LMFDB - Embedded newform 200.2.k.f.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:	$4$
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			107.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	200.107
Dual form			200.2.k.f.43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.00000i) q^{2} +(-2.22474 + 2.22474i) q^{3} +2.00000i q^{4} -4.44949 q^{6} +(-2.00000 + 2.00000i) q^{8} -6.89898i q^{9} +0.550510 q^{11} +(-4.44949 - 4.44949i) q^{12} -4.00000 q^{16} +(1.67423 + 1.67423i) q^{17} +(6.89898 - 6.89898i) q^{18} +8.34847i q^{19} +(0.550510 + 0.550510i) q^{22} -8.89898i q^{24} +(8.67423 + 8.67423i) q^{27} +(-4.00000 - 4.00000i) q^{32} +(-1.22474 + 1.22474i) q^{33} +3.34847i q^{34} +13.7980 q^{36} +(-8.34847 + 8.34847i) q^{38} +12.7980 q^{41} +(6.00000 - 6.00000i) q^{43} +1.10102i q^{44} +(8.89898 - 8.89898i) q^{48} +7.00000i q^{49} -7.44949 q^{51} +17.3485i q^{54} +(-18.5732 - 18.5732i) q^{57} -6.00000i q^{59} -8.00000i q^{64} -2.44949 q^{66} +(-5.57321 - 5.57321i) q^{67} +(-3.34847 + 3.34847i) q^{68} +(13.7980 + 13.7980i) q^{72} +(-7.22474 + 7.22474i) q^{73} -16.6969 q^{76} -17.8990 q^{81} +(12.7980 + 12.7980i) q^{82} +(10.0227 - 10.0227i) q^{83} +12.0000 q^{86} +(-1.10102 + 1.10102i) q^{88} -13.8990i q^{89} +17.7980 q^{96} +(12.0000 + 12.0000i) q^{97} +(-7.00000 + 7.00000i) q^{98} -3.79796i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

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

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
$3$	−2.22474	+	2.22474i	−1.28446	+	1.28446i	−0.346353	+	0.938104i	$0.612580\pi$
$3$	−2.22474	+	2.22474i	−1.28446	+	1.28446i	−0.938104	+	0.346353i	$0.887420\pi$
$4$			2.00000i			1.00000i
$5$		0			0
$5$		0			0
$6$	−4.44949			−1.81650
$7$		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
$9$		−	6.89898i		−	2.29966i
$10$		0			0
$11$	0.550510			0.165985			0.0829925	−	0.996550i	$-0.473552\pi$
$11$	0.550510			0.165985			0.0829925	+	0.996550i	$0.473552\pi$
$12$	−4.44949	−	4.44949i	−1.28446	−	1.28446i
$13$		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
$17$	1.67423	+	1.67423i	0.406062	+	0.406062i	0.880363	−	0.474301i	$-0.157299\pi$
$17$	1.67423	+	1.67423i	0.406062	+	0.406062i	−0.474301	+	0.880363i	$0.657299\pi$
$18$	6.89898	−	6.89898i	1.62611	−	1.62611i
$19$			8.34847i			1.91527i	0.287984	+	0.957635i	$0.407015\pi$
$19$			8.34847i			1.91527i	−0.287984	+	0.957635i	$0.592985\pi$
$20$		0			0
$21$		0			0
$22$	0.550510	+	0.550510i	0.117369	+	0.117369i
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		−	8.89898i		−	1.81650i
$25$		0			0
$26$		0			0
$27$	8.67423	+	8.67423i	1.66936	+	1.66936i
$28$		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
$34$			3.34847i			0.574258i
$35$		0			0
$36$	13.7980			2.29966
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$	−8.34847	+	8.34847i	−1.35430	+	1.35430i
$39$		0			0
$40$		0			0
$41$	12.7980			1.99871			0.999353	−	0.0359748i	$-0.0114536\pi$
$41$	12.7980			1.99871			0.999353	+	0.0359748i	$0.0114536\pi$
$42$		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	$0.0260248\pi$
$44$			1.10102i			0.165985i
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	8.89898	−	8.89898i	1.28446	−	1.28446i
$49$			7.00000i			1.00000i
$50$		0			0
$51$	−7.44949			−1.04314
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$			17.3485i			2.36083i
$55$		0			0
$56$		0			0
$57$	−18.5732	−	18.5732i	−2.46008	−	2.46008i
$58$		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
$66$	−2.44949			−0.301511
$67$	−5.57321	−	5.57321i	−0.680876	−	0.680876i	0.279321	−	0.960198i	$-0.409891\pi$
$67$	−5.57321	−	5.57321i	−0.680876	−	0.680876i	−0.960198	+	0.279321i	$0.909891\pi$
