# Properties

 Label 16.4.e.a Level 16 Weight 4 Character orbit 16.e Analytic conductor 0.944 Analytic rank 0 Dimension 10 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$16 = 2^{4}$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 16.e (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.944030560092$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{10}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{4} q^{2}$$ $$-\beta_{5} q^{3}$$ $$+ ( 1 - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{4}$$ $$+ ( -1 + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{5}$$ $$+ ( -3 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{6}$$ $$+ ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{7}$$ $$+ ( -6 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{8}$$ $$+ ( 1 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{4} q^{2}$$ $$-\beta_{5} q^{3}$$ $$+ ( 1 - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{4}$$ $$+ ( -1 + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{5}$$ $$+ ( -3 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{6}$$ $$+ ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{7}$$ $$+ ( -6 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{8}$$ $$+ ( 1 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{9}$$ $$+ ( -6 - 8 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{10}$$ $$+ ( 2 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - 6 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{11}$$ $$+ ( 10 + 2 \beta_{1} + 16 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{12}$$ $$+ ( 2 - \beta_{1} + \beta_{2} - 9 \beta_{4} + 4 \beta_{5} + 7 \beta_{6} - \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{13}$$ $$+ ( 20 + 2 \beta_{1} - 14 \beta_{2} - 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 4 \beta_{9} ) q^{14}$$ $$+ ( -9 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} - 14 \beta_{4} - \beta_{5} + 4 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} ) q^{15}$$ $$+ ( 28 - 6 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{16}$$ $$+ ( -3 + 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 16 \beta_{6} - 2 \beta_{8} - 5 \beta_{9} ) q^{17}$$ $$+ ( 14 - 2 \beta_{1} + 50 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 6 \beta_{9} ) q^{18}$$ $$+ ( 8 \beta_{1} + 10 \beta_{2} + 10 \beta_{4} + 3 \beta_{5} + 18 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{19}$$ $$+ ( -22 + 8 \beta_{1} - 48 \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 9 \beta_{6} - \beta_{7} - 5 \beta_{8} + 4 \beta_{9} ) q^{20}$$ $$+ ( 2 - 4 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} + 20 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} ) q^{21}$$ $$+ ( -59 - \beta_{1} + 11 \beta_{2} + \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 7 \beta_{7} + 5 \beta_{8} - 5 \beta_{9} ) q^{22}$$ $$+ ( -7 - 7 \beta_{1} - 27 \beta_{2} + 3 \beta_{3} + 12 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} - 