# Properties

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

# Related objects

## 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{1345})$$ Defining polynomial: $$x^{4} - x^{3} + 337 x^{2} + 336 x + 112896$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{2} + ( -4 + 4 \beta_{2} ) q^{4} + ( \beta_{1} + 2 \beta_{2} ) q^{5} + ( 1 - 3 \beta_{2} + \beta_{3} ) q^{7} -8 q^{8} +O(q^{10})$$ $$q + 2 \beta_{2} q^{2} + ( -4 + 4 \beta_{2} ) q^{4} + ( \beta_{1} + 2 \beta_{2} ) q^{5} + ( 1 - 3 \beta_{2} + \beta_{3} ) q^{7} -8 q^{8} + ( -6 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{10} + ( 33 + \beta_{1} - 34 \beta_{2} + \beta_{3} ) q^{11} + ( 20 + \beta_{3} ) q^{13} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{14} -16 \beta_{2} q^{16} + ( -48 + 4 \beta_{1} + 44 \beta_{2} + 4 \beta_{3} ) q^{17} + ( 3 \beta_{1} - 23 \beta_{2} ) q^{19} + ( -12 + 4 \beta_{3} ) q^{20} + ( 66 + 2 \beta_{3} ) q^{22} + ( -4 \beta_{1} + 76 \beta_{2} ) q^{23} + ( -220 + 5 \beta_{1} + 215 \beta_{2} + 5 \beta_{3} ) q^{25} + ( -2 \beta_{1} + 42 \beta_{2} ) q^{26} + ( 8 - 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{28} + ( -33 - 11 \beta_{3} ) q^{29} + ( 259 + 2 \beta_{1} - 261 \beta_{2} + 2 \beta_{3} ) q^{31} + ( 32 - 32 \beta_{2} ) q^{32} + ( -96 + 8 \beta_{3} ) q^{34} + ( 9 - 4 \beta_{1} - 338 \beta_{2} - 3 \beta_{3} ) q^{35} + ( -9 \beta_{1} + \beta_{2} ) q^{37} + ( 40 + 6 \beta_{1} - 46 \beta_{2} + 6 \beta_{3} ) q^{38} + ( -8 \beta_{1} - 16 \beta_{2} ) q^{40} + ( 216 - 6 \beta_{3} ) q^{41} + ( -46 - 15 \beta_{3} ) q^{43} + ( -4 \beta_{1} + 136 \beta_{2} ) q^{44} + ( -144 - 8 \beta_{1} + 152 \beta_{2} - 8 \beta_{3} ) q^{46} + ( 12 \beta_{1} + 282 \beta_{2} ) q^{47} + ( 328 + 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{49} + ( -440 + 10 \beta_{3} ) q^{50} + ( -80 - 4 \beta_{1} + 84 \beta_{2} - 4 \beta_{3} ) q^{52} + ( -123 + 3 \beta_{1} + 120 \beta_{2} + 3 \beta_{3} ) q^{53} + ( -237 - 31 \beta_{3} ) q^{55} + ( -8 + 24 \beta_{2} - 8 \beta_{3} ) q^{56} + ( 22 \beta_{1} - 88 \beta_{2} ) q^{58} + ( 9 - 25 \beta_{1} + 16 \beta_{2} - 25 \beta_{3} ) q^{59} + ( 4 \beta_{1} - 114 \beta_{2} ) q^{61} + ( 518 + 4 \beta_{3} ) q^{62} + 64 q^{64} + ( 18 \beta_{1} - 294 \beta_{2} ) q^{65} + ( -338 - 11 \beta_{1} + 349 \beta_{2} - 11 \beta_{3} ) q^{67} + ( -16 \beta_{1} - 176 \beta_{2} ) q^{68} + ( 684 - 2 \beta_{1} - 664 \beta_{2} - 8 \beta_{3} ) q^{70} + ( -246 + 20 \beta_{3} ) q^{71} + ( 478 - 35 \beta_{1} - 443 \beta_{2} - 35 \beta_{3} ) q^{73} + ( 16 - 18 \beta_{1} + 2 \beta_{2} - 18 \beta_{3} ) q^{74} + ( 80 + 12 \beta_{3} ) q^{76} + ( 270 + 35 \beta_{1} - 404 \beta_{2} + 32 \beta_{3} ) q^{77} + ( -8 \beta_{1} + 267 \beta_{2} ) q^{79} + ( 48 - 16 \beta_{1} - 32 \beta_{2} - 16 \beta_{3} ) q^{80} + ( 12 \beta_{1} + 420 \beta_{2} ) q^{82} + ( 123 - 25 \beta_{3} ) q^{83} + ( -1488 + 56 \beta_{3} ) q^{85} + ( 30 \beta_{1} - 122 \beta_{2} ) q^{86} + ( -264 - 8 \beta_{1} + 272 \beta_{2} - 8 \beta_{3} ) q^{88} + ( -42 \beta_{1} + 408 \beta_{2} ) q^{89} + ( 356 + 3 \beta_{1} - 63 \beta_{2} + 22 \beta_{3} ) q^{91} + ( -288 - 16 \beta_{3} ) q^{92} + ( -588 + 24 \beta_{1} + 564 \beta_{2} + 24 \beta_{3} ) q^{94} + ( -948 - 14 \beta_{1} + 962 \beta_{2} - 14 \beta_{3} ) q^{95} + ( 959 + 35 \beta_{3} ) q^{97} + ( -6 + 6 \beta_{1} + 656 \beta_{2} + 12 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 8 q^{4} + 5 q^{5} - 32 q^{8} + O(q^{10})$$ $$4 q + 4 q^{2} - 8 q^{4} + 5 q^{5} - 32 q^{8} - 10 q^{10} + 67 q^{11} + 82 q^{13} + 18 q^{14} - 32 q^{16} - 92 q^{17} - 43 q^{19} - 40 q^{20} + 268 q^{22} + 148 q^{23} - 435 q^{25} + 82 q^{26} + 36 q^{28} - 154 q^{29} + 520 q^{31} + 64 q^{32} - 368 q^{34} - 650 q^{35} - 7 q^{37} + 86 q^{38} - 40 q^{40} + 852 q^{41} - 214 q^{43} + 268 q^{44} - 296 q^{46} + 576 q^{47} + 1318 q^{49} - 1740 q^{50} - 164 q^{52} - 243 q^{53} - 1010 q^{55} - 154 q^{58} - 7 q^{59} - 224 q^{61} + 2080 q^{62} + 256 q^{64} - 570 q^{65} - 687 q^{67} - 368 q^{68} + 1390 q^{70} - 944 q^{71} + 921 q^{73} + 14 q^{74} + 344 q^{76} + 371 q^{77} + 526 q^{79} + 80 q^{80} + 852 q^{82} + 442 q^{83} - 5840 q^{85} - 214 q^{86} - 536 q^{88} + 774 q^{89} + 1345 q^{91} - 1184 q^{92} - 1152 q^{94} - 1910 q^{95} + 3906 q^{97} + 1318 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 337 x^{2} + 336 x + 112896$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 337 \nu^{2} - 337 \nu + 112896$$$$)/113232$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 673$$$$)/337$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 336 \beta_{2} + \beta_{1} - 337$$ $$\nu^{3}$$ $$=$$ $$337 \beta_{3} - 673$$

