# Properties

 Label 252.4.j.b Level $252$ Weight $4$ Character orbit 252.j Analytic conductor $14.868$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.j (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8684813214$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$9$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Defining polynomial: $$x^{18} - 7 x^{17} + 81 x^{16} - 470 x^{15} + 2687 x^{14} - 12243 x^{13} + 43732 x^{12} - 169303 x^{11} + 166449 x^{10} - 57172 x^{9} - 3758051 x^{8} + 73649097 x^{7} + 358194386 x^{6} + 3599409199 x^{5} + 26802380109 x^{4} + 114591208112 x^{3} + 655546044172 x^{2} + 1324018291596 x + 5500612092612$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{6}\cdot 3^{14}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{3} + ( -1 + \beta_{2} + \beta_{3} ) q^{5} -7 \beta_{2} q^{7} + ( -5 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{3} + ( -1 + \beta_{2} + \beta_{3} ) q^{5} -7 \beta_{2} q^{7} + ( -5 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{9} + ( -10 \beta_{2} + \beta_{5} + \beta_{6} - \beta_{12} ) q^{11} + ( -3 + 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{13} + ( 8 + \beta_{2} + \beta_{3} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{17} ) q^{15} + ( -3 + \beta_{2} - \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{17} ) q^{17} + ( -7 + 3 \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{17} ) q^{19} + ( -7 \beta_{2} - 7 \beta_{7} ) q^{21} + ( -28 + 28 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - 3 \beta_{11} + 3 \beta_{12} - 3 \beta_{14} + 2 \beta_{15} - 3 \beta_{16} - 2 \beta_{17} ) q^{23} + ( 2 - 2 \beta_{1} - 38 \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} - 5 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - 3 \beta_{12} - \beta_{13} - \beta_{16} - \beta_{17} ) q^{25} + ( -26 + 7 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 4 \beta_{7} - 2 \beta_{9} - 3 \beta_{10} - 2 \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} + 2 \beta_{16} ) q^{27} + ( -1 + 6 \beta_{1} - 33 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} + 6 \beta_{7} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} - 3 \beta_{16} - 3 \beta_{17} ) q^{29} + ( -8 + 7 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} + \beta_{6} + 8 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + \beta_{11} - 3 \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{31} + ( 8 - 2 \beta_{1} - 8 \beta_{2} - 9 \beta_{3} - 2 \beta_{4} - \beta_{5} + 9 \beta_{6} - 12 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} + 4 \beta_{15} - 4 \beta_{16} - 4 \beta_{17} ) q^{33} + ( 7 - 7 \beta_{3} + 7 \beta_{6} ) q^{35} + ( -3 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} - 5 \beta_{5} + 6 \beta_{6} - 8 \beta_{7} - \beta_{9} + 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} + 4 \beta_{14} + 2 \beta_{15} - 3 \beta_{16} - 2 \beta_{17} ) q^{37} + ( 10 + 8 \beta_{1} - 4 \beta_{2} + 9 \beta_{3} - 3 \beta_{5} - 9 \beta_{6} + 5 \beta_{7} - 6 \beta_{9} + 3 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} ) q^{39} + ( 1 - 15 \beta_{1} - 6 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} + 8 \beta_{6} - 18 \beta_{7} + 8 \beta_{8} + \beta_{9} - 3 \beta_{10} + 6 \beta_{11} - 8 \beta_{12} + 8 \beta_{14} - 3 \beta_{15} + 3 \beta_{16} + 2 \beta_{17} ) q^{41} + ( 9 - 29 \beta_{1} - 16 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} - 7 \beta_{8} - 7 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} + 6 \beta_{12} + \beta_{13} - 3 \beta_{16} - 3 \beta_{17} ) q^{43} + ( -33 - 11 \beta_{1} + 45 \beta_{2} - 15 \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 7 \beta_{7} - \beta_{8} - 8 \beta_{9} + 6 \beta_{10} - 3 \beta_{11} + 6 \beta_{12} - 3 \beta_{13} + 3 \beta_{15} - 6 \beta_{16} - 6 \beta_{17} ) q^{45} + ( 2 + 3 \beta_{1} - 123 \beta_{2} + 3 \beta_{4} + \beta_{5} + 6 \beta_{6} - 12 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{47} + ( -49 + 49 \beta_{2} ) q^{49} + ( 65 + 3 \beta_{1} - 50 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 13 \beta_{6} + 5 \beta_{8} - \beta_{9} - 3 \beta_{10} + 5 \beta_{11} - 9 \beta_{12} + 3 \beta_{14} - 6 \beta_{15} + 3 \beta_{16} + 5 \beta_{17} ) q^{51} + ( 70 + 21 \beta_{1} + 18 \beta_{2} + 13 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} - 16 \beta_{6} + 50 \beta_{7} + 9 \beta_{8} + 7 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + 5 \beta_{12} - 5 \beta_{13} - 4 \beta_{14} + 5 \beta_{15} - 9 \beta_{16} - 12 \beta_{17} ) q^{53} + ( -84 - 9 \beta_{1} - 18 \beta_{2} - 24 \beta_{3} - \beta_{4} + 4 \beta_{5} + 25 \beta_{6} - 44 \beta_{7} - 9 \beta_{8} - \beta_{10} - 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 11 \beta_{14} + 2 \beta_{15} - 3 \beta_{16} - 4 \beta_{17} ) q^{55} + ( -62 + 4 \beta_{1} - 25 \beta_{2} + 12 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} - \beta_{8} - 8 \beta_{9} + 3 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - 6 \beta_{15} - 3 \beta_{16} ) q^{57} + ( -39 - 27 \beta_{1} + 53 \beta_{2} + 6 \beta_{3} - 3 \beta_{6} + 5 \beta_{7} + 3 \beta_{8} + 5 \beta_{9} + \beta_{10} - 3 \beta_{11} + 5 \beta_{12} - 5 \beta_{14} - \beta_{15} - 3 \beta_{16} - 4 \beta_{17} ) q^{59} + ( 8 - 23 \beta_{1} - 38 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 7 \beta_{5} - 18 \beta_{6} - 28 \beta_{7} - 5 \beta_{8} + \beta_{9} - 5 \beta_{10} - 3 \beta_{11} - 6 \beta_{12} + 6 \beta_{13} + 4 \beta_{16} + 4 \beta_{17} ) q^{61} + ( -7 + 42 \beta_{2} - 7 \beta_{4} - 7 \beta_{5} - 7 \beta_{7} ) q^{63} + ( -32 + 78 \beta_{1} - 154 \beta_{2} + 6 \beta_{3} - 9 \beta_{4} - 8 \beta_{6} + 3 \beta_{7} + 13 \beta_{8} + 11 \beta_{9} - 9 \beta_{10} + 12 \beta_{11} - 12 \beta_{12} + 12 \beta_{13} + 6 \beta_{16} + 6 \beta_{17} ) q^{65} + ( -62 - 7 \beta_{1} + 80 \beta_{2} + 30 \beta_{3} - 6 \beta_{4} - \beta_{5} - \beta_{6} + 49 \beta_{7} - 9 \beta_{8} + 9 \beta_{9} + 8 \beta_{10} - 6 \beta_{11} + 9 \beta_{15} - 7 \beta_{16} + 7 \beta_{17} ) q^{67} + ( 12 + 22 \beta_{1} + 39 \beta_{2} - 7 \beta_{3} - 5 \beta_{4} - 7 \beta_{5} + 20 \beta_{6} + 24 \beta_{7} - 7 \beta_{8} - 13 \beta_{9} + 3 \beta_{10} + 10 \beta_{12} + 7 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - 4 \beta_{16} + 6 \beta_{17} ) q^{69} + ( 179 + 27 \beta_{1} + 18 \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} + 8 \beta_{6} + 47 \beta_{7} - 2 \beta_{9} - 9 \beta_{10} - 2 \beta_{11} - 7 \beta_{12} + 7 \beta_{13} + 3 \beta_{14} - 7 \beta_{15} + 18 \beta_{16} + 4 \beta_{17} ) q^{71} + ( -7 + 39 \beta_{1} - 9 \beta_{2} - 6 \beta_{3} + 10 \beta_{4} + 5 \beta_{5} + 8 \beta_{6} - 4 \beta_{7} - 9 \beta_{8} - 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + 9 \beta_{16} - 8 \beta_{17} ) q^{73} + ( -107 - 84 \beta_{2} + 21 \beta_{3} - 9 \beta_{4} - 9 \beta_{5} - 25 \beta_{7} - 2 \beta_{8} - \beta_{9} - 9 \beta_{10} - 6 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - 9 \beta_{14} + 12 \beta_{15} - 3 \beta_{16} - 3 \beta_{17} ) q^{75} + ( -70 + 70 \beta_{2} + 7 \beta_{4} - 7 \beta_{6} + 7 \beta_{12} - 7 \beta_{14} - 7 \beta_{17} ) q^{77} + ( 38 - 91 \beta_{1} - 15 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 11 \beta_{5} - 11 \beta_{6} - 48 \beta_{7} - 7 \beta_{8} + 8 \beta_{9} + \beta_{11} - 3 \beta_{12} - 7 \beta_{13} + 3 \beta_{16} + 3 \beta_{17} ) q^{79} + ( 26 + 36 \beta_{1} + 45 \beta_{2} + 2 \beta_{3} - 11 \beta_{4} + 3 \beta_{5} - 19 \beta_{6} + 9 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} + 3 \beta_{10} + 9 \beta_{11} - 11 \beta_{12} + \beta_{13} - 4 \beta_{14} + 4 \beta_{15} - 7 \beta_{16} + 6 \beta_{17} ) q^{81} + ( -4 + 24 \beta_{1} - 84 \beta_{2} + 9 \beta_{3} + 9 \beta_{4} - 7 \beta_{5} - 13 \beta_{6} - 21 \beta_{7} + 7 \beta_{9} - 2 \beta_{10} - 16 \beta_{11} + 7 \beta_{12} + 2 \beta_{13} + 9 \beta_{16} + 9 \beta_{17} ) q^{83} + ( -100 - 40 \beta_{1} + 135 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + 6 \beta_{5} - \beta_{6} + 58 \beta_{7} - 2 \beta_{8} + 12 \beta_{9} + 5 \beta_{10} - 2 \beta_{11} - 8 \beta_{12} + 8 \beta_{14} - 11 \beta_{15} + 4 \beta_{16} + 19 \beta_{17} ) q^{85} + ( 183 - 4 \beta_{1} - 53 \beta_{2} - 12 \beta_{3} + 8 \beta_{4} + 16 \beta_{5} + 9 \beta_{6} - 54 \beta_{7} - 14 \beta_{8} - 15 \beta_{9} + 3 \beta_{10} - 8 \beta_{11} + 2 \beta_{12} + 5 \beta_{13} - 5 \beta_{14} - 4 \beta_{15} + 16 \beta_{16} + 10 \beta_{17} ) q^{87} + ( 137 + 21 \beta_{1} + 30 \beta_{2} - 13 \beta_{4} - 19 \beta_{5} + 3 \beta_{6} + 74 \beta_{7} + 12 \beta_{8} + 10 \beta_{9} - 3 \beta_{10} - 8 \beta_{11} + 5 \beta_{12} - 5 \beta_{13} - 4 \beta_{14} + 5 \beta_{15} - 12 \beta_{16} ) q^{89} + ( 14 + 7 \beta_{1} - 7 \beta_{3} + 7 \beta_{5} + 7 \beta_{6} + 7 \beta_{11} - 7 \beta_{12} + 7 \beta_{13} + 7 \beta_{14} - 7 \beta_{15} + 7 \beta_{16} + 14 \beta_{17} ) q^{91} + ( -254 + 2 \beta_{1} + 209 \beta_{2} + 39 \beta_{3} + 6 \beta_{4} + 21 \beta_{5} - 33 \beta_{6} + 8 \beta_{7} - 3 \beta_{8} + 18 \beta_{10} - 3 \beta_{12} - 3 \beta_{13} - 6 \beta_{14} + 6 \beta_{15} - 6 \beta_{16} - 9 \beta_{17} ) q^{93} + ( 39 - 102 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 18 \beta_{4} + 6 \beta_{5} - 30 \beta_{7} + 26 \beta_{8} + 28 \beta_{9} + 3 \beta_{10} + 12 \beta_{11} - 21 \beta_{12} + 21 \beta_{14} - 3 \beta_{15} + 18 \beta_{16} + 30 \beta_{17} ) q^{95} + ( 48 - 117 \beta_{1} + 90 \beta_{2} - 5 \beta_{3} - \beta_{4} - 11 \beta_{5} + 5 \beta_{6} - 55 \beta_{7} - 14 \beta_{8} + \beta_{9} + 13 \beta_{10} - 7 \beta_{11} + 21 \beta_{12} - 7 \beta_{13} - 5 \beta_{16} - 5 \beta_{17} ) q^{97} + ( 261 + 5 \beta_{1} + 