Properties

 Label 3150.3.c.d Level $3150$ Weight $3$ Character orbit 3150.c Analytic conductor $85.831$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 3150.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$85.8312832735$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: 16.0.9671731157401600000000.1 Defining polynomial: $$x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256$$ x^16 + 7*x^12 + 753*x^8 + 112*x^4 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{28}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 630) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + 2 q^{4} - \beta_{2} q^{7} - 2 \beta_{5} q^{8}+O(q^{10})$$ q - b5 * q^2 + 2 * q^4 - b2 * q^7 - 2*b5 * q^8 $$q - \beta_{5} q^{2} + 2 q^{4} - \beta_{2} q^{7} - 2 \beta_{5} q^{8} + (\beta_{11} - \beta_{4}) q^{11} + ( - \beta_{14} - 2 \beta_{3}) q^{13} - \beta_{11} q^{14} + 4 q^{16} + (2 \beta_{10} - \beta_{9} - 2 \beta_{5}) q^{17} + (\beta_{13} - 2 \beta_{6} - 10) q^{19} + ( - \beta_{3} + 2 \beta_{2}) q^{22} + (\beta_{10} + 2 \beta_{9} + 2 \beta_{5} + \beta_1) q^{23} + (\beta_{8} - \beta_{7} - 4 \beta_{4}) q^{26} - 2 \beta_{2} q^{28} + (2 \beta_{11} + 2 \beta_{7} + \beta_{4}) q^{29} + (\beta_{15} - 4 \beta_{6} - 14) q^{31} - 4 \beta_{5} q^{32} + (\beta_{15} - \beta_{13} - 2 \beta_{6} + 4) q^{34} + (\beta_{14} - \beta_{12} + 2 \beta_{3} + 6 \beta_{2}) q^{37} + (4 \beta_{10} + 10 \beta_{5} - \beta_1) q^{38} + (2 \beta_{11} + \beta_{8} - 3 \beta_{7} - 8 \beta_{4}) q^{41} + (2 \beta_{14} + 8 \beta_{3} + 10 \beta_{2}) q^{43} + (2 \beta_{11} - 2 \beta_{4}) q^{44} + ( - 2 \beta_{15} - \beta_{6} - 4) q^{46} + (10 \beta_{10} - 3 \beta_{9} - 4 \beta_{5} - 2 \beta_1) q^{47} - 7 q^{49} + ( - 2 \beta_{14} - 4 \beta_{3}) q^{52} + (3 \beta_{10} + 4 \beta_{9} - 9 \beta_{5} + 3 \beta_1) q^{53} - 2 \beta_{11} q^{56} + (2 \beta_{14} + 2 \beta_{12} + \beta_{3} + 4 \beta_{2}) q^{58} + ( - \beta_{8} + 4 \beta_{7} + 6 \beta_{4}) q^{59} + ( - 2 \beta_{15} - 6 \beta_{6} - 10) q^{61} + (8 \beta_{10} - 2 \beta_{9} + 14 \beta_{5} - \beta_1) q^{62} + 8 q^{64} + ( - 3 \beta_{14} + 3 \beta_{12} - 8 \beta_{3} + 2 \beta_{2}) q^{67} + (4 \beta_{10} - 2 \beta_{9} - 4 \beta_{5}) q^{68} + (7 \beta_{11} - 2 \beta_{8} - 4 \beta_{7} + 23 \beta_{4}) q^{71} + (5 \beta_{14} + 20 \beta_{2}) q^{73} + (6 \beta_{11} - 2 \beta_{8} + 4 \beta_{4}) q^{74} + (2 \beta_{13} - 4 \beta_{6} - 20) q^{76} + (\beta_{10} - 7 \beta_{5}) q^{77} + ( - 