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Properties

Label 1323.2.c.d
Level $1323$
Weight $2$
Character orbit 1323.c
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(1322,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
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("1323.1322");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.c (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: 12.0.13026266817859584.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + ( - \beta_{6} - 1) q^{4} + \beta_1 q^{5} + (\beta_{10} - \beta_{8}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + ( - \beta_{6} - 1) q^{4} + \beta_1 q^{5} + (\beta_{10} - \beta_{8}) q^{8} + (\beta_{11} + \beta_{5}) q^{10} + ( - \beta_{10} + \beta_{8}) q^{11} + (\beta_{7} + \beta_{5}) q^{13} + (\beta_{9} + \beta_{6}) q^{16} - \beta_{3} q^{17} + (\beta_{11} - \beta_{5}) q^{19} - \beta_{2} q^{20} + ( - \beta_{9} - 3 \beta_{6} - 2) q^{22} + (\beta_{10} + \beta_{8} + \beta_{4}) q^{23} + ( - \beta_{9} - 2 \beta_{6} + 5) q^{25} + ( - 2 \beta_{3} - \beta_1) q^{26} + (\beta_{10} + \beta_{8}) q^{29} + (2 \beta_{11} + \beta_{7} - \beta_{5}) q^{31} + \beta_{4} q^{32} + (\beta_{11} - 3 \beta_{7} - \beta_{5}) q^{34} + ( - \beta_{9} - 1) q^{37} + (2 \beta_{3} - \beta_{2}) q^{38} + ( - \beta_{11} + 2 \beta_{5}) q^{40} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{41} + ( - \beta_{9} + 2) q^{43} + (2 \beta_{10} - 6 \beta_{8} - \beta_{4}) q^{44} + ( - \beta_{9} - 2) q^{46} + (2 \beta_{3} - \beta_1) q^{47} + (3 \beta_{10} + \beta_{8} - \beta_{4}) q^{50} + (\beta_{11} - 4 \beta_{7} - \beta_{5}) q^{52} + ( - \beta_{10} - 2 \beta_{8} - \beta_{4}) q^{53} + (\beta_{11} - 2 \beta_{5}) q^{55} + (\beta_{9} + \beta_{6} - 4) q^{58} + (\beta_{3} + \beta_1) q^{59} + (2 \beta_{11} + \beta_{7}) q^{61} + (2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{62} + (\beta_{6} + 2) q^{64} + (3 \beta_{10} + 3 \beta_{8} - \beta_{4}) q^{65} + ( - 2 \beta_{9} - \beta_{6} - 5) q^{67} + (3 \beta_{3} - \beta_{2}) q^{68} + (\beta_{8} + \beta_{4}) q^{71} + ( - \beta_{11} + 3 \beta_{7} - 2 \beta_{5}) q^{73} + (\beta_{10} - \beta_{8} - \beta_{4}) q^{74} + ( - 3 \beta_{11} + 6 \beta_{7}) q^{76} + ( - 2 \beta_{9} - \beta_{6} + 7) q^{79} + ( - 3 \beta_{3} - \beta_{2} - \beta_1) q^{80} + ( - \beta_{11} - 3 \beta_{7}) q^{82} + ( - 2 \beta_{3} - \beta_1) q^{83} + (2 \beta_{9} + \beta_{6} - 2) q^{85} + (\beta_{10} + 2 \beta_{8} - \beta_{4}) q^{86} + (2 \beta_{9} + 5 \beta_{6} + 10) q^{88} + ( - 2 \beta_{3} - \beta_{2}) q^{89} + (3 \beta_{10} + \beta_{4}) q^{92} + ( - 3 \beta_{11} + 6 \beta_{7} + \beta_{5}) q^{94} + ( - 3 \beta_{10} - \beta_{4}) q^{95} + ( - 4 \beta_{7} + 2 \beta_{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 16 q^{4} + 8 q^{16} - 40 q^{22} + 48 q^{25} - 16 q^{37} + 20 q^{43} - 28 q^{46} - 40 q^{58} + 28 q^{64} - 72 q^{67} + 72 q^{79} - 12 q^{85} + 148 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1298\nu^{11} - 10953\nu^{9} + 71803\nu^{7} - 202311\nu^{5} + 490451\nu^{3} - 438921\nu ) / 197955 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{11} - 117\nu^{9} + 850\nu^{7} - 3420\nu^{5} + 7895\nu^{3} - 7056\nu ) / 747 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3734\nu^{11} + 34254\nu^{9} - 224554\nu^{7} + 743958\nu^{5} - 1731773\nu^{3} + 1545408\nu ) / 197955 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1298\nu^{11} - 10953\nu^{9} + 71803\nu^{7} - 202311\nu^{5} + 490451\nu^{3} - 43011\nu ) / 65985 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 787\nu^{10} - 10047\nu^{8} + 58532\nu^{6} - 191649\nu^{4} + 281254\nu^{2} - 89064 ) / 65985 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 288\nu^{10} - 1888\nu^{8} + 9933\nu^{6} - 12896\nu^{4} + 6336\nu^{2} + 68439 ) / 21995 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2596\nu^{10} - 21906\nu^{8} + 143606\nu^{6} - 404622\nu^{4} + 980902\nu^{2} - 283977 ) / 197955 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5192\nu^{11} - 43812\nu^{9} + 287212\nu^{7} - 809244\nu^{5} + 1763849\nu^{3} - 172044\nu ) / 197955 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -333\nu^{10} + 2183\nu^{8} - 9423\nu^{6} + 14911\nu^{4} - 7326\nu^{2} + 48026 ) / 21995 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -88\nu^{11} + 783\nu^{9} - 4868\nu^{7} + 13716\nu^{5} - 26581\nu^{3} + 2916\nu ) / 2385 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -2353\nu^{10} + 20313\nu^{8} - 133163\nu^{6} + 393741\nu^{4} - 843586\nu^{2} + 248931 ) / 65985 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{11} - \beta_{9} + 9\beta_{7} - 2\beta_{6} + 10 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{8} + 4\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{11} + 2\beta_{9} + 12\beta_{7} + 3\beta_{6} + \beta_{5} - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{10} - 