Properties

Label 4016.2.a.l
Level 4016
Weight 2
Character orbit 4016.a
Self dual yes
Analytic conductor 32.068
Analytic rank 0
Dimension 19
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 2008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{4} q^{5} + ( 1 + \beta_{13} ) q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{4} q^{5} + ( 1 + \beta_{13} ) q^{7} + ( 1 + \beta_{2} ) q^{9} + ( 1 - \beta_{12} ) q^{11} + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{7} - \beta_{12} + \beta_{13} ) q^{13} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{7} - \beta_{13} - \beta_{15} ) q^{15} + ( -1 + \beta_{1} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{11} + \beta_{13} ) q^{17} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{19} + ( \beta_{1} + \beta_{7} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{21} + ( 1 + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{23} + ( -\beta_{3} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{25} + ( 2 + \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{14} + \beta_{15} + \beta_{18} ) q^{27} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} - \beta_{9} - 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{18} ) q^{29} + ( 3 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} - \beta_{9} - \beta_{14} - 2 \beta_{15} + \beta_{16} - \beta_{18} ) q^{31} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{33} + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{12} + \beta_{15} - \beta_{16} ) q^{35} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{18} ) q^{37} + ( 2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{39} + ( -1 + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{10} + 2 \beta_{12} + \beta_{15} - \beta_{16} - \beta_{18} ) q^{41} + ( 2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{43} + ( 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{16} + \beta_{18} ) q^{45} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} - 3 \beta_{7} + 2 \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{47} + ( 2 + \beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{49} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{14} + 2 \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{51} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{17} - \beta_{18} ) q^{53} + ( 3 + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} + \beta_{16} ) q^{55} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{8} + \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{57} + ( 3 + 2 \beta_{3} - \beta_{5} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{61} + ( 4 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{63} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{8} - \beta_{11} - 2 \beta_{14} - \beta_{15} + \beta_{17} ) q^{65} + ( 2 + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} ) q^{67} + ( -2 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{69} + ( 2 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} + \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{71} + ( 1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{15} + \beta_{16} ) q^{73} + ( -1 + 4 \beta_{1} - 4 