Properties

Label 1323.4.a.bf
Level $1323$
Weight $4$
Character orbit 1323.a
Self dual yes
Analytic conductor $78.060$
Analytic rank $1$
Dimension $6$
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: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 42x^{4} + 369x^{2} - 112 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 6) q^{4} + ( - \beta_{3} - 2 \beta_1) q^{5} + (\beta_{5} + 2 \beta_{3} + 8 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 6) q^{4} + ( - \beta_{3} - 2 \beta_1) q^{5} + (\beta_{5} + 2 \beta_{3} + 8 \beta_1) q^{8} + ( - \beta_{4} - 4 \beta_{2} - 30) q^{10} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{11} + ( - \beta_{4} - \beta_{2} - 18) q^{13} + (3 \beta_{4} + 8 \beta_{2} + 70) q^{16} + (3 \beta_{5} + 2 \beta_{3} + 7 \beta_1) q^{17} + (\beta_{4} - 5 \beta_{2} - 33) q^{19} + ( - 6 \beta_{5} - 8 \beta_{3} - 49 \beta_1) q^{20} + ( - 3 \beta_{2} - 14) q^{22} + ( - \beta_{5} - 5 \beta_{3} - 15 \beta_1) q^{23} + (3 \beta_{4} + 9 \beta_{2} + 70) q^{25} + ( - 3 \beta_{5} - 10 \beta_{3} - 23 \beta_1) q^{26} + ( - 7 \beta_{5} + 4 \beta_{3} - 21 \beta_1) q^{29} + (5 \beta_{4} - 7 \beta_{2} + 15) q^{31} + (6 \beta_{5} + 24 \beta_{3} + 71 \beta_1) q^{32} + (5 \beta_{4} + 23 \beta_{2} + 108) q^{34} + ( - 6 \beta_{4} - 4 \beta_{2} - 67) q^{37} + ( - 3 \beta_{5} - 2 \beta_{3} - 88 \beta_1) q^{38} + ( - 6 \beta_{4} - 57 \beta_{2} - 474) q^{40} + (6 \beta_{5} - 3 \beta_{3} - 54 \beta_1) q^{41} + ( - 3 \beta_{4} - 11 \beta_{2} - 110) q^{43} + (5 \beta_{5} - 14 \beta_{3} - 36 \beta_1) q^{44} + ( - 6 \beta_{4} - 29 \beta_{2} - 222) q^{46} + ( - 3 \beta_{5} + 4 \beta_{3} + 59 \beta_1) q^{47} + (15 \beta_{5} + 42 \beta_{3} + 145 \beta_1) q^{50} + ( - 5 \beta_{4} - 47 \beta_{2} - 204) q^{52} + ( - 8 \beta_{5} + 20 \beta_{3} - 20 \beta_1) q^{53} + ( - 4 \beta_{4} + 26 \beta_{2} - 141) q^{55} + ( - 3 \beta_{4} - 41 \beta_{2} - 300) q^{58} + ( - 3 \beta_{5} + 28 \beta_{3} - 109 \beta_1) q^{59} + (10 \beta_{4} - 2 \beta_{2} - 192) q^{61} + (3 \beta_{5} + 26 \beta_{3} - 80 \beta_1) q^{62} + (6 \beta_{4} + 79 \beta_{2} + 494) q^{64} + ( - 3 \beta_{5} + 24 \beta_{3} + 135 \beta_1) q^{65} + ( - 3 \beta_{4} + 9 \beta_{2} + 154) q^{67} + (9 \beta_{5} + 70 \beta_{3} + 257 \beta_1) q^{68} + ( - 21 \beta_{3} - 14 \beta_1) q^{71} + ( - 7 \beta_{4} - 55 \beta_{2} - 210) q^{73} + ( - 16 \beta_{5} - 56 \beta_{3} - 77 \beta_1) q^{74} + ( - 13 \beta_{4} - 64 \beta_{2} - 978) q^{76} + ( - 9 \beta_{4} - 45 \beta_{2} - 250) q^{79} + ( - 21 \beta_{5} - 98 \beta_{3} - 622 \beta_1) q^{80} + (3 \beta_{4} - 36 \beta_{2} - 750) q^{82} + (39 \beta_{5} + 52 \beta_{3} - 19 \beta_1) q^{83} + (3 \beta_{4} - 99 \beta_{2} - 372) q^{85} + ( - 17 \beta_{5} - 46 \beta_{3} - 205 \beta_1) q^{86} + ( - 9 \beta_{4} - 20 \beta_{2} - 410) q^{88} + (24 \beta_{5} - 67 \beta_{3} + 160 \beta_1) q^{89} + ( - 33 \beta_{5} - 66 \beta_{3} - 362 \beta_1) q^{92} + (\beta_{4} + 55 \beta_{2} + 828) q^{94} + (39 \beta_{5} + 39 \beta_{3} + 189 \beta_1) q^{95} + (10 \beta_{4} + 70 \beta_{2} - 552) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 36 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 36 q^{4} - 180 q^{10} - 108 q^{13} + 420 q^{16} - 198 q^{19} - 84 q^{22} + 420 q^{25} + 90 q^{31} + 648 q^{34} - 402 q^{37} - 2844 q^{40} - 660 q^{43} - 1332 q^{46} - 1224 q^{52} - 846 q^{55} - 1800 q^{58} - 1152 q^{61} + 2964 q^{64} + 924 q^{67} - 1260 q^{73} - 5868 q^{76} - 1500 q^{79} - 4500 q^{82} - 2232 q^{85} - 2460 q^{88} + 4968 q^{94} - 3312 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 42x^{4} + 369x^{2} - 112 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 38\nu^{3} + 265\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 32\nu^{2} + 106 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 44\nu^{3} - 409\nu ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{3} + 24\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{4} + 32\beta_{2} + 342 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 38\beta_{5} + 88\beta_{3} + 647\beta_1 \) 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
−5.45019
−3.46131
−0.560992
0.560992
3.46131
5.45019
−5.45019 0 21.7046 19.3432 0 0 −74.6929 0 −105.424
1.2 −3.46131 0 3.98067 −6.55599 0 0 13.9122 0 22.6923
1.3 −0.560992 0 −7.68529 12.9561 0 0 8.79932 0 −7.26827
1.4 0.560992 0 −7.68529 −12.9561 0 0 −8.79932 0 −7.26827
1.5 3.46131 0 3.98067 6.55599 0 0 −13.9122 0 22.6923
1.6 5.45019 0 21.7046 −19.3432 0 0 74.6929 0 −105.424
\(n\): 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
3.b odd 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.bf 6
3.b odd 2 1 inner 1323.4.a.bf 6
7.b odd 2 1 1323.4.a.bg yes 6
21.c even 2 1 1323.4.a.bg yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.4.a.bf 6 1.a even 1 1 trivial
1323.4.a.bf 6 3.b odd 2 1 inner
1323.4.a.bg yes 6 7.b odd 2 1
1323.4.a.bg yes 6 21.c even 2 1

