LMFDB - Embedded newform 360.6.f.b.289.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,6,Mod(289,360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("360.289");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$360 = 2^{3} \cdot 3^{2} \cdot 5$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	360.f (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:	$57.7381751327$
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:	$x^{8} + 41x^{6} + 460x^{4} + 969x^{2} + 9$ x^8 + 41x^6 + 460x^4 + 969*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{31}\cdot 3^{2}\cdot 5^{3}$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.7
Root			$4.73066i$ of defining polynomial
Character	$\chi$	$=$	360.289
Dual form			360.6.f.b.289.8

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(53.0051 - 17.7613i) q^{5} +188.968i q^{7} +501.871 q^{11} +1061.48i q^{13} -29.5861i q^{17} -1578.33 q^{19} +1295.86i q^{23} +(2494.07 - 1882.88i) q^{25} -3586.63 q^{29} -3526.32 q^{31} +(3356.31 + 10016.2i) q^{35} -8413.79i q^{37} -7015.12 q^{41} +22694.2i q^{43} -3501.99i q^{47} -18901.8 q^{49} -27309.1i q^{53} +(26601.7 - 8913.88i) q^{55} +7925.39 q^{59} -7020.54 q^{61} +(18853.3 + 56264.0i) q^{65} -17631.2i q^{67} -13432.9 q^{71} +39946.8i q^{73} +94837.3i q^{77} +93321.2 q^{79} +58448.5i q^{83} +(-525.488 - 1568.21i) q^{85} -13989.4 q^{89} -200586. q^{91} +(-83659.6 + 28033.2i) q^{95} +110640. i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 8 q^{5} + 736 q^{11} + 1376 q^{19} - 2136 q^{25} - 5872 q^{29} + 4224 q^{31} - 19232 q^{35} - 23600 q^{41} - 45000 q^{49} + 15008 q^{55} - 91680 q^{59} + 123856 q^{61} + 72064 q^{65} + 125632 q^{71}+ \cdots - 442592 q^{95}+O(q^{100})$ 8 * q - 8 * q^5 + 736 * q^11 + 1376 * q^19 - 2136 * q^25 - 5872 * q^29 + 4224 * q^31 - 19232 * q^35 - 23600 * q^41 - 45000 * q^49 + 15008 * q^55 - 91680 * q^59 + 123856 * q^61 + 72064 * q^65 + 125632 * q^71 + 43264 * q^79 - 293760 * q^85 + 41904 * q^89 - 487616 * q^91 - 442592 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$ .

$n$	$181$	$217$	$271$	$281$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	53.0051	−	17.7613i	0.948183	−	0.317724i
$5$	53.0051	−	17.7613i	0.948183	−	0.317724i
$6$		0			0
$7$			188.968i			1.45761i	0.684720	+	0.728807i	$0.259925\pi$
$7$			188.968i			1.45761i	−0.684720	+	0.728807i	$0.740075\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	501.871			1.25058			0.625288	−	0.780394i	$-0.284982\pi$
$11$	501.871			1.25058			0.625288	+	0.780394i	$0.284982\pi$
$12$		0			0
$13$			1061.48i			1.74203i	0.491259	+	0.871013i	$0.336537\pi$
$13$			1061.48i			1.74203i	−0.491259	+	0.871013i	$0.663463\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	29.5861i		−	0.0248294i	−0.999923	−	0.0124147i	$-0.996048\pi$
$17$		−	29.5861i		−	0.0248294i	0.999923	−	0.0124147i	$-0.00395182\pi$
$18$		0			0
$19$	−1578.33			−1.00303			−0.501515	−	0.865149i	$-0.667224\pi$
$19$	−1578.33			−1.00303			−0.501515	+	0.865149i	$0.667224\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1295.86i			0.510786i	0.966837	+	0.255393i	$0.0822048\pi$
$23$			1295.86i			0.510786i	−0.966837	+	0.255393i	$0.917795\pi$
$24$		0			0
$25$	2494.07	−	1882.88i	0.798103	−	0.602521i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3586.63			−0.791938			−0.395969	−	0.918264i	$-0.629591\pi$
$29$	−3586.63			−0.791938			−0.395969	+	0.918264i	$0.629591\pi$
$30$		0			0
$31$	−3526.32			−0.659048			−0.329524	−	0.944147i	$-0.606888\pi$
$31$	−3526.32			−0.659048			−0.329524	+	0.944147i	$0.606888\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3356.31	+	10016.2i	0.463118	+	1.38208i
$36$		0			0
$37$		−	8413.79i		−	1.01039i	−0.863006	−	0.505193i	$-0.831421\pi$
$37$		−	8413.79i		−	1.01039i	0.863006	−	0.505193i	$-0.168579\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7015.12			−0.651742			−0.325871	−	0.945414i	$-0.605657\pi$
$41$	−7015.12			−0.651742			−0.325871	+	0.945414i	$0.605657\pi$
$42$		0			0
$43$			22694.2i			1.87173i	0.352358	+	0.935865i	$0.385380\pi$
$43$			22694.2i			1.87173i	−0.352358	+	0.935865i	$0.614620\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3501.99i		−	0.231244i	−0.993293	−	0.115622i	$-0.963114\pi$
$47$		−	3501.99i		−	0.231244i	0.993293	−	0.115622i	$-0.0368861\pi$
$48$		0			0
$49$	−18901.8			−1.12464
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	27309.1i		−	1.33542i	−0.744422	−	0.667710i	$-0.767275\pi$
$53$		−	27309.1i		−	1.33542i	0.744422	−	0.667710i	$-0.232725\pi$
$54$		0			0
$55$	26601.7	−	8913.88i	1.18578	−	0.397338i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7925.39			0.296409			0.148204	−	0.988957i	$-0.452651\pi$
$59$	7925.39			0.296409			0.148204	+	0.988957i	$0.452651\pi$
$60$		0			0
$61$	−7020.54			−0.241572			−0.120786	−	0.992679i	$-0.538541\pi$
$61$	−7020.54			−0.241572			−0.120786	+	0.992679i	$0.538541\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	18853.3	+	56264.0i	0.553483	+	1.65176i
$66$		0			0
$67$		−	17631.2i		−	0.479837i	−0.970793	−	0.239919i	$-0.922879\pi$
$67$		−	17631.2i		−	0.479837i	0.970793	−	0.239919i	$-0.0771208\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−13432.9			−0.316246			−0.158123	−	0.987419i	$-0.550544\pi$
