Properties

Label 2400.2.f.a
Level 2400
Weight 2
Character orbit 2400.f
Analytic conductor 19.164
Analytic rank 1
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2400.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1640964851\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 96)
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.

\(f(q)\) \(=\) \( q + i q^{3} + 4 i q^{7} - q^{9} +O(q^{10})\) \( q + i q^{3} + 4 i q^{7} - q^{9} -4 q^{11} + 2 i q^{13} -6 i q^{17} -4 q^{19} -4 q^{21} -i q^{27} -2 q^{29} -4 q^{31} -4 i q^{33} -2 i q^{37} -2 q^{39} + 2 q^{41} + 4 i q^{43} -8 i q^{47} -9 q^{49} + 6 q^{51} -10 i q^{53} -4 i q^{57} -4 q^{59} + 6 q^{61} -4 i q^{63} -4 i q^{67} + 16 q^{71} + 6 i q^{73} -16 i q^{77} + 4 q^{79} + q^{81} + 12 i q^{83} -2 i q^{87} -10 q^{89} -8 q^{91} -4 i q^{93} -14 i q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 8q^{11} - 8q^{19} - 8q^{21} - 4q^{29} - 8q^{31} - 4q^{39} + 4q^{41} - 18q^{49} + 12q^{51} - 8q^{59} + 12q^{61} + 32q^{71} + 8q^{79} + 2q^{81} - 20q^{89} - 16q^{91} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
1.00000i
1.00000i
0 1.00000i 0 0 0 4.00000i 0 −1.00000 0
1249.2 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 2400.2.f.a 2
3.b odd 2 1 7200.2.f.x 2
4.b odd 2 1 2400.2.f.r 2
5.b even 2 1 inner 2400.2.f.a 2
5.c odd 4 1 96.2.a.a 1
5.c odd 4 1 2400.2.a.r 1
8.b even 2 1 4800.2.f.bh 2
8.d odd 2 1 4800.2.f.e 2
12.b even 2 1 7200.2.f.f 2
15.d odd 2 1 7200.2.f.x 2
15.e even 4 1 288.2.a.c 1
15.e even 4 1 7200.2.a.e 1
20.d odd 2 1 2400.2.f.r 2
20.e even 4 1 96.2.a.b yes 1
20.e even 4 1 2400.2.a.q 1
35.f even 4 1 4704.2.a.t 1
40.e odd 2 1 4800.2.f.e 2
40.f even 2 1 4800.2.f.bh 2
40.i odd 4 1 192.2.a.c 1
40.i odd 4 1 4800.2.a.f 1
40.k even 4 1 192.2.a.a 1
40.k even 4 1 4800.2.a.co 1
45.k odd 12 2 2592.2.i.b 2
45.l even 12 2 2592.2.i.q 2
60.h even 2 1 7200.2.f.f 2
60.l odd 4 1 288.2.a.b 1
60.l odd 4 1 7200.2.a.bx 1
80.i odd 4 1 768.2.d.a 2
80.j even 4 1 768.2.d.h 2
80.s even 4 1 768.2.d.h 2
80.t odd 4 1 768.2.d.a 2
120.q odd 4 1 576.2.a.g 1
120.w even 4 1 576.2.a.h 1
140.j odd 4 1 4704.2.a.e 1
180.v odd 12 2 2592.2.i.w 2
180.x even 12 2 2592.2.i.h 2
240.z odd 4 1 2304.2.d.s 2
240.bb even 4 1 2304.2.d.c 2
240.bd odd 4 1 2304.2.d.s 2
240.bf even 4 1 2304.2.d.c 2
280.s even 4 1 9408.2.a.bj 1
280.y odd 4 1 9408.2.a.ct 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.a.a 1 5.c odd 4 1
96.2.a.b yes 1 20.e even 4 1
192.2.a.a 1 40.k even 4 1
192.2.a.c 1 40.i odd 4 1
288.2.a.b 1 60.l odd 4 1
288.2.a.c 1 15.e even 4 1
576.2.a.g 1 120.q odd 4 1
576.2.a.h 1 120.w even 4 1
768.2.d.a 2 80.i odd 4 1
768.2.d.a 2 80.t odd 4 1
768.2.d.h 2 80.j even 4 1
768.2.d.h 2 80.s even 4 1
2304.2.d.c 2 240.bb even 4 1
2304.2.d.c 2 240.bf even 4 1
2304.2.d.s 2 240.z odd 4 1
2304.2.d.s 2 240.bd odd 4 1
2400.2.a.q 1 20.e even 4 1
2400.2.a.r 1 5.c odd 4 1
2400.2.f.a 2 1.a even 1 1 trivial
2400.2.f.a 2 5.b even 2 1 inner
2400.2.f.r 2 4.b odd 2 1
2400.2.f.r 2 20.d odd 2 1
2592.2.i.b 2 45.k odd 12 2
2592.2.i.h 2 180.x even 12 2
2592.2.i.q 2 45.l even 12 2
2592.2.i.w 2 180.v odd 12 2
4704.2.a.e 1 140.j odd 4 1
4704.2.a.t 1 35.f even 4 1
4800.2.a.f 1 40.i odd 4 1
4800.2.a.co 1 40.k even 4 1
4800.2.f.e 2 8.d odd 2 1
4800.2.f.e 2 40.e odd 2 1
4800.2.f.bh 2 8.b even 2 1
4800.2.f.bh 2 40.f even 2 1
7200.2.a.e 1 15.e even 4 1
7200.2.a.bx 1 60.l odd 4 1
7200.2.f.f 2 12.b even 2 1
7200.2.f.f 2 60.h even 2 1
7200.2.f.x 2 3.b odd 2 1
7200.2.f.x 2 15.d odd 2 1
9408.2.a.bj 1 280.s even 4 1
9408.2.a.ct 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 \(S_{2}^{\mathrm{new}}(2400, [\chi])\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} \)
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 4 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 + 2 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 30 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 6 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 6 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 16 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 10 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 2 T^{2} + 9409 T^{4} \)
show more
show less