# Properties

 Label 36.6.e.a Level $36$ Weight $6$ Character orbit 36.e Analytic conductor $5.774$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 36.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.77381751327$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 175 x^{8} + 8800 x^{6} + 124623 x^{4} + 498609 x^{2} + 442368$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}\cdot 3^{10}$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{3} + ( 4 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{5} + ( 6 + 6 \beta_{1} + \beta_{2} - \beta_{7} + \beta_{9} ) q^{7} + ( 21 + 41 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q + ( 2 + 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{3} + ( 4 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{5} + ( 6 + 6 \beta_{1} + \beta_{2} - \beta_{7} + \beta_{9} ) q^{7} + ( 21 + 41 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{9} + ( 33 + 34 \beta_{1} + \beta_{2} - 8 \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{11} + ( 37 \beta_{1} - 7 \beta_{3} + 3 \beta_{4} + \beta_{6} + 8 \beta_{7} - 3 \beta_{8} ) q^{13} + ( 73 + 119 \beta_{1} + 9 \beta_{2} + 7 \beta_{3} - \beta_{4} - 5 \beta_{5} - 3 \beta_{6} + 6 \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{15} + ( 222 + 8 \beta_{1} - 2 \beta_{2} - 56 \beta_{3} - 19 \beta_{4} + 7 \beta_{5} + 3 \beta_{6} + 8 \beta_{8} - 3 \beta_{9} ) q^{17} + ( -89 - 9 \beta_{1} - 17 \beta_{2} + 28 \beta_{3} - 33 \beta_{4} + 3 \beta_{5} - \beta_{6} - 9 \beta_{8} + \beta_{9} ) q^{19} + ( -126 - 211 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} - 31 \beta_{4} - 18 \beta_{5} + 3 \beta_{6} - 24 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} ) q^{21} + ( 20 - 63 \beta_{1} + 61 \beta_{3} + 120 \beta_{4} + 20 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} + 25 \beta_{8} ) q^{23} + ( -889 - 925 \beta_{1} + 33 \beta_{2} + 111 \beta_{3} + 189 \beta_{4} + 6 \beta_{5} - 33 \beta_{7} - 30 \beta_{8} - 12 \beta_{9} ) q^{25} + ( -422 + 584 \beta_{1} - 39 \beta_{2} + 31 \beta_{3} + 2 \beta_{4} - 38 \beta_{5} - 6 \beta_{6} + 21 \beta_{7} - 10 \beta_{8} - 9 \beta_{9} ) q^{27} + ( -1243 - 1243 \beta_{1} - 2 \beta_{2} + 65 \beta_{3} - 160 \beta_{4} + 45 \beta_{5} + 2 \beta_{7} + 45 \beta_{8} - 27 \beta_{9} ) q^{29} + ( 18 - 487 \beta_{1} - 255 \beta_{3} - 24 \beta_{4} + 18 \beta_{5} - 12 \beta_{6} - 6 \beta_{7} - 39 \beta_{8} ) q^{31} + ( 1867 + 1811 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} - 49 \beta_{4} - 65 \beta_{5} + 21 \beta_{6} - 24 \beta_{7} - 10 \beta_{8} - 27 \beta_{9} ) q^{33} + ( 3676 + 70 \beta_{1} + 7 \beta_{2} - 409 \beta_{3} - 110 \beta_{4} + 50 \beta_{5} - 21 \beta_{6} + 70 \beta_{8} + 21 \beta_{9} ) q^{35} + ( -1515 - 72 \beta_{1} + 23 \beta_{2} + 266 \beta_{3} - 159 \beta_{4} + 3 \beta_{5} + 13 \beta_{6} - 72 \beta_{8} - 13 \beta_{9} ) q^{37} + ( -883 - 2880 \beta_{1} + 33 \beta_{2} + 65 \beta_{3} - 21 \beta_{4} - 69 \beta_{5} - 30 \beta_{6} + 51 \beta_{7} - 18 \beta_{8} + 51 \beta_{9} ) q^{39} + ( 64 + 3750 \beta_{1} + 140 \beta_{3} + 426 \beta_{4} + 64 \beta_{5} + 51 \beta_{6} - 23 \beta_{7} + 71 \beta_{8} ) q^{41} + ( 328 + 283 \beta_{1} - 162 \beta_{2} + 432 \beta_{3} + 477 \beta_{4} + 18 \beta_{5} + 162 \beta_{7} - 27 \beta_{8} + 54 \beta_{9} ) q^{43} + ( -4734 + 3327 \beta_{1} + 210 \beta_{2} - 69 \beta_{3} - 225 \beta_{4} - 66 \beta_{5} + 3 \beta_{6} - 78 \beta_{7} + 9 \beta_{8} + 30 \beta_{9} ) q^{45} + ( -5140 - 5084 \beta_{1} - 33 \beta_{2} - 100 \beta_{3} - 278 \beta_{4} + 2 \beta_{5} + 33 \beta_{7} + 58 \beta_{8} + 87 \beta_{9} ) q^{47} + ( -36 + 814 \beta_{1} - 299 \beta_{3} - 291 \beta_{4} - 36 \beta_{5} + 53 \beta_{6} - 62 \beta_{7} - 69 \beta_{8} ) q^{49} + ( 13193 + 8353 \beta_{1} - 120 \beta_{2} - 46 \beta_{3} - 44 \beta_{4} - 13 \beta_{5} - 27 \beta_{6} + 45 \beta_{7} - 23 \beta_{8} + 84 \beta_{9} ) q^{51} + ( 11659 + 53 \beta_{2} - 162 \beta_{3} - 225 \beta_{4} + 45 \beta_{5} + 27 \beta_{6} - 27 \beta_{9} ) q^{53} + ( 1485 + 90 \beta_{1} + 264 \beta_{2} - 465 \beta_{3} - 105 \beta_{4} + 57 \beta_{5} - 66 \beta_{6} + 90 \beta_{8} + 66 \beta_{9} ) q^{55} + ( -5411 - 16139 \beta_{1} - 273 \beta_{2} + 169 \beta_{3} + 241 \beta_{4} + 24 \beta_{5} + 114 \beta_{6} + 249 \beta_{7} + 42 \beta_{8} - 138 \beta_{9} ) q^{57} + ( -125 + 17926 \beta_{1} + 57 \beta_{3} - 656 \beta_{4} - 125 \beta_{5} - 138 \beta_{6} + 182 \beta_{7} - 100 \beta_{8} ) q^{59} + ( 235 + 307 \beta_{1} + 120 \beta_{2} - 645 \beta_{3} - 486 \beta_{4} - 75 \beta_{5} - 120 \beta_{7} - 3 \beta_{8} - 87 \beta_{9} ) q^{61} + ( -16607 + 6441 \beta_{1} - 166 \beta_{2} - 252 \beta_{3} + 245 \beta_{4} + 173 \beta_{5} + 44 \beta_{6} - 118 \beta_{7} - 55 \beta_{8} - 40 \beta_{9} ) q^{63} + ( -29335 - 29499 \beta_{1} + 140 \beta_{2} + 371 \beta_{3} + 1008 \beta_{4} - 29 \beta_{5} - 140 \beta_{7} - 193 \beta_{8} - 87 \beta_{9} ) q^{65} + ( 81 - 2863 \beta_{1} + 1227 \beta_{3} + 720 \beta_{4} + 81 \beta_{5} - 75 \beta_{6} - 195 \beta_{7} + 333 \beta_{8} ) q^{67} + ( 29879 + 16801 \beta_{1} + 261 \beta_{2} - 25 \beta_{3} - 116 \beta_{4} + 275 \beta_{5} - 132 \beta_{6} - 222 \beta_{7} + 163 \beta_{8} - 57 \beta_{9} ) q^{69} + ( 23217 - 340 \beta_{1} - 347 \beta_{2} + 2542 \beta_{3} + 1505 \beta_{4} - 437 \beta_{5} + 129 \beta_{6} - 340 \beta_{8} - 129 \beta_{9} ) q^{71} + ( 1586 + 72 \beta_{1} - 690 \beta_{2} + 336 \beta_{3} + 1419 \beta_{4} - 255 \beta_{5} + 141 \beta_{6} + 72 \beta_{8} - 