LMFDB - Embedded newform 336.6.h.a.239.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(239,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("336.239");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.h (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:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 444940 x^{16} + 56262171366 x^{12} + \cdots + 77\!\cdots\!16 $ x^20 + 444940x^16 + 56262171366x^12 + 1699160102026524x^8 + 13319972960343660513x^4 + 772185348835324025616
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{35}\cdot 3^{14}\cdot 7^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.8
Root			$-7.46608 - 7.46608i$ of defining polynomial
Character	$\chi$	$=$	336.239
Dual form			336.6.h.a.239.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.46608 + 13.6842i) q^{3} +31.0107i q^{5} +49.0000i q^{7} +(-131.515 - 204.335i) q^{9} +O(q^{10})$	\(q+(-7.46608 + 13.6842i) q^{3} +31.0107i q^{5} +49.0000i q^{7} +(-131.515 - 204.335i) q^{9} +273.251 q^{11} +431.702 q^{13} +(-424.356 - 231.528i) q^{15} +648.939i q^{17} +2309.56i q^{19} +(-670.526 - 365.838i) q^{21} +3727.31 q^{23} +2163.34 q^{25} +(3778.07 - 274.100i) q^{27} +7005.12i q^{29} -1725.19i q^{31} +(-2040.11 + 3739.22i) q^{33} -1519.52 q^{35} -12257.2 q^{37} +(-3223.13 + 5907.51i) q^{39} -16640.7i q^{41} +9182.45i q^{43} +(6336.56 - 4078.37i) q^{45} +23236.3 q^{47} -2401.00 q^{49} +(-8880.22 - 4845.04i) q^{51} +19112.1i q^{53} +8473.68i q^{55} +(-31604.5 - 17243.4i) q^{57} +10334.1 q^{59} -7055.07 q^{61} +(10012.4 - 6444.24i) q^{63} +13387.4i q^{65} -46154.6i q^{67} +(-27828.4 + 51005.3i) q^{69} -75909.3 q^{71} -51116.5 q^{73} +(-16151.7 + 29603.6i) q^{75} +13389.3i q^{77} +10609.7i q^{79} +(-24456.5 + 53746.3i) q^{81} +67540.1 q^{83} -20124.0 q^{85} +(-95859.5 - 52300.8i) q^{87} +73333.0i q^{89} +21153.4i q^{91} +(23607.9 + 12880.4i) q^{93} -71621.1 q^{95} -134051. q^{97} +(-35936.6 - 55834.6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 140 q^{9}+O(q^{10}) $ 20 * q - 140 * q^9	$ 20 q - 140 q^{9} + 1048 q^{13} - 980 q^{21} + 5916 q^{25} - 26056 q^{33} + 61360 q^{37} - 92512 q^{45} - 48020 q^{49} - 20720 q^{57} + 46680 q^{61} - 28360 q^{69} - 54280 q^{73} + 152660 q^{81} - 150536 q^{85} + 41688 q^{93} - 421352 q^{97}+O(q^{100}) $ 20 * q - 140 * q^9 + 1048 * q^13 - 980 * q^21 + 5916 * q^25 - 26056 * q^33 + 61360 * q^37 - 92512 * q^45 - 48020 * q^49 - 20720 * q^57 + 46680 * q^61 - 28360 * q^69 - 54280 * q^73 + 152660 * q^81 - 150536 * q^85 + 41688 * q^93 - 421352 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−7.46608	+	13.6842i	−0.478949	+	0.877842i
$3$	−7.46608	+	13.6842i	−0.478949	+	0.877842i
$4$		0			0
$5$			31.0107i			0.554736i	0.960764	+	0.277368i	$0.0894621\pi$
$5$			31.0107i			0.554736i	−0.960764	+	0.277368i	$0.910538\pi$
$6$		0			0
$7$			49.0000i			0.377964i
$7$			49.0000i			0.377964i
$8$		0			0
$9$	−131.515	−	204.335i	−0.541215	−	0.840884i
$10$		0			0
$11$	273.251			0.680894			0.340447	−	0.940264i	$-0.389422\pi$
$11$	273.251			0.680894			0.340447	+	0.940264i	$0.389422\pi$
$12$		0			0
$13$	431.702			0.708478			0.354239	−	0.935155i	$-0.384740\pi$
$13$	431.702			0.708478			0.354239	+	0.935155i	$0.384740\pi$
$14$		0			0
$15$	−424.356	−	231.528i	−0.486970	−	0.265690i
$16$		0			0
$17$			648.939i			0.544605i	0.962212	+	0.272303i	$0.0877852\pi$
$17$			648.939i			0.544605i	−0.962212	+	0.272303i	$0.912215\pi$
$18$		0			0
$19$			2309.56i			1.46773i	0.679296	+	0.733864i	$0.262285\pi$
$19$			2309.56i			1.46773i	−0.679296	+	0.733864i	$0.737715\pi$
$20$		0			0
$21$	−670.526	−	365.838i	−0.331793	−	0.181026i
$22$		0			0
$23$	3727.31			1.46918			0.734592	−	0.678509i	$-0.237373\pi$
$23$	3727.31			1.46918			0.734592	+	0.678509i	$0.237373\pi$
$24$		0			0
$25$	2163.34			0.692268
$26$		0			0
$27$	3778.07	−	274.100i	0.997379	−	0.0723602i
$28$		0			0
$29$			7005.12i			1.54675i	0.633948	+	0.773376i	$0.281433\pi$
$29$			7005.12i			1.54675i	−0.633948	+	0.773376i	$0.718567\pi$
$30$		0			0
$31$		−	1725.19i		−	0.322429i	−0.986919	−	0.161214i	$-0.948459\pi$
$31$		−	1725.19i		−	0.322429i	0.986919	−	0.161214i	$-0.0515410\pi$
$32$		0			0
$33$	−2040.11	+	3739.22i	−0.326114	+	0.597718i
$34$		0			0
$35$	−1519.52			−0.209670
$36$		0			0
$37$	−12257.2			−1.47193			−0.735967	−	0.677018i	$-0.763272\pi$
$37$	−12257.2			−1.47193			−0.735967	+	0.677018i	$0.763272\pi$
$38$		0			0
$39$	−3223.13	+	5907.51i	−0.339325	+	0.621932i
$40$		0			0
$41$		−	16640.7i		−	1.54601i	−0.634401	−	0.773004i	$-0.718753\pi$
$41$		−	16640.7i		−	1.54601i	0.634401	−	0.773004i	$-0.281247\pi$
$42$		0			0
$43$			9182.45i			0.757334i	0.925533	+	0.378667i	$0.123617\pi$
$43$			9182.45i			0.757334i	−0.925533	+	0.378667i	$0.876383\pi$
$44$		0			0
$45$	6336.56	−	4078.37i	0.466469	−	0.300231i
$46$		0			0
$47$	23236.3			1.53434			0.767171	−	0.641443i	$-0.221664\pi$
$47$	23236.3			1.53434			0.767171	+	0.641443i	$0.221664\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	−8880.22	−	4845.04i	−0.478078	−	0.260838i
$52$		0			0
$53$			19112.1i			0.934584i	0.884103	+	0.467292i	$0.154770\pi$
$53$			19112.1i			0.934584i	−0.884103	+	0.467292i	$0.845230\pi$
$54$		0			0
$55$			8473.68i			0.377716i
$56$		0			0
$57$	−31604.5	−	17243.4i	−1.28843	−	0.702968i
$58$		0			0
$59$	10334.1			0.386495			0.193247	−	0.981150i	$-0.438098\pi$
$59$	10334.1			0.386495			0.193247	+	0.981150i	$0.438098\pi$
$60$		0			0
$61$	−7055.07			−0.242760			−0.121380	−	0.992606i	$-0.538732\pi$
$61$	−7055.07			−0.242760			−0.121380	+	0.992606i	$0.538732\pi$
$62$		0			0
$63$	10012.4	−	6444.24i	0.317824	−	0.204560i
$64$		0			0
$65$			13387.4i			0.393018i
$66$		0			0
$67$		−	46154.6i		−	1.25611i	−0.778168	−	0.628056i	$-0.783851\pi$
$67$		−	46154.6i		−	1.25611i	0.778168	−	0.628056i	$-0.216149\pi$
$68$		0			0
$69$	−27828.4	+	51005.3i	−0.703665	+	1.28971i
$70$		0			0
