LMFDB - Embedded newform 1200.2.v.f.593.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,2,Mod(257,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))

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

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

chi := DirichletCharacter("1200.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1200.v (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:	$9.58204824255$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			593.1
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.593
Dual form			1200.2.v.f.257.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(1.22474 - 1.22474i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(-1.22474 - 1.22474i) q^{3} +(1.22474 - 1.22474i) q^{7} +3.00000i q^{9} +(-3.67423 - 3.67423i) q^{13} -1.00000i q^{19} -3.00000 q^{21} +(3.67423 - 3.67423i) q^{27} -7.00000 q^{31} +(4.89898 - 4.89898i) q^{37} +9.00000i q^{39} +(-8.57321 - 8.57321i) q^{43} +4.00000i q^{49} +(-1.22474 + 1.22474i) q^{57} -13.0000 q^{61} +(3.67423 + 3.67423i) q^{63} +(-11.0227 + 11.0227i) q^{67} +(-9.79796 - 9.79796i) q^{73} +4.00000i q^{79} -9.00000 q^{81} -9.00000 q^{91} +(8.57321 + 8.57321i) q^{93} +(-13.4722 + 13.4722i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 12 q^{21} - 28 q^{31} - 52 q^{61} - 36 q^{81} - 36 q^{91}+O(q^{100}) $ 4 * q - 12 * q^21 - 28 * q^31 - 52 * q^61 - 36 * q^81 - 36 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$3$	−1.22474	−	1.22474i	−0.707107	−	0.707107i
$3$	−1.22474	−	1.22474i	−0.707107	−	0.707107i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.22474	−	1.22474i	0.462910	−	0.462910i	−0.436698	−	0.899608i	$-0.643852\pi$
$7$	1.22474	−	1.22474i	0.462910	−	0.462910i	0.899608	+	0.436698i	$0.143852\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−3.67423	−	3.67423i	−1.01905	−	1.01905i	−0.999815	−	0.0192343i	$-0.993877\pi$
$13$	−3.67423	−	3.67423i	−1.01905	−	1.01905i	−0.0192343	−	0.999815i	$-0.506123\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$		−	1.00000i		−	0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$		0			0
$21$	−3.00000			−0.654654
$22$		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
$27$	3.67423	−	3.67423i	0.707107	−	0.707107i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.89898	−	4.89898i	0.805387	−	0.805387i	−0.178545	−	0.983932i	$-0.557139\pi$
$37$	4.89898	−	4.89898i	0.805387	−	0.805387i	0.983932	+	0.178545i	$0.0571389\pi$
$38$		0			0
$39$			9.00000i			1.44115i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−8.57321	−	8.57321i	−1.30740	−	1.30740i	−0.923283	−	0.384120i	$-0.874505\pi$
$43$	−8.57321	−	8.57321i	−1.30740	−	1.30740i	−0.384120	−	0.923283i	$-0.625495\pi$
$44$		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
$49$			4.00000i			0.571429i
$50$		0			0
$51$		0			0
$52$		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
$57$	−1.22474	+	1.22474i	−0.162221	+	0.162221i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−13.0000			−1.66448			−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000			−1.66448			−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$	3.67423	+	3.67423i	0.462910	+	0.462910i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−11.0227	+	11.0227i	−1.34664	+	1.34664i	−0.457352	+	0.889286i	$0.651202\pi$
$67$	−11.0227	+	11.0227i	−1.34664	+	1.34664i	−0.889286	+	0.457352i	$0.848798\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−9.79796	−	9.79796i	−1.14676	−	1.14676i	−0.987185	−	0.159579i	$-0.948986\pi$
$73$	−9.79796	−	9.79796i	−1.14676	−	1.14676i	−0.159579	−	0.987185i	$-0.551014\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$			4.00000i			0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$			4.00000i			0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−9.00000			−0.943456
