Properties

Label 240.2.w.a
Level $240$
Weight $2$
Character orbit 240.w
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.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{3} + ( -2 + \zeta_{8}^{2} ) q^{5} + 4 \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{8} q^{3} + ( -2 + \zeta_{8}^{2} ) q^{5} + 4 \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11} + ( 3 - 3 \zeta_{8}^{2} ) q^{13} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{15} + ( -1 - \zeta_{8}^{2} ) q^{17} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{19} -4 q^{21} -4 \zeta_{8} q^{23} + ( 3 - 4 \zeta_{8}^{2} ) q^{25} + \zeta_{8}^{3} q^{27} + 4 \zeta_{8}^{2} q^{29} + ( -4 + 4 \zeta_{8}^{2} ) q^{33} + ( -4 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{35} + ( 5 + 5 \zeta_{8}^{2} ) q^{37} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{39} -4 \zeta_{8} q^{43} + ( -1 - 2 \zeta_{8}^{2} ) q^{45} + 4 \zeta_{8}^{3} q^{47} -9 \zeta_{8}^{2} q^{49} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{51} + ( 1 - \zeta_{8}^{2} ) q^{53} + ( -12 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{55} + ( 4 + 4 \zeta_{8}^{2} ) q^{57} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{59} + 4 q^{61} -4 \zeta_{8} q^{63} + ( -3 + 9 \zeta_{8}^{2} ) q^{65} -4 \zeta_{8}^{3} q^{67} -4 \zeta_{8}^{2} q^{69} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} + ( -3 + 3 \zeta_{8}^{2} ) q^{73} + ( 3 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{75} + ( -16 - 16 \zeta_{8}^{2} ) q^{77} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{79} - q^{81} -4 \zeta_{8} q^{83} + ( 3 + \zeta_{8}^{2} ) q^{85} + 4 \zeta_{8}^{3} q^{87} + 8 \zeta_{8}^{2} q^{89} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{91} + ( -4 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{95} + ( -3 - 3 \zeta_{8}^{2} ) q^{97} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{5} + O(q^{10}) \) \( 4 q - 8 q^{5} + 12 q^{13} - 4 q^{17} - 16 q^{21} + 12 q^{25} - 16 q^{33} + 20 q^{37} - 4 q^{45} + 4 q^{53} + 16 q^{57} + 16 q^{61} - 12 q^{65} - 12 q^{73} - 64 q^{77} - 4 q^{81} + 12 q^{85} - 12 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(-1\) \(\zeta_{8}^{2}\) \(1\) \(1\)

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
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 −0.707107 0.707107i 0 −2.00000 + 1.00000i 0 2.82843 2.82843i 0 1.00000i 0
127.2 0 0.707107 + 0.707107i 0 −2.00000 + 1.00000i 0 −2.82843 + 2.82843i 0 1.00000i 0
223.1 0 −0.707107 + 0.707107i 0 −2.00000 1.00000i 0 2.82843 + 2.82843i 0 1.00000i 0
223.2 0 0.707107 0.707107i 0 −2.00000 1.00000i 0 −2.82843 2.82843i 0 1.00000i 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
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.a 4
3.b odd 2 1 720.2.x.e 4
4.b odd 2 1 inner 240.2.w.a 4
5.b even 2 1 1200.2.w.a 4
5.c odd 4 1 inner 240.2.w.a 4
5.c odd 4 1 1200.2.w.a 4
8.b even 2 1 960.2.w.c 4
8.d odd 2 1 960.2.w.c 4
12.b even 2 1 720.2.x.e 4
15.d odd 2 1 3600.2.x.d 4
15.e even 4 1 720.2.x.e 4
15.e even 4 1 3600.2.x.d 4
20.d odd 2 1 1200.2.w.a 4
20.e even 4 1 inner 240.2.w.a 4
20.e even 4 1 1200.2.w.a 4
40.i odd 4 1 960.2.w.c 4
40.k even 4 1 960.2.w.c 4
60.h even 2 1 3600.2.x.d 4
60.l odd 4 1 720.2.x.e 4
60.l odd 4 1 3600.2.x.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.w.a 4 1.a even 1 1 trivial
240.2.w.a 4 4.b odd 2 1 inner
240.2.w.a 4 5.c odd 4 1 inner
240.2.w.a 4 20.e even 4 1 inner
720.2.x.e 4 3.b odd 2 1
720.2.x.e 4 12.b even 2 1
720.2.x.e 4 15.e even 4 1
720.2.x.e 4 60.l odd 4 1
960.2.w.c 4 8.b even 2 1
960.2.w.c 4 8.d odd 2 1
960.2.w.c 4 40.i odd 4 1
960.2.w.c 4 40.k even 4 1
1200.2.w.a 4 5.b even 2 1
1200.2.w.a 4 5.c odd 4 1
1200.2.w.a 4 20.d odd 2 1
1200.2.w.a 4 20.e even 4 1
3600.2.x.d 4 15.d odd 2 1
3600.2.x.d 4 15.e even 4 1
3600.2.x.d 4 60.h even 2 1
3600.2.x.d 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 256 \) acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + T^{4} \)
$5$ \( ( 5 + 4 T + T^{2} )^{2} \)
$7$ \( 256 + T^{4} \)
$11$ \( ( 32 + T^{2} )^{2} \)
$13$ \( ( 18 - 6 T + T^{2} )^{2} \)
$17$ \( ( 2 + 2 T + T^{2} )^{2} \)
$19$ \( ( -32 + T^{2} )^{2} \)
$23$ \( 256 + T^{4} \)
$29$ \( ( 16 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 50 - 10 T + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( 256 + T^{4} \)
$47$ \( 256 + T^{4} \)
$53$ \( ( 2 - 2 T + T^{2} )^{2} \)
$59$ \( ( -128 + T^{2} )^{2} \)
$61$ \( ( -4 + T )^{4} \)
$67$ \( 256 + T^{4} \)
$71$ \( ( 32 + T^{2} )^{2} \)
$73$ \( ( 18 + 6 T + T^{2} )^{2} \)
$79$ \( ( -32 + T^{2} )^{2} \)
$83$ \( 256 + T^{4} \)
$89$ \( ( 64 + T^{2} )^{2} \)
$97$ \( ( 18 + 6 T + T^{2} )^{2} \)
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