LMFDB - Embedded newform 24.22.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,22,Mod(1,24)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, 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("24.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	24.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:	$67.0745626289$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2295485x - 828958533 $ x^3 - x^2 - 2295485*x - 828958533
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{4}\cdot 5\cdot 7 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1671.27$ of defining polynomial
Character	$\chi$	$=$	24.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-59049.0 q^{3} +4.18179e7 q^{5} -6.87132e8 q^{7} +3.48678e9 q^{9} +O(q^{10})$	$q-59049.0 q^{3} +4.18179e7 q^{5} -6.87132e8 q^{7} +3.48678e9 q^{9} +1.49871e9 q^{11} -8.84777e9 q^{13} -2.46931e12 q^{15} -9.41434e12 q^{17} -4.23870e13 q^{19} +4.05745e13 q^{21} +1.09316e14 q^{23} +1.27190e15 q^{25} -2.05891e14 q^{27} +1.18227e15 q^{29} -1.95307e15 q^{31} -8.84971e13 q^{33} -2.87345e16 q^{35} +4.20325e16 q^{37} +5.22452e14 q^{39} -6.92189e16 q^{41} +8.95366e16 q^{43} +1.45810e17 q^{45} -5.26493e17 q^{47} -8.63955e16 q^{49} +5.55907e17 q^{51} -2.24753e18 q^{53} +6.26728e16 q^{55} +2.50291e18 q^{57} -2.41265e17 q^{59} +3.17652e18 q^{61} -2.39588e18 q^{63} -3.69996e17 q^{65} -2.04213e19 q^{67} -6.45498e18 q^{69} -3.92177e19 q^{71} -4.75993e19 q^{73} -7.51046e19 q^{75} -1.02981e18 q^{77} +8.63125e18 q^{79} +1.21577e19 q^{81} +2.65834e20 q^{83} -3.93688e20 q^{85} -6.98121e19 q^{87} -4.57167e20 q^{89} +6.07959e18 q^{91} +1.15327e20 q^{93} -1.77254e21 q^{95} -9.21686e20 q^{97} +5.22566e18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 177147 q^{3} + 2080026 q^{5} - 1205282064 q^{7} + 10460353203 q^{9}+O(q^{10}) $ 3 * q - 177147 * q^3 + 2080026 * q^5 - 1205282064 * q^7 + 10460353203 * q^9	$ 3 q - 177147 q^{3} + 2080026 q^{5} - 1205282064 q^{7} + 10460353203 q^{9} + 13839247500 q^{11} + 718855551690 q^{13} - 122823455274 q^{15} + 2135189843046 q^{17} - 40122324686988 q^{19} + 71170700597136 q^{21} + 278424417682632 q^{23} + 13\!\cdots\!01 q^{25}+ \cdots + 48\!\cdots\!00 q^{99}+O(q^{100}) $ 3 * q - 177147 * q^3 + 2080026 * q^5 - 1205282064 * q^7 + 10460353203 * q^9 + 13839247500 * q^11 + 718855551690 * q^13 - 122823455274 * q^15 + 2135189843046 * q^17 - 40122324686988 * q^19 + 71170700597136 * q^21 + 278424417682632 * q^23 + 1348043260553901 * q^25 - 617673396283947 * q^27 + 442708167991794 * q^29 + 8016070162990152 * q^31 - 817193725627500 * q^33 + 125384157242400 * q^35 + 27729341388737058 * q^37 - 42447701471742810 * q^39 - 125648125186340562 * q^41 - 229052541499074612 * q^43 + 7252602210474426 * q^45 - 448613782068047712 * q^47 + 365221903446092427 * q^49 - 126080825042023254 * q^51 + 1406206217208267066 * q^53 - 3437829264920292504 * q^55 + 2369183150441954412 * q^57 - 1844638981471622100 * q^59 - 3294066300350351382 * q^61 - 4202558699560283664 * q^63 - 19537666262756991444 * q^65 - 33491023693155020652 * q^67 - 16440683439741736968 * q^69 - 79431018431598881160 * q^71 - 46612822906958319618 * q^73 - 79600606492447300149 * q^75 - 255759806110876888128 * q^77 - 197919973704661098024 * q^79 + 36472996377170786403 * q^81 + 111848528551886940276 * q^83 - 531596971435705186956 * q^85 - 26141474611747443906 * q^87 - 731008175840243483058 * q^89 - 546304658547739915488 * q^91 - 473340927054405485448 * q^93 - 1952599110289053971304 * q^95 - 1594521920055126193722 * q^97 + 48254472304578247500 * q^99

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
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	0	0
$5$	4.18179e7	1.91504	0.957520	−	0.288368i	$-0.0931127\pi$
$5$	4.18179e7	1.91504	0.957520	+	0.288368i	$0.0931127\pi$
$6$	0	0
$7$	−6.87132e8	−0.919413	−0.459707	−	0.888071i	$-0.652045\pi$
$7$	−6.87132e8	−0.919413	−0.459707	+	0.888071i	$0.652045\pi$
$8$	0	0
$9$	3.48678e9	0.333333
$10$	0	0
$11$	1.49871e9	0.0174218	0.00871090	−	0.999962i	$-0.497227\pi$
$11$	1.49871e9	0.0174218	0.00871090	+	0.999962i	$0.497227\pi$
$12$	0	0
$13$	−8.84777e9	−0.0178004	−0.00890018	−	0.999960i	$-0.502833\pi$
$13$	−8.84777e9	−0.0178004	−0.00890018	+	0.999960i	$0.502833\pi$
$14$	0	0
$15$	−2.46931e12	−1.10565
$16$	0	0
$17$	−9.41434e12	−1.13260	−0.566299	−	0.824200i	$-0.691625\pi$
$17$	−9.41434e12	−1.13260	−0.566299	+	0.824200i	$0.691625\pi$
$18$	0	0
$19$	−4.23870e13	−1.58606	−0.793030	−	0.609183i	$-0.791498\pi$
$19$	−4.23870e13	−1.58606	−0.793030	+	0.609183i	$0.791498\pi$
$20$	0	0
$21$	4.05745e13	0.530823
$22$	0	0
$23$	1.09316e14	0.550225	0.275112	−	0.961412i	$-0.411285\pi$
$23$	1.09316e14	0.550225	0.275112	+	0.961412i	$0.411285\pi$
$24$	0	0
$25$	1.27190e15	2.66738
$26$	0	0
$27$	−2.05891e14	−0.192450
$28$	0	0
$29$	1.18227e15	0.521842	0.260921	−	0.965360i	$-0.415974\pi$
$29$	1.18227e15	0.521842	0.260921	+	0.965360i	$0.415974\pi$
$30$	0	0
$31$	−1.95307e15	−0.427977	−0.213988	−	0.976836i	$-0.568645\pi$
$31$	−1.95307e15	−0.427977	−0.213988	+	0.976836i	$0.568645\pi$
$32$	0	0
$33$	−8.84971e13	−0.0100585
$34$	0	0
$35$	−2.87345e16	−1.76071
$36$	0	0
$37$	4.20325e16	1.43704	0.718518	−	0.695509i	$-0.244821\pi$
$37$	4.20325e16	1.43704	0.718518	+	0.695509i	$0.244821\pi$
$38$	0	0
$39$	5.22452e14	0.0102770
