LMFDB - Embedded newform 33.6.d.a.32.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,6,Mod(32,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("33.32");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	33.d (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:	$5.29266605383$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 3 $ x^2 - x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			32.1
Root			$0.500000 + 1.65831i$ of defining polynomial
Character	$\chi$	$=$	33.32
Dual form			33.6.d.a.32.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(15.5000 - 1.65831i) q^{3} -32.0000 q^{4} -96.1821i q^{5} +(237.500 - 51.4077i) q^{9} +O(q^{10})$	$q+(15.5000 - 1.65831i) q^{3} -32.0000 q^{4} -96.1821i q^{5} +(237.500 - 51.4077i) q^{9} -401.312i q^{11} +(-496.000 + 53.0660i) q^{12} +(-159.500 - 1490.82i) q^{15} +1024.00 q^{16} +3077.83i q^{20} +4978.25i q^{23} -6126.00 q^{25} +(3596.00 - 1190.67i) q^{27} +7775.00 q^{31} +(-665.500 - 6220.33i) q^{33} +(-7600.00 + 1645.05i) q^{36} -1267.00 q^{37} +12842.0i q^{44} +(-4944.50 - 22843.3i) q^{45} -17518.4i q^{47} +(15872.0 - 1698.11i) q^{48} +16807.0 q^{49} -21478.5i q^{53} -38599.0 q^{55} +47364.7i q^{59} +(5104.00 + 47706.3i) q^{60} -32768.0 q^{64} +72917.0 q^{67} +(8255.50 + 77162.9i) q^{69} -53148.9i q^{71} +(-94953.0 + 10158.8i) q^{75} -98490.5i q^{80} +(53763.5 - 24418.7i) q^{81} +118486. i q^{89} -159304. i q^{92} +(120512. - 12893.4i) q^{93} -163183. q^{97} +(-20630.5 - 95311.5i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 31 q^{3} - 64 q^{4} + 475 q^{9}+O(q^{10}) $ 2 * q + 31 * q^3 - 64 * q^4 + 475 * q^9	$ 2 q + 31 q^{3} - 64 q^{4} + 475 q^{9} - 992 q^{12} - 319 q^{15} + 2048 q^{16} - 12252 q^{25} + 7192 q^{27} + 15550 q^{31} - 1331 q^{33} - 15200 q^{36} - 2534 q^{37} - 9889 q^{45} + 31744 q^{48} + 33614 q^{49} - 77198 q^{55} + 10208 q^{60} - 65536 q^{64} + 145834 q^{67} + 16511 q^{69} - 189906 q^{75} + 107527 q^{81} + 241025 q^{93} - 326366 q^{97} - 41261 q^{99}+O(q^{100}) $ 2 * q + 31 * q^3 - 64 * q^4 + 475 * q^9 - 992 * q^12 - 319 * q^15 + 2048 * q^16 - 12252 * q^25 + 7192 * q^27 + 15550 * q^31 - 1331 * q^33 - 15200 * q^36 - 2534 * q^37 - 9889 * q^45 + 31744 * q^48 + 33614 * q^49 - 77198 * q^55 + 10208 * q^60 - 65536 * q^64 + 145834 * q^67 + 16511 * q^69 - 189906 * q^75 + 107527 * q^81 + 241025 * q^93 - 326366 * q^97 - 41261 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$.

$n$	$13$	$23$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$	15.5000	−	1.65831i	0.994325	−	0.106381i
$3$	15.5000	−	1.65831i	0.994325	−	0.106381i
$4$	−32.0000			−1.00000
$5$		−	96.1821i		−	1.72056i	−0.509823	−	0.860279i	$-0.670289\pi$
$5$		−	96.1821i		−	1.72056i	0.509823	−	0.860279i	$-0.329711\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	237.500	−	51.4077i	0.977366	−	0.211554i
$10$		0			0
$11$		−	401.312i		−	1.00000i
$11$		−	401.312i		−	1.00000i
$12$	−496.000	+	53.0660i	−0.994325	+	0.106381i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−159.500	−	1490.82i	−0.183034	−	1.71079i
$16$	1024.00			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$			3077.83i			1.72056i
$21$		0			0
$22$		0			0
$23$			4978.25i			1.96226i	0.193339	+	0.981132i	$0.438068\pi$
$23$			4978.25i			1.96226i	−0.193339	+	0.981132i	$0.561932\pi$
$24$		0			0
$25$	−6126.00			−1.96032
$26$		0			0
$27$	3596.00	−	1190.67i	0.949315	−	0.314327i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	7775.00			1.45310			0.726551	−	0.687112i	$-0.241122\pi$
$31$	7775.00			1.45310			0.726551	+	0.687112i	$0.241122\pi$
$32$		0			0
$33$	−665.500	−	6220.33i	−0.106381	−	0.994325i
$34$		0			0
$35$		0			0
$36$	−7600.00	+	1645.05i	−0.977366	+	0.211554i
$37$	−1267.00			−0.152150			−0.0760751	−	0.997102i	$-0.524239\pi$
$37$	−1267.00			−0.152150			−0.0760751	+	0.997102i	$0.524239\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$			12842.0i			1.00000i
$45$	−4944.50	−	22843.3i	−0.363991	−	1.68162i
$46$		0			0
$47$		−	17518.4i		−	1.15678i	−0.815761	−	0.578389i	$-0.803682\pi$
