Properties

Label 7200.2.f.r
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2400)
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.

\(f(q)\) \(=\) \( q + i q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{7} + i q^{13} + 3 q^{19} + 4 i q^{23} + 4 q^{29} + 7 q^{31} + 6 i q^{37} - 6 q^{41} - 9 i q^{43} - 6 i q^{47} + 6 q^{49} - 2 i q^{53} - 10 q^{59} - q^{61} - 3 i q^{67} - 14 q^{71} + 10 i q^{73} + 8 q^{79} + 18 i q^{83} - q^{91} + 3 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{19} + 8 q^{29} + 14 q^{31} - 12 q^{41} + 12 q^{49} - 20 q^{59} - 2 q^{61} - 28 q^{71} + 16 q^{79} - 2 q^{91}+O(q^{100}) \) 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 1.00000i 0 0 0
6049.2 0 0 0 0 0 1.00000i 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.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.r 2
3.b odd 2 1 2400.2.f.k 2
4.b odd 2 1 7200.2.f.l 2
5.b even 2 1 inner 7200.2.f.r 2
5.c odd 4 1 7200.2.a.s 1
5.c odd 4 1 7200.2.a.bi 1
12.b even 2 1 2400.2.f.h 2
15.d odd 2 1 2400.2.f.k 2
15.e even 4 1 2400.2.a.g yes 1
15.e even 4 1 2400.2.a.bb yes 1
20.d odd 2 1 7200.2.f.l 2
20.e even 4 1 7200.2.a.r 1
20.e even 4 1 7200.2.a.bj 1
24.f even 2 1 4800.2.f.t 2
24.h odd 2 1 4800.2.f.q 2
60.h even 2 1 2400.2.f.h 2
60.l odd 4 1 2400.2.a.f 1
60.l odd 4 1 2400.2.a.bc yes 1
120.i odd 2 1 4800.2.f.q 2
120.m even 2 1 4800.2.f.t 2
120.q odd 4 1 4800.2.a.w 1
120.q odd 4 1 4800.2.a.bx 1
120.w even 4 1 4800.2.a.x 1
120.w even 4 1 4800.2.a.bw 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.2.a.f 1 60.l odd 4 1
2400.2.a.g yes 1 15.e even 4 1
2400.2.a.bb yes 1 15.e even 4 1
2400.2.a.bc yes 1 60.l odd 4 1
2400.2.f.h 2 12.b even 2 1
2400.2.f.h 2 60.h even 2 1
2400.2.f.k 2 3.b odd 2 1
2400.2.f.k 2 15.d odd 2 1
4800.2.a.w 1 120.q odd 4 1
4800.2.a.x 1 120.w even 4 1
4800.2.a.bw 1 120.w even 4 1
4800.2.a.bx 1 120.q odd 4 1
4800.2.f.q 2 24.h odd 2 1
4800.2.f.q 2 120.i odd 2 1
4800.2.f.t 2 24.f even 2 1
4800.2.f.t 2 120.m even 2 1
7200.2.a.r 1 20.e even 4 1
7200.2.a.s 1 5.c odd 4 1
7200.2.a.bi 1 5.c odd 4 1
7200.2.a.bj 1 20.e even 4 1
7200.2.f.l 2 4.b odd 2 1
7200.2.f.l 2 20.d odd 2 1
7200.2.f.r 2 1.a even 1 1 trivial
7200.2.f.r 2 5.b even 2 1 inner

Hecke kernels

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

\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} + 1 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{19} - 3 \) Copy content Toggle raw display
\( T_{29} - 4 \) Copy content Toggle raw display
\( T_{31} - 7 \) 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
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 4)^{2} \) Copy content Toggle raw display
$31$ \( (T - 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 81 \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( (T + 10)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 9 \) Copy content Toggle raw display
$71$ \( (T + 14)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 324 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 9 \) Copy content Toggle raw display
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