Properties

Label 3750.2.c.c
Level $3750$
Weight $2$
Character orbit 3750.c
Analytic conductor $29.944$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3750.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9439007580\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3750\mathbb{Z}\right)^\times\).

\(n\) \(2501\) \(3127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
1.61803i
0.618034i
1.61803i
0.618034i
1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 1.00000i −1.00000 0
1249.3 1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 1.00000i −1.00000 0
1249.4 1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 1.00000i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(3750, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 4 + T^{2} )^{2} \)
$11$ \( ( 4 + 6 T + T^{2} )^{2} \)
$13$ \( 81 + 27 T^{2} + T^{4} \)
$17$ \( 81 + 63 T^{2} + T^{4} \)
$19$ \( ( 20 - 10 T + T^{2} )^{2} \)
$23$ \( ( 36 + T^{2} )^{2} \)
$29$ \( ( 5 - 5 T + T^{2} )^{2} \)
$31$ \( ( -36 + 6 T + T^{2} )^{2} \)
$37$ \( 361 + 83 T^{2} + T^{4} \)
$41$ \( ( -31 + T + T^{2} )^{2} \)
$43$ \( 16 + 12 T^{2} + T^{4} \)
$47$ \( 1936 + 108 T^{2} + T^{4} \)
$53$ \( 9801 + 207 T^{2} + T^{4} \)
$59$ \( ( -80 + T^{2} )^{2} \)
$61$ \( ( 19 + 11 T + T^{2} )^{2} \)
$67$ \( 1296 + 108 T^{2} + T^{4} \)
$71$ \( ( -116 + 6 T + T^{2} )^{2} \)
$73$ \( 961 + 107 T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( 5 + 5 T + T^{2} )^{2} \)
$97$ \( 9801 + 243 T^{2} + T^{4} \)
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