LMFDB - Embedded newform 72.16.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,16,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, 16, names="a")

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

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 16 $
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:	$102.739323672$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{22}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 22 $ x^2 - 22
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.69042$ 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)

$f(q)$	$=$	$q-161936. q^{5} -2.16834e6 q^{7} +O(q^{10})$	$q-161936. q^{5} -2.16834e6 q^{7} +1.16723e8 q^{11} +37971.5 q^{13} -9.72730e8 q^{17} -7.00231e9 q^{19} -2.18197e10 q^{23} -4.29419e9 q^{25} +2.93511e10 q^{29} -1.25467e10 q^{31} +3.51133e11 q^{35} -6.88911e11 q^{37} +4.62373e11 q^{41} -1.44271e12 q^{43} +6.53547e12 q^{47} -4.58786e10 q^{49} +6.75320e12 q^{53} -1.89017e13 q^{55} -1.52767e13 q^{59} +3.29878e13 q^{61} -6.14897e9 q^{65} -1.34942e13 q^{67} -2.28420e13 q^{71} +7.39181e13 q^{73} -2.53094e14 q^{77} +1.69104e14 q^{79} +2.62044e14 q^{83} +1.57520e14 q^{85} +1.04120e14 q^{89} -8.23350e10 q^{91} +1.13393e15 q^{95} +1.54015e15 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18340 q^{5} - 680400 q^{7}+O(q^{10}) $ 2 * q + 18340 * q^5 - 680400 * q^7	$ 2 q + 18340 q^{5} - 680400 q^{7} + 77720632 q^{11} - 207881332 q^{13} + 1673313692 q^{17} - 2240887112 q^{19} - 16655801552 q^{23} - 2312200850 q^{25} + 139493975892 q^{29} - 43133623424 q^{31} + 619372303200 q^{35} - 1457105544996 q^{37} + 1377923837772 q^{41} - 2763654497656 q^{43} + 7396441682400 q^{47} - 2579485205486 q^{49} + 5824660261252 q^{53} - 25932807442960 q^{55} + 22792784775448 q^{59} + 14623417395116 q^{61} - 37489085612840 q^{65} + 68414350989784 q^{67} - 64404223302256 q^{71} + 229480150537460 q^{73} - 311126750827200 q^{77} - 8902155902944 q^{79} + 239472633418184 q^{83} + 634539486862840 q^{85} - 472605248938452 q^{89} - 309453044594400 q^{91} + 19\!\cdots\!60 q^{95}+ \cdots + 20\!\cdots\!48 q^{97}+O(q^{100}) $ 2 * q + 18340 * q^5 - 680400 * q^7 + 77720632 * q^11 - 207881332 * q^13 + 1673313692 * q^17 - 2240887112 * q^19 - 16655801552 * q^23 - 2312200850 * q^25 + 139493975892 * q^29 - 43133623424 * q^31 + 619372303200 * q^35 - 1457105544996 * q^37 + 1377923837772 * q^41 - 2763654497656 * q^43 + 7396441682400 * q^47 - 2579485205486 * q^49 + 5824660261252 * q^53 - 25932807442960 * q^55 + 22792784775448 * q^59 + 14623417395116 * q^61 - 37489085612840 * q^65 + 68414350989784 * q^67 - 64404223302256 * q^71 + 229480150537460 * q^73 - 311126750827200 * q^77 - 8902155902944 * q^79 + 239472633418184 * q^83 + 634539486862840 * q^85 - 472605248938452 * q^89 - 309453044594400 * q^91 + 1992299281656560 * q^95 + 2032136651194948 * 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
$5$	−161936.	−0.926978	−0.463489	−	0.886103i	$-0.653403\pi$
$5$	−161936.	−0.926978	−0.463489	+	0.886103i	$0.653403\pi$
$6$	0	0
$7$	−2.16834e6	−0.995156	−0.497578	−	0.867419i	$-0.665777\pi$
$7$	−2.16834e6	−0.995156	−0.497578	+	0.867419i	$0.665777\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.16723e8	1.80597	0.902984	−	0.429673i	$-0.141371\pi$
$11$	1.16723e8	1.80597	0.902984	+	0.429673i	$0.141371\pi$
$12$	0	0
$13$	37971.5	0.000167835 0	8.39175e−5	−	1.00000i	$-0.499973\pi$
$13$	37971.5	0.000167835 0	8.39175e−5		1.00000i	$0.499973\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−9.72730e8	−0.574944	−0.287472	−	0.957789i	$-0.592815\pi$
$17$	−9.72730e8	−0.574944	−0.287472	+	0.957789i	$0.592815\pi$
$18$	0	0
$19$	−7.00231e9	−1.79717	−0.898585	−	0.438800i	$-0.855404\pi$
$19$	−7.00231e9	−1.79717	−0.898585	+	0.438800i	$0.855404\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.18197e10	−1.33626	−0.668128	−	0.744047i	$-0.732904\pi$
$23$	−2.18197e10	−1.33626	−0.668128	+	0.744047i	$0.732904\pi$
$24$	0	0
$25$	−4.29419e9	−0.140712
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.93511e10	0.315965	0.157983	−	0.987442i	$-0.449501\pi$
$29$	2.93511e10	0.315965	0.157983	+	0.987442i	$0.449501\pi$
$30$	0	0
$31$	−1.25467e10	−0.0819059	−0.0409529	−	0.999161i	$-0.513039\pi$
$31$	−1.25467e10	−0.0819059	−0.0409529	+	0.999161i	$0.513039\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.51133e11	0.922488
$36$	0	0
$37$	−6.88911e11	−1.19303	−0.596514	−	0.802603i	$-0.703448\pi$
$37$	−6.88911e11	−1.19303	−0.596514	+	0.802603i	$0.703448\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.62373e11	0.370778	0.185389	−	0.982665i	$-0.440646\pi$
$41$	4.62373e11	0.370778	0.185389	+	0.982665i	$0.440646\pi$
$42$	0	0
$43$	−1.44271e12	−0.809404	−0.404702	−	0.914449i	$-0.632625\pi$
$43$	−1.44271e12	−0.809404	−0.404702	+	0.914449i	$0.632625\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.53547e12	1.88167	0.940833	−	0.338869i	$-0.110044\pi$
$47$	6.53547e12	1.88167	0.940833	+	0.338869i	$0.110044\pi$
$48$	0	0
$49$	−4.58786e10	−0.00966361
