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Properties

Label 63.3.m.e
Level $63$
Weight $3$
Character orbit 63.m
Analytic conductor $1.717$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,3,Mod(10,63)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 63.m (of order \(6\), 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: \(1.71662566547\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (9 \beta_1 - 9) q^{4} + (2 \beta_{3} + \beta_{2}) q^{5} + ( - 3 \beta_1 - 5) q^{7} - 5 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (9 \beta_1 - 9) q^{4} + (2 \beta_{3} + \beta_{2}) q^{5} + ( - 3 \beta_1 - 5) q^{7} - 5 \beta_{3} q^{8} + (13 \beta_1 + 13) q^{10} + (\beta_{3} + \beta_{2}) q^{11} + ( - 12 \beta_1 + 6) q^{13} + (3 \beta_{3} + 8 \beta_{2}) q^{14} - 29 \beta_1 q^{16} + (4 \beta_{3} - 4 \beta_{2}) q^{17} + (8 \beta_1 - 16) q^{19} + ( - 9 \beta_{3} - 18 \beta_{2}) q^{20} + 13 q^{22} + 2 \beta_{2} q^{23} + ( - 14 \beta_1 + 14) q^{25} + (12 \beta_{3} + 6 \beta_{2}) q^{26} + ( - 45 \beta_1 + 72) q^{28} - 7 \beta_{3} q^{29} + ( - 17 \beta_1 - 17) q^{31} + (9 \beta_{3} + 9 \beta_{2}) q^{32} + (104 \beta_1 - 52) q^{34} + ( - 13 \beta_{3} - 2 \beta_{2}) q^{35} - 8 \beta_1 q^{37} + ( - 8 \beta_{3} + 8 \beta_{2}) q^{38} + (65 \beta_1 - 130) q^{40} + (6 \beta_{3} + 12 \beta_{2}) q^{41} - 4 q^{43} - 9 \beta_{2} q^{44} + ( - 26 \beta_1 + 26) q^{46} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{47} + (39 \beta_1 + 16) q^{49} + 14 \beta_{3} q^{50} + (54 \beta_1 + 54) q^{52} + ( - 23 \beta_{3} - 23 \beta_{2}) q^{53} + ( - 26 \beta_1 + 13) q^{55} + (25 \beta_{3} - 15 \beta_{2}) q^{56} - 91 \beta_1 q^{58} + (13 \beta_{3} - 13 \beta_{2}) q^{59} + ( - 22 \beta_1 + 44) q^{61} + (17 \beta_{3} + 34 \beta_{2}) q^{62} + q^{64} + 18 \beta_{2} q^{65} + (104 \beta_1 - 104) q^{67} + ( - 72 \beta_{3} - 36 \beta_{2}) q^{68} + ( - 143 \beta_1 - 26) q^{70} + 26 \beta_{3} q^{71} + (28 \beta_1 + 28) q^{73} + (8 \beta_{3} + 8 \beta_{2}) q^{74} + ( - 144 \beta_1 + 72) q^{76} + ( - 8 \beta_{3} - 5 \beta_{2}) q^{77} - 11 \beta_1 q^{79} + ( - 29 \beta_{3} + 29 \beta_{2}) q^{80} + ( - 78 \beta_1 + 156) q^{82} + (21 \beta_{3} + 42 \beta_{2}) q^{83} + 156 q^{85} + 4 \beta_{2} q^{86} + (65 \beta_1 - 65) q^{88} + (44 \beta_{3} + 22 \beta_{2}) q^{89} + (78 \beta_1 - 66) q^{91} + 18 \beta_{3} q^{92} + ( - 26 \beta_1 - 26) q^{94} + ( - 24 \beta_{3} - 24 \beta_{2}) q^{95} + ( - 42 \beta_1 + 21) q^{97} + ( - 39 \beta_{3} - 55 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{4} - 26 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 18 q^{4} - 26 q^{7} + 78 q^{10} - 58 q^{16} - 48 q^{19} + 52 q^{22} + 28 q^{25} + 198 q^{28} - 102 q^{31} - 16 q^{37} - 390 q^{40} - 16 q^{43} + 52 q^{46} + 142 q^{49} + 324 q^{52} - 182 q^{58} + 132 q^{61} + 4 q^{64} - 208 q^{67} - 390 q^{70} + 168 q^{73} - 22 q^{79} + 468 q^{82} + 624 q^{85} - 130 q^{88} - 108 q^{91} - 156 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu + 9 ) / 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 4\nu^{2} + 28\nu - 9 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 5 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 7\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} - 5 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