$40$	0	0
$41$	−6.92189e16	−0.805367	−0.402684	−	0.915339i	$-0.631923\pi$
$41$	−6.92189e16	−0.805367	−0.402684	+	0.915339i	$0.631923\pi$
$42$	0	0
$43$	8.95366e16	0.631803	0.315902	−	0.948792i	$-0.397693\pi$
$43$	8.95366e16	0.631803	0.315902	+	0.948792i	$0.397693\pi$
$44$	0	0
$45$	1.45810e17	0.638346
$46$	0	0
$47$	−5.26493e17	−1.46004	−0.730021	−	0.683425i	$-0.760490\pi$
$47$	−5.26493e17	−1.46004	−0.730021	+	0.683425i	$0.760490\pi$
$48$	0	0
$49$	−8.63955e16	−0.154679
$50$	0	0
$51$	5.55907e17	0.653906
$52$	0	0
$53$	−2.24753e18	−1.76526	−0.882629	−	0.470070i	$-0.844229\pi$
$53$	−2.24753e18	−1.76526	−0.882629	+	0.470070i	$0.844229\pi$
$54$	0	0
$55$	6.26728e16	0.0333634
$56$	0	0
$57$	2.50291e18	0.915712
$58$	0	0
$59$	−2.41265e17	−0.0614538	−0.0307269	−	0.999528i	$-0.509782\pi$
$59$	−2.41265e17	−0.0614538	−0.0307269	+	0.999528i	$0.509782\pi$
$60$	0	0
$61$	3.17652e18	0.570148	0.285074	−	0.958505i	$-0.407982\pi$
$61$	3.17652e18	0.570148	0.285074	+	0.958505i	$0.407982\pi$
$62$	0	0
$63$	−2.39588e18	−0.306471
$64$	0	0
$65$	−3.69996e17	−0.0340884
$66$	0	0
$67$	−2.04213e19	−1.36867	−0.684334	−	0.729169i	$-0.739907\pi$
$67$	−2.04213e19	−1.36867	−0.684334	+	0.729169i	$0.739907\pi$
$68$	0	0
$69$	−6.45498e18	−0.317672
$70$	0	0
$71$	−3.92177e19	−1.42978	−0.714891	−	0.699236i	$-0.753524\pi$
$71$	−3.92177e19	−1.42978	−0.714891	+	0.699236i	$0.753524\pi$
$72$	0	0
$73$	−4.75993e19	−1.29631	−0.648157	−	0.761507i	$-0.724460\pi$
$73$	−4.75993e19	−1.29631	−0.648157	+	0.761507i	$0.724460\pi$
$74$	0	0
$75$	−7.51046e19	−1.54001
$76$	0	0
$77$	−1.02981e18	−0.0160178
$78$	0	0
$79$	8.63125e18	0.102563	0.0512813	−	0.998684i	$-0.483669\pi$
$79$	8.63125e18	0.102563	0.0512813	+	0.998684i	$0.483669\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	0	0
$83$	2.65834e20	1.88057	0.940287	−	0.340382i	$-0.110556\pi$
$83$	2.65834e20	1.88057	0.940287	+	0.340382i	$0.110556\pi$
$84$	0	0
$85$	−3.93688e20	−2.16897
$86$	0	0
$87$	−6.98121e19	−0.301286
$88$	0	0
$89$	−4.57167e20	−1.55410	−0.777051	−	0.629438i	$-0.783285\pi$
$89$	−4.57167e20	−1.55410	−0.777051	+	0.629438i	$0.783285\pi$
$90$	0	0
$91$	6.07959e18	0.0163659
$92$	0	0
$93$	1.15327e20	0.247093
$94$	0	0
$95$	−1.77254e21	−3.03737
$96$	0	0
$97$	−9.21686e20	−1.26905	−0.634527	−	0.772901i	$-0.718805\pi$
$97$	−9.21686e20	−1.26905	−0.634527	+	0.772901i	$0.718805\pi$
$98$	0	0
$99$	5.22566e18	0.00580727
$100$	0	0
$101$	1.57744e21	1.42095	0.710475	−	0.703722i	$-0.248480\pi$
$101$	1.57744e21	1.42095	0.710475	+	0.703722i	$0.248480\pi$
$102$	0	0
$103$	1.20014e21	0.879919	0.439959	−	0.898018i	$-0.354993\pi$
$103$	1.20014e21	0.879919	0.439959	+	0.898018i	$0.354993\pi$
$104$	0	0
$105$	1.69674e21	1.01655
$106$	0	0
$107$	−3.40780e21	−1.67473	−0.837365	−	0.546644i	$-0.815905\pi$
$107$	−3.40780e21	−1.67473	−0.837365	+	0.546644i	$0.815905\pi$
$108$	0	0
$109$	1.46243e21	0.591693	0.295846	−	0.955236i	$-0.404398\pi$
$109$	1.46243e21	0.591693	0.295846	+	0.955236i	$0.404398\pi$
$110$	0	0
$111$	−2.48198e21	−0.829673
$112$	0	0
$113$	−1.53398e21	−0.425105	−0.212552	−	0.977150i	$-0.568178\pi$
$113$	−1.53398e21	−0.425105	−0.212552	+	0.977150i	$0.568178\pi$
$114$	0	0
$115$	4.57136e21	1.05370
$116$	0	0
$117$	−3.08503e19	−0.00593345
$118$	0	0
$119$	6.46889e21	1.04133
$120$	0	0
$121$	−7.39800e21	−0.999696
$122$	0	0
$123$	4.08731e21	0.464979
$124$	0	0
$125$	3.32481e22	3.19309
$126$	0	0
$127$	1.28428e21	0.104405	0.0522024	−	0.998637i	$-0.483376\pi$
$127$	1.28428e21	0.104405	0.0522024	+	0.998637i	$0.483376\pi$
$128$	0	0
$129$	−5.28705e21	−0.364772
$130$	0	0
$131$	1.22923e22	0.721577	0.360789	−	0.932648i	$-0.382508\pi$
$131$	1.22923e22	0.721577	0.360789	+	0.932648i	$0.382508\pi$
$132$	0	0
$133$	2.91254e22	1.45824
$134$	0	0
$135$	−8.60995e21	−0.368549
$136$	0	0
$137$	−1.02076e22	−0.374420	−0.187210	−	0.982320i	$-0.559945\pi$
$137$	−1.02076e22	−0.374420	−0.187210	+	0.982320i	$0.559945\pi$
$138$	0	0
$139$	−5.48653e22	−1.72839	−0.864197	−	0.503154i	$-0.832173\pi$
$139$	−5.48653e22	−1.72839	−0.864197	+	0.503154i	$0.832173\pi$
$140$	0	0
$141$	3.10889e22	0.842955
$142$	0	0
$143$	−1.32602e19	−0.000310114 0
$144$	0	0
$145$	4.94403e22	0.999348
$146$	0	0
$147$	5.10157e21	0.0893041
$148$	0	0
$149$	−3.46082e22	−0.525682	−0.262841	−	0.964839i	$-0.584659\pi$
$149$	−3.46082e22	−0.525682	−0.262841	+	0.964839i	$0.584659\pi$
$150$	0	0
$151$	−3.56145e21	−0.0470294	−0.0235147	−	0.999723i	$-0.507486\pi$
$151$	−3.56145e21	−0.0470294	−0.0235147	+	0.999723i	$0.507486\pi$
$152$	0	0
$153$	−3.28258e22	−0.377533
$154$	0	0
$155$	−8.16734e22	−0.819592
$156$	0	0
$157$	2.21166e23	1.93987	0.969933	−	0.243374i	$-0.0782541\pi$
$157$	2.21166e23	1.93987	0.969933	+	0.243374i	$0.0782541\pi$
$158$	0	0
$159$	1.32714e23	1.01917
$160$	0	0
$161$	−7.51143e22	−0.505884
$162$	0	0
$163$	−3.21431e23	−1.90159	−0.950797	−	0.309814i	$-0.899733\pi$
$163$	−3.21431e23	−1.90159	−0.950797	+	0.309814i	$0.899733\pi$
$164$	0	0
$165$	−3.70077e21	−0.0192624
$166$	0	0
$167$	−1.92776e23	−0.884158	−0.442079	−	0.896976i	$-0.645759\pi$
$167$	−1.92776e23	−0.884158	−0.442079	+	0.896976i	$0.645759\pi$
$168$	0	0
$169$	−2.46986e23	−0.999683
$170$	0	0
$171$	−1.47794e23	−0.528687
$172$	0	0
$173$	1.48749e23	0.470946	0.235473	−	0.971881i	$-0.424336\pi$
$173$	1.48749e23	0.470946	0.235473	+	0.971881i	$0.424336\pi$
$174$	0	0
