LMFDB - Embedded newform 324.5.g.d.53.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,5,Mod(53,324)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$33.4918680392$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	324.53
Dual form			324.5.g.d.269.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.79423 + 4.50000i) q^{5} +(-2.50000 - 4.33013i) q^{7} +O(q^{10})$	$q+(7.79423 + 4.50000i) q^{5} +(-2.50000 - 4.33013i) q^{7} +(-101.325 + 58.5000i) q^{11} +(17.0000 - 29.4449i) q^{13} -450.000i q^{17} -64.0000 q^{19} +(530.008 + 306.000i) q^{23} +(-272.000 - 471.118i) q^{25} +(-919.719 + 531.000i) q^{29} +(348.500 - 603.620i) q^{31} -45.0000i q^{35} -748.000 q^{37} +(592.361 + 342.000i) q^{41} +(-1309.00 - 2267.25i) q^{43} +(2291.50 - 1323.00i) q^{47} +(1188.00 - 2057.68i) q^{49} +1071.00i q^{53} -1053.00 q^{55} +(-5035.07 - 2907.00i) q^{59} +(-3202.00 - 5546.03i) q^{61} +(265.004 - 153.000i) q^{65} +(2609.00 - 4518.92i) q^{67} +6570.00i q^{71} -4519.00 q^{73} +(506.625 + 292.500i) q^{77} +(-3751.00 - 6496.92i) q^{79} +(4746.69 - 2740.50i) q^{83} +(2025.00 - 3507.40i) q^{85} -8874.00i q^{89} -170.000 q^{91} +(-498.831 - 288.000i) q^{95} +(-5285.50 - 9154.75i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{7}+O(q^{10}) $ 4 * q - 10 * q^7	$ 4 q - 10 q^{7} + 68 q^{13} - 256 q^{19} - 1088 q^{25} + 1394 q^{31} - 2992 q^{37} - 5236 q^{43} + 4752 q^{49} - 4212 q^{55} - 12808 q^{61} + 10436 q^{67} - 18076 q^{73} - 15004 q^{79} + 8100 q^{85} - 680 q^{91} - 21142 q^{97}+O(q^{100}) $ 4 * q - 10 * q^7 + 68 * q^13 - 256 * q^19 - 1088 * q^25 + 1394 * q^31 - 2992 * q^37 - 5236 * q^43 + 4752 * q^49 - 4212 * q^55 - 12808 * q^61 + 10436 * q^67 - 18076 * q^73 - 15004 * q^79 + 8100 * q^85 - 680 * q^91 - 21142 * q^97

Display 100 coefficients 10 coefficients

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
$5$	7.79423	+	4.50000i	0.311769	+	0.180000i	0.647718	−	0.761880i	$-0.275724\pi$
$5$	7.79423	+	4.50000i	0.311769	+	0.180000i	−0.335949	+	0.941880i	$0.609057\pi$
$6$		0			0
$7$	−2.50000	−	4.33013i	−0.0510204	−	0.0883699i	0.839387	−	0.543534i	$-0.182914\pi$
$7$	−2.50000	−	4.33013i	−0.0510204	−	0.0883699i	−0.890408	+	0.455164i	$0.849581\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−101.325	+	58.5000i	−0.837396	+	0.483471i	−0.856378	−	0.516349i	$-0.827291\pi$
$11$	−101.325	+	58.5000i	−0.837396	+	0.483471i	0.0189819	+	0.999820i	$0.493957\pi$
$12$		0			0
$13$	17.0000	−	29.4449i	0.100592	−	0.174230i	−0.811337	−	0.584579i	$-0.801260\pi$
$13$	17.0000	−	29.4449i	0.100592	−	0.174230i	0.911929	+	0.410349i	$0.134593\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	450.000i		−	1.55709i	−0.627587	−	0.778547i	$-0.715957\pi$
$17$		−	450.000i		−	1.55709i	0.627587	−	0.778547i	$-0.284043\pi$
$18$		0			0
$19$	−64.0000			−0.177285			−0.0886427	−	0.996063i	$-0.528253\pi$
$19$	−64.0000			−0.177285			−0.0886427	+	0.996063i	$0.528253\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	530.008	+	306.000i	1.00190	+	0.578450i	0.908811	−	0.417207i	$-0.136991\pi$
$23$	530.008	+	306.000i	1.00190	+	0.578450i	0.0930934	+	0.995657i	$0.470324\pi$
$24$		0			0
$25$	−272.000	−	471.118i	−0.435200	−	0.753789i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−919.719	+	531.000i	−1.09360	+	0.631391i	−0.934533	−	0.355876i	$-0.884182\pi$
$29$	−919.719	+	531.000i	−1.09360	+	0.631391i	−0.159069	+	0.987268i	$0.550849\pi$
$30$		0			0
$31$	348.500	−	603.620i	0.362643	−	0.628116i	−0.625752	−	0.780022i	$-0.715208\pi$
$31$	348.500	−	603.620i	0.362643	−	0.628116i	0.988395	+	0.151906i	$0.0485411\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	45.0000i		−	0.0367347i
$36$		0			0
$37$	−748.000			−0.546384			−0.273192	−	0.961959i	$-0.588079\pi$
$37$	−748.000			−0.546384			−0.273192	+	0.961959i	$0.588079\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	592.361	+	342.000i	0.352386	+	0.203450i	0.665736	−	0.746188i	$-0.268118\pi$
$41$	592.361	+	342.000i	0.352386	+	0.203450i	−0.313349	+	0.949638i	$0.601451\pi$
$42$		0			0
$43$	−1309.00	−	2267.25i	−0.707950	−	1.22621i	−0.965616	−	0.259972i	$-0.916287\pi$
$43$	−1309.00	−	2267.25i	−0.707950	−	1.22621i	0.257666	−	0.966234i	$-0.417047\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2291.50	−	1323.00i	1.03735	−	0.598914i	0.118267	−	0.992982i	$-0.462266\pi$
$47$	2291.50	−	1323.00i	1.03735	−	0.598914i	0.919081	+	0.394068i	$0.128933\pi$
$48$		0			0
$49$	1188.00	−	2057.68i	0.494794	−	0.857008i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			1071.00i			0.381274i	0.981661	+	0.190637i	$0.0610554\pi$
$53$			1071.00i			0.381274i	−0.981661	+	0.190637i	$0.938945\pi$
$54$		0			0
$55$	−1053.00			−0.348099
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5035.07	−	2907.00i	−1.44644	−	0.835105i	−0.448177	−	0.893945i	$-0.647926\pi$
$59$	−5035.07	−	2907.00i	−1.44644	−	0.835105i	−0.998267	+	0.0588402i	$0.981260\pi$
$60$		0			0
$61$	−3202.00	−	5546.03i	−0.860521	−	1.49047i	−0.871426	−	0.490526i	$-0.836805\pi$
$61$	−3202.00	−	5546.03i	−0.860521	−	1.49047i	0.0109051	−	0.999941i	$-0.496529\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	265.004	−	153.000i	0.0627228	−	0.0362130i
$66$		0			0
$67$	2609.00	−	4518.92i	0.581198	−	1.00667i	−0.414139	−	0.910214i	$-0.635917\pi$
$67$	2609.00	−	4518.92i	0.581198	−	1.00667i	0.995338	−	0.0964517i	$-0.0307493\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			6570.00i			1.30331i	0.758514	+	0.651656i	$0.225926\pi$
$71$			6570.00i			1.30331i	−0.758514	+	0.651656i	$0.774074\pi$
$72$		0			0
$73$	−4519.00			−0.848002			−0.424001	−	0.905662i	$-0.639375\pi$
$73$	−4519.00			−0.848002			−0.424001	+	0.905662i	$0.639375\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	506.625	+	292.500i	0.0854486	+	0.0493338i
$78$		0			0
$79$	−3751.00	−	6496.92i	−0.601025	−	1.04101i	−0.992666	−	0.120888i	$-0.961426\pi$
