LMFDB - Embedded newform 20.18.c.a.9.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,18,Mod(9,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("20.9");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$N$	$=$	$20 = 2^{2} \cdot 5$
Weight:	$k$	$=$	$18$
Character orbit:	$[\chi]$	$=$	20.c (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:	$36.6444174689$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{7} - 10513788 x^{6} + 47438777752 x^{5} - 249513269598475 x^{4} + \cdots + 12\!\cdots\!68$ x^8 - x^7 - 10513788x^6 + 47438777752x^5 - 249513269598475x^4 + 2482001120988967319x^3 + 4452658897796563124186x^2 - 53508982948055868249424962x + 129749232836567172546933042968
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{35}\cdot 3^{4}\cdot 5^{12}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			9.8
Root			$3569.15 - 2557.70i$ of defining polynomial
Character	$\chi$	$=$	20.9
Dual form			20.18.c.a.9.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+20057.4i q^{3} +(873406. + 10106.1i) q^{5} +2.05068e7i q^{7} -2.73159e8 q^{9} +5.77076e8 q^{11} +3.44138e9i q^{13} +(-2.02702e8 + 1.75182e10i) q^{15} +3.27062e10i q^{17} +1.11593e11 q^{19} -4.11313e11 q^{21} -2.10264e11i q^{23} +(7.62735e11 + 1.76535e10i) q^{25} -2.88864e12i q^{27} +6.55026e11 q^{29} -8.68067e12 q^{31} +1.15746e13i q^{33} +(-2.07244e11 + 1.79107e13i) q^{35} -9.38770e12i q^{37} -6.90250e13 q^{39} +2.89250e13 q^{41} -1.01667e14i q^{43} +(-2.38578e14 - 2.76057e12i) q^{45} -4.83144e13i q^{47} -1.87898e14 q^{49} -6.56002e14 q^{51} +2.29506e13i q^{53} +(5.04021e14 + 5.83199e12i) q^{55} +2.23827e15i q^{57} -7.24317e14 q^{59} -1.42187e15 q^{61} -5.60161e15i q^{63} +(-3.47789e13 + 3.00572e15i) q^{65} -3.93647e15i q^{67} +4.21735e15 q^{69} +1.00594e16 q^{71} -6.74204e15i q^{73} +(-3.54082e14 + 1.52985e16i) q^{75} +1.18340e16i q^{77} +5.75650e15 q^{79} +2.26628e16 q^{81} -2.82990e16i q^{83} +(-3.30533e14 + 2.85658e16i) q^{85} +1.31381e16i q^{87} +3.29446e16 q^{89} -7.05716e16 q^{91} -1.74112e17i q^{93} +(9.74661e16 + 1.12777e15i) q^{95} +7.01625e16i q^{97} -1.57633e17 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 1276800 q^{5} - 376848248 q^{9} - 396200640 q^{11} + 5677983200 q^{15} + 11821646592 q^{19} - 420670059472 q^{21} + 1642212165000 q^{25} + 1543712861232 q^{29} - 13722543013312 q^{31} - 13325691076800 q^{35}+ \cdots - 23\!\cdots\!60 q^{99}+O(q^{100})$ 8 * q + 1276800 * q^5 - 376848248 * q^9 - 396200640 * q^11 + 5677983200 * q^15 + 11821646592 * q^19 - 420670059472 * q^21 + 1642212165000 * q^25 + 1543712861232 * q^29 - 13722543013312 * q^31 - 13325691076800 * q^35 - 79298226023808 * q^39 + 29824965017568 * q^41 - 238216983006800 * q^45 - 535615080275736 * q^49 - 764079320419264 * q^51 + 339212427988000 * q^55 + 94539545151744 * q^59 - 3380827226969024 * q^61 + 130929936568800 * q^65 + 5559752175554416 * q^69 + 20155606060098624 * q^71 + 938454087760000 * q^75 - 3634057920126144 * q^79 + 15949137882627512 * q^81 + 3575192471022400 * q^85 + 103487949043655568 * q^89 - 69783870139121088 * q^91 + 67219016711959200 * q^95 - 239420571619653760 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$ .

$n$	$11$	$17$
$\chi(n)$	$1$	$-1$

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
$3$			20057.4i			1.76500i	0.470316	+	0.882498i	$0.344140\pi$
$3$			20057.4i			1.76500i	−0.470316	+	0.882498i	$0.655860\pi$
$4$		0			0
$5$	873406.	+	10106.1i	0.999933	+	0.0115701i
$5$	873406.	+	10106.1i	0.999933	+	0.0115701i
$6$		0			0
$7$			2.05068e7i			1.34451i	0.740319	+	0.672256i	$0.234674\pi$
$7$			2.05068e7i			1.34451i	−0.740319	+	0.672256i	$0.765326\pi$
$8$		0			0
$9$	−2.73159e8			−2.11521
$10$		0			0
$11$	5.77076e8			0.811699			0.405850	−	0.913940i	$-0.366976\pi$
$11$	5.77076e8			0.811699			0.405850	+	0.913940i	$0.366976\pi$
$12$		0			0
$13$			3.44138e9i			1.17008i	0.811006	+	0.585038i	$0.198920\pi$
$13$			3.44138e9i			1.17008i	−0.811006	+	0.585038i	$0.801080\pi$
$14$		0			0
$15$	−2.02702e8	+	1.75182e10i	−0.0204213	+	1.76488i
$16$		0			0
$17$			3.27062e10i			1.13714i	0.822634	+	0.568571i	$0.192504\pi$
$17$			3.27062e10i			1.13714i	−0.822634	+	0.568571i	$0.807496\pi$
$18$		0			0
$19$	1.11593e11			1.50741			0.753706	−	0.657212i	$-0.228264\pi$
$19$	1.11593e11			1.50741			0.753706	+	0.657212i	$0.228264\pi$
$20$		0			0
$21$	−4.11313e11			−2.37306
$22$		0			0
$23$		−	2.10264e11i		−	0.559859i	−0.960020	−	0.279930i	$-0.909689\pi$
$23$		−	2.10264e11i		−	0.559859i	0.960020	−	0.279930i	$-0.0903112\pi$
$24$		0			0
$25$	7.62735e11	+	1.76535e10i	0.999732	+	0.0231387i
$26$		0			0
$27$		−	2.88864e12i		−	1.96834i
$28$		0			0
$29$	6.55026e11			0.243151			0.121575	−	0.992582i	$-0.461205\pi$
$29$	6.55026e11			0.243151			0.121575	+	0.992582i	$0.461205\pi$
$30$		0			0
$31$	−8.68067e12			−1.82801			−0.914006	−	0.405701i	$-0.867028\pi$
$31$	−8.68067e12			−1.82801			−0.914006	+	0.405701i	$0.867028\pi$
$32$		0			0
$33$			1.15746e13i			1.43265i
$34$		0			0
$35$	−2.07244e11	+	1.79107e13i	−0.0155562	+	1.34442i
$36$		0			0
$37$		−	9.38770e12i		−	0.439384i	−0.975569	−	0.219692i	$-0.929495\pi$
$37$		−	9.38770e12i		−	0.439384i	0.975569	−	0.219692i	$-0.0705053\pi$
$38$		0			0
$39$	−6.90250e13			−2.06518
$40$		0			0
$41$	2.89250e13			0.565733			0.282866	−	0.959159i	$-0.408715\pi$
$41$	2.89250e13			0.565733			0.282866	+	0.959159i	$0.408715\pi$
$42$		0			0
$43$		−	1.01667e14i		−	1.32647i	−0.748410	−	0.663237i	$-0.769182\pi$
$43$		−	1.01667e14i		−	1.32647i	0.748410	−	0.663237i	$-0.230818\pi$
$44$		0			0
$45$	−2.38578e14	−	2.76057e12i	−2.11507	−	0.0244733i
$46$		0			0
$47$		−	4.83144e13i		−	0.295968i	−0.988990	−	0.147984i	$-0.952722\pi$
$47$		−	4.83144e13i		−	0.295968i	0.988990	−	0.147984i	$-0.0472784\pi$
$48$		0			0
