LMFDB - Embedded newform 324.5.g.f.53.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [324,5,Mod(53,324)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("324.53"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	324.g (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,-52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$33.4918680392$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 6242x^{12} + 11954017x^{8} + 7856017920x^{4} + 1485512441856 $ x^16 + 6242x^12 + 11954017x^8 + 7856017920*x^4 + 1485512441856
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{44} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.1
Root			$-5.35887 - 5.35887i$ of defining polynomial
Character	$\chi$	$=$	324.53
Dual form			324.5.g.f.269.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-29.1904 - 16.8531i) q^{5} +(16.7081 + 28.9393i) q^{7} +(119.377 - 68.9223i) q^{11} +(-128.464 + 222.506i) q^{13} +56.9994i q^{17} -328.391 q^{19} +(56.9556 + 32.8833i) q^{23} +(255.552 + 442.629i) q^{25} +(400.370 - 231.153i) q^{29} +(916.519 - 1587.46i) q^{31} -1126.33i q^{35} +359.661 q^{37} +(2295.67 + 1325.41i) q^{41} +(-430.174 - 745.083i) q^{43} +(2744.30 - 1584.42i) q^{47} +(642.177 - 1112.28i) q^{49} -5281.61i q^{53} -4646.21 q^{55} +(-1327.28 - 766.308i) q^{59} +(3614.88 + 6261.16i) q^{61} +(7499.82 - 4330.02i) q^{65} +(295.046 - 511.035i) q^{67} +4706.23i q^{71} -2748.96 q^{73} +(3989.13 + 2303.12i) q^{77} +(2983.53 + 5167.62i) q^{79} +(10339.3 - 5969.42i) q^{83} +(960.615 - 1663.83i) q^{85} -9339.75i q^{89} -8585.57 q^{91} +(9585.87 + 5534.40i) q^{95} +(-1598.84 - 2769.27i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 52 q^{7} - 220 q^{13} - 616 q^{19} + 328 q^{25} + 1472 q^{31} + 2456 q^{37} - 388 q^{43} - 3432 q^{49} - 72 q^{55} - 1804 q^{61} + 2180 q^{67} - 9088 q^{73} + 5012 q^{79} + 5364 q^{85} - 3512 q^{91}+ \cdots + 2528 q^{97}+O(q^{100}) $ 16 * q - 52 * q^7 - 220 * q^13 - 616 * q^19 + 328 * q^25 + 1472 * q^31 + 2456 * q^37 - 388 * q^43 - 3432 * q^49 - 72 * q^55 - 1804 * q^61 + 2180 * q^67 - 9088 * q^73 + 5012 * q^79 + 5364 * q^85 - 3512 * q^91 + 2528 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−29.1904	−	16.8531i	−1.16761	−	0.674123i	−0.214498	−	0.976724i	$-0.568811\pi$
$5$	−29.1904	−	16.8531i	−1.16761	−	0.674123i	−0.953117	+	0.302602i	$0.902145\pi$
$6$		0			0
$7$	16.7081	+	28.9393i	0.340982	+	0.590598i	0.984615	−	0.174736i	$-0.0559071\pi$
$7$	16.7081	+	28.9393i	0.340982	+	0.590598i	−0.643633	+	0.765334i	$0.722574\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	119.377	−	68.9223i	0.986586	−	0.569606i	0.0823338	−	0.996605i	$-0.473763\pi$
$11$	119.377	−	68.9223i	0.986586	−	0.569606i	0.904252	+	0.426999i	$0.140429\pi$
$12$		0			0
$13$	−128.464	+	222.506i	−0.760142	+	1.31660i	0.182636	+	0.983181i	$0.441537\pi$
$13$	−128.464	+	222.506i	−0.760142	+	1.31660i	−0.942777	+	0.333423i	$0.891796\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			56.9994i			0.197230i	0.995126	+	0.0986149i	$0.0314412\pi$
$17$			56.9994i			0.197230i	−0.995126	+	0.0986149i	$0.968559\pi$
$18$		0			0
$19$	−328.391			−0.909672			−0.454836	−	0.890575i	$-0.650302\pi$
$19$	−328.391			−0.909672			−0.454836	+	0.890575i	$0.650302\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	56.9556	+	32.8833i	0.107667	+	0.0621613i	0.552866	−	0.833270i	$-0.313534\pi$
$23$	56.9556	+	32.8833i	0.107667	+	0.0621613i	−0.445200	+	0.895431i	$0.646867\pi$
$24$		0			0
$25$	255.552	+	442.629i	0.408883	+	0.708206i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	400.370	−	231.153i	0.476064	−	0.274855i	−0.242711	−	0.970099i	$-0.578037\pi$
$29$	400.370	−	231.153i	0.476064	−	0.274855i	0.718775	+	0.695243i	$0.244703\pi$
$30$		0			0
$31$	916.519	−	1587.46i	0.953714	−	1.65188i	0.216431	−	0.976298i	$-0.430558\pi$
$31$	916.519	−	1587.46i	0.953714	−	1.65188i	0.737284	−	0.675583i	$-0.236108\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	1126.33i		−	0.919455i
$36$		0			0
$37$	359.661			0.262718			0.131359	−	0.991335i	$-0.458066\pi$
$37$	359.661			0.262718			0.131359	+	0.991335i	$0.458066\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2295.67	+	1325.41i	1.36566	+	0.788463i	0.990370	−	0.138445i	$-0.0442105\pi$
$41$	2295.67	+	1325.41i	1.36566	+	0.788463i	0.375288	+	0.926908i	$0.377544\pi$
$42$		0			0
$43$	−430.174	−	745.083i	−0.232652	−	0.402965i	0.725936	−	0.687763i	$-0.241407\pi$
$43$	−430.174	−	745.083i	−0.232652	−	0.402965i	−0.958588	+	0.284797i	$0.908074\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2744.30	−	1584.42i	1.24233	−	0.717258i	0.272760	−	0.962082i	$-0.412064\pi$
$47$	2744.30	−	1584.42i	1.24233	−	0.717258i	0.969567	+	0.244824i	$0.0787302\pi$
$48$		0			0
$49$	642.177	−	1112.28i	0.267462	−	0.463258i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	5281.61i		−	1.88025i	−0.340836	−	0.940123i	$-0.610710\pi$
$53$		−	5281.61i		−	1.88025i	0.340836	−	0.940123i	$-0.389290\pi$
$54$		0			0
$55$	−4646.21			−1.53594
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1327.28	−	766.308i	−0.381294	−	0.220140i	0.297087	−	0.954850i	$-0.403985\pi$
$59$	−1327.28	−	766.308i	−0.381294	−	0.220140i	−0.678381	+	0.734710i	$0.737318\pi$
$60$		0			0
$61$	3614.88	+	6261.16i	0.971481	+	1.68265i	0.691091	+	0.722768i	$0.257130\pi$
$61$	3614.88	+	6261.16i	0.971481	+	1.68265i	0.280390	+	0.959886i	$0.409536\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7499.82	−	4330.02i	1.77511	−	1.02486i
$66$		0			0
$67$	295.046	−	511.035i	0.0657265	−	0.113842i	−0.831290	−	0.555840i	$-0.812397\pi$
$67$	295.046	−	511.035i	0.0657265	−	0.113842i	0.897016	+	0.441998i	$0.145730\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4706.23i			0.933591i	0.884365	+	0.466795i	$0.154592\pi$
$71$			4706.23i			0.933591i	−0.884365	+	0.466795i	$0.845408\pi$
$72$		0			0
