+	0.916832i	$0.369263\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.67718e12	0.442386	0.221193	−	0.975230i	$-0.429005\pi$
$17$	3.67718e12	0.442386	0.221193	+	0.975230i	$0.429005\pi$
$18$	0	0
$19$	−7.39341e12	−0.276651	−0.138325	−	0.990387i	$-0.544172\pi$
$19$	−7.39341e12	−0.276651	−0.138325	+	0.990387i	$0.544172\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.08605e14	−1.04998	−0.524992	−	0.851107i	$-0.675932\pi$
$23$	−2.08605e14	−1.04998	−0.524992	+	0.851107i	$0.675932\pi$
$24$	0	0
$25$	−4.62348e14	−0.969614
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.10405e15	0.487314	0.243657	−	0.969862i	$-0.421653\pi$
$29$	1.10405e15	0.487314	0.243657	+	0.969862i	$0.421653\pi$
$30$	0	0
$31$	−7.87819e15	−1.72635	−0.863175	−	0.504905i	$-0.831528\pi$
$31$	−7.87819e15	−1.72635	−0.863175	+	0.504905i	$0.831528\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.87689e15	−0.237557
$36$	0	0
$37$	1.14668e16	0.392034	0.196017	−	0.980601i	$-0.437199\pi$
$37$	1.14668e16	0.392034	0.196017	+	0.980601i	$0.437199\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.57394e15	−0.111394	−0.0556968	−	0.998448i	$-0.517738\pi$
$41$	−9.57394e15	−0.111394	−0.0556968	+	0.998448i	$0.517738\pi$
$42$	0	0
$43$	−1.54549e17	−1.09056	−0.545278	−	0.838255i	$-0.683576\pi$
$43$	−1.54549e17	−1.09056	−0.545278	+	0.838255i	$0.683576\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.63961e17	−0.732001	−0.366001	−	0.930615i	$-0.619273\pi$
$47$	−2.63961e17	−0.732001	−0.366001	+	0.930615i	$0.619273\pi$
$48$	0	0
$49$	4.78814e17	0.857250
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.43334e18	1.12577	0.562887	−	0.826534i	$-0.309691\pi$
$53$	1.43334e18	1.12577	0.562887	+	0.826534i	$0.309691\pi$
$54$	0	0
$55$	−4.04939e17	−0.215566
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.89337e18	1.50113	0.750564	−	0.660798i	$-0.229782\pi$
$59$	5.89337e18	1.50113	0.750564	+	0.660798i	$0.229782\pi$
$60$	0	0
$61$	2.96513e18	0.532207	0.266104	−	0.963944i	$-0.414264\pi$
$61$	2.96513e18	0.532207	0.266104	+	0.963944i	$0.414264\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.51086e18	0.139198
$66$	0	0
$67$	−1.75878e19	−1.17876	−0.589380	−	0.807856i	$-0.700628\pi$
$67$	−1.75878e19	−1.17876	−0.589380	+	0.807856i	$0.700628\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.35158e19	−1.22190	−0.610952	−	0.791668i	$-0.709213\pi$
$71$	−3.35158e19	−1.22190	−0.610952	+	0.791668i	$0.709213\pi$
$72$	0	0
$73$	−6.12183e19	−1.66721	−0.833607	−	0.552358i	$-0.813728\pi$
$73$	−6.12183e19	−1.66721	−0.833607	+	0.552358i	$0.813728\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.08352e20	1.68532
$78$	0	0
$79$	−5.83561e19	−0.693429	−0.346714	−	0.937971i	$-0.612703\pi$
$79$	−5.83561e19	−0.693429	−0.346714	+	0.937971i	$0.612703\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.25379e20	0.886958	0.443479	−	0.896285i	$-0.353744\pi$
$83$	1.25379e20	0.886958	0.443479	+	0.896285i	$0.353744\pi$
$84$	0	0
$85$	1.39970e19	0.0771143
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.73528e20	1.60972	0.804860	−	0.593465i	$-0.202240\pi$
$89$	4.73528e20	1.60972	0.804860	+	0.593465i	$0.202240\pi$
$90$	0	0
$91$	−4.04268e20	−1.08827
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.81426e19	−0.0482243
$96$	0	0
$97$	−2.83797e20	−0.390756	−0.195378	−	0.980728i	$-0.562593\pi$
$97$	−2.83797e20	−0.390756	−0.195378	+	0.980728i	$0.562593\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.60683e20	−0.414980	−0.207490	−	0.978237i	$-0.566530\pi$
$101$	−4.60683e20	−0.414980	−0.207490	+	0.978237i	$0.566530\pi$
$102$	0	0
$103$	9.01928e20	0.661273	0.330637	−	0.943758i	$-0.392737\pi$
$103$	9.01928e20	0.661273	0.330637	+	0.943758i	$0.392737\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.07173e21	1.01813	0.509065	−	0.860728i	$-0.329991\pi$
$107$	2.07173e21	1.01813	0.509065	+	0.860728i	$0.329991\pi$
$108$	0	0
$109$	−1.82773e21	−0.739492	−0.369746	−	0.929133i	$-0.620555\pi$
$109$	−1.82773e21	−0.739492	−0.369746	+	0.929133i	$0.620555\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.45088e21	−0.679201	−0.339601	−	0.940570i	$-0.610292\pi$
$113$	−2.45088e21	−0.679201	−0.339601	+	0.940570i	$0.610292\pi$
$114$	0	0
$115$	−7.94041e20	−0.183027
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.74524e21	−0.602888
$120$	0	0
$121$	3.91702e21	0.529309
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.57495e21	−0.343332
$126$	0	0
$127$	1.85246e22	1.50595	0.752976	−	0.658048i	$-0.228618\pi$
