LMFDB - Embedded newform 800.2.o.f.143.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(143,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, 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("800.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.o (of order $4$, degree $2$, not 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:	$6.38803216170$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			143.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	800.143
Dual form			800.2.o.f.207.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.22474 + 2.22474i) q^{3} +6.89898i q^{9} -0.550510 q^{11} +(1.67423 - 1.67423i) q^{17} +8.34847i q^{19} +(-8.67423 + 8.67423i) q^{27} +(-1.22474 - 1.22474i) q^{33} +12.7980 q^{41} +(-6.00000 - 6.00000i) q^{43} -7.00000i q^{49} +7.44949 q^{51} +(-18.5732 + 18.5732i) q^{57} -6.00000i q^{59} +(5.57321 - 5.57321i) q^{67} +(-7.22474 - 7.22474i) q^{73} -17.8990 q^{81} +(-10.0227 - 10.0227i) q^{83} +13.8990i q^{89} +(12.0000 - 12.0000i) q^{97} -3.79796i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 12 q^{11} - 8 q^{17} - 20 q^{27} + 12 q^{41} - 24 q^{43} + 20 q^{51} - 40 q^{57} - 12 q^{67} - 24 q^{73} - 52 q^{81} + 4 q^{83} + 48 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 - 12 * q^11 - 8 * q^17 - 20 * q^27 + 12 * q^41 - 24 * q^43 + 20 * q^51 - 40 * q^57 - 12 * q^67 - 24 * q^73 - 52 * q^81 + 4 * q^83 + 48 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	2.22474	+	2.22474i	1.28446	+	1.28446i	0.938104	+	0.346353i	$0.112580\pi$
$3$	2.22474	+	2.22474i	1.28446	+	1.28446i	0.346353	+	0.938104i	$0.387420\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$		0			0
$9$			6.89898i			2.29966i
$10$		0			0
$11$	−0.550510			−0.165985			−0.0829925	−	0.996550i	$-0.526448\pi$
$11$	−0.550510			−0.165985			−0.0829925	+	0.996550i	$0.526448\pi$
$12$		0			0
$13$		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
$16$		0			0
$17$	1.67423	−	1.67423i	0.406062	−	0.406062i	−0.474301	−	0.880363i	$-0.657299\pi$
$17$	1.67423	−	1.67423i	0.406062	−	0.406062i	0.880363	+	0.474301i	$0.157299\pi$
$18$		0			0
$19$			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			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$		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			$\pi$
$32$		0			0
$33$	−1.22474	−	1.22474i	−0.213201	−	0.213201i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$		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	$-0.473975\pi$
$43$	−6.00000	−	6.00000i	−0.914991	−	0.914991i	−0.996660	+	0.0816682i	$0.973975\pi$
$44$		0			0
$45$		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			0
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$	7.44949			1.04314
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		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
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.57321	−	5.57321i	0.680876	−	0.680876i	−0.279321	−	0.960198i	$-0.590109\pi$
$67$	5.57321	−	5.57321i	0.680876	−	0.680876i	0.960198	+	0.279321i	$0.0901094\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−7.22474	−	7.22474i	−0.845592	−	0.845592i	0.143987	−	0.989580i	$-0.454008\pi$
$73$	−7.22474	−	7.22474i	−0.845592	−	0.845592i	−0.989580	+	0.143987i	$0.954008\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		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
$82$		0			0
$83$	−10.0227	−	10.0227i	−1.10013	−	1.10013i	−0.994394	−	0.105741i	$-0.966279\pi$
$83$	−10.0227	−	10.0227i	−1.10013	−	1.10013i	−0.105741	−	0.994394i	$-0.533721\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			13.8990i			1.47329i	0.676280	+	0.736644i	$0.263591\pi$
$89$			13.8990i			1.47329i	−0.676280	+	0.736644i	$0.736409\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	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	$-0.0805058\pi$
$98$		0			0
$99$		−	3.79796i		−	0.381709i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		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
$107$	3.32577	−	3.32577i	0.321514	−	0.321514i	−0.527834	−	0.849348i	$-0.676996\pi$
$107$	3.32577	−	3.32577i	0.321514	−	0.321514i	0.849348	+	0.527834i	$0.176996\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	15.0227	+	15.0227i	1.41322	+	1.41322i	0.733148	+	0.680069i	$0.238050\pi$