## Character values

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

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

## 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
 −8.91856 + 15.4474i 9.41856 − 16.3134i −8.91856 − 15.4474i 9.41856 + 16.3134i
1.00000 1.73205i 0 −2.00000 3.46410i −7.91856 + 13.7153i 0 18.3371 + 2.59808i −8.00000 0 15.8371 + 27.4307i
37.2 1.00000 1.73205i 0 −2.00000 3.46410i 10.4186 18.0455i 0 −18.3371 + 2.59808i −8.00000 0 −20.8371 36.0910i
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −7.91856 13.7153i 0 18.3371 2.59808i −8.00000 0 15.8371 27.4307i
109.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 10.4186 + 18.0455i 0 −18.3371 2.59808i −8.00000 0 −20.8371 + 36.0910i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.g 4
3.b odd 2 1 42.4.e.c 4
7.b odd 2 1 882.4.g.bf 4
7.c even 3 1 inner 126.4.g.g 4
7.c even 3 1 882.4.a.v 2
7.d odd 6 1 882.4.a.z 2
7.d odd 6 1 882.4.g.bf 4
12.b even 2 1 336.4.q.j 4
21.c even 2 1 294.4.e.l 4
21.g even 6 1 294.4.a.m 2
21.g even 6 1 294.4.e.l 4
21.h odd 6 1 42.4.e.c 4
21.h odd 6 1 294.4.a.n 2
84.j odd 6 1 2352.4.a.ca 2
84.n even 6 1 336.4.q.j 4
84.n even 6 1 2352.4.a.bq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.c 4 3.b odd 2 1
42.4.e.c 4 21.h odd 6 1
126.4.g.g 4 1.a even 1 1 trivial
126.4.g.g 4 7.c even 3 1 inner
294.4.a.m 2 21.g even 6 1
294.4.a.n 2 21.h odd 6 1
294.4.e.l 4 21.c even 2 1
294.4.e.l 4 21.g even 6 1
336.4.q.j 4 12.b even 2 1
336.4.q.j 4 84.n even 6 1
882.4.a.v 2 7.c even 3 1
882.4.a.z 2 7.d odd 6 1
882.4.g.bf 4 7.b odd 2 1
882.4.g.bf 4 7.d odd 6 1
2352.4.a.bq 2 84.n even 6 1
2352.4.a.ca 2 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 5 T_{5}^{3} + 355 T_{5}^{2} + 1650 T_{5} + 108900$$ acting on $$S_{4}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$108900 + 1650 T + 355 T^{2} - 5 T^{3} + T^{4}$$
$7$ $$117649 - 659 T^{2} + T^{4}$$
$11$ $$617796 - 52662 T + 3703 T^{2} - 67 T^{3} + T^{4}$$
$13$ $$( 84 - 41 T + T^{2} )^{2}$$
$17$ $$10653696 - 300288 T + 11728 T^{2} + 92 T^{3} + T^{4}$$
$19$ $$6574096 - 110252 T + 4413 T^{2} + 43 T^{3} + T^{4}$$
$23$ $$9216 - 14208 T + 21808 T^{2} - 148 T^{3} + T^{4}$$
$29$ $$( -39204 + 77 T + T^{2} )^{2}$$
$31$ $$4389725025 - 34452600 T + 204145 T^{2} - 520 T^{3} + T^{4}$$
$37$ $$741146176 - 190568 T + 27273 T^{2} + 7 T^{3} + T^{4}$$
$41$ $$( 33264 - 426 T + T^{2} )^{2}$$
$43$ $$( -72794 + 107 T + T^{2} )^{2}$$
$47$ $$1191906576 - 19885824 T + 297252 T^{2} - 576 T^{3} + T^{4}$$
$53$ $$137733696 + 2851848 T + 47313 T^{2} + 243 T^{3} + T^{4}$$
$59$ $$44160500736 - 1471008 T + 210193 T^{2} + 7 T^{3} + T^{4}$$
$61$ $$51322896 + 1604736 T + 43012 T^{2} + 224 T^{3} + T^{4}$$
$67$ $$5976217636 + 53109222 T + 394663 T^{2} + 687 T^{3} + T^{4}$$
$71$ $$( -78804 + 472 T + T^{2} )^{2}$$
$73$ $$39938423716 + 184058166 T + 1048087 T^{2} - 921 T^{3} + T^{4}$$
$79$ $$2270427201 - 25063374 T + 229027 T^{2} - 526 T^{3} + T^{4}$$
$83$ $$( -197946 - 221 T + T^{2} )^{2}$$
$89$ $$196582277376 + 343173024 T + 1042452 T^{2} - 774 T^{3} + T^{4}$$
$97$ $$( 541646 - 1953 T + T^{2} )^{2}$$