$49$	−1.87898e14			−0.807710
$50$		0			0
$51$	−6.56002e14			−2.00705
$52$		0			0
$53$			2.29506e13i			0.0506349i	0.999679	+	0.0253174i	$0.00805965\pi$
$53$			2.29506e13i			0.0506349i	−0.999679	+	0.0253174i	$0.991940\pi$
$54$		0			0
$55$	5.04021e14	+	5.83199e12i	0.811645	+	0.00939148i
$56$		0			0
$57$			2.23827e15i			2.66058i
$58$		0			0
$59$	−7.24317e14			−0.642224			−0.321112	−	0.947041i	$-0.604057\pi$
$59$	−7.24317e14			−0.642224			−0.321112	+	0.947041i	$0.604057\pi$
$60$		0			0
$61$	−1.42187e15			−0.949636			−0.474818	−	0.880084i	$-0.657486\pi$
$61$	−1.42187e15			−0.949636			−0.474818	+	0.880084i	$0.657486\pi$
$62$		0			0
$63$		−	5.60161e15i		−	2.84393i
$64$		0			0
$65$	−3.47789e13	+	3.00572e15i	−0.0135379	+	1.17000i
$66$		0			0
$67$		−	3.93647e15i		−	1.18432i	−0.805819	−	0.592162i	$-0.798274\pi$
$67$		−	3.93647e15i		−	1.18432i	0.805819	−	0.592162i	$-0.201726\pi$
$68$		0			0
$69$	4.21735e15			0.988150
$70$		0			0
$71$	1.00594e16			1.84874			0.924371	−	0.381495i	$-0.124591\pi$
$71$	1.00594e16			1.84874			0.924371	+	0.381495i	$0.124591\pi$
$72$		0			0
$73$		−	6.74204e15i		−	0.978470i	−0.872152	−	0.489235i	$-0.837276\pi$
$73$		−	6.74204e15i		−	0.978470i	0.872152	−	0.489235i	$-0.162724\pi$
$74$		0			0
$75$	−3.54082e14	+	1.52985e16i	−0.0408398	+	1.76452i
$76$		0			0
$77$			1.18340e16i			1.09134i
$78$		0			0
$79$	5.75650e15			0.426902			0.213451	−	0.976954i	$-0.431530\pi$
$79$	5.75650e15			0.426902			0.213451	+	0.976954i	$0.431530\pi$
$80$		0			0
$81$	2.26628e16			1.35891
$82$		0			0
$83$		−	2.82990e16i		−	1.37914i	−0.724221	−	0.689568i	$-0.757800\pi$
$83$		−	2.82990e16i		−	1.37914i	0.724221	−	0.689568i	$-0.242200\pi$
$84$		0			0
$85$	−3.30533e14	+	2.85658e16i	−0.0131569	+	1.13707i
$86$		0			0
$87$			1.31381e16i			0.429160i
$88$		0			0
$89$	3.29446e16			0.887092			0.443546	−	0.896252i	$-0.353720\pi$
$89$	3.29446e16			0.887092			0.443546	+	0.896252i	$0.353720\pi$
$90$		0			0
$91$	−7.05716e16			−1.57318
$92$		0			0
$93$		−	1.74112e17i		−	3.22643i
$94$		0			0
$95$	9.74661e16	+	1.12777e15i	1.50731	+	0.0174410i
$96$		0			0
$97$			7.01625e16i			0.908961i	0.890757	+	0.454480i	$0.150175\pi$
$97$			7.01625e16i			0.908961i	−0.890757	+	0.454480i	$0.849825\pi$
$98$		0			0
$99$	−1.57633e17			−1.71692
$100$		0			0
$101$	−9.20483e16			−0.845832			−0.422916	−	0.906169i	$-0.638993\pi$
$101$	−9.20483e16			−0.845832			−0.422916	+	0.906169i	$0.638993\pi$
$102$		0			0
$103$			2.17615e17i			1.69267i	0.532651	+	0.846335i	$0.321196\pi$
$103$			2.17615e17i			1.69267i	−0.532651	+	0.846335i	$0.678804\pi$
$104$		0			0
$105$	−3.59243e17	−	4.15677e15i	−2.37290	−	0.0274566i
$106$		0			0
$107$			1.39995e17i			0.787679i	0.919179	+	0.393840i	$0.128853\pi$
$107$			1.39995e17i			0.787679i	−0.919179	+	0.393840i	$0.871147\pi$
$108$		0			0
$109$	1.29841e17			0.624145			0.312072	−	0.950058i	$-0.398977\pi$
$109$	1.29841e17			0.624145			0.312072	+	0.950058i	$0.398977\pi$
$110$		0			0
$111$	1.88293e17			0.775511
$112$		0			0
$113$			1.35449e17i			0.479303i	0.970859	+	0.239651i	$0.0770331\pi$
$113$			1.35449e17i			0.479303i	−0.970859	+	0.239651i	$0.922967\pi$
$114$		0			0
$115$	2.12495e15	−	1.83646e17i	0.00647765	−	0.559822i
$116$		0			0
$117$		−	9.40042e17i		−	2.47496i
$118$		0			0
$119$	−6.70700e17			−1.52890
$120$		0			0
$121$	−1.72431e17			−0.341145
$122$		0			0
$123$			5.80161e17i			0.998516i
$124$		0			0
$125$	6.65999e17	+	2.31269e16i	0.999398	+	0.0347042i
$126$		0			0
$127$		−	5.78568e17i		−	0.758617i	−0.925270	−	0.379309i	$-0.876162\pi$
$127$		−	5.78568e17i		−	0.758617i	0.925270	−	0.379309i	$-0.123838\pi$
$128$		0			0
$129$	2.03918e18			2.34122
$130$		0			0
$131$	9.65252e15			0.00972377			0.00486189	−	0.999988i	$-0.498452\pi$
$131$	9.65252e15			0.00972377			0.00486189	+	0.999988i	$0.498452\pi$
$132$		0			0
$133$			2.28842e18i			2.02673i
$134$		0			0
$135$	2.91929e16	−	2.52295e18i	0.0227740	−	1.96821i
$136$		0			0
$137$			2.62091e18i			1.80438i	0.431343	+	0.902188i	$0.358040\pi$
$137$			2.62091e18i			1.80438i	−0.431343	+	0.902188i	$0.641960\pi$
$138$		0			0
$139$	1.44661e18			0.880494			0.440247	−	0.897877i	$-0.354891\pi$
$139$	1.44661e18			0.880494			0.440247	+	0.897877i	$0.354891\pi$
$140$		0			0
$141$	9.69061e17			0.522382
$142$		0			0
$143$			1.98594e18i			0.949749i
$144$		0			0
$145$	5.72103e17	+	6.61976e15i	0.243134	+	0.00281329i
$146$		0			0
$147$		−	3.76874e18i		−	1.42561i
$148$		0			0
$149$	−6.92614e17			−0.233565			−0.116783	−	0.993158i	$-0.537258\pi$
$149$	−6.92614e17			−0.233565			−0.116783	+	0.993158i	$0.537258\pi$
$150$		0			0
$151$	−3.64449e17			−0.109732			−0.0548659	−	0.998494i	$-0.517473\pi$
$151$	−3.64449e17			−0.109732			−0.0548659	+	0.998494i	$0.517473\pi$
$152$		0			0
$153$		−	8.93399e18i		−	2.40530i
$154$		0			0
$155$	−7.58175e18	−	8.77278e16i	−1.82789	−	0.0211504i
$156$		0			0
$157$			2.54001e18i			0.549148i	0.961566	+	0.274574i	$0.0885368\pi$
$157$			2.54001e18i			0.549148i	−0.961566	+	0.274574i	$0.911463\pi$
$158$		0			0
$159$	−4.60330e17			−0.0893703
$160$		0			0
$161$	4.31185e18			0.752737
$162$		0			0
$163$			2.36995e18i			0.372516i	0.982501	+	0.186258i	$0.0596361\pi$
$163$			2.36995e18i			0.372516i	−0.982501	+	0.186258i	$0.940364\pi$
$164$		0			0
$165$	−1.16974e17	+	1.01094e19i	−0.0165759	+	1.43255i
$166$		0			0
$167$			7.21353e18i			0.922695i	0.887220	+	0.461347i	$0.152634\pi$
$167$			7.21353e18i			0.922695i	−0.887220	+	0.461347i	$0.847366\pi$
$168$		0			0
$169$	−3.19266e18			−0.369076
$170$		0			0
$171$	−3.04826e19			−3.18849
$172$		0			0
$173$		−	6.37728e18i		−	0.604288i	−0.953262	−	0.302144i	$-0.902298\pi$
$173$		−	6.37728e18i		−	0.604288i	0.953262	−	0.302144i	$-0.0977023\pi$
$174$		0			0
$175$	−3.62016e17	+	1.56413e19i	−0.0311103	+	1.34415i
$176$		0			0
