Properties

Label 60.4.h.a
Level $60$
Weight $4$
Character orbit 60.h
Analytic conductor $3.540$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 60.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.54011460034\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + ( \beta_{1} - 4 \beta_{2} ) q^{3} -8 q^{4} -5 \beta_{3} q^{5} + ( 14 + 2 \beta_{3} ) q^{6} + ( 22 \beta_{1} - 11 \beta_{2} ) q^{7} -16 \beta_{2} q^{8} + ( -22 - 7 \beta_{3} ) q^{9} +O(q^{10})\) \( q + 2 \beta_{2} q^{2} + ( \beta_{1} - 4 \beta_{2} ) q^{3} -8 q^{4} -5 \beta_{3} q^{5} + ( 14 + 2 \beta_{3} ) q^{6} + ( 22 \beta_{1} - 11 \beta_{2} ) q^{7} -16 \beta_{2} q^{8} + ( -22 - 7 \beta_{3} ) q^{9} + ( 20 \beta_{1} - 10 \beta_{2} ) q^{10} + ( -8 \beta_{1} + 32 \beta_{2} ) q^{12} + 44 \beta_{3} q^{14} + ( -35 \beta_{1} + 5 \beta_{2} ) q^{15} + 64 q^{16} + ( 28 \beta_{1} - 58 \beta_{2} ) q^{18} + 40 \beta_{3} q^{20} + ( 55 - 77 \beta_{3} ) q^{21} + 67 \beta_{2} q^{23} + ( -112 - 16 \beta_{3} ) q^{24} -125 q^{25} + ( -71 \beta_{1} + 95 \beta_{2} ) q^{27} + ( -176 \beta_{1} + 88 \beta_{2} ) q^{28} -28 \beta_{3} q^{29} + ( 50 - 70 \beta_{3} ) q^{30} + 128 \beta_{2} q^{32} -275 \beta_{2} q^{35} + ( 176 + 56 \beta_{3} ) q^{36} + ( -160 \beta_{1} + 80 \beta_{2} ) q^{40} + 206 \beta_{3} q^{41} + ( 308 \beta_{1} - 44 \beta_{2} ) q^{42} + ( 238 \beta_{1} - 119 \beta_{2} ) q^{43} + ( -175 + 110 \beta_{3} ) q^{45} -268 q^{46} + 301 \beta_{2} q^{47} + ( 64 \beta_{1} - 256 \beta_{2} ) q^{48} + 867 q^{49} -250 \beta_{2} q^{50} + ( -238 - 142 \beta_{3} ) q^{54} -352 \beta_{3} q^{56} + ( 112 \beta_{1} - 56 \beta_{2} ) q^{58} + ( 280 \beta_{1} - 40 \beta_{2} ) q^{60} -952 q^{61} + ( -484 \beta_{1} - 143 \beta_{2} ) q^{63} -512 q^{64} + ( 490 \beta_{1} - 245 \beta_{2} ) q^{67} + ( 469 + 67 \beta_{3} ) q^{69} + 1100 q^{70} + ( -224 \beta_{1} + 464 \beta_{2} ) q^{72} + ( -125 \beta_{1} + 500 \beta_{2} ) q^{75} -320 \beta_{3} q^{80} + ( 239 + 308 \beta_{3} ) q^{81} + ( -824 \beta_{1} + 412 \beta_{2} ) q^{82} -77 \beta_{2} q^{83} + ( -440 + 616 \beta_{3} ) q^{84} + 476 \beta_{3} q^{86} + ( -196 \beta_{1} + 28 \beta_{2} ) q^{87} -424 \beta_{3} q^{89} + ( -440 \beta_{1} - 130 \beta_{2} ) q^{90} -536 \beta_{2} q^{92} -1204 q^{94} + ( 896 + 128 \beta_{3} ) q^{96} + 1734 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 32q^{4} + 56q^{6} - 88q^{9} + O(q^{10}) \) \( 4q - 32q^{4} + 56q^{6} - 88q^{9} + 256q^{16} + 220q^{21} - 448q^{24} - 500q^{25} + 200q^{30} + 704q^{36} - 700q^{45} - 1072q^{46} + 3468q^{49} - 952q^{54} - 3808q^{61} - 2048q^{64} + 1876q^{69} + 4400q^{70} + 956q^{81} - 1760q^{84} - 4816q^{94} + 3584q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 4 x^{2} + 9\):

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
−1.58114 0.707107i
1.58114 0.707107i
−1.58114 + 0.707107i
1.58114 + 0.707107i
2.82843i −1.58114 + 4.94975i −8.00000 11.1803i 14.0000 + 4.47214i −34.7851 22.6274i −22.0000 15.6525i −31.6228
59.2 2.82843i 1.58114 + 4.94975i −8.00000 11.1803i 14.0000 4.47214i 34.7851 22.6274i −22.0000 + 15.6525i 31.6228
59.3 2.82843i −1.58114 4.94975i −8.00000 11.1803i 14.0000 4.47214i −34.7851 22.6274i −22.0000 + 15.6525i −31.6228
59.4 2.82843i 1.58114 4.94975i −8.00000 11.1803i 14.0000 + 4.47214i 34.7851 22.6274i −22.0000 15.6525i 31.6228
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.4.h.a 4
3.b odd 2 1 inner 60.4.h.a 4
4.b odd 2 1 inner 60.4.h.a 4
5.b even 2 1 inner 60.4.h.a 4
12.b even 2 1 inner 60.4.h.a 4
15.d odd 2 1 inner 60.4.h.a 4
20.d odd 2 1 CM 60.4.h.a 4
60.h even 2 1 inner 60.4.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.h.a 4 1.a even 1 1 trivial
60.4.h.a 4 3.b odd 2 1 inner
60.4.h.a 4 4.b odd 2 1 inner
60.4.h.a 4 5.b even 2 1 inner
60.4.h.a 4 12.b even 2 1 inner
60.4.h.a 4 15.d odd 2 1 inner
60.4.h.a 4 20.d odd 2 1 CM
60.4.h.a 4 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 1210 \) acting on \(S_{4}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 + T^{2} )^{2} \)
$3$ \( 729 + 44 T^{2} + T^{4} \)
$5$ \( ( 125 + T^{2} )^{2} \)
$7$ \( ( -1210 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 8978 + T^{2} )^{2} \)
$29$ \( ( 3920 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 212180 + T^{2} )^{2} \)
$43$ \( ( -141610 + T^{2} )^{2} \)
$47$ \( ( 181202 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 952 + T )^{4} \)
$67$ \( ( -600250 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 11858 + T^{2} )^{2} \)
$89$ \( ( 898880 + T^{2} )^{2} \)
$97$ \( T^{4} \)
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