Properties

Label 108.4.b.b
Level 108
Weight 4
Character orbit 108.b
Analytic conductor 6.372
Analytic rank 0
Dimension 12
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 108.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.37220628062\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -\beta_{6} q^{4} + \beta_{5} q^{5} -\beta_{4} q^{7} + ( -\beta_{1} + \beta_{9} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} -\beta_{6} q^{4} + \beta_{5} q^{5} -\beta_{4} q^{7} + ( -\beta_{1} + \beta_{9} ) q^{8} + ( 4 + \beta_{3} - \beta_{4} + \beta_{6} ) q^{10} + ( -\beta_{1} + \beta_{2} + \beta_{9} ) q^{11} + ( -6 - \beta_{6} + \beta_{10} ) q^{13} + ( -\beta_{1} + \beta_{2} + 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{14} + ( 8 + 2 \beta_{4} - \beta_{6} - \beta_{10} + \beta_{11} ) q^{16} + ( -6 \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{17} + ( -4 \beta_{3} - \beta_{6} - \beta_{10} - 2 \beta_{11} ) q^{19} + ( -7 \beta_{1} + \beta_{2} + 6 \beta_{5} - 3 \beta_{7} - \beta_{8} ) q^{20} + ( 4 + \beta_{3} + 5 \beta_{4} - 3 \beta_{6} - \beta_{10} - \beta_{11} ) q^{22} + ( -18 \beta_{1} + \beta_{2} - \beta_{7} - 4 \beta_{8} + \beta_{9} ) q^{23} + ( -24 + 17 \beta_{6} + 3 \beta_{10} + 4 \beta_{11} ) q^{25} + ( 6 \beta_{1} - 2 \beta_{2} + 4 \beta_{5} + 6 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{26} + ( -24 + 4 \beta_{3} + 4 \beta_{4} - \beta_{6} ) q^{28} + ( -10 \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{7} + 2 \beta_{8} - 3 \beta_{9} ) q^{29} + ( 8 - 4 \beta_{3} - 7 \beta_{4} + 11 \beta_{6} - \beta_{10} - 6 \beta_{11} ) q^{31} + ( -11 \beta_{1} - 8 \beta_{5} - 8 \beta_{7} + 8 \beta_{8} - \beta_{9} ) q^{32} + ( -32 + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{10} + 6 \beta_{11} ) q^{34} + ( 37 \beta_{1} - 2 \beta_{2} + 3 \beta_{7} + 8 \beta_{8} - 2 \beta_{9} ) q^{35} + ( -20 - 5 \beta_{6} + 5 \beta_{10} ) q^{37} + ( 16 \beta_{1} + 2 \beta_{2} - 20 \beta_{5} + 10 \beta_{7} - 6 \beta_{8} - 2 \beta_{9} ) q^{38} + ( 64 + 4 \beta_{3} - 6 \beta_{4} - 3 \beta_{6} - 3 \beta_{10} - 5 \beta_{11} ) q^{40} + ( 70 \beta_{1} + \beta_{2} - 8 \beta_{5} - \beta_{7} - 18 \beta_{8} - 3 \beta_{9} ) q^{41} + ( -16 - 8 \beta_{3} - 6 \beta_{4} - 26 \beta_{6} - 2 \beta_{10} + 4 \beta_{11} ) q^{43} + ( -3 \beta_{1} - 3 \beta_{2} - 10 \beta_{5} - 15 \beta_{7} - 13 \beta_{8} - 4 \beta_{9} ) q^{44} + ( 136 + 4 \beta_{4} - 20 \beta_{6} - 2 \beta_{10} - 2 \beta_{11} ) q^{46} + ( 34 \beta_{1} - 6 \beta_{2} - 4 \beta_{7} + 8 \beta_{8} - 6 \beta_{9} ) q^{47} + ( 16 - 23 \beta_{6} + 3 \beta_{10} - 4 \beta_{11} ) q^{49} + ( 32 \beta_{1} - 6 \beta_{2} + 12 \beta_{5} + 18 \beta_{7} + 26 \beta_{8} - 10 \beta_{9} ) q^{50} + ( 80 + 8 \beta_{3} + 14 \beta_{6} + 4 \beta_{10} + 12 \beta_{11} ) q^{52} + ( -62 \beta_{1} + 3 \beta_{2} - 19 \beta_{5} - 3 \beta_{7} + 14 \beta_{8} - 9 \beta_{9} ) q^{53} + ( 16 + 8 \beta_{3} - 15 \beta_{4} + 26 \beta_{6} + 2 \beta_{10} - 4 \beta_{11} ) q^{55} + ( 15 \beta_{1} - 4 \beta_{2} + 8 \beta_{5} - 20 \beta_{7} - 12 \beta_{8} - 3 \beta_{9} ) q^{56} + ( -80 + 2 \beta_{3} - 6 \beta_{4} - 10 \beta_{6} + 2 \beta_{10} - 6 \beta_{11} ) q^{58} + ( -114 \beta_{1} - 6 \beta_{2} + 8 \beta_{7} - 32 \beta_{8} - 6 \beta_{9} ) q^{59} + ( 28 + 44 \beta_{6} - 4 \beta_{10} + 8 \beta_{11} ) q^{61} + ( 25 \beta_{1} + 9 \beta_{2} - 6 \beta_{5} + 17 \beta_{7} - 31 \beta_{8} - 11 \beta_{9} ) q^{62} + ( -264 - 16 \beta_{3} - 2 \beta_{4} - 19 \beta_{6} - 7 \beta_{10} - 9 \beta_{11} ) q^{64} + ( -50 \beta_{1} - 3 \beta_{2} - 18 \beta_{5} + 3 \beta_{7} + 14 \beta_{8} + 9 \beta_{9} ) q^{65} + 22 \beta_{4} q^{67} + ( 6 \beta_{1} + 2 \beta_{2} - 4 \beta_{5} - 22 \beta_{7} + 46 \beta_{8} + 4 \beta_{9} ) q^{68} + ( -276 + \beta_{3} - 7 \beta_{4} + 41 \beta_{6} + 5 \beta_{10} + 5 \beta_{11} ) q^{70} + ( 88 \beta_{1} + 7 \beta_{2} - 17 \beta_{7} + 28 \beta_{8} + 7 \beta_{9} ) q^{71} + ( 5 - 11 \beta_{6} - 9 \beta_{10} - 4 \beta_{11} ) q^{73} + ( 20 \beta_{1} - 10 \beta_{2} + 20 \beta_{5} + 30 \beta_{7} - 10 \beta_{8} + 10 \beta_{9} ) q^{74} + ( 48 - 8 \beta_{3} + 32 \beta_{4} - 8 \beta_{6} + 12 \beta_{10} + 4 \beta_{11} ) q^{76} + ( 200 \beta_{1} - 4 \beta_{2} + 21 \beta_{5} + 4 \beta_{7} - 48 \beta_{8} + 12 \beta_{9} ) q^{77} + ( -8 + 12 \beta_{3} + 34 \beta_{4} - 9 \beta_{6} + 3 \beta_{10} + 10 \beta_{11} ) q^{79} + ( -73 \beta_{1} + 12 \beta_{2} + 16 \beta_{5} - 28 \beta_{7} - 36 \beta_{8} + \beta_{9} ) q^{80} + ( 520 - 8 \beta_{3} + 4 \beta_{4} + 60 \beta_{6} + 2 \beta_{10} - 6 \beta_{11} ) q^{82} + ( 109 \beta_{1} + 13 \beta_{2} + 26 \beta_{7} + 24 \beta_{8} + 13 \beta_{9} ) q^{83} + ( -30 - 27 \beta_{6} - 13 \beta_{10} - 8 \beta_{11} ) q^{85} + ( -6 \beta_{1} + 10 \beta_{2} - 28 \beta_{5} + 26 \beta_{7} + 58 \beta_{8} + 34 \beta_{9} ) q^{86} + ( 416 - 28 \beta_{3} - 22 \beta_{4} - 7 \beta_{6} - 11 \beta_{10} - 13 \beta_{11} ) q^{88} + ( -200 \beta_{1} + 4 \beta_{2} + 70 \beta_{5} - 4 \beta_{7} + 48 \beta_{8} - 12 \beta_{9} ) q^{89} + ( -16 + 12 \beta_{3} + 38 \beta_{4} - 21 \beta_{6} + 3 \beta_{10} + 14 \beta_{11} ) q^{91} + ( -148 \beta_{1} - 16 \beta_{5} - 16 \beta_{7} - 16 \beta_{8} + 12 \beta_{9} ) q^{92} + ( -264 - 10 \beta_{3} - 