$177$		−	1.45279e19i		−	1.13352i
$178$		0			0
$179$	1.26665e19			0.898268			0.449134	−	0.893464i	$-0.351733\pi$
$179$	1.26665e19			0.898268			0.449134	+	0.893464i	$0.351733\pi$
$180$		0			0
$181$	−2.79641e18			−0.180440			−0.0902199	−	0.995922i	$-0.528757\pi$
$181$	−2.79641e18			−0.180440			−0.0902199	+	0.995922i	$0.528757\pi$
$182$		0			0
$183$		−	2.85190e19i		−	1.67610i
$184$		0			0
$185$	9.48731e16	−	8.19927e18i	0.00508374	−	0.439355i
$186$		0			0
$187$			1.88740e19i			0.923017i
$188$		0			0
$189$	5.92367e19			2.64646
$190$		0			0
$191$	1.05592e19			0.431366			0.215683	−	0.976463i	$-0.430802\pi$
$191$	1.05592e19			0.431366			0.215683	+	0.976463i	$0.430802\pi$
$192$		0			0
$193$		−	2.71015e19i		−	1.01334i	−0.862140	−	0.506670i	$-0.830876\pi$
$193$		−	2.71015e19i		−	1.01334i	0.862140	−	0.506670i	$-0.169124\pi$
$194$		0			0
$195$	−6.02869e19	−	6.97574e17i	−2.06504	−	0.0238944i
$196$		0			0
$197$			2.05133e19i			0.644278i	0.946692	+	0.322139i	$0.104402\pi$
$197$			2.05133e19i			0.644278i	−0.946692	+	0.322139i	$0.895598\pi$
$198$		0			0
$199$	−6.93743e17			−0.0199962			−0.00999809	−	0.999950i	$-0.503183\pi$
$199$	−6.93743e17			−0.0199962			−0.00999809	+	0.999950i	$0.503183\pi$
$200$		0			0
$201$	7.89553e19			2.09033
$202$		0			0
$203$			1.34325e19i			0.326919i
$204$		0			0
$205$	2.52633e19	+	2.92320e17i	0.565695	+	0.00654561i
$206$		0			0
$207$			5.74355e19i			1.18422i
$208$		0			0
$209$	6.43977e19			1.22356
$210$		0			0
$211$	−5.59096e19			−0.979683			−0.489841	−	0.871812i	$-0.662945\pi$
$211$	−5.59096e19			−0.979683			−0.489841	+	0.871812i	$0.662945\pi$
$212$		0			0
$213$			2.01765e20i			3.26302i
$214$		0			0
$215$	1.02746e18	−	8.87966e19i	0.0153475	−	1.32638i
$216$		0			0
$217$		−	1.78013e20i		−	2.45778i
$218$		0			0
$219$	1.35228e20			1.72700
$220$		0			0
$221$	−1.12554e20			−1.33054
$222$		0			0
$223$		−	7.30921e19i		−	0.800349i	−0.916439	−	0.400174i	$-0.868950\pi$
$223$		−	7.30921e19i		−	0.800349i	0.916439	−	0.400174i	$-0.131050\pi$
$224$		0			0
$225$	−2.08348e20	−	4.82220e18i	−2.11465	−	0.0489433i
$226$		0			0
$227$		−	1.73698e20i		−	1.63522i	−0.575776	−	0.817608i	$-0.695300\pi$
$227$		−	1.73698e20i		−	1.63522i	0.575776	−	0.817608i	$-0.304700\pi$
$228$		0			0
$229$	2.02538e20			1.76972			0.884860	−	0.465857i	$-0.154254\pi$
$229$	2.02538e20			1.76972			0.884860	+	0.465857i	$0.154254\pi$
$230$		0			0
$231$	−2.37359e20			−1.92621
$232$		0			0
$233$			1.97092e20i			1.48643i	0.669052	+	0.743215i	$0.266700\pi$
$233$			1.97092e20i			1.48643i	−0.669052	+	0.743215i	$0.733300\pi$
$234$		0			0
$235$	4.88270e17	−	4.21981e19i	0.00342439	−	0.295948i
$236$		0			0
$237$			1.15460e20i			0.753480i
$238$		0			0
$239$	−2.80766e20			−1.70593			−0.852967	−	0.521964i	$-0.825199\pi$
$239$	−2.80766e20			−1.70593			−0.852967	+	0.521964i	$0.825199\pi$
$240$		0			0
$241$	−1.26963e20			−0.718677			−0.359339	−	0.933207i	$-0.616998\pi$
$241$	−1.26963e20			−0.718677			−0.359339	+	0.933207i	$0.616998\pi$
$242$		0			0
$243$			8.15169e19i			0.430124i
$244$		0			0
$245$	−1.64111e20	−	1.89892e18i	−0.807656	−	0.00934532i
$246$		0			0
$247$			3.84034e20i			1.76378i
$248$		0			0
$249$	5.67604e20			2.43417
$250$		0			0
$251$	2.72960e20			1.09363			0.546817	−	0.837252i	$-0.315839\pi$
$251$	2.72960e20			1.09363			0.546817	+	0.837252i	$0.315839\pi$
$252$		0			0
$253$		−	1.21338e20i		−	0.454437i
$254$		0			0
$255$	−5.72956e20	−	6.62962e18i	−2.00692	−	0.0232219i
$256$		0			0
$257$		−	4.96245e20i		−	1.62654i	−0.581888	−	0.813269i	$-0.697686\pi$
$257$		−	4.96245e20i		−	1.62654i	0.581888	−	0.813269i	$-0.302314\pi$
$258$		0			0
$259$	1.92512e20			0.590757
$260$		0			0
$261$	−1.78926e20			−0.514315
$262$		0			0
$263$		−	2.31553e20i		−	0.623773i	−0.950119	−	0.311886i	$-0.899039\pi$
$263$		−	2.31553e20i		−	0.623773i	0.950119	−	0.311886i	$-0.100961\pi$
$264$		0			0
$265$	−2.31941e17	+	2.00452e19i	−0.000585853	+	0.0506315i
$266$		0			0
$267$			6.60782e20i			1.56571i
$268$		0			0
$269$	1.16084e20			0.258154			0.129077	−	0.991635i	$-0.458798\pi$
$269$	1.16084e20			0.258154			0.129077	+	0.991635i	$0.458798\pi$
$270$		0			0
$271$	4.17435e20			0.871666			0.435833	−	0.900027i	$-0.356454\pi$
$271$	4.17435e20			0.871666			0.435833	+	0.900027i	$0.356454\pi$
$272$		0			0
$273$		−	1.41548e21i		−	2.77666i
$274$		0			0
$275$	4.40156e20	+	1.01874e19i	0.811482	+	0.0187817i
$276$		0			0
$277$			6.80268e18i			0.0117924i	0.999983	+	0.00589620i	$0.00187683\pi$
$277$			6.80268e18i			0.0117924i	−0.999983	+	0.00589620i	$0.998123\pi$
$278$		0			0
$279$	2.37120e21			3.86663
$280$		0			0
$281$	−7.39693e20			−1.13514			−0.567568	−	0.823327i	$-0.692116\pi$
$281$	−7.39693e20			−1.13514			−0.567568	+	0.823327i	$0.692116\pi$
$282$		0			0
$283$		−	3.04592e20i		−	0.440082i	−0.975491	−	0.220041i	$-0.929381\pi$
$283$		−	3.04592e20i		−	0.440082i	0.975491	−	0.220041i	$-0.0706191\pi$
$284$		0			0
$285$	−2.26202e19	+	1.95492e21i	−0.0307832	+	2.66040i
$286$		0			0
$287$			5.93160e20i			0.760634i
$288$		0			0
$289$	−2.42457e20			−0.293092
$290$		0			0
$291$	−1.40728e21			−1.60431
$292$		0			0
$293$			1.72745e21i			1.85794i	0.370159	+	0.928968i	$0.379303\pi$
$293$			1.72745e21i			1.85794i	−0.370159	+	0.928968i	$0.620697\pi$
$294$		0			0
$295$	−6.32623e20	−	7.32002e18i	−0.642181	−	0.00743063i
$296$		0			0
$297$		−	1.66696e21i		−	1.59770i
$298$		0			0
$299$	7.23599e20			0.655078
$300$		0			0
$301$	2.08487e21			1.78346
$302$		0			0
$303$		−	1.84625e21i		−	1.49289i
$304$		0			0
$305$	−1.24187e21	−	1.43696e19i	−0.949572	−	0.0109874i
$306$		0			0
$307$			6.77874e20i			0.490312i	0.969484	+	0.245156i	$0.0788392\pi$
$307$			6.77874e20i			0.490312i	−0.969484	+	0.245156i	$0.921161\pi$
$308$		0			0
$309$	−4.36479e21			−2.98756
$310$		0			0