$127$	1.85246e22	1.50595	0.752976	+	0.658048i	$0.228618\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.23574e22	−1.31242	−0.656210	−	0.754579i	$-0.727841\pi$
$131$	−2.23574e22	−1.31242	−0.656210	+	0.754579i	$0.727841\pi$
$132$	0	0
$133$	7.53025e21	0.377023
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.29591e22	0.842151	0.421075	−	0.907026i	$-0.361653\pi$
$137$	2.29591e22	0.842151	0.421075	+	0.907026i	$0.361653\pi$
$138$	0	0
$139$	4.16594e22	1.31237	0.656187	−	0.754598i	$-0.272168\pi$
$139$	4.16594e22	1.31237	0.656187	+	0.754598i	$0.272168\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.22256e22	−0.987523
$144$	0	0
$145$	4.20248e21	0.0849458
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.33948e22	0.659147	0.329573	−	0.944130i	$-0.393095\pi$
$149$	4.33948e22	0.659147	0.329573	+	0.944130i	$0.393095\pi$
$150$	0	0
$151$	−8.01728e22	−1.05869	−0.529346	−	0.848406i	$-0.677563\pi$
$151$	−8.01728e22	−1.05869	−0.529346	+	0.848406i	$0.677563\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.99878e22	−0.300928
$156$	0	0
$157$	−1.91718e23	−1.68157	−0.840787	−	0.541366i	$-0.817907\pi$
$157$	−1.91718e23	−1.68157	−0.840787	+	0.541366i	$0.817907\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.12466e23	1.43093
$162$	0	0
$163$	1.98873e23	1.17654	0.588269	−	0.808665i	$-0.299809\pi$
$163$	1.98873e23	1.17654	0.588269	+	0.808665i	$0.299809\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.01098e22	−0.183962	−0.0919808	−	0.995761i	$-0.529320\pi$
$167$	−4.01098e22	−0.183962	−0.0919808	+	0.995761i	$0.529320\pi$
$168$	0	0
$169$	−8.95177e22	−0.362325
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.87093e23	0.592344	0.296172	−	0.955135i	$-0.404290\pi$
$173$	1.87093e23	0.592344	0.296172	+	0.955135i	$0.404290\pi$
$174$	0	0
$175$	4.70906e23	1.32140
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.94050e23	1.75748	0.878742	−	0.477298i	$-0.158384\pi$
$179$	7.94050e23	1.75748	0.878742	+	0.477298i	$0.158384\pi$
$180$	0	0
$181$	−1.89362e23	−0.372964	−0.186482	−	0.982458i	$-0.559709\pi$
$181$	−1.89362e23	−0.372964	−0.186482	+	0.982458i	$0.559709\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.36476e22	0.0683371
$186$	0	0
$187$	−3.91189e23	−0.547078
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.17670e24	−1.31770	−0.658848	−	0.752276i	$-0.728956\pi$
$191$	−1.17670e24	−1.31770	−0.658848	+	0.752276i	$0.728956\pi$
$192$	0	0
$193$	1.30716e23	0.131213	0.0656064	−	0.997846i	$-0.479102\pi$
$193$	1.30716e23	0.131213	0.0656064	+	0.997846i	$0.479102\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.43917e23	0.602045	0.301023	−	0.953617i	$-0.402672\pi$
$197$	7.43917e23	0.602045	0.301023	+	0.953617i	$0.402672\pi$
$198$	0	0
$199$	2.02892e24	1.47675	0.738377	−	0.674388i	$-0.235592\pi$
$199$	2.02892e24	1.47675	0.738377	+	0.674388i	$0.235592\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.12448e24	−0.664116
$204$	0	0
$205$	−3.64426e22	−0.0194175
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.86531e23	0.342121
$210$	0	0
$211$	2.73877e24	1.07793	0.538964	−	0.842329i	$-0.318816\pi$
$211$	2.73877e24	1.07793	0.538964	+	0.842329i	$0.318816\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.88281e23	−0.190100
$216$	0	0
$217$	8.02401e24	2.35269
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.45955e24	0.353266
$222$	0	0
$223$	4.35130e24	0.958116	0.479058	−	0.877783i	$-0.340978\pi$
$223$	4.35130e24	0.958116	0.479058	+	0.877783i	$0.340978\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.67094e23	0.140145	0.0700724	−	0.997542i	$-0.477677\pi$
$227$	7.67094e23	0.140145	0.0700724	+	0.997542i	$0.477677\pi$
$228$	0	0
$229$	7.54631e24	1.25737	0.628684	−	0.777661i	$-0.283594\pi$
$229$	7.54631e24	1.25737	0.628684	+	0.777661i	$0.283594\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.10251e24	−0.292079	−0.146039	−	0.989279i	$-0.546653\pi$
$233$	−2.10251e24	−0.292079	−0.146039	+	0.989279i	$0.546653\pi$
$234$	0	0
$235$	−1.00475e24	−0.127598
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.45098e25	1.54342	0.771709	−	0.635976i	$-0.219402\pi$
$239$	1.45098e25	1.54342	0.771709	+	0.635976i	$0.219402\pi$
$240$	0	0
$241$	−3.56087e24	−0.347038	−0.173519	−	0.984831i	$-0.555514\pi$
$241$	−3.56087e24	−0.347038	−0.173519	+	0.984831i	$0.555514\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.82257e24	0.149431
$246$	0	0
$247$	−2.93461e24	−0.220918
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.01810e25	1.28342	0.641710	−	0.766948i	$-0.278225\pi$
$251$	2.01810e25	1.28342	0.641710	+	0.766948i	$0.278225\pi$
$252$	0	0
$253$	2.21920e25	1.29846
