Properties

Label 5625.2.a.c
Level $5625$
Weight $2$
Character orbit 5625.a
Self dual yes
Analytic conductor $44.916$
Analytic rank $1$
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: \(1\)
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 625)
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.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta - 1) q^{4} + (3 \beta - 1) q^{7} + (2 \beta - 1) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta - 1) q^{4} + (3 \beta - 1) q^{7} + (2 \beta - 1) q^{8} + ( - \beta + 1) q^{11} + ( - 4 \beta + 1) q^{13} + ( - 2 \beta - 3) q^{14} - 3 \beta q^{16} + (4 \beta - 5) q^{17} + ( - 3 \beta + 4) q^{19} + q^{22} + (3 \beta - 3) q^{23} + (3 \beta + 4) q^{26} + ( - \beta + 4) q^{28} + ( - 2 \beta + 6) q^{29} + 2 q^{31} + ( - \beta + 5) q^{32} + (\beta - 4) q^{34} + 3 q^{37} + ( - \beta + 3) q^{38} - 4 \beta q^{41} + ( - 4 \beta + 6) q^{43} + (\beta - 2) q^{44} - 3 q^{46} + ( - \beta - 10) q^{47} + (3 \beta + 3) q^{49} + (\beta - 5) q^{52} + ( - 4 \beta - 2) q^{53} + (\beta + 7) q^{56} + ( - 4 \beta + 2) q^{58} + (8 \beta + 1) q^{59} + ( - 3 \beta - 4) q^{61} - 2 \beta q^{62} + (2 \beta + 1) q^{64} + ( - 4 \beta - 5) q^{67} + ( - 5 \beta + 9) q^{68} + (7 \beta - 8) q^{71} + (3 \beta - 5) q^{73} - 3 \beta q^{74} + (4 \beta - 7) q^{76} + (\beta - 4) q^{77} + ( - 6 \beta + 3) q^{79} + (4 \beta + 4) q^{82} - 8 \beta q^{83} + ( - 2 \beta + 4) q^{86} + (\beta - 3) q^{88} + ( - \beta - 2) q^{89} + ( - 5 \beta - 13) q^{91} + ( - 3 \beta + 6) q^{92} + (11 \beta + 1) q^{94} + ( - \beta - 9) q^{97} + ( - 6 \beta - 3) q^{98} +O(q^{100}) \) 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 \( 2 q - q^{2} - q^{4} + q^{7} + q^{11} - 2 q^{13} - 8 q^{14} - 3 q^{16} - 6 q^{17} + 5 q^{19} + 2 q^{22} - 3 q^{23} + 11 q^{26} + 7 q^{28} + 10 q^{29} + 4 q^{31} + 9 q^{32} - 7 q^{34} + 6 q^{37} + 5 q^{38} - 4 q^{41} + 8 q^{43} - 3 q^{44} - 6 q^{46} - 21 q^{47} + 9 q^{49} - 9 q^{52} - 8 q^{53} + 15 q^{56} + 10 q^{59} - 11 q^{61} - 2 q^{62} + 4 q^{64} - 14 q^{67} + 13 q^{68} - 9 q^{71} - 7 q^{73} - 3 q^{74} - 10 q^{76} - 7 q^{77} + 12 q^{82} - 8 q^{83} + 6 q^{86} - 5 q^{88} - 5 q^{89} - 31 q^{91} + 9 q^{92} + 13 q^{94} - 19 q^{97} - 12 q^{98}+O(q^{100}) \) 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 3.85410 2.23607 0 0
1.2 0.618034 0 −1.61803 0 0 −2.85410 −2.23607 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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.c 2
3.b odd 2 1 625.2.a.d yes 2
5.b even 2 1 5625.2.a.e 2
12.b even 2 1 10000.2.a.b 2
15.d odd 2 1 625.2.a.a 2
15.e even 4 2 625.2.b.b 4
60.h even 2 1 10000.2.a.m 2
75.h odd 10 2 625.2.d.e 4
75.h odd 10 2 625.2.d.i 4
75.j odd 10 2 625.2.d.c 4
75.j odd 10 2 625.2.d.f 4
75.l even 20 4 625.2.e.e 8
75.l even 20 4 625.2.e.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.a 2 15.d odd 2 1
625.2.a.d yes 2 3.b odd 2 1
625.2.b.b 4 15.e even 4 2
625.2.d.c 4 75.j odd 10 2
625.2.d.e 4 75.h odd 10 2
625.2.d.f 4 75.j odd 10 2
625.2.d.i 4 75.h odd 10 2
625.2.e.e 8 75.l even 20 4
625.2.e.f 8 75.l even 20 4
5625.2.a.c 2 1.a even 1 1 trivial
5625.2.a.e 2 5.b even 2 1
10000.2.a.b 2 12.b even 2 1
10000.2.a.m 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5625))\):

\( T_{2}^{2} + T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 11 \) 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
$11$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T - 5 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T - 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 21T + 109 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 10T - 55 \) Copy content Toggle raw display
$61$ \( T^{2} + 11T + 19 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 29 \) Copy content Toggle raw display
$71$ \( T^{2} + 9T - 41 \) Copy content Toggle raw display
$73$ \( T^{2} + 7T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} - 45 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$89$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$97$ \( T^{2} + 19T + 89 \) Copy content Toggle raw display
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