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.75320e12	0.789661	0.394831	−	0.918754i	$-0.370803\pi$
$53$	6.75320e12	0.789661	0.394831	+	0.918754i	$0.370803\pi$
$54$	0	0
$55$	−1.89017e13	−1.67409
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.52767e13	−0.799173	−0.399586	−	0.916696i	$-0.630846\pi$
$59$	−1.52767e13	−0.799173	−0.399586	+	0.916696i	$0.630846\pi$
$60$	0	0
$61$	3.29878e13	1.34394	0.671970	−	0.740578i	$-0.265448\pi$
$61$	3.29878e13	1.34394	0.671970	+	0.740578i	$0.265448\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.14897e9	−0.000155579 0
$66$	0	0
$67$	−1.34942e13	−0.272010	−0.136005	−	0.990708i	$-0.543426\pi$
$67$	−1.34942e13	−0.272010	−0.136005	+	0.990708i	$0.543426\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.28420e13	−0.298055	−0.149028	−	0.988833i	$-0.547614\pi$
$71$	−2.28420e13	−0.298055	−0.149028	+	0.988833i	$0.547614\pi$
$72$	0	0
$73$	7.39181e13	0.783122	0.391561	−	0.920152i	$-0.371935\pi$
$73$	7.39181e13	0.783122	0.391561	+	0.920152i	$0.371935\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.53094e14	−1.79722
$78$	0	0
$79$	1.69104e14	0.990719	0.495360	−	0.868688i	$-0.335036\pi$
$79$	1.69104e14	0.990719	0.495360	+	0.868688i	$0.335036\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.62044e14	1.05996	0.529979	−	0.848011i	$-0.322200\pi$
$83$	2.62044e14	1.05996	0.529979	+	0.848011i	$0.322200\pi$
$84$	0	0
$85$	1.57520e14	0.532960
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.04120e14	0.249523	0.124761	−	0.992187i	$-0.460183\pi$
$89$	1.04120e14	0.249523	0.124761	+	0.992187i	$0.460183\pi$
$90$	0	0
$91$	−8.23350e10	−0.000167022 0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.13393e15	1.66594
$96$	0	0
$97$	1.54015e15	1.93542	0.967708	−	0.252075i	$-0.0811129\pi$
$97$	1.54015e15	1.93542	0.967708	+	0.252075i	$0.0811129\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.55394e15	−1.44220	−0.721098	−	0.692833i	$-0.756362\pi$
$101$	−1.55394e15	−1.44220	−0.721098	+	0.692833i	$0.756362\pi$
$102$	0	0
$103$	−1.61833e15	−1.29654	−0.648272	−	0.761409i	$-0.724508\pi$
$103$	−1.61833e15	−1.29654	−0.648272	+	0.761409i	$0.724508\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.51910e14	0.573083	0.286542	−	0.958068i	$-0.407494\pi$
$107$	9.51910e14	0.573083	0.286542	+	0.958068i	$0.407494\pi$
$108$	0	0
$109$	−3.25626e15	−1.70616	−0.853080	−	0.521780i	$-0.825268\pi$
$109$	−3.25626e15	−1.70616	−0.853080	+	0.521780i	$0.825268\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.65429e15	1.46122	0.730609	−	0.682797i	$-0.239236\pi$
$113$	3.65429e15	1.46122	0.730609	+	0.682797i	$0.239236\pi$
$114$	0	0
$115$	3.53340e15	1.23868
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.10921e15	0.572159
$120$	0	0
$121$	9.44694e15	2.26152
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.63729e15	1.05741
$126$	0	0
$127$	3.04466e15	0.507004	0.253502	−	0.967335i	$-0.418418\pi$
$127$	3.04466e15	0.507004	0.253502	+	0.967335i	$0.418418\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.78722e15	−0.499788	−0.249894	−	0.968273i	$-0.580396\pi$
$131$	−3.78722e15	−0.499788	−0.249894	+	0.968273i	$0.580396\pi$
$132$	0	0
$133$	1.51834e16	1.78847
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.26867e15	−0.496932	−0.248466	−	0.968641i	$-0.579926\pi$
$137$	−5.26867e15	−0.496932	−0.248466	+	0.968641i	$0.579926\pi$
$138$	0	0
$139$	1.57574e16	1.33313	0.666567	−	0.745445i	$-0.267763\pi$
$139$	1.57574e16	1.33313	0.666567	+	0.745445i	$0.267763\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.43214e12	0.000303105 0
$144$	0	0
$145$	−4.75301e15	−0.292893
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.55866e15	−0.128563	−0.0642816	−	0.997932i	$-0.520476\pi$
$149$	−2.55866e15	−0.128563	−0.0642816	+	0.997932i	$0.520476\pi$
$150$	0	0
$151$	2.68525e16	1.22084	0.610418	−	0.792080i	$-0.291002\pi$
$151$	2.68525e16	1.22084	0.610418	+	0.792080i	$0.291002\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.03176e15	0.0759249
$156$	0	0
$157$	−1.03910e16	−0.352703	−0.176351	−	0.984327i	$-0.556430\pi$
$157$	−1.03910e16	−0.352703	−0.176351	+	0.984327i	$0.556430\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.73124e16	1.32978
$162$	0	0
$163$	6.68200e16	1.71198	0.855992	−	0.516989i	$-0.172947\pi$
$163$	6.68200e16	1.71198	0.855992	+	0.516989i	$0.172947\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.21466e16	0.259467	0.129733	−	0.991549i	$-0.458588\pi$
$167$	1.21466e16	0.259467	0.129733	+	0.991549i	$0.458588\pi$
$168$	0	0
$169$	−5.11859e16	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.31211e16	−0.215092	−0.107546	−	0.994200i	$-0.534299\pi$
$173$	−1.31211e16	−0.215092	−0.107546	+	0.994200i	$0.534299\pi$
$174$	0	0
$175$	9.31125e15	0.140031
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.25282e17	1.59034	0.795169	−	0.606388i	$-0.207382\pi$
$179$	1.25282e17	1.59034	0.795169	+	0.606388i	$0.207382\pi$
$180$	0	0
$181$	7.85627e15	0.0917545	0.0458772	−	0.998947i	$-0.485392\pi$
$181$	7.85627e15	0.0917545	0.0458772	+	0.998947i	$0.485392\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.11560e17	1.10591
