# Properties

 Label 165.2.k.b Level $165$ Weight $2$ Character orbit 165.k Analytic conductor $1.318$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 165.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.31753163335$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{2} + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{4} + ( 2 - \zeta_{8}^{2} ) q^{5} + ( \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{6} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{7} + ( -1 - 3 \zeta_{8} - \zeta_{8}^{2} ) q^{8} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{2} + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{4} + ( 2 - \zeta_{8}^{2} ) q^{5} + ( \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{6} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{7} + ( -1 - 3 \zeta_{8} - \zeta_{8}^{2} ) q^{8} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} + ( 1 - \zeta_{8} - 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{10} + \zeta_{8}^{2} q^{11} + ( 1 + 3 \zeta_{8} + 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{12} + ( -2 + 2 \zeta_{8}^{2} ) q^{13} -2 q^{14} + ( 1 - \zeta_{8} + 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{15} -3 q^{16} + 4 \zeta_{8}^{3} q^{17} + ( -1 - 4 \zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{18} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{19} + ( -1 - 6 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{20} + ( 4 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{21} + ( 1 + \zeta_{8} + \zeta_{8}^{2} ) q^{22} + ( 1 + 6 \zeta_{8} + \zeta_{8}^{2} ) q^{23} + ( 4 + 2 \zeta_{8} + 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{24} + ( 3 - 4 \zeta_{8}^{2} ) q^{25} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{26} + ( \zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( 2 - 2 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{28} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{29} + ( 2 + 4 \zeta_{8} + 4 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{30} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{31} + ( -1 + \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{32} + ( -1 - \zeta_{8} - \zeta_{8}^{3} ) q^{33} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{34} + ( -6 + 4 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{35} + ( -8 - 4 \zeta_{8} - \zeta_{8}^{2} ) q^{36} + ( -3 - 4 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{37} + ( 2 + 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{38} + ( -2 - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{39} + ( -3 - 6 \zeta_{8} - \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{40} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{41} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{42} + ( -4 + 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{43} + ( 1 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{44} + ( 2 - 6 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{45} + ( 8 + 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{46} + ( 3 - 3 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{47} + ( 3 \zeta_{8} - 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{48} + ( -8 \zeta_{8} + 5 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{49} + ( -1 - 4 \zeta_{8} - 7 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{50} + ( 4 - 4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{51} + ( 2 + 8 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{52} + ( -1 - \zeta_{8}^{2} ) q^{53} + ( 6 + 5 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{54} + ( 1 + 2 \zeta_{8}^{2} ) q^{55} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{56} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{57} + ( 4 - 4 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{58} + 4 q^{59} + ( 6 + 5 \zeta_{8} + 7 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{60} + ( 2 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{61} + ( -8 + 8 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{62} + ( 2 + 2 \zeta_{8} - 6 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{63} + ( -2 \zeta_{8} - 7 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{64} + ( -2 + 6 \zeta_{8}^{2} ) q^{65} + ( -2 - 2 \zeta_{8} + \zeta_{8}^{3} ) q^{66} + ( -5 + 2 \zeta_{8} - 5 \zeta_{8}^{2} ) q^{67} + ( 8 + 4 \zeta_{8} + 8 \zeta_{8}^{2} ) q^{68} + ( -7 - 2 \zeta_{8} - 5 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{69} + ( -4 + 2 \zeta_{8}^{2} ) q^{70} + ( -6 \zeta_{8} - 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{71} + ( -7 - 3 \zeta_{8} + 5 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{72} + ( -10 - 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{74} + ( 4 + \zeta_{8} + 3 \zeta_{8}^{2} + 7 \zeta_{8}^{3} ) q^{75} + ( 8 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{76} + ( 2 - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{77} + ( -4 - 6 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{78} + ( -2 \zeta_{8} + 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{79} + ( -6 + 3 \zeta_{8}^{2} ) q^{80} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + ( -2 - 6 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{82} + ( 2 - 10 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{83} + ( 8 - 6 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{84} + ( 4 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{85} + ( 10 \zeta_{8} + 14 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{86} + ( -4 + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{87} + ( 1 - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{88} + ( 4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{89} + ( -5 - 9 \zeta_{8} - 5 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{90} + ( 8 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{91} + ( 13 - 13 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{92} + ( 4 + 4 \zeta_{8} - 6 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{93} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{94} + ( 6 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{95} + ( -4 + 3 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{96} + ( 5 - 8 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{97} + ( -3 - 11 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{98} + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 8q^{5} - 8q^{7} - 4q^{8} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 8q^{5} - 8q^{7} - 4q^{8} + 4q^{9} + 4q^{10} + 4q^{12} - 8q^{13} - 8q^{14} + 4q^{15} - 12q^{16} - 4q^{18} - 4q^{20} + 4q^{22} + 4q^{23} + 16q^{24} + 12q^{25} + 8q^{28} + 8q^{29} + 8q^{30} - 24q^{31} - 4q^{32} - 4q^{33} - 24q^{35} - 32q^{36} - 12q^{37} + 8q^{38} - 8q^{39} - 12q^{40} - 16q^{43} + 4q^{44} + 8q^{45} + 32q^{46} + 12q^{47} - 4q^{50} + 16q^{51} + 8q^{52} - 4q^{53} + 24q^{54} + 4q^{55} + 16q^{58} + 16q^{59} + 24q^{60} + 8q^{61} - 32q^{62} + 8q^{63} - 8q^{65} - 8q^{66} - 20q^{67} + 32q^{68} - 28q^{69} - 16q^{70} - 28q^{72} - 40q^{74} + 16q^{75} + 32q^{76} + 8q^{77} - 16q^{78} - 24q^{80} - 28q^{81} - 8q^{82} + 8q^{83} + 32q^{84} - 16q^{87} + 4q^{88} + 16q^{89} - 20q^{90} + 32q^{91} + 52q^{92} + 16q^{93} - 16q^{96} + 20q^{97} - 12q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/165\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$67$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
0.292893 + 0.292893i 1.41421 1.00000i 1.82843i 2.00000 + 1.00000i 0.707107 + 0.121320i −3.41421 + 3.41421i 1.12132 1.12132i 1.00000 2.82843i 0.292893 + 0.878680i
23.2 1.70711 + 1.70711i −1.41421 1.00000i 3.82843i 2.00000 + 1.00000i −0.707107 4.12132i −0.585786 + 0.585786i −3.12132 + 3.12132i 1.00000 + 2.82843i 1.70711 + 5.12132i
122.1 0.292893 0.292893i 1.41421 + 1.00000i 1.82843i 2.00000 1.00000i 0.707107 0.121320i −3.41421 3.41421i 1.12132 + 1.12132i 1.00000 + 2.82843i 0.292893 0.878680i
122.2 1.70711 1.70711i −1.41421 + 1.00000i 3.82843i 2.00000 1.00000i −0.707107 + 4.12132i −0.585786 0.585786i −3.12132 3.12132i 1.00000 2.82843i 1.70711 5.12132i
 $$n$$: 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
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.k.b yes 4
3.b odd 2 1 165.2.k.a 4
5.b even 2 1 825.2.k.c 4
5.c odd 4 1 165.2.k.a 4
5.c odd 4 1 825.2.k.f 4
15.d odd 2 1 825.2.k.f 4
15.e even 4 1 inner 165.2.k.b yes 4
15.e even 4 1 825.2.k.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.k.a 4 3.b odd 2 1
165.2.k.a 4 5.c odd 4 1
165.2.k.b yes 4 1.a even 1 1 trivial
165.2.k.b yes 4 15.e even 4 1 inner
825.2.k.c 4 5.b even 2 1
825.2.k.c 4 15.e even 4 1
825.2.k.f 4 5.c odd 4 1
825.2.k.f 4 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 4 T_{2}^{3} + 8 T_{2}^{2} - 4 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(165, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$3$ $$9 - 2 T^{2} + T^{4}$$
$5$ $$( 5 - 4 T + T^{2} )^{2}$$
$7$ $$16 + 32 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$11$ $$( 1 + T^{2} )^{2}$$
$13$ $$( 8 + 4 T + T^{2} )^{2}$$
$17$ $$256 + T^{4}$$
$19$ $$( 8 + T^{2} )^{2}$$
$23$ $$1156 + 136 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$( -4 - 4 T + T^{2} )^{2}$$
$31$ $$( 28 + 12 T + T^{2} )^{2}$$
$37$ $$4 + 24 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$41$ $$784 + 72 T^{2} + T^{4}$$
$43$ $$16 - 64 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$47$ $$324 + 216 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$53$ $$( 2 + 2 T + T^{2} )^{2}$$
$59$ $$( -4 + T )^{4}$$
$61$ $$( -68 - 4 T + T^{2} )^{2}$$
$67$ $$2116 + 920 T + 200 T^{2} + 20 T^{3} + T^{4}$$
$71$ $$1296 + 216 T^{2} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$3136 + 144 T^{2} + T^{4}$$
$83$ $$8464 + 736 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$89$ $$( -16 - 8 T + T^{2} )^{2}$$
$97$ $$196 + 280 T + 200 T^{2} - 20 T^{3} + T^{4}$$