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Properties

Label 1638.2.dm.a
Level $1638$
Weight $2$
Character orbit 1638.dm
Analytic conductor $13.079$
Analytic rank $0$
Dimension $4$
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] = [1638,2,Mod(415,1638)]
 
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, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.415");
 
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.dm (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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + 4 \zeta_{12} q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{7} - \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + 4 \zeta_{12} q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{7} - \zeta_{12}^{3} q^{8} - 4 \zeta_{12}^{2} q^{10} + (\zeta_{12}^{3} - \zeta_{12}) q^{11} + (2 \zeta_{12}^{3} + 3) q^{13} + (3 \zeta_{12}^{2} - 2) q^{14} + (\zeta_{12}^{2} - 1) q^{16} + \zeta_{12}^{2} q^{17} + \zeta_{12} q^{19} + 4 \zeta_{12}^{3} q^{20} + q^{22} + (6 \zeta_{12}^{2} - 6) q^{23} + 11 \zeta_{12}^{2} q^{25} + ( - 2 \zeta_{12}^{2} - 3 \zeta_{12} + 2) q^{26} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}) q^{28} + 3 q^{29} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{31} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{32} - \zeta_{12}^{3} q^{34} + ( - 12 \zeta_{12}^{2} + 8) q^{35} - 6 \zeta_{12} q^{37} - \zeta_{12}^{2} q^{38} + ( - 4 \zeta_{12}^{2} + 4) q^{40} - 2 \zeta_{12}^{3} q^{41} + 10 q^{43} - \zeta_{12} q^{44} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{46} - 7 \zeta_{12} q^{47} + (5 \zeta_{12}^{2} - 8) q^{49} - 11 \zeta_{12}^{3} q^{50} + (2 \zeta_{12}^{3} + \cdots - 2 \zeta_{12}) q^{52} + \cdots + ( - 5 \zeta_{12}^{3} + 8 \zeta_{12}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 8 q^{10} + 12 q^{13} - 2 q^{14} - 2 q^{16} + 2 q^{17} + 4 q^{22} - 12 q^{23} + 22 q^{25} + 4 q^{26} + 12 q^{29} + 8 q^{35} - 2 q^{38} + 8 q^{40} + 40 q^{43} - 22 q^{49} + 6 q^{52} + 22 q^{53} - 16 q^{55} - 10 q^{56} - 22 q^{61} - 32 q^{62} - 4 q^{64} - 16 q^{65} - 2 q^{68} + 12 q^{74} + 8 q^{77} - 4 q^{82} + 2 q^{88} + 20 q^{91} - 24 q^{92} + 14 q^{94} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-1\) \(-1 + \zeta_{12}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i 3.46410 + 2.00000i 0 −0.866025 − 2.50000i − 1.00000i 0 −2.00000 − 3.46410i
415.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −3.46410 − 2.00000i 0 0.866025 + 2.50000i 1.00000i 0 −2.00000 − 3.46410i
1117.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 3.46410 − 2.00000i 0 −0.866025 + 2.50000i 1.00000i 0 −2.00000 + 3.46410i
1117.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i −3.46410 + 2.00000i 0 0.866025 − 2.50000i − 1.00000i 0 −2.00000 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r 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.dm.a 4
3.b odd 2 1 182.2.n.a ✓ 4
7.c even 3 1 inner 1638.2.dm.a 4
13.b even 2 1 inner 1638.2.dm.a 4
21.c even 2 1 1274.2.n.f 4
21.g even 6 1 1274.2.d.b 2
21.g even 6 1 1274.2.n.f 4
21.h odd 6 1 182.2.n.a ✓ 4
21.h odd 6 1 1274.2.d.e 2
39.d odd 2 1 182.2.n.a ✓ 4
91.r even 6 1 inner 1638.2.dm.a 4
273.g even 2 1 1274.2.n.f 4
273.w odd 6 1 182.2.n.a ✓ 4
273.w odd 6 1 1274.2.d.e 2
273.ba even 6 1 1274.2.d.b 2
273.ba even 6 1 1274.2.n.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.n.a ✓ 4 3.b odd 2 1
182.2.n.a ✓ 4 21.h odd 6 1
182.2.n.a ✓ 4 39.d odd 2 1
182.2.n.a ✓ 4 273.w odd 6 1
1274.2.d.b 2 21.g even 6 1
1274.2.d.b 2 273.ba even 6 1
1274.2.d.e 2 21.h odd 6 1
1274.2.d.e 2 273.w odd 6 1
1274.2.n.f 4 21.c even 2 1
1274.2.n.f 4 21.g even 6 1
1274.2.n.f 4 273.g even 2 1
1274.2.n.f 4 273.ba even 6 1
1638.2.dm.a 4 1.a even 1 1 trivial
1638.2.dm.a 4 7.c even 3 1 inner
1638.2.dm.a 4 13.b even 2 1 inner
1638.2.dm.a 4 91.r even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 16T_{5}^{2} + 256 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$37$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$43$ \( (T - 10)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$53$ \( (T^{2} - 11 T + 121)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$61$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$71$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 324 T^{2} + 104976 \) Copy content Toggle raw display
$97$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
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