# Properties

 Label 126.3.n.a Level $126$ Weight $3$ Character orbit 126.n Analytic conductor $3.433$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 126.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43325133094$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{5} + ( -5 + 2 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{5} + ( -5 + 2 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + ( -8 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{10} -6 \beta_{2} q^{11} + ( -1 + 16 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} ) q^{13} + ( -8 - 5 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} ) q^{14} + ( -4 - 4 \beta_{2} ) q^{16} + ( 16 + 2 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{17} + ( -7 + 2 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{19} + ( -4 - 8 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{20} -6 \beta_{3} q^{22} + ( -12 + 18 \beta_{1} - 12 \beta_{2} ) q^{23} + ( -24 \beta_{1} - 11 \beta_{2} - 24 \beta_{3} ) q^{25} + ( -16 - \beta_{1} + 16 \beta_{2} - 2 \beta_{3} ) q^{26} + ( 10 - 8 \beta_{1} - 4 \beta_{3} ) q^{28} + 24 \beta_{3} q^{29} + ( 34 - 6 \beta_{1} + 17 \beta_{2} + 6 \beta_{3} ) q^{31} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{32} + ( 4 + 16 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{34} + ( 20 - 2 \beta_{1} - 14 \beta_{2} - 22 \beta_{3} ) q^{35} + ( 11 - 12 \beta_{1} + 11 \beta_{2} ) q^{37} + ( -8 - 7 \beta_{1} - 4 \beta_{2} + 7 \beta_{3} ) q^{38} + ( 8 - 4 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{40} + ( 26 - 8 \beta_{1} + 52 \beta_{2} - 4 \beta_{3} ) q^{41} + ( 7 - 6 \beta_{3} ) q^{43} + ( 12 + 12 \beta_{2} ) q^{44} + ( -12 \beta_{1} + 36 \beta_{2} - 12 \beta_{3} ) q^{46} + ( 22 + 2 \beta_{1} - 22 \beta_{2} + 4 \beta_{3} ) q^{47} + ( 20 \beta_{1} + \beta_{2} - 20 \beta_{3} ) q^{49} + ( 48 - 11 \beta_{3} ) q^{50} + ( 4 - 16 \beta_{1} + 2 \beta_{2} + 16 \beta_{3} ) q^{52} + ( -18 \beta_{1} + 60 \beta_{2} - 18 \beta_{3} ) q^{53} + ( 12 + 24 \beta_{1} + 24 \beta_{2} + 12 \beta_{3} ) q^{55} + ( 8 + 10 \beta_{1} - 8 \beta_{2} ) q^{56} + ( -48 - 48 \beta_{2} ) q^{58} + ( 8 - 14 \beta_{1} + 4 \beta_{2} + 14 \beta_{3} ) q^{59} + ( -12 + 8 \beta_{1} + 12 \beta_{2} + 16 \beta_{3} ) q^{61} + ( -12 + 34 \beta_{1} - 24 \beta_{2} + 17 \beta_{3} ) q^{62} + 8 q^{64} + ( -90 - 42 \beta_{1} - 90 \beta_{2} ) q^{65} + ( -42 \beta_{1} - 55 \beta_{2} - 42 \beta_{3} ) q^{67} + ( -16 + 4 \beta_{1} + 16 \beta_{2} + 8 \beta_{3} ) q^{68} + ( 44 + 20 \beta_{1} + 40 \beta_{2} - 14 \beta_{3} ) q^{70} + ( -78 - 42 \beta_{3} ) q^{71} + ( -22 - 40 \beta_{1} - 11 \beta_{2} + 40 \beta_{3} ) q^{73} + ( 11 \beta_{1} - 24 \beta_{2} + 11 \beta_{3} ) q^{74} + ( -14 - 8 \beta_{1} - 28 \beta_{2} - 4 \beta_{3} ) q^{76} + ( -30 + 24 \beta_{1} + 12 \beta_{3} ) q^{77} + ( -5 + 66 \beta_{1} - 5 \beta_{2} ) q^{79} + ( 16 + 8 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{80} + ( 8 + 26 \beta_{1} - 8 \beta_{2} + 52 \beta_{3} ) q^{82} + ( -10 + 76 \beta_{1} - 20 \beta_{2} + 38 \beta_{3} ) q^{83} + ( -72 + 60 \beta_{3} ) q^{85} + ( 12 + 7 \beta_{1} + 12 \beta_{2} ) q^{86} + ( 12 \beta_{1} + 12 \beta_{3} ) q^{88} + ( 12 - 12 \beta_{2} ) q^{89} + ( -101 - 34 \beta_{1} - 91 \beta_{2} - 80 \beta_{3} ) q^{91} + ( 24 + 36 \beta_{3} ) q^{92} + ( -8 + 22 \beta_{1} - 4 \beta_{2} - 22 \beta_{3} ) q^{94} + ( -54 \beta_{1} - 66 \beta_{2} - 54 \beta_{3} ) q^{95} + ( -12 - 8 \beta_{1} - 24 \beta_{2} - 4 \beta_{3} ) q^{97} + ( 40 + 80 \beta_{2} + \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 12 q^{5} - 10 q^{7} + O(q^{10})$$ $$4 q - 4 q^{4} - 12 q^{5} - 10 q^{7} - 24 q^{10} + 12 q^{11} - 24 q^{14} - 8 q^{16} + 48 q^{17} - 42 q^{19} - 24 q^{23} + 22 q^{25} - 96 q^{26} + 40 q^{28} + 102 q^{31} + 108 q^{35} + 22 q^{37} - 24 q^{38} + 48 q^{40} + 28 q^{43} + 24 q^{44} - 72 q^{46} + 132 q^{47} - 2 q^{49} + 192 q^{50} + 12 q^{52} - 120 q^{53} + 48 q^{56} - 96 q^{58} + 24 q^{59} - 72 q^{61} + 32 q^{64} - 180 q^{65} + 110 q^{67} - 96 q^{68} + 96 q^{70} - 312 q^{71} - 66 q^{73} + 48 q^{74} - 120 q^{77} - 10 q^{79} + 48 q^{80} + 48 q^{82} - 288 q^{85} + 24 q^{86} + 72 q^{89} - 222 q^{91} + 96 q^{92} - 24 q^{94} + 132 q^{95} + O(q^{100})$$

