Properties

Label 1148.2.a.e
Level $1148$
Weight $2$
Character orbit 1148.a
Self dual yes
Analytic conductor $9.167$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1935333.1
Defining polynomial: \(x^{5} - 2 x^{4} - 8 x^{3} + 10 x^{2} + 13 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 + \beta_{4} ) q^{5} + q^{7} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 + \beta_{4} ) q^{5} + q^{7} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} -\beta_{2} q^{11} + ( 1 + \beta_{3} ) q^{13} + ( 1 + \beta_{2} + \beta_{4} ) q^{15} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{17} + ( 2 - \beta_{1} + \beta_{3} ) q^{19} + \beta_{1} q^{21} + ( 2 + \beta_{1} ) q^{23} + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{25} + ( 4 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{27} + ( -1 - \beta_{2} - \beta_{4} ) q^{29} + ( 1 - \beta_{4} ) q^{31} + ( -1 - 2 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{33} + ( 1 + \beta_{4} ) q^{35} + ( -1 - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{37} + ( 1 + \beta_{1} - \beta_{2} ) q^{39} + q^{41} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{43} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{45} + ( 3 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{47} + q^{49} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{51} + ( -1 - \beta_{4} ) q^{53} + ( 3 - 2 \beta_{1} + \beta_{4} ) q^{55} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{57} + ( -1 - 2 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{59} + ( 4 + \beta_{2} - 4 \beta_{3} - 2 \beta_{4} ) q^{61} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{63} + ( 1 + 2 \beta_{3} + 3 \beta_{4} ) q^{65} + ( 5 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{67} + ( 4 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{69} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{71} + ( \beta_{2} - 2 \beta_{4} ) q^{73} + ( 2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{75} -\beta_{2} q^{77} + ( 4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{79} + ( 1 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{81} + ( 6 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{83} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{85} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{87} + ( -4 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{89} + ( 1 + \beta_{3} ) q^{91} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{93} + ( 1 - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{95} + ( 2 - 3 \beta_{1} ) q^{97} + ( -7 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 2q^{3} + 3q^{5} + 5q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 2q^{3} + 3q^{5} + 5q^{7} + 5q^{9} + 7q^{13} + 3q^{15} - 3q^{17} + 10q^{19} + 2q^{21} + 12q^{23} + 2q^{25} + 14q^{27} - 3q^{29} + 7q^{31} - 3q^{33} + 3q^{35} + q^{37} + 7q^{39} + 5q^{41} + 13q^{43} - 3q^{45} + 9q^{47} + 5q^{49} + 9q^{51} - 3q^{53} + 9q^{55} - 11q^{57} - 3q^{59} + 16q^{61} + 5q^{63} + 3q^{65} + 19q^{67} + 24q^{69} - 12q^{71} + 4q^{73} + 8q^{75} + 28q^{79} + 5q^{81} + 18q^{83} - 15q^{85} - 6q^{87} + 7q^{91} + q^{93} + 3q^{95} + 4q^{97} - 33q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 8 x^{3} + 10 x^{2} + 13 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 3 \nu^{3} - 2 \nu^{2} + 9 \nu - 8 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - 3 \nu^{3} - 5 \nu^{2} + 12 \nu + 4 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 11 \nu^{2} - 3 \nu + 16 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(3 \beta_{4} - 11 \beta_{3} + 11 \beta_{2} + 14 \beta_{1} + 28\)

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
−2.10409
−1.43444
0.704110
1.60064
3.23378
0 −2.10409 0 −1.26228 0 1.00000 0 1.42721 0
1.2 0 −1.43444 0 1.63444 0 1.00000 0 −0.942381 0
1.3 0 0.704110 0 3.89333 0 1.00000 0 −2.50423 0
1.4 0 1.60064 0 −2.47347 0 1.00000 0 −0.437946 0
1.5 0 3.23378 0 1.20798 0 1.00000 0 7.45735 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.e 5
4.b odd 2 1 4592.2.a.bd 5
7.b odd 2 1 8036.2.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.a.e 5 1.a even 1 1 trivial
4592.2.a.bd 5 4.b odd 2 1
8036.2.a.j 5 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 2 T_{3}^{4} - 8 T_{3}^{3} + 10 T_{3}^{2} + 13 T_{3} - 11 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1148))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( -11 + 13 T + 10 T^{2} - 8 T^{3} - 2 T^{4} + T^{5} \)
$5$ \( -24 + 12 T + 20 T^{2} - 9 T^{3} - 3 T^{4} + T^{5} \)
$7$ \( ( -1 + T )^{5} \)
$11$ \( -72 + 84 T + 4 T^{2} - 21 T^{3} + T^{5} \)
$13$ \( 29 - 80 T + 38 T^{2} + 7 T^{3} - 7 T^{4} + T^{5} \)
$17$ \( 81 + 540 T - 72 T^{2} - 45 T^{3} + 3 T^{4} + T^{5} \)
$19$ \( -3 - 81 T + 30 T^{2} + 22 T^{3} - 10 T^{4} + T^{5} \)
$23$ \( 3 + 21 T - 70 T^{2} + 48 T^{3} - 12 T^{4} + T^{5} \)
$29$ \( 264 + 84 T - 64 T^{2} - 21 T^{3} + 3 T^{4} + T^{5} \)
$31$ \( 8 - 32 T + 26 T^{2} + 7 T^{3} - 7 T^{4} + T^{5} \)
$37$ \( 188 + 10 T - 145 T^{2} - 56 T^{3} - T^{4} + T^{5} \)
$41$ \( ( -1 + T )^{5} \)
$43$ \( -593 - 112 T + 190 T^{2} + 19 T^{3} - 13 T^{4} + T^{5} \)
$47$ \( 12 + 78 T + 107 T^{2} - 66 T^{3} - 9 T^{4} + T^{5} \)
$53$ \( 24 + 12 T - 20 T^{2} - 9 T^{3} + 3 T^{4} + T^{5} \)
$59$ \( -792 + 1176 T - 178 T^{2} - 87 T^{3} + 3 T^{4} + T^{5} \)
$61$ \( 35992 - 21064 T + 3634 T^{2} - 131 T^{3} - 16 T^{4} + T^{5} \)
$67$ \( -904 - 1028 T + 692 T^{2} + 25 T^{3} - 19 T^{4} + T^{5} \)
$71$ \( 576 - 288 T - 444 T^{2} - 69 T^{3} + 12 T^{4} + T^{5} \)
$73$ \( -3832 + 1552 T + 254 T^{2} - 83 T^{3} - 4 T^{4} + T^{5} \)
$79$ \( 8352 - 5616 T + 528 T^{2} + 184 T^{3} - 28 T^{4} + T^{5} \)
$83$ \( 9000 - 6300 T + 1230 T^{2} + 3 T^{3} - 18 T^{4} + T^{5} \)
$89$ \( -3753 - 711 T + 1284 T^{2} - 228 T^{3} + T^{5} \)
$97$ \( 127 + 1157 T + 226 T^{2} - 80 T^{3} - 4 T^{4} + T^{5} \)
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