# Properties

 Label 35.6.e.a Level $35$ Weight $6$ Character orbit 35.e Analytic conductor $5.613$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 35.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.61343369345$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - x^{11} + 110 x^{10} + 27 x^{9} + 9714 x^{8} - 3257 x^{7} + 240269 x^{6} - 608504 x^{5} + 5067668 x^{4} - 7120352 x^{3} + 9029136 x^{2} - 1794240 x + 313600$$ x^12 - x^11 + 110*x^10 + 27*x^9 + 9714*x^8 - 3257*x^7 + 240269*x^6 - 608504*x^5 + 5067668*x^4 - 7120352*x^3 + 9029136*x^2 - 1794240*x + 313600 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} - \beta_1 + 1) q^{2} + (\beta_{6} - 3 \beta_{4} + \beta_{3} + \beta_1) q^{3} + (\beta_{9} - 5 \beta_{4} - \beta_{3} - \beta_1) q^{4} + ( - 25 \beta_{4} + 25) q^{5} + (\beta_{8} - \beta_{7} - \beta_{5} - 5 \beta_{3} + \beta_{2} + 16) q^{6} + ( - 2 \beta_{10} + 3 \beta_{8} - 3 \beta_{6} - \beta_{5} - 9 \beta_{4} + 7 \beta_{3} + \cdots + 3) q^{7}+ \cdots + ( - 2 \beta_{11} - 3 \beta_{10} + 4 \beta_{9} + 3 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} + \cdots - 65) q^{9}+O(q^{10})$$ q + (-b4 - b1 + 1) * q^2 + (b6 - 3*b4 + b3 + b1) * q^3 + (b9 - 5*b4 - b3 - b1) * q^4 + (-25*b4 + 25) * q^5 + (b8 - b7 - b5 - 5*b3 + b2 + 16) * q^6 + (-2*b10 + 3*b8 - 3*b6 - b5 - 9*b4 + 7*b3 + 7*b1 + 3) * q^7 + (2*b8 - 4*b5 - 5*b3 - 23) * q^8 + (-2*b11 - 3*b10 + 4*b9 + 3*b8 + 2*b7 - 3*b6 - 3*b5 + 65*b4 - 4*b2 - 2*b1 - 65) * q^9 $$q + ( - \beta_{4} - \beta_1 + 1) q^{2} + (\beta_{6} - 3 \beta_{4} + \beta_{3} + \beta_1) q^{3} + (\beta_{9} - 5 \beta_{4} - \beta_{3} - \beta_1) q^{4} + ( - 25 \beta_{4} + 25) q^{5} + (\beta_{8} - \beta_{7} - \beta_{5} - 5 \beta_{3} + \beta_{2} + 16) q^{6} + ( - 2 \beta_{10} + 3 \beta_{8} - 3 \beta_{6} - \beta_{5} - 9 \beta_{4} + 7 \beta_{3} + \cdots + 3) q^{7}+ \cdots + (461 \beta_{8} + 524 \beta_{7} - 3201 \beta_{5} - 1922 \beta_{3} + \cdots - 32765) q^{99}+O(q^{100})$$ q + (-b4 - b1 + 1) * q^2 + (b6 - 3*b4 + b3 + b1) * q^3 + (b9 - 5*b4 - b3 - b1) * q^4 + (-25*b4 + 25) * q^5 + (b8 - b7 - b5 - 5*b3 + b2 + 16) * q^6 + (-2*b10 + 3*b8 - 3*b6 - b5 - 9*b4 + 7*b3 + 7*b1 + 3) * q^7 + (2*b8 - 4*b5 - 5*b3 - 23) * q^8 + (-2*b11 - 3*b10 + 4*b9 + 3*b8 + 2*b7 - 3*b6 - 3*b5 + 65*b4 - 4*b2 - 2*b1 - 65) * q^9 + (-25*b4 - 25*b3 - 25*b1) * q^10 + (-4*b11 - b10 - 4*b9 + 9*b6 + 151*b4 - 32*b3 - 32*b1) * q^11 + (5*b11 - 9*b10 - 10*b9 + 9*b8 - 5*b7 + 13*b6 + 13*b5 + 64*b4 + 10*b2 + 27*b1 - 64) * q^12 + (-3*b8 + 10*b7 - 7*b5 - 4*b3 - 29) * q^13 + (7*b11 - 7*b10 - 14*b9 - 7*b6 - 49*b4 - 21*b3 - 7*b2 - 28*b1 + 301) * q^14 + (-25*b5 + 25*b3 - 75) * q^15 + (8*b11 - 8*b10 + 11*b9 + 8*b8 - 8*b7 - 8*b6 - 8*b5 + 69*b4 - 11*b2 - 19*b1 - 69) * q^16 + (4*b10 + 20*b9 - 34*b6 - 256*b4 + 2*b3 + 2*b1) * q^17 + (5*b11 + 7*b10 - 20*b9 + 31*b6 - 47*b4 - 56*b3 - 56*b1) * q^18 + (-8*b11 + 3*b10 + 8*b9 - 3*b8 + 8*b7 + 43*b6 + 43*b5 + 67*b4 - 8*b2 + 230*b1 - 67) * q^19 + (-25*b3 + 25*b2 - 125) * q^20 + (-14*b11 + 19*b10 - 11*b8 - 14*b7 - 31*b6 + 41*b5 - 352*b4 + 28*b3 - 56*b1 + 794) * q^21 + (15*b8 - 15*b7 - 39*b5 + 308*b3 + 50*b2 - 1063) * q^22 + (-20*b11 + 53*b10 + 60*b9 - 53*b8 + 20*b7 - 34*b6 - 34*b5 - 646*b4 - 60*b2 - 49*b1 + 646) * q^23 + (-17*b11 - 13*b10 + 5*b9 - 73*b6 + 1208*b4 + 47*b3 + 47*b1) * q^24 - 625*b4 * q^25 + (-19*b11 + 39*b10 - 6*b9 - 39*b8 + 19*b7 + 119*b6 + 119*b5 + 495*b4 + 6*b2 + 210*b1 - 495) * q^26 + (-6*b8 - 60*b7 + 13*b5 - 103*b3 + 60*b2 + 375) * q^27 + (46*b10 - 27*b8 + 35*b7 - 36*b6 + 93*b5 - 843*b4 + 252*b3 - 301*b1 - 48) * q^28 + (-103*b8 + 38*b7 - 147*b5 + 264*b3 - 112*b2 + 240) * q^29 + (25*b11 - 25*b10 - 25*b9 + 25*b8 - 25*b7 - 25*b6 - 25*b5 - 400*b4 + 25*b2 + 125*b1 + 400) * q^30 + (68*b11 - 18*b10 - 12*b9 - 148*b6 + 1206*b4 + 434*b3 + 434*b1) * q^31 + (40*b11 + 30*b10 - 40*b9 + 52*b6 - 1649*b4 - 611*b3 - 611*b1) * q^32 + (40*b11 + 40*b10 - 40*b8 - 40*b7 + 192*b6 + 192*b5 + 866*b4 + 704*b1 - 866) * q^33 + (14*b8 + 26*b7 - 54*b5 - 792*b3 - 74*b2 + 198) * q^34 + (-75*b10 + 25*b8 - 25*b6 + 50*b5 - 75*b4 + 175*b3 - 150) * q^35 + (81*b8 + 29*b7 - b5 + 588*b3 + 38*b2 - 4313) * q^36 + (10*b11 + 25*b10 - 220*b9 - 25*b8 - 10*b7 + 9*b6 + 9*b5 - 2445*b4 + 220*b2 - 14*b1 + 2445) * q^37 + (-65*b11 + 97*b10 - 140*b9 + 177*b6 + 7679*b4 - 136*b3 - 136*b1) * q^38 + (-36*b11 - 244*b10 + 100*b9 - 174*b6 + 2460*b4 - 994*b3 - 994*b1) * q^39 + (-50*b10 + 50*b8 - 100*b6 - 100*b5 + 575*b4 + 125*b1 - 575) * q^40 + (-62*b8 + 112*b7 - 198*b5 - 44*b3 - 168*b2 + 607) * q^41 + (-35*b11 + 7*b10 + 210*b9 + 56*b8 - 105*b6 - 280*b5 - 4480*b4 + 77*b3 - 140*b2 - 973*b1 + 3234) * q^42 + (112*b8 - 120*b7 + 193*b5 + 871*b3 + 40*b2 - 3909) * q^43 + (-29*b11 - 261*b10 + 92*b9 + 261*b8 + 29*b7 - 161*b6 - 161*b5 - 4355*b4 - 92*b2 + 1372*b1 + 4355) * q^44 + (-50*b11 - 75*b10 + 100*b9 - 75*b6 + 1625*b4 - 50*b3 - 50*b1) * q^45 + (-112*b11 + 272*b10 + 133*b9 + 552*b6 - 1279*b4 - 1741*b3 - 1741*b1) * q^46 + (-80*b11 - 135*b10 + 120*b9 + 135*b8 + 80*b7 - 59*b6 - 59*b5 + 7483*b4 - 120*b2 + 1402*b1 - 7483) * q^47 + (267*b8 - 95*b7 + 291*b5 + 451*b3 + 70*b2 + 1988) * q^48 + (70*b11 - 135*b10 - 196*b9 + 66*b8 - 56*b7 + 165*b6 - 246*b5 - 10852*b4 + 350*b3 + 392*b2 - 14*b1 + 4161) * q^49 + (-625*b3 - 625) * q^50 + (76*b11 - 6*b10 - 284*b9 + 6*b8 - 76*b7 + 4*b6 + 4*b5 - 7126*b4 + 284*b2 + 38*b1 + 7126) * q^51 + (85*b11 + 165*b10 + 190*b9 + 625*b6 + 6273*b4 + 938*b3 + 938*b1) * q^52 + (110*b11 + 155*b10 + 240*b9 - 567*b6 + 135*b4 - 1068*b3 - 1068*b1) * q^53 + (95*b11 - 335*b10 + 7*b9 + 335*b8 - 95*b7 - 935*b6 - 935*b5 + 3238*b4 - 7*b2 + 733*b1 - 3238) * q^54 + (-25*b8 - 100*b7 - 225*b5 - 800*b3 - 100*b2 + 3775) * q^55 + (7*b11 + 63*b10 + 175*b9 - 7*b8 - 63*b7 + 343*b6 + 287*b5 - 12320*b4 - 133*b3 + 14*b2 - 77*b1 + 11851) * q^56 + (-166*b8 + 220*b7 + 312*b5 - 30*b3 - 220*b2 - 16396) * q^57 + (-135*b11 + 435*b10 + 270*b9 - 435*b8 + 135*b7 + 963*b6 + 963*b5 - 6118*b4 - 270*b2 - 2071*b1 + 6118) * q^58 + (68*b11 - 598*b10 - 476*b9 + 342*b6 + 10892*b4 + 968*b3 + 968*b1) * q^59 + (125*b11 - 225*b10 - 250*b9 + 325*b6 + 1600*b4 + 675*b3 + 675*b1) * q^60 + (22*b11 + 443*b10 + 112*b9 - 443*b8 - 22*b7 - 727*b6 - 727*b5 + 11694*b4 - 112*b2 - 2444*b1 - 11694) * q^61 + (-480*b8 + 320*b7 + 1048*b5 + 772*b3 - 790*b2 + 18184) * q^62 + (140*b11 + 179*b10 - 420*b9 - 181*b8 + 615*b6 - 33*b5 - 23075*b4 + 3066*b3 + 280*b2 + 2674*b1 + 7071) * q^63 + (128*b8 - 288*b7 + 272*b5 - 155*b3 + 523*b2 - 26107) * q^64 + (-250*b11 + 75*b10 - 75*b8 + 250*b7 - 175*b6 - 175*b5 + 725*b4 + 100*b1 - 725) * q^65 + (-192*b11 + 112*b10 - 160*b9 - 208*b6 + 23226*b4 + 282*b3 + 282*b1) * q^66 + (-260*b11 + 234*b10 - 460*b9 - 1901*b6 + 23351*b4 - 239*b3 - 239*b1) * q^67 + (30*b11 + 298*b10 - 390*b9 - 298*b8 - 30*b7 - 562*b6 - 562*b5 + 20074*b4 + 390*b2 - 2156*b1 - 20074) * q^68 + (-219*b8 + 34*b7 - 451*b5 - 3328*b3 + 904*b2 + 2732) * q^69 + (175*b9 - 175*b8 + 175*b7 + 175*b5 - 7525*b4 - 700*b3 - 525*b2 - 175*b1 + 6300) * q^70 + (-114*b8 - 116*b7 + 1874*b5 - 3068*b3 - 396*b2 - 26588) * q^71 + (265*b11 + 101*b10 - 340*b9 - 101*b8 - 265*b7 + 1025*b6 + 1025*b5 - 19105*b4 + 340*b2 + 3416*b1 + 19105) * q^72 + (20*b11 + 1010*b10 + 500*b9 + 276*b6 + 882*b4 - 34*b3 - 34*b1) * q^73 + (-39*b11 - 421*b10 + 352*b9 - 941*b6 + 363*b4 + 4662*b3 + 4662*b1) * q^74 + (-625*b6 - 625*b5 + 1875*b4 - 625*b1 - 1875) * q^75 + (255*b8 - 245*b7 + 785*b5 + 6228*b3 + 762*b2 + 2537) * q^76 + (-210*b11 + 111*b10 + 140*b9 + 12*b8 + 219*b6 - 1320*b5 - 17137*b4 + 1414*b3 - 1400*b2 + 2450*b1 + 12122) * q^77 + (-318*b8 + 590*b7 - 170*b5 - 2084*b3 - 430*b2 - 36386) * q^78 + (380*b11 - 60*b10 + 1116*b9 + 60*b8 - 380*b7 - 820*b6 - 820*b5 - 37458*b4 - 1116*b2 - 2576*b1 + 37458) * q^79 + (200*b11 - 200*b10 + 275*b9 - 200*b6 + 1725*b4 - 475*b3 - 475*b1) * q^80 + (284*b11 - 84*b10 - 184*b9 + 2196*b6 + 8851*b4 + 8060*b3 + 8060*b1) * q^81 + (-150*b11 + 710*b10 - 360*b9 - 710*b8 + 150*b7 + 1942*b6 + 1942*b5 + 4861*b4 + 360*b2 - 3461*b1 - 4861) * q^82 + (222*b8 - 540*b7 - 1275*b5 + 1645*b3 + 1420*b2 + 31123) * q^83 + (-504*b11 + 144*b10 + 126*b9 - 27*b8 + 77*b7 - 2136*b6 - 1993*b5 - 25266*b4 - 7749*b3 + 455*b2 - 6762*b1 + 10648) * q^84 + (100*b8 + 850*b5 + 50*b3 + 500*b2 - 6400) * q^85 + (271*b11 - 591*b10 + 849*b9 + 591*b8 - 271*b7 - 1631*b6 - 1631*b5 - 34200*b4 - 849*b2 + 2115*b1 + 34200) * q^86 + (-160*b11 - 600*b10 + 2120*b9 - 37*b6 + 34477*b4 - 7721*b3 - 7721*b1) * q^87 + (145*b11 + 97*b10 - 1230*b9 + 1281*b6 + 7395*b4 + 936*b3 + 936*b1) * q^88 + (-186*b11 - 429*b10 - 140*b9 + 429*b8 + 186*b7 + 3531*b6 + 3531*b5 + 6542*b4 + 140*b2 - 6634*b1 - 6542) * q^89 + (175*b8 + 125*b7 - 775*b5 - 1400*b3 - 500*b2 - 1175) * q^90 + (-140*b11 + 60*b10 - 868*b9 + 323*b8 + 252*b7 + 2610*b6 + 5217*b5 - 10356*b4 - 2520*b3 + 1036*b2 + 6398*b1 + 1009) * q^91 + (114*b8 - 680*b7 - 4060*b5 - 1055*b3 + 1880*b2 - 47249) * q^92 + (-580*b11 - 310*b10 + 660*b9 + 310*b8 + 580*b7 - 56*b6 - 56*b5 - 34166*b4 - 660*b2 - 9790*b1 + 34166) * q^93 + (169*b11 + 231*b10 - 2180*b9 + 1111*b6 + 55563*b4 + 3666*b3 + 3666*b1) * q^94 + (-200*b11 + 75*b10 + 200*b9 + 1075*b6 + 1675*b4 + 5750*b3 + 5750*b1) * q^95 + (-111*b11 - 1179*b10 + 195*b9 + 1179*b8 + 111*b7 - 3999*b6 - 3999*b5 + 9528*b4 - 195*b2 - 2559*b1 - 9528) * q^96 + (1744*b8 + 120*b7 - 464*b5 + 2152*b3 - 1480*b2 - 12738) * q^97 + (490*b11 - 1386*b10 + 609*b8 - 245*b7 - 2618*b6 - 889*b5 - 9765*b4 - 7105*b3 + 7497*b1 - 5467) * q^98 + (461*b8 + 524*b7 - 3201*b5 - 1922*b3 - 2532*b2 - 32765) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 5 q^{2} - 20 q^{3} - 31 q^{4} + 150 q^{5} + 192 q^{6} - 20 q^{7} - 270 q^{8} - 378 q^{9}+O(q^{10})$$ 12 * q + 5 * q^2 - 20 * q^3 - 31 * q^4 + 150 * q^5 + 192 * q^6 - 20 * q^7 - 270 * q^8 - 378 * q^9 $$12 q + 5 q^{2} - 20 q^{3} - 31 q^{4} + 150 q^{5} + 192 q^{6} - 20 q^{7} - 270 q^{8} - 378 q^{9} - 125 q^{10} + 924 q^{11} - 370 q^{12} - 300 q^{13} + 3409 q^{14} - 1000 q^{15} - 435 q^{16} - 1540 q^{17} - 195 q^{18} - 92 q^{19} - 1550 q^{20} + 7288 q^{21} - 13710 q^{22} + 3920 q^{23} + 7200 q^{24} - 3750 q^{25} - 2635 q^{26} + 4120 q^{27} - 6015 q^{28} + 2528 q^{29} + 2400 q^{30} + 7160 q^{31} - 9105 q^{32} - 4460 q^{33} + 4332 q^{34} - 2500 q^{35} - 52750 q^{36} + 14170 q^{37} + 46215 q^{38} + 15376 q^{39} - 3375 q^{40} + 8196 q^{41} + 10500 q^{42} - 48920 q^{43} + 27873 q^{44} + 9450 q^{45} - 6815 q^{46} - 42940 q^{47} + 23220 q^{48} - 17856 q^{49} - 6250 q^{50} + 42008 q^{51} + 36115 q^{52} + 2450 q^{53} - 19566 q^{54} + 46200 q^{55} + 67998 q^{56} - 194200 q^{57} + 36110 q^{58} + 64600 q^{59} + 9250 q^{60} - 73620 q^{61} + 222880 q^{62} - 57780 q^{63} - 315994 q^{64} - 3750 q^{65} + 139138 q^{66} + 142620 q^{67} - 124330 q^{68} + 34688 q^{69} + 34475 q^{70} - 308512 q^{71} + 117495 q^{72} + 5120 q^{73} - 2785 q^{74} - 12500 q^{75} + 15550 q^{76} + 44230 q^{77} - 428180 q^{78} + 222504 q^{79} + 10875 q^{80} + 43986 q^{81} - 31665 q^{82} + 359160 q^{83} - 19966 q^{84} - 77000 q^{85} + 207160 q^{86} + 209300 q^{87} + 45145 q^{88} - 41648 q^{89} - 9750 q^{90} - 31376 q^{91} - 584370 q^{92} + 198520 q^{93} + 333699 q^{94} + 2300 q^{95} - 61824 q^{96} - 147960 q^{97} - 101815 q^{98} - 381544 q^{99}+O(q^{100})$$ 12 * q + 5 * q^2 - 20 * q^3 - 31 * q^4 + 150 * q^5 + 192 * q^6 - 20 * q^7 - 270 * q^8 - 378 * q^9 - 125 * q^10 + 924 * q^11 - 370 * q^12 - 300 * q^13 + 3409 * q^14 - 1000 * q^15 - 435 * q^16 - 1540 * q^17 - 195 * q^18 - 92 * q^19 - 1550 * q^20 + 7288 * q^21 - 13710 * q^22 + 3920 * q^23 + 7200 * q^24 - 3750 * q^25 - 2635 * q^26 + 4120 * q^27 - 6015 * q^28 + 2528 * q^29 + 2400 * q^30 + 7160 * q^31 - 9105 * q^32 - 4460 * q^33 + 4332 * q^34 - 2500 * q^35 - 52750 * q^36 + 14170 * q^37 + 46215 * q^38 + 15376 * q^39 - 3375 * q^40 + 8196 * q^41 + 10500 * q^42 - 48920 * q^43 + 27873 * q^44 + 9450 * q^45 - 6815 * q^46 - 42940 * q^47 + 23220 * q^48 - 17856 * q^49 - 6250 * q^50 + 42008 * q^51 + 36115 * q^52 + 2450 * q^53 - 19566 * q^54 + 46200 * q^55 + 67998 * q^56 - 194200 * q^57 + 36110 * q^58 + 64600 * q^59 + 9250 * q^60 - 73620 * q^61 + 222880 * q^62 - 57780 * q^63 - 315994 * q^64 - 3750 * q^65 + 139138 * q^66 + 142620 * q^67 - 124330 * q^68 + 34688 * q^69 + 34475 * q^70 - 308512 * q^71 + 117495 * q^72 + 5120 * q^73 - 2785 * q^74 - 12500 * q^75 + 15550 * q^76 + 44230 * q^77 - 428180 * q^78 + 222504 * q^79 + 10875 * q^80 + 43986 * q^81 - 31665 * q^82 + 359160 * q^83 - 19966 * q^84 - 77000 * q^85 + 207160 * q^86 + 209300 * q^87 + 45145 * q^88 - 41648 * q^89 - 9750 * q^90 - 31376 * q^91 - 584370 * q^92 + 198520 * q^93 + 333699 * q^94 + 2300 * q^95 - 61824 * q^96 - 147960 * q^97 - 101815 * q^98 - 381544 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 110 x^{10} + 27 x^{9} + 9714 x^{8} - 3257 x^{7} + 240269 x^{6} - 608504 x^{5} + 5067668 x^{4} - 7120352 x^{3} + 9029136 x^{2} - 1794240 x + 313600$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 28\!\cdots\!87 \nu^{11} + \cdots + 21\!\cdots\!60 ) / 59\!\cdots\!60$$ (28230181680367687*v^11 - 133725817906773464*v^10 + 3228942815539669144*v^9 - 7866915029237918170*v^8 + 267470564024133907693*v^7 - 844940438036854392342*v^6 + 7210253086068556811390*v^5 - 13733163183321376193458*v^4 + 172488893418565320606489*v^3 - 9848865216344630186408*v^2 + 1899336597051364106720*v + 21547821407039195027925760) / 595840743865600904598060 $$\beta_{3}$$ $$=$$ $$( 66\!\cdots\!49 \nu^{11} + \cdots - 11\!\cdots\!00 ) / 59\!\cdots\!60$$ (66388882378066849*v^11 + 28230181680367687*v^10 + 7197281425360947613*v^9 + 12218724065108421680*v^8 + 649253412456411874696*v^7 + 700495386575182055196*v^6 + 15806745328634071405235*v^5 - 17380902067880561667271*v^4 + 305322749531891334677403*v^3 + 5099431532023610873044*v^2 - 1155929670352927198960*v - 118374181391324226242000) / 595840743865600904598060 $$\beta_{4}$$ $$=$$ $$( - 60\!\cdots\!45 \nu^{11} + \cdots + 10\!\cdots\!60 ) / 95\!\cdots\!60$$ (-6039499050577766645*v^11 + 4977276932528697061*v^10 - 664796578470440213942*v^9 - 278222977171374861223*v^8 - 58863193362354159936410*v^7 + 9282593808429195967629*v^6 - 1462312323568472326910641*v^5 + 3422151405014628172065320*v^4 - 30328081641557240551657524*v^3 + 38118195151269240531420592*v^2 - 54613049204049911387937424*v + 10854805651234298860308160) / 9533451901849614473568960 $$\beta_{5}$$ $$=$$ $$( 52\!\cdots\!28 \nu^{11} + \cdots + 10\!\cdots\!20 ) / 14\!\cdots\!40$$ (52322101745624380228*v^11 + 30607787177058670459*v^10 + 5693107512848974435651*v^9 + 10379043110318982599270*v^8 + 512180378623485246907117*v^7 + 622066563111152923675782*v^6 + 12429499745875403113244285*v^5 - 13334354917553499228708277*v^4 + 214860347241695946183782106*v^3 + 4708257231455631913367548*v^2 - 1048147295193847137434320*v + 107410033366125948920869120) / 14300177852774421710353440 $$\beta_{6}$$ $$=$$ $$( 15\!\cdots\!47 \nu^{11} + \cdots - 26\!\cdots\!40 ) / 28\!\cdots\!80$$ (151076034291374185547*v^11 - 270100722912524149903*v^10 + 16513889933321721359462*v^9 - 8819865760350601583663*v^8 + 1440818444881405886231798*v^7 - 1659244044540004057540923*v^6 + 34645031716165556521494679*v^5 - 120088673516912456214643388*v^4 + 794420238005862000834919680*v^3 - 1546932437238343332569148616*v^2 + 1350502024155491915228513536*v - 268110604843501017027082240) / 28600355705548843420706880 $$\beta_{7}$$ $$=$$ $$( 70\!