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Properties

Label 126.2.f.d
Level $126$
Weight $2$
Character orbit 126.f
Analytic conductor $1.006$
Analytic rank $0$
Dimension $4$
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] = [126,2,Mod(43,126)]
 
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([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.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: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_1 q^{3} + (\beta_{2} - 1) q^{4} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{5} + (\beta_{3} - \beta_1) q^{6} - \beta_{2} q^{7} - q^{8} + (\beta_{3} + 3 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_1 q^{3} + (\beta_{2} - 1) q^{4} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{5} + (\beta_{3} - \beta_1) q^{6} - \beta_{2} q^{7} - q^{8} + (\beta_{3} + 3 \beta_{2}) q^{9} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{10} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{11} + \beta_{3} q^{12} + (2 \beta_{2} - 2) q^{13} + ( - \beta_{2} + 1) q^{14} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{15} - \beta_{2} q^{16} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{17} + (3 \beta_{2} + \beta_1 - 3) q^{18} + 5 q^{19} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{20} + ( - \beta_{3} + \beta_1) q^{21} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{22} + ( - 2 \beta_{3} - 5 \beta_{2} + \beta_1 + 4) q^{23} + \beta_1 q^{24} + (3 \beta_{3} - 7 \beta_{2} - 6 \beta_1 + 3) q^{25} - 2 q^{26} + (2 \beta_{3} - 2 \beta_1 - 3) q^{27} + q^{28} + ( - 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 2) q^{29} + (\beta_{3} + 3 \beta_{2} + \beta_1 + 3) q^{30} + (2 \beta_{2} - 2) q^{31} + ( - \beta_{2} + 1) q^{32} + ( - 2 \beta_{3} - 6 \beta_{2} + \beta_1 + 3) q^{33} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{34} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{35} + ( - \beta_{3} + \beta_1 - 3) q^{36} + 2 q^{37} + 5 \beta_{2} q^{38} + 2 \beta_{3} q^{39} + (2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{40} + (2 \beta_{3} + 8 \beta_{2} - \beta_1 - 7) q^{41} - \beta_{3} q^{42} + ( - 3 \beta_{3} + \beta_{2} + 6 \beta_1 - 3) q^{43} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{44} + ( - 4 \beta_{3} + 3 \beta_{2} - \beta_1 - 6) q^{45} + ( - \beta_{3} - \beta_{2} - \beta_1 + 5) q^{46} + ( - \beta_{3} + \beta_1) q^{48} + (\beta_{2} - 1) q^{49} + (6 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 7) q^{50} + ( - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 3) q^{51} - 2 \beta_{2} q^{52} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{53} + (2 \beta_{3} - 3 \beta_{2}) q^{54} - 6 q^{55} + \beta_{2} q^{56} - 5 \beta_1 q^{57} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{58} + (6 \beta_{3} - 3 \beta_1 + 3) q^{59} + ( - \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 3) q^{60} + (3 \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 