Properties

Label 147.4.e.f
Level $147$
Weight $4$
Character orbit 147.e
Analytic conductor $8.673$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,4,Mod(67,147)]
 
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, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.67");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.e (of order \(3\), 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: \(2\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} - 12 \zeta_{6} q^{5} + 3 q^{6} + 15 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} - 12 \zeta_{6} q^{5} + 3 q^{6} + 15 q^{8} - 9 \zeta_{6} q^{9} + ( - 12 \zeta_{6} + 12) q^{10} + (20 \zeta_{6} - 20) q^{11} - 21 \zeta_{6} q^{12} - 84 q^{13} - 36 q^{15} - 41 \zeta_{6} q^{16} + ( - 96 \zeta_{6} + 96) q^{17} + ( - 9 \zeta_{6} + 9) q^{18} - 12 \zeta_{6} q^{19} - 84 q^{20} - 20 q^{22} + 176 \zeta_{6} q^{23} + ( - 45 \zeta_{6} + 45) q^{24} + (19 \zeta_{6} - 19) q^{25} - 84 \zeta_{6} q^{26} - 27 q^{27} + 58 q^{29} - 36 \zeta_{6} q^{30} + ( - 264 \zeta_{6} + 264) q^{31} + ( - 161 \zeta_{6} + 161) q^{32} + 60 \zeta_{6} q^{33} + 96 q^{34} - 63 q^{36} - 258 \zeta_{6} q^{37} + ( - 12 \zeta_{6} + 12) q^{38} + (252 \zeta_{6} - 252) q^{39} - 180 \zeta_{6} q^{40} + 156 q^{43} + 140 \zeta_{6} q^{44} + (108 \zeta_{6} - 108) q^{45} + (176 \zeta_{6} - 176) q^{46} + 408 \zeta_{6} q^{47} - 123 q^{48} - 19 q^{50} - 288 \zeta_{6} q^{51} + (588 \zeta_{6} - 588) q^{52} + ( - 722 \zeta_{6} + 722) q^{53} - 27 \zeta_{6} q^{54} + 240 q^{55} - 36 q^{57} + 58 \zeta_{6} q^{58} + (492 \zeta_{6} - 492) q^{59} + (252 \zeta_{6} - 252) q^{60} + 492 \zeta_{6} q^{61} + 264 q^{62} - 167 q^{64} + 1008 \zeta_{6} q^{65} + (60 \zeta_{6} - 60) q^{66} + (412 \zeta_{6} - 412) q^{67} - 672 \zeta_{6} q^{68} + 528 q^{69} + 296 q^{71} - 135 \zeta_{6} q^{72} + (240 \zeta_{6} - 240) q^{73} + ( - 258 \zeta_{6} + 258) q^{74} + 57 \zeta_{6} q^{75} - 84 q^{76} - 252 q^{78} - 776 \zeta_{6} q^{79} + (492 \zeta_{6} - 492) q^{80} + (81 \zeta_{6} - 81) q^{81} + 924 q^{83} - 1152 q^{85} + 156 \zeta_{6} q^{86} + ( - 174 \zeta_{6} + 174) q^{87} + (300 \zeta_{6} - 300) q^{88} + 744 \zeta_{6} q^{89} - 108 q^{90} + 1232 q^{92} - 792 \zeta_{6} q^{93} + (408 \zeta_{6} - 408) q^{94} + (144 \zeta_{6} - 144) q^{95} - 483 \zeta_{6} q^{96} - 168 q^{97} + 180 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} + 7 q^{4} - 12 q^{5} + 6 q^{6} + 30 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 3 q^{3} + 7 q^{4} - 12 q^{5} + 6 q^{6} + 30 q^{8} - 9 q^{9} + 12 q^{10} - 20 q^{11} - 21 q^{12} - 168 q^{13} - 72 q^{15} - 41 q^{16} + 96 q^{17} + 9 q^{18} - 12 q^{19} - 168 q^{20} - 40 q^{22} + 176 q^{23} + 45 q^{24} - 19 q^{25} - 84 q^{26} - 54 q^{27} + 116 q^{29} - 36 q^{30} + 264 q^{31} + 161 q^{32} + 60 q^{33} + 192 q^{34} - 126 q^{36} - 258 q^{37} + 12 q^{38} - 252 q^{39} - 180 q^{40} + 312 q^{43} + 140 q^{44} - 108 q^{45} - 176 q^{46} + 408 q^{47} - 246 q^{48} - 38 q^{50} - 288 q^{51} - 588 q^{52} + 722 q^{53} - 27 