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Properties

Label 147.4.g.c
Level $147$
Weight $4$
Character orbit 147.g
Analytic conductor $8.673$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,4,Mod(68,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.68");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.g (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: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.443364212736.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 16x^{6} + 175x^{4} - 1296x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{7} - \beta_{5}) q^{3} + ( - 9 \beta_1 + 9) q^{4} + ( - \beta_{7} + 2 \beta_{6} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - 3 \beta_{3} - 24 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{7} - \beta_{5}) q^{3} + ( - 9 \beta_1 + 9) q^{4} + ( - \beta_{7} + 2 \beta_{6} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - 192 \beta_{4} - 408) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{4} - 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{4} - 96 q^{9} - 408 q^{15} + 220 q^{16} - 204 q^{18} + 1088 q^{22} + 92 q^{25} + 204 q^{30} - 1728 q^{36} - 920 q^{37} - 276 q^{39} + 352 q^{43} + 1496 q^{46} + 1224 q^{51} - 360 q^{57} - 952 q^{58} - 1836 q^{60} + 5048 q^{64} + 256 q^{67} + 204 q^{72} - 9384 q^{78} + 1768 q^{79} - 1692 q^{81} + 4896 q^{85} + 544 q^{88} - 1248 q^{93} - 3264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 16x^{6} + 175x^{4} - 1296x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -16\nu^{6} + 175\nu^{4} - 2800\nu^{2} + 20736 ) / 14175 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} - 49\nu^{5} + 217\nu^{3} - 1377\nu ) / 5103 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -47\nu^{6} + 1400\nu^{4} - 8225\nu^{2} + 60912 ) / 14175 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 104 ) / 175 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{7} - 175\nu^{5} + 2800\nu^{3} + 7614\nu ) / 14175 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -16\nu^{7} + 175\nu^{5} - 2800\nu^{3} + 34911\nu ) / 14175 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 193\nu^{7} - 1225\nu^{5} + 5425\nu^{3} - 51678\nu ) / 127575 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - 8\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -18\beta_{7} - 7\beta_{6} + 14\beta_{5} - 9\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 16\beta_{3} - 47\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -144\beta_{7} - 62\beta_{6} + 31\beta_{5} - 288\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -175\beta_{4} - 104 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1575\beta_{7} + 71\beta_{6} + 71\beta_{5} - 1575\beta_{2} ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
2.17132 + 2.07011i
−2.17132 − 2.07011i
2.87843 + 0.845366i
−2.87843 − 0.845366i
2.17132 − 2.07011i
−2.17132 + 2.07011i
2.87843 − 0.845366i
−2.87843 + 0.845366i
−3.57071 − 2.06155i −3.58554 − 3.76084i 4.50000 + 7.79423i 5.04975 − 8.74643i 5.04975 + 20.8207i 0 − 4.12311i −1.28786 + 26.9693i −36.0624 + 20.8207i
68.2 −3.57071 − 2.06155i 3.58554 + 3.76084i 4.50000 + 7.79423i −5.04975 + 8.74643i −5.04975 − 20.8207i 0 − 4.12311i −1.28786 + 26.9693i 36.0624 − 20.8207i
68.3 3.57071 + 2.06155i −1.46422 − 4.98559i 4.50000 + 7.79423i 5.04975 − 8.74643i 5.04975 − 20.8207i 0 4.12311i −22.7121 + 14.6000i 36.0624 − 20.8207i
68.4 3.57071 + 2.06155i 1.46422 + 4.98559i 4.50000 + 7.79423i −5.04975 + 8.74643i −5.04975 + 20.8207i 0 4.12311i −22.7121 + 14.6000i −36.0624 + 20.8207i
80.1 −3.57071 + 2.06155i −3.58554 + 3.76084i 4.50000 − 7.79423i 5.04975 + 8.74643i 5.04975 − 20.8207i 0 4.12311i −1.28786 − 26.9693i −36.0624 − 20.8207i
80.2 −3.57071 + 2.06155i 3.58554 − 3.76084i 4.50000 − 7.79423i −5.04975 − 8.74643i −5.04975 + 20.8207i 0 4.12311i −1.28786 − 26.9693i 36.0624 + 20.8207i
80.3 3.57071 − 2.06155i −1.46422 + 4.98559i 4.50000 − 7.79423i 5.04975 + 8.74643i 5.04975 + 20.8207i 0 − 4.12311i −22.7121 − 14.6000i 36.0624 + 20.8207i
80.4 3.57071 − 2.06155i 1.46422 − 4.98559i 4.50000 − 7.79423i −5.04975 − 8.74643i −5.04975 − 20.8207i 0 − 4.12311i −22.7121 − 14.6000i −36.0624 − 20.8207i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.4
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
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.g.c 8
3.b odd 2 1 inner 147.4.g.c 8
7.b odd 2 1 inner 147.4.g.c 8
7.c even 3 1 21.4.c.b ✓ 4
7.c even 3 1 inner 147.4.g.c 8
7.d odd 6 1 21.4.c.b ✓ 4
7.d odd 6 1 inner 147.4.g.c 8
21.c even 2 1 inner 147.4.g.c 8
21.g even 6 1 21.4.c.b ✓ 4
21.g even 6 1 inner 147.4.g.c 8
21.h odd 6 1 21.4.c.b ✓ 4
21.h odd 6 1 inner 147.4.g.c 8
28.f even 6 1 336.4.k.b 4
28.g odd 6 1 336.4.k.b 4
84.j odd 6 1 336.4.k.b 4
84.n even 6 1 336.4.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.c.b ✓ 4 7.c even 3 1
21.4.c.b ✓ 4 7.d odd 6 1
21.4.c.b ✓ 4 21.g even 6 1
21.4.c.b ✓ 4 21.h odd 6 1
147.4.g.c 8 1.a even 1 1 trivial
147.4.g.c 8 3.b odd 2 1 inner
147.4.g.c 8 7.b odd 2 1 inner
147.4.g.c 8 7.c even 3 1 inner
147.4.g.c 8 7.d odd 6 1 inner
147.4.g.c 8 21.c even 2 1 inner
147.4.g.c 8 21.g even 6 1 inner
147.4.g.c 8 21.h odd 6 1 inner
336.4.k.b 4 28.f even 6 1
336.4.k.b 4 28.g odd 6 1
336.4.k.b 4 84.j odd 6 1
336.4.k.b 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{4} - 17T_{2}^{2} + 289 \) Copy content Toggle raw display
\( T_{19}^{4} - 1350T_{19}^{2} + 1822500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 17 T^{2} + 289)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 48 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{4} + 102 T^{2} + 10404)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 1088 T^{2} + 1183744)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3174)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 3672 T^{2} + 13483584)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 1350 T^{2} + 1822500)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 8228 T^{2} + 67699984)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 3332)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 64896 T^{2} + 4211490816)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 230 T + 52900)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 19992)^{4} \) Copy content Toggle raw display
$43$ \( (T - 44)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 117912 T^{2} + 13903239744)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 42500 T^{2} + 1806250000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 17238 T^{2} + 297148644)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 5046 T^{2} + 25462116)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 64 T + 4096)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 213248)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 7776 T^{2} + 60466176)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 442 T + 195364)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 244902)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 235008 T^{2} + 55228760064)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1193496)^{4} \) Copy content Toggle raw display
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