Properties

Label 126.2.f.c
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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Newspace parameters

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

Self dual: no
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
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

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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} ) q^{2} + ( 1 - \beta_{3} ) q^{3} -\beta_{2} q^{4} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} ) q^{6} + ( 1 - \beta_{2} ) q^{7} + q^{8} + ( -1 - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} ) q^{2} + ( 1 - \beta_{3} ) q^{3} -\beta_{2} q^{4} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} ) q^{6} + ( 1 - \beta_{2} ) q^{7} + q^{8} + ( -1 - 2 \beta_{3} ) q^{9} + ( 1 + 2 \beta_{1} - \beta_{3} ) q^{10} + ( -2 + 2 \beta_{2} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{12} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{13} + \beta_{2} q^{14} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{15} + ( -1 + \beta_{2} ) q^{16} + 2 q^{17} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{18} + ( 5 + 2 \beta_{1} - \beta_{3} ) q^{19} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{20} + ( 1 - \beta_{1} - \beta_{2} ) q^{21} -2 \beta_{2} q^{22} + \beta_{2} q^{23} + ( 1 - \beta_{3} ) q^{24} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{25} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{26} + ( -5 - \beta_{3} ) q^{27} - q^{28} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{29} + ( 3 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{30} -6 \beta_{2} q^{31} -\beta_{2} q^{32} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{33} + ( -2 + 2 \beta_{2} ) q^{34} + ( -1 - 2 \beta_{1} + \beta_{3} ) q^{35} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{36} + ( 2 - 8 \beta_{1} + 4 \beta_{3} ) q^{37} + ( -5 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{38} + ( 8 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{39} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{40} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{41} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{42} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{43} + 2 q^{44} + ( -8 - \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{45} - q^{46} + ( 4 \beta_{1} - 8 \beta_{3} ) q^{47} + ( -1 + \beta_{1} + \beta_{2} ) q^{48} -\beta_{2} q^{49} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{50} + ( 2 - 2 \beta_{3} ) q^{51} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{52} + ( -6 + 4 \beta_{1} - 2 \beta_{3} ) q^{53} + ( 5 + \beta_{1} - 5 \beta_{2} ) q^{54} + ( 2 + 4 \beta_{1} - 2 \beta_{3} ) q^{55} + ( 1 - \beta_{2} ) q^{56} + ( 7 + 2 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{57} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{58} + 2 \beta_{2} q^{59} + ( 1 + 3 \beta_{2} + 2 \beta_{3} ) q^{60} + ( -9 - \beta_{1} + 9 \beta_{2} + 2 \beta_{3} ) q^{61} + 6 q^{62} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{63} + q^{64} + ( 12 + 2 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{65} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{66} + ( -2 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{67} -2 \beta_{2} q^{68} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{69} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{70} + ( 5 + 4 \beta_{1} - 2 \beta_{3} ) q^{71} + ( -1 - 2 \beta_{3} ) q^{72} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{73} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{74} + ( 2 + 6 \beta_{2} + 4 \beta_{3} ) q^{75} + ( -\beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{76} + 2 \beta_{2} q^{77} + ( -4 - 4 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{78} + ( -3 - 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{79} + ( 1 + 2 \beta_{1} - \beta_{3} ) q^{80} + ( -7 + 4 \beta_{3} ) q^{81} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{82} + ( -2 + 2 \beta_{2} ) q^{83} + ( -1 + \beta_{3} ) q^{84} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{85} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{86} + ( 6 - 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{87} + ( -2 + 2 \beta_{2} ) q^{88} + ( -12 + 4 \beta_{1} - 2 \beta_{3} ) q^{89} + ( 3 - 2 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{90} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{91} + ( 1 - \beta_{2} ) q^{92} + ( -6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{93} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{94} + ( -6 \beta_{1} - 11 \beta_{2} - 6 \beta_{3} ) q^{95} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{96} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{97} + q^{98} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{3} - 2q^{4} - 2q^{5} - 2q^{6} + 2q^{7} + 4q^{8} - 4q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{3} - 2q^{4} - 2q^{5} - 2q^{6} + 2q^{7} + 4q^{8} - 4q^{9} + 4q^{10} - 4q^{11} - 2q^{12} + 2q^{14} - 14q^{15} - 2q^{16} + 8q^{17} + 2q^{18} + 20q^{19} - 2q^{20} + 2q^{21} - 4q^{22} + 2q^{23} + 4q^{24} - 4q^{25} - 20q^{27} - 4q^{28} + 4q^{29} + 4q^{30} - 12q^{31} - 2q^{32} - 4q^{33} - 4q^{34} - 4q^{35} + 2q^{36} + 8q^{37} - 10q^{38} + 24q^{39} - 2q^{40} + 2q^{42} - 4q^{43} + 8q^{44} - 22q^{45} - 4q^{46} - 2q^{48} - 2q^{49} - 4q^{50} + 8q^{51} - 24q^{53} + 10q^{54} + 8q^{55} + 2q^{56} + 20q^{57} + 4q^{58} + 4q^{59} + 10q^{60} - 18q^{61} + 24q^{62} - 2q^{63} + 4q^{64} + 24q^{65} - 4q^{66} + 16q^{67} - 4q^{68} + 2q^{69} + 2q^{70} + 20q^{71} - 4q^{72} - 8q^{73} - 4q^{74} + 20q^{75} - 10q^{76} + 4q^{77} - 6q^{79} + 4q^{80} - 28q^{81} - 4q^{83} - 4q^{84} - 4q^{85} - 4q^{86} + 28q^{87} - 4q^{88} - 48q^{89} - 4q^{90} + 2q^{92} - 12q^{93} - 22q^{95} - 2q^{96} + 4q^{97} + 4q^{98} + 4q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
1.22474 + 0.707107i
−1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−0.500000 + 0.866025i 1.00000 1.41421i −0.500000 0.866025i −1.72474 2.98735i 0.724745 + 1.57313i 0.500000 0.866025i 1.00000 −1.00000 2.82843i 3.44949
43.2 −0.500000 + 0.866025i 1.00000 + 1.41421i −0.500000 0.866025i 0.724745 + 1.25529i −1.72474 + 0.158919i 0.500000 0.866025i 1.00000 −1.00000 + 2.82843i −1.44949
85.1 −0.500000 0.866025i 1.00000 1.41421i −0.500000 + 0.866025i 0.724745 1.25529i −1.72474 0.158919i 0.500000 + 0.866025i 1.00000 −1.00000 2.82843i −1.44949
85.2 −0.500000 0.866025i 1.00000 + 1.41421i −0.500000 + 0.866025i −1.72474 + 2.98735i 0.724745 1.57313i 0.500000 + 0.866025i 1.00000 −1.00000 + 2.82843i 3.44949
\(n\): 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.c 4
3.b odd 2 1 378.2.f.d 4
4.b odd 2 1 1008.2.r.e 4
7.b odd 2 1 882.2.f.j 4
7.c even 3 1 882.2.e.m 4
7.c even 3 1 882.2.h.k 4
7.d odd 6 1 882.2.e.n 4
7.d odd 6 1 882.2.h.l 4
9.c even 3 1 inner 126.2.f.c 4
9.c even 3 1 1134.2.a.p 2
9.d odd 6 1 378.2.f.d 4
9.d odd 6 1 1134.2.a.i 2
12.b even 2 1 3024.2.r.e 4
21.c even 2 1 2646.2.f.k 4
21.g even 6 1 2646.2.e.k 4
21.g even 6 1 2646.2.h.n 4
21.h odd 6 1 2646.2.e.l 4