6 \beta_{13} - 4 \beta_{6}) q^{79} + ( - 4 \beta_{14} - 2 \beta_{12} - 8 \beta_{3} + 4 \beta_{2}) q^{82} + (6 \beta_{10} - \beta_{9} - 12 \beta_{5} - 2 \beta_1) q^{83} + (10 \beta_{11} - 2 \beta_{8} + 2 \beta_{7} + 16 \beta_{4}) q^{86} + ( - 2 \beta_{3} + 4 \beta_{2}) q^{88} + ( - 16 \beta_{11} - 3 \beta_{8} - 26 \beta_{4}) q^{89} + (\beta_{15} + 2 \beta_{13} - 2 \beta_{6}) q^{91} + (2 \beta_{10} + 4 \beta_{9} + 4 \beta_{5} + 2 \beta_1) q^{92} + (3 \beta_{15} + \beta_{13} - 10 \beta_{6} + 8) q^{94} + (3 \beta_{14} - 6 \beta_{12} + 14 \beta_{3} + 12 \beta_{2}) q^{97} + 7 \beta_{5} q^{98}+O(q^{100})$$ q - b5 * q^2 + 2 * q^4 - b2 * q^7 - 2*b5 * q^8 + (b11 - b4) * q^11 + (-b14 - 2*b3) * q^13 - b11 * q^14 + 4 * q^16 + (2*b10 - b9 - 2*b5) * q^17 + (b13 - 2*b6 - 10) * q^19 + (-b3 + 2*b2) * q^22 + (b10 + 2*b9 + 2*b5 + b1) * q^23 + (b8 - b7 - 4*b4) * q^26 - 2*b2 * q^28 + (2*b11 + 2*b7 + b4) * q^29 + (b15 - 4*b6 - 14) * q^31 - 4*b5 * q^32 + (b15 - b13 - 2*b6 + 4) * q^34 + (b14 - b12 + 2*b3 + 6*b2) * q^37 + (4*b10 + 10*b5 - b1) * q^38 + (2*b11 + b8 - 3*b7 - 8*b4) * q^41 + (2*b14 + 8*b3 + 10*b2) * q^43 + (2*b11 - 2*b4) * q^44 + (-2*b15 - b6 - 4) * q^46 + (10*b10 - 3*b9 - 4*b5 - 2*b1) * q^47 - 7 * q^49 + (-2*b14 - 4*b3) * q^52 + (3*b10 + 4*b9 - 9*b5 + 3*b1) * q^53 - 2*b11 * q^56 + (2*b14 + 2*b12 + b3 + 4*b2) * q^58 + (-b8 + 4*b7 + 6*b4) * q^59 + (-2*b15 - 6*b6 - 10) * q^61 + (8*b10 - 2*b9 + 14*b5 - b1) * q^62 + 8 * q^64 + (-3*b14 + 3*b12 - 8*b3 + 2*b2) * q^67 + (4*b10 - 2*b9 - 4*b5) * q^68 + (7*b11 - 2*b8 - 4*b7 + 23*b4) * q^71 + (5*b14 + 20*b2) * q^73 + (6*b11 - 2*b8 + 4*b4) * q^74 + (2*b13 - 4*b6 - 20) * q^76 + (b10 - 7*b5) * q^77 + (-6*b13 - 4*b6) * q^79 + (-4*b14 - 2*b12 - 8*b3 + 4*b2) * q^82 + (6*b10 - b9 - 12*b5 - 2*b1) * q^83 + (10*b11 - 2*b8 + 2*b7 + 16*b4) * q^86 + (-2*b3 + 4*b2) * q^88 + (-16*b11 - 3*b8 - 26*b4) * q^89 + (b15 + 2*b13 - 2*b6) * q^91 + (2*b10 + 4*b9 + 4*b5 + 2*b1) * q^92 + (3*b15 + b13 - 10*b6 + 8) * q^94 + (3*b14 - 6*b12 + 14*b3 + 12*b2) * q^97 + 7*b5 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 32 q^{4}+O(q^{10})$$ 16 * q + 32 * q^4 $$16 q + 32 q^{4} + 64 q^{16} - 160 q^{19} - 224 q^{31} + 64 q^{34} - 64 q^{46} - 112 q^{49} - 160 q^{61} + 128 q^{64} - 320 q^{76} + 128 q^{94}+O(q^{100})$$ 16 * q + 32 * q^4 + 64 * q^16 - 160 * q^19 - 224 * q^31 + 64 * q^34 - 64 * q^46 - 112 * q^49 - 160 * q^61 + 128 * q^64 - 320 * q^76 + 128 