Properties

Label 3.40.a.a
Level $3$
Weight $40$
Character orbit 3.a
Self dual yes
Analytic conductor $28.902$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 40 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9018654169\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - 216694123 x - 94580724378\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{6}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -369000 - \beta_{1} ) q^{2} + 1162261467 q^{3} + ( 335300075200 + 785144 \beta_{1} + 16 \beta_{2} ) q^{4} + ( 3119016476430 + 32303812 \beta_{1} - 513 \beta_{2} ) q^{5} + ( -428874481323000 - 1162261467 \beta_{1} ) q^{6} + ( 4613961459153368 - 30823389604 \beta_{1} + 348985 \beta_{2} ) q^{7} + ( -508852326783711744 - 484502113856 \beta_{1} - 17712000 \beta_{2} ) q^{8} + 1350851717672992089 q^{9} +O(q^{10})\) \( q +(-369000 - \beta_{1}) q^{2} +1162261467 q^{3} +(335300075200 + 785144 \beta_{1} + 16 \beta_{2}) q^{4} +(3119016476430 + 32303812 \beta_{1} - 513 \beta_{2}) q^{5} +(-428874481323000 - 1162261467 \beta_{1}) q^{6} +(4613961459153368 - 30823389604 \beta_{1} + 348985 \beta_{2}) q^{7} +(-508852326783711744 - 484502113856 \beta_{1} - 17712000 \beta_{2}) q^{8} +1350851717672992089 q^{9} +(-25343199726347478960 - 4627593552014 \beta_{1} - 351748864 \beta_{2}) q^{10} +(-13350109203715747932 + 87947509241336 \beta_{1} + 4631921442 \beta_{2}) q^{11} +(\)\(38\!\cdots\!00\)\( + 912542617246248 \beta_{1} + 18596183472 \beta_{2}) q^{12} +(-\)\(46\!\cdots\!98\)\( - 569679887088520 \beta_{1} - 206263539326 \beta_{2}) q^{13} +(\)\(21\!\cdots\!32\)\( + 94200779733928 \beta_{1} + 380851317504 \beta_{2}) q^{14} +(\)\(36\!\cdots\!10\)\( + 37545475924812204 \beta_{1} - 596240132571 \beta_{2}) q^{15} +(\)\(36\!\cdots\!28\)\( + 690890446516379136 \beta_{1} + 4656654271488 \beta_{2}) q^{16} +(-\)\(56\!\cdots\!06\)\( + 926992962792897096 \beta_{1} - 6009628140594 \beta_{2}) q^{17} +(-\)\(49\!\cdots\!00\)\( - 1350851717672992089 \beta_{1}) q^{18} +(-\)\(23\!\cdots\!96\)\( + 2957557631558772120 \beta_{1} - 160826348093766 \beta_{2}) q^{19} +(\)\(11\!\cdots\!80\)\( + 17692841649648817552 \beta_{1} + 469278711728352 \beta_{2}) q^{20} +(\)\(53\!\cdots\!56\)\( - 35824838019057589068 \beta_{1} + 405611818060995 \beta_{2}) q^{21} +(-\)\(60\!\cdots\!32\)\( - \)\(13\!\cdots\!48\)\( \beta_{1} - 2897971855497728 \beta_{2}) q^{22} +(-\)\(85\!\cdots\!24\)\( - \)\(24\!