# Properties

 Label 630.2.a.f 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

# Related objects

## 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} + O(q^{10})$$ $$q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} + 2q^{13} - q^{14} + q^{16} + 6q^{17} - 4q^{19} + q^{20} + q^{25} - 2q^{26} + q^{28} + 6q^{29} - 4q^{31} - q^{32} - 6q^{34} + q^{35} + 2q^{37} + 4q^{38} - q^{40} - 6q^{41} + 8q^{43} + 12q^{47} + q^{49} - q^{50} + 2q^{52} - 6q^{53} - q^{56} - 6q^{58} + 12q^{59} + 2q^{61} + 4q^{62} + q^{64} + 2q^{65} + 8q^{67} + 6q^{68} - q^{70} + 14q^{73} - 2q^{74} - 4q^{76} - 16q^{79} + q^{80} + 6q^{82} - 12q^{83} + 6q^{85} - 8q^{86} - 6q^{89} + 2q^{91} - 12q^{94} - 4q^{95} + 14q^{97} - q^{98} + O(q^{100})$$

## 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

## 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.f 1
3.b odd 2 1 210.2.a.d 1
4.b odd 2 1 5040.2.a.ba 1
5.b even 2 1 3150.2.a.ba 1
5.c odd 4 2 3150.2.g.o 2
7.b odd 2 1 4410.2.a.f 1
12.b even 2 1 1680.2.a.b 1
15.d odd 2 1 1050.2.a.a 1
15.e even 4 2 1050.2.g.h 2
21.c even 2 1 1470.2.a.m 1
21.g even 6 2 1470.2.i.h 2
21.h odd 6 2 1470.2.i.d 2
24.f even 2 1 6720.2.a.cc 1
24.h odd 2 1 6720.2.a.bb 1
60.h even 2 1 8400.2.a.cn 1
105.g even 2 1 7350.2.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.d 1 3.b odd 2 1
630.2.a.f 1 1.a even 1 1 trivial
1050.2.a.a 1 15.d odd 2 1
1050.2.g.h 2 15.e even 4 2
1470.2.a.m 1 21.c even 2 1
1470.2.i.d 2 21.h odd 6 2
1470.2.i.h 2 21.g even 6 2
1680.2.a.b 1 12.b even 2 1
3150.2.a.ba 1 5.b even 2 1
3150.2.g.o 2 5.c odd 4 2
4410.2.a.f 1 7.b odd 2 1
5040.2.a.ba 1 4.b odd 2 1
6720.2.a.bb 1 24.h odd 2 1
6720.2.a.cc 1 24.f even 2 1
7350.2.a.bd 1 105.g even 2 1
8400.2.a.cn 1 60.h even 2 1

## Atkin-Lehner signs

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

## Hecke kernels

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

 $$T_{11}$$ $$T_{13} - 2$$ $$T_{17} - 6$$ $$T_{19} + 4$$ $$T_{29} - 6$$

## Hecke Characteristic Polynomials

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