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Properties

Label 1323.2.i.c
Level $1323$
Weight $2$
Character orbit 1323.i
Analytic conductor $10.564$
Analytic rank $0$
Dimension $12$
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] = [1323,2,Mod(521,1323)]
 
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([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.521");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.i (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: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 63)
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 basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{3}) q^{2} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_1) q^{4} - \beta_{10} q^{5} + (2 \beta_{8} + \beta_{5} + \beta_{4} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{3}) q^{2} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_1) q^{4} - \beta_{10} q^{5} + (2 \beta_{8} + \beta_{5} + \beta_{4} + \cdots - 1) q^{8}+ \cdots + (\beta_{10} + 2 \beta_{7} + \beta_{6}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{4} - 4 q^{16} - 10 q^{22} + 24 q^{23} - 30 q^{29} + 2 q^{37} - 10 q^{43} - 54 q^{44} + 20 q^{46} + 36 q^{50} + 12 q^{53} + 2 q^{58} + 16 q^{64} - 24 q^{67} - 12 q^{74} + 12 q^{79} - 6 q^{85} + 96 q^{86} + 34 q^{88} - 30 q^{92}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( -28\nu^{10} + 148\nu^{8} + 446\nu^{6} - 3807\nu^{4} + 17052\nu^{2} + 4446 ) / 12897 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -49\nu^{11} + 259\nu^{9} - 1369\nu^{7} + 861\nu^{5} - 252\nu^{3} - 15864\nu ) / 4299 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -164\nu^{10} + 1481\nu^{8} - 7214\nu^{6} + 17007\nu^{4} - 16197\nu^{2} - 3438 ) / 12897 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -175\nu^{10} + 925\nu^{8} - 3661\nu^{6} - 1224\nu^{4} + 16296\nu^{2} - 30249 ) / 12897 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 148\nu^{10} - 987\nu^{8} + 5217\nu^{6} - 10175\nu^{4} + 17343\nu^{2} - 777 ) / 4299 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\nu^{11} - 370\nu^{9} + 1751\nu^{7} - 1230\nu^{5} + 360\nu^{3} + 8947\nu ) / 1433 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -730\nu^{11} + 5701\nu^{9} - 30748\nu^{7} + 76698\nu^{5} - 122898\nu^{3} + 66168\nu ) / 12897 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -877\nu^{10} + 6478\nu^{8} - 34855\nu^{6} + 79281\nu^{4} - 123654\nu^{2} + 31473 ) / 12897 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 543\nu^{11} - 3689\nu^{9} + 19499\nu^{7} - 39839\nu^{5} + 64821\nu^{3} - 18972\nu ) / 4299 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2237\nu^{11} - 15509\nu^{9} + 81362\nu^{7} - 167049\nu^{5} + 249792\nu^{3} - 42912\nu ) / 12897 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -890\nu^{11} + 6342\nu^{9} - 33522\nu^{7} + 71935\nu^{5} - 111438\nu^{3} + 32616\nu ) / 4299 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} - 2\beta_{7} + \beta_{6} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 2\beta_{5} + \beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{11} - 8\beta_{10} + \beta_{9} - 4\beta_{7} + 4\beta_{6} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{8} + 6\beta_{5} - 6\beta_{3} + 5\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -19\beta_{11} - 16\beta_{10} - 2\beta_{9} + 16\beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{5} - 16\beta_{4} - 7\beta_{3} + 30\beta _1 - 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 67\beta_{10} + 134\beta_{7} - 88\beta_{6} - 23\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -67\beta_{8} - 118\beta_{5} - 67\beta_{4} + 104\beta_{3} + 37\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 400\beta_{11} + 578\beta_{10} + 134\beta_{9} + 289\beta_{7} - 400\beta_{6} - 134\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -289\beta_{8} - 333\beta_{5} + 645\beta_{3} - 467\beta _1 + 978 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1801\beta_{11} + 1267\beta_{10} + 668\beta_{9} - 1267\beta_{7} ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(\beta_{5}\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1
1.29589 + 0.748185i
−1.29589 − 0.748185i
−0.474636 − 0.274031i
0.474636 + 0.274031i
1.82904 + 1.05600i
−1.82904 − 1.05600i
1.82904 − 1.05600i
−1.82904 + 1.05600i
−0.474636 + 0.274031i
0.474636 − 0.274031i
1.29589 − 0.748185i
−1.29589 + 0.748185i
− 2.27639i 0 −3.18194 −0.717144 + 1.24213i 0 0 2.69056i 0 2.82757 + 1.63250i
521.2 − 2.27639i 0 −3.18194 0.717144 − 1.24213i 0 0 2.69056i 0 −2.82757 − 1.63250i
521.3 − 0.641589i 0 1.58836 −1.10552 + 1.91482i 0 0 − 2.30225i 0 1.22853 + 0.709292i
521.4 − 0.641589i 0 1.58836 1.10552 − 1.91482i 0 0 − 2.30225i 0 −1.22853 − 0.709292i
