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Properties

Label 144.5.q.c
Level $144$
Weight $5$
Character orbit 144.q
Analytic conductor $14.885$
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] = [144,5,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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 6x^{6} + 121x^{5} + 1104x^{4} - 1647x^{3} + 6529x^{2} + 85254x + 440076 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{7} + \beta_{3} - 1) q^{5} + (\beta_{7} + \beta_{5} + \beta_{4} + \cdots + 1) q^{7}+ \cdots + ( - 3 \beta_{7} - 2 \beta_{6} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{7} + \beta_{3} - 1) q^{5} + (\beta_{7} + \beta_{5} + \beta_{4} + \cdots + 1) q^{7}+ \cdots + ( - 81 \beta_{7} + 168 \beta_{6} + \cdots + 2814) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 9 q^{3} - 9 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 9 q^{3} - 9 q^{5} - 13 q^{7} + 21 q^{9} + 18 q^{11} - 5 q^{13} - 225 q^{15} - 562 q^{19} - 1167 q^{21} + 1719 q^{23} + 353 q^{25} - 648 q^{27} + 2115 q^{29} - 187 q^{31} + 3258 q^{33} + 16 q^{37} + 8265 q^{39} - 7920 q^{41} + 68 q^{43} + 5679 q^{45} - 13689 q^{47} - 327 q^{49} - 10449 q^{51} + 1818 q^{55} - 21861 q^{57} + 20052 q^{59} - 1937 q^{61} - 5559 q^{63} + 25965 q^{65} - 154 q^{67} + 21645 q^{69} - 7802 q^{73} + 30297 q^{75} - 25641 q^{77} + 2195 q^{79} + 19701 q^{81} - 37017 q^{83} - 3042 q^{85} - 22455 q^{87} - 15830 q^{91} - 36489 q^{93} + 37116 q^{95} + 7282 q^{97} + 10035 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 6x^{6} + 121x^{5} + 1104x^{4} - 1647x^{3} + 6529x^{2} + 85254x + 440076 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 115 \nu^{7} - 175 \nu^{6} + 4562 \nu^{5} - 26525 \nu^{4} - 65600 \nu^{3} + 13645 \nu^{2} + \cdots - 10751676 ) / 5629338 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 119 \nu^{7} - 2867 \nu^{6} - 8306 \nu^{5} + 100469 \nu^{4} - 747082 \nu^{3} - 2322289 \nu^{2} + \cdots + 44309382 ) / 5629338 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 505 \nu^{7} + 7625 \nu^{6} + 8618 \nu^{5} - 281663 \nu^{4} + 505282 \nu^{3} + 4989577 \nu^{2} + \cdots - 150628920 ) / 5629338 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 457 \nu^{7} + 5180 \nu^{6} - 10711 \nu^{5} - 33392 \nu^{4} - 143180 \nu^{3} + 5464828 \nu^{2} + \cdots - 17103723 ) / 2814669 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 665 \nu^{7} - 8053 \nu^{6} + 23477 \nu^{5} - 118565 \nu^{4} + 198040 \nu^{3} - 3976835 \nu^{2} + \cdots - 86664786 ) / 2814669 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -20\nu^{7} + 202\nu^{6} - 485\nu^{5} + 2360\nu^{4} - 6760\nu^{3} + 102320\nu^{2} + 396585\nu + 1802148 ) / 72171 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4613 \nu^{7} - 22861 \nu^{6} + 141194 \nu^{5} + 376069 \nu^{4} + 1408318 \nu^{3} + \cdots + 183190098 ) / 5629338 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} - 2\beta _1 + 2 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 4\beta_{5} + 9\beta_{4} - 9\beta_{3} - 9\beta_{2} + 10\beta _1 + 2 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{7} - 2\beta_{6} + \beta_{5} + 9\beta_{4} - 90\beta_{3} - 162\beta_{2} + 169\beta _1 - 382 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{7} - 20\beta_{6} - 116\beta_{5} - 18\beta_{4} - 225\beta_{3} - 234\beta_{2} + 