141 \beta_{9} ) q^{73} + ( -22545 - 54320 \beta_{1} + 558 \beta_{2} - 522 \beta_{3} - 92 \beta_{4} + 459 \beta_{5} - 171 \beta_{6} - 657 \beta_{7} - 72 \beta_{8} + 18 \beta_{9} ) q^{75} + ( -80 + 42586 \beta_{1} - 1743 \beta_{3} - 1331 \beta_{4} - 80 \beta_{5} + 24 \beta_{6} - 403 \beta_{7} - 316 \beta_{8} ) q^{77} + ( 5165 + 5579 \beta_{1} + 122 \beta_{2} - 1608 \beta_{3} - 3555 \beta_{4} + 141 \beta_{5} - 122 \beta_{7} + 555 \beta_{8} - 112 \beta_{9} ) q^{79} + ( -34143 + 12537 \beta_{1} - 264 \beta_{2} + 927 \beta_{3} + 987 \beta_{4} + 366 \beta_{5} - 66 \beta_{6} + 501 \beta_{7} + 294 \beta_{8} + 6 \beta_{9} ) q^{81} + ( -45496 - 45386 \beta_{1} + 21 \beta_{2} - 1810 \beta_{3} + 910 \beta_{4} - 610 \beta_{5} - 21 \beta_{7} - 500 \beta_{8} - 129 \beta_{9} ) q^{83} + ( -288 + 9189 \beta_{1} + 2004 \beta_{3} - 222 \beta_{4} - 288 \beta_{5} - 165 \beta_{6} + 1065 \beta_{7} + 15 \beta_{8} ) q^{85} + ( 57802 + 35435 \beta_{1} + 69 \beta_{2} + 880 \beta_{3} + 2228 \beta_{4} + 376 \beta_{5} + 399 \beta_{6} + 678 \beta_{7} - 289 \beta_{8} - 243 \beta_{9} ) q^{87} + ( 59847 - 280 \beta_{1} + 529 \beta_{2} + 1150 \beta_{3} - 1135 \beta_{4} + 115 \beta_{5} - 375 \beta_{6} - 280 \beta_{8} + 375 \beta_{9} ) q^{89} + ( 12301 + 666 \beta_{1} + 84 \beta_{2} - 3471 \beta_{3} + 27 \beta_{4} + 261 \beta_{5} + 24 \beta_{6} + 666 \beta_{8} - 24 \beta_{9} ) q^{91} + ( -7135 - 63531 \beta_{1} - 261 \beta_{2} - 649 \beta_{3} - 504 \beta_{4} + 207 \beta_{5} - 72 \beta_{6} - 234 \beta_{7} + 459 \beta_{8} + 495 \beta_{9} ) q^{93} + ( -776 + 77866 \beta_{1} + 384 \beta_{3} - 3428 \beta_{4} - 776 \beta_{5} + 438 \beta_{6} + 26 \beta_{7} - 394 \beta_{8} ) q^{95} + ( 8090 + 8414 \beta_{1} + 749 \beta_{2} - 2478 \beta_{3} - 855 \beta_{4} - 519 \beta_{5} - 749 \beta_{7} - 195 \beta_{8} + 605 \beta_{9} ) q^{97} + ( -51981 + 35769 \beta_{1} - 492 \beta_{2} - 210 \beta_{3} - 1083 \beta_{4} + 297 \beta_{5} - 87 \beta_{6} + 237 \beta_{7} - 345 \beta_{8} + 12 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 12q^{3} - 21q^{5} + 29q^{7} + 12q^{9} + O(q^{10})$$ $$10q + 12q^{3} - 21q^{5} + 29q^{7} + 12q^{9} + 177q^{11} - 181q^{13} + 117q^{15} + 2280q^{17} - 832q^{19} - 207q^{21} + 399q^{23} - 4778q^{25} - 7128q^{27} - 6033q^{29} + 2759q^{31} + 9603q^{33} + 37146q^{35} - 15172q^{37} + 5529q^{39} - 18435q^{41} + 1469q^{43} - 64089q^{45} - 25155q^{47} - 4056q^{49} + 90612q^{51} + 116844q^{53} + 14778q^{55} + 26934q^{57} - 90537q^{59} + 1403q^{61} - 198255q^{63} - 148407q^{65} + 13907q^{67} + 214425q^{69} + 229368q^{71} + 15200q^{73} + 44640q^{75} - 211983q^{77} + 29993q^{79} - 404172q^{81} - 228951q^{83} - 49662q^{85} + 397323q^{87} + 598332q^{89} + 124930q^{91} + 250041q^{93} - 