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Properties

Label 1617.2.c.a
Level $1617$
Weight $2$
Character orbit 1617.c
Analytic conductor $12.912$
Analytic rank $0$
Dimension $32$
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] = [1617,2,Mod(538,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1617.538");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(12.9118100068\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 231)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{4} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{4} - 32 q^{9} - 4 q^{11} + 8 q^{15} + 40 q^{16} + 8 q^{22} - 48 q^{23} + 24 q^{36} + 64 q^{37} + 56 q^{44} - 72 q^{53} - 24 q^{58} + 8 q^{64} - 40 q^{67} + 72 q^{71} - 48 q^{78} + 32 q^{81} - 128 q^{86} - 48 q^{88} - 16 q^{92} - 32 q^{93} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
538.1 − 2.58645i − 1.00000i −4.68975 1.50244i −2.58645 0 6.95692i −1.00000 3.88599
538.2 − 2.58645i 1.00000i −4.68975 − 1.50244i 2.58645 0 6.95692i −1.00000 −3.88599
538.3 − 2.50303i − 1.00000i −4.26518 − 0.980927i −2.50303 0 5.66981i −1.00000 −2.45529
538.4 − 2.50303i 1.00000i −4.26518 0.980927i 2.50303 0 5.66981i −1.00000 2.45529
538.5 − 1.98162i − 1.00000i −1.92681 0.364814i −1.98162 0 − 0.145025i −1.00000 0.722923
538.6 − 1.98162i 1.00000i −1.92681 − 0.364814i 1.98162 0 − 0.145025i −1.00000 −0.722923
538.7 − 1.72615i − 1.00000i −0.979587 − 1.65517i −1.72615 0 − 1.76138i −1.00000 −2.85707
538.8 − 1.72615i 1.00000i −0.979587 1.65517i 1.72615 0 − 1.76138i −1.00000 2.85707
538.9 − 1.07576i − 1.00000i 0.842743 3.36307i −1.07576 0 − 3.05811i −1.00000 3.61785
538.10 − 1.07576i 1.00000i 0.842743 − 3.36307i 1.07576 0 − 3.05811i −1.00000 −3.61785
538.11 − 0.655105i − 1.00000i 1.57084 − 3.40177i −0.655105 0 − 2.33927i −1.00000 −2.22852
538.12 − 0.655105i 1.00000i 1.57084 3.40177i 0.655105 0 − 2.33927i −1.00000 2.22852
538.13 − 0.615506i − 1.00000i 1.62115 3.28584i −0.615506 0 − 2.22884i −1.00000 2.02245
538.14 − 0.615506i 1.00000i 1.62115 − 3.28584i 0.615506 0 − 2.22884i −1.00000 −2.02245
538.15 − 0.416420i − 1.00000i 1.82659 − 0.478282i −0.416420 0 − 1.59347i −1.00000 −0.199166
538.16 − 0.416420i 1.00000i 1.82659 0.478282i 0.416420 0 − 1.59347i −1.00000 0.199166
538.17 0.416420i − 1.00000i 1.82659 − 0.478282i 0.416420 0 1.59347i −1.00000 0.199166
538.18 0.416420i 1.00000i 1.82659 0.478282i −0.416420 0 1.59347i −1.00000 −0.199166
538.19 0.615506i − 1.00000i 1.62115 3.28584i 0.615506 0 2.22884i −1.00000 −2.02245
538.20 0.615506i 1.00000i 1.62115 − 3.28584i −0.615506 0 2.22884i −1.00000 2.02245
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 538.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1617.2.c.a 32
7.b odd 2 1 inner 1617.2.c.a 32
7.c even 3 1 231.2.p.a ✓ 32
7.d odd 6 1 231.2.p.a ✓ 32
11.b odd 2 1 inner 1617.2.c.a 32
21.g even 6 1 693.2.bg.b 32
21.h odd 6 1 693.2.bg.b 32
77.b even 2 1 inner 1617.2.c.a 32
77.h odd 6 1 231.2.p.a ✓ 32
77.i even 6 1 231.2.p.a ✓ 32
231.k odd 6 1 693.2.bg.b 32
231.l even 6 1 693.2.bg.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.p.a ✓ 32 7.c even 3 1
231.2.p.a ✓ 32 7.d odd 6 1
231.2.p.a ✓ 32 77.h odd 6 1
231.2.p.a ✓ 32 77.i even 6 1
693.2.bg.b 32 21.g even 6 1
693.2.bg.b 32 21.h odd 6 1
693.2.bg.b 32 231.k odd 6 1
693.2.bg.b 32 231.l even 6 1
1617.2.c.a 32 1.a even 1 1 trivial
1617.2.c.a 32 7.b odd 2 1 inner
1617.2.c.a 32 11.b odd 2 1 inner
1617.2.c.a 32 77.b even 2 1 inner