# Properties

 Label 361.2.e.b Level $361$ Weight $2$ Character orbit 361.e Analytic conductor $2.883$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$361 = 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 361.e (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.88259951297$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/361\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i −0.173648 − 0.984808i 0.939693 + 0.342020i −0.766044 − 0.642788i
0.439693 + 2.49362i 0.500000 0.419550i −4.14543 + 1.50881i 1.26604 + 0.460802i 1.26604 + 1.06234i −0.766044 + 1.32683i −3.05303 5.28801i −0.446967 + 2.53487i −0.592396 + 3.35965i
54.1 −0.673648 0.565258i 0.500000 0.181985i −0.213011 1.20805i −0.439693 + 2.49362i −0.439693 0.160035i 0.939693 + 1.62760i −1.41875 + 2.45734i −2.08125 + 1.74638i 1.70574 1.43128i
62.1 −1.26604 + 0.460802i 0.500000 2.83564i −0.141559 + 0.118782i 0.673648 + 0.565258i 0.673648 + 3.82045i −0.173648 0.300767i 1.47178 2.54920i −4.97178 1.80958i −1.11334 0.405223i
99.1 −1.26604 0.460802i 0.500000 + 2.83564i −0.141559 0.118782i 0.673648 0.565258i 0.673648 3.82045i −0.173648 + 0.300767i 1.47178 + 2.54920i −4.97178 + 1.80958i −1.11334 + 0.405223i
234.1 −0.673648 + 0.565258i 0.500000 + 0.181985i −0.213011 + 1.20805i −0.439693 2.49362i −0.439693 + 0.160035i 0.939693 1.62760i −1.41875 2.45734i −2.08125 1.74638i 1.70574 + 1.43128i
245.1 0.439693 2.49362i 0.500000 + 0.419550i −4.14543 1.50881i 1.26604 0.460802i 1.26604 1.06234i −0.766044 1.32683i −3.05303 + 5.28801i −0.446967 2.53487i −0.592396 3.35965i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 245.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.2.e.b 6
19.b odd 2 1 361.2.e.f 6
19.c even 3 1 361.2.e.a 6
19.c even 3 1 361.2.e.h 6
19.d odd 6 1 19.2.e.a 6
19.d odd 6 1 361.2.e.g 6
19.e even 9 1 361.2.a.h 3
19.e even 9 2 361.2.c.h 6
19.e even 9 1 361.2.e.a 6
19.e even 9 1 inner 361.2.e.b 6
19.e even 9 1 361.2.e.h 6
19.f odd 18 1 19.2.e.a 6
19.f odd 18 1 361.2.a.g 3
19.f odd 18 2 361.2.c.i 6
19.f odd 18 1 361.2.e.f 6
19.f odd 18 1 361.2.e.g 6
57.f even 6 1 171.2.u.c 6
57.j even 18 1 171.2.u.c 6
57.j even 18 1 3249.2.a.z 3
57.l odd 18 1 3249.2.a.s 3
76.f even 6 1 304.2.u.b 6
76.k even 18 1 304.2.u.b 6
76.k even 18 1 5776.2.a.br 3
76.l odd 18 1 5776.2.a.bi 3
95.h odd 6 1 475.2.l.a 6
95.l even 12 2 475.2.u.a 12
95.o odd 18 1 475.2.l.a 6
95.o odd 18 1 9025.2.a.bd 3
95.p even 18 1 9025.2.a.x 3
95.r even 36 2 475.2.u.a 12
133.i even 6 1 931.2.v.a 6
133.j odd 6 1 931.2.v.b 6
133.n odd 6 1 931.2.x.a 6
133.p even 6 1 931.2.w.a 6
133.s even 6 1 931.2.x.b 6
133.ba even 18 1 931.2.w.a 6
133.bb even 18 1 931.2.x.b 6
133.bd odd 18 1 931.2.x.a 6
133.be odd 18 1 931.2.v.b 6
133.bf even 18 1 931.2.v.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 19.d odd 6 1
19.2.e.a 6 19.f odd 18 1
171.2.u.c 6 57.f even 6 1
171.2.u.c 6 57.j even 18 1
304.2.u.b 6 76.f even 6 1
304.2.u.b 6 76.k even 18 1
361.2.a.g 3 19.f odd 18 1
