LMFDB - Embedded newform 144.9.g.g.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,9,Mod(127,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("144.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	144.g (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$58.6625198488$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-35}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 9 $ x^2 - x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.1
Root			$0.500000 + 2.95804i$ of defining polynomial
Character	$\chi$	$=$	144.127
Dual form			144.9.g.g.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+510.000 q^{5} -2555.75i q^{7} -19168.1i q^{11} -27710.0 q^{13} -50370.0 q^{17} +108619. i q^{19} -176347. i q^{23} -130525. q^{25} -54978.0 q^{29} +1.17564e6i q^{31} -1.30343e6i q^{35} +793730. q^{37} +75582.0 q^{41} +499648. i q^{43} -2.86755e6i q^{47} -767039. q^{49} -1.11662e7 q^{53} -9.77573e6i q^{55} +2.18325e7i q^{59} -2.38266e7 q^{61} -1.41321e7 q^{65} +7.49473e6i q^{67} +1.00824e7i q^{71} +6.51661e6 q^{73} -4.89888e7 q^{77} -4.87892e7i q^{79} -7.34483e7i q^{83} -2.56887e7 q^{85} -8.67958e7 q^{89} +7.08197e7i q^{91} +5.53958e7i q^{95} -4.66703e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 1020 q^{5} - 55420 q^{13} - 100740 q^{17} - 261050 q^{25} - 109956 q^{29} + 1587460 q^{37} + 151164 q^{41} - 1534078 q^{49} - 22332420 q^{53} - 47653244 q^{61} - 28264200 q^{65} + 13033220 q^{73}+ \cdots - 93340540 q^{97}+O(q^{100}) $ 2 * q + 1020 * q^5 - 55420 * q^13 - 100740 * q^17 - 261050 * q^25 - 109956 * q^29 + 1587460 * q^37 + 151164 * q^41 - 1534078 * q^49 - 22332420 * q^53 - 47653244 * q^61 - 28264200 * q^65 + 13033220 * q^73 - 97977600 * q^77 - 51377400 * q^85 - 173591556 * q^89 - 93340540 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	510.000	0.816000	0.408000	−	0.912982i	$-0.366226\pi$
$5$	510.000	0.816000	0.408000	+	0.912982i	$0.366226\pi$
$6$	0	0
$7$	− 2555.75i	− 1.06445i	−0.846603	−	0.532225i	$-0.821356\pi$
$7$	− 2555.75i	− 1.06445i	0.846603	−	0.532225i	$-0.178644\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 19168.1i	− 1.30921i	−0.755972	−	0.654603i	$-0.772836\pi$
$11$	− 19168.1i	− 1.30921i	0.755972	−	0.654603i	$-0.227164\pi$
$12$	0	0
$13$	−27710.0	−0.970204	−0.485102	−	0.874458i	$-0.661218\pi$
$13$	−27710.0	−0.970204	−0.485102	+	0.874458i	$0.661218\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−50370.0	−0.603082	−0.301541	−	0.953453i	$-0.597501\pi$
$17$	−50370.0	−0.603082	−0.301541	+	0.953453i	$0.597501\pi$
$18$	0	0
$19$	108619.i	0.833474i	0.909027	+	0.416737i	$0.136826\pi$
$19$	108619.i	0.833474i	−0.909027	+	0.416737i	$0.863174\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 176347.i	− 0.630167i	−0.949064	−	0.315083i	$-0.897968\pi$
$23$	− 176347.i	− 0.630167i	0.949064	−	0.315083i	$-0.102032\pi$
$24$	0	0
$25$	−130525.	−0.334144
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−54978.0	−0.0777315	−0.0388657	−	0.999244i	$-0.512374\pi$
$29$	−54978.0	−0.0777315	−0.0388657	+	0.999244i	$0.512374\pi$
$30$	0	0
$31$	1.17564e6i	1.27300i	0.771276	+	0.636501i	$0.219619\pi$
$31$	1.17564e6i	1.27300i	−0.771276	+	0.636501i	$0.780381\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.30343e6i	− 0.868592i
$36$	0	0
$37$	793730.	0.423512	0.211756	−	0.977323i	$-0.432082\pi$
$37$	793730.	0.423512	0.211756	+	0.977323i	$0.432082\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	75582.0	0.0267475	0.0133737	−	0.999911i	$-0.495743\pi$
$41$	75582.0	0.0267475	0.0133737	+	0.999911i	$0.495743\pi$
$42$	0	0
$43$	499648.i	0.146147i	0.997327	+	0.0730736i	$0.0232808\pi$
$43$	499648.i	0.146147i	−0.997327	+	0.0730736i	$0.976719\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.86755e6i	− 0.587651i	−0.955859	−	0.293825i	$-0.905072\pi$
$47$	− 2.86755e6i	− 0.587651i	0.955859	−	0.293825i	$-0.0949284\pi$
$48$	0	0
$49$	−767039.	−0.133056
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.11662e7	−1.41515	−0.707575	−	0.706639i	$-0.750211\pi$
$53$	−1.11662e7	−1.41515	−0.707575	+	0.706639i	$0.750211\pi$
$54$	0	0
$55$	− 9.77573e6i	− 1.06831i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.18325e7i	1.80175i	0.434078	+	0.900875i	$0.357074\pi$
$59$	2.18325e7i	1.80175i	−0.434078	+	0.900875i	$0.642926\pi$
$60$	0	0
$61$	−2.38266e7	−1.72085	−0.860425	−	0.509577i	$-0.829802\pi$
$61$	−2.38266e7	−1.72085	−0.860425	+	0.509577i	$0.829802\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.41321e7	−0.791687
$66$	0	0
$67$	7.49473e6i	0.371926i	0.982557	+	0.185963i	$0.0595405\pi$
$67$	7.49473e6i	0.371926i	−0.982557	+	0.185963i	$0.940460\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.00824e7i	0.396763i	0.980125	+	0.198382i	$0.0635685\pi$
$71$	1.00824e7i	0.396763i	−0.980125	+	0.198382i	$0.936431\pi$
$72$	0	0
$73$	6.51661e6	0.229472	0.114736	−	0.993396i	$-0.463398\pi$
$73$	6.51661e6	0.229472	0.114736	+	0.993396i	$0.463398\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.89888e7	−1.39359