21.h odd 6 1 2646.2.h.m 4
36.f odd 6 1 1008.2.r.e 4
36.f odd 6 1 9072.2.a.bk 2
36.h even 6 1 3024.2.r.e 4
36.h even 6 1 9072.2.a.bd 2
63.g even 3 1 882.2.e.m 4
63.h even 3 1 882.2.h.k 4
63.i even 6 1 2646.2.h.n 4
63.j odd 6 1 2646.2.h.m 4
63.k odd 6 1 882.2.e.n 4
63.l odd 6 1 882.2.f.j 4
63.l odd 6 1 7938.2.a.bn 2
63.n odd 6 1 2646.2.e.l 4
63.o even 6 1 2646.2.f.k 4
63.o even 6 1 7938.2.a.bm 2
63.s even 6 1 2646.2.e.k 4
63.t odd 6 1 882.2.h.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 1.a even 1 1 trivial
126.2.f.c 4 9.c even 3 1 inner
378.2.f.d 4 3.b odd 2 1
378.2.f.d 4 9.d odd 6 1
882.2.e.m 4 7.c even 3 1
882.2.e.m 4 63.g even 3 1
882.2.e.n 4 7.d odd 6 1
882.2.e.n 4 63.k odd 6 1
882.2.f.j 4 7.b odd 2 1
882.2.f.j 4 63.l odd 6 1
882.2.h.k 4 7.c even 3 1
882.2.h.k 4 63.h even 3 1
882.2.h.l 4 7.d odd 6 1
882.2.h.l 4 63.t odd 6 1
1008.2.r.e 4 4.b odd 2 1
1008.2.r.e 4 36.f odd 6 1
1134.2.a.i 2 9.d odd 6 1
1134.2.a.p 2 9.c even 3 1
2646.2.e.k 4 21.g even 6 1
2646.2.e.k 4 63.s even 6 1
2646.2.e.l 4 21.h odd 6 1
2646.2.e.l 4 63.n odd 6 1
2646.2.f.k 4 21.c even 2 1
2646.2.f.k 4 63.o even 6 1
2646.2.h.m 4 21.h odd 6 1
2646.2.h.m 4 63.j odd 6 1
2646.2.h.n 4 21.g even 6 1
2646.2.h.n 4 63.i even 6 1
3024.2.r.e 4 12.b even 2 1
3024.2.r.e 4 36.h even 6 1
7938.2.a.bm 2 63.o even 6 1
7938.2.a.bn 2 63.l odd 6 1
9072.2.a.bd 2 36.h even 6 1
9072.2.a.bk 2 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2 T_{5}^{3} + 9 T_{5}^{2} - 10 T_{5} + 25 \) acting on \(S_{2}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 - 2 T + 3 T^{2} )^{2} \)
$5$ \( 1 + 2 T - T^{2} - 10 T^{3} - 20 T^{4} - 50 T^{5} - 25 T^{6} + 250 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - T + T^{2} )^{2} \)
$11$ \( ( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \)
$13$ \( 1 - 2 T^{2} - 165 T^{4} - 338 T^{6} + 28561 T^{8} \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{4} \)
$19$ \( ( 1 - 10 T + 57 T^{2} - 190 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} )^{2} \)
$29$ \( 1 - 4 T - 22 T^{2} + 80 T^{3} + 139 T^{4} + 2320 T^{5} - 18502 T^{6} - 97556 T^{7} + 707281 T^{8} \)
$31$ \( ( 1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 4 T - 18 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( 1 + 14 T^{2} - 1485 T^{4} + 23534 T^{6} + 2825761 T^{8} \)
$43$ \( 1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 3440 T^{5} - 92450 T^{6} + 318028 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 2 T^{2} - 2205 T^{4} + 4418 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 + 12 T + 118 T^{2} + 636 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 2 T - 55 T^{2} - 118 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( 1 + 18 T + 127 T^{2} + 1350 T^{3} + 15324 T^{4} + 82350 T^{5} + 472567 T^{6} + 4085658 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 16 T + 82 T^{2} - 640 T^{3} + 8635 T^{4} - 42880 T^{5} + 368098 T^{6} - 4812208 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 - 10 T + 143 T^{2} - 710 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 4 T + 126 T^{2} + 292 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 + 6 T - 107 T^{2} - 90 T^{3} + 11364 T^{4} - 7110 T^{5} - 667787 T^{6} + 2958234 T^{7} + 38950081 T^{8} \)
$83$ \( ( 1 + 2 T - 79 T^{2} + 166 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 24 T + 298 T^{2} + 2136 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 7760 T^{5} - 1486622 T^{6} - 3650692 T^{7} + 88529281 T^{8} \)
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