$73$	−2748.96			−0.515849			−0.257924	−	0.966165i	$-0.583039\pi$
$73$	−2748.96			−0.515849			−0.257924	+	0.966165i	$0.583039\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3989.13	+	2303.12i	0.672816	+	0.388451i
$78$		0			0
$79$	2983.53	+	5167.62i	0.478053	+	0.828011i	0.999683	−	0.0251600i	$-0.00800952\pi$
$79$	2983.53	+	5167.62i	0.478053	+	0.828011i	−0.521631	+	0.853171i	$0.674676\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10339.3	−	5969.42i	1.50085	−	0.866515i	0.500848	−	0.865535i	$-0.333021\pi$
$83$	10339.3	−	5969.42i	1.50085	−	0.866515i	1.00000		0.000979819i	$-0.000311886\pi$
$84$		0			0
$85$	960.615	−	1663.83i	0.132957	−	0.230288i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	9339.75i		−	1.17911i	−0.807727	−	0.589556i	$-0.799303\pi$
$89$		−	9339.75i		−	1.17911i	0.807727	−	0.589556i	$-0.200697\pi$
$90$		0			0
$91$	−8585.57			−1.03678
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9585.87	+	5534.40i	1.06215	+	0.613230i
$96$		0			0
$97$	−1598.84	−	2769.27i	−0.169927	−	0.294322i	0.768467	−	0.639889i	$-0.221020\pi$
$97$	−1598.84	−	2769.27i	−0.169927	−	0.294322i	−0.938394	+	0.345568i	$0.887686\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4275.71	+	2468.58i	−0.419146	+	0.241994i	−0.694712	−	0.719288i	$-0.744468\pi$
$101$	−4275.71	+	2468.58i	−0.419146	+	0.241994i	0.275566	+	0.961282i	$0.411135\pi$
$102$		0			0
$103$	7812.71	−	13532.0i	0.736423	−	1.27552i	−0.217673	−	0.976022i	$-0.569847\pi$
$103$	7812.71	−	13532.0i	0.736423	−	1.27552i	0.954096	−	0.299500i	$-0.0968200\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	15974.3i		−	1.39526i	−0.716460	−	0.697629i	$-0.754239\pi$
$107$		−	15974.3i		−	1.39526i	0.716460	−	0.697629i	$-0.245761\pi$
$108$		0			0
$109$	4163.62			0.350444			0.175222	−	0.984529i	$-0.443936\pi$
$109$	4163.62			0.350444			0.175222	+	0.984529i	$0.443936\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11971.5	−	6911.77i	−0.937547	−	0.541293i	−0.0483564	−	0.998830i	$-0.515398\pi$
$113$	−11971.5	−	6911.77i	−0.937547	−	0.541293i	−0.889190	+	0.457537i	$0.848732\pi$
$114$		0			0
$115$	−1108.37	−	1919.75i	−0.0838087	−	0.145161i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1649.52	+	952.353i	−0.116484	+	0.0672518i
$120$		0			0
$121$	2180.06	−	3775.97i	0.148901	−	0.257904i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			3839.00i			0.245696i
$126$		0			0
$127$	19659.0			1.21886			0.609432	−	0.792839i	$-0.291398\pi$
$127$	19659.0			1.21886			0.609432	+	0.792839i	$0.291398\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	18028.6	+	10408.8i	1.05055	+	0.606537i	0.922804	−	0.385269i	$-0.125891\pi$
$131$	18028.6	+	10408.8i	1.05055	+	0.606537i	0.127749	+	0.991806i	$0.459225\pi$
$132$		0			0
$133$	−5486.81	−	9503.43i	−0.310182	−	0.537251i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14851.4	−	8574.49i	0.791275	−	0.456843i	−0.0491359	−	0.998792i	$-0.515647\pi$
$137$	14851.4	−	8574.49i	0.791275	−	0.456843i	0.840411	+	0.541949i	$0.182313\pi$
$138$		0			0
$139$	−7958.44	+	13784.4i	−0.411906	+	0.713443i	−0.995098	−	0.0988916i	$-0.968470\pi$
$139$	−7958.44	+	13784.4i	−0.411906	+	0.713443i	0.583192	+	0.812335i	$0.301804\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			35416.1i			1.73192i
$144$		0			0
$145$	−15582.6			−0.741145
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−22913.0	−	13228.8i	−1.03207	−	0.595866i	−0.114494	−	0.993424i	$-0.536525\pi$
$149$	−22913.0	−	13228.8i	−1.03207	−	0.595866i	−0.917577	+	0.397558i	$0.869858\pi$
$150$		0			0
$151$	13380.7	+	23176.0i	0.586846	+	1.01645i	0.994643	+	0.103374i	$0.0329640\pi$
$151$	13380.7	+	23176.0i	0.586846	+	1.01645i	−0.407796	+	0.913073i	$0.633703\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−53507.1	+	30892.3i	−2.22714	+	1.28584i
$156$		0			0
$157$	−20572.9	+	35633.3i	−0.834635	+	1.44563i	0.0596925	+	0.998217i	$0.480988\pi$
$157$	−20572.9	+	35633.3i	−0.834635	+	1.44563i	−0.894327	+	0.447413i	$0.852345\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2197.68i			0.0847836i
$162$		0			0
$163$	−18225.9			−0.685982			−0.342991	−	0.939339i	$-0.611440\pi$
$163$	−18225.9			−0.685982			−0.342991	+	0.939339i	$0.611440\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−24824.3	−	14332.3i	−0.890111	−	0.513906i	−0.0161325	−	0.999870i	$-0.505135\pi$
$167$	−24824.3	−	14332.3i	−0.890111	−	0.513906i	−0.873979	+	0.485964i	$0.838469\pi$
$168$		0			0
$169$	−18725.5	−	32433.5i	−0.655631	−	1.13559i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4725.35	+	2728.18i	−0.157885	+	0.0911551i	−0.576861	−	0.816842i	$-0.695723\pi$
$173$	−4725.35	+	2728.18i	−0.157885	+	0.0911551i	0.418976	+	0.907997i	$0.362389\pi$
$174$		0			0
$175$	−8539.59	+	14791.0i	−0.278844	+	0.482971i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			41680.0i			1.30083i	0.759577	+	0.650417i	$0.225406\pi$
$179$			41680.0i			1.30083i	−0.759577	+	0.650417i	$0.774594\pi$
$180$		0			0
$181$	49304.0			1.50496			0.752480	−	0.658615i	$-0.228857\pi$
$181$	49304.0			1.50496			0.752480	+	0.658615i	$0.228857\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10498.6	−	6061.40i	−0.306754	−	0.177104i
$186$		0			0
$187$	3928.53	+	6804.41i	0.112343	+	0.194584i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−26518.8	+	15310.6i	−0.726921	+	0.419688i	−0.817295	−	0.576220i	$-0.804527\pi$
$191$	−26518.8	+	15310.6i	−0.726921	+	0.419688i	0.0903735	+	0.995908i	$0.471194\pi$
$192$		0			0
$193$	4884.57	−	8460.32i	0.131133	−	0.227129i	−0.792981	−	0.609247i	$-0.791472\pi$
$193$	4884.57	−	8460.32i	0.131133	−	0.227129i	0.924114	+	0.382118i	$0.124805\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	19789.3i		−	0.509916i	−0.966952	−	0.254958i	$-0.917938\pi$
$197$		−	19789.3i		−	0.509916i	0.966952	−	0.254958i	$-0.0820616\pi$
$198$		0			0
$199$	53235.2			1.34429			0.672144	−	0.740420i	$-0.265373\pi$
