# Properties

 Label 252.2.l.b Level 252 Weight 2 Character orbit 252.l Analytic conductor 2.012 Analytic rank 0 Dimension 14 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$14$$ Relative dimension: $$7$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -\beta_{5} + \beta_{9} ) q^{5} + \beta_{6} q^{7} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -\beta_{5} + \beta_{9} ) q^{5} + \beta_{6} q^{7} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} ) q^{9} + ( -1 + \beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{11} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{12} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{8} - \beta_{9} - \beta_{13} ) q^{15} + ( 1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{8} - \beta_{12} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{13} ) q^{19} + ( -2 \beta_{2} + \beta_{4} + \beta_{9} + \beta_{13} ) q^{21} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{12} ) q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} ) q^{25} + ( \beta_{1} - \beta_{2} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{27} + ( -\beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{29} + ( 1 - \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{31} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{10} + \beta_{11} + \beta_{13} ) q^{33} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{35} + ( -2 + \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{37} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{39} + ( -3 + 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{6} + \beta_{9} - \beta_{10} ) q^{41} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{13} ) q^{43} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{45} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{47} + ( 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{49} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} ) q^{51} + ( -3 + 3 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - 3 \beta_{12} ) q^{53} + ( -2 - 3 \beta_{1} - \beta_{2} + \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{12} + \beta_{13} ) q^{55} + ( -\beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{57} + ( 1 - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{59} + ( -1 + \beta_{2} + \beta_{5} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{61} + ( -1 - \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{7} - 4 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} ) q^{63} + ( -6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - \beta_{5} - 4 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + 3 \beta_{12} ) q^{65} + ( -2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{67} + ( -5 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{12} ) q^{69} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{12} ) q^{71} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{73} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{75} + ( -1 + 5 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{13} ) q^{77} + ( -3 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} ) q^{79} + ( \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{81} + ( 2 - 3 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{9} + \beta_{10} + 