$68$	−3.34847	+	3.34847i	−0.406062	+	0.406062i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	13.7980	+	13.7980i	1.62611	+	1.62611i
$73$	−7.22474	+	7.22474i	−0.845592	+	0.845592i	−0.989580	−	0.143987i	$-0.954008\pi$
$73$	−7.22474	+	7.22474i	−0.845592	+	0.845592i	0.143987	+	0.989580i	$0.454008\pi$
$74$		0			0
$75$		0			0
$76$	−16.6969			−1.91527
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−17.8990			−1.98878
$82$	12.7980	+	12.7980i	1.41330	+	1.41330i
$83$	10.0227	−	10.0227i	1.10013	−	1.10013i	0.105741	−	0.994394i	$-0.466279\pi$
$83$	10.0227	−	10.0227i	1.10013	−	1.10013i	0.994394	−	0.105741i	$-0.0337214\pi$
$84$		0			0
$85$		0			0
$86$	12.0000			1.29399
$87$		0			0
$88$	−1.10102	+	1.10102i	−0.117369	+	0.117369i
$89$		−	13.8990i		−	1.47329i	−0.676280	−	0.736644i	$-0.736409\pi$
$89$		−	13.8990i		−	1.47329i	0.676280	−	0.736644i	$-0.263591\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	17.7980			1.81650
$97$	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
$99$		−	3.79796i		−	0.381709i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−7.44949	−	7.44949i	−0.737609	−	0.737609i
$103$		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
$107$	−3.32577	−	3.32577i	−0.321514	−	0.321514i	0.527834	−	0.849348i	$-0.323004\pi$
$107$	−3.32577	−	3.32577i	−0.321514	−	0.321514i	−0.849348	+	0.527834i	$0.823004\pi$
$108$	−17.3485	+	17.3485i	−1.66936	+	1.66936i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	15.0227	−	15.0227i	1.41322	−	1.41322i	0.680069	−	0.733148i	$-0.261950\pi$
$113$	15.0227	−	15.0227i	1.41322	−	1.41322i	0.733148	−	0.680069i	$-0.238050\pi$
$114$		−	37.1464i		−	3.47908i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	6.00000	−	6.00000i	0.552345	−	0.552345i
$119$		0			0
$120$		0			0
$121$	−10.6969			−0.972449
$122$		0			0
$123$	−28.4722	+	28.4722i	−2.56725	+	2.56725i
$124$		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
$129$			26.6969i			2.35053i
$130$		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
$134$		−	11.1464i		−	0.962905i
$135$		0			0
$136$	−6.69694			−0.574258
$137$	11.6742	+	11.6742i	0.997397	+	0.997397i	0.999997	−	0.00259945i	$-0.000827431\pi$
$137$	11.6742	+	11.6742i	0.997397	+	0.997397i	−0.00259945	+	0.999997i	$0.500827\pi$
$138$		0			0
$139$			18.3485i			1.55630i	0.628080	+	0.778148i	$0.283841\pi$
$139$			18.3485i			1.55630i	−0.628080	+	0.778148i	$0.716159\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			27.5959i			2.29966i
$145$		0			0
$146$	−14.4495			−1.19585
$147$	−15.5732	−	15.5732i	−1.28446	−	1.28446i
$148$		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			$\pi$
$152$	−16.6969	−	16.6969i	−1.35430	−	1.35430i
$153$	11.5505	−	11.5505i	0.933803	−	0.933803i
$154$		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
$162$	−17.8990	−	17.8990i	−1.40628	−	1.40628i
$163$	7.77526	−	7.77526i	0.609005	−	0.609005i	−0.333681	−	0.942686i	$-0.608291\pi$
$163$	7.77526	−	7.77526i	0.609005	−	0.609005i	0.942686	+	0.333681i	$0.108291\pi$
$164$			25.5959i			1.99871i
$165$		0			0
$166$	20.0454			1.55583
$167$		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
$171$	57.5959			4.40447
$172$	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
$176$	−2.20204			−0.165985
$177$	13.3485	+	13.3485i	1.00333	+	1.00333i
$178$	13.8990	−	13.8990i	1.04177	−	1.04177i
$179$		−	26.1464i		−	1.95428i	−0.212607	−	0.977138i	$-0.568195\pi$
$179$		−	26.1464i		−	1.95428i	0.212607	−	0.977138i	$-0.431805\pi$
$180$		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
$187$	0.921683	+	0.921683i	0.0674002	+	0.0674002i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	17.7980	+	17.7980i	1.28446	+	1.28446i
