Properties

Label 625.2.a.b
Level $625$
Weight $2$
Character orbit 625.a
Self dual yes
Analytic conductor $4.991$
Analytic rank $1$
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: \(1\)
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 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.

\(f(q)\) \(=\) \( q -\beta q^{2} + q^{3} + ( -1 + \beta ) q^{4} -\beta q^{6} + ( -1 + \beta ) q^{7} + ( -1 + 2 \beta ) q^{8} -2 q^{9} +O(q^{10})\) \( q -\beta q^{2} + q^{3} + ( -1 + \beta ) q^{4} -\beta q^{6} + ( -1 + \beta ) q^{7} + ( -1 + 2 \beta ) q^{8} -2 q^{9} + ( -2 - 2 \beta ) q^{11} + ( -1 + \beta ) q^{12} + ( -3 + 3 \beta ) q^{13} - q^{14} -3 \beta q^{16} + ( -2 - 2 \beta ) q^{17} + 2 \beta q^{18} + ( -4 + 3 \beta ) q^{19} + ( -1 + \beta ) q^{21} + ( 2 + 4 \beta ) q^{22} + ( 7 - 2 \beta ) q^{23} + ( -1 + 2 \beta ) q^{24} -3 q^{26} -5 q^{27} + ( 2 - \beta ) q^{28} + ( -2 - \beta ) q^{29} -3 q^{31} + ( 5 - \beta ) q^{32} + ( -2 - 2 \beta ) q^{33} + ( 2 + 4 \beta ) q^{34} + ( 2 - 2 \beta ) q^{36} + ( 3 - 2 \beta ) q^{37} + ( -3 + \beta ) q^{38} + ( -3 + 3 \beta ) q^{39} + ( -4 + 2 \beta ) q^{41} - q^{42} -3 \beta q^{43} -2 \beta q^{44} + ( 2 - 5 \beta ) q^{46} + ( -1 + \beta ) q^{47} -3 \beta q^{48} + ( -5 - \beta ) q^{49} + ( -2 - 2 \beta ) q^{51} + ( 6 - 3 \beta ) q^{52} + ( 3 - 4 \beta ) q^{53} + 5 \beta q^{54} + ( 3 - \beta ) q^{56} + ( -4 + 3 \beta ) q^{57} + ( 1 + 3 \beta ) q^{58} + ( -6 - 3 \beta ) q^{59} + ( -1 + 6 \beta ) q^{61} + 3 \beta q^{62} + ( 2 - 2 \beta ) q^{63} + ( 1 + 2 \beta ) q^{64} + ( 2 + 4 \beta ) q^{66} + ( 8 - 2 \beta ) q^{67} -2 \beta q^{68} + ( 7 - 2 \beta ) q^{69} + ( -5 - \beta ) q^{71} + ( 2 - 4 \beta ) q^{72} -9 q^{73} + ( 2 - \beta ) q^{74} + ( 7 - 4 \beta ) q^{76} -2 \beta q^{77} -3 q^{78} -5 \beta q^{79} + q^{81} + ( -2 + 2 \beta ) q^{82} + ( -3 - 2 \beta ) q^{83} + ( 2 - \beta ) q^{84} + ( 3 + 3 \beta ) q^{86} + ( -2 - \beta ) q^{87} + ( -2 - 6 \beta ) q^{88} + ( 4 - 8 \beta ) q^{89} + ( 6 - 3 \beta ) q^{91} + ( -9 + 7 \beta ) q^{92} -3 q^{93} - q^{94} + ( 5 - \beta ) q^{96} + ( 1 - 3 \beta ) q^{97} + ( 1 + 6 \beta ) q^{98} + ( 4 + 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 2q^{3} - q^{4} - q^{6} - q^{7} - 4q^{9} + O(q^{10}) \) \( 2q - q^{2} + 2q^{3} - q^{4} - q^{6} - q^{7} - 4q^{9} - 6q^{11} - q^{12} - 3q^{13} - 2q^{14} - 3q^{16} - 6q^{17} + 2q^{18} - 5q^{19} - q^{21} + 8q^{22} + 12q^{23} - 6q^{26} - 10q^{27} + 3q^{28} - 5q^{29} - 6q^{31} + 9q^{32} - 6q^{33} + 8q^{34} + 2q^{36} + 4q^{37} - 5q^{38} - 3q^{39} - 6q^{41} - 2q^{42} - 3q^{43} - 2q^{44} - q^{46} - q^{47} - 3q^{48} - 11q^{49} - 6q^{51} + 9q^{52} + 2q^{53} + 5q^{54} + 5q^{56} - 5q^{57} + 5q^{58} - 15q^{59} + 4q^{61} + 3q^{62} + 2q^{63} + 4q^{64} + 8q^{66} + 14q^{67} - 2q^{68} + 12q^{69} - 11q^{71} - 18q^{73} + 3q^{74} + 10q^{76} - 2q^{77} - 6q^{78} - 5q^{79} + 2q^{81} - 2q^{82} - 8q^{83} + 3q^{84} + 9q^{86} - 5q^{87} - 10q^{88} + 9q^{91} - 11q^{92} - 6q^{93} - 2q^{94} + 9q^{96} - q^{97} + 8q^{98} + 12q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
−1.61803 1.00000 0.618034 0 −1.61803 0.618034 2.23607 −2.00000 0
1.2 0.618034 1.00000 −1.61803 0 0.618034 −1.61803 −2.23607 −2.00000 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.b 2
3.b odd 2 1 5625.2.a.f 2
4.b odd 2 1 10000.2.a.c 2
5.b even 2 1 625.2.a.c 2
5.c odd 4 2 625.2.b.a 4
15.d odd 2 1 5625.2.a.d 2
20.d odd 2 1 10000.2.a.l 2
25.d even 5 2 25.2.d.a 4
25.d even 5 2 625.2.d.h 4
25.e even 10 2 125.2.d.a 4
25.e even 10 2 625.2.d.b 4
25.f odd 20 4 125.2.e.a 8
25.f odd 20 4 625.2.e.c 8
75.j odd 10 2 225.2.h.b 4
100.j odd 10 2 400.2.u.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 25.d even 5 2
125.2.d.a 4 25.e even 10 2
125.2.e.a 8 25.f odd 20 4
225.2.h.b 4 75.j odd 10 2
400.2.u.b 4 100.j odd 10 2
625.2.a.b 2 1.a even 1 1 trivial
625.2.a.c 2 5.b even 2 1
625.2.b.a 4 5.c odd 4 2
625.2.d.b 4 25.e even 10 2
625.2.d.h 4 25.d even 5 2
625.2.e.c 8 25.f odd 20 4
5625.2.a.d 2 15.d odd 2 1
5625.2.a.f 2 3.b odd 2 1
10000.2.a.c 2 4.b odd 2 1
10000.2.a.l 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(625))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{3} - 1 \)

Hecke characteristic polynomials

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