LMFDB - Embedded newform 531.5.c.d.235.40

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,Mod(235,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("531.235");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	531.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 177)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.40
Character	$\chi$	$=$	531.235
Dual form			531.5.c.d.235.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.81880i q^{2} -45.1337 q^{4} -21.3172 q^{5} -30.7266 q^{7} -227.790i q^{8} +O(q^{10})$	\(q+7.81880i q^{2} -45.1337 q^{4} -21.3172 q^{5} -30.7266 q^{7} -227.790i q^{8} -166.675i q^{10} -207.480i q^{11} +206.020i q^{13} -240.245i q^{14} +1058.91 q^{16} +391.490 q^{17} -321.627 q^{19} +962.125 q^{20} +1622.24 q^{22} -287.967i q^{23} -170.575 q^{25} -1610.83 q^{26} +1386.80 q^{28} -1020.53 q^{29} +560.747i q^{31} +4634.75i q^{32} +3060.98i q^{34} +655.006 q^{35} -1257.89i q^{37} -2514.74i q^{38} +4855.86i q^{40} +1524.79 q^{41} -2155.53i q^{43} +9364.31i q^{44} +2251.55 q^{46} +245.854i q^{47} -1456.88 q^{49} -1333.69i q^{50} -9298.43i q^{52} +1142.98 q^{53} +4422.89i q^{55} +6999.22i q^{56} -7979.34i q^{58} +(3465.52 + 327.946i) q^{59} -518.066i q^{61} -4384.37 q^{62} -19295.6 q^{64} -4391.77i q^{65} +2495.71i q^{67} -17669.4 q^{68} +5121.36i q^{70} -9311.79 q^{71} +8005.70i q^{73} +9835.18 q^{74} +14516.2 q^{76} +6375.14i q^{77} +5130.68 q^{79} -22573.0 q^{80} +11922.1i q^{82} -13428.1i q^{83} -8345.49 q^{85} +16853.7 q^{86} -47261.8 q^{88} +3141.70i q^{89} -6330.29i q^{91} +12997.0i q^{92} -1922.28 q^{94} +6856.20 q^{95} +5322.39i q^{97} -11391.0i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) $ 40 * q - 320 * q^4 + 80 * q^7	$ 40 q - 320 q^{4} + 80 q^{7} + 3944 q^{16} + 528 q^{17} + 444 q^{19} - 444 q^{20} + 1304 q^{22} + 4880 q^{25} + 1452 q^{26} - 1160 q^{28} + 996 q^{29} - 10320 q^{35} + 5196 q^{41} - 10476 q^{46} + 5104 q^{49} + 2184 q^{53} + 11736 q^{59} - 15240 q^{62} - 81012 q^{64} - 29568 q^{68} + 5964 q^{71} - 14376 q^{74} + 3480 q^{76} + 19020 q^{79} - 33096 q^{80} + 20220 q^{85} + 65880 q^{86} - 14932 q^{88} - 17864 q^{94} - 11004 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 + 80 * q^7 + 3944 * q^16 + 528 * q^17 + 444 * q^19 - 444 * q^20 + 1304 * q^22 + 4880 * q^25 + 1452 * q^26 - 1160 * q^28 + 996 * q^29 - 10320 * q^35 + 5196 * q^41 - 10476 * q^46 + 5104 * q^49 + 2184 * q^53 + 11736 * q^59 - 15240 * q^62 - 81012 * q^64 - 29568 * q^68 + 5964 * q^71 - 14376 * q^74 + 3480 * q^76 + 19020 * q^79 - 33096 * q^80 + 20220 * q^85 + 65880 * q^86 - 14932 * q^88 - 17864 * q^94 - 11004 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$1$	$-1$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$			7.81880i			1.95470i	0.211629	+	0.977350i	$0.432123\pi$
$2$			7.81880i			1.95470i	−0.211629	+	0.977350i	$0.567877\pi$
$3$		0			0
$3$		0			0
$4$	−45.1337			−2.82085
$5$	−21.3172			−0.852690			−0.426345	−	0.904561i	$-0.640199\pi$
$5$	−21.3172			−0.852690			−0.426345	+	0.904561i	$0.640199\pi$
$6$		0			0
$7$	−30.7266			−0.627073			−0.313537	−	0.949576i	$-0.601514\pi$
$7$	−30.7266			−0.627073			−0.313537	+	0.949576i	$0.601514\pi$
$8$		−	227.790i		−	3.55922i
$9$		0			0
$10$		−	166.675i		−	1.66675i
$11$		−	207.480i		−	1.71471i	−0.514728	−	0.857353i	$-0.672107\pi$
$11$		−	207.480i		−	1.71471i	0.514728	−	0.857353i	$-0.327893\pi$
$12$		0			0
$13$			206.020i			1.21905i	0.792766	+	0.609526i	$0.208640\pi$
$13$			206.020i			1.21905i	−0.792766	+	0.609526i	$0.791360\pi$
$14$		−	240.245i		−	1.22574i
$15$		0			0
$16$	1058.91			4.13636
$17$	391.490			1.35464			0.677318	−	0.735690i	$-0.263142\pi$
$17$	391.490			1.35464			0.677318	+	0.735690i	$0.263142\pi$
$18$		0			0
$19$	−321.627			−0.890934			−0.445467	−	0.895298i	$-0.646962\pi$
$19$	−321.627			−0.890934			−0.445467	+	0.895298i	$0.646962\pi$
$20$	962.125			2.40531
$21$		0			0
$22$	1622.24			3.35174
$23$		−	287.967i		−	0.544360i	−0.962246	−	0.272180i	$-0.912255\pi$
$23$		−	287.967i		−	0.544360i	0.962246	−	0.272180i	$-0.0877447\pi$
$24$		0			0
$25$	−170.575			−0.272921
$26$	−1610.83			−2.38288
$27$		0			0
$28$	1386.80			1.76888
$29$	−1020.53			−1.21347			−0.606737	−	0.794902i	$-0.707522\pi$
$29$	−1020.53			−1.21347			−0.606737	+	0.794902i	$0.707522\pi$
$30$		0			0
$31$			560.747i			0.583504i	0.956494	+	0.291752i	$0.0942381\pi$
$31$			560.747i			0.583504i	−0.956494	+	0.291752i	$0.905762\pi$
$32$			4634.75i			4.52612i
$33$		0			0
$34$			3060.98i			2.64791i
$35$	655.006			0.534699
$36$		0			0
$37$		−	1257.89i		−	0.918838i	−0.888220	−	0.459419i	$-0.848058\pi$
$37$		−	1257.89i		−	0.918838i	0.888220	−	0.459419i	$-0.151942\pi$
$38$		−	2514.74i		−	1.74151i
$39$		0			0
$40$			4855.86i			3.03491i
$41$	1524.79			0.907076			0.453538	−	0.891237i	$-0.350162\pi$
$41$	1524.79			0.907076			0.453538	+	0.891237i	$0.350162\pi$
$42$		0			0
$43$		−	2155.53i		−	1.16578i	−0.812550	−	0.582892i	$-0.801921\pi$
$43$		−	2155.53i		−	1.16578i	0.812550	−	0.582892i	$-0.198079\pi$
$44$			9364.31i			4.83694i
$45$		0			0
$46$	2251.55			1.06406
$47$			245.854i			0.111296i	0.998450	+	0.0556482i	$0.0177225\pi$
$47$			245.854i			0.111296i	−0.998450	+	0.0556482i	$0.982277\pi$
$48$		0			0
$49$	−1456.88			−0.606779
$50$		−	1333.69i		−	0.533478i
$51$		0			0
$52$		−	9298.43i		−	3.43877i
$53$	1142.98			0.406899			0.203449	−	0.979085i	$-0.434785\pi$
$53$	1142.98			0.406899			0.203449	+	0.979085i	$0.434785\pi$
$54$		0			0
$55$			4422.89i			1.46211i
$56$			6999.22i			2.23189i
$57$		0			0
$58$		−	7979.34i		−	2.37198i
$59$	3465.52	+	327.946i	0.995552	+	0.0942101i
$59$	3465.52	+	327.946i	0.995552	+	0.0942101i
$60$		0			0
$61$		−	518.066i		−	0.139228i	−0.997574	−	0.0696138i	$-0.977823\pi$
