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Properties

Label 1323.2.h.b
Level $1323$
Weight $2$
Character orbit 1323.h
Analytic conductor $10.564$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,2,Mod(226,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1323.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.h (of order \(3\), 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: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{3} - 1) q^{2} + ( - 2 \beta_{4} - \beta_{3} + 1) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - \beta_1) q^{5} + (2 \beta_{4} - \beta_{3} - 2) q^{8} + ( - \beta_{5} + \beta_{4} + \cdots - 2 \beta_{2}) q^{10}+ \cdots + (7 \beta_{5} - 7 \beta_{4} + \cdots - \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} - 3 q^{5} - 12 q^{8} + 6 q^{11} - 3 q^{13} + 6 q^{16} - 6 q^{17} - 3 q^{19} + 6 q^{20} - 9 q^{22} + 12 q^{23} + 6 q^{25} + 3 q^{26} + 9 q^{29} + 6 q^{31} + 9 q^{34} + 3 q^{37} - 6 q^{38}+ \cdots - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{18}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{18}^{5} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{18}^{5} + \zeta_{18}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \) Copy content Toggle raw display

Display \(\zeta_{18}^j\) in terms of \(\beta_i\)

\(\zeta_{18}\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{2}\)\(=\) \( ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{18}^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{5}\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{18}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1 + \beta_{1}\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
−0.766044 + 0.642788i
−0.173648 − 0.984808i
0.939693 + 0.342020i
−0.766044 − 0.642788i
−0.173648 + 0.984808i
0.939693 − 0.342020i
−2.53209 0 4.41147 0.439693 + 0.761570i 0 0 −6.10607 0 −1.11334 − 1.92836i
226.2 −1.34730 0 −0.184793 −1.26604 − 2.19285i 0 0 2.94356 0 1.70574 + 2.95442i
226.3 0.879385 0 −1.22668 −0.673648 − 1.16679i 0 0 −2.83750 0 −0.592396 − 1.02606i
802.1 −2.53209 0 4.41147 0.439693 − 0.761570i 0 0 −6.10607 0 −1.11334 + 1.92836i
802.2 −1.34730 0 −0.184793 −1.26604 + 2.19285i 0 0 2.94356 0 1.70574 − 2.95442i
802.3 0.879385 0 −1.22668 −0.673648 + 1.16679i 0 0 −2.83750 0 −0.592396 + 1.02606i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.h.b 6
3.b odd 2 1 441.2.h.e 6
7.b odd 2 1 1323.2.h.c 6
7.c even 3 1 1323.2.f.d 6
7.c even 3 1 1323.2.g.e 6
7.d odd 6 1 189.2.f.b 6
7.d odd 6 1 1323.2.g.d 6
9.c even 3 1 1323.2.g.e 6
9.d odd 6 1 441.2.g.b 6
21.c even 2 1 441.2.h.d 6
21.g even 6 1 63.2.f.a ✓ 6
21.g even 6 1 441.2.g.c 6
21.h odd 6 1 441.2.f.c 6
21.h odd 6 1 441.2.g.b 6
28.f even 6 1 3024.2.r.k 6
63.g even 3 1 1323.2.f.d 6
63.h even 3 1 inner 1323.2.h.b 6
63.h even 3 1 3969.2.a.l 3
63.i even 6 1 441.2.h.d 6
63.i even 6 1 567.2.a.h 3
63.j odd 6 1 441.2.h.e 6
63.j odd 6 1 3969.2.a.q 3
63.k odd 6 1 189.2.f.b 6
63.l odd 6 1 1323.2.g.d 6
63.n odd 6 1 441.2.f.c 6
63.o even 6 1 441.2.g.c 6
63.s even 6 1 63.2.f.a ✓ 6
63.t odd 6 1 567.2.a.c 3
63.t odd 6 1 1323.2.h.c 6
84.j odd 6 1 1008.2.r.h 6
252.n even 6 1 3024.2.r.k 6
252.r odd 6 1 9072.2.a.ca 3
252.bj even 6 1 9072.2.a.bs 3
252.bn odd 6 1 1008.2.r.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a ✓ 6 21.g even 6 1
63.2.f.a ✓ 6 63.s even 6 1
189.2.f.b 6 7.d odd 6 1
189.2.f.b 6 63.k odd 6 1
441.2.f.c 6 21.h odd 6 1
441.2.f.c 6 63.n odd 6 1
441.2.g.b 6 9.d odd 6 1
441.2.g.b 6 21.h odd 6 1
441.2.g.c 6 21.g even 6 1
441.2.g.c 6 63.o even 6 1
441.2.h.d 6 21.c even 2 1
441.2.h.d 6 63.i even 6 1
441.2.h.e 6 3.b odd 2 1
441.2.h.e 6 63.j odd 6 1
567.2.a.c 3 63.t odd 6 1
567.2.a.h 3 63.i even 6 1
1008.2.r.h 6 84.j odd 6 1
1008.2.r.h 6 252.bn odd 6 1
1323.2.f.d 6 7.c even 3 1
1323.2.f.d 6 63.g even 3 1
1323.2.g.d 6 7.d odd 6 1
1323.2.g.d 6 63.l odd 6 1
1323.2.g.e 6 7.c even 3 1
1323.2.g.e 6 9.c even 3 1
1323.2.h.b 6 1.a even 1 1 trivial
1323.2.h.b 6 63.h even 3 1 inner
1323.2.h.c 6 7.b odd 2 1
1323.2.h.c 6 63.t odd 6 1
3024.2.r.k 6 28.f even 6 1
3024.2.r.k 6 252.n even 6 1
3969.2.a.l 3 63.h even 3 1
3969.2.a.q 3 63.j odd 6 1
9072.2.a.bs 3 252.bj even 6 1
9072.2.a.ca 3 252.r odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\):

\( T_{2}^{3} + 3T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{5}^{6} + 3T_{5}^{5} + 9T_{5}^{4} + 6T_{5}^{3} + 9T_{5}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} + 3 T^{2} - 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + \cdots + 11449 \) Copy content Toggle raw display
$17$ \( T^{6} + 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$23$ \( T^{6} - 12 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{6} - 9 T^{5} + \cdots + 110889 \) Copy content Toggle raw display
$31$ \( (T^{3} - 3 T^{2} + \cdots + 323)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} + \cdots + 104329 \) Copy content Toggle raw display
$41$ \( T^{6} + 9 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( (T^{3} - 3 T^{2} - 54 T - 51)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} - 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$59$ \( (T^{3} + 3 T^{2} - 72 T + 51)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 6 T^{2} - 15 T - 19)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 12 T^{2} + \cdots - 17)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 9 T^{2} - 54 T + 27)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 21 T^{5} + \cdots + 72361 \) Copy content Toggle raw display
$79$ \( (T^{3} + 21 T^{2} + \cdots + 181)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 18 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$89$ \( T^{6} + 12 T^{5} + \cdots + 660969 \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{5} + \cdots + 104329 \) Copy content Toggle raw display
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