# Properties

 Label 363.3.b.m Level $363$ Weight $3$ Character orbit 363.b Analytic conductor $9.891$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 363.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.89103359628$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 29 x^{6} + 282 x^{4} + 1061 x^{2} + 1331$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{2} + \beta_{4} + \beta_{5} ) q^{3} + ( -4 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{4} + ( \beta_{1} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{5} + ( 2 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{6} + ( 5 + 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{7} ) q^{8} + ( 3 + 2 \beta_{2} + \beta_{4} + 4 \beta_{5} + 3 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{2} + \beta_{4} + \beta_{5} ) q^{3} + ( -4 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{4} + ( \beta_{1} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{5} + ( 2 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{6} + ( 5 + 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{7} ) q^{8} + ( 3 + 2 \beta_{2} + \beta_{4} + 4 \beta_{5} + 3 \beta_{7} ) q^{9} + ( 2 + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{10} + ( 11 - \beta_{1} + 7 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{12} + ( 5 + 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} - 3 \beta_{7} ) q^{13} + ( 3 \beta_{1} + 5 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{14} + ( -4 + 2 \beta_{1} + 5 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{15} + ( 5 + 11 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{16} + ( 5 \beta_{1} + 2 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} ) q^{17} + ( -4 - \beta_{1} - 6 \beta_{2} + 10 \beta_{3} + \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} ) q^{18} + ( 5 - 9 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{19} + ( -2 \beta_{1} + 7 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{20} + ( -6 - 3 \beta_{1} - 9 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} ) q^{21} + ( 2 \beta_{1} - \beta_{3} + 6 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} ) q^{23} + ( -1 + 8 \beta_{1} - 7 \beta_{2} + 10 \beta_{3} - \beta_{4} - 8 \beta_{5} + 6 \beta_{6} ) q^{24} + ( 13 + 15 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{25} + ( 11 \beta_{1} - 7 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - 9 \beta_{7} ) q^{26} + ( -6 - 9 \beta_{1} - 7 \beta_{2} + \beta_{4} + 4 \beta_{5} + 3 \beta_{7} ) q^{27} + ( -30 - 23 \beta_{2} + 10 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} ) q^{28} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{29} + ( -5 - 8 \beta_{1} - 13 \beta_{2} + 8 \beta_{3} - 3 \beta_{4} - 8 \beta_{5} + 9 \beta_{6} + 3 \beta_{7} ) q^{30} + ( -6 - 15 \beta_{2} + 5 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} ) q^{31} + ( \beta_{1} + 9 \beta_{3} - 8 \beta_{4} - 12 \beta_{5} + 8 \beta_{6} + \beta_{7} ) q^{32} + ( -21 - 26 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{34} + ( -3 \beta_{1} + \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{35} + ( -16 - 10 \beta_{1} - 23 \beta_{2} + \beta_{3} + 6 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} - 9 \beta_{7} ) q^{36} + ( 21 + 16 \beta_{2} + 9 \beta_{3} - 9 \beta_{5} - 9 \beta_{7} ) q^{37} + ( 7 \beta_{1} - 4 \beta_{3} + 11 \beta_{4} + 16 \beta_{5} - 11 \beta_{6} + 7 \beta_{7} ) q^{38} + ( 10 + \beta_{1} + 15 \beta_{2} - \beta_{3} + 11 \beta_{4} + 12 \beta_{5} - 9 \beta_{6} ) q^{39} + ( -7 - 27 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} ) q^{40} + ( -10 \beta_{1} + 6 \beta_{3} + 13 \beta_{4} + 14 \beta_{5} - 13 \beta_{6} + 19 \beta_{7} ) q^{41} + ( 24 - 12 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 9 \beta_{6} - 6 \beta_{7} ) q^{42} + ( -15 - 6 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} - 7 \beta_{5} - 5 \beta_{6} - 7 \beta_{7} ) q^{43} + ( 2 - 7 \beta_{1} + 14 \beta_{2} + 7 \beta_{3} - 10 \beta_{4} - 11 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} ) q^{45} + ( -33 + 11 \beta_{2} + 11 \beta_{3} + 11 \beta_{4} + 11 \beta_{6} ) q^{46} + ( 23 \beta_{1} + 4 \beta_{3} - 18 \beta_{4} - 15 \beta_{5} + 18 \beta_{6} - 14 \beta_{7} ) q^{47} + ( -45 - 6 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} - 15 \beta_{5} + 3 \beta_{6} - 9 \beta_{7} ) q^{48} + ( -8 + 12 \beta_{2} - 9 \beta_{3} - 4 \beta_{4} + 5 \beta_{5} - 4 \beta_{6} + 5 \beta_{7} ) q^{49} + ( 9 \beta_{1} + 17 \beta_{3} - 16 \beta_{4} - 20 \beta_{5} + 16 \beta_{6} + \beta_{7} ) q^{50} + ( 13 + 4 \beta_{1} + 14 \beta_{2} - \beta_{3} - 9 \beta_{4} - 11 \beta_{5} + 9 \beta_{6} - 15 \beta_{7} ) q^{51} + ( 2 + 25 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} + 7 \beta_{5} + 5 \beta_{6} + 7 \beta_{7} ) q^{52} + ( 7 \beta_{1} + 6 \beta_{3} - 6 \beta_{4} - 15 \beta_{5} + 6 \beta_{6} ) q^{53} + ( 68 - 10 \beta_{1} + 3 \beta_{2} - 8 \beta_{3} + \beta_{4} + 3 \beta_{5} - 12 \beta_{6} + 3 \beta_{7} ) q^{54} + ( -18 \beta_{1} - 23 \beta_{3} + 10 \beta_{4} - 4 \beta_{5} - 10 \beta_{6} - 13 \beta_{7} ) q^{56} + ( -15 + 12 \beta_{1} - 28 \beta_{2} - 12 \beta_{3} + \beta_{4} + 7 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} ) q^{57} + ( -18 - 16 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{58} + ( 11 \beta_{1} + 7 \beta_{3} + 11 \beta_{4} + 12 \beta_{5} - 11 \beta_{6} + 18 \beta_{7} ) q^{59} + ( 23 - 7 \beta_{1} - 15 \beta_{2} - 5 \beta_{3} - 5 \beta_{4} - 9 \beta_{5} - 9 \beta_{6} - 12 \beta_{7} ) q^{60} + ( 15 + 17 \beta_{2} - 13 \beta_{3} - 11 \beta_{4} + 2 \beta_{5} - 11 \beta_{6} + 2 \beta_{7} ) q^{61} + ( -26 \beta_{1} - 5 \beta_{3} + 10 \beta_{4} - 10 \beta_{5} - 10 \beta_{6} + 5 \beta_{7} ) q^{62} + ( 33 + 6 \beta_{1} + 3 \beta_{3} - 6 \beta_{4} + 9 \beta_{5} + 9 \beta_{7} ) q^{63} + ( -15 - 14 \beta_{2} - 4 \beta_{3} - 14 \beta_{4} - 10 \beta_{5} - 14 \beta_{6} - 10 \beta_{7} ) q^{64} + ( 13 \beta_{1} - 15 \beta_{3} + 12 \beta_{4} + 13 \beta_{5} - 12 \beta_{6} - 3 \beta_{7} ) q^{65} + ( 17 - 21 \beta_{2} - 15 \beta_{3} - 12 \beta_{4} + 3 \beta_{5} - 12 \beta_{6} + 3 \beta_{7} ) q^{67} + ( -11 \beta_{1} - 15 