Properties

Label 60.3.g.a
Level $60$
Weight $3$
Character orbit 60.g
Analytic conductor $1.635$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.63488158616\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \(x^{2} + 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + \beta ) q^{3} + \beta q^{5} + 2 q^{7} + ( -1 + 4 \beta ) q^{9} +O(q^{10})\) \( q + ( 2 + \beta ) q^{3} + \beta q^{5} + 2 q^{7} + ( -1 + 4 \beta ) q^{9} -6 \beta q^{11} + 8 q^{13} + ( -5 + 2 \beta ) q^{15} -6 \beta q^{17} -34 q^{19} + ( 4 + 2 \beta ) q^{21} -18 \beta q^{23} -5 q^{25} + ( -22 + 7 \beta ) q^{27} + 18 \beta q^{29} + 14 q^{31} + ( 30 - 12 \beta ) q^{33} + 2 \beta q^{35} + 56 q^{37} + ( 16 + 8 \beta ) q^{39} + 12 \beta q^{41} + 8 q^{43} + ( -20 - \beta ) q^{45} + 18 \beta q^{47} -45 q^{49} + ( 30 - 12 \beta ) q^{51} + 18 \beta q^{53} + 30 q^{55} + ( -68 - 34 \beta ) q^{57} -6 \beta q^{59} -46 q^{61} + ( -2 + 8 \beta ) q^{63} + 8 \beta q^{65} + 32 q^{67} + ( 90 - 36 \beta ) q^{69} -24 \beta q^{71} -106 q^{73} + ( -10 - 5 \beta ) q^{75} -12 \beta q^{77} -22 q^{79} + ( -79 - 8 \beta ) q^{81} + 54 \beta q^{83} + 30 q^{85} + ( -90 + 36 \beta ) q^{87} -48 \beta q^{89} + 16 q^{91} + ( 28 + 14 \beta ) q^{93} -34 \beta q^{95} + 122 q^{97} + ( 120 + 6 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} + 4q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 4q^{3} + 4q^{7} - 2q^{9} + 16q^{13} - 10q^{15} - 68q^{19} + 8q^{21} - 10q^{25} - 44q^{27} + 28q^{31} + 60q^{33} + 112q^{37} + 32q^{39} + 16q^{43} - 40q^{45} - 90q^{49} + 60q^{51} + 60q^{55} - 136q^{57} - 92q^{61} - 4q^{63} + 64q^{67} + 180q^{69} - 212q^{73} - 20q^{75} - 44q^{79} - 158q^{81} + 60q^{85} - 180q^{87} + 32q^{91} + 56q^{93} + 244q^{97} + 240q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
2.23607i
2.23607i
0 2.00000 2.23607i 0 2.23607i 0 2.00000 0 −1.00000 8.94427i 0
41.2 0 2.00000 + 2.23607i 0 2.23607i 0 2.00000 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

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 - 4 T + T^{2} \)
$5$ \( 5 + T^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( 180 + T^{2} \)
$13$ \( ( -8 + T )^{2} \)
$17$ \( 180 + T^{2} \)
$19$ \( ( 34 + T )^{2} \)
$23$ \( 1620 + T^{2} \)
$29$ \( 1620 + T^{2} \)
$31$ \( ( -14 + T )^{2} \)
$37$ \( ( -56 + T )^{2} \)
$41$ \( 720 + T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( 1620 + T^{2} \)
$53$ \( 1620 + T^{2} \)
$59$ \( 180 + T^{2} \)
$61$ \( ( 46 + T )^{2} \)
$67$ \( ( -32 + T )^{2} \)
$71$ \( 2880 + T^{2} \)
$73$ \( ( 106 + T )^{2} \)
$79$ \( ( 22 + T )^{2} \)
$83$ \( 14580 + T^{2} \)
$89$ \( 11520 + T^{2} \)
$97$ \( ( -122 + T )^{2} \)
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