# Properties

 Label 3150.3.c.f 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: $$2^{28}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} + 2 q^{4} + \beta_{2} q^{7} - 2 \beta_{6} q^{8}+O(q^{10})$$ q - b6 * q^2 + 2 * q^4 + b2 * q^7 - 2*b6 * q^8 $$q - \beta_{6} q^{2} + 2 q^{4} + \beta_{2} q^{7} - 2 \beta_{6} q^{8} + (\beta_{12} + 2 \beta_{7} - \beta_{5}) q^{11} + (\beta_{14} + 2 \beta_{4} + 4 \beta_{2}) q^{13} - \beta_{12} q^{14} + 4 q^{16} + (\beta_{13} + 2 \beta_{11} + 6 \beta_{6} - \beta_1) q^{17} + ( - \beta_{15} + \beta_{9} + 10) q^{19} + ( - 2 \beta_{10} + \beta_{4} - 2 \beta_{2}) q^{22} + (3 \beta_{11} + 2 \beta_{6} + \beta_1) q^{23} + ( - 4 \beta_{12} - 4 \beta_{5} - \beta_{3}) q^{26} + 2 \beta_{2} q^{28} + ( - 2 \beta_{12} + 2 \beta_{7} + 13 \beta_{5} + 2 \beta_{3}) q^{29} + (2 \beta_{15} + 3 \beta_{9} - 2 \beta_{8} + 14) q^{31} - 4 \beta_{6} q^{32} + ( - \beta_{15} + 2 \beta_{9} + 2 \beta_{8} - 12) q^{34} + ( - 4 \beta_{14} + 3 \beta_{10} + 8 \beta_{4} + 10 \beta_{2}) q^{37} + (2 \beta_{13} - 10 \beta_{6} - \beta_1) q^{38} + ( - 6 \beta_{12} - 6 \beta_{7} + 20 \beta_{5} - \beta_{3}) q^{41} + (6 \beta_{14} + 2 \beta_{10} + 4 \beta_{4} - 2 \beta_{2}) q^{43} + (2 \beta_{12} + 4 \beta_{7} - 2 \beta_{5}) q^{44} + ( - 2 \beta_{9} + 3 \beta_{8} - 4) q^{46} + ( - 3 \beta_{13} - 2 \beta_{11} + 5 \beta_1) q^{47} - 7 q^{49} + (2 \beta_{14} + 4 \beta_{4} + 8 \beta_{2}) q^{52} + (2 \beta_{13} - 3 \beta_{11} + 15 \beta_{6} - 7 \beta_1) q^{53} - 2 \beta_{12} q^{56} + ( - 4 \beta_{14} - 2 \beta_{10} - 13 \beta_{4} + 4 \beta_{2}) q^{58} + ( - 12 \beta_{12} - 9 \beta_{7} + 2 \beta_{5} + 2 \beta_{3}) q^{59} + (2 \beta_{15} - 6 \beta_{9} + 6 \beta_{8} - 18) q^{61} + ( - 4 \beta_{13} - 4 \beta_{11} - 14 \beta_{6} - 3 \beta_1) q^{62} + 8 q^{64} + ( - 4 \beta_{14} - \beta_{10} - 4 \beta_{4} + 14 \beta_{2}) q^{67} + (2 \beta_{13} + 4 \beta_{11} + 12 \beta_{6} - 2 \beta_1) q^{68} + ( - 13 \beta_{12} + 4 \beta_{7} + 7 \beta_{5}) q^{71} + (3 \beta_{14} + 4 \beta_{10} + 24 \beta_{2}) q^{73} + ( - 10 \beta_{12} - 6 \beta_{7} - 16 \beta_{5} + 4 \beta_{3}) q^{74} + ( - 2 \beta_{15} + 2 \beta_{9} + 20) q^{76} + ( - 2 \beta_{13} - \beta_{11} - 7 \beta_{6}) q^{77} + ( - 2 \beta_{15} - 14 \beta_{9} - 12 \beta_{8} + 8) q^{79} + (2 \beta_{14} + 6 \beta_{10} - 20 \beta_{4} + 12 \beta_{2}) q^{82} + ( - \beta_{13} + 10 \beta_{11} - 16 \beta_{6} + 23 \beta_1) q^{83} + (2 \beta_{12} - 4 \beta_{7} - 8 \beta_{5} - 6 \beta_{3}) q^{86} + ( - 4 \beta_{10} + 2 \beta_{4} - 4 \beta_{2}) q^{88} + (\beta_{7} - 