# Properties

 Label 182.2.a.d Level $182$ Weight $2$ Character orbit 182.a Self dual yes Analytic conductor $1.453$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$182 = 2 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 182.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.45327731679$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} - 2q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} - 2q^{9} - 3q^{11} + q^{12} + q^{13} + q^{14} + q^{16} - 2q^{18} + 2q^{19} + q^{21} - 3q^{22} - 3q^{23} + q^{24} - 5q^{25} + q^{26} - 5q^{27} + q^{28} + 5q^{31} + q^{32} - 3q^{33} - 2q^{36} - 7q^{37} + 2q^{38} + q^{39} + 3q^{41} + q^{42} + 8q^{43} - 3q^{44} - 3q^{46} - 3q^{47} + q^{48} + q^{49} - 5q^{50} + q^{52} - 12q^{53} - 5q^{54} + q^{56} + 2q^{57} + 6q^{59} - q^{61} + 5q^{62} - 2q^{63} + q^{64} - 3q^{66} + 5q^{67} - 3q^{69} + 12q^{71} - 2q^{72} + 11q^{73} - 7q^{74} - 5q^{75} + 2q^{76} - 3q^{77} + q^{78} - q^{79} + q^{81} + 3q^{82} + 12q^{83} + q^{84} + 8q^{86} - 3q^{88} - 18q^{89} + q^{91} - 3q^{92} + 5q^{93} - 3q^{94} + q^{96} + 17q^{97} + q^{98} + 6q^{99} + O(q^{100})$$

## 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
1.00000 1.00000 1.00000 0 1.00000 1.00000 1.00000 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 182.2.a.d 1
3.b odd 2 1 1638.2.a.f 1
4.b odd 2 1 1456.2.a.d 1
5.b even 2 1 4550.2.a.c 1
7.b odd 2 1 1274.2.a.j 1
7.c even 3 2 1274.2.f.d 2
7.d odd 6 2 1274.2.f.i 2
8.b even 2 1 5824.2.a.k 1
8.d odd 2 1 5824.2.a.x 1
13.b even 2 1 2366.2.a.e 1
13.d odd 4 2 2366.2.d.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.a.d 1 1.a even 1 1 trivial
1274.2.a.j 1 7.b odd 2 1
1274.2.f.d 2 7.c even 3 2
1274.2.f.i 2 7.d odd 6 2
1456.2.a.d 1 4.b odd 2 1
1638.2.a.f 1 3.b odd 2 1
2366.2.a.e 1 13.b even 2 1
2366.2.d.e 2 13.d odd 4 2
4550.2.a.c 1 5.b even 2 1
5824.2.a.k 1 8.b even 2 1
5824.2.a.x 1 8.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(182))$$:

 $$T_{3} - 1$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$3 + T$$
$13$ $$-1 + T$$
$17$ $$T$$
$19$ $$-2 + T$$
$23$ $$3 + T$$
$29$ $$T$$
$31$ $$-5 + T$$
$37$ $$7 + T$$
$41$ $$-3 + T$$
$43$ $$-8 + T$$
$47$ $$3 + T$$
$53$ $$12 + T$$
$59$ $$-6 + T$$
$61$ $$1 + T$$
$67$ $$-5 + T$$
$71$ $$-12 + T$$
$73$ $$-11 + T$$
$79$ $$1 + T$$
$83$ $$-12 + T$$
$89$ $$18 + T$$
$97$ $$-17 + T$$