Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2400,2,Mod(1199,2400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2400.1199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.1640964851\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{3} + 1) q^{9} + \beta_{3} q^{11} + ( - 2 \beta_{2} - 2 \beta_1) q^{17} + 2 q^{19} + ( - 2 \beta_{2} + 3 \beta_1) q^{27} + ( - \beta_{2} + 3 \beta_1) q^{33} - 4 \beta_{3} q^{41} + ( - 5 \beta_{2} + 5 \beta_1) q^{43} - 7 q^{49} + (2 \beta_{3} + 8) q^{51} - 2 \beta_{2} q^{57} + 5 \beta_{3} q^{59} + ( - 7 \beta_{2} + 7 \beta_1) q^{67} + ( - \beta_{2} + \beta_1) q^{73} + (2 \beta_{3} - 7) q^{81} + (\beta_{2} + \beta_1) q^{83} - 2 \beta_{3} q^{89} + ( - 5 \beta_{2} + 5 \beta_1) q^{97} + (\beta_{3} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} + 8 q^{19} - 28 q^{49} + 32 q^{51} - 28 q^{81} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} - \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1199.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0 −1.41421 1.00000i 0 0 0 0 0 1.00000 + 2.82843i 0
1199.2 0 −1.41421 + 1.00000i 0 0 0 0 0 1.00000 2.82843i 0
1199.3 0 1.41421 1.00000i 0 0 0 0 0 1.00000 2.82843i 0
1199.4 0 1.41421 + 1.00000i 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
5.b even 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m 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.m.a 4
3.b odd 2 1 inner 2400.2.m.a 4
4.b odd 2 1 600.2.m.a 4
5.b even 2 1 inner 2400.2.m.a 4
5.c odd 4 1 96.2.f.a 2
5.c odd 4 1 2400.2.b.a 2
8.b even 2 1 600.2.m.a 4
8.d odd 2 1 CM 2400.2.m.a 4
12.b even 2 1 600.2.m.a 4
15.d odd 2 1 inner 2400.2.m.a 4
15.e even 4 1 96.2.f.a 2
15.e even 4 1 2400.2.b.a 2
20.d odd 2 1 600.2.m.a 4
20.e even 4 1 24.2.f.a 2
20.e even 4 1 600.2.b.a 2
24.f even 2 1 inner 2400.2.m.a 4
24.h odd 2 1 600.2.m.a 4
40.e odd 2 1 inner 2400.2.m.a 4
40.f even 2 1 600.2.m.a 4
40.i odd 4 1 24.2.f.a 2
40.i odd 4 1 600.2.b.a 2
40.k even 4 1 96.2.f.a 2
40.k even 4 1 2400.2.b.a 2
45.k odd 12 2 2592.2.p.b 4
45.l even 12 2 2592.2.p.b 4
60.h even 2 1 600.2.m.a 4
60.l odd 4 1 24.2.f.a 2
60.l odd 4 1 600.2.b.a 2
80.i odd 4 1 768.2.c.h 4
80.j even 4 1 768.2.c.h 4
80.s even 4 1 768.2.c.h 4
80.t odd 4 1 768.2.c.h 4
120.i odd 2 1 600.2.m.a 4
120.m even 2 1 inner 2400.2.m.a 4
120.q odd 4 1 96.2.f.a 2
120.q odd 4 1 2400.2.b.a 2
120.w even 4 1 24.2.f.a 2
120.w even 4 1 600.2.b.a 2
180.v odd 12 2 648.2.l.b 4
180.x even 12 2 648.2.l.b 4
240.z odd 4 1 768.2.c.h 4
240.bb even 4 1 768.2.c.h 4
240.bd odd 4 1 768.2.c.h 4
240.bf even 4 1 768.2.c.h 4
360.bo even 12 2 2592.2.p.b 4
360.br even 12 2 648.2.l.b 4
360.bt odd 12 2 2592.2.p.b 4
360.bu odd 12 2 648.2.l.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.f.a 2 20.e even 4 1
24.2.f.a 2 40.i odd 4 1
24.2.f.a 2 60.l odd 4 1
24.2.f.a 2 120.w even 4 1
96.2.f.a 2 5.c odd 4 1
96.2.f.a 2 15.e even 4 1
96.2.f.a 2 40.k even 4 1
96.2.f.a 2 120.q odd 4 1
600.2.b.a 2 20.e even 4 1
600.2.b.a 2 40.i odd 4 1
600.2.b.a 2 60.l odd 4 1
600.2.b.a 2 120.w even 4 1
600.2.m.a 4 4.b odd 2 1
600.2.m.a 4 8.b even 2 1
600.2.m.a 4 12.b even 2 1
600.2.m.a 4 20.d odd 2 1
600.2.m.a 4 24.h odd 2 1
600.2.m.a 4 40.f even 2 1
600.2.m.a 4 60.h even 2 1
600.2.m.a 4 120.i odd 2 1
648.2.l.b 4 180.v odd 12 2
648.2.l.b 4 180.x even 12 2
648.2.l.b 4 360.br even 12 2
648.2.l.b 4 360.bu odd 12 2
768.2.c.h 4 80.i odd 4 1
768.2.c.h 4 80.j even 4 1
768.2.c.h 4 80.s even 4 1
768.2.c.h 4 80.t odd 4 1
768.2.c.h 4 240.z odd 4 1
768.2.c.h 4 240.bb even 4 1
768.2.c.h 4 240.bd odd 4 1
768.2.c.h 4 240.bf even 4 1
2400.2.b.a 2 5.c odd 4 1
2400.2.b.a 2 15.e even 4 1
2400.2.b.a 2 40.k even 4 1
2400.2.b.a 2 120.q odd 4 1
2400.2.m.a 4 1.a even 1 1 trivial
2400.2.m.a 4 3.b odd 2 1 inner
2400.2.m.a 4 5.b even 2 1 inner
2400.2.m.a 4 8.d odd 2 1 CM
2400.2.m.a 4 15.d odd 2 1 inner
2400.2.m.a 4 24.f even 2 1 inner
2400.2.m.a 4 40.e odd 2 1 inner
2400.2.m.a 4 120.m even 2 1 inner
2592.2.p.b 4 45.k odd 12 2
2592.2.p.b 4 45.l even 12 2
2592.2.p.b 4 360.bo even 12 2
2592.2.p.b 4 360.bt odd 12 2

Hecke kernels

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

\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 8 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$19$ \( (T - 2)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 128)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 200)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
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