Properties

Label 230.3.d.a
Level $230$
Weight $3$
Character orbit 230.d
Analytic conductor $6.267$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.26704608029\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 78 x^{14} + 2165 x^{12} + 28310 x^{10} + 184804 x^{8} + 569634 x^{6} + 696037 x^{4} + 285578 x^{2} + 529\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{6} q^{2} -\beta_{5} q^{3} + 2 q^{4} + \beta_{2} q^{5} -\beta_{10} q^{6} + ( \beta_{1} + \beta_{7} ) q^{7} -2 \beta_{6} q^{8} + ( 5 + \beta_{6} - \beta_{10} - \beta_{11} - \beta_{14} ) q^{9} +O(q^{10})\) \( q -\beta_{6} q^{2} -\beta_{5} q^{3} + 2 q^{4} + \beta_{2} q^{5} -\beta_{10} q^{6} + ( \beta_{1} + \beta_{7} ) q^{7} -2 \beta_{6} q^{8} + ( 5 + \beta_{6} - \beta_{10} - \beta_{11} - \beta_{14} ) q^{9} -\beta_{8} q^{10} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{7} - \beta_{9} ) q^{11} -2 \beta_{5} q^{12} + ( 1 + \beta_{2} + \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{13} + ( 2 \beta_{2} + \beta_{8} - \beta_{9} - \beta_{12} ) q^{14} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{12} ) q^{15} + 4 q^{16} + ( -2 \beta_{2} - 3 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{12} ) q^{17} + ( -3 - 4 \beta_{6} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{18} + ( 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{7} + 4 \beta_{8} + \beta_{9} ) q^{19} + 2 \beta_{2} q^{20} + ( -2 \beta_{1} + \beta_{2} + 5 \beta_{3} - 4 \beta_{4} - \beta_{7} - \beta_{8} + 2 \beta_{9} + 3 \beta_{12} ) q^{21} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} - \beta_{12} ) q^{22} + ( 1 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{5} + 5 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{23} -2 \beta_{10} q^{24} -5 q^{25} + ( 6 + 2 \beta_{5} - \beta_{10} + 2 \beta_{11} + 2 \beta_{15} ) q^{26} + ( -8 + \beta_{2} + \beta_{4} - \beta_{5} - 8 \beta_{6} - \beta_{7} + 4 \beta_{10} - \beta_{12} - 2 \beta_{15} ) q^{27} + ( 2 \beta_{1} + 2 \beta_{7} ) q^{28} + ( -6 - \beta_{2} - \beta_{4} - \beta_{5} - 6 \beta_{6} + \beta_{7} - \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{29} + ( -\beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{30} + ( -8 - \beta_{2} - \beta_{4} - 5 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{10} + \beta_{12} - \beta_{14} + 3 \beta_{15} ) q^{31} -4 \beta_{6} q^{32} + ( -4 \beta_{1} - \beta_{2} + 9 \beta_{3} - 8 \beta_{4} - \beta_{7} + \beta_{8} + 4 \beta_{9} + 5 \beta_{12} ) q^{33} + ( \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{7} ) q^{34} + ( 4 + \beta_{2} + \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - \beta_{7} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{35} + ( 10 + 2 \beta_{6} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{14} ) q^{36} + ( 4 \beta_{1} + 3 \beta_{2} + \beta_{3} - 4 \beta_{4} + 2 \beta_{7} + 11 \beta_{8} + 5 \beta_{9} ) q^{37} + ( \beta_{1} - 4 \beta_{2} - \beta_{3} + 5 \beta_{4} - 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - \beta_{12} ) q^{38} + ( 18 - 2 \beta_{5} + 11 \beta_{6} - 2 \beta_{10} - 5 \beta_{11} + 4 \beta_{13} - \beta_{14} ) q^{39} -2 \beta_{8} q^{40} + ( -10 - 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + \beta_{10} - 2 \beta_{11} + 3 \beta_{12} - 4 \beta_{13} + \beta_{14} - \beta_{15} ) q^{41} + ( -\beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} + 4 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 3 \beta_{12} ) q^{42} + ( -\beta_{1} + 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 5 \beta_{12} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{9} ) q^{44} + ( 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{45} + ( -9 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 4 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - \beta_{14} - \beta_{15} ) q^{46} + ( -10 + 4 \beta_{2} + 4 \beta_{4} + \beta_{5} + 6 \beta_{6} - 4 \beta_{7} + 2 \beta_{10} + 4 \beta_{11} - 4 \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{47} -4 \beta_{5} q^{48} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} - 9 \beta_{6} + \beta_{7} - 3 \beta_{11} + \beta_{12} - 2 \beta_{13} - 3 \beta_{15} ) q^{49} + 5 \beta_{6} q^{50} + ( -2 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} - 6 \beta_{8} + 4 \beta_{9} ) q^{51} + ( 2 + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{52} + ( -6 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + \beta_{7} + \beta_{9} - \beta_{12} ) q^{53} + ( 16 - 2 \beta_{2} - 2 \beta_{4} + 6 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} - 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{54} + ( 2 + \beta_{2} + \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} - 2 \beta_{10} - \beta_{11} - \beta_{12} - 3 \beta_{14} ) q^{55} + ( 4 \beta_{2} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} ) q^{56} + ( -2 \beta_{1} + 3 \beta_{2} - 11 \beta_{3} + 10 \beta_{4} + 11 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 5 \beta_{12} ) q^{57} + ( 8 - \beta_{2} - \beta_{4} - 2 \beta_{5} + 5 \beta_{6} + \beta_{7} + 2 \beta_{10} + \beta_{12} + 2 \beta_{13} + 4 \beta_{14} ) q^{58} + ( 10 - 3 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} + 3 \beta_{11} + 3 \beta_{12} + 8 \beta_{14} + \beta_{15} ) q^{59} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{12} ) q^{60} + ( -4 \beta_{1} - 3 \beta_{2} + 9 \beta_{3} - 2 \beta_{4} - 9 \beta_{7} - 5 \beta_{8} + 8 \beta_{9} + 5 \beta_{12} ) q^{61} + ( 4 + 3 \beta_{2} + 3 \beta_{4} + 8 \beta_{5} + 9 \beta_{6} - 3 \beta_{7} - 2 \beta_{10} + 4 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} ) q^{62} + ( -2 \beta_{1} - 19 \beta_{2} + 11 \beta_{4} - 2 \beta_{7} - 8 \beta_{8} - 5 \beta_{9} ) q^{63} + 8 q^{64} + ( -2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 5 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{12} ) q^{65} + ( -3 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} + 5 \beta_{4} + 8 \beta_{7} - 3 \beta_{8} - 5 \beta_{9} - 5 \beta_{12} ) q^{66} + ( -5 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} + 3 \beta_{7} - 12 \beta_{8} - 6 \beta_{9} - 4 \beta_{12} ) q^{67} + ( -4 \beta_{2} - 6 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{12} ) q^{68} + ( -21 - 4 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} + 6 \beta_{7} + 9 \beta_{8} - 3 \beta_{9} + 