$79$	−3751.00	−	6496.92i	−0.601025	−	1.04101i	0.391641	−	0.920118i	$-0.371908\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4746.69	−	2740.50i	0.689024	−	0.397808i	−0.114222	−	0.993455i	$-0.536438\pi$
$83$	4746.69	−	2740.50i	0.689024	−	0.397808i	0.803246	+	0.595647i	$0.203104\pi$
$84$		0			0
$85$	2025.00	−	3507.40i	0.280277	−	0.485454i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	8874.00i		−	1.12031i	−0.828387	−	0.560157i	$-0.810741\pi$
$89$		−	8874.00i		−	1.12031i	0.828387	−	0.560157i	$-0.189259\pi$
$90$		0			0
$91$	−170.000			−0.0205289
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−498.831	−	288.000i	−0.0552721	−	0.0319114i
$96$		0			0
$97$	−5285.50	−	9154.75i	−0.561749	−	0.972978i	−0.997344	−	0.0728355i	$-0.976795\pi$
$97$	−5285.50	−	9154.75i	−0.561749	−	0.972978i	0.435595	−	0.900143i	$-0.356538\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−11356.2	+	6556.50i	−1.11324	+	0.642731i	−0.939667	−	0.342089i	$-0.888865\pi$
$101$	−11356.2	+	6556.50i	−1.11324	+	0.642731i	−0.173576	+	0.984821i	$0.555532\pi$
$102$		0			0
$103$	2915.00	−	5048.93i	0.274767	−	0.475910i	−0.695310	−	0.718710i	$-0.744733\pi$
$103$	2915.00	−	5048.93i	0.274767	−	0.475910i	0.970076	+	0.242801i	$0.0780660\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1089.00i		−	0.0951175i	−0.998868	−	0.0475587i	$-0.984856\pi$
$107$		−	1089.00i		−	0.0951175i	0.998868	−	0.0475587i	$-0.0151441\pi$
$108$		0			0
$109$	−5020.00			−0.422523			−0.211262	−	0.977430i	$-0.567757\pi$
$109$	−5020.00			−0.422523			−0.211262	+	0.977430i	$0.567757\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	18519.1	+	10692.0i	1.45032	+	0.837340i	0.998499	−	0.0547701i	$-0.0174426\pi$
$113$	18519.1	+	10692.0i	1.45032	+	0.837340i	0.451817	+	0.892111i	$0.350776\pi$
$114$		0			0
$115$	2754.00	+	4770.07i	0.208242	+	0.360686i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1948.56	+	1125.00i	−0.137600	+	0.0794435i
$120$		0			0
$121$	−476.000	+	824.456i	−0.0325114	+	0.0563115i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	10521.0i		−	0.673344i
$126$		0			0
$127$	9227.00			0.572075			0.286038	−	0.958218i	$-0.407662\pi$
$127$	9227.00			0.572075			0.286038	+	0.958218i	$0.407662\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3702.26	+	2137.50i	0.215737	+	0.124556i	0.603975	−	0.797003i	$-0.293583\pi$
$131$	3702.26	+	2137.50i	0.215737	+	0.124556i	−0.388238	+	0.921559i	$0.626916\pi$
$132$		0			0
$133$	160.000	+	277.128i	0.00904517	+	0.0156667i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9805.14	+	5661.00i	−0.522411	+	0.301614i	−0.737921	−	0.674887i	$-0.764192\pi$
$137$	−9805.14	+	5661.00i	−0.522411	+	0.301614i	0.215509	+	0.976502i	$0.430859\pi$
$138$		0			0
$139$	−4906.00	+	8497.44i	−0.253921	+	0.439803i	−0.964602	−	0.263711i	$-0.915054\pi$
$139$	−4906.00	+	8497.44i	−0.253921	+	0.439803i	0.710681	+	0.703514i	$0.248387\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			3978.00i			0.194533i
$144$		0			0
$145$	−9558.00			−0.454602
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−14271.2	−	8239.50i	−0.642819	−	0.371132i	0.142880	−	0.989740i	$-0.454364\pi$
$149$	−14271.2	−	8239.50i	−0.642819	−	0.371132i	−0.785700	+	0.618608i	$0.787697\pi$
$150$		0			0
$151$	12777.5	+	22131.3i	0.560392	+	0.970628i	0.997462	+	0.0712001i	$0.0226829\pi$
$151$	12777.5	+	22131.3i	0.560392	+	0.970628i	−0.437070	+	0.899428i	$0.643984\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5432.58	−	3136.50i	0.226122	−	0.130552i
$156$		0			0
$157$	−10582.0	+	18328.6i	−0.429307	+	0.743582i	−0.996812	−	0.0797880i	$-0.974576\pi$
$157$	−10582.0	+	18328.6i	−0.429307	+	0.743582i	0.567504	+	0.823370i	$0.307909\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	3060.00i		−	0.118051i
$162$		0			0
$163$	33830.0			1.27329			0.636644	−	0.771158i	$-0.280322\pi$
$163$	33830.0			1.27329			0.636644	+	0.771158i	$0.280322\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−23710.0	−	13689.0i	−0.850158	−	0.490839i	0.0105465	−	0.999944i	$-0.496643\pi$
$167$	−23710.0	−	13689.0i	−0.850158	−	0.490839i	−0.860704	+	0.509106i	$0.829976\pi$
$168$		0			0
$169$	13702.5	+	23733.4i	0.479763	+	0.830973i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	40007.8	−	23098.5i	1.33676	−	0.771777i	0.350431	−	0.936588i	$-0.386035\pi$
$173$	40007.8	−	23098.5i	1.33676	−	0.771777i	0.986325	+	0.164812i	$0.0527017\pi$
$174$		0			0
$175$	−1360.00	+	2355.59i	−0.0444082	+	0.0769172i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			20781.0i			0.648575i	0.945959	+	0.324288i	$0.105125\pi$
$179$			20781.0i			0.648575i	−0.945959	+	0.324288i	$0.894875\pi$
$180$		0			0
$181$	−19504.0			−0.595342			−0.297671	−	0.954669i	$-0.596210\pi$
$181$	−19504.0			−0.595342			−0.297671	+	0.954669i	$0.596210\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5830.08	−	3366.00i	−0.170346	−	0.0983492i
$186$		0			0
$187$	26325.0	+	45596.2i	0.752810	+	1.30390i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6095.09	+	3519.00i	−0.167076	+	0.0964612i	−0.581206	−	0.813756i	$-0.697419\pi$
$191$	−6095.09	+	3519.00i	−0.167076	+	0.0964612i	0.414131	+	0.910217i	$0.364086\pi$
$192$		0			0
$193$	−25763.5	+	44623.7i	−0.691656	+	1.19798i	0.279639	+	0.960105i	$0.409785\pi$
$193$	−25763.5	+	44623.7i	−0.691656	+	1.19798i	−0.971295	+	0.237878i	$0.923548\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			60435.0i			1.55724i	0.627495	+	0.778621i	$0.284080\pi$
$197$			60435.0i			1.55724i	−0.627495	+	0.778621i	$0.715920\pi$
$198$		0			0
$199$	45665.0			1.15313			0.576564	−	0.817052i	$-0.304393\pi$
$199$	45665.0			1.15313			0.576564	+	0.817052i	$0.304393\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4598.59	+	2655.00i	0.111592	+	0.0644277i
$204$		0			0
