Properties

Label 1502.2.a.h
Level 1502
Weight 2
Character orbit 1502.a
Self dual yes
Analytic conductor 11.994
Analytic rank 0
Dimension 19
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1502 = 2 \cdot 751 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1502.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.9935303836\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} -\beta_{3} q^{5} -\beta_{1} q^{6} + ( 1 - \beta_{13} ) q^{7} - q^{8} + ( 1 + \beta_{1} + \beta_{7} + \beta_{14} - \beta_{17} + \beta_{18} ) q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} -\beta_{3} q^{5} -\beta_{1} q^{6} + ( 1 - \beta_{13} ) q^{7} - q^{8} + ( 1 + \beta_{1} + \beta_{7} + \beta_{14} - \beta_{17} + \beta_{18} ) q^{9} + \beta_{3} q^{10} + \beta_{11} q^{11} + \beta_{1} q^{12} + ( 1 + \beta_{7} - \beta_{9} + \beta_{14} ) q^{13} + ( -1 + \beta_{13} ) q^{14} + ( -\beta_{4} - \beta_{5} + \beta_{6} + \beta_{12} + \beta_{14} ) q^{15} + q^{16} + ( 1 - \beta_{8} - \beta_{10} - \beta_{12} - \beta_{16} ) q^{17} + ( -1 - \beta_{1} - \beta_{7} - \beta_{14} + \beta_{17} - \beta_{18} ) q^{18} + ( -\beta_{3} + \beta_{4} - \beta_{6} + \beta_{10} - \beta_{15} ) q^{19} -\beta_{3} q^{20} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{21} -\beta_{11} q^{22} + ( -\beta_{2} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{14} + \beta_{16} ) q^{23} -\beta_{1} q^{24} + ( 1 - \beta_{2} - \beta_{7} - \beta_{10} - \beta_{11} + \beta_{15} + \beta_{16} ) q^{25} + ( -1 - \beta_{7} + \beta_{9} - \beta_{14} ) q^{26} + ( 1 + \beta_{1} + \beta_{2} + \beta_{8} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{27} + ( 1 - \beta_{13} ) q^{28} + ( \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{14} - \beta_{16} + \beta_{17} ) q^{29} + ( \beta_{4} + \beta_{5} - \beta_{6} - \beta_{12} - \beta_{14} ) q^{30} + ( 1 - \beta_{2} + \beta_{6} + \beta_{9} - \beta_{14} - \beta_{18} ) q^{31} - q^{32} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{14} - \beta_{16} ) q^{33} + ( -1 + \beta_{8} + \beta_{10} + \beta_{12} + \beta_{16} ) q^{34} + ( \beta_{2} - 2 \beta_{3} - \beta_{7} - \beta_{8} - \beta_{14} - \beta_{16} - \beta_{18} ) q^{35} + ( 1 + \beta_{1} + \beta_{7} + \beta_{14} - \beta_{17} + \beta_{18} ) q^{36} + ( 2 - \beta_{2} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{18} ) q^{37} + ( \beta_{3} - \beta_{4} + \beta_{6} - \beta_{10} + \beta_{15} ) q^{38} + ( -2 + 2 \beta_{1} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{15} + 2 \beta_{17} ) q^{39} + \beta_{3} q^{40} + ( 1 + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{16} ) q^{41} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{5} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{42} + ( 1 - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{43} + \beta_{11} q^{44} + ( 1 - \beta_{3} + 2 \beta_{6} + \beta_{8} + 3 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} ) q^{45} + ( \beta_{2} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{14} - \beta_{16} ) q^{46} + ( \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} - \beta_{18} ) q^{47} + \beta_{1} q^{48} + ( 3 + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{49} + ( -1 + \beta_{2} + \beta_{7} + \beta_{10} + \beta_{11} - \beta_{15} - \beta_{16} ) q^{50} + ( \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{12} + \beta_{14} + \beta_{16} - \beta_{18} ) q^{51} + ( 1 + \beta_{7} - \beta_{9} + \beta_{14} ) q^{52} + ( -1 - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{10} + \beta_{13} - \beta_{15} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} - \beta_{8} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{54} + ( 