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.05388e24	−0.399675	−0.199838	−	0.979829i	$-0.564041\pi$
$257$	−8.05388e24	−0.399675	−0.199838	+	0.979829i	$0.564041\pi$
$258$	0	0
$259$	−1.16790e25	−0.534267
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.45231e25	0.565618	0.282809	−	0.959176i	$-0.408734\pi$
$263$	1.45231e25	0.565618	0.282809	+	0.959176i	$0.408734\pi$
$264$	0	0
$265$	5.45590e24	0.196239
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.59528e25	1.71958	0.859792	−	0.510645i	$-0.170593\pi$
$269$	5.59528e25	1.71958	0.859792	+	0.510645i	$0.170593\pi$
$270$	0	0
$271$	3.65662e25	1.03969	0.519844	−	0.854261i	$-0.325990\pi$
$271$	3.65662e25	1.03969	0.519844	+	0.854261i	$0.325990\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.91858e25	1.19908
$276$	0	0
$277$	8.21467e25	1.85589	0.927946	−	0.372715i	$-0.121573\pi$
$277$	8.21467e25	1.85589	0.927946	+	0.372715i	$0.121573\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9.16736e25	−1.78167	−0.890837	−	0.454324i	$-0.849881\pi$
$281$	−9.16736e25	−1.78167	−0.890837	+	0.454324i	$0.849881\pi$
$282$	0	0
$283$	5.45902e25	0.984821	0.492411	−	0.870363i	$-0.336116\pi$
$283$	5.45902e25	0.984821	0.492411	+	0.870363i	$0.336116\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.75114e24	0.151808
$288$	0	0
$289$	−5.55702e25	−0.804294
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.07858e26	1.35127	0.675635	−	0.737237i	$-0.263870\pi$
$293$	1.07858e26	1.35127	0.675635	+	0.737237i	$0.263870\pi$
$294$	0	0
$295$	2.24327e25	0.261668
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.27999e25	−0.838460
$300$	0	0
$301$	1.57410e26	1.48622
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.12866e25	0.0927714
$306$	0	0
$307$	1.57928e25	0.121201	0.0606007	−	0.998162i	$-0.480698\pi$
$307$	1.57928e25	0.121201	0.0606007	+	0.998162i	$0.480698\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.16533e26	−1.45057	−0.725286	−	0.688447i	$-0.758293\pi$
$311$	−2.16533e26	−1.45057	−0.725286	+	0.688447i	$0.758293\pi$
$312$	0	0
$313$	2.08285e26	1.30450	0.652249	−	0.758005i	$-0.273826\pi$
$313$	2.08285e26	1.30450	0.652249	+	0.758005i	$0.273826\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.51043e26	−0.827900	−0.413950	−	0.910300i	$-0.635851\pi$
$317$	−1.51043e26	−0.827900	−0.413950	+	0.910300i	$0.635851\pi$
$318$	0	0
$319$	−1.17451e26	−0.602638
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.71869e25	−0.122387
$324$	0	0
$325$	−1.83516e26	−0.774281
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.68846e26	0.997579
$330$	0	0
$331$	3.02767e26	1.05418	0.527090	−	0.849809i	$-0.323283\pi$
$331$	3.02767e26	1.05418	0.527090	+	0.849809i	$0.323283\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.69467e25	−0.205475
$336$	0	0
$337$	−5.07259e26	−1.46257	−0.731284	−	0.682073i	$-0.761079\pi$
$337$	−5.07259e26	−1.46257	−0.731284	+	0.682073i	$0.761079\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.38103e26	2.13489
$342$	0	0
$343$	8.12080e25	0.194541
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.92246e26	−0.407753	−0.203876	−	0.978997i	$-0.565354\pi$
$347$	−1.92246e26	−0.407753	−0.203876	+	0.978997i	$0.565354\pi$
$348$	0	0
$349$	−4.69006e26	−0.936507	−0.468254	−	0.883594i	$-0.655117\pi$
$349$	−4.69006e26	−0.936507	−0.468254	+	0.883594i	$0.655117\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.16954e26	−0.915836	−0.457918	−	0.888994i	$-0.651405\pi$
$353$	−5.16954e26	−0.915836	−0.457918	+	0.888994i	$0.651405\pi$
$354$	0	0
$355$	−1.27576e26	−0.212995
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.26652e26	1.22696	0.613481	−	0.789710i	$-0.289769\pi$
$359$	8.26652e26	1.22696	0.613481	+	0.789710i	$0.289769\pi$
$360$	0	0
$361$	−6.59547e26	−0.923464
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.33023e26	−0.290619
$366$	0	0
$367$	−1.35866e27	−1.59999	−0.799995	−	0.600007i	$-0.795165\pi$
$367$	−1.35866e27	−1.59999	−0.799995	+	0.600007i	$0.795165\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.45986e27	−1.53422
$372$	0	0
$373$	−1.22128e27	−1.21303	−0.606517	−	0.795071i	$-0.707434\pi$
$373$	−1.22128e27	−1.21303	−0.606517	+	0.795071i	$0.707434\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.38220e26	0.389142
$378$	0	0
$379$	−6.57479e26	−0.552294	−0.276147	−	0.961115i	$-0.589058\pi$
$379$	−6.57479e26	−0.552294	−0.276147	+	0.961115i	$0.589058\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.90325e27	−1.43188	−0.715942	−	0.698159i	$-0.754003\pi$
$383$	−1.90325e27	−1.43188	−0.715942	+	0.698159i	$0.754003\pi$
$384$	0	0
