# Properties

 Label 126.3.c.b Level $126$ Weight $3$ Character orbit 126.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43325133094$$ Analytic rank: $$0$$ Dimension: $$4$$ 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: $$2^{2}$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 2 q^{4} + ( - \beta_{3} + 2 \beta_{2}) q^{5} + (2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{7} + 2 \beta_1 q^{8}+O(q^{10})$$ q + b1 * q^2 + 2 * q^4 + (-b3 + 2*b2) * q^5 + (2*b3 + b2 + 3*b1 + 2) * q^7 + 2*b1 * q^8 $$q + \beta_1 q^{2} + 2 q^{4} + ( - \beta_{3} + 2 \beta_{2}) q^{5} + (2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{7} + 2 \beta_1 q^{8} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{10} + ( - 3 \beta_1 + 6) q^{11} + ( - 2 \beta_{3} + 8 \beta_{2}) q^{13} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 6) q^{14} + 4 q^{16} + (11 \beta_{3} + 2 \beta_{2}) q^{17} + (8 \beta_{3} - 2 \beta_{2}) q^{19} + ( - 2 \beta_{3} + 4 \beta_{2}) q^{20} + (6 \beta_1 - 6) q^{22} + ( - 9 \beta_1 - 6) q^{23} + ( - 12 \beta_1 + 7) q^{25} + ( - 8 \beta_{3} + 4 \beta_{2}) q^{26} + (4 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 4) q^{28} - 30 q^{29} + ( - 12 \beta_{3} - 12 \beta_{2}) q^{31} + 4 \beta_1 q^{32} + ( - 2 \beta_{3} - 22 \beta_{2}) q^{34} + ( - 8 \beta_{3} + 10 \beta_{2} + 9 \beta_1 + 6) q^{35} + ( - 36 \beta_1 - 20) q^{37} + (2 \beta_{3} - 16 \beta_{2}) q^{38} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{40} + (7 \beta_{3} - 14 \beta_{2}) q^{41} + (30 \beta_1 - 32) q^{43} + ( - 6 \beta_1 + 12) q^{44} + ( - 6 \beta_1 - 18) q^{46} + ( - 4 \beta_{3} - 28 \beta_{2}) q^{47} + (2 \beta_{3} - 20 \beta_{2} + 24 \beta_1 - 5) q^{49} + (7 \beta_1 - 24) q^{50} + ( - 4 \beta_{3} + 16 \beta_{2}) q^{52} + ( - 12 \beta_1 + 54) q^{53} + 6 \beta_{2} q^{55} + ( - 2 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 12) q^{56} - 30 \beta_1 q^{58} + ( - 20 \beta_{3} + 28 \beta_{2}) q^{59} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{61} + (12 \beta_{3} + 24 \beta_{2}) q^{62} + 8 q^{64} + ( - 36 \beta_1 - 60) q^{65} + (12 \beta_1 + 44) q^{67} + (22 \beta_{3} + 4 \beta_{2}) q^{68} + ( - 10 \beta_{3} + 16 \beta_{2} + 6 \beta_1 + 18) q^{70} + (57 \beta_1 + 30) q^{71} + (26 \beta_{3} + 4 \beta_{2}) q^{73} + ( - 20 \beta_1 - 72) q^{74} + (16 \beta_{3} - 4 \beta_{2}) q^{76} + (15 \beta_{3} + 18 \beta_{2} + 12 \beta_1 - 6) q^{77} + ( - 72 \beta_1 + 32) q^{79} + ( - 4 \beta_{3} + 8 \beta_{2}) q^{80} + (14 \beta_{3} - 14 \beta_{2}) q^{82} + ( - 20 \beta_{3} - 32 \beta_{2}) q^{83} + (60 \beta_1 + 54) q^{85} + ( - 32 \beta_1 + 60) q^{86} + (12 \beta_1 - 12) q^{88} + (21 \beta_{3} + 54 \beta_{2}) q^{89} + ( - 28 \beta_{3} + 28 \beta_{2} + 