Properties

Label 720.6.f.d
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: \(\Q(\sqrt{-31}) \)
Defining polynomial: \(x^{2} - x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-31}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 5 - 5 \beta ) q^{5} -11 \beta q^{7} +O(q^{10})\) \( q + ( 5 - 5 \beta ) q^{5} -11 \beta q^{7} -100 q^{11} -66 \beta q^{13} + 88 \beta q^{17} -2244 q^{19} + 307 \beta q^{23} + ( -3075 - 50 \beta ) q^{25} -7854 q^{29} + 2144 q^{31} + ( -6820 - 55 \beta ) q^{35} -934 \beta q^{37} + 7414 q^{41} + 1595 \beta q^{43} + 847 \beta q^{47} + 1803 q^{49} -2178 \beta q^{53} + ( -500 + 500 \beta ) q^{55} -25972 q^{59} -3058 q^{61} + ( -40920 - 330 \beta ) q^{65} + 5279 \beta q^{67} + 37608 q^{71} -2156 \beta q^{73} + 1100 \beta q^{77} + 79728 q^{79} -1463 \beta q^{83} + ( 54560 + 440 \beta ) q^{85} -826 q^{89} -90024 q^{91} + ( -11220 + 11220 \beta ) q^{95} + 3376 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10q^{5} + O(q^{10}) \) \( 2q + 10q^{5} - 200q^{11} - 4488q^{19} - 6150q^{25} - 15708q^{29} + 4288q^{31} - 13640q^{35} + 14828q^{41} + 3606q^{49} - 1000q^{55} - 51944q^{59} - 6116q^{61} - 81840q^{65} + 75216q^{71} + 159456q^{79} + 109120q^{85} - 1652q^{89} - 180048q^{91} - 22440q^{95} + O(q^{100}) \)

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
0.500000 + 2.78388i
0.500000 2.78388i
0 0 0 5.00000 55.6776i 0 122.491i 0 0 0
289.2 0 0 0 5.00000 + 55.6776i 0 122.491i 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.d 2
3.b odd 2 1 80.6.c.b 2
4.b odd 2 1 180.6.d.b 2
5.b even 2 1 inner 720.6.f.d 2
12.b even 2 1 20.6.c.a 2
15.d odd 2 1 80.6.c.b 2
15.e even 4 2 400.6.a.s 2
20.d odd 2 1 180.6.d.b 2
20.e even 4 2 900.6.a.q 2
24.f even 2 1 320.6.c.e 2
24.h odd 2 1 320.6.c.d 2
60.h even 2 1 20.6.c.a 2
60.l odd 4 2 100.6.a.d 2
120.i odd 2 1 320.6.c.d 2
120.m even 2 1 320.6.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.c.a 2 12.b even 2 1
20.6.c.a 2 60.h even 2 1
80.6.c.b 2 3.b odd 2 1
80.6.c.b 2 15.d odd 2 1
100.6.a.d 2 60.l odd 4 2
180.6.d.b 2 4.b odd 2 1
180.6.d.b 2 20.d odd 2 1
320.6.c.d 2 24.h odd 2 1
320.6.c.d 2 120.i odd 2 1
320.6.c.e 2 24.f even 2 1
320.6.c.e 2 120.m even 2 1
400.6.a.s 2 15.e even 4 2
720.6.f.d 2 1.a even 1 1 trivial
720.6.f.d 2 5.b even 2 1 inner
900.6.a.q 2 20.e even 4 2

Hecke kernels

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

\( T_{7}^{2} + 15004 \)
\( T_{11} + 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 10 T + 3125 T^{2} \)
$7$ \( 1 - 18610 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 + 100 T + 161051 T^{2} )^{2} \)
$13$ \( 1 - 202442 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 - 1879458 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 + 2244 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 - 1185810 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 + 7854 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 - 2144 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 30515770 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 - 7414 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 + 21442214 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 - 369731298 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 - 248174170 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 + 25972 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 + 3058 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 + 755362070 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 - 37608 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 - 3569749522 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 - 79728 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 - 7612675530 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 + 826 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 - 15761405890 T^{2} + 73742412689492826049 T^{4} \)
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