# Properties

 Label 45.4.f.a Level $45$ Weight $4$ Character orbit 45.f Analytic conductor $2.655$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 45.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.65508595026$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944$$ x^12 - 16*x^10 - 14*x^8 - 512*x^6 + 3889*x^4 + 126224*x^2 + 506944 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}\cdot 3^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + (\beta_{5} - 6 \beta_1) q^{4} + (\beta_{11} + \beta_{10} + \beta_{9}) q^{5} + ( - \beta_{4} + 2 \beta_1 + 2) q^{7} + ( - 3 \beta_{10} - 7 \beta_{9} + 3 \beta_{8} + 2 \beta_{7}) q^{8}+O(q^{10})$$ q + b6 * q^2 + (b5 - 6*b1) * q^4 + (b11 + b10 + b9) * q^5 + (-b4 + 2*b1 + 2) * q^7 + (-3*b10 - 7*b9 + 3*b8 + 2*b7) * q^8 $$q + \beta_{6} q^{2} + (\beta_{5} - 6 \beta_1) q^{4} + (\beta_{11} + \beta_{10} + \beta_{9}) q^{5} + ( - \beta_{4} + 2 \beta_1 + 2) q^{7} + ( - 3 \beta_{10} - 7 \beta_{9} + 3 \beta_{8} + 2 \beta_{7}) q^{8} + ( - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 16) q^{10} + (\beta_{11} - 3 \beta_{10} + 7 \beta_{9} - \beta_{7} - 7 \beta_{6}) q^{11} + (2 \beta_{5} + 3 \beta_{3} - 2 \beta_{2} + 9 \beta_1 - 9) q^{13} + ( - \beta_{11} - 3 \beta_{9} - 7 \beta_{8} - \beta_{7} - 3 \beta_{6}) q^{14} + (5 \beta_{4} - 5 \beta_{3} + 9 \beta_{2} - 54) q^{16} + ( - 2 \beta_{11} + 12 \beta_{10} + 12 \beta_{8} + 8 \beta_{6}) q^{17} + (4 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 20 \beta_1) q^{19} + ( - 5 \beta_{11} + 10 \beta_{10} + 10 \beta_{9} - 21 \beta_{8} - \beta_{7} + 21 \beta_{6}) q^{20} + ( - 6 \beta_{5} + \beta_{4} - 6 \beta_{2} + 88 \beta_1 + 88) q^{22} + ( - 2 \beta_{10} + 8 \beta_{9} + 2 \beta_{8} - 12 \beta_{7}) q^{23} + ( - 6 \beta_{5} - \beta_{4} - 3 \beta_{3} + 8 \beta_{2} - 71 \beta_1 - 12) q^{25} + ( - 7 \beta_{11} - 33 \beta_{10} + 6 \beta_{9} + 7 \beta_{7} - 6 \beta_{6}) q^{26} + (6 \beta_{5} + \beta_{3} - 6 \beta_{2} + 48 \beta_1 - 48) q^{28} + (5 \beta_{11} - 25 \beta_{9} - 4 \beta_{8} + 5 \beta_{7} - 25 \beta_{6}) q^{29} + ( - 5 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 104) q^{31} + (12 \beta_{11} + 38 \beta_{10} + 38 \beta_{8} - 29 \beta_{6}) q^{32} + ( - 12 \beta_{5} - 10 \beta_{4} - 10 \beta_{3} - 56 \beta_1) q^{34} + (3 \beta_{11} + 8 \beta_{10} - 17 \beta_{9} - 33 \beta_{8} + 7 \beta_{7} - 17 \beta_{6}) q^{35} + (8 \beta_{5} - 7 \beta_{4} + 8 \beta_{2} + 69 \beta_1 + 69) q^{37} + ( - 47 \beta_{10} - 6 \beta_{9} + 47 \beta_{8} + 18 \beta_{7}) q^{38} + (26 \beta_{5} + 2 \beta_{4} + 11 \beta_{3} - 19 \beta_{2} - 168 \beta_1 + 214) q^{40} + (16 \beta_{11} - 29 \beta_{10} - 38 \beta_{9} - 16 \beta_{7} + 38 \beta_{6}) q^{41} + ( - 16 \beta_{5} - 14 \beta_{3} + 16 \beta_{2} + 8 \beta_1 - 8) q^{43} + ( - 3 \beta_{11} + 91 \beta_{9} - 53 \beta_{8} - 3 \beta_{7} + 91 \beta_{6}) q^{44} + ( - 10 \beta_{4} + 10 \beta_{3} - 28 \beta_{2} + 56) q^{46} + ( - 18 \beta_{11} + 23 \beta_{10} + 23 \beta_{8} + 58 \beta_{6}) q^{47} + (16 \beta_{5} - 111 \beta_1) q^{49} + (18 \beta_{11} + 63 \beta_{10} - 37 \beta_{9} - \beta_{8} - 16 \beta_{7} - 74 \beta_{6}) q^{50} + (25 \beta_{5} + 23 \beta_{4} + 25 \beta_{2} - 26 \beta_1 - 26) q^{52} + (5 \beta_{10} + 34 \beta_{9} - 5 \beta_{8} + 20 \beta_{7}) q^{53} + (27 \beta_{4} - 4 \beta_{3} + 28 \beta_{2} + 22 \beta_1 - 126) q^{55} + ( - 5 \beta_{11} + 13 \beta_{10} - 25 \beta_{9} + 5 \beta_{7} + 25 \beta_{6}) q^{56} + ( - 31 \beta_{5} - 6 \beta_{3} + 31 \beta_{2} + 322 \beta_1 - 322) q^{58} + ( - 17 \beta_{11} - 41 \beta_{9} + 31 \beta_{8} - 17 \beta_{7} - 41 \beta_{6}) q^{59} + ( - 15 \beta_{4} + 15 \beta_{3} - 48 \beta_{2} + 8) q^{61} + ( - 18 \beta_{11} - 47 \beta_{10} - 47 \beta_{8} - 118 \beta_{6}) q^{62} + ( - 57 \beta_{5} - 10 \beta_{4} - 10 \beta_{3} + 78 \beta_1) q^{64} + ( - 14 \beta_{11} - 64 \beta_{10} + 86 \beta_{9} - 19 \beta_{8} + 6 \beta_{7} - 56 \beta_{6}) q^{65} + ( - 44 \beta_{5} + 2 \beta_{4} - 44 \beta_{2} + 136 \beta_1 + 136) q^{67} + (10 \beta_{10} + 16 \beta_{9} - 10 \beta_{8} - 60 \beta_{7}) q^{68} + (2 \beta_{5} - 4 \beta_{4} + 23 \beta_{3} - 10 \beta_{2} + 176 \beta_1 - 128) q^{70} + ( - 32 \beta_{11} - 4 \beta_{10} + 16 \beta_{9} + 32 \beta_{7} - 16 \beta_{6}) q^{71} + (18 \beta_{5} - 12 \beta_{3} - 18 \beta_{2} - 331 \beta_1 + 331) q^{73} + (9 \beta_{11} - 38 \beta_{9} - \beta_{8} + 9 \beta_{7} - 38 \beta_{6}) q^{74} + (25 \beta_{4} - 25 \beta_{3} + 104 \beta_{2} - 40) q^{76} + (48 \beta_{11} + 12 \beta_{10} + 12 \beta_{8} + 32 \beta_{6}) q^{77} + (28 \beta_{5} + 25 \beta_{4} + 25 \beta_{3} + 672 \beta_1) q^{79} + ( - 39 \beta_{11} - 44 \beta_{10} - 169 \beta_{9} + 115 \beta_{8} + \cdots + 260 \beta_{6}) q^{80}+ \cdots + ( - 48 \beta_{10} - 255 \beta_{9} + 48 \beta_{8} + 32 \beta_{7}) q^{98}+O(q^{100})$$ q + b6 * q^2 + (b5 - 6*b1) * q^4 + (b11 + b10 + b9) * q^5 + (-b4 + 2*b1 + 2) * q^7 + (-3*b10 - 7*b9 + 3*b8 + 2*b7) * q^8 + (-3*b5 - 2*b4 - b3 - 2*b2 - 2*b1 + 16) * q^10 + (b11 - 3*b10 + 7*b9 - b7 - 7*b6) * q^11 + (2*b5 + 3*b3 - 2*b2 + 9*b1 - 9) * q^13 + (-b11 - 3*b9 - 7*b8 - b7 - 3*b6) * q^14 + (5*b4 - 5*b3 + 9*b2 - 54) * q^16 + (-2*b11 + 12*b10 + 12*b8 + 8*b6) * q^17 + (4*b5 + 5*b4 + 5*b3 - 20*b1) * q^19 + (-5*b11 + 10*b10 + 10*b9 - 21*b8 - b7 + 21*b6) * q^20 + (-6*b5 + b4 - 6*b2 + 88*b1 + 88) * q^22 + (-2*b10 + 8*b9 + 2*b8 - 12*b7) * q^23 + (-6*b5 - b4 - 3*b3 + 8*b2 - 71*b1 - 12) * q^25 + (-7*b11 - 33*b10 + 6*b9 + 7*b7 - 6*b6) * q^26 + (6*b5 + b3 - 6*b2 + 48*b1 - 48) * q^28 + (5*b11 - 25*b9 - 4*b8 + 5*b7 - 25*b6) * q^29 + (-5*b4 + 5*b3 - 4*b2 - 104) * q^31 + (12*b11 + 38*b10 + 38*b8 - 29*b6) * q^32 + (-12*b5 - 10*b4 - 10*b3 - 56*b1) * q^34 + (3*b11 + 8*b10 - 17*b9 - 33*b8 + 7*b7 - 17*b6) * q^35 + (8*b5 - 7*b4 + 8*b2 + 69*b1 + 69) * q^37 + (-47*b10 - 6*b9 + 47*b8 + 18*b7) * q^38 + (26*b5 + 2*b4 + 11*b3 - 19*b2 - 168*b1 + 214) * q^40 + (16*b11 - 29*b10 - 38*b9 - 16*b7 + 38*b6) * q^41 + (-16*b5 - 14*b3 + 16*b2 + 8*b1 - 8) * q^43 + (-3*b11 + 91*b9 - 53*b8 - 3*b7 + 91*b6) * q^44 + (-10*b4 + 10*b3 - 28*b2 + 56) * q^46 + (-18*b11 + 23*b10 + 23*b8 + 58*b6) * q^47 + (16*b5 - 111*b1) * q^49 + (18*b11 + 63*b10 - 37*b9 - b8 - 16*b7 - 74*b6) * q^50 + (25*b5 + 23*b4 + 25*b2 - 26*b1 - 26) * q^52 + (5*b10 + 34*b9 - 5*b8 + 20*b7) * q^53 + (27*b4 - 4*b3 + 28*b2 + 22*b1 - 126) * q^55 + (-5*b11 + 13*b10 - 25*b9 + 5*b7 + 25*b6) * q^56 + (-31*b5 - 6*b3 + 31*b2 + 322*b1 - 322) * q^58 + (-17*b11 - 41*b9 + 31*b8 - 17*b7 - 41*b6) * q^59 + (-15*b4 + 15*b3 - 48*b2 + 8) * q^61 + (-18*b11 - 47*b10 - 47*b8 - 118*b6) * q^62 + (-57*b5 - 10*b4 - 10*b3 + 78*b1) * q^64 + (-14*b11 - 64*b10 + 86*b9 - 19*b8 + 6*b7 - 56*b6) * q^65 + (-44*b5 + 2*b4 - 44*b2 + 136*b1 + 136) * q^67 + (10*b10 + 16*b9 - 10*b8 - 60*b7) * q^68 + (2*b5 - 4*b4 + 23*b3 - 10*b2 + 176*b1 - 128) * q^70 + (-32*b11 - 4*b10 + 16*b9 + 32*b7 - 16*b6) * q^71 + (18*b5 - 12*b3 - 18*b2 - 331*b1 + 331) * q^73 + (9*b11 - 38*b9 - b8 + 9*b7 - 38*b6) * q^74 + (25*b4 - 25*b3 + 104*b2 - 40) * q^76 + (48*b11 + 12*b10 + 12*b8 + 32*b6) * q^77 + (28*b5 + 25*b4 + 25*b3 + 672*b1) * q^79 + (-39*b11 - 44*b10 - 169*b9 + 115*b8 + 25*b7 + 260*b6) * q^80 + (35*b5 - 3*b4 + 35*b2 - 654*b1 - 654) * q^82 + (153*b10 + 78*b9 - 153*b8 - 2*b7) * q^83 + (8*b5 - 28*b4 - 44*b3 + 2*b2 - 28*b1 - 146) * q^85 + (46*b11 + 194*b10 + 82*b9 - 46*b7 - 82*b6) * q^86 + (102*b5 + 67*b3 - 102*b2 - 664*b1 + 664) * q^88 + (42*b11 + 36*b9 + 163*b8 + 42*b7 + 36*b6) * q^89 + (25*b4 - 25*b3 + 24*b2 + 396) * q^91 + (20*b11 - 170*b10 - 170*b8 + 144*b6) * q^92 + (48*b5 - 5*b4 - 5*b3 - 648*b1) * q^94 + (-165*b10 + 210*b9 + 65*b8 - 30*b7 + 60*b6) * q^95 + (-50*b5 - 78*b4 - 50*b2 + 231*b1 + 231) * q^97 + (-48*b10 - 255*b9 + 48*b8 + 32*b7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 24 q^{7}+O(q^{10})$$ 12 * q + 24 * q^7 $$12 q + 24 q^{7} + 192 q^{10} - 108 q^{13} - 648 q^{16} + 1056 q^{22} - 144 q^{25} - 576 q^{28} - 1248 q^{31} + 828 q^{37} + 2568 q^{40} - 96 q^{43} + 672 q^{46} - 312 q^{52} - 1512 q^{55} - 3864 q^{58} + 96 q^{61} + 1632 q^{67} - 1536 q^{70} + 3972 q^{73} - 480 q^{76} - 7848 q^{82} - 1752 q^{85} + 7968 q^{88} + 4752 q^{91} + 2772 q^{97}+O(q^{100})$$ 12 * q + 24 * q^7 + 192 * q^10 - 108 * q^13 - 648 * q^16 + 1056 * q^22 - 144 * q^25 - 576 * q^28 - 1248 * q^31 + 828 * q^37 + 2568 * q^40 - 96 * q^43 + 672 * q^46 - 312 * q^52 - 1512 * q^55 - 3864 * q^58 + 96 * q^61 + 1632 * q^67 - 1536 * q^70 + 3972 * q^73 - 480 * q^76 - 7848 * q^82 - 1752 * q^85 + 7968 * q^88 + 4752 * q^91 + 2772 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944$$ :

 $$\beta_{1}$$ $$=$$ $$( -3\nu^{10} - 56\nu^{8} + 634\nu^{6} + 3248\nu^{4} + 83197\nu^{2} + 418024 ) / 182400$$ (-3*v^10 - 56*v^8 + 634*v^6 + 3248*v^4 + 83197*v^2 + 418024) / 182400 $$\beta_{2}$$ $$=$$ $$( -71\nu^{10} + 2538\nu^{8} - 4502\nu^{6} - 356014\nu^{4} - 1896111\nu^{2} - 1578712 ) / 3620640$$ (-71*v^10 + 2538*v^8 - 4502*v^6 - 356014*v^4 - 1896111*v^2 - 1578712) / 3620640 $$\beta_{3}$$ $$=$$ $$( 1029\nu^{10} - 22972\nu^{8} + 158738\nu^{6} - 2560724\nu^{4} + 22729229\nu^{2} + 78379688 ) / 9051600$$ (1029*v^10 - 22972*v^8 + 158738*v^6 - 2560724*v^4 + 22729229*v^2 + 78379688) / 9051600 $$\beta_{4}$$ $$=$$ $$( 5723\nu^{10} - 110024\nu^{8} + 167406\nu^{6} - 2434608\nu^{4} + 44277523\nu^{2} + 559571096 ) / 18103200$$ (5723*v^10 - 110024*v^8 + 167406*v^6 - 2434608*v^4 + 44277523*v^2 + 559571096) / 18103200 $$\beta_{5}$$ $$=$$ $$( 1265\nu^{10} - 2936\nu^{8} - 23478\nu^{6} - 1362480\nu^{4} - 16244903\nu^{2} - 51726904 ) / 2896512$$ (1265*v^10 - 2936*v^8 - 23478*v^6 - 1362480*v^4 - 16244903*v^2 - 51726904) / 2896512 $$\beta_{6}$$ $$=$$ $$( 113621 \nu^{11} + 12179272 \nu^{9} - 76074438 \nu^{7} - 253832976 \nu^{5} - 12758183579 \nu^{3} - 93476563288 \nu ) / 25778956800$$ (113621*v^11 + 12179272*v^9 - 76074438*v^7 - 253832976*v^5 - 12758183579*v^3 - 93476563288*v) / 25778956800 $$\beta_{7}$$ $$=$$ $$( 28451 \nu^{11} + 1391000 \nu^{9} - 20705978 \nu^{7} + 10214960 \nu^{5} - 1219736029 \nu^{3} - 1295931240 \nu ) / 2577895680$$ (28451*v^11 + 1391000*v^9 - 20705978*v^7 + 10214960*v^5 - 1219736029*v^3 - 1295931240*v) / 2577895680 $$\beta_{8}$$ $$=$$ $$( -6773\nu^{11} + 146104\nu^{9} - 577306\nu^{7} + 5223568\nu^{5} - 79774373\nu^{3} - 554840616\nu ) / 226131200$$ (-6773*v^11 + 146104*v^9 - 577306*v^7 + 5223568*v^5 - 79774373*v^3 - 554840616*v) / 226131200 $$\beta_{9}$$ $$=$$ $$( - 1093919 \nu^{11} + 7804552 \nu^{9} - 15466318 \nu^{7} + 1380835984 \nu^{5} + 6018524881 \nu^{3} + 28260515592 \nu ) / 25778956800$$ (-1093919*v^11 + 7804552*v^9 - 15466318*v^7 + 1380835984*v^5 + 6018524881*v^3 + 28260515592*v) / 25778956800 $$\beta_{10}$$ $$=$$ $$( -1095\nu^{11} + 7908\nu^{9} + 18534\nu^{7} + 745404\nu^{5} + 8928141\nu^{3} - 11971272\nu ) / 21482464$$ (-1095*v^11 + 7908*v^9 + 18534*v^7 + 745404*v^5 + 8928141*v^3 - 11971272*v) / 21482464 $$\beta_{11}$$ $$=$$ $$( -9087\nu^{11} + 123320\nu^{9} - 524974\nu^{7} + 11892160\nu^{5} - 5216047\nu^{3} - 669132280\nu ) / 135678720$$ (-9087*v^11 + 123320*v^9 - 524974*v^7 + 11892160*v^5 - 5216047*v^3 - 669132280*v) / 135678720
 $$\nu$$ $$=$$ $$( -6\beta_{11} - \beta_{10} + 12\beta_{9} - \beta_{8} + 6\beta_{6} ) / 24$$ (-6*b11 - b10 + 12*b9 - b8 + 6*b6) / 24 $$\nu^{2}$$ $$=$$ $$( -4\beta_{5} - 3\beta_{4} + 11\beta_{3} - 20\beta_{2} - 64\beta _1 + 64 ) / 24$$ (-4*b5 - 3*b4 + 11*b3 - 20*b2 - 64*b1 + 64) / 24 $$\nu^{3}$$ $$=$$ $$( -12\beta_{11} + 53\beta_{10} + 66\beta_{9} - 137\beta_{8} + 6\beta_{7} + 120\beta_{6} ) / 24$$ (-12*b11 + 53*b10 + 66*b9 - 137*b8 + 6*b7 + 120*b6) / 24 $$\nu^{4}$$ $$=$$ $$( 12\beta_{5} - 35\beta_{4} + 5\beta_{3} - 268\beta_{2} + 1136 ) / 24$$ (12*b5 - 35*b4 + 5*b3 - 268*b2 + 1136) / 24 $$\nu^{5}$$ $$=$$ $$( -6\beta_{11} - 615\beta_{10} + 1704\beta_{9} - 1259\beta_{8} - 348\beta_{7} + 1518\beta_{6} ) / 24$$ (-6*b11 - 615*b10 + 1704*b9 - 1259*b8 - 348*b7 + 1518*b6) / 24 $$\nu^{6}$$ $$=$$ $$( 676\beta_{5} - 1157\beta_{4} + 909\beta_{3} - 3212\beta_{2} + 5824\beta _1 + 25216 ) / 24$$ (676*b5 - 1157*b4 + 909*b3 - 3212*b2 + 5824*b1 + 25216) / 24 $$\nu^{7}$$ $$=$$ $$( -5940\beta_{11} - 2133\beta_{10} + 21438\beta_{9} - 12439\beta_{8} - 6918\beta_{7} + 24264\beta_{6} ) / 24$$ (-5940*b11 - 2133*b10 + 21438*b9 - 12439*b8 - 