Properties

Label 1148.2.a.a
Level $1148$
Weight $2$
Character orbit 1148.a
Self dual yes
Analytic conductor $9.167$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.16682615204\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{3} + ( -1 - \beta ) q^{5} + q^{7} + ( 1 + 3 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta ) q^{3} + ( -1 - \beta ) q^{5} + q^{7} + ( 1 + 3 \beta ) q^{9} + q^{11} -5 q^{13} + ( 4 + 3 \beta ) q^{15} + ( -3 + 4 \beta ) q^{17} + ( -3 + 3 \beta ) q^{19} + ( -1 - \beta ) q^{21} + ( -1 - \beta ) q^{23} + ( -1 + 3 \beta ) q^{25} + ( -7 - 4 \beta ) q^{27} + ( 8 + \beta ) q^{29} + ( 7 - 3 \beta ) q^{31} + ( -1 - \beta ) q^{33} + ( -1 - \beta ) q^{35} + ( 4 - 2 \beta ) q^{37} + ( 5 + 5 \beta ) q^{39} - q^{41} + ( -7 - 2 \beta ) q^{43} + ( -10 - 7 \beta ) q^{45} + ( -10 + 2 \beta ) q^{47} + q^{49} + ( -9 - 5 \beta ) q^{51} + ( -5 + \beta ) q^{53} + ( -1 - \beta ) q^{55} + ( -6 - 3 \beta ) q^{57} + ( -1 + 3 \beta ) q^{59} + ( -1 - 4 \beta ) q^{61} + ( 1 + 3 \beta ) q^{63} + ( 5 + 5 \beta ) q^{65} + ( -5 + 3 \beta ) q^{67} + ( 4 + 3 \beta ) q^{69} + ( -3 - 4 \beta ) q^{71} + 7 q^{73} + ( -8 - 5 \beta ) q^{75} + q^{77} + ( -12 - 2 \beta ) q^{79} + ( 16 + 6 \beta ) q^{81} + ( 3 - 6 \beta ) q^{83} + ( -9 - 5 \beta ) q^{85} + ( -11 - 10 \beta ) q^{87} + ( 3 - 5 \beta ) q^{89} -5 q^{91} + ( 2 - \beta ) q^{93} + ( -6 - 3 \beta ) q^{95} + ( -7 + 7 \beta ) q^{97} + ( 1 + 3 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} - 3q^{5} + 2q^{7} + 5q^{9} + O(q^{10}) \) \( 2q - 3q^{3} - 3q^{5} + 2q^{7} + 5q^{9} + 2q^{11} - 10q^{13} + 11q^{15} - 2q^{17} - 3q^{19} - 3q^{21} - 3q^{23} + q^{25} - 18q^{27} + 17q^{29} + 11q^{31} - 3q^{33} - 3q^{35} + 6q^{37} + 15q^{39} - 2q^{41} - 16q^{43} - 27q^{45} - 18q^{47} + 2q^{49} - 23q^{51} - 9q^{53} - 3q^{55} - 15q^{57} + q^{59} - 6q^{61} + 5q^{63} + 15q^{65} - 7q^{67} + 11q^{69} - 10q^{71} + 14q^{73} - 21q^{75} + 2q^{77} - 26q^{79} + 38q^{81} - 23q^{85} - 32q^{87} + q^{89} - 10q^{91} + 3q^{93} - 15q^{95} - 7q^{97} + 5q^{99} + O(q^{100}) \)

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
2.30278
−1.30278
0 −3.30278 0 −3.30278 0 1.00000 0 7.90833 0
1.2 0 0.302776 0 0.302776 0 1.00000 0 −2.90833 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(1\)

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 1148.2.a.a 2
4.b odd 2 1 4592.2.a.o 2
7.b odd 2 1 8036.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.a.a 2 1.a even 1 1 trivial
4592.2.a.o 2 4.b odd 2 1
8036.2.a.g 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 3 T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1148))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -1 + 3 T + T^{2} \)
$5$ \( -1 + 3 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( 5 + T )^{2} \)
$17$ \( -51 + 2 T + T^{2} \)
$19$ \( -27 + 3 T + T^{2} \)
$23$ \( -1 + 3 T + T^{2} \)
$29$ \( 69 - 17 T + T^{2} \)
$31$ \( 1 - 11 T + T^{2} \)
$37$ \( -4 - 6 T + T^{2} \)
$41$ \( ( 1 + T )^{2} \)
$43$ \( 51 + 16 T + T^{2} \)
$47$ \( 68 + 18 T + T^{2} \)
$53$ \( 17 + 9 T + T^{2} \)
$59$ \( -29 - T + T^{2} \)
$61$ \( -43 + 6 T + T^{2} \)
$67$ \( -17 + 7 T + T^{2} \)
$71$ \( -27 + 10 T + T^{2} \)
$73$ \( ( -7 + T )^{2} \)
$79$ \( 156 + 26 T + T^{2} \)
$83$ \( -117 + T^{2} \)
$89$ \( -81 - T + T^{2} \)
$97$ \( -147 + 7 T + T^{2} \)
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