Properties

Label 625.2.a.d
Level $625$
Weight $2$
Character orbit 625.a
Self dual yes
Analytic conductor $4.991$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.99065012633\)
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: yes
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta + 1) q^{3} + (\beta - 1) q^{4} + (2 \beta + 1) q^{6} + (3 \beta - 1) q^{7} + ( - 2 \beta + 1) q^{8} + (3 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta + 1) q^{3} + (\beta - 1) q^{4} + (2 \beta + 1) q^{6} + (3 \beta - 1) q^{7} + ( - 2 \beta + 1) q^{8} + (3 \beta - 1) q^{9} + (\beta - 1) q^{11} + \beta q^{12} + ( - 4 \beta + 1) q^{13} + (2 \beta + 3) q^{14} - 3 \beta q^{16} + ( - 4 \beta + 5) q^{17} + (2 \beta + 3) q^{18} + ( - 3 \beta + 4) q^{19} + (5 \beta + 2) q^{21} + q^{22} + ( - 3 \beta + 3) q^{23} + ( - 3 \beta - 1) q^{24} + ( - 3 \beta - 4) q^{26} + (2 \beta - 1) q^{27} + ( - \beta + 4) q^{28} + (2 \beta - 6) q^{29} + 2 q^{31} + (\beta - 5) q^{32} + \beta q^{33} + (\beta - 4) q^{34} + ( - \beta + 4) q^{36} + 3 q^{37} + (\beta - 3) q^{38} + ( - 7 \beta - 3) q^{39} + 4 \beta q^{41} + (7 \beta + 5) q^{42} + ( - 4 \beta + 6) q^{43} + ( - \beta + 2) q^{44} - 3 q^{46} + (\beta + 10) q^{47} + ( - 6 \beta - 3) q^{48} + (3 \beta + 3) q^{49} + ( - 3 \beta + 1) q^{51} + (\beta - 5) q^{52} + (4 \beta + 2) q^{53} + (\beta + 2) q^{54} + ( - \beta - 7) q^{56} + ( - 2 \beta + 1) q^{57} + ( - 4 \beta + 2) q^{58} + ( - 8 \beta - 1) q^{59} + ( - 3 \beta - 4) q^{61} + 2 \beta q^{62} + (3 \beta + 10) q^{63} + (2 \beta + 1) q^{64} + (\beta + 1) q^{66} + ( - 4 \beta - 5) q^{67} + (5 \beta - 9) q^{68} - 3 \beta q^{69} + ( - 7 \beta + 8) q^{71} + ( - \beta - 7) q^{72} + (3 \beta - 5) q^{73} + 3 \beta q^{74} + (4 \beta - 7) q^{76} + ( - \beta + 4) q^{77} + ( - 10 \beta - 7) q^{78} + ( - 6 \beta + 3) q^{79} + ( - 6 \beta + 4) q^{81} + (4 \beta + 4) q^{82} + 8 \beta q^{83} + (2 \beta + 3) q^{84} + (2 \beta - 4) q^{86} + ( - 2 \beta - 4) q^{87} + (\beta - 3) q^{88} + (\beta + 2) q^{89} + ( - 5 \beta - 13) q^{91} + (3 \beta - 6) q^{92} + (2 \beta + 2) q^{93} + (11 \beta + 1) q^{94} + ( - 3 \beta - 4) q^{96} + ( - \beta - 9) q^{97} + (6 \beta + 3) q^{98} + ( - \beta + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} - q^{4} + 4 q^{6} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 3 q^{3} - q^{4} + 4 q^{6} + q^{7} + q^{9} - q^{11} + q^{12} - 2 q^{13} + 8 q^{14} - 3 q^{16} + 6 q^{17} + 8 q^{18} + 5 q^{19} + 9 q^{21} + 2 q^{22} + 3 q^{23} - 5 q^{24} - 11 q^{26} + 7 q^{28} - 10 q^{29} + 4 q^{31} - 9 q^{32} + q^{33} - 7 q^{34} + 7 q^{36} + 6 q^{37} - 5 q^{38} - 13 q^{39} + 4 q^{41} + 17 q^{42} + 8 q^{43} + 3 q^{44} - 6 q^{46} + 21 q^{47} - 12 q^{48} + 9 q^{49} - q^{51} - 9 q^{52} + 8 q^{53} + 5 q^{54} - 15 q^{56} - 10 q^{59} - 11 q^{61} + 2 q^{62} + 23 q^{63} + 4 q^{64} + 3 q^{66} - 14 q^{67} - 13 q^{68} - 3 q^{69} + 9 q^{71} - 15 q^{72} - 7 q^{73} + 3 q^{74} - 10 q^{76} + 7 q^{77} - 24 q^{78} + 2 q^{81} + 12 q^{82} + 8 q^{83} + 8 q^{84} - 6 q^{86} - 10 q^{87} - 5 q^{88} + 5 q^{89} - 31 q^{91} - 9 q^{92} + 6 q^{93} + 13 q^{94} - 11 q^{96} - 19 q^{97} + 12 q^{98} + 7 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
−0.618034 0.381966 −1.61803 0 −0.236068 −2.85410 2.23607 −2.85410 0
1.2 1.61803 2.61803 0.618034 0 4.23607 3.85410 −2.23607 3.85410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 625.2.a.d yes 2
3.b odd 2 1 5625.2.a.c 2
4.b odd 2 1 10000.2.a.b 2
5.b even 2 1 625.2.a.a 2
5.c odd 4 2 625.2.b.b 4
15.d odd 2 1 5625.2.a.e 2
20.d odd 2 1 10000.2.a.m 2
25.d even 5 2 625.2.d.c 4
25.d even 5 2 625.2.d.f 4
25.e even 10 2 625.2.d.e 4
25.e even 10 2 625.2.d.i 4
25.f odd 20 4 625.2.e.e 8
25.f odd 20 4 625.2.e.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.a 2 5.b even 2 1
625.2.a.d yes 2 1.a even 1 1 trivial
625.2.b.b 4 5.c odd 4 2
625.2.d.c 4 25.d even 5 2
625.2.d.e 4 25.e even 10 2
625.2.d.f 4 25.d even 5 2
625.2.d.i 4 25.e even 10 2
625.2.e.e 8 25.f odd 20 4
625.2.e.f 8 25.f odd 20 4
5625.2.a.c 2 3.b odd 2 1
5625.2.a.e 2 15.d odd 2 1
10000.2.a.b 2 4.b odd 2 1
10000.2.a.m 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(625))\):

\( T_{2}^{2} - T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 3T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 1 \) 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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