# Properties

 Label 405.4.b.e Level $405$ Weight $4$ Character orbit 405.b Analytic conductor $23.896$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 405.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.8957735523$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 91 x^{14} + 3268 x^{12} + 59128 x^{10} + 571975 x^{8} + 2881141 x^{6} + 6555196 x^{4} + 4069504 x^{2} + 614656$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{12}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -3 + \beta_{2} ) q^{4} + \beta_{9} q^{5} + ( -\beta_{1} - \beta_{13} ) q^{7} + ( -3 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -3 + \beta_{2} ) q^{4} + \beta_{9} q^{5} + ( -\beta_{1} - \beta_{13} ) q^{7} + ( -3 \beta_{1} + \beta_{3} ) q^{8} + ( -1 - \beta_{10} ) q^{10} + ( -5 + \beta_{2} + \beta_{5} ) q^{11} + ( 2 \beta_{1} - \beta_{3} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{13} + ( 5 - 2 \beta_{2} + \beta_{7} - \beta_{10} - \beta_{12} ) q^{14} + ( 8 - 3 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} ) q^{16} + ( \beta_{1} - \beta_{3} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{17} + ( -\beta_{2} + \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{19} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{12} + 3 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{20} + ( -11 \beta_{1} + 3 \beta_{3} - 3 \beta_{8} - 3 \beta_{9} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{22} + ( -\beta_{3} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - \beta_{14} + \beta_{15} ) q^{23} + ( -4 + 2 \beta_{1} - 3 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{25} + ( -30 + 6 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} - 3 \beta_{10} - 3 \beta_{12} ) q^{26} + ( 19 \beta_{1} - 2 \beta_{3} - 6 \beta_{8} - 6 \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{28} + ( 28 - 7 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{10} - 2 \beta_{12} ) q^{29} + ( 1 - 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} - \beta_{7} - 4 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} ) q^{31} + ( 2 \beta_{1} - \beta_{3} + \beta_{8} + \beta_{9} + \beta_{10} - 5 \beta_{11} - \beta_{12} + 3 \beta_{14} - \beta_{15} ) q^{32} + ( -7 + 5 \beta_{2} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - 9 \beta_{8} + 9 \beta_{9} ) q^{34} + ( -12 + 4 \beta_{1} + 10 \beta_{2} - \beta_{3} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} + 4 \beta_{14} ) q^{35} + ( -23 \beta_{1} + 6 \beta_{3} - 6 \beta_{8} - 6 \beta_{9} - \beta_{10} - 7 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{37} + ( -4 \beta_{1} - 2 \beta_{3} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - 9 \beta_{13} - \beta_{15} ) q^{38} + ( 4 - 7 \beta_{1} - 2 \beta_{3} + \beta_{4} + 5 \beta_{5} + 3 \beta_{6} - 13 \beta_{8} + \beta_{9} + 4 \beta_{10} - \beta_{11} + 3 \beta_{12} + 2 \beta_{13} - 5 \beta_{14} + \beta_{15} ) q^{40} + ( -38 - 2 \beta_{2} + 4 \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{9} ) q^{41} + ( 35 \beta_{1} + \beta_{3} - 9 \beta_{8} - 9 \beta_{9} - \beta_{10} - 3 \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{43} + ( 94 - 29 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} + 3 \beta_{10} + 3 \beta_{12} ) q^{44} + ( -4 + 6 \beta_{2} - 3 \beta_{4} + 7 \beta_{5} + 5 \beta_{6} + 7 \beta_{7} - 18 \beta_{8} + 18 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} ) q^{46} + ( -2 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 5 \beta_{11} - 3 \beta_{12} + 6 \beta_{13} + 8 \beta_{14} ) q^{47} + ( 5 - 3 \beta_{2} + 2 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - 11 \beta_{8} + 11 \beta_{9} ) q^{49} + ( -25 + 2 \beta_{1} + 25 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} + 9 \beta_{11} - \beta_{12} + 9 \beta_{13} - 3 \beta_{14} ) q^{50} + ( -46 \beta_{1} + 9 \beta_{3} - 18 \beta_{8} - 18 \beta_{9} + \beta_{10} + 12 \beta_{11} - \beta_{12} + 8 \beta_{13} - 18 \beta_{14} ) q^{52} + ( -\beta_{1} - 3 \beta_{3} + \beta_{8} + \beta_{9} + 2 \beta_{10} - 5 \beta_{11} - 2 \beta_{12} - 12 \beta_{13} + 5 \beta_{14} + \beta_{15} ) q^{53} + ( -2 - 27 \beta_{1} - 7 \beta_{3} - \beta_{4} - 7 \beta_{5} - \beta_{6} - 2 \beta_{7} - 11 \beta_{8} - 4 \beta_{9} + 13 \beta_{11} - 6 \beta_{12} - 4 \beta_{14} - \beta_{15} ) q^{55} + ( -136 + 20 \beta_{2} - \beta_{4} + 7 \beta_{5} + 4 \beta_{6} + 9 \beta_{7} - 7 \beta_{8} + 7 \beta_{9} + 3 \beta_{10} + 3 \beta_{12} ) q^{56} + ( 92 \beta_{1} - 21 \beta_{8} - 21 \beta_{9} - \beta_{10} - 5 \beta_{11} + \beta_{12} + 5 \beta_{13} - 11 \beta_{14} + 6 \beta_{15} ) q^{58} + ( 125 - 27 \beta_{2} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} ) q^{59} + ( 6 - 3 \beta_{2} + 6 \beta_{4} + \beta_{5} + 2 \beta_{6} - 8 \beta_{7} - 9 \beta_{8} + 9 \beta_{9} + 5 \beta_{10} + 5 \beta_{12} ) q^{61} + ( 5 \beta_{1} + 6 \beta_{3} - \beta_{8} - \beta_{9} - 4 \beta_{10} + 13 \beta_{11} + 4 \beta_{12} + 6 \beta_{13} - 31 \beta_{14} + 7 \beta_{15} ) q^{62} + ( 20 - 9 \beta_{2} - \beta_{4} - 7 \beta_{5} - \beta_{6} + 4 \beta_{7} - 20 \beta_{8} + 20 \beta_{9} - 4 \beta_{10} - 4 \beta_{12} ) q^{64} + ( -16 + 5 \beta_{1} + 30 \beta_{2} - 5 \beta_{3} - \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - \beta_{7} + 11 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} - 8 \beta_{11} - 2 \beta_{12} - 6 \beta_{13} - 2 \beta_{14} - 6 \beta_{15} ) q^{65} + ( -22 \beta_{1} - 5 \beta_{3} - 12 \beta_{8} - 12 \beta_{9} + \beta_{10} + 7 \beta_{11} - \beta_{12} - 7 \beta_{13} + 18 \beta_{14} + \beta_{15} ) q^{67} + ( -23 \beta_{1} + 7 \beta_{3} + 3 \beta_{8} + 3 \beta_{9} - 9 \beta_{10} + 11 \beta_{11} + 9 \beta_{12} - 3 \beta_{13} - \beta_{14} - 6 \beta_{15} ) q^{68} + ( -29 - 82 \beta_{1} + 15 \beta_{2} + 8 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 21 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 12 \beta_{11} + 4 \beta_{12} + 5 \beta_{13} + 16 \beta_{14} - \beta_{15} ) q^{70} + ( -191 - 2 \beta_{2} + 2 \beta_{4} - 9 \beta_{5} - 8 \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} - 3 \beta_{10} - 3 \beta_{12} ) q^{71} + ( 89 \beta_{1} + 6 \beta_{3} - 3 \beta_{8} - 3 \beta_{9} - 5 \beta_{10} + 7 \beta_{11} + 5 \beta_{12} - 7 \beta_{13} + 7 \beta_{14} - 6 \beta_{15} ) q^{73} + ( 235 - 58 \beta_{2} + \beta_{4} - 5 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + 4 \beta_{10} + 4 \beta_{12} ) q^{74} + ( -21 - 3 \beta_{2} + \beta_{4} + 5 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} - 19 \beta_{8} + 19 \beta_{9} - 3 \beta_{10} - 3 \beta_{12} ) q^{76} + ( \beta_{1} + 8 \beta_{3} - 5 \beta_{8} - 5 \beta_{9} - 3 \beta_{10} - 31 \beta_{11} + 3 \beta_{12} + 15 \beta_{13} + 2 \beta_{14} + 6 \beta_{15} ) q^{77} + ( 5 - 8 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - 9 \beta_{6} - 3 \beta_{8} + 3 \beta_{9} - 6 \beta_{10} - 6 \beta_{12} ) q^{79} + ( 95 + 4 \beta_{1} + 20 \beta_{2} + 4 \beta_{3} + 5 \beta_{4} + 7 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} + 3 \beta_{10} - 17 \beta_{11} + 18 \beta_{12} + 3 \beta_{13} + 19 \beta_{14} ) q^{80} + ( -46 \beta_{1} - 16 \beta_{3} - 21 \beta_{8} - 21 \beta_{9} + 7 \beta_{10} - 35 \beta_{11} - 7 \beta_{12} - 10 \beta_{13} + 17 \beta_{14} - \beta_{15} ) q^{82} + ( -22 \beta_{1} + 13 \beta_{3} - 11 \beta_{8} - 11 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} + 15 \beta_{13} - 19 \beta_{14} + 5 \beta_{15} ) q^{83} + ( 35 - 123 \beta_{1} - 10 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - 10 \beta_{5} - 3 \beta_{6} + 8 \beta_{8} + 4 \beta_{9} - 5 \beta_{10} - 14 \beta_{11} + 2 \beta_{12} - 12 \beta_{13} - 5 \beta_{14} - 6 \beta_{15} ) q^{85} + ( -367 + 35 \beta_{2} + \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + 9 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} + 9 \beta_{12} ) q^{86} + ( 208 \beta_{1} - 16 \beta_{3} - 18 \beta_{8} - 18 \beta_{9} + 5 \beta_{10} - 6 \beta_{11} - 5 \beta_{12} - 11 \beta_{13} + 28 \beta_{14} - 7 \beta_{15} ) q^{88} + ( 250 - 23 \beta_{2} - 11 \beta_{4} + 5 \beta_{5} + 11 \beta_{6} + 15 \beta_{7} - \beta_{8} + \beta_{9} - 3 \beta_{10} - 3 \beta_{12} ) q^{89} + ( -54 + 12 \beta_{2} + 5 \beta_{4} + 4 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} + 5 \beta_{10} + 5 \beta_{12} ) q^{91} + ( -20 \beta_{1} + \beta_{3} + 15 \beta_{8} + 15 \beta_{9} - 15 \beta_{10} + 11 \beta_{11} + 15 \beta_{12} - 21 \beta_{13} + 29 \beta_{14} ) q^{92} + ( 66 - 12 \beta_{2} - \beta_{4} - 15 \beta_{5} - 2 \beta_{6} - 13 \beta_{7} - 17 \beta_{8} + 17 \beta_{9} + 3 \beta_{10} + 3 \beta_{12} ) q^{94} + ( 117 + 26 \beta_{1} + 25 \beta_{2} - 14 \beta_{3} + 9 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 3 \beta_{8} - \beta_{9} + 5 \beta_{10} + 29 \beta_{11} - 24 \beta_{13} - 11 \beta_{14} - \beta_{15} ) q^{95} + ( 4 \beta_{1} - 25 \beta_{3} + 15 \beta_{8} + 15 \beta_{9} - 7 \beta_{10} + 22 \beta_{11} + 7 \beta_{12} - 23 \beta_{13} - 34 \beta_{14} - \beta_{15} ) q^{97} + ( 39 \beta_{1} - 11 \beta_{3} - 7 \beta_{8} - 7 \beta_{9} - 9 \beta_{10} - 7 \beta_{11} + 9 \beta_{12} + 15 \beta_{13} + 29 \beta_{14} - 6 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 54 q^{4} - 3 