# Properties

 Label 144.6.i.d Level 144 Weight 6 Character orbit 144.i Analytic conductor 23.095 Analytic rank 0 Dimension 10 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 144.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.0952700531$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}\cdot 3^{10}$$ Twist minimal: no (minimal twist has level 36) 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 $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$\beta_{1}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.11227i 9.84603i − 3.71922i − 7.64342i 2.13639i − 1.11227i − 9.84603i 3.71922i 7.64342i − 2.13639i
0 −13.7082 7.42194i 0 −55.1996 + 95.6086i 0 50.8724 + 88.1135i 0 132.829 + 203.483i 0
49.2 0 −8.67637 + 12.9507i 0 13.1603 22.7942i 0 31.6287 + 54.7826i 0 −92.4411 224.730i 0
49.3 0 −7.64564 13.5847i 0 40.7270 70.5412i 0 −89.6312 155.246i 0 −126.088 + 207.727i 0
49.4 0 11.7655 10.2260i 0 4.88422 8.45972i 0 68.3340 + 118.358i 0 33.8560 240.630i 0
49.5 0 12.2647 + 9.62174i 0 −14.0718 + 24.3731i 0 −75.7039 131.123i 0 57.8441 + 236.015i 0
97.1 0 −13.7082 + 7.42194i 0 −55.1996 95.6086i 0 50.8724 88.1135i 0 132.829 203.483i 0
97.2 0 −8.67637 12.9507i 0 13.1603 + 22.7942i 0 31.6287 54.7826i 0 −92.4411 + 224.730i 0
97.3 0 −7.64564 + 13.5847i 0 40.7270 + 70.5412i 0 −89.6312 + 155.246i 0 −126.088 207.727i 0
97.4 0 11.7655 + 10.2260i 0 4.88422 + 8.45972i 0 68.3340 118.358i 0 33.8560 + 240.630i 0
97.5 0 12.2647 9.62174i 0 −14.0718 24.3731i 0 −75.7039 + 131.123i 0 57.8441 236.015i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.5 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 144.6.i.d 10
3.b odd 2 1 432.6.i.d 10
4.b odd 2 1 36.6.e.a 10
9.c even 3 1 inner 144.6.i.d 10
9.d odd 6 1 432.6.i.d 10
12.b even 2 1 108.6.e.a 10
36.f odd 6 1 36.6.e.a 10
36.f odd 6 1 324.6.a.e 5
36.h even 6 1 108.6.e.a 10
36.h even 6 1 324.6.a.d 5

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} + \cdots$$ acting on $$S_{6}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 12 T + 66 T^{2} - 2160 T^{3} + 72981 T^{4} + 1732104 T^{5} + 17734383 T^{6} - 127545840 T^{7} + 947027862 T^{8} + 41841412812 T^{9} + 847288609443 T^{10}$$
$5$ $$1 + 21 T - 5203 T^{2} - 519930 T^{3} + 14035794 T^{4} + 2854822770 T^{5} + 76722872007 T^{6} - 9761967315441 T^{7} - 599011867854189 T^{8} + 11924309583255600 T^{9} + 2533145723872694124 T^{10} + 37263467447673750000 T^{11} -$$$$58\!\cdots\!25$$$$T^{12} -$$$$29\!\cdots\!25$$$$T^{13} +$$$$73\!\cdots\!75$$$$T^{14} +$$$$85\!\cdots\!50$$$$T^{15} +$$$$13\!\cdots\!50$$$$T^{16} -$$$$15\!\cdots\!50$$$$T^{17} -$$$$47\!\cdots\!75$$$$T^{18} +$$$$59\!\cdots\!25$$$$T^{19} +$$$$88\!\cdots\!25$$$$T^{20}$$
