Properties

Label 1456.2.a.t
Level $1456$
Weight $2$
Character orbit 1456.a
Self dual yes
Analytic conductor $11.626$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} + \beta_{2} ) q^{3} + ( 1 - \beta_{1} ) q^{5} + q^{7} + ( 3 - 2 \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} + \beta_{2} ) q^{3} + ( 1 - \beta_{1} ) q^{5} + q^{7} + ( 3 - 2 \beta_{1} ) q^{9} + ( -1 + \beta_{1} - \beta_{2} ) q^{11} + q^{13} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{15} + ( 1 + \beta_{1} + \beta_{2} ) q^{17} + ( 1 + \beta_{1} ) q^{19} + ( 1 - \beta_{1} + \beta_{2} ) q^{21} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{23} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{25} + ( 4 - 4 \beta_{1} ) q^{27} + ( 8 + \beta_{2} ) q^{29} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{31} + ( -6 + 2 \beta_{1} ) q^{33} + ( 1 - \beta_{1} ) q^{35} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{37} + ( 1 - \beta_{1} + \beta_{2} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -4 + 2 \beta_{1} + 3 \beta_{2} ) q^{43} + ( 9 - 5 \beta_{1} + 2 \beta_{2} ) q^{45} + ( 3 - \beta_{1} + 4 \beta_{2} ) q^{47} + q^{49} + ( 2 + 2 \beta_{1} ) q^{51} + ( 2 + 2 \beta_{1} - 3 \beta_{2} ) q^{53} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{55} + ( -1 + \beta_{1} + \beta_{2} ) q^{57} + ( 2 - 2 \beta_{1} - 4 \beta_{2} ) q^{59} -2 q^{61} + ( 3 - 2 \beta_{1} ) q^{63} + ( 1 - \beta_{1} ) q^{65} + ( 2 + 6 \beta_{1} - 4 \beta_{2} ) q^{67} + ( -5 + 9 \beta_{1} - 5 \beta_{2} ) q^{69} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{71} + ( -3 - \beta_{1} - 4 \beta_{2} ) q^{73} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{75} + ( -1 + \beta_{1} - \beta_{2} ) q^{77} + ( 6 - 4 \beta_{1} + \beta_{2} ) q^{79} + ( 3 - 6 \beta_{1} + 4 \beta_{2} ) q^{81} + ( 1 + 9 \beta_{1} - 4 \beta_{2} ) q^{83} + ( -3 - \beta_{1} - \beta_{2} ) q^{85} + ( 11 - 7 \beta_{1} + 7 \beta_{2} ) q^{87} + ( -1 + 5 \beta_{1} - 2 \beta_{2} ) q^{89} + q^{91} + ( -7 - \beta_{1} + 3 \beta_{2} ) q^{93} + ( -2 - \beta_{2} ) q^{95} + ( -3 - \beta_{1} ) q^{97} + ( -7 + 7 \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{3} + 2q^{5} + 3q^{7} + 7q^{9} + O(q^{10}) \) \( 3q + 2q^{3} + 2q^{5} + 3q^{7} + 7q^{9} - 2q^{11} + 3q^{13} + 6q^{15} + 4q^{17} + 4q^{19} + 2q^{21} - 10q^{23} - 5q^{25} + 8q^{27} + 24q^{29} + 4q^{31} - 16q^{33} + 2q^{35} + 2q^{39} + 2q^{41} - 10q^{43} + 22q^{45} + 8q^{47} + 3q^{49} + 8q^{51} + 8q^{53} - 6q^{55} - 2q^{57} + 4q^{59} - 6q^{61} + 7q^{63} + 2q^{65} + 12q^{67} - 6q^{69} + 6q^{71} - 10q^{73} + 16q^{75} - 2q^{77} + 14q^{79} + 3q^{81} + 12q^{83} - 10q^{85} + 26q^{87} + 2q^{89} + 3q^{91} - 22q^{93} - 6q^{95} - 10q^{97} - 14q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
0.470683
2.34292
−1.81361
0 −2.24914 0 0.529317 0 1.00000 0 2.05863 0
1.2 0 1.14637 0 −1.34292 0 1.00000 0 −1.68585 0
1.3 0 3.10278 0 2.81361 0 1.00000 0 6.62721 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 1456.2.a.t 3
4.b odd 2 1 91.2.a.d 3
8.b even 2 1 5824.2.a.bs 3
8.d odd 2 1 5824.2.a.by 3
12.b even 2 1 819.2.a.i 3
20.d odd 2 1 2275.2.a.m 3
28.d even 2 1 637.2.a.j 3
28.f even 6 2 637.2.e.i 6
28.g odd 6 2 637.2.e.j 6
52.b odd 2 1 1183.2.a.i 3
52.f even 4 2 1183.2.c.f 6
84.h odd 2 1 5733.2.a.x 3
364.h even 2 1 8281.2.a.bg 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 4.b odd 2 1
637.2.a.j 3 28.d even 2 1
637.2.e.i 6 28.f even 6 2
637.2.e.j 6 28.g odd 6 2
819.2.a.i 3 12.b even 2 1
1183.2.a.i 3 52.b odd 2 1
1183.2.c.f 6 52.f even 4 2
1456.2.a.t 3 1.a even 1 1 trivial
2275.2.a.m 3 20.d odd 2 1
5733.2.a.x 3 84.h odd 2 1
5824.2.a.bs 3 8.b even 2 1
5824.2.a.by 3 8.d odd 2 1
8281.2.a.bg 3 364.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1456))\):

\( T_{3}^{3} - 2 T_{3}^{2} - 6 T_{3} + 8 \)
\( T_{5}^{3} - 2 T_{5}^{2} - 3 T_{5} + 2 \)
\( T_{11}^{3} + 2 T_{11}^{2} - 6 T_{11} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 8 - 6 T - 2 T^{2} + T^{3} \)
$5$ \( 2 - 3 T - 2 T^{2} + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( -8 - 6 T + 2 T^{2} + T^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( -4 - 10 T - 4 T^{2} + T^{3} \)
$19$ \( 4 + T - 4 T^{2} + T^{3} \)
$23$ \( -136 + T + 10 T^{2} + T^{3} \)
$29$ \( -454 + 185 T - 24 T^{2} + T^{3} \)
$31$ \( -16 - 19 T - 4 T^{2} + T^{3} \)
$37$ \( -124 - 58 T + T^{3} \)
$41$ \( -8 - 28 T - 2 T^{2} + T^{3} \)
$43$ \( -628 - 71 T + 10 T^{2} + T^{3} \)
$47$ \( 544 - 79 T - 8 T^{2} + T^{3} \)
$53$ \( -22 - 35 T - 8 T^{2} + T^{3} \)
$59$ \( 688 - 156 T - 4 T^{2} + T^{3} \)
$61$ \( ( 2 + T )^{3} \)
$67$ \( 976 - 124 T - 12 T^{2} + T^{3} \)
$71$ \( -16 - 22 T - 6 T^{2} + T^{3} \)
$73$ \( -274 - 99 T + 10 T^{2} + T^{3} \)
$79$ \( 16 + 5 T - 14 T^{2} + T^{3} \)
$83$ \( 3268 - 271 T - 12 T^{2} + T^{3} \)
$89$ \( 422 - 95 T - 2 T^{2} + T^{3} \)
$97$ \( 22 + 29 T + 10 T^{2} + T^{3} \)
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