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Properties

Label 1575.2.bc.d
Level $1575$
Weight $2$
Character orbit 1575.bc
Analytic conductor $12.576$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,2,Mod(899,1575)]
 
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, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.899");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.bc (of order \(6\), degree \(2\), not 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{4} + 24 q^{11} + 24 q^{14} - 32 q^{16} - 12 q^{19} + 12 q^{31} + 48 q^{41} - 24 q^{44} - 8 q^{46} - 12 q^{49} - 120 q^{56} + 48 q^{59} + 112 q^{64} - 168 q^{74} - 36 q^{79} - 168 q^{86} - 24 q^{89} - 36 q^{91} - 24 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
899.1 −1.37068 − 2.37409i 0 −2.75754 + 4.77621i 0 0 −1.65853 − 2.06138i 9.63615 0 0
899.2 −1.23340 − 2.13631i 0 −2.04255 + 3.53780i 0 0 −1.88881 + 1.85267i 5.14351 0 0
899.3 −0.997835 − 1.72830i 0 −0.991350 + 1.71707i 0 0 −1.95727 − 1.78020i −0.0345244 0 0
899.4 −0.569823 − 0.986962i 0 0.350603 − 0.607263i 0 0 −1.48558 + 2.18931i −3.07842 0 0
899.5 −0.489834 − 0.848417i 0 0.520126 − 0.900884i 0 0 2.33346 + 1.24698i −2.97844 0 0
899.6 −0.199105 − 0.344861i 0 0.920714 − 1.59472i 0 0 1.30338 + 2.30243i −1.52970 0 0
899.7 0.199105 + 0.344861i 0 0.920714 − 1.59472i 0 0 −1.30338 − 2.30243i 1.52970 0 0
899.8 0.489834 + 0.848417i 0 0.520126 − 0.900884i 0 0 −2.33346 − 1.24698i 2.97844 0 0
899.9 0.569823 + 0.986962i 0 0.350603 − 0.607263i 0 0 1.48558 − 2.18931i 3.07842 0 0
899.10 0.997835 + 1.72830i 0 −0.991350 + 1.71707i 0 0 1.95727 + 1.78020i 0.0345244 0 0
899.11 1.23340 + 2.13631i 0 −2.04255 + 3.53780i 0 0 1.88881 − 1.85267i −5.14351 0 0
899.12 1.37068 + 2.37409i 0 −2.75754 + 4.77621i 0 0 1.65853 + 2.06138i −9.63615 0 0
1349.1 −1.37068 + 2.37409i 0 −2.75754 − 4.77621i 0 0 −1.65853 + 2.06138i 9.63615 0 0
1349.2 −1.23340 + 2.13631i 0 −2.04255 − 3.53780i 0 0 −1.88881 − 1.85267i 5.14351 0 0
1349.3 −0.997835 + 1.72830i 0 −0.991350 − 1.71707i 0 0 −1.95727 + 1.78020i −0.0345244 0 0
1349.4 −0.569823 + 0.986962i 0 0.350603 + 0.607263i 0 0 −1.48558 − 2.18931i −3.07842 0 0
1349.5 −0.489834 + 0.848417i 0 0.520126 + 0.900884i 0 0 2.33346 − 1.24698i −2.97844 0 0
1349.6 −0.199105 + 0.344861i 0 0.920714 + 1.59472i 0 0 1.30338 − 2.30243i −1.52970 0 0
1349.7 0.199105 − 0.344861i 0 0.920714 + 1.59472i 0 0 −1.30338 + 2.30243i 1.52970 0 0
1349.8 0.489834 − 0.848417i 0 0.520126 + 0.900884i 0 0 −2.33346 + 1.24698i 2.97844 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 899.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{2}^{24} + 20 T_{2}^{22} + 256 T_{2}^{20} + 1976 T_{2}^{18} + 11092 T_{2}^{16} + 41240 T_{2}^{14} + \cdots + 1296 \) Copy content Toggle raw display
\( T_{11}^{12} - 12 T_{11}^{11} + 38 T_{11}^{10} + 120 T_{11}^{9} - 660 T_{11}^{8} - 3000 T_{11}^{7} + \cdots + 19044 \) Copy content Toggle raw display