$92$		0			0
$93$	8.57321	+	8.57321i	0.889001	+	0.889001i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−13.4722	+	13.4722i	−1.36789	+	1.36789i	−0.504457	+	0.863437i	$0.668307\pi$
$97$	−13.4722	+	13.4722i	−1.36789	+	1.36789i	−0.863437	+	0.504457i	$0.831693\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−2.44949	−	2.44949i	−0.241355	−	0.241355i	0.576055	−	0.817411i	$-0.304591\pi$
$103$	−2.44949	−	2.44949i	−0.241355	−	0.241355i	−0.817411	+	0.576055i	$0.804591\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		−	19.0000i		−	1.81987i	−0.414751	−	0.909935i	$-0.636131\pi$
$109$		−	19.0000i		−	1.81987i	0.414751	−	0.909935i	$-0.363869\pi$
$110$		0			0
$111$	−12.0000			−1.13899
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	11.0227	−	11.0227i	1.01905	−	1.01905i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	7.34847	−	7.34847i	0.652071	−	0.652071i	−0.301420	−	0.953491i	$-0.597461\pi$
$127$	7.34847	−	7.34847i	0.652071	−	0.652071i	0.953491	+	0.301420i	$0.0974607\pi$
$128$		0			0
$129$			21.0000i			1.84895i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	−1.22474	−	1.22474i	−0.106199	−	0.106199i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$		−	16.0000i		−	1.35710i	−0.734553	−	0.678551i	$-0.762608\pi$
$139$		−	16.0000i		−	1.35710i	0.734553	−	0.678551i	$-0.237392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.89898	−	4.89898i	0.404061	−	0.404061i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	23.0000			1.87171			0.935857	−	0.352381i	$-0.114628\pi$
$151$	23.0000			1.87171			0.935857	+	0.352381i	$0.114628\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.22474	+	1.22474i	−0.0977453	+	0.0977453i	−0.754288	−	0.656543i	$-0.772018\pi$
$157$	−1.22474	+	1.22474i	−0.0977453	+	0.0977453i	0.656543	+	0.754288i	$0.272018\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	3.67423	+	3.67423i	0.287788	+	0.287788i	0.836205	−	0.548417i	$-0.184769\pi$
$163$	3.67423	+	3.67423i	0.287788	+	0.287788i	−0.548417	+	0.836205i	$0.684769\pi$
$164$		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
$169$			14.0000i			1.07692i
$170$		0			0
$171$	3.00000			0.229416
$172$		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
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	7.00000			0.520306			0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000			0.520306			0.260153	+	0.965567i	$0.416227\pi$
$182$		0			0
$183$	15.9217	+	15.9217i	1.17696	+	1.17696i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	9.00000i		−	0.654654i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	8.57321	+	8.57321i	0.617113	+	0.617113i	0.944790	−	0.327677i	$-0.106266\pi$
$193$	8.57321	+	8.57321i	0.617113	+	0.617113i	−0.327677	+	0.944790i	$0.606266\pi$
$194$		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
$199$		−	11.0000i		−	0.779769i	−0.920864	−	0.389885i	$-0.872515\pi$
$199$		−	11.0000i		−	0.779769i	0.920864	−	0.389885i	$-0.127485\pi$
$200$		0			0
$201$	27.0000			1.90443
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	13.0000			0.894957			0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000			0.894957			0.447478	+	0.894295i	$0.352322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−8.57321	+	8.57321i	−0.581988	+	0.581988i
$218$		0			0
$219$			24.0000i			1.62177i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−20.8207	−	20.8207i	−1.39425	−	1.39425i	−0.815506	−	0.578749i	$-0.803541\pi$
$223$	−20.8207	−	20.8207i	−1.39425	−	1.39425i	−0.578749	−	0.815506i	$-0.696459\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		−	29.0000i		−	1.91637i	−0.286143	−	0.958187i	$-0.592373\pi$
