# 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 $2$

# Learn more

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.00303247010$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 46x^{6} + 561x^{4} + 836x^{2} + 256$$ x^8 + 46*x^6 + 561*x^4 + 836*x^2 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1) q^{2} + \beta_{4} q^{3} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} - 5) q^{4} + ( - \beta_{7} - \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{5} + ( - \beta_{6} + 3 \beta_{5} - 3 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{6} + ( - 3 \beta_{6} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 13 \beta_{3} + \cdots - 5 \beta_1) q^{8}+ \cdots + (3 \beta_{5} - 3 \beta_{4} + \beta_{3} - 6 \beta_1) q^{9}+O(q^{10})$$ q + (-b3 + b1) * q^2 + b4 * q^3 + (b7 + b6 - b5 - b4 - b2 - 5) * q^4 + (-b7 - b4 + 2*b3 + b2 - b1 + 2) * q^5 + (-b6 + 3*b5 - 3*b3 + b2 + b1 + 3) * q^6 + (-3*b6 - b3 - b2 - b1 + 1) * q^7 + (-2*b7 + 2*b6 - 4*b5 + 4*b4 + 13*b3 - 5*b1) * q^8 + (3*b5 - 3*b4 + b3 - 6*b1) * q^9 $$q + ( - \beta_{3} + \beta_1) q^{2} + \beta_{4} q^{3} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} - 5) q^{4} + ( - \beta_{7} - \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{5} + ( - \beta_{6} + 3 \beta_{5} - 3 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{6} + ( - 3 \beta_{6} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 13 \beta_{3} + \cdots - 5 \beta_1) q^{8}+ \cdots + ( - 52 \beta_{7} + 189 \beta_{4} - 248 \beta_{3} - 144 \beta_{2} + \cdots - 248) q^{99}+O(q^{100})$$ q + (-b3 + b1) * q^2 + b4 * q^3 + (b7 + b6 - b5 - b4 - b2 - 5) * q^4 + (-b7 - b4 + 2*b3 + b2 - b1 + 2) * q^5 + (-b6 + 3*b5 - 3*b3 + b2 + b1 + 3) * q^6 + (-3*b6 - b3 - b2 - b1 + 1) * q^7 + (-2*b7 + 2*b6 - 4*b5 + 4*b4 + 13*b3 - 5*b1) * q^8 + (3*b5 - 3*b4 + b3 - 6*b1) * q^9 + (-b6 - b5 - 10*b3 + 4*b2 + 4*b1 + 10) * q^10 + (2*b6 + 3*b5 + 14*b3 - 14) * q^11 + (5*b7 - 3*b4 - 23*b3 - 9*b2 + 9*b1 - 23) * q^12 + (2*b7 + 2*b6 - 5*b5 - 5*b4 + 6*b2 - 12) * q^13 + (-2*b7 + 2*b4 + 14*b3 - 10*b2 + 10*b1 + 14) * q^14 + (-3*b7 + 3*b6 - 8*b5 + 8*b4 - 30*b3 + 2*b1) * q^15 + (-b7 - b6 + 9*b5 + 9*b4 + 25*b2 + 53) * q^16 + (4*b7 - b6 + 4*b5 + b4 + 25*b3 - 15*b2 - 9*b1 - 2) * q^17 + (-3*b7 - 3*b6 - 3*b5 - 3*b4 - 5*b2 + 55) * q^18 + (b7 - b6 - 2*b5 + 2*b4 + 6*b3 + 10*b1) * q^19 + (-b7 - 13*b4 - 38*b3 - 4*b2 + 4*b1 - 38) * q^20 + (-5*b7 - 5*b6 + 2*b5 + 2*b4 + 4*b2 - 30) * q^21 + (3*b7 - 9*b4 + 3*b3 + 17*b2 - 17*b1 + 3) * q^22 + (b6 + 6*b5 + 3*b3 + 9*b2 + 9*b1 - 3) * q^23 + (13*b6 - 3*b5 + 111*b3 - 33*b2 - 33*b1 - 111) * q^24 + (7*b7 - 7*b6 + 9*b5 - 9*b4 - 21*b3 - 10*b1) * q^25 + (b7 - b6 - 9*b5 + 9*b4 - 34*b3 - 34*b1) * q^26 + (6*b6 - 4*b5 - 66*b3 + 12*b2 + 12*b1 + 66) * q^27 + (-6*b6 - 14*b5 + 94*b3 + 14*b2 + 14*b1 - 94) * q^28 + (-25*b7 + 21*b4 + 12*b3 - b2 + b1 + 12) * q^29 + (-6*b7 - 6*b6 + 22*b5 + 22*b4 + 4*b2 - 12) * q^30 + (13*b7 - 16*b4 + 73*b3 - 31*b2 + 31*b1 + 73) * q^31 + (18*b7 - 18*b6 + 20*b5 - 20*b4 - 301*b3 + 37*b1) * q^32 + (4*b7 + 4*b6 - 25*b5 - 25*b4 + 14*b2 + 100) * q^33 + (-20*b7 + 5*b6 - 3*b5 + 12*b4 + 164*b3 + 7*b2 - 6*b1 + 129) * q^34 + (17*b7 + 17*b6 + 2*b5 + 2*b4 - 10*b2 + 138) * q^35 + (-8*b7 + 8*b6 + 10*b5 - 10*b4 + 37*b3 + 19*b1) * q^36 + (b7 + 33*b4 - 72*b3 + 15*b2 - 15*b1 - 72) * q^37 + (8*b7 + 8*b6 - 4*b5 - 4*b4 + 4*b2 - 100) * q^38 + (2*b7 + 26*b4 - 106*b3 - 28*b2 + 28*b1 - 106) * q^39 + (13*b6 - 55*b5 + 46*b3 - 16*b2 - 16*b1 - 46) * q^40 + (-8*b6 - 8*b5 + 119*b3 + 16*b2 + 16*b1 - 119) * q^41 + (6*b7 - 6*b6 + 10*b5 - 10*b4 - 20*b3 + 4*b1) * q^42 + (-4*b7 + 4*b6 + 3*b5 - 3*b4 - 92*b3 + 36*b1) * q^43 + (-9*b6 + 31*b5 - 71*b3 - 15*b2 - 15*b1 + 71) * q^44 + (-17*b6 + 49*b5 + 140*b3 - 25*b2 - 25*b1 - 140) * q^45 + (24*b7 - 36*b4 - 124*b3 - 124) * q^46 + (-12*b7 - 12*b6 - 20*b5 - 20*b4 + 52*b2 - 40) * q^47 + (-29*b7 + 51*b4 + 319*b3 + 81*b2 - 81*b1 + 319) * q^48 + (-31*b7 + 31*b6 + 20*b5 - 20*b4 - 91*b3 - 96*b1) * q^49 + (-b7 - b6 - 17*b5 - 17*b4 - 81*b2 + 59) * q^50 + (21*b7 - b6 - 47*b5 - 50*b4 + 42*b3 + 2*b2 + 8*b1 + 32) * q^51 + (-27*b7 - 27*b6 + 21*b5 + 21*b4 + 26*b2 + 334) * q^52 + (21*b7 - 21*b6 - 15*b5 + 15*b4 + 134*b3 - 86*b1) * q^53 + (20*b7 - 12*b4 - 204*b3 - 44*b2 + 44*b1 - 204) * q^54 + (-7*b7 - 7*b6 + 36*b5 + 36*b4 - 30*b2 - 246) * q^55 + (-2*b7 + 30*b4 + 86*b3 + 10*b2 - 10*b1 + 86) * q^56 + (-14*b6 + 28*b5 + 34*b3 - 34) * q^57 + (-19*b6 + 61*b5 - 38*b3 + 108*b2 + 108*b1 + 38) * q^58 + (-40*b7 + 40*b6 - 15*b5 + 15*b4 + 44*b3 + 80*b1) * q^59 + (2*b7 - 2*b6 + 6*b5 - 6*b4 - 396*b3 + 84*b1) * q^60 + (-11*b6 - 47*b5 - 160*b3 - 25*b2 - 25*b1 + 160) * q^61 + (78*b6 - 110*b5 + 334*b3 + 18*b2 + 18*b1 - 334) * q^62 + (57*b7 - 24*b4 + b3 + 53*b2 - 53*b1 + 1) * q^63 + (49*b7 + 49*b6 - 25*b5 - 25*b4 - 249*b2 - 405) * q^64 + (18*b7 - 72*b4 + 74*b3 + 20*b2 - 20*b1 + 74) * q^65 + (-11*b7 + 11*b6 - 61*b5 + 61*b4 - 126*b3 + 26*b1) * q^66 + (-5*b7 - 5*b6 + 81*b5 + 81*b4 - 90*b2 + 98) * q^67 + (30*b7 - 33*b6 + 81*b5 + 16*b4 - 25*b3 + 134*b2 + 111*b1 + 240) * q^68 + (-7*b7 - 7*b6 - 4*b5 - 4*b4 + 52*b2 + 230) * q^69 + (-8*b7 + 8*b6 - 4*b5 + 4*b4 - 64*b3 + 40*b1) * q^70 + (-59*b7 + 14*b4 + 173*b3 + 71*b2 - 71*b1 + 