# Properties

 Label 65.2.e.a 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: $$\Q(\sqrt{-3}, \sqrt{5})$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

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

## Character Values

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

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$

## 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
 0.809017 + 1.40126i −0.309017 − 0.535233i 0.809017 − 1.40126i −0.309017 + 0.535233i
−0.809017 1.40126i 1.11803 + 1.93649i −0.309017 + 0.535233i 1.00000 1.80902 3.13331i 0.118034 0.204441i −2.23607 −1.00000 + 1.73205i −0.809017 1.40126i
16.2 0.309017 + 0.535233i −1.11803 1.93649i 0.809017 1.40126i 1.00000 0.690983 1.19682i −2.11803 + 3.66854i 2.23607 −1.00000 + 1.73205i 0.309017 + 0.535233i
61.1 −0.809017 + 1.40126i 1.11803 1.93649i −0.309017 0.535233i 1.00000 1.80902 + 3.13331i 0.118034 + 0.204441i −2.23607 −1.00000 1.73205i −0.809017 + 1.40126i
61.2 0.309017 0.535233i −1.11803 + 1.93649i 0.809017 + 1.40126i 1.00000 0.690983 + 1.19682i −2.11803 3.66854i 2.23607 −1.00000 1.73205i 0.309017 0.535233i
 $$n$$: 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 $$T_{2}^{4} + T_{2}^{3} + 2 T_{2}^{2} - T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(65, [\chi])$$.