# Properties

 Label 441.2.p.a Level $441$ Weight $2$ Character orbit 441.p Analytic conductor $3.521$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( \beta_{1} - 2 \beta_{3} ) q^{5} -2 \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( \beta_{1} - 2 \beta_{3} ) q^{5} -2 \beta_{3} q^{8} + ( 4 - 2 \beta_{2} ) q^{10} + ( -\beta_{1} + \beta_{3} ) q^{11} + ( 3 - 6 \beta_{2} ) q^{13} + ( 4 - 4 \beta_{2} ) q^{16} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -1 - \beta_{2} ) q^{19} -2 q^{22} + 4 \beta_{1} q^{23} -\beta_{2} q^{25} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{26} + 2 \beta_{3} q^{29} + ( -2 + \beta_{2} ) q^{31} + ( -4 + 8 \beta_{2} ) q^{34} + ( 1 - \beta_{2} ) q^{37} + ( -\beta_{1} - \beta_{3} ) q^{38} + ( -4 - 4 \beta_{2} ) q^{40} + ( 6 \beta_{1} - 3 \beta_{3} ) q^{41} - q^{43} + 8 \beta_{2} q^{46} + ( -5 \beta_{1} + 10 \beta_{3} ) q^{47} -\beta_{3} q^{50} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -2 + 4 \beta_{2} ) q^{55} + ( -4 + 4 \beta_{2} ) q^{58} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{59} + ( 2 + 2 \beta_{2} ) q^{61} + ( -2 \beta_{1} + \beta_{3} ) q^{62} -8 q^{64} -9 \beta_{1} q^{65} -11 \beta_{2} q^{67} + 5 \beta_{3} q^{71} + ( -2 + \beta_{2} ) q^{73} + ( \beta_{1} - \beta_{3} ) q^{74} + ( -5 + 5 \beta_{2} ) q^{79} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{80} + ( 6 + 6 \beta_{2} ) q^{82} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{83} + 12 q^{85} -\beta_{1} q^{86} + 4 \beta_{2} q^{88} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{89} + ( -20 + 10 \beta_{2} ) q^{94} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{95} + ( 6 - 12 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 12q^{10} + 8q^{16} - 6q^{19} - 8q^{22} - 2q^{25} - 6q^{31} + 2q^{37} - 24q^{40} - 4q^{43} + 16q^{46} - 8q^{58} + 12q^{61} - 32q^{64} - 22q^{67} - 6q^{73} - 10q^{79} + 36q^{82} + 48q^{85} + 8q^{88} - 60q^{94} + O(q^{100})$$

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 $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 + 0.707107i 0 0 −1.22474 2.12132i 0 0 2.82843i 0 3.00000 + 1.73205i
80.2 1.22474 0.707107i 0 0 1.22474 + 2.12132i 0 0 2.82843i 0 3.00000 + 1.73205i
215.1 −1.22474 0.707107i 0 0 −1.22474 + 2.12132i 0 0 2.82843i 0 3.00000 1.73205i
215.2 1.22474 + 0.707107i 0 0 1.22474 2.12132i 0 0 2.82843i 0 3.00000 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.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 441.2.p.a 4
3.b odd 2 1 inner 441.2.p.a 4
7.b odd 2 1 63.2.p.a 4
7.c even 3 1 63.2.p.a 4
7.c even 3 1 441.2.c.a 4
7.d odd 6 1 441.2.c.a 4
7.d odd 6 1 inner 441.2.p.a 4