$71$	−75909.3			−1.78710			−0.893551	−	0.448962i	$-0.851794\pi$
$71$	−75909.3			−1.78710			−0.893551	+	0.448962i	$0.851794\pi$
$72$		0			0
$73$	−51116.5			−1.12267			−0.561337	−	0.827587i	$-0.689713\pi$
$73$	−51116.5			−1.12267			−0.561337	+	0.827587i	$0.689713\pi$
$74$		0			0
$75$	−16151.7	+	29603.6i	−0.331562	+	0.607703i
$76$		0			0
$77$			13389.3i			0.257354i
$78$		0			0
$79$			10609.7i			0.191264i	0.995417	+	0.0956320i	$0.0304872\pi$
$79$			10609.7i			0.191264i	−0.995417	+	0.0956320i	$0.969513\pi$
$80$		0			0
$81$	−24456.5	+	53746.3i	−0.414173	+	0.910198i
$82$		0			0
$83$	67540.1			1.07613			0.538067	−	0.842902i	$-0.319155\pi$
$83$	67540.1			1.07613			0.538067	+	0.842902i	$0.319155\pi$
$84$		0			0
$85$	−20124.0			−0.302112
$86$		0			0
$87$	−95859.5	−	52300.8i	−1.35780	−	0.740816i
$88$		0			0
$89$			73333.0i			0.981351i	0.871342	+	0.490675i	$0.163250\pi$
$89$			73333.0i			0.981351i	−0.871342	+	0.490675i	$0.836750\pi$
$90$		0			0
$91$			21153.4i			0.267779i
$92$		0			0
$93$	23607.9	+	12880.4i	0.283042	+	0.154427i
$94$		0			0
$95$	−71621.1			−0.814201
$96$		0			0
$97$	−134051.			−1.44657			−0.723285	−	0.690549i	$-0.757369\pi$
$97$	−134051.			−1.44657			−0.723285	+	0.690549i	$0.757369\pi$
$98$		0			0
$99$	−35936.6	−	55834.6i	−0.368510	−	0.572553i
$100$		0			0
$101$		−	34452.8i		−	0.336063i	−0.985782	−	0.168032i	$-0.946259\pi$
$101$		−	34452.8i		−	0.336063i	0.985782	−	0.168032i	$-0.0537411\pi$
$102$		0			0
$103$		−	41975.0i		−	0.389850i	−0.980818	−	0.194925i	$-0.937554\pi$
$103$		−	41975.0i		−	0.389850i	0.980818	−	0.194925i	$-0.0624464\pi$
$104$		0			0
$105$	11344.9	−	20793.5i	0.100422	−	0.184058i
$106$		0			0
$107$	716.196			0.00604745			0.00302373	−	0.999995i	$-0.499038\pi$
$107$	716.196			0.00604745			0.00302373	+	0.999995i	$0.499038\pi$
$108$		0			0
$109$	−176002.			−1.41890			−0.709450	−	0.704755i	$-0.751057\pi$
$109$	−176002.			−1.41890			−0.709450	+	0.704755i	$0.751057\pi$
$110$		0			0
$111$	91513.5	−	167731.i	0.704982	−	1.29213i
$112$		0			0
$113$			119037.i			0.876973i	0.898738	+	0.438487i	$0.144485\pi$
$113$			119037.i			0.876973i	−0.898738	+	0.438487i	$0.855515\pi$
$114$		0			0
$115$			115586.i			0.815009i
$116$		0			0
$117$	−56775.4	−	88211.9i	−0.383439	−	0.595748i
$118$		0			0
$119$	−31798.0			−0.205841
$120$		0			0
$121$	−86385.1			−0.536384
$122$		0			0
$123$	227715.	+	124241.i	1.35715	+	0.740460i
$124$		0			0
$125$			163995.i			0.938762i
$126$		0			0
$127$			119105.i			0.655272i	0.944804	+	0.327636i	$0.106252\pi$
$127$			119105.i			0.655272i	−0.944804	+	0.327636i	$0.893748\pi$
$128$		0			0
$129$	−125655.	−	68556.9i	−0.664820	−	0.362725i
$130$		0			0
$131$	19611.7			0.0998477			0.0499238	−	0.998753i	$-0.484102\pi$
$131$	19611.7			0.0998477			0.0499238	+	0.998753i	$0.484102\pi$
$132$		0			0
$133$	−113169.			−0.554749
$134$		0			0
$135$	8500.02	+	117160.i	0.0401408	+	0.553281i
$136$		0			0
$137$			111921.i			0.509462i	0.967012	+	0.254731i	$0.0819869\pi$
$137$			111921.i			0.509462i	−0.967012	+	0.254731i	$0.918013\pi$
$138$		0			0
$139$			151822.i			0.666495i	0.942839	+	0.333247i	$0.108144\pi$
$139$			151822.i			0.666495i	−0.942839	+	0.333247i	$0.891856\pi$
$140$		0			0
$141$	−173484.	+	317970.i	−0.734872	+	1.34691i
$142$		0			0
$143$	117963.			0.482398
$144$		0			0
$145$	−217233.			−0.858038
$146$		0			0
$147$	17926.1	−	32855.8i	0.0684214	−	0.125406i
$148$		0			0
$149$		−	224095.i		−	0.826926i	−0.910521	−	0.413463i	$-0.864319\pi$
$149$		−	224095.i		−	0.826926i	0.910521	−	0.413463i	$-0.135681\pi$
$150$		0			0
$151$		−	75904.1i		−	0.270909i	−0.990784	−	0.135454i	$-0.956751\pi$
$151$		−	75904.1i		−	0.270909i	0.990784	−	0.135454i	$-0.0432494\pi$
$152$		0			0
$153$	132601.	−	85345.4i	0.457950	−	0.294748i
$154$		0			0
$155$	53499.4			0.178863
$156$		0			0
$157$	539676.			1.74737			0.873684	−	0.486494i	$-0.161725\pi$
$157$	539676.			1.74737			0.873684	+	0.486494i	$0.161725\pi$
$158$		0			0
$159$	−261534.	−	142692.i	−0.820418	−	0.447619i
$160$		0			0
$161$			182638.i			0.555300i
$162$		0			0
$163$			278911.i			0.822236i	0.911582	+	0.411118i	$0.134862\pi$
$163$			278911.i			0.822236i	−0.911582	+	0.411118i	$0.865138\pi$
$164$		0			0
$165$	−115956.	−	63265.2i	−0.331575	−	0.180907i
$166$		0			0
$167$	140515.			0.389880			0.194940	−	0.980815i	$-0.437549\pi$
$167$	140515.			0.389880			0.194940	+	0.980815i	$0.437549\pi$
$168$		0			0
$169$	−184926.			−0.498060
$170$		0			0
$171$	471924.	−	303743.i	1.23419	−	0.794356i
$172$		0			0
$173$		−	383778.i		−	0.974910i	−0.873148	−	0.487455i	$-0.837925\pi$
$173$		−	383778.i		−	0.974910i	0.873148	−	0.487455i	$-0.162075\pi$
$174$		0			0
$175$			106004.i			0.261653i
$176$		0			0
$177$	−77155.4	+	141414.i	−0.185111	+	0.339281i
$178$		0			0
$179$	216235.			0.504421			0.252210	−	0.967672i	$-0.418843\pi$
$179$	216235.			0.504421			0.252210	+	0.967672i	$0.418843\pi$
$180$		0			0
$181$	17131.1			0.0388677			0.0194339	−	0.999811i	$-0.493814\pi$
$181$	17131.1			0.0388677			0.0194339	+	0.999811i	$0.493814\pi$
$182$		0			0
$183$	52673.7	−	96543.0i	0.116270	−	0.213105i
$184$		0			0
$185$		−	380105.i		−	0.816534i
$186$		0			0
$187$			177323.i			0.370818i
$188$		0			0
$189$	13430.9	+	185125.i	0.0273496	+	0.376974i
$190$		0			0
$191$	−563825.			−1.11831			−0.559153	−	0.829064i	$-0.688874\pi$
$191$	−563825.			−1.11831			−0.559153	+	0.829064i	$0.688874\pi$
$192$		0			0
$193$	111313.			0.215106			0.107553	−	0.994199i	$-0.465699\pi$
$193$	111313.			0.215106			0.107553	+	0.994199i	$0.465699\pi$
$194$		0			0
$195$	−183196.	−	99951.3i	−0.345008	−	0.188236i
$196$		0			0
$197$			175093.i			0.321442i	0.987000	+	0.160721i	$0.0513820\pi$
$197$			175093.i			0.321442i	−0.987000	+	0.160721i	$0.948618\pi$
$198$		0			0
$199$			25023.4i			0.0447934i	0.999749	+	0.0223967i	$0.00712968\pi$
$199$			25023.4i			0.0447934i	−0.999749	+	0.0223967i	$0.992870\pi$
$200$		0			0