42 \beta_1) q^{91} + ( - 18 \beta_1 - 12) q^{92} + (28 \beta_{3} + 8 \beta_{2}) q^{94} + (54 \beta_1 + 60) q^{95} + (10 \beta_{3} + 44 \beta_{2}) q^{97} + (20 \beta_{3} - 4 \beta_{2} - 5 \beta_1 + 48) q^{98}+O(q^{100})$$ q + b1 * q^2 + 2 * q^4 + (-b3 + 2*b2) * q^5 + (2*b3 + b2 + 3*b1 + 2) * q^7 + 2*b1 * q^8 + (-2*b3 + 2*b2) * q^10 + (-3*b1 + 6) * q^11 + (-2*b3 + 8*b2) * q^13 + (-b3 - 4*b2 + 2*b1 + 6) * q^14 + 4 * q^16 + (11*b3 + 2*b2) * q^17 + (8*b3 - 2*b2) * q^19 + (-2*b3 + 4*b2) * q^20 + (6*b1 - 6) * q^22 + (-9*b1 - 6) * q^23 + (-12*b1 + 7) * q^25 + (-8*b3 + 4*b2) * q^26 + (4*b3 + 2*b2 + 6*b1 + 4) * q^28 - 30 * q^29 + (-12*b3 - 12*b2) * q^31 + 4*b1 * q^32 + (-2*b3 - 22*b2) * q^34 + (-8*b3 + 10*b2 + 9*b1 + 6) * q^35 + (-36*b1 - 20) * q^37 + (2*b3 - 16*b2) * q^38 + (-4*b3 + 4*b2) * q^40 + (7*b3 - 14*b2) * q^41 + (30*b1 - 32) * q^43 + (-6*b1 + 12) * q^44 + (-6*b1 - 18) * q^46 + (-4*b3 - 28*b2) * q^47 + (2*b3 - 20*b2 + 24*b1 - 5) * q^49 + (7*b1 - 24) * q^50 + (-4*b3 + 16*b2) * q^52 + (-12*b1 + 54) * q^53 + 6*b2 * q^55 + (-2*b3 - 8*b2 + 4*b1 + 12) * q^56 - 30*b1 * q^58 + (-20*b3 + 28*b2) * q^59 + (-4*b3 + 4*b2) * q^61 + (12*b3 + 24*b2) * q^62 + 8 * q^64 + (-36*b1 - 60) * q^65 + (12*b1 + 44) * q^67 + (22*b3 + 4*b2) * q^68 + (-10*b3 + 16*b2 + 6*b1 + 18) * q^70 + (57*b1 + 30) * q^71 + (26*b3 + 4*b2) * q^73 + (-20*b1 - 72) * q^74 + (16*b3 - 4*b2) * q^76 + (15*b3 + 18*b2 + 12*b1 - 6) * q^77 + (-72*b1 + 32) * q^79 + (-4*b3 + 8*b2) * q^80 + (14*b3 - 14*b2) * q^82 + (-20*b3 - 32*b2) * q^83 + (60*b1 + 54) * q^85 + (-32*b1 + 60) * q^86 + (12*b1 - 12) * q^88 + (21*b3 + 54*b2) * q^89 + (-28*b3 + 28*b2 + 42*b1) * q^91 + (-18*b1 - 12) * q^92 + (28*b3 + 8*b2) * q^94 + (54*b1 + 60) * q^95 + (10*b3 + 44*b2) * q^97 + (20*b3 - 4*b2 - 5*b1 + 48) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} + 8 q^{7}+O(q^{10})$$ 4 * q + 8 * q^4 + 8 * q^7 $$4 q + 8 q^{4} + 8 q^{7} + 24 q^{11} + 24 q^{14} + 16 q^{16} - 24 q^{22} - 24 q^{23} + 28 q^{25} + 16 q^{28} - 120 q^{29} + 24 q^{35} - 80 q^{37} - 128 q^{43} + 48 q^{44} - 72 q^{46} - 20 q^{49} - 96 q^{50} + 216 q^{53} + 48 q^{56} + 32 q^{64} - 240 q^{65} + 176 q^{67} + 72 q^{70} + 120 q^{71} - 288 q^{74} - 24 q^{77} + 128 q^{79} + 216 q^{85} + 240 q^{86} - 48 q^{88} - 48 q^{92} + 240 q^{95} + 192 q^{98}+O(q^{100})$$ 4 * q + 8 * q^4 + 8 * q^7 + 24 * q^11 + 24 * q^14 + 16 * q^16 - 24 * q^22 - 24 * q^23 + 28 * q^25 + 16 * q^28 - 120 * q^29 + 24 * q^35 - 80 * q^37 - 128 * q^43 + 48 * q^44 - 72 * q^46 - 20 * q^49 - 96 * q^50 + 216 * q^53 + 48 * q^56 + 32 * q^64 - 240 * q^65 + 176 * q^67 + 72 * q^70 + 120 * q^71 - 288 * q^74 - 24 * q^77 + 128 * q^79 + 216 * q^85 + 240 * q^86 - 48 * q^88 - 48 * q^92 + 240 * q^95 + 192 * q^98

