Properties

 Label 2352.2.h.p Level $2352$ Weight $2$ Character orbit 2352.h Analytic conductor $18.781$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.h (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 12x^{14} + 97x^{12} - 432x^{10} + 1392x^{8} - 2502x^{6} + 3181x^{4} - 1650x^{2} + 625$$ x^16 - 12*x^14 + 97*x^12 - 432*x^10 + 1392*x^8 - 2502*x^6 + 3181*x^4 - 1650*x^2 + 625 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{16}\cdot 3^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - \beta_{9} q^{3} + \beta_{13} q^{5} - \beta_{3} q^{9}+O(q^{10})$$ q - b9 * q^3 + b13 * q^5 - b3 * q^9 $$q - \beta_{9} q^{3} + \beta_{13} q^{5} - \beta_{3} q^{9} - \beta_{7} q^{13} + \beta_1 q^{15} + (\beta_{15} - \beta_{13}) q^{17} + ( - \beta_{14} - \beta_{11} + 2 \beta_{9}) q^{19} + (\beta_{8} + \beta_1) q^{23} + (\beta_{4} - \beta_{3} - 3) q^{25} + \beta_{12} q^{27} + \beta_{10} q^{29} + (\beta_{14} + \beta_{9}) q^{31} + (\beta_{4} - \beta_{3} - 4) q^{37} + (\beta_{8} + \beta_{5} - \beta_{2}) q^{39} + (\beta_{15} + \beta_{13}) q^{41} + ( - \beta_{8} + 2 \beta_{2} + \beta_1) q^{43} + (\beta_{15} - \beta_{13} - \beta_{7} - 3 \beta_{6}) q^{45} + (\beta_{14} - 2 \beta_{12} - \beta_{9}) q^{47} + ( - \beta_{8} + 2 \beta_{5} - \beta_{2} - \beta_1) q^{51} + ( - \beta_{10} - \beta_{4} - \beta_{3}) q^{53} + (\beta_{10} + \beta_{4} + 2 \beta_{3} + 5) q^{57} + ( - \beta_{11} - 3 \beta_{9}) q^{59} + (\beta_{7} - 4 \beta_{6}) q^{61} + ( - \beta_{10} + 2 \beta_{4} + 2 \beta_{3}) q^{65} + ( - \beta_{5} + \beta_{2}) q^{67} + (\beta_{15} - 3 \beta_{13} - 2 \beta_{7} + \beta_{6}) q^{69} + ( - \beta_{8} - \beta_{5} - 3 \beta_{2} + \beta_1) q^{71} + ( - 4 \beta_{7} + \beta_{6}) q^{73} + (\beta_{12} - \beta_{11} + 3 \beta_{9}) q^{75} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{79} + ( - \beta_{10} + 2 \beta_{4} + 1) q^{81} + (\beta_{14} - 2 \beta_{12} - \beta_{11} - 4 \beta_{9}) q^{83} + ( - 3 \beta_{4} + 3 \beta_{3} + 10) q^{85} + ( - 3 \beta_{14} + \beta_{12} - \beta_{11} + \beta_{9}) q^{87} + (\beta_{15} - \beta_{13}) q^{89} + ( - \beta_{10} - \beta_{4} + \beta_{3} + 4) q^{93} + (\beta_{5} + 3 \beta_{2} - 2 \beta_1) q^{95} - 7 \beta_{6} q^{97}+O(q^{100})$$ q - b9 * q^3 + b13 * q^5 - b3 * q^9 - b7 * q^13 + b1 * q^15 + (b15 - b13) * q^17 + (-b14 - b11 + 2*b9) * q^19 + (b8 + b1) * q^23 + (b4 - b3 - 3) * q^25 + b12 * q^27 + b10 * q^29 + (b14 + b9) * q^31 + (b4 - b3 - 