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{12} + 105\nu^{8} + 15\nu^{4} + 42056 ) / 3132$$ (-v^12 + 105*v^8 + 15*v^4 + 42056) / 3132 $$\beta_{2}$$ $$=$$ $$( 14\nu^{12} + 96\nu^{8} + 10752\nu^{4} + 815 ) / 2349$$ (14*v^12 + 96*v^8 + 10752*v^4 + 815) / 2349 $$\beta_{3}$$ $$=$$ $$( -33\nu^{14} - 247\nu^{10} - 25025\nu^{6} - 18304\nu^{2} ) / 8352$$ (-33*v^14 - 247*v^10 - 25025*v^6 - 18304*v^2) / 8352 $$\beta_{4}$$ $$=$$ $$( 245 \nu^{15} - 374 \nu^{13} + 1419 \nu^{11} - 2490 \nu^{9} + 182157 \nu^{7} - 278358 \nu^{5} - 185272 \nu^{3} + 39712 \nu ) / 150336$$ (245*v^15 - 374*v^13 + 1419*v^11 - 2490*v^9 + 182157*v^7 - 278358*v^5 - 185272*v^3 + 39712*v) / 150336 $$\beta_{5}$$ $$=$$ $$( 245 \nu^{15} + 374 \nu^{13} + 1419 \nu^{11} + 2490 \nu^{9} + 182157 \nu^{7} + 278358 \nu^{5} - 185272 \nu^{3} - 39712 \nu ) / 150336$$ (245*v^15 + 374*v^13 + 1419*v^11 + 2490*v^9 + 182157*v^7 + 278358*v^5 - 185272*v^3 - 39712*v) / 150336 $$\beta_{6}$$ $$=$$ $$( 31\nu^{14} + 201\nu^{10} + 23295\nu^{6} - 10112\nu^{2} ) / 2592$$ (31*v^14 + 201*v^10 + 23295*v^6 - 10112*v^2) / 2592 $$\beta_{7}$$ $$=$$ $$( 23\nu^{14} + 72\nu^{12} + 177\nu^{10} + 504\nu^{8} + 17367\nu^{6} + 53064\nu^{4} + 12704\nu^{2} + 4032 ) / 2592$$ (23*v^14 + 72*v^12 + 177*v^10 + 504*v^8 + 17367*v^6 + 53064*v^4 + 12704*v^2 + 4032) / 2592 $$\beta_{8}$$ $$=$$ $$( 23\nu^{14} + 177\nu^{10} + 17367\nu^{6} + 12704\nu^{2} ) / 864$$ (23*v^14 + 177*v^10 + 17367*v^6 + 12704*v^2) / 864 $$\beta_{9}$$ $$=$$ $$( 2025 \nu^{14} + 8 \nu^{12} + 12879 \nu^{10} - 840 \nu^{8} + 1510569 \nu^{6} - 120 \nu^{4} - 655776 \nu^{2} - 336448 ) / 75168$$ (2025*v^14 + 8*v^12 + 12879*v^10 - 840*v^8 + 1510569*v^6 - 120*v^4 - 655776*v^2 - 336448) / 75168 $$\beta_{10}$$ $$=$$ $$( - 403 \nu^{15} + 122 \nu^{13} - 2925 \nu^{11} + 1110 \nu^{9} - 303675 \nu^{7} + 92826 \nu^{5} - 115960 \nu^{3} + 149024 \nu ) / 50112$$ (-403*v^15 + 122*v^13 - 2925*v^11 + 1110*v^9 - 303675*v^7 + 92826*v^5 - 115960*v^3 + 149024*v) / 50112 $$\beta_{11}$$ $$=$$ $$( 403 \nu^{15} + 122 \nu^{13} + 2925 \nu^{11} + 1110 \nu^{9} + 303675 \nu^{7} + 92826 \nu^{5} + 115960 \nu^{3} + 149024 \nu ) / 50112$$ (403*v^15 + 122*v^13 + 2925*v^11 + 1110*v^9 + 303675*v^7 + 92826*v^5 + 115960*v^3 + 149024*v) / 50112 $$\beta_{12}$$ $$=$$ $$( 257 \nu^{15} + 2051 \nu^{13} + 3291 \nu^{11} + 14325 \nu^{9} + 203901 \nu^{7} + 1556115 \nu^{5} + 1217444 \nu^{3} + 322064 \nu ) / 75168$$ (257*v^15 + 2051*v^13 + 3291*v^11 + 14325*v^9 + 