6 \beta_{10} - \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 6 \beta_{14} - 3 \beta_{15} ) q^{69} + ( -8 - \beta_{2} - \beta_{4} - 3 \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} - 2 \beta_{13} ) q^{70} + ( 16 - 3 \beta_{2} - 3 \beta_{4} + 13 \beta_{5} + 16 \beta_{6} + 3 \beta_{7} - 2 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} - 4 \beta_{13} + \beta_{14} - \beta_{15} ) q^{71} + ( -6 - 8 \beta_{6} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{72} + ( -3 - 3 \beta_{2} - 3 \beta_{4} + 7 \beta_{5} - 9 \beta_{6} + 3 \beta_{7} - 7 \beta_{10} - \beta_{11} + 3 \beta_{12} - 3 \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{73} + ( -2 \beta_{1} - 14 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 3 \beta_{8} - 3 \beta_{9} - 5 \beta_{12} ) q^{74} + 5 \beta_{5} q^{75} + ( 6 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} + 8 \beta_{8} + 2 \beta_{9} ) q^{76} + ( -62 + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + 4 \beta_{14} - 8 \beta_{15} ) q^{77} + ( -31 - 4 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} - 17 \beta_{6} + 4 \beta_{7} + 3 \beta_{10} + \beta_{11} + 4 \beta_{12} - \beta_{13} + 9 \beta_{14} - \beta_{15} ) q^{78} + ( -5 \beta_{1} - 9 \beta_{2} + 14 \beta_{3} - \beta_{4} - 17 \beta_{7} + 2 \beta_{8} - \beta_{9} + 6 \beta_{12} ) q^{79} + 4 \beta_{2} q^{80} + ( -5 + 2 \beta_{2} + 2 \beta_{4} + 6 \beta_{5} + 38 \beta_{6} - 2 \beta_{7} - 8 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 4 \beta_{13} + 2 \beta_{15} ) q^{81} + ( -2 + 3 \beta_{2} + 3 \beta_{4} + 6 \beta_{5} + 9 \beta_{6} - 3 \beta_{7} - 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{14} - 6 \beta_{15} ) q^{82} + ( -2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 7 \beta_{4} - 8 \beta_{7} + 16 \beta_{8} - 3 \beta_{9} - 8 \beta_{12} ) q^{83} + ( -4 \beta_{1} + 2 \beta_{2} + 10 \beta_{3} - 8 \beta_{4} - 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + 6 \beta_{12} ) q^{84} + ( 2 + \beta_{2} + \beta_{4} - 3 \beta_{5} + 5 \beta_{6} - \beta_{7} + 2 \beta_{10} + 3 \beta_{11} - \beta_{12} + 2 \beta_{13} + 3 \beta_{15} ) q^{85} + ( -9 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{7} - 11 \beta_{8} - 5 \beta_{9} + \beta_{12} ) q^{86} + ( -14 + 5 \beta_{2} + 5 \beta_{4} + 11 \beta_{5} - 8 \beta_{6} - 5 \beta_{7} + 6 \beta_{10} + 10 \beta_{11} - 5 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} + 6 \beta_{15} ) q^{87} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} ) q^{88} + ( -4 \beta_{1} - 13 \beta_{2} - 9 \beta_{3} + 10 \beta_{4} - 7 \beta_{7} - 7 \beta_{8} - 10 \beta_{9} + 3 \beta_{12} ) q^{89} + ( -\beta_{1} - 3 \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{7} - 6 \beta_{8} + \beta_{12} ) q^{90} + ( 3 \beta_{1} + 3 \beta_{2} + 16 \beta_{3} - 9 \beta_{4} + \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 4 \beta_{12} ) q^{91} + ( 2 + 2 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} + 2 \beta_{5} + 10 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{92} + ( 57 - 3 \beta_{2} - 3 \beta_{4} + 7 \beta_{5} + 41 \beta_{6} + 3 \beta_{7} - 3 \beta_{10} - 7 