Properties

Label 3360.2.ba.d
Level $3360$
Weight $2$
Character orbit 3360.ba
Analytic conductor $26.830$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.ba (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} + 8 x^{18} - 20 x^{17} + 50 x^{16} - 92 x^{15} + 144 x^{14} - 244 x^{13} + 481 x^{12} - 696 x^{11} + 848 x^{10} - 2088 x^{9} + 4329 x^{8} - 6588 x^{7} + 11664 x^{6} - 22356 x^{5} + 36450 x^{4} - 43740 x^{3} + 52488 x^{2} - 78732 x + 59049\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{11} q^{3} -\beta_{7} q^{5} -\beta_{7} q^{7} -\beta_{19} q^{9} +O(q^{10})\) \( q -\beta_{11} q^{3} -\beta_{7} q^{5} -\beta_{7} q^{7} -\beta_{19} q^{9} + ( 1 - \beta_{2} + \beta_{9} + \beta_{11} ) q^{11} + ( \beta_{1} + \beta_{9} - \beta_{13} + \beta_{17} ) q^{13} + \beta_{8} q^{15} + ( -\beta_{1} - \beta_{5} + \beta_{7} - \beta_{12} ) q^{17} + ( \beta_{3} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{19} + \beta_{8} q^{21} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{18} - \beta_{19} ) q^{23} - q^{25} + ( -2 + \beta_{1} + \beta_{3} - \beta_{7} + \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{17} ) q^{27} + ( -\beta_{5} - \beta_{6} + \beta_{7} - \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{29} + ( -\beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{18} - \beta_{19} ) q^{31} + ( -2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{16} ) q^{33} - q^{35} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{9} + 2 \beta_{13} - \beta_{16} - \beta_{17} ) q^{37} + ( -2 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{8} + 2 \beta_{10} - 2 \beta_{11} - \beta_{15} + \beta_{16} - 2 \beta_{18} + \beta_{19} ) q^{39} + ( -\beta_{1} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{41} + ( -\beta_{1} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{15} + \beta_{16} + \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{43} -\beta_{3} q^{45} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} - \beta_{17} ) q^{47} - q^{49} + ( \beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{51} + ( \beta_{5} + \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} + 3 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} + 3 \beta_{18} - 3 \beta_{19} ) q^{53} + ( -\beta_{7} - \beta_{8} + \beta_{10} - \beta_{15} ) q^{55} + ( -2 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} ) q^{57} + ( 2 - 2 \beta_{1} + 4 \beta_{2} + \beta_{5} - \beta_{6} - 3 \beta_{9} - \beta_{11} + 2 \beta_{13} - 2 \beta_{17} ) q^{59} + ( -3 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{9} - \beta_{13} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{61} -\beta_{3} q^{63} + ( \beta_{3} + \beta_{6} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{65} + ( -\beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{14} + \beta_{16} + \beta_{17} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} - 2 \beta_{19} ) q^{69} + ( -2 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{8} - \beta_{9} + 3 \beta_{10} - 2 \beta_{11} - \beta_{13} + \beta_{16} ) q^{71} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + \beta_{11} - 2 \beta_{13} + 2 \beta_{17} ) q^{73} + \beta_{11} q^{75} + ( -\beta_{7} - \beta_{8} + \beta_{10} - \beta_{15} ) q^{77} + ( \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - 4 \beta_{7} - 3 \beta_{8} + 4 \beta_{9} + 3 \beta_{10} - 4 \beta_{11} + 2 \beta_{14} - 4 \beta_{15} + 3 \beta_{16} + 3 \beta_{17} - 3 \beta_{18} + 3 \beta_{19} ) q^{79} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} - \beta_{18} ) q^{81} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} - 2 \beta_{18} - 2 \beta_{19} ) q^{83} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{13} + \beta_{16} ) q^{85} + ( 3 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} + 2 \beta_{19} ) q^{87} + ( -\beta_{1} - 2 \beta_{3} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{89} + ( \beta_{3} + \beta_{6} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{91} + ( 4 + \beta_{1} + 2 \beta_{3} + 2 \beta_{7} + 4 \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{17} + 2 \beta_{19} ) q^{93} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} - \beta_{17} ) q^{95} + ( 4 - \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{16} - \beta_{17} ) q^{97} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{10} + 3 \beta_{11} + \beta_{13} - \beta_{15} - 2 \beta_{17} - \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{3} + 8 q^{9} + O(q^{10}) \) \( 20 q + 4 q^{3} + 8 q^{9} + 8 q^{11} + 8 q^{13} + 4 q^{15} + 4 q^{21} - 8 q^{23} - 20 q^{25} - 20 q^{27} - 40 q^{33} - 20 q^{35} + 16 q^{37} - 4 q^{39} - 20 q^{49} - 4 q^{51} - 16 q^{57} + 64 q^{59} - 64 q^{61} + 8 q^{69} + 40 q^{71} + 8 q^{73} - 4 q^{75} + 8 q^{81} - 24 q^{83} + 8 q^{85} + 48 q^{87} + 72 q^{93} - 16 q^{95} + 88 q^{97} - 28 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 4 x^{19} + 8 x^{18} - 20 x^{17} + 50 x^{16} - 92 x^{15} + 144 x^{14} - 244 x^{13} + 481 x^{12} - 696 x^{11} + 848 x^{10} - 2088 x^{9} + 4329 x^{8} - 6588 x^{7} + 11664 x^{6} - 22356 x^{5} + 36450 x^{4} - 43740 x^{3} + 52488 x^{2} - 78732 x + 59049\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)
\(\beta_{2}\)\(=\)\((\)\(-3065 \nu^{19} - 4748 \nu^{18} + 24339 \nu^{17} - 14083 \nu^{16} + 15787 \nu^{15} - 152128 \nu^{14} + 411695 \nu^{13} - 176083 \nu^{12} - 173920 \nu^{11} - 1644688 \nu^{10} + 2615336 \nu^{9} + 1647184 \nu^{8} + 1445511 \nu^{7} - 16656228 \nu^{6} + 24614739 \nu^{5} - 6752403 \nu^{4} + 38178459 \nu^{3} - 125393832 \nu^{2} + 72864279 \nu + 74775717\)\()/33014952\)
\(\beta_{3}\)\(=\)\((\)\(4 \nu^{19} - 13 \nu^{18} + 20 \nu^{17} - 56 \nu^{16} + 140 \nu^{15} - 218 \nu^{14} + 300 \nu^{13} - 544 \nu^{12} + 1192 \nu^{11} - 1341 \nu^{10} + 1304 \nu^{9} - 5808 \nu^{8} + 11052 \nu^{7} - 13365 \nu^{6} + 26892 \nu^{5} - 54432 \nu^{4} + 78732 \nu^{3} - 65610 \nu^{2} + 78732 \nu - 157464\)\()/19683\)
