Properties

Label 2112.2.a.t
Level $2112$
Weight $2$
Character orbit 2112.a
Self dual yes
Analytic conductor $16.864$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.8644049069\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 132)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 2q^{5} + 2q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} - 2q^{5} + 2q^{7} + q^{9} + q^{11} - 6q^{13} - 2q^{15} - 4q^{17} + 2q^{19} + 2q^{21} - 8q^{23} - q^{25} + q^{27} + q^{33} - 4q^{35} + 6q^{37} - 6q^{39} - 10q^{43} - 2q^{45} - 3q^{49} - 4q^{51} - 14q^{53} - 2q^{55} + 2q^{57} + 12q^{59} + 14q^{61} + 2q^{63} + 12q^{65} - 4q^{67} - 8q^{69} + 6q^{73} - q^{75} + 2q^{77} + 2q^{79} + q^{81} - 16q^{83} + 8q^{85} - 14q^{89} - 12q^{91} - 4q^{95} - 2q^{97} + q^{99} + O(q^{100}) \)

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
0
0 1.00000 0 −2.00000 0 2.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(11\) \(-1\)

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 2112.2.a.t 1
3.b odd 2 1 6336.2.a.cf 1
4.b odd 2 1 2112.2.a.b 1
8.b even 2 1 132.2.a.a 1
8.d odd 2 1 528.2.a.h 1
12.b even 2 1 6336.2.a.bz 1
24.f even 2 1 1584.2.a.c 1
24.h odd 2 1 396.2.a.b 1
40.f even 2 1 3300.2.a.k 1
40.i odd 4 2 3300.2.c.c 2
56.h odd 2 1 6468.2.a.l 1
72.j odd 6 2 3564.2.i.g 2
72.n even 6 2 3564.2.i.b 2
88.b odd 2 1 1452.2.a.c 1
88.g even 2 1 5808.2.a.bf 1
88.o even 10 4 1452.2.i.k 4
88.p odd 10 4 1452.2.i.l 4
120.i odd 2 1 9900.2.a.g 1
120.w even 4 2 9900.2.c.k 2
264.m even 2 1 4356.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.2.a.a 1 8.b even 2 1
396.2.a.b 1 24.h odd 2 1
528.2.a.h 1 8.d odd 2 1
1452.2.a.c 1 88.b odd 2 1
1452.2.i.k 4 88.o even 10 4
1452.2.i.l 4 88.p odd 10 4
1584.2.a.c 1 24.f even 2 1
2112.2.a.b 1 4.b odd 2 1
2112.2.a.t 1 1.a even 1 1 trivial
3300.2.a.k 1 40.f even 2 1
3300.2.c.c 2 40.i odd 4 2
3564.2.i.b 2 72.n even 6 2
3564.2.i.g 2 72.j odd 6 2
4356.2.a.c 1 264.m even 2 1
5808.2.a.bf 1 88.g even 2 1
6336.2.a.bz 1 12.b even 2 1
6336.2.a.cf 1 3.b odd 2 1
6468.2.a.l 1 56.h odd 2 1
9900.2.a.g 1 120.i odd 2 1
9900.2.c.k 2 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2112))\):

\( T_{5} + 2 \)
\( T_{7} - 2 \)
\( T_{13} + 6 \)
\( T_{17} + 4 \)
\( T_{19} - 2 \)
\( T_{23} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T \)
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 - 2 T + 7 T^{2} \)
$11$ \( 1 - T \)
$13$ \( 1 + 6 T + 13 T^{2} \)
$17$ \( 1 + 4 T + 17 T^{2} \)
$19$ \( 1 - 2 T + 19 T^{2} \)
$23$ \( 1 + 8 T + 23 T^{2} \)
$29$ \( 1 + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 + 10 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 + 14 T + 53 T^{2} \)
$59$ \( 1 - 12 T + 59 T^{2} \)
$61$ \( 1 - 14 T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 6 T + 73 T^{2} \)
$79$ \( 1 - 2 T + 79 T^{2} \)
$83$ \( 1 + 16 T + 83 T^{2} \)
$89$ \( 1 + 14 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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