$193$	−19.4722	+	19.4722i	−1.40164	+	1.40164i	−0.606735	+	0.794904i	$0.707521\pi$
$193$	−19.4722	+	19.4722i	−1.40164	+	1.40164i	−0.794904	+	0.606735i	$0.792479\pi$
$194$			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	$0.250000\pi$
$198$	3.79796	−	3.79796i	0.269909	−	0.269909i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	24.7980			1.74911
$202$		0			0
$203$		0			0
$204$		−	14.8990i		−	1.04314i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			4.59592i			0.317906i
$210$		0			0
$211$	15.0454			1.03577			0.517884	−	0.855451i	$-0.326720\pi$
$211$	15.0454			1.03577			0.517884	+	0.855451i	$0.326720\pi$
$212$		0			0
$213$		0			0
$214$		−	6.65153i		−	0.454689i
$215$		0			0
$216$	−34.6969			−2.36083
$217$		0			0
$218$		0			0
$219$		−	32.1464i		−	2.17225i
$220$		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
$226$	30.0454			1.99859
$227$	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	$0.720078\pi$
$228$	37.1464	−	37.1464i	2.46008	−	2.46008i
$229$		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
$241$	−1.69694			−0.109309			−0.0546547	−	0.998505i	$-0.517406\pi$
$241$	−1.69694			−0.109309			−0.0546547	+	0.998505i	$0.517406\pi$
$242$	−10.6969	−	10.6969i	−0.687625	−	0.687625i
$243$	13.7980	−	13.7980i	0.885139	−	0.885139i
$244$		0			0
$245$		0			0
$246$	−56.9444			−3.63064
$247$		0			0
$248$		0			0
$249$			44.5959i			2.82615i
$250$		0			0
$251$	−23.9444			−1.51136			−0.755678	−	0.654943i	$-0.772693\pi$
$251$	−23.9444			−1.51136			−0.755678	+	0.654943i	$0.772693\pi$
$252$		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	$0.864792\pi$
$258$	−26.6969	+	26.6969i	−1.66208	+	1.66208i
$259$		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	$0.750000\pi$
$264$		−	4.89898i		−	0.301511i
$265$		0			0
$266$		0			0
$267$	30.9217	+	30.9217i	1.89238	+	1.89238i
$268$	11.1464	−	11.1464i	0.680876	−	0.680876i
$269$		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			$\pi$
$272$	−6.69694	−	6.69694i	−0.406062	−	0.406062i
$273$		0			0
$274$			23.3485i			1.41053i
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$	−18.3485	+	18.3485i	−1.10047	+	1.10047i
$279$		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
$283$	−4.47219	+	4.47219i	−0.265844	+	0.265844i	−0.827423	−	0.561579i	$-0.810194\pi$
$283$	−4.47219	+	4.47219i	−0.265844	+	0.265844i	0.561579	+	0.827423i	$0.310194\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−27.5959	+	27.5959i	−1.62611	+	1.62611i
$289$		−	11.3939i		−	0.670228i
$290$		0			0
$291$	−53.3939			−3.13000
$292$	−14.4495	−	14.4495i	−0.845592	−	0.845592i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		−	31.1464i		−	1.81650i
$295$		0			0
$296$		0			0
$297$	4.77526	+	4.77526i	0.277088	+	0.277088i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		−	33.3939i		−	1.91527i
$305$		0			0
$306$	23.1010			1.32060
$307$	−17.8207	−	17.8207i	−1.01708	−	1.01708i	−0.999852	−	0.0172273i	$-0.994516\pi$
$307$	−17.8207	−	17.8207i	−1.01708	−	1.01708i	−0.0172273	−	0.999852i	$-0.505484\pi$
$308$		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
$321$	14.7980			0.825942
$322$		0			0
$323$	−13.9773	+	13.9773i	−0.777718	+	0.777718i
$324$		−	35.7980i		−	1.98878i
$325$		0			0
$326$	15.5505			0.861263
$327$		0			0
$328$	−25.5959	+	25.5959i	−1.41330	+	1.41330i
$329$		0			0
$330$		0			0
$331$	35.0454			1.92627			0.963135	−	0.269019i	$-0.0866994\pi$
$331$	35.0454			1.92627			0.963135	+	0.269019i	$0.0866994\pi$
$332$	20.0454	+	20.0454i	1.10013	+	1.10013i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−20.5732	−	20.5732i	−1.12069	−	1.12069i	−0.991637	−	0.129057i	$-0.958805\pi$
$337$	−20.5732	−	20.5732i	−1.12069	−	1.12069i	−0.129057	−	0.991637i	$-0.541195\pi$