\(\beta_{4}\)\(=\)\((\)\(-1046 \nu^{19} + 4016 \nu^{18} - 7894 \nu^{17} + 15751 \nu^{16} - 41938 \nu^{15} + 83692 \nu^{14} - 102138 \nu^{13} + 189551 \nu^{12} - 411752 \nu^{11} + 457704 \nu^{10} - 637384 \nu^{9} + 1781520 \nu^{8} - 3473046 \nu^{7} + 5420520 \nu^{6} - 8602686 \nu^{5} + 19347903 \nu^{4} - 27190242 \nu^{3} + 24818076 \nu^{2} - 47777202 \nu + 50644359\)\()/2913084\)
\(\beta_{5}\)\(=\)\((\)\(14087 \nu^{19} - 29093 \nu^{18} + 54229 \nu^{17} - 179431 \nu^{16} + 360067 \nu^{15} - 583837 \nu^{14} + 904809 \nu^{13} - 1658027 \nu^{12} + 3477056 \nu^{11} - 3123552 \nu^{10} + 5679832 \nu^{9} - 18171696 \nu^{8} + 25724367 \nu^{7} - 42802749 \nu^{6} + 82142829 \nu^{5} - 152329167 \nu^{4} + 218398923 \nu^{3} - 200152053 \nu^{2} + 332623017 \nu - 453201075\)\()/5826168\)
\(\beta_{6}\)\(=\)\((\)\(15587 \nu^{19} - 35645 \nu^{18} + 68833 \nu^{17} - 206197 \nu^{16} + 432103 \nu^{15} - 741157 \nu^{14} + 1105533 \nu^{13} - 2020553 \nu^{12} + 4232480 \nu^{11} - 4054056 \nu^{10} + 7070632 \nu^{9} - 21251664 \nu^{8} + 31961835 \nu^{7} - 53587845 \nu^{6} + 98293257 \nu^{5} - 188147853 \nu^{4} + 271053135 \nu^{3} - 250435557 \nu^{2} + 424293309 \nu - 554489793\)\()/5826168\)
\(\beta_{7}\)\(=\)\((\)\(-10591 \nu^{19} + 23833 \nu^{18} - 42761 \nu^{17} + 135986 \nu^{16} - 287963 \nu^{15} + 467645 \nu^{14} - 706797 \nu^{13} + 1320118 \nu^{12} - 2750392 \nu^{11} + 2583984 \nu^{10} - 4423352 \nu^{9} + 14182848 \nu^{8} - 20852487 \nu^{7} + 33317433 \nu^{6} - 64515609 \nu^{5} + 121077666 \nu^{4} - 173340891 \nu^{3} + 160471125 \nu^{2} - 267616629 \nu + 359687142\)\()/2676888\)
\(\beta_{8}\)\(=\)\((\)\(-293635 \nu^{19} + 660817 \nu^{18} - 1211543 \nu^{17} + 3827339 \nu^{16} - 8045015 \nu^{15} + 12993041 \nu^{14} - 20057427 \nu^{13} + 37263211 \nu^{12} - 76363120 \nu^{11} + 72524424 \nu^{10} - 125550536 \nu^{9} + 397136352 \nu^{8} - 580180011 \nu^{7} + 934535457 \nu^{6} - 1831911471 \nu^{5} + 3388232835 \nu^{4} - 4839402591 \nu^{3} + 4574749833 \nu^{2} - 7522774803 \nu + 10175343363\)\()/66029904\)
\(\beta_{9}\)\(=\)\((\)\(305785 \nu^{19} - 688141 \nu^{18} + 1212575 \nu^{17} - 3888257 \nu^{16} + 8345501 \nu^{15} - 13197461 \nu^{14} + 20157699 \nu^{13} - 38418409 \nu^{12} + 78033088 \nu^{11} - 73221192 \nu^{10} + 127229240 \nu^{9} - 407991648 \nu^{8} + 597119985 \nu^{7} - 938597949 \nu^{6} + 1873732743 \nu^{5} - 3485452761 \nu^{4} + 4860234981 \nu^{3} - 4673721789 \nu^{2} + 7702528707 \nu - 10111609809\)\()/66029904\)
\(\beta_{10}\)\(=\)\((\)\(102563 \nu^{19} - 231171 \nu^{18} + 423719 \nu^{17} - 1335231 \nu^{16} + 2829207 \nu^{15} - 4582323 \nu^{14} + 6967331 \nu^{13} - 13151327 \nu^{12} + 27032208 \nu^{11} - 25350088 \nu^{10} + 43930408 \nu^{9} - 139983104 \nu^{8} + 204595323 \nu^{7} - 326457891 \nu^{6} + 639826479 \nu^{5} - 1203133095 \nu^{4} + 1699162191 \nu^{3} - 1584824859 \nu^{2} + 2687300307 \nu - 3603872007\)\()/22009968\)
\(\beta_{11}\)\(=\)\((\)\(-968387 \nu^{19} + 2225687 \nu^{18} - 3924853 \nu^{17} + 12426469 \nu^{16} - 26902543 \nu^{15} + 42749791 \nu^{14} - 64233105 \nu^{13} + 122717165 \nu^{12} - 251834000 \nu^{11} + 236462040 \nu^{10} - 404150536 \nu^{9} + 1309058304 \nu^{8} - 1937534571 \nu^{7} + 3025687095 \nu^{6} - 5963231421 \nu^{5} + 11242972125 \nu^{4} - 15733440567 \nu^{3} + 14777307495 \nu^{2} - 24628635873 \nu + 32540388309\)\()/ 198089712 \)