3 \beta_{11} - \beta_{12} - \beta_{13} ) q^{83} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{85} + ( 3 - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 5 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} - 3 \beta_{12} ) q^{87} + ( -2 \beta_{2} + 2 \beta_{6} + \beta_{9} + 2 \beta_{10} ) q^{89} + ( 1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} - \beta_{12} - \beta_{13} ) q^{91} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} + 4 \beta_{11} - \beta_{12} ) q^{93} + ( -2 + \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{12} ) q^{95} + ( 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{97} + ( -3 + 3 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q + 4q^{5} - 3q^{7} + 10q^{9} + O(q^{10})$$ $$14q + 4q^{5} - 3q^{7} + 10q^{9} - 4q^{11} + 2q^{13} + 7q^{15} + 2q^{17} + 7q^{19} - 2q^{21} - 22q^{23} + 18q^{25} + 9q^{27} + q^{29} - q^{31} + 5q^{33} - 19q^{35} + 10q^{37} - 20q^{39} - 33q^{41} + 7q^{43} + 5q^{45} - 3q^{47} - 13q^{49} + 20q^{51} - 15q^{53} - 28q^{55} - 18q^{57} - 14q^{59} - 10q^{61} - 39q^{63} + 15q^{65} + 6q^{67} - 43q^{69} + 2q^{71} + 21q^{73} + q^{75} + 19q^{77} - 10q^{79} + 22q^{81} - 25q^{83} + 8q^{85} - 2q^{87} - 6q^{89} + 2q^{91} + 16q^{93} - 28q^{95} - 18q^{97} + 7q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 5 x^{12} - 3 x^{11} + 7 x^{10} + 30 x^{9} - 117 x^{7} + 270 x^{5} + 189 x^{4} - 243 x^{3} - 1215 x^{2} + 2187$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$35 \nu^{13} - 72 \nu^{12} - 157 \nu^{11} - 312 \nu^{10} + 290 \nu^{9} + 1383 \nu^{8} - 1143 \nu^{7} - 3393 \nu^{6} + 2025 \nu^{5} + 8802 \nu^{4} + 9288 \nu^{3} - 23814 \nu^{2} - 63180 \nu + 46656$$$$)/43011$$ $$\beta_{3}$$ $$=$$ $$($$$$-8 \nu^{13} + 2 \nu^{12} - 23 \nu^{11} + 5 \nu^{10} + 37 \nu^{9} - 127 \nu^{8} + 78 \nu^{7} + 225 \nu^{6} - 72 \nu^{5} + 297 \nu^{4} - 1701 \nu^{3} - 2295 \nu^{2} + 5184 \nu - 8505$$$$)/4779$$ $$\beta_{4}$$ $$=$$ $$($$$$28 \nu^{13} - 15 \nu^{12} - 191 \nu^{11} - 135 \nu^{10} + 289 \nu^{9} + 546 \nu^{8} + 498 \nu^{7} - 2151 \nu^{6} - 1152 \nu^{5} + 8370 \nu^{4} + 2916 \nu^{3} + 1701 \nu^{2} - 24381 \nu - 40581$$$$)/14337$$ $$\beta_{5}$$ $$=$$ $$($$$$26 \nu^{13} + 7 \nu^{12} - 70 \nu^{11} - 266 \nu^{10} - 301 \nu^{9} + 955 \nu^{8} + 846 \nu^{7} - 2529 \nu^{6} - 2574 \nu^{5} - 864 \nu^{4} + 14985 \nu^{3} + 11124 \nu^{2} - 48438 \nu - 30618$$$$)/14337$$ $$\beta_{6}$$ $$=$$ $$($$$$-56 \nu^{13} + 441 \nu^{12} - 62 \nu^{11} - 1227 \nu^{10} - 761 \nu^{9} + 165 \nu^{8} + 8622 \nu^{7} + 1638 \nu^{6} - 26190 \nu^{5} + 2538 \nu^{4} + 35424 \nu^{3} + 61722 \nu^{2} + 22113 \nu - 293058$$$$)/43011$$ $$\beta_{7}$$ $$=$$ $$($$$$-83 \nu^{13} + 423 \nu^{12} + 406 \nu^{11} - 354 \nu^{10} - 2237 \nu^{9} - 2364 \nu^{8} + 7902 \nu^{7} + 7065 \nu^{6} - 17037 \nu^{5} - 20142 \nu^{4} + 7965 \nu^{3} + 116154 \nu^{2} + 87966 \nu - 237654$$$$)/43011$$ $$\beta_{8}$$ $$=$$ $$($$$$97 \nu^{13} - 675 \nu^{12} + \nu^{11} + 1491 \nu^{10} + 2002 \nu^{9} + 1209 \nu^{8} - 10584 \nu^{7} - 2736 \nu^{6} + 42714 \nu^{5} - 1269 \nu^{4} - 51003 \nu^{3} - 89424 \nu^{2} - 59292 \nu + 454896$$$$)/43011$$ $$\beta_{9}$$ $$=$$ $$($$$$-73 \nu^{13} - 360 \nu^{12} - 40 \nu^{11} + 804 \nu^{10} + 1703 \nu^{9} - 417 \nu^{8} - 8694 \nu^{7} + 1413 \nu^{6} + 19845 \nu^{5} - 2295 \nu^{4} - 32022 \nu^{3} - 98901 \nu^{2} - 11178 \nu + 239841$$$$)/43011$$ $$\beta_{10}$$ $$=$$ $$($$$$5 \nu^{13} + 203 \nu^{12} + 68 \nu^{11} - 463 \nu^{10} - 760 \nu^{9} - 247 \nu^{8} + 3780 \nu^{7} + 1044 \nu^{6} - 11700 \nu^{5} - 999 \nu^{4} + 21789 \nu^{3} + 41445 \nu^{2} + 11583 \nu - 116397$$$$)/14337$$ $$\beta_{11}$$ $$=$$ $$($$$$239 \nu^{13} - 324 \nu^{12} - 385 \nu^{11} + 93 \nu^{10} + 1133 \nu^{9} + 3687 \nu^{8} - 6129 \nu^{7} - 13221 \nu^{6} + 20358 \nu^{5} + 15120 \nu^{4} - 4239 \nu^{3} - 46413 \nu^{2} - 113724 \nu + 267543$$$$)/43011$$ $$\beta_{12}$$ $$=$$ $$($$$$179 \nu^{13} + 471 \nu^{12} - 769 \nu^{11} - 2352 \nu^{10} - 763 \nu^{9} + 5184 \nu^{8} + 9288 \nu^{7} - 16785 \nu^{6} - 31266 \nu^{5} + 33021 \nu^{4} + 101142 \nu^{3} + 55485 \nu^{2} - 162081 \nu - 324405$$$$)/43011$$ $$\beta_{13}$$ $$=$$ $$($$$$500 \nu^{13} - 81 \nu^{12} - 1096 \nu^{11} - 1392 \nu^{10} - 145 \nu^{9} + 10167 \nu^{8} - 27936 \nu^{6} + 5319 \nu^{5} + 40797 \nu^{4} + 89964 \nu^{3} + 243 \nu^{2} - 335097 \nu + 97686$$$$)/43011$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{2} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{12} - 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{12} - \beta_{11} + \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 8$$ $$\nu^{6}$$ $$=$$ $$5 \beta_{13} - 9 \beta_{12} - 2 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} - \beta_{8} + 6 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} + 6 \beta_{2} + 2 \beta_{1} - 1$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{13} + 5 \beta_{12} - 4 \beta_{11} - 2 \beta_{10} - 18 \beta_{9} + 10 \beta_{8} - 10 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 8 \beta_{4} + 10 \beta_{3} - 5 \beta_{2} - 10 \beta_{1} + 11$$ $$\nu^{8}$$ $$=$$ $$13 \beta_{13} - 13 \beta_{12} - 14 \beta_{11} - 22 \beta_{10} - \beta_{9} - 8 \beta_{8} + 2 \beta_{7} + 23 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - 23 \beta_{3} + 2 \beta_{2} + 27 \beta_{1} - 5$$ $$\nu^{9}$$ $$=$$ $$-9 \beta_{13} + 10 \beta_{12} + 14 \beta_{11} + 10 \beta_{10} + 22 \beta_{9} + 19 \beta_{8} - 9 \beta_{7} + 32 \beta_{6} + 9 \beta_{4} - 18 \beta_{3} - 58 \beta_{2} - 20 \beta_{1} - 9$$ $$\nu^{10}$$ $$=$$ $$21 \beta_{13} + 22 \beta_{12} - 65 \beta_{11} - 46 \beta_{10} - 28 \beta_{9} - 6 \beta_{8} - 32 \beta_{7} - 26 \beta_{6} - 14 \beta_{5} - 5 \beta_{4} - 35 \beta_{3} - 137 \beta_{2} - 44 \beta_{1} + 57$$ $$\nu^{11}$$ $$=$$ $$-54 \beta_{13} + 66 \beta_{12} - 10 \beta_{11} - 53 \beta_{10} - 115 \beta_{9} + 31 \beta_{8} - 41 \beta_{7} - 10 \beta_{6} - 38 \beta_{5} - 55 \beta_{4} - 115 \beta_{3} + 66 \beta_{2} - 54 \beta_{1} - 242$$ $$\nu^{12}$$ $$=$$ $$77 \beta_{13} + 81 \beta_{12} + 7 \beta_{11} - 58 \beta_{10} + 167 \beta_{9} - 19 \beta_{8} + 135 \beta_{7} + 60 \beta_{6} - 181 \beta_{5} - 78 \beta_{4} - 26 \beta_{3} - 183 \beta_{2} - 304 \beta_{1} - 109$$ $$\nu^{13}$$ $$=$$ $$51 \beta_{13} - 310 \beta_{12} + 329 \beta_{11} + 367 \beta_{10} - 117 \beta_{9} - 62 \beta_{8} - 325 \beta_{7} + 146 \beta_{6} + 38 \beta_{5} - 89 \beta_{4} + 307 \beta_{3} - 167 \beta_{2} - 325 \beta_{1} - 457$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 −1.73040 + 0.0755709i −1.58203 − 0.705117i −0.674693 − 1.59524i −0.473632 + 1.66604i 1.13119 − 1.31165i 1.64515 + 0.541745i 1.68442 − 0.403398i −1.73040 − 0.0755709i −1.58203 + 0.705117i −0.674693 + 1.59524i −0.473632 − 1.66604i 1.13119 + 1.31165i 1.64515 − 0.541745i 1.68442 + 0.403398i
