LMFDB - Embedded newform 1600.2.d.e.801.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1600,2,Mod(801,1600)] 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("1600.801"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			801.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1600.801
Dual form			1600.2.d.e.801.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +3.46410 q^{7} +2.00000 q^{9} +3.00000i q^{11} +3.46410i q^{13} -3.00000 q^{17} +1.00000i q^{19} +3.46410i q^{21} +5.00000i q^{27} -10.3923i q^{29} +6.92820 q^{31} -3.00000 q^{33} +10.3923i q^{37} -3.46410 q^{39} -9.00000 q^{41} -4.00000i q^{43} +10.3923 q^{47} +5.00000 q^{49} -3.00000i q^{51} -1.00000 q^{57} -12.0000i q^{59} +3.46410i q^{61} +6.92820 q^{63} +11.0000i q^{67} +10.3923 q^{71} -7.00000 q^{73} +10.3923i q^{77} -10.3923 q^{79} +1.00000 q^{81} +15.0000i q^{83} +10.3923 q^{87} -3.00000 q^{89} +12.0000i q^{91} +6.92820i q^{93} -14.0000 q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{9} - 12 q^{17} - 12 q^{33} - 36 q^{41} + 20 q^{49} - 4 q^{57} - 28 q^{73} + 4 q^{81} - 12 q^{89} - 56 q^{97}+O(q^{100}) $ 4 * q + 8 * q^9 - 12 * q^17 - 12 * q^33 - 36 * q^41 + 20 * q^49 - 4 * q^57 - 28 * q^73 + 4 * q^81 - 12 * q^89 - 56 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	0	0
$13$	3.46410i	0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$	3.46410i	0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	1.00000i	0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$	1.00000i	0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$	0	0
$21$	3.46410i	0.755929i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	− 10.3923i	− 1.92980i	−0.262613	−	0.964901i	$-0.584584\pi$
$29$	− 10.3923i	− 1.92980i	0.262613	−	0.964901i	$-0.415416\pi$
$30$	0	0
$31$	6.92820	1.24434	0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820	1.24434	0.622171	+	0.782881i	$0.286251\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.3923i	1.70848i	0.519875	+	0.854242i	$0.325978\pi$
$37$	10.3923i	1.70848i	−0.519875	+	0.854242i	$0.674022\pi$
$38$	0	0
$39$	−3.46410	−0.554700
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.3923	1.51587	0.757937	−	0.652328i	$-0.226208\pi$
$47$	10.3923	1.51587	0.757937	+	0.652328i	$0.226208\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	− 3.00000i	− 0.420084i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	− 12.0000i	− 1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$	− 12.0000i	− 1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$	0	0
$61$	3.46410i	0.443533i	0.975100	+	0.221766i	$0.0711822\pi$
$61$	3.46410i	0.443533i	−0.975100	+	0.221766i	$0.928818\pi$
$62$	0	0
$63$	6.92820	0.872872
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.0000i	1.34386i	0.740613	+	0.671932i	$0.234535\pi$
$67$	11.0000i	1.34386i	−0.740613	+	0.671932i	$0.765465\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.3923i	1.18431i
$78$	0	0
$79$	−10.3923	−1.16923	−0.584613	−	0.811312i	$-0.698754\pi$
$79$	−10.3923	−1.16923	−0.584613	+	0.811312i	$0.698754\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.0000i	1.64646i	0.567705	+	0.823232i	$0.307831\pi$