$338$	13.0000	−	13.0000i	0.707107	−	0.707107i
$339$			66.8434i			3.63043i
$340$		0			0
$341$		0			0
$342$	57.5959	+	57.5959i	3.11443	+	3.11443i
$343$		0			0
$344$			24.0000i			1.29399i
$345$		0			0
$346$		0			0
$347$	16.6742	+	16.6742i	0.895120	+	0.895120i	0.995000	−	0.0998797i	$-0.0318458\pi$
$347$	16.6742	+	16.6742i	0.895120	+	0.895120i	−0.0998797	+	0.995000i	$0.531846\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	−2.20204	−	2.20204i	−0.117369	−	0.117369i
$353$	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	$0.0443038\pi$
$354$			26.6969i			1.41893i
$355$		0			0
$356$	27.7980			1.47329
$357$		0			0
$358$	26.1464	−	26.1464i	1.38188	−	1.38188i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−50.6969			−2.66826
$362$		0			0
$363$	23.7980	−	23.7980i	1.24907	−	1.24907i
$364$		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
$369$		−	88.2929i		−	4.59634i
$370$		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	$0.750000\pi$
$374$			1.84337i			0.0953182i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	11.6515i		−	0.598499i	−0.954175	−	0.299249i	$-0.903264\pi$
$379$		−	11.6515i		−	0.598499i	0.954175	−	0.299249i	$-0.0967363\pi$
$380$		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	$0.750000\pi$
$384$			35.5959i			1.81650i
$385$		0			0
$386$	−38.9444			−1.98222
$387$	−41.3939	−	41.3939i	−2.10417	−	2.10417i
$388$	−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
$393$	40.0454	−	40.0454i	2.02002	−	2.02002i
$394$		0			0
$395$		0			0
$396$	7.59592			0.381709
$397$		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
$401$	37.2929			1.86232			0.931158	−	0.364615i	$-0.118800\pi$
$401$	37.2929			1.86232			0.931158	+	0.364615i	$0.118800\pi$
$402$	24.7980	+	24.7980i	1.23681	+	1.23681i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	14.8990	−	14.8990i	0.737609	−	0.737609i
$409$		−	18.3939i		−	0.909519i	−0.890614	−	0.454759i	$-0.849725\pi$
$409$		−	18.3939i		−	0.909519i	0.890614	−	0.454759i	$-0.150275\pi$
$410$		0			0
$411$	−51.9444			−2.56223
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−40.8207	−	40.8207i	−1.99900	−	1.99900i
$418$	−4.59592	+	4.59592i	−0.224794	+	0.224794i
$419$			22.8434i			1.11597i	0.829851	+	0.557986i	$0.188426\pi$
$419$			22.8434i			1.11597i	−0.829851	+	0.557986i	$0.811574\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	15.0454	+	15.0454i	0.732399	+	0.732399i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	6.65153	−	6.65153i	0.321514	−	0.321514i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−34.6969	−	34.6969i	−1.66936	−	1.66936i
$433$	17.2702	−	17.2702i	0.829951	−	0.829951i	−0.157559	−	0.987510i	$-0.550362\pi$
$433$	17.2702	−	17.2702i	0.829951	−	0.829951i	0.987510	+	0.157559i	$0.0503624\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	32.1464	−	32.1464i	1.53602	−	1.53602i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	48.2929			2.29966
$442$		0			0
$443$	−26.7196	+	26.7196i	−1.26949	+	1.26949i	−0.323136	+	0.946353i	$0.604737\pi$
$443$	−26.7196	+	26.7196i	−1.26949	+	1.26949i	−0.946353	+	0.323136i	$0.895263\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			16.1010i			0.759854i	0.925016	+	0.379927i	$0.124051\pi$
$449$			16.1010i			0.759854i	−0.925016	+	0.379927i	$0.875949\pi$
$450$		0			0
$451$	7.04541			0.331755
$452$	30.0454	+	30.0454i	1.41322	+	1.41322i
$453$		0			0
$454$			4.00000i			0.187729i
$455$		0			0
$456$	74.2929			3.47908
$457$	3.92168	+	3.92168i	0.183449	+	0.183449i	0.792857	−	0.609408i	$-0.208593\pi$
$457$	3.92168	+	3.92168i	0.183449	+	0.183449i	−0.609408	+	0.792857i	$0.708593\pi$
$458$		0			0
$459$			29.0454i			1.35572i
$460$		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
