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Properties

Label 775.2.ck.b
Level $775$
Weight $2$
Character orbit 775.ck
Analytic conductor $6.188$
Analytic rank $0$
Dimension $80$
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(49,775)]
 
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([15, 26]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.ck (of order \(30\), degree \(8\), 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: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{30})\)
Twist minimal: no (minimal twist has level 155)
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{4} - 24 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{4} - 24 q^{6} - 12 q^{9} + 32 q^{11} + 20 q^{14} - 32 q^{16} + 18 q^{19} - 48 q^{21} - 220 q^{24} - 8 q^{26} + 44 q^{29} - 12 q^{31} + 32 q^{34} - 2 q^{36} + 110 q^{39} - 48 q^{41} + 104 q^{44} - 100 q^{46} - 186 q^{49} - 84 q^{51} - 204 q^{54} + 10 q^{56} - 62 q^{59} + 136 q^{61} + 46 q^{64} + 96 q^{69} - 60 q^{71} - 100 q^{74} + 28 q^{76} - 98 q^{79} + 50 q^{81} + 36 q^{84} + 326 q^{86} - 8 q^{89} - 74 q^{91} + 188 q^{94} + 44 q^{96} - 34 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −1.62755 − 2.24013i −0.773962 − 1.73835i −1.75122 + 5.38971i 0 −2.63446 + 4.56302i 1.30736 − 1.17715i 9.65698 − 3.13774i −0.415444 + 0.461397i 0
49.2 −1.25115 − 1.72206i −1.33072 − 2.98885i −0.782088 + 2.40702i 0 −3.48206 + 6.03111i −1.20673 + 1.08654i 1.07474 − 0.349204i −5.15503 + 5.72524i 0
49.3 −1.13264 − 1.55894i 0.195676 + 0.439495i −0.529393 + 1.62930i 0 0.463517 − 0.802834i 2.77400 − 2.49772i −0.525691 + 0.170807i 1.85252 − 2.05744i 0
49.4 −0.529735 − 0.729118i 0.160755 + 0.361062i 0.367040 − 1.12963i 0 0.178099 − 0.308477i −2.99156 + 2.69361i −2.73233 + 0.887788i 1.90287 − 2.11335i 0
49.5 −0.499366 − 0.687318i 1.01988 + 2.29069i 0.394994 − 1.21567i 0 1.06514 − 1.84487i −2.11700 + 1.90616i −2.64878 + 0.860640i −2.19970 + 2.44301i 0
49.6 0.499366 + 0.687318i −1.01988 − 2.29069i 0.394994 − 1.21567i 0 1.06514 − 1.84487i 2.11700 − 1.90616i 2.64878 − 0.860640i −2.19970 + 2.44301i 0
49.7 0.529735 + 0.729118i −0.160755 − 0.361062i 0.367040 − 1.12963i 0 0.178099 − 0.308477i 2.99156 − 2.69361i 2.73233 − 0.887788i 1.90287 − 2.11335i 0
49.8 1.13264 + 1.55894i −0.195676 − 0.439495i −0.529393 + 1.62930i 0 0.463517 − 0.802834i −2.77400 + 2.49772i 0.525691 − 0.170807i 1.85252 − 2.05744i 0
49.9 1.25115 + 1.72206i 1.33072 + 2.98885i −0.782088 + 2.40702i 0 −3.48206 + 6.03111i 1.20673 − 1.08654i −1.07474 + 0.349204i −5.15503 + 5.72524i 0
49.10 1.62755 + 2.24013i 0.773962 + 1.73835i −1.75122 + 5.38971i 0 −2.63446 + 4.56302i −1.30736 + 1.17715i −9.65698 + 3.13774i −0.415444 + 0.461397i 0
174.1 −1.62755 + 2.24013i −0.773962 + 1.73835i −1.75122 − 5.38971i 0 −2.63446 − 4.56302i 1.30736 + 1.17715i 9.65698 + 3.13774i −0.415444 − 0.461397i 0
174.2 −1.25115 + 1.72206i −1.33072 + 2.98885i −0.782088 − 2.40702i 0 −3.48206 − 6.03111i −1.20673 − 1.08654i 1.07474 + 0.349204i −5.15503 − 5.72524i 0
174.3 −1.13264 + 1.55894i 0.195676 − 0.439495i −0.529393 − 1.62930i 0 0.463517 + 0.802834i 2.77400 + 2.49772i −0.525691 − 0.170807i 1.85252 + 2.05744i 0
174.4 −0.529735 + 0.729118i 0.160755 − 0.361062i 0.367040 + 1.12963i 0 0.178099 + 0.308477i −2.99156 − 2.69361i −2.73233 − 0.887788i 1.90287 + 2.11335i 0
174.5 −0.499366 + 0.687318i 1.01988 − 2.29069i 0.394994 + 1.21567i 0 1.06514 + 1.84487i −2.11700 − 1.90616i −2.64878 − 0.860640i −2.19970 − 2.44301i 0
174.6 0.499366 − 0.687318i −1.01988 + 2.29069i 0.394994 + 1.21567i 0 1.06514 + 1.84487i 2.11700 + 1.90616i 2.64878 + 0.860640i −2.19970 − 2.44301i 0
174.7 0.529735 − 0.729118i −0.160755 + 0.361062i 0.367040 + 1.12963i 0 0.178099 + 0.308477i 2.99156 + 2.69361i 2.73233 + 0.887788i 1.90287 + 2.11335i 0
174.8 1.13264 − 1.55894i −0.195676 + 0.439495i −0.529393 − 1.62930i 0 0.463517 + 0.802834i −2.77400 − 2.49772i 0.525691 + 0.170807i 1.85252 + 2.05744i 0
174.9 1.25115 − 1.72206i 1.33072 − 2.98885i −0.782088 − 2.40702i 0 −3.48206 − 6.03111i 1.20673 + 1.08654i −1.07474 − 0.349204i −5.15503 − 5.72524i 0
174.10 1.62755 − 2.24013i 0.773962 − 1.73835i −1.75122 − 5.38971i 0 −2.63446 − 4.56302i −1.30736 − 1.17715i −9.65698 − 3.13774i −0.415444 − 0.461397i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.10
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.ck.b 80
5.b even 2 1 inner 775.2.ck.b 80
5.c odd 4 1 155.2.q.b ✓ 40
5.c odd 4 1 775.2.bl.b 40
31.g even 15 1 inner 775.2.ck.b 80
155.u even 30 1 inner 775.2.ck.b 80
155.w odd 60 1 155.2.q.b ✓ 40
155.w odd 60 1 775.2.bl.b 40
155.w odd 60 1 4805.2.a.ba 20
155.x even 60 1 4805.2.a.z 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.q.b ✓ 40 5.c odd 4 1
155.2.q.b ✓ 40 155.w odd 60 1
775.2.bl.b 40 5.c odd 4 1
775.2.bl.b 40 155.w odd 60 1
775.2.ck.b 80 1.a even 1 1 trivial
775.2.ck.b 80 5.b even 2 1 inner
775.2.ck.b 80 31.g even 15 1 inner
775.2.ck.b 80 155.u even 30 1 inner
4805.2.a.z 20 155.x even 60 1
4805.2.a.ba 20 155.w odd 60 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} - 22 T_{2}^{78} + 341 T_{2}^{76} - 4351 T_{2}^{74} + 47992 T_{2}^{72} - 449369 T_{2}^{70} + \cdots + 707281 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display