1 + 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{12} + \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{55} + ( -1 + \beta_{13} ) q^{56} + ( 2 \beta_{1} + \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} - 2 \beta_{10} + \beta_{13} - \beta_{15} + \beta_{18} ) q^{57} + ( -\beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{14} + \beta_{16} - \beta_{17} ) q^{58} + ( -1 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{13} + \beta_{14} + \beta_{16} - 2 \beta_{17} ) q^{59} + ( -\beta_{4} - \beta_{5} + \beta_{6} + \beta_{12} + \beta_{14} ) q^{60} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{14} - \beta_{16} - \beta_{18} ) q^{61} + ( -1 + \beta_{2} - \beta_{6} - \beta_{9} + \beta_{14} + \beta_{18} ) q^{62} + ( 5 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{17} + 2 \beta_{18} ) q^{63} + q^{64} + ( -\beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{13} + \beta_{14} ) q^{65} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} - \beta_{14} + \beta_{16} ) q^{66} + ( 3 + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{17} ) q^{67} + ( 1 - \beta_{8} - \beta_{10} - \beta_{12} - \beta_{16} ) q^{68} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{16} - \beta_{17} ) q^{69} + ( -\beta_{2} + 2 \beta_{3} + \beta_{7} + \beta_{8} + \beta_{14} + \beta_{16} + \beta_{18} ) q^{70} + ( -1 - \beta_{2} + \beta_{3} - \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{15} + \beta_{16} ) q^{71} + ( -1 - \beta_{1} - \beta_{7} - \beta_{14} + \beta_{17} - \beta_{18} ) q^{72} + ( 4 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{15} + 2 \beta_{18} ) q^{73} + ( -2 + \beta_{2} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{18} ) q^{74} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{7} - 4 \beta_{10} - \beta_{11} + \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{18} ) q^{75} + ( -\beta_{3} + \beta_{4} - \beta_{6} + \beta_{10} - \beta_{15} ) q^{76} + ( -1 - \beta_{5} + 3 \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + 3 \beta_{16} - 2 \beta_{17} ) q^{77} + ( 2 - 2 \beta_{1} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{15} - 2 \beta_{17} ) q^{78} + ( 3 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} ) q^{79} -\beta_{3} q^{80} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{81} + ( -1 - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{16} ) q^{82} + ( -\beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{10} - \beta_{12} + \beta_{14} - \beta_{15} - \beta_{16} + \beta_{18} ) q^{83} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{84} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{6} + \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{13} - \beta_{15} + \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{85} + ( -1 + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{86} + ( -\beta_{1} + \beta_{3} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{16} + \beta_{17} + \beta_{18} ) q^{87} -\beta_{11} q^{88} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} + \beta_{13} - \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{89} + ( -1 + \beta_{3} - 2 \beta_{6} - \beta_{8} - 3 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} ) q^{90} + ( 4 + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - 3 \beta_{13} + \beta_{15} + 2 \beta_{17} ) q^{91} + ( -\beta_{2} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{14} + \beta_{16} ) q^{92} + ( 2 - 2 \beta_{2} - \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} + \beta_{13} - \beta_{14} + \beta_{15} - 2 \beta_{17} - \beta_{18} ) q^{93} + ( -\beta_{6} + \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{94} + ( -1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{18} ) q^{95} -\beta_{1} q^{96} + ( 5 - 2 \beta_{1} - \beta_{5} - \beta_{8} - \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} ) q^{97} + ( -3 - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{98} + ( -1 + \beta_{2} + \beta_{3} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19q - 19q^{2} + 6q^{3} + 19q^{4} + 2q^{5} - 6q^{6} + 13q^{7} - 19q^{8} + 25q^{9} + O(q^{10}) \) \( 19q - 19q^{2} + 6q^{3} + 19q^{4} + 2q^{5} - 6q^{6} + 13q^{7} - 19q^{8} + 25q^{9} - 2q^{10} - 7q^{11} + 6q^{12} + 19q^{13} - 13q^{14} + 6q^{15} + 19q^{16} + 11q^{17} - 25q^{18} + 7q^{19} + 2q^{20} + 7q^{21} + 7q^{22} + 12q^{23} - 6q^{24} + 37q^{25} - 19q^{26} + 24q^{27} + 13q^{28} - 8q^{29} - 6q^{30} + 32q^{31} - 19q^{32} + 24q^{33} - 11q^{34} - 19q^{35} + 25q^{36} + 41q^{37} - 7q^{38} - 2q^{39} - 2q^{40} + 15q^{41} - 7q^{42} + 15q^{43} - 7q^{44} + 28q^{45} - 12q^{46} - 6q^{47} + 6q^{48} + 42q^{49} - 37q^{50} + 8q^{51} + 19q^{52} + 6q^{53} - 24q^{54} + 22q^{55} - 13q^{56} + 24q^{57} + 8q^{58} - 4q^{59} + 6q^{60} + 13q^{61} - 32q^{62} + 37q^{63} + 19q^{64} + 20q^{65} - 24q^{66} + 47q^{67} + 11q^{68} + 15q^{69} + 19q^{70} - 25q^{72} + 64q^{73} - 41q^{74} - 3q^{75} + 7q^{76} - 4q^{77} + 2q^{78} + 34q^{79} + 2q^{80} + 27q^{81} - 15q^{82} + 4q^{83} + 7q^{84} + 21q^{85} - 15q^{86} + 7q^{88} + 18q^{89} - 28q^{90} + 28q^{91} + 12q^{92} + 43q^{93} + 6q^{94} - 8q^{95} - 6q^{96} + 82q^{97} - 42q^{98} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19} - 6 x^{18} - 23 x^{17} + 190 x^{16} + 128 x^{15} - 2394 x^{14} + 749 x^{13} + 15539 x^{12} - 11338 x^{11} - 56744 x^{10} + 50183 x^{9} + 120237 x^{8} - 102992 x^{7} - 148539 x^{6} + 99519 x^{5} + 104388 x^{4} - 41081 x^{3} - 35733 x^{2} + 5542 x + 4222\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-3760520192170430386 \nu^{18} + 30305082839094078765 \nu^{17} + 41200800710110050060 \nu^{16} - 886378677694456159036 \nu^{15} + 907949322179800655710 \nu^{14} + 9829864376744292242242 \nu^{13} - 19458254457701180205140 \nu^{12} - 51174904871571611337619 \nu^{11} + 141922316336666980879649 \nu^{10} + 121951098427378259989987 \nu^{9} - 501893339143834545238647 \nu^{8} - 80600003893061145349438 \nu^{7} + 896205871816710953186330 \nu^{6} - 122113529794623773103574 \nu^{5} - 760681740768789785859421 \nu^{4} + 150786020791875092571053 \nu^{3} + 280020833321376854132087 \nu^{2} - 39882937575483697112922 \nu - 31872343548678071734688\)\()/ \)\(17\!\cdots\!02\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-29076110460680765573 \nu^{18} + 206990547035840834085 \nu^{17} + 449984253396810000572 \nu^{16} - 6101941593449854293310 \nu^{15} + 2797270294997483076910 \nu^{14} + 68736174355783190880152 \nu^{13} - 96465481661643000176109 \nu^{12} - 370669531209545576582376 \nu^{11} + 744314723041094580645146 \nu^{10} + 974192543055183898521206 \nu^{9} - 2613862736587341024233861 \nu^{8} - 1051452713367576331560234 \nu^{7} + 4457592187429563078996866 \nu^{6} + 66062436696734714273185 \nu^{5} - 3452999863129750339575306 \nu^{4} + 327922948269301043803302 \nu^{3} + 1137688178086206599168283 \nu^{2} - 85125640017045839740932 \nu - 122836999681487683614610\)\()/ \)\(35\!\cdots\!04\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-30945253987065060989 \nu^{18} + 219306674910305950519 \nu^{17} + 521676309588807770152 \nu^{16} - 6637331055069694956182 \nu^{15} + 1812199387042442266854 \nu^{14} + 78055232816059923303620 \nu^{13} - 91332498446351700215337 \nu^{12} - 454285839667464592518246 \nu^{11} + 748467525203016066124524 \nu^{10} + 1388661333356872083903068 \nu^{9} - 2762612032560116272703091 \nu^{8} - 2178547418940168131940534 \nu^{7} + 5028250121350414105405834 \nu^{6} + 1615298067719303956571269 \nu^{5} - 4277654130079105760072216 \nu^{4} - 538411709022266322520764 \nu^{3} + 1492995420294699602344401 \nu^{2} + 41402129559144245656416 \nu - 151051038335736692981254\)\()/ \)\(35\!\cdots\!04\)\( \)