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{93} + ( -10 - 4 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} + 10 \beta_{6} + 4 \beta_{7} - 5 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 6 \beta_{15} ) q^{94} + ( -14 + 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} - 11 \beta_{6} - 3 \beta_{7} + 4 \beta_{10} + 7 \beta_{11} - 3 \beta_{12} + \beta_{14} ) q^{95} -4 \beta_{10} q^{96} + ( 10 \beta_{1} - 9 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 7 \beta_{7} - \beta_{8} + 10 \beta_{9} - 3 \beta_{12} ) q^{97} + ( 17 - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - 3 \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{14} - 5 \beta_{15} ) q^{98} + ( 4 \beta_{1} - 38 \beta_{2} - 8 \beta_{3} + 14 \beta_{4} + 4 \beta_{7} - 16 \beta_{8} - 6 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9} + O(q^{10}) \) \( 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9} + 24 q^{13} + 64 q^{16} - 32 q^{18} + 4 q^{23} - 16 q^{24} - 80 q^{25} + 96 q^{26} - 96 q^{27} - 108 q^{29} - 116 q^{31} + 60 q^{35} + 128 q^{36} + 248 q^{39} - 156 q^{41} - 124 q^{46} - 128 q^{47} - 28 q^{49} + 48 q^{52} + 224 q^{54} + 160 q^{58} + 204 q^{59} + 64 q^{62} + 128 q^{64} - 268 q^{69} - 120 q^{70} + 236 q^{71} - 64 q^{72} - 112 q^{73} - 936 q^{77} - 432 q^{78} - 136 q^{81} - 64 q^{82} + 60 q^{85} - 152 q^{87} + 8 q^{92} + 856 q^{93} - 216 q^{94} - 160 q^{95} - 32 q^{96} + 256 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 78 x^{14} + 2165 x^{12} + 28310 x^{10} + 184804 x^{8} + 569634 x^{6} + 696037 x^{4} + 285578 x^{2} + 529\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 249266 \nu^{15} + 19450521 \nu^{13} + 540300528 \nu^{11} + 7075107694 \nu^{9} + 46294879486 \nu^{7} + 143284317912 \nu^{5} + 176966386308 \nu^{3} + 76734676377 \nu \)\()/ 1473983980 \)
\(\beta_{3}\)\(=\)\((\)\(-3655930587 \nu^{15} - 279912880683 \nu^{13} - 7519929098840 \nu^{11} - 93142815057018 \nu^{9} - 550737169153146 \nu^{7} - 1359775399344612 \nu^{5} - 747373718410547 \nu^{3} + 192163143584185 \nu\)\()/ 13607820103360 \)
\(\beta_{4}\)\(=\)\((\)\(-2028139777 \nu^{15} - 159386627629 \nu^{13} - 4483504013264 \nu^{11} - 59956611391678 \nu^{9} - 407018725514790 \nu^{7} - 1350423473295780 \nu^{5} - 1906976451293273 \nu^{3} - 890813001228721 \nu\)\()/ 3401955025840 \)
\(\beta_{5}\)\(=\)\((\)\(3266665611 \nu^{14} + 251231034675 \nu^{12} + 6803101750504 \nu^{10} + 85389448480554 \nu^{8} + 517393068635482 \nu^{6} + 1353800445494324 \nu^{4} + 988318890938067 \nu^{2} + 38708131230591\)\()/ 6803910051680 \)
\(\beta_{6}\)\(=\)\((\)\( 328071 \nu^{14} + 25254583 \nu^{12} + 684448696 \nu^{10} + 8586649314 \nu^{8} + 51832164194 \nu^{6} + 133890163508 \nu^{4} + 92129883455 \nu^{2} + 885659459 \)\()/ 505359680 \)
\(\beta_{7}\)\(=\)\((\)\(651909559 \nu^{15} + 49847669703 \nu^{13} + 1334253632328 \nu^{11} + 16362993734546 \nu^{9} + 94189913828650 \nu^{7} + 212449351122740 \nu^{5} + 42951096904351 \nu^{3} - 93749086185413 \nu\)\()/ 485993575120 \)
\(\beta_{8}\)\(=\)\((\)\(191987899 \nu^{15} + 14808207659 \nu^{13} + 402812007992 \nu^{11} + 5087217548666 \nu^{9} + 31116072823834 \nu^{7} + 83034850253508 \nu^{5} + 65752777188627 \nu^{3} + 8504894308903 \nu\)\()/ 114351429440 \)