$78$	0	0
$79$	− 4.87892e7i	− 1.25261i	−0.779579	−	0.626304i	$-0.784567\pi$
$79$	− 4.87892e7i	− 1.25261i	0.779579	−	0.626304i	$-0.215433\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 7.34483e7i	− 1.54764i	−0.633407	−	0.773819i	$-0.718344\pi$
$83$	− 7.34483e7i	− 1.54764i	0.633407	−	0.773819i	$-0.281656\pi$
$84$	0	0
$85$	−2.56887e7	−0.492115
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.67958e7	−1.38337	−0.691685	−	0.722199i	$-0.743131\pi$
$89$	−8.67958e7	−1.38337	−0.691685	+	0.722199i	$0.743131\pi$
$90$	0	0
$91$	7.08197e7i	1.03273i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.53958e7i	0.680115i
$96$	0	0
$97$	−4.66703e7	−0.527173	−0.263587	−	0.964636i	$-0.584905\pi$
$97$	−4.66703e7	−0.527173	−0.263587	+	0.964636i	$0.584905\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.59910e7	−0.634161	−0.317080	−	0.948399i	$-0.602703\pi$
$101$	−6.59910e7	−0.634161	−0.317080	+	0.948399i	$0.602703\pi$
$102$	0	0
$103$	1.64884e8i	1.46497i	0.680782	+	0.732486i	$0.261640\pi$
$103$	1.64884e8i	1.46497i	−0.680782	+	0.732486i	$0.738360\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.27326e8i	− 0.971364i	−0.874136	−	0.485682i	$-0.838571\pi$
$107$	− 1.27326e8i	− 0.971364i	0.874136	−	0.485682i	$-0.161429\pi$
$108$	0	0
$109$	−1.56119e8	−1.10598	−0.552992	−	0.833186i	$-0.686514\pi$
$109$	−1.56119e8	−1.10598	−0.552992	+	0.833186i	$0.686514\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.36346e8	−1.44955	−0.724776	−	0.688984i	$-0.758057\pi$
$113$	−2.36346e8	−1.44955	−0.724776	+	0.688984i	$0.758057\pi$
$114$	0	0
$115$	− 8.99367e7i	− 0.514216i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.28733e8i	0.641951i
$120$	0	0
$121$	−1.53057e8	−0.714023
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.65786e8	−1.08866
$126$	0	0
$127$	− 3.67741e8i	− 1.41360i	−0.707412	−	0.706802i	$-0.750137\pi$
$127$	− 3.67741e8i	− 1.41360i	0.707412	−	0.706802i	$-0.249863\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2.16350e8i	− 0.734636i	−0.930095	−	0.367318i	$-0.880276\pi$
$131$	− 2.16350e8i	− 0.734636i	0.930095	−	0.367318i	$-0.119724\pi$
$132$	0	0
$133$	2.77603e8	0.887193
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.86442e8	1.09699	0.548494	−	0.836155i	$-0.315201\pi$
$137$	3.86442e8	1.09699	0.548494	+	0.836155i	$0.315201\pi$
$138$	0	0
$139$	3.51077e8i	0.940465i	0.882543	+	0.470232i	$0.155830\pi$
$139$	3.51077e8i	0.940465i	−0.882543	+	0.470232i	$0.844170\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.31148e8i	1.27020i
$144$	0	0
$145$	−2.80388e7	−0.0634289
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.54099e8	0.921308	0.460654	−	0.887580i	$-0.347615\pi$
$149$	4.54099e8	0.921308	0.460654	+	0.887580i	$0.347615\pi$
$150$	0	0
$151$	− 6.60188e8i	− 1.26987i	−0.772565	−	0.634936i	$-0.781027\pi$
$151$	− 6.60188e8i	− 1.26987i	0.772565	−	0.634936i	$-0.218973\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.99578e8i	1.03877i
$156$	0	0
$157$	4.35318e8	0.716486	0.358243	−	0.933628i	$-0.383376\pi$
$157$	4.35318e8	0.716486	0.358243	+	0.933628i	$0.383376\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.50697e8	−0.670782
$162$	0	0
$163$	2.44065e8i	0.345744i	0.984944	+	0.172872i	$0.0553047\pi$
$163$	2.44065e8i	0.345744i	−0.984944	+	0.172872i	$0.944695\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 6.71351e8i	− 0.863145i	−0.902078	−	0.431573i	$-0.857959\pi$
$167$	− 6.71351e8i	− 0.863145i	0.902078	−	0.431573i	$-0.142041\pi$
$168$	0	0
$169$	−4.78866e7	−0.0587040
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.76764e9	1.97337	0.986685	−	0.162644i	$-0.0520022\pi$
$173$	1.76764e9	1.97337	0.986685	+	0.162644i	$0.0520022\pi$
$174$	0	0
$175$	3.33589e8i	0.355680i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 7.56967e8i	− 0.737335i	−0.929561	−	0.368668i	$-0.879814\pi$
$179$	− 7.56967e8i	− 0.737335i	0.929561	−	0.368668i	$-0.120186\pi$
$180$	0	0
$181$	6.27094e8	0.584277	0.292138	−	0.956376i	$-0.405633\pi$
$181$	6.27094e8	0.584277	0.292138	+	0.956376i	$0.405633\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.04802e8	0.345586
$186$	0	0
$187$	9.65497e8i	0.789559i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.07924e9i	− 0.810933i	−0.914110	−	0.405466i	$-0.867109\pi$
$191$	− 1.07924e9i	− 0.810933i	0.914110	−	0.405466i	$-0.132891\pi$
$192$	0	0
$193$	−2.96757e7	−0.0213881	−0.0106940	−	0.999943i	$-0.503404\pi$
$193$	−2.96757e7	−0.0213881	−0.0106940	+	0.999943i	$0.503404\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.12484e9	0.746837	0.373419	−	0.927663i	$-0.378186\pi$
$197$	1.12484e9	0.746837	0.373419	+	0.927663i	$0.378186\pi$
$198$	0	0
$199$	1.04718e9i	0.667742i	0.942619	+	0.333871i	$0.108355\pi$
$199$	1.04718e9i	0.667742i	−0.942619	+	0.333871i	$0.891645\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.40510e8i	0.0827413i
$204$	0	0
$205$	3.85468e7	0.0218259