\(\beta_{5}\)\(=\)\((\)\(35514340071205248367 \nu^{18} - 242632159693174792561 \nu^{17} - 643305185654029693144 \nu^{16} + 7400901982469154931142 \nu^{15} - 817581316649995040554 \nu^{14} - 87979131560055003267392 \nu^{13} + 91618867761805274083903 \nu^{12} + 519709571191677577219486 \nu^{11} - 798377461348665252899432 \nu^{10} - 1619993422592754555171372 \nu^{9} + 3075165837805834008163477 \nu^{8} + 2598218311229720705507906 \nu^{7} - 5902151591410770828067210 \nu^{6} - 1931036361335959872817407 \nu^{5} + 5457929300039474936559668 \nu^{4} + 551106253558324847470664 \nu^{3} - 2161482440858026625606455 \nu^{2} - 15471705894405999436640 \nu + 261764691393999547176914\)\()/ \)\(35\!\cdots\!04\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-39225674615895018295 \nu^{18} + 281334979929612838243 \nu^{17} + 630357296227855912720 \nu^{16} - 8430966253267278684034 \nu^{15} + 3188656397738161913338 \nu^{14} + 97614079406512640434288 \nu^{13} - 125532268617349703729955 \nu^{12} - 553483373863916062832792 \nu^{11} + 999509744710647745384134 \nu^{10} + 1614712864453446395048338 \nu^{9} - 3630156788064959123947035 \nu^{8} - 2309127089727213676127182 \nu^{7} + 6513098057013774939701338 \nu^{6} + 1393894842156114864238095 \nu^{5} - 5474664469504159485153834 \nu^{4} - 329634434562021923542522 \nu^{3} + 1952055797874724118604737 \nu^{2} + 7043498701436966126708 \nu - 222304348635242813768410\)\()/ \)\(35\!\cdots\!04\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-11167946556581994711 \nu^{18} + 72037020882250373706 \nu^{17} + 218163849798413458391 \nu^{16} - 2198104257192141994412 \nu^{15} - 249022931837379544406 \nu^{14} + 26155264541052107927708 \nu^{13} - 22385598655266651602489 \nu^{12} - 154856996848546561721898 \nu^{11} + 209171645728351253723877 \nu^{10} + 485395555103885711253293 \nu^{9} - 815202509303176415989788 \nu^{8} - 790394723234322452109492 \nu^{7} + 1547205525833040825470336 \nu^{6} + 618091829102644345876033 \nu^{5} - 1382690686240782796797340 \nu^{4} - 216675894834980889747933 \nu^{3} + 521040532835462169581003 \nu^{2} + 23377396899492401072662 \nu - 61007845242638016139457\)\()/ \)\(88\!\cdots\!51\)\( \)
\(\beta_{8}\)\(=\)\((\)\(22935642479960504263 \nu^{18} - 163807917546078032844 \nu^{17} - 347968100546541242034 \nu^{16} + 4811256116260884393776 \nu^{15} - 2415755773575146538442 \nu^{14} - 53879781520772923569932 \nu^{13} + 78553475030489475053325 \nu^{12} + 287609418859943312771113 \nu^{11} - 601867975227054779042041 \nu^{10} - 740938814962541543108443 \nu^{9} + 2111650992028433615016010 \nu^{8} + 758295797594366651969202 \nu^{7} - 3615592773829894301652558 \nu^{6} + 12016842140245612195017 \nu^{5} + 2840490829818975545731353 \nu^{4} - 242616546224541959763249 \nu^{3} - 970625595458513263192090 \nu^{2} + 49046681426265715429604 \nu + 113592698179520862754802\)\()/ \)\(17\!\cdots\!02\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-53078987331776809413 \nu^{18} + 367515426498576955637 \nu^{17} + 920244526033840804776 \nu^{16} - 11088118004222127647018 \nu^{15} + 2323875647558612679514 \nu^{14} + 129631647355211696363740 \nu^{13} - 147215378581633743827297 \nu^{12} - 745654129012549222047316 \nu^{11} + 1227483135034000590929390 \nu^{10} + 2223128080408016070228306 \nu^{9} - 4564940454465900858809637 \nu^{8} - 3293148157720498569728226 \nu^{7} + 8377236903027542588927042 \nu^{6} + 2119801390785974910893953 \nu^{5} - 7282681049875190648528770 \nu^{4} - 555446990794750228761258 \nu^{3} + 2758013642041956597019251 \nu^{2} + 43217652384396165704284 \nu - 338893975678974074603630\)\()/ \)\(35\!\cdots\!04\)\( \)
