Properties

Label 7200.2.f.bh
Level $7200$
Weight $2$
Character orbit 7200.f
Analytic conductor $57.492$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{7} + \beta_{3} q^{11} - \beta_1 q^{13} + \beta_1 q^{17} - \beta_{2} q^{23} + 6 q^{29} + \beta_{3} q^{31} + 5 \beta_1 q^{37} - 2 q^{41} - 3 \beta_{2} q^{43} - \beta_{2} q^{47} - q^{49} - 3 \beta_1 q^{53} + 2 \beta_{3} q^{59} - 2 q^{61} + \beta_{2} q^{67} - \beta_{3} q^{71} - 3 \beta_1 q^{73} + 8 \beta_1 q^{77} - 2 \beta_{3} q^{79} - \beta_{2} q^{83} + 10 q^{89} + \beta_{3} q^{91} - \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{29} - 8 q^{41} - 4 q^{49} - 8 q^{61} + 40 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\zeta_{8}^{3} + 4\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + 2\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6049.1
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0 0 0 0 0 2.82843i 0 0 0
6049.2 0 0 0 0 0 2.82843i 0 0 0
6049.3 0 0 0 0 0 2.82843i 0 0 0
6049.4 0 0 0 0 0 2.82843i 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.f.bh 4
3.b odd 2 1 800.2.c.f 4
4.b odd 2 1 inner 7200.2.f.bh 4
5.b even 2 1 inner 7200.2.f.bh 4
5.c odd 4 1 1440.2.a.o 2
5.c odd 4 1 7200.2.a.cm 2
12.b even 2 1 800.2.c.f 4
15.d odd 2 1 800.2.c.f 4
15.e even 4 1 160.2.a.c 2
15.e even 4 1 800.2.a.m 2
20.d odd 2 1 inner 7200.2.f.bh 4
20.e even 4 1 1440.2.a.o 2
20.e even 4 1 7200.2.a.cm 2
24.f even 2 1 1600.2.c.n 4
24.h odd 2 1 1600.2.c.n 4
40.i odd 4 1 2880.2.a.bk 2
40.k even 4 1 2880.2.a.bk 2
60.h even 2 1 800.2.c.f 4
60.l odd 4 1 160.2.a.c 2
60.l odd 4 1 800.2.a.m 2
105.k odd 4 1 7840.2.a.bf 2
120.i odd 2 1 1600.2.c.n 4
120.m even 2 1 1600.2.c.n 4
120.q odd 4 1 320.2.a.g 2
120.q odd 4 1 1600.2.a.bc 2
120.w even 4 1 320.2.a.g 2
120.w even 4 1 1600.2.a.bc 2
240.z odd 4 1 1280.2.d.l 4
240.bb even 4 1 1280.2.d.l 4
240.bd odd 4 1 1280.2.d.l 4
240.bf even 4 1 1280.2.d.l 4
420.w even 4 1 7840.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.c 2 15.e even 4 1
160.2.a.c 2 60.l odd 4 1
320.2.a.g 2 120.q odd 4 1
320.2.a.g 2 120.w even 4 1
800.2.a.m 2 15.e even 4 1
800.2.a.m 2 60.l odd 4 1
800.2.c.f 4 3.b odd 2 1
800.2.c.f 4 12.b even 2 1
800.2.c.f 4 15.d odd 2 1
800.2.c.f 4 60.h even 2 1
1280.2.d.l 4 240.z odd 4 1
1280.2.d.l 4 240.bb even 4 1
1280.2.d.l 4 240.bd odd 4 1
1280.2.d.l 4 240.bf even 4 1
1440.2.a.o 2 5.c odd 4 1
1440.2.a.o 2 20.e even 4 1
1600.2.a.bc 2 120.q odd 4 1
1600.2.a.bc 2 120.w even 4 1
1600.2.c.n 4 24.f even 2 1
1600.2.c.n 4 24.h odd 2 1
1600.2.c.n 4 120.i odd 2 1
1600.2.c.n 4 120.m even 2 1
2880.2.a.bk 2 40.i odd 4 1
2880.2.a.bk 2 40.k even 4 1
7200.2.a.cm 2 5.c odd 4 1
7200.2.a.cm 2 20.e even 4 1
7200.2.f.bh 4 1.a even 1 1 trivial
7200.2.f.bh 4 4.b odd 2 1 inner
7200.2.f.bh 4 5.b even 2 1 inner
7200.2.f.bh 4 20.d odd 2 1 inner
7840.2.a.bf 2 105.k odd 4 1
7840.2.a.bf 2 420.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 \(S_{2}^{\mathrm{new}}(7200, [\chi])\):

\( T_{7}^{2} + 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 32 \) Copy content Toggle raw display
\( T_{13}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display
\( T_{31}^{2} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$41$ \( (T + 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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