# Properties

 Label 100.4.e.e Level $100$ Weight $4$ Character orbit 100.e Analytic conductor $5.900$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 100.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.90019100057$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096$$ x^12 - 7*x^10 + 44*x^8 - 156*x^6 + 704*x^4 - 1792*x^2 + 4096 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{15}$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + \beta_{3}) q^{2} + \beta_{9} q^{3} + ( - \beta_{11} + \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{4} + ( - \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{6} + ( - 2 \beta_{11} - \beta_{10} - \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \cdots - 1) q^{7}+ \cdots + (3 \beta_{5} + 3 \beta_{4} + 10 \beta_{3} + 3 \beta_{2}) q^{9}+O(q^{10})$$ q + (b5 + b3) * q^2 + b9 * q^3 + (-b11 + b5 + b4 - 2*b3) * q^4 + (-b10 + b9 + 2*b8 + b5 - b4 + b3 + b1) * q^6 + (-2*b11 - b10 - b8 - b7 - 2*b6 - b5 + 2*b4 + b3 + b2 + 2*b1 - 1) * q^7 + (-b11 - 2*b9 + 2*b8 + b7 - b6 + b5 - 3*b4 + b3 - b2 - b1) * q^8 + (3*b5 + 3*b4 + 10*b3 + 3*b2) * q^9 $$q + (\beta_{5} + \beta_{3}) q^{2} + \beta_{9} q^{3} + ( - \beta_{11} + \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{4} + ( - \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{6} + ( - 2 \beta_{11} - \beta_{10} - \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \cdots - 1) q^{7}+ \cdots + ( - 28 \beta_{11} + 29 \beta_{10} + 29 \beta_{9} + 45 \beta_{7} - 45 \beta_{6} + \cdots + 14 \beta_{2}) q^{99}+O(q^{100})$$ q + (b5 + b3) * q^2 + b9 * q^3 + (-b11 + b5 + b4 - 2*b3) * q^4 + (-b10 + b9 + 2*b8 + b5 - b4 + b3 + b1) * q^6 + (-2*b11 - b10 - b8 - b7 - 2*b6 - b5 + 2*b4 + b3 + b2 + 2*b1 - 1) * q^7 + (-b11 - 2*b9 + 2*b8 + b7 - b6 + b5 - 3*b4 + b3 - b2 - b1) * q^8 + (3*b5 + 3*b4 + 10*b3 + 3*b2) * q^9 + (-b10 + b9 - 2*b8 - b7 - b6 + 3*b5 - 3*b4 + 4*b3 + 4*b1 - 6) * q^11 + (-2*b11 + 2*b10 + 3*b8 + 8*b7 - 2*b6 + 6*b5 + 2*b4 + 8*b3 + 5*b2 + 2*b1 + 8) * q^12 + (-4*b8 - 3*b6 + 5*b4 + 10*b3 + 4*b2 - 10) * q^13 + (-b11 - 3*b10 - 3*b9 + 7*b7 - 7*b6 + 2*b5 + 2*b4 + 12*b3 + b2) * q^14 + (6*b10 - 6*b9 - 4*b8 - 4*b7 - 4*b6 - 4*b5 + 4*b4 + 2*b1 + 24) * q^16 + (b8 + 2*b5 + 29*b3 + b2 + 27) * q^17 + (-3*b11 + 6*b9 + 3*b7 - 3*b6 + 3*b5 + 10*b4 + 