# Properties

 Label 76.3.j.a Level $76$ Weight $3$ Character orbit 76.j Analytic conductor $2.071$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 76.j (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07085000914$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$3$$ over $$\Q(\zeta_{18})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ Defining polynomial: $$x^{18} + 93 x^{16} + 3429 x^{14} + 64261 x^{12} + 647217 x^{10} + 3386277 x^{8} + 8232133 x^{6} + 8319228 x^{4} + 2467872 x^{2} + 69312$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{4} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{12} ) q^{3} -\beta_{17} q^{5} + ( 2 - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{7} + ( 1 - \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{16} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{4} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{12} ) q^{3} -\beta_{17} q^{5} + ( 2 - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{7} + ( 1 - \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{16} ) q^{9} + ( \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{10} + 3 \beta_{11} - \beta_{12} + \beta_{13} ) q^{11} + ( 2 + \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + 2 \beta_{11} - 3 \beta_{12} - \beta_{14} - \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{13} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} + \beta_{15} + 2 \beta_{17} ) q^{15} + ( -4 - 3 \beta_{1} + \beta_{2} + 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{10} - 7 \beta_{11} + 5 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - \beta_{15} - \beta_{17} ) q^{17} + ( 2 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + \beta_{9} - 5 \beta_{10} - 6 \beta_{11} + 4 \beta_{12} + \beta_{13} - \beta_{14} ) q^{19} + ( -4 - \beta_{2} - 2 \beta_{3} + \beta_{7} - 2 \beta_{8} - 4 \beta_{9} - 4 \beta_{11} - 4 \beta_{12} - 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{17} ) q^{21} + ( 5 \beta_{1} - 4 \beta_{2} + 6 \beta_{4} - \beta_{5} + 5 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + 6 \beta_{9} + \beta_{10} + 6 \beta_{11} - 11 \beta_{12} + \beta_{13} + \beta_{17} ) q^{23} + ( -3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - \beta_{5} - \beta_{7} + 3 \beta_{8} + 6 \beta_{10} + 6 \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{17} ) q^{25} + ( -18 + 2 \beta_{2} + \beta_{3} + 9 \beta_{4} + 4 \beta_{6} - 4 \beta_{7} - 5 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} - 7 \beta_{12} - \beta_{13} - 2 \beta_{15} - 2 \beta_{16} ) q^{27} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + 6 \beta_{8} - 2 \beta_{11} - 4 \beta_{12} + 3 \beta_{13} + 3 \beta_{15} + \beta_{17} ) q^{29} + ( -6 + 5 \beta_{1} - 6 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + 8 \beta_{9} - 9 \beta_{10} + 4 \beta_{11} + 6 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} ) q^{31} + ( -12 - 2 \beta_{2} - \beta_{3} + 13 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 13 \beta_{9} + 14 \beta_{10} - \beta_{11} - \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + 3 \beta_{16} ) q^{33} + ( 16 - \beta_{2} - \beta_{3} - 22 \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{7} + 4 \beta_{8} + 16 \beta_{9} + 7 \beta_{10} - 10 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - 2 \beta_{16} ) q^{35} + ( -8 - 6 \beta_{1} + 6 \beta_{2} + 16 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} - 5 \beta_{8} - 14 \beta_{9} - 2 \beta_{10} - 14 \beta_{11} - 2 \beta_{12} - \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{37} + ( 2 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 7 \beta_{5} - 7 \beta_{6} - \beta_{7} - \beta_{8} + 6 \beta_{9} - 23 \beta_{10} - 6 \beta_{11} + 23 \beta_{12} - 3 \beta_{14} + 3 \beta_{15} + \beta_{16} - \beta_{17} ) q^{39} + ( 7 + 3 \beta_{1} - 10 \beta_{2} - 5 \beta_{4} + 7 \beta_{5} + \beta_{6} - 10 \beta_{7} + \beta_{8} - 7 \beta_{9} - 5 \beta_{10} - 8 \beta_{11} + 3 \beta_{12} - 2 \beta_{15} - 4 \beta_{17} ) q^{41} + ( -8 + \beta_{1} - \beta_{3} + 16 \beta_{4} - 6 \beta_{6} - \beta_{7} - 12 \beta_{8} + 16 \beta_{9} - 8 \beta_{10} - 7 \beta_{11} + 6 \beta_{12} + \beta_{14} + 2 \beta_{17} ) q^{43} + ( 32 - 4 \beta_{1} + 8 \beta_{2} - 32 \beta_{4} - 5 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} - \beta_{8} - 10 \beta_{9} + 11 \beta_{10} - 4 \beta_{11} + 16 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{45} + ( 20 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 8 \beta_{9} + 14 \beta_{10} + 20 \beta_{11} - 2 \beta_{17} ) q^{47} + ( 6 \beta_{1} - 3 \beta_{2} - 11 \beta_{4} - 6 \beta_{5} - 10 \beta_{6} - 4 \beta_{7} - 6 \beta_{8} + 12 \beta_{9} - 34 \beta_{10} + 40 \beta_{11} - 18 \beta_{12} + \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{49} + ( 23 