LMFDB - Embedded newform 33.4.d.a.32.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,4,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, 4, names="a")

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

chi := DirichletCharacter("33.32");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
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:	$1.94706303019$
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:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			32.2
Root			$0.500000 - 1.65831i$ of defining polynomial
Character	$\chi$	$=$	33.32
Dual form			33.4.d.a.32.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.00000 + 3.31662i) q^{3} -8.00000 q^{4} +13.2665i q^{5} +(5.00000 - 26.5330i) q^{9} +O(q^{10})$	$q+(-4.00000 + 3.31662i) q^{3} -8.00000 q^{4} +13.2665i q^{5} +(5.00000 - 26.5330i) q^{9} +36.4829i q^{11} +(32.0000 - 26.5330i) q^{12} +(-44.0000 - 53.0660i) q^{15} +64.0000 q^{16} -106.132i q^{20} +192.364i q^{23} -51.0000 q^{25} +(68.0000 + 122.715i) q^{27} -340.000 q^{31} +(-121.000 - 145.931i) q^{33} +(-40.0000 + 212.264i) q^{36} +434.000 q^{37} -291.863i q^{44} +(352.000 + 66.3325i) q^{45} -643.425i q^{47} +(-256.000 + 212.264i) q^{48} +343.000 q^{49} -225.530i q^{53} -484.000 q^{55} +550.560i q^{59} +(352.000 + 424.528i) q^{60} -512.000 q^{64} +416.000 q^{67} +(-638.000 - 769.457i) q^{69} +1028.15i q^{71} +(204.000 - 169.148i) q^{75} +849.056i q^{80} +(-679.000 - 265.330i) q^{81} +132.665i q^{89} -1538.91i q^{92} +(1360.00 - 1127.65i) q^{93} -34.0000 q^{97} +(968.000 + 182.414i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{3} - 16 q^{4} + 10 q^{9}+O(q^{10}) $ 2 * q - 8 * q^3 - 16 * q^4 + 10 * q^9	$ 2 q - 8 q^{3} - 16 q^{4} + 10 q^{9} + 64 q^{12} - 88 q^{15} + 128 q^{16} - 102 q^{25} + 136 q^{27} - 680 q^{31} - 242 q^{33} - 80 q^{36} + 868 q^{37} + 704 q^{45} - 512 q^{48} + 686 q^{49} - 968 q^{55} + 704 q^{60} - 1024 q^{64} + 832 q^{67} - 1276 q^{69} + 408 q^{75} - 1358 q^{81} + 2720 q^{93} - 68 q^{97} + 1936 q^{99}+O(q^{100}) $ 2 * q - 8 * q^3 - 16 * q^4 + 10 * q^9 + 64 * q^12 - 88 * q^15 + 128 * q^16 - 102 * q^25 + 136 * q^27 - 680 * q^31 - 242 * q^33 - 80 * q^36 + 868 * q^37 + 704 * q^45 - 512 * q^48 + 686 * q^49 - 968 * q^55 + 704 * q^60 - 1024 * q^64 + 832 * q^67 - 1276 * q^69 + 408 * q^75 - 1358 * q^81 + 2720 * q^93 - 68 * q^97 + 1936 * 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	$0.5\pi$
$3$	−4.00000	+	3.31662i	−0.769800	+	0.638285i
$3$	−4.00000	+	3.31662i	−0.769800	+	0.638285i
$4$	−8.00000			−1.00000
$5$			13.2665i			1.18659i	0.804984	+	0.593296i	$0.202174\pi$
$5$			13.2665i			1.18659i	−0.804984	+	0.593296i	$0.797826\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	5.00000	−	26.5330i	0.185185	−	0.982704i
$10$		0			0
$11$			36.4829i			1.00000i
$11$			36.4829i			1.00000i
$12$	32.0000	−	26.5330i	0.769800	−	0.638285i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−44.0000	−	53.0660i	−0.757383	−	0.913439i
$16$	64.0000			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			$\pi$
$20$		−	106.132i		−	1.18659i
$21$		0			0
$22$		0			0
$23$			192.364i			1.74394i	0.489556	+	0.871972i	$0.337159\pi$
$23$			192.364i			1.74394i	−0.489556	+	0.871972i	$0.662841\pi$
$24$		0			0
$25$	−51.0000			−0.408000
$26$		0			0
$27$	68.0000	+	122.715i	0.484689	+	0.874686i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−340.000			−1.96986			−0.984932	−	0.172940i	$-0.944673\pi$
$31$	−340.000			−1.96986			−0.984932	+	0.172940i	$0.944673\pi$
$32$		0			0
$33$	−121.000	−	145.931i	−0.638285	−	0.769800i
$34$		0			0
$35$		0			0
$36$	−40.0000	+	212.264i	−0.185185	+	0.982704i
$37$	434.000			1.92836			0.964178	−	0.265257i	$-0.0854567\pi$
$37$	434.000			1.92836			0.964178	+	0.265257i	$0.0854567\pi$
$38$		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			$\pi$
$44$		−	291.863i		−	1.00000i
$45$	352.000	+	66.3325i	1.16607	+	0.219739i
$46$		0			0
$47$		−	643.425i		−	1.99688i	−0.0558632	−	0.998438i	$-0.517791\pi$
