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Properties

Label 775.2.bl.f
Level $775$
Weight $2$
Character orbit 775.bl
Analytic conductor $6.188$
Analytic rank $0$
Dimension $112$
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] = [775,2,Mod(51,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.bl (of order \(15\), degree \(8\), 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(14\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 155)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 16 q^{4} - 26 q^{6} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 16 q^{4} - 26 q^{6} + 32 q^{9} - 10 q^{11} + 20 q^{14} - 44 q^{16} + 46 q^{19} + 26 q^{21} - 48 q^{24} - 78 q^{26} - 16 q^{29} - 26 q^{31} + 4 q^{34} - 4 q^{36} - 16 q^{41} + 86 q^{44} + 4 q^{46} - 42 q^{49} + 92 q^{51} - 32 q^{54} - 34 q^{56} - 26 q^{59} + 40 q^{61} - 24 q^{64} + 252 q^{66} + 4 q^{69} + 116 q^{71} + 18 q^{74} + 224 q^{76} + 134 q^{79} - 60 q^{81} + 20 q^{84} - 126 q^{86} - 82 q^{89} + 2 q^{91} + 244 q^{94} + 208 q^{96} - 94 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −0.820665 − 2.52575i −2.06664 − 2.29523i −4.08788 + 2.97002i 0 −4.10116 + 7.10343i 0.158118 + 1.50439i 6.55924 + 4.76557i −0.683521 + 6.50327i 0
51.2 −0.657871 − 2.02472i 0.0246032 + 0.0273246i −2.04866 + 1.48844i 0 0.0391389 − 0.0677905i 0.366291 + 3.48503i 0.916769 + 0.666072i 0.313444 − 2.98222i 0
51.3 −0.649000 − 1.99742i 0.966660 + 1.07358i −1.95043 + 1.41707i 0 1.51703 − 2.62758i 0.165452 + 1.57417i 0.698108 + 0.507205i 0.0954328 − 0.907983i 0
51.4 −0.450639 − 1.38693i −0.800290 − 0.888812i −0.102453 + 0.0744362i 0 −0.872074 + 1.51048i −0.451986 − 4.30036i −2.21017 − 1.60578i 0.164063 − 1.56095i 0
51.5 −0.338718 − 1.04247i −1.52374 − 1.69229i 0.646026 − 0.469365i 0 −1.24804 + 2.16166i −0.0753778 − 0.717171i −2.48167 − 1.80304i −0.228460 + 2.17366i 0
51.6 −0.306786 − 0.944189i 1.78101 + 1.97801i 0.820658 − 0.596243i 0 1.32123 − 2.28844i −0.0895820 − 0.852316i −2.42108 − 1.75902i −0.426951 + 4.06217i 0
51.7 −0.0521621 − 0.160538i −0.695671 − 0.772621i 1.59498 − 1.15882i 0 −0.0877476 + 0.151983i 0.401803 + 3.82290i −0.542357 − 0.394045i 0.200600 − 1.90859i 0
51.8 0.0521621 + 0.160538i 0.695671 + 0.772621i 1.59498 − 1.15882i 0 −0.0877476 + 0.151983i −0.401803 − 3.82290i 0.542357 + 0.394045i 0.200600 − 1.90859i 0
51.9 0.306786 + 0.944189i −1.78101 − 1.97801i 0.820658 − 0.596243i 0 1.32123 − 2.28844i 0.0895820 + 0.852316i 2.42108 + 1.75902i −0.426951 + 4.06217i 0
51.10 0.338718 + 1.04247i 1.52374 + 1.69229i 0.646026 − 0.469365i 0 −1.24804 + 2.16166i 0.0753778 + 0.717171i 2.48167 + 1.80304i −0.228460 + 2.17366i 0
51.11 0.450639 + 1.38693i 0.800290 + 0.888812i −0.102453 + 0.0744362i 0 −0.872074 + 1.51048i 0.451986 + 4.30036i 2.21017 + 1.60578i 0.164063 − 1.56095i 0
51.12 0.649000 + 1.99742i −0.966660 − 1.07358i −1.95043 + 1.41707i 0 1.51703 − 2.62758i −0.165452 − 1.57417i −0.698108 − 0.507205i 0.0954328 − 0.907983i 0
51.13 0.657871 + 2.02472i −0.0246032 − 0.0273246i −2.04866 + 1.48844i 0 0.0391389 − 0.0677905i −0.366291 − 3.48503i −0.916769 − 0.666072i 0.313444 − 2.98222i 0
51.14 0.820665 + 2.52575i 2.06664 + 2.29523i −4.08788 + 2.97002i 0 −4.10116 + 7.10343i −0.158118 − 1.50439i −6.55924 − 4.76557i −0.683521 + 6.50327i 0
76.1 −0.820665 + 2.52575i −2.06664 + 2.29523i −4.08788 − 2.97002i 0 −4.10116 − 7.10343i 0.158118 − 1.50439i 6.55924 − 4.76557i −0.683521 − 6.50327i 0
76.2 −0.657871 + 2.02472i 0.0246032 − 0.0273246i −2.04866 − 1.48844i 0 0.0391389 + 0.0677905i 0.366291 − 3.48503i 0.916769 − 0.666072i 0.313444 + 2.98222i 0
76.3 −0.649000 + 1.99742i 0.966660 − 1.07358i −1.95043 − 1.41707i 0 1.51703 + 2.62758i 0.165452 − 1.57417i 0.698108 − 0.507205i 0.0954328 + 0.907983i 0
76.4 −0.450639 + 1.38693i −0.800290 + 0.888812i −0.102453 − 0.0744362i 0 −0.872074 − 1.51048i −0.451986 + 4.30036i −2.21017 + 1.60578i 0.164063 + 1.56095i 0
76.5 −0.338718 + 1.04247i −1.52374 + 1.69229i 0.646026 + 0.469365i 0 −1.24804 − 2.16166i −0.0753778 + 0.717171i −2.48167 + 1.80304i −0.228460 − 2.17366i 0
76.6 −0.306786 + 0.944189i 1.78101 − 1.97801i 0.820658 + 0.596243i 0 1.32123 + 2.28844i −0.0895820 + 0.852316i −2.42108 + 1.75902i −0.426951 − 4.06217i 0
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.g even 15 1 inner
155.u even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.bl.f 112
5.b even 2 1 inner 775.2.bl.f 112
5.c odd 4 2 155.2.u.a ✓ 112
31.g even 15 1 inner 775.2.bl.f 112
155.u even 30 1 inner 775.2.bl.f 112
155.w odd 60 2 155.2.u.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.u.a ✓ 112 5.c odd 4 2
155.2.u.a ✓ 112 155.w odd 60 2
775.2.bl.f 112 1.a even 1 1 trivial
775.2.bl.f 112 5.b even 2 1 inner
775.2.bl.f 112 31.g even 15 1 inner
775.2.bl.f 112 155.u even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{112} + 36 T_{2}^{110} + 754 T_{2}^{108} + 12012 T_{2}^{106} + 160359 T_{2}^{104} + \cdots + 13305853201 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display