$229$		−	29.0000i		−	1.91637i	0.286143	−	0.958187i	$-0.407627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	4.89898	−	4.89898i	0.318223	−	0.318223i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	17.0000			1.09507			0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000			1.09507			0.547533	+	0.836784i	$0.315567\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.707107	+	0.707107i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3.67423	+	3.67423i	−0.233786	+	0.233786i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		−	12.0000i		−	0.745644i
$260$		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
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	28.0000			1.70088			0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000			1.70088			0.850439	+	0.526073i	$0.176336\pi$
$272$		0			0
$273$	11.0227	+	11.0227i	0.667124	+	0.667124i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	23.2702	−	23.2702i	1.39817	−	1.39817i	0.592869	−	0.805299i	$-0.297995\pi$
$277$	23.2702	−	23.2702i	1.39817	−	1.39817i	0.805299	−	0.592869i	$-0.202005\pi$
$278$		0			0
$279$		−	21.0000i		−	1.25724i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	15.9217	+	15.9217i	0.946446	+	0.946446i	0.998637	−	0.0521913i	$-0.0166205\pi$
$283$	15.9217	+	15.9217i	0.946446	+	0.946446i	−0.0521913	+	0.998637i	$0.516621\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$	33.0000			1.93449
$292$		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
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−21.0000			−1.21042
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.22474	−	1.22474i	0.0698999	−	0.0698999i	−0.671293	−	0.741192i	$-0.734261\pi$
$307$	1.22474	−	1.22474i	0.0698999	−	0.0698999i	0.741192	+	0.671293i	$0.234261\pi$
$308$		0			0
$309$			6.00000i			0.341328i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−3.67423	−	3.67423i	−0.207680	−	0.207680i	0.595601	−	0.803281i	$-0.296914\pi$
$313$	−3.67423	−	3.67423i	−0.207680	−	0.207680i	−0.803281	+	0.595601i	$0.796914\pi$
$314$		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
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−23.2702	+	23.2702i	−1.28684	+	1.28684i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−32.0000			−1.75888			−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000			−1.75888			−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$	14.6969	+	14.6969i	0.805387	+	0.805387i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−25.7196	+	25.7196i	−1.40104	+	1.40104i	−0.604223	+	0.796815i	$0.706516\pi$
$337$	−25.7196	+	25.7196i	−1.40104	+	1.40104i	−0.796815	+	0.604223i	$0.793484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	13.4722	+	13.4722i	0.727430	+	0.727430i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		−	14.0000i		−	0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$		−	14.0000i		−	0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$		0			0
$351$	−27.0000			−1.44115
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$		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
$361$	18.0000			0.947368
$362$		0			0
$363$	−13.4722	−	13.4722i	−0.707107	−	0.707107i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−11.0227	+	11.0227i	−0.575380	+	0.575380i	−0.933627	−	0.358247i	$-0.883375\pi$
$367$	−11.0227	+	11.0227i	−0.575380	+	0.575380i	0.358247	+	0.933627i	$0.383375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	20.8207	+	20.8207i	1.07805	+	1.07805i	0.996684	+	0.0813690i	$0.0259292\pi$
$373$	20.8207	+	20.8207i	1.07805	+	1.07805i	0.0813690	+	0.996684i	$0.474071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			29.0000i			1.48963i	0.667271	+	0.744815i	$0.267462\pi$
$379$			29.0000i			1.48963i	−0.667271	+	0.744815i	$0.732538\pi$
$380$		0			0
$381$	−18.0000			−0.922168
$382$		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
$387$	25.7196	−	25.7196i	1.30740	−	1.30740i
$388$		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
