# Properties

 Label 8036.2.a.r Level $8036$ Weight $2$ Character orbit 8036.a Self dual yes Analytic conductor $64.168$ Analytic rank $0$ Dimension $15$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8036 = 2^{2} \cdot 7^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8036.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1677830643$$ Analytic rank: $$0$$ Dimension: $$15$$ Coefficient field: $$\mathbb{Q}[x]/(x^{15} - \cdots)$$ Defining polynomial: $$x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + 13971 x^{7} - 20311 x^{6} - 22309 x^{5} + 38415 x^{4} + 8429 x^{3} - 22584 x^{2} - 399 x + 3381$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1148) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{14}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{3} q^{5} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{3} q^{5} + ( 2 + \beta_{2} ) q^{9} + ( 1 - \beta_{3} + \beta_{5} - \beta_{12} ) q^{11} -\beta_{14} q^{13} + ( \beta_{2} + \beta_{5} + \beta_{8} - \beta_{12} + \beta_{14} ) q^{15} + \beta_{9} q^{17} + ( -1 + \beta_{2} - \beta_{3} - \beta_{10} + \beta_{11} ) q^{19} + ( \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{23} + ( 2 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{25} + ( -1 + \beta_{1} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{27} + ( 1 + \beta_{1} - \beta_{5} - \beta_{6} ) q^{29} + ( -2 + \beta_{1} - \beta_{3} - \beta_{12} ) q^{31} + ( 1 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{33} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} + \beta_{10} - \beta_{12} ) q^{37} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{39} + q^{41} + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{43} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{9} + \beta_{10} + \beta_{13} ) q^{45} + ( 2 + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{12} - \beta_{13} ) q^{47} + ( 3 + 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{12} + \beta_{14} ) q^{51} + ( 1 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{53} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{55} + ( 2 + \beta_{1} - 2 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{57} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{10} - \beta_{13} ) q^{59} + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{61} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 3 \beta_{7} - \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{65} + ( 1 - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{14} ) q^{67} + ( -2 - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{69} + ( -\beta_{1} - \beta_{2} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{71} + ( 1 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{73} + ( 4 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{6} + \beta_{8} - \beta_{10} - 3 \beta_{11} + \beta_{12} + \beta_{14} ) q^{75} + ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - 2 \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{79} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} + \beta_{14} ) q^{81} + ( -1 - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{83} + ( 3 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{10} - 3 \beta_{11} + \beta_{12} - \beta_{13} ) q^{85} + ( 2 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{11} - \beta_{12} + \beta_{13} ) q^{87} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} + \beta_{13} + \beta_{14} ) q^{89} + ( 4 - 3 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{93} + ( -2 + 3 \beta_{1} - \beta_{2} - \beta_{5} - 3 \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} ) q^{95} + ( \beta_{1} + \beta_{3} + \beta_{5} - \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} ) q^{97} + ( \beta_{1} + \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} - 3 \beta_{13} + \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$15q + q^{3} - 3q^{5} + 30q^{9} + O(q^{10})$$ $$15q + q^{3} - 3q^{5} + 30q^{9} + 9q^{11} - 7q^{13} + 2q^{15} - 3q^{17} - 7q^{19} - q^{23} + 32q^{25} - 11q^{27} + 18q^{29} - 30q^{31} + 16q^{33} + 23q^{37} + 5q^{39} + 15q^{41} + 12q^{43} + 13q^{45} + 16q^{47} + 29q^{51} + 33q^{53} - 37q^{55} + 16q^{57} + 10q^{59} - q^{61} + 16q^{65} + 20q^{67} - 21q^{69} + 5q^{71} + 3q^{73} + 51q^{75} + 25q^{79} + 43q^{81} - 18q^{83} + 36q^{85} + 53q^{87} + 11q^{89} + 65q^{93} - 30q^{95} - 16q^{97} - 18q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + 13971 x^{7} - 20311 x^{6} - 22309 x^{5} + 38415 x^{4} + 8429 x^{3} - 22584 x^{2} - 399 x + 3381$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ $$\beta_{3}$$ $$=$$ $$($$$$21568415171 \nu^{14} + 33366538546 \nu^{13} - 780431594905 \nu^{12} - 1076896328346 \nu^{11} + 11058300703621 \nu^{10} + 12550722332099 \nu^{9} - 78350006830010 \nu^{8} - 63368265036347 \nu^{7} + 290964255861732 \nu^{6} + 125086901622927 \nu^{5} - 518478043052718 \nu^{4} - 34154729402677 \nu^{3} + 315331689321588 \nu^{2} - 33588397215636 \nu - 53909708702913$$$$)/ 8954190428232$$ $$\beta_{4}$$ $$=$$ $$($$$$207367915591 \nu^{14} + 75537732154 \nu^{13} - 6864511642829 \nu^{12} - 2264494596922 \nu^{11} + 84630564241705 \nu^{10} + 19997951254183 \nu^{9} - 471109920542042 \nu^{8} - 26134945978879 \nu^{7} + 1061905945918156 \nu^{6} - 337765061235149 \nu^{5} - 14256388065710 \nu^{4} + 1086025882031975 \nu^{3} - 2504020310672772 \nu^{2} - 382307010614964 \nu + 964719687046731$$$$)/ 53725142569392$$ $$\beta_{5}$$ $$=$$ $$($$$$375675670889 \nu^{14} + 197307815606 \nu^{13} - 13458627125155 \nu^{12} - 5416632772358 \nu^{11} + 187920186390263 \nu^{10} + 41971019614937 \nu^{9} - 1297715569983766 \nu^{8} + 1604986959871 \nu^{7} + 4597551204850484 \nu^{6} - 1062369256372771 \nu^{5} - 7504847710757362 \nu^{4} + 2957473201007065 \nu^{3} + 3676894843258596 \nu^{2} - 925571855654028 \nu - 394220916659787$$$$)/ 53725142569392$$ $$\beta_{6}$$ $$=$$ $$($$$$563847518531 \nu^{14} + 271104011042 \nu^{13} - 20576643380593 \nu^{12} - 7746814774274 \nu^{11} + 294326513071613 \nu^{10} + 65601128365283 \nu^{9} - 2097712145040178 \nu^{8} - 64560226435499 \nu^{7} + 7768694051421020 \nu^{6} - 1280840055975121 \nu^{5} - 13752243630567190 \nu^{4} + 4090409019404467 \nu^{3} + 8604345326302092 \nu^{2} - 1583632566957636 \nu - 1413258738046953$$$$)/ 53725142569392$$ $$\beta_{7}$$ $$=$$ $$($$$$82318075711 \nu^{14} + 110890419958 \nu^{13} - 2903659812299 \nu^{12} - 3508288802422 \nu^{11} + 39739641971875 \nu^{10} + 39661374923485 \nu^{9} - 268865569213064 \nu^{8} - 191585391007741 \nu^{7} + 943989351011626 \nu^{6} + 359635444491391 \nu^{5} - 1587383470520204 \nu^{4} - 134303162335843 \nu^{3} + 916971510040800 \nu^{2} + 51742333507248 \nu - 123529112322861$$$$)/ 6715642821174$$ $$\beta_{8}$$ $$=$$ $$($$$$57933555161 \nu^{14} + 19902931966 \nu^{13} - 2038445185159 \nu^{12} - 636604082118 \nu^{11} + 27895029041707 \nu^{10} + 6543610028837 \nu^{9} - 188732383570406 \nu^{8} - 21783985515293 \nu^{7} + 661794329691888 \nu^{6} - 12151856350959 \nu^{5} - 1127435686751094 \nu^{4} + 128904705950189 \nu^{3} + 743133409360908 \nu^{2} - 36292597607964 \nu - 132293240921499$$$$)/ 4477095214116$$ $$\beta_{9}$$ $$=$$ $$($$$$263299916363 \nu^{14} + 284951510738 \nu^{13} - 9037992446569 \nu^{12} - 9319581543506 \nu^{11} + 119443657067957 \nu^{10} + 110071063789547 \nu^{9} - 772667003836402 \nu^{8} - 576510242603315 \nu^{7} + 2574917570991548 \nu^{6} + 1337367111898247 \nu^{5} - 4158684124869190 \nu^{4} - 1188084234706517 \nu^{3} + 2549675888785068 \nu^{2} + 395328021911436 \nu - 369589303008177$$$$)/ 17908380856464$$ $$\beta_{10}$$ $$=$$ $$($$$$-803771605457 \nu^{14} + 55554951370 \nu^{13} + 29873873602171 \nu^{12} - 2773231651450 \nu^{11} - 437414324713775 \nu^{10} + 64143993774511 \nu^{9} + 3202662100683478 \nu^{8} - 708242318071255 \nu^{7} - 12199681840975508 \nu^{6} + 3575923666457563 \nu^{5} + 22280485010412706 \nu^{4} - 7066331684287153 \nu^{3} - 14659545336879396 \nu^{2} + 2212465125784428 \nu + 2285567947233123$$$$)/ 53725142569392$$ $$\beta_{11}$$ $$=$$ $$($$$$-527518007087 \nu^{14} - 247737754298 \nu^{13} + 19237736393941 \nu^{12} + 7202431283834 \nu^{11} - 274960302342017 \nu^{10} - 63910333187519 \nu^{9} + 1958752009116682 \nu^{8} + 105751315240871 \nu^{7} - 7259458623838364 \nu^{6} + 925907072139781 \nu^{5} + 12894254808423982 \nu^{4} - 3195696128916367 \nu^{3} - 8114892589426428 \nu^{2} + 1033679062742580 \nu + 1300628388320541$$$$)/ 17908380856464$$ $$\beta_{12}$$ $$=$$ $$($$$$-1991383769819 \nu^{14} - 970794976610 \nu^{13} + 72904083529129 \nu^{12} + 29524283624978 \nu^{11} - 1045949435856725 \nu^{10} - 291365982179867 \nu^{9} + 7472488513327378 \nu^{8} + 870143948269859 \nu^{7} - 27720511397395436 \nu^{6} + 1482781049982937 \nu^{5} + 49218218809824022 \nu^{4} - 8617048669258075 \nu^{3} - 31166858944757532 \nu^{2} + 2619386748959364 \nu + 5020217462879745$$$$)/ 53725142569392$$ $$\beta_{13}$$ $$=$$ $$($$$$154414062065 \nu^{14} + 69132074966 \nu^{13} - 5550942035752 \nu^{12} - 2103794805215 \nu^{11} + 77954228306660 \nu^{10} + 20374588096121 \nu^{9} - 543496021568218 \nu^{8} - 55206873249419 \nu^{7} + 1963368044552309 \nu^{6} - 139122061304389 \nu^{5} - 3386733041584858 \nu^{4} + 660872437651219 \nu^{3} + 2062137225183060 \nu^{2} - 196898900419467 \nu - 312802565393409$$$$)/ 3357821410587$$ $$\beta_{14}$$ $$=$$ $$($$$$-1366326190169 \nu^{14} - 650669688638 \nu^{13} + 49657832878051 \nu^{12} + 19718267961758 \nu^{11} - 706794387141431 \nu^{10} - 191670870598865 \nu^{9} + 5008459392835390 \nu^{8} + 533869176269825 \nu^{7} - 18440308332790964 \nu^{6} + 1233561484274971 \nu^{5} + 32538031186207330 \nu^{4} - 6160094617298017 \nu^{3} - 20547001848439524 \nu^{2} + 1854323154805428 \nu + 3416523056642787$$$$)/ 26862571284696$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 7 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{14} + \beta_{13} - \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} + 11 \beta_{2} + \beta_{1} + 38$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{14} + 2 \beta_{12} + \beta_{10} + 14 \beta_{9} - 16 \beta_{8} - 14 \beta_{7} + 12 \beta_{6} - \beta_{5} - 4 \beta_{3} + 59 \beta_{1} - 5$$ $$\nu^{6}$$ $$=$$ $$14 \beta_{14} + 14 \beta_{13} + \beta_{12} + 2 \beta_{11} - 17 \beta_{10} + 17 \beta_{9} - 32 \beta_{8} + \beta_{7} + 23 \beta_{6} - 19 \beta_{5} + \beta_{4} - 21 \beta_{3} + 116 \beta_{2} + 27 \beta_{1} + 333$$ $$\nu^{7}$$ $$=$$ $$-62 \beta_{14} - 10 \beta_{13} + 44 \beta_{12} - 7 \beta_{11} + 12 \beta_{10} + 170 \beta_{9} - 202 \beta_{8} - 166 \beta_{7} + 132 \beta_{6} - 29 \beta_{5} + 10 \beta_{4} - 67 \beta_{3} + 7 \beta_{2} + 558 \beta_{1} + 38$$ $$\nu^{8}$$ $$=$$ $$147 \beta_{14} + 152 \beta_{13} + 30 \beta_{12} + 60 \beta_{11} - 243 \beta_{10} + 234 \beta_{9} - 419 \beta_{8} + 27 \beta_{7} + 370 \beta_{6} - 263 \beta_{5} + 28 \beta_{4} - 320 \beta_{3} + 1240 \beta_{2} + 451 \beta_{1} + 3159$$ $$\nu^{9}$$ $$=$$ $$-927 \beta_{14} - 256 \beta_{13} + 