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 2$$ (v^3 + 4*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_1 ) / 2$$ (b3 - b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$2\beta_1$$ 2*b1

## Character values

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

 $$n$$ $$29$$ $$73$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i −0.707107 − 1.22474i
−1.41421 0 2.00000 1.01461i 0 −2.24264 6.63103i −2.82843 0 1.43488i
55.2 −1.41421 0 2.00000 1.01461i 0 −2.24264 + 6.63103i −2.82843 0 1.43488i
55.3 1.41421 0 2.00000 5.91359i 0 6.24264 + 3.16693i 2.82843 0 8.36308i
55.4 1.41421 0 2.00000 5.91359i 0 6.24264 3.16693i 2.82843 0 8.36308i
 $$n$$: 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.3.c.b 4
3.b odd 2 1 42.3.c.a 4
4.b odd 2 1 1008.3.f.g 4
7.b odd 2 1 inner 126.3.c.b 4
7.c even 3 1 882.3.n.a 4
7.c even 3 1 882.3.n.d 4
7.d odd 6 1 882.3.n.a 4
7.d odd 6 1 882.3.n.d 4
12.b even 2 1 336.3.f.c 4
15.d odd 2 1 1050.3.f.a 4
15.e even 4 2 1050.3.h.a 8
21.c even 2 1 42.3.c.a 4
21.g even 6 1 294.3.g.b 4
21.g even 6 1 294.3.g.c 4
21.h odd 6 1 294.3.g.b 4
21.h odd 6 1 294.3.g.c 4
24.f even 2 1 1344.3.f.e 4
24.h odd 2 1 1344.3.f.f 4
28.d even 2 1 1008.3.f.g 4
84.h odd 2 1 336.3.f.c 4
105.g even 2 1 1050.3.f.a 4
105.k odd 4 2 1050.3.h.a 8
168.e odd 2 1 1344.3.f.e 4
168.i even 2 1 1344.3.f.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a 4 3.b odd 2 1
42.3.c.a 4 21.c even 2 1
126.3.c.b 4 1.a even 1 1 trivial
126.3.c.b 4 7.b odd 2 1 inner
294.3.g.b 4 21.g even 6 1
294.3.g.b 4 21.h odd 6 1
294.3.g.c 4 21.g even 6 1
294.3.g.c 4 21.h odd 6 1
336.3.f.c 4 12.b even 2 1
336.3.f.c 4 84.h odd 2 1
882.3.n.a 4 7.c even 3 1
882.3.n.a 4 7.d odd 6 1
882.3.n.d 4 7.c even 3 1
882.3.n.d 4 7.d odd 6 1
1008.3.f.g 4 4.b odd 2 1
1008.3.f.g 4 28.d even 2 1
1050.3.f.a 4 15.d odd 2 1
1050.3.f.a 4 105.g even 2 1
1050.3.h.a 8 15.e even 4 2
1050.3.h.a 8 105.k odd 4 2
1344.3.f.e 4 24.f even 2 1
1344.3.f.e 4 168.e odd 2 1
1344.3.f.f 4 24.h odd 2 1
1344.3.f.f 4 168.i even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 36T^{2} + 36$$
$7$ $$T^{4} - 8 T^{3} + 42 T^{2} + \cdots + 2401$$
$11$ $$(T^{2} - 12 T + 18)^{2}$$
$13$ $$T^{4} + 432 T^{2} + 28224$$
$17$ $$T^{4} + 1476 T^{2} + 509796$$
$19$ $$T^{4} + 792 T^{2} + 138384$$
$23$ $$(T^{2} + 12 T - 126)^{2}$$
$29$ $$(T + 30)^{4}$$
$31$ $$T^{4} + 2592 T^{2} + 186624$$
$37$ $$(T^{2} + 40 T - 2192)^{2}$$
$41$ $$T^{4} + 1764 T^{2} + 86436$$
$43$ $$(T^{2} + 64 T - 776)^{2}$$
$47$ $$T^{4} + 4896 T^{2} + \cdots + 5089536$$
$53$ $$(T^{2} - 108 T + 2628)^{2}$$
$59$ $$T^{4} + 9504 T^{2} + 2304$$
$61$ $$T^{4} + 288T^{2} + 2304$$
$67$ $$(T^{2} - 88 T + 1648)^{2}$$
$71$ $$(T^{2} - 60 T - 5598)^{2}$$
$73$ $$T^{4} + 8208 T^{2} + \cdots + 16064064$$
$79$ $$(T^{2} - 64 T - 9344)^{2}$$
$83$ $$T^{4} + 10944 T^{2} + \cdots + 451584$$
$89$ $$T^{4} + 22788 T^{2} + \cdots + 37234404$$
$97$ $$T^{4} + 12816 T^{2} + \cdots + 27123264$$