34 \beta_{4} + 46 \beta_{6} + 2 \beta_{10} + 2 \beta_{11} ) q^{94} + ( -346 \beta_{1} + 11 \beta_{2} - 31 \beta_{7} - 76 \beta_{8} + 11 \beta_{9} ) q^{95} + ( 11 - 68 \beta_{6} - 12 \beta_{10} - 16 \beta_{11} ) q^{97} + ( -24 \beta_{1} - 6 \beta_{2} + 12 \beta_{5} + 18 \beta_{7} - 38 \beta_{8} + 22 \beta_{9} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 6q^{4} + O(q^{10}) \) \( 12q + 6q^{4} + 42q^{10} - 72q^{13} + 114q^{16} + 66q^{22} - 384q^{25} - 282q^{28} - 324q^{34} - 240q^{37} + 774q^{40} + 1752q^{46} + 288q^{49} + 924q^{52} - 948q^{58} + 144q^{61} - 3066q^{64} - 3558q^{70} + 156q^{73} + 576q^{76} + 5832q^{82} - 168q^{85} + 5022q^{88} - 3444q^{94} + 516q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + 6854 x^{2} - 888 x + 9496\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-33253968445 \nu^{11} - 2668198080286 \nu^{10} + 792575592373 \nu^{9} + 1425147997482 \nu^{8} + 24644484803959 \nu^{7} - 193768555419338 \nu^{6} - 61078234561525 \nu^{5} - 887616443102162 \nu^{4} - 85890745896528 \nu^{3} - 2267866287233700 \nu^{2} - 2389458067685830 \nu - 5634039587496684\)\()/ 2246281435673440 \)
\(\beta_{2}\)\(=\)\((\)\(16345777051 \nu^{11} - 1455335526278 \nu^{10} - 1946811804847 \nu^{9} - 1614515226786 \nu^{8} + 21894737772791 \nu^{7} - 72858396390586 \nu^{6} - 162793265999057 \nu^{5} - 670564599311566 \nu^{4} - 710732231788632 \nu^{3} - 1759491359890068 \nu^{2} - 2792533544858270 \nu - 2835784389936540\)\()/ 112314071783672 \)
\(\beta_{3}\)\(=\)\((\)\(2029314856 \nu^{11} - 20657394999 \nu^{10} - 72348050754 \nu^{9} - 44845772052 \nu^{8} + 509091530022 \nu^{7} - 18858252443 \nu^{6} - 5180674803238 \nu^{5} - 15414104133564 \nu^{4} - 24647091348478 \nu^{3} - 3230408502820 \nu^{2} - 21152396498056 \nu - 67923831391169\)\()/ 6381481351345 \)
\(\beta_{4}\)\(=\)\((\)\(-2849350 \nu^{11} + 6585588 \nu^{10} - 7900308 \nu^{9} + 8937888 \nu^{8} - 274261062 \nu^{7} + 514828262 \nu^{6} - 1348583642 \nu^{5} + 577023966 \nu^{4} - 3610618898 \nu^{3} - 4417697498 \nu^{2} - 20418808208 \nu - 3774567143\)\()/ 3046053151 \)
\(\beta_{5}\)\(=\)\((\)\(-454920367261 \nu^{11} - 699408466494 \nu^{10} - 909094946747 \nu^{9} + 6055960107450 \nu^{8} - 16604234843593 \nu^{7} - 44620628131914 \nu^{6} - 261918488231045 \nu^{5} - 346379732140290 \nu^{4} - 372084361441392 \nu^{3} - 4408588811940 \nu^{2} + 876761148010650 \nu + 873663409442004\)\()/ 449256287134688 \)
\(\beta_{6}\)\(=\)\((\)\(-23425387202 \nu^{11} - 19598582637 \nu^{10} - 48064283082 \nu^{9} + 315709019049 \nu^{8} - 1232017295874 \nu^{7} - 1577989182089 \nu^{6} - 13160711736994 \nu^{5} - 9640316846217 \nu^{4} - 29399170369744 \nu^{3} - 39045109376800 \nu^{2} - 83437255959808 \nu - 38283473678982\)\()/ 12762962702690 \)
