LMFDB - Embedded newform 200.2.k.d.107.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,2,Mod(43,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("200.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	200.k (of order $4$, degree $2$, 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:	$1.59700804043$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			107.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	200.107
Dual form			200.2.k.d.43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.00000i) q^{2} +(2.00000 - 2.00000i) q^{3} +2.00000i q^{4} +4.00000 q^{6} +(-2.00000 + 2.00000i) q^{8} -5.00000i q^{9} -6.00000 q^{11} +(4.00000 + 4.00000i) q^{12} -4.00000 q^{16} +(4.00000 + 4.00000i) q^{17} +(5.00000 - 5.00000i) q^{18} -2.00000i q^{19} +(-6.00000 - 6.00000i) q^{22} +8.00000i q^{24} +(-4.00000 - 4.00000i) q^{27} +(-4.00000 - 4.00000i) q^{32} +(-12.0000 + 12.0000i) q^{33} +8.00000i q^{34} +10.0000 q^{36} +(2.00000 - 2.00000i) q^{38} -6.00000 q^{41} +(6.00000 - 6.00000i) q^{43} -12.0000i q^{44} +(-8.00000 + 8.00000i) q^{48} +7.00000i q^{49} +16.0000 q^{51} -8.00000i q^{54} +(-4.00000 - 4.00000i) q^{57} -6.00000i q^{59} -8.00000i q^{64} -24.0000 q^{66} +(-6.00000 - 6.00000i) q^{67} +(-8.00000 + 8.00000i) q^{68} +(10.0000 + 10.0000i) q^{72} +(12.0000 - 12.0000i) q^{73} +4.00000 q^{76} -1.00000 q^{81} +(-6.00000 - 6.00000i) q^{82} +(2.00000 - 2.00000i) q^{83} +12.0000 q^{86} +(12.0000 - 12.0000i) q^{88} +18.0000i q^{89} -16.0000 q^{96} +(12.0000 + 12.0000i) q^{97} +(-7.00000 + 7.00000i) q^{98} +30.0000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{3} + 8 q^{6} - 4 q^{8} - 12 q^{11} + 8 q^{12} - 8 q^{16} + 8 q^{17} + 10 q^{18} - 12 q^{22} - 8 q^{27} - 8 q^{32} - 24 q^{33} + 20 q^{36} + 4 q^{38} - 12 q^{41} + 12 q^{43} - 16 q^{48}+ \cdots - 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^3 + 8 * q^6 - 4 * q^8 - 12 * q^11 + 8 * q^12 - 8 * q^16 + 8 * q^17 + 10 * q^18 - 12 * q^22 - 8 * q^27 - 8 * q^32 - 24 * q^33 + 20 * q^36 + 4 * q^38 - 12 * q^41 + 12 * q^43 - 16 * q^48 + 32 * q^51 - 8 * q^57 - 48 * q^66 - 12 * q^67 - 16 * q^68 + 20 * q^72 + 24 * q^73 + 8 * q^76 - 2 * q^81 - 12 * q^82 + 4 * q^83 + 24 * q^86 + 24 * q^88 - 32 * q^96 + 24 * q^97 - 14 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$3$	2.00000	−	2.00000i	1.15470	−	1.15470i	0.169102	−	0.985599i	$-0.445913\pi$
$3$	2.00000	−	2.00000i	1.15470	−	1.15470i	0.985599	−	0.169102i	$-0.0540867\pi$
$4$			2.00000i			1.00000i
$5$		0			0
$5$		0			0
$6$	4.00000			1.63299
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	−2.00000	+	2.00000i	−0.707107	+	0.707107i
$9$		−	5.00000i		−	1.66667i
$10$		0			0
$11$	−6.00000			−1.80907			−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000			−1.80907			−0.904534	+	0.426401i	$0.859781\pi$
$12$	4.00000	+	4.00000i	1.15470	+	1.15470i
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	4.00000	+	4.00000i	0.970143	+	0.970143i	0.999567	−	0.0294245i	$-0.00936746\pi$
$17$	4.00000	+	4.00000i	0.970143	+	0.970143i	−0.0294245	+	0.999567i	$0.509367\pi$
$18$	5.00000	−	5.00000i	1.17851	−	1.17851i
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$		0			0
$22$	−6.00000	−	6.00000i	−1.27920	−	1.27920i
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$			8.00000i			1.63299i
$25$		0			0
$26$		0			0
$27$	−4.00000	−	4.00000i	−0.769800	−	0.769800i
$28$		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			$\pi$
$32$	−4.00000	−	4.00000i	−0.707107	−	0.707107i
$33$	−12.0000	+	12.0000i	−2.08893	+	2.08893i
$34$			8.00000i			1.37199i
$35$		0			0
$36$	10.0000			1.66667
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$	2.00000	−	2.00000i	0.324443	−	0.324443i
$39$		0			0
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	6.00000	−	6.00000i	0.914991	−	0.914991i	−0.0816682	−	0.996660i	$-0.526025\pi$
$43$	6.00000	−	6.00000i	0.914991	−	0.914991i	0.996660	+	0.0816682i	$0.0260248\pi$
$44$		−	12.0000i		−	1.80907i
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	−8.00000	+	8.00000i	−1.15470	+	1.15470i
$49$			7.00000i			1.00000i
$50$		0			0
$51$	16.0000			2.24045
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		−	8.00000i		−	1.08866i
$55$		0			0
$56$		0			0
$57$	−4.00000	−	4.00000i	−0.529813	−	0.529813i
$58$		0			0
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$	−24.0000			−2.95420