$47$		−	17518.4i		−	1.15678i	0.815761	−	0.578389i	$-0.196318\pi$
$48$	15872.0	−	1698.11i	0.994325	−	0.106381i
$49$	16807.0			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	21478.5i		−	1.05030i	−0.851010	−	0.525150i	$-0.824009\pi$
$53$		−	21478.5i		−	1.05030i	0.851010	−	0.525150i	$-0.175991\pi$
$54$		0			0
$55$	−38599.0			−1.72056
$56$		0			0
$57$		0			0
$58$		0			0
$59$			47364.7i			1.77143i	0.464226	+	0.885717i	$0.346333\pi$
$59$			47364.7i			1.77143i	−0.464226	+	0.885717i	$0.653667\pi$
$60$	5104.00	+	47706.3i	0.183034	+	1.71079i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	−32768.0			−1.00000
$65$		0			0
$66$		0			0
$67$	72917.0			1.98446			0.992229	−	0.124427i	$-0.0397094\pi$
$67$	72917.0			1.98446			0.992229	+	0.124427i	$0.0397094\pi$
$68$		0			0
$69$	8255.50	+	77162.9i	0.208747	+	1.95113i
$70$		0			0
$71$		−	53148.9i		−	1.25126i	−0.780119	−	0.625631i	$-0.784841\pi$
$71$		−	53148.9i		−	1.25126i	0.780119	−	0.625631i	$-0.215159\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−94953.0	+	10158.8i	−1.94920	+	0.208540i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		−	98490.5i		−	1.72056i
$81$	53763.5	−	24418.7i	0.910490	−	0.413532i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			118486.i			1.58560i	0.609482	+	0.792800i	$0.291377\pi$
$89$			118486.i			1.58560i	−0.609482	+	0.792800i	$0.708623\pi$
$90$		0			0
$91$		0			0
$92$		−	159304.i		−	1.96226i
$93$	120512.	−	12893.4i	1.44486	−	0.154582i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−163183.			−1.76094			−0.880472	−	0.474098i	$-0.842774\pi$
$97$	−163183.			−1.76094			−0.880472	+	0.474098i	$0.842774\pi$
$98$		0			0
$99$	−20630.5	−	95311.5i	−0.211554	−	0.977366i
$100$	196032.			1.96032
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−180244.			−1.67405			−0.837024	−	0.547167i	$-0.815706\pi$
$103$	−180244.			−1.67405			−0.837024	+	0.547167i	$0.815706\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−115072.	+	38101.4i	−0.949315	+	0.314327i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	−19638.5	+	2101.08i	−0.151287	+	0.0161859i
$112$		0			0
$113$			191538.i			1.41111i	0.708657	+	0.705553i	$0.249301\pi$
$113$			191538.i			1.41111i	−0.708657	+	0.705553i	$0.750699\pi$
$114$		0			0
$115$	478819.			3.37619
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−161051.			−1.00000
$122$		0			0
$123$		0			0
$124$	−248800.			−1.45310
$125$			288643.i			1.65229i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	21296.0	+	199051.i	0.106381	+	0.994325i
$133$		0			0
$134$		0			0
$135$	−114521.	−	345871.i	−0.540817	−	1.63335i
$136$		0			0
$137$			351751.i			1.60116i	0.599227	+	0.800579i	$0.295475\pi$
$137$			351751.i			1.60116i	−0.599227	+	0.800579i	$0.704525\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	−29051.0	−	271535.i	−0.123059	−	1.15021i
$142$		0			0
$143$		0			0
$144$	243200.	−	52641.5i	0.977366	−	0.211554i
$145$		0			0
$146$		0			0
$147$	260508.	−	27871.3i	0.994325	−	0.106381i
$148$	40544.0			0.152150
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	747816.i		−	2.50015i
$156$		0			0
$157$	−280117.			−0.906965			−0.453482	−	0.891265i	$-0.649818\pi$
$157$	−280117.			−0.906965			−0.453482	+	0.891265i	$0.649818\pi$
$158$		0			0
$159$	−35618.0	−	332916.i	−0.111732	−	1.04434i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	164144.			0.483900			0.241950	−	0.970289i	$-0.422213\pi$
$163$	164144.			0.483900			0.241950	+	0.970289i	$0.422213\pi$
$164$		0			0
$165$	−598284.	+	64009.2i	−1.71079	+	0.183034i
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	371293.			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$		−	410943.i		−	1.00000i
$177$	78545.5	+	734153.i	0.188447	+	1.76138i
$178$		0			0
$179$			170723.i			0.398254i	0.979974	+	0.199127i	$0.0638106\pi$
$179$			170723.i			0.398254i	−0.979974	+	0.199127i	$0.936189\pi$
$180$	158224.	+	730984.i	0.363991	+	1.68162i
$181$	279875.			0.634991			0.317496	−	0.948260i	$-0.397158\pi$
