LMFDB - Embedded newform 1200.2.v.e.593.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,2,Mod(257,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1200.257"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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 300)
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.e.257.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(2.44949 - 2.44949i) q^{7} +3.00000i q^{9} +(4.89898 + 4.89898i) q^{13} +8.00000i q^{19} -6.00000 q^{21} +(3.67423 - 3.67423i) q^{27} -4.00000 q^{31} +(4.89898 - 4.89898i) q^{37} -12.0000i q^{39} +(7.34847 + 7.34847i) q^{43} -5.00000i q^{49} +(9.79796 - 9.79796i) q^{57} +14.0000 q^{61} +(7.34847 + 7.34847i) q^{63} +(2.44949 - 2.44949i) q^{67} +(-9.79796 - 9.79796i) q^{73} +4.00000i q^{79} -9.00000 q^{81} +24.0000 q^{91} +(4.89898 + 4.89898i) q^{93} +(9.79796 - 9.79796i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 24 q^{21} - 16 q^{31} + 56 q^{61} - 36 q^{81} + 96 q^{91}+O(q^{100}) $ 4 * q - 24 * q^21 - 16 * q^31 + 56 * q^61 - 36 * q^81 + 96 * q^91

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
$7$	2.44949	−	2.44949i	0.925820	−	0.925820i	−0.0716124	−	0.997433i	$-0.522814\pi$
$7$	2.44949	−	2.44949i	0.925820	−	0.925820i	0.997433	+	0.0716124i	$0.0228145\pi$
$8$		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
$13$	4.89898	+	4.89898i	1.35873	+	1.35873i	0.875482	+	0.483250i	$0.160544\pi$
$13$	4.89898	+	4.89898i	1.35873	+	1.35873i	0.483250	+	0.875482i	$0.339456\pi$
$14$		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
$19$			8.00000i			1.83533i	0.397360	+	0.917663i	$0.369927\pi$
$19$			8.00000i			1.83533i	−0.397360	+	0.917663i	$0.630073\pi$
$20$		0			0
$21$	−6.00000			−1.30931
$22$		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
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		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
$39$		−	12.0000i		−	1.92154i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	7.34847	+	7.34847i	1.12063	+	1.12063i	0.991647	+	0.128984i	$0.0411717\pi$
$43$	7.34847	+	7.34847i	1.12063	+	1.12063i	0.128984	+	0.991647i	$0.458828\pi$
$44$		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
$49$		−	5.00000i		−	0.714286i
$50$		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
$57$	9.79796	−	9.79796i	1.29777	−	1.29777i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	14.0000			1.79252			0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000			1.79252			0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$	7.34847	+	7.34847i	0.925820	+	0.925820i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.44949	−	2.44949i	0.299253	−	0.299253i	−0.541468	−	0.840721i	$-0.682131\pi$
$67$	2.44949	−	2.44949i	0.299253	−	0.299253i	0.840721	+	0.541468i	$0.182131\pi$
$68$		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
$91$	24.0000			2.51588
$92$		0			0
$93$	4.89898	+	4.89898i	0.508001	+	0.508001i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	9.79796	−	9.79796i	0.994832	−	0.994832i	−0.00515471	−	0.999987i	$-0.501641\pi$
$97$	9.79796	−	9.79796i	0.994832	−	0.994832i	0.999987	+	0.00515471i	$0.00164080\pi$
$98$		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
$109$			2.00000i			0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$			2.00000i			0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$		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
$117$	−14.6969	+	14.6969i	−1.35873	+	1.35873i
$118$		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
$129$		−	18.0000i		−	1.58481i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	19.5959	+	19.5959i	1.69918	+	1.69918i
$134$		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
$147$	−6.12372	+	6.12372i	−0.505076	+	0.505076i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−4.00000			−0.325515			−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000			−0.325515			−0.162758	+	0.986666i	$0.552039\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−14.6969	+	14.6969i	−1.17294	+	1.17294i	−0.191439	+	0.981505i	$0.561315\pi$