$61$		−	518.066i		−	0.139228i	0.997574	−	0.0696138i	$-0.0221767\pi$
$62$	−4384.37			−1.14057
$63$		0			0
$64$	−19295.6			−4.71085
$65$		−	4391.77i		−	1.03947i
$66$		0			0
$67$			2495.71i			0.555961i	0.960587	+	0.277980i	$0.0896650\pi$
$67$			2495.71i			0.555961i	−0.960587	+	0.277980i	$0.910335\pi$
$68$	−17669.4			−3.82123
$69$		0			0
$70$			5121.36i			1.04518i
$71$	−9311.79			−1.84721			−0.923606	−	0.383344i	$-0.874772\pi$
$71$	−9311.79			−1.84721			−0.923606	+	0.383344i	$0.874772\pi$
$72$		0			0
$73$			8005.70i			1.50229i	0.660138	+	0.751144i	$0.270498\pi$
$73$			8005.70i			1.50229i	−0.660138	+	0.751144i	$0.729502\pi$
$74$	9835.18			1.79605
$75$		0			0
$76$	14516.2			2.51319
$77$			6375.14i			1.07525i
$78$		0			0
$79$	5130.68			0.822093			0.411047	−	0.911614i	$-0.365163\pi$
$79$	5130.68			0.822093			0.411047	+	0.911614i	$0.365163\pi$
$80$	−22573.0			−3.52703
$81$		0			0
$82$			11922.1i			1.77306i
$83$		−	13428.1i		−	1.94921i	−0.223939	−	0.974603i	$-0.571892\pi$
$83$		−	13428.1i		−	1.94921i	0.223939	−	0.974603i	$-0.428108\pi$
$84$		0			0
$85$	−8345.49			−1.15508
$86$	16853.7			2.27876
$87$		0			0
$88$	−47261.8			−6.10302
$89$			3141.70i			0.396630i	0.980138	+	0.198315i	$0.0635468\pi$
$89$			3141.70i			0.396630i	−0.980138	+	0.198315i	$0.936453\pi$
$90$		0			0
$91$		−	6330.29i		−	0.764435i
$92$			12997.0i			1.53556i
$93$		0			0
$94$	−1922.28			−0.217551
$95$	6856.20			0.759690
$96$		0			0
$97$			5322.39i			0.565670i	0.959169	+	0.282835i	$0.0912749\pi$
$97$			5322.39i			0.565670i	−0.959169	+	0.282835i	$0.908725\pi$
$98$		−	11391.0i		−	1.18607i
$99$		0			0
$100$	7698.69			0.769869
$101$			6022.91i			0.590424i	0.955432	+	0.295212i	$0.0953902\pi$
$101$			6022.91i			0.590424i	−0.955432	+	0.295212i	$0.904610\pi$
$102$		0			0
$103$		−	3738.09i		−	0.352350i	−0.984359	−	0.176175i	$-0.943627\pi$
$103$		−	3738.09i		−	0.352350i	0.984359	−	0.176175i	$-0.0563725\pi$
$104$	46929.3			4.33888
$105$		0			0
$106$			8936.72i			0.795365i
$107$	−10702.1			−0.934764			−0.467382	−	0.884056i	$-0.654803\pi$
$107$	−10702.1			−0.934764			−0.467382	+	0.884056i	$0.654803\pi$
$108$		0			0
$109$			21298.0i			1.79261i	0.443442	+	0.896303i	$0.353757\pi$
$109$			21298.0i			1.79261i	−0.443442	+	0.896303i	$0.646243\pi$
$110$	−34581.7			−2.85799
$111$		0			0
$112$	−32536.6			−2.59380
$113$			16771.3i			1.31344i	0.754135	+	0.656720i	$0.228057\pi$
$113$			16771.3i			1.31344i	−0.754135	+	0.656720i	$0.771943\pi$
$114$		0			0
$115$			6138.65i			0.464170i
$116$	46060.4			3.42303
$117$		0			0
$118$	−2564.14	+	27096.2i	−0.184153	+	1.94601i
$119$	−12029.2			−0.849456
$120$		0			0
$121$	−28406.8			−1.94022
$122$	4050.65			0.272148
$123$		0			0
$124$		−	25308.6i		−	1.64598i
$125$	16959.5			1.08541
$126$		0			0
$127$	27235.6			1.68861			0.844305	−	0.535863i	$-0.180014\pi$
$127$	27235.6			1.68861			0.844305	+	0.535863i	$0.180014\pi$
$128$		−	76712.8i		−	4.68218i
$129$		0			0
$130$	34338.4			2.03186
$131$		−	1718.72i		−	0.100153i	−0.998745	−	0.0500763i	$-0.984054\pi$
$131$		−	1718.72i		−	0.100153i	0.998745	−	0.0500763i	$-0.0159465\pi$
$132$		0			0
$133$	9882.51			0.558681
$134$	−19513.4			−1.08674
$135$		0			0
$136$		−	89177.6i		−	4.82145i
$137$	−8888.26			−0.473560			−0.236780	−	0.971563i	$-0.576092\pi$
$137$	−8888.26			−0.473560			−0.236780	+	0.971563i	$0.576092\pi$
$138$		0			0
$139$	2771.10			0.143424			0.0717121	−	0.997425i	$-0.477154\pi$
$139$	2771.10			0.143424			0.0717121	+	0.997425i	$0.477154\pi$
$140$	−29562.8			−1.50831
$141$		0			0
$142$		−	72807.1i		−	3.61074i
$143$	42744.9			2.09032
$144$		0			0
$145$	21754.9			1.03472
$146$	−62594.9			−2.93652
$147$		0			0
$148$			56773.1i			2.59191i
$149$		−	8123.08i		−	0.365888i	−0.983123	−	0.182944i	$-0.941437\pi$
$149$		−	8123.08i		−	0.365888i	0.983123	−	0.182944i	$-0.0585627\pi$
$150$		0			0
$151$		−	2438.11i		−	0.106930i	−0.998570	−	0.0534651i	$-0.982973\pi$
$151$		−	2438.11i		−	0.106930i	0.998570	−	0.0534651i	$-0.0170266\pi$
$152$			73263.5i			3.17103i
$153$		0			0
$154$	−49845.9			−2.10179
$155$		−	11953.6i		−	0.497547i
$156$		0			0
$157$			13811.6i			0.560332i	0.959952	+	0.280166i	$0.0903895\pi$
$157$			13811.6i			0.560332i	−0.959952	+	0.280166i	$0.909611\pi$
$158$			40115.8i			1.60695i
$159$		0			0
$160$		−	98800.0i		−	3.85938i
$161$			8848.23i			0.341354i
$162$		0			0
$163$	−14070.7			−0.529593			−0.264796	−	0.964304i	$-0.585305\pi$
$163$	−14070.7			−0.529593			−0.264796	+	0.964304i	$0.585305\pi$
$164$	−68819.5			−2.55873
$165$		0			0
$166$	104991.			3.81011
$167$	37863.4			1.35765			0.678823	−	0.734302i	$-0.262490\pi$
$167$	37863.4			1.35765			0.678823	+	0.734302i	$0.262490\pi$
$168$		0			0
$169$	−13883.2			−0.486089
$170$		−	65251.7i		−	2.25784i
$171$		0			0
$172$			97287.2i			3.28851i
$173$		−	13221.6i		−	0.441765i	−0.975300	−	0.220883i	$-0.929106\pi$
$173$		−	13221.6i		−	0.441765i	0.975300	−	0.220883i	$-0.0708938\pi$
$174$		0			0
$175$	5241.20			0.171141
$176$		−	219702.i		−	7.09264i
$177$		0			0
$178$	−24564.3			−0.775292
$179$			398.257i			0.0124296i	0.999981	+	0.00621480i	$0.00197825\pi$
$179$			398.257i			0.0124296i	−0.999981	+	0.00621480i	$0.998022\pi$
$180$		0			0
$181$	46013.5			1.40452			0.702260	−	0.711920i	$-0.252174\pi$
$181$	46013.5			1.40452			0.702260	+	0.711920i	$0.252174\pi$
$182$	49495.3			1.49424
$183$		0			0
$184$	−65596.0			−1.93750
$185$			26814.7i			0.783483i
$186$		0			0
$187$		−	81226.2i		−	2.32281i
$188$		−	11096.3i		−	0.313951i
$189$		0			0
$190$			53607.3i			1.48497i
$191$			60179.4i			1.64961i	0.565418	+	0.824804i	$0.308715\pi$
$191$			60179.4i			1.64961i	−0.565418	+	0.824804i	$0.691285\pi$
$192$		0			0
$193$	−31955.1			−0.857879			−0.428939	−	0.903333i	$-0.641113\pi$
$193$	−31955.1			−0.857879			−0.428939	+	0.903333i	$0.641113\pi$