$181$	279875.			0.634991			0.317496	+	0.948260i	$0.397158\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			121863.i			0.261783i
$186$		0			0
$187$		0			0
$188$			560589.i			1.15678i
$189$		0			0
$190$		0			0
$191$		−	970886.i		−	1.92568i	−0.270069	−	0.962841i	$-0.587046\pi$
$191$		−	970886.i		−	1.92568i	0.270069	−	0.962841i	$-0.412954\pi$
$192$	−507904.	+	54339.6i	−0.994325	+	0.106381i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−537824.			−1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−799900.			−1.43187			−0.715934	−	0.698168i	$-0.753999\pi$
$199$	−799900.			−1.43187			−0.715934	+	0.698168i	$0.753999\pi$
$200$		0			0
$201$	1.13021e6	−	120919.i	1.97320	−	0.211108i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	255920.	+	1.18234e6i	0.415125	+	1.91785i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			687311.i			1.05030i
$213$	−88137.5	−	823808.i	−0.133110	−	1.24416i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	1.23517e6			1.72056
$221$		0			0
$222$		0			0
$223$	−247531.			−0.333325			−0.166662	−	0.986014i	$-0.553299\pi$
$223$	−247531.			−0.333325			−0.166662	+	0.986014i	$0.553299\pi$
$224$		0			0
$225$	−1.45492e6	+	314923.i	−1.91595	+	0.414714i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	1.17102e6			1.47563			0.737815	−	0.675003i	$-0.235858\pi$
$229$	1.17102e6			1.47563			0.737815	+	0.675003i	$0.235858\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	−1.68496e6			−1.99030
$236$		−	1.51567e6i		−	1.77143i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−163328.	−	1.52660e6i	−0.183034	−	1.71079i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	792840.	−	467646.i	0.861331	−	0.508044i
$244$		0			0
$245$		−	1.61653e6i		−	1.72056i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	754615.i		−	0.756034i	−0.925799	−	0.378017i	$-0.876606\pi$
$251$		−	754615.i		−	0.756034i	0.925799	−	0.378017i	$-0.123394\pi$
$252$		0			0
$253$	1.99783e6			1.96226
$254$		0			0
$255$		0			0
$256$	1.04858e6			1.00000
$257$			2.09024e6i			1.97408i	0.160485	+	0.987038i	$0.448694\pi$
$257$			2.09024e6i			1.97408i	−0.160485	+	0.987038i	$0.551306\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	−2.06584e6			−1.80710
$266$		0			0
$267$	196488.	+	1.83654e6i	0.168677	+	1.57660i
$268$	−2.33334e6			−1.98446
$269$			2.07038e6i			1.74450i	0.489064	+	0.872248i	$0.337339\pi$
$269$			2.07038e6i			1.74450i	−0.489064	+	0.872248i	$0.662661\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			2.45843e6i			1.96032i
$276$	−264176.	−	2.46921e6i	−0.208747	−	1.95113i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	1.84656e6	−	399695.i	1.42021	−	0.307410i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$			1.70077e6i			1.25126i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.41986e6			−1.00000
$290$		0			0
$291$	−2.52934e6	+	270608.i	−1.75095	+	0.187331i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	4.55564e6			3.04785
$296$		0			0
$297$	−477829.	−	1.44312e6i	−0.314327	−	0.949315i
$298$		0			0
$299$		0			0
$300$	3.03850e6	−	325082.i	1.94920	−	0.208540i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	−2.79378e6	+	298901.i	−1.66455	+	0.178086i
$310$		0			0
$311$		−	560344.i		−	0.328514i	−0.986418	−	0.164257i	$-0.947477\pi$
$311$		−	560344.i		−	0.328514i	0.986418	−	0.164257i	$-0.0525226\pi$
$312$		0			0
$313$	−1.04882e6			−0.605117			−0.302559	−	0.953131i	$-0.597841\pi$
$313$	−1.04882e6			−0.605117			−0.302559	+	0.953131i	$0.597841\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	1.42430e6i		−	0.796075i	−0.917369	−	0.398037i	$-0.869691\pi$
$317$		−	1.42430e6i		−	0.796075i	0.917369	−	0.398037i	$-0.130309\pi$
$318$		0			0
$319$		0			0
$320$			3.15170e6i			1.72056i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.72043e6	+	781397.i	−0.910490	+	0.413532i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	736925.			0.369703			0.184852	−	0.982766i	$-0.440820\pi$
$331$	736925.			0.369703			0.184852	+	0.982766i	$0.440820\pi$
