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Properties

Label 1638.2.r.v
Level $1638$
Weight $2$
Character orbit 1638.r
Analytic conductor $13.079$
Analytic rank $0$
Dimension $4$
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(757,1638)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.757");
 
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.r (of order \(3\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_1 - 1) q^{4} + ( - \beta_{3} - 2) q^{5} + ( - \beta_1 + 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_1 - 1) q^{4} + ( - \beta_{3} - 2) q^{5} + ( - \beta_1 + 1) q^{7} - q^{8} + ( - \beta_{2} - 2 \beta_1) q^{10} + (2 \beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{13} + q^{14} - \beta_1 q^{16} + (2 \beta_{3} - 2 \beta_{2}) q^{17} + (2 \beta_{3} - 2 \beta_{2}) q^{19} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{20} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{22}+ \cdots + ( - \beta_1 + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 8 q^{5} + 2 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} - 8 q^{5} + 2 q^{7} - 4 q^{8} - 4 q^{10} - 4 q^{11} - 4 q^{13} + 4 q^{14} - 2 q^{16} + 4 q^{20} + 4 q^{22} + 10 q^{23} + 8 q^{25} - 8 q^{26} + 2 q^{28} + 4 q^{29} + 16 q^{31} + 2 q^{32} - 4 q^{35} + 12 q^{37} + 8 q^{40} + 12 q^{41} - 8 q^{43} + 8 q^{44} - 10 q^{46} + 8 q^{47} - 2 q^{49} + 4 q^{50} - 4 q^{52} - 4 q^{55} - 2 q^{56} - 4 q^{58} + 16 q^{59} + 4 q^{61} + 8 q^{62} + 4 q^{64} + 26 q^{65} + 4 q^{67} - 8 q^{70} + 2 q^{71} - 16 q^{73} - 12 q^{74} - 8 q^{77} + 32 q^{79} + 4 q^{80} - 12 q^{82} - 12 q^{85} - 16 q^{86} + 4 q^{88} - 24 q^{89} + 4 q^{91} - 20 q^{92} + 4 q^{94} - 12 q^{95} + 20 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −3.73205 0 0.500000 − 0.866025i −1.00000 0 −1.86603 − 3.23205i
757.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.267949 0 0.500000 − 0.866025i −1.00000 0 −0.133975 − 0.232051i
1387.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −3.73205 0 0.500000 + 0.866025i −1.00000 0 −1.86603 + 3.23205i
1387.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.267949 0 0.500000 + 0.866025i −1.00000 0 −0.133975 + 0.232051i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.r.v 4
3.b odd 2 1 182.2.g.e ✓ 4
12.b even 2 1 1456.2.s.l 4
13.c even 3 1 inner 1638.2.r.v 4
21.c even 2 1 1274.2.g.l 4
21.g even 6 1 1274.2.e.p 4
21.g even 6 1 1274.2.h.p 4
21.h odd 6 1 1274.2.e.o 4
21.h odd 6 1 1274.2.h.o 4
39.h odd 6 1 2366.2.a.r 2
39.i odd 6 1 182.2.g.e ✓ 4
39.i odd 6 1 2366.2.a.t 2
39.k even 12 2 2366.2.d.l 4
156.p even 6 1 1456.2.s.l 4
273.r even 6 1 1274.2.h.p 4
273.s odd 6 1 1274.2.h.o 4
273.bf even 6 1 1274.2.e.p 4
273.bm odd 6 1 1274.2.e.o 4
273.bn even 6 1 1274.2.g.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.e ✓ 4 3.b odd 2 1
182.2.g.e ✓ 4 39.i odd 6 1
1274.2.e.o 4 21.h odd 6 1
1274.2.e.o 4 273.bm odd 6 1
1274.2.e.p 4 21.g even 6 1
1274.2.e.p 4 273.bf even 6 1
1274.2.g.l 4 21.c even 2 1
1274.2.g.l 4 273.bn even 6 1
1274.2.h.o 4 21.h odd 6 1
1274.2.h.o 4 273.s odd 6 1
1274.2.h.p 4 21.g even 6 1
1274.2.h.p 4 273.r even 6 1
1456.2.s.l 4 12.b even 2 1
1456.2.s.l 4 156.p even 6 1
1638.2.r.v 4 1.a even 1 1 trivial
1638.2.r.v 4 13.c even 3 1 inner
2366.2.a.r 2 39.h odd 6 1
2366.2.a.t 2 39.i odd 6 1
2366.2.d.l 4 39.k even 12 2

Hecke kernels

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

\( T_{5}^{2} + 4T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 4T_{11}^{3} + 24T_{11}^{2} - 32T_{11} + 64 \) Copy content Toggle raw display
\( T_{17}^{4} + 12T_{17}^{2} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$19$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$23$ \( T^{4} - 10 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 16 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$71$ \( T^{4} - 2 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$73$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + \cdots + 17424 \) Copy content Toggle raw display
$97$ \( T^{4} - 20 T^{3} + \cdots + 2704 \) Copy content Toggle raw display
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