Properties

Label 1872.2.d.d
Level $1872$
Weight $2$
Character orbit 1872.d
Analytic conductor $14.948$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1872,2,Mod(287,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 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("1872.287");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.d (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: \(14.9479952584\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} - \beta_1 q^{7} + \beta_{5} q^{11} + q^{13} - \beta_{3} q^{17} + ( - 2 \beta_{7} + \beta_1) q^{19} + ( - \beta_{5} + \beta_{4}) q^{23} + 3 q^{25} - 3 \beta_{2} q^{29} + ( - \beta_{7} - 2 \beta_1) q^{31} + ( - \beta_{5} + \beta_{4}) q^{35} + (\beta_{6} + 2) q^{37} + (2 \beta_{3} + \beta_{2}) q^{41} + ( - \beta_{7} - \beta_1) q^{43} + (\beta_{5} + 4 \beta_{4}) q^{47} + ( - \beta_{6} - 1) q^{49} + (\beta_{3} - 2 \beta_{2}) q^{53} + (\beta_{7} - \beta_1) q^{55} + 3 \beta_{4} q^{59} - \beta_{6} q^{61} - \beta_{2} q^{65} + (\beta_{7} - 2 \beta_1) q^{67} + ( - 2 \beta_{5} + 3 \beta_{4}) q^{71} + (\beta_{6} - 2) q^{73} + ( - \beta_{3} - 5 \beta_{2}) q^{77} - 2 \beta_{7} q^{79} + 3 \beta_{4} q^{83} - \beta_{6} q^{85} + ( - 2 \beta_{3} - \beta_{2}) q^{89} - \beta_1 q^{91} + (3 \beta_{5} + \beta_{4}) q^{95} + (\beta_{6} + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{13} + 24 q^{25} + 16 q^{37} - 8 q^{49} - 16 q^{73} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 14\nu^{4} + 56\nu^{2} + 27 ) / 63 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{7} - 7\nu^{5} + 35\nu^{3} - 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\nu^{7} + 49\nu^{5} + 133\nu^{3} + 801\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} + \nu^{5} - 5\nu^{3} - 63\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{7} - 7\nu^{5} - 19\nu^{3} - 81\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{6} - 8\nu^{4} + 4\nu^{2} - 36 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\nu^{6} + 14\nu^{4} + 56\nu^{2} + 423 ) / 63 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 3\beta _1 - 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{5} + 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{7} - 4\beta_{6} + 5\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{5} + 19\beta_{4} + 5\beta_{3} - 19\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7\beta_{7} - 7\beta _1 - 44 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -29\beta_{5} - 13\beta_{4} - 29\beta_{3} - 13\beta_{2} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
1.40294 + 1.01575i
0.178197 1.72286i
−0.178197 1.72286i
−1.40294 + 1.01575i
−1.40294 1.01575i
−0.178197 + 1.72286i
0.178197 + 1.72286i
1.40294 1.01575i
0 0 0 1.41421i 0 3.96812i 0 0 0
287.2 0 0 0 1.41421i 0 0.504017i 0 0 0
287.3 0 0 0 1.41421i 0 0.504017i 0 0 0
287.4 0 0 0 1.41421i 0 3.96812i 0 0 0
287.5 0 0 0 1.41421i 0 3.96812i 0 0 0
287.6 0 0 0 1.41421i 0 0.504017i 0 0 0
287.7 0 0 0 1.41421i 0 0.504017i 0 0 0
287.8 0 0 0 1.41421i 0 3.96812i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.2.d.d 8
3.b odd 2 1 inner 1872.2.d.d 8
4.b odd 2 1 inner 1872.2.d.d 8
8.b even 2 1 7488.2.d.g 8
8.d odd 2 1 7488.2.d.g 8
12.b even 2 1 inner 1872.2.d.d 8
24.f even 2 1 7488.2.d.g 8
24.h odd 2 1 7488.2.d.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1872.2.d.d 8 1.a even 1 1 trivial
1872.2.d.d 8 3.b odd 2 1 inner
1872.2.d.d 8 4.b odd 2 1 inner
1872.2.d.d 8 12.b even 2 1 inner
7488.2.d.g 8 8.b even 2 1
7488.2.d.g 8 8.d odd 2 1
7488.2.d.g 8 24.f even 2 1
7488.2.d.g 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1872, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 16 T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$13$ \( (T - 1)^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} + 30)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 96 T^{2} + 1764)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 32 T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 64 T^{2} + 484)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T - 56)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 244 T^{2} + 13924)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 212 T^{2} + 7396)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 76 T^{2} + 484)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 60)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 96 T^{2} + 1764)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 188 T^{2} + 196)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4 T - 56)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 64 T^{2} + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 244 T^{2} + 13924)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T - 24)^{4} \) Copy content Toggle raw display
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