# Properties

 Label 441.2.e.a Level $441$ Weight $2$ Character orbit 441.e Analytic conductor $3.521$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ 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 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} -2 \zeta_{6} q^{5} -3 q^{8} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} -2 \zeta_{6} q^{5} -3 q^{8} + ( -2 + 2 \zeta_{6} ) q^{10} + ( 4 - 4 \zeta_{6} ) q^{11} -2 q^{13} + \zeta_{6} q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} -2 q^{20} -4 q^{22} + ( 1 - \zeta_{6} ) q^{25} + 2 \zeta_{6} q^{26} + 2 q^{29} + ( -5 + 5 \zeta_{6} ) q^{32} + 6 q^{34} -6 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{38} + 6 \zeta_{6} q^{40} -2 q^{41} -4 q^{43} -4 \zeta_{6} q^{44} - q^{50} + ( -2 + 2 \zeta_{6} ) q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} -8 q^{55} -2 \zeta_{6} q^{58} + ( 12 - 12 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + 7 q^{64} + 4 \zeta_{6} q^{65} + ( -4 + 4 \zeta_{6} ) q^{67} + 6 \zeta_{6} q^{68} + ( 6 - 6 \zeta_{6} ) q^{73} + ( -6 + 6 \zeta_{6} ) q^{74} -4 q^{76} + 16 \zeta_{6} q^{79} + ( 2 - 2 \zeta_{6} ) q^{80} + 2 \zeta_{6} q^{82} + 12 q^{83} + 12 q^{85} + 4 \zeta_{6} q^{86} + ( -12 + 12 \zeta_{6} ) q^{88} -14 \zeta_{6} q^{89} + ( -8 + 8 \zeta_{6} ) q^{95} + 18 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{4} - 2q^{5} - 6q^{8} + O(q^{10})$$ $$2q - q^{2} + q^{4} - 2q^{5} - 6q^{8} - 2q^{10} + 4q^{11} - 4q^{13} + q^{16} - 6q^{17} - 4q^{19} - 4q^{20} - 8q^{22} + q^{25} + 2q^{26} + 4q^{29} - 5q^{32} + 12q^{34} - 6q^{37} - 4q^{38} + 6q^{40} - 4q^{41} - 8q^{43} - 4q^{44} - 2q^{50} - 2q^{52} + 6q^{53} - 16q^{55} - 2q^{58} + 12q^{59} + 2q^{61} + 14q^{64} + 4q^{65} - 4q^{67} + 6q^{68} + 6q^{73} - 6q^{74} - 8q^{76} + 16q^{79} + 2q^{80} + 2q^{82} + 24q^{83} + 24q^{85} + 4q^{86} - 12q^{88} - 14q^{89} - 8q^{95} + 36q^{97} + O(q^{100})$$

## 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 0.500000 + 0.866025i −1.00000 + 1.73205i 0 0 −3.00000 0 −1.00000 1.73205i
361.1 −0.500000 0.866025i 0 0.500000 0.866025i −1.00000 1.73205i 0 0 −3.00000 0 −1.00000 + 1.73205i
 $$n$$: 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 441.2.e.a 2
3.b odd 2 1 147.2.e.b 2
7.b odd 2 1 441.2.e.b 2
7.c even 3 1 63.2.a.a 1
7.c even 3 1 inner 441.2.e.a 2
7.d odd 6 1 441.2.a.f 1
7.d odd 6 1 441.2.e.b 2
12.b even 2 1 2352.2.q.x 2
21.c even 2 1 147.2.e.c 2
21.g even 6 1 147.2.a.a 1
21.g even 6 1 147.2.e.c 2
21.h odd 6 1 21.2.a.a 1
21.h odd 6 1 147.2.e.b 2
28.f even 6 1 7056.2.a.p 1
28.g odd 6 1 1008.2.a.l 1