685 \beta_{12} - 156 \beta_{11} + 52 \beta_{10} + 2001 \beta_{9} - 2390 \beta_{8} - 1858 \beta_{7} + 1452 \beta_{6} - 508 \beta_{5} + 250 \beta_{4} - 903 \beta_{3} + 228 \beta_{2} + 5654 \beta_{1} + 1342$$ $$\nu^{10}$$ $$=$$ $$1327 \beta_{14} + 1492 \beta_{13} + 627 \beta_{12} + 1096 \beta_{11} - 3269 \beta_{10} + 3039 \beta_{9} - 5210 \beta_{8} + 459 \beta_{7} + 5154 \beta_{6} - 3302 \beta_{5} + 521 \beta_{4} - 4318 \beta_{3} + 13447 \beta_{2} + 6366 \beta_{1} + 31539$$ $$\nu^{11}$$ $$=$$ $$-12261 \beta_{14} - 4464 \beta_{13} + 9349 \beta_{12} - 2401 \beta_{11} - 866 \beta_{10} + 23339 \beta_{9} - 27666 \beta_{8} - 20283 \beta_{7} + 16290 \beta_{6} - 7403 \beta_{5} + 4286 \beta_{4} - 11529 \beta_{3} + 4656 \beta_{2} + 59734 \beta_{1} + 23366$$ $$\nu^{12}$$ $$=$$ $$10196 \beta_{14} + 13647 \beta_{13} + 10931 \beta_{12} + 16261 \beta_{11} - 42318 \beta_{10} + 38527 \beta_{9} - 63562 \beta_{8} + 6361 \beta_{7} + 66741 \beta_{6} - 40030 \beta_{5} + 8268 \beta_{4} - 54924 \beta_{3} + 147413 \beta_{2} + 83308 \beta_{1} + 326176$$ $$\nu^{13}$$ $$=$$ $$-152923 \beta_{14} - 66240 \beta_{13} + 119702 \beta_{12} - 31483 \beta_{11} - 28360 \beta_{10} + 271267 \beta_{9} - 318032 \beta_{8} - 218879 \beta_{7} + 186813 \beta_{6} - 99003 \beta_{5} + 62906 \beta_{4} - 144488 \beta_{3} + 77660 \beta_{2} + 647792 \beta_{1} + 337884$$ $$\nu^{14}$$ $$=$$ $$56795 \beta_{14} + 115353 \beta_{13} + 170227 \beta_{12} + 216472 \beta_{11} - 532792 \beta_{10} + 481792 \beta_{9} - 768045 \beta_{8} + 78391 \beta_{7} + 829524 \beta_{6} - 478852 \beta_{5} + 120857 \beta_{4} - 676566 \beta_{3} + 1629219 \beta_{2} + 1048772 \beta_{1} + 3460210$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.24213 −2.90634 −2.73742 −2.47735 −2.11769 −0.741382 −0.449590 0.515082 1.27444 1.40418 1.72996 1.93225 2.29181 3.10047 3.42372
0 −3.24213 0 −1.49679 0 0 0 7.51143 0
1.2 0 −2.90634 0 −0.506103 0 0 0 5.44682 0
1.3 0 −2.73742 0 0.863054 0 0 0 4.49344 0
1.4 0 −2.47735 0 3.49844 0 0 0 3.13725 0
1.5 0 −2.11769 0 −3.93656 0 0 0 1.48461 0
1.6 0 −0.741382 0 2.48663 0 0 0 −2.45035 0
1.7 0 −0.449590 0 0.787803 0 0 0 −2.79787 0
1.8 0 0.515082 0 −2.19964 0 0 0 −2.73469 0
1.9 0 1.27444 0 −4.37429 0 0 0 −1.37580 0
1.10 0 1.40418 0 −3.61567 0 0 0 −1.02828 0
1.11 0 1.72996 0 0.138351 0 0 0 −0.00725198 0
1.12 0 1.93225 0 0.475465 0 0 0 0.733587 0
1.13 0 2.29181 0 3.92303 0 0 0 2.25239 0
1.14 0 3.10047 0 3.31369 0 0 0 6.61290 0
1.15 0 3.42372 0 −2.35740 0 0 0 8.72183 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$41$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8036.2.a.r 15
7.b odd 2 1 8036.2.a.q 15
7.d odd 6 2 1148.2.i.e 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.i.e 30 7.d odd 6 2
8036.2.a.q 15 7.b odd 2 1
8036.2.a.r 15 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8036))$$:

 $$T_{3}^{15} - \cdots$$ $$T_{5}^{15} + \cdots$$ $$T_{11}^{15} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T + 8 T^{2} - 3 T^{3} + 39 T^{4} - 31 T^{5} + 179 T^{6} - 213 T^{7} + 606 T^{8} - 916 T^{9} + 1919 T^{10} - 3696 T^{11} + 6947 T^{12} - 12504 T^{13} + 21048 T^{14} - 35139 T^{15} + 63144 T^{16} - 112536 T^{17} + 187569 T^{18} - 299376 T^{19} + 466317 T^{20} - 667764 T^{21} + 1325322 T^{22} - 1397493 T^{23} + 3523257 T^{24} - 1830519 T^{25} + 6908733 T^{26} - 1594323 T^{27} + 12754584 T^{28} - 4782969 T^{29} + 14348907 T^{30}$$
$5$ $$1 + 3 T + 26 T^{2} + 75 T^{3} + 351 T^{4} + 983 T^{5} + 3481 T^{6} + 9515 T^{7} + 29060 T^{8} + 75678 T^{9} + 209221 T^{10} + 513824 T^{11} + 1309803 T^{12} + 3067058 T^{13} + 7286756 T^{14} + 16275117 T^{15} + 36433780 T^{16} + 76676450 T^{17} + 163725375 T^{18} + 321140000 T^{19} + 653815625 T^{20} + 1182468750 T^{21} + 2270312500 T^{22} + 3716796875 T^{23} + 6798828125 T^{24} + 9599609375 T^{25} + 17138671875 T^{26} + 18310546875 T^{27} + 31738281250 T^{28} + 18310546875 T^{29} + 30517578125 T^{30}$$