$199$	53235.2			1.34429			0.672144	+	0.740420i	$0.265373\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13378.8	+	7724.28i	0.324658	+	0.187442i
$204$		0			0
$205$	−44674.3	−	77378.2i	−1.06304	−	1.84124i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−39202.4	+	22633.5i	−0.897469	+	0.518154i
$210$		0			0
$211$	1306.95	−	2263.71i	0.0293559	−	0.0508459i	−0.850974	−	0.525207i	$-0.823988\pi$
$211$	1306.95	−	2263.71i	0.0293559	−	0.0508459i	0.880330	+	0.474362i	$0.157321\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			28999.0i			0.627344i
$216$		0			0
$217$	61253.3			1.30080
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12682.7	−	7322.37i	−0.259673	−	0.149923i
$222$		0			0
$223$	−12064.6	−	20896.5i	−0.242607	−	0.420208i	0.718849	−	0.695166i	$-0.244669\pi$
$223$	−12064.6	−	20896.5i	−0.242607	−	0.420208i	−0.961456	+	0.274958i	$0.911336\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−69285.3	+	40001.9i	−1.34459	+	0.776299i	−0.987477	−	0.157762i	$-0.949572\pi$
$227$	−69285.3	+	40001.9i	−1.34459	+	0.776299i	−0.357112	+	0.934061i	$0.616239\pi$
$228$		0			0
$229$	−7932.10	+	13738.8i	−0.151258	+	0.261986i	−0.931690	−	0.363254i	$-0.881666\pi$
$229$	−7932.10	+	13738.8i	−0.151258	+	0.261986i	0.780432	+	0.625240i	$0.214999\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	44937.9i		−	0.827753i	−0.910333	−	0.413876i	$-0.864175\pi$
$233$		−	44937.9i		−	0.827753i	0.910333	−	0.413876i	$-0.135825\pi$
$234$		0			0
$235$	−106810.			−1.93408
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4829.94	−	2788.57i	−0.0845562	−	0.0488186i	0.457126	−	0.889402i	$-0.348879\pi$
$239$	−4829.94	−	2788.57i	−0.0845562	−	0.0488186i	−0.541682	+	0.840584i	$0.682212\pi$
$240$		0			0
$241$	23957.3	+	41495.2i	0.412480	+	0.714437i	0.995160	−	0.0982648i	$-0.0313292\pi$
$241$	23957.3	+	41495.2i	0.412480	+	0.714437i	−0.582680	+	0.812702i	$0.697996\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−37490.8	+	21645.3i	−0.624586	+	0.360605i
$246$		0			0
$247$	42186.5	−	73069.1i	0.691479	−	1.19768i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			79435.9i			1.26087i	0.776243	+	0.630434i	$0.217123\pi$
$251$			79435.9i			1.26087i	−0.776243	+	0.630434i	$0.782877\pi$
$252$		0			0
$253$	9065.58			0.141630
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18155.6	−	10482.2i	−0.274881	−	0.158703i	0.356223	−	0.934401i	$-0.384064\pi$
$257$	−18155.6	−	10482.2i	−0.274881	−	0.158703i	−0.631104	+	0.775698i	$0.717398\pi$
$258$		0			0
$259$	6009.27	+	10408.4i	0.0895822	+	0.155161i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	109171.	−	63030.1i	1.57833	−	0.911248i	0.583235	−	0.812303i	$-0.301787\pi$
$263$	109171.	−	63030.1i	1.57833	−	0.911248i	0.995093	−	0.0989448i	$-0.0315467\pi$
$264$		0			0
$265$	−89011.3	+	154172.i	−1.26752	+	2.19540i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			4162.75i			0.0575276i	0.999586	+	0.0287638i	$0.00915706\pi$
$269$			4162.75i			0.0575276i	−0.999586	+	0.0287638i	$0.990843\pi$
$270$		0			0
$271$	−1002.99			−0.0136571			−0.00682853	−	0.999977i	$-0.502174\pi$
$271$	−1002.99			−0.0136571			−0.00682853	+	0.999977i	$0.502174\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	61014.0	+	35226.4i	0.806796	+	0.465804i
$276$		0			0
$277$	20142.6	+	34888.1i	0.262517	+	0.454692i	0.966910	−	0.255118i	$-0.0821142\pi$
$277$	20142.6	+	34888.1i	0.262517	+	0.454692i	−0.704393	+	0.709810i	$0.748781\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	91512.8	−	52834.9i	1.15896	−	0.669127i	0.207907	−	0.978149i	$-0.433335\pi$
$281$	91512.8	−	52834.9i	1.15896	−	0.669127i	0.951055	+	0.309022i	$0.100002\pi$
$282$		0			0
$283$	14942.1	−	25880.4i	0.186568	−	0.323146i	−0.757535	−	0.652794i	$-0.773597\pi$
$283$	14942.1	−	25880.4i	0.186568	−	0.323146i	0.944104	+	0.329648i	$0.106930\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			88580.2i			1.07541i
$288$		0			0
$289$	80272.1			0.961100
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−95447.4	−	55106.6i	−1.11181	−	0.641902i	−0.172510	−	0.985008i	$-0.555188\pi$
$293$	−95447.4	−	55106.6i	−1.11181	−	0.641902i	−0.939297	+	0.343106i	$0.888521\pi$
$294$		0			0
$295$	25829.3	+	44737.6i	0.296803	+	0.514078i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−14633.5	+	8448.65i	−0.163684	+	0.0945028i
$300$		0			0
$301$	14374.8	−	24897.9i	0.158660	−	0.274808i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	243687.i		−	2.61959i
$306$		0			0
$307$	−172215.			−1.82723			−0.913616	−	0.406579i	$-0.866722\pi$
$307$	−172215.			−1.82723			−0.913616	+	0.406579i	$0.866722\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	38056.1	+	21971.7i	0.393462	+	0.227166i	0.683659	−	0.729801i	$-0.260387\pi$
$311$	38056.1	+	21971.7i	0.393462	+	0.227166i	−0.290197	+	0.956967i	$0.593721\pi$
$312$		0			0
$313$	27275.1	+	47241.9i	0.278406	+	0.482213i	0.970989	−	0.239125i	$-0.0768606\pi$
$313$	27275.1	+	47241.9i	0.278406	+	0.482213i	−0.692583	+	0.721338i	$0.743527\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	146541.	−	84605.7i	1.45828	−	0.841940i	0.459356	−	0.888252i	$-0.348080\pi$
$317$	146541.	−	84605.7i	1.45828	−	0.841940i	0.998927	+	0.0463126i	$0.0147470\pi$
$318$		0			0
$319$	31863.2	−	55188.8i	0.313118	−	0.542337i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	18718.1i		−	0.179414i
$324$		0			0
$325$	−131317.			−1.24324
$326$		0			0
$327$		0			0
$328$		0			0
$329$	91704.2	+	52945.5i	0.847223	+	0.489144i
$330$		0			0
$331$	−41266.0	−	71474.8i	−0.376648	−	0.652374i	0.613924	−	0.789365i	$-0.289590\pi$
$331$	−41266.0	−	71474.8i	−0.376648	−	0.652374i	−0.990572	+	0.136991i	$0.956257\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17225.0	+	9944.87i	−0.153486	+	0.0886155i
$336$		0			0
$337$	72007.4	−	124720.i	0.634041	−	1.09819i	−0.352677	−	0.935745i	$-0.614729\pi$