$47$		−	643.425i		−	1.99688i	0.0558632	−	0.998438i	$-0.482209\pi$
$48$	−256.000	+	212.264i	−0.769800	+	0.638285i
$49$	343.000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	225.530i		−	0.584509i	−0.956341	−	0.292255i	$-0.905595\pi$
$53$		−	225.530i		−	0.584509i	0.956341	−	0.292255i	$-0.0944055\pi$
$54$		0			0
$55$	−484.000			−1.18659
$56$		0			0
$57$		0			0
$58$		0			0
$59$			550.560i			1.21486i	0.794373	+	0.607430i	$0.207800\pi$
$59$			550.560i			1.21486i	−0.794373	+	0.607430i	$0.792200\pi$
$60$	352.000	+	424.528i	0.757383	+	0.913439i
$61$		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
$66$		0			0
$67$	416.000			0.758545			0.379272	−	0.925285i	$-0.376174\pi$
$67$	416.000			0.758545			0.379272	+	0.925285i	$0.376174\pi$
$68$		0			0
$69$	−638.000	−	769.457i	−1.11313	−	1.34249i
$70$		0			0
$71$			1028.15i			1.71858i	0.511486	+	0.859292i	$0.329095\pi$
$71$			1028.15i			1.71858i	−0.511486	+	0.859292i	$0.670905\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	204.000	−	169.148i	0.314079	−	0.260420i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$			849.056i			1.18659i
$81$	−679.000	−	265.330i	−0.931413	−	0.363964i
$82$		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
$89$			132.665i			0.158005i	0.996874	+	0.0790026i	$0.0251735\pi$
$89$			132.665i			0.158005i	−0.996874	+	0.0790026i	$0.974826\pi$
$90$		0			0
$91$		0			0
$92$		−	1538.91i		−	1.74394i
$93$	1360.00	−	1127.65i	1.51640	−	1.25733i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−34.0000			−0.0355895			−0.0177947	−	0.999842i	$-0.505665\pi$
$97$	−34.0000			−0.0355895			−0.0177947	+	0.999842i	$0.505665\pi$
$98$		0			0
$99$	968.000	+	182.414i	0.982704	+	0.185185i
$100$	408.000			0.408000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	1172.00			1.12117			0.560585	−	0.828097i	$-0.310576\pi$
$103$	1172.00			1.12117			0.560585	+	0.828097i	$0.310576\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−544.000	−	981.721i	−0.484689	−	0.874686i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	−1736.00	+	1439.42i	−1.48445	+	1.23084i
$112$		0			0
$113$			1087.85i			0.905634i	0.891604	+	0.452817i	$0.149581\pi$
$113$			1087.85i			0.905634i	−0.891604	+	0.452817i	$0.850419\pi$
$114$		0			0
$115$	−2552.00			−2.06935
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1331.00			−1.00000
$122$		0			0
$123$		0			0
$124$	2720.00			1.96986
$125$			981.721i			0.702462i
$126$		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	$0.5\pi$
$132$	968.000	+	1167.45i	0.638285	+	0.769800i
$133$		0			0
$134$		0			0
$135$	−1628.00	+	902.122i	−1.03790	+	0.575128i
$136$		0			0
$137$		−	2971.70i		−	1.85321i	−0.376042	−	0.926603i	$-0.622715\pi$
$137$		−	2971.70i		−	1.85321i	0.376042	−	0.926603i	$-0.377285\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	2134.00	+	2573.70i	1.27458	+	1.53720i
$142$		0			0
$143$		0			0
$144$	320.000	−	1698.11i	0.185185	−	0.982704i
$145$		0			0
$146$		0			0
$147$	−1372.00	+	1137.60i	−0.769800	+	0.638285i
$148$	−3472.00			−1.92836
$149$		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
$155$		−	4510.61i		−	2.33743i
$156$		0			0
$157$	1334.00			0.678120			0.339060	−	0.940765i	$-0.389891\pi$
$157$	1334.00			0.678120			0.339060	+	0.940765i	$0.389891\pi$
$158$		0			0
$159$	748.000	+	902.122i	0.373083	+	0.449955i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	3728.00			1.79141			0.895704	−	0.444651i	$-0.146672\pi$
$163$	3728.00			1.79141			0.895704	+	0.444651i	$0.146672\pi$
$164$		0			0
$165$	1936.00	−	1605.25i	0.913439	−	0.757383i
$166$		0			0
$167$		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
$171$		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
$176$			2334.90i			1.00000i
$177$	−1826.00	−	2202.24i	−0.775427	−	0.935200i
$178$		0			0
$179$		−	4344.78i		−	1.81421i	−0.420902	−	0.907106i	$-0.638286\pi$
$179$		−	4344.78i		−	1.81421i	0.420902	−	0.907106i	$-0.361714\pi$
$180$	−2816.00	−	530.660i	−1.16607	−	0.219739i
$181$	−2050.00			−0.841852			−0.420926	−	0.907095i	$-0.638295\pi$
