Properties

Label 2475.2.a.bd
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 3 + \beta_{2} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} ) q^{7} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 3 + \beta_{2} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} ) q^{7} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{8} - q^{11} + ( 2 + \beta_{2} ) q^{13} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{14} + ( 5 - 2 \beta_{1} + 3 \beta_{2} ) q^{16} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} ) q^{19} + ( -1 + \beta_{1} ) q^{22} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{23} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{26} + ( 7 + \beta_{1} + 2 \beta_{2} ) q^{28} -2 \beta_{1} q^{29} + ( 6 + \beta_{2} ) q^{31} + ( 13 - \beta_{1} + 4 \beta_{2} ) q^{32} + ( 1 - 3 \beta_{2} ) q^{34} + ( -1 - 3 \beta_{2} ) q^{37} + ( -7 - 3 \beta_{2} ) q^{38} + ( -1 - \beta_{1} ) q^{41} + ( 4 + 2 \beta_{1} - 3 \beta_{2} ) q^{43} + ( -3 - \beta_{2} ) q^{44} + ( -11 - \beta_{1} - 4 \beta_{2} ) q^{46} + ( 3 - \beta_{2} ) q^{47} + 2 \beta_{1} q^{49} + ( 12 - 2 \beta_{1} + 4 \beta_{2} ) q^{52} + ( 2 + 4 \beta_{1} + 4 \beta_{2} ) q^{53} + ( 9 - 4 \beta_{1} + \beta_{2} ) q^{56} + ( 8 + 2 \beta_{1} + 2 \beta_{2} ) q^{58} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{59} + ( -4 \beta_{1} - \beta_{2} ) q^{61} + ( 8 - 6 \beta_{1} + 2 \beta_{2} ) q^{62} + ( 15 - 8 \beta_{1} + 3 \beta_{2} ) q^{64} + ( 4 - 6 \beta_{1} - \beta_{2} ) q^{67} + ( -7 + \beta_{1} - 2 \beta_{2} ) q^{68} + ( -9 - \beta_{2} ) q^{71} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -7 + \beta_{1} - 6 \beta_{2} ) q^{74} + ( -11 + 5 \beta_{1} - 4 \beta_{2} ) q^{76} + ( -1 - \beta_{1} - \beta_{2} ) q^{77} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{79} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{82} + ( -4 + 6 \beta_{1} ) q^{83} + ( -10 - 6 \beta_{1} - 8 \beta_{2} ) q^{86} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{88} + ( -2 + 4 \beta_{1} ) q^{89} + ( 6 + \beta_{2} ) q^{91} + ( -13 + 8 \beta_{1} - 5 \beta_{2} ) q^{92} + ( 1 - 3 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 11 - 6 \beta_{1} - 2 \beta_{2} ) q^{97} + ( -8 - 2 \beta_{1} - 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{2} + 8q^{4} + 3q^{7} + 6q^{8} + O(q^{10}) \) \( 3q + 2q^{2} + 8q^{4} + 3q^{7} + 6q^{8} - 3q^{11} + 5q^{13} - 6q^{14} + 10q^{16} + 4q^{17} - q^{19} - 2q^{22} + 8q^{26} + 20q^{28} - 2q^{29} + 17q^{31} + 34q^{32} + 6q^{34} - 18q^{38} - 4q^{41} + 17q^{43} - 8q^{44} - 30q^{46} + 10q^{47} + 2q^{49} + 30q^{52} + 6q^{53} + 22q^{56} + 24q^{58} + 6q^{59} - 3q^{61} + 16q^{62} + 34q^{64} + 7q^{67} - 18q^{68} - 26q^{71} - 10q^{73} - 14q^{74} - 24q^{76} - 3q^{77} + 6q^{79} + 10q^{82} - 6q^{83} - 28q^{86} - 6q^{88} - 2q^{89} + 17q^{91} - 26q^{92} + 2q^{94} + 29q^{97} - 24q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x - 2\):

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

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
3.12489
−0.363328
−1.76156
−2.12489 0 2.51514 0 0 3.64002 −1.09461 0 0
1.2 1.36333 0 −0.141336 0 0 −2.50466 −2.91934 0 0
1.3 2.76156 0 5.62620 0 0 1.86464 10.0140 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

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 2475.2.a.bd 3
3.b odd 2 1 825.2.a.i 3
5.b even 2 1 2475.2.a.z 3
5.c odd 4 2 2475.2.c.q 6
15.d odd 2 1 825.2.a.m yes 3
15.e even 4 2 825.2.c.f 6
33.d even 2 1 9075.2.a.cj 3
165.d even 2 1 9075.2.a.cd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.i 3 3.b odd 2 1
825.2.a.m yes 3 15.d odd 2 1
825.2.c.f 6 15.e even 4 2
2475.2.a.z 3 5.b even 2 1
2475.2.a.bd 3 1.a even 1 1 trivial
2475.2.c.q 6 5.c odd 4 2
9075.2.a.cd 3 165.d even 2 1
9075.2.a.cj 3 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{3} - 2 T_{2}^{2} - 5 T_{2} + 8 \)
\( T_{7}^{3} - 3 T_{7}^{2} - 7 T_{7} + 17 \)
\( T_{29}^{3} + 2 T_{29}^{2} - 24 T_{29} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 - 5 T - 2 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( T^{3} \)
$7$ \( 17 - 7 T - 3 T^{2} + T^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( 8 - 5 T^{2} + T^{3} \)
$17$ \( -22 - 25 T - 4 T^{2} + T^{3} \)
$19$ \( 25 - 19 T + T^{2} + T^{3} \)
$23$ \( 58 - 43 T + T^{3} \)
$29$ \( 16 - 24 T + 2 T^{2} + T^{3} \)
$31$ \( -136 + 88 T - 17 T^{2} + T^{3} \)
$37$ \( 34 - 75 T + T^{3} \)
$41$ \( -2 - T + 4 T^{2} + T^{3} \)
$43$ \( 1100 - 32 T - 17 T^{2} + T^{3} \)
$47$ \( -8 + 25 T - 10 T^{2} + T^{3} \)
$53$ \( 824 - 148 T - 6 T^{2} + T^{3} \)
$59$ \( 136 - 31 T - 6 T^{2} + T^{3} \)
$61$ \( 244 - 88 T + 3 T^{2} + T^{3} \)
$67$ \( 1588 - 192 T - 7 T^{2} + T^{3} \)
$71$ \( 580 + 217 T + 26 T^{2} + T^{3} \)
$73$ \( -472 - 44 T + 10 T^{2} + T^{3} \)
$79$ \( 212 - 37 T - 6 T^{2} + T^{3} \)
$83$ \( -1328 - 216 T + 6 T^{2} + T^{3} \)
$89$ \( -328 - 100 T + 2 T^{2} + T^{3} \)
$97$ \( 2153 + 75 T - 29 T^{2} + T^{3} \)
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