173) * q^71 + (5*b7 + 5*b6 - 73*b5 - 73*b4 + 25*b2 + 173) * q^72 + (-30*b7 - 46*b4 - 209*b3 - 70*b2 + 70*b1 - 209) * q^73 + (-63*b6 + 129*b5 - 208*b3 - 42*b2 - 42*b1 + 208) * q^74 + (-18*b6 - 39*b5 - 166*b3 + 58*b2 + 58*b1 + 166) * q^75 + (8*b7 - 8*b6 - 24*b5 + 24*b4 + 108*b3 - 76*b1) * q^76 + (-b7 + b6 - 6*b5 + 6*b4 + 386*b3 + 88*b1) * q^77 + (30*b6 + 22*b5 + 362*b3 - 86*b2 - 86*b1 - 362) * q^78 + (-43*b6 + 2*b5 - 125*b3 - 123*b2 - 123*b1 + 125) * q^79 + (-95*b7 + 93*b4 + 86*b3 + 108*b2 - 108*b1 + 86) * q^80 + (15*b5 + 15*b4 + 150*b2 + 35) * q^81 + (24*b7 - 8*b4 - 41*b3 + 103*b2 - 103*b1 - 41) * q^82 + (76*b7 - 76*b6 + b5 - b4 - 100*b3 + 24*b1) * q^83 + (-26*b7 - 26*b6 - 18*b5 - 18*b4 - 44*b2 - 356) * q^84 + (-31*b7 + 46*b6 + 54*b5 + 39*b4 - 232*b3 - 109*b2 - 11*b1 - 316) * q^85 + (39*b7 + 39*b6 - 45*b5 - 45*b4 - 74*b2 - 550) * q^86 + (-49*b7 + 49*b6 + 28*b5 - 28*b4 + 382*b3 - 174*b1) * q^87 + (25*b7 - 135*b4 + 49*b3 + 7*b2 - 7*b1 + 49) * q^88 + (41*b7 + 41*b6 - 16*b5 - 16*b4 - 290) * q^89 + (-b7 - 97*b4 + 310*b3 + 40*b2 - 40*b1 + 310) * q^90 + (64*b6 - 56*b5 + 220*b3 + 106*b2 + 106*b1 - 220) * q^91 + (44*b6 - 60*b5 + 232*b3 - 160*b2 - 160*b1 - 232) * q^92 + (57*b7 - 57*b6 - 46*b5 + 46*b4 - 530*b3 + 184*b1) * q^93 + (32*b7 - 32*b6 - 8*b5 + 8*b4 - 440*b3 - 8*b1) * q^94 + (30*b6 - 24*b5 - 178*b3 + 26*b2 + 26*b1 + 178) * q^95 + (-109*b6 + 291*b5 - 527*b3 + 193*b2 + 193*b1 + 527) * q^96 + (-14*b7 - 24*b4 + 233*b3 - 168*b2 + 168*b1 + 233) * q^97 + (-76*b7 - 76*b6 + 36*b5 + 36*b4 + 55*b2 + 879) * q^98 + (-52*b7 + 189*b4 - 248*b3 - 144*b2 + 144*b1 - 248) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 36 q^{4} + 14 q^{5} + 22 q^{6} + 2 q^{7}+O(q^{10})$$ 8 * q - 36 * q^4 + 14 * q^5 + 22 * q^6 + 2 * q^7 $$8 q - 36 q^{4} + 14 q^{5} + 22 q^{6} + 2 q^{7} + 78 q^{10} - 108 q^{11} - 174 q^{12} - 88 q^{13} + 108 q^{14} + 420 q^{16} - 10 q^{17} + 428 q^{18} - 306 q^{20} - 260 q^{21} + 30 q^{22} - 22 q^{23} - 862 q^{24} + 540 q^{27} - 764 q^{28} + 46 q^{29} - 120 q^{30} + 610 q^{31} + 816 q^{33} + 1002 q^{34} + 1172 q^{35} - 574 q^{37} - 768 q^{38} - 844 q^{39} - 342 q^{40} - 968 q^{41} + 550 q^{44} - 1154 q^{45} - 944 q^{46} - 368 q^{47} + 2494 q^{48} + 468 q^{50} + 296 q^{51} + 2564 q^{52} - 1592 q^{54} - 1996 q^{55} + 684 q^{56} - 300 q^{57} + 266 q^{58} + 1258 q^{61} - 2516 q^{62} + 122 q^{63} - 3044 q^{64} + 628 q^{65} + 764 q^{67} + 1914 q^{68} + 1812 q^{69} + 1266 q^{71} + 1404 q^{72} - 1732 q^{73} + 1538 q^{74} + 1292 q^{75} - 2836 q^{78} + 914 q^{79} + 498 q^{80} + 280 q^{81} - 280 q^{82} - 2952 q^{84} - 2498 q^{85} - 4244 