$175$	−8.73966e23	−2.45242
$176$	0	0
$177$	1.42465e22	0.0354804
$178$	0	0
$179$	−3.04507e23	−0.673970	−0.336985	−	0.941510i	$-0.609407\pi$
$179$	−3.04507e23	−0.673970	−0.336985	+	0.941510i	$0.609407\pi$
$180$	0	0
$181$	−2.74401e23	−0.540457	−0.270228	−	0.962796i	$-0.587099\pi$
$181$	−2.74401e23	−0.540457	−0.270228	+	0.962796i	$0.587099\pi$
$182$	0	0
$183$	−1.87570e23	−0.329175
$184$	0	0
$185$	1.75771e24	2.75198
$186$	0	0
$187$	−1.41093e22	−0.0197319
$188$	0	0
$189$	1.41474e23	0.176941
$190$	0	0
$191$	3.88356e22	0.0434890	0.0217445	−	0.999764i	$-0.493078\pi$
$191$	3.88356e22	0.0434890	0.0217445	+	0.999764i	$0.493078\pi$
$192$	0	0
$193$	4.72989e23	0.474787	0.237394	−	0.971414i	$-0.423707\pi$
$193$	4.72989e23	0.474787	0.237394	+	0.971414i	$0.423707\pi$
$194$	0	0
$195$	2.18479e22	0.0196809
$196$	0	0
$197$	9.09009e22	0.0735653	0.0367826	−	0.999323i	$-0.488289\pi$
$197$	9.09009e22	0.0735653	0.0367826	+	0.999323i	$0.488289\pi$
$198$	0	0
$199$	9.81119e23	0.714109	0.357055	−	0.934084i	$-0.383781\pi$
$199$	9.81119e23	0.714109	0.357055	+	0.934084i	$0.383781\pi$
$200$	0	0
$201$	1.20586e24	0.790200
$202$	0	0
$203$	−8.12378e23	−0.479789
$204$	0	0
$205$	−2.89459e24	−1.54231
$206$	0	0
$207$	3.81160e23	0.183408
$208$	0	0
$209$	−6.35256e22	−0.0276320
$210$	0	0
$211$	1.28388e24	0.505312	0.252656	−	0.967556i	$-0.418696\pi$
$211$	1.28388e24	0.505312	0.252656	+	0.967556i	$0.418696\pi$
$212$	0	0
$213$	2.31577e24	0.825485
$214$	0	0
$215$	3.74424e24	1.20993
$216$	0	0
$217$	1.34202e24	0.393488
$218$	0	0
$219$	2.81069e24	0.748427
$220$	0	0
$221$	8.32959e22	0.0201607
$222$	0	0
$223$	1.11219e24	0.244893	0.122446	−	0.992475i	$-0.460926\pi$
$223$	1.11219e24	0.244893	0.122446	+	0.992475i	$0.460926\pi$
$224$	0	0
$225$	4.43485e24	0.889125
$226$	0	0
$227$	1.79086e24	0.327182	0.163591	−	0.986528i	$-0.447692\pi$
$227$	1.79086e24	0.327182	0.163591	+	0.986528i	$0.447692\pi$
$228$	0	0
$229$	−5.98363e24	−0.996993	−0.498496	−	0.866892i	$-0.666114\pi$
$229$	−5.98363e24	−0.996993	−0.498496	+	0.866892i	$0.666114\pi$
$230$	0	0
$231$	6.08092e22	0.00924790
$232$	0	0
$233$	8.20293e24	1.13955	0.569773	−	0.821802i	$-0.307031\pi$
$233$	8.20293e24	1.13955	0.569773	+	0.821802i	$0.307031\pi$
$234$	0	0
$235$	−2.20169e25	−2.79604
$236$	0	0
$237$	−5.09667e23	−0.0592146
$238$	0	0
$239$	7.01227e24	0.745900	0.372950	−	0.927851i	$-0.378346\pi$
$239$	7.01227e24	0.745900	0.372950	+	0.927851i	$0.378346\pi$
$240$	0	0
$241$	−1.03693e25	−1.01058	−0.505288	−	0.862951i	$-0.668614\pi$
$241$	−1.03693e25	−1.01058	−0.505288	+	0.862951i	$0.668614\pi$
$242$	0	0
$243$	−7.17898e23	−0.0641500
$244$	0	0
$245$	−3.61288e24	−0.296217
$246$	0	0
$247$	3.75030e23	0.0282324
$248$	0	0
$249$	−1.56972e25	−1.08575
$250$	0	0
$251$	2.28168e25	1.45104	0.725522	−	0.688199i	$-0.241598\pi$
$251$	2.28168e25	1.45104	0.725522	+	0.688199i	$0.241598\pi$
$252$	0	0
$253$	1.63832e23	0.00958591
$254$	0	0
$255$	2.32469e25	1.25226
$256$	0	0
$257$	−1.57165e25	−0.779935	−0.389968	−	0.920829i	$-0.627514\pi$
$257$	−1.57165e25	−0.779935	−0.389968	+	0.920829i	$0.627514\pi$
$258$	0	0
$259$	−2.88819e25	−1.32123
$260$	0	0
$261$	4.12233e24	0.173947
$262$	0	0
$263$	1.60373e24	0.0624593	0.0312296	−	0.999512i	$-0.490058\pi$
$263$	1.60373e24	0.0624593	0.0312296	+	0.999512i	$0.490058\pi$
$264$	0	0
$265$	−9.39870e25	−3.38054
$266$	0	0
$267$	2.69952e25	0.897261
$268$	0	0
$269$	−4.92003e25	−1.51206	−0.756029	−	0.654538i	$-0.772863\pi$
$269$	−4.92003e25	−1.51206	−0.756029	+	0.654538i	$0.772863\pi$
$270$	0	0
$271$	−7.98786e24	−0.227119	−0.113559	−	0.993531i	$-0.536225\pi$
$271$	−7.98786e24	−0.227119	−0.113559	+	0.993531i	$0.536225\pi$
$272$	0	0
$273$	−3.58993e23	−0.00944885
$274$	0	0
$275$	1.90621e24	0.0464705
$276$	0	0
$277$	2.26388e25	0.511465	0.255733	−	0.966748i	$-0.417683\pi$
$277$	2.26388e25	0.511465	0.255733	+	0.966748i	$0.417683\pi$
$278$	0	0
$279$	−6.80994e24	−0.142659
$280$	0	0
$281$	7.09532e24	0.137897	0.0689486	−	0.997620i	$-0.478036\pi$
$281$	7.09532e24	0.137897	0.0689486	+	0.997620i	$0.478036\pi$
$282$	0	0
$283$	1.98336e24	0.0357804	0.0178902	−	0.999840i	$-0.494305\pi$
$283$	1.98336e24	0.0357804	0.0178902	+	0.999840i	$0.494305\pi$
$284$	0	0
$285$	1.04666e26	1.75362
$286$	0	0
$287$	4.75625e25	0.740465
$288$	0	0
$289$	1.95378e25	0.282780
$290$	0	0
$291$	5.44246e25	0.732688
$292$	0	0
$293$	1.17672e26	1.47422	0.737109	−	0.675774i	$-0.236190\pi$
$293$	1.17672e26	1.47422	0.737109	+	0.675774i	$0.236190\pi$
$294$	0	0
$295$	−1.00892e25	−0.117687
$296$	0	0
$297$	−3.08570e23	−0.00335283
$298$	0	0
$299$	−9.67200e23	−0.00979419
$300$	0	0
$301$	−6.15235e25	−0.580888
$302$	0	0
$303$	−9.31464e25	−0.820386
$304$	0	0
$305$	1.32835e26	1.09186
$306$	0	0
$307$	7.71265e25	0.591903	0.295952	−	0.955203i	$-0.404363\pi$
$307$	7.71265e25	0.591903	0.295952	+	0.955203i	$0.404363\pi$
$308$	0	0
$309$	−7.08673e25	−0.508021
$310$	0	0
$311$	7.96322e25	0.533463	0.266732	−	0.963771i	$-0.414056\pi$
$311$	7.96322e25	0.533463	0.266732	+	0.963771i	$0.414056\pi$
$312$	0	0
$313$	−5.68844e25	−0.356269	−0.178134	−	0.984006i	$-0.557006\pi$
$313$	−5.68844e25	−0.356269	−0.178134	+	0.984006i	$0.557006\pi$
$314$	0	0
$315$	−1.00191e26	−0.586904
$316$	0	0
$317$	−1.51016e26	−0.827755	−0.413878	−	0.910333i	$-0.635826\pi$
$317$	−1.51016e26	−0.827755	−0.413878	+	0.910333i	$0.635826\pi$
$318$	0	0
