# Properties

 Label 169.4.e.g Level $169$ Weight $4$ Character orbit 169.e Analytic conductor $9.971$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.e (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.1731891456.1 Defining polynomial: $$x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256$$ x^8 - 9*x^6 + 65*x^4 - 144*x^2 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{3} - \beta_1) q^{2} + ( - 3 \beta_{5} + 3 \beta_{4} - 4 \beta_{2} - 1) q^{3} + ( - 5 \beta_{5} - 5 \beta_{2}) q^{4} + (10 \beta_{7} + 5 \beta_{6}) q^{5} + (14 \beta_{3} + 10 \beta_1) q^{6} + (7 \beta_{3} + \beta_1) q^{7} + ( - 4 \beta_{7} + 7 \beta_{6}) q^{8} + (15 \beta_{5} + 25 \beta_{2}) q^{9}+O(q^{10})$$ q + (-2*b7 + b6 - 2*b3 - b1) * q^2 + (-3*b5 + 3*b4 - 4*b2 - 1) * q^3 + (-5*b5 - 5*b2) * q^4 + (10*b7 + 5*b6) * q^5 + (14*b3 + 10*b1) * q^6 + (7*b3 + b1) * q^7 + (-4*b7 + 7*b6) * q^8 + (15*b5 + 25*b2) * q^9 $$q + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{3} - \beta_1) q^{2} + ( - 3 \beta_{5} + 3 \beta_{4} - 4 \beta_{2} - 1) q^{3} + ( - 5 \beta_{5} - 5 \beta_{2}) q^{4} + (10 \beta_{7} + 5 \beta_{6}) q^{5} + (14 \beta_{3} + 10 \beta_1) q^{6} + (7 \beta_{3} + \beta_1) q^{7} + ( - 4 \beta_{7} + 7 \beta_{6}) q^{8} + (15 \beta_{5} + 25 \beta_{2}) q^{9} + ( - 5 \beta_{5} + 5 \beta_{4} - 5 \beta_{2}) q^{10} + ( - \beta_{7} + 15 \beta_{6} - \beta_{3} - 15 \beta_1) q^{11} + (20 \beta_{4} - 60) q^{12} + (10 \beta_{4} - 18) q^{14} + (50 \beta_{7} + 10 \beta_{6} + 50 \beta_{3} - 10 \beta_1) q^{15} + (15 \beta_{5} - 15 \beta_{4} - 21 \beta_{2} - 36) q^{16} + (16 \beta_{5} - 27 \beta_{2}) q^{17} + (80 \beta_{7} - 55 \beta_{6}) q^{18} + (73 \beta_{3} - 5 \beta_1) q^{19} + (100 \beta_{3} - 25 \beta_1) q^{20} + (19 \beta_{7} - 25 \beta_{6}) q^{21} + ( - 46 \beta_{5} - 108 \beta_{2}) q^{22} + ( - 33 \beta_{5} + 33 \beta_{4} + 56 \beta_{2} + 89) q^{23} + (88 \beta_{7} - 40 \beta_{6} + 88 \beta_{3} + 40 \beta_1) q^{24} + ( - 75 \beta_{4} - 75) q^{25} + ( - 9 \beta_{4} + 163) q^{27} + (20 \beta_{7} - 40 \beta_{6} + 20 \beta_{3} + 40 \beta_1) q^{28} + ( - 60 \beta_{5} + 60 \beta_{4} - 47 \beta_{2} + 13) q^{29} + (20 \beta_{5} + 80 \beta_{2}) q^{30} + (120 \beta_{7} + 100 \beta_{6}) q^{31} + ( - 20 \beta_{3} - 65 \beta_1) q^{32} + (181 \beta_{3} + 63 \beta_1) q^{33} + ( - 22 \beta_{7} - 5 \beta_{6}) q^{34} + (30 \beta_{5} - 20 \beta_{2}) q^{35} + (125 \beta_{5} - 125 \beta_{4} + 425 \beta_{2} + 300) q^{36} + (73 \beta_{7} - 44 \beta_{6} + 73 \beta_{3} + 44 \beta_1) q^{37} + (58 \beta_{4} - 126) q^{38} + ( - 15 \beta_{4} - 100) q^{40} + (259 \beta_{7} - 20 \beta_{6} + 259 \beta_{3} + 20 \beta_1) q^{41} + (94 \beta_{5} - 94 \beta_{4} + 232 \beta_{2} + 138) q^{42} + ( - 97 \beta_{5} - 276 \beta_{2}) q^{43} + ( - 300 \beta_{7} + 80 \beta_{6}) q^{44} + ( - 200 \beta_{3} + 25 \beta_1) q^{45} + ( - 46 \beta_{3} + 10 \beta_1) q^{46} + (100 \beta_{7} + 140 \beta_{6}) q^{47} + (48 \beta_{5} - 96 \beta_{2}) q^{48} + (15 \beta_{5} - 15 \beta_{4} - 275 \beta_{2} - 290) q^{49} + ( - 150 \beta_{7} + 150 \beta_{6} - 150 \beta_{3} - 150 \beta_1) q^{50} + (65 \beta_{4} + 149) q^{51} + ( - 165 \beta_{4} + 190) q^{53} + ( - 362 \beta_{7} + 190 \beta_{6} - 362 \beta_{3} - 190 \beta_1) q^{54} + (70 \beta_{5} - 70 \beta_{4} - 220 \beta_{2} - 290) q^{55} + (60 \beta_{5} + 116 \beta_{2}) q^{56} + (13 \beta_{7} - 199 \beta_{6}) q^{57} + (214 \beta_{3} + 167 \beta_1) q^{58} + ( - 377 \beta_{3} - 55 \beta_1) q^{59} + ( - 200 \beta_{7} - 200 \beta_{6}) q^{60} + ( - 200 \beta_{5} + 151 \beta_{2}) q^{61} + ( - 180 \beta_{5} + 180 \beta_{4} - 340 \beta_{2} - 160) q^{62} + ( - 130 \beta_{7} + 130 \beta_{6} - 130 \beta_{3} - 130 \beta_1) q^{63} + ( - 95 \beta_{4} + 588) q^{64} + (370 \beta_{4} - 614) q^{66} + ( - 283 \beta_{7} - 91 \beta_{6} - 283 \beta_{3} + 91 \beta_1) q^{67} + ( - 135 \beta_{5} + 135 \beta_{4} + 185 \beta_{2} + 320) q^{68} + ( - 135 \beta_{5} + 172 \beta_{2}) q^{69} + (20 \beta_{7} - 40 \beta_{6}) q^{70} + ( - 11 \beta_{3} - 105 \beta_1) q^{71} + ( - 460 \beta_{3} - 235 \beta_1) q^{72} + (250 \beta_{7} - 85 \beta_{6}) q^{73} + (205 \beta_{5} + 527 \beta_{2}) q^{74} + ( - 75 \beta_{5} + 75 \beta_{4} - 900 \beta_{2} - 825) q^{75} + ( - 100 \beta_{7} - 340 \beta_{6} - 100 \beta_{3} + 340 \beta_1) q^{76} + (121 \beta_{4} - 67) q^{77} + (40 \beta_{4} + 140) q^{79} + ( - 660 \beta_{7} - 255 \beta_{6} - 660 \beta_{3} + 255 \beta_1) q^{80} + ( - 120 \beta_{5} + 120 \beta_{4} - 121 \beta_{2} - 1) q^{81} + (319 \beta_{5} + 917 \beta_{2}) q^{82} + ( - 180 \beta_{7} - 100 \beta_{6}) q^{83} + ( - 500 \beta_{3} - 220 \beta_1) q^{84} + ( - 750 \beta_{3} + 295 \beta_1) q^{85} + ( - 746 \beta_{7} + 470 \beta_{6}) q^{86} + (201 \beta_{5} + 908 \beta_{2}) q^{87} + ( - 172 \beta_{5} + 172 \beta_{4} - 596 \beta_{2} - 424) q^{88} + ( - 523 \beta_{7} - 125 \beta_{6} - 523 \beta_{3} + 