$194$	−41614.7			−1.10572
$195$		0			0
$196$	65754.2			1.71163
$197$	34993.6			0.901689			0.450844	−	0.892603i	$-0.351123\pi$
$197$	34993.6			0.901689			0.450844	+	0.892603i	$0.351123\pi$
$198$		0			0
$199$	3731.67			0.0942317			0.0471159	−	0.998889i	$-0.484997\pi$
$199$	3731.67			0.0942317			0.0471159	+	0.998889i	$0.484997\pi$
$200$			38855.4i			0.971385i
$201$		0			0
$202$	−47092.0			−1.15410
$203$	31357.5			0.760938
$204$		0			0
$205$	−32504.4			−0.773454
$206$	29227.4			0.688740
$207$		0			0
$208$			218156.i			5.04244i
$209$			66731.1i			1.52769i
$210$		0			0
$211$			13733.7i			0.308476i	0.988034	+	0.154238i	$0.0492923\pi$
$211$			13733.7i			0.308476i	−0.988034	+	0.154238i	$0.950708\pi$
$212$	−51586.8			−1.14780
$213$		0			0
$214$		−	83677.7i		−	1.82718i
$215$			45950.0i			0.994052i
$216$		0			0
$217$		−	17229.8i		−	0.365899i
$218$	−166524.			−3.50401
$219$		0			0
$220$		−	199621.i		−	4.12441i
$221$			80654.7i			1.65137i
$222$		0			0
$223$	−62492.9			−1.25667			−0.628335	−	0.777943i	$-0.716263\pi$
$223$	−62492.9			−1.25667			−0.628335	+	0.777943i	$0.716263\pi$
$224$		−	142410.i		−	2.83821i
$225$		0			0
$226$	−131132.			−2.56738
$227$		−	15746.2i		−	0.305579i	−0.988259	−	0.152790i	$-0.951174\pi$
$227$		−	15746.2i		−	0.305579i	0.988259	−	0.152790i	$-0.0488257\pi$
$228$		0			0
$229$			50347.4i			0.960078i	0.877247	+	0.480039i	$0.159377\pi$
$229$			50347.4i			0.960078i	−0.877247	+	0.480039i	$0.840623\pi$
$230$	−47996.9			−0.907314
$231$		0			0
$232$			232467.i			4.31903i
$233$			54304.7i			1.00029i	0.865942	+	0.500144i	$0.166720\pi$
$233$			54304.7i			1.00029i	−0.865942	+	0.500144i	$0.833280\pi$
$234$		0			0
$235$		−	5240.92i		−	0.0949012i
$236$	−156411.	−	14801.4i	−2.80831	−	0.265753i
$237$		0			0
$238$		−	94053.6i		−	1.66043i
$239$	−44247.5			−0.774628			−0.387314	−	0.921948i	$-0.626597\pi$
$239$	−44247.5			−0.774628			−0.387314	+	0.921948i	$0.626597\pi$
$240$		0			0
$241$	93615.1			1.61180			0.805902	−	0.592050i	$-0.201681\pi$
$241$	93615.1			1.61180			0.805902	+	0.592050i	$0.201681\pi$
$242$		−	222107.i		−	3.79255i
$243$		0			0
$244$			23382.2i			0.392741i
$245$	31056.6			0.517394
$246$		0			0
$247$		−	66261.6i		−	1.08610i
$248$	127733.			2.07682
$249$		0			0
$250$			132603.i			2.12164i
$251$	31025.2			0.492455			0.246228	−	0.969212i	$-0.420809\pi$
$251$	31025.2			0.492455			0.246228	+	0.969212i	$0.420809\pi$
$252$		0			0
$253$	−59747.2			−0.933419
$254$			212950.i			3.30073i
$255$		0			0
$256$	291072.			4.44141
$257$	51107.2			0.773777			0.386889	−	0.922126i	$-0.373550\pi$
$257$	51107.2			0.773777			0.386889	+	0.922126i	$0.373550\pi$
$258$		0			0
$259$			38650.6i			0.576179i
$260$			198217.i			2.93220i
$261$		0			0
$262$	13438.3			0.195768
$263$	−83386.8			−1.20555			−0.602776	−	0.797911i	$-0.705939\pi$
$263$	−83386.8			−1.20555			−0.602776	+	0.797911i	$0.705939\pi$
$264$		0			0
$265$	−24365.1			−0.346958
$266$			77269.3i			1.09205i
$267$		0			0
$268$		−	112640.i		−	1.56828i
$269$		−	55398.4i		−	0.765584i	−0.923835	−	0.382792i	$-0.874963\pi$
$269$		−	55398.4i		−	0.765584i	0.923835	−	0.382792i	$-0.125037\pi$
$270$		0			0
$271$	33799.6			0.460228			0.230114	−	0.973164i	$-0.426090\pi$
$271$	33799.6			0.460228			0.230114	+	0.973164i	$0.426090\pi$
$272$	414552.			5.60326
$273$		0			0
$274$		−	69495.5i		−	0.925669i
$275$			35390.9i			0.467979i
$276$		0			0
$277$	71567.4			0.932730			0.466365	−	0.884592i	$-0.345563\pi$
$277$	71567.4			0.932730			0.466365	+	0.884592i	$0.345563\pi$
$278$			21666.7i			0.280351i
$279$		0			0
$280$		−	149204.i		−	1.90311i
$281$	−121564.			−1.53955			−0.769774	−	0.638316i	$-0.779631\pi$
$281$	−121564.			−1.53955			−0.769774	+	0.638316i	$0.779631\pi$
$282$		0			0
$283$			142151.i			1.77491i	0.460890	+	0.887457i	$0.347530\pi$
$283$			142151.i			1.77491i	−0.460890	+	0.887457i	$0.652470\pi$
$284$	420275.			5.21071
$285$		0			0
$286$			334214.i			4.08594i
$287$	−46851.7			−0.568803
$288$		0			0
$289$	69743.4			0.835041
$290$			170097.i			2.02256i
$291$		0			0
$292$		−	361326.i		−	4.23774i
$293$	96135.1			1.11982			0.559908	−	0.828555i	$-0.310836\pi$
$293$	96135.1			1.11982			0.559908	+	0.828555i	$0.310836\pi$
$294$		0			0
$295$	−73875.3	−	6990.89i	−0.848897	−	0.0803320i
$296$	−286535.			−3.27035
$297$		0			0
$298$	63512.8			0.715202
$299$	59326.9			0.663604
$300$		0			0
$301$			66232.2i			0.731032i
$302$	19063.1			0.209016
$303$		0			0
$304$	−340574.			−3.68522
$305$			11043.7i			0.118718i
$306$		0			0
$307$	35837.3			0.380240			0.190120	−	0.981761i	$-0.439112\pi$
$307$	35837.3			0.380240			0.190120	+	0.981761i	$0.439112\pi$
$308$		−	287733.i		−	3.03311i
$309$		0			0
$310$	93462.6			0.972556
$311$	55265.8			0.571394			0.285697	−	0.958320i	$-0.407775\pi$
$311$	55265.8			0.571394			0.285697	+	0.958320i	$0.407775\pi$
$312$		0			0
$313$		−	96714.2i		−	0.987192i	−0.869691	−	0.493596i	$-0.835682\pi$
$313$		−	96714.2i		−	0.987192i	0.869691	−	0.493596i	$-0.164318\pi$
$314$	−107990.			−1.09528
$315$		0			0
$316$	−231566.			−2.31900
$317$	132005.			1.31363			0.656813	−	0.754053i	$-0.271904\pi$
$317$	132005.			1.31363			0.656813	+	0.754053i	$0.271904\pi$
$318$		0			0
$319$			211740.i			2.08075i
$320$	411330.			4.01689
$321$		0			0
$322$	−69182.6			−0.667245
$323$	−125914.			−1.20689
$324$		0			0
$325$		−	35141.9i		−	0.332705i
$326$		−	110016.i		−	1.03519i
$327$		0			0
$328$		−	347333.i		−	3.22848i
$329$		−	7554.24i		−	0.0697909i
$330$		0			0
$331$	53983.3			0.492724			0.246362	−	0.969178i	$-0.420765\pi$
$331$	53983.3			0.492724			0.246362	+	0.969178i	$0.420765\pi$
$332$			606058.i			5.49842i
$333$		0			0
$334$			296046.i			2.65379i
$335$		−	53201.6i		−	0.474062i
$336$		0			0
$337$		−	87020.3i		−	0.766233i	−0.923700	−	0.383116i	$-0.874851\pi$
