# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$182 = 2 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 182.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.45327731679$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{3} - q^{4} + 2 i q^{5} -i q^{6} + i q^{7} -i q^{8} -2 q^{9} +O(q^{10})$$ $$q + i q^{2} - q^{3} - q^{4} + 2 i q^{5} -i q^{6} + i q^{7} -i q^{8} -2 q^{9} -2 q^{10} + 5 i q^{11} + q^{12} + ( -3 - 2 i ) q^{13} - q^{14} -2 i q^{15} + q^{16} -2 q^{17} -2 i q^{18} + 4 i q^{19} -2 i q^{20} -i q^{21} -5 q^{22} + 9 q^{23} + i q^{24} + q^{25} + ( 2 - 3 i ) q^{26} + 5 q^{27} -i q^{28} + 2 q^{30} -5 i q^{31} + i q^{32} -5 i q^{33} -2 i q^{34} -2 q^{35} + 2 q^{36} + 3 i q^{37} -4 q^{38} + ( 3 + 2 i ) q^{39} + 2 q^{40} -5 i q^{41} + q^{42} + 4 q^{43} -5 i q^{44} -4 i q^{45} + 9 i q^{46} + 13 i q^{47} - q^{48} - q^{49} + i q^{50} + 2 q^{51} + ( 3 + 2 i ) q^{52} + 14 q^{53} + 5 i q^{54} -10 q^{55} + q^{56} -4 i q^{57} -6 i q^{59} + 2 i q^{60} -13 q^{61} + 5 q^{62} -2 i q^{63} - q^{64} + ( 4 - 6 i ) q^{65} + 5 q^{66} + 3 i q^{67} + 2 q^{68} -9 q^{69} -2 i q^{70} + 2 i q^{72} + i q^{73} -3 q^{74} - q^{75} -4 i q^{76} -5 q^{77} + ( -2 + 3 i ) q^{78} -15 q^{79} + 2 i q^{80} + q^{81} + 5 q^{82} + 6 i q^{83} + i q^{84} -4 i q^{85} + 4 i q^{86} + 5 q^{88} -6 i q^{89} + 4 q^{90} + ( 2 - 3 i ) q^{91} -9 q^{92} + 5 i q^{93} -13 q^{94} -8 q^{95} -i q^{96} -7 i q^{97} -i q^{98} -10 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{4} - 4q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{4} - 4q^{9} - 4q^{10} + 2q^{12} - 6q^{13} - 2q^{14} + 2q^{16} - 4q^{17} - 10q^{22} + 18q^{23} + 2q^{25} + 4q^{26} + 10q^{27} + 4q^{30} - 4q^{35} + 4q^{36} - 8q^{38} + 6q^{39} + 4q^{40} + 2q^{42} + 8q^{43} - 2q^{48} - 2q^{49} + 4q^{51} + 6q^{52} + 28q^{53} - 20q^{55} + 2q^{56} - 26q^{61} + 10q^{62} - 2q^{64} + 8q^{65} + 10q^{66} + 4q^{68} - 18q^{69} - 6q^{74} - 2q^{75} - 10q^{77} - 4q^{78} - 30q^{79} + 2q^{81} + 10q^{82} + 10q^{88} + 8q^{90} + 4q^{91} - 18q^{92} - 26q^{94} - 16q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/182\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$157$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
155.1
 − 1.00000i 1.00000i
1.00000i −1.00000 −1.00000 2.00000i 1.00000i 1.00000i 1.00000i −2.00000 −2.00000
155.2 1.00000i −1.00000 −1.00000 2.00000i 1.00000i 1.00000i 1.00000i −2.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 182.2.d.a 2
3.b odd 2 1 1638.2.c.a 2
4.b odd 2 1 1456.2.k.a 2
7.b odd 2 1 1274.2.d.d 2
7.c even 3 2 1274.2.n.e 4
7.d odd 6 2 1274.2.n.b 4
13.b even 2 1 inner 182.2.d.a 2
13.d odd 4 1 2366.2.a.c 1
13.d odd 4 1 2366.2.a.l 1
39.d odd 2 1 1638.2.c.a 2
52.b odd 2 1 1456.2.k.a 2
91.b odd 2 1 1274.2.d.d 2
91.r even 6 2 1274.2.n.e 4
91.s odd 6 2 1274.2.n.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.d.a 2 1.a even 1 1 trivial
182.2.d.a 2 13.b even 2 1 inner
1274.2.d.d 2 7.b odd 2 1
1274.2.d.d 2 91.b odd 2 1
1274.2.n.b 4 7.d odd 6 2
1274.2.n.b 4 91.s odd 6 2
1274.2.n.e 4 7.c even 3 2
1274.2.n.e 4 91.r even 6 2
1456.2.k.a 2 4.b odd 2 1
1456.2.k.a 2 52.b odd 2 1
1638.2.c.a 2 3.b odd 2 1
1638.2.c.a 2 39.d odd 2 1
2366.2.a.c 1 13.d odd 4 1
2366.2.a.l 1 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(182, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$4 + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$25 + T^{2}$$
$13$ $$13 + 6 T + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$( -9 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$25 + T^{2}$$
$37$ $$9 + T^{2}$$
$41$ $$25 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$169 + T^{2}$$
$53$ $$( -14 + T )^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$( 13 + T )^{2}$$
$67$ $$9 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$1 + T^{2}$$
$79$ $$( 15 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$36 + T^{2}$$
$97$ $$49 + T^{2}$$