Properties

Label 60.3.b.a
Level $60$
Weight $3$
Character orbit 60.b
Analytic conductor $1.635$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.63488158616\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + ( 3 \beta_{2} - 3 \beta_{3} ) q^{13} + ( 5 - \beta_{1} + 4 \beta_{2} ) q^{15} + ( -7 \beta_{2} - 7 \beta_{3} ) q^{17} -6 q^{19} + ( -16 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{21} + ( \beta_{2} + \beta_{3} ) q^{23} + ( -15 + 5 \beta_{2} - 5 \beta_{3} ) q^{25} + ( -9 \beta_{2} + 2 \beta_{3} ) q^{27} + ( 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{29} + 34 q^{31} + ( 9 \beta_{2} - \beta_{3} ) q^{33} + ( -4 \beta_{1} + 6 \beta_{2} + 10 \beta_{3} ) q^{35} + ( -11 \beta_{2} + 11 \beta_{3} ) q^{37} + ( 24 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{39} + ( -4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( -7 \beta_{2} + 7 \beta_{3} ) q^{43} + ( 40 + \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{45} + ( \beta_{2} + \beta_{3} ) q^{47} -15 q^{49} + ( -70 + 14 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} ) q^{51} + ( 9 \beta_{2} + 9 \beta_{3} ) q^{53} + ( -40 + 5 \beta_{2} - 5 \beta_{3} ) q^{55} -6 \beta_{3} q^{57} + ( -22 \beta_{1} - 11 \beta_{2} + 11 \beta_{3} ) q^{59} + 74 q^{61} + ( -18 \beta_{2} - 14 \beta_{3} ) q^{63} + ( 6 \beta_{1} - 9 \beta_{2} - 15 \beta_{3} ) q^{65} + ( 23 \beta_{2} - 23 \beta_{3} ) q^{67} + ( 10 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{69} + ( -12 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{71} + ( 14 \beta_{2} - 14 \beta_{3} ) q^{73} + ( 40 + 10 \beta_{1} + 5 \beta_{2} - 20 \beta_{3} ) q^{75} + ( 16 \beta_{2} + 16 \beta_{3} ) q^{77} -78 q^{79} + ( -79 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{81} + ( -23 \beta_{2} - 23 \beta_{3} ) q^{83} + ( -70 - 35 \beta_{2} + 35 \beta_{3} ) q^{85} + ( 27 \beta_{2} - 3 \beta_{3} ) q^{87} + ( 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{89} + 96 q^{91} + 34 \beta_{3} q^{93} + ( -6 \beta_{1} - 6 \beta_{2} ) q^{95} + ( 8 \beta_{2} - 8 \beta_{3} ) q^{97} + ( 80 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{9} + 20q^{15} - 24q^{19} - 64q^{21} - 60q^{25} + 136q^{31} + 96q^{39} + 160q^{45} - 60q^{49} - 280q^{51} - 160q^{55} + 296q^{61} + 40q^{69} + 160q^{75} - 312q^{79} - 316q^{81} - 280q^{85} + 384q^{91} + 320q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 2 \nu^{2} + 4 \nu + 3 \)
\(\beta_{3}\)\(=\)\( -2 \nu^{3} + 2 \nu^{2} - 4 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 6\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} - 2 \beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
1.61803i
1.61803i
0.618034i
0.618034i