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} + 3 \beta_{15} + \beta_{17} + \beta_{18} ) q^{75} + ( -2 - 2 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{77} + ( 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{79} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 4 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} - \beta_{16} + 2 \beta_{18} ) q^{81} + ( 4 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - \beta_{13} - \beta_{14} - 3 \beta_{15} + 2 \beta_{16} ) q^{83} + ( -2 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - \beta_{15} + 2 \beta_{16} - 3 \beta_{17} - \beta_{18} ) q^{85} + ( 3 + \beta_{1} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{87} + ( -1 + \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} ) q^{89} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + 3 \beta_{17} ) q^{91} + ( -1 + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} - 3 \beta_{15} + \beta_{16} - \beta_{17} - 3 \beta_{18} ) q^{93} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - 3 \beta_{13} - \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{95} + ( 3 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} - \beta_{15} - 2 \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{97} + ( 2 - 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} + \beta_{12} - 2 \beta_{13} - \beta_{16} - \beta_{18} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19q + 6q^{3} - 8q^{5} + 11q^{7} + 21q^{9} + O(q^{10}) \) \( 19q + 6q^{3} - 8q^{5} + 11q^{7} + 21q^{9} + 15q^{11} - 8q^{13} + 17q^{15} - 4q^{17} + 14q^{19} - 9q^{21} + 28q^{23} + 25q^{25} + 21q^{27} - 13q^{29} + 20q^{31} - 6q^{33} + 32q^{35} - 16q^{37} + 27q^{39} + 2q^{41} + 28q^{43} - 29q^{45} + 37q^{47} + 36q^{49} + 35q^{51} - 37q^{53} + 24q^{55} - 11q^{57} + 32q^{59} - 7q^{61} + 45q^{63} + q^{65} + 45q^{67} - 12q^{69} + 49q^{71} + 16q^{73} + 35q^{75} - 40q^{77} + 33q^{79} + 15q^{81} + 43q^{83} - 28q^{85} + 48q^{87} + 3q^{89} + 56q^{91} - 48q^{93} + 43q^{95} + 8q^{97} + 74q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19} - 6 x^{18} - 21 x^{17} + 179 x^{16} + 90 x^{15} - 2109 x^{14} + 926 x^{13} + 12681 x^{12} - 10845 x^{11} - 41921 x^{10} + 43551 x^{9} + 76260 x^{8} - 80907 x^{7} - 72526 x^{6} + 64793 x^{5} + 33209 x^{4} - 13777 x^{3} - 7607 x^{2} - 747 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(196521312495034 \nu^{18} - 1441579400369817 \nu^{17} - 2924913574354401 \nu^{16} + 41912041347671166 \nu^{15} - 17818217143386024 \nu^{14} - 475669164938282307 \nu^{13} + 595092550551867088 \nu^{12} + 2706294034633503240 \nu^{11} - 4580166278975943512 \nu^{10} - 8221073571569711829 \nu^{9} + 16553976937434110332 \nu^{8} + 13024341630912266379 \nu^{7} - 30398145885453740584 \nu^{6} - 9681015167402184673 \nu^{5} + 26640229615794321610 \nu^{4} + 2970308817404871287 \nu^{3} - 8517014376848128207 \nu^{2} - 1027269288097940330 \nu + 166591551288078888\)\()/ 77904810267289528 \)
