LMFDB - Embedded newform 200.2.k.f.43.2

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:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			43.2
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	200.43
Dual form			200.2.k.f.107.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.00000i) q^{2} +(0.224745 + 0.224745i) q^{3} -2.00000i q^{4} +0.449490 q^{6} +(-2.00000 - 2.00000i) q^{8} -2.89898i q^{9} +5.44949 q^{11} +(0.449490 - 0.449490i) q^{12} -4.00000 q^{16} +(-5.67423 + 5.67423i) q^{17} +(-2.89898 - 2.89898i) q^{18} +6.34847i q^{19} +(5.44949 - 5.44949i) q^{22} -0.898979i q^{24} +(1.32577 - 1.32577i) q^{27} +(-4.00000 + 4.00000i) q^{32} +(1.22474 + 1.22474i) q^{33} +11.3485i q^{34} -5.79796 q^{36} +(6.34847 + 6.34847i) q^{38} -6.79796 q^{41} +(6.00000 + 6.00000i) q^{43} -10.8990i q^{44} +(-0.898979 - 0.898979i) q^{48} -7.00000i q^{49} -2.55051 q^{51} -2.65153i q^{54} +(-1.42679 + 1.42679i) q^{57} +6.00000i q^{59} +8.00000i q^{64} +2.44949 q^{66} +(11.5732 - 11.5732i) q^{67} +(11.3485 + 11.3485i) q^{68} +(-5.79796 + 5.79796i) q^{72} +(-4.77526 - 4.77526i) q^{73} +12.6969 q^{76} -8.10102 q^{81} +(-6.79796 + 6.79796i) q^{82} +(-12.0227 - 12.0227i) q^{83} +12.0000 q^{86} +(-10.8990 - 10.8990i) q^{88} +4.10102i q^{89} -1.79796 q^{96} +(12.0000 - 12.0000i) q^{97} +(-7.00000 - 7.00000i) q^{98} -15.7980i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} - 8 q^{6} - 8 q^{8} + 12 q^{11} - 8 q^{12} - 16 q^{16} - 8 q^{17} + 8 q^{18} + 12 q^{22} + 20 q^{27} - 16 q^{32} + 16 q^{36} - 4 q^{38} + 12 q^{41} + 24 q^{43} + 16 q^{48} - 20 q^{51}+ \cdots - 28 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^3 - 8 * q^6 - 8 * q^8 + 12 * q^11 - 8 * q^12 - 16 * q^16 - 8 * q^17 + 8 * q^18 + 12 * q^22 + 20 * q^27 - 16 * q^32 + 16 * q^36 - 4 * q^38 + 12 * q^41 + 24 * q^43 + 16 * q^48 - 20 * q^51 - 40 * q^57 + 12 * q^67 + 16 * q^68 + 16 * q^72 - 24 * q^73 - 8 * q^76 - 52 * q^81 + 12 * q^82 - 4 * q^83 + 48 * q^86 - 24 * q^88 + 32 * q^96 + 48 * q^97 - 28 * q^98

Character values

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

$n$	$101$	$151$	$177$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$3$	0.224745	+	0.224745i	0.129757	+	0.129757i	0.769002	−	0.639246i	$-0.220753\pi$
$3$	0.224745	+	0.224745i	0.129757	+	0.129757i	−0.639246	+	0.769002i	$0.720753\pi$
$4$		−	2.00000i		−	1.00000i
$5$		0			0
$5$		0			0
$6$	0.449490			0.183503
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$		−	2.89898i		−	0.966326i
$10$		0			0
$11$	5.44949			1.64308			0.821541	−	0.570149i	$-0.193114\pi$
$11$	5.44949			1.64308			0.821541	+	0.570149i	$0.193114\pi$
$12$	0.449490	−	0.449490i	0.129757	−	0.129757i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−5.67423	+	5.67423i	−1.37620	+	1.37620i	−0.525266	+	0.850938i	$0.676034\pi$
$17$	−5.67423	+	5.67423i	−1.37620	+	1.37620i	−0.850938	+	0.525266i	$0.823966\pi$
$18$	−2.89898	−	2.89898i	−0.683296	−	0.683296i
$19$			6.34847i			1.45644i	0.685344	+	0.728219i	$0.259652\pi$
$19$			6.34847i			1.45644i	−0.685344	+	0.728219i	$0.740348\pi$
$20$		0			0
$21$		0			0
$22$	5.44949	−	5.44949i	1.16184	−	1.16184i
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		−	0.898979i		−	0.183503i
$25$		0			0
$26$		0			0
$27$	1.32577	−	1.32577i	0.255144	−	0.255144i
$28$		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
$33$	1.22474	+	1.22474i	0.213201	+	0.213201i
$34$			11.3485i			1.94625i
$35$		0			0
$36$	−5.79796			−0.966326
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	6.34847	+	6.34847i	1.02986	+	1.02986i
$39$		0			0
$40$		0			0
$41$	−6.79796			−1.06166			−0.530831	−	0.847477i	$-0.678120\pi$
$41$	−6.79796			−1.06166			−0.530831	+	0.847477i	$0.678120\pi$
$42$		0			0
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	0.996660	−	0.0816682i	$-0.0260248\pi$
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	−0.0816682	+	0.996660i	$0.526025\pi$
$44$		−	10.8990i		−	1.64308i
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	−0.898979	−	0.898979i	−0.129757	−	0.129757i
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$	−2.55051			−0.357143
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		−	2.65153i		−	0.360828i
$55$		0			0
$56$		0			0
$57$	−1.42679	+	1.42679i	−0.188982	+	0.188982i
$58$		0			0