$67$	−6.00000	−	6.00000i	−0.733017	−	0.733017i	0.238200	−	0.971216i	$-0.423443\pi$
$67$	−6.00000	−	6.00000i	−0.733017	−	0.733017i	−0.971216	+	0.238200i	$0.923443\pi$
$68$	−8.00000	+	8.00000i	−0.970143	+	0.970143i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	10.0000	+	10.0000i	1.17851	+	1.17851i
$73$	12.0000	−	12.0000i	1.40449	−	1.40449i	0.619486	−	0.785007i	$-0.287341\pi$
$73$	12.0000	−	12.0000i	1.40449	−	1.40449i	0.785007	−	0.619486i	$-0.212659\pi$
$74$		0			0
$75$		0			0
$76$	4.00000			0.458831
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$	−6.00000	−	6.00000i	−0.662589	−	0.662589i
$83$	2.00000	−	2.00000i	0.219529	−	0.219529i	−0.588771	−	0.808300i	$-0.700388\pi$
$83$	2.00000	−	2.00000i	0.219529	−	0.219529i	0.808300	+	0.588771i	$0.200388\pi$
$84$		0			0
$85$		0			0
$86$	12.0000			1.29399
$87$		0			0
$88$	12.0000	−	12.0000i	1.27920	−	1.27920i
$89$			18.0000i			1.90800i	0.299813	+	0.953998i	$0.403076\pi$
$89$			18.0000i			1.90800i	−0.299813	+	0.953998i	$0.596924\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−16.0000			−1.63299
$97$	12.0000	+	12.0000i	1.21842	+	1.21842i	0.968187	+	0.250229i	$0.0805058\pi$
$97$	12.0000	+	12.0000i	1.21842	+	1.21842i	0.250229	+	0.968187i	$0.419494\pi$
$98$	−7.00000	+	7.00000i	−0.707107	+	0.707107i
$99$			30.0000i			3.01511i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	16.0000	+	16.0000i	1.58424	+	1.58424i
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.0000	+	14.0000i	1.35343	+	1.35343i	0.881791	+	0.471640i	$0.156338\pi$
$107$	14.0000	+	14.0000i	1.35343	+	1.35343i	0.471640	+	0.881791i	$0.343662\pi$
$108$	8.00000	−	8.00000i	0.769800	−	0.769800i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.00000	+	8.00000i	−0.752577	+	0.752577i	−0.974959	−	0.222383i	$-0.928617\pi$
$113$	−8.00000	+	8.00000i	−0.752577	+	0.752577i	0.222383	+	0.974959i	$0.428617\pi$
$114$		−	8.00000i		−	0.749269i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	6.00000	−	6.00000i	0.552345	−	0.552345i
$119$		0			0
$120$		0			0
$121$	25.0000			2.27273
$122$		0			0
$123$	−12.0000	+	12.0000i	−1.08200	+	1.08200i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	8.00000	−	8.00000i	0.707107	−	0.707107i
$129$		−	24.0000i		−	2.11308i
$130$		0			0
$131$	−18.0000			−1.57267			−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000			−1.57267			−0.786334	+	0.617802i	$0.788023\pi$
$132$	−24.0000	−	24.0000i	−2.08893	−	2.08893i
$133$		0			0
$134$		−	12.0000i		−	1.03664i
$135$		0			0
$136$	−16.0000			−1.37199
$137$	−16.0000	−	16.0000i	−1.36697	−	1.36697i	−0.864723	−	0.502249i	$-0.832506\pi$
$137$	−16.0000	−	16.0000i	−1.36697	−	1.36697i	−0.502249	−	0.864723i	$-0.667494\pi$
$138$		0			0
$139$		−	22.0000i		−	1.86602i	−0.359856	−	0.933008i	$-0.617174\pi$
$139$		−	22.0000i		−	1.86602i	0.359856	−	0.933008i	$-0.382826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			20.0000i			1.66667i
$145$		0			0
$146$	24.0000			1.98625
$147$	14.0000	+	14.0000i	1.15470	+	1.15470i
$148$		0			0
$149$		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			$\pi$
$152$	4.00000	+	4.00000i	0.324443	+	0.324443i
$153$	20.0000	−	20.0000i	1.61690	−	1.61690i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−1.00000	−	1.00000i	−0.0785674	−	0.0785674i
$163$	−18.0000	+	18.0000i	−1.40987	+	1.40987i	−0.649550	+	0.760319i	$0.725042\pi$
$163$	−18.0000	+	18.0000i	−1.40987	+	1.40987i	−0.760319	+	0.649550i	$0.774958\pi$
$164$		−	12.0000i		−	0.937043i
$165$		0			0
$166$	4.00000			0.310460
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$	−10.0000			−0.764719
$172$	12.0000	+	12.0000i	0.914991	+	0.914991i
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$	24.0000			1.80907
$177$	−12.0000	−	12.0000i	−0.901975	−	0.901975i
$178$	−18.0000	+	18.0000i	−1.34916	+	1.34916i
$179$			18.0000i			1.34538i	0.739923	+	0.672692i	$0.234862\pi$
$179$			18.0000i			1.34538i	−0.739923	+	0.672692i	$0.765138\pi$
$180$		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
$187$	−24.0000	−	24.0000i	−1.75505	−	1.75505i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−16.0000	−	16.0000i	−1.15470	−	1.15470i
$193$	12.0000	−	12.0000i	0.863779	−	0.863779i	−0.127996	−	0.991775i	$-0.540854\pi$
$193$	12.0000	−	12.0000i	0.863779	−	0.863779i	0.991775	+	0.127996i	$0.0408544\pi$
$194$			24.0000i			1.72310i
$195$		0			0
