# Properties

 Label 960.2.b.c Level $960$ Weight $2$ Character orbit 960.b Analytic conductor $7.666$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 960.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.66563859404$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36$$ x^12 - 16*x^8 - 24*x^7 + 96*x^5 + 304*x^4 + 384*x^3 + 288*x^2 + 144*x + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{14}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{3} - q^{5} + (\beta_{6} + \beta_{5} - \beta_{4}) q^{7} + \beta_{8} q^{9}+O(q^{10})$$ q - b5 * q^3 - q^5 + (b6 + b5 - b4) * q^7 + b8 * q^9 $$q - \beta_{5} q^{3} - q^{5} + (\beta_{6} + \beta_{5} - \beta_{4}) q^{7} + \beta_{8} q^{9} + \beta_{11} q^{11} + (\beta_{10} - \beta_{8} + \beta_{7}) q^{13} + \beta_{5} q^{15} + ( - \beta_{10} + \beta_{2} + \beta_1) q^{17} + \beta_{9} q^{19} + ( - \beta_{8} + \beta_{2} + 2) q^{21} + (\beta_{9} - \beta_{5} - \beta_{4}) q^{23} + q^{25} + (\beta_{11} + \beta_{4} + \beta_{3}) q^{27} + ( - \beta_{8} - \beta_{7} + \beta_{2} - \beta_1 - 2) q^{29} + (\beta_{11} - \beta_{6}) q^{31} + (\beta_{10} - \beta_{8} - \beta_{7} + \beta_1 + 1) q^{33} + ( - \beta_{6} - \beta_{5} + \beta_{4}) q^{35} + (\beta_{10} - \beta_{8} + \beta_{7} + \beta_{2} + \beta_1) q^{37} + ( - \beta_{9} - 2 \beta_{4} - \beta_{3}) q^{39} + ( - \beta_{8} + \beta_{7}) q^{41} + ( - 3 \beta_{5} - 3 \beta_{4}) q^{43} - \beta_{8} q^{45} + ( - \beta_{9} - \beta_{5} - \beta_{4} + 2 \beta_{3}) q^{47} + (\beta_{8} + \beta_{7} - 2 \beta_{2} + 2 \beta_1 + 1) q^{49} + (2 \beta_{9} + 3 \beta_{6} - 2 \beta_{4} - \beta_{3}) q^{51} + ( - 2 \beta_{8} - 2 \beta_{7} - \beta_{2} + \beta_1 - 4) q^{53} - \beta_{11} q^{55} + (2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_1 - 1) q^{57} + (\beta_{11} - 2 \beta_{5} + 2 \beta_{4}) q^{59} + (\beta_{8} - \beta_{7} + \beta_{2} + \beta_1) q^{61} + (\beta_{9} - 3 \beta_{5} - \beta_{4} - 2 \beta_{3}) q^{63} + ( - \beta_{10} + \beta_{8} - \beta_{7}) q^{65} + (2 \beta_{9} - \beta_{5} - \beta_{4} - 2 \beta_{3}) q^{67} + (2 \beta_{10} + \beta_{7} - \beta_1 + 2) q^{69} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{71} + ( - 2 \beta_{8} - 2 \beta_{7} + \beta_{2} - \beta_1 - 4) q^{73} - \beta_{5} q^{75} + (\beta_{2} - \beta_1 - 2) q^{77} + ( - 3 \beta_{11} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4}) q^{79} + (2 \beta_{10} - \beta_{8} + \beta_{7} + 2 \beta_1 - 1) q^{81} + ( - 6 \beta_{6} - \beta_{5} + \beta_{4}) q^{83} + (\beta_{10} - \beta_{2} - \beta_1) q^{85} + (\beta_{9} - 3 \beta_{6} + 2 \beta_{4} - 2 \beta_{3}) q^{87} + ( - 4 \beta_{10} + 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{2} - 2 \beta_1) q^{89} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{91} + (\beta_{10} - \beta_{8} - \beta_{7} - \beta_{2} + \beta_1 + 2) q^{93} - \beta_{9} q^{95} + ( - \beta_{2} + \beta_1 + 4) q^{97} + ( - \beta_{9} + 3 \beta_{6} + 4 \beta_{4} - \beta_{3}) q^{99}+O(q^{100})$$ q - b5 * q^3 - q^5 + (b6 + b5 - b4) * q^7 + b8 * q^9 + b11 * q^11 + (b10 - b8 + b7) * q^13 + b5 * q^15 + (-b10 + b2 + b1) * q^17 + b9 * q^19 + (-b8 + b2 + 2) * q^21 + (b9 - b5 - b4) * q^23 + q^25 + (b11 + b4 + b3) * q^27 + (-b8 - b7 + b2 - b1 - 2) * q^29 + (b11 - b6) * q^31 + (b10 - b8 - b7 + b1 + 1) * q^33 + (-b6 - b5 + b4) * q^35 + (b10 - b8 + b7 + b2 + b1) * q^37 + (-b9 - 2*b4 - b3) * q^39 + (-b8 + b7) * q^41 + (-3*b5 - 3*b4) * q^43 - b8 * q^45 + (-b9 - b5 - b4 + 2*b3) * q^47 + (b8 + b7 - 2*b2 + 2*b1 + 1) * q^49 + (2*b9 + 3*b6 - 2*b4 - b3) * q^51 + (-2*b8 - 2*b7 - b2 + b1 - 4) * q^53 - b11 * q^55 + (2*b10 - b8 + b7 - b1 - 1) * q^57 + (b11 - 2*b5 + 2*b4) * q^59 + (b8 - b7 + b2 + b1) * q^61 + (b9 - 3*b5 - b4 - 2*b3) * q^63 + (-b10 + b8 - b7) * q^65 + (2*b9 - b5 - b4 - 2*b3) * q^67 + (2*b10 + b7 - b1 + 2) * q^69 + (-2*b5 - 2*b4 + 2*b3) * q^71 + (-2*b8 - 2*b7 + b2 - b1 - 4) * q^73 - b5 * q^75 + (b2 - b1 - 2) * q^77 + (-3*b11 - b6 + 2*b5 - 2*b4) * q^79 + (2*b10 - b8 + b7 + 2*b1 - 1) * q^81 + (-6*b6 - b5 + b4) * q^83 + (b10 - b2 - b1) * q^85 + (b9 - 3*b6 + 2*b4 - 2*b3) * q^87 + (-4*b10 + 2*b8 - 2*b7 - 2*b2 - 2*b1) * q^89 + (2*b5 + 2*b4 + 2*b3) * q^91 + (b10 - b8 - b7 - b2 + b1 + 2) * q^93 - b9 * q^95 + (-b2 + b1 + 4) * q^97 + (-b9 + 3*b6 + 4*b4 - b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{5} - 4 q^{9}+O(q^{10})$$ 12 * q - 12 * q^5 - 4 * q^9 $$12 q - 12 q^{5} - 4 q^{9} + 32 q^{21} + 12 q^{25} - 8 q^{29} + 16 q^{33} + 4 q^{45} - 12 q^{49} - 40 q^{53} - 8 q^{57} + 24 q^{69} - 24 q^{73} - 16 q^{77} - 20 q^{81} + 24 q^{93} + 40 q^{97}+O(q^{100})$$ 12 * q - 12 * q^5 - 4 * q^9 + 32 * q^21 + 12 * q^25 - 8 * q^29 + 16 * q^33 + 4 * q^45 - 12 * q^49 - 40 * q^53 - 8 * q^57 + 24 * q^69 - 24 * q^73 - 16 * q^77 - 20 * q^81 + 24 * q^93 + 40 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36$$ :

