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Properties

Label 775.2.bs.a
Level $775$
Weight $2$
Character orbit 775.bs
Analytic conductor $6.188$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,2,Mod(182,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.182");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.bs (of order \(20\), 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: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 20 q^{11} + 64 q^{16} - 80 q^{21} - 32 q^{31} - 312 q^{36} + 40 q^{46} - 56 q^{51} - 144 q^{56} + 264 q^{66} - 76 q^{71} - 28 q^{76} + 48 q^{81} + 120 q^{86} + 120 q^{91} + 540 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
182.1 −1.23927 + 2.43220i −2.53627 + 1.29229i −3.20425 − 4.41027i 0 − 7.77022i −3.10237 − 0.491368i 9.30534 − 1.47382i 2.99928 − 4.12816i 0
182.2 −1.03452 + 2.03035i 1.78424 − 0.909114i −1.87654 − 2.58283i 0 4.56312i −2.33045 − 0.369108i 2.68405 − 0.425112i 0.593657 − 0.817099i 0
182.3 −0.965204 + 1.89432i −0.184195 + 0.0938523i −1.48126 − 2.03878i 0 − 0.439512i 4.14122 + 0.655905i 1.09207 − 0.172967i −1.73824 + 2.39248i 0
182.4 −0.431258 + 0.846391i −1.16302 + 0.592587i 0.645176 + 0.888009i 0 − 1.23993i −4.02685 − 0.637790i −2.90630 + 0.460313i −0.761904 + 1.04867i 0
182.5 −0.419627 + 0.823565i −2.84187 + 1.44801i 0.673398 + 0.926853i 0 − 2.94809i 4.01818 + 0.636417i −2.87176 + 0.454842i 4.21617 − 5.80306i 0
182.6 −0.308328 + 0.605128i 1.24990 − 0.636857i 0.904457 + 1.24488i 0 0.952711i 0.984672 + 0.155957i −2.37376 + 0.375966i −0.606687 + 0.835033i 0
182.7 0.308328 − 0.605128i −1.24990 + 0.636857i 0.904457 + 1.24488i 0 0.952711i −0.984672 − 0.155957i 2.37376 − 0.375966i −0.606687 + 0.835033i 0
182.8 0.419627 − 0.823565i 2.84187 − 1.44801i 0.673398 + 0.926853i 0 − 2.94809i −4.01818 − 0.636417i 2.87176 − 0.454842i 4.21617 − 5.80306i 0
182.9 0.431258 − 0.846391i 1.16302 − 0.592587i 0.645176 + 0.888009i 0 − 1.23993i 4.02685 + 0.637790i 2.90630 − 0.460313i −0.761904 + 1.04867i 0
182.10 0.965204 − 1.89432i 0.184195 − 0.0938523i −1.48126 − 2.03878i 0 − 0.439512i −4.14122 − 0.655905i −1.09207 + 0.172967i −1.73824 + 2.39248i 0
182.11 1.03452 − 2.03035i −1.78424 + 0.909114i −1.87654 − 2.58283i 0 4.56312i 2.33045 + 0.369108i −2.68405 + 0.425112i 0.593657 − 0.817099i 0
182.12 1.23927 − 2.43220i 2.53627 − 1.29229i −3.20425 − 4.41027i 0 − 7.77022i 3.10237 + 0.491368i −9.30534 + 1.47382i 2.99928 − 4.12816i 0
232.1 −2.74630 − 0.434971i −0.271601 − 1.71482i 5.45083 + 1.77108i 0 4.82755i 0.313459 − 0.159715i −9.24428 − 4.71020i −0.0136815 + 0.00444540i 0
232.2 −2.47918 − 0.392663i 0.494392 + 3.12147i 4.09002 + 1.32893i 0 − 7.93281i −3.01577 + 1.53661i −5.14507 − 2.62154i −6.64598 + 2.15941i 0
232.3 −1.93427 − 0.306358i 0.0466583 + 0.294589i 1.74542 + 0.567122i 0 − 0.584108i 1.54035 − 0.784849i 0.287484 + 0.146480i 2.76856 − 0.899561i 0
232.4 −1.37690 − 0.218080i −0.404985 − 2.55698i −0.0538055 − 0.0174825i 0 3.60903i 2.30438 − 1.17414i 2.55452 + 1.30159i −3.52094 + 1.14402i 0
232.5 −1.21397 − 0.192273i 0.0241238 + 0.152312i −0.465371 − 0.151208i 0 − 0.189540i −3.78662 + 1.92938i 2.72614 + 1.38904i 2.83055 − 0.919702i 0
232.6 −0.538122 − 0.0852301i 0.388976 + 2.45590i −1.61980 − 0.526306i 0 − 1.35473i 2.08986 − 1.06484i 1.79769 + 0.915967i −3.02697 + 0.983521i 0
232.7 0.538122 + 0.0852301i −0.388976 − 2.45590i −1.61980 − 0.526306i 0 − 1.35473i −2.08986 + 1.06484i −1.79769 − 0.915967i −3.02697 + 0.983521i 0
232.8 1.21397 + 0.192273i −0.0241238 − 0.152312i −0.465371 − 0.151208i 0 − 0.189540i 3.78662 − 1.92938i −2.72614 − 1.38904i 2.83055 − 0.919702i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 182.12
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
5.c odd 4 2 inner
31.f odd 10 1 inner
155.m odd 10 1 inner
155.r even 20 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.bs.a ✓ 96
5.b even 2 1 inner 775.2.bs.a ✓ 96
5.c odd 4 2 inner 775.2.bs.a ✓ 96
31.f odd 10 1 inner 775.2.bs.a ✓ 96
155.m odd 10 1 inner 775.2.bs.a ✓ 96
155.r even 20 2 inner 775.2.bs.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.bs.a ✓ 96 1.a even 1 1 trivial
775.2.bs.a ✓ 96 5.b even 2 1 inner
775.2.bs.a ✓ 96 5.c odd 4 2 inner
775.2.bs.a ✓ 96 31.f odd 10 1 inner
775.2.bs.a ✓ 96 155.m odd 10 1 inner
775.2.bs.a ✓ 96 155.r even 20 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 130 T_{2}^{92} + 10207 T_{2}^{88} - 644515 T_{2}^{84} + 39392813 T_{2}^{80} + \cdots + 10\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display