↑

Properties

Label 144.4.s.e
Level $144$
Weight $4$
Character orbit 144.s
Analytic conductor $8.496$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,4,Mod(47,144)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.47");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 144.s (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: \(8.49627504083\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 7x^{10} + 175x^{8} - 1170x^{6} + 14175x^{4} - 45927x^{2} + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{9} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{3}) q^{3} + ( - \beta_{8} + \beta_{2} - 2 \beta_1 + 1) q^{5} + (\beta_{10} - \beta_{3}) q^{7} + ( - \beta_{11} + \beta_{5} - \beta_{2} + \cdots - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{3}) q^{3} + ( - \beta_{8} + \beta_{2} - 2 \beta_1 + 1) q^{5} + (\beta_{10} - \beta_{3}) q^{7} + ( - \beta_{11} + \beta_{5} - \beta_{2} + \cdots - 8) q^{9}+ \cdots + (45 \beta_{10} + 45 \beta_{9} + \cdots - 36 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{5} - 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{5} - 42 q^{9} - 24 q^{13} + 252 q^{21} + 294 q^{25} + 276 q^{29} + 414 q^{33} - 480 q^{37} - 738 q^{41} - 1980 q^{45} + 534 q^{49} + 1998 q^{57} - 840 q^{61} + 3576 q^{65} + 288 q^{69} + 156 q^{73} - 6912 q^{77} - 5418 q^{81} - 720 q^{85} + 4248 q^{93} + 606 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 7x^{10} + 175x^{8} - 1170x^{6} + 14175x^{4} - 45927x^{2} + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -7\nu^{10} + 49\nu^{8} - 496\nu^{6} + 3087\nu^{4} - 17577\nu^{2} - 209952 ) / 209952 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} + 16\nu^{8} - 319\nu^{6} + 4041\nu^{4} - 15552\nu^{2} + 174231 ) / 11664 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{11} + 49\nu^{9} - 496\nu^{7} + 3087\nu^{5} - 17577\nu^{3} - 419904\nu ) / 209952 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{11} + 49\nu^{9} - 496\nu^{7} + 3087\nu^{5} - 17577\nu^{3} + 209952\nu ) / 209952 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{10} - 221\nu^{8} + 1961\nu^{6} - 21474\nu^{4} + 114453\nu^{2} - 986337 ) / 23328 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\nu^{11} + 253\nu^{9} - 1465\nu^{7} + 48690\nu^{5} - 119637\nu^{3} + 2770929\nu ) / 209952 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 53\nu^{11} + 196\nu^{9} + 3119\nu^{7} - 6525\nu^{5} + 256284\nu^{3} + 1213785\nu ) / 629856 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 67\nu^{10} - 145\nu^{8} + 5812\nu^{6} + 30069\nu^{4} + 188649\nu^{2} + 3542940 ) / 209952 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -53\nu^{11} + 1505\nu^{9} - 15026\nu^{7} + 127053\nu^{5} - 1006425\nu^{3} + 4946994\nu ) / 629856 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -2\nu^{11} + 14\nu^{9} - 350\nu^{7} + 2340\nu^{5} - 8667\nu^{3} + 52488\nu ) / 19683 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26\nu^{10} - 857\nu^{8} + 8789\nu^{6} - 94842\nu^{4} + 681777\nu^{2} - 4144365 ) / 69984 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{5} + \beta_{2} + 6\beta _1 + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{10} - 2\beta_{9} + 4\beta_{7} + 4\beta_{4} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{8} + 2\beta_{5} + 9\beta_{2} + 49\beta _1 - 119 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{9} - 13\beta_{7} + 21\beta_{6} - 22\beta_{4} + 64\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -49\beta_{11} + 112\beta_{8} + 63\beta_{5} - 112\beta_{2} + 976\beta _1 + 521 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -336\beta_{10} - 56\beta_{9} - 