$196$	−14.0000			−1.00000
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$	−30.0000	+	30.0000i	−2.13201	+	2.13201i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	−24.0000			−1.69283
$202$		0			0
$203$		0			0
$204$			32.0000i			2.24045i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			12.0000i			0.830057i
$210$		0			0
$211$	14.0000			0.963800			0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000			0.963800			0.481900	+	0.876226i	$0.339947\pi$
$212$		0			0
$213$		0			0
$214$			28.0000i			1.91404i
$215$		0			0
$216$	16.0000			1.08866
$217$		0			0
$218$		0			0
$219$		−	48.0000i		−	3.24354i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$		0			0
$226$	−16.0000			−1.06430
$227$	2.00000	+	2.00000i	0.132745	+	0.132745i	0.770357	−	0.637613i	$-0.220078\pi$
$227$	2.00000	+	2.00000i	0.132745	+	0.132745i	−0.637613	+	0.770357i	$0.720078\pi$
$228$	8.00000	−	8.00000i	0.529813	−	0.529813i
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.00000	+	4.00000i	−0.262049	+	0.262049i	−0.825886	−	0.563837i	$-0.809325\pi$
$233$	−4.00000	+	4.00000i	−0.262049	+	0.262049i	0.563837	+	0.825886i	$0.309325\pi$
$234$		0			0
$235$		0			0
$236$	12.0000			0.781133
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−26.0000			−1.67481			−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000			−1.67481			−0.837404	+	0.546585i	$0.815928\pi$
$242$	25.0000	+	25.0000i	1.60706	+	1.60706i
$243$	10.0000	−	10.0000i	0.641500	−	0.641500i
$244$		0			0
$245$		0			0
$246$	−24.0000			−1.53018
$247$		0			0
$248$		0			0
$249$		−	8.00000i		−	0.506979i
$250$		0			0
$251$	−6.00000			−0.378717			−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000			−0.378717			−0.189358	+	0.981908i	$0.560641\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−8.00000	−	8.00000i	−0.499026	−	0.499026i	0.412108	−	0.911135i	$-0.364792\pi$
$257$	−8.00000	−	8.00000i	−0.499026	−	0.499026i	−0.911135	+	0.412108i	$0.864792\pi$
$258$	24.0000	−	24.0000i	1.49417	−	1.49417i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−18.0000	−	18.0000i	−1.11204	−	1.11204i
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		−	48.0000i		−	2.95420i
$265$		0			0
$266$		0			0
$267$	36.0000	+	36.0000i	2.20316	+	2.20316i
$268$	12.0000	−	12.0000i	0.733017	−	0.733017i
$269$		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			$\pi$
$272$	−16.0000	−	16.0000i	−0.970143	−	0.970143i
$273$		0			0
$274$		−	32.0000i		−	1.93319i
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$	22.0000	−	22.0000i	1.31947	−	1.31947i
$279$		0			0
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	−18.0000	+	18.0000i	−1.06999	+	1.06999i	−0.0726300	+	0.997359i	$0.523139\pi$
$283$	−18.0000	+	18.0000i	−1.06999	+	1.06999i	−0.997359	+	0.0726300i	$0.976861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−20.0000	+	20.0000i	−1.17851	+	1.17851i
$289$			15.0000i			0.882353i
$290$		0			0
$291$	48.0000			2.81381
$292$	24.0000	+	24.0000i	1.40449	+	1.40449i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$			28.0000i			1.63299i
$295$		0			0
$296$		0			0
$297$	24.0000	+	24.0000i	1.39262	+	1.39262i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$			8.00000i			0.458831i
$305$		0			0
$306$	40.0000			2.28665
$307$	−6.00000	−	6.00000i	−0.342438	−	0.342438i	0.514845	−	0.857283i	$-0.327849\pi$
$307$	−6.00000	−	6.00000i	−0.342438	−	0.342438i	−0.857283	+	0.514845i	$0.827849\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−24.0000	+	24.0000i	−1.35656	+	1.35656i	−0.478440	+	0.878120i	$0.658798\pi$
$313$	−24.0000	+	24.0000i	−1.35656	+	1.35656i	−0.878120	+	0.478440i	$0.841202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	56.0000			3.12562
$322$		0			0
$323$	8.00000	−	8.00000i	0.445132	−	0.445132i
$324$		−	2.00000i		−	0.111111i
$325$		0			0
$326$	−36.0000			−1.99386
$327$		0			0
$328$	12.0000	−	12.0000i	0.662589	−	0.662589i
$329$		0			0
$330$		0			0
$331$	−26.0000			−1.42909			−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000			−1.42909			−0.714545	+	0.699590i	$0.753366\pi$
$332$	4.00000	+	4.00000i	0.219529	+	0.219529i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	24.0000	+	24.0000i	1.30736	+	1.30736i	0.923312	+	0.384052i	$0.125472\pi$
$337$	24.0000	+	24.0000i	1.30736	+	1.30736i	0.384052	+	0.923312i	$0.374528\pi$
$338$	13.0000	−	13.0000i	0.707107	−	0.707107i