$181$	−2050.00			−0.841852			−0.420926	+	0.907095i	$0.638295\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			5757.66i			2.28817i
$186$		0			0
$187$		0			0
$188$			5147.40i			1.99688i
$189$		0			0
$190$		0			0
$191$			789.357i			0.299036i	0.988759	+	0.149518i	$0.0477722\pi$
$191$			789.357i			0.299036i	−0.988759	+	0.149518i	$0.952228\pi$
$192$	2048.00	−	1698.11i	0.769800	−	0.638285i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−2744.00			−1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−3940.00			−1.40351			−0.701757	−	0.712417i	$-0.747601\pi$
$199$	−3940.00			−1.40351			−0.701757	+	0.712417i	$0.747601\pi$
$200$		0			0
$201$	−1664.00	+	1379.72i	−0.583928	+	0.484167i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	5104.00	+	961.821i	1.71378	+	0.322953i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			1804.24i			0.584509i
$213$	−3410.00	−	4112.61i	−1.09695	−	1.32297i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	3872.00			1.18659
$221$		0			0
$222$		0			0
$223$	668.000			0.200595			0.100297	−	0.994958i	$-0.468021\pi$
$223$	668.000			0.200595			0.100297	+	0.994958i	$0.468021\pi$
$224$		0			0
$225$	−255.000	+	1353.18i	−0.0755556	+	0.400943i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	−3310.00			−0.955157			−0.477579	−	0.878589i	$-0.658485\pi$
$229$	−3310.00			−0.955157			−0.477579	+	0.878589i	$0.658485\pi$
$230$		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
$235$	8536.00			2.36948
$236$		−	4404.48i		−	1.21486i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−2816.00	−	3396.22i	−0.757383	−	0.913439i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	3596.00	−	1190.67i	0.949315	−	0.314327i
$244$		0			0
$245$			4550.41i			1.18659i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	7926.73i		−	1.99335i	−0.0814769	−	0.996675i	$-0.525964\pi$
$251$		−	7926.73i		−	1.99335i	0.0814769	−	0.996675i	$-0.474036\pi$
$252$		0			0
$253$	−7018.00			−1.74394
$254$		0			0
$255$		0			0
$256$	4096.00			1.00000
$257$		−	1777.71i		−	0.431481i	−0.976451	−	0.215740i	$-0.930784\pi$
$257$		−	1777.71i		−	0.431481i	0.976451	−	0.215740i	$-0.0692165\pi$
$258$		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
$265$	2992.00			0.693574
$266$		0			0
$267$	−440.000	−	530.660i	−0.100852	−	0.121632i
$268$	−3328.00			−0.758545
$269$			8371.16i			1.89739i	0.316188	+	0.948696i	$0.397597\pi$
$269$			8371.16i			1.89739i	−0.316188	+	0.948696i	$0.602403\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	1860.63i		−	0.408000i
$276$	5104.00	+	6155.66i	1.11313	+	1.34249i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	−1700.00	+	9021.22i	−0.364790	+	1.93579i
$280$		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			$\pi$
$284$		−	8225.23i		−	1.71858i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4913.00			−1.00000
$290$		0			0
$291$	136.000	−	112.765i	0.0273968	−	0.0227162i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	−7304.00			−1.44154
$296$		0			0
$297$	−4477.00	+	2480.84i	−0.874686	+	0.484689i
$298$		0			0
$299$		0			0
$300$	−1632.00	+	1353.18i	−0.314079	+	0.260420i
$301$		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
$309$	−4688.00	+	3887.08i	−0.863078	+	0.715626i
$310$		0			0
$311$		−	5538.76i		−	1.00989i	−0.863153	−	0.504943i	$-0.831514\pi$
$311$		−	5538.76i		−	1.00989i	0.863153	−	0.504943i	$-0.168486\pi$
$312$		0			0
$313$	10982.0			1.98319			0.991596	−	0.129370i	$-0.0412955\pi$
$313$	10982.0			1.98319			0.991596	+	0.129370i	$0.0412955\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			9565.15i			1.69474i	0.531004	+	0.847369i	$0.321815\pi$
$317$			9565.15i			1.69474i	−0.531004	+	0.847369i	$0.678185\pi$
$318$		0			0
$319$		0			0
$320$		−	6792.45i		−	1.18659i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	5432.00	+	2122.64i	0.931413	+	0.363964i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	8120.00			1.34839			0.674193	−	0.738555i	$-0.264492\pi$
