# Properties

 Label 17.4.c.a Level 17 Weight 4 Character orbit 17.c Analytic conductor 1.003 Analytic rank 0 Dimension 8 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$17$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 17.c (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.0030324701$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$+ ( \beta_{1} - \beta_{3} ) q^{2}$$ $$+ \beta_{4} q^{3}$$ $$+ ( -5 - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{4}$$ $$+ ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{7} ) q^{5}$$ $$+ ( 3 + \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - \beta_{6} ) q^{6}$$ $$+ ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{7}$$ $$+ ( -5 \beta_{1} + 13 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{8}$$ $$+ ( -6 \beta_{1} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( \beta_{1} - \beta_{3} ) q^{2}$$ $$+ \beta_{4} q^{3}$$ $$+ ( -5 - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{4}$$ $$+ ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{7} ) q^{5}$$ $$+ ( 3 + \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - \beta_{6} ) q^{6}$$ $$+ ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{7}$$ $$+ ( -5 \beta_{1} + 13 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{8}$$ $$+ ( -6 \beta_{1} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{9}$$ $$+ ( 10 + 4 \beta_{1} + 4 \beta_{2} - 10 \beta_{3} - \beta_{5} - \beta_{6} ) q^{10}$$ $$+ ( -14 + 14 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} ) q^{11}$$ $$+ ( -23 + 9 \beta_{1} - 9 \beta_{2} - 23 \beta_{3} - 3 \beta_{4} + 5 \beta_{7} ) q^{12}$$ $$+ ( -12 + 6 \beta_{2} - 5 \beta_{4} - 5 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{13}$$ $$+ ( 14 + 10 \beta_{1} - 10 \beta_{2} + 14 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} ) q^{14}$$ $$+ ( 2 \beta_{1} - 30 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{15}$$ $$+ ( 53 + 25 \beta_{2} + 9 \beta_{4} + 9 \beta_{5} - \beta_{6} - \beta_{7} ) q^{16}$$ $$+ ( -2 - 9 \beta_{1} - 15 \beta_{2} + 25 \beta_{3} + \beta_{4} + 4 \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{17}$$ $$+ ( 55 - 5 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{18}$$ $$+ ( 10 \beta_{1} + 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{19}$$ $$+ ( -38 + 4 \beta_{1} - 4 \beta_{2} - 38 \beta_{3} - 13 \beta_{4} - \beta_{7} ) q^{20}$$ $$+ ( -30 + 4 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} ) q^{21}$$ $$+ ( 3 - 17 \beta_{1} + 17 \beta_{2} + 3 \beta_{3} - 9 \beta_{4} + 3 \beta_{7} ) q^{22}$$ $$+ ( -3 + 9 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} + 6 \beta_{5} + \beta_{6} ) q^{23}$$ $$+ ( -111 - 33 \beta_{1} - 33 \beta_{2} + 111 \beta_{3} - 3 \beta_{5} + 13 \beta_{6} ) q^{24}$$ $$+ ( -10 \beta_{1} - 21 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} ) q^{25}$$ $$+ ( -34 \beta_{1} - 34 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} - \beta_{6} + \beta_{7} ) q^{26}$$ $$+ ( 66 + 12 \beta_{1} + 12 \beta_{2} - 66 \beta_{3} - 4 \beta_{5} + 6 \beta_{6} ) q^{27}$$ $$+ ( -94 + 14 \beta_{1} + 14 \beta_{2} + 94 \beta_{3} - 14 \beta_{5} - 6 \beta_{6} ) q^{28}$$ $$+ ( 12 + \beta_{1} - \beta_{2} + 12 \beta_{3} + 21 \beta_{4} - 25 \beta_{7} ) q^{29}$$ $$+ ( -12 + 4 \beta_{2} + 22 \beta_{4} + 22 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} ) q^{30}$$ $$+ ( 73 + 31 \beta_{1} - 31 \beta_{2} + 73 \beta_{3} - 16 \beta_{4} + 13 \beta_{7} ) q^{31}$$ $$+ ( 37 \beta_{1} - 301 \beta_{3} - 20 \beta_{4} + 20 \beta_{5} - 18 \beta_{6} + 18 \beta_{7} ) q^{32}$$ $$+ ( 100 + 14 \beta_{2} - 25 \beta_{4} - 25 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{33}$$ $$+ ( 129 - 6 \beta_{1} + 7 \beta_{2} + 164 \beta_{3} + 12 \beta_{4} - 3 \beta_{5} + 5 \beta_{6} - 20 \beta_{7} ) q^{34}$$ $$+ ( 138 - 10 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 17 \beta_{6} + 17 \beta_{7} ) q^{35}$$ $$+ ( 19 \beta_{1} + 37 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} ) q^{36}$$ $$+ ( -72 - 15 \beta_{1} + 15 \beta_{2} - 72 \beta_{3} + 33 \beta_{4} + \beta_{7} ) q^{37}$$ $$+ ( -100 + 4 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} ) q^{38}$$ $$+ ( -106 + 28 \beta_{1} - 28 \beta_{2} - 106 \beta_{3} + 26 \beta_{4} + 2 \beta_{7} ) q^{39}$$ $$+ ( -46 - 16 \beta_{1} - 16 \beta_{2} + 46 \beta_{3} - 55 \beta_{5} + 13 \beta_{6} ) q^{40}$$ $$+ ( -119 + 16 \beta_{1} + 16 \beta_{2} + 119 \beta_{3} - 8 \beta_{5} - 8 \beta_{6} ) q^{41}$$ $$+ ( 4 \beta_{1} - 20 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{42}$$ $$+ ( 36 \beta_{1} - 92 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{43}$$ $$+ ( 71 - 15 \beta_{1} - 15 \beta_{2} - 71 \beta_{3} + 31 \beta_{5} - 9 \beta_{6} ) q^{44}$$ $$+ ( -140 - 25 \beta_{1} - 25 \beta_{2} + 140 \beta_{3} + 49 \beta_{5} - 17 \beta_{6} ) q^{45}$$ $$+ ( -124 - 124 \beta_{3} - 36 \beta_{4} + 24 \beta_{7} ) q^{46}$$ $$+ ( -40 + 52 \beta_{2} - 20 \beta_{4} - 20 \beta_{5} - 12 \beta_{6} - 12 \beta_{7} ) q^{47}$$ $$+ ( 319 - 81 \beta_{1} + 81 \beta_{2} + 319 \beta_{3} + 51 \beta_{4} - 29 \beta_{7} ) q^{48}$$ $$+ ( -96 \beta_{1} - 91 \beta_{3} - 20 \beta_{4} + 20 \beta_{5} + 31 \beta_{6} - 31 \beta_{7} ) q^{49}$$ $$+ ( 59 - 81 \beta_{2} - 17 \beta_{4} - 17 \beta_{5} - \beta_{6} - \beta_{7} ) q^{50}$$ $$+ ( 32 + 8 \beta_{1} + 2 \beta_{2} + 42 \beta_{3} - 50 \beta_{4} - 47 \beta_{5} - \beta_{6} + 21 \beta_{7} ) q^{51}$$ $$+ ( 334 + 26 \beta_{2} + 21 \beta_{4} + 21 \beta_{5} - 27 \beta_{6} - 27 \beta_{7} ) q^{52}$$ $$+ ( -86 \beta_{1} + 134 \beta_{3} + 15 \beta_{4} - 15 \beta_{5} - 21 \beta_{6} + 21 \beta_{7} ) q^{53}$$ $$+ ( -204 + 44 \beta_{1} - 44 \beta_{2} - 204 \beta_{3} - 12 \beta_{4} + 20 \beta_{7} ) q^{54}$$ $$+ ( -246 - 30 \beta_{2} + 36 \beta_{4} + 36 \beta_{5} - 7 \beta_{6} - 7 \beta_{7} ) q^{55}$$ $$+ ( 86 - 10 \beta_{1} + 10 \beta_{2} + 86 \beta_{3} + 30 \beta_{4} - 2 \beta_{7} ) q^{56}$$ $$+ ( -34 + 34 \beta_{3} + 28 \beta_{5} - 14 \beta_{6} ) q^{57}$$ $$+ ( 38 + 108 \beta_{1} + 108 \beta_{2} - 38 \beta_{3} + 61 \beta_{5} - 19 \beta_{6} ) q^{58}$$ $$+ ( 80 \beta_{1} + 44 \beta_{3} + 15 \beta_{4} - 15 \beta_{5} + 40 \beta_{6} - 40 \beta_{7} ) q^{59}$$ $$+ ( 84 \beta_{1} - 396 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{60}$$ $$+ ( 160 - 25 \beta_{1} - 25 \beta_{2} - 160 \beta_{3} - 47 \beta_{5} - 11 \beta_{6} ) q^{61}$$ $$+ ( -334 + 18 \beta_{1} + 18 \beta_{2} + 334 \beta_{3} - 110 \beta_{5} + 78 \beta_{6} ) q^{62}$$ $$+ ( 1 - 53 \beta_{1} + 53 \beta_{2} + \beta_{3} - 24 \beta_{4} + 57 \beta_{7} ) q^{63}$$ $$+ ( -405 - 249 \beta_{2} - 25 \beta_{4} - 25 \beta_{5} + 49 \beta_{6} + 49 \beta_{7} ) q^{64}$$ $$+ ( 74 - 20 \beta_{1} + 20 \beta_{2} + 74 \beta_{3} - 72 \beta_{4} + 18 \beta_{7} ) q^{65}$$ $$+ ( 26 \beta_{1} - 126 \beta_{3} + 61 \beta_{4} - 61 \beta_{5} + 11 \beta_{6} - 11 \beta_{7} ) q^{66}$$ $$+ ( 98 - 90 \beta_{2} + 81 \beta_{4} + 81 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} ) q^{67}$$ $$+ ( 240 + 111 \beta_{1} + 134 \beta_{2} - 25 \beta_{3} + 16 \beta_{4} + 81 \beta_{5} - 33 \beta_{6} + 30 \beta_{7} ) q^{68}$$ $$+ ( 230 + 52 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} - 7 \beta_{6} - 7 \beta_{7} ) q^{69}$$ $$+ ( 40 \beta_{1} - 64 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} ) q^{70}$$ $$+ ( 173 - 71 \beta_{1} + 71 \beta_{2} + 173 \beta_{3} + 14 \beta_{4} - 59 \beta_{7} ) q^{71}$$ $$+ ( 173 + 25 \beta_{2} - 73 \beta_{4} - 73 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} ) q^{72}$$ $$+ ( -209 + 70 \beta_{1} - 70 \beta_{2} - 209 \beta_{3} - 46 \beta_{4} - 30 \beta_{7} ) q^{73}$$ $$+ ( 208 - 42 \beta_{1} - 42 \beta_{2} - 208 \beta_{3} + 129 \beta_{5} - 63 \beta_{6} ) q^{74}$$ $$+ ( 166 + 58 \beta_{1} + 58 \beta_{2} - 166 \beta_{3} - 39 \beta_{5} - 18 \beta_{6} ) q^{75}$$ $$+ ( -76 \beta_{1} + 108 \beta_{3} + 24 \beta_{4} - 24 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} ) q^{76}$$ $$+ ( 88 \beta_{1} + 386 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} + \beta_{6} - \beta_{7} ) q^{77}$$ $$+ ( -362 - 86 \beta_{1} - 86 \beta_{2} + 362 \beta_{3} + 22 \beta_{5} + 30 \beta_{6} ) q^{78}$$ $$+ ( 125 - 123 \beta_{1} - 123 \beta_{2} - 125 \beta_{3} + 2 \beta_{5} - 43 \beta_{6} ) q^{79}$$ $$+ ( 86 - 108 \beta_{1} + 108 \beta_{2} + 86 \beta_{3} + 93 \beta_{4} - 95 \beta_{7} ) q^{80}$$ $$+ ( 35 + 150 \beta_{2} + 15 \beta_{4} + 15 \beta_{5} ) q^{81}$$ $$+ ( -41 - 103 \beta_{1} + 103 \beta_{2} - 41 \beta_{3} - 8 \beta_{4} + 24 \beta_{7} ) q^{82}$$ $$+ ( 24 \beta_{1} - 100 \beta_{3} - \beta_{4} + \beta_{5} - 