LMFDB - Embedded newform 160.6.c.d.129.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(129,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("160.129");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	160.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:	$25.6614111701$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 18 x^{10} + 348 x^{9} + 21226 x^{8} - 87824 x^{7} + 205428 x^{6} + 2113880 x^{5} + \cdots + 1072562500 $ x^12 - 6x^11 + 18x^10 + 348x^9 + 21226x^8 - 87824x^7 + 205428x^6 + 2113880x^5 + 27827225x^4 - 59798850x^3 + 89646050x^2 + 438522500*x + 1072562500
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{46}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.8
Root			$-1.27837 + 1.27837i$ of defining polynomial
Character	$\chi$	$=$	160.129
Dual form			160.6.c.d.129.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.01877i q^{3} +(-46.1478 + 31.5496i) q^{5} -15.3895i q^{7} +217.812 q^{9} +O(q^{10})$	$q+5.01877i q^{3} +(-46.1478 + 31.5496i) q^{5} -15.3895i q^{7} +217.812 q^{9} +576.187 q^{11} -607.144i q^{13} +(-158.340 - 231.605i) q^{15} +2013.10i q^{17} -2421.47 q^{19} +77.2365 q^{21} +4356.71i q^{23} +(1134.25 - 2911.89i) q^{25} +2312.71i q^{27} -1224.89 q^{29} -7812.15 q^{31} +2891.75i q^{33} +(485.534 + 710.194i) q^{35} -3806.09i q^{37} +3047.12 q^{39} -4476.54 q^{41} +14713.6i q^{43} +(-10051.5 + 6871.88i) q^{45} +20493.5i q^{47} +16570.2 q^{49} -10103.3 q^{51} -10903.0i q^{53} +(-26589.8 + 18178.5i) q^{55} -12152.8i q^{57} -31131.9 q^{59} -4486.25 q^{61} -3352.02i q^{63} +(19155.2 + 28018.4i) q^{65} +45444.5i q^{67} -21865.3 q^{69} +10392.4 q^{71} +38979.1i q^{73} +(14614.1 + 5692.52i) q^{75} -8867.24i q^{77} -23211.9 q^{79} +41321.3 q^{81} -34715.0i q^{83} +(-63512.5 - 92900.2i) q^{85} -6147.46i q^{87} +97820.7 q^{89} -9343.66 q^{91} -39207.4i q^{93} +(111745. - 76396.3i) q^{95} +151178. i q^{97} +125500. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 60 q^{5} - 268 q^{9}+O(q^{10}) $ 12 * q - 60 * q^5 - 268 * q^9	$ 12 q - 60 q^{5} - 268 q^{9} + 12080 q^{21} - 16420 q^{25} - 7864 q^{29} + 65560 q^{41} - 60420 q^{45} + 17956 q^{49} - 118232 q^{61} + 11360 q^{65} + 204464 q^{69} - 346436 q^{81} + 96320 q^{85} + 354936 q^{89}+O(q^{100}) $ 12 * q - 60 * q^5 - 268 * q^9 + 12080 * q^21 - 16420 * q^25 - 7864 * q^29 + 65560 * q^41 - 60420 * q^45 + 17956 * q^49 - 118232 * q^61 + 11360 * q^65 + 204464 * q^69 - 346436 * q^81 + 96320 * q^85 + 354936 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			5.01877i			0.321954i	0.986958	+	0.160977i	$0.0514646\pi$
$3$			5.01877i			0.321954i	−0.986958	+	0.160977i	$0.948535\pi$
$4$		0			0
$5$	−46.1478	+	31.5496i	−0.825518	+	0.564376i
$5$	−46.1478	+	31.5496i	−0.825518	+	0.564376i
$6$		0			0
$7$		−	15.3895i		−	0.118708i	−0.998237	−	0.0593540i	$-0.981096\pi$
$7$		−	15.3895i		−	0.118708i	0.998237	−	0.0593540i	$-0.0189041\pi$
$8$		0			0
$9$	217.812			0.896345
$10$		0			0
$11$	576.187			1.43576			0.717880	−	0.696167i	$-0.245113\pi$
$11$	576.187			1.43576			0.717880	+	0.696167i	$0.245113\pi$
$12$		0			0
$13$		−	607.144i		−	0.996399i	−0.867062	−	0.498200i	$-0.833995\pi$
$13$		−	607.144i		−	0.996399i	0.867062	−	0.498200i	$-0.166005\pi$
$14$		0			0
$15$	−158.340	−	231.605i	−0.181703	−	0.265779i
$16$		0			0
$17$			2013.10i			1.68944i	0.535208	+	0.844720i	$0.320233\pi$
$17$			2013.10i			1.68944i	−0.535208	+	0.844720i	$0.679767\pi$
$18$		0			0
$19$	−2421.47			−1.53884			−0.769422	−	0.638741i	$-0.779456\pi$
$19$	−2421.47			−1.53884			−0.769422	+	0.638741i	$0.779456\pi$
$20$		0			0
$21$	77.2365			0.0382186
$22$		0			0
$23$			4356.71i			1.71727i	0.512586	+	0.858636i	$0.328687\pi$
$23$			4356.71i			1.71727i	−0.512586	+	0.858636i	$0.671313\pi$
$24$		0			0
$25$	1134.25	−	2911.89i	0.362959	−	0.931805i
$26$		0			0
$27$			2312.71i			0.610537i
$28$		0			0
$29$	−1224.89			−0.270460			−0.135230	−	0.990814i	$-0.543177\pi$
$29$	−1224.89			−0.270460			−0.135230	+	0.990814i	$0.543177\pi$
$30$		0			0
$31$	−7812.15			−1.46005			−0.730023	−	0.683423i	$-0.760491\pi$
$31$	−7812.15			−1.46005			−0.730023	+	0.683423i	$0.760491\pi$
$32$		0			0
$33$			2891.75i			0.462249i
$34$		0			0
$35$	485.534	+	710.194i	0.0669960	+	0.0979956i
$36$		0			0
$37$		−	3806.09i		−	0.457062i	−0.973537	−	0.228531i	$-0.926608\pi$
$37$		−	3806.09i		−	0.457062i	0.973537	−	0.228531i	$-0.0733922\pi$
$38$		0			0
$39$	3047.12			0.320795
$40$		0			0
$41$	−4476.54			−0.415895			−0.207947	−	0.978140i	$-0.566678\pi$
$41$	−4476.54			−0.415895			−0.207947	+	0.978140i	$0.566678\pi$
$42$		0			0
$43$			14713.6i			1.21352i	0.794884	+	0.606762i	$0.207532\pi$
$43$			14713.6i			1.21352i	−0.794884	+	0.606762i	$0.792468\pi$
$44$		0			0
$45$	−10051.5	+	6871.88i	−0.739949	+	0.505876i
$46$		0			0
$47$			20493.5i			1.35323i	0.736336	+	0.676616i	$0.236554\pi$
$47$			20493.5i			1.35323i	−0.736336	+	0.676616i	$0.763446\pi$
$48$		0			0
$49$	16570.2			0.985908
$50$		0			0
$51$	−10103.3			−0.543923
$52$		0			0
$53$		−	10903.0i		−	0.533157i	−0.963813	−	0.266579i	$-0.914107\pi$
$53$		−	10903.0i		−	0.533157i	0.963813	−	0.266579i	$-0.0858932\pi$
$54$		0			0
$55$	−26589.8	+	18178.5i	−1.18524	+	0.810309i
$56$		0			0
$57$		−	12152.8i		−	0.495437i
$58$		0			0
$59$	−31131.9			−1.16433			−0.582164	−	0.813071i	$-0.697794\pi$
$59$	−31131.9			−1.16433			−0.582164	+	0.813071i	$0.697794\pi$
$60$		0			0
$61$	−4486.25			−0.154369			−0.0771843	−	0.997017i	$-0.524593\pi$
$61$	−4486.25			−0.154369			−0.0771843	+	0.997017i	$0.524593\pi$
$62$		0			0
$63$		−	3352.02i		−	0.106403i
$64$		0			0
$65$	19155.2	+	28018.4i	0.562344	+	0.822545i
$66$		0			0
$67$			45444.5i			1.23679i	0.785869	+	0.618393i	$0.212216\pi$
$67$			45444.5i			1.23679i	−0.785869	+	0.618393i	$0.787784\pi$