$319$	1.77188e24	0.00909143
$320$	0	0
$321$	2.01227e26	0.966906
$322$	0	0
$323$	3.99045e26	1.79637
$324$	0	0
$325$	−1.12535e25	−0.0474802
$326$	0	0
$327$	−8.63549e25	−0.341614
$328$	0	0
$329$	3.61770e26	1.34238
$330$	0	0
$331$	−3.29882e26	−1.14859	−0.574295	−	0.818649i	$-0.694724\pi$
$331$	−3.29882e26	−1.14859	−0.574295	+	0.818649i	$0.694724\pi$
$332$	0	0
$333$	1.46558e26	0.479012
$334$	0	0
$335$	−8.53977e26	−2.62105
$336$	0	0
$337$	4.68433e26	1.35062	0.675310	−	0.737534i	$-0.264010\pi$
$337$	4.68433e26	1.35062	0.675310	+	0.737534i	$0.264010\pi$
$338$	0	0
$339$	9.05800e25	0.245434
$340$	0	0
$341$	−2.92708e24	−0.00745613
$342$	0	0
$343$	4.43160e26	1.06163
$344$	0	0
$345$	−2.69934e26	−0.608355
$346$	0	0
$347$	−6.05861e26	−1.28503	−0.642515	−	0.766273i	$-0.722109\pi$
$347$	−6.05861e26	−1.28503	−0.642515	+	0.766273i	$0.722109\pi$
$348$	0	0
$349$	8.48298e25	0.169388	0.0846938	−	0.996407i	$-0.473009\pi$
$349$	8.48298e25	0.169388	0.0846938	+	0.996407i	$0.473009\pi$
$350$	0	0
$351$	1.82168e24	0.00342568
$352$	0	0
$353$	−6.10413e26	−1.08141	−0.540704	−	0.841213i	$-0.681842\pi$
$353$	−6.10413e26	−1.08141	−0.540704	+	0.841213i	$0.681842\pi$
$354$	0	0
$355$	−1.64000e27	−2.73809
$356$	0	0
$357$	−3.81982e26	−0.601210
$358$	0	0
$359$	−6.20073e26	−0.920346	−0.460173	−	0.887829i	$-0.652213\pi$
$359$	−6.20073e26	−0.920346	−0.460173	+	0.887829i	$0.652213\pi$
$360$	0	0
$361$	1.08245e27	1.51558
$362$	0	0
$363$	4.36845e26	0.577175
$364$	0	0
$365$	−1.99050e27	−2.48249
$366$	0	0
$367$	1.05311e27	1.24017	0.620084	−	0.784536i	$-0.287099\pi$
$367$	1.05311e27	1.24017	0.620084	+	0.784536i	$0.287099\pi$
$368$	0	0
$369$	−2.41351e26	−0.268456
$370$	0	0
$371$	1.54435e27	1.62300
$372$	0	0
$373$	9.54206e26	0.947762	0.473881	−	0.880589i	$-0.342853\pi$
$373$	9.54206e26	0.947762	0.473881	+	0.880589i	$0.342853\pi$
$374$	0	0
$375$	−1.96326e27	−1.84353
$376$	0	0
$377$	−1.04605e25	−0.00928897
$378$	0	0
$379$	7.26138e26	0.609968	0.304984	−	0.952357i	$-0.401349\pi$
$379$	7.26138e26	0.609968	0.304984	+	0.952357i	$0.401349\pi$
$380$	0	0
$381$	−7.58355e25	−0.0602782
$382$	0	0
$383$	1.16288e27	0.874879	0.437439	−	0.899248i	$-0.355885\pi$
$383$	1.16288e27	0.874879	0.437439	+	0.899248i	$0.355885\pi$
$384$	0	0
$385$	−4.30645e25	−0.0306748
$386$	0	0
$387$	3.12195e26	0.210601
$388$	0	0
$389$	1.26901e27	0.810951	0.405476	−	0.914106i	$-0.367106\pi$
$389$	1.26901e27	0.810951	0.405476	+	0.914106i	$0.367106\pi$
$390$	0	0
$391$	−1.02913e27	−0.623184
$392$	0	0
$393$	−7.25845e26	−0.416603
$394$	0	0
$395$	3.60941e26	0.196412
$396$	0	0
$397$	3.15060e27	1.62590	0.812948	−	0.582337i	$-0.197862\pi$
$397$	3.15060e27	1.62590	0.812948	+	0.582337i	$0.197862\pi$
$398$	0	0
$399$	−1.71983e27	−0.841918
$400$	0	0
$401$	−2.77564e26	−0.128928	−0.0644640	−	0.997920i	$-0.520534\pi$
$401$	−2.77564e26	−0.128928	−0.0644640	+	0.997920i	$0.520534\pi$
$402$	0	0
$403$	1.72803e25	0.00761814
$404$	0	0
$405$	5.08409e26	0.212782
$406$	0	0
$407$	6.29944e25	0.0250357
$408$	0	0
$409$	−1.19865e27	−0.452478	−0.226239	−	0.974072i	$-0.572643\pi$
$409$	−1.19865e27	−0.452478	−0.226239	+	0.974072i	$0.572643\pi$
$410$	0	0
$411$	6.02751e26	0.216172
$412$	0	0
$413$	1.65781e26	0.0565015
$414$	0	0
$415$	1.11166e28	3.60137
$416$	0	0
$417$	3.23974e27	0.997888
$418$	0	0
$419$	5.80111e27	1.69928	0.849638	−	0.527366i	$-0.176820\pi$
$419$	5.80111e27	1.69928	0.849638	+	0.527366i	$0.176820\pi$
$420$	0	0
$421$	5.69555e26	0.158699	0.0793494	−	0.996847i	$-0.474716\pi$
$421$	5.69555e26	0.158699	0.0793494	+	0.996847i	$0.474716\pi$
$422$	0	0
$423$	−1.83577e27	−0.486680
$424$	0	0
$425$	−1.19741e28	−3.02107
$426$	0	0
$427$	−2.18269e27	−0.524202
$428$	0	0
$429$	7.83002e23	0.000179045 0
$430$	0	0
$431$	−1.82453e27	−0.397320	−0.198660	−	0.980069i	$-0.563659\pi$
$431$	−1.82453e27	−0.397320	−0.198660	+	0.980069i	$0.563659\pi$
$432$	0	0
$433$	4.12366e27	0.855381	0.427690	−	0.903925i	$-0.359327\pi$
$433$	4.12366e27	0.855381	0.427690	+	0.903925i	$0.359327\pi$
$434$	0	0
$435$	−2.91940e27	−0.576974
$436$	0	0
$437$	−4.63356e27	−0.872689
$438$	0	0
$439$	6.93325e27	1.24468	0.622342	−	0.782745i	$-0.286181\pi$
$439$	6.93325e27	1.24468	0.622342	+	0.782745i	$0.286181\pi$
$440$	0	0
$441$	−3.01242e26	−0.0515598
$442$	0	0
$443$	−1.53594e27	−0.250689	−0.125345	−	0.992113i	$-0.540004\pi$
$443$	−1.53594e27	−0.250689	−0.125345	+	0.992113i	$0.540004\pi$
$444$	0	0
$445$	−1.91178e28	−2.97617
$446$	0	0
$447$	2.04358e27	0.303502
$448$	0	0
$449$	−1.32087e27	−0.187186	−0.0935929	−	0.995611i	$-0.529835\pi$
$449$	−1.32087e27	−0.187186	−0.0935929	+	0.995611i	$0.529835\pi$
$450$	0	0
$451$	−1.03739e26	−0.0140310
$452$	0	0
$453$	2.10300e26	0.0271524
$454$	0	0
$455$	2.54236e26	0.0313413
$456$	0	0
$457$	2.28794e27	0.269354	0.134677	−	0.990890i	$-0.457000\pi$
$457$	2.28794e27	0.269354	0.134677	+	0.990890i	$0.457000\pi$
$458$	0	0
$459$	1.93833e27	0.217969
$460$	0	0
$461$	8.47625e27	0.910635	0.455317	−	0.890329i	$-0.349526\pi$
$461$	8.47625e27	0.910635	0.455317	+	0.890329i	$0.349526\pi$
$462$	0	0
$463$	−9.11529e27	−0.935772	−0.467886	−	0.883789i	$-0.654984\pi$
$463$	−9.11529e27	−0.935772	−0.467886	+	0.883789i	$0.654984\pi$
$464$	0	0
$465$	4.82273e27	0.473192
$466$	0	0
$467$	1.41181e28	1.32419	0.662093	−	0.749421i	$-0.269668\pi$
$467$	1.41181e28	1.32419	0.662093	+	0.749421i	$0.269668\pi$