15*b3 + 3*b2 - 3*b1 - 12) * q^18 + (-4*b11 + 2*b10 + 2*b9 - 5*b7 + 5*b6 - 17*b5 - 17*b4 + 2*b3 + 2*b2) * q^19 + (3*b8 + 11*b7 + 11*b6 - 8*b5 + 8*b4 - 19*b3 + 8) * q^21 + (-5*b11 - 6*b10 - 4*b8 + 9*b7 - 5*b6 + 7*b5 + 5*b4 - 29*b3 + b2 + 5*b1 - 32) * q^22 + (-6*b11 - 7*b9 + 3*b8 + 6*b7 + 7*b6 + 6*b5 - 15*b4 + 7*b3 + 3*b2 - 6*b1 - 1) * q^23 + (4*b11 + 8*b10 + 8*b9 + 12*b7 - 12*b6 + 12*b5 + 12*b4 - 60*b3) * q^24 + (-8*b10 + 8*b9 - 6*b8 - 6*b7 - 6*b6 - 13*b5 + 13*b4 - 7*b3 - 2*b1 - 8) * q^26 + (8*b10 - 12*b7 - 36*b5 - 12*b3 + 12) * q^27 + (2*b11 - 6*b9 - 11*b8 - 2*b7 - 12*b6 - 2*b5 + 34*b4 - 82*b3 + 9*b2 + 2*b1 + 80) * q^28 + (-19*b7 + 19*b6 + b5 + b4 + 46*b3 - 18*b2) * q^29 + (9*b10 - 9*b9 + 6*b8 + 17*b7 + 17*b6 + 33*b5 - 33*b4 + 16*b3 - 12*b1 - 10) * q^31 + (6*b11 + 4*b10 - 16*b8 - 6*b7 + 6*b6 + 2*b5 - 6*b4 + 26*b3 - 22*b2 - 6*b1 + 24) * q^32 + (23*b8 + 34*b6 - 12*b4 - 12*b3 - 23*b2 + 12) * q^33 + (-2*b11 + 2*b10 + 2*b9 + 2*b7 - 2*b6 + 29*b5 + 29*b4 + 37*b3 + 2*b2) * q^34 + (-6*b10 + 6*b9 + 19*b8 - 5*b7 - 5*b6 - 12*b5 + 12*b4 - 7*b3 - b1 + 52) * q^36 + (7*b8 - 27*b7 + 41*b5 - 6*b3 + 7*b2 - 74) * q^37 + (14*b11 - 6*b8 - 14*b7 - 22*b6 - 14*b5 - 18*b4 + 146*b3 - 8*b2 + 14*b1 - 160) * q^38 + (12*b11 - 17*b10 - 17*b9 - 19*b7 + 19*b6 - 51*b5 - 51*b4 - 6*b3 - 6*b2) * q^39 + (-7*b8 - 18*b7 - 18*b6 + 11*b5 - 11*b4 + 29*b3 - 86) * q^41 + (19*b11 + 6*b10 + 22*b8 + 25*b7 + 19*b6 + 11*b5 - 19*b4 - 109*b3 + 3*b2 - 19*b1 - 76) * q^42 + (20*b11 + 17*b9 - 10*b8 - 20*b7 + 6*b6 - 20*b5 - 38*b4 + 6*b3 - 10*b2 + 20*b1 - 26) * q^43 + (-4*b11 - 14*b10 - 14*b9 + 18*b7 - 18*b6 - 8*b5 - 8*b4 - 182*b3 + 10*b2) * q^44 + (13*b10 - 13*b9 - 16*b8 - 26*b7 - 26*b6 - 7*b5 + 7*b4 + 19*b3 + b1 - 144) * q^46 + (26*b11 - 25*b10 + 13*b8 - 7*b7 + 26*b6 - 47*b5 - 26*b4 - 33*b3 - 13*b2 - 26*b1 + 33) * q^47 + (8*b11 + 24*b9 + 12*b8 - 8*b7 + 16*b6 - 8*b5 - 56*b4 - 216*b3 - 20*b2 + 8*b1 + 208) * q^48 + (2*b7 - 2*b6 + 39*b5 + 39*b4 + 6*b3 + 41*b2) * q^49 + (-31*b10 + 31*b9 - 4*b7 - 4*b6 - 12*b5 + 12*b4 - 8*b3 + 8) * q^51 + (-b11 - 28*b10 + 17*b8 - 5*b7 - b6 - 7*b5 + b4 + 83*b3 + 18*b2 + b1 + 84) * q^52 + (-41*b8 - 49*b6 + 33*b4 - 32*b3 + 41*b2 + 32) * q^53 + (32*b11 + 8*b10 + 8*b9 - 8*b7 + 8*b6 - 40*b5 - 40*b4 + 224*b3 - 24*b2) * q^54 + (-16*b10 + 16*b9 - 12*b8 + 8*b7 + 8*b6 + 64*b5 - 64*b4 + 56*b3 - 28*b1 + 120) * q^56 + (-48*b8 - 14*b7 - 82*b5 + 30*b3 - 48*b2 + 98) * q^57 + (-20*b11 - 36*b9 + 38*b8 + 20*b7 + 56*b6 + 20*b5 + 8*b4 + 100*b3 - 18*b2 - 20*b1 - 80) * q^58 + (4*b11 + 54*b10 + 54*b9 + 9*b7 - 9*b6 + 29*b5 + 29*b4 - 2*b3 - 2*b2) * q^59 + (-7*b8 + 22*b7 + 22*b6 - 29*b5 + 29*b4 - 51*b3 - 26) * q^61 + (-7*b11 + 30*b10 - 28*b8 - 61*b7 - 7*b6 - 11*b5 + 7*b4 - 183*b3 - 21*b2 + 7*b1 - 240) * q^62 + (10*b11 - 19*b9 - 5*b8 - 10*b7 - 17*b6 - 10*b5 + 41*b4 - 17*b3 - 5*b2 + 10*b1 + 7) * q^63 + (-4*b11 - 28*b10 - 28*b9 - 60*b7 + 60*b6 + 12*b5 + 12*b4 - 164*b3 - 44*b2) * q^64 + (46*b10 - 46*b9 + 68*b8 + 68*b7 + 68*b6 + 46*b5 - 46*b4 - 22*b3 - 22*b1 - 88) * q^66 + (-20*b11 + 43*b10 - 10*b8 + 30*b7 - 20*b6 + 110*b5 + 20*b4 + 50*b3 + 10*b2 + 20*b1 - 50) * q^67 + (-25*b11 + 4*b9 + 31*b8 + 25*b7 + 21*b6 + 25*b5 + 33*b4 - 91*b3 - 6*b2 - 25*b1 + 116) * q^68 + (21*b7 - 21*b6 - 64*b5 - 64*b4 - 205*b3 - 43*b2) * q^69 + (45*b10 - 45*b9 - 2*b8 - 27*b7 - 27*b6 - 75*b5 + 75*b4 - 48*b3 + 4*b1 + 46) * q^71 + (b11 + 26*b10 + 38*b8 + 47*b7 + b6 + 51*b5 - b4 + 267*b3 + 37*b2 - b1 + 264) * q^72 + (-4*b8 + 22*b6 + 30*b4 + 95*b3 + 4*b2 - 95) * q^73 + (-68*b11 + 14*b10 + 14*b9 + 14*b7 - 14*b6 - 33*b5 - 33*b4 + 293*b3 + 14*b2) * q^74 + (-12*b10 + 12*b9 - 52*b8 + 4*b7 + 4*b6 - 168*b5 + 168*b4 - 172*b3 + 40*b1 + 208) * q^76 + (45*b8 + 80*b7 + 10*b5 - 280*b3 + 45*b2 - 210) * q^77 + (15*b11 - 22*b9 - 72*b8 - 15*b7 - 13*b6 - 15*b5 + 19*b4 + 321*b3 + 57*b2 + 15*b1 - 336) * q^78 + (-16*b11 - 72*b10 - 72*b9 + 60*b7 - 60*b6 + 172*b5 + 172*b4 + 8*b3 + 8*b2) * q^79 + (-15*b8 - 36*b7 - 36*b6 + 21*b5 - 21*b4 + 57*b3 + 437) * q^81 + (-29*b11 - 14*b10 - 36*b8 - 43*b7 - 29*b6 - 93*b5 + 29*b4 + 95*b3 - 7*b2 + 29*b1 + 116) * q^82 + (-80*b11 + 33*b9 + 40*b8 + 80*b7 - 28*b6 + 80*b5 + 164*b4 - 28*b3 + 40*b2 - 80*b1 + 108) * q^83 + (20*b11 + 50*b10 + 50*b9 - 38*b7 + 38*b6 - 96*b5 - 96*b4 - 222*b3 + 26*b2) * q^84 + (-37*b10 + 37*b9 - 18*b8 + 28*b7 + 28*b6 + 17*b5 - 17*b4 - 11*b3 + 49*b1 - 224) * q^86 + (-76*b11 - 68*b10 - 38*b8 + 34*b7 - 76*b6 + 178*b5 + 76*b4 + 110*b3 + 38*b2 + 76*b1 - 110) * q^87 + (12*b11 - 16*b9 - 60*b8 - 12*b7 - 68*b6 - 12*b5 - 108*b4 - 100*b3 + 48*b2 + 12*b1 + 88) * q^88 + (82*b7 - 82*b6 - 66*b5 - 66*b4 + 224*b3 + 16*b2) * q^89 + (-7*b10 + 7*b9 - 12*b8 - 10*b7 - 10*b6 + 6*b5 - 6*b4 + 16*b3 + 24*b1 - 28) * q^91 + (-6*b11 - 6*b10 - 61*b8 - 60*b7 - 6*b6 - 214*b5 + 6*b4 + 60*b3 - 55*b2 + 6*b1 + 208) * q^92 + (89*b8 - 18*b6 - 196*b4 - 196*b3 - 89*b2 + 196) * q^93 + (15*b11 + b10 + b9 - 53*b7 + 53*b6 + 10*b5 + 10*b4 + 320*b3 + 101*b2) * q^94 + (32*b8 + 16*b7 + 16*b6 + 256*b5 - 256*b4 + 240*b3 + 64*b1 - 160) * q^96 + (106*b8 + 160*b7 + 52*b5 + 379*b3 + 106*b2 + 487) * q^97 + (-37*b11 + 82*b9 - 4*b8 + 37*b7 - 45*b6 + 37*b5 + 10*b4 + 185*b3 + 41*b2 - 37*b1 - 148) * q^98 + (-28*b11 + 29*b10 + 29*b9 + 45*b7 - 45*b6 + 121*b5 + 121*b4 + 14*b3 + 14*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{2} + 8 q^{6} + 12 q^{8}+O(q^{10})$$ 12 * q + 6 * q^2 + 8 * q^6 + 12 * q^8 $$12 q + 6 q^{2} + 8 q^{6} + 12 q^{8} + 80 q^{12} - 116 q^{13} + 312 q^{16} + 332 q^{17} - 198 q^{18} - 144 q^{21} - 360 q^{22} - 164 q^{26} + 880 q^{28} + 376 q^{32} - 80 q^{33} + 460 q^{36} - 508 q^{37} - 1600 q^{38} - 656 q^{41} - 1160 q^{42} - 1432 q^{46} + 2720 q^{48} + 932 q^{52} + 644 q^{53} + 2048 q^{56} + 960 q^{57} - 1576 q^{58} - 896 q^{61} - 2440 q^{62} - 1680 q^{66} + 844 q^{68} + 3036 q^{72} - 1436 q^{73} + 800 q^{76} - 3120 q^{77} - 3720 q^{78} + 5988 q^{81} + 1352 q^{82} - 2552 q^{86} + 2400 q^{88} + 1840 q^{92} + 3280 q^{93} + 1088 q^{96} + 4772 q^{97} - 1698 q^{98}+O(q^{100})$$ 12 * q + 6 * q^2 + 8 * q^6 + 12 * q^8 + 80 * q^12 - 116 * q^13 + 312 * q^16 + 332 * q^17 - 198 * q^18 - 144 * q^21 - 360 * q^22 - 164 * q^26 + 880 * q^28 + 376 * q^32 - 80 * q^33 + 460 * q^36 - 508 * q^37 - 1600 * q^38 - 656 * q^41 - 1160 * q^42 - 1432 * q^46 + 2720 * q^48 + 932 * q^52 + 644 * q^53 + 2048 * q^56 + 960 * q^57 - 1576 * q^58 - 896 * q^61 - 2440 * q^62 - 1680 * q^66 + 844 * q^68 + 3036 * q^72 - 1436 * q^73 + 800 * q^76 - 3120 * q^77 - 3720 * q^78 + 5988 * q^81 + 1352 * q^82 - 2552 * q^86 + 2400 * q^88 + 1840 * q^92 + 3280 * q^93 + 1088 * q^96 + 4772 * q^97 - 1698 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu^{2} - 2$$ 2*v^2 - 2 $$\beta_{2}$$ $$=$$ $$( \nu^{11} - 23\nu^{9} + 92\nu^{7} - 668\nu^{5} + 1152\nu^{3} - 8704\nu ) / 1280$$ (v^11 - 23*v^9 + 92*v^7 - 668*v^5 + 1152*v^3 - 8704*v) / 1280 $$\beta_{3}$$ $$=$$ $$( \nu^{11} - 8\nu^{9} + 27\nu^{7} - 128\nu^{5} + 412\nu^{3} - 1504\nu ) / 640$$ (v^11 - 8*v^9 + 27*v^7 - 128*v^5 + 412*v^3 - 1504*v) / 640 $$\beta_{4}$$ $$=$$ $$( - 5 \nu^{11} + 32 \nu^{10} + 35 \nu^{9} - 256 \nu^{8} - 220 \nu^{7} + 864 \nu^{6} + 780 \nu^{5} - 4096 \nu^{4} - 3520 \nu^{3} + 13184 \nu^{2} + 8960 \nu - 48128 ) / 5120$$ (-5*v^11 + 32*v^10 + 35*v^9 - 256*v^8 - 220*v^7 + 864*v^6 + 780*v^5 - 4096*v^4 - 3520*v^3 + 13184*v^2 + 8960*v - 48128) / 5120 $$\beta_{5}$$ $$=$$ $$( - 13 \nu^{11} - 32 \nu^{10} + 99 \nu^{9} + 256 \nu^{8} - 436 \nu^{7} - 864 \nu^{6} + 1804 \nu^{5} + 4096 \nu^{4} - 6816 \nu^{3} - 13184 \nu^{2} + 20992 \nu + 48128 ) / 5120$$ (-13*v^11 - 32*v^10 + 99*v^9 + 256*v^8 - 436*v^7 - 864*v^6 + 1804*v^5 + 4096*v^4 - 6816*v^3 - 13184*v^2 + 20992*v + 48128) / 5120 $$\beta_{6}$$ $$=$$ $$( - 13 \nu^{11} + 48 \nu^{10} + 99 \nu^{9} + 16 \nu^{8} - 436 \nu^{7} + 416 \nu^{6} + 1804 \nu^{5} + 576 \nu^{4} - 6816 \nu^{3} + 8576 \nu^{2} + 10752 \nu + 22528 ) / 5120$$ (-13*v^11 + 48*v^10 + 99*v^9 + 16*v^8 - 436*v^7 + 416*v^6 + 1804*v^5 + 576*v^4 - 6816*v^3 + 8576*v^2 + 10752*v + 22528) / 5120 $$\beta_{7}$$ $$=$$ $$( 21 \nu^{11} + 48 \nu^{10} - 163 \nu^{9} + 16 \nu^{8} + 652 \nu^{7} + 416 \nu^{6} - 2828 \nu^{5} + 576 \nu^{4} + 10112 \nu^{3} + 8576 \nu^{2} - 22784 \nu + 22528 ) / 5120$$ (21*v^11 + 48*v^10 - 163*v^9 + 16*v^8 + 652*v^7 + 416*v^6 - 2828*v^5 + 576*v^4 + 10112*v^3 + 8576*v^2 - 22784*v + 22528) / 5120 $$\beta_{8}$$ $$=$$ $$( -17\nu^{10} + 31\nu^{8} - 324\nu^{6} + 636\nu^{4} - 4384\nu^{2} - 512 ) / 640$$ (-17*v^10 + 31*v^8 - 324*v^6 + 636*v^4 - 4384*v^2 - 512) / 640 $$\beta_{9}$$ $$=$$ $$( - 45 \nu^{11} - 12 \nu^{10} + 115 \nu^{9} - 44 \nu^{8} - 900 \nu^{7} - 144 \nu^{6} + 1740 \nu^{5} - 2224 \nu^{4} - 10720 \nu^{3} + 256 \nu^{2} + 2560 \nu - 28672 ) / 5120$$ (-45*v^11 - 12*v^10 + 115*v^9 - 44*v^8 - 900*v^7 - 144*v^6 + 1740*v^5 - 2224*v^4 - 10720*v^3 + 256*v^2 + 2560*v - 28672) / 5120 $$\beta_{10}$$ $$=$$ $$( - 45 \nu^{11} + 12 \nu^{10} + 115 \nu^{9} + 44 \nu^{8} - 900 \nu^{7} + 144 \nu^{6} + 1740 \nu^{5} + 2224 \nu^{4} - 10720 \nu^{3} - 256 \nu^{2} + 2560 \nu + 28672 ) / 5120$$ (-45*v^11 + 12*v^10 + 115*v^9 + 44*v^8 - 900*v^7 + 144*v^6 + 1740*v^5 + 2224*v^4 - 10720*v^3 - 256*v^2 + 2560*v + 28672) / 5120 $$\beta_{11}$$ $$=$$ $$( 3\nu^{11} - 7\nu^{9} + 50\nu^{7} - 156\nu^{5} + 824\nu^{3} - 448\nu ) / 256$$ (3*v^11 - 7*v^9 + 50*v^7 - 156*v^5 + 824*v^3 - 448*v) / 256