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 5 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + \beta_{10} + 18 \beta_{11} - 23 \beta_{12} - 2 \beta_{13} + \beta_{14} + 3 \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{51} + ( 4 - \beta_{1} - 5 \beta_{2} + 5 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + \beta_{6} - 5 \beta_{8} - 14 \beta_{9} + 26 \beta_{10} - 4 \beta_{11} + 7 \beta_{12} + 6 \beta_{13} - 2 \beta_{14} - 5 \beta_{15} - \beta_{16} - 6 \beta_{17} ) q^{53} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 4 \beta_{9} - 7 \beta_{10} - 34 \beta_{11} + 40 \beta_{12} - 6 \beta_{13} + 6 \beta_{14} + 2 \beta_{15} + 3 \beta_{16} + 2 \beta_{17} ) q^{55} + ( 24 + 2 \beta_{1} + 9 \beta_{2} - 7 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - \beta_{8} + 8 \beta_{9} - 15 \beta_{10} - 21 \beta_{11} + 6 \beta_{12} - 3 \beta_{13} + 4 \beta_{14} + 2 \beta_{15} + 3 \beta_{16} + 3 \beta_{17} ) q^{57} + ( -9 - 4 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} - 9 \beta_{4} - 4 \beta_{5} + 12 \beta_{7} + 5 \beta_{8} - 18 \beta_{9} + 13 \beta_{10} - 22 \beta_{11} - 22 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 6 \beta_{15} + 8 \beta_{16} - 6 \beta_{17} ) q^{59} + ( -4 + \beta_{2} + \beta_{3} + 10 \beta_{4} - 6 \beta_{5} + 12 \beta_{7} - \beta_{8} + 16 \beta_{9} + 6 \beta_{10} + 10 \beta_{11} - 6 \beta_{12} - 3 \beta_{13} + \beta_{15} + \beta_{16} - 3 \beta_{17} ) q^{61} + ( -6 - \beta_{1} + 2 \beta_{3} - 20 \beta_{4} - 5 \beta_{5} + \beta_{6} - 5 \beta_{7} + \beta_{8} + 7 \beta_{10} + 26 \beta_{11} - 7 \beta_{12} + 5 \beta_{14} - 2 \beta_{15} - 2 \beta_{17} ) q^{63} + ( -48 + \beta_{2} - \beta_{3} + 24 \beta_{4} - \beta_{5} + 6 \beta_{6} - 5 \beta_{7} + \beta_{8} - 16 \beta_{9} + 33 \beta_{10} + 32 \beta_{11} - 22 \beta_{12} + \beta_{13} + 5 \beta_{15} + 5 \beta_{16} ) q^{65} + ( 1 - 3 \beta_{1} + \beta_{2} - 7 \beta_{3} - 15 \beta_{4} - \beta_{6} - 7 \beta_{7} + 2 \beta_{8} - 12 \beta_{9} - 12 \beta_{10} - \beta_{11} + 16 \beta_{12} - 7 \beta_{13} + 3 \beta_{14} - 7 \beta_{15} - 3 \beta_{16} ) q^{67} + ( -12 + 6 \beta_{1} + 4 \beta_{3} - 12 \beta_{4} + 10 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 7 \beta_{8} + 56 \beta_{9} - 54 \beta_{10} - 12 \beta_{11} + 59 \beta_{12} + 2 \beta_{13} - 5 \beta_{14} + 4 \beta_{15} - 4 \beta_{16} - \beta_{17} ) q^{69} + ( -38 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 46 \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 46 \beta_{9} + 37 \beta_{10} - 4 \beta_{11} + 5 \beta_{12} + 7 \beta_{13} - 8 \beta_{14} - 7 \beta_{16} ) q^{71} + ( 11 + 8 \beta_{1} - 5 \beta_{2} + \beta_{3} - 34 \beta_{4} + 3 \beta_{5} + \beta_{6} + 5 \beta_{7} - \beta_{8} + 11 \beta_{9} + 20 \beta_{10} - 45 \beta_{12} - 5 \beta_{13} + \beta_{14} - \beta_{15} + 6 \beta_{16} ) q^{73} + ( -16 - 6 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 32 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 5 \beta_{9} - 34 \beta_{10} - 5 \beta_{11} - 34 \beta_{12} + 4 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} ) q^{75} + ( 6 - 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 20 \beta_{9} - 5 \beta_{10} - 20 \beta_{11} + 5 \beta_{12} + 5 \beta_{14} - 5 \beta_{15} - 7 \beta_{16} + 7 \beta_{17} ) q^{77} + ( 34 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 28 \beta_{4} - \beta_{5} + 5 \beta_{6} - 2 \beta_{7} + 5 \beta_{8} - 34 \beta_{9} + 7 \beta_{10} - 4 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 5 \beta_{15} + 10 \beta_{17} ) q^{79} + ( -17 + 11 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 51 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 11 \beta_{7} + 8 \beta_{8} + 51 \beta_{9} - 20 \beta_{10} - 11 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{16} + 5 \beta_{17} ) q^{81} + ( 24 - 24 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} + \beta_{8} - 29 \beta_{9} + 33 \beta_{10} + 8 \beta_{11} + 24 \beta_{12} + 12 \beta_{13} - 11 \beta_{14} - 8 \beta_{15} - 8 \beta_{16} + 3 \beta_{17} ) q^{83} + ( 22 - 8 \beta_{1} + 8 \beta_{2} + 9 \beta_{3} + 32 \beta_{4} + 2 \beta_{5} + 8 \beta_{6} - \beta_{7} - 10 \beta_{9} + 8 \beta_{10} + 22 \beta_{11} + 24 \beta_{12} + 4 \beta_{13} - 7 \beta_{14} - 4 \beta_{15} - 7 \beta_{16} + 5 \beta_{17} ) q^{85} + ( -8 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} + 8 \beta_{4} + 5 \beta_{5} + 12 \beta_{6} + 7 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} - 39 \beta_{10} + 34 \beta_{11} - 42 \beta_{12} - 5 \beta_{13} - 5 \beta_{14} - 8 \beta_{15} + 3 \beta_{16} - 5 \beta_{17} ) q^{87} + ( 54 + 10 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} - 30 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + 10 \beta_{8} + 26 \beta_{9} - 14 \beta_{10} + 24 \beta_{11} - 52 \beta_{12} + 5 \beta_{13} + 6 \beta_{14} + \beta_{15} + 12 \beta_{16} + 6 \beta_{17} ) q^{89} + ( -6 - 3 \beta_{1} + 9 \beta_{2} - 8 \beta_{3} + 40 \beta_{4} + 9 \beta_{5} + 3 \beta_{6} + 9 \beta_{8} - 32 \beta_{9} + 55 \beta_{10} - 40 \beta_{11} + 31 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} + 8 \beta_{15} + \beta_{16} + 4 \beta_{17} ) q^{91} + ( -30 + \beta_{1} + \beta_{2} + 6 \beta_{3} + 12 \beta_{4} - \beta_{5} + 10 \beta_{6} + 2 \beta_{7} + 5 \beta_{8} + 18 \beta_{9} + 13 \beta_{10} - 54 \beta_{11} + 44 \beta_{12} + \beta_{13} - \beta_{14} + 5 \beta_{15} - 6 \beta_{16} + 5 \beta_{17} ) q^{93} + ( -10 - 2 \beta_{1} - 10 \beta_{2} - 4 \beta_{3} + 20 \beta_{4} + 11 \beta_{5} + 8 \beta_{6} - 11 \beta_{7} + 2 \beta_{8} + 16 \beta_{9} - 37 \beta_{10} - 76 \beta_{11} + 6 \beta_{12} - \beta_{14} - 4 \beta_{15} - 8 \beta_{16} - 5 \beta_{17} ) q^{95} + ( -60 + 7 \beta_{1} + 10 \beta_{2} - 7 \beta_{3} + 3 \beta_{4} + 7 \beta_{5} - 17 \beta_{7} - 57 \beta_{9} - 10 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} - 8 \beta_{13} - 8 \beta_{14} - 2 \beta_{15} - 7 \beta_{16} + 2 \beta_{17} ) q^{97} + ( -24 + 3 \beta_{1} - 9 \beta_{2} - \beta_{3} + 21 \beta_{4} + 7 \beta_{5} + 3 \beta_{6} - 14 \beta_{7} + 9 \beta_{8} + 45 \beta_{9} - 7 \beta_{10} + 21 \beta_{11} + 2 \beta_{13} - \beta_{15} - 3 \beta_{16} + 2 \beta_{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q - 6q^{3} + 9q^{7} + 6q^{9} + O(q^{10})$$ $$18q - 6q^{3} + 9q^{7} + 6q^{9} - 15q^{11} + 51q^{13} + 21q^{15} - 45q^{17} + 30q^{19} - 63q^{21} + 48q^{23} - 54q^{25} - 198q^{27} - 39q^{29} - 108q^{31} - 105q^{33} + 51q^{35} + 48q^{39} + 54q^{41} + 75q^{43} + 288q^{45} + 339q^{47} - 24q^{49} + 360q^{51} + 69q^{53} - 51q^{55} + 510q^{57} - 483q^{59} - 36q^{61} - 267q^{63} - 585q^{65} - 87q^{67} - 351q^{69} - 234q^{71} - 132q^{73} + 108q^{77} + 363q^{79} + 258q^{81} + 279q^{83} + 666q^{85} + 600q^{89} + 270q^{91} - 456q^{93} - 39q^{95} - 801q^{97} - 267q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} + 93 x^{16} + 3429 x^{14} + 64261 x^{12} + 647217 x^{10} + 3386277 x^{8} + 8232133 x^{6} + 8319228 x^{4} + 2467872 x^{2} + 69312$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$209415 \nu^{16} + 21385203 \nu^{14} + 856101563 \nu^{12} + 16988391675 \nu^{10} + 173289673767 \nu^{8} + 838953142315 \nu^{6} + 1492988376027 \nu^{4} + 717644768940 \nu^{2} - 87012228696 \nu - 5641264464$$$$)/ 174024457392$$ $$\beta_{2}$$ $$=$$ $$($$$$209415 \nu^{16} + 21385203 \nu^{14} + 856101563 \nu^{12} + 16988391675 \nu^{10} + 173289673767 \nu^{8} + 838953142315 \nu^{6} + 1492988376027 \nu^{4} + 717644768940 \nu^{2} + 87012228696 \nu - 5641264464$$$$)/ 174024457392$$ $$\beta_{3}$$ $$=$$ $$($$$$103021303575977 \nu^{16} + 9142933787921503 \nu^{14} + 315299474772470975 \nu^{12} + 5341210410044580379 \nu^{10} + 45319436032825833851 \nu^{8} + 165108014281179112579 \nu^{6} + 101006975635179024567 \nu^{4} - 177533443631457549678 \nu^{2} + 56236356811614666648$$$$)/ 18595918563334754748$$ $$\beta_{4}$$ $$=$$ $$($$$$6185597 \nu^{17} + 591176061 \nu^{15} + 22835687541 \nu^{13} + 462556367605 \nu^{11} + 5294541300849 \nu^{9} + 34116160058661 \nu^{7} + 114681096004341 \nu^{5} + 164926508337168 \nu^{3} + 69806264079024 \nu + 6612929380896$$$$)/ 13225858761792$$ $$\beta_{5}$$ $$=$$ $$($$$$21307902360226751 \nu^{17} + 12923880049145062 \nu^{16} + 2029864741038824639 \nu^{15} + 1187806470401576590 \nu^{14} + 77156143119142574903 \nu^{13} + 43740144259416552670 \nu^{12} + 1503744656663588632375 \nu^{11} + 837441313983081944030 \nu^{10} + 15977086148660713783531 \nu^{9} + 8964501357540755862278 \nu^{8} + 90445956981209747924647 \nu^{7} + 52888979654682545521790 \nu^{6} + 247688622814187776629015 \nu^{5} + 153807669299209335425982 \nu^{4} + 276951487824653832085728 \nu^{3} + 163584759074688739803336 \nu^{2} + 85994899137728002418592 \nu + 18986345814851468476224$$$$)/$$$$16\!\cdots\!76$$ $$\beta_{6}$$ $$=$$ $$($$$$21307902360226751 \nu^{17} - 12923880049145062 \nu^{16} + 2029864741038824639 \nu^{15} - 1187806470401576590 \nu^{14} + 77156143119142574903 \nu^{13} - 43740144259416552670 \nu^{12} + 1503744656663588632375 \nu^{11} - 837441313983081944030 \nu^{10} + 15977086148660713783531 \nu^{9} - 8964501357540755862278 \nu^{8} + 90445956981209747924647 \nu^{7} - 52888979654682545521790 \nu^{6} + 247688622814187776629015 \nu^{5} - 153807669299209335425982 \nu^{4} + 276951487824653832085728 \nu^{3} - 163584759074688739803336 \nu^{2} + 85994899137728002418592 \nu - 18986345814851468476224$$$$)/$$$$16\!\cdots\!76$$ $$\beta_{7}$$ $$=$$ $$($$$$-356420916893601 \nu^{17} - 84585395212837 \nu^{16} - 32413949123345859 \nu^{15} - 5975400590311957 \nu^{14} - 1160708994558830091 \nu^{13} - 141560882685702949 \nu^{12} - 20900421407663018667 \nu^{11} - 1058533984963821029 \nu^{10} - 198011810669232279327 \nu^{9} + 6444753337028719351 \nu^{8} - 926382620163067424475 \nu^{7} + 128923463614356506923 \nu^{6} - 1740816180557103044331 \nu^{5} + 541373962530316733163 \nu^{4} - 966396343240290658194 \nu^{3} + 557831701929060994512 \nu^{2} + 10192478574256957800 \nu + 55795156588540909344$$$$)/$$$$22\!