$468$	0	0
$469$	1.40321e28	1.25837
$470$	0	0
$471$	−1.30596e28	−1.11998
$472$	0	0
$473$	1.34189e26	0.0110072
$474$	0	0
$475$	−5.39121e28	−4.23062
$476$	0	0
$477$	−7.83664e27	−0.588419
$478$	0	0
$479$	2.20208e28	1.58238	0.791189	−	0.611572i	$-0.209463\pi$
$479$	2.20208e28	1.58238	0.791189	+	0.611572i	$0.209463\pi$
$480$	0	0
$481$	−3.71894e26	−0.0255797
$482$	0	0
$483$	4.43542e27	0.292072
$484$	0	0
$485$	−3.85430e28	−2.43029
$486$	0	0
$487$	−1.76771e28	−1.06748	−0.533738	−	0.845650i	$-0.679213\pi$
$487$	−1.76771e28	−1.06748	−0.533738	+	0.845650i	$0.679213\pi$
$488$	0	0
$489$	1.89802e28	1.09789
$490$	0	0
$491$	−5.25882e27	−0.291429	−0.145715	−	0.989327i	$-0.546548\pi$
$491$	−5.25882e27	−0.291429	−0.145715	+	0.989327i	$0.546548\pi$
$492$	0	0
$493$	−1.11303e28	−0.591038
$494$	0	0
$495$	2.18527e26	0.0111211
$496$	0	0
$497$	2.69478e28	1.31456
$498$	0	0
$499$	−1.70312e28	−0.796508	−0.398254	−	0.917275i	$-0.630384\pi$
$499$	−1.70312e28	−0.796508	−0.398254	+	0.917275i	$0.630384\pi$
$500$	0	0
$501$	1.13832e28	0.510469
$502$	0	0
$503$	7.74942e27	0.333277	0.166639	−	0.986018i	$-0.446709\pi$
$503$	7.74942e27	0.333277	0.166639	+	0.986018i	$0.446709\pi$
$504$	0	0
$505$	6.59654e28	2.72118
$506$	0	0
$507$	1.45843e28	0.577167
$508$	0	0
$509$	−2.30034e28	−0.873483	−0.436742	−	0.899587i	$-0.643868\pi$
$509$	−2.30034e28	−0.873483	−0.436742	+	0.899587i	$0.643868\pi$
$510$	0	0
$511$	3.27070e28	1.19185
$512$	0	0
$513$	8.72710e27	0.305237
$514$	0	0
$515$	5.01876e28	1.68508
$516$	0	0
$517$	−7.89058e26	−0.0254366
$518$	0	0
$519$	−8.78350e27	−0.271901
$520$	0	0
$521$	−6.09418e27	−0.181184	−0.0905918	−	0.995888i	$-0.528876\pi$
$521$	−6.09418e27	−0.181184	−0.0905918	+	0.995888i	$0.528876\pi$
$522$	0	0
$523$	2.18205e28	0.623155	0.311578	−	0.950221i	$-0.399143\pi$
$523$	2.18205e28	0.623155	0.311578	+	0.950221i	$0.399143\pi$
$524$	0	0
$525$	5.16068e28	1.41591
$526$	0	0
$527$	1.83869e28	0.484726
$528$	0	0
$529$	−2.75217e28	−0.697253
$530$	0	0
$531$	−8.41241e26	−0.0204846
$532$	0	0
$533$	6.12433e26	0.0143358
$534$	0	0
$535$	−1.42507e29	−3.20718
$536$	0	0
$537$	1.79808e28	0.389117
$538$	0	0
$539$	−1.29481e26	−0.00269479
$540$	0	0
$541$	−6.59608e27	−0.132043	−0.0660214	−	0.997818i	$-0.521031\pi$
$541$	−6.59608e27	−0.132043	−0.0660214	+	0.997818i	$0.521031\pi$
$542$	0	0
$543$	1.62031e28	0.312033
$544$	0	0
$545$	6.11557e28	1.13311
$546$	0	0
$547$	4.42458e28	0.788870	0.394435	−	0.918924i	$-0.370940\pi$
$547$	4.42458e28	0.788870	0.394435	+	0.918924i	$0.370940\pi$
$548$	0	0
$549$	1.10758e28	0.190049
$550$	0	0
$551$	−5.01130e28	−0.827673
$552$	0	0
$553$	−5.93081e27	−0.0942975
$554$	0	0
$555$	−1.03791e29	−1.58886
$556$	0	0
$557$	4.40727e28	0.649666	0.324833	−	0.945771i	$-0.394692\pi$
$557$	4.40727e28	0.649666	0.324833	+	0.945771i	$0.394692\pi$
$558$	0	0
$559$	−7.92199e26	−0.0112463
$560$	0	0
$561$	8.33141e26	0.0113922
$562$	0	0
$563$	6.93890e28	0.914013	0.457006	−	0.889463i	$-0.348922\pi$
$563$	6.93890e28	0.914013	0.457006	+	0.889463i	$0.348922\pi$
$564$	0	0
$565$	−6.41479e28	−0.814093
$566$	0	0
$567$	−8.35392e27	−0.102157
$568$	0	0
$569$	−1.87409e28	−0.220858	−0.110429	−	0.993884i	$-0.535222\pi$
$569$	−1.87409e28	−0.220858	−0.110429	+	0.993884i	$0.535222\pi$
$570$	0	0
$571$	−3.73641e28	−0.424400	−0.212200	−	0.977226i	$-0.568063\pi$
$571$	−3.73641e28	−0.424400	−0.212200	+	0.977226i	$0.568063\pi$
$572$	0	0
$573$	−2.29320e27	−0.0251084
$574$	0	0
$575$	1.39039e29	1.46766
$576$	0	0
$577$	−1.05657e29	−1.07536	−0.537680	−	0.843149i	$-0.680699\pi$
$577$	−1.05657e29	−1.07536	−0.537680	+	0.843149i	$0.680699\pi$
$578$	0	0
$579$	−2.79295e28	−0.274118
$580$	0	0
$581$	−1.82663e29	−1.72903
$582$	0	0
$583$	−3.36838e27	−0.0307540
$584$	0	0
$585$	−1.29009e27	−0.0113628
$586$	0	0
$587$	−1.31403e29	−1.11662	−0.558310	−	0.829632i	$-0.688550\pi$
$587$	−1.31403e29	−1.11662	−0.558310	+	0.829632i	$0.688550\pi$
$588$	0	0
$589$	8.27847e28	0.678797
$590$	0	0
$591$	−5.36761e27	−0.0424729
$592$	0	0
$593$	−2.29428e29	−1.75215	−0.876075	−	0.482174i	$-0.839847\pi$
$593$	−2.29428e29	−1.75215	−0.876075	+	0.482174i	$0.839847\pi$
$594$	0	0
$595$	2.70516e29	1.99418
$596$	0	0
$597$	−5.79341e28	−0.412291
$598$	0	0
$599$	1.80553e27	0.0124058	0.00620288	−	0.999981i	$-0.498026\pi$
$599$	1.80553e27	0.0124058	0.00620288	+	0.999981i	$0.498026\pi$
$600$	0	0
$601$	9.79897e28	0.650128	0.325064	−	0.945692i	$-0.394614\pi$
$601$	9.79897e28	0.650128	0.325064	+	0.945692i	$0.394614\pi$
$602$	0	0
$603$	−7.12047e28	−0.456222
$604$	0	0
$605$	−3.09369e29	−1.91446
$606$	0	0
$607$	−9.51352e28	−0.568670	−0.284335	−	0.958725i	$-0.591773\pi$
$607$	−9.51352e28	−0.568670	−0.284335	+	0.958725i	$0.591773\pi$
$608$	0	0
$609$	4.79701e28	0.277006
$610$	0	0
$611$	4.65829e27	0.0259893
$612$	0	0
$613$	−1.06132e29	−0.572151	−0.286075	−	0.958207i	$-0.592351\pi$
$613$	−1.06132e29	−0.572151	−0.286075	+	0.958207i	$0.592351\pi$
$614$	0	0
$615$	1.70923e29	0.890453
$616$	0	0
$617$	1.24152e29	0.625113	0.312557	−	0.949899i	$-0.398815\pi$
$617$	1.24152e29	0.625113	0.312557	+	0.949899i	$0.398815\pi$
$618$	0	0
$619$	3.70280e28	0.180210	0.0901049	−	0.995932i	$-0.471280\pi$
$619$	3.70280e28	0.180210	0.0901049	+	0.995932i	$0.471280\pi$
$620$	0	0
$621$	−2.25071e28	−0.105891
$622$	0	0
$623$	3.14134e29	1.42886
$624$	0	0
$625$	7.83875e29	3.44752
