Properties

Label 1323.2.h.g
Level $1323$
Weight $2$
Character orbit 1323.h
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(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: \(12\)
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: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{5} + \beta_1 + 1) q^{4} + \beta_{2} q^{5} + ( - \beta_{5} - \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{5} + \beta_1 + 1) q^{4} + \beta_{2} q^{5} + ( - \beta_{5} - \beta_1 - 2) q^{8} + ( - \beta_{10} - 2 \beta_{2}) q^{10} + ( - \beta_{7} - \beta_{5} - \beta_{4}) q^{11} + (\beta_{11} - \beta_{9}) q^{13} + ( - \beta_{5} + 2 \beta_1) q^{16} + (\beta_{11} - \beta_{10} - \beta_{2}) q^{17} + ( - \beta_{10} + \beta_{8} + \beta_{3} + \beta_{2}) q^{19} + (\beta_{11} + 2 \beta_{10} + 3 \beta_{2}) q^{20} + \beta_{4} q^{22} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} + 1) q^{23} + (3 \beta_{6} + \beta_{4} - 3 \beta_1) q^{25} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8}) q^{26} + ( - 2 \beta_{7} + \beta_{6} + 4 \beta_{4} + 4) q^{29} + ( - 2 \beta_{8} + \beta_{3}) q^{31} + (\beta_1 - 3) q^{32} + (2 \beta_{11} + 3 \beta_{2}) q^{34} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{37} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{3} - \beta_{2}) q^{38} + ( - \beta_{11} - 2 \beta_{10} - 4 \beta_{2}) q^{40} + (\beta_{11} + 2 \beta_{10} - \beta_{9} - 2 \beta_{8} + \beta_{3} + \beta_{2}) q^{41} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 2) q^{43} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_1) q^{44} + (\beta_{7} - 4 \beta_{6} - \beta_{4} - 1) q^{46} + ( - 2 \beta_{9} - \beta_{8} + \beta_{3}) q^{47} + (3 \beta_{7} - 4 \beta_{6} + 3 \beta_{5} - 9 \beta_{4} + 4 \beta_1) q^{50} + ( - \beta_{10} + \beta_{8} + \beta_{3} + \beta_{2}) q^{52} + (2 \beta_{7} - 3 \beta_{6} + 5 \beta_{4} + 5) q^{53} + ( - \beta_{9} - \beta_{8} - \beta_{3}) q^{55} + (\beta_{7} - 7 \beta_{6} - \beta_{4} - 1) q^{58} + (\beta_{9} - \beta_{8} - 2 \beta_{3}) q^{59} + ( - 2 \beta_{9} + \beta_{8} - 2 \beta_{3}) q^{61} + (2 \beta_{9} + \beta_{8} - 4 \beta_{3}) q^{62} + (\beta_{5} - 2 \beta_1 - 3) q^{64} - 3 \beta_{5} q^{65} + ( - 3 \beta_{5} + 3 \beta_1 - 2) q^{67} + ( - 3 \beta_{10} - 4 \beta_{2}) q^{68} + ( - \beta_1 - 6) q^{71} + ( - \beta_{10} + \beta_{2}) q^{73} + (2 \beta_{7} + \beta_{6} + 2 \beta_{5} - 7 \beta_{4} - \beta_1) q^{74} + (2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \beta_{3} + \beta_{2}) q^{76} + (3 \beta_{5} + 3 \beta_1 - 1) q^{79} + ( - \beta_{11} + \beta_{10} + 4 \beta_{2}) q^{80} + ( - \beta_{11} - 2 \beta_{10} + \beta_{9} + 2 \beta_{8} - 4 \beta_{3} - 4 \beta_{2}) q^{82} + (\beta_{11} - \beta_{10} - 3 \beta_{2}) q^{83} + ( - 6 \beta_{6} - 3 \beta_{4} + 6 \beta_1) q^{85} + (2 \beta_{7} + \beta_{6} - 7 \beta_{4} - 7) q^{86} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_1) q^{88} + (\beta_{11} + 2 \beta_{10} - \beta_{9} - 2 \beta_{8} + 3 \beta_{3} + 3 \beta_{2}) q^{89} + (4 \beta_{6} + 9 \beta_{4} + 9) q^{92} + ( - \beta_{9} + 3 \beta_{8} - 3 \beta_{3}) q^{94} + (3 \beta_{5} - 9) q^{95} + ( - \beta_{11} - \beta_{10} - 2 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} + 12 q^{4} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} + 12 q^{4} - 24 q^{8} + 8 q^{11} + 12 q^{16} - 6 q^{22} + 4 q^{23} - 12 q^{25} + 22 q^{29} - 32 q^{32} + 6 q^{37} - 6 q^{43} - 14 q^{44} - 12 q^{46} + 56 q^{50} + 28 q^{53} - 18 q^{58} - 48 q^{64} + 12 q^{65} - 76 q^{71} + 36 q^{74} - 12 q^{79} + 30 q^{85} - 36 q^{86} + 6 q^{88} + 62 q^{92} - 120 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} ) / 3 \) 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

