Properties

Label 1344.4.a.bs
Level $1344$
Weight $4$
Character orbit 1344.a
Self dual yes
Analytic conductor $79.299$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,4,Mod(1,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1344.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.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: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.37341.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 57x - 148 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 672)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + (\beta_1 - 2) q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + (\beta_1 - 2) q^{5} + 7 q^{7} + 9 q^{9} + (\beta_{2} - \beta_1 - 16) q^{11} + (\beta_{2} + 4 \beta_1 - 2) q^{13} + ( - 3 \beta_1 + 6) q^{15} + (\beta_{2} - \beta_1 + 18) q^{17} + ( - \beta_{2} - 2 \beta_1 - 28) q^{19} - 21 q^{21} + (\beta_{2} + 3 \beta_1 + 16) q^{23} + (\beta_{2} + 4 \beta_1 + 31) q^{25} - 27 q^{27} + ( - 3 \beta_{2} + 10 \beta_1 - 6) q^{29} + (3 \beta_{2} + 2 \beta_1 + 24) q^{31} + ( - 3 \beta_{2} + 3 \beta_1 + 48) q^{33} + (7 \beta_1 - 14) q^{35} + (\beta_{2} + 4 \beta_1 - 70) q^{37} + ( - 3 \beta_{2} - 12 \beta_1 + 6) q^{39} + ( - 3 \beta_{2} + 5 \beta_1 + 138) q^{41} + (2 \beta_{2} + 8 \beta_1 - 56) q^{43} + (9 \beta_1 - 18) q^{45} + (26 \beta_1 + 24) q^{47} + 49 q^{49} + ( - 3 \beta_{2} + 3 \beta_1 - 54) q^{51} + ( - 3 \beta_{2} - 12 \beta_1 - 134) q^{53} + ( - 11 \beta_{2} - 10 \beta_1 - 152) q^{55} + (3 \beta_{2} + 6 \beta_1 + 84) q^{57} + ( - 38 \beta_1 - 180) q^{59} + (7 \beta_{2} - 30 \beta_1 - 266) q^{61} + 63 q^{63} + ( - 6 \beta_{2} + 34 \beta_1 + 580) q^{65} + ( - 2 \beta_{2} + 38 \beta_1 - 16) q^{67} + ( - 3 \beta_{2} - 9 \beta_1 - 48) q^{69} + ( - \beta_{2} + 21 \beta_1 - 152) q^{71} + (16 \beta_{2} + 18 \beta_1 + 410) q^{73} + ( - 3 \beta_{2} - 12 \beta_1 - 93) q^{75} + (7 \beta_{2} - 7 \beta_1 - 112) q^{77} + (8 \beta_{2} - 26 \beta_1 - 456) q^{79} + 81 q^{81} + ( - 10 \beta_{2} - 40 \beta_1 - 20) q^{83} + ( - 11 \beta_{2} + 24 \beta_1 - 220) q^{85} + (9 \beta_{2} - 30 \beta_1 + 18) q^{87} + (3 \beta_{2} - 31 \beta_1 + 914) q^{89} + (7 \beta_{2} + 28 \beta_1 - 14) q^{91} + ( - 9 \beta_{2} - 6 \beta_1 - 72) q^{93} + (8 \beta_{2} - 52 \beta_1 - 216) q^{95} + ( - 12 \beta_{2} - 26 \beta_1 + 650) q^{97} + (9 \beta_{2} - 9 \beta_1 - 144) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 9 q^{3} - 6 q^{5} + 21 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 9 q^{3} - 6 q^{5} + 21 q^{7} + 27 q^{9} - 48 q^{11} - 6 q^{13} + 18 q^{15} + 54 q^{17} - 84 q^{19} - 63 q^{21} + 48 q^{23} + 93 q^{25} - 81 q^{27} - 18 q^{29} + 72 q^{31} + 144 q^{33} - 42 q^{35} - 210 q^{37} + 18 q^{39} + 414 q^{41} - 168 q^{43} - 54 q^{45} + 72 q^{47} + 147 q^{49} - 162 q^{51} - 402 q^{53} - 456 q^{55} + 252 q^{57} - 540 q^{59} - 798 q^{61} + 189 q^{63} + 1740 q^{65} - 48 q^{67} - 144 q^{69} - 456 q^{71} + 1230 q^{73} - 279 q^{75} - 336 q^{77} - 1368 q^{79} + 243 q^{81} - 60 q^{83} - 660 q^{85} + 54 q^{87} + 2742 q^{89} - 42 q^{91} - 216 q^{93} - 648 q^{95} + 1950 q^{97} - 432 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 57x - 148 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 16\nu - 152 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 8\beta _1 + 152 ) / 4 \) 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