$205$	3078.00	+	5331.25i	0.0732421	+	0.126859i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6484.80	−	3744.00i	0.148458	−	0.0857123i
$210$		0			0
$211$	−6562.00	+	11365.7i	−0.147391	+	0.255289i	−0.930262	−	0.366895i	$-0.880421\pi$
$211$	−6562.00	+	11365.7i	−0.147391	+	0.255289i	0.782871	+	0.622184i	$0.213754\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	23562.0i		−	0.509724i
$216$		0			0
$217$	−3485.00			−0.0740088
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−13250.2	−	7650.00i	−0.271292	−	0.156631i
$222$		0			0
$223$	39425.0	+	68286.1i	0.792797	+	1.37316i	0.924229	+	0.381839i	$0.124709\pi$
$223$	39425.0	+	68286.1i	0.792797	+	1.37316i	−0.131432	+	0.991325i	$0.541957\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19314.1	+	11151.0i	−0.374820	+	0.216402i	−0.675562	−	0.737303i	$-0.736099\pi$
$227$	−19314.1	+	11151.0i	−0.374820	+	0.216402i	0.300742	+	0.953705i	$0.402766\pi$
$228$		0			0
$229$	−4435.00	+	7681.65i	−0.0845712	+	0.146482i	−0.905208	−	0.424968i	$-0.860285\pi$
$229$	−4435.00	+	7681.65i	−0.0845712	+	0.146482i	0.820637	+	0.571450i	$0.193619\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	26010.0i		−	0.479103i	−0.970884	−	0.239551i	$-0.923000\pi$
$233$		−	26010.0i		−	0.479103i	0.970884	−	0.239551i	$-0.0770003\pi$
$234$		0			0
$235$	23814.0			0.431218
$236$		0			0
$237$		0			0
$238$		0			0
$239$	55557.3	+	32076.0i	0.972624	+	0.561545i	0.900035	−	0.435817i	$-0.143541\pi$
$239$	55557.3	+	32076.0i	0.972624	+	0.561545i	0.0725889	+	0.997362i	$0.476874\pi$
$240$		0			0
$241$	−16711.0	−	28944.3i	−0.287719	−	0.498344i	0.685546	−	0.728029i	$-0.259564\pi$
$241$	−16711.0	−	28944.3i	−0.287719	−	0.498344i	−0.973265	+	0.229686i	$0.926230\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	18519.1	−	10692.0i	0.308523	−	0.178126i
$246$		0			0
$247$	−1088.00	+	1884.47i	−0.0178334	+	0.0308884i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	110466.i		−	1.75340i	−0.481037	−	0.876700i	$-0.659740\pi$
$251$		−	110466.i		−	1.75340i	0.481037	−	0.876700i	$-0.340260\pi$
$252$		0			0
$253$	−71604.0			−1.11866
$254$		0			0
$255$		0			0
$256$		0			0
$257$	13141.1	+	7587.00i	0.198959	+	0.114869i	0.596170	−	0.802858i	$-0.296688\pi$
$257$	13141.1	+	7587.00i	0.198959	+	0.114869i	−0.397211	+	0.917727i	$0.630022\pi$
$258$		0			0
$259$	1870.00	+	3238.94i	0.0278767	+	0.0482839i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−60576.7	+	34974.0i	−0.875779	+	0.505631i	−0.869264	−	0.494348i	$-0.835407\pi$
$263$	−60576.7	+	34974.0i	−0.875779	+	0.505631i	−0.00651453	+	0.999979i	$0.502074\pi$
$264$		0			0
$265$	−4819.50	+	8347.62i	−0.0686294	+	0.118870i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			50346.0i			0.695762i	0.937539	+	0.347881i	$0.113099\pi$
$269$			50346.0i			0.695762i	−0.937539	+	0.347881i	$0.886901\pi$
$270$		0			0
$271$	108323.			1.47497			0.737483	−	0.675366i	$-0.236014\pi$
$271$	108323.			1.47497			0.737483	+	0.675366i	$0.236014\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	55120.8	+	31824.0i	0.728870	+	0.420813i
$276$		0			0
$277$	−15358.0	−	26600.8i	−0.200159	−	0.346686i	0.748421	−	0.663224i	$-0.230813\pi$
$277$	−15358.0	−	26600.8i	−0.200159	−	0.346686i	−0.948580	+	0.316539i	$0.897479\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−111941.	+	64629.0i	−1.41767	+	0.818493i	−0.996094	−	0.0882999i	$-0.971857\pi$
$281$	−111941.	+	64629.0i	−1.41767	+	0.818493i	−0.421577	+	0.906793i	$0.638523\pi$
$282$		0			0
$283$	31988.0	−	55404.8i	0.399406	−	0.691791i	−0.594247	−	0.804283i	$-0.702550\pi$
$283$	31988.0	−	55404.8i	0.399406	−	0.691791i	0.993653	+	0.112492i	$0.0358832\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	3420.00i		−	0.0415205i
$288$		0			0
$289$	−118979.			−1.42454
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−94076.3	−	54315.0i	−1.09584	−	0.632681i	−0.160711	−	0.987001i	$-0.551379\pi$
$293$	−94076.3	−	54315.0i	−1.09584	−	0.632681i	−0.935124	+	0.354321i	$0.884712\pi$
$294$		0			0
$295$	−26163.0	−	45315.6i	−0.300638	−	0.520720i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	18020.3	−	10404.0i	0.201567	−	0.116375i
$300$		0			0
$301$	−6545.00	+	11336.3i	−0.0722398	+	0.125123i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	57636.0i		−	0.619575i
$306$		0			0
$307$	6410.00			0.0680113			0.0340057	−	0.999422i	$-0.489174\pi$
$307$	6410.00			0.0680113			0.0340057	+	0.999422i	$0.489174\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	61480.9	+	35496.0i	0.635652	+	0.366994i	0.782938	−	0.622100i	$-0.213720\pi$
$311$	61480.9	+	35496.0i	0.635652	+	0.366994i	−0.147286	+	0.989094i	$0.547054\pi$
$312$		0			0
$313$	−80480.5	−	139396.i	−0.821489	−	1.42286i	−0.904573	−	0.426319i	$-0.859810\pi$
$313$	−80480.5	−	139396.i	−0.821489	−	1.42286i	0.0830835	−	0.996543i	$-0.473523\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	168644.	−	97366.5i	1.67823	−	0.968927i	0.715442	−	0.698672i	$-0.246225\pi$
$317$	168644.	−	97366.5i	1.67823	−	0.968927i	0.962789	−	0.270255i	$-0.0871079\pi$
$318$		0			0
$319$	62127.0	−	107607.i	0.610519	−	1.05745i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			28800.0i			0.276050i
$324$		0			0
$325$	−18496.0			−0.175110
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−11457.5	−	6615.00i	−0.105852	−	0.0611136i
$330$		0			0
$331$	16643.0	+	28826.5i	0.151906	+	0.263109i	0.931928	−	0.362643i	$-0.118125\pi$
$331$	16643.0	+	28826.5i	0.151906	+	0.263109i	−0.780022	+	0.625752i	$0.784792\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	40670.3	−	23481.0i	0.362400	−	0.209231i
$336$		0			0
$337$	63845.0	−	110583.i	0.562169	−	0.973706i	−0.435138	−	0.900364i	$-0.643300\pi$
$337$	63845.0	−	110583.i	0.562169	−	0.973706i	0.997307	−	0.0733418i	$-0.0233664\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			81549.0i			0.701310i
$342$		0			0
$343$	−23885.0			−0.203019