$7$ 1
$11$ $$1 - 9 T + 89 T^{2} - 498 T^{3} + 2998 T^{4} - 12910 T^{5} + 62896 T^{6} - 244295 T^{7} + 1118943 T^{8} - 4242745 T^{9} + 18355959 T^{10} - 65389764 T^{11} + 257372053 T^{12} - 841875891 T^{13} + 3082353593 T^{14} - 9581848759 T^{15} + 33905889523 T^{16} - 101866982811 T^{17} + 342562202543 T^{18} - 957371534724 T^{19} + 2956245552909 T^{20} - 7516281574945 T^{21} + 21805033580253 T^{22} - 52366802833895 T^{23} + 148305477973136 T^{24} - 334852151598910 T^{25} + 855364388491778 T^{26} - 1562937331607058 T^{27} + 3072521380809859 T^{28} - 3417748502249169 T^{29} + 4177248169415651 T^{30}$$
$13$ $$1 + 7 T + 76 T^{2} + 375 T^{3} + 2621 T^{4} + 10793 T^{5} + 61717 T^{6} + 220271 T^{7} + 1122185 T^{8} + 3552915 T^{9} + 17109259 T^{10} + 48621973 T^{11} + 230086024 T^{12} + 601963629 T^{13} + 2936644665 T^{14} + 7551218442 T^{15} + 38176380645 T^{16} + 101731853301 T^{17} + 505498994728 T^{18} + 1388692170853 T^{19} + 6352548101887 T^{20} + 17149242098235 T^{21} + 70415444549645 T^{22} + 179681821645391 T^{23} + 654477887803441 T^{24} + 1487906702526257 T^{25} + 4697252392770977 T^{26} + 8736781920930375 T^{27} + 23018508101011228 T^{28} + 27561634699895023 T^{29} + 51185893014090757 T^{30}$$
$17$ $$1 + 3 T + 50 T^{2} + 128 T^{3} + 1899 T^{4} + 4363 T^{5} + 49157 T^{6} + 88690 T^{7} + 1034344 T^{8} + 1427757 T^{9} + 18105190 T^{10} + 15514456 T^{11} + 284744193 T^{12} + 98181573 T^{13} + 4458511390 T^{14} + 907579244 T^{15} + 75794693630 T^{16} + 28374474597 T^{17} + 1398948220209 T^{18} + 1295782879576 T^{19} + 25706780757830 T^{20} + 34462583102733 T^{21} + 424431344385512 T^{22} + 618679927442290 T^{23} + 5829424244963029 T^{24} + 8795781387658987 T^{25} + 65082331088195067 T^{26} + 74575646365409408 T^{27} + 495228901645296850 T^{28} + 505133479678202787 T^{29} + 2862423051509815793 T^{30}$$
$19$ $$1 + 7 T + 118 T^{2} + 440 T^{3} + 5385 T^{4} + 8099 T^{5} + 160785 T^{6} - 56883 T^{7} + 4342870 T^{8} - 8044488 T^{9} + 105632963 T^{10} - 341081490 T^{11} + 2256115895 T^{12} - 9680538834 T^{13} + 45674046668 T^{14} - 205914593365 T^{15} + 867806886692 T^{16} - 3494674519074 T^{17} + 15474698923805 T^{18} - 44450080858290 T^{19} + 261557674051337 T^{20} - 378460025153928 T^{21} + 3881968759150930 T^{22} - 966076016461203 T^{23} + 51883341487396515 T^{24} + 49655505621930299 T^{25} + 627300044166909315 T^{26} + 973858564389110840 T^{27} + 4962252048546332962 T^{28} + 5593046800480188847 T^{29} + 15181127029874798299 T^{30}$$
$23$ $$1 + T + 100 T^{2} - 240 T^{3} + 5823 T^{4} - 23139 T^{5} + 307392 T^{6} - 1276767 T^{7} + 12804893 T^{8} - 57452087 T^{9} + 449235292 T^{10} - 2003430672 T^{11} + 13871296909 T^{12} - 58567273319 T^{13} + 364132115290 T^{14} - 1474508609730 T^{15} + 8375038651670 T^{16} - 30982087585751 T^{17} + 168772069491803 T^{18} - 560642042683152 T^{19} + 2891432427017156 T^{20} - 8504970773950343 T^{21} + 43598425532512171 T^{22} - 99984881744266527 T^{23} + 553659918912434496 T^{24} - 958568042972624211 T^{25} + 5548211220332796921 T^{26} - 5259509863684877040 T^{27} + 50403636193646738300 T^{28} + 11592836324538749809 T^{29} +$$$$26\!\cdots\!07$$$$T^{30}$$