q^{86} + 442 q^{88} - 2156 q^{89} + 2478 q^{90} - 1632 q^{91} - 1768 q^{92} + 1484 q^{95} + 3998 q^{96} + 1836 q^{97} + 6728 q^{98} - 2088 q^{99}+O(q^{100})$$ 8 * q - 36 * q^4 + 14 * q^5 + 22 * q^6 + 2 * q^7 + 78 * q^10 - 108 * q^11 - 174 * q^12 - 88 * q^13 + 108 * q^14 + 420 * q^16 - 10 * q^17 + 428 * q^18 - 306 * q^20 - 260 * q^21 + 30 * q^22 - 22 * q^23 - 862 * q^24 + 540 * q^27 - 764 * q^28 + 46 * q^29 - 120 * q^30 + 610 * q^31 + 816 * q^33 + 1002 * q^34 + 1172 * q^35 - 574 * q^37 - 768 * q^38 - 844 * q^39 - 342 * q^40 - 968 * q^41 + 550 * q^44 - 1154 * q^45 - 944 * q^46 - 368 * q^47 + 2494 * q^48 + 468 * q^50 + 296 * q^51 + 2564 * q^52 - 1592 * q^54 - 1996 * q^55 + 684 * q^56 - 300 * q^57 + 266 * q^58 + 1258 * q^61 - 2516 * q^62 + 122 * q^63 - 3044 * q^64 + 628 * q^65 + 764 * q^67 + 1914 * q^68 + 1812 * q^69 + 1266 * q^71 + 1404 * q^72 - 1732 * q^73 + 1538 * q^74 + 1292 * q^75 - 2836 * q^78 + 914 * q^79 + 498 * q^80 + 280 * q^81 - 280 * q^82 - 2952 * q^84 - 2498 * q^85 - 4244 * q^86 + 442 * q^88 - 2156 * q^89 + 2478 * q^90 - 1632 * q^91 - 1768 * q^92 + 1484 * q^95 + 3998 * q^96 + 1836 * q^97 + 6728 * q^98 - 2088 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{4} - 23\nu^{2} - 16 ) / 10$$ (-v^4 - 23*v^2 - 16) / 10 $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 46\nu^{5} + 545\nu^{3} + 468\nu ) / 160$$ (v^7 + 46*v^5 + 545*v^3 + 468*v) / 160 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + 4\nu^{6} - 42\nu^{5} + 172\nu^{4} - 413\nu^{3} + 1864\nu^{2} + 396\nu + 1120 ) / 160$$ (-v^7 + 4*v^6 - 42*v^5 + 172*v^4 - 413*v^3 + 1864*v^2 + 396*v + 1120) / 160 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 4\nu^{6} + 42\nu^{5} + 172\nu^{4} + 413\nu^{3} + 1864\nu^{2} - 396\nu + 1120 ) / 160$$ (v^7 + 4*v^6 + 42*v^5 + 172*v^4 + 413*v^3 + 1864*v^2 - 396*v + 1120) / 160 $$\beta_{6}$$ $$=$$ $$( -5\nu^{7} + 4\nu^{6} - 226\nu^{5} + 180\nu^{4} - 2673\nu^{3} + 2128\nu^{2} - 3156\nu + 2208 ) / 160$$ (-5*v^7 + 4*v^6 - 226*v^5 + 180*v^4 - 2673*v^3 + 2128*v^2 - 3156*v + 2208) / 160 $$\beta_{7}$$ $$=$$ $$( 5\nu^{7} + 4\nu^{6} + 226\nu^{5} + 180\nu^{4} + 2673\nu^{3} + 2128\nu^{2} + 3156\nu + 2208 ) / 160$$ (5*v^7 + 4*v^6 + 226*v^5 + 180*v^4 + 2673*v^3 + 2128*v^2 + 3156*v + 2208) / 160