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−120210.	−	69403.5i	−0.998351	−	0.576398i	−0.0905907	−	0.995888i	$-0.528875\pi$
$347$	−120210.	−	69403.5i	−0.998351	−	0.576398i	−0.907760	+	0.419490i	$0.862209\pi$
$348$		0			0
$349$	−101896.	−	176489.i	−0.836578	−	1.44900i	−0.892739	−	0.450573i	$-0.851220\pi$
$349$	−101896.	−	176489.i	−0.836578	−	1.44900i	0.0561617	−	0.998422i	$-0.482114\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−27996.9	+	16164.0i	−0.224678	+	0.129718i	−0.608114	−	0.793849i	$-0.708074\pi$
$353$	−27996.9	+	16164.0i	−0.224678	+	0.129718i	0.383437	+	0.923567i	$0.374740\pi$
$354$		0			0
$355$	−29565.0	+	51208.1i	−0.234596	+	0.406333i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			185922.i			1.44259i	0.692630	+	0.721293i	$0.256452\pi$
$359$			185922.i			1.44259i	−0.692630	+	0.721293i	$0.743548\pi$
$360$		0			0
$361$	−126225.			−0.968570
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−35222.1	−	20335.5i	−0.264381	−	0.152640i
$366$		0			0
$367$	−75539.5	−	130838.i	−0.560844	−	0.971410i	−0.997423	−	0.0717442i	$-0.977143\pi$
$367$	−75539.5	−	130838.i	−0.560844	−	0.971410i	0.436579	−	0.899666i	$-0.356190\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4637.57	−	2677.50i	0.0336932	−	0.0194528i
$372$		0			0
$373$	3239.00	−	5610.11i	0.0232806	−	0.0403231i	−0.854150	−	0.520026i	$-0.825922\pi$
$373$	3239.00	−	5610.11i	0.0232806	−	0.0403231i	0.877431	+	0.479703i	$0.159256\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			36108.0i			0.254051i
$378$		0			0
$379$	99008.0			0.689274			0.344637	−	0.938736i	$-0.388002\pi$
$379$	99008.0			0.689274			0.344637	+	0.938736i	$0.388002\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−78862.0	−	45531.0i	−0.537614	−	0.310391i	0.206498	−	0.978447i	$-0.433793\pi$
$383$	−78862.0	−	45531.0i	−0.537614	−	0.310391i	−0.744111	+	0.668056i	$0.767127\pi$
$384$		0			0
$385$	2632.50	+	4559.62i	0.0177602	+	0.0307615i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	82548.7	−	47659.5i	0.545520	−	0.314956i	−0.201793	−	0.979428i	$-0.564677\pi$
$389$	82548.7	−	47659.5i	0.545520	−	0.314956i	0.747313	+	0.664472i	$0.231343\pi$
$390$		0			0
$391$	137700.	−	238503.i	0.900701	−	1.56006i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	67518.0i		−	0.432738i
$396$		0			0
$397$	−163438.			−1.03698			−0.518492	−	0.855083i	$-0.673506\pi$
$397$	−163438.			−1.03698			−0.518492	+	0.855083i	$0.673506\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	246485.	+	142308.i	1.53286	+	0.884994i	0.999228	+	0.0392745i	$0.0125047\pi$
$401$	246485.	+	142308.i	1.53286	+	0.884994i	0.533627	+	0.845720i	$0.320829\pi$
$402$		0			0
$403$	−11849.0	−	20523.1i	−0.0729578	−	0.126367i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	75791.1	−	43758.0i	0.457540	−	0.264161i
$408$		0			0
$409$	−53762.5	+	93119.4i	−0.321390	+	0.556664i	−0.980775	−	0.195141i	$-0.937483\pi$
$409$	−53762.5	+	93119.4i	−0.321390	+	0.556664i	0.659385	+	0.751806i	$0.270817\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			29070.0i			0.170430i
$414$		0			0
$415$	49329.0			0.286422
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−130148.	−	75141.0i	−0.741327	−	0.428005i	0.0812249	−	0.996696i	$-0.474117\pi$
$419$	−130148.	−	75141.0i	−0.741327	−	0.428005i	−0.822551	+	0.568691i	$0.807450\pi$
$420$		0			0
$421$	−67210.0	−	116411.i	−0.379201	−	0.656796i	0.611745	−	0.791055i	$-0.290468\pi$
$421$	−67210.0	−	116411.i	−0.379201	−	0.656796i	−0.990946	+	0.134259i	$0.957135\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−212003.	+	122400.i	−1.17372	+	0.677647i
$426$		0			0
$427$	−16010.0	+	27730.1i	−0.0878083	+	0.152088i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	237726.i		−	1.27974i	−0.768483	−	0.639871i	$-0.778988\pi$
$431$		−	237726.i		−	1.27974i	0.768483	−	0.639871i	$-0.221012\pi$
$432$		0			0
$433$	112187.			0.598366			0.299183	−	0.954196i	$-0.403286\pi$
$433$	112187.			0.598366			0.299183	+	0.954196i	$0.403286\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−33920.5	−	19584.0i	−0.177623	−	0.102551i
$438$		0			0
$439$	102322.	+	177226.i	0.530931	+	0.919599i	0.999348	+	0.0360919i	$0.0114909\pi$
$439$	102322.	+	177226.i	0.530931	+	0.919599i	−0.468418	+	0.883507i	$0.655176\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−43725.6	+	25245.0i	−0.222807	+	0.128638i	−0.607249	−	0.794511i	$-0.707727\pi$
$443$	−43725.6	+	25245.0i	−0.222807	+	0.128638i	0.384442	+	0.923149i	$0.374394\pi$
$444$		0			0
$445$	39933.0	−	69166.0i	0.201656	−	0.349279i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	363528.i		−	1.80321i	−0.432565	−	0.901603i	$-0.642391\pi$
$449$		−	363528.i		−	1.80321i	0.432565	−	0.901603i	$-0.357609\pi$
$450$		0			0
$451$	−80028.0			−0.393449
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1325.02	−	765.000i	−0.00640028	−	0.00369521i
$456$		0			0
$457$	−3338.50	−	5782.45i	−0.0159852	−	0.0276872i	0.857922	−	0.513780i	$-0.171755\pi$
$457$	−3338.50	−	5782.45i	−0.0159852	−	0.0276872i	−0.873907	+	0.486092i	$0.838422\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−198620.	+	114674.i	−0.934592	+	0.539587i	−0.888261	−	0.459339i	$-0.848086\pi$
$461$	−198620.	+	114674.i	−0.934592	+	0.539587i	−0.0463307	+	0.998926i	$0.514753\pi$
$462$		0			0
$463$	−119400.	+	206806.i	−0.556981	+	0.964720i	0.440765	+	0.897622i	$0.354707\pi$
$463$	−119400.	+	206806.i	−0.556981	+	0.964720i	−0.997746	+	0.0670973i	$0.978626\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			263133.i			1.20654i	0.797537	+	0.603270i	$0.206136\pi$
$467$			263133.i			1.20654i	−0.797537	+	0.603270i	$0.793864\pi$
$468$		0			0
$469$	−26090.0			−0.118612
$470$		0			0
$471$		0			0
$472$		0			0
$473$	265269.	+	153153.i	1.18567	+	0.684547i
$474$		0			0
$475$	17408.0	+	30151.5i	0.0771546	+	0.133636i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8090.41	+	4671.00i	−0.0352614	+	0.0203582i	−0.517527	−	0.855667i	$-0.673147\pi$