\(\beta_{9}\)\(=\)\((\)\(-229620557 \nu^{15} - 17699955559 \nu^{13} - 480902133464 \nu^{11} - 6059544792898 \nu^{9} - 36877647447520 \nu^{7} - 97036544153580 \nu^{5} - 71790826408533 \nu^{3} - 4166619657821 \nu\)\()/ 121498393780 \)
\(\beta_{10}\)\(=\)\((\)\(14365389855 \nu^{14} + 1105876709199 \nu^{12} + 29972322851736 \nu^{10} + 375976473404450 \nu^{8} + 2268003524367106 \nu^{6} + 5840712005282452 \nu^{4} + 3940822109052359 \nu^{2} - 44834433467221\)\()/ 6803910051680 \)
\(\beta_{11}\)\(=\)\((\)\(45221309975 \nu^{15} - 21002456926 \nu^{14} + 3479923444575 \nu^{13} - 1617708410646 \nu^{12} + 94261526625960 \nu^{11} - 43888163873888 \nu^{10} + 1181548675553650 \nu^{9} - 551450943599924 \nu^{8} + 7119963817216130 \nu^{7} - 3335789410501956 \nu^{6} + 18295459522336580 \nu^{5} - 8640632102007032 \nu^{4} + 12205236948457855 \nu^{3} - 5987597192457438 \nu^{2} - 277533627658085 \nu - 82437933075262\)\()/ 13607820103360 \)
\(\beta_{12}\)\(=\)\((\)\(33188908451 \nu^{15} + 2553066418411 \nu^{13} + 69108994984088 \nu^{11} + 865212237602154 \nu^{9} + 5200964736292826 \nu^{7} + 13281722071508452 \nu^{5} + 8606845528408123 \nu^{3} - 392465157363513 \nu\)\()/ 6803910051680 \)
\(\beta_{13}\)\(=\)\((\)\(45221309975 \nu^{15} - 28278344419 \nu^{14} + 3479923444575 \nu^{13} - 2174316072571 \nu^{12} + 94261526625960 \nu^{11} - 58816507643080 \nu^{10} + 1181548675553650 \nu^{9} - 735781705952346 \nu^{8} + 7119963817216130 \nu^{7} - 4420040893973322 \nu^{6} + 18295459522336580 \nu^{5} - 11297228352398164 \nu^{4} + 12205236948457855 \nu^{3} - 7477536169138859 \nu^{2} - 277533627658085 \nu + 28012979344985\)\()/ 13607820103360 \)
\(\beta_{14}\)\(=\)\((\)\(-45221309975 \nu^{15} + 40620036785 \nu^{14} - 3479923444575 \nu^{13} + 3124787270393 \nu^{12} - 94261526625960 \nu^{11} + 84613633553592 \nu^{10} - 1181548675553650 \nu^{9} + 1060681057319710 \nu^{8} - 7119963817216130 \nu^{7} + 6398166721254382 \nu^{6} - 18295459522336580 \nu^{5} + 16500140438754044 \nu^{4} - 12205236948457855 \nu^{3} + 11210557588727913 \nu^{2} + 277533627658085 \nu + 6504254102973\)\()/ 13607820103360 \)
\(\beta_{15}\)\(=\)\((\)\(-45221309975 \nu^{15} + 68821388859 \nu^{14} - 3479923444575 \nu^{13} + 5299632441843 \nu^{12} - 94261526625960 \nu^{11} + 143717482361288 \nu^{10} - 1181548675553650 \nu^{9} + 1804651216508106 \nu^{8} - 7119963817216130 \nu^{7} + 10906087984591930 \nu^{6} - 18295459522336580 \nu^{5} + 28188544007262900 \nu^{4} - 12205236948457855 \nu^{3} + 19272909387076131 \nu^{2} + 277533627658085 \nu + 78181554830047\)\()/ 13607820103360 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{14} + \beta_{12} - \beta_{11} - 3 \beta_{10} + \beta_{7} + 5 \beta_{6} - 3 \beta_{5} - \beta_{4} - \beta_{2} - 18\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{12} + 5 \beta_{9} - 19 \beta_{8} + 2 \beta_{7} - 7 \beta_{4} + 5 \beta_{3} + 10 \beta_{2} - 25 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(14 \beta_{15} - 47 \beta_{14} - 14 \beta_{13} - 31 \beta_{12} + 43 \beta_{11} + 79 \beta_{10} - 31 \beta_{7} - 179 \beta_{6} + 117 \beta_{5} + 31 \beta_{4} + 31 \beta_{2} + 400\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-349 \beta_{12} - 193 \beta_{9} + 929 \beta_{8} - 28 \beta_{7} + 313 \beta_{4} - 223 \beta_{3} - 342 \beta_{2} + 834 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(-359 \beta_{15} + 898 \beta_{14} + 325 \beta_{13} + 543 \beta_{12} - 872 \beta_{11} - 1235 \beta_{10} + 543 \beta_{7} + 3226 \beta_{6} - 2129 \beta_{5} - 543 \beta_{4} - 543 \beta_{2} - 5998\)
