Properties

Label 720.2.w.d
Level $720$
Weight $2$
Character orbit 720.w
Analytic conductor $5.749$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.w (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
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.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( 2 + 2 \zeta_{8}^{2} ) q^{7} +O(q^{10})\) \( q + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( 2 + 2 \zeta_{8}^{2} ) q^{7} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{11} + ( 1 - \zeta_{8}^{2} ) q^{13} + 4 \zeta_{8}^{3} q^{17} -4 \zeta_{8} q^{23} + ( 4 + 3 \zeta_{8}^{2} ) q^{25} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{29} + 4 q^{31} + ( 6 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{35} + ( 1 + \zeta_{8}^{2} ) q^{37} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{41} + ( 8 - 8 \zeta_{8}^{2} ) q^{43} + 8 \zeta_{8}^{3} q^{47} + \zeta_{8}^{2} q^{49} + 4 \zeta_{8} q^{53} + ( -2 + 6 \zeta_{8}^{2} ) q^{55} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{59} + 8 q^{61} + ( \zeta_{8} - 3 \zeta_{8}^{3} ) q^{65} + ( -4 - 4 \zeta_{8}^{2} ) q^{67} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} + ( 1 - \zeta_{8}^{2} ) q^{73} + 8 \zeta_{8}^{3} q^{77} + 12 \zeta_{8}^{2} q^{79} -4 \zeta_{8} q^{83} + ( -8 + 4 \zeta_{8}^{2} ) q^{85} + ( 9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{89} + 4 q^{91} + ( -11 - 11 \zeta_{8}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{7} + O(q^{10}) \) \( 4q + 8q^{7} + 4q^{13} + 16q^{25} + 16q^{31} + 4q^{37} + 32q^{43} - 8q^{55} + 32q^{61} - 16q^{67} + 4q^{73} - 32q^{85} + 16q^{91} - 44q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{8}^{2}\) \(-1\)

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 0 −2.12132 0.707107i 0 2.00000 + 2.00000i 0 0 0
17.2 0 0 0 2.12132 + 0.707107i 0 2.00000 + 2.00000i 0 0 0
593.1 0 0 0 −2.12132 + 0.707107i 0 2.00000 2.00000i 0 0 0
593.2 0 0 0 2.12132 0.707107i 0 2.00000 2.00000i 0 0 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 720.2.w.d 4
3.b odd 2 1 inner 720.2.w.d 4
4.b odd 2 1 45.2.f.a 4
5.b even 2 1 3600.2.w.b 4
5.c odd 4 1 inner 720.2.w.d 4
5.c odd 4 1 3600.2.w.b 4
8.b even 2 1 2880.2.w.k 4
8.d odd 2 1 2880.2.w.b 4
12.b even 2 1 45.2.f.a 4
15.d odd 2 1 3600.2.w.b 4
15.e even 4 1 inner 720.2.w.d 4
15.e even 4 1 3600.2.w.b 4
20.d odd 2 1 225.2.f.a 4
20.e even 4 1 45.2.f.a 4
20.e even 4 1 225.2.f.a 4
24.f even 2 1 2880.2.w.b 4
24.h odd 2 1 2880.2.w.k 4
36.f odd 6 2 405.2.m.a 8
36.h even 6 2 405.2.m.a 8
40.i odd 4 1 2880.2.w.k 4
40.k even 4 1 2880.2.w.b 4
60.h even 2 1 225.2.f.a 4
60.l odd 4 1 45.2.f.a 4
60.l odd 4 1 225.2.f.a 4
120.q odd 4 1 2880.2.w.b 4
120.w even 4 1 2880.2.w.k 4
180.v odd 12 2 405.2.m.a 8
180.x even 12 2 405.2.m.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.f.a 4 4.b odd 2 1
45.2.f.a 4 12.b even 2 1
45.2.f.a 4 20.e even 4 1
45.2.f.a 4 60.l odd 4 1
225.2.f.a 4 20.d odd 2 1
225.2.f.a 4 20.e even 4 1
225.2.f.a 4 60.h even 2 1
225.2.f.a 4 60.l odd 4 1
405.2.m.a 8 36.f odd 6 2
405.2.m.a 8 36.h even 6 2
405.2.m.a 8 180.v odd 12 2
405.2.m.a 8 180.x even 12 2
720.2.w.d 4 1.a even 1 1 trivial
720.2.w.d 4 3.b odd 2 1 inner
720.2.w.d 4 5.c odd 4 1 inner
720.2.w.d 4 15.e even 4 1 inner
2880.2.w.b 4 8.d odd 2 1
2880.2.w.b 4 24.f even 2 1
2880.2.w.b 4 40.k even 4 1
2880.2.w.b 4 120.q odd 4 1
2880.2.w.k 4 8.b even 2 1
2880.2.w.k 4 24.h odd 2 1
2880.2.w.k 4 40.i odd 4 1
2880.2.w.k 4 120.w even 4 1
3600.2.w.b 4 5.b even 2 1
3600.2.w.b 4 5.c odd 4 1
3600.2.w.b 4 15.d odd 2 1
3600.2.w.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 \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{2} - 4 T_{7} + 8 \)
\( T_{13}^{2} - 2 T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 - 8 T^{2} + T^{4} \)
$7$ \( ( 8 - 4 T + T^{2} )^{2} \)
$11$ \( ( 8 + T^{2} )^{2} \)
$13$ \( ( 2 - 2 T + T^{2} )^{2} \)
$17$ \( 256 + T^{4} \)
$19$ \( T^{4} \)
$23$ \( 256 + T^{4} \)
$29$ \( ( -18 + T^{2} )^{2} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( ( 2 - 2 T + T^{2} )^{2} \)
$41$ \( ( 2 + T^{2} )^{2} \)
$43$ \( ( 128 - 16 T + T^{2} )^{2} \)
$47$ \( 4096 + T^{4} \)
$53$ \( 256 + T^{4} \)
$59$ \( ( -72 + T^{2} )^{2} \)
$61$ \( ( -8 + T )^{4} \)
$67$ \( ( 32 + 8 T + T^{2} )^{2} \)
$71$ \( ( 32 + T^{2} )^{2} \)
$73$ \( ( 2 - 2 T + T^{2} )^{2} \)
$79$ \( ( 144 + T^{2} )^{2} \)
$83$ \( 256 + T^{4} \)
$89$ \( ( -162 + T^{2} )^{2} \)
$97$ \( ( 242 + 22 T + T^{2} )^{2} \)
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