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Properties

Label 240.2.o.b
Level $240$
Weight $2$
Character orbit 240.o
Analytic conductor $1.916$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{3} + ( -\zeta_{24} - 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{5} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{3} + ( -\zeta_{24} - 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{5} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{9} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{11} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{13} + ( 1 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{15} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{17} + ( 2 - 4 \zeta_{24}^{4} ) q^{19} -6 \zeta_{24}^{6} q^{23} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{25} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{27} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{29} + ( -2 + 4 \zeta_{24}^{4} ) q^{31} + ( 2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{33} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{37} + ( -4 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{39} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{41} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{43} + ( -4 - 3 \zeta_{24} - 2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{45} -6 \zeta_{24}^{6} q^{47} -7 q^{49} + ( 2 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{51} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{53} + ( -4 - 6 \zeta_{24} - 6 \zeta_{24}^{3} + 8 \zeta_{24}^{4} + 6 \zeta_{24}^{5} ) q^{55} + ( -2 \zeta_{24} + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{57} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{59} -2 q^{61} + ( 6 \zeta_{24} - 8 \zeta_{24}^{2} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{65} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{67} + ( 6 + 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{69} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{71} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{73} + ( 4 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + \zeta_{24}^{5} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{75} + ( 6 - 12 \zeta_{24}^{4} ) q^{79} + ( -7 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{81} -6 \zeta_{24}^{6} q^{83} + ( 6 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{85} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{87} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{89} + ( 2 \zeta_{24} - 4 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{93} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{95} + ( -8 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 16 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9} + O(q^{10}) \) \( 8 q + 8 q^{9} + 8 q^{25} - 32 q^{45} - 56 q^{49} - 16 q^{61} + 48 q^{69} - 56 q^{81} + 48 q^{85} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
−0.965926 + 0.258819i
0.258819 − 0.965926i
−0.965926 − 0.258819i
0.258819 + 0.965926i
0.965926 − 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.258819 − 0.965926i
0 −1.41421 − 1.00000i 0 −1.73205 + 1.41421i 0 0 0 1.00000 + 2.82843i 0
239.2 0 −1.41421 − 1.00000i 0 1.73205 + 1.41421i 0 0 0 1.00000 + 2.82843i 0
239.3 0 −1.41421 + 1.00000i 0 −1.73205 − 1.41421i 0 0 0 1.00000 − 2.82843i 0
239.4 0 −1.41421 + 1.00000i 0 1.73205 − 1.41421i 0 0 0 1.00000 − 2.82843i 0
239.5 0 1.41421 − 1.00000i 0 −1.73205 − 1.41421i 0 0 0 1.00000 − 2.82843i 0
239.6 0 1.41421 − 1.00000i 0 1.73205 − 1.41421i 0 0 0 1.00000 − 2.82843i 0
239.7 0 1.41421 + 1.00000i 0 −1.73205 + 1.41421i 0 0 0 1.00000 + 2.82843i 0
239.8 0 1.41421 + 1.00000i 0 1.73205 + 1.41421i 0 0 0 1.00000 + 2.82843i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.o.b ✓ 8
3.b odd 2 1 inner 240.2.o.b ✓ 8
4.b odd 2 1 inner 240.2.o.b ✓ 8
5.b even 2 1 inner 240.2.o.b ✓ 8
5.c odd 4 1 1200.2.h.j 4
5.c odd 4 1 1200.2.h.n 4
8.b even 2 1 960.2.o.d 8
8.d odd 2 1 960.2.o.d 8
12.b even 2 1 inner 240.2.o.b ✓ 8
15.d odd 2 1 inner 240.2.o.b ✓ 8
15.e even 4 1 1200.2.h.j 4
15.e even 4 1 1200.2.h.n 4
20.d odd 2 1 inner 240.2.o.b ✓ 8
20.e even 4 1 1200.2.h.j 4
20.e even 4 1 1200.2.h.n 4
24.f even 2 1 960.2.o.d 8
24.h odd 2 1 960.2.o.d 8
40.e odd 2 1 960.2.o.d 8
40.f even 2 1 960.2.o.d 8
60.h even 2 1 inner 240.2.o.b ✓ 8
60.l odd 4 1 1200.2.h.j 4
60.l odd 4 1 1200.2.h.n 4
120.i odd 2 1 960.2.o.d 8
120.m even 2 1 960.2.o.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.o.b ✓ 8 1.a even 1 1 trivial
240.2.o.b ✓ 8 3.b odd 2 1 inner
240.2.o.b ✓ 8 4.b odd 2 1 inner
240.2.o.b ✓ 8 5.b even 2 1 inner
240.2.o.b ✓ 8 12.b even 2 1 inner
240.2.o.b ✓ 8 15.d odd 2 1 inner
240.2.o.b ✓ 8 20.d odd 2 1 inner
240.2.o.b ✓ 8 60.h even 2 1 inner
960.2.o.d 8 8.b even 2 1
960.2.o.d 8 8.d odd 2 1
960.2.o.d 8 24.f even 2 1
960.2.o.d 8 24.h odd 2 1
960.2.o.d 8 40.e odd 2 1
960.2.o.d 8 40.f even 2 1
960.2.o.d 8 120.i odd 2 1
960.2.o.d 8 120.m even 2 1
1200.2.h.j 4 5.c odd 4 1
1200.2.h.j 4 15.e even 4 1
1200.2.h.j 4 20.e even 4 1
1200.2.h.j 4 60.l odd 4 1
1200.2.h.n 4 5.c odd 4 1
1200.2.h.n 4 15.e even 4 1
1200.2.h.n 4 20.e even 4 1
1200.2.h.n 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 9 - 2 T^{2} + T^{4} )^{2} \)
$5$ \( ( 25 - 2 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( -24 + T^{2} )^{4} \)
$13$ \( ( 24 + T^{2} )^{4} \)
$17$ \( ( -12 + T^{2} )^{4} \)
$19$ \( ( 12 + T^{2} )^{4} \)
$23$ \( ( 36 + T^{2} )^{4} \)
$29$ \( ( 8 + T^{2} )^{4} \)
$31$ \( ( 12 + T^{2} )^{4} \)
$37$ \( ( 24 + T^{2} )^{4} \)
$41$ \( ( 32 + T^{2} )^{4} \)
$43$ \( ( -72 + T^{2} )^{4} \)
$47$ \( ( 36 + T^{2} )^{4} \)
$53$ \( ( -108 + T^{2} )^{4} \)
$59$ \( ( -24 + T^{2} )^{4} \)
$61$ \( ( 2 + T )^{8} \)
$67$ \( ( -72 + T^{2} )^{4} \)
$71$ \( ( -96 + T^{2} )^{4} \)
$73$ \( ( 96 + T^{2} )^{4} \)
$79$ \( ( 108 + T^{2} )^{4} \)
$83$ \( ( 36 + T^{2} )^{4} \)
$89$ \( ( 32 + T^{2} )^{4} \)
$97$ \( T^{8} \)
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