Properties

 Label 99.3.k.b Level $99$ Weight $3$ Character orbit 99.k Analytic conductor $2.698$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 99.k (of order $$10$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.69755461717$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 21x^{14} + 227x^{12} - 1488x^{10} + 24225x^{8} - 62832x^{6} + 64372x^{4} + 7986x^{2} + 14641$$ x^16 - 21*x^14 + 227*x^12 - 1488*x^10 + 24225*x^8 - 62832*x^6 + 64372*x^4 + 7986*x^2 + 14641 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + ( - \beta_{12} + \beta_{11} - \beta_{6} - \beta_{4} - 3 \beta_{2} - 1) q^{4} + (\beta_{15} - \beta_{14} + \beta_{9} + \beta_{8} + \beta_{3} - \beta_1) q^{5} + (\beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} - 2 \beta_{6} + 3) q^{7} + ( - 2 \beta_{14} + 3 \beta_{9} + 5 \beta_{8} + \beta_{3} - 2 \beta_1) q^{8}+O(q^{10})$$ q + b5 * q^2 + (-b12 + b11 - b6 - b4 - 3*b2 - 1) * q^4 + (b15 - b14 + b9 + b8 + b3 - b1) * q^5 + (b12 + b11 - b10 + b7 - 2*b6 + 3) * q^7 + (-2*b14 + 3*b9 + 5*b8 + b3 - 2*b1) * q^8 $$q + \beta_{5} q^{2} + ( - \beta_{12} + \beta_{11} - \beta_{6} - \beta_{4} - 3 \beta_{2} - 1) q^{4} + (\beta_{15} - \beta_{14} + \beta_{9} + \beta_{8} + \beta_{3} - \beta_1) q^{5} + (\beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} - 2 \beta_{6} + 3) q^{7} + ( - 2 \beta_{14} + 3 \beta_{9} + 5 \beta_{8} + \beta_{3} - 2 \beta_1) q^{8} + ( - 2 \beta_{11} - 2 \beta_{7} + 3 \beta_{6} + 3 \beta_{2}) q^{10} + (2 \beta_{15} - \beta_{14} + \beta_{8} + 2 \beta_{3} - 3 \beta_1) q^{11} + ( - \beta_{11} + \beta_{10} + 3 \beta_{7} - 2 \beta_{6} + \beta_{4} + 2 \beta_{2} - 2) q^{13} + ( - \beta_{15} + \beta_{13} + 3 \beta_{9} + 6 \beta_{5} - \beta_{3}) q^{14} + ( - 3 \beta_{11} + 6 \beta_{10} - 6 \beta_{7} + 13 \beta_{6} + 8 \beta_{4} + \cdots - 13) q^{16}+ \cdots + ( - 6 \beta_{15} - 2 \beta_{14} + 5 \beta_{13} - 13 \beta_{9} + 14 \beta_{8} + \cdots - 14 \beta_1) q^{98}+O(q^{100})$$ q + b5 * q^2 + (-b12 + b11 - b6 - b4 - 3*b2 - 1) * q^4 + (b15 - b14 + b9 + b8 + b3 - b1) * q^5 + (b12 + b11 - b10 + b7 - 2*b6 + 3) * q^7 + (-2*b14 + 3*b9 + 5*b8 + b3 - 2*b1) * q^8 + (-2*b11 - 2*b7 + 3*b6 + 3*b2) * q^10 + (2*b15 - b14 + b8 + 2*b3 - 3*b1) * q^11 + (-b11 + b10 + 3*b7 - 2*b6 + b4 + 2*b2 - 2) * q^13 + (-b15 + b13 + 3*b9 + 6*b5 - b3) * q^14 + (-3*b11 + 6*b10 - 6*b7 + 13*b6 + 8*b4 + 3*b2 - 13) * q^16 + (b15 + b14 - 2*b13 - 3*b9 - 3*b8 + 3*b5 - b3 + 6*b1) * q^17 + (3*b12 - 4*b10 + 3*b7 + 2*b6 - 9*b4 + 5*b2) * q^19 + (-2*b15 + 2*b14 - 9*b9 - 8*b8 - 4*b5 + b1) * q^20 + (-b12 - 2*b11 - 5*b10 + 2*b7 - 18*b6 + 2*b4 + 3*b2 + 12) * q^22 + (b14 + 2*b13 + 2*b9 + 5*b8 - 3*b5 - b3 + b1) * q^23 + (5*b12 + 6*b11 - 3*b10 + b7 - 4*b6 + 9*b4 + 3*b2 + 3) * q^25 + (-3*b15 + b14 - b13 + 11*b9 - 2*b8 - b5 - 3*b3 - 5*b1) * q^26 + (-5*b12 + 4*b11 + 5*b10 + 10*b6 - 15*b4 - 34*b2 - 20) * q^28 + (-b15 + 2*b14 + 2*b13 - 2*b9 + 9*b8 - 11*b5 - 3*b3 - 6*b1) * q^29 + (-10*b12 - 3*b11 + 7*b10 - 5*b7 - 4*b6 - b4 + 19*b2 + 9) * q^31 + (-2*b15 + 3*b14 - 2*b13 - 21*b9 - 21*b8 - 18*b5 + b3 + 21*b1) * q^32 + (2*b12 + 3*b11 - 4*b10 + b7 + 15*b6 - 17*b2 + 15) * q^34 + (2*b14 - 4*b13 - 6*b9 - 2*b8 - 3*b5 + 6*b1) * q^35 + (-2*b12 + 2*b11 + b10 + b7 - 3*b6 + 4*b4 - 15*b2 + 3) * q^37 + (-3*b15 + 3*b14 + 4*b13 + 9*b9 - 15*b8 + 15*b5 - 7*b3) * q^38 + (6*b12 + 6*b11 - 9*b10 + 9*b7 - 53*b6 + 7*b4 + 19*b2 + 52) * q^40 + (b15 + 6*b14 + b13 - 8*b9 + 3*b8 - 3*b3 + 6*b1) * q^41 + (-7*b11 - 7*b7 + 23*b6 - 30*b4 - 7*b2 - 15) * q^43 + (-6*b15 + b14 + b13 + 10*b9 + 10*b8 + 9*b5 - 5*b3 - 25*b1) * q^44 + (-3*b11 + 3*b10 - 4*b7 + 22*b6 + 18*b4 + 20*b2 - 20) * q^46 + (4*b15 - 4*b13 - 3*b9 - 8*b8 + 10*b5 + 7*b3 + 16*b1) * q^47 + (b12 + 3*b11 - 5*b10 + 6*b7 - 7*b6 + 32*b4 - 3*b2 + 7) * q^49 + (-b15 - b14 + 3*b13 - 10*b9 - 10*b8 + 6*b5 + 2*b3 + 16*b1) * q^50 + (9*b10 + 35*b6 - 57*b4 - 13*b2 - 79) * q^52 + (4*b15 - 8*b14 - 7*b9 + 4*b8 - 13*b5 - 19*b1) * q^53 + (7*b12 + 3*b11 + 2*b10 - 3*b7 - 28*b6 + 19*b4 - 10*b2 - 18) * q^55 + (-5*b14 - 5*b13 + 11*b9 + 27*b8 - 6*b5 + 5*b3 + 5*b1) * q^56 + (-10*b12 - 14*b11 + 7*b10 - 4*b7 - 38*b6 + 54*b4 + 68*b2 - 7) * q^58 + (7*b15 - 6*b14 + 6*b13 + 36*b9 + 17*b8 + 11*b5 + 7*b3 - 21*b1) * q^59 + (3*b12 - 11*b11 - 3*b10 + 42*b6 - 30*b4 - 49*b2 - 18) * q^61 + (5*b15 - 7*b14 - 7*b13 + 7*b9 + 15*b8 - 8*b5 + 9*b3 - 21*b1) * q^62 + (20*b12 + 2*b11 - 18*b10 + 10*b7 + 19*b6 - 9*b4 + 69*b2 + 89) * q^64 + (8*b15 - 9*b14 + 5*b13 + 4*b9 - 4*b8 - 5*b5 - b3 + 4*b1) * q^65 + (-7*b12 - 7*b11 + 14*b10 - 7*b7 + 29*b6 - 22*b2 - 1) * q^67 + (3*b15 - 9*b14 + 12*b13 + 2*b9 + 9*b8 + 13*b5 + 6*b3 - 2*b1) * q^68 + (7*b12 - 7*b11 - 4*b10 - 4*b7 + 11*b6 + 27*b4 + 31*b2 + 31) * q^70 + (b15 - b14 - 13*b13 - 2*b9 + 4*b8 + 7*b5 + 14*b3 - 11*b1) * q^71 + (-13*b12 - 13*b11 + 25*b10 - 25*b7 - 40*b6 + 22*b4 + 39*b2 + 5) * q^73 + (-b15 - 4*b14 - b13 + 5*b9 + 27*b8 + 2*b3 - 4*b1) * q^74 + (2*b12 + 29*b11 + 29*b7 - 33*b6 - 78*b4 - 113*b2 - 39) * q^76 + (2*b15 + 4*b14 - 3*b13 - 18*b9 - 14*b8 + 3*b5 - 2*b3 + 9*b1) * q^77 + (21*b11 - 21*b10 + 8*b7 + 25*b6 + 33*b4 - 12*b2 + 12) * q^79 + (-b15 + b13 + 30*b9 + 9*b8 + 42*b5 - 12*b3 - 18*b1) * q^80 + (-5*b12 + 5*b11 - 15*b10 + 10*b7 - 26*b6 + 48*b4 - 5*b2 + 26) * q^82 + (-8*b15 - 8*b14 + 10*b13 + 12*b9 + 12*b8 + 21*b5 + 2*b3 + 9*b1) * q^83 + (-15*b12 - 9*b10 - 15*b7 + 35*b6 - 80*b4 + 10*b2 - 61) * q^85 + (7*b15 + 7*b14 - 14*b9 - 23*b8 + b5 + 16*b1) * q^86 + (-19*b12 + 6*b11 + 26*b10 - 6*b7 - 67*b6 - 28*b4 - 75*b2 - 58) * q^88 + (5*b14 - 2*b13 - 6*b9 - 23*b8 + b5 - 5*b3 - 11*b1) * q^89 + (-3*b12 + 2*b11 - b10 + 5*b7 - 63*b6 + 24*b4 + 22*b2 + 1) * q^91 + (8*b15 + 11*b14 - 11*b13 - 33*b9 - 32*b8 - 21*b5 + 8*b3 + 22*b1) * q^92 + (25*b12 + 10*b11 - 25*b10 + 79*b6 - 15*b4 - 40*b2 + 49) * q^94 + (-8*b15 + 3*b14 + 3*b13 - 3*b9 + 10*b8 - 13*b5 + 2*b3 + 14*b1) * q^95 + (6*b12 + 13*b11 + 7*b10 + 3*b7 - 19*b6 + 22*b4 + 80*b2 + 86) * q^97 + (-6*b15 - 2*b14 + 5*b13 - 13*b9 + 14*b8 - 15*b5 - 8*b3 - 14*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 4 q^{4} + 30 q^{7}+O(q^{10})$$ 16 * q - 4 * q^4 + 30 * q^7 $$16 q - 4 q^{4} + 30 q^{7} - 30 q^{13} - 176 q^{16} + 90 q^{22} - 74 q^{25} - 50 q^{28} + 130 q^{31} + 328 q^{34} + 90 q^{37} + 450 q^{40} - 370 q^{46} - 54 q^{49} - 790 q^{52} - 476 q^{55} - 630 q^{58} + 210 q^{61} + 1104 q^{64} + 300 q^{67} + 268 q^{70} - 170 q^{73} + 30 q^{79} + 90 q^{82} - 610 q^{85} - 600 q^{88} - 402 q^{91} + 1030 q^{94} + 870 q^{97}+O(q^{100})$$ 16 * q - 4 * q^4 + 30 * q^7 - 30 * q^13 - 176 * q^16 + 90 * q^22 - 74 * q^25 - 50 * q^28 + 130 * q^31 + 328 * q^34 + 90 * q^37 + 450 * q^40 - 370 * q^46 - 54 * q^49 - 790 * q^52 - 476 * q^55 - 630 * q^58 + 210 * q^61 + 1104 * q^64 + 300 * q^67 + 268 * q^70 - 170 * q^73 + 30 * q^79 + 90 * q^82 - 610 * q^85 - 600 * q^88 - 402 * q^91 + 1030 * q^94 + 870 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 21x^{14} + 227x^{12} - 1488x^{10} + 24225x^{8} - 62832x^{6} + 64372x^{4} + 7986x^{2} + 14641$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 95551278 \nu^{14} - 126822633412 \nu^{12} + 2507387635134 \nu^{10} - 25685652894720 \nu^{8} + 157339424200095 \nu^{6} + \cdots - 29\!\cdots\!58 ) / 39\!\cdots\!45$$ (95551278*v^14 - 126822633412*v^12 + 2507387635134*v^10 - 25685652894720*v^8 + 157339424200095*v^6 - 2848556551427906*v^4 + 4847148905801004*v^2 - 2957606944853858) / 3911331152625545 $$\beta_{3}$$ $$=$$ $$( - 32130978280 \nu^{15} + 31948404670 \nu^{13} + 5924614934588 \nu^{11} - 93193803417051 \nu^{9} + 133796302845113 \nu^{7} + \cdots - 43\!