$385$	4.12433e26	0.293776
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.12948e26	−0.327795	−0.163897	−	0.986477i	$-0.552407\pi$
$389$	−5.12948e26	−0.327795	−0.163897	+	0.986477i	$0.552407\pi$
$390$	0	0
$391$	−7.67079e26	−0.464498
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.22129e26	−0.120875
$396$	0	0
$397$	2.36102e27	1.21843	0.609213	−	0.793007i	$-0.291485\pi$
$397$	2.36102e27	1.21843	0.609213	+	0.793007i	$0.291485\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.27592e27	1.52166	0.760829	−	0.648952i	$-0.224792\pi$
$401$	3.27592e27	1.52166	0.760829	+	0.648952i	$0.224792\pi$
$402$	0	0
$403$	−3.12703e27	−1.37857
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.21987e27	−0.484809
$408$	0	0
$409$	4.16749e27	1.57318	0.786592	−	0.617473i	$-0.211844\pi$
$409$	4.16749e27	1.57318	0.786592	+	0.617473i	$0.211844\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.00245e27	−2.04575
$414$	0	0
$415$	4.77245e26	0.154610
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.60990e27	−0.764497	−0.382249	−	0.924060i	$-0.624850\pi$
$419$	−2.60990e27	−0.764497	−0.382249	+	0.924060i	$0.624850\pi$
$420$	0	0
$421$	−2.10137e27	−0.585518	−0.292759	−	0.956186i	$-0.594573\pi$
$421$	−2.10137e27	−0.585518	−0.292759	+	0.956186i	$0.594573\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.70014e27	−0.428944
$426$	0	0
$427$	−3.02001e27	−0.725297
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.31265e27	−0.721380	−0.360690	−	0.932686i	$-0.617459\pi$
$431$	−3.31265e27	−0.721380	−0.360690	+	0.932686i	$0.617459\pi$
$432$	0	0
$433$	−4.60260e27	−0.954730	−0.477365	−	0.878705i	$-0.658408\pi$
$433$	−4.60260e27	−0.954730	−0.477365	+	0.878705i	$0.658408\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.54230e27	0.290479
$438$	0	0
$439$	−2.40231e27	−0.431273	−0.215637	−	0.976474i	$-0.569183\pi$
$439$	−2.40231e27	−0.431273	−0.215637	+	0.976474i	$0.569183\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.24244e27	1.50851	0.754254	−	0.656583i	$-0.227999\pi$
$443$	9.24244e27	1.50851	0.754254	+	0.656583i	$0.227999\pi$
$444$	0	0
$445$	1.80245e27	0.280597
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.68141e27	0.946851	0.473426	−	0.880834i	$-0.343017\pi$
$449$	6.68141e27	0.946851	0.473426	+	0.880834i	$0.343017\pi$
$450$	0	0
$451$	1.01850e27	0.137755
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.53882e27	−0.189700
$456$	0	0
$457$	1.53406e27	0.180602	0.0903011	−	0.995915i	$-0.471217\pi$
$457$	1.53406e27	0.180602	0.0903011	+	0.995915i	$0.471217\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.04683e27	0.864500	0.432250	−	0.901754i	$-0.357720\pi$
$461$	8.04683e27	0.864500	0.432250	+	0.901754i	$0.357720\pi$
$462$	0	0
$463$	−1.43789e28	−1.47613	−0.738067	−	0.674727i	$-0.764261\pi$
$463$	−1.43789e28	−1.47613	−0.738067	+	0.674727i	$0.764261\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.22594e27	−0.302573	−0.151286	−	0.988490i	$-0.548342\pi$
$467$	−3.22594e27	−0.302573	−0.151286	+	0.988490i	$0.548342\pi$
$468$	0	0
$469$	1.79133e28	1.60642
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.64414e28	1.34864
$474$	0	0
$475$	3.41833e27	0.268245
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.52220e27	0.253099	0.126550	−	0.991960i	$-0.459610\pi$
$479$	3.52220e27	0.253099	0.126550	+	0.991960i	$0.459610\pi$
$480$	0	0
$481$	4.55141e27	0.313057
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.08026e27	−0.0681143
$486$	0	0
$487$	5.18350e27	0.313018	0.156509	−	0.987677i	$-0.449976\pi$
$487$	5.18350e27	0.313018	0.156509	+	0.987677i	$0.449976\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.74020e28	−1.51854	−0.759270	−	0.650776i	$-0.774444\pi$
$491$	−2.74020e28	−1.51854	−0.759270	+	0.650776i	$0.774444\pi$
$492$	0	0
$493$	4.05978e27	0.215581
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.41361e28	1.66522
$498$	0	0
$499$	1.08162e28	0.505848	0.252924	−	0.967486i	$-0.418608\pi$
$499$	1.08162e28	0.505848	0.252924	+	0.967486i	$0.418608\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.89398e28	−1.67467	−0.837337	−	0.546687i	$-0.815889\pi$
$503$	−3.89398e28	−1.67467	−0.837337	+	0.546687i	$0.815889\pi$
$504$	0	0
$505$	−1.75356e27	−0.0723371
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.89861e28	1.10066	0.550329	−	0.834948i	$-0.314502\pi$
$509$	2.89861e28	1.10066	0.550329	+	0.834948i	$0.314502\pi$
$510$	0	0
$511$	6.23514e28	2.27209
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.43313e27	0.115269
$516$	0	0
$517$	2.80808e28	0.905231