$626$	0	0
$627$	3.75112e27	0.0159534
$628$	0	0
$629$	−3.95708e29	−1.62758
$630$	0	0
$631$	−3.98518e29	−1.58540	−0.792702	−	0.609610i	$-0.791326\pi$
$631$	−3.98518e29	−1.58540	−0.792702	+	0.609610i	$0.791326\pi$
$632$	0	0
$633$	−7.58120e28	−0.291742
$634$	0	0
$635$	5.37060e28	0.199939
$636$	0	0
$637$	7.64408e26	0.00275335
$638$	0	0
$639$	−1.36744e29	−0.476594
$640$	0	0
$641$	3.11810e29	1.05167	0.525836	−	0.850586i	$-0.323753\pi$
$641$	3.11810e29	1.05167	0.525836	+	0.850586i	$0.323753\pi$
$642$	0	0
$643$	5.91629e29	1.93123	0.965616	−	0.259973i	$-0.0837138\pi$
$643$	5.91629e29	1.93123	0.965616	+	0.259973i	$0.0837138\pi$
$644$	0	0
$645$	−2.21093e29	−0.698552
$646$	0	0
$647$	2.57623e29	0.787934	0.393967	−	0.919125i	$-0.371102\pi$
$647$	2.57623e29	0.787934	0.393967	+	0.919125i	$0.371102\pi$
$648$	0	0
$649$	−3.61586e26	−0.00107064
$650$	0	0
$651$	−7.92448e28	−0.227180
$652$	0	0
$653$	−4.06486e29	−1.12838	−0.564192	−	0.825644i	$-0.690812\pi$
$653$	−4.06486e29	−1.12838	−0.564192	+	0.825644i	$0.690812\pi$
$654$	0	0
$655$	5.14037e29	1.38185
$656$	0	0
$657$	−1.65968e29	−0.432105
$658$	0	0
$659$	−1.76184e29	−0.444294	−0.222147	−	0.975013i	$-0.571306\pi$
$659$	−1.76184e29	−0.444294	−0.222147	+	0.975013i	$0.571306\pi$
$660$	0	0
$661$	−2.99970e29	−0.732760	−0.366380	−	0.930465i	$-0.619403\pi$
$661$	−2.99970e29	−0.732760	−0.366380	+	0.930465i	$0.619403\pi$
$662$	0	0
$663$	−4.91854e27	−0.0116398
$664$	0	0
$665$	1.21797e30	2.79259
$666$	0	0
$667$	1.29241e29	0.287130
$668$	0	0
$669$	−6.56734e28	−0.141389
$670$	0	0
$671$	4.76066e27	0.00993301
$672$	0	0
$673$	5.25225e29	1.06215	0.531077	−	0.847324i	$-0.321788\pi$
$673$	5.25225e29	1.06215	0.531077	+	0.847324i	$0.321788\pi$
$674$	0	0
$675$	−2.61874e29	−0.513337
$676$	0	0
$677$	3.62662e29	0.689162	0.344581	−	0.938757i	$-0.388021\pi$
$677$	3.62662e29	0.689162	0.344581	+	0.938757i	$0.388021\pi$
$678$	0	0
$679$	6.33320e29	1.16678
$680$	0	0
$681$	−1.05748e29	−0.188898
$682$	0	0
$683$	−2.66967e29	−0.462424	−0.231212	−	0.972903i	$-0.574269\pi$
$683$	−2.66967e29	−0.462424	−0.231212	+	0.972903i	$0.574269\pi$
$684$	0	0
$685$	−4.26863e29	−0.717030
$686$	0	0
$687$	3.53327e29	0.575614
$688$	0	0
$689$	1.98856e28	0.0314222
$690$	0	0
$691$	−5.15096e29	−0.789531	−0.394765	−	0.918782i	$-0.629174\pi$
$691$	−5.15096e29	−0.789531	−0.394765	+	0.918782i	$0.629174\pi$
$692$	0	0
$693$	−3.59072e27	−0.00533928
$694$	0	0
$695$	−2.29436e30	−3.30994
$696$	0	0
$697$	6.51650e29	0.912158
$698$	0	0
$699$	−4.84375e29	−0.657917
$700$	0	0
$701$	−4.60995e29	−0.607655	−0.303827	−	0.952727i	$-0.598265\pi$
$701$	−4.60995e29	−0.607655	−0.303827	+	0.952727i	$0.598265\pi$
$702$	0	0
$703$	−1.78163e30	−2.27922
$704$	0	0
$705$	1.30007e30	1.61429
$706$	0	0
$707$	−1.08391e30	−1.30644
$708$	0	0
$709$	−1.05247e30	−1.23147	−0.615733	−	0.787955i	$-0.711140\pi$
$709$	−1.05247e30	−1.23147	−0.615733	+	0.787955i	$0.711140\pi$
$710$	0	0
$711$	3.00953e28	0.0341875
$712$	0	0
$713$	−2.13501e29	−0.235483
$714$	0	0
$715$	−5.54514e26	−0.000593881 0
$716$	0	0
$717$	−4.14068e29	−0.430646
$718$	0	0
$719$	2.88703e29	0.291607	0.145804	−	0.989314i	$-0.453423\pi$
$719$	2.88703e29	0.291607	0.145804	+	0.989314i	$0.453423\pi$
$720$	0	0
$721$	−8.24658e29	−0.809009
$722$	0	0
$723$	6.12295e29	0.583456
$724$	0	0
$725$	1.50374e30	1.39195
$726$	0	0
$727$	1.96219e30	1.76454	0.882268	−	0.470748i	$-0.156016\pi$
$727$	1.96219e30	1.76454	0.882268	+	0.470748i	$0.156016\pi$
$728$	0	0
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−8.42928e29	−0.715580
$732$	0	0
$733$	−7.69033e28	−0.0634386	−0.0317193	−	0.999497i	$-0.510098\pi$
$733$	−7.69033e28	−0.0634386	−0.0317193	+	0.999497i	$0.510098\pi$
$734$	0	0
$735$	2.13337e29	0.171021
$736$	0	0
$737$	−3.06055e28	−0.0238447
$738$	0	0
$739$	−4.37247e29	−0.331101	−0.165550	−	0.986201i	$-0.552940\pi$
$739$	−4.37247e29	−0.331101	−0.165550	+	0.986201i	$0.552940\pi$
$740$	0	0
$741$	−2.21452e28	−0.0163000
$742$	0	0
$743$	1.93922e30	1.38754	0.693770	−	0.720197i	$-0.255948\pi$
$743$	1.93922e30	1.38754	0.693770	+	0.720197i	$0.255948\pi$
$744$	0	0
$745$	−1.44724e30	−1.00670
$746$	0	0
$747$	9.26906e29	0.626858
$748$	0	0
$749$	2.34161e30	1.53977
$750$	0	0
$751$	−9.91312e29	−0.633857	−0.316928	−	0.948449i	$-0.602651\pi$
$751$	−9.91312e29	−0.633857	−0.316928	+	0.948449i	$0.602651\pi$
$752$	0	0
$753$	−1.34731e30	−0.837761
$754$	0	0
$755$	−1.48933e29	−0.0900631
$756$	0	0
$757$	−2.12609e30	−1.25048	−0.625238	−	0.780434i	$-0.714998\pi$
$757$	−2.12609e30	−1.25048	−0.625238	+	0.780434i	$0.714998\pi$
$758$	0	0
$759$	−9.67411e27	−0.00553442
$760$	0	0
$761$	−8.13487e29	−0.452701	−0.226351	−	0.974046i	$-0.572680\pi$
$761$	−8.13487e29	−0.452701	−0.226351	+	0.974046i	$0.572680\pi$
$762$	0	0
$763$	−1.00488e30	−0.544010
$764$	0	0
$765$	−1.37271e30	−0.722990
$766$	0	0
$767$	2.13466e27	0.00109390
$768$	0	0
$769$	3.51201e30	1.75117	0.875587	−	0.483061i	$-0.160475\pi$
$769$	3.51201e30	1.75117	0.875587	+	0.483061i	$0.160475\pi$
$770$	0	0
$771$	9.28045e29	0.450296
$772$	0	0
$773$	−1.85796e30	−0.877308	−0.438654	−	0.898656i	$-0.644545\pi$
$773$	−1.85796e30	−0.877308	−0.438654	+	0.898656i	$0.644545\pi$
$774$	0	0
$775$	−2.48412e30	−1.14157
$776$	0	0
$777$	1.70545e30	0.762812
$778$	0	0
$779$	2.93398e30	1.27736
$780$	0	0
$781$	−5.87758e28	−0.0249094
$782$	0	0
$783$	−2.43420e29	−0.100429
