Properties

Label 60.2.h.b
Level $60$
Weight $2$
Character orbit 60.h
Analytic conductor $0.479$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.479102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + \beta_{3} q^{4} + ( \beta_{1} + \beta_{2} ) q^{5} + ( 2 - \beta_{3} ) q^{6} + ( 2 \beta_{1} - \beta_{2} ) q^{8} -3 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + \beta_{3} q^{4} + ( \beta_{1} + \beta_{2} ) q^{5} + ( 2 - \beta_{3} ) q^{6} + ( 2 \beta_{1} - \beta_{2} ) q^{8} -3 q^{9} + ( -2 - \beta_{3} ) q^{10} + ( -2 \beta_{1} - \beta_{2} ) q^{12} + ( -1 + 2 \beta_{3} ) q^{15} + ( -4 + \beta_{3} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{17} + 3 \beta_{2} q^{18} + ( 2 - 4 \beta_{3} ) q^{19} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{20} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{23} + ( 4 + \beta_{3} ) q^{24} + 5 q^{25} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{27} + ( 4 \beta_{1} - \beta_{2} ) q^{30} + ( -2 + 4 \beta_{3} ) q^{31} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{32} + ( 4 + 2 \beta_{3} ) q^{34} -3 \beta_{3} q^{36} + ( -8 \beta_{1} + 2 \beta_{2} ) q^{38} + ( 4 - 3 \beta_{3} ) q^{40} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{45} + ( -4 + 2 \beta_{3} ) q^{46} + ( -6 \beta_{1} + 6 \beta_{2} ) q^{47} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{48} -7 q^{49} -5 \beta_{2} q^{50} + ( 2 - 4 \beta_{3} ) q^{51} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -6 + 3 \beta_{3} ) q^{54} + ( 6 \beta_{1} + 6 \beta_{2} ) q^{57} + ( -8 + \beta_{3} ) q^{60} -2 q^{61} + ( 8 \beta_{1} - 2 \beta_{2} ) q^{62} + ( -4 - 3 \beta_{3} ) q^{64} + ( 4 \beta_{1} - 6 \beta_{2} ) q^{68} + 6 q^{69} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{72} + ( -5 \beta_{1} + 5 \beta_{2} ) q^{75} + ( 16 - 2 \beta_{3} ) q^{76} + ( -2 + 4 \beta_{3} ) q^{79} + ( -6 \beta_{1} - \beta_{2} ) q^{80} + 9 q^{81} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{83} -10 q^{85} + ( 6 + 3 \beta_{3} ) q^{90} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{92} + ( -6 \beta_{1} - 6 \beta_{2} ) q^{93} + ( 12 - 6 \beta_{3} ) q^{94} + ( 10 \beta_{1} - 10 \beta_{2} ) q^{95} + ( -4 + 5 \beta_{3} ) q^{96} + 7 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 6q^{6} - 12q^{9} + O(q^{10}) \) \( 4q + 2q^{4} + 6q^{6} - 12q^{9} - 10q^{10} - 14q^{16} + 18q^{24} + 20q^{25} + 20q^{34} - 6q^{36} + 10q^{40} - 12q^{46} - 28q^{49} - 18q^{54} - 30q^{60} - 8q^{61} - 22q^{64} + 24q^{69} + 60q^{76} + 36q^{81} - 40q^{85} + 30q^{90} + 36q^{94} - 6q^{96} + O(q^{100}) \)

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

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

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
−0.309017 + 0.535233i
−0.309017 0.535233i
0.809017 1.40126i
0.809017 + 1.40126i
−1.11803 0.866025i 1.73205i 0.500000 + 1.93649i 2.23607 1.50000 1.93649i 0 1.11803 2.59808i −3.00000 −2.50000 1.93649i
59.2 −1.11803 + 0.866025i 1.73205i 0.500000 1.93649i 2.23607 1.50000 + 1.93649i 0 1.11803 + 2.59808i −3.00000 −2.50000 + 1.93649i
59.3 1.11803 0.866025i 1.73205i 0.500000 1.93649i −2.23607 1.50000 + 1.93649i 0 −1.11803 2.59808i −3.00000 −2.50000 + 1.93649i
59.4 1.11803 + 0.866025i 1.73205i 0.500000 + 1.93649i −2.23607 1.50000 1.93649i 0 −1.11803 + 2.59808i −3.00000 −2.50000 1.93649i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
20.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 60.2.h.b 4
3.b odd 2 1 inner 60.2.h.b 4
4.b odd 2 1 inner 60.2.h.b 4
5.b even 2 1 inner 60.2.h.b 4
5.c odd 4 2 300.2.e.a 4
8.b even 2 1 960.2.o.a 4
8.d odd 2 1 960.2.o.a 4
12.b even 2 1 inner 60.2.h.b 4
15.d odd 2 1 CM 60.2.h.b 4
15.e even 4 2 300.2.e.a 4
20.d odd 2 1 inner 60.2.h.b 4
20.e even 4 2 300.2.e.a 4
24.f even 2 1 960.2.o.a 4
24.h odd 2 1 960.2.o.a 4
40.e odd 2 1 960.2.o.a 4
40.f even 2 1 960.2.o.a 4
60.h even 2 1 inner 60.2.h.b 4
60.l odd 4 2 300.2.e.a 4
120.i odd 2 1 960.2.o.a 4
120.m even 2 1 960.2.o.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.h.b 4 1.a even 1 1 trivial
60.2.h.b 4 3.b odd 2 1 inner
60.2.h.b 4 4.b odd 2 1 inner
60.2.h.b 4 5.b even 2 1 inner
60.2.h.b 4 12.b even 2 1 inner
60.2.h.b 4 15.d odd 2 1 CM
60.2.h.b 4 20.d odd 2 1 inner
60.2.h.b 4 60.h even 2 1 inner
300.2.e.a 4 5.c odd 4 2
300.2.e.a 4 15.e even 4 2
300.2.e.a 4 20.e even 4 2
300.2.e.a 4 60.l odd 4 2
960.2.o.a 4 8.b even 2 1
960.2.o.a 4 8.d odd 2 1
960.2.o.a 4 24.f even 2 1
960.2.o.a 4 24.h odd 2 1
960.2.o.a 4 40.e odd 2 1
960.2.o.a 4 40.f even 2 1
960.2.o.a 4 120.i odd 2 1
960.2.o.a 4 120.m even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - T^{2} + T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -20 + T^{2} )^{2} \)
$19$ \( ( 60 + T^{2} )^{2} \)
$23$ \( ( 12 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 60 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( ( 108 + T^{2} )^{2} \)
$53$ \( ( -20 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( ( 2 + T )^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 60 + T^{2} )^{2} \)
$83$ \( ( 12 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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