394764q^{95} + 40541q^{97} - 697239q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 175 x^{8} + 8800 x^{6} + 124623 x^{4} + 498609 x^{2} + 442368$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$99 \nu^{9} + 23021 \nu^{7} + 1847072 \nu^{5} + 56550029 \nu^{3} + 389674035 \nu - 207097728$$$$)/ 414195456$$ $$\beta_{2}$$ $$=$$ $$($$$$-47225 \nu^{8} - 9472484 \nu^{6} - 609253100 \nu^{4} - 13835630451 \nu^{2} - 61649294112$$$$)/ 427139064$$ $$\beta_{3}$$ $$=$$ $$($$$$338269 \nu^{9} - 1280856 \nu^{8} + 58481419 \nu^{7} - 217698528 \nu^{6} + 2848921216 \nu^{5} - 10174989216 \nu^{4} + 35591574291 \nu^{3} - 110092975560 \nu^{2} + 89923920885 \nu - 198138592512$$$$)/ 6834225024$$ $$\beta_{4}$$ $$=$$ $$($$$$-1356343 \nu^{9} + 375936 \nu^{8} - 234685369 \nu^{7} + 64407552 \nu^{6} - 11456638240 \nu^{5} + 2918015232 \nu^{4} - 144232448121 \nu^{3} + 22460575104 \nu^{2} - 331549576551 \nu + 3943792512$$$$)/ 13668450048$$ $$\beta_{5}$$ $$=$$ $$($$$$-4752101 \nu^{9} + 5053584 \nu^{8} - 822538331 \nu^{7} + 855338304 \nu^{6} - 40189663904 \nu^{5} + 40623828672 \nu^{4} - 507612794859 \nu^{3} + 503333827632 \nu^{2} - 1118204357445 \nu + 1147556557440$$$$)/ 13668450048$$ $$\beta_{6}$$ $$=$$ $$($$$$-5525341 \nu^{9} + 5651680 \nu^{8} - 929563411 \nu^{7} + 984567424 \nu^{6} - 42541455328 \nu^{5} + 48528039040 \nu^{4} - 423981840819 \nu^{3} + 598444064160 \nu^{2} - 305274835917 \nu + 696632164992$$$$)/ 13668450048$$ $$\beta_{7}$$ $$=$$ $$($$$$5368449 \nu^{9} + 1430176 \nu^{8} + 927771087 \nu^{7} + 219429760 \nu^{6} + 45271460064 \nu^{5} + 7683913600 \nu^{4} + 580333224015 \nu^{3} - 23644711200 \nu^{2} + 1852676757009 \nu - 594055313280$$$$)/ 13668450048$$ $$\beta_{8}$$ $$=$$ $$($$$$5415571 \nu^{9} + 6627168 \nu^{8} + 936462397 \nu^{7} + 1128424320 \nu^{6} + 45643692832 \nu^{5} + 52372017792 \nu^{4} + 571331339613 \nu^{3} + 530214202656 \nu^{2} + 1369631277027 \nu + 815163765120$$$$)/ 13668450048$$ $$\beta_{9}$$ $$=$$ $$($$$$-6201879 \nu^{9} - 3841840 \nu^{8} - 1046526249 \nu^{7} - 677985472 \nu^{6} - 48239297760 \nu^{5} - 34014091072 \nu^{4} - 495164989401 \nu^{3} - 423179263248 \nu^{2} - 485122677687 \nu - 308242564992$$$$)/ 13668450048$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{8} + \beta_{5} + 3 \beta_{4} + 5 \beta_{3} + 7 \beta_{1} + 4$$$$)/18$$ $$\nu^{2}$$ $$=$$ $$($$$$9 \beta_{9} + 6 \beta_{8} - 9 \beta_{6} - \beta_{5} + 17 \beta_{4} - 12 \beta_{3} - 9 \beta_{2} + 6 \beta_{1} - 1261$$$$)/36$$ $$\nu^{3}$$ $$=$$ $$($$$$6 \beta_{9} - 83 \beta_{8} + 42 \beta_{7} + 6 \beta_{6} - 74 \beta_{5} - 132 \beta_{4} - 340 \beta_{3} - 21 \beta_{2} - 1925 \beta_{1} - 995$$$$)/18$$ $$\nu^{4}$$ $$=$$ $$($$$$-396 \beta_{9} - 329 \beta_{8} + 396 \beta_{6} + 147 \beta_{5} - 1393 \beta_{4} + 479 \beta_{3} + 378 \beta_{2} - 329 \beta_{1} + 46848$$$$)/18$$ $$\nu^{5}$$ $$=$$ $$($$$$-1851 \beta_{9} + 14124 \beta_{8} - 8790 \beta_{7} - 1851 \beta_{6} + 12955 \beta_{5} + 21697 \beta_{4} + 56472 \beta_{3} + 4395 \beta_{2} + 544428 \beta_{1} + 278107$$$$)/36$$ $$\nu^{6}$$ $$=$$ $$($$$$33687 \beta_{9} + 32633 \beta_{8} - 33687 \beta_{6} - 20593 \beta_{5} + 168231 \beta_{4} - 35066 \beta_{3} - 36090 \beta_{2} + 32633 \beta_{1} - 3998422$$$$)/18$$ $$\nu^{7}$$ $$=$$ $$($$$$111930 \beta_{9} - 625543 \beta_{8} + 426012 \beta_{7} + 111930 \beta_{6} - 595221 \beta_{5} - 1014503 \beta_{4} - 2498807 \beta_{3} - 213006 \beta_{2} - 30838657 \beta_{1} - 15701778$$$$)/18$$ $$\nu^{8}$$ $$=$$ $$($$$$-5933115 \beta_{9} - 6360114 \beta_{8} + 5933115 \beta_{6} + 4761227 \beta_{5} - 36526363 \beta_{4} + 5223660 \beta_{3} + 7035939 \beta_{2} - 6360114 \beta_{1} + 717686687$$$$)/36$$ $$\nu^{9}$$ $$=$$ $$($$$$-12187632 \beta_{9} + 57177589 \beta_{8} - 41054970 \beta_{7} - 12187632 \beta_{6} + 55890928 \beta_{5} + 97095906 \beta_{4} + 228785360 \beta_{3} + 20527485 \beta_{2} + 3239641123 \beta_{1} + 1647122695$$$$)/18$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/36\mathbb{Z}\right)^\times$$.

 $$n$$ $$19$$ $$29$$ $$\chi(n)$$ $$1$$ $$\beta_{1}$$

## 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
 2.13639i − 7.64342i − 3.71922i 9.84603i 1.11227i − 2.13639i 7.64342i 3.71922i − 9.84603i − 1.11227i
0 −12.2647 9.62174i 0 −14.0718 + 24.3731i 0 75.7039 + 131.123i 0 57.8441 + 236.015i 0
13.2 0 −11.7655 + 10.2260i 0 4.88422 8.45972i 0 −68.3340 118.358i 0 33.8560 240.630i 0
13.3 0 7.64564 + 13.5847i 0 40.7270 70.5412i 0 89.6312 + 155.246i 0 −126.088 + 207.727i 0
13.4 0 8.67637 12.9507i 0 13.1603 22.7942i 0 −31.6287 54.7826i 0 −92.4411 224.730i 0
13.5 0 13.7082 + 7.42194i 0 −55.1996 + 95.6086i 0 −50.8724 88.1135i 0 132.829 + 203.483i 0
25.1 0 −12.2647 + 9.62174i 0 −14.0718 24.3731i 0 75.7039 131.123i 0 57.8441 236.015i 0
25.2 0 −11.7655 10.2260i 0 4.88422 + 8.45972i 0 −68.3340 + 118.358i 0 33.8560 + 240.630i 0
25.3 0 7.64564 13.5847i 0 40.7270 + 70.5412i 0 89.6312 155.246i 0 −126.088 207.727i 0
25.4 0 8.67637 + 12.9507i 0 13.1603 + 22.7942i 0 −31.6287 + 54.7826i 0 −92.4411 + 224.730i 0
25.5 0 13.7082 7.42194i 0 −55.1996 95.6086i 0 −50.8724 + 88.1135i 0 132.829 203.483i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 25.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.6.e.a 10
3.b odd 2 1 108.6.e.a 10
4.b odd 2 1 144.6.i.d 10
9.c even 3 1 inner 36.6.e.a 10
9.c even 3 1 324.6.a.e 5
9.d odd 6 1 108.6.e.a 10
9.d odd 6 1 324.6.a.d 5
12.b even 2 1 432.6.i.d 10
36.f odd 6 1 144.6.i.d 10
36.h even 6 1 432.6.i.d 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.6.e.a 10 1.a even 1 1 trivial
