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: \( x^{2} + 5 \) Copy content Toggle raw display
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.

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

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