\cdots\!29 \nu^{11} + \cdots + 10\!\cdots\!40 ) / 10\!\cdots\!60$$ (7091894478336914729*v^11 + 1411801999708260032*v^10 + 772990456491140565773*v^9 + 1161485286890362349770*v^8 + 69260803279607131194011*v^7 + 63758324818681303471356*v^6 + 1693912209577006520603770*v^5 - 1926490887512260700141126*v^4 + 31400176434228223257936348*v^3 + 412482034194436777839944*v^2 - 97167571241683005764960*v + 107743678670267150252505440) / 1021441275198172979310960 $$\beta_{8}$$ $$=$$ $$( 16\!\cdots\!16 \nu^{11} + \cdots + 73\!\cdots\!76 ) / 14\!\cdots\!44$$ (16063048464078134816*v^11 + 7977603534081794105*v^10 + 1746180598725624158405*v^9 + 3059198080549517329546*v^8 + 157157152306574834673947*v^7 + 179372545053722983092954*v^6 + 3820655373495828539523319*v^5 - 4155449062191167306088143*v^4 + 68158744233247905816711078*v^3 + 1328428462363840694290484*v^2 - 298502823994606888776560*v + 73522585881312782070480176) / 1430017785277442171035344 $$\beta_{9}$$ $$=$$ $$( - 54\!\cdots\!01 \nu^{11} + \cdots + 98\!\cdots\!40 ) / 23\!\cdots\!40$$ (-54621046984712167201*v^11 + 44682571666036802801*v^10 - 6011958331935405715930*v^9 - 2552881690802807437727*v^8 - 532365753911013086926474*v^7 + 80741362729562035487877*v^6 - 13224037893430787227816709*v^5 + 30868886253403175795256964*v^4 - 274174025772142730303627328*v^3 + 345426721610757473957685392*v^2 - 493896182093230194401033216*v + 98166747586673986647741440) / 2383362975462403618392240 $$\beta_{10}$$ $$=$$ $$( - 13\!\cdots\!81 \nu^{11} + \cdots + 24\!\cdots\!60 ) / 40\!\cdots\!40$$ (-139478356307493384181*v^11 + 175345975795757180573*v^10 - 15334702064029637541934*v^9 + 149384528847355829017*v^8 - 1348948846630186060003354*v^7 + 804616953571597380973005*v^6 - 33179340034742427577681121*v^5 + 93437579537306714970742864*v^4 - 716539518823875975101698236*v^3 + 1148800577474268071998274336*v^2 - 1257218836543510135952592944*v + 249753470635519944424976960) / 4085765100792691917243840 $$\beta_{11}$$ $$=$$ $$( - 96\!\cdots\!41 \nu^{11} + \cdots + 17\!\cdots\!60 ) / 14\!\cdots\!40$$ (-969388706845078941841*v^11 + 907065566243025426803*v^10 - 106691226944671427981434*v^9 - 33047141121934547844173*v^8 - 9430977795125532582265174*v^7 + 2545918467646447090830315*v^6 - 233861202257717465115639581*v^5 + 575145419155729177351723384*v^4 - 4897614943386828431458750476*v^3 + 6713400393754856774999004416*v^2 - 8760165758735224833443687024*v + 1740925601580401309248804160) / 14300177852774421710353440
