# Properties

 Label 65.2.e.b Level 65 Weight 2 Character orbit 65.e Analytic conductor 0.519 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$65 = 5 \cdot 13$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 65.e (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.519027613138$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{13})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{2} ) q^{3} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} - q^{5} + ( 1 - \beta_{1} - \beta_{3} ) q^{6} -\beta_{2} q^{7} -3 q^{8} -2 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{2} ) q^{3} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} - q^{5} + ( 1 - \beta_{1} - \beta_{3} ) q^{6} -\beta_{2} q^{7} -3 q^{8} -2 \beta_{2} q^{9} -\beta_{1} q^{10} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{11} + ( 2 - \beta_{3} ) q^{12} + ( -1 + 2 \beta_{3} ) q^{13} + ( 1 - \beta_{3} ) q^{14} + ( 1 + \beta_{2} ) q^{15} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{16} + ( -2 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 2 - 2 \beta_{3} ) q^{18} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{19} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{20} - q^{21} + ( -1 + \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{22} + ( 3 + 3 \beta_{2} ) q^{23} + ( 3 + 3 \beta_{2} ) q^{24} + q^{25} + ( -6 - \beta_{1} - 6 \beta_{2} ) q^{26} -5 q^{27} + ( 1 + \beta_{1} + \beta_{2} ) q^{28} + ( -3 + 4 \beta_{1} - 3 \beta_{2} ) q^{29} + ( -1 + \beta_{1} + \beta_{3} ) q^{30} -4 q^{31} + ( -1 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{32} + ( -2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{33} + ( -11 + 5 \beta_{3} ) q^{34} + \beta_{2} q^{35} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{36} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{37} + ( 9 - 3 \beta_{3} ) q^{38} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{39} + 3 q^{40} + ( -3 - 3 \beta_{2} ) q^{41} -\beta_{1} q^{42} + ( 4 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{43} + ( 4 - \beta_{3} ) q^{44} + 2 \beta_{2} q^{45} + ( -3 + 3 \beta_{1} + 3 \beta_{3} ) q^{46} + ( 4 - 4 \beta_{3} ) q^{47} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{48} + ( 6 + 6 \beta_{2} ) q^{49} + \beta_{1} q^{50} + ( 5 - 2 \beta_{3} ) q^{51} + ( 3 - 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{52} + ( 2 + 4 \beta_{3} ) q^{53} -5 \beta_{1} q^{54} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{55} + 3 \beta_{2} q^{56} + ( -3 + 2 \beta_{3} ) q^{57} + ( -1 + \beta_{1} + 12 \beta_{2} + \beta_{3} ) q^{58} + ( 6 - 6 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{59} + ( -2 + \beta_{3} ) q^{60} -\beta_{2} q^{61} -4 \beta_{1} q^{62} + ( -2 - 2 \beta_{2} ) q^{63} + ( -5 + 6 \beta_{3} ) q^{64} + ( 1 - 2 \beta_{3} ) q^{65} + ( -5 - \beta_{3} ) q^{66} + ( 7 + 7 \beta_{2} ) q^{67} + ( -9 - 7 \beta_{1} - 9 \beta_{2} ) q^{68} -3 \beta_{2} q^{69} + ( -1 + \beta_{3} ) q^{70} + ( -6 + 6 \beta_{1} - 9 \beta_{2} + 6 \beta_{3} ) q^{71} + 6 \beta_{2} q^{72} + ( -6 - 4 \beta_{3} ) q^{73} + ( 1 - \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{74} + ( -1 - \beta_{2} ) q^{75} + ( 7 + 5 \beta_{1} + 7 \beta_{2} ) q^{76} + ( 1 + 2 \beta_{3} ) q^{77} + ( -1 + \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{78} -4 \beta_{3} q^{79} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{80} + ( -1 - \beta_{2} ) q^{81} + ( 3 - 3 \beta_{1} - 3 \beta_{3} ) q^{82} + ( -4 + 4 \beta_{3} ) q^{83} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{84} + ( 2 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{85} + ( 17 - 5 \beta_{3} ) q^{86} + ( 4 - 4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{87} + ( -9 + 6 \beta_{1} - 9 \beta_{2} ) q^{88} + ( -3 + 4 \beta_{1} - 3 \beta_{2} ) q^{89} + ( -2 + 2 \beta_{3} ) q^{90} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{91} + ( -6 + 3 \beta_{3} ) q^{92} + ( 4 + 4 \beta_{2} ) q^{93} + ( 12 + 4 \beta_{1} + 12 \beta_{2} ) q^{94} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{95} + ( 4 - \beta_{3} ) q^{96} + ( -2 + 2 \beta_{1} - 13 \beta_{2} + 2 \beta_{3} ) q^{97} + ( -6 + 6 \beta_{1} + 6 \beta_{3} ) q^{98} + ( 2 + 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} - 2q^{3} - 3q^{4} - 4q^{5} + q^{6} + 2q^{7} - 12q^{8} + 4q^{9} + O(q^{10})$$ $$4q + q^{2} - 2q^{3} - 3q^{4} - 4q^{5} + q^{6} + 2q^{7} - 12q^{8} + 4q^{9} - q^{10} + 4q^{11} + 6q^{12} + 2q^{14} + 2q^{15} + 3q^{16} - 8q^{17} + 4q^{18} + 4q^{19} + 3q^{20} - 4q^{21} + 11q^{22} + 6q^{23} + 6q^{24} + 4q^{25} - 13q^{26} - 20q^{27} + 3q^{28} - 2q^{29} - q^{30} - 16q^{31} - 7q^{32} + 4q^{33} - 34q^{34} - 2q^{35} + 6q^{36} + 30q^{38} + 12q^{40} - 6q^{41} - q^{42} + 6q^{43} + 14q^{44} - 4q^{45} - 3q^{46} + 8q^{47} + 3q^{48} + 12q^{49} + q^{50} + 16q^{51} + 13q^{52} + 16q^{53} - 5q^{54} - 4q^{55} - 6q^{56} - 8q^{57} - 25q^{58} - 6q^{60} + 2q^{61} - 4q^{62} - 4q^{63} - 8q^{64} - 22q^{66} + 14q^{67} - 25q^{68} + 6q^{69} - 2q^{70} + 12q^{71} - 12q^{72} - 32q^{73} + 13q^{74} - 2q^{75} + 19q^{76} + 8q^{77} - 13q^{78} - 8q^{79} - 3q^{80} - 2q^{81} + 3q^{82} - 8q^{83} + 3q^{84} + 8q^{85} + 58q^{86} - 2q^{87} - 12q^{88} - 2q^{89} - 4q^{90} - 18q^{92} + 8q^{93} + 28q^{94} - 4q^{95} + 14q^{96} + 24q^{97} - 6q^{98} + 16q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 4 x^{2} + 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu^{2} - 4 \nu - 3$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 7$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{3} - 7$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/65\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 −0.651388 − 1.12824i 1.15139 + 1.99426i −0.651388 + 1.12824i 1.15139 − 1.99426i
−0.651388 1.12824i −0.500000 0.866025i 0.151388 0.262211i −1.00000 −0.651388 + 1.12824i 0.500000 0.866025i −3.00000 1.00000 1.73205i 0.651388 + 1.12824i
16.2 1.15139 + 1.99426i −0.500000 0.866025i −1.65139 + 2.86029i −1.00000 1.15139 1.99426i 0.500000 0.866025i −3.00000 1.00000 1.73205i −1.15139 1.99426i
61.1 −0.651388 + 1.12824i −0.500000 + 0.866025i 0.151388 + 0.262211i −1.00000 −0.651388 1.12824i 0.500000 + 0.866025i −3.00000 1.00000 + 1.73205i 0.651388 1.12824i
61.2 1.15139 1.99426i −0.500000 + 0.866025i −1.65139 2.86029i −1.00000 1.15139 + 1.99426i 0.500000 + 0.866025i −3.00000 1.00000 + 1.73205i −1.15139 + 1.99426i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
13.c Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{3} + 4 T_{2}^{2} + 3 T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(65, [\chi])$$.