\cdots\!76$$ $$\beta_{8}$$ $$=$$ $$($$$$356420916893601 \nu^{17} - 84585395212837 \nu^{16} + 32413949123345859 \nu^{15} - 5975400590311957 \nu^{14} + 1160708994558830091 \nu^{13} - 141560882685702949 \nu^{12} + 20900421407663018667 \nu^{11} - 1058533984963821029 \nu^{10} + 198011810669232279327 \nu^{9} + 6444753337028719351 \nu^{8} + 926382620163067424475 \nu^{7} + 128923463614356506923 \nu^{6} + 1740816180557103044331 \nu^{5} + 541373962530316733163 \nu^{4} + 966396343240290658194 \nu^{3} + 557831701929060994512 \nu^{2} - 10192478574256957800 \nu + 55795156588540909344$$$$)/$$$$22\!\cdots\!76$$ $$\beta_{9}$$ $$=$$ $$($$$$-30578865294080117 \nu^{17} - 9292132113830094 \nu^{16} - 2848706449870389401 \nu^{15} - 817494977563699038 \nu^{14} - 105313087778809730849 \nu^{13} - 27767346851853045678 \nu^{12} - 1981734855138083887105 \nu^{11} - 456696393282690219726 \nu^{10} - 20085455778755268339325 \nu^{9} - 3613930626675568410102 \nu^{8} - 106018973858087065310881 \nu^{7} - 10213969077236531364270 \nu^{6} - 259031416490425683173601 \nu^{5} + 12885382854331920928818 \nu^{4} - 245772295640201960461668 \nu^{3} + 54906988656881342162448 \nu^{2} - 40233077119486778882880 \nu + 17641491875231716199520$$$$)/$$$$16\!\cdots\!76$$ $$\beta_{10}$$ $$=$$ $$($$$$30600034474056845 \nu^{17} + 17795857570083582 \nu^{16} + 2847359718602523677 \nu^{15} + 1645965155810586246 \nu^{14} + 104923489970995620581 \nu^{13} + 60446536734618041238 \nu^{12} + 1960441049946278852101 \nu^{11} + 1131735633699699198966 \nu^{10} + 19591016775336282193633 \nu^{9} + 11434966984186084818750 \nu^{8} + 100659926058446279288917 \nu^{7} + 60191110055156592895830 \nu^{6} + 234803239959855855700197 \nu^{5} + 145187412576671752297974 \nu^{4} + 222044499167772489923280 \nu^{3} + 128353110743143432185192 \nu^{2} + 68353407262496286219072 \nu + 16866863503588187406720$$$$)/$$$$16\!\cdots\!76$$ $$\beta_{11}$$ $$=$$ $$($$$$-30578865294080117 \nu^{17} + 9292132113830094 \nu^{16} - 2848706449870389401 \nu^{15} + 817494977563699038 \nu^{14} - 105313087778809730849 \nu^{13} + 27767346851853045678 \nu^{12} - 1981734855138083887105 \nu^{11} + 456696393282690219726 \nu^{10} - 20085455778755268339325 \nu^{9} + 3613930626675568410102 \nu^{8} - 106018973858087065310881 \nu^{7} + 10213969077236531364270 \nu^{6} - 259031416490425683173601 \nu^{5} - 12885382854331920928818 \nu^{4} - 245772295640201960461668 \nu^{3} - 54906988656881342162448 \nu^{2} - 40233077119486778882880 \nu - 17641491875231716199520$$$$)/$$$$16\!\cdots\!76$$ $$\beta_{12}$$ $$=$$ $$($$$$30600034474056845 \nu^{17} - 17795857570083582 \nu^{16} + 2847359718602523677 \nu^{15} - 1645965155810586246 \nu^{14} + 104923489970995620581 \nu^{13} - 60446536734618041238 \nu^{12} + 1960441049946278852101 \nu^{11} - 1131735633699699198966 \nu^{10} + 19591016775336282193633 \nu^{9} - 11434966984186084818750 \nu^{8} + 100659926058446279288917 \nu^{7} - 60191110055156592895830 \nu^{6} + 234803239959855855700197 \nu^{5} - 145187412576671752297974 \nu^{4} + 222044499167772489923280 \nu^{3} - 128353110743143432185192 \nu^{2} + 68353407262496286219072 \nu - 16866863503588187406720$$$$)/$$$$16\!\cdots\!76$$ $$\beta_{13}$$ $$=$$ $$($$$$-2992452061703166 \nu^{17} - 1957404767943563 \nu^{16} - 270240639017877677 \nu^{15} - 173715741970508557 \nu^{14} - 9555418270538650385 \nu^{13} - 5990690020676948525 \nu^{12} - 168102984777232484709 \nu^{11} - 101482997790847027201 \nu^{10} - 1520496906844321281001 \nu^{9} - 861069284623690843169 \nu^{8} - 6345815016952600651489 \nu^{7} - 3137052271342403139001 \nu^{6} - 7249653970449891821421 \nu^{5} - 1919132537068401466773 \nu^{4} + 9057175479137712478173 \nu^{3} + 3373135428997693443882 \nu^{2} + 11804400309444777341088 \nu - 1068490779420678666312$$$$)/$$$$70\!\cdots\!24$$ $$\beta_{14}$$ $$=$$ $$($$$$-40207947127666487 \nu^{17} - 47898861085310355 \nu^{16} - 3683209991497786079 \nu^{15} - 4418432327712763071 \nu^{14} - 132892631528273117975 \nu^{13} - 160742476080266035815 \nu^{12} - 2411226129183011287999 \nu^{11} - 2945351338600272256287 \nu^{10} - 23049392895410291223331 \nu^{9} - 28513867486948769040867 \nu^{8} - 109667478341841075751015 \nu^{7} - 138252907082485063877031 \nu^{6} - 218913924589800259930839 \nu^{5} - 284524779447566533302111 \nu^{4} - 167854559979994870942272 \nu^{3} - 208178000999139775297860 \nu^{2} - 56316967455250952729832 \nu - 17046979348526320544784$$$$)/$$$$42\!