LMFDB - Embedded newform 24.4.f.a.11.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,4,Mod(11,24)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("24.11");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	24.f (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:	$1.41604584014$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
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			11.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	24.11
Dual form			24.4.f.a.11.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} +(5.00000 + 1.41421i) q^{3} -8.00000 q^{4} +(-4.00000 + 14.1421i) q^{6} -22.6274i q^{8} +(23.0000 + 14.1421i) q^{9} +O(q^{10})$	\(q+2.82843i q^{2} +(5.00000 + 1.41421i) q^{3} -8.00000 q^{4} +(-4.00000 + 14.1421i) q^{6} -22.6274i q^{8} +(23.0000 + 14.1421i) q^{9} -70.7107i q^{11} +(-40.0000 - 11.3137i) q^{12} +64.0000 q^{16} +107.480i q^{17} +(-40.0000 + 65.0538i) q^{18} -106.000 q^{19} +200.000 q^{22} +(32.0000 - 113.137i) q^{24} -125.000 q^{25} +(95.0000 + 103.238i) q^{27} +181.019i q^{32} +(100.000 - 353.553i) q^{33} -304.000 q^{34} +(-184.000 - 113.137i) q^{36} -299.813i q^{38} +56.5685i q^{41} +290.000 q^{43} +565.685i q^{44} +(320.000 + 90.5097i) q^{48} +343.000 q^{49} -353.553i q^{50} +(-152.000 + 537.401i) q^{51} +(-292.000 + 268.701i) q^{54} +(-530.000 - 149.907i) q^{57} -325.269i q^{59} -512.000 q^{64} +(1000.00 + 282.843i) q^{66} -70.0000 q^{67} -859.842i q^{68} +(320.000 - 520.431i) q^{72} -430.000 q^{73} +(-625.000 - 176.777i) q^{75} +848.000 q^{76} +(329.000 + 650.538i) q^{81} -160.000 q^{82} -681.651i q^{83} +820.244i q^{86} -1600.00 q^{88} +1329.36i q^{89} +(-256.000 + 905.097i) q^{96} +1910.00 q^{97} +970.151i q^{98} +(1000.00 - 1626.35i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{3} - 16 q^{4} - 8 q^{6} + 46 q^{9}+O(q^{10}) $ 2 * q + 10 * q^3 - 16 * q^4 - 8 * q^6 + 46 * q^9	$ 2 q + 10 q^{3} - 16 q^{4} - 8 q^{6} + 46 q^{9} - 80 q^{12} + 128 q^{16} - 80 q^{18} - 212 q^{19} + 400 q^{22} + 64 q^{24} - 250 q^{25} + 190 q^{27} + 200 q^{33} - 608 q^{34} - 368 q^{36} + 580 q^{43} + 640 q^{48} + 686 q^{49} - 304 q^{51} - 584 q^{54} - 1060 q^{57} - 1024 q^{64} + 2000 q^{66} - 140 q^{67} + 640 q^{72} - 860 q^{73} - 1250 q^{75} + 1696 q^{76} + 658 q^{81} - 320 q^{82} - 3200 q^{88} - 512 q^{96} + 3820 q^{97} + 2000 q^{99}+O(q^{100}) $ 2 * q + 10 * q^3 - 16 * q^4 - 8 * q^6 + 46 * q^9 - 80 * q^12 + 128 * q^16 - 80 * q^18 - 212 * q^19 + 400 * q^22 + 64 * q^24 - 250 * q^25 + 190 * q^27 + 200 * q^33 - 608 * q^34 - 368 * q^36 + 580 * q^43 + 640 * q^48 + 686 * q^49 - 304 * q^51 - 584 * q^54 - 1060 * q^57 - 1024 * q^64 + 2000 * q^66 - 140 * q^67 + 640 * q^72 - 860 * q^73 - 1250 * q^75 + 1696 * q^76 + 658 * q^81 - 320 * q^82 - 3200 * q^88 - 512 * q^96 + 3820 * q^97 + 2000 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$.

$n$	$7$	$13$	$17$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$			2.82843i			1.00000i
$2$			2.82843i			1.00000i
$3$	5.00000	+	1.41421i	0.962250	+	0.272166i
$3$	5.00000	+	1.41421i	0.962250	+	0.272166i
$4$	−8.00000			−1.00000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	−4.00000	+	14.1421i	−0.272166	+	0.962250i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	22.6274i		−	1.00000i
$9$	23.0000	+	14.1421i	0.851852	+	0.523783i
$10$		0			0
$11$		−	70.7107i		−	1.93819i	−0.246691	−	0.969094i	$-0.579343\pi$
$11$		−	70.7107i		−	1.93819i	0.246691	−	0.969094i	$-0.420657\pi$
$12$	−40.0000	−	11.3137i	−0.962250	−	0.272166i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	64.0000			1.00000
$17$			107.480i			1.53340i	0.642006	+	0.766700i	$0.278102\pi$
$17$			107.480i			1.53340i	−0.642006	+	0.766700i	$0.721898\pi$
$18$	−40.0000	+	65.0538i	−0.523783	+	0.851852i
$19$	−106.000			−1.27990			−0.639949	−	0.768417i	$-0.721045\pi$
$19$	−106.000			−1.27990			−0.639949	+	0.768417i	$0.721045\pi$
$20$		0			0
$21$		0			0
$22$	200.000			1.93819
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	32.0000	−	113.137i	0.272166	−	0.962250i
$25$	−125.000			−1.00000
$26$		0			0
$27$	95.0000	+	103.238i	0.677139	+	0.735855i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			181.019i			1.00000i
$33$	100.000	−	353.553i	0.527508	−	1.86502i
$34$	−304.000			−1.53340
$35$		0			0
$36$	−184.000	−	113.137i	−0.851852	−	0.523783i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		−	299.813i		−	1.27990i
$39$		0			0
$40$		0			0
$41$			56.5685i			0.215476i	0.994179	+	0.107738i	$0.0343608\pi$