6918*b7 + 24264*b6) / 24 $$\nu^{8}$$ $$=$$ $$( 252\beta_{5} - 4255\beta_{4} + 5385\beta_{3} - 14588\beta_{2} - 20480\beta _1 + 129968 ) / 8$$ (252*b5 - 4255*b4 + 5385*b3 - 14588*b2 - 20480*b1 + 129968) / 8 $$\nu^{9}$$ $$=$$ $$( -88602\beta_{11} + 24551\beta_{10} + 292920\beta_{9} - 220021\beta_{8} - 38724\beta_{7} + 359538\beta_{6} ) / 24$$ (-88602*b11 + 24551*b10 + 292920*b9 - 220021*b8 - 38724*b7 + 359538*b6) / 24 $$\nu^{10}$$ $$=$$ $$( 30812\beta_{5} - 127323\beta_{4} + 201011\beta_{3} - 706676\beta_{2} - 856384\beta _1 + 4399744 ) / 24$$ (30812*b5 - 127323*b4 + 201011*b3 - 706676*b2 - 856384*b1 + 4399744) / 24 $$\nu^{11}$$ $$=$$ $$( - 776748 \beta_{11} - 305227 \beta_{10} + 4045218 \beta_{9} - 3762665 \beta_{8} - 584730 \beta_{7} + 4953432 \beta_{6} ) / 24$$ (-776748*b11 - 305227*b10 + 4045218*b9 - 3762665*b8 - 584730*b7 + 4953432*b6) / 24

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$-1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 −3.78139 − 0.0336790i −0.347140 + 2.27426i 2.02004 + 2.30794i −2.02004 − 2.30794i 0.347140 − 2.27426i 3.78139 + 0.0336790i −3.78139 + 0.0336790i −0.347140 − 2.27426i 2.02004 − 2.30794i −2.02004 + 2.30794i 0.347140 + 2.27426i 3.78139 − 0.0336790i
−3.74771 3.74771i 0 20.0907i −0.572391 + 11.1657i 0 1.80948 1.80948i 45.3126 45.3126i 0 43.9909 39.7006i
8.2 −2.62140 2.62140i 0 5.74350i −8.38994 7.38978i 0 −10.8652 + 10.8652i −5.91519 + 5.91519i 0 2.62183 + 41.3650i
8.3 −0.287902 0.287902i 0 7.83422i −9.93887 5.12043i 0 15.0557 15.0557i −4.55871 + 4.55871i 0 1.38724 + 4.33560i
8.4 0.287902 + 0.287902i 0 7.83422i 9.93887 + 5.12043i 0 15.0557 15.0557i 4.55871 4.55871i 0 1.38724 + 4.33560i
8.5 2.62140 + 2.62140i 0 5.74350i 8.38994 + 7.38978i 0 −10.8652 + 10.8652i 5.91519 5.91519i 0 2.62183 + 41.3650i
8.6 3.74771 + 3.74771i 0 20.0907i 0.572391 11.1657i 0 1.80948 1.80948i −45.3126 + 45.3126i 0 43.9909 39.7006i
17.1 −3.74771 + 3.74771i 0 20.0907i −0.572391 11.1657i 0 1.80948 + 1.80948i 45.3126 + 45.3126i 0 43.9909 + 39.7006i
17.2 −2.62140 + 2.62140i 0 5.74350i −8.38994 + 7.38978i 0 −10.8652 10.8652i −5.91519 5.91519i 0 2.62183 41.3650i
17.3 −0.287902 + 0.287902i 0 7.83422i −9.93887 + 5.12043i 0 15.0557 + 15.0557i −4.55871 4.55871i 0 1.38724 4.33560i
17.4 0.287902 0.287902i 0 7.83422i 9.93887 5.12043i 0 15.0557 + 15.0557i 4.55871 + 4.55871i 0 1.38724 4.33560i
17.5 2.62140 2.62140i 0 5.74350i 8.38994 7.38978i 0 −10.8652 10.8652i 5.91519 + 5.91519i 0 2.62183 41.3650i
