Properties

Label 1148.2.a.d
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: 5.5.287349.1
Defining polynomial: \(x^{5} - 6 x^{3} + 7 x - 1\)
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 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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( -1 + \beta_{2} - \beta_{3} ) q^{5} - q^{7} + ( \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( -1 + \beta_{2} - \beta_{3} ) q^{5} - q^{7} + ( \beta_{2} + \beta_{4} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{11} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( 2 - \beta_{1} + \beta_{4} ) q^{15} + ( -\beta_{3} - \beta_{4} ) q^{17} + ( 3 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{19} -\beta_{2} q^{21} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{23} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{25} + ( 2 + \beta_{1} - \beta_{2} ) q^{27} + ( 2 - 3 \beta_{1} - \beta_{4} ) q^{29} + ( 3 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} + ( 1 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{33} + ( 1 - \beta_{2} + \beta_{3} ) q^{35} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{37} + ( 4 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{39} - q^{41} + ( 6 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{43} + ( 2 - \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{45} + ( 2 + 3 \beta_{4} ) q^{47} + q^{49} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{51} + ( 5 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{53} + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{55} + ( 3 + 5 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{57} + ( 1 - 2 \beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{59} + ( -3 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} ) q^{61} + ( -\beta_{2} - \beta_{4} ) q^{63} + ( -1 + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{65} + ( 3 - 4 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{67} + ( 3 - 8 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{69} + ( 3 + 7 \beta_{1} + \beta_{2} + \beta_{3} - 5 \beta_{4} ) q^{71} + ( -3 + 5 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{73} + ( -3 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{75} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{77} + ( 2 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{79} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{81} + ( -1 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{83} + ( 3 - \beta_{2} - \beta_{3} ) q^{85} + ( 1 - 7 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{87} + ( -5 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{89} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{91} + ( -2 + 9 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} ) q^{93} + ( 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{95} + ( -6 + 4 \beta_{1} + \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{97} + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 2q^{3} - 3q^{5} - 5q^{7} + q^{9} + O(q^{10}) \) \( 5q + 2q^{3} - 3q^{5} - 5q^{7} + q^{9} + 6q^{11} + 7q^{13} + 9q^{15} + q^{17} + 2q^{19} - 2q^{21} + 4q^{23} + 2q^{25} + 8q^{27} + 11q^{29} + 13q^{31} + 11q^{33} + 3q^{35} + 5q^{37} + 23q^{39} - 5q^{41} + 29q^{43} + 11q^{45} + 7q^{47} + 5q^{49} - 3q^{51} + 21q^{53} + 19q^{55} + 9q^{57} + 3q^{59} - 8q^{61} - q^{63} - 5q^{65} + 3q^{67} + 10q^{69} + 22q^{71} - 16q^{73} - 18q^{75} - 6q^{77} + 4q^{79} - 15q^{81} - 6q^{83} + 13q^{85} + 6q^{87} - 20q^{89} - 7q^{91} - 5q^{93} - 7q^{95} - 24q^{97} + 27q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 6 x^{3} + 7 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 5 \beta_{2} + 7\)

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
0.145487
1.14203
−1.36870
−2.05679
2.13797
0 −1.97883 0 −2.39996 0 −1.00000 0 0.915782 0
1.2 0 −0.695770 0 1.38288 0 −1.00000 0 −2.51590 0
1.3 0 −0.126667 0 −4.03743 0 −1.00000 0 −2.98396 0
1.4 0 2.23037 0 1.70418 0 −1.00000 0 1.97454 0
1.5 0 2.57090 0 0.350332 0 −1.00000 0 3.60954 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.d 5
4.b odd 2 1 4592.2.a.bc 5
7.b odd 2 1 8036.2.a.k 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.a.d 5 1.a even 1 1 trivial
4592.2.a.bc 5 4.b odd 2 1
8036.2.a.k 5 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 2 T_{3}^{4} - 6 T_{3}^{3} + 8 T_{3}^{2} + 9 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1148))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( 1 + 9 T + 8 T^{2} - 6 T^{3} - 2 T^{4} + T^{5} \)
$5$ \( -8 + 28 T - 12 T^{2} - 9 T^{3} + 3 T^{4} + T^{5} \)
$7$ \( ( 1 + T )^{5} \)
$11$ \( 24 - 84 T + 68 T^{2} - 7 T^{3} - 6 T^{4} + T^{5} \)
$13$ \( -27 + 18 T + 36 T^{2} - 5 T^{3} - 7 T^{4} + T^{5} \)
$17$ \( 1 + 18 T - 14 T^{2} - 17 T^{3} - T^{4} + T^{5} \)
$19$ \( -3643 + 1167 T + 176 T^{2} - 68 T^{3} - 2 T^{4} + T^{5} \)
$23$ \( -5721 + 2109 T + 310 T^{2} - 92 T^{3} - 4 T^{4} + T^{5} \)
$29$ \( -216 + 396 T + 400 T^{2} - 31 T^{3} - 11 T^{4} + T^{5} \)
$31$ \( 1176 - 2520 T + 782 T^{2} - 25 T^{3} - 13 T^{4} + T^{5} \)
$37$ \( -1748 + 670 T + 223 T^{2} - 68 T^{3} - 5 T^{4} + T^{5} \)
$41$ \( ( 1 + T )^{5} \)
$43$ \( -2161 + 3008 T - 1450 T^{2} + 307 T^{3} - 29 T^{4} + T^{5} \)
$47$ \( -1532 + 1010 T + 289 T^{2} - 74 T^{3} - 7 T^{4} + T^{5} \)
$53$ \( -152 - 284 T + 240 T^{2} + 89 T^{3} - 21 T^{4} + T^{5} \)
$59$ \( -55448 + 14408 T + 586 T^{2} - 263 T^{3} - 3 T^{4} + T^{5} \)
$61$ \( 11944 + 6936 T - 878 T^{2} - 167 T^{3} + 8 T^{4} + T^{5} \)
$67$ \( 51208 + 28924 T + 108 T^{2} - 345 T^{3} - 3 T^{4} + T^{5} \)
$71$ \( -308544 + 6336 T + 5596 T^{2} - 203 T^{3} - 22 T^{4} + T^{5} \)
$73$ \( 2664 - 192 T - 514 T^{2} - 27 T^{3} + 16 T^{4} + T^{5} \)
$79$ \( -100896 + 17616 T + 1408 T^{2} - 288 T^{3} - 4 T^{4} + T^{5} \)
$83$ \( 5272 + 1604 T - 326 T^{2} - 89 T^{3} + 6 T^{4} + T^{5} \)
$89$ \( -5297 - 4239 T - 818 T^{2} + 46 T^{3} + 20 T^{4} + T^{5} \)
$97$ \( -122173 - 55351 T - 6952 T^{2} - 122 T^{3} + 24 T^{4} + T^{5} \)
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