Properties

Label 1323.4.a.be
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: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 40x^{4} + 453x^{2} - 1278 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 189)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 5) q^{4} - \beta_{4} q^{5} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 5) q^{4} - \beta_{4} q^{5} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{8} + ( - 3 \beta_{5} + \beta_{3} + 4) q^{10} + (\beta_{4} - 2 \beta_{2} - 4 \beta_1) q^{11} + (5 \beta_{5} + 2 \beta_{3} - 11) q^{13} + (4 \beta_{5} + 3 \beta_{3} - 27) q^{16} + (5 \beta_{4} - 4 \beta_1) q^{17} + ( - 11 \beta_{3} + 14) q^{19} + (\beta_{4} - 2 \beta_{2} + 8 \beta_1) q^{20} + ( - 11 \beta_{5} - 23 \beta_{3} - 48) q^{22} + (6 \beta_{4} + 4 \beta_{2} - 10 \beta_1) q^{23} + ( - 2 \beta_{5} - 12 \beta_{3} - 3) q^{25} + (8 \beta_{4} + 7 \beta_{2} - 3 \beta_1) q^{26} + ( - 9 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{29} + ( - 11 \beta_{5} + 13 \beta_{3} + 13) q^{31} + (13 \beta_{4} - \beta_{2} - 23 \beta_1) q^{32} + (15 \beta_{5} - 9 \beta_{3} - 72) q^{34} + (\beta_{5} - 12 \beta_{3} - 185) q^{37} + (11 \beta_{4} - 11 \beta_{2} - 30 \beta_1) q^{38} + (13 \beta_{5} - 19 \beta_{3} + 76) q^{40} + (15 \beta_{4} + 4 \beta_{2} + 52 \beta_1) q^{41} + (8 \beta_{5} - 2 \beta_{3} - 263) q^{43} + ( - 7 \beta_{4} - 18 \beta_{2} - 108 \beta_1) q^{44} + (46 \beta_{5} + 20 \beta_{3} - 170) q^{46} + (29 \beta_{4} - 14 \beta_{2} - 26 \beta_1) q^{47} + (8 \beta_{4} - 14 \beta_{2} - 51 \beta_1) q^{50} + (33 \beta_{5} + 36 \beta_{3} - 11) q^{52} + ( - 8 \beta_{4} + 12 \beta_{2} - 26 \beta_1) q^{53} + ( - 22 \beta_{5} + 24 \beta_{3} - 38) q^{55} + ( - 41 \beta_{5} - 11 \beta_{3} + 18) q^{58} + ( - 23 \beta_{4} + 26 \beta_{2} - 40 \beta_1) q^{59} + ( - 63 \beta_{5} + \beta_{3} - 127) q^{61} + ( - 35 \beta_{4} + 2 \beta_{2} + 65 \beta_1) q^{62} + ( - 69 \beta_{3} - 131) q^{64} + (13 \beta_{4} + 16 \beta_{2} - 94 \beta_1) q^{65} + ( - 54 \beta_{5} - 19 \beta_{3} - 323) q^{67} + ( - \beta_{4} + 6 \beta_{2} - 76 \beta_1) q^{68} + ( - 51 \beta_{4} + 18 \beta_{2} - 12 \beta_1) q^{71} + ( - 20 \beta_{5} + 55 \beta_{3} + 388) q^{73} + (14 \beta_{4} - 11 \beta_{2} - 233 \beta_1) q^{74} + ( - 44 \beta_{5} - 52 \beta_{3} - 502) q^{76} + (61 \beta_{5} - 64 \beta_{3} + 165) q^{79} + (37 \beta_{4} + 10 \beta_{2} - 64 \beta_1) q^{80} + (73 \beta_{5} + 73 \beta_{3} + 600) q^{82} + (28 \beta_{4} - 14 \beta_{2} + 318 \beta_1) q^{83} + (22 \beta_{5} + 56 \beta_{3} - 626) q^{85} + (18 \beta_{4} + 6 \beta_{2} - 271 \beta_1) q^{86} + ( - 59 \beta_{5} - 79 \beta_{3} - 920) q^{88} + ( - 51 \beta_{4} - 26 \beta_{2} - 174 \beta_1) q^{89} + (24 \beta_{4} + 34 \beta_{2} - 10 \beta_1) q^{92} + ( - 11 \beta_{5} - 181 \beta_{3} - 398) q^{94} + ( - 80 \beta_{4} + 22 \beta_{2} - 88 \beta_1) q^{95} + ( - 72 \beta_{5} - 154 \beta_{3} + 189) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 32 q^{4} + 20 q^{10} - 52 q^{13} - 148 q^{16} + 62 q^{19} - 356 q^{22} - 46 q^{25} + 82 q^{31} - 420 q^{34} - 1132 q^{37} + 444 q^{40} - 1566 q^{43} - 888 q^{46} + 72 q^{52} - 224 q^{55} + 4 q^{58} - 886 q^{61} - 924 q^{64} - 2084 q^{67} + 2398 q^{73} - 3204 q^{76} + 984 q^{79} + 3892 q^{82} - 3600 q^{85} - 5796 q^{88} - 2772 q^{94} + 682 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 40x^{4} + 453x^{2} - 1278 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 19\nu^{3} - 6\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 31\nu^{3} + 198\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} - 27\nu^{2} + 130 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{2} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{5} + 27\beta_{3} + 221 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -19\beta_{4} + 31\beta_{2} + 329\beta_1 \) 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
−4.70753
−3.68757
−2.05936
2.05936
3.68757
4.70753
−4.70753 0 14.1608 0.830453 0 0 −29.0024 0 −3.90938
1.2 −3.68757 0 5.59820 −11.8719 0 0 8.85681 0 43.7785
1.3 −2.05936 0 −3.75905 14.5041 0 0 24.2161 0 −29.8691
1.4 2.05936 0 −3.75905 −14.5041 0 0 −24.2161 0 −29.8691
1.5 3.68757 0 5.59820 11.8719 0 0 −8.85681 0 43.7785
1.6 4.70753 0 14.1608 −0.830453 0 0 29.0024 0 −3.90938
\(n\): 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.be 6
3.b odd 2 1 inner 1323.4.a.be 6
7.b odd 2 1 1323.4.a.bd 6
7.d odd 6 2 189.4.e.e 12
21.c even 2 1 1323.4.a.bd 6
21.g even 6 2 189.4.e.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.e 12 7.d odd 6 2
189.4.e.e 12 21.g even 6 2
1323.4.a.bd 6 7.b odd 2 1
1323.4.a.bd 6 21.c even 2 1
1323.4.a.be 6 1.a even 1 1 trivial
1323.4.a.be 6 3.b odd 2 1 inner