\cdots\!96\)\( \beta_{1} + 7342398742559922 \beta_{2}) q^{23} +(-\)\(59\!\cdots\!48\)\( - \)\(56\!\cdots\!52\)\( \beta_{1} - 20585975103504000 \beta_{2}) q^{24} +(-\)\(74\!\cdots\!25\)\( - \)\(53\!\cdots\!20\)\( \beta_{1} + 8936618308826980 \beta_{2}) q^{25} +(\)\(21\!\cdots\!52\)\( + \)\(96\!\cdots\!66\)\( \beta_{1} + 75502035906725376 \beta_{2}) q^{26} +\)\(15\!\cdots\!63\)\( q^{27} +(-\)\(10\!\cdots\!16\)\( - \)\(13\!\cdots\!64\)\( \beta_{1} - 315943026832013952 \beta_{2}) q^{28} +(-\)\(15\!\cdots\!62\)\( + \)\(18\!\cdots\!44\)\( \beta_{1} + 119886169875921837 \beta_{2}) q^{29} +(-\)\(29\!\cdots\!20\)\( - \)\(53\!\cdots\!38\)\( \beta_{1} - 408824150688223488 \beta_{2}) q^{30} +(-\)\(94\!\cdots\!96\)\( + \)\(90\!\cdots\!60\)\( \beta_{1} + 2713599485454965797 \beta_{2}) q^{31} +(-\)\(37\!\cdots\!92\)\( - \)\(49\!\cdots\!28\)\( \beta_{1} - 2815744285881851904 \beta_{2}) q^{32} +(-\)\(15\!\cdots\!44\)\( + \)\(10\!\cdots\!12\)\( \beta_{1} + 5383503810207675414 \beta_{2}) q^{33} +(-\)\(67\!\cdots\!00\)\( - \)\(18\!\cdots\!46\)\( \beta_{1} - 12897652529867331072 \beta_{2}) q^{34} +(-\)\(92\!\cdots\!00\)\( + \)\(64\!\cdots\!20\)\( \beta_{1} - 14321872082373654330 \beta_{2}) q^{35} +(\)\(45\!\cdots\!00\)\( + \)\(10\!\cdots\!16\)\( \beta_{1} + 21613627482767873424 \beta_{2}) q^{36} +(-\)\(17\!\cdots\!62\)\( + \)\(19\!\cdots\!00\)\( \beta_{1} + 85573013785692403224 \beta_{2}) q^{37} +(-\)\(13\!\cdots\!88\)\( + \)\(48\!\cdots\!24\)\( \beta_{1} + 4442002987126795776 \beta_{2}) q^{38} +(-\)\(53\!\cdots\!66\)\( - \)\(66\!\cdots\!40\)\( \beta_{1} - \)\(23\!\cdots\!42\)\( \beta_{2}) q^{39} +(-\)\(34\!\cdots\!80\)\( - \)\(26\!\cdots\!12\)\( \beta_{1} - \)\(24\!\cdots\!12\)\( \beta_{2}) q^{40} +(\)\(15\!\cdots\!94\)\( - \)\(38\!\cdots\!24\)\( \beta_{1} + \)\(53\!\cdots\!38\)\( \beta_{2}) q^{41} +(\)\(24\!\cdots\!44\)\( + \)\(10\!\cdots\!76\)\( \beta_{1} + \)\(44\!\cdots\!68\)\( \beta_{2}) q^{42} +(-\)\(49\!\cdots\!96\)\( - \)\(68\!\cdots\!24\)\( \beta_{1} - \)\(61\!\cdots\!18\)\( \beta_{2}) q^{43} +(\)\(12\!\cdots\!16\)\( + \)\(13\!\cdots\!40\)\( \beta_{1} + \)\(48\!\cdots\!40\)\( \beta_{2}) q^{44} +(\)\(42\!\cdots\!70\)\( + \)\(43\!\cdots\!68\)\( \beta_{1} - \)\(69\!\cdots\!57\)\( \beta_{2}) q^{45} +(\)\(21\!\cdots\!24\)\( + \)\(17\!\cdots\!12\)\( \beta_{1} + \)\(15\!\cdots\!04\)\( \beta_{2}) q^{46} +(-\)\(43\!\cdots\!48\)\( - \)\(20\!\cdots\!40\)\( \beta_{1} - \)\(99\!\cdots\!42\)\( \beta_{2}) q^{47} +(\)\(42\!