LMFDB - Embedded newform 200.2.k.c.107.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [200,2,Mod(43,200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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.c.43.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(2.00000 - 2.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} +2.00000 q^{11} +(-4.00000 - 4.00000i) q^{13} -4.00000i q^{14} -4.00000 q^{16} +(3.00000 + 3.00000i) q^{18} +6.00000i q^{19} +(2.00000 - 2.00000i) q^{22} +(6.00000 + 6.00000i) q^{23} -8.00000 q^{26} +(-4.00000 - 4.00000i) q^{28} +(-4.00000 + 4.00000i) q^{32} +6.00000 q^{36} +(-8.00000 + 8.00000i) q^{37} +(6.00000 + 6.00000i) q^{38} +2.00000 q^{41} -4.00000i q^{44} +12.0000 q^{46} +(2.00000 - 2.00000i) q^{47} -1.00000i q^{49} +(-8.00000 + 8.00000i) q^{52} +(-4.00000 - 4.00000i) q^{53} -8.00000 q^{56} -14.0000i q^{59} +(6.00000 + 6.00000i) q^{63} +8.00000i q^{64} +(6.00000 - 6.00000i) q^{72} +16.0000i q^{74} +12.0000 q^{76} +(4.00000 - 4.00000i) q^{77} -9.00000 q^{81} +(2.00000 - 2.00000i) q^{82} +(-4.00000 - 4.00000i) q^{88} -14.0000i q^{89} -16.0000 q^{91} +(12.0000 - 12.0000i) q^{92} -4.00000i q^{94} +(-1.00000 - 1.00000i) q^{98} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{7} - 4 q^{8} + 4 q^{11} - 8 q^{13} - 8 q^{16} + 6 q^{18} + 4 q^{22} + 12 q^{23} - 16 q^{26} - 8 q^{28} - 8 q^{32} + 12 q^{36} - 16 q^{37} + 12 q^{38} + 4 q^{41} + 24 q^{46} + 4 q^{47}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^7 - 4 * q^8 + 4 * q^11 - 8 * q^13 - 8 * q^16 + 6 * q^18 + 4 * q^22 + 12 * q^23 - 16 * q^26 - 8 * q^28 - 8 * q^32 + 12 * q^36 - 16 * q^37 + 12 * q^38 + 4 * q^41 + 24 * q^46 + 4 * q^47 - 16 * q^52 - 8 * q^53 - 16 * q^56 + 12 * q^63 + 12 * q^72 + 24 * q^76 + 8 * q^77 - 18 * q^81 + 4 * q^82 - 8 * q^88 - 32 * q^91 + 24 * q^92 - 2 * 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
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	2.00000i		−	1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.00000	−	2.00000i	0.755929	−	0.755929i	−0.219650	−	0.975579i	$-0.570491\pi$
$7$	2.00000	−	2.00000i	0.755929	−	0.755929i	0.975579	+	0.219650i	$0.0704915\pi$
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$			3.00000i			1.00000i
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	−4.00000	−	4.00000i	−1.10940	−	1.10940i	−0.993229	−	0.116171i	$-0.962938\pi$
$13$	−4.00000	−	4.00000i	−1.10940	−	1.10940i	−0.116171	−	0.993229i	$-0.537062\pi$
$14$		−	4.00000i		−	1.06904i
$15$		0			0
$16$	−4.00000			−1.00000
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$	3.00000	+	3.00000i	0.707107	+	0.707107i
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$		0			0
$22$	2.00000	−	2.00000i	0.426401	−	0.426401i
$23$	6.00000	+	6.00000i	1.25109	+	1.25109i	0.955233	+	0.295853i	$0.0956039\pi$
$23$	6.00000	+	6.00000i	1.25109	+	1.25109i	0.295853	+	0.955233i	$0.404396\pi$
$24$		0			0
$25$		0			0
$26$	−8.00000			−1.56893
$27$		0			0
$28$	−4.00000	−	4.00000i	−0.755929	−	0.755929i
$29$		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
$33$		0			0
$34$		0			0
$35$		0			0
$36$	6.00000			1.00000
$37$	−8.00000	+	8.00000i	−1.31519	+	1.31519i	−0.397658	+	0.917534i	$0.630177\pi$
$37$	−8.00000	+	8.00000i	−1.31519	+	1.31519i	−0.917534	+	0.397658i	$0.869823\pi$
$38$	6.00000	+	6.00000i	0.973329	+	0.973329i
$39$		0			0
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		−	4.00000i		−	0.603023i
$45$		0			0
$46$	12.0000			1.76930
$47$	2.00000	−	2.00000i	0.291730	−	0.291730i	−0.546033	−	0.837763i	$-0.683863\pi$
$47$	2.00000	−	2.00000i	0.291730	−	0.291730i	0.837763	+	0.546033i	$0.183863\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$		0			0
$51$		0			0
$52$	−8.00000	+	8.00000i	−1.10940	+	1.10940i
$53$	−4.00000	−	4.00000i	−0.549442	−	0.549442i	0.376837	−	0.926279i	$-0.377012\pi$
$53$	−4.00000	−	4.00000i	−0.549442	−	0.549442i	−0.926279	+	0.376837i	$0.877012\pi$
$54$		0			0
$55$		0			0
$56$	−8.00000			−1.06904
$57$		0			0
$58$		0			0
$59$		−	14.0000i		−	1.82264i	−0.411693	−	0.911322i	$-0.635063\pi$