$332$		0			0
$333$	−300912.	+	65133.5i	−0.148706	+	0.0321880i
$334$		0			0
$335$		−	7.01331e6i		−	3.41437i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	317630.	+	2.96885e6i	0.150115	+	1.40310i
$340$		0			0
$341$		−	3.12020e6i		−	1.45310i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	7.42169e6	−	794031.i	3.35703	−	0.359162i
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	1.65599e6i		−	0.707330i	−0.935372	−	0.353665i	$-0.884935\pi$
$353$		−	1.65599e6i		−	0.707330i	0.935372	−	0.353665i	$-0.115065\pi$
$354$		0			0
$355$	−5.11197e6			−2.15287
$356$		−	3.79157e6i		−	1.58560i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	2.47610e6			1.00000
$362$		0			0
$363$	−2.49629e6	+	267073.i	−0.994325	+	0.106381i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−1.31317e6			−0.508926			−0.254463	−	0.967082i	$-0.581899\pi$
$367$	−1.31317e6			−0.508926			−0.254463	+	0.967082i	$0.581899\pi$
$368$			5.09773e6i			1.96226i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−3.85640e6	+	412588.i	−1.44486	+	0.154582i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	478660.	+	4.47396e6i	0.175772	+	1.64291i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.88862e6			0.675379			0.337690	−	0.941258i	$-0.390355\pi$
$379$	1.88862e6			0.675379			0.337690	+	0.941258i	$0.390355\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			2.36310e6i			0.823161i	0.911373	+	0.411581i	$0.135023\pi$
$383$			2.36310e6i			0.823161i	−0.911373	+	0.411581i	$0.864977\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	5.22186e6			1.76094
$389$			2.21192e6i			0.741132i	0.928806	+	0.370566i	$0.120836\pi$
$389$			2.21192e6i			0.741132i	−0.928806	+	0.370566i	$0.879164\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	660176.	+	3.04997e6i	0.211554	+	0.977366i
$397$	1.56024e6			0.496839			0.248420	−	0.968653i	$-0.420089\pi$
$397$	1.56024e6			0.496839			0.248420	+	0.968653i	$0.420089\pi$
$398$		0			0
$399$		0			0
$400$	−6.27302e6			−1.96032
$401$			2.96902e6i			0.922044i	0.887389	+	0.461022i	$0.152517\pi$
$401$			2.96902e6i			0.922044i	−0.887389	+	0.461022i	$0.847483\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−2.34864e6	−	5.17109e6i	−0.711506	−	1.56655i
$406$		0			0
$407$			508462.i			0.152150i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	583314.	+	5.45214e6i	0.170332	+	1.59207i
$412$	5.76781e6			1.67405
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	7.18696e6i		−	1.99991i	−0.00955353	−	0.999954i	$-0.503041\pi$
$419$		−	7.18696e6i		−	1.99991i	0.00955353	−	0.999954i	$-0.496959\pi$
$420$		0			0
$421$	−6.85705e6			−1.88552			−0.942762	−	0.333466i	$-0.891782\pi$
$421$	−6.85705e6			−1.88552			−0.942762	+	0.333466i	$0.891782\pi$
$422$		0			0
$423$	−900581.	−	4.16062e6i	−0.244721	−	1.13060i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	3.68230e6	−	1.21924e6i	0.949315	−	0.314327i
$433$	−5.10513e6			−1.30854			−0.654270	−	0.756261i	$-0.727024\pi$
$433$	−5.10513e6			−1.30854			−0.654270	+	0.756261i	$0.727024\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	3.99166e6	−	864009.i	0.977366	−	0.211554i
$442$		0			0
$443$			7.12726e6i			1.72549i	0.505637	+	0.862746i	$0.331258\pi$
$443$			7.12726e6i			1.72549i	−0.505637	+	0.862746i	$0.668742\pi$
$444$	628432.	−	67234.6i	0.151287	−	0.0161859i
$445$	1.13963e7			2.72812
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	7.73196e6i		−	1.80998i	−0.425432	−	0.904990i	$-0.639878\pi$
$449$		−	7.73196e6i		−	1.80998i	0.425432	−	0.904990i	$-0.360122\pi$
$450$		0			0
$451$		0			0
$452$		−	6.12923e6i		−	1.41111i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$	−1.53222e7			−3.37619
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	7.10982e6			1.54137			0.770684	−	0.637218i	$-0.219915\pi$
$463$	7.10982e6			1.54137			0.770684	+	0.637218i	$0.219915\pi$
$464$		0			0
$465$	−1.24011e6	−	1.15911e7i	−0.265968	−	2.48596i
$466$		0			0
$467$			8.86299e6i			1.88056i	0.340396	+	0.940282i	$0.389439\pi$