$186$	0	0
$187$	−1.13540e17	−1.03833
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.13840e16	−0.556994	−0.278497	−	0.960437i	$-0.589836\pi$
$191$	−7.13840e16	−0.556994	−0.278497	+	0.960437i	$0.589836\pi$
$192$	0	0
$193$	1.04593e17	0.754782	0.377391	−	0.926054i	$-0.376821\pi$
$193$	1.04593e17	0.754782	0.377391	+	0.926054i	$0.376821\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.07387e17	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$197$	−2.07387e17	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$198$	0	0
$199$	−2.52600e17	−1.44889	−0.724443	−	0.689334i	$-0.757903\pi$
$199$	−2.52600e17	−1.44889	−0.724443	+	0.689334i	$0.757903\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.36431e16	−0.314435
$204$	0	0
$205$	−7.48750e16	−0.343703
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.17328e17	−3.24563
$210$	0	0
$211$	4.07629e17	1.50711	0.753557	−	0.657383i	$-0.228336\pi$
$211$	4.07629e17	1.50711	0.753557	+	0.657383i	$0.228336\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.33627e17	0.750300
$216$	0	0
$217$	2.72054e16	0.0815092
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.69360e13	−9.64957e−5 0
$222$	0	0
$223$	−2.57353e17	−0.628410	−0.314205	−	0.949355i	$-0.601738\pi$
$223$	−2.57353e17	−0.628410	−0.314205	+	0.949355i	$0.601738\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.41239e17	1.58403	0.792017	−	0.610500i	$-0.209031\pi$
$227$	7.41239e17	1.58403	0.792017	+	0.610500i	$0.209031\pi$
$228$	0	0
$229$	−5.40714e17	−1.08194	−0.540968	−	0.841043i	$-0.681942\pi$
$229$	−5.40714e17	−1.08194	−0.540968	+	0.841043i	$0.681942\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.34981e17	−0.764365	−0.382182	−	0.924087i	$-0.624827\pi$
$233$	−4.34981e17	−0.764365	−0.382182	+	0.924087i	$0.624827\pi$
$234$	0	0
$235$	−1.05833e18	−1.74426
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.12267e17	−1.03433	−0.517164	−	0.855886i	$-0.673012\pi$
$239$	−7.12267e17	−1.03433	−0.517164	+	0.855886i	$0.673012\pi$
$240$	0	0
$241$	−1.08573e16	−0.0148113	−0.00740567	−	0.999973i	$-0.502357\pi$
$241$	−1.08573e16	−0.0148113	−0.00740567	+	0.999973i	$0.502357\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	7.42941e15	0.00895795
$246$	0	0
$247$	−2.65888e14	−0.000301628 0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.97060e17	0.499869	0.249934	−	0.968263i	$-0.419591\pi$
$251$	4.97060e17	0.499869	0.249934	+	0.968263i	$0.419591\pi$
$252$	0	0
$253$	−2.54685e18	−2.41324
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.81572e17	−0.237187	−0.118594	−	0.992943i	$-0.537839\pi$
$257$	−2.81572e17	−0.237187	−0.118594	+	0.992943i	$0.537839\pi$
$258$	0	0
$259$	1.49379e18	1.18725
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.25206e18	1.59556	0.797779	−	0.602950i	$-0.206008\pi$
$263$	2.25206e18	1.59556	0.797779	+	0.602950i	$0.206008\pi$
$264$	0	0
$265$	−1.09359e18	−0.731998
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−8.99999e17	−0.538394	−0.269197	−	0.963085i	$-0.586758\pi$
$269$	−8.99999e17	−0.538394	−0.269197	+	0.963085i	$0.586758\pi$
$270$	0	0
$271$	−4.90226e17	−0.277413	−0.138707	−	0.990334i	$-0.544294\pi$
$271$	−4.90226e17	−0.277413	−0.138707	+	0.990334i	$0.544294\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.01230e17	−0.254122
$276$	0	0
$277$	6.65132e17	0.319381	0.159691	−	0.987167i	$-0.448950\pi$
$277$	6.65132e17	0.319381	0.159691	+	0.987167i	$0.448950\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.52418e18	−1.08849	−0.544244	−	0.838927i	$-0.683183\pi$
$281$	−2.52418e18	−1.08849	−0.544244	+	0.838927i	$0.683183\pi$
$282$	0	0
$283$	4.36532e18	1.78492	0.892458	−	0.451130i	$-0.148979\pi$
$283$	4.36532e18	1.78492	0.892458	+	0.451130i	$0.148979\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.00258e18	−0.368982
$288$	0	0
$289$	−1.91622e18	−0.669440
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.59279e18	0.817073	0.408536	−	0.912742i	$-0.366039\pi$
$293$	2.59279e18	0.817073	0.408536	+	0.912742i	$0.366039\pi$
$294$	0	0
$295$	2.47386e18	0.740815
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.28526e14	−0.000224270 0
$300$	0	0
$301$	3.12828e18	0.805484
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.34193e18	−1.24580
$306$	0	0
$307$	5.08485e18	1.12912	0.564560	−	0.825392i	$-0.309046\pi$
$307$	5.08485e18	1.12912	0.564560	+	0.825392i	$0.309046\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.67003e18	1.34408	0.672040	−	0.740515i	$-0.265418\pi$
$311$	6.67003e18	1.34408	0.672040	+	0.740515i	$0.265418\pi$
$312$	0	0
$313$	8.20097e18	1.57501	0.787504	−	0.616310i	$-0.211373\pi$
$313$	8.20097e18	1.57501	0.787504	+	0.616310i	$0.211373\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.92096e18	1.55764	0.778820	−	0.627248i	$-0.215819\pi$
$317$	8.92096e18	1.55764	0.778820	+	0.627248i	$0.215819\pi$
$318$	0	0
$319$	3.42594e18	0.570624
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.81135e18	1.03327
$324$	0	0