$59$		−	14.0000i		−	1.82264i	0.411693	−	0.911322i	$-0.364937\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	6.00000	+	6.00000i	0.755929	+	0.755929i
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	6.00000	−	6.00000i	0.707107	−	0.707107i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$			16.0000i			1.85996i
$75$		0			0
$76$	12.0000			1.37649
$77$	4.00000	−	4.00000i	0.455842	−	0.455842i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$	2.00000	−	2.00000i	0.220863	−	0.220863i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−4.00000	−	4.00000i	−0.426401	−	0.426401i
$89$		−	14.0000i		−	1.48400i	−0.670402	−	0.741999i	$-0.733878\pi$
$89$		−	14.0000i		−	1.48400i	0.670402	−	0.741999i	$-0.266122\pi$
$90$		0			0
$91$	−16.0000			−1.67726
$92$	12.0000	−	12.0000i	1.25109	−	1.25109i
$93$		0			0
$94$		−	4.00000i		−	0.412568i
$95$		0			0
$96$		0			0
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	−1.00000	−	1.00000i	−0.101015	−	0.101015i
$99$			6.00000i			0.603023i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−14.0000	−	14.0000i	−1.37946	−	1.37946i	−0.845525	−	0.533936i	$-0.820712\pi$
$103$	−14.0000	−	14.0000i	−1.37946	−	1.37946i	−0.533936	−	0.845525i	$-0.679288\pi$
$104$			16.0000i			1.56893i
$105$		0			0
$106$	−8.00000			−0.777029
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$	−8.00000	+	8.00000i	−0.755929	+	0.755929i
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	12.0000	−	12.0000i	1.10940	−	1.10940i
$118$	−14.0000	−	14.0000i	−1.28880	−	1.28880i
$119$		0			0
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	12.0000			1.06904
$127$	2.00000	−	2.00000i	0.177471	−	0.177471i	−0.612781	−	0.790253i	$-0.709949\pi$
$127$	2.00000	−	2.00000i	0.177471	−	0.177471i	0.790253	+	0.612781i	$0.209949\pi$
$128$	8.00000	+	8.00000i	0.707107	+	0.707107i
$129$		0			0
$130$		0			0
$131$	22.0000			1.92215			0.961074	−	0.276289i	$-0.0891049\pi$
$131$	22.0000			1.92215			0.961074	+	0.276289i	$0.0891049\pi$
$132$		0			0
$133$	12.0000	+	12.0000i	1.04053	+	1.04053i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$		−	14.0000i		−	1.18746i	−0.804663	−	0.593732i	$-0.797654\pi$
$139$		−	14.0000i		−	1.18746i	0.804663	−	0.593732i	$-0.202346\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.00000	−	8.00000i	−0.668994	−	0.668994i
$144$		−	12.0000i		−	1.00000i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	16.0000	+	16.0000i	1.31519	+	1.31519i
$149$		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			$\pi$
$152$	12.0000	−	12.0000i	0.973329	−	0.973329i
$153$		0			0
$154$		−	8.00000i		−	0.644658i
$155$		0			0
$156$		0			0
$157$	−8.00000	+	8.00000i	−0.638470	+	0.638470i	−0.950178	−	0.311708i	$-0.899099\pi$
$157$	−8.00000	+	8.00000i	−0.638470	+	0.638470i	0.311708	+	0.950178i	$0.399099\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	24.0000			1.89146
$162$	−9.00000	+	9.00000i	−0.707107	+	0.707107i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		−	4.00000i		−	0.312348i
$165$		0			0
$166$		0			0
$167$	−18.0000	+	18.0000i	−1.39288	+	1.39288i	−0.574089	+	0.818793i	$0.694644\pi$
$167$	−18.0000	+	18.0000i	−1.39288	+	1.39288i	−0.818793	+	0.574089i	$0.805356\pi$
$168$		0			0
$169$			19.0000i			1.46154i
$170$		0			0
$171$	−18.0000			−1.37649
$172$		0			0
$173$	16.0000	+	16.0000i	1.21646	+	1.21646i	0.968864	+	0.247593i	$0.0796397\pi$
$173$	16.0000	+	16.0000i	1.21646	+	1.21646i	0.247593	+	0.968864i	$0.420360\pi$
$174$		0			0
$175$		0			0
$176$	−8.00000			−0.603023
$177$		0			0
$178$	−14.0000	−	14.0000i	−1.04934	−	1.04934i
$179$			26.0000i			1.94333i	0.236360	+	0.971666i	$0.424046\pi$
$179$			26.0000i			1.94333i	−0.236360	+	0.971666i	$0.575954\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$	−16.0000	+	16.0000i	−1.18600	+	1.18600i
$183$		0			0
$184$		−	24.0000i		−	1.76930i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−4.00000	−	4.00000i	−0.291730	−	0.291730i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$	−2.00000			−0.142857