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.08202e9	1.09119
$210$	0	0
$211$	− 2.77676e9i	− 1.40090i	−0.713699	−	0.700452i	$-0.752982\pi$
$211$	− 2.77676e9i	− 1.40090i	0.713699	−	0.700452i	$-0.247018\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.54821e8i	0.119256i
$216$	0	0
$217$	3.00465e9	1.35505
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.39575e9	0.585113
$222$	0	0
$223$	2.27822e9i	0.921248i	0.887595	+	0.460624i	$0.152374\pi$
$223$	2.27822e9i	0.921248i	−0.887595	+	0.460624i	$0.847626\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.48863e9i	0.560639i	0.959907	+	0.280319i	$0.0904404\pi$
$227$	1.48863e9i	0.560639i	−0.959907	+	0.280319i	$0.909560\pi$
$228$	0	0
$229$	1.69447e9	0.616157	0.308079	−	0.951361i	$-0.400314\pi$
$229$	1.69447e9	0.616157	0.308079	+	0.951361i	$0.400314\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.21423e8	−0.176916	−0.0884580	−	0.996080i	$-0.528194\pi$
$233$	−5.21423e8	−0.176916	−0.0884580	+	0.996080i	$0.528194\pi$
$234$	0	0
$235$	− 1.46245e9i	− 0.479523i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.40690e9i	1.35065i	0.737522	+	0.675323i	$0.235996\pi$
$239$	4.40690e9i	1.35065i	−0.737522	+	0.675323i	$0.764004\pi$
$240$	0	0
$241$	−1.62148e9	−0.480666	−0.240333	−	0.970691i	$-0.577257\pi$
$241$	−1.62148e9	−0.480666	−0.240333	+	0.970691i	$0.577257\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.91190e8	−0.108573
$246$	0	0
$247$	− 3.00984e9i	− 0.808640i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.31321e8i	0.0330855i	0.999863	+	0.0165428i	$0.00526597\pi$
$251$	1.31321e8i	0.0330855i	−0.999863	+	0.0165428i	$0.994734\pi$
$252$	0	0
$253$	−3.38023e9	−0.825019
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.27789e9	−1.20984	−0.604920	−	0.796287i	$-0.706795\pi$
$257$	−5.27789e9	−1.20984	−0.604920	+	0.796287i	$0.706795\pi$
$258$	0	0
$259$	− 2.02857e9i	− 0.450808i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 7.38745e9i	− 1.54409i	−0.635570	−	0.772044i	$-0.719235\pi$
$263$	− 7.38745e9i	− 1.54409i	0.635570	−	0.772044i	$-0.280765\pi$
$264$	0	0
$265$	−5.69477e9	−1.15476
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.46450e8	−0.0661655	−0.0330828	−	0.999453i	$-0.510532\pi$
$269$	−3.46450e8	−0.0661655	−0.0330828	+	0.999453i	$0.510532\pi$
$270$	0	0
$271$	6.51715e6i	0.00120832i	1.00000		0.000604158i	$0.000192310\pi$
$271$	6.51715e6i	0.00120832i	−1.00000		0.000604158i	$0.999808\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.50192e9i	0.437464i
$276$	0	0
$277$	2.15061e9	0.365293	0.182647	−	0.983179i	$-0.441534\pi$
$277$	2.15061e9	0.365293	0.182647	+	0.983179i	$0.441534\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.04256e10	−1.67215	−0.836074	−	0.548616i	$-0.815155\pi$
$281$	−1.04256e10	−1.67215	−0.836074	+	0.548616i	$0.815155\pi$
$282$	0	0
$283$	1.28042e9i	0.199622i	0.995006	+	0.0998108i	$0.0318238\pi$
$283$	1.28042e9i	0.199622i	−0.995006	+	0.0998108i	$0.968176\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.93168e8i	− 0.0284714i
$288$	0	0
$289$	−4.43862e9	−0.636292
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.13786e9	0.290074	0.145037	−	0.989426i	$-0.453670\pi$
$293$	2.13786e9	0.290074	0.145037	+	0.989426i	$0.453670\pi$
$294$	0	0
$295$	1.11346e10i	1.47023i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.88656e9i	0.611390i
$300$	0	0
$301$	1.27697e9	0.155567
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.21516e10	−1.40421
$306$	0	0
$307$	5.45140e9i	0.613698i	0.951758	+	0.306849i	$0.0992747\pi$
$307$	5.45140e9i	0.613698i	−0.951758	+	0.306849i	$0.900725\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 1.07550e10i	− 1.14965i	−0.818275	−	0.574827i	$-0.805069\pi$
$311$	− 1.07550e10i	− 1.14965i	0.818275	−	0.574827i	$-0.194931\pi$
$312$	0	0
$313$	−2.99804e8	−0.0312364	−0.0156182	−	0.999878i	$-0.504972\pi$
$313$	−2.99804e8	−0.0312364	−0.0156182	+	0.999878i	$0.504972\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.31172e9	−0.526015	−0.263007	−	0.964794i	$-0.584714\pi$
$317$	−5.31172e9	−0.526015	−0.263007	+	0.964794i	$0.584714\pi$
$318$	0	0
$319$	1.05382e9i	0.101767i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 5.47115e9i	− 0.502653i
$324$	0	0
$325$	3.61685e9	0.324188
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.32872e9	−0.625525
$330$	0	0
$331$	− 1.01004e10i	− 0.841446i	−0.907189	−	0.420723i	$-0.861776\pi$
$331$	− 1.01004e10i	− 0.841446i	0.907189	−	0.420723i	$-0.138224\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.82231e9i	0.303492i
$336$	0	0
$337$	−1.84359e10	−1.42937	−0.714684	−	0.699448i	$-0.753429\pi$
$337$	−1.84359e10	−1.42937	−0.714684	+	0.699448i	$0.753429\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.25348e10	1.66662
$342$	0	0
$343$	− 1.27730e10i	− 0.922820i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.27822e10i	− 0.881636i	−0.897597	−	0.440818i	$-0.854688\pi$
$347$	− 1.27822e10i	− 0.881636i	0.897597	−	0.440818i	$-0.145312\pi$
$348$	0	0
