Properties

Label 60.3.k.a
Level $60$
Weight $3$
Character orbit 60.k
Analytic conductor $1.635$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 60.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.63488158616\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{5} + ( 5 - 5 \beta_{2} + 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{5} + ( 5 - 5 \beta_{2} + 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} + ( -4 + 5 \beta_{1} - 5 \beta_{3} ) q^{11} -10 \beta_{1} q^{13} + ( -6 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{15} + ( -10 + 10 \beta_{2} + 2 \beta_{3} ) q^{17} + ( -10 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} ) q^{19} + ( -6 + 5 \beta_{1} - 5 \beta_{3} ) q^{21} + ( -10 - 6 \beta_{1} - 10 \beta_{2} ) q^{23} + ( -4 + 2 \beta_{1} - 3 \beta_{2} + 14 \beta_{3} ) q^{25} + 3 \beta_{3} q^{27} + ( -5 \beta_{1} + 22 \beta_{2} - 5 \beta_{3} ) q^{29} + ( 4 + 10 \beta_{1} - 10 \beta_{3} ) q^{31} + ( 15 - 4 \beta_{1} + 15 \beta_{2} ) q^{33} + ( 14 + 13 \beta_{1} - 22 \beta_{2} + 11 \beta_{3} ) q^{35} + 6 \beta_{3} q^{37} -30 \beta_{2} q^{39} + ( 50 + 10 \beta_{1} - 10 \beta_{3} ) q^{41} + ( 30 + 4 \beta_{1} + 30 \beta_{2} ) q^{43} + ( -3 - 6 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} ) q^{45} + 18 \beta_{3} q^{47} + ( 20 \beta_{1} - 13 \beta_{2} + 20 \beta_{3} ) q^{49} + ( -6 - 10 \beta_{1} + 10 \beta_{3} ) q^{51} + ( -50 - 12 \beta_{1} - 50 \beta_{2} ) q^{53} + ( -27 + 16 \beta_{1} + 41 \beta_{2} - 18 \beta_{3} ) q^{55} + ( 30 - 30 \beta_{2} - 10 \beta_{3} ) q^{57} + ( -15 \beta_{1} + 52 \beta_{2} - 15 \beta_{3} ) q^{59} + ( -78 + 10 \beta_{1} - 10 \beta_{3} ) q^{61} + ( 15 - 6 \beta_{1} + 15 \beta_{2} ) q^{63} + ( 60 - 30 \beta_{1} - 30 \beta_{2} - 10 \beta_{3} ) q^{65} + ( -10 + 10 \beta_{2} - 12 \beta_{3} ) q^{67} + ( -10 \beta_{1} - 18 \beta_{2} - 10 \beta_{3} ) q^{69} + ( -20 - 40 \beta_{1} + 40 \beta_{3} ) q^{71} + ( -5 + 32 \beta_{1} - 5 \beta_{2} ) q^{73} + ( -42 - 4 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{75} + ( -50 + 50 \beta_{2} - 58 \beta_{3} ) q^{77} + ( -10 \beta_{1} + 24 \beta_{2} - 10 \beta_{3} ) q^{79} -9 q^{81} + ( -60 + 34 \beta_{1} - 60 \beta_{2} ) q^{83} + ( -46 - 32 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{85} + ( 15 - 15 \beta_{2} + 22 \beta_{3} ) q^{87} + ( 60 \beta_{1} - 10 \beta_{2} + 60 \beta_{3} ) q^{89} + ( 60 - 50 \beta_{1} + 50 \beta_{3} ) q^{91} + ( 30 + 4 \beta_{1} + 30 \beta_{2} ) q^{93} + ( 100 - 50 \beta_{3} ) q^{95} + ( 75 - 75 \beta_{2} - 16 \beta_{3} ) q^{97} + ( 15 \beta_{1} - 12 \beta_{2} + 15 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{5} + 20 q^{7} + O(q^{10}) \) \( 4 q + 12 q^{5} + 20 q^{7} - 16 q^{11} - 24 q^{15} - 40 q^{17} - 24 q^{21} - 40 q^{23} - 16 q^{25} + 16 q^{31} + 60 q^{33} + 56 q^{35} + 200 q^{41} + 120 q^{43} - 12 q^{45} - 24 q^{51} - 200 q^{53} - 108 q^{55} + 120 q^{57} - 312 q^{61} + 60 q^{63} + 240 q^{65} - 40 q^{67} - 80 q^{71} - 20 q^{73} - 168 q^{75} - 200 q^{77} - 36 q^{81} - 240 q^{83} - 184 q^{85} + 60 q^{87} + 240 q^{91} + 120 q^{93} + 400 q^{95} + 300 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
