Properties

Label 684.5.h.d
Level $684$
Weight $5$
Character orbit 684.h
Self dual yes
Analytic conductor $70.705$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,5,Mod(37,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.37");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 684.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: yes
Analytic conductor: \(70.7050547493\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 11x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 2 \beta_1) q^{5} + (5 \beta_{3} - 34) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 2 \beta_1) q^{5} + (5 \beta_{3} - 34) q^{7} + ( - 13 \beta_{2} - 9 \beta_1) q^{11} + ( - 33 \beta_{2} + \beta_1) q^{17} - 361 q^{19} + (50 \beta_{2} - 55 \beta_1) q^{23} + (93 \beta_{3} + 527) q^{25} + (29 \beta_{2} - 47 \beta_1) q^{35} + ( - 85 \beta_{3} - 1806) q^{43} + ( - 177 \beta_{2} + 74 \beta_1) q^{47} + ( - 365 \beta_{3} + 1955) q^{49} + (427 \beta_{3} + 6272) q^{55} + ( - 515 \beta_{3} + 1326) q^{61} + (275 \beta_{3} + 5154) q^{73} + (887 \beta_{2} - 339 \beta_1) q^{77} + ( - 600 \beta_{2} + 660 \beta_1) q^{83} + ( - 13 \beta_{3} + 3712) q^{85} + (361 \beta_{2} + 722 \beta_1) q^{95}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 146 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 146 q^{7} - 1444 q^{19} + 1922 q^{25} - 7054 q^{43} + 8550 q^{49} + 24234 q^{55} + 6334 q^{61} + 20066 q^{73} + 14874 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 11x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} - 5\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{3} + 61\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} - 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{2} - 7\beta_1 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 17 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -10\beta_{2} - 61\beta_1 ) / 48 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−3.04547
1.31342
−1.31342
3.04547
0 0 0 −46.4618 0 20.1238 0 0 0
37.2 0 0 0 −7.23174 0 −93.1238 0 0 0
37.3 0 0 0 7.23174 0 −93.1238 0 0 0
37.4 0 0 0 46.4618 0 20.1238 0 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
3.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.5.h.d 4
3.b odd 2 1 inner 684.5.h.d 4
19.b odd 2 1 CM 684.5.h.d 4
57.d even 2 1 inner 684.5.h.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.5.h.d 4 1.a even 1 1 trivial
684.5.h.d 4 3.b odd 2 1 inner
684.5.h.d 4 19.b odd 2 1 CM
684.5.h.d 4 57.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 2211T_{5}^{2} + 112896 \) acting on \(S_{5}^{\mathrm{new}}(684, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2211 T^{2} + 112896 \) Copy content Toggle raw display
$7$ \( (T^{2} + 73 T - 1874)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 83571 T^{2} + \cdots + 1571963904 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 291651 T^{2} + \cdots + 1688223744 \) Copy content Toggle raw display
$19$ \( (T + 361)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 1094400)^{2} \) 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
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 3527 T + 2183326)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 11216211 T^{2} + \cdots + 11715778900224 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 3167 T - 31507634)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 10033 T + 15466366)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 157593600)^{2} \) 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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