LMFDB - Embedded newform 592.6.g.c.369.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,6,Mod(369,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("592.369");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	592.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$94.9472213293$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 390 x^{14} + 60701 x^{12} + 4799932 x^{10} + 203487156 x^{8} + 4519465040 x^{6} + \cdots + 178006118400 $ x^16 + 390x^14 + 60701x^12 + 4799932x^10 + 203487156x^8 + 4519465040x^6 + 48993644736x^4 + 211923220224*x^2 + 178006118400
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 37)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.16
Root			$-10.0606i$ of defining polynomial
Character	$\chi$	$=$	592.369
Dual form			592.6.g.c.369.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+25.7580 q^{3} +77.4051i q^{5} -168.370 q^{7} +420.475 q^{9} +O(q^{10})$	$q+25.7580 q^{3} +77.4051i q^{5} -168.370 q^{7} +420.475 q^{9} +603.572 q^{11} -268.415i q^{13} +1993.80i q^{15} +873.888i q^{17} +706.696i q^{19} -4336.87 q^{21} +141.099i q^{23} -2866.55 q^{25} +4571.40 q^{27} +6352.26i q^{29} +5756.36i q^{31} +15546.8 q^{33} -13032.7i q^{35} +(-3768.82 - 7425.63i) q^{37} -6913.84i q^{39} -4153.25 q^{41} -8999.12i q^{43} +32546.9i q^{45} +22240.1 q^{47} +11541.4 q^{49} +22509.6i q^{51} -13348.3 q^{53} +46719.6i q^{55} +18203.1i q^{57} +7567.74i q^{59} +53741.6i q^{61} -70795.3 q^{63} +20776.7 q^{65} +1077.89 q^{67} +3634.43i q^{69} -41521.0 q^{71} -50604.9 q^{73} -73836.7 q^{75} -101623. q^{77} -9485.04i q^{79} +15574.8 q^{81} -1476.60 q^{83} -67643.4 q^{85} +163622. i q^{87} -11185.2i q^{89} +45193.0i q^{91} +148272. i q^{93} -54701.9 q^{95} +17542.0i q^{97} +253787. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 18 q^{3} - 190 q^{7} + 1394 q^{9}+O(q^{10}) $ 16 * q + 18 * q^3 - 190 * q^7 + 1394 * q^9	$ 16 q + 18 q^{3} - 190 q^{7} + 1394 q^{9} + 1110 q^{11} - 7010 q^{21} - 12052 q^{25} - 4266 q^{27} - 2478 q^{33} - 11400 q^{37} + 3918 q^{41} - 3822 q^{47} - 32618 q^{49} - 24126 q^{53} - 219268 q^{63} + 98976 q^{65} - 23560 q^{67} + 50046 q^{71} - 196274 q^{73} - 214054 q^{75} - 239574 q^{77} + 317312 q^{81} + 215814 q^{83} - 346472 q^{85} + 132504 q^{95} + 574860 q^{99}+O(q^{100}) $ 16 * q + 18 * q^3 - 190 * q^7 + 1394 * q^9 + 1110 * q^11 - 7010 * q^21 - 12052 * q^25 - 4266 * q^27 - 2478 * q^33 - 11400 * q^37 + 3918 * q^41 - 3822 * q^47 - 32618 * q^49 - 24126 * q^53 - 219268 * q^63 + 98976 * q^65 - 23560 * q^67 + 50046 * q^71 - 196274 * q^73 - 214054 * q^75 - 239574 * q^77 + 317312 * q^81 + 215814 * q^83 - 346472 * q^85 + 132504 * q^95 + 574860 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/592\mathbb{Z}\right)^\times$.

$n$	$113$	$149$	$223$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	25.7580			1.65238			0.826188	−	0.563394i	$-0.190505\pi$
$3$	25.7580			1.65238			0.826188	+	0.563394i	$0.190505\pi$
$4$		0			0
$5$			77.4051i			1.38466i	0.721579	+	0.692332i	$0.243417\pi$
$5$			77.4051i			1.38466i	−0.721579	+	0.692332i	$0.756583\pi$
$6$		0			0
$7$	−168.370			−1.29873			−0.649366	−	0.760476i	$-0.724966\pi$
$7$	−168.370			−1.29873			−0.649366	+	0.760476i	$0.724966\pi$
$8$		0			0
$9$	420.475			1.73035
$10$		0			0
$11$	603.572			1.50400			0.751999	−	0.659164i	$-0.229090\pi$
$11$	603.572			1.50400			0.751999	+	0.659164i	$0.229090\pi$
$12$		0			0
$13$		−	268.415i		−	0.440503i	−0.975443	−	0.220251i	$-0.929312\pi$
$13$		−	268.415i		−	0.440503i	0.975443	−	0.220251i	$-0.0706878\pi$
$14$		0			0
$15$			1993.80i			2.28799i
$16$		0			0
$17$			873.888i			0.733388i	0.930342	+	0.366694i	$0.119510\pi$
$17$			873.888i			0.733388i	−0.930342	+	0.366694i	$0.880490\pi$
$18$		0			0
$19$			706.696i			0.449106i	0.974462	+	0.224553i	$0.0720921\pi$
$19$			706.696i			0.449106i	−0.974462	+	0.224553i	$0.927908\pi$
$20$		0			0
$21$	−4336.87			−2.14599
$22$		0			0
$23$			141.099i			0.0556167i	0.999613	+	0.0278083i	$0.00885281\pi$
$23$			141.099i			0.0556167i	−0.999613	+	0.0278083i	$0.991147\pi$
$24$		0			0
$25$	−2866.55			−0.917296
$26$		0			0
$27$	4571.40			1.20681
$28$		0			0
$29$			6352.26i			1.40260i	0.712867	+	0.701300i	$0.247396\pi$
$29$			6352.26i			1.40260i	−0.712867	+	0.701300i	$0.752604\pi$
$30$		0			0
$31$			5756.36i			1.07583i	0.842999	+	0.537915i	$0.180788\pi$
$31$			5756.36i			1.07583i	−0.842999	+	0.537915i	$0.819212\pi$
$32$		0			0
$33$	15546.8			2.48517
$34$		0			0
$35$		−	13032.7i		−	1.79831i
$36$		0			0
$37$	−3768.82	−	7425.63i	−0.452586	−	0.891721i
$37$	−3768.82	−	7425.63i	−0.452586	−	0.891721i
$38$		0			0
$39$		−	6913.84i		−	0.727876i
$40$		0			0
$41$	−4153.25			−0.385859			−0.192929	−	0.981213i	$-0.561799\pi$
$41$	−4153.25			−0.385859			−0.192929	+	0.981213i	$0.561799\pi$
$42$		0			0
$43$		−	8999.12i		−	0.742214i	−0.928590	−	0.371107i	$-0.878978\pi$
$43$		−	8999.12i		−	0.742214i	0.928590	−	0.371107i	$-0.121022\pi$
$44$		0			0
$45$			32546.9i			2.39595i
$46$		0			0
$47$	22240.1			1.46856			0.734280	−	0.678846i	$-0.237520\pi$
$47$	22240.1			1.46856			0.734280	+	0.678846i	$0.237520\pi$
$48$		0			0
$49$	11541.4			0.686703
$50$		0			0
$51$			22509.6i			1.21183i
$52$		0			0
$53$	−13348.3			−0.652736			−0.326368	−	0.945243i	$-0.605825\pi$
$53$	−13348.3			−0.652736			−0.326368	+	0.945243i	$0.605825\pi$
$54$		0			0
$55$			46719.6i			2.08253i
$56$		0			0
$57$			18203.1i			0.742092i
$58$		0			0
$59$			7567.74i			0.283032i	0.989936	+	0.141516i	$0.0451977\pi$
$59$			7567.74i			0.283032i	−0.989936	+	0.141516i	$0.954802\pi$
