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: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{2} - \beta_1) q^{3} + \beta_{3} q^{4} + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{3} + 2) q^{6} + ( - \beta_{2} + 2 \beta_1) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{2} - \beta_1) q^{3} + \beta_{3} q^{4} + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{3} + 2) q^{6} + ( - \beta_{2} + 2 \beta_1) q^{8} - 3 q^{9} + ( - \beta_{3} - 2) q^{10} + ( - \beta_{2} - 2 \beta_1) q^{12} + (2 \beta_{3} - 1) q^{15} + (\beta_{3} - 4) q^{16} + ( - 2 \beta_{2} - 2 \beta_1) q^{17} + 3 \beta_{2} q^{18} + ( - 4 \beta_{3} + 2) q^{19} + (3 \beta_{2} - 2 \beta_1) q^{20} + ( - 2 \beta_{2} + 2 \beta_1) q^{23} + (\beta_{3} + 4) q^{24} + 5 q^{25} + ( - 3 \beta_{2} + 3 \beta_1) q^{27} + ( - \beta_{2} + 4 \beta_1) q^{30} + (4 \beta_{3} - 2) q^{31} + (3 \beta_{2} + 2 \beta_1) q^{32} + (2 \beta_{3} + 4) q^{34} - 3 \beta_{3} q^{36} + (2 \beta_{2} - 8 \beta_1) q^{38} + ( - 3 \beta_{3} + 4) q^{40} + ( - 3 \beta_{2} - 3 \beta_1) q^{45} + (2 \beta_{3} - 4) q^{46} + (6 \beta_{2} - 6 \beta_1) q^{47} + ( - 5 \beta_{2} + 2 \beta_1) q^{48} - 7 q^{49} - 5 \beta_{2} q^{50} + ( - 4 \beta_{3} + 2) q^{51} + (2 \beta_{2} + 2 \beta_1) q^{53} + (3 \beta_{3} - 6) q^{54} + (6 \beta_{2} + 6 \beta_1) q^{57} + (\beta_{3} - 8) q^{60} - 2 q^{61} + ( - 2 \beta_{2} + 8 \beta_1) q^{62} + ( - 3 \beta_{3} - 4) q^{64} + ( - 6 \beta_{2} + 4 \beta_1) q^{68} + 6 q^{69} + (3 \beta_{2} - 6 \beta_1) q^{72} + (5 \beta_{2} - 5 \beta_1) q^{75} + ( - 2 \beta_{3} + 16) q^{76} + (4 \beta_{3} - 2) q^{79} + ( - \beta_{2} - 6 \beta_1) q^{80} + 9 q^{81} + (2 \beta_{2} - 2 \beta_1) q^{83} - 10 q^{85} + (3 \beta_{3} + 6) q^{90} + (2 \beta_{2} + 4 \beta_1) q^{92} + ( - 6 \beta_{2} - 6 \beta_1) q^{93} + ( - 6 \beta_{3} + 12) q^{94} + ( - 10 \beta_{2} + 10 \beta_1) q^{95} + (5 \beta_{3} - 4) q^{96} + 7 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 6 q^{6} - 12 q^{9} - 10 q^{10} - 14 q^{16} + 18 q^{24} + 20 q^{25} + 20 q^{34} - 6 q^{36} + 10 q^{40} - 12 q^{46} - 28 q^{49} - 18 q^{54} - 30 q^{60} - 8 q^{61} - 22 q^{64} + 24 q^{69} + 60 q^{76} + 36 q^{81} - 40 q^{85} + 30 q^{90} + 36 q^{94} - 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} + \nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} + 3\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} + 2\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(60, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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