\(\beta_{7}\)\(=\)\((\)\(-1144977542195 \nu^{11} + 4352732408014 \nu^{10} - 29629791457 \nu^{9} + 26260811026362 \nu^{8} - 128693237482831 \nu^{7} + 247175465899922 \nu^{6} - 505821450280655 \nu^{5} + 2483737463807558 \nu^{4} - 214719710377848 \nu^{3} + 7168378187690340 \nu^{2} + 893868308969470 \nu + 19027629960424476\)\()/ 561570358918360 \)
\(\beta_{8}\)\(=\)\((\)\(159260070855 \nu^{11} + 117861110662 \nu^{10} + 438728620059 \nu^{9} - 2345594945694 \nu^{8} + 10405960829647 \nu^{7} + 5462455990466 \nu^{6} + 93693140746805 \nu^{5} + 30241542078854 \nu^{4} + 390731460563916 \nu^{3} + 122017097284380 \nu^{2} + 719519253923230 \nu - 385589858796132\)\()/ 70196294864795 \)
\(\beta_{9}\)\(=\)\((\)\(716000535355 \nu^{11} - 22806431518 \nu^{10} + 1186916251229 \nu^{9} - 10373131256454 \nu^{8} + 49370641188207 \nu^{7} + 1347814842646 \nu^{6} + 379075967096835 \nu^{5} - 39097470243746 \nu^{4} + 1094945871786096 \nu^{3} - 524652329852580 \nu^{2} + 2831947063781450 \nu - 2432191580688972\)\()/ 102103701621520 \)
\(\beta_{10}\)\(=\)\((\)\(-114943514282 \nu^{11} + 101958949683 \nu^{10} + 310110651558 \nu^{9} + 1675099805049 \nu^{8} - 9122517442074 \nu^{7} + 1428113010271 \nu^{6} - 26263858992874 \nu^{5} + 63325710114663 \nu^{4} + 4591871758616 \nu^{3} + 245754664595480 \nu^{2} - 29530710450688 \nu + 1169684312559898\)\()/ 12762962702690 \)
\(\beta_{11}\)\(=\)\((\)\(29835469322 \nu^{11} + 31182551757 \nu^{10} + 48231870066 \nu^{9} - 289609225533 \nu^{8} + 1601507488746 \nu^{7} + 2210589724721 \nu^{6} + 14942761872730 \nu^{5} + 20689356220701 \nu^{4} + 49348940923840 \nu^{3} + 69766706628088 \nu^{2} + 108997215408544 \nu + 58683428040738\)\()/ 2552592540538 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{11} - \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} - 9 \beta_{6} + 8 \beta_{5} - \beta_{2} + 14 \beta_{1} - 4\)\()/72\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{10} - 6 \beta_{9} + 6 \beta_{8} - 2 \beta_{7} - 11 \beta_{6} + 6 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{1} - 24\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(13 \beta_{11} - \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 13 \beta_{7} + 30 \beta_{6} + 46 \beta_{5} + 48 \beta_{4} - 6 \beta_{3} - 5 \beta_{2} + 92 \beta_{1} + 224\)\()/72\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{11} + 3 \beta_{10} + 18 \beta_{9} - 32 \beta_{8} + 26 \beta_{7} - 61 \beta_{6} + 44 \beta_{5} + 14 \beta_{4} - 28 \beta_{3} + 26 \beta_{2} - 40 \beta_{1} - 850\)\()/36\)