$157$	−14.6969	+	14.6969i	−1.17294	+	1.17294i	−0.981505	+	0.191439i	$0.938685\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−17.1464	−	17.1464i	−1.34301	−	1.34301i	−0.893045	−	0.449966i	$-0.851436\pi$
$163$	−17.1464	−	17.1464i	−1.34301	−	1.34301i	−0.449966	−	0.893045i	$-0.648564\pi$
$164$		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
$169$			35.0000i			2.69231i
$170$		0			0
$171$	−24.0000			−1.83533
$172$		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
$181$	−26.0000			−1.93256			−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000			−1.93256			−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$	−17.1464	−	17.1464i	−1.26750	−	1.26750i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	18.0000i		−	1.30931i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−19.5959	−	19.5959i	−1.41055	−	1.41055i	−0.756171	−	0.654374i	$-0.772932\pi$
$193$	−19.5959	−	19.5959i	−1.41055	−	1.41055i	−0.654374	−	0.756171i	$-0.727068\pi$
$194$		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
$199$			28.0000i			1.98487i	0.122782	+	0.992434i	$0.460818\pi$
$199$			28.0000i			1.98487i	−0.122782	+	0.992434i	$0.539182\pi$
$200$		0			0
$201$	−6.00000			−0.423207
$202$		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
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−9.79796	+	9.79796i	−0.665129	+	0.665129i
$218$		0			0
$219$			24.0000i			1.62177i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	7.34847	+	7.34847i	0.492090	+	0.492090i	0.908964	−	0.416874i	$-0.136874\pi$
$223$	7.34847	+	7.34847i	0.492090	+	0.492090i	−0.416874	+	0.908964i	$0.636874\pi$
$224$		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
$229$			22.0000i			1.45380i	0.686743	+	0.726900i	$0.259040\pi$
$229$			22.0000i			1.45380i	−0.686743	+	0.726900i	$0.740960\pi$
$230$		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
$241$	14.0000			0.901819			0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000			0.901819			0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.707107	+	0.707107i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−39.1918	+	39.1918i	−2.49372	+	2.49372i
$248$		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
$259$		−	24.0000i		−	1.49129i
$260$		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
$273$	−29.3939	−	29.3939i	−1.77900	−	1.77900i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−14.6969	+	14.6969i	−0.883053	+	0.883053i	−0.993844	−	0.110790i	$-0.964662\pi$
$277$	−14.6969	+	14.6969i	−0.883053	+	0.883053i	0.110790	+	0.993844i	$0.464662\pi$
$278$		0			0
$279$		−	12.0000i		−	0.718421i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	7.34847	+	7.34847i	0.436821	+	0.436821i	0.890941	−	0.454120i	$-0.150046\pi$
$283$	7.34847	+	7.34847i	0.436821	+	0.436821i	−0.454120	+	0.890941i	$0.650046\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$	−24.0000			−1.40690
$292$		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
$301$	36.0000			2.07501
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−22.0454	+	22.0454i	−1.25820	+	1.25820i	−0.306245	+	0.951953i	$0.599073\pi$
$307$	−22.0454	+	22.0454i	−1.25820	+	1.25820i	−0.951953	+	0.306245i	$0.900927\pi$
$308$		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
$313$	−19.5959	−	19.5959i	−1.10763	−	1.10763i	−0.993462	−	0.114165i	$-0.963581\pi$
$313$	−19.5959	−	19.5959i	−1.10763	−	1.10763i	−0.114165	−	0.993462i	$-0.536419\pi$
$314$		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
$327$	2.44949	−	2.44949i	0.135457	−	0.135457i
$328$		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
$337$	9.79796	−	9.79796i	0.533729	−	0.533729i	−0.387951	−	0.921680i	$-0.626817\pi$
$337$	9.79796	−	9.79796i	0.533729	−	0.533729i	0.921680	+	0.387951i	$0.126817\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	4.89898	+	4.89898i	0.264520	+	0.264520i
$344$		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
$351$	36.0000			1.92154
$352$		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
$361$	−45.0000			−2.36842
$362$		0			0