0 −1.73040 + 0.0755709i 0 0.967857 0 −1.11482 2.39941i 0 2.98858 0.261536i 0
193.2 0 −1.58203 0.705117i 0 −2.52026 0 1.98143 + 1.75326i 0 2.00562 + 2.23103i 0
193.3 0 −0.674693 1.59524i 0 4.14520 0 0.190437 + 2.63889i 0 −2.08958 + 2.15260i 0
193.4 0 −0.473632 + 1.66604i 0 −1.90301 0 −2.43415 + 1.03677i 0 −2.55135 1.57817i 0
193.5 0 1.13119 1.31165i 0 −1.52940 0 2.53654 0.752299i 0 −0.440838 2.96743i 0
193.6 0 1.64515 + 0.541745i 0 −0.763837 0 −1.05641 + 2.42569i 0 2.41302 + 1.78250i 0
193.7 0 1.68442 0.403398i 0 3.60346 0 −1.60302 2.10483i 0 2.67454 1.35898i 0
205.1 0 −1.73040 0.0755709i 0 0.967857 0 −1.11482 + 2.39941i 0 2.98858 + 0.261536i 0
205.2 0 −1.58203 + 0.705117i 0 −2.52026 0 1.98143 1.75326i 0 2.00562 2.23103i 0
205.3 0 −0.674693 + 1.59524i 0 4.14520 0 0.190437 2.63889i 0 −2.08958 2.15260i 0
205.4 0 −0.473632 1.66604i 0 −1.90301 0 −2.43415 1.03677i 0 −2.55135 + 1.57817i 0
205.5 0 1.13119 + 1.31165i 0 −1.52940 0 2.53654 + 0.752299i 0 −0.440838 + 2.96743i 0
205.6 0 1.64515 0.541745i 0 −0.763837 0 −1.05641 2.42569i 0 2.41302 1.78250i 0
205.7 0 1.68442 + 0.403398i 0 3.60346 0 −1.60302 + 2.10483i 0 2.67454 + 1.35898i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 205.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.l.b yes 14
3.b odd 2 1 756.2.l.b 14
4.b odd 2 1 1008.2.t.j 14
7.b odd 2 1 1764.2.l.i 14
7.c even 3 1 252.2.i.b 14
7.c even 3 1 1764.2.j.g 14
7.d odd 6 1 1764.2.i.i 14
7.d odd 6 1 1764.2.j.h 14
9.c even 3 1 252.2.i.b 14
9.c even 3 1 2268.2.k.e 14
9.d odd 6 1 756.2.i.b 14
9.d odd 6 1 2268.2.k.f 14
12.b even 2 1 3024.2.t.j 14
21.c even 2 1 5292.2.l.i 14
21.g even 6 1 5292.2.i.i 14
21.g even 6 1 5292.2.j.g 14
21.h odd 6 1 756.2.i.b 14
21.h odd 6 1 5292.2.j.h 14
28.g odd 6 1 1008.2.q.j 14
36.f odd 6 1 1008.2.q.j 14
36.h even 6 1 3024.2.q.j 14
63.g even 3 1 inner 252.2.l.b yes 14
63.h even 3 1 1764.2.j.g 14
63.h even 3 1 2268.2.k.e 14
63.i even 6 1 5292.2.j.g 14
63.j odd 6 1 2268.2.k.f 14
63.j odd 6 1 5292.2.j.h 14
63.k odd 6 1 1764.2.l.i 14
63.l odd 6 1 1764.2.i.i 14
63.n odd 6 1 756.2.l.b 14
63.o even 6 1 5292.2.i.i 14
63.s even 6 1 5292.2.l.i 14
63.t odd 6 1 1764.2.j.h 14
84.n even 6 1 3024.2.q.j 14
252.o even 6 1 3024.2.t.j 14
252.bl odd 6 1 1008.2.t.j 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.b 14 7.c even 3 1
252.2.i.b 14 9.c even 3 1
252.2.l.b yes 14 1.a even 1 1 trivial
252.2.l.b yes 14 63.g even 3 1 inner
756.2.i.b 14 9.d odd 6 1
756.2.i.b 14 21.h odd 6 1
756.2.l.b 14 3.b odd 2 1
756.2.l.b 14 63.n odd 6 1
1008.2.q.j 14 28.g odd 6 1
1008.2.q.j 14 36.f odd 6 1
1008.2.t.j 14 4.b odd 2 1
1008.2.t.j 14 252.bl odd 6 1
1764.2.i.i 14 7.d odd 6 1
1764.2.i.i 14 63.l odd 6 1
1764.2.j.g 14 7.c even 3 1
1764.2.j.g 14 63.h even 3 1
1764.2.j.h 14 7.d odd 6 1
1764.2.j.h 14 63.t odd 6 1
1764.2.l.i 14 7.b odd 2 1
1764.2.l.i 14 63.k odd 6 1
2268.2.k.e 14 9.c even 3 1
2268.2.k.e 14 63.h even 3 1
2268.2.k.f 14 9.d odd 6 1
2268.2.k.f 14 63.j odd 6 1
3024.2.q.j 14 36.h even 6 1
3024.2.q.j 14 84.n even 6 1
3024.2.t.j 14 12.b even 2 1