$7$ $$1 + 29 T - 39569 T^{2} - 3762444 T^{3} + 440397336 T^{4} + 77352503496 T^{5} - 2769093584103 T^{6} + 560172784984473 T^{7} + 238615372451780007 T^{8} - 16031898530170676332 T^{9} -$$$$69\!\cdots\!60$$$$T^{10} -$$$$26\!\cdots\!24$$$$T^{11} +$$$$67\!\cdots\!43$$$$T^{12} +$$$$26\!\cdots\!39$$$$T^{13} -$$$$22\!\cdots\!03$$$$T^{14} +$$$$10\!\cdots\!72$$$$T^{15} +$$$$99\!\cdots\!64$$$$T^{16} -$$$$14\!\cdots\!92$$$$T^{17} -$$$$25\!\cdots\!69$$$$T^{18} +$$$$31\!\cdots\!03$$$$T^{19} +$$$$17\!\cdots\!49$$$$T^{20}$$
$11$ $$1 + 177 T - 396232 T^{2} + 71434269 T^{3} + 104816625882 T^{4} - 33726096455301 T^{5} - 6913987980717606 T^{6} + 8552599160812456257 T^{7} -$$$$10\!\cdots\!51$$$$T^{8} -$$$$50\!\cdots\!82$$$$T^{9} +$$$$47\!\cdots\!16$$$$T^{10} -$$$$81\!\cdots\!82$$$$T^{11} -$$$$27\!\cdots\!51$$$$T^{12} +$$$$35\!\cdots\!07$$$$T^{13} -$$$$46\!\cdots\!06$$$$T^{14} -$$$$36\!\cdots\!51$$$$T^{15} +$$$$18\!\cdots\!82$$$$T^{16} +$$$$20\!\cdots\!19$$$$T^{17} -$$$$17\!\cdots\!32$$$$T^{18} +$$$$12\!\cdots\!27$$$$T^{19} +$$$$11\!\cdots\!01$$$$T^{20}$$
$13$ $$1 + 181 T - 1012331 T^{2} + 14482182 T^{3} + 446454243174 T^{4} - 84043375137762 T^{5} - 192479505557683773 T^{6} - 802846347498861897 T^{7} +$$$$98\!\cdots\!51$$$$T^{8} +$$$$77\!\cdots\!72$$$$T^{9} -$$$$41\!\cdots\!24$$$$T^{10} +$$$$28\!\cdots\!96$$$$T^{11} +$$$$13\!\cdots\!99$$$$T^{12} -$$$$41\!\cdots\!29$$$$T^{13} -$$$$36\!\cdots\!73$$$$T^{14} -$$$$59\!\cdots\!66$$$$T^{15} +$$$$11\!\cdots\!26$$$$T^{16} +$$$$14\!\cdots\!74$$$$T^{17} -$$$$36\!\cdots\!31$$$$T^{18} +$$$$24\!\cdots\!33$$$$T^{19} +$$$$49\!\cdots\!49$$$$T^{20}$$
$17$ $$( 1 - 1140 T + 4980550 T^{2} - 3443850354 T^{3} + 10068870522169 T^{4} - 5069379208548852 T^{5} + 14296356292995309833 T^{6} -$$$$69\!\cdots\!46$$$$T^{7} +$$$$14\!\cdots\!50$$$$T^{8} -$$$$46\!\cdots\!40$$$$T^{9} +$$$$57\!\cdots\!57$$$$T^{10} )^{2}$$
$19$ $$( 1 - 416 T + 5046258 T^{2} - 6215761044 T^{3} + 20272296121125 T^{4} - 15898268281316088 T^{5} + 50196212153221491375 T^{6} -$$$$38\!\cdots\!44$$$$T^{7} +$$$$76\!\cdots\!42$$$$T^{8} -$$$$15\!\cdots\!16$$$$T^{9} +$$$$93\!\cdots\!99$$$$T^{10} )^{2}$$
$23$ $$1 + 399 T - 16077241 T^{2} + 38108825820 T^{3} + 155650662506976 T^{4} - 562944417983120520 T^{5} - 48522958516353490863 T^{6} +$$$$48\!\cdots\!51$$$$T^{7} -$$$$71\!\cdots\!41$$$$T^{8} -$$$$11\!\cdots\!52$$$$T^{9} +$$$$81\!\cdots\!80$$$$T^{10} -$$$$74\!\cdots\!36$$$$T^{11} -$$$$29\!\cdots\!09$$$$T^{12} +$$$$12\!\cdots\!57$$$$T^{13} -$$$$83\!\cdots\!63$$$$T^{14} -$$$$62\!\cdots\!60$$$$T^{15} +$$$$11\!\cdots\!24$$$$T^{16} +$$$$17\!\cdots\!40$$$$T^{17} -$$$$47\!\cdots\!41$$$$T^{18} +$$$$75\!\cdots\!57$$$$T^{19} +$$$$12\!\cdots\!49$$$$T^{20}$$