$339$			32.0000i			1.73800i
$340$		0			0
$341$		0			0
$342$	−10.0000	−	10.0000i	−0.540738	−	0.540738i
$343$		0			0
$344$			24.0000i			1.29399i
$345$		0			0
$346$		0			0
$347$	−26.0000	−	26.0000i	−1.39575	−	1.39575i	−0.811755	−	0.583998i	$-0.801488\pi$
$347$	−26.0000	−	26.0000i	−1.39575	−	1.39575i	−0.583998	−	0.811755i	$-0.698512\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	24.0000	+	24.0000i	1.27920	+	1.27920i
$353$	16.0000	−	16.0000i	0.851594	−	0.851594i	−0.138735	−	0.990329i	$-0.544304\pi$
$353$	16.0000	−	16.0000i	0.851594	−	0.851594i	0.990329	+	0.138735i	$0.0443038\pi$
$354$		−	24.0000i		−	1.27559i
$355$		0			0
$356$	−36.0000			−1.90800
$357$		0			0
$358$	−18.0000	+	18.0000i	−0.951330	+	0.951330i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	50.0000	−	50.0000i	2.62432	−	2.62432i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$			30.0000i			1.56174i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		−	48.0000i		−	2.48202i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			38.0000i			1.95193i	0.217930	+	0.975964i	$0.430070\pi$
$379$			38.0000i			1.95193i	−0.217930	+	0.975964i	$0.569930\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		−	32.0000i		−	1.63299i
$385$		0			0
$386$	24.0000			1.22157
$387$	−30.0000	−	30.0000i	−1.52499	−	1.52499i
$388$	−24.0000	+	24.0000i	−1.21842	+	1.21842i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	−14.0000	−	14.0000i	−0.707107	−	0.707107i
$393$	−36.0000	+	36.0000i	−1.81596	+	1.81596i
$394$		0			0
$395$		0			0
$396$	−60.0000			−3.01511
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.00000			−0.299626			−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000			−0.299626			−0.149813	+	0.988714i	$0.547867\pi$
$402$	−24.0000	−	24.0000i	−1.19701	−	1.19701i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−32.0000	+	32.0000i	−1.58424	+	1.58424i
$409$		−	22.0000i		−	1.08783i	−0.839140	−	0.543915i	$-0.816941\pi$
$409$		−	22.0000i		−	1.08783i	0.839140	−	0.543915i	$-0.183059\pi$
$410$		0			0
$411$	−64.0000			−3.15689
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−44.0000	−	44.0000i	−2.15469	−	2.15469i
$418$	−12.0000	+	12.0000i	−0.586939	+	0.586939i
$419$			18.0000i			0.879358i	0.898155	+	0.439679i	$0.144908\pi$
$419$			18.0000i			0.879358i	−0.898155	+	0.439679i	$0.855092\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	14.0000	+	14.0000i	0.681509	+	0.681509i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−28.0000	+	28.0000i	−1.35343	+	1.35343i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	16.0000	+	16.0000i	0.769800	+	0.769800i
$433$	12.0000	−	12.0000i	0.576683	−	0.576683i	−0.357305	−	0.933988i	$-0.616304\pi$
$433$	12.0000	−	12.0000i	0.576683	−	0.576683i	0.933988	+	0.357305i	$0.116304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	48.0000	−	48.0000i	2.29353	−	2.29353i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	35.0000			1.66667
$442$		0			0
$443$	2.00000	−	2.00000i	0.0950229	−	0.0950229i	−0.657997	−	0.753020i	$-0.728596\pi$
$443$	2.00000	−	2.00000i	0.0950229	−	0.0950229i	0.753020	+	0.657997i	$0.228596\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	42.0000i		−	1.98210i	−0.133482	−	0.991051i	$-0.542616\pi$
$449$		−	42.0000i		−	1.98210i	0.133482	−	0.991051i	$-0.457384\pi$
$450$		0			0
$451$	36.0000			1.69517
$452$	−16.0000	−	16.0000i	−0.752577	−	0.752577i
$453$		0			0
$454$			4.00000i			0.187729i
$455$		0			0
$456$	16.0000			0.749269
$457$	24.0000	+	24.0000i	1.12267	+	1.12267i	0.991338	+	0.131335i	$0.0419262\pi$
$457$	24.0000	+	24.0000i	1.12267	+	1.12267i	0.131335	+	0.991338i	$0.458074\pi$
$458$		0			0
$459$		−	32.0000i		−	1.49363i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	−8.00000			−0.370593
$467$	22.0000	+	22.0000i	1.01804	+	1.01804i	0.999834	+	0.0182043i	$0.00579493\pi$
$467$	22.0000	+	22.0000i	1.01804	+	1.01804i	0.0182043	+	0.999834i	$0.494205\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	12.0000	+	12.0000i	0.552345	+	0.552345i
$473$	−36.0000	+	36.0000i	−1.65528	+	1.65528i
$474$		0			0
$475$		0			0
$476$		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
$482$	−26.0000	−	26.0000i	−1.18427	−	1.18427i
$483$		0			0
$484$			50.0000i			2.27273i
$485$		0			0
$486$	20.0000			0.907218
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$		0			0