361.2.a.h 3 19.e even 9 1
361.2.c.h 6 19.e even 9 2
361.2.c.i 6 19.f odd 18 2
361.2.e.a 6 19.c even 3 1
361.2.e.a 6 19.e even 9 1
361.2.e.b 6 1.a even 1 1 trivial
361.2.e.b 6 19.e even 9 1 inner
361.2.e.f 6 19.b odd 2 1
361.2.e.f 6 19.f odd 18 1
361.2.e.g 6 19.d odd 6 1
361.2.e.g 6 19.f odd 18 1
361.2.e.h 6 19.c even 3 1
361.2.e.h 6 19.e even 9 1
475.2.l.a 6 95.h odd 6 1
475.2.l.a 6 95.o odd 18 1
475.2.u.a 12 95.l even 12 2
475.2.u.a 12 95.r even 36 2
931.2.v.a 6 133.i even 6 1
931.2.v.a 6 133.bf even 18 1
931.2.v.b 6 133.j odd 6 1
931.2.v.b 6 133.be odd 18 1
931.2.w.a 6 133.p even 6 1
931.2.w.a 6 133.ba even 18 1
931.2.x.a 6 133.n odd 6 1
931.2.x.a 6 133.bd odd 18 1
931.2.x.b 6 133.s even 6 1
931.2.x.b 6 133.bb even 18 1
3249.2.a.s 3 57.l odd 18 1
3249.2.a.z 3 57.j even 18 1
5776.2.a.bi 3 76.l odd 18 1
5776.2.a.br 3 76.k even 18 1
9025.2.a.x 3 95.p even 18 1
9025.2.a.bd 3 95.o odd 18 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(361, [\chi])$$:

 $$T_{2}^{6} + 3 T_{2}^{5} + 9 T_{2}^{4} + 24 T_{2}^{3} + 36 T_{2}^{2} + 27 T_{2} + 9$$ $$T_{3}^{6} - 3 T_{3}^{5} + 12 T_{3}^{4} - 19 T_{3}^{3} + 15 T_{3}^{2} - 6 T_{3} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 27 T + 36 T^{2} + 24 T^{3} + 9 T^{4} + 3 T^{5} + T^{6}$$
$3$ $$1 - 6 T + 15 T^{2} - 19 T^{3} + 12 T^{4} - 3 T^{5} + T^{6}$$
$5$ $$9 - 27 T + 36 T^{2} - 24 T^{3} + 9 T^{4} - 3 T^{5} + T^{6}$$
$7$ $$1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6}$$
$11$ $$81 - 81 T + 81 T^{2} - 18 T^{3} + 9 T^{4} + T^{6}$$
$13$ $$1369 + 1443 T + 885 T^{2} + 352 T^{3} + 96 T^{4} + 15 T^{5} + T^{6}$$
$17$ $$9 + 27 T + 9 T^{2} - 24 T^{3} + 18 T^{4} - 3 T^{5} + T^{6}$$
$19$ $$T^{6}$$
$23$ $$576 + 144 T^{2} + 24 T^{3} - 6 T^{5} + T^{6}$$
$29$ $$12321 - 2997 T + 522 T^{2} + 84 T^{3} + 6 T^{5} + T^{6}$$
$31$ $$2809 - 318 T + 513 T^{2} + 160 T^{3} + 75 T^{4} + 9 T^{5} + T^{6}$$
$37$ $$( 17 - 21 T + T^{3} )^{2}$$
$41$ $$12321 + 4995 T + 2412 T^{2} + 483 T^{3} - 9 T^{4} - 6 T^{5} + T^{6}$$
$43$ $$26569 - 18093 T + 6672 T^{2} - 1531 T^{3} + 249 T^{4} - 24 T^{5} + T^{6}$$
$47$ $$9 - 27 T + 63 T^{2} - 84 T^{3} + 72 T^{4} - 15 T^{5} + T^{6}$$
$53$ $$2601 - 3213 T + 1764 T^{2} - 537 T^{3} + 99 T^{4} - 12 T^{5} + T^{6}$$
$59$ $$71289 + 4806 T + 3006 T^{2} + 699 T^{3} - 18 T^{4} - 6 T^{5} + T^{6}$$
$61$ $$32761 - 21720 T + 7500 T^{2} - 1765 T^{3} + 276 T^{4} - 24 T^{5} + T^{6}$$
$67$ $$179776 - 40704 T + 13296 T^{2} - 1448 T^{3} - 48 T^{4} + 6 T^{5} + T^{6}$$
$71$ $$788544 + 63936 T - 7632 T^{2} + 456 T^{3} + 144 T^{4} - 6 T^{5} + T^{6}$$
$73$ $$4096 - 3072 T + 3072 T^{2} - 1216 T^{3} + 240 T^{4} - 24 T^{5} + T^{6}$$
$79$ $$654481 + 259689 T + 44382 T^{2} + 4528 T^{3} + 366 T^{4} + 24 T^{5} + T^{6}$$
$83$ $$210681 + 86751 T + 35721 T^{2} + 918 T^{3} + 189 T^{4} + T^{6}$$
$89$ $$3249 + 513 T - 504 T^{2} + 84 T^{3} + 99 T^{4} - 3 T^{5} + T^{6}$$
$97$ $$16129 - 16002 T + 7236 T^{2} - 1061 T^{3} + 144 T^{4} - 18 T^{5} + T^{6}$$