# Properties

 Label 441.4.e.i Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $2$ CM discriminant -3 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) 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.

 $$f(q)$$ $$=$$ $$q + ( 8 - 8 \zeta_{6} ) q^{4} +O(q^{10})$$ $$q + ( 8 - 8 \zeta_{6} ) q^{4} -70 q^{13} -64 \zeta_{6} q^{16} -56 \zeta_{6} q^{19} + ( 125 - 125 \zeta_{6} ) q^{25} + ( -308 + 308 \zeta_{6} ) q^{31} -110 \zeta_{6} q^{37} -520 q^{43} + ( -560 + 560 \zeta_{6} ) q^{52} -182 \zeta_{6} q^{61} -512 q^{64} + ( 880 - 880 \zeta_{6} ) q^{67} + ( -1190 + 1190 \zeta_{6} ) q^{73} -448 q^{76} -884 \zeta_{6} q^{79} -1330 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{4} + O(q^{10})$$ $$2q + 8q^{4} - 140q^{13} - 64q^{16} - 56q^{19} + 125q^{25} - 308q^{31} - 110q^{37} - 1040q^{43} - 560q^{52} - 182q^{61} - 1024q^{64} + 880q^{67} - 1190q^{73} - 896q^{76} - 884q^{79} - 2660q^{97} + O(q^{100})$$

## 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
0 0 4.00000 + 6.92820i 0 0 0 0 0 0
361.1 0 0 4.00000 6.92820i 0 0 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.c even 3 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.e.i 2
3.b odd 2 1 CM 441.4.e.i 2
7.b odd 2 1 441.4.e.j 2
7.c even 3 1 9.4.a.a 1
7.c even 3 1 inner 441.4.e.i 2
7.d odd 6 1 441.4.a.f 1
7.d odd 6 1 441.4.e.j 2
21.c even 2 1 441.4.e.j 2
21.g even 6 1 441.4.a.f 1
21.g even 6 1 441.4.e.j 2
21.h odd 6 1 9.4.a.a 1
21.h odd 6 1 inner 441.4.e.i 2
28.g odd 6 1 144.4.a.d 1
35.j even 6 1 225.4.a.d 1
35.l odd 12 2 225.4.b.g 2
56.k odd 6 1 576.4.a.l 1
56.p even 6 1 576.4.a.m 1
63.g even 3 1 81.4.c.b 2
63.h even 3 1 81.4.c.b 2
63.j odd 6 1 81.4.c.b 2
63.n odd 6 1 81.4.c.b 2
77.h odd 6 1 1089.4.a.g 1
84.n even 6 1 144.4.a.d 1
91.r even 6 1 1521.4.a.g 1
105.o odd 6 1 225.4.a.d 1
105.x even 12 2 225.4.b.g 2
168.s odd 6 1 576.4.a.m 1
168.v even 6 1 576.4.a.l 1
231.l even 6 1 1089.4.a.g 1
273.w odd 6 1 1521.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 7.c even 3 1
9.4.a.a 1 21.h odd 6 1
81.4.c.b 2 63.g even 3 1
81.4.c.b 2 63.h even 3 1
81.4.c.b 2 63.j odd 6 1
81.4.c.b 2 63.n odd 6 1
144.4.a.d 1 28.g odd 6 1
144.4.a.d 1 84.n even 6 1
225.4.a.d 1 35.j even 6 1
225.4.a.d 1 105.o odd 6 1
225.4.b.g 2 35.l odd 12 2
225.4.b.g 2 105.x even 12 2
441.4.a.f 1 7.d odd 6 1
441.4.a.f 1 21.g even 6 1
441.4.e.i 2 1.a even 1 1 trivial
441.4.e.i 2 3.b odd 2 1 CM
441.4.e.i 2 7.c even 3 1 inner
441.4.e.i 2 21.h odd 6 1 inner
441.4.e.j 2 7.b odd 2 1
441.4.e.j 2 7.d odd 6 1
441.4.e.j 2 21.c even 2 1
441.4.e.j 2 21.g even 6 1
576.4.a.l 1 56.k odd 6 1
576.4.a.l 1 168.v even 6 1
576.4.a.m 1 56.p even 6 1
576.4.a.m 1 168.s odd 6 1
1089.4.a.g 1 77.h odd 6 1
1089.4.a.g 1 231.l even 6 1
1521.4.a.g 1 91.r even 6 1
1521.4.a.g 1 273.w odd 6 1

## Hecke kernels

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

 $$T_{2}$$ $$T_{5}$$ $$T_{13} + 70$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$( 70 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$3136 + 56 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$94864 + 308 T + T^{2}$$
$37$ $$12100 + 110 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 520 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$33124 + 182 T + T^{2}$$
$67$ $$774400 - 880 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$1416100 + 1190 T + T^{2}$$
$79$ $$781456 + 884 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( 1330 + T )^{2}$$