Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{3} q^{5} + ( -2 + 4 \zeta_{12}^{2} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{3} q^{5} + ( -2 + 4 \zeta_{12}^{2} ) q^{7} + 3 q^{9} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{11} + 4 q^{13} + ( -1 + 2 \zeta_{12}^{2} ) q^{15} + 6 \zeta_{12}^{3} q^{17} + ( 2 - 4 \zeta_{12}^{2} ) q^{19} -6 \zeta_{12}^{3} q^{21} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{23} - q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + 6 \zeta_{12}^{3} q^{29} + ( -2 + 4 \zeta_{12}^{2} ) q^{31} -6 q^{33} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{35} -4 q^{37} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{39} -12 \zeta_{12}^{3} q^{41} + ( 4 - 8 \zeta_{12}^{2} ) q^{43} -3 \zeta_{12}^{3} q^{45} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{47} -5 q^{49} + ( 6 - 12 \zeta_{12}^{2} ) q^{51} + 6 \zeta_{12}^{3} q^{53} + ( 2 - 4 \zeta_{12}^{2} ) q^{55} + 6 \zeta_{12}^{3} q^{57} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{59} -10 q^{61} + ( -6 + 12 \zeta_{12}^{2} ) q^{63} -4 \zeta_{12}^{3} q^{65} + ( 4 - 8 \zeta_{12}^{2} ) q^{67} -6 q^{69} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{71} -2 q^{73} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{75} + 12 \zeta_{12}^{3} q^{77} + ( -6 + 12 \zeta_{12}^{2} ) q^{79} + 9 q^{81} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{83} + 6 q^{85} + ( 6 - 12 \zeta_{12}^{2} ) q^{87} + ( -8 + 16 \zeta_{12}^{2} ) q^{91} -6 \zeta_{12}^{3} q^{93} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{95} + 10 q^{97} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9} + O(q^{10}) \) \( 4 q + 12 q^{9} + 16 q^{13} - 4 q^{25} - 24 q^{33} - 16 q^{37} - 20 q^{49} - 40 q^{61} - 24 q^{69} - 8 q^{73} + 36 q^{81} + 24 q^{85} + 40 q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0 −1.73205 0 1.00000i 0 3.46410i 0 3.00000 0
191.2 0 −1.73205 0 1.00000i 0 3.46410i 0 3.00000 0
191.3 0 1.73205 0 1.00000i 0 3.46410i 0 3.00000 0
191.4 0 1.73205 0 1.00000i 0 3.46410i 0 3.00000 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
4.b odd 2 1 inner
12.b 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.h.b 4
3.b odd 2 1 inner 240.2.h.b 4
4.b odd 2 1 inner 240.2.h.b 4
5.b even 2 1 1200.2.h.m 4
5.c odd 4 1 1200.2.o.a 4
5.c odd 4 1 1200.2.o.b 4
8.b even 2 1 960.2.h.d 4
8.d odd 2 1 960.2.h.d 4
12.b even 2 1 inner 240.2.h.b 4
15.d odd 2 1 1200.2.h.m 4
15.e even 4 1 1200.2.o.a 4
15.e even 4 1 1200.2.o.b 4
20.d odd 2 1 1200.2.h.m 4
20.e even 4 1 1200.2.o.a 4
20.e even 4 1 1200.2.o.b 4
24.f even 2 1 960.2.h.d 4
24.h odd 2 1 960.2.h.d 4
60.h even 2 1 1200.2.h.m 4
60.l odd 4 1 1200.2.o.a 4
60.l odd 4 1 1200.2.o.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.h.b 4 1.a even 1 1 trivial
240.2.h.b 4 3.b odd 2 1 inner
240.2.h.b 4 4.b odd 2 1 inner
240.2.h.b 4 12.b even 2 1 inner
960.2.h.d 4 8.b even 2 1
960.2.h.d 4 8.d odd 2 1
960.2.h.d 4 24.f even 2 1
960.2.h.d 4 24.h odd 2 1
1200.2.h.m 4 5.b even 2 1
1200.2.h.m 4 15.d odd 2 1
1200.2.h.m 4 20.d odd 2 1
1200.2.h.m 4 60.h even 2 1
1200.2.o.a 4 5.c odd 4 1
1200.2.o.a 4 15.e even 4 1
1200.2.o.a 4 20.e even 4 1
1200.2.o.a 4 60.l odd 4 1
1200.2.o.b 4 5.c odd 4 1
1200.2.o.b 4 15.e even 4 1
1200.2.o.b 4 20.e even 4 1
1200.2.o.b 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 12 \) acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( 12 + T^{2} )^{2} \)
$11$ \( ( -12 + T^{2} )^{2} \)
$13$ \( ( -4 + T )^{4} \)
$17$ \( ( 36 + T^{2} )^{2} \)
$19$ \( ( 12 + T^{2} )^{2} \)
$23$ \( ( -12 + T^{2} )^{2} \)
$29$ \( ( 36 + T^{2} )^{2} \)
$31$ \( ( 12 + T^{2} )^{2} \)
$37$ \( ( 4 + T )^{4} \)
$41$ \( ( 144 + T^{2} )^{2} \)
$43$ \( ( 48 + T^{2} )^{2} \)
$47$ \( ( -12 + T^{2} )^{2} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( -12 + T^{2} )^{2} \)
$61$ \( ( 10 + T )^{4} \)
$67$ \( ( 48 + T^{2} )^{2} \)
$71$ \( ( -192 + T^{2} )^{2} \)
$73$ \( ( 2 + T )^{4} \)
$79$ \( ( 108 + T^{2} )^{2} \)
$83$ \( ( -108 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( -10 + T )^{4} \)
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