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−13.4722	+	13.4722i	−0.676150	+	0.676150i	−0.959127	−	0.282977i	$-0.908678\pi$
$397$	−13.4722	+	13.4722i	−0.676150	+	0.676150i	0.282977	+	0.959127i	$0.408678\pi$
$398$		0			0
$399$			3.00000i			0.150188i
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	25.7196	+	25.7196i	1.28119	+	1.28119i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			31.0000i			1.53285i	0.642333	+	0.766426i	$0.277967\pi$
$409$			31.0000i			1.53285i	−0.642333	+	0.766426i	$0.722033\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−19.5959	+	19.5959i	−0.959616	+	0.959616i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−15.9217	+	15.9217i	−0.770504	+	0.770504i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−15.9217	−	15.9217i	−0.765147	−	0.765147i	0.212101	−	0.977248i	$-0.431970\pi$
$433$	−15.9217	−	15.9217i	−0.765147	−	0.765147i	−0.977248	+	0.212101i	$0.931970\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	41.0000i		−	1.95682i	−0.206666	−	0.978412i	$-0.566261\pi$
$439$		−	41.0000i		−	1.95682i	0.206666	−	0.978412i	$-0.433739\pi$
$440$		0			0
$441$	−12.0000			−0.571429
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−28.1691	−	28.1691i	−1.32350	−	1.32350i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	29.3939	−	29.3939i	1.37499	−	1.37499i	0.522108	−	0.852879i	$-0.325146\pi$
$457$	29.3939	−	29.3939i	1.37499	−	1.37499i	0.852879	−	0.522108i	$-0.174854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−26.9444	−	26.9444i	−1.25221	−	1.25221i	−0.954726	−	0.297486i	$-0.903852\pi$
$463$	−26.9444	−	26.9444i	−1.25221	−	1.25221i	−0.297486	−	0.954726i	$-0.596148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$			27.0000i			1.24674i
$470$		0			0
$471$	3.00000			0.138233
$472$		0			0
$473$		0			0
$474$		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
$481$	−36.0000			−1.64146
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	25.7196	−	25.7196i	1.16547	−	1.16547i	0.182208	−	0.983260i	$-0.441675\pi$
$487$	25.7196	−	25.7196i	1.16547	−	1.16547i	0.983260	−	0.182208i	$-0.0583245\pi$
$488$		0			0
$489$		−	9.00000i		−	0.406994i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	11.0000i		−	0.492428i	−0.969216	−	0.246214i	$-0.920813\pi$
$499$		−	11.0000i		−	0.492428i	0.969216	−	0.246214i	$-0.0791865\pi$
$500$		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
$507$	17.1464	−	17.1464i	0.761500	−	0.761500i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−24.0000			−1.06170
$512$		0			0
$513$	−3.67423	−	3.67423i	−0.162221	−	0.162221i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−20.8207	−	20.8207i	−0.910424	−	0.910424i	0.0858814	−	0.996305i	$-0.472629\pi$
$523$	−20.8207	−	20.8207i	−0.910424	−	0.910424i	−0.996305	+	0.0858814i	$0.972629\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			23.0000i			1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$	−8.57321	−	8.57321i	−0.367912	−	0.367912i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−17.1464	+	17.1464i	−0.733128	+	0.733128i	−0.971238	−	0.238110i	$-0.923472\pi$
$547$	−17.1464	+	17.1464i	−0.733128	+	0.733128i	0.238110	+	0.971238i	$0.423472\pi$
$548$		0			0
$549$		−	39.0000i		−	1.66448i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	4.89898	+	4.89898i	0.208326	+	0.208326i
$554$		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
$559$			63.0000i			2.66462i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−11.0227	+	11.0227i	−0.462910	+	0.462910i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	−47.0000			−1.96689			−0.983444	−	0.181210i	$-0.941999\pi$
$571$	−47.0000			−1.96689			−0.983444	+	0.181210i	$0.941999\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	23.2702	−	23.2702i	0.968749	−	0.968749i	−0.0307771	−	0.999526i	$-0.509798\pi$
