# Properties

 Label 245.4.e.q Level $245$ Weight $4$ Character orbit 245.e Analytic conductor $14.455$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 245.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.4554679514$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} + 27 x^{10} + 22 x^{9} + 399 x^{8} + 492 x^{7} + 4046 x^{6} + 8784 x^{5} + 22536 x^{4} + 22736 x^{3} + 18792 x^{2} + 4256 x + 784$$ x^12 - 2*x^11 + 27*x^10 + 22*x^9 + 399*x^8 + 492*x^7 + 4046*x^6 + 8784*x^5 + 22536*x^4 + 22736*x^3 + 18792*x^2 + 4256*x + 784 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}\cdot 7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + ( - \beta_{9} - \beta_{7} + 3 \beta_{3} + 3) q^{3} + ( - \beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - 3 \beta_{5} + \beta_{4} + \cdots - 2) q^{4}+ \cdots + ( - 3 \beta_{11} - \beta_{10} - 6 \beta_{7} - 5 \beta_{5} + \beta_{4} + 16 \beta_{3}) q^{9}+O(q^{10})$$ q - b5 * q^2 + (-b9 - b7 + 3*b3 + 3) * q^3 + (-b11 - 2*b10 + b9 + b8 + b7 - b6 - 3*b5 + b4 - 2*b3 - 3*b2 + 2*b1 - 2) * q^4 + 5*b3 * q^5 + (b6 + 4*b2 + 2*b1 - 6) * q^6 + (2*b9 + 3*b8 - 5*b6 - 9*b2 + b1 - 8) * q^8 + (-3*b11 - b10 - 6*b7 - 5*b5 + b4 + 16*b3) * q^9 $$q - \beta_{5} q^{2} + ( - \beta_{9} - \beta_{7} + 3 \beta_{3} + 3) q^{3} + ( - \beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - 3 \beta_{5} + \beta_{4} + \cdots - 2) q^{4}+ \cdots + (26 \beta_{9} + 244 \beta_{8} - 140 \beta_{6} + 116 \beta_{2} + 308 \beta_1 - 536) q^{99}+O(q^{100})$$ q - b5 * q^2 + (-b9 - b7 + 3*b3 + 3) * q^3 + (-b11 - 2*b10 + b9 + b8 + b7 - b6 - 3*b5 + b4 - 2*b3 - 3*b2 + 2*b1 - 2) * q^4 + 5*b3 * q^5 + (b6 + 4*b2 + 2*b1 - 6) * q^6 + (2*b9 + 3*b8 - 5*b6 - 9*b2 + b1 - 8) * q^8 + (-3*b11 - b10 - 6*b7 - 5*b5 + b4 + 16*b3) * q^9 + (5*b5 + 5*b2) * q^10 + (-5*b11 + 5*b10 + 2*b9 + 5*b8 + 2*b7 - b6 - 5*b5 + b4 + 7*b3 - 5*b2 - 5*b1 + 7) * q^11 + (-5*b11 - 13*b10 + 2*b7 - 6*b5 + 8*b4 - 28*b3) * q^12 + (5*b8 - 8*b6 - 2*b2 + 4*b1 - 27) * q^13 + (5*b9 - 15) * q^15 + (3*b11 + b10 - 12*b7 + 27*b5 - 15*b4 + 44*b3) * q^16 + (2*b11 - 10*b10 - b9 - 2*b8 - b7 - 4*b6 - 16*b5 + 4*b4 + b3 - 16*b2 + 10*b1 + 1) * q^17 + (-3*b11 - 21*b10 + 8*b9 + 3*b8 + 8*b7 - 3*b6 + b5 + 3*b4 - 68*b3 + b2 + 21*b1 - 68) * q^18 + (3*b11 + 10*b10 + 3*b7 + 6*b5 - 4*b4 - 52*b3) * q^19 + (-5*b9 - 5*b8 + 5*b6 + 15*b2 - 10*b1 + 10) * q^20 + (22*b9 + b8 - 19*b6 - 12*b2 + 17*b1 - 48) * q^22 + (-8*b11 - 11*b10 - 6*b7 + 3*b5 - 9*b4 - 56*b3) * q^23 + (-9*b11 - 6*b10 + b9 + 9*b8 + b7 - 8*b6 - 50*b5 + 8*b4 + 2*b3 - 50*b2 + 6*b1 + 2) * q^24 + (-25*b3 - 25) * q^25 + (5*b11 + 4*b10 - 15*b7 + 58*b5 - 19*b4 - 24*b3) * q^26 + (12*b9 - 21*b8 + 16*b6 + 50*b2 - 10*b1 - 185) * q^27 + (-8*b9 - b8 - 13*b6 + 39*b2 - 7*b1 + 21) * q^29 + (-10*b10 + 20*b5 - 5*b4 - 30*b3) * q^30 + (29*b10 + 10*b9 + 10*b7 + 17*b6 + 11*b5 - 17*b4 + 68*b3 + 11*b2 - 29*b1 + 68) * q^31 + (15*b11 + 35*b10 - 42*b9 - 15*b8 - 42*b7 + 31*b6 + 99*b5 - 31*b4 + 116*b3 + 99*b2 - 35*b1 + 116) * q^32 + (23*b11 + 44*b10 - 8*b7 - 26*b5 - 8*b4 - 13*b3) * q^33 + (2*b9 + 22*b8 - 11*b6 - 62*b2 + 18*b1 - 114) * q^34 + (14*b9 - 25*b8 + 13*b6 - 63*b2 - 65*b1 + 68) * q^36 + (-28*b11 - 11*b10 + 4*b7 - 33*b5 - 9*b4 - 8*b3) * q^37 + (7*b11 + 2*b10 + 3*b9 - 7*b8 + 3*b7 + 4*b6 - 16*b5 - 4*b4 + 6*b3 - 16*b2 - 2*b1 + 6) * q^38 + (-31*b11 - 57*b10 + 6*b9 + 31*b8 + 6*b7 - 11*b6 - 39*b5 + 11*b4 - 21*b3 - 39*b2 + 57*b1 - 21) * q^39 + (-15*b11 - 5*b10 + 10*b7 - 45*b5 + 25*b4 - 40*b3) * q^40 + (-37*b9 + 15*b8 - 3*b6 - 11*b2 - 7*b1 - 86) * q^41 + (2*b9 - 34*b8 + 16*b6 - 8*b1 - 74) * q^43 + (-10*b11 + 5*b10 - b7 + 122*b5 - 48*b4 - 62*b3) * q^44 + (15*b11 + 5*b10 + 30*b9 - 15*b8 + 30*b7 + 5*b6 + 25*b5 - 5*b4 - 80*b3 + 25*b2 - 5*b1 - 80) * q^45 + (20*b11 + 25*b10 - 35*b9 - 20*b8 - 35*b7 + 30*b6 - 42*b5 - 30*b4 + 92*b3 - 42*b2 - 25*b1 + 92) * q^46 + (-9*b11 + 14*b10 + 2*b7 - 50*b5 - 32*b4 - 81*b3) * q^47 + (47*b9 + 9*b8 - 6*b6 - 132*b2 - 10*b1 - 198) * q^48 - 25*b2 * q^50 + (-47*b11 - 59*b10 - 10*b7 - 65*b5 + 41*b4 + 145*b3) * q^51 + (32*b11 + 65*b10 - 93*b9 - 32*b8 - 93*b7 + 48*b6 + 196*b5 - 48*b4 + 230*b3 + 196*b2 - 65*b1 + 230) * q^52 + (-60*b11 - 35*b10 - 2*b9 + 60*b8 - 2*b7 - 19*b6 - 73*b5 + 19*b4 + 146*b3 - 73*b2 + 35*b1 + 146) * q^53 + (-45*b11 - 120*b10 + 83*b7 - 16*b5 + 81*b4 - 428*b3) * q^54 + (-10*b9 - 25*b8 + 5*b6 + 25*b2 + 25*b1 - 35) * q^55 + (-72*b9 + 48*b8 - 38*b6 - 30*b2 + 86*b1 + 266) * q^57 + (-25*b11 - 35*b10 - 128*b5 + 15*b4 - 296*b3) * q^58 + (-59*b11 - 33*b10 - 11*b9 + 59*b8 - 11*b7 - 47*b6 + 49*b5 + 47*b4 + 162*b3 + 49*b2 + 33*b1 + 162) * q^59 + (25*b11 + 65*b10 - 10*b9 - 25*b8 - 10*b7 + 40*b6 + 30*b5 - 40*b4 + 140*b3 + 30*b2 - 65*b1 + 140) * q^60 + (-10*b11 - 48*b10 - 24*b7 - 132*b5 + 4*b4 - 84*b3) * q^61 + (13*b9 - 28*b8 + 6*b6 + 154*b2 - b1 - 4) * q^62 + (-10*b9 - 91*b8 + 63*b6 + 351*b2 - 67*b1 + 116) * q^64 + (-25*b11 - 20*b10 - 10*b5 + 40*b4 - 135*b3) * q^65 + (-41*b11 - 148*b10 + 75*b9 + 41*b8 + 75*b7 - 23*b6 + 23*b4 - 664*b3 + 148*b1 - 664) * q^66 + (12*b11 - 31*b10 + 12*b9 - 12*b8 + 12*b7 - 35*b6 + 51*b5 + 35*b4 - 330*b3 + 51*b2 + 31*b1 - 330) * q^67 + (67*b11 + 15*b10 - 78*b7 + 236*b5 - 58*b4 + 420*b3) * q^68 + (-48*b9 - 16*b8 + 53*b6 - 77*b2 - 81*b1 + 238) * q^69 + (-14*b9 + 92*b6 - 116*b2 - 60*b1 + 26) * q^71 + (51*b11 + 47*b10 - 78*b7 + 27*b5 - 67*b4 + 492*b3) * q^72 + (85*b11 + 60*b10 - 11*b9 - 85*b8 - 11*b7 - 46*b6 + 64*b5 + 46*b4 + 414*b3 + 64*b2 - 60*b1 + 414) * q^73 + (4*b11 - 27*b10 + 21*b9 - 4*b8 + 21*b7 - 36*b6 - 132*b5 + 36*b4 - 128*b3 - 132*b2 + 27*b1 - 