$41$			56.5685i			0.215476i	−0.994179	+	0.107738i	$0.965639\pi$
$42$		0			0
$43$	290.000			1.02848			0.514239	−	0.857647i	$-0.328074\pi$
$43$	290.000			1.02848			0.514239	+	0.857647i	$0.328074\pi$
$44$			565.685i			1.93819i
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	320.000	+	90.5097i	0.962250	+	0.272166i
$49$	343.000			1.00000
$50$		−	353.553i		−	1.00000i
$51$	−152.000	+	537.401i	−0.417338	+	1.47551i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−292.000	+	268.701i	−0.735855	+	0.677139i
$55$		0			0
$56$		0			0
$57$	−530.000	−	149.907i	−1.23158	−	0.348344i
$58$		0			0
$59$		−	325.269i		−	0.717736i	−0.933388	−	0.358868i	$-0.883163\pi$
$59$		−	325.269i		−	0.717736i	0.933388	−	0.358868i	$-0.116837\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	−512.000			−1.00000
$65$		0			0
$66$	1000.00	+	282.843i	1.86502	+	0.527508i
$67$	−70.0000			−0.127640			−0.0638199	−	0.997961i	$-0.520328\pi$
$67$	−70.0000			−0.127640			−0.0638199	+	0.997961i	$0.520328\pi$
$68$		−	859.842i		−	1.53340i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	320.000	−	520.431i	0.523783	−	0.851852i
$73$	−430.000			−0.689420			−0.344710	−	0.938709i	$-0.612023\pi$
$73$	−430.000			−0.689420			−0.344710	+	0.938709i	$0.612023\pi$
$74$		0			0
$75$	−625.000	−	176.777i	−0.962250	−	0.272166i
$76$	848.000			1.27990
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	329.000	+	650.538i	0.451303	+	0.892371i
$82$	−160.000			−0.215476
$83$		−	681.651i		−	0.901457i	−0.892661	−	0.450728i	$-0.851164\pi$
$83$		−	681.651i		−	0.901457i	0.892661	−	0.450728i	$-0.148836\pi$
$84$		0			0
$85$		0			0
$86$			820.244i			1.02848i
$87$		0			0
$88$	−1600.00			−1.93819
$89$			1329.36i			1.58328i	0.610988	+	0.791640i	$0.290773\pi$
$89$			1329.36i			1.58328i	−0.610988	+	0.791640i	$0.709227\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−256.000	+	905.097i	−0.272166	+	0.962250i
$97$	1910.00			1.99929			0.999645	−	0.0266459i	$-0.00848265\pi$
$97$	1910.00			1.99929			0.999645	+	0.0266459i	$0.00848265\pi$
$98$			970.151i			1.00000i
$99$	1000.00	−	1626.35i	1.01519	−	1.65105i
$100$	1000.00			1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	−1520.00	−	429.921i	−1.47551	−	0.417338i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1405.73i			1.27006i	0.772486	+	0.635032i	$0.219013\pi$
$107$			1405.73i			1.27006i	−0.772486	+	0.635032i	$0.780987\pi$
$108$	−760.000	−	825.901i	−0.677139	−	0.735855i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2387.19i		−	1.98733i	−0.112387	−	0.993665i	$-0.535850\pi$
$113$		−	2387.19i		−	1.98733i	0.112387	−	0.993665i	$-0.464150\pi$
$114$	424.000	−	1499.07i	0.348344	−	1.23158i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	920.000			0.717736
$119$		0			0
$120$		0			0
$121$	−3669.00			−2.75657
$122$		0			0
$123$	−80.0000	+	282.843i	−0.0586452	+	0.207342i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	1448.15i		−	1.00000i
$129$	1450.00	+	410.122i	0.989654	+	0.279916i
$130$		0			0
$131$			2729.43i			1.82039i	0.414176	+	0.910197i	$0.364070\pi$
$131$			2729.43i			1.82039i	−0.414176	+	0.910197i	$0.635930\pi$
$132$	−800.000	+	2828.43i	−0.527508	+	1.86502i
$133$		0			0
$134$		−	197.990i		−	0.127640i
$135$		0			0
$136$	2432.00			1.53340
$137$		−	2285.37i		−	1.42520i	−0.701571	−	0.712599i	$-0.747518\pi$
$137$		−	2285.37i		−	1.42520i	0.701571	−	0.712599i	$-0.252482\pi$
$138$		0			0
$139$	−1474.00			−0.899446			−0.449723	−	0.893168i	$-0.648477\pi$
$139$	−1474.00			−0.899446			−0.449723	+	0.893168i	$0.648477\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1472.00	+	905.097i	0.851852	+	0.523783i
$145$		0			0
$146$		−	1216.22i		−	0.689420i
$147$	1715.00	+	485.075i	0.962250	+	0.272166i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	500.000	−	1767.77i	0.272166	−	0.962250i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$			2398.51i			1.27990i
$153$	−1520.00	+	2472.05i	−0.803168	+	1.30623i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−1840.00	+	930.553i	−0.892371	+	0.451303i
$163$	−970.000			−0.466112			−0.233056	−	0.972463i	$-0.574873\pi$
$163$	−970.000			−0.466112			−0.233056	+	0.972463i	$0.574873\pi$
$164$		−	452.548i		−	0.215476i
$165$		0			0
$166$	1928.00			0.901457
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	2197.00			1.00000
$170$		0			0
$171$	−2438.00	−	1499.07i	−1.09028	−	0.670389i
$172$	−2320.00			−1.02848
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$		−	4525.48i		−	1.93819i
$177$	460.000	−	1626.35i	0.195343	−	0.690642i