$479$	−8090.41	+	4671.00i	−0.0352614	+	0.0203582i	0.482266	+	0.876025i	$0.339814\pi$
$480$		0			0
$481$	−12716.0	+	22024.8i	−0.0549617	+	0.0951965i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	95139.0i		−	0.404460i
$486$		0			0
$487$	331262.			1.39673			0.698367	−	0.715740i	$-0.253910\pi$
$487$	331262.			1.39673			0.698367	+	0.715740i	$0.253910\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	343203.	+	198148.i	1.42360	+	0.821917i	0.996605	−	0.0823364i	$-0.0262382\pi$
$491$	343203.	+	198148.i	1.42360	+	0.821917i	0.426997	+	0.904253i	$0.359572\pi$
$492$		0			0
$493$	238950.	+	413874.i	0.983135	+	1.70284i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	28448.9	−	16425.0i	0.115174	−	0.0664956i
$498$		0			0
$499$	−22525.0	+	39014.4i	−0.0904615	+	0.156684i	−0.907705	−	0.419608i	$-0.862168\pi$
$499$	−22525.0	+	39014.4i	−0.0904615	+	0.156684i	0.817244	+	0.576292i	$0.195501\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			233172.i			0.921596i	0.887505	+	0.460798i	$0.152437\pi$
$503$			233172.i			0.921596i	−0.887505	+	0.460798i	$0.847563\pi$
$504$		0			0
$505$	−118017.			−0.462766
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−141551.	−	81724.5i	−0.546358	−	0.315440i	0.201294	−	0.979531i	$-0.435485\pi$
$509$	−141551.	−	81724.5i	−0.546358	−	0.315440i	−0.747652	+	0.664091i	$0.768819\pi$
$510$		0			0
$511$	11297.5	+	19567.8i	0.0432654	+	0.0749378i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	45440.4	−	26235.0i	0.171328	−	0.0989160i
$516$		0			0
$517$	−154791.	+	268106.i	−0.579115	+	1.00306i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	384606.i		−	1.41690i	−0.705759	−	0.708452i	$-0.749394\pi$
$521$		−	384606.i		−	1.41690i	0.705759	−	0.708452i	$-0.250606\pi$
$522$		0			0
$523$	−214642.			−0.784714			−0.392357	−	0.919813i	$-0.628340\pi$
$523$	−214642.			−0.784714			−0.392357	+	0.919813i	$0.628340\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−271629.	−	156825.i	−0.978036	−	0.564669i
$528$		0			0
$529$	47351.5	+	82015.2i	0.169209	+	0.293078i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	20140.3	−	11628.0i	0.0708943	−	0.0409308i
$534$		0			0
$535$	4900.50	−	8487.91i	0.0171211	−	0.0296547i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			277992.i			0.956874i
$540$		0			0
$541$	545156.			1.86263			0.931314	−	0.364217i	$-0.118663\pi$
$541$	545156.			1.86263			0.931314	+	0.364217i	$0.118663\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−39127.0	−	22590.0i	−0.131730	−	0.0760542i
$546$		0			0
$547$	245711.	+	425584.i	0.821202	+	1.42236i	0.904788	+	0.425863i	$0.140029\pi$
$547$	245711.	+	425584.i	0.821202	+	1.42236i	−0.0835858	+	0.996501i	$0.526637\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	58862.0	−	33984.0i	0.193880	−	0.111936i
$552$		0			0
$553$	−18755.0	+	32484.6i	−0.0613291	+	0.106225i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			445077.i			1.43458i	0.696775	+	0.717290i	$0.254618\pi$
$557$			445077.i			1.43458i	−0.696775	+	0.717290i	$0.745382\pi$
$558$		0			0
$559$	−89012.0			−0.284856
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−133164.	−	76882.5i	−0.420118	−	0.242555i	0.275010	−	0.961441i	$-0.411319\pi$
$563$	−133164.	−	76882.5i	−0.420118	−	0.242555i	−0.695128	+	0.718886i	$0.744652\pi$
$564$		0			0
$565$	96228.0	+	166672.i	0.301443	+	0.522114i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	233577.	−	134856.i	0.721450	−	0.416529i	−0.0938360	−	0.995588i	$-0.529913\pi$
$569$	233577.	−	134856.i	0.721450	−	0.416529i	0.815286	+	0.579058i	$0.196580\pi$
$570$		0			0
$571$	−294859.	+	510711.i	−0.904362	+	1.56640i	−0.0825900	+	0.996584i	$0.526319\pi$
$571$	−294859.	+	510711.i	−0.904362	+	1.56640i	−0.821772	+	0.569817i	$0.807014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	332928.i		−	1.00697i
$576$		0			0
$577$	184094.			0.552953			0.276476	−	0.961021i	$-0.410833\pi$
$577$	184094.			0.552953			0.276476	+	0.961021i	$0.410833\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−23733.4	−	13702.5i	−0.0703086	−	0.0405927i
$582$		0			0
$583$	−62653.5	−	108519.i	−0.184335	−	0.319278i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	356407.	−	205772.i	1.03436	−	0.597185i	0.116126	−	0.993235i	$-0.462952\pi$
$587$	356407.	−	205772.i	1.03436	−	0.597185i	0.918229	+	0.396049i	$0.129619\pi$
$588$		0			0
$589$	−22304.0	+	38631.7i	−0.0642913	+	0.111356i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			339966.i			0.966777i	0.875406	+	0.483388i	$0.160594\pi$
$593$			339966.i			0.966777i	−0.875406	+	0.483388i	$0.839406\pi$
$594$		0			0
$595$	−20250.0			−0.0571994
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−259439.	−	149787.i	−0.723071	−	0.417465i	0.0928108	−	0.995684i	$-0.470415\pi$
$599$	−259439.	−	149787.i	−0.723071	−	0.417465i	−0.815882	+	0.578218i	$0.803748\pi$
$600$		0			0
$601$	258058.	+	446969.i	0.714443	+	1.23745i	0.963174	+	0.268879i	$0.0866531\pi$
$601$	258058.	+	446969.i	0.714443	+	1.23745i	−0.248731	+	0.968573i	$0.580014\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7420.11	+	4284.00i	−0.0202721	+	0.0117041i
$606$		0			0
$607$	243287.	−	421385.i	0.660300	−	1.14367i	−0.320236	−	0.947338i	$-0.603762\pi$
$607$	243287.	−	421385.i	0.660300	−	1.14367i	0.980537	−	0.196336i	$-0.0629043\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	89964.0i		−	0.240983i
$612$		0			0
$613$	−189550.			−0.504432			−0.252216	−	0.967671i	$-0.581159\pi$
$613$	−189550.			−0.504432			−0.252216	+	0.967671i	$0.581159\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−128387.	−	74124.0i	−0.337248	−	0.194710i	0.321807	−	0.946805i	$-0.395710\pi$
$617$	−128387.	−	74124.0i	−0.337248	−	0.194710i	−0.659054	+	0.752095i	$0.729043\pi$
$618$		0			0