−5.47374
−3.13924
8.61298
0 −3.00000 0 −12.9475 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 −8.27848 0 7.00000 0 9.00000 0
1.3 0 −3.00000 0 15.2260 0 7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.a.bs 3
4.b odd 2 1 1344.4.a.bu 3
8.b even 2 1 672.4.a.r yes 3
8.d odd 2 1 672.4.a.p 3
24.f even 2 1 2016.4.a.u 3
24.h odd 2 1 2016.4.a.v 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.p 3 8.d odd 2 1
672.4.a.r yes 3 8.b even 2 1
1344.4.a.bs 3 1.a even 1 1 trivial
1344.4.a.bu 3 4.b odd 2 1
2016.4.a.u 3 24.f even 2 1
2016.4.a.v 3 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5}^{3} + 6T_{5}^{2} - 216T_{5} - 1632 \) Copy content Toggle raw display
\( T_{11}^{3} + 48T_{11}^{2} - 3060T_{11} - 95488 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 6 T^{2} - 216 T - 1632 \) Copy content Toggle raw display
$7$ \( (T - 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 48 T^{2} - 3060 T - 95488 \) Copy content Toggle raw display
$13$ \( T^{3} + 6 T^{2} - 6756 T + 63656 \) Copy content Toggle raw display
$17$ \( T^{3} - 54 T^{2} - 2856 T + 24736 \) Copy content Toggle raw display
$19$ \( T^{3} + 84 T^{2} - 1872 T - 200256 \) Copy content Toggle raw display
$23$ \( T^{3} - 48 T^{2} - 4500 T + 187648 \) Copy content Toggle raw display
$29$ \( T^{3} + 18 T^{2} - 57108 T + 4848088 \) Copy content Toggle raw display
$31$ \( T^{3} - 72 T^{2} - 30144 T + 2342912 \) Copy content Toggle raw display
$37$ \( T^{3} + 210 T^{2} + 7932 T - 53576 \) Copy content Toggle raw display
$41$ \( T^{3} - 414 T^{2} + 18456 T + 4957664 \) Copy content Toggle raw display
$43$ \( T^{3} + 168 T^{2} - 17664 T - 722944 \) Copy content Toggle raw display
$47$ \( T^{3} - 72 T^{2} - 152400 T - 17124736 \) Copy content Toggle raw display
$53$ \( T^{3} + 402 T^{2} - 7044 T - 7840072 \) Copy content Toggle raw display
$59$ \( T^{3} + 540 T^{2} + \cdots + 11538688 \) Copy content Toggle raw display
$61$ \( T^{3} + 798 T^{2} + \cdots - 170037272 \) Copy content Toggle raw display
$67$ \( T^{3} + 48 T^{2} - 349776 T - 44046592 \) Copy content Toggle raw display
$71$ \( T^{3} + 456 T^{2} + \cdots - 19646112 \) Copy content Toggle raw display
$73$ \( T^{3} - 1230 T^{2} + \cdots + 640250616 \) Copy content Toggle raw display
$79$ \( T^{3} + 1368 T^{2} + \cdots - 182717824 \) Copy content Toggle raw display
$83$ \( T^{3} + 60 T^{2} - 675600 T - 90712000 \) Copy content Toggle raw display
$89$ \( T^{3} - 2742 T^{2} + \cdots - 524337504 \) Copy content Toggle raw display
$97$ \( T^{3} - 1950 T^{2} + \cdots - 50085448 \) Copy content Toggle raw display
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