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		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
$66$	2.44949			0.301511
$67$	11.5732	−	11.5732i	1.41389	−	1.41389i	0.691895	−	0.721998i	$-0.256776\pi$
$67$	11.5732	−	11.5732i	1.41389	−	1.41389i	0.721998	−	0.691895i	$-0.243224\pi$
$68$	11.3485	+	11.3485i	1.37620	+	1.37620i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−5.79796	+	5.79796i	−0.683296	+	0.683296i
$73$	−4.77526	−	4.77526i	−0.558901	−	0.558901i	0.370093	−	0.928995i	$-0.379326\pi$
$73$	−4.77526	−	4.77526i	−0.558901	−	0.558901i	−0.928995	+	0.370093i	$0.879326\pi$
$74$		0			0
$75$		0			0
$76$	12.6969			1.45644
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−8.10102			−0.900113
$82$	−6.79796	+	6.79796i	−0.750709	+	0.750709i
$83$	−12.0227	−	12.0227i	−1.31966	−	1.31966i	−0.914041	−	0.405622i	$-0.867055\pi$
$83$	−12.0227	−	12.0227i	−1.31966	−	1.31966i	−0.405622	−	0.914041i	$-0.632945\pi$
$84$		0			0
$85$		0			0
$86$	12.0000			1.29399
$87$		0			0
$88$	−10.8990	−	10.8990i	−1.16184	−	1.16184i
$89$			4.10102i			0.434707i	0.976093	+	0.217354i	$0.0697425\pi$
$89$			4.10102i			0.434707i	−0.976093	+	0.217354i	$0.930258\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−1.79796			−0.183503
$97$	12.0000	−	12.0000i	1.21842	−	1.21842i	0.250229	−	0.968187i	$-0.419494\pi$
$97$	12.0000	−	12.0000i	1.21842	−	1.21842i	0.968187	−	0.250229i	$-0.0805058\pi$
$98$	−7.00000	−	7.00000i	−0.707107	−	0.707107i
$99$		−	15.7980i		−	1.58775i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−2.55051	+	2.55051i	−0.252538	+	0.252538i
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.6742	+	10.6742i	−1.03192	+	1.03192i	−0.0324436	+	0.999474i	$0.510329\pi$
$107$	−10.6742	+	10.6742i	−1.03192	+	1.03192i	−0.999474	+	0.0324436i	$0.989671\pi$
$108$	−2.65153	−	2.65153i	−0.255144	−	0.255144i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.02270	−	7.02270i	−0.660640	−	0.660640i	0.294891	−	0.955531i	$-0.404717\pi$
$113$	−7.02270	−	7.02270i	−0.660640	−	0.660640i	−0.955531	+	0.294891i	$0.904717\pi$
$114$			2.85357i			0.267261i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	6.00000	+	6.00000i	0.552345	+	0.552345i
$119$		0			0
$120$		0			0
$121$	18.6969			1.69972
$122$		0			0
$123$	−1.52781	−	1.52781i	−0.137758	−	0.137758i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	8.00000	+	8.00000i	0.707107	+	0.707107i
$129$			2.69694i			0.237452i
$130$		0			0
$131$	−18.0000			−1.57267			−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000			−1.57267			−0.786334	+	0.617802i	$0.788023\pi$
$132$	2.44949	−	2.44949i	0.213201	−	0.213201i
$133$		0			0
$134$		−	23.1464i		−	1.99955i
$135$		0			0
$136$	22.6969			1.94625
$137$	4.32577	−	4.32577i	0.369575	−	0.369575i	−0.497747	−	0.867322i	$-0.665839\pi$
$137$	4.32577	−	4.32577i	0.369575	−	0.369575i	0.867322	+	0.497747i	$0.165839\pi$
$138$		0			0
$139$		−	3.65153i		−	0.309719i	−0.987937	−	0.154859i	$-0.950508\pi$
$139$		−	3.65153i		−	0.309719i	0.987937	−	0.154859i	$-0.0494925\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			11.5959i			0.966326i
$145$		0			0
$146$	−9.55051			−0.790406
$147$	1.57321	−	1.57321i	0.129757	−	0.129757i
$148$		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			$\pi$
$152$	12.6969	−	12.6969i	1.02986	−	1.02986i
$153$	16.4495	+	16.4495i	1.32986	+	1.32986i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−8.10102	+	8.10102i	−0.636476	+	0.636476i
$163$	10.2247	+	10.2247i	0.800864	+	0.800864i	0.983231	−	0.182367i	$-0.0583758\pi$
$163$	10.2247	+	10.2247i	0.800864	+	0.800864i	−0.182367	+	0.983231i	$0.558376\pi$
$164$			13.5959i			1.06166i
$165$		0			0
$166$	−24.0454			−1.86629
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$	18.4041			1.40740
$172$	12.0000	−	12.0000i	0.914991	−	0.914991i
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$	−21.7980			−1.64308
$177$	−1.34847	+	1.34847i	−0.101357	+	0.101357i
$178$	4.10102	+	4.10102i	0.307384	+	0.307384i
$179$		−	8.14643i		−	0.608893i	−0.952529	−	0.304446i	$-0.901529\pi$
$179$		−	8.14643i		−	0.608893i	0.952529	−	0.304446i	$-0.0984714\pi$
$180$		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
$187$	−30.9217	+	30.9217i	−2.26122	+	2.26122i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−1.79796	+	1.79796i	−0.129757	+	0.129757i