\cdots\!44$$ $$\beta_{15}$$ $$=$$ $$($$$$-40207947127666487 \nu^{17} + 47898861085310355 \nu^{16} - 3683209991497786079 \nu^{15} + 4418432327712763071 \nu^{14} - 132892631528273117975 \nu^{13} + 160742476080266035815 \nu^{12} - 2411226129183011287999 \nu^{11} + 2945351338600272256287 \nu^{10} - 23049392895410291223331 \nu^{9} + 28513867486948769040867 \nu^{8} - 109667478341841075751015 \nu^{7} + 138252907082485063877031 \nu^{6} - 218913924589800259930839 \nu^{5} + 284524779447566533302111 \nu^{4} - 167854559979994870942272 \nu^{3} + 208178000999139775297860 \nu^{2} - 56316967455250952729832 \nu + 17046979348526320544784$$$$)/$$$$42\!\cdots\!44$$ $$\beta_{16}$$ $$=$$ $$($$$$-89899552734606139 \nu^{17} - 1576548808318068 \nu^{16} - 8362310334460095151 \nu^{15} - 102976667688246492 \nu^{14} - 308342462440461549319 \nu^{13} - 1734324174272048796 \nu^{12} - 5776125910255860834167 \nu^{11} + 17795603693081960484 \nu^{10} - 58075520974252197437579 \nu^{9} + 888360430575354176388 \nu^{8} - 302078464085178974875847 \nu^{7} + 10705277145937685961348 \nu^{6} - 717975899925076248816807 \nu^{5} + 56241196451553784914612 \nu^{4} - 651798809527250310940500 \nu^{3} + 117906430648705249114296 \nu^{2} - 74264982370409236974576 \nu + 30151239574730085354240$$$$)/$$$$84\!\cdots\!88$$ $$\beta_{17}$$ $$=$$ $$($$$$-89899552734606139 \nu^{17} + 1576548808318068 \nu^{16} - 8362310334460095151 \nu^{15} + 102976667688246492 \nu^{14} - 308342462440461549319 \nu^{13} + 1734324174272048796 \nu^{12} - 5776125910255860834167 \nu^{11} - 17795603693081960484 \nu^{10} - 58075520974252197437579 \nu^{9} - 888360430575354176388 \nu^{8} - 302078464085178974875847 \nu^{7} - 10705277145937685961348 \nu^{6} - 717975899925076248816807 \nu^{5} - 56241196451553784914612 \nu^{4} - 651798809527250310940500 \nu^{3} - 117906430648705249114296 \nu^{2} - 74264982370409236974576 \nu - 30151239574730085354240$$$$)/$$$$84\!\cdots\!88$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{2} - \beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{12} - 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - \beta_{1} - 8$$ $$\nu^{3}$$ $$=$$ $$\beta_{17} + \beta_{16} + \beta_{15} + \beta_{14} - 2 \beta_{13} + 13 \beta_{12} + 13 \beta_{10} - 2 \beta_{8} + 2 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} + 8 \beta_{4} - \beta_{3} - 19 \beta_{2} + 19 \beta_{1} - 4$$ $$\nu^{4}$$ $$=$$ $$\beta_{17} - \beta_{16} - 2 \beta_{15} + 2 \beta_{14} + 18 \beta_{12} + 22 \beta_{11} - 18 \beta_{10} - 22 \beta_{9} - 58 \beta_{8} - 58 \beta_{7} - 26 \beta_{6} + 26 \beta_{5} - 18 \beta_{3} + 31 \beta_{2} + 31 \beta_{1} + 132$$ $$\nu^{5}$$ $$=$$ $$-29 \beta_{17} - 29 \beta_{16} - 38 \beta_{15} - 38 \beta_{14} + 60 \beta_{13} - 459 \beta_{12} + 40 \beta_{11} - 459 \beta_{10} + 40 \beta_{9} + 85 \beta_{8} - 85 \beta_{7} + 225 \beta_{6} + 225 \beta_{5} - 252 \beta_{4} + 30 \beta_{3} + 426 \beta_{2} - 426 \beta_{1} + 126$$ $$\nu^{6}$$ $$=$$ $$-89 \beta_{17} + 89 \beta_{16} + 77 \beta_{15} - 77 \beta_{14} - 296 \beta_{12} - 20 \beta_{11} + 296 \beta_{10} + 20 \beta_{9} + 1680 \beta_{8} + 1680 \beta_{7} + 618 \beta_{6} - 618 \beta_{5} + 379 \beta_{3} - 910 \beta_{2} - 910 \beta_{1} - 2544$$ $$\nu^{7}$$ $$=$$ $$896 \beta_{17} + 896 \beta_{16} + 1224 \beta_{15} + 1224 \beta_{14} - 1642 \beta_{13} + 14421 \beta_{12} - 2380 \beta_{11} + 14421 \beta_{10} - 2380 \beta_{9} - 2827 \beta_{8} + 2827 \beta_{7} - 7835 \beta_{6} - 7835 \beta_{5} + 7176 \beta_{4} - 821 \beta_{3} - 10574 \beta_{2} + 10574 \beta_{1} - 3588$$ $$\nu^{8}$$ $$=$$ $$3694 \beta_{17} - 3694 \beta_{16} - 2424 \beta_{15} + 2424 \beta_{14} + 4566 \beta_{12} - 9828 \beta_{11} - 4566 \beta_{10} + 9828 \beta_{9} - 49104 \beta_{8} - 49104 \beta_{7} - 14868 \beta_{6} + 14868 \beta_{5} - 9022 \beta_{3} + 26527 \beta_{2} + 26527 \beta_{1} + 54666$$ $$\nu^{9}$$ $$=$$ $$-28118 \beta_{17} - 28118 \beta_{16} - 37658 \beta_{15} - 37658 \beta_{14} + 45666 \beta_{13} - 438743 \beta_{12} + 97194 \beta_{11} - 438743 \beta_{10} + 97194 \beta_{9} + 88198 \beta_{8} - 88198 \beta_{7} + 250771 \beta_{6} + 250771 \beta_{5} - 203544 \beta_{4} + 22833 \beta_{3} + 281249 \beta_{2} - 281249 \beta_{1} + 101772$$ $$\nu^{10}$$ $$=$$ $$-126447 \beta_{17} + 126447 \beta_{16} + 73373 \beta_{15} - 73373 \beta_{14} - 56837 \beta_{12} + 459788 \beta_{11} + 56837 \beta_{10} - 459788 \beta_{9} + 1443824 \beta_{8} + 1443824 \beta_{7} + 372529 \beta_{6} - 372529 \beta_{5} + 234051 \beta_{3} - 776255 \beta_{2} - 776255 \beta_{1} - 1291752$$ $$\nu^{11}$$ $$=$$ $$871475 \beta_{17} + 871475 \beta_{16} + 1136400 \beta_{15} + 1136400 \beta_{14} - 1299616 \beta_{13} + 13174994 \beta_{12} - 3409206 \beta_{11} + 13174994 \beta_{10} - 3409206 \beta_{9} - 2690432 \beta_{8} + 2690432 \beta_{7} - 7739754 \beta_{6} - 7739754 \beta_{5} + 5845504 \beta_{4} - 649808 \beta_{3} - 7820631 \beta_{2} + 7820631 \beta_{1} - 2922752$$ $$\nu^{12}$$ $$=$$ $$4014631 \beta_{17} - 4014631 \beta_{16} - 2203840 \beta_{15} + 2203840 \beta_{14} + 