$201$	631590.	+	344594.i	1.10267	+	0.601614i
$202$		0			0
$203$	−343251.			−0.584617
$204$		0			0
$205$	516039.			0.857626
$206$		0			0
$207$	−490198.	−	761620.i	−0.795145	−	1.23541i
$208$		0			0
$209$			631089.i			0.999367i
$210$		0			0
$211$			543279.i			0.840073i	0.907507	+	0.420037i	$0.137983\pi$
$211$			543279.i			0.840073i	−0.907507	+	0.420037i	$0.862017\pi$
$212$		0			0
$213$	566745.	−	1.03876e6i	0.855932	−	1.56879i
$214$		0			0
$215$	−284754.			−0.420120
$216$		0			0
$217$	84534.5			0.121867
$218$		0			0
$219$	381640.	−	699489.i	0.537705	−	0.985532i
$220$		0			0
$221$			280149.i			0.385841i
$222$		0			0
$223$		−	93393.8i		−	0.125764i	−0.998021	−	0.0628819i	$-0.979971\pi$
$223$		−	93393.8i		−	0.125764i	0.998021	−	0.0628819i	$-0.0200292\pi$
$224$		0			0
$225$	−284512.	−	442046.i	−0.374666	−	0.582118i
$226$		0			0
$227$	−968469.			−1.24744			−0.623722	−	0.781646i	$-0.714380\pi$
$227$	−968469.			−1.24744			−0.623722	+	0.781646i	$0.714380\pi$
$228$		0			0
$229$	698135.			0.879733			0.439867	−	0.898063i	$-0.355026\pi$
$229$	698135.			0.879733			0.439867	+	0.898063i	$0.355026\pi$
$230$		0			0
$231$	−183222.	−	99965.5i	−0.225916	−	0.123259i
$232$		0			0
$233$		−	1.39409e6i		−	1.68228i	−0.540814	−	0.841142i	$-0.681884\pi$
$233$		−	1.39409e6i		−	1.68228i	0.540814	−	0.841142i	$-0.318116\pi$
$234$		0			0
$235$			720572.i			0.851154i
$236$		0			0
$237$	−145185.	−	79212.5i	−0.167900	−	0.0916058i
$238$		0			0
$239$	1.42313e6			1.61157			0.805783	−	0.592210i	$-0.201745\pi$
$239$	1.42313e6			1.61157			0.805783	+	0.592210i	$0.201745\pi$
$240$		0			0
$241$	846607.			0.938943			0.469471	−	0.882948i	$-0.344445\pi$
$241$	846607.			0.938943			0.469471	+	0.882948i	$0.344445\pi$
$242$		0			0
$243$	−552881.	−	735942.i	−0.600643	−	0.799518i
$244$		0			0
$245$		−	74456.6i		−	0.0792479i
$246$		0			0
$247$			997044.i			1.03985i
$248$		0			0
$249$	−504260.	+	924233.i	−0.515414	+	0.944677i
$250$		0			0
$251$	96135.7			0.0963164			0.0481582	−	0.998840i	$-0.484665\pi$
$251$	96135.7			0.0963164			0.0481582	+	0.998840i	$0.484665\pi$
$252$		0			0
$253$	1.01849e6			1.00036
$254$		0			0
$255$	150248.	−	275382.i	0.144696	−	0.265207i
$256$		0			0
$257$		−	695936.i		−	0.657259i	−0.944459	−	0.328630i	$-0.893413\pi$
$257$		−	695936.i		−	0.657259i	0.944459	−	0.328630i	$-0.106587\pi$
$258$		0			0
$259$		−	600604.i		−	0.556338i
$260$		0			0
$261$	1.43139e6	−	921279.i	1.30064	−	0.837125i
$262$		0			0
$263$	−303934.			−0.270950			−0.135475	−	0.990781i	$-0.543256\pi$
$263$	−303934.			−0.270950			−0.135475	+	0.990781i	$0.543256\pi$
$264$		0			0
$265$	−592679.			−0.518447
$266$		0			0
$267$	−1.00350e6	−	547510.i	−0.861471	−	0.470017i
$268$		0			0
$269$		−	2.20360e6i		−	1.85675i	−0.371648	−	0.928374i	$-0.621207\pi$
$269$		−	2.20360e6i		−	1.85675i	0.371648	−	0.928374i	$-0.378793\pi$
$270$		0			0
$271$			1.28050e6i			1.05915i	0.848263	+	0.529575i	$0.177648\pi$
$271$			1.28050e6i			1.05915i	−0.848263	+	0.529575i	$0.822352\pi$
$272$		0			0
$273$	−289468.	−	157933.i	−0.235068	−	0.128253i
$274$		0			0
$275$	591134.			0.471361
$276$		0			0
$277$	749476.			0.586892			0.293446	−	0.955976i	$-0.405198\pi$
$277$	749476.			0.586892			0.293446	+	0.955976i	$0.405198\pi$
$278$		0			0
$279$	−352517.	+	226889.i	−0.271125	+	0.174503i
$280$		0			0
$281$			1.94049e6i			1.46604i	0.680208	+	0.733019i	$0.261889\pi$
$281$			1.94049e6i			1.46604i	−0.680208	+	0.733019i	$0.738111\pi$
$282$		0			0
$283$		−	577899.i		−	0.428930i	−0.976732	−	0.214465i	$-0.931199\pi$
$283$		−	577899.i		−	0.428930i	0.976732	−	0.214465i	$-0.0688007\pi$
$284$		0			0
$285$	534729.	−	980078.i	0.389961	−	0.714740i
$286$		0			0
$287$	815394.			0.584336
$288$		0			0
$289$	998735.			0.703405
$290$		0			0
$291$	1.00083e6	−	1.83438e6i	0.692834	−	1.26986i
$292$		0			0
$293$		−	914960.i		−	0.622634i	−0.950306	−	0.311317i	$-0.899230\pi$
$293$		−	914960.i		−	0.622634i	0.950306	−	0.311317i	$-0.100770\pi$
$294$		0			0
$295$			320468.i			0.214402i
$296$		0			0
$297$	1.03236e6	−	74898.0i	0.679109	−	0.0492696i
$298$		0			0
$299$	1.60909e6			1.04088
$300$		0			0
$301$	−449940.			−0.286245
$302$		0			0
$303$	471460.	+	257228.i	0.295011	+	0.160957i
$304$		0			0
$305$		−	218782.i		−	0.134667i
$306$		0			0
$307$			2.55889e6i			1.54955i	0.632237	+	0.774775i	$0.282137\pi$
$307$			2.55889e6i			1.54955i	−0.632237	+	0.774775i	$0.717863\pi$
$308$		0			0
$309$	574395.	+	313389.i	0.342227	+	0.186719i
$310$		0			0
$311$	−160879.			−0.0943189			−0.0471595	−	0.998887i	$-0.515017\pi$
$311$	−160879.			−0.0943189			−0.0471595	+	0.998887i	$0.515017\pi$
$312$		0			0
$313$	25303.8			0.0145991			0.00729954	−	0.999973i	$-0.497676\pi$
$313$	25303.8			0.0145991			0.00729954	+	0.999973i	$0.497676\pi$
$314$		0			0
$315$	199840.	+	310491.i	0.113477	+	0.176309i
$316$		0			0
$317$			2.07422e6i			1.15933i	0.814856	+	0.579663i	$0.196816\pi$
$317$			2.07422e6i			1.15933i	−0.814856	+	0.579663i	$0.803184\pi$
$318$		0			0
$319$			1.91415e6i			1.05317i
$320$		0			0
$321$	−5347.18	+	9800.57i	−0.00289642	+	0.00530871i
$322$		0			0
$323$	−1.49877e6			−0.799333
$324$		0			0
$325$	933918.			0.490457
$326$		0			0
$327$	1.31405e6	−	2.40845e6i	0.679582	−	1.24557i
$328$		0			0
$329$			1.13858e6i			0.579926i
$330$		0			0
$331$			3.79890e6i			1.90585i	0.303211	+	0.952924i	$0.401941\pi$
$331$			3.79890e6i			1.90585i	−0.303211	+	0.952924i	$0.598059\pi$
$332$		0			0
$333$	1.61201e6	+	2.50458e6i	0.796632	+	1.23773i
$334$		0			0
$335$	1.43129e6			0.696810
$336$		0			0
$337$	−3.26997e6			−1.56845			−0.784223	−	0.620479i	$-0.786938\pi$
$337$	−3.26997e6			−1.56845			−0.784223	+	0.620479i	$0.786938\pi$
$338$		0			0
$339$	−1.62893e6	−	888741.i	−0.769844	−	0.420026i
$340$		0			0
$341$		−	471410.i		−	0.219540i
$342$		0			0
$343$		−	117649.i		−	0.0539949i
$344$		0			0
$345$	−1.58171e6	−	862978.i	−0.715450	−	0.390348i