$68$		0			0
$69$	−21865.3			−0.552883
$70$		0			0
$71$	10392.4			0.244664			0.122332	−	0.992489i	$-0.460963\pi$
$71$	10392.4			0.244664			0.122332	+	0.992489i	$0.460963\pi$
$72$		0			0
$73$			38979.1i			0.856101i	0.903755	+	0.428051i	$0.140800\pi$
$73$			38979.1i			0.856101i	−0.903755	+	0.428051i	$0.859200\pi$
$74$		0			0
$75$	14614.1	+	5692.52i	0.299999	+	0.116856i
$76$		0			0
$77$		−	8867.24i		−	0.170436i
$78$		0			0
$79$	−23211.9			−0.418449			−0.209225	−	0.977868i	$-0.567094\pi$
$79$	−23211.9			−0.418449			−0.209225	+	0.977868i	$0.567094\pi$
$80$		0			0
$81$	41321.3			0.699780
$82$		0			0
$83$		−	34715.0i		−	0.553123i	−0.960996	−	0.276561i	$-0.910805\pi$
$83$		−	34715.0i		−	0.553123i	0.960996	−	0.276561i	$-0.0891949\pi$
$84$		0			0
$85$	−63512.5	−	92900.2i	−0.953480	−	1.39466i
$86$		0			0
$87$		−	6147.46i		−	0.0870758i
$88$		0			0
$89$	97820.7			1.30905			0.654524	−	0.756041i	$-0.272869\pi$
$89$	97820.7			1.30905			0.654524	+	0.756041i	$0.272869\pi$
$90$		0			0
$91$	−9343.66			−0.118281
$92$		0			0
$93$		−	39207.4i		−	0.470068i
$94$		0			0
$95$	111745.	−	76396.3i	1.27034	−	0.868487i
$96$		0			0
$97$			151178.i			1.63139i	0.578480	+	0.815696i	$0.303646\pi$
$97$			151178.i			1.63139i	−0.578480	+	0.815696i	$0.696354\pi$
$98$		0			0
$99$	125500.			1.28694
$100$		0			0
$101$	−145004.			−1.41441			−0.707207	−	0.707006i	$-0.750045\pi$
$101$	−145004.			−1.41441			−0.707207	+	0.707006i	$0.750045\pi$
$102$		0			0
$103$		−	66042.5i		−	0.613382i	−0.951809	−	0.306691i	$-0.900778\pi$
$103$		−	66042.5i		−	0.613382i	0.951809	−	0.306691i	$-0.0992218\pi$
$104$		0			0
$105$	−3564.30	+	2436.78i	−0.0315501	+	0.0215697i
$106$		0			0
$107$		−	52323.5i		−	0.441812i	−0.975295	−	0.220906i	$-0.929099\pi$
$107$		−	52323.5i		−	0.441812i	0.975295	−	0.220906i	$-0.0709014\pi$
$108$		0			0
$109$	−39690.3			−0.319976			−0.159988	−	0.987119i	$-0.551146\pi$
$109$	−39690.3			−0.319976			−0.159988	+	0.987119i	$0.551146\pi$
$110$		0			0
$111$	19101.9			0.147153
$112$		0			0
$113$		−	88601.8i		−	0.652749i	−0.945240	−	0.326375i	$-0.894173\pi$
$113$		−	88601.8i		−	0.652749i	0.945240	−	0.326375i	$-0.105827\pi$
$114$		0			0
$115$	−137452.	−	201053.i	−0.969187	−	1.41764i
$116$		0			0
$117$		−	132243.i		−	0.893118i
$118$		0			0
$119$	30980.6			0.200550
$120$		0			0
$121$	170940.			1.06140
$122$		0			0
$123$		−	22466.8i		−	0.133899i
$124$		0			0
$125$	39526.0	+	170162.i	0.226260	+	0.974067i
$126$		0			0
$127$			73729.9i			0.405634i	0.979217	+	0.202817i	$0.0650096\pi$
$127$			73729.9i			0.405634i	−0.979217	+	0.202817i	$0.934990\pi$
$128$		0			0
$129$	−73844.3			−0.390699
$130$		0			0
$131$	220952.			1.12492			0.562459	−	0.826825i	$-0.309856\pi$
$131$	220952.			1.12492			0.562459	+	0.826825i	$0.309856\pi$
$132$		0			0
$133$			37265.2i			0.182673i
$134$		0			0
$135$	−72965.1	−	106727.i	−0.344572	−	0.504009i
$136$		0			0
$137$		−	98725.5i		−	0.449395i	−0.974429	−	0.224697i	$-0.927861\pi$
$137$		−	98725.5i		−	0.449395i	0.974429	−	0.224697i	$-0.0721393\pi$
$138$		0			0
$139$	100712.			0.442126			0.221063	−	0.975260i	$-0.429047\pi$
$139$	100712.			0.442126			0.221063	+	0.975260i	$0.429047\pi$
$140$		0			0
$141$	−102852.			−0.435679
$142$		0			0
$143$		−	349828.i		−	1.43059i
$144$		0			0
$145$	56526.1	−	38644.9i	0.223270	−	0.152641i
$146$		0			0
$147$			83161.9i			0.317417i
$148$		0			0
$149$	108845.			0.401647			0.200823	−	0.979627i	$-0.435638\pi$
$149$	108845.			0.401647			0.200823	+	0.979627i	$0.435638\pi$
$150$		0			0
$151$	180847.			0.645459			0.322730	−	0.946491i	$-0.395400\pi$
$151$	180847.			0.645459			0.322730	+	0.946491i	$0.395400\pi$
$152$		0			0
$153$			438477.i			1.51432i
$154$		0			0
$155$	360514.	−	246470.i	1.20529	−	0.824015i
$156$		0			0
$157$		−	68451.0i		−	0.221631i	−0.993841	−	0.110815i	$-0.964654\pi$
$157$		−	68451.0i		−	0.221631i	0.993841	−	0.110815i	$-0.0353463\pi$
$158$		0			0
$159$	54719.6			0.171652
$160$		0			0
$161$	67047.7			0.203854
$162$		0			0
$163$		−	150595.i		−	0.443958i	−0.975051	−	0.221979i	$-0.928748\pi$
$163$		−	150595.i		−	0.443958i	0.975051	−	0.221979i	$-0.0712516\pi$
$164$		0			0
$165$	−91233.6	−	133448.i	−0.260882	−	0.381595i
$166$		0			0
$167$			642469.i			1.78263i	0.453384	+	0.891315i	$0.350217\pi$
$167$			642469.i			1.78263i	−0.453384	+	0.891315i	$0.649783\pi$
$168$		0			0
$169$	2668.94			0.00718823
$170$		0			0
$171$	−527424.			−1.37934
$172$		0			0
$173$		−	266842.i		−	0.677857i	−0.940812	−	0.338929i	$-0.889935\pi$
$173$		−	266842.i		−	0.677857i	0.940812	−	0.338929i	$-0.110065\pi$
$174$		0			0
$175$	−44812.6	−	17455.5i	−0.110613	−	0.0430861i
$176$		0			0
$177$		−	156244.i		−	0.374860i
$178$		0			0
$179$	−526108.			−1.22728			−0.613638	−	0.789587i	$-0.710295\pi$
$179$	−526108.			−1.22728			−0.613638	+	0.789587i	$0.710295\pi$
$180$		0			0
$181$	−52351.7			−0.118778			−0.0593888	−	0.998235i	$-0.518915\pi$
$181$	−52351.7			−0.118778			−0.0593888	+	0.998235i	$0.518915\pi$
$182$		0			0
$183$		−	22515.5i		−	0.0496996i
$184$		0			0
$185$	120081.	+	175643.i	0.257955	+	0.377313i
$186$		0			0
$187$			1.15992e6i			2.42563i
$188$		0			0
$189$	35591.5			0.0724756
$190$		0			0
$191$	−528420.			−1.04808			−0.524041	−	0.851693i	$-0.675576\pi$
$191$	−528420.			−1.04808			−0.524041	+	0.851693i	$0.675576\pi$
$192$		0			0
$193$		−	472653.i		−	0.913375i	−0.889627	−	0.456688i	$-0.849036\pi$
$193$		−	472653.i		−	0.913375i	0.889627	−	0.456688i	$-0.150964\pi$
$194$		0			0
$195$	−140618.	+	96135.3i	−0.264822	+	0.181049i
$196$		0			0
$197$			853368.i			1.56665i	0.621615	+	0.783323i	$0.286477\pi$
$197$			853368.i			1.56665i	−0.621615	+	0.783323i	$0.713523\pi$
$198$		0			0
$199$	−187602.			−0.335818			−0.167909	−	0.985802i	$-0.553702\pi$