\cdots\!77 \nu ) / 39\!\cdots\!45$$ (-32130978280*v^15 + 31948404670*v^13 + 5924614934588*v^11 - 93193803417051*v^9 + 133796302845113*v^7 - 13315710113037464*v^5 + 32652886804323412*v^3 - 43479585817911177*v) / 3911331152625545 $$\beta_{4}$$ $$=$$ $$( - 5854282560 \nu^{14} + 125428263227 \nu^{12} - 1388833610400 \nu^{10} + 9410848624160 \nu^{8} - 146776049383200 \nu^{6} + \cdots + 88541185404320 ) / 355575559329595$$ (-5854282560*v^14 + 125428263227*v^12 - 1388833610400*v^10 + 9410848624160*v^8 - 146776049383200*v^6 + 435164502510080*v^4 - 723417785330064*v^2 + 88541185404320) / 355575559329595 $$\beta_{5}$$ $$=$$ $$( - 5854282560 \nu^{15} + 125428263227 \nu^{13} - 1388833610400 \nu^{11} + 9410848624160 \nu^{9} - 146776049383200 \nu^{7} + \cdots + 88541185404320 \nu ) / 355575559329595$$ (-5854282560*v^15 + 125428263227*v^13 - 1388833610400*v^11 + 9410848624160*v^9 - 146776049383200*v^7 + 435164502510080*v^5 - 723417785330064*v^3 + 88541185404320*v) / 355575559329595 $$\beta_{6}$$ $$=$$ $$( 7530891822 \nu^{14} - 161922628410 \nu^{12} + 1773071437239 \nu^{10} - 11760003820535 \nu^{8} + 185084618505135 \nu^{6} + \cdots + 243765923221832 ) / 355575559329595$$ (7530891822*v^14 - 161922628410*v^12 + 1773071437239*v^10 - 11760003820535*v^8 + 185084618505135*v^6 - 549582649580084*v^4 + 383464959034245*v^2 + 243765923221832) / 355575559329595 $$\beta_{7}$$ $$=$$ $$( 115291499597 \nu^{14} - 2410283199611 \nu^{12} + 25350978529100 \nu^{10} - 157438624000303 \nu^{8} + \cdots + 13\!\cdots\!07 ) / 39\!\cdots\!45$$ (115291499597*v^14 - 2410283199611*v^12 + 25350978529100*v^10 - 157438624000303*v^8 + 2656571559543129*v^6 - 6231838875170656*v^4 - 5628083390615697*v^2 + 13794053936060607) / 3911331152625545 $$\beta_{8}$$ $$=$$ $$( - 147141366924 \nu^{15} + 3034037174595 \nu^{13} - 32273567888895 \nu^{11} + 207193723996925 \nu^{9} + \cdots - 753747908220945 \nu ) / 39\!\cdots\!45$$ (-147141366924*v^15 + 3034037174595*v^13 - 32273567888895*v^11 + 207193723996925*v^9 - 3493127922571590*v^7 + 7983662121563898*v^5 - 7328561282206395*v^3 - 753747908220945*v) / 3911331152625545 $$\beta_{9}$$ $$=$$ $$( 13385174382 \nu^{15} - 287350891637 \nu^{13} + 3161905047639 \nu^{11} - 21170852444695 \nu^{9} + 331860667888335 \nu^{7} + \cdots + 155224737817512 \nu ) / 355575559329595$$ (13385174382*v^15 - 287350891637*v^13 + 3161905047639*v^11 - 21170852444695*v^9 + 331860667888335*v^7 - 984747152090164*v^5 + 1106882744364309*v^3 + 155224737817512*v) / 355575559329595 $$\beta_{10}$$ $$=$$ $$( - 298410148268 \nu^{14} + 6273343728321 \nu^{12} - 67973023670350 \nu^{10} + 448617912246188 \nu^{8} + \cdots + 12\!