162 \beta_{2} - 6 \beta_{3} - 7 \beta_{4} - 20 \beta_{5} + 9 \beta_{6} - 17 \beta_{7} - 11 \beta_{8} - 28 \beta_{9} - 15 \beta_{10} + 6 \beta_{11} + 12 \beta_{12} + 3 \beta_{13} + 6 \beta_{15} + 6 \beta_{16} - 9 \beta_{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q + 11q^{3} - 6q^{5} - 63q^{7} - 109q^{9} + O(q^{10})$$ $$18q + 11q^{3} - 6q^{5} - 63q^{7} - 109q^{9} - 93q^{11} - 18q^{13} + 162q^{15} - 54q^{17} - 90q^{19} - 49q^{21} - 246q^{23} - 315q^{25} - 394q^{27} - 318q^{29} - 18q^{31} + 33q^{33} + 84q^{35} - 72q^{37} + 268q^{39} - 57q^{41} - 171q^{43} - 318q^{45} - 1056q^{47} - 441q^{49} + 705q^{51} + 1512q^{53} - 1800q^{55} - 1271q^{57} - 411q^{59} - 198q^{61} + 308q^{63} - 1326q^{65} - 441q^{67} + 642q^{69} + 3516q^{71} + 54q^{73} - 2497q^{75} - 651q^{77} + 72q^{79} + 1163q^{81} - 558q^{83} - 1008q^{85} + 2766q^{87} + 2784q^{89} + 252q^{91} - 2618q^{93} - 156q^{95} + 909q^{97} + 6318q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 7 x^{17} + 81 x^{16} - 470 x^{15} + 2687 x^{14} - 12243 x^{13} + 43732 x^{12} - 169303 x^{11} + 166449 x^{10} - 57172 x^{9} - 3758051 x^{8} + 73649097 x^{7} + 358194386 x^{6} + 3599409199 x^{5} + 26802380109 x^{4} + 114591208112 x^{3} + 655546044172 x^{2} + 1324018291596 x + 5500612092612$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{17} + 6 \nu^{16} - 75 \nu^{15} + 395 \nu^{14} - 2292 \nu^{13} + 9951 \nu^{12} - 33781 \nu^{11} + 135522 \nu^{10} - 30927 \nu^{9} + 26245 \nu^{8} + 3784296 \nu^{7} - 69864801 \nu^{6} - 428059187 \nu^{5} - 4027468386 \nu^{4} - 30829848495 \nu^{3} - 145421056607 \nu^{2} - 800967100779 \nu - 1842555855894$$$$)/ 282429536481$$ $$\beta_{2}$$ $$=$$ $$($$$$-13315800857 \nu^{17} - 468924789381 \nu^{16} + 3153303989877 \nu^{15} - 26780596660724 \nu^{14} + 144361057555383 \nu^{13} - 639270116555289 \nu^{12} + 2482336757105530 \nu^{11} - 5357935068192207 \nu^{10} + 27019261047320319 \nu^{9} + 50256977890674110 \nu^{8} - 590608070231663955 \nu^{7} - 412547145864241521 \nu^{6} - 29543405127212658064 \nu^{5} - 245152146093031333293 \nu^{4} - 1615394096202361273185 \nu^{3} - 10383177724615955168890 \nu^{2} - 31868395054573493278170 \nu - 116580749049536069497368$$$$)/$$$$28\!\cdots\!60$$ $$\beta_{3}$$ $$=$$ $$($$$$-9988724986 \nu^{17} - 2313647830569 \nu^{16} + 16444663818747 \nu^{15} - 124540717085305 \nu^{14} + 629547150035334 \nu^{13} - 2680955378956353 \nu^{12} + 10141182689344577 \nu^{11} - 22536757427037804 \nu^{10} + 99047133357274653 \nu^{9} + 265594891716414313 \nu^{8} - 2629925749006371672 \nu^{7} + 458413780959705585 \nu^{6} - 99814262517300381347 \nu^{5} - 909433856547434475246 \nu^{4} - 5850450196503191312541 \nu^{3} - 38670595329535605423791 \nu^{2} - 114461929263992641354566 \nu - 464396716562452808324190$$$$)/$$$$14\!\cdots\!80$$ $$\beta_{4}$$ $$=$$ $$($$$$87164256743 \nu^{17} - 1343101290501 \nu^{16} + 12045789087477 \nu^{15} - 85460950299124 \nu^{14} + 462832600118583 \nu^{13} - 2172786707640729 \nu^{12} + 8042682088562810 \nu^{11} - 25973820358208847 \nu^{10} + 59894778453147519 \nu^{9} + 68997473241627070 \nu^{8} - 942356588222826195 \nu^{7} + 7662634801042297359 \nu^{6} - 4430882560499337104 \nu^{5} + 25498408347055087827 \nu^{4} + 255724465807000035615 \nu^{3} - 5361947343016804551290 \nu^{2} - 430033984255033874010 \nu - 152377127259409722530328$$$$)/$$$$28\!\cdots\!60$$ $$\beta_{5}$$ $$=$$ $$($$$$-133776214024 \nu^{17} + 6794583925713 \nu^{16} - 48031849953471 \nu^{15} + 359420053452287 \nu^{14} - 1855377197024424 \nu^{13} + 8022621727128417 \nu^{12} - 29312743281552835 \nu^{11} + 71486092324490706 \nu^{10} - 263233836597346257 \nu^{9} - 818942256410141795 \nu^{8} + 6695540736339845550 \nu^{7} - 10310230434880559517 \nu^{6} + 220985470955047942957 \nu^{5} + 1845505158809816982444 \nu^{4} + 12290445346717856888085 \nu^{3} + 93994409721886472385205 \nu^{2} + 275503805564435814349710 \nu + 1245115109834095486888434$$$$)/$$$$14\!\cdots\!80$$ $$\beta_{6}$$ $$=$$ $$($$$$52446298994 \nu^{17} - 1450917493401 \nu^{16} + 10638858706899 \nu^{15} - 76601942339581 \nu^{14} + 387846820484274 \nu^{13} - 1648336188413505 \nu^{12} + 5851869770644781 \nu^{11} - 14845223982533040 \nu^{10} + 43270860967811925 \nu^{9} + 201857339992033189 \nu^{8} - 1310508564781732476 \nu^{7} + 3150072701260064121 \nu^{6} - 27391848724391882687 \nu^{5} - 279317041563965678850 \nu^{4} - 1727428316646712567533 \nu^{3} - 15709959719493607025843 \nu^{2} - 44475396597257734186038 \nu - 235405553302236589374102$$$$)/$$$$47\!\cdots\!60$$ $$\beta_{7}$$ $$=$$ $$($$$$-21375320078 \nu^{17} + 163188313455 \nu^{16} - 1262052587301 \nu^{15} + 6901775515771 \nu^{14} - 30681598044558 \nu^{13} + 117512906055303 \nu^{12} - 295874142739883 \nu^{11} + 1133477226885024 \nu^{10} + 1809063074792397 \nu^{9} - 25891808439339187 \nu^{8} + 28821123162823668 \nu^{7} - 976695734708706351 \nu^{6} - 7267831316077867351 \nu^{5} - 47077444909330366434 \nu^{4} - 326430947106959972421 \nu^{3} - 819795063291250302091 \nu^{2} - 3559151763419097933126 \nu + 3813712563302326219002$$$$)/$$$$10\!