4) * q^37 + (b8 + b5 - b2) * q^39 + (b15 + b13) * q^41 + (-b8 + 2*b2 + b1) * q^43 + (b15 - b13 - b7 - 3*b6) * q^45 + (b14 - 2*b12 - b9) * q^47 + (-b8 + 2*b5 - b2 - b1) * q^51 + (-b10 - b4 - b3) * q^53 + (b10 + b4 + 2*b3 + 5) * q^57 + (-b11 - 3*b9) * q^59 + (b7 - 4*b6) * q^61 + (-b10 + 2*b4 + 2*b3) * q^65 + (-b5 + b2) * q^67 + (b15 - 3*b13 - 2*b7 + b6) * q^69 + (-b8 - b5 - 3*b2 + b1) * q^71 + (-4*b7 + b6) * q^73 + (b12 - b11 + 3*b9) * q^75 + (-2*b5 + 2*b2) * q^79 + (-b10 + 2*b4 + 1) * q^81 + (b14 - 2*b12 - b11 - 4*b9) * q^83 + (-3*b4 + 3*b3 + 10) * q^85 + (-3*b14 + b12 - b11 + b9) * q^87 + (b15 - b13) * q^89 + (-b10 - b4 + b3 + 4) * q^93 + (b5 + 3*b2 - 2*b1) * q^95 - 7*b6 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{9}+O(q^{10})$$ 16 * q + 8 * q^9 $$16 q + 8 q^{9} - 32 q^{25} - 48 q^{37} + 72 q^{57} + 32 q^{81} + 112 q^{85} + 48 q^{93}+O(q^{100})$$ 16 * q + 8 * q^9 - 32 * q^25 - 48 * q^37 + 72 * q^57 + 32 * q^81 + 112 * q^85 + 48 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 12x^{14} + 97x^{12} - 432x^{10} + 1392x^{8} - 2502x^{6} + 3181x^{4} - 1650x^{2} + 625$$ :

 $$\beta_{1}$$ $$=$$ $$( 1983 \nu^{14} - 260575 \nu^{12} + 3547200 \nu^{10} - 27979150 \nu^{8} + 125308500 \nu^{6} - 341211350 \nu^{4} + 429906927 \nu^{2} - 140609075 ) / 45226950$$ (1983*v^14 - 260575*v^12 + 3547200*v^10 - 27979150*v^8 + 125308500*v^6 - 341211350*v^4 + 429906927*v^2 - 140609075) / 45226950 $$\beta_{2}$$ $$=$$ $$( 20867 \nu^{14} - 344205 \nu^{12} + 2915280 \nu^{10} - 16261730 \nu^{8} + 53541630 \nu^{6} - 131508430 \nu^{4} + 115210403 \nu^{2} - 156096575 ) / 45226950$$ (20867*v^14 - 344205*v^12 + 2915280*v^10 - 16261730*v^8 + 53541630*v^6 - 131508430*v^4 + 115210403*v^2 - 156096575) / 45226950 $$\beta_{3}$$ $$=$$ $$( - 739 \nu^{14} + 3643 \nu^{12} - 18358 \nu^{10} - 76302 \nu^{8} + 441787 \nu^{6} - 2227272 \nu^{4} + 2610991 \nu^{2} - 2784875 ) / 611175$$ (-739*v^14 + 3643*v^12 - 18358*v^10 - 76302*v^8 + 441787*v^6 - 2227272*v^4 + 2610991*v^2 - 2784875) / 611175 $$\beta_{4}$$ $$=$$ $$( 2786 \nu^{14} - 33307 \nu^{12} + 266042 \nu^{10} - 1141452 \nu^{8} + 3419887 \nu^{6} - 4966422 \nu^{4} + 4049116 \nu^{2} + 265475 ) / 611175$$ (2786*v^14 - 33307*v^12 + 266042*v^10 - 1141452*v^8 + 3419887*v^6 - 4966422*v^4 + 4049116*v^2 + 265475) / 611175 $$\beta_{5}$$ $$=$$ $$( 41947 \nu^{14} - 498361 \nu^{12} + 4000176 \nu^{10} - 17653306 \nu^{8} + 56108646 \nu^{6} - 99287606 \nu^{4} + 