\(\beta_{12}\)\(=\)\((\)\(549287 \nu^{19} - 1274081 \nu^{18} + 2313757 \nu^{17} - 7172269 \nu^{16} + 15447271 \nu^{15} - 25071541 \nu^{14} + 37856421 \nu^{13} - 71320049 \nu^{12} + 145845104 \nu^{11} - 139013880 \nu^{10} + 237644584 \nu^{9} - 751537584 \nu^{8} + 1118015487 \nu^{7} - 1777344849 \nu^{6} + 3468762549 \nu^{5} - 6521421861 \nu^{4} + 9193313295 \nu^{3} - 8733353661 \nu^{2} + 14369462613 \nu - 19060761321\)\()/99044856\)
\(\beta_{13}\)\(=\)\((\)\(-663485 \nu^{19} + 1581179 \nu^{18} - 2792950 \nu^{17} + 8748385 \nu^{16} - 18956245 \nu^{15} + 30199939 \nu^{14} - 45769074 \nu^{13} + 86518085 \nu^{12} - 177234440 \nu^{11} + 169137432 \nu^{10} - 286904296 \nu^{9} + 919036608 \nu^{8} - 1366827309 \nu^{7} + 2138304771 \nu^{6} - 4240349190 \nu^{5} + 7915890969 \nu^{4} - 11076581277 \nu^{3} + 10519699635 \nu^{2} - 17539455690 \nu + 22939493301\)\()/99044856\)
\(\beta_{14}\)\(=\)\((\)\(-531505 \nu^{19} + 1205809 \nu^{18} - 2179559 \nu^{17} + 6904502 \nu^{16} - 14734325 \nu^{15} + 23648777 \nu^{14} - 36120075 \nu^{13} + 68097682 \nu^{12} - 138845320 \nu^{11} + 131232912 \nu^{10} - 226809056 \nu^{9} + 721251744 \nu^{8} - 1058973705 \nu^{7} + 1685196009 \nu^{6} - 3322516887 \nu^{5} + 6192507510 \nu^{4} - 8731079181 \nu^{3} + 8311446369 \nu^{2} - 13755339891 \nu + 18199807218\)\()/49522428\)
\(\beta_{15}\)\(=\)\((\)\(282421 \nu^{19} - 639424 \nu^{18} + 1147703 \nu^{17} - 3678638 \nu^{16} + 7835453 \nu^{15} - 12456995 \nu^{14} + 19121007 \nu^{13} - 36561016 \nu^{12} + 73787356 \nu^{11} - 68721426 \nu^{10} + 121420604 \nu^{9} - 386221818 \nu^{8} + 560087973 \nu^{7} - 885419262 \nu^{6} + 1772161983 \nu^{5} - 3313332216 \nu^{4} + 4606044345 \nu^{3} - 4398404733 \nu^{2} + 7447542003 \nu - 9713481768\)\()/24761214\)
\(\beta_{16}\)\(=\)\((\)\(-2291315 \nu^{19} + 5299301 \nu^{18} - 9366223 \nu^{17} + 29478187 \nu^{16} - 64143871 \nu^{15} + 102304909 \nu^{14} - 152328483 \nu^{13} + 294249803 \nu^{12} - 603971984 \nu^{11} + 561437160 \nu^{10} - 961793416 \nu^{9} + 3114010368 \nu^{8} - 4627932219 \nu^{7} + 7173674037 \nu^{6} - 14165139303 \nu^{5} + 27005640003 \nu^{4} - 37524230343 \nu^{3} + 34990003269 \nu^{2} - 59634785763 \nu + 78213648195\)\()/ 198089712 \)
\(\beta_{17}\)\(=\)\((\)\(-766795 \nu^{19} + 1739871 \nu^{18} - 3059047 \nu^{17} + 9863967 \nu^{16} - 20921895 \nu^{15} + 33007287 \nu^{14} - 51060139 \nu^{13} + 96244639 \nu^{12} - 194648112 \nu^{11} + 184751144 \nu^{10} - 321604616 \nu^{9} + 1025856832 \nu^{8} - 1492420323 \nu^{7} + 2371715199 \nu^{6} - 4737575439 \nu^{5} + 8671734855 \nu^{4} - 12146144319 \nu^{3} + 11724599583 \nu^{2} - 19157387355 \nu + 25041020967\)\()/66029904\)
\(\beta_{18}\)\(=\)\((\)\(-397 \nu^{19} + 894 \nu^{18} - 1593 \nu^{17} + 5183 \nu^{16} - 10945 \nu^{15} + 17338 \nu^{14} - 26885 \nu^{13} + 50939 \nu^{12} - 102960 \nu^{11} + 96088 \nu^{10} - 169096 \nu^{9} + 540304 \nu^{8} - 778573 \nu^{7} + 1240134 \nu^{6} - 2488617 \nu^{5} + 4600935 \nu^{4} - 6447681 \nu^{3} + 6154218 \nu^{2} - 10291293 \nu + 13509099\)\()/33048\)