36.6.e.a 10 9.c even 3 1 inner
108.6.e.a 10 3.b odd 2 1
108.6.e.a 10 9.d odd 6 1
144.6.i.d 10 4.b odd 2 1
144.6.i.d 10 36.f odd 6 1
324.6.a.d 5 9.d odd 6 1
324.6.a.e 5 9.c even 3 1
432.6.i.d 10 12.b even 2 1
432.6.i.d 10 36.h even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(36, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$847288609443 - 41841412812 T + 947027862 T^{2} + 127545840 T^{3} + 17734383 T^{4} - 1732104 T^{5} + 72981 T^{6} + 2160 T^{7} + 66 T^{8} - 12 T^{9} + T^{10}$$
$5$ $$4234048679706624 - 457491118694400 T + 53123350023936 T^{2} - 900082524816 T^{3} + 74758828257 T^{4} - 926555355 T^{5} + 91398294 T^{6} - 323055 T^{7} + 10422 T^{8} + 21 T^{9} + T^{10}$$
$7$ $$56\!\cdots\!04$$$$+ 11838881253749666512 T + 258259673848446996 T^{2} + 1826372997396420 T^{3} + 22593265190433 T^{4} + 75213929775 T^{5} + 1392244974 T^{6} + 2300235 T^{7} + 44466 T^{8} - 29 T^{9} + T^{10}$$
$11$ $$17\!\cdots\!25$$$$-$$$$97\!\cdots\!45$$$$T +$$$$15\!\cdots\!99$$$$T^{2} - 931807571894691318 T^{3} + 12558885570039933 T^{4} - 43348856880519 T^{5} + 161733498741 T^{6} - 156952350 T^{7} + 409023 T^{8} - 177 T^{9} + T^{10}$$
$13$ $$14\!\cdots\!36$$$$+$$$$58\!\cdots\!92$$$$T +$$$$30\!\cdots\!56$$$$T^{2} + 33136080731853487872 T^{3} + 151974583529204709 T^{4} + 128853638078853 T^{5} + 536032392354 T^{6} + 216094281 T^{7} + 844134 T^{8} + 181 T^{9} + T^{10}$$
$17$ $$( 113704762586184 - 1066166319636 T + 3030697566 T^{2} - 2118735 T^{3} - 1140 T^{4} + T^{5} )^{2}$$
$19$ $$( -9782364002768192 + 13442524247504 T + 2095532308 T^{2} - 7334237 T^{3} + 416 T^{4} + T^{5} )^{2}$$
$23$ $$46\!\cdots\!96$$$$-$$$$17\!\cdots\!24$$$$T +$$$$62\!\cdots\!80$$$$T^{2} -$$$$48\!\cdots\!00$$$$T^{3} +$$$$79\!\cdots\!49$$$$T^{4} - 431345646852301611 T^{5} + 256756407066486 T^{6} - 45813128391 T^{7} + 16104474 T^{8} - 399 T^{9} + T^{10}$$
$29$ $$96\!\cdots\!36$$$$+$$$$11\!\cdots\!48$$$$T +$$$$14\!\cdots\!56$$$$T^{2} +$$$$27\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!05$$$$T^{4} + 27319508978657387769 T^{5} + 6982404272464842 T^{6} + 402872482485 T^{7} + 106207902 T^{8} + 6033 T^{9} + T^{10}$$
$31$ $$69\!\cdots\!00$$$$-$$$$16\!\cdots\!00$$$$T +$$$$12\!\cdots\!00$$$$T^{2} -$$$$22\!\cdots\!80$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4} - 25438125963411614607 T^{5} + 6788262652151238 T^{6} - 426408133899 T^{7} + 88243278 T^{8} - 2759 T^{9} + T^{10}$$
$37$ $$( 21920819628401968000 + 1428492776519168 T - 1301026358312 T^{2} - 145518692 T^{3} + 7586 T^{4} + T^{5} )^{2}$$
$41$ $$85\!\cdots\!09$$$$+$$$$12\!\cdots\!55$$$$T +$$$$19\!\cdots\!35$$$$T^{2} +$$$$19\!\cdots\!22$$$$T^{3} +$$$$44\!\cdots\!25$$$$T^{4} +$$$$26\!\cdots\!37$$$$T^{5} + 54390089528524629 T^{6} + 2062935126630 T^{7} + 461601963 T^{8} + 18435 T^{9} + T^{10}$$