$29$ $$1 - 18 T + 351 T^{2} - 3938 T^{3} + 45998 T^{4} - 391789 T^{5} + 3532456 T^{6} - 25324251 T^{7} + 196317762 T^{8} - 1257094469 T^{9} + 8801517748 T^{10} - 51641736178 T^{11} + 332842157595 T^{12} - 1810862813908 T^{13} + 10895718454521 T^{14} - 55671549974722 T^{15} + 315975835181109 T^{16} - 1522935626496628 T^{17} + 8117687381584455 T^{18} - 36525218805712018 T^{19} + 180529241955372452 T^{20} - 747749106861311549 T^{21} + 3386457111759700458 T^{22} - 12668365723674017211 T^{23} + 51245854845334304264 T^{24} -$$$$16\!\cdots\!89$$$$T^{25} +$$$$56\!\cdots\!42$$$$T^{26} -$$$$13\!\cdots\!58$$$$T^{27} +$$$$36\!\cdots\!39$$$$T^{28} -$$$$53\!\cdots\!58$$$$T^{29} +$$$$86\!\cdots\!49$$$$T^{30}$$
$31$ $$1 + 30 T + 729 T^{2} + 12702 T^{3} + 192287 T^{4} + 2469077 T^{5} + 28541588 T^{6} + 295094480 T^{7} + 2801338765 T^{8} + 24357725643 T^{9} + 196723756650 T^{10} + 1473344847030 T^{11} + 10319381498616 T^{12} + 67463739856866 T^{13} + 413955859752302 T^{14} + 2377693087445112 T^{15} + 12832631652321362 T^{16} + 64832654002448226 T^{17} + 307424694225269256 T^{18} + 1360664906473992630 T^{19} + 5632034134420104150 T^{20} + 21617571168950591883 T^{21} + 77072152435630312915 T^{22} +$$$$25\!\cdots\!80$$$$T^{23} +$$$$75\!\cdots\!48$$$$T^{24} +$$$$20\!\cdots\!77$$$$T^{25} +$$$$48\!\cdots\!97$$$$T^{26} +$$$$10\!\cdots\!22$$$$T^{27} +$$$$17\!\cdots\!39$$$$T^{28} +$$$$22\!\cdots\!30$$$$T^{29} +$$$$23\!\cdots\!51$$$$T^{30}$$
$37$ $$1 - 23 T + 400 T^{2} - 5145 T^{3} + 58100 T^{4} - 567646 T^{5} + 5181174 T^{6} - 43727378 T^{7} + 355029388 T^{8} - 2734178148 T^{9} + 20402693048 T^{10} - 144984533863 T^{11} + 997358220188 T^{12} - 6561885310287 T^{13} + 41976916551657 T^{14} - 258407011673340 T^{15} + 1553145912411309 T^{16} - 8983220989782903 T^{17} + 50519185927182764 T^{18} - 271724358969213943 T^{19} + 1414803469404710936 T^{20} - 7015153081234310532 T^{21} + 33703606240220184604 T^{22} -$$$$15\!\cdots\!38$$$$T^{23} +$$$$67\!\cdots\!98$$$$T^{24} -$$$$27\!\cdots\!54$$$$T^{25} +$$$$10\!\cdots\!00$$$$T^{26} -$$$$33\!\cdots\!45$$$$T^{27} +$$$$97\!\cdots\!00$$$$T^{28} -$$$$20\!\cdots\!47$$$$T^{29} +$$$$33\!\cdots\!93$$$$T^{30}$$
$41$ $$( 1 - T )^{15}$$
$43$ $$1 - 12 T + 269 T^{2} - 2316 T^{3} + 34068 T^{4} - 253607 T^{5} + 3153596 T^{6} - 21954583 T^{7} + 240692362 T^{8} - 1568196355 T^{9} + 15411762010 T^{10} - 94482346296 T^{11} + 854049383983 T^{12} - 4960571657906 T^{13} + 41710893999939 T^{14} - 228266308319210 T^{15} + 1793568441997377 T^{16} - 9172096995468194 T^{17} + 67902904372336381 T^{18} - 323016339999111096 T^{19} + 2265659136976650430 T^{20} - 9913138492073486395 T^{21} + 65424663542903264734 T^{22} -$$$$25\!\cdots\!83$$$$T^{23} +$$$$15\!\cdots\!28$$$$T^{24} -$$$$54\!\cdots\!43$$$$T^{25} +$$$$31\!\cdots\!76$$$$T^{26} -$$$$92\!\cdots\!16$$$$T^{27} +$$$$46\!\cdots\!67$$$$T^{28} -$$$$88\!\cdots\!88$$$$T^{29} +$$$$31\!\cdots\!07$$$$T^{30}$$
$47$ $$1 - 16 T + 460 T^{2} - 6182 T^{3} + 103105 T^{4} - 1195645 T^{5} + 15130362 T^{6} - 154307107 T^{7} + 1634434125 T^{8} - 14876354705 T^{9} + 137984024424 T^{10} - 1134183253390 T^{11} + 9424749493183 T^{12} - 70531837481534 T^{13} + 531469788689268 T^{14} - 3633615760597546 T^{15} + 24979080068395596 T^{16} - 155804828996708606 T^{17} + 978505766630738609 T^{18} - 5534452472085368590 T^{19} + 31645947047410450968 T^{20} -$$$$16\!\cdots\!45$$$$T^{21} +$$$$82\!\cdots\!75$$$$T^{22} -$$$$36\!\cdots\!27$$$$T^{23} +$$$$16\!\cdots\!54$$$$T^{24} -$$$$62\!\cdots\!05$$$$T^{25} +$$$$25\!\cdots\!15$$$$T^{26} -$$$$71\!\cdots\!62$$$$T^{27} +$$$$25\!\cdots\!20$$$$T^{28} -$$$$41\!\cdots\!04$$$$T^{29} +$$$$12\!\cdots\!43$$$$T^{30}$$
