Properties

Label 5625.2.a.f
Level $5625$
Weight $2$
Character orbit 5625.a
Self dual yes
Analytic conductor $44.916$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.9158511370\)
Analytic rank: \(0\)
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 25)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -1 + \beta ) q^{4} + ( -1 + \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( -1 + \beta ) q^{4} + ( -1 + \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + ( 2 + 2 \beta ) q^{11} + ( -3 + 3 \beta ) q^{13} + q^{14} -3 \beta q^{16} + ( 2 + 2 \beta ) q^{17} + ( -4 + 3 \beta ) q^{19} + ( 2 + 4 \beta ) q^{22} + ( -7 + 2 \beta ) q^{23} + 3 q^{26} + ( 2 - \beta ) q^{28} + ( 2 + \beta ) q^{29} -3 q^{31} + ( -5 + \beta ) q^{32} + ( 2 + 4 \beta ) q^{34} + ( 3 - 2 \beta ) q^{37} + ( 3 - \beta ) q^{38} + ( 4 - 2 \beta ) q^{41} -3 \beta q^{43} + 2 \beta q^{44} + ( 2 - 5 \beta ) q^{46} + ( 1 - \beta ) q^{47} + ( -5 - \beta ) q^{49} + ( 6 - 3 \beta ) q^{52} + ( -3 + 4 \beta ) q^{53} + ( -3 + \beta ) q^{56} + ( 1 + 3 \beta ) q^{58} + ( 6 + 3 \beta ) q^{59} + ( -1 + 6 \beta ) q^{61} -3 \beta q^{62} + ( 1 + 2 \beta ) q^{64} + ( 8 - 2 \beta ) q^{67} + 2 \beta q^{68} + ( 5 + \beta ) q^{71} -9 q^{73} + ( -2 + \beta ) q^{74} + ( 7 - 4 \beta ) q^{76} + 2 \beta q^{77} -5 \beta q^{79} + ( -2 + 2 \beta ) q^{82} + ( 3 + 2 \beta ) q^{83} + ( -3 - 3 \beta ) q^{86} + ( -2 - 6 \beta ) q^{88} + ( -4 + 8 \beta ) q^{89} + ( 6 - 3 \beta ) q^{91} + ( 9 - 7 \beta ) q^{92} - q^{94} + ( 1 - 3 \beta ) q^{97} + ( -1 - 6 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - q^{7} + O(q^{10}) \) \( 2 q + q^{2} - q^{4} - q^{7} + 6 q^{11} - 3 q^{13} + 2 q^{14} - 3 q^{16} + 6 q^{17} - 5 q^{19} + 8 q^{22} - 12 q^{23} + 6 q^{26} + 3 q^{28} + 5 q^{29} - 6 q^{31} - 9 q^{32} + 8 q^{34} + 4 q^{37} + 5 q^{38} + 6 q^{41} - 3 q^{43} + 2 q^{44} - q^{46} + q^{47} - 11 q^{49} + 9 q^{52} - 2 q^{53} - 5 q^{56} + 5 q^{58} + 15 q^{59} + 4 q^{61} - 3 q^{62} + 4 q^{64} + 14 q^{67} + 2 q^{68} + 11 q^{71} - 18 q^{73} - 3 q^{74} + 10 q^{76} + 2 q^{77} - 5 q^{79} - 2 q^{82} + 8 q^{83} - 9 q^{86} - 10 q^{88} + 9 q^{91} + 11 q^{92} - 2 q^{94} - q^{97} - 8 q^{98} + O(q^{100}) \)

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
−0.618034
1.61803
−0.618034 0 −1.61803 0 0 −1.61803 2.23607 0 0
1.2 1.61803 0 0.618034 0 0 0.618034 −2.23607 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

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 5625.2.a.f 2
3.b odd 2 1 625.2.a.b 2
5.b even 2 1 5625.2.a.d 2
12.b even 2 1 10000.2.a.c 2
15.d odd 2 1 625.2.a.c 2
15.e even 4 2 625.2.b.a 4
25.d even 5 2 225.2.h.b 4
60.h even 2 1 10000.2.a.l 2
75.h odd 10 2 125.2.d.a 4
75.h odd 10 2 625.2.d.b 4
75.j odd 10 2 25.2.d.a 4
75.j odd 10 2 625.2.d.h 4
75.l even 20 4 125.2.e.a 8
75.l even 20 4 625.2.e.c 8
300.n even 10 2 400.2.u.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 75.j odd 10 2
125.2.d.a 4 75.h odd 10 2
125.2.e.a 8 75.l even 20 4
225.2.h.b 4 25.d even 5 2
400.2.u.b 4 300.n even 10 2
625.2.a.b 2 3.b odd 2 1
625.2.a.c 2 15.d odd 2 1
625.2.b.a 4 15.e even 4 2
625.2.d.b 4 75.h odd 10 2
625.2.d.h 4 75.j odd 10 2
625.2.e.c 8 75.l even 20 4
5625.2.a.d 2 5.b even 2 1
5625.2.a.f 2 1.a even 1 1 trivial
10000.2.a.c 2 12.b even 2 1
10000.2.a.l 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5625))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -1 + T + T^{2} \)
$11$ \( 4 - 6 T + T^{2} \)
$13$ \( -9 + 3 T + T^{2} \)
$17$ \( 4 - 6 T + T^{2} \)
$19$ \( -5 + 5 T + T^{2} \)
$23$ \( 31 + 12 T + T^{2} \)
$29$ \( 5 - 5 T + T^{2} \)
$31$ \( ( 3 + T )^{2} \)
$37$ \( -1 - 4 T + T^{2} \)
$41$ \( 4 - 6 T + T^{2} \)
$43$ \( -9 + 3 T + T^{2} \)
$47$ \( -1 - T + T^{2} \)
$53$ \( -19 + 2 T + T^{2} \)
$59$ \( 45 - 15 T + T^{2} \)
$61$ \( -41 - 4 T + T^{2} \)
$67$ \( 44 - 14 T + T^{2} \)
$71$ \( 29 - 11 T + T^{2} \)
$73$ \( ( 9 + T )^{2} \)
$79$ \( -25 + 5 T + T^{2} \)
$83$ \( 11 - 8 T + T^{2} \)
$89$ \( -80 + T^{2} \)
$97$ \( -11 + T + T^{2} \)
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