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Properties

Label 144.4.i.d
Level $144$
Weight $4$
Character orbit 144.i
Analytic conductor $8.496$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,4,Mod(49,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 144.i (of order \(3\), 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: \(8.49627504083\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.6831243.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 13x^{4} + 49x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{3} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{5} + ( - 2 \beta_{5} + \beta_{4} - 3 \beta_{2} + \cdots + 4) q^{7} + ( - 3 \beta_{5} + \beta_{3} - \beta_{2} + \cdots + 6) q^{9}+ \cdots + (9 \beta_{5} + 54 \beta_{4} + \cdots + 225) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 6 q^{5} + 6 q^{7} + 39 q^{9} - 51 q^{11} + 12 q^{13} + 180 q^{15} - 222 q^{17} - 30 q^{19} - 120 q^{21} - 210 q^{23} - 3 q^{25} - 648 q^{27} + 456 q^{29} - 48 q^{31} - 603 q^{33} + 1104 q^{35}+ \cdots + 1854 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 13x^{4} + 49x^{2} + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 9\nu^{3} + 17\nu + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 5\nu^{3} + 20\nu^{2} + 15\nu - 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 4\nu^{4} + 13\nu^{3} + 44\nu^{2} + 25\nu + 100 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 9\nu^{3} + 12\nu^{2} - 5\nu + 52 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{4} + 3\nu^{3} - 21\nu^{2} + 18\nu - 20 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} + 4\beta _1 - 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta_{2} - 26 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - 3\beta_{4} + 4\beta_{3} - \beta_{2} - 11\beta _1 + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{5} - 7\beta_{4} - 5\beta_{3} + 11\beta_{2} + 2\beta _1 + 138 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -18\beta_{5} + 37\beta_{4} - 55\beta_{3} + \beta_{2} + 178\beta _1 - 62 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
− 2.63162i
2.13353i
− 1.23396i
2.63162i
− 2.13353i
1.23396i
0 −5.16718 + 0.547914i 0 2.44901 − 4.24182i 0 −5.32725 − 9.22708i 0 26.3996 − 5.66234i 0
49.2 0 2.51979 − 4.54430i 0 −6.37096 + 11.0348i 0 −7.02674 − 12.1707i 0 −14.3013 − 22.9014i 0
49.3 0 4.14739 + 3.13036i 0 6.92194 − 11.9892i 0 15.3540 + 26.5939i 0 7.40171 + 25.9656i 0
97.1 0 −5.16718 − 0.547914i 0 2.44901 + 4.24182i 0 −5.32725 + 9.22708i 0 26.3996 + 5.66234i 0
97.2 0 2.51979 + 4.54430i 0 −6.37096 − 11.0348i 0 −7.02674 + 12.1707i 0 −14.3013 + 22.9014i 0
97.3 0 4.14739 − 3.13036i 0 6.92194 + 11.9892i 0 15.3540 − 26.5939i 0 7.40171 − 25.9656i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.4.i.d 6
3.b odd 2 1 432.4.i.d 6
4.b odd 2 1 36.4.e.a ✓ 6
9.c even 3 1 inner 144.4.i.d 6
9.c even 3 1 1296.4.a.v 3
9.d odd 6 1 432.4.i.d 6
9.d odd 6 1 1296.4.a.w 3
12.b even 2 1 108.4.e.a 6
36.f odd 6 1 36.4.e.a ✓ 6
36.f odd 6 1 324.4.a.c 3
36.h even 6 1 108.4.e.a 6
36.h even 6 1 324.4.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.e.a ✓ 6 4.b odd 2 1
36.4.e.a ✓ 6 36.f odd 6 1
108.4.e.a 6 12.b even 2 1
108.4.e.a 6 36.h even 6 1
144.4.i.d 6 1.a even 1 1 trivial
144.4.i.d 6 9.c even 3 1 inner
324.4.a.c 3 36.f odd 6 1
324.4.a.d 3 36.h even 6 1
432.4.i.d 6 3.b odd 2 1
432.4.i.d 6 9.d odd 6 1
1296.4.a.v 3 9.c even 3 1
1296.4.a.w 3 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 6T_{5}^{5} + 207T_{5}^{4} - 702T_{5}^{3} + 34425T_{5}^{2} - 147744T_{5} + 746496 \) acting on \(S_{4}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{6} - 6 T^{5} + \cdots + 746496 \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} + \cdots + 21141604 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 4386545361 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 4155865156 \) Copy content Toggle raw display
$17$ \( (T^{3} + 111 T^{2} + \cdots - 577476)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 15 T^{2} + \cdots + 216368)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 210 T^{5} + \cdots + 5391684 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 8292430196964 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 9331729724944 \) Copy content Toggle raw display
$37$ \( (T^{3} + 48 T^{2} + \cdots - 682352)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 139158426750849 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 2031049672201 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 41\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{3} + 1104 T^{2} + \cdots + 11853648)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 93\!\cdots\!69 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 1463461189696 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 9579414973969 \) Copy content Toggle raw display
$71$ \( (T^{3} + 60 T^{2} + \cdots - 113211648)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 375 T^{2} + \cdots + 158369284)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 318578661907984 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 12101965948944 \) Copy content Toggle raw display
$89$ \( (T^{3} + 462 T^{2} + \cdots + 170122248)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 74\!\cdots\!29 \) Copy content Toggle raw display
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