$113$	15.0227	+	15.0227i	1.41322	+	1.41322i	0.680069	+	0.733148i	$0.261950\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		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	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$		0			0
$129$		−	26.6969i		−	2.35053i
$130$		0			0
$131$	18.0000			1.57267			0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000			1.57267			0.786334	+	0.617802i	$0.211977\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.6742	−	11.6742i	0.997397	−	0.997397i	−0.00259945	−	0.999997i	$-0.500827\pi$
$137$	11.6742	−	11.6742i	0.997397	−	0.997397i	0.999997	+	0.00259945i	$0.000827431\pi$
$138$		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
$144$		0			0
$145$		0			0
$146$		0			0
$147$	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			$\pi$
$152$		0			0
$153$	11.5505	+	11.5505i	0.933803	+	0.933803i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−7.77526	−	7.77526i	−0.609005	−	0.609005i	0.333681	−	0.942686i	$-0.391709\pi$
$163$	−7.77526	−	7.77526i	−0.609005	−	0.609005i	−0.942686	+	0.333681i	$0.891709\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		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
$171$	−57.5959			−4.40447
$172$		0			0
$173$		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
$176$		0			0
$177$	13.3485	−	13.3485i	1.00333	−	1.00333i
$178$		0			0
$179$		−	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			$\pi$
$192$		0			0
$193$	−19.4722	−	19.4722i	−1.40164	−	1.40164i	−0.794904	−	0.606735i	$-0.792479\pi$
$193$	−19.4722	−	19.4722i	−1.40164	−	1.40164i	−0.606735	−	0.794904i	$-0.707521\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		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
$204$		0			0
$205$		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	$-0.673280\pi$
$211$	−15.0454			−1.03577			−0.517884	+	0.855451i	$0.673280\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		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	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.00000	+	2.00000i	−0.132745	+	0.132745i	−0.770357	−	0.637613i	$-0.779922\pi$
$227$	−2.00000	+	2.00000i	−0.132745	+	0.132745i	0.637613	+	0.770357i	$0.279922\pi$
$228$		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
$233$	−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
$236$		0			0
$237$		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	$0.517406\pi$
$242$		0			0
$243$	−13.7980	−	13.7980i	−0.885139	−	0.885139i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		−	44.5959i		−	2.82615i
$250$		0			0
$251$	23.9444			1.51136			0.755678	−	0.654943i	$-0.227307\pi$
$251$	23.9444			1.51136			0.755678	+	0.654943i	$0.227307\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−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	$0.364792\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−30.9217	+	30.9217i	−1.89238	+	1.89238i
$268$		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
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$		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	$-0.189806\pi$
$283$	4.47219	+	4.47219i	0.265844	+	0.265844i	−0.561579	+	0.827423i	$0.689806\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			11.3939i			0.670228i
$290$		0			0
$291$	53.3939			3.13000
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		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
$304$		0			0
$305$		0			0
$306$		0			0
$307$	17.8207	−	17.8207i	1.01708	−	1.01708i	0.0172273	−	0.999852i	$-0.494516\pi$
$307$	17.8207	−	17.8207i	1.01708	−	1.01708i	0.999852	−	0.0172273i	$-0.00548391\pi$
$308$		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
$321$	14.7980			0.825942
$322$		0			0
$323$	13.9773	+	13.9773i	0.777718	+	0.777718i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−35.0454			−1.92627			−0.963135	−	0.269019i	$-0.913301\pi$
$331$	−35.0454			−1.92627			−0.963135	+	0.269019i	$0.913301\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−20.5732	+	20.5732i	−1.12069	+	1.12069i	−0.129057	+	0.991637i	$0.541195\pi$
$337$	−20.5732	+	20.5732i	−1.12069	+	1.12069i	−0.991637	+	0.129057i	$0.958805\pi$
$338$		0			0