$346$		0			0
$347$	−2.21278e6			−0.986542			−0.493271	−	0.869876i	$-0.664199\pi$
$347$	−2.21278e6			−0.986542			−0.493271	+	0.869876i	$0.664199\pi$
$348$		0			0
$349$	3.11399e6			1.36853			0.684264	−	0.729234i	$-0.260124\pi$
$349$	3.11399e6			1.36853			0.684264	+	0.729234i	$0.260124\pi$
$350$		0			0
$351$	1.63100e6	−	118330.i	0.706620	−	0.0512656i
$352$		0			0
$353$		−	1.19521e6i		−	0.510513i	−0.966873	−	0.255256i	$-0.917840\pi$
$353$		−	1.19521e6i		−	0.510513i	0.966873	−	0.255256i	$-0.0821599\pi$
$354$		0			0
$355$		−	2.35400e6i		−	0.991369i
$356$		0			0
$357$	237407.	−	435131.i	0.0985877	−	0.180696i
$358$		0			0
$359$	4.53735e6			1.85809			0.929045	−	0.369968i	$-0.120631\pi$
$359$	4.53735e6			1.85809			0.929045	+	0.369968i	$0.120631\pi$
$360$		0			0
$361$	−2.85798e6			−1.15423
$362$		0			0
$363$	644959.	−	1.18211e6i	0.256901	−	0.470860i
$364$		0			0
$365$		−	1.58516e6i		−	0.622788i
$366$		0			0
$367$			1.10438e6i			0.428008i	0.976833	+	0.214004i	$0.0686506\pi$
$367$			1.10438e6i			0.428008i	−0.976833	+	0.214004i	$0.931349\pi$
$368$		0			0
$369$	−3.40028e6	+	2.18850e6i	−1.30001	+	0.836723i
$370$		0			0
$371$	−936492.			−0.353240
$372$		0			0
$373$	−665317.			−0.247603			−0.123802	−	0.992307i	$-0.539509\pi$
$373$	−665317.			−0.247603			−0.123802	+	0.992307i	$0.539509\pi$
$374$		0			0
$375$	−2.24414e6	−	1.22440e6i	−0.824085	−	0.449619i
$376$		0			0
$377$			3.02413e6i			1.09584i
$378$		0			0
$379$			1.17286e6i			0.419420i	0.977764	+	0.209710i	$0.0672521\pi$
$379$			1.17286e6i			0.419420i	−0.977764	+	0.209710i	$0.932748\pi$
$380$		0			0
$381$	−1.62986e6	−	889250.i	−0.575226	−	0.313842i
$382$		0			0
$383$	−3.41559e6			−1.18979			−0.594893	−	0.803805i	$-0.702805\pi$
$383$	−3.41559e6			−1.18979			−0.594893	+	0.803805i	$0.702805\pi$
$384$		0			0
$385$	−415210.			−0.142763
$386$		0			0
$387$	1.87629e6	−	1.20763e6i	0.636830	−	0.409880i
$388$		0			0
$389$		−	3.49028e6i		−	1.16946i	−0.811227	−	0.584731i	$-0.801200\pi$
$389$		−	3.49028e6i		−	1.16946i	0.811227	−	0.584731i	$-0.198800\pi$
$390$		0			0
$391$			2.41880e6i			0.800126i
$392$		0			0
$393$	−146423.	+	268371.i	−0.0478220	+	0.0876505i
$394$		0			0
$395$	−329012.			−0.106101
$396$		0			0
$397$	1.87522e6			0.597139			0.298570	−	0.954388i	$-0.403490\pi$
$397$	1.87522e6			0.597139			0.298570	+	0.954388i	$0.403490\pi$
$398$		0			0
$399$	844926.	−	1.54862e6i	0.265697	−	0.486982i
$400$		0			0
$401$		−	921836.i		−	0.286281i	−0.989702	−	0.143141i	$-0.954280\pi$
$401$		−	921836.i		−	0.286281i	0.989702	−	0.143141i	$-0.0457201\pi$
$402$		0			0
$403$		−	744770.i		−	0.228433i
$404$		0			0
$405$	−1.66671e6	−	758413.i	−0.504919	−	0.229757i
$406$		0			0
$407$	−3.34930e6			−1.00223
$408$		0			0
$409$	−1.95540e6			−0.578000			−0.289000	−	0.957329i	$-0.593323\pi$
$409$	−1.95540e6			−0.578000			−0.289000	+	0.957329i	$0.593323\pi$
$410$		0			0
$411$	−1.53156e6	−	835615.i	−0.447227	−	0.244007i
$412$		0			0
$413$			506372.i			0.146081i
$414$		0			0
$415$			2.09446e6i			0.596970i
$416$		0			0
$417$	−2.07756e6	−	1.13351e6i	−0.585077	−	0.319217i
$418$		0			0
$419$	−6.10515e6			−1.69887			−0.849437	−	0.527690i	$-0.823058\pi$
$419$	−6.10515e6			−1.69887			−0.849437	+	0.527690i	$0.823058\pi$
$420$		0			0
$421$	−1.22827e6			−0.337746			−0.168873	−	0.985638i	$-0.554013\pi$
$421$	−1.22827e6			−0.337746			−0.168873	+	0.985638i	$0.554013\pi$
$422$		0			0
$423$	−3.05592e6	−	4.74798e6i	−0.830408	−	1.29020i
$424$		0			0
$425$			1.40388e6i			0.377013i
$426$		0			0
$427$		−	345698.i		−	0.0917546i
$428$		0			0
$429$	−880721.	+	1.61423e6i	−0.231044	+	0.423469i
$430$		0			0
$431$	−14052.6			−0.00364387			−0.00182193	−	0.999998i	$-0.500580\pi$
$431$	−14052.6			−0.00364387			−0.00182193	+	0.999998i	$0.500580\pi$
$432$		0			0
$433$	−6.42099e6			−1.64582			−0.822909	−	0.568172i	$-0.807651\pi$
$433$	−6.42099e6			−1.64582			−0.822909	+	0.568172i	$0.807651\pi$
$434$		0			0
$435$	1.62188e6	−	2.97267e6i	0.410957	−	0.753222i
$436$		0			0
$437$			8.60846e6i			2.15636i
$438$		0			0
$439$		−	1.88754e6i		−	0.467450i	−0.972303	−	0.233725i	$-0.924908\pi$
$439$		−	1.88754e6i		−	0.467450i	0.972303	−	0.233725i	$-0.0750916\pi$
$440$		0			0
$441$	315768.	+	490608.i	0.0773164	+	0.120126i
$442$		0			0
$443$	2.58093e6			0.624836			0.312418	−	0.949945i	$-0.398861\pi$
$443$	2.58093e6			0.624836			0.312418	+	0.949945i	$0.398861\pi$
$444$		0			0
$445$	−2.27410e6			−0.544390
$446$		0			0
$447$	3.06656e6	+	1.67311e6i	0.725911	+	0.396056i
$448$		0			0
$449$			5.08339e6i			1.18998i	0.803735	+	0.594988i	$0.202843\pi$
$449$			5.08339e6i			1.18998i	−0.803735	+	0.594988i	$0.797157\pi$
$450$		0			0
$451$		−	4.54708e6i		−	1.05267i
$452$		0			0
$453$	1.03869e6	+	566706.i	0.237815	+	0.129752i
$454$		0			0
$455$	−655981.			−0.148547
$456$		0			0
$457$	−4.34884e6			−0.974055			−0.487027	−	0.873387i	$-0.661919\pi$
$457$	−4.34884e6			−0.974055			−0.487027	+	0.873387i	$0.661919\pi$
$458$		0			0
$459$	177874.	+	2.45174e6i	0.0394077	+	0.543178i
$460$		0			0
$461$			3.77026e6i			0.826265i	0.910671	+	0.413133i	$0.135565\pi$
$461$			3.77026e6i			0.826265i	−0.910671	+	0.413133i	$0.864435\pi$
$462$		0			0
$463$			3.26552e6i			0.707946i	0.935256	+	0.353973i	$0.115170\pi$
$463$			3.26552e6i			0.707946i	−0.935256	+	0.353973i	$0.884830\pi$
$464$		0			0
$465$	−399431.	+	732097.i	−0.0856662	+	0.157013i
$466$		0			0
$467$	1.02593e6			0.217683			0.108842	−	0.994059i	$-0.465286\pi$
$467$	1.02593e6			0.217683			0.108842	+	0.994059i	$0.465286\pi$
$468$		0			0
$469$	2.26158e6			0.474766
$470$		0			0
$471$	−4.02927e6	+	7.38504e6i	−0.836901	+	1.53391i
$472$		0			0
$473$			2.50911e6i			0.515664i
$474$		0			0
$475$			4.99637e6i			1.01606i
$476$		0			0
$477$	3.90527e6	−	2.51353e6i	0.785877	−	0.505811i
$478$		0			0
$479$	−776699.			−0.154673			−0.0773364	−	0.997005i	$-0.524642\pi$
$479$	−776699.			−0.154673			−0.0773364	+	0.997005i	$0.524642\pi$
