Properties

Label 60.4.i.a
Level $60$
Weight $4$
Character orbit 60.i
Analytic conductor $3.540$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.54011460034\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - 5 \zeta_{8} + \zeta_{8}^{2} ) q^{3} + ( 5 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{5} + ( 23 + 23 \zeta_{8}^{2} ) q^{7} + ( 10 \zeta_{8} + 23 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - 5 \zeta_{8} + \zeta_{8}^{2} ) q^{3} + ( 5 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{5} + ( 23 + 23 \zeta_{8}^{2} ) q^{7} + ( 10 \zeta_{8} + 23 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{9} + ( -5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{11} + ( -24 + 24 \zeta_{8}^{2} ) q^{13} + ( 50 - 15 \zeta_{8} - 25 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{15} + 110 \zeta_{8}^{3} q^{17} -68 \zeta_{8}^{2} q^{19} + ( -46 - 115 \zeta_{8} - 115 \zeta_{8}^{3} ) q^{21} + 140 \zeta_{8} q^{23} + ( -100 - 75 \zeta_{8}^{2} ) q^{25} + ( -73 - 73 \zeta_{8}^{2} - 95 \zeta_{8}^{3} ) q^{27} + ( 95 \zeta_{8} - 95 \zeta_{8}^{3} ) q^{29} + 94 q^{31} + ( -25 + 10 \zeta_{8} + 25 \zeta_{8}^{2} ) q^{33} + ( -115 \zeta_{8} + 345 \zeta_{8}^{3} ) q^{35} + ( -66 - 66 \zeta_{8}^{2} ) q^{37} + ( 120 \zeta_{8} - 48 \zeta_{8}^{2} - 120 \zeta_{8}^{3} ) q^{39} + ( -140 \zeta_{8} - 140 \zeta_{8}^{3} ) q^{41} + ( 126 - 126 \zeta_{8}^{2} ) q^{43} + ( -50 - 230 \zeta_{8} + 150 \zeta_{8}^{2} + 115 \zeta_{8}^{3} ) q^{45} + 120 \zeta_{8}^{3} q^{47} + 715 \zeta_{8}^{2} q^{49} + ( 550 - 110 \zeta_{8} - 110 \zeta_{8}^{3} ) q^{51} + 460 \zeta_{8} q^{53} + ( 75 + 25 \zeta_{8}^{2} ) q^{55} + ( 68 + 68 \zeta_{8}^{2} + 340 \zeta_{8}^{3} ) q^{57} + ( 35 \zeta_{8} - 35 \zeta_{8}^{3} ) q^{59} -126 q^{61} + ( -529 + 460 \zeta_{8} + 529 \zeta_{8}^{2} ) q^{63} + ( -360 \zeta_{8} - 120 \zeta_{8}^{3} ) q^{65} + ( 68 + 68 \zeta_{8}^{2} ) q^{67} + ( -140 \zeta_{8} - 700 \zeta_{8}^{2} + 140 \zeta_{8}^{3} ) q^{69} + ( -670 \zeta_{8} - 670 \zeta_{8}^{3} ) q^{71} + ( 403 - 403 \zeta_{8}^{2} ) q^{73} + ( 175 + 500 \zeta_{8} - 25 \zeta_{8}^{2} + 375 \zeta_{8}^{3} ) q^{75} -230 \zeta_{8}^{3} q^{77} -234 \zeta_{8}^{2} q^{79} + ( -329 + 460 \zeta_{8} + 460 \zeta_{8}^{3} ) q^{81} -1020 \zeta_{8} q^{83} + ( -550 - 1100 \zeta_{8}^{2} ) q^{85} + ( -475 - 475 \zeta_{8}^{2} + 190 \zeta_{8}^{3} ) q^{87} + ( 730 \zeta_{8} - 730 \zeta_{8}^{3} ) q^{89} -1104 q^{91} + ( -94 - 470 \zeta_{8} + 94 \zeta_{8}^{2} ) q^{93} + ( 680 \zeta_{8} - 340 \zeta_{8}^{3} ) q^{95} + ( 723 + 723 \zeta_{8}^{2} ) q^{97} + ( 115 \zeta_{8} - 100 \zeta_{8}^{2} - 115 