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Properties

Label 430.2.g.b
Level $430$
Weight $2$
Character orbit 430.g
Analytic conductor $3.434$
Analytic rank $0$
Dimension $40$
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] = [430,2,Mod(257,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.g (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: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{10} + 16 q^{11} - 24 q^{13} + 8 q^{15} - 40 q^{16} - 12 q^{17} - 16 q^{21} + 44 q^{23} + 24 q^{25} + 32 q^{31} - 64 q^{35} + 48 q^{36} - 28 q^{38} - 4 q^{40} + 8 q^{41} - 16 q^{43} - 28 q^{47} + 24 q^{52} - 80 q^{53} + 24 q^{56} + 64 q^{57} + 12 q^{58} + 24 q^{67} - 12 q^{68} + 40 q^{78} - 120 q^{81} + 48 q^{83} + 28 q^{87} - 44 q^{92} - 16 q^{95} - 60 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1 −0.707107 + 0.707107i −2.27071 − 2.27071i − 1.00000i −1.31680 + 1.80721i 3.21127 −0.136059 + 0.136059i 0.707107 + 0.707107i 7.31224i −0.346772 − 2.20902i
257.2 −0.707107 + 0.707107i −1.88501 − 1.88501i − 1.00000i 1.60089 − 1.56114i 2.66581 3.34265 − 3.34265i 0.707107 + 0.707107i 4.10653i −0.0281087 + 2.23589i
257.3 −0.707107 + 0.707107i −1.12833 − 1.12833i − 1.00000i −1.89019 − 1.19465i 1.59570 0.600193 − 0.600193i 0.707107 + 0.707107i − 0.453751i 2.18131 − 0.491820i
257.4 −0.707107 + 0.707107i −0.515696 − 0.515696i − 1.00000i 0.417939 + 2.19666i 0.729304 −1.42532 + 1.42532i 0.707107 + 0.707107i − 2.46812i −1.84880 − 1.25775i
257.5 −0.707107 + 0.707107i −0.325226 − 0.325226i − 1.00000i 2.06254 − 0.863667i 0.459939 −2.30721 + 2.30721i 0.707107 + 0.707107i − 2.78846i −0.847732 + 2.06914i
257.6 −0.707107 + 0.707107i 0.146163 + 0.146163i − 1.00000i 1.92089 + 1.14462i −0.206706 2.59530 − 2.59530i 0.707107 + 0.707107i − 2.95727i −2.16765 + 0.548906i
257.7 −0.707107 + 0.707107i 0.427501 + 0.427501i − 1.00000i −2.20958 + 0.343186i −0.604578 0.978156 − 0.978156i 0.707107 + 0.707107i − 2.63449i 1.31974 − 1.80508i
257.8 −0.707107 + 0.707107i 1.53278 + 1.53278i − 1.00000i −1.06852 + 1.96425i −2.16769 −1.64724 + 1.64724i 0.707107 + 0.707107i 1.69886i −0.633377 − 2.14449i
257.9 −0.707107 + 0.707107i 1.91446 + 1.91446i − 1.00000i 2.21975 + 0.269648i −2.70746 −1.02061 + 1.02061i 0.707107 + 0.707107i 4.33033i −1.76027 + 1.37893i
257.10 −0.707107 + 0.707107i 2.10406 + 2.10406i − 1.00000i −1.02982 − 1.98481i −2.97559 3.26278 − 3.26278i 0.707107 + 0.707107i 5.85412i 2.13167 + 0.675277i
257.11 0.707107 − 0.707107i −2.10406 − 2.10406i − 1.00000i 1.02982 + 1.98481i −2.97559 −3.26278 + 3.26278i −0.707107 − 0.707107i 5.85412i 2.13167 + 0.675277i
257.12 0.707107 − 0.707107i −1.91446 − 1.91446i − 1.00000i −2.21975 − 0.269648i −2.70746 1.02061 − 1.02061i −0.707107 − 0.707107i 4.33033i −1.76027 + 1.37893i
257.13 0.707107 − 0.707107i −1.53278 − 1.53278i − 1.00000i 1.06852 − 1.96425i −2.16769 1.64724 − 1.64724i −0.707107 − 0.707107i 1.69886i −0.633377 − 2.14449i
257.14 0.707107 − 0.707107i −0.427501 − 0.427501i − 1.00000i 2.20958 − 0.343186i −0.604578 −0.978156 + 0.978156i −0.707107 − 0.707107i − 2.63449i 1.31974 − 1.80508i
257.15 0.707107 − 0.707107i −0.146163 − 0.146163i − 1.00000i −1.92089 − 1.14462i −0.206706 −2.59530 + 2.59530i −0.707107 − 0.707107i − 2.95727i −2.16765 + 0.548906i
257.16 0.707107 − 0.707107i 0.325226 + 0.325226i − 1.00000i −2.06254 + 0.863667i 0.459939 2.30721 − 2.30721i −0.707107 − 0.707107i − 2.78846i −0.847732 + 2.06914i
257.17 0.707107 − 0.707107i 0.515696 + 0.515696i − 1.00000i −0.417939 − 2.19666i 0.729304 1.42532 − 1.42532i −0.707107 − 0.707107i − 2.46812i −1.84880 − 1.25775i
257.18 0.707107 − 0.707107i 1.12833 + 1.12833i − 1.00000i 1.89019 + 1.19465i 1.59570 −0.600193 + 0.600193i −0.707107 − 0.707107i − 0.453751i 2.18131 − 0.491820i
257.19 0.707107 − 0.707107i 1.88501 + 1.88501i − 1.00000i −1.60089 + 1.56114i 2.66581 −3.34265 + 3.34265i −0.707107 − 0.707107i 4.10653i −0.0281087 + 2.23589i
257.20 0.707107 − 0.707107i 2.27071 + 2.27071i − 1.00000i 1.31680 − 1.80721i 3.21127 0.136059 − 0.136059i −0.707107 − 0.707107i 7.31224i −0.346772 − 2.20902i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
43.b odd 2 1 inner
215.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.g.b ✓ 40
5.c odd 4 1 inner 430.2.g.b ✓ 40
43.b odd 2 1 inner 430.2.g.b ✓ 40
215.g even 4 1 inner 430.2.g.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.g.b ✓ 40 1.a even 1 1 trivial
430.2.g.b ✓ 40 5.c odd 4 1 inner
430.2.g.b ✓ 40 43.b odd 2 1 inner
430.2.g.b ✓ 40 215.g even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 318 T_{3}^{36} + 38851 T_{3}^{32} + 2295270 T_{3}^{28} + 67158577 T_{3}^{24} + 873086360 T_{3}^{20} + \cdots + 10000 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display