# Properties

 Label 630.2.a.a Level $630$ Weight $2$ Character orbit 630.a Self dual yes Analytic conductor $5.031$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 630.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.03057532734$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + q^{10} + 4q^{11} - 2q^{13} + q^{14} + q^{16} - 2q^{17} + 4q^{19} - q^{20} - 4q^{22} + 8q^{23} + q^{25} + 2q^{26} - q^{28} + 2q^{29} - q^{32} + 2q^{34} + q^{35} + 6q^{37} - 4q^{38} + q^{40} + 6q^{41} - 4q^{43} + 4q^{44} - 8q^{46} + q^{49} - q^{50} - 2q^{52} + 10q^{53} - 4q^{55} + q^{56} - 2q^{58} - 12q^{59} + 14q^{61} + q^{64} + 2q^{65} - 12q^{67} - 2q^{68} - q^{70} + 8q^{71} + 10q^{73} - 6q^{74} + 4q^{76} - 4q^{77} + 16q^{79} - q^{80} - 6q^{82} + 12q^{83} + 2q^{85} + 4q^{86} - 4q^{88} - 10q^{89} + 2q^{91} + 8q^{92} - 4q^{95} + 2q^{97} - q^{98} + O(q^{100})$$

## 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
 0
−1.00000 0 1.00000 −1.00000 0 −1.00000 −1.00000 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$

## 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 630.2.a.a 1
3.b odd 2 1 210.2.a.e 1
4.b odd 2 1 5040.2.a.k 1
5.b even 2 1 3150.2.a.bp 1
5.c odd 4 2 3150.2.g.q 2
7.b odd 2 1 4410.2.a.t 1
12.b even 2 1 1680.2.a.j 1
15.d odd 2 1 1050.2.a.c 1
15.e even 4 2 1050.2.g.g 2
21.c even 2 1 1470.2.a.j 1
21.g even 6 2 1470.2.i.j 2
21.h odd 6 2 1470.2.i.a 2
24.f even 2 1 6720.2.a.bq 1
24.h odd 2 1 6720.2.a.j 1
60.h even 2 1 8400.2.a.ce 1
105.g even 2 1 7350.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.e 1 3.b odd 2 1
630.2.a.a 1 1.a even 1 1 trivial
1050.2.a.c 1 15.d odd 2 1
1050.2.g.g 2 15.e even 4 2
1470.2.a.j 1 21.c even 2 1
1470.2.i.a 2 21.h odd 6 2
1470.2.i.j 2 21.g even 6 2
1680.2.a.j 1 12.b even 2 1
3150.2.a.bp 1 5.b even 2 1
3150.2.g.q 2 5.c odd 4 2
4410.2.a.t 1 7.b odd 2 1
5040.2.a.k 1 4.b odd 2 1
6720.2.a.j 1 24.h odd 2 1
6720.2.a.bq 1 24.f even 2 1
7350.2.a.w 1 105.g even 2 1
8400.2.a.ce 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(630))$$:

 $$T_{11} - 4$$ $$T_{13} + 2$$ $$T_{17} + 2$$ $$T_{19} - 4$$ $$T_{29} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$1 + T$$
$11$ $$-4 + T$$
$13$ $$2 + T$$
$17$ $$2 + T$$
$19$ $$-4 + T$$
$23$ $$-8 + T$$
$29$ $$-2 + T$$
$31$ $$T$$
$37$ $$-6 + T$$
$41$ $$-6 + T$$
$43$ $$4 + T$$
$47$ $$T$$
$53$ $$-10 + T$$
$59$ $$12 + T$$
$61$ $$-14 + T$$
$67$ $$12 + T$$
$71$ $$-8 + T$$
$73$ $$-10 + T$$
$79$ $$-16 + T$$
$83$ $$-12 + T$$
$89$ $$10 + T$$
$97$ $$-2 + T$$
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