$311$	7.52932e20			0.487857			0.243928	−	0.969793i	$-0.421564\pi$
$311$	7.52932e20			0.487857			0.243928	+	0.969793i	$0.421564\pi$
$312$		0			0
$313$			1.23121e20i			0.0755451i	0.999286	+	0.0377725i	$0.0120262\pi$
$313$			1.23121e20i			0.0755451i	−0.999286	+	0.0377725i	$0.987974\pi$
$314$		0			0
$315$	5.66105e19	−	4.89248e21i	0.0329046	−	2.84374i
$316$		0			0
$317$		−	1.05878e21i		−	0.583179i	−0.956544	−	0.291590i	$-0.905816\pi$
$317$		−	1.05878e21i		−	0.583179i	0.956544	−	0.291590i	$-0.0941841\pi$
$318$		0			0
$319$	3.77999e20			0.197365
$320$		0			0
$321$	−2.80793e21			−1.39025
$322$		0			0
$323$			3.64979e21i			1.71414i
$324$		0			0
$325$	−6.07522e19	+	2.62486e21i	−0.0270741	+	1.16976i
$326$		0			0
$327$			2.60426e21i			1.10161i
$328$		0			0
$329$	9.90773e20			0.397932
$330$		0			0
$331$	−1.44177e20			−0.0549992			−0.0274996	−	0.999622i	$-0.508755\pi$
$331$	−1.44177e20			−0.0549992			−0.0274996	+	0.999622i	$0.508755\pi$
$332$		0			0
$333$			2.56433e21i			0.929391i
$334$		0			0
$335$	3.97824e19	−	3.43813e21i	0.0137028	−	1.18425i
$336$		0			0
$337$		−	5.32303e20i		−	0.174303i	−0.996195	−	0.0871515i	$-0.972224\pi$
$337$		−	5.32303e20i		−	0.174303i	0.996195	−	0.0871515i	$-0.0277764\pi$
$338$		0			0
$339$	−2.71676e21			−0.845968
$340$		0			0
$341$	−5.00941e21			−1.48380
$342$		0			0
$343$			9.17320e20i			0.258536i
$344$		0			0
$345$	3.68346e21	+	4.26210e19i	0.988084	+	0.0114330i
$346$		0			0
$347$			3.51236e20i			0.0897012i	0.998994	+	0.0448506i	$0.0142812\pi$
$347$			3.51236e20i			0.0897012i	−0.998994	+	0.0448506i	$0.985719\pi$
$348$		0			0
$349$	4.27665e21			1.04013			0.520065	−	0.854127i	$-0.325908\pi$
$349$	4.27665e21			1.04013			0.520065	+	0.854127i	$0.325908\pi$
$350$		0			0
$351$	9.94089e21			2.30311
$352$		0			0
$353$			5.56822e21i			1.22922i	0.788829	+	0.614612i	$0.210687\pi$
$353$			5.56822e21i			1.22922i	−0.788829	+	0.614612i	$0.789313\pi$
$354$		0			0
$355$	8.78594e21	+	1.01661e20i	1.84862	+	0.0213902i
$356$		0			0
$357$		−	1.34525e22i		−	2.69850i
$358$		0			0
$359$	−8.60603e21			−1.64627			−0.823133	−	0.567848i	$-0.807776\pi$
$359$	−8.60603e21			−1.64627			−0.823133	+	0.567848i	$0.807776\pi$
$360$		0			0
$361$	6.97264e21			1.27229
$362$		0			0
$363$		−	3.45851e21i		−	0.602119i
$364$		0			0
$365$	6.81358e19	−	5.88854e21i	0.0113210	−	0.978404i
$366$		0			0
$367$			2.83082e21i			0.449005i	0.974473	+	0.224503i	$0.0720757\pi$
$367$			2.83082e21i			0.449005i	−0.974473	+	0.224503i	$0.927924\pi$
$368$		0			0
$369$	−7.90113e21			−1.19664
$370$		0			0
$371$	−4.70644e20			−0.0680791
$372$		0			0
$373$		−	4.63579e21i		−	0.640617i	−0.947313	−	0.320308i	$-0.896213\pi$
$373$		−	4.63579e21i		−	0.640617i	0.947313	−	0.320308i	$-0.103787\pi$
$374$		0			0
$375$	−4.63865e20	+	1.33582e22i	−0.0612528	+	1.76393i
$376$		0			0
$377$			2.25419e21i			0.284505i
$378$		0			0
$379$	4.78697e21			0.577600			0.288800	−	0.957389i	$-0.406744\pi$
$379$	4.78697e21			0.577600			0.288800	+	0.957389i	$0.406744\pi$
$380$		0			0
$381$	1.16046e22			1.33896
$382$		0			0
$383$		−	1.06724e22i		−	1.17781i	−0.808203	−	0.588903i	$-0.799560\pi$
$383$		−	1.06724e22i		−	1.17781i	0.808203	−	0.588903i	$-0.200440\pi$
$384$		0			0
$385$	−1.19595e20	+	1.03359e22i	−0.0126269	+	1.09127i
$386$		0			0
$387$			2.77713e22i			2.80577i
$388$		0			0
$389$	−1.07226e22			−1.03688			−0.518440	−	0.855114i	$-0.673487\pi$
$389$	−1.07226e22			−1.03688			−0.518440	+	0.855114i	$0.673487\pi$
$390$		0			0
$391$	6.87695e21			0.636640
$392$		0			0
$393$			1.93604e20i			0.0171624i
$394$		0			0
$395$	5.02776e21	+	5.81758e19i	0.426873	+	0.00493932i
$396$		0			0
$397$		−	8.40574e21i		−	0.683686i	−0.939757	−	0.341843i	$-0.888949\pi$
$397$		−	8.40574e21i		−	0.683686i	0.939757	−	0.341843i	$-0.111051\pi$
$398$		0			0
$399$	−4.58997e22			−3.57717
$400$		0			0
$401$	−1.31897e22			−0.985161			−0.492581	−	0.870267i	$-0.663946\pi$
$401$	−1.31897e22			−0.985161			−0.492581	+	0.870267i	$0.663946\pi$
$402$		0			0
$403$		−	2.98735e22i		−	2.13891i
$404$		0			0
$405$	1.97938e22	+	2.29032e20i	1.35882	+	0.0157228i
$406$		0			0
$407$		−	5.41741e21i		−	0.356648i
$408$		0			0
$409$	2.13563e22			1.34858			0.674290	−	0.738466i	$-0.264450\pi$
$409$	2.13563e22			1.34858			0.674290	+	0.738466i	$0.264450\pi$
$410$		0			0
$411$	−5.25686e22			−3.18472
$412$		0			0
$413$		−	1.48534e22i		−	0.863477i
$414$		0			0
$415$	2.85993e20	−	2.47165e22i	0.0159568	−	1.37904i
$416$		0			0
$417$			2.90153e22i			1.55407i
$418$		0			0
$419$	−8.29552e21			−0.426604			−0.213302	−	0.976986i	$-0.568422\pi$
$419$	−8.29552e21			−0.426604			−0.213302	+	0.976986i	$0.568422\pi$
$420$		0			0
$421$	1.92376e22			0.950064			0.475032	−	0.879968i	$-0.342436\pi$
$421$	1.92376e22			0.950064			0.475032	+	0.879968i	$0.342436\pi$
$422$		0			0
$423$			1.31975e22i			0.626035i
$424$		0			0
$425$	−5.77378e20	+	2.49462e22i	−0.0263120	+	1.13684i
$426$		0			0
$427$		−	2.91580e22i		−	1.27680i
$428$		0			0
$429$	−3.98327e22			−1.67630
$430$		0			0
$431$	3.99847e22			1.61747			0.808736	−	0.588172i	$-0.200152\pi$
$431$	3.99847e22			1.61747			0.808736	+	0.588172i	$0.200152\pi$
$432$		0			0
$433$			4.69877e22i			1.82741i	0.406373	+	0.913707i	$0.366793\pi$
$433$			4.69877e22i			1.82741i	−0.406373	+	0.913707i	$0.633207\pi$
$434$		0			0
$435$	−1.32775e20	+	1.14749e22i	−0.00496544	+	0.429131i
$436$		0			0
$437$		−	2.34641e22i		−	0.843939i
$438$		0			0
$439$	4.23882e21			0.146655			0.0733274	−	0.997308i	$-0.476638\pi$
$439$	4.23882e21			0.146655			0.0733274	+	0.997308i	$0.476638\pi$
$440$		0			0
$441$	5.13260e22			1.70848
$442$		0			0
$443$			2.78410e22i			0.891770i	0.895090	+	0.445885i	$0.147111\pi$
$443$			2.78410e22i			0.891770i	−0.895090	+	0.445885i	$0.852889\pi$
$444$		0			0
$445$	2.87740e22	+	3.32941e20i	0.887032	+	0.0102638i
