Properties

Label 720.6.f.a
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 10)
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.

\(f(q)\) \(=\) \( q + ( -55 - 10 i ) q^{5} + 158 i q^{7} +O(q^{10})\) \( q + ( -55 - 10 i ) q^{5} + 158 i q^{7} -148 q^{11} + 684 i q^{13} + 2048 i q^{17} + 2220 q^{19} -1246 i q^{23} + ( 2925 + 1100 i ) q^{25} -270 q^{29} + 2048 q^{31} + ( 1580 - 8690 i ) q^{35} + 4372 i q^{37} + 2398 q^{41} -2294 i q^{43} + 10682 i q^{47} -8157 q^{49} -2964 i q^{53} + ( 8140 + 1480 i ) q^{55} + 39740 q^{59} -42298 q^{61} + ( 6840 - 37620 i ) q^{65} + 32098 i q^{67} -4248 q^{71} + 30104 i q^{73} -23384 i q^{77} + 35280 q^{79} -27826 i q^{83} + ( 20480 - 112640 i ) q^{85} -85210 q^{89} -108072 q^{91} + ( -122100 - 22200 i ) q^{95} + 97232 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 110q^{5} + O(q^{10}) \) \( 2q - 110q^{5} - 296q^{11} + 4440q^{19} + 5850q^{25} - 540q^{29} + 4096q^{31} + 3160q^{35} + 4796q^{41} - 16314q^{49} + 16280q^{55} + 79480q^{59} - 84596q^{61} + 13680q^{65} - 8496q^{71} + 70560q^{79} + 40960q^{85} - 170420q^{89} - 216144q^{91} - 244200q^{95} + O(q^{100}) \)

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 0 158.000i 0 0 0
289.2 0 0 0 −55.0000 + 10.0000i 0 158.000i 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.a 2
3.b odd 2 1 80.6.c.c 2
4.b odd 2 1 90.6.c.a 2
5.b even 2 1 inner 720.6.f.a 2
12.b even 2 1 10.6.b.a 2
15.d odd 2 1 80.6.c.c 2
15.e even 4 1 400.6.a.c 1
15.e even 4 1 400.6.a.k 1
20.d odd 2 1 90.6.c.a 2
20.e even 4 1 450.6.a.c 1
20.e even 4 1 450.6.a.w 1
24.f even 2 1 320.6.c.b 2
24.h odd 2 1 320.6.c.a 2
60.h even 2 1 10.6.b.a 2
60.l odd 4 1 50.6.a.c 1
60.l odd 4 1 50.6.a.e 1
120.i odd 2 1 320.6.c.a 2
120.m even 2 1 320.6.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.b.a 2 12.b even 2 1
10.6.b.a 2 60.h even 2 1
50.6.a.c 1 60.l odd 4 1
50.6.a.e 1 60.l odd 4 1
80.6.c.c 2 3.b odd 2 1
80.6.c.c 2 15.d odd 2 1
90.6.c.a 2 4.b odd 2 1
90.6.c.a 2 20.d odd 2 1
320.6.c.a 2 24.h odd 2 1
320.6.c.a 2 120.i odd 2 1
320.6.c.b 2 24.f even 2 1
320.6.c.b 2 120.m even 2 1
400.6.a.c 1 15.e even 4 1
400.6.a.k 1 15.e even 4 1
450.6.a.c 1 20.e even 4 1
450.6.a.w 1 20.e even 4 1
720.6.f.a 2 1.a even 1 1 trivial
720.6.f.a 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 \(S_{6}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{2} + 24964 \)
\( T_{11} + 148 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 110 T + 3125 T^{2} \)
$7$ \( 1 - 8650 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 + 148 T + 161051 T^{2} )^{2} \)
$13$ \( 1 - 274730 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 + 1354590 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 - 2220 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 - 11320170 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 + 270 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 - 2048 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 119573530 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 - 2398 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 - 288754450 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 - 344584890 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 - 827605690 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 - 39740 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 + 42298 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 - 1669968610 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 + 4248 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 - 3239892370 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 - 35280 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 - 7103795010 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 + 85210 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 - 7720618690 T^{2} + 73742412689492826049 T^{4} \)
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