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Properties

Label 1323.2.f.g
Level $1323$
Weight $2$
Character orbit 1323.f
Analytic conductor $10.564$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,2,Mod(442,1323)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.442");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.f (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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{5} \)
Twist minimal: no (minimal twist has level 441)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{4}+ \cdots + ( - \beta_{5} - \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{4}+ \cdots + (\beta_{11} + \beta_{10} + 2 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 6 q^{4} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} - 6 q^{4} - 24 q^{8} + 8 q^{11} - 6 q^{16} - 6 q^{22} + 4 q^{23} - 12 q^{25} + 22 q^{29} + 16 q^{32} - 12 q^{37} - 6 q^{43} + 28 q^{44} + 24 q^{46} + 56 q^{50} - 56 q^{53} - 18 q^{58} - 48 q^{64} - 6 q^{65} - 76 q^{71} + 36 q^{74} + 6 q^{79} + 30 q^{85} - 36 q^{86} + 6 q^{88} + 62 q^{92} + 60 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -49\nu^{10} + 259\nu^{8} - 1369\nu^{6} + 861\nu^{4} - 252\nu^{2} - 7266 ) / 4299 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -148\nu^{11} + 987\nu^{9} - 5217\nu^{7} + 10175\nu^{5} - 17343\nu^{3} - 3522\nu ) / 4299 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -148\nu^{11} + 987\nu^{9} - 5217\nu^{7} + 10175\nu^{5} - 17343\nu^{3} + 9375\nu ) / 4299 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 148\nu^{10} - 987\nu^{8} + 5217\nu^{6} - 10175\nu^{4} + 17343\nu^{2} - 5076 ) / 4299 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 161\nu^{10} - 851\nu^{8} + 3884\nu^{6} - 2829\nu^{4} + 828\nu^{2} + 6678 ) / 4299 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -296\nu^{10} + 1974\nu^{8} - 10434\nu^{6} + 20350\nu^{4} - 30387\nu^{2} + 1554 ) / 4299 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -120\nu^{10} + 839\nu^{8} - 4230\nu^{6} + 8250\nu^{4} - 10034\nu^{2} + 630 ) / 1433 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -494\nu^{11} + 3430\nu^{9} - 18130\nu^{7} + 38978\nu^{5} - 64569\nu^{3} + 34836\nu ) / 4299 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 532\nu^{11} - 4245\nu^{9} + 23052\nu^{7} - 58070\nu^{5} + 93015\nu^{3} - 50082\nu ) / 4299 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 641\nu^{11} - 4207\nu^{9} + 22237\nu^{7} - 41561\nu^{5} + 65325\nu^{3} + 12756\nu ) / 4299 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -1162\nu^{11} + 7575\nu^{9} - 38811\nu^{7} + 69140\nu^{5} - 96255\nu^{3} - 17544\nu ) / 4299 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{10} + \beta_{8} - 4\beta_{3} - 8\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + 5\beta_{6} - \beta_{5} + 7\beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{11} - 6\beta_{10} + 2\beta_{9} + 12\beta_{8} - 34\beta_{3} - 17\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{5} - 23\beta _1 - 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{11} + 30\beta_{10} + 7\beta_{9} + 30\beta_{8} - 74\beta_{3} + 74\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 37\beta_{7} - 104\beta_{6} - 118\beta_{4} - 118 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 74\beta_{11} + 282\beta_{10} - 37\beta_{9} - 141\beta_{8} + 326\beta_{3} + 652\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 178\beta_{7} - 467\beta_{6} + 178\beta_{5} - 511\beta_{4} + 467\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 178\beta_{11} + 645\beta_{10} - 356\beta_{9} - 1290\beta_{8} + 2890\beta_{3} + 1445\beta_{2} ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
442.1
−0.474636 − 0.274031i
0.474636 + 0.274031i
−1.29589 − 0.748185i
1.29589 + 0.748185i
−1.82904 − 1.05600i
1.82904 + 1.05600i
−0.474636 + 0.274031i
0.474636 − 0.274031i
−1.29589 + 0.748185i
1.29589 − 0.748185i
−1.82904 + 1.05600i
1.82904 − 1.05600i
−0.849814 − 1.47192i 0 −0.444368 + 0.769668i −0.474636 + 0.822093i 0 0 −1.88874 0 1.61341
442.2 −0.849814 − 1.47192i 0 −0.444368 + 0.769668i 0.474636 − 0.822093i 0 0 −1.88874 0 −1.61341
442.3 0.119562 + 0.207087i 0 0.971410 − 1.68253i −1.29589 + 2.24456i 0 0 0.942820 0 −0.619757
442.4 0.119562 + 0.207087i 0 0.971410 − 1.68253i 1.29589 − 2.24456i 0 0 0.942820 0 0.619757
442.5 1.23025 + 2.13086i 0 −2.02704 + 3.51094i −1.82904 + 3.16799i 0 0 −5.05408 0 −9.00071
442.6 1.23025 + 2.13086i 0 −2.02704 + 3.51094i 1.82904 − 3.16799i 0 0 −5.05408 0 9.00071
883.1 −0.849814 + 1.47192i 0 −0.444368 − 0.769668i −0.474636 − 0.822093i 0 0 −1.88874 0 1.61341
883.2 −0.849814 + 1.47192i 0 −0.444368 − 0.769668i 0.474636 + 0.822093i 0 0 −1.88874 0 −1.61341
883.3 0.119562 − 0.207087i 0 0.971410 + 1.68253i −1.29589 − 2.24456i 0 0 0.942820 0 −0.619757
883.4 0.119562 − 0.207087i 0 0.971410 + 1.68253i 1.29589 + 2.24456i 0 0 0.942820 0 0.619757
883.5 1.23025 − 2.13086i 0 −2.02704 − 3.51094i −1.82904 − 3.16799i 0 0 −5.05408 0 −9.00071
883.6 1.23025 − 2.13086i 0 −2.02704 − 3.51094i 1.82904 + 3.16799i 0 0 −5.05408 0 9.00071
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 442.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.c even 3 1 inner
63.l odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.f.g 12
3.b odd 2 1 441.2.f.g ✓ 12
7.b odd 2 1 inner 1323.2.f.g 12
7.c even 3 1 1323.2.g.g 12
7.c even 3 1 1323.2.h.g 12
7.d odd 6 1 1323.2.g.g 12
7.d odd 6 1 1323.2.h.g 12
9.c even 3 1 inner 1323.2.f.g 12
9.c even 3 1 3969.2.a.bd 6
9.d odd 6 1 441.2.f.g ✓ 12
9.d odd 6 1 3969.2.a.be 6
21.c even 2 1 441.2.f.g ✓ 12
21.g even 6 1 441.2.g.g 12
21.g even 6 1 441.2.h.g 12
21.h odd 6 1 441.2.g.g 12
21.h odd 6 1 441.2.h.g 12
63.g even 3 1 1323.2.h.g 12
63.h even 3 1 1323.2.g.g 12
63.i even 6 1 441.2.g.g 12
63.j odd 6 1 441.2.g.g 12
63.k odd 6 1 1323.2.h.g 12
63.l odd 6 1 inner 1323.2.f.g 12
63.l odd 6 1 3969.2.a.bd 6
63.n odd 6 1 441.2.h.g 12
63.o even 6 1 441.2.f.g ✓ 12
63.o even 6 1 3969.2.a.be 6
63.s even 6 1 441.2.h.g 12
63.t odd 6 1 1323.2.g.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.g ✓ 12 3.b odd 2 1
441.2.f.g ✓ 12 9.d odd 6 1
441.2.f.g ✓ 12 21.c even 2 1
441.2.f.g ✓ 12 63.o even 6 1
441.2.g.g 12 21.g even 6 1
441.2.g.g 12 21.h odd 6 1
441.2.g.g 12 63.i even 6 1
441.2.g.g 12 63.j odd 6 1
441.2.h.g 12 21.g even 6 1
441.2.h.g 12 21.h odd 6 1
441.2.h.g 12 63.n odd 6 1
441.2.h.g 12 63.s even 6 1
1323.2.f.g 12 1.a even 1 1 trivial
1323.2.f.g 12 7.b odd 2 1 inner
1323.2.f.g 12 9.c even 3 1 inner
1323.2.f.g 12 63.l odd 6 1 inner
1323.2.g.g 12 7.c even 3 1
1323.2.g.g 12 7.d odd 6 1
1323.2.g.g 12 63.h even 3 1
1323.2.g.g 12 63.t odd 6 1
1323.2.h.g 12 7.c even 3 1
1323.2.h.g 12 7.d odd 6 1
1323.2.h.g 12 63.g even 3 1
1323.2.h.g 12 63.k odd 6 1
3969.2.a.bd 6 9.c even 3 1
3969.2.a.bd 6 63.l odd 6 1
3969.2.a.be 6 9.d odd 6 1
3969.2.a.be 6 63.o even 6 1