\(\nu^{5}\)\(=\)\((\)\(119 \beta_{11} + 23 \beta_{10} - 129 \beta_{9} - 149 \beta_{8} - 387 \beta_{7} + 392 \beta_{6} + 86 \beta_{5} - 100 \beta_{4} - 130 \beta_{3} - 7 \beta_{2} - 1704 \beta_{1} - 572\)\()/72\)
\(\nu^{6}\)\(=\)\((\)\(167 \beta_{11} + 172 \beta_{10} + 297 \beta_{9} - 633 \beta_{8} - 183 \beta_{7} + 627 \beta_{6} - 6 \beta_{5} + 384 \beta_{4} + 204 \beta_{3} + 57 \beta_{2} - 552 \beta_{1} - 5210\)\()/36\)
\(\nu^{7}\)\(=\)\((\)\(-75 \beta_{11} + 683 \beta_{10} + 525 \beta_{9} + 543 \beta_{8} - 249 \beta_{7} + 2386 \beta_{6} - 2682 \beta_{5} - 2996 \beta_{4} + 154 \beta_{3} + 1239 \beta_{2} - 16092 \beta_{1} - 35724\)\()/72\)
\(\nu^{8}\)\(=\)\((\)\(1272 \beta_{11} + 529 \beta_{10} + 72 \beta_{9} - 1448 \beta_{8} - 3032 \beta_{7} + 11519 \beta_{6} - 5200 \beta_{5} - 1918 \beta_{4} + 2228 \beta_{3} - 1528 \beta_{2} - 6176 \beta_{1} + 30702\)\()/36\)
\(\nu^{9}\)\(=\)\((\)\(-7085 \beta_{11} + 5021 \beta_{10} + 15741 \beta_{9} + 1455 \beta_{8} + 20007 \beta_{7} - 7938 \beta_{6} - 32010 \beta_{5} - 3564 \beta_{4} + 17118 \beta_{3} + 2295 \beta_{2} + 64596 \beta_{1} - 156028\)\()/72\)
\(\nu^{10}\)\(=\)\((\)\(-12715 \beta_{11} - 7950 \beta_{10} - 17487 \beta_{9} + 56011 \beta_{8} + 8305 \beta_{7} - 461 \beta_{6} - 44218 \beta_{5} - 46676 \beta_{4} + 1624 \beta_{3} - 6211 \beta_{2} - 40964 \beta_{1} + 415114\)\()/36\)
\(\nu^{11}\)\(=\)\((\)\(-22857 \beta_{11} - 49075 \beta_{10} + 2019 \beta_{9} + 26941 \beta_{8} + 102273 \beta_{7} - 60590 \beta_{6} - 49390 \beta_{5} + 130636 \beta_{4} + 96910 \beta_{3} - 115975 \beta_{2} + 1521828 \beta_{1} + 3750276\)\()/72\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
2.61836 + 1.60260i
2.61836 1.60260i
−1.29835 1.36719i
−1.29835 + 1.36719i
−0.453986 + 2.07664i
−0.453986 2.07664i
−2.18604 + 2.07664i
−2.18604 2.07664i
0.433705 1.36719i
0.433705 + 1.36719i
0.886307 + 1.60260i
0.886307 1.60260i
−2.72048 0.773954i 0 6.80199 + 4.21105i 20.8488i 0 13.9048i −15.2455 16.7205i 0 16.1360 56.7186i
107.2 −2.72048 + 0.773954i 0 6.80199 4.21105i 20.8488i 0 13.9048i −15.2455 + 16.7205i 0 16.1360 + 56.7186i
107.3 −2.27435 1.68146i 0 2.34537 + 7.64848i 5.83890i 0 8.83113i 7.52641 21.3390i 0 −9.81789 + 13.2797i
107.4 −2.27435 + 1.68146i 0 2.34537 7.64848i 5.83890i 0 8.83113i 7.52641 + 21.3390i 0 −9.81789 13.2797i
107.5 −0.419903 2.79708i 0 −7.64736 + 2.34901i 1.49508i 0 26.1852i 9.78152 + 20.4040i 0 4.18188 0.627790i
107.6 −0.419903 + 2.79708i 0 −7.64736 2.34901i 1.49508i 0 26.1852i 9.78152 20.4040i 0 4.18188 + 0.627790i
