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: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_1 q^{5} + (\beta_{4} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_1 q^{5} + (\beta_{4} - \beta_1 + 1) q^{9} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{11} + \beta_{6} q^{13} + ( - \beta_{7} - \beta_{3}) q^{15} + ( - \beta_{4} - \beta_1) q^{17} - \beta_{3} q^{19} + (\beta_{7} + \beta_{5} - 2 \beta_{2}) q^{23} + ( - \beta_{6} + 1) q^{25} + ( - \beta_{7} - \beta_{5} + \beta_{2}) q^{27} + ( - \beta_{4} + \beta_1) q^{29} + \beta_{3} q^{31} + (\beta_{6} + 2 \beta_{4} + 2 \beta_1) q^{33} - \beta_{6} q^{37} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{39} + ( - 2 \beta_{4} + 2 \beta_1) q^{41} + (\beta_{7} + \beta_{5} + 4 \beta_{2}) q^{43} + ( - \beta_{6} - \beta_1 - 4) q^{45} + (\beta_{7} + \beta_{5} - 2 \beta_{2}) q^{47} - 7 q^{49} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3}) q^{51} + (3 \beta_{4} + 3 \beta_1) q^{53} + ( - \beta_{7} - \beta_{5} + 2 \beta_{3} - 4 \beta_{2}) q^{55} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{57} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{59} - 2 q^{61} + ( - 5 \beta_{4} + \beta_1) q^{65} + ( - \beta_{7} - \beta_{5} - 4 \beta_{2}) q^{67} + ( - 3 \beta_{4} + 3 \beta_1 + 6) q^{69} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3}) q^{71} + 2 \beta_{6} q^{73} + (\beta_{7} - \beta_{5} - \beta_{3} + \beta_{2}) q^{75} - 3 \beta_{3} q^{79} + (2 \beta_{4} - 2 \beta_1 - 7) q^{81} + (\beta_{7} + \beta_{5} - 2 \beta_{2}) q^{83} + ( - \beta_{6} + 6) q^{85} + (\beta_{7} + \beta_{5}) q^{87} + (2 \beta_{4} - 2 \beta_1) q^{89} + (\beta_{6} - \beta_{4} - \beta_1) q^{93} + ( - 2 \beta_{5} + \beta_{3} + 2 \beta_{2}) q^{95} + (\beta_{7} - \beta_{5} + 5 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} + 8 q^{25} - 32 q^{45} - 56 q^{49} - 16 q^{61} + 48 q^{69} - 56 q^{81} + 48 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\zeta_{24}^{4} - 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}^{2} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{5} + 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 4\zeta_{24}^{7} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{4} + 2\zeta_{24}^{3} + 2\zeta_{24} + 1 \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( 4\beta_{7} + 3\beta_{6} - 2\beta_{5} - 3\beta_{4} + 3\beta_{3} + 4\beta_{2} + 3\beta_1 ) / 24 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( -\beta_{7} - \beta_{5} + 3\beta_{4} + 2\beta_{2} + 3\beta_1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} + 3\beta_{4} + 4\beta_{2} - 3\beta_1 ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{3} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} + 3\beta_{6} - 4\beta_{5} + 3\beta_{4} + 3\beta_{3} - 4\beta_{2} - 3\beta_1 ) / 24 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{7} - \beta_{5} + 2\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -2\beta_{7} + 3\beta_{6} + 4\beta_{5} + 3\beta_{4} - 3\beta_{3} + 4\beta_{2} - 3\beta_1 ) / 24 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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