Properties

Label 63.2.f.a
Level $63$
Weight $2$
Character orbit 63.f
Analytic conductor $0.503$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,2,Mod(22,63)]
 
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([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 63.f (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: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_1 - 1) q^{2} + (\beta_{5} - \beta_{3}) q^{3} + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - \beta_1) q^{4}+ \cdots + ( - 2 \beta_{5} + \beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_1 - 1) q^{2} + (\beta_{5} - \beta_{3}) q^{3} + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - \beta_1) q^{4}+ \cdots + (3 \beta_{5} - 6 \beta_{4} - 3 \beta_{3} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} - 3 q^{5} - 9 q^{6} - 3 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 3 q^{4} - 3 q^{5} - 9 q^{6} - 3 q^{7} + 12 q^{8} - 6 q^{11} + 18 q^{12} + 3 q^{13} - 3 q^{14} - 9 q^{15} - 3 q^{16} + 12 q^{17} + 9 q^{18} - 6 q^{19} + 6 q^{20} - 9 q^{22} - 12 q^{23} - 27 q^{24} + 6 q^{25} - 6 q^{26} + 27 q^{27} + 6 q^{28} - 9 q^{29} + 18 q^{30} + 3 q^{31} - 9 q^{34} + 6 q^{35} - 27 q^{36} - 6 q^{37} - 6 q^{38} - 18 q^{39} + 9 q^{40} + 9 q^{42} + 3 q^{43} + 30 q^{44} + 9 q^{45} - 3 q^{47} - 3 q^{49} + 6 q^{50} + 9 q^{51} + 21 q^{52} + 12 q^{53} - 27 q^{54} - 6 q^{56} - 9 q^{57} + 9 q^{58} + 3 q^{59} - 6 q^{61} - 60 q^{62} + 24 q^{64} - 15 q^{65} + 36 q^{66} + 12 q^{67} - 6 q^{68} - 9 q^{69} + 18 q^{71} - 36 q^{72} - 42 q^{73} + 30 q^{74} - 9 q^{75} - 15 q^{76} - 6 q^{77} + 54 q^{78} + 21 q^{79} - 30 q^{80} - 18 q^{82} + 18 q^{83} - 9 q^{84} - 9 q^{85} - 6 q^{86} + 9 q^{87} - 27 q^{88} + 24 q^{89} + 27 q^{90} - 6 q^{91} - 3 q^{92} - 27 q^{93} + 18 q^{94} + 12 q^{95} + 27 q^{96} + 3 q^{97} + 6 q^{98} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{18}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{18}^{5} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{18}^{5} + \zeta_{18}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \) Copy content Toggle raw display
\(\zeta_{18}\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{2}\)\(=\) \( ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{18}^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{5}\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(10\) \(29\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
−0.766044 0.642788i
−0.173648 + 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
−0.173648 0.984808i
0.939693 + 0.342020i
−1.26604 2.19285i −1.11334 1.32683i −2.20574 + 3.82045i 0.439693 0.761570i −1.50000 + 4.12122i −0.500000 0.866025i 6.10607 −0.520945 + 2.95442i −2.22668
22.2 −0.673648 1.16679i 1.70574 0.300767i 0.0923963 0.160035i −1.26604 + 2.19285i −1.50000 1.78763i −0.500000 0.866025i −2.94356 2.81908 1.02606i 3.41147
22.3 0.439693 + 0.761570i −0.592396 + 1.62760i 0.613341 1.06234i −0.673648 + 1.16679i −1.50000 + 0.264490i −0.500000 0.866025i 2.83750 −2.29813 1.92836i −1.18479
43.1 −1.26604 + 2.19285i −1.11334 + 1.32683i −2.20574 3.82045i 0.439693 + 0.761570i −1.50000 4.12122i −0.500000 + 0.866025i 6.10607 −0.520945 2.95442i −2.22668
43.2 −0.673648 + 1.16679i 1.70574 + 0.300767i 0.0923963 + 0.160035i −1.26604 2.19285i −1.50000 + 1.78763i −0.500000 + 0.866025i −2.94356 2.81908 + 1.02606i 3.41147
43.3 0.439693 0.761570i −0.592396 1.62760i 0.613341 + 1.06234i −0.673648 1.16679i −1.50000 0.264490i −0.500000 + 0.866025i 2.83750 −2.29813 + 1.92836i −1.18479