$60$		0			0
$61$			53741.6i			1.84921i	0.380929	+	0.924604i	$0.375604\pi$
$61$			53741.6i			1.84921i	−0.380929	+	0.924604i	$0.624396\pi$
$62$		0			0
$63$	−70795.3			−2.24726
$64$		0			0
$65$	20776.7			0.609949
$66$		0			0
$67$	1077.89			0.0293351			0.0146676	−	0.999892i	$-0.495331\pi$
$67$	1077.89			0.0293351			0.0146676	+	0.999892i	$0.495331\pi$
$68$		0			0
$69$			3634.43i			0.0918997i
$70$		0			0
$71$	−41521.0			−0.977512			−0.488756	−	0.872421i	$-0.662549\pi$
$71$	−41521.0			−0.977512			−0.488756	+	0.872421i	$0.662549\pi$
$72$		0			0
$73$	−50604.9			−1.11144			−0.555719	−	0.831370i	$-0.687557\pi$
$73$	−50604.9			−1.11144			−0.555719	+	0.831370i	$0.687557\pi$
$74$		0			0
$75$	−73836.7			−1.51572
$76$		0			0
$77$	−101623.			−1.95329
$78$		0			0
$79$		−	9485.04i		−	0.170990i	−0.996339	−	0.0854952i	$-0.972753\pi$
$79$		−	9485.04i		−	0.170990i	0.996339	−	0.0854952i	$-0.0272472\pi$
$80$		0			0
$81$	15574.8			0.263760
$82$		0			0
$83$	−1476.60			−0.0235271			−0.0117636	−	0.999931i	$-0.503745\pi$
$83$	−1476.60			−0.0235271			−0.0117636	+	0.999931i	$0.503745\pi$
$84$		0			0
$85$	−67643.4			−1.01550
$86$		0			0
$87$			163622.i			2.31762i
$88$		0			0
$89$		−	11185.2i		−	0.149682i	−0.997195	−	0.0748409i	$-0.976155\pi$
$89$		−	11185.2i		−	0.149682i	0.997195	−	0.0748409i	$-0.0238449\pi$
$90$		0			0
$91$			45193.0i			0.572095i
$92$		0			0
$93$			148272.i			1.77768i
$94$		0			0
$95$	−54701.9			−0.621861
$96$		0			0
$97$			17542.0i			0.189300i	0.995511	+	0.0946500i	$0.0301732\pi$
$97$			17542.0i			0.189300i	−0.995511	+	0.0946500i	$0.969827\pi$
$98$		0			0
$99$	253787.			2.60244
$100$		0			0
$101$	−21138.2			−0.206188			−0.103094	−	0.994672i	$-0.532874\pi$
$101$	−21138.2			−0.206188			−0.103094	+	0.994672i	$0.532874\pi$
$102$		0			0
$103$			21654.8i			0.201123i	0.994931	+	0.100561i	$0.0320639\pi$
$103$			21654.8i			0.201123i	−0.994931	+	0.100561i	$0.967936\pi$
$104$		0			0
$105$		−	335696.i		−	2.97148i
$106$		0			0
$107$	75852.3			0.640485			0.320243	−	0.947336i	$-0.396236\pi$
$107$	75852.3			0.640485			0.320243	+	0.947336i	$0.396236\pi$
$108$		0			0
$109$			224063.i			1.80636i	0.429265	+	0.903179i	$0.358773\pi$
$109$			224063.i			1.80636i	−0.429265	+	0.903179i	$0.641227\pi$
$110$		0			0
$111$	−97077.3	−	191269.i	−0.747843	−	1.47346i
$112$		0			0
$113$		−	84675.4i		−	0.623823i	−0.950111	−	0.311911i	$-0.899031\pi$
$113$		−	84675.4i		−	0.623823i	0.950111	−	0.311911i	$-0.100969\pi$
$114$		0			0
$115$	−10921.8			−0.0770104
$116$		0			0
$117$		−	112862.i		−	0.762224i
$118$		0			0
$119$		−	147136.i		−	0.952473i
$120$		0			0
$121$	203248.			1.26201
$122$		0			0
$123$	−106979.			−0.637584
$124$		0			0
$125$			20005.2i			0.114517i
$126$		0			0
$127$	−256661.			−1.41205			−0.706026	−	0.708185i	$-0.749514\pi$
$127$	−256661.			−1.41205			−0.706026	+	0.708185i	$0.749514\pi$
$128$		0			0
$129$		−	231799.i		−	1.22642i
$130$		0			0
$131$			79025.9i			0.402338i	0.979557	+	0.201169i	$0.0644741\pi$
$131$			79025.9i			0.402338i	−0.979557	+	0.201169i	$0.935526\pi$
$132$		0			0
$133$		−	118986.i		−	0.583268i
$134$		0			0
$135$			353850.i			1.67103i
$136$		0			0
$137$	55006.4			0.250387			0.125194	−	0.992132i	$-0.460045\pi$
$137$	55006.4			0.250387			0.125194	+	0.992132i	$0.460045\pi$
$138$		0			0
$139$	305471.			1.34101			0.670507	−	0.741903i	$-0.266077\pi$
$139$	305471.			1.34101			0.670507	+	0.741903i	$0.266077\pi$
$140$		0			0
$141$	572860.			2.42662
$142$		0			0
$143$		−	162008.i		−	0.662515i
$144$		0			0
$145$	−491698.			−1.94213
$146$		0			0
$147$	297284.			1.13469
$148$		0			0
$149$	304779.			1.12465			0.562327	−	0.826915i	$-0.309906\pi$
$149$	304779.			1.12465			0.562327	+	0.826915i	$0.309906\pi$
$150$		0			0
$151$	−16190.3			−0.0577845			−0.0288923	−	0.999583i	$-0.509198\pi$
$151$	−16190.3			−0.0577845			−0.0288923	+	0.999583i	$0.509198\pi$
$152$		0			0
$153$			367448.i			1.26902i
$154$		0			0
$155$	−445572.			−1.48967
$156$		0			0
$157$	−312830.			−1.01288			−0.506441	−	0.862275i	$-0.669039\pi$
$157$	−312830.			−1.01288			−0.506441	+	0.862275i	$0.669039\pi$
$158$		0			0
$159$	−343827.			−1.07857
$160$		0			0
$161$		−	23756.9i		−	0.0722311i
$162$		0			0
$163$			436231.i			1.28602i	0.765858	+	0.643010i	$0.222315\pi$
$163$			436231.i			1.28602i	−0.765858	+	0.643010i	$0.777685\pi$
$164$		0			0
$165$			1.20340e6i			3.44113i
$166$		0			0
$167$		−	587366.i		−	1.62974i	−0.579646	−	0.814868i	$-0.696809\pi$
$167$		−	587366.i		−	1.62974i	0.579646	−	0.814868i	$-0.303191\pi$
$168$		0			0
$169$	299246.			0.805957
$170$		0			0
$171$			297148.i			0.777110i
$172$		0			0
$173$	−693665.			−1.76212			−0.881059	−	0.473007i	$-0.843169\pi$
$173$	−693665.			−1.76212			−0.881059	+	0.473007i	$0.843169\pi$
$174$		0			0
$175$	482641.			1.19132
$176$		0			0
$177$			194930.i			0.467676i
$178$		0			0
$179$		−	359738.i		−	0.839177i	−0.907714	−	0.419588i	$-0.862174\pi$
$179$		−	359738.i		−	0.839177i	0.907714	−	0.419588i	$-0.137826\pi$
$180$		0			0
$181$	259702.			0.589222			0.294611	−	0.955617i	$-0.404810\pi$
$181$	259702.			0.589222			0.294611	+	0.955617i	$0.404810\pi$
$182$		0			0
$183$			1.38428e6i			3.05559i
$184$		0			0
$185$	574782.	−	291726.i	1.23473	−	0.626680i
$186$		0			0
$187$			527454.i			1.10301i
$188$		0			0
$189$	−769686.			−1.56733
$190$		0			0
$191$		−	42814.5i		−	0.0849195i	−0.999098	−	0.0424598i	$-0.986481\pi$
$191$		−	42814.5i		−	0.0849195i	0.999098	−	0.0424598i	$-0.0135194\pi$
$192$		0			0
$193$			923411.i			1.78444i	0.451601	+	0.892220i	$0.350853\pi$
