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Properties

Label 750.2.e.a
Level $750$
Weight $2$
Character orbit 750.e
Analytic conductor $5.989$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [750,2,Mod(443,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([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("750.443");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.e (of order \(4\), degree \(2\), 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: \(5.98878015160\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{6} - 32 q^{16} - 16 q^{21} + 56 q^{31} + 28 q^{36} + 48 q^{46} + 96 q^{51} - 152 q^{61} - 16 q^{66} - 32 q^{76} - 76 q^{81} + 80 q^{91} + 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
443.1 −0.707107 − 0.707107i −1.65880 + 0.498383i 1.00000i 0 1.52536 + 0.820539i 0.812400 − 0.812400i 0.707107 − 0.707107i 2.50323 − 1.65343i 0
443.2 −0.707107 − 0.707107i −1.38609 − 1.03864i 1.00000i 0 0.245684 + 1.71454i −0.163947 + 0.163947i 0.707107 − 0.707107i 0.842471 + 2.87928i 0
443.3 −0.707107 − 0.707107i −0.498383 + 1.65880i 1.00000i 0 1.52536 − 0.820539i −0.812400 + 0.812400i 0.707107 − 0.707107i −2.50323 − 1.65343i 0
443.4 −0.707107 − 0.707107i 0.559477 − 1.63920i 1.00000i 0 −1.55470 + 0.763481i 2.32981 − 2.32981i 0.707107 − 0.707107i −2.37397 − 1.83419i 0
443.5 −0.707107 − 0.707107i 1.03864 + 1.38609i 1.00000i 0 0.245684 − 1.71454i 0.163947 − 0.163947i 0.707107 − 0.707107i −0.842471 + 2.87928i 0
443.6 −0.707107 − 0.707107i 1.04905 − 1.37822i 1.00000i 0 −1.71634 + 0.232753i −3.22259 + 3.22259i 0.707107 − 0.707107i −0.798968 − 2.89165i 0
443.7 −0.707107 − 0.707107i 1.37822 − 1.04905i 1.00000i 0 −1.71634 − 0.232753i 3.22259 − 3.22259i 0.707107 − 0.707107i 0.798968 − 2.89165i 0
443.8 −0.707107 − 0.707107i 1.63920 − 0.559477i 1.00000i 0 −1.55470 − 0.763481i −2.32981 + 2.32981i 0.707107 − 0.707107i 2.37397 − 1.83419i 0
443.9 0.707107 + 0.707107i −1.63920 + 0.559477i 1.00000i 0 −1.55470 − 0.763481i 2.32981 − 2.32981i −0.707107 + 0.707107i 2.37397 − 1.83419i 0
443.10 0.707107 + 0.707107i −1.37822 + 1.04905i 1.00000i 0 −1.71634 − 0.232753i −3.22259 + 3.22259i −0.707107 + 0.707107i 0.798968 − 2.89165i 0
443.11 0.707107 + 0.707107i −1.04905 + 1.37822i 1.00000i 0 −1.71634 + 0.232753i 3.22259 − 3.22259i −0.707107 + 0.707107i −0.798968 − 2.89165i 0
443.12 0.707107 + 0.707107i −1.03864 − 1.38609i 1.00000i 0 0.245684 − 1.71454i −0.163947 + 0.163947i −0.707107 + 0.707107i −0.842471 + 2.87928i 0
443.13 0.707107 + 0.707107i −0.559477 + 1.63920i 1.00000i 0 −1.55470 + 0.763481i −2.32981 + 2.32981i −0.707107 + 0.707107i −2.37397 − 1.83419i 0
443.14 0.707107 + 0.707107i 0.498383 − 1.65880i 1.00000i 0 1.52536 − 0.820539i 0.812400 − 0.812400i −0.707107 + 0.707107i −2.50323 − 1.65343i 0
443.15 0.707107 + 0.707107i 1.38609 + 1.03864i 1.00000i 0 0.245684 + 1.71454i 0.163947 − 0.163947i −0.707107 + 0.707107i 0.842471 + 2.87928i 0
443.16 0.707107 + 0.707107i 1.65880 − 0.498383i 1.00000i 0 1.52536 + 0.820539i −0.812400 + 0.812400i −0.707107 + 0.707107i 2.50323 − 1.65343i 0
557.1 −0.707107 + 0.707107i −1.65880 − 0.498383i − 1.00000i 0 1.52536 − 0.820539i 0.812400 + 0.812400i 0.707107 + 0.707107i 2.50323 + 1.65343i 0
557.2 −0.707107 + 0.707107i −1.38609 + 1.03864i − 1.00000i 0 0.245684 − 1.71454i −0.163947 − 0.163947i 0.707107 + 0.707107i 0.842471 − 2.87928i 0
557.3 −0.707107 + 0.707107i −0.498383 − 1.65880i − 1.00000i 0 1.52536 + 0.820539i −0.812400 − 0.812400i 0.707107 + 0.707107i −2.50323 + 1.65343i 0
557.4 −0.707107 + 0.707107i 0.559477 + 1.63920i − 1.00000i 0 −1.55470 − 0.763481i 2.32981 + 2.32981i 0.707107 + 0.707107i −2.37397 + 1.83419i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 443.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 750.2.e.a ✓ 32
3.b odd 2 1 inner 750.2.e.a ✓ 32
5.b even 2 1 inner 750.2.e.a ✓ 32
5.c odd 4 2 inner 750.2.e.a ✓ 32
15.d odd 2 1 inner 750.2.e.a ✓ 32
15.e even 4 2 inner 750.2.e.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
750.2.e.a ✓ 32 1.a even 1 1 trivial
750.2.e.a ✓ 32 3.b odd 2 1 inner
750.2.e.a ✓ 32 5.b even 2 1 inner
750.2.e.a ✓ 32 5.c odd 4 2 inner
750.2.e.a ✓ 32 15.d odd 2 1 inner
750.2.e.a ✓ 32 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 551T_{7}^{12} + 51801T_{7}^{8} + 88736T_{7}^{4} + 256 \) acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\). Copy content Toggle raw display