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Properties

Label 144.5.q.a
Level $144$
Weight $5$
Character orbit 144.q
Analytic conductor $14.885$
Analytic rank $0$
Dimension $6$
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] = [144,5,Mod(65,144)]
 
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, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 144.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: \(14.8852746841\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.39400128.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 9)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 3 \beta_{3} + \beta_{2} - 1) q^{3} + ( - 3 \beta_{4} + \beta_{2}) q^{5} + (2 \beta_{5} - 4 \beta_{4} + 5 \beta_{3} + \cdots + 7) q^{7}+ \cdots + (3 \beta_{5} - 45 \beta_{3} + 6 \beta_{2} + \cdots + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - 3 \beta_{3} + \beta_{2} - 1) q^{3} + ( - 3 \beta_{4} + \beta_{2}) q^{5} + (2 \beta_{5} - 4 \beta_{4} + 5 \beta_{3} + \cdots + 7) q^{7}+ \cdots + ( - 165 \beta_{5} - 3 \beta_{4} + \cdots + 2625) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 12 q^{5} - 12 q^{7} + 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 12 q^{5} - 12 q^{7} + 99 q^{9} - 483 q^{11} - 6 q^{13} + 1026 q^{15} + 258 q^{19} + 480 q^{21} + 282 q^{23} - 273 q^{25} - 54 q^{27} - 1056 q^{29} - 1290 q^{31} + 279 q^{33} + 12 q^{37} - 1974 q^{39} + 7629 q^{41} + 285 q^{43} - 4212 q^{45} + 9642 q^{47} - 1863 q^{49} - 2457 q^{51} - 2016 q^{55} + 5367 q^{57} - 6225 q^{59} + 3630 q^{61} + 7578 q^{63} - 7158 q^{65} + 5055 q^{67} - 13878 q^{69} - 14622 q^{73} - 21021 q^{75} + 2580 q^{77} - 4764 q^{79} + 18387 q^{81} + 1866 q^{83} + 12366 q^{85} + 21564 q^{87} - 34836 q^{91} + 19254 q^{93} + 13362 q^{95} - 28959 q^{97} + 9126 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 11\nu^{4} - 121\nu^{3} + 98\nu^{2} + 1118\nu - 220 ) / 1098 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 11\nu^{4} + 121\nu^{3} - 98\nu^{2} + 529\nu + 220 ) / 549 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 55\nu^{5} - 56\nu^{4} + 616\nu^{3} + 649\nu^{2} + 5488\nu + 22 ) / 1098 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 373\nu^{5} - 260\nu^{4} + 3958\nu^{3} + 6817\nu^{2} + 37558\nu + 15082 ) / 1098 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -406\nu^{5} + 623\nu^{4} - 4657\nu^{3} - 3583\nu^{2} - 36898\nu + 6206 ) / 1098 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} - 21\beta_{3} + 2\beta_{2} + \beta _1 - 22 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} + 11\beta_{2} - 11\beta _1 - 26 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 22\beta_{5} - 11\beta_{4} + 237\beta_{3} - 23\beta_{2} - 46\beta _1 + 22 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23\beta_{5} - 46\beta_{4} + 549\beta_{3} - 270\beta_{2} - 135\beta _1 + 572 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−0.102534 − 0.177594i
1.89154 + 3.27625i
−1.28901 − 2.23263i
−0.102534 + 0.177594i
1.89154 − 3.27625i
−1.28901 + 2.23263i
0 −8.32172 − 3.42768i 0 −30.0804 + 17.3669i 0 −15.6054 + 27.0294i 0 57.5020 + 57.0484i 0
65.2 0 1.11837 + 8.93024i 0 10.2044 − 5.89150i 0 −26.6364 + 46.1356i 0 −78.4985 + 19.9746i 0
65.3 0 8.70335 + 2.29167i 0 13.8760 − 8.01130i 0 36.2418 − 62.7727i 0 70.4965 + 39.8904i 0
113.1 0 −8.32172 + 3.42768i 0 −30.0804 − 17.3669i 0 −15.6054 − 27.0294i 0 57.5020 − 57.0484i 0
113.2 0 1.11837 − 8.93024i 0 10.2044 + 5.89150i 0 −26.6364 − 46.1356i 0 −78.4985 − 19.9746i 0
113.3 0 8.70335 − 2.29167i 0 13.8760 + 8.01130i 0 36.2418 + 62.7727i 0 70.4965 − 39.8904i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 144.5.q.a 6
3.b odd 2 1 432.5.q.a 6
4.b odd 2 1 9.5.d.a ✓ 6
9.c even 3 1 432.5.q.a 6
9.c even 3 1 1296.5.e.c 6
9.d odd 6 1 inner 144.5.q.a 6
9.d odd 6 1 1296.5.e.c 6
12.b even 2 1 27.5.d.a 6
36.f odd 6 1 27.5.d.a 6
36.f odd 6 1 81.5.b.a 6
36.h even 6 1 9.5.d.a ✓ 6
36.h even 6 1 81.5.b.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.5.d.a ✓ 6 4.b odd 2 1
9.5.d.a ✓ 6 36.h even 6 1
27.5.d.a 6 12.b even 2 1
27.5.d.a 6 36.f odd 6 1
81.5.b.a 6 36.f odd 6 1
81.5.b.a 6 36.h even 6 1
144.5.q.a 6 1.a even 1 1 trivial
144.5.q.a 6 9.d odd 6 1 inner
432.5.q.a 6 3.b odd 2 1
432.5.q.a 6 9.c even 3 1
1296.5.e.c 6 9.c even 3 1
1296.5.e.c 6 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 12T_{5}^{5} - 729T_{5}^{4} - 9324T_{5}^{3} + 649161T_{5}^{2} - 8825166T_{5} + 43001388 \) acting on \(S_{5}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{6} + 12 T^{5} + \cdots + 43001388 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 14524588324 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 481294471563 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 708378089104 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 47166451632 \) Copy content Toggle raw display
$19$ \( (T^{3} - 129 T^{2} + \cdots + 1195028)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 10049071819968 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 92\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 63\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T^{3} - 6 T^{2} + \cdots + 1276743376)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 32\!\cdots\!63 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 65\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 34\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 74\!\cdots\!87 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 26\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( (T^{3} + 7311 T^{2} + \cdots - 15741832472)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 53\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 52\!\cdots\!89 \) Copy content Toggle raw display
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