$337$		−	87020.3i		−	0.766233i	0.923700	−	0.383116i	$-0.125149\pi$
$338$		−	108550.i		−	0.950158i
$339$		0			0
$340$	376662.			3.25832
$341$	116344.			1.00054
$342$		0			0
$343$	118539.			1.00757
$344$	−491010.			−4.14928
$345$		0			0
$346$	103377.			0.863519
$347$			74094.0i			0.615353i	0.951491	+	0.307677i	$0.0995515\pi$
$347$			74094.0i			0.615353i	−0.951491	+	0.307677i	$0.900449\pi$
$348$		0			0
$349$		−	27934.6i		−	0.229347i	−0.993403	−	0.114673i	$-0.963418\pi$
$349$		−	27934.6i		−	0.229347i	0.993403	−	0.114673i	$-0.0365821\pi$
$350$			40979.9i			0.334530i
$351$		0			0
$352$	961615.			7.76097
$353$			140969.i			1.13129i	0.824648	+	0.565647i	$0.191373\pi$
$353$			140969.i			1.13129i	−0.824648	+	0.565647i	$0.808627\pi$
$354$		0			0
$355$	198502.			1.57510
$356$		−	141797.i		−	1.11883i
$357$		0			0
$358$	−3113.89			−0.0242961
$359$	29736.8			0.230731			0.115365	−	0.993323i	$-0.463196\pi$
$359$	29736.8			0.230731			0.115365	+	0.993323i	$0.463196\pi$
$360$		0			0
$361$	−26877.0			−0.206237
$362$			359770.i			2.74542i
$363$		0			0
$364$			285709.i			2.15636i
$365$		−	170659.i		−	1.28099i
$366$		0			0
$367$			52026.5i			0.386272i	0.981172	+	0.193136i	$0.0618658\pi$
$367$			52026.5i			0.386272i	−0.981172	+	0.193136i	$0.938134\pi$
$368$		−	304930.i		−	2.25167i
$369$		0			0
$370$	−209659.			−1.53148
$371$	−35119.8			−0.255155
$372$		0			0
$373$	70424.5			0.506181			0.253091	−	0.967443i	$-0.418553\pi$
$373$	70424.5			0.506181			0.253091	+	0.967443i	$0.418553\pi$
$374$	635091.			4.54039
$375$		0			0
$376$	56003.0			0.396128
$377$		−	210250.i		−	1.47929i
$378$		0			0
$379$	−39937.6			−0.278038			−0.139019	−	0.990290i	$-0.544395\pi$
$379$	−39937.6			−0.278038			−0.139019	+	0.990290i	$0.544395\pi$
$380$	−309445.			−2.14297
$381$		0			0
$382$	−470531.			−3.22449
$383$	−155977.			−1.06332			−0.531659	−	0.846959i	$-0.678431\pi$
$383$	−155977.			−1.06332			−0.531659	+	0.846959i	$0.678431\pi$
$384$		0			0
$385$		−	135900.i		−	0.916852i
$386$		−	249851.i		−	1.67690i
$387$		0			0
$388$		−	240219.i		−	1.59567i
$389$	184572.			1.21974			0.609870	−	0.792501i	$-0.291222\pi$
$389$	184572.			1.21974			0.609870	+	0.792501i	$0.291222\pi$
$390$		0			0
$391$		−	112736.i		−	0.737411i
$392$			331862.i			2.15966i
$393$		0			0
$394$			273608.i			1.76253i
$395$	−109372.			−0.700990
$396$		0			0
$397$			216054.i			1.37082i	0.728155	+	0.685412i	$0.240378\pi$
$397$			216054.i			1.37082i	−0.728155	+	0.685412i	$0.759622\pi$
$398$			29177.2i			0.184195i
$399$		0			0
$400$	−180624.			−1.12890
$401$		−	186079.i		−	1.15720i	−0.815611	−	0.578601i	$-0.803599\pi$
$401$		−	186079.i		−	1.15720i	0.815611	−	0.578601i	$-0.196401\pi$
$402$		0			0
$403$	−115525.			−0.711321
$404$		−	271836.i		−	1.66550i
$405$		0			0
$406$			245178.i			1.48741i
$407$	−260986.			−1.57554
$408$		0			0
$409$			142032.i			0.849064i	0.905413	+	0.424532i	$0.139561\pi$
$409$			142032.i			0.849064i	−0.905413	+	0.424532i	$0.860439\pi$
$410$		−	254145.i		−	1.51187i
$411$		0			0
$412$			168713.i			0.993929i
$413$	−106484.	−	10076.6i	−0.624284	−	0.0590767i
$414$		0			0
$415$			286250.i			1.66207i
$416$	−954850.			−5.51758
$417$		0			0
$418$	−521757.			−2.98618
$419$			305281.i			1.73889i	0.494032	+	0.869444i	$0.335523\pi$
$419$			305281.i			1.73889i	−0.494032	+	0.869444i	$0.664477\pi$
$420$		0			0
$421$			40020.9i			0.225799i	0.993606	+	0.112900i	$0.0360139\pi$
$421$			40020.9i			0.225799i	−0.993606	+	0.112900i	$0.963986\pi$
$422$	−107381.			−0.602978
$423$		0			0
$424$		−	260359.i		−	1.44824i
$425$	−66778.6			−0.369708
$426$		0			0
$427$			15918.4i			0.0873059i
$428$	483025.			2.63683
$429$		0			0
$430$	−359274.			−1.94307
$431$			199954.i			1.07641i	0.842815	+	0.538203i	$0.180897\pi$
$431$			199954.i			1.07641i	−0.842815	+	0.538203i	$0.819103\pi$
$432$		0			0
$433$	−173422.			−0.924972			−0.462486	−	0.886627i	$-0.653042\pi$
$433$	−173422.			−0.924972			−0.462486	+	0.886627i	$0.653042\pi$
$434$	134717.			0.715224
$435$		0			0
$436$		−	961254.i		−	5.05668i
$437$			92617.9i			0.484989i
$438$		0			0
$439$	236050.			1.22483			0.612415	−	0.790536i	$-0.290198\pi$
$439$	236050.			1.22483			0.612415	+	0.790536i	$0.290198\pi$
$440$	1.00749e6			5.20398
$441$		0			0
$442$	−630623.			−3.22794
$443$		−	390579.i		−	1.99022i	−0.0987629	−	0.995111i	$-0.531489\pi$
$443$		−	390579.i		−	1.99022i	0.0987629	−	0.995111i	$-0.468511\pi$
$444$		0			0
$445$		−	66972.4i		−	0.338202i
$446$		−	488620.i		−	2.45641i
$447$		0			0
$448$	592889.			2.95405
$449$	−297658.			−1.47647			−0.738236	−	0.674543i	$-0.764341\pi$
$449$	−297658.			−1.47647			−0.738236	+	0.674543i	$0.764341\pi$
$450$		0			0
$451$		−	316364.i		−	1.55537i
$452$		−	756951.i		−	3.70502i
$453$		0			0
$454$	123116.			0.597316
$455$			134944.i			0.651826i
$456$		0			0
$457$			138561.i			0.663449i	0.943376	+	0.331725i	$0.107630\pi$
$457$			138561.i			0.663449i	−0.943376	+	0.331725i	$0.892370\pi$
$458$	−393657.			−1.87666
$459$		0			0
$460$		−	277060.i		−	1.30936i
$461$	249929.			1.17602			0.588010	−	0.808853i	$-0.299911\pi$
$461$	249929.			1.17602			0.588010	+	0.808853i	$0.299911\pi$
$462$		0			0
$463$		−	101021.i		−	0.471248i	−0.971844	−	0.235624i	$-0.924287\pi$
$463$		−	101021.i		−	0.471248i	0.971844	−	0.235624i	$-0.0757134\pi$
$464$	−1.08065e6			−5.01937
$465$		0			0
$466$	−424597.			−1.95526
$467$		−	290164.i		−	1.33048i	−0.746628	−	0.665242i	$-0.768328\pi$
$467$		−	290164.i		−	1.33048i	0.746628	−	0.665242i	$-0.231672\pi$
$468$		0			0
$469$		−	76684.6i		−	0.348628i
$470$	40977.7			0.185503
$471$		0			0
$472$	74702.8	−	789411.i	0.335315	−	3.54339i
$473$	−447229.			−1.99898
$474$		0			0
$475$	54861.7			0.243154
$476$	542920.			2.39619
$477$		0			0
$478$		−	345963.i		−	1.51417i
