Properties

Label 1323.4.a.bb
Level $1323$
Weight $4$
Character orbit 1323.a
Self dual yes
Analytic conductor $78.060$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,4,Mod(1,1323)]
 
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([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

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: yes
Analytic conductor: \(78.0595269376\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.346909504.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 12x^{4} + 2x^{3} + 39x^{2} + 25x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} + (\beta_{4} - 4) q^{5} + ( - \beta_{5} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} + (\beta_{4} - 4) q^{5} + ( - \beta_{5} + 2 \beta_1) q^{8} + ( - \beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{10} + (\beta_{5} + 3 \beta_{3} + 2 \beta_1) q^{11} + (2 \beta_{5} + \beta_{3} - 6 \beta_1) q^{13} + (3 \beta_{4} + 2 \beta_{2} - 5) q^{16} + ( - 2 \beta_{4} - 6 \beta_{2} - 7) q^{17} + (\beta_{5} - 7 \beta_{3} - 6 \beta_1) q^{19} + ( - \beta_{4} + 3 \beta_{2} - 2) q^{20} + ( - 9 \beta_{4} - 3 \beta_{2} + 22) q^{22} + (\beta_{5} + 6 \beta_{3} + 12 \beta_1) q^{23} + ( - 15 \beta_{4} - 6 \beta_{2} + 37) q^{25} + ( - 8 \beta_{4} - 21 \beta_{2} - 61) q^{26} + ( - 2 \beta_{5} - 9 \beta_{3} - 6 \beta_1) q^{29} + ( - 4 \beta_{5} + \beta_{3} - 30 \beta_1) q^{31} + (3 \beta_{5} - 6 \beta_{3} - 4 \beta_1) q^{32} + (8 \beta_{5} + 4 \beta_{3} - 51 \beta_1) q^{34} + (3 \beta_{4} - 10 \beta_{2} - 52) q^{37} + (11 \beta_{4} - 21 \beta_{2} - 56) q^{38} + (6 \beta_{5} + 18 \beta_{3} + 42 \beta_1) q^{40} + (3 \beta_{4} + 24 \beta_{2} - 60) q^{41} + ( - 12 \beta_{4} + 22 \beta_{2} + 109) q^{43} + (4 \beta_{5} - 6 \beta_{3} - 24 \beta_1) q^{44} + ( - 15 \beta_{4} + 10 \beta_{2} + 129) q^{46} + (23 \beta_{4} - 30 \beta_{2} - 302) q^{47} + (21 \beta_{5} + 30 \beta_{3} - 20 \beta_1) q^{50} + (13 \beta_{5} + 8 \beta_{3} - 168 \beta_1) q^{52} + ( - 19 \beta_{5} + 28 \beta_1) q^{53} + ( - 25 \beta_{5} - 29 \beta_{3} - 126 \beta_1) q^{55} + (24 \beta_{4} + \beta_{2} - 63) q^{58} + ( - 10 \beta_{4} - 48 \beta_{2} + 1) q^{59} + ( - 5 \beta_{5} + 5 \beta_{3} + 126 \beta_1) q^{61} + (10 \beta_{4} + 3 \beta_{2} - 343) q^{62} + ( - 21 \beta_{4} - 50 \beta_{2} + 11) q^{64} + ( - 15 \beta_{5} - 33 \beta_{3} - 102 \beta_1) q^{65} + ( - 21 \beta_{4} + 6 \beta_{2} + 7) q^{67} + ( - 16 \beta_{4} - 63 \beta_{2} - 485) q^{68} + ( - 27 \beta_{5} - 12 \beta_{3} + 52 \beta_1) q^{71} + (2 \beta_{5} + 22 \beta_{3} - 192 \beta_1) q^{73} + (7 \beta_{5} - 6 \beta_{3} - 119 \beta_1) q^{74} + (2 \beta_{5} + 34 \beta_{3} - 144 \beta_1) q^{76} + ( - 27 \beta_{4} + 72 \beta_{2} + 326) q^{79} + ( - 46 \beta_{4} - 12 \beta_{2} + 478) q^{80} + ( - 27 \beta_{5} - 6 \beta_{3} + 111 \beta_1) q^{82} + ( - 43 \beta_{4} + 30 \beta_{2} - 482) q^{83} + (39 \beta_{4} - 6 \beta_{2} - 324) q^{85} + ( - 10 \beta_{5} + 24 \beta_{3} + 251 \beta_1) q^{86} + (72 \beta_{4} - 38 \beta_{2} - 422) q^{88} + ( - 59 \beta_{4} + 42 \beta_{2} - 253) q^{89} + ( - 3 \beta_{5} - 18 \beta_{3} + 88 \beta_1) q^{92} + (7 \beta_{5} - 46 \beta_{3} - 489 \beta_1) q^{94} + (33 \beta_{5} - 3 \beta_{3} + 138 \beta_1) q^{95} + ( - 23 \beta_{5} + 5 \beta_{3} - 270 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{4} - 24 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{4} - 24 q^{5} - 30 q^{16} - 42 q^{17} - 12 q^{20} + 132 q^{22} + 222 q^{25} - 366 q^{26} - 312 q^{37} - 336 q^{38} - 360 q^{41} + 654 q^{43} + 774 q^{46} - 1812 q^{47} - 378 q^{58} + 6 q^{59} - 2058 q^{62} + 66 q^{64} + 42 q^{67} - 2910 q^{68} + 1956 q^{79} + 2868 q^{80} - 2892 q^{83} - 1944 q^{85} - 2532 q^{88} - 1518 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 12x^{4} + 2x^{3} + 39x^{2} + 25x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 10\nu^{2} + 27\nu + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{5} - 6\nu^{4} - 24\nu^{3} + 30\nu^{2} + 51\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} + 16\nu^{3} - 22\nu^{2} - 57\nu + 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{5} + 6\nu^{4} + 36\nu^{3} - 30\nu^{2} - 135\nu - 46 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{5} + 2\nu^{4} - 68\nu^{3} - 22\nu^{2} + 201\nu + 76 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{2} - 6\beta _1 + 2 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} - 4\beta _1 + 25 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{4} + 9\beta_{2} - 42\beta _1 + 62 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} - \beta_{4} - 6\beta_{3} + 10\beta_{2} - 44\beta _1 + 173 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{5} - 27\beta_{4} - 4\beta_{3} + 83\beta_{2} - 330\beta _1 + 662 ) / 12 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.04600
2.69129
−0.673097
−2.24419
−0.0935508
−1.72645
−4.77246 0 14.7764 1.85905 0 0 −32.3399 0 −8.87225
1.2 −2.78484 0 −0.244684 −20.8311 0 0 22.9601 0 58.0112
1.3 −1.57109 0 −5.53167 6.97204 0 0 21.2595 0 −10.9537
1.4 1.57109 0 −5.53167 6.97204 0 0 −21.2595 0 10.9537
1.5 2.78484 0 −0.244684 −20.8311 0 0 −22.9601 0 −58.0112
1.6 4.77246 0 14.7764 1.85905 0 0 32.3399 0 8.87225
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 1323.4.a.bb 6
3.b odd 2 1 1323.4.a.bc yes 6
7.b odd 2 1 1323.4.a.bc yes 6
21.c even 2 1 inner 1323.4.a.bb 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.4.a.bb 6 1.a even 1 1 trivial
1323.4.a.bb 6 21.c even 2 1 inner
1323.4.a.bc yes 6 3.b odd 2 1
1323.4.a.bc yes 6 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1323))\):