125 \beta_1) q^{89} + ( - 125 \beta_{4} + 300) q^{90} + ( - 280 \beta_{4} - 660) q^{92} + (1080 \beta_{7} - 40 \beta_{6} + 1080 \beta_{3} + 40 \beta_1) q^{93} + ( - 320 \beta_{5} + 320 \beta_{4} - 680 \beta_{2} - 360) q^{94} + (390 \beta_{5} - 440 \beta_{2}) q^{95} + ( - 800 \beta_{7} + 320 \beta_{6}) q^{96} + ( - 27 \beta_{3} + 469 \beta_1) q^{97} + (520 \beta_{3} + 245 \beta_1) q^{98} + (910 \beta_{7} - 390 \beta_{6}) q^{99}+O(q^{100})$$ q + (-2*b7 + b6 - 2*b3 - b1) * q^2 + (-3*b5 + 3*b4 - 4*b2 - 1) * q^3 + (-5*b5 - 5*b2) * q^4 + (10*b7 + 5*b6) * q^5 + (14*b3 + 10*b1) * q^6 + (7*b3 + b1) * q^7 + (-4*b7 + 7*b6) * q^8 + (15*b5 + 25*b2) * q^9 + (-5*b5 + 5*b4 - 5*b2) * q^10 + (-b7 + 15*b6 - b3 - 15*b1) * q^11 + (20*b4 - 60) * q^12 + (10*b4 - 18) * q^14 + (50*b7 + 10*b6 + 50*b3 - 10*b1) * q^15 + (15*b5 - 15*b4 - 21*b2 - 36) * q^16 + (16*b5 - 27*b2) * q^17 + (80*b7 - 55*b6) * q^18 + (73*b3 - 5*b1) * q^19 + (100*b3 - 25*b1) * q^20 + (19*b7 - 25*b6) * q^21 + (-46*b5 - 108*b2) * q^22 + (-33*b5 + 33*b4 + 56*b2 + 89) * q^23 + (88*b7 - 40*b6 + 88*b3 + 40*b1) * q^24 + (-75*b4 - 75) * q^25 + (-9*b4 + 163) * q^27 + (20*b7 - 40*b6 + 20*b3 + 40*b1) * q^28 + (-60*b5 + 60*b4 - 47*b2 + 13) * q^29 + (20*b5 + 80*b2) * q^30 + (120*b7 + 100*b6) * q^31 + (-20*b3 - 65*b1) * q^32 + (181*b3 + 63*b1) * q^33 + (-22*b7 - 5*b6) * q^34 + (30*b5 - 20*b2) * q^35 + (125*b5 - 125*b4 + 425*b2 + 300) * q^36 + (73*b7 - 44*b6 + 73*b3 + 44*b1) * q^37 + (58*b4 - 126) * q^38 + (-15*b4 - 100) * q^40 + (259*b7 - 20*b6 + 259*b3 + 20*b1) * q^41 + (94*b5 - 94*b4 + 232*b2 + 138) * q^42 + (-97*b5 - 276*b2) * q^43 + (-300*b7 + 80*b6) * q^44 + (-200*b3 + 25*b1) * q^45 + (-46*b3 + 10*b1) * q^46 + (100*b7 + 140*b6) * q^47 + (48*b5 - 96*b2) * q^48 + (15*b5 - 15*b4 - 275*b2 - 290) * q^49 + (-150*b7 + 150*b6 - 150*b3 - 150*b1) * q^50 + (65*b4 + 149) * q^51 + (-165*b4 + 190) * q^53 + (-362*b7 + 190*b6 - 362*b3 - 190*b1) * q^54 + (70*b5 - 70*b4 - 220*b2 - 290) * q^55 + (60*b5 + 116*b2) * q^56 + (13*b7 - 199*b6) * q^57 + (214*b3 + 167*b1) * q^58 + (-377*b3 - 55*b1) * q^59 + (-200*b7 - 200*b6) * q^60 + (-200*b5 + 151*b2) * q^61 + (-180*b5 + 180*b4 - 340*b2 - 160) * q^62 + (-130*b7 + 130*b6 - 130*b3 - 130*b1) * q^63 + (-95*b4 + 588) * q^64 + (370*b4 - 614) * q^66 + (-283*b7 - 91*b6 - 283*b3 + 91*b1) * q^67 + (-135*b5 + 135*b4 + 185*b2 + 320) * q^68 + (-135*b5 + 172*b2) * q^69 + (20*b7 - 40*b6) * q^70 + (-11*b3 - 105*b1) * q^71 + (-460*b3 - 235*b1) * q^72 + (250*b7 - 85*b6) * q^73 + (205*b5 + 527*b2) * q^74 + (-75*b5 + 75*b4 - 900*b2 - 825) * q^75 + (-100*b7 - 340*b6 - 100*b3 + 340*b1) * q^76 + (121*b4 - 67) * q^77 + (40*b4 + 140) * q^79 + (-660*b7 - 255*b6 - 660*b3 + 255*b1) * q^80 + (-120*b5 + 120*b4 - 121*b2 - 1) * q^81 + (319*b5 + 917*b2) * q^82 + (-180*b7 - 100*b6) * q^83 + (-500*b3 - 220*b1) * q^84 + (-750*b3 + 295*b1) * q^85 + (-746*b7 + 470*b6) * q^86 + (201*b5 + 908*b2) * q^87 + (-172*b5 + 172*b4 - 596*b2 - 424) * q^88 + (-523*b7 - 125*b6 - 523*b3 + 125*b1) * q^89 + (-125*b4 + 300) * q^90 + (-280*b4 - 660) * q^92 + (1080*b7 - 40*b6 + 1080*b3 + 40*b1) * q^93 + (-320*b5 + 320*b4 - 680*b2 - 360) * q^94 + (390*b5 - 440*b2) * q^95 + (-800*b7 + 320*b6) * q^96 + (-27*b3 + 469*b1) * q^97 + (520*b3 + 245*b1) * q^98 + (910*b7 - 390*b6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 10 q^{3} + 10 q^{4} - 70 q^{9}+O(q^{10})$$ 8 * q - 10 * q^3 + 10 * q^4 - 70 * q^9 $$8 q - 10 q^{3} + 10 q^{4} - 70 q^{9} - 10 q^{10} - 560 q^{12} - 184 q^{14} - 114 q^{16} + 140 q^{17} + 340 q^{22} + 290 q^{23} - 300 q^{25} + 1340 q^{27} - 68 q^{29} - 280 q^{30} + 140 q^{35} + 1450 q^{36} - 1240 q^{38} - 740 q^{40} + 740 q^{42} + 910 q^{43} + 480 q^{48} - 1130 q^{49} + 932 q^{51} + 2180 q^{53} - 1020 q^{55} - 344 q^{56} - 1004 q^{61} - 1000 q^{62} + 5084 q^{64} - 6392 q^{66} + 1010 q^{68} - 958 q^{69} - 1698 q^{74} - 3450 q^{75} - 1020 q^{77} + 960 q^{79} - 244 q^{81} - 3030 q^{82} - 3230 q^{87} - 2040 q^{88} + 2900 q^{90} - 4160 q^{92} - 2080 q^{94} + 2540 q^{95}+O(q^{100})$$ 8 * q - 10 * q^3 + 10 * q^4 - 70 * q^9 - 10 * q^10 - 560 * q^12 - 184 * q^14 - 114 * q^16 + 140 * q^17 + 340 * q^22 + 290 * q^23 - 300 * q^25 + 1340 * q^27 - 68 * q^29 - 280 * q^30 + 140 * q^35 + 1450 * q^36 - 1240 * q^38 - 740 * q^40 + 740 * q^42 + 910 * q^43 + 480 * q^48 - 1130 * q^49 + 932 * q^51 + 2180 * q^53 - 1020 * q^55 - 344 * q^56 - 1004 * q^61 - 1000 * q^62 + 5084 * q^64 - 6392 * q^66 + 1010 * q^68 - 958 * q^69 - 1698 * q^74 - 3450 * q^75 - 1020 * q^77 + 960 * q^79 - 244 * q^81 - 3030 * q^82 - 3230 * q^87 - 2040 * q^88 + 2900 * q^90 - 4160 * q^92 - 2080 * q^94 + 2540 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040$$ (9*v^6 - 65*v^4 + 585*v^2 - 1296) / 1040 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} - 181\nu ) / 260$$ (-v^7 - 181*v) / 260 $$\beta_{4}$$ $$=$$ $$( \nu^{6} + 116 ) / 65$$ (v^6 + 116) / 65 $$\beta_{5}$$ $$=$$ $$( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040$$ (-29*v^6 + 325*v^4 - 1885*v^2 + 4176) / 1040 $$\beta_{6}$$ $$=$$ $$( 9\nu^{7} - 65\nu^{5} + 585\nu^{3} - 256\nu ) / 1040$$ (9*v^7 - 65*v^5 + 585*v^3 - 256*v) / 1040 $$\beta_{7}$$ $$=$$ $$( 9\nu^{7} - 65\nu^{5} + 377\nu^{3} - 256\nu ) / 832$$ (9*v^7 - 65*v^5 + 377*v^3 - 256*v) / 832
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + 5\beta_{2} + 4$$ b5 - b4 + 5*b2 + 4 $$\nu^{3}$$ $$=$$ $$-4\beta_{7} + 5\beta_{6}$$ -4*b7 + 5*b6 $$\nu^{4}$$ $$=$$ $$9\beta_{5} + 29\beta_{2}$$ 9*b5 + 29*b2 $$\nu^{5}$$ $$=$$ $$-36\beta_{7} + 29\beta_{6} - 36\beta_{3} - 29\beta_1$$ -36*b7 + 29*b6 - 36*b3 - 29*b1 $$\nu^{6}$$ $$=$$ $$65\beta_{4} - 116$$ 65*b4 - 116 $$\nu^{7}$$ $$=$$ $$-260\beta_{3} - 181\beta_1$$ -260*b3 - 181*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 2.21837 + 1.28078i −1.35234 − 0.780776i 1.35234 + 0.780776i −2.21837 − 1.28078i 2.21837 − 1.28078i −1.35234 + 0.780776i 1.35234 − 0.780776i −2.21837 + 1.28078i
−3.95042 + 2.28078i −4.34233 7.52113i 6.40388 11.0918i 2.80776i 34.3081 + 19.8078i 8.28055 + 4.78078i 21.9309i −24.2116 + 41.9358i −6.40388 11.0918i
23.2 −0.379706 + 0.219224i 1.84233 + 3.19101i −3.90388 + 6.76172i 17.8078i −1.39909 0.807764i 4.70983 + 2.71922i 6.93087i 6.71165 11.6249i 3.90388 + 6.76172i
23.3 0.379706 0.219224i 1.84233 + 3.19101i −3.90388 + 6.76172i 17.8078i 1.39909 + 0.807764i −4.70983 2.71922i 6.93087i 6.71165 11.6249i 3.90388 + 6.76172i
23.4 3.95042 2.28078i −4.34233 7.52113i 6.40388 11.0918i 2.80776i −34.3081 19.8078i −8.28055 4.78078i 21.9309i −24.2116 + 41.9358i −6.40388 11.0918i
147.1 −3.95042 2.28078i −4.34233 + 7.52113i 6.40388 + 11.0918i 2.80776i 34.3081 19.8078i 8.28055 4.78078i 21.9309i −24.2116 41.9358i −6.40388 + 11.0918i
147.2 −0.379706 0.219224i 1.84233 3.19101i −3.90388 6.76172i 17.8078i −1.39909 + 0.807764i 4.70983 2.71922i 6.93087i 6.71165 + 11.6249i 3.90388 6.76172i
147.3 0.379706 + 0.219224i 1.84233 3.19101i −3.90388 6.76172i 17.8078i 1.39909 0.807764i −4.70983 + 2.71922i 6.93087i 6.71165 + 11.6249i 3.90388 6.76172i
147.4 3.95042 + 2.28078i −4.34233 + 7.52113i 6.40388 + 11.0918i 2.80776i −34.3081 + 19.8078i −8.28055 + 4.78078i 21.9309i −24.2116 41.9358i −6.40388 + 11.0918i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 147.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.e.g 8
13.b even 2 1 inner 169.4.e.g 8
13.c even 3 1 169.4.b.e 4
