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Properties

Label 930.2.s.c
Level $930$
Weight $2$
Character orbit 930.s
Analytic conductor $7.426$
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] = [930,2,Mod(439,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.439");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.s (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: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} + 4 q^{5} - 12 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} + 4 q^{5} - 12 q^{6} + 12 q^{9} - 4 q^{11} + 12 q^{14} + 24 q^{16} - 4 q^{19} - 4 q^{20} - 12 q^{21} + 12 q^{24} - 4 q^{25} + 16 q^{26} + 32 q^{29} - 8 q^{30} + 24 q^{31} - 20 q^{34} + 48 q^{35} - 12 q^{36} + 32 q^{39} - 12 q^{41} + 4 q^{44} - 4 q^{45} + 56 q^{46} + 24 q^{49} - 16 q^{50} + 20 q^{51} - 24 q^{54} + 36 q^{55} - 12 q^{56} - 4 q^{59} - 40 q^{61} - 24 q^{64} - 4 q^{65} + 8 q^{66} + 28 q^{69} - 32 q^{70} - 56 q^{71} + 12 q^{74} + 16 q^{75} + 4 q^{76} - 72 q^{79} + 4 q^{80} - 12 q^{81} + 12 q^{84} + 40 q^{85} - 24 q^{86} + 128 q^{89} - 160 q^{91} + 56 q^{94} - 56 q^{95} - 12 q^{96} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
439.1 − 1.00000i −0.866025 − 0.500000i −1.00000 −1.89131 + 1.19288i −0.500000 + 0.866025i 2.10840 + 1.21729i 1.00000i 0.500000 + 0.866025i 1.19288 + 1.89131i
439.2 − 1.00000i −0.866025 − 0.500000i −1.00000 −0.968769 + 2.01531i −0.500000 + 0.866025i −3.61895 − 2.08940i 1.00000i 0.500000 + 0.866025i 2.01531 + 0.968769i
439.3 − 1.00000i −0.866025 − 0.500000i −1.00000 −0.291362 − 2.21700i −0.500000 + 0.866025i 3.11266 + 1.79709i 1.00000i 0.500000 + 0.866025i −2.21700 + 0.291362i
439.4 − 1.00000i −0.866025 − 0.500000i −1.00000 0.875319 − 2.05762i −0.500000 + 0.866025i 0.765960 + 0.442227i 1.00000i 0.500000 + 0.866025i −2.05762 − 0.875319i
439.5 − 1.00000i −0.866025 − 0.500000i −1.00000 1.24897 + 1.85475i −0.500000 + 0.866025i −0.669647 − 0.386621i 1.00000i 0.500000 + 0.866025i 1.85475 − 1.24897i
439.6 − 1.00000i −0.866025 − 0.500000i −1.00000 2.02715 + 0.943742i −0.500000 + 0.866025i 3.49773 + 2.01941i 1.00000i 0.500000 + 0.866025i 0.943742 − 2.02715i
439.7 1.00000i 0.866025 + 0.500000i −1.00000 −2.21961 − 0.270763i −0.500000 + 0.866025i −0.765960 − 0.442227i − 1.00000i 0.500000 + 0.866025i 0.270763 − 2.21961i
439.8 1.00000i 0.866025 + 0.500000i −1.00000 −1.77430 − 1.36083i −0.500000 + 0.866025i −3.11266 − 1.79709i − 1.00000i 0.500000 + 0.866025i 1.36083 − 1.77430i
439.9 1.00000i 0.866025 + 0.500000i −1.00000 −0.196273 + 2.22744i −0.500000 + 0.866025i −3.49773 − 2.01941i − 1.00000i 0.500000 + 0.866025i −2.22744 − 0.196273i
439.10 1.00000i 0.866025 + 0.500000i −1.00000 0.981775 + 2.00901i −0.500000 + 0.866025i 0.669647 + 0.386621i − 1.00000i 0.500000 + 0.866025i −2.00901 + 0.981775i
439.11 1.00000i 0.866025 + 0.500000i −1.00000 1.97872 − 1.04148i −0.500000 + 0.866025i −2.10840 − 1.21729i − 1.00000i 0.500000 + 0.866025i 1.04148 + 1.97872i
439.12 1.00000i 0.866025 + 0.500000i −1.00000 2.22970 + 0.168678i −0.500000 + 0.866025i 3.61895 + 2.08940i − 1.00000i 0.500000 + 0.866025i −0.168678 + 2.22970i
769.1 − 1.00000i 0.866025 − 0.500000i −1.00000 −2.21961 + 0.270763i −0.500000 − 0.866025i −0.765960 + 0.442227i 1.00000i 0.500000 − 0.866025i 0.270763 + 2.21961i
769.2 − 1.00000i 0.866025 − 0.500000i −1.00000 −1.77430 + 1.36083i −0.500000 − 0.866025i −3.11266 + 1.79709i 1.00000i 0.500000 − 0.866025i 1.36083 + 1.77430i
769.3 − 1.00000i 0.866025 − 0.500000i −1.00000 −0.196273 − 2.22744i −0.500000 − 0.866025i −3.49773 + 2.01941i 1.00000i 0.500000 − 0.866025i −2.22744 + 0.196273i
769.4 − 1.00000i 0.866025 − 0.500000i −1.00000 0.981775 − 2.00901i −0.500000 − 0.866025i 0.669647 − 0.386621i 1.00000i 0.500000 − 0.866025i −2.00901 − 0.981775i
769.5 − 1.00000i 0.866025 − 0.500000i −1.00000 1.97872 + 1.04148i −0.500000 − 0.866025i −2.10840 + 1.21729i 1.00000i 0.500000 − 0.866025i 1.04148 − 1.97872i
769.6 − 1.00000i 0.866025 − 0.500000i −1.00000 2.22970 − 0.168678i −0.500000 − 0.866025i 3.61895 − 2.08940i 1.00000i 0.500000 − 0.866025i −0.168678 − 2.22970i
769.7 1.00000i −0.866025 + 0.500000i −1.00000 −1.89131 − 1.19288i −0.500000 − 0.866025i 2.10840 − 1.21729i − 1.00000i 0.500000 − 0.866025i 1.19288 − 1.89131i
769.8 1.00000i −0.866025 + 0.500000i −1.00000 −0.968769 − 2.01531i −0.500000 − 0.866025i −3.61895 + 2.08940i − 1.00000i 0.500000 − 0.866025i 2.01531 − 0.968769i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 439.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.c even 3 1 inner
155.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.s.c ✓ 24
5.b even 2 1 inner 930.2.s.c ✓ 24
31.c even 3 1 inner 930.2.s.c ✓ 24
155.j even 6 1 inner 930.2.s.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.s.c ✓ 24 1.a even 1 1 trivial
930.2.s.c ✓ 24 5.b even 2 1 inner
930.2.s.c ✓ 24 31.c even 3 1 inner
930.2.s.c ✓ 24 155.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} - 54 T_{7}^{22} + 1845 T_{7}^{20} - 39122 T_{7}^{18} + 608562 T_{7}^{16} - 6462558 T_{7}^{14} + \cdots + 104060401 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display