$199$	−187602.			−0.335818			−0.167909	+	0.985802i	$0.553702\pi$
$200$		0			0
$201$	−228076.			−0.398188
$202$		0			0
$203$			18850.5i			0.0321058i
$204$		0			0
$205$	206583.	−	141233.i	0.343328	−	0.234721i
$206$		0			0
$207$			948943.i			1.53927i
$208$		0			0
$209$	−1.39522e6			−2.20941
$210$		0			0
$211$	494910.			0.765280			0.382640	−	0.923898i	$-0.375015\pi$
$211$	494910.			0.765280			0.382640	+	0.923898i	$0.375015\pi$
$212$		0			0
$213$			52157.0i			0.0787705i
$214$		0			0
$215$	−464209.	−	679001.i	−0.684884	−	1.00179i
$216$		0			0
$217$			120225.i			0.173319i
$218$		0			0
$219$	−195627.			−0.275626
$220$		0			0
$221$	1.22224e6			1.68336
$222$		0			0
$223$			176639.i			0.237862i	0.992903	+	0.118931i	$0.0379467\pi$
$223$			176639.i			0.237862i	−0.992903	+	0.118931i	$0.962053\pi$
$224$		0			0
$225$	247052.	−	634245.i	0.325336	−	0.835219i
$226$		0			0
$227$			889730.i			1.14602i	0.819547	+	0.573012i	$0.194225\pi$
$227$			889730.i			1.14602i	−0.819547	+	0.573012i	$0.805775\pi$
$228$		0			0
$229$	837396.			1.05522			0.527609	−	0.849487i	$-0.323089\pi$
$229$	837396.			1.05522			0.527609	+	0.849487i	$0.323089\pi$
$230$		0			0
$231$	44502.7			0.0548727
$232$		0			0
$233$		−	688718.i		−	0.831097i	−0.909571	−	0.415548i	$-0.863590\pi$
$233$		−	688718.i		−	0.831097i	0.909571	−	0.415548i	$-0.136410\pi$
$234$		0			0
$235$	−646563.	−	945732.i	−0.763732	−	1.11712i
$236$		0			0
$237$		−	116495.i		−	0.134722i
$238$		0			0
$239$	785117.			0.889078			0.444539	−	0.895760i	$-0.353368\pi$
$239$	785117.			0.889078			0.444539	+	0.895760i	$0.353368\pi$
$240$		0			0
$241$	−966622.			−1.07205			−0.536024	−	0.844203i	$-0.680074\pi$
$241$	−966622.			−1.07205			−0.536024	+	0.844203i	$0.680074\pi$
$242$		0			0
$243$			769371.i			0.835834i
$244$		0			0
$245$	−764677.	+	522782.i	−0.813885	+	0.556423i
$246$		0			0
$247$			1.47018e6i			1.53330i
$248$		0			0
$249$	174227.			0.178080
$250$		0			0
$251$	−170904.			−0.171226			−0.0856128	−	0.996328i	$-0.527285\pi$
$251$	−170904.			−0.171226			−0.0856128	+	0.996328i	$0.527285\pi$
$252$		0			0
$253$			2.51028e6i			2.46559i
$254$		0			0
$255$	466245.	−	318755.i	0.449018	−	0.306977i
$256$		0			0
$257$		−	972840.i		−	0.918774i	−0.888236	−	0.459387i	$-0.848069\pi$
$257$		−	972840.i		−	0.918774i	0.888236	−	0.459387i	$-0.151931\pi$
$258$		0			0
$259$	−58574.0			−0.0542570
$260$		0			0
$261$	−266796.			−0.242426
$262$		0			0
$263$		−	1.12993e6i		−	1.00730i	−0.863906	−	0.503652i	$-0.831989\pi$
$263$		−	1.12993e6i		−	1.00730i	0.863906	−	0.503652i	$-0.168011\pi$
$264$		0			0
$265$	343985.	+	503149.i	0.300901	+	0.440131i
$266$		0			0
$267$			490940.i			0.421454i
$268$		0			0
$269$	1.52961e6			1.28884			0.644420	−	0.764672i	$-0.277099\pi$
$269$	1.52961e6			1.28884			0.644420	+	0.764672i	$0.277099\pi$
$270$		0			0
$271$	−627262.			−0.518831			−0.259415	−	0.965766i	$-0.583530\pi$
$271$	−627262.			−0.518831			−0.259415	+	0.965766i	$0.583530\pi$
$272$		0			0
$273$		−	46893.7i		−	0.0380810i
$274$		0			0
$275$	653537.	−	1.67779e6i	0.521121	−	1.33785i
$276$		0			0
$277$		−	1.78572e6i		−	1.39834i	−0.714954	−	0.699172i	$-0.753552\pi$
$277$		−	1.78572e6i		−	1.39834i	0.714954	−	0.699172i	$-0.246448\pi$
$278$		0			0
$279$	−1.70158e6			−1.30870
$280$		0			0
$281$	639884.			0.483432			0.241716	−	0.970347i	$-0.422290\pi$
$281$	639884.			0.483432			0.241716	+	0.970347i	$0.422290\pi$
$282$		0			0
$283$		−	1.21056e6i		−	0.898505i	−0.893405	−	0.449253i	$-0.851690\pi$
$283$		−	1.21056e6i		−	0.898505i	0.893405	−	0.449253i	$-0.148310\pi$
$284$		0			0
$285$	383416.	+	560825.i	0.279613	+	0.408992i
$286$		0			0
$287$			68891.9i			0.0493700i
$288$		0			0
$289$	−2.63271e6			−1.85421
$290$		0			0
$291$	−758726.			−0.525234
$292$		0			0
$293$		−	2.15389e6i		−	1.46573i	−0.680373	−	0.732866i	$-0.738182\pi$
$293$		−	2.15389e6i		−	1.46573i	0.680373	−	0.732866i	$-0.261818\pi$
$294$		0			0
$295$	1.43667e6	−	982198.i	0.961173	−	0.657119i
$296$		0			0
$297$			1.33255e6i			0.876584i
$298$		0			0
$299$	2.64515e6			1.71109
$300$		0			0
$301$	226436.			0.144055
$302$		0			0
$303$		−	727742.i		−	0.455377i
$304$		0			0
$305$	207031.	−	141539.i	0.127434	−	0.0871220i
$306$		0			0
$307$			1.74569e6i			1.05711i	0.848898	+	0.528557i	$0.177267\pi$
$307$			1.74569e6i			1.05711i	−0.848898	+	0.528557i	$0.822733\pi$
$308$		0			0
$309$	331452.			0.197481
$310$		0			0
$311$	3.01625e6			1.76834			0.884172	−	0.467161i	$-0.154723\pi$
$311$	3.01625e6			1.76834			0.884172	+	0.467161i	$0.154723\pi$
$312$		0			0
$313$		−	245464.i		−	0.141621i	−0.997490	−	0.0708103i	$-0.977442\pi$
$313$		−	245464.i		−	0.141621i	0.997490	−	0.0708103i	$-0.0225585\pi$
$314$		0			0
$315$	105755.	+	154689.i	0.0600516	+	0.0878379i
$316$		0			0
$317$			2.72385e6i			1.52242i	0.648506	+	0.761210i	$0.275394\pi$
$317$			2.72385e6i			1.52242i	−0.648506	+	0.761210i	$0.724606\pi$
$318$		0			0
$319$	−705767.			−0.388315
$320$		0			0
$321$	262600.			0.142243
$322$		0			0
$323$		−	4.87465e6i		−	2.59978i
$324$		0			0
$325$	−1.76794e6	−	688651.i	−0.928450	−	0.361652i
$326$		0			0
$327$		−	199196.i		−	0.103018i
$328$		0			0
$329$	315386.			0.160640
$330$		0			0
$331$	1.61205e6			0.808737			0.404368	−	0.914596i	$-0.367491\pi$
$331$	1.61205e6			0.808737			0.404368	+	0.914596i	$0.367491\pi$
$332$		0			0
$333$		−	829013.i		−	0.409685i
$334$		0			0
$335$	−1.43376e6	−	2.09716e6i	−0.698012	−	1.02099i
$336$		0			0
$337$		−	1.48681e6i		−	0.713151i	−0.934267	−	0.356575i	$-0.883944\pi$
$337$		−	1.48681e6i		−	0.713151i	0.934267	−	0.356575i	$-0.116056\pi$
$338$		0			0
$339$	444672.			0.210155
$340$		0			0
$341$	−4.50126e6			−2.09627
$342$		0			0
$343$		−	513659.i		−	0.235743i
$344$		0			0
$345$	1.00904e6	−	689842.i	0.456415	−	0.312034i
$346$		0			0
