# Properties

 Label 210.2.a.b Level 210 Weight 2 Character orbit 210.a Self dual yes Analytic conductor 1.677 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 210.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + q^{7} - q^{8} + q^{9} - q^{10} + q^{12} + 2q^{13} - q^{14} + q^{15} + q^{16} - 6q^{17} - q^{18} + 8q^{19} + q^{20} + q^{21} - q^{24} + q^{25} - 2q^{26} + q^{27} + q^{28} + 6q^{29} - q^{30} - 4q^{31} - q^{32} + 6q^{34} + q^{35} + q^{36} - 10q^{37} - 8q^{38} + 2q^{39} - q^{40} - 6q^{41} - q^{42} - 4q^{43} + q^{45} + q^{48} + q^{49} - q^{50} - 6q^{51} + 2q^{52} - 6q^{53} - q^{54} - q^{56} + 8q^{57} - 6q^{58} - 12q^{59} + q^{60} - 10q^{61} + 4q^{62} + q^{63} + q^{64} + 2q^{65} - 4q^{67} - 6q^{68} - q^{70} + 12q^{71} - q^{72} - 10q^{73} + 10q^{74} + q^{75} + 8q^{76} - 2q^{78} + 8q^{79} + q^{80} + q^{81} + 6q^{82} + 12q^{83} + q^{84} - 6q^{85} + 4q^{86} + 6q^{87} - 6q^{89} - q^{90} + 2q^{91} - 4q^{93} + 8q^{95} - q^{96} - 10q^{97} - q^{98} + O(q^{100})$$

## 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 1.00000 1.00000 1.00000 −1.00000 1.00000 −1.00000 1.00000 −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 210.2.a.b 1
3.b odd 2 1 630.2.a.h 1
4.b odd 2 1 1680.2.a.g 1
5.b even 2 1 1050.2.a.k 1
5.c odd 4 2 1050.2.g.c 2
7.b odd 2 1 1470.2.a.b 1
7.c even 3 2 1470.2.i.l 2
7.d odd 6 2 1470.2.i.s 2
8.b even 2 1 6720.2.a.n 1
8.d odd 2 1 6720.2.a.bi 1
12.b even 2 1 5040.2.a.g 1
15.d odd 2 1 3150.2.a.f 1
15.e even 4 2 3150.2.g.i 2
20.d odd 2 1 8400.2.a.cm 1
21.c even 2 1 4410.2.a.bi 1
35.c odd 2 1 7350.2.a.cs 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.b 1 1.a even 1 1 trivial
630.2.a.h 1 3.b odd 2 1
1050.2.a.k 1 5.b even 2 1
1050.2.g.c 2 5.c odd 4 2
1470.2.a.b 1 7.b odd 2 1
1470.2.i.l 2 7.c even 3 2
1470.2.i.s 2 7.d odd 6 2
1680.2.a.g 1 4.b odd 2 1
3150.2.a.f 1 15.d odd 2 1
3150.2.g.i 2 15.e even 4 2
4410.2.a.bi 1 21.c even 2 1
5040.2.a.g 1 12.b even 2 1
6720.2.a.n 1 8.b even 2 1
6720.2.a.bi 1 8.d odd 2 1
7350.2.a.cs 1 35.c odd 2 1
8400.2.a.cm 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(210))$$:

 $$T_{11}$$ $$T_{17} + 6$$ $$T_{19} - 8$$

## Hecke characteristic polynomials

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