$29$ $$1 + 6033 T + 3652157 T^{2} + 31641196734 T^{3} + 283528398607854 T^{4} + 668469168127712358 T^{5} +$$$$64\!\cdots\!39$$$$T^{6} +$$$$82\!\cdots\!55$$$$T^{7} -$$$$90\!\cdots\!17$$$$T^{8} +$$$$14\!\cdots\!60$$$$T^{9} +$$$$58\!\cdots\!16$$$$T^{10} +$$$$29\!\cdots\!40$$$$T^{11} -$$$$38\!\cdots\!17$$$$T^{12} +$$$$71\!\cdots\!95$$$$T^{13} +$$$$11\!\cdots\!39$$$$T^{14} +$$$$24\!\cdots\!42$$$$T^{15} +$$$$21\!\cdots\!54$$$$T^{16} +$$$$48\!\cdots\!66$$$$T^{17} +$$$$11\!\cdots\!57$$$$T^{18} +$$$$38\!\cdots\!17$$$$T^{19} +$$$$13\!\cdots\!01$$$$T^{20}$$
$31$ $$1 + 2759 T - 54902477 T^{2} + 189444651072 T^{3} + 2052158291804100 T^{4} - 13274031992302596720 T^{5} -$$$$32\!\cdots\!47$$$$T^{6} +$$$$42\!\cdots\!39$$$$T^{7} -$$$$13\!\cdots\!49$$$$T^{8} -$$$$36\!\cdots\!48$$$$T^{9} +$$$$63\!\cdots\!24$$$$T^{10} -$$$$10\!\cdots\!48$$$$T^{11} -$$$$11\!\cdots\!49$$$$T^{12} +$$$$99\!\cdots\!89$$$$T^{13} -$$$$21\!\cdots\!47$$$$T^{14} -$$$$25\!\cdots\!20$$$$T^{15} +$$$$11\!\cdots\!00$$$$T^{16} +$$$$29\!\cdots\!72$$$$T^{17} -$$$$24\!\cdots\!77$$$$T^{18} +$$$$35\!\cdots\!09$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20}$$
$37$ $$( 1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} +$$$$13\!\cdots\!82$$$$T^{6} +$$$$38\!\cdots\!04$$$$T^{7} +$$$$67\!\cdots\!49$$$$T^{8} +$$$$17\!\cdots\!86$$$$T^{9} +$$$$16\!\cdots\!57$$$$T^{10} )^{2}$$
$41$ $$1 + 18435 T - 117679042 T^{2} - 4344492069675 T^{3} - 505249106564622 T^{4} +$$$$52\!\cdots\!97$$$$T^{5} +$$$$16\!\cdots\!40$$$$T^{6} -$$$$39\!\cdots\!43$$$$T^{7} -$$$$31\!\cdots\!03$$$$T^{8} +$$$$10\!\cdots\!42$$$$T^{9} +$$$$29\!\cdots\!40$$$$T^{10} +$$$$11\!\cdots\!42$$$$T^{11} -$$$$41\!\cdots\!03$$$$T^{12} -$$$$62\!\cdots\!43$$$$T^{13} +$$$$30\!\cdots\!40$$$$T^{14} +$$$$10\!\cdots\!97$$$$T^{15} -$$$$12\!\cdots\!22$$$$T^{16} -$$$$12\!\cdots\!75$$$$T^{17} -$$$$38\!\cdots\!42$$$$T^{18} +$$$$69\!\cdots\!35$$$$T^{19} +$$$$43\!\cdots\!01$$$$T^{20}$$
$43$ $$1 + 1469 T - 271863536 T^{2} - 4016430594327 T^{3} + 12129147672135834 T^{4} +$$$$75\!\cdots\!27$$$$T^{5} +$$$$55\!\cdots\!62$$$$T^{6} +$$$$12\!\cdots\!57$$$$T^{7} -$$$$29\!\cdots\!39$$$$T^{8} -$$$$61\!\cdots\!62$$$$T^{9} -$$$$11\!\cdots\!84$$$$T^{10} -$$$$90\!\cdots\!66$$$$T^{11} -$$$$64\!\cdots\!11$$$$T^{12} +$$$$38\!\cdots\!99$$$$T^{13} +$$$$26\!\cdots\!62$$$$T^{14} +$$$$52\!\cdots\!61$$$$T^{15} +$$$$12\!\cdots\!66$$$$T^{16} -$$$$59\!\cdots\!89$$$$T^{17} -$$$$59\!\cdots\!36$$$$T^{18} +$$$$47\!\cdots\!67$$$$T^{19} +$$$$47\!\cdots\!49$$$$T^{20}$$