$347$		−	1.46436e6i		−	0.652866i	−0.945220	−	0.326433i	$-0.894153\pi$
$347$		−	1.46436e6i		−	0.652866i	0.945220	−	0.326433i	$-0.105847\pi$
$348$		0			0
$349$	4.33263e6			1.90409			0.952045	−	0.305957i	$-0.0989763\pi$
$349$	4.33263e6			1.90409			0.952045	+	0.305957i	$0.0989763\pi$
$350$		0			0
$351$	1.40415e6			0.608338
$352$		0			0
$353$			2.70452e6i			1.15519i	0.816323	+	0.577595i	$0.196009\pi$
$353$			2.70452e6i			1.15519i	−0.816323	+	0.577595i	$0.803991\pi$
$354$		0			0
$355$	−479586.	+	327876.i	−0.201974	+	0.138082i
$356$		0			0
$357$			155485.i			0.0645680i
$358$		0			0
$359$	4.24919e6			1.74008			0.870041	−	0.492979i	$-0.164092\pi$
$359$	4.24919e6			1.74008			0.870041	+	0.492979i	$0.164092\pi$
$360$		0			0
$361$	3.38740e6			1.36804
$362$		0			0
$363$			857910.i			0.341724i
$364$		0			0
$365$	−1.22978e6	−	1.79880e6i	−0.483163	−	0.706727i
$366$		0			0
$367$			2.20652e6i			0.855153i	0.903979	+	0.427576i	$0.140632\pi$
$367$			2.20652e6i			0.855153i	−0.903979	+	0.427576i	$0.859368\pi$
$368$		0			0
$369$	−975045.			−0.372785
$370$		0			0
$371$	−167792.			−0.0632901
$372$		0			0
$373$		−	744943.i		−	0.277237i	−0.990346	−	0.138618i	$-0.955734\pi$
$373$		−	744943.i		−	0.277237i	0.990346	−	0.138618i	$-0.0442661\pi$
$374$		0			0
$375$	−854007.	+	198372.i	−0.313605	+	0.0728454i
$376$		0			0
$377$			743687.i			0.269486i
$378$		0			0
$379$	−4.37944e6			−1.56610			−0.783052	−	0.621956i	$-0.786338\pi$
$379$	−4.37944e6			−1.56610			−0.783052	+	0.621956i	$0.786338\pi$
$380$		0			0
$381$	−370033.			−0.130596
$382$		0			0
$383$		−	756975.i		−	0.263684i	−0.991271	−	0.131842i	$-0.957911\pi$
$383$		−	756975.i		−	0.263684i	0.991271	−	0.131842i	$-0.0420892\pi$
$384$		0			0
$385$	279758.	+	409204.i	0.0961902	+	0.140698i
$386$		0			0
$387$			3.20480e6i			1.08774i
$388$		0			0
$389$	−585747.			−0.196262			−0.0981310	−	0.995174i	$-0.531286\pi$
$389$	−585747.			−0.196262			−0.0981310	+	0.995174i	$0.531286\pi$
$390$		0			0
$391$	−8.77049e6			−2.90123
$392$		0			0
$393$			1.10891e6i			0.362172i
$394$		0			0
$395$	1.07118e6	−	732326.i	0.345437	−	0.236163i
$396$		0			0
$397$			1.09423e6i			0.348443i	0.984707	+	0.174221i	$0.0557408\pi$
$397$			1.09423e6i			0.348443i	−0.984707	+	0.174221i	$0.944259\pi$
$398$		0			0
$399$	−187026.			−0.0588124
$400$		0			0
$401$	1.89977e6			0.589985			0.294993	−	0.955500i	$-0.404683\pi$
$401$	1.89977e6			0.589985			0.294993	+	0.955500i	$0.404683\pi$
$402$		0			0
$403$			4.74310e6i			1.45479i
$404$		0			0
$405$	−1.90689e6	+	1.30367e6i	−0.577681	+	0.394940i
$406$		0			0
$407$		−	2.19302e6i		−	0.656231i
$408$		0			0
$409$	−1.93174e6			−0.571004			−0.285502	−	0.958378i	$-0.592160\pi$
$409$	−1.93174e6			−0.571004			−0.285502	+	0.958378i	$0.592160\pi$
$410$		0			0
$411$	495481.			0.144685
$412$		0			0
$413$			479105.i			0.138215i
$414$		0			0
$415$	1.09524e6	+	1.60202e6i	0.312170	+	0.456613i
$416$		0			0
$417$			505453.i			0.142344i
$418$		0			0
$419$	282578.			0.0786327			0.0393163	−	0.999227i	$-0.487482\pi$
$419$	282578.			0.0786327			0.0393163	+	0.999227i	$0.487482\pi$
$420$		0			0
$421$	−2.75800e6			−0.758384			−0.379192	−	0.925318i	$-0.623798\pi$
$421$	−2.75800e6			−0.758384			−0.379192	+	0.925318i	$0.623798\pi$
$422$		0			0
$423$			4.46374e6i			1.21296i
$424$		0			0
$425$	5.86193e6	+	2.28335e6i	1.57423	+	0.613197i
$426$		0			0
$427$			69041.3i			0.0183248i
$428$		0			0
$429$	1.75571e6			0.460585
$430$		0			0
$431$	4.19567e6			1.08795			0.543974	−	0.839102i	$-0.316919\pi$
$431$	4.19567e6			1.08795			0.543974	+	0.839102i	$0.316919\pi$
$432$		0			0
$433$		−	5.64763e6i		−	1.44759i	−0.690014	−	0.723796i	$-0.742396\pi$
$433$		−	5.64763e6i		−	1.44759i	0.690014	−	0.723796i	$-0.257604\pi$
$434$		0			0
$435$	193950.	+	283692.i	0.0491435	+	0.0718826i
$436$		0			0
$437$		−	1.05496e7i		−	2.64261i
$438$		0			0
$439$	−196052.			−0.0485522			−0.0242761	−	0.999705i	$-0.507728\pi$
$439$	−196052.			−0.0485522			−0.0242761	+	0.999705i	$0.507728\pi$
$440$		0			0
$441$	3.60918e6			0.883714
$442$		0			0
$443$			1.04783e6i			0.253678i	0.991923	+	0.126839i	$0.0404832\pi$
$443$			1.04783e6i			0.253678i	−0.991923	+	0.126839i	$0.959517\pi$
$444$		0			0
$445$	−4.51421e6	+	3.08620e6i	−1.08064	+	0.738796i
$446$		0			0
$447$			546270.i			0.129312i
$448$		0			0
$449$	−2.90131e6			−0.679170			−0.339585	−	0.940575i	$-0.610287\pi$
$449$	−2.90131e6			−0.679170			−0.339585	+	0.940575i	$0.610287\pi$
$450$		0			0
$451$	−2.57933e6			−0.597124
$452$		0			0
$453$			907629.i			0.207808i
$454$		0			0
$455$	431190.	−	294789.i	0.0976428	−	0.0667548i
$456$		0			0
$457$			5.56310e6i			1.24602i	0.782212	+	0.623012i	$0.214091\pi$
$457$			5.56310e6i			1.24602i	−0.782212	+	0.623012i	$0.785909\pi$
$458$		0			0
$459$	−4.65571e6			−1.03147
$460$		0			0
$461$	−1.79359e6			−0.393070			−0.196535	−	0.980497i	$-0.562969\pi$
$461$	−1.79359e6			−0.393070			−0.196535	+	0.980497i	$0.562969\pi$
$462$		0			0
$463$			5.33589e6i			1.15679i	0.815757	+	0.578395i	$0.196321\pi$
$463$			5.33589e6i			1.15679i	−0.815757	+	0.578395i	$0.803679\pi$
$464$		0			0
$465$	1.23698e6	+	1.80934e6i	0.265295	+	0.388049i
$466$		0			0
$467$		−	3.58740e6i		−	0.761180i	−0.924744	−	0.380590i	$-0.875721\pi$
$467$		−	3.58740e6i		−	0.761180i	0.924744	−	0.380590i	$-0.124279\pi$
$468$		0			0
$469$	699369.			0.146816
$470$		0			0
$471$	343540.			0.0713550
$472$		0			0
$473$			8.47779e6i			1.74233i
$474$		0			0
$475$	−2.74654e6	+	7.05105e6i	−0.558537	+	1.43390i
$476$		0			0
$477$		−	2.37480e6i		−	0.477893i
$478$		0			0
$479$	−8.13513e6			−1.62004			−0.810020	−	0.586402i	$-0.800544\pi$
$479$	−8.13513e6			−1.62004			−0.810020	+	0.586402i	$0.800544\pi$
$480$		0			0
$481$	−2.31085e6			−0.455416
$482$		0			0
$483$			336497.i			0.0656317i
$484$		0			0