$489$			72.0000i			3.25595i
$490$		0			0
$491$	42.0000			1.89543			0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000			1.89543			0.947717	+	0.319113i	$0.103385\pi$
$492$	−24.0000	−	24.0000i	−1.08200	−	1.08200i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	8.00000	−	8.00000i	0.358489	−	0.358489i
$499$			14.0000i			0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$		0			0
$501$		0			0
$502$	−6.00000	−	6.00000i	−0.267793	−	0.267793i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−26.0000	−	26.0000i	−1.15470	−	1.15470i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	+	16.0000i	0.707107	+	0.707107i
$513$	−8.00000	+	8.00000i	−0.353209	+	0.353209i
$514$		−	16.0000i		−	0.705730i
$515$		0			0
$516$	48.0000			2.11308
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$	−18.0000	+	18.0000i	−0.787085	+	0.787085i	−0.981015	−	0.193930i	$-0.937876\pi$
$523$	−18.0000	+	18.0000i	−0.787085	+	0.787085i	0.193930	+	0.981015i	$0.437876\pi$
$524$		−	36.0000i		−	1.57267i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	48.0000	−	48.0000i	2.08893	−	2.08893i
$529$		−	23.0000i		−	1.00000i
$530$		0			0
$531$	−30.0000			−1.30189
$532$		0			0
$533$		0			0
$534$			72.0000i			3.11574i
$535$		0			0
$536$	24.0000			1.03664
$537$	36.0000	+	36.0000i	1.55351	+	1.55351i
$538$		0			0
$539$		−	42.0000i		−	1.80907i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		−	32.0000i		−	1.37199i
$545$		0			0
$546$		0			0
$547$	−6.00000	−	6.00000i	−0.256541	−	0.256541i	0.567104	−	0.823646i	$-0.308064\pi$
$547$	−6.00000	−	6.00000i	−0.256541	−	0.256541i	−0.823646	+	0.567104i	$0.808064\pi$
$548$	32.0000	−	32.0000i	1.36697	−	1.36697i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	44.0000			1.86602
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−96.0000			−4.05312
$562$	−18.0000	−	18.0000i	−0.759284	−	0.759284i
$563$	26.0000	−	26.0000i	1.09577	−	1.09577i	0.100870	−	0.994900i	$-0.467837\pi$
$563$	26.0000	−	26.0000i	1.09577	−	1.09577i	0.994900	−	0.100870i	$-0.0321625\pi$
$564$		0			0
$565$		0			0
$566$	−36.0000			−1.51319
$567$		0			0
$568$		0			0
$569$		−	42.0000i		−	1.76073i	−0.474295	−	0.880366i	$-0.657297\pi$
$569$		−	42.0000i		−	1.76073i	0.474295	−	0.880366i	$-0.342703\pi$
$570$		0			0
$571$	22.0000			0.920671			0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000			0.920671			0.460336	+	0.887745i	$0.347729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−40.0000			−1.66667
$577$	24.0000	+	24.0000i	0.999133	+	0.999133i	1.00000		0.000866551i	$-0.000275832\pi$
$577$	24.0000	+	24.0000i	0.999133	+	0.999133i	−0.000866551		1.00000i	$0.500276\pi$
$578$	−15.0000	+	15.0000i	−0.623918	+	0.623918i
$579$		−	48.0000i		−	1.99481i
$580$		0			0
$581$		0			0
$582$	48.0000	+	48.0000i	1.98966	+	1.98966i
$583$		0			0
$584$			48.0000i			1.98625i
$585$		0			0
$586$		0			0
$587$	34.0000	+	34.0000i	1.40333	+	1.40333i	0.789221	+	0.614109i	$0.210484\pi$
$587$	34.0000	+	34.0000i	1.40333	+	1.40333i	0.614109	+	0.789221i	$0.289516\pi$
$588$	−28.0000	+	28.0000i	−1.15470	+	1.15470i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	32.0000	−	32.0000i	1.31408	−	1.31408i	0.395705	−	0.918378i	$-0.370500\pi$
$593$	32.0000	−	32.0000i	1.31408	−	1.31408i	0.918378	−	0.395705i	$-0.129500\pi$
$594$			48.0000i			1.96946i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−46.0000			−1.87638			−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000			−1.87638			−0.938190	+	0.346122i	$0.887498\pi$
$602$		0			0
$603$	−30.0000	+	30.0000i	−1.22169	+	1.22169i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$	−8.00000	+	8.00000i	−0.324443	+	0.324443i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	40.0000	+	40.0000i	1.61690	+	1.61690i
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		−	12.0000i		−	0.484281i
$615$		0			0
$616$		0			0
$617$	−28.0000	−	28.0000i	−1.12724	−	1.12724i	−0.990624	−	0.136613i	$-0.956378\pi$
$617$	−28.0000	−	28.0000i	−1.12724	−	1.12724i	−0.136613	−	0.990624i	$-0.543622\pi$
$618$		0			0
$619$		−	26.0000i		−	1.04503i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$		−	26.0000i		−	1.04503i	0.852631	−	0.522514i	$-0.175006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$	−48.0000			−1.91847