\cdots\!73 ) / 39\!\cdots\!45$$ (-298410148268*v^14 + 6273343728321*v^12 - 67973023670350*v^10 + 448617912246188*v^8 - 7274459802857679*v^6 + 19128698026990893*v^4 - 21604623242768698*v^2 + 12113219634382273) / 3911331152625545 $$\beta_{11}$$ $$=$$ $$( - 312289527222 \nu^{14} + 6157499382802 \nu^{12} - 62836567910801 \nu^{10} + 380954707458452 \nu^{8} + \cdots - 13\!\cdots\!26 ) / 39\!\cdots\!45$$ (-312289527222*v^14 + 6157499382802*v^12 - 62836567910801*v^10 + 380954707458452*v^8 - 7033948088071471*v^6 + 10251501399039197*v^4 - 2385515339301303*v^2 - 13973111793682426) / 3911331152625545 $$\beta_{12}$$ $$=$$ $$( - 88180182 \nu^{14} + 1926186986 \nu^{12} - 21449350233 \nu^{10} + 145597396588 \nu^{8} - 2222070588469 \nu^{6} + \cdots - 484613427771 ) / 924883223605$$ (-88180182*v^14 + 1926186986*v^12 - 21449350233*v^10 + 145597396588*v^8 - 2222070588469*v^6 + 7173219951326*v^4 - 7225398476758*v^2 - 484613427771) / 924883223605 $$\beta_{13}$$ $$=$$ $$( 598241835319 \nu^{15} - 13289199325505 \nu^{13} + 150985563543754 \nu^{11} + \cdots - 26\!\cdots\!82 \nu ) / 39\!\cdots\!45$$ (598241835319*v^15 - 13289199325505*v^13 + 150985563543754*v^11 - 1053101453448714*v^9 + 15549320783909747*v^7 - 54993468657667254*v^5 + 81927088464326798*v^3 - 26696886947325782*v) / 3911331152625545 $$\beta_{14}$$ $$=$$ $$( - 689220968556 \nu^{15} + 13753077377678 \nu^{13} - 141872714227952 \nu^{11} + 873330891688747 \nu^{9} + \cdots - 27\!\cdots\!29 \nu ) / 39\!\cdots\!45$$ (-689220968556*v^15 + 13753077377678*v^13 - 141872714227952*v^11 + 873330891688747*v^9 - 15741455888370946*v^7 + 26567121684692105*v^5 - 11566454641488780*v^3 - 27988577807897729*v) / 3911331152625545 $$\beta_{15}$$ $$=$$ $$( 699614355640 \nu^{15} - 14626335417890 \nu^{13} + 156853985883961 \nu^{11} + \cdots + 42\!\cdots\!97 \nu ) / 39\!\cdots\!45$$ (699614355640*v^15 - 14626335417890*v^13 + 156853985883961*v^11 - 1013981993431611*v^9 + 16721614333533753*v^7 - 41532830958531704*v^5 + 27101892635701972*v^3 + 42075022138460597*v) / 3911331152625545