$467$			8.86299e6i			1.88056i	−0.340396	+	0.940282i	$0.610561\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−4.34181e6	+	464521.i	−0.901818	+	0.0964836i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.10416e6	−	5.10113e6i	−0.222196	−	1.02653i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	5.15363e6			1.00000
$485$			1.56953e7i			3.02981i
$486$		0			0
$487$	−4.40023e6			−0.840724			−0.420362	−	0.907357i	$-0.638097\pi$
$487$	−4.40023e6			−0.840724			−0.420362	+	0.907357i	$0.638097\pi$
$488$		0			0
$489$	2.54423e6	−	272202.i	0.481155	−	0.0514777i
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−9.16726e6	+	1.98429e6i	−1.68162	+	0.363991i
$496$	7.96160e6			1.45310
$497$		0			0
$498$		0			0
$499$	7.47980e6			1.34474			0.672370	−	0.740215i	$-0.265276\pi$
$499$	7.47980e6			1.34474			0.672370	+	0.740215i	$0.265276\pi$
$500$		−	9.23656e6i		−	1.65229i
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	5.75504e6	−	615720.i	0.994325	−	0.106381i
$508$		0			0
$509$			4.20392e6i			0.719218i	0.933103	+	0.359609i	$0.117090\pi$
$509$			4.20392e6i			0.719218i	−0.933103	+	0.359609i	$0.882910\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$			1.73362e7i			2.88030i
$516$		0			0
$517$	−7.03034e6			−1.15678
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	1.27663e6i		−	0.206048i	−0.994679	−	0.103024i	$-0.967148\pi$
$521$		−	1.27663e6i		−	0.206048i	0.994679	−	0.103024i	$-0.0328519\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−681472.	−	6.36962e6i	−0.106381	−	0.994325i
$529$	−1.83467e7			−2.85048
$530$		0			0
$531$	2.43491e6	+	1.12491e7i	0.374754	+	1.73134i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	283112.	+	2.64621e6i	0.0423666	+	0.395994i
$538$		0			0
$539$		−	6.74484e6i		−	1.00000i
$540$	3.66467e6	+	1.10679e7i	0.540817	+	1.63335i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	4.33806e6	−	464120.i	0.631388	−	0.0675509i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		−	1.12560e7i		−	1.60116i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	202086.	+	1.88887e6i	0.0278487	+	0.260298i
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	929632.	+	8.68913e6i	0.123059	+	1.15021i
$565$	1.84226e7			2.42789
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−1.61003e6	−	1.50487e7i	−0.204856	−	1.91475i
$574$		0			0
$575$		−	3.04968e7i		−	3.84667i
$576$	−7.78240e6	+	1.68453e6i	−0.977366	+	0.211554i
$577$	8.05397e6			1.00709			0.503547	−	0.863968i	$-0.332028\pi$
$577$	8.05397e6			1.00709			0.503547	+	0.863968i	$0.332028\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−8.61956e6			−1.05030
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.05843e7i		−	1.26785i	−0.773396	−	0.633923i	$-0.781444\pi$
$587$		−	1.05843e7i		−	1.26785i	0.773396	−	0.633923i	$-0.218556\pi$
$588$	−8.33627e6	+	891880.i	−0.994325	+	0.106381i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.29741e6			−0.152150
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.23984e7	+	1.32648e6i	−1.42374	+	0.152323i
$598$		0			0
$599$		−	9.63463e6i		−	1.09715i	−0.836100	−	0.548577i	$-0.815170\pi$
$599$		−	9.63463e6i		−	1.09715i	0.836100	−	0.548577i	$-0.184830\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	1.73178e7	−	3.74849e6i	1.93954	−	0.419820i
$604$		0			0
$605$			1.54902e7i			1.72056i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	6.02885e6i		−	0.637561i	−0.947829	−	0.318781i	$-0.896727\pi$
$617$		−	6.02885e6i		−	0.637561i	0.947829	−	0.318781i	$-0.103273\pi$
$618$		0			0
$619$	−1.91271e6			−0.200642			−0.100321	−	0.994955i	$-0.531987\pi$
$619$	−1.91271e6			−0.200642			−0.100321	+	0.994955i	$0.531987\pi$
$620$			2.39301e7i			2.50015i
$621$	5.92745e6	+	1.79018e7i	0.616792	+	1.86281i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	8.61850e6			0.882535
$626$		0			0
$627$		0			0
$628$	8.96374e6			0.906965
$629$		0			0
$630$		0			0
$631$	−1.28703e7			−1.28681			−0.643405	−	0.765526i	$-0.722479\pi$