$325$	−1.63057e14	−2.36164e−5 0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.41711e19	−1.87255
$330$	0	0
$331$	−1.45337e19	−1.83513	−0.917564	−	0.397588i	$-0.869847\pi$
$331$	−1.45337e19	−1.83513	−0.917564	+	0.397588i	$0.869847\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.18520e18	0.252148
$336$	0	0
$337$	1.35321e18	0.149328	0.0746641	−	0.997209i	$-0.476212\pi$
$337$	1.35321e18	0.149328	0.0746641	+	0.997209i	$0.476212\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.46448e18	−0.147919
$342$	0	0
$343$	1.03938e19	1.00477
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.45477e19	1.28921	0.644603	−	0.764518i	$-0.277023\pi$
$347$	1.45477e19	1.28921	0.644603	+	0.764518i	$0.277023\pi$
$348$	0	0
$349$	−2.00760e19	−1.70407	−0.852034	−	0.523486i	$-0.824631\pi$
$349$	−2.00760e19	−1.70407	−0.852034	+	0.523486i	$0.824631\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.21308e18	0.717951	0.358975	−	0.933347i	$-0.383126\pi$
$353$	9.21308e18	0.717951	0.358975	+	0.933347i	$0.383126\pi$
$354$	0	0
$355$	3.69896e18	0.276291
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.20098e19	0.824758	0.412379	−	0.911012i	$-0.364698\pi$
$359$	1.20098e19	0.824758	0.412379	+	0.911012i	$0.364698\pi$
$360$	0	0
$361$	3.38512e19	2.22982
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.19700e19	−0.725936
$366$	0	0
$367$	−2.49016e18	−0.144955	−0.0724773	−	0.997370i	$-0.523090\pi$
$367$	−2.49016e18	−0.144955	−0.0724773	+	0.997370i	$0.523090\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.46432e19	−0.785836
$372$	0	0
$373$	−3.29030e19	−1.69597	−0.847987	−	0.530017i	$-0.822186\pi$
$373$	−3.29030e19	−1.69597	−0.847987	+	0.530017i	$0.822186\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.11451e15	5.30301e−5 0
$378$	0	0
$379$	−1.89533e19	−0.866743	−0.433372	−	0.901215i	$-0.642676\pi$
$379$	−1.89533e19	−0.866743	−0.433372	+	0.901215i	$0.642676\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.02672e18	0.127933	0.0639663	−	0.997952i	$-0.479625\pi$
$383$	3.02672e18	0.127933	0.0639663	+	0.997952i	$0.479625\pi$
$384$	0	0
$385$	4.09851e19	1.66598
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.09103e18	−0.153890	−0.0769451	−	0.997035i	$-0.524517\pi$
$389$	−4.09103e18	−0.153890	−0.0769451	+	0.997035i	$0.524517\pi$
$390$	0	0
$391$	2.12247e19	0.768272
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.73841e19	−0.918375
$396$	0	0
$397$	−9.49752e18	−0.306677	−0.153339	−	0.988174i	$-0.549003\pi$
$397$	−9.49752e18	−0.306677	−0.153339	+	0.988174i	$0.549003\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.20346e19	−0.659967	−0.329983	−	0.943987i	$-0.607043\pi$
$401$	−2.20346e19	−0.659967	−0.329983	+	0.943987i	$0.607043\pi$
$402$	0	0
$403$	−4.76415e14	−1.37467e−5 0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.04116e19	−2.15457
$408$	0	0
$409$	−5.54538e19	−1.43221	−0.716105	−	0.697993i	$-0.754077\pi$
$409$	−5.54538e19	−1.43221	−0.716105	+	0.697993i	$0.754077\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.31251e19	0.795302
$414$	0	0
$415$	−4.24345e19	−0.982558
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.94382e19	−0.418843	−0.209422	−	0.977825i	$-0.567158\pi$
$419$	−1.94382e19	−0.418843	−0.209422	+	0.977825i	$0.567158\pi$
$420$	0	0
$421$	−1.88886e19	−0.392722	−0.196361	−	0.980532i	$-0.562912\pi$
$421$	−1.88886e19	−0.392722	−0.196361	+	0.980532i	$0.562912\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.17709e18	0.0809015
$426$	0	0
$427$	−7.15287e19	−1.33743
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.00494e18	−0.104696	−0.0523479	−	0.998629i	$-0.516670\pi$
$431$	−6.00494e18	−0.104696	−0.0523479	+	0.998629i	$0.516670\pi$
$432$	0	0
$433$	−3.07741e19	−0.518235	−0.259118	−	0.965846i	$-0.583432\pi$
$433$	−3.07741e19	−0.518235	−0.259118	+	0.965846i	$0.583432\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.52788e20	2.40148
$438$	0	0
$439$	−4.66145e19	−0.708006	−0.354003	−	0.935244i	$-0.615180\pi$
$439$	−4.66145e19	−0.708006	−0.354003	+	0.935244i	$0.615180\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.79254e19	−1.38953	−0.694764	−	0.719238i	$-0.744491\pi$
$443$	−9.79254e19	−1.38953	−0.694764	+	0.719238i	$0.744491\pi$
$444$	0	0
$445$	−1.68608e19	−0.231302
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.72775e19	0.221633	0.110816	−	0.993841i	$-0.464653\pi$
$449$	1.72775e19	0.221633	0.110816	+	0.993841i	$0.464653\pi$
$450$	0	0
$451$	5.39695e19	0.669613
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.33330e16	0.000154826 0
$456$	0	0
$457$	4.42260e19	0.496942	0.248471	−	0.968639i	$-0.420072\pi$
$457$	4.42260e19	0.496942	0.248471	+	0.968639i	$0.420072\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.68186e20	1.77024	0.885120	−	0.465364i	$-0.154076\pi$
$461$	1.68186e20	1.77024	0.885120	+	0.465364i	$0.154076\pi$
$462$	0	0
$463$	−4.67287e19	−0.476131	−0.238065	−	0.971249i	$-0.576513\pi$
$463$	−4.67287e19	−0.476131	−0.238065	+	0.971249i	$0.576513\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.35648e20	1.29579	0.647897	−	0.761728i	$-0.275649\pi$