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} ) / 4$$ (b7 - b6 + b5 + b4 - 2*b3) / 4 $$\nu^{2}$$ $$=$$ $$( \beta _1 + 2 ) / 2$$ (b1 + 2) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{11} + \beta_{10} + \beta_{9} - 3\beta_{5} - 3\beta_{4} - 2\beta_{3} - 2\beta_{2} ) / 2$$ (b11 + b10 + b9 - 3*b5 - 3*b4 - 2*b3 - 2*b2) / 2 $$\nu^{4}$$ $$=$$ $$( 3\beta_{10} - 3\beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} + \beta _1 - 12 ) / 2$$ (3*b10 - 3*b9 + b8 - b5 + b4 - b3 + b1 - 12) / 2 $$\nu^{5}$$ $$=$$ $$( - 3 \beta_{11} - \beta_{10} - \beta_{9} - 5 \beta_{7} + 5 \beta_{6} - 10 \beta_{5} - 10 \beta_{4} + 14 \beta_{3} - 8 \beta_{2} ) / 2$$ (-3*b11 - b10 - b9 - 5*b7 + 5*b6 - 10*b5 - 10*b4 + 14*b3 - 8*b2) / 2 $$\nu^{6}$$ $$=$$ $$( 7 \beta_{10} - 7 \beta_{9} - 11 \beta_{8} - 14 \beta_{7} - 14 \beta_{6} + 5 \beta_{5} - 5 \beta_{4} + 19 \beta_{3} - \beta _1 - 60 ) / 2$$ (7*b10 - 7*b9 - 11*b8 - 14*b7 - 14*b6 + 5*b5 - 5*b4 + 19*b3 - b1 - 60) / 2 $$\nu^{7}$$ $$=$$ $$( - 25 \beta_{11} - 19 \beta_{10} - 19 \beta_{9} + 3 \beta_{7} - 3 \beta_{6} - 12 \beta_{5} - 12 \beta_{4} - 74 \beta_{3} + 16 \beta_{2} ) / 2$$ (-25*b11 - 19*b10 - 19*b9 + 3*b7 - 3*b6 - 12*b5 - 12*b4 - 74*b3 + 16*b2) / 2 $$\nu^{8}$$ $$=$$ $$( - 35 \beta_{10} + 35 \beta_{9} - 41 \beta_{8} - 18 \beta_{7} - 18 \beta_{6} + 47 \beta_{5} - 47 \beta_{4} + 65 \beta_{3} + 9 \beta _1 - 348 ) / 2$$ (-35*b10 + 35*b9 - 41*b8 - 18*b7 - 18*b6 + 47*b5 - 47*b4 + 65*b3 + 9*b1 - 348) / 2 $$\nu^{9}$$ $$=$$ $$( 49 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} - 47 \beta_{7} + 47 \beta_{6} - 80 \beta_{5} - 80 \beta_{4} - 358 \beta_{3} + 88 \beta_{2} ) / 2$$ (49*b11 + 3*b10 + 3*b9 - 47*b7 + 47*b6 - 80*b5 - 80*b4 - 358*b3 + 88*b2) / 2 $$\nu^{10}$$ $$=$$ $$( - 85 \beta_{10} + 85 \beta_{9} + 97 \beta_{8} + 234 \beta_{7} + 234 \beta_{6} - 47 \beta_{5} + 47 \beta_{4} - 281 \beta_{3} - 185 \beta _1 - 516 ) / 2$$ (-85*b10 + 85*b9 + 97*b8 + 234*b7 + 234*b6 - 47*b5 + 47*b4 - 281*b3 - 185*b1 - 516) / 2 $$\nu^{11}$$ $$=$$ $$( 271 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 345 \beta_{7} + 345 \beta_{6} + 392 \beta_{5} + 392 \beta_{4} + 1526 \beta_{3} + 72 \beta_{2} ) / 2$$ (271*b11 - 3*b10 - 3*b9 - 345*b7 + 345*b6 + 392*b5 + 392*b4 + 1526*b3 + 72*b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$77$$ $$\chi(n)$$ $$-1$$ $$\beta_{3}$$

