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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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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 2 \beta_{1} + 3\)

Character values

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

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

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.

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 \( T_{5}^{4} + 3 T_{5}^{3} + 15 T_{5}^{2} - 18 T_{5} + 36 \) acting on \(S_{2}^{\mathrm{new}}(126, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \)
$5$ \( ( 1 + 3 T + 5 T^{2} )^{2}( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} ) \)
$7$ \( ( 1 + T + T^{2} )^{2} \)
$11$ \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 198 T^{5} - 847 T^{6} + 3993 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 + 3 T + 28 T^{2} + 51 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 5 T + 19 T^{2} )^{4} \)
$23$ \( ( 1 - 9 T + 23 T^{2} )^{2}( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} ) \)
$29$ \( 1 - 6 T + 2 T^{2} + 144 T^{3} - 729 T^{4} + 4176 T^{5} + 1682 T^{6} - 146334 T^{7} + 707281 T^{8} \)
$31$ \( ( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{4} \)
$41$ \( 1 + 15 T + 95 T^{2} + 720 T^{3} + 5994 T^{4} + 29520 T^{5} + 159695 T^{6} + 1033815 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 3182 T^{5} - 20339 T^{6} + 79507 T^{7} + 3418801 T^{8} \)
$47$ \( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 6 T + 82 T^{2} + 318 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 3 T - 37 T^{2} + 216 T^{3} - 1896 T^{4} + 12744 T^{5} - 128797 T^{6} - 616137 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 11 T + 43 T^{2} + 484 T^{3} - 5018 T^{4} + 29524 T^{5} + 160003 T^{6} - 2496791 T^{7} + 13845841 T^{8} \)
$67$ \( ( 1 + 13 T + 67 T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} ) \)
$71$ \( ( 1 - 3 T + 70 T^{2} - 213 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 7 T + 84 T^{2} - 511 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 + 7 T - 47 T^{2} - 434 T^{3} - 896 T^{4} - 34286 T^{5} - 293327 T^{6} + 3451273 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 12 T + 74 T^{2} - 1152 T^{3} - 13941 T^{4} - 95616 T^{5} + 509786 T^{6} + 6861444 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 - 18 T + 226 T^{2} - 1602 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 + T - 119 T^{2} - 74 T^{3} + 4894 T^{4} - 7178 T^{5} - 1119671 T^{6} + 912673 T^{7} + 88529281 T^{8} \)
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