$71$	−13432.9			−0.316246			−0.158123	+	0.987419i	$0.550544\pi$
$72$		0			0
$73$			39946.8i			0.877353i	0.898645	+	0.438677i	$0.144553\pi$
$73$			39946.8i			0.877353i	−0.898645	+	0.438677i	$0.855447\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			94837.3i			1.82286i
$78$		0			0
$79$	93321.2			1.68234			0.841168	−	0.540774i	$-0.181869\pi$
$79$	93321.2			1.68234			0.841168	+	0.540774i	$0.181869\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			58448.5i			0.931275i	0.884975	+	0.465638i	$0.154175\pi$
$83$			58448.5i			0.931275i	−0.884975	+	0.465638i	$0.845825\pi$
$84$		0			0
$85$	−525.488	−	1568.21i	−0.00788888	−	0.0235428i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13989.4			−0.187208			−0.0936038	−	0.995610i	$-0.529839\pi$
$89$	−13989.4			−0.187208			−0.0936038	+	0.995610i	$0.529839\pi$
$90$		0			0
$91$	−200586.			−2.53920
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−83659.6	+	28033.2i	−0.951057	+	0.318687i
$96$		0			0
$97$			110640.i			1.19394i	0.802264	+	0.596969i	$0.203629\pi$
$97$			110640.i			1.19394i	−0.802264	+	0.596969i	$0.796371\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−198468.			−1.93592			−0.967960	−	0.251103i	$-0.919207\pi$
$101$	−198468.			−1.93592			−0.967960	+	0.251103i	$0.919207\pi$
$102$		0			0
$103$			134780.i			1.25179i	0.779906	+	0.625897i	$0.215267\pi$
$103$			134780.i			1.25179i	−0.779906	+	0.625897i	$0.784733\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			39785.5i			0.335943i	0.985792	+	0.167971i	$0.0537216\pi$
$107$			39785.5i			0.335943i	−0.985792	+	0.167971i	$0.946278\pi$
$108$		0			0
$109$	−92692.6			−0.747272			−0.373636	−	0.927575i	$-0.621889\pi$
$109$	−92692.6			−0.747272			−0.373636	+	0.927575i	$0.621889\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			66923.1i			0.493037i	0.969138	+	0.246519i	$0.0792866\pi$
$113$			66923.1i			0.493037i	−0.969138	+	0.246519i	$0.920713\pi$
$114$		0			0
$115$	23016.2	+	68687.2i	0.162289	+	0.484318i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5590.81			0.0361916
$120$		0			0
$121$	90823.1			0.563940
$122$		0			0
$123$		0			0
$124$		0			0
$125$	98756.1	−	144100.i	0.565313	−	0.824877i
$126$		0			0
$127$			58998.9i			0.324589i	0.986742	+	0.162295i	$0.0518895\pi$
$127$			58998.9i			0.324589i	−0.986742	+	0.162295i	$0.948110\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	209380.			1.06600			0.533001	−	0.846115i	$-0.321064\pi$
$131$	209380.			1.06600			0.533001	+	0.846115i	$0.321064\pi$
$132$		0			0
$133$		−	298254.i		−	1.46203i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			326432.i			1.48591i	0.669344	+	0.742953i	$0.266575\pi$
$137$			326432.i			1.48591i	−0.669344	+	0.742953i	$0.733425\pi$
$138$		0			0
$139$	233648.			1.02571			0.512856	−	0.858475i	$-0.328588\pi$
$139$	233648.			1.02571			0.512856	+	0.858475i	$0.328588\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			532727.i			2.17854i
$144$		0			0
$145$	−190109.	+	63703.2i	−0.750902	+	0.251618i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	145336.			0.536299			0.268149	−	0.963377i	$-0.413588\pi$
$149$	145336.			0.536299			0.268149	+	0.963377i	$0.413588\pi$
$150$		0			0
$151$	287343.			1.02555			0.512777	−	0.858522i	$-0.328617\pi$
$151$	287343.			1.02555			0.512777	+	0.858522i	$0.328617\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−186913.	+	62632.0i	−0.624899	+	0.209395i
$156$		0			0
$157$			42626.2i			0.138015i	0.997616	+	0.0690077i	$0.0219833\pi$
$157$			42626.2i			0.138015i	−0.997616	+	0.0690077i	$0.978017\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−244876.			−0.744528
$162$		0			0
$163$		−	221599.i		−	0.653279i	−0.945149	−	0.326639i	$-0.894084\pi$
$163$		−	221599.i		−	0.653279i	0.945149	−	0.326639i	$-0.105916\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	334835.i		−	0.929051i	−0.885560	−	0.464525i	$-0.846225\pi$
$167$		−	334835.i		−	0.929051i	0.885560	−	0.464525i	$-0.153775\pi$
$168$		0			0
$169$	−755454.			−2.03466
$170$		0			0
$171$		0			0
$172$		0			0
$173$			268475.i			0.682006i	0.940062	+	0.341003i	$0.110767\pi$
$173$			268475.i			0.682006i	−0.940062	+	0.341003i	$0.889233\pi$
$174$		0			0
$175$	355803.	+	471299.i	0.878242	+	1.16333i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−268052.			−0.625297			−0.312649	−	0.949869i	$-0.601216\pi$
$179$	−268052.			−0.625297			−0.312649	+	0.949869i	$0.601216\pi$
$180$		0			0
$181$	−90565.5			−0.205479			−0.102739	−	0.994708i	$-0.532761\pi$
$181$	−90565.5			−0.205479			−0.102739	+	0.994708i	$0.532761\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−149440.	−	445973.i	−0.321024	−	0.958031i
$186$		0			0
$187$		−	14848.4i		−	0.0310510i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	706725.			1.40174			0.700870	−	0.713290i	$-0.252796\pi$
$191$	706725.			1.40174			0.700870	+	0.713290i	$0.252796\pi$
$192$		0			0
$193$			236999.i			0.457988i	0.973428	+	0.228994i	$0.0735435\pi$
$193$			236999.i			0.457988i	−0.973428	+	0.228994i	$0.926456\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	607901.i		−	1.11601i	−0.829838	−	0.558004i	$-0.811567\pi$
$197$		−	607901.i		−	1.11601i	0.829838	−	0.558004i	$-0.188433\pi$