$363$	−13.4722	−	13.4722i	−0.707107	−	0.707107i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	26.9444	−	26.9444i	1.40649	−	1.40649i	0.629421	−	0.777064i	$-0.283292\pi$
$367$	26.9444	−	26.9444i	1.40649	−	1.40649i	0.777064	−	0.629421i	$-0.216708\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	4.89898	+	4.89898i	0.253660	+	0.253660i	0.822469	−	0.568810i	$-0.192596\pi$
$373$	4.89898	+	4.89898i	0.253660	+	0.253660i	−0.568810	+	0.822469i	$0.692596\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			8.00000i			0.410932i	0.978664	+	0.205466i	$0.0658711\pi$
$379$			8.00000i			0.410932i	−0.978664	+	0.205466i	$0.934129\pi$
$380$		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
$387$	−22.0454	+	22.0454i	−1.12063	+	1.12063i
$388$		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
$397$	−14.6969	+	14.6969i	−0.737618	+	0.737618i	−0.972117	−	0.234498i	$-0.924655\pi$
$397$	−14.6969	+	14.6969i	−0.737618	+	0.737618i	0.234498	+	0.972117i	$0.424655\pi$
$398$		0			0
$399$		−	48.0000i		−	2.40301i
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−19.5959	−	19.5959i	−0.976142	−	0.976142i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	38.0000i		−	1.87898i	−0.342578	−	0.939490i	$-0.611300\pi$
$409$		−	38.0000i		−	1.87898i	0.342578	−	0.939490i	$-0.388700\pi$
$410$		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
$427$	34.2929	−	34.2929i	1.65955	−	1.65955i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	29.3939	+	29.3939i	1.41258	+	1.41258i	0.740271	+	0.672308i	$0.234697\pi$
$433$	29.3939	+	29.3939i	1.41258	+	1.41258i	0.672308	+	0.740271i	$0.265303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			28.0000i			1.33637i	0.743996	+	0.668184i	$0.232928\pi$
$439$			28.0000i			1.33637i	−0.743996	+	0.668184i	$0.767072\pi$
$440$		0			0
$441$	15.0000			0.714286
$442$		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
$453$	4.89898	+	4.89898i	0.230174	+	0.230174i
$454$		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
$469$		−	12.0000i		−	0.554109i
$470$		0			0
$471$	36.0000			1.65879
$472$		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
$481$	48.0000			2.18861
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	2.44949	−	2.44949i	0.110997	−	0.110997i	−0.649427	−	0.760424i	$-0.724991\pi$
$487$	2.44949	−	2.44949i	0.110997	−	0.110997i	0.760424	+	0.649427i	$0.224991\pi$
$488$		0			0
$489$			42.0000i			1.89931i
$490$		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
$499$		−	32.0000i		−	1.43252i	−0.697835	−	0.716258i	$-0.745853\pi$
$499$		−	32.0000i		−	1.43252i	0.697835	−	0.716258i	$-0.254147\pi$
$500$		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
$507$	42.8661	−	42.8661i	1.90375	−	1.90375i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−48.0000			−2.12339
$512$		0			0
$513$	29.3939	+	29.3939i	1.29777	+	1.29777i
$514$		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
$523$	31.8434	+	31.8434i	1.39241	+	1.39241i	0.819885	+	0.572528i	$0.194037\pi$
$523$	31.8434	+	31.8434i	1.39241	+	1.39241i	0.572528	+	0.819885i	$0.305963\pi$
$524$		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
$541$	−46.0000			−1.97769			−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−46.0000			−1.97769			−0.988847	+	0.148933i	$0.952416\pi$
$542$		0			0
$543$	31.8434	+	31.8434i	1.36653	+	1.36653i
$544$		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
$549$			42.0000i			1.79252i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	9.79796	+	9.79796i	0.416652	+	0.416652i
$554$		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
$559$			72.0000i			3.04528i
$560$		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
$567$	−22.0454	+	22.0454i	−0.925820	+	0.925820i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	16.0000			0.669579			0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000			0.669579			0.334790	+	0.942293i	$0.391335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	9.79796	−	9.79796i	0.407894	−	0.407894i	−0.473109	−	0.881004i	$-0.656868\pi$
$577$	9.79796	−	9.79796i	0.407894	−	0.407894i	0.881004	+	0.473109i	$0.156868\pi$