q^{5} + O(q^{10})$$ $$16 q - 54 q^{4} - 3 q^{5} - 10 q^{10} - 90 q^{11} + 102 q^{14} + 146 q^{16} - 4 q^{19} + 6 q^{20} - 71 q^{25} - 468 q^{26} + 516 q^{29} + 38 q^{31} - 212 q^{34} - 267 q^{35} - 44 q^{40} - 576 q^{41} + 1644 q^{44} - 290 q^{46} + 4 q^{49} - 558 q^{50} + 15 q^{55} - 2430 q^{56} + 2202 q^{59} + 20 q^{61} + 322 q^{64} - 339 q^{65} - 636 q^{70} - 2952 q^{71} + 4080 q^{74} - 396 q^{76} + 218 q^{79} + 1266 q^{80} + 704 q^{85} - 6108 q^{86} + 4074 q^{89} - 942 q^{91} + 1078 q^{94} + 1692 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 91 x^{14} + 3268 x^{12} + 59128 x^{10} + 571975 x^{8} + 2881141 x^{6} + 6555196 x^{4} + 4069504 x^{2} + 614656$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 11$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 19 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$-13 \nu^{14} - 1022 \nu^{12} - 29466 \nu^{10} - 370126 \nu^{8} - 1713581 \nu^{6} + 1226376 \nu^{4} + 22574480 \nu^{2} + 15782656$$$$)/176256$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{14} + 94 \nu^{12} + 11574 \nu^{10} + 387218 \nu^{8} + 5621959 \nu^{6} + 36266220 \nu^{4} + 85224032 \nu^{2} + 17199616$$$$)/352512$$ $$\beta_{6}$$ $$=$$ $$($$$$27 \nu^{14} + 2222 \nu^{12} + 68574 \nu^{10} + 968298 \nu^{8} + 5902643 \nu^{6} + 8366676 \nu^{4} - 30262848 \nu^{2} - 13692160$$$$)/117504$$ $$\beta_{7}$$ $$=$$ $$($$$$-73 \nu^{14} - 6194 \nu^{12} - 200394 \nu^{10} - 3084622 \nu^{8} - 22981937 \nu^{6} - 73691748 \nu^{4} - 64228384 \nu^{2} - 8294528$$$$)/176256$$ $$\beta_{8}$$ $$=$$ $$($$$$-849 \nu^{15} - 1225 \nu^{14} - 75642 \nu^{13} - 104762 \nu^{12} - 2626650 \nu^{11} - 3454794 \nu^{10} - 45044382 \nu^{9} - 55246030 \nu^{8} - 399823161 \nu^{7} - 443406929 \nu^{6} - 1749552876 \nu^{5} - 1674133020 \nu^{4} - 3094154400 \nu^{3} - 2430051904 \nu^{2} - 864930048 \nu - 833385728$$$$)/34546176$$ $$\beta_{9}$$ $$=$$ $$($$$$-849 \nu^{15} + 1225 \nu^{14} - 75642 \nu^{13} + 104762 \nu^{12} - 2626650 \nu^{11} + 3454794 \nu^{10} - 45044382 \nu^{9} + 55246030 \nu^{8} - 399823161 \nu^{7} + 443406929 \nu^{6} - 1749552876 \nu^{5} + 1674133020 \nu^{4} - 3094154400 \nu^{3} + 2430051904 \nu^{2} - 864930048 \nu + 833385728$$$$)/34546176$$ $$\beta_{10}$$ $$=$$ $$($$$$-25 \nu^{15} - 33 \nu^{14} - 2138 \nu^{13} - 3018 \nu^{12} - 70506 \nu^{11} - 105210 \nu^{10} - 1127470 \nu^{9} - 1750686 \nu^{8} - 9049121 \nu^{7} - 14215017 \nu^{6} - 34165980 \nu^{5} - 50432796 \nu^{4} - 49592896 \nu^{3} - 52858752 \nu^{2} - 17007872 \nu - 11354880$$$$)/705024$$ $$\beta_{11}$$ $$=$$ $$($$$$-101 \nu^{15} - 8407 \nu^{13} - 263820 \nu^{11} - 3850424 \nu^{9} - 25802659 \nu^{7} - 63425913 \nu^{5} - 13608404 \nu^{3} - 100002400 \nu$$$$)/2878848$$ $$\beta_{12}$$ $$=$$ $$($$$$25 \nu^{15} - 33 \nu^{14} + 2138 \nu^{13} - 3018 \nu^{12} + 70506 \nu^{11} - 105210 \nu^{10} + 1127470 \nu^{9} - 1750686 \nu^{8} + 9049121 \nu^{7} - 14215017 \nu^{6} + 34165980 \nu^{5} - 50432796 \nu^{4} + 49592896 \nu^{3} - 52858752 \nu^{2} + 17007872 \nu - 11354880$$$$)/705024$$ $$\beta_{13}$$ $$=$$ $$($$$$-895 \nu^{15} - 