$467$	1.35648e20	1.29579	0.647897	+	0.761728i	$0.275649\pi$
$468$	0	0
$469$	2.92599e19	0.270693
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.68397e20	−1.46176
$474$	0	0
$475$	3.00692e19	0.252883
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.70533e19	−0.213650	−0.106825	−	0.994278i	$-0.534068\pi$
$479$	−2.70533e19	−0.213650	−0.106825	+	0.994278i	$0.534068\pi$
$480$	0	0
$481$	−2.61590e16	−0.000200232 0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.49406e20	−1.79409
$486$	0	0
$487$	6.46030e19	0.450594	0.225297	−	0.974290i	$-0.427665\pi$
$487$	6.46030e19	0.450594	0.225297	+	0.974290i	$0.427665\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.06362e20	−1.35369	−0.676845	−	0.736125i	$-0.736653\pi$
$491$	−2.06362e20	−1.35369	−0.676845	+	0.736125i	$0.736653\pi$
$492$	0	0
$493$	−2.85507e19	−0.181662
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.95292e19	0.296612
$498$	0	0
$499$	9.81301e19	0.570227	0.285114	−	0.958494i	$-0.407969\pi$
$499$	9.81301e19	0.570227	0.285114	+	0.958494i	$0.407969\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.13875e19	−0.117058	−0.0585290	−	0.998286i	$-0.518641\pi$
$503$	−2.13875e19	−0.117058	−0.0585290	+	0.998286i	$0.518641\pi$
$504$	0	0
$505$	2.51640e20	1.33688
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.92005e20	0.961455	0.480728	−	0.876870i	$-0.340373\pi$
$509$	1.92005e20	0.961455	0.480728	+	0.876870i	$0.340373\pi$
$510$	0	0
$511$	−1.60279e20	−0.779328
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.62066e20	1.20187
$516$	0	0
$517$	7.62838e20	3.39823
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.29054e20	0.542612	0.271306	−	0.962493i	$-0.412545\pi$
$521$	1.29054e20	0.542612	0.271306	+	0.962493i	$0.412545\pi$
$522$	0	0
$523$	5.30149e19	0.216588	0.108294	−	0.994119i	$-0.465461\pi$
$523$	5.30149e19	0.216588	0.108294	+	0.994119i	$0.465461\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.22045e19	0.0470913
$528$	0	0
$529$	2.09463e20	0.785579
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.75570e16	6.22295e−5 0
$534$	0	0
$535$	−1.54149e20	−0.531235
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.35507e18	−0.0174522
$540$	0	0
$541$	−1.68157e20	−0.533009	−0.266504	−	0.963834i	$-0.585869\pi$
$541$	−1.68157e20	−0.533009	−0.266504	+	0.963834i	$0.585869\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.27306e20	1.58157
$546$	0	0
$547$	1.54065e20	0.449572	0.224786	−	0.974408i	$-0.427832\pi$
$547$	1.54065e20	0.449572	0.224786	+	0.974408i	$0.427832\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.05525e20	−0.567843
$552$	0	0
$553$	−3.66674e20	−0.985921
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.37962e20	−0.606168	−0.303084	−	0.952964i	$-0.598016\pi$
$557$	−2.37962e20	−0.606168	−0.303084	+	0.952964i	$0.598016\pi$
$558$	0	0
$559$	−5.47818e16	−0.000135846 0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.11320e20	0.966867	0.483433	−	0.875381i	$-0.339390\pi$
$563$	4.11320e20	0.966867	0.483433	+	0.875381i	$0.339390\pi$
$564$	0	0
$565$	−5.91763e20	−1.35452
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.48215e20	1.19017	0.595084	−	0.803663i	$-0.297119\pi$
$569$	5.48215e20	1.19017	0.595084	+	0.803663i	$0.297119\pi$
$570$	0	0
$571$	8.42478e19	0.178151	0.0890754	−	0.996025i	$-0.471609\pi$
$571$	8.42478e19	0.178151	0.0890754	+	0.996025i	$0.471609\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9.36979e19	0.188027
$576$	0	0
$577$	−5.78619e20	−1.13129	−0.565645	−	0.824649i	$-0.691373\pi$
$577$	−5.78619e20	−1.13129	−0.565645	+	0.824649i	$0.691373\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.68200e20	−1.05482
$582$	0	0
$583$	7.88252e20	1.42610
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.85378e20	0.318620	0.159310	−	0.987229i	$-0.449073\pi$
$587$	1.85378e20	0.318620	0.159310	+	0.987229i	$0.449073\pi$
$588$	0	0
$589$	8.78555e19	0.147199
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.85659e20	−0.773431	−0.386716	−	0.922199i	$-0.626390\pi$
$593$	−4.85659e20	−0.773431	−0.386716	+	0.922199i	$0.626390\pi$
$594$	0	0
$595$	−3.41557e20	−0.530379
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.12745e20	−0.461838	−0.230919	−	0.972973i	$-0.574173\pi$
$599$	−3.12745e20	−0.461838	−0.230919	+	0.972973i	$0.574173\pi$
$600$	0	0
$601$	−1.20401e20	−0.173409	−0.0867047	−	0.996234i	$-0.527634\pi$
$601$	−1.20401e20	−0.173409	−0.0867047	+	0.996234i	$0.527634\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.52980e21	−2.09638
$606$	0	0
$607$	6.42029e20	0.858301	0.429151	−	0.903233i	$-0.358813\pi$
$607$	6.42029e20	0.858301	0.429151	+	0.903233i	$0.358813\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.48161e17	0.000315810 0
$612$	0	0
$613$	−5.95059e19	−0.0738935	−0.0369467	−	0.999317i	$-0.511763\pi$
$613$	−5.95059e19	−0.0738935	−0.0369467	+	0.999317i	$0.511763\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.36774e20	−0.161758	−0.0808790	−	0.996724i	$-0.525773\pi$
$617$	−1.36774e20	−0.161758	−0.0808790	+	0.996724i	$0.525773\pi$
$618$	0	0