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - 12$$ b7 + b6 - b5 - b4 + b2 - 12 $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 8\beta_{3} - 21\beta_1$$ b7 - b6 - b5 + b4 - 8*b3 - 21*b1 $$\nu^{4}$$ $$=$$ $$-23\beta_{7} - 23\beta_{6} + 23\beta_{5} + 23\beta_{4} - 33\beta_{2} + 260$$ -23*b7 - 23*b6 + 23*b5 + 23*b4 - 33*b2 + 260 $$\nu^{5}$$ $$=$$ $$-33\beta_{7} + 33\beta_{6} + 13\beta_{5} - 13\beta_{4} + 304\beta_{3} + 477\beta_1$$ -33*b7 + 33*b6 + 13*b5 - 13*b4 + 304*b3 + 477*b1 $$\nu^{6}$$ $$=$$ $$523\beta_{7} + 523\beta_{6} - 503\beta_{5} - 503\beta_{4} + 953\beta_{2} - 5868$$ 523*b7 + 523*b6 - 503*b5 - 503*b4 + 953*b2 - 5868 $$\nu^{7}$$ $$=$$ $$973\beta_{7} - 973\beta_{6} - 53\beta_{5} + 53\beta_{4} - 9464\beta_{3} - 10965\beta_1$$ 973*b7 - 973*b6 - 53*b5 + 53*b4 - 9464*b3 - 10965*b1

## 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 Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.4.c.a 8
3.b odd 2 1 153.4.f.a 8
4.b odd 2 1 272.4.o.e 8
17.c even 4 1 inner 17.4.c.a 8
17.d even 8 2 289.4.a.f 8
17.d even 8 2 289.4.b.c 8
51.f odd 4 1 153.4.f.a 8
68.f odd 4 1 272.4.o.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.c.a 8 1.a even 1 1 trivial
17.4.c.a 8 17.c even 4 1 inner
153.4.f.a 8 3.b odd 2 1
153.4.f.a 8 51.f odd 4 1
272.4.o.e 8 4.b odd 2 1
272.4.o.e 8 68.f odd 4 1
289.4.a.f 8 17.d even 8 2
289.4.b.c 8 17.d even 8 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(17, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 50 T^{6} + 673 T^{4} + \cdots + 256$$
$3$ $$T^{8} - 180 T^{5} + 3008 T^{4} + \cdots + 4096$$
$5$ $$T^{8} - 14 T^{7} + 98 T^{6} + \cdots + 6270016$$
$7$ $$T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 330366976$$
$11$ $$T^{8} + 108 T^{7} + \cdots + 40571627776$$
$13$ $$(T^{4} + 44 T^{3} - 2336 T^{2} + \cdots - 468640)^{2}$$
$17$ $$T^{8} + \cdots + 582622237229761$$
$19$ $$T^{8} + 5324 T^{6} + \cdots + 2286918209536$$
$23$ $$T^{8} + 22 T^{7} + \cdots + 23983351398400$$
$29$ $$T^{8} - 46 T^{7} + \cdots + 70\!\cdots\!16$$
$31$ $$T^{8} - 610 T^{7} + \cdots + 49\!\cdots\!76$$
$37$ $$T^{8} + 574 T^{7} + \cdots + 15\!\cdots\!00$$
$41$ $$T^{8} + 968 T^{7} + \cdots + 14\!\cdots\!36$$
$43$ $$T^{8} + 111120 T^{6} + \cdots + 16\!\cdots\!76$$
$47$ $$(T^{4} + 184 T^{3} - 111664 T^{2} + \cdots + 1730640896)^{2}$$
$53$ $$T^{8} + 761540 T^{6} + \cdots + 50\!\cdots\!36$$
$59$ $$T^{8} + 839904 T^{6} + \cdots + 48\!\cdots\!76$$
$61$ $$T^{8} - 1258 T^{7} + \cdots + 22\!\cdots\!96$$
$67$ $$(T^{4} - 382 T^{3} + \cdots - 80371889536)^{2}$$
$71$ $$T^{8} - 1266 T^{7} + \cdots + 14\!\cdots\!36$$
$73$ $$T^{8} + 1732 T^{7} + \cdots + 45\!\cdots\!96$$
$79$ $$T^{8} - 914 T^{7} + \cdots + 16\!\cdots\!16$$
$83$ $$T^{8} + 2153216 T^{6} + \cdots + 12\!\cdots\!36$$
$89$ $$(T^{4} + 1078 T^{3} + \cdots - 22878545920)^{2}$$
$97$ $$T^{8} - 1836 T^{7} + \cdots + 81\!\cdots\!00$$
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