Hecke kernels

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

\( T_{2}^{6} - T_{2}^{5} + 5T_{2}^{4} + 2T_{2}^{3} + 17T_{2}^{2} - 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{12} + 21T_{5}^{10} + 333T_{5}^{8} + 2106T_{5}^{6} + 9963T_{5}^{4} + 8748T_{5}^{2} + 6561 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{5} + 5 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 21 T^{10} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 4 T^{5} + 17 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 39 T^{10} + \cdots + 6561 \) Copy content Toggle raw display
$17$ \( (T^{6} - 84 T^{4} + \cdots - 3969)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 75 T^{4} + \cdots - 3969)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 2 T^{5} + \cdots + 3481)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 11 T^{5} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 6059221281 \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} - 24 T + 27)^{4} \) Copy content Toggle raw display
$41$ \( T^{12} + 162 T^{10} + \cdots + 43046721 \) Copy content Toggle raw display
$43$ \( (T^{6} + 3 T^{5} + \cdots + 729)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 37822859361 \) Copy content Toggle raw display
$53$ \( (T^{3} + 14 T^{2} + \cdots - 263)^{4} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 22430753361 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 311374044081 \) Copy content Toggle raw display
$67$ \( (T^{6} + 111 T^{4} + \cdots + 124609)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 19 T^{2} + \cdots + 227)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} - 75 T^{4} + \cdots - 3969)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 3 T^{5} + \cdots + 11449)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 51769445841 \) Copy content Toggle raw display
$89$ \( (T^{6} - 246 T^{4} + \cdots - 3969)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 111 T^{10} + \cdots + 96059601 \) Copy content Toggle raw display
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