$577$	23.2702	−	23.2702i	0.968749	−	0.968749i	0.999526	+	0.0307771i	$0.00979822\pi$
$578$		0			0
$579$		−	21.0000i		−	0.872730i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$			7.00000i			0.288430i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−13.4722	+	13.4722i	−0.551380	+	0.551380i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−23.0000			−0.938190			−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000			−0.938190			−0.469095	+	0.883148i	$0.655420\pi$
$602$		0			0
$603$	−33.0681	−	33.0681i	−1.34664	−	1.34664i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	31.8434	−	31.8434i	1.29248	−	1.29248i	0.359235	−	0.933247i	$-0.383038\pi$
$607$	31.8434	−	31.8434i	1.29248	−	1.29248i	0.933247	−	0.359235i	$-0.116962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−34.2929	−	34.2929i	−1.38508	−	1.38508i	−0.835337	−	0.549739i	$-0.814727\pi$
$613$	−34.2929	−	34.2929i	−1.38508	−	1.38508i	−0.549739	−	0.835337i	$-0.685273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$			49.0000i			1.96948i	0.174042	+	0.984738i	$0.444317\pi$
$619$			49.0000i			1.96948i	−0.174042	+	0.984738i	$0.555683\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	43.0000			1.71180			0.855901	−	0.517139i	$-0.173003\pi$
$631$	43.0000			1.71180			0.855901	+	0.517139i	$0.173003\pi$
$632$		0			0
$633$	−15.9217	−	15.9217i	−0.632830	−	0.632830i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	14.6969	−	14.6969i	0.582314	−	0.582314i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	22.0454	+	22.0454i	0.869386	+	0.869386i	0.992404	−	0.123018i	$-0.0392574\pi$
$643$	22.0454	+	22.0454i	0.869386	+	0.869386i	−0.123018	+	0.992404i	$0.539257\pi$
$644$		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
$649$		0			0
$650$		0			0
$651$	21.0000			0.823055
$652$		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
$657$	29.3939	−	29.3939i	1.14676	−	1.14676i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−38.0000			−1.47803			−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000			−1.47803			−0.739014	+	0.673690i	$0.764708\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			51.0000i			1.97177i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−9.79796	−	9.79796i	−0.377684	−	0.377684i	0.492582	−	0.870266i	$-0.336053\pi$
$673$	−9.79796	−	9.79796i	−0.377684	−	0.377684i	−0.870266	+	0.492582i	$0.836053\pi$
$674$		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
$679$			33.0000i			1.26642i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−35.5176	+	35.5176i	−1.35508	+	1.35508i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−4.89898	−	4.89898i	−0.184769	−	0.184769i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			31.0000i			1.16423i	0.813107	+	0.582115i	$0.197775\pi$
$709$			31.0000i			1.16423i	−0.813107	+	0.582115i	$0.802225\pi$
$710$		0			0
$711$	−12.0000			−0.450035
$712$		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
$721$	−6.00000			−0.223452
$722$		0			0
$723$	−20.8207	−	20.8207i	−0.774329	−	0.774329i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	37.9671	−	37.9671i	1.40812	−	1.40812i	0.638498	−	0.769624i	$-0.279556\pi$
$727$	37.9671	−	37.9671i	1.40812	−	1.40812i	0.769624	−	0.638498i	$-0.220444\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	14.6969	+	14.6969i	0.542844	+	0.542844i	0.924362	−	0.381518i	$-0.124598\pi$
$733$	14.6969	+	14.6969i	0.542844	+	0.542844i	−0.381518	+	0.924362i	$0.624598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	16.0000i		−	0.588570i	−0.955718	−	0.294285i	$-0.904919\pi$
$739$		−	16.0000i		−	0.588570i	0.955718	−	0.294285i	$-0.0950814\pi$
$740$		0			0
$741$	9.00000			0.330623
$742$		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