$178$	−3760.00			−1.58328
$179$		−	2870.85i		−	1.19876i	−0.800465	−	0.599379i	$-0.795414\pi$
$179$		−	2870.85i		−	1.19876i	0.800465	−	0.599379i	$-0.204586\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	7600.00			2.97202
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−2560.00	−	724.077i	−0.962250	−	0.272166i
$193$	2090.00			0.779490			0.389745	−	0.920923i	$-0.372563\pi$
$193$	2090.00			0.779490			0.389745	+	0.920923i	$0.372563\pi$
$194$			5402.30i			1.99929i
$195$		0			0
$196$	−2744.00			−1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	4600.00	+	2828.43i	1.65105	+	1.01519i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			2828.43i			1.00000i
$201$	−350.000	−	98.9949i	−0.122821	−	0.0347391i
$202$		0			0
$203$		0			0
$204$	1216.00	−	4299.21i	0.417338	−	1.47551i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			7495.33i			2.48068i
$210$		0			0
$211$	−6118.00			−1.99612			−0.998058	−	0.0622910i	$-0.980159\pi$
$211$	−6118.00			−1.99612			−0.998058	+	0.0622910i	$0.980159\pi$
$212$		0			0
$213$		0			0
$214$	−3976.00			−1.27006
$215$		0			0
$216$	2336.00	−	2149.60i	0.735855	−	0.677139i
$217$		0			0
$218$		0			0
$219$	−2150.00	−	608.112i	−0.663395	−	0.187636i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−2875.00	−	1767.77i	−0.851852	−	0.523783i
$226$	6752.00			1.98733
$227$		−	1903.53i		−	0.556572i	−0.960498	−	0.278286i	$-0.910234\pi$
$227$		−	1903.53i		−	0.556572i	0.960498	−	0.278286i	$-0.0897663\pi$
$228$	4240.00	+	1199.25i	1.23158	+	0.348344i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3773.12i			1.06088i	0.847722	+	0.530441i	$0.177974\pi$
$233$			3773.12i			1.06088i	−0.847722	+	0.530441i	$0.822026\pi$
$234$		0			0
$235$		0			0
$236$			2602.15i			0.717736i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−1222.00			−0.326622			−0.163311	−	0.986575i	$-0.552217\pi$
$241$	−1222.00			−0.326622			−0.163311	+	0.986575i	$0.552217\pi$
$242$		−	10377.5i		−	2.75657i
$243$	725.000	+	3717.97i	0.191394	+	0.981513i
$244$		0			0
$245$		0			0
$246$	−800.000	−	226.274i	−0.207342	−	0.0586452i
$247$		0			0
$248$		0			0
$249$	964.000	−	3408.25i	0.245345	−	0.867427i
$250$		0			0
$251$		−	6689.23i		−	1.68215i	−0.540916	−	0.841077i	$-0.681922\pi$
$251$		−	6689.23i		−	1.68215i	0.540916	−	0.841077i	$-0.318078\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4096.00			1.00000
$257$		−	7274.71i		−	1.76570i	−0.469658	−	0.882849i	$-0.655623\pi$
$257$		−	7274.71i		−	1.76570i	0.469658	−	0.882849i	$-0.344377\pi$
$258$	−1160.00	+	4101.22i	−0.279916	+	0.989654i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−7720.00			−1.82039
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	−8000.00	−	2262.74i	−1.86502	−	0.527508i
$265$		0			0
$266$		0			0
$267$	−1880.00	+	6646.80i	−0.430914	+	1.52351i
$268$	560.000			0.127640
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$			6878.73i			1.53340i
$273$		0			0
$274$	6464.00			1.42520
$275$			8838.83i			1.93819i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		−	4169.10i		−	0.899446i
$279$		0			0
$280$		0			0
$281$		−	1216.22i		−	0.258199i	−0.991632	−	0.129099i	$-0.958791\pi$
$281$		−	1216.22i		−	0.258199i	0.991632	−	0.129099i	$-0.0412086\pi$
$282$		0			0
$283$	8030.00			1.68669			0.843346	−	0.537371i	$-0.180582\pi$
$283$	8030.00			1.68669			0.843346	+	0.537371i	$0.180582\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−2560.00	+	4163.44i	−0.523783	+	0.851852i
$289$	−6639.00			−1.35131
$290$		0			0
$291$	9550.00	+	2701.15i	1.92382	+	0.544138i
$292$	3440.00			0.689420
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−1372.00	+	4850.75i	−0.272166	+	0.962250i
$295$		0			0
$296$		0			0
$297$	7300.00	−	6717.51i	1.42623	−	1.31242i
$298$		0			0
$299$		0			0
$300$	5000.00	+	1414.21i	0.962250	+	0.272166i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−6784.00			−1.27990
$305$		0			0
$306$	−6992.00	−	4299.21i	−1.30623	−	0.803168i
$307$	−7990.00			−1.48539			−0.742693	−	0.669632i	$-0.766452\pi$
$307$	−7990.00			−1.48539			−0.742693	+	0.669632i	$0.766452\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	8390.00			1.51511			0.757557	−	0.652769i	$-0.226393\pi$
$313$	8390.00			1.51511			0.757557	+	0.652769i	$0.226393\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−1988.00	+	7028.64i	−0.345668	+	1.22212i
$322$		0			0
$323$		−	11392.9i		−	1.96259i
$324$	−2632.00	−	5204.31i	−0.451303	−	0.892371i
$325$		0			0
$326$		−	2743.57i		−	0.466112i
$327$		0			0
$328$	1280.00			0.215476
$329$		0			0
