Properties

Label 63.4.e.b
Level $63$
Weight $4$
Character orbit 63.e
Analytic conductor $3.717$
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] = [63,4,Mod(37,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), degree \(2\), 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: \(3.71712033036\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{2} + 4 \zeta_{6} q^{4} + ( - 7 \zeta_{6} + 7) q^{5} + ( - 14 \zeta_{6} + 21) q^{7} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{2} + 4 \zeta_{6} q^{4} + ( - 7 \zeta_{6} + 7) q^{5} + ( - 14 \zeta_{6} + 21) q^{7} + 24 q^{8} - 14 \zeta_{6} q^{10} - 5 \zeta_{6} q^{11} - 14 q^{13} + ( - 42 \zeta_{6} + 14) q^{14} + ( - 16 \zeta_{6} + 16) q^{16} - 21 \zeta_{6} q^{17} + (49 \zeta_{6} - 49) q^{19} + 28 q^{20} - 10 q^{22} + (159 \zeta_{6} - 159) q^{23} + 76 \zeta_{6} q^{25} + (28 \zeta_{6} - 28) q^{26} + (28 \zeta_{6} + 56) q^{28} - 58 q^{29} - 147 \zeta_{6} q^{31} + 160 \zeta_{6} q^{32} - 42 q^{34} + ( - 147 \zeta_{6} + 49) q^{35} + (219 \zeta_{6} - 219) q^{37} + 98 \zeta_{6} q^{38} + ( - 168 \zeta_{6} + 168) q^{40} - 350 q^{41} - 124 q^{43} + ( - 20 \zeta_{6} + 20) q^{44} + 318 \zeta_{6} q^{46} + ( - 525 \zeta_{6} + 525) q^{47} + ( - 392 \zeta_{6} + 245) q^{49} + 152 q^{50} - 56 \zeta_{6} q^{52} + 303 \zeta_{6} q^{53} - 35 q^{55} + ( - 336 \zeta_{6} + 504) q^{56} + (116 \zeta_{6} - 116) q^{58} - 105 \zeta_{6} q^{59} + ( - 413 \zeta_{6} + 413) q^{61} - 294 q^{62} + 448 q^{64} + (98 \zeta_{6} - 98) q^{65} - 415 \zeta_{6} q^{67} + ( - 84 \zeta_{6} + 84) q^{68} + ( - 98 \zeta_{6} - 196) q^{70} + 432 q^{71} + 1113 \zeta_{6} q^{73} + 438 \zeta_{6} q^{74} - 196 q^{76} + ( - 35 \zeta_{6} - 70) q^{77} + ( - 103 \zeta_{6} + 103) q^{79} - 112 \zeta_{6} q^{80} + (700 \zeta_{6} - 700) q^{82} - 1092 q^{83} - 147 q^{85} + (248 \zeta_{6} - 248) q^{86} - 120 \zeta_{6} q^{88} + (329 \zeta_{6} - 329) q^{89} + (196 \zeta_{6} - 294) q^{91} - 636 q^{92} - 1050 \zeta_{6} q^{94} + 343 \zeta_{6} q^{95} - 882 q^{97} + ( - 490 \zeta_{6} - 294) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{4} + 7 q^{5} + 28 q^{7} + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{4} + 7 q^{5} + 28 q^{7} + 48 q^{8} - 14 q^{10} - 5 q^{11} - 28 q^{13} - 14 q^{14} + 16 q^{16} - 21 q^{17} - 49 q^{19} + 56 q^{20} - 20 q^{22} - 159 q^{23} + 76 q^{25} - 28 q^{26} + 140 q^{28} - 116 q^{29} - 147 q^{31} + 160 q^{32} - 84 q^{34} - 49 q^{35} - 219 q^{37} + 98 q^{38} + 168 q^{40} - 700 q^{41} - 248 q^{43} + 20 q^{44} + 318 q^{46} + 525 q^{47} + 98 q^{49} + 304 q^{50} - 56 q^{52} + 303 q^{53} - 70 q^{55} + 672 q^{56} - 116 q^{58} - 105 q^{59} + 413 q^{61} - 588 q^{62} + 896 q^{64} - 98 q^{65} - 415 q^{67} + 84 q^{68} - 490 q^{70} + 864 q^{71} + 1113 q^{73} + 438 q^{74} - 392 q^{76} - 175 q^{77} + 103 q^{79} - 112 q^{80} - 700 q^{82} - 2184 q^{83} - 294 q^{85} - 248 q^{86} - 120 q^{88} - 329 q^{89} - 392 q^{91} - 1272 q^{92} - 1050 q^{94} + 343 q^{95} - 1764 q^{97} - 1078 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 1.73205i 0 2.00000 + 3.46410i 3.50000 6.06218i 0 14.0000 12.1244i 24.0000 0 −7.00000 12.1244i