\(\beta_{4}\)\(=\)\((\)\(209544026239557 \nu^{18} - 1435803919056848 \nu^{17} - 3871628649572152 \nu^{16} + 43016014683542135 \nu^{15} + 3243544109696637 \nu^{14} - 509405238497308593 \nu^{13} + 372341170757747395 \nu^{12} + 3079098801819725010 \nu^{11} - 3266269544593699975 \nu^{10} - 10210653079904899581 \nu^{9} + 11898633774999893396 \nu^{8} + 18475628716806910011 \nu^{7} - 20363135569951304978 \nu^{6} - 17067609862621199939 \nu^{5} + 14446277403237981944 \nu^{4} + 7316444036690675014 \nu^{3} - 2070932208707001660 \nu^{2} - 1800728050905473725 \nu - 242077884568730940\)\()/ 77904810267289528 \)
\(\beta_{5}\)\(=\)\((\)\(325315462336008 \nu^{18} - 1976289568186404 \nu^{17} - 6464820675496617 \nu^{16} + 57670999367330371 \nu^{15} + 19676691355848852 \nu^{14} - 658008988812589996 \nu^{13} + 394842138098307975 \nu^{12} + 3769404406780378010 \nu^{11} - 3952782696839577594 \nu^{10} - 11513787143158348190 \nu^{9} + 15060880540333249297 \nu^{8} + 18022917578774451206 \nu^{7} - 26919821315765788637 \nu^{6} - 11837322142659575168 \nu^{5} + 20551698355449594859 \nu^{4} + 912217879033491144 \nu^{3} - 3803526816531730763 \nu^{2} - 12176117032348777 \nu - 47302041294770420\)\()/ 77904810267289528 \)
\(\beta_{6}\)\(=\)\((\)\(346551847676251 \nu^{18} - 1668795238793348 \nu^{17} - 9554492206151678 \nu^{16} + 52690311990224015 \nu^{15} + 99268216390112767 \nu^{14} - 672831196084720603 \nu^{13} - 480330669882914705 \nu^{12} + 4531249610863146126 \nu^{11} + 1011790432893158971 \nu^{10} - 17538233581202119839 \nu^{9} - 194355797236659890 \nu^{8} + 39545581195445075145 \nu^{7} - 2411607128241522824 \nu^{6} - 49510080137058458545 \nu^{5} + 2855882369841072346 \nu^{4} + 29568840637183398650 \nu^{3} - 308192843555754662 \nu^{2} - 5219928908967227585 \nu - 498446106796647004\)\()/ 77904810267289528 \)
\(\beta_{7}\)\(=\)\((\)\(-498188113457669 \nu^{18} + 2523550044875629 \nu^{17} + 12303917053161873 \nu^{16} - 76188368510367967 \nu^{15} - 98542977570888683 \nu^{14} + 909421304377374380 \nu^{13} + 150593332231996867 \nu^{12} - 5531017263463239106 \nu^{11} + 1903876238659999871 \nu^{10} + 18353616170262585332 \nu^{9} - 11013728035806195900 \nu^{8} - 32825601884278571158 \nu^{7} + 23418204013206537144 \nu^{6} + 29170695205661226350 \nu^{5} - 20413894654710609824 \nu^{4} - 11312765533213992939 \nu^{3} + 4920039105746828385 \nu^{2} + 2667131240989337419 \nu - 4071618367057092\)\()/ 77904810267289528 \)
\(\beta_{8}\)\(=\)\((\)\(1007706538158390 \nu^{18} - 6067612651914337 \nu^{17} - 19728860249241516 \nu^{16} + 175301510143739749 \nu^{15} + 51557910709593326 \nu^{14} - 1963532947901539733 \nu^{13} + 1333989630839258891 \nu^{12} + 10860911091044278840 \nu^{11} - 12827190398504637216 \nu^{10} - 30953673001625573677 \nu^{9} + 47778169234604056599 \nu^{8} + 41554386482378362657 \nu^{7} - 82552732261583313803 \nu^{6} - 16139376886382327193 \nu^{5} + 58885576401540912769 \nu^{4} - 7789723814283011947 \nu^{3} - 8788558770665667400 \nu^{2} + 701698033358479469 \nu + 10329328802699356\)\()/ 77904810267289528 \)
\(\beta_{9}\)\(=\)\((\)\(-1027594418075442 \nu^{18} + 6762123352898153 \nu^{17} + 17708158431050788 \nu^{16} - 195121779192913155 \nu^{15} + 22140646155625798 \nu^{14} + 2178765600486528667 \nu^{13} - 2280762022625764797 \nu^{12} - 11959308405924989292 \nu^{11} + 18873883305931959538 \nu^{10} + 33399622787542986827 \nu^{9} - 68537331180212418241 \nu^{8} - 41967676115055920471 \nu^{7} + 119852409134664945067 \nu^{6} + 9494499845645210107 \nu^{5} - 88948367973291215153 \nu^{4} + 15105377852532462783 \nu^{3} + 14887804797132335270 \nu^{2} - 817380982643208987 \nu - 90972821278178156\)\()/ 77904810267289528 \)