75908 \nu^{13} - 2465730 \nu^{11} - 38436238 \nu^{9} - 295408283 \nu^{7} - 1022927202 \nu^{5} - 1116316120 \nu^{3} + 79369792 \nu$$$$)/17273088$$ $$\beta_{14}$$ $$=$$ $$($$$$-365 \nu^{15} - 31990 \nu^{13} - 1088058 \nu^{11} - 18206894 \nu^{9} - 157923085 \nu^{7} - 689377056 \nu^{5} - 1314355088 \nu^{3} - 640762624 \nu$$$$)/5757696$$ $$\beta_{15}$$ $$=$$ $$($$$$-137 \nu^{15} - 12712 \nu^{13} - 468198 \nu^{11} - 8743514 \nu^{9} - 87674005 \nu^{7} - 455445594 \nu^{5} - 1030303640 \nu^{3} - 475010752 \nu$$$$)/1016064$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 11$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 19 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} - 27 \beta_{2} + 208$$ $$\nu^{5}$$ $$=$$ $$-\beta_{15} + 3 \beta_{14} - \beta_{12} - 5 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 33 \beta_{3} + 418 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-4 \beta_{12} - 4 \beta_{10} - 20 \beta_{9} + 20 \beta_{8} + 4 \beta_{7} - \beta_{6} + 33 \beta_{5} - 41 \beta_{4} + 687 \beta_{2} - 4588$$ $$\nu^{7}$$ $$=$$ $$45 \beta_{15} - 143 \beta_{14} + 30 \beta_{13} + 57 \beta_{12} + 211 \beta_{11} - 57 \beta_{10} - 39 \beta_{9} - 39 \beta_{8} + 918 \beta_{3} - 9830 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$200 \beta_{12} + 200 \beta_{10} + 235 \beta_{9} - 235 \beta_{8} - 212 \beta_{7} + 29 \beta_{6} - 912 \beta_{5} + 1288 \beta_{4} - 17333 \beta_{2} + 108335$$ $$\nu^{9}$$ $$=$$ $$-1476 \beta_{15} + 4732 \beta_{14} - 1518 \beta_{13} - 2112 \beta_{12} - 6662 \beta_{11} + 2112 \beta_{10} + 1254 \beta_{9} + 1254 \beta_{8} - 24314 \beta_{3} + 238799 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-6868 \beta_{12} - 6868 \beta_{10} + 212 \beta_{9} - 212 \beta_{8} + 7672 \beta_{7} - 508 \beta_{6} + 24260 \beta_{5} - 36676 \beta_{4} + 437433 \beta_{2} - 2640943$$ $$\nu^{11}$$ $$=$$ $$42740 \beta_{15} - 136604 \beta_{14} + 52824 \beta_{13} + 66188 \beta_{12} + 189356 \beta_{11} - 66188 \beta_{10} - 36404 \beta_{9} - 36404 \beta_{8} + 631557 \beta_{3} - 5902367 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$202384 \beta_{12} + 202384 \beta_{10} - 125815 \beta_{9} + 125815 \beta_{8} - 237952 \beta_{7} + 4228 \beta_{6} - 637341 \beta_{5} + 996029 \beta_{4} - 11058627 \beta_{2} + 65437500$$ $$\nu^{13}$$ $$=$$ $$-1162845 \beta_{15} + 3697319 \beta_{14} - 1577928 \beta_{13} - 1911261 \beta_{12} - 5122729 \beta_{11} + 1911261 \beta_{10} + 983541 \beta_{9} + 983541 \beta_{8} - 16250517 \beta_{3} + 147362278 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-5510372 \beta_{12} - 5510372 \beta_{10} + 5450348 \beta_{9} - 5450348 \beta_{8} + 6825860 \beta_{7} + 125203 \beta_{6} + 16638285 \beta_{5} - 26358613 \beta_{4} + 280010935 \beta_{2} - 1636326120$$ $$\nu^{15}$$ $$=$$ $$30553497 \beta_{15} - 96685123 \beta_{14} + 43569078 \beta_{13} + 52782405 \beta_{12} + 134973239 \beta_{11} - 52782405 \beta_{10} - 25248315 \beta_{9} - 25248315 \beta_{8} + 415965778 \beta_{3} - 3702009370 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$$.