$47$ $$1 - 25155 T - 401246233 T^{2} + 14349179861244 T^{3} + 97557609874842960 T^{4} -$$$$41\!\cdots\!12$$$$T^{5} -$$$$25\!\cdots\!27$$$$T^{6} +$$$$55\!\cdots\!85$$$$T^{7} +$$$$12\!\cdots\!11$$$$T^{8} -$$$$42\!\cdots\!56$$$$T^{9} -$$$$37\!\cdots\!60$$$$T^{10} -$$$$96\!\cdots\!92$$$$T^{11} +$$$$67\!\cdots\!39$$$$T^{12} +$$$$67\!\cdots\!55$$$$T^{13} -$$$$71\!\cdots\!27$$$$T^{14} -$$$$26\!\cdots\!84$$$$T^{15} +$$$$14\!\cdots\!40$$$$T^{16} +$$$$47\!\cdots\!92$$$$T^{17} -$$$$30\!\cdots\!33$$$$T^{18} -$$$$44\!\cdots\!85$$$$T^{19} +$$$$40\!\cdots\!49$$$$T^{20}$$
$53$ $$( 1 - 58422 T + 3354568213 T^{2} - 110313236959296 T^{3} + 3390725554692289246 T^{4} -$$$$71\!\cdots\!28$$$$T^{5} +$$$$14\!\cdots\!78$$$$T^{6} -$$$$19\!\cdots\!04$$$$T^{7} +$$$$24\!\cdots\!41$$$$T^{8} -$$$$17\!\cdots\!22$$$$T^{9} +$$$$12\!\cdots\!93$$$$T^{10} )^{2}$$
$59$ $$1 - 90537 T + 2831117840 T^{2} - 13805150996349 T^{3} - 966660594685472478 T^{4} +$$$$10\!\cdots\!09$$$$T^{5} +$$$$69\!\cdots\!78$$$$T^{6} -$$$$39\!\cdots\!93$$$$T^{7} +$$$$84\!\cdots\!01$$$$T^{8} +$$$$17\!\cdots\!74$$$$T^{9} -$$$$12\!\cdots\!20$$$$T^{10} +$$$$12\!\cdots\!26$$$$T^{11} +$$$$42\!\cdots\!01$$$$T^{12} -$$$$14\!\cdots\!07$$$$T^{13} +$$$$18\!\cdots\!78$$$$T^{14} +$$$$19\!\cdots\!91$$$$T^{15} -$$$$12\!\cdots\!78$$$$T^{16} -$$$$13\!\cdots\!51$$$$T^{17} +$$$$19\!\cdots\!40$$$$T^{18} -$$$$44\!\cdots\!63$$$$T^{19} +$$$$34\!\cdots\!01$$$$T^{20}$$
$61$ $$1 - 1403 T - 3536905883 T^{2} - 452840008146 T^{3} + 7065863261737144698 T^{4} +$$$$54\!\cdots\!90$$$$T^{5} -$$$$10\!\cdots\!33$$$$T^{6} -$$$$70\!\cdots\!89$$$$T^{7} +$$$$11\!\cdots\!67$$$$T^{8} +$$$$32\!\cdots\!04$$$$T^{9} -$$$$10\!\cdots\!12$$$$T^{10} +$$$$27\!\cdots\!04$$$$T^{11} +$$$$80\!\cdots\!67$$$$T^{12} -$$$$42\!\cdots\!89$$$$T^{13} -$$$$51\!\cdots\!33$$$$T^{14} +$$$$23\!\cdots\!90$$$$T^{15} +$$$$25\!\cdots\!98$$$$T^{16} -$$$$13\!\cdots\!46$$$$T^{17} -$$$$91\!\cdots\!83$$$$T^{18} -$$$$30\!\cdots\!03$$$$T^{19} +$$$$18\!\cdots\!01$$$$T^{20}$$
$67$ $$1 + 13907 T - 3876685544 T^{2} + 77425491657903 T^{3} + 10014688417385231130 T^{4} -$$$$30\!\cdots\!39$$$$T^{5} -$$$$79\!\cdots\!54$$$$T^{6} +$$$$69\!\cdots\!51$$$$T^{7} -$$$$33\!\cdots\!67$$$$T^{8} -$$$$37\!\cdots\!46$$$$T^{9} +$$$$22\!\cdots\!76$$$$T^{10} -$$$$51\!\cdots\!22$$$$T^{11} -$$$$60\!\cdots\!83$$$$T^{12} +$$$$16\!\cdots\!93$$$$T^{13} -$$$$26\!\cdots\!54$$$$T^{14} -$$$$13\!\cdots\!73$$$$T^{15} +$$$$60\!\cdots\!70$$$$T^{16} +$$$$63\!\cdots\!29$$$$T^{17} -$$$$42\!\cdots\!44$$$$T^{18} +$$$$20\!\cdots\!49$$$$T^{19} +$$$$20\!\cdots\!49$$$$T^{20}$$
