Properties

Label 832.2.w.a
Level $832$
Weight $2$
Character orbit 832.w
Analytic conductor $6.644$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [832,2,Mod(257,832)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(832, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("832.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 832 = 2^{6} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 832.w (of order \(6\), degree \(2\), not 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: \(6.64355344817\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{3} + (2 \zeta_{6} - 1) q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{3} + (2 \zeta_{6} - 1) q^{5} - \zeta_{6} q^{9} + (3 \zeta_{6} + 1) q^{13} + ( - 2 \zeta_{6} - 2) q^{15} + 3 \zeta_{6} q^{17} + (2 \zeta_{6} - 4) q^{19} + (6 \zeta_{6} - 6) q^{23} + 2 q^{25} - 4 q^{27} + ( - 3 \zeta_{6} + 3) q^{29} + ( - 4 \zeta_{6} + 2) q^{31} + ( - 5 \zeta_{6} - 5) q^{37} + (2 \zeta_{6} - 8) q^{39} + ( - 3 \zeta_{6} - 3) q^{41} - 8 \zeta_{6} q^{43} + ( - \zeta_{6} + 2) q^{45} + (4 \zeta_{6} - 2) q^{47} + (7 \zeta_{6} - 7) q^{49} - 6 q^{51} + 3 q^{53} + ( - 8 \zeta_{6} + 4) q^{57} + ( - 4 \zeta_{6} + 8) q^{59} + \zeta_{6} q^{61} + (5 \zeta_{6} - 7) q^{65} + (2 \zeta_{6} + 2) q^{67} - 12 \zeta_{6} q^{69} + (2 \zeta_{6} - 4) q^{71} + (2 \zeta_{6} - 1) q^{73} + (4 \zeta_{6} - 4) q^{75} - 4 q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + (16 \zeta_{6} - 8) q^{83} + (3 \zeta_{6} - 6) q^{85} + 6 \zeta_{6} q^{87} + ( - 4 \zeta_{6} - 4) q^{89} + (4 \zeta_{6} + 4) q^{93} - 6 \zeta_{6} q^{95} + ( - 4 \zeta_{6} + 8) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - q^{9} + 5 q^{13} - 6 q^{15} + 3 q^{17} - 6 q^{19} - 6 q^{23} + 4 q^{25} - 8 q^{27} + 3 q^{29} - 15 q^{37} - 14 q^{39} - 9 q^{41} - 8 q^{43} + 3 q^{45} - 7 q^{49} - 12 q^{51} + 6 q^{53} + 12 q^{59} + q^{61} - 9 q^{65} + 6 q^{67} - 12 q^{69} - 6 q^{71} - 4 q^{75} - 8 q^{79} + 11 q^{81} - 9 q^{85} + 6 q^{87} - 12 q^{89} + 12 q^{93} - 6 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/832\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(703\) \(769\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 1.73205i 0 1.73205i 0 0 0 −0.500000 + 0.866025i 0
641.1 0 −1.00000 + 1.73205i 0 1.73205i 0 0 0 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 832.2.w.a 2
4.b odd 2 1 832.2.w.d 2
8.b even 2 1 208.2.w.b 2
8.d odd 2 1 13.2.e.a 2
13.e even 6 1 inner 832.2.w.a 2
24.f even 2 1 117.2.q.c 2
24.h odd 2 1 1872.2.by.d 2
40.e odd 2 1 325.2.n.a 2
40.k even 4 2 325.2.m.a 4
52.i odd 6 1 832.2.w.d 2
56.e even 2 1 637.2.q.a 2
56.k odd 6 1 637.2.k.a 2
56.k odd 6 1 637.2.u.c 2
56.m even 6 1 637.2.k.c 2
56.m even 6 1 637.2.u.b 2
104.h odd 2 1 169.2.e.a 2
104.m even 4 2 169.2.c.a 4
104.n odd 6 1 169.2.b.a 2
104.n odd 6 1 169.2.e.a 2
104.p odd 6 1 13.2.e.a 2
104.p odd 6 1 169.2.b.a 2
104.r even 6 1 2704.2.f.b 2
104.s even 6 1 208.2.w.b 2
104.s even 6 1 2704.2.f.b 2
104.u even 12 2 169.2.a.a 2
104.u even 12 2 169.2.c.a 4
104.x odd 12 2 2704.2.a.o 2
312.ba even 6 1 117.2.q.c 2
312.ba even 6 1 1521.2.b.a 2
312.bg odd 6 1 1872.2.by.d 2
312.bn even 6 1 1521.2.b.a 2
312.bq odd 12 2 1521.2.a.k 2
520.cd odd 6 1 325.2.n.a 2
520.cs even 12 2 325.2.m.a 4
520.cz even 12 2 4225.2.a.v 2
728.bh even 6 1 637.2.k.c 2
728.br odd 6 1 637.2.u.c 2
728.ci even 6 1 637.2.q.a 2
728.cv even 6 1 637.2.u.b 2
728.dd odd 6 1 637.2.k.a 2
728.ec odd 12 2 8281.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 8.d odd 2 1
13.2.e.a 2 104.p odd 6 1
117.2.q.c 2 24.f even 2 1
117.2.q.c 2 312.ba even 6 1
169.2.a.a 2 104.u even 12 2
169.2.b.a 2 104.n odd 6 1
169.2.b.a 2 104.p odd 6 1
169.2.c.a 4 104.m even 4 2
169.2.c.a 4 104.u even 12 2
169.2.e.a 2 104.h odd 2 1
169.2.e.a 2 104.n odd 6 1
208.2.w.b 2 8.b even 2 1
208.2.w.b 2 104.s even 6 1
325.2.m.a 4 40.k even 4 2
325.2.m.a 4 520.cs even 12 2
325.2.n.a 2 40.e odd 2 1
325.2.n.a 2 520.cd odd 6 1
637.2.k.a 2 56.k odd 6 1
637.2.k.a 2 728.dd odd 6 1
637.2.k.c 2 56.m even 6 1
637.2.k.c 2 728.bh even 6 1
637.2.q.a 2 56.e even 2 1
637.2.q.a 2 728.ci even 6 1
637.2.u.b 2 56.m even 6 1
637.2.u.b 2 728.cv even 6 1
637.2.u.c 2 56.k odd 6 1
637.2.u.c 2 728.br odd 6 1
832.2.w.a 2 1.a even 1 1 trivial
832.2.w.a 2 13.e even 6 1 inner
832.2.w.d 2 4.b odd 2 1
832.2.w.d 2 52.i odd 6 1
1521.2.a.k 2 312.bq odd 12 2
1521.2.b.a 2 312.ba even 6 1
1521.2.b.a 2 312.bn even 6 1
1872.2.by.d 2 24.h odd 2 1
1872.2.by.d 2 312.bg odd 6 1
2704.2.a.o 2 104.x odd 12 2
2704.2.f.b 2 104.r even 6 1
2704.2.f.b 2 104.s even 6 1
4225.2.a.v 2 520.cz even 12 2
8281.2.a.q 2 728.ec odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(832, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 3 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 12 \) Copy content Toggle raw display
$37$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 3 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 192 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
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