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Properties

Label 126.3.s.b
Level $126$
Weight $3$
Character orbit 126.s
Analytic conductor $3.433$
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] = [126,3,Mod(53,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 126.s (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: \(3.43325133094\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + \beta_1 q^{5} + (3 \beta_{2} + 5) q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + \beta_1 q^{5} + (3 \beta_{2} + 5) q^{7} + 2 \beta_{3} q^{8} + 2 \beta_{2} q^{10} + (5 \beta_{3} - 5 \beta_1) q^{11} + 15 q^{13} + (3 \beta_{3} + 5 \beta_1) q^{14} + (4 \beta_{2} - 4) q^{16} + (8 \beta_{3} - 8 \beta_1) q^{17} + ( - 13 \beta_{2} + 13) q^{19} + 2 \beta_{3} q^{20} - 10 q^{22} - 16 \beta_1 q^{23} - 23 \beta_{2} q^{25} + 15 \beta_1 q^{26} + (16 \beta_{2} - 6) q^{28} - 16 \beta_{3} q^{29} - 3 \beta_{2} q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} - 16 q^{34} + (3 \beta_{3} + 5 \beta_1) q^{35} + (17 \beta_{2} - 17) q^{37} + ( - 13 \beta_{3} + 13 \beta_1) q^{38} + (4 \beta_{2} - 4) q^{40} - 57 \beta_{3} q^{41} - 85 q^{43} - 10 \beta_1 q^{44} - 32 \beta_{2} q^{46} + 51 \beta_1 q^{47} + (39 \beta_{2} + 16) q^{49} - 23 \beta_{3} q^{50} + 30 \beta_{2} q^{52} + ( - 24 \beta_{3} + 24 \beta_1) q^{53} - 10 q^{55} + (16 \beta_{3} - 6 \beta_1) q^{56} + ( - 32 \beta_{2} + 32) q^{58} + (64 \beta_{3} - 64 \beta_1) q^{59} + ( - 72 \beta_{2} + 72) q^{61} - 3 \beta_{3} q^{62} - 8 q^{64} + 15 \beta_1 q^{65} - 43 \beta_{2} q^{67} - 16 \beta_1 q^{68} + (16 \beta_{2} - 6) q^{70} + 37 \beta_{3} q^{71} + 95 \beta_{2} q^{73} + (17 \beta_{3} - 17 \beta_1) q^{74} + 26 q^{76} + (25 \beta_{3} - 40 \beta_1) q^{77} + (69 \beta_{2} - 69) q^{79} + (4 \beta_{3} - 4 \beta_1) q^{80} + ( - 114 \beta_{2} + 114) q^{82} + 43 \beta_{3} q^{83} - 16 q^{85} - 85 \beta_1 q^{86} - 20 \beta_{2} q^{88} + 96 \beta_1 q^{89} + (45 \beta_{2} + 75) q^{91} - 32 \beta_{3} q^{92} + 102 \beta_{2} q^{94} + ( - 13 \beta_{3} + 13 \beta_1) q^{95} + 16 q^{97} + (39 \beta_{3} + 16 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 26 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 26 q^{7} + 4 q^{10} + 60 q^{13} - 8 q^{16} + 26 q^{19} - 40 q^{22} - 46 q^{25} + 8 q^{28} - 6 q^{31} - 64 q^{34} - 34 q^{37} - 8 q^{40} - 340 q^{43} - 64 q^{46} + 142 q^{49} + 60 q^{52} - 40 q^{55} + 64 q^{58} + 144 q^{61} - 32 q^{64} - 86 q^{67} + 8 q^{70} + 190 q^{73} + 104 q^{76} - 138 q^{79} + 228 q^{82} - 64 q^{85} - 40 q^{88} + 390 q^{91} + 204 q^{94} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.22474 + 0.707107i
1.22474 − 0.707107i
−1.22474 − 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 − 1.73205i −1.22474 + 0.707107i 0 6.50000 − 2.59808i 2.82843i 0 1.00000 − 1.73205i
53.2 1.22474 − 0.707107i 0 1.00000 − 1.73205i 1.22474 − 0.707107i 0 6.50000 − 2.59808i − 2.82843i 0 1.00000 − 1.73205i
107.1 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −1.22474 − 0.707107i 0 6.50000 + 2.59808i − 2.82843i 0 1.00000 + 1.73205i
107.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.22474 + 0.707107i 0 6.50000 + 2.59808i 2.82843i 0 1.00000 + 1.73205i
\(n\): 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.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.3.s.b ✓ 4
3.b odd 2 1 inner 126.3.s.b ✓ 4
4.b odd 2 1 1008.3.dc.a 4
7.b odd 2 1 882.3.s.c 4
7.c even 3 1 inner 126.3.s.b ✓ 4
7.c even 3 1 882.3.b.c 2
7.d odd 6 1 882.3.b.d 2
7.d odd 6 1 882.3.s.c 4
12.b even 2 1 1008.3.dc.a 4
21.c even 2 1 882.3.s.c 4
21.g even 6 1 882.3.b.d 2
21.g even 6 1 882.3.s.c 4
21.h odd 6 1 inner 126.3.s.b ✓ 4
21.h odd 6 1 882.3.b.c 2
28.g odd 6 1 1008.3.dc.a 4
84.n even 6 1 1008.3.dc.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.s.b ✓ 4 1.a even 1 1 trivial
126.3.s.b ✓ 4 3.b odd 2 1 inner
126.3.s.b ✓ 4 7.c even 3 1 inner
126.3.s.b ✓ 4 21.h odd 6 1 inner
882.3.b.c 2 7.c even 3 1
882.3.b.c 2 21.h odd 6 1
882.3.b.d 2 7.d odd 6 1
882.3.b.d 2 21.g even 6 1
882.3.s.c 4 7.b odd 2 1
882.3.s.c 4 7.d odd 6 1
882.3.s.c 4 21.c even 2 1
882.3.s.c 4 21.g even 6 1
1008.3.dc.a 4 4.b odd 2 1
1008.3.dc.a 4 12.b even 2 1
1008.3.dc.a 4 28.g odd 6 1
1008.3.dc.a 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 2T_{5}^{2} + 4 \) acting on \(S_{3}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T^{2} - 13 T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 50T^{2} + 2500 \) Copy content Toggle raw display
$13$ \( (T - 15)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 128 T^{2} + 16384 \) Copy content Toggle raw display
$19$ \( (T^{2} - 13 T + 169)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 512 T^{2} + 262144 \) Copy content Toggle raw display
$29$ \( (T^{2} + 512)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 17 T + 289)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 6498)^{2} \) Copy content Toggle raw display
$43$ \( (T + 85)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 5202 T^{2} + 27060804 \) Copy content Toggle raw display
$53$ \( T^{4} - 1152 T^{2} + 1327104 \) Copy content Toggle raw display
$59$ \( T^{4} - 8192 T^{2} + 67108864 \) Copy content Toggle raw display
$61$ \( (T^{2} - 72 T + 5184)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 43 T + 1849)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2738)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 95 T + 9025)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 69 T + 4761)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 3698)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 18432 T^{2} + 339738624 \) Copy content Toggle raw display
$97$ \( (T - 16)^{4} \) Copy content Toggle raw display
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