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{12} + \beta_{11} - \beta_{10} + 2\beta_{7} - 2\beta_{6} - 5\beta_{4} - \beta_{2} + 2$$ b12 + b11 - b10 + 2*b7 - 2*b6 - 5*b4 - b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{15} - 2\beta_{13} - \beta_{9} - 11\beta_{5} + 2\beta_{3} + \beta_1$$ b15 - 2*b13 - b9 - 11*b5 + 2*b3 + b1 $$\nu^{4}$$ $$=$$ $$-12\beta_{12} + 6\beta_{11} - 3\beta_{10} + 18\beta_{7} - 82\beta_{6} - 19\beta_{4} - 25\beta_{2} + 3$$ -12*b12 + 6*b11 - 3*b10 + 18*b7 - 82*b6 - 19*b4 - 25*b2 + 3 $$\nu^{5}$$ $$=$$ $$30\beta_{15} + 3\beta_{14} - 18\beta_{13} - 149\beta_{9} - 37\beta_{8} - 146\beta_{5} + 33\beta_{3} + 37\beta_1$$ 30*b15 + 3*b14 - 18*b13 - 149*b9 - 37*b8 - 146*b5 + 33*b3 + 37*b1 $$\nu^{6}$$ $$=$$ $$-212\beta_{12} + 212\beta_{11} + 70\beta_{10} + 70\beta_{7} - 282\beta_{6} - 364\beta_{4} - 1191\beta_{2} - 434$$ -212*b12 + 212*b11 + 70*b10 + 70*b7 - 282*b6 - 364*b4 - 1191*b2 - 434 $$\nu^{7}$$ $$=$$ $$282 \beta_{15} + 282 \beta_{14} - 70 \beta_{13} - 2321 \beta_{9} - 2321 \beta_{8} - 714 \beta_{5} + 212 \beta_{3} + 1607 \beta_1$$ 282*b15 + 282*b14 - 70*b13 - 2321*b9 - 2321*b8 - 714*b5 + 212*b3 + 1607*b1 $$\nu^{8}$$ $$=$$ $$- 2556 \beta_{12} + 1749 \beta_{11} + 4305 \beta_{10} - 1278 \beta_{7} + 6413 \beta_{6} - 7691 \beta_{4} - 15755 \beta_{2} - 18311$$ -2556*b12 + 1749*b11 + 4305*b10 - 1278*b7 + 6413*b6 - 7691*b4 - 15755*b2 - 18311 $$\nu^{9}$$ $$=$$ $$1278\beta_{15} + 6054\beta_{14} + 1278\beta_{13} - 18759\beta_{9} - 35195\beta_{8} - 3027\beta_{3} + 6054\beta_1$$ 1278*b15 + 6054*b14 + 1278*b13 - 18759*b9 - 35195*b8 - 3027*b3 + 6054*b1 $$\nu^{10}$$ $$=$$ $$- 43805 \beta_{12} - 22490 \beta_{11} + 87610 \beta_{10} - 65120 \beta_{7} + 124340 \beta_{6} - 80535 \beta_{2} - 291103$$ -43805*b12 - 22490*b11 + 87610*b10 - 65120*b7 + 124340*b6 - 80535*b2 - 291103 $$\nu^{11}$$ $$=$$ $$- 21315 \beta_{15} + 65120 \beta_{14} + 65120 \beta_{13} - 65120 \beta_{9} - 274720 \beta_{8} + 209600 \beta_{5} - 108925 \beta_{3} - 147798 \beta_1$$ -21315*b15 + 65120*b14 + 65120*b13 - 65120*b9 - 274720*b8 + 209600*b5 - 108925*b3 - 147798*b1 $$\nu^{12}$$ $$=$$ $$- 300528 \beta_{12} - 640368 \beta_{11} + 980208 \beta_{10} - 1280736 \beta_{7} + 2603456 \beta_{6} + 1360625 \beta_{4} + 640368 \beta_{2} - 2603456$$ -300528*b12 - 640368*b11 + 980208*b10 - 1280736*b7 + 2603456*b6 + 1360625*b4 + 640368*b2 - 2603456 $$\nu^{13}$$ $$=$$ $$- 980208 \beta_{15} + 339840 \beta_{14} + 1280736 \beta_{13} + 2982608 \beta_{9} - 339840 \beta_{8} + 6185713 \beta_{5} - 1960416 \beta_{3} - 2982608 \beta_1$$ -980208*b15 + 339840*b14 + 1280736*b13 + 2982608*b9 - 339840*b8 + 6185713*b5 - 1960416*b3 - 2982608*b1 $$\nu^{14}$$ $$=$$ $$4144001 \beta_{12} - 10565728 \beta_{11} + 5282864 \beta_{10} - 14709729 \beta_{7} + 37812551 \beta_{6} + 25058897 \beta_{4} + 35624625 \beta_{2} - 5282864$$ 4144001*b12 - 10565728*b11 + 5282864*b10 - 14709729*b7 + 37812551*b6 + 25058897*b4 + 35624625*b2 - 5282864 $$\nu^{15}$$ $$=$$ $$- 18853730 \beta_{15} - 5282864 \beta_{14} + 14709729 \beta_{13} + 96434907 \beta_{9} + 56756081 \beta_{8} + 91152043 \beta_{5} - 24136594 \beta_{3} - 56756081 \beta_1$$ -18853730*b15 - 5282864*b14 + 14709729*b13 + 96434907*b9 + 56756081*b8 + 91152043*b5 - 24136594*b3 - 56756081*b1

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 3.67414 + 1.19380i 1.32111 + 0.429256i −1.32111 − 0.429256i −3.67414 − 1.19380i −1.83190 − 2.52140i −0.386583 − 0.532086i 0.386583 + 0.532086i 1.83190 + 2.52140i −1.83190 + 2.52140i −0.386583 + 0.532086i 0.386583 − 0.532086i 1.83190 − 2.52140i 3.67414 − 1.19380i 1.32111 − 0.429256i −1.32111 + 0.429256i −3.67414 + 1.19380i
−2.27075 3.12541i 0 −3.37586 + 10.3898i 3.35774 + 2.43954i 0 7.08028 + 2.30052i 25.4416 8.26648i 0 16.0339i
19.2 −0.816494 1.12381i 0 0.639787 1.96906i −3.53770 2.57029i 0 0.582836 + 0.189375i −8.01969 + 2.60575i 0 6.07432i
19.3 0.816494 + 1.12381i 0 0.639787 1.96906i 3.53770 + 2.57029i 0 0.582836 + 0.189375i 8.01969 2.60575i 0 6.07432i
19.4 2.27075 + 3.12541i 0 −3.37586 + 10.3898i −3.35774 2.43954i 0 7.08028 + 2.30052i −25.4416 + 8.26648i 0 16.0339i
28.1 −2.96408 + 0.963089i 0 4.62219 3.35821i −0.439256 + 1.35189i 0 2.23863 + 3.08121i −3.13867 + 4.32000i 0 4.43016i
28.2 −0.625505 + 0.203239i 0 −2.88612 + 2.09689i 2.50346 7.70484i 0 −2.40175 3.30573i 2.92545 4.02653i 0 5.32822i
28.3 0.625505 0.203239i 0 −2.88612 + 2.09689i −2.50346 + 7.70484i 0 −2.40175 3.30573i −2.92545 + 4.02653i 0 5.32822i
28.4 2.96408 0.963089i 0 4.62219 3.35821i 0.439256 1.35189i 0 2.23863 + 3.08121i 3.13867 4.32000i 0 4.43016i
46.1 −2.96408 0.963089i 0 4.62219 + 3.35821i −0.439256 1.35189i 0 2.23863 3.08121i −3.13867 4.32000i 0 4.43016i
46.2 −0.625505 0.203239i 0 −2.88612 2.09689i 2.50346 + 7.70484i 0 −2.40175 + 3.30573i 2.92545 + 4.02653i 0 5.32822i
46.3 0.625505 + 0.203239i 0 −2.88612 2.09689i −2.50346 7.70484i 0 −2.40175 + 3.30573i −2.92545 4.02653i 0 5.32822i
46.4 2.96408 + 0.963089i 0 4.62219 + 3.35821i 0.439256 + 1.35189i 0 2.23863 3.08121i 3.13867 + 4.32000i 0 4.43016i
73.1 −2.27075 + 3.12541i 0 −3.37586 10.3898i 3.35774 2.43954i 0 7.08028 2.30052i 25.4416 + 8.26648i 0 16.0339i
73.2 −0.816494 + 1.12381i 0 0.639787 + 1.96906i −3.53770 + 2.57029i 0 0.582836 0.189375i −8.01969 2.60575i 0 6.07432i