26 \beta_{5} - 4 \beta_{3}) q^{89} + ( - 7 \beta_{9} - 2 \beta_{8} - 28) q^{91} + (6 \beta_{11} + 4 \beta_{6} + 2 \beta_1) q^{92} + (3 \beta_{15} - 10 \beta_{9} - 2 \beta_{8}) q^{94} + ( - 5 \beta_{14} - 2 \beta_{10} - 2 \beta_{4} + 32 \beta_{2}) q^{97} + 7 \beta_{6} q^{98}+O(q^{100})$$ q - b6 * q^2 + 2 * q^4 + b2 * q^7 - 2*b6 * q^8 + (b12 + 2*b7 - b5) * q^11 + (b14 + 2*b4 + 4*b2) * q^13 - b12 * q^14 + 4 * q^16 + (b13 + 2*b11 + 6*b6 - b1) * q^17 + (-b15 + b9 + 10) * q^19 + (-2*b10 + b4 - 2*b2) * q^22 + (3*b11 + 2*b6 + b1) * q^23 + (-4*b12 - 4*b5 - b3) * q^26 + 2*b2 * q^28 + (-2*b12 + 2*b7 + 13*b5 + 2*b3) * q^29 + (2*b15 + 3*b9 - 2*b8 + 14) * q^31 - 4*b6 * q^32 + (-b15 + 2*b9 + 2*b8 - 12) * q^34 + (-4*b14 + 3*b10 + 8*b4 + 10*b2) * q^37 + (2*b13 - 10*b6 - b1) * q^38 + (-6*b12 - 6*b7 + 20*b5 - b3) * q^41 + (6*b14 + 2*b10 + 4*b4 - 2*b2) * q^43 + (2*b12 + 4*b7 - 2*b5) * q^44 + (-2*b9 + 3*b8 - 4) * q^46 + (-3*b13 - 2*b11 + 5*b1) * q^47 - 7 * q^49 + (2*b14 + 4*b4 + 8*b2) * q^52 + (2*b13 - 3*b11 + 15*b6 - 7*b1) * q^53 - 2*b12 * q^56 + (-4*b14 - 2*b10 - 13*b4 + 4*b2) * q^58 + (-12*b12 - 9*b7 + 2*b5 + 2*b3) * q^59 + (2*b15 - 6*b9 + 6*b8 - 18) * q^61 + (-4*b13 - 4*b11 - 14*b6 - 3*b1) * q^62 + 8 * q^64 + (-4*b14 - b10 - 4*b4 + 14*b2) * q^67 + (2*b13 + 4*b11 + 12*b6 - 2*b1) * q^68 + (-13*b12 + 4*b7 + 7*b5) * q^71 + (3*b14 + 4*b10 + 24*b2) * q^73 + (-10*b12 - 6*b7 - 16*b5 + 4*b3) * q^74 + (-2*b15 + 2*b9 + 20) * q^76 + (-2*b13 - b11 - 7*b6) * q^77 + (-2*b15 - 14*b9 - 12*b8 + 8) * q^79 + (2*b14 + 6*b10 - 20*b4 + 12*b2) * q^82 + (-b13 + 10*b11 - 16*b6 + 23*b1) * q^83 + (2*b12 - 4*b7 - 8*b5 - 6*b3) * q^86 + (-4*b10 + 2*b4 - 4*b2) * q^88 + (b7 - 26*b5 - 4*b3) * q^89 + (-7*b9 - 2*b8 - 28) * q^91 + (6*b11 + 4*b6 + 2*b1) * q^92 + (3*b15 - 10*b9 - 2*b8) * q^94 + (-5*b14 - 2*b10 - 2*b4 + 32*b2) * q^97 + 7*b6 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 32 q^{4}+O(q^{10})$$ 16 * q + 32 * q^4 $$16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100})$$ 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

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

 $$\beta_{1}$$ $$=$$ $$( -\nu^{12} + 105\nu^{8} + 15\nu^{4} + 42056 ) / 9396$$ (-v^12 + 105*v^8 + 15*v^4 + 42056) / 9396 $$\beta_{2}$$ $$=$$ $$( -14\nu^{12} - 96\nu^{8} - 10752\nu^{4} - 815 ) / 2349$$ (-14*v^12 - 96*v^8 - 10752*v^4 - 815) / 2349 $$\beta_{3}$$ $$=$$ $$( -\nu^{12} - 7\nu^{8} - 737\nu^{4} - 56 ) / 36$$ (-v^12 - 7*v^8 - 737*v^4 - 56) / 36 $$\beta_{4}$$ $$=$$ $$( 33\nu^{14} + 247\nu^{10} + 25025\nu^{6} + 18304\nu^{2} ) / 