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
244.1
 − 5.05435i − 5.02371i − 4.07626i − 3.08740i − 2.67506i − 2.46385i − 0.785333i − 0.473990i 0.473990i 0.785333i 2.46385i 2.67506i 3.08740i 4.07626i 5.02371i 5.05435i
5.05435i 0 −17.5465 −7.77185 + 8.03731i 0 21.0117i 48.2513i 0 40.6234 + 39.2817i
244.2 5.02371i 0 −17.2377 5.75603 9.58479i 0 5.38197i 46.4074i 0 −48.1512 28.9166i
244.3 4.07626i 0 −8.61587 10.3994 + 4.10522i 0 13.3430i 2.51043i 0 16.7339 42.3906i
244.4 3.08740i 0 −1.53204 −5.83763 9.53531i 0 31.3204i 19.9692i 0 −29.4393 + 18.0231i
244.5 2.67506i 0 0.844033 −10.9788 2.11350i 0 15.4153i 23.6584i 0 −5.65374 + 29.3689i
244.6 2.46385i 0 1.92944 −0.0773702 + 11.1801i 0 19.2401i 24.4647i 0 27.5460 + 0.190629i
244.7 0.785333i 0 7.38325 −3.61462 10.5799i 0 20.9136i 12.0810i 0 −8.30875 + 2.83868i
244.8 0.473990i 0 7.77533 10.6248 + 3.48041i 0 8.20657i 7.47735i 0 1.64968 5.03606i
244.9 0.473990i 0 7.77533 10.6248 3.48041i 0 8.20657i 7.47735i 0 1.64968 + 5.03606i
244.10 0.785333i 0 7.38325 −3.61462 + 10.5799i 0 20.9136i 12.0810i 0 −8.30875 2.83868i
244.11 2.46385i 0 1.92944 −0.0773702 11.1801i 0 19.2401i 24.4647i 0 27.5460 0.190629i
244.12 2.67506i 0 0.844033 −10.9788 + 2.11350i 0 15.4153i 23.6584i 0 −5.65374 29.3689i
244.13 3.08740i 0 −1.53204 −5.83763 + 9.53531i 0 31.3204i 19.9692i 0 −29.4393 18.0231i
244.14 4.07626i 0 −8.61587 10.3994 4.10522i 0 13.3430i 2.51043i 0 16.7339 + 42.3906i
244.15 5.02371i 0 −17.2377 5.75603 + 9.58479i 0 5.38197i 46.4074i 0 −48.1512 + 28.9166i
244.16 5.05435i 0 −17.5465 −7.77185 8.03731i 0 21.0117i 48.2513i 0 40.6234 39.2817i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 244.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.4.b.e 16
3.b odd 2 1 405.4.b.f 16
5.b even 2 1 inner 405.4.b.e 16
5.c odd 4 2 2025.4.a.bk 16
9.c even 3 2 45.4.j.a 32
9.d odd 6 2 135.4.j.a 32
15.d odd 2 1 405.4.b.f 16
15.e even 4 2 2025.4.a.bl 16
45.h odd 6 2 135.4.j.a 32
45.j even 6 2 45.4.j.a 32
45.k odd 12 4 225.4.e.g 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.j.a 32 9.c even 3 2
45.4.j.a 32 45.j even 6 2
135.4.j.a 32 9.d odd 6 2
135.4.j.a 32 45.h odd 6 2
225.4.e.g 32 45.k odd 12 4
405.4.b.e 16 1.a even 1 1 trivial
405.4.b.e 16 5.b even 2 1 inner
405.4.b.f 16 3.b odd 2 1
405.4.b.f 16 15.d odd 2 1
2025.4.a.bk 16 5.c odd 4 2