$479$	90383.7			0.393930			0.196965	−	0.980411i	$-0.436891\pi$
$479$	90383.7			0.393930			0.196965	+	0.980411i	$0.436891\pi$
$480$		0			0
$481$	259150.			1.12011
$482$			731958.i			3.15059i
$483$		0			0
$484$	1.28210e6			5.47308
$485$		−	113459.i		−	0.482341i
$486$		0			0
$487$	151820.			0.640132			0.320066	−	0.947395i	$-0.396295\pi$
$487$	151820.			0.640132			0.320066	+	0.947395i	$0.396295\pi$
$488$	−118010.			−0.495542
$489$		0			0
$490$			242825.i			1.01135i
$491$	103785.			0.430497			0.215249	−	0.976559i	$-0.430944\pi$
$491$	103785.			0.430497			0.215249	+	0.976559i	$0.430944\pi$
$492$		0			0
$493$	−399528.			−1.64382
$494$	518086.			2.12299
$495$		0			0
$496$			593779.i			2.41358i
$497$	286120.			1.15834
$498$		0			0
$499$	469746.			1.88652			0.943262	−	0.332049i	$-0.107740\pi$
$499$	469746.			1.88652			0.943262	+	0.332049i	$0.107740\pi$
$500$	−765443.			−3.06177
$501$		0			0
$502$			242580.i			0.962602i
$503$		−	138093.i		−	0.545801i	−0.962042	−	0.272901i	$-0.912017\pi$
$503$		−	138093.i		−	0.545801i	0.962042	−	0.272901i	$-0.0879831\pi$
$504$		0			0
$505$		−	128392.i		−	0.503448i
$506$		−	467151.i		−	1.82455i
$507$		0			0
$508$	−1.22924e6			−4.76332
$509$		−	123336.i		−	0.476051i	−0.971259	−	0.238026i	$-0.923500\pi$
$509$		−	123336.i		−	0.476051i	0.971259	−	0.238026i	$-0.0765002\pi$
$510$		0			0
$511$		−	245988.i		−	0.942045i
$512$			1.04843e6i			3.99944i
$513$		0			0
$514$			399597.i			1.51250i
$515$			79685.7i			0.300446i
$516$		0			0
$517$	51009.6			0.190841
$518$	−302202.			−1.12626
$519$		0			0
$520$	−1.00040e6			−3.69972
$521$	68858.9			0.253679			0.126840	−	0.991923i	$-0.459517\pi$
$521$	68858.9			0.253679			0.126840	+	0.991923i	$0.459517\pi$
$522$		0			0
$523$	269285.			0.984482			0.492241	−	0.870459i	$-0.336178\pi$
$523$	269285.			0.984482			0.492241	+	0.870459i	$0.336178\pi$
$524$			77572.1i			0.282516i
$525$		0			0
$526$		−	651985.i		−	2.35649i
$527$			219527.i			0.790435i
$528$		0			0
$529$	196916.			0.703672
$530$		−	190506.i		−	0.678199i
$531$		0			0
$532$	−446034.			−1.57596
$533$			314138.i			1.10577i
$534$		0			0
$535$	228139.			0.797063
$536$	568498.			1.97879
$537$		0			0
$538$	433149.			1.49649
$539$			302272.i			1.04045i
$540$		0			0
$541$			110046.i			0.375993i	0.982170	+	0.187997i	$0.0601994\pi$
$541$			110046.i			0.375993i	−0.982170	+	0.187997i	$0.939801\pi$
$542$			264272.i			0.899607i
$543$		0			0
$544$			1.81446e6i			6.13125i
$545$		−	454014.i		−	1.52854i
$546$		0			0
$547$	163036.			0.544889			0.272444	−	0.962172i	$-0.412168\pi$
$547$	163036.			0.544889			0.272444	+	0.962172i	$0.412168\pi$
$548$	401159.			1.33584
$549$		0			0
$550$	−276714.			−0.914758
$551$	328231.			1.08113
$552$		0			0
$553$	−157648.			−0.515513
$554$			559572.i			1.82321i
$555$		0			0
$556$	−125070.			−0.404578
$557$	447290.			1.44171			0.720857	−	0.693084i	$-0.243749\pi$
$557$	447290.			1.44171			0.720857	+	0.693084i	$0.243749\pi$
$558$		0			0
$559$	444083.			1.42115
$560$	693591.			2.21171
$561$		0			0
$562$		−	950487.i		−	3.00936i
$563$		−	50595.3i		−	0.159622i	−0.996810	−	0.0798111i	$-0.974568\pi$
$563$		−	50595.3i		−	0.159622i	0.996810	−	0.0798111i	$-0.0254317\pi$
$564$		0			0
$565$		−	357518.i		−	1.11996i
$566$	−1.11145e6			−3.46943
$567$		0			0
$568$			2.12114e6i			6.57464i
$569$		−	596739.i		−	1.84315i	−0.388204	−	0.921573i	$-0.626904\pi$
$569$		−	596739.i		−	1.84315i	0.388204	−	0.921573i	$-0.373096\pi$
$570$		0			0
$571$		−	2297.80i		−	0.00704759i	−0.999994	−	0.00352379i	$-0.998878\pi$
$571$		−	2297.80i		−	0.00704759i	0.999994	−	0.00352379i	$-0.00112166\pi$
$572$	−1.92923e6			−5.89648
$573$		0			0
$574$		−	366324.i		−	1.11184i
$575$			49120.0i			0.148567i
$576$		0			0
$577$	−635182.			−1.90786			−0.953931	−	0.300027i	$-0.903004\pi$
$577$	−635182.			−1.90786			−0.953931	+	0.300027i	$0.903004\pi$
$578$			545310.i			1.63225i
$579$		0			0
$580$	−981880.			−2.91879
$581$			412599.i			1.22229i
$582$		0			0
$583$		−	237145.i		−	0.697712i
$584$	1.82362e6			5.34698
$585$		0			0
$586$			751661.i			2.18891i
$587$			46453.2i			0.134815i	0.997726	+	0.0674077i	$0.0214728\pi$
$587$			46453.2i			0.134815i	−0.997726	+	0.0674077i	$0.978527\pi$
$588$		0			0
$589$		−	180351.i		−	0.519863i
$590$	54660.4	−	577616.i	0.157025	−	1.65934i
$591$		0			0
$592$		−	1.33199e6i		−	3.80064i
$593$	292298.			0.831221			0.415611	−	0.909543i	$-0.363568\pi$
$593$	292298.			0.831221			0.415611	+	0.909543i	$0.363568\pi$
$594$		0			0
$595$	256428.			0.724323
$596$			366624.i			1.03212i
$597$		0			0
$598$			463865.i			1.29715i
$599$	−239592.			−0.667758			−0.333879	−	0.942616i	$-0.608358\pi$
$599$	−239592.			−0.667758			−0.333879	+	0.942616i	$0.608358\pi$
$600$		0			0
$601$		−	235688.i		−	0.652512i	−0.945281	−	0.326256i	$-0.894213\pi$
$601$		−	235688.i		−	0.652512i	0.945281	−	0.326256i	$-0.105787\pi$
$602$	−517857.			−1.42895
$603$		0			0
$604$			110041.i			0.301634i
$605$	605554.			1.65441
$606$		0			0
$607$	423021.			1.14811			0.574056	−	0.818816i	$-0.305369\pi$
$607$	423021.			1.14811			0.574056	+	0.818816i	$0.305369\pi$
$608$		−	1.49066e6i		−	4.03247i
$609$		0			0
$610$	−86348.7			−0.232058
$611$	−50650.7			−0.135676
$612$		0			0
$613$			210522.i			0.560243i	0.959965	+	0.280122i	$0.0903748\pi$
$613$			210522.i			0.560243i	−0.959965	+	0.280122i	$0.909625\pi$
$614$			280205.i			0.743256i
$615$		0			0
$616$	1.45219e6			3.82704
$617$	342587.			0.899914			0.449957	−	0.893050i	$-0.351439\pi$
$617$	342587.			0.899914			0.449957	+	0.893050i	$0.351439\pi$
$618$		0			0
$619$	−202870.			−0.529463			−0.264732	−	0.964322i	$-0.585283\pi$
$619$	−202870.			−0.529463			−0.264732	+	0.964322i	$0.585283\pi$
$620$			539508.i			1.40351i
$621$		0			0
$622$			432112.i			1.11690i
$623$		−	96533.8i		−	0.248716i