$349$	−6.39381e8	−0.0430981	−0.0215490	−	0.999768i	$-0.506860\pi$
$349$	−6.39381e8	−0.0430981	−0.0215490	+	0.999768i	$0.506860\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.59837e10	−1.67341	−0.836705	−	0.547653i	$-0.815521\pi$
$353$	−2.59837e10	−1.67341	−0.836705	+	0.547653i	$0.815521\pi$
$354$	0	0
$355$	5.14203e9i	0.323759i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.22541e10i	1.33978i	0.742461	+	0.669889i	$0.233658\pi$
$359$	2.22541e10i	1.33978i	−0.742461	+	0.669889i	$0.766342\pi$
$360$	0	0
$361$	5.18543e9	0.305320
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.32347e9	0.187249
$366$	0	0
$367$	− 1.86229e10i	− 1.02656i	−0.858221	−	0.513280i	$-0.828430\pi$
$367$	− 1.86229e10i	− 1.02656i	0.858221	−	0.513280i	$-0.171570\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.85380e10i	1.50636i
$372$	0	0
$373$	1.19680e10	0.618283	0.309141	−	0.951016i	$-0.399958\pi$
$373$	1.19680e10	0.618283	0.309141	+	0.951016i	$0.399958\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.52344e9	0.0754154
$378$	0	0
$379$	2.30787e10i	1.11855i	0.828982	+	0.559275i	$0.188920\pi$
$379$	2.30787e10i	1.11855i	−0.828982	+	0.559275i	$0.811080\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.43419e10i	0.666518i	0.942835	+	0.333259i	$0.108148\pi$
$383$	1.43419e10i	0.666518i	−0.942835	+	0.333259i	$0.891852\pi$
$384$	0	0
$385$	−2.49843e10	−1.13717
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.73457e10	1.19424	0.597119	−	0.802152i	$-0.296312\pi$
$389$	2.73457e10	1.19424	0.597119	+	0.802152i	$0.296312\pi$
$390$	0	0
$391$	8.88257e9i	0.380042i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 2.48825e10i	− 1.02213i
$396$	0	0
$397$	3.99456e10	1.60808	0.804039	−	0.594576i	$-0.202680\pi$
$397$	3.99456e10	1.60808	0.804039	+	0.594576i	$0.202680\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.12767e10	0.822863	0.411431	−	0.911441i	$-0.365029\pi$
$401$	2.12767e10	0.822863	0.411431	+	0.911441i	$0.365029\pi$
$402$	0	0
$403$	− 3.25771e10i	− 1.23507i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.52143e10i	− 0.554465i
$408$	0	0
$409$	1.14283e10	0.408404	0.204202	−	0.978929i	$-0.434540\pi$
$409$	1.14283e10	0.408404	0.204202	+	0.978929i	$0.434540\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.57982e10	1.91788
$414$	0	0
$415$	− 3.74586e10i	− 1.26287i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.10009e10i	− 0.356922i	−0.983947	−	0.178461i	$-0.942888\pi$
$419$	− 1.10009e10i	− 0.356922i	0.983947	−	0.178461i	$-0.0571119\pi$
$420$	0	0
$421$	2.28766e10	0.728220	0.364110	−	0.931356i	$-0.381373\pi$
$421$	2.28766e10	0.728220	0.364110	+	0.931356i	$0.381373\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.57454e9	0.201516
$426$	0	0
$427$	6.08948e10i	1.83176i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.55108e8i	0.0276786i	0.999904	+	0.0138393i	$0.00440532\pi$
$431$	9.55108e8i	0.0276786i	−0.999904	+	0.0138393i	$0.995595\pi$
$432$	0	0
$433$	3.82225e10	1.08735	0.543673	−	0.839297i	$-0.317033\pi$
$433$	3.82225e10	1.08735	0.543673	+	0.839297i	$0.317033\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.91546e10	0.525228
$438$	0	0
$439$	6.40288e10i	1.72392i	0.506976	+	0.861960i	$0.330763\pi$
$439$	6.40288e10i	1.72392i	−0.506976	+	0.861960i	$0.669237\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 7.47659e10i	− 1.94128i	−0.240533	−	0.970641i	$-0.577322\pi$
$443$	− 7.47659e10i	− 1.94128i	0.240533	−	0.970641i	$-0.422678\pi$
$444$	0	0
$445$	−4.42658e10	−1.12883
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.51987e10	−0.620001	−0.310000	−	0.950736i	$-0.600329\pi$
$449$	−2.51987e10	−0.620001	−0.310000	+	0.950736i	$0.600329\pi$
$450$	0	0
$451$	− 1.44876e9i	− 0.0350180i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.61181e10i	0.842711i
$456$	0	0
$457$	4.66828e9	0.107027	0.0535133	−	0.998567i	$-0.482958\pi$
$457$	4.66828e9	0.107027	0.0535133	+	0.998567i	$0.482958\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.88096e10	0.859281	0.429641	−	0.903000i	$-0.358640\pi$
$461$	3.88096e10	0.859281	0.429641	+	0.903000i	$0.358640\pi$
$462$	0	0
$463$	− 3.23432e10i	− 0.703817i	−0.936034	−	0.351908i	$-0.885533\pi$
$463$	− 3.23432e10i	− 0.703817i	0.936034	−	0.351908i	$-0.114467\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.31902e10i	0.487570i	0.969829	+	0.243785i	$0.0783891\pi$
$467$	2.31902e10i	0.487570i	−0.969829	+	0.243785i	$0.921611\pi$
$468$	0	0
$469$	1.91546e10	0.395897
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.57731e9	0.191337
$474$	0	0
$475$	− 1.41775e10i	− 0.278500i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.83542e9i	0.0918528i	0.998945	+	0.0459264i	$0.0146240\pi$
$479$	4.83542e9i	0.0918528i	−0.998945	+	0.0459264i	$0.985376\pi$
$480$	0	0
$481$	−2.19943e10	−0.410893
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.38018e10	−0.430173
$486$	0	0
$487$	3.03878e10i	0.540236i	0.962827	+	0.270118i	$0.0870628\pi$