$198$		0			0
$199$	−621060.			−1.11173			−0.555867	−	0.831272i	$-0.687613\pi$
$199$	−621060.			−1.11173			−0.555867	+	0.831272i	$0.687613\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	677756.i		−	1.15434i
$204$		0			0
$205$	−371837.	+	124598.i	−0.617971	+	0.207074i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−792118.			−1.25437
$210$		0			0
$211$	1.06177e6			1.64181			0.820906	−	0.571063i	$-0.193469\pi$
$211$	1.06177e6			1.64181			0.820906	+	0.571063i	$0.193469\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	403078.	+	1.20291e6i	0.594693	+	1.77474i
$216$		0			0
$217$		−	666360.i		−	0.960638i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	31405.2			0.0432534
$222$		0			0
$223$			240721.i			0.324154i	0.986778	+	0.162077i	$0.0518193\pi$
$223$			240721.i			0.324154i	−0.986778	+	0.162077i	$0.948181\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	202494.i		−	0.260824i	−0.991460	−	0.130412i	$-0.958370\pi$
$227$		−	202494.i		−	0.260824i	0.991460	−	0.130412i	$-0.0416299\pi$
$228$		0			0
$229$	284279.			0.358225			0.179112	−	0.983829i	$-0.442677\pi$
$229$	284279.			0.358225			0.179112	+	0.983829i	$0.442677\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	345405.i		−	0.416811i	−0.978042	−	0.208406i	$-0.933173\pi$
$233$		−	345405.i		−	0.416811i	0.978042	−	0.208406i	$-0.0668274\pi$
$234$		0			0
$235$	−62199.9	−	185623.i	−0.0734717	−	0.219262i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.59920e6			1.81095			0.905476	−	0.424398i	$-0.139514\pi$
$239$	1.59920e6			1.81095			0.905476	+	0.424398i	$0.139514\pi$
$240$		0			0
$241$	−42246.4			−0.0468541			−0.0234270	−	0.999726i	$-0.507458\pi$
$241$	−42246.4			−0.0468541			−0.0234270	+	0.999726i	$0.507458\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.00189e6	+	335720.i	−1.06636	+	0.357324i
$246$		0			0
$247$		−	1.67537e6i		−	1.74731i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−802841.			−0.804351			−0.402175	−	0.915563i	$-0.631746\pi$
$251$	−802841.			−0.804351			−0.402175	+	0.915563i	$0.631746\pi$
$252$		0			0
$253$			650354.i			0.638776i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	941153.i		−	0.888848i	−0.895817	−	0.444424i	$-0.853408\pi$
$257$		−	941153.i		−	0.888848i	0.895817	−	0.444424i	$-0.146592\pi$
$258$		0			0
$259$	1.58993e6			1.47275
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1.53130e6i		−	1.36512i	−0.730830	−	0.682560i	$-0.760867\pi$
$263$		−	1.53130e6i		−	1.36512i	0.730830	−	0.682560i	$-0.239133\pi$
$264$		0			0
$265$	−485045.	−	1.44752e6i	−0.424294	−	1.26622i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.25712e6			1.05924			0.529621	−	0.848234i	$-0.322334\pi$
$269$	1.25712e6			1.05924			0.529621	+	0.848234i	$0.322334\pi$
$270$		0			0
$271$	843976.			0.698083			0.349041	−	0.937107i	$-0.386507\pi$
$271$	843976.			0.698083			0.349041	+	0.937107i	$0.386507\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.25170e6	−	944961.i	0.998089	−	0.753498i
$276$		0			0
$277$		−	1.52070e6i		−	1.19082i	−0.803423	−	0.595408i	$-0.796990\pi$
$277$		−	1.52070e6i		−	1.19082i	0.803423	−	0.595408i	$-0.203010\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.97975e6			1.49570			0.747848	−	0.663870i	$-0.231087\pi$
$281$	1.97975e6			1.49570			0.747848	+	0.663870i	$0.231087\pi$
$282$		0			0
$283$			2.02901e6i			1.50597i	0.658036	+	0.752986i	$0.271387\pi$
$283$			2.02901e6i			1.50597i	−0.658036	+	0.752986i	$0.728613\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	1.32563e6i		−	0.949987i
$288$		0			0
$289$	1.41898e6			0.999384
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	2.23140e6i		−	1.51848i	−0.650813	−	0.759238i	$-0.725572\pi$
$293$		−	2.23140e6i		−	1.51848i	0.650813	−	0.759238i	$-0.274428\pi$
$294$		0			0
$295$	420086.	−	140765.i	0.281050	−	0.0941761i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.37553e6			−0.889802
$300$		0			0
$301$	−4.28847e6			−2.72826
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−372124.	+	124694.i	−0.229054	+	0.0767531i
$306$		0			0
$307$			334318.i			0.202448i	0.994864	+	0.101224i	$0.0322759\pi$
$307$			334318.i			0.202448i	−0.994864	+	0.101224i	$0.967724\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.39090e6			−1.40172			−0.700858	−	0.713300i	$-0.747200\pi$
$311$	−2.39090e6			−1.40172			−0.700858	+	0.713300i	$0.747200\pi$
$312$		0			0
$313$		−	1.88563e6i		−	1.08791i	−0.839113	−	0.543957i	$-0.816925\pi$
$313$		−	1.88563e6i		−	1.08791i	0.839113	−	0.543957i	$-0.183075\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			179636.i			0.100403i	0.998739	+	0.0502014i	$0.0159863\pi$
$317$			179636.i			0.100403i	−0.998739	+	0.0502014i	$0.984014\pi$
$318$		0			0
$319$	−1.80002e6			−0.990378
$320$		0			0
$321$		0			0
$322$		0			0
$323$			46696.7i			0.0249046i
$324$		0			0
$325$	1.99864e6	+	2.64742e6i	1.04961	+	1.39032i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	661763.			0.337064
$330$		0			0
$331$	168620.			0.0845937			0.0422968	−	0.999105i	$-0.486532\pi$
$331$	168620.			0.0845937			0.0422968	+	0.999105i	$0.486532\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−313152.	−	934541.i	−0.152456	−	0.454974i