$631$	−1.28703e7			−1.28681			−0.643405	+	0.765526i	$0.722479\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	1.13978e6	+	1.06533e7i	0.111732	+	1.04434i
$637$		0			0
$638$		0			0
$639$	−2.73226e6	−	1.26229e7i	−0.264710	−	1.22294i
$640$		0			0
$641$			1.43345e7i			1.37796i	0.724779	+	0.688982i	$0.241942\pi$
$641$			1.43345e7i			1.37796i	−0.724779	+	0.688982i	$0.758058\pi$
$642$		0			0
$643$	−2.09678e7			−1.99998			−0.999988	−	0.00489569i	$-0.998442\pi$
$643$	−2.09678e7			−1.99998			−0.999988	+	0.00489569i	$0.998442\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	2.00464e7i		−	1.88267i	−0.337469	−	0.941337i	$-0.609571\pi$
$647$		−	2.00464e7i		−	1.88267i	0.337469	−	0.941337i	$-0.390429\pi$
$648$		0			0
$649$	1.90080e7			1.77143
$650$		0			0
$651$		0			0
$652$	−5.25261e6			−0.483900
$653$			6.95248e6i			0.638053i	0.947746	+	0.319026i	$0.103356\pi$
$653$			6.95248e6i			0.638053i	−0.947746	+	0.319026i	$0.896644\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$	1.91451e7	−	2.04829e6i	1.71079	−	0.183034i
$661$	−2.15101e7			−1.91487			−0.957433	−	0.288657i	$-0.906791\pi$
$661$	−2.15101e7			−1.91487			−0.957433	+	0.288657i	$0.906791\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−3.83673e6	+	410484.i	−0.331433	+	0.0354593i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−2.20291e7	+	7.29403e6i	−1.86096	+	0.616181i
$676$	−1.18814e7			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.77354e7i			1.45475i	0.686238	+	0.727377i	$0.259261\pi$
$683$			1.77354e7i			1.45475i	−0.686238	+	0.727377i	$0.740739\pi$
$684$		0			0
$685$	3.38322e7			2.75489
$686$		0			0
$687$	1.81509e7	−	1.94193e6i	1.46726	−	0.156979i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	2.50313e7			1.99429			0.997146	−	0.0754971i	$-0.0240544\pi$
$691$	2.50313e7			1.99429			0.997146	+	0.0754971i	$0.0240544\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$			1.31502e7i			1.00000i
$705$	−2.61168e7	+	2.79419e6i	−1.97901	+	0.211730i
$706$		0			0
$707$		0			0
$708$	−2.51346e6	−	2.34929e7i	−0.188447	−	1.76138i
$709$	−2.59156e7			−1.93618			−0.968091	−	0.250598i	$-0.919373\pi$
$709$	−2.59156e7			−1.93618			−0.968091	+	0.250598i	$0.919373\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			3.87059e7i			2.85137i
$714$		0			0
$715$		0			0
$716$		−	5.46314e6i		−	0.398254i
$717$		0			0
$718$		0			0
$719$		−	2.77237e7i		−	2.00000i	−0.00108351	−	0.999999i	$-0.500345\pi$
$719$		−	2.77237e7i		−	2.00000i	0.00108351	−	0.999999i	$-0.499655\pi$
$720$	−5.06317e6	−	2.33915e7i	−0.363991	−	1.68162i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−8.95600e6			−0.634991
$725$		0			0
$726$		0			0
$727$	1.70878e7			1.19908			0.599542	−	0.800343i	$-0.295349\pi$
$727$	1.70878e7			1.19908			0.599542	+	0.800343i	$0.295349\pi$
$728$		0			0
$729$	1.15135e7	−	8.56329e6i	0.802397	−	0.596790i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	−2.68072e6	−	2.50563e7i	−0.183034	−	1.71079i
$736$		0			0
$737$		−	2.92624e7i		−	1.98446i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		−	3.89961e6i		−	0.261783i
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	2.56094e7			1.65691			0.828455	−	0.560055i	$-0.189220\pi$
$751$	2.56094e7			1.65691			0.828455	+	0.560055i	$0.189220\pi$
$752$		−	1.79389e7i		−	1.15678i
$753$	−1.25139e6	−	1.16965e7i	−0.0804275	−	0.751744i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.95760e7			−1.24161			−0.620805	−	0.783965i	$-0.713194\pi$
$757$	−1.95760e7			−1.24161			−0.620805	+	0.783965i	$0.713194\pi$
$758$		0			0
$759$	3.09664e7	−	3.31303e6i	1.95113	−	0.208747i
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$			3.10683e7i			1.92568i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.62529e7	−	1.73887e6i	0.994325	−	0.106381i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	3.46628e6	+	3.23988e7i	0.210004	+	1.96287i
$772$		0			0
$773$			3.10221e7i			1.86734i	0.358137	+	0.933669i	$0.383412\pi$
$773$			3.10221e7i			1.86734i	−0.358137	+	0.933669i	$0.616588\pi$
$774$		0			0
$775$	−4.76296e7			−2.84855