\(\beta_{10}\)\(=\)\((\)\(1083690820042657 \nu^{18} - 6973741260199556 \nu^{17} - 20471590021611702 \nu^{16} + 205820671089989371 \nu^{15} + 29617048668770011 \nu^{14} - 2387368572651253167 \nu^{13} + 1798484996667138455 \nu^{12} + 14027991543956850112 \nu^{11} - 16481112740348660517 \nu^{10} - 44788609308024169219 \nu^{9} + 62515975677943484674 \nu^{8} + 77126164322063243227 \nu^{7} - 114320497483603387644 \nu^{6} - 67030073201629306155 \nu^{5} + 92460718390550759524 \nu^{4} + 26968630910803904880 \nu^{3} - 21158349312845396762 \nu^{2} - 6540496659385454301 \nu - 355943311226414196\)\()/ 77904810267289528 \)
\(\beta_{11}\)\(=\)\((\)\(1092816357613181 \nu^{18} - 7459601496024570 \nu^{17} - 18319998171235509 \nu^{16} + 216809473764291880 \nu^{15} - 38084178623578787 \nu^{14} - 2452948852753917143 \nu^{13} + 2587612671357407948 \nu^{12} + 13809479980337519780 \nu^{11} - 21008885863502095005 \nu^{10} - 40699585786793027837 \nu^{9} + 76099498049635194623 \nu^{8} + 58998155312108487979 \nu^{7} - 134698131321362958305 \nu^{6} - 31609719806534699185 \nu^{5} + 105275876872953735805 \nu^{4} - 1609513481442969668 \nu^{3} - 23331520877142669999 \nu^{2} - 1314890913361810684 \nu + 316411219035010592\)\()/ 77904810267289528 \)
\(\beta_{12}\)\(=\)\((\)\(-1134567769505457 \nu^{18} + 7266804211212438 \nu^{17} + 21427225666173599 \nu^{16} - 213680552404619654 \nu^{15} - 31574624618047029 \nu^{14} + 2462130918680185093 \nu^{13} - 1860936011562635530 \nu^{12} - 14289017657424015228 \nu^{11} + 16964033962802824291 \nu^{10} + 44531006101484114293 \nu^{9} - 63643830928625667821 \nu^{8} - 72903731379051850569 \nu^{7} + 114267928088836612117 \nu^{6} + 56413798233430700797 \nu^{5} - 90005221911718988253 \nu^{4} - 17474895021582544010 \nu^{3} + 20358214123365373471 \nu^{2} + 4862472104716683620 \nu - 12617454281865112\)\()/ 77904810267289528 \)
\(\beta_{13}\)\(=\)\((\)\(1372031837712026 \nu^{18} - 8632704946027114 \nu^{17} - 26488322998174035 \nu^{16} + 254137595755781765 \nu^{15} + 54680083798284962 \nu^{14} - 2933166057645453758 \nu^{13} + 2068849535716815029 \nu^{12} + 17062702713254432978 \nu^{11} - 19537063246727999042 \nu^{10} - 53342262959495678406 \nu^{9} + 74236047532850061909 \nu^{8} + 87637830612841723000 \nu^{7} - 134126430474056569771 \nu^{6} - 67877444561900174468 \nu^{5} + 104965677147146286421 \nu^{4} + 20562863803007876548 \nu^{3} - 21757429321116481975 \nu^{2} - 5320072813767842773 \nu - 364829756993871316\)\()/ 77904810267289528 \)
\(\beta_{14}\)\(=\)\((\)\(-1426213381500197 \nu^{18} + 8799653434478100 \nu^{17} + 28760184662207652 \nu^{16} - 261180023056501289 \nu^{15} - 93169367682922185 \nu^{14} + 3052987536609974455 \nu^{13} - 1727735731011966637 \nu^{12} - 18126754975236576178 \nu^{11} + 17826983466581622477 \nu^{10} + 58672937274694536375 \nu^{9} - 69370865243696314662 \nu^{8} - 102777671504280390989 \nu^{7} + 126833847987152230770 \nu^{6} + 90812240101347596701 \nu^{5} - 100607382747545578412 \nu^{4} - 36001730803095990114 \nu^{3} + 22114795187908175950 \nu^{2} + 7656742856427456895 \nu + 266086065397816924\)\()/ 77904810267289528 \)