$336$		0			0
$337$		−	1.74240e6i		−	0.835744i	−0.908506	−	0.417872i	$-0.862776\pi$
$337$		−	1.74240e6i		−	0.835744i	0.908506	−	0.417872i	$-0.137224\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.76976e6			−0.824190
$342$		0			0
$343$		−	395842.i		−	0.181672i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			2.89447e6i			1.29046i	0.763987	+	0.645232i	$0.223239\pi$
$347$			2.89447e6i			1.29046i	−0.763987	+	0.645232i	$0.776761\pi$
$348$		0			0
$349$	−63803.8			−0.0280403			−0.0140202	−	0.999902i	$-0.504463\pi$
$349$	−63803.8			−0.0280403			−0.0140202	+	0.999902i	$0.504463\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			2.79206e6i			1.19258i	0.802769	+	0.596290i	$0.203359\pi$
$353$			2.79206e6i			1.19258i	−0.802769	+	0.596290i	$0.796641\pi$
$354$		0			0
$355$	−712014.	+	238586.i	−0.299859	+	0.100479i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.48391e6			0.607674			0.303837	−	0.952724i	$-0.401732\pi$
$359$	1.48391e6			0.607674			0.303837	+	0.952724i	$0.401732\pi$
$360$		0			0
$361$	15032.0			0.00607083
$362$		0			0
$363$		0			0
$364$		0			0
$365$	709507.	+	2.11738e6i	0.278756	+	0.831892i
$366$		0			0
$367$		−	1.08188e6i		−	0.419289i	−0.977778	−	0.209644i	$-0.932769\pi$
$367$		−	1.08188e6i		−	0.419289i	0.977778	−	0.209644i	$-0.0672306\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.16053e6			1.94652
$372$		0			0
$373$		−	127750.i		−	0.0475434i	−0.999717	−	0.0237717i	$-0.992433\pi$
$373$		−	127750.i		−	0.0475434i	0.999717	−	0.0237717i	$-0.00756748\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	3.80714e6i		−	1.37958i
$378$		0			0
$379$	1.61295e6			0.576798			0.288399	−	0.957510i	$-0.406877\pi$
$379$	1.61295e6			0.576798			0.288399	+	0.957510i	$0.406877\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			3.96656e6i			1.38171i	0.722993	+	0.690855i	$0.242766\pi$
$383$			3.96656e6i			1.38171i	−0.722993	+	0.690855i	$0.757234\pi$
$384$		0			0
$385$	1.68443e6	+	5.02686e6i	0.579165	+	1.72840i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.54852e6			−1.18898			−0.594488	−	0.804104i	$-0.702645\pi$
$389$	−3.54852e6			−1.18898			−0.594488	+	0.804104i	$0.702645\pi$
$390$		0			0
$391$	38339.5			0.0126825
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.94650e6	−	1.65751e6i	1.59516	−	0.534518i
$396$		0			0
$397$			580854.i			0.184965i	0.995714	+	0.0924827i	$0.0294803\pi$
$397$			580854.i			0.184965i	−0.995714	+	0.0924827i	$0.970520\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.10413e6			−0.653449			−0.326725	−	0.945120i	$-0.605945\pi$
$401$	−2.10413e6			−0.653449			−0.326725	+	0.945120i	$0.605945\pi$
$402$		0			0
$403$		−	3.74313e6i		−	1.14808i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4.22263e6i		−	1.26356i
$408$		0			0
$409$	−3.23687e6			−0.956790			−0.478395	−	0.878145i	$-0.658781\pi$
$409$	−3.23687e6			−0.956790			−0.478395	+	0.878145i	$0.658781\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1.49764e6i			0.432049i
$414$		0			0
$415$	1.03812e6	+	3.09807e6i	0.295888	+	0.883020i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.03059e6			−0.565049			−0.282525	−	0.959260i	$-0.591172\pi$
$419$	−2.03059e6			−0.565049			−0.282525	+	0.959260i	$0.591172\pi$
$420$		0			0
$421$	2.15883e6			0.593625			0.296813	−	0.954936i	$-0.404076\pi$
$421$	2.15883e6			0.593625			0.296813	+	0.954936i	$0.404076\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−55707.0	−	73789.9i	−0.0149602	−	0.0198164i
$426$		0			0
$427$		−	1.32665e6i		−	0.352118i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.49812e6			−1.42568			−0.712838	−	0.701328i	$-0.752591\pi$
$431$	−5.49812e6			−1.42568			−0.712838	+	0.701328i	$0.752591\pi$
$432$		0			0
$433$			3.87201e6i			0.992470i	0.868188	+	0.496235i	$0.165284\pi$
$433$			3.87201e6i			0.992470i	−0.868188	+	0.496235i	$0.834716\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	2.04530e6i		−	0.512334i
$438$		0			0
$439$	−3.15465e6			−0.781250			−0.390625	−	0.920550i	$-0.627741\pi$
$439$	−3.15465e6			−0.781250			−0.390625	+	0.920550i	$0.627741\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			4.07021e6i			0.985389i	0.870202	+	0.492694i	$0.163988\pi$
$443$			4.07021e6i			0.985389i	−0.870202	+	0.492694i	$0.836012\pi$
$444$		0			0
$445$	−741508.	+	248470.i	−0.177507	+	0.0594803i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	957718.			0.224193			0.112096	−	0.993697i	$-0.464243\pi$
$449$	957718.			0.224193			0.112096	+	0.993697i	$0.464243\pi$
$450$		0			0
$451$	−3.52068e6			−0.815052
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.06321e7	+	3.56267e6i	−2.40763	+	0.806765i
$456$		0			0
$457$		−	7.38167e6i		−	1.65335i	−0.562682	−	0.826673i	$-0.690231\pi$
$457$		−	7.38167e6i		−	1.65335i	0.562682	−	0.826673i	$-0.309769\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.96862e6			0.650582			0.325291	−	0.945614i	$-0.394538\pi$
$461$	2.96862e6			0.650582			0.325291	+	0.945614i	$0.394538\pi$
$462$		0			0
$463$			6.84593e6i			1.48416i	0.670313	+	0.742079i	$0.266160\pi$
$463$			6.84593e6i			1.48416i	−0.670313	+	0.742079i	$0.733840\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			4.20816e6i			0.892895i	0.894810	+	0.446447i	$0.147311\pi$
