# Properties

 Label 441.4.p.b Level $441$ Weight $4$ Character orbit 441.p Analytic conductor $26.020$ Analytic rank $0$ Dimension $8$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.12745506816.5 Defining polynomial: $$x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( 8 + 5 \beta_{1} - 8 \beta_{2} + 5 \beta_{4} ) q^{4} + ( 10 \beta_{3} - 5 \beta_{5} + 10 \beta_{6} - 5 \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( 8 + 5 \beta_{1} - 8 \beta_{2} + 5 \beta_{4} ) q^{4} + ( 10 \beta_{3} - 5 \beta_{5} + 10 \beta_{6} - 5 \beta_{7} ) q^{8} + ( 13 \beta_{6} + \beta_{7} ) q^{11} + ( -111 \beta_{2} + 40 \beta_{4} ) q^{16} + ( -205 - 59 \beta_{1} ) q^{22} + ( 7 \beta_{3} - 25 \beta_{5} ) q^{23} + ( 125 - 125 \beta_{2} ) q^{25} + ( 11 \beta_{3} + 37 \beta_{5} + 11 \beta_{6} + 37 \beta_{7} ) q^{29} + 111 \beta_{6} q^{32} + 4 \beta_{4} q^{37} + 202 \beta_{1} q^{43} + ( -219 \beta_{3} + 67 \beta_{5} ) q^{44} + ( 187 + 185 \beta_{1} - 187 \beta_{2} + 185 \beta_{4} ) q^{46} + ( 125 \beta_{3} + 125 \beta_{6} ) q^{50} + ( 67 \beta_{6} + 85 \beta_{7} ) q^{53} + ( -65 \beta_{2} - 167 \beta_{4} ) q^{58} + ( -888 - 235 \beta_{1} ) q^{64} + ( -740 + 740 \beta_{2} ) q^{67} + ( 53 \beta_{3} + 141 \beta_{5} + 53 \beta_{6} + 141 \beta_{7} ) q^{71} + ( 8 \beta_{6} - 4 \beta_{7} ) q^{74} + 1384 \beta_{2} q^{79} + ( 404 \beta_{3} - 202 \beta_{5} ) q^{86} + ( -2065 - 1025 \beta_{1} + 2065 \beta_{2} - 1025 \beta_{4} ) q^{88} + ( 501 \beta_{3} + 15 \beta_{5} + 501 \beta_{6} + 15 \beta_{7} ) q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 32q^{4} + O(q^{10})$$ $$8q + 32q^{4} - 444q^{16} - 1640q^{22} + 500q^{25} + 748q^{46} - 260q^{58} - 7104q^{64} - 2960q^{67} + 5536q^{79} - 8260q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} - 148$$$$)/55$$ $$\beta_{2}$$ $$=$$ $$($$$$-8 \nu^{6} + 55 \nu^{4} - 440 \nu^{2} + 576$$$$)/495$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} - 241 \nu$$$$)/165$$ $$\beta_{4}$$ $$=$$ $$($$$$-23 \nu^{6} + 220 \nu^{4} - 1265 \nu^{2} + 1656$$$$)/495$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 368 \nu$$$$)/55$$ $$\beta_{6}$$ $$=$$ $$($$$$-38 \nu^{7} + 385 \nu^{5} - 2090 \nu^{3} + 2736 \nu$$$$)/1485$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + 8 \nu^{5} - 55 \nu^{3} + 72 \nu$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{5} + 3 \beta_{3}$$$$)/9$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - 4 \beta_{2} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{7} + 24 \beta_{6} - 7 \beta_{5} + 24 \beta_{3}$$$$)/9$$ $$\nu^{4}$$ $$=$$ $$8 \beta_{4} - 23 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$($$$$-38 \beta_{7} + 165 \beta_{6}$$$$)/9$$ $$\nu^{6}$$ $$=$$ $$-55 \beta_{1} - 148$$ $$\nu^{7}$$ $$=$$ $$($$$$241 \beta_{5} - 1104 \beta_{3}$$$$)/9$$