$627$	24.0000	+	24.0000i	0.958468	+	0.958468i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	28.0000	−	28.0000i	1.11290	−	1.11290i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	42.0000			1.65890			0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000			1.65890			0.829450	+	0.558581i	$0.188654\pi$
$642$	56.0000	+	56.0000i	2.21014	+	2.21014i
$643$	6.00000	−	6.00000i	0.236617	−	0.236617i	−0.578831	−	0.815448i	$-0.696491\pi$
$643$	6.00000	−	6.00000i	0.236617	−	0.236617i	0.815448	+	0.578831i	$0.196491\pi$
$644$		0			0
$645$		0			0
$646$	16.0000			0.629512
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	2.00000	−	2.00000i	0.0785674	−	0.0785674i
$649$			36.0000i			1.41312i
$650$		0			0
$651$		0			0
$652$	−36.0000	−	36.0000i	−1.40987	−	1.40987i
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$	24.0000			0.937043
$657$	−60.0000	−	60.0000i	−2.34082	−	2.34082i
$658$		0			0
$659$			18.0000i			0.701180i	0.936529	+	0.350590i	$0.114019\pi$
$659$			18.0000i			0.701180i	−0.936529	+	0.350590i	$0.885981\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−26.0000	−	26.0000i	−1.01052	−	1.01052i
$663$		0			0
$664$			8.00000i			0.310460i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	36.0000	−	36.0000i	1.38770	−	1.38770i	0.557564	−	0.830134i	$-0.311736\pi$
$673$	36.0000	−	36.0000i	1.38770	−	1.38770i	0.830134	−	0.557564i	$-0.188264\pi$
$674$			48.0000i			1.84889i
$675$		0			0
$676$	26.0000			1.00000
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$	−32.0000	+	32.0000i	−1.22895	+	1.22895i
$679$		0			0
$680$		0			0
$681$	8.00000			0.306561
$682$		0			0
$683$	22.0000	−	22.0000i	0.841807	−	0.841807i	−0.147287	−	0.989094i	$-0.547054\pi$
$683$	22.0000	−	22.0000i	0.841807	−	0.841807i	0.989094	+	0.147287i	$0.0470541\pi$
$684$		−	20.0000i		−	0.764719i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−24.0000	+	24.0000i	−0.914991	+	0.914991i
$689$		0			0
$690$		0			0
$691$	−46.0000			−1.74992			−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−46.0000			−1.74992			−0.874961	+	0.484193i	$0.839113\pi$
$692$		0			0
$693$		0			0
$694$		−	52.0000i		−	1.97389i
$695$		0			0
$696$		0			0
$697$	−24.0000	−	24.0000i	−0.909065	−	0.909065i
$698$		0			0
$699$			16.0000i			0.605176i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$			48.0000i			1.80907i
$705$		0			0
$706$	32.0000			1.20434
$707$		0			0
$708$	24.0000	−	24.0000i	0.901975	−	0.901975i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$	−36.0000	−	36.0000i	−1.34916	−	1.34916i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−36.0000			−1.34538
$717$		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
$722$	15.0000	+	15.0000i	0.558242	+	0.558242i
$723$	−52.0000	+	52.0000i	−1.93390	+	1.93390i
$724$		0			0
$725$		0			0
$726$	100.000			3.71135
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$		−	43.0000i		−	1.59259i
$730$		0			0
$731$	48.0000			1.77534
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	36.0000	+	36.0000i	1.32608	+	1.32608i
$738$	−30.0000	+	30.0000i	−1.10432	+	1.10432i
$739$			34.0000i			1.25071i	0.780340	+	0.625355i	$0.215046\pi$
$739$			34.0000i			1.25071i	−0.780340	+	0.625355i	$0.784954\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−10.0000	−	10.0000i	−0.365881	−	0.365881i
$748$	48.0000	−	48.0000i	1.75505	−	1.75505i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−12.0000	+	12.0000i	−0.437304	+	0.437304i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	−38.0000	+	38.0000i	−1.38022	+	1.38022i
$759$		0			0
$760$		0			0
$761$	54.0000			1.95750			0.978749	−	0.205061i	$-0.0657392\pi$
$761$	54.0000			1.95750			0.978749	+	0.205061i	$0.0657392\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	32.0000	−	32.0000i	1.15470	−	1.15470i
$769$		−	22.0000i		−	0.793340i	−0.917961	−	0.396670i	$-0.870166\pi$
$769$		−	22.0000i		−	0.793340i	0.917961	−	0.396670i	$-0.129834\pi$
$770$		0			0
$771$	−32.0000			−1.15245
$772$	24.0000	+	24.0000i	0.863779	+	0.863779i
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		−	60.0000i		−	2.15666i
$775$		0			0
$776$	−48.0000			−1.72310
$777$		0			0
$778$		0			0
$779$			12.0000i			0.429945i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	28.0000i		−	1.00000i
$785$		0			0
$786$	−72.0000			−2.56815
$787$	−18.0000	−	18.0000i	−0.641631	−	0.641631i	0.309326	−	0.950956i	$-0.399897\pi$