\cdots\!60$$ $$\beta_{8}$$ $$=$$ $$($$$$-541960125439 \nu^{17} + 4635014468847 \nu^{16} - 35551554518235 \nu^{15} + 203299474367714 \nu^{14} - 925913453445519 \nu^{13} + 3608609672794371 \nu^{12} - 9715051369935808 \nu^{11} + 33114499458330843 \nu^{10} + 24017204040634599 \nu^{9} - 674901471527606372 \nu^{8} + 1030133185737863223 \nu^{7} - 23349046402765226049 \nu^{6} - 167508341581618323878 \nu^{5} - 990121454810876979363 \nu^{4} - 7117181122299752079501 \nu^{3} - 12207206554862892581756 \nu^{2} - 70753648528273560695226 \nu + 199640860819024447248348$$$$)/$$$$14\!\cdots\!80$$ $$\beta_{9}$$ $$=$$ $$($$$$-1127380069015 \nu^{17} + 8981968251993 \nu^{16} - 70210606342593 \nu^{15} + 391725238900304 \nu^{14} - 1776555936941415 \nu^{13} + 6877912344986733 \nu^{12} - 17966061030501766 \nu^{11} + 64956241358289639 \nu^{10} + 74121405106170957 \nu^{9} - 1298754835431025034 \nu^{8} + 1583732185260697371 \nu^{7} - 49216645720002454899 \nu^{6} - 377463511820240057180 \nu^{5} - 2346799132337023002099 \nu^{4} - 16779091826574927425787 \nu^{3} - 39181165677579006501362 \nu^{2} - 171979375753517328954702 \nu + 212969668982302808498208$$$$)/$$$$28\!\cdots\!60$$ $$\beta_{10}$$ $$=$$ $$($$$$-808888063876 \nu^{17} - 3731737888569 \nu^{16} + 24220753702287 \nu^{15} - 311340631185475 \nu^{14} + 1833523278864924 \nu^{13} - 8439376098995193 \nu^{12} + 36712882086841727 \nu^{11} - 66460058210998494 \nu^{10} + 482786678102726193 \nu^{9} + 428973285056160463 \nu^{8} - 11085954006236786682 \nu^{7} - 24896233798916969715 \nu^{6} - 723407552850215487017 \nu^{5} - 5464271441064704744856 \nu^{4} - 35944720093817458645821 \nu^{3} - 197573386582095247391561 \nu^{2} - 635242897576427569591086 \nu - 1996318482980846059703850$$$$)/$$$$14\!\cdots\!80$$ $$\beta_{11}$$ $$=$$ $$($$$$345092458312 \nu^{17} - 4149165254847 \nu^{16} + 29911777199121 \nu^{15} - 185999581724705 \nu^{14} + 898024623511272 \nu^{13} - 3741100769912559 \nu^{12} + 11953379544648061 \nu^{11} - 34763086329172542 \nu^{10} + 44368602058199199 \nu^{9} + 579877899585645149 \nu^{8} - 2077338202865906466 \nu^{7} + 15143000982058328595 \nu^{6} + 51466066663262063309 \nu^{5} + 179151384508592013372 \nu^{4} + 1253947125370374952197 \nu^{3} - 14364505320239100864523 \nu^{2} - 24138398717057057304978 \nu - 373784798998292943161550$$$$)/$$$$47\!\cdots\!60$$ $$\beta_{12}$$ $$=$$ $$($$$$835642665467 \nu^{17} + 295562583375 \nu^{16} + 865211681409 \nu^{15} + 117611454367916 \nu^{14} - 825805139295093 \nu^{13} + 4192191575165643 \nu^{12} - 22011683320753438 \nu^{11} + 39542883518250909 \nu^{10} - 404904968846267133 \nu^{9} + 248718149382532438 \nu^{8} + 6462536782200803913 \nu^{7} + 30908442870423963219 \nu^{6} + 588423154770848237824 \nu^{5} + 4317944756837326367991 \nu^{4} + 28787394897827919952419 \nu^{3} + 144664848834399811941214 \nu^{2} + 481867745763076562621934 \nu + 1337459081995976438887752$$$$)/$$$$70\!\cdots\!40$$ $$\beta_{13}$$ $$=$$ $$($$$$1808290611835 \nu^{17} + 24078182823471 \nu^{16} - 162934882103391 \nu^{15} + 1558368178227628 \nu^{14} - 8613482756881845 \nu^{13} + 38742333615652971 \nu^{12} - 159342061943449502 \nu^{11} + 344231952665490333 \nu^{10} - 1940546540977081341 \nu^{9} - 2415443478244567018 \nu^{8} + 41024331746091476457 \nu^{7} + 50932386163964856147 \nu^{6} + 2310366631251866702240 \nu^{5} + 18100214786711034759927 \nu^{4} + 119476843497180014922051 \nu^{3} + 717480309616191297923966 \nu^{2} + 2249302688825050345047246 \nu + 7960137035704649359775016$$$$)/$$$$14\!\cdots\!80$$ $$\beta_{14}$$ $$=$$ $$($$$$985735662991 \nu^{17} - 2977298291364 \nu^{16} + 27222686370246 \nu^{15} - 59978426067029 \nu^{14} + 29980675117311 \nu^{13} + 683596606436760 \nu^{12} - 9682825997402441 \nu^{11} + 6353319825171315 \nu^{10} - 329791798074569730 \nu^{9} + 656276595717482651 \nu^{8} + 3837274780301664171 \nu^{7} + 40398257464466373474 \nu^{6} + 558273296575425407357 \nu^{5} + 3878159588309020057455 \nu^{4} + 26992325841508720959108 \nu^{3} + 121499479051532528986043 \nu^{2} + 432691915800409583607828 \nu + 924100941905741907002202$$$$)/$$$$70\!\cdots\!40$$ $$\beta_{15}$$ $$=$$ $$($$$$-4518126258467 \nu^{17} + 74078356177587 \nu^{16} - 553906196437551 \nu^{15} + 3667883539275130 \nu^{14} - 17941342908716787 \nu^{13} + 74639272418675319 \nu^{12} - 250750351048317536 \nu^{11} + 664898634204218127 \nu^{10} - 1278989065022301429 \nu^{9} - 10412611390104433204 \nu^{8} + 51706224065871399291 \nu^{7} - 231941394755648806845 \nu^{6} + 157323199889094758546 \nu^{5} + 4523888170846831016073 \nu^{4} + 24184958895951049879143 \nu^{3} + 468263194752432099244628 \nu^{2} + 1084363512444308889389358 \nu + 8722946418932296918730220$$$$)/$$$$28\!