\(\nu^{7}\)\(=\)\((\)\(13910 \beta_{12} + 6966 \beta_{9} - 36654 \beta_{8} + 52 \beta_{7} - 11798 \beta_{4} + 8514 \beta_{3} + 12044 \beta_{2} - 29577 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(28822 \beta_{15} - 66115 \beta_{14} - 25026 \beta_{13} - 39353 \beta_{12} + 66439 \beta_{11} + 84671 \beta_{10} - 39353 \beta_{7} - 232271 \beta_{6} + 154427 \beta_{5} + 39353 \beta_{4} + 39353 \beta_{2} + 406476\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-520505 \beta_{12} - 250339 \beta_{9} + 1366801 \beta_{8} + 12974 \beta_{7} + 430885 \beta_{4} - 313151 \beta_{3} - 434946 \beta_{2} + 1065837 \beta_{1}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-1081278 \beta_{15} + 2410481 \beta_{14} + 923762 \beta_{13} + 1431797 \beta_{12} - 2458153 \beta_{11} - 3010165 \beta_{10} + 1431797 \beta_{7} + 8384705 \beta_{6} - 5600419 \beta_{5} - 1431797 \beta_{4} - 1431797 \beta_{2} - 14401944\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(19083917 \beta_{12} + 9029821 \beta_{9} - 50060453 \beta_{8} - 679532 \beta_{7} - 15643385 \beta_{4} + 11400387 \beta_{3} + 15800198 \beta_{2} - 38587470 \beta_{1}\)\()/2\)
\(\nu^{12}\)\(=\)\(19844966 \beta_{15} - 43809734 \beta_{14} - 16848718 \beta_{13} - 26024800 \beta_{12} + 44933550 \beta_{11} + 54218408 \beta_{10} - 26024800 \beta_{7} - 151719058 \beta_{6} + 101571480 \beta_{5} + 26024800 \beta_{4} + 26024800 \beta_{2} + 259136963\)
\(\nu^{13}\)\(=\)\((\)\(-694927048 \beta_{12} - 326698624 \beta_{9} + 1822403440 \beta_{8} + 27487912 \beta_{7} + 567460000 \beta_{4} - 413915040 \beta_{3} - 574139472 \beta_{2} + 1399069645 \beta_{1}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-1446079476 \beta_{15} + 3181503425 \beta_{14} + 1224795972 \beta_{13} + 1890378909 \beta_{12} - 3270129841 \beta_{11} - 3924573439 \beta_{10} + 1890378909 \beta_{7} + 10997064301 \beta_{6} - 7369713143 \beta_{5} - 1890378909 \beta_{4} - 1890378909 \beta_{2} - 18751391662\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(25246432839 \beta_{12} + 11839142477 \beta_{9} - 66202882547 \beta_{8} - 1035217254 \beta_{7} - 20585412727 \beta_{4} + 15018968005 \beta_{3} + 20850847362 \beta_{2} - 50751416961 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
0.0431371i
0.0431371i
2.98291i
2.98291i
1.01877i
1.01877i
3.47734i
3.47734i
6.02373i
6.02373i
3.68124i
3.68124i
2.26343i
2.26343i
1.00527i
1.00527i
−1.41421 −5.41949 2.00000 2.23607i 7.66432 8.24199i −2.82843 20.3709 3.16228i
91.2 −1.41421 −5.41949 2.00000 2.23607i 7.66432 8.24199i −2.82843 20.3709 3.16228i
91.3 −1.41421 −0.278523 2.00000 2.23607i 0.393890 8.51262i −2.82843 −8.92243 3.16228i
91.4 −1.41421 −0.278523 2.00000 2.23607i 0.393890 8.51262i −2.82843 −8.92243 3.16228i