$446$		0			0
$447$		−	1.38920e22i		−	0.412242i
$448$		0			0
$449$	−8.27920e21			−0.236534			−0.118267	−	0.992982i	$-0.537734\pi$
$449$	−8.27920e21			−0.236534			−0.118267	+	0.992982i	$0.537734\pi$
$450$		0			0
$451$	1.66919e22			0.459205
$452$		0			0
$453$		−	7.30989e21i		−	0.193676i
$454$		0			0
$455$	−6.16376e22	−	7.13204e20i	−1.57307	−	0.0182019i
$456$		0			0
$457$			1.78740e22i			0.439476i	0.975559	+	0.219738i	$0.0705202\pi$
$457$			1.78740e22i			0.439476i	−0.975559	+	0.219738i	$0.929480\pi$
$458$		0			0
$459$	9.44765e22			2.23829
$460$		0			0
$461$	−3.01542e22			−0.688478			−0.344239	−	0.938882i	$-0.611863\pi$
$461$	−3.01542e22			−0.688478			−0.344239	+	0.938882i	$0.611863\pi$
$462$		0			0
$463$			7.46737e22i			1.64335i	0.569958	+	0.821674i	$0.306960\pi$
$463$			7.46737e22i			1.64335i	−0.569958	+	0.821674i	$0.693040\pi$
$464$		0			0
$465$	1.75959e21	−	1.52070e23i	0.0373303	−	3.22622i
$466$		0			0
$467$		−	4.87795e22i		−	0.997801i	−0.866659	−	0.498901i	$-0.833737\pi$
$467$		−	4.87795e22i		−	0.997801i	0.866659	−	0.498901i	$-0.166263\pi$
$468$		0			0
$469$	8.07243e22			1.59234
$470$		0			0
$471$	−5.09461e22			−0.969244
$472$		0			0
$473$		−	5.86696e22i		−	1.07670i
$474$		0			0
$475$	8.51160e22	+	1.97000e21i	1.50701	+	0.0348796i
$476$		0			0
$477$		−	6.26916e21i		−	0.107103i
$478$		0			0
$479$	−2.90336e22			−0.478684			−0.239342	−	0.970935i	$-0.576932\pi$
$479$	−2.90336e22			−0.478684			−0.239342	+	0.970935i	$0.576932\pi$
$480$		0			0
$481$	3.23066e22			0.514113
$482$		0			0
$483$			8.64844e22i			1.32858i
$484$		0			0
$485$	−7.09069e20	+	6.12803e22i	−0.0105168	+	0.908900i
$486$		0			0
$487$			9.23980e21i			0.132332i	0.997809	+	0.0661662i	$0.0210768\pi$
$487$			9.23980e21i			0.132332i	−0.997809	+	0.0661662i	$0.978923\pi$
$488$		0			0
$489$	−4.75351e22			−0.657490
$490$		0			0
$491$	−9.28966e22			−1.24110			−0.620550	−	0.784167i	$-0.713091\pi$
$491$	−9.28966e22			−1.24110			−0.620550	+	0.784167i	$0.713091\pi$
$492$		0			0
$493$			2.14234e22i			0.276497i
$494$		0			0
$495$	−1.37678e23	−	1.59306e21i	−1.71680	−	0.0198650i
$496$		0			0
$497$			2.06286e23i			2.48565i
$498$		0			0
$499$	5.39863e21			0.0628679			0.0314340	−	0.999506i	$-0.489993\pi$
$499$	5.39863e21			0.0628679			0.0314340	+	0.999506i	$0.489993\pi$
$500$		0			0
$501$	−1.44685e23			−1.62855
$502$		0			0
$503$		−	1.41403e22i		−	0.153862i	−0.997036	−	0.0769308i	$-0.975488\pi$
$503$		−	1.41403e22i		−	0.153862i	0.997036	−	0.0769308i	$-0.0245121\pi$
$504$		0			0
$505$	−8.03955e22	−	9.30250e20i	−0.845775	−	0.00978640i
$506$		0			0
$507$		−	6.40364e22i		−	0.651418i
$508$		0			0
$509$	8.67443e22			0.853375			0.426687	−	0.904399i	$-0.359680\pi$
$509$	8.67443e22			0.853375			0.426687	+	0.904399i	$0.359680\pi$
$510$		0			0
$511$	1.38258e23			1.31556
$512$		0			0
$513$		−	3.22352e23i		−	2.96711i
$514$		0			0
$515$	−2.19924e21	+	1.90066e23i	−0.0195844	+	1.69256i
$516$		0			0
$517$		−	2.78811e22i		−	0.240237i
$518$		0			0
$519$	1.27912e23			1.06657
$520$		0			0
$521$	−1.96915e23			−1.58912			−0.794562	−	0.607182i	$-0.792300\pi$
$521$	−1.96915e23			−1.58912			−0.794562	+	0.607182i	$0.792300\pi$
$522$		0			0
$523$		−	1.20199e23i		−	0.938936i	−0.882949	−	0.469468i	$-0.844446\pi$
$523$		−	1.20199e23i		−	0.938936i	0.882949	−	0.469468i	$-0.155554\pi$
$524$		0			0
$525$	−3.13723e23	−	7.26109e21i	−2.37242	−	0.0549095i
$526$		0			0
$527$		−	2.83912e23i		−	2.07871i
$528$		0			0
$529$	9.68389e22			0.686557
$530$		0			0
$531$	1.97854e23			1.35844
$532$		0			0
$533$			9.95420e22i			0.661950i
$534$		0			0
$535$	−1.41480e21	+	1.22272e23i	−0.00911356	+	0.787627i
$536$		0			0
$537$			2.54057e23i			1.58544i
$538$		0			0
$539$	−1.08431e23			−0.655617
$540$		0			0
$541$	−8.28201e22			−0.485243			−0.242621	−	0.970121i	$-0.578007\pi$
$541$	−8.28201e22			−0.485243			−0.242621	+	0.970121i	$0.578007\pi$
$542$		0			0
$543$		−	5.60886e22i		−	0.318476i
$544$		0			0
$545$	1.13403e23	+	1.31218e21i	0.624103	+	0.00722144i
$546$		0			0
$547$		−	6.13233e22i		−	0.327140i	−0.986532	−	0.163570i	$-0.947699\pi$
$547$		−	6.13233e22i		−	0.327140i	0.986532	−	0.163570i	$-0.0523009\pi$
$548$		0			0
$549$	3.88397e23			2.00868
$550$		0			0
$551$	7.30964e22			0.366528
$552$		0			0
$553$			1.18047e23i			0.573974i
$554$		0			0
$555$	1.64456e23	+	1.90291e21i	0.775460	+	0.00897278i
$556$		0			0
$557$			1.95058e23i			0.892060i	0.895018	+	0.446030i	$0.147163\pi$
$557$			1.95058e23i			0.892060i	−0.895018	+	0.446030i	$0.852837\pi$
$558$		0			0
$559$	3.49875e23			1.55207
$560$		0			0
$561$	−3.78563e23			−1.62912
$562$		0			0
$563$		−	3.03504e23i		−	1.26719i	−0.773663	−	0.633597i	$-0.781578\pi$
$563$		−	3.03504e23i		−	1.26719i	0.773663	−	0.633597i	$-0.218422\pi$
$564$		0			0
$565$	−1.36886e21	+	1.18302e23i	−0.00554560	+	0.479271i
$566$		0			0
$567$			4.64741e23i			1.82707i
$568$		0			0
$569$	1.09368e23			0.417286			0.208643	−	0.977992i	$-0.433095\pi$
$569$	1.09368e23			0.417286			0.208643	+	0.977992i	$0.433095\pi$
$570$		0			0
$571$	−7.74427e21			−0.0286796			−0.0143398	−	0.999897i	$-0.504565\pi$
$571$	−7.74427e21			−0.0286796			−0.0143398	+	0.999897i	$0.504565\pi$
$572$		0			0
$573$			2.11789e23i			0.761360i
$574$		0			0
$575$	3.71189e21	−	1.60376e23i	0.0129544	−	0.559710i
$576$		0			0
$577$		−	3.69728e23i		−	1.25282i	−0.779495	−	0.626408i	$-0.784524\pi$
$577$		−	3.69728e23i		−	1.25282i	0.779495	−	0.626408i	$-0.215476\pi$
$578$		0			0
$579$	5.43585e23			1.78854
$580$		0			0
$581$	5.80321e23			1.85426
$582$		0			0
$583$			1.32442e22i			0.0411003i
$584$		0			0
$585$	9.50017e21	−	8.21038e23i	0.0286356	−	2.47479i
$586$		0			0
$587$		−	3.11591e23i		−	0.912350i	−0.889890	−	0.456175i	$-0.849219\pi$
$587$		−	3.11591e23i		−	0.912350i	0.889890	−	0.456175i	$-0.150781\pi$
$588$		0			0
