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Properties

Label 1638.2.m.f
Level $1638$
Weight $2$
Character orbit 1638.m
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(289,1638)]
 
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, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.289");
 
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.m (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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.4740147.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 10x^{4} + 25x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + (\beta_{5} + \beta_{4} + \cdots + \beta_{2}) q^{5}+ \cdots + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + (\beta_{5} + \beta_{4} + \cdots + \beta_{2}) q^{5}+ \cdots + (3 \beta_1 - 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 6 q^{4} - 3 q^{7} + 6 q^{8} + 4 q^{11} - 4 q^{13} - 3 q^{14} + 6 q^{16} + 2 q^{17} + 3 q^{19} + 4 q^{22} + 8 q^{23} - 15 q^{25} - 4 q^{26} - 3 q^{28} + q^{29} + 13 q^{31} + 6 q^{32} + 2 q^{34} + 4 q^{37} + 3 q^{38} + 3 q^{41} - 7 q^{43} + 4 q^{44} + 8 q^{46} - 13 q^{47} - 39 q^{49} - 15 q^{50} - 4 q^{52} - 13 q^{53} + 9 q^{55} - 3 q^{56} + q^{58} - 18 q^{59} - 11 q^{61} + 13 q^{62} + 6 q^{64} - 39 q^{65} - 6 q^{67} + 2 q^{68} + 15 q^{71} + 22 q^{73} + 4 q^{74} + 3 q^{76} + 16 q^{77} + q^{79} + 3 q^{82} + 28 q^{83} - 12 q^{85} - 7 q^{86} + 4 q^{88} - 34 q^{89} + 2 q^{91} + 8 q^{92} - 13 q^{94} + 27 q^{97} - 39 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 5\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} - 6\nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 8\nu^{3} - \nu^{2} + 15\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 6\nu^{2} - 12\nu + 3 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -10\beta_{5} - 10\beta_{4} - 5\beta_{3} - 5\beta_{2} + 6\beta _1 - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{3} - 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 50\beta_{5} + 56\beta_{4} + 25\beta_{3} + 28\beta_{2} - 48\beta _1 + 24 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
2.39248i
− 0.355387i
− 2.03709i
− 2.39248i
0.355387i
2.03709i
1.00000 0 1.00000 −2.07195 + 3.58872i 0 −0.500000 + 2.59808i 1.00000 0 −2.07195 + 3.58872i
289.2 1.00000 0 1.00000 0.307774 − 0.533081i 0 −0.500000 + 2.59808i 1.00000 0 0.307774 − 0.533081i
289.3 1.00000 0 1.00000 1.76417 − 3.05564i 0 −0.500000 + 2.59808i 1.00000 0 1.76417 − 3.05564i
1621.1 1.00000 0 1.00000 −2.07195 − 3.58872i 0 −0.500000 − 2.59808i 1.00000 0 −2.07195 − 3.58872i
1621.2 1.00000 0 1.00000 0.307774 + 0.533081i 0 −0.500000 − 2.59808i 1.00000 0 0.307774 + 0.533081i
1621.3 1.00000 0 1.00000 1.76417 + 3.05564i 0 −0.500000 − 2.59808i 1.00000 0 1.76417 + 3.05564i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h 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.m.f 6
3.b odd 2 1 182.2.e.c ✓ 6
7.c even 3 1 1638.2.p.f 6
13.c even 3 1 1638.2.p.f 6
21.c even 2 1 1274.2.e.q 6
21.g even 6 1 1274.2.g.m 6
21.g even 6 1 1274.2.h.q 6
21.h odd 6 1 182.2.h.c yes 6
21.h odd 6 1 1274.2.g.n 6
39.i odd 6 1 182.2.h.c yes 6
91.h even 3 1 inner 1638.2.m.f 6
273.r even 6 1 1274.2.e.q 6
273.s odd 6 1 182.2.e.c ✓ 6
273.bf even 6 1 1274.2.g.m 6
273.bm odd 6 1 1274.2.g.n 6
273.bn even 6 1 1274.2.h.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.e.c ✓ 6 3.b odd 2 1
182.2.e.c ✓ 6 273.s odd 6 1
182.2.h.c yes 6 21.h odd 6 1
182.2.h.c yes 6 39.i odd 6 1
1274.2.e.q 6 21.c even 2 1
1274.2.e.q 6 273.r even 6 1
1274.2.g.m 6 21.g even 6 1
1274.2.g.m 6 273.bf even 6 1
1274.2.g.n 6 21.h odd 6 1
1274.2.g.n 6 273.bm odd 6 1
1274.2.h.q 6 21.g even 6 1
1274.2.h.q 6 273.bn even 6 1
1638.2.m.f 6 1.a even 1 1 trivial
1638.2.m.f 6 91.h even 3 1 inner
1638.2.p.f 6 7.c even 3 1
1638.2.p.f 6 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 15T_{5}^{4} - 18T_{5}^{3} + 225T_{5}^{2} - 135T_{5} + 81 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 15 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} - 4 T^{5} + \cdots + 225 \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} - T^{2} - 30 T - 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$23$ \( (T^{3} - 4 T^{2} - 45 T - 27)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - T^{5} + \cdots + 31329 \) Copy content Toggle raw display
$31$ \( T^{6} - 13 T^{5} + \cdots + 961 \) Copy content Toggle raw display
$37$ \( (T^{3} - 2 T^{2} - 31 T + 89)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 3 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} + 7 T^{5} + \cdots + 1521 \) Copy content Toggle raw display
$47$ \( T^{6} + 13 T^{5} + \cdots + 1521 \) Copy content Toggle raw display
$53$ \( T^{6} + 13 T^{5} + \cdots + 1521 \) Copy content Toggle raw display
$59$ \( (T + 3)^{6} \) Copy content Toggle raw display
$61$ \( T^{6} + 11 T^{5} + \cdots + 51529 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$71$ \( T^{6} - 15 T^{5} + \cdots + 342225 \) Copy content Toggle raw display
$73$ \( T^{6} - 22 T^{5} + \cdots + 36481 \) Copy content Toggle raw display
$79$ \( T^{6} - T^{5} + \cdots + 169 \) Copy content Toggle raw display
$83$ \( (T^{3} - 14 T^{2} + \cdots + 225)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 17 T^{2} + 66 T - 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 27 T^{5} + \cdots + 49729 \) Copy content Toggle raw display
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