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−52.0000			−1.89751			−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−52.0000			−1.89751			−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.22474	+	1.22474i	−0.0445141	+	0.0445141i	−0.729013	−	0.684499i	$-0.760021\pi$
$757$	−1.22474	+	1.22474i	−0.0445141	+	0.0445141i	0.684499	+	0.729013i	$0.260021\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−23.2702	−	23.2702i	−0.842436	−	0.842436i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	49.0000i		−	1.76699i	−0.468445	−	0.883493i	$-0.655186\pi$
$769$		−	49.0000i		−	1.76699i	0.468445	−	0.883493i	$-0.344814\pi$
$770$		0			0
$771$		0			0
$772$		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
$777$	−14.6969	+	14.6969i	−0.527250	+	0.527250i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−35.5176	+	35.5176i	−1.26607	+	1.26607i	−0.317962	+	0.948103i	$0.602999\pi$
$787$	−35.5176	+	35.5176i	−1.26607	+	1.26607i	−0.948103	+	0.317962i	$0.897001\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	47.7650	+	47.7650i	1.69619	+	1.69619i
$794$		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
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	−37.0000			−1.29925			−0.649623	−	0.760257i	$-0.725073\pi$
$811$	−37.0000			−1.29925			−0.649623	+	0.760257i	$0.725073\pi$
$812$		0			0
$813$	−34.2929	−	34.2929i	−1.20270	−	1.20270i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−8.57321	+	8.57321i	−0.299939	+	0.299939i
$818$		0			0
$819$		−	27.0000i		−	0.943456i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	40.4166	+	40.4166i	1.40883	+	1.40883i	0.766037	+	0.642796i	$0.222226\pi$
$823$	40.4166	+	40.4166i	1.40883	+	1.40883i	0.642796	+	0.766037i	$0.277774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$			46.0000i			1.59765i	0.601566	+	0.798823i	$0.294544\pi$
$829$			46.0000i			1.59765i	−0.601566	+	0.798823i	$0.705456\pi$
$830$		0			0
$831$	−57.0000			−1.97731
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−25.7196	+	25.7196i	−0.889001	+	0.889001i
$838$		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
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	13.4722	−	13.4722i	0.462910	−	0.462910i
$848$		0			0
$849$		−	39.0000i		−	1.33848i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	33.0681	+	33.0681i	1.13223	+	1.13223i	0.989806	+	0.142425i	$0.0454900\pi$
$853$	33.0681	+	33.0681i	1.13223	+	1.13223i	0.142425	+	0.989806i	$0.454510\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		−	56.0000i		−	1.91070i	−0.295484	−	0.955348i	$-0.595481\pi$
$859$		−	56.0000i		−	1.91070i	0.295484	−	0.955348i	$-0.404519\pi$
$860$		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
$867$	−20.8207	+	20.8207i	−0.707107	+	0.707107i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	81.0000			2.74458
$872$		0			0
$873$	−40.4166	−	40.4166i	−1.36789	−	1.36789i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−37.9671	+	37.9671i	−1.28206	+	1.28206i	−0.342563	+	0.939495i	$0.611295\pi$
$877$	−37.9671	+	37.9671i	−1.28206	+	1.28206i	−0.939495	+	0.342563i	$0.888705\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	15.9217	+	15.9217i	0.535807	+	0.535807i	0.922295	−	0.386487i	$-0.126312\pi$
$883$	15.9217	+	15.9217i	0.535807	+	0.535807i	−0.386487	+	0.922295i	$0.626312\pi$
$884$		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
$889$		−	18.0000i		−	0.603701i
$890$		0			0
$891$		0			0
$892$		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
$903$	25.7196	+	25.7196i	0.855896	+	0.855896i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	31.8434	−	31.8434i	1.05734	−	1.05734i	0.0590889	−	0.998253i	$-0.481180\pi$
$907$	31.8434	−	31.8434i	1.05734	−	1.05734i	0.998253	−	0.0590889i	$-0.0188195\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	1.00000i		−	0.0329870i	−0.999864	−	0.0164935i	$-0.994750\pi$