\(\beta_{19}\)\(=\)\((\)\(10769 \nu^{19} - 24802 \nu^{18} + 44829 \nu^{17} - 140687 \nu^{16} + 301253 \nu^{15} - 485006 \nu^{14} + 733417 \nu^{13} - 1399115 \nu^{12} + 2841424 \nu^{11} - 2665784 \nu^{10} + 4664584 \nu^{9} - 14741272 \nu^{8} + 21732945 \nu^{7} - 34386570 \nu^{6} + 67777965 \nu^{5} - 127556127 \nu^{4} + 177543333 \nu^{3} - 168181758 \nu^{2} + 285961185 \nu - 370112571\)\()/892296\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{14} - 2 \beta_{12} - \beta_{4}\)\()/4\)
\(\nu^{2}\)\(=\)\(\beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{19} - 2 \beta_{18} + 2 \beta_{17} - 2 \beta_{15} + 3 \beta_{14} + 2 \beta_{13} - 6 \beta_{11} - 2 \beta_{10} - 6 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 6\)\()/4\)
\(\nu^{4}\)\(=\)\(\beta_{19} - \beta_{15} - \beta_{14} + \beta_{11} - 2 \beta_{10} - \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{3} - \beta_{2} + 1\)
\(\nu^{5}\)\(=\)\((\)\(16 \beta_{19} - 12 \beta_{18} + 10 \beta_{17} + 6 \beta_{16} - 24 \beta_{15} + 11 \beta_{14} - 2 \beta_{13} + 6 \beta_{12} - 6 \beta_{11} + 16 \beta_{10} + 28 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 12 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} - 10 \beta_{3} + 4 \beta_{2} + 2 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\(\beta_{19} + 3 \beta_{18} + \beta_{17} - \beta_{16} + 3 \beta_{15} + 2 \beta_{14} + 2 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} + 2 \beta_{8} - 8 \beta_{7} - 6 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 5\)
\(\nu^{7}\)\(=\)\((\)\(40 \beta_{19} - 4 \beta_{18} + 20 \beta_{17} + 28 \beta_{16} - 20 \beta_{15} - 7 \beta_{14} - 14 \beta_{13} - 34 \beta_{12} + 2 \beta_{11} - 46 \beta_{10} + 64 \beta_{9} - 44 \beta_{8} - 2 \beta_{7} + 24 \beta_{6} - 30 \beta_{5} + 13 \beta_{4} + 34 \beta_{3} - 36 \beta_{2} + 2 \beta_{1} + 80\)\()/4\)
\(\nu^{8}\)\(=\)\(2 \beta_{19} - 3 \beta_{18} + 2 \beta_{17} - 10 \beta_{16} - 7 \beta_{15} - 3 \beta_{14} - 12 \beta_{12} + 15 \beta_{11} - 8 \beta_{10} + 9 \beta_{9} - 8 \beta_{8} + 5 \beta_{7} + 9 \beta_{6} - 3 \beta_{5} + 8 \beta_{4} + 6 \beta_{3} + 5 \beta_{2} - 21\)
\(\nu^{9}\)\(=\)\((\)\(18 \beta_{19} + 62 \beta_{18} + 22 \beta_{17} - 8 \beta_{16} + 70 \beta_{15} + 57 \beta_{14} - 26 \beta_{13} + 36 \beta_{12} - 6 \beta_{11} - 74 \beta_{10} - 4 \beta_{9} - 14 \beta_{8} - 126 \beta_{7} + 2 \beta_{6} - 18 \beta_{5} + 129 \beta_{4} - 30 \beta_{3} + 66 \beta_{2} + 36 \beta_{1} - 98\)\()/4\)
\(\nu^{10}\)\(=\)\(40 \beta_{19} - 24 \beta_{18} + 38 \beta_{17} + 30 \beta_{16} - 28 \beta_{15} + 28 \beta_{14} - 20 \beta_{13} + 44 \beta_{12} - 8 \beta_{11} - 108 \beta_{10} + 76 \beta_{9} - 92 \beta_{8} + 4 \beta_{7} - 18 \beta_{6} + 10 \beta_{5} - 18 \beta_{4} - 43 \beta_{3} - 46 \beta_{2} + 32 \beta_{1} + 18\)
\(\nu^{11}\)\(=\)\((\)\(-116 \beta_{19} + 36 \beta_{18} - 24 \beta_{17} - 164 \beta_{16} - 36 \beta_{15} - 43 \beta_{14} - 236 \beta_{13} + 126 \beta_{12} + 672 \beta_{11} - 156 \beta_{10} + 156 \beta_{9} - 164 \beta_{8} + 120 \beta_{7} + 150 \beta_{6} + 6 \beta_{5} + 87 \beta_{4} - 206 \beta_{3} - 200 \beta_{2} - 26 \beta_{1} + 752\)\()/4\)