$624$		0			0
$625$	−254919.			−0.652594
$626$	756189.			1.92966
$627$		0			0
$628$		−	623369.i		−	1.58061i
$629$		−	492451.i		−	1.24469i
$630$		0			0
$631$	−41966.6			−0.105401			−0.0527006	−	0.998610i	$-0.516783\pi$
$631$	−41966.6			−0.105401			−0.0527006	+	0.998610i	$0.516783\pi$
$632$		−	1.16872e6i		−	2.92601i
$633$		0			0
$634$			1.03212e6i			2.56775i
$635$	−580587.			−1.43986
$636$		0			0
$637$		−	300146.i		−	0.739696i
$638$	−1.65555e6			−4.06725
$639$		0			0
$640$			1.63531e6i			3.99245i
$641$	336370.			0.818656			0.409328	−	0.912387i	$-0.365763\pi$
$641$	336370.			0.818656			0.409328	+	0.912387i	$0.365763\pi$
$642$		0			0
$643$	−387046.			−0.936140			−0.468070	−	0.883691i	$-0.655051\pi$
$643$	−387046.			−0.936140			−0.468070	+	0.883691i	$0.655051\pi$
$644$		−	399353.i		−	0.962909i
$645$		0			0
$646$		−	984495.i		−	2.35911i
$647$	−640773.			−1.53072			−0.765360	−	0.643602i	$-0.777439\pi$
$647$	−640773.			−1.53072			−0.765360	+	0.643602i	$0.777439\pi$
$648$		0			0
$649$	68042.0	−	719024.i	0.161543	−	1.70708i
$650$	274768.			0.650338
$651$		0			0
$652$	635064.			1.49390
$653$	−59695.3			−0.139995			−0.0699977	−	0.997547i	$-0.522299\pi$
$653$	−59695.3			−0.139995			−0.0699977	+	0.997547i	$0.522299\pi$
$654$		0			0
$655$			36638.4i			0.0853991i
$656$	1.61462e6			3.75199
$657$		0			0
$658$	59065.1			0.136420
$659$		−	277090.i		−	0.638043i	−0.947747	−	0.319022i	$-0.896646\pi$
$659$		−	277090.i		−	0.638043i	0.947747	−	0.319022i	$-0.103354\pi$
$660$		0			0
$661$	−170596.			−0.390450			−0.195225	−	0.980758i	$-0.562544\pi$
$661$	−170596.			−0.390450			−0.195225	+	0.980758i	$0.562544\pi$
$662$			422085.i			0.963128i
$663$		0			0
$664$	−3.05879e6			−6.93766
$665$	−210668.			−0.476381
$666$		0			0
$667$			293879.i			0.660568i
$668$	−1.70891e6			−3.82972
$669$		0			0
$670$	415973.			0.926649
$671$	−107488.			−0.238734
$672$		0			0
$673$		−	124494.i		−	0.274864i	−0.990511	−	0.137432i	$-0.956115\pi$
$673$		−	124494.i		−	0.274864i	0.990511	−	0.137432i	$-0.0438848\pi$
$674$	680394.			1.49775
$675$		0			0
$676$	626599.			1.37119
$677$	−470982.			−1.02761			−0.513804	−	0.857908i	$-0.671764\pi$
$677$	−470982.			−1.02761			−0.513804	+	0.857908i	$0.671764\pi$
$678$		0			0
$679$		−	163539.i		−	0.354716i
$680$			1.90102e6i			4.11120i
$681$		0			0
$682$			909667.i			1.95575i
$683$		−	543532.i		−	1.16516i	−0.812775	−	0.582578i	$-0.802044\pi$
$683$		−	543532.i		−	1.16516i	0.812775	−	0.582578i	$-0.197956\pi$
$684$		0			0
$685$	189473.			0.403800
$686$			926836.i			1.96949i
$687$		0			0
$688$		−	2.28251e6i		−	4.82210i
$689$			235476.i			0.496031i
$690$		0			0
$691$		−	466845.i		−	0.977725i	−0.872361	−	0.488862i	$-0.837412\pi$
$691$		−	466845.i		−	0.977725i	0.872361	−	0.488862i	$-0.162588\pi$
$692$			596739.i			1.24615i
$693$		0			0
$694$	−579327.			−1.20283
$695$	−59072.1			−0.122296
$696$		0			0
$697$	596942.			1.22876
$698$	218415.			0.448304
$699$		0			0
$700$	−236554.			−0.482764
$701$			661106.i			1.34535i	0.739939	+	0.672674i	$0.234855\pi$
$701$			661106.i			1.34535i	−0.739939	+	0.672674i	$0.765145\pi$
$702$		0			0
$703$			404571.i			0.818624i
$704$			4.00345e6i			8.07773i
$705$		0			0
$706$	−1.10221e6			−2.21134
$707$		−	185064.i		−	0.370239i
$708$		0			0
$709$	−64729.8			−0.128769			−0.0643845	−	0.997925i	$-0.520508\pi$
$709$	−64729.8			−0.128769			−0.0643845	+	0.997925i	$0.520508\pi$
$710$			1.55205e6i			3.07884i
$711$		0			0
$712$	715649.			1.41169
$713$	161476.			0.317636
$714$		0			0
$715$	−911203.			−1.78239
$716$		−	17974.8i		−	0.0350621i
$717$		0			0
$718$			232506.i			0.451009i
$719$		−	648569.i		−	1.25458i	−0.778786	−	0.627290i	$-0.784164\pi$
$719$		−	648569.i		−	1.25458i	0.778786	−	0.627290i	$-0.215836\pi$
$720$		0			0
$721$			114859.i			0.220950i
$722$		−	210146.i		−	0.403131i
$723$		0			0
$724$	−2.07676e6			−3.96195
$725$	174078.			0.331182
$726$		0			0
$727$	201809.			0.381831			0.190916	−	0.981606i	$-0.438854\pi$
$727$	201809.			0.381831			0.190916	+	0.981606i	$0.438854\pi$
$728$	−1.44198e6			−2.72079
$729$		0			0
$730$	1.33435e6			2.50394
$731$		−	843870.i		−	1.57921i
$732$		0			0
$733$	−517718.			−0.963575			−0.481787	−	0.876288i	$-0.660012\pi$
$733$	−517718.			−0.963575			−0.481787	+	0.876288i	$0.660012\pi$
$734$	−406785.			−0.755045
$735$		0			0
$736$	1.33465e6			2.46384
$737$	517808.			0.953310
$738$		0			0
$739$		−	1.01019e6i		−	1.84976i	−0.380263	−	0.924879i	$-0.624166\pi$
$739$		−	1.01019e6i		−	1.84976i	0.380263	−	0.924879i	$-0.375834\pi$
$740$		−	1.21025e6i		−	2.21009i
$741$		0			0
$742$		−	274595.i		−	0.498752i
$743$	521556.			0.944765			0.472382	−	0.881394i	$-0.343394\pi$
$743$	521556.			0.944765			0.472382	+	0.881394i	$0.343394\pi$
$744$		0			0
$745$			173162.i			0.311989i
$746$			550635.i			0.989432i
$747$		0			0
$748$			3.66603e6i			6.55229i
$749$	328839.			0.586165
$750$		0			0
$751$			879845.i			1.56001i	0.625776	+	0.780003i	$0.284783\pi$
$751$			879845.i			1.56001i	−0.625776	+	0.780003i	$0.715217\pi$
$752$			260336.i			0.460362i
$753$		0			0
$754$	1.64390e6			2.89157
$755$			51973.9i			0.0911782i
$756$		0			0
$757$	−1.05871e6			−1.84750			−0.923748	−	0.383000i	$-0.874891\pi$
$757$	−1.05871e6			−1.84750			−0.923748	+	0.383000i	$0.874891\pi$
$758$		−	312264.i		−	0.543481i
$759$		0			0
$760$		−	1.56178e6i		−	2.70391i
$761$	319446.			0.551604			0.275802	−	0.961214i	$-0.411057\pi$
$761$	319446.			0.551604			0.275802	+	0.961214i	$0.411057\pi$
$762$		0			0
$763$		−	654414.i		−	1.12410i
$764$		−	2.71612e6i		−	4.65330i
$765$		0			0
$766$		−	1.21955e6i		−	2.07847i
$767$	−67563.3	+	713966.i	−0.114847	+	1.21363i
$768$		0			0
$769$			547254.i			0.925414i	0.886511	+	0.462707i	$0.153122\pi$
$769$			547254.i			0.925414i	−0.886511	+	0.462707i	$0.846878\pi$