## 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
 −1.76129 + 0.947553i −1.83244 − 0.801352i −1.13579 − 1.64620i 1.76129 + 0.947553i 1.83244 − 0.801352i 1.13579 − 1.64620i −1.76129 − 0.947553i −1.83244 + 0.801352i −1.13579 + 1.64620i 1.76129 − 0.947553i 1.83244 + 0.801352i 1.13579 + 1.64620i
−2.70884 + 0.813737i 2.61822 + 2.61822i 6.67566 4.40857i 0 −9.22289 4.96181i 17.7783 17.7783i −14.4959 + 17.3744i 13.2899i 0
7.2 −1.03109 + 2.63379i −5.55970 5.55970i −5.87372 5.43134i 0 20.3756 8.91056i 1.14202 1.14202i 20.3613 9.86997i 34.8205i 0
7.3 0.510409 + 2.78199i 4.02923 + 4.02923i −7.47897 + 2.83991i 0 −9.15273 + 13.2658i −14.4440 + 14.4440i −11.7179 19.3569i 5.46937i 0
7.4 0.813737 2.70884i −2.61822 2.61822i −6.67566 4.40857i 0 −9.22289 + 4.96181i −17.7783 + 17.7783i −17.3744 + 14.4959i 13.2899i 0
7.5 2.63379 1.03109i 5.55970 + 5.55970i 5.87372 5.43134i 0 20.3756 + 8.91056i −1.14202 + 1.14202i 9.86997 20.3613i 34.8205i 0
7.6 2.78199 + 0.510409i −4.02923 4.02923i 7.47897 + 2.83991i 0 −9.15273 13.2658i 14.4440 14.4440i 19.3569 + 11.7179i 5.46937i 0
43.1 −2.70884 0.813737i 2.61822 2.61822i 6.67566 + 4.40857i 0 −9.22289 + 4.96181i 17.7783 + 17.7783i −14.4959 17.3744i 13.2899i 0
43.2 −1.03109 2.63379i −5.55970 + 5.55970i −5.87372 + 5.43134i 0 20.3756 + 8.91056i 1.14202 + 1.14202i 20.3613 + 9.86997i 34.8205i 0
43.3 0.510409 2.78199i 4.02923 4.02923i −7.47897 2.83991i 0 −9.15273 13.2658i −14.4440 14.4440i −11.7179 + 19.3569i 5.46937i 0
43.4 0.813737 + 2.70884i −2.61822 + 2.61822i −6.67566 + 4.40857i 0 −9.22289 4.96181i −17.7783 17.7783i −17.3744 14.4959i 13.2899i 0
43.5 2.63379 + 1.03109i 5.55970 5.55970i 5.87372 + 5.43134i 0 20.3756 8.91056i −1.14202 1.14202i 9.86997 + 20.3613i 34.8205i 0
43.6 2.78199 0.510409i −4.02923 + 4.02923i 7.47897 2.83991i 0 −9.15273 + 13.2658i 14.4440 + 14.4440i 19.3569 11.7179i 5.46937i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.4.e.e 12
4.b odd 2 1 inner 100.4.e.e 12
5.b even 2 1 20.4.e.b 12
5.c odd 4 1 20.4.e.b 12
5.c odd 4 1 inner 100.4.e.e 12
15.d odd 2 1 180.4.k.e 12
15.e even 4 1 180.4.k.e 12
20.d odd 2 1 20.4.e.b 12
20.e even 4 1 20.4.e.b 12
20.e even 4 1 inner 100.4.e.e 12
