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

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

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

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