$193$	7.47219	+	7.47219i	0.537860	+	0.537860i	0.922900	−	0.385040i	$-0.125812\pi$
$193$	7.47219	+	7.47219i	0.537860	+	0.537860i	−0.385040	+	0.922900i	$0.625812\pi$
$194$		−	24.0000i		−	1.72310i
$195$		0			0
$196$	−14.0000			−1.00000
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$	−15.7980	−	15.7980i	−1.12271	−	1.12271i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	5.20204			0.366924
$202$		0			0
$203$		0			0
$204$			5.10102i			0.357143i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			34.5959i			2.39305i
$210$		0			0
$211$	−29.0454			−1.99957			−0.999784	−	0.0207756i	$-0.993386\pi$
$211$	−29.0454			−1.99957			−0.999784	+	0.0207756i	$0.993386\pi$
$212$		0			0
$213$		0			0
$214$			21.3485i			1.45935i
$215$		0			0
$216$	−5.30306			−0.360828
$217$		0			0
$218$		0			0
$219$		−	2.14643i		−	0.145042i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$		0			0
$226$	−14.0454			−0.934287
$227$	2.00000	−	2.00000i	0.132745	−	0.132745i	−0.637613	−	0.770357i	$-0.720078\pi$
$227$	2.00000	−	2.00000i	0.132745	−	0.132745i	0.770357	+	0.637613i	$0.220078\pi$
$228$	2.85357	+	2.85357i	0.188982	+	0.188982i
$229$		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	0.563837	−	0.825886i	$-0.309325\pi$
$233$	−4.00000	−	4.00000i	−0.262049	−	0.262049i	−0.825886	+	0.563837i	$0.809325\pi$
$234$		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
$241$	27.6969			1.78412			0.892058	−	0.451920i	$-0.149261\pi$
$241$	27.6969			1.78412			0.892058	+	0.451920i	$0.149261\pi$
$242$	18.6969	−	18.6969i	1.20188	−	1.20188i
$243$	−5.79796	−	5.79796i	−0.371939	−	0.371939i
$244$		0			0
$245$		0			0
$246$	−3.05561			−0.194819
$247$		0			0
$248$		0			0
$249$		−	5.40408i		−	0.342470i
$250$		0			0
$251$	29.9444			1.89007			0.945036	−	0.326965i	$-0.106026\pi$
$251$	29.9444			1.89007			0.945036	+	0.326965i	$0.106026\pi$
$252$		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	−0.911135	−	0.412108i	$-0.864792\pi$
$257$	−8.00000	+	8.00000i	−0.499026	+	0.499026i	0.412108	+	0.911135i	$0.364792\pi$
$258$	2.69694	+	2.69694i	0.167904	+	0.167904i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−18.0000	+	18.0000i	−1.11204	+	1.11204i
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		−	4.89898i		−	0.301511i
$265$		0			0
$266$		0			0
$267$	−0.921683	+	0.921683i	−0.0564061	+	0.0564061i
$268$	−23.1464	−	23.1464i	−1.41389	−	1.41389i
$269$		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			$\pi$
$272$	22.6969	−	22.6969i	1.37620	−	1.37620i
$273$		0			0
$274$		−	8.65153i		−	0.522658i
$275$		0			0
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	−3.65153	−	3.65153i	−0.219004	−	0.219004i
$279$		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
$283$	22.4722	+	22.4722i	1.33583	+	1.33583i	0.900053	+	0.435780i	$0.143527\pi$
$283$	22.4722	+	22.4722i	1.33583	+	1.33583i	0.435780	+	0.900053i	$0.356473\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	11.5959	+	11.5959i	0.683296	+	0.683296i
$289$		−	47.3939i		−	2.78788i
$290$		0			0
$291$	5.39388			0.316195
$292$	−9.55051	+	9.55051i	−0.558901	+	0.558901i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		−	3.14643i		−	0.183503i
$295$		0			0
$296$		0			0
$297$	7.22474	−	7.22474i	0.419222	−	0.419222i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		−	25.3939i		−	1.45644i
$305$		0			0
$306$	32.8990			1.88071
$307$	23.8207	−	23.8207i	1.35952	−	1.35952i	0.485006	−	0.874511i	$-0.338817\pi$
$307$	23.8207	−	23.8207i	1.35952	−	1.35952i	0.874511	−	0.485006i	$-0.161183\pi$
$308$		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	−0.878120	−	0.478440i	$-0.841202\pi$
$313$	−24.0000	−	24.0000i	−1.35656	−	1.35656i	−0.478440	−	0.878120i	$-0.658798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−4.79796			−0.267796
$322$		0			0
$323$	−36.0227	−	36.0227i	−2.00436	−	2.00436i
$324$			16.2020i			0.900113i
$325$		0			0
$326$	20.4495			1.13259
$327$		0			0
$328$	13.5959	+	13.5959i	0.750709	+	0.750709i
$329$		0			0
$330$		0			0
$331$	−9.04541			−0.497181			−0.248590	−	0.968609i	$-0.579967\pi$
$331$	−9.04541			−0.497181			−0.248590	+	0.968609i	$0.579967\pi$
$332$	−24.0454	+	24.0454i	−1.31966	+	1.31966i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3.42679	+	3.42679i	−0.186669	+	0.186669i	−0.794254	−	0.607585i	$-0.792138\pi$
