Properties

Label 1425.2.a.u
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: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
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.

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

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

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

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
−2.14510
−0.523976
2.66908
−2.14510 −1.00000 2.60147 0 2.14510 −2.74657 −1.29021 1.00000 0
1.2 −0.523976 −1.00000 −1.72545 0 0.523976 3.20147 1.95205 1.00000 0
1.3 2.66908 −1.00000 5.12398 0 −2.66908 −0.454904 8.33816 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.u 3
3.b odd 2 1 4275.2.a.be 3
5.b even 2 1 1425.2.a.v yes 3
5.c odd 4 2 1425.2.c.p 6
15.d odd 2 1 4275.2.a.bh 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1425.2.a.u 3 1.a even 1 1 trivial
1425.2.a.v yes 3 5.b even 2 1
1425.2.c.p 6 5.c odd 4 2
4275.2.a.be 3 3.b odd 2 1
4275.2.a.bh 3 15.d odd 2 1

Hecke kernels

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

\( T_{2}^{3} - 6 T_{2} - 3 \)
\( T_{7}^{3} - 9 T_{7} - 4 \)
\( T_{11}^{3} + 3 T_{11}^{2} - 6 T_{11} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 - 6 T + T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( -4 - 9 T + T^{3} \)
$11$ \( -12 - 6 T + 3 T^{2} + T^{3} \)
$13$ \( -64 - 12 T + 6 T^{2} + T^{3} \)
$17$ \( T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -12 - 24 T - 3 T^{2} + T^{3} \)
$29$ \( ( -3 + T )^{3} \)
$31$ \( 16 - 9 T^{2} + T^{3} \)
$37$ \( -184 + 12 T^{2} + T^{3} \)
$41$ \( 426 - 123 T + T^{3} \)
$43$ \( ( 8 + T )^{3} \)
$47$ \( -96 - 60 T - 6 T^{2} + T^{3} \)
$53$ \( -69 - 69 T + 9 T^{2} + T^{3} \)
$59$ \( -444 + 183 T - 24 T^{2} + T^{3} \)
$61$ \( -113 - 9 T + 9 T^{2} + T^{3} \)
$67$ \( 92 - 66 T + 9 T^{2} + T^{3} \)
$71$ \( 522 - 81 T - 6 T^{2} + T^{3} \)
$73$ \( -97 - 93 T - 3 T^{2} + T^{3} \)
$79$ \( 916 - 48 T - 15 T^{2} + T^{3} \)
$83$ \( -84 + 66 T - 15 T^{2} + T^{3} \)
$89$ \( -3 - 33 T + 3 T^{2} + T^{3} \)
$97$ \( 1448 - 168 T - 12 T^{2} + T^{3} \)
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