3024.2.t.j 14 252.o even 6 1
5292.2.i.i 14 21.g even 6 1
5292.2.i.i 14 63.o even 6 1
5292.2.j.g 14 21.g even 6 1
5292.2.j.g 14 63.i even 6 1
5292.2.j.h 14 21.h odd 6 1
5292.2.j.h 14 63.j odd 6 1
5292.2.l.i 14 21.c even 2 1
5292.2.l.i 14 63.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{7} - 2 T_{5}^{6} - 20 T_{5}^{5} + 12 T_{5}^{4} + 129 T_{5}^{3} + 81 T_{5}^{2} - 108 T_{5} - 81$$ acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 5 T^{2} - 3 T^{3} + 7 T^{4} + 30 T^{5} - 117 T^{7} + 270 T^{9} + 189 T^{10} - 243 T^{11} - 1215 T^{12} + 2187 T^{14}$$
$5$ $$( 1 - 2 T + 15 T^{2} - 48 T^{3} + 154 T^{4} - 429 T^{5} + 1202 T^{6} - 2471 T^{7} + 6010 T^{8} - 10725 T^{9} + 19250 T^{10} - 30000 T^{11} + 46875 T^{12} - 31250 T^{13} + 78125 T^{14} )^{2}$$
$7$ $$1 + 3 T + 11 T^{2} - 5 T^{3} - 15 T^{4} - 88 T^{5} + 312 T^{6} + 753 T^{7} + 2184 T^{8} - 4312 T^{9} - 5145 T^{10} - 12005 T^{11} + 184877 T^{12} + 352947 T^{13} + 823543 T^{14}$$
$11$ $$( 1 + 2 T + 36 T^{2} + 57 T^{3} + 460 T^{4} + 402 T^{5} + 2501 T^{6} + 455 T^{7} + 27511 T^{8} + 48642 T^{9} + 612260 T^{10} + 834537 T^{11} + 5797836 T^{12} + 3543122 T^{13} + 19487171 T^{14} )^{2}$$
$13$ $$1 - 2 T - 25 T^{2} - 20 T^{3} + 172 T^{4} + 1281 T^{5} + 1882 T^{6} - 1142 T^{7} - 27931 T^{8} - 309997 T^{9} + 9092 T^{10} + 1859069 T^{11} + 4530175 T^{12} + 5185603 T^{13} - 73974588 T^{14} + 67412839 T^{15} + 765599575 T^{16} + 4084374593 T^{17} + 259676612 T^{18} - 115099716121 T^{19} - 134817602179 T^{20} - 71658806414 T^{21} + 1535205216922 T^{22} + 13584363696813 T^{23} + 23711660598028 T^{24} - 35843207880740 T^{25} - 582452128062025 T^{26} - 605750213184506 T^{27} + 3937376385699289 T^{28}$$
$17$ $$1 - 2 T - 65 T^{2} + 210 T^{3} + 2087 T^{4} - 9143 T^{5} - 37340 T^{6} + 240381 T^{7} + 293834 T^{8} - 4176065 T^{9} + 3382763 T^{10} + 47662233 T^{11} - 157347285 T^{12} - 273515658 T^{13} + 3170983122 T^{14} - 4649766186 T^{15} - 45473365365 T^{16} + 234164550729 T^{17} + 282531748523 T^{18} - 5929415122705 T^{19} + 7092438449546 T^{20} + 98637620554413 T^{21} - 260474782846940 T^{22} - 1084248954812071 T^{23} + 4207379270237063 T^{24} + 7197098224602930 T^{25} - 37870445419934465 T^{26} - 19809156065811874 T^{27} + 168377826559400929 T^{28}$$
$19$ $$1 - 7 T - 54 T^{2} + 381 T^{3} + 1875 T^{4} - 9873 T^{5} - 65652 T^{6} + 221430 T^{7} + 1870425 T^{8} - 4319703 T^{9} - 46476858 T^{10} + 61637031 T^{11} + 1073881146 T^{12} - 457871775 T^{13} - 21789737442 T^{14} - 8699563725 T^{15} + 387671093706 T^{16} + 422768395629 T^{17} - 6056910611418 T^{18} - 10696012278597 T^{19} + 87995791969425 T^{20} + 197930019166770 T^{21} - 1115004880767732 T^{22} - 3185895640172067 T^{23} + 11495749233376875 T^{24} + 44382788640221439 T^{25} - 119519005629572694 T^{26} - 294370884235799413 T^{27} + 799006685782884121 T^{28}$$
$23$ $$( 1 + 11 T + 129 T^{2} + 876 T^{3} + 6604 T^{4} + 35691 T^{5} + 217568 T^{6} + 992939 T^{7} + 5004064 T^{8} + 18880539 T^{9} + 80350868 T^{10} + 245140716 T^{11} + 830288247 T^{12} + 1628394779 T^{13} + 3404825447 T^{14} )^{2}$$
