Properties

Label 2475.2.a.s
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -1 + \beta ) q^{4} + ( -2 + 3 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( -1 + \beta ) q^{4} + ( -2 + 3 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} - q^{11} + ( -3 - 2 \beta ) q^{13} + ( 3 + \beta ) q^{14} -3 \beta q^{16} + ( 1 - \beta ) q^{17} + ( 3 - 6 \beta ) q^{19} -\beta q^{22} + ( 4 - 5 \beta ) q^{23} + ( -2 - 5 \beta ) q^{26} + ( 5 - 2 \beta ) q^{28} + ( 3 - \beta ) q^{29} -3 q^{31} + ( -5 + \beta ) q^{32} - q^{34} + ( -7 - 2 \beta ) q^{37} + ( -6 - 3 \beta ) q^{38} + 3 q^{41} + 6 q^{43} + ( 1 - \beta ) q^{44} + ( -5 - \beta ) q^{46} + ( -1 + 8 \beta ) q^{47} + ( 6 - 3 \beta ) q^{49} + ( 1 - 3 \beta ) q^{52} + ( -2 + 7 \beta ) q^{53} + ( -8 + \beta ) q^{56} + ( -1 + 2 \beta ) q^{58} + ( -7 + 4 \beta ) q^{59} + ( -8 + 5 \beta ) q^{61} -3 \beta q^{62} + ( 1 + 2 \beta ) q^{64} -8 q^{67} + ( -2 + \beta ) q^{68} + ( 8 - 10 \beta ) q^{71} + ( -12 + \beta ) q^{73} + ( -2 - 9 \beta ) q^{74} + ( -9 + 3 \beta ) q^{76} + ( 2 - 3 \beta ) q^{77} + ( 1 + 3 \beta ) q^{79} + 3 \beta q^{82} + ( -15 + 3 \beta ) q^{83} + 6 \beta q^{86} + ( -1 + 2 \beta ) q^{88} + ( 15 - 5 \beta ) q^{89} -11 \beta q^{91} + ( -9 + 4 \beta ) q^{92} + ( 8 + 7 \beta ) q^{94} -\beta q^{97} + ( -3 + 3 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - q^{7} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - q^{7} - 2q^{11} - 8q^{13} + 7q^{14} - 3q^{16} + q^{17} - q^{22} + 3q^{23} - 9q^{26} + 8q^{28} + 5q^{29} - 6q^{31} - 9q^{32} - 2q^{34} - 16q^{37} - 15q^{38} + 6q^{41} + 12q^{43} + q^{44} - 11q^{46} + 6q^{47} + 9q^{49} - q^{52} + 3q^{53} - 15q^{56} - 10q^{59} - 11q^{61} - 3q^{62} + 4q^{64} - 16q^{67} - 3q^{68} + 6q^{71} - 23q^{73} - 13q^{74} - 15q^{76} + q^{77} + 5q^{79} + 3q^{82} - 27q^{83} + 6q^{86} + 25q^{89} - 11q^{91} - 14q^{92} + 23q^{94} - q^{97} - 3q^{98} + O(q^{100}) \)

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
−0.618034
1.61803
−0.618034 0 −1.61803 0 0 −3.85410 2.23607 0 0
1.2 1.61803 0 0.618034 0 0 2.85410 −2.23607 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.s 2
3.b odd 2 1 275.2.a.d 2
5.b even 2 1 2475.2.a.n 2
5.c odd 4 2 2475.2.c.p 4
12.b even 2 1 4400.2.a.bv 2
15.d odd 2 1 275.2.a.g yes 2
15.e even 4 2 275.2.b.e 4
33.d even 2 1 3025.2.a.m 2
60.h even 2 1 4400.2.a.bg 2
60.l odd 4 2 4400.2.b.x 4
165.d even 2 1 3025.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.d 2 3.b odd 2 1
275.2.a.g yes 2 15.d odd 2 1
275.2.b.e 4 15.e even 4 2
2475.2.a.n 2 5.b even 2 1
2475.2.a.s 2 1.a even 1 1 trivial
2475.2.c.p 4 5.c odd 4 2
3025.2.a.i 2 165.d even 2 1
3025.2.a.m 2 33.d even 2 1
4400.2.a.bg 2 60.h even 2 1
4400.2.a.bv 2 12.b even 2 1
4400.2.b.x 4 60.l odd 4 2

Hecke kernels

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

\( T_{2}^{2} - T_{2} - 1 \)
\( T_{7}^{2} + T_{7} - 11 \)
\( T_{29}^{2} - 5 T_{29} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -11 + T + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 11 + 8 T + T^{2} \)
$17$ \( -1 - T + T^{2} \)
$19$ \( -45 + T^{2} \)
$23$ \( -29 - 3 T + T^{2} \)
$29$ \( 5 - 5 T + T^{2} \)
$31$ \( ( 3 + T )^{2} \)
$37$ \( 59 + 16 T + T^{2} \)
$41$ \( ( -3 + T )^{2} \)
$43$ \( ( -6 + T )^{2} \)
$47$ \( -71 - 6 T + T^{2} \)
$53$ \( -59 - 3 T + T^{2} \)
$59$ \( 5 + 10 T + T^{2} \)
$61$ \( -1 + 11 T + T^{2} \)
$67$ \( ( 8 + T )^{2} \)
$71$ \( -116 - 6 T + T^{2} \)
$73$ \( 131 + 23 T + T^{2} \)
$79$ \( -5 - 5 T + T^{2} \)
$83$ \( 171 + 27 T + T^{2} \)
$89$ \( 125 - 25 T + T^{2} \)
$97$ \( -1 + T + T^{2} \)
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