73.3 0.816494 1.12381i 0 0.639787 + 1.96906i 3.53770 2.57029i 0 0.582836 0.189375i 8.01969 + 2.60575i 0 6.07432i
73.4 2.27075 3.12541i 0 −3.37586 10.3898i −3.35774 + 2.43954i 0 7.08028 2.30052i −25.4416 8.26648i 0 16.0339i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.3.k.b 16
3.b odd 2 1 inner 99.3.k.b 16
11.c even 5 1 1089.3.c.l 16
11.d odd 10 1 inner 99.3.k.b 16
11.d odd 10 1 1089.3.c.l 16
33.f even 10 1 inner 99.3.k.b 16
33.f even 10 1 1089.3.c.l 16
33.h odd 10 1 1089.3.c.l 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.k.b 16 1.a even 1 1 trivial
99.3.k.b 16 3.b odd 2 1 inner
99.3.k.b 16 11.d odd 10 1 inner
99.3.k.b 16 33.f even 10 1 inner
1089.3.c.l 16 11.c even 5 1
1089.3.c.l 16 11.d odd 10 1
1089.3.c.l 16 33.f even 10 1
1089.3.c.l 16 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - 6T_{2}^{14} + 172T_{2}^{12} - 2568T_{2}^{10} + 20265T_{2}^{8} + 1848T_{2}^{6} + 71027T_{2}^{4} - 51909T_{2}^{2} + 14641$$ acting on $$S_{3}^{\mathrm{new}}(99, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 6 T^{14} + 172 T^{12} + \cdots + 14641$$
$3$ $$T^{16}$$
$5$ $$T^{16} + 87 T^{14} + \cdots + 1908029761$$
$7$ $$(T^{8} - 15 T^{7} + 77 T^{6} - 200 T^{5} + \cdots + 5041)^{2}$$
$11$ $$T^{16} + 417 T^{14} + \cdots + 45\!\cdots\!61$$
$13$ $$(T^{8} + 15 T^{7} - 403 T^{6} + \cdots + 441168016)^{2}$$
$17$ $$T^{16} + \cdots + 803437664440576$$
$19$ $$(T^{8} - 640 T^{6} - 16235 T^{5} + \cdots + 5628000400)^{2}$$
$23$ $$(T^{8} - 1430 T^{6} + \cdots + 1301766400)^{2}$$
$29$ $$T^{16} - 1115 T^{14} + \cdots + 18\!\cdots\!00$$
$31$ $$(T^{8} - 65 T^{7} + \cdots + 350813367025)^{2}$$
$37$ $$(T^{8} - 45 T^{7} + 1615 T^{6} + \cdots + 474368400)^{2}$$
$41$ $$T^{16} - 5155 T^{14} + \cdots + 10\!\cdots\!00$$
$43$ $$(T^{8} + 5102 T^{6} + \cdots + 431696305296)^{2}$$
$47$ $$T^{16} + 2455 T^{14} + \cdots + 11\!\cdots\!00$$
$53$ $$T^{16} + 12905 T^{14} + \cdots + 17\!\cdots\!25$$
$59$ $$T^{16} + 4320 T^{14} + \cdots + 41\!\cdots\!25$$
$61$ $$(T^{8} - 105 T^{7} + \cdots + 3114589632400)^{2}$$
$67$ $$(T^{4} - 75 T^{3} - 3655 T^{2} + \cdots - 3482380)^{4}$$
$71$ $$T^{16} + 1575 T^{14} + \cdots + 16\!\cdots\!00$$
$73$ $$(T^{8} + 85 T^{7} + \cdots + 494466201667216)^{2}$$
$79$ $$(T^{8} - 15 T^{7} + \cdots + 436852010010025)^{2}$$
$83$ $$T^{16} - 23291 T^{14} + \cdots + 21\!\cdots\!01$$
$89$ $$(T^{8} - 29246 T^{6} + \cdots + 33083847444736)^{2}$$
$97$ $$(T^{8} - 435 T^{7} + \cdots + 20\!\cdots\!25)^{2}$$