$619$	−5695.00	−	9864.03i	−0.0148632	−	0.0257438i	0.858498	−	0.512817i	$-0.171398\pi$
$619$	−5695.00	−	9864.03i	−0.0148632	−	0.0257438i	−0.873361	+	0.487073i	$0.838065\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−38425.5	+	22185.0i	−0.0990020	+	0.0571588i
$624$		0			0
$625$	−122656.	+	212446.i	−0.313998	+	0.543861i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			336600.i			0.850771i
$630$		0			0
$631$	−6715.00			−0.0168650			−0.00843252	−	0.999964i	$-0.502684\pi$
$631$	−6715.00			−0.0168650			−0.00843252	+	0.999964i	$0.502684\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	71917.3	+	41521.5i	0.178355	+	0.102974i
$636$		0			0
$637$	−40392.0	−	69961.0i	−0.0995443	−	0.172416i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−382011.	+	220554.i	−0.929736	+	0.536783i	−0.886728	−	0.462292i	$-0.847027\pi$
$641$	−382011.	+	220554.i	−0.929736	+	0.536783i	−0.0430077	+	0.999075i	$0.513694\pi$
$642$		0			0
$643$	−273724.	+	474104.i	−0.662050	+	1.14670i	0.318026	+	0.948082i	$0.396980\pi$
$643$	−273724.	+	474104.i	−0.662050	+	1.14670i	−0.980076	+	0.198623i	$0.936353\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	42228.0i		−	0.100877i	−0.998727	−	0.0504385i	$-0.983938\pi$
$647$		−	42228.0i		−	0.100877i	0.998727	−	0.0504385i	$-0.0160619\pi$
$648$		0			0
$649$	680238.			1.61500
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−224295.	−	129496.i	−0.526008	−	0.303691i	0.213381	−	0.976969i	$-0.431552\pi$
$653$	−224295.	−	129496.i	−0.526008	−	0.303691i	−0.739389	+	0.673278i	$0.764886\pi$
$654$		0			0
$655$	19237.5	+	33320.3i	0.0448400	+	0.0776652i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−271138.	+	156542.i	−0.624337	+	0.360461i	−0.778556	−	0.627575i	$-0.784047\pi$
$659$	−271138.	+	156542.i	−0.624337	+	0.360461i	0.154218	+	0.988037i	$0.450714\pi$
$660$		0			0
$661$	343160.	−	594371.i	0.785405	−	1.36036i	−0.143352	−	0.989672i	$-0.545788\pi$
$661$	343160.	−	594371.i	0.785405	−	1.36036i	0.928757	−	0.370690i	$-0.120879\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			2880.00i			0.00651252i
$666$		0			0
$667$	−649944.			−1.46091
$668$		0			0
$669$		0			0
$670$		0			0
$671$	648885.	+	374634.i	1.44120	+	0.832074i
$672$		0			0
$673$	−107364.	−	185959.i	−0.237043	−	0.410570i	0.722822	−	0.691035i	$-0.242845\pi$
$673$	−107364.	−	185959.i	−0.237043	−	0.410570i	−0.959864	+	0.280465i	$0.909511\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	297599.	−	171819.i	0.649314	−	0.374881i	−0.138880	−	0.990309i	$-0.544350\pi$
$677$	297599.	−	171819.i	0.649314	−	0.374881i	0.788193	+	0.615428i	$0.211017\pi$
$678$		0			0
$679$	−26427.5	+	45773.8i	−0.0573214	+	0.0992835i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	626238.i		−	1.34245i	−0.741254	−	0.671225i	$-0.765769\pi$
$683$		−	626238.i		−	1.34245i	0.741254	−	0.671225i	$-0.234231\pi$
$684$		0			0
$685$	−101898.			−0.217162
$686$		0			0
$687$		0			0
$688$		0			0
$689$	31535.4	+	18207.0i	0.0664294	+	0.0383531i
$690$		0			0
$691$	−342238.	−	592774.i	−0.716757	−	1.24146i	−0.962278	−	0.272069i	$-0.912292\pi$
$691$	−342238.	−	592774.i	−0.716757	−	1.24146i	0.245521	−	0.969391i	$-0.421041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−76477.0	+	44154.0i	−0.158329	+	0.0914114i
$696$		0			0
$697$	153900.	−	266563.i	0.316791	−	0.548698i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	809523.i		−	1.64738i	−0.567042	−	0.823689i	$-0.691912\pi$
$701$		−	809523.i		−	1.64738i	0.567042	−	0.823689i	$-0.308088\pi$
$702$		0			0
$703$	47872.0			0.0968659
$704$		0			0
$705$		0			0
$706$		0			0
$707$	56781.0	+	32782.5i	0.113596	+	0.0655848i
$708$		0			0
$709$	159824.	+	276823.i	0.317943	+	0.550694i	0.980059	−	0.198708i	$-0.0636745\pi$
$709$	159824.	+	276823.i	0.317943	+	0.550694i	−0.662116	+	0.749402i	$0.730341\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	369415.	−	213282.i	0.726668	−	0.419542i
$714$		0			0
$715$	−17901.0	+	31005.4i	−0.0350159	+	0.0606493i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	55836.0i		−	0.108008i	−0.998541	−	0.0540041i	$-0.982802\pi$
$719$		−	55836.0i		−	0.108008i	0.998541	−	0.0540041i	$-0.0171984\pi$
$720$		0			0
$721$	−29150.0			−0.0560748
$722$		0			0
$723$		0			0
$724$		0			0
$725$	500327.	+	288864.i	0.951871	+	0.549563i
$726$		0			0
$727$	−431936.	−	748136.i	−0.817243	−	1.41551i	−0.907706	−	0.419606i	$-0.862168\pi$
$727$	−431936.	−	748136.i	−0.817243	−	1.41551i	0.0904635	−	0.995900i	$-0.471165\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.02026e6	+	589050.i	−1.90932	+	1.10234i
$732$		0			0
$733$	−103804.	+	179794.i	−0.193200	+	0.334631i	−0.946309	−	0.323264i	$-0.895220\pi$
$733$	−103804.	+	179794.i	−0.193200	+	0.334631i	0.753109	+	0.657895i	$0.228553\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			610506.i			1.12397i
$738$		0			0
$739$	−803590.			−1.47145			−0.735725	−	0.677280i	$-0.763159\pi$
$739$	−803590.			−1.47145			−0.735725	+	0.677280i	$0.763159\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	536633.	+	309825.i	0.972074	+	0.561227i	0.899868	−	0.436162i	$-0.143663\pi$
$743$	536633.	+	309825.i	0.972074	+	0.561227i	0.0722063	+	0.997390i	$0.476996\pi$
$744$		0			0
$745$	−74155.5	−	128441.i	−0.133607	−	0.231415i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4715.51	+	2722.50i	−0.00840553	+	0.00485293i
$750$		0			0
$751$	532992.	−	923170.i	0.945020	−	1.63682i	0.189310	−	0.981917i	$-0.439375\pi$
$751$	532992.	−	923170.i	0.945020	−	1.63682i	0.755710	−	0.654906i	$-0.227292\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			229995.i			0.403482i
$756$		0			0
$757$	750410.			1.30950			0.654752	−	0.755844i	$-0.272773\pi$
$757$	750410.			1.30950			0.654752	+	0.755844i	$0.272773\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−357849.	−	206604.i	−0.617917	−	0.356754i	0.158141	−	0.987417i	$-0.449450\pi$