521.5 1.18593i 0 0.593579 −1.41899 + 2.45776i 0 0 3.07579i 0 −2.91472 − 1.68281i
521.6 1.18593i 0 0.593579 1.41899 − 2.45776i 0 0 3.07579i 0 2.91472 + 1.68281i
1097.1 − 1.18593i 0 0.593579 −1.41899 − 2.45776i 0 0 − 3.07579i 0 −2.91472 + 1.68281i
1097.2 − 1.18593i 0 0.593579 1.41899 + 2.45776i 0 0 − 3.07579i 0 2.91472 − 1.68281i
1097.3 0.641589i 0 1.58836 −1.10552 − 1.91482i 0 0 2.30225i 0 1.22853 − 0.709292i
1097.4 0.641589i 0 1.58836 1.10552 + 1.91482i 0 0 2.30225i 0 −1.22853 + 0.709292i
1097.5 2.27639i 0 −3.18194 −0.717144 − 1.24213i 0 0 − 2.69056i 0 2.82757 − 1.63250i
1097.6 2.27639i 0 −3.18194 0.717144 + 1.24213i 0 0 − 2.69056i 0 −2.82757 + 1.63250i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.i even 6 1 inner
63.j 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.i.c 12
3.b odd 2 1 441.2.i.c 12
7.b odd 2 1 inner 1323.2.i.c 12
7.c even 3 1 189.2.o.a 12
7.c even 3 1 1323.2.s.c 12
7.d odd 6 1 189.2.o.a 12
7.d odd 6 1 1323.2.s.c 12
9.c even 3 1 441.2.s.c 12
9.d odd 6 1 1323.2.s.c 12
21.c even 2 1 441.2.i.c 12
21.g even 6 1 63.2.o.a ✓ 12
21.g even 6 1 441.2.s.c 12
21.h odd 6 1 63.2.o.a ✓ 12
21.h odd 6 1 441.2.s.c 12
28.f even 6 1 3024.2.cc.a 12
28.g odd 6 1 3024.2.cc.a 12
63.g even 3 1 63.2.o.a ✓ 12
63.h even 3 1 441.2.i.c 12
63.h even 3 1 567.2.c.c 12
63.i even 6 1 567.2.c.c 12
63.i even 6 1 inner 1323.2.i.c 12
63.j odd 6 1 567.2.c.c 12
63.j odd 6 1 inner 1323.2.i.c 12
63.k odd 6 1 63.2.o.a ✓ 12
63.l odd 6 1 441.2.s.c 12
63.n odd 6 1 189.2.o.a 12
63.o even 6 1 1323.2.s.c 12
63.s even 6 1 189.2.o.a 12
63.t odd 6 1 441.2.i.c 12
63.t odd 6 1 567.2.c.c 12
84.j odd 6 1 1008.2.cc.a 12
84.n even 6 1 1008.2.cc.a 12
252.n even 6 1 1008.2.cc.a 12
252.o even 6 1 3024.2.cc.a 12
252.bl odd 6 1 1008.2.cc.a 12
252.bn odd 6 1 3024.2.cc.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.o.a ✓ 12 21.g even 6 1
63.2.o.a ✓ 12 21.h odd 6 1
63.2.o.a ✓ 12 63.g even 3 1
63.2.o.a ✓ 12 63.k odd 6 1
189.2.o.a 12 7.c even 3 1
189.2.o.a 12 7.d odd 6 1
189.2.o.a 12 63.n odd 6 1
189.2.o.a 12 63.s even 6 1
441.2.i.c 12 3.b odd 2 1
441.2.i.c 12 21.c even 2 1
441.2.i.c 12 63.h even 3 1
441.2.i.c 12 63.t odd 6 1
441.2.s.c 12 9.c even 3 1
441.2.s.c 12 21.g even 6 1
441.2.s.c 12 21.h odd 6 1
441.2.s.c 12 63.l odd 6 1
567.2.c.c 12 63.h even 3 1
567.2.c.c 12 63.i even 6 1
567.2.c.c 12 63.j odd 6 1
567.2.c.c 12 63.t odd 6 1
1008.2.cc.a 12 84.j odd 6 1
1008.2.cc.a 12 84.n even 6 1
1008.2.cc.a 12 252.n even 6 1
1008.2.cc.a 12 252.bl odd 6 1
1323.2.i.c 12 1.a even 1 1 trivial
1323.2.i.c 12 7.b odd 2 1 inner
1323.2.i.c 12 63.i even 6 1 inner
1323.2.i.c 12 63.j odd 6 1 inner
1323.2.s.c 12 7.c even 3 1
1323.2.s.c 12 7.d odd 6 1
1323.2.s.c 12 9.d odd 6 1
1323.2.s.c 12 63.o even 6 1
3024.2.cc.a 12 28.f even 6 1
3024.2.cc.a 12 28.g odd 6 1
3024.2.cc.a 12 252.o even 6 1
3024.2.cc.a 12 252.bn odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 7T_{2}^{4} + 10T_{2}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 7 T^{4} + 10 T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 15 T^{10} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 11 T^{4} + 121 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 45 T^{10} + \cdots + 1750329 \) Copy content Toggle raw display
$17$ \( T^{12} + 54 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$19$ \( T^{12} - 63 T^{10} + \cdots + 4782969 \) Copy content Toggle raw display
$23$ \( (T^{6} - 12 T^{5} + \cdots + 27)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 15 T^{5} + \cdots + 243)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 129 T^{4} + \cdots + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - T^{5} + 5 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 72 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$43$ \( (T^{6} + 5 T^{5} + \cdots + 38809)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 93 T^{4} + \cdots - 81)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 6 T^{5} + \cdots + 20667)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 219 T^{4} + \cdots - 136161)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 24 T^{4} + \cdots + 243)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 6 T^{2} - 15 T + 7)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + 163 T^{4} + \cdots + 363)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 279 T^{10} + \cdots + 4782969 \) Copy content Toggle raw display
$79$ \( (T^{3} - 3 T^{2} - 24 T + 79)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 132211504881 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 282429536481 \) Copy content Toggle raw display
$97$ \( T^{12} - 69 T^{10} + \cdots + 10673289 \) Copy content Toggle raw display
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