Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 480)
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.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{11} + \beta q^{13} - \beta q^{17} - 8 q^{19} - 2 \beta q^{23} - 6 q^{29} - \beta q^{37} + 6 q^{41} + 2 \beta q^{43} - 6 \beta q^{47} + 7 q^{49} + 3 \beta q^{53} + 12 q^{59} + 14 q^{61} + 6 \beta q^{67} + \beta q^{73} + 8 q^{79} + 2 \beta q^{83} + 2 q^{89} + 7 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{11} - 16 q^{19} - 12 q^{29} + 12 q^{41} + 14 q^{49} + 24 q^{59} + 28 q^{61} + 16 q^{79} + 4 q^{89}+O(q^{100}) \) 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
1.00000i
1.00000i
0 0 0 0 0 0 0 0 0
6049.2 0 0 0 0 0 0 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 7200.2.f.b 2
3.b odd 2 1 2400.2.f.n 2
4.b odd 2 1 7200.2.f.bb 2
5.b even 2 1 inner 7200.2.f.b 2
5.c odd 4 1 1440.2.a.j 1
5.c odd 4 1 7200.2.a.u 1
12.b even 2 1 2400.2.f.e 2
15.d odd 2 1 2400.2.f.n 2
15.e even 4 1 480.2.a.e yes 1
15.e even 4 1 2400.2.a.j 1
20.d odd 2 1 7200.2.f.bb 2
20.e even 4 1 1440.2.a.k 1
20.e even 4 1 7200.2.a.bg 1
24.f even 2 1 4800.2.f.ba 2
24.h odd 2 1 4800.2.f.j 2
40.i odd 4 1 2880.2.a.j 1
40.k even 4 1 2880.2.a.i 1
60.h even 2 1 2400.2.f.e 2
60.l odd 4 1 480.2.a.b 1
60.l odd 4 1 2400.2.a.y 1
120.i odd 2 1 4800.2.f.j 2
120.m even 2 1 4800.2.f.ba 2
120.q odd 4 1 960.2.a.o 1
120.q odd 4 1 4800.2.a.u 1
120.w even 4 1 960.2.a.f 1
120.w even 4 1 4800.2.a.ca 1
240.z odd 4 1 3840.2.k.p 2
240.bb even 4 1 3840.2.k.k 2
240.bd odd 4 1 3840.2.k.p 2
240.bf even 4 1 3840.2.k.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.b 1 60.l odd 4 1
480.2.a.e yes 1 15.e even 4 1
960.2.a.f 1 120.w even 4 1
960.2.a.o 1 120.q odd 4 1
1440.2.a.j 1 5.c odd 4 1
1440.2.a.k 1 20.e even 4 1
2400.2.a.j 1 15.e even 4 1
2400.2.a.y 1 60.l odd 4 1
2400.2.f.e 2 12.b even 2 1
2400.2.f.e 2 60.h even 2 1
2400.2.f.n 2 3.b odd 2 1
2400.2.f.n 2 15.d odd 2 1
2880.2.a.i 1 40.k even 4 1
2880.2.a.j 1 40.i odd 4 1
3840.2.k.k 2 240.bb even 4 1
3840.2.k.k 2 240.bf even 4 1
3840.2.k.p 2 240.z odd 4 1
3840.2.k.p 2 240.bd odd 4 1
4800.2.a.u 1 120.q odd 4 1
4800.2.a.ca 1 120.w even 4 1
4800.2.f.j 2 24.h odd 2 1
4800.2.f.j 2 120.i odd 2 1
4800.2.f.ba 2 24.f even 2 1
4800.2.f.ba 2 120.m even 2 1
7200.2.a.u 1 5.c odd 4 1
7200.2.a.bg 1 20.e even 4 1
7200.2.f.b 2 1.a even 1 1 trivial
7200.2.f.b 2 5.b even 2 1 inner
7200.2.f.bb 2 4.b odd 2 1
7200.2.f.bb 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(7200, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display
\( T_{19} + 8 \) Copy content Toggle raw display
\( T_{29} + 6 \) Copy content Toggle raw display
\( T_{31} \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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