Properties

Label 720.4.f.f
Level $720$
Weight $4$
Character orbit 720.f
Analytic conductor $42.481$
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: \( k \) \(=\) \( 4 \)
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: \(2\)
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.

\(f(q)\) \(=\) \( q + ( 5 - 10 i ) q^{5} + 26 i q^{7} +O(q^{10})\) \( q + ( 5 - 10 i ) q^{5} + 26 i q^{7} -28 q^{11} + 12 i q^{13} -64 i q^{17} -60 q^{19} -58 i q^{23} + ( -75 - 100 i ) q^{25} + 90 q^{29} + 128 q^{31} + ( 260 + 130 i ) q^{35} -236 i q^{37} -242 q^{41} -362 i q^{43} -226 i q^{47} -333 q^{49} + 108 i q^{53} + ( -140 + 280 i ) q^{55} + 20 q^{59} + 542 q^{61} + ( 120 + 60 i ) q^{65} -434 i q^{67} -1128 q^{71} + 632 i q^{73} -728 i q^{77} -720 q^{79} -478 i q^{83} + ( -640 - 320 i ) q^{85} -490 q^{89} -312 q^{91} + ( -300 + 600 i ) q^{95} -1456 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10q^{5} + O(q^{10}) \) \( 2q + 10q^{5} - 56q^{11} - 120q^{19} - 150q^{25} + 180q^{29} + 256q^{31} + 520q^{35} - 484q^{41} - 666q^{49} - 280q^{55} + 40q^{59} + 1084q^{61} + 240q^{65} - 2256q^{71} - 1440q^{79} - 1280q^{85} - 980q^{89} - 624q^{91} - 600q^{95} + O(q^{100}) \)

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 5.00000 10.0000i 0 26.0000i 0 0 0
289.2 0 0 0 5.00000 + 10.0000i 0 26.0000i 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.f 2
3.b odd 2 1 80.4.c.a 2
4.b odd 2 1 90.4.c.b 2
5.b even 2 1 inner 720.4.f.f 2
12.b even 2 1 10.4.b.a 2
15.d odd 2 1 80.4.c.a 2
15.e even 4 1 400.4.a.h 1
15.e even 4 1 400.4.a.n 1
20.d odd 2 1 90.4.c.b 2
20.e even 4 1 450.4.a.j 1
20.e even 4 1 450.4.a.k 1
24.f even 2 1 320.4.c.d 2
24.h odd 2 1 320.4.c.c 2
60.h even 2 1 10.4.b.a 2
60.l odd 4 1 50.4.a.b 1
60.l odd 4 1 50.4.a.d 1
84.h odd 2 1 490.4.c.b 2
120.i odd 2 1 320.4.c.c 2
120.m even 2 1 320.4.c.d 2
120.q odd 4 1 1600.4.a.u 1
120.q odd 4 1 1600.4.a.bh 1
120.w even 4 1 1600.4.a.t 1
120.w even 4 1 1600.4.a.bg 1
420.o odd 2 1 490.4.c.b 2
420.w even 4 1 2450.4.a.o 1
420.w even 4 1 2450.4.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 12.b even 2 1
10.4.b.a 2 60.h even 2 1
50.4.a.b 1 60.l odd 4 1
50.4.a.d 1 60.l odd 4 1
80.4.c.a 2 3.b odd 2 1
80.4.c.a 2 15.d odd 2 1
90.4.c.b 2 4.b odd 2 1
90.4.c.b 2 20.d odd 2 1
320.4.c.c 2 24.h odd 2 1
320.4.c.c 2 120.i odd 2 1
320.4.c.d 2 24.f even 2 1
320.4.c.d 2 120.m even 2 1
400.4.a.h 1 15.e even 4 1
400.4.a.n 1 15.e even 4 1
450.4.a.j 1 20.e even 4 1
450.4.a.k 1 20.e even 4 1
490.4.c.b 2 84.h odd 2 1
490.4.c.b 2 420.o odd 2 1
720.4.f.f 2 1.a even 1 1 trivial
720.4.f.f 2 5.b even 2 1 inner
1600.4.a.t 1 120.w even 4 1
1600.4.a.u 1 120.q odd 4 1
1600.4.a.bg 1 120.w even 4 1
1600.4.a.bh 1 120.q odd 4 1
2450.4.a.o 1 420.w even 4 1
2450.4.a.bb 1 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{2} + 676 \)
\( T_{11} + 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 125 - 10 T + T^{2} \)
$7$ \( 676 + T^{2} \)
$11$ \( ( 28 + T )^{2} \)
$13$ \( 144 + T^{2} \)
$17$ \( 4096 + T^{2} \)
$19$ \( ( 60 + T )^{2} \)
$23$ \( 3364 + T^{2} \)
$29$ \( ( -90 + T )^{2} \)
$31$ \( ( -128 + T )^{2} \)
$37$ \( 55696 + T^{2} \)
$41$ \( ( 242 + T )^{2} \)
$43$ \( 131044 + T^{2} \)
$47$ \( 51076 + T^{2} \)
$53$ \( 11664 + T^{2} \)
$59$ \( ( -20 + T )^{2} \)
$61$ \( ( -542 + T )^{2} \)
$67$ \( 188356 + T^{2} \)
$71$ \( ( 1128 + T )^{2} \)
$73$ \( 399424 + T^{2} \)
$79$ \( ( 720 + T )^{2} \)
$83$ \( 228484 + T^{2} \)
$89$ \( ( 490 + T )^{2} \)
$97$ \( 2119936 + T^{2} \)
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