# Properties

 Label 65.2.m.a Level $65$ Weight $2$ Character orbit 65.m Analytic conductor $0.519$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.519027613138$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.22581504.2 Defining polynomial: $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + \nu^{5} + 4 \nu^{4} - 3 \nu^{3} - 2 \nu^{2} + 8 \nu - 8$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} - \nu^{5} - 4 \nu^{4} + 3 \nu^{3} + 10 \nu^{2} - 16 \nu + 8$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 7 \nu^{3} - 3 \nu^{2} + 18 \nu - 16$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{7} - 5 \nu^{6} + 2 \nu^{5} + 7 \nu^{4} - 8 \nu^{3} - 9 \nu^{2} + 28 \nu - 20$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} + 7 \nu^{6} - 3 \nu^{5} - 11 \nu^{4} + 15 \nu^{3} + 11 \nu^{2} - 40 \nu + 32$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$7 \nu^{7} - 20 \nu^{6} + 11 \nu^{5} + 30 \nu^{4} - 45 \nu^{3} - 28 \nu^{2} + 116 \nu - 88$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + 2 \beta_{2} + \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} + \beta_{1} - 1$$ $$\nu^{5}$$ $$=$$ $$\beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 1$$ $$\nu^{6}$$ $$=$$ $$-\beta_{7} - 3 \beta_{6} - 5 \beta_{5} + \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 4 \beta_{1} - 1$$ $$\nu^{7}$$ $$=$$ $$-6 \beta_{7} - 8 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{2} + \beta_{1} + 6$$