$487$	3.03878e10i	0.540236i	−0.962827	+	0.270118i	$0.912937\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.56483e10i	0.957472i	0.877959	+	0.478736i	$0.158905\pi$
$491$	5.56483e10i	0.957472i	−0.877959	+	0.478736i	$0.841095\pi$
$492$	0	0
$493$	2.76924e9	0.0468784
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.57681e10	0.422335
$498$	0	0
$499$	− 7.88458e10i	− 1.27168i	−0.771822	−	0.635838i	$-0.780655\pi$
$499$	− 7.88458e10i	− 1.27168i	0.771822	−	0.635838i	$-0.219345\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 4.41092e10i	− 0.689061i	−0.938775	−	0.344530i	$-0.888038\pi$
$503$	− 4.41092e10i	− 0.689061i	0.938775	−	0.344530i	$-0.111962\pi$
$504$	0	0
$505$	−3.36554e10	−0.517475
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.05927e10	−0.157811	−0.0789055	−	0.996882i	$-0.525143\pi$
$509$	−1.05927e10	−0.157811	−0.0789055	+	0.996882i	$0.525143\pi$
$510$	0	0
$511$	− 1.66548e10i	− 0.244262i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.40908e10i	1.19542i
$516$	0	0
$517$	−5.49654e10	−0.769356
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.24958e11	−1.69595	−0.847973	−	0.530039i	$-0.822177\pi$
$521$	−1.24958e11	−1.69595	−0.847973	+	0.530039i	$0.822177\pi$
$522$	0	0
$523$	2.80408e10i	0.374786i	0.982285	+	0.187393i	$0.0600038\pi$
$523$	2.80408e10i	0.374786i	−0.982285	+	0.187393i	$0.939996\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 5.92172e10i	− 0.767724i
$528$	0	0
$529$	4.72129e10	0.602890
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.09438e9	−0.0259505
$534$	0	0
$535$	− 6.49363e10i	− 0.792633i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.47027e10i	0.174197i
$540$	0	0
$541$	1.44659e11	1.68871	0.844356	−	0.535782i	$-0.179983\pi$
$541$	1.44659e11	1.68871	0.844356	+	0.535782i	$0.179983\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.96205e10	−0.902483
$546$	0	0
$547$	1.03774e10i	0.115915i	0.998319	+	0.0579573i	$0.0184587\pi$
$547$	1.03774e10i	0.115915i	−0.998319	+	0.0579573i	$0.981541\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 5.97167e9i	− 0.0647872i
$552$	0	0
$553$	−1.24693e11	−1.33334
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.47312e9	0.0568610	0.0284305	−	0.999596i	$-0.490949\pi$
$557$	5.47312e9	0.0568610	0.0284305	+	0.999596i	$0.490949\pi$
$558$	0	0
$559$	− 1.38453e10i	− 0.141793i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.36118e10i	0.434081i	0.976163	+	0.217040i	$0.0696403\pi$
$563$	4.36118e10i	0.434081i	−0.976163	+	0.217040i	$0.930360\pi$
$564$	0	0
$565$	−1.20536e11	−1.18283
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.27822e10	−0.121943	−0.0609716	−	0.998140i	$-0.519420\pi$
$569$	−1.27822e10	−0.121943	−0.0609716	+	0.998140i	$0.519420\pi$
$570$	0	0
$571$	− 7.59455e10i	− 0.714427i	−0.934023	−	0.357213i	$-0.883727\pi$
$571$	− 7.59455e10i	− 0.714427i	0.934023	−	0.357213i	$-0.116273\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.30176e10i	0.210566i
$576$	0	0
$577$	2.13827e10	0.192912	0.0964560	−	0.995337i	$-0.469249\pi$
$577$	2.13827e10	0.192912	0.0964560	+	0.995337i	$0.469249\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.87715e11	−1.64738
$582$	0	0
$583$	2.14035e11i	1.85272i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.07298e10i	− 0.511504i	−0.966742	−	0.255752i	$-0.917677\pi$
$587$	− 6.07298e10i	− 0.511504i	0.966742	−	0.255752i	$-0.0823231\pi$
$588$	0	0
$589$	−1.27697e11	−1.06101
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.15978e11	−0.937899	−0.468949	−	0.883225i	$-0.655367\pi$
$593$	−1.15978e11	−0.937899	−0.468949	+	0.883225i	$0.655367\pi$
$594$	0	0
$595$	6.56538e10i	0.523832i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 2.40647e11i	− 1.86927i	−0.355607	−	0.934636i	$-0.615726\pi$
$599$	− 2.40647e11i	− 1.86927i	0.355607	−	0.934636i	$-0.384274\pi$
$600$	0	0
$601$	−1.92942e11	−1.47887	−0.739434	−	0.673229i	$-0.764907\pi$
$601$	−1.92942e11	−1.47887	−0.739434	+	0.673229i	$0.764907\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−7.80591e10	−0.582643
$606$	0	0
$607$	1.62042e11i	1.19364i	0.802376	+	0.596819i	$0.203569\pi$
$607$	1.62042e11i	1.19364i	−0.802376	+	0.596819i	$0.796431\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.94597e10i	0.570141i
$612$	0	0
$613$	1.76424e11	1.24944	0.624722	−	0.780847i	$-0.285212\pi$
$613$	1.76424e11	1.24944	0.624722	+	0.780847i	$0.285212\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.84986e10	0.679656	0.339828	−	0.940488i	$-0.389631\pi$
$617$	9.84986e10	0.679656	0.339828	+	0.940488i	$0.389631\pi$
$618$	0	0
$619$	− 1.28596e10i	− 0.0875923i	−0.999040	−	0.0437961i	$-0.986055\pi$
$619$	− 1.28596e10i	− 0.0875923i	0.999040	−	0.0437961i	$-0.0139452\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.21828e11i	1.47253i
$624$	0	0
$625$	−8.45648e10	−0.554204
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.99802e10	−0.255413
$630$	0	0
$631$	− 1.73463e11i	− 1.09418i	−0.837073	−	0.547091i	$-0.815735\pi$