$337$	72007.4	−	124720.i	0.634041	−	1.09819i	0.986718	−	0.162445i	$-0.0519381\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	252674.i		−	2.17296i
$342$		0			0
$343$	123151.			1.04676
$344$		0			0
$345$		0			0
$346$		0			0
$347$	92627.8	+	53478.7i	0.769277	+	0.444142i	0.832616	−	0.553850i	$-0.186842\pi$
$347$	92627.8	+	53478.7i	0.769277	+	0.444142i	−0.0633399	+	0.997992i	$0.520175\pi$
$348$		0			0
$349$	30438.7	+	52721.4i	0.249905	+	0.432849i	0.963499	−	0.267711i	$-0.0862672\pi$
$349$	30438.7	+	52721.4i	0.249905	+	0.432849i	−0.713594	+	0.700559i	$0.752934\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	17924.0	−	10348.4i	0.143842	−	0.0830471i	−0.426352	−	0.904557i	$-0.640201\pi$
$353$	17924.0	−	10348.4i	0.143842	−	0.0830471i	0.570194	+	0.821510i	$0.306868\pi$
$354$		0			0
$355$	79314.4	−	137377.i	0.629355	−	1.09007i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			123377.i			0.957290i	0.878008	+	0.478645i	$0.158872\pi$
$359$			123377.i			0.957290i	−0.878008	+	0.478645i	$0.841128\pi$
$360$		0			0
$361$	−22480.0			−0.172497
$362$		0			0
$363$		0			0
$364$		0			0
$365$	80243.1	+	46328.4i	0.602313	+	0.347746i
$366$		0			0
$367$	48990.2	+	84853.6i	0.363729	+	0.629996i	0.988571	−	0.150754i	$-0.0481703\pi$
$367$	48990.2	+	84853.6i	0.363729	+	0.629996i	−0.624843	+	0.780751i	$0.714837\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	152846.	−	88245.8i	1.11047	−	0.641130i
$372$		0			0
$373$	45516.0	−	78835.9i	0.327149	−	0.566639i	−0.654796	−	0.755806i	$-0.727245\pi$
$373$	45516.0	−	78835.9i	0.327149	−	0.566639i	0.981945	+	0.189167i	$0.0605787\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			118780.i			0.835716i
$378$		0			0
$379$	−85165.4			−0.592905			−0.296452	−	0.955048i	$-0.595804\pi$
$379$	−85165.4			−0.592905			−0.296452	+	0.955048i	$0.595804\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	173292.	+	100050.i	1.18135	+	0.682055i	0.956327	−	0.292297i	$-0.0944197\pi$
$383$	173292.	+	100050.i	1.18135	+	0.682055i	0.225027	+	0.974353i	$0.427753\pi$
$384$		0			0
$385$	−77629.4	−	134458.i	−0.523727	−	0.907122i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	133789.	−	77243.4i	0.884143	−	0.510460i	0.0121210	−	0.999927i	$-0.496142\pi$
$389$	133789.	−	77243.4i	0.884143	−	0.510460i	0.872022	+	0.489466i	$0.162808\pi$
$390$		0			0
$391$	−1874.33	+	3246.44i	−0.0122601	+	0.0212350i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	201126.i		−	1.28906i
$396$		0			0
$397$	154482.			0.980162			0.490081	−	0.871677i	$-0.336967\pi$
$397$	154482.			0.980162			0.490081	+	0.871677i	$0.336967\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−172894.	−	99820.5i	−1.07521	−	0.620771i	−0.145607	−	0.989343i	$-0.546513\pi$
$401$	−172894.	−	99820.5i	−1.07521	−	0.620771i	−0.929599	+	0.368572i	$0.879847\pi$
$402$		0			0
$403$	235479.	+	407862.i	1.44992	+	2.51133i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	42935.2	−	24788.7i	0.259194	−	0.149646i
$408$		0			0
$409$	−5038.24	+	8726.49i	−0.0301184	+	0.0521667i	−0.880692	−	0.473690i	$-0.842922\pi$
$409$	−5038.24	+	8726.49i	−0.0301184	+	0.0521667i	0.850573	+	0.525857i	$0.176255\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	51214.3i		−	0.300255i
$414$		0			0
$415$	−402412.			−2.33655
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−236663.	−	136638.i	−1.34804	−	0.778292i	−0.360069	−	0.932926i	$-0.617247\pi$
$419$	−236663.	−	136638.i	−1.34804	−	0.778292i	−0.987972	+	0.154634i	$0.950580\pi$
$420$		0			0
$421$	−100883.	−	174734.i	−0.569183	−	0.985854i	−0.996647	−	0.0818216i	$-0.973926\pi$
$421$	−100883.	−	174734.i	−0.569183	−	0.985854i	0.427464	−	0.904032i	$-0.359407\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−25229.6	+	14566.3i	−0.139679	+	0.0806439i
$426$		0			0
$427$	−120796.	+	209224.i	−0.662515	+	1.14751i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			44036.2i			0.237058i	0.992951	+	0.118529i	$0.0378179\pi$
$431$			44036.2i			0.237058i	−0.992951	+	0.118529i	$0.962182\pi$
$432$		0			0
$433$	−139788.			−0.745580			−0.372790	−	0.927916i	$-0.621599\pi$
$433$	−139788.			−0.745580			−0.372790	+	0.927916i	$0.621599\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−18703.7	−	10798.6i	−0.0979412	−	0.0565464i
$438$		0			0
$439$	23825.0	+	41266.1i	0.123624	+	0.214124i	0.921194	−	0.389103i	$-0.127215\pi$
$439$	23825.0	+	41266.1i	0.123624	+	0.214124i	−0.797570	+	0.603226i	$0.793882\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−110727.	+	63928.0i	−0.564215	+	0.325749i	−0.754835	−	0.655914i	$-0.772283\pi$
$443$	−110727.	+	63928.0i	−0.564215	+	0.325749i	0.190621	+	0.981664i	$0.438950\pi$
$444$		0			0
$445$	−157404.	+	272631.i	−0.794867	+	1.37675i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			77490.5i			0.384376i	0.981358	+	0.192188i	$0.0615583\pi$
$449$			77490.5i			0.384376i	−0.981358	+	0.192188i	$0.938442\pi$
$450$		0			0
$451$	365400.			1.79645
$452$		0			0
$453$		0			0
$454$		0			0
$455$	250616.	+	144693.i	1.21056	+	0.698916i
$456$		0			0
$457$	−48887.2	−	84675.2i	−0.234079	−	0.405437i	0.724925	−	0.688827i	$-0.241874\pi$
$457$	−48887.2	−	84675.2i	−0.234079	−	0.405437i	−0.959005	+	0.283390i	$0.908541\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	171170.	−	98825.2i	0.805427	−	0.465014i	−0.0399380	−	0.999202i	$-0.512716\pi$
$461$	171170.	−	98825.2i	0.805427	−	0.465014i	0.845365	+	0.534188i	$0.179383\pi$
$462$		0			0
$463$	38041.2	−	65889.3i	0.177457	−	0.307364i	−0.763552	−	0.645746i	$-0.776546\pi$
$463$	38041.2	−	65889.3i	0.177457	−	0.307364i	0.941009	+	0.338382i	$0.109880\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			78375.5i			0.359374i	0.983724	+	0.179687i	$0.0575085\pi$
$467$			78375.5i			0.359374i	−0.983724	+	0.179687i	$0.942492\pi$
$468$		0			0
$469$	19718.7			0.0896462
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−102706.	−	59297.1i	−0.459062	−	0.265040i