$467$			4.20816e6i			0.892895i	−0.894810	+	0.446447i	$0.852689\pi$
$468$		0			0
$469$	3.33172e6			0.699417
$470$		0			0
$471$		0			0
$472$		0			0
$473$			1.13895e7i			2.34074i
$474$		0			0
$475$	−3.93647e6	+	2.97181e6i	−0.800522	+	0.604347i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.73077e6			−1.14123			−0.570617	−	0.821217i	$-0.693296\pi$
$479$	−5.73077e6			−1.14123			−0.570617	+	0.821217i	$0.693296\pi$
$480$		0			0
$481$	8.93110e6			1.76012
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.96511e6	+	5.86447e6i	0.379343	+	1.13207i
$486$		0			0
$487$		−	6.63249e6i		−	1.26723i	−0.773650	−	0.633613i	$-0.781571\pi$
$487$		−	6.63249e6i		−	1.26723i	0.773650	−	0.633613i	$-0.218429\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.26463e6			0.611125			0.305563	−	0.952172i	$-0.401156\pi$
$491$	3.26463e6			0.611125			0.305563	+	0.952172i	$0.401156\pi$
$492$		0			0
$493$			106114.i			0.0196633i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	2.53839e6i		−	0.460965i
$498$		0			0
$499$	8.51527e6			1.53090			0.765450	−	0.643495i	$-0.222516\pi$
$499$	8.51527e6			1.53090			0.765450	+	0.643495i	$0.222516\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	3.24174e6i		−	0.571292i	−0.958335	−	0.285646i	$-0.907792\pi$
$503$		−	3.24174e6i		−	0.571292i	0.958335	−	0.285646i	$-0.0922082\pi$
$504$		0			0
$505$	−1.05198e7	+	3.52505e6i	−1.83561	+	0.615088i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.04569e7			1.78900			0.894499	−	0.447070i	$-0.147533\pi$
$509$	1.04569e7			1.78900			0.894499	+	0.447070i	$0.147533\pi$
$510$		0			0
$511$	−7.54865e6			−1.27884
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.39387e6	+	7.14403e6i	0.397725	+	1.18693i
$516$		0			0
$517$		−	1.75755e6i		−	0.289188i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	328142.			0.0529625			0.0264812	−	0.999649i	$-0.491570\pi$
$521$	328142.			0.0529625			0.0264812	+	0.999649i	$0.491570\pi$
$522$		0			0
$523$			9.42102e6i			1.50606i	0.657984	+	0.753032i	$0.271410\pi$
$523$			9.42102e6i			1.50606i	−0.657984	+	0.753032i	$0.728590\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			104330.i			0.0163637i
$528$		0			0
$529$	4.75709e6			0.739098
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	7.44643e6i		−	1.13535i
$534$		0			0
$535$	706642.	+	2.10883e6i	0.106737	+	0.318535i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9.48624e6			−1.40644
$540$		0			0
$541$	6.08041e6			0.893182			0.446591	−	0.894738i	$-0.352638\pi$
$541$	6.08041e6			0.893182			0.446591	+	0.894738i	$0.352638\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.91318e6	+	1.64634e6i	−0.708551	+	0.237426i
$546$		0			0
$547$		−	4.32182e6i		−	0.617587i	−0.951129	−	0.308794i	$-0.900075\pi$
$547$		−	4.32182e6i		−	0.617587i	0.951129	−	0.308794i	$-0.0999253\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.66089e6			0.794338
$552$		0			0
$553$			1.76347e7i			2.45220i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			7.41019e6i			1.01203i	0.862526	+	0.506013i	$0.168881\pi$
$557$			7.41019e6i			1.01203i	−0.862526	+	0.506013i	$0.831119\pi$
$558$		0			0
$559$	−2.40895e7			−3.26061
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	1.08039e7i		−	1.43652i	−0.695775	−	0.718260i	$-0.744939\pi$
$563$		−	1.08039e7i		−	1.43652i	0.695775	−	0.718260i	$-0.255061\pi$
$564$		0			0
$565$	1.18864e6	+	3.54726e6i	0.156650	+	0.467490i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.77404e6			0.229712			0.114856	−	0.993382i	$-0.463359\pi$
$569$	1.77404e6			0.229712			0.114856	+	0.993382i	$0.463359\pi$
$570$		0			0
$571$	−70981.0			−0.00911070			−0.00455535	−	0.999990i	$-0.501450\pi$
$571$	−70981.0			−0.00911070			−0.00455535	+	0.999990i	$0.501450\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.43995e6	+	3.23197e6i	0.307759	+	0.407660i
$576$		0			0
$577$			4.04410e6i			0.505688i	0.967507	+	0.252844i	$0.0813659\pi$
$577$			4.04410e6i			0.505688i	−0.967507	+	0.252844i	$0.918634\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.10449e7			−1.35744
$582$		0			0
$583$		−	1.37056e7i		−	1.67004i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	2.86862e6i		−	0.343620i	−0.985130	−	0.171810i	$-0.945038\pi$
$587$		−	2.86862e6i		−	0.343620i	0.985130	−	0.171810i	$-0.0549615\pi$
$588$		0			0
$589$	5.56570e6			0.661046
$590$		0			0
$591$		0			0
$592$		0			0
$593$			3.23131e6i			0.377347i	0.982040	+	0.188674i	$0.0604188\pi$
$593$			3.23131e6i			0.377347i	−0.982040	+	0.188674i	$0.939581\pi$
$594$		0			0
$595$	296341.	−	99300.1i	0.0343163	−	0.0114989i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3.80078e6			0.432819			0.216409	−	0.976303i	$-0.430565\pi$
$599$	3.80078e6			0.432819			0.216409	+	0.976303i	$0.430565\pi$
$600$		0			0
$601$	5.45289e6			0.615801			0.307900	−	0.951419i	$-0.400374\pi$
$601$	5.45289e6			0.615801			0.307900	+	0.951419i	$0.400374\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.81408e6	−	1.61314e6i	0.534719	−	0.179177i
$606$		0			0
$607$		−	1.50483e7i		−	1.65774i	−0.559441	−	0.828870i	$-0.688984\pi$
$607$		−	1.50483e7i		−	1.65774i	0.559441	−	0.828870i	$-0.311016\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.71731e6			0.402833