$197$	12.0000	−	12.0000i	0.854965	−	0.854965i	−0.135775	−	0.990740i	$-0.543352\pi$
$197$	12.0000	−	12.0000i	0.854965	−	0.854965i	0.990740	+	0.135775i	$0.0433525\pi$
$198$	6.00000	+	6.00000i	0.426401	+	0.426401i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−28.0000			−1.95085
$207$	−18.0000	+	18.0000i	−1.25109	+	1.25109i
$208$	16.0000	+	16.0000i	1.10940	+	1.10940i
$209$			12.0000i			0.830057i
$210$		0			0
$211$	22.0000			1.51454			0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000			1.51454			0.757271	+	0.653101i	$0.226532\pi$
$212$	−8.00000	+	8.00000i	−0.549442	+	0.549442i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−14.0000	−	14.0000i	−0.937509	−	0.937509i	0.0606498	−	0.998159i	$-0.480683\pi$
$223$	−14.0000	−	14.0000i	−0.937509	−	0.937509i	−0.998159	+	0.0606498i	$0.980683\pi$
$224$			16.0000i			1.06904i
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		−	24.0000i		−	1.56893i
$235$		0			0
$236$	−28.0000			−1.82264
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$	−7.00000	+	7.00000i	−0.449977	+	0.449977i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	24.0000	−	24.0000i	1.52708	−	1.52708i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.00000			0.126239			0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000			0.126239			0.0631194	+	0.998006i	$0.479895\pi$
$252$	12.0000	−	12.0000i	0.755929	−	0.755929i
$253$	12.0000	+	12.0000i	0.754434	+	0.754434i
$254$		−	4.00000i		−	0.250982i
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$			32.0000i			1.98838i
$260$		0			0
$261$		0			0
$262$	22.0000	−	22.0000i	1.35916	−	1.35916i
$263$	6.00000	+	6.00000i	0.369976	+	0.369976i	0.867468	−	0.497492i	$-0.165746\pi$
$263$	6.00000	+	6.00000i	0.369976	+	0.369976i	−0.497492	+	0.867468i	$0.665746\pi$
$264$		0			0
$265$		0			0
$266$	24.0000			1.47153
$267$		0			0
$268$		0			0
$269$		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			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−8.00000	+	8.00000i	−0.480673	+	0.480673i	−0.905347	−	0.424673i	$-0.860389\pi$
$277$	−8.00000	+	8.00000i	−0.480673	+	0.480673i	0.424673	+	0.905347i	$0.360389\pi$
$278$	−14.0000	−	14.0000i	−0.839664	−	0.839664i
$279$		0			0
$280$		0			0
$281$	22.0000			1.31241			0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000			1.31241			0.656205	+	0.754583i	$0.272161\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$	−16.0000			−0.946100
$287$	4.00000	−	4.00000i	0.236113	−	0.236113i
$288$	−12.0000	−	12.0000i	−0.707107	−	0.707107i
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.0000	−	24.0000i	−1.40209	−	1.40209i	−0.793419	−	0.608676i	$-0.791701\pi$
$293$	−24.0000	−	24.0000i	−1.40209	−	1.40209i	−0.608676	−	0.793419i	$-0.708299\pi$
$294$		0			0
$295$		0			0
$296$	32.0000			1.85996
$297$		0			0
$298$		0			0
$299$		−	48.0000i		−	2.77591i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		−	24.0000i		−	1.37649i
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$	−8.00000	−	8.00000i	−0.455842	−	0.455842i
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$			16.0000i			0.902932i
$315$		0			0
$316$		0			0
$317$	12.0000	−	12.0000i	0.673987	−	0.673987i	−0.284646	−	0.958633i	$-0.591876\pi$
$317$	12.0000	−	12.0000i	0.673987	−	0.673987i	0.958633	+	0.284646i	$0.0918759\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	24.0000	−	24.0000i	1.33747	−	1.33747i
$323$		0			0
$324$			18.0000i			1.00000i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−4.00000	−	4.00000i	−0.220863	−	0.220863i
$329$		−	8.00000i		−	0.441054i
$330$		0			0
$331$	−18.0000			−0.989369			−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000			−0.989369			−0.494685	+	0.869072i	$0.664716\pi$
$332$		0			0
$333$	−24.0000	−	24.0000i	−1.31519	−	1.31519i
$334$			36.0000i			1.96983i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	19.0000	+	19.0000i	1.03346	+	1.03346i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−18.0000	+	18.0000i	−0.973329	+	0.973329i
$343$	12.0000	+	12.0000i	0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$	32.0000			1.72033