$474$		0			0
$475$	−83921.1	−	145356.i	−0.371949	−	0.644235i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−178440.	+	103022.i	−0.777714	+	0.449014i	−0.835620	−	0.549309i	$-0.814891\pi$
$479$	−178440.	+	103022.i	−0.777714	+	0.449014i	0.0579053	+	0.998322i	$0.481558\pi$
$480$		0			0
$481$	−46203.5	+	80026.8i	−0.199703	+	0.345896i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			107781.i			0.458206i
$486$		0			0
$487$	200708.			0.846264			0.423132	−	0.906068i	$-0.360931\pi$
$487$	200708.			0.846264			0.423132	+	0.906068i	$0.360931\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−181227.	−	104631.i	−0.751726	−	0.434009i	0.0745913	−	0.997214i	$-0.476235\pi$
$491$	−181227.	−	104631.i	−0.751726	−	0.434009i	−0.826317	+	0.563205i	$0.809568\pi$
$492$		0			0
$493$	13175.6	+	22820.8i	0.0542097	+	0.0938939i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−136195.	+	78632.3i	−0.551377	+	0.318338i
$498$		0			0
$499$	177056.	−	306670.i	0.711066	−	1.23160i	−0.253391	−	0.967364i	$-0.581546\pi$
$499$	177056.	−	306670.i	0.711066	−	1.23160i	0.964457	−	0.264239i	$-0.0851206\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			99176.2i			0.391987i	0.980605	+	0.195993i	$0.0627931\pi$
$503$			99176.2i			0.391987i	−0.980605	+	0.195993i	$0.937207\pi$
$504$		0			0
$505$	166413.			0.652535
$506$		0			0
$507$		0			0
$508$		0			0
$509$	48435.5	+	27964.2i	0.186951	+	0.107936i	0.590554	−	0.806998i	$-0.298909\pi$
$509$	48435.5	+	27964.2i	0.186951	+	0.107936i	−0.403603	+	0.914934i	$0.632242\pi$
$510$		0			0
$511$	−45930.0	−	79553.0i	−0.175895	−	0.304660i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−456112.	+	263336.i	−1.71972	+	0.992879i
$516$		0			0
$517$	218404.	−	378287.i	0.817108	−	1.41527i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			217215.i			0.800229i	0.916465	+	0.400114i	$0.131030\pi$
$521$			217215.i			0.800229i	−0.916465	+	0.400114i	$0.868970\pi$
$522$		0			0
$523$	104089.			0.380542			0.190271	−	0.981732i	$-0.439063\pi$
$523$	104089.			0.380542			0.190271	+	0.981732i	$0.439063\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	90484.1	+	52241.0i	0.325800	+	0.188101i
$528$		0			0
$529$	−137758.	−	238604.i	−0.492272	−	0.852640i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−589822.	+	340534.i	−2.07619	+	1.19869i
$534$		0			0
$535$	−269216.	+	466296.i	−0.940575	+	1.62912i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	177041.i		−	0.609392i
$540$		0			0
$541$	−116397.			−0.397693			−0.198846	−	0.980031i	$-0.563720\pi$
$541$	−116397.			−0.397693			−0.198846	+	0.980031i	$0.563720\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−121538.	−	70169.8i	−0.409183	−	0.236242i
$546$		0			0
$547$	−199324.	−	345240.i	−0.666171	−	1.15384i	−0.978966	−	0.204022i	$-0.934599\pi$
$547$	−199324.	−	345240.i	−0.666171	−	1.15384i	0.312795	−	0.949821i	$-0.398735\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−131478.	+	75908.8i	−0.433062	+	0.250028i
$552$		0			0
$553$	−99698.2	+	172682.i	−0.326015	+	0.564674i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			203998.i			0.657530i	0.944412	+	0.328765i	$0.106632\pi$
$557$			203998.i			0.657530i	−0.944412	+	0.328765i	$0.893368\pi$
$558$		0			0
$559$	221047.			0.707394
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−328609.	−	189722.i	−1.03672	−	0.598551i	−0.117819	−	0.993035i	$-0.537590\pi$
$563$	−328609.	−	189722.i	−1.03672	−	0.598551i	−0.918903	+	0.394484i	$0.870923\pi$
$564$		0			0
$565$	232969.	+	403514.i	0.729796	+	1.26404i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	79665.0	−	45994.6i	0.246061	−	0.142063i	−0.371898	−	0.928274i	$-0.621293\pi$
$569$	79665.0	−	45994.6i	0.246061	−	0.142063i	0.617959	+	0.786210i	$0.287960\pi$
$570$		0			0
$571$	39717.0	−	68791.9i	0.121816	−	0.210992i	−0.798668	−	0.601772i	$-0.794462\pi$
$571$	39717.0	−	68791.9i	0.121816	−	0.210992i	0.920484	+	0.390781i	$0.127795\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			33613.6i			0.101667i
$576$		0			0
$577$	41340.4			0.124172			0.0620860	−	0.998071i	$-0.480225\pi$
$577$	41340.4			0.124172			0.0620860	+	0.998071i	$0.480225\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	345502.	+	199476.i	1.02352	+	0.590932i
$582$		0			0
$583$	−364021.	−	630502.i	−1.07100	−	1.85502i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	215802.	−	124593.i	0.626295	−	0.361592i	−0.153021	−	0.988223i	$-0.548900\pi$
$587$	215802.	−	124593.i	0.626295	−	0.361592i	0.779316	+	0.626631i	$0.215567\pi$
$588$		0			0
$589$	−300977.	+	521308.i	−0.867567	+	1.50267i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	37868.4i		−	0.107688i	−0.998549	−	0.0538440i	$-0.982853\pi$
$593$		−	37868.4i		−	0.107688i	0.998549	−	0.0538440i	$-0.0171474\pi$
$594$		0			0
$595$	64200.3			0.181344
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−251228.	−	145046.i	−0.700186	−	0.404253i	0.107231	−	0.994234i	$-0.465802\pi$
$599$	−251228.	−	145046.i	−0.700186	−	0.404253i	−0.807417	+	0.589981i	$0.799135\pi$
$600$		0			0
$601$	−4261.24	−	7380.69i	−0.0117974	−	0.0204337i	0.860066	−	0.510182i	$-0.170422\pi$
$601$	−4261.24	−	7380.69i	−0.0117974	−	0.0204337i	−0.871864	+	0.489748i	$0.837089\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−127274.	+	73481.4i	−0.347718	+	0.200755i
$606$		0			0
$607$	−347609.	+	602076.i	−0.943439	+	1.63408i	−0.184591	+	0.982815i	$0.559096\pi$
$607$	−347609.	+	602076.i	−0.943439	+	1.63408i	−0.758848	+	0.651268i	$0.774237\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			814165.i			2.18087i
$612$		0			0
$613$	457316.			1.21701			0.608506	−	0.793549i	$-0.291769\pi$
$613$	457316.			1.21701			0.608506	+	0.793549i	$0.291769\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−171987.	−	99296.6i	−0.451778	−	0.260834i	0.256803	−	0.966464i	$-0.417331\pi$