$480$		0			0
$481$	−5.29148e6			−1.04283
$482$		0			0
$483$	−2.49926e6	−	1.36359e6i	−0.487466	−	0.265960i
$484$		0			0
$485$		−	4.15700e6i		−	0.802464i
$486$		0			0
$487$		−	9.66548e6i		−	1.84672i	−0.383935	−	0.923360i	$-0.625431\pi$
$487$		−	9.66548e6i		−	1.84672i	0.383935	−	0.923360i	$-0.374569\pi$
$488$		0			0
$489$	−3.81667e6	−	2.08237e6i	−0.721793	−	0.393809i
$490$		0			0
$491$	7.98460e6			1.49469			0.747343	−	0.664439i	$-0.231329\pi$
$491$	7.98460e6			1.49469			0.747343	+	0.664439i	$0.231329\pi$
$492$		0			0
$493$	−4.54590e6			−0.842369
$494$		0			0
$495$	1.73147e6	−	1.11442e6i	0.317616	−	0.204426i
$496$		0			0
$497$		−	3.71956e6i		−	0.675461i
$498$		0			0
$499$			1.05004e7i			1.88779i	0.330244	+	0.943896i	$0.392869\pi$
$499$			1.05004e7i			1.88779i	−0.330244	+	0.943896i	$0.607131\pi$
$500$		0			0
$501$	−1.04910e6	+	1.92283e6i	−0.186733	+	0.342253i
$502$		0			0
$503$	5.98180e6			1.05417			0.527086	−	0.849812i	$-0.323285\pi$
$503$	5.98180e6			1.05417			0.527086	+	0.849812i	$0.323285\pi$
$504$		0			0
$505$	1.06840e6			0.186426
$506$		0			0
$507$	1.38067e6	−	2.53057e6i	0.238545	−	0.437218i
$508$		0			0
$509$		−	6.18196e6i		−	1.05762i	−0.848739	−	0.528812i	$-0.822638\pi$
$509$		−	6.18196e6i		−	1.05762i	0.848739	−	0.528812i	$-0.177362\pi$
$510$		0			0
$511$		−	2.50471e6i		−	0.424331i
$512$		0			0
$513$	633051.	+	8.72568e6i	0.106205	+	1.46388i
$514$		0			0
$515$	1.30167e6			0.216264
$516$		0			0
$517$	6.34933e6			1.04472
$518$		0			0
$519$	5.25169e6	+	2.86532e6i	0.855817	+	0.466933i
$520$		0			0
$521$			8.27286e6i			1.33525i	0.744500	+	0.667623i	$0.232688\pi$
$521$			8.27286e6i			1.33525i	−0.744500	+	0.667623i	$0.767312\pi$
$522$		0			0
$523$		−	1.75488e6i		−	0.280539i	−0.990113	−	0.140269i	$-0.955203\pi$
$523$		−	1.75488e6i		−	0.280539i	0.990113	−	0.140269i	$-0.0447969\pi$
$524$		0			0
$525$	−1.45058e6	−	791432.i	−0.229690	−	0.125319i
$526$		0			0
$527$	1.11955e6			0.175596
$528$		0			0
$529$	7.45653e6			1.15850
$530$		0			0
$531$	−1.35909e6	−	2.11162e6i	−0.209177	−	0.324997i
$532$		0			0
$533$		−	7.18383e6i		−	1.09531i
$534$		0			0
$535$			22209.7i			0.00335474i
$536$		0			0
$537$	−1.61443e6	+	2.95900e6i	−0.241592	+	0.442802i
$538$		0			0
$539$	−656075.			−0.0972705
$540$		0			0
$541$	−597706.			−0.0877999			−0.0439000	−	0.999036i	$-0.513978\pi$
$541$	−597706.			−0.0877999			−0.0439000	+	0.999036i	$0.513978\pi$
$542$		0			0
$543$	−127902.	+	234426.i	−0.0186157	+	0.0341198i
$544$		0			0
$545$		−	5.45795e6i		−	0.787115i
$546$		0			0
$547$			9.85127e6i			1.40775i	0.710326	+	0.703873i	$0.248547\pi$
$547$			9.85127e6i			1.40775i	−0.710326	+	0.703873i	$0.751453\pi$
$548$		0			0
$549$	927849.	+	1.44160e6i	0.131385	+	0.204133i
$550$		0			0
$551$	−1.61788e7			−2.27021
$552$		0			0
$553$	−519873.			−0.0722910
$554$		0			0
$555$	5.20144e6	+	2.83789e6i	0.716788	+	0.391078i
$556$		0			0
$557$		−	7.17315e6i		−	0.979652i	−0.871820	−	0.489826i	$-0.837060\pi$
$557$		−	7.17315e6i		−	0.979652i	0.871820	−	0.489826i	$-0.162940\pi$
$558$		0			0
$559$			3.96409e6i			0.536554i
$560$		0			0
$561$	−2.42653e6	−	1.32391e6i	−0.325520	−	0.177603i
$562$		0			0
$563$	527120.			0.0700872			0.0350436	−	0.999386i	$-0.488843\pi$
$563$	527120.			0.0700872			0.0350436	+	0.999386i	$0.488843\pi$
$564$		0			0
$565$	−3.69142e6			−0.486488
$566$		0			0
$567$	−2.63357e6	−	1.19837e6i	−0.344023	−	0.156543i
$568$		0			0
$569$			6.75497e6i			0.874667i	0.899299	+	0.437334i	$0.144077\pi$
$569$			6.75497e6i			0.874667i	−0.899299	+	0.437334i	$0.855923\pi$
$570$		0			0
$571$		−	6.26648e6i		−	0.804329i	−0.915568	−	0.402164i	$-0.868258\pi$
$571$		−	6.26648e6i		−	0.804329i	0.915568	−	0.402164i	$-0.131742\pi$
$572$		0			0
$573$	4.20956e6	−	7.71550e6i	0.535612	−	0.981697i
$574$		0			0
$575$	8.06344e6			1.01707
$576$		0			0
$577$	−1.10962e7			−1.38750			−0.693751	−	0.720215i	$-0.744043\pi$
$577$	−1.10962e7			−1.38750			−0.693751	+	0.720215i	$0.744043\pi$
$578$		0			0
$579$	−831070.	+	1.52323e6i	−0.103025	+	0.188829i
$580$		0			0
$581$			3.30947e6i			0.406741i
$582$		0			0
$583$			5.22239e6i			0.636353i
$584$		0			0
$585$	2.73551e6	−	1.76064e6i	0.330482	−	0.212707i
$586$		0			0
$587$	1.48039e7			1.77329			0.886645	−	0.462450i	$-0.153030\pi$
$587$	1.48039e7			1.77329			0.886645	+	0.462450i	$0.153030\pi$
$588$		0			0
$589$	3.98444e6			0.473238
$590$		0			0
$591$	−2.39601e6	−	1.30726e6i	−0.282176	−	0.153955i
$592$		0			0
$593$			1.45962e7i			1.70452i	0.523116	+	0.852262i	$0.324770\pi$
$593$			1.45962e7i			1.70452i	−0.523116	+	0.852262i	$0.675230\pi$
$594$		0			0
$595$		−	986078.i		−	0.114188i
$596$		0			0
$597$	−342426.	−	186827.i	−0.0393215	−	0.0214538i
$598$		0			0
$599$	3.89256e6			0.443270			0.221635	−	0.975130i	$-0.428861\pi$
$599$	3.89256e6			0.443270			0.221635	+	0.975130i	$0.428861\pi$
$600$		0			0
$601$	1.15955e7			1.30949			0.654746	−	0.755849i	$-0.272776\pi$
$601$	1.15955e7			1.30949			0.654746	+	0.755849i	$0.272776\pi$
$602$		0			0
$603$	−9.43101e6	+	6.07004e6i	−1.05625	+	0.679827i
$604$		0			0
$605$		−	2.67886e6i		−	0.297551i
$606$		0			0
$607$		−	1.63844e7i		−	1.80492i	−0.430775	−	0.902459i	$-0.641760\pi$
$607$		−	1.63844e7i		−	1.80492i	0.430775	−	0.902459i	$-0.358240\pi$
$608$		0			0
$609$	2.56274e6	−	4.69712e6i	0.280002	−	0.513202i
$610$		0			0
$611$	1.00312e7			1.08705
$612$		0			0
$613$	9.08825e6			0.976853			0.488426	−	0.872605i	$-0.337571\pi$
$613$	9.08825e6			0.976853			0.488426	+	0.872605i	$0.337571\pi$
$614$		0			0
$615$	−3.85279e6	+	7.06159e6i	−0.410760	+	0.752861i
$616$		0			0
$617$		−	1.26021e7i		−	1.33270i	−0.745641	−	0.666348i	$-0.767857\pi$
$617$		−	1.26021e7i		−	1.33270i	0.745641	−	0.666348i	$-0.232143\pi$
$618$		0			0
$619$		−	1.79908e7i		−	1.88723i	−0.331047	−	0.943614i	$-0.607402\pi$
$619$		−	1.79908e7i		−	1.88723i	0.331047	−	0.943614i	$-0.392598\pi$
$620$		0			0