$43$ $$66\!\cdots\!21$$$$+$$$$12\!\cdots\!87$$$$T +$$$$25\!\cdots\!51$$$$T^{2} +$$$$17\!\cdots\!82$$$$T^{3} +$$$$23\!\cdots\!29$$$$T^{4} +$$$$10\!\cdots\!13$$$$T^{5} + 163643432526985509 T^{6} + 3368564386026 T^{7} + 463178679 T^{8} - 1469 T^{9} + T^{10}$$
$47$ $$18\!\cdots\!24$$$$-$$$$67\!\cdots\!72$$$$T +$$$$22\!\cdots\!80$$$$T^{2} -$$$$46\!\cdots\!36$$$$T^{3} +$$$$10\!\cdots\!81$$$$T^{4} -$$$$46\!\cdots\!21$$$$T^{5} + 70002824199553998 T^{6} + 2958341092011 T^{7} + 745478802 T^{8} + 25155 T^{9} + T^{10}$$
$53$ $$( -86411235527486540928 + 56566983606862464 T - 12585968591112 T^{2} + 1263590748 T^{3} - 58422 T^{4} + T^{5} )^{2}$$
$59$ $$48\!\cdots\!81$$$$+$$$$12\!\cdots\!97$$$$T +$$$$26\!\cdots\!79$$$$T^{2} +$$$$20\!\cdots\!82$$$$T^{3} +$$$$17\!\cdots\!05$$$$T^{4} +$$$$58\!\cdots\!23$$$$T^{5} + 5843326331426559669 T^{6} + 207986454772038 T^{7} + 6405739335 T^{8} + 90537 T^{9} + T^{10}$$
$61$ $$45\!\cdots\!24$$$$-$$$$36\!\cdots\!00$$$$T +$$$$31\!\cdots\!04$$$$T^{2} +$$$$43\!\cdots\!68$$$$T^{3} +$$$$43\!\cdots\!13$$$$T^{4} -$$$$16\!\cdots\!43$$$$T^{5} + 416930581745459910 T^{6} - 4007745839055 T^{7} + 686075622 T^{8} - 1403 T^{9} + T^{10}$$
$67$ $$20\!\cdots\!49$$$$-$$$$97\!\cdots\!99$$$$T +$$$$33\!\cdots\!95$$$$T^{2} -$$$$50\!\cdots\!66$$$$T^{3} +$$$$55\!\cdots\!89$$$$T^{4} -$$$$24\!\cdots\!65$$$$T^{5} + 9050009696855720325 T^{6} - 133754061247050 T^{7} + 2873939991 T^{8} - 13907 T^{9} + T^{10}$$
$71$ $$($$$$18\!\cdots\!64$$$$- 6456975319152617472 T + 401418831672000 T^{2} - 1262766096 T^{3} - 114684 T^{4} + T^{5} )^{2}$$
$73$ $$( -$$$$17\!\cdots\!80$$$$+ 11344870273180522892 T + 31964902867486 T^{2} - 6758523719 T^{3} - 7600 T^{4} + T^{5} )^{2}$$
$79$ $$13\!\cdots\!00$$$$-$$$$18\!\cdots\!40$$$$T +$$$$19\!\cdots\!44$$$$T^{2} -$$$$43\!\cdots\!40$$$$T^{3} +$$$$24\!\cdots\!81$$$$T^{4} -$$$$42\!\cdots\!57$$$$T^{5} + 92009757017835850806 T^{6} - 635783520754389 T^{7} + 10032930366 T^{8} - 29993 T^{9} + T^{10}$$
$83$ $$16\!\cdots\!64$$$$+$$$$84\!\cdots\!16$$$$T +$$$$33\!\cdots\!76$$$$T^{2} +$$$$72\!\cdots\!80$$$$T^{3} +$$$$14\!\cdots\!45$$$$T^{4} +$$$$20\!\cdots\!23$$$$T^{5} +$$$$34\!\cdots\!02$$$$T^{6} + 3907824185416635 T^{7} + 41098635198 T^{8} + 228951 T^{9} + T^{10}$$
$89$ $$( -$$$$18\!\cdots\!72$$$$- 48110869394595006192 T + 21981512500848 T^{2} + 24695946936 T^{3} - 299166 T^{4} + T^{5} )^{2}$$
$97$ $$17\!\cdots\!25$$$$+$$$$11\!\cdots\!75$$$$T +$$$$74\!\cdots\!91$$$$T^{2} +$$$$50\!\cdots\!02$$$$T^{3} +$$$$20\!\cdots\!05$$$$T^{4} +$$$$11\!\cdots\!05$$$$T^{5} +$$$$45\!\cdots\!77$$$$T^{6} + 1218915608584182 T^{7} + 25043205147 T^{8} - 40541 T^{9} + T^{10}$$