$193$			923411.i			1.78444i	−0.451601	+	0.892220i	$0.649147\pi$
$194$		0			0
$195$	535166.			1.00786
$196$		0			0
$197$	−152054.			−0.279146			−0.139573	−	0.990212i	$-0.544573\pi$
$197$	−152054.			−0.279146			−0.139573	+	0.990212i	$0.544573\pi$
$198$		0			0
$199$		−	769973.i		−	1.37830i	−0.724620	−	0.689149i	$-0.757985\pi$
$199$		−	769973.i		−	1.37830i	0.724620	−	0.689149i	$-0.242015\pi$
$200$		0			0
$201$	27764.3			0.0484726
$202$		0			0
$203$		−	1.06953e6i		−	1.82160i
$204$		0			0
$205$		−	321483.i		−	0.534285i
$206$		0			0
$207$			59328.7i			0.0962363i
$208$		0			0
$209$			426542.i			0.675454i
$210$		0			0
$211$	330604.			0.511213			0.255607	−	0.966781i	$-0.417725\pi$
$211$	330604.			0.511213			0.255607	+	0.966781i	$0.417725\pi$
$212$		0			0
$213$	−1.06950e6			−1.61522
$214$		0			0
$215$	696578.			1.02772
$216$		0			0
$217$		−	969198.i		−	1.39722i
$218$		0			0
$219$	−1.30348e6			−1.83651
$220$		0			0
$221$	234565.			0.323059
$222$		0			0
$223$	523244.			0.704600			0.352300	−	0.935887i	$-0.385400\pi$
$223$	523244.			0.704600			0.352300	+	0.935887i	$0.385400\pi$
$224$		0			0
$225$	−1.20531e6			−1.58724
$226$		0			0
$227$			1.12179e6i			1.44492i	0.691410	+	0.722462i	$0.256990\pi$
$227$			1.12179e6i			1.44492i	−0.691410	+	0.722462i	$0.743010\pi$
$228$		0			0
$229$	−1.14774e6			−1.44629			−0.723146	−	0.690696i	$-0.757304\pi$
$229$	−1.14774e6			−1.44629			−0.723146	+	0.690696i	$0.757304\pi$
$230$		0			0
$231$	−2.61761e6			−3.22757
$232$		0			0
$233$	989725.			1.19433			0.597165	−	0.802118i	$-0.296294\pi$
$233$	989725.			1.19433			0.597165	+	0.802118i	$0.296294\pi$
$234$		0			0
$235$			1.72150e6i			2.03346i
$236$		0			0
$237$		−	244316.i		−	0.282540i
$238$		0			0
$239$		−	1.13545e6i		−	1.28580i	−0.765950	−	0.642900i	$-0.777731\pi$
$239$		−	1.13545e6i		−	1.28580i	0.765950	−	0.642900i	$-0.222269\pi$
$240$		0			0
$241$			157029.i			0.174156i	0.996202	+	0.0870778i	$0.0277529\pi$
$241$			157029.i			0.174156i	−0.996202	+	0.0870778i	$0.972247\pi$
$242$		0			0
$243$	−709676.			−0.770982
$244$		0			0
$245$			893365.i			0.950853i
$246$		0			0
$247$	189688.			0.197832
$248$		0			0
$249$	−38034.4			−0.0388757
$250$		0			0
$251$		−	98057.8i		−	0.0982421i	−0.998793	−	0.0491211i	$-0.984358\pi$
$251$		−	98057.8i		−	0.0982421i	0.998793	−	0.0491211i	$-0.0156420\pi$
$252$		0			0
$253$			85163.5i			0.0836474i
$254$		0			0
$255$	−1.74236e6			−1.67798
$256$		0			0
$257$		−	1.81434e6i		−	1.71350i	−0.515730	−	0.856751i	$-0.672479\pi$
$257$		−	1.81434e6i		−	1.71350i	0.515730	−	0.856751i	$-0.327521\pi$
$258$		0			0
$259$	634556.	+	1.25025e6i	0.587788	+	1.15811i
$260$		0			0
$261$			2.67097e6i			2.42699i
$262$		0			0
$263$	797108.			0.710604			0.355302	−	0.934752i	$-0.384378\pi$
$263$	797108.			0.710604			0.355302	+	0.934752i	$0.384378\pi$
$264$		0			0
$265$		−	1.03323e6i		−	0.903821i
$266$		0			0
$267$		−	288109.i		−	0.247331i
$268$		0			0
$269$	1.28791e6			1.08519			0.542593	−	0.839996i	$-0.317443\pi$
$269$	1.28791e6			1.08519			0.542593	+	0.839996i	$0.317443\pi$
$270$		0			0
$271$	1.27896e6			1.05787			0.528936	−	0.848662i	$-0.322591\pi$
$271$	1.27896e6			1.05787			0.528936	+	0.848662i	$0.322591\pi$
$272$		0			0
$273$			1.16408e6i			0.945316i
$274$		0			0
$275$	−1.73017e6			−1.37961
$276$		0			0
$277$			1.40148e6i			1.09746i	0.836000	+	0.548729i	$0.184888\pi$
$277$			1.40148e6i			1.09746i	−0.836000	+	0.548729i	$0.815112\pi$
$278$		0			0
$279$			2.42041e6i			1.86156i
$280$		0			0
$281$			494703.i			0.373748i	0.982384	+	0.186874i	$0.0598356\pi$
$281$			494703.i			0.373748i	−0.982384	+	0.186874i	$0.940164\pi$
$282$		0			0
$283$		−	832541.i		−	0.617930i	−0.951073	−	0.308965i	$-0.900017\pi$
$283$		−	832541.i		−	0.617930i	0.951073	−	0.308965i	$-0.0999827\pi$
$284$		0			0
$285$	−1.40901e6			−1.02755
$286$		0			0
$287$	699282.			0.501127
$288$		0			0
$289$	656177.			0.462143
$290$		0			0
$291$			451848.i			0.312795i
$292$		0			0
$293$	951351.			0.647398			0.323699	−	0.946160i	$-0.395073\pi$
$293$	951351.			0.647398			0.323699	+	0.946160i	$0.395073\pi$
$294$		0			0
$295$	−585781.			−0.391905
$296$		0			0
$297$	2.75917e6			1.81504
$298$		0			0
$299$	37873.2			0.0244993
$300$		0			0
$301$			1.51518e6i			0.963936i
$302$		0			0
$303$	−544478.			−0.340701
$304$		0			0
$305$	−4.15987e6			−2.56053
$306$		0			0
$307$	458968.			0.277931			0.138965	−	0.990297i	$-0.455622\pi$
$307$	458968.			0.277931			0.138965	+	0.990297i	$0.455622\pi$
$308$		0			0
$309$			557785.i			0.332331i
$310$		0			0
$311$			1.66612e6i			0.976799i	0.872620	+	0.488400i	$0.162419\pi$
$311$			1.66612e6i			0.976799i	−0.872620	+	0.488400i	$0.837581\pi$
$312$		0			0
$313$		−	1.75761e6i		−	1.01405i	−0.861931	−	0.507026i	$-0.830745\pi$
$313$		−	1.75761e6i		−	1.01405i	0.861931	−	0.507026i	$-0.169255\pi$
$314$		0			0
$315$		−	5.47992e6i		−	3.11170i
$316$		0			0
$317$	1.17532e6			0.656912			0.328456	−	0.944519i	$-0.393472\pi$
$317$	1.17532e6			0.656912			0.328456	+	0.944519i	$0.393472\pi$
$318$		0			0
$319$			3.83405e6i			2.10951i
$320$		0			0
$321$	1.95380e6			1.05832
$322$		0			0
$323$	−617573.			−0.329368
$324$		0			0
$325$			769426.i			0.404072i
$326$		0			0
$327$			5.77141e6i			2.98478i
$328$		0			0
$329$	−3.74456e6			−1.90727
$330$		0			0
$331$			65616.6i			0.0329188i	0.999865	+	0.0164594i	$0.00523942\pi$
$331$			65616.6i			0.0329188i	−0.999865	+	0.0164594i	$0.994761\pi$
$332$		0			0
$333$	−1.58470e6	−	3.12229e6i	−0.783132	−	1.54299i
$334$		0			0
$335$			83434.2i			0.0406193i
$336$		0			0
$337$	2.60182e6			1.24797			0.623983	−	0.781438i	$-0.285514\pi$