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} - 12 \beta_{7} ) q^{68} + ( -11 - 17 \beta_{1} + 11 \beta_{2} - 19 \beta_{3} + 15 \beta_{4} + 25 \beta_{5} - 15 \beta_{6} + 18 \beta_{7} ) q^{69} + ( 44 - 11 \beta_{2} - 8 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} ) q^{70} + ( 3 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - 11 \beta_{5} + 2 \beta_{6} - 5 \beta_{7} ) q^{71} + ( 80 - 10 \beta_{1} + 32 \beta_{2} - 26 \beta_{3} + 11 \beta_{4} + 7 \beta_{5} - 12 \beta_{6} - 3 \beta_{7} ) q^{72} + ( -56 - 18 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{73} + ( 39 \beta_{1} - 11 \beta_{3} - 16 \beta_{4} - 7 \beta_{5} + 16 \beta_{6} - 27 \beta_{7} ) q^{74} + ( -9 - 18 \beta_{1} + 18 \beta_{3} + 12 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{75} + ( -59 - 6 \beta_{2} + 5 \beta_{3} + 11 \beta_{4} + 6 \beta_{5} + 11 \beta_{6} + 6 \beta_{7} ) q^{76} + ( -8 + 16 \beta_{1} + 43 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 7 \beta_{5} + 27 \beta_{6} - 6 \beta_{7} ) q^{78} + ( -1 + 48 \beta_{2} - 17 \beta_{3} - 8 \beta_{4} + 9 \beta_{5} - 8 \beta_{6} + 9 \beta_{7} ) q^{79} + ( -21 \beta_{1} - 2 \beta_{3} + 8 \beta_{4} - 9 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} ) q^{80} + ( -15 + 9 \beta_{1} - 34 \beta_{2} + 9 \beta_{3} - 17 \beta_{4} + 4 \beta_{5} + 3 \beta_{7} ) q^{81} + ( -37 + 5 \beta_{2} + 34 \beta_{3} + 10 \beta_{4} - 24 \beta_{5} + 10 \beta_{6} - 24 \beta_{7} ) q^{82} + ( -12 \beta_{1} + 6 \beta_{3} + 7 \beta_{4} + 10 \beta_{5} - 7 \beta_{6} + 13 \beta_{7} ) q^{83} + ( 93 + 18 \beta_{1} + 57 \beta_{2} - 18 \beta_{3} - 15 \beta_{4} + 18 \beta_{5} - 15 \beta_{6} + 15 \beta_{7} ) q^{84} + ( -7 + 28 \beta_{2} - \beta_{3} - 8 \beta_{4} - 7 \beta_{5} - 8 \beta_{6} - 7 \beta_{7} ) q^{85} + ( 9 \beta_{1} - 22 \beta_{3} + 11 \beta_{4} + 33 \beta_{5} - 11 \beta_{6} - 11 \beta_{7} ) q^{86} + ( 10 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 6 \beta_{7} ) q^{87} + ( 16 \beta_{1} + 6 \beta_{3} - 5 \beta_{4} - 8 \beta_{5} + 5 \beta_{6} + \beta_{7} ) q^{89} + ( 52 + 10 \beta_{1} - 43 \beta_{2} + 8 \beta_{3} - 21 \beta_{4} - 11 \beta_{5} + 3 \beta_{6} - 9 \beta_{7} ) q^{90} + ( -7 + 28 \beta_{2} + 7 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{91} + ( -47 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - 15 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{92} + ( 45 - 19 \beta_{2} - 11 \beta_{4} + 4 \beta_{5} + 15 \beta_{6} + 15 \beta_{7} ) q^{93} + ( -95 - 82 \beta_{2} - \beta_{3} + 12 \beta_{4} + 13 \beta_{5} + 12 \beta_{6} + 13 \beta_{7} ) q^{94} + ( 19 \beta_{1} - 18 \beta_{3} + 4 \beta_{4} + 27 \beta_{5} - 4 \beta_{6} - 14 \beta_{7} ) q^{95} + ( 41 - 4 \beta_{1} - 10 \beta_{2} - 5 \beta_{3} - 10 \beta_{4} - 23 \beta_{5} - 3 \beta_{6} - 24 \beta_{7} ) q^{96} + ( -41 + 5 \beta_{2} + 10 \beta_{3} + 8 \beta_{4} - 2 \beta_{5} + 8 \beta_{6} - 2 \beta_{7} ) q^{97} + ( -10 \beta_{1} + 31 \beta_{3} - 8 \beta_{4} - \beta_{5} + 8 \beta_{6} + 23 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 5 q^{3} - 26 q^{4} + q^{6} + 28 q^{7} + 11 q^{9} + O(q^{10})$$ $$8 q + 5 q^{3} - 26 q^{4} + q^{6} + 28 q^{7} + 11 q^{9} - 6 q^{10} + 53 q^{12} + 44 q^{13} - 54 q^{15} - 14 q^{16} + q^{18} + 68 q^{19} - 6 q^{21} + 33 q^{24} + 42 q^{25} - 25 q^{27} - 118 q^{28} + 10 q^{30} + 2 q^{31} - 66 q^{34} - 7 q^{36} + 140 q^{37} + 38 q^{39} + 58 q^{40} + 174 q^{42} - 78 q^{43} - 36 q^{45} - 286 q^{46} - 285 q^{48} - 140 q^{49} + 58 q^{51} - 102 q^{52} + 523 q^{54} - 22 q^{57} - 68 q^{58} + 262 q^{60} + 22 q^{61} + 246 q^{63} - 52 q^{64} + 184 q^{67} - 176 q^{69} + 374 q^{70} + 489 q^{72} - 378 q^{73} - 33 q^{75} - 450 q^{76} - 246 q^{78} - 252 q^{79} + 11 q^{81} - 200 q^{82} + 450 q^{84} - 156 q^{85} + 66 q^{87} + 598 q^{90} - 148 q^{91} + 380 q^{93} - 460 q^{94} + 399 q^{96} - 324 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 29 x^{6} + 282 x^{4} + 1061 x^{2} + 1331$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 30 \nu^{4} + 260 \nu^{2} + 541$$$$)/52$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} + 22 \nu^{6} - 47 \nu^{5} + 517 \nu^{4} - 377 \nu^{3} + 3575 \nu^{2} - 1121 \nu + 6897$$$$)/286$$ $$\beta_{4}$$ $$=$$ $$($$$$-27 \nu^{7} + 11 \nu^{6} - 706 \nu^{5} + 330 \nu^{4} - 5304 \nu^{3} + 3432 \nu^{2} - 9199 \nu + 10527$$$$)/1144$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{7} + 94 \nu^{5} + 611 \nu^{3} + 812 \nu$$$$)/143$$ $$\beta_{6}$$ $$=$$ $$($$$$35 \nu^{7} - 77 \nu^{6} + 894 \nu^{5} - 1738 \nu^{4} + 6812 \nu^{3} - 10868 \nu^{2} + 13683 \nu - 17061$$$$)/1144$$ $$\beta_{7}$$ $$=$$ $$($$$$-10 \nu^{7} - 22 \nu^{6} - 235 \nu^{5} - 517 \nu^{4} - 1599 \nu^{3} - 3575 \nu^{2} - 2745 \nu - 6897$$$$)/286$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - 8$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{7} - 3 \beta_{5} - 2 \beta_{3} - 10 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} - 15 \beta_{6} + \beta_{5} - 15 \beta_{4} - 16 \beta_{3} + 23 \beta_{2} + 85$$ $$\nu^{5}$$ $$=$$ $$33 \beta_{7} + 8 \beta_{6} + 36 \beta_{5} - 8 \beta_{4} + 41 \beta_{3} + 113 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-30 \beta_{7} + 190 \beta_{6} - 30 \beta_{5} + 190 \beta_{4} + 220 \beta_{3} - 378 \beta_{2} - 1011$$ $$\nu^{7}$$ $$=$$ $$-470 \beta_{7} - 188 \beta_{6} - 352 \beta_{5} + 188 \beta_{4} - 658 \beta_{3} - 1331 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
122.1
 − 3.58727i − 3.06025i − 2.00420i − 1.65816i 1.65816i 2.00420i 3.06025i 3.58727i
3.58727i −2.74329 + 1.21424i −8.86854 2.08639i 4.35581 + 9.84093i 8.86303 17.4648i 6.05125 6.66201i 7.48447
122.2 3.06025i 0.126437 2.99733i −5.36516 6.63932i −9.17261 0.386930i −3.06298 4.17774i −8.96803 0.757950i −20.3180
122.3 2.00420i 2.80061 + 1.07544i −0.0168066 5.48438i 2.15539 5.61298i 5.59084 7.98311i 6.68687 + 6.02376i 10.9918
122.4 1.65816i 2.31624 + 1.90658i 1.25051 0.698503i 3.16141 3.84069i 2.60911 8.70618i 1.72990 + 8.83218i −1.15823
122.5 1.65816i 2.31624 1.90658i 1.25051 0.698503i 3.16141 + 3.84069i 2.60911 8.70618i 1.72990 8.83218i −1.15823
122.6 2.00420i 2.80061 1.07544i −0.0168066 5.48438i 2.15539 + 5.61298i 5.59084 7.98311i 6.68687 6.02376i 10.9918