174619 \beta_{12} - 16329884 \beta_{11} - 174619 \beta_{10} + 16329884 \beta_{9} - 42611265 \beta_{8} - 42611265 \beta_{7} - 9769965 \beta_{6} + 9769965 \beta_{5} - 6419306 \beta_{3} + 22824200 \beta_{2} + 22824200 \beta_{1} + 32971146$$ $$\nu^{13}$$ $$=$$ $$-26622829 \beta_{17} - 26622829 \beta_{16} - 33988119 \beta_{15} - 33988119 \beta_{14} + 37619138 \beta_{13} - 393303354 \beta_{12} + 110991596 \beta_{11} - 393303354 \beta_{10} + 110991596 \beta_{9} + 81225494 \beta_{8} - 81225494 \beta_{7} + 234668188 \beta_{6} + 234668188 \beta_{5} - 169868256 \beta_{4} + 18809569 \beta_{3} + 223460740 \beta_{2} - 223460740 \beta_{1} + 84934128$$ $$\nu^{14}$$ $$=$$ $$-123188846 \beta_{17} + 123188846 \beta_{16} + 66046944 \beta_{15} - 66046944 \beta_{14} + 23991495 \beta_{12} + 527304288 \beta_{11} - 23991495 \beta_{10} - 527304288 \beta_{9} + 1260497793 \beta_{8} + 1260497793 \beta_{7} + 266577351 \beta_{6} - 266577351 \beta_{5} + 182107331 \beta_{3} - 673513940 \beta_{2} - 673513940 \beta_{1} - 890242260$$ $$\nu^{15}$$ $$=$$ $$804684652 \beta_{17} + 804684652 \beta_{16} + 1012343518 \beta_{15} + 1012343518 \beta_{14} - 1100650188 \beta_{13} + 11708483164 \beta_{12} - 3472747716 \beta_{11} + 11708483164 \beta_{10} - 3472747716 \beta_{9} - 2437831736 \beta_{8} + 2437831736 \beta_{7} - 7049083706 \beta_{6} - 7049083706 \beta_{5} + 4977288708 \beta_{4} - 550325094 \beta_{3} - 6488784751 \beta_{2} + 6488784751 \beta_{1} - 2488644354$$ $$\nu^{16}$$ $$=$$ $$3718260648 \beta_{17} - 3718260648 \beta_{16} - 1975813312 \beta_{15} + 1975813312 \beta_{14} - 1270453589 \beta_{12} - 16326763294 \beta_{11} + 1270453589 \beta_{10} + 16326763294 \beta_{9} - 37340903662 \beta_{8} - 37340903662 \beta_{7} - 7493728439 \beta_{6} + 7493728439 \beta_{5} - 5268782367 \beta_{3} + 19921386799 \beta_{2} + 19921386799 \beta_{1} + 24955550316$$ $$\nu^{17}$$ $$=$$ $$-24153101263 \beta_{17} - 24153101263 \beta_{16} - 30096307983 \beta_{15} - 30096307983 \beta_{14} + 32406252302 \beta_{13} - 348091131067 \beta_{12} + 106278734832 \beta_{11} - 348091131067 \beta_{10} + 106278734832 \beta_{9} + 72892657834 \beta_{8} - 72892657834 \beta_{7} + 210669512067 \beta_{6} + 210669512067 \beta_{5} - 146602278296 \beta_{4} + 16203126151 \beta_{3} + 190197764553 \beta_{2} - 190197764553 \beta_{1} + 73301139148$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$\beta_{9}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 3.09175i 0.656794i − 4.19502i − 1.57014i − 0.176873i 5.44868i 1.57014i 0.176873i − 5.44868i 3.84460i 1.29756i − 4.09415i − 3.09175i − 0.656794i 4.19502i − 3.84460i − 1.29756i 4.09415i
0 −0.617749 + 1.69725i 0 0.757040 + 4.29338i 0 −2.41011 + 4.17443i 0 4.39535 + 3.68814i 0
13.2 0 0.215056 0.590861i 0 −1.30596 7.40644i 0 3.03221 5.25195i 0 6.59153 + 5.53095i 0
13.3 0 1.87447 5.15008i 0 0.722564 + 4.09786i 0 2.33853 4.05046i 0 −16.1152 13.5223i 0
21.1 0 −2.27531 + 2.71161i 0 −5.68966 + 2.07087i 0 −6.17078 10.6881i 0 −0.612961 3.47627i 0
21.2 0 −1.37974 + 1.64431i 0 3.12714 1.13819i 0 5.96525 + 10.3321i 0 0.762765 + 4.32585i 0
21.3 0 2.23630 2.66512i 0 1.62283 0.590661i 0 −1.87959 3.25555i 0 −0.538989 3.05676i 0
29.1 0 −2.27531 2.71161i 0 −5.68966 2.07087i 0 −6.17078 + 10.6881i 0 −0.612961 + 3.47627i 0
29.2 0 −1.37974 1.64431i 0 3.12714 + 1.13819i 0 5.96525 10.3321i 0 0.762765 4.32585i 0
29.3 0 2.23630 + 2.66512i 0 1.62283 + 0.590661i 0 −1.87959 + 3.25555i 0 −0.538989 + 3.05676i 0
33.1 0 −4.45984 0.786390i 0 5.56146 + 4.66662i 0 4.24641 + 7.35499i 0 10.8145 + 3.93617i 0
33.2 0 −1.95150 0.344102i 0 −6.26982 5.26101i 0 −0.733695 1.27080i 0 −4.76730 1.73516i 0
33.3 0 3.35830 + 0.592159i 0 1.47441 + 1.23717i 0 0.111774 + 0.193599i 0 2.47031 + 0.899120i 0
41.1 0 −0.617749 1.69725i 0 0.757040 4.29338i 0 −2.41011 4.17443i 0 4.39535 3.68814i 0
41.2 0 0.215056 + 0.590861i 0 −1.30596 + 7.40644i 0 3.03221 + 5.25195i 0 6.59153 5.53095i 0
41.3 0 1.87447 + 5.15008i 0 0.722564 4.09786i 0 2.33853 + 4.05046i 0 −16.1152 + 13.5223i 0
53.1 0 −4.45984 + 0.786390i 0 5.56146 4.66662i 0 4.24641 7.35499i 0 10.8145 3.93617i 0
53.2 0 −1.95150 + 0.344102i 0 −6.26982 + 5.26101i 0 −0.733695 + 1.27080i 0 −4.76730 + 1.73516i 0
53.3 0 3.35830 0.592159i 0 1.47441 1.23717i 0 0.111774 0.193599i 0 2.47031 0.899120i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.j.a 18
4.b odd 2 1 304.3.z.b 18
19.e even 9 1 1444.3.c.c 18
19.f odd 18 1 inner 76.3.j.a 18
19.f odd 18 1 1444.3.c.c 18
76.k even 18 1 304.3.z.b 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.j.a 18 1.a even 1 1 trivial
76.3.j.a 18 19.f odd 18 1 inner
304.3.z.b 18 4.b odd 2 1
304.3.z.b 18 76.k even 18 1
1444.3.c.c 18 19.e even 9 1
1444.3.c.c 18 19.f odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$25351947 + 15462333 T + 52829334 T^{2} + 86993568 T^{3} + 75029787 T^{4} + 38132874 T^{5} + 8815600 T^{6} - 1037244 T^{7} - 1122870 T^{8} - 134565 T^{9} + 49944 T^{10} - 10329 T^{11} - 9158 T^{12} - 663 T^{13} + 579 T^{14} + 138 T^{15} + 15 T^{16} + 6 T^{17} + T^{18}$$