124734379 \nu^{2} - 65760095 ) / 9045390$$ (41947*v^14 - 498361*v^12 + 4000176*v^10 - 17653306*v^8 + 56108646*v^6 - 99287606*v^4 + 124734379*v^2 - 65760095) / 9045390 $$\beta_{6}$$ $$=$$ $$( 203 \nu^{15} - 2381 \nu^{13} + 18686 \nu^{11} - 81166 \nu^{9} + 254896 \nu^{7} - 473576 \nu^{5} + 642763 \nu^{3} - 561525 \nu ) / 210750$$ (203*v^15 - 2381*v^13 + 18686*v^11 - 81166*v^9 + 254896*v^7 - 473576*v^5 + 642763*v^3 - 561525*v) / 210750 $$\beta_{7}$$ $$=$$ $$( 1797 \nu^{15} - 13115 \nu^{13} + 74590 \nu^{11} - 7940 \nu^{9} - 684460 \nu^{7} + 4437260 \nu^{5} - 6536117 \nu^{3} + 6056225 \nu ) / 1222350$$ (1797*v^15 - 13115*v^13 + 74590*v^11 - 7940*v^9 - 684460*v^7 + 4437260*v^5 - 6536117*v^3 + 6056225*v) / 1222350 $$\beta_{8}$$ $$=$$ $$( - 448577 \nu^{14} + 5115739 \nu^{12} - 39578784 \nu^{10} + 162926554 \nu^{8} - 477286824 \nu^{6} + 688243394 \nu^{4} - 612107777 \nu^{2} + \cdots + 19459775 ) / 45226950$$ (-448577*v^14 + 5115739*v^12 - 39578784*v^10 + 162926554*v^8 - 477286824*v^6 + 688243394*v^4 - 612107777*v^2 + 19459775) / 45226950 $$\beta_{9}$$ $$=$$ $$( 396239 \nu^{15} - 5143443 \nu^{13} + 42079208 \nu^{11} - 196431398 \nu^{9} + 625998838 \nu^{7} - 1165072378 \nu^{5} + 1301286409 \nu^{3} + \cdots - 859155425 \nu ) / 226134750$$ (396239*v^15 - 5143443*v^13 + 42079208*v^11 - 196431398*v^9 + 625998838*v^7 - 1165072378*v^5 + 1301286409*v^3 - 859155425*v) / 226134750 $$\beta_{10}$$ $$=$$ $$( 6854 \nu^{14} - 78723 \nu^{12} + 627888 \nu^{10} - 2676528 \nu^{8} + 8475618 \nu^{6} - 14170608 \nu^{4} + 17841074 \nu^{2} - 6203925 ) / 611175$$ (6854*v^14 - 78723*v^12 + 627888*v^10 - 2676528*v^8 + 8475618*v^6 - 14170608*v^4 + 17841074*v^2 - 6203925) / 611175 $$\beta_{11}$$ $$=$$ $$( - 148127 \nu^{15} + 1702799 \nu^{13} - 12387244 \nu^{11} + 45773164 \nu^{9} - 97649834 \nu^{7} - 1728946 \nu^{5} + 335203713 \nu^{3} + \cdots - 596954175 \nu ) / 75378250$$ (-148127*v^15 + 1702799*v^13 - 12387244*v^11 + 45773164*v^9 - 97649834*v^7 - 1728946*v^5 + 335203713*v^3 - 596954175*v) / 75378250 $$\beta_{12}$$ $$=$$ $$( - 934049 \nu^{15} + 12587563 \nu^{13} - 106255328 \nu^{11} + 526954868 \nu^{9} - 1802844658 \nu^{7} + 3811230298 \nu^{5} + \cdots + 2462381325 \nu ) / 226134750$$ (-934049*v^15 + 12587563*v^13 - 106255328*v^11 + 526954868*v^9 - 1802844658*v^7 + 3811230298*v^5 - 4791091069*v^3 + 2462381325*v) / 226134750 $$\beta_{13}$$ $$=$$ $$( 26459 \nu^{15} - 313703 \nu^{13} + 2495718 \nu^{11} - 10811208 \nu^{9} + 33158448 \nu^{7} - 53305788 \nu^{5} + 50468549 \nu^{3} - 12014375 \nu ) / 6111750$$ (26459*v^15 - 313703*v^13 + 2495718*v^11 - 10811208*v^9 + 33158448*v^7 - 53305788*v^5 + 50468549*v^3 - 12014375*v) / 6111750 $$\beta_{14}$$ $$=$$ $$( - 2209813 \nu^{15} + 25402981 \nu^{13} - 203257436 \nu^{11} + 872075666 \nu^{9} - 2767434646 \nu^{7} + 4607661826 \nu^{5} + \cdots + 1635066675 \nu ) / 226134750$$ (-2209813*v^15 + 25402981*v^13 - 203257436*v^11 + 872075666*v^9 - 2767434646*v^7 + 4607661826*v^5 - 5646101103*v^3 + 1635066675*v) / 226134750 $$\beta_{15}$$ $$=$$ $$( 32879 \nu^{15} - 382733 \nu^{13} + 3044898 \nu^{11} - 13003413 \nu^{9} + 40454928 \nu^{7} - 65035668 \nu^{5} + 75654434 \nu^{3} - 14658125 \nu ) / 3055875$$ (32879*v^15 - 382733*v^13 + 3044898*v^11 - 13003413*v^9 + 40454928*v^7 - 65035668*v^5 + 75654434*v^3 - 14658125*v) / 3055875
 $$\nu$$ $$=$$ $$( -3\beta_{13} - 3\beta_{12} - \beta_{11} + 3\beta_{7} - 6\beta_{6} ) / 12$$ (-3*b13 - 3*b12 - b11 + 3*b7 - 6*b6) / 12 $$\nu^{2}$$ $$=$$ $$( -3\beta_{10} + \beta_{8} + 10\beta_{5} - 6\beta_{2} - \beta _1 + 18 ) / 12$$ (-3*b10 + b8 + 10*b5 - 6*b2 - b1 + 18) / 12 $$\nu^{3}$$ $$=$$ $$( -3\beta_{15} - 6\beta_{14} - 12\beta_{13} - 5\beta_{11} + 9\beta_{9} ) / 6$$ (-3*b15 - 6*b14 - 12*b13 - 5*b11 + 9*b9) / 6 $$\nu^{4}$$ $$=$$ $$( -6\beta_{10} - 3\beta_{8} + 11\beta_{5} - \beta_{4} + \beta_{3} - 17\beta_{2} + 3\beta _1 - 24 ) / 4$$ (-6*b10 - 3*b8 + 11*b5 - b4 + b3 - 17*b2 + 3*b1 - 24) / 4 $$\nu^{5}$$ $$=$$ $$( - 21 \beta_{15} - 57 \beta_{14} - 51 \beta_{13} + 51 \beta_{12} - 47 \beta_{11} + 9 \beta_{9} - 78 \beta_{7} + 111 \beta_{6} ) / 12$$ (-21*b15 - 57*b14 - 51*b13 + 51*b12 - 47*b11 + 9*b9 - 78*b7 + 111*b6) / 12 $$\nu^{6}$$ $$=$$ $$( -35\beta_{8} - 32\beta_{5} - 27\beta_{4} + 27\beta_{3} - 96\beta_{2} + 29\beta _1 - 324 ) / 6$$ (-35*b8 - 32*b5 - 27*b4 + 27*b3 - 96*b2 + 29*b1 - 324) / 6 $$\nu^{7}$$ $$=$$ $$( 126 \beta_{15} + 57 \beta_{14} + 231 \beta_{13} + 231 \beta_{12} + 7 \beta_{11} - 489 \beta_{9} - 399 \beta_{7} + 636 \beta_{6} ) / 12$$ (126*b15 + 57*b14 + 231*b13 + 231*b12 + 7*b11 - 489*b9 - 399*b7 + 636*b6) / 12 $$\nu^{8}$$ $$=$$ $$( 168\beta_{10} - 7\beta_{8} - 374\beta_{5} - 25\beta_{4} + 97\beta_{3} + 28\beta_{2} - 17\beta _1 - 514 ) / 4$$ (168*b10 - 7*b8 - 374*b5 - 25*b4 + 97*b3 + 28*b2 - 17*b1 - 514) / 4 $$\nu^{9}$$ $$=$$ $$( 723\beta_{15} + 948\beta_{14} + 1092\beta_{13} + 