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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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.t (of order \(6\), degree \(2\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 24q - 6q^{5} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 6q^{5} + 8q^{9} + 30q^{13} + 6q^{17} + 24q^{21} + 6q^{25} + 42q^{29} - 14q^{33} - 24q^{41} + 24q^{49} - 18q^{53} - 42q^{57} + 18q^{61} - 20q^{65} - 16q^{73} + 52q^{81} - 78q^{85} + 14q^{89} + 60q^{93} - 4q^{97} + O(q^{100}) \)

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.

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 1185.12
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 \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{12} - 11 T_{3}^{10} + 90 T_{3}^{8} - 323 T_{3}^{6} + 862 T_{3}^{4} - 279 T_{3}^{2} + 81 \)
\(T_{5}^{12} + \cdots\)