Properties

Label 147.3.b.c
Level $147$
Weight $3$
Character orbit 147.b
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(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: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) 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: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - \beta - 2) q^{3} - q^{4} + \beta q^{5} + ( - 2 \beta + 5) q^{6} + 3 \beta q^{8} + (4 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + ( - \beta - 2) q^{3} - q^{4} + \beta q^{5} + ( - 2 \beta + 5) q^{6} + 3 \beta q^{8} + (4 \beta - 1) q^{9} - 5 q^{10} + 5 \beta q^{11} + (\beta + 2) q^{12} + 2 q^{13} + ( - 2 \beta + 5) q^{15} - 19 q^{16} + 12 \beta q^{17} + ( - \beta - 20) q^{18} - 16 q^{19} - \beta q^{20} - 25 q^{22} - 6 \beta q^{23} + ( - 6 \beta + 15) q^{24} + 20 q^{25} + 2 \beta q^{26} + ( - 7 \beta + 22) q^{27} - 7 \beta q^{29} + (5 \beta + 10) q^{30} + 3 q^{31} - 7 \beta q^{32} + ( - 10 \beta + 25) q^{33} - 60 q^{34} + ( - 4 \beta + 1) q^{36} + 12 q^{37} - 16 \beta q^{38} + ( - 2 \beta - 4) q^{39} - 15 q^{40} + 14 \beta q^{41} + 44 q^{43} - 5 \beta q^{44} + ( - \beta - 20) q^{45} + 30 q^{46} - 6 \beta q^{47} + (19 \beta + 38) q^{48} + 20 \beta q^{50} + ( - 24 \beta + 60) q^{51} - 2 q^{52} - 9 \beta q^{53} + (22 \beta + 35) q^{54} - 25 q^{55} + (16 \beta + 32) q^{57} + 35 q^{58} - 9 \beta q^{59} + (2 \beta - 5) q^{60} + 26 q^{61} + 3 \beta q^{62} - 41 q^{64} + 2 \beta q^{65} + (25 \beta + 50) q^{66} + 52 q^{67} - 12 \beta q^{68} + (12 \beta - 30) q^{69} - 42 \beta q^{71} + ( - 3 \beta - 60) q^{72} - 18 q^{73} + 12 \beta q^{74} + ( - 20 \beta - 40) q^{75} + 16 q^{76} + ( - 4 \beta + 10) q^{78} - 79 q^{79} - 19 \beta q^{80} + ( - 8 \beta - 79) q^{81} - 70 q^{82} - 63 \beta q^{83} - 60 q^{85} + 44 \beta q^{86} + (14 \beta - 35) q^{87} - 75 q^{88} + 22 \beta q^{89} + ( - 20 \beta + 5) q^{90} + 6 \beta q^{92} + ( - 3 \beta - 6) q^{93} + 30 q^{94} - 16 \beta q^{95} + (14 \beta - 35) q^{96} + 93 q^{97} + ( - 5 \beta - 100) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} - 2 q^{4} + 10 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} - 2 q^{4} + 10 q^{6} - 2 q^{9} - 10 q^{10} + 4 q^{12} + 4 q^{13} + 10 q^{15} - 38 q^{16} - 40 q^{18} - 32 q^{19} - 50 q^{22} + 30 q^{24} + 40 q^{25} + 44 q^{27} + 20 q^{30} + 6 q^{31} + 50 q^{33} - 120 q^{34} + 2 q^{36} + 24 q^{37} - 8 q^{39} - 30 q^{40} + 88 q^{43} - 40 q^{45} + 60 q^{46} + 76 q^{48} + 120 q^{51} - 4 q^{52} + 70 q^{54} - 50 q^{55} + 64 q^{57} + 70 q^{58} - 10 q^{60} + 52 q^{61} - 82 q^{64} + 100 q^{66} + 104 q^{67} - 60 q^{69} - 120 q^{72} - 36 q^{73} - 80 q^{75} + 32 q^{76} + 20 q^{78} - 158 q^{79} - 158 q^{81} - 140 q^{82} - 120 q^{85} - 70 q^{87} - 150 q^{88} + 10 q^{90} - 12 q^{93} + 60 q^{94} - 70 q^{96} + 186 q^{97} - 200 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
2.23607i
2.23607i
2.23607i −2.00000 + 2.23607i −1.00000 2.23607i 5.00000 + 4.47214i 0 6.70820i −1.00000 8.94427i −5.00000
50.2 2.23607i −2.00000 2.23607i −1.00000 2.23607i 5.00000 4.47214i 0 6.70820i −1.00000 + 8.94427i −5.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 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.c 2
3.b odd 2 1 inner 147.3.b.c 2
7.b odd 2 1 147.3.b.d 2
7.c even 3 2 147.3.h.b 4
7.d odd 6 2 21.3.h.b 4
21.c even 2 1 147.3.b.d 2
21.g even 6 2 21.3.h.b 4
21.h odd 6 2 147.3.h.b 4
28.f even 6 2 336.3.bn.f 4
84.j odd 6 2 336.3.bn.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.h.b 4 7.d odd 6 2
21.3.h.b 4 21.g even 6 2
147.3.b.c 2 1.a even 1 1 trivial
147.3.b.c 2 3.b odd 2 1 inner
147.3.b.d 2 7.b odd 2 1
147.3.b.d 2 21.c even 2 1
147.3.h.b 4 7.c even 3 2
147.3.h.b 4 21.h odd 6 2
336.3.bn.f 4 28.f even 6 2
336.3.bn.f 4 84.j odd 6 2

Hecke kernels

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

\( T_{2}^{2} + 5 \) Copy content Toggle raw display
\( T_{5}^{2} + 5 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 5 \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 125 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 720 \) Copy content Toggle raw display
$19$ \( (T + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 180 \) Copy content Toggle raw display
$29$ \( T^{2} + 245 \) Copy content Toggle raw display
$31$ \( (T - 3)^{2} \) Copy content Toggle raw display
$37$ \( (T - 12)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 980 \) Copy content Toggle raw display
$43$ \( (T - 44)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 180 \) Copy content Toggle raw display
$53$ \( T^{2} + 405 \) Copy content Toggle raw display
$59$ \( T^{2} + 405 \) Copy content Toggle raw display
$61$ \( (T - 26)^{2} \) Copy content Toggle raw display
$67$ \( (T - 52)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 8820 \) Copy content Toggle raw display
$73$ \( (T + 18)^{2} \) Copy content Toggle raw display
$79$ \( (T + 79)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 19845 \) Copy content Toggle raw display
$89$ \( T^{2} + 2420 \) Copy content Toggle raw display
$97$ \( (T - 93)^{2} \) Copy content Toggle raw display
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