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Properties

Label 775.2.bf.d
Level $775$
Weight $2$
Character orbit 775.bf
Analytic conductor $6.188$
Analytic rank $0$
Dimension $48$
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(349,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.bf (of order \(10\), degree \(4\), not 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: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 155)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 28 q^{4} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 28 q^{4} + 20 q^{9} + 10 q^{11} + 4 q^{14} + 4 q^{19} + 12 q^{21} - 32 q^{24} - 52 q^{26} + 34 q^{29} + 4 q^{31} - 8 q^{34} + 316 q^{36} + 4 q^{39} + 24 q^{41} - 2 q^{44} - 80 q^{46} + 30 q^{49} - 60 q^{51} + 18 q^{54} - 112 q^{56} + 44 q^{59} - 32 q^{61} - 50 q^{64} - 180 q^{66} - 96 q^{69} - 6 q^{71} + 82 q^{74} - 154 q^{76} + 26 q^{79} - 4 q^{81} + 118 q^{84} - 44 q^{86} - 10 q^{89} - 20 q^{91} + 196 q^{94} - 134 q^{96} - 332 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1 −2.59225 − 0.842272i 2.82307 − 0.917270i 4.39229 + 3.19119i 0 −8.09068 1.18334 − 1.62873i −5.49387 − 7.56166i 4.70127 − 3.41567i 0
349.2 −2.26450 − 0.735779i −2.92347 + 0.949892i 2.96854 + 2.15677i 0 7.31909 1.57539 − 2.16834i −2.33626 − 3.21559i 5.21732 − 3.79060i 0
349.3 −2.00598 − 0.651781i −1.75513 + 0.570277i 1.98109 + 1.43934i 0 3.89245 −1.84099 + 2.53391i −0.556355 − 0.765757i 0.328223 − 0.238468i 0
349.4 −1.66497 − 0.540981i −0.518412 + 0.168442i 0.861421 + 0.625859i 0 0.954263 −2.65722 + 3.65734i 0.962353 + 1.32457i −2.18667 + 1.58871i 0
349.5 −1.04535 − 0.339655i 1.26147 − 0.409877i −0.640644 − 0.465455i 0 −1.45790 0.257347 − 0.354207i 1.80373 + 2.48262i −1.00374 + 0.729258i 0
349.6 −0.175269 − 0.0569485i 1.96706 − 0.639136i −1.59056 − 1.15561i 0 −0.381163 0.542768 − 0.747057i 0.429611 + 0.591308i 1.03378 − 0.751082i 0
349.7 0.175269 + 0.0569485i −1.96706 + 0.639136i −1.59056 − 1.15561i 0 −0.381163 −0.542768 + 0.747057i −0.429611 − 0.591308i 1.03378 − 0.751082i 0
349.8 1.04535 + 0.339655i −1.26147 + 0.409877i −0.640644 − 0.465455i 0 −1.45790 −0.257347 + 0.354207i −1.80373 − 2.48262i −1.00374 + 0.729258i 0
349.9 1.66497 + 0.540981i 0.518412 − 0.168442i 0.861421 + 0.625859i 0 0.954263 2.65722 − 3.65734i −0.962353 − 1.32457i −2.18667 + 1.58871i 0
349.10 2.00598 + 0.651781i 1.75513 − 0.570277i 1.98109 + 1.43934i 0 3.89245 1.84099 − 2.53391i 0.556355 + 0.765757i 0.328223 − 0.238468i 0
349.11 2.26450 + 0.735779i 2.92347 − 0.949892i 2.96854 + 2.15677i 0 7.31909 −1.57539 + 2.16834i 2.33626 + 3.21559i 5.21732 − 3.79060i 0
349.12 2.59225 + 0.842272i −2.82307 + 0.917270i 4.39229 + 3.19119i 0 −8.09068 −1.18334 + 1.62873i 5.49387 + 7.56166i 4.70127 − 3.41567i 0
374.1 −1.41112 + 1.94224i −1.42062 − 1.95532i −1.16301 − 3.57938i 0 5.80237 −1.41771 + 0.460643i 4.02670 + 1.30835i −0.878047 + 2.70235i 0
374.2 −1.39656 + 1.92221i 1.67618 + 2.30707i −1.12645 − 3.46686i 0 −6.77556 2.36476 − 0.768358i 3.71780 + 1.20799i −1.58593 + 4.88098i 0
374.3 −0.843189 + 1.16055i −0.376801 − 0.518622i −0.0178745 − 0.0550119i 0 0.919601 −0.218119 + 0.0708711i −2.64970 − 0.860940i 0.800061 − 2.46234i 0
374.4 −0.662266 + 0.911531i 1.73132 + 2.38295i 0.225742 + 0.694762i 0 −3.31873 −0.255014 + 0.0828589i −2.92593 − 0.950694i −1.75396 + 5.39812i 0
374.5 −0.290274 + 0.399527i −1.30510 − 1.79631i 0.542671 + 1.67017i 0 1.09651 4.30102 − 1.39749i −1.76415 − 0.573206i −0.596411 + 1.83556i 0
374.6 −0.239362 + 0.329453i −0.0573558 − 0.0789435i 0.566789 + 1.74440i 0 0.0397370 −4.51039 + 1.46551i −1.48496 − 0.482491i 0.924109 − 2.84411i 0
374.7 0.239362 − 0.329453i 0.0573558 + 0.0789435i 0.566789 + 1.74440i 0 0.0397370 4.51039 − 1.46551i 1.48496 + 0.482491i 0.924109 − 2.84411i 0
374.8 0.290274 − 0.399527i 1.30510 + 1.79631i 0.542671 + 1.67017i 0 1.09651 −4.30102 + 1.39749i 1.76415 + 0.573206i −0.596411 + 1.83556i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.12
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 775.2.bf.d 48
5.b even 2 1 inner 775.2.bf.d 48
5.c odd 4 1 155.2.h.a ✓ 24
5.c odd 4 1 775.2.k.e 24
31.d even 5 1 inner 775.2.bf.d 48
155.n even 10 1 inner 775.2.bf.d 48
155.r even 20 1 4805.2.a.s 12
155.s odd 20 1 155.2.h.a ✓ 24
155.s odd 20 1 775.2.k.e 24
155.s odd 20 1 4805.2.a.t 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.h.a ✓ 24 5.c odd 4 1
155.2.h.a ✓ 24 155.s odd 20 1
775.2.k.e 24 5.c odd 4 1
775.2.k.e 24 155.s odd 20 1
775.2.bf.d 48 1.a even 1 1 trivial
775.2.bf.d 48 5.b even 2 1 inner
775.2.bf.d 48 31.d even 5 1 inner
775.2.bf.d 48 155.n even 10 1 inner
4805.2.a.s 12 155.r even 20 1
4805.2.a.t 12 155.s odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 26 T_{2}^{46} + 359 T_{2}^{44} - 3535 T_{2}^{42} + 29425 T_{2}^{40} - 210003 T_{2}^{38} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display