Properties

Label 4800.2.f.i
Level $4800$
Weight $2$
Character orbit 4800.f
Analytic conductor $38.328$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(38.3281929702\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q -i q^{3} - q^{9} +O(q^{10})\) \( q -i q^{3} - q^{9} -4 q^{11} + 6 i q^{13} -6 i q^{17} + 4 q^{19} + i q^{27} -2 q^{29} + 8 q^{31} + 4 i q^{33} + 2 i q^{37} + 6 q^{39} -6 q^{41} -12 i q^{43} -8 i q^{47} + 7 q^{49} -6 q^{51} + 6 i q^{53} -4 i q^{57} -12 q^{59} -14 q^{61} + 4 i q^{67} -8 q^{71} + 6 i q^{73} -8 q^{79} + q^{81} + 12 i q^{83} + 2 i q^{87} -10 q^{89} -8 i q^{93} + 2 i q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 8q^{11} + 8q^{19} - 4q^{29} + 16q^{31} + 12q^{39} - 12q^{41} + 14q^{49} - 12q^{51} - 24q^{59} - 28q^{61} - 16q^{71} - 16q^{79} + 2q^{81} - 20q^{89} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(4351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3649.1
1.00000i
1.00000i
0 1.00000i 0 0 0 0 0 −1.00000 0
3649.2 0 1.00000i 0 0 0 0 0 −1.00000 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 4800.2.f.i 2
4.b odd 2 1 4800.2.f.bc 2
5.b even 2 1 inner 4800.2.f.i 2
5.c odd 4 1 960.2.a.j 1
5.c odd 4 1 4800.2.a.r 1
8.b even 2 1 1200.2.f.g 2
8.d odd 2 1 600.2.f.b 2
15.e even 4 1 2880.2.a.bb 1
20.d odd 2 1 4800.2.f.bc 2
20.e even 4 1 960.2.a.c 1
20.e even 4 1 4800.2.a.cd 1
24.f even 2 1 1800.2.f.j 2
24.h odd 2 1 3600.2.f.c 2
40.e odd 2 1 600.2.f.b 2
40.f even 2 1 1200.2.f.g 2
40.i odd 4 1 240.2.a.c 1
40.i odd 4 1 1200.2.a.o 1
40.k even 4 1 120.2.a.b 1
40.k even 4 1 600.2.a.c 1
60.l odd 4 1 2880.2.a.x 1
80.i odd 4 1 3840.2.k.j 2
80.j even 4 1 3840.2.k.o 2
80.s even 4 1 3840.2.k.o 2
80.t odd 4 1 3840.2.k.j 2
120.i odd 2 1 3600.2.f.c 2
120.m even 2 1 1800.2.f.j 2
120.q odd 4 1 360.2.a.b 1
120.q odd 4 1 1800.2.a.n 1
120.w even 4 1 720.2.a.d 1
120.w even 4 1 3600.2.a.t 1
280.y odd 4 1 5880.2.a.a 1
360.bo even 12 2 3240.2.q.g 2
360.bt odd 12 2 3240.2.q.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.b 1 40.k even 4 1
240.2.a.c 1 40.i odd 4 1
360.2.a.b 1 120.q odd 4 1
600.2.a.c 1 40.k even 4 1
600.2.f.b 2 8.d odd 2 1
600.2.f.b 2 40.e odd 2 1
720.2.a.d 1 120.w even 4 1
960.2.a.c 1 20.e even 4 1
960.2.a.j 1 5.c odd 4 1
1200.2.a.o 1 40.i odd 4 1
1200.2.f.g 2 8.b even 2 1
1200.2.f.g 2 40.f even 2 1
1800.2.a.n 1 120.q odd 4 1
1800.2.f.j 2 24.f even 2 1
1800.2.f.j 2 120.m even 2 1
2880.2.a.x 1 60.l odd 4 1
2880.2.a.bb 1 15.e even 4 1
3240.2.q.g 2 360.bo even 12 2
3240.2.q.q 2 360.bt odd 12 2
3600.2.a.t 1 120.w even 4 1
3600.2.f.c 2 24.h odd 2 1
3600.2.f.c 2 120.i odd 2 1
3840.2.k.j 2 80.i odd 4 1
3840.2.k.j 2 80.t odd 4 1
3840.2.k.o 2 80.j even 4 1
3840.2.k.o 2 80.s even 4 1
4800.2.a.r 1 5.c odd 4 1
4800.2.a.cd 1 20.e even 4 1
4800.2.f.i 2 1.a even 1 1 trivial
4800.2.f.i 2 5.b even 2 1 inner
4800.2.f.bc 2 4.b odd 2 1
4800.2.f.bc 2 20.d odd 2 1
5880.2.a.a 1 280.y odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4800, [\chi])\):

\( T_{7} \)
\( T_{11} + 4 \)
\( T_{13}^{2} + 36 \)
\( T_{19} - 4 \)
\( T_{23} \)
\( T_{31} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 144 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( ( 14 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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