Properties

Label 1449.2.a.j
Level $1449$
Weight $2$
Character orbit 1449.a
Self dual yes
Analytic conductor $11.570$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1449.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.5703232529\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 1 + \beta ) q^{4} + ( 3 - \beta ) q^{5} - q^{7} + 3 q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( 1 + \beta ) q^{4} + ( 3 - \beta ) q^{5} - q^{7} + 3 q^{8} + ( -3 + 2 \beta ) q^{10} + 5 q^{11} + \beta q^{13} -\beta q^{14} + ( -2 + \beta ) q^{16} + ( 1 + 2 \beta ) q^{17} + ( 3 - 2 \beta ) q^{19} + \beta q^{20} + 5 \beta q^{22} - q^{23} + ( 7 - 5 \beta ) q^{25} + ( 3 + \beta ) q^{26} + ( -1 - \beta ) q^{28} + ( 3 - 4 \beta ) q^{29} + 3 q^{31} + ( -3 - \beta ) q^{32} + ( 6 + 3 \beta ) q^{34} + ( -3 + \beta ) q^{35} -9 q^{37} + ( -6 + \beta ) q^{38} + ( 9 - 3 \beta ) q^{40} + ( 3 + 4 \beta ) q^{41} + ( -6 + 5 \beta ) q^{43} + ( 5 + 5 \beta ) q^{44} -\beta q^{46} + ( 4 + 2 \beta ) q^{47} + q^{49} + ( -15 + 2 \beta ) q^{50} + ( 3 + 2 \beta ) q^{52} + ( 1 + 5 \beta ) q^{53} + ( 15 - 5 \beta ) q^{55} -3 q^{56} + ( -12 - \beta ) q^{58} + ( 3 - 3 \beta ) q^{59} + ( -8 + 3 \beta ) q^{61} + 3 \beta q^{62} + ( 1 - 6 \beta ) q^{64} + ( -3 + 2 \beta ) q^{65} + ( -5 - 3 \beta ) q^{67} + ( 7 + 5 \beta ) q^{68} + ( 3 - 2 \beta ) q^{70} + ( 6 - 3 \beta ) q^{71} + ( 7 - 4 \beta ) q^{73} -9 \beta q^{74} + ( -3 - \beta ) q^{76} -5 q^{77} - q^{79} + ( -9 + 4 \beta ) q^{80} + ( 12 + 7 \beta ) q^{82} + ( -1 - 2 \beta ) q^{83} + ( -3 + 3 \beta ) q^{85} + ( 15 - \beta ) q^{86} + 15 q^{88} + ( -4 - 3 \beta ) q^{89} -\beta q^{91} + ( -1 - \beta ) q^{92} + ( 6 + 6 \beta ) q^{94} + ( 15 - 7 \beta ) q^{95} + ( -13 - 2 \beta ) q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{4} + 5 q^{5} - 2 q^{7} + 6 q^{8} + O(q^{10}) \) \( 2 q + q^{2} + 3 q^{4} + 5 q^{5} - 2 q^{7} + 6 q^{8} - 4 q^{10} + 10 q^{11} + q^{13} - q^{14} - 3 q^{16} + 4 q^{17} + 4 q^{19} + q^{20} + 5 q^{22} - 2 q^{23} + 9 q^{25} + 7 q^{26} - 3 q^{28} + 2 q^{29} + 6 q^{31} - 7 q^{32} + 15 q^{34} - 5 q^{35} - 18 q^{37} - 11 q^{38} + 15 q^{40} + 10 q^{41} - 7 q^{43} + 15 q^{44} - q^{46} + 10 q^{47} + 2 q^{49} - 28 q^{50} + 8 q^{52} + 7 q^{53} + 25 q^{55} - 6 q^{56} - 25 q^{58} + 3 q^{59} - 13 q^{61} + 3 q^{62} - 4 q^{64} - 4 q^{65} - 13 q^{67} + 19 q^{68} + 4 q^{70} + 9 q^{71} + 10 q^{73} - 9 q^{74} - 7 q^{76} - 10 q^{77} - 2 q^{79} - 14 q^{80} + 31 q^{82} - 4 q^{83} - 3 q^{85} + 29 q^{86} + 30 q^{88} - 11 q^{89} - q^{91} - 3 q^{92} + 18 q^{94} + 23 q^{95} - 28 q^{97} + q^{98} + O(q^{100}) \)

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
−1.30278
2.30278
−1.30278 0 −0.302776 4.30278 0 −1.00000 3.00000 0 −5.60555
1.2 2.30278 0 3.30278 0.697224 0 −1.00000 3.00000 0 1.60555
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(23\) \(1\)

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 1449.2.a.j 2
3.b odd 2 1 483.2.a.f 2
12.b even 2 1 7728.2.a.x 2
21.c even 2 1 3381.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.f 2 3.b odd 2 1
1449.2.a.j 2 1.a even 1 1 trivial
3381.2.a.p 2 21.c even 2 1
7728.2.a.x 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1449))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 3 - 5 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -5 + T )^{2} \)
$13$ \( -3 - T + T^{2} \)
$17$ \( -9 - 4 T + T^{2} \)
$19$ \( -9 - 4 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -51 - 2 T + T^{2} \)
$31$ \( ( -3 + T )^{2} \)
$37$ \( ( 9 + T )^{2} \)
$41$ \( -27 - 10 T + T^{2} \)
$43$ \( -69 + 7 T + T^{2} \)
$47$ \( 12 - 10 T + T^{2} \)
$53$ \( -69 - 7 T + T^{2} \)
$59$ \( -27 - 3 T + T^{2} \)
$61$ \( 13 + 13 T + T^{2} \)
$67$ \( 13 + 13 T + T^{2} \)
$71$ \( -9 - 9 T + T^{2} \)
$73$ \( -27 - 10 T + T^{2} \)
$79$ \( ( 1 + T )^{2} \)
$83$ \( -9 + 4 T + T^{2} \)
$89$ \( 1 + 11 T + T^{2} \)
$97$ \( 183 + 28 T + T^{2} \)
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