203901*v^7 + 1556115*v^5 + 1217444*v^3 + 322064*v) / 75168 $$\beta_{13}$$ $$=$$ $$( 91\nu^{15} + 137\nu^{13} + 537\nu^{11} + 927\nu^{9} + 67887\nu^{7} + 103737\nu^{5} - 69044\nu^{3} - 14800\nu ) / 8352$$ (91*v^15 + 137*v^13 + 537*v^11 + 927*v^9 + 67887*v^7 + 103737*v^5 - 69044*v^3 - 14800*v) / 8352 $$\beta_{14}$$ $$=$$ $$( 1895 \nu^{15} - 415 \nu^{13} + 12957 \nu^{11} - 2361 \nu^{9} + 1425867 \nu^{7} - 311151 \nu^{5} - 25348 \nu^{3} + 588464 \nu ) / 75168$$ (1895*v^15 - 415*v^13 + 12957*v^11 - 2361*v^9 + 1425867*v^7 - 311151*v^5 - 25348*v^3 + 588464*v) / 75168 $$\beta_{15}$$ $$=$$ $$( - 2425 \nu^{15} + 1225 \nu^{13} - 17859 \nu^{11} + 9183 \nu^{9} - 1833429 \nu^{7} + 933753 \nu^{5} - 984964 \nu^{3} + 954928 \nu ) / 75168$$ (-2425*v^15 + 1225*v^13 - 17859*v^11 + 9183*v^9 - 1833429*v^7 + 933753*v^5 - 984964*v^3 + 954928*v) / 75168
 $$\nu$$ $$=$$ $$( 3\beta_{15} + 5\beta_{14} - 3\beta_{13} + \beta_{12} - 6\beta_{11} - 6\beta_{10} + 6\beta_{5} - 6\beta_{4} ) / 48$$ (3*b15 + 5*b14 - 3*b13 + b12 - 6*b11 - 6*b10 + 6*b5 - 6*b4) / 48 $$\nu^{2}$$ $$=$$ $$( 3\beta_{9} - 3\beta_{8} - 9\beta_{6} - 27\beta_{3} + \beta_1 ) / 24$$ (3*b9 - 3*b8 - 9*b6 - 27*b3 + b1) / 24 $$\nu^{3}$$ $$=$$ $$( -3\beta_{15} - \beta_{14} - 9\beta_{13} + 7\beta_{12} - 12\beta_{11} + 12\beta_{10} + 60\beta_{5} + 60\beta_{4} ) / 24$$ (-3*b15 - b14 - 9*b13 + 7*b12 - 12*b11 + 12*b10 + 60*b5 + 60*b4) / 24 $$\nu^{4}$$ $$=$$ $$( 3\beta_{8} - 9\beta_{7} + 42\beta_{2} + \beta _1 - 14 ) / 8$$ (3*b8 - 9*b7 + 42*b2 + b1 - 14) / 8 $$\nu^{5}$$ $$=$$ $$( 15 \beta_{15} - 35 \beta_{14} + 105 \beta_{13} + 65 \beta_{12} - 66 \beta_{11} - 66 \beta_{10} - 726 \beta_{5} + 726 \beta_{4} ) / 48$$ (15*b15 - 35*b14 + 105*b13 + 65*b12 - 66*b11 - 66*b10 - 726*b5 + 726*b4) / 48 $$\nu^{6}$$ $$=$$ $$( -60\beta_{9} - 36\beta_{8} + 135\beta_{6} - 243\beta_{3} - 20\beta_1 ) / 12$$ (-60*b9 - 36*b8 + 135*b6 - 243*b3 - 20*b1) / 12 $$\nu^{7}$$ $$=$$ $$( - 273 \beta_{15} + 533 \beta_{14} + 429 \beta_{13} + 13 \beta_{12} - 1218 \beta_{11} + 1218 \beta_{10} - 2262 \beta_{5} - 2262 \beta_{4} ) / 48$$ (-273*b15 + 533*b14 + 429*b13 + 13*b12 - 1218*b11 + 1218*b10 - 2262*b5 - 2262*b4) / 48 $$\nu^{8}$$ $$=$$ $$( -21\beta_{8} + 63\beta_{7} - 282\beta_{2} + 217\beta _1 - 2914 ) / 8$$ (-21*b8 + 63*b7 - 282*b2 + 217*b1 - 2914) / 8 $$\nu^{9}$$ $$=$$ $$( - 867 \beta_{15} - 1241 \beta_{14} + 459 \beta_{13} - 493 \beta_{12} + 3876 \beta_{11} + 3876 \beta_{10} - 1140 \beta_{5} + 1140 \beta_{4} ) / 24$$ (-867*b15 - 1241*b14 + 459*b13 - 493*b12 + 3876*b11 + 3876*b10 - 1140*b5 + 1140*b4) / 24 $$\nu^{10}$$ $$=$$ $$( -1815\beta_{9} + 3135\beta_{8} + 4059\beta_{6} + 21033\beta_{3} - 605\beta_1 ) / 24$$ (-1815*b9 + 3135*b8 + 4059*b6 + 21033*b3 - 605*b1) / 24 $$\nu^{11}$$ $$=$$ $$( 6141 \beta_{15} - 2225 \beta_{14} + 9879 \beta_{13} - 10057 \beta_{12} + 27462 \beta_{11} - 27462 \beta_{10} - 79998 \beta_{5} - 79998 \beta_{4} ) / 48$$ (6141*b15 - 2225*b14 + 9879*b13 - 10057*b12 + 27462*b11 - 27462*b10 - 79998*b5 - 79998*b4) / 48 $$\nu^{12}$$ $$=$$ $$( -540\beta_{8} + 1620\beta_{7} - 7245\beta_{2} - 564\beta _1 + 7567 ) / 2$$ (-540*b8 + 1620*b7 - 7245*b2 - 564*b1 + 7567) / 2 $$\nu^{13}$$ $$=$$ $$( 699 \beta_{15} + 43105 \beta_{14} - 84579 \beta_{13} - 41707 \beta_{12} - 3126 \beta_{11} - 3126 \beta_{10} + 565806 \beta_{5} - 565806 \beta_{4} ) / 48$$ (699*b15 + 43105*b14 - 84579*b13 - 41707*b12 - 3126*b11 - 3126*b10 + 565806*b5 - 565806*b4) / 48 $$\nu^{14}$$ $$=$$ $$( 102921\beta_{9} + 32799\beta_{8} - 230139\beta_{6} + 220023\beta_{3} + 34307\beta_1 ) / 24$$ (102921*b9 + 32799*b8 - 230139*b6 + 220023*b3 + 34307*b1) / 24 $$\nu^{15}$$ $$=$$ $$( 81435 \beta_{15} - 192455 \beta_{14} - 194895 \beta_{13} + 29585 \beta_{12} + 364188 \beta_{11} - 364188 \beta_{10} + 1125300 \beta_{5} + 1125300 \beta_{4} ) / 24$$ (81435*b15 - 192455*b14 - 194895*b13 + 29585*b12 + 364188*b11 - 364188*b10 + 1125300*b5 + 1125300*b4) / 24

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 2.08559 + 0.941471i −0.796626 − 0.359610i −0.941471 + 2.08559i 0.359610 − 0.796626i 0.359610 + 0.796626i −0.941471 − 2.08559i −0.796626 + 0.359610i 2.08559 − 0.941471i −0.359610 + 0.796626i 0.941471 − 2.08559i 0.796626 + 0.359610i −2.08559 − 0.941471i −2.08559 + 0.941471i 0.796626 − 0.359610i 0.941471 + 2.08559i −0.359610 − 0.796626i
−1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.2 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.3 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.4 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.5 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.6 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.7 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.8 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.9 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.10 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.11 