76 \beta_{6} + 76 \beta_{7} ) q^{83}$$ $$+ ( -356 - 44 \beta_{2} - 18 \beta_{4} - 18 \beta_{5} - 26 \beta_{6} - 26 \beta_{7} ) q^{84}$$ $$+ ( -316 - 11 \beta_{1} - 109 \beta_{2} - 232 \beta_{3} + 39 \beta_{4} + 54 \beta_{5} + 46 \beta_{6} - 31 \beta_{7} ) q^{85}$$ $$+ ( -550 - 74 \beta_{2} - 45 \beta_{4} - 45 \beta_{5} + 39 \beta_{6} + 39 \beta_{7} ) q^{86}$$ $$+ ( -174 \beta_{1} + 382 \beta_{3} - 28 \beta_{4} + 28 \beta_{5} + 49 \beta_{6} - 49 \beta_{7} ) q^{87}$$ $$+ ( 49 - 7 \beta_{1} + 7 \beta_{2} + 49 \beta_{3} - 135 \beta_{4} + 25 \beta_{7} ) q^{88}$$ $$+ ( -290 - 16 \beta_{4} - 16 \beta_{5} + 41 \beta_{6} + 41 \beta_{7} ) q^{89}$$ $$+ ( 310 - 40 \beta_{1} + 40 \beta_{2} + 310 \beta_{3} - 97 \beta_{4} - \beta_{7} ) q^{90}$$ $$+ ( -220 + 106 \beta_{1} + 106 \beta_{2} + 220 \beta_{3} - 56 \beta_{5} + 64 \beta_{6} ) q^{91}$$ $$+ ( -232 - 160 \beta_{1} - 160 \beta_{2} + 232 \beta_{3} - 60 \beta_{5} + 44 \beta_{6} ) q^{92}$$ $$+ ( 184 \beta_{1} - 530 \beta_{3} + 46 \beta_{4} - 46 \beta_{5} - 57 \beta_{6} + 57 \beta_{7} ) q^{93}$$ $$+ ( -8 \beta_{1} - 440 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} - 32 \beta_{6} + 32 \beta_{7} ) q^{94}$$ $$+ ( 178 + 26 \beta_{1} + 26 \beta_{2} - 178 \beta_{3} - 24 \beta_{5} + 30 \beta_{6} ) q^{95}$$ $$+ ( 527 + 193 \beta_{1} + 193 \beta_{2} - 527 \beta_{3} + 291 \beta_{5} - 109 \beta_{6} ) q^{96}$$ $$+ ( 233 + 168 \beta_{1} - 168 \beta_{2} + 233 \beta_{3} - 24 \beta_{4} - 14 \beta_{7} ) q^{97}$$ $$+ ( 879 + 55 \beta_{2} + 36 \beta_{4} + 36 \beta_{5} - 76 \beta_{6} - 76 \beta_{7} ) q^{98}$$ $$+ ( -248 + 144 \beta_{1} - 144 \beta_{2} - 248 \beta_{3} + 189 \beta_{4} - 52 \beta_{7} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut -\mathstrut 36q^{4}$$ $$\mathstrut +\mathstrut 14q^{5}$$ $$\mathstrut +\mathstrut 22q^{6}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut -\mathstrut 36q^{4}$$ $$\mathstrut +\mathstrut 14q^{5}$$ $$\mathstrut +\mathstrut 22q^{6}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut 78q^{10}$$ $$\mathstrut -\mathstrut 108q^{11}$$ $$\mathstrut -\mathstrut 174q^{12}$$ $$\mathstrut -\mathstrut 88q^{13}$$ $$\mathstrut +\mathstrut 108q^{14}$$ $$\mathstrut +\mathstrut 420q^{16}$$ $$\mathstrut -\mathstrut 10q^{17}$$ $$\mathstrut +\mathstrut 428q^{18}$$ $$\mathstrut -\mathstrut 306q^{20}$$ $$\mathstrut -\mathstrut 260q^{21}$$ $$\mathstrut +\mathstrut 30q^{22}$$ $$\mathstrut -\mathstrut 22q^{23}$$ $$\mathstrut -\mathstrut 862q^{24}$$ $$\mathstrut +\mathstrut 540q^{27}$$ $$\mathstrut -\mathstrut 764q^{28}$$ $$\mathstrut +\mathstrut 46q^{29}$$ $$\mathstrut -\mathstrut 120q^{30}$$ $$\mathstrut +\mathstrut 610q^{31}$$ $$\mathstrut +\mathstrut 816q^{33}$$ $$\mathstrut +\mathstrut 1002q^{34}$$ $$\mathstrut +\mathstrut 1172q^{35}$$ $$\mathstrut -\mathstrut 574q^{37}$$ $$\mathstrut -\mathstrut 768q^{38}$$ $$\mathstrut -\mathstrut 844q^{39}$$ $$\mathstrut -\mathstrut 342q^{40}$$ $$\mathstrut -\mathstrut 968q^{41}$$ $$\mathstrut +\mathstrut 550q^{44}$$ $$\mathstrut -\mathstrut 1154q^{45}$$ $$\mathstrut -\mathstrut 944q^{46}$$ $$\mathstrut -\mathstrut 368q^{47}$$ $$\mathstrut +\mathstrut 2494q^{48}$$ $$\mathstrut +\mathstrut 468q^{50}$$ $$\mathstrut +\mathstrut 296q^{51}$$ $$\mathstrut +\mathstrut 2564q^{52}$$ $$\mathstrut -\mathstrut 1592q^{54}$$ $$\mathstrut -\mathstrut 1996q^{55}$$ $$\mathstrut +\mathstrut 684q^{56}$$ $$\mathstrut -\mathstrut 300q^{57}$$ $$\mathstrut +\mathstrut 266q^{58}$$ $$\mathstrut +\mathstrut 1258q^{61}$$ $$\mathstrut -\mathstrut 2516q^{62}$$ $$\mathstrut +\mathstrut 122q^{63}$$ $$\mathstrut -\mathstrut 3044q^{64}$$ $$\mathstrut +\mathstrut 628q^{65}$$ $$\mathstrut +\mathstrut 764q^{67}$$ $$\mathstrut +\mathstrut 1914q^{68}$$ $$\mathstrut +\mathstrut 1812q^{69}$$ $$\mathstrut +\mathstrut 1266q^{71}$$ $$\mathstrut +\mathstrut 1404q^{72}$$ $$\mathstrut -\mathstrut 1732q^{73}$$ $$\mathstrut +\mathstrut 1538q^{74}$$ $$\mathstrut +\mathstrut 1292q^{75}$$ $$\mathstrut -\mathstrut 2836q^{78}$$ $$\mathstrut +\mathstrut 914q^{79}$$ $$\mathstrut +\mathstrut 498q^{80}$$ $$\mathstrut +\mathstrut 280q^{81}$$ $$\mathstrut -\mathstrut 280q^{82}$$ $$\mathstrut -\mathstrut 2952q^{84}$$ $$\mathstrut -\mathstrut 2498q^{85}$$ $$\mathstrut -\mathstrut 4244q^{86}$$ $$\mathstrut +\mathstrut 442q^{88}$$ $$\mathstrut -\mathstrut 2156q^{89}$$ $$\mathstrut +\mathstrut 2478q^{90}$$ $$\mathstrut -\mathstrut 1632q^{91}$$ $$\mathstrut -\mathstrut 1768q^{92}$$ $$\mathstrut +\mathstrut 1484q^{95}$$ $$\mathstrut +\mathstrut 3998q^{96}$$ $$\mathstrut +\mathstrut 1836q^{97}$$ $$\mathstrut +\mathstrut 6728q^{98}$$ $$\mathstrut -\mathstrut 2088q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8}\mathstrut +\mathstrut$$ $$46$$ $$x^{6}\mathstrut +\mathstrut$$ $$561$$ $$x^{4}\mathstrut +\mathstrut$$ $$836$$ $$x^{2}\mathstrut +\mathstrut$$ $$256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{4} - 23 \nu^{2} - 16$$$$)/10$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 46 \nu^{5} + 545 \nu^{3} + 468 \nu$$$$)/160$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 172 \nu^{4} - 413 \nu^{3} + 1864 \nu^{2} + 396 \nu + 1120$$$$)/160$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 4 \nu^{6} + 42 \nu^{5} + 172 \nu^{4} + 413 \nu^{3} + 1864 \nu^{2} - 396 \nu + 1120$$$$)/160$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{7} + 4 \nu^{6} - 226 \nu^{5} + 180 \nu^{4} - 2673 \nu^{3} + 2128 \nu^{2} - 3156 \nu + 2208$$$$)/160$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 4 \nu^{6} + 226 \nu^{5} + 180 \nu^{4} + 2673 \nu^{3} + 2128 \nu^{2} + 3156 \nu + 2208$$$$)/160$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$12$$ $$\nu^{3}$$ $$=$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$8$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$21$$ $$\beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-$$$$23$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$23$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$23$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$23$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$33$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$260$$ $$\nu^{5}$$ $$=$$ $$-$$$$33$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$33$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$13$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$13$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$304$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$477$$ $$\beta_{1}$$ $$\nu^{6}$$ $$=$$ $$523$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$523$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$503$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$503$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$953$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$5868$$ $$\nu^{7}$$ $$=$$ $$973$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$973$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$53$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$53$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$9464$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$10965$$ $$\beta_{1}$$

## Character Values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 − 4.93651i − 0.648995i 1.11783i 4.46767i − 4.46767i − 1.11783i 0.648995i 4.93651i
3.93651i −0.299807 + 0.299807i −7.49613 1.37942 1.37942i 1.18019 + 1.18019i 17.9849 + 17.9849i 1.98349i 26.8202i −5.43011 5.43011i
4.2 0.351005i 2.28193 2.28193i 7.87680 −9.32676 + 9.32676i 0.800971 + 0.800971i −23.5385 23.5385i 5.57284i 16.5856i −3.27374 3.27374i
4.3 2.11783i −5.92758 + 5.92758i 3.51478 10.1567 10.1567i −12.5536 12.5536i 3.21600 + 3.21600i 24.3864i 43.2725i 21.5102 + 21.5102i
4.4 5.46767i 3.94546 3.94546i −21.8954 4.79064 4.79064i 21.5725 + 21.5725i 3.33761 + 3.33761i 75.9757i 4.13329i 26.1937 + 26.1937i
13.1 5.46767i 3.94546 + 3.94546i −21.8954 4.79064 + 4.79064i 21.5725 21.5725i 3.33761 3.33761i 75.9757i 4.13329i 26.1937 26.1937i
13.2 2.11783i −5.92758 5.92758i 3.51478 10.1567 + 10.1567i −12.5536 + 12.5536i 3.21600 3.21600i 24.3864i 43.2725i 21.5102 21.5102i
13.3 0.351005i 2.28193 + 2.28193i 7.87680 −9.32676 9.32676i 0.800971 0.800971i −23.5385 + 23.5385i 5.57284i 16.5856i −3.27374 + 3.27374i
13.4 3.93651i −0.299807 0.299807i −7.49613 1.37942 + 1.37942i 1.18019 1.18019i 17.9849 17.9849i 1.98349i 26.8202i −5.43011 + 5.43011i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.4 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.c Even 1 yes

## Hecke kernels

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