$621$	1.40820e7	−	1.02166e6i	1.46533	−	0.106310i
$622$		0			0
$623$	−3.59332e6			−0.370916
$624$		0			0
$625$	1.67484e6			0.171504
$626$		0			0
$627$	−8.63596e6	−	4.71177e6i	−0.877287	−	0.478646i
$628$		0			0
$629$		−	7.95420e6i		−	0.801623i
$630$		0			0
$631$		−	1.52790e7i		−	1.52764i	−0.645428	−	0.763821i	$-0.723321\pi$
$631$		−	1.52790e7i		−	1.52764i	0.645428	−	0.763821i	$-0.276679\pi$
$632$		0			0
$633$	−7.43435e6	−	4.05617e6i	−0.737452	−	0.402353i
$634$		0			0
$635$	−3.69353e6			−0.363503
$636$		0			0
$637$	−1.03652e6			−0.101211
$638$		0			0
$639$	9.98323e6	+	1.55109e7i	0.967206	+	1.50275i
$640$		0			0
$641$		−	1.52864e7i		−	1.46947i	−0.678354	−	0.734736i	$-0.737306\pi$
$641$		−	1.52864e7i		−	1.46947i	0.678354	−	0.734736i	$-0.262694\pi$
$642$		0			0
$643$			2.31579e6i			0.220888i	0.993882	+	0.110444i	$0.0352272\pi$
$643$			2.31579e6i			0.220888i	−0.993882	+	0.110444i	$0.964773\pi$
$644$		0			0
$645$	2.12600e6	−	3.89663e6i	0.201216	−	0.368799i
$646$		0			0
$647$	−1.83598e7			−1.72428			−0.862140	−	0.506671i	$-0.830876\pi$
$647$	−1.83598e7			−1.72428			−0.862140	+	0.506671i	$0.830876\pi$
$648$		0			0
$649$	2.82380e6			0.263162
$650$		0			0
$651$	−631142.	+	1.15679e6i	−0.0583679	+	0.106980i
$652$		0			0
$653$			9.96514e6i			0.914535i	0.889329	+	0.457267i	$0.151172\pi$
$653$			9.96514e6i			0.914535i	−0.889329	+	0.457267i	$0.848828\pi$
$654$		0			0
$655$			608173.i			0.0553891i
$656$		0			0
$657$	6.72260e6	+	1.04449e7i	0.607608	+	0.944040i
$658$		0			0
$659$	−1.49253e7			−1.33878			−0.669390	−	0.742911i	$-0.733444\pi$
$659$	−1.49253e7			−1.33878			−0.669390	+	0.742911i	$0.733444\pi$
$660$		0			0
$661$	1.95642e7			1.74164			0.870822	−	0.491598i	$-0.163587\pi$
$661$	1.95642e7			1.74164			0.870822	+	0.491598i	$0.163587\pi$
$662$		0			0
$663$	−3.83361e6	−	2.09161e6i	−0.338707	−	0.184798i
$664$		0			0
$665$		−	3.50943e6i		−	0.307739i
$666$		0			0
$667$			2.61103e7i			2.27246i
$668$		0			0
$669$	1.27802e6	+	697286.i	0.110401	+	0.0602345i
$670$		0			0
$671$	−1.92780e6			−0.165294
$672$		0			0
$673$	7.47877e6			0.636491			0.318246	−	0.948008i	$-0.396906\pi$
$673$	7.47877e6			0.636491			0.318246	+	0.948008i	$0.396906\pi$
$674$		0			0
$675$	8.17323e6	−	592971.i	0.690454	−	0.0500927i
$676$		0			0
$677$		−	5.33357e6i		−	0.447245i	−0.974676	−	0.223623i	$-0.928212\pi$
$677$		−	5.33357e6i		−	0.447245i	0.974676	−	0.223623i	$-0.0717884\pi$
$678$		0			0
$679$		−	6.56848e6i		−	0.546752i
$680$		0			0
$681$	7.23067e6	−	1.32527e7i	0.597463	−	1.09506i
$682$		0			0
$683$	−9.72244e6			−0.797487			−0.398743	−	0.917063i	$-0.630554\pi$
$683$	−9.72244e6			−0.797487			−0.398743	+	0.917063i	$0.630554\pi$
$684$		0			0
$685$	−3.47076e6			−0.282617
$686$		0			0
$687$	−5.21234e6	+	9.55343e6i	−0.421348	+	0.772267i
$688$		0			0
$689$			8.25073e6i			0.662132i
$690$		0			0
$691$		−	1.54033e7i		−	1.22721i	−0.789612	−	0.613606i	$-0.789718\pi$
$691$		−	1.54033e7i		−	1.22721i	0.789612	−	0.613606i	$-0.210282\pi$
$692$		0			0
$693$	2.73590e6	−	1.76089e6i	0.216405	−	0.139284i
$694$		0			0
$695$	−4.70809e6			−0.369728
$696$		0			0
$697$	1.07988e7			0.841965
$698$		0			0
$699$	1.90770e7	+	1.04084e7i	1.47678	+	0.805729i
$700$		0			0
$701$			47107.8i			0.00362075i	0.999998	+	0.00181037i	$0.000576260\pi$
$701$			47107.8i			0.00362075i	−0.999998	+	0.00181037i	$0.999424\pi$
$702$		0			0
$703$		−	2.83088e7i		−	2.16040i
$704$		0			0
$705$	−9.86046e6	−	5.37985e6i	−0.747179	−	0.407660i
$706$		0			0
$707$	1.68819e6			0.127020
$708$		0			0
$709$	−1.79314e6			−0.133967			−0.0669836	−	0.997754i	$-0.521338\pi$
$709$	−1.79314e6			−0.133967			−0.0669836	+	0.997754i	$0.521338\pi$
$710$		0			0
$711$	2.16792e6	−	1.39533e6i	0.160831	−	0.103515i
$712$		0			0
$713$		−	6.43034e6i		−	0.473707i
$714$		0			0
$715$			3.65811e6i			0.267603i
$716$		0			0
$717$	−1.06252e7	+	1.94743e7i	−0.771859	+	1.41470i
$718$		0			0
$719$	1.45978e7			1.05309			0.526544	−	0.850148i	$-0.323487\pi$
$719$	1.45978e7			1.05309			0.526544	+	0.850148i	$0.323487\pi$
$720$		0			0
$721$	2.05678e6			0.147350
$722$		0			0
$723$	−6.32084e6	+	1.15851e7i	−0.449706	+	0.824244i
$724$		0			0
$725$			1.51544e7i			1.07077i
$726$		0			0
$727$		−	1.77708e7i		−	1.24701i	−0.781818	−	0.623507i	$-0.785707\pi$
$727$		−	1.77708e7i		−	1.24701i	0.781818	−	0.623507i	$-0.214293\pi$
$728$		0			0
$729$	1.41986e7	−	2.07114e6i	0.989528	−	0.144341i
$730$		0			0
$731$	−5.95885e6			−0.412448
$732$		0			0
$733$	1.27593e7			0.877138			0.438569	−	0.898697i	$-0.355485\pi$
$733$	1.27593e7			0.877138			0.438569	+	0.898697i	$0.355485\pi$
$734$		0			0
$735$	1.01888e6	+	555899.i	0.0695672	+	0.0379558i
$736$		0			0
$737$		−	1.26118e7i		−	0.855279i
$738$		0			0
$739$		−	9.57582e6i		−	0.645008i	−0.946568	−	0.322504i	$-0.895475\pi$
$739$		−	9.57582e6i		−	0.645008i	0.946568	−	0.322504i	$-0.104525\pi$
$740$		0			0
$741$	−1.36438e7	−	7.44401e6i	−0.912827	−	0.498037i
$742$		0			0
$743$	−9.65898e6			−0.641888			−0.320944	−	0.947098i	$-0.604000\pi$
$743$	−9.65898e6			−0.641888			−0.320944	+	0.947098i	$0.604000\pi$
$744$		0			0
$745$	6.94934e6			0.458725
$746$		0			0
$747$	−8.88255e6	−	1.38008e7i	−0.582420	−	0.904905i
$748$		0			0
$749$			35093.6i			0.00228572i
$750$		0			0
$751$			1.44433e7i			0.934475i	0.884132	+	0.467238i	$0.154751\pi$
$751$			1.44433e7i			0.934475i	−0.884132	+	0.467238i	$0.845249\pi$
$752$		0			0
$753$	−717757.	+	1.31554e6i	−0.0461307	+	0.0845507i
$754$		0			0
$755$	2.35384e6			0.150283
$756$		0			0
$757$	1.70339e7			1.08037			0.540187	−	0.841545i	$-0.318354\pi$
$757$	1.70339e7			1.08037			0.540187	+	0.841545i	$0.318354\pi$
$758$		0			0
$759$	−7.60414e6	+	1.39372e7i	−0.479121	+	0.878157i
$760$		0			0
$761$		−	1.06579e7i		−	0.667128i	−0.942727	−	0.333564i	$-0.891749\pi$
$761$		−	1.06579e7i		−	0.667128i	0.942727	−	0.333564i	$-0.108251\pi$
$762$		0			0
$763$		−	8.62411e6i		−	0.536294i