$331$	8120.00			1.34839			0.674193	+	0.738555i	$0.264492\pi$
$332$		0			0
$333$	2170.00	−	11515.3i	0.357103	−	1.89500i
$334$		0			0
$335$			5518.86i			0.900083i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	−3608.00	−	4351.41i	−0.578052	−	0.697157i
$340$		0			0
$341$		−	12404.2i		−	1.96986i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	10208.0	−	8464.03i	1.59299	−	1.32083i
$346$		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
$353$			9923.34i			1.49622i	0.663574	+	0.748111i	$0.269039\pi$
$353$			9923.34i			1.49622i	−0.663574	+	0.748111i	$0.730961\pi$
$354$		0			0
$355$	−13640.0			−2.03926
$356$		−	1061.32i		−	0.158005i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	6859.00			1.00000
$362$		0			0
$363$	5324.00	−	4414.43i	0.769800	−	0.638285i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−9916.00			−1.41038			−0.705192	−	0.709016i	$-0.749139\pi$
$367$	−9916.00			−1.41038			−0.705192	+	0.709016i	$0.749139\pi$
$368$			12311.3i			1.74394i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−10880.0	+	9021.22i	−1.51640	+	1.25733i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−3256.00	−	3926.88i	−0.448371	−	0.540756i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	12800.0			1.73481			0.867403	−	0.497605i	$-0.165787\pi$
$379$	12800.0			1.73481			0.867403	+	0.497605i	$0.165787\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			3774.32i			0.503547i	0.967786	+	0.251774i	$0.0810139\pi$
$383$			3774.32i			0.503547i	−0.967786	+	0.251774i	$0.918986\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	272.000			0.0355895
$389$			5983.19i			0.779845i	0.920848	+	0.389923i	$0.127498\pi$
$389$			5983.19i			0.779845i	−0.920848	+	0.389923i	$0.872502\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−7744.00	−	1459.31i	−0.982704	−	0.185185i
$397$	−2374.00			−0.300120			−0.150060	−	0.988677i	$-0.547947\pi$
$397$	−2374.00			−0.300120			−0.150060	+	0.988677i	$0.547947\pi$
$398$		0			0
$399$		0			0
$400$	−3264.00			−0.408000
$401$		−	13240.0i		−	1.64881i	−0.566001	−	0.824404i	$-0.691510\pi$
$401$		−	13240.0i		−	1.64881i	0.566001	−	0.824404i	$-0.308490\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	3520.00	−	9007.95i	0.431877	−	1.10521i
$406$		0			0
$407$			15833.6i			1.92836i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	9856.00	+	11886.8i	1.18287	+	1.42660i
$412$	−9376.00			−1.12117
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			5207.10i			0.607121i	0.952812	+	0.303560i	$0.0981754\pi$
$419$			5207.10i			0.607121i	−0.952812	+	0.303560i	$0.901825\pi$
$420$		0			0
$421$	11630.0			1.34635			0.673173	−	0.739485i	$-0.264931\pi$
$421$	11630.0			1.34635			0.673173	+	0.739485i	$0.264931\pi$
$422$		0			0
$423$	−17072.0	−	3217.13i	−1.96234	−	0.369792i
$424$		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	$0.5\pi$
$432$	4352.00	+	7853.77i	0.484689	+	0.874686i
$433$	−13282.0			−1.47412			−0.737058	−	0.675830i	$-0.763786\pi$
$433$	−13282.0			−1.47412			−0.737058	+	0.675830i	$0.763786\pi$
$434$		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
$441$	1715.00	−	9100.82i	0.185185	−	0.982704i
$442$		0			0
$443$			72.9657i			0.00782552i	0.999992	+	0.00391276i	$0.00124547\pi$
$443$			72.9657i			0.00782552i	−0.999992	+	0.00391276i	$0.998755\pi$
$444$	13888.0	−	11515.3i	1.48445	−	1.23084i
$445$	−1760.00			−0.187488
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	17777.1i		−	1.86849i	−0.356627	−	0.934247i	$-0.616073\pi$
$449$		−	17777.1i		−	1.86849i	0.356627	−	0.934247i	$-0.383927\pi$
$450$		0			0
$451$		0			0
$452$		−	8702.82i		−	0.905634i
$453$		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
$460$	20416.0			2.06935
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	13268.0			1.33178			0.665892	−	0.746048i	$-0.268051\pi$
$463$	13268.0			1.33178			0.665892	+	0.746048i	$0.268051\pi$
$464$		0			0
$465$	14960.0	+	18042.4i	1.49194	+	1.79935i
$466$		0			0
$467$		−	19747.2i		−	1.95673i	−0.206897	−	0.978363i	$-0.566337\pi$