10 \beta_{7} + \beta_{9} ) q^{23}$$ $$+ ( -80 - 2 \beta_{1} + 28 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 12 \beta_{6} + 6 \beta_{7} - 4 \beta_{8} - 6 \beta_{9} ) q^{24}$$ $$+ ( 5 \beta_{2} - 12 \beta_{3} + 16 \beta_{4} - 12 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} - 8 \beta_{9} ) q^{25}$$ $$+ ( -28 - 6 \beta_{1} - 74 \beta_{2} - 10 \beta_{3} + 5 \beta_{4} + 8 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} + 4 \beta_{9} ) q^{26}$$ $$+ ( 14 + 6 \beta_{1} + 16 \beta_{2} + 4 \beta_{3} - 6 \beta_{4} - 18 \beta_{6} - 10 \beta_{7} - 10 \beta_{8} + 8 \beta_{9} ) q^{27}$$ $$+ ( 34 + 8 \beta_{1} + 94 \beta_{2} + 10 \beta_{3} - 22 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} ) q^{28}$$ $$+ ( -16 - 3 \beta_{1} + 13 \beta_{2} - 11 \beta_{4} - 24 \beta_{5} - 11 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} ) q^{29}$$ $$+ ( 122 - 40 \beta_{2} - 6 \beta_{3} + 10 \beta_{4} + 20 \beta_{5} + 14 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} - 2 \beta_{9} ) q^{30}$$ $$+ ( 34 + 2 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} - 12 \beta_{8} + 6 \beta_{9} ) q^{31}$$ $$+ ( 96 + 4 \beta_{1} - 64 \beta_{2} - 6 \beta_{3} - 30 \beta_{4} + 6 \beta_{5} - 14 \beta_{6} - 2 \beta_{7} - 10 \beta_{8} ) q^{32}$$ $$+ ( 7 - 3 \beta_{1} - 3 \beta_{2} + 14 \beta_{3} - 2 \beta_{4} - 14 \beta_{5} + 10 \beta_{8} - 3 \beta_{9} ) q^{33}$$ $$+ ( 50 - 2 \beta_{1} + 106 \beta_{2} + 18 \beta_{3} - 12 \beta_{4} - 2 \beta_{5} + 6 \beta_{6} - 14 \beta_{7} + 10 \beta_{8} + 2 \beta_{9} ) q^{34}$$ $$+ ( 48 - 8 \beta_{1} - 52 \beta_{2} - 12 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} - 12 \beta_{7} + 12 \beta_{8} + 4 \beta_{9} ) q^{35}$$ $$+ ( -43 - 4 \beta_{1} - 131 \beta_{2} + 3 \beta_{3} - 19 \beta_{4} - 20 \beta_{5} + 36 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} ) q^{36}$$ $$+ ( -5 + 4 \beta_{1} - 10 \beta_{2} - 28 \beta_{3} - 9 \beta_{4} - 7 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{37}$$ $$+ ( -127 + \beta_{1} + 97 \beta_{2} - 23 \beta_{3} + 24 \beta_{4} - 19 \beta_{5} - 4 \beta_{6} + 17 \beta_{7} - 5 \beta_{8} + 3 \beta_{9} ) q^{38}$$ $$+ ( 11 + 11 \beta_{1} + 79 \beta_{2} + 7 \beta_{3} - 12 \beta_{4} + 7 \beta_{5} + 22 \beta_{6} + 10 \beta_{7} - 5 \beta_{9} ) q^{39}$$ $$+ ( -144 + 2 \beta_{1} + 52 \beta_{2} + 12 \beta_{3} + 16 \beta_{4} - 14 \beta_{5} - 54 \beta_{6} - 8 \beta_{7} - 2 \beta_{8} + 10 \beta_{9} ) q^{40}$$ $$+ ( 2 + 2 \beta_{1} + 6 \beta_{2} + 24 \beta_{3} - 16 \beta_{4} + 24 \beta_{5} - 12 \beta_{6} + 4 \beta_{7} + 6 \beta_{9} ) q^{41}$$ $$+ ( -76 + 4 \beta_{1} - 156 \beta_{2} - 12 \beta_{3} - 6 \beta_{4} - 12 \beta_{5} - 18 \beta_{6} - 12 \beta_{7} - 20 \beta_{8} - 4 \beta_{9} ) q^{42}$$ $$+ ( -72 - 8 \beta_{1} - 80 \beta_{2} + 3 \beta_{3} - 8 \beta_{4} + 40 \beta_{6} + 8 \beta_{7} + 8 \beta_{8} - 16 \beta_{9} ) q^{43}$$ $$+ ( 72 - 14 \beta_{1} + 134 \beta_{2} + 5 \beta_{3} + 63 \beta_{4} + 19 \beta_{5} + 27 \beta_{6} - 9 \beta_{7} - \beta_{8} + 6 \beta_{9} ) q^{44}$$ $$+ ( 2 + 5 \beta_{1} - 9 \beta_{2} + 37 \beta_{4} + 44 \beta_{5} - 11 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 12 \beta_{9} ) q^{45}$$ $$+ ( 104 - 10 \beta_{1} - 34 \beta_{2} + 20 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - 30 \beta_{6} - 12 \beta_{7} - 22 \beta_{8} + 16 \beta_{9} ) q^{46}$$ $$+ ( -102 + 10 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 36 \beta_{4} + 4 \beta_{5} - 24 \beta_{6} + 20 \beta_{8} - 2 \beta_{9} ) q^{47}$$ $$+ ( 168 + 16 \beta_{1} - 16 \beta_{2} - 26 \beta_{3} + 86 \beta_{4} - 2 \beta_{5} + 34 \beta_{6} + 10 \beta_{7} + 6 \beta_{8} + 4 \beta_{9} ) q^{48}$$ $$+ ( 1 - 10 \beta_{1} + 22 \beta_{2} - 36 \beta_{3} + 20 \beta_{4} + 36 \beta_{5} + 64 \beta_{6} - 4 \beta_{8} + 22 \beta_{9} ) q^{49}$$ $$+ ( 80 + 8 \beta_{1} + 120 \beta_{2} + 8 \beta_{4} - 24 \beta_{5} - 11 \beta_{6} + 32 \beta_{7} - 8 \beta_{8} - 32 \beta_{9} ) q^{50}$$ $$+ ( -160 - 16 \beta_{1} + 130 \beta_{2} - 22 \beta_{4} - 22 \beta_{5} - 78 \beta_{6} + 18 \beta_{7} - 18 \beta_{8} - 14 \beta_{9} ) q^{51}$$ $$+ ( -56 - 28 \beta_{1} - 122 \beta_{2} - 21 \beta_{3} + 41 \beta_{4} - \beta_{5} - 85 \beta_{6} - 3 \beta_{7} + 15 \beta_{8} - 16 \beta_{9} ) q^{52}$$ $$+ ( -5 + 12 \beta_{1} - 38 \beta_{2} + 68 \beta_{3} - 85 \beta_{4} + 37 \beta_{6} - 19 \beta_{7} - 19 \beta_{8} - 21 \beta_{9} ) q^{53}$$ $$+ ( -120 + 4 \beta_{1} + 12 \beta_{2} + 44 \beta_{3} - 36 \beta_{4} + 24 \beta_{6} - 12 \beta_{7} - 8 \beta_{8} + 24 \beta_{9} ) q^{54}$$ $$+ ( 15 + 15 \beta_{1} - 157 \beta_{2} - 25 \beta_{3} - 60 \beta_{4} - 25 \beta_{5} - 18 \beta_{6} + 18 \beta_{7} + 15 \beta_{9} ) q^{55}$$ $$+ ( -20 + 16 \beta_{1} + 60 \beta_{2} - 14 \beta_{3} - 38 \beta_{4} + 14 \beta_{5} + 114 \beta_{6} - 18 \beta_{7} + 22 \beta_{8} + 12 \beta_{9} ) q^{56}$$ $$+ ( -9 - 9 \beta_{1} - 21 \beta_{2} - 18 \beta_{3} - 64 \beta_{4} - 18 \beta_{5} - 54 \beta_{6} - 46 \beta_{7} + 41 \beta_{9} ) q^{57}$$ $$+ ( 16 + 34 \beta_{1} - 106 \beta_{2} + 30 \beta_{3} - 3 \beta_{4} + 20 \beta_{5} + 19 \beta_{6} + 18 \beta_{7} - 8 \beta_{8} - 24 \beta_{9} ) q^{58}$$ $$+ ( 164 - 36 \beta_{1} + 192 \beta_{2} - 29 \beta_{3} + 84 \beta_{4} + 60 \beta_{6} + 28 \beta_{7} + 28 \beta_{8} - 8 \beta_{9} ) q^{59}$$ $$+ ( -22 - 36 \beta_{1} + 58 \beta_{2} - 44 \beta_{3} - 96 \beta_{4} - 14 \beta_{5} - 50 \beta_{6} - 12 \beta_{7} + 18 \beta_{8} ) q^{60}$$ $$+ ( 94 + 19 \beta_{1} - 79 \beta_{2} + 67 \beta_{4} - 12 \beta_{5} + 51 \beta_{6} - 21 \beta_{7} + 21 \beta_{8} - 4 \beta_{9} ) q^{61}$$ $$+ ( 12 + 4 \beta_{1} - 68 \beta_{2} + 4 \beta_{3} - 36 \beta_{4} - 36 \beta_{5} - 4 \beta_{6} + 20 \beta_{7} + 4 \beta_{8} + 12 \beta_{9} ) q^{62}$$ $$+ ( 267 + 15 \beta_{1} - 7 \beta_{2} + 33 \beta_{3} + 26 \beta_{4} - 33 \beta_{5} - 44 \beta_{6} + 10 \beta_{8} - 7 \beta_{9} ) q^{63}$$ $$+ ( 80 - 4 \beta_{1} + 32 \beta_{2} + 28 \beta_{3} - 116 \beta_{4} - 32 \beta_{6} - 4 \beta_{7} + 40 \beta_{8} + 12 \beta_{9} ) q^{64}$$ $$+ ( -44 + 18 \beta_{1} + 26 \beta_{2} + 40 \beta_{3} + 36 \beta_{4} - 40 \beta_{5} + 16 \beta_{6} - 52 \beta_{8} + 26 \beta_{9} ) q^{65}$$ $$+ ( -58 + 18 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} + 50 \beta_{5} + 2 \beta_{6} - 10 \beta_{7} - 26 \beta_{8} - 10 \beta_{9} ) q^{66}$$ $$+ ( 176 - 202 \beta_{2} + 30 \beta_{4} - 13 \beta_{5} - 74 \beta_{6} + 22 \beta_{7} - 22 \beta_{8} - 26 \beta_{9} ) q^{67}$$ $$+ ( -78 - 4 \beta_{1} + 38 \beta_{2} - 4 \beta_{3} - 36 \beta_{4} + 66 \beta_{5} + 130 \beta_{6} - 20 \beta_{7} - 22 \beta_{8} + 20 \beta_{9} ) q^{68}$$ $$+ ( 58 - 12 \beta_{1} + 54 \beta_{2} - 52 \beta_{3} + 48 \beta_{6} + 8 \beta_{7} + 8 \beta_{8} - 16 \beta_{9} ) q^{69}$$ $$+ ( 2 - 30 \beta_{1} - 62 \beta_{2} + 2 \beta_{3} - 40 \beta_{4} + 42 \beta_{5} - 32 \beta_{6} - 14 \beta_{7} + 6 \beta_{8} + 6 \beta_{9} ) q^{70}$$ $$+ ( 1 + \beta_{1} + 333 \beta_{2} - 21 \beta_{3} - 20 \beta_{4} - 21 \beta_{5} + 10 \beta_{6} - 26 \beta_{7} + 9 \beta_{9} ) q^{71}$$ $$+ ( 70 + 24 \beta_{1} - 14 \beta_{2} - 33 \beta_{3} + 79 \beta_{4} + 45 \beta_{5} - 113 \beta_{6} + 49 \beta_{7} + \beta_{8} - 26 \beta_{9} ) q^{72}$$ $$+ ( -29 - 29 \beta_{1} + 37 \beta_{2} + 14 \beta_{3} + 64 \beta_{4} + 14 \beta_{5} - 62 \beta_{6} + 10 \beta_{7} - 3 \beta_{9} ) q^{73}$$ $$+ ( -30 - 28 \beta_{1} + 140 \beta_{2} + 32 \beta_{3} + 3 \beta_{4} - 18 \beta_{5} - \beta_{6} + 20 \beta_{7} + 38 \beta_{8} - 30 \beta_{9} ) q^{74}$$ $$+ ( -312 - 288 \beta_{2} + 13 \beta_{3} + 16 \beta_{4} - 16 \beta_{6} - 32 \beta_{7} - 32 \beta_{8} + 24 \beta_{9} ) q^{75}$$ $$+ ( -110 + 18 \beta_{1} - 104 \beta_{2} + 17 \beta_{3} + 119 \beta_{4} - 55 \beta_{5} + 53 \beta_{6} + 43 \beta_{7} - 3 \beta_{8} - 54 \beta_{9} ) q^{76}$$ $$+ ( -42 - 8 \beta_{1} + 14 \beta_{2} - 8 \beta_{4} - 12 \beta_{5} - 88 \beta_{6} + 32 \beta_{7} - 32 \beta_{8} - 20 \beta_{9} ) q^{77}$$ $$+ ( -68 + 10 \beta_{1} + 138 \beta_{2} - 56 \beta_{3} + 22 \beta_{4} + 2 \beta_{5} + 86 \beta_{6} + 8 \beta_{7} + 22 \beta_{8} + 4 \beta_{9} ) q^{78}$$ $$+ ( -452 + 12 \beta_{1} + 36 \beta_{2} - 8 \beta_{3} + 40 \beta_{4} + 8 \beta_{5} + 48 \beta_{6} - 56 \beta_{8} + 36 \beta_{9} ) q^{79}$$ $$+ ( -280 + 20 \beta_{1} - 112 \beta_{2} + 74 \beta_{3} + 90 \beta_{4} + 14 \beta_{5} + 34 \beta_{6} - 50 \beta_{7} - 50 \beta_{8} + 8 \beta_{9} ) q^{80}$$ $$+ ( 50 - 15 \beta_{1} - 15 \beta_{2} - 18 \beta_{3} + 6 \beta_{4} + 18 \beta_{5} + 66 \beta_{8} - 15 \beta_{9} ) q^{81}$$ $$+ ( -84 + 4 \beta_{1} - 228 \beta_{2} - 44 \beta_{3} - 12 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} - 60 \beta_{7} + 20 \beta_{8} + 44 \beta_{9} ) q^{82}$$ $$+ ( -240 - 24 \beta_{1} + 256 \beta_{2} - 56 \beta_{4} + 55 \beta_{5} + 104 \beta_{6} - 72 \beta_{7} + 72 \beta_{8} + 40 \beta_{9} ) q^{83}$$ $$+ ( 188 + 8 \beta_{1} + 232 \beta_{2} + 86 \beta_{3} + 42 \beta_{4} - 22 \beta_{5} - 134 \beta_{6} + 26 \beta_{7} - 6 \beta_{8} + 8 \beta_{9} ) q^{84}$$ $$+ ( -22 + 20 \beta_{1} - 18 \beta_{2} - 28 \beta_{3} + 