0 −2.23607 2.00000i 0 −2.23607 4.47214i 0 8.00000i 0 1.00000 + 8.94427i 0
29.2 0 −2.23607 + 2.00000i 0 −2.23607 + 4.47214i 0 8.00000i 0 1.00000 8.94427i 0
29.3 0 2.23607 2.00000i 0 2.23607 + 4.47214i 0 8.00000i 0 1.00000 8.94427i 0
29.4 0 2.23607 + 2.00000i 0 2.23607 4.47214i 0 8.00000i 0 1.00000 + 8.94427i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.3.b.a 4
3.b odd 2 1 inner 60.3.b.a 4
4.b odd 2 1 240.3.c.d 4
5.b even 2 1 inner 60.3.b.a 4
5.c odd 4 1 300.3.g.e 2
5.c odd 4 1 300.3.g.h 2
8.b even 2 1 960.3.c.h 4
8.d odd 2 1 960.3.c.g 4
9.c even 3 2 1620.3.t.b 8
9.d odd 6 2 1620.3.t.b 8
12.b even 2 1 240.3.c.d 4
15.d odd 2 1 inner 60.3.b.a 4
15.e even 4 1 300.3.g.e 2
15.e even 4 1 300.3.g.h 2
20.d odd 2 1 240.3.c.d 4
20.e even 4 1 1200.3.l.h 2
20.e even 4 1 1200.3.l.q 2
24.f even 2 1 960.3.c.g 4
24.h odd 2 1 960.3.c.h 4
40.e odd 2 1 960.3.c.g 4
40.f even 2 1 960.3.c.h 4
45.h odd 6 2 1620.3.t.b 8
45.j even 6 2 1620.3.t.b 8
60.h even 2 1 240.3.c.d 4
60.l odd 4 1 1200.3.l.h 2
60.l odd 4 1 1200.3.l.q 2
120.i odd 2 1 960.3.c.h 4
120.m even 2 1 960.3.c.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.b.a 4 1.a even 1 1 trivial
60.3.b.a 4 3.b odd 2 1 inner
60.3.b.a 4 5.b even 2 1 inner
60.3.b.a 4 15.d odd 2 1 inner
240.3.c.d 4 4.b odd 2 1
240.3.c.d 4 12.b even 2 1
240.3.c.d 4 20.d odd 2 1
240.3.c.d 4 60.h even 2 1
300.3.g.e 2 5.c odd 4 1
300.3.g.e 2 15.e even 4 1
300.3.g.h 2 5.c odd 4 1
300.3.g.h 2 15.e even 4 1
960.3.c.g 4 8.d odd 2 1
960.3.c.g 4 24.f even 2 1
960.3.c.g 4 40.e odd 2 1
960.3.c.g 4 120.m even 2 1
960.3.c.h 4 8.b even 2 1
960.3.c.h 4 24.h odd 2 1
960.3.c.h 4 40.f even 2 1
960.3.c.h 4 120.i odd 2 1
1200.3.l.h 2 20.e even 4 1
1200.3.l.h 2 60.l odd 4 1
1200.3.l.q 2 20.e even 4 1
1200.3.l.q 2 60.l odd 4 1
1620.3.t.b 8 9.c even 3 2
1620.3.t.b 8 9.d odd 6 2
1620.3.t.b 8 45.h odd 6 2
1620.3.t.b 8 45.j even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 81 - 2 T^{2} + T^{4} \)
$5$ \( 625 + 30 T^{2} + T^{4} \)
$7$ \( ( 64 + T^{2} )^{2} \)
$11$ \( ( 80 + T^{2} )^{2} \)
$13$ \( ( 144 + T^{2} )^{2} \)
$17$ \( ( -980 + T^{2} )^{2} \)
$19$ \( ( 6 + T )^{4} \)
$23$ \( ( -20 + T^{2} )^{2} \)
$29$ \( ( 720 + T^{2} )^{2} \)
$31$ \( ( -34 + T )^{4} \)
$37$ \( ( 1936 + T^{2} )^{2} \)
$41$ \( ( 320 + T^{2} )^{2} \)
$43$ \( ( 784 + T^{2} )^{2} \)
$47$ \( ( -20 + T^{2} )^{2} \)
$53$ \( ( -1620 + T^{2} )^{2} \)
$59$ \( ( 9680 + T^{2} )^{2} \)
$61$ \( ( -74 + T )^{4} \)
$67$ \( ( 8464 + T^{2} )^{2} \)
$71$ \( ( 2880 + T^{2} )^{2} \)
$73$ \( ( 3136 + T^{2} )^{2} \)
$79$ \( ( 78 + T )^{4} \)
$83$ \( ( -10580 + T^{2} )^{2} \)
$89$ \( ( 320 + T^{2} )^{2} \)
$97$ \( ( 1024 + T^{2} )^{2} \)
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