$485$	−4.76960e6	−	6.97652e6i	−0.920719	−	1.34674i
$486$		0			0
$487$			3.09262e6i			0.590886i	0.955360	+	0.295443i	$0.0954673\pi$
$487$			3.09262e6i			0.590886i	−0.955360	+	0.295443i	$0.904533\pi$
$488$		0			0
$489$	755802.			0.142934
$490$		0			0
$491$	1.43937e6			0.269443			0.134722	−	0.990883i	$-0.456986\pi$
$491$	1.43937e6			0.269443			0.134722	+	0.990883i	$0.456986\pi$
$492$		0			0
$493$		−	2.46583e6i		−	0.456926i
$494$		0			0
$495$	−5.79157e6	+	3.95949e6i	−1.06239	+	0.726316i
$496$		0			0
$497$		−	159934.i		−	0.0290436i
$498$		0			0
$499$	4.10009e6			0.737126			0.368563	−	0.929603i	$-0.379850\pi$
$499$	4.10009e6			0.737126			0.368563	+	0.929603i	$0.379850\pi$
$500$		0			0
$501$	−3.22441e6			−0.573926
$502$		0			0
$503$		−	602734.i		−	0.106220i	−0.998589	−	0.0531099i	$-0.983087\pi$
$503$		−	602734.i		−	0.106220i	0.998589	−	0.0531099i	$-0.0169134\pi$
$504$		0			0
$505$	6.69162e6	−	4.57482e6i	1.16762	−	0.798262i
$506$		0			0
$507$			13394.8i			0.00231428i
$508$		0			0
$509$	6.05028e6			1.03510			0.517548	−	0.855654i	$-0.326845\pi$
$509$	6.05028e6			1.03510			0.517548	+	0.855654i	$0.326845\pi$
$510$		0			0
$511$	599871.			0.101626
$512$		0			0
$513$		−	5.60015e6i		−	0.939521i
$514$		0			0
$515$	2.08362e6	+	3.04772e6i	0.346178	+	0.506357i
$516$		0			0
$517$			1.18081e7i			1.94292i
$518$		0			0
$519$	1.33922e6			0.218239
$520$		0			0
$521$	5.55881e6			0.897196			0.448598	−	0.893734i	$-0.351923\pi$
$521$	5.55881e6			0.897196			0.448598	+	0.893734i	$0.351923\pi$
$522$		0			0
$523$		−	2.83998e6i		−	0.454006i	−0.973894	−	0.227003i	$-0.927107\pi$
$523$		−	2.83998e6i		−	0.454006i	0.973894	−	0.227003i	$-0.0728927\pi$
$524$		0			0
$525$	87605.2	−	224904.i	0.0138718	−	0.0356123i
$526$		0			0
$527$		−	1.57266e7i		−	2.46666i
$528$		0			0
$529$	−1.25446e7			−1.94902
$530$		0			0
$531$	−6.78090e6			−1.04364
$532$		0			0
$533$			2.71791e6i			0.414397i
$534$		0			0
$535$	1.65079e6	+	2.41462e6i	0.249348	+	0.364723i
$536$		0			0
$537$		−	2.64042e6i		−	0.395127i
$538$		0			0
$539$	9.54751e6			1.41553
$540$		0			0
$541$	9.13760e6			1.34227			0.671134	−	0.741336i	$-0.265808\pi$
$541$	9.13760e6			1.34227			0.671134	+	0.741336i	$0.265808\pi$
$542$		0			0
$543$		−	262741.i		−	0.0382410i
$544$		0			0
$545$	1.83162e6	−	1.25221e6i	0.264146	−	0.180587i
$546$		0			0
$547$		−	1.05091e7i		−	1.50174i	−0.660448	−	0.750872i	$-0.729633\pi$
$547$		−	1.05091e7i		−	1.50174i	0.660448	−	0.750872i	$-0.270367\pi$
$548$		0			0
$549$	−977159.			−0.138368
$550$		0			0
$551$	2.96604e6			0.416196
$552$		0			0
$553$			357220.i			0.0496733i
$554$		0			0
$555$	−881512.	+	602658.i	−0.121477	+	0.0830497i
$556$		0			0
$557$			7.65599e6i			1.04560i	0.852457	+	0.522798i	$0.175112\pi$
$557$			7.65599e6i			1.04560i	−0.852457	+	0.522798i	$0.824888\pi$
$558$		0			0
$559$	8.93328e6			1.20915
$560$		0			0
$561$	−5.82138e6			−0.780942
$562$		0			0
$563$		−	3.79662e6i		−	0.504807i	−0.967622	−	0.252404i	$-0.918779\pi$
$563$		−	3.79662e6i		−	0.504807i	0.967622	−	0.252404i	$-0.0812211\pi$
$564$		0			0
$565$	2.79535e6	+	4.08878e6i	0.368396	+	0.538856i
$566$		0			0
$567$		−	635916.i		−	0.0830696i
$568$		0			0
$569$	−1.02221e7			−1.32360			−0.661802	−	0.749679i	$-0.730208\pi$
$569$	−1.02221e7			−1.32360			−0.661802	+	0.749679i	$0.730208\pi$
$570$		0			0
$571$	3.67046e6			0.471119			0.235559	−	0.971860i	$-0.424308\pi$
$571$	3.67046e6			0.471119			0.235559	+	0.971860i	$0.424308\pi$
$572$		0			0
$573$		−	2.65202e6i		−	0.337435i
$574$		0			0
$575$	1.26863e7	+	4.94158e6i	1.60016	+	0.623298i
$576$		0			0
$577$		−	8.98729e6i		−	1.12380i	−0.827205	−	0.561900i	$-0.810071\pi$
$577$		−	8.98729e6i		−	1.12380i	0.827205	−	0.561900i	$-0.189929\pi$
$578$		0			0
$579$	2.37214e6			0.294065
$580$		0			0
$581$	−534247.			−0.0656602
$582$		0			0
$583$		−	6.28215e6i		−	0.765486i
$584$		0			0
$585$	4.17222e6	+	6.10274e6i	0.504055	+	0.737285i
$586$		0			0
$587$		−	1.46226e7i		−	1.75157i	−0.482698	−	0.875787i	$-0.660343\pi$
$587$		−	1.46226e7i		−	1.75157i	0.482698	−	0.875787i	$-0.339657\pi$
$588$		0			0
$589$	1.89169e7			2.24678
$590$		0			0
$591$	−4.28286e6			−0.504388
$592$		0			0
$593$		−	2.63114e6i		−	0.307261i	−0.988128	−	0.153630i	$-0.950903\pi$
$593$		−	2.63114e6i		−	0.307261i	0.988128	−	0.153630i	$-0.0490965\pi$
$594$		0			0
$595$	−1.42969e6	+	977427.i	−0.165558	+	0.113186i
$596$		0			0
$597$		−	941531.i		−	0.108118i
$598$		0			0
$599$	−7.14762e6			−0.813944			−0.406972	−	0.913441i	$-0.633415\pi$
$599$	−7.14762e6			−0.813944			−0.406972	+	0.913441i	$0.633415\pi$
$600$		0			0
$601$	−3.98447e6			−0.449971			−0.224986	−	0.974362i	$-0.572234\pi$
$601$	−3.98447e6			−0.449971			−0.224986	+	0.974362i	$0.572234\pi$
$602$		0			0
$603$			9.89835e6i			1.10859i
$604$		0			0
$605$	−7.88852e6	+	5.39310e6i	−0.876208	+	0.599032i
$606$		0			0
$607$		−	1.57671e7i		−	1.73692i	−0.495761	−	0.868459i	$-0.665111\pi$
$607$		−	1.57671e7i		−	1.73692i	0.495761	−	0.868459i	$-0.334889\pi$
$608$		0			0
$609$	−94606.5			−0.0103366
$610$		0			0
$611$	1.24425e7			1.34836
$612$		0			0
$613$		−	2.48018e6i		−	0.266583i	−0.991077	−	0.133291i	$-0.957445\pi$
$613$		−	2.48018e6i		−	0.266583i	0.991077	−	0.133291i	$-0.0425546\pi$
$614$		0			0
$615$	708817.	+	1.03679e6i	0.0755695	+	0.110536i
$616$		0			0
$617$		−	5.49933e6i		−	0.581563i	−0.956790	−	0.290781i	$-0.906085\pi$
$617$		−	5.49933e6i		−	0.581563i	0.956790	−	0.290781i	$-0.0939152\pi$
$618$		0			0
$619$	−4.76882e6			−0.500247			−0.250124	−	0.968214i	$-0.580471\pi$
$619$	−4.76882e6			−0.500247			−0.250124	+	0.968214i	$0.580471\pi$
$620$		0			0
$621$	−1.00758e7			−1.04846
$622$		0			0
$623$		−	1.50541e6i		−	0.155395i
$624$		0			0
$625$	−7.19260e6	−	6.60560e6i	−0.736522	−	0.676413i
$626$		0			0
$627$		−	7.00228e6i		−	0.711329i