$631$	− 1.73463e11i	− 1.09418i	0.837073	−	0.547091i	$-0.184265\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.87548e11i	− 1.15350i
$636$	0	0
$637$	2.12547e10	0.129091
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.13903e11	0.674690	0.337345	−	0.941381i	$-0.390471\pi$
$641$	1.13903e11	0.674690	0.337345	+	0.941381i	$0.390471\pi$
$642$	0	0
$643$	− 7.04067e10i	− 0.411879i	−0.978565	−	0.205940i	$-0.933975\pi$
$643$	− 7.04067e10i	− 0.411879i	0.978565	−	0.205940i	$-0.0660250\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.99175e11i	− 1.13663i	−0.822812	−	0.568314i	$-0.807596\pi$
$647$	− 1.99175e11i	− 1.13663i	0.822812	−	0.568314i	$-0.192404\pi$
$648$	0	0
$649$	4.18487e11	2.35886
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.49972e9	−0.0357472	−0.0178736	−	0.999840i	$-0.505690\pi$
$653$	−6.49972e9	−0.0357472	−0.0178736	+	0.999840i	$0.505690\pi$
$654$	0	0
$655$	− 1.10339e11i	− 0.599463i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2.20982e11i	− 1.17170i	−0.810421	−	0.585848i	$-0.800761\pi$
$659$	− 2.20982e11i	− 1.17170i	0.810421	−	0.585848i	$-0.199239\pi$
$660$	0	0
$661$	−2.69549e11	−1.41199	−0.705995	−	0.708217i	$-0.749500\pi$
$661$	−2.69549e11	−1.41199	−0.705995	+	0.708217i	$0.749500\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.41578e11	0.723949
$666$	0	0
$667$	9.69518e9i	0.0489838i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.56711e11i	2.25295i
$672$	0	0
$673$	9.44470e10	0.460392	0.230196	−	0.973144i	$-0.426063\pi$
$673$	9.44470e10	0.460392	0.230196	+	0.973144i	$0.426063\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.02735e10	−0.382136	−0.191068	−	0.981577i	$-0.561195\pi$
$677$	−8.02735e10	−0.382136	−0.191068	+	0.981577i	$0.561195\pi$
$678$	0	0
$679$	1.19277e11i	0.561150i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.00783e11i	1.38220i	0.722760	+	0.691099i	$0.242873\pi$
$683$	3.00783e11i	1.38220i	−0.722760	+	0.691099i	$0.757127\pi$
$684$	0	0
$685$	1.97085e11	0.895142
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.09416e11	1.37298
$690$	0	0
$691$	− 3.06208e11i	− 1.34309i	−0.740964	−	0.671544i	$-0.765631\pi$
$691$	− 3.06208e11i	− 1.34309i	0.740964	−	0.671544i	$-0.234369\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.79049e11i	0.767419i
$696$	0	0
$697$	−3.80707e9	−0.0161309
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.73603e11	1.13305	0.566524	−	0.824045i	$-0.308288\pi$
$701$	2.73603e11	1.13305	0.566524	+	0.824045i	$0.308288\pi$
$702$	0	0
$703$	8.62143e10i	0.352987i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.68656e11i	0.675033i
$708$	0	0
$709$	−1.76662e11	−0.699129	−0.349564	−	0.936912i	$-0.613670\pi$
$709$	−1.76662e11	−0.699129	−0.349564	+	0.936912i	$0.613670\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.07321e11	0.802203
$714$	0	0
$715$	2.70885e11i	1.03648i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 2.25510e11i	− 0.843821i	−0.906637	−	0.421911i	$-0.861360\pi$
$719$	− 2.25510e11i	− 0.843821i	0.906637	−	0.421911i	$-0.138640\pi$
$720$	0	0
$721$	4.21402e11	1.55939
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.17600e9	0.0259735
$726$	0	0
$727$	− 2.87080e11i	− 1.02770i	−0.857881	−	0.513849i	$-0.828219\pi$
$727$	− 2.87080e11i	− 1.02770i	0.857881	−	0.513849i	$-0.171781\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 2.51673e10i	− 0.0881388i
$732$	0	0
$733$	−2.94176e11	−1.01904	−0.509520	−	0.860459i	$-0.670177\pi$
$733$	−2.94176e11	−1.01904	−0.509520	+	0.860459i	$0.670177\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.43660e11	0.486928
$738$	0	0
$739$	9.69888e10i	0.325195i	0.986692	+	0.162598i	$0.0519872\pi$
$739$	9.69888e10i	0.325195i	−0.986692	+	0.162598i	$0.948013\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1.01567e11i	− 0.333271i	−0.986019	−	0.166635i	$-0.946710\pi$
$743$	− 1.01567e11i	− 0.333271i	0.986019	−	0.166635i	$-0.0532902\pi$
$744$	0	0
$745$	2.31590e11	0.751788
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.25413e11	−1.03397
$750$	0	0
$751$	− 4.17899e11i	− 1.31375i	−0.754001	−	0.656873i	$-0.771879\pi$
$751$	− 4.17899e11i	− 1.31375i	0.754001	−	0.656873i	$-0.228121\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 3.36696e11i	− 1.03621i
$756$	0	0
$757$	1.82006e11	0.554244	0.277122	−	0.960835i	$-0.410619\pi$
$757$	1.82006e11	0.554244	0.277122	+	0.960835i	$0.410619\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.27419e11	1.27443	0.637213	−	0.770687i	$-0.280087\pi$
$761$	4.27419e11	1.27443	0.637213	+	0.770687i	$0.280087\pi$
$762$	0	0
$763$	3.99000e11i	1.17727i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 6.04978e11i	− 1.74807i
$768$	0	0
$769$	−5.09969e11	−1.45827	−0.729136	−	0.684368i	$-0.760078\pi$
$769$	−5.09969e11	−1.45827	−0.729136	+	0.684368i	$0.760078\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.49408e11	0.418462	0.209231	−	0.977866i	$-0.432904\pi$
$773$	1.49408e11	0.418462	0.209231	+	0.977866i	$0.432904\pi$