$29$ $$1 - T - 89 T^{2} + 606 T^{3} + 3413 T^{4} - 45595 T^{5} + 49603 T^{6} + 1643802 T^{7} - 7893892 T^{8} - 19552444 T^{9} + 271585946 T^{10} - 420585531 T^{11} - 3441402105 T^{12} + 10956432363 T^{13} + 10513309734 T^{14} + 317736538527 T^{15} - 2894219170305 T^{16} - 10257660515559 T^{17} + 192087579472826 T^{18} - 401043092198156 T^{19} - 4695471055055332 T^{20} + 28355381176486818 T^{21} + 24813722822104483 T^{22} - 661453320769747055 T^{23} + 1435873787253586013 T^{24} + 7393508918017732374 T^{25} - 31489515705286744649 T^{26} - 10260628712958602189 T^{27} +$$$$29\!\cdots\!81$$$$T^{28}$$
$31$ $$1 + T - 85 T^{2} - 302 T^{3} + 2323 T^{4} + 16596 T^{5} - 8720 T^{6} - 121502 T^{7} + 336059 T^{8} - 5031700 T^{9} - 52734649 T^{10} - 249456712 T^{11} - 356372186 T^{12} + 8588183707 T^{13} + 82956488229 T^{14} + 266233694917 T^{15} - 342473670746 T^{16} - 7431564907192 T^{17} - 48701555779129 T^{18} - 144053299086700 T^{19} + 298253599533179 T^{20} - 3342837639714722 T^{21} - 7437209846485520 T^{22} + 438791969378495916 T^{23} + 1903996510656400723 T^{24} - 7673360022714258962 T^{25} - 66951336622026729685 T^{26} + 24417546297445042591 T^{27} +$$$$75\!\cdots\!21$$$$T^{28}$$
$37$ $$1 - 10 T - 84 T^{2} + 1296 T^{3} + 1134 T^{4} - 66630 T^{5} + 77382 T^{6} + 1851174 T^{7} - 1251993 T^{8} - 21837810 T^{9} - 268575252 T^{10} - 774884982 T^{11} + 27666538119 T^{12} + 26159905206 T^{13} - 1381876923558 T^{14} + 967916492622 T^{15} + 37875490684911 T^{16} - 39250248993246 T^{17} - 503353262863572 T^{18} - 1514320157614170 T^{19} - 3212271503983137 T^{20} + 175735422719804142 T^{21} + 271802685103314822 T^{22} - 8659350722545980510 T^{23} + 5452934678321840766 T^{24} +$$$$23\!\cdots\!48$$$$T^{25} -$$$$55\!\cdots\!04$$$$T^{26} -$$$$24\!\cdots\!70$$$$T^{27} +$$$$90\!\cdots\!89$$$$T^{28}$$
$41$ $$1 + 33 T + 463 T^{2} + 3882 T^{3} + 26359 T^{4} + 177381 T^{5} + 987377 T^{6} + 3338436 T^{7} - 1489480 T^{8} - 146370792 T^{9} - 1532374696 T^{10} - 10553354379 T^{11} - 66691501599 T^{12} - 460808734581 T^{13} - 3110830401306 T^{14} - 18893158117821 T^{15} - 112108414187919 T^{16} - 727347737155059 T^{17} - 4330124653343656 T^{18} - 16957963898481192 T^{19} - 7075185264884680 T^{20} + 650174679078190116 T^{21} + 7884131517953805617 T^{22} + 58071334904735196141 T^{23} +$$$$35\!\cdots\!59$$$$T^{24} +$$$$21\!\cdots\!62$$$$T^{25} +$$$$10\!\cdots\!03$$$$T^{26} +$$$$30\!\cdots\!93$$$$T^{27} +$$$$37\!\cdots\!61$$$$T^{28}$$
$43$ $$1 - 7 T - 222 T^{2} + 1221 T^{3} + 30003 T^{4} - 121065 T^{5} - 2964828 T^{6} + 8400318 T^{7} + 232132089 T^{8} - 430114695 T^{9} - 14993117802 T^{10} + 15328887375 T^{11} + 817956723570 T^{12} - 259275458991 T^{13} - 38016795252930 T^{14} - 11148844736613 T^{15} + 1512401981880930 T^{16} + 1218753848524125 T^{17} - 51258486134595402 T^{18} - 63230491623369885 T^{19} + 1467391209891779361 T^{20} + 2283362771617132026 T^{21} - 34653503452639217628 T^{22} - 60846374564133897795 T^{23} +$$$$64\!\cdots\!47$$$$T^{24} +$$$$11\!\cdots\!47$$$$T^{25} -$$$$88\!\cdots\!22$$$$T^{26} -$$$$12\!\cdots\!01$$$$T^{27} +$$$$73\!\cdots\!49$$$$T^{28}$$
$47$ $$1 + 3 T - 215 T^{2} - 750 T^{3} + 23197 T^{4} + 82998 T^{5} - 1793695 T^{6} - 5388753 T^{7} + 119401781 T^{8} + 227010657 T^{9} - 7448336422 T^{10} - 6436178352 T^{11} + 428559183834 T^{12} + 95001497937 T^{13} - 21681767924409 T^{14} + 4465070403039 T^{15} + 946687237089306 T^{16} - 668223345039696 T^{17} - 36345505720041382 T^{18} + 52063760718739599 T^{19} + 1287057508065100949 T^{20} - 2730066860264352639 T^{21} - 42710185828767396895 T^{22} + 92885591006583455466 T^{23} +$$$$12\!\cdots\!53$$$$T^{24} -$$$$18\!\cdots\!50$$$$T^{25} -$$$$24\!\cdots\!15$$$$T^{26} +$$$$16\!\cdots\!81$$$$T^{27} +$$$$25\!\cdots\!69$$$$T^{28}$$
