# Properties

 Label 441.4.e.l Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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 147) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{2} + ( -8 + 8 \zeta_{6} ) q^{4} -18 \zeta_{6} q^{5} +O(q^{10})$$ $$q + 4 \zeta_{6} q^{2} + ( -8 + 8 \zeta_{6} ) q^{4} -18 \zeta_{6} q^{5} + ( 72 - 72 \zeta_{6} ) q^{10} + ( -50 + 50 \zeta_{6} ) q^{11} + 36 q^{13} + 64 \zeta_{6} q^{16} + ( -126 + 126 \zeta_{6} ) q^{17} -72 \zeta_{6} q^{19} + 144 q^{20} -200 q^{22} + 14 \zeta_{6} q^{23} + ( -199 + 199 \zeta_{6} ) q^{25} + 144 \zeta_{6} q^{26} -158 q^{29} + ( -36 + 36 \zeta_{6} ) q^{31} + ( -256 + 256 \zeta_{6} ) q^{32} -504 q^{34} + 162 \zeta_{6} q^{37} + ( 288 - 288 \zeta_{6} ) q^{38} -270 q^{41} -324 q^{43} -400 \zeta_{6} q^{44} + ( -56 + 56 \zeta_{6} ) q^{46} + 72 \zeta_{6} q^{47} -796 q^{50} + ( -288 + 288 \zeta_{6} ) q^{52} + ( -22 + 22 \zeta_{6} ) q^{53} + 900 q^{55} -632 \zeta_{6} q^{58} + ( -468 + 468 \zeta_{6} ) q^{59} + 792 \zeta_{6} q^{61} -144 q^{62} -512 q^{64} -648 \zeta_{6} q^{65} + ( -232 + 232 \zeta_{6} ) q^{67} -1008 \zeta_{6} q^{68} + 734 q^{71} + ( 180 - 180 \zeta_{6} ) q^{73} + ( -648 + 648 \zeta_{6} ) q^{74} + 576 q^{76} -236 \zeta_{6} q^{79} + ( 1152 - 1152 \zeta_{6} ) q^{80} -1080 \zeta_{6} q^{82} + 36 q^{83} + 2268 q^{85} -1296 \zeta_{6} q^{86} -234 \zeta_{6} q^{89} -112 q^{92} + ( -288 + 288 \zeta_{6} ) q^{94} + ( -1296 + 1296 \zeta_{6} ) q^{95} -468 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} - 8q^{4} - 18q^{5} + O(q^{10})$$ $$2q + 4q^{2} - 8q^{4} - 18q^{5} + 72q^{10} - 50q^{11} + 72q^{13} + 64q^{16} - 126q^{17} - 72q^{19} + 288q^{20} - 400q^{22} + 14q^{23} - 199q^{25} + 144q^{26} - 316q^{29} - 36q^{31} - 256q^{32} - 1008q^{34} + 162q^{37} + 288q^{38} - 540q^{41} - 648q^{43} - 400q^{44} - 56q^{46} + 72q^{47} - 1592q^{50} - 288q^{52} - 22q^{53} + 1800q^{55} - 632q^{58} - 468q^{59} + 792q^{61} - 288q^{62} - 1024q^{64} - 648q^{65} - 232q^{67} - 1008q^{68} + 1468q^{71} + 180q^{73} - 648q^{74} + 1152q^{76} - 236q^{79} + 1152q^{80} - 1080q^{82} + 72q^{83} + 4536q^{85} - 1296q^{86} - 234q^{89} - 224q^{92} - 288q^{94} - 1296q^{95} - 936q^{97} + O(q^{100})$$

## 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
2.00000 3.46410i 0 −4.00000 6.92820i −9.00000 + 15.5885i 0 0 0 0 36.0000 + 62.3538i
361.1 2.00000 + 3.46410i 0 −4.00000 + 6.92820i −9.00000 15.5885i 0 0 0 0 36.0000 62.3538i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.l 2
3.b odd 2 1 147.4.e.d 2
7.b odd 2 1 441.4.e.o 2
7.c even 3 1 441.4.a.c 1
7.c even 3 1 inner 441.4.e.l 2
7.d odd 6 1 441.4.a.a 1
7.d odd 6 1 441.4.e.o 2
21.c even 2 1 147.4.e.a 2
21.g even 6 1 147.4.a.h yes 1
21.g even 6 1 147.4.e.a 2
21.h odd 6 1 147.4.a.f 1
21.h odd 6 1 147.4.e.d 2
84.j odd 6 1 2352.4.a.s 1
84.n even 6 1 2352.4.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.f 1 21.h odd 6 1
147.4.a.h yes 1 21.g even 6 1
147.4.e.a 2 21.c even 2 1
147.4.e.a 2 21.g even 6 1
147.4.e.d 2 3.b odd 2 1
147.4.e.d 2 21.h odd 6 1
441.4.a.a 1 7.d odd 6 1
441.4.a.c 1 7.c even 3 1
441.4.e.l 2 1.a even 1 1 trivial
441.4.e.l 2 7.c even 3 1 inner
441.4.e.o 2 7.b odd 2 1
441.4.e.o 2 7.d odd 6 1
2352.4.a.s 1 84.j odd 6 1
2352.4.a.t 1 84.n even 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 4 T_{2} + 16$$ $$T_{5}^{2} + 18 T_{5} + 324$$ $$T_{13} - 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 4 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$324 + 18 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$2500 + 50 T + T^{2}$$
$13$ $$( -36 + T )^{2}$$
$17$ $$15876 + 126 T + T^{2}$$
$19$ $$5184 + 72 T + T^{2}$$
$23$ $$196 - 14 T + T^{2}$$
$29$ $$( 158 + T )^{2}$$
$31$ $$1296 + 36 T + T^{2}$$
$37$ $$26244 - 162 T + T^{2}$$
$41$ $$( 270 + T )^{2}$$
$43$ $$( 324 + T )^{2}$$
$47$ $$5184 - 72 T + T^{2}$$
$53$ $$484 + 22 T + T^{2}$$
$59$ $$219024 + 468 T + T^{2}$$
$61$ $$627264 - 792 T + T^{2}$$
$67$ $$53824 + 232 T + T^{2}$$
$71$ $$( -734 + T )^{2}$$
$73$ $$32400 - 180 T + T^{2}$$
$79$ $$55696 + 236 T + T^{2}$$
$83$ $$( -36 + T )^{2}$$
$89$ $$54756 + 234 T + T^{2}$$
$97$ $$( 468 + T )^{2}$$