Properties

Label 7200.2.f.m
Level $7200$
Weight $2$
Character orbit 7200.f
Analytic conductor $57.492$
Analytic rank $1$
Dimension $2$
CM discriminant -4
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q + 6 i q^{13} + 2 i q^{17} -10 q^{29} + 2 i q^{37} -10 q^{41} + 7 q^{49} -14 i q^{53} -10 q^{61} -6 i q^{73} + 10 q^{89} -18 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 20q^{29} - 20q^{41} + 14q^{49} - 20q^{61} + 20q^{89} + O(q^{100}) \)

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 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
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.m 2
3.b odd 2 1 800.2.c.e 2
4.b odd 2 1 CM 7200.2.f.m 2
5.b even 2 1 inner 7200.2.f.m 2
5.c odd 4 1 288.2.a.d 1
5.c odd 4 1 7200.2.a.v 1
12.b even 2 1 800.2.c.e 2
15.d odd 2 1 800.2.c.e 2
15.e even 4 1 32.2.a.a 1
15.e even 4 1 800.2.a.d 1
20.d odd 2 1 inner 7200.2.f.m 2
20.e even 4 1 288.2.a.d 1
20.e even 4 1 7200.2.a.v 1
24.f even 2 1 1600.2.c.l 2
24.h odd 2 1 1600.2.c.l 2
40.i odd 4 1 576.2.a.c 1
40.k even 4 1 576.2.a.c 1
45.k odd 12 2 2592.2.i.e 2
45.l even 12 2 2592.2.i.t 2
60.h even 2 1 800.2.c.e 2
60.l odd 4 1 32.2.a.a 1
60.l odd 4 1 800.2.a.d 1
80.i odd 4 1 2304.2.d.j 2
80.j even 4 1 2304.2.d.j 2
80.s even 4 1 2304.2.d.j 2
80.t odd 4 1 2304.2.d.j 2
105.k odd 4 1 1568.2.a.e 1
105.w odd 12 2 1568.2.i.f 2
105.x even 12 2 1568.2.i.g 2
120.i odd 2 1 1600.2.c.l 2
120.m even 2 1 1600.2.c.l 2
120.q odd 4 1 64.2.a.a 1
120.q odd 4 1 1600.2.a.n 1
120.w even 4 1 64.2.a.a 1
120.w even 4 1 1600.2.a.n 1
165.l odd 4 1 3872.2.a.f 1
180.v odd 12 2 2592.2.i.t 2
180.x even 12 2 2592.2.i.e 2
195.s even 4 1 5408.2.a.g 1
240.z odd 4 1 256.2.b.b 2
240.bb even 4 1 256.2.b.b 2
240.bd odd 4 1 256.2.b.b 2
240.bf even 4 1 256.2.b.b 2
255.o even 4 1 9248.2.a.f 1
420.w even 4 1 1568.2.a.e 1
420.bp odd 12 2 1568.2.i.g 2
420.br even 12 2 1568.2.i.f 2
480.bq odd 8 2 1024.2.e.j 4
480.br even 8 2 1024.2.e.j 4
480.ca odd 8 2 1024.2.e.j 4
480.cb even 8 2 1024.2.e.j 4
660.q even 4 1 3872.2.a.f 1
780.w odd 4 1 5408.2.a.g 1
840.bm even 4 1 3136.2.a.m 1
840.bp odd 4 1 3136.2.a.m 1
1020.x odd 4 1 9248.2.a.f 1
1320.bn odd 4 1 7744.2.a.v 1
1320.bt even 4 1 7744.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 15.e even 4 1
32.2.a.a 1 60.l odd 4 1
64.2.a.a 1 120.q odd 4 1
64.2.a.a 1 120.w even 4 1
256.2.b.b 2 240.z odd 4 1
256.2.b.b 2 240.bb even 4 1
256.2.b.b 2 240.bd odd 4 1
256.2.b.b 2 240.bf even 4 1
288.2.a.d 1 5.c odd 4 1
288.2.a.d 1 20.e even 4 1
576.2.a.c 1 40.i odd 4 1
576.2.a.c 1 40.k even 4 1
800.2.a.d 1 15.e even 4 1
800.2.a.d 1 60.l odd 4 1
800.2.c.e 2 3.b odd 2 1
800.2.c.e 2 12.b even 2 1
800.2.c.e 2 15.d odd 2 1
800.2.c.e 2 60.h even 2 1
1024.2.e.j 4 480.bq odd 8 2
1024.2.e.j 4 480.br even 8 2
1024.2.e.j 4 480.ca odd 8 2
1024.2.e.j 4 480.cb even 8 2
1568.2.a.e 1 105.k odd 4 1
1568.2.a.e 1 420.w even 4 1
1568.2.i.f 2 105.w odd 12 2
1568.2.i.f 2 420.br even 12 2
1568.2.i.g 2 105.x even 12 2
1568.2.i.g 2 420.bp odd 12 2
1600.2.a.n 1 120.q odd 4 1
1600.2.a.n 1 120.w even 4 1
1600.2.c.l 2 24.f even 2 1
1600.2.c.l 2 24.h odd 2 1
1600.2.c.l 2 120.i odd 2 1
1600.2.c.l 2 120.m even 2 1
2304.2.d.j 2 80.i odd 4 1
2304.2.d.j 2 80.j even 4 1
2304.2.d.j 2 80.s even 4 1
2304.2.d.j 2 80.t odd 4 1
2592.2.i.e 2 45.k odd 12 2
2592.2.i.e 2 180.x even 12 2
2592.2.i.t 2 45.l even 12 2
2592.2.i.t 2 180.v odd 12 2
3136.2.a.m 1 840.bm even 4 1
3136.2.a.m 1 840.bp odd 4 1
3872.2.a.f 1 165.l odd 4 1
3872.2.a.f 1 660.q even 4 1
5408.2.a.g 1 195.s even 4 1
5408.2.a.g 1 780.w odd 4 1
7200.2.a.v 1 5.c odd 4 1
7200.2.a.v 1 20.e even 4 1
7200.2.f.m 2 1.a even 1 1 trivial
7200.2.f.m 2 4.b odd 2 1 CM
7200.2.f.m 2 5.b even 2 1 inner
7200.2.f.m 2 20.d odd 2 1 inner
7744.2.a.v 1 1320.bn odd 4 1
7744.2.a.v 1 1320.bt even 4 1
9248.2.a.f 1 255.o even 4 1
9248.2.a.f 1 1020.x odd 4 1

Hecke kernels

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

\( T_{7} \)
\( T_{11} \)
\( T_{13}^{2} + 36 \)
\( T_{17}^{2} + 4 \)
\( T_{19} \)
\( T_{29} + 10 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 10 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 196 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( 324 + T^{2} \)
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