Properties

Label 2475.2.a.r
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.73205 0 1.00000 0 0 −2.00000 1.73205 0 0
1.2 1.73205 0 1.00000 0 0 −2.00000 −1.73205 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.r 2
3.b odd 2 1 825.2.a.e 2
5.b even 2 1 495.2.a.c 2
5.c odd 4 2 2475.2.c.n 4
15.d odd 2 1 165.2.a.b 2
15.e even 4 2 825.2.c.c 4
20.d odd 2 1 7920.2.a.bz 2
33.d even 2 1 9075.2.a.bh 2
55.d odd 2 1 5445.2.a.s 2
60.h even 2 1 2640.2.a.x 2
105.g even 2 1 8085.2.a.bd 2
165.d even 2 1 1815.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.b 2 15.d odd 2 1
495.2.a.c 2 5.b even 2 1
825.2.a.e 2 3.b odd 2 1
825.2.c.c 4 15.e even 4 2
1815.2.a.i 2 165.d even 2 1
2475.2.a.r 2 1.a even 1 1 trivial
2475.2.c.n 4 5.c odd 4 2
2640.2.a.x 2 60.h even 2 1
5445.2.a.s 2 55.d odd 2 1
7920.2.a.bz 2 20.d odd 2 1
8085.2.a.bd 2 105.g even 2 1
9075.2.a.bh 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{2} - 3 \)
\( T_{7} + 2 \)
\( T_{29}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -8 + 4 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( -8 - 4 T + T^{2} \)
$23$ \( -48 + T^{2} \)
$29$ \( -12 + T^{2} \)
$31$ \( -32 + 8 T + T^{2} \)
$37$ \( -44 + 4 T + T^{2} \)
$41$ \( -12 + T^{2} \)
$43$ \( -44 + 4 T + T^{2} \)
$47$ \( -48 + T^{2} \)
$53$ \( -12 + 12 T + T^{2} \)
$59$ \( -48 + T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( ( 8 + T )^{2} \)
$71$ \( -192 + T^{2} \)
$73$ \( -104 + 4 T + T^{2} \)
$79$ \( 88 + 20 T + T^{2} \)
$83$ \( 132 - 24 T + T^{2} \)
$89$ \( -12 - 12 T + T^{2} \)
$97$ \( ( -10 + T )^{2} \)
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