2025.4.a.bl 16 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{16} + \cdots$$ $$T_{11}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$614656 + 4069504 T^{2} + 6555196 T^{4} + 2881141 T^{6} + 571975 T^{8} + 59128 T^{10} + 3268 T^{12} + 91 T^{14} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$59604644775390625 + 1430511474609375 T + 152587890625000 T^{2} - 46783447265625 T^{3} - 1390625000000 T^{4} + 36298828125 T^{5} + 11978125000 T^{6} + 2217328125 T^{7} + 14938750 T^{8} + 17738625 T^{9} + 766600 T^{10} + 18585 T^{11} - 5696 T^{12} - 1533 T^{13} + 40 T^{14} + 3 T^{15} + T^{16}$$
$7$ $$5787171908657455716 + 390455956286750700 T^{2} + 8888477253929469 T^{4} + 93497855511042 T^{6} + 525057427827 T^{8} + 1665664668 T^{10} + 2969739 T^{12} + 2742 T^{14} + T^{16}$$
$11$ $$( 137637673416 - 26757336852 T - 809472312 T^{2} + 158752116 T^{3} + 4235328 T^{4} - 160551 T^{5} - 3975 T^{6} + 45 T^{7} + T^{8} )^{2}$$
$13$ $$53\!\cdots\!96$$$$+$$$$53\!\cdots\!48$$$$T^{2} + 78699104335949476032 T^{4} + 303501872607601680 T^{6} + 478016713374336 T^{8} + 349151939964 T^{10} + 121119660 T^{12} + 18483 T^{14} + T^{16}$$
$17$ $$18\!\cdots\!56$$$$+$$$$60\!\cdots\!56$$$$T^{2} +$$$$29\!\cdots\!64$$$$T^{4} + 33539383820058451168 T^{6} + 16572039849099712 T^{8} + 4131909482668 T^{10} + 547814617 T^{12} + 36958 T^{14} + T^{16}$$
$19$ $$( 11815095359200 + 5704507496360 T - 241429092524 T^{2} - 3427815988 T^{3} + 150069100 T^{4} + 445916 T^{5} - 24341 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$23$ $$30\!\cdots\!64$$$$+$$$$35\!\cdots\!40$$$$T^{2} +$$$$59\!\cdots\!01$$$$T^{4} +$$$$17\!\cdots\!26$$$$T^{6} + 2343990096556801783 T^{8} + 162205058882176 T^{10} + 6186573343 T^{12} + 123130 T^{14} + T^{16}$$
$29$ $$( -1696760290826316 - 389207524968612 T - 24156887205807 T^{2} - 280639491210 T^{3} + 2316962691 T^{4} + 17565264 T^{5} - 80745 T^{6} - 258 T^{7} + T^{8} )^{2}$$
$31$ $$( -127965906581806160 + 5383914360201224 T - 53884731790376 T^{2} - 311543511460 T^{3} + 5330728816 T^{4} + 5857382 T^{5} - 137654 T^{6} - 19 T^{7} + T^{8} )^{2}$$
$37$ $$12\!\cdots\!24$$$$+$$$$75\!\cdots\!00$$$$T^{2} +$$$$16\!\cdots\!00$$$$T^{4} +$$$$17\!\cdots\!92$$$$T^{6} + 95318152008399860976 T^{8} + 2923068725661888 T^{10} + 47940529104 T^{12} + 369900 T^{14} + T^{16}$$
$41$ $$( -50914609409062035 + 9102671053428924 T - 366728419206576 T^{2} + 3271477051440 T^{3} + 19704394530 T^{4} - 65723148 T^{5} - 270816 T^{6} + 288 T^{7} + T^{8} )^{2}$$