\cdots\!76\)\( + \)\(80\!\cdots\!12\)\( \beta_{1} + \)\(54\!\cdots\!96\)\( \beta_{2}) q^{48} +(-\)\(44\!\cdots\!71\)\( - \)\(74\!\cdots\!80\)\( \beta_{1} + \)\(16\!\cdots\!36\)\( \beta_{2}) q^{49} +(\)\(67\!\cdots\!00\)\( + \)\(75\!\cdots\!65\)\( \beta_{1} + \)\(57\!\cdots\!40\)\( \beta_{2}) q^{50} +(-\)\(65\!\cdots\!02\)\( + \)\(10\!\cdots\!32\)\( \beta_{1} - \)\(69\!\cdots\!98\)\( \beta_{2}) q^{51} +(-\)\(54\!\cdots\!76\)\( - \)\(76\!\cdots\!84\)\( \beta_{1} - \)\(65\!\cdots\!24\)\( \beta_{2}) q^{52} +(-\)\(36\!\cdots\!14\)\( + \)\(10\!\cdots\!24\)\( \beta_{1} + \)\(18\!\cdots\!37\)\( \beta_{2}) q^{53} +(-\)\(57\!\cdots\!00\)\( - \)\(15\!\cdots\!63\)\( \beta_{1}) q^{54} +(-\)\(50\!\cdots\!20\)\( + \)\(25\!\cdots\!52\)\( \beta_{1} + \)\(11\!\cdots\!52\)\( \beta_{2}) q^{55} +(\)\(21\!\cdots\!00\)\( + \)\(23\!\cdots\!44\)\( \beta_{1} + \)\(10\!\cdots\!84\)\( \beta_{2}) q^{56} +(-\)\(26\!\cdots\!32\)\( + \)\(34\!\cdots\!40\)\( \beta_{1} - \)\(18\!\cdots\!22\)\( \beta_{2}) q^{57} +(-\)\(79\!\cdots\!76\)\( + \)\(50\!\cdots\!70\)\( \beta_{1} - \)\(32\!\cdots\!76\)\( \beta_{2}) q^{58} +(\)\(11\!\cdots\!16\)\( - \)\(40\!\cdots\!52\)\( \beta_{1} + \)\(26\!\cdots\!44\)\( \beta_{2}) q^{59} +(\)\(12\!\cdots\!60\)\( + \)\(20\!\cdots\!84\)\( \beta_{1} + \)\(54\!\cdots\!84\)\( \beta_{2}) q^{60} +(-\)\(87\!\cdots\!34\)\( - \)\(59\!\cdots\!00\)\( \beta_{1} - \)\(18\!\cdots\!56\)\( \beta_{2}) q^{61} +(-\)\(32\!\cdots\!04\)\( - \)\(64\!\cdots\!80\)\( \beta_{1} - \)\(23\!\cdots\!92\)\( \beta_{2}) q^{62} +(\)\(62\!\cdots\!52\)\( - \)\(41\!\cdots\!56\)\( \beta_{1} + \)\(47\!\cdots\!65\)\( \beta_{2}) q^{63} +(\)\(30\!\cdots\!68\)\( + \)\(26\!\cdots\!08\)\( \beta_{1} + \)\(62\!\cdots\!28\)\( \beta_{2}) q^{64} +(\)\(86\!\cdots\!20\)\( - \)\(30\!\cdots\!52\)\( \beta_{1} - \)\(71\!\cdots\!02\)\( \beta_{2}) q^{65} +(-\)\(70\!\cdots\!44\)\( - \)\(15\!\cdots\!16\)\( \beta_{1} - \)\(33\!\cdots\!76\)\( \beta_{2}) q^{66} +(-\)\(66\!\cdots\!96\)\( + \)\(26\!\cdots\!20\)\( \beta_{1} - \)\(14\!\cdots\!60\)\( \beta_{2}) q^{67} +(\)\(42\!\cdots\!00\)\( + \)\(54\!\cdots\!12\)\( \beta_{1} + \)\(10\!\cdots\!40\)\( \beta_{2}) q^{68} +(-\)\(99\!\cdots\!08\)\( - \)\(28\!\cdots\!32\)\( \beta_{1} + \)\(85\!\cdots\!74\)\( \beta_{2}) q^{69} +(-\)\(14\!\cdots\!00\)\( + \)\(98\!\cdots\!60\)\( \beta_{1} - \)\(57\!\cdots\!40\)\( \beta_{2}) q^{70} +(-\)\(10\!\cdots\!48\)\( - \)\(25\!\cdots\!16\)\( \beta_{1} + \)\(16\!