$337$	2.60182e6			1.24797			0.623983	+	0.781438i	$0.285514\pi$
$338$		0			0
$339$		−	2.18107e6i		−	1.03079i
$340$		0			0
$341$			3.47438e6i			1.61805i
$342$		0			0
$343$	886566.			0.406889
$344$		0			0
$345$	−281324.			−0.127250
$346$		0			0
$347$		−	908062.i		−	0.404848i	−0.979298	−	0.202424i	$-0.935118\pi$
$347$		−	908062.i		−	0.404848i	0.979298	−	0.202424i	$-0.0648819\pi$
$348$		0			0
$349$	3.30403e6			1.45205			0.726023	−	0.687670i	$-0.241367\pi$
$349$	3.30403e6			1.45205			0.726023	+	0.687670i	$0.241367\pi$
$350$		0			0
$351$		−	1.22703e6i		−	0.531604i
$352$		0			0
$353$		−	3.27707e6i		−	1.39974i	−0.714269	−	0.699872i	$-0.753241\pi$
$353$		−	3.27707e6i		−	1.39974i	0.714269	−	0.699872i	$-0.246759\pi$
$354$		0			0
$355$		−	3.21394e6i		−	1.35353i
$356$		0			0
$357$		−	3.78994e6i		−	1.57384i
$358$		0			0
$359$	2.67944e6			1.09726			0.548628	−	0.836067i	$-0.315151\pi$
$359$	2.67944e6			1.09726			0.548628	+	0.836067i	$0.315151\pi$
$360$		0			0
$361$	1.97668e6			0.798304
$362$		0			0
$363$	5.23527e6			2.08532
$364$		0			0
$365$		−	3.91708e6i		−	1.53897i
$366$		0			0
$367$	−1.54222e6			−0.597695			−0.298848	−	0.954301i	$-0.596602\pi$
$367$	−1.54222e6			−0.597695			−0.298848	+	0.954301i	$0.596602\pi$
$368$		0			0
$369$	−1.74634e6			−0.667671
$370$		0			0
$371$	2.24746e6			0.847729
$372$		0			0
$373$	2.18057e6			0.811516			0.405758	−	0.913981i	$-0.367008\pi$
$373$	2.18057e6			0.811516			0.405758	+	0.913981i	$0.367008\pi$
$374$		0			0
$375$			515295.i			0.189225i
$376$		0			0
$377$	1.70504e6			0.617849
$378$		0			0
$379$	562144.			0.201025			0.100512	−	0.994936i	$-0.467952\pi$
$379$	562144.			0.201025			0.100512	+	0.994936i	$0.467952\pi$
$380$		0			0
$381$	−6.61108e6			−2.33324
$382$		0			0
$383$		−	3.80268e6i		−	1.32462i	−0.749228	−	0.662312i	$-0.769575\pi$
$383$		−	3.80268e6i		−	1.32462i	0.749228	−	0.662312i	$-0.230425\pi$
$384$		0			0
$385$		−	7.86617e6i		−	2.70465i
$386$		0			0
$387$		−	3.78390e6i		−	1.28429i
$388$		0			0
$389$			2.54132e6i			0.851503i	0.904840	+	0.425751i	$0.139990\pi$
$389$			2.54132e6i			0.851503i	−0.904840	+	0.425751i	$0.860010\pi$
$390$		0			0
$391$	−123305.			−0.0407886
$392$		0			0
$393$			2.03555e6i			0.664814i
$394$		0			0
$395$	734191.			0.236764
$396$		0			0
$397$	3.65631e6			1.16430			0.582152	−	0.813080i	$-0.302211\pi$
$397$	3.65631e6			1.16430			0.582152	+	0.813080i	$0.302211\pi$
$398$		0			0
$399$		−	3.06485e6i		−	0.963778i
$400$		0			0
$401$			1.28914e6i			0.400350i	0.979760	+	0.200175i	$0.0641511\pi$
$401$			1.28914e6i			0.400350i	−0.979760	+	0.200175i	$0.935849\pi$
$402$		0			0
$403$	1.54510e6			0.473906
$404$		0			0
$405$			1.20557e6i			0.365219i
$406$		0			0
$407$	−2.27476e6	−	4.48190e6i	−0.680689	−	1.34115i
$408$		0			0
$409$			966332.i			0.285640i	0.989749	+	0.142820i	$0.0456169\pi$
$409$			966332.i			0.285640i	−0.989749	+	0.142820i	$0.954383\pi$
$410$		0			0
$411$	1.41686e6			0.413734
$412$		0			0
$413$		−	1.27418e6i		−	0.367583i
$414$		0			0
$415$		−	114297.i		−	0.0325772i
$416$		0			0
$417$	7.86833e6			2.21586
$418$		0			0
$419$	1.23157e6			0.342709			0.171354	−	0.985209i	$-0.445186\pi$
$419$	1.23157e6			0.342709			0.171354	+	0.985209i	$0.445186\pi$
$420$		0			0
$421$		−	3.22979e6i		−	0.888115i	−0.895998	−	0.444057i	$-0.853539\pi$
$421$		−	3.22979e6i		−	0.888115i	0.895998	−	0.444057i	$-0.146461\pi$
$422$		0			0
$423$	9.35140e6			2.54112
$424$		0			0
$425$		−	2.50505e6i		−	0.672734i
$426$		0			0
$427$		−	9.04846e6i		−	2.40162i
$428$		0			0
$429$		−	4.17300e6i		−	1.09473i
$430$		0			0
$431$		−	1.57396e6i		−	0.408131i	−0.978957	−	0.204065i	$-0.934584\pi$
$431$		−	1.57396e6i		−	0.408131i	0.978957	−	0.204065i	$-0.0654155\pi$
$432$		0			0
$433$	−1.33511e6			−0.342213			−0.171106	−	0.985253i	$-0.554734\pi$
$433$	−1.33511e6			−0.342213			−0.171106	+	0.985253i	$0.554734\pi$
$434$		0			0
$435$	−1.26652e7			−3.20913
$436$		0			0
$437$	−99714.2			−0.0249778
$438$		0			0
$439$		−	6.63631e6i		−	1.64348i	−0.569861	−	0.821741i	$-0.693003\pi$
$439$		−	6.63631e6i		−	1.64348i	0.569861	−	0.821741i	$-0.306997\pi$
$440$		0			0
$441$	4.85288e6			1.18824
$442$		0			0
$443$	−5.21572e6			−1.26271			−0.631357	−	0.775493i	$-0.717502\pi$
$443$	−5.21572e6			−1.26271			−0.631357	+	0.775493i	$0.717502\pi$
$444$		0			0
$445$	865792.			0.207259
$446$		0			0
$447$	7.85049e6			1.85835
$448$		0			0
$449$			6.96431e6i			1.63028i	0.579264	+	0.815140i	$0.303340\pi$
$449$			6.96431e6i			1.63028i	−0.579264	+	0.815140i	$0.696660\pi$
$450$		0			0
$451$	−2.50679e6			−0.580331
$452$		0			0
$453$	−417029.			−0.0954819
$454$		0			0
$455$	−3.49817e6			−0.792159
$456$		0			0
$457$		−	3.29043e6i		−	0.736990i	−0.929630	−	0.368495i	$-0.879873\pi$
$457$		−	3.29043e6i		−	0.736990i	0.929630	−	0.368495i	$-0.120127\pi$
$458$		0			0
$459$			3.99489e6i			0.885061i
$460$		0			0
$461$			1.03373e6i			0.226546i	0.993564	+	0.113273i	$0.0361335\pi$
$461$			1.03373e6i			0.226546i	−0.993564	+	0.113273i	$0.963866\pi$
$462$		0			0
$463$			421953.i			0.0914770i	0.998953	+	0.0457385i	$0.0145641\pi$
$463$			421953.i			0.0914770i	−0.998953	+	0.0457385i	$0.985436\pi$
$464$		0			0
$465$	−1.14770e7			−2.46149
$466$		0			0
$467$		−	3.68933e6i		−	0.782808i	−0.920219	−	0.391404i	$-0.871990\pi$
$467$		−	3.68933e6i		−	0.782808i	0.920219	−	0.391404i	$-0.128010\pi$
$468$		0			0
$469$	−181484.			−0.0380984
$470$		0			0
$471$	−8.05787e6			−1.67366
$472$		0			0
$473$		−	5.43162e6i		−	1.11629i
$474$		0			0
$475$		−	2.02578e6i		−	0.411963i
$476$		0			0
$477$	−5.61264e6			−1.12946
$478$		0			0
