# Properties

 Label 126.3.n.c 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: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 4 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{5} + ( - 5 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 3) q^{7} + 2 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 2*b2 * q^4 + (-4*b3 - b2 - 2*b1 + 1) * q^5 + (-5*b3 + 2*b2 - 4*b1 + 3) * q^7 + 2*b3 * q^8 $$q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 4 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{5} + ( - 5 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 3) q^{7} + 2 \beta_{3} q^{8} + ( - \beta_{3} + 4 \beta_{2} + \beta_1 + 8) q^{10} + ( - 3 \beta_{3} + 9 \beta_{2} - 3 \beta_1) q^{11} + (2 \beta_{3} + 12 \beta_{2} + 4 \beta_1 + 6) q^{13} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 10) q^{14} + ( - 4 \beta_{2} - 4) q^{16} + (2 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 10) q^{17} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{19} + (4 \beta_{3} + 4 \beta_{2} + 8 \beta_1 + 2) q^{20} + (9 \beta_{3} + 6) q^{22} + ( - 15 \beta_{2} - 9 \beta_1 - 15) q^{23} + ( - 12 \beta_{3} - 2 \beta_{2} - 12 \beta_1) q^{25} + (12 \beta_{3} + 4 \beta_{2} + 6 \beta_1 - 4) q^{26} + (2 \beta_{3} + 2 \beta_{2} + 10 \beta_1 - 4) q^{28} + ( - 6 \beta_{3} - 12) q^{29} + (15 \beta_{3} - 7 \beta_{2} - 15 \beta_1 - 14) q^{31} + ( - 4 \beta_{3} - 4 \beta_1) q^{32} + (5 \beta_{3} - 8 \beta_{2} + 10 \beta_1 - 4) q^{34} + ( - 14 \beta_{3} - 35 \beta_{2} - 7 \beta_1 - 7) q^{35} + ( - 31 \beta_{2} + 24 \beta_1 - 31) q^{37} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{38} + (4 \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 8) q^{40} + (10 \beta_{3} + 4 \beta_{2} + 20 \beta_1 + 2) q^{41} + (6 \beta_{3} - 2) q^{43} + ( - 18 \beta_{2} + 6 \beta_1 - 18) q^{44} + ( - 15 \beta_{3} - 18 \beta_{2} - 15 \beta_1) q^{46} + (2 \beta_{3} + 29 \beta_{2} + \beta_1 - 29) q^{47} + ( - 26 \beta_{3} - 40 \beta_{2} - 4 \beta_1 - 25) q^{49} + ( - 2 \beta_{3} + 24) q^{50} + (4 \beta_{3} - 12 \beta_{2} - 4 \beta_1 - 24) q^{52} + ( - 12 \beta_{3} - 39 \beta_{2} - 12 \beta_1) q^{53} + (15 \beta_{3} - 6 \beta_{2} + 30 \beta_1 - 3) q^{55} + (2 \beta_{3} + 16 \beta_{2} - 4 \beta_1 - 4) q^{56} + (12 \beta_{2} - 12 \beta_1 + 12) q^{58} + (25 \beta_{3} + 13 \beta_{2} - 25 \beta_1 + 26) q^{59} + (64 \beta_{3} + 7 \beta_{2} + 32 \beta_1 - 7) q^{61} + ( - 7 \beta_{3} - 60 \beta_{2} - 14 \beta_1 - 30) q^{62} + 8 q^{64} + (42 \beta_{2} + 42 \beta_1 + 42) q^{65} + ( - 45 \beta_{3} + 29 \beta_{2} - 45 \beta_1) q^{67} + ( - 8 \beta_{3} + 10 \beta_{2} - 4 \beta_1 - 10) q^{68} + ( - 35 \beta_{3} + 14 \beta_{2} - 7 \beta_1 + 28) q^{70} + (30 \beta_{3} + 6) q^{71} + (16 \beta_{3} + 53 \beta_{2} - 16 \beta_1 + 106) q^{73} + ( - 31 \beta_{3} + 48 \beta_{2} - 31 \beta_1) q^{74} + (2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 2) q^{76} + ( - 21 \beta_{2} + 42 \beta_1 - 42) q^{77} + (55 \beta_{2} - 15 \beta_1 + 55) q^{79} + (8 \beta_{3} - 4 \beta_{2} - 8 \beta_1 - 8) q^{80} + (4 \beta_{3} + 20 \beta_{2} + 2 \beta_1 - 20) q^{82} + (4 \beta_{3} + 136 \beta_{2} + 8 \beta_1 + 68) q^{83} + ( - 24 \beta_{3} - 9) q^{85} + ( - 12 \beta_{2} - 2 \beta_1 - 12) q^{86} + ( - 18 \beta_{3} + 12 \beta_{2} - 18 \beta_1) q^{88} + (48 \beta_{3} - 63 \beta_{2} + 24 \beta_1 + 63) q^{89} + ( - 8 \beta_{3} + 48 \beta_{2} + 44 \beta_1 + 30) q^{91} + ( - 18 \beta_{3} + 30) q^{92} + (29 \beta_{3} - 2 \beta_{2} - 29 \beta_1 - 4) q^{94} + ( - 9 \beta_{3} - 15 \beta_{2} - 9 \beta_1) q^{95} + (26 \beta_{3} + 44 \beta_{2} + 52 \beta_1 + 22) q^{97} + ( - 40 \beta_{3} + 44 \beta_{2} - 25 \beta_1 + 52) q^{98}+O(q^{100})$$ q + b1 * q^2 + 2*b2 * q^4 + (-4*b3 - b2 - 2*b1 + 1) * q^5 + (-5*b3 + 2*b2 - 4*b1 + 3) * q^7 + 2*b3 * q^8 + (-b3 + 4*b2 + b1 + 8) * q^10 + (-3*b3 + 9*b2 - 3*b1) * q^11 + (2*b3 + 12*b2 + 4*b1 + 6) * q^13 + (2*b3 + 2*b2 + 3*b1 + 10) * q^14 + (-4*b2 - 4) * q^16 + (2*b3 + 5*b2 - 2*b1 + 10) * q^17 + (-2*b3 - b2 - b1 + 1) * q^19 + (4*b3 + 4*b2 + 8*b1 + 2) * q^20 + (9*b3 + 6) * q^22 + (-15*b2 - 9*b1 - 15) * q^23 + (-12*b3 - 2*b2 - 12*b1) * q^25 + (12*b3 + 4*b2 + 6*b1 - 4) * q^26 + (2*b3 + 2*b2 + 10*b1 - 4) * q^28 + (-6*b3 - 12) * q^29 + (15*b3 - 7*b2 - 15*b1 - 14) * q^31 + (-4*b3 - 4*b1) * q^32 + (5*b3 - 8*b2 + 10*b1 - 4) * q^34 + (-14*b3 - 35*b2 - 7*b1 - 7) * q^35 + (-31*b2 + 24*b1 - 31) * q^37 + (-b3 + 2*b2 + b1 + 4) * q^38 + (4*b3 + 8*b2 + 2*b1 - 8) * q^40 + (10*b3 + 4*b2 + 20*b1 + 2) * q^41 + (6*b3 - 2) * q^43 + (-18*b2 + 6*b1 - 18) * q^44 + (-15*b3 - 18*b2 - 15*b1) * q^46 + (2*b3 + 29*b2 + b1 - 29) * q^47 + (-26*b3 - 40*b2 - 4*b1 - 25) * q^49 + (-2*b3 + 24) * q^50 + (4*b3 - 12*b2 - 4*b1 - 24) * q^52 + (-12*b3 - 39*b2 - 12*b1) * q^53 + (15*b3 - 6*b2 + 30*b1 - 3) * q^55 + (2*b3 + 16*b2 - 4*b1 - 4) * q^56 + (12*b2 - 12*b1 + 12) * q^58 + (25*b3 + 13*b2 - 25*b1 + 26) * q^59 + (64*b3 + 7*b2 + 32*b1 - 7) * q^61 + (-7*b3 - 60*b2 - 14*b1 - 30) * q^62 + 8 * q^64 + (42*b2 + 42*b1 + 42) * q^65 + (-45*b3 + 29*b2 - 45*b1) * q^67 + (-8*b3 + 10*b2 - 4*b1 - 10) * q^68 + (-35*b3 + 14*b2 - 7*b1 + 28) * q^70 + (30*b3 + 6) * q^71 + (16*b3 + 53*b2 - 16*b1 + 106) * q^73 + (-31*b3 + 48*b2 - 31*b1) * q^74 + (2*b3 + 4*b2 + 4*b1 + 2) * q^76 + (-21*b2 + 42*b1 - 