\cdots\!74\)\( \beta_{2}) q^{71} +(-\)\(68\!\cdots\!16\)\( - \)\(65\!\cdots\!84\)\( \beta_{1} - \)\(23\!\cdots\!00\)\( \beta_{2}) q^{72} +(-\)\(12\!\cdots\!74\)\( - \)\(16\!\cdots\!88\)\( \beta_{1} + \)\(37\!\cdots\!20\)\( \beta_{2}) q^{73} +(-\)\(84\!\cdots\!08\)\( - \)\(10\!\cdots\!50\)\( \beta_{1} - \)\(59\!\cdots\!44\)\( \beta_{2}) q^{74} +(-\)\(86\!\cdots\!75\)\( - \)\(62\!\cdots\!40\)\( \beta_{1} + \)\(10\!\cdots\!60\)\( \beta_{2}) q^{75} +(-\)\(18\!\cdots\!56\)\( - \)\(23\!\cdots\!16\)\( \beta_{1} + \)\(97\!\cdots\!68\)\( \beta_{2}) q^{76} +(-\)\(33\!\cdots\!24\)\( - \)\(20\!\cdots\!64\)\( \beta_{1} - \)\(58\!\cdots\!76\)\( \beta_{2}) q^{77} +(\)\(24\!\cdots\!84\)\( + \)\(11\!\cdots\!22\)\( \beta_{1} + \)\(87\!\cdots\!92\)\( \beta_{2}) q^{78} +(-\)\(34\!\cdots\!00\)\( + \)\(73\!\cdots\!44\)\( \beta_{1} - \)\(36\!\cdots\!91\)\( \beta_{2}) q^{79} +(\)\(15\!\cdots\!20\)\( + \)\(10\!\cdots\!88\)\( \beta_{1} + \)\(24\!\cdots\!88\)\( \beta_{2}) q^{80} +\)\(18\!\cdots\!21\)\( q^{81} +(-\)\(55\!\cdots\!84\)\( - \)\(28\!\cdots\!82\)\( \beta_{1} - \)\(16\!\cdots\!44\)\( \beta_{2}) q^{82} +(\)\(23\!\cdots\!80\)\( + \)\(12\!\cdots\!08\)\( \beta_{1} - \)\(53\!\cdots\!30\)\( \beta_{2}) q^{83} +(-\)\(12\!\cdots\!72\)\( - \)\(15\!\cdots\!88\)\( \beta_{1} - \)\(36\!\cdots\!84\)\( \beta_{2}) q^{84} +(\)\(25\!\cdots\!20\)\( - \)\(13\!\cdots\!52\)\( \beta_{1} + \)\(44\!\cdots\!98\)\( \beta_{2}) q^{85} +(\)\(53\!\cdots\!68\)\( + \)\(47\!\cdots\!36\)\( \beta_{1} + \)\(12\!\cdots\!92\)\( \beta_{2}) q^{86} +(-\)\(17\!\cdots\!54\)\( + \)\(21\!\cdots\!48\)\( \beta_{1} + \)\(13\!\cdots\!79\)\( \beta_{2}) q^{87} +(-\)\(11\!\cdots\!84\)\( - \)\(12\!\cdots\!72\)\( \beta_{1} - \)\(71\!\cdots\!16\)\( \beta_{2}) q^{88} +(\)\(45\!\cdots\!14\)\( - \)\(22\!\cdots\!72\)\( \beta_{1} + \)\(34\!\cdots\!12\)\( \beta_{2}) q^{89} +(-\)\(34\!\cdots\!40\)\( - \)\(62\!\cdots\!46\)\( \beta_{1} - \)\(47\!\cdots\!96\)\( \beta_{2}) q^{90} +(-\)\(86\!\cdots\!36\)\( + \)\(28\!\cdots\!04\)\( \beta_{1} - \)\(90\!\cdots\!14\)\( \beta_{2}) q^{91} +(-\)\(45\!\cdots\!12\)\( - \)\(12\!\cdots\!56\)\( \beta_{1} - \)\(48\!\cdots\!52\)\( \beta_{2}) q^{92} +(-\)\(10\!\cdots\!32\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} + \)\(31\!\cdots\!99\)\( \beta_{2}) q^{93} +(\)\(15\!\cdots\!84\)\( + \)\(31\!\cdots\!04\)\( \beta_{1} + \)\(64\!\cdots\!92\)\( \beta_{2}) q^{94} +(\)\(15\!\cdots\!80\)\( - \)\(22\!\cdots\!08\)\( \beta_{1} + \)\(61\!