\(\beta_{15}\)\(=\)\((\)\(-741068081655316 \nu^{18} + 4595827585193675 \nu^{17} + 14706488707551162 \nu^{16} - 135847265665372052 \nu^{15} - 41251315050697064 \nu^{14} + 1577439360532800966 \nu^{13} - 983893018901486730 \nu^{12} - 9260428297436121382 \nu^{11} + 9800513948355364761 \nu^{10} + 29356222080806208834 \nu^{9} - 37923471534717328802 \nu^{8} - 49289123370662958684 \nu^{7} + 69525859366441445447 \nu^{6} + 39555619296920917818 \nu^{5} - 55645495433960070451 \nu^{4} - 12687570570163284801 \nu^{3} + 12445583835862380927 \nu^{2} + 3058107440142454636 \nu - 18145595185768772\)\()/ 38952405133644764 \)
\(\beta_{16}\)\(=\)\((\)\(-1788992130708019 \nu^{18} + 11616156032253848 \nu^{17} + 32536276431610078 \nu^{16} - 339399271749954761 \nu^{15} - 11582361937006627 \nu^{14} + 3869531900522449111 \nu^{13} - 3397394588597605237 \nu^{12} - 22042316492135969886 \nu^{11} + 29610043641973866985 \nu^{10} + 66273975827091356685 \nu^{9} - 109891497445168112760 \nu^{8} - 100149415988085444129 \nu^{7} + 196461996303947941470 \nu^{6} + 61210305129251803963 \nu^{5} - 153155685352928992056 \nu^{4} - 4144258394197102016 \nu^{3} + 32758860467229390110 \nu^{2} + 1617997536471123617 \nu - 577666401596282284\)\()/ 77904810267289528 \)
\(\beta_{17}\)\(=\)\((\)\(-2405904627279449 \nu^{18} + 15658637998627299 \nu^{17} + 44007385279792924 \nu^{16} - 458604356413412660 \nu^{15} - 24696620256853403 \nu^{14} + 5253377279220361316 \nu^{13} - 4439755889786174314 \nu^{12} - 30218043560971088396 \nu^{11} + 38921551280019068667 \nu^{10} + 92817108697767086806 \nu^{9} - 144639030879757924651 \nu^{8} - 147891536233421275612 \nu^{7} + 259387225697679758869 \nu^{6} + 107332204391096892346 \nu^{5} - 204811156944413783707 \nu^{4} - 26845110033970579257 \nu^{3} + 46303300320837429364 \nu^{2} + 6903621981002853558 \nu - 440523358723489168\)\()/ 77904810267289528 \)
\(\beta_{18}\)\(=\)\((\)\(2792699668717361 \nu^{18} - 16416910358581216 \nu^{17} - 59062739523594601 \nu^{16} + 486780419195914490 \nu^{15} + 264505957024420359 \nu^{14} - 5678792202416026997 \nu^{13} + 2420943149231987866 \nu^{12} + 33581661759727533114 \nu^{11} - 29233897625988081233 \nu^{10} - 107791343874420349483 \nu^{9} + 117984898012449172935 \nu^{8} + 185420740613861372835 \nu^{7} - 219246910241231539663 \nu^{6} - 157164230605325737913 \nu^{5} + 175208861191622404069 \nu^{4} + 57037782759786383952 \nu^{3} - 37801683907989238741 \nu^{2} - 12531931956461062566 \nu - 498882579668991336\)\()/ 77904810267289528 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{18} + \beta_{15} + \beta_{14} - \beta_{12} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} + 7 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(2 \beta_{18} - \beta_{16} - 2 \beta_{15} + 3 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{8} + 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{3} + 11 \beta_{2} - \beta_{1} + 30\)
\(\nu^{5}\)\(=\)\(14 \beta_{18} - \beta_{17} - \beta_{16} + 12 \beta_{15} + 14 \beta_{14} - 2 \beta_{13} - 10 \beta_{12} - \beta_{11} - 12 \beta_{10} - 12 \beta_{9} - 12 \beta_{8} + 14 \beta_{7} + 12 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + 13 \beta_{3} + 2 \beta_{2} + 54 \beta_{1} + 26\)