$5$ $$294796874304 - 671961738816 T + 683484210432 T^{2} - 399293058240 T^{3} + 147174168600 T^{4} - 35720207550 T^{5} + 5657797405 T^{6} - 302688396 T^{7} - 83459277 T^{8} + 14779771 T^{9} + 188496 T^{10} - 533835 T^{11} + 170814 T^{12} - 2484 T^{13} + 2178 T^{14} - 83 T^{15} + 27 T^{16} + T^{18}$$
$7$ $$44446758976 - 169679588160 T + 783664154352 T^{2} + 475238221088 T^{3} + 485179613676 T^{4} + 34843057218 T^{5} + 45616766057 T^{6} - 746346624 T^{7} + 3336259878 T^{8} - 200995858 T^{9} + 127986468 T^{10} - 10826829 T^{11} + 3682079 T^{12} - 281592 T^{13} + 45999 T^{14} - 1822 T^{15} + 273 T^{16} - 9 T^{17} + T^{18}$$
$11$ $$423492482169 - 2470947761763 T + 13248443642949 T^{2} - 7401073144248 T^{3} + 5009592611223 T^{4} - 223389995625 T^{5} + 387770036833 T^{6} - 16818901638 T^{7} + 18303286542 T^{8} + 67444360 T^{9} + 414602055 T^{10} + 7290945 T^{11} + 6761001 T^{12} + 214452 T^{13} + 64593 T^{14} + 2167 T^{15} + 426 T^{16} + 15 T^{17} + T^{18}$$
$13$ $$645464049285312 - 2360547930919104 T + 3329945737897248 T^{2} - 2397004301983608 T^{3} + 1066748494874544 T^{4} - 327113111886696 T^{5} + 72384369790113 T^{6} - 11915537845851 T^{7} + 1588031957610 T^{8} - 201302250453 T^{9} + 27310862961 T^{10} - 3224525283 T^{11} + 276469353 T^{12} - 18339759 T^{13} + 1042569 T^{14} - 48738 T^{15} + 1935 T^{16} - 51 T^{17} + T^{18}$$
$17$ $$844355393617120641 - 162252857912465187 T + 277463374023439476 T^{2} - 109022044548264858 T^{3} + 13880154865976928 T^{4} + 4351310619267249 T^{5} - 454032987887159 T^{6} - 74722715188377 T^{7} + 4919104806102 T^{8} + 412136029720 T^{9} + 60414425451 T^{10} + 2422855899 T^{11} + 50303340 T^{12} + 4947858 T^{13} + 722286 T^{14} + 35119 T^{15} + 1209 T^{16} + 45 T^{17} + T^{18}$$
$19$ $$10\!\cdots\!41$$$$-$$$$86\!\cdots\!30$$$$T -$$$$57\!\cdots\!83$$$$T^{2} + 73911438407295380434 T^{3} + 1213013065907154447 T^{4} - 328794885147165354 T^{5} - 189400695033232 T^{6} + 1013564740339056 T^{7} - 1327681807701 T^{8} - 2828685593204 T^{9} - 3677788941 T^{10} + 7777447536 T^{11} - 4025872 T^{12} - 19359594 T^{13} + 197847 T^{14} + 33394 T^{15} - 723 T^{16} - 30 T^{17} + T^{18}$$
$23$ $$73\!\cdots\!76$$$$-$$$$38\!\cdots\!28$$$$T +$$$$57\!\cdots\!40$$$$T^{2} + 59024013099126257664 T^{3} - 16682523779288984832 T^{4} + 2098602780737853312 T^{5} - 18868237951436864 T^{6} - 11086215369190944 T^{7} + 1086752007221424 T^{8} - 52854358461464 T^{9} + 2377656507600 T^{10} - 93721539618 T^{11} + 2569758375 T^{12} - 67422195 T^{13} + 3153474 T^{14} - 81353 T^{15} + 2163 T^{16} - 48 T^{17} + T^{18}$$
$29$ $$17\!\cdots\!88$$$$+$$$$46\!\cdots\!40$$$$T +$$$$11\!\cdots\!32$$$$T^{2} -$$$$61\!\cdots\!00$$$$T^{3} +$$$$89\!\cdots\!08$$$$T^{4} - 74914150891253746968 T^{5} + 4057817608990769841 T^{6} - 145389172302725589 T^{7} + 6172883514996129 T^{8} - 403919052145461 T^{9} + 15326693953236 T^{10} - 240185599992 T^{11} - 572499921 T^{12} + 142404948 T^{13} - 230616 T^{14} + 99033 T^{15} + 1335 T^{16} + 39 T^{17} + T^{18}$$
$31$ $$96\!\cdots\!68$$$$+$$$$37\!\cdots\!72$$$$T +$$$$31\!\cdots\!28$$$$T^{2} -$$$$72\!\cdots\!72$$$$T^{3} - 82760862563443982628 T^{4} + 22178080427697923616 T^{5} + 5354215705390311369 T^{6} + 424558272892598847 T^{7} + 12674212527711321 T^{8} - 149738997345822 T^{9} - 16216987277529 T^{10} + 15907769334 T^{11} + 20472912201 T^{12} + 377504235 T^{13} - 6335910 T^{14} - 207360 T^{15} + 1968 T^{16} + 108 T^{17} + T^{18}$$
$37$ $$21\!\cdots\!28$$$$+$$$$67\!\cdots\!20$$$$T^{2} +$$$$52\!\cdots\!72$$$$T^{4} +$$$$17\!\cdots\!52$$$$T^{6} + 314798978583602208 T^{8} + 326575392438564 T^{10} + 201689909991 T^{12} + 71932221 T^{14} + 13410 T^{16} + T^{18}$$
$41$ $$48\!\cdots\!67$$$$+$$$$11\!\cdots\!72$$$$T -$$$$19\!\cdots\!08$$$$T^{2} +$$$$66\!\cdots\!90$$$$T^{3} +$$$$81\!\cdots\!24$$$$T^{4} +$$$$19\!\cdots\!66$$$$T^{5} - 40415730969263718213 T^{6} + 7030708955884272060 T^{7} + 368765386446744396 T^{8} - 909237046544370 T^{9} + 33772677718662 T^{10} + 2578911821712 T^{11} - 57116908503 T^{12} - 67117248 T^{13} + 1040184 T^{14} - 255267 T^{15} + 4122 T^{16} - 54 T^{17} + T^{18}$$
$43$ $$71\!\cdots\!49$$$$-$$$$10\!\cdots\!90$$$$T +$$$$53\!\cdots\!30$$$$T^{2} -$$$$72\!\cdots\!51$$$$T^{3} +$$$$52\!\cdots\!80$$$$T^{4} -$$$$23\!\cdots\!95$$$$T^{5} + 99340798471506768717 T^{6} - 3545694812094779316 T^{7} + 84304552742754621 T^{8} - 2381679245987638 T^{9} + 95052244080633 T^{10} - 1843321234188 T^{11} + 82861156017 T^{12} - 820786947 T^{13} + 18371001 T^{14} - 123417 T^{15} + 537 T^{16} - 75 T^{17} + T^{18}$$
