Properties

Label 126.4.g.d
Level 126
Weight 4
Character orbit 126.g
Analytic conductor 7.434
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.43424066072\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + ( -9 + 9 \zeta_{6} ) q^{5} + ( -7 - 14 \zeta_{6} ) q^{7} -8 q^{8} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + ( -9 + 9 \zeta_{6} ) q^{5} + ( -7 - 14 \zeta_{6} ) q^{7} -8 q^{8} + 18 \zeta_{6} q^{10} -57 \zeta_{6} q^{11} -70 q^{13} + ( -42 + 14 \zeta_{6} ) q^{14} + ( -16 + 16 \zeta_{6} ) q^{16} + 51 \zeta_{6} q^{17} + ( -5 + 5 \zeta_{6} ) q^{19} + 36 q^{20} -114 q^{22} + ( 69 - 69 \zeta_{6} ) q^{23} + 44 \zeta_{6} q^{25} + ( -140 + 140 \zeta_{6} ) q^{26} + ( -56 + 84 \zeta_{6} ) q^{28} -114 q^{29} -23 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + 102 q^{34} + ( 189 - 63 \zeta_{6} ) q^{35} + ( 253 - 253 \zeta_{6} ) q^{37} + 10 \zeta_{6} q^{38} + ( 72 - 72 \zeta_{6} ) q^{40} + 42 q^{41} -124 q^{43} + ( -228 + 228 \zeta_{6} ) q^{44} -138 \zeta_{6} q^{46} + ( 201 - 201 \zeta_{6} ) q^{47} + ( -147 + 392 \zeta_{6} ) q^{49} + 88 q^{50} + 280 \zeta_{6} q^{52} -393 \zeta_{6} q^{53} + 513 q^{55} + ( 56 + 112 \zeta_{6} ) q^{56} + ( -228 + 228 \zeta_{6} ) q^{58} + 219 \zeta_{6} q^{59} + ( 709 - 709 \zeta_{6} ) q^{61} -46 q^{62} + 64 q^{64} + ( 630 - 630 \zeta_{6} ) q^{65} -419 \zeta_{6} q^{67} + ( 204 - 204 \zeta_{6} ) q^{68} + ( 252 - 378 \zeta_{6} ) q^{70} + 96 q^{71} + 313 \zeta_{6} q^{73} -506 \zeta_{6} q^{74} + 20 q^{76} + ( -798 + 1197 \zeta_{6} ) q^{77} + ( -461 + 461 \zeta_{6} ) q^{79} -144 \zeta_{6} q^{80} + ( 84 - 84 \zeta_{6} ) q^{82} + 588 q^{83} -459 q^{85} + ( -248 + 248 \zeta_{6} ) q^{86} + 456 \zeta_{6} q^{88} + ( -1017 + 1017 \zeta_{6} ) q^{89} + ( 490 + 980 \zeta_{6} ) q^{91} -276 q^{92} -402 \zeta_{6} q^{94} -45 \zeta_{6} q^{95} -1834 q^{97} + ( 490 + 294 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{4} - 9q^{5} - 28q^{7} - 16q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{4} - 9q^{5} - 28q^{7} - 16q^{8} + 18q^{10} - 57q^{11} - 140q^{13} - 70q^{14} - 16q^{16} + 51q^{17} - 5q^{19} + 72q^{20} - 228q^{22} + 69q^{23} + 44q^{25} - 140q^{26} - 28q^{28} - 228q^{29} - 23q^{31} + 32q^{32} + 204q^{34} + 315q^{35} + 253q^{37} + 10q^{38} + 72q^{40} + 84q^{41} - 248q^{43} - 228q^{44} - 138q^{46} + 201q^{47} + 98q^{49} + 176q^{50} + 280q^{52} - 393q^{53} + 1026q^{55} + 224q^{56} - 228q^{58} + 219q^{59} + 709q^{61} - 92q^{62} + 128q^{64} + 630q^{65} - 419q^{67} + 204q^{68} + 126q^{70} + 192q^{71} + 313q^{73} - 506q^{74} + 40q^{76} - 399q^{77} - 461q^{79} - 144q^{80} + 84q^{82} + 1176q^{83} - 918q^{85} - 248q^{86} + 456q^{88} - 1017q^{89} + 1960q^{91} - 552q^{92} - 402q^{94} - 45q^{95} - 3668q^{97} + 1274q^{98} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 1.73205i 0 −2.00000 3.46410i −4.50000 + 7.79423i 0 −14.0000 12.1244i −8.00000 0 9.00000 + 15.5885i
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −4.50000 7.79423i 0 −14.0000 + 12.1244i −8.00000 0 9.00000 15.5885i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.4.g.d 2
3.b odd 2 1 14.4.c.a 2