$330$		0			0
$331$	−8242.00			−1.36864			−0.684322	−	0.729180i	$-0.739902\pi$
$331$	−8242.00			−1.36864			−0.684322	+	0.729180i	$0.739902\pi$
$332$			5453.21i			0.901457i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−11410.0			−1.84434			−0.922170	−	0.386786i	$-0.873585\pi$
$337$	−11410.0			−1.84434			−0.922170	+	0.386786i	$0.873585\pi$
$338$			6214.05i			1.00000i
$339$	3376.00	−	11936.0i	0.540882	−	1.91231i
$340$		0			0
$341$		0			0
$342$	4240.00	−	6895.71i	0.670389	−	1.09028i
$343$		0			0
$344$		−	6561.95i		−	1.02848i
$345$		0			0
$346$		0			0
$347$			11435.3i			1.76911i	0.466437	+	0.884554i	$0.345537\pi$
$347$			11435.3i			1.76911i	−0.466437	+	0.884554i	$0.654463\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	12800.0			1.93819
$353$			12377.2i			1.86621i	0.359605	+	0.933104i	$0.382911\pi$
$353$			12377.2i			1.86621i	−0.359605	+	0.933104i	$0.617089\pi$
$354$	4600.00	+	1301.08i	0.690642	+	0.195343i
$355$		0			0
$356$		−	10634.9i		−	1.58328i
$357$		0			0
$358$	8120.00			1.19876
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	4377.00			0.638140
$362$		0			0
$363$	−18345.0	−	5188.75i	−2.65251	−	0.750244i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−800.000	+	1301.08i	−0.112863	+	0.183554i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$			21496.0i			2.97202i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	11666.0			1.58111			0.790557	−	0.612389i	$-0.209791\pi$
$379$	11666.0			1.58111			0.790557	+	0.612389i	$0.209791\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	2048.00	−	7240.77i	0.272166	−	0.962250i
$385$		0			0
$386$			5911.41i			0.779490i
$387$	6670.00	+	4101.22i	0.876112	+	0.538699i
$388$	−15280.0			−1.99929
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		−	7761.20i		−	1.00000i
$393$	−3860.00	+	13647.2i	−0.495448	+	1.75167i
$394$		0			0
$395$		0			0
$396$	−8000.00	+	13010.8i	−1.01519	+	1.65105i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−8000.00			−1.00000
$401$		−	14453.3i		−	1.79990i	−0.435989	−	0.899952i	$-0.643601\pi$
$401$		−	14453.3i		−	1.79990i	0.435989	−	0.899952i	$-0.356399\pi$
$402$	280.000	−	989.949i	0.0347391	−	0.122821i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	12160.0	+	3439.37i	1.47551	+	0.417338i
$409$	16346.0			1.97618			0.988090	−	0.153877i	$-0.0491758\pi$
$409$	16346.0			1.97618			0.988090	+	0.153877i	$0.0491758\pi$
$410$		0			0
$411$	3232.00	−	11426.8i	0.387890	−	1.37140i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−7370.00	−	2084.55i	−0.865493	−	0.244798i
$418$	−21200.0			−2.48068
$419$			3493.11i			0.407278i	0.979046	+	0.203639i	$0.0652769\pi$
$419$			3493.11i			0.407278i	−0.979046	+	0.203639i	$0.934723\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		−	17304.3i		−	1.99612i
$423$		0			0
$424$		0			0
$425$		−	13435.0i		−	1.53340i
$426$		0			0
$427$		0			0
$428$		−	11245.8i		−	1.27006i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	6080.00	+	6607.21i	0.677139	+	0.735855i
$433$	5510.00			0.611533			0.305766	−	0.952107i	$-0.401087\pi$
$433$	5510.00			0.611533			0.305766	+	0.952107i	$0.401087\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	1720.00	−	6081.12i	0.187636	−	0.663395i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	7889.00	+	4850.75i	0.851852	+	0.523783i
$442$		0			0
$443$		−	3736.35i		−	0.400721i	−0.979722	−	0.200361i	$-0.935789\pi$
$443$		−	3736.35i		−	0.400721i	0.979722	−	0.200361i	$-0.0642114\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			7438.76i			0.781864i	0.920420	+	0.390932i	$0.127847\pi$
$449$			7438.76i			0.781864i	−0.920420	+	0.390932i	$0.872153\pi$
$450$	5000.00	−	8131.73i	0.523783	−	0.851852i
$451$	4000.00			0.417633
$452$			19097.5i			1.98733i
$453$		0			0
$454$	5384.00			0.556572
$455$		0			0
$456$	−3392.00	+	11992.5i	−0.348344	+	1.23158i
$457$	−18070.0			−1.84963			−0.924813	−	0.380422i	$-0.875779\pi$
$457$	−18070.0			−1.84963			−0.924813	+	0.380422i	$0.875779\pi$
$458$		0			0
$459$	−11096.0	+	10210.6i	−1.12836	+	1.03832i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−10672.0			−1.06088
$467$			13471.8i			1.33490i	0.744653	+	0.667452i	$0.232615\pi$
$467$			13471.8i			1.33490i	−0.744653	+	0.667452i	$0.767385\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−7360.00			−0.717736
$473$		−	20506.1i		−	1.99339i
$474$		0			0
$475$	13250.0			1.27990
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		−	3456.34i		−	0.326622i
$483$		0			0
$484$	29352.0			2.75657
$485$		0			0