$628$		0			0
$629$	7.66204e6			0.772179
$630$		0			0
$631$	8.81040e6			0.880891			0.440445	−	0.897779i	$-0.354821\pi$
$631$	8.81040e6			0.880891			0.440445	+	0.897779i	$0.354821\pi$
$632$		0			0
$633$			2.48384e6i			0.246385i
$634$		0			0
$635$	−2.32615e6	−	3.40247e6i	−0.228930	−	0.334858i
$636$		0			0
$637$		−	1.00605e7i		−	0.982359i
$638$		0			0
$639$	2.26359e6			0.219303
$640$		0			0
$641$	1.36273e7			1.30998			0.654991	−	0.755637i	$-0.272673\pi$
$641$	1.36273e7			1.30998			0.654991	+	0.755637i	$0.272673\pi$
$642$		0			0
$643$			8.82742e6i			0.841989i	0.907063	+	0.420994i	$0.138319\pi$
$643$			8.82742e6i			0.841989i	−0.907063	+	0.420994i	$0.861681\pi$
$644$		0			0
$645$	3.40775e6	−	2.32976e6i	0.322529	−	0.220501i
$646$		0			0
$647$		−	1.37606e7i		−	1.29234i	−0.763194	−	0.646170i	$-0.776370\pi$
$647$		−	1.37606e7i		−	1.29234i	0.763194	−	0.646170i	$-0.223630\pi$
$648$		0			0
$649$	−1.79378e7			−1.67169
$650$		0			0
$651$	−603383.			−0.0558009
$652$		0			0
$653$			1.40149e7i			1.28619i	0.765785	+	0.643096i	$0.222350\pi$
$653$			1.40149e7i			1.28619i	−0.765785	+	0.643096i	$0.777650\pi$
$654$		0			0
$655$	−1.01965e7	+	6.97096e6i	−0.928639	+	0.634877i
$656$		0			0
$657$			8.49012e6i			0.767362i
$658$		0			0
$659$	−1.14800e7			−1.02974			−0.514871	−	0.857268i	$-0.672160\pi$
$659$	−1.14800e7			−1.02974			−0.514871	+	0.857268i	$0.672160\pi$
$660$		0			0
$661$	2.21158e6			0.196879			0.0984395	−	0.995143i	$-0.468615\pi$
$661$	2.21158e6			0.196879			0.0984395	+	0.995143i	$0.468615\pi$
$662$		0			0
$663$			6.13415e6i			0.541964i
$664$		0			0
$665$	−1.17570e6	−	1.71971e6i	−0.103096	−	0.150800i
$666$		0			0
$667$		−	5.33650e6i		−	0.464453i
$668$		0			0
$669$	−886511.			−0.0765807
$670$		0			0
$671$	−2.58492e6			−0.221636
$672$		0			0
$673$			1.10100e7i			0.937018i	0.883459	+	0.468509i	$0.155209\pi$
$673$			1.10100e7i			0.937018i	−0.883459	+	0.468509i	$0.844791\pi$
$674$		0			0
$675$	6.73436e6	+	2.62318e6i	0.568901	+	0.221600i
$676$		0			0
$677$		−	4.18326e6i		−	0.350787i	−0.984498	−	0.175394i	$-0.943880\pi$
$677$		−	4.18326e6i		−	0.350787i	0.984498	−	0.175394i	$-0.0561198\pi$
$678$		0			0
$679$	2.32655e6			0.193659
$680$		0			0
$681$	−4.46535e6			−0.368968
$682$		0			0
$683$			4.16261e6i			0.341440i	0.985320	+	0.170720i	$0.0546093\pi$
$683$			4.16261e6i			0.341440i	−0.985320	+	0.170720i	$0.945391\pi$
$684$		0			0
$685$	3.11475e6	+	4.55597e6i	0.253628	+	0.370983i
$686$		0			0
$687$			4.20270e6i			0.339732i
$688$		0			0
$689$	−6.61968e6			−0.531238
$690$		0			0
$691$	3.95374e6			0.315002			0.157501	−	0.987519i	$-0.449656\pi$
$691$	3.95374e6			0.315002			0.157501	+	0.987519i	$0.449656\pi$
$692$		0			0
$693$		−	1.93139e6i		−	0.152770i
$694$		0			0
$695$	−4.64766e6	+	3.17744e6i	−0.364983	+	0.249526i
$696$		0			0
$697$		−	9.01173e6i		−	0.702629i
$698$		0			0
$699$	3.45652e6			0.267575
$700$		0			0
$701$	5.82473e6			0.447693			0.223847	−	0.974624i	$-0.428139\pi$
$701$	5.82473e6			0.447693			0.223847	+	0.974624i	$0.428139\pi$
$702$		0			0
$703$			9.21633e6i			0.703347i
$704$		0			0
$705$	4.74642e6	−	3.24495e6i	0.359661	−	0.245887i
$706$		0			0
$707$			2.23154e6i			0.167902i
$708$		0			0
$709$	1.98273e7			1.48131			0.740657	−	0.671883i	$-0.234514\pi$
$709$	1.98273e7			1.48131			0.740657	+	0.671883i	$0.234514\pi$
$710$		0			0
$711$	−5.05583e6			−0.375075
$712$		0			0
$713$		−	3.40353e7i		−	2.50729i
$714$		0			0
$715$	1.10369e7	+	1.61438e7i	0.807391	+	1.18098i
$716$		0			0
$717$			3.94032e6i			0.286243i
$718$		0			0
$719$	2.33346e7			1.68336			0.841681	−	0.539975i	$-0.181566\pi$
$719$	2.33346e7			1.68336			0.841681	+	0.539975i	$0.181566\pi$
$720$		0			0
$721$	−1.01636e6			−0.0728133
$722$		0			0
$723$		−	4.85125e6i		−	0.345150i
$724$		0			0
$725$	−1.38933e6	+	3.56675e6i	−0.0981658	+	0.252016i
$726$		0			0
$727$		−	426490.i		−	0.0299277i	−0.999888	−	0.0149638i	$-0.995237\pi$
$727$		−	426490.i		−	0.0299277i	0.999888	−	0.0149638i	$-0.00476332\pi$
$728$		0			0
$729$	6.17979e6			0.430680
$730$		0			0
$731$	−2.96200e7			−2.05018
$732$		0			0
$733$			1.12572e7i			0.773876i	0.922106	+	0.386938i	$0.126467\pi$
$733$			1.12572e7i			0.773876i	−0.922106	+	0.386938i	$0.873533\pi$
$734$		0			0
$735$	−2.62372e6	−	3.83774e6i	−0.179143	−	0.262034i
$736$		0			0
$737$			2.61845e7i			1.77573i
$738$		0			0
$739$	−1.03427e7			−0.696662			−0.348331	−	0.937372i	$-0.613251\pi$
$739$	−1.03427e7			−0.696662			−0.348331	+	0.937372i	$0.613251\pi$
$740$		0			0
$741$	−7.37850e6			−0.493654
$742$		0			0
$743$			7.76663e6i			0.516132i	0.966127	+	0.258066i	$0.0830852\pi$
$743$			7.76663e6i			0.516132i	−0.966127	+	0.258066i	$0.916915\pi$
$744$		0			0
$745$	−5.02298e6	+	3.43403e6i	−0.331566	+	0.226680i
$746$		0			0
$747$		−	7.56134e6i		−	0.495789i
$748$		0			0
$749$	−805234.			−0.0524466
$750$		0			0
$751$	−1.21299e7			−0.784796			−0.392398	−	0.919795i	$-0.628354\pi$
$751$	−1.21299e7			−0.784796			−0.392398	+	0.919795i	$0.628354\pi$
$752$		0			0
$753$		−	857729.i		−	0.0551268i
$754$		0			0
$755$	−8.34569e6	+	5.70565e6i	−0.532838	+	0.364282i
$756$		0			0
$757$			2.43566e7i			1.54482i	0.635125	+	0.772409i	$0.280948\pi$
$757$			2.43566e7i			1.54482i	−0.635125	+	0.772409i	$0.719052\pi$
$758$		0			0
$759$	−1.25985e7			−0.793807
$760$		0			0
$761$	−1.14858e7			−0.718951			−0.359475	−	0.933155i	$-0.617044\pi$
$761$	−1.14858e7			−0.718951			−0.359475	+	0.933155i	$0.617044\pi$
$762$		0			0
$763$			610815.i			0.0379838i
$764$		0			0
$765$	−1.38338e7	−	2.02348e7i	−0.854648	−	1.25010i
$766$		0			0
$767$			1.89015e7i			1.16014i
$768$		0			0
$769$	3.97185e6			0.242202			0.121101	−	0.992640i	$-0.461358\pi$
$769$	3.97185e6			0.242202			0.121101	+	0.992640i	$0.461358\pi$
$770$		0			0
$771$	4.88246e6			0.295803
