# Properties

 Label 15.5.f.a Level $15$ Weight $5$ Character orbit 15.f Analytic conductor $1.551$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 15.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.55054944626$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 60x^{5} + 1973x^{4} - 3300x^{3} + 1800x^{2} + 31560x + 276676$$ x^8 - 60*x^5 + 1973*x^4 - 3300*x^3 + 1800*x^2 + 31560*x + 276676 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} - 12 \beta_{2} - \beta_1) q^{4} + ( - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} + 8 \beta_{2} + \cdots - 10) q^{5}+ \cdots + 27 \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^2 + b5 * q^3 + (b6 + b5 + b4 - 2*b3 - 12*b2 - b1) * q^4 + (-b7 + 2*b6 - 3*b5 + 2*b4 - b3 + 8*b2 + b1 - 10) * q^5 + (3*b7 + 2*b5 - 3*b4 + 2*b3 - 3*b1 + 3) * q^6 + (-5*b7 - 5*b6 + 3*b3 + 4*b1 + 5) * q^7 + (5*b7 - 5*b6 - 2*b5 - 7*b4 + 5*b3 + 25*b2 + 20) * q^8 + 27*b2 * q^9 $$q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} - 12 \beta_{2} - \beta_1) q^{4} + ( - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} + 8 \beta_{2} + \cdots - 10) q^{5}+ \cdots + ( - 108 \beta_{6} + 432 \beta_{5} - 243 \beta_{4} - 324 \beta_{3} + \cdots + 243 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + b5 * q^3 + (b6 + b5 + b4 - 2*b3 - 12*b2 - b1) * q^4 + (-b7 + 2*b6 - 3*b5 + 2*b4 - b3 + 8*b2 + b1 - 10) * q^5 + (3*b7 + 2*b5 - 3*b4 + 2*b3 - 3*b1 + 3) * q^6 + (-5*b7 - 5*b6 + 3*b3 + 4*b1 + 5) * q^7 + (5*b7 - 5*b6 - 2*b5 - 7*b4 + 5*b3 + 25*b2 + 20) * q^8 + 27*b2 * q^9 + (-4*b7 + 3*b6 - 17*b5 + 13*b4 - 14*b3 - 43*b2 - 11*b1 + 15) * q^10 + (4*b7 + 16*b5 - 9*b4 + 16*b3 - 9*b1 - 38) * q^11 + (10*b3 + 45*b2 + 9*b1 - 45) * q^12 + (-5*b7 + 5*b6 - 14*b5 - 8*b4 - 5*b3 - 45*b2 - 40) * q^13 + (8*b6 + 28*b5 + 13*b4 - 36*b3 - 10*b2 - 13*b1) * q^14 + (6*b7 + 3*b6 - 7*b5 - 12*b4 - 9*b3 - 33*b2 + 9*b1 + 15) * q^15 + (-7*b7 + 7*b5 + 37*b4 + 7*b3 + 37*b1 + 81) * q^16 + (5*b7 + 5*b6 + 29*b3 - 115*b2 + 110) * q^17 + 27*b4 * q^18 + (-18*b6 + 42*b5 - 48*b4 - 24*b3 + 48*b2 + 48*b1) * q^19 + (b7 - 27*b6 - 32*b5 - 42*b4 + b3 + 207*b2 + 49*b1 + 70) * q^20 + (-18*b7 - 2*b5 - 27*b4 - 2*b3 - 27*b1 + 108) * q^21 + (35*b7 + 35*b6 + 63*b3 + 120*b2 - 100*b1 - 155) * q^22 + (-30*b7 + 30*b6 - 72*b5 + 94*b4 - 30*b3 - 210*b2 - 180) * q^23 + (-9*b6 + 21*b5 + 36*b4 - 12*b3 - 216*b2 - 36*b1) * q^24 + (39*b7 - 23*b6 + 22*b5 - 8*b4 + 19*b3 + 58*b2 - 94*b1 - 170) * q^25 + (-40*b7 - 40*b5 - 10*b4 - 40*b3 - 10*b1 - 358) * q^26 - 27*b3 * q^27 + (25*b7 - 25*b6 + 22*b5 - 164*b4 + 25*b3 + 460*b2 + 435) * q^28 + (-46*b6 - 26*b5 - b4 + 72*b3 + 386*b2 + b1) * q^29 + (-36*b7 - 18*b6 + 7*b5 - 18*b4 + 19*b3 - 297*b2 + 81*b1 - 315) * q^30 + (48*b7 + 72*b5 + 102*b4 + 72*b3 + 102*b1 - 88) * q^31 + (-15*b7 - 15*b6 - 27*b3 - 615*b2 + 49*b1 + 630) * q^32 + (-15*b7 + 15*b6 - 62*b5 + 66*b4 - 15*b3 + 330*b2 + 345) * q^33 + (92*b6 - 88*b5 - 28*b4 - 4*b3 - 294*b2 + 28*b1) * q^34 + (10*b7 + 110*b6 + 80*b5 + 55*b4 - 110*b3 + 260*b2 - 65*b1 + 820) * q^35 + (27*b7 - 27*b5 - 27*b4 - 27*b3 - 27*b1 + 297) * q^36 + (-75*b7 - 75*b6 - 207*b3 + 515*b2 + 144*b1 - 440) * q^37 + (60*b7 - 60*b6 + 336*b5 - 54*b4 + 60*b3 - 930*b2 - 990) * q^38 + (54*b6 - 71*b5 + 9*b4 + 17*b3 - 306*b2 - 9*b1) * q^39 + (-118*b7 + 61*b6 + 11*b5 + 221*b4 + 92*b3 - 806*b2 + 33*b1 - 310) * q^40 + (118*b7 + 22*b5 - 168*b4 + 22*b3 - 168*b1 - 398) * q^41 + (-15*b7 - 15*b6 - 11*b3 + 930*b2 + 102*b1 - 915) * q^42 + (-10*b7 + 10*b6 + 32*b5 + 68*b4 - 10*b3 - 160*b2 - 150) * q^43 + (60*b6 - 180*b5 + 65*b4 + 120*b3 + 1038*b2 - 65*b1) * q^44 + (54*b7 + 27*b6 + 27*b5 + 27*b4 + 54*b3 - 297*b2 - 54*b1 - 270) * q^45 + (-62*b7 - 358*b5 - 178*b4 - 358*b3 - 178*b1 + 1722) * q^46 + (-60*b7 - 60*b6 - 168*b3 - 570*b2 + 118*b1 + 630) * q^47 + (90*b7 - 90*b6 + 208*b5 - 243*b4 + 90*b3 + 495*b2 + 405) * q^48 + (-182*b6 - 182*b5 + 178*b4 + 364*b3 - 1059*b2 - 178*b1) * q^49 + (-4*b7 - 122*b6 + 58*b5 - 37*b4 + 416*b3 + 2512*b2 + 34*b1 + 470) * q^50 + (30*b7 + 125*b5 + 15*b4 + 125*b3 + 15*b1 + 768) * q^51 + (-10*b7 - 10*b6 - 66*b3 + 160*b2 - 130*b1 - 150) * q^52 + (-25*b7 + 25*b6 - 362*b5 - 186*b4 - 25*b3 + 115*b2 + 140) * q^53 + (-81*b6 + 54*b5 - 81*b4 + 27*b3 + 162*b2 + 81*b1) * q^54 + (39*b7 - 23*b6 + 262*b5 - 398*b4 - 221*b3 - 1292*b2 + 146*b1 - 475) * q^55 + (-20*b7 - 140*b5 + 425*b4 - 140*b3 + 425*b1 - 3830) * q^56 + (90*b7 + 90*b6 + 18*b3 + 630*b2 - 342*b1 - 720) * q^57 + (-125*b7 + 125*b6 + 82*b5 + 682*b4 - 125*b3 + 230*b2 + 355) * q^58 + (148*b6 - 52*b5 - 