Properties

Label 2475.2.a.ba
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} ) q^{7} + ( -1 - \beta_{2} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} ) q^{7} + ( -1 - \beta_{2} ) q^{8} - q^{11} + ( 1 - \beta_{1} - \beta_{2} ) q^{13} + ( -1 - 3 \beta_{1} - \beta_{2} ) q^{14} + ( -1 - 2 \beta_{2} ) q^{16} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{17} + ( -2 - 2 \beta_{2} ) q^{19} + \beta_{1} q^{22} -4 q^{23} + ( 1 + \beta_{1} + \beta_{2} ) q^{26} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{28} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -2 - 2 \beta_{1} ) q^{31} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{32} + ( -5 - 3 \beta_{1} - 3 \beta_{2} ) q^{34} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{37} + ( -2 + 4 \beta_{1} ) q^{38} + ( -2 - 2 \beta_{1} - 6 \beta_{2} ) q^{41} + ( 5 - 3 \beta_{1} - 3 \beta_{2} ) q^{43} + ( -\beta_{1} - \beta_{2} ) q^{44} + 4 \beta_{1} q^{46} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( -3 + 4 \beta_{1} + 2 \beta_{2} ) q^{49} + ( -3 - \beta_{1} + \beta_{2} ) q^{52} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -3 - \beta_{1} - \beta_{2} ) q^{56} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{58} + ( -4 + 4 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -6 - 4 \beta_{1} ) q^{61} + ( 4 + 4 \beta_{1} + 2 \beta_{2} ) q^{62} + ( -2 - 5 \beta_{1} + \beta_{2} ) q^{64} + ( 4 - 4 \beta_{2} ) q^{67} + ( 5 + 5 \beta_{1} + \beta_{2} ) q^{68} + ( 4 + 6 \beta_{2} ) q^{71} + ( -5 + 5 \beta_{1} + \beta_{2} ) q^{73} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{74} + ( -4 - 2 \beta_{1} ) q^{76} + ( -1 - \beta_{1} - \beta_{2} ) q^{77} + ( -2 - 4 \beta_{1} + 6 \beta_{2} ) q^{79} + ( -2 + 10 \beta_{1} + 2 \beta_{2} ) q^{82} + ( 3 + \beta_{1} + 5 \beta_{2} ) q^{83} + ( 3 + \beta_{1} + 3 \beta_{2} ) q^{86} + ( 1 + \beta_{2} ) q^{88} + ( -8 + 6 \beta_{1} ) q^{89} + ( -2 - 2 \beta_{1} ) q^{91} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{92} + ( -6 + 6 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 4 + 4 \beta_{1} + 8 \beta_{2} ) q^{97} + ( -6 - 3 \beta_{1} - 4 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + q^{4} + 4q^{7} - 3q^{8} + O(q^{10}) \) \( 3q - q^{2} + q^{4} + 4q^{7} - 3q^{8} - 3q^{11} + 2q^{13} - 6q^{14} - 3q^{16} - 6q^{19} + q^{22} - 12q^{23} + 4q^{26} + 12q^{28} - 8q^{29} - 8q^{31} + 3q^{32} - 18q^{34} + 4q^{37} - 2q^{38} - 8q^{41} + 12q^{43} - q^{44} + 4q^{46} - 16q^{47} - 5q^{49} - 10q^{52} - 16q^{53} - 10q^{56} + 20q^{58} - 8q^{59} - 22q^{61} + 16q^{62} - 11q^{64} + 12q^{67} + 20q^{68} + 12q^{71} - 10q^{73} + 8q^{74} - 14q^{76} - 4q^{77} - 10q^{79} + 4q^{82} + 10q^{83} + 10q^{86} + 3q^{88} - 18q^{89} - 8q^{91} - 4q^{92} - 12q^{94} + 16q^{97} - 21q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

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

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
2.17009
0.311108
−1.48119
−2.17009 0 2.70928 0 0 3.70928 −1.53919 0 0
1.2 −0.311108 0 −1.90321 0 0 −0.903212 1.21432 0 0
1.3 1.48119 0 0.193937 0 0 1.19394 −2.67513 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.ba 3
3.b odd 2 1 825.2.a.l 3
5.b even 2 1 2475.2.a.bc 3
5.c odd 4 2 495.2.c.e 6
15.d odd 2 1 825.2.a.j 3
15.e even 4 2 165.2.c.b 6
33.d even 2 1 9075.2.a.cg 3
60.l odd 4 2 2640.2.d.h 6
165.d even 2 1 9075.2.a.ch 3
165.l odd 4 2 1815.2.c.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.c.b 6 15.e even 4 2
495.2.c.e 6 5.c odd 4 2
825.2.a.j 3 15.d odd 2 1
825.2.a.l 3 3.b odd 2 1
1815.2.c.e 6 165.l odd 4 2
2475.2.a.ba 3 1.a even 1 1 trivial
2475.2.a.bc 3 5.b even 2 1
2640.2.d.h 6 60.l odd 4 2
9075.2.a.cg 3 33.d even 2 1
9075.2.a.ch 3 165.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{3} + T_{2}^{2} - 3 T_{2} - 1 \)
\( T_{7}^{3} - 4 T_{7}^{2} + 4 \)
\( T_{29}^{3} + 8 T_{29}^{2} - 16 T_{29} - 160 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 3 T + T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( T^{3} \)
$7$ \( 4 - 4 T^{2} + T^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( 4 - 4 T - 2 T^{2} + T^{3} \)
$17$ \( -52 - 28 T + T^{3} \)
$19$ \( -40 - 4 T + 6 T^{2} + T^{3} \)
$23$ \( ( 4 + T )^{3} \)
$29$ \( -160 - 16 T + 8 T^{2} + T^{3} \)
$31$ \( -16 + 8 T + 8 T^{2} + T^{3} \)
$37$ \( 32 - 16 T - 4 T^{2} + T^{3} \)
$41$ \( -928 - 112 T + 8 T^{2} + T^{3} \)
$43$ \( 148 - 12 T^{2} + T^{3} \)
$47$ \( 32 + 48 T + 16 T^{2} + T^{3} \)
$53$ \( 16 + 32 T + 16 T^{2} + T^{3} \)
$59$ \( 80 - 64 T + 8 T^{2} + T^{3} \)
$61$ \( 8 + 108 T + 22 T^{2} + T^{3} \)
$67$ \( 64 - 16 T - 12 T^{2} + T^{3} \)
$71$ \( 944 - 96 T - 12 T^{2} + T^{3} \)
$73$ \( -388 - 44 T + 10 T^{2} + T^{3} \)
$79$ \( -1720 - 212 T + 10 T^{2} + T^{3} \)
$83$ \( 604 - 60 T - 10 T^{2} + T^{3} \)
$89$ \( -520 - 12 T + 18 T^{2} + T^{3} \)
$97$ \( 2432 - 160 T - 16 T^{2} + T^{3} \)
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