$761$	−357849.	−	206604.i	−0.617917	−	0.356754i	−0.776058	+	0.630662i	$0.782783\pi$
$762$		0			0
$763$	12550.0	+	21737.2i	0.0215573	+	0.0373384i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−171192.	+	98838.0i	−0.291001	+	0.168009i
$768$		0			0
$769$	366690.	−	635127.i	0.620079	−	1.07401i	−0.369392	−	0.929274i	$-0.620434\pi$
$769$	366690.	−	635127.i	0.620079	−	1.07401i	0.989471	−	0.144734i	$-0.0462328\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	231822.i		−	0.387968i	−0.981005	−	0.193984i	$-0.937859\pi$
$773$		−	231822.i		−	0.387968i	0.981005	−	0.193984i	$-0.0621410\pi$
$774$		0			0
$775$	−379168.			−0.631289
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−37911.1	−	21888.0i	−0.0624729	−	0.0360688i
$780$		0			0
$781$	−384345.	−	665705.i	−0.630114	−	1.09139i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−164957.	+	95238.0i	−0.267690	+	0.154551i
$786$		0			0
$787$	−251362.	+	435372.i	−0.405836	+	0.702928i	−0.994418	−	0.105509i	$-0.966353\pi$
$787$	−251362.	+	435372.i	−0.405836	+	0.702928i	0.588583	+	0.808437i	$0.299686\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	106920.i		−	0.170886i
$792$		0			0
$793$	−217736.			−0.346245
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−516360.	−	298120.i	−0.812898	−	0.469327i	0.0350635	−	0.999385i	$-0.488837\pi$
$797$	−516360.	−	298120.i	−0.812898	−	0.469327i	−0.847961	+	0.530058i	$0.822170\pi$
$798$		0			0
$799$	−595350.	−	1.03118e6i	−0.932564	−	1.61525i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	457888.	−	264362.i	0.710113	−	0.409984i
$804$		0			0
$805$	13770.0	−	23850.3i	0.0212492	−	0.0368047i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			376038.i			0.574559i	0.957847	+	0.287280i	$0.0927509\pi$
$809$			376038.i			0.574559i	−0.957847	+	0.287280i	$0.907249\pi$
$810$		0			0
$811$	−331072.			−0.503362			−0.251681	−	0.967810i	$-0.580983\pi$
$811$	−331072.			−0.503362			−0.251681	+	0.967810i	$0.580983\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	263679.	+	152235.i	0.396972	+	0.229192i
$816$		0			0
$817$	83776.0	+	145104.i	0.125509	+	0.217388i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	237833.	−	137313.i	0.352847	−	0.203716i	−0.313092	−	0.949723i	$-0.601365\pi$
$821$	237833.	−	137313.i	0.352847	−	0.203716i	0.665938	+	0.746007i	$0.268031\pi$
$822$		0			0
$823$	270598.	−	468689.i	0.399507	−	0.691966i	−0.594158	−	0.804348i	$-0.702515\pi$
$823$	270598.	−	468689.i	0.399507	−	0.691966i	0.993665	+	0.112382i	$0.0358481\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	358362.i		−	0.523975i	−0.965071	−	0.261988i	$-0.915622\pi$
$827$		−	358362.i		−	0.523975i	0.965071	−	0.261988i	$-0.0843780\pi$
$828$		0			0
$829$	39626.0			0.0576595			0.0288298	−	0.999584i	$-0.490822\pi$
$829$	39626.0			0.0576595			0.0288298	+	0.999584i	$0.490822\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−925954.	−	534600.i	−1.33444	−	0.770440i
$834$		0			0
$835$	−123201.	−	213390.i	−0.176702	−	0.306057i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	914029.	−	527715.i	1.29848	−	0.749679i	0.318341	−	0.947976i	$-0.396874\pi$
$839$	914029.	−	527715.i	1.29848	−	0.749679i	0.980142	+	0.198297i	$0.0635411\pi$
$840$		0			0
$841$	210282.	−	364218.i	0.297310	−	0.514956i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			246645.i			0.345429i
$846$		0			0
$847$	4760.00			0.00663499
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−396446.	−	228888.i	−0.547425	−	0.316056i
$852$		0			0
$853$	16517.0	+	28608.3i	0.0227004	+	0.0393182i	0.877152	−	0.480212i	$-0.159440\pi$
$853$	16517.0	+	28608.3i	0.0227004	+	0.0393182i	−0.854452	+	0.519530i	$0.826107\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	395121.	−	228123.i	0.537982	−	0.310604i	−0.206278	−	0.978493i	$-0.566135\pi$
$857$	395121.	−	228123.i	0.537982	−	0.310604i	0.744261	+	0.667889i	$0.232802\pi$
$858$		0			0
$859$	−171802.	+	297570.i	−0.232832	+	0.403276i	−0.958640	−	0.284620i	$-0.908132\pi$
$859$	−171802.	+	297570.i	−0.232832	+	0.403276i	0.725809	+	0.687897i	$0.241466\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			74556.0i			0.100106i	0.998747	+	0.0500531i	$0.0159391\pi$
$863$			74556.0i			0.100106i	−0.998747	+	0.0500531i	$0.984061\pi$
$864$		0			0
$865$	415773.			0.555679
$866$		0			0
$867$		0			0
$868$		0			0
$869$	760140.	+	438867.i	1.00659	+	0.581157i
$870$		0			0
$871$	−88706.0	−	153643.i	−0.116928	−	0.202524i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−45557.3	+	26302.5i	−0.0595034	+	0.0343543i
$876$		0			0
$877$	348047.	−	602835.i	0.452521	−	0.783789i	−0.546021	−	0.837772i	$-0.683858\pi$
$877$	348047.	−	602835.i	0.452521	−	0.783789i	0.998542	+	0.0539822i	$0.0171914\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			672426.i			0.866349i	0.901310	+	0.433174i	$0.142607\pi$
$881$			672426.i			0.866349i	−0.901310	+	0.433174i	$0.857393\pi$
$882$		0			0
$883$	−1.44813e6			−1.85731			−0.928657	−	0.370938i	$-0.879036\pi$
$883$	−1.44813e6			−1.85731			−0.928657	+	0.370938i	$0.879036\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.30739e6	−	754821.i	−1.66172	−	0.959393i	−0.971895	−	0.235416i	$-0.924355\pi$
$887$	−1.30739e6	−	754821.i	−1.66172	−	0.959393i	−0.689823	−	0.723978i	$-0.742312\pi$
$888$		0			0
$889$	−23067.5	−	39954.1i	−0.0291875	−	0.0505542i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−146656.	+	84672.0i	−0.183907	+	0.106179i
$894$		0			0
$895$	−93514.5	+	161972.i	−0.116744	+	0.202206i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			740214.i			0.915879i
$900$		0			0
$901$	481950.			0.593680
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−152019.	−	87768.0i	−0.185609	−	0.107162i
$906$		0			0
$907$	−187414.	−	324611.i	−0.227818	−	0.394592i	0.729343	−	0.684148i	$-0.239826\pi$
