Properties

Label 2400.2.b.a
Level 2400
Weight 2
Character orbit 2400.b
Analytic conductor 19.164
Analytic rank 0
Dimension 2
CM discriminant -8
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(19.1640964851\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{3} + ( -1 - 2 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{3} + ( -1 - 2 \beta ) q^{9} + 2 \beta q^{11} -4 \beta q^{17} -2 q^{19} + ( 5 + \beta ) q^{27} + ( -4 - 2 \beta ) q^{33} -8 \beta q^{41} -10 q^{43} + 7 q^{49} + ( 8 + 4 \beta ) q^{51} + ( 2 - 2 \beta ) q^{57} -10 \beta q^{59} + 14 q^{67} -2 q^{73} + ( -7 + 4 \beta ) q^{81} -2 \beta q^{83} + 4 \beta q^{89} + 10 q^{97} + ( 8 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{9} - 4q^{19} + 10q^{27} - 8q^{33} - 20q^{43} + 14q^{49} + 16q^{51} + 4q^{57} + 28q^{67} - 4q^{73} - 14q^{81} + 20q^{97} + 16q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2351.1
1.41421i
1.41421i
0 −1.00000 1.41421i 0 0 0 0 0 −1.00000 + 2.82843i 0
2351.2 0 −1.00000 + 1.41421i 0 0 0 0 0 −1.00000 2.82843i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
3.b odd 2 1 inner
24.f 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.b.a 2
3.b odd 2 1 inner 2400.2.b.a 2
4.b odd 2 1 600.2.b.a 2
5.b even 2 1 96.2.f.a 2
5.c odd 4 2 2400.2.m.a 4
8.b even 2 1 600.2.b.a 2
8.d odd 2 1 CM 2400.2.b.a 2
12.b even 2 1 600.2.b.a 2
15.d odd 2 1 96.2.f.a 2
15.e even 4 2 2400.2.m.a 4
20.d odd 2 1 24.2.f.a 2
20.e even 4 2 600.2.m.a 4
24.f even 2 1 inner 2400.2.b.a 2
24.h odd 2 1 600.2.b.a 2
40.e odd 2 1 96.2.f.a 2
40.f even 2 1 24.2.f.a 2
40.i odd 4 2 600.2.m.a 4
40.k even 4 2 2400.2.m.a 4
45.h odd 6 2 2592.2.p.b 4
45.j even 6 2 2592.2.p.b 4
60.h even 2 1 24.2.f.a 2
60.l odd 4 2 600.2.m.a 4
80.k odd 4 2 768.2.c.h 4
80.q even 4 2 768.2.c.h 4
120.i odd 2 1 24.2.f.a 2
120.m even 2 1 96.2.f.a 2
120.q odd 4 2 2400.2.m.a 4
120.w even 4 2 600.2.m.a 4
180.n even 6 2 648.2.l.b 4
180.p odd 6 2 648.2.l.b 4
240.t even 4 2 768.2.c.h 4
240.bm odd 4 2 768.2.c.h 4
360.z odd 6 2 2592.2.p.b 4
360.bd even 6 2 2592.2.p.b 4
360.bh odd 6 2 648.2.l.b 4
360.bk even 6 2 648.2.l.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.f.a 2 20.d odd 2 1
24.2.f.a 2 40.f even 2 1
24.2.f.a 2 60.h even 2 1
24.2.f.a 2 120.i odd 2 1
96.2.f.a 2 5.b even 2 1
96.2.f.a 2 15.d odd 2 1
96.2.f.a 2 40.e odd 2 1
96.2.f.a 2 120.m even 2 1
600.2.b.a 2 4.b odd 2 1
600.2.b.a 2 8.b even 2 1
600.2.b.a 2 12.b even 2 1
600.2.b.a 2 24.h odd 2 1
600.2.m.a 4 20.e even 4 2
600.2.m.a 4 40.i odd 4 2
600.2.m.a 4 60.l odd 4 2
600.2.m.a 4 120.w even 4 2
648.2.l.b 4 180.n even 6 2
648.2.l.b 4 180.p odd 6 2
648.2.l.b 4 360.bh odd 6 2
648.2.l.b 4 360.bk even 6 2
768.2.c.h 4 80.k odd 4 2
768.2.c.h 4 80.q even 4 2
768.2.c.h 4 240.t even 4 2
768.2.c.h 4 240.bm odd 4 2
2400.2.b.a 2 1.a even 1 1 trivial
2400.2.b.a 2 3.b odd 2 1 inner
2400.2.b.a 2 8.d odd 2 1 CM
2400.2.b.a 2 24.f even 2 1 inner
2400.2.m.a 4 5.c odd 4 2
2400.2.m.a 4 15.e even 4 2
2400.2.m.a 4 40.k even 4 2
2400.2.m.a 4 120.q odd 4 2
2592.2.p.b 4 45.h odd 6 2
2592.2.p.b 4 45.j even 6 2
2592.2.p.b 4 360.z odd 6 2
2592.2.p.b 4 360.bd even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2400, [\chi])\):

\( T_{7} \)
\( T_{11}^{2} + 8 \)
\( T_{23} \)
\( T_{43} + 10 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 2 T + 3 T^{2} \)
$5$ 1
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} ) \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} ) \)
$19$ \( ( 1 + 2 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{2} \)
$37$ \( ( 1 - 37 T^{2} )^{2} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )( 1 + 6 T + 41 T^{2} ) \)
$43$ \( ( 1 + 10 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 6 T + 59 T^{2} )( 1 + 6 T + 59 T^{2} ) \)
$61$ \( ( 1 - 61 T^{2} )^{2} \)
$67$ \( ( 1 - 14 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 2 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 79 T^{2} )^{2} \)
$83$ \( ( 1 - 18 T + 83 T^{2} )( 1 + 18 T + 83 T^{2} ) \)
$89$ \( ( 1 - 18 T + 89 T^{2} )( 1 + 18 T + 89 T^{2} ) \)
$97$ \( ( 1 - 10 T + 97 T^{2} )^{2} \)
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