32 \beta_{4} - 112 \beta_{6} + 24 \beta_{7} + 24 \beta_{8} + 24 \beta_{9} ) q^{85}$$ $$+ ( 333 + 3 \beta_{1} + 87 \beta_{2} - 67 \beta_{3} + 112 \beta_{4} + 19 \beta_{5} - 64 \beta_{6} + 29 \beta_{7} + 13 \beta_{8} - 13 \beta_{9} ) q^{86}$$ $$+ ( 17 + 17 \beta_{1} - 547 \beta_{2} + 83 \beta_{3} + 44 \beta_{4} + 83 \beta_{5} + 106 \beta_{6} + 6 \beta_{7} - 39 \beta_{9} ) q^{87}$$ $$+ ( 172 - 22 \beta_{1} - 176 \beta_{2} - 32 \beta_{3} - 60 \beta_{4} - 70 \beta_{5} + 74 \beta_{6} + 4 \beta_{7} - 58 \beta_{8} + 34 \beta_{9} ) q^{88}$$ $$+ ( 53 + 53 \beta_{1} + 35 \beta_{2} - 14 \beta_{3} - 14 \beta_{5} + 110 \beta_{6} + 102 \beta_{7} - 53 \beta_{9} ) q^{89}$$ $$+ ( 172 - 50 \beta_{1} + 290 \beta_{2} - 30 \beta_{3} - 5 \beta_{4} - 72 \beta_{5} - 31 \beta_{6} - 58 \beta_{7} + 4 \beta_{8} + 44 \beta_{9} ) q^{90}$$ $$+ ( 392 + 64 \beta_{1} + 288 \beta_{2} + 70 \beta_{3} - 240 \beta_{4} - 16 \beta_{6} - 32 \beta_{7} - 32 \beta_{8} - 40 \beta_{9} ) q^{91}$$ $$+ ( 98 + 32 \beta_{1} - 434 \beta_{2} + 58 \beta_{3} - 166 \beta_{4} - 4 \beta_{5} - 32 \beta_{6} - 26 \beta_{7} - 28 \beta_{8} + 52 \beta_{9} ) q^{92}$$ $$+ ( -212 - 36 \beta_{1} + 184 \beta_{2} - 100 \beta_{4} - 40 \beta_{5} - 68 \beta_{6} + 12 \beta_{7} - 12 \beta_{8} + 8 \beta_{9} ) q^{93}$$ $$+ ( -372 - 28 \beta_{1} + 252 \beta_{2} + 4 \beta_{3} + 108 \beta_{4} - 4 \beta_{5} - 36 \beta_{6} - 44 \beta_{7} - 28 \beta_{8} - 20 \beta_{9} ) q^{94}$$ $$+ ( 673 - 43 \beta_{1} - 21 \beta_{2} - 107 \beta_{3} - 74 \beta_{4} + 107 \beta_{5} + 44 \beta_{6} + 54 \beta_{8} - 21 \beta_{9} ) q^{95}$$ $$+ ( -432 - 20 \beta_{1} + 304 \beta_{2} - 32 \beta_{3} - 112 \beta_{4} - 108 \beta_{5} - 116 \beta_{6} + 56 \beta_{7} - 52 \beta_{8} - 44 \beta_{9} ) q^{96}$$ $$+ ( 25 - 49 \beta_{1} - 73 \beta_{2} - 10 \beta_{3} - 158 \beta_{4} + 10 \beta_{5} - 48 \beta_{6} + 86 \beta_{8} - 73 \beta_{9} ) q^{97}$$ $$+ ( 36 - 68 \beta_{1} - 364 \beta_{2} - 60 \beta_{3} + 49 \beta_{4} - 100 \beta_{5} - 20 \beta_{6} + 68 \beta_{7} + 52 \beta_{8} + 4 \beta_{9} ) q^{98}$$ $$+ ( 544 + 56 \beta_{1} - 436 \beta_{2} - 12 \beta_{4} + 23 \beta_{5} + 196 \beta_{6} + 20 \beta_{7} - 20 \beta_{8} + 52 \beta_{9} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 2q^{5}$$ $$\mathstrut -\mathstrut 32q^{6}$$ $$\mathstrut -\mathstrut 44q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$10q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 2q^{5}$$ $$\mathstrut -\mathstrut 32q^{6}$$ $$\mathstrut -\mathstrut 44q^{8}$$ $$\mathstrut -\mathstrut 68q^{10}$$ $$\mathstrut +\mathstrut 18q^{11}$$ $$\mathstrut +\mathstrut 100q^{12}$$ $$\mathstrut -\mathstrut 2q^{13}$$ $$\mathstrut +\mathstrut 188q^{14}$$ $$\mathstrut -\mathstrut 124q^{15}$$ $$\mathstrut +\mathstrut 280q^{16}$$ $$\mathstrut -\mathstrut 4q^{17}$$ $$\mathstrut +\mathstrut 174q^{18}$$ $$\mathstrut -\mathstrut 26q^{19}$$ $$\mathstrut -\mathstrut 196q^{20}$$ $$\mathstrut +\mathstrut 52q^{21}$$ $$\mathstrut -\mathstrut 588q^{22}$$ $$\mathstrut -\mathstrut 