$339$			66.8434i			3.63043i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−16.6742	+	16.6742i	−0.895120	+	0.895120i	−0.995000	−	0.0998797i	$-0.968154\pi$
$347$	−16.6742	+	16.6742i	−0.895120	+	0.895120i	0.0998797	+	0.995000i	$0.468154\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	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	$0.544304\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−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	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$			88.2929i			4.59634i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		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	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	41.3939	−	41.3939i	2.10417	−	2.10417i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	40.0454	+	40.0454i	2.02002	+	2.02002i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		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
$401$	37.2929			1.86232			0.931158	−	0.364615i	$-0.118800\pi$
$401$	37.2929			1.86232			0.931158	+	0.364615i	$0.118800\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			18.3939i			0.909519i	0.890614	+	0.454759i	$0.150275\pi$
$409$			18.3939i			0.909519i	−0.890614	+	0.454759i	$0.849725\pi$
$410$		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
$418$		0			0
$419$			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			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	17.2702	+	17.2702i	0.829951	+	0.829951i	0.987510	−	0.157559i	$-0.0503624\pi$
$433$	17.2702	+	17.2702i	0.829951	+	0.829951i	−0.157559	+	0.987510i	$0.550362\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		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	0.946353	+	0.323136i	$0.104737\pi$
$443$	26.7196	+	26.7196i	1.26949	+	1.26949i	0.323136	+	0.946353i	$0.395263\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	16.1010i		−	0.759854i	−0.925016	−	0.379927i	$-0.875949\pi$
$449$		−	16.1010i		−	0.759854i	0.925016	−	0.379927i	$-0.124051\pi$
$450$		0			0
$451$	−7.04541			−0.331755
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.92168	−	3.92168i	0.183449	−	0.183449i	−0.609408	−	0.792857i	$-0.708593\pi$
$457$	3.92168	−	3.92168i	0.183449	−	0.183449i	0.792857	+	0.609408i	$0.208593\pi$
$458$		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	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−22.0000	+	22.0000i	−1.01804	+	1.01804i	−0.0182043	+	0.999834i	$0.505795\pi$
$467$	−22.0000	+	22.0000i	−1.01804	+	1.01804i	−0.999834	+	0.0182043i	$0.994205\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	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
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$		−	34.5959i		−	1.56448i
$490$		0			0
$491$	−42.0000			−1.89543			−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000			−1.89543			−0.947717	+	0.319113i	$0.896615\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			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
$502$		0			0
$503$		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
$507$	−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
$512$		0			0
$513$	−72.4166	−	72.4166i	−3.19727	−	3.19727i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		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	−0.752619	−	0.658456i	$-0.771210\pi$
$523$	−32.2702	−	32.2702i	−1.41108	−	1.41108i	−0.658456	−	0.752619i	$-0.728790\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			23.0000i			1.00000i
$530$		0			0
$531$	41.3939			1.79634
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	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
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−31.1691	+	31.1691i	−1.33270	+	1.33270i	−0.429746	+	0.902950i	$0.641397\pi$
$547$	−31.1691	+	31.1691i	−1.33270	+	1.33270i	−0.902950	+	0.429746i	$0.858603\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		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
$561$	−4.10102			−0.173145
$562$		0			0
$563$	−26.0000	−	26.0000i	−1.09577	−	1.09577i	−0.994900	−	0.100870i	$-0.967837\pi$
$563$	−26.0000	−	26.0000i	−1.09577	−	1.09577i	−0.100870	−	0.994900i	$-0.532163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	40.5959i		−	1.70187i	−0.525271	−	0.850935i	$-0.676036\pi$
