Properties

Label 936.2.dg.a
Level $936$
Weight $2$
Character orbit 936.dg
Analytic conductor $7.474$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [936,2,Mod(829,936)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(936, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 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("936.829");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.dg (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: \(7.47399762919\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} - 2 \zeta_{12} - 2) q^{5} + (2 \zeta_{12}^{2} - 4) q^{7} + ( - 2 \zeta_{12}^{3} + 2) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} - 2 \zeta_{12} - 2) q^{5} + (2 \zeta_{12}^{2} - 4) q^{7} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \cdots + 2) q^{10} + \cdots + (5 \zeta_{12}^{3} + 5) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 8 q^{5} - 12 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 8 q^{5} - 12 q^{7} + 8 q^{8} + 2 q^{10} - 4 q^{11} - 12 q^{14} + 8 q^{16} + 6 q^{17} + 6 q^{19} + 12 q^{20} - 8 q^{22} + 6 q^{23} + 8 q^{25} + 4 q^{26} - 8 q^{32} + 6 q^{34} + 24 q^{35} + 12 q^{37} - 16 q^{40} + 18 q^{41} - 12 q^{43} + 12 q^{46} + 10 q^{49} - 20 q^{50} + 28 q^{52} + 8 q^{55} - 24 q^{56} + 12 q^{58} - 8 q^{59} + 12 q^{62} + 18 q^{65} + 18 q^{67} + 12 q^{70} + 6 q^{71} + 24 q^{74} - 16 q^{80} - 16 q^{82} + 8 q^{83} - 24 q^{86} - 8 q^{88} - 12 q^{89} - 36 q^{92} - 16 q^{94} - 6 q^{95} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.366025 1.36603i 0 −1.73205 + 1.00000i −3.73205 0 −3.00000 1.73205i 2.00000 + 2.00000i 0 1.36603 + 5.09808i
829.2 1.36603 0.366025i 0 1.73205 1.00000i −0.267949 0 −3.00000 1.73205i 2.00000 2.00000i 0 −0.366025 + 0.0980762i
901.1 −0.366025 + 1.36603i 0 −1.73205 1.00000i −3.73205 0 −3.00000 + 1.73205i 2.00000 2.00000i 0 1.36603 5.09808i
901.2 1.36603 + 0.366025i 0 1.73205 + 1.00000i −0.267949 0 −3.00000 + 1.73205i 2.00000 + 2.00000i 0 −0.366025 0.0980762i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
104.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.dg.a 4
3.b odd 2 1 104.2.s.b yes 4
8.b even 2 1 936.2.dg.b 4
12.b even 2 1 416.2.ba.a 4
13.e even 6 1 936.2.dg.b 4
24.f even 2 1 416.2.ba.b 4
24.h odd 2 1 104.2.s.a 4
39.h odd 6 1 104.2.s.a 4
104.s even 6 1 inner 936.2.dg.a 4
156.r even 6 1 416.2.ba.b 4
312.ba even 6 1 416.2.ba.a 4
312.bg odd 6 1 104.2.s.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.s.a 4 24.h odd 2 1
104.2.s.a 4 39.h odd 6 1
104.2.s.b yes 4 3.b odd 2 1
104.2.s.b yes 4 312.bg odd 6 1
416.2.ba.a 4 12.b even 2 1
416.2.ba.a 4 312.ba even 6 1
416.2.ba.b 4 24.f even 2 1
416.2.ba.b 4 156.r even 6 1
936.2.dg.a 4 1.a even 1 1 trivial
936.2.dg.a 4 104.s even 6 1 inner
936.2.dg.b 4 8.b even 2 1
936.2.dg.b 4 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 4T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 4 \) 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
$7$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$29$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$31$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} + \cdots + 1089 \) Copy content Toggle raw display
$41$ \( T^{4} - 18 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$43$ \( T^{4} + 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$47$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$53$ \( T^{4} + 114T^{2} + 1521 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$61$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + \cdots + 6084 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + \cdots + 13924 \) Copy content Toggle raw display
$73$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$97$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
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