$486$	−10516.0	+	2050.61i	−0.981513	+	0.191394i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−4850.00	−	1371.79i	−0.448517	−	0.126860i
$490$		0			0
$491$			18002.9i			1.65471i	0.561681	+	0.827354i	$0.310155\pi$
$491$			18002.9i			1.65471i	−0.561681	+	0.827354i	$0.689845\pi$
$492$	640.000	−	2262.74i	0.0586452	−	0.207342i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	9640.00	+	2726.60i	0.867427	+	0.245345i
$499$	−18214.0			−1.63401			−0.817005	−	0.576631i	$-0.804367\pi$
$499$	−18214.0			−1.63401			−0.817005	+	0.576631i	$0.804367\pi$
$500$		0			0
$501$		0			0
$502$	18920.0			1.68215
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	10985.0	+	3107.03i	0.962250	+	0.272166i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$			11585.2i			1.00000i
$513$	−10070.0	−	10943.2i	−0.866669	−	0.941819i
$514$	20576.0			1.76570
$515$		0			0
$516$	−11600.0	−	3280.98i	−0.989654	−	0.279916i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			21948.6i			1.84565i	0.385215	+	0.922827i	$0.374127\pi$
$521$			21948.6i			1.84565i	−0.385215	+	0.922827i	$0.625873\pi$
$522$		0			0
$523$	−4750.00			−0.397138			−0.198569	−	0.980087i	$-0.563629\pi$
$523$	−4750.00			−0.397138			−0.198569	+	0.980087i	$0.563629\pi$
$524$		−	21835.5i		−	1.82039i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	6400.00	−	22627.4i	0.527508	−	1.86502i
$529$	−12167.0			−1.00000
$530$		0			0
$531$	4600.00	−	7481.19i	0.375938	−	0.611405i
$532$		0			0
$533$		0			0
$534$	−18800.0	−	5317.44i	−1.52351	−	0.430914i
$535$		0			0
$536$			1583.92i			0.127640i
$537$	4060.00	−	14354.3i	0.326261	−	1.15351i
$538$		0			0
$539$		−	24253.8i		−	1.93819i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	−19456.0			−1.53340
$545$		0			0
$546$		0			0
$547$	−21850.0			−1.70793			−0.853966	−	0.520329i	$-0.825809\pi$
$547$	−21850.0			−1.70793			−0.853966	+	0.520329i	$0.825809\pi$
$548$			18283.0i			1.42520i
$549$		0			0
$550$	−25000.0			−1.93819
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	11792.0			0.899446
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	38000.0	+	10748.0i	2.85982	+	0.808880i
$562$	3440.00			0.258199
$563$		−	12391.3i		−	0.927589i	−0.885943	−	0.463795i	$-0.846488\pi$
$563$		−	12391.3i		−	0.927589i	0.885943	−	0.463795i	$-0.153512\pi$
$564$		0			0
$565$		0			0
$566$			22712.3i			1.68669i
$567$		0			0
$568$		0			0
$569$			27039.8i			1.99221i	0.0881913	+	0.996104i	$0.471891\pi$
$569$			27039.8i			1.99221i	−0.0881913	+	0.996104i	$0.528109\pi$
$570$		0			0
$571$	27038.0			1.98162			0.990810	−	0.135261i	$-0.0431872\pi$
$571$	27038.0			1.98162			0.990810	+	0.135261i	$0.0431872\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−11776.0	−	7240.77i	−0.851852	−	0.523783i
$577$	19550.0			1.41053			0.705266	−	0.708943i	$-0.250827\pi$
$577$	19550.0			1.41053			0.705266	+	0.708943i	$0.250827\pi$
$578$		−	18777.9i		−	1.35131i
$579$	10450.0	+	2955.71i	0.750064	+	0.212150i
$580$		0			0
$581$		0			0
$582$	−7640.00	+	27011.5i	−0.544138	+	1.92382i
$583$		0			0
$584$			9729.79i			0.689420i
$585$		0			0
$586$		0			0
$587$		−	26493.9i		−	1.86289i	−0.363876	−	0.931447i	$-0.618547\pi$
$587$		−	26493.9i		−	1.86289i	0.363876	−	0.931447i	$-0.381453\pi$
$588$	−13720.0	−	3880.60i	−0.962250	−	0.272166i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			12173.6i			0.843015i	0.906825	+	0.421507i	$0.138499\pi$
$593$			12173.6i			0.843015i	−0.906825	+	0.421507i	$0.861501\pi$
$594$	19000.0	+	20647.5i	1.31242	+	1.42623i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−4000.00	+	14142.1i	−0.272166	+	0.962250i
$601$	−14398.0			−0.977216			−0.488608	−	0.872503i	$-0.662495\pi$
$601$	−14398.0			−0.977216			−0.488608	+	0.872503i	$0.662495\pi$
$602$		0			0
$603$	−1610.00	−	989.949i	−0.108730	−	0.0668555i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		−	19188.0i		−	1.27990i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	12160.0	−	19776.4i	0.803168	−	1.30623i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		−	22599.1i		−	1.48539i
$615$		0			0
$616$		0			0
$617$			11206.2i			0.731192i	0.930774	+	0.365596i	$0.119135\pi$
$617$			11206.2i			0.731192i	−0.930774	+	0.365596i	$0.880865\pi$
$618$		0			0
$619$	−30706.0			−1.99383			−0.996913	−	0.0785136i	$-0.974983\pi$
$619$	−30706.0			−1.99383			−0.996913	+	0.0785136i	$0.974983\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15625.0			1.00000
$626$			23730.5i			1.51511i
