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Properties

Label 1710.2.d.g
Level $1710$
Weight $2$
Character orbit 1710.d
Analytic conductor $13.654$
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] = [1710,2,Mod(1369,1710)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1710, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1710.1369");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.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: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{4} + \beta_{7} q^{5} + ( - \beta_{6} + \beta_{3} + \beta_{2}) q^{7} - \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{4} + \beta_{7} q^{5} + ( - \beta_{6} + \beta_{3} + \beta_{2}) q^{7} - \beta_1 q^{8} + \beta_{3} q^{10} + (\beta_{7} + \beta_{5} - \beta_{4}) q^{11} + ( - \beta_{6} + \beta_{3} + \beta_{2}) q^{13} - \beta_{4} q^{14} + q^{16} + ( - \beta_{7} + \beta_{5} - 4 \beta_1) q^{17} + q^{19} - \beta_{7} q^{20} - \beta_{2} q^{22} + ( - \beta_{7} + \beta_{5} + 2 \beta_1) q^{23} + (\beta_{6} - \beta_{3} - 2 \beta_{2} - 1) q^{25} - \beta_{4} q^{26} + (\beta_{6} - \beta_{3} - \beta_{2}) q^{28} + (3 \beta_{7} + 3 \beta_{5} - \beta_{4}) q^{29} + ( - 2 \beta_{6} - 2 \beta_{3} - 4) q^{31} + \beta_1 q^{32} + ( - \beta_{6} - \beta_{3} + 4) q^{34} + (2 \beta_{5} + \beta_{4} + 2 \beta_1) q^{35} + (\beta_{6} - \beta_{3} + \beta_{2}) q^{37} + \beta_1 q^{38} - \beta_{3} q^{40} + ( - 2 \beta_{7} - 2 \beta_{5} + \beta_{4}) q^{41} + ( - 2 \beta_{6} + 2 \beta_{3} + \beta_{2}) q^{43} + ( - \beta_{7} - \beta_{5} + \beta_{4}) q^{44} + ( - \beta_{6} - \beta_{3} - 2) q^{46} + ( - \beta_{7} + \beta_{5} - 6 \beta_1) q^{47} + ( - 2 \beta_{6} - 2 \beta_{3} - 1) q^{49} + ( - \beta_{7} - \beta_{5} + \cdots - \beta_1) q^{50}+ \cdots + (2 \beta_{7} - 2 \beta_{5} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 8 q^{16} + 8 q^{19} - 8 q^{25} - 32 q^{31} + 32 q^{34} - 16 q^{46} - 8 q^{49} + 16 q^{55} - 16 q^{61} - 8 q^{64} - 16 q^{70} - 8 q^{76} + 32 q^{79} + 48 q^{85} - 64 q^{91} + 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{7} - 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}^{2} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{7} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{5} + 2\zeta_{24}^{4} + \zeta_{24}^{3} + \zeta_{24} - 1 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} - 2\zeta_{24}^{4} + \zeta_{24}^{3} + \zeta_{24} + 1 \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( -\beta_{7} + \beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1369.1
0.258819 − 0.965926i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.965926 − 0.258819i
−0.965926 − 0.258819i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.258819 − 0.965926i
− 1.00000i 0 −1.00000 −1.41421 − 1.73205i 0 − 1.03528i 1.00000i 0 −1.73205 + 1.41421i
1369.2 − 1.00000i 0 −1.00000 −1.41421 + 1.73205i 0 3.86370i 1.00000i 0 1.73205 + 1.41421i
1369.3 − 1.00000i 0 −1.00000 1.41421 − 1.73205i 0 1.03528i 1.00000i 0 −1.73205 − 1.41421i
1369.4 − 1.00000i 0 −1.00000 1.41421 + 1.73205i 0 − 3.86370i 1.00000i 0 1.73205 − 1.41421i
1369.5 1.00000i 0 −1.00000 −1.41421 − 1.73205i 0 − 3.86370i − 1.00000i 0 1.73205 − 1.41421i
1369.6 1.00000i 0 −1.00000 −1.41421 + 1.73205i 0 1.03528i − 1.00000i 0 −1.73205 − 1.41421i
1369.7 1.00000i 0 −1.00000 1.41421 − 1.73205i 0 3.86370i − 1.00000i 0 1.73205 + 1.41421i
1369.8 1.00000i 0 −1.00000 1.41421 + 1.73205i 0 − 1.03528i − 1.00000i 0 −1.73205 + 1.41421i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1369.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.d.g ✓ 8
3.b odd 2 1 inner 1710.2.d.g ✓ 8
5.b even 2 1 inner 1710.2.d.g ✓ 8
5.c odd 4 1 8550.2.a.cu 4
5.c odd 4 1 8550.2.a.cv 4
15.d odd 2 1 inner 1710.2.d.g ✓ 8
15.e even 4 1 8550.2.a.cu 4
15.e even 4 1 8550.2.a.cv 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.d.g ✓ 8 1.a even 1 1 trivial
1710.2.d.g ✓ 8 3.b odd 2 1 inner
1710.2.d.g ✓ 8 5.b even 2 1 inner
1710.2.d.g ✓ 8 15.d odd 2 1 inner
8550.2.a.cu 4 5.c odd 4 1
8550.2.a.cu 4 15.e even 4 1
8550.2.a.cv 4 5.c odd 4 1
8550.2.a.cv 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7}^{4} + 16T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} - 16T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} + 16T_{13}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 56 T^{2} + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 32 T^{2} + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 112 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T - 32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 96 T^{2} + 576)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 208 T^{2} + 7744)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 192 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 256 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 192 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 288 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 304 T^{2} + 1936)^{2} \) Copy content Toggle raw display
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