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Properties

Label 155.2.n.a
Level $155$
Weight $2$
Character orbit 155.n
Analytic conductor $1.238$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [155,2,Mod(4,155)] 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("155.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(155, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 155.n (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.23768123133\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 2 q^{4} - 10 q^{5} - 4 q^{6} + 8 q^{9} + 4 q^{10} - 14 q^{11} - 16 q^{14} + 21 q^{15} - 22 q^{16} - 2 q^{19} - 29 q^{20} - 38 q^{21} - 30 q^{24} - 10 q^{25} + 12 q^{26} + 20 q^{29} - 30 q^{30} + 26 q^{31}+ \cdots - 52 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.53334 + 0.823132i 0.851532 + 0.276679i 4.12224 − 2.99498i −1.95835 − 1.07930i −2.38496 −1.09214 − 1.50319i −4.84638 + 6.67047i −1.77850 − 1.29215i 5.84956 + 1.12224i
4.2 −1.90735 + 0.619737i 0.725535 + 0.235741i 1.63589 − 1.18855i 2.06323 − 0.862016i −1.52995 −0.105855 − 0.145698i −0.0260217 + 0.0358157i −1.95622 − 1.42128i −3.40109 + 2.92283i
4.3 −1.58285 + 0.514300i −2.21876 − 0.720920i 0.622883 − 0.452551i −2.23096 + 0.151008i 3.88274 1.16442 + 1.60269i 1.20333 − 1.65624i 1.97614 + 1.43575i 3.45362 − 1.38641i
4.4 −1.53412 + 0.498465i −0.906662 − 0.294592i 0.487018 − 0.353839i 0.594355 + 2.15563i 1.53777 −0.381739 − 0.525419i 1.32551 − 1.82441i −1.69180 − 1.22916i −1.98632 − 3.01073i
4.5 −1.03006 + 0.334686i 2.61471 + 0.849570i −0.669031 + 0.486079i −1.31324 − 1.80981i −2.97764 2.94133 + 4.04839i 1.79968 − 2.47704i 3.68788 + 2.67940i 1.95843 + 1.42468i
4.6 −0.276467 + 0.0898296i −0.803274 − 0.260999i −1.54967 + 1.12590i −0.0371642 − 2.23576i 0.245524 −1.44375 − 1.98715i 0.669025 − 0.920834i −1.84992 − 1.34405i 0.211112 + 0.614775i
4.7 −0.124167 + 0.0403444i 2.85207 + 0.926694i −1.60424 + 1.16555i 2.19114 + 0.445972i −0.391521 −2.49647 − 3.43610i 0.305651 − 0.420692i 4.84849 + 3.52264i −0.290061 + 0.0330252i
4.8 0.124167 − 0.0403444i −2.85207 − 0.926694i −1.60424 + 1.16555i 2.19114 − 0.445972i −0.391521 2.49647 + 3.43610i −0.305651 + 0.420692i 4.84849 + 3.52264i 0.254076 − 0.143776i
4.9 0.276467 − 0.0898296i 0.803274 + 0.260999i −1.54967 + 1.12590i −0.0371642 + 2.23576i 0.245524 1.44375 + 1.98715i −0.669025 + 0.920834i −1.84992 − 1.34405i 0.190563 + 0.621452i
4.10 1.03006 − 0.334686i −2.61471 − 0.849570i −0.669031 + 0.486079i −1.31324 + 1.80981i −2.97764 −2.94133 − 4.04839i −1.79968 + 2.47704i 3.68788 + 2.67940i −0.746995 + 2.30373i
4.11 1.53412 − 0.498465i 0.906662 + 0.294592i 0.487018 − 0.353839i 0.594355 − 2.15563i 1.53777 0.381739 + 0.525419i −1.32551 + 1.82441i −1.69180 − 1.22916i −0.162696 − 3.60326i
4.12 1.58285 − 0.514300i 2.21876 + 0.720920i 0.622883 − 0.452551i −2.23096 − 0.151008i 3.88274 −1.16442 − 1.60269i −1.20333 + 1.65624i 1.97614 + 1.43575i −3.60895 + 0.908360i
4.13 1.90735 − 0.619737i −0.725535 − 0.235741i 1.63589 − 1.18855i 2.06323 + 0.862016i −1.52995 0.105855 + 0.145698i 0.0260217 − 0.0358157i −1.95622 − 1.42128i 4.46954 + 0.365508i
4.14 2.53334 − 0.823132i −0.851532 − 0.276679i 4.12224 − 2.99498i −1.95835 + 1.07930i −2.38496 1.09214 + 1.50319i 4.84638 − 6.67047i −1.77850 − 1.29215i −4.07276 + 4.34620i
39.1 −2.53334 − 0.823132i 0.851532 − 0.276679i 4.12224 + 2.99498i −1.95835 + 1.07930i −2.38496 −1.09214 + 1.50319i −4.84638 − 6.67047i −1.77850 + 1.29215i 5.84956 − 1.12224i
39.2 −1.90735 − 0.619737i 0.725535 − 0.235741i 1.63589 + 1.18855i 2.06323 + 0.862016i −1.52995 −0.105855 + 0.145698i −0.0260217 − 0.0358157i −1.95622 + 1.42128i −3.40109 − 2.92283i
39.3 −1.58285 − 0.514300i −2.21876 + 0.720920i 0.622883 + 0.452551i −2.23096 − 0.151008i 3.88274 1.16442 − 1.60269i 1.20333 + 1.65624i 1.97614 − 1.43575i 3.45362 + 1.38641i
39.4 −1.53412 − 0.498465i −0.906662 + 0.294592i 0.487018 + 0.353839i 0.594355 − 2.15563i 1.53777 −0.381739 + 0.525419i 1.32551 + 1.82441i −1.69180 + 1.22916i −1.98632 + 3.01073i
39.5 −1.03006 − 0.334686i 2.61471 − 0.849570i −0.669031 − 0.486079i −1.31324 + 1.80981i −2.97764 2.94133 − 4.04839i 1.79968 + 2.47704i 3.68788 − 2.67940i 1.95843 − 1.42468i
39.6 −0.276467 − 0.0898296i −0.803274 + 0.260999i −1.54967 − 1.12590i −0.0371642 + 2.23576i 0.245524 −1.44375 + 1.98715i 0.669025 + 0.920834i −1.84992 + 1.34405i 0.211112 − 0.614775i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.d even 5 1 inner
155.n even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.2.n.a ✓ 56
5.b even 2 1 inner 155.2.n.a ✓ 56
5.c odd 4 2 775.2.k.h 56
31.d even 5 1 inner 155.2.n.a ✓ 56
155.n even 10 1 inner 155.2.n.a ✓ 56
155.s odd 20 2 775.2.k.h 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.n.a ✓ 56 1.a even 1 1 trivial
155.2.n.a ✓ 56 5.b even 2 1 inner
155.2.n.a ✓ 56 31.d even 5 1 inner
155.2.n.a ✓ 56 155.n even 10 1 inner
775.2.k.h 56 5.c odd 4 2
775.2.k.h 56 155.s odd 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(155, [\chi])\).