\cdots\!92\)\( \beta_{2}) q^{95} +(-\)\(43\!\cdots\!64\)\( - \)\(57\!\cdots\!76\)\( \beta_{1} - \)\(32\!\cdots\!68\)\( \beta_{2}) q^{96} +(-\)\(44\!\cdots\!06\)\( + \)\(35\!\cdots\!36\)\( \beta_{1} - \)\(27\!\cdots\!24\)\( \beta_{2}) q^{97} +(\)\(57\!\cdots\!28\)\( - \)\(30\!\cdots\!77\)\( \beta_{1} + \)\(66\!\cdots\!64\)\( \beta_{2}) q^{98} +(-\)\(18\!\cdots\!48\)\( + \)\(11\!\cdots\!04\)\( \beta_{1} + \)\(62\!\cdots\!38\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 1107000q^{2} + 3486784401q^{3} + 1005900225600q^{4} + 9357049429290q^{5} - 1286623443969000q^{6} + 13841884377460104q^{7} - 1526556980351135232q^{8} + 4052555153018976267q^{9} + O(q^{10}) \) \( 3q - 1107000q^{2} + 3486784401q^{3} + 1005900225600q^{4} + 9357049429290q^{5} - 1286623443969000q^{6} + 13841884377460104q^{7} - 1526556980351135232q^{8} + 4052555153018976267q^{9} - 76029599179042436880q^{10} - 40050327611147243796q^{11} + \)\(11\!\cdots\!00\)\(q^{12} - \)\(13\!\cdots\!94\)\(q^{13} + \)\(64\!\cdots\!96\)\(q^{14} + \)\(10\!\cdots\!30\)\(q^{15} + \)\(10\!\cdots\!84\)\(q^{16} - \)\(16\!\cdots\!18\)\(q^{17} - \)\(14\!\cdots\!00\)\(q^{18} - \)\(69\!\cdots\!88\)\(q^{19} + \)\(33\!\cdots\!40\)\(q^{20} + \)\(16\!\cdots\!68\)\(q^{21} - \)\(18\!\cdots\!96\)\(q^{22} - \)\(25\!\cdots\!72\)\(q^{23} - \)\(17\!\cdots\!44\)\(q^{24} - \)\(22\!\cdots\!75\)\(q^{25} + \)\(64\!\cdots\!56\)\(q^{26} + \)\(47\!\cdots\!89\)\(q^{27} - \)\(31\!\cdots\!48\)\(q^{28} - \)\(46\!\cdots\!86\)\(q^{29} - \)\(88\!\cdots\!60\)\(q^{30} - \)\(28\!\cdots\!88\)\(q^{31} - \)\(11\!\cdots\!76\)\(q^{32} - \)\(46\!\cdots\!32\)\(q^{33} - \)\(20\!\cdots\!00\)\(q^{34} - \)\(27\!\cdots\!00\)\(q^{35} + \)\(13\!\cdots\!00\)\(q^{36} - \)\(52\!\cdots\!86\)\(q^{37} - \)\(40\!\cdots\!64\)\(q^{38} - \)\(16\!\cdots\!98\)\(q^{39} - \)\(10\!\cdots\!40\)\(q^{40} + \)\(47\!\cdots\!82\)\(q^{41} + \)\(74\!\cdots\!32\)\(q^{42} - \)\(14\!\cdots\!88\)\(q^{43} + \)\(38\!\cdots\!48\)\(q^{44} + \)\(12\!\cdots\!10\)\(q^{45} + \)\(64\!\cdots\!72\)\(q^{46} - \)\(13\!\cdots\!44\)\(q^{47} + \)\(12\!\cdots\!28\)\(q^{48} - \)\(13\!\cdots\!13\)\(q^{49} + \)\(20\!\cdots\!00\)\(q^{50} - \)\(19\!\cdots\!06\)\(q^{51} - \)\(16\!\cdots\!28\)\(q^{52} - \)\(11\!\cdots\!42\)\(q^{53} - \)\(17\!\cdots\!00\)\(q^{54} - \)\(15\!\cdots\!60\)\(q^{55} + \)\(63\!\cdots\!00\)\(q^{56} - \)\(80\!\cdots\!96\)\(q^{57} - \)\(23\!\cdots\!28\)\(q^{58} + \)\(35\!\cdots\!48\)\(q^{59} + \)\(38\!