Properties

Label 1296.4.a.x
Level $1296$
Weight $4$
Character orbit 1296.a
Self dual yes
Analytic conductor $76.466$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1296,4,Mod(1,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1296.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.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: \(76.4664753674\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.29952.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 8x^{2} + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 648)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{5} + (\beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{11} + (2 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{13} + ( - \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 4) q^{17} + ( - 5 \beta_{2} - 3 \beta_1 - 20) q^{19} + ( - 7 \beta_{3} + 6 \beta_{2} + 5 \beta_1 + 50) q^{23} + ( - 16 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 2) q^{25} + (4 \beta_{3} - 4 \beta_{2} + 9 \beta_1 - 54) q^{29} + ( - 16 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 20) q^{31} + (7 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 102) q^{35} + (22 \beta_{3} - 17 \beta_{2} - \beta_1 - 69) q^{37} + ( - 20 \beta_{3} + 4 \beta_{2} + 12 \beta_1 - 96) q^{41} + (56 \beta_{3} - 3 \beta_{2} + 19 \beta_1 - 40) q^{43} + (12 \beta_{3} + 16 \beta_1 + 192) q^{47} + (44 \beta_{3} + 8 \beta_{2} - 4 \beta_1 - 67) q^{49} + (34 \beta_{3} - 24 \beta_{2} + 10 \beta_1 - 236) q^{53} + ( - 32 \beta_{3} + \beta_{2} - \beta_1 - 76) q^{55} + ( - 28 \beta_{3} - 24 \beta_{2} - 12 \beta_1 + 248) q^{59} + ( - 114 \beta_{3} + \beta_{2} - 3 \beta_1 - 137) q^{61} + (21 \beta_{3} - 6 \beta_1 - 332) q^{65} + ( - 96 \beta_{3} + 21 \beta_{2} + 19 \beta_1 - 116) q^{67} + (43 \beta_{3} - 6 \beta_{2} + 27 \beta_1 + 430) q^{71} + (36 \beta_{3} + 40 \beta_{2} - 44 \beta_1 - 191) q^{73} + ( - 84 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 432) q^{77} + (160 \beta_{3} + 23 \beta_{2} - 23 \beta_1 - 172) q^{79} + (6 \beta_{3} + 20 \beta_{2} - 38 \beta_1 + 532) q^{83} + (126 \beta_{3} - 7 \beta_{2} - 11 \beta_1 - 331) q^{85} + (55 \beta_{3} + 56 \beta_{2} - 40 \beta_1 - 528) q^{89} + ( - 64 \beta_{3} - 3 \beta_{2} - 5 \beta_1 - 444) q^{91} + ( - 163 \beta_{3} + 6 \beta_{2} + 37 \beta_1 + 514) q^{95} + ( - 204 \beta_{3} + 4 \beta_{2} - 40 \beta_1 - 454) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{5} + 8 q^{11} + 4 q^{13} - 16 q^{17} - 80 q^{19} + 200 q^{23} + 8 q^{25} - 216 q^{29} - 80 q^{31} + 408 q^{35} - 276 q^{37} - 384 q^{41} - 160 q^{43} + 768 q^{47} - 268 q^{49} - 944 q^{53} - 304 q^{55} + 992 q^{59} - 548 q^{61} - 1328 q^{65} - 464 q^{67} + 1720 q^{71} - 764 q^{73} - 1728 q^{77} - 688 q^{79} + 2128 q^{83} - 1324 q^{85} - 2112 q^{89} - 1776 q^{91} + 2056 q^{95} - 1816 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 8x^{2} + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{3} - \nu^{2} - 14\nu + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{3} + \nu^{2} - 16\nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} - 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} - 2\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{3} + 7\beta_{2} - 8\beta_1 ) / 12 \) 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
−1.50597
−2.39417
2.39417
1.50597
0 0 0 −17.9847 0 −7.28309 0 0 0
1.2 0 0 0 −6.33932 0 −19.1946 0 0 0
1.3 0 0 0 5.80342 0 26.1228 0 0 0
1.4 0 0 0 10.5206 0 0.354888 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1296.4.a.x 4
3.b odd 2 1 1296.4.a.bb 4
4.b odd 2 1 648.4.a.g 4
12.b even 2 1 648.4.a.j yes 4
36.f odd 6 2 648.4.i.v 8
36.h even 6 2 648.4.i.u 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.g 4 4.b odd 2 1
648.4.a.j yes 4 12.b even 2 1
648.4.i.u 8 36.h even 6 2
648.4.i.v 8 36.f odd 6 2
1296.4.a.x 4 1.a even 1 1 trivial
1296.4.a.bb 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 8T_{5}^{3} - 222T_{5}^{2} - 376T_{5} + 6961 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1296))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 8 T^{3} - 222 T^{2} + \cdots + 6961 \) Copy content Toggle raw display
$7$ \( T^{4} - 552 T^{2} - 3456 T + 1296 \) Copy content Toggle raw display
$11$ \( T^{4} - 8 T^{3} - 1560 T^{2} + \cdots + 61072 \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} - 2274 T^{2} + \cdots + 1237417 \) Copy content Toggle raw display
$17$ \( T^{4} + 16 T^{3} - 7782 T^{2} + \cdots + 14319721 \) Copy content Toggle raw display
$19$ \( T^{4} + 80 T^{3} - 10824 T^{2} + \cdots + 31790992 \) Copy content Toggle raw display
$23$ \( T^{4} - 200 T^{3} + \cdots + 48919312 \) Copy content Toggle raw display
$29$ \( T^{4} + 216 T^{3} + \cdots + 54863217 \) Copy content Toggle raw display
$31$ \( T^{4} + 80 T^{3} - 15936 T^{2} + \cdots + 48805888 \) Copy content Toggle raw display
$37$ \( T^{4} + 276 T^{3} + \cdots + 948334041 \) Copy content Toggle raw display
$41$ \( T^{4} + 384 T^{3} + \cdots - 2045323008 \) Copy content Toggle raw display
$43$ \( T^{4} + 160 T^{3} + \cdots + 513055504 \) Copy content Toggle raw display
$47$ \( T^{4} - 768 T^{3} + \cdots - 1142228736 \) Copy content Toggle raw display
$53$ \( T^{4} + 944 T^{3} + \cdots - 22310316032 \) Copy content Toggle raw display
$59$ \( T^{4} - 992 T^{3} + \cdots + 593268736 \) Copy content Toggle raw display
$61$ \( T^{4} + 548 T^{3} + \cdots + 104164416841 \) Copy content Toggle raw display
$67$ \( T^{4} + 464 T^{3} + \cdots - 30700073072 \) Copy content Toggle raw display
$71$ \( T^{4} - 1720 T^{3} + \cdots - 9807661424 \) Copy content Toggle raw display
$73$ \( T^{4} + 764 T^{3} + \cdots - 194677777919 \) Copy content Toggle raw display
$79$ \( T^{4} + 688 T^{3} + \cdots + 320004558736 \) Copy content Toggle raw display
$83$ \( T^{4} - 2128 T^{3} + \cdots + 6134511616 \) Copy content Toggle raw display
$89$ \( T^{4} + 2112 T^{3} + \cdots - 1312975087623 \) Copy content Toggle raw display
$97$ \( T^{4} + 1816 T^{3} + \cdots + 2989754128 \) Copy content Toggle raw display
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