$619$	4.06727e20	0.469486	0.234743	−	0.972057i	$-0.424575\pi$
$619$	4.06727e20	0.469486	0.234743	+	0.972057i	$0.424575\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.25768e20	−0.248314
$624$	0	0
$625$	−7.81834e20	−0.839488
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.70125e20	0.685924
$630$	0	0
$631$	−1.51198e21	−1.51122	−0.755609	−	0.655023i	$-0.772659\pi$
$631$	−1.51198e21	−1.51122	−0.755609	+	0.655023i	$0.772659\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.93042e20	−0.469982
$636$	0	0
$637$	−1.74208e15	−1.62189e−6 0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.19317e21	1.05990	0.529952	−	0.848028i	$-0.322210\pi$
$641$	1.19317e21	1.05990	0.529952	+	0.848028i	$0.322210\pi$
$642$	0	0
$643$	2.81273e19	0.0244088	0.0122044	−	0.999926i	$-0.496115\pi$
$643$	2.81273e19	0.0244088	0.0122044	+	0.999926i	$0.496115\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.11423e20	0.423641	0.211821	−	0.977309i	$-0.432061\pi$
$647$	5.11423e20	0.423641	0.211821	+	0.977309i	$0.432061\pi$
$648$	0	0
$649$	−1.78314e21	−1.44328
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.75704e21	−1.35810	−0.679052	−	0.734090i	$-0.737609\pi$
$653$	−1.75704e21	−1.35810	−0.679052	+	0.734090i	$0.737609\pi$
$654$	0	0
$655$	6.13289e20	0.463293
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.07640e21	−0.776842	−0.388421	−	0.921482i	$-0.626979\pi$
$659$	−1.07640e21	−0.776842	−0.388421	+	0.921482i	$0.626979\pi$
$660$	0	0
$661$	1.39157e21	0.981734	0.490867	−	0.871235i	$-0.336680\pi$
$661$	1.39157e21	0.981734	0.490867	+	0.871235i	$0.336680\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.45874e21	−1.65787
$666$	0	0
$667$	−6.40432e20	−0.422210
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.85043e21	2.42711
$672$	0	0
$673$	−1.31402e21	−0.810004	−0.405002	−	0.914316i	$-0.632729\pi$
$673$	−1.31402e21	−0.810004	−0.405002	+	0.914316i	$0.632729\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.68867e21	−0.995700	−0.497850	−	0.867263i	$-0.665877\pi$
$677$	−1.68867e21	−0.995700	−0.497850	+	0.867263i	$0.665877\pi$
$678$	0	0
$679$	−3.33956e21	−1.92604
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.76934e21	1.52835	0.764173	−	0.645011i	$-0.223147\pi$
$683$	2.76934e21	1.52835	0.764173	+	0.645011i	$0.223147\pi$
$684$	0	0
$685$	8.53190e20	0.460645
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.56429e17	0.000132533 0
$690$	0	0
$691$	−2.59811e21	−1.31393	−0.656965	−	0.753921i	$-0.728160\pi$
$691$	−2.59811e21	−1.31393	−0.656965	+	0.753921i	$0.728160\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.55170e21	−1.23579
$696$	0	0
$697$	−4.49764e20	−0.213176
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.28826e21	1.03901	0.519506	−	0.854467i	$-0.326116\pi$
$701$	2.28826e21	1.03901	0.519506	+	0.854467i	$0.326116\pi$
$702$	0	0
$703$	4.82397e21	2.14407
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.36947e21	1.43521
$708$	0	0
$709$	1.16499e21	0.485819	0.242909	−	0.970049i	$-0.421898\pi$
$709$	1.16499e21	0.485819	0.242909	+	0.970049i	$0.421898\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.73764e20	0.109447
$714$	0	0
$715$	−7.17724e17	−0.000280971 0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−5.72261e20	−0.214846	−0.107423	−	0.994213i	$-0.534260\pi$
$719$	−5.72261e20	−0.214846	−0.107423	+	0.994213i	$0.534260\pi$
$720$	0	0
$721$	3.50908e21	1.29026
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.26039e20	−0.0444601
$726$	0	0
$727$	−2.31038e21	−0.798316	−0.399158	−	0.916882i	$-0.630697\pi$
$727$	−2.31038e21	−0.798316	−0.399158	+	0.916882i	$0.630697\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.40337e21	0.465362
$732$	0	0
$733$	−2.17298e21	−0.705954	−0.352977	−	0.935632i	$-0.614831\pi$
$733$	−2.17298e21	−0.705954	−0.352977	+	0.935632i	$0.614831\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.57508e21	−0.491242
$738$	0	0
$739$	6.25136e21	1.91048	0.955238	−	0.295839i	$-0.0955993\pi$
$739$	6.25136e21	1.91048	0.955238	+	0.295839i	$0.0955993\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.54644e21	−0.453856	−0.226928	−	0.973912i	$-0.572868\pi$
$743$	−1.54644e21	−0.453856	−0.226928	+	0.973912i	$0.572868\pi$
$744$	0	0
$745$	4.14341e20	0.119175
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.06406e21	−0.570307
$750$	0	0
$751$	4.82569e21	1.30695	0.653476	−	0.756947i	$-0.273310\pi$
$751$	4.82569e21	1.30695	0.653476	+	0.756947i	$0.273310\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.34839e21	−1.13169
$756$	0	0
$757$	2.08704e21	0.532490	0.266245	−	0.963905i	$-0.414217\pi$
$757$	2.08704e21	0.532490	0.266245	+	0.963905i	$0.414217\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.97877e21	0.975807	0.487903	−	0.872898i	$-0.337762\pi$
$761$	3.97877e21	0.975807	0.487903	+	0.872898i	$0.337762\pi$
$762$	0	0
$763$	7.06066e21	1.69790
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.80081e17	−0.000134129 0
$768$	0	0
$769$	5.34029e21	1.21092	0.605462	−	0.795874i	$-0.292988\pi$