$337$	−3.42679	+	3.42679i	−0.186669	+	0.186669i	0.607585	+	0.794254i	$0.292138\pi$
$338$	13.0000	+	13.0000i	0.707107	+	0.707107i
$339$		−	3.15663i		−	0.171445i
$340$		0			0
$341$		0			0
$342$	18.4041	−	18.4041i	0.995179	−	0.995179i
$343$		0			0
$344$		−	24.0000i		−	1.29399i
$345$		0			0
$346$		0			0
$347$	9.32577	−	9.32577i	0.500633	−	0.500633i	−0.411001	−	0.911635i	$-0.634821\pi$
$347$	9.32577	−	9.32577i	0.500633	−	0.500633i	0.911635	+	0.411001i	$0.134821\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	−21.7980	+	21.7980i	−1.16184	+	1.16184i
$353$	16.0000	+	16.0000i	0.851594	+	0.851594i	0.990329	−	0.138735i	$-0.0443038\pi$
$353$	16.0000	+	16.0000i	0.851594	+	0.851594i	−0.138735	+	0.990329i	$0.544304\pi$
$354$			2.69694i			0.143341i
$355$		0			0
$356$	8.20204			0.434707
$357$		0			0
$358$	−8.14643	−	8.14643i	−0.430552	−	0.430552i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−21.3031			−1.12121
$362$		0			0
$363$	4.20204	+	4.20204i	0.220550	+	0.220550i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$			19.7071i			1.02591i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$			61.8434i			3.19784i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			26.3485i			1.35343i	0.736245	+	0.676715i	$0.236597\pi$
$379$			26.3485i			1.35343i	−0.736245	+	0.676715i	$0.763403\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$			3.59592i			0.183503i
$385$		0			0
$386$	14.9444			0.760649
$387$	17.3939	−	17.3939i	0.884180	−	0.884180i
$388$	−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
$393$	−4.04541	−	4.04541i	−0.204064	−	0.204064i
$394$		0			0
$395$		0			0
$396$	−31.5959			−1.58775
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−31.2929			−1.56269			−0.781345	−	0.624099i	$-0.785466\pi$
$401$	−31.2929			−1.56269			−0.781345	+	0.624099i	$0.785466\pi$
$402$	5.20204	−	5.20204i	0.259454	−	0.259454i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	5.10102	+	5.10102i	0.252538	+	0.252538i
$409$		−	40.3939i		−	1.99735i	−0.0514740	−	0.998674i	$-0.516392\pi$
$409$		−	40.3939i		−	1.99735i	0.0514740	−	0.998674i	$-0.483608\pi$
$410$		0			0
$411$	1.94439			0.0959095
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.820663	−	0.820663i	0.0401880	−	0.0401880i
$418$	34.5959	+	34.5959i	1.69214	+	1.69214i
$419$			40.8434i			1.99533i	0.0683046	+	0.997665i	$0.478241\pi$
$419$			40.8434i			1.99533i	−0.0683046	+	0.997665i	$0.521759\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	−29.0454	+	29.0454i	−1.41391	+	1.41391i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	21.3485	+	21.3485i	1.03192	+	1.03192i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−5.30306	+	5.30306i	−0.255144	+	0.255144i
$433$	−29.2702	−	29.2702i	−1.40663	−	1.40663i	−0.776429	−	0.630205i	$-0.782971\pi$
$433$	−29.2702	−	29.2702i	−1.40663	−	1.40663i	−0.630205	−	0.776429i	$-0.717029\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−2.14643	−	2.14643i	−0.102560	−	0.102560i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−20.2929			−0.966326
$442$		0			0
$443$	24.7196	+	24.7196i	1.17447	+	1.17447i	0.981133	+	0.193332i	$0.0619295\pi$
$443$	24.7196	+	24.7196i	1.17447	+	1.17447i	0.193332	+	0.981133i	$0.438070\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	25.8990i		−	1.22225i	−0.791535	−	0.611124i	$-0.790718\pi$
$449$		−	25.8990i		−	1.22225i	0.791535	−	0.611124i	$-0.209282\pi$
$450$		0			0
$451$	−37.0454			−1.74440
$452$	−14.0454	+	14.0454i	−0.660640	+	0.660640i
$453$		0			0
$454$		−	4.00000i		−	0.187729i
$455$		0			0
$456$	5.70714			0.267261
$457$	−27.9217	+	27.9217i	−1.30612	+	1.30612i	−0.381930	+	0.924191i	$0.624740\pi$
$457$	−27.9217	+	27.9217i	−1.30612	+	1.30612i	−0.924191	+	0.381930i	$0.875260\pi$
$458$		0			0
$459$			15.0454i			0.702259i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$	−8.00000			−0.370593
$467$	22.0000	−	22.0000i	1.01804	−	1.01804i	0.0182043	−	0.999834i	$-0.494205\pi$
$467$	22.0000	−	22.0000i	1.01804	−	1.01804i	0.999834	−	0.0182043i	$-0.00579493\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	12.0000	−	12.0000i	0.552345	−	0.552345i
$473$	32.6969	+	32.6969i	1.50341	+	1.50341i
$474$		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
$482$	27.6969	−	27.6969i	1.26156	−	1.26156i
$483$		0			0
$484$		−	37.3939i		−	1.69972i