 $$\beta_{1}$$ $$=$$ $$( 3193 \nu^{11} - 2244 \nu^{10} + 318 \nu^{9} + 2790 \nu^{8} - 55642 \nu^{7} - 38316 \nu^{6} + 52914 \nu^{5} + 250740 \nu^{4} + 773740 \nu^{3} + 629112 \nu^{2} + \cdots + 244884 ) / 51972$$ (3193*v^11 - 2244*v^10 + 318*v^9 + 2790*v^8 - 55642*v^7 - 38316*v^6 + 52914*v^5 + 250740*v^4 + 773740*v^3 + 629112*v^2 + 341484*v + 244884) / 51972 $$\beta_{2}$$ $$=$$ $$( 4689 \nu^{11} - 3876 \nu^{10} + 3854 \nu^{9} - 5534 \nu^{8} - 68264 \nu^{7} - 56268 \nu^{6} + 37850 \nu^{5} + 438544 \nu^{4} + 1102444 \nu^{3} + 911136 \nu^{2} + \cdots + 48876 ) / 51972$$ (4689*v^11 - 3876*v^10 + 3854*v^9 - 5534*v^8 - 68264*v^7 - 56268*v^6 + 37850*v^5 + 438544*v^4 + 1102444*v^3 + 911136*v^2 + 502080*v + 48876) / 51972 $$\beta_{3}$$ $$=$$ $$( 34 \nu^{11} + 4 \nu^{10} - 17 \nu^{9} + 12 \nu^{8} - 546 \nu^{7} - 896 \nu^{6} + 196 \nu^{5} + 3468 \nu^{4} + 10454 \nu^{3} + 13152 \nu^{2} + 7260 \nu + 2736 ) / 366$$ (34*v^11 + 4*v^10 - 17*v^9 + 12*v^8 - 546*v^7 - 896*v^6 + 196*v^5 + 3468*v^4 + 10454*v^3 + 13152*v^2 + 7260*v + 2736) / 366 $$\beta_{4}$$ $$=$$ $$( - 9709 \nu^{11} + 6990 \nu^{10} - 6270 \nu^{9} + 4284 \nu^{8} + 155434 \nu^{7} + 116508 \nu^{6} - 57894 \nu^{5} - 856824 \nu^{4} - 2383876 \nu^{3} - 2154324 \nu^{2} + \cdots - 596232 ) / 103944$$ (-9709*v^11 + 6990*v^10 - 6270*v^9 + 4284*v^8 + 155434*v^7 + 116508*v^6 - 57894*v^5 - 856824*v^4 - 2383876*v^3 - 2154324*v^2 - 1602804*v - 596232) / 103944 $$\beta_{5}$$ $$=$$ $$( 12807 \nu^{11} - 7336 \nu^{10} + 6002 \nu^{9} - 4224 \nu^{8} - 204280 \nu^{7} - 188332 \nu^{6} + 79370 \nu^{5} + 1157448 \nu^{4} + 3257084 \nu^{3} + 3239028 \nu^{2} + \cdots + 820728 ) / 103944$$ (12807*v^11 - 7336*v^10 + 6002*v^9 - 4224*v^8 - 204280*v^7 - 188332*v^6 + 79370*v^5 + 1157448*v^4 + 3257084*v^3 + 3239028*v^2 + 2201280*v + 820728) / 103944 $$\beta_{6}$$ $$=$$ $$( - 1688 \nu^{11} + 1054 \nu^{10} - 840 \nu^{9} + 684 \nu^{8} + 26378 \nu^{7} + 24587 \nu^{6} - 12648 \nu^{5} - 151968 \nu^{4} - 420176 \nu^{3} - 400856 \nu^{2} + \cdots - 105120 ) / 12993$$ (-1688*v^11 + 1054*v^10 - 840*v^9 + 684*v^8 + 26378*v^7 + 24587*v^6 - 12648*v^5 - 151968*v^4 - 420176*v^3 - 400856*v^2 - 282372*v - 105120) / 12993 $$\beta_{7}$$ $$=$$ $$( - 2365 \nu^{11} + 1728 \nu^{10} - 620 \nu^{9} - 408 \nu^{8} + 38369 \nu^{7} + 28380 \nu^{6} - 29398 \nu^{5} - 210938 \nu^{4} - 548176 \nu^{3} - 448716 \nu^{2} + \cdots - 128100 ) / 17324$$ (-2365*v^11 + 1728*v^10 - 620*v^9 - 408*v^8 + 38369*v^7 + 28380*v^6 - 29398*v^5 - 210938*v^4 - 548176*v^3 - 448716*v^2 - 245094*v - 128100) / 17324 $$\beta_{8}$$ $$=$$ $$( 7568 \nu^{11} - 5700 \nu^{10} + 2978 \nu^{9} - 1790 \nu^{8} - 119671 \nu^{7} - 90816 \nu^{6} + 85724 \nu^{5} + 699454 \nu^{4} + 1757628 \nu^{3} + 1442196 \nu^{2} + \cdots + 138132 ) / 51972$$ (7568*v^11 - 5700*v^10 + 2978*v^9 - 1790*v^8 - 119671*v^7 - 90816*v^6 + 85724*v^5 + 699454*v^4 + 1757628*v^3 + 1442196*v^2 + 789498*v + 138132) / 51972 $$\beta_{9}$$ $$=$$ $$( 8153 \nu^{11} + 4002 \nu^{10} - 8316 \nu^{9} + 6516 \nu^{8} - 135194 \nu^{7} - 253752 \nu^{6} + 29934 \nu^{5} + 882480 \nu^{4} + 2784512 \nu^{3} + 3554004 \nu^{2} + \cdots + 743400 ) / 51972$$ (8153*v^11 + 4002*v^10 - 8316*v^9 + 6516*v^8 - 135194*v^7 - 253752*v^6 + 29934*v^5 + 882480*v^4 + 2784512*v^3 + 3554004*v^2 + 1962852*v + 743400) / 51972 $$\beta_{10}$$ $$=$$ $$( - 118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} - 10408 \nu^{4} - 27758 \nu^{3} - 22776 \nu^{2} - 12468 \nu - 4320 ) / 426$$ (-118*v^11 + 90*v^10 - 53*v^9 + 32*v^8 + 1866*v^7 + 1416*v^6 - 1364*v^5 - 10408*v^4 - 27758*v^3 - 22776*v^2 - 12468*v - 4320) / 426 $$\beta_{11}$$ $$=$$ $$( 17023 \nu^{11} - 10696 \nu^{10} + 8738 \nu^{9} - 6744 \nu^{8} - 267980 \nu^{7} - 238924 \nu^{6} + 119690 \nu^{5} + 1529352 \nu^{4} + 4246428 \nu^{3} + 4054116 \nu^{2} + \cdots + 1063800 ) / 51972$$ (17023*v^11 - 10696*v^10 + 8738*v^9 - 6744*v^8 - 267980*v^7 - 238924*v^6 + 119690*v^5 + 1529352*v^4 + 4246428*v^3 + 4054116*v^2 + 2857032*v + 1063800) / 51972
 $$\nu$$ $$=$$ $$( \beta_{11} + \beta_{8} + \beta_{7} - 2\beta_{5} - \beta_{2} + 1 ) / 4$$ (b11 + b8 + b7 - 2*b5 - b2 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{11} - \beta_{9} + 3\beta_{6} - 2\beta_{5} - 2\beta_{4} + 3\beta_{3} ) / 4$$ (b11 - b9 + 3*b6 - 2*b5 - 2*b4 + 3*b3) / 4 $$\nu^{3}$$ $$=$$ $$( \beta_{10} - 4\beta_{5} - 4\beta_{4} + \beta_{3} + 2\beta_{2} + 2\beta_1 ) / 2$$ (b10 - 4*b5 - 4*b4 + b3 + 2*b2 + 2*b1) / 2 $$\nu^{4}$$ $$=$$ $$( 4\beta_{10} + 5\beta_{8} + \beta_{7} + 3\beta_{2} + 4\beta _1 + 13 ) / 2$$ (4*b10 + 5*b8 + b7 + 3*b2 + 4*b1 + 13) / 2 $$\nu^{5}$$ $$=$$ $$( 19 \beta_{11} + 6 \beta_{10} + \beta_{9} + 20 \beta_{8} + 18 \beta_{7} + 21 \beta_{6} + 2 \beta_{5} + 34 \beta_{4} - 7 \beta_{3} + \beta_{2} + 17 \beta _1 + 58 ) / 4$$ (19*b11 + 6*b10 + b9 + 20*b8 + 18*b7 + 21*b6 + 2*b5 + 34*b4 - 7*b3 + b2 + 17*b1 + 58) / 4 $$\nu^{6}$$ $$=$$ $$16\beta_{11} + 27\beta_{6} - 8\beta_{5} + 8\beta_{4}$$ 16*b11 + 27*b6 - 8*b5 + 8*b4 $$\nu^{7}$$ $$=$$ $$( 47 \beta_{11} + 16 \beta_{10} - 4 \beta_{9} - 43 \beta_{8} - 51 \beta_{7} + 60 \beta_{6} - 78 \beta_{5} - 8 \beta_{4} + 20 \beta_{3} + 39 \beta_{2} + 4 \beta _1 - 155 ) / 2$$ (47*b11 + 16*b10 - 4*b9 - 43*b8 - 51*b7 + 60*b6 - 78*b5 - 8*b4 + 20*b3 + 39*b2 + 4*b1 - 155) / 2 $$\nu^{8}$$ $$=$$ $$( 94\beta_{10} - 48\beta_{8} - 118\beta_{7} + 131\beta_{2} + 83\beta _1 - 306 ) / 2$$ (94*b10 - 48*b8 - 118*b7 + 131*b2 + 83*b1 - 306) / 2 $$\nu^{9}$$ $$=$$ $$83 \beta_{10} + 24 \beta_{9} + 24 \beta_{8} - 24 \beta_{7} + 212 \beta_{5} + 212 \beta_{4} - 107 \beta_{3} + 106 \beta_{2} + 106 \beta_1$$ 83*b10 + 24*b9 + 24*b8 - 24*b7 + 212*b5 + 212*b4 - 107*b3 + 106*b2 + 106*b1 $$\nu^{10}$$ $$=$$ $$213\beta_{11} + 82\beta_{9} + 318\beta_{6} + 426\beta_{5} + 688\beta_{4} - 318\beta_{3}$$ 213*b11 + 82*b9 + 318*b6 + 426*b5 + 688*b4 - 318*b3 $$\nu^{11}$$ $$=$$ $$( 1193 \beta_{11} - 426 \beta_{10} + 131 \beta_{9} - 1324 \beta_{8} - 1062 \beta_{7} + 1671 \beta_{6} + 262 \beta_{5} + 1862 \beta_{4} - 557 \beta_{3} - 131 \beta_{2} - 931 \beta _1 - 4142 ) / 2$$ (1193*b11 - 426*b10 + 131*b9 - 1324*b8 - 1062*b7 + 1671*b6 + 262*b5 + 1862*b4 - 557*b3 - 131*b2 - 931*b1 - 4142) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$577$$ $$641$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
671.1
 0.583700 − 2.17840i 0.583700 + 2.17840i −1.50511 − 0.403293i −1.50511 + 0.403293i −0.673288 + 0.180407i −0.673288 − 0.180407i −0.180407 + 0.673288i −0.180407 − 0.673288i −0.403293 − 1.50511i −0.403293 + 1.50511i 2.17840 − 0.583700i 2.17840 + 0.583700i
0 −1.59470 0.675970i 0 −1.00000 0 0.648061i 0 2.08613 + 2.15594i 0
671.2 0 −1.59470 + 0.675970i 0 −1.00000 0 0.648061i 0 2.08613 2.15594i 0
671.3 0 −1.10182 1.33641i 0 −1.00000 0 4.67282i 0 −0.571993 + 2.94497i 0
671.4 0 −1.10182 + 1.33641i 0 −1.00000 0 4.67282i 0 −0.571993 2.94497i 0
671.5 0 −0.492881 1.66044i 0 −1.00000 0 1.32088i 0 −2.51414 + 1.63680i 0
671.6 0 −0.492881 + 1.66044i 0 −1.00000 0 1.32088i 0 −2.51414 1.63680i 0
671.7 0 0.492881 1.66044i 0 −1.00000 0 1.32088i 0 −2.51414 1.63680i 0
671.8 0 0.492881 + 1.66044i 0 −1.00000 0 1.32088i 0 −2.51414 + 1.63680i 0
671.9 0 1.10182 1.33641i 0 −1.00000 0 4.67282i 0 −0.571993 2.94497i 0
671.10 0 1.10182 + 1.33641i 0 −1.00000 0 4.67282i 0 −0.571993 + 2.94497i 0