$627$	−10600.0	+	37476.7i	−0.675157	+	2.38704i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−30590.0	−	8652.16i	−1.92076	−	0.543274i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	31763.2i		−	1.95721i	−0.205745	−	0.978606i	$-0.565962\pi$
$641$		−	31763.2i		−	1.95721i	0.205745	−	0.978606i	$-0.434038\pi$
$642$	−19880.0	−	5622.91i	−1.22212	−	0.345668i
$643$	28550.0			1.75101			0.875507	−	0.483205i	$-0.160528\pi$
$643$	28550.0			1.75101			0.875507	+	0.483205i	$0.160528\pi$
$644$		0			0
$645$		0			0
$646$	32224.0			1.96259
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	14720.0	−	7444.42i	0.892371	−	0.451303i
$649$	−23000.0			−1.39111
$650$		0			0
$651$		0			0
$652$	7760.00			0.466112
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$			3620.39i			0.215476i
$657$	−9890.00	−	6081.12i	−0.587284	−	0.361107i
$658$		0			0
$659$		−	16107.9i		−	0.952161i	−0.879402	−	0.476081i	$-0.842057\pi$
$659$		−	16107.9i		−	0.952161i	0.879402	−	0.476081i	$-0.157943\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		−	23311.9i		−	1.36864i
$663$		0			0
$664$	−15424.0			−0.901457
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	19190.0			1.09914			0.549569	−	0.835448i	$-0.314792\pi$
$673$	19190.0			1.09914			0.549569	+	0.835448i	$0.314792\pi$
$674$		−	32272.4i		−	1.84434i
$675$	−11875.0	−	12904.7i	−0.677139	−	0.735855i
$676$	−17576.0			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	33760.0	+	9548.77i	1.91231	+	0.540882i
$679$		0			0
$680$		0			0
$681$	2692.00	−	9517.66i	0.151480	−	0.535562i
$682$		0			0
$683$			33632.8i			1.88422i	0.335300	+	0.942112i	$0.391162\pi$
$683$			33632.8i			1.88422i	−0.335300	+	0.942112i	$0.608838\pi$
$684$	19504.0	+	11992.5i	1.09028	+	0.670389i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	18560.0			1.02848
$689$		0			0
$690$		0			0
$691$	−1978.00			−0.108895			−0.0544477	−	0.998517i	$-0.517340\pi$
$691$	−1978.00			−0.108895			−0.0544477	+	0.998517i	$0.517340\pi$
$692$		0			0
$693$		0			0
$694$	−32344.0			−1.76911
$695$		0			0
$696$		0			0
$697$	−6080.00			−0.330411
$698$		0			0
$699$	−5336.00	+	18865.6i	−0.288735	+	1.02083i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$			36203.9i			1.93819i
$705$		0			0
$706$	−35008.0			−1.86621
$707$		0			0
$708$	−3680.00	+	13010.8i	−0.195343	+	0.690642i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	30080.0			1.58328
$713$		0			0
$714$		0			0
$715$		0			0
$716$			22966.8i			1.19876i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$			12380.0i			0.638140i
$723$	−6110.00	−	1728.17i	−0.314292	−	0.0888953i
$724$		0			0
$725$		0			0
$726$	14676.0	−	51887.5i	0.750244	−	2.65251i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−1633.00	+	19615.1i	−0.0829650	+	0.996552i
$730$		0			0
$731$			31169.3i			1.57707i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			4949.75i			0.247390i
$738$	−3680.00	−	2262.74i	−0.183554	−	0.112863i
$739$	36074.0			1.79567			0.897837	−	0.440327i	$-0.145138\pi$
$739$	36074.0			1.79567			0.897837	+	0.440327i	$0.145138\pi$
$740$		0			0
$741$		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
$747$	9640.00	−	15678.0i	0.472168	−	0.767908i
$748$	−60800.0			−2.97202
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	9460.00	−	33446.2i	0.457824	−	1.61865i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$			32996.4i			1.58111i
$759$		0			0
$760$		0			0
$761$		−	24381.0i		−	1.16138i	−0.814124	−	0.580691i	$-0.802782\pi$
$761$		−	24381.0i		−	1.16138i	0.814124	−	0.580691i	$-0.197218\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	20480.0	+	5792.62i	0.962250	+	0.272166i
$769$	40106.0			1.88070			0.940351	−	0.340207i	$-0.110497\pi$
$769$	40106.0			1.88070			0.940351	+	0.340207i	$0.110497\pi$
$770$		0			0
$771$	10288.0	−	36373.6i	0.480562	−	1.69904i
$772$	−16720.0			−0.779490
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	−11600.0	+	18865.6i	−0.538699	+	0.876112i
$775$		0			0
$776$		−	43218.4i		−	1.99929i
$777$		0			0
$778$		0			0
$779$		−	5996.27i		−	0.275788i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	21952.0			1.00000
$785$		0			0
$786$	−38600.0	−	10917.7i	−1.75167	−	0.495448i
$787$	6950.00			0.314791			0.157396	−	0.987536i	$-0.449690\pi$
$787$	6950.00			0.314791			0.157396	+	0.987536i	$0.449690\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−36800.0	−	22627.4i	−1.65105	−	1.01519i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		−	22627.4i		−	1.00000i
$801$	−18800.0	+	30575.3i	−0.829295	+	1.34872i
$802$	40880.0			1.79990
$803$			30405.6i			1.33623i