\cdots\!60$$ $$\beta_{16}$$ $$=$$ $$($$$$-5301535780511 \nu^{17} + 24231887336811 \nu^{16} - 201658791233823 \nu^{15} + 797135476182310 \nu^{14} - 2780896817102511 \nu^{13} + 8267812217474367 \nu^{12} + 4562272683547852 \nu^{11} + 99409102814805651 \nu^{10} + 1201700843187897963 \nu^{9} - 4075878816920270872 \nu^{8} - 12987579818889790977 \nu^{7} - 219120055231244881485 \nu^{6} - 2583581727335824053082 \nu^{5} - 17930646704571786099411 \nu^{4} - 122247938831166756829281 \nu^{3} - 500800627337151902170576 \nu^{2} - 1814204302357830528047346 \nu - 2890887473171211341138220$$$$)/$$$$28\!\cdots\!60$$ $$\beta_{17}$$ $$=$$ $$($$$$1807606890121 \nu^{17} - 12687264062385 \nu^{16} + 99424585969077 \nu^{15} - 519472989416822 \nu^{14} + 2272911163990521 \nu^{13} - 8673974267785341 \nu^{12} + 20575975123994056 \nu^{11} - 85306117915969533 \nu^{10} - 208908924981034569 \nu^{9} + 2010316302768900164 \nu^{8} - 972733387961635041 \nu^{7} + 78923554987170922287 \nu^{6} + 675768281049567568562 \nu^{5} + 4447081725194280935973 \nu^{4} + 30646604467595396861907 \nu^{3} + 91141537205666036636012 \nu^{2} + 381974508700326322267302 \nu - 3260417988187224655284$$$$)/$$$$94\!\cdots\!20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} - \beta_{8} - \beta_{2} - 3 \beta_{1} + 5$$$$)/9$$ $$\nu^{2}$$ $$=$$ $$($$$$-6 \beta_{17} - 6 \beta_{16} + 3 \beta_{11} + 3 \beta_{10} + \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} - 6 \beta_{3} + 5 \beta_{2} - 58$$$$)/9$$ $$\nu^{3}$$ $$=$$ $$($$$$-21 \beta_{17} - 30 \beta_{16} + 9 \beta_{15} + 18 \beta_{14} - 18 \beta_{13} + 9 \beta_{12} + 15 \beta_{11} + 15 \beta_{10} + \beta_{9} + 29 \beta_{8} + 9 \beta_{7} - 30 \beta_{6} - 57 \beta_{3} + 59 \beta_{2} + 60 \beta_{1} - 13$$$$)/9$$ $$\nu^{4}$$ $$=$$ $$($$$$-72 \beta_{17} - 180 \beta_{16} + 27 \beta_{15} - 54 \beta_{14} - 108 \beta_{13} + 162 \beta_{12} + 81 \beta_{10} + 85 \beta_{9} + 107 \beta_{8} - 96 \beta_{7} - 252 \beta_{6} + 81 \beta_{5} + 81 \beta_{4} + 9 \beta_{3} - 223 \beta_{2} - 78 \beta_{1} + 1217$$$$)/9$$ $$\nu^{5}$$ $$=$$ $$($$$$-456 \beta_{17} - 240 \beta_{16} + 27 \beta_{15} - 243 \beta_{14} - 162 \beta_{13} + 459 \beta_{12} + 147 \beta_{11} + 66 \beta_{10} + 187 \beta_{9} + 77 \beta_{8} - 3054 \beta_{7} + 93 \beta_{6} + 1134 \beta_{5} + 972 \beta_{4} + 1029 \beta_{3} + 6092 \beta_{2} - 2259 \beta_{1} - 2449$$$$)/9$$ $$\nu^{6}$$ $$=$$ $$($$$$-1587 \beta_{17} + 1428 \beta_{16} - 666 \beta_{15} - 252 \beta_{14} + 576 \beta_{13} - 72 \beta_{12} - 93 \beta_{11} - 741 \beta_{10} + 1828 \beta_{9} + 2513 \beta_{8} - 34371 \beta_{7} - 3378 \beta_{6} + 4455 \beta_{5} + 5184 \beta_{4} + 3561 \beta_{3} + 93551 \beta_{2} - 11460 \beta_{1} - 55219$$$$)/9$$ $$\nu^{7}$$ $$=$$ $$($$$$7461 \beta_{17} + 4302 \beta_{16} - 3159 \beta_{15} + 162 \beta_{14} + 3888 \beta_{13} - 6885 \beta_{12} - 13878 \beta_{11} - 9909 \beta_{10} + 21397 \beta_{9} + 22265 \beta_{8} - 100185 \beta_{7} - 5733 \beta_{6} + 14418 \beta_{5} + 1944 \beta_{4} - 12303 \beta_{3} + 974873 \beta_{2} + 10155 \beta_{1} - 384460$$$$)/9$$ $$\nu^{8}$$ $$=$$ $$($$$$20982 \beta_{17} - 11013 \beta_{16} - 14175 \beta_{15} + 12042 \beta_{14} - 3618 \beta_{13} + 7992 \beta_{12} - 93138 \beta_{11} - 87954 \beta_{10} + 154141 \beta_{9} + 165236 \beta_{8} - 112404 \beta_{7} + 267474 \beta_{6} - 17901 \beta_{5} - 92826 \beta_{4} - 212595 \beta_{3} + 3236345 \beta_{2} - 161748 \beta_{1} - 103510$$$$)/9$$ $$\nu^{9}$$ $$=$$ $$($$$$-439068 \beta_{17} + 71520 \beta_{16} - 309060 \beta_{15} - 5706 \beta_{14} - 73107 \beta_{13} + 278325 \beta_{12} - 489279 \beta_{11} - 938910 \beta_{10} + 25750 \beta_{9} + 473690 \beta_{8} - 347841 \beta_{7} + 1678128 \beta_{6} - 273213 \beta_{5} - 206874 \beta_{4} - 1267815 \beta_{3} + 5198198 \beta_{2} - 2428257 \beta_{1} - 3245269$$$$)/9$$ $$\nu^{10}$$ $$=$$ $$($$$$-3824217 \beta_{17} + 1765701 \beta_{16} - 3699297 \beta_{15} - 903258 \beta_{14} + 91395 \beta_{13} + 1582011 \beta_{12} - 1618569 \beta_{11} - 5368950 \beta_{10} - 4407683 \beta_{9} + 1289294 \beta_{8} - 3132960 \beta_{7} + 3134448 \beta_{6} + 1712502 \beta_{5} - 98010 \beta_{4} - 7645554 \beta_{3} + 4030166 \beta_{2} - 6183375 \beta_{1} - 61187920$$$$)/9$$ $$\nu^{11}$$ $$=$$ $$($$$$-11087304 \beta_{17} + 15749805 \beta_{16} - 20902563 \beta_{15} - 6591834 \beta_{14} + 2960442 \beta_{13} - 2786508 \beta_{12} - 535371 \beta_{11} - 24976554 \beta_{10} - 28066232 \beta_{9} + 10038164 \beta_{8} - 45480612 \beta_{7} - 14331804 \beta_{6} + 28553472 \beta_{5} - 13455639 \beta_{4} - 29070897 \beta_{3} - 38298403 \beta_{2} - 67424283 \beta_{1} - 232243624$$$$)/9$$ $$\nu^{12}$$ $$=$$ $$($$$$14676342 \beta_{17} + 89633085 \beta_{16} - 59481288 \beta_{15} - 44294508 \beta_{14} + 16015383 \beta_{13} - 30119841 \beta_{12} - 20932977 \beta_{11} - 86198160 \beta_{10} - 101222738 \beta_{9} + 41233835 \beta_{8} - 484290846 \beta_{7} - 128002845 \beta_{6} + 150773967 \beta_{5} - 153791541 \beta_{4} + 22779870 \beta_{3} + 533706026 \beta_{2} - 1112502801 \beta_{1} - 2427640285$$$$)/9$$ $$\nu^{13}$$ $$=$$ $$($$$$442774980 \beta_{17} + 605352321 \beta_{16} - 97572249 \beta_{15} - 214361937 \beta_{14} + 194301396 \beta_{13} - 179294634 \beta_{12} - 304087149 \beta_{11} - 335895525 \beta_{10} - 450825008 \beta_{9} + 374121791 \beta_{8} - 2537411847 \beta_{7} - 124945614 \beta_{6} + 448483797 \beta_{5} - 732315654 \beta_{4} + 664496928 \beta_{3} + 12013801076 \beta_{2} - 7849739103 \beta_{1} - 32994122641$$$$)/9$$ $$\nu^{14}$$ $$=$$ $$($$$$5315664396 \beta_{17} + 5932936452 \beta_{16} - 444688353 \beta_{15} - 344211282 \beta_{14} + 1885348467 \beta_{13} - 1845801297 \beta_{12} - 3059341827 \beta_{11} - 3390002700 \beta_{10} - 3255101021 \beta_{9} + 3745302377 \beta_{8} - 4790670267 \beta_{7} + 2706708243 \beta_{6} - 585870165 \beta_{5} - 2905300953 \beta_{4} + 5029695705 \beta_{3} + 75493251251 \beta_{2} - 25398103086 \beta_{1} - 235211094478$$$$)/9$$ $$\nu^{15}$$ $$=$$ $$($$$$46755578436 \beta_{17} + 54143189529 \beta_{16} - 7754425749 \beta_{15} - 3121162803 \beta_{14} + 15791601087 \beta_{13} - 14637076155 \beta_{12} - 31011815370 \beta_{11} - 34996952595 \beta_{10} - 29200518287 \beta_{9} + 9885844775 \beta_{8} + 40695123627 \beta_{7} + 33465305544 \beta_{6} - 16850363013 \beta_{5} - 15679987776 \beta_{4} + 40228344789 \beta_{3} + 185571522149 \beta_{2} - 18773224239 \beta_{1} - 879993183487$$$$)/9$$ $$\nu^{16}$$ $$=$$ $$($$$$307650105540 \beta_{17} + 421537425192 \beta_{16} - 87575030820 \beta_{15} - 42696660708 \beta_{14} + 139680491472 \beta_{13} - 126907103160 \beta_{12} - 207498198906 \beta_{11} - 270332000370 \beta_{10} - 226581201086 \beta_{9} - 127261637902 \beta_{8} + 507516190230 \beta_{7} + 333167406516 \beta_{6} - 137642880906 \beta_{5} - 123360306669 \beta_{4} + 239540087196 \beta_{3} - 1080442733461 \beta_{2} + 136803439137 \beta_{1} - 1342007900404$$$$)/9$$ $$\nu^{17}$$ $$=$$ $$($$$$1584764063562 \beta_{17} + 2562826191528 \beta_{16} - 637015741470 \beta_{15} - 221969986308 \beta_{14} + 1060721820504 \beta_{13} - 1128824636166 \beta_{12} - 835114817544 \beta_{11} - 1520222039718 \beta_{10} - 1521513519203 \beta_{9} - 1586620895899 \beta_{8} + 4091738679078 \beta_{7} + 2112977781948 \beta_{6} - 1113904952676 \beta_{5} - 1215565501095 \beta_{4} + 947794190880 \beta_{3} - 17342726370418 \beta_{2} + 759631317543 \beta_{1} + 2952115076105$$$$)/9$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
85.1
 6.01410 + 1.36338i 4.38317 − 3.94388i 2.54993 + 4.95961i 0.0652281 + 5.11138i −0.0672952 − 5.08536i −0.208655 − 5.05363i −2.16791 + 4.11878i −2.97222 − 3.34985i −4.09635 + 1.01354i 6.01410 − 1.36338i 4.38317 + 3.94388i 2.54993 − 4.95961i 0.0652281 − 5.11138i −0.0672952 + 5.08536i −0.208655 + 5.05363i −2.16791 − 4.11878i −2.97222 + 3.34985i −4.09635 − 1.01354i
0 −5.01410 + 1.36338i 0 −0.333146 + 0.577025i 0 −3.50000 6.06218i 0 23.2824 13.6722i 0
85.2 0 −3.38317 3.94388i 0 −8.26221 + 14.3106i 0 −3.50000 6.06218i 0 −4.10831 + 26.6856i 0
85.3 0 −1.54993 + 4.95961i 0 3.85902 6.68401i 0 −3.50000 6.06218i 0 −22.1955 15.3741i 0
85.4 0 0.934772 + 5.11138i 0 −7.13443 + 12.3572i 0 −3.50000 6.06218i 0 −25.2524 + 9.55595i 0
85.5 0 1.06730 5.08536i 0 1.60891 2.78671i 0 −3.50000 6.06218i 0 −24.7218 10.8552i 0
85.6 0 1.20866 5.05363i 0 7.66838 13.2820i 0 −3.50000 6.06218i 0 −24.0783 12.2162i 0
85.7 0 3.16791 + 4.11878i 0 8.76630 15.1837i 0 −3.50000 6.06218i 0 −6.92866 + 26.0959i 0
85.8 0 3.97222 3.34985i 0 −9.35724 + 16.2072i 0 −3.50000 6.06218i 0 4.55703 26.6127i 0
85.9 0 5.09635 + 1.01354i 0 0.184423 0.319430i 0 −3.50000 6.06218i 0 24.9455 + 10.3307i 0
169.1 0 −5.01410 1.36338i 0 −0.333146 0.577025i 0 −3.50000 + 6.06218i 0 23.2824 + 13.6722i 0
169.2 0 −3.38317 + 3.94388i 0 −8.26221 14.3106i 0 −3.50000 + 6.06218i 0 −4.10831 26.6856i 0
169.3 0 −1.54993 4.95961i 0 3.85902 + 6.68401i 0 −3.50000 + 6.06218i 0 −22.1955 + 15.3741i 0
169.4 0 0.934772 5.11138i 0 −7.13443 12.3572i 0 −3.50000 + 6.06218i 0 −25.2524 9.55595i 0
169.5 0 1.06730 + 5.08536i 0 1.60891 + 2.78671i 0 −3.50000 + 6.06218i 0 −24.7218 + 10.8552i 0
169.6 0 1.20866 + 5.05363i 0 7.66838 + 13.2820i 0 −3.50000 + 6.06218i 0 −24.0783 + 12.2162i 0
169.7 0 3.16791 4.11878i 0 8.76630 + 15.1837i 0 −3.50000 + 6.06218i 0 −6.92866 26.0959i 0
169.8 0 3.97222 + 3.34985i 0 −9.35724 16.2072i 0 −3.50000 + 6.06218i 0 4.55703 + 26.6127i 0
169.9 0 5.09635 1.01354i 0 0.184423 + 0.319430i 0 −3.50000 + 6.06218i 0 24.9455 10.3307i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 169.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.j.b 18