18\beta_{8} + 17\beta_{4} + 18\beta_{3} + 3\beta_{2} + 51\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 32\beta_{9} + 37\beta_{6} - 185 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 27\beta_{10} + 96\beta_{8} - 74\beta_{4} + 96\beta_{3} + 27\beta_{2} + 222\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -106\beta_{11} + 55\beta_{9} - 222\beta_{7} + 51\beta_{6} - 59\beta_{5} - 277 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 177\beta_{10} + 495\beta_{8} - 328\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -1479\beta_{11} - 835\beta_{9} - 2952\beta_{7} - 644\beta_{6} - 1026\beta_{5} + 3787 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1026\beta_{10} + 2505\beta_{8} - 1477\beta_{4} - 2505\beta_{3} - 1026\beta_{2} - 4431\beta_1 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1322.1
0.617942 − 0.356769i
−0.617942 − 0.356769i
1.90412 + 1.09935i
−1.90412 + 1.09935i
1.65604 − 0.956115i
−1.65604 − 0.956115i
1.65604 + 0.956115i
−1.65604 + 0.956115i
1.90412 − 1.09935i
−1.90412 − 1.09935i
0.617942 + 0.356769i
−0.617942 + 0.356769i
− 2.49086i 0 −4.20440 −1.23588 0 0 5.49086i 0 3.07842i
1322.2 − 2.49086i 0 −4.20440 1.23588 0 0 5.49086i 0 − 3.07842i
1322.3 − 1.83424i 0 −1.36445 −3.80824 0 0 − 1.16576i 0 6.98525i
1322.4 − 1.83424i 0 −1.36445 3.80824 0 0 − 1.16576i 0 − 6.98525i
1322.5 − 0.656620i 0 1.56885 −3.31208 0 0 − 2.34338i 0 2.17478i
1322.6 − 0.656620i 0 1.56885 3.31208 0 0 − 2.34338i 0 − 2.17478i
1322.7 0.656620i 0 1.56885 −3.31208 0 0 2.34338i 0 − 2.17478i
1322.8 0.656620i 0 1.56885 3.31208 0 0 2.34338i 0 2.17478i
1322.9 1.83424i 0 −1.36445 −3.80824 0 0 1.16576i 0 − 6.98525i
1322.10 1.83424i 0 −1.36445 3.80824 0 0 1.16576i 0 6.98525i
1322.11 2.49086i 0 −4.20440 −1.23588 0 0 − 5.49086i 0 − 3.07842i
1322.12 2.49086i 0 −4.20440 1.23588 0 0 − 5.49086i 0 3.07842i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1322.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.c.d 12
3.b odd 2 1 inner 1323.2.c.d 12
7.b odd 2 1 inner 1323.2.c.d 12
7.c even 3 1 189.2.p.d ✓ 12
7.d odd 6 1 189.2.p.d ✓ 12
21.c even 2 1 inner 1323.2.c.d 12
21.g even 6 1 189.2.p.d ✓ 12
21.h odd 6 1 189.2.p.d ✓ 12
63.g even 3 1 567.2.s.f 12
63.h even 3 1 567.2.i.f 12
63.i even 6 1 567.2.i.f 12
63.j odd 6 1 567.2.i.f 12
63.k odd 6 1 567.2.s.f 12
63.n odd 6 1 567.2.s.f 12
63.s even 6 1 567.2.s.f 12
63.t odd 6 1 567.2.i.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.d ✓ 12 7.c even 3 1
189.2.p.d ✓ 12 7.d odd 6 1
189.2.p.d ✓ 12 21.g even 6 1
189.2.p.d ✓ 12 21.h odd 6 1
567.2.i.f 12 63.h even 3 1
567.2.i.f 12 63.i even 6 1
567.2.i.f 12 63.j odd 6 1
567.2.i.f 12 63.t odd 6 1
567.2.s.f 12 63.g even 3 1
567.2.s.f 12 63.k odd 6 1
567.2.s.f 12 63.n odd 6 1
567.2.s.f 12 63.s even 6 1
1323.2.c.d 12 1.a even 1 1 trivial
1323.2.c.d 12 3.b odd 2 1 inner
1323.2.c.d 12 7.b odd 2 1 inner
1323.2.c.d 12 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 10T_{2}^{4} + 25T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 10 T^{4} + 25 T^{2} + 9)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 27 T^{4} + \cdots - 243)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + 37 T^{4} + \cdots + 225)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 42 T^{4} + \cdots + 675)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 30 T^{4} + \cdots - 243)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 51 T^{4} + \cdots + 243)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 94 T^{4} + \cdots + 729)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 37 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 138 T^{4} + \cdots + 64827)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 4 T^{2} - 19 T - 67)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} - 114 T^{4} + \cdots - 243)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 5 T^{2} - 16 T - 1)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} - 135 T^{4} + \cdots - 19683)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 118 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 63 T^{4} + \cdots - 2187)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 129 T^{4} + \cdots + 49923)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 18 T^{2} + \cdots - 677)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + 85 T^{4} + \cdots + 19881)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 195 T^{4} + \cdots + 177147)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 18 T^{2} + 15 T + 7)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 159 T^{4} + \cdots - 6075)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 201 T^{4} + \cdots - 87723)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 204 T^{4} + \cdots + 1728)^{2} \) Copy content Toggle raw display
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