40.e odd 2 1 320.4.n.k 12
40.f even 2 1 320.4.n.k 12
40.i odd 4 1 320.4.n.k 12
40.k even 4 1 320.4.n.k 12
60.h even 2 1 180.4.k.e 12
60.l odd 4 1 180.4.k.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.e.b 12 5.b even 2 1
20.4.e.b 12 5.c odd 4 1
20.4.e.b 12 20.d odd 2 1
20.4.e.b 12 20.e even 4 1
100.4.e.e 12 1.a even 1 1 trivial
100.4.e.e 12 4.b odd 2 1 inner
100.4.e.e 12 5.c odd 4 1 inner
100.4.e.e 12 20.e even 4 1 inner
180.4.k.e 12 15.d odd 2 1
180.4.k.e 12 15.e even 4 1
180.4.k.e 12 60.h even 2 1
180.4.k.e 12 60.l odd 4 1
320.4.n.k 12 40.e odd 2 1
320.4.n.k 12 40.f even 2 1
320.4.n.k 12 40.i odd 4 1
320.4.n.k 12 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 5064T_{3}^{8} + 4945680T_{3}^{4} + 757350400$$ acting on $$S_{4}^{\mathrm{new}}(100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 6 T^{11} + 18 T^{10} + \cdots + 262144$$
$3$ $$T^{12} + 5064 T^{8} + \cdots + 757350400$$
$5$ $$T^{12}$$
$7$ $$T^{12} + 573704 T^{8} + \cdots + 473344000000$$
$11$ $$(T^{6} + 3000 T^{4} + 1778960 T^{2} + \cdots + 88064000)^{2}$$
$13$ $$(T^{6} + 58 T^{5} + 1682 T^{4} + \cdots + 7296200)^{2}$$
$17$ $$(T^{6} - 166 T^{5} + 13778 T^{4} + \cdots + 3024864200)^{2}$$
$19$ $$(T^{6} - 24160 T^{4} + \cdots - 148035584000)^{2}$$
$23$ $$T^{12} + 150061064 T^{8} + \cdots + 43\!\cdots\!00$$
$29$ $$(T^{6} + 65648 T^{4} + \cdots + 234782887936)^{2}$$
$31$ $$(T^{6} + 75640 T^{4} + \cdots + 4998782336000)^{2}$$
$37$ $$(T^{6} + 254 T^{5} + \cdots + 6862252857800)^{2}$$
$41$ $$(T^{3} + 164 T^{2} - 18428 T - 1791008)^{4}$$
$43$ $$T^{12} + 14590421064 T^{8} + \cdots + 22\!\cdots\!00$$
$47$ $$T^{12} + 61550198664 T^{8} + \cdots + 28\!\cdots\!00$$
$53$ $$(T^{6} - 322 T^{5} + \cdots + 273645700473800)^{2}$$
$59$ $$(T^{6} - 688480 T^{4} + \cdots - 742151346176000)^{2}$$
$61$ $$(T^{3} + 224 T^{2} - 55468 T - 11698768)^{4}$$
$67$ $$T^{12} + 442279393224 T^{8} + \cdots + 16\!\cdots\!00$$
$71$ $$(T^{6} + 715960 T^{4} + \cdots + 29\!\cdots\!00)^{2}$$
$73$ $$(T^{6} + 718 T^{5} + \cdots + 17037736128200)^{2}$$
$79$ $$(T^{6} - 2383360 T^{4} + \cdots - 42\!\cdots\!00)^{2}$$
$83$ $$T^{12} + 2795286470344 T^{8} + \cdots + 13\!\cdots\!00$$
$89$ $$(T^{6} + 1431168 T^{4} + \cdots + 84\!\cdots\!16)^{2}$$
$97$ $$(T^{6} - 2386 T^{5} + \cdots + 60\!\cdots\!00)^{2}$$