$569$		−	40.5959i		−	1.70187i	0.525271	−	0.850935i	$-0.323964\pi$
$570$		0			0
$571$	−22.0000			−0.920671			−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000			−0.920671			−0.460336	+	0.887745i	$0.652271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−32.8207	+	32.8207i	−1.36634	+	1.36634i	−0.500750	+	0.865592i	$0.666942\pi$
$577$	−32.8207	+	32.8207i	−1.36634	+	1.36634i	−0.865592	+	0.500750i	$0.833058\pi$
$578$		0			0
$579$		−	86.6413i		−	3.60069i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.3258	−	13.3258i	0.550013	−	0.550013i	−0.376431	−	0.926445i	$-0.622849\pi$
$587$	13.3258	−	13.3258i	0.550013	−	0.550013i	0.926445	+	0.376431i	$0.122849\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.97730	−	4.97730i	−0.204393	−	0.204393i	0.597486	−	0.801879i	$-0.296166\pi$
$593$	−4.97730	−	4.97730i	−0.204393	−	0.204393i	−0.801879	+	0.597486i	$0.796166\pi$
$594$		0			0
$595$		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	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		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	$-0.824994\pi$
$619$		−	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
$626$		0			0
$627$	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	$0.188654\pi$
$642$		0			0
$643$	−6.00000	−	6.00000i	−0.236617	−	0.236617i	0.578831	−	0.815448i	$-0.303509\pi$
$643$	−6.00000	−	6.00000i	−0.236617	−	0.236617i	−0.815448	+	0.578831i	$0.803509\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$			3.30306i			0.129657i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		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
$656$		0			0
$657$	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			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		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	$0.311736\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−8.89898			−0.341010
$682$		0			0
$683$	36.7196	+	36.7196i	1.40504	+	1.40504i	0.782937	+	0.622101i	$0.213721\pi$
$683$	36.7196	+	36.7196i	1.40504	+	1.40504i	0.622101	+	0.782937i	$0.286279\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−45.0454			−1.71361			−0.856804	−	0.515642i	$-0.827553\pi$
$691$	−45.0454			−1.71361			−0.856804	+	0.515642i	$0.827553\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		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
$704$		0			0
$705$		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
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		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
$722$		0			0
$723$	−3.77526	−	3.77526i	−0.140403	−	0.140403i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		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	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.06811	+	3.06811i	−0.113015	+	0.113015i
$738$		0			0
$739$			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	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	69.1464	−	69.1464i	2.52994	−	2.52994i
$748$		0			0
$749$		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	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		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
$768$		0			0
$769$		−	55.0908i		−	1.98663i	−0.115454	−	0.993313i	$-0.536832\pi$
$769$		−	55.0908i		−	1.98663i	0.115454	−	0.993313i	$-0.463168\pi$
$770$		0			0
$771$	−35.5959			−1.28196
$772$		0			0
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			106.843i			3.82806i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	18.0000	−	18.0000i	0.641631	−	0.641631i	−0.309326	−	0.950956i	$-0.600103\pi$
$787$	18.0000	−	18.0000i	0.641631	−	0.641631i	0.950956	+	0.309326i	$0.100103\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		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
$801$	−95.8888			−3.38806
$802$		0			0
$803$	3.97730	+	3.97730i	0.140356	+	0.140356i
$804$		0			0
$805$		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	$-0.267501\pi$
$811$	38.0000			1.33436			0.667180	+	0.744896i	$0.267501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	50.0908	−	50.0908i	1.75246	−	1.75246i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		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
$827$	40.0681	−	40.0681i	1.39330	−	1.39330i	0.575510	−	0.817795i	$-0.304804\pi$
