Properties

Label 1148.2.a.b
Level $1148$
Weight $2$
Character orbit 1148.a
Self dual yes
Analytic conductor $9.167$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{5} + q^{7} + ( -2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{5} + q^{7} + ( -2 \beta_{1} + \beta_{2} ) q^{9} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{11} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{13} + ( -2 + 4 \beta_{1} - 4 \beta_{2} ) q^{15} + ( 1 + \beta_{2} ) q^{17} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{19} + ( -1 + \beta_{1} ) q^{21} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{23} + ( 3 - 4 \beta_{1} ) q^{25} -3 \beta_{2} q^{27} + ( -4 + 2 \beta_{1} ) q^{29} + ( -4 - 4 \beta_{1} + 4 \beta_{2} ) q^{31} + ( 2 - 8 \beta_{1} + 6 \beta_{2} ) q^{33} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -7 - 3 \beta_{1} + \beta_{2} ) q^{37} + ( -5 \beta_{1} + 4 \beta_{2} ) q^{39} - q^{41} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{43} + ( 6 - 4 \beta_{1} + 2 \beta_{2} ) q^{45} + ( -1 + 5 \beta_{1} + \beta_{2} ) q^{47} + q^{49} + ( 2 \beta_{1} - \beta_{2} ) q^{51} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -12 + 8 \beta_{1} ) q^{55} + ( 8 - 4 \beta_{1} + \beta_{2} ) q^{57} + ( -2 - 4 \beta_{1} + 4 \beta_{2} ) q^{59} + ( -10 + 2 \beta_{1} ) q^{61} + ( -2 \beta_{1} + \beta_{2} ) q^{63} + ( -8 + 4 \beta_{1} + 2 \beta_{2} ) q^{65} + ( -6 - 4 \beta_{1} + 4 \beta_{2} ) q^{67} + ( -5 + \beta_{2} ) q^{69} + ( 4 + 2 \beta_{1} - 6 \beta_{2} ) q^{71} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{73} + ( -11 + 7 \beta_{1} - 4 \beta_{2} ) q^{75} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{77} + ( 4 - 6 \beta_{1} + 6 \beta_{2} ) q^{79} + ( -3 + 3 \beta_{1} ) q^{81} + ( -6 + 4 \beta_{1} ) q^{83} + ( 2 - 2 \beta_{1} ) q^{85} + ( 8 - 6 \beta_{1} + 2 \beta_{2} ) q^{87} + ( -1 + 5 \beta_{1} - 7 \beta_{2} ) q^{89} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{91} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{93} + ( -2 + 10 \beta_{1} - 12 \beta_{2} ) q^{95} + ( 1 + 5 \beta_{1} - 4 \beta_{2} ) q^{97} + ( -6 + 10 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 3q^{7} + O(q^{10}) \) \( 3q - 3q^{3} + 3q^{7} - 6q^{11} - 3q^{13} - 6q^{15} + 3q^{17} - 9q^{19} - 3q^{21} + 3q^{23} + 9q^{25} - 12q^{29} - 12q^{31} + 6q^{33} - 21q^{37} - 3q^{41} - 3q^{43} + 18q^{45} - 3q^{47} + 3q^{49} + 18q^{53} - 36q^{55} + 24q^{57} - 6q^{59} - 30q^{61} - 24q^{65} - 18q^{67} - 15q^{69} + 12q^{71} - 12q^{73} - 33q^{75} - 6q^{77} + 12q^{79} - 9q^{81} - 18q^{83} + 6q^{85} + 24q^{87} - 3q^{89} - 3q^{91} - 6q^{95} + 3q^{97} - 18q^{99} + O(q^{100}) \)

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
−1.53209
−0.347296
1.87939
0 −2.53209 0 3.75877 0 1.00000 0 3.41147 0
1.2 0 −1.34730 0 −3.06418 0 1.00000 0 −1.18479 0
1.3 0 0.879385 0 −0.694593 0 1.00000 0 −2.22668 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.b 3
4.b odd 2 1 4592.2.a.v 3
7.b odd 2 1 8036.2.a.h 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.a.b 3 1.a even 1 1 trivial
4592.2.a.v 3 4.b odd 2 1
8036.2.a.h 3 7.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -3 + 3 T^{2} + T^{3} \)
$5$ \( -8 - 12 T + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( -136 - 24 T + 6 T^{2} + T^{3} \)
$13$ \( -57 - 18 T + 3 T^{2} + T^{3} \)
$17$ \( 3 - 3 T^{2} + T^{3} \)
$19$ \( -73 + 6 T + 9 T^{2} + T^{3} \)
$23$ \( 37 - 18 T - 3 T^{2} + T^{3} \)
$29$ \( 8 + 36 T + 12 T^{2} + T^{3} \)
$31$ \( -192 + 12 T^{2} + T^{3} \)
$37$ \( 179 + 126 T + 21 T^{2} + T^{3} \)
$41$ \( ( 1 + T )^{3} \)
$43$ \( -19 - 36 T + 3 T^{2} + T^{3} \)
$47$ \( -381 - 90 T + 3 T^{2} + T^{3} \)
$53$ \( -72 + 72 T - 18 T^{2} + T^{3} \)
$59$ \( -152 - 36 T + 6 T^{2} + T^{3} \)
$61$ \( 872 + 288 T + 30 T^{2} + T^{3} \)
$67$ \( -136 + 60 T + 18 T^{2} + T^{3} \)
$71$ \( -24 - 36 T - 12 T^{2} + T^{3} \)
$73$ \( -64 + 12 T^{2} + T^{3} \)
$79$ \( 152 - 60 T - 12 T^{2} + T^{3} \)
$83$ \( -136 + 60 T + 18 T^{2} + T^{3} \)
$89$ \( -269 - 114 T + 3 T^{2} + T^{3} \)
$97$ \( 233 - 60 T - 3 T^{2} + T^{3} \)
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