257*b4 - 96*b3 + 382*b2 + 257*b1) * q^59 + (69*b7 + 42*b6 + 87*b5 - 18*b4 - 121*b3 - 1452*b2 - 354*b1 - 165) * q^60 + (-234*b7 + 234*b5 - 36*b4 + 234*b3 - 36*b1 - 478) * q^61 + (270*b7 + 270*b6 + 6*b3 - 3720*b2 - 532*b1 + 3450) * q^62 + (-135*b7 + 135*b6 - 54*b5 + 108*b4 - 135*b3 + 135) * q^63 + (65*b6 + 305*b5 - 55*b4 - 370*b3 + 334*b2 + 55*b1) * q^64 + (-43*b7 - 229*b6 + 306*b5 + 76*b4 - 163*b3 + 1229*b2 + 38*b1 - 130) * q^65 + (-90*b7 - 250*b5 + 405*b4 - 250*b3 + 405*b1 + 1296) * q^66 + (230*b7 + 230*b6 + 234*b3 + 1000*b2 + 380*b1 - 1230) * q^67 + (-120*b7 + 120*b6 - 120*b5 + 14*b4 - 120*b3 - 300*b2 - 180) * q^68 + (-102*b6 - 82*b5 - 372*b4 + 184*b3 - 660*b2 + 372*b1) * q^69 + (395*b7 - 185*b6 - 400*b5 - 430*b4 + 395*b3 + 740*b2 + 630*b1 + 725) * q^70 + (-380*b7 + 280*b5 + 240*b4 + 280*b3 + 240*b1 + 1132) * q^71 + (-135*b7 - 135*b6 + 189*b3 + 675*b2 + 189*b1 - 540) * q^72 + (150*b7 - 150*b6 - 372*b5 - 156*b4 + 150*b3 + 1525*b2 + 1375) * q^73 + (-552*b6 + 1008*b5 + 38*b4 - 456*b3 - 990*b2 - 38*b1) * q^74 + (-234*b7 - 162*b6 - 257*b5 + 423*b4 - 114*b3 - 198*b2 + 189*b1 - 630) * q^75 + (546*b7 + 294*b5 - 936*b4 + 294*b3 - 936*b1 + 270) * q^76 + (-90*b7 - 90*b6 + 366*b3 + 90*b2 - 622*b1) * q^77 + (-150*b7 + 150*b6 - 518*b5 - 60*b4 - 150*b3 - 660*b2 - 510) * q^78 + (260*b6 + 80*b5 + 650*b4 - 340*b3 + 2000*b2 - 650*b1) * q^79 + (b7 + 533*b6 - 1132*b5 + 33*b4 - 419*b3 - 193*b2 - 856*b1 + 1330) * q^80 - 729 * q^81 + (-220*b7 - 220*b6 + 1116*b3 + 3510*b2 + 278*b1 - 3290) * q^82 + (400*b7 - 400*b6 + 140*b5 - 334*b4 + 400*b3 - 3850*b2 - 4250) * q^83 + (342*b6 + 182*b5 + 567*b4 - 524*b3 - 990*b2 - 567*b1) * q^84 + (-467*b7 + 209*b6 - 416*b5 - 176*b4 - 707*b3 + 1081*b2 - 18*b1 - 1720) * q^85 + (184*b7 - 44*b5 - 354*b4 - 44*b3 - 354*b1 + 1732) * q^86 + (-135*b7 - 135*b6 - 339*b3 - 1260*b2 - 144*b1 + 1395) * q^87 + (145*b7 - 145*b6 + 478*b5 + 208*b4 + 145*b3 - 1720*b2 - 1865) * q^88 + (582*b6 - 558*b5 + 162*b4 - 24*b3 + 2388*b2 - 162*b1) * q^89 + (81*b7 + 108*b6 - 297*b5 - 297*b4 + 351*b3 + 297*b2 - 351*b1 + 1080) * q^90 + (388*b7 + 632*b5 + 122*b4 + 632*b3 + 122*b1 + 1872) * q^91 + (-710*b7 - 710*b6 + 538*b3 + 4330*b2 + 2474*b1 - 3620) * q^92 + (450*b7 - 450*b6 + 464*b5 - 468*b4 + 450*b3 + 1980*b2 + 1530) * q^93 + (-446*b6 + 814*b5 - 956*b4 - 368*b3 - 856*b2 + 956*b1) * q^94 + (-24*b7 + 108*b6 - 552*b5 + 1128*b4 - 924*b3 - 4218*b2 + 774*b1 + 2370) * q^95 + (57*b7 + 683*b5 - 192*b4 + 683*b3 - 192*b1 - 537) * q^96 + (70*b7 + 70*b6 - 1146*b3 - 7365*b2 - 140*b1 + 7295) * q^97 + (-550*b7 + 550*b6 - 356*b5 + 223*b4 - 550*b3 + 5530*b2 + 6080) * q^98 + (-108*b6 + 432*b5 - 243*b4 - 324*b3 - 918*b2 + 243*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 84 q^{5} + 36 q^{6} + 20 q^{7} + 180 q^{8}+O(q^{10})$$ 8 * q - 84 * q^5 + 36 * q^6 + 20 * q^7 + 180 * q^8 $$8 q - 84 q^{5} + 36 q^{6} + 20 q^{7} + 180 q^{8} + 104 q^{10} - 288 q^{11} - 360 q^{12} - 340 q^{13} + 144 q^{15} + 620 q^{16} + 900 q^{17} + 564 q^{20} + 792 q^{21} - 1100 q^{22} - 1560 q^{23} - 1204 q^{25} - 3024 q^{26} + 3580 q^{28} - 2664 q^{30} - 512 q^{31} + 4980 q^{32} + 2700 q^{33} + 6600 q^{35} + 2484 q^{36} - 3820 q^{37} - 7680 q^{38} - 2952 q^{40} - 2712 q^{41} - 7380 q^{42} - 1240 q^{43} - 1944 q^{45} + 13528 q^{46} + 4800 q^{47} + 3600 q^{48} + 3744 q^{50} + 6264 q^{51} - 1240 q^{52} + 1020 q^{53} - 3644 q^{55} - 30720 q^{56} - 5400 q^{57} + 2340 q^{58} - 1044 q^{60} - 4760 q^{61} + 28680 q^{62} + 540 q^{63} - 1212 q^{65} + 10008 q^{66} - 8920 q^{67} - 1920 q^{68} + 7380 q^{70} + 7536 q^{71} - 4860 q^{72} + 11600 q^{73} - 5976 q^{75} + 4344 q^{76} - 360 q^{77} - 4680 q^{78} + 10644 q^{80} - 5832 q^{81} - 27200 q^{82} - 32400 q^{83} - 15628 q^{85} + 14592 q^{86} + 10620 q^{87} - 14340 q^{88} + 8964 q^{90} + 16528 q^{91} - 31800 q^{92} + 14040 q^{93} + 18864 q^{95} - 4068 q^{96} + 58640 q^{97} + 46440 q^{98}+O(q^{100})$$ 8 * q - 84 * q^5 + 36 * q^6 + 20 * q^7 + 180 * q^8 + 104 * q^10 - 288 * q^11 - 360 * q^12 - 340 * q^13 + 144 * q^15 + 620 * q^16 + 900 * q^17 + 564 * q^20 + 792 * q^21 - 1100 * q^22 - 1560 * q^23 - 1204 * q^25 - 3024 * q^26 + 3580 * q^28 - 2664 * q^30 - 512 * q^31 + 4980 * q^32 + 2700 * q^33 + 6600 * q^35 + 2484 * q^36 - 3820 * q^37 - 7680 * q^38 - 2952 * q^40 - 2712 * q^41 - 7380 * q^42 - 1240 * q^43 - 1944 * q^45 + 13528 * q^46 + 4800 * q^47 + 3600 * q^48 + 3744 * q^50 + 6264 * q^51 - 1240 * q^52 + 1020 * q^53 - 3644 * q^55 - 30720 * q^56 - 5400 * q^57 + 2340 * q^58 - 1044 * q^60 - 4760 * q^61 + 