$827$	40.0681	−	40.0681i	1.39330	−	1.39330i	0.817795	−	0.575510i	$-0.195196\pi$
$828$		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
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		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
$844$		0			0
$845$		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	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−25.0681	+	25.0681i	−0.856310	+	0.856310i	−0.990901	−	0.134591i	$-0.957028\pi$
$857$	−25.0681	+	25.0681i	−0.856310	+	0.856310i	0.134591	+	0.990901i	$0.457028\pi$
$858$		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	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−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
$876$		0			0
$877$		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	$0.598061\pi$
$882$		0			0
$883$	22.2247	+	22.2247i	0.747922	+	0.747922i	0.974089	−	0.226166i	$-0.0726193\pi$
$883$	22.2247	+	22.2247i	0.747922	+	0.747922i	−0.226166	+	0.974089i	$0.572619\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		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
$891$	9.85357			0.330107
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−42.0000	+	42.0000i	−1.39459	+	1.39459i	−0.579898	+	0.814689i	$0.696908\pi$
$907$	−42.0000	+	42.0000i	−1.39459	+	1.39459i	−0.814689	+	0.579898i	$0.803092\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	5.51760	+	5.51760i	0.182606	+	0.182606i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		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	$-0.653582\pi$
$929$		−	54.0000i		−	1.77168i	0.463988	−	0.885841i	$-0.346418\pi$
$930$		0			0
$931$	58.4393			1.91527
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−2.82066	+	2.82066i	−0.0921470	+	0.0921470i	−0.751678	−	0.659531i	$-0.770755\pi$
$937$	−2.82066	+	2.82066i	−0.0921470	+	0.0921470i	0.659531	+	0.751678i	$0.270755\pi$
$938$		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
$944$		0			0
$945$		0			0
$946$		0			0
$947$	38.0000	−	38.0000i	1.23483	−	1.23483i	0.272749	−	0.962085i	$-0.412067\pi$
$947$	38.0000	−	38.0000i	1.23483	−	1.23483i	0.962085	−	0.272749i	$-0.0879328\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−41.7196	−	41.7196i	−1.35143	−	1.35143i	−0.884062	−	0.467370i	$-0.845202\pi$
$953$	−41.7196	−	41.7196i	−1.35143	−	1.35143i	−0.467370	−	0.884062i	$-0.654798\pi$
$954$		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
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$		0			0
$969$			62.1918i			1.99789i
$970$		0			0
$971$	53.9444			1.73116			0.865579	−	0.500773i	$-0.166951\pi$
$971$	53.9444			1.73116			0.865579	+	0.500773i	$0.166951\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.3258	+	18.3258i	−0.586293	+	0.586293i	−0.936625	−	0.350332i	$-0.886069\pi$
$977$	−18.3258	+	18.3258i	−0.586293	+	0.586293i	0.350332	+	0.936625i	$0.386069\pi$
$978$		0			0
$979$		−	7.65153i		−	0.244544i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		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
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		0			0

Twists

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

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

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.o (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.22474 + 2.22474i) q^{3} +6.89898i q^{9} -0.550510 q^{11} +(1.67423 - 1.67423i) q^{17} +8.34847i q^{19} +(-8.67423 + 8.67423i) q^{27} +(-1.22474 - 1.22474i) q^{33} +12.7980 q^{41} +(-6.00000 - 6.00000i) q^{43} -7.00000i q^{49} +7.44949 q^{51} +(-18.5732 + 18.5732i) q^{57} -6.00000i q^{59} +(5.57321 - 5.57321i) q^{67} +(-7.22474 - 7.22474i) q^{73} -17.8990 q^{81} +(-10.0227 - 10.0227i) q^{83} +13.8990i q^{89} +(12.0000 - 12.0000i) q^{97} -3.79796i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 12 q^{11} - 8 q^{17} - 20 q^{27} + 12 q^{41} - 24 q^{43} + 20 q^{51} - 40 q^{57} - 12 q^{67} - 24 q^{73} - 52 q^{81} + 4 q^{83} + 48 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 - 12 * q^11 - 8 * q^17 - 20 * q^27 + 12 * q^41 - 24 * q^43 + 20 * q^51 - 40 * q^57 - 12 * q^67 - 24 * q^73 - 52 * q^81 + 4 * q^83 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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