\(\beta_{10}\)\(=\)\((\)\(65269994698618887489 \nu^{18} - 440295935608668625411 \nu^{17} - 1191848115016691865296 \nu^{16} + 13359502166014633995958 \nu^{15} - 1037892829669082123282 \nu^{14} - 157625826309534025609356 \nu^{13} + 159778105580821061796529 \nu^{12} + 921037838225169820617946 \nu^{11} - 1388161951758038181773760 \nu^{10} - 2825851115465353418466920 \nu^{9} + 5260830789395044175743819 \nu^{8} + 4431949649138633371027590 \nu^{7} - 9808515862489234476993834 \nu^{6} - 3227048649923522292410445 \nu^{5} + 8655297992211802941703232 \nu^{4} + 1006944353978834637406320 \nu^{3} - 3252303057734070582886901 \nu^{2} - 76711896678285078567952 \nu + 382673437272331542837754\)\()/ \)\(35\!\cdots\!04\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-67545888761056757405 \nu^{18} + 474152691703187535957 \nu^{17} + 1126972007891038411660 \nu^{16} - 14225479148011647792746 \nu^{15} + 4245207934669709738222 \nu^{14} + 164933151038642750448804 \nu^{13} - 201754926509586295161889 \nu^{12} - 936527712809011504410176 \nu^{11} + 1640288763668577806266250 \nu^{10} + 2733320468157187722611582 \nu^{9} - 6025875980509308282125301 \nu^{8} - 3890467207784257373413434 \nu^{7} + 10950927147978590859975302 \nu^{6} + 2281174130293399234730025 \nu^{5} - 9407015588513958382958478 \nu^{4} - 481304642842418863805234 \nu^{3} + 3472206780176237103939911 \nu^{2} + 23882619695694000437704 \nu - 407486381451920829795090\)\()/ \)\(35\!\cdots\!04\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-69985589363716372003 \nu^{18} + 452117294523362098281 \nu^{17} + 1325535324928609159564 \nu^{16} - 13563574351565447656442 \nu^{15} - 639254816371122214178 \nu^{14} + 157208079346273319443112 \nu^{13} - 145053125102334518860899 \nu^{12} - 891743569213297416919646 \nu^{11} + 1282552727139659807366264 \nu^{10} + 2595492162936333398605104 \nu^{9} - 4739692707772156654829369 \nu^{8} - 3671011968791771277480298 \nu^{7} + 8342749374284723742911106 \nu^{6} + 2133527757290567026500795 \nu^{5} - 6635766416826814603288452 \nu^{4} - 482217178587693989306880 \nu^{3} + 2185026163458131991656443 \nu^{2} + 28482245194557351640248 \nu - 221905016240633020395442\)\()/ \)\(35\!\cdots\!04\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-77065122442043030315 \nu^{18} + 532774562852707940065 \nu^{17} + 1309795665184004200420 \nu^{16} - 15947559575155028954070 \nu^{15} + 4066746067386985452170 \nu^{14} + 184141182031040943735476 \nu^{13} - 220131935427482750177679 \nu^{12} - 1037180746879497347929890 \nu^{11} + 1804220288341667306366968 \nu^{10} + 2973324222468615128719668 \nu^{9} - 6624784111405967801651453 \nu^{8} - 4032909788018291328760214 \nu^{7} + 11979292489662292335093490 \nu^{6} + 1963412732253456474997411 \nu^{5} - 10207686105122915636409520 \nu^{4} - 72364707768952431818544 \nu^{3} + 3775019299987169463270923 \nu^{2} - 93743526820595676419864 \nu - 448047703088435065929750\)\()/ \)\(35\!\cdots\!04\)\( \)