42) * q^77 + (55*b2 - 15*b1 + 55) * q^79 + (8*b3 - 4*b2 - 8*b1 - 8) * q^80 + (4*b3 + 20*b2 + 2*b1 - 20) * q^82 + (4*b3 + 136*b2 + 8*b1 + 68) * q^83 + (-24*b3 - 9) * q^85 + (-12*b2 - 2*b1 - 12) * q^86 + (-18*b3 + 12*b2 - 18*b1) * q^88 + (48*b3 - 63*b2 + 24*b1 + 63) * q^89 + (-8*b3 + 48*b2 + 44*b1 + 30) * q^91 + (-18*b3 + 30) * q^92 + (29*b3 - 2*b2 - 29*b1 - 4) * q^94 + (-9*b3 - 15*b2 - 9*b1) * q^95 + (26*b3 + 44*b2 + 52*b1 + 22) * q^97 + (-40*b3 + 44*b2 - 25*b1 + 52) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 6 q^{5} + 8 q^{7}+O(q^{10})$$ 4 * q - 4 * q^4 + 6 * q^5 + 8 * q^7 $$4 q - 4 q^{4} + 6 q^{5} + 8 q^{7} + 24 q^{10} - 18 q^{11} + 36 q^{14} - 8 q^{16} + 30 q^{17} + 6 q^{19} + 24 q^{22} - 30 q^{23} + 4 q^{25} - 24 q^{26} - 20 q^{28} - 48 q^{29} - 42 q^{31} + 42 q^{35} - 62 q^{37} + 12 q^{38} - 48 q^{40} - 8 q^{43} - 36 q^{44} + 36 q^{46} - 174 q^{47} - 20 q^{49} + 96 q^{50} - 72 q^{52} + 78 q^{53} - 48 q^{56} + 24 q^{58} + 78 q^{59} - 42 q^{61} + 32 q^{64} + 84 q^{65} - 58 q^{67} - 60 q^{68} + 84 q^{70} + 24 q^{71} + 318 q^{73} - 96 q^{74} - 126 q^{77} + 110 q^{79} - 24 q^{80} - 120 q^{82} - 36 q^{85} - 24 q^{86} - 24 q^{88} + 378 q^{89} + 24 q^{91} + 120 q^{92} - 12 q^{94} + 30 q^{95} + 120 q^{98}+O(q^{100})$$ 4 * q - 4 * q^4 + 6 * q^5 + 8 * q^7 + 24 * q^10 - 18 * q^11 + 36 * q^14 - 8 * q^16 + 30 * q^17 + 6 * q^19 + 24 * q^22 - 30 * q^23 + 4 * q^25 - 24 * q^26 - 20 * q^28 - 48 * q^29 - 42 * q^31 + 42 * q^35 - 62 * q^37 + 12 * q^38 - 48 * q^40 - 8 * q^43 - 36 * q^44 + 36 * q^46 - 174 * q^47 - 20 * q^49 + 96 * q^50 - 72 * q^52 + 78 * q^53 - 48 * q^56 + 24 * q^58 + 78 * q^59 - 42 * q^61 + 32 * q^64 + 84 * q^65 - 58 * q^67 - 60 * q^68 + 84 * q^70 + 24 * q^71 + 318 * q^73 - 96 * q^74 - 126 * q^77 + 110 * q^79 - 24 * q^80 - 120 * q^82 - 36 * q^85 - 24 * q^86 - 24 * q^88 + 378 * q^89 + 24 * q^91 + 120 * q^92 - 12 * q^94 + 30 * q^95 + 120 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## 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 −2.74264 1.58346i 0 −2.24264 6.63103i 2.82843 0 3.87868 2.23936i
19.2 0.707107 1.22474i 0 −1.00000 1.73205i 5.74264 + 3.31552i 0 6.24264 + 3.16693i −2.82843 0 8.12132 4.68885i
73.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i −2.74264 + 1.58346i 0 −2.24264 + 6.63103i 2.82843 0 3.87868 + 2.23936i
73.2 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 5.74264 3.31552i 0 6.24264 3.16693i −2.82843 0 8.12132 + 4.68885i
 $$n$$: 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.c 4
3.b odd 2 1 14.3.d.a 4
4.b odd 2 1 1008.3.cg.l 4
7.b odd 2 1 882.3.n.b 4
7.c even 3 1 882.3.c.f 4
7.c even 3 1 882.3.n.b 4
