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Properties

Label 975.2.s.e
Level $975$
Weight $2$
Character orbit 975.s
Analytic conductor $7.785$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(818,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.818");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.s (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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{3} + 24 q^{12} + 24 q^{16} - 24 q^{22} + 16 q^{27} - 8 q^{36} - 12 q^{42} + 64 q^{43} + 20 q^{48} + 16 q^{51} + 72 q^{52} + 8 q^{61} - 72 q^{66} - 84 q^{78} + 112 q^{81} - 48 q^{82} - 20 q^{87}+ \cdots - 40 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
818.1 −1.57280 + 1.57280i −0.0375613 + 1.73164i − 2.94741i 0 −2.66446 − 2.78261i −3.19872 − 3.19872i 1.49009 + 1.49009i −2.99718 − 0.130086i 0
818.2 −1.57280 + 1.57280i 1.73164 − 0.0375613i − 2.94741i 0 −2.66446 + 2.78261i 3.19872 + 3.19872i 1.49009 + 1.49009i 2.99718 − 0.130086i 0
818.3 −1.25684 + 1.25684i −1.36721 + 1.06336i − 1.15927i 0 0.381882 − 3.05483i 1.60133 + 1.60133i −1.05665 − 1.05665i 0.738515 − 2.90768i 0
818.4 −1.25684 + 1.25684i 1.06336 − 1.36721i − 1.15927i 0 0.381882 + 3.05483i −1.60133 − 1.60133i −1.05665 − 1.05665i −0.738515 − 2.90768i 0
818.5 −0.597551 + 0.597551i −0.458083 + 1.67038i 1.28586i 0 −0.724408 − 1.27186i 2.39819 + 2.39819i −1.96347 − 1.96347i −2.58032 − 1.53034i 0
818.6 −0.597551 + 0.597551i 1.67038 − 0.458083i 1.28586i 0 −0.724408 + 1.27186i −2.39819 − 2.39819i −1.96347 − 1.96347i 2.58032 − 1.53034i 0
818.7 −0.299314 + 0.299314i −1.72753 + 0.125002i 1.82082i 0 0.479660 − 0.554490i −0.976029 − 0.976029i −1.14363 − 1.14363i 2.96875 − 0.431892i 0
818.8 −0.299314 + 0.299314i 0.125002 − 1.72753i 1.82082i 0 0.479660 + 0.554490i 0.976029 + 0.976029i −1.14363 − 1.14363i −2.96875 − 0.431892i 0
818.9 0.299314 − 0.299314i −1.72753 + 0.125002i 1.82082i 0 −0.479660 + 0.554490i 0.976029 + 0.976029i 1.14363 + 1.14363i 2.96875 − 0.431892i 0
818.10 0.299314 − 0.299314i 0.125002 − 1.72753i 1.82082i 0 −0.479660 − 0.554490i −0.976029 − 0.976029i 1.14363 + 1.14363i −2.96875 − 0.431892i 0
818.11 0.597551 − 0.597551i −0.458083 + 1.67038i 1.28586i 0 0.724408 + 1.27186i −2.39819 − 2.39819i 1.96347 + 1.96347i −2.58032 − 1.53034i 0
818.12 0.597551 − 0.597551i 1.67038 − 0.458083i 1.28586i 0 0.724408 − 1.27186i 2.39819 + 2.39819i 1.96347 + 1.96347i 2.58032 − 1.53034i 0
818.13 1.25684 − 1.25684i −1.36721 + 1.06336i − 1.15927i 0 −0.381882 + 3.05483i −1.60133 − 1.60133i 1.05665 + 1.05665i 0.738515 − 2.90768i 0
818.14 1.25684 − 1.25684i 1.06336 − 1.36721i − 1.15927i 0 −0.381882 − 3.05483i 1.60133 + 1.60133i 1.05665 + 1.05665i −0.738515 − 2.90768i 0
818.15 1.57280 − 1.57280i −0.0375613 + 1.73164i − 2.94741i 0 2.66446 + 2.78261i 3.19872 + 3.19872i −1.49009 − 1.49009i −2.99718 − 0.130086i 0
818.16 1.57280 − 1.57280i 1.73164 − 0.0375613i − 2.94741i 0 2.66446 − 2.78261i −3.19872 − 3.19872i −1.49009 − 1.49009i 2.99718 − 0.130086i 0
857.1 −1.57280 − 1.57280i −0.0375613 − 1.73164i 2.94741i 0 −2.66446 + 2.78261i −3.19872 + 3.19872i 1.49009 − 1.49009i −2.99718 + 0.130086i 0
857.2 −1.57280 − 1.57280i 1.73164 + 0.0375613i 2.94741i 0 −2.66446 − 2.78261i 3.19872 − 3.19872i 1.49009 − 1.49009i 2.99718 + 0.130086i 0
857.3 −1.25684 − 1.25684i −1.36721 − 1.06336i 1.15927i 0 0.381882 + 3.05483i 1.60133 − 1.60133i −1.05665 + 1.05665i 0.738515 + 2.90768i 0
857.4 −1.25684 − 1.25684i 1.06336 + 1.36721i 1.15927i 0 0.381882 − 3.05483i −1.60133 + 1.60133i −1.05665 + 1.05665i −0.738515 + 2.90768i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 818.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
13.b even 2 1 inner
15.e even 4 1 inner
39.d odd 2 1 inner
65.h odd 4 1 inner
195.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.s.e 32
3.b odd 2 1 inner 975.2.s.e 32
5.b even 2 1 195.2.s.b ✓ 32
5.c odd 4 1 195.2.s.b ✓ 32
5.c odd 4 1 inner 975.2.s.e 32
13.b even 2 1 inner 975.2.s.e 32
15.d odd 2 1 195.2.s.b ✓ 32
15.e even 4 1 195.2.s.b ✓ 32
15.e even 4 1 inner 975.2.s.e 32
39.d odd 2 1 inner 975.2.s.e 32
65.d even 2 1 195.2.s.b ✓ 32
65.h odd 4 1 195.2.s.b ✓ 32
65.h odd 4 1 inner 975.2.s.e 32
195.e odd 2 1 195.2.s.b ✓ 32
195.s even 4 1 195.2.s.b ✓ 32
195.s even 4 1 inner 975.2.s.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.s.b ✓ 32 5.b even 2 1
195.2.s.b ✓ 32 5.c odd 4 1
195.2.s.b ✓ 32 15.d odd 2 1
195.2.s.b ✓ 32 15.e even 4 1
195.2.s.b ✓ 32 65.d even 2 1
195.2.s.b ✓ 32 65.h odd 4 1
195.2.s.b ✓ 32 195.e odd 2 1
195.2.s.b ✓ 32 195.s even 4 1
975.2.s.e 32 1.a even 1 1 trivial
975.2.s.e 32 3.b odd 2 1 inner
975.2.s.e 32 5.c odd 4 1 inner
975.2.s.e 32 13.b even 2 1 inner
975.2.s.e 32 15.e even 4 1 inner
975.2.s.e 32 39.d odd 2 1 inner
975.2.s.e 32 65.h odd 4 1 inner
975.2.s.e 32 195.s even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{16} + 35T_{2}^{12} + 263T_{2}^{8} + 133T_{2}^{4} + 4 \) Copy content Toggle raw display
\( T_{7}^{16} + 581T_{7}^{12} + 71996T_{7}^{8} + 1711024T_{7}^{4} + 5290000 \) Copy content Toggle raw display