$804$	2800.00	+	791.960i	0.122821	+	0.0347391i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	43727.5i		−	1.90034i	−0.311730	−	0.950171i	$-0.600908\pi$
$809$		−	43727.5i		−	1.90034i	0.311730	−	0.950171i	$-0.399092\pi$
$810$		0			0
$811$	−37582.0			−1.62723			−0.813614	−	0.581405i	$-0.802503\pi$
$811$	−37582.0			−1.62723			−0.813614	+	0.581405i	$0.802503\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−9728.00	+	34393.7i	−0.417338	+	1.47551i
$817$	−30740.0			−1.31635
$818$			46233.5i			1.97618i
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	32320.0	+	9141.48i	1.37140	+	0.387890i
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	−12500.0	+	44194.2i	−0.527508	+	1.86502i
$826$		0			0
$827$		−	41360.1i		−	1.73909i	−0.493850	−	0.869547i	$-0.664411\pi$
$827$		−	41360.1i		−	1.73909i	0.493850	−	0.869547i	$-0.335589\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			36865.7i			1.53340i
$834$	5896.00	−	20845.5i	0.244798	−	0.865493i
$835$		0			0
$836$		−	59962.7i		−	2.48068i
$837$		0			0
$838$	−9880.00			−0.407278
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−24389.0			−1.00000
$842$		0			0
$843$	1720.00	−	6081.12i	0.0702728	−	0.248452i
$844$	48944.0			1.99612
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	40150.0	+	11356.1i	1.62302	+	0.459060i
$850$	38000.0			1.53340
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	31808.0			1.27006
$857$			46589.9i			1.85704i	0.371289	+	0.928518i	$0.378916\pi$
$857$			46589.9i			1.85704i	−0.371289	+	0.928518i	$0.621084\pi$
$858$		0			0
$859$	−45646.0			−1.81306			−0.906532	−	0.422138i	$-0.861280\pi$
$859$	−45646.0			−1.81306			−0.906532	+	0.422138i	$0.861280\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	−18688.0	+	17196.8i	−0.735855	+	0.677139i
$865$		0			0
$866$			15584.6i			0.611533i
$867$	−33195.0	−	9388.96i	−1.30030	−	0.367781i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	43930.0	+	27011.5i	1.70310	+	1.04719i
$874$		0			0
$875$		0			0
$876$	17200.0	+	4864.89i	0.663395	+	0.187636i
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	31508.7i		−	1.20494i	−0.798141	−	0.602471i	$-0.794183\pi$
$881$		−	31508.7i		−	1.20494i	0.798141	−	0.602471i	$-0.205817\pi$
$882$	−13720.0	+	22313.5i	−0.523783	+	0.851852i
$883$	−5290.00			−0.201611			−0.100806	−	0.994906i	$-0.532142\pi$
$883$	−5290.00			−0.201611			−0.100806	+	0.994906i	$0.532142\pi$
$884$		0			0
$885$		0			0
$886$	10568.0			0.400721
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	46000.0	−	23263.8i	1.72958	−	0.874711i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−21040.0			−0.781864
$899$		0			0
$900$	23000.0	+	14142.1i	0.851852	+	0.523783i
$901$		0			0
$902$			11313.7i			0.417633i
$903$		0			0
$904$	−54016.0			−1.98733
$905$		0			0
$906$		0			0
$907$	26210.0			0.959525			0.479762	−	0.877399i	$-0.340723\pi$
$907$	26210.0			0.959525			0.479762	+	0.877399i	$0.340723\pi$
$908$			15228.3i			0.556572i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−33920.0	−	9594.02i	−1.23158	−	0.348344i
$913$	−48200.0			−1.74719
$914$		−	51109.7i		−	1.84963i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−28880.0	−	31384.2i	−1.03832	−	1.12836i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−39950.0	−	11299.6i	−1.42931	−	0.404271i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	56200.8i		−	1.98481i	−0.123007	−	0.992406i	$-0.539254\pi$
$929$		−	56200.8i		−	1.98481i	0.123007	−	0.992406i	$-0.460746\pi$
$930$		0			0
$931$	−36358.0			−1.27990
$932$		−	30185.0i		−	1.06088i
$933$		0			0
$934$	−38104.0			−1.33490
$935$		0			0
$936$		0			0
$937$	56270.0			1.96186			0.980929	−	0.194367i	$-0.0622652\pi$
$937$	56270.0			1.96186			0.980929	+	0.194367i	$0.0622652\pi$
$938$		0			0
$939$	41950.0	+	11865.3i	1.45792	+	0.412362i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		−	20817.2i		−	0.717736i
$945$		0			0
$946$	58000.0			1.99339
$947$			2525.79i			0.0866705i	0.999061	+	0.0433353i	$0.0137984\pi$
$947$			2525.79i			0.0866705i	−0.999061	+	0.0433353i	$0.986202\pi$
$948$		0			0
$949$		0			0
$950$			37476.7i			1.27990i
$951$		0			0
$952$		0			0
$953$		−	36701.7i		−	1.24752i	−0.781617	−	0.623759i	$-0.785605\pi$
$953$		−	36701.7i		−	1.24752i	0.781617	−	0.623759i	$-0.214395\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	29791.0			1.00000
$962$		0			0
$963$	−19880.0	+	32331.8i	−0.665238	+	1.08191i
$964$	9776.00			0.326622
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$			83020.0i			2.75657i
$969$	16112.0	−	56964.5i	0.534151	−	1.88851i
$970$		0			0