35.j even 6 1 1575.2.a.c 1
35.l odd 12 2 1575.2.d.a 2
56.k odd 6 1 4032.2.a.k 1
56.p even 6 1 4032.2.a.h 1
63.g even 3 1 567.2.f.b 2
63.h even 3 1 567.2.f.b 2
63.j odd 6 1 567.2.f.g 2
63.n odd 6 1 567.2.f.g 2
77.h odd 6 1 7623.2.a.g 1
84.h odd 2 1 2352.2.q.e 2
84.j odd 6 1 2352.2.a.v 1
84.j odd 6 1 2352.2.q.e 2
84.n even 6 1 336.2.a.a 1
84.n even 6 1 2352.2.q.x 2
105.o odd 6 1 525.2.a.d 1
105.p even 6 1 3675.2.a.n 1
105.x even 12 2 525.2.d.a 2
168.s odd 6 1 1344.2.a.g 1
168.v even 6 1 1344.2.a.s 1
168.ba even 6 1 9408.2.a.bv 1
168.be odd 6 1 9408.2.a.m 1
231.l even 6 1 2541.2.a.j 1
273.w odd 6 1 3549.2.a.c 1
336.bt odd 12 2 5376.2.c.r 2
336.bu even 12 2 5376.2.c.l 2
357.q odd 6 1 6069.2.a.b 1
399.w even 6 1 7581.2.a.d 1
420.ba even 6 1 8400.2.a.bn 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 21.h odd 6 1
63.2.a.a 1 7.c even 3 1
147.2.a.a 1 21.g even 6 1
147.2.e.b 2 3.b odd 2 1
147.2.e.b 2 21.h odd 6 1
147.2.e.c 2 21.c even 2 1
147.2.e.c 2 21.g even 6 1
336.2.a.a 1 84.n even 6 1
441.2.a.f 1 7.d odd 6 1
441.2.e.a 2 1.a even 1 1 trivial
441.2.e.a 2 7.c even 3 1 inner
441.2.e.b 2 7.b odd 2 1
441.2.e.b 2 7.d odd 6 1
525.2.a.d 1 105.o odd 6 1
525.2.d.a 2 105.x even 12 2
567.2.f.b 2 63.g even 3 1
567.2.f.b 2 63.h even 3 1
567.2.f.g 2 63.j odd 6 1
567.2.f.g 2 63.n odd 6 1
1008.2.a.l 1 28.g odd 6 1
1344.2.a.g 1 168.s odd 6 1
1344.2.a.s 1 168.v even 6 1
1575.2.a.c 1 35.j even 6 1
1575.2.d.a 2 35.l odd 12 2
2352.2.a.v 1 84.j odd 6 1
2352.2.q.e 2 84.h odd 2 1
2352.2.q.e 2 84.j odd 6 1
2352.2.q.x 2 12.b even 2 1
2352.2.q.x 2 84.n even 6 1
2541.2.a.j 1 231.l even 6 1
3549.2.a.c 1 273.w odd 6 1
3675.2.a.n 1 105.p even 6 1
4032.2.a.h 1 56.p even 6 1
4032.2.a.k 1 56.k odd 6 1
5376.2.c.l 2 336.bu even 12 2
5376.2.c.r 2 336.bt odd 12 2
6069.2.a.b 1 357.q odd 6 1
7056.2.a.p 1 28.f even 6 1
7581.2.a.d 1 399.w even 6 1
7623.2.a.g 1 77.h odd 6 1
8400.2.a.bn 1 420.ba even 6 1
9408.2.a.m 1 168.be odd 6 1
9408.2.a.bv 1 168.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{2} + T_{2} + 1$$ $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 - 4 T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$16 + 4 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$36 + 6 T + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$144 - 12 T + T^{2}$$
$61$ $$4 - 2 T + T^{2}$$
$67$ $$16 + 4 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$36 - 6 T + T^{2}$$
$79$ $$256 - 16 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$196 + 14 T + T^{2}$$
$97$ $$( -18 + T )^{2}$$
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