\(\nu^{12}\)\(=\)\(-6 \beta_{19} + 38 \beta_{18} + 74 \beta_{17} + 58 \beta_{16} + 42 \beta_{15} + 34 \beta_{14} - 104 \beta_{13} - 24 \beta_{12} - 62 \beta_{11} + 82 \beta_{10} + 174 \beta_{9} + 118 \beta_{8} - 70 \beta_{7} - 58 \beta_{6} + 126 \beta_{5} + 12 \beta_{4} + 6 \beta_{3} + 88 \beta_{2} + 46 \beta_{1} - 137\)
\(\nu^{13}\)\(=\)\((\)\(-828 \beta_{19} + 524 \beta_{18} + 16 \beta_{17} + 324 \beta_{16} + 1348 \beta_{15} - 511 \beta_{14} - 684 \beta_{13} + 270 \beta_{12} + 1360 \beta_{11} - 2052 \beta_{10} + 780 \beta_{9} - 604 \beta_{8} + 104 \beta_{7} + 632 \beta_{6} + 556 \beta_{5} - 915 \beta_{4} + 708 \beta_{3} - 1064 \beta_{2} + 1948 \beta_{1} + 1872\)\()/4\)
\(\nu^{14}\)\(=\)\(-320 \beta_{19} + 16 \beta_{18} - 70 \beta_{17} - 158 \beta_{16} + 164 \beta_{15} - 92 \beta_{14} - 140 \beta_{13} - 436 \beta_{12} + 56 \beta_{11} - 428 \beta_{10} + 68 \beta_{9} - 748 \beta_{8} + 676 \beta_{7} + 646 \beta_{6} - 190 \beta_{5} - 62 \beta_{4} + 88 \beta_{3} - 306 \beta_{2} + 59 \beta_{1} + 1070\)
\(\nu^{15}\)\(=\)\((\)\(-2622 \beta_{19} + 1838 \beta_{18} - 474 \beta_{17} - 1468 \beta_{16} + 2582 \beta_{15} - 3171 \beta_{14} - 534 \beta_{13} - 740 \beta_{12} - 490 \beta_{11} - 2514 \beta_{10} - 4172 \beta_{9} + 778 \beta_{8} + 666 \beta_{7} - 398 \beta_{6} + 1878 \beta_{5} - 269 \beta_{4} - 2186 \beta_{3} + 1190 \beta_{2} + 2644 \beta_{1} - 6710\)\()/4\)
\(\nu^{16}\)\(=\)\(-223 \beta_{19} - 130 \beta_{18} + 510 \beta_{17} + 682 \beta_{16} - 37 \beta_{15} + 447 \beta_{14} - 496 \beta_{13} + 1460 \beta_{12} - 211 \beta_{11} - 1052 \beta_{10} + 2075 \beta_{9} - 1308 \beta_{8} + 615 \beta_{7} - 197 \beta_{6} + 15 \beta_{5} - 864 \beta_{4} - 1464 \beta_{3} - 1043 \beta_{2} + 2162 \beta_{1} + 2195\)
\(\nu^{17}\)\(=\)\((\)\(-3780 \beta_{19} + 4848 \beta_{18} - 2538 \beta_{17} - 5258 \beta_{16} + 3140 \beta_{15} + 165 \beta_{14} + 862 \beta_{13} + 3614 \beta_{12} + 3926 \beta_{11} + 2388 \beta_{10} - 8600 \beta_{9} - 10754 \beta_{8} + 7606 \beta_{7} - 7228 \beta_{6} + 6198 \beta_{5} - 3857 \beta_{4} - 7098 \beta_{3} + 1892 \beta_{2} - 574 \beta_{1} + 20464\)\()/4\)
\(\nu^{18}\)\(=\)\(251 \beta_{19} + 1913 \beta_{18} + 373 \beta_{17} + 427 \beta_{16} + 21 \beta_{15} - 5082 \beta_{14} - 320 \beta_{13} - 2610 \beta_{12} + 296 \beta_{11} + 2802 \beta_{10} - 756 \beta_{9} + 3586 \beta_{8} + 1288 \beta_{7} - 428 \beta_{6} + 576 \beta_{5} + 1791 \beta_{4} + 722 \beta_{3} + 505 \beta_{2} + 876 \beta_{1} + 6105\)
\(\nu^{19}\)\(=\)\((\)\(-5180 \beta_{19} + 4520 \beta_{18} + 548 \beta_{17} - 3216 \beta_{16} + 976 \beta_{15} - 1433 \beta_{14} + 11394 \beta_{13} - 12970 \beta_{12} + 4926 \beta_{11} + 14850 \beta_{10} + 35868 \beta_{9} + 920 \beta_{8} + 1146 \beta_{7} - 2760 \beta_{6} + 5098 \beta_{5} - 9857 \beta_{4} + 7306 \beta_{3} + 9052 \beta_{2} + 20138 \beta_{1} + 2752\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times\).