848q^{24}$$ $$\mathstrut -\mathstrut 264q^{26}$$ $$\mathstrut +\mathstrut 184q^{27}$$ $$\mathstrut +\mathstrut 280q^{28}$$ $$\mathstrut -\mathstrut 202q^{29}$$ $$\mathstrut +\mathstrut 1236q^{30}$$ $$\mathstrut +\mathstrut 368q^{31}$$ $$\mathstrut +\mathstrut 968q^{32}$$ $$\mathstrut -\mathstrut 4q^{33}$$ $$\mathstrut +\mathstrut 436q^{34}$$ $$\mathstrut +\mathstrut 476q^{35}$$ $$\mathstrut -\mathstrut 596q^{36}$$ $$\mathstrut -\mathstrut 10q^{37}$$ $$\mathstrut -\mathstrut 1232q^{38}$$ $$\mathstrut -\mathstrut 1336q^{40}$$ $$\mathstrut -\mathstrut 680q^{42}$$ $$\mathstrut -\mathstrut 838q^{43}$$ $$\mathstrut +\mathstrut 868q^{44}$$ $$\mathstrut +\mathstrut 194q^{45}$$ $$\mathstrut +\mathstrut 1132q^{46}$$ $$\mathstrut -\mathstrut 944q^{47}$$ $$\mathstrut +\mathstrut 1768q^{48}$$ $$\mathstrut +\mathstrut 94q^{49}$$ $$\mathstrut +\mathstrut 726q^{50}$$ $$\mathstrut -\mathstrut 1500q^{51}$$ $$\mathstrut -\mathstrut 236q^{52}$$ $$\mathstrut -\mathstrut 378q^{53}$$ $$\mathstrut -\mathstrut 1376q^{54}$$ $$\mathstrut -\mathstrut 488q^{56}$$ $$\mathstrut +\mathstrut 8q^{58}$$ $$\mathstrut +\mathstrut 1706q^{59}$$ $$\mathstrut -\mathstrut 192q^{60}$$ $$\mathstrut +\mathstrut 910q^{61}$$ $$\mathstrut -\mathstrut 80q^{62}$$ $$\mathstrut +\mathstrut 2628q^{63}$$ $$\mathstrut +\mathstrut 512q^{64}$$ $$\mathstrut -\mathstrut 492q^{65}$$ $$\mathstrut -\mathstrut 428q^{66}$$ $$\mathstrut +\mathstrut 1942q^{67}$$ $$\mathstrut -\mathstrut 880q^{68}$$ $$\mathstrut +\mathstrut 580q^{69}$$ $$\mathstrut +\mathstrut 160q^{70}$$ $$\mathstrut +\mathstrut 1092q^{72}$$ $$\mathstrut -\mathstrut 452q^{74}$$ $$\mathstrut -\mathstrut 2954q^{75}$$ $$\mathstrut -\mathstrut 1228q^{76}$$ $$\mathstrut -\mathstrut 268q^{77}$$ $$\mathstrut -\mathstrut 772q^{78}$$ $$\mathstrut -\mathstrut 4416q^{79}$$ $$\mathstrut -\mathstrut 2648q^{80}$$ $$\mathstrut +\mathstrut 482q^{81}$$ $$\mathstrut -\mathstrut 704q^{82}$$ $$\mathstrut -\mathstrut 2562q^{83}$$ $$\mathstrut +\mathstrut 1960q^{84}$$ $$\mathstrut -\mathstrut 12q^{85}$$ $$\mathstrut +\mathstrut 3764q^{86}$$ $$\mathstrut +\mathstrut 1528q^{88}$$ $$\mathstrut +\mathstrut 1896q^{90}$$ $$\mathstrut +\mathstrut 3332q^{91}$$ $$\mathstrut +\mathstrut 632q^{92}$$ $$\mathstrut -\mathstrut 2192q^{93}$$ $$\mathstrut -\mathstrut 3248q^{94}$$ $$\mathstrut +\mathstrut 6900q^{95}$$ $$\mathstrut -\mathstrut 4432q^{96}$$ $$\mathstrut -\mathstrut 4q^{97}$$ $$\mathstrut +\mathstrut 314q^{98}$$ $$\mathstrut +\mathstrut 4958q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10}\mathstrut -\mathstrut$$ $$2$$ $$x^{9}\mathstrut -\mathstrut$$ $$x^{8}\mathstrut +\mathstrut$$ $$6$$ $$x^{7}\mathstrut +\mathstrut$$ $$14$$ $$x^{6}\mathstrut -\mathstrut$$ $$80$$ $$x^{5}\mathstrut +\mathstrut$$ $$56$$ $$x^{4}\mathstrut +\mathstrut$$ $$96$$ $$x^{3}\mathstrut -\mathstrut$$ $$64$$ $$x^{2}\mathstrut -\mathstrut$$ $$512$$ $$x\mathstrut +\mathstrut$$ $$1024$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{9} - 14 \nu^{8} + 7 \nu^{7} + 82 \nu^{6} - 170 \nu^{5} - 120 \nu^{4} + 536 \nu^{3} + 384 \nu^{2} - 2752 \nu + 3072$$$$)/1280$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{9} - 2 \nu^{8} + 101 \nu^{7} - 114 \nu^{6} + 210 \nu^{5} - 120 \nu^{4} + 8 \nu^{3} - 3008 \nu^{2} + 3264 \nu - 7424$$$$)/1280$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{9} - 18 \nu^{8} + 49 \nu^{7} + 14 \nu^{6} - 230 \nu^{5} + 120 \nu^{4} + 872 \nu^{3} - 1472 \nu^{2} - 1984 \nu + 7424$$$$)/1280$$ $$\beta_{5}$$ $$=$$ $$($$$$-17 \nu^{9} + 38 \nu^{8} - 39 \nu^{7} + 86 \nu^{6} + 90 \nu^{5} + 520 \nu^{4} - 792 \nu^{3} + 2752 \nu^{2} - 1856 \nu - 4864$$$$)/1280$$ $$\beta_{6}$$ $$=$$ $$($$$$17 \nu^{9} - 38 \nu^{8} + 39 \nu^{7} + 74 \nu^{6} - 90 \nu^{5} - 680 \nu^{4} + 1432 \nu^{3} - 512 \nu^{2} - 2624 \nu + 2304$$$$)/1280$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{9} + 6 \nu^{8} + 29 \nu^{7} - 58 \nu^{6} + 18 \nu^{5} + 184 \nu^{4} + 136 \nu^{3} - 1216 \nu^{2} + 1088 \nu - 1280$$$$)/256$$ $$\beta_{8}$$ $$=$$ $$($$$$-37 \nu^{9} + 158 \nu^{8} - 179 \nu^{7} + 46 \nu^{6} - 350 \nu^{5} + 1800 \nu^{4} - 4472 \nu^{3} + 3712 \nu^{2} - 3776 \nu + 5376$$$$)/1280$$ $$\beta_{9}$$ $$=$$ $$($$$$-49 \nu^{9} + 46 \nu^{8} + 297 \nu^{7} - 818 \nu^{6} + 10 \nu^{5} + 2040 \nu^{4} + 616 \nu^{3} - 11136 \nu^{2} + 17088 \nu - 7168$$$$)/1280$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1}\mathstrut +\mathstrut$$ $$1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{9}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$2$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-$$$$\beta_{9}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$8$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$14$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$34$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$4$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$18$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$73$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-$$$$15$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$20$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$28$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$14$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$14$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$27$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$24$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$34$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$20$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$46$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$30$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$50$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$12$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$64$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$42$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$207$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$7$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$58$