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−2.13293e7			−1.25126
$782$		0			0
$783$		0			0
$784$	1.72104e7			1.00000
$785$			2.69422e7i			1.56049i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−3.20206e7	+	3.42581e6i	−1.79685	+	0.192241i
$796$	2.55968e7			1.43187
$797$		−	3.46060e7i		−	1.92977i	−0.262670	−	0.964886i	$-0.584603\pi$
$797$		−	3.46060e7i		−	1.92977i	0.262670	−	0.964886i	$-0.415397\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	6.09111e6	+	2.81405e7i	0.335440	+	1.54971i
$802$		0			0
$803$		0			0
$804$	−3.61668e7	+	3.86941e6i	−1.97320	+	0.211108i
$805$		0			0
$806$		0			0
$807$	3.43334e6	+	3.20909e7i	0.185581	+	1.73460i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	1.57877e7i		−	0.832579i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$	3.57048e7			1.83750			0.918749	−	0.394843i	$-0.129201\pi$
$823$	3.57048e7			1.83750			0.918749	+	0.394843i	$0.129201\pi$
$824$		0			0
$825$	4.07685e6	+	3.81057e7i	0.208540	+	1.94920i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−8.18946e6	−	3.78347e7i	−0.415125	−	1.91785i
$829$	1.90687e7			0.963684			0.481842	−	0.876258i	$-0.339968\pi$
$829$	1.90687e7			0.963684			0.481842	+	0.876258i	$0.339968\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	2.79589e7	−	9.25745e6i	1.37945	−	0.456749i
$838$		0			0
$839$			1.06668e7i			0.523155i	0.965182	+	0.261577i	$0.0842427\pi$
$839$			1.06668e7i			0.523155i	−0.965182	+	0.261577i	$0.915757\pi$
$840$		0			0
$841$	−2.05111e7			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	3.57117e7i		−	1.72056i
$846$		0			0
$847$		0			0
$848$		−	2.19939e7i		−	1.05030i
$849$		0			0
$850$		0			0
$851$		−	6.30745e6i		−	0.298559i
$852$	2.82040e6	+	2.63619e7i	0.133110	+	1.24416i
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−1.50484e7			−0.695835			−0.347918	−	0.937525i	$-0.613111\pi$
$859$	−1.50484e7			−0.695835			−0.347918	+	0.937525i	$0.613111\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			4.32270e7i			1.97573i	0.155310	+	0.987866i	$0.450362\pi$
$863$			4.32270e7i			1.97573i	−0.155310	+	0.987866i	$0.549638\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−2.20078e7	+	2.35457e6i	−0.994325	+	0.106381i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−3.87560e7	+	8.38886e6i	−1.72109	+	0.372535i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$	−3.95254e7			−1.72056
$881$		−	4.55216e7i		−	1.97596i	−0.154589	−	0.987979i	$-0.549405\pi$
$881$		−	4.55216e7i		−	1.97596i	0.154589	−	0.987979i	$-0.450595\pi$
$882$		0			0
$883$	−6.29994e6			−0.271916			−0.135958	−	0.990715i	$-0.543411\pi$
$883$	−6.29994e6			−0.271916			−0.135958	+	0.990715i	$0.543411\pi$
$884$		0			0
$885$	7.06124e7	−	7.55467e6i	3.03056	−	0.324233i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−9.79949e6	−	2.15759e7i	−0.413532	−	0.910490i
$892$	7.92099e6			0.333325
$893$		0			0
$894$		0			0
$895$	1.64205e7			0.685219
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	4.65576e7	−	1.00776e7i	1.91595	−	0.414714i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	2.69190e7i		−	1.09254i
$906$		0			0
$907$	3.06168e7			1.23578			0.617891	−	0.786264i	$-0.287987\pi$
$907$	3.06168e7			1.23578			0.617891	+	0.786264i	$0.287987\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	2.62088e7i		−	1.04629i	−0.852245	−	0.523143i	$-0.824759\pi$
$911$		−	2.62088e7i		−	1.04629i	0.852245	−	0.523143i	$-0.175241\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−3.74728e7			−1.47563
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	7.76164e6			0.298263
$926$		0			0
$927$	−4.28080e7	+	9.26593e6i	−1.63616	+	0.354152i
$928$		0			0
$929$		−	4.43239e7i		−	1.68499i	−0.538702	−	0.842497i	$-0.681085\pi$
$929$		−	4.43239e7i		−	1.68499i	0.538702	−	0.842497i	$-0.318915\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−929225.	−	8.68533e6i	−0.0349476	−	0.326650i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	−1.62567e7	+	1.73927e6i	−0.601684	+	0.0643729i
$940$	5.39187e7			1.99030
