Properties

Label 144.4.i.a
Level $144$
Weight $4$
Character orbit 144.i
Analytic conductor $8.496$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 6 \zeta_{6} + 3) q^{3} + 9 \zeta_{6} q^{5} + (31 \zeta_{6} - 31) q^{7} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 6 \zeta_{6} + 3) q^{3} + 9 \zeta_{6} q^{5} + (31 \zeta_{6} - 31) q^{7} - 27 q^{9} + (15 \zeta_{6} - 15) q^{11} + 37 \zeta_{6} q^{13} + ( - 27 \zeta_{6} + 54) q^{15} - 42 q^{17} + 28 q^{19} + (93 \zeta_{6} + 93) q^{21} + 195 \zeta_{6} q^{23} + ( - 44 \zeta_{6} + 44) q^{25} + (162 \zeta_{6} - 81) q^{27} + (111 \zeta_{6} - 111) q^{29} - 205 \zeta_{6} q^{31} + (45 \zeta_{6} + 45) q^{33} - 279 q^{35} - 166 q^{37} + ( - 111 \zeta_{6} + 222) q^{39} + 261 \zeta_{6} q^{41} + (43 \zeta_{6} - 43) q^{43} - 243 \zeta_{6} q^{45} + ( - 177 \zeta_{6} + 177) q^{47} - 618 \zeta_{6} q^{49} + (252 \zeta_{6} - 126) q^{51} + 114 q^{53} - 135 q^{55} + ( - 168 \zeta_{6} + 84) q^{57} + 159 \zeta_{6} q^{59} + (191 \zeta_{6} - 191) q^{61} + ( - 837 \zeta_{6} + 837) q^{63} + (333 \zeta_{6} - 333) q^{65} - 421 \zeta_{6} q^{67} + ( - 585 \zeta_{6} + 1170) q^{69} - 156 q^{71} + 182 q^{73} + ( - 132 \zeta_{6} - 132) q^{75} - 465 \zeta_{6} q^{77} + ( - 1133 \zeta_{6} + 1133) q^{79} + 729 q^{81} + (1083 \zeta_{6} - 1083) q^{83} - 378 \zeta_{6} q^{85} + (333 \zeta_{6} + 333) q^{87} - 1050 q^{89} - 1147 q^{91} + (615 \zeta_{6} - 1230) q^{93} + 252 \zeta_{6} q^{95} + ( - 901 \zeta_{6} + 901) q^{97} + ( - 405 \zeta_{6} + 405) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{5} - 31 q^{7} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 9 q^{5} - 31 q^{7} - 54 q^{9} - 15 q^{11} + 37 q^{13} + 81 q^{15} - 84 q^{17} + 56 q^{19} + 279 q^{21} + 195 q^{23} + 44 q^{25} - 111 q^{29} - 205 q^{31} + 135 q^{33} - 558 q^{35} - 332 q^{37} + 333 q^{39} + 261 q^{41} - 43 q^{43} - 243 q^{45} + 177 q^{47} - 618 q^{49} + 228 q^{53} - 270 q^{55} + 159 q^{59} - 191 q^{61} + 837 q^{63} - 333 q^{65} - 421 q^{67} + 1755 q^{69} - 312 q^{71} + 364 q^{73} - 396 q^{75} - 465 q^{77} + 1133 q^{79} + 1458 q^{81} - 1083 q^{83} - 378 q^{85} + 999 q^{87} - 2100 q^{89} - 2294 q^{91} - 1845 q^{93} + 252 q^{95} + 901 q^{97} + 405 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 5.19615i 0 4.50000 7.79423i 0 −15.5000 26.8468i 0 −27.0000 0
97.1 0 5.19615i 0 4.50000 + 7.79423i 0 −15.5000 + 26.8468i 0 −27.0000 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
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.a 2
3.b odd 2 1 432.4.i.a 2
4.b odd 2 1 18.4.c.a 2
9.c even 3 1 inner 144.4.i.a 2
9.c even 3 1 1296.4.a.b 1
9.d odd 6 1 432.4.i.a 2
9.d odd 6 1 1296.4.a.g 1
12.b even 2 1 54.4.c.a 2
36.f odd 6 1 18.4.c.a 2
36.f odd 6 1 162.4.a.d 1
36.h even 6 1 54.4.c.a 2
36.h even 6 1 162.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.4.c.a 2 4.b odd 2 1
18.4.c.a 2 36.f odd 6 1
54.4.c.a 2 12.b even 2 1
54.4.c.a 2 36.h even 6 1
144.4.i.a 2 1.a even 1 1 trivial
144.4.i.a 2 9.c even 3 1 inner
162.4.a.a 1 36.h even 6 1
162.4.a.d 1 36.f odd 6 1
432.4.i.a 2 3.b odd 2 1
432.4.i.a 2 9.d odd 6 1
1296.4.a.b 1 9.c even 3 1
1296.4.a.g 1 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 9T_{5} + 81 \) acting on \(S_{4}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 27 \) Copy content Toggle raw display
$5$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$7$ \( T^{2} + 31T + 961 \) Copy content Toggle raw display
$11$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$13$ \( T^{2} - 37T + 1369 \) Copy content Toggle raw display
$17$ \( (T + 42)^{2} \) Copy content Toggle raw display
$19$ \( (T - 28)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 195T + 38025 \) Copy content Toggle raw display
$29$ \( T^{2} + 111T + 12321 \) Copy content Toggle raw display
$31$ \( T^{2} + 205T + 42025 \) Copy content Toggle raw display
$37$ \( (T + 166)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 261T + 68121 \) Copy content Toggle raw display
$43$ \( T^{2} + 43T + 1849 \) Copy content Toggle raw display
$47$ \( T^{2} - 177T + 31329 \) Copy content Toggle raw display
$53$ \( (T - 114)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 159T + 25281 \) Copy content Toggle raw display
$61$ \( T^{2} + 191T + 36481 \) Copy content Toggle raw display
$67$ \( T^{2} + 421T + 177241 \) Copy content Toggle raw display
$71$ \( (T + 156)^{2} \) Copy content Toggle raw display
$73$ \( (T - 182)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 1133 T + 1283689 \) Copy content Toggle raw display
$83$ \( T^{2} + 1083 T + 1172889 \) Copy content Toggle raw display
$89$ \( (T + 1050)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 901T + 811801 \) Copy content Toggle raw display
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