8352$$ (33*v^14 + 247*v^10 + 25025*v^6 + 18304*v^2) / 8352 $$\beta_{5}$$ $$=$$ $$( 245 \nu^{15} - 374 \nu^{13} + 1419 \nu^{11} - 2490 \nu^{9} + 182157 \nu^{7} - 278358 \nu^{5} - 185272 \nu^{3} + 39712 \nu ) / 150336$$ (245*v^15 - 374*v^13 + 1419*v^11 - 2490*v^9 + 182157*v^7 - 278358*v^5 - 185272*v^3 + 39712*v) / 150336 $$\beta_{6}$$ $$=$$ $$( 245 \nu^{15} + 374 \nu^{13} + 1419 \nu^{11} + 2490 \nu^{9} + 182157 \nu^{7} + 278358 \nu^{5} - 185272 \nu^{3} - 39712 \nu ) / 150336$$ (245*v^15 + 374*v^13 + 1419*v^11 + 2490*v^9 + 182157*v^7 + 278358*v^5 - 185272*v^3 - 39712*v) / 150336 $$\beta_{7}$$ $$=$$ $$( 23\nu^{14} + 177\nu^{10} + 17367\nu^{6} + 12704\nu^{2} ) / 2592$$ (23*v^14 + 177*v^10 + 17367*v^6 + 12704*v^2) / 2592 $$\beta_{8}$$ $$=$$ $$( 31\nu^{14} + 201\nu^{10} + 23295\nu^{6} - 10112\nu^{2} ) / 2592$$ (31*v^14 + 201*v^10 + 23295*v^6 - 10112*v^2) / 2592 $$\beta_{9}$$ $$=$$ $$( 91 \nu^{15} + 137 \nu^{13} + 537 \nu^{11} + 927 \nu^{9} + 67887 \nu^{7} + 103737 \nu^{5} - 69044 \nu^{3} - 14800 \nu ) / 25056$$ (91*v^15 + 137*v^13 + 537*v^11 + 927*v^9 + 67887*v^7 + 103737*v^5 - 69044*v^3 - 14800*v) / 25056 $$\beta_{10}$$ $$=$$ $$( 91 \nu^{15} - 137 \nu^{13} + 537 \nu^{11} - 927 \nu^{9} + 67887 \nu^{7} - 103737 \nu^{5} - 69044 \nu^{3} + 14800 \nu ) / 12528$$ (91*v^15 - 137*v^13 + 537*v^11 - 927*v^9 + 67887*v^7 - 103737*v^5 - 69044*v^3 + 14800*v) / 12528 $$\beta_{11}$$ $$=$$ $$( 403 \nu^{15} - 122 \nu^{13} + 2925 \nu^{11} - 1110 \nu^{9} + 303675 \nu^{7} - 92826 \nu^{5} + 115960 \nu^{3} - 149024 \nu ) / 50112$$ (403*v^15 - 122*v^13 + 2925*v^11 - 1110*v^9 + 303675*v^7 - 92826*v^5 + 115960*v^3 - 149024*v) / 50112 $$\beta_{12}$$ $$=$$ $$( 403 \nu^{15} + 122 \nu^{13} + 2925 \nu^{11} + 1110 \nu^{9} + 303675 \nu^{7} + 92826 \nu^{5} + 115960 \nu^{3} + 149024 \nu ) / 50112$$ (403*v^15 + 122*v^13 + 2925*v^11 + 1110*v^9 + 303675*v^7 + 92826*v^5 + 115960*v^3 + 149024*v) / 50112 $$\beta_{13}$$ $$=$$ $$( 25\nu^{14} + 159\nu^{10} + 18649\nu^{6} - 8096\nu^{2} ) / 928$$ (25*v^14 + 159*v^10 + 18649*v^6 - 8096*v^2) / 928 $$\beta_{14}$$ $$=$$ $$( 1349 \nu^{15} + 407 \nu^{13} + 9735 \nu^{11} + 3201 \nu^{9} + 1018545 \nu^{7} + 311271 \nu^{5} + 388916 \nu^{3} + 499664 \nu ) / 75168$$ (1349*v^15 + 407*v^13 + 9735*v^11 + 3201*v^9 + 1018545*v^7 + 311271*v^5 + 388916*v^3 + 499664*v) / 75168 $$\beta_{15}$$ $$=$$ $$( - 1349 \nu^{15} + 407 \nu^{13} - 9735 \nu^{11} + 3201 \nu^{9} - 1018545 \nu^{7} + 311271 \nu^{5} - 388916 \nu^{3} + 499664 \nu ) / 37584$$ (-1349*v^15 + 407*v^13 - 9735*v^11 + 3201*v^9 - 1018545*v^7 + 311271*v^5 - 388916*v^3 + 499664*v) / 37584