17.6 3.74771 3.74771i 0 20.0907i 0.572391 + 11.1657i 0 1.80948 + 1.80948i −45.3126 45.3126i 0 43.9909 + 39.7006i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.4.f.a 12
3.b odd 2 1 inner 45.4.f.a 12
4.b odd 2 1 720.4.w.d 12
5.b even 2 1 225.4.f.c 12
5.c odd 4 1 inner 45.4.f.a 12
5.c odd 4 1 225.4.f.c 12
12.b even 2 1 720.4.w.d 12
15.d odd 2 1 225.4.f.c 12
15.e even 4 1 inner 45.4.f.a 12
15.e even 4 1 225.4.f.c 12
20.e even 4 1 720.4.w.d 12
60.l odd 4 1 720.4.w.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.f.a 12 1.a even 1 1 trivial
45.4.f.a 12 3.b odd 2 1 inner
45.4.f.a 12 5.c odd 4 1 inner
45.4.f.a 12 15.e even 4 1 inner
225.4.f.c 12 5.b even 2 1
225.4.f.c 12 5.c odd 4 1
225.4.f.c 12 15.d odd 2 1
225.4.f.c 12 15.e even 4 1
720.4.w.d 12 4.b odd 2 1
720.4.w.d 12 12.b even 2 1
720.4.w.d 12 20.e even 4 1
720.4.w.d 12 60.l odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(45, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 978 T^{8} + 149073 T^{4} + \cdots + 4096$$
$3$ $$T^{12}$$
$5$ $$T^{12} + 72 T^{10} + \cdots + 3814697265625$$
$7$ $$(T^{6} - 12 T^{5} + 72 T^{4} + \cdots + 700928)^{2}$$
$11$ $$(T^{6} + 4344 T^{4} + \cdots + 2390999552)^{2}$$
$13$ $$(T^{6} + 54 T^{5} + 1458 T^{4} + \cdots + 40033352)^{2}$$
$17$ $$T^{12} + 138301968 T^{8} + \cdots + 81\!\cdots\!96$$
$19$ $$(T^{6} + 26640 T^{4} + \cdots + 604661760000)^{2}$$
$23$ $$T^{12} + 946426368 T^{8} + \cdots + 44\!\cdots\!96$$
$29$ $$(T^{6} - 58854 T^{4} + \cdots - 3276052522632)^{2}$$
$31$ $$(T^{3} + 312 T^{2} + 19728 T + 333184)^{4}$$
$37$ $$(T^{6} - 414 T^{5} + \cdots + 431026413512)^{2}$$
$41$ $$(T^{6} + 367686 T^{4} + \cdots + 13\!\cdots\!88)^{2}$$
$43$ $$(T^{6} + 48 T^{5} + \cdots + 12363096915968)^{2}$$
$47$ $$T^{12} + 22535002368 T^{8} + \cdots + 11\!\cdots\!96$$
$53$ $$T^{12} + 14798523408 T^{8} + \cdots + 95\!\cdots\!76$$
$59$ $$(T^{6} - 421176 T^{4} + \cdots - 19\!\cdots\!28)^{2}$$
$61$ $$(T^{3} - 24 T^{2} - 341088 T - 70019072)^{4}$$
$67$ $$(T^{6} - 816 T^{5} + \cdots + 25\!\cdots\!72)^{2}$$
$71$ $$(T^{6} + 602976 T^{4} + \cdots + 826210599600128)^{2}$$
$73$ $$(T^{6} - 1986 T^{5} + \cdots + 35\!\cdots\!72)^{2}$$
$79$ $$(T^{6} + 1996512 T^{4} + \cdots + 23\!\cdots\!44)^{2}$$
$83$ $$T^{12} + 4125230408448 T^{8} + \cdots + 40\!\cdots\!76$$
$89$ $$(T^{6} - 2356326 T^{4} + \cdots - 17\!\cdots\!28)^{2}$$
$97$ $$(T^{6} - 1386 T^{5} + \cdots + 13\!\cdots\!12)^{2}$$