$467$		−	19747.2i		−	1.95673i	0.206897	−	0.978363i	$-0.433663\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−5336.00	+	4424.38i	−0.522017	+	0.432833i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−5984.00	−	1127.65i	−0.574399	−	0.108242i
$478$		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
$484$	10648.0			1.00000
$485$		−	451.061i		−	0.0422302i
$486$		0			0
$487$	−16684.0			−1.55241			−0.776206	−	0.630480i	$-0.782858\pi$
$487$	−16684.0			−1.55241			−0.776206	+	0.630480i	$0.782858\pi$
$488$		0			0
$489$	−14912.0	+	12364.4i	−1.37903	+	1.14343i
$490$		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
$495$	−2420.00	+	12842.0i	−0.219739	+	1.16607i
$496$	−21760.0			−1.96986
$497$		0			0
$498$		0			0
$499$	−4120.00			−0.369612			−0.184806	−	0.982775i	$-0.559166\pi$
$499$	−4120.00			−0.369612			−0.184806	+	0.982775i	$0.559166\pi$
$500$		−	7853.77i		−	0.702462i
$501$		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
$507$	−8788.00	+	7286.62i	−0.769800	+	0.638285i
$508$		0			0
$509$			5028.00i			0.437843i	0.975742	+	0.218922i	$0.0702539\pi$
$509$			5028.00i			0.437843i	−0.975742	+	0.218922i	$0.929746\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$			15548.3i			1.33037i
$516$		0			0
$517$	23474.0			1.99688
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	12762.4i		−	1.07319i	−0.843841	−	0.536593i	$-0.819711\pi$
$521$		−	12762.4i		−	1.07319i	0.843841	−	0.536593i	$-0.180289\pi$
$522$		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
$528$	−7744.00	−	9339.62i	−0.638285	−	0.769800i
$529$	−24837.0			−2.04134
$530$		0			0
$531$	14608.0	+	2752.80i	1.19385	+	0.224974i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	14410.0	+	17379.1i	1.15798	+	1.39658i
$538$		0			0
$539$			12513.6i			1.00000i
$540$	13024.0	−	7216.98i	1.03790	−	0.575128i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	8200.00	−	6799.08i	0.648058	−	0.537342i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$			23773.6i			1.85321i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−19096.0	−	23030.6i	−1.46050	−	1.76143i
$556$		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	$0.5\pi$
$564$	−17072.0	−	20589.6i	−1.27458	−	1.53720i
$565$	−14432.0			−1.07462
$566$		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
$573$	−2618.00	−	3157.43i	−0.190870	−	0.230198i
$574$		0			0
$575$		−	9810.58i		−	0.711529i
$576$	−2560.00	+	13584.9i	−0.185185	+	0.982704i
$577$	22466.0			1.62092			0.810461	−	0.585793i	$-0.199217\pi$
$577$	22466.0			1.62092			0.810461	+	0.585793i	$0.199217\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	8228.00			0.584509
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	11389.3i		−	0.800828i	−0.916334	−	0.400414i	$-0.868866\pi$
$587$		−	11389.3i		−	0.800828i	0.916334	−	0.400414i	$-0.131134\pi$
$588$	10976.0	−	9100.82i	0.769800	−	0.638285i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	27776.0			1.92836
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	15760.0	−	13067.5i	1.08043	−	0.895841i
$598$		0			0
$599$			23116.9i			1.57684i	0.615134	+	0.788422i	$0.289102\pi$
$599$			23116.9i			1.57684i	−0.615134	+	0.788422i	$0.710898\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	2080.00	−	11037.7i	0.140471	−	0.745425i
$604$		0			0
$605$		−	17657.7i		−	1.18659i
$606$		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
$617$		−	30433.3i		−	1.98574i	−0.119209	−	0.992869i	$-0.538036\pi$
$617$		−	30433.3i		−	1.98574i	0.119209	−	0.992869i	$-0.461964\pi$
$618$		0			0
$619$	1856.00			0.120515			0.0602576	−	0.998183i	$-0.480808\pi$
$619$	1856.00			0.120515			0.0602576	+	0.998183i	$0.480808\pi$
$620$			36084.9i			2.33743i
$621$	−23606.0	+	13080.8i	−1.52540	+	0.845271i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−19399.0			−1.24154
$626$		0			0
$627$		0			0
$628$	−10672.0			−0.678120
$629$		0			0
$630$		0			0
$631$	12908.0			0.814357			0.407179	−	0.913349i	$-0.366513\pi$
