# Properties

 Label 441.2.a.i Level $441$ Weight $2$ Character orbit 441.a Self dual yes Analytic conductor $3.521$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} + ( -2 + \beta ) q^{5} + ( 3 + \beta ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} + ( -2 + \beta ) q^{5} + ( 3 + \beta ) q^{8} -\beta q^{10} + 2 q^{11} + ( 4 + \beta ) q^{13} + 3 q^{16} + ( -2 - 3 \beta ) q^{17} + 2 \beta q^{19} + ( 2 - 3 \beta ) q^{20} + ( 2 + 2 \beta ) q^{22} + ( 2 - 4 \beta ) q^{23} + ( 1 - 4 \beta ) q^{25} + ( 6 + 5 \beta ) q^{26} + ( 4 - 2 \beta ) q^{29} + ( -4 - 2 \beta ) q^{31} + ( -3 + \beta ) q^{32} + ( -8 - 5 \beta ) q^{34} -4 q^{37} + ( 4 + 2 \beta ) q^{38} + ( -4 + \beta ) q^{40} + ( -2 + 3 \beta ) q^{41} -4 \beta q^{43} + ( 2 + 4 \beta ) q^{44} + ( -6 - 2 \beta ) q^{46} -2 \beta q^{47} + ( -7 - 3 \beta ) q^{50} + ( 8 + 9 \beta ) q^{52} + 2 q^{53} + ( -4 + 2 \beta ) q^{55} + 2 \beta q^{58} + ( 4 + 2 \beta ) q^{59} + ( 8 - 3 \beta ) q^{61} + ( -8 - 6 \beta ) q^{62} + ( -7 - 2 \beta ) q^{64} + ( -6 + 2 \beta ) q^{65} + 4 \beta q^{67} + ( -14 - 7 \beta ) q^{68} + ( 2 + 8 \beta ) q^{71} + ( 4 - 7 \beta ) q^{73} + ( -4 - 4 \beta ) q^{74} + ( 8 + 2 \beta ) q^{76} + ( 8 - 4 \beta ) q^{79} + ( -6 + 3 \beta ) q^{80} + ( 4 + \beta ) q^{82} + ( 4 + 8 \beta ) q^{83} + ( -2 + 4 \beta ) q^{85} + ( -8 - 4 \beta ) q^{86} + ( 6 + 2 \beta ) q^{88} + ( 10 - 3 \beta ) q^{89} -14 q^{92} + ( -4 - 2 \beta ) q^{94} + ( 4 - 4 \beta ) q^{95} + ( 4 + \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{4} - 4q^{5} + 6q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{4} - 4q^{5} + 6q^{8} + 4q^{11} + 8q^{13} + 6q^{16} - 4q^{17} + 4q^{20} + 4q^{22} + 4q^{23} + 2q^{25} + 12q^{26} + 8q^{29} - 8q^{31} - 6q^{32} - 16q^{34} - 8q^{37} + 8q^{38} - 8q^{40} - 4q^{41} + 4q^{44} - 12q^{46} - 14q^{50} + 16q^{52} + 4q^{53} - 8q^{55} + 8q^{59} + 16q^{61} - 16q^{62} - 14q^{64} - 12q^{65} - 28q^{68} + 4q^{71} + 8q^{73} - 8q^{74} + 16q^{76} + 16q^{79} - 12q^{80} + 8q^{82} + 8q^{83} - 4q^{85} - 16q^{86} + 12q^{88} + 20q^{89} - 28q^{92} - 8q^{94} + 8q^{95} + 8q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
−0.414214 0 −1.82843 −3.41421 0 0 1.58579 0 1.41421
1.2 2.41421 0 3.82843 −0.585786 0 0 4.41421 0 −1.41421
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.a.i 2
3.b odd 2 1 147.2.a.e yes 2
4.b odd 2 1 7056.2.a.cf 2
7.b odd 2 1 441.2.a.j 2
7.c even 3 2 441.2.e.g 4
7.d odd 6 2 441.2.e.f 4
12.b even 2 1 2352.2.a.bc 2
15.d odd 2 1 3675.2.a.bd 2
21.c even 2 1 147.2.a.d 2
21.g even 6 2 147.2.e.e 4
21.h odd 6 2 147.2.e.d 4
24.f even 2 1 9408.2.a.dt 2
24.h odd 2 1 9408.2.a.di 2
28.d even 2 1 7056.2.a.cv 2
84.h odd 2 1 2352.2.a.be 2
84.j odd 6 2 2352.2.q.bb 4
84.n even 6 2 2352.2.q.bd 4
105.g even 2 1 3675.2.a.bf 2
168.e odd 2 1 9408.2.a.dq 2
168.i even 2 1 9408.2.a.ef 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d 2 21.c even 2 1
147.2.a.e yes 2 3.b odd 2 1
147.2.e.d 4 21.h odd 6 2
147.2.e.e 4 21.g even 6 2
441.2.a.i 2 1.a even 1 1 trivial
441.2.a.j 2 7.b odd 2 1
441.2.e.f 4 7.d odd 6 2
441.2.e.g 4 7.c even 3 2
2352.2.a.bc 2 12.b even 2 1
2352.2.a.be 2 84.h odd 2 1
2352.2.q.bb 4 84.j odd 6 2
2352.2.q.bd 4 84.n even 6 2
3675.2.a.bd 2 15.d odd 2 1
3675.2.a.bf 2 105.g even 2 1
7056.2.a.cf 2 4.b odd 2 1
7056.2.a.cv 2 28.d even 2 1
9408.2.a.di 2 24.h odd 2 1
9408.2.a.dq 2 168.e odd 2 1
9408.2.a.dt 2 24.f even 2 1
9408.2.a.ef 2 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{2} - 2 T_{2} - 1$$ $$T_{5}^{2} + 4 T_{5} + 2$$ $$T_{13}^{2} - 8 T_{13} + 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$2 + 4 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$14 - 8 T + T^{2}$$
$17$ $$-14 + 4 T + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$-28 - 4 T + T^{2}$$
$29$ $$8 - 8 T + T^{2}$$
$31$ $$8 + 8 T + T^{2}$$
$37$ $$( 4 + T )^{2}$$
$41$ $$-14 + 4 T + T^{2}$$
$43$ $$-32 + T^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$8 - 8 T + T^{2}$$
$61$ $$46 - 16 T + T^{2}$$
$67$ $$-32 + T^{2}$$
$71$ $$-124 - 4 T + T^{2}$$
$73$ $$-82 - 8 T + T^{2}$$
$79$ $$32 - 16 T + T^{2}$$
$83$ $$-112 - 8 T + T^{2}$$
$89$ $$82 - 20 T + T^{2}$$
$97$ $$14 - 8 T + T^{2}$$
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