$589$	−9.68704e23			−2.75557
$590$		0			0
$591$	−4.11444e23			−1.13715
$592$		0			0
$593$		−	7.15618e21i		−	0.0192184i	−0.999954	−	0.00960918i	$-0.996941\pi$
$593$		−	7.15618e21i		−	0.0192184i	0.999954	−	0.00960918i	$-0.00305875\pi$
$594$		0			0
$595$	−5.85793e23	−	6.77816e21i	−1.52880	−	0.0176896i
$596$		0			0
$597$		−	1.39147e22i		−	0.0352932i
$598$		0			0
$599$	−3.08044e23			−0.759425			−0.379712	−	0.925105i	$-0.623977\pi$
$599$	−3.08044e23			−0.759425			−0.379712	+	0.925105i	$0.623977\pi$
$600$		0			0
$601$	4.96702e23			1.19032			0.595159	−	0.803608i	$-0.297089\pi$
$601$	4.96702e23			1.19032			0.595159	+	0.803608i	$0.297089\pi$
$602$		0			0
$603$			1.07528e24i			2.50510i
$604$		0			0
$605$	−1.50602e23	−	1.74260e21i	−0.341122	−	0.00394709i
$606$		0			0
$607$		−	5.84823e23i		−	1.28801i	−0.765020	−	0.644007i	$-0.777271\pi$
$607$		−	5.84823e23i		−	1.28801i	0.765020	−	0.644007i	$-0.222729\pi$
$608$		0			0
$609$	−2.69420e23			−0.577010
$610$		0			0
$611$	1.66268e23			0.346305
$612$		0			0
$613$			8.18237e23i			1.65754i	0.559587	+	0.828772i	$0.310960\pi$
$613$			8.18237e23i			1.65754i	−0.559587	+	0.828772i	$0.689040\pi$
$614$		0			0
$615$	−5.86317e21	+	5.06716e23i	−0.0115530	+	0.998449i
$616$		0			0
$617$		−	5.30463e23i		−	1.01679i	−0.861124	−	0.508395i	$-0.830239\pi$
$617$		−	5.30463e23i		−	1.01679i	0.861124	−	0.508395i	$-0.169761\pi$
$618$		0			0
$619$	2.83076e23			0.527877			0.263939	−	0.964540i	$-0.414978\pi$
$619$	2.83076e23			0.527877			0.263939	+	0.964540i	$0.414978\pi$
$620$		0			0
$621$	−6.07378e23			−1.10200
$622$		0			0
$623$			6.75587e23i			1.19270i
$624$		0			0
$625$	5.81453e23	+	2.69298e22i	0.998929	+	0.0462651i
$626$		0			0
$627$			1.29165e24i			2.15959i
$628$		0			0
$629$	3.07036e23			0.499642
$630$		0			0
$631$	−1.17627e24			−1.86319			−0.931596	−	0.363495i	$-0.881583\pi$
$631$	−1.17627e24			−1.86319			−0.931596	+	0.363495i	$0.881583\pi$
$632$		0			0
$633$		−	1.12140e24i		−	1.72914i
$634$		0			0
$635$	5.84707e21	−	5.05324e23i	0.00877731	−	0.758566i
$636$		0			0
$637$		−	6.46628e23i		−	0.945081i
$638$		0			0
$639$	−2.74781e24			−3.91048
$640$		0			0
$641$	−7.67432e23			−1.06352			−0.531761	−	0.846894i	$-0.678470\pi$
$641$	−7.67432e23			−1.06352			−0.531761	+	0.846894i	$0.678470\pi$
$642$		0			0
$643$		−	1.17773e24i		−	1.58948i	−0.606952	−	0.794738i	$-0.707608\pi$
$643$		−	1.17773e24i		−	1.58948i	0.606952	−	0.794738i	$-0.292392\pi$
$644$		0			0
$645$	1.78103e24	+	2.06081e22i	2.34106	+	0.0270883i
$646$		0			0
$647$		−	6.99769e23i		−	0.895918i	−0.894054	−	0.447959i	$-0.852151\pi$
$647$		−	6.99769e23i		−	0.895918i	0.894054	−	0.447959i	$-0.147849\pi$
$648$		0			0
$649$	−4.17986e23			−0.521293
$650$		0			0
$651$	3.57047e24			4.33798
$652$		0			0
$653$			1.70215e23i			0.201482i	0.994913	+	0.100741i	$0.0321213\pi$
$653$			1.70215e23i			0.201482i	−0.994913	+	0.100741i	$0.967879\pi$
$654$		0			0
$655$	8.43057e21	+	9.75494e19i	0.00972312	+	0.000112505i
$656$		0			0
$657$			1.84165e24i			2.06967i
$658$		0			0
$659$	1.04323e24			1.14249			0.571246	−	0.820779i	$-0.306460\pi$
$659$	1.04323e24			1.14249			0.571246	+	0.820779i	$0.306460\pi$
$660$		0			0
$661$	−7.48506e23			−0.798882			−0.399441	−	0.916759i	$-0.630796\pi$
$661$	−7.48506e23			−0.798882			−0.399441	+	0.916759i	$0.630796\pi$
$662$		0			0
$663$		−	2.25755e24i		−	2.34840i
$664$		0			0
$665$	−2.31270e22	+	1.99872e24i	−0.0234496	+	2.02660i
$666$		0			0
$667$		−	1.37729e23i		−	0.136130i
$668$		0			0
$669$	1.46604e24			1.41261
$670$		0			0
$671$	−8.20528e23			−0.770818
$672$		0			0
$673$			3.36822e23i			0.308512i	0.988031	+	0.154256i	$0.0492981\pi$
$673$			3.36822e23i			0.308512i	−0.988031	+	0.154256i	$0.950702\pi$
$674$		0			0
$675$	5.09944e22	−	2.20327e24i	0.0455450	−	1.96782i
$676$		0			0
$677$		−	1.02218e24i		−	0.890271i	−0.895463	−	0.445136i	$-0.853155\pi$
$677$		−	1.02218e24i		−	0.890271i	0.895463	−	0.445136i	$-0.146845\pi$
$678$		0			0
$679$	−1.43881e24			−1.22211
$680$		0			0
$681$	3.48393e24			2.88615
$682$		0			0
$683$			1.82259e24i			1.47269i	0.676605	+	0.736346i	$0.263450\pi$
$683$			1.82259e24i			1.47269i	−0.676605	+	0.736346i	$0.736550\pi$
$684$		0			0
$685$	−2.64872e22	+	2.28912e24i	−0.0208769	+	1.80425i
$686$		0			0
$687$			4.06238e24i			3.12355i
$688$		0			0
$689$	−7.89817e22			−0.0592466
$690$		0			0
$691$	−1.16712e22			−0.00854186			−0.00427093	−	0.999991i	$-0.501359\pi$
$691$	−1.16712e22			−0.00854186			−0.00427093	+	0.999991i	$0.501359\pi$
$692$		0			0
$693$		−	3.23255e24i		−	2.30841i
$694$		0			0
$695$	1.26348e24	+	1.46196e22i	0.880435	+	0.0101874i
$696$		0			0
$697$			9.46029e23i			0.643318i
$698$		0			0
$699$	−3.95316e24			−2.62354
$700$		0			0
$701$	2.55946e24			1.65785			0.828924	−	0.559361i	$-0.188953\pi$
$701$	2.55946e24			1.65785			0.828924	+	0.559361i	$0.188953\pi$
$702$		0			0
$703$		−	1.04760e24i		−	0.662333i
$704$		0			0
$705$	8.46383e23	+	9.79343e21i	0.522347	+	0.00604404i
$706$		0			0
$707$		−	1.88761e24i		−	1.13723i
$708$		0			0
$709$	−1.01642e24			−0.597836			−0.298918	−	0.954279i	$-0.596626\pi$
$709$	−1.01642e24			−0.597836			−0.298918	+	0.954279i	$0.596626\pi$
$710$		0			0
$711$	−1.57244e24			−0.902988
$712$		0			0
$713$			1.82524e24i			1.02343i
$714$		0			0
$715$	−2.00701e22	+	1.73453e24i	−0.0109887	+	0.949685i
$716$		0			0
$717$		−	5.63143e24i		−	3.01097i
$718$		0			0
$719$	3.69187e23			0.192775			0.0963874	−	0.995344i	$-0.469271\pi$
$719$	3.69187e23			0.192775			0.0963874	+	0.995344i	$0.469271\pi$
$720$		0			0
$721$	−4.46259e24			−2.27581
$722$		0			0
$723$		−	2.54656e24i		−	1.26846i
$724$		0			0
$725$	4.99611e23	+	1.15635e22i	0.243086	+	0.00562620i
$726$		0			0
$727$			6.96764e22i			0.0331164i	0.999863	+	0.0165582i	$0.00527088\pi$