$770$	1.06258e6			1.79217
$771$		0			0
$772$	1.44225e6			2.41995
$773$			385413.i			0.645011i	0.946568	+	0.322505i	$0.104525\pi$
$773$			385413.i			0.645011i	−0.946568	+	0.322505i	$0.895475\pi$
$774$		0			0
$775$		−	95649.6i		−	0.159250i
$776$	1.21239e6			2.01335
$777$		0			0
$778$			1.44313e6i			2.38423i
$779$	−490415.			−0.808144
$780$		0			0
$781$			1.93201e6i			3.16743i
$782$	881461.			1.44142
$783$		0			0
$784$	−1.54270e6			−2.50986
$785$		−	294426.i		−	0.477789i
$786$		0			0
$787$	1.03990e6			1.67896			0.839481	−	0.543389i	$-0.182859\pi$
$787$	1.03990e6			1.67896			0.839481	+	0.543389i	$0.182859\pi$
$788$	−1.57939e6			−2.54353
$789$		0			0
$790$		−	855158.i		−	1.37023i
$791$		−	515325.i		−	0.823623i
$792$		0			0
$793$	106732.			0.169726
$794$	−1.68929e6			−2.67955
$795$		0			0
$796$	−168424.			−0.265814
$797$			436583.i			0.687306i	0.939097	+	0.343653i	$0.111664\pi$
$797$			436583.i			0.687306i	−0.939097	+	0.343653i	$0.888336\pi$
$798$		0			0
$799$			96249.2i			0.150766i
$800$		−	790574.i		−	1.23527i
$801$		0			0
$802$	1.45492e6			2.26198
$803$	1.66102e6			2.57598
$804$		0			0
$805$		−	188620.i		−	0.291069i
$806$		−	903267.i		−	1.39042i
$807$		0			0
$808$	1.37196e6			2.10145
$809$			43381.1i			0.0662832i	0.999451	+	0.0331416i	$0.0105512\pi$
$809$			43381.1i			0.0662832i	−0.999451	+	0.0331416i	$0.989449\pi$
$810$		0			0
$811$			112796.i			0.171495i	0.996317	+	0.0857476i	$0.0273279\pi$
$811$			112796.i			0.171495i	−0.996317	+	0.0857476i	$0.972672\pi$
$812$	−1.41528e6			−2.14649
$813$		0			0
$814$		−	2.04060e6i		−	3.07970i
$815$	299949.			0.451578
$816$		0			0
$817$			693279.i			1.03864i
$818$	−1.11052e6			−1.65967
$819$		0			0
$820$	1.46704e6			2.18180
$821$		−	606308.i		−	0.899512i	−0.893152	−	0.449756i	$-0.851511\pi$
$821$		−	606308.i		−	0.899512i	0.893152	−	0.449756i	$-0.148489\pi$
$822$		0			0
$823$		−	883912.i		−	1.30500i	−0.757790	−	0.652499i	$-0.773721\pi$
$823$		−	883912.i		−	1.30500i	0.757790	−	0.652499i	$-0.226279\pi$
$824$	−851500.			−1.25409
$825$		0			0
$826$	78787.3	−	832574.i	0.115477	−	1.22029i
$827$	4182.48			0.00611537			0.00305769	−	0.999995i	$-0.499027\pi$
$827$	4182.48			0.00611537			0.00305769	+	0.999995i	$0.499027\pi$
$828$		0			0
$829$	1.03471e6			1.50561			0.752803	−	0.658246i	$-0.228701\pi$
$829$	1.03471e6			1.50561			0.752803	+	0.658246i	$0.228701\pi$
$830$	−2.23813e6			−3.24884
$831$		0			0
$832$		−	3.97529e6i		−	5.74277i
$833$	−570353.			−0.821965
$834$		0			0
$835$	−807143.			−1.15765
$836$		−	3.01182e6i		−	4.30939i
$837$		0			0
$838$	−2.38693e6			−3.39900
$839$			1.28432e6i			1.82452i	0.409611	+	0.912260i	$0.365664\pi$
$839$			1.28432e6i			1.82452i	−0.409611	+	0.912260i	$0.634336\pi$
$840$		0			0
$841$	334205.			0.472521
$842$	−312916.			−0.441370
$843$		0			0
$844$		−	619851.i		−	0.870166i
$845$	295951.			0.414483
$846$		0			0
$847$	872843.			1.21666
$848$	1.21031e6			1.68308
$849$		0			0
$850$		−	522128.i		−	0.722669i
$851$	−362230.			−0.500179
$852$		0			0
$853$	131299.			0.180452			0.0902262	−	0.995921i	$-0.471241\pi$
$853$	131299.			0.180452			0.0902262	+	0.995921i	$0.471241\pi$
$854$	−124463.			−0.170657
$855$		0			0
$856$			2.43784e6i			3.32703i
$857$			41212.7i			0.0561138i	0.999606	+	0.0280569i	$0.00893196\pi$
$857$			41212.7i			0.0561138i	−0.999606	+	0.0280569i	$0.991068\pi$
$858$		0			0
$859$			794929.i			1.07731i	0.842525	+	0.538657i	$0.181068\pi$
$859$			794929.i			1.07731i	−0.842525	+	0.538657i	$0.818932\pi$
$860$		−	2.07389e6i		−	2.80407i
$861$		0			0
$862$	−1.56340e6			−2.10405
$863$		−	675152.i		−	0.906525i	−0.891377	−	0.453263i	$-0.850260\pi$
$863$		−	675152.i		−	0.906525i	0.891377	−	0.453263i	$-0.149740\pi$
$864$		0			0
$865$			281848.i			0.376689i
$866$		−	1.35595e6i		−	1.80804i
$867$		0			0
$868$			777646.i			1.03215i
$869$		−	1.06451e6i		−	1.40965i
$870$		0			0
$871$	−514165.			−0.677745
$872$	4.85147e6			6.38028
$873$		0			0
$874$	−724161.			−0.948009
$875$	−521107.			−0.680629
$876$		0			0
$877$	461113.			0.599526			0.299763	−	0.954014i	$-0.403092\pi$
$877$	461113.			0.599526			0.299763	+	0.954014i	$0.403092\pi$
$878$			1.84563e6i			2.39418i
$879$		0			0
$880$			4.68343e6i			6.04782i
$881$		−	39808.0i		−	0.0512883i	−0.999671	−	0.0256442i	$-0.991836\pi$
$881$		−	39808.0i		−	0.0512883i	0.999671	−	0.0256442i	$-0.00816368\pi$
$882$		0			0
$883$	−206390.			−0.264708			−0.132354	−	0.991203i	$-0.542254\pi$
$883$	−206390.			−0.264708			−0.132354	+	0.991203i	$0.542254\pi$
$884$		−	3.64024e6i		−	4.65828i
$885$		0			0
$886$	3.05386e6			3.89029
$887$		−	488379.i		−	0.620740i	−0.950616	−	0.310370i	$-0.899547\pi$
$887$		−	488379.i		−	0.620740i	0.950616	−	0.310370i	$-0.100453\pi$
$888$		0			0
$889$	−836857.			−1.05888
$890$	523644.			0.661083
$891$		0			0
$892$	2.82053e6			3.54488
$893$		−	79073.2i		−	0.0991577i
$894$		0			0
$895$		−	8489.74i		−	0.0105986i
$896$			2.35712e6i			2.93607i
$897$		0			0
$898$		−	2.32733e6i		−	2.88606i
$899$		−	572260.i		−	0.708067i
$900$		0			0
$901$	447464.			0.551200
$902$	2.47358e6			3.04028
$903$		0			0
$904$	3.82034e6			4.67482
$905$	−980880.			−1.19762
$906$		0			0
$907$	−664689.			−0.807986			−0.403993	−	0.914762i	$-0.632378\pi$
$907$	−664689.			−0.807986			−0.403993	+	0.914762i	$0.632378\pi$
$908$			710683.i			0.861994i
$909$		0			0
$910$	−1.05510e6			−1.27412
$911$	−467295.			−0.563060			−0.281530	−	0.959552i	$-0.590842\pi$
$911$	−467295.			−0.563060			−0.281530	+	0.959552i	$0.590842\pi$
$912$		0			0
$913$	−2.78605e6			−3.34232
$914$	−1.08338e6			−1.29684
$915$		0			0
$916$		−	2.27236e6i		−	2.70824i
$917$			52810.4i			0.0628030i
$918$		0			0
$919$		−	32821.0i		−	0.0388616i	−0.999811	−	0.0194308i	$-0.993815\pi$
$919$		−	32821.0i		−	0.0388616i	0.999811	−	0.0194308i	$-0.00618541\pi$