$769$	5.34029e21	1.21092	0.605462	+	0.795874i	$0.292988\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.44754e21	1.62430	0.812150	−	0.583448i	$-0.198297\pi$
$773$	7.44754e21	1.62430	0.812150	+	0.583448i	$0.198297\pi$
$774$	0	0
$775$	5.38777e19	0.0115251
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.23768e21	−0.666351
$780$	0	0
$781$	−2.66618e21	−0.538279
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.68268e21	0.326948
$786$	0	0
$787$	−2.01188e21	−0.383522	−0.191761	−	0.981442i	$-0.561420\pi$
$787$	−2.01188e21	−0.383522	−0.191761	+	0.981442i	$0.561420\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.92373e21	−1.45414
$792$	0	0
$793$	1.25260e18	0.000225560 0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.58422e21	1.48855	0.744275	−	0.667873i	$-0.232795\pi$
$797$	8.58422e21	1.48855	0.744275	+	0.667873i	$0.232795\pi$
$798$	0	0
$799$	−6.35725e21	−1.08185
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.62792e21	1.41429
$804$	0	0
$805$	−7.66160e21	−1.23268
$806$	0	0
$807$	0	0
$808$	0	0
$809$	9.80336e19	0.0151971	0.00759856	−	0.999971i	$-0.497581\pi$
$809$	9.80336e19	0.0151971	0.00759856	+	0.999971i	$0.497581\pi$
$810$	0	0
$811$	−7.63811e21	−1.16233	−0.581165	−	0.813786i	$-0.697403\pi$
$811$	−7.63811e21	−1.16233	−0.581165	+	0.813786i	$0.697403\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.08206e22	−1.58697
$816$	0	0
$817$	1.01023e22	1.45464
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.24216e20	−0.114411	−0.0572054	−	0.998362i	$-0.518219\pi$
$821$	−8.24216e20	−0.114411	−0.0572054	+	0.998362i	$0.518219\pi$
$822$	0	0
$823$	1.59084e21	0.216835	0.108417	−	0.994105i	$-0.465422\pi$
$823$	1.59084e21	0.216835	0.108417	+	0.994105i	$0.465422\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.33001e20	0.109485	0.0547424	−	0.998501i	$-0.482566\pi$
$827$	8.33001e20	0.109485	0.0547424	+	0.998501i	$0.482566\pi$
$828$	0	0
$829$	9.94770e20	0.128400	0.0641998	−	0.997937i	$-0.479551\pi$
$829$	9.94770e20	0.128400	0.0641998	+	0.997937i	$0.479551\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.46275e19	0.00555603
$834$	0	0
$835$	−1.96698e21	−0.240520
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4.65119e21	0.548718	0.274359	−	0.961627i	$-0.411534\pi$
$839$	4.65119e21	0.548718	0.274359	+	0.961627i	$0.411534\pi$
$840$	0	0
$841$	−7.76770e21	−0.900166
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.28886e21	0.926978
$846$	0	0
$847$	−2.04842e22	−2.25057
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.50318e22	1.59419
$852$	0	0
$853$	9.26300e21	0.965237	0.482619	−	0.875831i	$-0.339686\pi$
$853$	9.26300e21	0.965237	0.482619	+	0.875831i	$0.339686\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.37621e21	−0.540904	−0.270452	−	0.962733i	$-0.587173\pi$
$857$	−5.37621e21	−0.540904	−0.270452	+	0.962733i	$0.587173\pi$
$858$	0	0
$859$	1.13208e22	1.11925	0.559627	−	0.828745i	$-0.310944\pi$
$859$	1.13208e22	1.11925	0.559627	+	0.828745i	$0.310944\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.84122e22	−1.75802	−0.879011	−	0.476802i	$-0.841796\pi$
$863$	−1.84122e22	−1.75802	−0.879011	+	0.476802i	$0.841796\pi$
$864$	0	0
$865$	2.12478e21	0.199386
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.97383e22	1.78921
$870$	0	0
$871$	−5.12394e17	−4.56529e−5 0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.22235e22	−1.05229
$876$	0	0
$877$	−1.24428e22	−1.05298	−0.526492	−	0.850180i	$-0.676493\pi$
$877$	−1.24428e22	−1.05298	−0.526492	+	0.850180i	$0.676493\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.76504e22	−1.44356	−0.721780	−	0.692122i	$-0.756676\pi$
$881$	−1.76504e22	−1.44356	−0.721780	+	0.692122i	$0.756676\pi$
$882$	0	0
$883$	5.50057e21	0.442285	0.221143	−	0.975241i	$-0.429021\pi$
$883$	5.50057e21	0.442285	0.221143	+	0.975241i	$0.429021\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.45506e21	−0.501734	−0.250867	−	0.968022i	$-0.580716\pi$
$887$	−6.45506e21	−0.501734	−0.250867	+	0.968022i	$0.580716\pi$
$888$	0	0
$889$	−6.60186e21	−0.504548
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.57633e22	−3.38167
$894$	0	0
$895$	−2.02876e22	−1.47421
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.68258e20	−0.0258794
$900$	0	0
$901$	−6.56904e21	−0.454011
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.27222e21	−0.0850544
$906$	0	0
$907$	−1.01966e22	−0.670504	−0.335252	−	0.942129i	$-0.608821\pi$
$907$	−1.01966e22	−0.670504	−0.335252	+	0.942129i	$0.608821\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.59652e22	−1.01575	−0.507873	−	0.861432i	$-0.669568\pi$
$911$	−1.59652e22	−1.01575	−0.507873	+	0.861432i	$0.669568\pi$
$912$	0	0
$913$	3.05865e22	1.91425
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.21197e21	0.497368
$918$	0	0
$919$	−6.79447e21	−0.404845	−0.202423	−	0.979298i	$-0.564881\pi$
$919$	−6.79447e21	−0.404845	−0.202423	+	0.979298i	$0.564881\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.67346e17	−5.00241e−5 0
$924$	0	0
$925$	2.95832e21	0.167873
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.30089e22	1.26409	0.632044	−	0.774933i	$-0.282216\pi$