$473$	3.30306	−	3.30306i	0.151875	−	0.151875i
$474$		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
$482$	−1.69694	−	1.69694i	−0.0772934	−	0.0772934i
$483$		0			0
$484$		−	21.3939i		−	0.972449i
$485$		0			0
$486$	27.5959			1.25178
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$		0			0
$489$			34.5959i			1.56448i
$490$		0			0
$491$	42.0000			1.89543			0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000			1.89543			0.947717	+	0.319113i	$0.103385\pi$
$492$	−56.9444	−	56.9444i	−2.56725	−	2.56725i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−44.5959	+	44.5959i	−1.99839	+	1.99839i
$499$			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
$502$	−23.9444	−	23.9444i	−1.06869	−	1.06869i
$503$		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
$507$	28.9217	+	28.9217i	1.28446	+	1.28446i
$508$		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
$513$	−72.4166	+	72.4166i	−3.19727	+	3.19727i
$514$		−	16.0000i		−	0.705730i
$515$		0			0
$516$	−53.3939			−2.35053
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−36.1918			−1.58559			−0.792797	−	0.609486i	$-0.791376\pi$
$521$	−36.1918			−1.58559			−0.792797	+	0.609486i	$0.791376\pi$
$522$		0			0
$523$	32.2702	−	32.2702i	1.41108	−	1.41108i	0.658456	−	0.752619i	$-0.271210\pi$
$523$	32.2702	−	32.2702i	1.41108	−	1.41108i	0.752619	−	0.658456i	$-0.228790\pi$
$524$		−	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
$531$	−41.3939			−1.79634
$532$		0			0
$533$		0			0
$534$			61.8434i			2.67622i
$535$		0			0
$536$	22.2929			0.962905
$537$	58.1691	+	58.1691i	2.51018	+	2.51018i
$538$		0			0
$539$			3.85357i			0.165985i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		−	13.3939i		−	0.574258i
$545$		0			0
$546$		0			0
$547$	31.1691	+	31.1691i	1.33270	+	1.33270i	0.902950	+	0.429746i	$0.141397\pi$
$547$	31.1691	+	31.1691i	1.33270	+	1.33270i	0.429746	+	0.902950i	$0.358603\pi$
$548$	−23.3485	+	23.3485i	−0.997397	+	0.997397i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−36.6969			−1.55630
$557$		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
$561$	−4.10102			−0.173145
$562$	−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
$566$	−8.94439			−0.375961
$567$		0			0
$568$		0			0
$569$			40.5959i			1.70187i	0.525271	+	0.850935i	$0.323964\pi$
$569$			40.5959i			1.70187i	−0.525271	+	0.850935i	$0.676036\pi$
$570$		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
$576$	−55.1918			−2.29966
$577$	−32.8207	−	32.8207i	−1.36634	−	1.36634i	−0.865592	−	0.500750i	$-0.833058\pi$
$577$	−32.8207	−	32.8207i	−1.36634	−	1.36634i	−0.500750	−	0.865592i	$-0.666942\pi$
$578$	11.3939	−	11.3939i	0.473923	−	0.473923i
$579$		−	86.6413i		−	3.60069i
$580$		0			0
$581$		0			0
$582$	−53.3939	−	53.3939i	−2.21325	−	2.21325i
$583$		0			0
$584$		−	28.8990i		−	1.19585i
$585$		0			0
$586$		0			0
$587$	−13.3258	−	13.3258i	−0.550013	−	0.550013i	0.376431	−	0.926445i	$-0.377151\pi$
$587$	−13.3258	−	13.3258i	−0.550013	−	0.550013i	−0.926445	+	0.376431i	$0.877151\pi$
$588$	31.1464	−	31.1464i	1.28446	−	1.28446i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.97730	+	4.97730i	−0.204393	+	0.204393i	−0.801879	−	0.597486i	$-0.796166\pi$
$593$	−4.97730	+	4.97730i	−0.204393	+	0.204393i	0.597486	+	0.801879i	$0.296166\pi$
$594$			9.55051i			0.391862i
$595$		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
$601$	8.30306			0.338689			0.169344	−	0.985557i	$-0.445835\pi$
$601$	8.30306			0.338689			0.169344	+	0.985557i	$0.445835\pi$
$602$		0			0
$603$	−38.4495	+	38.4495i	−1.56578	+	1.56578i
$604$		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	$0.250000\pi$
