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Properties

Label 1575.2.bk.i
Level $1575$
Weight $2$
Character orbit 1575.bk
Analytic conductor $12.576$
Analytic rank $0$
Dimension $24$
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] = [1575,2,Mod(26,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.bk (of order \(6\), 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: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4} + 36 q^{19} - 60 q^{31} - 24 q^{46} - 36 q^{49} + 48 q^{61} - 48 q^{64} + 60 q^{79} + 60 q^{91} - 48 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −2.22514 − 1.28469i 0 2.30084 + 3.98517i 0 0 0.151755 + 2.64140i − 6.68468i 0 0
26.2 −2.22514 − 1.28469i 0 2.30084 + 3.98517i 0 0 −0.151755 − 2.64140i − 6.68468i 0 0
26.3 −1.65683 − 0.956572i 0 0.830062 + 1.43771i 0 0 2.39757 − 1.11878i 0.650234i 0 0
26.4 −1.65683 − 0.956572i 0 0.830062 + 1.43771i 0 0 −2.39757 + 1.11878i 0.650234i 0 0
26.5 −1.14177 − 0.659204i 0 −0.130901 − 0.226727i 0 0 −1.57437 + 2.12635i 2.98198i 0 0
26.6 −1.14177 − 0.659204i 0 −0.130901 − 0.226727i 0 0 1.57437 − 2.12635i 2.98198i 0 0
26.7 1.14177 + 0.659204i 0 −0.130901 − 0.226727i 0 0 1.57437 − 2.12635i − 2.98198i 0 0
26.8 1.14177 + 0.659204i 0 −0.130901 − 0.226727i 0 0 −1.57437 + 2.12635i − 2.98198i 0 0
26.9 1.65683 + 0.956572i 0 0.830062 + 1.43771i 0 0 −2.39757 + 1.11878i − 0.650234i 0 0
26.10 1.65683 + 0.956572i 0 0.830062 + 1.43771i 0 0 2.39757 − 1.11878i − 0.650234i 0 0
26.11 2.22514 + 1.28469i 0 2.30084 + 3.98517i 0 0 −0.151755 − 2.64140i 6.68468i 0 0
26.12 2.22514 + 1.28469i 0 2.30084 + 3.98517i 0 0 0.151755 + 2.64140i 6.68468i 0 0
1151.1 −2.22514 + 1.28469i 0 2.30084 − 3.98517i 0 0 0.151755 − 2.64140i 6.68468i 0 0
1151.2 −2.22514 + 1.28469i 0 2.30084 − 3.98517i 0 0 −0.151755 + 2.64140i 6.68468i 0 0
1151.3 −1.65683 + 0.956572i 0 0.830062 − 1.43771i 0 0 2.39757 + 1.11878i − 0.650234i 0 0
1151.4 −1.65683 + 0.956572i 0 0.830062 − 1.43771i 0 0 −2.39757 − 1.11878i − 0.650234i 0 0
1151.5 −1.14177 + 0.659204i 0 −0.130901 + 0.226727i 0 0 −1.57437 − 2.12635i − 2.98198i 0 0
1151.6 −1.14177 + 0.659204i 0 −0.130901 + 0.226727i 0 0 1.57437 + 2.12635i − 2.98198i 0 0
1151.7 1.14177 − 0.659204i 0 −0.130901 + 0.226727i 0 0 1.57437 + 2.12635i 2.98198i 0 0
1151.8 1.14177 − 0.659204i 0 −0.130901 + 0.226727i 0 0 −1.57437 − 2.12635i 2.98198i 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.12
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
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.bk.i 24
3.b odd 2 1 inner 1575.2.bk.i 24
5.b even 2 1 inner 1575.2.bk.i 24
5.c odd 4 2 315.2.bb.b ✓ 24
7.d odd 6 1 inner 1575.2.bk.i 24
15.d odd 2 1 inner 1575.2.bk.i 24
15.e even 4 2 315.2.bb.b ✓ 24
21.g even 6 1 inner 1575.2.bk.i 24
35.i odd 6 1 inner 1575.2.bk.i 24
35.k even 12 2 315.2.bb.b ✓ 24
35.k even 12 2 2205.2.g.b 24
35.l odd 12 2 2205.2.g.b 24
105.p even 6 1 inner 1575.2.bk.i 24
105.w odd 12 2 315.2.bb.b ✓ 24
105.w odd 12 2 2205.2.g.b 24
105.x even 12 2 2205.2.g.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bb.b ✓ 24 5.c odd 4 2
315.2.bb.b ✓ 24 15.e even 4 2
315.2.bb.b ✓ 24 35.k even 12 2
315.2.bb.b ✓ 24 105.w odd 12 2
1575.2.bk.i 24 1.a even 1 1 trivial
1575.2.bk.i 24 3.b odd 2 1 inner
1575.2.bk.i 24 5.b even 2 1 inner
1575.2.bk.i 24 7.d odd 6 1 inner
1575.2.bk.i 24 15.d odd 2 1 inner
1575.2.bk.i 24 21.g even 6 1 inner
1575.2.bk.i 24 35.i odd 6 1 inner
1575.2.bk.i 24 105.p even 6 1 inner
2205.2.g.b 24 35.k even 12 2
2205.2.g.b 24 35.l odd 12 2
2205.2.g.b 24 105.w odd 12 2
2205.2.g.b 24 105.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1575, [\chi])\):

\( T_{2}^{12} - 12T_{2}^{10} + 102T_{2}^{8} - 420T_{2}^{6} + 1260T_{2}^{4} - 1764T_{2}^{2} + 1764 \) Copy content Toggle raw display
\( T_{11}^{12} - 48T_{11}^{10} + 1728T_{11}^{8} - 23804T_{11}^{6} + 239520T_{11}^{4} - 1107072T_{11}^{2} + 3694084 \) Copy content Toggle raw display
\( T_{37}^{12} + 27T_{37}^{10} + 540T_{37}^{8} + 4725T_{37}^{6} + 30618T_{37}^{4} + 35721T_{37}^{2} + 35721 \) Copy content Toggle raw display