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} - 36\beta_{4} + \beta_{3} + \beta_1$$ b9 - 36*b4 + b3 + b1 $$\nu^{3}$$ $$=$$ $$-2\beta_{8} + 4\beta_{5} + 69\beta_{3} + 3\beta_{2} - 22$$ -2*b8 + 4*b5 + 69*b3 + 3*b2 - 22 $$\nu^{4}$$ $$=$$ $$8\beta_{11} - 91\beta_{9} - 8\beta_{7} + 8\beta_{6} + 8\beta_{5} + 2468\beta_{4} + 91\beta_{2} - 197\beta _1 - 2468$$ 8*b11 - 91*b9 - 8*b7 + 8*b6 + 8*b5 + 2468*b4 + 91*b2 - 197*b1 - 2468 $$\nu^{5}$$ $$=$$ $$206\beta_{10} - 395\beta_{9} + 460\beta_{6} + 6066\beta_{4} - 5449\beta_{3} - 5449\beta_1$$ 206*b10 - 395*b9 + 460*b6 + 6066*b4 - 5449*b3 - 5449*b1 $$\nu^{6}$$ $$=$$ $$124\beta_{8} + 872\beta_{7} - 1168\beta_{5} - 25089\beta_{3} - 7983\beta_{2} + 194128$$ 124*b8 + 872*b7 - 1168*b5 - 25089*b3 - 7983*b2 + 194128 $$\nu^{7}$$ $$=$$ $$- 544 \beta_{11} - 18162 \beta_{10} + 43887 \beta_{9} + 18162 \beta_{8} + 544 \beta_{7} - 42148 \beta_{6} - 42148 \beta_{5} - 820902 \beta_{4} - 43887 \beta_{2} + 461245 \beta _1 + 820902$$ -544*b11 - 18162*b10 + 43887*b9 + 18162*b8 + 544*b7 - 42148*b6 - 42148*b5 - 820902*b4 - 43887*b2 + 461245*b1 + 820902 $$\nu^{8}$$ $$=$$ $$- 77928 \beta_{11} - 29640 \beta_{10} + 705963 \beta_{9} - 145752 \beta_{6} - 16333132 \beta_{4} + 2758077 \beta_{3} + 2758077 \beta_1$$ -77928*b11 - 29640*b10 + 705963*b9 - 145752*b6 - 16333132*b4 + 2758077*b3 + 2758077*b1 $$\nu^{9}$$ $$=$$ $$-1548246\beta_{8} - 127104\beta_{7} + 3699708\beta_{5} + 40566289\beta_{3} + 4580067\beta_{2} - 92353626$$ -1548246*b8 - 127104*b7 + 3699708*b5 + 40566289*b3 + 4580067*b2 - 92353626 $$\nu^{10}$$ $$=$$ $$6669096 \beta_{11} + 4420596 \beta_{10} - 63318823 \beta_{9} - 4420596 \beta_{8} - 6669096 \beta_{7} + 16749024 \beta_{6} + 16749024 \beta_{5} + 1428426792 \beta_{4} + \cdots - 1428426792$$ 6669096*b11 + 4420596*b10 - 63318823*b9 - 4420596*b8 - 6669096*b7 + 16749024*b6 + 16749024*b5 + 1428426792*b4 + 63318823*b2 - 284237401*b1 - 1428426792 $$\nu^{11}$$ $$=$$ $$18921120 \beta_{11} + 132144410 \beta_{10} - 462055479 \beta_{9} + 324463252 \beta_{6} + 9631351150 \beta_{4} - 3651786501 \beta_{3} - 3651786501 \beta_1$$ 18921120*b11 + 132144410*b10 - 462055479*b9 + 324463252*b6 + 9631351150*b4 - 3651786501*b3 - 3651786501*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 4.86278 + 8.42258i 2.06025 + 3.56846i 0.675239 + 1.16955i 0.101659 + 0.176079i −3.11366 − 5.39303i −4.08626 − 7.07762i 4.86278 − 8.42258i 2.06025 − 3.56846i 0.675239 − 1.16955i 0.101659 − 0.176079i −3.11366 + 5.39303i −4.08626 + 7.07762i
−4.36278 7.55656i −2.36982 + 4.10465i −22.0677 + 38.2224i 12.5000 + 21.6506i 41.3561 8.78435 + 129.344i 105.888 110.268 + 190.990i 109.070 188.914i
11.2 −1.56025 2.70243i −13.9359 + 24.1376i 11.1312 19.2799i 12.5000 + 21.6506i 86.9736 −89.4698 93.8198i −169.326 −266.916 462.312i 39.0062 67.5608i
11.3 −0.175239 0.303524i 10.7955 18.6984i 15.9386 27.6064i 12.5000 + 21.6506i −7.56720 −129.103 11.8023i −22.3876 −111.586 193.273i 4.38099 7.58809i
11.4 0.398341 + 0.689946i −5.28741 + 9.15807i 15.6826 27.1631i 12.5000 + 21.6506i −8.42477 103.426 + 78.1675i 50.4819 65.5865 + 113.599i −9.95851 + 17.2487i
11.5 3.61366 + 6.25905i 7.96046 13.7879i −10.1171 + 17.5234i 12.5000 + 21.6506i 115.066 102.346 + 79.5760i 85.0347 −5.23796 9.07242i −90.3416 + 156.476i
11.6 4.58626 + 7.94364i −7.16289 + 12.4065i −26.0676 + 45.1505i 12.5000 + 21.6506i −131.404 −5.98242 129.504i −184.691 18.8861 + 32.7118i −114.657 + 198.591i