\cdots\!80\)\(q^{60} - \)\(26\!\cdots\!02\)\(q^{61} - \)\(98\!\cdots\!12\)\(q^{62} + \)\(18\!\cdots\!56\)\(q^{63} + \)\(92\!\cdots\!04\)\(q^{64} + \)\(26\!\cdots\!60\)\(q^{65} - \)\(21\!\cdots\!32\)\(q^{66} - \)\(20\!\cdots\!88\)\(q^{67} + \)\(12\!\cdots\!00\)\(q^{68} - \)\(29\!\cdots\!24\)\(q^{69} - \)\(43\!\cdots\!00\)\(q^{70} - \)\(31\!\cdots\!44\)\(q^{71} - \)\(20\!\cdots\!48\)\(q^{72} - \)\(38\!\cdots\!22\)\(q^{73} - \)\(25\!\cdots\!24\)\(q^{74} - \)\(25\!\cdots\!25\)\(q^{75} - \)\(55\!\cdots\!68\)\(q^{76} - \)\(99\!\cdots\!72\)\(q^{77} + \)\(74\!\cdots\!52\)\(q^{78} - \)\(10\!\cdots\!00\)\(q^{79} + \)\(45\!\cdots\!60\)\(q^{80} + \)\(54\!\cdots\!63\)\(q^{81} - \)\(16\!\cdots\!52\)\(q^{82} + \)\(69\!\cdots\!40\)\(q^{83} - \)\(36\!\cdots\!16\)\(q^{84} + \)\(76\!\cdots\!60\)\(q^{85} + \)\(15\!\cdots\!04\)\(q^{86} - \)\(53\!\cdots\!62\)\(q^{87} - \)\(34\!\cdots\!52\)\(q^{88} + \)\(13\!\cdots\!42\)\(q^{89} - \)\(10\!\cdots\!20\)\(q^{90} - \)\(25\!\cdots\!08\)\(q^{91} - \)\(13\!\cdots\!36\)\(q^{92} - \)\(32\!\cdots\!96\)\(q^{93} + \)\(45\!\cdots\!52\)\(q^{94} + \)\(46\!\cdots\!40\)\(q^{95} - \)\(12\!\cdots\!92\)\(q^{96} - \)\(13\!\cdots\!18\)\(q^{97} + \)\(17\!\cdots\!84\)\(q^{98} - \)\(54\!\cdots\!44\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 216694123 x - 94580724378\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 72 \nu \)
\(\beta_{2}\)\(=\)\( 324 \nu^{2} - 212148 \nu - 46805930568 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/72\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{2} + 5893 \beta_{1} + 93611861136\)\()/648\)

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
14934.1
−436.856
−14497.2
−1.44426e6 1.16226e9 1.53612e12 2.64208e13 −1.67860e15 −2.07513e16 −1.42456e18 1.35085e18 −3.81584e19
1.2 −337546. 1.16226e9 −4.35818e11 2.60351e13 −3.92317e14 −1.06972e16 3.32677e17 1.35085e18 −8.78806e18
1.3 674802. 1.16226e9 −9.43987e10 −4.30989e13 7.84296e14 4.52904e16 −4.34676e17 1.35085e18 −2.90832e19
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

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 3.40.a.a 3
3.b odd 2 1 9.40.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.40.a.a 3 1.a even 1 1 trivial
9.40.a.d 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 1107000 T_{2}^{2} - 714859333632 T_{2} - \)328967845897568256

