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Properties

Label 786.2.c.a
Level $786$
Weight $2$
Character orbit 786.c
Analytic conductor $6.276$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [786,2,Mod(785,786)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(786, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("786.785");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 786 = 2 \cdot 3 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 786.c (of order \(2\), degree \(1\), 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: \(6.27624159887\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 22 q^{2} + q^{3} + 22 q^{4} - q^{6} - 2 q^{7} - 22 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 22 q^{2} + q^{3} + 22 q^{4} - q^{6} - 2 q^{7} - 22 q^{8} + q^{9} + q^{12} - 4 q^{13} + 2 q^{14} + 4 q^{15} + 22 q^{16} + 10 q^{17} - q^{18} + 2 q^{21} + 8 q^{23} - q^{24} - 16 q^{25} + 4 q^{26} - 11 q^{27} - 2 q^{28} - 22 q^{29} - 4 q^{30} - 22 q^{32} - 9 q^{33} - 10 q^{34} + q^{36} + q^{39} - 2 q^{42} - 14 q^{43} - 19 q^{45} - 8 q^{46} + 12 q^{47} + q^{48} + 8 q^{49} + 16 q^{50} - 4 q^{52} + 11 q^{54} - 8 q^{55} + 2 q^{56} - 10 q^{57} + 22 q^{58} + 4 q^{60} + 12 q^{61} - 9 q^{63} + 22 q^{64} + 9 q^{66} + 10 q^{68} - 2 q^{69} - 36 q^{71} - q^{72} - 20 q^{75} - q^{78} + 25 q^{81} + 24 q^{83} + 2 q^{84} + 14 q^{86} - 12 q^{87} + 19 q^{90} + 28 q^{91} + 8 q^{92} - 4 q^{93} - 12 q^{94} - 40 q^{95} - q^{96} - 8 q^{98} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
785.1 −1.00000 −1.73135 − 0.0491919i 1.00000 − 2.94334i 1.73135 + 0.0491919i −1.34901 −1.00000 2.99516 + 0.170337i 2.94334i
785.2 −1.00000 −1.73135 + 0.0491919i 1.00000 2.94334i 1.73135 − 0.0491919i −1.34901 −1.00000 2.99516 − 0.170337i − 2.94334i
785.3 −1.00000 −1.64178 − 0.551867i 1.00000 − 1.95151i 1.64178 + 0.551867i 3.22239 −1.00000 2.39089 + 1.81209i 1.95151i
785.4 −1.00000 −1.64178 + 0.551867i 1.00000 1.95151i 1.64178 − 0.551867i 3.22239 −1.00000 2.39089 − 1.81209i − 1.95151i
785.5 −1.00000 −1.40604 − 1.01146i 1.00000 0.821228i 1.40604 + 1.01146i −3.12036 −1.00000 0.953902 + 2.84431i − 0.821228i
785.6 −1.00000 −1.40604 + 1.01146i 1.00000 − 0.821228i 1.40604 − 1.01146i −3.12036 −1.00000 0.953902 − 2.84431i 0.821228i
785.7 −1.00000 −0.546763 − 1.64349i 1.00000 2.07196i 0.546763 + 1.64349i −3.72685 −1.00000 −2.40210 + 1.79720i − 2.07196i
785.8 −1.00000 −0.546763 + 1.64349i 1.00000 − 2.07196i 0.546763 − 1.64349i −3.72685 −1.00000 −2.40210 − 1.79720i 2.07196i
785.9 −1.00000 −0.535808 − 1.64709i 1.00000 0.0777980i 0.535808 + 1.64709i 3.16036 −1.00000 −2.42582 + 1.76505i − 0.0777980i
785.10 −1.00000 −0.535808 + 1.64709i 1.00000 − 0.0777980i 0.535808 − 1.64709i 3.16036 −1.00000 −2.42582 − 1.76505i 0.0777980i
785.11 −1.00000 0.216821 − 1.71843i 1.00000 3.92465i −0.216821 + 1.71843i 0.0863337 −1.00000 −2.90598 − 0.745183i − 3.92465i
785.12 −1.00000 0.216821 + 1.71843i 1.00000 − 3.92465i −0.216821 − 1.71843i 0.0863337 −1.00000 −2.90598 + 0.745183i 3.92465i
785.13 −1.00000 0.622365 − 1.61637i 1.00000 − 3.59080i −0.622365 + 1.61637i 1.80489 −1.00000 −2.22532 − 2.01195i 3.59080i
785.14 −1.00000 0.622365 + 1.61637i 1.00000 3.59080i −0.622365 − 1.61637i 1.80489 −1.00000 −2.22532 + 2.01195i − 3.59080i
785.15 −1.00000 0.841374 − 1.51396i 1.00000 − 1.03393i −0.841374 + 1.51396i 1.51546 −1.00000 −1.58418 − 2.54762i 1.03393i
785.16 −1.00000 0.841374 + 1.51396i 1.00000 1.03393i −0.841374 − 1.51396i 1.51546 −1.00000 −1.58418 + 2.54762i − 1.03393i
785.17 −1.00000 1.38518 − 1.03984i 1.00000 − 2.51864i −1.38518 + 1.03984i −4.47400 −1.00000 0.837445 − 2.88074i 2.51864i
785.18 −1.00000 1.38518 + 1.03984i 1.00000 2.51864i −1.38518 − 1.03984i −4.47400 −1.00000 0.837445 + 2.88074i − 2.51864i
785.19 −1.00000 1.62370 − 0.603002i 1.00000 3.09367i −1.62370 + 0.603002i 2.91971 −1.00000 2.27278 − 1.95818i − 3.09367i
785.20 −1.00000 1.62370 + 0.603002i 1.00000 − 3.09367i −1.62370 − 0.603002i 2.91971 −1.00000 2.27278 + 1.95818i 3.09367i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 785.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
393.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 786.2.c.a ✓ 22
3.b odd 2 1 786.2.c.b yes 22
131.b odd 2 1 786.2.c.b yes 22
393.d even 2 1 inner 786.2.c.a ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.c.a ✓ 22 1.a even 1 1 trivial
786.2.c.a ✓ 22 393.d even 2 1 inner
786.2.c.b yes 22 3.b odd 2 1
786.2.c.b yes 22 131.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{11} - 5 T_{17}^{10} - 97 T_{17}^{9} + 497 T_{17}^{8} + 2996 T_{17}^{7} - 16318 T_{17}^{6} + \cdots + 1084752 \) acting on \(S_{2}^{\mathrm{new}}(786, [\chi])\). Copy content Toggle raw display