$612$		0			0
$613$		−	2.30782e6i		−	0.248056i	−0.992279	−	0.124028i	$-0.960419\pi$
$613$		−	2.30782e6i		−	0.248056i	0.992279	−	0.124028i	$-0.0395813\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			381761.i			0.0403718i	0.999796	+	0.0201859i	$0.00642581\pi$
$617$			381761.i			0.0403718i	−0.999796	+	0.0201859i	$0.993574\pi$
$618$		0			0
$619$	1.07208e7			1.12460			0.562302	−	0.826932i	$-0.309916\pi$
$619$	1.07208e7			1.12460			0.562302	+	0.826932i	$0.309916\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	2.64354e6i		−	0.272876i
$624$		0			0
$625$	2.67517e6	−	9.39207e6i	0.273937	−	0.961748i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−248931.			−0.0250872
$630$		0			0
$631$	−4.56189e6			−0.456112			−0.228056	−	0.973648i	$-0.573237\pi$
$631$	−4.56189e6			−0.456112			−0.228056	+	0.973648i	$0.573237\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.04790e6	+	3.12724e6i	0.103130	+	0.307770i
$636$		0			0
$637$		−	2.00639e7i		−	1.95915i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.70983e6			0.260494			0.130247	−	0.991482i	$-0.458423\pi$
$641$	2.70983e6			0.260494			0.130247	+	0.991482i	$0.458423\pi$
$642$		0			0
$643$			6.10481e6i			0.582297i	0.956678	+	0.291149i	$0.0940374\pi$
$643$			6.10481e6i			0.582297i	−0.956678	+	0.291149i	$0.905963\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.22081e7i		−	1.14654i	−0.819367	−	0.573269i	$-0.805675\pi$
$647$		−	1.22081e7i		−	1.14654i	0.819367	−	0.573269i	$-0.194325\pi$
$648$		0			0
$649$	3.97752e6			0.370681
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	6.26514e6i		−	0.574973i	−0.957785	−	0.287487i	$-0.907180\pi$
$653$		−	6.26514e6i		−	0.574973i	0.957785	−	0.287487i	$-0.0928197\pi$
$654$		0			0
$655$	1.10982e7	−	3.71887e6i	1.01076	−	0.338694i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.85107e7			−1.66039			−0.830194	−	0.557474i	$-0.811771\pi$
$659$	−1.85107e7			−1.66039			−0.830194	+	0.557474i	$0.811771\pi$
$660$		0			0
$661$	−8.46041e6			−0.753161			−0.376581	−	0.926384i	$-0.622900\pi$
$661$	−8.46041e6			−0.753161			−0.376581	+	0.926384i	$0.622900\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.29737e6	−	1.58089e7i	−0.464522	−	1.38627i
$666$		0			0
$667$		−	4.64777e6i		−	0.404511i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.52340e6			−0.302104
$672$		0			0
$673$		−	7.25182e6i		−	0.617177i	−0.951196	−	0.308588i	$-0.900143\pi$
$673$		−	7.25182e6i		−	0.617177i	0.951196	−	0.308588i	$-0.0998566\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	4.74893e6i		−	0.398221i	−0.979977	−	0.199110i	$-0.936195\pi$
$677$		−	4.74893e6i		−	0.398221i	0.979977	−	0.199110i	$-0.0638052\pi$
$678$		0			0
$679$	−2.09073e7			−1.74030
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.37712e7i			1.12959i	0.825233	+	0.564793i	$0.191044\pi$
$683$			1.37712e7i			1.12959i	−0.825233	+	0.564793i	$0.808956\pi$
$684$		0			0
$685$	5.79786e6	+	1.73025e7i	0.472108	+	1.40891i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.89882e7			2.32634
$690$		0			0
$691$	1.64305e7			1.30905			0.654526	−	0.756040i	$-0.272868\pi$
$691$	1.64305e7			1.30905			0.654526	+	0.756040i	$0.272868\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.23845e7	−	4.14990e6i	0.972563	−	0.325893i
$696$		0			0
$697$			207550.i			0.0161823i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.05170e7			0.808342			0.404171	−	0.914683i	$-0.367560\pi$
$701$	1.05170e7			0.808342			0.404171	+	0.914683i	$0.367560\pi$
$702$		0			0
$703$			1.32797e7i			1.01345i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	3.75041e7i		−	2.82182i
$708$		0			0
$709$	1.32691e6			0.0991349			0.0495674	−	0.998771i	$-0.484216\pi$
$709$	1.32691e6			0.0991349			0.0495674	+	0.998771i	$0.484216\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	4.56962e6i		−	0.336633i
$714$		0			0
$715$	9.46193e6	+	2.82372e7i	0.692173	+	2.06565i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.65753e6			0.335995			0.167998	−	0.985787i	$-0.446270\pi$
$719$	4.65753e6			0.335995			0.167998	+	0.985787i	$0.446270\pi$
$720$		0			0
$721$	−2.54691e7			−1.82463
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.94531e6	+	6.75318e6i	−0.632048	+	0.477159i
$726$		0			0
$727$		−	1.47884e6i		−	0.103773i	−0.998653	−	0.0518867i	$-0.983477\pi$
$727$		−	1.47884e6i		−	0.103773i	0.998653	−	0.0518867i	$-0.0165235\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	671432.			0.0464739
$732$		0			0
$733$		−	1.41118e7i		−	0.970115i	−0.874482	−	0.485057i	$-0.838799\pi$
$733$		−	1.41118e7i		−	0.970115i	0.874482	−	0.485057i	$-0.161201\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	8.84856e6i		−	0.600073i
$738$		0			0
$739$	5.94045e6			0.400137			0.200068	−	0.979782i	$-0.435884\pi$
$739$	5.94045e6			0.400137			0.200068	+	0.979782i	$0.435884\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1.46589e7i			0.974156i	0.873359	+	0.487078i	$0.161937\pi$
$743$			1.46589e7i			0.974156i	−0.873359	+	0.487078i	$0.838063\pi$
$744$		0			0
$745$	7.70353e6	−	2.58135e6i	0.508510	−	0.170395i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.51817e6			−0.489674
$750$		0			0