$53$ $$1 - 33 T + 863 T^{2} - 15722 T^{3} + 246347 T^{4} - 3183252 T^{5} + 36912998 T^{6} - 375975233 T^{7} + 3571763601 T^{8} - 31328295324 T^{9} + 268476858640 T^{10} - 2219388789806 T^{11} + 18375669724390 T^{12} - 146996823158855 T^{13} + 1154099940434404 T^{14} - 8551045524129902 T^{15} + 61167296843023412 T^{16} - 412914076253223695 T^{17} + 2735714581558010030 T^{18} - 17512045077577236686 T^{19} +$$$$11\!\cdots\!20$$$$T^{20} -$$$$69\!\cdots\!96$$$$T^{21} +$$$$41\!\cdots\!37$$$$T^{22} -$$$$23\!\cdots\!13$$$$T^{23} +$$$$12\!\cdots\!34$$$$T^{24} -$$$$55\!\cdots\!48$$$$T^{25} +$$$$22\!\cdots\!59$$$$T^{26} -$$$$77\!\cdots\!02$$$$T^{27} +$$$$22\!\cdots\!99$$$$T^{28} -$$$$45\!\cdots\!77$$$$T^{29} +$$$$73\!\cdots\!57$$$$T^{30}$$
$59$ $$1 - 10 T + 438 T^{2} - 3184 T^{3} + 90647 T^{4} - 466961 T^{5} + 11795042 T^{6} - 34131003 T^{7} + 1053566807 T^{8} + 72516125 T^{9} + 66451314548 T^{10} + 316500930128 T^{11} + 3005689076311 T^{12} + 40055060945190 T^{13} + 114526107347490 T^{14} + 2931515365957526 T^{15} + 6757040333501910 T^{16} + 139431667150206390 T^{17} + 617305416803676869 T^{18} + 3835156027196752208 T^{19} + 47507659470857401852 T^{20} + 3058768850077461125 T^{21} +$$$$26\!\cdots\!33$$$$T^{22} -$$$$50\!\cdots\!63$$$$T^{23} +$$$$10\!\cdots\!38$$$$T^{24} -$$$$23\!\cdots\!61$$$$T^{25} +$$$$27\!\cdots\!73$$$$T^{26} -$$$$56\!\cdots\!04$$$$T^{27} +$$$$45\!\cdots\!02$$$$T^{28} -$$$$61\!\cdots\!10$$$$T^{29} +$$$$36\!\cdots\!99$$$$T^{30}$$
$61$ $$1 + T + 523 T^{2} + 129 T^{3} + 136932 T^{4} - 51712 T^{5} + 23976824 T^{6} - 20622714 T^{7} + 3155814965 T^{8} - 3736521011 T^{9} + 331449399927 T^{10} - 447634864256 T^{11} + 28677224204791 T^{12} - 39659467357703 T^{13} + 2076688162703281 T^{14} - 2726716307441572 T^{15} + 126677977924900141 T^{16} - 147572878038012863 T^{17} + 6509185027227665971 T^{18} - 6197881156545159296 T^{19} +$$$$27\!\cdots\!27$$$$T^{20} -$$$$19\!\cdots\!71$$$$T^{21} +$$$$99\!\cdots\!65$$$$T^{22} -$$$$39\!\cdots\!34$$$$T^{23} +$$$$28\!\cdots\!84$$$$T^{24} -$$$$36\!\cdots\!12$$$$T^{25} +$$$$59\!\cdots\!52$$$$T^{26} +$$$$34\!\cdots\!09$$$$T^{27} +$$$$84\!\cdots\!63$$$$T^{28} +$$$$98\!\cdots\!41$$$$T^{29} +$$$$60\!\cdots\!01$$$$T^{30}$$
$67$ $$1 - 20 T + 429 T^{2} - 5822 T^{3} + 73632 T^{4} - 722379 T^{5} + 6792071 T^{6} - 53130812 T^{7} + 435608371 T^{8} - 3414484347 T^{9} + 30999954694 T^{10} - 278288722590 T^{11} + 2671766006547 T^{12} - 23401501064390 T^{13} + 206689542780553 T^{14} - 1689647377929920 T^{15} + 13848199366297051 T^{16} - 105049338278046710 T^{17} + 803568359427095361 T^{18} - 5607829721846523390 T^{19} + 41853817148231902258 T^{20} -$$$$30\!\cdots\!43$$$$T^{21} +$$$$26\!\cdots\!33$$$$T^{22} -$$$$21\!\cdots\!92$$$$T^{23} +$$$$18\!\cdots\!37$$$$T^{24} -$$$$13\!\cdots\!71$$$$T^{25} +$$$$89\!\cdots\!56$$$$T^{26} -$$$$47\!\cdots\!42$$$$T^{27} +$$$$23\!\cdots\!23$$$$T^{28} -$$$$73\!\cdots\!80$$$$T^{29} +$$$$24\!\cdots\!43$$$$T^{30}$$
$71$ $$1 - 5 T + 673 T^{2} - 3977 T^{3} + 223612 T^{4} - 1485542 T^{5} + 49080265 T^{6} - 348399794 T^{7} + 8003819994 T^{8} - 57819268023 T^{9} + 1028517214635 T^{10} - 7247014954358 T^{11} + 107290099585961 T^{12} - 713261170497997 T^{13} + 9210290678641452 T^{14} - 56332695950174480 T^{15} + 653930638183543092 T^{16} - 3595549560480402877 T^{17} + 38400306832910887471 T^{18} -$$$$18\!\cdots\!98$$$$T^{19} +$$$$18\!\cdots\!85$$$$T^{20} -$$$$74\!\cdots\!83$$$$T^{21} +$$$$72\!\cdots\!54$$$$T^{22} -$$$$22\!\cdots\!34$$$$T^{23} +$$$$22\!\cdots\!15$$$$T^{24} -$$$$48\!\cdots\!42$$$$T^{25} +$$$$51\!\cdots\!52$$$$T^{26} -$$$$65\!\cdots\!57$$$$T^{27} +$$$$78\!\cdots\!03$$$$T^{28} -$$$$41\!\cdots\!05$$$$T^{29} +$$$$58\!\cdots\!51$$$$T^{30}$$