$764$		0			0
$765$	2.64662e6	+	4.11204e6i	0.163507	+	0.254041i
$766$		0			0
$767$	4.46126e6			0.273823
$768$		0			0
$769$	−1.31211e6			−0.0800118			−0.0400059	−	0.999199i	$-0.512738\pi$
$769$	−1.31211e6			−0.0800118			−0.0400059	+	0.999199i	$0.512738\pi$
$770$		0			0
$771$	9.52334e6	+	5.19592e6i	0.576970	+	0.314794i
$772$		0			0
$773$			3.80328e6i			0.228934i	0.993427	+	0.114467i	$0.0365160\pi$
$773$			3.80328e6i			0.228934i	−0.993427	+	0.114467i	$0.963484\pi$
$774$		0			0
$775$		−	3.73218e6i		−	0.223207i
$776$		0			0
$777$	8.21880e6	+	4.48416e6i	0.488378	+	0.266458i
$778$		0			0
$779$	3.84327e7			2.26912
$780$		0			0
$781$	−2.07423e7			−1.21683
$782$		0			0
$783$	1.92010e6	+	2.64658e7i	0.111923	+	1.54270i
$784$		0			0
$785$			1.67357e7i			0.969327i
$786$		0			0
$787$		−	7.73357e6i		−	0.445085i	−0.974923	−	0.222543i	$-0.928564\pi$
$787$		−	7.73357e6i		−	0.445085i	0.974923	−	0.222543i	$-0.0714357\pi$
$788$		0			0
$789$	2.26919e6	−	4.15909e6i	0.129771	−	0.237852i
$790$		0			0
$791$	−5.83282e6			−0.331465
$792$		0			0
$793$	−3.04569e6			−0.171990
$794$		0			0
$795$	4.42499e6	−	8.11034e6i	0.248310	−	0.455115i
$796$		0			0
$797$			3.37650e6i			0.188287i	0.995559	+	0.0941437i	$0.0300113\pi$
$797$			3.37650e6i			0.188287i	−0.995559	+	0.0941437i	$0.969989\pi$
$798$		0			0
$799$			1.50789e7i			0.835610i
$800$		0			0
$801$	1.49845e7	−	9.64440e6i	0.825203	−	0.531122i
$802$		0			0
$803$	−1.39676e7			−0.764422
$804$		0			0
$805$	−5.66374e6			−0.308044
$806$		0			0
$807$	3.01546e7	+	1.64523e7i	1.62993	+	0.889288i
$808$		0			0
$809$		−	3.08570e7i		−	1.65761i	−0.559537	−	0.828805i	$-0.689021\pi$
$809$		−	3.08570e7i		−	1.65761i	0.559537	−	0.828805i	$-0.310979\pi$
$810$		0			0
$811$			2.39558e7i			1.27897i	0.768805	+	0.639483i	$0.220852\pi$
$811$			2.39558e7i			1.27897i	−0.768805	+	0.639483i	$0.779148\pi$
$812$		0			0
$813$	−1.75227e7	−	9.56034e6i	−0.929766	−	0.507279i
$814$		0			0
$815$	−8.64921e6			−0.456123
$816$		0			0
$817$	−2.12074e7			−1.11156
$818$		0			0
$819$	4.32238e6	−	2.78200e6i	0.225171	−	0.144926i
$820$		0			0
$821$			1.08818e6i			0.0563435i	0.999603	+	0.0281718i	$0.00896854\pi$
$821$			1.08818e6i			0.0563435i	−0.999603	+	0.0281718i	$0.991031\pi$
$822$		0			0
$823$		−	1.75123e7i		−	0.901248i	−0.892714	−	0.450624i	$-0.851202\pi$
$823$		−	1.75123e7i		−	0.901248i	0.892714	−	0.450624i	$-0.148798\pi$
$824$		0			0
$825$	−4.41345e6	+	8.08920e6i	−0.225758	+	0.413781i
$826$		0			0
$827$	2.23971e7			1.13875			0.569375	−	0.822078i	$-0.307185\pi$
$827$	2.23971e7			1.13875			0.569375	+	0.822078i	$0.307185\pi$
$828$		0			0
$829$	−1.84316e7			−0.931486			−0.465743	−	0.884920i	$-0.654213\pi$
$829$	−1.84316e7			−0.931486			−0.465743	+	0.884920i	$0.654213\pi$
$830$		0			0
$831$	−5.59565e6	+	1.02560e7i	−0.281092	+	0.515199i
$832$		0			0
$833$		−	1.55810e6i		−	0.0778008i
$834$		0			0
$835$			4.35746e6i			0.216280i
$836$		0			0
$837$	−472876.	−	6.51789e6i	−0.0233310	−	0.321583i
$838$		0			0
$839$	2.98579e7			1.46438			0.732192	−	0.681098i	$-0.238497\pi$
$839$	2.98579e7			1.46438			0.732192	+	0.681098i	$0.238497\pi$
$840$		0			0
$841$	−2.85605e7			−1.39244
$842$		0			0
$843$	−2.65540e7	−	1.44878e7i	−1.28695	−	0.702158i
$844$		0			0
$845$		−	5.73468e6i		−	0.276291i
$846$		0			0
$847$		−	4.23287e6i		−	0.202734i
$848$		0			0
$849$	7.90810e6	+	4.31464e6i	0.376533	+	0.205436i
$850$		0			0
$851$	−4.56866e7			−2.16254
$852$		0			0
$853$	1.57574e7			0.741501			0.370750	−	0.928733i	$-0.379101\pi$
$853$	1.57574e7			0.741501			0.370750	+	0.928733i	$0.379101\pi$
$854$		0			0
$855$	9.41926e6	+	1.46347e7i	0.440658	+	0.684649i
$856$		0			0
$857$		−	1.87916e7i		−	0.874001i	−0.899461	−	0.437000i	$-0.856041\pi$
$857$		−	1.87916e7i		−	0.874001i	0.899461	−	0.437000i	$-0.143959\pi$
$858$		0			0
$859$		−	978284.i		−	0.0452358i	−0.999744	−	0.0226179i	$-0.992800\pi$
$859$		−	978284.i		−	0.0452358i	0.999744	−	0.0226179i	$-0.00720011\pi$
$860$		0			0
$861$	−6.08780e6	+	1.11580e7i	−0.279868	+	0.512955i
$862$		0			0
$863$	5.09831e6			0.233023			0.116512	−	0.993189i	$-0.462829\pi$
$863$	5.09831e6			0.233023			0.116512	+	0.993189i	$0.462829\pi$
$864$		0			0
$865$	1.19012e7			0.540817
$866$		0			0
$867$	−7.45664e6	+	1.36669e7i	−0.336895	+	0.617479i
$868$		0			0
$869$			2.89909e6i			0.130231i
$870$		0			0
$871$		−	1.99251e7i		−	0.889927i
$872$		0			0
$873$	1.76297e7	+	2.73912e7i	0.782906	+	1.21640i
$874$		0			0
$875$	−8.03575e6			−0.354819
$876$		0			0
$877$	−3.76613e7			−1.65347			−0.826734	−	0.562592i	$-0.809804\pi$
$877$	−3.76613e7			−1.65347			−0.826734	+	0.562592i	$0.809804\pi$
$878$		0			0
$879$	1.25205e7	+	6.83117e6i	0.546575	+	0.298210i
$880$		0			0
$881$			3.15933e7i			1.37137i	0.727898	+	0.685686i	$0.240498\pi$
$881$			3.15933e7i			1.37137i	−0.727898	+	0.685686i	$0.759502\pi$
$882$		0			0
$883$		−	3.57917e7i		−	1.54483i	−0.635119	−	0.772415i	$-0.719049\pi$
$883$		−	3.57917e7i		−	1.54483i	0.635119	−	0.772415i	$-0.280951\pi$
$884$		0			0
$885$	−4.38535e6	−	2.39264e6i	−0.188211	−	0.102688i
$886$		0			0
$887$	2.24956e7			0.960039			0.480019	−	0.877258i	$-0.340630\pi$
$887$	2.24956e7			0.960039			0.480019	+	0.877258i	$0.340630\pi$
$888$		0			0
$889$	−5.83616e6			−0.247670
$890$		0			0
$891$	−6.68275e6	+	1.46862e7i	−0.282008	+	0.619748i
$892$		0			0
$893$			5.36656e7i			2.25200i
$894$		0			0
$895$			6.70558e6i			0.279820i
$896$		0			0
$897$	−1.20136e7	+	2.20191e7i	−0.498531	+	0.913732i
$898$		0			0
$899$	1.20852e7			0.498717
$900$		0			0
$901$	−1.24026e7			−0.508980
$902$		0			0
$903$	3.35929e6	−	6.15707e6i	0.137097	−	0.251278i
$904$		0			0
$905$			531247.i			0.0215613i
$906$		0			0
$907$		−	1.85191e6i		−	0.0747485i	−0.999301	−	0.0373743i	$-0.988101\pi$
$907$		−	1.85191e6i		−	0.0747485i	0.999301	−	0.0373743i	$-0.0118994\pi$
$908$		0			0
$909$	−7.03991e6	+	4.53107e6i	−0.282590	+	0.181883i
$910$		0			0
