Properties

Label 147.3.f.c
Level $147$
Weight $3$
Character orbit 147.f
Analytic conductor $4.005$
Analytic rank $0$
Dimension $2$
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] = [147,3,Mod(19,147)]
 
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([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 147.f (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: \(4.00545988610\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + ( - \zeta_{6} - 1) q^{3} + ( - 3 \zeta_{6} + 3) q^{4} + ( - 3 \zeta_{6} + 6) q^{5} + (2 \zeta_{6} - 1) q^{6} - 7 q^{8} + 3 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + ( - \zeta_{6} - 1) q^{3} + ( - 3 \zeta_{6} + 3) q^{4} + ( - 3 \zeta_{6} + 6) q^{5} + (2 \zeta_{6} - 1) q^{6} - 7 q^{8} + 3 \zeta_{6} q^{9} + ( - 3 \zeta_{6} - 3) q^{10} + ( - 11 \zeta_{6} + 11) q^{11} + (3 \zeta_{6} - 6) q^{12} + (8 \zeta_{6} - 4) q^{13} - 9 q^{15} - 5 \zeta_{6} q^{16} + ( - 14 \zeta_{6} - 14) q^{17} + ( - 3 \zeta_{6} + 3) q^{18} + ( - 2 \zeta_{6} + 4) q^{19} + ( - 18 \zeta_{6} + 9) q^{20} - 11 q^{22} - 28 \zeta_{6} q^{23} + (7 \zeta_{6} + 7) q^{24} + ( - 2 \zeta_{6} + 2) q^{25} + ( - 4 \zeta_{6} + 8) q^{26} + ( - 6 \zeta_{6} + 3) q^{27} + 25 q^{29} + 9 \zeta_{6} q^{30} + (19 \zeta_{6} + 19) q^{31} + (33 \zeta_{6} - 33) q^{32} + (11 \zeta_{6} - 22) q^{33} + (28 \zeta_{6} - 14) q^{34} + 9 q^{36} + 58 \zeta_{6} q^{37} + ( - 2 \zeta_{6} - 2) q^{38} + ( - 12 \zeta_{6} + 12) q^{39} + (21 \zeta_{6} - 42) q^{40} + (4 \zeta_{6} - 2) q^{41} + 26 q^{43} - 33 \zeta_{6} q^{44} + (9 \zeta_{6} + 9) q^{45} + (28 \zeta_{6} - 28) q^{46} + ( - 44 \zeta_{6} + 88) q^{47} + (10 \zeta_{6} - 5) q^{48} - 2 q^{50} + 42 \zeta_{6} q^{51} + (12 \zeta_{6} + 12) q^{52} + (31 \zeta_{6} - 31) q^{53} + (3 \zeta_{6} - 6) q^{54} + ( - 66 \zeta_{6} + 33) q^{55} - 6 q^{57} - 25 \zeta_{6} q^{58} + (5 \zeta_{6} + 5) q^{59} + (27 \zeta_{6} - 27) q^{60} + (8 \zeta_{6} - 16) q^{61} + ( - 38 \zeta_{6} + 19) q^{62} + 13 q^{64} + 36 \zeta_{6} q^{65} + (11 \zeta_{6} + 11) q^{66} + ( - 52 \zeta_{6} + 52) q^{67} + (42 \zeta_{6} - 84) q^{68} + (56 \zeta_{6} - 28) q^{69} + 64 q^{71} - 21 \zeta_{6} q^{72} + ( - 4 \zeta_{6} - 4) q^{73} + ( - 58 \zeta_{6} + 58) q^{74} + (2 \zeta_{6} - 4) q^{75} + ( - 12 \zeta_{6} + 6) q^{76} - 12 q^{78} - 17 \zeta_{6} q^{79} + ( - 15 \zeta_{6} - 15) q^{80} + (9 \zeta_{6} - 9) q^{81} + ( - 2 \zeta_{6} + 4) q^{82} + (62 \zeta_{6} - 31) q^{83} - 126 q^{85} - 26 \zeta_{6} q^{86} + ( - 25 \zeta_{6} - 25) q^{87} + (77 \zeta_{6} - 77) q^{88} + ( - 46 \zeta_{6} + 92) q^{89} + ( - 18 \zeta_{6} + 9) q^{90} - 84 q^{92} - 57 \zeta_{6} q^{93} + ( - 44 \zeta_{6} - 44) q^{94} + ( - 18 \zeta_{6} + 18) q^{95} + ( - 33 \zeta_{6} + 66) q^{96} + (106 \zeta_{6} - 53) q^{97} + 33 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 3 q^{3} + 3 q^{4} + 9 q^{5} - 14 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 3 q^{3} + 3 q^{4} + 9 q^{5} - 14 q^{8} + 3 q^{9} - 9 q^{10} + 11 q^{11} - 9 q^{12} - 18 q^{15} - 5 q^{16} - 42 q^{17} + 3 q^{18} + 6 q^{19} - 22 q^{22} - 28 q^{23} + 21 q^{24} + 2 q^{25} + 12 q^{26} + 50 q^{29} + 9 q^{30} + 57 q^{31} - 33 q^{32} - 33 q^{33} + 18 q^{36} + 58 q^{37} - 6 q^{38} + 12 q^{39} - 63 q^{40} + 52 q^{43} - 33 