$787$	−18.0000	−	18.0000i	−0.641631	−	0.641631i	−0.950956	+	0.309326i	$0.899897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−60.0000	−	60.0000i	−2.13201	−	2.13201i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	90.0000			3.17999
$802$	−6.00000	−	6.00000i	−0.211867	−	0.211867i
$803$	−72.0000	+	72.0000i	−2.54082	+	2.54082i
$804$		−	48.0000i		−	1.69283i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	6.00000i		−	0.210949i	−0.994422	−	0.105474i	$-0.966364\pi$
$809$		−	6.00000i		−	0.210949i	0.994422	−	0.105474i	$-0.0336361\pi$
$810$		0			0
$811$	−38.0000			−1.33436			−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000			−1.33436			−0.667180	+	0.744896i	$0.732499\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−64.0000			−2.24045
$817$	−12.0000	−	12.0000i	−0.419827	−	0.419827i
$818$	22.0000	−	22.0000i	0.769212	−	0.769212i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−64.0000	−	64.0000i	−2.23226	−	2.23226i
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14.0000	+	14.0000i	0.486828	+	0.486828i	0.907304	−	0.420476i	$-0.138137\pi$
$827$	14.0000	+	14.0000i	0.486828	+	0.486828i	−0.420476	+	0.907304i	$0.638137\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−28.0000	+	28.0000i	−0.970143	+	0.970143i
$834$		−	88.0000i		−	3.04719i
$835$		0			0
$836$	−24.0000			−0.830057
$837$		0			0
$838$	−18.0000	+	18.0000i	−0.621800	+	0.621800i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	−36.0000	+	36.0000i	−1.23991	+	1.23991i
$844$			28.0000i			0.963800i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$			72.0000i			2.47103i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$	−56.0000			−1.91404
$857$	−16.0000	−	16.0000i	−0.546550	−	0.546550i	0.378892	−	0.925441i	$-0.376305\pi$
$857$	−16.0000	−	16.0000i	−0.546550	−	0.546550i	−0.925441	+	0.378892i	$0.876305\pi$
$858$		0			0
$859$			58.0000i			1.97893i	0.144757	+	0.989467i	$0.453760\pi$
$859$			58.0000i			1.97893i	−0.144757	+	0.989467i	$0.546240\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$			32.0000i			1.08866i
$865$		0			0
$866$	24.0000			0.815553
$867$	30.0000	+	30.0000i	1.01885	+	1.01885i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	60.0000	−	60.0000i	2.03069	−	2.03069i
$874$		0			0
$875$		0			0
$876$	96.0000			3.24354
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$	35.0000	+	35.0000i	1.17851	+	1.17851i
$883$	42.0000	−	42.0000i	1.41341	−	1.41341i	0.682910	−	0.730502i	$-0.260714\pi$
$883$	42.0000	−	42.0000i	1.41341	−	1.41341i	0.730502	−	0.682910i	$-0.239286\pi$
$884$		0			0
$885$		0			0
$886$	4.00000			0.134383
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	6.00000			0.201008
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	42.0000	−	42.0000i	1.40156	−	1.40156i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	36.0000	+	36.0000i	1.19867	+	1.19867i
$903$		0			0
$904$		−	32.0000i		−	1.06430i
$905$		0			0
$906$		0			0
$907$	42.0000	+	42.0000i	1.39459	+	1.39459i	0.814689	+	0.579898i	$0.196908\pi$
$907$	42.0000	+	42.0000i	1.39459	+	1.39459i	0.579898	+	0.814689i	$0.303092\pi$
$908$	−4.00000	+	4.00000i	−0.132745	+	0.132745i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	16.0000	+	16.0000i	0.529813	+	0.529813i
$913$	−12.0000	+	12.0000i	−0.397142	+	0.397142i
$914$			48.0000i			1.58770i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	32.0000	−	32.0000i	1.05616	−	1.05616i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−24.0000			−0.790827
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			54.0000i			1.77168i	0.463988	+	0.885841i	$0.346418\pi$
$929$			54.0000i			1.77168i	−0.463988	+	0.885841i	$0.653582\pi$
$930$		0			0
$931$	14.0000			0.458831
$932$	−8.00000	−	8.00000i	−0.262049	−	0.262049i
$933$		0			0
$934$			44.0000i			1.43972i
$935$		0			0
$936$		0			0
$937$	−36.0000	−	36.0000i	−1.17607	−	1.17607i	−0.980737	−	0.195331i	$-0.937422\pi$
$937$	−36.0000	−	36.0000i	−1.17607	−	1.17607i	−0.195331	−	0.980737i	$-0.562578\pi$
$938$		0			0
$939$			96.0000i			3.13284i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$			24.0000i			0.781133i
$945$		0			0
$946$	−72.0000			−2.34092
$947$	−38.0000	−	38.0000i	−1.23483	−	1.23483i	−0.962085	−	0.272749i	$-0.912067\pi$
$947$	−38.0000	−	38.0000i	−1.23483	−	1.23483i	−0.272749	−	0.962085i	$-0.587933\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	32.0000	−	32.0000i	1.03658	−	1.03658i	0.0372767	−	0.999305i	$-0.488132\pi$
