Properties

Label 720.4.f.j
Level $720$
Weight $4$
Character orbit 720.f
Analytic conductor $42.481$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(42.4813752041\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
Defining polynomial: \(x^{4} + 21 x^{2} + 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} - \beta_{3} ) q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{7} +O(q^{10})\) \( q + ( -1 - \beta_{2} - \beta_{3} ) q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{7} + ( -20 - \beta_{2} - 2 \beta_{3} ) q^{11} + ( -\beta_{1} - 4 \beta_{2} ) q^{13} + ( -8 \beta_{1} - 3 \beta_{2} ) q^{17} + ( 32 - 4 \beta_{2} - 8 \beta_{3} ) q^{19} + ( -\beta_{1} + 3 \beta_{2} ) q^{23} + ( 61 + 5 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{25} + ( -166 + 7 \beta_{2} + 14 \beta_{3} ) q^{29} + ( -28 + 2 \beta_{2} + 4 \beta_{3} ) q^{31} + ( -80 - 15 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{35} + ( 33 \beta_{1} + 6 \beta_{2} ) q^{37} + ( 202 + 2 \beta_{2} + 4 \beta_{3} ) q^{41} + ( 41 \beta_{1} - 7 \beta_{2} ) q^{43} + ( -27 \beta_{1} + 19 \beta_{2} ) q^{47} + ( -41 + 6 \beta_{2} + 12 \beta_{3} ) q^{49} + ( -14 \beta_{1} + 27 \beta_{2} ) q^{53} + ( 204 + 5 \beta_{1} + 14 \beta_{2} + 24 \beta_{3} ) q^{55} + ( 92 + \beta_{2} + 2 \beta_{3} ) q^{59} + ( 154 + 16 \beta_{2} + 32 \beta_{3} ) q^{61} + ( -248 + 30 \beta_{1} - 13 \beta_{2} + 22 \beta_{3} ) q^{65} + ( -53 \beta_{1} - 23 \beta_{2} ) q^{67} + ( -48 + 30 \beta_{2} + 60 \beta_{3} ) q^{71} + ( -96 \beta_{1} - 42 \beta_{2} ) q^{73} + ( -66 \beta_{1} - 12 \beta_{2} ) q^{77} + ( 140 + 50 \beta_{2} + 100 \beta_{3} ) q^{79} + ( 119 \beta_{1} + 15 \beta_{2} ) q^{83} + ( -128 + 95 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} ) q^{85} + ( 582 - 24 \beta_{2} - 48 \beta_{3} ) q^{89} + ( -528 + 42 \beta_{2} + 84 \beta_{3} ) q^{91} + ( 704 + 20 \beta_{1} - 56 \beta_{2} - 16 \beta_{3} ) q^{95} + ( -122 \beta_{1} - 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} + O(q^{10}) \) \( 4 q - 6 q^{5} - 84 q^{11} + 112 q^{19} + 256 q^{25} - 636 q^{29} - 104 q^{31} - 300 q^{35} + 816 q^{41} - 140 q^{49} + 864 q^{55} + 372 q^{59} + 680 q^{61} - 948 q^{65} - 72 q^{71} + 760 q^{79} - 508 q^{85} + 2232 q^{89} - 1944 q^{91} + 2784 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 21 x^{2} + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{3} + 32 \nu \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{3} + 34 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{3} + 15 \nu^{2} - 17 \nu + 160 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + 2 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + \beta_{2} - 64\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(16 \beta_{2} - 17 \beta_{1}\)\()/6\)

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
2.70156i
2.70156i
3.70156i
3.70156i
0 0 0 −11.1047 1.29844i 0 16.2094i 0 0 0
289.2 0 0 0 −11.1047 + 1.29844i 0 16.2094i 0 0 0
289.3 0 0 0 8.10469 7.70156i 0 22.2094i 0 0 0
289.4 0 0 0 8.10469 + 7.70156i 0 22.2094i 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.4.f.j 4
3.b odd 2 1 240.4.f.f 4
4.b odd 2 1 45.4.b.b 4
5.b even 2 1 inner 720.4.f.j 4
12.b even 2 1 15.4.b.a 4
15.d odd 2 1 240.4.f.f 4
15.e even 4 1 1200.4.a.bn 2
15.e even 4 1 1200.4.a.bt 2
20.d odd 2 1 45.4.b.b 4
20.e even 4 1 225.4.a.i 2
20.e even 4 1 225.4.a.o 2
24.f even 2 1 960.4.f.q 4
24.h odd 2 1 960.4.f.p 4
60.h even 2 1 15.4.b.a 4
60.l odd 4 1 75.4.a.c 2
60.l odd 4 1 75.4.a.f 2
120.i odd 2 1 960.4.f.p 4
120.m even 2 1 960.4.f.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 12.b even 2 1
15.4.b.a 4 60.h even 2 1
45.4.b.b 4 4.b odd 2 1
45.4.b.b 4 20.d odd 2 1
75.4.a.c 2 60.l odd 4 1
75.4.a.f 2 60.l odd 4 1
225.4.a.i 2 20.e even 4 1
225.4.a.o 2 20.e even 4 1
240.4.f.f 4 3.b odd 2 1
240.4.f.f 4 15.d odd 2 1
720.4.f.j 4 1.a even 1 1 trivial
720.4.f.j 4 5.b even 2 1 inner
960.4.f.p 4 24.h odd 2 1
960.4.f.p 4 120.i odd 2 1
960.4.f.q 4 24.f even 2 1
960.4.f.q 4 120.m even 2 1
1200.4.a.bn 2 15.e even 4 1
1200.4.a.bt 2 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 \(S_{4}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{4} + 756 T_{7}^{2} + 129600 \)
\( T_{11}^{2} + 42 T_{11} + 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 15625 + 750 T - 110 T^{2} + 6 T^{3} + T^{4} \)
$7$ \( 129600 + 756 T^{2} + T^{4} \)
$11$ \( ( 72 + 42 T + T^{2} )^{2} \)
$13$ \( 1327104 + 3780 T^{2} + T^{4} \)
$17$ \( 2483776 + 7252 T^{2} + T^{4} \)
$19$ \( ( -5120 - 56 T + T^{2} )^{2} \)
$23$ \( 6400 + 2464 T^{2} + T^{4} \)
$29$ \( ( 7200 + 318 T + T^{2} )^{2} \)
$31$ \( ( -800 + 52 T + T^{2} )^{2} \)
$37$ \( 41990400 + 106596 T^{2} + T^{4} \)
$41$ \( ( 40140 - 408 T + T^{2} )^{2} \)
$43$ \( 8256266496 + 196128 T^{2} + T^{4} \)
$47$ \( 6186766336 + 189712 T^{2} + T^{4} \)
$53$ \( 813390400 + 218644 T^{2} + T^{4} \)
$59$ \( ( 8280 - 186 T + T^{2} )^{2} \)
$61$ \( ( -65564 - 340 T + T^{2} )^{2} \)
$67$ \( 9419867136 + 341712 T^{2} + T^{4} \)
$71$ \( ( -331776 + 36 T + T^{2} )^{2} \)
$73$ \( 104976000000 + 1126224 T^{2} + T^{4} \)
$79$ \( ( -886400 - 380 T + T^{2} )^{2} \)
$83$ \( 40558737664 + 1371040 T^{2} + T^{4} \)
$89$ \( ( 98820 - 1116 T + T^{2} )^{2} \)
$97$ \( 196199387136 + 1475712 T^{2} + T^{4} \)
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