$479$		−	320565.i		−	0.0638378i	−0.999490	−	0.0319189i	$-0.989838\pi$
$479$		−	320565.i		−	0.0638378i	0.999490	−	0.0319189i	$-0.0101618\pi$
$480$		0			0
$481$	−1.99315e6	+	1.01161e6i	−0.392805	+	0.199365i
$482$		0			0
$483$		−	611929.i		−	0.119353i
$484$		0			0
$485$	−1.35784e6			−0.262117
$486$		0			0
$487$		−	2.00859e6i		−	0.383768i	−0.981418	−	0.191884i	$-0.938540\pi$
$487$		−	2.00859e6i		−	0.383768i	0.981418	−	0.191884i	$-0.0614597\pi$
$488$		0			0
$489$			1.12365e7i			2.12499i
$490$		0			0
$491$	7.62309e6			1.42701			0.713505	−	0.700650i	$-0.247106\pi$
$491$	7.62309e6			1.42701			0.713505	+	0.700650i	$0.247106\pi$
$492$		0			0
$493$	−5.55117e6			−1.02865
$494$		0			0
$495$			1.96444e7i			3.60351i
$496$		0			0
$497$	6.99089e6			1.26953
$498$		0			0
$499$			2.42605e6i			0.436163i	0.975931	+	0.218082i	$0.0699799\pi$
$499$			2.42605e6i			0.436163i	−0.975931	+	0.218082i	$0.930020\pi$
$500$		0			0
$501$		−	1.51294e7i		−	2.69294i
$502$		0			0
$503$			5.77617e6i			1.01793i	0.860786	+	0.508967i	$0.169973\pi$
$503$			5.77617e6i			1.01793i	−0.860786	+	0.508967i	$0.830027\pi$
$504$		0			0
$505$		−	1.63620e6i		−	0.285502i
$506$		0			0
$507$	7.70799e6			1.33175
$508$		0			0
$509$	−6.24081e6			−1.06769			−0.533847	−	0.845581i	$-0.679254\pi$
$509$	−6.24081e6			−1.06769			−0.533847	+	0.845581i	$0.679254\pi$
$510$		0			0
$511$	8.52034e6			1.44346
$512$		0			0
$513$			3.23059e6i			0.541986i
$514$		0			0
$515$	−1.67619e6			−0.278488
$516$		0			0
$517$	1.34235e7			2.20871
$518$		0			0
$519$	−1.78674e7			−2.91168
$520$		0			0
$521$	522409.			0.0843173			0.0421586	−	0.999111i	$-0.486577\pi$
$521$	522409.			0.0843173			0.0421586	+	0.999111i	$0.486577\pi$
$522$		0			0
$523$		−	205708.i		−	0.0328849i	−0.999865	−	0.0164425i	$-0.994766\pi$
$523$		−	205708.i		−	0.0328849i	0.999865	−	0.0164425i	$-0.00523404\pi$
$524$		0			0
$525$	1.24319e7			1.96851
$526$		0			0
$527$	−5.03042e6			−0.789001
$528$		0			0
$529$	6.41643e6			0.996907
$530$		0			0
$531$			3.18204e6i			0.489745i
$532$		0			0
$533$			1.11480e6i			0.169972i
$534$		0			0
$535$			5.87135e6i			0.886857i
$536$		0			0
$537$		−	9.26613e6i		−	1.38664i
$538$		0			0
$539$	6.96608e6			1.03280
$540$		0			0
$541$		−	1.28578e6i		−	0.188875i	−0.995531	−	0.0944374i	$-0.969895\pi$
$541$		−	1.28578e6i		−	0.188875i	0.995531	−	0.0944374i	$-0.0301052\pi$
$542$		0			0
$543$	6.68941e6			0.973617
$544$		0			0
$545$	−1.73436e7			−2.50120
$546$		0			0
$547$			986186.i			0.140926i	0.997514	+	0.0704629i	$0.0224476\pi$
$547$			986186.i			0.140926i	−0.997514	+	0.0704629i	$0.977552\pi$
$548$		0			0
$549$			2.25970e7i			3.19978i
$550$		0			0
$551$	−4.48912e6			−0.629915
$552$		0			0
$553$			1.59700e6i			0.222070i
$554$		0			0
$555$	1.48052e7	−	7.51428e6i	2.04025	−	1.03551i
$556$		0			0
$557$		−	1.36157e7i		−	1.85952i	−0.368163	−	0.929761i	$-0.620013\pi$
$557$		−	1.36157e7i		−	1.85952i	0.368163	−	0.929761i	$-0.379987\pi$
$558$		0			0
$559$	−2.41550e6			−0.326947
$560$		0			0
$561$			1.35862e7i			1.82259i
$562$		0			0
$563$			6.75543e6i			0.898218i	0.893477	+	0.449109i	$0.148259\pi$
$563$			6.75543e6i			0.898218i	−0.893477	+	0.449109i	$0.851741\pi$
$564$		0			0
$565$	6.55431e6			0.863785
$566$		0			0
$567$	−2.62232e6			−0.342553
$568$		0			0
$569$			1.28082e7i			1.65846i	0.558905	+	0.829232i	$0.311222\pi$
$569$			1.28082e7i			1.65846i	−0.558905	+	0.829232i	$0.688778\pi$
$570$		0			0
$571$	−414554.			−0.0532097			−0.0266049	−	0.999646i	$-0.508470\pi$
$571$	−414554.			−0.0532097			−0.0266049	+	0.999646i	$0.508470\pi$
$572$		0			0
$573$		−	1.10282e6i		−	0.140319i
$574$		0			0
$575$		−	404468.i		−	0.0510170i
$576$		0			0
$577$			3.27944e6i			0.410072i	0.978754	+	0.205036i	$0.0657311\pi$
$577$			3.27944e6i			0.410072i	−0.978754	+	0.205036i	$0.934269\pi$
$578$		0			0
$579$			2.37852e7i			2.94857i
$580$		0			0
$581$	248616.			0.0305554
$582$		0			0
$583$	−8.05669e6			−0.981714
$584$		0			0
$585$	8.73608e6			1.05542
$586$		0			0
$587$		−	1.55344e7i		−	1.86080i	−0.366550	−	0.930398i	$-0.619461\pi$
$587$		−	1.55344e7i		−	1.86080i	0.366550	−	0.930398i	$-0.380539\pi$
$588$		0			0
$589$	−4.06800e6			−0.483162
$590$		0			0
$591$	−3.91661e6			−0.461255
$592$		0			0
$593$	−9.41131e6			−1.09904			−0.549520	−	0.835481i	$-0.685189\pi$
$593$	−9.41131e6			−1.09904			−0.549520	+	0.835481i	$0.685189\pi$
$594$		0			0
$595$	1.13891e7			1.31886
$596$		0			0
$597$		−	1.98330e7i		−	2.27747i
$598$		0			0
$599$	1.54540e7			1.75984			0.879920	−	0.475122i	$-0.157596\pi$
$599$	1.54540e7			1.75984			0.879920	+	0.475122i	$0.157596\pi$
$600$		0			0
$601$	−5.18219e6			−0.585231			−0.292616	−	0.956230i	$-0.594526\pi$
$601$	−5.18219e6			−0.585231			−0.292616	+	0.956230i	$0.594526\pi$
$602$		0			0
$603$	453226.			0.0507600
$604$		0			0
$605$			1.57324e7i			1.74746i
$606$		0			0
$607$			8.72049e6i			0.960659i	0.877088	+	0.480330i	$0.159483\pi$
$607$			8.72049e6i			0.960659i	−0.877088	+	0.480330i	$0.840517\pi$
$608$		0			0
$609$		−	2.75490e7i		−	3.00997i
$610$		0			0
$611$		−	5.96958e6i		−	0.646905i
$612$		0			0
$613$	4.99682e6			0.537084			0.268542	−	0.963268i	$-0.413458\pi$
$613$	4.99682e6			0.537084			0.268542	+	0.963268i	$0.413458\pi$
$614$		0			0
$615$		−	8.28076e6i		−	0.882841i
$616$		0			0
$617$	−1.00772e7			−1.06568			−0.532840	−	0.846216i	$-0.678875\pi$
$617$	−1.00772e7			−1.06568			−0.532840	+	0.846216i	$0.678875\pi$
$618$		0			0
$619$	−1.33209e7			−1.39735			−0.698676	−	0.715438i	$-0.746227\pi$
$619$	−1.33209e7			−1.39735			−0.698676	+	0.715438i	$0.746227\pi$
$620$		0			0
