Properties

Label 720.6.f.g
Level $720$
Weight $6$
Character orbit 720.f
Analytic conductor $115.476$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 720.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(115.476350265\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 55 - 10 i ) q^{5} + 4 i q^{7} +O(q^{10})\) \( q + ( 55 - 10 i ) q^{5} + 4 i q^{7} -500 q^{11} + 288 i q^{13} -1516 i q^{17} -1344 q^{19} + 4100 i q^{23} + ( 2925 - 1100 i ) q^{25} -2646 q^{29} + 5612 q^{31} + ( 40 + 220 i ) q^{35} -7288 i q^{37} + 18986 q^{41} -2404 i q^{43} + 8900 i q^{47} + 16791 q^{49} + 39804 i q^{53} + ( -27500 + 5000 i ) q^{55} + 28300 q^{59} + 18290 q^{61} + ( 2880 + 15840 i ) q^{65} -65956 i q^{67} -28800 q^{71} + 30808 i q^{73} -2000 i q^{77} + 60228 q^{79} + 2468 i q^{83} + ( -15160 - 83380 i ) q^{85} + 22678 q^{89} -1152 q^{91} + ( -73920 + 13440 i ) q^{95} -36968 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 110q^{5} + O(q^{10}) \) \( 2q + 110q^{5} - 1000q^{11} - 2688q^{19} + 5850q^{25} - 5292q^{29} + 11224q^{31} + 80q^{35} + 37972q^{41} + 33582q^{49} - 55000q^{55} + 56600q^{59} + 36580q^{61} + 5760q^{65} - 57600q^{71} + 120456q^{79} - 30320q^{85} + 45356q^{89} - 2304q^{91} - 147840q^{95} + O(q^{100}) \)

Character values

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

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

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
1.00000i
1.00000i
0 0 0 55.0000 10.0000i 0 4.00000i 0 0 0
289.2 0 0 0 55.0000 + 10.0000i 0 4.00000i 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
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 720.6.f.g 2
3.b odd 2 1 240.6.f.a 2
4.b odd 2 1 90.6.c.b 2
5.b even 2 1 inner 720.6.f.g 2
12.b even 2 1 30.6.c.a 2
15.d odd 2 1 240.6.f.a 2
20.d odd 2 1 90.6.c.b 2
20.e even 4 1 450.6.a.f 1
20.e even 4 1 450.6.a.s 1
60.h even 2 1 30.6.c.a 2
60.l odd 4 1 150.6.a.f 1
60.l odd 4 1 150.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.a 2 12.b even 2 1
30.6.c.a 2 60.h even 2 1
90.6.c.b 2 4.b odd 2 1
90.6.c.b 2 20.d odd 2 1
150.6.a.f 1 60.l odd 4 1
150.6.a.j 1 60.l odd 4 1
240.6.f.a 2 3.b odd 2 1
240.6.f.a 2 15.d odd 2 1
450.6.a.f 1 20.e even 4 1
450.6.a.s 1 20.e even 4 1
720.6.f.g 2 1.a even 1 1 trivial
720.6.f.g 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{2} + 16 \)
\( T_{11} + 500 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 3125 - 110 T + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( ( 500 + T )^{2} \)
$13$ \( 82944 + T^{2} \)
$17$ \( 2298256 + T^{2} \)
$19$ \( ( 1344 + T )^{2} \)
$23$ \( 16810000 + T^{2} \)
$29$ \( ( 2646 + T )^{2} \)
$31$ \( ( -5612 + T )^{2} \)
$37$ \( 53114944 + T^{2} \)
$41$ \( ( -18986 + T )^{2} \)
$43$ \( 5779216 + T^{2} \)
$47$ \( 79210000 + T^{2} \)
$53$ \( 1584358416 + T^{2} \)
$59$ \( ( -28300 + T )^{2} \)
$61$ \( ( -18290 + T )^{2} \)
$67$ \( 4350193936 + T^{2} \)
$71$ \( ( 28800 + T )^{2} \)
$73$ \( 949132864 + T^{2} \)
$79$ \( ( -60228 + T )^{2} \)
$83$ \( 6091024 + T^{2} \)
$89$ \( ( -22678 + T )^{2} \)
$97$ \( 1366633024 + T^{2} \)
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