↑

Properties

Label 775.2.k.h
Level $775$
Weight $2$
Character orbit 775.k
Analytic conductor $6.188$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,2,Mod(101,775)]
 
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([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.k (of order \(5\), degree \(4\), 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: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 155)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{4} - 4 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{4} - 4 q^{6} - 8 q^{9} - 14 q^{11} + 16 q^{14} - 22 q^{16} + 2 q^{19} - 38 q^{21} + 30 q^{24} + 12 q^{26} - 20 q^{29} + 26 q^{31} + 2 q^{34} - 44 q^{36} + 18 q^{39} - 20 q^{41} - 62 q^{44} - 70 q^{46} + 42 q^{49} + 10 q^{51} + 26 q^{54} + 208 q^{56} + 50 q^{59} + 56 q^{61} - 12 q^{64} + 60 q^{66} - 34 q^{69} - 8 q^{71} + 84 q^{74} - 20 q^{76} - 122 q^{79} + 42 q^{81} + 40 q^{84} + 78 q^{86} + 28 q^{89} - 20 q^{91} - 160 q^{94} - 70 q^{96} + 52 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −0.823132 + 2.53334i 0.276679 + 0.851532i −4.12224 − 2.99498i 0 −2.38496 1.50319 + 1.09214i 6.67047 − 4.84638i 1.77850 − 1.29215i 0
101.2 −0.619737 + 1.90735i 0.235741 + 0.725535i −1.63589 − 1.18855i 0 −1.52995 0.145698 + 0.105855i 0.0358157 − 0.0260217i 1.95622 − 1.42128i 0
101.3 −0.514300 + 1.58285i −0.720920 − 2.21876i −0.622883 − 0.452551i 0 3.88274 −1.60269 − 1.16442i −1.65624 + 1.20333i −1.97614 + 1.43575i 0
101.4 −0.498465 + 1.53412i −0.294592 − 0.906662i −0.487018 − 0.353839i 0 1.53777 0.525419 + 0.381739i −1.82441 + 1.32551i 1.69180 − 1.22916i 0
101.5 −0.334686 + 1.03006i 0.849570 + 2.61471i 0.669031 + 0.486079i 0 −2.97764 −4.04839 − 2.94133i −2.47704 + 1.79968i −3.68788 + 2.67940i 0
101.6 −0.0898296 + 0.276467i −0.260999 − 0.803274i 1.54967 + 1.12590i 0 0.245524 1.98715 + 1.44375i −0.920834 + 0.669025i 1.84992 − 1.34405i 0
101.7 −0.0403444 + 0.124167i 0.926694 + 2.85207i 1.60424 + 1.16555i 0 −0.391521 3.43610 + 2.49647i −0.420692 + 0.305651i −4.84849 + 3.52264i 0
101.8 0.0403444 − 0.124167i −0.926694 − 2.85207i 1.60424 + 1.16555i 0 −0.391521 −3.43610 − 2.49647i 0.420692 − 0.305651i −4.84849 + 3.52264i 0
101.9 0.0898296 − 0.276467i 0.260999 + 0.803274i 1.54967 + 1.12590i 0 0.245524 −1.98715 − 1.44375i 0.920834 − 0.669025i 1.84992 − 1.34405i 0
101.10 0.334686 − 1.03006i −0.849570 − 2.61471i 0.669031 + 0.486079i 0 −2.97764 4.04839 + 2.94133i 2.47704 − 1.79968i −3.68788 + 2.67940i 0
101.11 0.498465 − 1.53412i 0.294592 + 0.906662i −0.487018 − 0.353839i 0 1.53777 −0.525419 − 0.381739i 1.82441 − 1.32551i 1.69180 − 1.22916i 0
101.12 0.514300 − 1.58285i 0.720920 + 2.21876i −0.622883 − 0.452551i 0 3.88274 1.60269 + 1.16442i 1.65624 − 1.20333i −1.97614 + 1.43575i 0
101.13 0.619737 − 1.90735i −0.235741 − 0.725535i −1.63589 − 1.18855i 0 −1.52995 −0.145698 − 0.105855i −0.0358157 + 0.0260217i 1.95622 − 1.42128i 0
101.14 0.823132 − 2.53334i −0.276679 − 0.851532i −4.12224 − 2.99498i 0 −2.38496 −1.50319 − 1.09214i −6.67047 + 4.84638i 1.77850 − 1.29215i 0
126.1 −2.19260 − 1.59301i 0.0871848 − 0.0633434i 1.65175 + 5.08355i 0 −0.292068 0.412403 + 1.26925i 2.80156 − 8.62233i −0.923462 + 2.84212i 0
126.2 −1.90114 − 1.38126i −2.47598 + 1.79890i 1.08842 + 3.34981i 0 7.19192 0.895504 + 2.75608i 1.10537 − 3.40198i 1.96736 − 6.05492i 0
126.3 −1.63933 − 1.19104i 1.87252 − 1.36047i 0.650782 + 2.00290i 0 −4.69005 −0.656165 − 2.01947i 0.0663574 − 0.204227i 0.728419 − 2.24184i 0
126.4 −1.38993 − 1.00985i 1.55949 − 1.13303i 0.294093 + 0.905124i 0 −3.31177 1.09854 + 3.38096i −0.556548 + 1.71288i 0.221182 − 0.680729i 0
126.5 −1.10757 − 0.804693i −1.35958 + 0.987794i −0.0388646 − 0.119613i 0 2.30070 0.0224427 + 0.0690714i −0.899311 + 2.76779i −0.0543250 + 0.167195i 0
126.6 −0.467495 − 0.339655i −0.0531777 + 0.0386358i −0.514848 − 1.58454i 0 0.0379832 −0.844608 − 2.59944i −0.654642 + 2.01478i −0.925716 + 2.84906i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.14
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.k.h 56
5.b even 2 1 inner 775.2.k.h 56
5.c odd 4 2 155.2.n.a ✓ 56
31.d even 5 1 inner 775.2.k.h 56
155.n even 10 1 inner 775.2.k.h 56
155.s odd 20 2 155.2.n.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.n.a ✓ 56 5.c odd 4 2
155.2.n.a ✓ 56 155.s odd 20 2
775.2.k.h 56 1.a even 1 1 trivial
775.2.k.h 56 5.b even 2 1 inner
775.2.k.h 56 31.d even 5 1 inner
775.2.k.h 56 155.n even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} + 15 T_{2}^{54} + 179 T_{2}^{52} + 1824 T_{2}^{50} + 16285 T_{2}^{48} + 113106 T_{2}^{46} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display