128) * q^74 + (25*b7 - 75*b3) * q^75 + (-19*b9 + 43*b8 - 38*b6 - 76*b2 + 106*b1 + 234) * q^76 + (-22*b9 + 19*b8 - 41*b6 - 254*b2 - 37*b1 + 112) * q^78 + (55*b11 + 25*b10 + 10*b7 + 67*b5 + 65*b4 + 237*b3) * q^79 + (-15*b11 - 5*b10 + 60*b9 + 15*b8 + 60*b7 - 75*b6 - 135*b5 + 75*b4 - 220*b3 - 135*b2 + 5*b1 - 220) * q^80 + (12*b11 + 34*b10 + 112*b9 - 12*b8 + 112*b7 + 70*b6 + 170*b5 - 70*b4 - 827*b3 + 170*b2 - 34*b1 - 827) * q^81 + (-b11 + 113*b10 - 46*b7 + 96*b5 - 2*b4 + 314*b3) * q^82 + (69*b9 + 41*b8 - 154*b6 + 32*b2 + 80*b1 - 70) * q^83 + (5*b9 + 10*b8 + 20*b6 + 80*b2 - 50*b1 - 5) * q^85 + (18*b11 - 4*b10 + 50*b7 + 2*b5 + 56*b4 + 172*b3) * q^86 + (-43*b11 - 68*b10 - 70*b9 + 43*b8 - 70*b7 + 66*b6 + 16*b5 - 66*b4 + 333*b3 + 16*b2 + 68*b1 + 333) * q^87 + (172*b11 + 144*b10 - 22*b9 - 172*b8 - 22*b7 + 52*b6 + 300*b5 - 52*b4 + 840*b3 + 300*b2 - 144*b1 + 840) * q^88 + (-79*b11 + 62*b10 + 33*b7 - 74*b5 + 80*b4 + 378*b3) * q^89 + (-40*b9 - 15*b8 + 15*b6 - 5*b2 - 105*b1 + 340) * q^90 + (60*b9 - 32*b8 - 24*b6 + 82*b2 + 86*b1 - 412) * q^92 + (168*b11 + 271*b10 - 4*b7 + 109*b5 - 91*b4 - 266*b3) * q^93 + (-9*b11 - 92*b10 + 23*b9 + 9*b8 + 23*b7 - 11*b6 - 164*b5 + 11*b4 - 536*b3 - 164*b2 + 92*b1 - 536) * q^94 + (-15*b11 - 50*b10 - 15*b9 + 15*b8 - 15*b7 - 20*b6 - 30*b5 + 20*b4 + 260*b3 - 30*b2 + 50*b1 + 260) * q^95 + (57*b11 + 148*b10 - 165*b7 + 252*b5 - 146*b4 + 1078*b3) * q^96 + (77*b9 + 84*b8 + 14*b6 - 134*b2 + 72*b1 - 63) * q^97 + (26*b9 + 244*b8 - 140*b6 + 116*b2 + 308*b1 - 536) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 2 q^{2} + 16 q^{3} - 14 q^{4} - 30 q^{5} - 48 q^{6} - 132 q^{8} - 70 q^{9}+O(q^{10})$$ 12 * q + 2 * q^2 + 16 * q^3 - 14 * q^4 - 30 * q^5 - 48 * q^6 - 132 * q^8 - 70 * q^9 $$12 q + 2 q^{2} + 16 q^{3} - 14 q^{4} - 30 q^{5} - 48 q^{6} - 132 q^{8} - 70 q^{9} + 10 q^{10} + 16 q^{11} + 160 q^{12} - 336 q^{13} - 160 q^{15} - 298 q^{16} - 4 q^{17} - 354 q^{18} + 308 q^{19} + 140 q^{20} - 472 q^{22} + 336 q^{23} - 92 q^{24} - 150 q^{25} + 56 q^{26} - 1928 q^{27} + 352 q^{29} + 120 q^{30} + 392 q^{31} + 770 q^{32} + 188 q^{33} - 1624 q^{34} + 460 q^{36} + 140 q^{37} + 20 q^{38} - 140 q^{39} + 330 q^{40} - 1312 q^{41} - 776 q^{43} + 160 q^{44} - 350 q^{45} + 388 q^{46} + 628 q^{47} - 2792 q^{48} - 100 q^{50} - 744 q^{51} + 1520 q^{52} + 676 q^{53} + 2284 q^{54} - 160 q^{55} + 2936 q^{57} + 2012 q^{58} + 996 q^{59} + 800 q^{60} + 740 q^{61} + 728 q^{62} + 2852 q^{64} + 840 q^{65} - 3620 q^{66} - 1768 q^{67} - 2940 q^{68} + 2096 q^{69} - 448 q^{71} - 2858 q^{72} + 2640 q^{73} - 928 q^{74} + 400 q^{75} + 2680 q^{76} + 16 q^{78} - 1636 q^{79} - 1490 q^{80} - 4442 q^{81} - 1756 q^{82} - 280 q^{83} + 40 q^{85} - 1180 q^{86} + 1940 q^{87} + 5652 q^{88} - 1904 q^{89} + 3540 q^{90} - 3904 