679\beta_{11} - 1089\beta_{9} ) / 6$$ (723*b15 + 948*b14 + 1092*b13 + 679*b11 - 1089*b9) / 6 $$\nu^{10}$$ $$=$$ $$( 2649 \beta_{10} + 2299 \beta_{8} - 4031 \beta_{5} + 1998 \beta_{4} - 252 \beta_{3} + 5313 \beta_{2} - 1717 \beta _1 + 7578 ) / 12$$ (2649*b10 + 2299*b8 - 4031*b5 + 1998*b4 - 252*b3 + 5313*b2 - 1717*b1 + 7578) / 12 $$\nu^{11}$$ $$=$$ $$( 4065 \beta_{15} + 7854 \beta_{14} + 5313 \beta_{13} - 5313 \beta_{12} + 7553 \beta_{11} + 4272 \beta_{9} + 10464 \beta_{7} - 20721 \beta_{6} ) / 12$$ (4065*b15 + 7854*b14 + 5313*b13 - 5313*b12 + 7553*b11 + 4272*b9 + 10464*b7 - 20721*b6) / 12 $$\nu^{12}$$ $$=$$ $$1110\beta_{8} + 777\beta_{5} + 1101\beta_{4} - 1101\beta_{3} + 2331\beta_{2} - 444\beta _1 + 6341$$ 1110*b8 + 777*b5 + 1101*b4 - 1101*b3 + 2331*b2 - 444*b1 + 6341 $$\nu^{13}$$ $$=$$ $$( - 22590 \beta_{15} - 15210 \beta_{14} - 26403 \beta_{13} - 26403 \beta_{12} + 4519 \beta_{11} + 80370 \beta_{9} + 54213 \beta_{7} - 116256 \beta_{6} ) / 12$$ (-22590*b15 - 15210*b14 - 26403*b13 - 26403*b12 + 4519*b11 + 80370*b9 + 54213*b7 - 116256*b6) / 12 $$\nu^{14}$$ $$=$$ $$( -74193\beta_{10} - 469\beta_{8} + 159560\beta_{5} - 75600\beta_{3} + 14964\beta_{2} + 25669\beta _1 + 194058 ) / 12$$ (-74193*b10 - 469*b8 + 159560*b5 - 75600*b3 + 14964*b2 + 25669*b1 + 194058) / 12 $$\nu^{15}$$ $$=$$ $$( -124593\beta_{15} - 154911\beta_{14} - 133422\beta_{13} - 94405\beta_{11} + 128304\beta_{9} ) / 6$$ (-124593*b15 - 154911*b14 - 133422*b13 - 94405*b11 + 128304*b9) / 6

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2255.1
 2.01367 + 1.16260i −0.667172 + 0.385192i −0.667172 − 0.385192i 2.01367 − 1.16260i −1.75344 + 1.01235i −1.19392 − 0.689309i −1.19392 + 0.689309i −1.75344 − 1.01235i 1.75344 + 1.01235i 1.19392 − 0.689309i 1.19392 + 0.689309i 1.75344 − 1.01235i −2.01367 + 1.16260i 0.667172 + 0.385192i 0.667172 − 0.385192i −2.01367 − 1.16260i
0 −1.70166 0.323042i 0 1.55481i 0 0 0 2.79129 + 1.09941i 0
2255.2 0 −1.70166 0.323042i 0 1.55481i 0 0 0 2.79129 + 1.09941i 0
2255.3 0 −1.70166 + 0.323042i 0 1.55481i 0 0 0 2.79129 1.09941i 0
2255.4 0 −1.70166 + 0.323042i 0 1.55481i 0 0 0 2.79129 1.09941i 0
2255.5 0 −0.777403 1.54779i 0 3.40332i 0 0 0 −1.79129 + 2.40651i 0
2255.6 0 −0.777403 1.54779i 0 3.40332i 0 0 0 −1.79129 + 2.40651i 0
2255.7 0 −0.777403 + 1.54779i 0 3.40332i 0 0 0 −1.79129 2.40651i 0
2255.8 0 −0.777403 + 1.54779i 0 3.40332i 0 0 0 −1.79129 2.40651i 0