$617$	−171987.	−	99296.6i	−0.451778	−	0.260834i	−0.708581	+	0.705630i	$0.750664\pi$
$618$		0			0
$619$	346327.	+	599856.i	0.903868	+	1.56554i	0.822430	+	0.568866i	$0.192618\pi$
$619$	346327.	+	599856.i	0.903868	+	1.56554i	0.0814376	+	0.996678i	$0.474049\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	270286.	−	156050.i	0.696382	−	0.402056i
$624$		0			0
$625$	224419.	−	388705.i	0.574512	−	0.995085i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			20500.5i			0.0518158i
$630$		0			0
$631$	−314958.			−0.791032			−0.395516	−	0.918459i	$-0.629434\pi$
$631$	−314958.			−0.791032			−0.395516	+	0.918459i	$0.629434\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−573855.	−	331315.i	−1.42316	−	0.821663i
$636$		0			0
$637$	164993.	+	285777.i	0.406619	+	0.704284i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−598138.	+	345335.i	−1.45575	+	0.840475i	−0.998798	−	0.0490188i	$-0.984391\pi$
$641$	−598138.	+	345335.i	−1.45575	+	0.840475i	−0.456947	+	0.889494i	$0.651057\pi$
$642$		0			0
$643$	−31620.3	+	54768.0i	−0.0764794	+	0.132466i	−0.901729	−	0.432302i	$-0.857701\pi$
$643$	−31620.3	+	54768.0i	−0.0764794	+	0.132466i	0.825249	+	0.564769i	$0.191035\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	597558.i		−	1.42748i	−0.700409	−	0.713742i	$-0.746999\pi$
$647$		−	597558.i		−	1.42748i	0.700409	−	0.713742i	$-0.253001\pi$
$648$		0			0
$649$	−211263.			−0.501572
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−492287.	−	284222.i	−1.15449	−	0.666548i	−0.204516	−	0.978863i	$-0.565562\pi$
$653$	−492287.	−	284222.i	−1.15449	−	0.666548i	−0.949978	+	0.312315i	$0.898895\pi$
$654$		0			0
$655$	−350840.	−	607673.i	−0.817761	−	1.41640i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	362129.	−	209075.i	0.833859	−	0.481429i	−0.0213128	−	0.999773i	$-0.506785\pi$
$659$	362129.	−	209075.i	0.833859	−	0.481429i	0.855172	+	0.518344i	$0.173451\pi$
$660$		0			0
$661$	135403.	−	234525.i	0.309902	−	0.536766i	−0.668439	−	0.743767i	$-0.733037\pi$
$661$	135403.	−	234525.i	0.309902	−	0.536766i	0.978341	+	0.207001i	$0.0663703\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			369878.i			0.836403i
$666$		0			0
$667$	30404.4			0.0683415
$668$		0			0
$669$		0			0
$670$		0			0
$671$	863066.	+	498291.i	1.91690	+	1.10672i
$672$		0			0
$673$	268438.	+	464949.i	0.592672	+	1.02654i	0.993871	+	0.110547i	$0.0352601\pi$
$673$	268438.	+	464949.i	0.592672	+	1.02654i	−0.401199	+	0.915991i	$0.631407\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−225624.	+	130264.i	−0.492275	+	0.284215i	−0.725518	−	0.688204i	$-0.758400\pi$
$677$	−225624.	+	130264.i	−0.492275	+	0.284215i	0.233243	+	0.972418i	$0.425066\pi$
$678$		0			0
$679$	53427.2	−	92538.7i	0.115884	−	0.200717i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	427175.i		−	0.915723i	−0.889023	−	0.457862i	$-0.848615\pi$
$683$		−	427175.i		−	0.915723i	0.889023	−	0.457862i	$-0.151385\pi$
$684$		0			0
$685$	−578026.			−1.23187
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.17519e6	+	678496.i	2.47554	+	1.42925i
$690$		0			0
$691$	102795.	+	178046.i	0.215285	+	0.372885i	0.953361	−	0.301833i	$-0.0975985\pi$
$691$	102795.	+	178046.i	0.215285	+	0.372885i	−0.738075	+	0.674718i	$0.764265\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	464620.	−	268248.i	0.961896	−	0.555351i
$696$		0			0
$697$	−75547.3	+	130852.i	−0.155508	+	0.269348i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			176791.i			0.359769i	0.983688	+	0.179884i	$0.0575724\pi$
$701$			176791.i			0.359769i	−0.983688	+	0.179884i	$0.942428\pi$
$702$		0			0
$703$	−118110.			−0.238987
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−142878.	−	82490.8i	−0.285843	−	0.165031i
$708$		0			0
$709$	−376584.	−	652262.i	−0.749150	−	1.29757i	−0.948230	−	0.317583i	$-0.897129\pi$
$709$	−376584.	−	652262.i	−0.749150	−	1.29757i	0.199080	−	0.979983i	$-0.436205\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	104402.	−	60276.4i	0.205366	−	0.118568i
$714$		0			0
$715$	596870.	−	1.03381e6i	1.16753	−	2.02222i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	653122.i		−	1.26339i	−0.775218	−	0.631694i	$-0.782360\pi$
$719$		−	653122.i		−	1.26339i	0.775218	−	0.631694i	$-0.217640\pi$
$720$		0			0
$721$	522143.			1.00443
$722$		0			0
$723$		0			0
$724$		0			0
$725$	204630.	+	118143.i	0.389309	+	0.224767i
$726$		0			0
$727$	−187285.	−	324386.i	−0.354351	−	0.613753i	0.632656	−	0.774433i	$-0.281965\pi$
$727$	−187285.	−	324386.i	−0.354351	−	0.613753i	−0.987007	+	0.160680i	$0.948631\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	42469.3	−	24519.6i	0.0794767	−	0.0458859i
$732$		0			0
$733$	349362.	−	605114.i	0.650232	−	1.12623i	−0.332835	−	0.942985i	$-0.608005\pi$
$733$	349362.	−	605114.i	0.650232	−	1.12623i	0.983066	−	0.183249i	$-0.0586616\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	81341.0i		−	0.149753i
$738$		0			0
$739$	4561.08			0.00835177			0.00417588	−	0.999991i	$-0.498671\pi$
$739$	4561.08			0.00835177			0.00417588	+	0.999991i	$0.498671\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−264662.	−	152803.i	−0.479417	−	0.276792i	0.240756	−	0.970586i	$-0.422605\pi$
$743$	−264662.	−	152803.i	−0.479417	−	0.276792i	−0.720174	+	0.693794i	$0.755938\pi$
$744$		0			0
$745$	445893.	+	772309.i	0.803374	+	1.39149i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	462285.	−	266901.i	0.824037	−	0.475758i
$750$		0			0
$751$	−362582.	+	628011.i	−0.642875	+	1.11349i	0.341912	+	0.939732i	$0.388925\pi$
$751$	−362582.	+	628011.i	−0.642875	+	1.11349i	−0.984788	+	0.173761i	$0.944408\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	902022.i		−	1.58243i
$756$		0			0
$757$	−273369.			−0.477043			−0.238521	−	0.971137i	$-0.576663\pi$
$757$	−273369.			−0.477043			−0.238521	+	0.971137i	$0.576663\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	425109.	+	245437.i	0.734060	+	0.423810i	0.819906	−	0.572499i	$-0.194026\pi$