$47$ $$13\!\cdots\!96$$$$-$$$$17\!\cdots\!04$$$$T +$$$$24\!\cdots\!72$$$$T^{2} -$$$$14\!\cdots\!48$$$$T^{3} + 48137582427306142176 T^{4} + 8302715013168275472 T^{5} + 1196708353734020545 T^{6} - 595870402277707551 T^{7} + 80308132596990966 T^{8} - 6693646881818969 T^{9} + 404690058903465 T^{10} - 17319592988919 T^{11} + 560958849279 T^{12} - 14680313577 T^{13} + 308361303 T^{14} - 4853438 T^{15} + 52263 T^{16} - 339 T^{17} + T^{18}$$
$53$ $$19\!\cdots\!52$$$$+$$$$39\!\cdots\!80$$$$T -$$$$18\!\cdots\!64$$$$T^{2} -$$$$99\!\cdots\!24$$$$T^{3} +$$$$98\!\cdots\!72$$$$T^{4} +$$$$12\!\cdots\!08$$$$T^{5} - 54603349243185742407 T^{6} - 13112289135675289188 T^{7} + 311215057521045174 T^{8} - 4180305303033966 T^{9} + 249822131396760 T^{10} + 1617822724437 T^{11} + 56176452618 T^{12} + 2833724790 T^{13} - 12734496 T^{14} + 327804 T^{15} + 7329 T^{16} - 69 T^{17} + T^{18}$$
$59$ $$34\!\cdots\!43$$$$-$$$$23\!\cdots\!83$$$$T +$$$$77\!\cdots\!18$$$$T^{2} +$$$$12\!\cdots\!46$$$$T^{3} +$$$$34\!\cdots\!62$$$$T^{4} +$$$$37\!\cdots\!03$$$$T^{5} +$$$$18\!\cdots\!71$$$$T^{6} +$$$$52\!\cdots\!91$$$$T^{7} +$$$$91\!\cdots\!76$$$$T^{8} + 8473708809754091676 T^{9} - 8124192694903797 T^{10} - 1407982873724235 T^{11} - 19345053280296 T^{12} - 106694397882 T^{13} + 403319538 T^{14} + 12286623 T^{15} + 105843 T^{16} + 483 T^{17} + T^{18}$$
$61$ $$70\!\cdots\!56$$$$+$$$$77\!\cdots\!80$$$$T +$$$$11\!\cdots\!92$$$$T^{2} + 21354990946436880344 T^{3} +$$$$31\!\cdots\!84$$$$T^{4} - 19104835694586512172 T^{5} + 1601932520781218909 T^{6} + 834921765294599427 T^{7} + 46665141704703270 T^{8} - 315436133715460 T^{9} - 62012230572300 T^{10} + 124506141918 T^{11} + 48997897463 T^{12} - 11766384 T^{13} - 14495964 T^{14} - 73906 T^{15} + 5883 T^{16} + 36 T^{17} + T^{18}$$
$67$ $$58\!\cdots\!28$$$$-$$$$11\!\cdots\!04$$$$T +$$$$19\!\cdots\!44$$$$T^{2} -$$$$11\!\cdots\!44$$$$T^{3} +$$$$57\!\cdots\!24$$$$T^{4} +$$$$43\!\cdots\!56$$$$T^{5} +$$$$66\!\cdots\!29$$$$T^{6} -$$$$20\!\cdots\!54$$$$T^{7} - 7049210264074818360 T^{8} - 3002876719961526 T^{9} + 30465397480680 T^{10} - 10602211029219 T^{11} + 690099100614 T^{12} + 13716786654 T^{13} + 116267382 T^{14} + 977376 T^{15} + 4743 T^{16} + 87 T^{17} + T^{18}$$
$71$ $$29\!\cdots\!52$$$$+$$$$29\!\cdots\!56$$$$T -$$$$20\!\cdots\!24$$$$T^{2} +$$$$54\!\cdots\!00$$$$T^{3} +$$$$26\!\cdots\!32$$$$T^{4} -$$$$13\!\cdots\!20$$$$T^{5} +$$$$46\!\cdots\!04$$$$T^{6} -$$$$24\!\cdots\!48$$$$T^{7} + 16131991759816923792 T^{8} - 463925649216453696 T^{9} + 5067525400591572 T^{10} + 24190687424916 T^{11} + 381524550651 T^{12} + 5037400611 T^{13} + 80957286 T^{14} + 3258459 T^{15} + 36315 T^{16} + 234 T^{17} + T^{18}$$
$73$ $$36\!\cdots\!24$$$$-$$$$11\!\cdots\!32$$$$T -$$$$80\!\cdots\!76$$$$T^{2} +$$$$34\!\cdots\!64$$$$T^{3} +$$$$12\!\cdots\!36$$$$T^{4} +$$$$28\!\cdots\!04$$$$T^{5} +$$$$49\!\cdots\!41$$$$T^{6} +$$$$56\!\cdots\!46$$$$T^{7} + 70204047682945482105 T^{8} + 1232705094305879666 T^{9} + 16747431784113048 T^{10} + 228670600280226 T^{11} + 2547209096199 T^{12} + 12473361984 T^{13} + 192630477 T^{14} + 1702710 T^{15} + 7962 T^{16} + 132 T^{17} + T^{18}$$
$79$ $$13\!\cdots\!52$$$$-$$$$17\!\cdots\!84$$$$T +$$$$15\!\cdots\!12$$$$T^{2} +$$$$28\!\cdots\!92$$$$T^{3} +$$$$54\!\cdots\!64$$$$T^{4} +$$$$47\!\cdots\!42$$$$T^{5} +$$$$30\!\cdots\!55$$$$T^{6} + 11317415770209495213 T^{7} + 597253243992207273 T^{8} + 24995769687700227 T^{9} + 1630819473994998 T^{10} + 27533065444200 T^{11} + 578980461963 T^{12} - 21029496066 T^{13} + 581690304 T^{14} - 8440755 T^{15} + 68253 T^{16} - 363 T^{17} + T^{18}$$
$83$ $$61\!\cdots\!41$$$$-$$$$23\!\cdots\!81$$$$T +$$$$12\!\cdots\!11$$$$T^{2} -$$$$32\!\cdots\!78$$$$T^{3} +$$$$12\!\cdots\!73$$$$T^{4} -$$$$29\!\cdots\!75$$$$T^{5} +$$$$74\!\cdots\!03$$$$T^{6} -$$$$11\!\cdots\!82$$$$T^{7} +$$$$20\!\cdots\!58$$$$T^{8} - 25677271736409973550 T^{9} + 350337898026756297 T^{10} - 3537648561707325 T^{11} + 36202550235279 T^{12} - 267656564628 T^{13} + 2114634591 T^{14} - 12125327 T^{15} + 77082 T^{16} - 279 T^{17} + T^{18}$$
$89$ $$26\!\cdots\!03$$$$+$$$$18\!\cdots\!22$$$$T +$$$$41\!\cdots\!97$$$$T^{2} -$$$$93\!\cdots\!56$$$$T^{3} +$$$$93\!\cdots\!24$$$$T^{4} -$$$$57\!\cdots\!52$$$$T^{5} +$$$$23\!\cdots\!12$$$$T^{6} -$$$$70\!\cdots\!86$$$$T^{7} +$$$$15\!\cdots\!10$$$$T^{8} -$$$$25\!\cdots\!48$$$$T^{9} + 3254649743532377529 T^{10} - 33026936297135535 T^{11} + 273347777080500 T^{12} - 1891417265613 T^{13} + 11075436900 T^{14} - 54211440 T^{15} + 212943 T^{16} - 600 T^{17} + T^{18}$$
$97$ $$14\!\cdots\!07$$$$-$$$$37\!\cdots\!12$$$$T -$$$$14\!\cdots\!54$$$$T^{2} +$$$$10\!\cdots\!97$$$$T^{3} +$$$$20\!\cdots\!32$$$$T^{4} -$$$$55\!\cdots\!95$$$$T^{5} -$$$$97\!\cdots\!93$$$$T^{6} -$$$$19\!\cdots\!80$$$$T^{7} +$$$$29\!\cdots\!77$$$$T^{8} + 82061661031865153118 T^{9} + 1525760111860451127 T^{10} + 22059938962834452 T^{11} + 253969331531589 T^{12} + 2298167624205 T^{13} + 16099781319 T^{14} + 85169637 T^{15} + 323469 T^{16} + 801 T^{17} + T^{18}$$