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Properties

Label 2898.2.g.b
Level $2898$
Weight $2$
Character orbit 2898.g
Analytic conductor $23.141$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2898,2,Mod(2575,2898)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2898, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2898.2575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.g (of order \(2\), degree \(1\), 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: \(23.1406465058\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2312.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 322)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 2 q^{5} + (\beta_1 - 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + 2 q^{5} + (\beta_1 - 1) q^{7} - q^{8} - 2 q^{10} + \beta_{2} q^{11} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{13} + ( - \beta_1 + 1) q^{14} + q^{16} + ( - \beta_{3} + \beta_1 - 3) q^{17} + 4 q^{19} + 2 q^{20} - \beta_{2} q^{22} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{23} - q^{25} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{26} + (\beta_1 - 1) q^{28} + 2 q^{29} + ( - 3 \beta_{3} - 4 \beta_{2} + \cdots + 3) q^{31}+ \cdots + ( - 3 \beta_{3} - 2 \beta_{2} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} + 8 q^{5} - 2 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} + 8 q^{5} - 2 q^{7} - 4 q^{8} - 8 q^{10} + 2 q^{14} + 4 q^{16} - 12 q^{17} + 16 q^{19} + 8 q^{20} - 2 q^{23} - 4 q^{25} - 2 q^{28} + 8 q^{29} - 4 q^{32} + 12 q^{34} - 4 q^{35} - 16 q^{38} - 8 q^{40} + 2 q^{46} + 8 q^{49} + 4 q^{50} + 2 q^{56} - 8 q^{58} + 16 q^{61} + 4 q^{64} - 12 q^{68} + 4 q^{70} - 12 q^{71} + 16 q^{76} + 8 q^{77} + 8 q^{80} - 16 q^{83} - 24 q^{85} - 8 q^{89} - 12 q^{91} - 2 q^{92} + 32 q^{95} - 16 q^{97} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x + 4 \) : Copy content Toggle raw display

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

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} + \beta_{2} + \beta _1 + 2 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2898\mathbb{Z}\right)^\times\).

\(n\) \(829\) \(1289\) \(1891\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2575.1
−0.780776 + 1.17915i
−0.780776 − 1.17915i
1.28078 − 0.599676i
1.28078 + 0.599676i
−1.00000 0 1.00000 2.00000 0 −2.56155 − 0.662153i −1.00000 0 −2.00000
2575.2 −1.00000 0 1.00000 2.00000 0 −2.56155 + 0.662153i −1.00000 0 −2.00000
2575.3 −1.00000 0 1.00000 2.00000 0 1.56155 − 2.13578i −1.00000 0 −2.00000
2575.4 −1.00000 0 1.00000 2.00000 0 1.56155 + 2.13578i −1.00000 0 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2898.2.g.b 4
3.b odd 2 1 322.2.c.c ✓ 4
7.b odd 2 1 2898.2.g.a 4
12.b even 2 1 2576.2.f.b 4
21.c even 2 1 322.2.c.d yes 4
23.b odd 2 1 2898.2.g.a 4
69.c even 2 1 322.2.c.d yes 4
84.h odd 2 1 2576.2.f.e 4
161.c even 2 1 inner 2898.2.g.b 4
276.h odd 2 1 2576.2.f.e 4
483.c odd 2 1 322.2.c.c ✓ 4
1932.b even 2 1 2576.2.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.c.c ✓ 4 3.b odd 2 1
322.2.c.c ✓ 4 483.c odd 2 1
322.2.c.d yes 4 21.c even 2 1
322.2.c.d yes 4 69.c even 2 1
2576.2.f.b 4 12.b even 2 1
2576.2.f.b 4 1932.b even 2 1
2576.2.f.e 4 84.h odd 2 1
2576.2.f.e 4 276.h odd 2 1
2898.2.g.a 4 7.b odd 2 1
2898.2.g.a 4 23.b odd 2 1
2898.2.g.b 4 1.a even 1 1 trivial
2898.2.g.b 4 161.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2898, [\chi])\):

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{11}^{4} + 10T_{11}^{2} + 8 \) Copy content Toggle raw display
\( T_{13}^{4} + 14T_{13}^{2} + 32 \) Copy content Toggle raw display
\( T_{29} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T - 2)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 10T^{2} + 8 \) Copy content Toggle raw display
$13$ \( T^{4} + 14T^{2} + 32 \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$29$ \( (T - 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 148T^{2} + 5408 \) Copy content Toggle raw display
$37$ \( T^{4} + 14T^{2} + 32 \) Copy content Toggle raw display
$41$ \( T^{4} + 56T^{2} + 512 \) Copy content Toggle raw display
$43$ \( T^{4} + 10T^{2} + 8 \) Copy content Toggle raw display
$47$ \( T^{4} + 92T^{2} + 2048 \) Copy content Toggle raw display
$53$ \( T^{4} + 190T^{2} + 32 \) Copy content Toggle raw display
$59$ \( T^{4} + 90T^{2} + 648 \) Copy content Toggle raw display
$61$ \( (T^{2} - 8 T - 52)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 58T^{2} + 8 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T - 144)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 160T^{2} + 2048 \) Copy content Toggle raw display
$79$ \( T^{4} + 180T^{2} + 2592 \) Copy content Toggle raw display
$83$ \( (T + 4)^{4} \) Copy content Toggle raw display
$89$ \( (T + 2)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 52)^{2} \) Copy content Toggle raw display
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