$53$ $$1 + 15 T - 44 T^{2} - 1563 T^{3} + 1621 T^{4} + 132831 T^{5} + 280796 T^{6} - 6875916 T^{7} - 32544895 T^{8} + 334617081 T^{9} + 3332374934 T^{10} - 9404405181 T^{11} - 219667861866 T^{12} + 268474558113 T^{13} + 14089335458730 T^{14} + 14229151579989 T^{15} - 617047023981594 T^{16} - 1400099630131737 T^{17} + 26294041101603254 T^{18} + 139935355155015933 T^{19} - 721336805685386455 T^{20} - 8077215121783465692 T^{21} + 17482272028748523356 T^{22} +$$$$43\!\cdots\!23$$$$T^{23} +$$$$28\!\cdots\!29$$$$T^{24} -$$$$14\!\cdots\!11$$$$T^{25} -$$$$21\!\cdots\!04$$$$T^{26} +$$$$39\!\cdots\!95$$$$T^{27} +$$$$13\!\cdots\!69$$$$T^{28}$$
$59$ $$1 + 14 T - 41 T^{2} - 1932 T^{3} - 9268 T^{4} + 67958 T^{5} + 1000462 T^{6} + 4853283 T^{7} - 9914932 T^{8} - 321187219 T^{9} - 1782148489 T^{10} - 6021692682 T^{11} - 34607406657 T^{12} + 638777810472 T^{13} + 11047710270819 T^{14} + 37687890817848 T^{15} - 120468382573017 T^{16} - 1236729221336478 T^{17} - 21594936596817529 T^{18} - 229624547391334481 T^{19} - 418217122774227412 T^{20} + 12078129944196810777 T^{21} +$$$$14\!\cdots\!02$$$$T^{22} +$$$$58\!\cdots\!62$$$$T^{23} -$$$$47\!\cdots\!68$$$$T^{24} -$$$$58\!\cdots\!88$$$$T^{25} -$$$$72\!\cdots\!21$$$$T^{26} +$$$$14\!\cdots\!06$$$$T^{27} +$$$$61\!\cdots\!61$$$$T^{28}$$
$61$ $$1 + 10 T - 277 T^{2} - 2192 T^{3} + 49786 T^{4} + 275394 T^{5} - 6629804 T^{6} - 25018331 T^{7} + 694393718 T^{8} + 1681932119 T^{9} - 61000901305 T^{10} - 82163567860 T^{11} + 4597121803909 T^{12} + 1969709923870 T^{13} - 300133324238415 T^{14} + 120152305356070 T^{15} + 17105890232345389 T^{16} - 18649568796430660 T^{17} - 844608780325722505 T^{18} + 1420553646240491819 T^{19} + 35775424305286664198 T^{20} - 78626180519452100951 T^{21} -$$$$12\!\cdots\!24$$$$T^{22} +$$$$32\!\cdots\!54$$$$T^{23} +$$$$35\!\cdots\!86$$$$T^{24} -$$$$95\!\cdots\!12$$$$T^{25} -$$$$73\!\cdots\!17$$$$T^{26} +$$$$16\!\cdots\!10$$$$T^{27} +$$$$98\!\cdots\!41$$$$T^{28}$$
$67$ $$1 - 6 T - 308 T^{2} + 164 T^{3} + 62908 T^{4} + 146213 T^{5} - 7475396 T^{6} - 42326464 T^{7} + 603793243 T^{8} + 5085070936 T^{9} - 25351846493 T^{10} - 374911832519 T^{11} + 131329441787 T^{12} + 10392590626498 T^{13} + 54642703536353 T^{14} + 696303571975366 T^{15} + 589537864181843 T^{16} - 112759607483911997 T^{17} - 510868126253868653 T^{18} + 6865481941569590152 T^{19} + 54618159926353884067 T^{20} -$$$$25\!\cdots\!72$$$$T^{21} -$$$$30\!\cdots\!36$$$$T^{22} +$$$$39\!\cdots\!11$$$$T^{23} +$$$$11\!\cdots\!92$$$$T^{24} +$$$$20\!\cdots\!12$$$$T^{25} -$$$$25\!\cdots\!88$$$$T^{26} -$$$$32\!\cdots\!22$$$$T^{27} +$$$$36\!\cdots\!29$$$$T^{28}$$
$71$ $$( 1 - T + 381 T^{2} - 417 T^{3} + 66850 T^{4} - 68550 T^{5} + 7142942 T^{6} - 6246700 T^{7} + 507148882 T^{8} - 345560550 T^{9} + 23926350350 T^{10} - 10596670977 T^{11} + 687411382731 T^{12} - 128100283921 T^{13} + 9095120158391 T^{14} )^{2}$$
