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Properties

Label 775.2.bf.c
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{4} - 32 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{4} - 32 q^{6} + 12 q^{9} + 2 q^{11} - 12 q^{14} - 48 q^{16} - 4 q^{19} - 20 q^{21} + 64 q^{24} + 44 q^{26} - 30 q^{29} + 12 q^{31} + 40 q^{34} - 116 q^{36} + 20 q^{39} - 48 q^{41} - 82 q^{44} + 96 q^{46} + 46 q^{49} + 52 q^{51} - 22 q^{54} - 96 q^{56} - 20 q^{59} - 80 q^{61} - 18 q^{64} - 4 q^{66} - 128 q^{69} - 14 q^{71} + 2 q^{74} + 86 q^{76} - 62 q^{79} + 68 q^{81} - 226 q^{84} - 44 q^{86} - 26 q^{89} + 108 q^{91} + 20 q^{94} + 202 q^{96} + 340 q^{99}+O(q^{100}) \) 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 −2.49377 − 0.810275i 0.251039 − 0.0815674i 3.94431 + 2.86571i 0 −0.692125 −0.0919008 + 0.126491i −4.43171 − 6.09973i −2.37068 + 1.72240i 0
349.2 −2.35953 − 0.766659i −1.47405 + 0.478948i 3.36160 + 2.44235i 0 3.84526 2.32590 − 3.20132i −3.14282 − 4.32573i −0.483619 + 0.351369i 0
349.3 −1.58691 − 0.515618i 1.63919 − 0.532605i 0.634386 + 0.460908i 0 −2.87586 −1.13381 + 1.56056i 1.19247 + 1.64129i −0.0237792 + 0.0172766i 0
349.4 −1.29445 − 0.420591i 2.99486 − 0.973089i −0.119342 − 0.0867070i 0 −4.28595 −0.538593 + 0.741309i 1.71804 + 2.36467i 5.59523 − 4.06517i 0
349.5 −1.08409 − 0.352241i −0.301289 + 0.0978947i −0.566864 − 0.411851i 0 0.361106 1.53999 − 2.11961i 1.80946 + 2.49051i −2.34586 + 1.70437i 0
349.6 −0.612984 − 0.199171i −2.77947 + 0.903105i −1.28195 − 0.931393i 0 1.88364 −1.75828 + 2.42006i 1.35800 + 1.86913i 4.48281 − 3.25695i 0
349.7 0.612984 + 0.199171i 2.77947 − 0.903105i −1.28195 − 0.931393i 0 1.88364 1.75828 − 2.42006i −1.35800 − 1.86913i 4.48281 − 3.25695i 0
349.8 1.08409 + 0.352241i 0.301289 − 0.0978947i −0.566864 − 0.411851i 0 0.361106 −1.53999 + 2.11961i −1.80946 − 2.49051i −2.34586 + 1.70437i 0
349.9 1.29445 + 0.420591i −2.99486 + 0.973089i −0.119342 − 0.0867070i 0 −4.28595 0.538593 − 0.741309i −1.71804 − 2.36467i 5.59523 − 4.06517i 0
349.10 1.58691 + 0.515618i −1.63919 + 0.532605i 0.634386 + 0.460908i 0 −2.87586 1.13381 − 1.56056i −1.19247 − 1.64129i −0.0237792 + 0.0172766i 0
349.11 2.35953 + 0.766659i 1.47405 − 0.478948i 3.36160 + 2.44235i 0 3.84526 −2.32590 + 3.20132i 3.14282 + 4.32573i −0.483619 + 0.351369i 0
349.12 2.49377 + 0.810275i −0.251039 + 0.0815674i 3.94431 + 2.86571i 0 −0.692125 0.0919008 − 0.126491i 4.43171 + 6.09973i −2.37068 + 1.72240i 0
374.1 −1.60853 + 2.21395i 1.57127 + 2.16266i −1.69618 − 5.22031i 0 −7.31546 −2.03047 + 0.659740i 9.08058 + 2.95046i −1.28118 + 3.94306i 0
374.2 −1.45352 + 2.00060i −0.459612 − 0.632602i −1.27165 − 3.91373i 0 1.93364 4.09364 − 1.33011i 4.97449 + 1.61631i 0.738109 − 2.27167i 0
374.3 −1.16387 + 1.60193i 0.901066 + 1.24021i −0.593549 − 1.82676i 0 −3.03545 −4.25435 + 1.38232i −0.149212 − 0.0484821i 0.200848 − 0.618148i 0
374.4 −1.13414 + 1.56101i −0.607992 − 0.836830i −0.532443 − 1.63869i 0 1.99585 −1.21385 + 0.394404i −0.508276 − 0.165149i 0.596422 − 1.83560i 0
374.5 −0.309624 + 0.426161i −1.20435 − 1.65764i 0.532288 + 1.63821i 0 1.07932 −0.684749 + 0.222489i −1.86492 − 0.605948i −0.370272 + 1.13958i 0
374.6 −0.178926 + 0.246271i 1.72617 + 2.37586i 0.589399 + 1.81398i 0 −0.893963 2.80544 − 0.911543i −1.13121 − 0.367552i −1.73803 + 5.34911i 0
374.7 0.178926 − 0.246271i −1.72617 − 2.37586i 0.589399 + 1.81398i 0 −0.893963 −2.80544 + 0.911543i 1.13121 + 0.367552i −1.73803 + 5.34911i 0
374.8 0.309624 − 0.426161i 1.20435 + 1.65764i 0.532288 + 1.63821i 0 1.07932 0.684749 − 0.222489i 1.86492 + 0.605948i −0.370272 + 1.13958i 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.c 48
5.b even 2 1 inner 775.2.bf.c 48
5.c odd 4 1 155.2.h.b ✓ 24
5.c odd 4 1 775.2.k.d 24
31.d even 5 1 inner 775.2.bf.c 48
155.n even 10 1 inner 775.2.bf.c 48
155.r even 20 1 4805.2.a.v 12
155.s odd 20 1 155.2.h.b ✓ 24
155.s odd 20 1 775.2.k.d 24
155.s odd 20 1 4805.2.a.u 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.h.b ✓ 24 5.c odd 4 1
155.2.h.b ✓ 24 155.s odd 20 1
775.2.k.d 24 5.c odd 4 1
775.2.k.d 24 155.s odd 20 1
775.2.bf.c 48 1.a even 1 1 trivial
775.2.bf.c 48 5.b even 2 1 inner
775.2.bf.c 48 31.d even 5 1 inner
775.2.bf.c 48 155.n even 10 1 inner
4805.2.a.u 12 155.s odd 20 1
4805.2.a.v 12 155.r even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 18 T_{2}^{46} + 231 T_{2}^{44} - 2503 T_{2}^{42} + 23881 T_{2}^{40} - 182603 T_{2}^{38} + \cdots + 4100625 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display