$485$		0			0
$486$	−11.5959			−0.526002
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$			4.59592i			0.207835i
$490$		0			0
$491$	42.0000			1.89543			0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000			1.89543			0.947717	+	0.319113i	$0.103385\pi$
$492$	−3.05561	+	3.05561i	−0.137758	+	0.137758i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−5.40408	−	5.40408i	−0.242163	−	0.242163i
$499$		−	14.0000i		−	0.626726i	−0.949633	−	0.313363i	$-0.898544\pi$
$499$		−	14.0000i		−	0.626726i	0.949633	−	0.313363i	$-0.101456\pi$
$500$		0			0
$501$		0			0
$502$	29.9444	−	29.9444i	1.33648	−	1.33648i
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.92168	+	2.92168i	−0.129757	+	0.129757i
$508$		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
$513$	8.41658	+	8.41658i	0.371601	+	0.371601i
$514$			16.0000i			0.705730i
$515$		0			0
$516$	5.39388			0.237452
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	42.1918			1.84846			0.924229	−	0.381839i	$-0.124709\pi$
$521$	42.1918			1.84846			0.924229	+	0.381839i	$0.124709\pi$
$522$		0			0
$523$	−14.2702	−	14.2702i	−0.623990	−	0.623990i	0.322559	−	0.946549i	$-0.395457\pi$
$523$	−14.2702	−	14.2702i	−0.623990	−	0.623990i	−0.946549	+	0.322559i	$0.895457\pi$
$524$			36.0000i			1.57267i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−4.89898	−	4.89898i	−0.213201	−	0.213201i
$529$			23.0000i			1.00000i
$530$		0			0
$531$	17.3939			0.754830
$532$		0			0
$533$		0			0
$534$			1.84337i			0.0797703i
$535$		0			0
$536$	−46.2929			−1.99955
$537$	1.83087	−	1.83087i	0.0790078	−	0.0790078i
$538$		0			0
$539$		−	38.1464i		−	1.64308i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		−	45.3939i		−	1.94625i
$545$		0			0
$546$		0			0
$547$	−25.1691	+	25.1691i	−1.07615	+	1.07615i	−0.0793039	+	0.996850i	$0.525270\pi$
$547$	−25.1691	+	25.1691i	−1.07615	+	1.07615i	−0.996850	+	0.0793039i	$0.974730\pi$
$548$	−8.65153	−	8.65153i	−0.369575	−	0.369575i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−7.30306			−0.309719
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−13.8990			−0.586815
$562$	−18.0000	+	18.0000i	−0.759284	+	0.759284i
$563$	26.0000	+	26.0000i	1.09577	+	1.09577i	0.994900	+	0.100870i	$0.0321625\pi$
$563$	26.0000	+	26.0000i	1.09577	+	1.09577i	0.100870	+	0.994900i	$0.467837\pi$
$564$		0			0
$565$		0			0
$566$	44.9444			1.88915
$567$		0			0
$568$		0			0
$569$		−	1.40408i		−	0.0588622i	−0.999567	−	0.0294311i	$-0.990630\pi$
$569$		−	1.40408i		−	0.0588622i	0.999567	−	0.0294311i	$-0.00936956\pi$
$570$		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
$576$	23.1918			0.966326
$577$	8.82066	−	8.82066i	0.367209	−	0.367209i	−0.499249	−	0.866458i	$-0.666391\pi$
$577$	8.82066	−	8.82066i	0.367209	−	0.367209i	0.866458	+	0.499249i	$0.166391\pi$
$578$	−47.3939	−	47.3939i	−1.97133	−	1.97133i
$579$			3.35867i			0.139582i
$580$		0			0
$581$		0			0
$582$	5.39388	−	5.39388i	0.223583	−	0.223583i
$583$		0			0
$584$			19.1010i			0.790406i
$585$		0			0
$586$		0			0
$587$	−20.6742	+	20.6742i	−0.853317	+	0.853317i	−0.990540	−	0.137223i	$-0.956182\pi$
$587$	−20.6742	+	20.6742i	−0.853317	+	0.853317i	0.137223	+	0.990540i	$0.456182\pi$
$588$	−3.14643	−	3.14643i	−0.129757	−	0.129757i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−27.0227	−	27.0227i	−1.10969	−	1.10969i	−0.993191	−	0.116499i	$-0.962833\pi$
$593$	−27.0227	−	27.0227i	−1.10969	−	1.10969i	−0.116499	−	0.993191i	$-0.537167\pi$
$594$		−	14.4495i		−	0.592870i
$595$		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
$601$	37.6969			1.53769			0.768845	−	0.639435i	$-0.220832\pi$
$601$	37.6969			1.53769			0.768845	+	0.639435i	$0.220832\pi$
$602$		0			0
$603$	−33.5505	−	33.5505i	−1.36628	−	1.36628i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$	−25.3939	−	25.3939i	−1.02986	−	1.02986i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	32.8990	−	32.8990i	1.32986	−	1.32986i
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		−	47.6413i		−	1.92265i
$615$		0			0
$616$		0			0
$617$	−28.0000	+	28.0000i	−1.12724	+	1.12724i	−0.136613	+	0.990624i	$0.543622\pi$
$617$	−28.0000	+	28.0000i	−1.12724	+	1.12724i	−0.990624	+	0.136613i	$0.956378\pi$
$618$		0			0
$619$			26.0000i			1.04503i	0.852631	+	0.522514i	$0.175006\pi$