## Character values

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

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
36.1
 1.20036 + 0.747754i −1.27597 + 0.609843i 1.40994 − 0.109843i 0.665665 − 1.24775i 1.20036 − 0.747754i −1.27597 − 0.609843i 1.40994 + 0.109843i 0.665665 + 1.24775i
−2.16117 + 1.24775i 1.41342 + 2.44811i 2.11378 3.66117i 1.00000i −6.10929 3.52720i −1.64996 0.952606i 5.55889i −2.49551 + 4.32235i −1.24775 2.16117i
36.2 −0.190254 + 0.109843i 0.800098 + 1.38581i −0.975869 + 1.69025i 1.00000i −0.304444 0.175771i −0.287734 0.166123i 0.868145i 0.219687 0.380509i 0.109843 + 0.190254i
36.3 1.05628 0.609843i −1.16612 2.01978i −0.256182 + 0.443720i 1.00000i −2.46350 1.42231i 3.11786 + 1.80010i 3.06430i −1.21969 + 2.11256i −0.609843 1.05628i
36.4 1.29515 0.747754i −0.0473938 0.0820885i 0.118272 0.204852i 1.00000i −0.122764 0.0708778i −4.18016 2.41342i 2.63726i 1.49551 2.59030i 0.747754 + 1.29515i
56.1 −2.16117 1.24775i 1.41342 2.44811i 2.11378 + 3.66117i 1.00000i −6.10929 + 3.52720i −1.64996 + 0.952606i 5.55889i −2.49551 4.32235i −1.24775 + 2.16117i
56.2 −0.190254 0.109843i 0.800098 1.38581i −0.975869 1.69025i 1.00000i −0.304444 + 0.175771i −0.287734 + 0.166123i 0.868145i 0.219687 + 0.380509i 0.109843 0.190254i
56.3 1.05628 + 0.609843i −1.16612 + 2.01978i −0.256182 0.443720i 1.00000i −2.46350 + 1.42231i 3.11786 1.80010i 3.06430i −1.21969 2.11256i −0.609843 + 1.05628i
56.4 1.29515 + 0.747754i −0.0473938 + 0.0820885i 0.118272 + 0.204852i 1.00000i −0.122764 + 0.0708778i −4.18016 + 2.41342i 2.63726i 1.49551 + 2.59030i 0.747754 1.29515i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 56.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.m.a 8
3.b odd 2 1 585.2.bu.c 8
4.b odd 2 1 1040.2.da.b 8
5.b even 2 1 325.2.n.d 8
5.c odd 4 1 325.2.m.b 8
5.c odd 4 1 325.2.m.c 8
13.b even 2 1 845.2.m.g 8
13.c even 3 1 845.2.c.g 8
13.c even 3 1 845.2.m.g 8
13.d odd 4 1 845.2.e.m 8
13.d odd 4 1 845.2.e.n 8
13.e even 6 1 inner 65.2.m.a 8
13.e even 6 1 845.2.c.g 8
13.f odd 12 1 845.2.a.l 4
13.f odd 12 1 845.2.a.m 4
13.f odd 12 1 845.2.e.m 8
13.f odd 12 1 845.2.e.n 8
39.h odd 6 1 585.2.bu.c 8
39.k even 12 1 7605.2.a.cf 4
39.k even 12 1 7605.2.a.cj 4
52.i odd 6 1 1040.2.da.b 8
65.l even 6 1 325.2.n.d 8
65.r odd 12 1 325.2.m.b 8
65.r odd 12 1 325.2.m.c 8
65.s odd 12 1 4225.2.a.bi 4
65.s odd 12 1 4225.2.a.bl 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 1.a even 1 1 trivial
65.2.m.a 8 13.e even 6 1 inner
325.2.m.b 8 5.c odd 4 1
325.2.m.b 8 65.r odd 12 1
325.2.m.c 8 5.c odd 4 1
325.2.m.c 8 65.r odd 12 1
325.2.n.d 8 5.b even 2 1
325.2.n.d 8 65.l even 6 1
585.2.bu.c 8 3.b odd 2 1
585.2.bu.c 8 39.h odd 6 1
845.2.a.l 4 13.f odd 12 1
845.2.a.m 4 13.f odd 12 1
845.2.c.g 8 13.c even 3 1
845.2.c.g 8 13.e even 6 1
845.2.e.m 8 13.d odd 4 1
845.2.e.m 8 13.f odd 12 1
845.2.e.n 8 13.d odd 4 1
845.2.e.n 8 13.f odd 12 1
845.2.m.g 8 13.b even 2 1
845.2.m.g 8 13.c even 3 1
1040.2.da.b 8 4.b odd 2 1
1040.2.da.b 8 52.i odd 6 1
4225.2.a.bi 4 65.s odd 12 1
4225.2.a.bl 4 65.s odd 12 1
7605.2.a.cf 4 39.k even 12 1
7605.2.a.cj 4 39.k even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(65, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T + 7 T^{2} - 30 T^{3} + 24 T^{4} - 5 T^{6} + T^{8}$$
$3$ $$1 + 10 T + 106 T^{2} - 56 T^{3} + 55 T^{4} - 8 T^{5} + 10 T^{6} - 2 T^{7} + T^{8}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$121 + 726 T + 1606 T^{2} + 924 T^{3} + 75 T^{4} - 84 T^{5} - 2 T^{6} + 6 T^{7} + T^{8}$$
$11$ $$1089 - 990 T^{2} + 867 T^{4} - 30 T^{6} + T^{8}$$
$13$ $$28561 + 2704 T^{2} + 1248 T^{3} + 30 T^{4} + 96 T^{5} + 16 T^{6} + T^{8}$$
$17$ $$169 - 130 T + 334 T^{2} + 128 T^{3} + 331 T^{4} - 16 T^{5} + 22 T^{6} + 2 T^{7} + T^{8}$$
$19$ $$( 169 + 78 T - T^{2} - 6 T^{3} + T^{4} )^{2}$$
$23$ $$89401 + 43654 T + 23110 T^{2} + 5104 T^{3} + 1795 T^{4} + 352 T^{5} + 94 T^{6} + 10 T^{7} + T^{8}$$
$29$ $$1 - 40 T + 1618 T^{2} + 704 T^{3} + 643 T^{4} - 64 T^{5} + 82 T^{6} + 8 T^{7} + T^{8}$$
$31$ $$( 64 + 32 T^{2} + T^{4} )^{2}$$
$37$ $$1 + 66 T + 1402 T^{2} - 3300 T^{3} + 2367 T^{4} + 300 T^{5} - 38 T^{6} - 6 T^{7} + T^{8}$$
$41$ $$( 1 + 6 T + 11 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$43$ $$169 - 130 T + 334 T^{2} + 128 T^{3} + 331 T^{4} - 16 T^{5} + 22 T^{6} + 2 T^{7} + T^{8}$$
$47$ $$1763584 + 350464 T^{2} + 14304 T^{4} + 208 T^{6} + T^{8}$$
$53$ $$( -48 + 36 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$59$ $$9 + 108 T + 486 T^{2} + 648 T^{3} + 183 T^{4} - 216 T^{5} + 30 T^{6} + 12 T^{7} + T^{8}$$
$61$ $$1590121 + 1215604 T + 603958 T^{2} + 178096 T^{3} + 38311 T^{4} + 5296 T^{5} + 526 T^{6} + 28 T^{7} + T^{8}$$
$67$ $$7667361 + 847314 T - 284454 T^{2} - 34884 T^{3} + 9615 T^{4} + 684 T^{5} - 102 T^{6} - 6 T^{7} + T^{8}$$
$71$ $$109767529 + 2263032 T - 2268434 T^{2} - 47088 T^{3} + 37047 T^{4} - 218 T^{6} + T^{8}$$
$73$ $$2930944 + 404608 T^{2} + 16944 T^{4} + 232 T^{6} + T^{8}$$
$79$ $$( 4432 - 640 T - 132 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$83$ $$36864 + 73728 T^{2} + 7104 T^{4} + 192 T^{6} + T^{8}$$
$89$ $$78375609 + 34420464 T + 3179718 T^{2} - 816480 T^{3} + 4143 T^{4} + 5040 T^{5} - 18 T^{6} - 24 T^{7} + T^{8}$$
$97$ $$196249 + 71766 T - 16946 T^{2} - 9396 T^{3} + 2187 T^{4} + 1740 T^{5} + 358 T^{6} + 30 T^{7} + T^{8}$$