 $$\nu$$ $$=$$ $$( \beta_{15} + 2\beta_{14} - 2\beta_{12} + 2\beta_{11} + \beta_{10} - 2\beta_{9} + 2\beta_{6} - 2\beta_{5} ) / 16$$ (b15 + 2*b14 - 2*b12 + 2*b11 + b10 - 2*b9 + 2*b6 - 2*b5) / 16 $$\nu^{2}$$ $$=$$ $$( \beta_{13} - 3\beta_{8} - 3\beta_{7} + 9\beta_{4} ) / 8$$ (b13 - 3*b8 - 3*b7 + 9*b4) / 8 $$\nu^{3}$$ $$=$$ $$( -\beta_{15} + 2\beta_{14} - 4\beta_{12} - 4\beta_{11} - 5\beta_{10} - 10\beta_{9} + 20\beta_{6} + 20\beta_{5} ) / 8$$ (-b15 + 2*b14 - 4*b12 - 4*b11 - 5*b10 - 10*b9 + 20*b6 + 20*b5) / 8 $$\nu^{4}$$ $$=$$ $$( 9\beta_{3} - 42\beta_{2} + 3\beta _1 - 14 ) / 8$$ (9*b3 - 42*b2 + 3*b1 - 14) / 8 $$\nu^{5}$$ $$=$$ $$( 5 \beta_{15} + 10 \beta_{14} - 22 \beta_{12} + 22 \beta_{11} - 55 \beta_{10} + 110 \beta_{9} - 242 \beta_{6} + 242 \beta_{5} ) / 16$$ (5*b15 + 10*b14 - 22*b12 + 22*b11 - 55*b10 + 110*b9 - 242*b6 + 242*b5) / 16 $$\nu^{6}$$ $$=$$ $$( -20\beta_{13} + 45\beta_{8} - 36\beta_{7} + 81\beta_{4} ) / 4$$ (-20*b13 + 45*b8 - 36*b7 + 81*b4) / 4 $$\nu^{7}$$ $$=$$ $$( - 91 \beta_{15} + 182 \beta_{14} - 406 \beta_{12} - 406 \beta_{11} + 169 \beta_{10} + 338 \beta_{9} - 754 \beta_{6} - 754 \beta_{5} ) / 16$$ (-91*b15 + 182*b14 - 406*b12 - 406*b11 + 169*b10 + 338*b9 - 754*b6 - 754*b5) / 16 $$\nu^{8}$$ $$=$$ $$( -63\beta_{3} + 282\beta_{2} + 651\beta _1 - 2914 ) / 8$$ (-63*b3 + 282*b2 + 651*b1 - 2914) / 8 $$\nu^{9}$$ $$=$$ $$( - 289 \beta_{15} - 578 \beta_{14} + 1292 \beta_{12} - 1292 \beta_{11} - 85 \beta_{10} + 170 \beta_{9} - 380 \beta_{6} + 380 \beta_{5} ) / 8$$ (-289*b15 - 578*b14 + 1292*b12 - 1292*b11 - 85*b10 + 170*b9 - 380*b6 + 380*b5) / 8 $$\nu^{10}$$ $$=$$ $$( -605\beta_{13} + 1353\beta_{8} + 3135\beta_{7} - 7011\beta_{4} ) / 8$$ (-605*b13 + 1353*b8 + 3135*b7 - 7011*b4) / 8 $$\nu^{11}$$ $$=$$ $$( 2047 \beta_{15} - 4094 \beta_{14} + 9154 \beta_{12} + 9154 \beta_{11} + 5963 \beta_{10} + 11926 \beta_{9} - 26666 \beta_{6} - 26666 \beta_{5} ) / 16$$ (2047*b15 - 4094*b14 + 9154*b12 + 9154*b11 + 5963*b10 + 11926*b9 - 26666*b6 - 26666*b5) / 16 $$\nu^{12}$$ $$=$$ $$( -1620\beta_{3} + 7245\beta_{2} - 1692\beta _1 + 7567 ) / 2$$ (-1620*b3 + 7245*b2 - 1692*b1 + 7567) / 2 $$\nu^{13}$$ $$=$$ $$( 233 \beta_{15} + 466 \beta_{14} - 1042 \beta_{12} + 1042 \beta_{11} + 42173 \beta_{10} - 84346 \beta_{9} + 188602 \beta_{6} - 188602 \beta_{5} ) / 16$$ (233*b15 + 466*b14 - 1042*b12 + 1042*b11 + 42173*b10 - 84346*b9 + 188602*b6 - 188602*b5) / 16 $$\nu^{14}$$ $$=$$ $$( 34307\beta_{13} - 76713\beta_{8} + 32799\beta_{7} - 73341\beta_{4} ) / 8$$ (34307*b13 - 76713*b8 + 32799*b7 - 73341*b4) / 8 $$\nu^{15}$$ $$=$$ $$( 27145 \beta_{15} - 54290 \beta_{14} + 121396 \beta_{12} + 121396 \beta_{11} - 83875 \beta_{10} - 167750 \beta_{9} + 375100 \beta_{6} + 375100 \beta_{5} ) / 8$$ (27145*b15 - 54290*b14 + 121396*b12 + 121396*b11 - 83875*b10 - 167750*b9 + 375100*b6 + 375100*b5) / 8