107.7 0.419903 2.79708i 0 −7.64736 2.34901i 1.49508i 0 26.1852i −9.78152 + 20.4040i 0 4.18188 + 0.627790i
107.8 0.419903 + 2.79708i 0 −7.64736 + 2.34901i 1.49508i 0 26.1852i −9.78152 20.4040i 0 4.18188 0.627790i
107.9 2.27435 1.68146i 0 2.34537 7.64848i 5.83890i 0 8.83113i −7.52641 21.3390i 0 −9.81789 13.2797i
107.10 2.27435 + 1.68146i 0 2.34537 + 7.64848i 5.83890i 0 8.83113i −7.52641 + 21.3390i 0 −9.81789 + 13.2797i
107.11 2.72048 0.773954i 0 6.80199 4.21105i 20.8488i 0 13.9048i 15.2455 16.7205i 0 16.1360 + 56.7186i
107.12 2.72048 + 0.773954i 0 6.80199 + 4.21105i 20.8488i 0 13.9048i 15.2455 + 16.7205i 0 16.1360 56.7186i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.b.b 12
3.b odd 2 1 inner 108.4.b.b 12
4.b odd 2 1 inner 108.4.b.b 12
8.b even 2 1 1728.4.c.j 12
8.d odd 2 1 1728.4.c.j 12
12.b even 2 1 inner 108.4.b.b 12
24.f even 2 1 1728.4.c.j 12
24.h odd 2 1 1728.4.c.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.b.b 12 1.a even 1 1 trivial
108.4.b.b 12 3.b odd 2 1 inner
108.4.b.b 12 4.b odd 2 1 inner
108.4.b.b 12 12.b even 2 1 inner
1728.4.c.j 12 8.b even 2 1
1728.4.c.j 12 8.d odd 2 1
1728.4.c.j 12 24.f even 2 1
1728.4.c.j 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 471 T_{5}^{4} + 15867 T_{5}^{2} + 33125 \) acting on \(S_{4}^{\mathrm{new}}(108, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T^{2} - 24 T^{4} + 592 T^{6} - 1536 T^{8} - 12288 T^{10} + 262144 T^{12} \)
$3$ 1
$5$ \( ( 1 - 279 T^{2} + 14742 T^{4} + 1160125 T^{6} + 230343750 T^{8} - 68115234375 T^{10} + 3814697265625 T^{12} )^{2} \)
$7$ \( ( 1 - 1101 T^{2} + 652854 T^{4} - 259162985 T^{6} + 76807620246 T^{8} - 15239257208301 T^{10} + 1628413597910449 T^{12} )^{2} \)
$11$ \( ( 1 + 3057 T^{2} + 6556998 T^{4} + 9226981429 T^{6} + 11616121933878 T^{8} + 9594175547636097 T^{10} + 5559917313492231481 T^{12} )^{2} \)
$13$ \( ( 1 + 18 T + 3915 T^{2} + 8332 T^{3} + 8601255 T^{4} + 86882562 T^{5} + 10604499373 T^{6} )^{4} \)
$17$ \( ( 1 - 11946 T^{2} + 72737391 T^{4} - 332667352460 T^{6} + 1755703794142479 T^{8} - 6960005245946724906 T^{10} + \)\(14\!\cdots\!09\)\( T^{12} )^{2} \)
$19$ \( ( 1 - 8322 T^{2} + 146058135 T^{4} - 772095813500 T^{6} + 6871433638291935 T^{8} - 18419206756468591842 T^{10} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \)
$23$ \( ( 1 + 52314 T^{2} + 1249837599 T^{4} + 18511268917228 T^{6} + 185020820073590511 T^{8} + \)\(11\!\cdots\!94\)\( T^{10} + \)\(32\!\cdots\!69\)\( T^{12} )^{2} \)
$29$ \( ( 1 - 117954 T^{2} + 6353615223 T^{4} - 198355607069564 T^{6} + 3779278507301015583 T^{8} - \)\(41\!\cdots\!14\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \)