'>\(32\!\cdots\!56\)\( \) acting on \(S_{40}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 1107000 T + 934408108032 T^{2} + 888191526050463744 T^{3} + \)\(51\!\cdots\!16\)\( T^{4} + \)\(33\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!72\)\( T^{6} \)
$3$ \( ( 1 - 1162261467 T )^{3} \)
$5$ \( 1 - 9357049429290 T + \)\(38\!\cdots\!75\)\( T^{2} - \)\(43\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!75\)\( T^{4} - \)\(30\!\cdots\!50\)\( T^{5} + \)\(60\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 13841884377460104 T + \)\(15\!\cdots\!29\)\( T^{2} - \)\(35\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!47\)\( T^{4} - \)\(11\!\cdots\!96\)\( T^{5} + \)\(75\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 + 40050327611147243796 T + \)\(80\!\cdots\!73\)\( T^{2} + \)\(36\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!43\)\( T^{4} + \)\(67\!\cdots\!76\)\( T^{5} + \)\(69\!\cdots\!71\)\( T^{6} \)
$13$ \( 1 + \)\(13\!\cdots\!94\)\( T + \)\(78\!\cdots\!31\)\( T^{2} + \)\(32\!\cdots\!08\)\( T^{3} + \)\(21\!\cdots\!87\)\( T^{4} + \)\(10\!\cdots\!26\)\( T^{5} + \)\(21\!\cdots\!33\)\( T^{6} \)
$17$ \( 1 + \)\(16\!\cdots\!18\)\( T + \)\(19\!\cdots\!67\)\( T^{2} + \)\(51\!\cdots\!24\)\( T^{3} + \)\(18\!\cdots\!51\)\( T^{4} + \)\(16\!\cdots\!62\)\( T^{5} + \)\(91\!\cdots\!77\)\( T^{6} \)
$19$ \( 1 + \)\(69\!\cdots\!88\)\( T + \)\(18\!\cdots\!53\)\( T^{2} + \)\(90\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!87\)\( T^{4} + \)\(38\!\cdots\!08\)\( T^{5} + \)\(41\!\cdots\!39\)\( T^{6} \)
$23$ \( 1 + \)\(25\!\cdots\!72\)\( T + \)\(25\!\cdots\!61\)\( T^{2} + \)\(38\!\cdots\!28\)\( T^{3} + \)\(32\!\cdots\!07\)\( T^{4} + \)\(42\!\cdots\!68\)\( T^{5} + \)\(20\!\cdots\!03\)\( T^{6} \)
$29$ \( 1 + \)\(46\!\cdots\!86\)\( T + \)\(35\!\cdots\!11\)\( T^{2} + \)\(95\!\cdots\!08\)\( T^{3} + \)\(38\!\cdots\!59\)\( T^{4} + \)\(53\!\cdots\!46\)\( T^{5} + \)\(12\!\cdots\!09\)\( T^{6} \)
$31$ \( 1 + \)\(28\!\cdots\!88\)\( T + \)\(49\!\cdots\!13\)\( T^{2} + \)\(65\!\cdots\!96\)\( T^{3} + \)\(71\!\cdots\!23\)\( T^{4} + \)\(59\!\cdots\!08\)\( T^{5} + \)\(30\!\cdots\!11\)\( T^{6} \)
$37$ \( 1 + \)\(52\!\cdots\!86\)\( T + \)\(36\!\cdots\!59\)\( T^{2} + \)\(12\!\cdots\!56\)\( T^{3} + \)\(52\!\cdots\!07\)\( T^{4} + \)\(10\!\cdots\!94\)\( T^{5} + \)\(30\!\cdots\!17\)\( T^{6} \)
$41$ \( 1 - \)\(47\!\cdots\!82\)\( T + \)\(26\!\cdots\!23\)\( T^{2} - \)\(68\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!03\)\( T^{4} - \)\(29\!\cdots\!22\)\( T^{5} + \)\(49\!\cdots\!81\)\( T^{6} \)