$751$	−6.08263e6			−0.393543			−0.196771	−	0.980449i	$-0.563046\pi$
$751$	−6.08263e6			−0.393543			−0.196771	+	0.980449i	$0.563046\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.52306e7	−	5.10359e6i	0.972413	−	0.325843i
$756$		0			0
$757$			8.11757e6i			0.514856i	0.966297	+	0.257428i	$0.0828751\pi$
$757$			8.11757e6i			0.514856i	−0.966297	+	0.257428i	$0.917125\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.63567e7			−1.02384			−0.511921	−	0.859033i	$-0.671066\pi$
$761$	−1.63567e7			−1.02384			−0.511921	+	0.859033i	$0.671066\pi$
$762$		0			0
$763$		−	1.75159e7i		−	1.08923i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			8.41267e6i			0.516352i
$768$		0			0
$769$	2.09586e7			1.27805			0.639024	−	0.769187i	$-0.279339\pi$
$769$	2.09586e7			1.27805			0.639024	+	0.769187i	$0.279339\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	8.95651e6i		−	0.539126i	−0.962983	−	0.269563i	$-0.913121\pi$
$773$		−	8.95651e6i		−	0.539126i	0.962983	−	0.269563i	$-0.0868793\pi$
$774$		0			0
$775$	−8.79489e6	+	6.63963e6i	−0.525989	+	0.397090i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.10722e7			0.653717
$780$		0			0
$781$	−6.74160e6			−0.395490
$782$		0			0
$783$		0			0
$784$		0			0
$785$	757097.	+	2.25940e6i	0.0438508	+	0.130864i
$786$		0			0
$787$		−	2.42009e7i		−	1.39282i	−0.717643	−	0.696411i	$-0.754779\pi$
$787$		−	2.42009e7i		−	1.39282i	0.717643	−	0.696411i	$-0.245221\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.26463e7			−0.718657
$792$		0			0
$793$		−	7.45219e6i		−	0.420824i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	9.91659e6i		−	0.552989i	−0.961016	−	0.276494i	$-0.910827\pi$
$797$		−	9.91659e6i		−	0.552989i	0.961016	−	0.276494i	$-0.0891728\pi$
$798$		0			0
$799$	−103610.			−0.00574164
$800$		0			0
$801$		0			0
$802$		0			0
$803$			2.00481e7i			1.09720i
$804$		0			0
$805$	−1.29797e7	+	4.34931e6i	−0.705949	+	0.236554i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−9.15657e6			−0.491883			−0.245941	−	0.969285i	$-0.579097\pi$
$809$	−9.15657e6			−0.491883			−0.245941	+	0.969285i	$0.579097\pi$
$810$		0			0
$811$	1.49145e7			0.796264			0.398132	−	0.917328i	$-0.369659\pi$
$811$	1.49145e7			0.796264			0.398132	+	0.917328i	$0.369659\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.93588e6	−	1.17459e7i	−0.207562	−	0.619428i
$816$		0			0
$817$		−	3.58189e7i		−	1.87740i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.39549e6			−0.486476			−0.243238	−	0.969967i	$-0.578210\pi$
$821$	−9.39549e6			−0.486476			−0.243238	+	0.969967i	$0.578210\pi$
$822$		0			0
$823$		−	1.59508e7i		−	0.820884i	−0.911887	−	0.410442i	$-0.865374\pi$
$823$		−	1.59508e7i		−	0.820884i	0.911887	−	0.410442i	$-0.134626\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1.68518e7i		−	0.856805i	−0.903588	−	0.428403i	$-0.859077\pi$
$827$		−	1.68518e7i		−	0.856805i	0.903588	−	0.428403i	$-0.140923\pi$
$828$		0			0
$829$	3.92620e6			0.198420			0.0992101	−	0.995067i	$-0.468368\pi$
$829$	3.92620e6			0.198420			0.0992101	+	0.995067i	$0.468368\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			559229.i			0.0279240i
$834$		0			0
$835$	−5.94710e6	−	1.77479e7i	−0.295182	−	0.880910i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.23626e7			−0.606322			−0.303161	−	0.952939i	$-0.598042\pi$
$839$	−1.23626e7			−0.606322			−0.303161	+	0.952939i	$0.598042\pi$
$840$		0			0
$841$	−7.64726e6			−0.372834
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.00429e7	+	1.34179e7i	−1.92923	+	0.646459i
$846$		0			0
$847$			1.71626e7i			0.822006i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.09031e7			0.516091
$852$		0			0
$853$			3.17612e6i			0.149460i	0.997204	+	0.0747299i	$0.0238095\pi$
$853$			3.17612e6i			0.149460i	−0.997204	+	0.0747299i	$0.976191\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.97233e7i		−	0.917334i	−0.888608	−	0.458667i	$-0.848327\pi$
$857$		−	1.97233e7i		−	0.917334i	0.888608	−	0.458667i	$-0.151673\pi$
$858$		0			0
$859$	−4.18456e6			−0.193494			−0.0967470	−	0.995309i	$-0.530844\pi$
$859$	−4.18456e6			−0.193494			−0.0967470	+	0.995309i	$0.530844\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	2.89684e7i		−	1.32403i	−0.749490	−	0.662015i	$-0.769701\pi$
$863$		−	2.89684e7i		−	1.32403i	0.749490	−	0.662015i	$-0.230299\pi$
$864$		0			0
$865$	4.76846e6	+	1.42305e7i	0.216689	+	0.646667i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.68352e7			2.10389
$870$		0			0
$871$	1.87152e7			0.835889
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.72302e7	+	1.86617e7i	1.20235	+	0.824007i
$876$		0			0
$877$			2.78801e7i			1.22404i	0.790843	+	0.612019i	$0.209642\pi$
$877$			2.78801e7i			1.22404i	−0.790843	+	0.612019i	$0.790358\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	4.21471e7			1.82948			0.914741	−	0.404040i	$-0.132394\pi$
$881$	4.21471e7			1.82948			0.914741	+	0.404040i	$0.132394\pi$
$882$		0			0
$883$			4.38615e7i			1.89314i	0.322505	+	0.946568i	$0.395475\pi$
$883$			4.38615e7i			1.89314i	−0.322505	+	0.946568i	$0.604525\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			2.32386e7i			0.991747i	0.868395	+	0.495874i	$0.165152\pi$
$887$			2.32386e7i			0.991747i	−0.868395	+	0.495874i	$0.834848\pi$
