# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 126.s (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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + \beta_1 q^{5} + (3 \beta_{2} + 5) q^{7} + 2 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 2*b2 * q^4 + b1 * q^5 + (3*b2 + 5) * q^7 + 2*b3 * q^8 $$q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + \beta_1 q^{5} + (3 \beta_{2} + 5) q^{7} + 2 \beta_{3} q^{8} + 2 \beta_{2} q^{10} + (5 \beta_{3} - 5 \beta_1) q^{11} + 15 q^{13} + (3 \beta_{3} + 5 \beta_1) q^{14} + (4 \beta_{2} - 4) q^{16} + (8 \beta_{3} - 8 \beta_1) q^{17} + ( - 13 \beta_{2} + 13) q^{19} + 2 \beta_{3} q^{20} - 10 q^{22} - 16 \beta_1 q^{23} - 23 \beta_{2} q^{25} + 15 \beta_1 q^{26} + (16 \beta_{2} - 6) q^{28} - 16 \beta_{3} q^{29} - 3 \beta_{2} q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} - 16 q^{34} + (3 \beta_{3} + 5 \beta_1) q^{35} + (17 \beta_{2} - 17) q^{37} + ( - 13 \beta_{3} + 13 \beta_1) q^{38} + (4 \beta_{2} - 4) q^{40} - 57 \beta_{3} q^{41} - 85 q^{43} - 10 \beta_1 q^{44} - 32 \beta_{2} q^{46} + 51 \beta_1 q^{47} + (39 \beta_{2} + 16) q^{49} - 23 \beta_{3} q^{50} + 30 \beta_{2} q^{52} + ( - 24 \beta_{3} + 24 \beta_1) q^{53} - 10 q^{55} + (16 \beta_{3} - 6 \beta_1) q^{56} + ( - 32 \beta_{2} + 32) q^{58} + (64 \beta_{3} - 64 \beta_1) q^{59} + ( - 72 \beta_{2} + 72) q^{61} - 3 \beta_{3} q^{62} - 8 q^{64} + 15 \beta_1 q^{65} - 43 \beta_{2} q^{67} - 16 \beta_1 q^{68} + (16 \beta_{2} - 6) q^{70} + 37 \beta_{3} q^{71} + 95 \beta_{2} q^{73} + (17 \beta_{3} - 17 \beta_1) q^{74} + 26 q^{76} + (25 \beta_{3} - 40 \beta_1) q^{77} + (69 \beta_{2} - 69) q^{79} + (4 \beta_{3} - 4 \beta_1) q^{80} + ( - 114 \beta_{2} + 114) q^{82} + 43 \beta_{3} q^{83} - 16 q^{85} - 85 \beta_1 q^{86} - 20 \beta_{2} q^{88} + 96 \beta_1 q^{89} + (45 \beta_{2} + 75) q^{91} - 32 \beta_{3} q^{92} + 102 \beta_{2} q^{94} + ( - 13 \beta_{3} + 13 \beta_1) q^{95} + 16 q^{97} + (39 \beta_{3} + 16 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + 2*b2 * q^4 + b1 * q^5 + (3*b2 + 5) * q^7 + 2*b3 * q^8 + 2*b2 * q^10 + (5*b3 - 5*b1) * q^11 + 15 * q^13 + (3*b3 + 5*b1) * q^14 + (4*b2 - 4) * q^16 + (8*b3 - 8*b1) * q^17 + (-13*b2 + 13) * q^19 + 2*b3 * q^20 - 10 * q^22 - 16*b1 * q^23 - 23*b2 * q^25 + 15*b1 * q^26 + (16*b2 - 6) * q^28 - 16*b3 * q^29 - 3*b2 * q^31 + (4*b3 - 4*b1) * q^32 - 16 * q^34 + (3*b3 + 5*b1) * q^35 + (17*b2 - 17) * q^37 + (-13*b3 + 13*b1) * q^38 + (4*b2 - 4) * q^40 - 57*b3 * q^41 - 85 * q^43 - 10*b1 * q^44 - 32*b2 * q^46 + 51*b1 * q^47 + (39*b2 + 16) * q^49 - 23*b3 * q^50 + 30*b2 * q^52 + (-24*b3 + 24*b1) * q^53 - 10 * q^55 + (16*b3 - 6*b1) * q^56 + (-32*b2 + 32) * q^58 + (64*b3 - 64*b1) * q^59 + (-72*b2 + 72) * q^61 - 3*b3 * q^62 - 8 * q^64 + 15*b1 * q^65 - 43*b2 * q^67 - 16*b1 * q^68 + (16*b2 - 6) * q^70 + 37*b3 * q^71 + 95*b2 * q^73 + (17*b3 - 17*b1) * q^74 + 26 * q^76 + (25*b3 - 40*b1) * q^77 + (69*b2 - 69) * q^79 + (4*b3 - 4*b1) * q^80 + (-114*b2 + 114) * q^82 + 43*b3 * q^83 - 16 * q^85 - 85*b1 * q^86 - 20*b2 * q^88 + 96*b1 * q^89 + (45*b2 + 75) * q^91 - 32*b3 * q^92 + 102*b2 * q^94 + (-13*b3 + 13*b1) * q^95 + 16 * q^97 + (39*b3 + 16*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + 26 q^{7}+O(q^{10})$$ 4 * q + 4 * q^4 + 26 * q^7 $$4 q + 4 q^{4} + 26 q^{7} + 4 q^{10} + 60 q^{13} - 8 q^{16} + 26 q^{19} - 40 q^{22} - 46 q^{25} + 8 q^{28} - 6 q^{31} - 64 q^{34} - 34 q^{37} - 8 q^{40} - 340 q^{43} - 64 q^{46} + 142 q^{49} + 60 q^{52} - 40 q^{55} + 64 q^{58} + 144 q^{61} - 32 q^{64} - 86 q^{67} + 8 q^{70} + 190 q^{73} + 104 q^{76} - 138 q^{79} + 228 q^{82} - 64 q^{85} - 40 q^{88} + 390 q^{91} + 204 q^{94} + 64 q^{97}+O(q^{100})$$ 4 * q + 4 * q^4 + 26 * q^7 + 4 * q^10 + 60 * q^13 - 8 * q^16 + 26 * q^19 - 40 * q^22 - 46 * q^25 + 8 * q^28 - 6 * q^31 - 64 * q^34 - 34 * q^37 - 8 * q^40 - 340 * q^43 - 64 * q^46 + 142 * q^49 + 60 * q^52 - 40 * q^55 + 64 * q^58 + 144 * q^61 - 32 * q^64 - 86 * q^67 + 8 * q^70 + 190 * q^73 + 104 * q^76 - 138 * q^79 + 228 * q^82 - 64 * q^85 - 40 * q^88 + 390 * q^91 + 204 * q^94 + 64 * q^97

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i −1.22474 + 0.707107i 0 6.50000 2.59808i 2.82843i 0 1.00000 1.73205i
53.2 1.22474 0.707107i 0 1.00000 1.73205i 1.22474 0.707107i 0 6.50000 2.59808i 2.82843i 0 1.00000 1.73205i
107.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −1.22474 0.707107i 0 6.50000 + 2.59808i 2.82843i 0 1.00000 + 1.73205i
107.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.22474 + 0.707107i 0 6.50000 + 2.59808i 2.82843i 0 1.00000 + 1.73205i
 $$n$$: 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
3.b odd 2 1 inner
7.c even 3 1 inner
21.h 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.s.b 4
3.b odd 2 1 inner 126.3.s.b 4
4.b odd 2 1 1008.3.dc.a 4
7.b odd 2 1 882.3.s.c 4
7.c even 3 1 inner 126.3.s.b 4
7.c even 3 1 882.3.b.c 2
7.d odd 6 1 882.3.b.d 2
7.d odd 6 1 882.3.s.c 4
12.b even 2 1 1008.3.dc.a 4
21.c even 2 1 882.3.s.c 4
21.g even 6 1 882.3.b.d 2
21.g even 6 1 882.3.s.c 4
21.h odd 6 1 inner 126.3.s.b 4
21.h odd 6 1 882.3.b.c 2
28.g odd 6 1 1008.3.dc.a 4
84.n even 6 1 1008.3.dc.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.s.b 4 1.a even 1 1 trivial
126.3.s.b 4 3.b odd 2 1 inner
126.3.s.b 4 7.c even 3 1 inner
126.3.s.b 4 21.h odd 6 1 inner
882.3.b.c 2 7.c even 3 1
882.3.b.c 2 21.h odd 6 1
882.3.b.d 2 7.d odd 6 1
882.3.b.d 2 21.g even 6 1
882.3.s.c 4 7.b odd 2 1
882.3.s.c 4 7.d odd 6 1
882.3.s.c 4 21.c even 2 1
882.3.s.c 4 21.g even 6 1
1008.3.dc.a 4 4.b odd 2 1
1008.3.dc.a 4 12.b even 2 1
1008.3.dc.a 4 28.g odd 6 1
1008.3.dc.a 4 84.n even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 2T^{2} + 4$$
$7$ $$(T^{2} - 13 T + 49)^{2}$$
$11$ $$T^{4} - 50T^{2} + 2500$$
$13$ $$(T - 15)^{4}$$
$17$ $$T^{4} - 128 T^{2} + 16384$$
$19$ $$(T^{2} - 13 T + 169)^{2}$$
$23$ $$T^{4} - 512 T^{2} + 262144$$
$29$ $$(T^{2} + 512)^{2}$$
$31$ $$(T^{2} + 3 T + 9)^{2}$$
$37$ $$(T^{2} + 17 T + 289)^{2}$$
$41$ $$(T^{2} + 6498)^{2}$$
$43$ $$(T + 85)^{4}$$
$47$ $$T^{4} - 5202 T^{2} + \cdots + 27060804$$
$53$ $$T^{4} - 1152 T^{2} + \cdots + 1327104$$
$59$ $$T^{4} - 8192 T^{2} + \cdots + 67108864$$
$61$ $$(T^{2} - 72 T + 5184)^{2}$$
$67$ $$(T^{2} + 43 T + 1849)^{2}$$
$71$ $$(T^{2} + 2738)^{2}$$
$73$ $$(T^{2} - 95 T + 9025)^{2}$$
$79$ $$(T^{2} + 69 T + 4761)^{2}$$
$83$ $$(T^{2} + 3698)^{2}$$
$89$ $$T^{4} - 18432 T^{2} + \cdots + 339738624$$
$97$ $$(T - 16)^{4}$$
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