$784$	0	0
$785$	9.24870e30	3.71492
$786$	0	0
$787$	−3.95934e30	−1.54842	−0.774209	−	0.632930i	$-0.781852\pi$
$787$	−3.95934e30	−1.54842	−0.774209	+	0.632930i	$0.781852\pi$
$788$	0	0
$789$	−9.46989e28	−0.0360609
$790$	0	0
$791$	1.05405e30	0.390847
$792$	0	0
$793$	−2.81051e28	−0.0101488
$794$	0	0
$795$	5.54984e30	1.95175
$796$	0	0
$797$	1.59540e30	0.546460	0.273230	−	0.961949i	$-0.411908\pi$
$797$	1.59540e30	0.546460	0.273230	+	0.961949i	$0.411908\pi$
$798$	0	0
$799$	4.95658e30	1.65364
$800$	0	0
$801$	−1.59404e30	−0.518034
$802$	0	0
$803$	−7.13373e28	−0.0225841
$804$	0	0
$805$	−3.14112e30	−0.968787
$806$	0	0
$807$	2.90523e30	0.872987
$808$	0	0
$809$	−1.96445e30	−0.575152	−0.287576	−	0.957758i	$-0.592849\pi$
$809$	−1.96445e30	−0.575152	−0.287576	+	0.957758i	$0.592849\pi$
$810$	0	0
$811$	5.85777e30	1.67114	0.835571	−	0.549382i	$-0.185137\pi$
$811$	5.85777e30	1.67114	0.835571	+	0.549382i	$0.185137\pi$
$812$	0	0
$813$	4.71675e29	0.131127
$814$	0	0
$815$	−1.34416e31	−3.64163
$816$	0	0
$817$	−3.79518e30	−1.00208
$818$	0	0
$819$	2.11982e28	0.00545529
$820$	0	0
$821$	−4.07728e30	−1.02274	−0.511372	−	0.859359i	$-0.670863\pi$
$821$	−4.07728e30	−1.02274	−0.511372	+	0.859359i	$0.670863\pi$
$822$	0	0
$823$	7.57530e30	1.85226	0.926129	−	0.377206i	$-0.123115\pi$
$823$	7.57530e30	1.85226	0.926129	+	0.377206i	$0.123115\pi$
$824$	0	0
$825$	−1.12560e29	−0.0268298
$826$	0	0
$827$	8.09644e30	1.88142	0.940711	−	0.339209i	$-0.110159\pi$
$827$	8.09644e30	1.88142	0.940711	+	0.339209i	$0.110159\pi$
$828$	0	0
$829$	−4.67371e30	−1.05886	−0.529431	−	0.848353i	$-0.677594\pi$
$829$	−4.67371e30	−1.05886	−0.529431	+	0.848353i	$0.677594\pi$
$830$	0	0
$831$	−1.33680e30	−0.295295
$832$	0	0
$833$	8.13356e29	0.175190
$834$	0	0
$835$	−8.06149e30	−1.69320
$836$	0	0
$837$	4.02120e29	0.0823642
$838$	0	0
$839$	−1.78537e30	−0.356637	−0.178319	−	0.983973i	$-0.557066\pi$
$839$	−1.78537e30	−0.356637	−0.178319	+	0.983973i	$0.557066\pi$
$840$	0	0
$841$	−3.73507e30	−0.727681
$842$	0	0
$843$	−4.18972e29	−0.0796150
$844$	0	0
$845$	−1.03285e31	−1.91443
$846$	0	0
$847$	5.08341e30	0.919134
$848$	0	0
$849$	−1.17116e29	−0.0206578
$850$	0	0
$851$	4.59481e30	0.790692
$852$	0	0
$853$	−3.78820e29	−0.0636017	−0.0318008	−	0.999494i	$-0.510124\pi$
$853$	−3.78820e29	−0.0636017	−0.0318008	+	0.999494i	$0.510124\pi$
$854$	0	0
$855$	−6.18045e30	−1.01246
$856$	0	0
$857$	6.68323e30	1.06829	0.534144	−	0.845394i	$-0.320634\pi$
$857$	6.68323e30	1.06829	0.534144	+	0.845394i	$0.320634\pi$
$858$	0	0
$859$	1.08420e30	0.169114	0.0845569	−	0.996419i	$-0.473053\pi$
$859$	1.08420e30	0.169114	0.0845569	+	0.996419i	$0.473053\pi$
$860$	0	0
$861$	−2.80852e30	−0.427508
$862$	0	0
$863$	2.95381e29	0.0438803	0.0219401	−	0.999759i	$-0.493016\pi$
$863$	2.95381e29	0.0438803	0.0219401	+	0.999759i	$0.493016\pi$
$864$	0	0
$865$	6.22039e30	0.901880
$866$	0	0
$867$	−1.15369e30	−0.163263
$868$	0	0
$869$	1.29357e28	0.00178683
$870$	0	0
$871$	1.80683e29	0.0243628
$872$	0	0
$873$	−3.21372e30	−0.423018
$874$	0	0
$875$	−2.28458e31	−2.93577
$876$	0	0
$877$	−1.13789e31	−1.42759	−0.713797	−	0.700353i	$-0.753026\pi$
$877$	−1.13789e31	−1.42759	−0.713797	+	0.700353i	$0.753026\pi$
$878$	0	0
$879$	−6.94839e30	−0.851140
$880$	0	0
$881$	1.24451e30	0.148851	0.0744257	−	0.997227i	$-0.476288\pi$
$881$	1.24451e30	0.148851	0.0744257	+	0.997227i	$0.476288\pi$
$882$	0	0
$883$	3.51456e30	0.410472	0.205236	−	0.978712i	$-0.434204\pi$
$883$	3.51456e30	0.410472	0.205236	+	0.978712i	$0.434204\pi$
$884$	0	0
$885$	5.95759e29	0.0679463
$886$	0	0
$887$	9.29130e30	1.03485	0.517426	−	0.855728i	$-0.326890\pi$
$887$	9.29130e30	1.03485	0.517426	+	0.855728i	$0.326890\pi$
$888$	0	0
$889$	−8.82470e29	−0.0959912
$890$	0	0
$891$	1.82208e28	0.00193576
$892$	0	0
$893$	2.23164e31	2.31571
$894$	0	0
$895$	−1.27339e31	−1.29068
$896$	0	0
$897$	5.71122e28	0.00565468
$898$	0	0
$899$	−2.30906e30	−0.223336
$900$	0	0
$901$	2.11590e31	1.99933
$902$	0	0
$903$	3.63290e30	0.335376
$904$	0	0
$905$	−1.14749e31	−1.03500
$906$	0	0
$907$	−1.98499e29	−0.0174937	−0.00874685	−	0.999962i	$-0.502784\pi$
$907$	−1.98499e29	−0.0174937	−0.00874685	+	0.999962i	$0.502784\pi$
$908$	0	0
$909$	5.50020e30	0.473650
$910$	0	0
$911$	−7.61422e29	−0.0640741	−0.0320371	−	0.999487i	$-0.510199\pi$
$911$	−7.61422e29	−0.0640741	−0.0320371	+	0.999487i	$0.510199\pi$
$912$	0	0
$913$	3.98407e29	0.0327630
$914$	0	0
$915$	−7.84380e30	−0.630383
$916$	0	0
$917$	−8.44640e30	−0.663428
$918$	0	0
$919$	2.14472e31	1.64648	0.823242	−	0.567691i	$-0.192163\pi$
$919$	2.14472e31	1.64648	0.823242	+	0.567691i	$0.192163\pi$
$920$	0	0
$921$	−4.55424e30	−0.341736
$922$	0	0
$923$	3.46989e29	0.0254506
$924$	0	0
$925$	5.34613e31	3.83311
$926$	0	0
$927$	4.18464e30	0.293306
$928$	0	0
$929$	−1.20678e31	−0.826916	−0.413458	−	0.910523i	$-0.635679\pi$
$929$	−1.20678e31	−0.826916	−0.413458	+	0.910523i	$0.635679\pi$
$930$	0	0
$931$	3.66204e30	0.245331
$932$	0	0
$933$	−4.70220e30	−0.307995
$934$	0	0
$935$	−5.90023e29	−0.0377874
$936$	0	0
$937$	1.48955e31	0.932803	0.466401	−	0.884573i	$-0.345550\pi$
$937$	1.48955e31	0.932803	0.466401	+	0.884573i	$0.345550\pi$
$938$	0	0
$939$	3.35897e30	0.205692
$940$	0	0
$941$	−8.63339e29	−0.0517000	−0.0258500	−	0.999666i	$-0.508229\pi$
$941$	−8.63339e29	−0.0517000	−0.0258500	+	0.999666i	$0.508229\pi$