3) q^{61} - 2 q^{62} + ( - 3 \beta_{2} - \beta_1 + 3) q^{63} + q^{64} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 2) q^{65} + ( - \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{66} + (6 \beta_{3} + 8 \beta_{2} - 3 \beta_1 - 5) q^{67} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{68} + ( - 4 \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{69} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{70} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{71} + ( - \beta_{3} - 3 \beta_{2}) q^{72} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 2) q^{73} + 2 \beta_{2} q^{74} + ( - 4 \beta_{3} + 18 \beta_{2} + 7 \beta_1 - 9) q^{75} + (5 \beta_{2} - 5) q^{76} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{77} + 2 \beta_1 q^{78} + ( - 3 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 3) q^{79} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{80} + (6 \beta_{2} + 5 \beta_1 - 6) q^{81} + (\beta_{3} + \beta_{2} + \beta_1 - 8) q^{82} + (4 \beta_{3} - 8 \beta_{2} - 8 \beta_1 + 4) q^{83} - \beta_1 q^{84} + (6 \beta_{2} - 6) q^{85} + ( - 6 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{86} + (2 \beta_{3} - 12 \beta_{2} - 4 \beta_1 + 6) q^{87} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{88} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 8) q^{89} + (\beta_{3} - 3 \beta_{2} - 5 \beta_1 - 3) q^{90} + 2 q^{91} + (\beta_{3} + 4 \beta_{2} - 2 \beta_1 + 1) q^{92} + 2 \beta_{3} q^{93} + ( - 10 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 10) q^{95} - \beta_{3} q^{96} + ( - 3 \beta_{3} + \beta_{2} + 6 \beta_1 - 3) q^{97} - q^{98} + ( - 5 \beta_{3} - 3 \beta_{2} + \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - q^{3} - 2 q^{4} - 3 q^{5} - 2 q^{6} - 2 q^{7} - 4 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - q^{3} - 2 q^{4} - 3 q^{5} - 2 q^{6} - 2 q^{7} - 4 q^{8} + 5 q^{9} - 6 q^{10} - 3 q^{11} - q^{12} - 4 q^{13} + 2 q^{14} + 15 q^{15} - 2 q^{16} - 6 q^{17} - 5 q^{18} + 20 q^{19} - 3 q^{20} + 2 q^{21} + 3 q^{22} + 9 q^{23} + q^{24} - 11 q^{25} - 8 q^{26} - 16 q^{27} + 4 q^{28} + 6 q^{29} + 18 q^{30} - 4 q^{31} + 2 q^{32} + 3 q^{33} - 3 q^{34} + 6 q^{35} - 10 q^{36} + 8 q^{37} + 10 q^{38} - 2 q^{39} + 3 q^{40} - 15 q^{41} + q^{42} - q^{43} + 6 q^{44} - 15 q^{45} + 18 q^{46} + 2 q^{48} - 2 q^{49} + 11 q^{50} - 15 q^{51} - 4 q^{52} - 12 q^{53} - 8 q^{54} - 24 q^{55} + 2 q^{56} - 5 q^{57} - 6 q^{58} + 3 q^{59} + 3 q^{60} + 11 q^{61} - 8 q^{62} + 5 q^{63} + 4 q^{64} - 6 q^{65} + 18 q^{66} - 13 q^{67} + 3 q^{68} + 21 q^{69} + 3 q^{70} + 6 q^{71} - 5 q^{72} + 14 q^{73} + 4 q^{74} + 11 q^{75} - 10 q^{76} - 3 q^{77} + 2 q^{78} - 7 q^{79} + 6 q^{80} - 7 q^{81} - 30 q^{82} - 12 q^{83} - q^{84} - 12 q^{85} + q^{86} - 6 q^{87} + 3 q^{88} + 36 q^{89} - 24 q^{90} + 8 q^{91} + 9 q^{92} - 2 q^{93} - 15 q^{95} + q^{96} - q^{97} - 4 q^{98} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display