Character values

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

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

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
1.82904 + 1.05600i
−1.82904 1.05600i
1.29589 + 0.748185i
−1.29589 0.748185i
0.474636 + 0.274031i
−0.474636 0.274031i
1.82904 1.05600i
−1.82904 + 1.05600i
1.29589 0.748185i
−1.29589 + 0.748185i
0.474636 0.274031i
−0.474636 + 0.274031i
−2.46050 0 4.05408 −1.82904 3.16799i 0 0 −5.05408 0 4.50036 + 7.79485i
226.2 −2.46050 0 4.05408 1.82904 + 3.16799i 0 0 −5.05408 0 −4.50036 7.79485i
226.3 −0.239123 0 −1.94282 −1.29589 2.24456i 0 0 0.942820 0 0.309879 + 0.536725i
226.4 −0.239123 0 −1.94282 1.29589 + 2.24456i 0 0 0.942820 0 −0.309879 0.536725i
226.5 1.69963 0 0.888736 −0.474636 0.822093i 0 0 −1.88874 0 −0.806704 1.39725i
226.6 1.69963 0 0.888736 0.474636 + 0.822093i 0 0 −1.88874 0 0.806704 + 1.39725i
802.1 −2.46050 0 4.05408 −1.82904 + 3.16799i 0 0 −5.05408 0 4.50036 7.79485i
802.2 −2.46050 0 4.05408 1.82904 3.16799i 0 0 −5.05408 0 −4.50036 + 7.79485i
802.3 −0.239123 0 −1.94282 −1.29589 + 2.24456i 0 0 0.942820 0 0.309879 0.536725i
802.4 −0.239123 0 −1.94282 1.29589 2.24456i 0 0 0.942820 0 −0.309879 + 0.536725i
802.5 1.69963 0 0.888736 −0.474636 + 0.822093i 0 0 −1.88874 0 −0.806704 + 1.39725i
802.6 1.69963 0 0.888736 0.474636 0.822093i 0 0 −1.88874 0 0.806704 1.39725i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.6
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{2}^{3} + T_{2}^{2} - 4T_{2} - 1 \) 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

$p$ $F_p(T)$
$2$ \( (T^{3} + T^{2} - 4 T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 21 T^{10} + 333 T^{8} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 4 T^{5} + 17 T^{4} + 2 T^{3} + 5 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 39 T^{10} + 1170 T^{8} + \cdots + 6561 \) Copy content Toggle raw display
$17$ \( T^{12} + 84 T^{10} + 5463 T^{8} + \cdots + 15752961 \) Copy content Toggle raw display
$19$ \( T^{12} + 75 T^{10} + 4329 T^{8} + \cdots + 15752961 \) Copy content Toggle raw display
$23$ \( (T^{6} - 2 T^{5} + 29 T^{4} - 68 T^{3} + \cdots + 3481)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 11 T^{5} + 107 T^{4} - 332 T^{3} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 129 T^{4} + 5508 T^{2} + \cdots - 77841)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 3 T^{5} + 33 T^{4} + 126 T^{3} + \cdots + 729)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 162 T^{10} + \cdots + 43046721 \) Copy content Toggle raw display
$43$ \( (T^{6} + 3 T^{5} + 33 T^{4} - 126 T^{3} + \cdots + 729)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 183 T^{4} + 10584 T^{2} + \cdots - 194481)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 14 T^{5} + 185 T^{4} + \cdots + 69169)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 183 T^{4} + 9450 T^{2} + \cdots - 149769)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 264 T^{4} + 21519 T^{2} + \cdots - 558009)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 111 T + 353)^{4} \) Copy content Toggle raw display
$71$ \( (T^{3} + 19 T^{2} + 116 T + 227)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + 75 T^{10} + 4329 T^{8} + \cdots + 15752961 \) Copy content Toggle raw display
$79$ \( (T^{3} + 3 T^{2} - 78 T - 107)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + 228 T^{10} + \cdots + 51769445841 \) Copy content Toggle raw display
$89$ \( T^{12} + 246 T^{10} + \cdots + 15752961 \) Copy content Toggle raw display
$97$ \( T^{12} + 111 T^{10} + \cdots + 96059601 \) Copy content Toggle raw display
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