$631$	12908.0			0.814357			0.407179	+	0.913349i	$0.366513\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−5984.00	−	7216.98i	−0.373083	−	0.449955i
$637$		0			0
$638$		0			0
$639$	27280.0	+	5140.77i	1.68886	+	0.318256i
$640$		0			0
$641$			32131.5i			1.97990i	0.141415	+	0.989950i	$0.454835\pi$
$641$			32131.5i			1.97990i	−0.141415	+	0.989950i	$0.545165\pi$
$642$		0			0
$643$	−10168.0			−0.623619			−0.311809	−	0.950145i	$-0.600935\pi$
$643$	−10168.0			−0.623619			−0.311809	+	0.950145i	$0.600935\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			3535.52i			0.214831i	0.994214	+	0.107416i	$0.0342575\pi$
$647$			3535.52i			0.214831i	−0.994214	+	0.107416i	$0.965742\pi$
$648$		0			0
$649$	−20086.0			−1.21486
$650$		0			0
$651$		0			0
$652$	−29824.0			−1.79141
$653$			6460.79i			0.387182i	0.981082	+	0.193591i	$0.0620135\pi$
$653$			6460.79i			0.387182i	−0.981082	+	0.193591i	$0.937986\pi$
$654$		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	$0.5\pi$
$660$	−15488.0	+	12842.0i	−0.913439	+	0.757383i
$661$	23582.0			1.38765			0.693823	−	0.720146i	$-0.255925\pi$
$661$	23582.0			1.38765			0.693823	+	0.720146i	$0.255925\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−2672.00	+	2215.51i	−0.154418	+	0.128036i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−3468.00	−	6258.47i	−0.197753	−	0.356872i
$676$	−17576.0			−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
$683$			4968.30i			0.278341i	0.990268	+	0.139170i	$0.0444436\pi$
$683$			4968.30i			0.278341i	−0.990268	+	0.139170i	$0.955556\pi$
$684$		0			0
$685$	39424.0			2.19900
$686$		0			0
$687$	13240.0	−	10978.0i	0.735280	−	0.609662i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−30328.0			−1.66965			−0.834827	−	0.550512i	$-0.814433\pi$
$691$	−30328.0			−1.66965			−0.834827	+	0.550512i	$0.814433\pi$
$692$		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
$704$		−	18679.2i		−	1.00000i
$705$	−34144.0	+	28310.7i	−1.82402	+	1.51240i
$706$		0			0
$707$		0			0
$708$	14608.0	+	17617.9i	0.775427	+	0.935200i
$709$	33554.0			1.77736			0.888679	−	0.458530i	$-0.151624\pi$
$709$	33554.0			1.77736			0.888679	+	0.458530i	$0.151624\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	65403.8i		−	3.43533i
$714$		0			0
$715$		0			0
$716$			34758.2i			1.81421i
$717$		0			0
$718$		0			0
$719$		−	31209.4i		−	1.61880i	−0.587259	−	0.809399i	$-0.699793\pi$
$719$		−	31209.4i		−	1.61880i	0.587259	−	0.809399i	$-0.300207\pi$
$720$	22528.0	+	4245.28i	1.16607	+	0.219739i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	16400.0			0.841852
$725$		0			0
$726$		0			0
$727$	33284.0			1.69799			0.848993	−	0.528405i	$-0.177210\pi$
$727$	33284.0			1.69799			0.848993	+	0.528405i	$0.177210\pi$
$728$		0			0
$729$	−10435.0	+	16689.3i	−0.530153	+	0.847902i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	−15092.0	−	18201.6i	−0.757383	−	0.913439i
$736$		0			0
$737$			15176.9i			0.758545i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		−	46061.3i		−	2.28817i
$741$		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
$751$	−39652.0			−1.92666			−0.963330	−	0.268319i	$-0.913532\pi$
$751$	−39652.0			−1.92666			−0.963330	+	0.268319i	$0.913532\pi$
$752$		−	41179.2i		−	1.99688i
$753$	26290.0	+	31706.9i	1.27233	+	1.53448i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−31426.0			−1.50885			−0.754424	−	0.656388i	$-0.772084\pi$
$757$	−31426.0			−1.50885			−0.754424	+	0.656388i	$0.772084\pi$
$758$		0			0
$759$	28072.0	−	23276.1i	1.34249	−	1.11313i
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		−	6314.85i		−	0.299036i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−16384.0	+	13584.9i	−0.769800	+	0.638285i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	5896.00	+	7110.84i	0.275408	+	0.332154i
$772$		0			0
$773$			28430.1i			1.32285i	0.750013	+	0.661423i	$0.230047\pi$
$773$			28430.1i			1.32285i	−0.750013	+	0.661423i	$0.769953\pi$
$774$		0			0
$775$	17340.0			0.803705
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−37510.0			−1.71858
$782$		0			0