## Character values

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

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

## 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
 −2.23256 + 1.28897i 1.00781 − 0.581861i −1.00781 + 0.581861i 2.23256 − 1.28897i −2.23256 − 1.28897i 1.00781 + 0.581861i −1.00781 − 0.581861i 2.23256 + 1.28897i
−4.68205 + 2.70318i 0 10.6144 18.3846i 0 0 0 71.5195i 0 0
80.2 −1.44168 + 0.832353i 0 −2.61438 + 4.52824i 0 0 0 22.0220i 0 0
80.3 1.44168 0.832353i 0 −2.61438 + 4.52824i 0 0 0 22.0220i 0 0
80.4 4.68205 2.70318i 0 10.6144 18.3846i 0 0 0 71.5195i 0 0
215.1 −4.68205 2.70318i 0 10.6144 + 18.3846i 0 0 0 71.5195i 0 0
215.2 −1.44168 0.832353i 0 −2.61438 4.52824i 0 0 0 22.0220i 0 0
215.3 1.44168 + 0.832353i 0 −2.61438 4.52824i 0 0 0 22.0220i 0 0
215.4 4.68205 + 2.70318i 0 10.6144 + 18.3846i 0 0 0 71.5195i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 215.4 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 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.p.b 8
3.b odd 2 1 inner 441.4.p.b 8
7.b odd 2 1 CM 441.4.p.b 8
7.c even 3 1 63.4.c.a 4
7.c even 3 1 inner 441.4.p.b 8
7.d odd 6 1 63.4.c.a 4
7.d odd 6 1 inner 441.4.p.b 8
21.c even 2 1 inner 441.4.p.b 8
21.g even 6 1 63.4.c.a 4
21.g even 6 1 inner 441.4.p.b 8
21.h odd 6 1 63.4.c.a 4
21.h odd 6 1 inner 441.4.p.b 8
28.f even 6 1 1008.4.k.a 4
28.g odd 6 1 1008.4.k.a 4
84.j odd 6 1 1008.4.k.a 4
84.n even 6 1 1008.4.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.c.a 4 7.c even 3 1
63.4.c.a 4 7.d odd 6 1
63.4.c.a 4 21.g even 6 1
63.4.c.a 4 21.h odd 6 1
441.4.p.b 8 1.a even 1 1 trivial
441.4.p.b 8 3.b odd 2 1 inner
441.4.p.b 8 7.b odd 2 1 CM
441.4.p.b 8 7.c even 3 1 inner
441.4.p.b 8 7.d odd 6 1 inner
441.4.p.b 8 21.c even 2 1 inner
441.4.p.b 8 21.g even 6 1 inner
441.4.p.b 8 21.h odd 6 1 inner
1008.4.k.a 4 28.f even 6 1
1008.4.k.a 4 28.g odd 6 1
1008.4.k.a 4 84.j odd 6 1
1008.4.k.a 4 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 32 T_{2}^{6} + 943 T_{2}^{4} - 2592 T_{2}^{2} + 6561$$ acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$6561 - 2592 T^{2} + 943 T^{4} - 32 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$14818219109136 - 20494439856 T^{2} + 24495532 T^{4} - 5324 T^{6} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$267118165009579536 - 25153313905008 T^{2} + 1851739468 T^{4} - 48668 T^{6} + T^{8}$$
$29$ $$( 450373284 + 97556 T^{2} + T^{4} )^{2}$$
$31$ $$T^{8}$$
$37$ $$( 12544 + 112 T^{2} + T^{4} )^{2}$$
$41$ $$T^{8}$$
$43$ $$( -285628 + T^{2} )^{4}$$
$47$ $$T^{8}$$
$53$ $$6424804038679120656 - 1509445868635728 T^{2} + 352095058348 T^{4} - 595508 T^{6} + T^{8}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$( 547600 + 740 T + T^{2} )^{4}$$
$71$ $$( 58795580484 + 1431644 T^{2} + T^{4} )^{2}$$
$73$ $$T^{8}$$
$79$ $$( 1915456 - 1384 T + T^{2} )^{4}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$