Properties

Label 1425.2.a.w
Level $1425$
Weight $2$
Character orbit 1425.a
Self dual yes
Analytic conductor $11.379$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \(x^{3} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + ( -\beta_{1} - \beta_{2} ) q^{7} + ( -1 - 2 \beta_{1} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + ( -\beta_{1} - \beta_{2} ) q^{7} + ( -1 - 2 \beta_{1} ) q^{8} + q^{9} + ( -1 - \beta_{1} + \beta_{2} ) q^{11} + ( 2 + \beta_{2} ) q^{12} + 4 q^{13} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{14} + ( 4 + \beta_{1} ) q^{16} + ( -2 - 2 \beta_{2} ) q^{17} -\beta_{1} q^{18} - q^{19} + ( -\beta_{1} - \beta_{2} ) q^{21} + ( 3 - \beta_{1} + \beta_{2} ) q^{22} + ( -3 + \beta_{1} + \beta_{2} ) q^{23} + ( -1 - 2 \beta_{1} ) q^{24} -4 \beta_{1} q^{26} + q^{27} + ( -9 - 5 \beta_{1} ) q^{28} -5 q^{29} + ( 5 + \beta_{1} - \beta_{2} ) q^{31} + ( -2 - \beta_{2} ) q^{32} + ( -1 - \beta_{1} + \beta_{2} ) q^{33} + ( 2 + 6 \beta_{1} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( 4 + 2 \beta_{1} ) q^{37} + \beta_{1} q^{38} + 4 q^{39} + ( 4 - \beta_{1} + \beta_{2} ) q^{41} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{42} + 4 q^{43} + ( 5 - 3 \beta_{1} - \beta_{2} ) q^{44} + ( -5 + \beta_{1} - \beta_{2} ) q^{46} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 4 + \beta_{1} ) q^{48} + ( 7 + 5 \beta_{1} - \beta_{2} ) q^{49} + ( -2 - 2 \beta_{2} ) q^{51} + ( 8 + 4 \beta_{2} ) q^{52} + ( -3 - 4 \beta_{1} ) q^{53} -\beta_{1} q^{54} + ( 10 + 5 \beta_{1} + 3 \beta_{2} ) q^{56} - q^{57} + 5 \beta_{1} q^{58} + ( 4 + 3 \beta_{1} - \beta_{2} ) q^{59} + ( 1 - 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{62} + ( -\beta_{1} - \beta_{2} ) q^{63} + ( -7 + 2 \beta_{1} ) q^{64} + ( 3 - \beta_{1} + \beta_{2} ) q^{66} + ( 5 - \beta_{1} + \beta_{2} ) q^{67} + ( -20 - 2 \beta_{1} - 2 \beta_{2} ) q^{68} + ( -3 + \beta_{1} + \beta_{2} ) q^{69} + ( 6 + 3 \beta_{1} - \beta_{2} ) q^{71} + ( -1 - 2 \beta_{1} ) q^{72} + ( 1 + 4 \beta_{1} ) q^{73} + ( -8 - 4 \beta_{1} - 2 \beta_{2} ) q^{74} + ( -2 - \beta_{2} ) q^{76} + ( -4 + 4 \beta_{2} ) q^{77} -4 \beta_{1} q^{78} + ( 1 - 3 \beta_{1} - 3 \beta_{2} ) q^{79} + q^{81} + ( 3 - 6 \beta_{1} + \beta_{2} ) q^{82} + ( 1 - \beta_{1} + \beta_{2} ) q^{83} + ( -9 - 5 \beta_{1} ) q^{84} -4 \beta_{1} q^{86} -5 q^{87} + ( 7 - \beta_{1} + \beta_{2} ) q^{88} + ( -1 - 2 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{91} + ( 3 + 5 \beta_{1} - 3 \beta_{2} ) q^{92} + ( 5 + \beta_{1} - \beta_{2} ) q^{93} + ( -6 - 2 \beta_{2} ) q^{94} + ( -2 - \beta_{2} ) q^{96} + ( -4 - 4 \beta_{1} + 2 \beta_{2} ) q^{97} + ( -19 - 5 \beta_{1} - 5 \beta_{2} ) q^{98} + ( -1 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 6q^{4} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 6q^{4} - 3q^{8} + 3q^{9} - 3q^{11} + 6q^{12} + 12q^{13} + 15q^{14} + 12q^{16} - 6q^{17} - 3q^{19} + 9q^{22} - 9q^{23} - 3q^{24} + 3q^{27} - 27q^{28} - 15q^{29} + 15q^{31} - 6q^{32} - 3q^{33} + 6q^{34} + 6q^{36} + 12q^{37} + 12q^{39} + 12q^{41} + 15q^{42} + 12q^{43} + 15q^{44} - 15q^{46} + 12q^{47} + 12q^{48} + 21q^{49} - 6q^{51} + 24q^{52} - 9q^{53} + 30q^{56} - 3q^{57} + 12q^{59} + 3q^{61} - 9q^{62} - 21q^{64} + 9q^{66} + 15q^{67} - 60q^{68} - 9q^{69} + 18q^{71} - 3q^{72} + 3q^{73} - 24q^{74} - 6q^{76} - 12q^{77} + 3q^{79} + 3q^{81} + 9q^{82} + 3q^{83} - 27q^{84} - 15q^{87} + 21q^{88} - 3q^{89} + 9q^{92} + 15q^{93} - 18q^{94} - 6q^{96} - 12q^{97} - 57q^{98} - 3q^{99} + O(q^{100}) \)