$727$			6.96764e22i			0.0331164i	−0.999863	+	0.0165582i	$0.994729\pi$
$728$		0			0
$729$	1.29166e24			0.599741
$730$		0			0
$731$	3.32515e24			1.50839
$732$		0			0
$733$			2.64270e24i			1.17129i	0.810567	+	0.585645i	$0.199159\pi$
$733$			2.64270e24i			1.17129i	−0.810567	+	0.585645i	$0.800841\pi$
$734$		0			0
$735$	3.80873e22	−	3.29164e24i	0.0164945	−	1.42551i
$736$		0			0
$737$		−	2.27164e24i		−	0.961315i
$738$		0			0
$739$	−1.47868e24			−0.611501			−0.305751	−	0.952112i	$-0.598907\pi$
$739$	−1.47868e24			−0.611501			−0.305751	+	0.952112i	$0.598907\pi$
$740$		0			0
$741$	−7.70272e24			−3.11307
$742$		0			0
$743$			4.25140e24i			1.67929i	0.543133	+	0.839647i	$0.317238\pi$
$743$			4.25140e24i			1.67929i	−0.543133	+	0.839647i	$0.682762\pi$
$744$		0			0
$745$	−6.04933e23	−	6.99963e21i	−0.233550	−	0.00270238i
$746$		0			0
$747$			7.73012e24i			2.91716i
$748$		0			0
$749$	−2.87084e24			−1.05904
$750$		0			0
$751$	−1.54644e24			−0.557691			−0.278846	−	0.960336i	$-0.589952\pi$
$751$	−1.54644e24			−0.557691			−0.278846	+	0.960336i	$0.589952\pi$
$752$		0			0
$753$			5.47486e24i			1.93026i
$754$		0			0
$755$	−3.18312e23	−	3.68316e21i	−0.109724	−	0.00126961i
$756$		0			0
$757$		−	1.47583e24i		−	0.497417i	−0.968578	−	0.248708i	$-0.919994\pi$
$757$		−	1.47583e24i		−	0.497417i	0.968578	−	0.248708i	$-0.0800061\pi$
$758$		0			0
$759$	2.43373e24			0.802080
$760$		0			0
$761$	7.26711e23			0.234203			0.117101	−	0.993120i	$-0.462640\pi$
$761$	7.26711e23			0.234203			0.117101	+	0.993120i	$0.462640\pi$
$762$		0			0
$763$			2.66261e24i			0.839169i
$764$		0			0
$765$	9.02879e22	−	7.80300e24i	0.0278296	−	2.40513i
$766$		0			0
$767$		−	2.49265e24i		−	0.751450i
$768$		0			0
$769$	4.71635e23			0.139070			0.0695348	−	0.997580i	$-0.477849\pi$
$769$	4.71635e23			0.139070			0.0695348	+	0.997580i	$0.477849\pi$
$770$		0			0
$771$	9.95337e24			2.87083
$772$		0			0
$773$		−	2.01954e24i		−	0.569805i	−0.958556	−	0.284903i	$-0.908039\pi$
$773$		−	2.01954e24i		−	0.569805i	0.958556	−	0.284903i	$-0.0919613\pi$
$774$		0			0
$775$	−6.62106e24	−	1.53244e23i	−1.82752	−	0.0422979i
$776$		0			0
$777$			3.86128e24i			1.04268i
$778$		0			0
$779$	3.22784e24			0.852792
$780$		0			0
$781$	5.80504e24			1.50062
$782$		0			0
$783$		−	1.89213e24i		−	0.478604i
$784$		0			0
$785$	−2.56697e22	+	2.21846e24i	−0.00635372	+	0.549111i
$786$		0			0
$787$		−	6.84973e24i		−	1.65916i	−0.558389	−	0.829579i	$-0.688580\pi$
$787$		−	6.84973e24i		−	1.65916i	0.558389	−	0.829579i	$-0.311420\pi$
$788$		0			0
$789$	4.64435e24			1.10096
$790$		0			0
$791$	−2.77763e24			−0.644428
$792$		0			0
$793$		−	4.89320e24i		−	1.11115i
$794$		0			0
$795$	−4.02054e23	−	4.65214e21i	−0.0893643	−	0.00103403i
$796$		0			0
$797$			3.79244e24i			0.825130i	0.910928	+	0.412565i	$0.135367\pi$
$797$			3.79244e24i			0.825130i	−0.910928	+	0.412565i	$0.864633\pi$
$798$		0			0
$799$	1.58018e24			0.336558
$800$		0			0
$801$	−8.99909e24			−1.87639
$802$		0			0
$803$		−	3.89067e24i		−	0.794223i
$804$		0			0
$805$	3.76599e24	+	4.35760e22i	0.752687	+	0.00870928i
$806$		0			0
$807$			2.32835e24i			0.455642i
$808$		0			0
$809$	3.08318e24			0.590795			0.295397	−	0.955375i	$-0.404548\pi$
$809$	3.08318e24			0.590795			0.295397	+	0.955375i	$0.404548\pi$
$810$		0			0
$811$	4.19581e24			0.787297			0.393648	−	0.919261i	$-0.371213\pi$
$811$	4.19581e24			0.787297			0.393648	+	0.919261i	$0.371213\pi$
$812$		0			0
$813$			8.37266e24i			1.53849i
$814$		0			0
$815$	−2.39510e22	+	2.06993e24i	−0.00431007	+	0.372491i
$816$		0			0
$817$		−	1.13454e25i		−	1.99954i
$818$		0			0
$819$	1.92773e25			3.32761
$820$		0			0
$821$	−6.34936e24			−1.07353			−0.536764	−	0.843733i	$-0.680353\pi$
$821$	−6.34936e24			−1.07353			−0.536764	+	0.843733i	$0.680353\pi$
$822$		0			0
$823$			3.83115e24i			0.634498i	0.948342	+	0.317249i	$0.102759\pi$
$823$			3.83115e24i			0.634498i	−0.948342	+	0.317249i	$0.897241\pi$
$824$		0			0
$825$	−2.04332e23	+	8.82838e24i	−0.0331496	+	1.43226i
$826$		0			0
$827$		−	1.93569e24i		−	0.307637i	−0.988099	−	0.153818i	$-0.950843\pi$
$827$		−	1.93569e24i		−	0.307637i	0.988099	−	0.153818i	$-0.0491571\pi$
$828$		0			0
$829$	−5.74799e24			−0.894958			−0.447479	−	0.894294i	$-0.647678\pi$
$829$	−5.74799e24			−0.894958			−0.447479	+	0.894294i	$0.647678\pi$
$830$		0			0
$831$	−1.36444e23			−0.0208135
$832$		0			0
$833$		−	6.14544e24i		−	0.918481i
$834$		0			0
$835$	−7.29007e22	+	6.30034e24i	−0.0106757	+	0.922633i
$836$		0			0
$837$			2.50753e25i			3.59816i
$838$		0			0
$839$	−1.81747e24			−0.255558			−0.127779	−	0.991803i	$-0.540785\pi$
$839$	−1.81747e24			−0.255558			−0.127779	+	0.991803i	$0.540785\pi$
$840$		0			0
$841$	−6.82809e24			−0.940878
$842$		0			0
$843$		−	1.48363e25i		−	2.00351i
$844$		0			0
$845$	−2.78849e24	−	3.22654e22i	−0.369051	−	0.00427026i
$846$		0			0
$847$		−	3.53600e24i		−	0.458673i
$848$		0			0
$849$	6.10931e24			0.776742
$850$		0			0
$851$	−1.97390e24			−0.245993
$852$		0			0
$853$			3.54939e24i			0.433598i	0.976216	+	0.216799i	$0.0695616\pi$
$853$			3.54939e24i			0.433598i	−0.976216	+	0.216799i	$0.930438\pi$
$854$		0			0
$855$	−2.66237e25	−	3.08061e23i	−3.18828	−	0.0368913i
$856$		0			0
$857$		−	8.57068e24i		−	1.00619i	−0.864232	−	0.503093i	$-0.832195\pi$
$857$		−	8.57068e24i		−	1.00619i	0.864232	−	0.503093i	$-0.167805\pi$
$858$		0			0
$859$	8.48867e24			0.977007			0.488503	−	0.872562i	$-0.337543\pi$
$859$	8.48867e24			0.977007			0.488503	+	0.872562i	$0.337543\pi$
$860$		0			0
$861$	−1.18972e25			−1.34252
$862$		0			0
$863$		−	1.68443e25i		−	1.86364i	−0.362927	−	0.931818i	$-0.618223\pi$
$863$		−	1.68443e25i		−	1.86364i	0.362927	−	0.931818i	$-0.381777\pi$
$864$		0			0
$865$	6.44495e22	−	5.56995e24i	0.00699169	−	0.604247i
$866$		0			0
$867$		−	4.86306e24i		−	0.517306i
$868$		0			0