## 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 0.359610 − 0.796626i −0.796626 − 0.359610i −0.941471 + 2.08559i −0.941471 − 2.08559i −0.796626 + 0.359610i 0.359610 + 0.796626i 2.08559 − 0.941471i −0.359610 + 0.796626i −2.08559 − 0.941471i 0.941471 − 2.08559i 0.796626 + 0.359610i 0.796626 − 0.359610i 0.941471 + 2.08559i −2.08559 + 0.941471i −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.f 16
3.b odd 2 1 inner 3150.3.c.f 16
5.b even 2 1 inner 3150.3.c.f 16
5.c odd 4 1 630.3.e.b 8
5.c odd 4 1 3150.3.e.f 8
15.d odd 2 1 inner 3150.3.c.f 16
15.e even 4 1 630.3.e.b 8
15.e even 4 1 3150.3.e.f 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{8} + 384T_{11}^{6} + 44832T_{11}^{4} + 1673216T_{11}^{2} + 15872256$$ acting on $$S_{3}^{\mathrm{new}}(3150, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{8}$$
$3$ $$T^{16}$$
$5$ $$T^{16}$$
$7$ $$(T^{2} + 7)^{8}$$
$11$ $$(T^{8} + 384 T^{6} + 44832 T^{4} + \cdots + 15872256)^{2}$$
$13$ $$(T^{8} + 792 T^{6} + 149208 T^{4} + \cdots + 14470416)^{2}$$
$17$ $$(T^{8} - 1152 T^{6} + 117888 T^{4} + \cdots + 331776)^{2}$$
$19$ $$(T^{4} - 40 T^{3} + 20 T^{2} + 7600 T + 24900)^{4}$$
$23$ $$(T^{8} - 616 T^{6} + 112792 T^{4} + \cdots + 80353296)^{2}$$
$29$ $$(T^{8} + 4136 T^{6} + \cdots + 182633150736)^{2}$$
$31$ $$(T^{4} - 56 T^{3} - 1468 T^{2} + \cdots - 53724)^{4}$$
$37$ $$(T^{8} + 9744 T^{6} + \cdots + 17685591019776)^{2}$$
$41$ $$(T^{8} + 8656 T^{6} + \cdots + 11776317808896)^{2}$$
$43$ $$(T^{8} + 11088 T^{6} + \cdots + 23646201405696)^{2}$$
$47$ $$(T^{8} - 7264 T^{6} + \cdots + 147161235456)^{2}$$
$53$ $$(T^{8} - 8464 T^{6} + \cdots + 8432658441216)^{2}$$
$59$ $$(T^{8} + 16816 T^{6} + \cdots + 7297000079616)^{2}$$
$61$ $$(T^{4} + 72 T^{3} - 3032 T^{2} + \cdots + 5776)^{4}$$
$67$ $$(T^{8} + 10384 T^{6} + \cdots + 913966592256)^{2}$$
$71$ $$(T^{8} + 11136 T^{6} + \cdots + 13463498301696)^{2}$$
$73$ $$(T^{8} + 21208 T^{6} + \cdots + 36190379032336)^{2}$$
$79$ $$(T^{4} - 32 T^{3} - 13840 T^{2} + \cdots - 14801856)^{4}$$
$83$ $$(T^{8} - 50528 T^{6} + \cdots + 36\!\cdots\!96)^{2}$$
$89$ $$(T^{8} + 14448 T^{6} + \cdots + 416365629696)^{2}$$
$97$ $$(T^{8} + 36376 T^{6} + \cdots + 615175117820176)^{2}$$