Properties

Label 3750.2.a.f
Level 3750
Weight 2
Character orbit 3750.a
Self dual yes
Analytic conductor 29.944
Analytic rank 0
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + ( 1 + \beta ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + ( 1 + \beta ) q^{7} + q^{8} + q^{9} + ( 2 + \beta ) q^{11} - q^{12} -4 \beta q^{13} + ( 1 + \beta ) q^{14} + q^{16} + ( -2 + 2 \beta ) q^{17} + q^{18} + ( 4 - 6 \beta ) q^{19} + ( -1 - \beta ) q^{21} + ( 2 + \beta ) q^{22} + ( -2 + 4 \beta ) q^{23} - q^{24} -4 \beta q^{26} - q^{27} + ( 1 + \beta ) q^{28} + ( 2 + 4 \beta ) q^{29} + ( 5 + \beta ) q^{31} + q^{32} + ( -2 - \beta ) q^{33} + ( -2 + 2 \beta ) q^{34} + q^{36} + 8 q^{37} + ( 4 - 6 \beta ) q^{38} + 4 \beta q^{39} + ( -4 + 6 \beta ) q^{41} + ( -1 - \beta ) q^{42} + ( 2 - 6 \beta ) q^{43} + ( 2 + \beta ) q^{44} + ( -2 + 4 \beta ) q^{46} + ( -8 + 6 \beta ) q^{47} - q^{48} + ( -5 + 3 \beta ) q^{49} + ( 2 - 2 \beta ) q^{51} -4 \beta q^{52} + ( 6 - 5 \beta ) q^{53} - q^{54} + ( 1 + \beta ) q^{56} + ( -4 + 6 \beta ) q^{57} + ( 2 + 4 \beta ) q^{58} + ( -2 - \beta ) q^{59} + ( 6 - 2 \beta ) q^{61} + ( 5 + \beta ) q^{62} + ( 1 + \beta ) q^{63} + q^{64} + ( -2 - \beta ) q^{66} + ( -8 + 4 \beta ) q^{67} + ( -2 + 2 \beta ) q^{68} + ( 2 - 4 \beta ) q^{69} + ( 12 - 4 \beta ) q^{71} + q^{72} + ( 10 - 4 \beta ) q^{73} + 8 q^{74} + ( 4 - 6 \beta ) q^{76} + ( 3 + 4 \beta ) q^{77} + 4 \beta q^{78} + ( -4 - \beta ) q^{79} + q^{81} + ( -4 + 6 \beta ) q^{82} + ( 7 - 3 \beta ) q^{83} + ( -1 - \beta ) q^{84} + ( 2 - 6 \beta ) q^{86} + ( -2 - 4 \beta ) q^{87} + ( 2 + \beta ) q^{88} + ( 10 - 4 \beta ) q^{89} + ( -4 - 8 \beta ) q^{91} + ( -2 + 4 \beta ) q^{92} + ( -5 - \beta ) q^{93} + ( -8 + 6 \beta ) q^{94} - q^{96} + ( -5 + \beta ) q^{97} + ( -5 + 3 \beta ) q^{98} + ( 2 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 3q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 3q^{7} + 2q^{8} + 2q^{9} + 5q^{11} - 2q^{12} - 4q^{13} + 3q^{14} + 2q^{16} - 2q^{17} + 2q^{18} + 2q^{19} - 3q^{21} + 5q^{22} - 2q^{24} - 4q^{26} - 2q^{27} + 3q^{28} + 8q^{29} + 11q^{31} + 2q^{32} - 5q^{33} - 2q^{34} + 2q^{36} + 16q^{37} + 2q^{38} + 4q^{39} - 2q^{41} - 3q^{42} - 2q^{43} + 5q^{44} - 10q^{47} - 2q^{48} - 7q^{49} + 2q^{51} - 4q^{52} + 7q^{53} - 2q^{54} + 3q^{56} - 2q^{57} + 8q^{58} - 5q^{59} + 10q^{61} + 11q^{62} + 3q^{63} + 2q^{64} - 5q^{66} - 12q^{67} - 2q^{68} + 20q^{71} + 2q^{72} + 16q^{73} + 16q^{74} + 2q^{76} + 10q^{77} + 4q^{78} - 9q^{79} + 2q^{81} - 2q^{82} + 11q^{83} - 3q^{84} - 2q^{86} - 8q^{87} + 5q^{88} + 16q^{89} - 16q^{91} - 11q^{93} - 10q^{94} - 2q^{96} - 9q^{97} - 7q^{98} + 5q^{99} + O(q^{100}) \)

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

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

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 3 T_{7} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3750))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( \)
$7$ \( 1 - 3 T + 15 T^{2} - 21 T^{3} + 49 T^{4} \)
$11$ \( 1 - 5 T + 27 T^{2} - 55 T^{3} + 121 T^{4} \)
$13$ \( 1 + 4 T + 10 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( 1 + 2 T + 30 T^{2} + 34 T^{3} + 289 T^{4} \)
$19$ \( 1 - 2 T - 6 T^{2} - 38 T^{3} + 361 T^{4} \)
$23$ \( 1 + 26 T^{2} + 529 T^{4} \)
$29$ \( 1 - 8 T + 54 T^{2} - 232 T^{3} + 841 T^{4} \)
$31$ \( 1 - 11 T + 91 T^{2} - 341 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 8 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 2 T + 38 T^{2} + 82 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 2 T + 42 T^{2} + 86 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 10 T + 74 T^{2} + 470 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 7 T + 87 T^{2} - 371 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 5 T + 123 T^{2} + 295 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 10 T + 142 T^{2} - 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 12 T + 150 T^{2} + 804 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 20 T + 222 T^{2} - 1420 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 16 T + 190 T^{2} - 1168 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 9 T + 177 T^{2} + 711 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 11 T + 185 T^{2} - 913 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 16 T + 222 T^{2} - 1424 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 9 T + 213 T^{2} + 873 T^{3} + 9409 T^{4} \)
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