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.12 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.13 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.14 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.15 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.16 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.16 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.b even 2 1 inner
15.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.3.c.d 16
3.b odd 2 1 inner 3150.3.c.d 16
5.b even 2 1 inner 3150.3.c.d 16
5.c odd 4 1 630.3.e.a 8
5.c odd 4 1 3150.3.e.i 8
15.d odd 2 1 inner 3150.3.c.d 16
15.e even 4 1 630.3.e.a 8
15.e even 4 1 3150.3.e.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.3.e.a 8 5.c odd 4 1
630.3.e.a 8 15.e even 4 1
3150.3.c.d 16 1.a even 1 1 trivial
3150.3.c.d 16 3.b odd 2 1 inner
3150.3.c.d 16 5.b even 2 1 inner
3150.3.c.d 16 15.d odd 2 1 inner
3150.3.e.i 8 5.c odd 4 1
3150.3.e.i 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 32T_{11}^{2} + 144$$ acting on $$S_{3}^{\mathrm{new}}(3150, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{8}$$
$3$ $$T^{16}$$
$5$ $$T^{16}$$
$7$ $$(T^{2} + 7)^{8}$$
$11$ $$(T^{4} + 32 T^{2} + 144)^{4}$$
$13$ $$(T^{8} + 504 T^{6} + 58776 T^{4} + \cdots + 5588496)^{2}$$
$17$ $$(T^{8} - 896 T^{6} + 139392 T^{4} + \cdots + 26873856)^{2}$$
$19$ $$(T^{4} + 40 T^{3} + 196 T^{2} + \cdots - 29916)^{4}$$
$23$ $$(T^{8} - 2408 T^{6} + \cdots + 75666805776)^{2}$$
$29$ $$(T^{8} + 2792 T^{6} + \cdots + 19925016336)^{2}$$
$31$ $$(T^{4} + 56 T^{3} - 300 T^{2} + \cdots - 111676)^{4}$$
$37$ $$(T^{8} + 2512 T^{6} + 1562208 T^{4} + \cdots + 58736896)^{2}$$
$41$ $$(T^{8} + 5776 T^{6} + \cdots + 1274875842816)^{2}$$
$43$ $$(T^{8} + 5584 T^{6} + \cdots + 7578747136)^{2}$$
$47$ $$(T^{8} - 11488 T^{6} + \cdots + 3582449565696)^{2}$$
$53$ $$(T^{8} - 12112 T^{6} + \cdots + 58107995136)^{2}$$
$59$ $$(T^{8} + 9328 T^{6} + \cdots + 21234991124736)^{2}$$
$61$ $$(T^{4} + 40 T^{3} - 3736 T^{2} + \cdots - 42096)^{4}$$
$67$ $$(T^{8} + 14096 T^{6} + \cdots + 75851279421696)^{2}$$
$71$ $$(T^{8} + 23936 T^{6} + \cdots + 842087039037696)^{2}$$
$73$ $$(T^{8} + 22200 T^{6} + \cdots + 48965006250000)^{2}$$
$79$ $$(T^{4} - 7376 T^{2} + 7795264)^{4}$$
$83$ $$(T^{8} - 5728 T^{6} + \cdots + 2702420361216)^{2}$$
$89$ $$(T^{8} + 26224 T^{6} + \cdots + 70331973325056)^{2}$$
$97$ $$(T^{8} + 45688 T^{6} + \cdots + 348149175467536)^{2}$$