$774$	0	0
$775$	− 1.53451e11i	− 0.425366i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.20966e9i	0.0222933i
$780$	0	0
$781$	1.93261e11	0.519445
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.22012e11	0.584652
$786$	0	0
$787$	7.33252e11i	1.91141i	0.294323	+	0.955706i	$0.404906\pi$
$787$	7.33252e11i	1.91141i	−0.294323	+	0.955706i	$0.595094\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.04040e11i	1.54298i
$792$	0	0
$793$	6.60236e11	1.66958
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.02703e11	−0.750212	−0.375106	−	0.926982i	$-0.622394\pi$
$797$	−3.02703e11	−0.750212	−0.375106	+	0.926982i	$0.622394\pi$
$798$	0	0
$799$	1.44438e11i	0.354401i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.24911e11i	− 0.300427i
$804$	0	0
$805$	−2.29855e11	−0.547358
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.84316e11	1.36412	0.682062	−	0.731295i	$-0.261084\pi$
$809$	5.84316e11	1.36412	0.682062	+	0.731295i	$0.261084\pi$
$810$	0	0
$811$	− 1.21470e11i	− 0.280793i	−0.990095	−	0.140396i	$-0.955162\pi$
$811$	− 1.21470e11i	− 0.280793i	0.990095	−	0.140396i	$-0.0448377\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.24473e11i	0.282127i
$816$	0	0
$817$	−5.42714e10	−0.121810
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.52470e11	−0.995903	−0.497952	−	0.867205i	$-0.665914\pi$
$821$	−4.52470e11	−0.995903	−0.497952	+	0.867205i	$0.665914\pi$
$822$	0	0
$823$	3.06704e11i	0.668528i	0.942479	+	0.334264i	$0.108488\pi$
$823$	3.06704e11i	0.668528i	−0.942479	+	0.334264i	$0.891512\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.93276e11i	− 0.626982i	−0.949591	−	0.313491i	$-0.898501\pi$
$827$	− 2.93276e11i	− 0.626982i	0.949591	−	0.313491i	$-0.101499\pi$
$828$	0	0
$829$	3.35532e11	0.710421	0.355210	−	0.934786i	$-0.384409\pi$
$829$	3.35532e11	0.710421	0.355210	+	0.934786i	$0.384409\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.86358e10	0.0802434
$834$	0	0
$835$	− 3.42389e11i	− 0.704326i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.42844e11i	0.691908i	0.938252	+	0.345954i	$0.112445\pi$
$839$	3.42844e11i	0.691908i	−0.938252	+	0.345954i	$0.887555\pi$
$840$	0	0
$841$	−4.97224e11	−0.993958
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.44222e10	−0.0479024
$846$	0	0
$847$	3.91175e11i	0.760042i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.39972e11i	− 0.266883i
$852$	0	0
$853$	−5.08662e11	−0.960801	−0.480400	−	0.877049i	$-0.659509\pi$
$853$	−5.08662e11	−0.960801	−0.480400	+	0.877049i	$0.659509\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.06764e11	−1.12486	−0.562428	−	0.826846i	$-0.690133\pi$
$857$	−6.06764e11	−1.12486	−0.562428	+	0.826846i	$0.690133\pi$
$858$	0	0
$859$	− 9.49431e11i	− 1.74378i	−0.489705	−	0.871888i	$-0.662895\pi$
$859$	− 9.49431e11i	− 1.74378i	0.489705	−	0.871888i	$-0.337105\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.99836e10i	0.0540556i	0.999635	+	0.0270278i	$0.00860426\pi$
$863$	2.99836e10i	0.0540556i	−0.999635	+	0.0270278i	$0.991396\pi$
$864$	0	0
$865$	9.01494e11	1.61027
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9.35196e11	−1.63992
$870$	0	0
$871$	− 2.07679e11i	− 0.360844i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.79283e11i	1.15883i
$876$	0	0
$877$	−8.80195e11	−1.48792	−0.743961	−	0.668223i	$-0.767055\pi$
$877$	−8.80195e11	−1.48792	−0.743961	+	0.668223i	$0.767055\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.04085e12	−1.72776	−0.863879	−	0.503699i	$-0.831972\pi$
$881$	−1.04085e12	−1.72776	−0.863879	+	0.503699i	$0.831972\pi$
$882$	0	0
$883$	− 7.60446e11i	− 1.25091i	−0.780261	−	0.625454i	$-0.784914\pi$
$883$	− 7.60446e11i	− 1.25091i	0.780261	−	0.625454i	$-0.215086\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 3.34097e11i	− 0.539732i	−0.962898	−	0.269866i	$-0.913021\pi$
$887$	− 3.34097e11i	− 0.539732i	0.962898	−	0.269866i	$-0.0869794\pi$
$888$	0	0
$889$	−9.39853e11	−1.50471
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.11471e11	0.489792
$894$	0	0
$895$	− 3.86053e11i	− 0.601666i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 6.46345e10i	− 0.0989523i
$900$	0	0
$901$	5.62442e11	0.853451
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.19818e11	0.476770
$906$	0	0
$907$	7.65213e11i	1.13071i	0.824846	+	0.565357i	$0.191262\pi$
$907$	7.65213e11i	1.13071i	−0.824846	+	0.565357i	$0.808738\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.83541e11i	0.556851i	0.960458	+	0.278425i	$0.0898125\pi$
$911$	3.83541e11i	0.556851i	−0.960458	+	0.278425i	$0.910188\pi$
$912$	0	0
$913$	−1.40786e12	−2.02618
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.52937e11	−0.781984
$918$	0	0
$919$	6.82775e11i	0.957229i	0.878025	+	0.478615i	$0.158861\pi$
$919$	6.82775e11i	0.957229i	−0.878025	+	0.478615i	$0.841139\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 2.79384e11i	− 0.384941i
$924$	0	0
$925$	−1.03602e11	−0.141514