\( T_{2}^{6} - 33T_{2}^{4} + 252T_{2}^{2} - 436 \) Copy content Toggle raw display
\( T_{5}^{3} + 12T_{5}^{2} - 171T_{5} + 270 \) Copy content Toggle raw display
\( T_{13}^{6} - 8505T_{13}^{4} + 17399448T_{13}^{2} - 5371563600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 33 T^{4} + \cdots - 436 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} + 12 T^{2} + \cdots + 270)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 5580 T^{4} + \cdots - 879150400 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots - 5371563600 \) Copy content Toggle raw display
$17$ \( (T^{3} + 21 T^{2} + \cdots + 91719)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 232895741184 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 103079719936 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 149059723600 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 2588939818704 \) Copy content Toggle raw display
$37$ \( (T^{3} + 156 T^{2} + \cdots - 706050)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 180 T^{2} + \cdots - 10490310)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 327 T^{2} + \cdots + 15007815)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + 906 T^{2} + \cdots - 63550440)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 12\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{3} - 3 T^{2} + \cdots + 60900795)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 20403087817536 \) Copy content Toggle raw display
$67$ \( (T^{3} - 21 T^{2} + \cdots - 4249360)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{3} - 978 T^{2} + \cdots + 324650908)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 1446 T^{2} + \cdots - 66493008)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 759 T^{2} + \cdots - 79092180)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
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