$920$	1.39833e6			1.65209
$921$		0			0
$922$			1.95415e6i			2.29877i
$923$		−	1.91841e6i		−	2.25185i
$924$		0			0
$925$			214565.i			0.250770i
$926$	789863.			0.921149
$927$		0			0
$928$		−	4.72991e6i		−	5.49233i
$929$			1.32545e6i			1.53579i	0.640577	+	0.767894i	$0.278695\pi$
$929$			1.32545e6i			1.53579i	−0.640577	+	0.767894i	$0.721305\pi$
$930$		0			0
$931$	468571.			0.540600
$932$		−	2.45097e6i		−	2.82167i
$933$		0			0
$934$	2.26873e6			2.60070
$935$			1.73152e6i			1.98063i
$936$		0			0
$937$		−	45039.7i		−	0.0512998i	−0.999671	−	0.0256499i	$-0.991834\pi$
$937$		−	45039.7i		−	0.0512998i	0.999671	−	0.0256499i	$-0.00816552\pi$
$938$	599582.			0.681463
$939$		0			0
$940$			236542.i			0.267702i
$941$		−	517189.i		−	0.584077i	−0.956407	−	0.292038i	$-0.905667\pi$
$941$		−	517189.i		−	0.584077i	0.956407	−	0.292038i	$-0.0943335\pi$
$942$		0			0
$943$		−	439090.i		−	0.493776i
$944$	3.66966e6	+	347264.i	4.11796	+	0.389687i
$945$		0			0
$946$		−	3.49680e6i		−	3.90740i
$947$	−510466.			−0.569203			−0.284601	−	0.958646i	$-0.591861\pi$
$947$	−510466.			−0.569203			−0.284601	+	0.958646i	$0.591861\pi$
$948$		0			0
$949$	−1.64933e6			−1.83137
$950$			428953.i			0.475294i
$951$		0			0
$952$			2.74012e6i			3.02340i
$953$	26017.8			0.0286473			0.0143237	−	0.999897i	$-0.495440\pi$
$953$	26017.8			0.0286473			0.0143237	+	0.999897i	$0.495440\pi$
$954$		0			0
$955$		−	1.28286e6i		−	1.40660i
$956$	1.99705e6			2.18511
$957$		0			0
$958$			706692.i			0.770015i
$959$	273106.			0.296957
$960$		0			0
$961$	609084.			0.659524
$962$			2.02624e6i			2.18948i
$963$		0			0
$964$	−4.22519e6			−4.54666
$965$	681195.			0.731504
$966$		0			0
$967$			262570.i			0.280797i	0.990095	+	0.140399i	$0.0448384\pi$
$967$			262570.i			0.280797i	−0.990095	+	0.140399i	$0.955162\pi$
$968$			6.47078e6i			6.90567i
$969$		0			0
$970$	887110.			0.942832
$971$	787423.			0.835160			0.417580	−	0.908640i	$-0.362878\pi$
$971$	787423.			0.835160			0.417580	+	0.908640i	$0.362878\pi$
$972$		0			0
$973$	−85146.4			−0.0899374
$974$			1.18705e6i			1.25127i
$975$		0			0
$976$		−	548584.i		−	0.575895i
$977$			252812.i			0.264855i	0.991193	+	0.132428i	$0.0422772\pi$
$977$			252812.i			0.264855i	−0.991193	+	0.132428i	$0.957723\pi$
$978$		0			0
$979$	651839.			0.680103
$980$	−1.40170e6			−1.45949
$981$		0			0
$982$			811472.i			0.841493i
$983$			1.29505e6i			1.34023i	0.742256	+	0.670116i	$0.233756\pi$
$983$			1.29505e6i			1.34023i	−0.742256	+	0.670116i	$0.766244\pi$
$984$		0			0
$985$	−745968.			−0.768861
$986$		−	3.12383e6i		−	3.21317i
$987$		0			0
$988$			2.99063e6i			3.06372i
$989$	−620722.			−0.634607
$990$		0			0
$991$			1.07822e6i			1.09789i	0.835857	+	0.548947i	$0.184971\pi$
$991$			1.07822e6i			1.09789i	−0.835857	+	0.548947i	$0.815029\pi$
$992$	−2.59892e6			−2.64101
$993$		0			0
$994$			2.23711e6i			2.26420i
$995$	−79548.9			−0.0803504
$996$		0			0
$997$	565271.			0.568678			0.284339	−	0.958724i	$-0.408226\pi$
$997$	565271.			0.568678			0.284339	+	0.958724i	$0.408226\pi$
$998$			3.67285e6i			3.68759i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.c.d.235.40		40
3.2	odd	2		177.5.c.a.58.1	✓	40
59.58	odd	2	inner	531.5.c.d.235.1		40
177.176	even	2		177.5.c.a.58.40	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.1	✓	40	3.2	odd	2
177.5.c.a.58.40	yes	40	177.176	even	2
531.5.c.d.235.1		40	59.58	odd	2	inner
531.5.c.d.235.40		40	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.81880i q^{2} -45.1337 q^{4} -21.3172 q^{5} -30.7266 q^{7} -227.790i q^{8} +O(q^{10})\)	\(q+7.81880i q^{2} -45.1337 q^{4} -21.3172 q^{5} -30.7266 q^{7} -227.790i q^{8} -166.675i q^{10} -207.480i q^{11} +206.020i q^{13} -240.245i q^{14} +1058.91 q^{16} +391.490 q^{17} -321.627 q^{19} +962.125 q^{20} +1622.24 q^{22} -287.967i q^{23} -170.575 q^{25} -1610.83 q^{26} +1386.80 q^{28} -1020.53 q^{29} +560.747i q^{31} +4634.75i q^{32} +3060.98i q^{34} +655.006 q^{35} -1257.89i q^{37} -2514.74i q^{38} +4855.86i q^{40} +1524.79 q^{41} -2155.53i q^{43} +9364.31i q^{44} +2251.55 q^{46} +245.854i q^{47} -1456.88 q^{49} -1333.69i q^{50} -9298.43i q^{52} +1142.98 q^{53} +4422.89i q^{55} +6999.22i q^{56} -7979.34i q^{58} +(3465.52 + 327.946i) q^{59} -518.066i q^{61} -4384.37 q^{62} -19295.6 q^{64} -4391.77i q^{65} +2495.71i q^{67} -17669.4 q^{68} +5121.36i q^{70} -9311.79 q^{71} +8005.70i q^{73} +9835.18 q^{74} +14516.2 q^{76} +6375.14i q^{77} +5130.68 q^{79} -22573.0 q^{80} +11922.1i q^{82} -13428.1i q^{83} -8345.49 q^{85} +16853.7 q^{86} -47261.8 q^{88} +3141.70i q^{89} -6330.29i q^{91} +12997.0i q^{92} -1922.28 q^{94} +6856.20 q^{95} +5322.39i q^{97} -11391.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) \) 40 * q - 320 * q^4 + 80 * q^7	\( 40 q - 320 q^{4} + 80 q^{7} + 3944 q^{16} + 528 q^{17} + 444 q^{19} - 444 q^{20} + 1304 q^{22} + 4880 q^{25} + 1452 q^{26} - 1160 q^{28} + 996 q^{29} - 10320 q^{35} + 5196 q^{41} - 10476 q^{46} + 5104 q^{49} + 2184 q^{53} + 11736 q^{59} - 15240 q^{62} - 81012 q^{64} - 29568 q^{68} + 5964 q^{71} - 14376 q^{74} + 3480 q^{76} + 19020 q^{79} - 33096 q^{80} + 20220 q^{85} + 65880 q^{86} - 14932 q^{88} - 17864 q^{94} - 11004 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 + 80 * q^7 + 3944 * q^16 + 528 * q^17 + 444 * q^19 - 444 * q^20 + 1304 * q^22 + 4880 * q^25 + 1452 * q^26 - 1160 * q^28 + 996 * q^29 - 10320 * q^35 + 5196 * q^41 - 10476 * q^46 + 5104 * q^49 + 2184 * q^53 + 11736 * q^59 - 15240 * q^62 - 81012 * q^64 - 29568 * q^68 + 5964 * q^71 - 14376 * q^74 + 3480 * q^76 + 19020 * q^79 - 33096 * q^80 + 20220 * q^85 + 65880 * q^86 - 14932 * q^88 - 17864 * q^94 - 11004 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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