$578$		0			0
$579$			48.0000i			1.99481i
$580$		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
$589$		−	32.0000i		−	1.31854i
$590$		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
$597$	34.2929	−	34.2929i	1.40351	−	1.40351i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$	7.34847	+	7.34847i	0.299253	+	0.299253i
$604$		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
$619$		−	32.0000i		−	1.28619i	−0.765787	−	0.643094i	$-0.777650\pi$
$619$		−	32.0000i		−	1.28619i	0.765787	−	0.643094i	$-0.222350\pi$
$620$		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
$631$	−44.0000			−1.75161			−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000			−1.75161			−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$	−19.5959	−	19.5959i	−0.778868	−	0.778868i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	24.4949	−	24.4949i	0.970523	−	0.970523i
$638$		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
$651$	24.0000			0.940634
$652$		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
$669$		−	18.0000i		−	0.695920i
$670$		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
$679$		−	48.0000i		−	1.84207i
$680$		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
$687$	26.9444	−	26.9444i	1.02799	−	1.02799i
$688$		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
$703$	39.1918	+	39.1918i	1.47815	+	1.47815i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			22.0000i			0.826227i	0.910679	+	0.413114i	$0.135559\pi$
$709$			22.0000i			0.826227i	−0.910679	+	0.413114i	$0.864441\pi$
$710$		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
$721$	−12.0000			−0.446903
$722$		0			0
$723$	−17.1464	−	17.1464i	−0.637683	−	0.637683i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−22.0454	+	22.0454i	−0.817619	+	0.817619i	−0.985763	−	0.168144i	$-0.946223\pi$
$727$	−22.0454	+	22.0454i	−0.817619	+	0.817619i	0.168144	+	0.985763i	$0.446223\pi$
$728$		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
$741$	96.0000			3.52665
$742$		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
$757$	34.2929	−	34.2929i	1.24640	−	1.24640i	0.289095	−	0.957301i	$-0.406646\pi$
$757$	34.2929	−	34.2929i	1.24640	−	1.24640i	0.957301	−	0.289095i	$-0.0933542\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	4.89898	+	4.89898i	0.177355	+	0.177355i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			2.00000i			0.0721218i	0.999350	+	0.0360609i	$0.0114810\pi$
$769$			2.00000i			0.0721218i	−0.999350	+	0.0360609i	$0.988519\pi$
$770$		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
$777$	−29.3939	+	29.3939i	−1.05450	+	1.05450i
$778$		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
$787$	2.44949	−	2.44949i	0.0873149	−	0.0873149i	−0.662100	−	0.749415i	$-0.730335\pi$
$787$	2.44949	−	2.44949i	0.0873149	−	0.0873149i	0.749415	+	0.662100i	$0.230335\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	68.5857	+	68.5857i	2.43555	+	2.43555i
$794$		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
$811$	56.0000			1.96643			0.983213	−	0.182462i	$-0.0584065\pi$
$811$	56.0000			1.96643			0.983213	+	0.182462i	$0.0584065\pi$
$812$		0			0
$813$	−34.2929	−	34.2929i	−1.20270	−	1.20270i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−58.7878	+	58.7878i	−2.05672	+	2.05672i
$818$		0			0
$819$			72.0000i			2.51588i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−17.1464	−	17.1464i	−0.597687	−	0.597687i	0.342010	−	0.939696i	$-0.388892\pi$
$823$	−17.1464	−	17.1464i	−0.597687	−	0.597687i	−0.939696	+	0.342010i	$0.888892\pi$
$824$		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
$831$	36.0000			1.24883
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−14.6969	+	14.6969i	−0.508001	+	0.508001i
$838$		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
$847$	26.9444	−	26.9444i	0.925820	−	0.925820i
$848$		0			0
$849$		−	18.0000i		−	0.617758i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	4.89898	+	4.89898i	0.167738	+	0.167738i	0.785984	−	0.618246i	$-0.212157\pi$