1.15139 + 1.99426i
−0.651388 − 1.12824i
1.15139 − 1.99426i
−0.651388 + 1.12824i
−1.80278 − 3.12250i 0 −4.50000 + 7.79423i −5.40833 + 3.12250i 0 −6.50000 − 2.59808i 18.0278 0 19.5000 + 11.2583i
10.2 1.80278 + 3.12250i 0 −4.50000 + 7.79423i 5.40833 − 3.12250i 0 −6.50000 − 2.59808i −18.0278 0 19.5000 + 11.2583i
19.1 −1.80278 + 3.12250i 0 −4.50000 − 7.79423i −5.40833 − 3.12250i 0 −6.50000 + 2.59808i 18.0278 0 19.5000 − 11.2583i
19.2 1.80278 − 3.12250i 0 −4.50000 − 7.79423i 5.40833 + 3.12250i 0 −6.50000 + 2.59808i −18.0278 0 19.5000 − 11.2583i
\(n\): 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
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.3.m.e ✓ 4
3.b odd 2 1 inner 63.3.m.e ✓ 4
4.b odd 2 1 1008.3.cg.k 4
7.b odd 2 1 441.3.m.h 4
7.c even 3 1 441.3.d.f 4
7.c even 3 1 441.3.m.h 4
7.d odd 6 1 inner 63.3.m.e ✓ 4
7.d odd 6 1 441.3.d.f 4
12.b even 2 1 1008.3.cg.k 4
21.c even 2 1 441.3.m.h 4
21.g even 6 1 inner 63.3.m.e ✓ 4
21.g even 6 1 441.3.d.f 4
21.h odd 6 1 441.3.d.f 4
21.h odd 6 1 441.3.m.h 4
28.f even 6 1 1008.3.cg.k 4
84.j odd 6 1 1008.3.cg.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.m.e ✓ 4 1.a even 1 1 trivial
63.3.m.e ✓ 4 3.b odd 2 1 inner
63.3.m.e ✓ 4 7.d odd 6 1 inner
63.3.m.e ✓ 4 21.g even 6 1 inner
441.3.d.f 4 7.c even 3 1
441.3.d.f 4 7.d odd 6 1
441.3.d.f 4 21.g even 6 1
441.3.d.f 4 21.h odd 6 1
441.3.m.h 4 7.b odd 2 1
441.3.m.h 4 7.c even 3 1
441.3.m.h 4 21.c even 2 1
441.3.m.h 4 21.h odd 6 1
1008.3.cg.k 4 4.b odd 2 1
1008.3.cg.k 4 12.b even 2 1
1008.3.cg.k 4 28.f even 6 1
1008.3.cg.k 4 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 13T_{2}^{2} + 169 \) acting on \(S_{3}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 39T^{2} + 1521 \) Copy content Toggle raw display
$7$ \( (T^{2} + 13 T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$13$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 624 T^{2} + 389376 \) Copy content Toggle raw display
$19$ \( (T^{2} + 24 T + 192)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 52T^{2} + 2704 \) Copy content Toggle raw display
$29$ \( (T^{2} - 637)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 51 T + 867)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1404)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 156 T^{2} + 24336 \) Copy content Toggle raw display
$53$ \( T^{4} + 6877 T^{2} + 47293129 \) Copy content Toggle raw display
$59$ \( T^{4} - 6591 T^{2} + 43441281 \) Copy content Toggle raw display
$61$ \( (T^{2} - 66 T + 1452)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 104 T + 10816)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 8788)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 84 T + 2352)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 17199)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 18876 T^{2} + 356303376 \) Copy content Toggle raw display
$97$ \( (T^{2} + 1323)^{2} \) Copy content Toggle raw display
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