# Properties

 Label 441.4.e.f 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( 7 - 7 \zeta_{6} ) q^{4} -12 \zeta_{6} q^{5} -15 q^{8} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( 7 - 7 \zeta_{6} ) q^{4} -12 \zeta_{6} q^{5} -15 q^{8} + ( -12 + 12 \zeta_{6} ) q^{10} + ( 20 - 20 \zeta_{6} ) q^{11} + 84 q^{13} -41 \zeta_{6} q^{16} + ( 96 - 96 \zeta_{6} ) q^{17} + 12 \zeta_{6} q^{19} -84 q^{20} -20 q^{22} -176 \zeta_{6} q^{23} + ( -19 + 19 \zeta_{6} ) q^{25} -84 \zeta_{6} q^{26} -58 q^{29} + ( -264 + 264 \zeta_{6} ) q^{31} + ( -161 + 161 \zeta_{6} ) q^{32} -96 q^{34} -258 \zeta_{6} q^{37} + ( 12 - 12 \zeta_{6} ) q^{38} + 180 \zeta_{6} q^{40} + 156 q^{43} -140 \zeta_{6} q^{44} + ( -176 + 176 \zeta_{6} ) q^{46} + 408 \zeta_{6} q^{47} + 19 q^{50} + ( 588 - 588 \zeta_{6} ) q^{52} + ( -722 + 722 \zeta_{6} ) q^{53} -240 q^{55} + 58 \zeta_{6} q^{58} + ( -492 + 492 \zeta_{6} ) q^{59} -492 \zeta_{6} q^{61} + 264 q^{62} -167 q^{64} -1008 \zeta_{6} q^{65} + ( -412 + 412 \zeta_{6} ) q^{67} -672 \zeta_{6} q^{68} -296 q^{71} + ( 240 - 240 \zeta_{6} ) q^{73} + ( -258 + 258 \zeta_{6} ) q^{74} + 84 q^{76} -776 \zeta_{6} q^{79} + ( -492 + 492 \zeta_{6} ) q^{80} + 924 q^{83} -1152 q^{85} -156 \zeta_{6} q^{86} + ( -300 + 300 \zeta_{6} ) q^{88} + 744 \zeta_{6} q^{89} -1232 q^{92} + ( 408 - 408 \zeta_{6} ) q^{94} + ( 144 - 144 \zeta_{6} ) q^{95} + 168 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 7q^{4} - 12q^{5} - 30q^{8} + O(q^{10})$$ $$2q - q^{2} + 7q^{4} - 12q^{5} - 30q^{8} - 12q^{10} + 20q^{11} + 168q^{13} - 41q^{16} + 96q^{17} + 12q^{19} - 168q^{20} - 40q^{22} - 176q^{23} - 19q^{25} - 84q^{26} - 116q^{29} - 264q^{31} - 161q^{32} - 192q^{34} - 258q^{37} + 12q^{38} + 180q^{40} + 312q^{43} - 140q^{44} - 176q^{46} + 408q^{47} + 38q^{50} + 588q^{52} - 722q^{53} - 480q^{55} + 58q^{58} - 492q^{59} - 492q^{61} + 528q^{62} - 334q^{64} - 1008q^{65} - 412q^{67} - 672q^{68} - 592q^{71} + 240q^{73} - 258q^{74} + 168q^{76} - 776q^{79} - 492q^{80} + 1848q^{83} - 2304q^{85} - 156q^{86} - 300q^{88} + 744q^{89} - 2464q^{92} + 408q^{94} + 144q^{95} + 336q^{97} + O(q^{100})$$

## 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
−0.500000 + 0.866025i 0 3.50000 + 6.06218i −6.00000 + 10.3923i 0 0 −15.0000 0 −6.00000 10.3923i
361.1 −0.500000 0.866025i 0 3.50000 6.06218i −6.00000 10.3923i 0 0 −15.0000 0 −6.00000 + 10.3923i
 $$n$$: 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.f 2
3.b odd 2 1 147.4.e.e 2
7.b odd 2 1 441.4.e.g 2
7.c even 3 1 441.4.a.h 1
7.c even 3 1 inner 441.4.e.f 2
7.d odd 6 1 441.4.a.g 1
7.d odd 6 1 441.4.e.g 2
21.c even 2 1 147.4.e.f 2
21.g even 6 1 147.4.a.d 1
21.g even 6 1 147.4.e.f 2
21.h odd 6 1 147.4.a.e yes 1
21.h odd 6 1 147.4.e.e 2
84.j odd 6 1 2352.4.a.bi 1
84.n even 6 1 2352.4.a.b 1

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

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} + 1$$ $$T_{5}^{2} + 12 T_{5} + 144$$ $$T_{13} - 84$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$144 + 12 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$400 - 20 T + T^{2}$$
$13$ $$( -84 + T )^{2}$$
$17$ $$9216 - 96 T + T^{2}$$
$19$ $$144 - 12 T + T^{2}$$
$23$ $$30976 + 176 T + T^{2}$$
$29$ $$( 58 + T )^{2}$$
$31$ $$69696 + 264 T + T^{2}$$
$37$ $$66564 + 258 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -156 + T )^{2}$$
$47$ $$166464 - 408 T + T^{2}$$
$53$ $$521284 + 722 T + T^{2}$$
$59$ $$242064 + 492 T + T^{2}$$
$61$ $$242064 + 492 T + T^{2}$$
$67$ $$169744 + 412 T + T^{2}$$
$71$ $$( 296 + T )^{2}$$
$73$ $$57600 - 240 T + T^{2}$$
$79$ $$602176 + 776 T + T^{2}$$
$83$ $$( -924 + T )^{2}$$
$89$ $$553536 - 744 T + T^{2}$$
$97$ $$( -168 + T )^{2}$$