# Properties

 Label 441.6.c.a Level $441$ Weight $6$ Character orbit 441.c Analytic conductor $70.729$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$70.7292645375$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 63) 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.

 $$f(q)$$ $$=$$ $$q -2 \beta_{1} q^{2} + 24 q^{4} -3 \beta_{3} q^{5} -112 \beta_{1} q^{8} +O(q^{10})$$ $$q -2 \beta_{1} q^{2} + 24 q^{4} -3 \beta_{3} q^{5} -112 \beta_{1} q^{8} + 12 \beta_{2} q^{10} + 409 \beta_{1} q^{11} + 95 \beta_{2} q^{13} + 320 q^{16} -22 \beta_{3} q^{17} -143 \beta_{2} q^{19} -72 \beta_{3} q^{20} + 1636 q^{22} -3260 \beta_{1} q^{23} -479 q^{25} + 190 \beta_{3} q^{26} -3254 \beta_{1} q^{29} + 187 \beta_{2} q^{31} -4224 \beta_{1} q^{32} + 88 \beta_{2} q^{34} + 203 q^{37} -286 \beta_{3} q^{38} + 672 \beta_{2} q^{40} + 5 \beta_{3} q^{41} + 4697 q^{43} + 9816 \beta_{1} q^{44} -13040 q^{46} -1399 \beta_{3} q^{47} + 958 \beta_{1} q^{50} + 2280 \beta_{2} q^{52} -5246 \beta_{1} q^{53} -2454 \beta_{2} q^{55} -13016 q^{58} -290 \beta_{3} q^{59} -1438 \beta_{2} q^{61} + 374 \beta_{3} q^{62} -6656 q^{64} -41895 \beta_{1} q^{65} + 30197 q^{67} -528 \beta_{3} q^{68} -2801 \beta_{1} q^{71} -4399 \beta_{2} q^{73} -406 \beta_{1} q^{74} -3432 \beta_{2} q^{76} -39385 q^{79} -960 \beta_{3} q^{80} -20 \beta_{2} q^{82} -3707 \beta_{3} q^{83} + 19404 q^{85} -9394 \beta_{1} q^{86} + 91616 q^{88} -6146 \beta_{3} q^{89} -78240 \beta_{1} q^{92} + 5596 \beta_{2} q^{94} + 63063 \beta_{1} q^{95} -170 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 96 q^{4} + O(q^{10})$$ $$4 q + 96 q^{4} + 1280 q^{16} + 6544 q^{22} - 1916 q^{25} + 812 q^{37} + 18788 q^{43} - 52160 q^{46} - 52064 q^{58} - 26624 q^{64} + 120788 q^{67} - 157540 q^{79} + 77616 q^{85} + 366464 q^{88} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$7 \nu^{2} - 7$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{3} + 28 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 7 \beta_{1}$$$$)/14$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 7$$$$)/7$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
440.1
 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i
2.82843i 0 24.0000 −51.4393 0 0 158.392i 0 145.492i
440.2 2.82843i 0 24.0000 51.4393 0 0 158.392i 0 145.492i
440.3 2.82843i 0 24.0000 −51.4393 0 0 158.392i 0 145.492i
440.4 2.82843i 0 24.0000 51.4393 0 0 158.392i 0 145.492i
 $$n$$: 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 441.6.c.a 4
3.b odd 2 1 inner 441.6.c.a 4
7.b odd 2 1 inner 441.6.c.a 4
7.c even 3 1 63.6.p.a 4
7.d odd 6 1 63.6.p.a 4
21.c even 2 1 inner 441.6.c.a 4
21.g even 6 1 63.6.p.a 4
21.h odd 6 1 63.6.p.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.p.a 4 7.c even 3 1
63.6.p.a 4 7.d odd 6 1
63.6.p.a 4 21.g even 6 1
63.6.p.a 4 21.h odd 6 1
441.6.c.a 4 1.a even 1 1 trivial
441.6.c.a 4 3.b odd 2 1 inner
441.6.c.a 4 7.b odd 2 1 inner
441.6.c.a 4 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 8$$ acting on $$S_{6}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 8 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( -2646 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( 334562 + T^{2} )^{2}$$
$13$ $$( 1326675 + T^{2} )^{2}$$
$17$ $$( -142296 + T^{2} )^{2}$$
$19$ $$( 3006003 + T^{2} )^{2}$$
$23$ $$( 21255200 + T^{2} )^{2}$$
$29$ $$( 21177032 + T^{2} )^{2}$$
$31$ $$( 5140443 + T^{2} )^{2}$$
$37$ $$( -203 + T )^{4}$$
$41$ $$( -7350 + T^{2} )^{2}$$
$43$ $$( -4697 + T )^{4}$$
$47$ $$( -575417094 + T^{2} )^{2}$$
$53$ $$( 55041032 + T^{2} )^{2}$$
$59$ $$( -24725400 + T^{2} )^{2}$$
$61$ $$( 303973068 + T^{2} )^{2}$$
$67$ $$( -30197 + T )^{4}$$
$71$ $$( 15691202 + T^{2} )^{2}$$
$73$ $$( 2844626547 + T^{2} )^{2}$$
$79$ $$( 39385 + T )^{4}$$
$83$ $$( -4040103606 + T^{2} )^{2}$$
$89$ $$( -11105354904 + T^{2} )^{2}$$
$97$ $$( 4248300 + T^{2} )^{2}$$