21.c even 2 1 63.2.p.a 4
21.g even 6 1 441.2.c.a 4
21.g even 6 1 inner 441.2.p.a 4
21.h odd 6 1 63.2.p.a 4
21.h odd 6 1 441.2.c.a 4
28.d even 2 1 1008.2.bt.b 4
28.f even 6 1 7056.2.k.b 4
28.g odd 6 1 1008.2.bt.b 4
28.g odd 6 1 7056.2.k.b 4
35.c odd 2 1 1575.2.bk.c 4
35.f even 4 2 1575.2.bc.a 8
35.j even 6 1 1575.2.bk.c 4
35.l odd 12 2 1575.2.bc.a 8
63.g even 3 1 567.2.i.d 4
63.h even 3 1 567.2.s.d 4
63.j odd 6 1 567.2.s.d 4
63.l odd 6 1 567.2.i.d 4
63.l odd 6 1 567.2.s.d 4
63.n odd 6 1 567.2.i.d 4
63.o even 6 1 567.2.i.d 4
63.o even 6 1 567.2.s.d 4
84.h odd 2 1 1008.2.bt.b 4
84.j odd 6 1 7056.2.k.b 4
84.n even 6 1 1008.2.bt.b 4
84.n even 6 1 7056.2.k.b 4
105.g even 2 1 1575.2.bk.c 4
105.k odd 4 2 1575.2.bc.a 8
105.o odd 6 1 1575.2.bk.c 4
105.x even 12 2 1575.2.bc.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.p.a 4 7.b odd 2 1
63.2.p.a 4 7.c even 3 1
63.2.p.a 4 21.c even 2 1
63.2.p.a 4 21.h odd 6 1
441.2.c.a 4 7.c even 3 1
441.2.c.a 4 7.d odd 6 1
441.2.c.a 4 21.g even 6 1
441.2.c.a 4 21.h odd 6 1
441.2.p.a 4 1.a even 1 1 trivial
441.2.p.a 4 3.b odd 2 1 inner
441.2.p.a 4 7.d odd 6 1 inner
441.2.p.a 4 21.g even 6 1 inner
567.2.i.d 4 63.g even 3 1
567.2.i.d 4 63.l odd 6 1
567.2.i.d 4 63.n odd 6 1
567.2.i.d 4 63.o even 6 1
567.2.s.d 4 63.h even 3 1
567.2.s.d 4 63.j odd 6 1
567.2.s.d 4 63.l odd 6 1
567.2.s.d 4 63.o even 6 1
1008.2.bt.b 4 28.d even 2 1
1008.2.bt.b 4 28.g odd 6 1
1008.2.bt.b 4 84.h odd 2 1
1008.2.bt.b 4 84.n even 6 1
1575.2.bc.a 8 35.f even 4 2
1575.2.bc.a 8 35.l odd 12 2
1575.2.bc.a 8 105.k odd 4 2
1575.2.bc.a 8 105.x even 12 2
1575.2.bk.c 4 35.c odd 2 1
1575.2.bk.c 4 35.j even 6 1
1575.2.bk.c 4 105.g even 2 1
1575.2.bk.c 4 105.o odd 6 1
7056.2.k.b 4 28.f even 6 1
7056.2.k.b 4 28.g odd 6 1
7056.2.k.b 4 84.j odd 6 1
7056.2.k.b 4 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2 T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$36 + 6 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$4 - 2 T^{2} + T^{4}$$
$13$ $$( 27 + T^{2} )^{2}$$
$17$ $$576 + 24 T^{2} + T^{4}$$
$19$ $$( 3 + 3 T + T^{2} )^{2}$$
$23$ $$1024 - 32 T^{2} + T^{4}$$
$29$ $$( 8 + T^{2} )^{2}$$
$31$ $$( 3 + 3 T + T^{2} )^{2}$$
$37$ $$( 1 - T + T^{2} )^{2}$$
$41$ $$( -54 + T^{2} )^{2}$$
$43$ $$( 1 + T )^{4}$$
$47$ $$22500 + 150 T^{2} + T^{4}$$
$53$ $$64 - 8 T^{2} + T^{4}$$
$59$ $$576 + 24 T^{2} + T^{4}$$
$61$ $$( 12 - 6 T + T^{2} )^{2}$$
$67$ $$( 121 + 11 T + T^{2} )^{2}$$
$71$ $$( 50 + T^{2} )^{2}$$
$73$ $$( 3 + 3 T + T^{2} )^{2}$$
$79$ $$( 25 + 5 T + T^{2} )^{2}$$
$83$ $$( -54 + T^{2} )^{2}$$
$89$ $$576 + 24 T^{2} + T^{4}$$
$97$ $$( 108 + T^{2} )^{2}$$