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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - 5 \beta + 5) q^{5} + 13 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \beta + 5) q^{5} + 13 \beta q^{7} - 28 q^{11} + 6 \beta q^{13} - 32 \beta q^{17} - 60 q^{19} - 29 \beta q^{23} + ( - 50 \beta - 75) q^{25} + 90 q^{29} + 128 q^{31} + (65 \beta + 260) q^{35} - 118 \beta q^{37} - 242 q^{41} - 181 \beta q^{43} - 113 \beta q^{47} - 333 q^{49} + 54 \beta q^{53} + (140 \beta - 140) q^{55} + 20 q^{59} + 542 q^{61} + (30 \beta + 120) q^{65} - 217 \beta q^{67} - 1128 q^{71} + 316 \beta q^{73} - 364 \beta q^{77} - 720 q^{79} - 239 \beta q^{83} + ( - 160 \beta - 640) q^{85} - 490 q^{89} - 312 q^{91} + (300 \beta - 300) q^{95} - 728 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{5} - 56 q^{11} - 120 q^{19} - 150 q^{25} + 180 q^{29} + 256 q^{31} + 520 q^{35} - 484 q^{41} - 666 q^{49} - 280 q^{55} + 40 q^{59} + 1084 q^{61} + 240 q^{65} - 2256 q^{71} - 1440 q^{79} - 1280 q^{85} - 980 q^{89} - 624 q^{91} - 600 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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 \(S_{4}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{2} + 676 \) Copy content Toggle raw display
\( T_{11} + 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

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