$761$	425109.	+	245437.i	0.734060	+	0.423810i	−0.0858458	+	0.996308i	$0.527359\pi$
$762$		0			0
$763$	69566.3	+	120492.i	0.119495	+	0.206971i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	341016.	−	196886.i	0.579675	−	0.334675i
$768$		0			0
$769$	330326.	−	572141.i	0.558586	−	0.967499i	−0.439029	−	0.898473i	$-0.644677\pi$
$769$	330326.	−	572141.i	0.558586	−	0.967499i	0.997615	−	0.0690264i	$-0.0219893\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			69649.7i			0.116563i	0.998300	+	0.0582814i	$0.0185621\pi$
$773$			69649.7i			0.116563i	−0.998300	+	0.0582814i	$0.981438\pi$
$774$		0			0
$775$	936873.			1.55983
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−753879.	−	435252.i	−1.24230	−	0.717242i
$780$		0			0
$781$	324364.	+	561815.i	0.531778	+	0.921067i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.20106e6	−	693434.i	1.94906	−	1.12529i
$786$		0			0
$787$	138523.	−	239929.i	0.223652	−	0.387376i	−0.732262	−	0.681023i	$-0.761536\pi$
$787$	138523.	−	239929.i	0.223652	−	0.387376i	0.955914	+	0.293646i	$0.0948688\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	461931.i		−	0.738285i
$792$		0			0
$793$	−1.85753e6			−2.95385
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13243.6	−	7646.20i	−0.0208492	−	0.0120373i	0.489539	−	0.871981i	$-0.337165\pi$
$797$	−13243.6	−	7646.20i	−0.0208492	−	0.0120373i	−0.510388	+	0.859944i	$0.670498\pi$
$798$		0			0
$799$	90311.1	+	156423.i	0.141465	+	0.245024i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−328162.	+	189465.i	−0.508929	+	0.293830i
$804$		0			0
$805$	37037.6	−	64151.0i	0.0571546	−	0.0989946i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			264170.i			0.403632i	0.979423	+	0.201816i	$0.0646843\pi$
$809$			264170.i			0.403632i	−0.979423	+	0.201816i	$0.935316\pi$
$810$		0			0
$811$	−303869.			−0.462002			−0.231001	−	0.972953i	$-0.574200\pi$
$811$	−303869.			−0.462002			−0.231001	+	0.972953i	$0.574200\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	532019.	+	307162.i	0.800963	+	0.462436i
$816$		0			0
$817$	141265.	+	244679.i	0.211637	+	0.366566i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	716783.	−	413835.i	1.06341	−	0.613961i	0.137037	−	0.990566i	$-0.456242\pi$
$821$	716783.	−	413835.i	1.06341	−	0.613961i	0.926374	+	0.376605i	$0.122909\pi$
$822$		0			0
$823$	−337494.	+	584557.i	−0.498272	+	0.863033i	−0.999998	−	0.00199378i	$-0.999365\pi$
$823$	−337494.	+	584557.i	−0.498272	+	0.863033i	0.501726	+	0.865027i	$0.332699\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1.05229e6i		−	1.53860i	−0.638887	−	0.769301i	$-0.720605\pi$
$827$		−	1.05229e6i		−	1.53860i	0.638887	−	0.769301i	$-0.279395\pi$
$828$		0			0
$829$	−622368.			−0.905603			−0.452802	−	0.891611i	$-0.649575\pi$
$829$	−622368.			−0.905603			−0.452802	+	0.891611i	$0.649575\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	63399.5	+	36603.7i	0.0913683	+	0.0527515i
$834$		0			0
$835$	483087.	+	836732.i	0.692872	+	1.20009i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	566354.	−	326985.i	0.804571	−	0.464519i	−0.0404961	−	0.999180i	$-0.512894\pi$
$839$	566354.	−	326985.i	0.804571	−	0.464519i	0.845067	+	0.534660i	$0.179560\pi$
$840$		0			0
$841$	−246777.	+	427430.i	−0.348909	+	0.604328i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			1.26233e6i			1.76790i
$846$		0			0
$847$	145699.			0.203090
$848$		0			0
$849$		0			0
$850$		0			0
$851$	20484.7	+	11826.9i	0.0282860	+	0.0163309i
$852$		0			0
$853$	−526137.	−	911296.i	−0.723104	−	1.25245i	−0.959750	−	0.280857i	$-0.909381\pi$
$853$	−526137.	−	911296.i	−0.723104	−	1.25245i	0.236646	−	0.971596i	$-0.423952\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	291064.	−	168046.i	0.396303	−	0.228806i	−0.288584	−	0.957454i	$-0.593185\pi$
$857$	291064.	−	168046.i	0.396303	−	0.228806i	0.684888	+	0.728649i	$0.259851\pi$
$858$		0			0
$859$	−464892.	+	805216.i	−0.630036	+	1.09126i	0.357507	+	0.933910i	$0.383627\pi$
$859$	−464892.	+	805216.i	−0.630036	+	1.09126i	−0.987544	+	0.157345i	$0.949707\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			760939.i			1.02171i	0.859666	+	0.510856i	$0.170671\pi$
$863$			760939.i			1.02171i	−0.859666	+	0.510856i	$0.829329\pi$
$864$		0			0
$865$	183913.			0.245799
$866$		0			0
$867$		0			0
$868$		0			0
$869$	712328.	+	411263.i	0.943280	+	0.544603i
$870$		0			0
$871$	75805.6	+	131299.i	0.0999229	+	0.173072i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−111098.	+	64142.5i	−0.145108	+	0.0837779i
$876$		0			0
$877$	25097.0	−	43469.2i	0.0326304	−	0.0565174i	−0.849249	−	0.527992i	$-0.822945\pi$
$877$	25097.0	−	43469.2i	0.0326304	−	0.0565174i	0.881879	+	0.471475i	$0.156278\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	421710.i		−	0.543328i	−0.962392	−	0.271664i	$-0.912426\pi$
$881$		−	421710.i		−	0.543328i	0.962392	−	0.271664i	$-0.0875740\pi$
$882$		0			0
$883$	−712187.			−0.913424			−0.456712	−	0.889615i	$-0.650973\pi$
$883$	−712187.			−0.913424			−0.456712	+	0.889615i	$0.650973\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	937657.	+	541356.i	1.19178	+	0.688075i	0.958710	−	0.284385i	$-0.0917892\pi$
$887$	937657.	+	541356.i	1.19178	+	0.688075i	0.233071	+	0.972460i	$0.425123\pi$
$888$		0			0
$889$	328466.	+	568919.i	0.415611	+	0.719859i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−901205.	+	520311.i	−1.13011	+	0.652469i
$894$		0			0
$895$	702437.	−	1.21666e6i	0.876922	−	1.51887i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	847426.i		−	1.04853i
$900$		0			0
$901$	301048.			0.370840
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.43920e6	−	830924.i	−1.75721	−	1.01453i
$906$		0			0
$907$	36755.6	+	63662.5i	0.0446795	+	0.0773872i	0.887500	−	0.460807i	$-0.152440\pi$