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.41212e28	1.01444	0.507222	−	0.861816i	$-0.330673\pi$
$521$	3.41212e28	1.01444	0.507222	+	0.861816i	$0.330673\pi$
$522$	0	0
$523$	−1.37991e28	−0.394078	−0.197039	−	0.980396i	$-0.563133\pi$
$523$	−1.37991e28	−0.394078	−0.197039	+	0.980396i	$0.563133\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.89696e28	−0.763714
$528$	0	0
$529$	4.04446e27	0.102465
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.80011e27	−0.0889529
$534$	0	0
$535$	7.88589e27	0.177475
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.09375e28	−1.06012
$540$	0	0
$541$	5.75791e28	1.15264	0.576320	−	0.817224i	$-0.304488\pi$
$541$	5.75791e28	1.15264	0.576320	+	0.817224i	$0.304488\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.95713e27	−0.128904
$546$	0	0
$547$	7.41040e28	1.32122	0.660609	−	0.750730i	$-0.270298\pi$
$547$	7.41040e28	1.32122	0.660609	+	0.750730i	$0.270298\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.16268e27	−0.134816
$552$	0	0
$553$	5.94362e28	0.945012
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.21534e28	1.06360	0.531799	−	0.846871i	$-0.321516\pi$
$557$	7.21534e28	1.06360	0.531799	+	0.846871i	$0.321516\pi$
$558$	0	0
$559$	−6.13439e28	−0.870859
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.75204e27	0.102112	0.0510561	−	0.998696i	$-0.483741\pi$
$563$	7.75204e27	0.102112	0.0510561	+	0.998696i	$0.483741\pi$
$564$	0	0
$565$	−9.32911e27	−0.118395
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.42718e29	1.68190	0.840952	−	0.541109i	$-0.181995\pi$
$569$	1.42718e29	1.68190	0.840952	+	0.541109i	$0.181995\pi$
$570$	0	0
$571$	−1.19587e29	−1.35833	−0.679165	−	0.733986i	$-0.737658\pi$
$571$	−1.19587e29	−1.35833	−0.679165	+	0.733986i	$0.737658\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9.64481e28	1.01808
$576$	0	0
$577$	−1.02439e29	−1.04261	−0.521304	−	0.853371i	$-0.674554\pi$
$577$	−1.02439e29	−1.04261	−0.521304	+	0.853371i	$0.674554\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.27699e29	−1.20876
$582$	0	0
$583$	−1.52482e29	−1.39219
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.48071e28	−0.720663	−0.360332	−	0.932824i	$-0.617336\pi$
$587$	−8.48071e28	−0.720663	−0.360332	+	0.932824i	$0.617336\pi$
$588$	0	0
$589$	5.82468e28	0.477597
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.14783e29	−0.876601	−0.438300	−	0.898829i	$-0.644419\pi$
$593$	−1.14783e29	−0.876601	−0.438300	+	0.898829i	$0.644419\pi$
$594$	0	0
$595$	−1.42560e28	−0.105092
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.72380e29	1.18442	0.592208	−	0.805785i	$-0.298256\pi$
$599$	1.72380e29	1.18442	0.592208	+	0.805785i	$0.298256\pi$
$600$	0	0
$601$	−2.73184e29	−1.81248	−0.906241	−	0.422761i	$-0.861061\pi$
$601$	−2.73184e29	−1.81248	−0.906241	+	0.422761i	$0.861061\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.49099e28	0.0922662
$606$	0	0
$607$	−2.31744e29	−1.38525	−0.692625	−	0.721298i	$-0.743546\pi$
$607$	−2.31744e29	−1.38525	−0.692625	+	0.721298i	$0.743546\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.04772e29	−0.584536
$612$	0	0
$613$	2.48641e29	1.34041	0.670205	−	0.742176i	$-0.266206\pi$
$613$	2.48641e29	1.34041	0.670205	+	0.742176i	$0.266206\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.47882e29	1.24810	0.624052	−	0.781383i	$-0.285486\pi$
$617$	2.47882e29	1.24810	0.624052	+	0.781383i	$0.285486\pi$
$618$	0	0
$619$	−5.18780e28	−0.252483	−0.126241	−	0.992000i	$-0.540291\pi$
$619$	−5.18780e28	−0.252483	−0.126241	+	0.992000i	$0.540291\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.82292e29	−2.19374
$624$	0	0
$625$	2.06857e29	0.909767
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.21655e28	0.173430
$630$	0	0
$631$	2.91890e29	1.16121	0.580605	−	0.814186i	$-0.302816\pi$
$631$	2.91890e29	1.16121	0.580605	+	0.814186i	$0.302816\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.05129e28	0.262509
$636$	0	0
$637$	1.90051e29	0.684553
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.18026e29	−1.07264	−0.536319	−	0.844015i	$-0.680186\pi$
$641$	−3.18026e29	−1.07264	−0.536319	+	0.844015i	$0.680186\pi$
$642$	0	0
$643$	−4.41840e29	−1.44228	−0.721140	−	0.692790i	$-0.756381\pi$
$643$	−4.41840e29	−1.44228	−0.721140	+	0.692790i	$0.756381\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.85233e28	0.178992	0.0894960	−	0.995987i	$-0.471474\pi$
$647$	5.85233e28	0.178992	0.0894960	+	0.995987i	$0.471474\pi$
$648$	0	0
$649$	−6.26952e29	−1.85637
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.75541e29	−1.87527	−0.937634	−	0.347623i	$-0.886989\pi$