\(\nu^{6}\)\(=\)\(32 \beta_{18} - \beta_{17} - 16 \beta_{16} - 30 \beta_{15} + 47 \beta_{14} - 64 \beta_{13} - 40 \beta_{12} - 33 \beta_{11} - 9 \beta_{10} - 28 \beta_{9} - 13 \beta_{8} + 59 \beta_{7} + 24 \beta_{6} - 29 \beta_{5} + 11 \beta_{4} + 15 \beta_{3} + 114 \beta_{2} - 23 \beta_{1} + 259\)
\(\nu^{7}\)\(=\)\(166 \beta_{18} - 11 \beta_{17} - 20 \beta_{16} + 120 \beta_{15} + 164 \beta_{14} - 42 \beta_{13} - 96 \beta_{12} - 21 \beta_{11} - 123 \beta_{10} - 127 \beta_{9} - 124 \beta_{8} + 171 \beta_{7} + 123 \beta_{6} - 41 \beta_{5} + 57 \beta_{4} + 149 \beta_{3} + 34 \beta_{2} + 444 \beta_{1} + 293\)
\(\nu^{8}\)\(=\)\(406 \beta_{18} - 20 \beta_{17} - 197 \beta_{16} - 358 \beta_{15} + 568 \beta_{14} - 810 \beta_{13} - 436 \beta_{12} - 423 \beta_{11} - 60 \beta_{10} - 319 \beta_{9} - 139 \beta_{8} + 703 \beta_{7} + 223 \beta_{6} - 339 \beta_{5} + 230 \beta_{4} + 185 \beta_{3} + 1167 \beta_{2} - 341 \beta_{1} + 2390\)
\(\nu^{9}\)\(=\)\(1866 \beta_{18} - 79 \beta_{17} - 301 \beta_{16} + 1162 \beta_{15} + 1811 \beta_{14} - 646 \beta_{13} - 941 \beta_{12} - 315 \beta_{11} - 1206 \beta_{10} - 1311 \beta_{9} - 1243 \beta_{8} + 1973 \beta_{7} + 1215 \beta_{6} - 583 \beta_{5} + 807 \beta_{4} + 1644 \beta_{3} + 449 \beta_{2} + 3798 \beta_{1} + 3163\)
\(\nu^{10}\)\(=\)\(4762 \beta_{18} - 277 \beta_{17} - 2224 \beta_{16} - 3940 \beta_{15} + 6298 \beta_{14} - 9446 \beta_{13} - 4498 \beta_{12} - 4928 \beta_{11} - 300 \beta_{10} - 3441 \beta_{9} - 1427 \beta_{8} + 7849 \beta_{7} + 1901 \beta_{6} - 3719 \beta_{5} + 3387 \beta_{4} + 2136 \beta_{3} + 11916 \beta_{2} - 4299 \beta_{1} + 22878\)
\(\nu^{11}\)\(=\)\(20481 \beta_{18} - 392 \beta_{17} - 3957 \beta_{16} + 11201 \beta_{15} + 19469 \beta_{14} - 8838 \beta_{13} - 9396 \beta_{12} - 4154 \beta_{11} - 11610 \beta_{10} - 13476 \beta_{9} - 12397 \beta_{8} + 22055 \beta_{7} + 11820 \beta_{6} - 7246 \beta_{5} + 10246 \beta_{4} + 17718 \beta_{3} + 5541 \beta_{2} + 33286 \beta_{1} + 33634\)
\(\nu^{12}\)\(=\)\(53886 \beta_{18} - 3302 \beta_{17} - 24225 \beta_{16} - 41620 \beta_{15} + 67437 \beta_{14} - 105894 \beta_{13} - 45613 \beta_{12} - 54679 \beta_{11} - 388 \beta_{10} - 36426 \beta_{9} - 14628 \beta_{8} + 85296 \beta_{7} + 15588 \beta_{6} - 39822 \beta_{5} + 43424 \beta_{4} + 23880 \beta_{3} + 121645 \beta_{2} - 50134 \beta_{1} + 223931\)
\(\nu^{13}\)\(=\)\(222039 \beta_{18} - 343 \beta_{17} - 48163 \beta_{16} + 108145 \beta_{15} + 206563 \beta_{14} - 113512 \beta_{13} - 95038 \beta_{12} - 51238 \beta_{11} - 110758 \beta_{10} - 138617 \beta_{9} - 123924 \beta_{8} + 241896 \beta_{7} + 113917 \beta_{6} - 84495 \beta_{5} + 123217 \beta_{4} + 187688 \beta_{3} + 66692 \beta_{2} + 296007 \beta_{1} + 355806\)
\(\nu^{14}\)\(=\)\(598270 \beta_{18} - 36416 \beta_{17} - 259618 \beta_{16} - 428934 \beta_{15} + 711545 \beta_{14} - 1160924 \beta_{13} - 460874 \beta_{12} - 589931 \beta_{11} + 18141 \beta_{10} - 382725 \beta_{9} - 151458 \beta_{8} + 914878 \beta_{7} + 124921 \beta_{6} - 421960 \beta_{5} + 519212 \beta_{4} + 261957 \beta_{3} + 1242366 \beta_{2} - 560071 \beta_{1} + 2224383\)