161\beta_{7} + 147\beta_{6} + 604\beta_{4} + 449\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 98\beta_{11} + 288\beta_{8} - 512\beta_{5} - 1351\beta_{2} + 6006\beta _1 + 5482 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -1029\beta_{10} + 758\beta_{9} + 2669\beta_{7} - 459\beta_{6} + 3929\beta_{4} + 3112\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1647\beta_{11} - 2833\beta_{8} - 4655\beta_{5} - 63\beta_{2} - 110551\beta _1 - 161089 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 9072\beta_{10} + 8563\beta_{9} + 14314\beta_{7} - 4368\beta_{6} - 95027\beta_{4} - 16822\beta_{3} ) / 3 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−1.79922 − 2.40058i
2.08777 − 2.15435i
−2.76659 − 1.16016i
2.76659 + 1.16016i
−2.08777 + 2.15435i
1.79922 + 2.40058i
−1.79922 + 2.40058i
2.08777 + 2.15435i
−2.76659 + 1.16016i
2.76659 − 1.16016i
−2.08777 − 2.15435i
1.79922 − 2.40058i
0 −4.15793 + 3.11634i 0 −9.07680 + 5.24049i 0 13.3065 + 7.68253i 0 7.57680 − 25.9151i 0
47.2 0 −3.73145 − 3.61612i 0 −2.34737 + 1.35526i 0 −10.3523 − 5.97691i 0 0.847372 + 26.9867i 0
47.3 0 −2.00946 + 4.79188i 0 17.4242 − 10.0598i 0 −26.2253 − 15.1412i 0 −18.9242 − 19.2581i 0
47.4 0 2.00946 − 4.79188i 0 17.4242 − 10.0598i 0 26.2253 + 15.1412i 0 −18.9242 − 19.2581i 0
47.5 0 3.73145 + 3.61612i 0 −2.34737 + 1.35526i 0 10.3523 + 5.97691i 0 0.847372 + 26.9867i 0
47.6 0 4.15793 − 3.11634i 0 −9.07680 + 5.24049i 0 −13.3065 − 7.68253i 0 7.57680 − 25.9151i 0
95.1 0 −4.15793 − 3.11634i 0 −9.07680 − 5.24049i 0 13.3065 − 7.68253i 0 7.57680 + 25.9151i 0
95.2 0 −3.73145 + 3.61612i 0 −2.34737 − 1.35526i 0 −10.3523 + 5.97691i 0 0.847372 − 26.9867i 0
95.3 0 −2.00946 − 4.79188i 0 17.4242 + 10.0598i 0 −26.2253 + 15.1412i 0 −18.9242 + 19.2581i 0
95.4 0 2.00946 + 4.79188i 0 17.4242 + 10.0598i 0 26.2253 − 15.1412i 0 −18.9242 + 19.2581i 0
95.5 0 3.73145 − 3.61612i 0 −2.34737 − 1.35526i 0 10.3523 − 5.97691i 0 0.847372 − 26.9867i 0
95.6 0 4.15793 + 3.11634i 0 −9.07680 − 5.24049i 0 −13.3065 + 7.68253i 0 7.57680 + 25.9151i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(144, [\chi])\):

\( T_{5}^{6} - 12T_{5}^{5} - 189T_{5}^{4} + 2844T_{5}^{3} + 60129T_{5}^{2} + 234630T_{5} + 326700 \) Copy content Toggle raw display
\( T_{7}^{12} - 1296 T_{7}^{10} + 1298349 T_{7}^{8} - 432250344 T_{7}^{6} + 105271671465 T_{7}^{4} + \cdots + 957026443992336 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( (T^{6} - 12 T^{5} + \cdots + 326700)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 957026443992336 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 151336128515625 \) Copy content Toggle raw display
$13$ \( (T^{6} + 12 T^{5} + \cdots + 3124810000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 16479 T^{4} + \cdots + 130942699200)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 13851 T^{4} + \cdots + 4269315600)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{6} - 138 T^{5} + \cdots + 53625465612)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{3} + 120 T^{2} + \cdots + 2226160)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 369 T^{5} + \cdots + 5342773203)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 59\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 80621568000000)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 91\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 39 T^{2} + \cdots - 21753640)^{4} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 24\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 16\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 257935645140625)^{2} \) Copy content Toggle raw display
show more
show less