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

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

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
1.68614 − 0.396143i
−1.18614 + 1.26217i
1.68614 + 0.396143i
−1.18614 − 1.26217i
0.500000 − 0.866025i −1.68614 + 0.396143i −0.500000 − 0.866025i −2.18614 − 3.78651i −0.500000 + 1.65831i −0.500000 + 0.866025i −1.00000 2.68614 − 1.33591i −4.37228
43.2 0.500000 − 0.866025i 1.18614 − 1.26217i −0.500000 − 0.866025i 0.686141 + 1.18843i −0.500000 − 1.65831i −0.500000 + 0.866025i −1.00000 −0.186141 − 2.99422i 1.37228
85.1 0.500000 + 0.866025i −1.68614 − 0.396143i −0.500000 + 0.866025i −2.18614 + 3.78651i −0.500000 − 1.65831i −0.500000 − 0.866025i −1.00000 2.68614 + 1.33591i −4.37228
85.2 0.500000 + 0.866025i 1.18614 + 1.26217i −0.500000 + 0.866025i 0.686141 − 1.18843i −0.500000 + 1.65831i −0.500000 − 0.866025i −1.00000 −0.186141 + 2.99422i 1.37228
\(n\): e.g. 2-40 or 990-1000
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 126.2.f.d ✓ 4
3.b odd 2 1 378.2.f.c 4
4.b odd 2 1 1008.2.r.f 4
7.b odd 2 1 882.2.f.k 4
7.c even 3 1 882.2.e.l 4
7.c even 3 1 882.2.h.m 4
7.d odd 6 1 882.2.e.k 4
7.d odd 6 1 882.2.h.n 4
9.c even 3 1 inner 126.2.f.d ✓ 4
9.c even 3 1 1134.2.a.k 2
9.d odd 6 1 378.2.f.c 4
9.d odd 6 1 1134.2.a.n 2
12.b even 2 1 3024.2.r.f 4
21.c even 2 1 2646.2.f.j 4
21.g even 6 1 2646.2.e.m 4
21.g even 6 1 2646.2.h.l 4
21.h odd 6 1 2646.2.e.n 4
21.h odd 6 1 2646.2.h.k 4
36.f odd 6 1 1008.2.r.f 4
36.f odd 6 1 9072.2.a.bm 2
36.h even 6 1 3024.2.r.f 4
36.h even 6 1 9072.2.a.bb 2
63.g even 3 1 882.2.e.l 4
63.h even 3 1 882.2.h.m 4
63.i even 6 1 2646.2.h.l 4
63.j odd 6 1 2646.2.h.k 4
63.k odd 6 1 882.2.e.k 4
63.l odd 6 1 882.2.f.k 4
63.l odd 6 1 7938.2.a.bh 2
63.n odd 6 1 2646.2.e.n 4
63.o even 6 1 2646.2.f.j 4
63.o even 6 1 7938.2.a.bs 2
63.s even 6 1 2646.2.e.m 4
63.t odd 6 1 882.2.h.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d ✓ 4 1.a even 1 1 trivial
126.2.f.d ✓ 4 9.c even 3 1 inner
378.2.f.c 4 3.b odd 2 1
378.2.f.c 4 9.d odd 6 1
882.2.e.k 4 7.d odd 6 1
882.2.e.k 4 63.k odd 6 1
882.2.e.l 4 7.c even 3 1
882.2.e.l 4 63.g even 3 1
882.2.f.k 4 7.b odd 2 1
882.2.f.k 4 63.l odd 6 1
882.2.h.m 4 7.c even 3 1
882.2.h.m 4 63.h even 3 1
882.2.h.n 4 7.d odd 6 1
882.2.h.n 4 63.t odd 6 1
1008.2.r.f 4 4.b odd 2 1
1008.2.r.f 4 36.f odd 6 1
1134.2.a.k 2 9.c even 3 1
1134.2.a.n 2 9.d odd 6 1
2646.2.e.m 4 21.g even 6 1
2646.2.e.m 4 63.s even 6 1
2646.2.e.n 4 21.h odd 6 1
2646.2.e.n 4 63.n odd 6 1
2646.2.f.j 4 21.c even 2 1
2646.2.f.j 4 63.o even 6 1
2646.2.h.k 4 21.h odd 6 1
2646.2.h.k 4 63.j odd 6 1
2646.2.h.l 4 21.g even 6 1
2646.2.h.l 4 63.i even 6 1
3024.2.r.f 4 12.b even 2 1
3024.2.r.f 4 36.h even 6 1
7938.2.a.bh 2 63.l odd 6 1
7938.2.a.bs 2 63.o even 6 1
9072.2.a.bb 2 36.h even 6 1
9072.2.a.bm 2 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 3T_{5}^{3} + 15T_{5}^{2} - 18T_{5} + 36 \) acting on \(S_{2}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} - 2 T^{2} + 3 T + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T - 6)^{2} \) Copy content Toggle raw display
$19$ \( (T - 5)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 9 T^{3} + 69 T^{2} - 108 T + 144 \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} + 60 T^{2} + 144 T + 576 \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 15 T^{3} + 177 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$43$ \( T^{4} + T^{3} + 75 T^{2} - 74 T + 5476 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 24)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 3 T^{3} + 81 T^{2} + \cdots + 5184 \) Copy content Toggle raw display
$61$ \( T^{4} - 11 T^{3} + 165 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$67$ \( T^{4} + 13 T^{3} + 201 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( (T^{2} - 3 T - 72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 7 T - 62)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 7 T^{3} + 111 T^{2} + \cdots + 3844 \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} + 240 T^{2} + \cdots + 9216 \) Copy content Toggle raw display
$89$ \( (T^{2} - 18 T + 48)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + T^{3} + 75 T^{2} - 74 T + 5476 \) Copy content Toggle raw display
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