$621$			645021.i			0.0671189i
$622$		0			0
$623$			1.88325e6i			0.194396i
$624$		0			0
$625$	−1.05065e7			−1.07586
$626$		0			0
$627$			1.09869e7i			1.11610i
$628$		0			0
$629$	6.48917e6	−	3.29353e6i	0.653977	−	0.331921i
$630$		0			0
$631$		−	1.50505e7i		−	1.50479i	−0.658710	−	0.752397i	$-0.728898\pi$
$631$		−	1.50505e7i		−	1.50479i	0.658710	−	0.752397i	$-0.271102\pi$
$632$		0			0
$633$	8.51571e6			0.844717
$634$		0			0
$635$		−	1.98669e7i		−	1.95522i
$636$		0			0
$637$		−	3.09789e6i		−	0.302495i
$638$		0			0
$639$	−1.74585e7			−1.69144
$640$		0			0
$641$	1.20473e7			1.15810			0.579048	−	0.815293i	$-0.303424\pi$
$641$	1.20473e7			1.15810			0.579048	+	0.815293i	$0.303424\pi$
$642$		0			0
$643$			1.63347e6i			0.155806i	0.996961	+	0.0779029i	$0.0248224\pi$
$643$			1.63347e6i			0.155806i	−0.996961	+	0.0779029i	$0.975178\pi$
$644$		0			0
$645$	1.79425e7			1.69818
$646$		0			0
$647$		−	9.26473e6i		−	0.870106i	−0.900405	−	0.435053i	$-0.856730\pi$
$647$		−	9.26473e6i		−	0.870106i	0.900405	−	0.435053i	$-0.143270\pi$
$648$		0			0
$649$			4.56767e6i			0.425680i
$650$		0			0
$651$		−	2.49646e7i		−	2.30873i
$652$		0			0
$653$			3.31509e6i			0.304238i	0.988362	+	0.152119i	$0.0486096\pi$
$653$			3.31509e6i			0.304238i	−0.988362	+	0.152119i	$0.951390\pi$
$654$		0			0
$655$	−6.11701e6			−0.557104
$656$		0			0
$657$	−2.12781e7			−1.92318
$658$		0			0
$659$	−245037.			−0.0219796			−0.0109898	−	0.999940i	$-0.503498\pi$
$659$	−245037.			−0.0219796			−0.0109898	+	0.999940i	$0.503498\pi$
$660$		0			0
$661$			751834.i			0.0669296i	0.999440	+	0.0334648i	$0.0106542\pi$
$661$			751834.i			0.0669296i	−0.999440	+	0.0334648i	$0.989346\pi$
$662$		0			0
$663$	6.04192e6			0.533816
$664$		0			0
$665$	9.21015e6			0.807630
$666$		0			0
$667$	−896300.			−0.0780079
$668$		0			0
$669$	1.34777e7			1.16426
$670$		0			0
$671$			3.24369e7i			2.78121i
$672$		0			0
$673$	−1.60295e6			−0.136422			−0.0682108	−	0.997671i	$-0.521729\pi$
$673$	−1.60295e6			−0.136422			−0.0682108	+	0.997671i	$0.521729\pi$
$674$		0			0
$675$	−1.31042e7			−1.10701
$676$		0			0
$677$	1.08159e7			0.906964			0.453482	−	0.891266i	$-0.350182\pi$
$677$	1.08159e7			0.906964			0.453482	+	0.891266i	$0.350182\pi$
$678$		0			0
$679$		−	2.95355e6i		−	0.245850i
$680$		0			0
$681$			2.88950e7i			2.38756i
$682$		0			0
$683$			1.52237e7i			1.24873i	0.781132	+	0.624366i	$0.214642\pi$
$683$			1.52237e7i			1.24873i	−0.781132	+	0.624366i	$0.785358\pi$
$684$		0			0
$685$			4.25778e6i			0.346702i
$686$		0			0
$687$	−2.95636e7			−2.38982
$688$		0			0
$689$			3.58290e6i			0.287532i
$690$		0			0
$691$	−3.49028e6			−0.278077			−0.139039	−	0.990287i	$-0.544401\pi$
$691$	−3.49028e6			−0.278077			−0.139039	+	0.990287i	$0.544401\pi$
$692$		0			0
$693$	−4.27301e7			−3.37987
$694$		0			0
$695$			2.36450e7i			1.85685i
$696$		0			0
$697$		−	3.62948e6i		−	0.282984i
$698$		0			0
$699$	2.54933e7			1.97348
$700$		0			0
$701$			1.12118e7i			0.861746i	0.902413	+	0.430873i	$0.141794\pi$
$701$			1.12118e7i			0.861746i	−0.902413	+	0.430873i	$0.858206\pi$
$702$		0			0
$703$	5.24766e6	−	2.66341e6i	0.400477	−	0.203259i
$704$		0			0
$705$			4.43423e7i			3.36005i
$706$		0			0
$707$	3.55903e6			0.267783
$708$		0			0
$709$		−	1.73632e7i		−	1.29722i	−0.761122	−	0.648609i	$-0.775351\pi$
$709$		−	1.73632e7i		−	1.29722i	0.761122	−	0.648609i	$-0.224649\pi$
$710$		0			0
$711$		−	3.98822e6i		−	0.295873i
$712$		0			0
$713$	−812218.			−0.0598341
$714$		0			0
$715$	1.25402e7			0.917362
$716$		0			0
$717$		−	2.92469e7i		−	2.12463i
$718$		0			0
$719$	−1.29054e7			−0.930997			−0.465498	−	0.885049i	$-0.654125\pi$
$719$	−1.29054e7			−0.930997			−0.465498	+	0.885049i	$0.654125\pi$
$720$		0			0
$721$		−	3.64602e6i		−	0.261204i
$722$		0			0
$723$			4.04475e6i			0.287771i
$724$		0			0
$725$		−	1.82091e7i		−	1.28660i
$726$		0			0
$727$		−	5.33510e6i		−	0.374375i	−0.982324	−	0.187187i	$-0.940063\pi$
$727$		−	5.33510e6i		−	0.374375i	0.982324	−	0.187187i	$-0.0599372\pi$
$728$		0			0
$729$	−2.20645e7			−1.53771
$730$		0			0
$731$	7.86422e6			0.544330
$732$		0			0
$733$	−2.34611e7			−1.61283			−0.806415	−	0.591349i	$-0.798595\pi$
$733$	−2.34611e7			−1.61283			−0.806415	+	0.591349i	$0.798595\pi$
$734$		0			0
$735$			2.30113e7i			1.57117i
$736$		0			0
$737$	650584.			0.0441199
$738$		0			0
$739$	1.21379e7			0.817585			0.408792	−	0.912627i	$-0.365950\pi$
$739$	1.21379e7			0.817585			0.408792	+	0.912627i	$0.365950\pi$
$740$		0			0
$741$	4.88598e6			0.326893
$742$		0			0
$743$	2.85192e7			1.89524			0.947622	−	0.319393i	$-0.103479\pi$
$743$	2.85192e7			1.89524			0.947622	+	0.319393i	$0.103479\pi$
$744$		0			0
$745$			2.35914e7i			1.55727i
$746$		0			0
$747$	−620875.			−0.0407101
$748$		0			0
$749$	−1.27712e7			−0.831818
$750$		0			0
$751$	−3.04518e7			−1.97022			−0.985108	−	0.171937i	$-0.944997\pi$
$751$	−3.04518e7			−1.97022			−0.985108	+	0.171937i	$0.944997\pi$
$752$		0			0
$753$		−	2.52577e6i		−	0.162333i
$754$		0			0
$755$		−	1.25321e6i		−	0.0800122i
$756$		0			0
$757$		−	1.86061e7i		−	1.18009i	−0.807369	−	0.590046i	$-0.799109\pi$
$757$		−	1.86061e7i		−	1.18009i	0.807369	−	0.590046i	$-0.200891\pi$
$758$		0			0
$759$			2.19364e6i			0.138217i
$760$		0			0
$761$	1.70681e7			1.06837			0.534186	−	0.845367i	$-0.320618\pi$
$761$	1.70681e7			1.06837			0.534186	+	0.845367i	$0.320618\pi$
$762$		0			0
$763$		−	3.77254e7i		−	2.34597i
$764$		0			0
$765$	−2.84424e7			−1.75716
$766$		0			0
$767$	2.03129e6			0.124676
$768$		0			0
$769$		−	8.20099e6i		−	0.500092i	−0.968234	−	0.250046i	$-0.919554\pi$
