Properties

 Label 17.4.d.a Level 17 Weight 4 Character orbit 17.d Analytic conductor 1.003 Analytic rank 0 Dimension 12 CM No Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$17$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 17.d (of order $$8$$ and degree $$4$$)

Newform invariants

 Self dual: No Analytic conductor: $$1.0030324701$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -\beta_{3} - \beta_{10} ) q^{2}$$ $$+ \beta_{8} q^{3}$$ $$+ ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{4}$$ $$+ ( -2 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{5}$$ $$+ ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{6}$$ $$+ ( -2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{7}$$ $$+ ( 2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} ) q^{8}$$ $$+ ( -4 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 4 \beta_{9} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -\beta_{3} - \beta_{10} ) q^{2}$$ $$+ \beta_{8} q^{3}$$ $$+ ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{4}$$ $$+ ( -2 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{5}$$ $$+ ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{6}$$ $$+ ( -2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{7}$$ $$+ ( 2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} ) q^{8}$$ $$+ ( -4 + 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 4 \beta_{9} ) q^{9}$$ $$+ ( -10 - 4 \beta_{1} + 3 \beta_{2} + 10 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} ) q^{10}$$ $$+ ( 2 - 7 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} - 5 \beta_{7} - \beta_{8} - 8 \beta_{9} + \beta_{10} - 7 \beta_{11} ) q^{11}$$ $$+ ( 5 + 4 \beta_{1} + 5 \beta_{2} - 19 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - 3 \beta_{7} + 19 \beta_{9} - 4 \beta_{10} - \beta_{11} ) q^{12}$$ $$+ ( 6 \beta_{1} - 11 \beta_{2} - 11 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} ) q^{13}$$ $$+ ( -7 - 4 \beta_{1} + 7 \beta_{2} + 11 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} + 6 \beta_{8} + 11 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} ) q^{14}$$ $$+ ( 20 - 5 \beta_{1} + 12 \beta_{3} + 5 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 20 \beta_{9} + 2 \beta_{10} ) q^{15}$$ $$+ ( 21 + 9 \beta_{2} - 9 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 8 \beta_{10} + 8 \beta_{11} ) q^{16}$$ $$+ ( 12 - 4 \beta_{1} - 6 \beta_{2} - 10 \beta_{3} + \beta_{4} - 8 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + \beta_{8} - 31 \beta_{9} - 15 \beta_{10} + 8 \beta_{11} ) q^{17}$$ $$+ ( 7 + 16 \beta_{2} - 16 \beta_{3} + 13 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{18}$$ $$+ ( -14 + 19 \beta_{1} - 16 \beta_{3} - 19 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 14 \beta_{9} + 14 \beta_{10} ) q^{19}$$ $$+ ( 38 + 8 \beta_{1} - 38 \beta_{2} + 29 \beta_{3} + 12 \beta_{4} - 6 \beta_{5} + 6 \beta_{7} - 9 \beta_{8} + 29 \beta_{9} + 8 \beta_{10} - 12 \beta_{11} ) q^{20}$$ $$+ ( -16 \beta_{1} + 40 \beta_{2} + 40 \beta_{3} + \beta_{5} - \beta_{6} - 5 \beta_{7} - 5 \beta_{8} + 10 \beta_{9} - 5 \beta_{10} - 5 \beta_{11} ) q^{21}$$ $$+ ( -65 - 7 \beta_{1} - 65 \beta_{2} - 13 \beta_{3} - 16 \beta_{4} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} + 13 \beta_{9} + 7 \beta_{10} - 16 \beta_{11} ) q^{22}$$ $$+ ( -20 + 9 \beta_{1} + 24 \beta_{2} - 20 \beta_{3} + 24 \beta_{4} + 24 \beta_{9} + 24 \beta_{10} + 9 \beta_{11} ) q^{23}$$ $$+ ( -21 - 4 \beta_{1} + 47 \beta_{2} + 21 \beta_{3} - 13 \beta_{4} + 5 \beta_{5} + \beta_{6} - \beta_{8} - 47 \beta_{9} + 13 \beta_{10} + 4 \beta_{11} ) q^{24}$$ $$+ ( -27 - 7 \beta_{1} - 47 \beta_{2} - 7 \beta_{4} + 10 