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	−8.00000	+	8.00000i	−0.426401	+	0.426401i
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$	−28.0000			−1.48400
$357$		0			0
$358$	26.0000	+	26.0000i	1.37414	+	1.37414i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$		0			0
$364$			32.0000i			1.67726i
$365$		0			0
$366$		0			0
$367$	22.0000	−	22.0000i	1.14839	−	1.14839i	0.161521	−	0.986869i	$-0.448360\pi$
$367$	22.0000	−	22.0000i	1.14839	−	1.14839i	0.986869	−	0.161521i	$-0.0516401\pi$
$368$	−24.0000	−	24.0000i	−1.25109	−	1.25109i
$369$			6.00000i			0.312348i
$370$		0			0
$371$	−16.0000			−0.830679
$372$		0			0
$373$	16.0000	+	16.0000i	0.828449	+	0.828449i	0.987302	−	0.158854i	$-0.0507798\pi$
$373$	16.0000	+	16.0000i	0.828449	+	0.828449i	−0.158854	+	0.987302i	$0.550780\pi$
$374$		0			0
$375$		0			0
$376$	−8.00000			−0.412568
$377$		0			0
$378$		0			0
$379$		−	34.0000i		−	1.74646i	−0.487306	−	0.873231i	$-0.662020\pi$
$379$		−	34.0000i		−	1.74646i	0.487306	−	0.873231i	$-0.337980\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	26.0000	+	26.0000i	1.32854	+	1.32854i	0.906646	+	0.421892i	$0.138634\pi$
$383$	26.0000	+	26.0000i	1.32854	+	1.32854i	0.421892	+	0.906646i	$0.361366\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	−2.00000	+	2.00000i	−0.101015	+	0.101015i
$393$		0			0
$394$		−	24.0000i		−	1.20910i
$395$		0			0
$396$	12.0000			0.603023
$397$	−28.0000	+	28.0000i	−1.40528	+	1.40528i	−0.623285	+	0.781995i	$0.714202\pi$
$397$	−28.0000	+	28.0000i	−1.40528	+	1.40528i	−0.781995	+	0.623285i	$0.785798\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−38.0000			−1.89763			−0.948815	−	0.315833i	$-0.897716\pi$
$401$	−38.0000			−1.89763			−0.948815	+	0.315833i	$0.897716\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−16.0000	+	16.0000i	−0.793091	+	0.793091i
$408$		0			0
$409$		−	14.0000i		−	0.692255i	−0.938187	−	0.346128i	$-0.887496\pi$
$409$		−	14.0000i		−	0.692255i	0.938187	−	0.346128i	$-0.112504\pi$
$410$		0			0
$411$		0			0
$412$	−28.0000	+	28.0000i	−1.37946	+	1.37946i
$413$	−28.0000	−	28.0000i	−1.37779	−	1.37779i
$414$			36.0000i			1.76930i
$415$		0			0
$416$	32.0000			1.56893
$417$		0			0
$418$	12.0000	+	12.0000i	0.586939	+	0.586939i
$419$			26.0000i			1.27018i	0.772437	+	0.635092i	$0.219038\pi$
$419$			26.0000i			1.27018i	−0.772437	+	0.635092i	$0.780962\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	22.0000	−	22.0000i	1.07094	−	1.07094i
$423$	6.00000	+	6.00000i	0.291730	+	0.291730i
$424$			16.0000i			0.777029i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−36.0000	+	36.0000i	−1.72211	+	1.72211i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$		0			0
$446$	−28.0000			−1.32584
$447$		0			0
$448$	16.0000	+	16.0000i	0.755929	+	0.755929i
$449$		−	34.0000i		−	1.60456i	−0.596948	−	0.802280i	$-0.703620\pi$
$449$		−	34.0000i		−	1.60456i	0.596948	−	0.802280i	$-0.296380\pi$
$450$		0			0
$451$	4.00000			0.188353
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	26.0000	+	26.0000i	1.20832	+	1.20832i	0.971570	+	0.236752i	$0.0760830\pi$
$463$	26.0000	+	26.0000i	1.20832	+	1.20832i	0.236752	+	0.971570i	$0.423917\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	−24.0000	−	24.0000i	−1.10940	−	1.10940i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−28.0000	+	28.0000i	−1.28880	+	1.28880i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	12.0000	−	12.0000i	0.549442	−	0.549442i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	64.0000			2.91815
$482$	−18.0000	+	18.0000i	−0.819878	+	0.819878i
$483$		0			0
$484$			14.0000i			0.636364i
$485$		0			0
$486$		0			0
$487$	22.0000	−	22.0000i	0.996915	−	0.996915i	−0.00308010	−	0.999995i	$-0.500980\pi$
$487$	22.0000	−	22.0000i	0.996915	−	0.996915i	0.999995	+	0.00308010i	$0.000980427\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.00000			0.0902587			0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000			0.0902587			0.0451294	+	0.998981i	$0.485630\pi$
$492$		0			0
$493$		0			0