Hecke kernels

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

\( T_{2}^{6} - 40T_{2}^{4} + 453T_{2}^{2} - 1278 \) Copy content Toggle raw display
\( T_{5}^{6} - 352T_{5}^{4} + 29892T_{5}^{2} - 20448 \) Copy content Toggle raw display
\( T_{13}^{3} + 26T_{13}^{2} - 3211T_{13} + 35818 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 40 T^{4} + \cdots - 1278 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 352 T^{4} + \cdots - 20448 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 2041855488 \) Copy content Toggle raw display
$13$ \( (T^{3} + 26 T^{2} + \cdots + 35818)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 4889546208 \) Copy content Toggle raw display
$19$ \( (T^{3} - 31 T^{2} + \cdots + 71044)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 54061976448 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 1224041183712 \) Copy content Toggle raw display
$31$ \( (T^{3} - 41 T^{2} + \cdots + 2204979)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 566 T^{2} + \cdots + 4245738)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 69555350983392 \) Copy content Toggle raw display
$43$ \( (T^{3} + 783 T^{2} + \cdots + 15519697)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 554088448396512 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 248222191554432 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 26\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( (T^{3} + 443 T^{2} + \cdots - 219938513)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 1042 T^{2} + \cdots - 181536376)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 17\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( (T^{3} - 1199 T^{2} + \cdots + 138593088)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 492 T^{2} + \cdots - 61449092)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 28\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 51\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( (T^{3} - 341 T^{2} + \cdots + 1582590381)^{2} \) Copy content Toggle raw display
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