# Properties

 Label 65.2.b.a Level 65 Weight 2 Character orbit 65.b Analytic conductor 0.519 Analytic rank 0 Dimension 6 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$65 = 5 \cdot 13$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 65.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.519027613138$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$-14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5 \beta_{1} - 7$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 3 \beta_{2} + 8 \beta_{1} - 9$$

## Character Values

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

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1
 0.403032 − 0.403032i −0.854638 + 0.854638i 1.45161 + 1.45161i 1.45161 − 1.45161i −0.854638 − 0.854638i 0.403032 + 0.403032i
2.67513i 0.481194i −5.15633 1.67513 + 1.48119i −1.28726 0.806063i 8.44358i 2.76845 3.96239 4.48119i
14.2 1.53919i 3.17009i −0.369102 0.539189 2.17009i 4.87936 1.70928i 2.51026i −7.04945 −3.34017 0.829914i
14.3 1.21432i 1.31111i 0.525428 −2.21432 + 0.311108i −1.59210 2.90321i 3.06668i 1.28100 0.377784 + 2.68889i
14.4 1.21432i 1.31111i 0.525428 −2.21432 0.311108i −1.59210 2.90321i 3.06668i 1.28100 0.377784 2.68889i
14.5 1.53919i 3.17009i −0.369102 0.539189 + 2.17009i 4.87936 1.70928i 2.51026i −7.04945 −3.34017 + 0.829914i
14.6 2.67513i 0.481194i −5.15633 1.67513 1.48119i −1.28726 0.806063i 8.44358i 2.76845 3.96239 + 4.48119i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 14.6 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

## Hecke kernels

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