Properties

Label 4800.2.f.w
Level 4800
Weight 2
Character orbit 4800.f
Analytic conductor 38.328
Analytic rank 0
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: \(0\)
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 30)
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.

\(f(q)\) \(=\) \( q -i q^{3} + 4 i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} + 4 i q^{7} - q^{9} + 2 i q^{13} + 6 i q^{17} + 4 q^{19} + 4 q^{21} + i q^{27} -6 q^{29} -8 q^{31} -2 i q^{37} + 2 q^{39} -6 q^{41} + 4 i q^{43} -9 q^{49} + 6 q^{51} -6 i q^{53} -4 i q^{57} + 10 q^{61} -4 i q^{63} -4 i q^{67} -2 i q^{73} + 8 q^{79} + q^{81} -12 i q^{83} + 6 i q^{87} -18 q^{89} -8 q^{91} + 8 i q^{93} + 2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 8q^{19} + 8q^{21} - 12q^{29} - 16q^{31} + 4q^{39} - 12q^{41} - 18q^{49} + 12q^{51} + 20q^{61} + 16q^{79} + 2q^{81} - 36q^{89} - 16q^{91} + O(q^{100}) \)

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 4.00000i 0 −1.00000 0
3649.2 0 1.00000i 0 0 0 4.00000i 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.w 2
4.b odd 2 1 4800.2.f.p 2
5.b even 2 1 inner 4800.2.f.w 2
5.c odd 4 1 960.2.a.p 1
5.c odd 4 1 4800.2.a.d 1
8.b even 2 1 1200.2.f.e 2
8.d odd 2 1 150.2.c.a 2
15.e even 4 1 2880.2.a.q 1
20.d odd 2 1 4800.2.f.p 2
20.e even 4 1 960.2.a.e 1
20.e even 4 1 4800.2.a.cq 1
24.f even 2 1 450.2.c.b 2
24.h odd 2 1 3600.2.f.i 2
40.e odd 2 1 150.2.c.a 2
40.f even 2 1 1200.2.f.e 2
40.i odd 4 1 240.2.a.b 1
40.i odd 4 1 1200.2.a.k 1
40.k even 4 1 30.2.a.a 1
40.k even 4 1 150.2.a.b 1
60.l odd 4 1 2880.2.a.a 1
80.i odd 4 1 3840.2.k.f 2
80.j even 4 1 3840.2.k.y 2
80.s even 4 1 3840.2.k.y 2
80.t odd 4 1 3840.2.k.f 2
120.i odd 2 1 3600.2.f.i 2
120.m even 2 1 450.2.c.b 2
120.q odd 4 1 90.2.a.c 1
120.q odd 4 1 450.2.a.d 1
120.w even 4 1 720.2.a.j 1
120.w even 4 1 3600.2.a.f 1
280.y odd 4 1 1470.2.a.d 1
280.y odd 4 1 7350.2.a.ct 1
280.bp odd 12 2 1470.2.i.q 2
280.br even 12 2 1470.2.i.o 2
360.bo even 12 2 810.2.e.l 2
360.bt odd 12 2 810.2.e.b 2
440.w odd 4 1 3630.2.a.w 1
520.x odd 4 1 5070.2.b.k 2
520.bc even 4 1 5070.2.a.w 1
520.bk odd 4 1 5070.2.b.k 2
680.u even 4 1 8670.2.a.g 1
840.bm even 4 1 4410.2.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 40.k even 4 1
90.2.a.c 1 120.q odd 4 1
150.2.a.b 1 40.k even 4 1
150.2.c.a 2 8.d odd 2 1
150.2.c.a 2 40.e odd 2 1
240.2.a.b 1 40.i odd 4 1
450.2.a.d 1 120.q odd 4 1
450.2.c.b 2 24.f even 2 1
450.2.c.b 2 120.m even 2 1
720.2.a.j 1 120.w even 4 1
810.2.e.b 2 360.bt odd 12 2
810.2.e.l 2 360.bo even 12 2
960.2.a.e 1 20.e even 4 1
960.2.a.p 1 5.c odd 4 1
1200.2.a.k 1 40.i odd 4 1
1200.2.f.e 2 8.b even 2 1
1200.2.f.e 2 40.f even 2 1
1470.2.a.d 1 280.y odd 4 1
1470.2.i.o 2 280.br even 12 2
1470.2.i.q 2 280.bp odd 12 2
2880.2.a.a 1 60.l odd 4 1
2880.2.a.q 1 15.e even 4 1
3600.2.a.f 1 120.w even 4 1
3600.2.f.i 2 24.h odd 2 1
3600.2.f.i 2 120.i odd 2 1
3630.2.a.w 1 440.w odd 4 1
3840.2.k.f 2 80.i odd 4 1
3840.2.k.f 2 80.t odd 4 1
3840.2.k.y 2 80.j even 4 1
3840.2.k.y 2 80.s even 4 1
4410.2.a.z 1 840.bm even 4 1
4800.2.a.d 1 5.c odd 4 1
4800.2.a.cq 1 20.e even 4 1
4800.2.f.p 2 4.b odd 2 1
4800.2.f.p 2 20.d odd 2 1
4800.2.f.w 2 1.a even 1 1 trivial
4800.2.f.w 2 5.b even 2 1 inner
5070.2.a.w 1 520.bc even 4 1
5070.2.b.k 2 520.x odd 4 1
5070.2.b.k 2 520.bk odd 4 1
7350.2.a.ct 1 280.y odd 4 1
8670.2.a.g 1 680.u even 4 1

Hecke kernels

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

\( T_{7}^{2} + 16 \)
\( T_{11} \)
\( T_{13}^{2} + 4 \)
\( T_{19} - 4 \)
\( T_{23} \)
\( T_{31} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} \)
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 47 T^{2} )^{2} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 10 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 18 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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