13.c even 3 1 inner 169.4.e.g 8
13.d odd 4 1 13.4.c.b 4
13.d odd 4 1 169.4.c.f 4
13.e even 6 1 169.4.b.e 4
13.e even 6 1 inner 169.4.e.g 8
13.f odd 12 1 13.4.c.b 4
13.f odd 12 1 169.4.a.f 2
13.f odd 12 1 169.4.a.j 2
13.f odd 12 1 169.4.c.f 4
39.f even 4 1 117.4.g.d 4
39.k even 12 1 117.4.g.d 4
39.k even 12 1 1521.4.a.l 2
39.k even 12 1 1521.4.a.t 2
52.f even 4 1 208.4.i.e 4
52.l even 12 1 208.4.i.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 13.d odd 4 1
13.4.c.b 4 13.f odd 12 1
117.4.g.d 4 39.f even 4 1
117.4.g.d 4 39.k even 12 1
169.4.a.f 2 13.f odd 12 1
169.4.a.j 2 13.f odd 12 1
169.4.b.e 4 13.c even 3 1
169.4.b.e 4 13.e even 6 1
169.4.c.f 4 13.d odd 4 1
169.4.c.f 4 13.f odd 12 1
169.4.e.g 8 1.a even 1 1 trivial
169.4.e.g 8 13.b even 2 1 inner
169.4.e.g 8 13.c even 3 1 inner
169.4.e.g 8 13.e even 6 1 inner
208.4.i.e 4 52.f even 4 1
208.4.i.e 4 52.l even 12 1
1521.4.a.l 2 39.k even 12 1
1521.4.a.t 2 39.k even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 21T_{2}^{6} + 437T_{2}^{4} - 84T_{2}^{2} + 16$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 21 T^{6} + 437 T^{4} + \cdots + 16$$
$3$ $$(T^{4} + 5 T^{3} + 57 T^{2} - 160 T + 1024)^{2}$$
$5$ $$(T^{4} + 325 T^{2} + 2500)^{2}$$
$7$ $$T^{8} - 121 T^{6} + 11937 T^{4} + \cdots + 7311616$$
$11$ $$T^{8} - 2057 T^{6} + \cdots + 610673479936$$
$13$ $$T^{8}$$
$17$ $$(T^{4} - 70 T^{3} + 4763 T^{2} + \cdots + 18769)^{2}$$
$19$ $$T^{8} + \cdots + 559724432982016$$
$23$ $$(T^{4} - 145 T^{3} + 20397 T^{2} + \cdots + 394384)^{2}$$
$29$ $$(T^{4} + 34 T^{3} + 16167 T^{2} + \cdots + 225330121)^{2}$$
$31$ $$(T^{4} + 94800 T^{2} + \cdots + 1413760000)^{2}$$
$37$ $$T^{8} - 34506 T^{6} + \cdots + 403490473681$$
$41$ $$T^{8} - 148122 T^{6} + \cdots + 24\!\cdots\!41$$
$43$ $$(T^{4} - 455 T^{3} + 195257 T^{2} + \cdots + 138485824)^{2}$$
$47$ $$(T^{4} + 168400 T^{2} + \cdots + 6789760000)^{2}$$
$53$ $$(T^{2} - 545 T - 41450)^{4}$$
$59$ $$T^{8} - 352953 T^{6} + \cdots + 51\!\cdots\!16$$
$61$ $$(T^{4} + 502 T^{3} + \cdots + 11448786001)^{2}$$
$67$ $$T^{8} - 183201 T^{6} + \cdots + 20\!\cdots\!36$$
$71$ $$T^{8} - 101777 T^{6} + \cdots + 33\!\cdots\!76$$
$73$ $$(T^{4} + 232525 T^{2} + \cdots + 3008522500)^{2}$$
$79$ $$(T^{2} - 240 T + 7600)^{4}$$
$83$ $$(T^{4} + 118800 T^{2} + \cdots + 655360000)^{2}$$
$89$ $$T^{8} - 556933 T^{6} + \cdots + 45\!\cdots\!56$$
$97$ $$T^{8} - 1955781 T^{6} + \cdots + 63\!\cdots\!56$$