$783$		0			0
$784$	21952.0			1.00000
$785$			17697.5i			0.804651i
$786$		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
$795$	−11968.0	+	9923.34i	−0.533913	+	0.442698i
$796$	31520.0			1.40351
$797$			44429.5i			1.97462i	0.158798	+	0.987311i	$0.449238\pi$
$797$			44429.5i			1.97462i	−0.158798	+	0.987311i	$0.550762\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	3520.00	+	663.325i	0.155272	+	0.0292602i
$802$		0			0
$803$		0			0
$804$	13312.0	−	11037.7i	0.583928	−	0.484167i
$805$		0			0
$806$		0			0
$807$	−27764.0	−	33484.6i	−1.21108	−	1.46061i
$808$		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
$815$			49457.5i			2.12567i
$816$		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
$823$	3332.00			0.141125			0.0705627	−	0.997507i	$-0.477521\pi$
$823$	3332.00			0.141125			0.0705627	+	0.997507i	$0.477521\pi$
$824$		0			0
$825$	6171.00	+	7442.51i	0.260420	+	0.314079i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−40832.0	−	7694.57i	−1.71378	−	0.322953i
$829$	−47734.0			−1.99984			−0.999922	−	0.0125057i	$-0.996019\pi$
$829$	−47734.0			−1.99984			−0.999922	+	0.0125057i	$0.996019\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−23120.0	−	41723.1i	−0.954772	−	1.72301i
$838$		0			0
$839$		−	43268.7i		−	1.78045i	−0.455517	−	0.890227i	$-0.650546\pi$
$839$		−	43268.7i		−	1.78045i	0.455517	−	0.890227i	$-0.349454\pi$
$840$		0			0
$841$	−24389.0			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$			29146.5i			1.18659i
$846$		0			0
$847$		0			0
$848$		−	14434.0i		−	0.584509i
$849$		0			0
$850$		0			0
$851$			83486.1i			3.36294i
$852$	27280.0	+	32900.9i	1.09695	+	1.32297i
$853$		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
$859$	50096.0			1.98982			0.994909	−	0.100779i	$-0.0321334\pi$
$859$	50096.0			1.98982			0.994909	+	0.100779i	$0.0321334\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	20105.4i		−	0.793042i	−0.918026	−	0.396521i	$-0.870217\pi$
$863$		−	20105.4i		−	0.793042i	0.918026	−	0.396521i	$-0.129783\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	19652.0	−	16294.6i	0.769800	−	0.638285i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−170.000	+	902.122i	−0.00659064	+	0.0349739i
$874$		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
$880$	−30976.0			−1.18659
$881$		−	39268.8i		−	1.50170i	−0.660471	−	0.750852i	$-0.729643\pi$
$881$		−	39268.8i		−	1.50170i	0.660471	−	0.750852i	$-0.270357\pi$
$882$		0			0
$883$	27272.0			1.03938			0.519692	−	0.854354i	$-0.326047\pi$
$883$	27272.0			1.03938			0.519692	+	0.854354i	$0.326047\pi$
$884$		0			0
$885$	29216.0	−	24224.6i	1.10970	−	0.920115i
$886$		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
$891$	9680.00	−	24771.9i	0.363964	−	0.931413i
$892$	−5344.00			−0.200595
$893$		0			0
$894$		0			0
$895$	57640.0			2.15273
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	2040.00	−	10825.5i	0.0755556	−	0.400943i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	27196.3i		−	0.998935i
$906$		0			0
$907$	−21256.0			−0.778163			−0.389082	−	0.921203i	$-0.627208\pi$
$907$	−21256.0			−0.778163			−0.389082	+	0.921203i	$0.627208\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	17836.8i		−	0.648694i	−0.945938	−	0.324347i	$-0.894856\pi$
$911$		−	17836.8i		−	0.648694i	0.945938	−	0.324347i	$-0.105144\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	26480.0			0.955157
$917$		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
$925$	−22134.0			−0.786769
$926$		0			0
$927$	5860.00	−	31096.7i	0.207624	−	1.10178i
$928$		0			0
$929$		−	1538.91i		−	0.0543489i	−0.999631	−	0.0271744i	$-0.991349\pi$
$929$		−	1538.91i		−	0.0543489i	0.999631	−	0.0271744i	$-0.00865096\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	18370.0	+	22155.1i	0.644595	+	0.777410i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	−43928.0	+	36423.2i	−1.52666	+	1.26584i
$940$	−68288.0			−2.36948
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$			35235.8i			1.21486i
$945$		0			0
$946$		0			0
