Properties

Label 125.2.a.b
Level $125$
Weight $2$
Character orbit 125.a
Self dual yes
Analytic conductor $0.998$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 125.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.998130025266\)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - \beta + 2) q^{3} + (\beta - 1) q^{4} + (\beta - 1) q^{6} + 3 q^{7} + ( - 2 \beta + 1) q^{8} + ( - 3 \beta + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + ( - \beta + 2) q^{3} + (\beta - 1) q^{4} + (\beta - 1) q^{6} + 3 q^{7} + ( - 2 \beta + 1) q^{8} + ( - 3 \beta + 2) q^{9} - 3 q^{11} + (2 \beta - 3) q^{12} + 3 \beta q^{13} + 3 \beta q^{14} - 3 \beta q^{16} + ( - 2 \beta - 1) q^{17} + ( - \beta - 3) q^{18} + ( - \beta - 2) q^{19} + ( - 3 \beta + 6) q^{21} - 3 \beta q^{22} + (2 \beta - 2) q^{23} + ( - 3 \beta + 4) q^{24} + (3 \beta + 3) q^{26} + ( - 2 \beta + 1) q^{27} + (3 \beta - 3) q^{28} + (6 \beta - 3) q^{29} + (5 \beta - 3) q^{31} + (\beta - 5) q^{32} + (3 \beta - 6) q^{33} + ( - 3 \beta - 2) q^{34} + (2 \beta - 5) q^{36} + ( - 6 \beta + 6) q^{37} + ( - 3 \beta - 1) q^{38} + (3 \beta - 3) q^{39} - 3 q^{41} + (3 \beta - 3) q^{42} + 9 q^{43} + ( - 3 \beta + 3) q^{44} + 2 q^{46} + (7 \beta - 3) q^{47} + ( - 3 \beta + 3) q^{48} + 2 q^{49} - \beta q^{51} + 3 q^{52} + ( - \beta - 3) q^{53} + ( - \beta - 2) q^{54} + ( - 6 \beta + 3) q^{56} + (\beta - 3) q^{57} + (3 \beta + 6) q^{58} + ( - 3 \beta + 9) q^{59} + ( - 5 \beta + 2) q^{61} + (2 \beta + 5) q^{62} + ( - 9 \beta + 6) q^{63} + (2 \beta + 1) q^{64} + ( - 3 \beta + 3) q^{66} + (3 \beta + 9) q^{67} + ( - \beta - 1) q^{68} + (4 \beta - 6) q^{69} - 3 q^{71} + ( - \beta + 8) q^{72} + ( - 3 \beta + 3) q^{73} - 6 q^{74} + ( - 2 \beta + 1) q^{76} - 9 q^{77} + 3 q^{78} + ( - 4 \beta + 7) q^{79} + (6 \beta - 2) q^{81} - 3 \beta q^{82} + ( - 4 \beta + 6) q^{83} + (6 \beta - 9) q^{84} + 9 \beta q^{86} + (9 \beta - 12) q^{87} + (6 \beta - 3) q^{88} + ( - 12 \beta + 6) q^{89} + 9 \beta q^{91} + ( - 2 \beta + 4) q^{92} + (8 \beta - 11) q^{93} + (4 \beta + 7) q^{94} + (6 \beta - 11) q^{96} + ( - 3 \beta - 3) q^{97} + 2 \beta q^{98} + (9 \beta - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} - q^{4} - q^{6} + 6 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 3 q^{3} - q^{4} - q^{6} + 6 q^{7} + q^{9} - 6 q^{11} - 4 q^{12} + 3 q^{13} + 3 q^{14} - 3 q^{16} - 4 q^{17} - 7 q^{18} - 5 q^{19} + 9 q^{21} - 3 q^{22} - 2 q^{23} + 5 q^{24} + 9 q^{26} - 3 q^{28} - q^{31} - 9 q^{32} - 9 q^{33} - 7 q^{34} - 8 q^{36} + 6 q^{37} - 5 q^{38} - 3 q^{39} - 6 q^{41} - 3 q^{42} + 18 q^{43} + 3 q^{44} + 4 q^{46} + q^{47} + 3 q^{48} + 4 q^{49} - q^{51} + 6 q^{52} - 7 q^{53} - 5 q^{54} - 5 q^{57} + 15 q^{58} + 15 q^{59} - q^{61} + 