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 10 \beta_{8} - 27 \beta_{9} + 22 \beta_{11} ) q^{25}$$ $$+ ( -57 + 4 \beta_{1} + 64 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 57 \beta_{9} - 2 \beta_{11} ) q^{26}$$ $$+ ( -60 - 4 \beta_{1} - 26 \beta_{2} + 60 \beta_{3} - 18 \beta_{5} + 11 \beta_{6} - 11 \beta_{8} + 26 \beta_{9} + 4 \beta_{11} ) q^{27}$$ $$+ ( 35 - 2 \beta_{1} - 31 \beta_{2} + 35 \beta_{3} + 12 \beta_{4} + 3 \beta_{6} + 12 \beta_{7} + 3 \beta_{8} - 31 \beta_{9} + 12 \beta_{10} - 2 \beta_{11} ) q^{28}$$ $$+ ( 51 + 10 \beta_{1} + 51 \beta_{2} - 88 \beta_{3} - 3 \beta_{4} + 14 \beta_{5} - 9 \beta_{6} + 14 \beta_{7} + 88 \beta_{9} - 10 \beta_{10} - 3 \beta_{11} ) q^{29}$$ $$+ ( 20 \beta_{1} - 62 \beta_{2} - 62 \beta_{3} - 11 \beta_{5} + 11 \beta_{6} - 7 \beta_{7} - 7 \beta_{8} + 42 \beta_{9} - 22 \beta_{10} - 22 \beta_{11} ) q^{30}$$ $$+ ( 6 + 3 \beta_{1} - 6 \beta_{2} + 74 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} + 6 \beta_{7} - 14 \beta_{8} + 74 \beta_{9} + 3 \beta_{10} + 4 \beta_{11} ) q^{31}$$ $$+ ( 72 - 6 \beta_{1} - 3 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + 10 \beta_{6} - 10 \beta_{7} + 2 \beta_{8} - 72 \beta_{9} - 15 \beta_{10} ) q^{32}$$ $$+ ( 130 + 56 \beta_{2} - 56 \beta_{3} + 6 \beta_{4} - 11 \beta_{5} - 11 \beta_{6} - 16 \beta_{7} + 16 \beta_{8} - 12 \beta_{10} + 12 \beta_{11} ) q^{33}$$ $$+ ( 56 - 13 \beta_{1} - 45 \beta_{2} + 10 \beta_{3} - 18 \beta_{4} + 25 \beta_{5} - 21 \beta_{6} + 2 \beta_{7} - \beta_{8} - 122 \beta_{9} - 2 \beta_{10} - 8 \beta_{11} ) q^{34}$$ $$+ ( -44 + 48 \beta_{2} - 48 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 11 \beta_{10} + 11 \beta_{11} ) q^{35}$$ $$+ ( 45 - 5 \beta_{1} - 67 \beta_{3} + 5 \beta_{4} - 13 \beta_{5} - \beta_{6} + \beta_{7} + 13 \beta_{8} - 45 \beta_{9} - \beta_{10} ) q^{36}$$ $$+ ( 98 - 26 \beta_{1} - 98 \beta_{2} + 73 \beta_{3} + 15 \beta_{4} + 12 \beta_{5} - 12 \beta_{7} + 9 \beta_{8} + 73 \beta_{9} - 26 \beta_{10} - 15 \beta_{11} ) q^{37}$$ $$+ ( -10 \beta_{1} + 146 \beta_{2} + 146 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} + 16 \beta_{7} + 16 \beta_{8} + 58 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{38}$$ $$+ ( -86 - 4 \beta_{1} - 86 \beta_{2} - 32 \beta_{3} + 14 \beta_{4} + 26 \beta_{5} - 4 \beta_{6} + 26 \beta_{7} + 32 \beta_{9} + 4 \beta_{10} + 14 \beta_{11} ) q^{39}$$ $$+ ( -139 + 6 \beta_{1} + 20 \beta_{2} - 139 \beta_{3} - 30 \beta_{4} + 11 \beta_{6} - 21 \beta_{7} + 11 \beta_{8} + 20 \beta_{9} - 30 \beta_{10} + 6 \beta_{11} ) q^{40}$$ $$+ ( -1 + 21 \beta_{1} + 44 \beta_{2} + \beta_{3} - 7 \beta_{4} - 12 \beta_{5} - 23 \beta_{6} + 23 \beta_{8} - 44 \beta_{9} + 7 \beta_{10} - 21 \beta_{11} ) q^{41}$$ $$+ ( 6 + 26 \beta_{1} - 120 \beta_{2} + 26 \beta_{4} - 8 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 8 \beta_{8} + 6 \beta_{9} - 4 \beta_{11} ) q^{42}$$ $$+ ( -136 + 6 \beta_{1} + 116 \beta_{2} + 6 \beta_{4} - 28 \beta_{5} - \beta_{6} - \beta_{7} - 28 \beta_{8} - 136 \beta_{9} - 18 \beta_{11} ) q^{43}$$ $$+ ( -115 - 7 \beta_{1} - 51 \beta_{2} + 115 \beta_{3} - 22 \beta_{4} + 33 \beta_{5} - 4 \beta_{6} + 4 \beta_{8} + 51 \beta_{9} + 22 \beta_{10} + 7 \beta_{11} ) q^{44}$$ $$+ ( -11 - 10 \beta_{1} - 54 \beta_{2} - 11 \beta_{3} - 13 \beta_{4} - 3 \beta_{6} + 9 \beta_{7} - 3 \beta_{8} - 54 \beta_{9} - 13 \beta_{10} - 10 \beta_{11} ) q^{45}$$ $$+ ( 63 - 20 \beta_{1} + 63 \beta_{2} - 187 \beta_{3} + 24 \beta_{4} - 15 \beta_{5} + 48 \beta_{6} - 15 \beta_{7} + 187 \beta_{9} + 20 \beta_{10} + 24 \beta_{11} ) q^{46}$$ $$+ ( -30 \beta_{1} + 6 \beta_{5} - 6 \beta_{6} - 22 \beta_{7} - 22 \beta_{8} - 112 \beta_{9} + 52 \beta_{10} + 52 \beta_{11} ) q^{47}$$ $$+ ( -51 + 5 \beta_{1} + 51 \beta_{2} + 149 \beta_{3} + 24 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} - 19 \beta_{8} + 149 \beta_{9} + 5 \beta_{10} - 24 \beta_{11} ) q^{48}$$ $$+ ( 58 + 43 \beta_{1} + 125 \beta_{3} - 43 \beta_{4} - \beta_{5} - 5 \beta_{6} + 5 \beta_{7} + \beta_{8} - 58 \beta_{9} + 34 \beta_{10} ) q^{49}$$ $$+ ( 157 - 111 \beta_{2} + 111 \beta_{3} - 13 \beta_{4} - 17 \beta_{5} - 17 \beta_{6} + 52 \beta_{7} - 52 \beta_{8} + 46 \beta_{10} - 46 \beta_{11} ) q^{50}$$ $$+ ( 34 + 34 \beta_{1} - 34 \beta_{2} + 68 \beta_{3} + 34 \beta_{4} + 34 \beta_{6} + 17 \beta_{7} - 136 \beta_{9} + 51 \beta_{10} + 17 \beta_{11} ) q^{51}$$ $$+ ( 22 + 59 \beta_{2} - 59 \beta_{3} + 18 \beta_{4} - 30 \beta_{5} - 30 \beta_{6} + 25 \beta_{7} - 25 \beta_{8} + 20 \beta_{10} - 20 \beta_{11} ) q^{52}$$ $$+ ( 29 - 39 \beta_{1} - 236 \beta_{3} + 39 \beta_{4} + 13 \beta_{5} + 40 \beta_{6} - 40 \beta_{7} - 13 \beta_{8} - 29 \beta_{9} - 58 \beta_{10} ) q^{53}$$ $$+ ( -8 + 45 \beta_{1} + 8 \beta_{2} + 60 \beta_{3} - 73 \beta_{4} + 25 \beta_{5} - 25 \beta_{7} + 32 \beta_{8} + 60 \beta_{9} + 45 \beta_{10} + 73 \beta_{11} ) q^{54}$$ $$+ ( 66 \beta_{1} + 122 \beta_{2} + 122 \beta_{3} - 22 \beta_{5} + 22 \beta_{6} - 6 \beta_{7} - 6 \beta_{8} + 132 \beta_{9} + 15 \beta_{10} + 15 \beta_{11} ) q^{55}$$ $$+ ( -127 + 40 \beta_{1} - 127 \beta_{2} - 101 \beta_{3} + 22 \beta_{4} - 15 \beta_{5} - 18 \beta_{6} - 15 \beta_{7} + 101 \beta_{9} - 40 \beta_{10} + 22 \beta_{11} ) q^{56}$$ $$+ ( -56 - 71 \beta_{1} + 14 \beta_{2} - 56 \beta_{3} - 21 \beta_{4} - 6 \beta_{6} + 48 \beta_{7} - 6 \beta_{8} + 14 \beta_{9} - 21 \beta_{10} - 71 \beta_{11} ) q^{57}$$ $$+ ( 39 - 56 \beta_{1} + 172 \beta_{2} - 39 \beta_{3} + 84 \beta_{4} - 17 \beta_{5} + 10 \beta_{6} - 10 \beta_{8} - 172 \beta_{9} - 84 \beta_{10} + 56 \beta_{11} ) q^{58}$$ $$+ ( -94 - 54 \beta_{1} - 356 \beta_{2} - 54 \beta_{4} + 14 \beta_{5} + 15 \beta_{6} + 15 \beta_{7} + 14 \beta_{8} - 94 \beta_{9} - 110 \beta_{11} ) q^{59}$$ $$+ ( -112 - 54 \beta_{1} + 158 \beta_{2} - 54 \beta_{4} + 55 \beta_{5} - 33 \beta_{6} - 33 \beta_{7} + 55 \beta_{8} - 112 \beta_{9} + 56 \beta_{11} ) q^{60}$$ $$+ ( -54 + 68 \beta_{1} - 51 \beta_{2} + 54 \beta_{3} + 85 \beta_{4} - \beta_{5} - 29 \beta_{6} + 29 \beta_{8} + 51 \beta_{9} - 85 \beta_{10} - 68 \beta_{11} ) q^{61}$$ $$+ ( 13 + 78 \beta_{1} + 89 \beta_{2} + 13 \beta_{3} - 40 \beta_{4} + 19 \beta_{6} - 18 \beta_{7} + 19 \beta_{8} + 89 \beta_{9} - 40 \beta_{10} + 78 \beta_{11} ) q^{62}$$ $$+ ( 106 - 8 \beta_{1} + 106 \beta_{2} - 54 \beta_{3} - 35 \beta_{4} - 18 \beta_{5} - 42 \beta_{6} - 18 \beta_{7} + 54 \beta_{9} + 8 \beta_{10} - 35 \beta_{11} ) q^{63}$$ $$+ ( -79 \beta_{1} - 29 \beta_{2} - 29 \beta_{3} + 15 \beta_{5} - 15 \beta_{6} + 44 \beta_{7} + 44 \beta_{8} + 41 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} ) q^{64}$$ $$+ ( 69 - 12 \beta_{1} - 69 \beta_{2} + 23 \beta_{3} - 36 \beta_{4} + 23 \beta_{5} - 23 \beta_{7} + 48 \beta_{8} + 23 \beta_{9} - 12 \beta_{10} + 36 \beta_{11} ) q^{65}$$ $$+ ( 124 - 39 \beta_{1} - 100 \beta_{3} + 39 \beta_{4} + 28 \beta_{5} - 27 \beta_{6} + 27 \beta_{7} - 28 \beta_{8} - 124 \beta_{9} - 44 \beta_{10} ) q^{66}$$ $$+ ( 144 + 128 \beta_{2} - 128 \beta_{3} + 50 \beta_{4} + 25 \beta_{5} + 25 \beta_{6} - 66 \beta_{7} + 66 \beta_{8} + 35 \beta_{10} - 35 \beta_{11} ) q^{67}$$ $$+ ( 175 + 38 \beta_{1} - 181 \beta_{2} + 27 \beta_{3} + 16 \beta_{4} - 43 \beta_{5} - 4 \beta_{6} - 32 \beta_{7} - 18 \beta_{8} + 82 \beta_{9} - 2 \beta_{10} - 93 \beta_{11} ) q^{68}$$ $$+ ( -66 - 18 \beta_{2} + 18 \beta_{3} - 114 \beta_{4} + 9 \beta_{5} + 9 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} + 