↑

Properties

Label 580.2.z.a
Level $580$
Weight $2$
Character orbit 580.z
Analytic conductor $4.631$
Analytic rank $0$
Dimension $24$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [580,2,Mod(121,580)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("580.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(580, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 0, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 580 = 2^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 580.z (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.63132331723\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{5} + 4 q^{7} - 4 q^{9} + 21 q^{21} - 2 q^{23} - 4 q^{25} - 21 q^{27} + 7 q^{29} + 7 q^{31} - 6 q^{33} + 4 q^{35} + 14 q^{37} - 35 q^{39} + 35 q^{43} - 4 q^{45} - 28 q^{47} + 22 q^{49} - 26 q^{51}+ \cdots - 35 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 0 −2.80806 + 0.640922i 0 0.623490 + 0.781831i 0 −0.162335 − 0.711236i 0 4.77153 − 2.29785i 0
121.2 0 −0.619217 + 0.141332i 0 0.623490 + 0.781831i 0 −0.746977 − 3.27272i 0 −2.33945 + 1.12662i 0
121.3 0 0.463763 − 0.105851i 0 0.623490 + 0.781831i 0 0.256968 + 1.12585i 0 −2.49903 + 1.20347i 0
121.4 0 2.11751 − 0.483307i 0 0.623490 + 0.781831i 0 0.696417 + 3.05120i 0 1.54734 − 0.745160i 0
241.1 0 −1.67989 + 1.33966i 0 −0.900969 + 0.433884i 0 0.152059 + 0.190676i 0 0.359752 − 1.57618i 0
241.2 0 0.169179 − 0.134916i 0 −0.900969 + 0.433884i 0 −0.790469 − 0.991216i 0 −0.657144 + 2.87913i 0
241.3 0 0.935391 − 0.745950i 0 −0.900969 + 0.433884i 0 −2.14291 − 2.68712i 0 −0.349047 + 1.52927i 0
241.4 0 2.09977 − 1.67451i 0 −0.900969 + 0.433884i 0 1.81182 + 2.27195i 0 0.937491 − 4.10742i 0
341.1 0 −1.10016 + 2.28451i 0 −0.222521 − 0.974928i 0 1.69370 + 0.815642i 0 −2.13815 − 2.68115i 0
341.2 0 −0.891680 + 1.85159i 0 −0.222521 − 0.974928i 0 −2.95029 − 1.42079i 0 −0.762835 − 0.956565i 0
341.3 0 0.141986 − 0.294836i 0 −0.222521 − 0.974928i 0 4.21521 + 2.02994i 0 1.80370 + 2.26177i 0
341.4 0 1.17141 − 2.43245i 0 −0.222521 − 0.974928i 0 −0.0331920 − 0.0159844i 0 −2.67416 − 3.35329i 0
361.1 0 −1.67989 − 1.33966i 0 −0.900969 − 0.433884i 0 0.152059 − 0.190676i 0 0.359752 + 1.57618i 0
361.2 0 0.169179 + 0.134916i 0 −0.900969 − 0.433884i 0 −0.790469 + 0.991216i 0 −0.657144 − 2.87913i 0
361.3 0 0.935391 + 0.745950i 0 −0.900969 − 0.433884i 0 −2.14291 + 2.68712i 0 −0.349047 − 1.52927i 0
361.4 0 2.09977 + 1.67451i 0 −0.900969 − 0.433884i 0 1.81182 − 2.27195i 0 0.937491 + 4.10742i 0
381.1 0 −1.10016 − 2.28451i 0 −0.222521 + 0.974928i 0 1.69370 − 0.815642i 0 −2.13815 + 2.68115i 0
381.2 0 −0.891680 − 1.85159i 0 −0.222521 + 0.974928i 0 −2.95029 + 1.42079i 0 −0.762835 + 0.956565i 0
381.3 0 0.141986 + 0.294836i 0 −0.222521 + 0.974928i 0 4.21521 − 2.02994i 0 1.80370 − 2.26177i 0
381.4 0 1.17141 + 2.43245i 0 −0.222521 + 0.974928i 0 −0.0331920 + 0.0159844i 0 −2.67416 + 3.35329i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 121.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 580.2.z.a ✓ 24
29.e even 14 1 inner 580.2.z.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.z.a ✓ 24 1.a even 1 1 trivial
580.2.z.a ✓ 24 29.e even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 4 T_{3}^{22} + 7 T_{3}^{21} + 26 T_{3}^{20} + 35 T_{3}^{19} - 228 T_{3}^{18} - 70 T_{3}^{17} + \cdots + 169 \) acting on \(S_{2}^{\mathrm{new}}(580, [\chi])\). Copy content Toggle raw display