12 q^{62} + 3 q^{63} + 4 q^{64} + 3 q^{66} + 21 q^{67} - 3 q^{68} - 8 q^{69} - 6 q^{71} + 15 q^{72} + 3 q^{73} - 12 q^{74} - 18 q^{77} + 6 q^{78} + 10 q^{79} + 2 q^{81} - 3 q^{82} + 8 q^{83} - 12 q^{84} + 9 q^{86} - 15 q^{87} + 9 q^{91} + 6 q^{92} - 14 q^{93} + 18 q^{94} - 16 q^{96} - 9 q^{97} + 2 q^{98} - 3 q^{99}+O(q^{100}) \) 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 2.61803 −1.61803 0 −1.61803 3.00000 2.23607 3.85410 0
1.2 1.61803 0.381966 0.618034 0 0.618034 3.00000 −2.23607 −2.85410 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 125.2.a.b yes 2
3.b odd 2 1 1125.2.a.c 2
4.b odd 2 1 2000.2.a.a 2
5.b even 2 1 125.2.a.a 2
5.c odd 4 2 125.2.b.b 4
7.b odd 2 1 6125.2.a.g 2
8.b even 2 1 8000.2.a.d 2
8.d odd 2 1 8000.2.a.u 2
15.d odd 2 1 1125.2.a.d 2
15.e even 4 2 1125.2.b.f 4
20.d odd 2 1 2000.2.a.l 2
20.e even 4 2 2000.2.c.e 4
25.d even 5 2 625.2.d.a 4
25.d even 5 2 625.2.d.g 4
25.e even 10 2 625.2.d.d 4
25.e even 10 2 625.2.d.j 4
25.f odd 20 4 625.2.e.d 8
25.f odd 20 4 625.2.e.g 8
35.c odd 2 1 6125.2.a.d 2
40.e odd 2 1 8000.2.a.c 2
40.f even 2 1 8000.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.a.a 2 5.b even 2 1
125.2.a.b yes 2 1.a even 1 1 trivial
125.2.b.b 4 5.c odd 4 2
625.2.d.a 4 25.d even 5 2
625.2.d.d 4 25.e even 10 2
625.2.d.g 4 25.d even 5 2
625.2.d.j 4 25.e even 10 2
625.2.e.d 8 25.f odd 20 4
625.2.e.g 8 25.f odd 20 4
1125.2.a.c 2 3.b odd 2 1
1125.2.a.d 2 15.d odd 2 1
1125.2.b.f 4 15.e even 4 2
2000.2.a.a 2 4.b odd 2 1
2000.2.a.l 2 20.d odd 2 1
2000.2.c.e 4 20.e even 4 2
6125.2.a.d 2 35.c odd 2 1
6125.2.a.g 2 7.b odd 2 1
8000.2.a.c 2 40.e odd 2 1
8000.2.a.d 2 8.b even 2 1
8000.2.a.u 2 8.d odd 2 1
8000.2.a.v 2 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(125))\). 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
$7$ \( (T - 3)^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 45 \) Copy content Toggle raw display
$31$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$41$ \( (T + 3)^{2} \) Copy content Toggle raw display
$43$ \( (T - 9)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - T - 61 \) Copy content Toggle raw display
$53$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$59$ \( T^{2} - 15T + 45 \) Copy content Toggle raw display
$61$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$67$ \( T^{2} - 21T + 99 \) Copy content Toggle raw display
$71$ \( (T + 3)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 5 \) Copy content Toggle raw display
$83$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$89$ \( T^{2} - 180 \) Copy content Toggle raw display
$97$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
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