q^{44} + 27 q^{45} - 28 q^{46} + 132 q^{47} - 4 q^{50} + 42 q^{51} + 36 q^{52} - 31 q^{53} - 9 q^{54} - 12 q^{57} - 25 q^{58} + 15 q^{59} - 27 q^{60} - 24 q^{61} + 26 q^{64} + 36 q^{65} + 33 q^{66} + 52 q^{67} - 126 q^{68} + 128 q^{71} - 21 q^{72} - 12 q^{73} + 58 q^{74} - 6 q^{75} - 24 q^{78} - 17 q^{79} - 45 q^{80} - 9 q^{81} + 6 q^{82} - 252 q^{85} - 26 q^{86} - 75 q^{87} - 77 q^{88} + 138 q^{89} - 168 q^{92} - 57 q^{93} - 132 q^{94} + 18 q^{95} + 99 q^{96} + 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i −1.50000 + 0.866025i 1.50000 + 2.59808i 4.50000 + 2.59808i 1.73205i 0 −7.00000 1.50000 2.59808i −4.50000 + 2.59808i
31.1 −0.500000 0.866025i −1.50000 0.866025i 1.50000 2.59808i 4.50000 2.59808i 1.73205i 0 −7.00000 1.50000 + 2.59808i −4.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.3.f.c 2
3.b odd 2 1 441.3.m.e 2
7.b odd 2 1 21.3.f.b 2
7.c even 3 1 21.3.f.b 2
7.c even 3 1 147.3.d.b 2
7.d odd 6 1 147.3.d.b 2
7.d odd 6 1 inner 147.3.f.c 2
21.c even 2 1 63.3.m.c 2
21.g even 6 1 441.3.d.b 2
21.g even 6 1 441.3.m.e 2
21.h odd 6 1 63.3.m.c 2
21.h odd 6 1 441.3.d.b 2
28.d even 2 1 336.3.bh.a 2
28.f even 6 1 2352.3.f.d 2
28.g odd 6 1 336.3.bh.a 2
28.g odd 6 1 2352.3.f.d 2
35.c odd 2 1 525.3.o.g 2
35.f even 4 2 525.3.s.c 4
35.j even 6 1 525.3.o.g 2
35.l odd 12 2 525.3.s.c 4
84.h odd 2 1 1008.3.cg.g 2
84.n even 6 1 1008.3.cg.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.b 2 7.b odd 2 1
21.3.f.b 2 7.c even 3 1
63.3.m.c 2 21.c even 2 1
63.3.m.c 2 21.h odd 6 1
147.3.d.b 2 7.c even 3 1
147.3.d.b 2 7.d odd 6 1
147.3.f.c 2 1.a even 1 1 trivial
147.3.f.c 2 7.d odd 6 1 inner
336.3.bh.a 2 28.d even 2 1
336.3.bh.a 2 28.g odd 6 1
441.3.d.b 2 21.g even 6 1
441.3.d.b 2 21.h odd 6 1
441.3.m.e 2 3.b odd 2 1
441.3.m.e 2 21.g even 6 1
525.3.o.g 2 35.c odd 2 1
525.3.o.g 2 35.j even 6 1
525.3.s.c 4 35.f even 4 2
525.3.s.c 4 35.l odd 12 2
1008.3.cg.g 2 84.h odd 2 1
1008.3.cg.g 2 84.n even 6 1
2352.3.f.d 2 28.f even 6 1
2352.3.f.d 2 28.g odd 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 9T_{5} + 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$13$ \( T^{2} + 48 \) Copy content Toggle raw display
$17$ \( T^{2} + 42T + 588 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 28T + 784 \) Copy content Toggle raw display
$29$ \( (T - 25)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 57T + 1083 \) Copy content Toggle raw display
$37$ \( T^{2} - 58T + 3364 \) Copy content Toggle raw display
$41$ \( T^{2} + 12 \) Copy content Toggle raw display
$43$ \( (T - 26)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 132T + 5808 \) Copy content Toggle raw display
$53$ \( T^{2} + 31T + 961 \) Copy content Toggle raw display
$59$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$61$ \( T^{2} + 24T + 192 \) Copy content Toggle raw display
$67$ \( T^{2} - 52T + 2704 \) Copy content Toggle raw display
$71$ \( (T - 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$79$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$83$ \( T^{2} + 2883 \) Copy content Toggle raw display
$89$ \( T^{2} - 138T + 6348 \) Copy content Toggle raw display
$97$ \( T^{2} + 8427 \) Copy content Toggle raw display
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