Properties

Label 400.3.b.a
Level $400$
Weight $3$
Character orbit 400.b
Self dual yes
Analytic conductor $10.899$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,3,Mod(351,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.351");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.b (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(10.8992105744\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{9} - 10 q^{13} + 30 q^{17} + 42 q^{29} + 70 q^{37} + 18 q^{41} + 49 q^{49} - 90 q^{53} - 22 q^{61} + 110 q^{73} + 81 q^{81} - 78 q^{89} - 130 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
0
0 0 0 0 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.b.a 1
3.b odd 2 1 3600.3.e.c 1
4.b odd 2 1 CM 400.3.b.a 1
5.b even 2 1 16.3.c.a 1
5.c odd 4 2 400.3.h.a 2
8.b even 2 1 1600.3.b.b 1
8.d odd 2 1 1600.3.b.b 1
12.b even 2 1 3600.3.e.c 1
15.d odd 2 1 144.3.g.a 1
15.e even 4 2 3600.3.j.a 2
20.d odd 2 1 16.3.c.a 1
20.e even 4 2 400.3.h.a 2
35.c odd 2 1 784.3.d.b 1
35.i odd 6 2 784.3.r.d 2
35.j even 6 2 784.3.r.e 2
40.e odd 2 1 64.3.c.a 1
40.f even 2 1 64.3.c.a 1
40.i odd 4 2 1600.3.h.b 2
40.k even 4 2 1600.3.h.b 2
45.h odd 6 2 1296.3.o.b 2
45.j even 6 2 1296.3.o.o 2
60.h even 2 1 144.3.g.a 1
60.l odd 4 2 3600.3.j.a 2
80.k odd 4 2 256.3.d.b 2
80.q even 4 2 256.3.d.b 2
120.i odd 2 1 576.3.g.b 1
120.m even 2 1 576.3.g.b 1
140.c even 2 1 784.3.d.b 1
140.p odd 6 2 784.3.r.e 2
140.s even 6 2 784.3.r.d 2
180.n even 6 2 1296.3.o.b 2
180.p odd 6 2 1296.3.o.o 2
240.t even 4 2 2304.3.b.f 2
240.bm odd 4 2 2304.3.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.c.a 1 5.b even 2 1
16.3.c.a 1 20.d odd 2 1
64.3.c.a 1 40.e odd 2 1
64.3.c.a 1 40.f even 2 1
144.3.g.a 1 15.d odd 2 1
144.3.g.a 1 60.h even 2 1
256.3.d.b 2 80.k odd 4 2
256.3.d.b 2 80.q even 4 2
400.3.b.a 1 1.a even 1 1 trivial
400.3.b.a 1 4.b odd 2 1 CM
400.3.h.a 2 5.c odd 4 2
400.3.h.a 2 20.e even 4 2
576.3.g.b 1 120.i odd 2 1
576.3.g.b 1 120.m even 2 1
784.3.d.b 1 35.c odd 2 1
784.3.d.b 1 140.c even 2 1
784.3.r.d 2 35.i odd 6 2
784.3.r.d 2 140.s even 6 2
784.3.r.e 2 35.j even 6 2
784.3.r.e 2 140.p odd 6 2
1296.3.o.b 2 45.h odd 6 2
1296.3.o.b 2 180.n even 6 2
1296.3.o.o 2 45.j even 6 2
1296.3.o.o 2 180.p odd 6 2
1600.3.b.b 1 8.b even 2 1
1600.3.b.b 1 8.d odd 2 1
1600.3.h.b 2 40.i odd 4 2
1600.3.h.b 2 40.k even 4 2
2304.3.b.f 2 240.t even 4 2
2304.3.b.f 2 240.bm odd 4 2
3600.3.e.c 1 3.b odd 2 1
3600.3.e.c 1 12.b even 2 1
3600.3.j.a 2 15.e even 4 2
3600.3.j.a 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{13} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 10 \) Copy content Toggle raw display
$17$ \( T - 30 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 42 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 70 \) Copy content Toggle raw display
$41$ \( T - 18 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 90 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 22 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 110 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 78 \) Copy content Toggle raw display
$97$ \( T + 130 \) Copy content Toggle raw display
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