Properties

Label 240.2.w.b
Level $240$
Weight $2$
Character orbit 240.w
Analytic conductor $1.916$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24} - \zeta_{24}^{5} ) q^{3} + ( 1 + \zeta_{24} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{5} + ( 1 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{7} -\zeta_{24}^{6} q^{9} +O(q^{10})\) \( q + ( \zeta_{24} - \zeta_{24}^{5} ) q^{3} + ( 1 + \zeta_{24} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{5} + ( 1 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{7} -\zeta_{24}^{6} q^{9} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{11} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{13} + ( \zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{15} + ( 2 - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{17} + ( -2 \zeta_{24} - 4 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{19} + ( 2 - \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{21} + ( 2 + 2 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{23} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{25} -\zeta_{24}^{3} q^{27} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{29} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{31} + ( -1 - \zeta_{24}^{6} ) q^{33} + ( -2 + 5 \zeta_{24} - 4 \zeta_{24}^{2} - \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 5 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{35} + ( -4 - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{37} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{39} + ( -6 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{41} + ( -2 - 4 \zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{43} + ( -1 + \zeta_{24}^{3} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{45} + ( -4 + 8 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{47} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{49} + ( -2 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{51} + ( -2 - 2 \zeta_{24}^{6} ) q^{53} + ( 1 + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{55} + ( -2 - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{57} + ( -7 \zeta_{24} - 7 \zeta_{24}^{3} + 7 \zeta_{24}^{5} ) q^{59} + ( -2 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{61} + ( -1 + 2 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{63} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{65} + ( 2 - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{67} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{69} + ( 4 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{71} + ( 3 + 3 \zeta_{24}^{6} ) q^{73} + ( 2 + 4 \zeta_{24}^{2} + \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{75} + ( 2 - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{77} + ( -2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{79} - q^{81} + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{83} + ( -6 + 4 \zeta_{24} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{85} + ( 3 - 6 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{87} -2 \zeta_{24}^{6} q^{89} + ( -4 + 6 \zeta_{24} - 6 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 6 \zeta_{24}^{5} ) q^{91} + ( 6 + 6 \zeta_{24}^{6} ) q^{93} + ( -4 \zeta_{24} - 8 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{95} + ( 3 - 3 \zeta_{24}^{6} ) q^{97} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} + O(q^{10}) \) \( 8 q + 8 q^{5} + 16 q^{17} + 16 q^{21} - 8 q^{33} - 32 q^{37} - 48 q^{41} - 8 q^{45} - 16 q^{53} - 16 q^{57} - 16 q^{61} + 24 q^{73} + 16 q^{77} - 8 q^{81} - 48 q^{85} + 48 q^{93} + 24 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(-1\) \(-\zeta_{24}^{3}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
0.965926 + 0.258819i
0 −0.707107 0.707107i 0 −0.224745 + 2.22474i 0 −3.14626 + 3.14626i 0 1.00000i 0
127.2 0 −0.707107 0.707107i 0 2.22474 0.224745i 0 0.317837 0.317837i 0 1.00000i 0
127.3 0 0.707107 + 0.707107i 0 −0.224745 + 2.22474i 0 3.14626 3.14626i 0 1.00000i 0
127.4 0 0.707107 + 0.707107i 0 2.22474 0.224745i 0 −0.317837 + 0.317837i 0 1.00000i 0
223.1 0 −0.707107 + 0.707107i 0 −0.224745 2.22474i 0 −3.14626 3.14626i 0 1.00000i 0
223.2 0 −0.707107 + 0.707107i 0 2.22474 + 0.224745i 0 0.317837 + 0.317837i 0 1.00000i 0
223.3 0 0.707107 0.707107i 0 −0.224745 2.22474i 0 3.14626 + 3.14626i 0 1.00000i 0
223.4 0 0.707107 0.707107i 0 2.22474 + 0.224745i 0 −0.317837 0.317837i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.w.b 8
3.b odd 2 1 720.2.x.f 8
4.b odd 2 1 inner 240.2.w.b 8
5.b even 2 1 1200.2.w.b 8
5.c odd 4 1 inner 240.2.w.b 8
5.c odd 4 1 1200.2.w.b 8
8.b even 2 1 960.2.w.d 8
8.d odd 2 1 960.2.w.d 8
12.b even 2 1 720.2.x.f 8
15.d odd 2 1 3600.2.x.n 8
15.e even 4 1 720.2.x.f 8
15.e even 4 1 3600.2.x.n 8
20.d odd 2 1 1200.2.w.b 8
20.e even 4 1 inner 240.2.w.b 8
20.e even 4 1 1200.2.w.b 8
40.i odd 4 1 960.2.w.d 8
40.k even 4 1 960.2.w.d 8
60.h even 2 1 3600.2.x.n 8
60.l odd 4 1 720.2.x.f 8
60.l odd 4 1 3600.2.x.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.w.b 8 1.a even 1 1 trivial
240.2.w.b 8 4.b odd 2 1 inner
240.2.w.b 8 5.c odd 4 1 inner
240.2.w.b 8 20.e even 4 1 inner
720.2.x.f 8 3.b odd 2 1
720.2.x.f 8 12.b even 2 1
720.2.x.f 8 15.e even 4 1
720.2.x.f 8 60.l odd 4 1
960.2.w.d 8 8.b even 2 1
960.2.w.d 8 8.d odd 2 1
960.2.w.d 8 40.i odd 4 1
960.2.w.d 8 40.k even 4 1
1200.2.w.b 8 5.b even 2 1
1200.2.w.b 8 5.c odd 4 1
1200.2.w.b 8 20.d odd 2 1
1200.2.w.b 8 20.e even 4 1
3600.2.x.n 8 15.d odd 2 1
3600.2.x.n 8 15.e even 4 1
3600.2.x.n 8 60.h even 2 1
3600.2.x.n 8 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 392 T_{7}^{4} + 16 \) acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + T^{4} )^{2} \)
$5$ \( ( 25 - 20 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$7$ \( 16 + 392 T^{4} + T^{8} \)
$11$ \( ( 2 + T^{2} )^{4} \)
$13$ \( ( 144 + T^{4} )^{2} \)
$17$ \( ( 16 + 32 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$19$ \( ( 16 - 40 T^{2} + T^{4} )^{2} \)
$23$ \( 160000 + 2336 T^{4} + T^{8} \)
$29$ \( ( 2500 + 116 T^{2} + T^{4} )^{2} \)
$31$ \( ( 72 + T^{2} )^{4} \)
$37$ \( ( 400 + 320 T + 128 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$41$ \( ( 12 + 12 T + T^{2} )^{4} \)
$43$ \( 4096 + 6272 T^{4} + T^{8} \)
$47$ \( 71639296 + 23072 T^{4} + T^{8} \)
$53$ \( ( 8 + 4 T + T^{2} )^{4} \)
$59$ \( ( -98 + T^{2} )^{4} \)
$61$ \( ( -20 + 4 T + T^{2} )^{4} \)
$67$ \( 4096 + 6272 T^{4} + T^{8} \)
$71$ \( ( 256 + 160 T^{2} + T^{4} )^{2} \)
$73$ \( ( 18 - 6 T + T^{2} )^{4} \)
$79$ \( ( 1600 - 112 T^{2} + T^{4} )^{2} \)
$83$ \( ( 16 + T^{4} )^{2} \)
$89$ \( ( 4 + T^{2} )^{4} \)
$97$ \( ( 18 - 6 T + T^{2} )^{4} \)
show more
show less