Properties

Label 3312.2.a.t
Level $3312$
Weight $2$
Character orbit 3312.a
Self dual yes
Analytic conductor $26.446$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3312.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.4464531494\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{5} +O(q^{10})\) \( q -2 q^{5} + ( 2 - 2 \beta ) q^{11} + ( 3 - \beta ) q^{13} -2 \beta q^{17} + ( -2 + 2 \beta ) q^{19} - q^{23} - q^{25} + ( -1 - \beta ) q^{29} + ( 5 - \beta ) q^{31} + ( -2 + 4 \beta ) q^{37} + ( -3 + 5 \beta ) q^{41} + 8 q^{43} + ( 7 - 3 \beta ) q^{47} -7 q^{49} -2 q^{53} + ( -4 + 4 \beta ) q^{55} + 4 \beta q^{59} + ( 6 - 4 \beta ) q^{61} + ( -6 + 2 \beta ) q^{65} + ( 2 - 2 \beta ) q^{67} + ( 11 + \beta ) q^{71} + ( -7 - 3 \beta ) q^{73} + ( 2 - 2 \beta ) q^{79} + ( 4 + 4 \beta ) q^{83} + 4 \beta q^{85} + ( 4 - 6 \beta ) q^{89} + ( 4 - 4 \beta ) q^{95} + ( -4 + 6 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} + O(q^{10}) \) \( 2 q - 4 q^{5} + 2 q^{11} + 5 q^{13} - 2 q^{17} - 2 q^{19} - 2 q^{23} - 2 q^{25} - 3 q^{29} + 9 q^{31} - q^{41} + 16 q^{43} + 11 q^{47} - 14 q^{49} - 4 q^{53} - 4 q^{55} + 4 q^{59} + 8 q^{61} - 10 q^{65} + 2 q^{67} + 23 q^{71} - 17 q^{73} + 2 q^{79} + 12 q^{83} + 4 q^{85} + 2 q^{89} + 4 q^{95} - 2 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.56155
−1.56155
0 0 0 −2.00000 0 0 0 0 0
1.2 0 0 0 −2.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3312.2.a.t 2
3.b odd 2 1 368.2.a.i 2
4.b odd 2 1 1656.2.a.j 2
12.b even 2 1 184.2.a.e 2
15.d odd 2 1 9200.2.a.br 2
24.f even 2 1 1472.2.a.u 2
24.h odd 2 1 1472.2.a.p 2
60.h even 2 1 4600.2.a.s 2
60.l odd 4 2 4600.2.e.o 4
69.c even 2 1 8464.2.a.bd 2
84.h odd 2 1 9016.2.a.w 2
276.h odd 2 1 4232.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.a.e 2 12.b even 2 1
368.2.a.i 2 3.b odd 2 1
1472.2.a.p 2 24.h odd 2 1
1472.2.a.u 2 24.f even 2 1
1656.2.a.j 2 4.b odd 2 1
3312.2.a.t 2 1.a even 1 1 trivial
4232.2.a.o 2 276.h odd 2 1
4600.2.a.s 2 60.h even 2 1
4600.2.e.o 4 60.l odd 4 2
8464.2.a.bd 2 69.c even 2 1
9016.2.a.w 2 84.h odd 2 1
9200.2.a.br 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3312))\):

\( T_{5} + 2 \)
\( T_{7} \)
\( T_{11}^{2} - 2 T_{11} - 16 \)
\( T_{13}^{2} - 5 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -16 - 2 T + T^{2} \)
$13$ \( 2 - 5 T + T^{2} \)
$17$ \( -16 + 2 T + T^{2} \)
$19$ \( -16 + 2 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -2 + 3 T + T^{2} \)
$31$ \( 16 - 9 T + T^{2} \)
$37$ \( -68 + T^{2} \)
$41$ \( -106 + T + T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( -8 - 11 T + T^{2} \)
$53$ \( ( 2 + T )^{2} \)
$59$ \( -64 - 4 T + T^{2} \)
$61$ \( -52 - 8 T + T^{2} \)
$67$ \( -16 - 2 T + T^{2} \)
$71$ \( 128 - 23 T + T^{2} \)
$73$ \( 34 + 17 T + T^{2} \)
$79$ \( -16 - 2 T + T^{2} \)
$83$ \( -32 - 12 T + T^{2} \)
$89$ \( -152 - 2 T + T^{2} \)
$97$ \( -152 + 2 T + T^{2} \)
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