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Properties

Label 1296.3.g.j
Level $1296$
Weight $3$
Character orbit 1296.g
Analytic conductor $35.313$
Analytic rank $0$
Dimension $8$
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] = [1296,3,Mod(1135,1296)]
 
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([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.1135");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.g (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(35.3134422611\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.856615824.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{5} + \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{5} + \beta_{6} q^{7} + (\beta_{6} + \beta_{2} - \beta_1) q^{11} + (\beta_{4} + \beta_{3} - 1) q^{13} + ( - \beta_{7} - \beta_{4} + 1) q^{17} + (\beta_{6} + 3 \beta_{5} - \beta_{2}) q^{19} + (\beta_{5} + 3 \beta_{2} + \beta_1) q^{23} + ( - \beta_{7} - 2 \beta_{4} + 4 \beta_{3} + 7) q^{25} + (3 \beta_{4} + \beta_{3} - 17) q^{29} + (2 \beta_{6} + 3 \beta_{5} + \cdots - 3 \beta_1) q^{31}+ \cdots + (4 \beta_{7} + 3 \beta_{4} + 6 \beta_{3} + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{5} - 10 q^{13} + 6 q^{17} + 46 q^{25} - 138 q^{29} - 20 q^{37} - 108 q^{41} - 82 q^{49} - 252 q^{53} - 14 q^{61} - 186 q^{65} + 74 q^{73} - 414 q^{77} + 60 q^{85} - 168 q^{89} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{5} + 7\nu^{3} + 10\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{6} + 8\nu^{4} + 14\nu^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{6} - 5\nu^{4} - 2\nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} + 30\nu^{5} + 87\nu^{3} + 66\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} - 32\nu^{5} - 95\nu^{3} - 50\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 3\nu^{6} + 30\nu^{4} + 84\nu^{2} + 43 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{2} - \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{4} - 5\beta_{3} - 51 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{6} - 2\beta_{5} - 2\beta_{2} + 3\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{7} + 7\beta_{4} + 13\beta_{3} + 114 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{6} + 9\beta_{5} + 12\beta_{2} - 16\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{7} - 14\beta_{4} - 20\beta_{3} - 185 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -43\beta_{6} - 41\beta_{5} - 73\beta_{2} + 84\beta_1 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1135.1
− 2.06288i
2.06288i
− 2.33086i
2.33086i
0.385731i
− 0.385731i
− 1.07834i
1.07834i
0 0 0 −9.23321 0 − 6.15562i 0 0 0
1135.2 0 0 0 −9.23321 0 6.15562i 0 0 0
1135.3 0 0 0 −0.710609 0 − 3.12324i 0 0 0
1135.4 0 0 0 −0.710609 0 3.12324i 0 0 0
1135.5 0 0 0 0.909226 0 − 7.05186i 0 0 0
1135.6 0 0 0 0.909226 0 7.05186i 0 0 0
1135.7 0 0 0 6.03459 0 − 11.8163i 0 0 0
1135.8 0 0 0 6.03459 0 11.8163i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1135.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.g.j 8
3.b odd 2 1 1296.3.g.k 8
4.b odd 2 1 inner 1296.3.g.j 8
9.c even 3 1 144.3.o.a ✓ 8
9.c even 3 1 144.3.o.c yes 8
9.d odd 6 1 432.3.o.a 8
9.d odd 6 1 432.3.o.b 8
12.b even 2 1 1296.3.g.k 8
36.f odd 6 1 144.3.o.a ✓ 8
36.f odd 6 1 144.3.o.c yes 8
36.h even 6 1 432.3.o.a 8
36.h even 6 1 432.3.o.b 8
72.j odd 6 1 1728.3.o.e 8
72.j odd 6 1 1728.3.o.f 8
72.l even 6 1 1728.3.o.e 8
72.l even 6 1 1728.3.o.f 8
72.n even 6 1 576.3.o.d 8
72.n even 6 1 576.3.o.f 8
72.p odd 6 1 576.3.o.d 8
72.p odd 6 1 576.3.o.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.o.a ✓ 8 9.c even 3 1
144.3.o.a ✓ 8 36.f odd 6 1
144.3.o.c yes 8 9.c even 3 1
144.3.o.c yes 8 36.f odd 6 1
432.3.o.a 8 9.d odd 6 1
432.3.o.a 8 36.h even 6 1
432.3.o.b 8 9.d odd 6 1
432.3.o.b 8 36.h even 6 1
576.3.o.d 8 72.n even 6 1
576.3.o.d 8 72.p odd 6 1
576.3.o.f 8 72.n even 6 1
576.3.o.f 8 72.p odd 6 1
1296.3.g.j 8 1.a even 1 1 trivial
1296.3.g.j 8 4.b odd 2 1 inner
1296.3.g.k 8 3.b odd 2 1
1296.3.g.k 8 12.b even 2 1
1728.3.o.e 8 72.j odd 6 1
1728.3.o.e 8 72.l even 6 1
1728.3.o.f 8 72.j odd 6 1
1728.3.o.f 8 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5}^{4} + 3T_{5}^{3} - 57T_{5}^{2} + 9T_{5} + 36 \) Copy content Toggle raw display
\( T_{17}^{4} - 3T_{17}^{3} - 822T_{17}^{2} + 1908T_{17} + 84168 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 3 T^{3} - 57 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 237 T^{6} + \cdots + 2566404 \) Copy content Toggle raw display
$11$ \( T^{8} + 396 T^{6} + \cdots + 12131289 \) Copy content Toggle raw display
$13$ \( (T^{4} + 5 T^{3} + \cdots - 3194)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 3 T^{3} + \cdots + 84168)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 2931572736 \) Copy content Toggle raw display
$23$ \( T^{8} + 1053 T^{6} + \cdots + 19131876 \) Copy content Toggle raw display
$29$ \( (T^{4} + 69 T^{3} + \cdots - 2011626)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 2673 T^{6} + \cdots + 944784 \) Copy content Toggle raw display
$37$ \( (T^{4} + 10 T^{3} + \cdots - 613568)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 54 T^{3} + \cdots - 2330613)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 29016737649 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 28643839776036 \) Copy content Toggle raw display
$53$ \( (T^{4} + 126 T^{3} + \cdots - 6508512)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 48359409452649 \) Copy content Toggle raw display
$61$ \( (T^{4} + 7 T^{3} + \cdots - 556736)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 68036119056801 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 726110197530624 \) Copy content Toggle raw display
$73$ \( (T^{4} - 37 T^{3} + \cdots + 416536)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 240627852449856 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{4} + 84 T^{3} + \cdots - 1161936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 10 T^{3} + \cdots - 5516309)^{2} \) Copy content Toggle raw display
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