Properties

Label 155.2.n.a
Level $155$
Weight $2$
Character orbit 155.n
Analytic conductor $1.238$
Analytic rank $0$
Dimension $56$
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] = [155,2,Mod(4,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 155.n (of order \(10\), degree \(4\), 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: \(1.23768123133\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 2 q^{4} - 10 q^{5} - 4 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 2 q^{4} - 10 q^{5} - 4 q^{6} + 8 q^{9} + 4 q^{10} - 14 q^{11} - 16 q^{14} + 21 q^{15} - 22 q^{16} - 2 q^{19} - 29 q^{20} - 38 q^{21} - 30 q^{24} - 10 q^{25} + 12 q^{26} + 20 q^{29} - 30 q^{30} + 26 q^{31} - 2 q^{34} + 24 q^{35} - 44 q^{36} - 18 q^{39} + 36 q^{40} - 20 q^{41} + 62 q^{44} + 8 q^{45} - 70 q^{46} - 42 q^{49} - 11 q^{50} + 10 q^{51} - 26 q^{54} - 30 q^{55} + 208 q^{56} - 50 q^{59} - 67 q^{60} + 56 q^{61} + 12 q^{64} - 27 q^{65} + 60 q^{66} + 34 q^{69} + 5 q^{70} - 8 q^{71} - 84 q^{74} - 38 q^{75} - 20 q^{76} + 122 q^{79} - 80 q^{80} + 42 q^{81} - 40 q^{84} - 18 q^{85} + 78 q^{86} - 28 q^{89} + 144 q^{90} - 20 q^{91} + 160 q^{94} + 9 q^{95} - 70 q^{96} - 52 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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:
Format:

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 \(S_{2}^{\mathrm{new}}(155, [\chi])\).