q^{54} + 480 q^{55} - 72 q^{57} + 58 q^{58} - 492 q^{59} - 252 q^{60} + 492 q^{61} + 528 q^{62} - 334 q^{64} + 1008 q^{65} - 60 q^{66} - 412 q^{67} - 672 q^{68} + 1056 q^{69} + 592 q^{71} - 135 q^{72} - 240 q^{73} + 258 q^{74} + 57 q^{75} - 168 q^{76} - 504 q^{78} - 776 q^{79} - 492 q^{80} - 81 q^{81} + 1848 q^{83} - 2304 q^{85} + 156 q^{86} + 174 q^{87} - 300 q^{88} + 744 q^{89} - 216 q^{90} + 2464 q^{92} - 792 q^{93} - 408 q^{94} - 144 q^{95} - 483 q^{96} - 336 q^{97} + 360 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 1.50000 2.59808i 3.50000 6.06218i −6.00000 10.3923i 3.00000 0 15.0000 −4.50000 7.79423i 6.00000 10.3923i
79.1 0.500000 0.866025i 1.50000 + 2.59808i 3.50000 + 6.06218i −6.00000 + 10.3923i 3.00000 0 15.0000 −4.50000 + 7.79423i 6.00000 + 10.3923i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.e.f 2
3.b odd 2 1 441.4.e.g 2
7.b odd 2 1 147.4.e.e 2
7.c even 3 1 147.4.a.d 1
7.c even 3 1 inner 147.4.e.f 2
7.d odd 6 1 147.4.a.e yes 1
7.d odd 6 1 147.4.e.e 2
21.c even 2 1 441.4.e.f 2
21.g even 6 1 441.4.a.h 1
21.g even 6 1 441.4.e.f 2
21.h odd 6 1 441.4.a.g 1
21.h odd 6 1 441.4.e.g 2
28.f even 6 1 2352.4.a.b 1
28.g odd 6 1 2352.4.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.d 1 7.c even 3 1
147.4.a.e yes 1 7.d odd 6 1
147.4.e.e 2 7.b odd 2 1
147.4.e.e 2 7.d odd 6 1
147.4.e.f 2 1.a even 1 1 trivial
147.4.e.f 2 7.c even 3 1 inner
441.4.a.g 1 21.h odd 6 1
441.4.a.h 1 21.g even 6 1
441.4.e.f 2 21.c even 2 1
441.4.e.f 2 21.g even 6 1
441.4.e.g 2 3.b odd 2 1
441.4.e.g 2 21.h odd 6 1
2352.4.a.b 1 28.f even 6 1
2352.4.a.bi 1 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 \(S_{4}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 12T_{5} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$13$ \( (T + 84)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 96T + 9216 \) Copy content Toggle raw display
$19$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$23$ \( T^{2} - 176T + 30976 \) Copy content Toggle raw display
$29$ \( (T - 58)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 264T + 69696 \) Copy content Toggle raw display
$37$ \( T^{2} + 258T + 66564 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 156)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 408T + 166464 \) Copy content Toggle raw display
$53$ \( T^{2} - 722T + 521284 \) Copy content Toggle raw display
$59$ \( T^{2} + 492T + 242064 \) Copy content Toggle raw display
$61$ \( T^{2} - 492T + 242064 \) Copy content Toggle raw display
$67$ \( T^{2} + 412T + 169744 \) Copy content Toggle raw display
$71$ \( (T - 296)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 240T + 57600 \) Copy content Toggle raw display
$79$ \( T^{2} + 776T + 602176 \) Copy content Toggle raw display
$83$ \( (T - 924)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 744T + 553536 \) Copy content Toggle raw display
$97$ \( (T + 168)^{2} \) Copy content Toggle raw display
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