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

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

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
2.52892
−0.167449
−2.36147
−2.52892 1.00000 4.39543 0 −2.52892 −4.92434 −6.05784 1.00000 0
1.2 0.167449 1.00000 −1.97196 0 0.167449 4.13941 −0.665102 1.00000 0
1.3 2.36147 1.00000 3.57653 0 2.36147 0.784934 3.72294 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(19\) \(1\)

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 1425.2.a.w yes 3
3.b odd 2 1 4275.2.a.bg 3
5.b even 2 1 1425.2.a.t 3
5.c odd 4 2 1425.2.c.o 6
15.d odd 2 1 4275.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1425.2.a.t 3 5.b even 2 1
1425.2.a.w yes 3 1.a even 1 1 trivial
1425.2.c.o 6 5.c odd 4 2
4275.2.a.bf 3 15.d odd 2 1
4275.2.a.bg 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1425))\):

\( T_{2}^{3} - 6 T_{2} + 1 \)
\( T_{7}^{3} - 21 T_{7} + 16 \)
\( T_{11}^{3} + 3 T_{11}^{2} - 12 T_{11} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T + T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( 16 - 21 T + T^{3} \)
$11$ \( -16 - 12 T + 3 T^{2} + T^{3} \)
$13$ \( ( -4 + T )^{3} \)
$17$ \( -208 - 36 T + 6 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( -52 + 6 T + 9 T^{2} + T^{3} \)
$29$ \( ( 5 + T )^{3} \)
$31$ \( -48 + 60 T - 15 T^{2} + T^{3} \)
$37$ \( 24 + 24 T - 12 T^{2} + T^{3} \)
$41$ \( -6 + 33 T - 12 T^{2} + T^{3} \)
$43$ \( ( -4 + T )^{3} \)
$47$ \( 192 - 12 T - 12 T^{2} + T^{3} \)
$53$ \( -197 - 69 T + 9 T^{2} + T^{3} \)
$59$ \( 320 - 9 T - 12 T^{2} + T^{3} \)
$61$ \( 43 - 57 T - 3 T^{2} + T^{3} \)
$67$ \( -52 + 60 T - 15 T^{2} + T^{3} \)
$71$ \( 282 + 51 T - 18 T^{2} + T^{3} \)
$73$ \( 31 - 93 T - 3 T^{2} + T^{3} \)
$79$ \( 620 - 186 T - 3 T^{2} + T^{3} \)
$83$ \( 12 - 12 T - 3 T^{2} + T^{3} \)
$89$ \( 45 - 81 T + 3 T^{2} + T^{3} \)
$97$ \( -904 - 72 T + 12 T^{2} + T^{3} \)
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