Properties

Label 147.3.b.g
Level $147$
Weight $3$
Character orbit 147.b
Analytic conductor $4.005$
Analytic rank $0$
Dimension $8$
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] = [147,3,Mod(50,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.50");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 147.b (of order \(2\), degree \(1\), 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: \(4.00545988610\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2271266013184.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 10x^{6} + 35x^{4} - 8x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{7} - \beta_{5}) q^{3} + (\beta_{6} + \beta_{3} - 2) q^{4} + ( - \beta_{7} + \beta_{4} - \beta_1) q^{5} + (2 \beta_{4} - \beta_1) q^{6} + (\beta_{6} - \beta_{3} - 3 \beta_{2} - 1) q^{8} + ( - \beta_{6} - 2 \beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{7} - \beta_{5}) q^{3} + (\beta_{6} + \beta_{3} - 2) q^{4} + ( - \beta_{7} + \beta_{4} - \beta_1) q^{5} + (2 \beta_{4} - \beta_1) q^{6} + (\beta_{6} - \beta_{3} - 3 \beta_{2} - 1) q^{8} + ( - \beta_{6} - 2 \beta_{3} + \beta_{2}) q^{9} + (3 \beta_{7} + 7 \beta_{5} - 3 \beta_1) q^{10} + ( - 2 \beta_{6} + 2 \beta_{3} - 4 \beta_{2} + 2) q^{11} + (\beta_{7} + 10 \beta_{5} - \beta_{4} - 4 \beta_1) q^{12} + ( - \beta_{7} + 3 \beta_{5} + \beta_1) q^{13} + ( - \beta_{6} - \beta_{3} + 5 \beta_{2} - 8) q^{15} + 9 q^{16} + ( - 2 \beta_{7} + 3 \beta_{4} - 2 \beta_1) q^{17} + ( - \beta_{6} + 2 \beta_{3} + 8 \beta_{2} - 5) q^{18} + (3 \beta_{7} - 8 \beta_{5} - 3 \beta_1) q^{19} + ( - \beta_{7} - 9 \beta_{4} - \beta_1) q^{20} + ( - 2 \beta_{6} - 2 \beta_{3} + 26) q^{22} + (3 \beta_{6} - 3 \beta_{3} - 10 \beta_{2} - 3) q^{23} + (2 \beta_{7} - 7 \beta_{5} - 7 \beta_{4} - \beta_1) q^{24} + (\beta_{6} + \beta_{3} - 2) q^{25} + ( - \beta_{7} - \beta_{4} - \beta_1) q^{26} + (2 \beta_{7} - 16 \beta_{5} + 3 \beta_{4} + 3 \beta_1) q^{27} + ( - 3 \beta_{6} + 3 \beta_{3} + 4 \beta_{2} + 3) q^{29} + (4 \beta_{6} + 6 \beta_{3} - 3 \beta_{2} - 29) q^{30} + ( - 8 \beta_{7} - 2 \beta_{5} + 8 \beta_1) q^{31} + (4 \beta_{6} - 4 \beta_{3} - 3 \beta_{2} - 4) q^{32} + ( - 4 \beta_{7} + 14 \beta_{5} - 6 \beta_{4} + 12 \beta_1) q^{33} + (8 \beta_{7} + 21 \beta_{5} - 8 \beta_1) q^{34} + (6 \beta_{6} + \beta_{3} - 5 \beta_{2} - 47) q^{36} + ( - 3 \beta_{6} - 3 \beta_{3} + 29) q^{37} + (3 \beta_{7} + 2 \beta_{4} + 3 \beta_1) q^{38} + (4 \beta_{6} - 2 \beta_{3} + \beta_{2} - 1) q^{39} + ( - 5 \beta_{7} - 35 \beta_{5} + 5 \beta_1) q^{40} + (4 \beta_{7} + 9 \beta_{4} + 4 \beta_1) q^{41} + (7 \beta_{6} + 7 \beta_{3} + 9) q^{43} + ( - 10 \beta_{6} + 10 \beta_{3} + 20 \beta_{2} + 10) q^{44} + (9 \beta_{7} + 11 \beta_{4} - \beta_1) q^{45} + ( - 13 \beta_{6} - 13 \beta_{3} + 57) q^{46} + (10 \beta_{7} + 10 \beta_{4} + 10 \beta_1) q^{47} + ( - 9 \beta_{7} - 9 \beta_{5}) q^{48} + (\beta_{6} - \beta_{3} - 7 \beta_{2} - 1) q^{50} + ( - 2 \beta_{6} - \beta_{3} + 14 \beta_{2} - 15) q^{51} + ( - 5 \beta_{7} + 5 \beta_{5} + 5 \beta_1) q^{52} + (4 \beta_{6} - 4 \beta_{3} + 2 \beta_{2} - 4) q^{53} + (3 \beta_{7} + 21 \beta_{5} + 17 \beta_{4} - 4 \beta_1) q^{54} + ( - 20 \beta_{7} + 20 \beta_1) q^{55} + ( - 11 \beta_{6} + 6 \beta_{3} - 3 \beta_{2} + 1) q^{57} + (7 \beta_{6} + 7 \beta_{3} - 21) q^{58} + (9 \beta_{7} - 16 \beta_{4} + 9 \beta_1) q^{59} + ( - \beta_{6} - 11 \beta_{3} - 35 \beta_{2} - 18) q^{60} + ( - 11 \beta_{7} - 43 \beta_{5} + 11 \beta_1) q^{61} + ( - 8 \beta_{7} + 18 \beta_{4} - 8 \beta_1) q^{62} + ( - 7 \beta_{6} - 7 \beta_{3} + 50) q^{64} + (5 \beta_{6} - 5 \beta_{3} - 5) q^{65} + ( - 24 \beta_{7} - 42 \beta_{5} + 2 \beta_{4} + 8 \beta_1) q^{66} + (8 \beta_{6} + 8 \beta_{3} - 54) q^{67} - 25 \beta_{4} q^{68} + (6 \beta_{7} - 21 \beta_{5} - 23 \beta_{4} - 2 \beta_1) q^{69} + (3 \beta_{6} - 3 \beta_{3} - 4 \beta_{2} - 3) q^{71} + ( - 8 \beta_{6} - 3 \beta_{3} - 30 \beta_{2} + 4) q^{72} + (20 \beta_{7} + 5 \beta_{5} - 20 \beta_1) q^{73} + ( - 3 \beta_{6} + 3 \beta_{3} + 44 \beta_{2} + 3) q^{74} + (\beta_{7} + 10 \beta_{5} - \beta_{4} - 4 \beta_1) q^{75} + (13 \beta_{7} - 18 \beta_{5} - 13 \beta_1) q^{76} + ( - \beta_{6} - 3 \beta_{3} - 3 \beta_{2} - 10) q^{78} + (4 \beta_{6} + 4 \beta_{3} + 8) q^{79} + ( - 9 \beta_{7} + 9 \beta_{4} - 9 \beta_1) q^{80} + ( - 13 \beta_{6} + 7 \beta_{3} + 10 \beta_{2} + 60) q^{81} + (14 \beta_{7} + 63 \beta_{5} - 14 \beta_1) q^{82} + (15 \beta_{7} - 10 \beta_{4} + 15 \beta_1) q^{83} + (5 \beta_{6} + 5 \beta_{3} - 65) q^{85} + (7 \beta_{6} - 7 \beta_{3} - 26 \beta_{2} - 7) q^{86} + ( - 6 \beta_{7} + 21 \beta_{5} + 11 \beta_{4} + 8 \beta_1) q^{87} + (22 \beta_{6} + 22 \beta_{3} - 6) q^{88} + ( - 14 \beta_{7} - 9 \beta_{4} - 14 \beta_1) q^{89} + (23 \beta_{7} + 77 \beta_{5} - 10 \beta_{4} - 13 \beta_1) q^{90} + ( - \beta_{6} + \beta_{3} + 82 \beta_{2} + 1) q^{92} + (6 \beta_{6} - 16 \beta_{3} + 8 \beta_{2} + 44) q^{93} + (10 \beta_{7} + 70 \beta_{5} - 10 \beta_1) q^{94} + ( - 14 \beta_{6} + 14 \beta_{3} - 2 \beta_{2} + 14) q^{95} + (8 \beta_{7} - 28 \beta_{5} - 10 \beta_{4} - 13 \beta_1) q^{96} + (20 \beta_{7} + 45 \beta_{5} - 20 \beta_1) q^{97} + (26 \beta_{6} - 14 \beta_{3} - 20 \beta_{2} + 42) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 4 q^{9} - 64 q^{15} + 72 q^{16} - 52 q^{18} + 208 q^{22} - 16 q^{25} - 240 q^{30} - 356 q^{36} + 232 q^{37} + 16 q^{39} + 72 q^{43} + 456 q^{46} - 124 q^{51} - 60 q^{57} - 168 q^{58} - 104 q^{60} + 400 q^{64} - 432 q^{67} + 12 q^{72} - 72 q^{78} + 64 q^{79} + 400 q^{81} - 520 q^{85} - 48 q^{88} + 440 q^{93} + 496 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 10x^{6} + 35x^{4} - 8x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 35\nu^{5} + 340\nu^{3} + 1082\nu ) / 285 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} - 13\nu^{4} - 53\nu^{2} + 2 ) / 57 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{6} + 61\nu^{4} + 275\nu^{2} + 109 ) / 57 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} + 100\nu^{5} + 320\nu^{3} + 217\nu ) / 285 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{7} - 100\nu^{5} - 320\nu^{3} + 353\nu ) / 285 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} - 29\nu^{4} - 89\nu^{2} + 60 ) / 19 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} + 74\nu^{5} + 271\nu^{3} - 7\nu ) / 57 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} - 2\beta_{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - 5\beta_{5} - 2\beta_{4} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{6} - \beta_{3} + 20\beta_{2} + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 24\beta_{7} + 79\beta_{5} + 5\beta_{4} - 26\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{6} - 10\beta_{3} - 67\beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -160\beta_{7} - 447\beta_{5} + 103\beta_{4} + 120\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1
0.707107 + 2.38591i
−0.707107 2.38591i
−0.707107 0.554453i
0.707107 + 0.554453i
