Properties

Label 60.2.h.a
Level $60$
Weight $2$
Character orbit 60.h
Analytic conductor $0.479$
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: \( k \) \(=\) \( 2 \)
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: \(\Q(\sqrt{-2}, \sqrt{-5})\)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} -\beta_{1} q^{3} -2 q^{4} -\beta_{3} q^{5} + ( -1 + \beta_{3} ) q^{6} + ( 2 \beta_{1} - \beta_{2} ) q^{7} + 2 \beta_{2} q^{8} + ( 2 + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} -\beta_{1} q^{3} -2 q^{4} -\beta_{3} q^{5} + ( -1 + \beta_{3} ) q^{6} + ( 2 \beta_{1} - \beta_{2} ) q^{7} + 2 \beta_{2} q^{8} + ( 2 + \beta_{3} ) q^{9} + ( -2 \beta_{1} + \beta_{2} ) q^{10} + 2 \beta_{1} q^{12} -2 \beta_{3} q^{14} + ( -\beta_{1} + 3 \beta_{2} ) q^{15} + 4 q^{16} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{18} + 2 \beta_{3} q^{20} + ( -5 - \beta_{3} ) q^{21} + \beta_{2} q^{23} + ( 2 - 2 \beta_{3} ) q^{24} -5 q^{25} + ( -\beta_{1} - 3 \beta_{2} ) q^{27} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{28} + 4 \beta_{3} q^{29} + ( 5 + \beta_{3} ) q^{30} -4 \beta_{2} q^{32} -5 \beta_{2} q^{35} + ( -4 - 2 \beta_{3} ) q^{36} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{40} -2 \beta_{3} q^{41} + ( -2 \beta_{1} + 6 \beta_{2} ) q^{42} + ( 2 \beta_{1} - \beta_{2} ) q^{43} + ( 5 - 2 \beta_{3} ) q^{45} + 2 q^{46} + 7 \beta_{2} q^{47} -4 \beta_{1} q^{48} + 3 q^{49} + 5 \beta_{2} q^{50} + ( -7 + \beta_{3} ) q^{54} + 4 \beta_{3} q^{56} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{58} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{60} + 8 q^{61} + ( 4 \beta_{1} + 3 \beta_{2} ) q^{63} -8 q^{64} + ( -10 \beta_{1} + 5 \beta_{2} ) q^{67} + ( 1 - \beta_{3} ) q^{69} -10 q^{70} + ( -4 \beta_{1} + 6 \beta_{2} ) q^{72} + 5 \beta_{1} q^{75} -4 \beta_{3} q^{80} + ( -1 + 4 \beta_{3} ) q^{81} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{82} -11 \beta_{2} q^{83} + ( 10 + 2 \beta_{3} ) q^{84} -2 \beta_{3} q^{86} + ( 4 \beta_{1} - 12 \beta_{2} ) q^{87} -8 \beta_{3} q^{89} + ( -4 \beta_{1} - 3 \beta_{2} ) q^{90} -2 \beta_{2} q^{92} + 14 q^{94} + ( -4 + 4 \beta_{3} ) q^{96} -3 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} - 4q^{6} + 8q^{9} + O(q^{10}) \) \( 4q - 8q^{4} - 4q^{6} + 8q^{9} + 16q^{16} - 20q^{21} + 8q^{24} - 20q^{25} + 20q^{30} - 16q^{36} + 20q^{45} + 8q^{46} + 12q^{49} - 28q^{54} + 32q^{61} - 32q^{64} + 4q^{69} - 40q^{70} - 4q^{81} + 40q^{84} + 56q^{94} - 16q^{96} + O(q^{100}) \)

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
1.41421i −1.58114 0.707107i −2.00000 2.23607i −1.00000 + 2.23607i 3.16228 2.82843i 2.00000 + 2.23607i −3.16228
59.2 1.41421i 1.58114 0.707107i −2.00000 2.23607i −1.00000 2.23607i −3.16228 2.82843i 2.00000 2.23607i 3.16228
59.3 1.41421i −1.58114 + 0.707107i −2.00000 2.23607i −1.00000 2.23607i 3.16228 2.82843i 2.00000 2.23607i −3.16228
59.4 1.41421i 1.58114 + 0.707107i −2.00000 2.23607i −1.00000 + 2.23607i −3.16228 2.82843i 2.00000 + 2.23607i 3.16228
\(n\): 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 60.2.h.a 4
3.b odd 2 1 inner 60.2.h.a 4
4.b odd 2 1 inner 60.2.h.a 4
5.b even 2 1 inner 60.2.h.a 4
5.c odd 4 2 300.2.e.b 4
8.b even 2 1 960.2.o.c 4
8.d odd 2 1 960.2.o.c 4
12.b even 2 1 inner 60.2.h.a 4
15.d odd 2 1 inner 60.2.h.a 4
15.e even 4 2 300.2.e.b 4
20.d odd 2 1 CM 60.2.h.a 4
20.e even 4 2 300.2.e.b 4
24.f even 2 1 960.2.o.c 4
24.h odd 2 1 960.2.o.c 4
40.e odd 2 1 960.2.o.c 4
40.f even 2 1 960.2.o.c 4
60.h even 2 1 inner 60.2.h.a 4
60.l odd 4 2 300.2.e.b 4
120.i odd 2 1 960.2.o.c 4
120.m even 2 1 960.2.o.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.h.a 4 1.a even 1 1 trivial
60.2.h.a 4 3.b odd 2 1 inner
60.2.h.a 4 4.b odd 2 1 inner
60.2.h.a 4 5.b even 2 1 inner
60.2.h.a 4 12.b even 2 1 inner
60.2.h.a 4 15.d odd 2 1 inner
60.2.h.a 4 20.d odd 2 1 CM
60.2.h.a 4 60.h even 2 1 inner
300.2.e.b 4 5.c odd 4 2
300.2.e.b 4 15.e even 4 2
300.2.e.b 4 20.e even 4 2
300.2.e.b 4 60.l odd 4 2
960.2.o.c 4 8.b even 2 1
960.2.o.c 4 8.d odd 2 1
960.2.o.c 4 24.f even 2 1
960.2.o.c 4 24.h odd 2 1
960.2.o.c 4 40.e odd 2 1
960.2.o.c 4 40.f even 2 1
960.2.o.c 4 120.i odd 2 1
960.2.o.c 4 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 10 \) acting on \(S_{2}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( 9 - 4 T^{2} + T^{4} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( ( -10 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 2 + T^{2} )^{2} \)
$29$ \( ( 80 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 20 + T^{2} )^{2} \)
$43$ \( ( -10 + T^{2} )^{2} \)
$47$ \( ( 98 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -8 + T )^{4} \)
$67$ \( ( -250 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 242 + T^{2} )^{2} \)
$89$ \( ( 320 + T^{2} )^{2} \)
$97$ \( T^{4} \)
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