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
−1.15110 + 1.29420i
−1.15110 1.29420i
0.871943 1.49657i
0.871943 + 1.49657i
1.73205 0.00281084i
1.73205 + 0.00281084i
0.842277 + 1.51346i
0.842277 1.51346i
−1.61735 0.619827i
−1.61735 + 0.619827i
1.64300 + 0.548239i
1.64300 0.548239i
−0.504688 + 1.65689i
−0.504688 1.65689i
−0.433306 1.67698i
−0.433306 + 1.67698i
1.58473 0.699012i
1.58473 + 0.699012i
−0.967551 1.43661i
−0.967551 + 1.43661i
0 −1.72909 0.101185i 0 1.00000i 0 1.00000i 0 2.97952 + 0.349915i 0
2591.2 0 −1.72909 + 0.101185i 0 1.00000i 0 1.00000i 0 2.97952 0.349915i 0
2591.3 0 −1.67479 0.441676i 0 1.00000i 0 1.00000i 0 2.60984 + 1.47943i 0
2591.4 0 −1.67479 + 0.441676i 0 1.00000i 0 1.00000i 0 2.60984 1.47943i 0
2591.5 0 −1.22276 1.22673i 0 1.00000i 0 1.00000i 0 −0.00973701 + 2.99998i 0
2591.6 0 −1.22276 + 1.22673i 0 1.00000i 0 1.00000i 0 −0.00973701 2.99998i 0
2591.7 0 −0.474600 1.66576i 0 1.00000i 0 1.00000i 0 −2.54951 + 1.58114i 0
2591.8 0 −0.474600 + 1.66576i 0 1.00000i 0 1.00000i 0 −2.54951 1.58114i 0
2591.9 0 0.705354 1.58192i 0 1.00000i 0 1.00000i 0 −2.00495 2.23163i 0
2591.10 0 0.705354 + 1.58192i 0 1.00000i 0 1.00000i 0 −2.00495 + 2.23163i 0
2591.11 0 0.774110 1.54944i 0 1.00000i 0 1.00000i 0 −1.80151 2.39887i 0
2591.12 0 0.774110 + 1.54944i 0 1.00000i 0 1.00000i 0 −1.80151 + 2.39887i 0
2591.13 0 0.814731 1.52847i 0 1.00000i 0 1.00000i 0 −1.67243 2.49058i 0
2591.14 0 0.814731 + 1.52847i 0 1.00000i 0 1.00000i 0 −1.67243 + 2.49058i 0
2591.15 0 1.49219 0.879407i 0 1.00000i 0 1.00000i 0 1.45329 2.62449i 0
2591.16 0 1.49219 + 0.879407i 0 1.00000i 0 1.00000i 0 1.45329 + 2.62449i 0
2591.17 0 1.61485 0.626300i 0 1.00000i 0 1.00000i 0 2.21550 2.02277i 0
2591.18 0 1.61485 + 0.626300i 0 1.00000i 0 1.00000i 0 2.21550 + 2.02277i 0
2591.19 0 1.70000 0.331674i 0 1.00000i 0 1.00000i 0 2.77998 1.12769i 0
2591.20 0 1.70000 + 0.331674i 0 1.00000i 0 1.00000i 0 2.77998 + 1.12769i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 3360.2.ba.d yes 20
3.b odd 2 1 3360.2.ba.c 20
4.b odd 2 1 3360.2.ba.c 20
12.b even 2 1 inner 3360.2.ba.d yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.ba.c 20 3.b odd 2 1
3360.2.ba.c 20 4.b odd 2 1
3360.2.ba.d yes 20 1.a even 1 1 trivial
3360.2.ba.d yes 20 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3360, [\chi])\):

\(T_{11}^{10} - \cdots\)
\(T_{23}^{10} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( 59049 - 78732 T + 26244 T^{2} + 26244 T^{3} - 30618 T^{4} + 8748 T^{5} + 3726 T^{6} - 2700 T^{7} + 369 T^{8} - 72 T^{9} + 124 T^{10} - 24 T^{11} + 41 T^{12} - 100 T^{13} + 46 T^{14} + 36 T^{15} - 42 T^{16} + 12 T^{17} + 4 T^{18} - 4 T^{19} + T^{20} \)
$5$ \( ( 1 + T^{2} )^{10} \)
$7$ \( ( 1 + T^{2} )^{10} \)
$11$ \( ( -576 + 192 T + 3280 T^{2} + 80 T^{3} - 5040 T^{4} - 1984 T^{5} + 897 T^{6} + 204 T^{7} - 58 T^{8} - 4 T^{9} + T^{10} )^{2} \)
$13$ \( ( 38336 - 110528 T + 24752 T^{2} + 37280 T^{3} - 10028 T^{4} - 4460 T^{5} + 1193 T^{6} + 224 T^{7} - 58 T^{8} - 4 T^{9} + T^{10} )^{2} \)
$17$ \( 67949891584 + 87410407424 T^{2} + 44795933952 T^{4} + 12090813824 T^{6} + 1939989920 T^{8} + 196035824 T^{10} + 12730657 T^{12} + 525860 T^{14} + 13190 T^{16} + 180 T^{18} + T^{20} \)