$73$ $$1 - 21 T - 107 T^{2} + 4532 T^{3} + 8593 T^{4} - 574123 T^{5} - 1508435 T^{6} + 52395764 T^{7} + 342478696 T^{8} - 4227610868 T^{9} - 46283154524 T^{10} + 253115011813 T^{11} + 4673586532751 T^{12} - 7744045394141 T^{13} - 369527474872486 T^{14} - 565315313772293 T^{15} + 24905542633030079 T^{16} + 98466042550457821 T^{17} - 1314360176412792284 T^{18} - 8764139996708872724 T^{19} + 51828748479625639144 T^{20} +$$$$57\!\cdots\!08$$$$T^{21} -$$$$12\!\cdots\!35$$$$T^{22} -$$$$33\!\cdots\!99$$$$T^{23} +$$$$36\!\cdots\!57$$$$T^{24} +$$$$14\!\cdots\!64$$$$T^{25} -$$$$24\!\cdots\!47$$$$T^{26} -$$$$35\!\cdots\!93$$$$T^{27} +$$$$12\!\cdots\!09$$$$T^{28}$$
$79$ $$1 + 10 T - 226 T^{2} - 4280 T^{3} + 9610 T^{4} + 654627 T^{5} + 3201292 T^{6} - 42192356 T^{7} - 519045919 T^{8} - 901423894 T^{9} + 24953123963 T^{10} + 320449224329 T^{11} + 1694205545953 T^{12} - 14615843505140 T^{13} - 287108375549019 T^{14} - 1154651636906060 T^{15} + 10573536812292673 T^{16} + 157993965113945831 T^{17} + 971926199561891003 T^{18} - 2773732161244197706 T^{19} -$$$$12\!\cdots\!99$$$$T^{20} -$$$$81\!\cdots\!04$$$$T^{21} +$$$$48\!\cdots\!12$$$$T^{22} +$$$$78\!\cdots\!13$$$$T^{23} +$$$$90\!\cdots\!10$$$$T^{24} -$$$$32\!\cdots\!20$$$$T^{25} -$$$$13\!\cdots\!66$$$$T^{26} +$$$$46\!\cdots\!90$$$$T^{27} +$$$$36\!\cdots\!81$$$$T^{28}$$
$83$ $$1 + 25 T + 157 T^{2} - 750 T^{3} - 6622 T^{4} + 69964 T^{5} + 1466905 T^{6} + 20424981 T^{7} + 74872112 T^{8} - 997051343 T^{9} + 7203133487 T^{10} + 269189728764 T^{11} + 1307717205141 T^{12} - 2467734607815 T^{13} - 39825488981322 T^{14} - 204821972448645 T^{15} + 9008863826216349 T^{16} + 153919187440781268 T^{17} + 341848621231895327 T^{18} - 3927425763234733549 T^{19} + 24478716252205585328 T^{20} +$$$$55\!\cdots\!87$$$$T^{21} +$$$$33\!\cdots\!05$$$$T^{22} +$$$$13\!\cdots\!92$$$$T^{23} -$$$$10\!\cdots\!78$$$$T^{24} -$$$$96\!\cdots\!50$$$$T^{25} +$$$$16\!\cdots\!77$$$$T^{26} +$$$$22\!\cdots\!75$$$$T^{27} +$$$$73\!\cdots\!29$$$$T^{28}$$
$89$ $$1 + 6 T - 395 T^{2} - 1770 T^{3} + 83413 T^{4} + 244017 T^{5} - 12791578 T^{6} - 19742595 T^{7} + 1607888462 T^{8} + 933579267 T^{9} - 173735706451 T^{10} - 11863751967 T^{11} + 16888438123377 T^{12} - 583089996186 T^{13} - 1538438146646466 T^{14} - 51895009660554 T^{15} + 133773318375269217 T^{16} - 8363577360424023 T^{17} - 10900567564453896691 T^{18} + 5213162127281843883 T^{19} +$$$$79\!\cdots\!82$$$$T^{20} -$$$$87\!\cdots\!55$$$$T^{21} -$$$$50\!\cdots\!18$$$$T^{22} +$$$$85\!\cdots\!53$$$$T^{23} +$$$$26\!\cdots\!13$$$$T^{24} -$$$$49\!\cdots\!30$$$$T^{25} -$$$$97\!\cdots\!95$$$$T^{26} +$$$$13\!\cdots\!14$$$$T^{27} +$$$$19\!\cdots\!41$$$$T^{28}$$
$97$ $$1 + 18 T - 332 T^{2} - 7282 T^{3} + 70792 T^{4} + 1694642 T^{5} - 11251793 T^{6} - 262778647 T^{7} + 1667779429 T^{8} + 30411590872 T^{9} - 218639545013 T^{10} - 2436064838528 T^{11} + 25890765026051 T^{12} + 94489464293929 T^{13} - 2645836565169718 T^{14} + 9165478036511113 T^{15} + 243606208130113859 T^{16} - 2223330604373865344 T^{17} - 19356001718168025653 T^{18} +$$$$26\!\cdots\!04$$$$T^{19} +$$$$13\!\cdots\!41$$$$T^{20} -$$$$21\!\cdots\!11$$$$T^{21} -$$$$88\!\cdots\!73$$$$T^{22} +$$$$12\!\cdots\!14$$$$T^{23} +$$$$52\!\cdots\!08$$$$T^{24} -$$$$52\!\cdots\!46$$$$T^{25} -$$$$23\!\cdots\!12$$$$T^{26} +$$$$12\!\cdots\!86$$$$T^{27} +$$$$65\!\cdots\!69$$$$T^{28}$$