7.b odd 2 1 882.4.g.u 2
7.c even 3 1 inner 126.4.g.d 2
7.c even 3 1 882.4.a.f 1
7.d odd 6 1 882.4.a.c 1
7.d odd 6 1 882.4.g.u 2
12.b even 2 1 112.4.i.a 2
15.d odd 2 1 350.4.e.e 2
15.e even 4 2 350.4.j.b 4
21.c even 2 1 98.4.c.a 2
21.g even 6 1 98.4.a.f 1
21.g even 6 1 98.4.c.a 2
21.h odd 6 1 14.4.c.a 2
21.h odd 6 1 98.4.a.d 1
24.f even 2 1 448.4.i.e 2
24.h odd 2 1 448.4.i.b 2
84.j odd 6 1 784.4.a.c 1
84.n even 6 1 112.4.i.a 2
84.n even 6 1 784.4.a.p 1
105.o odd 6 1 350.4.e.e 2
105.o odd 6 1 2450.4.a.q 1
105.p even 6 1 2450.4.a.d 1
105.x even 12 2 350.4.j.b 4
168.s odd 6 1 448.4.i.b 2
168.v even 6 1 448.4.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 3.b odd 2 1
14.4.c.a 2 21.h odd 6 1
98.4.a.d 1 21.h odd 6 1
98.4.a.f 1 21.g even 6 1
98.4.c.a 2 21.c even 2 1
98.4.c.a 2 21.g even 6 1
112.4.i.a 2 12.b even 2 1
112.4.i.a 2 84.n even 6 1
126.4.g.d 2 1.a even 1 1 trivial
126.4.g.d 2 7.c even 3 1 inner
350.4.e.e 2 15.d odd 2 1
350.4.e.e 2 105.o odd 6 1
350.4.j.b 4 15.e even 4 2
350.4.j.b 4 105.x even 12 2
448.4.i.b 2 24.h odd 2 1
448.4.i.b 2 168.s odd 6 1
448.4.i.e 2 24.f even 2 1
448.4.i.e 2 168.v even 6 1
784.4.a.c 1 84.j odd 6 1
784.4.a.p 1 84.n even 6 1
882.4.a.c 1 7.d odd 6 1
882.4.a.f 1 7.c even 3 1
882.4.g.u 2 7.b odd 2 1
882.4.g.u 2 7.d odd 6 1
2450.4.a.d 1 105.p even 6 1
2450.4.a.q 1 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 9 T_{5} + 81 \) acting on \(S_{4}^{\mathrm{new}}(126, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 4 T^{2} \)
$3$ \( \)
$5$ \( 1 + 9 T - 44 T^{2} + 1125 T^{3} + 15625 T^{4} \)
$7$ \( 1 + 28 T + 343 T^{2} \)
$11$ \( 1 + 57 T + 1918 T^{2} + 75867 T^{3} + 1771561 T^{4} \)
$13$ \( ( 1 + 70 T + 2197 T^{2} )^{2} \)
$17$ \( 1 - 51 T - 2312 T^{2} - 250563 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 5 T - 6834 T^{2} + 34295 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 69 T - 7406 T^{2} - 839523 T^{3} + 148035889 T^{4} \)
$29$ \( ( 1 + 114 T + 24389 T^{2} )^{2} \)
$31$ \( 1 + 23 T - 29262 T^{2} + 685193 T^{3} + 887503681 T^{4} \)
$37$ \( 1 - 253 T + 13356 T^{2} - 12815209 T^{3} + 2565726409 T^{4} \)
$41$ \( ( 1 - 42 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 + 124 T + 79507 T^{2} )^{2} \)
$47$ \( 1 - 201 T - 63422 T^{2} - 20868423 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 393 T + 5572 T^{2} + 58508661 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 219 T - 157418 T^{2} - 44978001 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 709 T + 275700 T^{2} - 160929529 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 419 T - 125202 T^{2} + 126019697 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 - 96 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 313 T - 291048 T^{2} - 121762321 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 461 T - 280518 T^{2} + 227290979 T^{3} + 243087455521 T^{4} \)
$83$ \( ( 1 - 588 T + 571787 T^{2} )^{2} \)
$89$ \( 1 + 1017 T + 329320 T^{2} + 716953473 T^{3} + 496981290961 T^{4} \)
$97$ \( ( 1 + 1834 T + 912673 T^{2} )^{2} \)
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