$494$		−	48.0000i		−	2.15962i
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			6.00000i			0.268597i	0.990941	+	0.134298i	$0.0428781\pi$
$499$			6.00000i			0.268597i	−0.990941	+	0.134298i	$0.957122\pi$
$500$		0			0
$501$		0			0
$502$	2.00000	−	2.00000i	0.0892644	−	0.0892644i
$503$	−14.0000	−	14.0000i	−0.624229	−	0.624229i	0.322381	−	0.946610i	$-0.395517\pi$
$503$	−14.0000	−	14.0000i	−0.624229	−	0.624229i	−0.946610	+	0.322381i	$0.895517\pi$
$504$		−	24.0000i		−	1.06904i
$505$		0			0
$506$	24.0000			1.06693
$507$		0			0
$508$	−4.00000	−	4.00000i	−0.177471	−	0.177471i
$509$		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
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	4.00000	−	4.00000i	0.175920	−	0.175920i
$518$	32.0000	+	32.0000i	1.40600	+	1.40600i
$519$		0			0
$520$		0			0
$521$	−38.0000			−1.66481			−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000			−1.66481			−0.832405	+	0.554168i	$0.813037\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		−	44.0000i		−	1.92215i
$525$		0			0
$526$	12.0000			0.523225
$527$		0			0
$528$		0			0
$529$			49.0000i			2.13043i
$530$		0			0
$531$	42.0000			1.82264
$532$	24.0000	−	24.0000i	1.04053	−	1.04053i
$533$	−8.00000	−	8.00000i	−0.346518	−	0.346518i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	2.00000i		−	0.0861461i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$			16.0000i			0.679775i
$555$		0			0
$556$	−28.0000			−1.18746
$557$	32.0000	−	32.0000i	1.35588	−	1.35588i	0.476957	−	0.878927i	$-0.341740\pi$
$557$	32.0000	−	32.0000i	1.35588	−	1.35588i	0.878927	−	0.476957i	$-0.158260\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	22.0000	−	22.0000i	0.928014	−	0.928014i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−18.0000	+	18.0000i	−0.755929	+	0.755929i
$568$		0			0
$569$			46.0000i			1.92842i	0.265139	+	0.964210i	$0.414582\pi$
$569$			46.0000i			1.92842i	−0.265139	+	0.964210i	$0.585418\pi$
$570$		0			0
$571$	−18.0000			−0.753277			−0.376638	−	0.926360i	$-0.622920\pi$
$571$	−18.0000			−0.753277			−0.376638	+	0.926360i	$0.622920\pi$
$572$	−16.0000	+	16.0000i	−0.668994	+	0.668994i
$573$		0			0
$574$		−	8.00000i		−	0.333914i
$575$		0			0
$576$	−24.0000			−1.00000
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	−17.0000	−	17.0000i	−0.707107	−	0.707107i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−8.00000	−	8.00000i	−0.331326	−	0.331326i
$584$		0			0
$585$		0			0
$586$	−48.0000			−1.98286
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	32.0000	−	32.0000i	1.31519	−	1.31519i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−48.0000	−	48.0000i	−1.96287	−	1.96287i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	42.0000			1.71322			0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000			1.71322			0.856608	+	0.515968i	$0.172568\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	2.00000	−	2.00000i	0.0811775	−	0.0811775i	−0.665352	−	0.746530i	$-0.731719\pi$
$607$	2.00000	−	2.00000i	0.0811775	−	0.0811775i	0.746530	+	0.665352i	$0.231719\pi$
$608$	−24.0000	−	24.0000i	−0.973329	−	0.973329i
$609$		0			0
$610$		0			0
$611$	−16.0000			−0.647291
$612$		0			0
$613$	−4.00000	−	4.00000i	−0.161558	−	0.161558i	0.621698	−	0.783257i	$-0.286443\pi$
$613$	−4.00000	−	4.00000i	−0.161558	−	0.161558i	−0.783257	+	0.621698i	$0.786443\pi$
$614$		0			0
$615$		0			0
$616$	−16.0000			−0.644658
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$			46.0000i			1.84890i	0.381308	+	0.924448i	$0.375474\pi$
$619$			46.0000i			1.84890i	−0.381308	+	0.924448i	$0.624526\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−28.0000	−	28.0000i	−1.12180	−	1.12180i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$	16.0000	+	16.0000i	0.638470	+	0.638470i
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		−	24.0000i		−	0.953162i
$635$		0			0
$636$		0			0
$637$	−4.00000	+	4.00000i	−0.158486	+	0.158486i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		−	48.0000i		−	1.89146i
$645$		0			0
$646$		0			0