\(\beta_{14}\)\(=\)\((\)\(81246465791995337029 \nu^{18} - 538011472036996170959 \nu^{17} - 1500040891301178160836 \nu^{16} + 16239099098267565245182 \nu^{15} - 682749423888030571250 \nu^{14} - 190058582570187933079240 \nu^{13} + 189727613796164956802949 \nu^{12} + 1096000891963530403875206 \nu^{11} - 1656270139070271276372200 \nu^{10} - 3286800831286664099437744 \nu^{9} + 6237831720989536635596223 \nu^{8} + 4936824531945407856072402 \nu^{7} - 11476874966336529882099158 \nu^{6} - 3283786466870076576469109 \nu^{5} + 9927586089252498464492112 \nu^{4} + 881333673434794369135432 \nu^{3} - 3681466686787631497562293 \nu^{2} - 47898124722861190738136 \nu + 432163679569144496110306\)\()/ \)\(35\!\cdots\!04\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-47097892268680919528 \nu^{18} + 319743617906841950481 \nu^{17} + 837922334641744413414 \nu^{16} - 9633843421968099827224 \nu^{15} + 1369200205499709724344 \nu^{14} + 112406080492905974955444 \nu^{13} - 121720435612565003591624 \nu^{12} - 644497773229142363627215 \nu^{11} + 1031071031664700952304797 \nu^{10} + 1910019844408435737913565 \nu^{9} - 3845750444490116295482927 \nu^{8} - 2787652479313939222193926 \nu^{7} + 7047960430634142289825628 \nu^{6} + 1696565935560911310021946 \nu^{5} - 6094487564041673976730955 \nu^{4} - 326408167210916415098145 \nu^{3} + 2271252467228465080858335 \nu^{2} - 14489750421562796486650 \nu - 271400599812957179459358\)\()/ \)\(17\!\cdots\!02\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-95971349547274628625 \nu^{18} + 656629516719657132985 \nu^{17} + 1661993776848826883504 \nu^{16} - 19670759651570058856394 \nu^{15} + 4075172162552562465278 \nu^{14} + 227474470867889711197860 \nu^{13} - 261446093195608246486625 \nu^{12} - 1285291669865683973133804 \nu^{11} + 2162449824953311239457686 \nu^{10} + 3712796511016076787639470 \nu^{9} - 7933994510972599758883509 \nu^{8} - 5155450907022371409992946 \nu^{7} + 14246641802290284696365898 \nu^{6} + 2801906095971079748896029 \nu^{5} - 11951769961894352195903022 \nu^{4} - 450996964601953574266394 \nu^{3} + 4322881678425745449196591 \nu^{2} - 22864874827629484464976 \nu - 507524935709983375201054\)\()/ \)\(35\!\cdots\!04\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-111180934604728577169 \nu^{18} + 754491441872043631203 \nu^{17} + 1983118773266591338776 \nu^{16} - 22754311006284932937834 \nu^{15} + 3094221402718334038530 \nu^{14} + 265927825342822788548944 \nu^{13} - 285979842814695639634801 \nu^{12} - 1529488751053174975282250 \nu^{11} + 2428243393839957414429780 \nu^{10} + 4563172264386688163453912 \nu^{9} - 9069817744293933606095027 \nu^{8} - 6775592737231410424291170 \nu^{7} + 16641663522586577497468190 \nu^{6} + 4371124193456046186265857 \nu^{5} - 14402873239663841582626060 \nu^{4} - 1084873524382554347794788 \nu^{3} + 5368607597817917780318525 \nu^{2} + 46591581792452054756420 \nu - 636544201841169330766818\)\()/ \)\(35\!\cdots\!04\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-73877807085197967677 \nu^{18} + 502177415190019153669 \nu^{17} + 1305252132687057833024 \nu^{16} - 15100496537891965102684 \nu^{15} + 2386531276977941393702 \nu^{14} + 175682674874401144958676 \nu^{13} - 193082530994896995013897 \nu^{12} - 1003030827811259566134932 \nu^{11} + 1623913474998411837953236 \nu^{10} + 2954195437628904708939242 \nu^{9} - 6023419714035382288866049 \nu^{8} - 4275419188119764235962802 \nu^{7} + 10964858192795472038843002 \nu^{6} + 2591271671957772689615417 \nu^{5} - 9399848291976604429964406 \nu^{4} - 549751809238712578969244 \nu^{3} + 3484720189061687655479305 \nu^{2} - 1274052971165535098948 \nu - 419394699939230303963256\)\()/ \)\(17\!\cdots\!02\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{18} - \beta_{17} + \beta_{14} + \beta_{7} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{18} - \beta_{17} + \beta_{16} - \beta_{15} + \beta_{8} + \beta_{2} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(10 \beta_{18} - 10 \beta_{17} - 2 \beta_{16} + 7 \beta_{14} + \beta_{12} + \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + \beta_{8} + 9 \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 10 \beta_{1} + 30\)
