Properties

Label 1250.2.a.d
Level $1250$
Weight $2$
Character orbit 1250.a
Self dual yes
Analytic conductor $9.981$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1250 = 2 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1250.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.98130025266\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
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.

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

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
−0.618034
1.61803
1.00000 0.381966 1.00000 0 0.381966 3.00000 1.00000 −2.85410 0
1.2 1.00000 2.61803 1.00000 0 2.61803 3.00000 1.00000 3.85410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

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 1250.2.a.d 2
4.b odd 2 1 10000.2.a.a 2
5.b even 2 1 1250.2.a.a 2
5.c odd 4 2 1250.2.b.b 4
20.d odd 2 1 10000.2.a.n 2
25.d even 5 2 250.2.d.a 4
25.e even 10 2 50.2.d.a 4
25.f odd 20 4 250.2.e.b 8
75.h odd 10 2 450.2.h.a 4
100.h odd 10 2 400.2.u.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.a 4 25.e even 10 2
250.2.d.a 4 25.d even 5 2
250.2.e.b 8 25.f odd 20 4
400.2.u.c 4 100.h odd 10 2
450.2.h.a 4 75.h odd 10 2
1250.2.a.a 2 5.b even 2 1
1250.2.a.d 2 1.a even 1 1 trivial
1250.2.b.b 4 5.c odd 4 2
10000.2.a.a 2 4.b odd 2 1
10000.2.a.n 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1250))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( 1 - 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -3 + T )^{2} \)
$11$ \( -1 - 4 T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 9 + 9 T + T^{2} \)
$19$ \( -5 - 5 T + T^{2} \)
$23$ \( 11 - 8 T + T^{2} \)
$29$ \( 5 + 10 T + T^{2} \)
$31$ \( -1 - 4 T + T^{2} \)
$37$ \( -61 - T + T^{2} \)
$41$ \( -11 + 6 T + T^{2} \)
$43$ \( 11 - 8 T + T^{2} \)
$47$ \( -71 - 6 T + T^{2} \)
$53$ \( 16 + 12 T + T^{2} \)
$59$ \( -20 + T^{2} \)
$61$ \( 19 + 11 T + T^{2} \)
$67$ \( 59 - 16 T + T^{2} \)
$71$ \( ( 3 + T )^{2} \)
$73$ \( -44 + 2 T + T^{2} \)
$79$ \( 20 + 10 T + T^{2} \)
$83$ \( 61 - 18 T + T^{2} \)
$89$ \( -20 + T^{2} \)
$97$ \( -101 - T + T^{2} \)
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