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Properties

Label 1296.3.q.g
Level $1296$
Weight $3$
Character orbit 1296.q
Analytic conductor $35.313$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1296,3,Mod(593,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.593");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.q (of order \(6\), degree \(2\), not 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - 3 \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} - 3 \beta_{2} q^{7} + ( - 7 \beta_{3} + 7 \beta_1) q^{11} + (7 \beta_{2} - 7) q^{13} - 5 \beta_{3} q^{17} + 19 q^{19} - \beta_1 q^{23} - 17 \beta_{2} q^{25} + (18 \beta_{3} - 18 \beta_1) q^{29} + (10 \beta_{2} - 10) q^{31} - 3 \beta_{3} q^{35} + 63 q^{37} - 18 \beta_1 q^{41} - 50 \beta_{2} q^{43} + ( - 15 \beta_{3} + 15 \beta_1) q^{47} + ( - 40 \beta_{2} + 40) q^{49} + 26 \beta_{3} q^{53} + 56 q^{55} + 35 \beta_1 q^{59} - 79 \beta_{2} q^{61} + (7 \beta_{3} - 7 \beta_1) q^{65} + ( - 77 \beta_{2} + 77) q^{67} - 28 \beta_{3} q^{71} - 17 q^{73} - 21 \beta_1 q^{77} - 11 \beta_{2} q^{79} + ( - 14 \beta_{3} + 14 \beta_1) q^{83} + ( - 40 \beta_{2} + 40) q^{85} + 15 \beta_{3} q^{89} + 21 q^{91} + 19 \beta_1 q^{95} + 97 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{7} - 14 q^{13} + 76 q^{19} - 34 q^{25} - 20 q^{31} + 252 q^{37} - 100 q^{43} + 80 q^{49} + 224 q^{55} - 158 q^{61} + 154 q^{67} - 68 q^{73} - 22 q^{79} + 80 q^{85} + 84 q^{91} + 194 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
−1.22474 + 0.707107i
1.22474 − 0.707107i
−1.22474 − 0.707107i
1.22474 + 0.707107i
0 0 0 −2.44949 + 1.41421i 0 −1.50000 + 2.59808i 0 0 0
593.2 0 0 0 2.44949 − 1.41421i 0 −1.50000 + 2.59808i 0 0 0
1025.1 0 0 0 −2.44949 − 1.41421i 0 −1.50000 − 2.59808i 0 0 0
1025.2 0 0 0 2.44949 + 1.41421i 0 −1.50000 − 2.59808i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.q.g 4
3.b odd 2 1 inner 1296.3.q.g 4
4.b odd 2 1 648.3.m.c 4
9.c even 3 1 432.3.e.f 2
9.c even 3 1 inner 1296.3.q.g 4
9.d odd 6 1 432.3.e.f 2
9.d odd 6 1 inner 1296.3.q.g 4
12.b even 2 1 648.3.m.c 4
36.f odd 6 1 216.3.e.a ✓ 2
36.f odd 6 1 648.3.m.c 4
36.h even 6 1 216.3.e.a ✓ 2
36.h even 6 1 648.3.m.c 4
72.j odd 6 1 1728.3.e.j 2
72.l even 6 1 1728.3.e.h 2
72.n even 6 1 1728.3.e.j 2
72.p odd 6 1 1728.3.e.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.e.a ✓ 2 36.f odd 6 1
216.3.e.a ✓ 2 36.h even 6 1
432.3.e.f 2 9.c even 3 1
432.3.e.f 2 9.d odd 6 1
648.3.m.c 4 4.b odd 2 1
648.3.m.c 4 12.b even 2 1
648.3.m.c 4 36.f odd 6 1
648.3.m.c 4 36.h even 6 1
1296.3.q.g 4 1.a even 1 1 trivial
1296.3.q.g 4 3.b odd 2 1 inner
1296.3.q.g 4 9.c even 3 1 inner
1296.3.q.g 4 9.d odd 6 1 inner
1728.3.e.h 2 72.l even 6 1
1728.3.e.h 2 72.p odd 6 1
1728.3.e.j 2 72.j odd 6 1
1728.3.e.j 2 72.n even 6 1

Hecke kernels

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

\( T_{5}^{4} - 8T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{2} + 3T_{7} + 9 \) 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
$5$ \( T^{4} - 8T^{2} + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 392 T^{2} + 153664 \) Copy content Toggle raw display
$13$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 200)^{2} \) Copy content Toggle raw display
$19$ \( (T - 19)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 8T^{2} + 64 \) Copy content Toggle raw display
$29$ \( T^{4} - 2592 T^{2} + 6718464 \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$37$ \( (T - 63)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 2592 T^{2} + 6718464 \) Copy content Toggle raw display
$43$ \( (T^{2} + 50 T + 2500)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 1800 T^{2} + 3240000 \) Copy content Toggle raw display
$53$ \( (T^{2} + 5408)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 9800 T^{2} + 96040000 \) Copy content Toggle raw display
$61$ \( (T^{2} + 79 T + 6241)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 77 T + 5929)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 6272)^{2} \) Copy content Toggle raw display
$73$ \( (T + 17)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 1568 T^{2} + 2458624 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1800)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 97 T + 9409)^{2} \) Copy content Toggle raw display
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