$71$ $$( 1 + 114684 T + 7758380659 T^{2} + 426246123888336 T^{3} + 19260501229393543450 T^{4} +$$$$77\!\cdots\!40$$$$T^{5} +$$$$34\!\cdots\!50$$$$T^{6} +$$$$13\!\cdots\!36$$$$T^{7} +$$$$45\!\cdots\!09$$$$T^{8} +$$$$12\!\cdots\!84$$$$T^{9} +$$$$19\!\cdots\!51$$$$T^{10} )^{2}$$
$73$ $$( 1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} -$$$$80\!\cdots\!84$$$$T^{5} +$$$$25\!\cdots\!33$$$$T^{6} -$$$$13\!\cdots\!86$$$$T^{7} +$$$$32\!\cdots\!22$$$$T^{8} -$$$$14\!\cdots\!00$$$$T^{9} +$$$$38\!\cdots\!93$$$$T^{10} )^{2}$$
$79$ $$1 + 29993 T - 5352351629 T^{2} + 358913063028768 T^{3} + 26234825811851125236 T^{4} -$$$$21\!\cdots\!52$$$$T^{5} +$$$$27\!\cdots\!85$$$$T^{6} +$$$$84\!\cdots\!45$$$$T^{7} -$$$$35\!\cdots\!45$$$$T^{8} -$$$$81\!\cdots\!80$$$$T^{9} +$$$$16\!\cdots\!00$$$$T^{10} -$$$$25\!\cdots\!20$$$$T^{11} -$$$$33\!\cdots\!45$$$$T^{12} +$$$$24\!\cdots\!55$$$$T^{13} +$$$$24\!\cdots\!85$$$$T^{14} -$$$$60\!\cdots\!48$$$$T^{15} +$$$$22\!\cdots\!36$$$$T^{16} +$$$$93\!\cdots\!32$$$$T^{17} -$$$$43\!\cdots\!29$$$$T^{18} +$$$$74\!\cdots\!07$$$$T^{19} +$$$$76\!\cdots\!01$$$$T^{20}$$
$83$ $$1 - 228951 T + 21403431983 T^{2} - 1202282302650156 T^{3} + 62567029919071222368 T^{4} -$$$$36\!\cdots\!68$$$$T^{5} +$$$$11\!\cdots\!01$$$$T^{6} +$$$$11\!\cdots\!41$$$$T^{7} -$$$$18\!\cdots\!73$$$$T^{8} +$$$$14\!\cdots\!84$$$$T^{9} -$$$$88\!\cdots\!72$$$$T^{10} +$$$$55\!\cdots\!12$$$$T^{11} -$$$$28\!\cdots\!77$$$$T^{12} +$$$$67\!\cdots\!87$$$$T^{13} +$$$$28\!\cdots\!01$$$$T^{14} -$$$$34\!\cdots\!24$$$$T^{15} +$$$$23\!\cdots\!32$$$$T^{16} -$$$$17\!\cdots\!92$$$$T^{17} +$$$$12\!\cdots\!83$$$$T^{18} -$$$$52\!\cdots\!93$$$$T^{19} +$$$$89\!\cdots\!49$$$$T^{20}$$
$89$ $$( 1 - 299166 T + 52616244181 T^{2} - 6660261403977288 T^{3} +$$$$67\!\cdots\!10$$$$T^{4} -$$$$55\!\cdots\!64$$$$T^{5} +$$$$37\!\cdots\!90$$$$T^{6} -$$$$20\!\cdots\!88$$$$T^{7} +$$$$91\!\cdots\!69$$$$T^{8} -$$$$29\!\cdots\!66$$$$T^{9} +$$$$54\!\cdots\!49$$$$T^{10} )^{2}$$
$97$ $$1 - 40541 T - 17893496138 T^{2} + 2263333692661293 T^{3} + 99710157551726941410 T^{4} -$$$$30\!\cdots\!95$$$$T^{5} +$$$$10\!\cdots\!20$$$$T^{6} +$$$$21\!\cdots\!29$$$$T^{7} -$$$$20\!\cdots\!15$$$$T^{8} -$$$$65\!\cdots\!06$$$$T^{9} +$$$$19\!\cdots\!00$$$$T^{10} -$$$$56\!\cdots\!42$$$$T^{11} -$$$$15\!\cdots\!35$$$$T^{12} +$$$$13\!\cdots\!97$$$$T^{13} +$$$$56\!\cdots\!20$$$$T^{14} -$$$$14\!\cdots\!15$$$$T^{15} +$$$$39\!\cdots\!90$$$$T^{16} +$$$$77\!\cdots\!49$$$$T^{17} -$$$$52\!\cdots\!38$$$$T^{18} -$$$$10\!\cdots\!37$$$$T^{19} +$$$$21\!\cdots\!49$$$$T^{20}$$