$73$ $$1 - 3 T + 583 T^{2} - 555 T^{3} + 162403 T^{4} + 132936 T^{5} + 29712937 T^{6} + 66363985 T^{7} + 4114280860 T^{8} + 13241903644 T^{9} + 463723711956 T^{10} + 1765474643243 T^{11} + 44185799832094 T^{12} + 179873270172127 T^{13} + 3652540420576382 T^{14} + 14631397178604766 T^{15} + 266635450702075886 T^{16} + 958544656747264783 T^{17} + 17189027293281711598 T^{18} + 50136374398203735563 T^{19} +$$$$96\!\cdots\!08$$$$T^{20} +$$$$20\!\cdots\!16$$$$T^{21} +$$$$45\!\cdots\!20$$$$T^{22} +$$$$53\!\cdots\!85$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24} +$$$$57\!\cdots\!64$$$$T^{25} +$$$$50\!\cdots\!31$$$$T^{26} -$$$$12\!\cdots\!55$$$$T^{27} +$$$$97\!\cdots\!39$$$$T^{28} -$$$$36\!\cdots\!27$$$$T^{29} +$$$$89\!\cdots\!57$$$$T^{30}$$
$79$ $$1 - 25 T + 867 T^{2} - 15594 T^{3} + 324404 T^{4} - 4707655 T^{5} + 75412400 T^{6} - 941871353 T^{7} + 12777037715 T^{8} - 142497901288 T^{9} + 1711757913945 T^{10} - 17371634352920 T^{11} + 188916919425901 T^{12} - 1760893570473008 T^{13} + 17547537073774455 T^{14} - 150880781779786061 T^{15} + 1386255428828181945 T^{16} - 10989736773322042928 T^{17} + 93143409036826803139 T^{18} -$$$$67\!\cdots\!20$$$$T^{19} +$$$$52\!\cdots\!55$$$$T^{20} -$$$$34\!\cdots\!48$$$$T^{21} +$$$$24\!\cdots\!85$$$$T^{22} -$$$$14\!\cdots\!33$$$$T^{23} +$$$$90\!\cdots\!00$$$$T^{24} -$$$$44\!\cdots\!55$$$$T^{25} +$$$$24\!\cdots\!16$$$$T^{26} -$$$$92\!\cdots\!54$$$$T^{27} +$$$$40\!\cdots\!13$$$$T^{28} -$$$$92\!\cdots\!25$$$$T^{29} +$$$$29\!\cdots\!99$$$$T^{30}$$
$83$ $$1 + 18 T + 957 T^{2} + 13278 T^{3} + 406532 T^{4} + 4526403 T^{5} + 104640076 T^{6} + 950016557 T^{7} + 18638437942 T^{8} + 138413158803 T^{9} + 2492847506614 T^{10} + 15191914173358 T^{11} + 268012339072371 T^{12} + 1377333735196800 T^{13} + 24729349092135191 T^{14} + 115390514275133166 T^{15} + 2052535974647220853 T^{16} + 9488452101770755200 T^{17} +$$$$15\!\cdots\!77$$$$T^{18} +$$$$72\!\cdots\!18$$$$T^{19} +$$$$98\!\cdots\!02$$$$T^{20} +$$$$45\!\cdots\!07$$$$T^{21} +$$$$50\!\cdots\!34$$$$T^{22} +$$$$21\!\cdots\!37$$$$T^{23} +$$$$19\!\cdots\!28$$$$T^{24} +$$$$70\!\cdots\!47$$$$T^{25} +$$$$52\!\cdots\!44$$$$T^{26} +$$$$14\!\cdots\!58$$$$T^{27} +$$$$84\!\cdots\!91$$$$T^{28} +$$$$13\!\cdots\!22$$$$T^{29} +$$$$61\!\cdots\!07$$$$T^{30}$$
$89$ $$1 - 11 T + 645 T^{2} - 8338 T^{3} + 217386 T^{4} - 3018655 T^{5} + 52577251 T^{6} - 707493399 T^{7} + 10081479552 T^{8} - 123380688221 T^{9} + 1563746987133 T^{10} - 17312911230666 T^{11} + 197999214628080 T^{12} - 2021153054738361 T^{13} + 20843092796609928 T^{14} - 197072121359103194 T^{15} + 1855035258898283592 T^{16} - 16009553346582557481 T^{17} +$$$$13\!\cdots\!20$$$$T^{18} -$$$$10\!\cdots\!06$$$$T^{19} +$$$$87\!\cdots\!17$$$$T^{20} -$$$$61\!\cdots\!81$$$$T^{21} +$$$$44\!\cdots\!08$$$$T^{22} -$$$$27\!\cdots\!19$$$$T^{23} +$$$$18\!\cdots\!59$$$$T^{24} -$$$$94\!\cdots\!55$$$$T^{25} +$$$$60\!\cdots\!54$$$$T^{26} -$$$$20\!\cdots\!98$$$$T^{27} +$$$$14\!\cdots\!05$$$$T^{28} -$$$$21\!\cdots\!51$$$$T^{29} +$$$$17\!\cdots\!49$$$$T^{30}$$
$97$ $$1 + 16 T + 870 T^{2} + 13555 T^{3} + 394150 T^{4} + 5742512 T^{5} + 120559588 T^{6} + 1613534841 T^{7} + 27451908702 T^{8} + 335768070768 T^{9} + 4892169113610 T^{10} + 54643508317377 T^{11} + 702367249354834 T^{12} + 7159961181650448 T^{13} + 82579753815804149 T^{14} + 766573984600929574 T^{15} + 8010236120133002453 T^{16} + 67368074758149065232 T^{17} +$$$$64\!\cdots\!82$$$$T^{18} +$$$$48\!\cdots\!37$$$$T^{19} +$$$$42\!\cdots\!70$$$$T^{20} +$$$$27\!\cdots\!72$$$$T^{21} +$$$$22\!\cdots\!26$$$$T^{22} +$$$$12\!\cdots\!01$$$$T^{23} +$$$$91\!\cdots\!96$$$$T^{24} +$$$$42\!\cdots\!88$$$$T^{25} +$$$$28\!\cdots\!50$$$$T^{26} +$$$$94\!\cdots\!55$$$$T^{27} +$$$$58\!\cdots\!90$$$$T^{28} +$$$$10\!\cdots\!04$$$$T^{29} +$$$$63\!\cdots\!93$$$$T^{30}$$