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Properties

Label 775.2.bl.c
Level $775$
Weight $2$
Character orbit 775.bl
Analytic conductor $6.188$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [775,2,Mod(51,775)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.bl (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 155)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 2 q^{2} + q^{3} - 10 q^{4} + 4 q^{6} - 4 q^{7} - 13 q^{8} + 2 q^{9} - 28 q^{11} + 13 q^{12} + 6 q^{13} - 6 q^{14} + 8 q^{16} + 10 q^{17} + 27 q^{18} - 13 q^{19} - 8 q^{21} - 2 q^{22} + 12 q^{23}+ \cdots + 53 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −0.646517 − 1.98977i −1.02325 − 1.13643i −1.92319 + 1.39728i 0 −1.59970 + 2.77076i 0.316994 + 3.01600i 0.638431 + 0.463847i 0.0691433 − 0.657854i 0
51.2 −0.322946 − 0.993927i 0.608786 + 0.676125i 0.734438 − 0.533600i 0 0.475414 − 0.823441i −0.0578457 − 0.550365i −2.45851 − 1.78621i 0.227060 − 2.16033i 0
51.3 0.102407 + 0.315176i 2.23911 + 2.48678i 1.52919 − 1.11102i 0 −0.554473 + 0.960376i 0.215933 + 2.05447i 1.04297 + 0.757765i −0.856892 + 8.15278i 0
51.4 0.260242 + 0.800942i −1.08404 − 1.20394i 1.04425 − 0.758693i 0 0.682177 − 1.18157i −0.454442 − 4.32373i 2.24207 + 1.62896i 0.0392388 − 0.373333i 0
51.5 0.542213 + 1.66876i −0.0714789 − 0.0793853i −0.872728 + 0.634074i 0 0.0937182 − 0.162325i 0.352980 + 3.35838i 1.30774 + 0.950129i 0.312393 − 2.97222i 0
76.1 −0.646517 + 1.98977i −1.02325 + 1.13643i −1.92319 − 1.39728i 0 −1.59970 − 2.77076i 0.316994 − 3.01600i 0.638431 − 0.463847i 0.0691433 + 0.657854i 0
76.2 −0.322946 + 0.993927i 0.608786 − 0.676125i 0.734438 + 0.533600i 0 0.475414 + 0.823441i −0.0578457 + 0.550365i −2.45851 + 1.78621i 0.227060 + 2.16033i 0
76.3 0.102407 − 0.315176i 2.23911 − 2.48678i 1.52919 + 1.11102i 0 −0.554473 − 0.960376i 0.215933 − 2.05447i 1.04297 − 0.757765i −0.856892 − 8.15278i 0
76.4 0.260242 − 0.800942i −1.08404 + 1.20394i 1.04425 + 0.758693i 0 0.682177 + 1.18157i −0.454442 + 4.32373i 2.24207 − 1.62896i 0.0392388 + 0.373333i 0
76.5 0.542213 − 1.66876i −0.0714789 + 0.0793853i −0.872728 − 0.634074i 0 0.0937182 + 0.162325i 0.352980 − 3.35838i 1.30774 − 0.950129i 0.312393 + 2.97222i 0
226.1 −0.756580 + 2.32851i −1.41886 − 0.301589i −3.23153 − 2.34784i 0 1.77574 − 3.07567i −0.504362 + 0.224557i 3.95039 − 2.87013i −0.818419 − 0.364384i 0
226.2 −0.319323 + 0.982776i 0.636758 + 0.135347i 0.754152 + 0.547924i 0 −0.336348 + 0.582571i −2.42078 + 1.07780i −2.45130 + 1.78098i −2.35349 − 1.04784i 0
226.3 0.132371 − 0.407396i −1.24842 − 0.265360i 1.46958 + 1.06772i 0 −0.273361 + 0.473476i 2.12201 − 0.944779i 1.32262 − 0.960937i −1.25250 − 0.557647i 0
226.4 0.686411 − 2.11256i −1.89602 − 0.403011i −2.37370 − 1.72459i 0 −2.15283 + 3.72881i −0.441192 + 0.196431i −1.67853 + 1.21953i 0.691836 + 0.308025i 0
226.5 0.821724 − 2.52901i 2.94840 + 0.626702i −4.10261 − 2.98072i 0 4.00770 − 6.94154i 0.988739 − 0.440215i −6.60686 + 4.80017i 5.55966 + 2.47532i 0
276.1 −1.25001 + 0.908182i −0.149301 + 1.42050i 0.119685 − 0.368352i 0 −1.10345 − 1.91123i 0.351398 − 0.0746919i −0.769995 − 2.36980i 0.938900 + 0.199569i 0
276.2 −1.16431 + 0.845919i 0.267371 − 2.54387i 0.0220000 − 0.0677090i 0 1.84060 + 3.18802i 0.455497 − 0.0968189i −0.857791 − 2.64001i −3.46532 − 0.736577i 0
276.3 0.0856516 − 0.0622295i −0.0569592 + 0.541931i −0.614570 + 1.89145i 0 0.0288454 + 0.0499618i −2.44643 + 0.520005i 0.130497 + 0.401629i 2.64400 + 0.561999i 0
276.4 1.78555 − 1.29728i −0.276475 + 2.63048i 0.887232 − 2.73062i 0 2.91882 + 5.05554i 3.81001 − 0.809843i −0.594138 − 1.82857i −3.90856 − 0.830791i 0
276.5 2.12578 − 1.54447i 0.110836 − 1.05453i 1.51553 − 4.66432i 0 −1.39308 − 2.41289i −4.56217 + 0.969719i −2.35827 − 7.25800i 1.83469 + 0.389976i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.5
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.bl.c 40
5.b even 2 1 155.2.q.a ✓ 40
5.c odd 4 2 775.2.ck.c 80
31.g even 15 1 inner 775.2.bl.c 40
155.u even 30 1 155.2.q.a ✓ 40
155.u even 30 1 4805.2.a.x 20
155.v odd 30 1 4805.2.a.y 20
155.w odd 60 2 775.2.ck.c 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.q.a ✓ 40 5.b even 2 1
155.2.q.a ✓ 40 155.u even 30 1
775.2.bl.c 40 1.a even 1 1 trivial
775.2.bl.c 40 31.g even 15 1 inner
775.2.ck.c 80 5.c odd 4 2
775.2.ck.c 80 155.w odd 60 2
4805.2.a.x 20 155.u even 30 1
4805.2.a.y 20 155.v odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 2 T_{2}^{39} + 17 T_{2}^{38} - 23 T_{2}^{37} + 152 T_{2}^{36} - 133 T_{2}^{35} + 1042 T_{2}^{34} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display