Properties

Label 126.2.e.a
Level $126$
Weight $2$
Character orbit 126.e
Analytic conductor $1.006$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

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

Character values

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

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(-1 + \zeta_{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 1.73205i 1.00000 −1.50000 + 2.59808i 1.73205i −2.00000 1.73205i −1.00000 −3.00000 1.50000 2.59808i
121.1 −1.00000 1.73205i 1.00000 −1.50000 2.59808i 1.73205i −2.00000 + 1.73205i −1.00000 −3.00000 1.50000 + 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h 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.e.a 2
3.b odd 2 1 378.2.e.b 2
4.b odd 2 1 1008.2.q.a 2
7.b odd 2 1 882.2.e.c 2
7.c even 3 1 126.2.h.b yes 2
7.c even 3 1 882.2.f.i 2
7.d odd 6 1 882.2.f.g 2
7.d odd 6 1 882.2.h.i 2
9.c even 3 1 126.2.h.b yes 2
9.c even 3 1 1134.2.g.e 2
9.d odd 6 1 378.2.h.a 2
9.d odd 6 1 1134.2.g.c 2
12.b even 2 1 3024.2.q.f 2
21.c even 2 1 2646.2.e.g 2
21.g even 6 1 2646.2.f.a 2
21.g even 6 1 2646.2.h.d 2
21.h odd 6 1 378.2.h.a 2
21.h odd 6 1 2646.2.f.d 2
28.g odd 6 1 1008.2.t.f 2
36.f odd 6 1 1008.2.t.f 2
36.h even 6 1 3024.2.t.a 2
63.g even 3 1 882.2.f.i 2
63.g even 3 1 1134.2.g.e 2
63.h even 3 1 inner 126.2.e.a 2
63.h even 3 1 7938.2.a.m 1
63.i even 6 1 2646.2.e.g 2
63.i even 6 1 7938.2.a.be 1
63.j odd 6 1 378.2.e.b 2
63.j odd 6 1 7938.2.a.t 1
63.k odd 6 1 882.2.f.g 2
63.l odd 6 1 882.2.h.i 2
63.n odd 6 1 1134.2.g.c 2
63.n odd 6 1 2646.2.f.d 2
63.o even 6 1 2646.2.h.d 2
63.s even 6 1 2646.2.f.a 2
63.t odd 6 1 882.2.e.c 2
63.t odd 6 1 7938.2.a.b 1
84.n even 6 1 3024.2.t.a 2
252.u odd 6 1 1008.2.q.a 2
252.bb even 6 1 3024.2.q.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.a 2 1.a even 1 1 trivial
126.2.e.a 2 63.h even 3 1 inner
126.2.h.b yes 2 7.c even 3 1
126.2.h.b yes 2 9.c even 3 1
378.2.e.b 2 3.b odd 2 1
378.2.e.b 2 63.j odd 6 1
378.2.h.a 2 9.d odd 6 1
378.2.h.a 2 21.h odd 6 1
882.2.e.c 2 7.b odd 2 1
882.2.e.c 2 63.t odd 6 1
882.2.f.g 2 7.d odd 6 1
882.2.f.g 2 63.k odd 6 1
882.2.f.i 2 7.c even 3 1
882.2.f.i 2 63.g even 3 1
882.2.h.i 2 7.d odd 6 1
882.2.h.i 2 63.l odd 6 1
1008.2.q.a 2 4.b odd 2 1
1008.2.q.a 2 252.u odd 6 1
1008.2.t.f 2 28.g odd 6 1
1008.2.t.f 2 36.f odd 6 1
1134.2.g.c 2 9.d odd 6 1
1134.2.g.c 2 63.n odd 6 1
1134.2.g.e 2 9.c even 3 1
1134.2.g.e 2 63.g even 3 1
2646.2.e.g 2 21.c even 2 1
2646.2.e.g 2 63.i even 6 1
2646.2.f.a 2 21.g even 6 1
2646.2.f.a 2 63.s even 6 1
2646.2.f.d 2 21.h odd 6 1
2646.2.f.d 2 63.n odd 6 1
2646.2.h.d 2 21.g even 6 1
2646.2.h.d 2 63.o even 6 1
3024.2.q.f 2 12.b even 2 1
3024.2.q.f 2 252.bb even 6 1
3024.2.t.a 2 36.h even 6 1
3024.2.t.a 2 84.n even 6 1
7938.2.a.b 1 63.t odd 6 1
7938.2.a.m 1 63.h even 3 1
7938.2.a.t 1 63.j odd 6 1
7938.2.a.be 1 63.i even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$79$ \( (T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} - T + 1 \) Copy content Toggle raw display
show more
show less