Properties

Label 147.6.a.m
Level $147$
Weight $6$
Character orbit 147.a
Self dual yes
Analytic conductor $23.576$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,6,Mod(1,147)]
 
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([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

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: yes
Analytic conductor: \(23.5764215125\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 97x^{2} + 7x + 294 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + 9 q^{3} + (\beta_{2} - \beta_1 + 18) q^{4} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{5} + ( - 9 \beta_1 + 9) q^{6} + ( - 2 \beta_{3} + \beta_{2} + \cdots + 38) q^{8}+ \cdots + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + 9 q^{3} + (\beta_{2} - \beta_1 + 18) q^{4} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{5} + ( - 9 \beta_1 + 9) q^{6} + ( - 2 \beta_{3} + \beta_{2} + \cdots + 38) q^{8}+ \cdots + (243 \beta_{3} + 729 \beta_{2} + \cdots + 8667) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 36 q^{3} + 69 q^{4} + 27 q^{6} + 123 q^{8} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 36 q^{3} + 69 q^{4} + 27 q^{6} + 123 q^{8} + 324 q^{9} + 283 q^{10} + 402 q^{11} + 621 q^{12} + 462 q^{13} + 3273 q^{16} + 276 q^{17} + 243 q^{18} + 510 q^{19} + 4719 q^{20} + 1375 q^{22} + 6900 q^{23} + 1107 q^{24} + 2814 q^{25} - 15138 q^{26} + 2916 q^{27} + 540 q^{29} + 2547 q^{30} - 6410 q^{31} + 15519 q^{32} + 3618 q^{33} + 21144 q^{34} + 5589 q^{36} + 15250 q^{37} - 41250 q^{38} + 4158 q^{39} - 8547 q^{40} + 4308 q^{41} + 29198 q^{43} + 70743 q^{44} + 61800 q^{46} - 15060 q^{47} + 29457 q^{48} - 7302 q^{50} + 2484 q^{51} - 47476 q^{52} + 13692 q^{53} + 2187 q^{54} + 73124 q^{55} + 4590 q^{57} + 52309 q^{58} + 34830 q^{59} + 42471 q^{60} - 5364 q^{61} + 16029 q^{62} - 73487 q^{64} + 66864 q^{65} + 12375 q^{66} - 5994 q^{67} - 58272 q^{68} + 62100 q^{69} + 89268 q^{71} + 9963 q^{72} + 59638 q^{73} - 185442 q^{74} + 25326 q^{75} + 21308 q^{76} - 136242 q^{78} - 44062 q^{79} - 33381 q^{80} + 26244 q^{81} + 57596 q^{82} - 208446 q^{83} + 36324 q^{85} - 136968 q^{86} + 4860 q^{87} + 87597 q^{88} - 77520 q^{89} + 22923 q^{90} + 158256 q^{92} - 57690 q^{93} - 73722 q^{94} - 221376 q^{95} + 139671 q^{96} - 188630 q^{97} + 32562 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 97x^{2} + 7x + 294 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 49 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 89\nu + 52 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 2\beta_{2} + 91\beta _1 + 46 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
10.1812
1.79080
−1.74818
−9.22385
−9.18123 9.00000 52.2950 22.0716 −82.6311 0 −186.333 81.0000 −202.644
1.2 −0.790805 9.00000 −31.3746 −104.192 −7.11724 0 50.1170 81.0000 82.3953
1.3 2.74818 9.00000 −24.4475 58.3673 24.7336 0 −155.128 81.0000 160.404
1.4 10.2239 9.00000 72.5272 23.7528 92.0147 0 414.344 81.0000 242.845
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.6.a.m 4
3.b odd 2 1 441.6.a.w 4
7.b odd 2 1 147.6.a.l 4
7.c even 3 2 21.6.e.c 8
7.d odd 6 2 147.6.e.o 8
21.c even 2 1 441.6.a.v 4
21.h odd 6 2 63.6.e.e 8
28.g odd 6 2 336.6.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c 8 7.c even 3 2
63.6.e.e 8 21.h odd 6 2
147.6.a.l 4 7.b odd 2 1
147.6.a.m 4 1.a even 1 1 trivial
147.6.e.o 8 7.d odd 6 2
336.6.q.j 8 28.g odd 6 2
441.6.a.v 4 21.c even 2 1
441.6.a.w 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2}^{4} - 3T_{2}^{3} - 94T_{2}^{2} + 186T_{2} + 204 \) Copy content Toggle raw display
\( T_{5}^{4} - 7657T_{5}^{2} + 302700T_{5} - 3188244 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} + \cdots + 204 \) Copy content Toggle raw display
$3$ \( (T - 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 7657 T^{2} + \cdots - 3188244 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 1682132124 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 149501563456 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots - 50104147968 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 7391138416576 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 3007939608576 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 408027025117872 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 86716089209547 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 50\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 18\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 991662745581932 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 270685655359056 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 85\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 25\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 16\!\cdots\!59 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 41\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 47\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 11\!\cdots\!44 \) Copy content Toggle raw display
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