$869$	3.32194e24			0.346516
$870$		0			0
$871$	1.35469e25			1.38575
$872$		0			0
$873$		−	1.91655e25i		−	1.92264i
$874$		0			0
$875$	−4.74259e23	+	1.36575e25i	−0.0466602	+	1.34370i
$876$		0			0
$877$			4.53773e24i			0.437867i	0.975740	+	0.218934i	$0.0702578\pi$
$877$			4.53773e24i			0.437867i	−0.975740	+	0.218934i	$0.929742\pi$
$878$		0			0
$879$	−3.46482e25			−3.27925
$880$		0			0
$881$	−1.06498e25			−0.988658			−0.494329	−	0.869275i	$-0.664586\pi$
$881$	−1.06498e25			−0.988658			−0.494329	+	0.869275i	$0.664586\pi$
$882$		0			0
$883$			1.94112e25i			1.76761i	0.467853	+	0.883806i	$0.345028\pi$
$883$			1.94112e25i			1.76761i	−0.467853	+	0.883806i	$0.654972\pi$
$884$		0			0
$885$	1.46821e23	−	1.26888e25i	0.0131150	−	1.13345i
$886$		0			0
$887$			9.02260e24i			0.790644i	0.918543	+	0.395322i	$0.129367\pi$
$887$			9.02260e24i			0.790644i	−0.918543	+	0.395322i	$0.870633\pi$
$888$		0			0
$889$	1.18646e25			1.01997
$890$		0			0
$891$	1.30781e25			1.10302
$892$		0			0
$893$		−	5.39155e24i		−	0.446146i
$894$		0			0
$895$	1.10630e25	+	1.28009e23i	0.898208	+	0.0103931i
$896$		0			0
$897$			1.45135e25i			1.15621i
$898$		0			0
$899$	−5.68606e24			−0.444482
$900$		0			0
$901$	−7.50628e23			−0.0575790
$902$		0			0
$903$			4.18170e25i			3.14780i
$904$		0			0
$905$	−2.44240e24	−	2.82608e22i	−0.180428	−	0.00208772i
$906$		0			0
$907$			6.65115e24i			0.482209i	0.970499	+	0.241104i	$0.0775096\pi$
$907$			6.65115e24i			0.482209i	−0.970499	+	0.241104i	$0.922490\pi$
$908$		0			0
$909$	2.51438e25			1.78911
$910$		0			0
$911$	7.54898e24			0.527208			0.263604	−	0.964631i	$-0.415089\pi$
$911$	7.54898e24			0.527208			0.263604	+	0.964631i	$0.415089\pi$
$912$		0			0
$913$		−	1.63307e25i		−	1.11944i
$914$		0			0
$915$	2.88217e23	−	2.49087e25i	0.0193928	−	1.67599i
$916$		0			0
$917$			1.97942e23i			0.0130737i
$918$		0			0
$919$	−8.14021e23			−0.0527781			−0.0263891	−	0.999652i	$-0.508401\pi$
$919$	−8.14021e23			−0.0527781			−0.0263891	+	0.999652i	$0.508401\pi$
$920$		0			0
$921$	−1.35964e25			−0.865399
$922$		0			0
$923$			3.46182e25i			2.16317i
$924$		0			0
$925$	1.65725e23	−	7.16033e24i	0.0101668	−	0.439267i
$926$		0			0
$927$		−	5.94434e25i		−	3.58036i
$928$		0			0
$929$	2.04441e25			1.20902			0.604512	−	0.796596i	$-0.293368\pi$
$929$	2.04441e25			1.20902			0.604512	+	0.796596i	$0.293368\pi$
$930$		0			0
$931$	−2.09681e25			−1.21755
$932$		0			0
$933$			1.51019e25i			0.861066i
$934$		0			0
$935$	−1.90742e23	+	1.64846e25i	−0.0106794	+	0.922955i
$936$		0			0
$937$		−	6.84554e24i		−	0.376375i	−0.982133	−	0.188188i	$-0.939739\pi$
$937$		−	6.84554e24i		−	0.376375i	0.982133	−	0.188188i	$-0.0602613\pi$
$938$		0			0
$939$	−2.46949e24			−0.133337
$940$		0			0
$941$	1.17514e25			0.623127			0.311564	−	0.950225i	$-0.399147\pi$
$941$	1.17514e25			0.623127			0.311564	+	0.950225i	$0.399147\pi$
$942$		0			0
$943$		−	6.08191e24i		−	0.316731i
$944$		0			0
$945$	5.17377e25	+	5.98652e23i	2.64628	+	0.0306199i
$946$		0			0
$947$		−	2.80544e25i		−	1.40937i	−0.709519	−	0.704687i	$-0.751088\pi$
$947$		−	2.80544e25i		−	1.40937i	0.709519	−	0.704687i	$-0.248912\pi$
$948$		0			0
$949$	2.32019e25			1.14488
$950$		0			0
$951$	2.12364e25			1.02931
$952$		0			0
$953$			1.01517e25i			0.483337i	0.970359	+	0.241668i	$0.0776946\pi$
$953$			1.01517e25i			0.483337i	−0.970359	+	0.241668i	$0.922305\pi$
$954$		0			0
$955$	9.22244e24	+	1.06712e23i	0.431337	+	0.00499097i
$956$		0			0
$957$			7.58168e24i			0.348349i
$958$		0			0
$959$	−5.37464e25			−2.42600
$960$		0			0
$961$	5.28040e25			2.34163
$962$		0			0
$963$		−	3.82408e25i		−	1.66611i
$964$		0			0
$965$	2.73890e23	−	2.36706e25i	0.0117245	−	1.01327i
$966$		0			0
$967$			1.61523e25i			0.679376i	0.940538	+	0.339688i	$0.110321\pi$
$967$			1.61523e25i			0.679376i	−0.940538	+	0.339688i	$0.889679\pi$
$968$		0			0
$969$	−7.32053e25			−3.02545
$970$		0			0
$971$	7.66908e24			0.311444			0.155722	−	0.987801i	$-0.450230\pi$
$971$	7.66908e24			0.311444			0.155722	+	0.987801i	$0.450230\pi$
$972$		0			0
$973$			2.96654e25i			1.18383i
$974$		0			0
$975$	−5.26478e25	−	1.21853e24i	−2.06463	−	0.0477856i
$976$		0			0
$977$			4.23271e25i			1.63123i	0.578597	+	0.815614i	$0.303600\pi$
$977$			4.23271e25i			1.63123i	−0.578597	+	0.815614i	$0.696400\pi$
$978$		0			0
$979$	1.90115e25			0.720052
$980$		0			0
$981$	−3.54671e25			−1.32020
$982$		0			0
$983$			1.51838e25i			0.555491i	0.960655	+	0.277745i	$0.0895871\pi$
$983$			1.51838e25i			0.555491i	−0.960655	+	0.277745i	$0.910413\pi$
$984$		0			0
$985$	−2.07310e23	+	1.79165e25i	−0.00745439	+	0.644235i
$986$		0			0
$987$			1.98723e25i			0.702349i
$988$		0			0
$989$	−2.13770e25			−0.742639
$990$		0			0
$991$	4.48387e25			1.53118			0.765592	−	0.643327i	$-0.222446\pi$
$991$	4.48387e25			1.53118			0.765592	+	0.643327i	$0.222446\pi$
$992$		0			0
$993$		−	2.89180e24i		−	0.0970734i
$994$		0			0
$995$	−6.05919e23	−	7.01103e21i	−0.0199948	−	0.000231359i
$996$		0			0
$997$			1.64100e25i			0.532353i	0.963924	+	0.266176i	$0.0857604\pi$
$997$			1.64100e25i			0.532353i	−0.963924	+	0.266176i	$0.914240\pi$
$998$		0			0
$999$	−2.71177e25			−0.864859

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.18.c.a.9.8	yes	8
4.3	odd	2		80.18.c.c.49.1		8
5.2	odd	4		100.18.a.f.1.8		8
5.3	odd	4		100.18.a.f.1.1		8
5.4	even	2	inner	20.18.c.a.9.1	✓	8
20.19	odd	2		80.18.c.c.49.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.18.c.a.9.1	✓	8	5.4	even	2	inner
20.18.c.a.9.8	yes	8	1.1	even	1	trivial
80.18.c.c.49.1		8	4.3	odd	2
80.18.c.c.49.8		8	20.19	odd	2
100.18.a.f.1.1		8	5.3	odd	4
100.18.a.f.1.8		8	5.2	odd	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	20.18.c.a.9.8

Level	$20$
Weight	$18$
Character	20.9
Analytic conductor	$36.644$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$