671.11 0 1.59470 0.675970i 0 −1.00000 0 0.648061i 0 2.08613 2.15594i 0
671.12 0 1.59470 + 0.675970i 0 −1.00000 0 0.648061i 0 2.08613 + 2.15594i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 671.12 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
24.f even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.b.c 12
3.b odd 2 1 960.2.b.d yes 12
4.b odd 2 1 inner 960.2.b.c 12
8.b even 2 1 960.2.b.d yes 12
8.d odd 2 1 960.2.b.d yes 12
12.b even 2 1 960.2.b.d yes 12
24.f even 2 1 inner 960.2.b.c 12
24.h odd 2 1 inner 960.2.b.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.b.c 12 1.a even 1 1 trivial
960.2.b.c 12 4.b odd 2 1 inner
960.2.b.c 12 24.f even 2 1 inner
960.2.b.c 12 24.h odd 2 1 inner
960.2.b.d yes 12 3.b odd 2 1
960.2.b.d yes 12 8.b even 2 1
960.2.b.d yes 12 8.d odd 2 1
960.2.b.d yes 12 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(960, [\chi])$$:

 $$T_{7}^{6} + 24T_{7}^{4} + 48T_{7}^{2} + 16$$ T7^6 + 24*T7^4 + 48*T7^2 + 16 $$T_{29}^{3} + 2T_{29}^{2} - 44T_{29} - 72$$ T29^3 + 2*T29^2 - 44*T29 - 72

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 2 T^{10} + 7 T^{8} + 12 T^{6} + \cdots + 729$$
$5$ $$(T + 1)^{12}$$
$7$ $$(T^{6} + 24 T^{4} + 48 T^{2} + 16)^{2}$$
$11$ $$(T^{6} + 44 T^{4} + 496 T^{2} + 576)^{2}$$
$13$ $$(T^{6} + 52 T^{4} + 304 T^{2} + 192)^{2}$$
$17$ $$(T^{6} + 100 T^{4} + 2608 T^{2} + \cdots + 15552)^{2}$$
$19$ $$(T^{6} - 52 T^{4} + 304 T^{2} - 192)^{2}$$
$23$ $$(T^{6} - 60 T^{4} + 1008 T^{2} + \cdots - 3888)^{2}$$
$29$ $$(T^{3} + 2 T^{2} - 44 T - 72)^{4}$$
$31$ $$(T^{6} + 64 T^{4} + 768 T^{2} + 2304)^{2}$$
$37$ $$(T^{6} + 100 T^{4} + 3184 T^{2} + \cdots + 32448)^{2}$$
$41$ $$(T^{6} + 64 T^{4} + 1216 T^{2} + \cdots + 6912)^{2}$$
$43$ $$(T^{6} - 144 T^{4} + 5184 T^{2} + \cdots - 34992)^{2}$$
$47$ $$(T^{6} - 172 T^{4} + 6256 T^{2} + \cdots - 3888)^{2}$$
$53$ $$(T^{3} + 10 T^{2} - 116 T - 1128)^{4}$$
$59$ $$(T^{6} + 92 T^{4} + 2224 T^{2} + \cdots + 5184)^{2}$$
$61$ $$(T^{6} + 144 T^{4} + 3456 T^{2} + \cdots + 6912)^{2}$$
$67$ $$(T^{6} - 256 T^{4} + 12736 T^{2} + \cdots - 178608)^{2}$$
$71$ $$(T^{6} - 208 T^{4} + 5632 T^{2} + \cdots - 27648)^{2}$$
$73$ $$(T^{3} + 6 T^{2} - 84 T + 104)^{4}$$
$79$ $$(T^{6} + 384 T^{4} + 27840 T^{2} + \cdots + 565504)^{2}$$
$83$ $$(T^{6} + 404 T^{4} + 49168 T^{2} + \cdots + 1838736)^{2}$$
$89$ $$(T^{6} + 640 T^{4} + 114688 T^{2} + \cdots + 3981312)^{2}$$
$97$ $$(T^{3} - 10 T^{2} - 4 T + 136)^{4}$$