$653$	−6.75541e29	−1.87527	−0.937634	+	0.347623i	$0.886989\pi$
$654$	0	0
$655$	−8.51020e28	−0.228774
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5.16262e28	−0.130189	−0.0650943	−	0.997879i	$-0.520735\pi$
$659$	−5.16262e28	−0.130189	−0.0650943	+	0.997879i	$0.520735\pi$
$660$	0	0
$661$	1.39834e29	0.341583	0.170792	−	0.985307i	$-0.445367\pi$
$661$	1.39834e29	0.341583	0.170792	+	0.985307i	$0.445367\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.86634e28	0.0657205
$666$	0	0
$667$	−2.30310e29	−0.511671
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.15439e29	−0.658155
$672$	0	0
$673$	−8.25080e29	−1.66854	−0.834272	−	0.551353i	$-0.814112\pi$
$673$	−8.25080e29	−1.66854	−0.834272	+	0.551353i	$0.814112\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.97987e29	−1.13635	−0.568173	−	0.822909i	$-0.692350\pi$
$677$	−5.97987e29	−1.13635	−0.568173	+	0.822909i	$0.692350\pi$
$678$	0	0
$679$	2.89050e29	0.532525
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.71862e29	0.990545	0.495272	−	0.868738i	$-0.335068\pi$
$683$	5.71862e29	0.990545	0.495272	+	0.868738i	$0.335068\pi$
$684$	0	0
$685$	8.73925e28	0.146799
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.68922e29	0.898982
$690$	0	0
$691$	9.55681e29	1.46485	0.732426	−	0.680847i	$-0.238388\pi$
$691$	9.55681e29	1.46485	0.732426	+	0.680847i	$0.238388\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.58574e29	0.228766
$696$	0	0
$697$	−3.52052e28	−0.0492790
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.42010e27	−0.00846258	−0.00423129	−	0.999991i	$-0.501347\pi$
$701$	−6.42010e27	−0.00846258	−0.00423129	+	0.999991i	$0.501347\pi$
$702$	0	0
$703$	−8.47787e28	−0.108457
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.69209e29	0.565539
$708$	0	0
$709$	−6.52437e29	−0.763402	−0.381701	−	0.924286i	$-0.624662\pi$
$709$	−6.52437e29	−0.763402	−0.381701	+	0.924286i	$0.624662\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.64343e30	1.81264
$714$	0	0
$715$	−1.60729e29	−0.172140
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.22078e29	0.123306	0.0616530	−	0.998098i	$-0.480363\pi$
$719$	1.22078e29	0.123306	0.0616530	+	0.998098i	$0.480363\pi$
$720$	0	0
$721$	−9.18621e29	−0.901189
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.10454e29	−0.472506
$726$	0	0
$727$	1.04525e30	0.939959	0.469980	−	0.882677i	$-0.344261\pi$
$727$	1.04525e30	0.939959	0.469980	+	0.882677i	$0.344261\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.68306e29	−0.482447
$732$	0	0
$733$	8.69967e29	0.717647	0.358824	−	0.933405i	$-0.383178\pi$
$733$	8.69967e29	0.717647	0.358824	+	0.933405i	$0.383178\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.87103e30	1.45772
$738$	0	0
$739$	1.11587e29	0.0844980	0.0422490	−	0.999107i	$-0.486548\pi$
$739$	1.11587e29	0.0844980	0.0422490	+	0.999107i	$0.486548\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.69344e29	0.693578	0.346789	−	0.937943i	$-0.387272\pi$
$743$	9.69344e29	0.693578	0.346789	+	0.937943i	$0.387272\pi$
$744$	0	0
$745$	1.65180e29	0.114899
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.11007e30	−1.38752
$750$	0	0
$751$	−7.68531e29	−0.491408	−0.245704	−	0.969345i	$-0.579019\pi$
$751$	−7.68531e29	−0.491408	−0.245704	+	0.969345i	$0.579019\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.05173e29	−0.184545
$756$	0	0
$757$	−2.16553e29	−0.127367	−0.0636836	−	0.997970i	$-0.520285\pi$
$757$	−2.16553e29	−0.127367	−0.0636836	+	0.997970i	$0.520285\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.88897e28	0.0550316	0.0275158	−	0.999621i	$-0.491240\pi$
$761$	9.88897e28	0.0550316	0.0275158	+	0.999621i	$0.491240\pi$
$762$	0	0
$763$	1.86156e30	1.00779
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.33921e30	1.19872
$768$	0	0
$769$	6.21082e29	0.309686	0.154843	−	0.987939i	$-0.450513\pi$
$769$	6.21082e29	0.309686	0.154843	+	0.987939i	$0.450513\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3.43242e30	−1.62075	−0.810373	−	0.585914i	$-0.800736\pi$
$773$	−3.43242e30	−1.62075	−0.810373	+	0.585914i	$0.800736\pi$
$774$	0	0
$775$	3.64247e30	1.67389
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.07841e28	0.0308172
$780$	0	0
$781$	3.56550e30	1.51107
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.29761e29	−0.293123
$786$	0	0
$787$	3.39619e30	1.32818	0.664091	−	0.747652i	$-0.268819\pi$
$787$	3.39619e30	1.32818	0.664091	+	0.747652i	$0.268819\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.49624e30	0.925622
$792$	0	0
$793$	1.17693e30	0.424992
