# Properties

 Label 210.4.a.a Level 210 Weight 4 Character orbit 210.a Self dual yes Analytic conductor 12.390 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 210.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.3904011012$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 2q^{2} - 3q^{3} + 4q^{4} - 5q^{5} + 6q^{6} + 7q^{7} - 8q^{8} + 9q^{9} + O(q^{10})$$ $$q - 2q^{2} - 3q^{3} + 4q^{4} - 5q^{5} + 6q^{6} + 7q^{7} - 8q^{8} + 9q^{9} + 10q^{10} + 12q^{11} - 12q^{12} + 2q^{13} - 14q^{14} + 15q^{15} + 16q^{16} - 18q^{17} - 18q^{18} + 56q^{19} - 20q^{20} - 21q^{21} - 24q^{22} - 156q^{23} + 24q^{24} + 25q^{25} - 4q^{26} - 27q^{27} + 28q^{28} - 186q^{29} - 30q^{30} - 52q^{31} - 32q^{32} - 36q^{33} + 36q^{34} - 35q^{35} + 36q^{36} - 178q^{37} - 112q^{38} - 6q^{39} + 40q^{40} - 138q^{41} + 42q^{42} - 412q^{43} + 48q^{44} - 45q^{45} + 312q^{46} - 456q^{47} - 48q^{48} + 49q^{49} - 50q^{50} + 54q^{51} + 8q^{52} - 198q^{53} + 54q^{54} - 60q^{55} - 56q^{56} - 168q^{57} + 372q^{58} + 348q^{59} + 60q^{60} + 110q^{61} + 104q^{62} + 63q^{63} + 64q^{64} - 10q^{65} + 72q^{66} - 196q^{67} - 72q^{68} + 468q^{69} + 70q^{70} - 936q^{71} - 72q^{72} + 542q^{73} + 356q^{74} - 75q^{75} + 224q^{76} + 84q^{77} + 12q^{78} + 992q^{79} - 80q^{80} + 81q^{81} + 276q^{82} - 276q^{83} - 84q^{84} + 90q^{85} + 824q^{86} + 558q^{87} - 96q^{88} + 630q^{89} + 90q^{90} + 14q^{91} - 624q^{92} + 156q^{93} + 912q^{94} - 280q^{95} + 96q^{96} + 110q^{97} - 98q^{98} + 108q^{99} + O(q^{100})$$

## 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
−2.00000 −3.00000 4.00000 −5.00000 6.00000 7.00000 −8.00000 9.00000 10.0000
 $$n$$: 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 210.4.a.a 1
3.b odd 2 1 630.4.a.v 1
4.b odd 2 1 1680.4.a.n 1
5.b even 2 1 1050.4.a.t 1
5.c odd 4 2 1050.4.g.o 2
7.b odd 2 1 1470.4.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.a.a 1 1.a even 1 1 trivial
630.4.a.v 1 3.b odd 2 1
1050.4.a.t 1 5.b even 2 1
1050.4.g.o 2 5.c odd 4 2
1470.4.a.n 1 7.b odd 2 1
1680.4.a.n 1 4.b odd 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(210))$$:

 $$T_{11} - 12$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T$$
$3$ $$1 + 3 T$$
$5$ $$1 + 5 T$$
$7$ $$1 - 7 T$$
$11$ $$1 - 12 T + 1331 T^{2}$$
$13$ $$1 - 2 T + 2197 T^{2}$$
$17$ $$1 + 18 T + 4913 T^{2}$$
$19$ $$1 - 56 T + 6859 T^{2}$$
$23$ $$1 + 156 T + 12167 T^{2}$$
$29$ $$1 + 186 T + 24389 T^{2}$$
$31$ $$1 + 52 T + 29791 T^{2}$$
$37$ $$1 + 178 T + 50653 T^{2}$$
$41$ $$1 + 138 T + 68921 T^{2}$$
$43$ $$1 + 412 T + 79507 T^{2}$$
$47$ $$1 + 456 T + 103823 T^{2}$$
$53$ $$1 + 198 T + 148877 T^{2}$$
$59$ $$1 - 348 T + 205379 T^{2}$$
$61$ $$1 - 110 T + 226981 T^{2}$$
$67$ $$1 + 196 T + 300763 T^{2}$$
$71$ $$1 + 936 T + 357911 T^{2}$$
$73$ $$1 - 542 T + 389017 T^{2}$$
$79$ $$1 - 992 T + 493039 T^{2}$$
$83$ $$1 + 276 T + 571787 T^{2}$$
$89$ $$1 - 630 T + 704969 T^{2}$$
$97$ $$1 - 110 T + 912673 T^{2}$$