$853$	4.89898	+	4.89898i	0.167738	+	0.167738i	−0.618246	+	0.785984i	$0.712157\pi$
$854$		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
$871$	24.0000			0.813209
$872$		0			0
$873$	29.3939	+	29.3939i	0.994832	+	0.994832i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	34.2929	−	34.2929i	1.15799	−	1.15799i	0.173080	−	0.984908i	$-0.444628\pi$
$877$	34.2929	−	34.2929i	1.15799	−	1.15799i	0.984908	−	0.173080i	$-0.0553718\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−41.6413	−	41.6413i	−1.40134	−	1.40134i	−0.795855	−	0.605487i	$-0.792978\pi$
$883$	−41.6413	−	41.6413i	−1.40134	−	1.40134i	−0.605487	−	0.795855i	$-0.707022\pi$
$884$		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
$889$		−	36.0000i		−	1.20740i
$890$		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
$903$	−44.0908	−	44.0908i	−1.46725	−	1.46725i
$904$		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
$919$		−	52.0000i		−	1.71532i	−0.514216	−	0.857661i	$-0.671917\pi$
$919$		−	52.0000i		−	1.71532i	0.514216	−	0.857661i	$-0.328083\pi$
$920$		0			0
$921$	54.0000			1.77936
$922$		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
$931$	40.0000			1.31095
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−39.1918	+	39.1918i	−1.28034	+	1.28034i	−0.339869	+	0.940473i	$0.610383\pi$
$937$	−39.1918	+	39.1918i	−1.28034	+	1.28034i	−0.940473	+	0.339869i	$0.889617\pi$
$938$		0			0
$939$			48.0000i			1.56642i
$940$		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
$949$		−	96.0000i		−	3.11629i
$950$		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
$961$	−15.0000			−0.483871
$962$		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
$973$	−39.1918	−	39.1918i	−1.25643	−	1.25643i
$974$		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
$981$	−6.00000			−0.191565
$982$		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
$991$	−44.0000			−1.39771			−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000			−1.39771			−0.698853	+	0.715265i	$0.746306\pi$
$992$		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.e.593.1		4
3.2	odd	2	CM	1200.2.v.e.593.1		4
4.3	odd	2		300.2.i.b.293.2	yes	4
5.2	odd	4	inner	1200.2.v.e.257.1		4
5.3	odd	4	inner	1200.2.v.e.257.2		4
5.4	even	2	inner	1200.2.v.e.593.2		4
12.11	even	2		300.2.i.b.293.2	yes	4
15.2	even	4	inner	1200.2.v.e.257.1		4
15.8	even	4	inner	1200.2.v.e.257.2		4
15.14	odd	2	inner	1200.2.v.e.593.2		4
20.3	even	4		300.2.i.b.257.1	✓	4
20.7	even	4		300.2.i.b.257.2	yes	4
20.19	odd	2		300.2.i.b.293.1	yes	4
60.23	odd	4		300.2.i.b.257.1	✓	4
60.47	odd	4		300.2.i.b.257.2	yes	4
60.59	even	2		300.2.i.b.293.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.i.b.257.1	✓	4	20.3	even	4
300.2.i.b.257.1	✓	4	60.23	odd	4
300.2.i.b.257.2	yes	4	20.7	even	4
300.2.i.b.257.2	yes	4	60.47	odd	4
300.2.i.b.293.1	yes	4	20.19	odd	2
300.2.i.b.293.1	yes	4	60.59	even	2
300.2.i.b.293.2	yes	4	4.3	odd	2
300.2.i.b.293.2	yes	4	12.11	even	2
1200.2.v.e.257.1		4	5.2	odd	4	inner
1200.2.v.e.257.1		4	15.2	even	4	inner
1200.2.v.e.257.2		4	5.3	odd	4	inner
1200.2.v.e.257.2		4	15.8	even	4	inner
1200.2.v.e.593.1		4	1.1	even	1	trivial
1200.2.v.e.593.1		4	3.2	odd	2	CM
1200.2.v.e.593.2		4	5.4	even	2	inner
1200.2.v.e.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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(2.44949 - 2.44949i) q^{7} +3.00000i q^{9} +(4.89898 + 4.89898i) q^{13} +8.00000i q^{19} -6.00000 q^{21} +(3.67423 - 3.67423i) q^{27} -4.00000 q^{31} +(4.89898 - 4.89898i) q^{37} -12.0000i q^{39} +(7.34847 + 7.34847i) q^{43} -5.00000i q^{49} +(9.79796 - 9.79796i) q^{57} +14.0000 q^{61} +(7.34847 + 7.34847i) q^{63} +(2.44949 - 2.44949i) q^{67} +(-9.79796 - 9.79796i) q^{73} +4.00000i q^{79} -9.00000 q^{81} +24.0000 q^{91} +(4.89898 + 4.89898i) q^{93} +(9.79796 - 9.79796i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 24 q^{21} - 16 q^{31} + 56 q^{61} - 36 q^{81} + 96 q^{91}+O(q^{100}) \) 4 * q - 24 * q^21 - 16 * q^31 + 56 * q^61 - 36 * q^81 + 96 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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