$608$	33.3939	−	33.3939i	1.35430	−	1.35430i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	23.1010	+	23.1010i	0.933803	+	0.933803i
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		−	35.6413i		−	1.43837i
$615$		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
$627$	−10.2247	−	10.2247i	−0.408337	−	0.408337i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−33.4722	+	33.4722i	−1.33040	+	1.33040i
$634$		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	$0.188654\pi$
$642$	14.7980	+	14.7980i	0.584029	+	0.584029i
$643$	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
$646$	−27.9546			−1.09986
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	35.7980	−	35.7980i	1.40628	−	1.40628i
$649$		−	3.30306i		−	0.129657i
$650$		0			0
$651$		0			0
$652$	15.5505	+	15.5505i	0.609005	+	0.609005i
$653$		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
$656$	−51.1918			−1.99871
$657$	49.8434	+	49.8434i	1.94457	+	1.94457i
$658$		0			0
$659$		−	50.6413i		−	1.97271i	−0.164644	−	0.986353i	$-0.552648\pi$
$659$		−	50.6413i		−	1.97271i	0.164644	−	0.986353i	$-0.447352\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	35.0454	+	35.0454i	1.36208	+	1.36208i
$663$		0			0
$664$			40.0908i			1.55583i
$665$		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	$-0.188264\pi$
$674$		−	41.1464i		−	1.58490i
$675$		0			0
$676$	26.0000			1.00000
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$	−66.8434	+	66.8434i	−2.56710	+	2.56710i
$679$		0			0
$680$		0			0
$681$	−8.89898			−0.341010
$682$		0			0
$683$	−36.7196	+	36.7196i	−1.40504	+	1.40504i	−0.622101	+	0.782937i	$0.713721\pi$
$683$	−36.7196	+	36.7196i	−1.40504	+	1.40504i	−0.782937	+	0.622101i	$0.786279\pi$
$684$			115.192i			4.40447i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−24.0000	+	24.0000i	−0.914991	+	0.914991i
$689$		0			0
$690$		0			0
$691$	45.0454			1.71361			0.856804	−	0.515642i	$-0.172447\pi$
$691$	45.0454			1.71361			0.856804	+	0.515642i	$0.172447\pi$
$692$		0			0
$693$		0			0
$694$			33.3485i			1.26589i
$695$		0			0
$696$		0			0
$697$	21.4268	+	21.4268i	0.811597	+	0.811597i
$698$		0			0
$699$		−	17.7980i		−	0.673181i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		−	4.40408i		−	0.165985i
$705$		0			0
$706$	32.0000			1.20434
$707$		0			0
$708$	−26.6969	+	26.6969i	−1.00333	+	1.00333i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$	27.7980	+	27.7980i	1.04177	+	1.04177i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	52.2929			1.95428
$717$		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
$722$	−50.6969	−	50.6969i	−1.88674	−	1.88674i
$723$	3.77526	−	3.77526i	0.140403	−	0.140403i
$724$		0			0
$725$		0			0
$726$	47.5959			1.76645
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$			7.69694i			0.285072i
$730$		0			0
$731$	20.0908			0.743086
$732$		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
$737$	−3.06811	−	3.06811i	−0.113015	−	0.113015i
$738$	88.2929	−	88.2929i	3.25010	−	3.25010i
$739$			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
$747$	−69.1464	−	69.1464i	−2.52994	−	2.52994i
$748$	−1.84337	+	1.84337i	−0.0674002	+	0.0674002i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	53.2702	−	53.2702i	1.94127	−	1.94127i
$754$		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	$0.250000\pi$
$758$	11.6515	−	11.6515i	0.423203	−	0.423203i
$759$		0			0
$760$		0			0
$761$	−17.2020			−0.623574			−0.311787	−	0.950152i	$-0.600927\pi$
$761$	−17.2020			−0.623574			−0.311787	+	0.950152i	$0.600927\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−35.5959	+	35.5959i	−1.28446	+	1.28446i
$769$			55.0908i			1.98663i	0.115454	+	0.993313i	$0.463168\pi$
$769$			55.0908i			1.98663i	−0.115454	+	0.993313i	$0.536832\pi$