$919$		−	1.00000i		−	0.0329870i	0.999864	−	0.0164935i	$-0.00525028\pi$
$920$		0			0
$921$	−3.00000			−0.0988534
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	7.34847	−	7.34847i	0.241355	−	0.241355i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	4.00000			0.131095
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	35.5176	−	35.5176i	1.16031	−	1.16031i	0.175902	−	0.984408i	$-0.443716\pi$
$937$	35.5176	−	35.5176i	1.16031	−	1.16031i	0.984408	−	0.175902i	$-0.0562841\pi$
$938$		0			0
$939$			9.00000i			0.293704i
$940$		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
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$			72.0000i			2.33722i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	18.0000			0.580645
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−41.6413	+	41.6413i	−1.33909	+	1.33909i	−0.442157	+	0.896938i	$0.645787\pi$
$967$	−41.6413	+	41.6413i	−1.33909	+	1.33909i	−0.896938	+	0.442157i	$0.854213\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	−19.5959	−	19.5959i	−0.628216	−	0.628216i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	57.0000			1.81987
$982$		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
$991$	−17.0000			−0.540023			−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000			−0.540023			−0.270011	+	0.962857i	$0.587027\pi$
$992$		0			0
$993$	39.1918	+	39.1918i	1.24372	+	1.24372i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−44.0908	+	44.0908i	−1.39637	+	1.39637i	−0.586214	+	0.810157i	$0.699382\pi$
$997$	−44.0908	+	44.0908i	−1.39637	+	1.39637i	−0.810157	+	0.586214i	$0.800618\pi$
$998$		0			0
$999$		−	36.0000i		−	1.13899i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.2.v.f.593.1		4
3.2	odd	2	CM	1200.2.v.f.593.1		4
4.3	odd	2		75.2.e.b.68.2	yes	4
5.2	odd	4	inner	1200.2.v.f.257.1		4
5.3	odd	4	inner	1200.2.v.f.257.2		4
5.4	even	2	inner	1200.2.v.f.593.2		4
12.11	even	2		75.2.e.b.68.2	yes	4
15.2	even	4	inner	1200.2.v.f.257.1		4
15.8	even	4	inner	1200.2.v.f.257.2		4
15.14	odd	2	inner	1200.2.v.f.593.2		4
20.3	even	4		75.2.e.b.32.1	✓	4
20.7	even	4		75.2.e.b.32.2	yes	4
20.19	odd	2		75.2.e.b.68.1	yes	4
60.23	odd	4		75.2.e.b.32.1	✓	4
60.47	odd	4		75.2.e.b.32.2	yes	4
60.59	even	2		75.2.e.b.68.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.e.b.32.1	✓	4	20.3	even	4
75.2.e.b.32.1	✓	4	60.23	odd	4
75.2.e.b.32.2	yes	4	20.7	even	4
75.2.e.b.32.2	yes	4	60.47	odd	4
75.2.e.b.68.1	yes	4	20.19	odd	2
75.2.e.b.68.1	yes	4	60.59	even	2
75.2.e.b.68.2	yes	4	4.3	odd	2
75.2.e.b.68.2	yes	4	12.11	even	2
1200.2.v.f.257.1		4	5.2	odd	4	inner
1200.2.v.f.257.1		4	15.2	even	4	inner
1200.2.v.f.257.2		4	5.3	odd	4	inner
1200.2.v.f.257.2		4	15.8	even	4	inner
1200.2.v.f.593.1		4	1.1	even	1	trivial
1200.2.v.f.593.1		4	3.2	odd	2	CM
1200.2.v.f.593.2		4	5.4	even	2	inner
1200.2.v.f.593.2		4	15.14	odd	2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1200.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(1.22474 - 1.22474i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 - 1.22474i) q^{3} +(1.22474 - 1.22474i) q^{7} +3.00000i q^{9} +(-3.67423 - 3.67423i) q^{13} -1.00000i q^{19} -3.00000 q^{21} +(3.67423 - 3.67423i) q^{27} -7.00000 q^{31} +(4.89898 - 4.89898i) q^{37} +9.00000i q^{39} +(-8.57321 - 8.57321i) q^{43} +4.00000i q^{49} +(-1.22474 + 1.22474i) q^{57} -13.0000 q^{61} +(3.67423 + 3.67423i) q^{63} +(-11.0227 + 11.0227i) q^{67} +(-9.79796 - 9.79796i) q^{73} +4.00000i q^{79} -9.00000 q^{81} -9.00000 q^{91} +(8.57321 + 8.57321i) q^{93} +(-13.4722 + 13.4722i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 12 q^{21} - 28 q^{31} - 52 q^{61} - 36 q^{81} - 36 q^{91}+O(q^{100}) \) 4 * q - 12 * q^21 - 28 * q^31 - 52 * q^61 - 36 * q^81 - 36 * q^91

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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