Properties

Label 3750.2.c.b
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: \(\Q(i, \sqrt{5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} -\beta_{3} q^{3} - q^{4} - q^{6} + ( \beta_{1} - \beta_{3} ) q^{7} + \beta_{3} q^{8} - q^{9} +O(q^{10})\) \( q -\beta_{3} q^{2} -\beta_{3} q^{3} - q^{4} - q^{6} + ( \beta_{1} - \beta_{3} ) q^{7} + \beta_{3} q^{8} - q^{9} + ( 2 - \beta_{2} ) q^{11} + \beta_{3} q^{12} + 4 \beta_{1} q^{13} + ( -1 + \beta_{2} ) q^{14} + q^{16} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{17} + \beta_{3} q^{18} + ( -4 - 6 \beta_{2} ) q^{19} + ( -1 + \beta_{2} ) q^{21} + ( \beta_{1} - 2 \beta_{3} ) q^{22} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{23} + q^{24} + 4 \beta_{2} q^{26} + \beta_{3} q^{27} + ( -\beta_{1} + \beta_{3} ) q^{28} + ( -2 + 4 \beta_{2} ) q^{29} + ( 5 - \beta_{2} ) q^{31} -\beta_{3} q^{32} + ( \beta_{1} - 2 \beta_{3} ) q^{33} + ( 2 + 2 \beta_{2} ) q^{34} + q^{36} -8 \beta_{3} q^{37} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{38} + 4 \beta_{2} q^{39} + ( -4 - 6 \beta_{2} ) q^{41} + ( -\beta_{1} + \beta_{3} ) q^{42} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -2 + \beta_{2} ) q^{44} + ( -2 - 4 \beta_{2} ) q^{46} + ( 6 \beta_{1} + 8 \beta_{3} ) q^{47} -\beta_{3} q^{48} + ( 5 + 3 \beta_{2} ) q^{49} + ( 2 + 2 \beta_{2} ) q^{51} -4 \beta_{1} q^{52} + ( 5 \beta_{1} + 6 \beta_{3} ) q^{53} + q^{54} + ( 1 - \beta_{2} ) q^{56} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{57} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{58} + ( 2 - \beta_{2} ) q^{59} + ( 6 + 2 \beta_{2} ) q^{61} + ( \beta_{1} - 5 \beta_{3} ) q^{62} + ( -\beta_{1} + \beta_{3} ) q^{63} - q^{64} + ( -2 + \beta_{2} ) q^{66} + ( 4 \beta_{1} + 8 \beta_{3} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{68} + ( -2 - 4 \beta_{2} ) q^{69} + ( 12 + 4 \beta_{2} ) q^{71} -\beta_{3} q^{72} + ( 4 \beta_{1} + 10 \beta_{3} ) q^{73} -8 q^{74} + ( 4 + 6 \beta_{2} ) q^{76} + ( 4 \beta_{1} - 3 \beta_{3} ) q^{77} -4 \beta_{1} q^{78} + ( 4 - \beta_{2} ) q^{79} + q^{81} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{82} + ( 3 \beta_{1} + 7 \beta_{3} ) q^{83} + ( 1 - \beta_{2} ) q^{84} + ( 2 + 6 \beta_{2} ) q^{86} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{87} + ( -\beta_{1} + 2 \beta_{3} ) q^{88} + ( -10 - 4 \beta_{2} ) q^{89} + ( -4 + 8 \beta_{2} ) q^{91} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{92} + ( \beta_{1} - 5 \beta_{3} ) q^{93} + ( 8 + 6 \beta_{2} ) q^{94} - q^{96} + ( \beta_{1} + 5 \beta_{3} ) q^{97} + ( -3 \beta_{1} - 5 \beta_{3} ) q^{98} + ( -2 + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{6} - 4q^{9} + 10q^{11} - 6q^{14} + 4q^{16} - 4q^{19} - 6q^{21} + 4q^{24} - 8q^{26} - 16q^{29} + 22q^{31} + 4q^{34} + 4q^{36} - 8q^{39} - 4q^{41} - 10q^{44} + 14q^{49} + 4q^{51} + 4q^{54} + 6q^{56} + 10q^{59} + 20q^{61} - 4q^{64} - 10q^{66} + 40q^{71} - 32q^{74} + 4q^{76} + 18q^{79} + 4q^{81} + 6q^{84} - 4q^{86} - 32q^{89} - 32q^{91} + 20q^{94} - 4q^{96} - 10q^{99} + O(q^{100}) \)

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
0.618034i
1.61803i
1.00000i 1.00000i −1.00000 0 −1.00000 2.61803i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 −1.00000 0.381966i 1.00000i −1.00000 0
1249.3 1.00000i 1.00000i −1.00000 0 −1.00000 0.381966i 1.00000i −1.00000 0
1249.4 1.00000i 1.00000i −1.00000 0 −1.00000 2.61803i 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.b 4
5.b even 2 1 inner 3750.2.c.b 4
5.c odd 4 1 3750.2.a.d 2
5.c odd 4 1 3750.2.a.f 2
25.d even 5 2 750.2.h.b 8
25.e even 10 2 750.2.h.b 8
25.f odd 20 2 150.2.g.a 4
25.f odd 20 2 750.2.g.b 4
75.l even 20 2 450.2.h.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.a 4 25.f odd 20 2
450.2.h.c 4 75.l even 20 2
750.2.g.b 4 25.f odd 20 2
750.2.h.b 8 25.d even 5 2
750.2.h.b 8 25.e even 10 2
3750.2.a.d 2 5.c odd 4 1
3750.2.a.f 2 5.c odd 4 1
3750.2.c.b 4 1.a even 1 1 trivial
3750.2.c.b 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 7 T_{7}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(3750, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( \)
$7$ \( 1 - 21 T^{2} + 197 T^{4} - 1029 T^{6} + 2401 T^{8} \)
$11$ \( ( 1 - 5 T + 27 T^{2} - 55 T^{3} + 121 T^{4} )^{2} \)
$13$ \( 1 - 4 T^{2} + 22 T^{4} - 676 T^{6} + 28561 T^{8} \)
$17$ \( 1 - 56 T^{2} + 1342 T^{4} - 16184 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 + 2 T - 6 T^{2} + 38 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 26 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 8 T + 54 T^{2} + 232 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 11 T + 91 T^{2} - 341 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 2 T + 38 T^{2} + 82 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 - 80 T^{2} + 5118 T^{4} - 147920 T^{6} + 3418801 T^{8} \)
$47$ \( 1 - 48 T^{2} + 494 T^{4} - 106032 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 125 T^{2} + 7993 T^{4} - 351125 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 5 T + 123 T^{2} - 295 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 10 T + 142 T^{2} - 610 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 - 156 T^{2} + 12182 T^{4} - 700284 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 - 20 T + 222 T^{2} - 1420 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 - 124 T^{2} + 9382 T^{4} - 660796 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 - 9 T + 177 T^{2} - 711 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 249 T^{2} + 27917 T^{4} - 1715361 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 16 T + 222 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 - 345 T^{2} + 48473 T^{4} - 3246105 T^{6} + 88529281 T^{8} \)
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