$769$		−	8.20099e6i		−	0.500092i	0.968234	−	0.250046i	$-0.0804458\pi$
$770$		0			0
$771$		−	4.67337e7i		−	2.83135i
$772$		0			0
$773$	6.10300e6			0.367363			0.183681	−	0.982986i	$-0.441199\pi$
$773$	6.10300e6			0.367363			0.183681	+	0.982986i	$0.441199\pi$
$774$		0			0
$775$		−	1.65009e7i		−	0.986856i
$776$		0			0
$777$	1.63449e7	+	3.22040e7i	0.971247	+	1.91363i
$778$		0			0
$779$		−	2.93509e6i		−	0.173291i
$780$		0			0
$781$	−2.50609e7			−1.47018
$782$		0			0
$783$			2.90387e7i			1.69267i
$784$		0			0
$785$		−	2.42146e7i		−	1.40250i
$786$		0			0
$787$	2.58372e7			1.48699			0.743496	−	0.668740i	$-0.233166\pi$
$787$	2.58372e7			1.48699			0.743496	+	0.668740i	$0.233166\pi$
$788$		0			0
$789$	2.05319e7			1.17419
$790$		0			0
$791$			1.42568e7i			0.810178i
$792$		0			0
$793$	1.44251e7			0.814581
$794$		0			0
$795$		−	2.66140e7i		−	1.49345i
$796$		0			0
$797$			9.74891e6i			0.543639i	0.962348	+	0.271819i	$0.0876253\pi$
$797$			9.74891e6i			0.543639i	−0.962348	+	0.271819i	$0.912375\pi$
$798$		0			0
$799$			1.94354e7i			1.07702i
$800$		0			0
$801$		−	4.70310e6i		−	0.259002i
$802$		0			0
$803$	−3.05437e7			−1.67160
$804$		0			0
$805$	1.83890e6			0.100016
$806$		0			0
$807$	3.31739e7			1.79314
$808$		0			0
$809$		−	1.05888e7i		−	0.568822i	−0.958702	−	0.284411i	$-0.908202\pi$
$809$		−	1.05888e7i		−	0.568822i	0.958702	−	0.284411i	$-0.0917980\pi$
$810$		0			0
$811$	−2.21309e7			−1.18153			−0.590767	−	0.806842i	$-0.701175\pi$
$811$	−2.21309e7			−1.18153			−0.590767	+	0.806842i	$0.701175\pi$
$812$		0			0
$813$	3.29434e7			1.74800
$814$		0			0
$815$	−3.37665e7			−1.78071
$816$		0			0
$817$	6.35964e6			0.333332
$818$		0			0
$819$			1.90025e7i			0.989924i
$820$		0			0
$821$	3.09814e7			1.60414			0.802071	−	0.597229i	$-0.203732\pi$
$821$	3.09814e7			1.60414			0.802071	+	0.597229i	$0.203732\pi$
$822$		0			0
$823$	−1.92131e7			−0.988776			−0.494388	−	0.869241i	$-0.664608\pi$
$823$	−1.92131e7			−0.988776			−0.494388	+	0.869241i	$0.664608\pi$
$824$		0			0
$825$	−4.45657e7			−2.27964
$826$		0			0
$827$		−	5.33826e6i		−	0.271416i	−0.990749	−	0.135708i	$-0.956669\pi$
$827$		−	5.33826e6i		−	0.271416i	0.990749	−	0.135708i	$-0.0433310\pi$
$828$		0			0
$829$		−	1.38953e7i		−	0.702235i	−0.936331	−	0.351117i	$-0.885802\pi$
$829$		−	1.38953e7i		−	0.702235i	0.936331	−	0.351117i	$-0.114198\pi$
$830$		0			0
$831$			3.60994e7i			1.81342i
$832$		0			0
$833$			1.00859e7i			0.503619i
$834$		0			0
$835$	4.54651e7			2.25664
$836$		0			0
$837$			2.63146e7i			1.29833i
$838$		0			0
$839$	1.11135e7			0.545064			0.272532	−	0.962147i	$-0.412139\pi$
$839$	1.11135e7			0.545064			0.272532	+	0.962147i	$0.412139\pi$
$840$		0			0
$841$	−1.98401e7			−0.967285
$842$		0			0
$843$			1.27426e7i			0.617572i
$844$		0			0
$845$			2.31632e7i			1.11598i
$846$		0			0
$847$	−3.42209e7			−1.63901
$848$		0			0
$849$		−	2.14446e7i		−	1.02105i
$850$		0			0
$851$	1.04775e6	−	531778.i	0.0495945	−	0.0251713i
$852$		0			0
$853$			3.40951e7i			1.60443i	0.597038	+	0.802213i	$0.296344\pi$
$853$			3.40951e7i			1.60443i	−0.597038	+	0.802213i	$0.703656\pi$
$854$		0			0
$855$	−2.30008e7			−1.07604
$856$		0			0
$857$			1.49337e7i			0.694568i	0.937760	+	0.347284i	$0.112896\pi$
$857$			1.49337e7i			0.694568i	−0.937760	+	0.347284i	$0.887104\pi$
$858$		0			0
$859$			7.59887e6i			0.351371i	0.984446	+	0.175686i	$0.0562142\pi$
$859$			7.59887e6i			0.351371i	−0.984446	+	0.175686i	$0.943786\pi$
$860$		0			0
$861$	1.80121e7			0.828051
$862$		0			0
$863$	7.69720e6			0.351808			0.175904	−	0.984407i	$-0.443715\pi$
$863$	7.69720e6			0.351808			0.175904	+	0.984407i	$0.443715\pi$
$864$		0			0
$865$		−	5.36933e7i		−	2.43994i
$866$		0			0
$867$	1.69018e7			0.763634
$868$		0			0
$869$		−	5.72491e6i		−	0.257169i
$870$		0			0
$871$		−	289322.i		−	0.0129222i
$872$		0			0
$873$			7.37599e6i			0.327555i
$874$		0			0
$875$		−	3.36828e6i		−	0.148726i
$876$		0			0
$877$	8.61090e6			0.378050			0.189025	−	0.981972i	$-0.439467\pi$
$877$	8.61090e6			0.378050			0.189025	+	0.981972i	$0.439467\pi$
$878$		0			0
$879$	2.45049e7			1.06975
$880$		0			0
$881$	2.80482e7			1.21749			0.608744	−	0.793367i	$-0.291674\pi$
$881$	2.80482e7			1.21749			0.608744	+	0.793367i	$0.291674\pi$
$882$		0			0
$883$			4.43487e7i			1.91416i	0.289816	+	0.957082i	$0.406406\pi$
$883$			4.43487e7i			1.91416i	−0.289816	+	0.957082i	$0.593594\pi$
$884$		0			0
$885$	−1.50886e7			−0.647574
$886$		0			0
$887$	3.99164e7			1.70350			0.851749	−	0.523949i	$-0.175542\pi$
$887$	3.99164e7			1.70350			0.851749	+	0.523949i	$0.175542\pi$
$888$		0			0
$889$	4.32140e7			1.83388
$890$		0			0
$891$	9.40049e6			0.396695
$892$		0			0
$893$			1.57170e7i			0.659539i
$894$		0			0
$895$	2.78455e7			1.16198
$896$		0			0
$897$	975537.			0.0404821
$898$		0			0
$899$	−3.65660e7			−1.50896
$900$		0			0
$901$		−	1.16650e7i		−	0.478709i
$902$		0			0
$903$			3.90280e7i			1.59279i
$904$		0			0
$905$			2.01023e7i			0.815875i
$906$		0			0
$907$		−	3.02380e6i		−	0.122049i	−0.998136	−	0.0610247i	$-0.980563\pi$
$907$		−	3.02380e6i		−	0.122049i	0.998136	−	0.0610247i	$-0.0194368\pi$
$908$		0			0
$909$	−8.88808e6			−0.356778
$910$		0			0
$911$			31968.4i			0.00127622i	1.00000		0.000638108i	$0.000203116\pi$
$911$			31968.4i			0.00127622i	−1.00000		0.000638108i	$0.999797\pi$
$912$		0			0
$913$	−891237.			−0.0353847
$914$		0			0
$915$	−1.07150e8			−4.23097
$916$		0			0
$917$		−	1.33056e7i		−	0.522529i
$918$		0			0
$919$			3.42133e7i			1.33631i	0.744024	+	0.668153i	$0.232915\pi$
$919$			3.42133e7i			1.33631i	−0.744024	+	0.668153i	$0.767085\pi$