Hecke kernels

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

\( T_{2}^{6} - 42T_{2}^{4} + 369T_{2}^{2} - 112 \) Copy content Toggle raw display
\( T_{5}^{6} - 585T_{5}^{4} + 86103T_{5}^{2} - 2699487 \) Copy content Toggle raw display
\( T_{13}^{3} + 54T_{13}^{2} - 684T_{13} - 48168 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 42 T^{4} + \cdots - 112 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 585 T^{4} + \cdots - 2699487 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 2457 T^{4} + \cdots - 1539727 \) Copy content Toggle raw display
$13$ \( (T^{3} + 54 T^{2} + \cdots - 48168)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 143993106432 \) Copy content Toggle raw display
$19$ \( (T^{3} + 99 T^{2} + \cdots - 450387)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 16248512287 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 1629346543552 \) Copy content Toggle raw display
$31$ \( (T^{3} - 45 T^{2} + \cdots - 1675863)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 201 T^{2} + \cdots - 8763417)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 249467996315727 \) Copy content Toggle raw display
$43$ \( (T^{3} + 330 T^{2} + \cdots - 819288)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 135023220900288 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 38601195716608 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( (T^{3} + 576 T^{2} + \cdots - 9597312)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 462 T^{2} + \cdots + 4527656)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 232630513987207 \) Copy content Toggle raw display
$73$ \( (T^{3} + 630 T^{2} + \cdots + 78997464)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 750 T^{2} + \cdots + 20170072)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 99\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 12\!\cdots\!07 \) Copy content Toggle raw display
$97$ \( (T^{3} + 1656 T^{2} + \cdots - 913012992)^{2} \) Copy content Toggle raw display
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