# Properties

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

# 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: 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{5} + ( 7 + 7 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{5} + ( 7 + 7 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + ( -8 - 4 \beta_{2} ) q^{10} + ( 12 \beta_{1} + 12 \beta_{3} ) q^{11} + ( -1 - 2 \beta_{2} ) q^{13} + ( 7 \beta_{1} + 7 \beta_{3} ) q^{14} + ( -4 - 4 \beta_{2} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{17} + ( 17 - 17 \beta_{2} ) q^{19} + ( -8 \beta_{1} - 4 \beta_{3} ) q^{20} -24 q^{22} -6 \beta_{1} q^{23} + \beta_{2} q^{25} + ( -\beta_{1} - 2 \beta_{3} ) q^{26} -14 q^{28} -24 \beta_{3} q^{29} + ( -14 - 7 \beta_{2} ) q^{31} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{32} + ( 4 + 8 \beta_{2} ) q^{34} + ( -14 \beta_{1} + 14 \beta_{3} ) q^{35} + ( 47 + 47 \beta_{2} ) q^{37} + ( 17 \beta_{1} - 17 \beta_{3} ) q^{38} + ( 8 - 8 \beta_{2} ) q^{40} + ( -56 \beta_{1} - 28 \beta_{3} ) q^{41} + 31 q^{43} -24 \beta_{1} q^{44} -12 \beta_{2} q^{46} + ( -34 \beta_{1} - 68 \beta_{3} ) q^{47} + 49 \beta_{2} q^{49} + \beta_{3} q^{50} + ( 4 + 2 \beta_{2} ) q^{52} + ( 54 \beta_{1} + 54 \beta_{3} ) q^{53} + ( -48 - 96 \beta_{2} ) q^{55} -14 \beta_{1} q^{56} + ( 48 + 48 \beta_{2} ) q^{58} + ( 34 \beta_{1} - 34 \beta_{3} ) q^{59} + ( -48 + 48 \beta_{2} ) q^{61} + ( -14 \beta_{1} - 7 \beta_{3} ) q^{62} + 8 q^{64} + 6 \beta_{1} q^{65} -31 \beta_{2} q^{67} + ( 4 \beta_{1} + 8 \beta_{3} ) q^{68} + ( -28 - 56 \beta_{2} ) q^{70} + 42 \beta_{3} q^{71} + ( -94 - 47 \beta_{2} ) q^{73} + ( 47 \beta_{1} + 47 \beta_{3} ) q^{74} + ( 34 + 68 \beta_{2} ) q^{76} + 84 \beta_{3} q^{77} + ( -41 - 41 \beta_{2} ) q^{79} + ( 8 \beta_{1} - 8 \beta_{3} ) q^{80} + ( 56 - 56 \beta_{2} ) q^{82} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{83} -24 q^{85} + 31 \beta_{1} q^{86} -48 \beta_{2} q^{88} + ( 24 \beta_{1} + 48 \beta_{3} ) q^{89} + ( 7 - 7 \beta_{2} ) q^{91} -12 \beta_{3} q^{92} + ( 136 + 68 \beta_{2} ) q^{94} + ( 102 \beta_{1} + 102 \beta_{3} ) q^{95} + ( 24 + 48 \beta_{2} ) q^{97} + 49 \beta_{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 14 q^{7} + O(q^{10})$$ $$4 q - 4 q^{4} + 14 q^{7} - 24 q^{10} - 8 q^{16} + 102 q^{19} - 96 q^{22} - 2 q^{25} - 56 q^{28} - 42 q^{31} + 94 q^{37} + 48 q^{40} + 124 q^{43} + 24 q^{46} - 98 q^{49} + 12 q^{52} + 96 q^{58} - 288 q^{61} + 32 q^{64} + 62 q^{67} - 282 q^{73} - 82 q^{79} + 336 q^{82} - 96 q^{85} + 96 q^{88} + 42 q^{91} + 408 q^{94} + O(q^{100})$$

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 4.24264 + 2.44949i 0 3.50000 6.06218i 2.82843 0 −6.00000 + 3.46410i
19.2 0.707107 1.22474i 0 −1.00000 1.73205i −4.24264 2.44949i 0 3.50000 6.06218i −2.82843 0 −6.00000 + 3.46410i
73.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i 4.24264 2.44949i 0 3.50000 + 6.06218i 2.82843 0 −6.00000 3.46410i
73.2 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −4.24264 + 2.44949i 0 3.50000 + 6.06218i −2.82843 0 −6.00000 3.46410i
 $$n$$: 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.d odd 6 1 inner
21.g even 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.b 4
3.b odd 2 1 inner 126.3.n.b 4
4.b odd 2 1 1008.3.cg.i 4
7.b odd 2 1 882.3.n.c 4
7.c even 3 1 882.3.c.c 4
7.c even 3 1 882.3.n.c 4
7.d odd 6 1 inner 126.3.n.b 4
7.d odd 6 1 882.3.c.c 4
12.b even 2 1 1008.3.cg.i 4
21.c even 2 1 882.3.n.c 4
21.g even 6 1 inner 126.3.n.b 4
21.g even 6 1 882.3.c.c 4
21.h odd 6 1 882.3.c.c 4
21.h odd 6 1 882.3.n.c 4
28.f even 6 1 1008.3.cg.i 4
84.j odd 6 1 1008.3.cg.i 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 24 T_{5}^{2} + 576$$ acting on $$S_{3}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$576 - 24 T^{2} + T^{4}$$
$7$ $$( 49 - 7 T + T^{2} )^{2}$$
$11$ $$82944 + 288 T^{2} + T^{4}$$
$13$ $$( 3 + T^{2} )^{2}$$
$17$ $$576 - 24 T^{2} + T^{4}$$
$19$ $$( 867 - 51 T + T^{2} )^{2}$$
$23$ $$5184 + 72 T^{2} + T^{4}$$
$29$ $$( -1152 + T^{2} )^{2}$$
$31$ $$( 147 + 21 T + T^{2} )^{2}$$
$37$ $$( 2209 - 47 T + T^{2} )^{2}$$
$41$ $$( 4704 + T^{2} )^{2}$$
$43$ $$( -31 + T )^{4}$$
$47$ $$48108096 - 6936 T^{2} + T^{4}$$
$53$ $$34012224 + 5832 T^{2} + T^{4}$$
$59$ $$48108096 - 6936 T^{2} + T^{4}$$
$61$ $$( 6912 + 144 T + T^{2} )^{2}$$
$67$ $$( 961 - 31 T + T^{2} )^{2}$$
$71$ $$( -3528 + T^{2} )^{2}$$
$73$ $$( 6627 + 141 T + T^{2} )^{2}$$
$79$ $$( 1681 + 41 T + T^{2} )^{2}$$
$83$ $$( 24 + T^{2} )^{2}$$
$89$ $$11943936 - 3456 T^{2} + T^{4}$$
$97$ $$( 1728 + T^{2} )^{2}$$