q^{92} + 1592 q^{93} - 3332 q^{94} + 1540 q^{95} - 6460 q^{96} - 1032 q^{97} - 5608 q^{99}+O(q^{100})$$ 12 * q + 2 * q^2 + 16 * q^3 - 14 * q^4 - 30 * q^5 - 48 * q^6 - 132 * q^8 - 70 * q^9 + 10 * q^10 + 16 * q^11 + 160 * q^12 - 336 * q^13 - 160 * q^15 - 298 * q^16 - 4 * q^17 - 354 * q^18 + 308 * q^19 + 140 * q^20 - 472 * q^22 + 336 * q^23 - 92 * q^24 - 150 * q^25 + 56 * q^26 - 1928 * q^27 + 352 * q^29 + 120 * q^30 + 392 * q^31 + 770 * q^32 + 188 * q^33 - 1624 * q^34 + 460 * q^36 + 140 * q^37 + 20 * q^38 - 140 * q^39 + 330 * q^40 - 1312 * q^41 - 776 * q^43 + 160 * q^44 - 350 * q^45 + 388 * q^46 + 628 * q^47 - 2792 * q^48 - 100 * q^50 - 744 * q^51 + 1520 * q^52 + 676 * q^53 + 2284 * q^54 - 160 * q^55 + 2936 * q^57 + 2012 * q^58 + 996 * q^59 + 800 * q^60 + 740 * q^61 + 728 * q^62 + 2852 * q^64 + 840 * q^65 - 3620 * q^66 - 1768 * q^67 - 2940 * q^68 + 2096 * q^69 - 448 * q^71 - 2858 * q^72 + 2640 * q^73 - 928 * q^74 + 400 * q^75 + 2680 * q^76 + 16 * q^78 - 1636 * q^79 - 1490 * q^80 - 4442 * q^81 - 1756 * q^82 - 280 * q^83 + 40 * q^85 - 1180 * q^86 + 1940 * q^87 + 5652 * q^88 - 1904 * q^89 + 3540 * q^90 - 3904 * q^92 + 1592 * q^93 - 3332 * q^94 + 1540 * q^95 - 6460 * q^96 - 1032 * q^97 - 5608 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 27 x^{10} + 22 x^{9} + 399 x^{8} + 492 x^{7} + 4046 x^{6} + 8784 x^{5} + 22536 x^{4} + 22736 x^{3} + 18792 x^{2} + 4256 x + 784$$ :

 $$\beta_{1}$$ $$=$$ $$( 28982856459 \nu^{11} - 100035188374 \nu^{10} + 862940207031 \nu^{9} - 394352390218 \nu^{8} + 10272625618891 \nu^{7} + \cdots + 17\!\cdots\!20 ) / 294724713250240$$ (28982856459*v^11 - 100035188374*v^10 + 862940207031*v^9 - 394352390218*v^8 + 10272625618891*v^7 - 1211833121440*v^6 + 95016036065672*v^5 + 120757074689168*v^4 + 196278374198728*v^3 + 45858625791056*v^2 + 9705619482888*v + 1781182410801920) / 294724713250240 $$\beta_{2}$$ $$=$$ $$( 64037594762 \nu^{11} - 188357432449 \nu^{10} + 1870129815820 \nu^{9} - 266932180717 \nu^{8} + 24649539006832 \nu^{7} + \cdots - 492665609549624 ) / 294724713250240$$ (64037594762*v^11 - 188357432449*v^10 + 1870129815820*v^9 - 266932180717*v^8 + 24649539006832*v^7 + 6993759652627*v^6 + 234876183172976*v^5 + 316994272040168*v^4 + 904173032154720*v^3 + 137017830949064*v^2 + 29452011210496*v - 492665609549624) / 294724713250240 $$\beta_{3}$$ $$=$$ $$( - 13582777089 \nu^{11} + 30056420270 \nu^{10} - 375018884157 \nu^{9} - 215463576678 \nu^{8} - 5427523732713 \nu^{7} + \cdots - 56425021720928 ) / 45342263576960$$ (-13582777089*v^11 + 30056420270*v^10 - 375018884157*v^9 - 215463576678*v^8 - 5427523732713*v^7 - 5556869540436*v^6 - 54619586026392*v^5 - 108540738554160*v^4 - 291454667038776*v^3 - 259213002261264*v^2 - 248822668409304*v - 56425021720928) / 45342263576960 $$\beta_{4}$$ $$=$$ $$( 111467791035 \nu^{11} + 52968478226 \nu^{10} + 2081845750599 \nu^{9} + 10853659471934 \nu^{8} + 41174948636763 \nu^{7} + \cdots + 572806666180672 ) / 294724713250240$$ (111467791035*v^11 + 52968478226*v^10 + 2081845750599*v^9 + 10853659471934*v^8 + 41174948636763*v^7 + 154097397736104*v^6 + 471066642895320*v^5 + 1902350626575408*v^4 + 3699234445246632*v^3 + 5197700243289872*v^2 + 2558030746151304*v + 572806666180672) / 294724713250240 $$\beta_{5}$$ $$=$$ $$( 76723499462 \nu^{11} - 139795554401 \nu^{10} + 2025141954592 \nu^{9} + 2061540815475 \nu^{8} + 30440356962676 \nu^{7} + \cdots + 331284635996552 ) / 147362356625120$$ (76723499462*v^11 - 139795554401*v^10 + 2025141954592*v^9 + 2061540815475*v^8 + 30440356962676*v^7 + 42629924047259*v^6 + 308340799348936*v^5 + 719247529422952*v^4 + 1787034414602656*v^3 + 1919551124767560*v^2 + 1464181233023008*v + 331284635996552) / 147362356625120 $$\beta_{6}$$ $$=$$ $$( - 236742116523 \nu^{11} + 798546202123 \nu^{10} - 7023150156897 \nu^{9} + 2877553801651 \nu^{8} + \cdots + 334278957216120 ) / 294724713250240$$ (-236742116523*v^11 + 798546202123*v^10 - 7023150156897*v^9 + 2877553801651*v^8 - 85020409137217*v^7 - 8151791041195*v^6 - 790304136830984*v^5 - 1014919239461336*v^4 - 2634619565272216*v^3 - 394884420582392*v^2 - 83831921404056*v + 334278957216120) / 294724713250240 $$\beta_{7}$$ $$=$$ $$( - 329010546210 \nu^{11} + 296347940193 \nu^{10} - 7628430332148 \nu^{9} - 18009638432243 \nu^{8} + \cdots - 15\!\cdots\!24 ) / 294724713250240$$ (-329010546210*v^11 + 296347940193*v^10 - 7628430332148*v^9 - 18009638432243*v^8 - 126233356787736*v^7 - 289673773266243*v^6 - 1339512180024800*v^5 - 4070737252260456*v^4 - 8911464618228864*v^3 - 11326738888198984*v^2 - 6747896857596288*v - 1519783792580424) / 294724713250240 $$\beta_{8}$$ $$=$$ $$( 12353465669 \nu^{11} - 35572783963 \nu^{10} + 357053140915 \nu^{9} - 37375371979 \nu^{8} + 4800734849059 \nu^{7} + \cdots - 127876575513208 ) / 10525882616080$$ (12353465669*v^11 - 35572783963*v^10 + 357053140915*v^9 - 37375371979*v^8 + 4800734849059*v^7 + 1420926501199*v^6 + 45892420865912*v^5 + 62323465083416*v^4 + 194497769222960*v^3 + 27265839251768*v^2 + 5868627652552*v - 127876575513208) / 10525882616080 $$\beta_{9}$$ $$=$$ $$( - 188811098545 \nu^{11} + 604767616393 \nu^{10} - 5588134013203 \nu^{9} + 1701043628257 \nu^{8} + \cdots + 503193043313576 ) / 147362356625120$$ (-188811098545*v^11 + 604767616393*v^10 - 5588134013203*v^9 + 1701043628257*v^8 - 69725462805091*v^7 - 13421629244633*v^6 - 654804748029080*v^5 - 858750029074376*v^4 - 2337235074058424*v^3 - 350010629941544*v^2 - 74727962170568*v + 503193043313576) / 147362356625120 $$\beta_{10}$$ $$=$$ $$( 534066780106 \nu^{11} - 1229497377337 \nu^{10} + 14908874191660 \nu^{9} + 7132996238619 \nu^{8} + 214032451057936 \nu^{7} + \cdots + 22\!