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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2400,2,Mod(2351,2400)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2400.2351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 600)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{3} + (\beta_{3} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{3} + (\beta_{3} + \beta_1) q^{9} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{11} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{17} + ( - 2 \beta_{3} + 3 \beta_{2} + \cdots + 3) q^{19}+ \cdots + ( - 2 \beta_{3} + 12 \beta_{2} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 2 q^{9} + 4 q^{19} - 20 q^{27} + 22 q^{33} + 40 q^{43} + 28 q^{49} - 10 q^{51} + 34 q^{57} + 28 q^{67} - 4 q^{73} + 14 q^{81} - 40 q^{97} + 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu + 1 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + 2\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{3} + 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2351.1
1.22474 + 0.707107i
1.22474 − 0.707107i
−1.22474 − 0.707107i
−1.22474 + 0.707107i
0 −1.72474 − 0.158919i 0 0 0 0 0 2.94949 + 0.548188i 0
2351.2 0 −1.72474 + 0.158919i 0 0 0 0 0 2.94949 − 0.548188i 0
2351.3 0 0.724745 − 1.57313i 0 0 0 0 0 −1.94949 − 2.28024i 0
2351.4 0 0.724745 + 1.57313i 0 0 0 0 0 −1.94949 + 2.28024i 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.b 4
3.b odd 2 1 inner 2400.2.b.b 4
4.b odd 2 1 600.2.b.d yes 4
5.b even 2 1 2400.2.b.d 4
5.c odd 4 2 2400.2.m.b 8
8.b even 2 1 600.2.b.d yes 4
8.d odd 2 1 CM 2400.2.b.b 4
12.b even 2 1 600.2.b.d yes 4
15.d odd 2 1 2400.2.b.d 4
15.e even 4 2 2400.2.m.b 8
20.d odd 2 1 600.2.b.b ✓ 4
20.e even 4 2 600.2.m.b 8
24.f even 2 1 inner 2400.2.b.b 4
24.h odd 2 1 600.2.b.d yes 4
40.e odd 2 1 2400.2.b.d 4
40.f even 2 1 600.2.b.b ✓ 4
40.i odd 4 2 600.2.m.b 8
40.k even 4 2 2400.2.m.b 8
60.h even 2 1 600.2.b.b ✓ 4
60.l odd 4 2 600.2.m.b 8
120.i odd 2 1 600.2.b.b ✓ 4
120.m even 2 1 2400.2.b.d 4
120.q odd 4 2 2400.2.m.b 8
120.w even 4 2 600.2.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.b.b ✓ 4 20.d odd 2 1
600.2.b.b ✓ 4 40.f even 2 1
600.2.b.b ✓ 4 60.h even 2 1
600.2.b.b ✓ 4 120.i odd 2 1
600.2.b.d yes 4 4.b odd 2 1
600.2.b.d yes 4 8.b even 2 1
600.2.b.d yes 4 12.b even 2 1
600.2.b.d yes 4 24.h odd 2 1
600.2.m.b 8 20.e even 4 2
600.2.m.b 8 40.i odd 4 2
600.2.m.b 8 60.l odd 4 2
600.2.m.b 8 120.w even 4 2
2400.2.b.b 4 1.a even 1 1 trivial
2400.2.b.b 4 3.b odd 2 1 inner
2400.2.b.b 4 8.d odd 2 1 CM
2400.2.b.b 4 24.f even 2 1 inner
2400.2.b.d 4 5.b even 2 1
2400.2.b.d 4 15.d odd 2 1
2400.2.b.d 4 40.e odd 2 1
2400.2.b.d 4 120.m even 2 1
2400.2.m.b 8 5.c odd 4 2
2400.2.m.b 8 15.e even 4 2
2400.2.m.b 8 40.k even 4 2
2400.2.m.b 8 120.q odd 4 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
\( T_{11}^{4} + 58T_{11}^{2} + 625 \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display
\( T_{43} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 58T^{2} + 625 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 70T^{2} + 361 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 53)^{2} \) 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
$41$ \( T^{4} + 118T^{2} + 25 \) Copy content Toggle raw display
$43$ \( (T - 10)^{4} \) 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
$67$ \( (T^{2} - 14 T - 5)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 215)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 490 T^{2} + 58081 \) Copy content Toggle raw display
$89$ \( T^{4} + 502 T^{2} + 55225 \) Copy content Toggle raw display
$97$ \( (T + 10)^{4} \) Copy content Toggle raw display
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