46.1 1.00000 + 1.73205i 0 2.00000 3.46410i 3.50000 + 6.06218i 0 14.0000 + 12.1244i 24.0000 0 −7.00000 + 12.1244i
\(n\): 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 63.4.e.b 2
3.b odd 2 1 7.4.c.a 2
7.b odd 2 1 441.4.e.k 2
7.c even 3 1 inner 63.4.e.b 2
7.c even 3 1 441.4.a.d 1
7.d odd 6 1 441.4.a.e 1
7.d odd 6 1 441.4.e.k 2
12.b even 2 1 112.4.i.c 2
15.d odd 2 1 175.4.e.a 2
15.e even 4 2 175.4.k.a 4
21.c even 2 1 49.4.c.a 2
21.g even 6 1 49.4.a.c 1
21.g even 6 1 49.4.c.a 2
21.h odd 6 1 7.4.c.a 2
21.h odd 6 1 49.4.a.d 1
24.f even 2 1 448.4.i.a 2
24.h odd 2 1 448.4.i.f 2
84.j odd 6 1 784.4.a.r 1
84.n even 6 1 112.4.i.c 2
84.n even 6 1 784.4.a.b 1
105.o odd 6 1 175.4.e.a 2
105.o odd 6 1 1225.4.a.c 1
105.p even 6 1 1225.4.a.d 1
105.x even 12 2 175.4.k.a 4
168.s odd 6 1 448.4.i.f 2
168.v even 6 1 448.4.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 3.b odd 2 1
7.4.c.a 2 21.h odd 6 1
49.4.a.c 1 21.g even 6 1
49.4.a.d 1 21.h odd 6 1
49.4.c.a 2 21.c even 2 1
49.4.c.a 2 21.g even 6 1
63.4.e.b 2 1.a even 1 1 trivial
63.4.e.b 2 7.c even 3 1 inner
112.4.i.c 2 12.b even 2 1
112.4.i.c 2 84.n even 6 1
175.4.e.a 2 15.d odd 2 1
175.4.e.a 2 105.o odd 6 1
175.4.k.a 4 15.e even 4 2
175.4.k.a 4 105.x even 12 2
441.4.a.d 1 7.c even 3 1
441.4.a.e 1 7.d odd 6 1
441.4.e.k 2 7.b odd 2 1
441.4.e.k 2 7.d odd 6 1
448.4.i.a 2 24.f even 2 1
448.4.i.a 2 168.v even 6 1
448.4.i.f 2 24.h odd 2 1
448.4.i.f 2 168.s odd 6 1
784.4.a.b 1 84.n even 6 1
784.4.a.r 1 84.j odd 6 1
1225.4.a.c 1 105.o odd 6 1
1225.4.a.d 1 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} + 4 \) acting on \(S_{4}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$7$ \( T^{2} - 28T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$13$ \( (T + 14)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 21T + 441 \) Copy content Toggle raw display
$19$ \( T^{2} + 49T + 2401 \) Copy content Toggle raw display
$23$ \( T^{2} + 159T + 25281 \) Copy content Toggle raw display
$29$ \( (T + 58)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 147T + 21609 \) Copy content Toggle raw display
$37$ \( T^{2} + 219T + 47961 \) Copy content Toggle raw display
$41$ \( (T + 350)^{2} \) Copy content Toggle raw display
$43$ \( (T + 124)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 525T + 275625 \) Copy content Toggle raw display
$53$ \( T^{2} - 303T + 91809 \) Copy content Toggle raw display
$59$ \( T^{2} + 105T + 11025 \) Copy content Toggle raw display
$61$ \( T^{2} - 413T + 170569 \) Copy content Toggle raw display
$67$ \( T^{2} + 415T + 172225 \) Copy content Toggle raw display
$71$ \( (T - 432)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 1113 T + 1238769 \) Copy content Toggle raw display
$79$ \( T^{2} - 103T + 10609 \) Copy content Toggle raw display
$83$ \( (T + 1092)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 329T + 108241 \) Copy content Toggle raw display
$97$ \( (T + 882)^{2} \) Copy content Toggle raw display
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