$929$	2.30089e22	1.26409	0.632044	+	0.774933i	$0.282216\pi$
$930$	0	0
$931$	3.21256e20	0.0173671
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.83862e22	0.962509
$936$	0	0
$937$	−7.54775e21	−0.388840	−0.194420	−	0.980918i	$-0.562282\pi$
$937$	−7.54775e21	−0.388840	−0.194420	+	0.980918i	$0.562282\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.93039e22	0.963216	0.481608	−	0.876387i	$-0.340053\pi$
$941$	1.93039e22	0.963216	0.481608	+	0.876387i	$0.340053\pi$
$942$	0	0
$943$	−1.00888e22	−0.495454
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.44024e22	0.685187	0.342593	−	0.939484i	$-0.388695\pi$
$947$	1.44024e22	0.685187	0.342593	+	0.939484i	$0.388695\pi$
$948$	0	0
$949$	2.80678e18	0.000131435 0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.65091e22	−0.749077	−0.374539	−	0.927211i	$-0.622199\pi$
$953$	−1.65091e22	−0.749077	−0.374539	+	0.927211i	$0.622199\pi$
$954$	0	0
$955$	1.15597e22	0.516321
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.14243e22	0.494525
$960$	0	0
$961$	−2.33078e22	−0.993291
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.69374e22	−0.699666
$966$	0	0
$967$	1.85951e22	0.756311	0.378155	−	0.925742i	$-0.376558\pi$
$967$	1.85951e22	0.756311	0.378155	+	0.925742i	$0.376558\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.84934e22	1.51790	0.758948	−	0.651151i	$-0.225714\pi$
$971$	3.84934e22	1.51790	0.758948	+	0.651151i	$0.225714\pi$
$972$	0	0
$973$	−3.41674e22	−1.32668
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.69863e21	0.289870	0.144935	−	0.989441i	$-0.453703\pi$
$977$	7.69863e21	0.289870	0.144935	+	0.989441i	$0.453703\pi$
$978$	0	0
$979$	1.21532e22	0.450630
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.45537e22	−1.24263	−0.621316	−	0.783560i	$-0.713402\pi$
$983$	−3.45537e22	−1.24263	−0.621316	+	0.783560i	$0.713402\pi$
$984$	0	0
$985$	3.35834e22	1.18947
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.14795e22	1.08157
$990$	0	0
$991$	3.38548e22	1.14569	0.572846	−	0.819663i	$-0.305839\pi$
$991$	3.38548e22	1.14569	0.572846	+	0.819663i	$0.305839\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.09050e22	1.34309
$996$	0	0
$997$	−3.03253e22	−0.980826	−0.490413	−	0.871490i	$-0.663154\pi$
$997$	−3.03253e22	−0.980826	−0.490413	+	0.871490i	$0.663154\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.16.a.f.1.1		2
3.2	odd	2		24.16.a.c.1.2	✓	2
4.3	odd	2		144.16.a.u.1.1		2
12.11	even	2		48.16.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.16.a.c.1.2	✓	2	3.2	odd	2
48.16.a.h.1.2		2	12.11	even	2
72.16.a.f.1.1		2	1.1	even	1	trivial
144.16.a.u.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

\(f(q)\)	\(=\)	\(q-161936. q^{5} -2.16834e6 q^{7} +O(q^{10})\)	\(q-161936. q^{5} -2.16834e6 q^{7} +1.16723e8 q^{11} +37971.5 q^{13} -9.72730e8 q^{17} -7.00231e9 q^{19} -2.18197e10 q^{23} -4.29419e9 q^{25} +2.93511e10 q^{29} -1.25467e10 q^{31} +3.51133e11 q^{35} -6.88911e11 q^{37} +4.62373e11 q^{41} -1.44271e12 q^{43} +6.53547e12 q^{47} -4.58786e10 q^{49} +6.75320e12 q^{53} -1.89017e13 q^{55} -1.52767e13 q^{59} +3.29878e13 q^{61} -6.14897e9 q^{65} -1.34942e13 q^{67} -2.28420e13 q^{71} +7.39181e13 q^{73} -2.53094e14 q^{77} +1.69104e14 q^{79} +2.62044e14 q^{83} +1.57520e14 q^{85} +1.04120e14 q^{89} -8.23350e10 q^{91} +1.13393e15 q^{95} +1.54015e15 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18340 q^{5} - 680400 q^{7}+O(q^{10}) \) 2 * q + 18340 * q^5 - 680400 * q^7	\( 2 q + 18340 q^{5} - 680400 q^{7} + 77720632 q^{11} - 207881332 q^{13} + 1673313692 q^{17} - 2240887112 q^{19} - 16655801552 q^{23} - 2312200850 q^{25} + 139493975892 q^{29} - 43133623424 q^{31} + 619372303200 q^{35} - 1457105544996 q^{37} + 1377923837772 q^{41} - 2763654497656 q^{43} + 7396441682400 q^{47} - 2579485205486 q^{49} + 5824660261252 q^{53} - 25932807442960 q^{55} + 22792784775448 q^{59} + 14623417395116 q^{61} - 37489085612840 q^{65} + 68414350989784 q^{67} - 64404223302256 q^{71} + 229480150537460 q^{73} - 311126750827200 q^{77} - 8902155902944 q^{79} + 239472633418184 q^{83} + 634539486862840 q^{85} - 472605248938452 q^{89} - 309453044594400 q^{91} + 19\!\cdots\!60 q^{95}+ \cdots + 20\!\cdots\!48 q^{97}+O(q^{100}) \) 2 * q + 18340 * q^5 - 680400 * q^7 + 77720632 * q^11 - 207881332 * q^13 + 1673313692 * q^17 - 2240887112 * q^19 - 16655801552 * q^23 - 2312200850 * q^25 + 139493975892 * q^29 - 43133623424 * q^31 + 619372303200 * q^35 - 1457105544996 * q^37 + 1377923837772 * q^41 - 2763654497656 * q^43 + 7396441682400 * q^47 - 2579485205486 * q^49 + 5824660261252 * q^53 - 25932807442960 * q^55 + 22792784775448 * q^59 + 14623417395116 * q^61 - 37489085612840 * q^65 + 68414350989784 * q^67 - 64404223302256 * q^71 + 229480150537460 * q^73 - 311126750827200 * q^77 - 8902155902944 * q^79 + 239472633418184 * q^83 + 634539486862840 * q^85 - 472605248938452 * q^89 - 309453044594400 * q^91 + 1992299281656560 * q^95 + 2032136651194948 * q^97

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