28680 * q^62 + 540 * q^63 - 1212 * q^65 + 10008 * q^66 - 8920 * q^67 - 1920 * q^68 + 7380 * q^70 + 7536 * q^71 - 4860 * q^72 + 11600 * q^73 - 5976 * q^75 + 4344 * q^76 - 360 * q^77 - 4680 * q^78 + 10644 * q^80 - 5832 * q^81 - 27200 * q^82 - 32400 * q^83 - 15628 * q^85 + 14592 * q^86 + 10620 * q^87 - 14340 * q^88 + 8964 * q^90 + 16528 * q^91 - 31800 * q^92 + 14040 * q^93 + 18864 * q^95 - 4068 * q^96 + 58640 * q^97 + 46440 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 60x^{5} + 1973x^{4} - 3300x^{3} + 1800x^{2} + 31560x + 276676$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 19857 \nu^{7} - 350053 \nu^{6} - 190938 \nu^{5} - 1295568 \nu^{4} + 60124233 \nu^{3} - 934582407 \nu^{2} + 716887878 \nu + 368125308 ) / 10440376750$$ (19857*v^7 - 350053*v^6 - 190938*v^5 - 1295568*v^4 + 60124233*v^3 - 934582407*v^2 + 716887878*v + 368125308) / 10440376750 $$\beta_{3}$$ $$=$$ $$( - 1224717 \nu^{7} + 40833643 \nu^{6} + 212097928 \nu^{5} + 1138297958 \nu^{4} - 4285577073 \nu^{3} + 47520018367 \nu^{2} + \cdots + 621224634602 ) / 83523014000$$ (-1224717*v^7 + 40833643*v^6 + 212097928*v^5 + 1138297958*v^4 - 4285577073*v^3 + 47520018367*v^2 + 89682965932*v + 621224634602) / 83523014000 $$\beta_{4}$$ $$=$$ $$( - 1331 \nu^{7} - 726 \nu^{6} - 396 \nu^{5} + 79644 \nu^{4} - 3304389 \nu^{3} + 2589906 \nu^{2} - 983124 \nu - 20889564 ) / 39697250$$ (-1331*v^7 - 726*v^6 - 396*v^5 + 79644*v^4 - 3304389*v^3 + 2589906*v^2 - 983124*v - 20889564) / 39697250 $$\beta_{5}$$ $$=$$ $$( - 6913292 \nu^{7} - 45272557 \nu^{6} - 24694122 \nu^{5} + 1350453158 \nu^{4} - 5477763248 \nu^{3} - 37379548933 \nu^{2} + \cdots + 599396712902 ) / 83523014000$$ (-6913292*v^7 - 45272557*v^6 - 24694122*v^5 + 1350453158*v^4 - 5477763248*v^3 - 37379548933*v^2 + 47175232482*v + 599396712902) / 83523014000 $$\beta_{6}$$ $$=$$ $$( 468490 \nu^{7} + 2002219 \nu^{6} + 16278122 \nu^{5} + 18734582 \nu^{4} + 693074870 \nu^{3} + 45879131 \nu^{2} + 15134603638 \nu + 30778570718 ) / 3340920560$$ (468490*v^7 + 2002219*v^6 + 16278122*v^5 + 18734582*v^4 + 693074870*v^3 + 45879131*v^2 + 15134603638*v + 30778570718) / 3340920560 $$\beta_{7}$$ $$=$$ $$( - 47399 \nu^{7} - 25854 \nu^{6} + 707666 \nu^{5} + 4673476 \nu^{4} - 77977231 \nu^{3} + 70577574 \nu^{2} + 999020754 \nu - 1154530656 ) / 317578000$$ (-47399*v^7 - 25854*v^6 + 707666*v^5 + 4673476*v^4 - 77977231*v^3 + 70577574*v^2 + 999020754*v - 1154530656) / 317578000
