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Properties

Label 608.2.s.c
Level $608$
Weight $2$
Character orbit 608.s
Analytic conductor $4.855$
Analytic rank $0$
Dimension $28$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [608,2,Mod(335,608)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(608, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 5])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("608.335"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.85490444289\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 6 q^{3} + 8 q^{9} + 16 q^{11} - 22 q^{17} - 4 q^{19} + 16 q^{25} + 36 q^{33} + 28 q^{35} + 6 q^{41} - 30 q^{43} - 68 q^{49} + 42 q^{51} - 26 q^{57} + 18 q^{59} - 78 q^{67} + 14 q^{73} + 6 q^{81}+ \cdots + 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
335.1 0 −2.07624 + 1.19872i 0 −0.418341 + 0.241529i 0 − 1.56686i 0 1.37385 − 2.37958i 0
335.2 0 −2.07624 + 1.19872i 0 0.418341 − 0.241529i 0 1.56686i 0 1.37385 − 2.37958i 0
335.3 0 −1.27630 + 0.736872i 0 −2.34524 + 1.35403i 0 3.01597i 0 −0.414040 + 0.717138i 0
335.4 0 −1.27630 + 0.736872i 0 2.34524 − 1.35403i 0 − 3.01597i 0 −0.414040 + 0.717138i 0
335.5 0 −0.705625 + 0.407393i 0 −3.59735 + 2.07693i 0 − 3.24695i 0 −1.16806 + 2.02314i 0
335.6 0 −0.705625 + 0.407393i 0 3.59735 − 2.07693i 0 3.24695i 0 −1.16806 + 2.02314i 0
335.7 0 −0.179130 + 0.103421i 0 −0.520463 + 0.300489i 0 − 4.27429i 0 −1.47861 + 2.56102i 0
335.8 0 −0.179130 + 0.103421i 0 0.520463 − 0.300489i 0 4.27429i 0 −1.47861 + 2.56102i 0
335.9 0 1.05104 − 0.606818i 0 −2.45005 + 1.41453i 0 0.450769i 0 −0.763544 + 1.32250i 0
335.10 0 1.05104 − 0.606818i 0 2.45005 − 1.41453i 0 − 0.450769i 0 −0.763544 + 1.32250i 0
335.11 0 2.03079 − 1.17247i 0 −1.50560 + 0.869259i 0 2.63359i 0 1.24939 − 2.16401i 0
335.12 0 2.03079 − 1.17247i 0 1.50560 − 0.869259i 0 − 2.63359i 0 1.24939 − 2.16401i 0
335.13 0 2.65547 − 1.53314i 0 −2.25688 + 1.30301i 0 − 4.30088i 0 3.20101 − 5.54431i 0
335.14 0 2.65547 − 1.53314i 0 2.25688 − 1.30301i 0 4.30088i 0 3.20101 − 5.54431i 0
559.1 0 −2.07624 − 1.19872i 0 −0.418341 − 0.241529i 0 1.56686i 0 1.37385 + 2.37958i 0
559.2 0 −2.07624 − 1.19872i 0 0.418341 + 0.241529i 0 − 1.56686i 0 1.37385 + 2.37958i 0
559.3 0 −1.27630 − 0.736872i 0 −2.34524 − 1.35403i 0 − 3.01597i 0 −0.414040 − 0.717138i 0
559.4 0 −1.27630 − 0.736872i 0 2.34524 + 1.35403i 0 3.01597i 0 −0.414040 − 0.717138i 0
559.5 0 −0.705625 − 0.407393i 0 −3.59735 − 2.07693i 0 3.24695i 0 −1.16806 − 2.02314i 0
559.6 0 −0.705625 − 0.407393i 0 3.59735 + 2.07693i 0 − 3.24695i 0 −1.16806 − 2.02314i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 335.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.d odd 6 1 inner
152.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.2.s.c 28
4.b odd 2 1 152.2.o.c ✓ 28
8.b even 2 1 152.2.o.c ✓ 28
8.d odd 2 1 inner 608.2.s.c 28
19.d odd 6 1 inner 608.2.s.c 28
76.f even 6 1 152.2.o.c ✓ 28
152.l odd 6 1 152.2.o.c ✓ 28
152.o even 6 1 inner 608.2.s.c 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.o.c ✓ 28 4.b odd 2 1
152.2.o.c ✓ 28 8.b even 2 1
152.2.o.c ✓ 28 76.f even 6 1
152.2.o.c ✓ 28 152.l odd 6 1
608.2.s.c 28 1.a even 1 1 trivial
608.2.s.c 28 8.d odd 2 1 inner
608.2.s.c 28 19.d odd 6 1 inner
608.2.s.c 28 152.o even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 3 T_{3}^{13} - 8 T_{3}^{12} + 33 T_{3}^{11} + 65 T_{3}^{10} - 210 T_{3}^{9} - 201 T_{3}^{8} + \cdots + 27 \) acting on \(S_{2}^{\mathrm{new}}(608, [\chi])\). Copy content Toggle raw display