Properties

Label 3750.2.a.g
Level $3750$
Weight $2$
Character orbit 3750.a
Self dual yes
Analytic conductor $29.944$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3750.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9439007580\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
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.

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

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
1.00000 1.00000 1.00000 0 1.00000 −2.00000 1.00000 1.00000 0
1.2 1.00000 1.00000 1.00000 0 1.00000 −2.00000 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

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 3750.2.a.g 2
5.b even 2 1 3750.2.a.b 2
5.c odd 4 2 3750.2.c.c 4
25.d even 5 2 750.2.g.a 4
25.e even 10 2 150.2.g.b 4
25.f odd 20 4 750.2.h.a 8
75.h odd 10 2 450.2.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.b 4 25.e even 10 2
450.2.h.b 4 75.h odd 10 2
750.2.g.a 4 25.d even 5 2
750.2.h.a 8 25.f odd 20 4
3750.2.a.b 2 5.b even 2 1
3750.2.a.g 2 1.a even 1 1 trivial
3750.2.c.c 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3750))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( 4 + 6 T + T^{2} \)
$13$ \( -9 - 3 T + T^{2} \)
$17$ \( 9 + 9 T + T^{2} \)
$19$ \( 20 + 10 T + T^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( 5 + 5 T + T^{2} \)
$31$ \( -36 + 6 T + T^{2} \)
$37$ \( 19 - 11 T + T^{2} \)
$41$ \( -31 + T + T^{2} \)
$43$ \( -4 + 2 T + T^{2} \)
$47$ \( 44 + 14 T + T^{2} \)
$53$ \( -99 - 3 T + T^{2} \)
$59$ \( -80 + T^{2} \)
$61$ \( 19 + 11 T + T^{2} \)
$67$ \( -36 - 6 T + T^{2} \)
$71$ \( -116 + 6 T + T^{2} \)
$73$ \( 31 - 13 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( 5 - 5 T + T^{2} \)
$97$ \( 99 - 21 T + T^{2} \)
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