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 $$\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.707107 + 1.22474i 0 −1.00000 1.73205i 1.24264 + 0.717439i 0 1.74264 + 6.77962i 2.82843 0 −1.75736 + 1.01461i
19.2 0.707107 1.22474i 0 −1.00000 1.73205i −7.24264 4.18154i 0 −6.74264 + 1.88064i −2.82843 0 −10.2426 + 5.91359i
73.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i 1.24264 0.717439i 0 1.74264 6.77962i 2.82843 0 −1.75736 1.01461i
73.2 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −7.24264 + 4.18154i 0 −6.74264 1.88064i −2.82843 0 −10.2426 5.91359i
 $$n$$: 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.3.n.a 4
3.b odd 2 1 42.3.g.a 4
4.b odd 2 1 1008.3.cg.h 4
7.b odd 2 1 882.3.n.e 4
7.c even 3 1 882.3.c.b 4
7.c even 3 1 882.3.n.e 4
7.d odd 6 1 inner 126.3.n.a 4
7.d odd 6 1 882.3.c.b 4
12.b even 2 1 336.3.bh.e 4
15.d odd 2 1 1050.3.p.a 4
15.e even 4 2 1050.3.q.a 8
21.c even 2 1 294.3.g.a 4
21.g even 6 1 42.3.g.a 4
21.g even 6 1 294.3.c.a 4
21.h odd 6 1 294.3.c.a 4
21.h odd 6 1 294.3.g.a 4
28.f even 6 1 1008.3.cg.h 4
84.j odd 6 1 336.3.bh.e 4
84.j odd 6 1 2352.3.f.e 4
84.n even 6 1 2352.3.f.e 4
105.p even 6 1 1050.3.p.a 4
105.w odd 12 2 1050.3.q.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a 4 3.b odd 2 1
42.3.g.a 4 21.g even 6 1
126.3.n.a 4 1.a even 1 1 trivial
126.3.n.a 4 7.d odd 6 1 inner
294.3.c.a 4 21.g even 6 1
294.3.c.a 4 21.h odd 6 1
294.3.g.a 4 21.c even 2 1
294.3.g.a 4 21.h odd 6 1
336.3.bh.e 4 12.b even 2 1
336.3.bh.e 4 84.j odd 6 1
882.3.c.b 4 7.c even 3 1
882.3.c.b 4 7.d odd 6 1
882.3.n.e 4 7.b odd 2 1
882.3.n.e 4 7.c even 3 1
1008.3.cg.h 4 4.b odd 2 1
1008.3.cg.h 4 28.f even 6 1
1050.3.p.a 4 15.d odd 2 1
1050.3.p.a 4 105.p even 6 1
1050.3.q.a 8 15.e even 4 2
1050.3.q.a 8 105.w odd 12 2
2352.3.f.e 4 84.j odd 6 1
2352.3.f.e 4 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 12 T_{5}^{3} + 36 T_{5}^{2} - 144 T_{5} + 144$$ acting on $$S_{3}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$144 - 144 T + 36 T^{2} + 12 T^{3} + T^{4}$$
$7$ $$2401 + 490 T + 51 T^{2} + 10 T^{3} + T^{4}$$
$11$ $$( 36 - 6 T + T^{2} )^{2}$$
$13$ $$145161 + 774 T^{2} + T^{4}$$
$17$ $$28224 - 8064 T + 936 T^{2} - 48 T^{3} + T^{4}$$
$19$ $$15129 + 5166 T + 711 T^{2} + 42 T^{3} + T^{4}$$
$23$ $$254016 - 12096 T + 1080 T^{2} + 24 T^{3} + T^{4}$$
$29$ $$( -1152 + T^{2} )^{2}$$
$31$ $$423801 - 66402 T + 4119 T^{2} - 102 T^{3} + T^{4}$$
$37$ $$27889 + 3674 T + 651 T^{2} - 22 T^{3} + T^{4}$$
$41$ $$3732624 + 4248 T^{2} + T^{4}$$
$43$ $$( -23 - 14 T + T^{2} )^{2}$$
$47$ $$2039184 - 188496 T + 7236 T^{2} - 132 T^{3} + T^{4}$$
$53$ $$8714304 + 354240 T + 11448 T^{2} + 120 T^{3} + T^{4}$$
$59$ $$1272384 + 27072 T - 936 T^{2} - 24 T^{3} + T^{4}$$
$61$ $$2304 + 3456 T + 1776 T^{2} + 72 T^{3} + T^{4}$$
$67$ $$253009 + 55330 T + 12603 T^{2} - 110 T^{3} + T^{4}$$
$71$ $$( 2556 + 156 T + T^{2} )^{2}$$
$73$ $$85322169 - 609642 T - 7785 T^{2} + 66 T^{3} + T^{4}$$
$79$ $$75463969 - 86870 T + 8787 T^{2} + 10 T^{3} + T^{4}$$
$83$ $$69956496 + 17928 T^{2} + T^{4}$$
$89$ $$( 432 - 36 T + T^{2} )^{2}$$
$97$ $$112896 + 1056 T^{2} + T^{4}$$