3.b odd 2 1 756.4.j.b 18
9.c even 3 1 inner 252.4.j.b 18
9.c even 3 1 2268.4.a.i 9
9.d odd 6 1 756.4.j.b 18
9.d odd 6 1 2268.4.a.h 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.j.b 18 1.a even 1 1 trivial
252.4.j.b 18 9.c even 3 1 inner
756.4.j.b 18 3.b odd 2 1
756.4.j.b 18 9.d odd 6 1
2268.4.a.h 9 9.d odd 6 1
2268.4.a.i 9 9.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$18\!\cdots\!29$$$$T_{5}^{8} -$$$$10\!\cdots\!08$$$$T_{5}^{7} +$$$$11\!\cdots\!80$$$$T_{5}^{6} -$$$$29\!\cdots\!68$$$$T_{5}^{5} +$$$$81\!\cdots\!25$$$$T_{5}^{4} +$$$$13\!\cdots\!58$$$$T_{5}^{3} +$$$$32\!\cdots\!21$$$$T_{5}^{2} -$$$$83\!\cdots\!98$$$$T_{5} +$$$$52\!\cdots\!96$$">$$T_{5}^{18} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$7625597484987 - 3106724901291 T + 1202940618345 T^{2} - 267320137410 T^{3} + 52473952899 T^{4} - 7064445213 T^{5} + 746143164 T^{6} - 26276805 T^{7} - 12866121 T^{8} + 3004938 T^{9} - 476523 T^{10} - 36045 T^{11} + 37908 T^{12} - 13293 T^{13} + 3657 T^{14} - 690 T^{15} + 115 T^{16} - 11 T^{17} + T^{18}$$
$5$ $$52444912836996 - 83327262188598 T + 321478263979821 T^{2} + 139000635955458 T^{3} + 811474750097325 T^{4} - 296960546362968 T^{5} + 116976039417180 T^{6} - 10572760083708 T^{7} + 1852923139029 T^{8} - 47486544840 T^{9} + 17780999634 T^{10} - 185166486 T^{11} + 94605300 T^{12} + 281502 T^{13} + 352323 T^{14} + 1368 T^{15} + 738 T^{16} + 6 T^{17} + T^{18}$$
$7$ $$( 49 + 7 T + T^{2} )^{9}$$
$11$ $$77\!\cdots\!00$$$$+$$$$34\!\cdots\!40$$$$T +$$$$10\!\cdots\!96$$$$T^{2} +$$$$26\!\cdots\!88$$$$T^{3} +$$$$86\!\cdots\!41$$$$T^{4} - 19063479501353149731 T^{5} + 39837921078165127722 T^{6} - 581485291342630263 T^{7} + 117038019847293144 T^{8} - 423765439794549 T^{9} + 113434509923559 T^{10} + 822053996856 T^{11} + 84577294503 T^{12} + 819025911 T^{13} + 36467685 T^{14} + 443070 T^{15} + 11295 T^{16} + 93 T^{17} + T^{18}$$
$13$ $$18\!\cdots\!16$$$$+$$$$13\!\cdots\!52$$$$T +$$$$46\!\cdots\!12$$$$T^{2} -$$$$11\!\cdots\!16$$$$T^{3} +$$$$70\!\cdots\!60$$$$T^{4} +$$$$67\!\cdots\!74$$$$T^{5} +$$$$15\!\cdots\!21$$$$T^{6} + 40192093077678623934 T^{7} + 2900296938662436159 T^{8} + 64205162926551164 T^{9} + 2041089434001882 T^{10} + 24495151585302 T^{11} + 593820288987 T^{12} + 4655524284 T^{13} + 126995994 T^{14} + 487086 T^{15} + 13059 T^{16} + 18 T^{17} + T^{18}$$
$17$ $$( -109582832291940 + 34269083897688 T - 836515485513 T^{2} - 176770160919 T^{3} + 2669633127 T^{4} + 114370515 T^{5} - 541854 T^{6} - 19674 T^{7} + 27 T^{8} + T^{9} )^{2}$$
$19$ $$( -20220704685815975 + 716445769363245 T + 39282983879373 T^{2} - 1646353799745 T^{3} - 470801511 T^{4} + 396854109 T^{5} - 798270 T^{6} - 34344 T^{7} + 45 T^{8} + T^{9} )^{2}$$
$23$ $$43\!\cdots\!96$$$$+$$$$34\!\cdots\!96$$$$T +$$$$23\!\cdots\!53$$$$T^{2} +$$$$61\!\cdots\!54$$$$T^{3} +$$$$20\!\cdots\!72$$$$T^{4} +$$$$40\!\cdots\!12$$$$T^{5} +$$$$11\!\cdots\!79$$$$T^{6} +$$$$17\!\cdots\!48$$$$T^{7} +$$$$26\!\cdots\!02$$$$T^{8} +$$$$23\!\cdots\!22$$$$T^{9} + 2390586686863371234 T^{10} + 15062096275811658 T^{11} + 137108277880299 T^{12} + 661439414430 T^{13} + 4661765190 T^{14} + 15551010 T^{15} + 100017 T^{16} + 246 T^{17} + T^{18}$$
$29$ $$32\!\cdots\!84$$$$+$$$$19\!\cdots\!64$$$$T +$$$$96\!\cdots\!52$$$$T^{2} +$$$$17\!\cdots\!24$$$$T^{3} +$$$$30\!\cdots\!68$$$$T^{4} +$$$$31\!\cdots\!80$$$$T^{5} +$$$$38\!\cdots\!73$$$$T^{6} +$$$$30\!\cdots\!92$$$$T^{7} +$$$$31\!\cdots\!42$$$$T^{8} +$$$$19\!\cdots\!36$$$$T^{9} + 15738690561753929391 T^{10} + 73062467773557066 T^{11} + 550414206558864 T^{12} + 1977861946158 T^{13} + 12060569223 T^{14} + 30018114 T^{15} + 168003 T^{16} + 318 T^{17} + T^{18}$$
$31$ $$20\!\cdots\!76$$$$+$$$$40\!\cdots\!08$$$$T +$$$$35\!\cdots\!64$$$$T^{2} +$$$$74\!\cdots\!12$$$$T^{3} +$$$$40\!\cdots\!64$$$$T^{4} +$$$$28\!\cdots\!82$$$$T^{5} +$$$$71\!\cdots\!01$$$$T^{6} +$$$$43\!\cdots\!26$$$$T^{7} +$$$$81\!\cdots\!88$$$$T^{8} +$$$$46\!\cdots\!30$$$$T^{9} + 46875106710908871471 T^{10} + 140087072237122548 T^{11} + 1079139911002962 T^{12} + 2024715680964 T^{13} + 17954662077 T^{14} + 17293614 T^{15} + 157887 T^{16} + 18 T^{17} + T^{18}$$
$37$ $$( -$$$$12\!\cdots\!48$$$$+ 3162568796978179212 T + 2473738641105426 T^{2} - 367302778249191 T^{3} + 312442901280 T^{4} + 14778748422 T^{5} - 6106560 T^{6} - 215775 T^{7} + 36 T^{8} + T^{9} )^{2}$$
$41$ $$19\!\cdots\!96$$$$+$$$$77\!\cdots\!36$$$$T +$$$$32\!\cdots\!64$$$$T^{2} +$$$$98\!\cdots\!64$$$$T^{3} +$$$$33\!\cdots\!65$$$$T^{4} +$$$$21\!\cdots\!71$$$$T^{5} +$$$$27\!\cdots\!76$$$$T^{6} -$$$$69\!\cdots\!23$$$$T^{7} +$$$$80\!\cdots\!38$$$$T^{8} +$$$$21\!\cdots\!33$$$$T^{9} +$$$$16\!\cdots\!11$$$$T^{10} + 293925351642190272 T^{11} + 16965162465133689 T^{12} + 6898136018103 T^{13} + 121212833319 T^{14} + 27404892 T^{15} + 414189 T^{16} + 57 T^{17} + T^{18}$$
