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Properties

Label 1216.2.t.d
Level $1216$
Weight $2$
Character orbit 1216.t
Analytic conductor $9.710$
Analytic rank $0$
Dimension $24$
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] = [1216,2,Mod(353,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.353");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.t (of order \(6\), degree \(2\), 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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{5} + 8 q^{9} + 30 q^{13} + 6 q^{17} + 24 q^{21} + 6 q^{25} + 42 q^{29} - 14 q^{33} - 24 q^{41} + 24 q^{49} - 18 q^{53} - 42 q^{57} + 18 q^{61} - 20 q^{65} - 16 q^{73} + 52 q^{81} - 78 q^{85} + 14 q^{89} + 60 q^{93} - 4 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1 0 −2.17731 − 1.25707i 0 0.891744 + 0.514849i 0 −4.18166 0 1.66044 + 2.87597i 0
353.2 0 −2.17731 − 1.25707i 0 −2.95556 − 1.70639i 0 2.54486 0 1.66044 + 2.87597i 0
353.3 0 −1.80664 − 1.04307i 0 −0.410474 − 0.236987i 0 −3.88259 0 0.675970 + 1.17081i 0
353.4 0 −1.80664 − 1.04307i 0 −1.98453 − 1.14577i 0 1.72665 0 0.675970 + 1.17081i 0
353.5 0 −0.495361 − 0.285997i 0 3.31453 + 1.91364i 0 2.37943 0 −1.33641 − 2.31473i 0
353.6 0 −0.495361 − 0.285997i 0 −0.355705 − 0.205367i 0 0.565533 0 −1.33641 − 2.31473i 0
353.7 0 0.495361 + 0.285997i 0 3.31453 + 1.91364i 0 −2.37943 0 −1.33641 − 2.31473i 0
353.8 0 0.495361 + 0.285997i 0 −0.355705 − 0.205367i 0 −0.565533 0 −1.33641 − 2.31473i 0
353.9 0 1.80664 + 1.04307i 0 −0.410474 − 0.236987i 0 3.88259 0 0.675970 + 1.17081i 0
353.10 0 1.80664 + 1.04307i 0 −1.98453 − 1.14577i 0 −1.72665 0 0.675970 + 1.17081i 0
353.11 0 2.17731 + 1.25707i 0 0.891744 + 0.514849i 0 4.18166 0 1.66044 + 2.87597i 0
353.12 0 2.17731 + 1.25707i 0 −2.95556 − 1.70639i 0 −2.54486 0 1.66044 + 2.87597i 0
1185.1 0 −2.17731 + 1.25707i 0 0.891744 − 0.514849i 0 −4.18166 0 1.66044 − 2.87597i 0
1185.2 0 −2.17731 + 1.25707i 0 −2.95556 + 1.70639i 0 2.54486 0 1.66044 − 2.87597i 0
1185.3 0 −1.80664 + 1.04307i 0 −0.410474 + 0.236987i 0 −3.88259 0 0.675970 − 1.17081i 0
1185.4 0 −1.80664 + 1.04307i 0 −1.98453 + 1.14577i 0 1.72665 0 0.675970 − 1.17081i 0
1185.5 0 −0.495361 + 0.285997i 0 3.31453 − 1.91364i 0 2.37943 0 −1.33641 + 2.31473i 0
1185.6 0 −0.495361 + 0.285997i 0 −0.355705 + 0.205367i 0 0.565533 0 −1.33641 + 2.31473i 0
1185.7 0 0.495361 − 0.285997i 0 3.31453 − 1.91364i 0 −2.37943 0 −1.33641 + 2.31473i 0
1185.8 0 0.495361 − 0.285997i 0 −0.355705 + 0.205367i 0 −0.565533 0 −1.33641 + 2.31473i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
152.k odd 6 1 inner
152.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.t.d ✓ 24
4.b odd 2 1 inner 1216.2.t.d ✓ 24
8.b even 2 1 1216.2.t.e yes 24
8.d odd 2 1 1216.2.t.e yes 24
19.c even 3 1 1216.2.t.e yes 24
76.g odd 6 1 1216.2.t.e yes 24
152.k odd 6 1 inner 1216.2.t.d ✓ 24
152.p even 6 1 inner 1216.2.t.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.t.d ✓ 24 1.a even 1 1 trivial
1216.2.t.d ✓ 24 4.b odd 2 1 inner
1216.2.t.d ✓ 24 152.k odd 6 1 inner
1216.2.t.d ✓ 24 152.p even 6 1 inner
1216.2.t.e yes 24 8.b even 2 1
1216.2.t.e yes 24 8.d odd 2 1
1216.2.t.e yes 24 19.c even 3 1
1216.2.t.e yes 24 76.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{12} - 11T_{3}^{10} + 90T_{3}^{8} - 323T_{3}^{6} + 862T_{3}^{4} - 279T_{3}^{2} + 81 \) Copy content Toggle raw display
\( T_{5}^{12} + 3 T_{5}^{11} - 12 T_{5}^{10} - 45 T_{5}^{9} + 160 T_{5}^{8} + 723 T_{5}^{7} + 603 T_{5}^{6} + \cdots + 36 \) Copy content Toggle raw display