\cdots\!08 ) / 294724713250240$$ (534066780106*v^11 - 1229497377337*v^10 + 14908874191660*v^9 + 7132996238619*v^8 + 214032451057936*v^7 + 201511222516091*v^6 + 2153529434603008*v^5 + 4110419162617384*v^4 + 11259141242188480*v^3 + 9892936870094152*v^2 + 9706700273783488*v + 2202314025449608) / 294724713250240 $$\beta_{11}$$ $$=$$ $$( - 1105671611289 \nu^{11} + 2180810036423 \nu^{10} - 29871225937975 \nu^{9} - 25079310240201 \nu^{8} + \cdots - 47\!\cdots\!72 ) / 294724713250240$$ (-1105671611289*v^11 + 2180810036423*v^10 - 29871225937975*v^9 - 25079310240201*v^8 - 443031842634559*v^7 - 562190917108619*v^6 - 4495121409160232*v^5 - 9906573606536536*v^4 - 25234295742080200*v^3 - 26506179743948888*v^2 - 20960996836434472*v - 4746227706689672) / 294724713250240
 $$\nu$$ $$=$$ $$( \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 6\beta_{5} + \beta_{4} + 6\beta_{2} - \beta_1 ) / 7$$ (b11 + b10 + b9 - b8 + b7 - b6 + 6*b5 + b4 + 6*b2 - b1) / 7 $$\nu^{2}$$ $$=$$ $$( \beta_{11} + 8\beta_{10} + \beta_{7} + 13\beta_{5} + \beta_{4} + 56\beta_{3} ) / 7$$ (b11 + 8*b10 + b7 + 13*b5 + b4 + 56*b3) / 7 $$\nu^{3}$$ $$=$$ $$( -15\beta_{9} + 8\beta_{8} + 15\beta_{6} - 83\beta_{2} + 15\beta _1 - 98 ) / 7$$ (-15*b9 + 8*b8 + 15*b6 - 83*b2 + 15*b1 - 98) / 7 $$\nu^{4}$$ $$=$$ $$( - 29 \beta_{11} - 127 \beta_{10} - 36 \beta_{9} + 29 \beta_{8} - 36 \beta_{7} + 43 \beta_{6} - 265 \beta_{5} - 43 \beta_{4} - 784 \beta_{3} - 265 \beta_{2} + 127 \beta _1 - 784 ) / 7$$ (-29*b11 - 127*b10 - 36*b9 + 29*b8 - 36*b7 + 43*b6 - 265*b5 - 43*b4 - 784*b3 - 265*b2 + 127*b1 - 784) / 7 $$\nu^{5}$$ $$=$$ $$( -106\beta_{11} - 323\beta_{10} - 267\beta_{7} - 1315\beta_{5} - 267\beta_{4} - 2254\beta_{3} ) / 7$$ (-106*b11 - 323*b10 - 267*b7 - 1315*b5 - 267*b4 - 2254*b3) / 7 $$\nu^{6}$$ $$=$$ $$( 876\beta_{9} - 477\beta_{8} - 1037\beta_{6} + 4815\beta_{2} - 2003\beta _1 + 12544 ) / 7$$ (876*b9 - 477*b8 - 1037*b6 + 4815*b2 - 2003*b1 + 12544) / 7 $$\nu^{7}$$ $$=$$ $$( 1632 \beta_{11} + 6357 \beta_{10} + 4971 \beta_{9} - 1632 \beta_{8} + 4971 \beta_{7} - 5209 \beta_{6} + 21769 \beta_{5} + 5209 \beta_{4} + 43918 \beta_{3} + 21769 \beta_{2} - 6357 \beta _1 + 43918 ) / 7$$ (1632*b11 + 6357*b10 + 4971*b9 - 1632*b8 + 4971*b7 - 5209*b6 + 21769*b5 + 5209*b4 + 43918*b3 + 21769*b2 - 6357*b1 + 43918) / 7 $$\nu^{8}$$ $$=$$ $$( 7253\beta_{11} + 32845\beta_{10} + 19034\beta_{7} + 84713\beta_{5} + 22135\beta_{4} + 212100\beta_{3} ) / 7$$ (7253*b11 + 32845*b10 + 19034*b7 + 84713*b5 + 22135*b4 + 212100*b3) / 7 $$\nu^{9}$$ $$=$$ $$( -94879\beta_{9} + 25362\beta_{8} + 103559\beta_{6} - 367219\beta_{2} + 118399\beta _1 - 814758 ) / 7$$ (-94879*b9 + 25362*b8 + 103559*b6 - 367219*b2 + 118399*b1 - 814758) / 7 $$\nu^{10}$$ $$=$$ $$( - 107577 \beta_{11} - 554835 \beta_{10} - 391392 \beta_{9} + 107577 \beta_{8} - 391392 \beta_{7} + 452229 \beta_{6} - 1472799 \beta_{5} - 452229 \beta_{4} - 3684464 \beta_{3} + \cdots - 3684464 ) / 7$$ (-107577*b11 - 554835*b10 - 391392*b9 + 107577*b8 - 391392*b7 + 452229*b6 - 1472799*b5 - 452229*b4 - 3684464*b3 - 1472799*b2 + 554835*b1 - 3684464) / 7 $$\nu^{11}$$ $$=$$ $$( - 384644 \beta_{11} - 2146229 \beta_{10} - 1830431 \beta_{7} - 6253953 \beta_{5} - 2053409 \beta_{4} - 14829990 \beta_{3} ) / 7$$ (-384644*b11 - 2146229*b10 - 1830431*b7 - 6253953*b5 - 2053409*b4 - 14829990*b3) / 7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 −1.52662 + 2.64418i −0.120924 + 0.209447i −1.02943 + 1.78303i −0.522449 + 0.904909i 2.14754 − 3.71965i 2.05188 − 3.55396i −1.52662 − 2.64418i −0.120924 − 0.209447i −1.02943 − 1.78303i −0.522449 − 0.904909i 2.14754 + 3.71965i 2.05188 + 3.55396i
−2.23372 3.86892i 4.90460 8.49501i −5.97903 + 10.3560i −2.50000 4.33013i −43.8221 0 17.6824 −34.6102 59.9466i −11.1686 + 19.3446i
116.2 −0.828031 1.43419i −0.166444 + 0.288289i 2.62873 4.55309i −2.50000 4.33013i 0.551283 0 −21.9552 13.4446 + 23.2867i −4.14016 + 7.17096i
116.3 −0.322324 0.558282i −2.09344 + 3.62594i 3.79221 6.56831i −2.50000 4.33013i 2.69906 0 −10.0465 4.73504 + 8.20133i −1.61162 + 2.79141i
116.4 0.184657 + 0.319836i 4.87035 8.43569i 3.93180 6.81008i −2.50000 4.33013i 3.59738 0 5.85867 −33.9406 58.7868i 0.923287 1.59918i
116.5 1.44043 + 2.49490i −1.44526 + 2.50327i −0.149696 + 0.259281i −2.50000 4.33013i −8.32721 0 22.1844 9.32244 + 16.1469i 7.20217 12.4745i
116.6 2.75899 + 4.77871i 1.93020 3.34320i −11.2240 + 19.4406i −2.50000 4.33013i 21.3015 0 −79.7239 6.04869 + 10.4766i 13.7949 23.8935i
226.1 −2.23372 + 3.86892i 4.90460 + 8.49501i −5.97903 10.3560i −2.50000 + 4.33013i −43.8221 0 17.6824 −34.6102 + 59.9466i −11.1686 19.3446i
226.2 −0.828031 + 1.43419i −0.166444 0.288289i 2.62873 + 4.55309i −2.50000 + 4.33013i 0.551283 0 −21.9552 13.4446 23.2867i −4.14016 7.17096i
226.3 −0.322324 + 0.558282i −2.09344 3.62594i 3.79221 + 6.56831i −2.50000 + 4.33013i 2.69906 0 −10.0465 4.73504 8.20133i −1.61162 2.79141i
226.4 0.184657 0.319836i 4.87035 + 8.43569i 3.93180 + 6.81008i −2.50000 + 4.33013i 3.59738 0 5.85867 −33.9406 + 58.7868i 0.923287 + 1.59918i
226.5 1.44043 2.49490i −1.44526 2.50327i −0.149696 0.259281i −2.50000 + 4.33013i −8.32721 0 22.1844 9.32244 16.1469i 7.20217 + 12.4745i
226.6 2.75899 4.77871i 1.93020 + 3.34320i −11.2240 19.4406i −2.50000 + 4.33013i 21.3015 0 −79.7239 6.04869 10.4766i 13.7949 + 23.8935i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 226.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.e.q 12