$619$			26.0000i			1.04503i	−0.852631	+	0.522514i	$0.824994\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$	−48.0000			−1.91847
$627$	−7.77526	+	7.77526i	−0.310514	+	0.310514i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−6.52781	−	6.52781i	−0.259457	−	0.259457i
$634$		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	$0.188654\pi$
$642$	−4.79796	+	4.79796i	−0.189360	+	0.189360i
$643$	6.00000	+	6.00000i	0.236617	+	0.236617i	0.815448	−	0.578831i	$-0.196491\pi$
$643$	6.00000	+	6.00000i	0.236617	+	0.236617i	−0.578831	+	0.815448i	$0.696491\pi$
$644$		0			0
$645$		0			0
$646$	−72.0454			−2.83459
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	16.2020	+	16.2020i	0.636476	+	0.636476i
$649$			32.6969i			1.28347i
$650$		0			0
$651$		0			0
$652$	20.4495	−	20.4495i	0.800864	−	0.800864i
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$	27.1918			1.06166
$657$	−13.8434	+	13.8434i	−0.540081	+	0.540081i
$658$		0			0
$659$		−	32.6413i		−	1.27153i	−0.771885	−	0.635763i	$-0.780686\pi$
$659$		−	32.6413i		−	1.27153i	0.771885	−	0.635763i	$-0.219314\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−9.04541	+	9.04541i	−0.351560	+	0.351560i
$663$		0			0
$664$			48.0908i			1.86629i
$665$		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	0.830134	+	0.557564i	$0.188264\pi$
$673$	36.0000	+	36.0000i	1.38770	+	1.38770i	0.557564	+	0.830134i	$0.311736\pi$
$674$			6.85357i			0.263990i
$675$		0			0
$676$	26.0000			1.00000
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$	−3.15663	−	3.15663i	−0.121230	−	0.121230i
$679$		0			0
$680$		0			0
$681$	0.898979			0.0344490
$682$		0			0
$683$	14.7196	+	14.7196i	0.563231	+	0.563231i	0.930224	−	0.366992i	$-0.119613\pi$
$683$	14.7196	+	14.7196i	0.563231	+	0.563231i	−0.366992	+	0.930224i	$0.619613\pi$
$684$		−	36.8082i		−	1.40740i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−24.0000	−	24.0000i	−0.914991	−	0.914991i
$689$		0			0
$690$		0			0
$691$	0.954592			0.0363144			0.0181572	−	0.999835i	$-0.494220\pi$
$691$	0.954592			0.0363144			0.0181572	+	0.999835i	$0.494220\pi$
$692$		0			0
$693$		0			0
$694$		−	18.6515i		−	0.708002i
$695$		0			0
$696$		0			0
$697$	38.5732	−	38.5732i	1.46106	−	1.46106i
$698$		0			0
$699$		−	1.79796i		−	0.0680051i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$			43.5959i			1.64308i
$705$		0			0
$706$	32.0000			1.20434
$707$		0			0
$708$	2.69694	+	2.69694i	0.101357	+	0.101357i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$	8.20204	−	8.20204i	0.307384	−	0.307384i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−16.2929			−0.608893
$717$		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
$722$	−21.3031	+	21.3031i	−0.792818	+	0.792818i
$723$	6.22474	+	6.22474i	0.231501	+	0.231501i
$724$		0			0
$725$		0			0
$726$	8.40408			0.311905
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$			21.6969i			0.803590i
$730$		0			0
$731$	−68.0908			−2.51843
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	63.0681	−	63.0681i	2.32314	−	2.32314i
$738$	19.7071	+	19.7071i	0.725430	+	0.725430i
$739$		−	34.0000i		−	1.25071i	−0.780340	−	0.625355i	$-0.784954\pi$
$739$		−	34.0000i		−	1.25071i	0.780340	−	0.625355i	$-0.215046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−34.8536	+	34.8536i	−1.27523	+	1.27523i
$748$	61.8434	+	61.8434i	2.26122	+	2.26122i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	6.72985	+	6.72985i	0.245249	+	0.245249i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	26.3485	+	26.3485i	0.957019	+	0.957019i
$759$		0			0
$760$		0			0
$761$	−36.7980			−1.33392			−0.666962	−	0.745091i	$-0.732406\pi$
$761$	−36.7980			−1.33392			−0.666962	+	0.745091i	$0.732406\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	3.59592	+	3.59592i	0.129757	+	0.129757i
$769$			33.0908i			1.19329i	0.802507	+	0.596643i	$0.203499\pi$
$769$			33.0908i			1.19329i	−0.802507	+	0.596643i	$0.796501\pi$
$770$		0			0
$771$	−3.59592			−0.129504
$772$	14.9444	−	14.9444i	0.537860	−	0.537860i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		−	34.7878i		−	1.25042i
$775$		0			0
$776$	−48.0000			−1.72310
$777$		0			0
$778$		0			0
$779$		−	43.1566i		−	1.54625i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			28.0000i			1.00000i
$785$		0			0
$786$	−8.09082			−0.288590
$787$	−18.0000	+	18.0000i	−0.641631	+	0.641631i	−0.950956	−	0.309326i	$-0.899897\pi$
