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Properties

Label 1638.2.c.i
Level $1638$
Weight $2$
Character orbit 1638.c
Analytic conductor $13.079$
Analytic rank $0$
Dimension $6$
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(883,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.883");
 
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.c (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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.30647296.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 8x^{3} + 25x^{2} - 10x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - q^{4} + ( - \beta_{2} - \beta_1) q^{5} + \beta_{4} q^{7} - \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - q^{4} + ( - \beta_{2} - \beta_1) q^{5} + \beta_{4} q^{7} - \beta_{4} q^{8} + (\beta_{2} - \beta_1) q^{10} + ( - \beta_{5} - 2 \beta_{4} + \cdots + \beta_1) q^{11}+ \cdots - \beta_{4} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} - 4 q^{10} - 10 q^{13} - 6 q^{14} + 6 q^{16} + 4 q^{17} + 14 q^{22} + 6 q^{23} - 18 q^{25} + 4 q^{26} - 16 q^{29} - 4 q^{35} + 8 q^{38} + 4 q^{40} + 16 q^{43} - 6 q^{49} + 10 q^{52} + 36 q^{53} + 60 q^{55} + 6 q^{56} + 10 q^{61} - 14 q^{62} - 6 q^{64} + 20 q^{65} - 4 q^{68} - 18 q^{74} + 14 q^{77} - 14 q^{79} + 30 q^{82} - 14 q^{88} + 4 q^{91} - 6 q^{92} + 34 q^{94} - 40 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 8x^{3} + 25x^{2} - 10x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{5} - 18\nu^{4} + 49\nu^{3} + 28\nu^{2} + 159\nu - 62 ) / 173 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{5} - 53\nu^{4} + 77\nu^{3} + 44\nu^{2} - 22\nu - 567 ) / 173 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -31\nu^{5} + 55\nu^{4} - 44\nu^{3} - 297\nu^{2} - 803\nu + 151 ) / 173 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -100\nu^{5} + 183\nu^{4} - 181\nu^{3} - 746\nu^{2} - 2741\nu + 515 ) / 173 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 2\beta_{4} - \beta_{3} + 7\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{3} + 11\beta_{2} - 11\beta _1 - 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{5} + 26\beta_{4} - 11\beta_{3} - 55\beta _1 - 26 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
2.08433 + 2.08433i
0.188470 + 0.188470i
−1.27280 − 1.27280i
−1.27280 + 1.27280i
0.188470 − 0.188470i
2.08433 − 2.08433i
− 1.00000i 0 −1.00000 − 4.16867i 0 − 1.00000i 1.00000i 0 −4.16867
883.2 − 1.00000i 0 −1.00000 − 0.376939i 0 − 1.00000i 1.00000i 0 −0.376939
883.3 − 1.00000i 0 −1.00000 2.54561i 0 − 1.00000i 1.00000i 0 2.54561
883.4 1.00000i 0 −1.00000 − 2.54561i 0 1.00000i − 1.00000i 0 2.54561
883.5 1.00000i 0 −1.00000 0.376939i 0 1.00000i − 1.00000i 0 −0.376939
883.6 1.00000i 0 −1.00000 4.16867i 0 1.00000i − 1.00000i 0 −4.16867
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.c.i 6
3.b odd 2 1 182.2.d.b ✓ 6
12.b even 2 1 1456.2.k.b 6
13.b even 2 1 inner 1638.2.c.i 6
21.c even 2 1 1274.2.d.l 6
21.g even 6 2 1274.2.n.n 12
21.h odd 6 2 1274.2.n.m 12
39.d odd 2 1 182.2.d.b ✓ 6
39.f even 4 1 2366.2.a.x 3
39.f even 4 1 2366.2.a.bc 3
156.h even 2 1 1456.2.k.b 6
273.g even 2 1 1274.2.d.l 6
273.w odd 6 2 1274.2.n.m 12
273.ba even 6 2 1274.2.n.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.d.b ✓ 6 3.b odd 2 1
182.2.d.b ✓ 6 39.d odd 2 1
1274.2.d.l 6 21.c even 2 1
1274.2.d.l 6 273.g even 2 1
1274.2.n.m 12 21.h odd 6 2
1274.2.n.m 12 273.w odd 6 2
1274.2.n.n 12 21.g even 6 2
1274.2.n.n 12 273.ba even 6 2
1456.2.k.b 6 12.b even 2 1
1456.2.k.b 6 156.h even 2 1
1638.2.c.i 6 1.a even 1 1 trivial
1638.2.c.i 6 13.b even 2 1 inner
2366.2.a.x 3 39.f even 4 1
2366.2.a.bc 3 39.f even 4 1

Hecke kernels

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

\( T_{5}^{6} + 24T_{5}^{4} + 116T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{6} + 65T_{11}^{4} + 1296T_{11}^{2} + 7744 \) Copy content Toggle raw display
\( T_{17}^{3} - 2T_{17}^{2} - 44T_{17} + 56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 24 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 65 T^{4} + \cdots + 7744 \) Copy content Toggle raw display
$13$ \( T^{6} + 10 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} - 2 T^{2} - 44 T + 56)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 28 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{3} - 3 T^{2} - 16 T + 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 8 T^{2} + \cdots - 176)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 65 T^{4} + \cdots + 7744 \) Copy content Toggle raw display
$37$ \( T^{6} + 65 T^{4} + \cdots + 1936 \) Copy content Toggle raw display
$41$ \( T^{6} + 113 T^{4} + \cdots + 1936 \) Copy content Toggle raw display
$43$ \( (T^{3} - 8 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 145 T^{4} + \cdots + 7744 \) Copy content Toggle raw display
$53$ \( (T - 6)^{6} \) Copy content Toggle raw display
$59$ \( T^{6} + 288 T^{4} + \cdots + 781456 \) Copy content Toggle raw display
$61$ \( (T^{3} - 5 T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 113 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( T^{6} + 336 T^{4} + \cdots + 802816 \) Copy content Toggle raw display
$73$ \( T^{6} + 89 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$79$ \( (T^{3} + 7 T^{2} + \cdots + 148)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 352 T^{4} + \cdots + 1567504 \) Copy content Toggle raw display
$89$ \( T^{6} + 316 T^{4} + \cdots + 53824 \) Copy content Toggle raw display
$97$ \( T^{6} + 185 T^{4} + \cdots + 38416 \) Copy content Toggle raw display
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