7.b odd 2 1 245.4.e.p 12
7.c even 3 1 245.4.a.o 6
7.c even 3 1 inner 245.4.e.q 12
7.d odd 6 1 245.4.a.p yes 6
7.d odd 6 1 245.4.e.p 12
21.g even 6 1 2205.4.a.ca 6
21.h odd 6 1 2205.4.a.bz 6
35.i odd 6 1 1225.4.a.bi 6
35.j even 6 1 1225.4.a.bj 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.o 6 7.c even 3 1
245.4.a.p yes 6 7.d odd 6 1
245.4.e.p 12 7.b odd 2 1
245.4.e.p 12 7.d odd 6 1
245.4.e.q 12 1.a even 1 1 trivial
245.4.e.q 12 7.c even 3 1 inner
1225.4.a.bi 6 35.i odd 6 1
1225.4.a.bj 6 35.j even 6 1
2205.4.a.bz 6 21.h odd 6 1
2205.4.a.ca 6 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(245, [\chi])$$:

 $$T_{2}^{12} - 2 T_{2}^{11} + 33 T_{2}^{10} + 2 T_{2}^{9} + 763 T_{2}^{8} - 252 T_{2}^{7} + 4662 T_{2}^{6} + 5200 T_{2}^{5} + 16472 T_{2}^{4} + 4784 T_{2}^{3} + 4328 T_{2}^{2} - 672 T_{2} + 784$$ T2^12 - 2*T2^11 + 33*T2^10 + 2*T2^9 + 763*T2^8 - 252*T2^7 + 4662*T2^6 + 5200*T2^5 + 16472*T2^4 + 4784*T2^3 + 4328*T2^2 - 672*T2 + 784 $$T_{3}^{12} - 16 T_{3}^{11} + 244 T_{3}^{10} - 1320 T_{3}^{9} + 9523 T_{3}^{8} - 9236 T_{3}^{7} + 245080 T_{3}^{6} - 64988 T_{3}^{5} + 2763073 T_{3}^{4} + 3324828 T_{3}^{3} + 21039206 T_{3}^{2} + \cdots + 2208196$$ T3^12 - 16*T3^11 + 244*T3^10 - 1320*T3^9 + 9523*T3^8 - 9236*T3^7 + 245080*T3^6 - 64988*T3^5 + 2763073*T3^4 + 3324828*T3^3 + 21039206*T3^2 + 6900984*T3 + 2208196

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 2 T^{11} + 33 T^{10} + 2 T^{9} + \cdots + 784$$
$3$ $$T^{12} - 16 T^{11} + 244 T^{10} + \cdots + 2208196$$
$5$ $$(T^{2} + 5 T + 25)^{6}$$
$7$ $$T^{12}$$
$11$ $$T^{12} - 16 T^{11} + \cdots + 85\!\cdots\!00$$
$13$ $$(T^{6} + 168 T^{5} + \cdots + 12513937372)^{2}$$
$17$ $$T^{12} + 4 T^{11} + \cdots + 14\!\cdots\!44$$
$19$ $$T^{12} - 308 T^{11} + \cdots + 26\!\cdots\!36$$
$23$ $$T^{12} - 336 T^{11} + \cdots + 89\!\cdots\!16$$
$29$ $$(T^{6} - 176 T^{5} + \cdots - 544215793700)^{2}$$
$31$ $$T^{12} - 392 T^{11} + \cdots + 27\!\cdots\!64$$
$37$ $$T^{12} - 140 T^{11} + \cdots + 73\!\cdots\!96$$
$41$ $$(T^{6} + 656 T^{5} + \cdots + 144691772208184)^{2}$$
$43$ $$(T^{6} + 388 T^{5} + \cdots + 440374360000)^{2}$$
$47$ $$T^{12} - 628 T^{11} + \cdots + 63\!\cdots\!00$$
$53$ $$T^{12} - 676 T^{11} + \cdots + 34\!\cdots\!04$$
$59$ $$T^{12} - 996 T^{11} + \cdots + 20\!\cdots\!16$$
$61$ $$T^{12} - 740 T^{11} + \cdots + 70\!\cdots\!04$$
$67$ $$T^{12} + 1768 T^{11} + \cdots + 78\!\cdots\!44$$
$71$ $$(T^{6} + 224 T^{5} + \cdots + 12\!\cdots\!88)^{2}$$
$73$ $$T^{12} - 2640 T^{11} + \cdots + 73\!\cdots\!64$$
$79$ $$T^{12} + 1636 T^{11} + \cdots + 15\!\cdots\!24$$
$83$ $$(T^{6} + 140 T^{5} + \cdots + 22\!\cdots\!00)^{2}$$
$89$ $$T^{12} + 1904 T^{11} + \cdots + 20\!\cdots\!76$$
$97$ $$(T^{6} + 516 T^{5} + \cdots - 510868966648482)^{2}$$