Properties

Label 240.2.v.e
Level $240$
Weight $2$
Character orbit 240.v
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.v (of order \(4\), degree \(2\), not 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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
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.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( 1 - \zeta_{8}^{2} ) q^{7} + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( 1 - \zeta_{8}^{2} ) q^{7} + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{9} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{11} + ( -2 - \zeta_{8} - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{15} + 2 \zeta_{8} q^{17} + 4 \zeta_{8}^{2} q^{19} + ( 2 - \zeta_{8} - \zeta_{8}^{3} ) q^{21} + 4 \zeta_{8}^{3} q^{23} + ( -4 + 3 \zeta_{8}^{2} ) q^{25} + ( 1 + 5 \zeta_{8} - \zeta_{8}^{2} ) q^{27} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{29} + 2 q^{31} + ( -1 - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{33} + ( -3 \zeta_{8} + \zeta_{8}^{3} ) q^{35} + ( 6 - 6 \zeta_{8}^{2} ) q^{37} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41} + ( -6 - 6 \zeta_{8}^{2} ) q^{43} + ( -2 + \zeta_{8} - 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{45} + 5 \zeta_{8}^{2} q^{49} + ( 2 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{51} + 4 \zeta_{8}^{3} q^{53} + ( -3 + \zeta_{8}^{2} ) q^{55} + ( -4 + 4 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{57} + ( -7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{59} -6 q^{61} + ( 1 + \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{63} + ( 4 - 4 \zeta_{8}^{2} ) q^{67} + ( -4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{69} + ( 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{71} + ( -5 - 5 \zeta_{8}^{2} ) q^{73} + ( -7 + 3 \zeta_{8} - \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{75} -2 \zeta_{8} q^{77} + 6 \zeta_{8}^{2} q^{79} + ( 7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} -12 \zeta_{8}^{3} q^{83} + ( 2 - 4 \zeta_{8}^{2} ) q^{85} + ( -5 - 10 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{87} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{89} + ( 2 + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{93} + ( 4 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{95} + ( 3 - 3 \zeta_{8}^{2} ) q^{97} + ( \zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{7} + O(q^{10}) \) \( 4 q + 4 q^{3} + 4 q^{7} - 8 q^{15} + 8 q^{21} - 16 q^{25} + 4 q^{27} + 8 q^{31} - 4 q^{33} + 24 q^{37} - 24 q^{43} - 8 q^{45} + 8 q^{51} - 12 q^{55} - 16 q^{57} - 24 q^{61} + 4 q^{63} + 16 q^{67} - 20 q^{73} - 28 q^{75} + 28 q^{81} + 8 q^{85} - 20 q^{87} + 8 q^{93} + 12 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 0.292893 1.70711i 0 0.707107 2.12132i 0 1.00000 + 1.00000i 0 −2.82843 1.00000i 0
17.2 0 1.70711 0.292893i 0 −0.707107 + 2.12132i 0 1.00000 + 1.00000i 0 2.82843 1.00000i 0
113.1 0 0.292893 + 1.70711i 0 0.707107 + 2.12132i 0 1.00000 1.00000i 0 −2.82843 + 1.00000i 0
113.2 0 1.70711 + 0.292893i 0 −0.707107 2.12132i 0 1.00000 1.00000i 0 2.82843 + 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
3.b odd 2 1 inner
5.c odd 4 1 inner
15.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.v.e 4
3.b odd 2 1 inner 240.2.v.e 4
4.b odd 2 1 30.2.e.a 4
5.b even 2 1 1200.2.v.b 4
5.c odd 4 1 inner 240.2.v.e 4
5.c odd 4 1 1200.2.v.b 4
8.b even 2 1 960.2.v.c 4
8.d odd 2 1 960.2.v.k 4
12.b even 2 1 30.2.e.a 4
15.d odd 2 1 1200.2.v.b 4
15.e even 4 1 inner 240.2.v.e 4
15.e even 4 1 1200.2.v.b 4
20.d odd 2 1 150.2.e.a 4
20.e even 4 1 30.2.e.a 4
20.e even 4 1 150.2.e.a 4
24.f even 2 1 960.2.v.k 4
24.h odd 2 1 960.2.v.c 4
36.f odd 6 2 810.2.m.f 8
36.h even 6 2 810.2.m.f 8
40.i odd 4 1 960.2.v.c 4
40.k even 4 1 960.2.v.k 4
60.h even 2 1 150.2.e.a 4
60.l odd 4 1 30.2.e.a 4
60.l odd 4 1 150.2.e.a 4
120.q odd 4 1 960.2.v.k 4
120.w even 4 1 960.2.v.c 4
180.v odd 12 2 810.2.m.f 8
180.x even 12 2 810.2.m.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.e.a 4 4.b odd 2 1
30.2.e.a 4 12.b even 2 1
30.2.e.a 4 20.e even 4 1
30.2.e.a 4 60.l odd 4 1
150.2.e.a 4 20.d odd 2 1
150.2.e.a 4 20.e even 4 1
150.2.e.a 4 60.h even 2 1
150.2.e.a 4 60.l odd 4 1
240.2.v.e 4 1.a even 1 1 trivial
240.2.v.e 4 3.b odd 2 1 inner
240.2.v.e 4 5.c odd 4 1 inner
240.2.v.e 4 15.e even 4 1 inner
810.2.m.f 8 36.f odd 6 2
810.2.m.f 8 36.h even 6 2
810.2.m.f 8 180.v odd 12 2
810.2.m.f 8 180.x even 12 2
960.2.v.c 4 8.b even 2 1
960.2.v.c 4 24.h odd 2 1
960.2.v.c 4 40.i odd 4 1
960.2.v.c 4 120.w even 4 1
960.2.v.k 4 8.d odd 2 1
960.2.v.k 4 24.f even 2 1
960.2.v.k 4 40.k even 4 1
960.2.v.k 4 120.q odd 4 1
1200.2.v.b 4 5.b even 2 1
1200.2.v.b 4 5.c odd 4 1
1200.2.v.b 4 15.d odd 2 1
1200.2.v.b 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{2} - 2 T_{7} + 2 \)
\( T_{11}^{2} + 2 \)
\( T_{17}^{4} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 - 12 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$5$ \( 25 + 8 T^{2} + T^{4} \)
$7$ \( ( 2 - 2 T + T^{2} )^{2} \)
$11$ \( ( 2 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( 16 + T^{4} \)
$19$ \( ( 16 + T^{2} )^{2} \)
$23$ \( 256 + T^{4} \)
$29$ \( ( -50 + T^{2} )^{2} \)
$31$ \( ( -2 + T )^{4} \)
$37$ \( ( 72 - 12 T + T^{2} )^{2} \)
$41$ \( ( 32 + T^{2} )^{2} \)
$43$ \( ( 72 + 12 T + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( 256 + T^{4} \)
$59$ \( ( -98 + T^{2} )^{2} \)
$61$ \( ( 6 + T )^{4} \)
$67$ \( ( 32 - 8 T + T^{2} )^{2} \)
$71$ \( ( 200 + T^{2} )^{2} \)
$73$ \( ( 50 + 10 T + T^{2} )^{2} \)
$79$ \( ( 36 + T^{2} )^{2} \)
$83$ \( 20736 + T^{4} \)
$89$ \( ( -8 + T^{2} )^{2} \)
$97$ \( ( 18 - 6 T + T^{2} )^{2} \)
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