16.1 −4.36278 + 7.55656i −2.36982 4.10465i −22.0677 38.2224i 12.5000 21.6506i 41.3561 8.78435 129.344i 105.888 110.268 190.990i 109.070 + 188.914i
16.2 −1.56025 + 2.70243i −13.9359 24.1376i 11.1312 + 19.2799i 12.5000 21.6506i 86.9736 −89.4698 + 93.8198i −169.326 −266.916 + 462.312i 39.0062 + 67.5608i
16.3 −0.175239 + 0.303524i 10.7955 + 18.6984i 15.9386 + 27.6064i 12.5000 21.6506i −7.56720 −129.103 + 11.8023i −22.3876 −111.586 + 193.273i 4.38099 + 7.58809i
16.4 0.398341 0.689946i −5.28741 9.15807i 15.6826 + 27.1631i 12.5000 21.6506i −8.42477 103.426 78.1675i 50.4819 65.5865 113.599i −9.95851 17.2487i
16.5 3.61366 6.25905i 7.96046 + 13.7879i −10.1171 17.5234i 12.5000 21.6506i 115.066 102.346 79.5760i 85.0347 −5.23796 + 9.07242i −90.3416 156.476i
16.6 4.58626 7.94364i −7.16289 12.4065i −26.0676 45.1505i 12.5000 21.6506i −131.404 −5.98242 + 129.504i −184.691 18.8861 32.7118i −114.657 198.591i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 16.6 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 35.6.e.a 12
7.c even 3 1 inner 35.6.e.a 12
7.c even 3 1 245.6.a.i 6
7.d odd 6 1 245.6.a.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.e.a 12 1.a even 1 1 trivial
35.6.e.a 12 7.c even 3 1 inner
245.6.a.h 6 7.d odd 6 1
245.6.a.i 6 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 5 T_{2}^{11} + 124 T_{2}^{10} - 275 T_{2}^{9} + 10044 T_{2}^{8} - 22195 T_{2}^{7} + 309235 T_{2}^{6} + 471890 T_{2}^{5} + 3125728 T_{2}^{4} - 1125720 T_{2}^{3} + 1657728 T_{2}^{2} + \cdots + 254016$$ acting on $$S_{6}^{\mathrm{new}}(35, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 5 T^{11} + 124 T^{10} + \cdots + 254016$$
$3$ $$T^{12} + 20 T^{11} + \cdots + 47324438284176$$
$5$ $$(T^{2} - 25 T + 625)^{6}$$
$7$ $$T^{12} + 20 T^{11} + \cdots + 22\!\cdots\!49$$
$11$ $$T^{12} - 924 T^{11} + \cdots + 54\!\cdots\!00$$
$13$ $$(T^{6} + 150 T^{5} + \cdots + 37\!\cdots\!36)^{2}$$
$17$ $$T^{12} + 1540 T^{11} + \cdots + 35\!\cdots\!16$$
$19$ $$T^{12} + 92 T^{11} + \cdots + 23\!\cdots\!16$$
$23$ $$T^{12} - 3920 T^{11} + \cdots + 30\!\cdots\!01$$
$29$ $$(T^{6} - 1264 T^{5} + \cdots - 10\!\cdots\!00)^{2}$$
$31$ $$T^{12} - 7160 T^{11} + \cdots + 36\!\cdots\!36$$
$37$ $$T^{12} - 14170 T^{11} + \cdots + 46\!\cdots\!56$$
$41$ $$(T^{6} - 4098 T^{5} + \cdots + 10\!\cdots\!25)^{2}$$
$43$ $$(T^{6} + 24460 T^{5} + \cdots + 19\!\cdots\!56)^{2}$$
$47$ $$T^{12} + 42940 T^{11} + \cdots + 97\!\cdots\!00$$
$53$ $$T^{12} - 2450 T^{11} + \cdots + 72\!\cdots\!56$$
$59$ $$T^{12} - 64600 T^{11} + \cdots + 13\!\cdots\!00$$
$61$ $$T^{12} + 73620 T^{11} + \cdots + 45\!\cdots\!16$$
$67$ $$T^{12} - 142620 T^{11} + \cdots + 33\!\cdots\!00$$
$71$ $$(T^{6} + 154256 T^{5} + \cdots + 99\!\cdots\!24)^{2}$$
$73$ $$T^{12} - 5120 T^{11} + \cdots + 26\!\cdots\!16$$
$79$ $$T^{12} - 222504 T^{11} + \cdots + 51\!\cdots\!96$$
$83$ $$(T^{6} - 179580 T^{5} + \cdots + 35\!\cdots\!64)^{2}$$
$89$ $$T^{12} + 41648 T^{11} + \cdots + 22\!\cdots\!36$$
$97$ $$(T^{6} + 73980 T^{5} + \cdots + 99\!\cdots\!84)^{2}$$