$770$		0			0
$771$	35.5959			1.28196
$772$	−38.9444	−	38.9444i	−1.40164	−	1.40164i
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		−	82.7878i		−	2.97574i
$775$		0			0
$776$	−48.0000			−1.72310
$777$		0			0
$778$		0			0
$779$			106.843i			3.82806i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		−	28.0000i		−	1.00000i
$785$		0			0
$786$	80.0908			2.85674
$787$	−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
$792$	7.59592	+	7.59592i	0.269909	+	0.269909i
$793$		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
$801$	−95.8888			−3.38806
$802$	37.2929	+	37.2929i	1.31686	+	1.31686i
$803$	−3.97730	+	3.97730i	−0.140356	+	0.140356i
$804$			49.5959i			1.74911i
$805$		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
$816$	29.7980			1.04314
$817$	50.0908	+	50.0908i	1.75246	+	1.75246i
$818$	18.3939	−	18.3939i	0.643127	−	0.643127i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−51.9444	−	51.9444i	−1.81177	−	1.81177i
$823$		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
$827$	−40.0681	−	40.0681i	−1.39330	−	1.39330i	−0.817795	−	0.575510i	$-0.804804\pi$
$827$	−40.0681	−	40.0681i	−1.39330	−	1.39330i	−0.575510	−	0.817795i	$-0.695196\pi$
$828$		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
$833$	−11.7196	+	11.7196i	−0.406062	+	0.406062i
$834$		−	81.6413i		−	2.82701i
$835$		0			0
$836$	−9.19184			−0.317906
$837$		0			0
$838$	−22.8434	+	22.8434i	−0.789111	+	0.789111i
$839$		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
$843$	40.0454	−	40.0454i	1.37924	−	1.37924i
$844$			30.0908i			1.03577i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		−	19.8990i		−	0.682931i
$850$		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
$856$	13.3031			0.454689
$857$	−25.0681	−	25.0681i	−0.856310	−	0.856310i	0.134591	−	0.990901i	$-0.457028\pi$
$857$	−25.0681	−	25.0681i	−0.856310	−	0.856310i	−0.990901	+	0.134591i	$0.957028\pi$
$858$		0			0
$859$		−	21.6515i		−	0.738741i	−0.929282	−	0.369370i	$-0.879573\pi$
$859$		−	21.6515i		−	0.738741i	0.929282	−	0.369370i	$-0.120427\pi$
$860$		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	$0.750000\pi$
$864$		−	69.3939i		−	2.36083i
$865$		0			0
$866$	34.5403			1.17373
$867$	25.3485	+	25.3485i	0.860879	+	0.860879i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	82.7878	−	82.7878i	2.80194	−	2.80194i
$874$		0			0
$875$		0			0
$876$	64.2929			2.17225
$877$		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	$0.598061\pi$
$882$	48.2929	+	48.2929i	1.62611	+	1.62611i
$883$	−22.2247	+	22.2247i	−0.747922	+	0.747922i	−0.974089	−	0.226166i	$-0.927381\pi$
$883$	−22.2247	+	22.2247i	−0.747922	+	0.747922i	0.226166	+	0.974089i	$0.427381\pi$
$884$		0			0
$885$		0			0
$886$	−53.4393			−1.79533
$887$		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
$891$	−9.85357			−0.330107
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−16.1010	+	16.1010i	−0.537298	+	0.537298i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	7.04541	+	7.04541i	0.234586	+	0.234586i
$903$		0			0
$904$			60.0908i			1.99859i
$905$		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			$\pi$
$912$	74.2929	+	74.2929i	2.46008	+	2.46008i
$913$	5.51760	−	5.51760i	0.182606	−	0.182606i
$914$			7.84337i			0.259436i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−29.0454	+	29.0454i	−0.958641	+	0.958641i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	79.2929			2.61279
$922$		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
$931$	−58.4393			−1.91527
$932$	−8.00000	−	8.00000i	−0.262049	−	0.262049i
$933$		0			0
$934$			44.0000i			1.43972i
$935$		0			0
$936$		0			0