$911$	2.53681e7			1.01272			0.506362	−	0.862321i	$-0.330990\pi$
$911$	2.53681e7			1.01272			0.506362	+	0.862321i	$0.330990\pi$
$912$		0			0
$913$	1.84554e7			0.732733
$914$		0			0
$915$	2.99386e6	+	1.63345e6i	0.118217	+	0.0644989i
$916$		0			0
$917$			960976.i			0.0377389i
$918$		0			0
$919$			2.36040e7i			0.921929i	0.887419	+	0.460964i	$0.152496\pi$
$919$			2.36040e7i			0.921929i	−0.887419	+	0.460964i	$0.847504\pi$
$920$		0			0
$921$	−3.50164e7	−	1.91049e7i	−1.36026	−	0.742156i
$922$		0			0
$923$	−3.27702e7			−1.26612
$924$		0			0
$925$	−2.65165e7			−1.01897
$926$		0			0
$927$	−8.57696e6	+	5.52035e6i	−0.327819	+	0.210993i
$928$		0			0
$929$			3.77392e7i			1.43467i	0.696726	+	0.717337i	$0.254639\pi$
$929$			3.77392e7i			1.43467i	−0.696726	+	0.717337i	$0.745361\pi$
$930$		0			0
$931$		−	5.54526e6i		−	0.209675i
$932$		0			0
$933$	1.20114e6	−	2.20150e6i	0.0451740	−	0.0827972i
$934$		0			0
$935$	−5.49891e6			−0.205706
$936$		0			0
$937$	2.25373e7			0.838595			0.419297	−	0.907849i	$-0.362276\pi$
$937$	2.25373e7			0.838595			0.419297	+	0.907849i	$0.362276\pi$
$938$		0			0
$939$	−188921.	+	346263.i	−0.00699223	+	0.0128157i
$940$		0			0
$941$		−	2.35680e7i		−	0.867657i	−0.900996	−	0.433828i	$-0.857162\pi$
$941$		−	2.35680e7i		−	0.867657i	0.900996	−	0.433828i	$-0.142838\pi$
$942$		0			0
$943$		−	6.20251e7i		−	2.27137i
$944$		0			0
$945$	−5.74086e6	+	416501.i	−0.209121	+	0.0151718i
$946$		0			0
$947$	−1.44399e7			−0.523228			−0.261614	−	0.965173i	$-0.584255\pi$
$947$	−1.44399e7			−0.523228			−0.261614	+	0.965173i	$0.584255\pi$
$948$		0			0
$949$	−2.20671e7			−0.795390
$950$		0			0
$951$	−2.83840e7	−	1.54863e7i	−1.01771	−	0.555259i
$952$		0			0
$953$			3.00105e6i			0.107039i	0.998567	+	0.0535193i	$0.0170439\pi$
$953$			3.00105e6i			0.107039i	−0.998567	+	0.0535193i	$0.982956\pi$
$954$		0			0
$955$		−	1.74846e7i		−	0.620365i
$956$		0			0
$957$	−2.61937e7	−	1.42912e7i	−0.924520	−	0.504417i
$958$		0			0
$959$	−5.48415e6			−0.192559
$960$		0			0
$961$	2.56529e7			0.896040
$962$		0			0
$963$	−94190.6	−	146344.i	−0.00327297	−	0.00508521i
$964$		0			0
$965$			3.45188e6i			0.119327i
$966$		0			0
$967$			3.14615e7i			1.08196i	0.841034	+	0.540982i	$0.181947\pi$
$967$			3.14615e7i			1.08196i	−0.841034	+	0.540982i	$0.818053\pi$
$968$		0			0
$969$	1.11899e7	−	2.05094e7i	0.382840	−	0.701688i
$970$		0			0
$971$	−3.01639e7			−1.02669			−0.513345	−	0.858182i	$-0.671594\pi$
$971$	−3.01639e7			−1.02669			−0.513345	+	0.858182i	$0.671594\pi$
$972$		0			0
$973$	−7.43926e6			−0.251911
$974$		0			0
$975$	−6.97271e6	+	1.27799e7i	−0.234904	+	0.430544i
$976$		0			0
$977$			4.28234e7i			1.43531i	0.696401	+	0.717653i	$0.254784\pi$
$977$			4.28234e7i			1.43531i	−0.696401	+	0.717653i	$0.745216\pi$
$978$		0			0
$979$			2.00383e7i			0.668196i
$980$		0			0
$981$	2.31470e7	+	3.59634e7i	0.767930	+	1.19313i
$982$		0			0
$983$	7.59677e6			0.250752			0.125376	−	0.992109i	$-0.459986\pi$
$983$	7.59677e6			0.250752			0.125376	+	0.992109i	$0.459986\pi$
$984$		0			0
$985$	−5.42975e6			−0.178315
$986$		0			0
$987$	−1.55805e7	−	8.50072e6i	−0.509084	−	0.277755i
$988$		0			0
$989$			3.42259e7i			1.11266i
$990$		0			0
$991$		−	4.05487e7i		−	1.31157i	−0.754947	−	0.655786i	$-0.772337\pi$
$991$		−	4.05487e7i		−	1.31157i	0.754947	−	0.655786i	$-0.227663\pi$
$992$		0			0
$993$	−5.19850e7	−	2.83629e7i	−1.67303	−	0.912805i
$994$		0			0
$995$	−775993.			−0.0248485
$996$		0			0
$997$	−3.62460e7			−1.15484			−0.577421	−	0.816447i	$-0.695941\pi$
$997$	−3.62460e7			−1.15484			−0.577421	+	0.816447i	$0.695941\pi$
$998$		0			0
$999$	−4.63086e7	+	3.35971e6i	−1.46807	+	0.106509i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.h.a.239.8	yes	20
3.2	odd	2	inner	336.6.h.a.239.14	yes	20
4.3	odd	2	inner	336.6.h.a.239.13	yes	20
12.11	even	2	inner	336.6.h.a.239.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.6.h.a.239.7	✓	20	12.11	even	2	inner
336.6.h.a.239.8	yes	20	1.1	even	1	trivial
336.6.h.a.239.13	yes	20	4.3	odd	2	inner
336.6.h.a.239.14	yes	20	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-7.46608 + 13.6842i) q^{3} +31.0107i q^{5} +49.0000i q^{7} +(-131.515 - 204.335i) q^{9} +O(q^{10})\)	\(q+(-7.46608 + 13.6842i) q^{3} +31.0107i q^{5} +49.0000i q^{7} +(-131.515 - 204.335i) q^{9} +273.251 q^{11} +431.702 q^{13} +(-424.356 - 231.528i) q^{15} +648.939i q^{17} +2309.56i q^{19} +(-670.526 - 365.838i) q^{21} +3727.31 q^{23} +2163.34 q^{25} +(3778.07 - 274.100i) q^{27} +7005.12i q^{29} -1725.19i q^{31} +(-2040.11 + 3739.22i) q^{33} -1519.52 q^{35} -12257.2 q^{37} +(-3223.13 + 5907.51i) q^{39} -16640.7i q^{41} +9182.45i q^{43} +(6336.56 - 4078.37i) q^{45} +23236.3 q^{47} -2401.00 q^{49} +(-8880.22 - 4845.04i) q^{51} +19112.1i q^{53} +8473.68i q^{55} +(-31604.5 - 17243.4i) q^{57} +10334.1 q^{59} -7055.07 q^{61} +(10012.4 - 6444.24i) q^{63} +13387.4i q^{65} -46154.6i q^{67} +(-27828.4 + 51005.3i) q^{69} -75909.3 q^{71} -51116.5 q^{73} +(-16151.7 + 29603.6i) q^{75} +13389.3i q^{77} +10609.7i q^{79} +(-24456.5 + 53746.3i) q^{81} +67540.1 q^{83} -20124.0 q^{85} +(-95859.5 - 52300.8i) q^{87} +73333.0i q^{89} +21153.4i q^{91} +(23607.9 + 12880.4i) q^{93} -71621.1 q^{95} -134051. q^{97} +(-35936.6 - 55834.6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 140 q^{9}+O(q^{10}) \) 20 * q - 140 * q^9	\( 20 q - 140 q^{9} + 1048 q^{13} - 980 q^{21} + 5916 q^{25} - 26056 q^{33} + 61360 q^{37} - 92512 q^{45} - 48020 q^{49} - 20720 q^{57} + 46680 q^{61} - 28360 q^{69} - 54280 q^{73} + 152660 q^{81} - 150536 q^{85} + 41688 q^{93} - 421352 q^{97}+O(q^{100}) \) 20 * q - 140 * q^9 + 1048 * q^13 - 980 * q^21 + 5916 * q^25 - 26056 * q^33 + 61360 * q^37 - 92512 * q^45 - 48020 * q^49 - 20720 * q^57 + 46680 * q^61 - 28360 * q^69 - 54280 * q^73 + 152660 * q^81 - 150536 * q^85 + 41688 * q^93 - 421352 * q^97

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