$920$		0			0
$921$	1.18221e7			0.459246
$922$		0			0
$923$			1.11449e7i			0.430597i
$924$		0			0
$925$	1.08035e7	+	2.12859e7i	0.415156	+	0.817972i
$926$		0			0
$927$			9.10530e6i			0.348013i
$928$		0			0
$929$	743746.			0.0282739			0.0141369	−	0.999900i	$-0.495500\pi$
$929$	743746.			0.0282739			0.0141369	+	0.999900i	$0.495500\pi$
$930$		0			0
$931$			8.15627e6i			0.308402i
$932$		0			0
$933$			4.29159e7i			1.61404i
$934$		0			0
$935$	−4.08277e7			−1.52730
$936$		0			0
$937$	−2.62123e7			−0.975341			−0.487670	−	0.873028i	$-0.662153\pi$
$937$	−2.62123e7			−0.975341			−0.487670	+	0.873028i	$0.662153\pi$
$938$		0			0
$939$		−	4.52724e7i		−	1.67560i
$940$		0			0
$941$	−8.93489e6			−0.328939			−0.164469	−	0.986382i	$-0.552591\pi$
$941$	−8.93489e6			−0.328939			−0.164469	+	0.986382i	$0.552591\pi$
$942$		0			0
$943$		−	586020.i		−	0.0214602i
$944$		0			0
$945$		−	5.95776e7i		−	2.17022i
$946$		0			0
$947$			4.14310e7i			1.50124i	0.660734	+	0.750620i	$0.270245\pi$
$947$			4.14310e7i			1.50124i	−0.660734	+	0.750620i	$0.729755\pi$
$948$		0			0
$949$			1.35831e7i			0.489592i
$950$		0			0
$951$	3.02738e7			1.08547
$952$		0			0
$953$	3.96432e6			0.141396			0.0706979	−	0.997498i	$-0.477477\pi$
$953$	3.96432e6			0.141396			0.0706979	+	0.997498i	$0.477477\pi$
$954$		0			0
$955$	3.31406e6			0.117585
$956$		0			0
$957$			9.87575e7i			3.48570i
$958$		0			0
$959$	−9.26142e6			−0.325185
$960$		0			0
$961$	−4.50658e6			−0.157412
$962$		0			0
$963$	3.18940e7			1.10826
$964$		0			0
$965$	−7.14768e7			−2.47085
$966$		0			0
$967$		−	5.27385e7i		−	1.81368i	−0.421472	−	0.906841i	$-0.638487\pi$
$967$		−	5.27385e7i		−	1.81368i	0.421472	−	0.906841i	$-0.361513\pi$
$968$		0			0
$969$	−1.59075e7			−0.544241
$970$		0			0
$971$	−4.09048e7			−1.39228			−0.696140	−	0.717907i	$-0.745101\pi$
$971$	−4.09048e7			−1.39228			−0.696140	+	0.717907i	$0.745101\pi$
$972$		0			0
$973$	−5.14321e7			−1.74162
$974$		0			0
$975$			1.98189e7i			0.667679i
$976$		0			0
$977$		−	2.65016e7i		−	0.888251i	−0.895965	−	0.444126i	$-0.853514\pi$
$977$		−	2.65016e7i		−	0.888251i	0.895965	−	0.444126i	$-0.146486\pi$
$978$		0			0
$979$		−	6.75108e6i		−	0.225121i
$980$		0			0
$981$			9.42128e7i			3.12563i
$982$		0			0
$983$	−1.41878e7			−0.468308			−0.234154	−	0.972200i	$-0.575232\pi$
$983$	−1.41878e7			−0.468308			−0.234154	+	0.972200i	$0.575232\pi$
$984$		0			0
$985$		−	1.17698e7i		−	0.386524i
$986$		0			0
$987$	−9.64524e7			−3.15152
$988$		0			0
$989$	1.26977e6			0.0412794
$990$		0			0
$991$		−	2.37057e7i		−	0.766775i	−0.923588	−	0.383388i	$-0.874757\pi$
$991$		−	2.37057e7i		−	0.766775i	0.923588	−	0.383388i	$-0.125243\pi$
$992$		0			0
$993$			1.69015e6i			0.0543942i
$994$		0			0
$995$	5.95999e7			1.90848
$996$		0			0
$997$		−	5.24739e7i		−	1.67188i	−0.548819	−	0.835941i	$-0.684923\pi$
$997$		−	5.24739e7i		−	1.67188i	0.548819	−	0.835941i	$-0.315077\pi$
$998$		0			0
$999$	−1.72288e7	−	3.39455e7i	−0.546187	−	1.07614i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.6.g.c.369.16		16
4.3	odd	2		37.6.b.a.36.16	yes	16
12.11	even	2		333.6.c.d.73.1		16
37.36	even	2	inner	592.6.g.c.369.15		16
148.147	odd	2		37.6.b.a.36.1	✓	16
444.443	even	2		333.6.c.d.73.16		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.b.a.36.1	✓	16	148.147	odd	2
37.6.b.a.36.16	yes	16	4.3	odd	2
333.6.c.d.73.1		16	12.11	even	2
333.6.c.d.73.16		16	444.443	even	2
592.6.g.c.369.15		16	37.36	even	2	inner
592.6.g.c.369.16		16	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:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	592.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+25.7580 q^{3} +77.4051i q^{5} -168.370 q^{7} +420.475 q^{9} +O(q^{10})\)	\(q+25.7580 q^{3} +77.4051i q^{5} -168.370 q^{7} +420.475 q^{9} +603.572 q^{11} -268.415i q^{13} +1993.80i q^{15} +873.888i q^{17} +706.696i q^{19} -4336.87 q^{21} +141.099i q^{23} -2866.55 q^{25} +4571.40 q^{27} +6352.26i q^{29} +5756.36i q^{31} +15546.8 q^{33} -13032.7i q^{35} +(-3768.82 - 7425.63i) q^{37} -6913.84i q^{39} -4153.25 q^{41} -8999.12i q^{43} +32546.9i q^{45} +22240.1 q^{47} +11541.4 q^{49} +22509.6i q^{51} -13348.3 q^{53} +46719.6i q^{55} +18203.1i q^{57} +7567.74i q^{59} +53741.6i q^{61} -70795.3 q^{63} +20776.7 q^{65} +1077.89 q^{67} +3634.43i q^{69} -41521.0 q^{71} -50604.9 q^{73} -73836.7 q^{75} -101623. q^{77} -9485.04i q^{79} +15574.8 q^{81} -1476.60 q^{83} -67643.4 q^{85} +163622. i q^{87} -11185.2i q^{89} +45193.0i q^{91} +148272. i q^{93} -54701.9 q^{95} +17542.0i q^{97} +253787. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 18 q^{3} - 190 q^{7} + 1394 q^{9}+O(q^{10}) \) 16 * q + 18 * q^3 - 190 * q^7 + 1394 * q^9	\( 16 q + 18 q^{3} - 190 q^{7} + 1394 q^{9} + 1110 q^{11} - 7010 q^{21} - 12052 q^{25} - 4266 q^{27} - 2478 q^{33} - 11400 q^{37} + 3918 q^{41} - 3822 q^{47} - 32618 q^{49} - 24126 q^{53} - 219268 q^{63} + 98976 q^{65} - 23560 q^{67} + 50046 q^{71} - 196274 q^{73} - 214054 q^{75} - 239574 q^{77} + 317312 q^{81} + 215814 q^{83} - 346472 q^{85} + 132504 q^{95} + 574860 q^{99}+O(q^{100}) \) 16 * q + 18 * q^3 - 190 * q^7 + 1394 * q^9 + 1110 * q^11 - 7010 * q^21 - 12052 * q^25 - 4266 * q^27 - 2478 * q^33 - 11400 * q^37 + 3918 * q^41 - 3822 * q^47 - 32618 * q^49 - 24126 * q^53 - 219268 * q^63 + 98976 * q^65 - 23560 * q^67 + 50046 * q^71 - 196274 * q^73 - 214054 * q^75 - 239574 * q^77 + 317312 * q^81 + 215814 * q^83 - 346472 * q^85 + 132504 * q^95 + 574860 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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