# Properties

 Label 65.2.l.a Level 65 Weight 2 Character orbit 65.l Analytic conductor 0.519 Analytic rank 0 Dimension 8 CM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$65 = 5 \cdot 13$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 65.l (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.519027613138$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.49787136.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} + 5 \nu^{4} - 5 \nu^{2} - 12$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 5 \nu^{5} - 5 \nu^{3} - 12 \nu$$$$)/40$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 36$$$$)/20$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu$$$$)/40$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 7$$$$)/5$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 3 \nu^{5} + 5 \nu^{3} + 12 \nu$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{4} - \beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2 \beta_{5} - \beta_{3} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 3 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} + 5 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$-5 \beta_{6} - 7$$ $$\nu^{7}$$ $$=$$ $$-10 \beta_{5} - 10 \beta_{3} - 7 \beta_{1}$$

## Character Values

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

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-1$$ $$1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 −1.09445 + 0.895644i −0.228425 + 1.39564i 0.228425 − 1.39564i 1.09445 − 0.895644i −1.09445 − 0.895644i −0.228425 − 1.39564i 0.228425 + 1.39564i 1.09445 + 0.895644i
−1.09445 + 1.89564i 0.866025 + 0.500000i −1.39564 2.41733i 0.456850 + 2.18890i −1.89564 + 1.09445i −0.866025 1.50000i 1.73205 −1.00000 1.73205i −4.64938 1.52962i
4.2 −0.228425 + 0.395644i −0.866025 0.500000i 0.895644 + 1.55130i 2.18890 0.456850i 0.395644 0.228425i 0.866025 + 1.50000i −1.73205 −1.00000 1.73205i −0.319250 + 0.970381i
4.3 0.228425 0.395644i 0.866025 + 0.500000i 0.895644 + 1.55130i −2.18890 0.456850i 0.395644 0.228425i −0.866025 1.50000i 1.73205 −1.00000 1.73205i −0.680750 + 0.761669i
4.4 1.09445 1.89564i −0.866025 0.500000i −1.39564 2.41733i −0.456850 + 2.18890i −1.89564 + 1.09445i 0.866025 + 1.50000i −1.73205 −1.00000 1.73205i 3.64938 + 3.26167i
49.1 −1.09445 1.89564i 0.866025 0.500000i −1.39564 + 2.41733i 0.456850 2.18890i −1.89564 1.09445i −0.866025 + 1.50000i 1.73205 −1.00000 + 1.73205i −4.64938 + 1.52962i
49.2 −0.228425 0.395644i −0.866025 + 0.500000i 0.895644 1.55130i 2.18890 + 0.456850i 0.395644 + 0.228425i 0.866025 1.50000i −1.73205 −1.00000 + 1.73205i −0.319250 0.970381i
49.3 0.228425 + 0.395644i 0.866025 0.500000i 0.895644 1.55130i −2.18890 + 0.456850i 0.395644 + 0.228425i −0.866025 + 1.50000i 1.73205 −1.00000 + 1.73205i −0.680750 0.761669i
49.4 1.09445 + 1.89564i −0.866025 + 0.500000i −1.39564 + 2.41733i −0.456850 2.18890i −1.89564 1.09445i 0.866025 1.50000i −1.73205 −1.00000 + 1.73205i 3.64938 3.26167i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.4 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
5.b Even 1 yes
13.e Even 1 yes
65.l Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(65, [\chi])$$.