\(\nu^{5}\)\(=\)\(13 \beta_{18} - 15 \beta_{17} + 10 \beta_{16} - 10 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + \beta_{12} + 4 \beta_{11} + 6 \beta_{10} + 3 \beta_{9} + 15 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + 13 \beta_{2} + 58 \beta_{1} + 18\)
\(\nu^{6}\)\(=\)\(96 \beta_{18} - 101 \beta_{17} - 30 \beta_{16} + 4 \beta_{15} + 46 \beta_{14} - 5 \beta_{13} + 15 \beta_{12} + 16 \beta_{11} + 62 \beta_{10} + 34 \beta_{9} + 18 \beta_{8} + 80 \beta_{7} + 22 \beta_{6} + 9 \beta_{4} - 12 \beta_{3} + 32 \beta_{2} + 99 \beta_{1} + 270\)
\(\nu^{7}\)\(=\)\(145 \beta_{18} - 193 \beta_{17} + 85 \beta_{16} - 79 \beta_{15} - 46 \beta_{14} - 41 \beta_{13} + 21 \beta_{12} + 66 \beta_{11} + 115 \beta_{10} + 62 \beta_{9} + 177 \beta_{8} + 33 \beta_{7} + 69 \beta_{6} - 5 \beta_{5} - 38 \beta_{4} + 29 \beta_{3} + 149 \beta_{2} + 525 \beta_{1} + 257\)
\(\nu^{8}\)\(=\)\(930 \beta_{18} - 1072 \beta_{17} - 351 \beta_{16} + 94 \beta_{15} + 278 \beta_{14} - 106 \beta_{13} + 186 \beta_{12} + 211 \beta_{11} + 768 \beta_{10} + 451 \beta_{9} + 254 \beta_{8} + 736 \beta_{7} + 349 \beta_{6} - 11 \beta_{5} + 42 \beta_{4} - 115 \beta_{3} + 401 \beta_{2} + 1012 \beta_{1} + 2634\)
\(\nu^{9}\)\(=\)\(1573 \beta_{18} - 2397 \beta_{17} + 666 \beta_{16} - 530 \beta_{15} - 738 \beta_{14} - 590 \beta_{13} + 327 \beta_{12} + 852 \beta_{11} + 1664 \beta_{10} + 964 \beta_{9} + 1961 \beta_{8} + 444 \beta_{7} + 1115 \beta_{6} - 119 \beta_{5} - 538 \beta_{4} + 310 \beta_{3} + 1648 \beta_{2} + 5036 \beta_{1} + 3380\)
\(\nu^{10}\)\(=\)\(9175 \beta_{18} - 11769 \beta_{17} - 3807 \beta_{16} + 1508 \beta_{15} + 1251 \beta_{14} - 1589 \beta_{13} + 2189 \beta_{12} + 2612 \beta_{11} + 8964 \beta_{10} + 5554 \beta_{9} + 3307 \beta_{8} + 6985 \beta_{7} + 4848 \beta_{6} - 307 \beta_{5} - 183 \beta_{4} - 1029 \beta_{3} + 4663 \beta_{2} + 10683 \beta_{1} + 26870\)
\(\nu^{11}\)\(=\)\(17145 \beta_{18} - 29325 \beta_{17} + 4745 \beta_{16} - 2619 \beta_{15} - 10236 \beta_{14} - 7453 \beta_{13} + 4519 \beta_{12} + 10217 \beta_{11} + 21795 \beta_{10} + 13362 \beta_{9} + 21420 \beta_{8} + 5710 \beta_{7} + 15698 \beta_{6} - 1975 \beta_{5} - 6883 \beta_{4} + 2992 \beta_{3} + 18039 \beta_{2} + 50367 \beta_{1} + 42768\)
\(\nu^{12}\)\(=\)\(92445 \beta_{18} - 131860 \beta_{17} - 40241 \beta_{16} + 20806 \beta_{15} - 1416 \beta_{14} - 20898 \beta_{13} + 25311 \beta_{12} + 31401 \beta_{11} + 102868 \beta_{10} + 66431 \beta_{9} + 41484 \beta_{8} + 68177 \beta_{7} + 63006 \beta_{6} - 5642 \beta_{5} - 8331 \beta_{4} - 8937 \beta_{3} + 52854 \beta_{2} + 115912 \beta_{1} + 282144\)
\(\nu^{13}\)\(=\)\(189444 \beta_{18} - 355071 \beta_{17} + 28095 \beta_{16} + 467 \beta_{15} - 131894 \beta_{14} - 88916 \beta_{13} + 58752 \beta_{12} + 119285 \beta_{11} + 272586 \beta_{10} + 174263 \beta_{9} + 234511 \beta_{8} + 72291 \beta_{7} + 206215 \beta_{6} - 28423 \beta_{5} - 84230 \beta_{4} + 27678 \beta_{3} + 197598 \beta_{2} + 520148 \beta_{1} + 529320\)
\(\nu^{14}\)\(=\)\(951790 \beta_{18} - 1495641 \beta_{17} - 422941 \beta_{16} + 266482 \beta_{15} - 148463 \beta_{14} - 258558 \beta_{13} + 291063 \beta_{12} + 371861 \beta_{11} + 1176880 \beta_{10} + 784039 \beta_{9} + 509590 \beta_{8} + 683060 \beta_{7} + 788675 \beta_{6} - 86676 \beta_{5} - 151568 \beta_{4} - 76389 \beta_{3} + 595243 \beta_{2} + 1285093 \beta_{1} + 3025986\)