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$			4.85015e7i			1.77143i
$945$		0			0
$946$		0			0
$947$			5.15961e7i			1.86957i	0.355213	+	0.934785i	$0.384408\pi$
$947$			5.15961e7i			1.86957i	−0.355213	+	0.934785i	$0.615592\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−2.36194e6	−	2.20767e7i	−0.0846870	−	0.791557i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−9.33818e7			−3.31325
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	5.22650e6	+	4.88513e7i	0.183034	+	1.71079i
$961$	3.18215e7			1.11151
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	3.68798e7i		−	1.25528i	−0.778504	−	0.627639i	$-0.784021\pi$
$971$		−	3.68798e7i		−	1.25528i	0.778504	−	0.627639i	$-0.215979\pi$
$972$	−2.53709e7	+	1.49647e7i	−0.861331	+	0.508044i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			3.66898e7i			1.22973i	0.788633	+	0.614864i	$0.210789\pi$
$977$			3.66898e7i			1.22973i	−0.788633	+	0.614864i	$0.789211\pi$
$978$		0			0
$979$	4.75500e7			1.58560
$980$			5.17291e7i			1.72056i
$981$		0			0
$982$		0			0
$983$		−	2.23060e6i		−	0.0736271i	−0.999322	−	0.0368136i	$-0.988279\pi$
$983$		−	2.23060e6i		−	0.0736271i	0.999322	−	0.0368136i	$-0.0117208\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	6.17681e7			1.99793			0.998965	−	0.0454936i	$-0.0144861\pi$
$991$	6.17681e7			1.99793			0.998965	+	0.0454936i	$0.0144861\pi$
$992$		0			0
$993$	1.14223e7	−	1.22205e6i	0.367605	−	0.0393293i
$994$		0			0
$995$			7.69361e7i			2.46361i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$	−4.55613e6	+	1.50858e6i	−0.144438	+	0.0478249i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.d.a.32.1	✓	2
3.2	odd	2	inner	33.6.d.a.32.2	yes	2
11.10	odd	2	CM	33.6.d.a.32.1	✓	2
33.32	even	2	inner	33.6.d.a.32.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.d.a.32.1	✓	2	1.1	even	1	trivial
33.6.d.a.32.1	✓	2	11.10	odd	2	CM
33.6.d.a.32.2	yes	2	3.2	odd	2	inner
33.6.d.a.32.2	yes	2	33.32	even	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:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	33.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(15.5000 - 1.65831i) q^{3} -32.0000 q^{4} -96.1821i q^{5} +(237.500 - 51.4077i) q^{9} +O(q^{10})\)	\(q+(15.5000 - 1.65831i) q^{3} -32.0000 q^{4} -96.1821i q^{5} +(237.500 - 51.4077i) q^{9} -401.312i q^{11} +(-496.000 + 53.0660i) q^{12} +(-159.500 - 1490.82i) q^{15} +1024.00 q^{16} +3077.83i q^{20} +4978.25i q^{23} -6126.00 q^{25} +(3596.00 - 1190.67i) q^{27} +7775.00 q^{31} +(-665.500 - 6220.33i) q^{33} +(-7600.00 + 1645.05i) q^{36} -1267.00 q^{37} +12842.0i q^{44} +(-4944.50 - 22843.3i) q^{45} -17518.4i q^{47} +(15872.0 - 1698.11i) q^{48} +16807.0 q^{49} -21478.5i q^{53} -38599.0 q^{55} +47364.7i q^{59} +(5104.00 + 47706.3i) q^{60} -32768.0 q^{64} +72917.0 q^{67} +(8255.50 + 77162.9i) q^{69} -53148.9i q^{71} +(-94953.0 + 10158.8i) q^{75} -98490.5i q^{80} +(53763.5 - 24418.7i) q^{81} +118486. i q^{89} -159304. i q^{92} +(120512. - 12893.4i) q^{93} -163183. q^{97} +(-20630.5 - 95311.5i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 31 q^{3} - 64 q^{4} + 475 q^{9}+O(q^{10}) \) 2 * q + 31 * q^3 - 64 * q^4 + 475 * q^9	\( 2 q + 31 q^{3} - 64 q^{4} + 475 q^{9} - 992 q^{12} - 319 q^{15} + 2048 q^{16} - 12252 q^{25} + 7192 q^{27} + 15550 q^{31} - 1331 q^{33} - 15200 q^{36} - 2534 q^{37} - 9889 q^{45} + 31744 q^{48} + 33614 q^{49} - 77198 q^{55} + 10208 q^{60} - 65536 q^{64} + 145834 q^{67} + 16511 q^{69} - 189906 q^{75} + 107527 q^{81} + 241025 q^{93} - 326366 q^{97} - 41261 q^{99}+O(q^{100}) \) 2 * q + 31 * q^3 - 64 * q^4 + 475 * q^9 - 992 * q^12 - 319 * q^15 + 2048 * q^16 - 12252 * q^25 + 7192 * q^27 + 15550 * q^31 - 1331 * q^33 - 15200 * q^36 - 2534 * q^37 - 9889 * q^45 + 31744 * q^48 + 33614 * q^49 - 77198 * q^55 + 10208 * q^60 - 65536 * q^64 + 145834 * q^67 + 16511 * q^69 - 189906 * q^75 + 107527 * q^81 + 241025 * q^93 - 326366 * q^97 - 41261 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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