−0.707107 + 0.554453i
0.707107 0.554453i
0.707107 2.38591i
−0.707107 + 2.38591i
3.37419i −2.96460 + 0.459485i −7.38516 5.69080i 1.55039 + 10.0031i 0 11.4222i 8.57775 2.72438i 19.2018
50.2 3.37419i 2.96460 0.459485i −7.38516 5.69080i −1.55039 10.0031i 0 11.4222i 8.57775 2.72438i −19.2018
50.3 0.784114i −0.843283 + 2.87904i 3.38516 4.64918i 2.25750 + 0.661230i 0 5.79081i −7.57775 4.85569i 3.64549
50.4 0.784114i 0.843283 2.87904i 3.38516 4.64918i −2.25750 0.661230i 0 5.79081i −7.57775 4.85569i −3.64549
50.5 0.784114i −0.843283 2.87904i 3.38516 4.64918i 2.25750 0.661230i 0 5.79081i −7.57775 + 4.85569i 3.64549
50.6 0.784114i 0.843283 + 2.87904i 3.38516 4.64918i −2.25750 + 0.661230i 0 5.79081i −7.57775 + 4.85569i −3.64549
50.7 3.37419i −2.96460 0.459485i −7.38516 5.69080i 1.55039 10.0031i 0 11.4222i 8.57775 + 2.72438i 19.2018
50.8 3.37419i 2.96460 + 0.459485i −7.38516 5.69080i −1.55039 + 10.0031i 0 11.4222i 8.57775 + 2.72438i −19.2018
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.3.b.g 8
3.b odd 2 1 inner 147.3.b.g 8
7.b odd 2 1 inner 147.3.b.g 8
7.c even 3 2 147.3.h.f 16
7.d odd 6 2 147.3.h.f 16
21.c even 2 1 inner 147.3.b.g 8
21.g even 6 2 147.3.h.f 16
21.h odd 6 2 147.3.h.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.3.b.g 8 1.a even 1 1 trivial
147.3.b.g 8 3.b odd 2 1 inner
147.3.b.g 8 7.b odd 2 1 inner
147.3.b.g 8 21.c even 2 1 inner
147.3.h.f 16 7.c even 3 2
147.3.h.f 16 7.d odd 6 2
147.3.h.f 16 21.g even 6 2
147.3.h.f 16 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{4} + 12T_{2}^{2} + 7 \) Copy content Toggle raw display
\( T_{5}^{4} + 54T_{5}^{2} + 700 \) Copy content Toggle raw display
\( T_{13}^{4} - 78T_{13}^{2} + 100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 12 T^{2} + 7)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{6} - 98 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{4} + 54 T^{2} + 700)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 496 T^{2} + 44800)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 78 T^{2} + 100)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 328 T^{2} + 17500)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 622 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1692 T^{2} + 389872)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 756 T^{2} + 137200)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 1872 T^{2} + 846400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 58 T + 580)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 2632 T^{2} + 1680700)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 18 T - 1340)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 6200 T^{2} + 7000000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1168 T^{2} + 2800)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 8322 T^{2} + 8316700)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 9134 T^{2} + 1119364)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 108 T + 1060)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 756 T^{2} + 137200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 11700 T^{2} + 33062500)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 16 T - 400)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 9450 T^{2} + 21437500)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 9112 T^{2} + 7426300)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 16500 T^{2} + 11222500)^{2} \) Copy content Toggle raw display
show more
show less