$83$	15.0000i	1.64646i	−0.567705	+	0.823232i	$0.692169\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.3923	1.11417
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	12.0000i	1.25794i
$92$	0	0
$93$	6.92820i	0.718421i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	6.00000i	0.603023i

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.d.e.801.4	yes	4
4.3	odd	2	inner	1600.2.d.e.801.1	✓	4
5.2	odd	4		1600.2.f.g.1249.4		4
5.3	odd	4		1600.2.f.c.1249.2		4
5.4	even	2		1600.2.d.f.801.1	yes	4
8.3	odd	2	inner	1600.2.d.e.801.3	yes	4
8.5	even	2	inner	1600.2.d.e.801.2	yes	4
16.3	odd	4		6400.2.a.cf.1.2		2
16.5	even	4		6400.2.a.cf.1.1		2
16.11	odd	4		6400.2.a.bb.1.2		2
16.13	even	4		6400.2.a.bb.1.1		2
20.3	even	4		1600.2.f.g.1249.3		4
20.7	even	4		1600.2.f.c.1249.1		4
20.19	odd	2		1600.2.d.f.801.4	yes	4
40.3	even	4		1600.2.f.c.1249.4		4
40.13	odd	4		1600.2.f.g.1249.1		4
40.19	odd	2		1600.2.d.f.801.2	yes	4
40.27	even	4		1600.2.f.g.1249.2		4
40.29	even	2		1600.2.d.f.801.3	yes	4
40.37	odd	4		1600.2.f.c.1249.3		4
80.19	odd	4		6400.2.a.ba.1.1		2
80.29	even	4		6400.2.a.cg.1.2		2
80.59	odd	4		6400.2.a.cg.1.1		2
80.69	even	4		6400.2.a.ba.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1600.2.d.e.801.1	✓	4	4.3	odd	2	inner
1600.2.d.e.801.2	yes	4	8.5	even	2	inner
1600.2.d.e.801.3	yes	4	8.3	odd	2	inner
1600.2.d.e.801.4	yes	4	1.1	even	1	trivial
1600.2.d.f.801.1	yes	4	5.4	even	2
1600.2.d.f.801.2	yes	4	40.19	odd	2
1600.2.d.f.801.3	yes	4	40.29	even	2
1600.2.d.f.801.4	yes	4	20.19	odd	2
1600.2.f.c.1249.1		4	20.7	even	4
1600.2.f.c.1249.2		4	5.3	odd	4
1600.2.f.c.1249.3		4	40.37	odd	4
1600.2.f.c.1249.4		4	40.3	even	4
1600.2.f.g.1249.1		4	40.13	odd	4
1600.2.f.g.1249.2		4	40.27	even	4
1600.2.f.g.1249.3		4	20.3	even	4
1600.2.f.g.1249.4		4	5.2	odd	4
6400.2.a.ba.1.1		2	80.19	odd	4
6400.2.a.ba.1.2		2	80.69	even	4
6400.2.a.bb.1.1		2	16.13	even	4
6400.2.a.bb.1.2		2	16.11	odd	4
6400.2.a.cf.1.1		2	16.5	even	4
6400.2.a.cf.1.2		2	16.3	odd	4
6400.2.a.cg.1.1		2	80.59	odd	4
6400.2.a.cg.1.2		2	80.29	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +3.46410 q^{7} +2.00000 q^{9} +3.00000i q^{11} +3.46410i q^{13} -3.00000 q^{17} +1.00000i q^{19} +3.46410i q^{21} +5.00000i q^{27} -10.3923i q^{29} +6.92820 q^{31} -3.00000 q^{33} +10.3923i q^{37} -3.46410 q^{39} -9.00000 q^{41} -4.00000i q^{43} +10.3923 q^{47} +5.00000 q^{49} -3.00000i q^{51} -1.00000 q^{57} -12.0000i q^{59} +3.46410i q^{61} +6.92820 q^{63} +11.0000i q^{67} +10.3923 q^{71} -7.00000 q^{73} +10.3923i q^{77} -10.3923 q^{79} +1.00000 q^{81} +15.0000i q^{83} +10.3923 q^{87} -3.00000 q^{89} +12.0000i q^{91} +6.92820i q^{93} -14.0000 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{9} - 12 q^{17} - 12 q^{33} - 36 q^{41} + 20 q^{49} - 4 q^{57} - 28 q^{73} + 4 q^{81} - 12 q^{89} - 56 q^{97}+O(q^{100}) \) 4 * q + 8 * q^9 - 12 * q^17 - 12 * q^33 - 36 * q^41 + 20 * q^49 - 4 * q^57 - 28 * q^73 + 4 * q^81 - 12 * q^89 - 56 * q^97

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