# Properties

 Label 441.4.e.c 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q -3 \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} -21 q^{8} +O(q^{10})$$ $$q -3 \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} -21 q^{8} + ( 9 - 9 \zeta_{6} ) q^{10} + ( -15 + 15 \zeta_{6} ) q^{11} + 64 q^{13} + 71 \zeta_{6} q^{16} + ( -84 + 84 \zeta_{6} ) q^{17} -16 \zeta_{6} q^{19} -3 q^{20} + 45 q^{22} -84 \zeta_{6} q^{23} + ( 116 - 116 \zeta_{6} ) q^{25} -192 \zeta_{6} q^{26} + 297 q^{29} + ( -253 + 253 \zeta_{6} ) q^{31} + ( 45 - 45 \zeta_{6} ) q^{32} + 252 q^{34} + 316 \zeta_{6} q^{37} + ( -48 + 48 \zeta_{6} ) q^{38} -63 \zeta_{6} q^{40} + 360 q^{41} + 26 q^{43} -15 \zeta_{6} q^{44} + ( -252 + 252 \zeta_{6} ) q^{46} + 30 \zeta_{6} q^{47} -348 q^{50} + ( -64 + 64 \zeta_{6} ) q^{52} + ( 363 - 363 \zeta_{6} ) q^{53} -45 q^{55} -891 \zeta_{6} q^{58} + ( 15 - 15 \zeta_{6} ) q^{59} -118 \zeta_{6} q^{61} + 759 q^{62} + 433 q^{64} + 192 \zeta_{6} q^{65} + ( 370 - 370 \zeta_{6} ) q^{67} -84 \zeta_{6} q^{68} + 342 q^{71} + ( 362 - 362 \zeta_{6} ) q^{73} + ( 948 - 948 \zeta_{6} ) q^{74} + 16 q^{76} -467 \zeta_{6} q^{79} + ( -213 + 213 \zeta_{6} ) q^{80} -1080 \zeta_{6} q^{82} + 477 q^{83} -252 q^{85} -78 \zeta_{6} q^{86} + ( 315 - 315 \zeta_{6} ) q^{88} -906 \zeta_{6} q^{89} + 84 q^{92} + ( 90 - 90 \zeta_{6} ) q^{94} + ( 48 - 48 \zeta_{6} ) q^{95} -503 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} - q^{4} + 3q^{5} - 42q^{8} + O(q^{10})$$ $$2q - 3q^{2} - q^{4} + 3q^{5} - 42q^{8} + 9q^{10} - 15q^{11} + 128q^{13} + 71q^{16} - 84q^{17} - 16q^{19} - 6q^{20} + 90q^{22} - 84q^{23} + 116q^{25} - 192q^{26} + 594q^{29} - 253q^{31} + 45q^{32} + 504q^{34} + 316q^{37} - 48q^{38} - 63q^{40} + 720q^{41} + 52q^{43} - 15q^{44} - 252q^{46} + 30q^{47} - 696q^{50} - 64q^{52} + 363q^{53} - 90q^{55} - 891q^{58} + 15q^{59} - 118q^{61} + 1518q^{62} + 866q^{64} + 192q^{65} + 370q^{67} - 84q^{68} + 684q^{71} + 362q^{73} + 948q^{74} + 32q^{76} - 467q^{79} - 213q^{80} - 1080q^{82} + 954q^{83} - 504q^{85} - 78q^{86} + 315q^{88} - 906q^{89} + 168q^{92} + 90q^{94} + 48q^{95} - 1006q^{97} + O(q^{100})$$

## 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
−1.50000 + 2.59808i 0 −0.500000 0.866025i 1.50000 2.59808i 0 0 −21.0000 0 4.50000 + 7.79423i
361.1 −1.50000 2.59808i 0 −0.500000 + 0.866025i 1.50000 + 2.59808i 0 0 −21.0000 0 4.50000 7.79423i
 $$n$$: 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.c 2
3.b odd 2 1 147.4.e.h 2
7.b odd 2 1 63.4.e.a 2
7.c even 3 1 441.4.a.k 1
7.c even 3 1 inner 441.4.e.c 2
7.d odd 6 1 63.4.e.a 2
7.d odd 6 1 441.4.a.l 1
21.c even 2 1 21.4.e.a 2
21.g even 6 1 21.4.e.a 2
21.g even 6 1 147.4.a.b 1
21.h odd 6 1 147.4.a.a 1
21.h odd 6 1 147.4.e.h 2
84.h odd 2 1 336.4.q.e 2
84.j odd 6 1 336.4.q.e 2
84.j odd 6 1 2352.4.a.i 1
84.n even 6 1 2352.4.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 21.c even 2 1
21.4.e.a 2 21.g even 6 1
63.4.e.a 2 7.b odd 2 1
63.4.e.a 2 7.d odd 6 1
147.4.a.a 1 21.h odd 6 1
147.4.a.b 1 21.g even 6 1
147.4.e.h 2 3.b odd 2 1
147.4.e.h 2 21.h odd 6 1
336.4.q.e 2 84.h odd 2 1
336.4.q.e 2 84.j odd 6 1
441.4.a.k 1 7.c even 3 1
441.4.a.l 1 7.d odd 6 1
441.4.e.c 2 1.a even 1 1 trivial
441.4.e.c 2 7.c even 3 1 inner
2352.4.a.i 1 84.j odd 6 1
2352.4.a.bd 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 $$S_{4}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{2} + 3 T_{2} + 9$$ $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{13} - 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 3 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$225 + 15 T + T^{2}$$
$13$ $$( -64 + T )^{2}$$
$17$ $$7056 + 84 T + T^{2}$$
$19$ $$256 + 16 T + T^{2}$$
$23$ $$7056 + 84 T + T^{2}$$
$29$ $$( -297 + T )^{2}$$
$31$ $$64009 + 253 T + T^{2}$$
$37$ $$99856 - 316 T + T^{2}$$
$41$ $$( -360 + T )^{2}$$
$43$ $$( -26 + T )^{2}$$
$47$ $$900 - 30 T + T^{2}$$
$53$ $$131769 - 363 T + T^{2}$$
$59$ $$225 - 15 T + T^{2}$$
$61$ $$13924 + 118 T + T^{2}$$
$67$ $$136900 - 370 T + T^{2}$$
$71$ $$( -342 + T )^{2}$$
$73$ $$131044 - 362 T + T^{2}$$
$79$ $$218089 + 467 T + T^{2}$$
$83$ $$( -477 + T )^{2}$$
$89$ $$820836 + 906 T + T^{2}$$
$97$ $$( 503 + T )^{2}$$