Properties

Label 625.2.a.c
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: \( x^{2} - x - 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} - \beta q^{6} + ( - \beta + 1) q^{7} + ( - 2 \beta + 1) q^{8} - 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} - \beta q^{6} + ( - \beta + 1) q^{7} + ( - 2 \beta + 1) q^{8} - 2 q^{9} + ( - 2 \beta - 2) q^{11} + ( - \beta + 1) q^{12} + ( - 3 \beta + 3) q^{13} - q^{14} - 3 \beta q^{16} + (2 \beta + 2) q^{17} - 2 \beta q^{18} + (3 \beta - 4) q^{19} + (\beta - 1) q^{21} + ( - 4 \beta - 2) q^{22} + (2 \beta - 7) q^{23} + (2 \beta - 1) q^{24} - 3 q^{26} + 5 q^{27} + (\beta - 2) q^{28} + ( - \beta - 2) q^{29} - 3 q^{31} + (\beta - 5) q^{32} + (2 \beta + 2) q^{33} + (4 \beta + 2) q^{34} + ( - 2 \beta + 2) q^{36} + (2 \beta - 3) q^{37} + ( - \beta + 3) q^{38} + (3 \beta - 3) q^{39} + (2 \beta - 4) q^{41} + q^{42} + 3 \beta q^{43} - 2 \beta q^{44} + ( - 5 \beta + 2) q^{46} + ( - \beta + 1) q^{47} + 3 \beta q^{48} + ( - \beta - 5) q^{49} + ( - 2 \beta - 2) q^{51} + (3 \beta - 6) q^{52} + (4 \beta - 3) q^{53} + 5 \beta q^{54} + ( - \beta + 3) q^{56} + ( - 3 \beta + 4) q^{57} + ( - 3 \beta - 1) q^{58} + ( - 3 \beta - 6) q^{59} + (6 \beta - 1) q^{61} - 3 \beta q^{62} + (2 \beta - 2) q^{63} + (2 \beta + 1) q^{64} + (4 \beta + 2) q^{66} + (2 \beta - 8) q^{67} + 2 \beta q^{68} + ( - 2 \beta + 7) q^{69} + ( - \beta - 5) q^{71} + (4 \beta - 2) q^{72} + 9 q^{73} + ( - \beta + 2) q^{74} + ( - 4 \beta + 7) q^{76} + 2 \beta q^{77} + 3 q^{78} - 5 \beta q^{79} + q^{81} + ( - 2 \beta + 2) q^{82} + (2 \beta + 3) q^{83} + ( - \beta + 2) q^{84} + (3 \beta + 3) q^{86} + (\beta + 2) q^{87} + (6 \beta + 2) q^{88} + ( - 8 \beta + 4) q^{89} + ( - 3 \beta + 6) q^{91} + ( - 7 \beta + 9) q^{92} + 3 q^{93} - q^{94} + ( - \beta + 5) q^{96} + (3 \beta - 1) q^{97} + ( - 6 \beta - 1) q^{98} + (4 \beta + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + q^{7} - 4 q^{9} - 6 q^{11} + q^{12} + 3 q^{13} - 2 q^{14} - 3 q^{16} + 6 q^{17} - 2 q^{18} - 5 q^{19} - q^{21} - 8 q^{22} - 12 q^{23} - 6 q^{26} + 10 q^{27} - 3 q^{28} - 5 q^{29} - 6 q^{31} - 9 q^{32} + 6 q^{33} + 8 q^{34} + 2 q^{36} - 4 q^{37} + 5 q^{38} - 3 q^{39} - 6 q^{41} + 2 q^{42} + 3 q^{43} - 2 q^{44} - q^{46} + q^{47} + 3 q^{48} - 11 q^{49} - 6 q^{51} - 9 q^{52} - 2 q^{53} + 5 q^{54} + 5 q^{56} + 5 q^{57} - 5 q^{58} - 15 q^{59} + 4 q^{61} - 3 q^{62} - 2 q^{63} + 4 q^{64} + 8 q^{66} - 14 q^{67} + 2 q^{68} + 12 q^{69} - 11 q^{71} + 18 q^{73} + 3 q^{74} + 10 q^{76} + 2 q^{77} + 6 q^{78} - 5 q^{79} + 2 q^{81} + 2 q^{82} + 8 q^{83} + 3 q^{84} + 9 q^{86} + 5 q^{87} + 10 q^{88} + 9 q^{91} + 11 q^{92} + 6 q^{93} - 2 q^{94} + 9 q^{96} + q^{97} - 8 q^{98} + 12 q^{99}+O(q^{100}) \) 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 −1.00000 −1.61803 0 0.618034 1.61803 2.23607 −2.00000 0
1.2 1.61803 −1.00000 0.618034 0 −1.61803 −0.618034 −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.c 2
3.b odd 2 1 5625.2.a.d 2
4.b odd 2 1 10000.2.a.l 2
5.b even 2 1 625.2.a.b 2
5.c odd 4 2 625.2.b.a 4
15.d odd 2 1 5625.2.a.f 2
20.d odd 2 1 10000.2.a.c 2
25.d even 5 2 125.2.d.a 4
25.d even 5 2 625.2.d.b 4
25.e even 10 2 25.2.d.a 4
25.e even 10 2 625.2.d.h 4
25.f odd 20 4 125.2.e.a 8
25.f odd 20 4 625.2.e.c 8
75.h odd 10 2 225.2.h.b 4
100.h 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.e even 10 2
125.2.d.a 4 25.d even 5 2
125.2.e.a 8 25.f odd 20 4
225.2.h.b 4 75.h odd 10 2
400.2.u.b 4 100.h odd 10 2
625.2.a.b 2 5.b even 2 1
625.2.a.c 2 1.a even 1 1 trivial
625.2.b.a 4 5.c odd 4 2
625.2.d.b 4 25.d even 5 2
625.2.d.h 4 25.e even 10 2
625.2.e.c 8 25.f odd 20 4
5625.2.a.d 2 3.b odd 2 1
5625.2.a.f 2 15.d odd 2 1
10000.2.a.c 2 20.d odd 2 1
10000.2.a.l 2 4.b 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
\( T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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