$787$	−18.0000	+	18.0000i	−0.641631	+	0.641631i	0.309326	+	0.950956i	$0.399897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−31.5959	+	31.5959i	−1.12271	+	1.12271i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	11.8888			0.420069
$802$	−31.2929	+	31.2929i	−1.10499	+	1.10499i
$803$	−26.0227	−	26.0227i	−0.918321	−	0.918321i
$804$		−	10.4041i		−	0.366924i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			6.00000i			0.210949i	0.994422	+	0.105474i	$0.0336361\pi$
$809$			6.00000i			0.210949i	−0.994422	+	0.105474i	$0.966364\pi$
$810$		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
$816$	10.2020			0.357143
$817$	−38.0908	+	38.0908i	−1.33263	+	1.33263i
$818$	−40.3939	−	40.3939i	−1.41234	−	1.41234i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	1.94439	−	1.94439i	0.0678183	−	0.0678183i
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	26.0681	−	26.0681i	0.906477	−	0.906477i	−0.0895090	−	0.995986i	$-0.528530\pi$
$827$	26.0681	−	26.0681i	0.906477	−	0.906477i	0.995986	+	0.0895090i	$0.0285298\pi$
$828$		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
$833$	39.7196	+	39.7196i	1.37620	+	1.37620i
$834$		−	1.64133i		−	0.0568345i
$835$		0			0
$836$	69.1918			2.39305
$837$		0			0
$838$	40.8434	+	40.8434i	1.41091	+	1.41091i
$839$		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
$843$	−4.04541	−	4.04541i	−0.139331	−	0.139331i
$844$			58.0908i			1.99957i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$			10.1010i			0.346666i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$	42.6969			1.45935
$857$	41.0681	−	41.0681i	1.40286	−	1.40286i	0.612010	−	0.790850i	$-0.290361\pi$
$857$	41.0681	−	41.0681i	1.40286	−	1.40286i	0.790850	−	0.612010i	$-0.209639\pi$
$858$		0			0
$859$			36.3485i			1.24019i	0.784525	+	0.620097i	$0.212907\pi$
$859$			36.3485i			1.24019i	−0.784525	+	0.620097i	$0.787093\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$			10.6061i			0.360828i
$865$		0			0
$866$	−58.5403			−1.98928
$867$	10.6515	−	10.6515i	0.361745	−	0.361745i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−34.7878	−	34.7878i	−1.17739	−	1.17739i
$874$		0			0
$875$		0			0
$876$	−4.29286			−0.145042
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$		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	$0.598061\pi$
$882$	−20.2929	+	20.2929i	−0.683296	+	0.683296i
$883$	−19.7753	−	19.7753i	−0.665490	−	0.665490i	0.291178	−	0.956669i	$-0.405953\pi$
$883$	−19.7753	−	19.7753i	−0.665490	−	0.665490i	−0.956669	+	0.291178i	$0.905953\pi$
$884$		0			0
$885$		0			0
$886$	49.4393			1.66095
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−44.1464			−1.47896
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−25.8990	−	25.8990i	−0.864260	−	0.864260i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−37.0454	+	37.0454i	−1.23348	+	1.23348i
$903$		0			0
$904$			28.0908i			0.934287i
$905$		0			0
$906$		0			0
$907$	42.0000	−	42.0000i	1.39459	−	1.39459i	0.579898	−	0.814689i	$-0.303092\pi$
$907$	42.0000	−	42.0000i	1.39459	−	1.39459i	0.814689	−	0.579898i	$-0.196908\pi$
$908$	−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			$\pi$
$912$	5.70714	−	5.70714i	0.188982	−	0.188982i
$913$	−65.5176	−	65.5176i	−2.16832	−	2.16832i
$914$			55.8434i			1.84713i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	15.0454	+	15.0454i	0.496572	+	0.496572i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	10.7071			0.352812
$922$		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	$-0.653582\pi$
$929$		−	54.0000i		−	1.77168i	0.463988	−	0.885841i	$-0.346418\pi$
$930$		0			0
$931$	44.4393			1.45644
$932$	−8.00000	+	8.00000i	−0.262049	+	0.262049i
$933$		0			0
$934$		−	44.0000i		−	1.43972i
$935$		0			0
$936$		0			0
$937$	38.8207	−	38.8207i	1.26822	−	1.26822i	0.321207	−	0.947009i	$-0.395912\pi$
$937$	38.8207	−	38.8207i	1.26822	−	1.26822i	0.947009	−	0.321207i	$-0.104088\pi$
$938$		0			0
$939$		−	10.7878i		−	0.352045i
$940$		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
$946$	65.3939			2.12614
$947$	−38.0000	+	38.0000i	−1.23483	+	1.23483i	−0.272749	+	0.962085i	$0.587933\pi$
$947$	−38.0000	+	38.0000i	−1.23483	+	1.23483i	−0.962085	+	0.272749i	$0.912067\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9.71964	+	9.71964i	0.314850	+	0.314850i	0.846785	−	0.531935i	$-0.178535\pi$