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.2.f.a 6
3.b odd 2 1 189.2.f.b 6
4.b odd 2 1 1008.2.r.h 6
7.b odd 2 1 441.2.f.c 6
7.c even 3 1 441.2.g.c 6
7.c even 3 1 441.2.h.d 6
7.d odd 6 1 441.2.g.b 6
7.d odd 6 1 441.2.h.e 6
9.c even 3 1 inner 63.2.f.a 6
9.c even 3 1 567.2.a.h 3
9.d odd 6 1 189.2.f.b 6
9.d odd 6 1 567.2.a.c 3
12.b even 2 1 3024.2.r.k 6
21.c even 2 1 1323.2.f.d 6
21.g even 6 1 1323.2.g.e 6
21.g even 6 1 1323.2.h.b 6
21.h odd 6 1 1323.2.g.d 6
21.h odd 6 1 1323.2.h.c 6
36.f odd 6 1 1008.2.r.h 6
36.f odd 6 1 9072.2.a.ca 3
36.h even 6 1 3024.2.r.k 6
36.h even 6 1 9072.2.a.bs 3
63.g even 3 1 441.2.h.d 6
63.h even 3 1 441.2.g.c 6
63.i even 6 1 1323.2.g.e 6
63.j odd 6 1 1323.2.g.d 6
63.k odd 6 1 441.2.h.e 6
63.l odd 6 1 441.2.f.c 6
63.l odd 6 1 3969.2.a.q 3
63.n odd 6 1 1323.2.h.c 6
63.o even 6 1 1323.2.f.d 6
63.o even 6 1 3969.2.a.l 3
63.s even 6 1 1323.2.h.b 6
63.t odd 6 1 441.2.g.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 1.a even 1 1 trivial
63.2.f.a 6 9.c even 3 1 inner
189.2.f.b 6 3.b odd 2 1
189.2.f.b 6 9.d odd 6 1
441.2.f.c 6 7.b odd 2 1
441.2.f.c 6 63.l odd 6 1
441.2.g.b 6 7.d odd 6 1
441.2.g.b 6 63.t odd 6 1
441.2.g.c 6 7.c even 3 1
441.2.g.c 6 63.h even 3 1
441.2.h.d 6 7.c even 3 1
441.2.h.d 6 63.g even 3 1
441.2.h.e 6 7.d odd 6 1
441.2.h.e 6 63.k odd 6 1
567.2.a.c 3 9.d odd 6 1
567.2.a.h 3 9.c even 3 1
1008.2.r.h 6 4.b odd 2 1
1008.2.r.h 6 36.f odd 6 1
1323.2.f.d 6 21.c even 2 1
1323.2.f.d 6 63.o even 6 1
1323.2.g.d 6 21.h odd 6 1
1323.2.g.d 6 63.j odd 6 1
1323.2.g.e 6 21.g even 6 1
1323.2.g.e 6 63.i even 6 1
1323.2.h.b 6 21.g even 6 1
1323.2.h.b 6 63.s even 6 1
1323.2.h.c 6 21.h odd 6 1
1323.2.h.c 6 63.n odd 6 1
3024.2.r.k 6 12.b even 2 1
3024.2.r.k 6 36.h even 6 1
3969.2.a.l 3 63.o even 6 1
3969.2.a.q 3 63.l odd 6 1
9072.2.a.bs 3 36.h even 6 1
9072.2.a.ca 3 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3T_{2}^{5} + 9T_{2}^{4} + 6T_{2}^{3} + 9T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} - 9T^{3} + 27 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} - 3 T^{5} + \cdots + 11449 \) Copy content Toggle raw display
$17$ \( (T^{3} - 6 T^{2} + 9 T - 3)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 3 T^{2} - 6 T - 17)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{6} + 9 T^{5} + \cdots + 110889 \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} + \cdots + 104329 \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} + \cdots - 323)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 9 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{6} + 3 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$53$ \( (T^{3} - 6 T^{2} - 9 T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 3 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$67$ \( T^{6} - 12 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$71$ \( (T^{3} - 9 T^{2} - 54 T - 27)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 21 T^{2} + \cdots - 269)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 21 T^{5} + \cdots + 32761 \) Copy content Toggle raw display
$83$ \( T^{6} - 18 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$89$ \( (T^{3} - 12 T^{2} + \cdots + 813)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 3 T^{5} + \cdots + 104329 \) Copy content Toggle raw display
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