$647$	22.0000	−	22.0000i	0.864909	−	0.864909i	−0.126994	−	0.991903i	$-0.540533\pi$
$647$	22.0000	−	22.0000i	0.864909	−	0.864909i	0.991903	+	0.126994i	$0.0405330\pi$
$648$	18.0000	+	18.0000i	0.707107	+	0.707107i
$649$		−	28.0000i		−	1.09910i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	36.0000	+	36.0000i	1.40879	+	1.40879i	0.766269	+	0.642520i	$0.222111\pi$
$653$	36.0000	+	36.0000i	1.40879	+	1.40879i	0.642520	+	0.766269i	$0.277889\pi$
$654$		0			0
$655$		0			0
$656$	−8.00000			−0.312348
$657$		0			0
$658$	−8.00000	−	8.00000i	−0.311872	−	0.311872i
$659$			26.0000i			1.01282i	0.862294	+	0.506408i	$0.169027\pi$
$659$			26.0000i			1.01282i	−0.862294	+	0.506408i	$0.830973\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−18.0000	+	18.0000i	−0.699590	+	0.699590i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−48.0000			−1.85996
$667$		0			0
$668$	36.0000	+	36.0000i	1.39288	+	1.39288i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$		0			0
$676$	38.0000			1.46154
$677$	12.0000	−	12.0000i	0.461197	−	0.461197i	−0.437850	−	0.899048i	$-0.644260\pi$
$677$	12.0000	−	12.0000i	0.461197	−	0.461197i	0.899048	+	0.437850i	$0.144260\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$			36.0000i			1.37649i
$685$		0			0
$686$	24.0000			0.916324
$687$		0			0
$688$		0			0
$689$			32.0000i			1.21910i
$690$		0			0
$691$	42.0000			1.59776			0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000			1.59776			0.798878	+	0.601494i	$0.205427\pi$
$692$	32.0000	−	32.0000i	1.21646	−	1.21646i
$693$	12.0000	+	12.0000i	0.455842	+	0.455842i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−48.0000	−	48.0000i	−1.81035	−	1.81035i
$704$			16.0000i			0.603023i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$	−28.0000	+	28.0000i	−1.04934	+	1.04934i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	52.0000			1.94333
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−56.0000			−2.08555
$722$	−17.0000	+	17.0000i	−0.632674	+	0.632674i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−38.0000	+	38.0000i	−1.40934	+	1.40934i	−0.646030	+	0.763312i	$0.723572\pi$
$727$	−38.0000	+	38.0000i	−1.40934	+	1.40934i	−0.763312	+	0.646030i	$0.776428\pi$
$728$	32.0000	+	32.0000i	1.18600	+	1.18600i
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	16.0000	+	16.0000i	0.590973	+	0.590973i	0.937894	−	0.346921i	$-0.112773\pi$
$733$	16.0000	+	16.0000i	0.590973	+	0.590973i	−0.346921	+	0.937894i	$0.612773\pi$
$734$		−	44.0000i		−	1.62407i
$735$		0			0
$736$	−48.0000			−1.76930
$737$		0			0
$738$	6.00000	+	6.00000i	0.220863	+	0.220863i
$739$		−	54.0000i		−	1.98642i	−0.116326	−	0.993211i	$-0.537112\pi$
$739$		−	54.0000i		−	1.98642i	0.116326	−	0.993211i	$-0.462888\pi$
$740$		0			0
$741$		0			0
$742$	−16.0000	+	16.0000i	−0.587378	+	0.587378i
$743$	26.0000	+	26.0000i	0.953847	+	0.953847i	0.998981	−	0.0451335i	$-0.0143713\pi$
$743$	26.0000	+	26.0000i	0.953847	+	0.953847i	−0.0451335	+	0.998981i	$0.514371\pi$
$744$		0			0
$745$		0			0
$746$	32.0000			1.17160
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	−8.00000	+	8.00000i	−0.291730	+	0.291730i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	32.0000	−	32.0000i	1.16306	−	1.16306i	0.179258	−	0.983802i	$-0.442630\pi$
$757$	32.0000	−	32.0000i	1.16306	−	1.16306i	0.983802	−	0.179258i	$-0.0573696\pi$
$758$	−34.0000	−	34.0000i	−1.23494	−	1.23494i
$759$		0			0
$760$		0			0
$761$	22.0000			0.797499			0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000			0.797499			0.398750	+	0.917060i	$0.369444\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	52.0000			1.87884
$767$	−56.0000	+	56.0000i	−2.02204	+	2.02204i
$768$		0			0
$769$		−	54.0000i		−	1.94729i	−0.228069	−	0.973645i	$-0.573241\pi$
$769$		−	54.0000i		−	1.94729i	0.228069	−	0.973645i	$-0.426759\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	36.0000	+	36.0000i	1.29483	+	1.29483i	0.931763	+	0.363067i	$0.118270\pi$
$773$	36.0000	+	36.0000i	1.29483	+	1.29483i	0.363067	+	0.931763i	$0.381730\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			12.0000i			0.429945i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			4.00000i			0.142857i
