Properties

Label 2475.2.a.q
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( 2 + \beta ) q^{4} + q^{7} + ( -4 - \beta ) q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( 2 + \beta ) q^{4} + q^{7} + ( -4 - \beta ) q^{8} + q^{11} + ( -3 + \beta ) q^{13} -\beta q^{14} + 3 \beta q^{16} + ( -4 + \beta ) q^{17} + 3 q^{19} -\beta q^{22} + ( -4 - \beta ) q^{23} + ( -4 + 2 \beta ) q^{26} + ( 2 + \beta ) q^{28} -2 \beta q^{29} + ( -3 + 3 \beta ) q^{31} + ( -4 - \beta ) q^{32} + ( -4 + 3 \beta ) q^{34} + ( 2 - 5 \beta ) q^{37} -3 \beta q^{38} + 3 \beta q^{41} + ( -3 + 3 \beta ) q^{43} + ( 2 + \beta ) q^{44} + ( 4 + 5 \beta ) q^{46} + ( -8 + \beta ) q^{47} -6 q^{49} -2 q^{52} + ( -4 + 2 \beta ) q^{53} + ( -4 - \beta ) q^{56} + ( 8 + 2 \beta ) q^{58} + ( -8 - \beta ) q^{59} + ( -5 - \beta ) q^{61} -12 q^{62} + ( 4 - \beta ) q^{64} + ( 1 - 3 \beta ) q^{67} + ( -4 - \beta ) q^{68} + ( 4 - 5 \beta ) q^{71} + ( 6 + 4 \beta ) q^{73} + ( 20 + 3 \beta ) q^{74} + ( 6 + 3 \beta ) q^{76} + q^{77} + ( -8 - 3 \beta ) q^{79} + ( -12 - 3 \beta ) q^{82} -12 q^{86} + ( -4 - \beta ) q^{88} + ( 12 + 2 \beta ) q^{89} + ( -3 + \beta ) q^{91} + ( -12 - 7 \beta ) q^{92} + ( -4 + 7 \beta ) q^{94} + ( -3 + 2 \beta ) q^{97} + 6 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 5 q^{4} + 2 q^{7} - 9 q^{8} + O(q^{10}) \) \( 2 q - q^{2} + 5 q^{4} + 2 q^{7} - 9 q^{8} + 2 q^{11} - 5 q^{13} - q^{14} + 3 q^{16} - 7 q^{17} + 6 q^{19} - q^{22} - 9 q^{23} - 6 q^{26} + 5 q^{28} - 2 q^{29} - 3 q^{31} - 9 q^{32} - 5 q^{34} - q^{37} - 3 q^{38} + 3 q^{41} - 3 q^{43} + 5 q^{44} + 13 q^{46} - 15 q^{47} - 12 q^{49} - 4 q^{52} - 6 q^{53} - 9 q^{56} + 18 q^{58} - 17 q^{59} - 11 q^{61} - 24 q^{62} + 7 q^{64} - q^{67} - 9 q^{68} + 3 q^{71} + 16 q^{73} + 43 q^{74} + 15 q^{76} + 2 q^{77} - 19 q^{79} - 27 q^{82} - 24 q^{86} - 9 q^{88} + 26 q^{89} - 5 q^{91} - 31 q^{92} - q^{94} - 4 q^{97} + 6 q^{98} + O(q^{100}) \)

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
2.56155
−1.56155
−2.56155 0 4.56155 0 0 1.00000 −6.56155 0 0
1.2 1.56155 0 0.438447 0 0 1.00000 −2.43845 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.q yes 2
3.b odd 2 1 2475.2.a.v yes 2
5.b even 2 1 2475.2.a.u yes 2
5.c odd 4 2 2475.2.c.j 4
15.d odd 2 1 2475.2.a.p 2
15.e even 4 2 2475.2.c.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.2.a.p 2 15.d odd 2 1
2475.2.a.q yes 2 1.a even 1 1 trivial
2475.2.a.u yes 2 5.b even 2 1
2475.2.a.v yes 2 3.b odd 2 1
2475.2.c.i 4 15.e even 4 2
2475.2.c.j 4 5.c odd 4 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( 2 + 5 T + T^{2} \)
$17$ \( 8 + 7 T + T^{2} \)
$19$ \( ( -3 + T )^{2} \)
$23$ \( 16 + 9 T + T^{2} \)
$29$ \( -16 + 2 T + T^{2} \)
$31$ \( -36 + 3 T + T^{2} \)
$37$ \( -106 + T + T^{2} \)
$41$ \( -36 - 3 T + T^{2} \)
$43$ \( -36 + 3 T + T^{2} \)
$47$ \( 52 + 15 T + T^{2} \)
$53$ \( -8 + 6 T + T^{2} \)
$59$ \( 68 + 17 T + T^{2} \)
$61$ \( 26 + 11 T + T^{2} \)
$67$ \( -38 + T + T^{2} \)
$71$ \( -104 - 3 T + T^{2} \)
$73$ \( -4 - 16 T + T^{2} \)
$79$ \( 52 + 19 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( 152 - 26 T + T^{2} \)
$97$ \( -13 + 4 T + T^{2} \)
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