$31$ \( ( 1 - 45741 T^{2} + 1957054854 T^{4} - 51110094297449 T^{6} + 1736893386843917574 T^{8} - \)\(36\!\cdots\!01\)\( T^{10} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \)
$37$ \( ( 1 + 60 T + 83559 T^{2} - 2089640 T^{3} + 4232514027 T^{4} + 153943584540 T^{5} + 129961739795077 T^{6} )^{4} \)
$41$ \( ( 1 - 189606 T^{2} + 21529296543 T^{4} - 1820253257865236 T^{6} + \)\(10\!\cdots\!63\)\( T^{8} - \)\(42\!\cdots\!86\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \)
$43$ \( ( 1 + 27042 T^{2} + 15896713575 T^{4} + 267144985943548 T^{6} + \)\(10\!\cdots\!75\)\( T^{8} + \)\(10\!\cdots\!42\)\( T^{10} + \)\(25\!\cdots\!49\)\( T^{12} )^{2} \)
$47$ \( ( 1 + 377142 T^{2} + 74876518767 T^{4} + 9593174285133748 T^{6} + \)\(80\!\cdots\!43\)\( T^{8} + \)\(43\!\cdots\!22\)\( T^{10} + \)\(12\!\cdots\!89\)\( T^{12} )^{2} \)
$53$ \( ( 1 - 398679 T^{2} + 82394238966 T^{4} - 12639200632897475 T^{6} + \)\(18\!\cdots\!14\)\( T^{8} - \)\(19\!\cdots\!39\)\( T^{10} + \)\(10\!\cdots\!89\)\( T^{12} )^{2} \)
$59$ \( ( 1 + 275982 T^{2} + 30860919687 T^{4} - 3157264383153500 T^{6} + \)\(13\!\cdots\!67\)\( T^{8} + \)\(49\!\cdots\!42\)\( T^{10} + \)\(75\!\cdots\!21\)\( T^{12} )^{2} \)
$61$ \( ( 1 - 36 T + 351135 T^{2} - 84569384 T^{3} + 79700973435 T^{4} - 1854733476996 T^{5} + 11694146092834141 T^{6} )^{4} \)
$67$ \( ( 1 - 1341390 T^{2} + 846750752247 T^{4} - 319903491280410596 T^{6} + \)\(76\!\cdots\!43\)\( T^{8} - \)\(10\!\cdots\!90\)\( T^{10} + \)\(74\!\cdots\!09\)\( T^{12} )^{2} \)
$71$ \( ( 1 + 1203654 T^{2} + 624403431231 T^{4} + 229814346307125076 T^{6} + \)\(79\!\cdots\!51\)\( T^{8} + \)\(19\!\cdots\!14\)\( T^{10} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \)
$73$ \( ( 1 - 39 T + 955110 T^{2} - 15631571 T^{3} + 371554026870 T^{4} - 5902034825271 T^{5} + 58871586708267913 T^{6} )^{4} \)
$79$ \( ( 1 - 1716006 T^{2} + 1350264931311 T^{4} - 731914440695115860 T^{6} + \)\(32\!\cdots\!31\)\( T^{8} - \)\(10\!\cdots\!46\)\( T^{10} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \)
$83$ \( ( 1 + 1600737 T^{2} + 1256006483670 T^{4} + 724491266389094437 T^{6} + \)\(41\!\cdots\!30\)\( T^{8} + \)\(17\!\cdots\!57\)\( T^{10} + \)\(34\!\cdots\!09\)\( T^{12} )^{2} \)
$89$ \( ( 1 + 123846 T^{2} + 872233472127 T^{4} - 77031279330160556 T^{6} + \)\(43\!\cdots\!47\)\( T^{8} + \)\(30\!\cdots\!66\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \)
$97$ \( ( 1 - 129 T + 1814286 T^{2} - 533624789 T^{3} + 1655849846478 T^{4} - 107453388635841 T^{5} + 760231058654565217 T^{6} )^{4} \)
show more
show less