$43$ \( 1 + \)\(14\!\cdots\!88\)\( T + \)\(93\!\cdots\!97\)\( T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(47\!\cdots\!79\)\( T^{4} + \)\(38\!\cdots\!12\)\( T^{5} + \)\(13\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 + \)\(13\!\cdots\!44\)\( T + \)\(28\!\cdots\!93\)\( T^{2} - \)\(30\!\cdots\!60\)\( T^{3} + \)\(45\!\cdots\!19\)\( T^{4} + \)\(34\!\cdots\!16\)\( T^{5} + \)\(43\!\cdots\!87\)\( T^{6} \)
$53$ \( 1 + \)\(11\!\cdots\!42\)\( T + \)\(91\!\cdots\!71\)\( T^{2} + \)\(43\!\cdots\!28\)\( T^{3} + \)\(16\!\cdots\!07\)\( T^{4} + \)\(34\!\cdots\!38\)\( T^{5} + \)\(54\!\cdots\!13\)\( T^{6} \)
$59$ \( 1 - \)\(35\!\cdots\!48\)\( T + \)\(19\!\cdots\!33\)\( T^{2} - \)\(81\!\cdots\!04\)\( T^{3} + \)\(22\!\cdots\!87\)\( T^{4} - \)\(47\!\cdots\!08\)\( T^{5} + \)\(15\!\cdots\!19\)\( T^{6} \)
$61$ \( 1 + \)\(26\!\cdots\!02\)\( T + \)\(89\!\cdots\!79\)\( T^{2} + \)\(22\!\cdots\!76\)\( T^{3} + \)\(37\!\cdots\!39\)\( T^{4} + \)\(47\!\cdots\!62\)\( T^{5} + \)\(76\!\cdots\!21\)\( T^{6} \)
$67$ \( 1 + \)\(20\!\cdots\!88\)\( T + \)\(10\!\cdots\!57\)\( T^{2} + \)\(28\!\cdots\!64\)\( T^{3} + \)\(16\!\cdots\!71\)\( T^{4} + \)\(54\!\cdots\!92\)\( T^{5} + \)\(44\!\cdots\!27\)\( T^{6} \)
$71$ \( 1 + \)\(31\!\cdots\!44\)\( T + \)\(74\!\cdots\!05\)\( T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!55\)\( T^{4} + \)\(78\!\cdots\!84\)\( T^{5} + \)\(39\!\cdots\!91\)\( T^{6} \)
$73$ \( 1 + \)\(38\!\cdots\!22\)\( T + \)\(13\!\cdots\!91\)\( T^{2} + \)\(26\!\cdots\!28\)\( T^{3} + \)\(62\!\cdots\!67\)\( T^{4} + \)\(83\!\cdots\!18\)\( T^{5} + \)\(10\!\cdots\!53\)\( T^{6} \)
$79$ \( 1 + \)\(10\!\cdots\!00\)\( T + \)\(24\!\cdots\!57\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!83\)\( T^{4} + \)\(10\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!59\)\( T^{6} \)
$83$ \( 1 - \)\(69\!\cdots\!40\)\( T + \)\(32\!\cdots\!33\)\( T^{2} - \)\(10\!\cdots\!88\)\( T^{3} + \)\(22\!\cdots\!51\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{5} + \)\(34\!\cdots\!23\)\( T^{6} \)
$89$ \( 1 - \)\(13\!\cdots\!42\)\( T + \)\(37\!\cdots\!43\)\( T^{2} - \)\(29\!\cdots\!56\)\( T^{3} + \)\(39\!\cdots\!87\)\( T^{4} - \)\(15\!\cdots\!02\)\( T^{5} + \)\(11\!\cdots\!29\)\( T^{6} \)
$97$ \( 1 + \)\(13\!\cdots\!18\)\( T + \)\(13\!\cdots\!07\)\( T^{2} + \)\(85\!\cdots\!04\)\( T^{3} + \)\(41\!\cdots\!31\)\( T^{4} + \)\(12\!\cdots\!02\)\( T^{5} + \)\(28\!\cdots\!37\)\( T^{6} \)
show more
show less