$43$ $$16\!\cdots\!84$$$$-$$$$91\!\cdots\!24$$$$T +$$$$50\!\cdots\!56$$$$T^{2} -$$$$62\!\cdots\!20$$$$T^{3} +$$$$18\!\cdots\!97$$$$T^{4} -$$$$10\!\cdots\!57$$$$T^{5} +$$$$57\!\cdots\!47$$$$T^{6} +$$$$28\!\cdots\!38$$$$T^{7} +$$$$44\!\cdots\!45$$$$T^{8} -$$$$10\!\cdots\!71$$$$T^{9} +$$$$25\!\cdots\!07$$$$T^{10} - 2910821490327368988 T^{11} + 24876572339675211 T^{12} - 2297422764567 T^{13} + 159013505232 T^{14} - 2222373 T^{15} + 484650 T^{16} + 171 T^{17} + T^{18}$$
$47$ $$20\!\cdots\!00$$$$+$$$$63\!\cdots\!60$$$$T +$$$$13\!\cdots\!16$$$$T^{2} +$$$$18\!\cdots\!08$$$$T^{3} +$$$$19\!\cdots\!24$$$$T^{4} +$$$$16\!\cdots\!76$$$$T^{5} +$$$$12\!\cdots\!57$$$$T^{6} +$$$$75\!\cdots\!00$$$$T^{7} +$$$$44\!\cdots\!16$$$$T^{8} +$$$$22\!\cdots\!32$$$$T^{9} +$$$$10\!\cdots\!05$$$$T^{10} + 4268803747522463136 T^{11} + 16209185204018394 T^{12} + 53188307887266 T^{13} + 160506226917 T^{14} + 388597500 T^{15} + 791037 T^{16} + 1056 T^{17} + T^{18}$$
$53$ $$( -62301759208236367632 +$$$$22\!\cdots\!40$$$$T - 4657590260398827288 T^{2} + 33418789409301885 T^{3} - 83768677280406 T^{4} - 64475218863 T^{5} + 591720030 T^{6} - 530190 T^{7} - 756 T^{8} + T^{9} )^{2}$$
$59$ $$51\!\cdots\!84$$$$-$$$$11\!\cdots\!16$$$$T +$$$$71\!\cdots\!56$$$$T^{2} +$$$$31\!\cdots\!92$$$$T^{3} +$$$$68\!\cdots\!41$$$$T^{4} +$$$$25\!\cdots\!67$$$$T^{5} +$$$$28\!\cdots\!97$$$$T^{6} +$$$$96\!\cdots\!72$$$$T^{7} +$$$$80\!\cdots\!35$$$$T^{8} +$$$$22\!\cdots\!79$$$$T^{9} +$$$$13\!\cdots\!35$$$$T^{10} + 2793313783245825900 T^{11} + 14857318993235877 T^{12} + 25342123646511 T^{13} + 104415352506 T^{14} + 120083769 T^{15} + 447732 T^{16} + 411 T^{17} + T^{18}$$
$61$ $$73\!\cdots\!00$$$$-$$$$36\!\cdots\!20$$$$T +$$$$12\!\cdots\!01$$$$T^{2} -$$$$21\!\cdots\!48$$$$T^{3} +$$$$27\!\cdots\!76$$$$T^{4} -$$$$17\!\cdots\!28$$$$T^{5} +$$$$92\!\cdots\!21$$$$T^{6} -$$$$28\!\cdots\!36$$$$T^{7} +$$$$86\!\cdots\!78$$$$T^{8} -$$$$14\!\cdots\!10$$$$T^{9} +$$$$48\!\cdots\!56$$$$T^{10} - 54020123556156808488 T^{11} + 199392027518913783 T^{12} - 49665914201520 T^{13} + 575464222890 T^{14} - 16829238 T^{15} + 947781 T^{16} + 198 T^{17} + T^{18}$$
$67$ $$49\!\cdots\!44$$$$-$$$$15\!\cdots\!08$$$$T +$$$$82\!\cdots\!40$$$$T^{2} +$$$$11\!\cdots\!68$$$$T^{3} +$$$$17\!\cdots\!13$$$$T^{4} +$$$$44\!\cdots\!37$$$$T^{5} +$$$$15\!\cdots\!54$$$$T^{6} +$$$$20\!\cdots\!01$$$$T^{7} +$$$$59\!\cdots\!84$$$$T^{8} +$$$$59\!\cdots\!03$$$$T^{9} +$$$$15\!\cdots\!47$$$$T^{10} +$$$$10\!\cdots\!28$$$$T^{11} + 2478491353014195003 T^{12} + 1285363077853743 T^{13} + 2964434570613 T^{14} + 947654070 T^{15} + 2146527 T^{16} + 441 T^{17} + T^{18}$$
$71$ $$($$$$35\!\cdots\!40$$$$-$$$$50\!\cdots\!29$$$$T + 203460654955330530 T^{2} + 156923398516730841 T^{3} - 368291298026556 T^{4} - 429865646247 T^{5} + 1620019710 T^{6} - 223272 T^{7} - 1758 T^{8} + T^{9} )^{2}$$
$73$ $$( -$$$$11\!\cdots\!96$$$$+$$$$25\!\cdots\!28$$$$T + 25410297757157940465 T^{2} - 323320820611206057 T^{3} - 75459232145700 T^{4} + 1247418011052 T^{5} + 72296355 T^{6} - 1908261 T^{7} - 27 T^{8} + T^{9} )^{2}$$
$79$ $$62\!\cdots\!00$$$$+$$$$49\!\cdots\!90$$$$T +$$$$41\!\cdots\!69$$$$T^{2} +$$$$12\!\cdots\!16$$$$T^{3} +$$$$57\!\cdots\!37$$$$T^{4} +$$$$92\!\cdots\!02$$$$T^{5} +$$$$49\!\cdots\!94$$$$T^{6} +$$$$46\!\cdots\!96$$$$T^{7} +$$$$27\!\cdots\!27$$$$T^{8} +$$$$34\!\cdots\!50$$$$T^{9} +$$$$10\!\cdots\!74$$$$T^{10} - 69989531243487909354 T^{11} + 2154468238268276346 T^{12} - 561746335946058 T^{13} + 3161959790319 T^{14} - 452362836 T^{15} + 2100060 T^{16} - 72 T^{17} + T^{18}$$
$83$ $$82\!\cdots\!00$$$$-$$$$46\!\cdots\!80$$$$T +$$$$41\!\cdots\!56$$$$T^{2} +$$$$10\!\cdots\!40$$$$T^{3} +$$$$23\!\cdots\!56$$$$T^{4} +$$$$17\!\cdots\!72$$$$T^{5} +$$$$90\!\cdots\!73$$$$T^{6} +$$$$30\!\cdots\!16$$$$T^{7} +$$$$81\!\cdots\!35$$$$T^{8} +$$$$15\!\cdots\!60$$$$T^{9} +$$$$26\!\cdots\!01$$$$T^{10} +$$$$35\!\cdots\!98$$$$T^{11} + 4716204326181416502 T^{12} + 4752602234467776 T^{13} + 5154073228431 T^{14} + 2829975624 T^{15} + 2381256 T^{16} + 558 T^{17} + T^{18}$$
$89$ $$($$$$28\!\cdots\!40$$$$-$$$$21\!\cdots\!36$$$$T - 58849744796899349382 T^{2} + 595604595645477675 T^{3} - 869276690497962 T^{4} - 1125822437505 T^{5} + 3408377562 T^{6} - 1536354 T^{7} - 1392 T^{8} + T^{9} )^{2}$$
$97$ $$82\!\cdots\!00$$$$-$$$$57\!\cdots\!80$$$$T +$$$$49\!\cdots\!44$$$$T^{2} -$$$$86\!\cdots\!80$$$$T^{3} +$$$$59\!\cdots\!25$$$$T^{4} -$$$$39\!\cdots\!23$$$$T^{5} +$$$$56\!\cdots\!53$$$$T^{6} +$$$$14\!\cdots\!24$$$$T^{7} +$$$$24\!\cdots\!87$$$$T^{8} +$$$$73\!\cdots\!79$$$$T^{9} +$$$$78\!\cdots\!05$$$$T^{10} +$$$$25\!\cdots\!04$$$$T^{11} + 10701152650072270815 T^{12} - 659184318836811 T^{13} + 8336210340366 T^{14} - 774335487 T^{15} + 3955824 T^{16} - 909 T^{17} + T^{18}$$