664\beta _1 - 6385 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 270\beta_{7} - 560\beta_{6} - 1205\beta_{5} - 720\beta_{4} + 720\beta_{3} + 450\beta_{2} + 6586\beta _1 - 7978 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1395 \beta_{7} - 353 \beta_{6} - 5165 \beta_{5} - 7470 \beta_{4} + 16380 \beta_{3} + 16605 \beta_{2} + \cdots + 11714 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11646 \beta_{7} + 799 \beta_{6} + 3322 \beta_{5} - 17109 \beta_{4} + 105804 \beta_{3} + 137898 \beta_{2} + \cdots + 548321 ) / 9 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
4.23522 + 4.06612i
3.72537 − 4.42407i
−3.41053 + 2.74723i
−3.05006 − 3.25531i
4.23522 − 4.06612i
3.72537 + 4.42407i
−3.41053 − 2.74723i
−3.05006 + 3.25531i
0 −8.37420 + 3.29740i 0 10.6364 − 6.14094i 0 −7.14202 + 12.3703i 0 59.2543 − 55.2261i 0
65.2 0 −0.256692 − 8.99634i 0 −7.67992 + 4.43400i 0 30.9381 − 53.5864i 0 −80.8682 + 4.61857i 0
65.3 0 4.23662 + 7.94047i 0 −34.8718 + 20.1332i 0 7.38688 − 12.7945i 0 −45.1021 + 67.2815i 0
65.4 0 8.89427 − 1.37550i 0 27.4152 − 15.8282i 0 −37.6830 + 65.2688i 0 77.2160 − 24.4682i 0
113.1 0 −8.37420 − 3.29740i 0 10.6364 + 6.14094i 0 −7.14202 − 12.3703i 0 59.2543 + 55.2261i 0
113.2 0 −0.256692 + 8.99634i 0 −7.67992 − 4.43400i 0 30.9381 + 53.5864i 0 −80.8682 − 4.61857i 0
113.3 0 4.23662 − 7.94047i 0 −34.8718 − 20.1332i 0 7.38688 + 12.7945i 0 −45.1021 − 67.2815i 0
113.4 0 8.89427 + 1.37550i 0 27.4152 + 15.8282i 0 −37.6830 − 65.2688i 0 77.2160 + 24.4682i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.4
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.c 8
3.b odd 2 1 432.5.q.c 8
4.b odd 2 1 36.5.g.a ✓ 8
9.c even 3 1 432.5.q.c 8
9.c even 3 1 1296.5.e.g 8
9.d odd 6 1 inner 144.5.q.c 8
9.d odd 6 1 1296.5.e.g 8
12.b even 2 1 108.5.g.a 8
36.f odd 6 1 108.5.g.a 8
36.f odd 6 1 324.5.c.a 8
36.h even 6 1 36.5.g.a ✓ 8
36.h even 6 1 324.5.c.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.5.g.a ✓ 8 4.b odd 2 1
36.5.g.a ✓ 8 36.h even 6 1
108.5.g.a 8 12.b even 2 1
108.5.g.a 8 36.f odd 6 1
144.5.q.c 8 1.a even 1 1 trivial
144.5.q.c 8 9.d odd 6 1 inner
324.5.c.a 8 36.f odd 6 1
324.5.c.a 8 36.h even 6 1
432.5.q.c 8 3.b odd 2 1
432.5.q.c 8 9.c even 3 1
1296.5.e.g 8 9.c even 3 1
1296.5.e.g 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 9 T_{5}^{7} - 1386 T_{5}^{6} - 12717 T_{5}^{5} + 1875474 T_{5}^{4} - 8355069 T_{5}^{3} + \cdots + 19274879556 \) acting on \(S_{5}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 9 T^{7} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 19274879556 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 968464619236 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{4} + 281 T^{3} + \cdots - 16594049624)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 52\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 59\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} + \cdots + 400007987296)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 40\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 31\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 49\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 523952945824816)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 51\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display
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