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Properties

Label 775.2.df.c
Level $775$
Weight $2$
Character orbit 775.df
Analytic conductor $6.188$
Analytic rank $0$
Dimension $352$
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(43,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([45, 38]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.df (of order \(60\), degree \(16\), 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: \(352\)
Relative dimension: \(22\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 352 q - 60 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 352 q - 60 q^{6} - 8 q^{11} + 80 q^{16} - 64 q^{21} + 48 q^{26} + 144 q^{31} + 236 q^{36} - 24 q^{41} - 80 q^{46} + 40 q^{51} + 84 q^{56} + 360 q^{66} + 104 q^{71} - 148 q^{76} - 380 q^{81} - 324 q^{86} - 160 q^{91} - 192 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −2.67851 + 0.424234i −0.601385 + 0.230850i 5.09233 − 1.65460i 0 1.51288 − 0.873463i 0.228028 + 4.35104i −8.10529 + 4.12985i −1.92106 + 1.72973i 0
43.2 −2.48056 + 0.392882i −1.49940 + 0.575565i 4.09671 − 1.33110i 0 3.49322 − 2.01681i −0.0945218 − 1.80358i −5.16367 + 2.63102i −0.312515 + 0.281390i 0
43.3 −2.23714 + 0.354328i 3.18207 − 1.22148i 2.97714 − 0.967332i 0 −6.68593 + 3.86012i 0.0262436 + 0.500759i −2.28123 + 1.16234i 6.40409 − 5.76627i 0
43.4 −1.98120 + 0.313792i −2.40646 + 0.923755i 1.92459 − 0.625337i 0 4.47783 − 2.58528i −0.131355 − 2.50641i −0.0422373 + 0.0215210i 2.70831 − 2.43858i 0
43.5 −1.85793 + 0.294267i 2.31081 − 0.887037i 1.46319 − 0.475420i 0 −4.03230 + 2.32805i −0.249544 − 4.76158i 0.773514 − 0.394125i 2.32358 − 2.09216i 0
43.6 −1.79787 + 0.284754i 1.28530 − 0.493382i 1.24912 − 0.405864i 0 −2.17031 + 1.25303i 0.107123 + 2.04403i 1.11357 − 0.567394i −0.820856 + 0.739102i 0
43.7 −1.44690 + 0.229167i −2.12185 + 0.814503i 0.138896 − 0.0451302i 0 2.88346 − 1.66477i 0.225334 + 4.29963i 2.41991 − 1.23301i 1.60941 − 1.44912i 0
43.8 −1.33666 + 0.211705i 0.667197 − 0.256113i −0.160284 + 0.0520794i 0 −0.837592 + 0.483584i 0.103992 + 1.98428i 2.61485 − 1.33233i −1.84988 + 1.66564i 0
43.9 −0.638547 + 0.101136i −2.35765 + 0.905015i −1.50460 + 0.488874i 0 1.41394 − 0.816338i −0.245240 − 4.67946i 2.06340 − 1.05135i 2.51000 − 2.26002i 0
43.10 −0.421829 + 0.0668111i 0.610017 − 0.234163i −1.72864 + 0.561668i 0 −0.241678 + 0.139533i 0.0140049 + 0.267229i 1.45274 − 0.740207i −1.91215 + 1.72170i 0
43.11 −0.410121 + 0.0649567i −1.68060 + 0.645121i −1.73813 + 0.564754i 0 0.647343 − 0.373744i −0.0533580 − 1.01813i 1.41611 − 0.721544i 0.178796 − 0.160988i 0
43.12 0.410121 − 0.0649567i 1.68060 − 0.645121i −1.73813 + 0.564754i 0 0.647343 − 0.373744i 0.0533580 + 1.01813i −1.41611 + 0.721544i 0.178796 − 0.160988i 0
43.13 0.421829 − 0.0668111i −0.610017 + 0.234163i −1.72864 + 0.561668i 0 −0.241678 + 0.139533i −0.0140049 − 0.267229i −1.45274 + 0.740207i −1.91215 + 1.72170i 0
43.14 0.638547 − 0.101136i 2.35765 − 0.905015i −1.50460 + 0.488874i 0 1.41394 − 0.816338i 0.245240 + 4.67946i −2.06340 + 1.05135i 2.51000 − 2.26002i 0
43.15 1.33666 − 0.211705i −0.667197 + 0.256113i −0.160284 + 0.0520794i 0 −0.837592 + 0.483584i −0.103992 − 1.98428i −2.61485 + 1.33233i −1.84988 + 1.66564i 0
43.16 1.44690 − 0.229167i 2.12185 − 0.814503i 0.138896 − 0.0451302i 0 2.88346 − 1.66477i −0.225334 − 4.29963i −2.41991 + 1.23301i 1.60941 − 1.44912i 0
43.17 1.79787 − 0.284754i −1.28530 + 0.493382i 1.24912 − 0.405864i 0 −2.17031 + 1.25303i −0.107123 − 2.04403i −1.11357 + 0.567394i −0.820856 + 0.739102i 0
43.18 1.85793 − 0.294267i −2.31081 + 0.887037i 1.46319 − 0.475420i 0 −4.03230 + 2.32805i 0.249544 + 4.76158i −0.773514 + 0.394125i 2.32358 − 2.09216i 0
43.19 1.98120 − 0.313792i 2.40646 − 0.923755i 1.92459 − 0.625337i 0 4.47783 − 2.58528i 0.131355 + 2.50641i 0.0422373 − 0.0215210i 2.70831 − 2.43858i 0
43.20 2.23714 − 0.354328i −3.18207 + 1.22148i 2.97714 − 0.967332i 0 −6.68593 + 3.86012i −0.0262436 − 0.500759i 2.28123 − 1.16234i 6.40409 − 5.76627i 0
See next 80 embeddings (of 352 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.22
Significant digits:
Format:

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.h odd 30 1 inner
155.v odd 30 1 inner
155.x even 60 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.df.c ✓ 352
5.b even 2 1 inner 775.2.df.c ✓ 352
5.c odd 4 2 inner 775.2.df.c ✓ 352
31.h odd 30 1 inner 775.2.df.c ✓ 352
155.v odd 30 1 inner 775.2.df.c ✓ 352
155.x even 60 2 inner 775.2.df.c ✓ 352
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.df.c ✓ 352 1.a even 1 1 trivial
775.2.df.c ✓ 352 5.b even 2 1 inner
775.2.df.c ✓ 352 5.c odd 4 2 inner
775.2.df.c ✓ 352 31.h odd 30 1 inner
775.2.df.c ✓ 352 155.v odd 30 1 inner
775.2.df.c ✓ 352 155.x even 60 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{352} - 304 T_{2}^{348} + 53728 T_{2}^{344} - 7207907 T_{2}^{340} + 809890608 T_{2}^{336} + \cdots + 18\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display