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$36$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$292$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$56$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$54$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$22$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$179$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$48$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$326$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$4$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$28$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$98$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$210$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$94$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$372$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$674$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$35$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1111$$$$)/4$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$15$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 1.28199 − 1.53509i 1.97476 − 0.316760i −1.62580 − 1.16481i 0.932438 + 1.76934i −1.56339 + 1.24732i 1.28199 + 1.53509i 1.97476 + 0.316760i −1.62580 + 1.16481i 0.932438 − 1.76934i −1.56339 − 1.24732i
−2.81708 0.253099i 5.49618 5.49618i 7.87188 + 1.42600i −4.66372 4.66372i −16.8743 + 14.0921i 24.8965i −21.8148 6.00953i 33.4160i 11.9577 + 14.3185i
5.2 −2.29152 + 1.65800i −5.96513 + 5.96513i 2.50210 7.59865i 8.67959 + 8.67959i 3.77903 23.5594i 1.63924i 6.86495 + 21.5609i 44.1656i −34.2802 5.49869i
5.3 0.460984 2.79061i 0.756776 0.756776i −7.57499 2.57285i 8.22587 + 8.22587i −1.76300 2.46073i 2.67171i −10.6718 + 19.9528i 25.8546i 26.7472 19.1632i
5.4 0.836901 + 2.70178i 1.98356 1.98356i −6.59919 + 4.52224i −0.596848 0.596848i 7.01918 + 3.69910i 29.0828i −17.7410 14.0449i 19.1310i 1.11305 2.11205i
5.5 2.81071 0.316066i −3.27139 + 3.27139i 7.80020 1.77674i −12.6449 12.6449i −8.16095 + 10.2289i 13.8754i 21.3626 7.45928i 5.59607i −39.5378 31.5445i
13.1 −2.81708 + 0.253099i 5.49618 + 5.49618i 7.87188 1.42600i −4.66372 + 4.66372i −16.8743 14.0921i 24.8965i −21.8148 + 6.00953i 33.4160i 11.9577 14.3185i
13.2 −2.29152 1.65800i −5.96513 5.96513i 2.50210 + 7.59865i 8.67959 8.67959i 3.77903 + 23.5594i 1.63924i 6.86495 21.5609i 44.1656i −34.2802 + 5.49869i
13.3 0.460984 + 2.79061i 0.756776 + 0.756776i −7.57499 + 2.57285i 8.22587 8.22587i −1.76300 + 2.46073i 2.67171i −10.6718 19.9528i 25.8546i 26.7472 + 19.1632i
13.4 0.836901 2.70178i 1.98356 + 1.98356i −6.59919 4.52224i −0.596848 + 0.596848i 7.01918 3.69910i 29.0828i −17.7410 + 14.0449i 19.1310i 1.11305 + 2.11205i
13.5 2.81071 + 0.316066i −3.27139 3.27139i 7.80020 + 1.77674i −12.6449 + 12.6449i −8.16095 10.2289i 13.8754i 21.3626 + 7.45928i 5.59607i −39.5378 + 31.5445i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
16.e Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{4}^{\mathrm{new}}(16, [\chi])$$.