0 −1.22474 1.22474i 0 4.22474 2.67423i 0 7.44949 7.44949i 0 3.00000i 0
13.2 0 1.22474 + 1.22474i 0 1.77526 + 4.67423i 0 2.55051 2.55051i 0 3.00000i 0
37.1 0 −1.22474 + 1.22474i 0 4.22474 + 2.67423i 0 7.44949 + 7.44949i 0 3.00000i 0
37.2 0 1.22474 1.22474i 0 1.77526 4.67423i 0 2.55051 + 2.55051i 0 3.00000i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.3.k.a 4
3.b odd 2 1 180.3.l.b 4
4.b odd 2 1 240.3.bg.d 4
5.b even 2 1 300.3.k.a 4
5.c odd 4 1 inner 60.3.k.a 4
5.c odd 4 1 300.3.k.a 4
8.b even 2 1 960.3.bg.b 4
8.d odd 2 1 960.3.bg.a 4
12.b even 2 1 720.3.bh.f 4
15.d odd 2 1 900.3.l.b 4
15.e even 4 1 180.3.l.b 4
15.e even 4 1 900.3.l.b 4
20.d odd 2 1 1200.3.bg.o 4
20.e even 4 1 240.3.bg.d 4
20.e even 4 1 1200.3.bg.o 4
40.i odd 4 1 960.3.bg.b 4
40.k even 4 1 960.3.bg.a 4
60.l odd 4 1 720.3.bh.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.k.a 4 1.a even 1 1 trivial
60.3.k.a 4 5.c odd 4 1 inner
180.3.l.b 4 3.b odd 2 1
180.3.l.b 4 15.e even 4 1
240.3.bg.d 4 4.b odd 2 1
240.3.bg.d 4 20.e even 4 1
300.3.k.a 4 5.b even 2 1
300.3.k.a 4 5.c odd 4 1
720.3.bh.f 4 12.b even 2 1
720.3.bh.f 4 60.l odd 4 1
900.3.l.b 4 15.d odd 2 1
900.3.l.b 4 15.e even 4 1
960.3.bg.a 4 8.d odd 2 1
960.3.bg.a 4 40.k even 4 1
960.3.bg.b 4 8.b even 2 1
960.3.bg.b 4 40.i odd 4 1
1200.3.bg.o 4 20.d odd 2 1
1200.3.bg.o 4 20.e even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + T^{4} \)
$5$ \( 625 - 300 T + 80 T^{2} - 12 T^{3} + T^{4} \)
$7$ \( 1444 - 760 T + 200 T^{2} - 20 T^{3} + T^{4} \)
$11$ \( ( -134 + 8 T + T^{2} )^{2} \)
$13$ \( 90000 + T^{4} \)
$17$ \( 35344 + 7520 T + 800 T^{2} + 40 T^{3} + T^{4} \)
$19$ \( 250000 + 1400 T^{2} + T^{4} \)
$23$ \( 8464 + 3680 T + 800 T^{2} + 40 T^{3} + T^{4} \)
$29$ \( 111556 + 1268 T^{2} + T^{4} \)
$31$ \( ( -584 - 8 T + T^{2} )^{2} \)
$37$ \( 11664 + T^{4} \)
$41$ \( ( 1900 - 100 T + T^{2} )^{2} \)
$43$ \( 3069504 - 210240 T + 7200 T^{2} - 120 T^{3} + T^{4} \)
$47$ \( 944784 + T^{4} \)
$53$ \( 20866624 + 913600 T + 20000 T^{2} + 200 T^{3} + T^{4} \)
$59$ \( 1833316 + 8108 T^{2} + T^{4} \)
$61$ \( ( 5484 + 156 T + T^{2} )^{2} \)
$67$ \( 53824 - 9280 T + 800 T^{2} + 40 T^{3} + T^{4} \)
$71$ \( ( -9200 + 40 T + T^{2} )^{2} \)
$73$ \( 9132484 - 60440 T + 200 T^{2} + 20 T^{3} + T^{4} \)
$79$ \( 576 + 2352 T^{2} + T^{4} \)
$83$ \( 13927824 + 895680 T + 28800 T^{2} + 240 T^{3} + T^{4} \)
$89$ \( 462250000 + 43400 T^{2} + T^{4} \)
$97$ \( 109872324 - 3144600 T + 45000 T^{2} - 300 T^{3} + T^{4} \)
show more
show less