3 \beta_{10} - 3 \beta_{11} ) q^{69}$$ $$+ ( 160 - 68 \beta_{1} + 154 \beta_{3} + 68 \beta_{4} + 13 \beta_{5} - 33 \beta_{6} + 33 \beta_{7} - 13 \beta_{8} - 160 \beta_{9} + 40 \beta_{10} ) q^{70}$$ $$+ ( -16 - 43 \beta_{1} + 16 \beta_{2} + 50 \beta_{3} - 44 \beta_{4} - 42 \beta_{5} + 42 \beta_{7} - 14 \beta_{8} + 50 \beta_{9} - 43 \beta_{10} + 44 \beta_{11} ) q^{71}$$ $$+ ( 67 \beta_{1} + 70 \beta_{2} + 70 \beta_{3} + 23 \beta_{5} - 23 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 19 \beta_{9} + \beta_{10} + \beta_{11} ) q^{72}$$ $$+ ( 139 - 5 \beta_{1} + 139 \beta_{2} - 42 \beta_{3} + 73 \beta_{4} - 41 \beta_{5} + 70 \beta_{6} - 41 \beta_{7} + 42 \beta_{9} + 5 \beta_{10} + 73 \beta_{11} ) q^{73}$$ $$+ ( -198 + 46 \beta_{1} - 109 \beta_{2} - 198 \beta_{3} - 68 \beta_{4} - 32 \beta_{6} + 3 \beta_{7} - 32 \beta_{8} - 109 \beta_{9} - 68 \beta_{10} + 46 \beta_{11} ) q^{74}$$ $$+ ( 30 + 77 \beta_{1} - 190 \beta_{2} - 30 \beta_{3} + 13 \beta_{4} + 41 \beta_{5} + 50 \beta_{6} - 50 \beta_{8} + 190 \beta_{9} - 13 \beta_{10} - 77 \beta_{11} ) q^{75}$$ $$+ ( 92 + 38 \beta_{1} + 142 \beta_{2} + 38 \beta_{4} - 44 \beta_{5} + 22 \beta_{6} + 22 \beta_{7} - 44 \beta_{8} + 92 \beta_{9} - 6 \beta_{11} ) q^{76}$$ $$+ ( -88 + 35 \beta_{1} + 14 \beta_{2} + 35 \beta_{4} - 9 \beta_{5} + 67 \beta_{6} + 67 \beta_{7} - 9 \beta_{8} - 88 \beta_{9} + 32 \beta_{11} ) q^{77}$$ $$+ ( 64 + 2 \beta_{1} - 22 \beta_{2} - 64 \beta_{3} - 12 \beta_{4} - 64 \beta_{5} + 48 \beta_{6} - 48 \beta_{8} + 22 \beta_{9} + 12 \beta_{10} - 2 \beta_{11} ) q^{78}$$ $$+ ( -42 + 27 \beta_{1} + 6 \beta_{2} - 42 \beta_{3} + 6 \beta_{4} - 64 \beta_{6} - 24 \beta_{7} - 64 \beta_{8} + 6 \beta_{9} + 6 \beta_{10} + 27 \beta_{11} ) q^{79}$$ $$+ ( -139 - 96 \beta_{1} - 139 \beta_{2} + 132 \beta_{3} - 32 \beta_{4} + 20 \beta_{5} - 31 \beta_{6} + 20 \beta_{7} - 132 \beta_{9} + 96 \beta_{10} - 32 \beta_{11} ) q^{80}$$ $$+ ( -28 \beta_{1} + 62 \beta_{2} + 62 \beta_{3} - 12 \beta_{5} + 12 \beta_{6} - 7 \beta_{7} - 7 \beta_{8} - 165 \beta_{9} + 86 \beta_{10} + 86 \beta_{11} ) q^{81}$$ $$+ ( -150 + 82 \beta_{1} + 150 \beta_{2} + 87 \beta_{3} + 43 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} + 8 \beta_{8} + 87 \beta_{9} + 82 \beta_{10} - 43 \beta_{11} ) q^{82}$$ $$+ ( -118 - 16 \beta_{1} + 272 \beta_{3} + 16 \beta_{4} - 19 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} + 19 \beta_{8} + 118 \beta_{9} + 94 \beta_{10} ) q^{83}$$ $$+ ( -308 - 110 \beta_{2} + 110 \beta_{3} - 32 \beta_{4} + 74 \beta_{5} + 74 \beta_{6} - 34 \beta_{7} + 34 \beta_{8} - 42 \beta_{10} + 42 \beta_{11} ) q^{84}$$ $$+ ( -279 - 60 \beta_{1} + 165 \beta_{2} + 122 \beta_{3} - 53 \beta_{4} - \beta_{5} - 42 \beta_{6} + 21 \beta_{7} + 15 \beta_{8} - 108 \beta_{9} + 98 \beta_{10} + 69 \beta_{11} ) q^{85}$$ $$+ ( -98 - 72 \beta_{2} + 72 \beta_{3} + 2 \beta_{4} + 34 \beta_{5} + 34 \beta_{6} - 79 \beta_{7} + 79 \beta_{8} + 111 \beta_{10} - 111 \beta_{11} ) q^{86}$$ $$+ ( -266 - \beta_{1} - 72 \beta_{3} + \beta_{4} - 20 \beta_{5} - 96 \beta_{6} + 96 \beta_{7} + 20 \beta_{8} + 266 \beta_{9} - 22 \beta_{10} ) q^{87}$$ $$+ ( -37 - 34 \beta_{1} + 37 \beta_{2} - 269 \beta_{3} + 75 \beta_{4} - 34 \beta_{5} + 34 \beta_{7} - 63 \beta_{8} - 269 \beta_{9} - 34 \beta_{10} - 75 \beta_{11} ) q^{88}$$ $$+ ( 96 \beta_{1} - 249 \beta_{2} - 249 \beta_{3} + 21 \beta_{5} - 21 \beta_{6} - 53 \beta_{7} - 53 \beta_{8} + 194 \beta_{9} + 7 \beta_{10} + 7 \beta_{11} ) q^{89}$$ $$+ ( -108 + 10 \beta_{1} - 108 \beta_{2} + 109 \beta_{3} - 54 \beta_{4} - 9 \beta_{5} - 11 \beta_{6} - 9 \beta_{7} - 109 \beta_{9} - 10 \beta_{10} - 54 \beta_{11} ) q^{90}$$ $$+ ( 242 - 34 \beta_{1} + 140 \beta_{2} + 