$785$		0			0
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−24.0000	−	24.0000i	−0.854965	−	0.854965i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	12.0000	−	12.0000i	0.426401	−	0.426401i
$793$		0			0
$794$			56.0000i			1.98737i
$795$		0			0
$796$		0			0
$797$	−28.0000	+	28.0000i	−0.991811	+	0.991811i	−0.999967	−	0.00815585i	$-0.997404\pi$
$797$	−28.0000	+	28.0000i	−0.991811	+	0.991811i	0.00815585	+	0.999967i	$0.497404\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	42.0000			1.48400
$802$	−38.0000	+	38.0000i	−1.34183	+	1.34183i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			26.0000i			0.914111i	0.889438	+	0.457056i	$0.151096\pi$
$809$			26.0000i			0.914111i	−0.889438	+	0.457056i	$0.848904\pi$
$810$		0			0
$811$	2.00000			0.0702295			0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000			0.0702295			0.0351147	+	0.999383i	$0.488820\pi$
$812$		0			0
$813$		0			0
$814$			32.0000i			1.12160i
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−14.0000	−	14.0000i	−0.489499	−	0.489499i
$819$		−	48.0000i		−	1.67726i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−34.0000	−	34.0000i	−1.18517	−	1.18517i	−0.978388	−	0.206778i	$-0.933702\pi$
$823$	−34.0000	−	34.0000i	−1.18517	−	1.18517i	−0.206778	−	0.978388i	$-0.566298\pi$
$824$			56.0000i			1.95085i
$825$		0			0
$826$	−56.0000			−1.94849
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$	36.0000	+	36.0000i	1.25109	+	1.25109i
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$	32.0000	−	32.0000i	1.10940	−	1.10940i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	24.0000			0.830057
$837$		0			0
$838$	26.0000	+	26.0000i	0.898155	+	0.898155i
$839$		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
$843$		0			0
$844$		−	44.0000i		−	1.51454i
$845$		0			0
$846$	12.0000			0.412568
$847$	−14.0000	+	14.0000i	−0.481046	+	0.481046i
$848$	16.0000	+	16.0000i	0.549442	+	0.549442i
$849$		0			0
$850$		0			0
$851$	−96.0000			−3.29084
$852$		0			0
$853$	−4.00000	−	4.00000i	−0.136957	−	0.136957i	0.635304	−	0.772262i	$-0.280875\pi$
$853$	−4.00000	−	4.00000i	−0.136957	−	0.136957i	−0.772262	+	0.635304i	$0.780875\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		−	14.0000i		−	0.477674i	−0.971060	−	0.238837i	$-0.923234\pi$
$859$		−	14.0000i		−	0.477674i	0.971060	−	0.238837i	$-0.0767661\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.00000	+	6.00000i	0.204242	+	0.204242i	0.801815	−	0.597573i	$-0.203868\pi$
$863$	6.00000	+	6.00000i	0.204242	+	0.204242i	−0.597573	+	0.801815i	$0.703868\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$			72.0000i			2.43544i
$875$		0			0
$876$		0			0
$877$	−8.00000	+	8.00000i	−0.270141	+	0.270141i	−0.829157	−	0.559016i	$-0.811179\pi$
$877$	−8.00000	+	8.00000i	−0.270141	+	0.270141i	0.559016	+	0.829157i	$0.311179\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−58.0000			−1.95407			−0.977035	−	0.213080i	$-0.931651\pi$
$881$	−58.0000			−1.95407			−0.977035	+	0.213080i	$0.931651\pi$
$882$	3.00000	−	3.00000i	0.101015	−	0.101015i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	42.0000	−	42.0000i	1.41022	−	1.41022i	0.652022	−	0.758200i	$-0.273921\pi$
$887$	42.0000	−	42.0000i	1.41022	−	1.41022i	0.758200	−	0.652022i	$-0.226079\pi$
$888$		0			0
$889$		−	8.00000i		−	0.268311i
$890$		0			0
$891$	−18.0000			−0.603023
$892$	−28.0000	+	28.0000i	−0.937509	+	0.937509i
$893$	12.0000	+	12.0000i	0.401565	+	0.401565i
$894$		0			0
$895$		0			0
$896$	32.0000			1.06904
$897$		0			0
$898$	−34.0000	−	34.0000i	−1.13459	−	1.13459i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	4.00000	−	4.00000i	0.133185	−	0.133185i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	44.0000	−	44.0000i	1.45301	−	1.45301i
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	52.0000			1.70883
$927$	42.0000	−	42.0000i	1.37946	−	1.37946i
$928$		0			0
$929$		−	34.0000i		−	1.11550i	−0.830008	−	0.557752i	$-0.811664\pi$