$942$	0	0
$943$	−7.56671e30	−0.443133
$944$	0	0
$945$	5.91617e30	0.338849
$946$	0	0
$947$	−1.29303e31	−0.724328	−0.362164	−	0.932114i	$-0.617962\pi$
$947$	−1.29303e31	−0.724328	−0.362164	+	0.932114i	$0.617962\pi$
$948$	0	0
$949$	4.21147e29	0.0230749
$950$	0	0
$951$	8.91737e30	0.477905
$952$	0	0
$953$	−1.85893e31	−0.974513	−0.487257	−	0.873259i	$-0.662002\pi$
$953$	−1.85893e31	−0.974513	−0.487257	+	0.873259i	$0.662002\pi$
$954$	0	0
$955$	1.62402e30	0.0832831
$956$	0	0
$957$	−1.04628e29	−0.00524894
$958$	0	0
$959$	7.01400e30	0.344247
$960$	0	0
$961$	−1.70110e31	−0.816836
$962$	0	0
$963$	−1.18823e31	−0.558244
$964$	0	0
$965$	1.97794e31	0.909236
$966$	0	0
$967$	−2.45390e31	−1.10377	−0.551886	−	0.833919i	$-0.686092\pi$
$967$	−2.45390e31	−1.10377	−0.551886	+	0.833919i	$0.686092\pi$
$968$	0	0
$969$	−2.35632e31	−1.03713
$970$	0	0
$971$	3.38258e31	1.45696	0.728478	−	0.685069i	$-0.240228\pi$
$971$	3.38258e31	1.45696	0.728478	+	0.685069i	$0.240228\pi$
$972$	0	0
$973$	3.76997e31	1.58911
$974$	0	0
$975$	6.64509e29	0.0274127
$976$	0	0
$977$	−6.12303e30	−0.247214	−0.123607	−	0.992331i	$-0.539446\pi$
$977$	−6.12303e30	−0.247214	−0.123607	+	0.992331i	$0.539446\pi$
$978$	0	0
$979$	−6.85158e29	−0.0270753
$980$	0	0
$981$	5.09917e30	0.197231
$982$	0	0
$983$	−2.33411e31	−0.883710	−0.441855	−	0.897087i	$-0.645679\pi$
$983$	−2.33411e31	−0.883710	−0.441855	+	0.897087i	$0.645679\pi$
$984$	0	0
$985$	3.80129e30	0.140880
$986$	0	0
$987$	−2.13622e31	−0.775024
$988$	0	0
$989$	9.78775e30	0.347634
$990$	0	0
$991$	−1.93811e31	−0.673913	−0.336957	−	0.941520i	$-0.609398\pi$
$991$	−1.93811e31	−0.673913	−0.336957	+	0.941520i	$0.609398\pi$
$992$	0	0
$993$	1.94792e31	0.663138
$994$	0	0
$995$	4.10284e31	1.36755
$996$	0	0
$997$	−4.93033e31	−1.60908	−0.804538	−	0.593901i	$-0.797587\pi$
$997$	−4.93033e31	−1.60908	−0.804538	+	0.593901i	$0.797587\pi$
$998$	0	0
$999$	−8.65412e30	−0.276558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	24.22.a.c.1.3	✓	3
3.2	odd	2		72.22.a.d.1.1		3
4.3	odd	2		48.22.a.l.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.22.a.c.1.3	✓	3	1.1	even	1	trivial
48.22.a.l.1.3		3	4.3	odd	2
72.22.a.d.1.1		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 24 = 2^{3} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	24.a (trivial)

\(f(q)\)	\(=\)	\(q-59049.0 q^{3} +4.18179e7 q^{5} -6.87132e8 q^{7} +3.48678e9 q^{9} +O(q^{10})\)	\(q-59049.0 q^{3} +4.18179e7 q^{5} -6.87132e8 q^{7} +3.48678e9 q^{9} +1.49871e9 q^{11} -8.84777e9 q^{13} -2.46931e12 q^{15} -9.41434e12 q^{17} -4.23870e13 q^{19} +4.05745e13 q^{21} +1.09316e14 q^{23} +1.27190e15 q^{25} -2.05891e14 q^{27} +1.18227e15 q^{29} -1.95307e15 q^{31} -8.84971e13 q^{33} -2.87345e16 q^{35} +4.20325e16 q^{37} +5.22452e14 q^{39} -6.92189e16 q^{41} +8.95366e16 q^{43} +1.45810e17 q^{45} -5.26493e17 q^{47} -8.63955e16 q^{49} +5.55907e17 q^{51} -2.24753e18 q^{53} +6.26728e16 q^{55} +2.50291e18 q^{57} -2.41265e17 q^{59} +3.17652e18 q^{61} -2.39588e18 q^{63} -3.69996e17 q^{65} -2.04213e19 q^{67} -6.45498e18 q^{69} -3.92177e19 q^{71} -4.75993e19 q^{73} -7.51046e19 q^{75} -1.02981e18 q^{77} +8.63125e18 q^{79} +1.21577e19 q^{81} +2.65834e20 q^{83} -3.93688e20 q^{85} -6.98121e19 q^{87} -4.57167e20 q^{89} +6.07959e18 q^{91} +1.15327e20 q^{93} -1.77254e21 q^{95} -9.21686e20 q^{97} +5.22566e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 177147 q^{3} + 2080026 q^{5} - 1205282064 q^{7} + 10460353203 q^{9}+O(q^{10}) \) 3 * q - 177147 * q^3 + 2080026 * q^5 - 1205282064 * q^7 + 10460353203 * q^9	\( 3 q - 177147 q^{3} + 2080026 q^{5} - 1205282064 q^{7} + 10460353203 q^{9} + 13839247500 q^{11} + 718855551690 q^{13} - 122823455274 q^{15} + 2135189843046 q^{17} - 40122324686988 q^{19} + 71170700597136 q^{21} + 278424417682632 q^{23} + 13\!\cdots\!01 q^{25}+ \cdots + 48\!\cdots\!00 q^{99}+O(q^{100}) \) 3 * q - 177147 * q^3 + 2080026 * q^5 - 1205282064 * q^7 + 10460353203 * q^9 + 13839247500 * q^11 + 718855551690 * q^13 - 122823455274 * q^15 + 2135189843046 * q^17 - 40122324686988 * q^19 + 71170700597136 * q^21 + 278424417682632 * q^23 + 1348043260553901 * q^25 - 617673396283947 * q^27 + 442708167991794 * q^29 + 8016070162990152 * q^31 - 817193725627500 * q^33 + 125384157242400 * q^35 + 27729341388737058 * q^37 - 42447701471742810 * q^39 - 125648125186340562 * q^41 - 229052541499074612 * q^43 + 7252602210474426 * q^45 - 448613782068047712 * q^47 + 365221903446092427 * q^49 - 126080825042023254 * q^51 + 1406206217208267066 * q^53 - 3437829264920292504 * q^55 + 2369183150441954412 * q^57 - 1844638981471622100 * q^59 - 3294066300350351382 * q^61 - 4202558699560283664 * q^63 - 19537666262756991444 * q^65 - 33491023693155020652 * q^67 - 16440683439741736968 * q^69 - 79431018431598881160 * q^71 - 46612822906958319618 * q^73 - 79600606492447300149 * q^75 - 255759806110876888128 * q^77 - 197919973704661098024 * q^79 + 36472996377170786403 * q^81 + 111848528551886940276 * q^83 - 531596971435705186956 * q^85 - 26141474611747443906 * q^87 - 731008175840243483058 * q^89 - 546304658547739915488 * q^91 - 473340927054405485448 * q^93 - 1952599110289053971304 * q^95 - 1594521920055126193722 * q^97 + 48254472304578247500 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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