7.d odd 6 1 inner 126.3.n.c 4
7.d odd 6 1 882.3.c.f 4
12.b even 2 1 112.3.s.b 4
15.d odd 2 1 350.3.k.a 4
15.e even 4 2 350.3.i.a 8
21.c even 2 1 98.3.d.a 4
21.g even 6 1 14.3.d.a 4
21.g even 6 1 98.3.b.b 4
21.h odd 6 1 98.3.b.b 4
21.h odd 6 1 98.3.d.a 4
24.f even 2 1 448.3.s.c 4
24.h odd 2 1 448.3.s.d 4
28.f even 6 1 1008.3.cg.l 4
84.h odd 2 1 784.3.s.c 4
84.j odd 6 1 112.3.s.b 4
84.j odd 6 1 784.3.c.e 4
84.n even 6 1 784.3.c.e 4
84.n even 6 1 784.3.s.c 4
105.p even 6 1 350.3.k.a 4
105.w odd 12 2 350.3.i.a 8
168.ba even 6 1 448.3.s.d 4
168.be odd 6 1 448.3.s.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a 4 3.b odd 2 1
14.3.d.a 4 21.g even 6 1
98.3.b.b 4 21.g even 6 1
98.3.b.b 4 21.h odd 6 1
98.3.d.a 4 21.c even 2 1
98.3.d.a 4 21.h odd 6 1
112.3.s.b 4 12.b even 2 1
112.3.s.b 4 84.j odd 6 1
126.3.n.c 4 1.a even 1 1 trivial
126.3.n.c 4 7.d odd 6 1 inner
350.3.i.a 8 15.e even 4 2
350.3.i.a 8 105.w odd 12 2
350.3.k.a 4 15.d odd 2 1
350.3.k.a 4 105.p even 6 1
448.3.s.c 4 24.f even 2 1
448.3.s.c 4 168.be odd 6 1
448.3.s.d 4 24.h odd 2 1
448.3.s.d 4 168.ba even 6 1
784.3.c.e 4 84.j odd 6 1
784.3.c.e 4 84.n even 6 1
784.3.s.c 4 84.h odd 2 1
784.3.s.c 4 84.n even 6 1
882.3.c.f 4 7.c even 3 1
882.3.c.f 4 7.d odd 6 1
882.3.n.b 4 7.b odd 2 1
882.3.n.b 4 7.c even 3 1
1008.3.cg.l 4 4.b odd 2 1
1008.3.cg.l 4 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 6T_{5}^{3} - 9T_{5}^{2} + 126T_{5} + 441$$ acting on $$S_{3}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 6 T^{3} - 9 T^{2} + 126 T + 441$$
$7$ $$T^{4} - 8 T^{3} + 42 T^{2} + \cdots + 2401$$
$11$ $$T^{4} + 18 T^{3} + 261 T^{2} + \cdots + 3969$$
$13$ $$T^{4} + 264T^{2} + 7056$$
$17$ $$T^{4} - 30 T^{3} + 351 T^{2} + \cdots + 2601$$
$19$ $$T^{4} - 6 T^{3} + 9 T^{2} + 18 T + 9$$
$23$ $$T^{4} + 30 T^{3} + 837 T^{2} + \cdots + 3969$$
$29$ $$(T^{2} + 24 T + 72)^{2}$$
$31$ $$T^{4} + 42 T^{3} - 615 T^{2} + \cdots + 1447209$$
$37$ $$T^{4} + 62 T^{3} + 4035 T^{2} + \cdots + 36481$$
$41$ $$T^{4} + 1224 T^{2} + 345744$$
$43$ $$(T^{2} + 4 T - 68)^{2}$$
$47$ $$T^{4} + 174 T^{3} + 12609 T^{2} + \cdots + 6335289$$
$53$ $$T^{4} - 78 T^{3} + 4851 T^{2} + \cdots + 1520289$$
$59$ $$T^{4} - 78 T^{3} - 1215 T^{2} + \cdots + 10517049$$
$61$ $$T^{4} + 42 T^{3} - 5409 T^{2} + \cdots + 35964009$$
$67$ $$T^{4} + 58 T^{3} + 6573 T^{2} + \cdots + 10297681$$
$71$ $$(T^{2} - 12 T - 1764)^{2}$$
$73$ $$T^{4} - 318 T^{3} + \cdots + 47485881$$
$79$ $$T^{4} - 110 T^{3} + 9525 T^{2} + \cdots + 6630625$$
$83$ $$T^{4} + 27936 T^{2} + \cdots + 189778176$$
$89$ $$T^{4} - 378 T^{3} + \cdots + 71419401$$
$97$ $$T^{4} + 11016 T^{2} + \cdots + 6780816$$