$929$		−	34.0000i		−	1.11550i	0.830008	−	0.557752i	$-0.188336\pi$
$930$		0			0
$931$	6.00000			0.196642
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−48.0000			−1.56893
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$	12.0000	+	12.0000i	0.390774	+	0.390774i
$944$			56.0000i			1.82264i
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		−	24.0000i		−	0.777029i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$	64.0000	−	64.0000i	2.06344	−	2.06344i
$963$		0			0
$964$			36.0000i			1.15948i
$965$		0			0
$966$		0			0
$967$	−38.0000	+	38.0000i	−1.22200	+	1.22200i	−0.255077	+	0.966921i	$0.582101\pi$
$967$	−38.0000	+	38.0000i	−1.22200	+	1.22200i	−0.966921	+	0.255077i	$0.917899\pi$
$968$	14.0000	+	14.0000i	0.449977	+	0.449977i
$969$		0			0
$970$		0			0
$971$	62.0000			1.98967			0.994837	−	0.101482i	$-0.0323585\pi$
$971$	62.0000			1.98967			0.994837	+	0.101482i	$0.0323585\pi$
$972$		0			0
$973$	−28.0000	−	28.0000i	−0.897639	−	0.897639i
$974$		−	44.0000i		−	1.40985i
$975$		0			0
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		−	28.0000i		−	0.894884i
$980$		0			0
$981$		0			0
$982$	2.00000	−	2.00000i	0.0638226	−	0.0638226i
$983$	−34.0000	−	34.0000i	−1.08443	−	1.08443i	−0.996090	−	0.0883413i	$-0.971843\pi$
$983$	−34.0000	−	34.0000i	−1.08443	−	1.08443i	−0.0883413	−	0.996090i	$-0.528157\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−48.0000	−	48.0000i	−1.52708	−	1.52708i
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−28.0000	+	28.0000i	−0.886769	+	0.886769i	−0.994211	−	0.107442i	$-0.965734\pi$
$997$	−28.0000	+	28.0000i	−0.886769	+	0.886769i	0.107442	+	0.994211i	$0.465734\pi$
$998$	6.00000	+	6.00000i	0.189927	+	0.189927i
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.k.b.43.1	✓	2	5.3	odd	4
200.2.k.b.43.1	✓	2	40.27	even	4
200.2.k.b.107.1	yes	2	5.4	even	2
200.2.k.b.107.1	yes	2	8.3	odd	2
200.2.k.c.43.1	yes	2	5.2	odd	4	inner
200.2.k.c.43.1	yes	2	40.3	even	4	inner
200.2.k.c.107.1	yes	2	1.1	even	1	trivial
200.2.k.c.107.1	yes	2	40.19	odd	2	CM
800.2.o.b.143.1		2	20.7	even	4
800.2.o.b.143.1		2	40.13	odd	4
800.2.o.b.207.1		2	4.3	odd	2
800.2.o.b.207.1		2	40.29	even	2
800.2.o.c.143.1		2	20.3	even	4
800.2.o.c.143.1		2	40.37	odd	4
800.2.o.c.207.1		2	8.5	even	2
800.2.o.c.207.1		2	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	200.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(2.00000 - 2.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} +2.00000 q^{11} +(-4.00000 - 4.00000i) q^{13} -4.00000i q^{14} -4.00000 q^{16} +(3.00000 + 3.00000i) q^{18} +6.00000i q^{19} +(2.00000 - 2.00000i) q^{22} +(6.00000 + 6.00000i) q^{23} -8.00000 q^{26} +(-4.00000 - 4.00000i) q^{28} +(-4.00000 + 4.00000i) q^{32} +6.00000 q^{36} +(-8.00000 + 8.00000i) q^{37} +(6.00000 + 6.00000i) q^{38} +2.00000 q^{41} -4.00000i q^{44} +12.0000 q^{46} +(2.00000 - 2.00000i) q^{47} -1.00000i q^{49} +(-8.00000 + 8.00000i) q^{52} +(-4.00000 - 4.00000i) q^{53} -8.00000 q^{56} -14.0000i q^{59} +(6.00000 + 6.00000i) q^{63} +8.00000i q^{64} +(6.00000 - 6.00000i) q^{72} +16.0000i q^{74} +12.0000 q^{76} +(4.00000 - 4.00000i) q^{77} -9.00000 q^{81} +(2.00000 - 2.00000i) q^{82} +(-4.00000 - 4.00000i) q^{88} -14.0000i q^{89} -16.0000 q^{91} +(12.0000 - 12.0000i) q^{92} -4.00000i q^{94} +(-1.00000 - 1.00000i) q^{98} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{7} - 4 q^{8} + 4 q^{11} - 8 q^{13} - 8 q^{16} + 6 q^{18} + 4 q^{22} + 12 q^{23} - 16 q^{26} - 8 q^{28} - 8 q^{32} + 12 q^{36} - 16 q^{37} + 12 q^{38} + 4 q^{41} + 24 q^{46} + 4 q^{47}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^7 - 4 * q^8 + 4 * q^11 - 8 * q^13 - 8 * q^16 + 6 * q^18 + 4 * q^22 + 12 * q^23 - 16 * q^26 - 8 * q^28 - 8 * q^32 + 12 * q^36 - 16 * q^37 + 12 * q^38 + 4 * q^41 + 24 * q^46 + 4 * q^47 - 16 * q^52 - 8 * q^53 - 16 * q^56 + 12 * q^63 + 12 * q^72 + 24 * q^76 + 8 * q^77 - 18 * q^81 + 4 * q^82 - 8 * q^88 - 32 * q^91 + 24 * q^92 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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