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Properties

Label 1638.2.bj.f
Level $1638$
Weight $2$
Character orbit 1638.bj
Analytic conductor $13.079$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1638,2,Mod(127,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.bj (of order \(6\), degree \(2\), 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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.195105024.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_{4} q^{4} + (2 \beta_{6} - \beta_{5} - \beta_{3} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{6} - \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - \beta_{4} q^{4} + (2 \beta_{6} - \beta_{5} - \beta_{3} + \cdots + 1) q^{5}+ \cdots + \beta_{6} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 6 q^{10} - 6 q^{11} + 12 q^{13} - 8 q^{14} - 4 q^{16} - 2 q^{17} - 12 q^{19} + 6 q^{20} - 4 q^{22} - 8 q^{23} - 24 q^{25} - 6 q^{26} - 2 q^{29} + 6 q^{35} + 18 q^{37} + 4 q^{38} + 12 q^{40} - 12 q^{41} - 8 q^{43} + 18 q^{46} + 4 q^{49} - 12 q^{50} + 12 q^{52} + 12 q^{53} - 22 q^{55} - 4 q^{56} - 24 q^{58} - 18 q^{59} - 8 q^{61} - 8 q^{62} - 8 q^{64} - 46 q^{65} + 18 q^{67} + 2 q^{68} - 6 q^{71} + 6 q^{74} - 12 q^{76} + 8 q^{77} - 4 q^{79} + 6 q^{80} + 10 q^{82} - 54 q^{85} + 4 q^{88} + 18 q^{89} + 6 q^{91} - 16 q^{92} - 2 q^{94} + 50 q^{95} - 54 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} + 2\nu^{5} + 10\nu^{4} + 10\nu^{3} + 42\nu^{2} - 45\nu - 27 ) / 216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 2\nu^{5} - 10\nu^{4} - 10\nu^{3} + 30\nu^{2} - 27\nu + 27 ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 8\nu^{6} - 4\nu^{5} - 26\nu^{4} + 52\nu^{3} + 18\nu^{2} - 153\nu + 162 ) / 108 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{7} + 13\nu^{6} - 2\nu^{5} - 22\nu^{4} + 26\nu^{3} + 54\nu^{2} - 225\nu + 243 ) / 108 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\nu^{7} - 49\nu^{6} + 14\nu^{5} + 130\nu^{4} - 218\nu^{3} - 150\nu^{2} + 801\nu - 891 ) / 216 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 14\nu^{6} - 10\nu^{5} - 32\nu^{4} + 58\nu^{3} + 24\nu^{2} - 207\nu + 270 ) / 36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + 2\beta_{4} - \beta_{3} + 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 3\beta_{6} + 3\beta_{5} - 3\beta_{3} + 6\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{7} - 9\beta_{6} + 2\beta_{5} - 4\beta_{4} - 5\beta_{3} - 6\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} - 6\beta_{6} + 6\beta_{5} - 24\beta_{4} - 8\beta_{3} - 6\beta_{2} + 8\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} - 18\beta_{6} - 18\beta_{5} - 36\beta_{4} + 6\beta_{2} - 9\beta _1 + 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-\beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.72124 − 0.193255i
−1.58726 + 0.693255i
1.30512 + 1.13871i
0.560908 − 1.63871i
−1.58726 − 0.693255i
1.72124 + 0.193255i
0.560908 + 1.63871i
1.30512 − 1.13871i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i − 3.05596i 0 0.866025 + 0.500000i 1.00000i 0 1.52798 + 2.64654i
127.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.78801i 0 0.866025 + 0.500000i 1.00000i 0 −0.894007 − 1.54846i
127.3 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.332808i 0 −0.866025 − 0.500000i − 1.00000i 0 0.166404 + 0.288220i
127.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i 4.39924i 0 −0.866025 − 0.500000i − 1.00000i 0 2.19962 + 3.80986i
1135.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i − 1.78801i 0 0.866025 − 0.500000i − 1.00000i 0 −0.894007 + 1.54846i
1135.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 3.05596i 0 0.866025 − 0.500000i − 1.00000i 0 1.52798 − 2.64654i
1135.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i − 4.39924i 0 −0.866025 + 0.500000i 1.00000i 0 2.19962 − 3.80986i
1135.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i − 0.332808i 0 −0.866025 + 0.500000i 1.00000i 0 0.166404 − 0.288220i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.bj.f 8
3.b odd 2 1 546.2.s.e ✓ 8
13.e even 6 1 inner 1638.2.bj.f 8
39.h odd 6 1 546.2.s.e ✓ 8
39.k even 12 1 7098.2.a.cn 4
39.k even 12 1 7098.2.a.co 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.e ✓ 8 3.b odd 2 1
546.2.s.e ✓ 8 39.h odd 6 1
1638.2.bj.f 8 1.a even 1 1 trivial
1638.2.bj.f 8 13.e even 6 1 inner
7098.2.a.cn 4 39.k even 12 1
7098.2.a.co 4 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5}^{8} + 32T_{5}^{6} + 276T_{5}^{4} + 608T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{8} + 6T_{11}^{7} + 3T_{11}^{6} - 54T_{11}^{5} - 31T_{11}^{4} + 648T_{11}^{3} + 2016T_{11}^{2} + 2304T_{11} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 32 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{8} - 12 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + \cdots + 37249 \) Copy content Toggle raw display
$19$ \( T^{8} + 12 T^{7} + \cdots + 234256 \) Copy content Toggle raw display
$23$ \( T^{8} + 8 T^{7} + \cdots + 183184 \) Copy content Toggle raw display
$29$ \( T^{8} + 2 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$31$ \( T^{8} + 114 T^{6} + \cdots + 47524 \) Copy content Toggle raw display
$37$ \( T^{8} - 18 T^{7} + \cdots + 5184 \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + \cdots + 20164 \) Copy content Toggle raw display
$47$ \( T^{8} + 378 T^{6} + \cdots + 59783824 \) Copy content Toggle raw display
$53$ \( (T^{4} - 6 T^{3} + \cdots + 2461)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 18 T^{7} + \cdots + 4477456 \) Copy content Toggle raw display
$61$ \( T^{8} + 8 T^{7} + \cdots + 2283121 \) Copy content Toggle raw display
$67$ \( T^{8} - 18 T^{7} + \cdots + 11916304 \) Copy content Toggle raw display
$71$ \( T^{8} + 6 T^{7} + \cdots + 324 \) Copy content Toggle raw display
$73$ \( T^{8} + 288 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$79$ \( (T^{4} + 2 T^{3} + \cdots - 104)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 382 T^{6} + \cdots + 22146436 \) Copy content Toggle raw display
$89$ \( T^{8} - 18 T^{7} + \cdots + 20584369 \) Copy content Toggle raw display
$97$ \( T^{8} + 54 T^{7} + \cdots + 1290496 \) Copy content Toggle raw display
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