$971$			60514.2i			1.99999i	0.00267705	+	0.999996i	$0.499148\pi$
$971$			60514.2i			1.99999i	−0.00267705	+	0.999996i	$0.500852\pi$
$972$	−5800.00	−	29743.7i	−0.191394	−	0.981513i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			58554.1i			1.91741i	0.284399	+	0.958706i	$0.408206\pi$
$977$			58554.1i			1.91741i	−0.284399	+	0.958706i	$0.591794\pi$
$978$	3880.00	−	13717.9i	0.126860	−	0.448517i
$979$	94000.0			3.06870
$980$		0			0
$981$		0			0
$982$	−50920.0			−1.65471
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	6400.00	+	1810.19i	0.207342	+	0.0586452i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−41210.0	−	11655.9i	−1.31698	−	0.372498i
$994$		0			0
$995$		0			0
$996$	−7712.00	+	27266.0i	−0.245345	+	0.867427i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		−	51517.0i		−	1.63401i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	24.4.f.a.11.2	yes	2
3.2	odd	2	inner	24.4.f.a.11.1	✓	2
4.3	odd	2		96.4.f.a.47.1		2
8.3	odd	2	CM	24.4.f.a.11.2	yes	2
8.5	even	2		96.4.f.a.47.1		2
12.11	even	2		96.4.f.a.47.2		2
16.3	odd	4		768.4.c.l.767.3		4
16.5	even	4		768.4.c.l.767.3		4
16.11	odd	4		768.4.c.l.767.2		4
16.13	even	4		768.4.c.l.767.2		4
24.5	odd	2		96.4.f.a.47.2		2
24.11	even	2	inner	24.4.f.a.11.1	✓	2
48.5	odd	4		768.4.c.l.767.1		4
48.11	even	4		768.4.c.l.767.4		4
48.29	odd	4		768.4.c.l.767.4		4
48.35	even	4		768.4.c.l.767.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.4.f.a.11.1	✓	2	3.2	odd	2	inner
24.4.f.a.11.1	✓	2	24.11	even	2	inner
24.4.f.a.11.2	yes	2	1.1	even	1	trivial
24.4.f.a.11.2	yes	2	8.3	odd	2	CM
96.4.f.a.47.1		2	4.3	odd	2
96.4.f.a.47.1		2	8.5	even	2
96.4.f.a.47.2		2	12.11	even	2
96.4.f.a.47.2		2	24.5	odd	2
768.4.c.l.767.1		4	48.5	odd	4
768.4.c.l.767.1		4	48.35	even	4
768.4.c.l.767.2		4	16.11	odd	4
768.4.c.l.767.2		4	16.13	even	4
768.4.c.l.767.3		4	16.3	odd	4
768.4.c.l.767.3		4	16.5	even	4
768.4.c.l.767.4		4	48.11	even	4
768.4.c.l.767.4		4	48.29	odd	4

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:	\( N \)	\(=\)	\( 24 = 2^{3} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	24.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} +(5.00000 + 1.41421i) q^{3} -8.00000 q^{4} +(-4.00000 + 14.1421i) q^{6} -22.6274i q^{8} +(23.0000 + 14.1421i) q^{9} +O(q^{10})\)	\(q+2.82843i q^{2} +(5.00000 + 1.41421i) q^{3} -8.00000 q^{4} +(-4.00000 + 14.1421i) q^{6} -22.6274i q^{8} +(23.0000 + 14.1421i) q^{9} -70.7107i q^{11} +(-40.0000 - 11.3137i) q^{12} +64.0000 q^{16} +107.480i q^{17} +(-40.0000 + 65.0538i) q^{18} -106.000 q^{19} +200.000 q^{22} +(32.0000 - 113.137i) q^{24} -125.000 q^{25} +(95.0000 + 103.238i) q^{27} +181.019i q^{32} +(100.000 - 353.553i) q^{33} -304.000 q^{34} +(-184.000 - 113.137i) q^{36} -299.813i q^{38} +56.5685i q^{41} +290.000 q^{43} +565.685i q^{44} +(320.000 + 90.5097i) q^{48} +343.000 q^{49} -353.553i q^{50} +(-152.000 + 537.401i) q^{51} +(-292.000 + 268.701i) q^{54} +(-530.000 - 149.907i) q^{57} -325.269i q^{59} -512.000 q^{64} +(1000.00 + 282.843i) q^{66} -70.0000 q^{67} -859.842i q^{68} +(320.000 - 520.431i) q^{72} -430.000 q^{73} +(-625.000 - 176.777i) q^{75} +848.000 q^{76} +(329.000 + 650.538i) q^{81} -160.000 q^{82} -681.651i q^{83} +820.244i q^{86} -1600.00 q^{88} +1329.36i q^{89} +(-256.000 + 905.097i) q^{96} +1910.00 q^{97} +970.151i q^{98} +(1000.00 - 1626.35i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{3} - 16 q^{4} - 8 q^{6} + 46 q^{9}+O(q^{10}) \) 2 * q + 10 * q^3 - 16 * q^4 - 8 * q^6 + 46 * q^9	\( 2 q + 10 q^{3} - 16 q^{4} - 8 q^{6} + 46 q^{9} - 80 q^{12} + 128 q^{16} - 80 q^{18} - 212 q^{19} + 400 q^{22} + 64 q^{24} - 250 q^{25} + 190 q^{27} + 200 q^{33} - 608 q^{34} - 368 q^{36} + 580 q^{43} + 640 q^{48} + 686 q^{49} - 304 q^{51} - 584 q^{54} - 1060 q^{57} - 1024 q^{64} + 2000 q^{66} - 140 q^{67} + 640 q^{72} - 860 q^{73} - 1250 q^{75} + 1696 q^{76} + 658 q^{81} - 320 q^{82} - 3200 q^{88} - 512 q^{96} + 3820 q^{97} + 2000 q^{99}+O(q^{100}) \) 2 * q + 10 * q^3 - 16 * q^4 - 8 * q^6 + 46 * q^9 - 80 * q^12 + 128 * q^16 - 80 * q^18 - 212 * q^19 + 400 * q^22 + 64 * q^24 - 250 * q^25 + 190 * q^27 + 200 * q^33 - 608 * q^34 - 368 * q^36 + 580 * q^43 + 640 * q^48 + 686 * q^49 - 304 * q^51 - 584 * q^54 - 1060 * q^57 - 1024 * q^64 + 2000 * q^66 - 140 * q^67 + 640 * q^72 - 860 * q^73 - 1250 * q^75 + 1696 * q^76 + 658 * q^81 - 320 * q^82 - 3200 * q^88 - 512 * q^96 + 3820 * q^97 + 2000 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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