122.7 3.06025i 0.126437 + 2.99733i −5.36516 6.63932i −9.17261 + 0.386930i −3.06298 4.17774i −8.96803 + 0.757950i −20.3180
122.8 3.58727i −2.74329 1.21424i −8.86854 2.08639i 4.35581 9.84093i 8.86303 17.4648i 6.05125 + 6.66201i 7.48447
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 122.8 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.m 8
3.b odd 2 1 inner 363.3.b.m 8
11.b odd 2 1 363.3.b.l 8
11.c even 5 2 33.3.h.b 16
11.c even 5 2 363.3.h.o 16
11.d odd 10 2 363.3.h.j 16
11.d odd 10 2 363.3.h.n 16
33.d even 2 1 363.3.b.l 8
33.f even 10 2 363.3.h.j 16
33.f even 10 2 363.3.h.n 16
33.h odd 10 2 33.3.h.b 16
33.h odd 10 2 363.3.h.o 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.h.b 16 11.c even 5 2
33.3.h.b 16 33.h odd 10 2
363.3.b.l 8 11.b odd 2 1
363.3.b.l 8 33.d even 2 1
363.3.b.m 8 1.a even 1 1 trivial
363.3.b.m 8 3.b odd 2 1 inner
363.3.h.j 16 11.d odd 10 2
363.3.h.j 16 33.f even 10 2
363.3.h.n 16 11.d odd 10 2
363.3.h.n 16 33.f even 10 2
363.3.h.o 16 11.c even 5 2
363.3.h.o 16 33.h odd 10 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}^{8} + 29 T_{2}^{6} + 282 T_{2}^{4} + 1061 T_{2}^{2} + 1331$$ $$T_{5}^{8} + 79 T_{5}^{6} + 1687 T_{5}^{4} + 6576 T_{5}^{2} + 2816$$ $$T_{7}^{4} - 14 T_{7}^{3} + 35 T_{7}^{2} + 138 T_{7} - 396$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1331 + 1061 T^{2} + 282 T^{4} + 29 T^{6} + T^{8}$$
$3$ $$6561 - 3645 T + 567 T^{2} + 135 T^{3} - 72 T^{4} + 15 T^{5} + 7 T^{6} - 5 T^{7} + T^{8}$$
$5$ $$2816 + 6576 T^{2} + 1687 T^{4} + 79 T^{6} + T^{8}$$
$7$ $$( -396 + 138 T + 35 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$11$ $$T^{8}$$
$13$ $$( 4444 + 2626 T - 85 T^{2} - 22 T^{3} + T^{4} )^{2}$$
$17$ $$8306771 + 12909824 T^{2} + 172761 T^{4} + 734 T^{6} + T^{8}$$
$19$ $$( -64999 + 8694 T + 7 T^{2} - 34 T^{3} + T^{4} )^{2}$$
$23$ $$25255373616 + 307285308 T^{2} + 1188583 T^{4} + 1848 T^{6} + T^{8}$$
$29$ $$340736 + 93696 T^{2} + 7312 T^{4} + 184 T^{6} + T^{8}$$
$31$ $$( -97204 - 30786 T - 1709 T^{2} - T^{3} + T^{4} )^{2}$$
$37$ $$( -698036 + 107680 T - 1111 T^{2} - 70 T^{3} + T^{4} )^{2}$$
$41$ $$3927187055891 + 26651833143 T^{2} + 27887368 T^{4} + 9433 T^{6} + T^{8}$$
$43$ $$( -684409 - 73219 T - 1528 T^{2} + 39 T^{3} + T^{4} )^{2}$$
$47$ $$14778028911536 + 51690193292 T^{2} + 40526643 T^{4} + 11237 T^{6} + T^{8}$$
$53$ $$1733384590256 + 8491109372 T^{2} + 11081859 T^{4} + 5633 T^{6} + T^{8}$$
$59$ $$162457590843251 + 315187108454 T^{2} + 162295647 T^{4} + 23636 T^{6} + T^{8}$$
$61$ $$( 1180476 - 62286 T - 4549 T^{2} - 11 T^{3} + T^{4} )^{2}$$
$67$ $$( -11977619 + 552568 T - 3927 T^{2} - 92 T^{3} + T^{4} )^{2}$$
$71$ $$1657701936 + 269041212 T^{2} + 2752883 T^{4} + 3227 T^{6} + T^{8}$$
$73$ $$( 2336356 + 316506 T + 12357 T^{2} + 189 T^{3} + T^{4} )^{2}$$
$79$ $$( -27267284 - 928896 T - 3903 T^{2} + 126 T^{3} + T^{4} )^{2}$$
$83$ $$46820268891 + 892608003 T^{2} + 4637264 T^{4} + 4757 T^{6} + T^{8}$$
$89$ $$3248250664131 + 15048846351 T^{2} + 19103392 T^{4} + 8529 T^{6} + T^{8}$$
$97$ $$( -3010169 + 18612 T + 7293 T^{2} + 162 T^{3} + T^{4} )^{2}$$