$43$ $$22\!\cdots\!56$$$$+$$$$63\!\cdots\!08$$$$T^{2} +$$$$19\!\cdots\!24$$$$T^{4} +$$$$15\!\cdots\!44$$$$T^{6} + 36326817147213144288 T^{8} + 1855032885866763 T^{10} + 37172815443 T^{12} + 322017 T^{14} + T^{16}$$
$47$ $$89\!\cdots\!44$$$$+$$$$10\!\cdots\!48$$$$T^{2} +$$$$11\!\cdots\!01$$$$T^{4} +$$$$23\!\cdots\!58$$$$T^{6} +$$$$19\!\cdots\!19$$$$T^{8} + 7114794582860476 T^{10} + 103054946131 T^{12} + 594514 T^{14} + T^{16}$$
$53$ $$45\!\cdots\!44$$$$+$$$$42\!\cdots\!08$$$$T^{2} +$$$$58\!\cdots\!68$$$$T^{4} +$$$$33\!\cdots\!04$$$$T^{6} +$$$$94\!\cdots\!72$$$$T^{8} + 14763428906730784 T^{10} + 127074664420 T^{12} + 564124 T^{14} + T^{16}$$
$59$ $$( -7915694152982015232 - 133548560466028992 T + 4567215170808288 T^{2} - 15464340977472 T^{3} - 76233486096 T^{4} + 336624651 T^{5} - 10707 T^{6} - 1101 T^{7} + T^{8} )^{2}$$
$61$ $$( -49277468581095097322 - 1204794883975463200 T - 6710949081021689 T^{2} + 20182587065930 T^{3} + 151586239273 T^{4} - 61355824 T^{5} - 750167 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$67$ $$26\!\cdots\!89$$$$+$$$$97\!\cdots\!40$$$$T^{2} +$$$$57\!\cdots\!72$$$$T^{4} +$$$$13\!\cdots\!04$$$$T^{6} +$$$$13\!\cdots\!82$$$$T^{8} + 682238458110843504 T^{10} + 1756284593724 T^{12} + 2156100 T^{14} + T^{16}$$
$71$ $$( -$$$$15\!\cdots\!24$$$$- 29482081723807632 T + 9514852456002840 T^{2} + 19250853383952 T^{3} - 121562934084 T^{4} - 336942828 T^{5} + 324162 T^{6} + 1476 T^{7} + T^{8} )^{2}$$
$73$ $$43\!\cdots\!96$$$$+$$$$16\!\cdots\!60$$$$T^{2} +$$$$24\!\cdots\!52$$$$T^{4} +$$$$18\!\cdots\!28$$$$T^{6} +$$$$80\!\cdots\!20$$$$T^{8} + 2129210402936425920 T^{10} + 3323642992497 T^{12} + 2819826 T^{14} + T^{16}$$
$79$ $$($$$$78\!\cdots\!32$$$$+ 1762270868813547680 T - 42665226765100928 T^{2} - 476180853712 T^{3} + 405337308784 T^{4} + 44693534 T^{5} - 1142846 T^{6} - 109 T^{7} + T^{8} )^{2}$$
$83$ $$15\!\cdots\!76$$$$+$$$$14\!\cdots\!84$$$$T^{2} +$$$$25\!\cdots\!77$$$$T^{4} +$$$$12\!\cdots\!34$$$$T^{6} +$$$$26\!\cdots\!39$$$$T^{8} + 282388628108496040 T^{10} + 1370593986895 T^{12} + 2411074 T^{14} + T^{16}$$
$89$ $$($$$$15\!\cdots\!00$$$$+ 9171892537616518200 T + 101208493454651010 T^{2} - 328435007142033 T^{3} - 908745918615 T^{4} + 2429925534 T^{5} - 266796 T^{6} - 2037 T^{7} + T^{8} )^{2}$$
$97$ $$17\!\cdots\!96$$$$+$$$$21\!\cdots\!68$$$$T^{2} +$$$$27\!\cdots\!20$$$$T^{4} +$$$$13\!\cdots\!08$$$$T^{6} +$$$$31\!\cdots\!24$$$$T^{8} + 39332894338327456611 T^{10} + 25767039322287 T^{12} + 8277621 T^{14} + T^{16}$$
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