\(\nu^{15}\)\(=\)\(2390667 \beta_{18} + 24909 \beta_{17} - 559761 \beta_{16} + 1047245 \beta_{15} + 2178127 \beta_{14} - 1399172 \beta_{13} - 970862 \beta_{12} - 607311 \beta_{11} - 1051467 \beta_{10} - 1428413 \beta_{9} - 1244605 \beta_{8} + 2622420 \beta_{7} + 1090639 \beta_{6} - 951943 \beta_{5} + 1435219 \beta_{4} + 1963587 \beta_{3} + 790976 \beta_{2} + 2654266 \beta_{1} + 3759565\)
\(\nu^{16}\)\(=\)\(6568957 \beta_{18} - 383804 \beta_{17} - 2760886 \beta_{16} - 4349501 \beta_{15} + 7464710 \beta_{14} - 12553596 \beta_{13} - 4665496 \beta_{12} - 6258198 \beta_{11} + 366287 \beta_{10} - 4008652 \beta_{9} - 1587514 \beta_{8} + 9746604 \beta_{7} + 981564 \beta_{6} - 4449797 \beta_{5} + 5966974 \beta_{4} + 2839020 \beta_{3} + 12697934 \beta_{2} - 6097556 \beta_{1} + 22327897\)
\(\nu^{17}\)\(=\)\(25636785 \beta_{18} + 460550 \beta_{17} - 6322641 \beta_{16} + 10172705 \beta_{15} + 22913109 \beta_{14} - 16748054 \beta_{13} - 10001968 \beta_{12} - 7016597 \beta_{11} - 9954693 \beta_{10} - 14748904 \beta_{9} - 12566728 \beta_{8} + 28228483 \beta_{7} + 10391540 \beta_{6} - 10511760 \beta_{5} + 16371454 \beta_{4} + 20367469 \beta_{3} + 9267444 \beta_{2} + 23894792 \beta_{1} + 39746289\)
\(\nu^{18}\)\(=\)\(71626570 \beta_{18} - 3932010 \beta_{17} - 29251285 \beta_{16} - 43615626 \beta_{15} + 78189603 \beta_{14} - 134550690 \beta_{13} - 47424330 \beta_{12} - 65709230 \beta_{11} + 5223217 \beta_{10} - 41938766 \beta_{9} - 16824361 \beta_{8} + 103464713 \beta_{7} + 7529642 \beta_{6} - 46824177 \beta_{5} + 66932680 \beta_{4} + 30526998 \beta_{3} + 129912594 \beta_{2} - 65303830 \beta_{1} + 225892133\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.12892
−2.63379
−2.08912
−1.99202
−1.53958
−1.31693
−0.455245
−0.312548
−0.157639
0.00508866
0.749639
1.27063
1.56031
2.01204
2.43794
2.50185
2.79120
3.03361
3.26349
0 −3.12892 0 −3.70734 0 −1.25809 0 6.79011 0
1.2 0 −2.63379 0 −2.05129 0 4.98850 0 3.93685 0
1.3 0 −2.08912 0 −2.23992 0 1.92906 0 1.36442 0
1.4 0 −1.99202 0 2.61608 0 4.58459 0 0.968146 0
1.5 0 −1.53958 0 −0.551255 0 −1.16112 0 −0.629704 0
1.6 0 −1.31693 0 −1.28443 0 1.02036 0 −1.26569 0
1.7 0 −0.455245 0 2.37571 0 1.84768 0 −2.79275 0
1.8 0 −0.312548 0 0.780222 0 −4.05339 0 −2.90231 0
1.9 0 −0.157639 0 −0.360777 0 −0.00399840 0 −2.97515 0
1.10 0 0.00508866 0 −3.22565 0 −4.03772 0 −2.99997 0
1.11 0 0.749639 0 0.796495 0 1.70851 0 −2.43804 0
1.12 0 1.27063 0 −4.07798 0 2.09947 0 −1.38551 0
1.13 0 1.56031 0 3.44545 0 1.77285 0 −0.565441 0
1.14 0 2.01204 0 3.16436 0 2.95084 0 1.04829 0
1.15 0 2.43794 0 −3.80787 0 −4.24792 0 2.94357 0
1.16 0 2.50185 0 0.708265 0 −3.63078 0 3.25924 0
1.17 0 2.79120 0 −3.17175 0 1.46800 0 4.79080 0
1.18 0 3.03361 0 −0.0964405 0 5.16490 0 6.20277 0
1.19 0 3.26349 0 2.68811 0 −0.141739 0 7.65038 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4016.2.a.l 19
4.b odd 2 1 2008.2.a.c 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.2.a.c 19 4.b odd 2 1
4016.2.a.l 19 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(251\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{19} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).