# Properties

 Label 441.4.e.h 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 7) 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( 7 - 7 \zeta_{6} ) q^{4} + 16 \zeta_{6} q^{5} -15 q^{8} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( 7 - 7 \zeta_{6} ) q^{4} + 16 \zeta_{6} q^{5} -15 q^{8} + ( 16 - 16 \zeta_{6} ) q^{10} + ( -8 + 8 \zeta_{6} ) q^{11} + 28 q^{13} -41 \zeta_{6} q^{16} + ( 54 - 54 \zeta_{6} ) q^{17} + 110 \zeta_{6} q^{19} + 112 q^{20} + 8 q^{22} + 48 \zeta_{6} q^{23} + ( -131 + 131 \zeta_{6} ) q^{25} -28 \zeta_{6} q^{26} + 110 q^{29} + ( -12 + 12 \zeta_{6} ) q^{31} + ( -161 + 161 \zeta_{6} ) q^{32} -54 q^{34} + 246 \zeta_{6} q^{37} + ( 110 - 110 \zeta_{6} ) q^{38} -240 \zeta_{6} q^{40} -182 q^{41} + 128 q^{43} + 56 \zeta_{6} q^{44} + ( 48 - 48 \zeta_{6} ) q^{46} + 324 \zeta_{6} q^{47} + 131 q^{50} + ( 196 - 196 \zeta_{6} ) q^{52} + ( -162 + 162 \zeta_{6} ) q^{53} -128 q^{55} -110 \zeta_{6} q^{58} + ( 810 - 810 \zeta_{6} ) q^{59} + 488 \zeta_{6} q^{61} + 12 q^{62} -167 q^{64} + 448 \zeta_{6} q^{65} + ( -244 + 244 \zeta_{6} ) q^{67} -378 \zeta_{6} q^{68} + 768 q^{71} + ( 702 - 702 \zeta_{6} ) q^{73} + ( 246 - 246 \zeta_{6} ) q^{74} + 770 q^{76} -440 \zeta_{6} q^{79} + ( 656 - 656 \zeta_{6} ) q^{80} + 182 \zeta_{6} q^{82} + 1302 q^{83} + 864 q^{85} -128 \zeta_{6} q^{86} + ( 120 - 120 \zeta_{6} ) q^{88} + 730 \zeta_{6} q^{89} + 336 q^{92} + ( 324 - 324 \zeta_{6} ) q^{94} + ( -1760 + 1760 \zeta_{6} ) q^{95} + 294 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 7q^{4} + 16q^{5} - 30q^{8} + O(q^{10})$$ $$2q - q^{2} + 7q^{4} + 16q^{5} - 30q^{8} + 16q^{10} - 8q^{11} + 56q^{13} - 41q^{16} + 54q^{17} + 110q^{19} + 224q^{20} + 16q^{22} + 48q^{23} - 131q^{25} - 28q^{26} + 220q^{29} - 12q^{31} - 161q^{32} - 108q^{34} + 246q^{37} + 110q^{38} - 240q^{40} - 364q^{41} + 256q^{43} + 56q^{44} + 48q^{46} + 324q^{47} + 262q^{50} + 196q^{52} - 162q^{53} - 256q^{55} - 110q^{58} + 810q^{59} + 488q^{61} + 24q^{62} - 334q^{64} + 448q^{65} - 244q^{67} - 378q^{68} + 1536q^{71} + 702q^{73} + 246q^{74} + 1540q^{76} - 440q^{79} + 656q^{80} + 182q^{82} + 2604q^{83} + 1728q^{85} - 128q^{86} + 120q^{88} + 730q^{89} + 672q^{92} + 324q^{94} - 1760q^{95} + 588q^{97} + O(q^{100})$$

## 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 8.00000 13.8564i 0 0 −15.0000 0 8.00000 + 13.8564i
361.1 −0.500000 0.866025i 0 3.50000 6.06218i 8.00000 + 13.8564i 0 0 −15.0000 0 8.00000 13.8564i
 $$n$$: 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.h 2
3.b odd 2 1 49.4.c.c 2
7.b odd 2 1 441.4.e.e 2
7.c even 3 1 63.4.a.b 1
7.c even 3 1 inner 441.4.e.h 2
7.d odd 6 1 441.4.a.i 1
7.d odd 6 1 441.4.e.e 2
21.c even 2 1 49.4.c.b 2
21.g even 6 1 49.4.a.b 1
21.g even 6 1 49.4.c.b 2
21.h odd 6 1 7.4.a.a 1
21.h odd 6 1 49.4.c.c 2
28.g odd 6 1 1008.4.a.c 1
35.j even 6 1 1575.4.a.e 1
84.j odd 6 1 784.4.a.g 1
84.n even 6 1 112.4.a.f 1
105.o odd 6 1 175.4.a.b 1
105.p even 6 1 1225.4.a.j 1
105.x even 12 2 175.4.b.b 2
168.s odd 6 1 448.4.a.i 1
168.v even 6 1 448.4.a.e 1
231.l even 6 1 847.4.a.b 1
273.w odd 6 1 1183.4.a.b 1
357.q odd 6 1 2023.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 21.h odd 6 1
49.4.a.b 1 21.g even 6 1
49.4.c.b 2 21.c even 2 1
49.4.c.b 2 21.g even 6 1
49.4.c.c 2 3.b odd 2 1
49.4.c.c 2 21.h odd 6 1
63.4.a.b 1 7.c even 3 1
112.4.a.f 1 84.n even 6 1
175.4.a.b 1 105.o odd 6 1
175.4.b.b 2 105.x even 12 2
441.4.a.i 1 7.d odd 6 1
441.4.e.e 2 7.b odd 2 1
441.4.e.e 2 7.d odd 6 1
441.4.e.h 2 1.a even 1 1 trivial
441.4.e.h 2 7.c even 3 1 inner
448.4.a.e 1 168.v even 6 1
448.4.a.i 1 168.s odd 6 1
784.4.a.g 1 84.j odd 6 1
847.4.a.b 1 231.l even 6 1
1008.4.a.c 1 28.g odd 6 1
1183.4.a.b 1 273.w odd 6 1
1225.4.a.j 1 105.p even 6 1
1575.4.a.e 1 35.j even 6 1
2023.4.a.a 1 357.q odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} + 1$$ $$T_{5}^{2} - 16 T_{5} + 256$$ $$T_{13} - 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$256 - 16 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$64 + 8 T + T^{2}$$
$13$ $$( -28 + T )^{2}$$
$17$ $$2916 - 54 T + T^{2}$$
$19$ $$12100 - 110 T + T^{2}$$
$23$ $$2304 - 48 T + T^{2}$$
$29$ $$( -110 + T )^{2}$$
$31$ $$144 + 12 T + T^{2}$$
$37$ $$60516 - 246 T + T^{2}$$
$41$ $$( 182 + T )^{2}$$
$43$ $$( -128 + T )^{2}$$
$47$ $$104976 - 324 T + T^{2}$$
$53$ $$26244 + 162 T + T^{2}$$
$59$ $$656100 - 810 T + T^{2}$$
$61$ $$238144 - 488 T + T^{2}$$
$67$ $$59536 + 244 T + T^{2}$$
$71$ $$( -768 + T )^{2}$$
$73$ $$492804 - 702 T + T^{2}$$
$79$ $$193600 + 440 T + T^{2}$$
$83$ $$( -1302 + T )^{2}$$
$89$ $$532900 - 730 T + T^{2}$$
$97$ $$( -294 + T )^{2}$$