$888$		0			0
$889$	−1.11489e7			−0.473126
$890$		0			0
$891$		0			0
$892$		0			0
$893$			5.52731e6i			0.231945i
$894$		0			0
$895$	−1.42081e7	+	4.76095e6i	−0.592896	+	0.198672i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.26476e7			0.521925
$900$		0			0
$901$	−807970.			−0.0331576
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.80043e6	+	1.60856e6i	−0.194831	+	0.0652854i
$906$		0			0
$907$		−	7.74916e6i		−	0.312778i	−0.987696	−	0.156389i	$-0.950015\pi$
$907$		−	7.74916e6i		−	0.312778i	0.987696	−	0.156389i	$-0.0499853\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.54221e7			−0.615668			−0.307834	−	0.951440i	$-0.599604\pi$
$911$	−1.54221e7			−0.615668			−0.307834	+	0.951440i	$0.599604\pi$
$912$		0			0
$913$			2.93336e7i			1.16463i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			3.95661e7i			1.55382i
$918$		0			0
$919$	2.40744e7			0.940302			0.470151	−	0.882586i	$-0.344199\pi$
$919$	2.40744e7			0.940302			0.470151	+	0.882586i	$0.344199\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	1.42588e7i		−	0.550909i
$924$		0			0
$925$	−1.58421e7	−	2.09846e7i	−0.608779	−	0.806392i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.51619e7			−0.576387			−0.288193	−	0.957572i	$-0.593055\pi$
$929$	−1.51619e7			−0.576387			−0.288193	+	0.957572i	$0.593055\pi$
$930$		0			0
$931$	2.98333e7			1.12804
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−263727.	−	787040.i	−0.00986564	−	0.0294420i
$936$		0			0
$937$			4.90390e6i			0.182470i	0.995829	+	0.0912352i	$0.0290815\pi$
$937$			4.90390e6i			0.182470i	−0.995829	+	0.0912352i	$0.970918\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.62296e7			0.597495			0.298748	−	0.954332i	$-0.403431\pi$
$941$	1.62296e7			0.597495			0.298748	+	0.954332i	$0.403431\pi$
$942$		0			0
$943$		−	9.09062e6i		−	0.332900i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			4.16817e7i			1.51033i	0.655537	+	0.755163i	$0.272442\pi$
$947$			4.16817e7i			1.51033i	−0.655537	+	0.755163i	$0.727558\pi$
$948$		0			0
$949$	−4.24028e7			−1.52837
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1.61135e7i			0.574723i	0.957822	+	0.287362i	$0.0927781\pi$
$953$			1.61135e7i			0.574723i	−0.957822	+	0.287362i	$0.907222\pi$
$954$		0			0
$955$	3.74600e7	−	1.25524e7i	1.32911	−	0.445366i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.16851e7			−2.16588
$960$		0			0
$961$	−1.61942e7			−0.565655
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.20942e6	+	1.25622e7i	0.145514	+	0.434256i
$966$		0			0
$967$			1.51707e7i			0.521722i	0.965376	+	0.260861i	$0.0840064\pi$
$967$			1.51707e7i			0.521722i	−0.965376	+	0.260861i	$0.915994\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.82884e7			0.962854			0.481427	−	0.876486i	$-0.340119\pi$
$971$	2.82884e7			0.962854			0.481427	+	0.876486i	$0.340119\pi$
$972$		0			0
$973$			4.41519e7i			1.49509i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			2.88218e7i			0.966017i	0.875616	+	0.483009i	$0.160456\pi$
$977$			2.88218e7i			0.966017i	−0.875616	+	0.483009i	$0.839544\pi$
$978$		0			0
$979$	−7.02086e6			−0.234117
$980$		0			0
$981$		0			0
$982$		0			0
$983$			2.26448e6i			0.0747453i	0.999301	+	0.0373727i	$0.0118989\pi$
$983$			2.26448e6i			0.0747453i	−0.999301	+	0.0373727i	$0.988101\pi$
$984$		0			0
$985$	−1.07971e7	−	3.22218e7i	−0.354582	−	1.05818i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.94085e7			−0.956053
$990$		0			0
$991$	995592.			0.0322031			0.0161015	−	0.999870i	$-0.494875\pi$
$991$	995592.			0.0322031			0.0161015	+	0.999870i	$0.494875\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.29193e7	+	1.10308e7i	−1.05413	+	0.353224i
$996$		0			0
$997$		−	5.57133e7i		−	1.77509i	−0.460718	−	0.887547i	$-0.652408\pi$
$997$		−	5.57133e7i		−	1.77509i	0.460718	−	0.887547i	$-0.347592\pi$
$998$		0			0
$999$		0			0

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	360.6.f.b.289.7		8
3.2	odd	2		40.6.c.a.9.3	✓	8
4.3	odd	2		720.6.f.n.289.7		8
5.4	even	2	inner	360.6.f.b.289.8		8
12.11	even	2		80.6.c.d.49.6		8
15.2	even	4		200.6.a.k.1.2		4
15.8	even	4		200.6.a.j.1.3		4
15.14	odd	2		40.6.c.a.9.6	yes	8
20.19	odd	2		720.6.f.n.289.8		8
24.5	odd	2		320.6.c.j.129.6		8
24.11	even	2		320.6.c.i.129.3		8
60.23	odd	4		400.6.a.ba.1.2		4
60.47	odd	4		400.6.a.z.1.3		4
60.59	even	2		80.6.c.d.49.3		8
120.29	odd	2		320.6.c.j.129.3		8
120.59	even	2		320.6.c.i.129.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.c.a.9.3	✓	8	3.2	odd	2
40.6.c.a.9.6	yes	8	15.14	odd	2
80.6.c.d.49.3		8	60.59	even	2
80.6.c.d.49.6		8	12.11	even	2
200.6.a.j.1.3		4	15.8	even	4
200.6.a.k.1.2		4	15.2	even	4
320.6.c.i.129.3		8	24.11	even	2
320.6.c.i.129.6		8	120.59	even	2
320.6.c.j.129.3		8	120.29	odd	2
320.6.c.j.129.6		8	24.5	odd	2
360.6.f.b.289.7		8	1.1	even	1	trivial
360.6.f.b.289.8		8	5.4	even	2	inner
400.6.a.z.1.3		4	60.47	odd	4
400.6.a.ba.1.2		4	60.23	odd	4
720.6.f.n.289.7		8	4.3	odd	2
720.6.f.n.289.8		8	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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	360.6.f.b.289.7

Level	$360$
Weight	$6$
Character	360.289
Analytic conductor	$57.738$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$