# Properties

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.43325133094$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-33})$$ Defining polynomial: $$x^{4} + 32 x^{2} + 289$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 2 q^{4} -\beta_{2} q^{5} + ( -4 + \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 2 q^{4} -\beta_{2} q^{5} + ( -4 + \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} + 2 \beta_{3} q^{10} + 9 \beta_{1} q^{11} -4 \beta_{3} q^{13} + ( -4 \beta_{1} - \beta_{2} ) q^{14} + 4 q^{16} -\beta_{2} q^{17} -2 \beta_{3} q^{19} -2 \beta_{2} q^{20} + 18 q^{22} + 15 \beta_{1} q^{23} -41 q^{25} + 4 \beta_{2} q^{26} + ( -8 + 2 \beta_{3} ) q^{28} -24 \beta_{1} q^{29} + 4 \beta_{1} q^{32} + 2 \beta_{3} q^{34} + ( -33 \beta_{1} + 4 \beta_{2} ) q^{35} + 16 q^{37} + 2 \beta_{2} q^{38} + 4 \beta_{3} q^{40} + 7 \beta_{2} q^{41} + 52 q^{43} + 18 \beta_{1} q^{44} + 30 q^{46} -4 \beta_{2} q^{47} + ( -17 - 8 \beta_{3} ) q^{49} -41 \beta_{1} q^{50} -8 \beta_{3} q^{52} -12 \beta_{1} q^{53} + 18 \beta_{3} q^{55} + ( -8 \beta_{1} - 2 \beta_{2} ) q^{56} -48 q^{58} + 4 \beta_{2} q^{59} + 4 \beta_{3} q^{61} + 8 q^{64} + 132 \beta_{1} q^{65} -52 q^{67} -2 \beta_{2} q^{68} + ( -66 - 8 \beta_{3} ) q^{70} -63 \beta_{1} q^{71} -8 \beta_{3} q^{73} + 16 \beta_{1} q^{74} -4 \beta_{3} q^{76} + ( -36 \beta_{1} - 9 \beta_{2} ) q^{77} + 104 q^{79} -4 \beta_{2} q^{80} -14 \beta_{3} q^{82} -20 \beta_{2} q^{83} -66 q^{85} + 52 \beta_{1} q^{86} + 36 q^{88} + 9 \beta_{2} q^{89} + ( 132 + 16 \beta_{3} ) q^{91} + 30 \beta_{1} q^{92} + 8 \beta_{3} q^{94} + 66 \beta_{1} q^{95} -16 \beta_{3} q^{97} + ( -17 \beta_{1} + 8 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} - 16 q^{7} + O(q^{10})$$ $$4 q + 8 q^{4} - 16 q^{7} + 16 q^{16} + 72 q^{22} - 164 q^{25} - 32 q^{28} + 64 q^{37} + 208 q^{43} + 120 q^{46} - 68 q^{49} - 192 q^{58} + 32 q^{64} - 208 q^{67} - 264 q^{70} + 416 q^{79} - 264 q^{85} + 144 q^{88} + 528 q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 32 x^{2} + 289$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 15 \nu$$$$)/17$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 49 \nu$$$$)/17$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 16$$ $$\nu^{3}$$ $$=$$ $$($$$$-15 \beta_{2} + 49 \beta_{1}$$$$)/2$$

## 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 + 4.06202i 0.707107 − 4.06202i −0.707107 + 4.06202i −0.707107 − 4.06202i
−1.41421 0 2.00000 8.12404i 0 −4.00000 + 5.74456i −2.82843 0 11.4891i
55.2 −1.41421 0 2.00000 8.12404i 0 −4.00000 5.74456i −2.82843 0 11.4891i
55.3 1.41421 0 2.00000 8.12404i 0 −4.00000 5.74456i 2.82843 0 11.4891i
55.4 1.41421 0 2.00000 8.12404i 0 −4.00000 + 5.74456i 2.82843 0 11.4891i
 $$n$$: 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.b odd 2 1 inner
21.c even 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.a 4
3.b odd 2 1 inner 126.3.c.a 4
4.b odd 2 1 1008.3.f.i 4
7.b odd 2 1 inner 126.3.c.a 4
7.c even 3 2 882.3.n.i 8
7.d odd 6 2 882.3.n.i 8
12.b even 2 1 1008.3.f.i 4
21.c even 2 1 inner 126.3.c.a 4
21.g even 6 2 882.3.n.i 8
21.h odd 6 2 882.3.n.i 8
28.d even 2 1 1008.3.f.i 4
84.h odd 2 1 1008.3.f.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.c.a 4 1.a even 1 1 trivial
126.3.c.a 4 3.b odd 2 1 inner
126.3.c.a 4 7.b odd 2 1 inner
126.3.c.a 4 21.c even 2 1 inner
882.3.n.i 8 7.c even 3 2
882.3.n.i 8 7.d odd 6 2
882.3.n.i 8 21.g even 6 2
882.3.n.i 8 21.h odd 6 2
1008.3.f.i 4 4.b odd 2 1
1008.3.f.i 4 12.b even 2 1
1008.3.f.i 4 28.d even 2 1
1008.3.f.i 4 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 66$$ acting on $$S_{3}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 66 + T^{2} )^{2}$$
$7$ $$( 49 + 8 T + T^{2} )^{2}$$
$11$ $$( -162 + T^{2} )^{2}$$
$13$ $$( 528 + T^{2} )^{2}$$
$17$ $$( 66 + T^{2} )^{2}$$
$19$ $$( 132 + T^{2} )^{2}$$
$23$ $$( -450 + T^{2} )^{2}$$
$29$ $$( -1152 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( -16 + T )^{4}$$
$41$ $$( 3234 + T^{2} )^{2}$$
$43$ $$( -52 + T )^{4}$$
$47$ $$( 1056 + T^{2} )^{2}$$
$53$ $$( -288 + T^{2} )^{2}$$
$59$ $$( 1056 + T^{2} )^{2}$$
$61$ $$( 528 + T^{2} )^{2}$$
$67$ $$( 52 + T )^{4}$$
$71$ $$( -7938 + T^{2} )^{2}$$
$73$ $$( 2112 + T^{2} )^{2}$$
$79$ $$( -104 + T )^{4}$$
$83$ $$( 26400 + T^{2} )^{2}$$
$89$ $$( 5346 + T^{2} )^{2}$$
$97$ $$( 8448 + T^{2} )^{2}$$