$953$	32.0000	−	32.0000i	1.03658	−	1.03658i	0.999305	−	0.0372767i	$-0.0118683\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	70.0000	−	70.0000i	2.25572	−	2.25572i
$964$		−	52.0000i		−	1.67481i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	−50.0000	+	50.0000i	−1.60706	+	1.60706i
$969$		−	32.0000i		−	1.02799i
$970$		0			0
$971$	54.0000			1.73294			0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000			1.73294			0.866471	+	0.499227i	$0.166383\pi$
$972$	20.0000	+	20.0000i	0.641500	+	0.641500i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	44.0000	+	44.0000i	1.40768	+	1.40768i	0.771709	+	0.635975i	$0.219402\pi$
$977$	44.0000	+	44.0000i	1.40768	+	1.40768i	0.635975	+	0.771709i	$0.280598\pi$
$978$	−72.0000	+	72.0000i	−2.30231	+	2.30231i
$979$		−	108.000i		−	3.45169i
$980$		0			0
$981$		0			0
$982$	42.0000	+	42.0000i	1.34027	+	1.34027i
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		−	48.0000i		−	1.53018i
$985$		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
$993$	−52.0000	+	52.0000i	−1.65017	+	1.65017i
$994$		0			0
$995$		0			0
$996$	16.0000			0.506979
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$	−14.0000	+	14.0000i	−0.443162	+	0.443162i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.2.k.d.107.1	yes	2
4.3	odd	2		800.2.o.a.207.1		2
5.2	odd	4		200.2.k.a.43.1	✓	2
5.3	odd	4	inner	200.2.k.d.43.1	yes	2
5.4	even	2		200.2.k.a.107.1	yes	2
8.3	odd	2	CM	200.2.k.d.107.1	yes	2
8.5	even	2		800.2.o.a.207.1		2
20.3	even	4		800.2.o.a.143.1		2
20.7	even	4		800.2.o.d.143.1		2
20.19	odd	2		800.2.o.d.207.1		2
40.3	even	4	inner	200.2.k.d.43.1	yes	2
40.13	odd	4		800.2.o.a.143.1		2
40.19	odd	2		200.2.k.a.107.1	yes	2
40.27	even	4		200.2.k.a.43.1	✓	2
40.29	even	2		800.2.o.d.207.1		2
40.37	odd	4		800.2.o.d.143.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.k.a.43.1	✓	2	5.2	odd	4
200.2.k.a.43.1	✓	2	40.27	even	4
200.2.k.a.107.1	yes	2	5.4	even	2
200.2.k.a.107.1	yes	2	40.19	odd	2
200.2.k.d.43.1	yes	2	5.3	odd	4	inner
200.2.k.d.43.1	yes	2	40.3	even	4	inner
200.2.k.d.107.1	yes	2	1.1	even	1	trivial
200.2.k.d.107.1	yes	2	8.3	odd	2	CM
800.2.o.a.143.1		2	20.3	even	4
800.2.o.a.143.1		2	40.13	odd	4
800.2.o.a.207.1		2	4.3	odd	2
800.2.o.a.207.1		2	8.5	even	2
800.2.o.d.143.1		2	20.7	even	4
800.2.o.d.143.1		2	40.37	odd	4
800.2.o.d.207.1		2	20.19	odd	2
800.2.o.d.207.1		2	40.29	even	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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	200.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{2} +(2.00000 - 2.00000i) q^{3} +2.00000i q^{4} +4.00000 q^{6} +(-2.00000 + 2.00000i) q^{8} -5.00000i q^{9} -6.00000 q^{11} +(4.00000 + 4.00000i) q^{12} -4.00000 q^{16} +(4.00000 + 4.00000i) q^{17} +(5.00000 - 5.00000i) q^{18} -2.00000i q^{19} +(-6.00000 - 6.00000i) q^{22} +8.00000i q^{24} +(-4.00000 - 4.00000i) q^{27} +(-4.00000 - 4.00000i) q^{32} +(-12.0000 + 12.0000i) q^{33} +8.00000i q^{34} +10.0000 q^{36} +(2.00000 - 2.00000i) q^{38} -6.00000 q^{41} +(6.00000 - 6.00000i) q^{43} -12.0000i q^{44} +(-8.00000 + 8.00000i) q^{48} +7.00000i q^{49} +16.0000 q^{51} -8.00000i q^{54} +(-4.00000 - 4.00000i) q^{57} -6.00000i q^{59} -8.00000i q^{64} -24.0000 q^{66} +(-6.00000 - 6.00000i) q^{67} +(-8.00000 + 8.00000i) q^{68} +(10.0000 + 10.0000i) q^{72} +(12.0000 - 12.0000i) q^{73} +4.00000 q^{76} -1.00000 q^{81} +(-6.00000 - 6.00000i) q^{82} +(2.00000 - 2.00000i) q^{83} +12.0000 q^{86} +(12.0000 - 12.0000i) q^{88} +18.0000i q^{89} -16.0000 q^{96} +(12.0000 + 12.0000i) q^{97} +(-7.00000 + 7.00000i) q^{98} +30.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{3} + 8 q^{6} - 4 q^{8} - 12 q^{11} + 8 q^{12} - 8 q^{16} + 8 q^{17} + 10 q^{18} - 12 q^{22} - 8 q^{27} - 8 q^{32} - 24 q^{33} + 20 q^{36} + 4 q^{38} - 12 q^{41} + 12 q^{43} - 16 q^{48}+ \cdots - 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^3 + 8 * q^6 - 4 * q^8 - 12 * q^11 + 8 * q^12 - 8 * q^16 + 8 * q^17 + 10 * q^18 - 12 * q^22 - 8 * q^27 - 8 * q^32 - 24 * q^33 + 20 * q^36 + 4 * q^38 - 12 * q^41 + 12 * q^43 - 16 * q^48 + 32 * q^51 - 8 * q^57 - 48 * q^66 - 12 * q^67 - 16 * q^68 + 20 * q^72 + 24 * q^73 + 8 * q^76 - 2 * q^81 - 12 * q^82 + 4 * q^83 + 24 * q^86 + 24 * q^88 - 32 * q^96 + 24 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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