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Properties

Label 1872.2.n.d
Level $1872$
Weight $2$
Character orbit 1872.n
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(1871,1872)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1872.1871");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.n (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: 8.0.764411904.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7} \)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{7} + 3\nu^{5} + 39\nu^{3} - 189\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} - 18\nu^{2} + 33 ) / 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - 12\nu^{2} + 27 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 3\nu^{5} + 15\nu^{3} - 27\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} - 12\nu^{5} + 42\nu^{3} - 54\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{6} + 12\nu^{4} - 24\nu^{2} + 54 ) / 45 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -14\nu^{7} + 66\nu^{5} - 132\nu^{3} + 432\nu ) / 135 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 2\beta_{3} - 6\beta_{2} + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 2\beta_{5} + \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{6} + 12\beta_{3} - 12\beta_{2} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} - 3\beta_{5} + 18\beta_{4} + 12\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -15\beta_{6} + 6\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{7} - 21\beta_{5} + 66\beta_{4} - 24\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1871.1
−1.27970 − 1.16721i
−1.27970 + 1.16721i
−1.69185 + 0.370982i
−1.69185 − 0.370982i
1.69185 + 0.370982i
1.69185 − 0.370982i
1.27970 − 1.16721i
1.27970 + 1.16721i
0 0 0 −3.30136 0 3.30136 0 0 0
1871.2 0 0 0 −3.30136 0 3.30136 0 0 0
1871.3 0 0 0 −1.04930 0 1.04930 0 0 0
1871.4 0 0 0 −1.04930 0 1.04930 0 0 0
1871.5 0 0 0 1.04930 0 −1.04930 0 0 0
1871.6 0 0 0 1.04930 0 −1.04930 0 0 0
1871.7 0 0 0 3.30136 0 −3.30136 0 0 0
1871.8 0 0 0 3.30136 0 −3.30136 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1871.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
13.b even 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 1872.2.n.d ✓ 8
3.b odd 2 1 1872.2.n.e yes 8
4.b odd 2 1 1872.2.n.e yes 8
12.b even 2 1 inner 1872.2.n.d ✓ 8
13.b even 2 1 inner 1872.2.n.d ✓ 8
39.d odd 2 1 1872.2.n.e yes 8
52.b odd 2 1 1872.2.n.e yes 8
156.h even 2 1 inner 1872.2.n.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1872.2.n.d ✓ 8 1.a even 1 1 trivial
1872.2.n.d ✓ 8 12.b even 2 1 inner
1872.2.n.d ✓ 8 13.b even 2 1 inner
1872.2.n.d ✓ 8 156.h even 2 1 inner
1872.2.n.e yes 8 3.b odd 2 1
1872.2.n.e yes 8 4.b odd 2 1
1872.2.n.e yes 8 39.d odd 2 1
1872.2.n.e yes 8 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5}^{4} - 12T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{7}^{4} - 12T_{7}^{2} + 12 \) Copy content Toggle raw display
\( T_{23} + 6 \) 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
$5$ \( (T^{4} - 12 T^{2} + 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 12 T^{2} + 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 12 T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 28 T^{2} + 100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 12 T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T + 6)^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 36 T^{2} + 108)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 24 T^{2} + 48)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 108 T^{2} + 2700)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 168 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 108 T^{2} + 2700)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 28 T^{2} + 100)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 132 T^{2} + 4332)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 324 T^{2} + 24300)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 132 T^{2} + 300)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 264 T^{2} + 17328)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 228 T^{2} + 10092)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 108 T^{2} + 12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 72 T^{2} + 432)^{2} \) Copy content Toggle raw display
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