Properties

Label 720.3.bh.c
Level $720$
Weight $3$
Character orbit 720.bh
Analytic conductor $19.619$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 720.bh (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.6185790339\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + 5 i q^{5} + ( -2 - 2 i ) q^{7} +O(q^{10})\) \( q + 5 i q^{5} + ( -2 - 2 i ) q^{7} -8 q^{11} + ( 3 - 3 i ) q^{13} + ( -7 - 7 i ) q^{17} + 20 i q^{19} + ( -2 + 2 i ) q^{23} -25 q^{25} -40 i q^{29} -52 q^{31} + ( 10 - 10 i ) q^{35} + ( -3 - 3 i ) q^{37} + 8 q^{41} + ( 42 - 42 i ) q^{43} + ( -18 - 18 i ) q^{47} -41 i q^{49} + ( -53 + 53 i ) q^{53} -40 i q^{55} -20 i q^{59} -48 q^{61} + ( 15 + 15 i ) q^{65} + ( -62 - 62 i ) q^{67} -28 q^{71} + ( -47 + 47 i ) q^{73} + ( 16 + 16 i ) q^{77} + ( 18 - 18 i ) q^{83} + ( 35 - 35 i ) q^{85} -80 i q^{89} -12 q^{91} -100 q^{95} + ( -63 - 63 i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} + O(q^{10}) \) \( 2q - 4q^{7} - 16q^{11} + 6q^{13} - 14q^{17} - 4q^{23} - 50q^{25} - 104q^{31} + 20q^{35} - 6q^{37} + 16q^{41} + 84q^{43} - 36q^{47} - 106q^{53} - 96q^{61} + 30q^{65} - 124q^{67} - 56q^{71} - 94q^{73} + 32q^{77} + 36q^{83} + 70q^{85} - 24q^{91} - 200q^{95} - 126q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
1.00000i
1.00000i
0 0 0 5.00000i 0 −2.00000 + 2.00000i 0 0 0
577.1 0 0 0 5.00000i 0 −2.00000 2.00000i 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.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.3.bh.c 2
3.b odd 2 1 80.3.p.c 2
4.b odd 2 1 90.3.g.b 2
5.c odd 4 1 inner 720.3.bh.c 2
12.b even 2 1 10.3.c.a 2
15.d odd 2 1 400.3.p.b 2
15.e even 4 1 80.3.p.c 2
15.e even 4 1 400.3.p.b 2
20.d odd 2 1 450.3.g.b 2
20.e even 4 1 90.3.g.b 2
20.e even 4 1 450.3.g.b 2
24.f even 2 1 320.3.p.h 2
24.h odd 2 1 320.3.p.a 2
60.h even 2 1 50.3.c.c 2
60.l odd 4 1 10.3.c.a 2
60.l odd 4 1 50.3.c.c 2
84.h odd 2 1 490.3.f.b 2
120.q odd 4 1 320.3.p.h 2
120.w even 4 1 320.3.p.a 2
420.w even 4 1 490.3.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.3.c.a 2 12.b even 2 1
10.3.c.a 2 60.l odd 4 1
50.3.c.c 2 60.h even 2 1
50.3.c.c 2 60.l odd 4 1
80.3.p.c 2 3.b odd 2 1
80.3.p.c 2 15.e even 4 1
90.3.g.b 2 4.b odd 2 1
90.3.g.b 2 20.e even 4 1
320.3.p.a 2 24.h odd 2 1
320.3.p.a 2 120.w even 4 1
320.3.p.h 2 24.f even 2 1
320.3.p.h 2 120.q odd 4 1
400.3.p.b 2 15.d odd 2 1
400.3.p.b 2 15.e even 4 1
450.3.g.b 2 20.d odd 2 1
450.3.g.b 2 20.e even 4 1
490.3.f.b 2 84.h odd 2 1
490.3.f.b 2 420.w even 4 1
720.3.bh.c 2 1.a even 1 1 trivial
720.3.bh.c 2 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{2} + 4 T_{7} + 8 \)
\( T_{11} + 8 \)