\(\nu^{15}\)\(=\)\(2123444 \beta_{18} - 4263971 \beta_{17} + 84246 \beta_{16} + 270917 \beta_{15} - 1630800 \beta_{14} - 1033607 \beta_{13} + 737547 \beta_{12} + 1379282 \beta_{11} + 3325956 \beta_{10} + 2191644 \beta_{9} + 2587562 \beta_{8} + 905228 \beta_{7} + 2603486 \beta_{6} - 380335 \beta_{5} - 1008395 \beta_{4} + 251131 \beta_{3} + 2177452 \beta_{2} + 5510799 \beta_{1} + 6456756\)
\(\nu^{16}\)\(=\)\(10004896 \beta_{18} - 17096882 \beta_{17} - 4461039 \beta_{16} + 3277301 \beta_{15} - 2835698 \beta_{14} - 3101641 \beta_{13} + 3344826 \beta_{12} + 4369096 \beta_{11} + 13486035 \beta_{10} + 9191526 \beta_{9} + 6176669 \beta_{8} + 7011607 \beta_{7} + 9648683 \beta_{6} - 1209261 \beta_{5} - 2237308 \beta_{4} - 646233 \beta_{3} + 6713785 \beta_{2} + 14477716 \beta_{1} + 32999887\)
\(\nu^{17}\)\(=\)\(24090869 \beta_{18} - 50866953 \beta_{17} - 1189598 \beta_{16} + 5419328 \beta_{15} - 19678325 \beta_{14} - 11883466 \beta_{13} + 9060210 \beta_{12} + 15905311 \beta_{11} + 39994481 \beta_{10} + 26934897 \beta_{9} + 28809860 \beta_{8} + 11215969 \beta_{7} + 32068711 \beta_{6} - 4876496 \beta_{5} - 11932955 \beta_{4} + 2258019 \beta_{3} + 24189418 \beta_{2} + 59605928 \beta_{1} + 77950265\)
\(\nu^{18}\)\(=\)\(107185878 \beta_{18} - 196451573 \beta_{17} - 47431680 \beta_{16} + 39376536 \beta_{15} - 42400156 \beta_{14} - 36615716 \beta_{13} + 38484796 \beta_{12} + 51124315 \beta_{11} + 154989621 \beta_{10} + 107374656 \beta_{9} + 74180123 \beta_{8} + 73579411 \beta_{7} + 116313191 \beta_{6} - 15923475 \beta_{5} - 30123224 \beta_{4} - 5421373 \beta_{3} + 76069064 \beta_{2} + 164981093 \beta_{1} + 364797939\)

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
−3.01539
−2.98144
−2.13636
−1.94953
−1.64365
−0.862574
−0.803709
−0.725860
−0.443463
0.493679
0.748702
1.28683
1.74260
2.07470
2.33529
2.66959
2.88063
2.91512
3.41483
−1.00000 −3.01539 1.00000 −2.64906 3.01539 2.56889 −1.00000 6.09259 2.64906
1.2 −1.00000 −2.98144 1.00000 3.72248 2.98144 −0.567696 −1.00000 5.88899 −3.72248
1.3 −1.00000 −2.13636 1.00000 −2.42355 2.13636 4.87888 −1.00000 1.56402 2.42355
1.4 −1.00000 −1.94953 1.00000 2.00911 1.94953 0.0633867 −1.00000 0.800679 −2.00911
1.5 −1.00000 −1.64365 1.00000 −0.210142 1.64365 −4.29873 −1.00000 −0.298402 0.210142
1.6 −1.00000 −0.862574 1.00000 −3.34357 0.862574 −0.0695492 −1.00000 −2.25597 3.34357
1.7 −1.00000 −0.803709 1.00000 2.36229 0.803709 4.01698 −1.00000 −2.35405 −2.36229
1.8 −1.00000 −0.725860 1.00000 3.71965 0.725860 −1.04918 −1.00000 −2.47313 −3.71965
1.9 −1.00000 −0.443463 1.00000 −2.41711 0.443463 −1.05600 −1.00000 −2.80334 2.41711
1.10 −1.00000 0.493679 1.00000 0.378923 −0.493679 2.20566 −1.00000 −2.75628 −0.378923
1.11 −1.00000 0.748702 1.00000 −1.18128 −0.748702 −4.67174 −1.00000 −2.43945 1.18128
1.12 −1.00000 1.28683 1.00000 −3.54885 −1.28683 3.93701 −1.00000 −1.34406 3.54885
1.13 −1.00000 1.74260 1.00000 1.47569 −1.74260 4.24779 −1.00000 0.0366584 −1.47569
1.14 −1.00000 2.07470 1.00000 4.37927 −2.07470 2.25344 −1.00000 1.30439 −4.37927
1.15 −1.00000 2.33529 1.00000 −4.05193 −2.33529 0.675791 −1.00000 2.45359 4.05193
1.16 −1.00000 2.66959 1.00000 1.32740 −2.66959 −3.70251 −1.00000 4.12668 −1.32740
1.17 −1.00000 2.88063 1.00000 2.69966 −2.88063 −2.78599 −1.00000 5.29806 −2.69966
1.18 −1.00000 2.91512 1.00000 −1.49286 −2.91512 3.25503 −1.00000 5.49794 1.49286
1.19 −1.00000 3.41483 1.00000 1.24390 −3.41483 3.09855 −1.00000 8.66107 −1.24390
\(n\): 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 1502.2.a.h 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1502.2.a.h 19 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(751\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{19} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1502))\).