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: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + i q^{7} +O(q^{10})\) \( q + i q^{7} + i q^{13} + 3 q^{19} + 4 i q^{23} + 4 q^{29} + 7 q^{31} + 6 i q^{37} -6 q^{41} -9 i q^{43} -6 i q^{47} + 6 q^{49} -2 i q^{53} -10 q^{59} - q^{61} -3 i q^{67} -14 q^{71} + 10 i q^{73} + 8 q^{79} + 18 i q^{83} - q^{91} + 3 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 6q^{19} + 8q^{29} + 14q^{31} - 12q^{41} + 12q^{49} - 20q^{59} - 2q^{61} - 28q^{71} + 16q^{79} - 2q^{91} + O(q^{100}) \)

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 \(S_{2}^{\mathrm{new}}(7200, [\chi])\):

\( T_{7}^{2} + 1 \)
\( T_{11} \)
\( T_{13}^{2} + 1 \)
\( T_{17} \)
\( T_{19} - 3 \)
\( T_{29} - 4 \)
\( T_{31} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -3 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( -4 + T )^{2} \)
$31$ \( ( -7 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 81 + T^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( 10 + T )^{2} \)
$61$ \( ( 1 + T )^{2} \)
$67$ \( 9 + T^{2} \)
$71$ \( ( 14 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 324 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 9 + T^{2} \)
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