Properties

Label 1456.2.a.q
Level $1456$
Weight $2$
Character orbit 1456.a
Self dual yes
Analytic conductor $11.626$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( 3 + \beta ) q^{5} - q^{7} - q^{9} +O(q^{10})\) \( q + \beta q^{3} + ( 3 + \beta ) q^{5} - q^{7} - q^{9} + 3 \beta q^{11} - q^{13} + ( 2 + 3 \beta ) q^{15} -\beta q^{17} + ( 3 - 3 \beta ) q^{19} -\beta q^{21} + ( 3 - 2 \beta ) q^{23} + ( 6 + 6 \beta ) q^{25} -4 \beta q^{27} + ( 3 + 2 \beta ) q^{29} + ( 1 + 3 \beta ) q^{31} + 6 q^{33} + ( -3 - \beta ) q^{35} + ( -2 - 3 \beta ) q^{37} -\beta q^{39} + ( 6 - 2 \beta ) q^{41} + 5 q^{43} + ( -3 - \beta ) q^{45} + ( -3 - \beta ) q^{47} + q^{49} -2 q^{51} + ( -3 - 2 \beta ) q^{53} + ( 6 + 9 \beta ) q^{55} + ( -6 + 3 \beta ) q^{57} + ( -6 - 4 \beta ) q^{59} + 6 q^{61} + q^{63} + ( -3 - \beta ) q^{65} + ( 6 - 6 \beta ) q^{67} + ( -4 + 3 \beta ) q^{69} + ( 6 - 5 \beta ) q^{71} + ( -5 + 3 \beta ) q^{73} + ( 12 + 6 \beta ) q^{75} -3 \beta q^{77} + ( -7 + 6 \beta ) q^{79} -5 q^{81} + ( -9 + 3 \beta ) q^{83} + ( -2 - 3 \beta ) q^{85} + ( 4 + 3 \beta ) q^{87} + ( 3 + \beta ) q^{89} + q^{91} + ( 6 + \beta ) q^{93} + ( 3 - 6 \beta ) q^{95} + ( -1 - 9 \beta ) q^{97} -3 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{5} - 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 6q^{5} - 2q^{7} - 2q^{9} - 2q^{13} + 4q^{15} + 6q^{19} + 6q^{23} + 12q^{25} + 6q^{29} + 2q^{31} + 12q^{33} - 6q^{35} - 4q^{37} + 12q^{41} + 10q^{43} - 6q^{45} - 6q^{47} + 2q^{49} - 4q^{51} - 6q^{53} + 12q^{55} - 12q^{57} - 12q^{59} + 12q^{61} + 2q^{63} - 6q^{65} + 12q^{67} - 8q^{69} + 12q^{71} - 10q^{73} + 24q^{75} - 14q^{79} - 10q^{81} - 18q^{83} - 4q^{85} + 8q^{87} + 6q^{89} + 2q^{91} + 12q^{93} + 6q^{95} - 2q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 −1.41421 0 1.58579 0 −1.00000 0 −1.00000 0
1.2 0 1.41421 0 4.41421 0 −1.00000 0 −1.00000 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.q 2
4.b odd 2 1 91.2.a.c 2
8.b even 2 1 5824.2.a.bk 2
8.d odd 2 1 5824.2.a.bl 2
12.b even 2 1 819.2.a.h 2
20.d odd 2 1 2275.2.a.j 2
28.d even 2 1 637.2.a.g 2
28.f even 6 2 637.2.e.g 4
28.g odd 6 2 637.2.e.f 4
52.b odd 2 1 1183.2.a.d 2
52.f even 4 2 1183.2.c.d 4
84.h odd 2 1 5733.2.a.s 2
364.h even 2 1 8281.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.c 2 4.b odd 2 1
637.2.a.g 2 28.d even 2 1
637.2.e.f 4 28.g odd 6 2
637.2.e.g 4 28.f even 6 2
819.2.a.h 2 12.b even 2 1
1183.2.a.d 2 52.b odd 2 1
1183.2.c.d 4 52.f even 4 2
1456.2.a.q 2 1.a even 1 1 trivial
2275.2.a.j 2 20.d odd 2 1
5733.2.a.s 2 84.h odd 2 1
5824.2.a.bk 2 8.b even 2 1
5824.2.a.bl 2 8.d odd 2 1
8281.2.a.v 2 364.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1456))\):

\( T_{3}^{2} - 2 \)
\( T_{5}^{2} - 6 T_{5} + 7 \)
\( T_{11}^{2} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( 7 - 6 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -18 + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -2 + T^{2} \)
$19$ \( -9 - 6 T + T^{2} \)
$23$ \( 1 - 6 T + T^{2} \)
$29$ \( 1 - 6 T + T^{2} \)
$31$ \( -17 - 2 T + T^{2} \)
$37$ \( -14 + 4 T + T^{2} \)
$41$ \( 28 - 12 T + T^{2} \)
$43$ \( ( -5 + T )^{2} \)
$47$ \( 7 + 6 T + T^{2} \)
$53$ \( 1 + 6 T + T^{2} \)
$59$ \( 4 + 12 T + T^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( -36 - 12 T + T^{2} \)
$71$ \( -14 - 12 T + T^{2} \)
$73$ \( 7 + 10 T + T^{2} \)
$79$ \( -23 + 14 T + T^{2} \)
$83$ \( 63 + 18 T + T^{2} \)
$89$ \( 7 - 6 T + T^{2} \)
$97$ \( -161 + 2 T + T^{2} \)
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