2255.9 0 0.777403 1.54779i 0 3.40332i 0 0 0 −1.79129 2.40651i 0
2255.10 0 0.777403 1.54779i 0 3.40332i 0 0 0 −1.79129 2.40651i 0
2255.11 0 0.777403 + 1.54779i 0 3.40332i 0 0 0 −1.79129 + 2.40651i 0
2255.12 0 0.777403 + 1.54779i 0 3.40332i 0 0 0 −1.79129 + 2.40651i 0
2255.13 0 1.70166 0.323042i 0 1.55481i 0 0 0 2.79129 1.09941i 0
2255.14 0 1.70166 0.323042i 0 1.55481i 0 0 0 2.79129 1.09941i 0
2255.15 0 1.70166 + 0.323042i 0 1.55481i 0 0 0 2.79129 + 1.09941i 0
2255.16 0 1.70166 + 0.323042i 0 1.55481i 0 0 0 2.79129 + 1.09941i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2255.16 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
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.h.p 16
3.b odd 2 1 inner 2352.2.h.p 16
4.b odd 2 1 inner 2352.2.h.p 16
7.b odd 2 1 inner 2352.2.h.p 16
12.b even 2 1 inner 2352.2.h.p 16
21.c even 2 1 inner 2352.2.h.p 16
28.d even 2 1 inner 2352.2.h.p 16
84.h odd 2 1 inner 2352.2.h.p 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.h.p 16 1.a even 1 1 trivial
2352.2.h.p 16 3.b odd 2 1 inner
2352.2.h.p 16 4.b odd 2 1 inner
2352.2.h.p 16 7.b odd 2 1 inner
2352.2.h.p 16 12.b even 2 1 inner
2352.2.h.p 16 21.c even 2 1 inner
2352.2.h.p 16 28.d even 2 1 inner
2352.2.h.p 16 84.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2352, [\chi])$$:

 $$T_{5}^{4} + 14T_{5}^{2} + 28$$ T5^4 + 14*T5^2 + 28 $$T_{11}$$ T11 $$T_{13}^{4} - 22T_{13}^{2} + 100$$ T13^4 - 22*T13^2 + 100 $$T_{47}^{4} - 168T_{47}^{2} + 4032$$ T47^4 - 168*T47^2 + 4032

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$(T^{8} - 2 T^{6} - 2 T^{4} - 18 T^{2} + \cdots + 81)^{2}$$
$5$ $$(T^{4} + 14 T^{2} + 28)^{4}$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$(T^{4} - 22 T^{2} + 100)^{4}$$
$17$ $$(T^{4} + 56 T^{2} + 700)^{4}$$
$19$ $$(T^{4} + 66 T^{2} + 900)^{4}$$
$23$ $$(T^{4} - 84 T^{2} + 1008)^{4}$$
$29$ $$(T^{4} + 84 T^{2} + 1008)^{4}$$
$31$ $$(T^{2} + 24)^{8}$$
$37$ $$(T^{2} + 6 T - 12)^{8}$$
$41$ $$(T^{4} + 56 T^{2} + 28)^{4}$$
$43$ $$(T^{4} + 132 T^{2} + 3600)^{4}$$
$47$ $$(T^{4} - 168 T^{2} + 4032)^{4}$$
$53$ $$(T^{4} + 112 T^{2} + 112)^{4}$$
$59$ $$(T^{4} - 126 T^{2} + 2268)^{4}$$
$61$ $$(T^{4} - 70 T^{2} + 196)^{4}$$
$67$ $$(T^{2} + 12)^{8}$$
$71$ $$(T^{4} - 252 T^{2} + 9072)^{4}$$
$73$ $$(T^{4} - 340 T^{2} + 27556)^{4}$$
$79$ $$(T^{2} + 48)^{8}$$
$83$ $$(T^{4} - 294 T^{2} + 6300)^{4}$$
$89$ $$(T^{4} + 56 T^{2} + 700)^{4}$$
$97$ $$(T^{2} - 98)^{8}$$