$907$	36755.6	+	63662.5i	0.0446795	+	0.0773872i	−0.842821	+	0.538194i	$0.819107\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.03923e6	−	599998.i	1.25220	−	0.722958i	0.280655	−	0.959809i	$-0.409448\pi$
$911$	1.03923e6	−	599998.i	1.25220	−	0.722958i	0.971546	+	0.236851i	$0.0761152\pi$
$912$		0			0
$913$	822852.	−	1.42522e6i	0.987143	−	1.70978i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			695645.i			0.827274i
$918$		0			0
$919$	311383.			0.368692			0.184346	−	0.982861i	$-0.440983\pi$
$919$	311383.			0.368692			0.184346	+	0.982861i	$0.440983\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.04716e6	−	604581.i	−1.22917	−	0.709661i
$924$		0			0
$925$	91912.1	+	159196.i	0.107421	+	0.186059i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.13249e6	+	653842.i	−1.31221	+	0.757602i	−0.982461	−	0.186468i	$-0.940296\pi$
$929$	−1.13249e6	+	653842.i	−1.31221	+	0.757602i	−0.329745	+	0.944070i	$0.606963\pi$
$930$		0			0
$931$	−210885.	+	365264.i	−0.243303	+	0.421413i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	264831.i		−	0.302932i
$936$		0			0
$937$	860787.			0.980430			0.490215	−	0.871601i	$-0.336918\pi$
$937$	860787.			0.980430			0.490215	+	0.871601i	$0.336918\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	225220.	+	130031.i	0.254347	+	0.146848i	0.621753	−	0.783213i	$-0.286421\pi$
$941$	225220.	+	130031.i	0.254347	+	0.146848i	−0.367406	+	0.930061i	$0.619754\pi$
$942$		0			0
$943$	87167.6	+	150979.i	0.0980238	+	0.169782i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−546051.	+	315263.i	−0.608882	+	0.351538i	−0.772528	−	0.634981i	$-0.781008\pi$
$947$	−546051.	+	315263.i	−0.608882	+	0.351538i	0.163646	+	0.986519i	$0.447675\pi$
$948$		0			0
$949$	353142.	−	611660.i	0.392118	−	0.679169i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	619620.i		−	0.682244i	−0.940019	−	0.341122i	$-0.889193\pi$
$953$		−	619620.i		−	0.682244i	0.940019	−	0.341122i	$-0.110807\pi$
$954$		0			0
$955$	1.03213e6			1.13169
$956$		0			0
$957$		0			0
$958$		0			0
$959$	496280.	+	286527.i	0.539622	+	0.311551i
$960$		0			0
$961$	−1.21825e6	−	2.11008e6i	−1.31914	−	2.28482i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−285165.	+	164640.i	−0.306225	+	0.176799i
$966$		0			0
$967$	−567000.	+	982073.i	−0.606359	+	1.05025i	0.385476	+	0.922718i	$0.374037\pi$
$967$	−567000.	+	982073.i	−0.606359	+	1.05025i	−0.991835	+	0.127527i	$0.959296\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	510281.i		−	0.541216i	−0.962690	−	0.270608i	$-0.912775\pi$
$971$		−	510281.i		−	0.541216i	0.962690	−	0.270608i	$-0.0872247\pi$
$972$		0			0
$973$	−531883.			−0.561811
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.42896e6	+	825012.i	1.49703	+	0.864313i	0.999994	−	0.00341413i	$-0.00108675\pi$
$977$	1.42896e6	+	825012.i	1.49703	+	0.864313i	0.497040	+	0.867727i	$0.334420\pi$
$978$		0			0
$979$	−643717.	−	1.11495e6i	−0.671629	−	1.16330i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−418594.	+	241676.i	−0.433198	+	0.250107i	−0.700708	−	0.713448i	$-0.747132\pi$
$983$	−418594.	+	241676.i	−0.433198	+	0.250107i	0.267510	+	0.963555i	$0.413799\pi$
$984$		0			0
$985$	−333511.	+	577658.i	−0.343746	+	0.595385i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	56582.2i		−	0.0578478i
$990$		0			0
$991$	1.47602e6			1.50295			0.751477	−	0.659759i	$-0.229342\pi$
$991$	1.47602e6			1.50295			0.751477	+	0.659759i	$0.229342\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.55395e6	−	897176.i	−1.56961	−	0.906216i
$996$		0			0
$997$	−11327.9	−	19620.6i	−0.0113962	−	0.0197388i	0.860271	−	0.509837i	$-0.170294\pi$
$997$	−11327.9	−	19620.6i	−0.0113962	−	0.0197388i	−0.871667	+	0.490098i	$0.836961\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.g.f.53.1		16
3.2	odd	2	inner	324.5.g.f.53.8		16
9.2	odd	6	inner	324.5.g.f.269.1		16
9.4	even	3		324.5.c.b.161.8	yes	8
9.5	odd	6		324.5.c.b.161.1	✓	8
9.7	even	3	inner	324.5.g.f.269.8		16
36.23	even	6		1296.5.e.f.161.1		8
36.31	odd	6		1296.5.e.f.161.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.5.c.b.161.1	✓	8	9.5	odd	6
324.5.c.b.161.8	yes	8	9.4	even	3
324.5.g.f.53.1		16	1.1	even	1	trivial
324.5.g.f.53.8		16	3.2	odd	2	inner
324.5.g.f.269.1		16	9.2	odd	6	inner
324.5.g.f.269.8		16	9.7	even	3	inner
1296.5.e.f.161.1		8	36.23	even	6
1296.5.e.f.161.8		8	36.31	odd	6

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

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.g (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-29.1904 - 16.8531i) q^{5} +(16.7081 + 28.9393i) q^{7} +(119.377 - 68.9223i) q^{11} +(-128.464 + 222.506i) q^{13} +56.9994i q^{17} -328.391 q^{19} +(56.9556 + 32.8833i) q^{23} +(255.552 + 442.629i) q^{25} +(400.370 - 231.153i) q^{29} +(916.519 - 1587.46i) q^{31} -1126.33i q^{35} +359.661 q^{37} +(2295.67 + 1325.41i) q^{41} +(-430.174 - 745.083i) q^{43} +(2744.30 - 1584.42i) q^{47} +(642.177 - 1112.28i) q^{49} -5281.61i q^{53} -4646.21 q^{55} +(-1327.28 - 766.308i) q^{59} +(3614.88 + 6261.16i) q^{61} +(7499.82 - 4330.02i) q^{65} +(295.046 - 511.035i) q^{67} +4706.23i q^{71} -2748.96 q^{73} +(3989.13 + 2303.12i) q^{77} +(2983.53 + 5167.62i) q^{79} +(10339.3 - 5969.42i) q^{83} +(960.615 - 1663.83i) q^{85} -9339.75i q^{89} -8585.57 q^{91} +(9585.87 + 5534.40i) q^{95} +(-1598.84 - 2769.27i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 52 q^{7} - 220 q^{13} - 616 q^{19} + 328 q^{25} + 1472 q^{31} + 2456 q^{37} - 388 q^{43} - 3432 q^{49} - 72 q^{55} - 1804 q^{61} + 2180 q^{67} - 9088 q^{73} + 5012 q^{79} + 5364 q^{85} - 3512 q^{91}+ \cdots + 2528 q^{97}+O(q^{100}) \) 16 * q - 52 * q^7 - 220 * q^13 - 616 * q^19 + 328 * q^25 + 1472 * q^31 + 2456 * q^37 - 388 * q^43 - 3432 * q^49 - 72 * q^55 - 1804 * q^61 + 2180 * q^67 - 9088 * q^73 + 5012 * q^79 + 5364 * q^85 - 3512 * q^91 + 2528 * q^97

Introduction

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