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} - 28\beta_{2} - \beta_1$$ b6 + b5 + b4 - 2*b3 - 28*b2 - b1 $$\nu^{3}$$ $$=$$ $$5\beta_{7} - 5\beta_{6} - 2\beta_{5} - 39\beta_{4} + 5\beta_{3} + 25\beta_{2} + 20$$ 5*b7 - 5*b6 - 2*b5 - 39*b4 + 5*b3 + 25*b2 + 20 $$\nu^{4}$$ $$=$$ $$-55\beta_{7} + 55\beta_{5} + 85\beta_{4} + 55\beta_{3} + 85\beta _1 - 959$$ -55*b7 + 55*b5 + 85*b4 + 55*b3 + 85*b1 - 959 $$\nu^{5}$$ $$=$$ $$305\beta_{7} + 305\beta_{6} - 475\beta_{3} - 2215\beta_{2} - 1679\beta _1 + 1910$$ 305*b7 + 305*b6 - 475*b3 - 2215*b2 - 1679*b1 + 1910 $$\nu^{6}$$ $$=$$ $$-2799\beta_{6} - 2559\beta_{5} - 5319\beta_{4} + 5358\beta_{3} + 42542\beta_{2} + 5319\beta_1$$ -2799*b6 - 2559*b5 - 5319*b4 + 5358*b3 + 42542*b2 + 5319*b1 $$\nu^{7}$$ $$=$$ $$-15795\beta_{7} + 15795\beta_{6} + 11598\beta_{5} + 76931\beta_{4} - 15795\beta_{3} - 139095\beta_{2} - 123300$$ -15795*b7 + 15795*b6 + 11598*b5 + 76931*b4 - 15795*b3 - 139095*b2 - 123300

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −5.02811 − 5.02811i −2.08045 − 2.08045i 3.30519 + 3.30519i 3.80336 + 3.80336i −5.02811 + 5.02811i −2.08045 + 2.08045i 3.30519 − 3.30519i 3.80336 − 3.80336i
−5.02811 5.02811i −3.67423 + 3.67423i 34.5637i −23.8949 7.35070i 36.9489 −38.5593 38.5593i 93.3405 93.3405i 27.0000i 83.1861 + 157.106i
7.2 −2.08045 2.08045i 3.67423 3.67423i 7.34348i −8.43390 23.5344i −15.2881 65.1093 + 65.1093i −48.5649 + 48.5649i 27.0000i −31.4158 + 66.5084i
7.3 3.30519 + 3.30519i 3.67423 3.67423i 5.84858i −16.2403 + 19.0066i 24.2881 −33.1649 33.1649i 33.5524 33.5524i 27.0000i −116.498 + 9.14312i
7.4 3.80336 + 3.80336i −3.67423 + 3.67423i 12.9311i 6.56915 24.1215i −27.9489 16.6149 + 16.6149i 11.6720 11.6720i 27.0000i 116.728 66.7579i
13.1 −5.02811 + 5.02811i −3.67423 3.67423i 34.5637i −23.8949 + 7.35070i 36.9489 −38.5593 + 38.5593i 93.3405 + 93.3405i 27.0000i 83.1861 157.106i
13.2 −2.08045 + 2.08045i 3.67423 + 3.67423i 7.34348i −8.43390 + 23.5344i −15.2881 65.1093 65.1093i −48.5649 48.5649i 27.0000i −31.4158 66.5084i
