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Properties

Label 144.3.o.a
Level $144$
Weight $3$
Character orbit 144.o
Analytic conductor $3.924$
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] = [144,3,Mod(31,144)]
 
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([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.o (of order \(6\), degree \(2\), 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: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{5}+ \cdots + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{5}+ \cdots + ( - 6 \beta_{7} - 3 \beta_{6} + \cdots + 87) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{3} + 3 q^{5} + 3 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{3} + 3 q^{5} + 3 q^{7} - 3 q^{9} + 18 q^{11} + 5 q^{13} - 21 q^{15} + 6 q^{17} - 33 q^{21} - 81 q^{23} - 23 q^{25} + 108 q^{27} + 69 q^{29} + 45 q^{31} + 72 q^{33} - 20 q^{37} - 141 q^{39} + 54 q^{41} - 117 q^{45} + 207 q^{47} + 41 q^{49} - 141 q^{51} - 252 q^{53} - 273 q^{57} - 306 q^{59} + 7 q^{61} + 441 q^{63} + 93 q^{65} + 12 q^{67} + 189 q^{69} + 74 q^{73} - 387 q^{75} + 207 q^{77} + 33 q^{79} + 117 q^{81} + 549 q^{83} - 30 q^{85} - 87 q^{87} - 168 q^{89} - 27 q^{93} - 684 q^{95} - 10 q^{97} + 585 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 7\nu^{3} + 10\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 10\nu^{5} + 4\nu^{4} + 31\nu^{3} + 22\nu^{2} + 30\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 11\nu^{5} - 2\nu^{4} + 32\nu^{3} - 14\nu^{2} + 16\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 12\nu^{5} + 20\nu^{4} + 39\nu^{3} + 50\nu^{2} + 14\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 11\nu^{5} + 34\nu^{4} - 32\nu^{3} + 70\nu^{2} - 16\nu + 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{6} + 11\nu^{5} - 10\nu^{4} + 35\nu^{3} - 28\nu^{2} + 22\nu - 14 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} + 31\nu^{5} + 88\nu^{3} + 6\nu^{2} + 52\nu + 16 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} - \beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{3} - \beta_{2} + \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{6} + \beta_{5} + \beta_{4} - 5\beta_{3} - \beta_{2} - 2\beta _1 - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 11\beta_{3} + 7\beta_{2} - 6\beta _1 + 48 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -16\beta_{6} - 7\beta_{5} - 2\beta_{4} + 25\beta_{3} + 2\beta_{2} + 16\beta _1 + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 34\beta_{7} + 8\beta_{6} + 11\beta_{5} - 8\beta_{4} - 57\beta_{3} - 42\beta_{2} + 34\beta _1 - 244 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2\beta_{7} + 86\beta_{6} + 43\beta_{5} - 125\beta_{3} + 2\beta_{2} - 126\beta _1 - 20 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.07834i
0.385731i
− 2.06288i
2.33086i
− 1.07834i
− 0.385731i
2.06288i
− 2.33086i
0 −2.64956 − 1.40707i 0 −3.01729 + 5.22611i 0 10.2332 − 5.90815i 0 5.04032 + 7.45622i 0
31.2 0 −1.28651 + 2.71015i 0 −0.454613 + 0.787412i 0 −6.10709 + 3.52593i 0 −5.68980 − 6.97325i 0
31.3 0 −0.456412 − 2.96508i 0 4.61660 − 7.99619i 0 −5.33093 + 3.07781i 0 −8.58338 + 2.70659i 0
31.4 0 2.89248 + 0.795973i 0 0.355304 − 0.615405i 0 2.70480 − 1.56162i 0 7.73285 + 4.60467i 0
79.1 0 −2.64956 + 1.40707i 0 −3.01729 − 5.22611i 0 10.2332 + 5.90815i 0 5.04032 − 7.45622i 0
79.2 0 −1.28651 − 2.71015i 0 −0.454613 − 0.787412i 0 −6.10709 − 3.52593i 0 −5.68980 + 6.97325i 0
79.3 0 −0.456412 + 2.96508i 0 4.61660 + 7.99619i 0 −5.33093 − 3.07781i 0 −8.58338 − 2.70659i 0
79.4 0 2.89248 − 0.795973i 0 0.355304 + 0.615405i 0 2.70480 + 1.56162i 0 7.73285 − 4.60467i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.o.a ✓ 8
3.b odd 2 1 432.3.o.b 8
4.b odd 2 1 144.3.o.c yes 8
8.b even 2 1 576.3.o.f 8
8.d odd 2 1 576.3.o.d 8
9.c even 3 1 144.3.o.c yes 8
9.c even 3 1 1296.3.g.j 8
9.d odd 6 1 432.3.o.a 8
9.d odd 6 1 1296.3.g.k 8
12.b even 2 1 432.3.o.a 8
24.f even 2 1 1728.3.o.e 8
24.h odd 2 1 1728.3.o.f 8
36.f odd 6 1 inner 144.3.o.a ✓ 8
36.f odd 6 1 1296.3.g.j 8
36.h even 6 1 432.3.o.b 8
36.h even 6 1 1296.3.g.k 8
72.j odd 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.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 1.a even 1 1 trivial
144.3.o.a ✓ 8 36.f odd 6 1 inner
144.3.o.c yes 8 4.b odd 2 1
144.3.o.c yes 8 9.c even 3 1
432.3.o.a 8 9.d odd 6 1
432.3.o.a 8 12.b even 2 1
432.3.o.b 8 3.b odd 2 1
432.3.o.b 8 36.h even 6 1
576.3.o.d 8 8.d odd 2 1
576.3.o.d 8 72.n even 6 1
576.3.o.f 8 8.b even 2 1
576.3.o.f 8 72.p odd 6 1
1296.3.g.j 8 9.c even 3 1
1296.3.g.j 8 36.f odd 6 1
1296.3.g.k 8 9.d odd 6 1
1296.3.g.k 8 36.h even 6 1
1728.3.o.e 8 24.f even 2 1
1728.3.o.e 8 72.j odd 6 1
1728.3.o.f 8 24.h odd 2 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 \(S_{3}^{\mathrm{new}}(144, [\chi])\):

\( T_{5}^{8} - 3T_{5}^{7} + 66T_{5}^{6} + 189T_{5}^{5} + 3186T_{5}^{4} + 729T_{5}^{3} + 2133T_{5}^{2} - 324T_{5} + 1296 \) Copy content Toggle raw display
\( T_{7}^{8} - 3T_{7}^{7} - 114T_{7}^{6} + 351T_{7}^{5} + 12366T_{7}^{4} + 32643T_{7}^{3} - 161487T_{7}^{2} - 446958T_{7} + 2566404 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 3 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} - 3 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 2566404 \) Copy content Toggle raw display
$11$ \( T^{8} - 18 T^{7} + \cdots + 12131289 \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{7} + \cdots + 10201636 \) 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
$23$ \( T^{8} + 81 T^{7} + \cdots + 19131876 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 4046639163876 \) Copy content Toggle raw display
$31$ \( T^{8} - 45 T^{7} + \cdots + 944784 \) Copy content Toggle raw display
$37$ \( (T^{4} + 10 T^{3} + \cdots - 613568)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 5431756955769 \) 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
$61$ \( T^{8} + \cdots + 309954973696 \) 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
$97$ \( T^{8} + \cdots + 30429664983481 \) Copy content Toggle raw display
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