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.15218e30	−1.07969	−0.539844	−	0.841765i	$-0.681517\pi$
$797$	−3.15218e30	−1.07969	−0.539844	+	0.841765i	$0.681517\pi$
$798$	0	0
$799$	−9.70633e29	−0.323827
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.51256e30	2.06176
$804$	0	0
$805$	8.08738e29	0.249431
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.77781e30	−1.69162	−0.845811	−	0.533482i	$-0.820883\pi$
$809$	−5.77781e30	−1.69162	−0.845811	+	0.533482i	$0.820883\pi$
$810$	0	0
$811$	−7.71903e29	−0.220213	−0.110107	−	0.993920i	$-0.535119\pi$
$811$	−7.71903e29	−0.220213	−0.110107	+	0.993920i	$0.535119\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.56998e29	0.205088
$816$	0	0
$817$	1.14265e30	0.301703
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.63389e30	0.409845	0.204923	−	0.978778i	$-0.434306\pi$
$821$	1.63389e30	0.409845	0.204923	+	0.978778i	$0.434306\pi$
$822$	0	0
$823$	4.56986e29	0.111739	0.0558695	−	0.998438i	$-0.482207\pi$
$823$	4.56986e29	0.111739	0.0558695	+	0.998438i	$0.482207\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.30833e30	0.536401	0.268201	−	0.963363i	$-0.413571\pi$
$827$	2.30833e30	0.536401	0.268201	+	0.963363i	$0.413571\pi$
$828$	0	0
$829$	−3.52497e29	−0.0798607	−0.0399303	−	0.999202i	$-0.512714\pi$
$829$	−3.52497e29	−0.0798607	−0.0399303	+	0.999202i	$0.512714\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.76069e30	0.379236
$834$	0	0
$835$	−1.52675e29	−0.0320672
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7.70025e29	0.153817	0.0769084	−	0.997038i	$-0.475495\pi$
$839$	7.70025e29	0.153817	0.0769084	+	0.997038i	$0.475495\pi$
$840$	0	0
$841$	−3.91392e30	−0.762525
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.40743e29	−0.0631585
$846$	0	0
$847$	−3.98951e30	−0.721347
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.39203e30	−0.411629
$852$	0	0
$853$	−6.31709e30	−1.06060	−0.530301	−	0.847810i	$-0.677921\pi$
$853$	−6.31709e30	−1.06060	−0.530301	+	0.847810i	$0.677921\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.16894e30	0.506541	0.253271	−	0.967395i	$-0.418494\pi$
$857$	3.16894e30	0.506541	0.253271	+	0.967395i	$0.418494\pi$
$858$	0	0
$859$	−8.21661e30	−1.28164	−0.640818	−	0.767693i	$-0.721405\pi$
$859$	−8.21661e30	−1.28164	−0.640818	+	0.767693i	$0.721405\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.74681e30	−0.705162	−0.352581	−	0.935781i	$-0.614696\pi$
$863$	−4.74681e30	−0.705162	−0.352581	+	0.935781i	$0.614696\pi$
$864$	0	0
$865$	7.12158e29	0.103254
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.20808e30	0.857530
$870$	0	0
$871$	−6.98097e30	−0.941293
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.64111e30	0.467897
$876$	0	0
$877$	6.72283e30	0.843444	0.421722	−	0.906725i	$-0.361426\pi$
$877$	6.72283e30	0.843444	0.421722	+	0.906725i	$0.361426\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.20529e31	−1.44160	−0.720800	−	0.693143i	$-0.756225\pi$
$881$	−1.20529e31	−1.44160	−0.720800	+	0.693143i	$0.756225\pi$
$882$	0	0
$883$	2.79063e30	0.325923	0.162962	−	0.986632i	$-0.447895\pi$
$883$	2.79063e30	0.325923	0.162962	+	0.986632i	$0.447895\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.22230e30	0.470274	0.235137	−	0.971962i	$-0.424446\pi$
$887$	4.22230e30	0.470274	0.235137	+	0.971962i	$0.424446\pi$
$888$	0	0
$889$	−1.88675e31	−2.05232
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.95157e30	0.202509
$894$	0	0
$895$	3.02250e30	0.306355
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.69790e30	−0.841274
$900$	0	0
$901$	5.27064e30	0.498027
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.20793e29	−0.0650130
$906$	0	0
$907$	8.17027e30	0.720045	0.360023	−	0.932944i	$-0.382769\pi$
$907$	8.17027e30	0.720045	0.360023	+	0.932944i	$0.382769\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8.02621e30	−0.675410	−0.337705	−	0.941252i	$-0.609651\pi$
$911$	−8.02621e30	−0.675410	−0.337705	+	0.941252i	$0.609651\pi$
$912$	0	0
$913$	−1.33381e31	−1.09686
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.27712e31	1.78858
$918$	0	0
$919$	−3.31927e30	−0.254818	−0.127409	−	0.991850i	$-0.540666\pi$
$919$	−3.31927e30	−0.254818	−0.127409	+	0.991850i	$0.540666\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.33031e31	−0.975746
$924$	0	0
$925$	−5.30165e30	−0.380122
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.23059e31	−1.52846	−0.764232	−	0.644941i	$-0.776882\pi$
$929$	−2.23059e31	−1.52846	−0.764232	+	0.644941i	$0.776882\pi$
$930$	0	0
$931$	−3.54007e30	−0.237159
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.48903e30	−0.0953636