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 92q^{7} + O(q^{10}) \) \( 4q - 4q^{3} + 92q^{7} - 96q^{13} + 200q^{15} - 184q^{21} - 400q^{25} - 292q^{27} + 376q^{31} - 100q^{33} - 264q^{37} + 504q^{43} - 200q^{45} + 2200q^{51} + 300q^{55} + 272q^{57} - 504q^{61} - 2116q^{63} + 272q^{67} + 1612q^{73} + 700q^{75} - 1316q^{81} - 2200q^{85} - 1900q^{87} - 4416q^{91} - 376q^{93} + 2892q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(\zeta_{8}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 −4.53553 2.53553i 0 −3.53553 + 10.6066i 0 23.0000 + 23.0000i 0 14.1421 + 23.0000i 0
17.2 0 2.53553 + 4.53553i 0 3.53553 10.6066i 0 23.0000 + 23.0000i 0 −14.1421 + 23.0000i 0
53.1 0 −4.53553 + 2.53553i 0 −3.53553 10.6066i 0 23.0000 23.0000i 0 14.1421 23.0000i 0
53.2 0 2.53553 4.53553i 0 3.53553 + 10.6066i 0 23.0000 23.0000i 0 −14.1421 23.0000i 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.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.4.i.a 4
3.b odd 2 1 inner 60.4.i.a 4
4.b odd 2 1 240.4.v.a 4
5.b even 2 1 300.4.i.c 4
5.c odd 4 1 inner 60.4.i.a 4
5.c odd 4 1 300.4.i.c 4
12.b even 2 1 240.4.v.a 4
15.d odd 2 1 300.4.i.c 4
15.e even 4 1 inner 60.4.i.a 4
15.e even 4 1 300.4.i.c 4
20.e even 4 1 240.4.v.a 4
60.l odd 4 1 240.4.v.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.i.a 4 1.a even 1 1 trivial
60.4.i.a 4 3.b odd 2 1 inner
60.4.i.a 4 5.c odd 4 1 inner
60.4.i.a 4 15.e even 4 1 inner
240.4.v.a 4 4.b odd 2 1
240.4.v.a 4 12.b even 2 1
240.4.v.a 4 20.e even 4 1
240.4.v.a 4 60.l odd 4 1
300.4.i.c 4 5.b even 2 1
300.4.i.c 4 5.c odd 4 1
300.4.i.c 4 15.d odd 2 1
300.4.i.c 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 46 T_{7} + 1058 \) acting on \(S_{4}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 729 + 108 T + 8 T^{2} + 4 T^{3} + T^{4} \)
$5$ \( 15625 + 200 T^{2} + T^{4} \)
$7$ \( ( 1058 - 46 T + T^{2} )^{2} \)
$11$ \( ( 50 + T^{2} )^{2} \)
$13$ \( ( 1152 + 48 T + T^{2} )^{2} \)
$17$ \( 146410000 + T^{4} \)
$19$ \( ( 4624 + T^{2} )^{2} \)
$23$ \( 384160000 + T^{4} \)
$29$ \( ( -18050 + T^{2} )^{2} \)
$31$ \( ( -94 + T )^{4} \)
$37$ \( ( 8712 + 132 T + T^{2} )^{2} \)
$41$ \( ( 39200 + T^{2} )^{2} \)
$43$ \( ( 31752 - 252 T + T^{2} )^{2} \)
$47$ \( 207360000 + T^{4} \)
$53$ \( 44774560000 + T^{4} \)
$59$ \( ( -2450 + T^{2} )^{2} \)
$61$ \( ( 126 + T )^{4} \)
$67$ \( ( 9248 - 136 T + T^{2} )^{2} \)
$71$ \( ( 897800 + T^{2} )^{2} \)
$73$ \( ( 324818 - 806 T + T^{2} )^{2} \)
$79$ \( ( 54756 + T^{2} )^{2} \)
$83$ \( 1082432160000 + T^{4} \)
$89$ \( ( -1065800 + T^{2} )^{2} \)
$97$ \( ( 1045458 - 1446 T + T^{2} )^{2} \)
show more
show less