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Properties

Label 1530.2.j.b
Level $1530$
Weight $2$
Character orbit 1530.j
Analytic conductor $12.217$
Analytic rank $0$
Dimension $28$
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] = [1530,2,Mod(863,1530)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1530, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("1530.863");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1530.j (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: \(12.2171115093\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{10} + 4 q^{13} - 28 q^{16} + 32 q^{19} - 16 q^{22} + 32 q^{25} - 16 q^{28} - 8 q^{31} + 16 q^{34} - 72 q^{37} + 8 q^{40} + 16 q^{43} - 32 q^{46} + 20 q^{49} + 4 q^{52} - 16 q^{55} + 24 q^{58} + 68 q^{61} + 32 q^{67} - 16 q^{70} + 8 q^{79} + 72 q^{82} - 28 q^{85} - 8 q^{91} + 48 q^{94} + 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
863.1 −0.707107 + 0.707107i 0 − 1.00000i −0.106302 − 2.23354i 0 − 2.17409i 0.707107 + 0.707107i 0 1.65452 + 1.50418i
863.2 −0.707107 + 0.707107i 0 − 1.00000i −2.21431 − 0.311205i 0 1.72799i 0.707107 + 0.707107i 0 1.78581 − 1.34570i
863.3 −0.707107 + 0.707107i 0 − 1.00000i 1.91988 + 1.14633i 0 3.68707i 0.707107 + 0.707107i 0 −2.16814 + 0.546983i
863.4 −0.707107 + 0.707107i 0 − 1.00000i 1.44202 − 1.70897i 0 0.242995i 0.707107 + 0.707107i 0 0.188762 + 2.22809i
863.5 −0.707107 + 0.707107i 0 − 1.00000i 1.02838 + 1.98555i 0 − 3.32441i 0.707107 + 0.707107i 0 −2.13118 − 0.676822i
863.6 −0.707107 + 0.707107i 0 − 1.00000i 2.23504 − 0.0677563i 0 − 0.869141i 0.707107 + 0.707107i 0 −1.53250 + 1.62832i
863.7 −0.707107 + 0.707107i 0 − 1.00000i −2.18339 + 0.482481i 0 − 3.29041i 0.707107 + 0.707107i 0 1.20273 − 1.88506i
863.8 0.707107 − 0.707107i 0 − 1.00000i −1.91988 − 1.14633i 0 3.68707i −0.707107 − 0.707107i 0 −2.16814 + 0.546983i
863.9 0.707107 − 0.707107i 0 − 1.00000i −2.23504 + 0.0677563i 0 − 0.869141i −0.707107 − 0.707107i 0 −1.53250 + 1.62832i
863.10 0.707107 − 0.707107i 0 − 1.00000i −1.02838 − 1.98555i 0 − 3.32441i −0.707107 − 0.707107i 0 −2.13118 − 0.676822i
863.11 0.707107 − 0.707107i 0 − 1.00000i −1.44202 + 1.70897i 0 0.242995i −0.707107 − 0.707107i 0 0.188762 + 2.22809i
863.12 0.707107 − 0.707107i 0 − 1.00000i 2.18339 − 0.482481i 0 − 3.29041i −0.707107 − 0.707107i 0 1.20273 − 1.88506i
863.13 0.707107 − 0.707107i 0 − 1.00000i 2.21431 + 0.311205i 0 1.72799i −0.707107 − 0.707107i 0 1.78581 − 1.34570i
863.14 0.707107 − 0.707107i 0 − 1.00000i 0.106302 + 2.23354i 0 − 2.17409i −0.707107 − 0.707107i 0 1.65452 + 1.50418i
1007.1 −0.707107 − 0.707107i 0 1.00000i −0.106302 + 2.23354i 0 2.17409i 0.707107 − 0.707107i 0 1.65452 − 1.50418i
1007.2 −0.707107 − 0.707107i 0 1.00000i −2.21431 + 0.311205i 0 − 1.72799i 0.707107 − 0.707107i 0 1.78581 + 1.34570i
1007.3 −0.707107 − 0.707107i 0 1.00000i 1.91988 − 1.14633i 0 − 3.68707i 0.707107 − 0.707107i 0 −2.16814 − 0.546983i
1007.4 −0.707107 − 0.707107i 0 1.00000i 1.44202 + 1.70897i 0 − 0.242995i 0.707107 − 0.707107i 0 0.188762 − 2.22809i
1007.5 −0.707107 − 0.707107i 0 1.00000i 1.02838 − 1.98555i 0 3.32441i 0.707107 − 0.707107i 0 −2.13118 + 0.676822i
1007.6 −0.707107 − 0.707107i 0 1.00000i 2.23504 + 0.0677563i 0 0.869141i 0.707107 − 0.707107i 0 −1.53250 − 1.62832i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
85.f odd 4 1 inner
255.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1530.2.j.b ✓ 28
3.b odd 2 1 inner 1530.2.j.b ✓ 28
5.c odd 4 1 1530.2.u.b yes 28
15.e even 4 1 1530.2.u.b yes 28
17.c even 4 1 1530.2.u.b yes 28
51.f odd 4 1 1530.2.u.b yes 28
85.f odd 4 1 inner 1530.2.j.b ✓ 28
255.k even 4 1 inner 1530.2.j.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1530.2.j.b ✓ 28 1.a even 1 1 trivial
1530.2.j.b ✓ 28 3.b odd 2 1 inner
1530.2.j.b ✓ 28 85.f odd 4 1 inner
1530.2.j.b ✓ 28 255.k even 4 1 inner
1530.2.u.b yes 28 5.c odd 4 1
1530.2.u.b yes 28 15.e even 4 1
1530.2.u.b yes 28 17.c even 4 1
1530.2.u.b yes 28 51.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{14} + 44T_{7}^{12} + 740T_{7}^{10} + 5920T_{7}^{8} + 22816T_{7}^{6} + 38208T_{7}^{4} + 19520T_{7}^{2} + 1024 \) acting on \(S_{2}^{\mathrm{new}}(1530, [\chi])\). Copy content Toggle raw display