$937$	−2.82066	−	2.82066i	−0.0921470	−	0.0921470i	0.659531	−	0.751678i	$-0.270755\pi$
$937$	−2.82066	−	2.82066i	−0.0921470	−	0.0921470i	−0.751678	+	0.659531i	$0.770755\pi$
$938$		0			0
$939$		−	106.788i		−	3.48489i
$940$		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
$946$	6.60612			0.214784
$947$	−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
$953$	−41.7196	+	41.7196i	−1.35143	+	1.35143i	−0.467370	+	0.884062i	$0.654798\pi$
$953$	−41.7196	+	41.7196i	−1.35143	+	1.35143i	−0.884062	+	0.467370i	$0.845202\pi$
$954$		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
$963$	−22.9444	+	22.9444i	−0.739373	+	0.739373i
$964$		−	3.39388i		−	0.109309i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	21.3939	−	21.3939i	0.687625	−	0.687625i
$969$		−	62.1918i		−	1.99789i
$970$		0			0
$971$	−53.9444			−1.73116			−0.865579	−	0.500773i	$-0.833049\pi$
$971$	−53.9444			−1.73116			−0.865579	+	0.500773i	$0.833049\pi$
$972$	27.5959	+	27.5959i	0.885139	+	0.885139i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.3258	−	18.3258i	−0.586293	−	0.586293i	0.350332	−	0.936625i	$-0.386069\pi$
$977$	−18.3258	−	18.3258i	−0.586293	−	0.586293i	−0.936625	+	0.350332i	$0.886069\pi$
$978$	−34.5959	+	34.5959i	−1.10626	+	1.10626i
$979$		−	7.65153i		−	0.244544i
$980$		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	$0.750000\pi$
$984$		−	113.889i		−	3.63064i
$985$		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
$993$	−77.9671	+	77.9671i	−2.47421	+	2.47421i
$994$		0			0
$995$		0			0
$996$	−89.1918			−2.82615
$997$		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.f.107.1	yes	4
4.3	odd	2		800.2.o.f.207.2		4
5.2	odd	4		200.2.k.e.43.2	✓	4
5.3	odd	4	inner	200.2.k.f.43.1	yes	4
5.4	even	2		200.2.k.e.107.2	yes	4
8.3	odd	2	CM	200.2.k.f.107.1	yes	4
8.5	even	2		800.2.o.f.207.2		4
20.3	even	4		800.2.o.f.143.2		4
20.7	even	4		800.2.o.e.143.1		4
20.19	odd	2		800.2.o.e.207.1		4
40.3	even	4	inner	200.2.k.f.43.1	yes	4
40.13	odd	4		800.2.o.f.143.2		4
40.19	odd	2		200.2.k.e.107.2	yes	4
40.27	even	4		200.2.k.e.43.2	✓	4
40.29	even	2		800.2.o.e.207.1		4
40.37	odd	4		800.2.o.e.143.1		4

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

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

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{2} +(-2.22474 + 2.22474i) q^{3} +2.00000i q^{4} -4.44949 q^{6} +(-2.00000 + 2.00000i) q^{8} -6.89898i q^{9} +0.550510 q^{11} +(-4.44949 - 4.44949i) q^{12} -4.00000 q^{16} +(1.67423 + 1.67423i) q^{17} +(6.89898 - 6.89898i) q^{18} +8.34847i q^{19} +(0.550510 + 0.550510i) q^{22} -8.89898i q^{24} +(8.67423 + 8.67423i) q^{27} +(-4.00000 - 4.00000i) q^{32} +(-1.22474 + 1.22474i) q^{33} +3.34847i q^{34} +13.7980 q^{36} +(-8.34847 + 8.34847i) q^{38} +12.7980 q^{41} +(6.00000 - 6.00000i) q^{43} +1.10102i q^{44} +(8.89898 - 8.89898i) q^{48} +7.00000i q^{49} -7.44949 q^{51} +17.3485i q^{54} +(-18.5732 - 18.5732i) q^{57} -6.00000i q^{59} -8.00000i q^{64} -2.44949 q^{66} +(-5.57321 - 5.57321i) q^{67} +(-3.34847 + 3.34847i) q^{68} +(13.7980 + 13.7980i) q^{72} +(-7.22474 + 7.22474i) q^{73} -16.6969 q^{76} -17.8990 q^{81} +(12.7980 + 12.7980i) q^{82} +(10.0227 - 10.0227i) q^{83} +12.0000 q^{86} +(-1.10102 + 1.10102i) q^{88} -13.8990i q^{89} +17.7980 q^{96} +(12.0000 + 12.0000i) q^{97} +(-7.00000 + 7.00000i) q^{98} -3.79796i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} - 8 q^{6} - 8 q^{8} + 12 q^{11} - 8 q^{12} - 16 q^{16} - 8 q^{17} + 8 q^{18} + 12 q^{22} + 20 q^{27} - 16 q^{32} + 16 q^{36} - 4 q^{38} + 12 q^{41} + 24 q^{43} + 16 q^{48} - 20 q^{51}+ \cdots - 28 q^{98}+O(q^{100}) \) 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

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