$947$			53444.1i			1.83390i	0.399007	+	0.916948i	$0.369355\pi$
$947$			53444.1i			1.83390i	−0.399007	+	0.916948i	$0.630645\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−31724.0	−	38260.6i	−1.08173	−	1.30461i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−10472.0			−0.354833
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	22528.0	+	27169.8i	0.757383	+	0.913439i
$961$	85809.0			2.88037
$962$		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
$971$			45444.4i			1.50194i	0.660339	+	0.750968i	$0.270413\pi$
$971$			45444.4i			1.50194i	−0.660339	+	0.750968i	$0.729587\pi$
$972$	−28768.0	+	9525.35i	−0.949315	+	0.314327i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			13982.9i			0.457884i	0.973440	+	0.228942i	$0.0735266\pi$
$977$			13982.9i			0.457884i	−0.973440	+	0.228942i	$0.926473\pi$
$978$		0			0
$979$	−4840.00			−0.158005
$980$		−	36403.3i		−	1.18659i
$981$		0			0
$982$		0			0
$983$		−	59029.3i		−	1.91530i	−0.287931	−	0.957651i	$-0.592968\pi$
$983$		−	59029.3i		−	1.91530i	0.287931	−	0.957651i	$-0.407032\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−51460.0			−1.64953			−0.824763	−	0.565478i	$-0.808692\pi$
$991$	−51460.0			−1.64953			−0.824763	+	0.565478i	$0.808692\pi$
$992$		0			0
$993$	−32480.0	+	26931.0i	−1.03799	+	0.860654i
$994$		0			0
$995$		−	52270.0i		−	1.66540i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$	29512.0	+	53258.4i	0.934653	+	1.68671i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.4.d.a.32.2	yes	2
3.2	odd	2	inner	33.4.d.a.32.1	✓	2
4.3	odd	2		528.4.b.a.65.1		2
11.10	odd	2	CM	33.4.d.a.32.2	yes	2
12.11	even	2		528.4.b.a.65.2		2
33.32	even	2	inner	33.4.d.a.32.1	✓	2
44.43	even	2		528.4.b.a.65.1		2
132.131	odd	2		528.4.b.a.65.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.d.a.32.1	✓	2	3.2	odd	2	inner
33.4.d.a.32.1	✓	2	33.32	even	2	inner
33.4.d.a.32.2	yes	2	1.1	even	1	trivial
33.4.d.a.32.2	yes	2	11.10	odd	2	CM
528.4.b.a.65.1		2	4.3	odd	2
528.4.b.a.65.1		2	44.43	even	2
528.4.b.a.65.2		2	12.11	even	2
528.4.b.a.65.2		2	132.131	odd	2

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	33.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.00000 + 3.31662i) q^{3} -8.00000 q^{4} +13.2665i q^{5} +(5.00000 - 26.5330i) q^{9} +O(q^{10})\)	\(q+(-4.00000 + 3.31662i) q^{3} -8.00000 q^{4} +13.2665i q^{5} +(5.00000 - 26.5330i) q^{9} +36.4829i q^{11} +(32.0000 - 26.5330i) q^{12} +(-44.0000 - 53.0660i) q^{15} +64.0000 q^{16} -106.132i q^{20} +192.364i q^{23} -51.0000 q^{25} +(68.0000 + 122.715i) q^{27} -340.000 q^{31} +(-121.000 - 145.931i) q^{33} +(-40.0000 + 212.264i) q^{36} +434.000 q^{37} -291.863i q^{44} +(352.000 + 66.3325i) q^{45} -643.425i q^{47} +(-256.000 + 212.264i) q^{48} +343.000 q^{49} -225.530i q^{53} -484.000 q^{55} +550.560i q^{59} +(352.000 + 424.528i) q^{60} -512.000 q^{64} +416.000 q^{67} +(-638.000 - 769.457i) q^{69} +1028.15i q^{71} +(204.000 - 169.148i) q^{75} +849.056i q^{80} +(-679.000 - 265.330i) q^{81} +132.665i q^{89} -1538.91i q^{92} +(1360.00 - 1127.65i) q^{93} -34.0000 q^{97} +(968.000 + 182.414i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{3} - 16 q^{4} + 10 q^{9}+O(q^{10}) \) 2 * q - 8 * q^3 - 16 * q^4 + 10 * q^9	\( 2 q - 8 q^{3} - 16 q^{4} + 10 q^{9} + 64 q^{12} - 88 q^{15} + 128 q^{16} - 102 q^{25} + 136 q^{27} - 680 q^{31} - 242 q^{33} - 80 q^{36} + 868 q^{37} + 704 q^{45} - 512 q^{48} + 686 q^{49} - 968 q^{55} + 704 q^{60} - 1024 q^{64} + 832 q^{67} - 1276 q^{69} + 408 q^{75} - 1358 q^{81} + 2720 q^{93} - 68 q^{97} + 1936 q^{99}+O(q^{100}) \) 2 * q - 8 * q^3 - 16 * q^4 + 10 * q^9 + 64 * q^12 - 88 * q^15 + 128 * q^16 - 102 * q^25 + 136 * q^27 - 680 * q^31 - 242 * q^33 - 80 * q^36 + 868 * q^37 + 704 * q^45 - 512 * q^48 + 686 * q^49 - 968 * q^55 + 704 * q^60 - 1024 * q^64 + 832 * q^67 - 1276 * q^69 + 408 * q^75 - 1358 * q^81 + 2720 * q^93 - 68 * q^97 + 1936 * q^99

Introduction

L-functions

Modular forms

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