$936$	0	0
$937$	8.24036e30	0.516037	0.258018	−	0.966140i	$-0.416931\pi$
$937$	8.24036e30	0.516037	0.258018	+	0.966140i	$0.416931\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.13254e31	−0.678208	−0.339104	−	0.940749i	$-0.610124\pi$
$941$	−1.13254e31	−0.678208	−0.339104	+	0.940749i	$0.610124\pi$
$942$	0	0
$943$	1.99717e30	0.116961
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.85285e31	−1.59810	−0.799050	−	0.601264i	$-0.794664\pi$
$947$	−2.85285e31	−1.59810	−0.799050	+	0.601264i	$0.794664\pi$
$948$	0	0
$949$	−2.42989e31	−1.33135
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.28550e30	0.119814	0.0599068	−	0.998204i	$-0.480920\pi$
$953$	2.28550e30	0.119814	0.0599068	+	0.998204i	$0.480920\pi$
$954$	0	0
$955$	−4.47903e30	−0.229693
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.33841e31	−1.14769
$960$	0	0
$961$	4.12404e31	1.98029
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.97561e29	0.0228723
$966$	0	0
$967$	2.86814e31	1.29010	0.645049	−	0.764141i	$-0.276837\pi$
$967$	2.86814e31	1.29010	0.645049	+	0.764141i	$0.276837\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.70387e31	1.59534	0.797670	−	0.603094i	$-0.206066\pi$
$971$	3.70387e31	1.59534	0.797670	+	0.603094i	$0.206066\pi$
$972$	0	0
$973$	−4.24305e31	−1.78852
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.87685e31	0.757770	0.378885	−	0.925444i	$-0.376308\pi$
$977$	1.87685e31	0.757770	0.378885	+	0.925444i	$0.376308\pi$
$978$	0	0
$979$	−5.03751e31	−1.99066
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.00084e31	−0.757531	−0.378766	−	0.925493i	$-0.623651\pi$
$983$	−2.00084e31	−0.757531	−0.378766	+	0.925493i	$0.623651\pi$
$984$	0	0
$985$	2.83167e30	0.104945
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.22397e31	1.14507
$990$	0	0
$991$	−1.44667e31	−0.503034	−0.251517	−	0.967853i	$-0.580929\pi$
$991$	−1.44667e31	−0.503034	−0.251517	+	0.967853i	$0.580929\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.72297e30	0.257420
$996$	0	0
$997$	3.92902e31	1.28229	0.641144	−	0.767421i	$-0.278460\pi$
$997$	3.92902e31	1.28229	0.641144	+	0.767421i	$0.278460\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.22.a.a.1.2		2
3.2	odd	2		24.22.a.a.1.1	✓	2
12.11	even	2		48.22.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.22.a.a.1.1	✓	2	3.2	odd	2
48.22.a.h.1.1		2	12.11	even	2
72.22.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:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

\(f(q)\)	\(=\)	\(q+3.80644e6 q^{5} -1.01851e9 q^{7} +O(q^{10})\)	\(q+3.80644e6 q^{5} -1.01851e9 q^{7} -1.06383e11 q^{11} +3.96922e11 q^{13} +3.67718e12 q^{17} -7.39341e12 q^{19} -2.08605e14 q^{23} -4.62348e14 q^{25} +1.10405e15 q^{29} -7.87819e15 q^{31} -3.87689e15 q^{35} +1.14668e16 q^{37} -9.57394e15 q^{41} -1.54549e17 q^{43} -2.63961e17 q^{47} +4.78814e17 q^{49} +1.43334e18 q^{53} -4.04939e17 q^{55} +5.89337e18 q^{59} +2.96513e18 q^{61} +1.51086e18 q^{65} -1.75878e19 q^{67} -3.35158e19 q^{71} -6.12183e19 q^{73} +1.08352e20 q^{77} -5.83561e19 q^{79} +1.25379e20 q^{83} +1.39970e19 q^{85} +4.73528e20 q^{89} -4.04268e20 q^{91} -2.81426e19 q^{95} -2.83797e20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 21948620 q^{5} - 659451408 q^{7}+O(q^{10}) \) 2 * q - 21948620 * q^5 - 659451408 * q^7	\( 2 q - 21948620 q^{5} - 659451408 q^{7} + 29910896872 q^{11} + 4468866812 q^{13} + 17665404721820 q^{17} - 22467979297496 q^{19} - 120995273049584 q^{23} - 275862509554850 q^{25} + 33\!\cdots\!84 q^{29}+ \cdots - 12\!\cdots\!20 q^{97}+O(q^{100}) \) 2 * q - 21948620 * q^5 - 659451408 * q^7 + 29910896872 * q^11 + 4468866812 * q^13 + 17665404721820 * q^17 - 22467979297496 * q^19 - 120995273049584 * q^23 - 275862509554850 * q^25 + 3318133806711684 * q^29 - 6983846005801568 * q^31 - 13124419284174240 * q^35 - 23423034370792788 * q^37 - 152179406345334996 * q^41 + 35137855888657880 * q^43 + 84350534118008928 * q^47 + 49189617082449970 * q^49 - 389728341256520684 * q^53 - 3915186255016270960 * q^55 + 3799253986972943944 * q^59 + 1500415393700389916 * q^61 + 11618501316836358040 * q^65 - 15727134260135911736 * q^67 + 15483319715102645936 * q^71 - 75370240511639281708 * q^73 + 157288770168613331136 * q^77 - 53554722400730974336 * q^79 + 299101821652772652632 * q^83 - 346270420314818644520 * q^85 + 614802488682208004364 * q^89 - 545181052826294967264 * q^91 + 360103706555542618640 * q^95 - 1201555028804369106620 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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