# Properties

 Label 441.3.m.a Level $441$ Weight $3$ Character orbit 441.m Analytic conductor $12.016$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$12.0163796583$$ 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 7) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -3 \zeta_{6} q^{2} + ( -5 + 5 \zeta_{6} ) q^{4} + 3 q^{8} +O(q^{10})$$ $$q -3 \zeta_{6} q^{2} + ( -5 + 5 \zeta_{6} ) q^{4} + 3 q^{8} + ( -6 + 6 \zeta_{6} ) q^{11} + 11 \zeta_{6} q^{16} + 18 q^{22} + 18 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} + 54 q^{29} + ( 45 - 45 \zeta_{6} ) q^{32} + 38 \zeta_{6} q^{37} + 58 q^{43} -30 \zeta_{6} q^{44} + ( 54 - 54 \zeta_{6} ) q^{46} + 75 q^{50} + ( -6 + 6 \zeta_{6} ) q^{53} -162 \zeta_{6} q^{58} -91 q^{64} + ( 118 - 118 \zeta_{6} ) q^{67} -114 q^{71} + ( 114 - 114 \zeta_{6} ) q^{74} + 94 \zeta_{6} q^{79} -174 \zeta_{6} q^{86} + ( -18 + 18 \zeta_{6} ) q^{88} -90 q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} - 5q^{4} + 6q^{8} + O(q^{10})$$ $$2q - 3q^{2} - 5q^{4} + 6q^{8} - 6q^{11} + 11q^{16} + 36q^{22} + 18q^{23} - 25q^{25} + 108q^{29} + 45q^{32} + 38q^{37} + 116q^{43} - 30q^{44} + 54q^{46} + 150q^{50} - 6q^{53} - 162q^{58} - 182q^{64} + 118q^{67} - 228q^{71} + 114q^{74} + 94q^{79} - 174q^{86} - 18q^{88} - 180q^{92} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.50000 + 2.59808i 0 −2.50000 4.33013i 0 0 0 3.00000 0 0
325.1 −1.50000 2.59808i 0 −2.50000 + 4.33013i 0 0 0 3.00000 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 441.3.m.a 2
3.b odd 2 1 49.3.d.a 2
7.b odd 2 1 CM 441.3.m.a 2
7.c even 3 1 63.3.d.a 1
7.c even 3 1 inner 441.3.m.a 2
7.d odd 6 1 63.3.d.a 1
7.d odd 6 1 inner 441.3.m.a 2
12.b even 2 1 784.3.s.a 2
21.c even 2 1 49.3.d.a 2
21.g even 6 1 7.3.b.a 1
21.g even 6 1 49.3.d.a 2
21.h odd 6 1 7.3.b.a 1
21.h odd 6 1 49.3.d.a 2
28.f even 6 1 1008.3.f.a 1
28.g odd 6 1 1008.3.f.a 1
84.h odd 2 1 784.3.s.a 2
84.j odd 6 1 112.3.c.a 1
84.j odd 6 1 784.3.s.a 2
84.n even 6 1 112.3.c.a 1
84.n even 6 1 784.3.s.a 2
105.o odd 6 1 175.3.d.a 1
105.p even 6 1 175.3.d.a 1
105.w odd 12 2 175.3.c.a 2
105.x even 12 2 175.3.c.a 2
168.s odd 6 1 448.3.c.a 1
168.v even 6 1 448.3.c.b 1
168.ba even 6 1 448.3.c.a 1
168.be odd 6 1 448.3.c.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.3.b.a 1 21.g even 6 1
7.3.b.a 1 21.h odd 6 1
49.3.d.a 2 3.b odd 2 1
49.3.d.a 2 21.c even 2 1
49.3.d.a 2 21.g even 6 1
49.3.d.a 2 21.h odd 6 1
63.3.d.a 1 7.c even 3 1
63.3.d.a 1 7.d odd 6 1
112.3.c.a 1 84.j odd 6 1
112.3.c.a 1 84.n even 6 1
175.3.c.a 2 105.w odd 12 2
175.3.c.a 2 105.x even 12 2
175.3.d.a 1 105.o odd 6 1
175.3.d.a 1 105.p even 6 1
441.3.m.a 2 1.a even 1 1 trivial
441.3.m.a 2 7.b odd 2 1 CM
441.3.m.a 2 7.c even 3 1 inner
441.3.m.a 2 7.d odd 6 1 inner
448.3.c.a 1 168.s odd 6 1
448.3.c.a 1 168.ba even 6 1
448.3.c.b 1 168.v even 6 1
448.3.c.b 1 168.be odd 6 1
784.3.s.a 2 12.b even 2 1
784.3.s.a 2 84.h odd 2 1
784.3.s.a 2 84.j odd 6 1
784.3.s.a 2 84.n even 6 1
1008.3.f.a 1 28.f even 6 1
1008.3.f.a 1 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{2} + 3 T_{2} + 9$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 5 T^{2} + 12 T^{3} + 16 T^{4}$$
$3$ 1
$5$ $$( 1 - 5 T + 25 T^{2} )( 1 + 5 T + 25 T^{2} )$$
$7$ 1
$11$ $$1 + 6 T - 85 T^{2} + 726 T^{3} + 14641 T^{4}$$
$13$ $$( 1 - 13 T )^{2}( 1 + 13 T )^{2}$$
$17$ $$( 1 - 17 T + 289 T^{2} )( 1 + 17 T + 289 T^{2} )$$
$19$ $$( 1 - 19 T + 361 T^{2} )( 1 + 19 T + 361 T^{2} )$$
$23$ $$1 - 18 T - 205 T^{2} - 9522 T^{3} + 279841 T^{4}$$
$29$ $$( 1 - 54 T + 841 T^{2} )^{2}$$
$31$ $$( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} )$$
$37$ $$1 - 38 T + 75 T^{2} - 52022 T^{3} + 1874161 T^{4}$$
$41$ $$( 1 - 41 T )^{2}( 1 + 41 T )^{2}$$
$43$ $$( 1 - 58 T + 1849 T^{2} )^{2}$$
$47$ $$( 1 - 47 T + 2209 T^{2} )( 1 + 47 T + 2209 T^{2} )$$
$53$ $$1 + 6 T - 2773 T^{2} + 16854 T^{3} + 7890481 T^{4}$$
$59$ $$( 1 - 59 T + 3481 T^{2} )( 1 + 59 T + 3481 T^{2} )$$
$61$ $$( 1 - 61 T + 3721 T^{2} )( 1 + 61 T + 3721 T^{2} )$$
$67$ $$1 - 118 T + 9435 T^{2} - 529702 T^{3} + 20151121 T^{4}$$
$71$ $$( 1 + 114 T + 5041 T^{2} )^{2}$$
$73$ $$( 1 - 73 T + 5329 T^{2} )( 1 + 73 T + 5329 T^{2} )$$
$79$ $$1 - 94 T + 2595 T^{2} - 586654 T^{3} + 38950081 T^{4}$$
$83$ $$( 1 - 83 T )^{2}( 1 + 83 T )^{2}$$
$89$ $$( 1 - 89 T + 7921 T^{2} )( 1 + 89 T + 7921 T^{2} )$$
$97$ $$( 1 - 97 T )^{2}( 1 + 97 T )^{2}$$