# Properties

 Label 33.2.f.a Level $33$ Weight $2$ Character orbit 33.f Analytic conductor $0.264$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 33.f (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.263506326670$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$\zeta_{20}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 −0.951057 − 0.309017i 0.951057 + 0.309017i 0.587785 + 0.809017i −0.587785 − 0.809017i −0.951057 + 0.309017i 0.951057 − 0.309017i 0.587785 − 0.809017i −0.587785 + 0.809017i
−0.587785 + 1.80902i −1.67229 + 0.451057i −1.30902 0.951057i 2.48990 0.809017i 0.166977 3.29032i 0.427051 0.587785i −0.587785 + 0.427051i 2.59310 1.50859i 4.97980i
2.2 0.587785 1.80902i −0.945746 + 1.45106i −1.30902 0.951057i −2.48990 + 0.809017i 2.06909 + 2.56378i 0.427051 0.587785i 0.587785 0.427051i −1.21113 2.74466i 4.97980i
8.1 −0.951057 + 0.690983i 1.34786 1.08779i −0.190983 + 0.587785i −0.224514 + 0.309017i −0.530249 + 1.96589i −2.92705 0.951057i −0.951057 2.92705i 0.633446 2.93236i 0.449028i
8.2 0.951057 0.690983i −1.72982 0.0877853i −0.190983 + 0.587785i 0.224514 0.309017i −1.70582 + 1.11179i −2.92705 0.951057i 0.951057 + 2.92705i 2.98459 + 0.303706i 0.449028i
17.1 −0.587785 1.80902i −1.67229 0.451057i −1.30902 + 0.951057i 2.48990 + 0.809017i 0.166977 + 3.29032i 0.427051 + 0.587785i −0.587785 0.427051i 2.59310 + 1.50859i 4.97980i
17.2 0.587785 + 1.80902i −0.945746 1.45106i −1.30902 + 0.951057i −2.48990 0.809017i 2.06909 2.56378i 0.427051 + 0.587785i 0.587785 + 0.427051i −1.21113 + 2.74466i 4.97980i
29.1 −0.951057 0.690983i 1.34786 + 1.08779i −0.190983 0.587785i −0.224514 0.309017i −0.530249 1.96589i −2.92705 + 0.951057i −0.951057 + 2.92705i 0.633446 + 2.93236i 0.449028i
29.2 0.951057 + 0.690983i −1.72982 + 0.0877853i −0.190983 0.587785i 0.224514 + 0.309017i −1.70582 1.11179i −2.92705 + 0.951057i 0.951057 2.92705i 2.98459 0.303706i 0.449028i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.2.f.a 8
3.b odd 2 1 inner 33.2.f.a 8
4.b odd 2 1 528.2.bn.c 8
5.b even 2 1 825.2.bi.b 8
5.c odd 4 1 825.2.bs.a 8
5.c odd 4 1 825.2.bs.d 8
9.c even 3 2 891.2.u.a 16
9.d odd 6 2 891.2.u.a 16
11.b odd 2 1 363.2.f.b 8
11.c even 5 1 363.2.d.f 8
11.c even 5 1 363.2.f.b 8
11.c even 5 1 363.2.f.d 8
11.c even 5 1 363.2.f.e 8
11.d odd 10 1 inner 33.2.f.a 8
11.d odd 10 1 363.2.d.f 8
11.d odd 10 1 363.2.f.d 8
11.d odd 10 1 363.2.f.e 8
12.b even 2 1 528.2.bn.c 8
15.d odd 2 1 825.2.bi.b 8
15.e even 4 1 825.2.bs.a 8
15.e even 4 1 825.2.bs.d 8
33.d even 2 1 363.2.f.b 8
33.f even 10 1 inner 33.2.f.a 8
33.f even 10 1 363.2.d.f 8
33.f even 10 1 363.2.f.d 8
33.f even 10 1 363.2.f.e 8
33.h odd 10 1 363.2.d.f 8
33.h odd 10 1 363.2.f.b 8
33.h odd 10 1 363.2.f.d 8
33.h odd 10 1 363.2.f.e 8
44.g even 10 1 528.2.bn.c 8
55.h odd 10 1 825.2.bi.b 8
55.l even 20 1 825.2.bs.a 8
55.l even 20 1 825.2.bs.d 8
99.o odd 30 2 891.2.u.a 16
99.p even 30 2 891.2.u.a 16
132.n odd 10 1 528.2.bn.c 8
165.r even 10 1 825.2.bi.b 8
165.u odd 20 1 825.2.bs.a 8
165.u odd 20 1 825.2.bs.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.f.a 8 1.a even 1 1 trivial
33.2.f.a 8 3.b odd 2 1 inner
33.2.f.a 8 11.d odd 10 1 inner
33.2.f.a 8 33.f even 10 1 inner
363.2.d.f 8 11.c even 5 1
363.2.d.f 8 11.d odd 10 1
363.2.d.f 8 33.f even 10 1
363.2.d.f 8 33.h odd 10 1
363.2.f.b 8 11.b odd 2 1
363.2.f.b 8 11.c even 5 1
363.2.f.b 8 33.d even 2 1
363.2.f.b 8 33.h odd 10 1
363.2.f.d 8 11.c even 5 1
363.2.f.d 8 11.d odd 10 1
363.2.f.d 8 33.f even 10 1
363.2.f.d 8 33.h odd 10 1
363.2.f.e 8 11.c even 5 1
363.2.f.e 8 11.d odd 10 1
363.2.f.e 8 33.f even 10 1
363.2.f.e 8 33.h odd 10 1
528.2.bn.c 8 4.b odd 2 1
528.2.bn.c 8 12.b even 2 1