$19$ \( 4194304 + 977272832 T^{2} + 2200174592 T^{4} + 1518010368 T^{6} + 463147008 T^{8} + 73811968 T^{10} + 6629632 T^{12} + 345472 T^{14} + 10272 T^{16} + 160 T^{18} + T^{20} \)
$23$ \( ( -2048 + 14336 T + 2560 T^{2} - 29184 T^{3} + 1472 T^{4} + 8960 T^{5} + 1184 T^{6} - 496 T^{7} - 92 T^{8} + 4 T^{9} + T^{10} )^{2} \)
$29$ \( 12131700736 + 79746191360 T^{2} + 161807557120 T^{4} + 114860512128 T^{6} + 23543119184 T^{8} + 2195178520 T^{10} + 108468001 T^{12} + 2982412 T^{14} + 44950 T^{16} + 340 T^{18} + T^{20} \)
$31$ \( 84934656 + 4369416192 T^{2} + 23354146816 T^{4} + 21383053312 T^{6} + 6516744192 T^{8} + 889239552 T^{10} + 59862272 T^{12} + 2076288 T^{14} + 36720 T^{16} + 312 T^{18} + T^{20} \)
$37$ \( ( 14336 - 12288 T - 32768 T^{2} + 28160 T^{3} + 16064 T^{4} - 14784 T^{5} + 656 T^{6} + 1040 T^{7} - 120 T^{8} - 8 T^{9} + T^{10} )^{2} \)
$41$ \( 82796609536 + 169213952000 T^{2} + 116459569152 T^{4} + 37359353856 T^{6} + 6395895808 T^{8} + 626825216 T^{10} + 36258048 T^{12} + 1240448 T^{14} + 24304 T^{16} + 248 T^{18} + T^{20} \)
$43$ \( 109650720587776 + 84287921586176 T^{2} + 24070126567424 T^{4} + 3478827237376 T^{6} + 285645361152 T^{8} + 14005714944 T^{10} + 417419776 T^{12} + 7513728 T^{14} + 78864 T^{16} + 440 T^{18} + T^{20} \)
$47$ \( ( -170432 + 21696 T + 308656 T^{2} + 49168 T^{3} - 91368 T^{4} - 4808 T^{5} + 6721 T^{6} + 120 T^{7} - 166 T^{8} + T^{10} )^{2} \)
$53$ \( 5940468068122624 + 3189982671929344 T^{2} + 639940365385728 T^{4} + 62411273666560 T^{6} + 3364583641088 T^{8} + 107201794048 T^{10} + 2080108032 T^{12} + 24626560 T^{14} + 172240 T^{16} + 648 T^{18} + T^{20} \)
$59$ \( ( -1082331136 + 731316224 T - 105102336 T^{2} - 27008512 T^{3} + 8513472 T^{4} - 401472 T^{5} - 81456 T^{6} + 8656 T^{7} + 32 T^{8} - 32 T^{9} + T^{10} )^{2} \)
$61$ \( ( 5874688 + 3552256 T - 4887808 T^{2} - 424192 T^{3} + 690368 T^{4} + 33600 T^{5} - 27904 T^{6} - 2640 T^{7} + 200 T^{8} + 32 T^{9} + T^{10} )^{2} \)
$67$ \( 143735929176064 + 188496501276672 T^{2} + 78007174168576 T^{4} + 14093634502656 T^{6} + 1220216950784 T^{8} + 55356850176 T^{10} + 1386139136 T^{12} + 19499392 T^{14} + 152464 T^{16} + 616 T^{18} + T^{20} \)
$71$ \( ( 123715584 - 311083008 T + 98510848 T^{2} + 34213888 T^{3} - 3130944 T^{4} - 792896 T^{5} + 44464 T^{6} + 6816 T^{7} - 332 T^{8} - 20 T^{9} + T^{10} )^{2} \)
$73$ \( ( 278528 - 360448 T - 825344 T^{2} + 485376 T^{3} + 70720 T^{4} - 73984 T^{5} + 8608 T^{6} + 1136 T^{7} - 196 T^{8} - 4 T^{9} + T^{10} )^{2} \)
$79$ \( 399543654143758336 + 489721569750177792 T^{2} + 86545078109933312 T^{4} + 5682647686646016 T^{6} + 179344774272752 T^{8} + 3094030259928 T^{10} + 31222633025 T^{12} + 188451116 T^{14} + 666326 T^{16} + 1268 T^{18} + T^{20} \)
$83$ \( ( 81019904 + 72505344 T + 965632 T^{2} - 10759424 T^{3} - 1040896 T^{4} + 375488 T^{5} + 38416 T^{6} - 3824 T^{7} - 364 T^{8} + 12 T^{9} + T^{10} )^{2} \)
$89$ \( 20979624443904 + 30803223379968 T^{2} + 16735357370368 T^{4} + 4250379124736 T^{6} + 522859421696 T^{8} + 29913569280 T^{10} + 887898368 T^{12} + 14413184 T^{14} + 127216 T^{16} + 568 T^{18} + T^{20} \)
$97$ \( ( -25700288 + 14235520 T + 8082016 T^{2} - 6210384 T^{3} + 703052 T^{4} + 220484 T^{5} - 48479 T^{6} + 184 T^{7} + 566 T^{8} - 44 T^{9} + T^{10} )^{2} \)
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