# Properties

 Label 441.4.e.a Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 49) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -5 \zeta_{6} q^{2} + ( -17 + 17 \zeta_{6} ) q^{4} + 45 q^{8} +O(q^{10})$$ $$q -5 \zeta_{6} q^{2} + ( -17 + 17 \zeta_{6} ) q^{4} + 45 q^{8} + ( -68 + 68 \zeta_{6} ) q^{11} -89 \zeta_{6} q^{16} + 340 q^{22} -40 \zeta_{6} q^{23} + ( 125 - 125 \zeta_{6} ) q^{25} + 166 q^{29} + ( -85 + 85 \zeta_{6} ) q^{32} -450 \zeta_{6} q^{37} -180 q^{43} -1156 \zeta_{6} q^{44} + ( -200 + 200 \zeta_{6} ) q^{46} -625 q^{50} + ( 590 - 590 \zeta_{6} ) q^{53} -830 \zeta_{6} q^{58} -287 q^{64} + ( 740 - 740 \zeta_{6} ) q^{67} -688 q^{71} + ( -2250 + 2250 \zeta_{6} ) q^{74} + 1384 \zeta_{6} q^{79} + 900 \zeta_{6} q^{86} + ( -3060 + 3060 \zeta_{6} ) q^{88} + 680 q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 5q^{2} - 17q^{4} + 90q^{8} + O(q^{10})$$ $$2q - 5q^{2} - 17q^{4} + 90q^{8} - 68q^{11} - 89q^{16} + 680q^{22} - 40q^{23} + 125q^{25} + 332q^{29} - 85q^{32} - 450q^{37} - 360q^{43} - 1156q^{44} - 200q^{46} - 1250q^{50} + 590q^{53} - 830q^{58} - 574q^{64} + 740q^{67} - 1376q^{71} - 2250q^{74} + 1384q^{79} + 900q^{86} - 3060q^{88} + 1360q^{92} + O(q^{100})$$

## Character values

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

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 0.5 − 0.866025i 0.5 + 0.866025i
−2.50000 + 4.33013i 0 −8.50000 14.7224i 0 0 0 45.0000 0 0
361.1 −2.50000 4.33013i 0 −8.50000 + 14.7224i 0 0 0 45.0000 0 0
 $$n$$: 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
7.c even 3 1 inner
7.d 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.e.a 2
3.b odd 2 1 49.4.c.d 2
7.b odd 2 1 CM 441.4.e.a 2
7.c even 3 1 441.4.a.m 1
7.c even 3 1 inner 441.4.e.a 2
7.d odd 6 1 441.4.a.m 1
7.d odd 6 1 inner 441.4.e.a 2
21.c even 2 1 49.4.c.d 2
21.g even 6 1 49.4.a.a 1
21.g even 6 1 49.4.c.d 2
21.h odd 6 1 49.4.a.a 1
21.h odd 6 1 49.4.c.d 2
84.j odd 6 1 784.4.a.k 1
84.n even 6 1 784.4.a.k 1
105.o odd 6 1 1225.4.a.l 1
105.p even 6 1 1225.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.a 1 21.g even 6 1
49.4.a.a 1 21.h odd 6 1
49.4.c.d 2 3.b odd 2 1
49.4.c.d 2 21.c even 2 1
49.4.c.d 2 21.g even 6 1
49.4.c.d 2 21.h odd 6 1
441.4.a.m 1 7.c even 3 1
441.4.a.m 1 7.d odd 6 1
441.4.e.a 2 1.a even 1 1 trivial
441.4.e.a 2 7.b odd 2 1 CM
441.4.e.a 2 7.c even 3 1 inner
441.4.e.a 2 7.d odd 6 1 inner
784.4.a.k 1 84.j odd 6 1
784.4.a.k 1 84.n even 6 1
1225.4.a.l 1 105.o odd 6 1
1225.4.a.l 1 105.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{2} + 5 T_{2} + 25$$ $$T_{5}$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$25 + 5 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$4624 + 68 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$1600 + 40 T + T^{2}$$
$29$ $$( -166 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$202500 + 450 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 180 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$348100 - 590 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$547600 - 740 T + T^{2}$$
$71$ $$( 688 + T )^{2}$$
$73$ $$T^{2}$$
$79$ $$1915456 - 1384 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$