$772$		0			0
$773$		−	1.28970e7i		−	0.776320i	−0.921592	−	0.388160i	$-0.873111\pi$
$773$		−	1.28970e7i		−	0.776320i	0.921592	−	0.388160i	$-0.126889\pi$
$774$		0			0
$775$	−8.86090e6	+	2.27481e7i	−0.529936	+	1.36048i
$776$		0			0
$777$		−	293969.i		−	0.0174683i
$778$		0			0
$779$	1.08398e7			0.639997
$780$		0			0
$781$	5.98796e6			0.351278
$782$		0			0
$783$		−	2.83282e6i		−	0.165126i
$784$		0			0
$785$	2.15960e6	+	3.15886e6i	0.125083	+	0.182960i
$786$		0			0
$787$			2.72579e7i			1.56876i	0.620283	+	0.784378i	$0.287018\pi$
$787$			2.72579e7i			1.56876i	−0.620283	+	0.784378i	$0.712982\pi$
$788$		0			0
$789$	5.67084e6			0.324306
$790$		0			0
$791$	−1.36354e6			−0.0774866
$792$		0			0
$793$			2.72380e6i			0.153813i
$794$		0			0
$795$	−2.52519e6	+	1.72638e6i	−0.141702	+	0.0968765i
$796$		0			0
$797$		−	6.68540e6i		−	0.372805i	−0.982473	−	0.186402i	$-0.940317\pi$
$797$		−	6.68540e6i		−	0.372805i	0.982473	−	0.186402i	$-0.0596828\pi$
$798$		0			0
$799$	−4.12555e7			−2.28620
$800$		0			0
$801$	2.13065e7			1.17336
$802$		0			0
$803$			2.24593e7i			1.22916i
$804$		0			0
$805$	−3.09411e6	+	2.11533e6i	−0.168285	+	0.115050i
$806$		0			0
$807$			7.67674e6i			0.414947i
$808$		0			0
$809$	1.68522e7			0.905283			0.452642	−	0.891693i	$-0.350482\pi$
$809$	1.68522e7			0.905283			0.452642	+	0.891693i	$0.350482\pi$
$810$		0			0
$811$	−697618.			−0.0372448			−0.0186224	−	0.999827i	$-0.505928\pi$
$811$	−697618.			−0.0372448			−0.0186224	+	0.999827i	$0.505928\pi$
$812$		0			0
$813$		−	3.14809e6i		−	0.167040i
$814$		0			0
$815$	4.75121e6	+	6.94963e6i	0.250559	+	0.366495i
$816$		0			0
$817$		−	3.56285e7i		−	1.86742i
$818$		0			0
$819$	−2.03516e6			−0.106020
$820$		0			0
$821$	−1.56284e7			−0.809203			−0.404602	−	0.914493i	$-0.632590\pi$
$821$	−1.56284e7			−0.809203			−0.404602	+	0.914493i	$0.632590\pi$
$822$		0			0
$823$		−	2.04220e7i		−	1.05099i	−0.850797	−	0.525494i	$-0.823881\pi$
$823$		−	2.04220e7i		−	1.05099i	0.850797	−	0.525494i	$-0.176119\pi$
$824$		0			0
$825$	8.42046e6	+	3.27995e6i	0.430726	+	0.167777i
$826$		0			0
$827$		−	2.98366e6i		−	0.151700i	−0.997119	−	0.0758501i	$-0.975833\pi$
$827$		−	2.98366e6i		−	0.151700i	0.997119	−	0.0758501i	$-0.0241670\pi$
$828$		0			0
$829$	−3.27893e7			−1.65709			−0.828544	−	0.559924i	$-0.810830\pi$
$829$	−3.27893e7			−1.65709			−0.828544	+	0.559924i	$0.810830\pi$
$830$		0			0
$831$	8.96212e6			0.450203
$832$		0			0
$833$			3.33574e7i			1.66563i
$834$		0			0
$835$	−2.02697e7	−	2.96486e7i	−1.00607	−	1.47159i
$836$		0			0
$837$		−	1.80672e7i		−	0.891411i
$838$		0			0
$839$	2.04377e6			0.100237			0.0501183	−	0.998743i	$-0.484040\pi$
$839$	2.04377e6			0.100237			0.0501183	+	0.998743i	$0.484040\pi$
$840$		0			0
$841$	−1.90108e7			−0.926851
$842$		0			0
$843$			3.21143e6i			0.155643i
$844$		0			0
$845$	−123166.	+	84204.0i	−0.00593401	+	0.00405687i
$846$		0			0
$847$		−	2.63069e6i		−	0.125997i
$848$		0			0
$849$	6.07553e6			0.289278
$850$		0			0
$851$	1.65820e7			0.784899
$852$		0			0
$853$			5.10077e6i			0.240029i	0.992772	+	0.120014i	$0.0382940\pi$
$853$			5.10077e6i			0.240029i	−0.992772	+	0.120014i	$0.961706\pi$
$854$		0			0
$855$	2.43395e7	−	1.66400e7i	1.13867	−	0.778464i
$856$		0			0
$857$			2.28599e6i			0.106322i	0.998586	+	0.0531608i	$0.0169296\pi$
$857$			2.28599e6i			0.106322i	−0.998586	+	0.0531608i	$0.983070\pi$
$858$		0			0
$859$	59406.4			0.00274695			0.00137347	−	0.999999i	$-0.499563\pi$
$859$	59406.4			0.00274695			0.00137347	+	0.999999i	$0.499563\pi$
$860$		0			0
$861$	−345753.			−0.0158949
$862$		0			0
$863$			5.40855e6i			0.247203i	0.992332	+	0.123602i	$0.0394445\pi$
$863$			5.40855e6i			0.247203i	−0.992332	+	0.123602i	$0.960556\pi$
$864$		0			0
$865$	8.41875e6	+	1.23142e7i	0.382567	+	0.559583i
$866$		0			0
$867$		−	1.32130e7i		−	0.596970i
$868$		0			0
$869$	−1.33744e7			−0.600793
$870$		0			0
$871$	2.75914e7			1.23233
$872$		0			0
$873$			3.29283e7i			1.46229i
$874$		0			0
$875$	2.61872e6	−	608287.i	0.115630	−	0.0268589i
$876$		0			0
$877$		−	1.48058e7i		−	0.650029i	−0.945709	−	0.325014i	$-0.894631\pi$
$877$		−	1.48058e7i		−	0.650029i	0.945709	−	0.325014i	$-0.105369\pi$
$878$		0			0
$879$	1.08099e7			0.471899
$880$		0			0
$881$	1.41273e7			0.613225			0.306612	−	0.951834i	$-0.400805\pi$
$881$	1.41273e7			0.613225			0.306612	+	0.951834i	$0.400805\pi$
$882$		0			0
$883$		−	2.84795e7i		−	1.22922i	−0.788830	−	0.614611i	$-0.789313\pi$
$883$		−	2.84795e7i		−	1.22922i	0.788830	−	0.614611i	$-0.210687\pi$
$884$		0			0
$885$	4.92943e6	+	7.21031e6i	0.211562	+	0.309454i
$886$		0			0
$887$			1.34648e7i			0.574635i	0.957835	+	0.287318i	$0.0927635\pi$
$887$			1.34648e7i			0.574635i	−0.957835	+	0.287318i	$0.907236\pi$
$888$		0			0
$889$	1.13467e6			0.0481520
$890$		0			0
$891$	2.38088e7			1.00472
$892$		0			0
$893$		−	4.96244e7i		−	2.08241i
$894$		0			0
$895$	2.42787e7	−	1.65985e7i	1.01314	−	0.692646i
$896$		0			0
$897$			1.32754e7i			0.550892i
$898$		0			0
$899$	9.56905e6			0.394884
$900$		0			0
$901$	2.19488e7			0.900738
$902$		0			0
$903$			1.13643e6i			0.0463791i
$904$		0			0
$905$	2.41592e6	−	1.65168e6i	0.0980530	−	0.0670353i
$906$		0			0
$907$			1.26283e7i			0.509713i	0.966979	+	0.254857i	$0.0820283\pi$
$907$			1.26283e7i			0.509713i	−0.966979	+	0.254857i	$0.917972\pi$
$908$		0			0
$909$	−3.15836e7			−1.26780
$910$		0			0
$911$	−3.41674e7			−1.36401			−0.682003	−	0.731349i	$-0.738891\pi$
$911$	−3.41674e7			−1.36401			−0.682003	+	0.731349i	$0.738891\pi$
$912$		0			0
$913$		−	2.00023e7i		−	0.794151i
$914$		0			0
$915$	710354.	+	1.03904e6i	0.0280493	+	0.0410279i
$916$		0			0
$917$		−	3.40036e6i		−	0.133537i
$918$		0			0
$919$	2.64593e7			1.03345			0.516726	−	0.856151i	$-0.327151\pi$
$919$	2.64593e7			1.03345			0.516726	+	0.856151i	$0.327151\pi$