13.3 3.30519 3.30519i 3.67423 + 3.67423i 5.84858i −16.2403 19.0066i 24.2881 −33.1649 + 33.1649i 33.5524 + 33.5524i 27.0000i −116.498 9.14312i
13.4 3.80336 3.80336i −3.67423 3.67423i 12.9311i 6.56915 + 24.1215i −27.9489 16.6149 16.6149i 11.6720 + 11.6720i 27.0000i 116.728 + 66.7579i
 $$n$$: 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.5.f.a 8
3.b odd 2 1 45.5.g.e 8
4.b odd 2 1 240.5.bg.c 8
5.b even 2 1 75.5.f.e 8
5.c odd 4 1 inner 15.5.f.a 8
5.c odd 4 1 75.5.f.e 8
15.d odd 2 1 225.5.g.m 8
15.e even 4 1 45.5.g.e 8
15.e even 4 1 225.5.g.m 8
20.e even 4 1 240.5.bg.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.5.f.a 8 1.a even 1 1 trivial
15.5.f.a 8 5.c odd 4 1 inner
45.5.g.e 8 3.b odd 2 1
45.5.g.e 8 15.e even 4 1
75.5.f.e 8 5.b even 2 1
75.5.f.e 8 5.c odd 4 1
225.5.g.m 8 15.d odd 2 1
225.5.g.m 8 15.e even 4 1
240.5.bg.c 8 4.b odd 2 1
240.5.bg.c 8 20.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 60 T^{5} + 1973 T^{4} + \cdots + 276676$$
$3$ $$(T^{4} + 729)^{2}$$
$5$ $$T^{8} + 84 T^{7} + \cdots + 152587890625$$
$7$ $$T^{8} - 20 T^{7} + \cdots + 30620728960000$$
$11$ $$(T^{4} + 144 T^{3} - 16742 T^{2} + \cdots + 27154144)^{2}$$
$13$ $$T^{8} + 340 T^{7} + \cdots + 158448579136$$
$17$ $$T^{8} + \cdots + 125636659418176$$
$19$ $$T^{8} + 567288 T^{6} + \cdots + 82\!\cdots\!00$$
$23$ $$T^{8} + 1560 T^{7} + \cdots + 40\!\cdots\!00$$
$29$ $$T^{8} + 1665012 T^{6} + \cdots + 23\!\cdots\!00$$
$31$ $$(T^{4} + 256 T^{3} + \cdots + 388673200000)^{2}$$
$37$ $$T^{8} + 3820 T^{7} + \cdots + 60\!\cdots\!00$$
$41$ $$(T^{4} + 1356 T^{3} + \cdots - 1195607024000)^{2}$$
$43$ $$T^{8} + 1240 T^{7} + \cdots + 37\!\cdots\!76$$
$47$ $$T^{8} - 4800 T^{7} + \cdots + 32\!\cdots\!76$$
$53$ $$T^{8} - 1020 T^{7} + \cdots + 29\!\cdots\!00$$
$59$ $$T^{8} + 28672428 T^{6} + \cdots + 12\!\cdots\!00$$
$61$ $$(T^{4} + 2380 T^{3} + \cdots - 51045861284864)^{2}$$
$67$ $$T^{8} + 8920 T^{7} + \cdots + 37\!\cdots\!96$$
$71$ $$(T^{4} - 3768 T^{3} + \cdots - 4392786466304)^{2}$$
$73$ $$T^{8} - 11600 T^{7} + \cdots + 48\!\cdots\!00$$
$79$ $$T^{8} + 118621200 T^{6} + \cdots + 60\!\cdots\!00$$
$83$ $$T^{8} + 32400 T^{7} + \cdots + 23\!\cdots\!56$$
$89$ $$T^{8} + 128964168 T^{6} + \cdots + 22\!\cdots\!00$$
$97$ $$T^{8} - 58640 T^{7} + \cdots + 23\!\cdots\!96$$