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.94973e11	−0.396021	−0.198011	−	0.980200i	$-0.563448\pi$
$929$	−2.94973e11	−0.396021	−0.198011	+	0.980200i	$0.563448\pi$
$930$	0	0
$931$	− 8.33152e10i	− 0.110898i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.92404e11i	0.644280i
$936$	0	0
$937$	1.03941e12	1.34843	0.674217	−	0.738533i	$-0.264481\pi$
$937$	1.03941e12	1.34843	0.674217	+	0.738533i	$0.264481\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.26891e11	−0.161836	−0.0809178	−	0.996721i	$-0.525785\pi$
$941$	−1.26891e11	−0.161836	−0.0809178	+	0.996721i	$0.525785\pi$
$942$	0	0
$943$	− 1.33286e10i	− 0.0168554i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.13064e11i	0.762265i	0.924520	+	0.381133i	$0.124466\pi$
$947$	6.13064e11i	0.762265i	−0.924520	+	0.381133i	$0.875534\pi$
$948$	0	0
$949$	−1.80575e11	−0.222635
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−6.58227e11	−0.798002	−0.399001	−	0.916951i	$-0.630643\pi$
$953$	−6.58227e11	−0.798002	−0.399001	+	0.916951i	$0.630643\pi$
$954$	0	0
$955$	− 5.50413e11i	− 0.661721i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 9.87647e11i	− 1.16769i
$960$	0	0
$961$	−5.29246e11	−0.620532
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.51346e10	−0.0174527
$966$	0	0
$967$	− 5.41485e11i	− 0.619271i	−0.950855	−	0.309635i	$-0.899793\pi$
$967$	− 5.41485e11i	− 0.619271i	0.950855	−	0.309635i	$-0.100207\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.54981e11i	0.399327i	0.979865	+	0.199663i	$0.0639849\pi$
$971$	3.54981e11i	0.399327i	−0.979865	+	0.199663i	$0.936015\pi$
$972$	0	0
$973$	8.97263e11	1.00108
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.02238e11	−0.660982	−0.330491	−	0.943809i	$-0.607214\pi$
$977$	−6.02238e11	−0.660982	−0.330491	+	0.943809i	$0.607214\pi$
$978$	0	0
$979$	1.66371e12i	1.81112i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.37465e12i	− 1.47223i	−0.676854	−	0.736117i	$-0.736657\pi$
$983$	− 1.37465e12i	− 1.47223i	0.676854	−	0.736117i	$-0.263343\pi$
$984$	0	0
$985$	5.73668e11	0.609419
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.81113e10	0.0920972
$990$	0	0
$991$	1.01081e12i	1.04803i	0.851709	+	0.524015i	$0.175567\pi$
$991$	1.01081e12i	1.04803i	−0.851709	+	0.524015i	$0.824433\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.34061e11i	0.544877i
$996$	0	0
$997$	−3.28556e11	−0.332528	−0.166264	−	0.986081i	$-0.553170\pi$
$997$	−3.28556e11	−0.332528	−0.166264	+	0.986081i	$0.553170\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.9.g.g.127.1		2
3.2	odd	2		16.9.c.a.15.2	yes	2
4.3	odd	2	inner	144.9.g.g.127.2		2
12.11	even	2		16.9.c.a.15.1	✓	2
15.2	even	4		400.9.h.b.399.4		4
15.8	even	4		400.9.h.b.399.2		4
15.14	odd	2		400.9.b.c.351.1		2
24.5	odd	2		64.9.c.d.63.1		2
24.11	even	2		64.9.c.d.63.2		2
48.5	odd	4		256.9.d.f.127.3		4
48.11	even	4		256.9.d.f.127.1		4
48.29	odd	4		256.9.d.f.127.2		4
48.35	even	4		256.9.d.f.127.4		4
60.23	odd	4		400.9.h.b.399.3		4
60.47	odd	4		400.9.h.b.399.1		4
60.59	even	2		400.9.b.c.351.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.9.c.a.15.1	✓	2	12.11	even	2
16.9.c.a.15.2	yes	2	3.2	odd	2
64.9.c.d.63.1		2	24.5	odd	2
64.9.c.d.63.2		2	24.11	even	2
144.9.g.g.127.1		2	1.1	even	1	trivial
144.9.g.g.127.2		2	4.3	odd	2	inner
256.9.d.f.127.1		4	48.11	even	4
256.9.d.f.127.2		4	48.29	odd	4
256.9.d.f.127.3		4	48.5	odd	4
256.9.d.f.127.4		4	48.35	even	4
400.9.b.c.351.1		2	15.14	odd	2
400.9.b.c.351.2		2	60.59	even	2
400.9.h.b.399.1		4	60.47	odd	4
400.9.h.b.399.2		4	15.8	even	4
400.9.h.b.399.3		4	60.23	odd	4
400.9.h.b.399.4		4	15.2	even	4

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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	144.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+510.000 q^{5} -2555.75i q^{7} -19168.1i q^{11} -27710.0 q^{13} -50370.0 q^{17} +108619. i q^{19} -176347. i q^{23} -130525. q^{25} -54978.0 q^{29} +1.17564e6i q^{31} -1.30343e6i q^{35} +793730. q^{37} +75582.0 q^{41} +499648. i q^{43} -2.86755e6i q^{47} -767039. q^{49} -1.11662e7 q^{53} -9.77573e6i q^{55} +2.18325e7i q^{59} -2.38266e7 q^{61} -1.41321e7 q^{65} +7.49473e6i q^{67} +1.00824e7i q^{71} +6.51661e6 q^{73} -4.89888e7 q^{77} -4.87892e7i q^{79} -7.34483e7i q^{83} -2.56887e7 q^{85} -8.67958e7 q^{89} +7.08197e7i q^{91} +5.53958e7i q^{95} -4.66703e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1020 q^{5} - 55420 q^{13} - 100740 q^{17} - 261050 q^{25} - 109956 q^{29} + 1587460 q^{37} + 151164 q^{41} - 1534078 q^{49} - 22332420 q^{53} - 47653244 q^{61} - 28264200 q^{65} + 13033220 q^{73}+ \cdots - 93340540 q^{97}+O(q^{100}) \) 2 * q + 1020 * q^5 - 55420 * q^13 - 100740 * q^17 - 261050 * q^25 - 109956 * q^29 + 1587460 * q^37 + 151164 * q^41 - 1534078 * q^49 - 22332420 * q^53 - 47653244 * q^61 - 28264200 * q^65 + 13033220 * q^73 - 97977600 * q^77 - 51377400 * q^85 - 173591556 * q^89 - 93340540 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more