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 $2$

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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: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
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: 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:

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 \(53\!\cdots\!36\)\( T_{5}^{2} - \)\(45\!\cdots\!00\)\( T_{5} + \)\(42\!\cdots\!24\)\( \)">\(T_{5}^{10} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

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