242 \beta_{3} - 26 \beta_{4} + 22 \beta_{6} - 102 \beta_{7} + 22 \beta_{8} + 140 \beta_{9} - 26 \beta_{10} - 34 \beta_{11} ) q^{91}$$ $$+ ( 77 - 106 \beta_{1} + 191 \beta_{2} - 77 \beta_{3} - 6 \beta_{4} + 38 \beta_{5} - 19 \beta_{6} + 19 \beta_{8} - 191 \beta_{9} + 6 \beta_{10} + 106 \beta_{11} ) q^{92}$$ $$+ ( -82 - 33 \beta_{1} + 246 \beta_{2} - 33 \beta_{4} - 3 \beta_{5} - 77 \beta_{6} - 77 \beta_{7} - 3 \beta_{8} - 82 \beta_{9} + 2 \beta_{11} ) q^{93}$$ $$+ ( 402 - 60 \beta_{1} - 512 \beta_{2} - 60 \beta_{4} - 72 \beta_{5} + 58 \beta_{6} + 58 \beta_{7} - 72 \beta_{8} + 402 \beta_{9} - 180 \beta_{11} ) q^{94}$$ $$+ ( 244 - 55 \beta_{1} + 64 \beta_{2} - 244 \beta_{3} + 27 \beta_{4} + 80 \beta_{5} - 28 \beta_{6} + 28 \beta_{8} - 64 \beta_{9} - 27 \beta_{10} + 55 \beta_{11} ) q^{95}$$ $$+ ( 187 + 17 \beta_{1} + 15 \beta_{2} + 187 \beta_{3} + 48 \beta_{4} + 53 \beta_{6} - 21 \beta_{7} + 53 \beta_{8} + 15 \beta_{9} + 48 \beta_{10} + 17 \beta_{11} ) q^{96}$$ $$+ ( 22 + 104 \beta_{1} + 22 \beta_{2} + 331 \beta_{3} + 108 \beta_{4} + 75 \beta_{5} - 28 \beta_{6} + 75 \beta_{7} - 331 \beta_{9} - 104 \beta_{10} + 108 \beta_{11} ) q^{97}$$ $$+ ( 147 \beta_{1} + 254 \beta_{2} + 254 \beta_{3} + 38 \beta_{7} + 38 \beta_{8} + 323 \beta_{9} - 60 \beta_{10} - 60 \beta_{11} ) q^{98}$$ $$+ ( 272 - 93 \beta_{1} - 272 \beta_{2} - 536 \beta_{3} - 17 \beta_{4} - 72 \beta_{5} + 72 \beta_{7} + 47 \beta_{8} - 536 \beta_{9} - 93 \beta_{10} + 17 \beta_{11} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q$$ $$\mathstrut -\mathstrut 4q^{2}$$ $$\mathstrut -\mathstrut 4q^{3}$$ $$\mathstrut -\mathstrut 20q^{5}$$ $$\mathstrut +\mathstrut 20q^{6}$$ $$\mathstrut -\mathstrut 4q^{7}$$ $$\mathstrut +\mathstrut 28q^{8}$$ $$\mathstrut -\mathstrut 64q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$12q$$ $$\mathstrut -\mathstrut 4q^{2}$$ $$\mathstrut -\mathstrut 4q^{3}$$ $$\mathstrut -\mathstrut 20q^{5}$$ $$\mathstrut +\mathstrut 20q^{6}$$ $$\mathstrut -\mathstrut 4q^{7}$$ $$\mathstrut +\mathstrut 28q^{8}$$ $$\mathstrut -\mathstrut 64q^{9}$$ $$\mathstrut -\mathstrut 116q^{10}$$ $$\mathstrut +\mathstrut 40q^{11}$$ $$\mathstrut +\mathstrut 56q^{12}$$ $$\mathstrut -\mathstrut 132q^{14}$$ $$\mathstrut +\mathstrut 244q^{15}$$ $$\mathstrut +\mathstrut 184q^{16}$$ $$\mathstrut +\mathstrut 52q^{17}$$ $$\mathstrut -\mathstrut 12q^{19}$$ $$\mathstrut +\mathstrut 572q^{20}$$ $$\mathstrut -\mathstrut 620q^{22}$$ $$\mathstrut -\mathstrut 276q^{23}$$ $$\mathstrut -\mathstrut 184q^{24}$$ $$\mathstrut -\mathstrut 464q^{25}$$ $$\mathstrut -\mathstrut 708q^{26}$$ $$\mathstrut -\mathstrut 664q^{27}$$ $$\mathstrut +\mathstrut 452q^{28}$$ $$\mathstrut +\mathstrut 632q^{29}$$ $$\mathstrut +\mathstrut 188q^{31}$$ $$\mathstrut +\mathstrut 700q^{32}$$ $$\mathstrut +\mathstrut 1400q^{33}$$ $$\mathstrut +\mathstrut 764q^{34}$$ $$\mathstrut -\mathstrut 632q^{35}$$ $$\mathstrut +\mathstrut 524q^{36}$$ $$\mathstrut +\mathstrut 940q^{37}$$ $$\mathstrut -\mathstrut 1112q^{39}$$ $$\mathstrut -\mathstrut 1864q^{40}$$ $$\mathstrut +\mathstrut 176q^{41}$$ $$\mathstrut +\mathstrut 48q^{42}$$ $$\mathstrut -\mathstrut 1360q^{43}$$ $$\mathstrut -\mathstrut 1364q^{44}$$ $$\mathstrut -\mathstrut 32q^{45}$$ $$\mathstrut +\mathstrut 452q^{46}$$ $$\mathstrut -\mathstrut 540q^{48}$$ $$\mathstrut +\mathstrut 1044q^{49}$$ $$\mathstrut +\mathstrut 2856q^{50}$$ $$\mathstrut +\mathstrut 340q^{51}$$ $$\mathstrut +\mathstrut 792q^{52}$$ $$\mathstrut -\mathstrut 360q^{53}$$ $$\mathstrut -\mathstrut 244q^{54}$$ $$\mathstrut -\mathstrut 1788q^{56}$$ $$\mathstrut -\mathstrut 148q^{57}$$ $$\mathstrut -\mathstrut 360q^{58}$$ $$\mathstrut -\mathstrut 584q^{59}$$ $$\mathstrut -\mathstrut 1792q^{60}$$ $$\mathstrut -\mathstrut 1052q^{61}$$ $$\mathstrut -\mathstrut 380q^{62}$$ $$\mathstrut +\mathstrut 1752q^{63}$$ $$\mathstrut +\mathstrut 404q^{65}$$ $$\mathstrut +\mathstrut 1372q^{66}$$ $$\mathstrut +\mathstrut 1080q^{67}$$ $$\mathstrut +\mathstrut 2532q^{68}$$ $$\mathstrut -\mathstrut 344q^{69}$$ $$\mathstrut +\mathstrut 2072q^{70}$$ $$\mathstrut +\mathstrut 28q^{71}$$ $$\mathstrut +\mathstrut 824q^{73}$$ $$\mathstrut -\mathstrut 2292q^{74}$$ $$\mathstrut +\mathstrut 400q^{75}$$ $$\mathstrut +\mathstrut 1328q^{76}$$ $$\mathstrut -\mathstrut 1252q^{77}$$ $$\mathstrut +\mathstrut 1128q^{78}$$ $$\mathstrut -\mathstrut 196q^{79}$$ $$\mathstrut -\mathstrut 904q^{80}$$ $$\mathstrut -\mathstrut 1528q^{82}$$ $$\mathstrut -\mathstrut 1008q^{83}$$ $$\mathstrut -\mathstrut 4768q^{84}$$ $$\mathstrut -\mathstrut 2824q^{85}$$ $$\mathstrut -\mathstrut 1200q^{86}$$ $$\mathstrut -\mathstrut 2516q^{87}$$ $$\mathstrut -\mathstrut 56q^{88}$$ $$\mathstrut -\mathstrut 860q^{90}$$ $$\mathstrut +\mathstrut 2456q^{91}$$ $$\mathstrut +\mathstrut 396q^{92}$$ $$\mathstrut -\mathstrut 836q^{93}$$ $$\mathstrut +\mathstrut 6360q^{94}$$ $$\mathstrut +\mathstrut 2172q^{95}$$ $$\mathstrut +\mathstrut 1668q^{96}$$ $$\mathstrut -\mathstrut 904q^{97}$$ $$\mathstrut +\mathstrut 3280q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12}\mathstrut +\mathstrut$$ $$54$$ $$x^{10}\mathstrut +\mathstrut$$ $$1085$$ $$x^{8}\mathstrut +\mathstrut$$ $$9836$$ $$x^{6}\mathstrut +\mathstrut$$ $$38276$$ $$x^{4}\mathstrut +\mathstrut$$ $$49664$$ $$x^{2}\mathstrut +\mathstrut$$ $$16384$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{11} - 408 \nu^{10} + 10 \nu^{9} - 17680 \nu^{8} - 1725 \nu^{7} - 268600 \nu^{6} - 73516 \nu^{5} - 1728288 \nu^{4} - 729732 \nu^{3} - 4623456 \nu^{2} - 1741056 \nu - 2889728$$$$)/1392640$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{11} + 408 \nu^{10} + 10 \nu^{9} + 17680 \nu^{8} - 1725 \nu^{7} + 268600 \nu^{6} - 73516 \nu^{5} + 1728288 \nu^{4} - 729732 \nu^{3} + 4623456 \nu^{2} - 1741056 \nu + 2889728$$$$)/1392640$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{10} + 41 \nu^{8} + 569 \nu^{6} + 3051 \nu^{4} + 5498 \nu^{2} + 2432$$$$)/544$$ $$\beta_{5}$$ $$=$$ $$($$$$241 \nu^{11} - 280 \nu^{10} + 19350 \nu^{9} + 2800 \nu^{8} + 502765 \nu^{7} + 387400 \nu^{6} + 5292396 \nu^{5} + 5701600 \nu^{4} + 20629572 \nu^{3} + 23023520 \nu^{2} + 17121536 \nu + 12462080$$$$)/1392640$$ $$\beta_{6}$$ $$=$$ $$($$$$-241 \nu^{11} - 280 \nu^{10} - 19350 \nu^{9} + 2800 \nu^{8} - 502765 \nu^{7} + 387400 \nu^{6} - 5292396 \nu^{5} + 5701600 \nu^{4} - 20629572 \nu^{3} + 23023520 \nu^{2} - 17121536 \nu + 12462080$$$$)/1392640$$ $$\beta_{7}$$ $$=$$ $$($$$$-325 \nu^{11} + 872 \nu^{10} - 18510 \nu^{9} + 34800 \nu^{8} - 386545 \nu^{7} + 459720 \nu^{6} - 3460060 \nu^{5} + 2176992 \nu^{4} - 10989460 \nu^{3} + 1717664 \nu^{2} - 1301760 \nu - 5347328$$$$)/1392640$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$325$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$872$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$18510$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$34800$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$386545$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$459720$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$3460060$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$2176992$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$10989460$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$1717664$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$1301760$$ $$\nu\mathstrut +\mathstrut$$ $$5347328$$$$)/1392640$$ $$\beta_{9}$$ $$=$$ $$($$$$19 \nu^{11} + 898 \nu^{9} + 15367 \nu^{7} + 114052 \nu^{5} + 336716 \nu^{3} + 239872 \nu$$$$)/69632$$ $$\beta_{10}$$ $$=$$ $$($$$$-51 \nu^{11} + 8 \nu^{10} - 2210 \nu^{9} - 80 \nu^{8} - 33575 \nu^{7} - 7960 \nu^{6} - 216036 \nu^{5} - 86432 \nu^{4} - 577932 \nu^{3} - 211424 \nu^{2} - 361216 \nu + 2048$$$$)/174080$$ $$\beta_{11}$$ $$=$$ $$($$$$-51 \nu^{11} - 8 \nu^{10} - 2210 \nu^{9} + 80 \nu^{8} - 33575 \nu^{7} + 7960 \nu^{6} - 216036 \nu^{5} + 86432 \nu^{4} - 577932 \nu^{3} + 211424 \nu^{2} - 361216 \nu - 2048$$$$)/174080$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$8$$ $$\nu^{3}$$ $$=$$ $$2$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$13$$ $$\beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-$$$$16$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$16$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$21$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$31$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$31$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$106$$ $$\nu^{5}$$ $$=$$ $$-$$$$50$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$50$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$138$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$19$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$19$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$189$$ $$\beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-$$$$28$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$28$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$250$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$250$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$31$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$31$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$373$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$619$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$619$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1574$$ $$\nu^{7}$$ $$=$$ $$990$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$990$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$2602$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$191$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$191$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$120$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$120$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$535$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$535$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2885$$ $$\beta_{1}$$ $$\nu^{8}$$ $$=$$ $$1072$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$1072$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$3946$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$3946$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$679$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$679$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$6349$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$11187$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$11187$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$24438$$ $$\nu^{9}$$ $$=$$ $$-$$$$18242$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$18242$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$46402$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$2195$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$2195$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$2796$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$2796$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$13567$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$13567$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$45213$$ $$\beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-$$$$28020$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$28020$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$62854$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$62854$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$13251$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$13251$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$107189$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$195539$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$195539$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$388206$$ $$\nu^{11}$$ $$=$$ $$326166$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$326166$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$814346$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$21583$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$21583$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$59104$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$59104$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$304847$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$304847$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$720309$$ $$\beta_{1}$$

Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/17\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\beta_{2}$$

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
 − 3.86166i − 0.705468i 4.15292i − 3.68604i − 1.22788i 2.49971i 3.86166i 0.705468i − 4.15292i 3.68604i 1.22788i − 2.49971i
−3.43772 3.43772i −4.67995 + 1.93850i 15.6358i −7.10390 17.1503i 22.7523 + 9.42432i 5.36561 12.9537i 26.2496 26.2496i −0.947753 + 0.947753i −34.5367 + 83.3791i
2.2 −1.20595 1.20595i 4.10553 1.70057i 5.09138i 2.60601 + 6.29147i −7.00185 2.90026i −5.31013 + 12.8198i −15.7875 + 15.7875i −5.12843 + 5.12843i 4.44447 10.7299i
2.3 2.22945 + 2.22945i −1.83980 + 0.762069i 1.94089i −1.91633 4.62643i −5.80073 2.40274i 1.06584 2.57316i 13.5085 13.5085i −16.2878 + 16.2878i 6.04203 14.5867i
8.1 −1.89932 + 1.89932i −1.65755 + 4.00167i 0.785167i 1.92782 + 0.798529i −4.45224 10.7487i 23.0956 9.56650i −16.6858 16.6858i 5.82599 + 5.82599i −5.17821 + 2.14488i
8.2 −0.161134 + 0.161134i 3.15299 7.61199i 7.94807i 2.54200 + 1.05293i 0.718496 + 1.73460i −19.8837 + 8.23610i −2.56978 2.56978i −28.9092 28.9092i −0.579266 + 0.239940i
8.3 2.47467 2.47467i −1.08123 + 2.61032i 4.24796i −8.05561 3.33674i 3.78400 + 9.13537i −6.33320 + 2.62330i 9.28506 + 9.28506i 13.4472 + 13.4472i −28.1923 + 11.6776i
9.1 −3.43772 + 3.43772i −4.67995 1.93850i 15.6358i −7.10390 + 17.1503i 22.7523 9.42432i 5.36561 + 12.9537i 26.2496 + 26.2496i −0.947753 0.947753i −34.5367 83.3791i
9.2 −1.20595 + 1.20595i 4.10553 + 1.70057i 5.09138i 2.60601 6.29147i −7.00185 + 2.90026i −5.31013 12.8198i −15.7875 15.7875i −5.12843 5.12843i 4.44447 + 10.7299i
9.3 2.22945 2.22945i −1.83980 0.762069i 1.94089i −1.91633 + 4.62643i −5.80073 + 2.40274i 1.06584 + 2.57316i 13.5085 + 13.5085i −16.2878 16.2878i 6.04203 + 14.5867i
15.1 −1.89932 1.89932i −1.65755 4.00167i 0.785167i 1.92782 0.798529i −4.45224 + 10.7487i 23.0956 + 9.56650i −16.6858 + 16.6858i 5.82599 5.82599i −5.17821 2.14488i
15.2 −0.161134 0.161134i 3.15299 + 7.61199i 7.94807i 2.54200 1.05293i 0.718496 1.73460i −19.8837 8.23610i −2.56978 + 2.56978i −28.9092 + 28.9092i −0.579266 0.239940i
15.3 2.47467 + 2.47467i −1.08123 2.61032i 4.24796i −8.05561 + 3.33674i 3.78400 9.13537i −6.33320 2.62330i 9.28506 9.28506i 13.4472 13.4472i −28.1923 11.6776i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 15.3 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
17.d Even 1 yes

Hecke kernels

There are no other newforms in $$S_{4}^{\mathrm{new}}(17, [\chi])$$.