Properties

Label 468.2.h.b
Level $468$
Weight $2$
Character orbit 468.h
Analytic conductor $3.737$
Analytic rank $0$
Dimension $4$
CM discriminant -52
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [468,2,Mod(467,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("468.467");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.h (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: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - x^{2} + 2x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - 2 q^{4} + (\beta_{3} + 1) q^{7} + 2 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - 2 q^{4} + (\beta_{3} + 1) q^{7} + 2 \beta_{2} q^{8} + \beta_1 q^{11} + \beta_{3} q^{13} + ( - \beta_{2} - \beta_1) q^{14} + 4 q^{16} + (2 \beta_{2} - \beta_1) q^{17} + (\beta_{3} - 5) q^{19} + 2 \beta_{3} q^{22} - 5 q^{25} - \beta_1 q^{26} + ( - 2 \beta_{3} - 2) q^{28} + ( - 4 \beta_{2} - \beta_1) q^{29} + (\beta_{3} + 7) q^{31} - 4 \beta_{2} q^{32} + ( - 2 \beta_{3} + 4) q^{34} + (5 \beta_{2} - \beta_1) q^{38} - 2 \beta_1 q^{44} + \beta_1 q^{47} + (2 \beta_{3} + 7) q^{49} + 5 \beta_{2} q^{50} - 2 \beta_{3} q^{52} + ( - \beta_{2} + 2 \beta_1) q^{53} + (2 \beta_{2} + 2 \beta_1) q^{56} + ( - 2 \beta_{3} - 8) q^{58} + \beta_{2} q^{59} - 4 \beta_{3} q^{61} + ( - 7 \beta_{2} - \beta_1) q^{62} - 8 q^{64} + (\beta_{3} - 11) q^{67} + ( - 4 \beta_{2} + 2 \beta_1) q^{68} - 5 \beta_{2} q^{71} + ( - 2 \beta_{3} + 10) q^{76} + (13 \beta_{2} + \beta_1) q^{77} + 7 \beta_{2} q^{83} - 4 \beta_{3} q^{88} + (\beta_{3} + 13) q^{91} + 2 \beta_{3} q^{94} + ( - 7 \beta_{2} - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 4 q^{7} + 16 q^{16} - 20 q^{19} - 20 q^{25} - 8 q^{28} + 28 q^{31} + 16 q^{34} + 28 q^{49} - 32 q^{58} - 32 q^{64} - 44 q^{67} + 40 q^{76} + 52 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - x^{2} + 2x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} + 7\nu - 3 ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{3} + 6\nu^{2} + 28\nu - 15 ) / 21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 2\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 17\beta_{2} + 3\beta _1 + 4 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
467.1
−1.30278 + 1.41421i
2.30278 + 1.41421i
−1.30278 1.41421i
2.30278 1.41421i
1.41421i 0 −2.00000 0 0 −2.60555 2.82843i 0 0
467.2 1.41421i 0 −2.00000 0 0 4.60555 2.82843i 0 0
467.3 1.41421i 0 −2.00000 0 0 −2.60555 2.82843i 0 0
467.4 1.41421i 0 −2.00000 0 0 4.60555 2.82843i 0 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
52.b odd 2 1 CM by \(\Q(\sqrt{-13}) \)
3.b odd 2 1 inner
156.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.h.b yes 4
3.b odd 2 1 inner 468.2.h.b yes 4
4.b odd 2 1 468.2.h.a 4
12.b even 2 1 468.2.h.a 4
13.b even 2 1 468.2.h.a 4
39.d odd 2 1 468.2.h.a 4
52.b odd 2 1 CM 468.2.h.b yes 4
156.h even 2 1 inner 468.2.h.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.2.h.a 4 4.b odd 2 1
468.2.h.a 4 12.b even 2 1
468.2.h.a 4 13.b even 2 1
468.2.h.a 4 39.d odd 2 1
468.2.h.b yes 4 1.a even 1 1 trivial
468.2.h.b yes 4 3.b odd 2 1 inner
468.2.h.b yes 4 52.b odd 2 1 CM
468.2.h.b yes 4 156.h 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 \(S_{2}^{\mathrm{new}}(468, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T - 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 26)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 68T^{2} + 324 \) Copy content Toggle raw display
$19$ \( (T^{2} + 10 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 116T^{2} + 36 \) Copy content Toggle raw display
$31$ \( (T^{2} - 14 T + 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 26)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 212 T^{2} + 10404 \) Copy content Toggle raw display
$59$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 208)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 22 T + 108)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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