$907$	−187414.	−	324611.i	−0.227818	−	0.394592i	−0.957161	+	0.289556i	$0.906492\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.04918e6	+	605745.i	−1.26419	+	0.729883i	−0.973883	−	0.227050i	$-0.927092\pi$
$911$	−1.04918e6	+	605745.i	−1.26419	+	0.729883i	−0.290311	+	0.956932i	$0.593759\pi$
$912$		0			0
$913$	−320638.	+	555362.i	−0.384657	+	0.666246i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	21375.0i		−	0.0254195i
$918$		0			0
$919$	656777.			0.777655			0.388827	−	0.921311i	$-0.372880\pi$
$919$	656777.			0.777655			0.388827	+	0.921311i	$0.372880\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	193453.	+	111690.i	0.227076	+	0.131102i
$924$		0			0
$925$	203456.	+	352396.i	0.237786	+	0.411858i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	501418.	−	289494.i	0.580990	−	0.335435i	−0.180537	−	0.983568i	$-0.557783\pi$
$929$	501418.	−	289494.i	0.580990	−	0.335435i	0.761527	+	0.648133i	$0.224450\pi$
$930$		0			0
$931$	−76032.0	+	131691.i	−0.0877197	+	0.151935i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			473850.i			0.542023i
$936$		0			0
$937$	195173.			0.222301			0.111150	−	0.993804i	$-0.464547\pi$
$937$	195173.			0.222301			0.111150	+	0.993804i	$0.464547\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.14256e6	+	659660.i	1.29033	+	0.744973i	0.978713	−	0.205234i	$-0.0657956\pi$
$941$	1.14256e6	+	659660.i	1.29033	+	0.744973i	0.311618	+	0.950207i	$0.399129\pi$
$942$		0			0
$943$	209304.	+	362525.i	0.235372	+	0.407676i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	842314.	−	486310.i	0.939235	−	0.542268i	0.0495146	−	0.998773i	$-0.484233\pi$
$947$	842314.	−	486310.i	0.939235	−	0.542268i	0.889720	+	0.456506i	$0.150899\pi$
$948$		0			0
$949$	−76823.0	+	133061.i	−0.0853019	+	0.147747i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	534384.i		−	0.588393i	−0.955745	−	0.294197i	$-0.904948\pi$
$953$		−	534384.i		−	0.588393i	0.955745	−	0.294197i	$-0.0950520\pi$
$954$		0			0
$955$	−63342.0			−0.0694520
$956$		0			0
$957$		0			0
$958$		0			0
$959$	49025.7	+	28305.0i	0.0533073	+	0.0307770i
$960$		0			0
$961$	218856.	+	379070.i	0.236980	+	0.410461i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−401613.	+	231872.i	−0.431274	+	0.248996i
$966$		0			0
$967$	391810.	−	678634.i	0.419008	−	0.725743i	−0.576832	−	0.816863i	$-0.695711\pi$
$967$	391810.	−	678634.i	0.419008	−	0.725743i	0.995840	+	0.0911200i	$0.0290447\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	276777.i		−	0.293556i	−0.989169	−	0.146778i	$-0.953110\pi$
$971$		−	276777.i		−	0.293556i	0.989169	−	0.146778i	$-0.0468904\pi$
$972$		0			0
$973$	49060.0			0.0518205
$974$		0			0
$975$		0			0
$976$		0			0
$977$	561387.	+	324117.i	0.588130	+	0.339557i	0.764358	−	0.644792i	$-0.223056\pi$
$977$	561387.	+	324117.i	0.588130	+	0.339557i	−0.176228	+	0.984349i	$0.556390\pi$
$978$		0			0
$979$	519129.	+	899158.i	0.541639	+	0.938146i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−251894.	+	145431.i	−0.260682	+	0.150505i	−0.624646	−	0.780908i	$-0.714757\pi$
$983$	−251894.	+	145431.i	−0.260682	+	0.150505i	0.363964	+	0.931413i	$0.381423\pi$
$984$		0			0
$985$	−271958.	+	471044.i	−0.280304	+	0.485500i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1.60222e6i		−	1.63806i
$990$		0			0
$991$	881243.			0.897322			0.448661	−	0.893702i	$-0.351901\pi$
$991$	881243.			0.897322			0.448661	+	0.893702i	$0.351901\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	355923.	+	205492.i	0.359510	+	0.207563i
$996$		0			0
$997$	345083.	+	597701.i	0.347163	+	0.601304i	0.985744	−	0.168251i	$-0.0538118\pi$
$997$	345083.	+	597701.i	0.347163	+	0.601304i	−0.638581	+	0.769554i	$0.720478\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.g.d.53.2		4
3.2	odd	2	inner	324.5.g.d.53.1		4
9.2	odd	6	inner	324.5.g.d.269.2		4
9.4	even	3		108.5.c.c.53.1	✓	2
9.5	odd	6		108.5.c.c.53.2	yes	2
9.7	even	3	inner	324.5.g.d.269.1		4
36.23	even	6		432.5.e.d.161.2		2
36.31	odd	6		432.5.e.d.161.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.c.c.53.1	✓	2	9.4	even	3
108.5.c.c.53.2	yes	2	9.5	odd	6
324.5.g.d.53.1		4	3.2	odd	2	inner
324.5.g.d.53.2		4	1.1	even	1	trivial
324.5.g.d.269.1		4	9.7	even	3	inner
324.5.g.d.269.2		4	9.2	odd	6	inner
432.5.e.d.161.1		2	36.31	odd	6
432.5.e.d.161.2		2	36.23	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(7.79423 + 4.50000i) q^{5} +(-2.50000 - 4.33013i) q^{7} +O(q^{10})\)	\(q+(7.79423 + 4.50000i) q^{5} +(-2.50000 - 4.33013i) q^{7} +(-101.325 + 58.5000i) q^{11} +(17.0000 - 29.4449i) q^{13} -450.000i q^{17} -64.0000 q^{19} +(530.008 + 306.000i) q^{23} +(-272.000 - 471.118i) q^{25} +(-919.719 + 531.000i) q^{29} +(348.500 - 603.620i) q^{31} -45.0000i q^{35} -748.000 q^{37} +(592.361 + 342.000i) q^{41} +(-1309.00 - 2267.25i) q^{43} +(2291.50 - 1323.00i) q^{47} +(1188.00 - 2057.68i) q^{49} +1071.00i q^{53} -1053.00 q^{55} +(-5035.07 - 2907.00i) q^{59} +(-3202.00 - 5546.03i) q^{61} +(265.004 - 153.000i) q^{65} +(2609.00 - 4518.92i) q^{67} +6570.00i q^{71} -4519.00 q^{73} +(506.625 + 292.500i) q^{77} +(-3751.00 - 6496.92i) q^{79} +(4746.69 - 2740.50i) q^{83} +(2025.00 - 3507.40i) q^{85} -8874.00i q^{89} -170.000 q^{91} +(-498.831 - 288.000i) q^{95} +(-5285.50 - 9154.75i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{7}+O(q^{10}) \) 4 * q - 10 * q^7	\( 4 q - 10 q^{7} + 68 q^{13} - 256 q^{19} - 1088 q^{25} + 1394 q^{31} - 2992 q^{37} - 5236 q^{43} + 4752 q^{49} - 4212 q^{55} - 12808 q^{61} + 10436 q^{67} - 18076 q^{73} - 15004 q^{79} + 8100 q^{85} - 680 q^{91} - 21142 q^{97}+O(q^{100}) \) 4 * q - 10 * q^7 + 68 * q^13 - 256 * q^19 - 1088 * q^25 + 1394 * q^31 - 2992 * q^37 - 5236 * q^43 + 4752 * q^49 - 4212 * q^55 - 12808 * q^61 + 10436 * q^67 - 18076 * q^73 - 15004 * q^79 + 8100 * q^85 - 680 * q^91 - 21142 * q^97

Introduction

L-functions

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