$920$		0			0
$921$	−8.76124e6			−0.340343
$922$		0			0
$923$		−	6.30968e6i		−	0.243783i
$924$		0			0
$925$	−1.10829e7	−	4.31704e6i	−0.425893	−	0.165895i
$926$		0			0
$927$		−	1.43849e7i		−	0.549802i
$928$		0			0
$929$	4.40572e7			1.67486			0.837428	−	0.546548i	$-0.184058\pi$
$929$	4.40572e7			1.67486			0.837428	+	0.546548i	$0.184058\pi$
$930$		0			0
$931$	−4.01241e7			−1.51716
$932$		0			0
$933$			1.51379e7i			0.569326i
$934$		0			0
$935$	−3.65950e7	−	5.35279e7i	−1.36897	−	2.00240i
$936$		0			0
$937$			3.49390e7i			1.30005i	0.759911	+	0.650027i	$0.225243\pi$
$937$			3.49390e7i			1.30005i	−0.759911	+	0.650027i	$0.774757\pi$
$938$		0			0
$939$	1.23193e6			0.0455954
$940$		0			0
$941$	−2.82105e7			−1.03857			−0.519285	−	0.854601i	$-0.673802\pi$
$941$	−2.82105e7			−1.03857			−0.519285	+	0.854601i	$0.673802\pi$
$942$		0			0
$943$		−	1.95030e7i		−	0.714204i
$944$		0			0
$945$	−1.64247e6	+	1.12290e6i	−0.0598299	+	0.0409035i
$946$		0			0
$947$			2.66274e7i			0.964838i	0.875941	+	0.482419i	$0.160242\pi$
$947$			2.66274e7i			0.964838i	−0.875941	+	0.482419i	$0.839758\pi$
$948$		0			0
$949$	2.36660e7			0.853019
$950$		0			0
$951$	−1.36704e7			−0.490150
$952$		0			0
$953$		−	1.81765e7i		−	0.648302i	−0.946005	−	0.324151i	$-0.894921\pi$
$953$		−	1.81765e7i		−	0.648302i	0.946005	−	0.324151i	$-0.105079\pi$
$954$		0			0
$955$	2.43854e7	−	1.66714e7i	0.865210	−	0.591513i
$956$		0			0
$957$		−	3.54208e6i		−	0.125020i
$958$		0			0
$959$	−1.51934e6			−0.0533468
$960$		0			0
$961$	3.24005e7			1.13173
$962$		0			0
$963$		−	1.13967e7i		−	0.396016i
$964$		0			0
$965$	1.49120e7	+	2.18119e7i	0.515487	+	0.754007i
$966$		0			0
$967$			6.99989e6i			0.240727i	0.992730	+	0.120364i	$0.0384060\pi$
$967$			6.99989e6i			0.240727i	−0.992730	+	0.120364i	$0.961594\pi$
$968$		0			0
$969$	2.44648e7			0.837012
$970$		0			0
$971$	−459462.			−0.0156387			−0.00781936	−	0.999969i	$-0.502489\pi$
$971$	−459462.			−0.0156387			−0.00781936	+	0.999969i	$0.502489\pi$
$972$		0			0
$973$		−	1.54992e6i		−	0.0524840i
$974$		0			0
$975$	3.45618e6	−	8.87288e6i	0.116435	−	0.298919i
$976$		0			0
$977$			2.28009e7i			0.764215i	0.924118	+	0.382108i	$0.124802\pi$
$977$			2.28009e7i			0.764215i	−0.924118	+	0.382108i	$0.875198\pi$
$978$		0			0
$979$	5.63630e7			1.87948
$980$		0			0
$981$	−8.64501e6			−0.286809
$982$		0			0
$983$			6.81107e6i			0.224818i	0.993662	+	0.112409i	$0.0358567\pi$
$983$			6.81107e6i			0.224818i	−0.993662	+	0.112409i	$0.964143\pi$
$984$		0			0
$985$	−2.69234e7	−	3.93811e7i	−0.884177	−	1.29329i
$986$		0			0
$987$			1.58285e6i			0.0517186i
$988$		0			0
$989$	−6.41029e7			−2.08395
$990$		0			0
$991$	5.95561e7			1.92638			0.963190	−	0.268821i	$-0.0866341\pi$
$991$	5.95561e7			1.92638			0.963190	+	0.268821i	$0.0866341\pi$
$992$		0			0
$993$			8.09049e6i			0.260376i
$994$		0			0
$995$	8.65742e6	−	5.91876e6i	0.277224	−	0.189528i
$996$		0			0
$997$			1.22887e7i			0.391531i	0.980651	+	0.195766i	$0.0627192\pi$
$997$			1.22887e7i			0.391531i	−0.980651	+	0.195766i	$0.937281\pi$
$998$		0			0
$999$	8.80239e6			0.279053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.c.d.129.8	yes	12
4.3	odd	2	inner	160.6.c.d.129.6	yes	12
5.2	odd	4		800.6.a.bb.1.4		6
5.3	odd	4		800.6.a.w.1.4		6
5.4	even	2	inner	160.6.c.d.129.5	✓	12
8.3	odd	2		320.6.c.l.129.7		12
8.5	even	2		320.6.c.l.129.5		12
20.3	even	4		800.6.a.bb.1.3		6
20.7	even	4		800.6.a.w.1.3		6
20.19	odd	2	inner	160.6.c.d.129.7	yes	12
40.19	odd	2		320.6.c.l.129.6		12
40.29	even	2		320.6.c.l.129.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.c.d.129.5	✓	12	5.4	even	2	inner
160.6.c.d.129.6	yes	12	4.3	odd	2	inner
160.6.c.d.129.7	yes	12	20.19	odd	2	inner
160.6.c.d.129.8	yes	12	1.1	even	1	trivial
320.6.c.l.129.5		12	8.5	even	2
320.6.c.l.129.6		12	40.19	odd	2
320.6.c.l.129.7		12	8.3	odd	2
320.6.c.l.129.8		12	40.29	even	2
800.6.a.w.1.3		6	20.7	even	4
800.6.a.w.1.4		6	5.3	odd	4
800.6.a.bb.1.3		6	20.3	even	4
800.6.a.bb.1.4		6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	160.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.01877i q^{3} +(-46.1478 + 31.5496i) q^{5} -15.3895i q^{7} +217.812 q^{9} +O(q^{10})\)	\(q+5.01877i q^{3} +(-46.1478 + 31.5496i) q^{5} -15.3895i q^{7} +217.812 q^{9} +576.187 q^{11} -607.144i q^{13} +(-158.340 - 231.605i) q^{15} +2013.10i q^{17} -2421.47 q^{19} +77.2365 q^{21} +4356.71i q^{23} +(1134.25 - 2911.89i) q^{25} +2312.71i q^{27} -1224.89 q^{29} -7812.15 q^{31} +2891.75i q^{33} +(485.534 + 710.194i) q^{35} -3806.09i q^{37} +3047.12 q^{39} -4476.54 q^{41} +14713.6i q^{43} +(-10051.5 + 6871.88i) q^{45} +20493.5i q^{47} +16570.2 q^{49} -10103.3 q^{51} -10903.0i q^{53} +(-26589.8 + 18178.5i) q^{55} -12152.8i q^{57} -31131.9 q^{59} -4486.25 q^{61} -3352.02i q^{63} +(19155.2 + 28018.4i) q^{65} +45444.5i q^{67} -21865.3 q^{69} +10392.4 q^{71} +38979.1i q^{73} +(14614.1 + 5692.52i) q^{75} -8867.24i q^{77} -23211.9 q^{79} +41321.3 q^{81} -34715.0i q^{83} +(-63512.5 - 92900.2i) q^{85} -6147.46i q^{87} +97820.7 q^{89} -9343.66 q^{91} -39207.4i q^{93} +(111745. - 76396.3i) q^{95} +151178. i q^{97} +125500. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 60 q^{5} - 268 q^{9}+O(q^{10}) \) 12 * q - 60 * q^5 - 268 * q^9	\( 12 q - 60 q^{5} - 268 q^{9} + 12080 q^{21} - 16420 q^{25} - 7864 q^{29} + 65560 q^{41} - 60420 q^{45} + 17956 q^{49} - 118232 q^{61} + 11360 q^{65} + 204464 q^{69} - 346436 q^{81} + 96320 q^{85} + 354936 q^{89}+O(q^{100}) \) 12 * q - 60 * q^5 - 268 * q^9 + 12080 * q^21 - 16420 * q^25 - 7864 * q^29 + 65560 * q^41 - 60420 * q^45 + 17956 * q^49 - 118232 * q^61 + 11360 * q^65 + 204464 * q^69 - 346436 * q^81 + 96320 * q^85 + 354936 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more