528.2.bn.c 8 44.g even 10 1
528.2.bn.c 8 132.n odd 10 1
825.2.bi.b 8 5.b even 2 1
825.2.bi.b 8 15.d odd 2 1
825.2.bi.b 8 55.h odd 10 1
825.2.bi.b 8 165.r even 10 1
825.2.bs.a 8 5.c odd 4 1
825.2.bs.a 8 15.e even 4 1
825.2.bs.a 8 55.l even 20 1
825.2.bs.a 8 165.u odd 20 1
825.2.bs.d 8 5.c odd 4 1
825.2.bs.d 8 15.e even 4 1
825.2.bs.d 8 55.l even 20 1
825.2.bs.d 8 165.u odd 20 1
891.2.u.a 16 9.c even 3 2
891.2.u.a 16 9.d odd 6 2
891.2.u.a 16 99.o odd 30 2
891.2.u.a 16 99.p even 30 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2} + 2 T^{4} + 8 T^{6} + 25 T^{8} + 32 T^{10} + 32 T^{12} + 64 T^{14} + 256 T^{16}$$
$3$ $$1 + 6 T + 13 T^{2} + 10 T^{3} + T^{4} + 30 T^{5} + 117 T^{6} + 162 T^{7} + 81 T^{8}$$
$5$ $$1 - T^{2} + 51 T^{4} + 109 T^{6} + 1136 T^{8} + 2725 T^{10} + 31875 T^{12} - 15625 T^{14} + 390625 T^{16}$$
$7$ $$( 1 + 5 T + 12 T^{2} - 5 T^{3} - 51 T^{4} - 35 T^{5} + 588 T^{6} + 1715 T^{7} + 2401 T^{8} )^{2}$$
$11$ $$1 + 19 T^{2} + 301 T^{4} + 2299 T^{6} + 14641 T^{8}$$
$13$ $$( 1 + 5 T + 18 T^{2} - 5 T^{3} - 21 T^{4} - 65 T^{5} + 3042 T^{6} + 10985 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$1 - 34 T^{2} + 267 T^{4} + 8548 T^{6} - 277795 T^{8} + 2470372 T^{10} + 22300107 T^{12} - 820677346 T^{14} + 6975757441 T^{16}$$
$19$ $$( 1 - 10 T + 69 T^{2} - 410 T^{3} + 1911 T^{4} - 7790 T^{5} + 24909 T^{6} - 68590 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 50 T^{2} + 1363 T^{4} - 26450 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$1 - 58 T^{2} + 1523 T^{4} - 6056 T^{6} - 679595 T^{8} - 5093096 T^{10} + 1077188963 T^{12} - 34499752618 T^{14} + 500246412961 T^{16}$$
$31$ $$( 1 + 10 T + 9 T^{2} - 130 T^{3} - 409 T^{4} - 4030 T^{5} + 8649 T^{6} + 297910 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 7215 T^{5} - 38332 T^{6} + 151959 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$1 - 87 T^{2} + 4308 T^{4} - 137029 T^{6} + 5286975 T^{8} - 230345749 T^{10} + 12173378388 T^{12} - 413259068967 T^{14} + 7984925229121 T^{16}$$
$43$ $$( 1 - 122 T^{2} + 6919 T^{4} - 225578 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$1 + 15 T^{2} - 969 T^{4} - 93835 T^{6} + 1517976 T^{8} - 207281515 T^{10} - 4728410889 T^{12} + 161688229935 T^{14} + 23811286661761 T^{16}$$
$53$ $$1 + 70 T^{2} + 6051 T^{4} + 452660 T^{6} + 19250861 T^{8} + 1271521940 T^{10} + 47745300531 T^{12} + 1551505279030 T^{14} + 62259690411361 T^{16}$$
$59$ $$1 + 122 T^{2} + 9663 T^{4} + 604444 T^{6} + 40851365 T^{8} + 2104069564 T^{10} + 117090059343 T^{12} + 5146025104202 T^{14} + 146830437604321 T^{16}$$
$61$ $$( 1 + 5 T + 61 T^{2} - 305 T^{3} + 796 T^{4} - 18605 T^{5} + 226981 T^{6} + 1134905 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + T + 73 T^{2} + 67 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$1 - 13 T^{2} + 5103 T^{4} - 100031 T^{6} + 31068680 T^{8} - 504256271 T^{10} + 129675808143 T^{12} - 1665303690973 T^{14} + 645753531245761 T^{16}$$
$73$ $$( 1 + 73 T^{2} - 360 T^{3} + 2449 T^{4} - 26280 T^{5} + 389017 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 25 T + 304 T^{2} - 2435 T^{3} + 19501 T^{4} - 192365 T^{5} + 1897264 T^{6} - 12325975 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$1 + 149 T^{2} + 16452 T^{4} + 1844407 T^{6} + 200466815 T^{8} + 12706119823 T^{10} + 780784297092 T^{12} + 48714115631981 T^{14} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 266 T^{2} + 31531 T^{4} - 2106986 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 3 T - 63 T^{2} + 835 T^{3} + 4236 T^{4} + 80995 T^{5} - 592767 T^{6} - 2738019 T^{7} + 88529281 T^{8} )^{2}$$
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