Properties

Label 2475.2.a.n
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
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.

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

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
1.61803
−0.618034
−1.61803 0 0.618034 0 0 −2.85410 2.23607 0 0
1.2 0.618034 0 −1.61803 0 0 3.85410 −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.n 2
3.b odd 2 1 275.2.a.g yes 2
5.b even 2 1 2475.2.a.s 2
5.c odd 4 2 2475.2.c.p 4
12.b even 2 1 4400.2.a.bg 2
15.d odd 2 1 275.2.a.d 2
15.e even 4 2 275.2.b.e 4
33.d even 2 1 3025.2.a.i 2
60.h even 2 1 4400.2.a.bv 2
60.l odd 4 2 4400.2.b.x 4
165.d even 2 1 3025.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.d 2 15.d odd 2 1
275.2.a.g yes 2 3.b odd 2 1
275.2.b.e 4 15.e even 4 2
2475.2.a.n 2 1.a even 1 1 trivial
2475.2.a.s 2 5.b even 2 1
2475.2.c.p 4 5.c odd 4 2
3025.2.a.i 2 33.d even 2 1
3025.2.a.m 2 165.d even 2 1
4400.2.a.bg 2 12.b even 2 1
4400.2.a.bv 2 60.h 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{2} + T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 11 \) Copy content Toggle raw display
\( T_{29}^{2} - 5T_{29} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

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