$953$	9.71964	+	9.71964i	0.314850	+	0.314850i	−0.531935	+	0.846785i	$0.678535\pi$
$954$		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
$963$	30.9444	+	30.9444i	0.997169	+	0.997169i
$964$		−	55.3939i		−	1.78412i
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	−37.3939	−	37.3939i	−1.20188	−	1.20188i
$969$		−	16.1918i		−	0.520157i
$970$		0			0
$971$	−0.0556128			−0.00178470			−0.000892350	−	1.00000i	$-0.500284\pi$
$971$	−0.0556128			−0.00178470			−0.000892350		1.00000i	$0.500284\pi$
$972$	−11.5959	+	11.5959i	−0.371939	+	0.371939i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25.6742	+	25.6742i	−0.821392	+	0.821392i	−0.986308	−	0.164916i	$-0.947265\pi$
$977$	−25.6742	+	25.6742i	−0.821392	+	0.821392i	0.164916	+	0.986308i	$0.447265\pi$
$978$	4.59592	+	4.59592i	0.146961	+	0.146961i
$979$			22.3485i			0.714260i
$980$		0			0
$981$		0			0
$982$	42.0000	−	42.0000i	1.34027	−	1.34027i
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$			6.11123i			0.194819i
$985$		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
$993$	−2.03291	−	2.03291i	−0.0645124	−	0.0645124i
$994$		0			0
$995$		0			0
$996$	−10.8082			−0.342470
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$	−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.f.43.2	yes	4
4.3	odd	2		800.2.o.f.143.1		4
5.2	odd	4	inner	200.2.k.f.107.2	yes	4
5.3	odd	4		200.2.k.e.107.1	yes	4
5.4	even	2		200.2.k.e.43.1	✓	4
8.3	odd	2	CM	200.2.k.f.43.2	yes	4
8.5	even	2		800.2.o.f.143.1		4
20.3	even	4		800.2.o.e.207.2		4
20.7	even	4		800.2.o.f.207.1		4
20.19	odd	2		800.2.o.e.143.2		4
40.3	even	4		200.2.k.e.107.1	yes	4
40.13	odd	4		800.2.o.e.207.2		4
40.19	odd	2		200.2.k.e.43.1	✓	4
40.27	even	4	inner	200.2.k.f.107.2	yes	4
40.29	even	2		800.2.o.e.143.2		4
40.37	odd	4		800.2.o.f.207.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.k.e.43.1	✓	4	5.4	even	2
200.2.k.e.43.1	✓	4	40.19	odd	2
200.2.k.e.107.1	yes	4	5.3	odd	4
200.2.k.e.107.1	yes	4	40.3	even	4
200.2.k.f.43.2	yes	4	1.1	even	1	trivial
200.2.k.f.43.2	yes	4	8.3	odd	2	CM
200.2.k.f.107.2	yes	4	5.2	odd	4	inner
200.2.k.f.107.2	yes	4	40.27	even	4	inner
800.2.o.e.143.2		4	20.19	odd	2
800.2.o.e.143.2		4	40.29	even	2
800.2.o.e.207.2		4	20.3	even	4
800.2.o.e.207.2		4	40.13	odd	4
800.2.o.f.143.1		4	4.3	odd	2
800.2.o.f.143.1		4	8.5	even	2
800.2.o.f.207.1		4	20.7	even	4
800.2.o.f.207.1		4	40.37	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} +(0.224745 + 0.224745i) q^{3} -2.00000i q^{4} +0.449490 q^{6} +(-2.00000 - 2.00000i) q^{8} -2.89898i q^{9} +5.44949 q^{11} +(0.449490 - 0.449490i) q^{12} -4.00000 q^{16} +(-5.67423 + 5.67423i) q^{17} +(-2.89898 - 2.89898i) q^{18} +6.34847i q^{19} +(5.44949 - 5.44949i) q^{22} -0.898979i q^{24} +(1.32577 - 1.32577i) q^{27} +(-4.00000 + 4.00000i) q^{32} +(1.22474 + 1.22474i) q^{33} +11.3485i q^{34} -5.79796 q^{36} +(6.34847 + 6.34847i) q^{38} -6.79796 q^{41} +(6.00000 + 6.00000i) q^{43} -10.8990i q^{44} +(-0.898979 - 0.898979i) q^{48} -7.00000i q^{49} -2.55051 q^{51} -2.65153i q^{54} +(-1.42679 + 1.42679i) q^{57} +6.00000i q^{59} +8.00000i q^{64} +2.44949 q^{66} +(11.5732 - 11.5732i) q^{67} +(11.3485 + 11.3485i) q^{68} +(-5.79796 + 5.79796i) q^{72} +(-4.77526 - 4.77526i) q^{73} +12.6969 q^{76} -8.10102 q^{81} +(-6.79796 + 6.79796i) q^{82} +(-12.0227 - 12.0227i) q^{83} +12.0000 q^{86} +(-10.8990 - 10.8990i) q^{88} +4.10102i q^{89} -1.79796 q^{96} +(12.0000 - 12.0000i) q^{97} +(-7.00000 - 7.00000i) q^{98} -15.7980i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} - 8 q^{6} - 8 q^{8} + 12 q^{11} - 8 q^{12} - 16 q^{16} - 8 q^{17} + 8 q^{18} + 12 q^{22} + 20 q^{27} - 16 q^{32} + 16 q^{36} - 4 q^{38} + 12 q^{41} + 24 q^{43} + 16 q^{48} - 20 q^{51}+ \cdots - 28 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 - 8 * q^6 - 8 * q^8 + 12 * q^11 - 8 * q^12 - 16 * q^16 - 8 * q^17 + 8 * q^18 + 12 * q^22 + 20 * q^27 - 16 * q^32 + 16 * q^36 - 4 * q^38 + 12 * q^41 + 24 * q^43 + 16 * q^48 - 20 * q^51 - 40 * q^57 + 12 * q^67 + 16 * q^68 + 16 * q^72 - 24 * q^73 - 8 * q^76 - 52 * q^81 + 12 * q^82 - 4 * q^83 + 48 * q^86 - 24 * q^88 + 32 * q^96 + 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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