91.5 −1.41421 2.34854 2.00000 2.23607i −3.32134 7.61815i −2.82843 −3.48436 3.16228i
91.6 −1.41421 2.34854 2.00000 2.23607i −3.32134 7.61815i −2.82843 −3.48436 3.16228i
91.7 −1.41421 4.76369 2.00000 2.23607i −6.73687 7.05858i −2.82843 13.6927 3.16228i
91.8 −1.41421 4.76369 2.00000 2.23607i −6.73687 7.05858i −2.82843 13.6927 3.16228i
91.9 1.41421 −3.79379 2.00000 2.23607i −5.36524 7.10180i 2.82843 5.39287 3.16228i
91.10 1.41421 −3.79379 2.00000 2.23607i −5.36524 7.10180i 2.82843 5.39287 3.16228i
91.11 1.41421 −3.36596 2.00000 2.23607i −4.76019 1.16919i 2.82843 2.32968 3.16228i
91.12 1.41421 −3.36596 2.00000 2.23607i −4.76019 1.16919i 2.82843 2.32968 3.16228i
91.13 1.41421 1.43837 2.00000 2.23607i 2.03417 10.1866i 2.82843 −6.93108 3.16228i
91.14 1.41421 1.43837 2.00000 2.23607i 2.03417 10.1866i 2.82843 −6.93108 3.16228i
91.15 1.41421 4.30716 2.00000 2.23607i 6.09125 1.47532i 2.82843 9.55167 3.16228i
91.16 1.41421 4.30716 2.00000 2.23607i 6.09125 1.47532i 2.82843 9.55167 3.16228i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.3.d.a 16
3.b odd 2 1 2070.3.c.a 16
4.b odd 2 1 1840.3.k.d 16
5.b even 2 1 1150.3.d.b 16
5.c odd 4 2 1150.3.c.c 32
23.b odd 2 1 inner 230.3.d.a 16
69.c even 2 1 2070.3.c.a 16
92.b even 2 1 1840.3.k.d 16
115.c odd 2 1 1150.3.d.b 16
115.e even 4 2 1150.3.c.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.d.a 16 1.a even 1 1 trivial
230.3.d.a 16 23.b odd 2 1 inner
1150.3.c.c 32 5.c odd 4 2
1150.3.c.c 32 115.e even 4 2
1150.3.d.b 16 5.b even 2 1
1150.3.d.b 16 115.c odd 2 1
1840.3.k.d 16 4.b odd 2 1
1840.3.k.d 16 92.b even 2 1
2070.3.c.a 16 3.b odd 2 1
2070.3.c.a 16 69.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(230, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{8} \)
$3$ \( ( 1336 + 3704 T - 4114 T^{2} - 456 T^{3} + 829 T^{4} + 16 T^{5} - 52 T^{6} + T^{8} )^{2} \)
$5$ \( ( 5 + T^{2} )^{8} \)
$7$ \( 221645107264 + 285091956800 T^{2} + 100475819056 T^{4} + 8103516608 T^{6} + 299518572 T^{8} + 6012916 T^{10} + 67901 T^{12} + 406 T^{14} + T^{16} \)
$11$ \( 21342021632671744 + 1858395528359936 T^{2} + 66311017788672 T^{4} + 1277708760960 T^{6} + 14656413328 T^{8} + 103091648 T^{10} + 435976 T^{12} + 1016 T^{14} + T^{16} \)
$13$ \( ( -343464224 - 219072992 T - 35098468 T^{2} + 212612 T^{3} + 299633 T^{4} + 5112 T^{5} - 898 T^{6} - 12 T^{7} + T^{8} )^{2} \)
$17$ \( 17115756259840000 + 7339457014528000 T^{2} + 359932722681600 T^{4} + 7295803595520 T^{6} + 75620799984 T^{8} + 424642616 T^{10} + 1272985 T^{12} + 1858 T^{14} + T^{16} \)
$19$ \( \)\(11\!\cdots\!24\)\( + 7426573318442000384 T^{2} + 117628148722456832 T^{4} + 821281616987520 T^{6} + 2997759304848 T^{8} + 6125065152 T^{10} + 7019336 T^{12} + 4184 T^{14} + T^{16} \)
$23$ \( \)\(61\!\cdots\!61\)\( - 46371345298154999236 T - 13981530387628964798 T^{2} - 1118018684633959212 T^{3} + 24625359187522136 T^{4} + 2483084721289948 T^{5} + 124487227993214 T^{6} - 4023615463020 T^{7} - 284833923122 T^{8} - 7606078380 T^{9} + 444849854 T^{10} + 16773532 T^{11} + 314456 T^{12} - 26988 T^{13} - 638 T^{14} - 4 T^{15} + T^{16} \)
$29$ \( ( -61767459836 - 2966407060 T + 1296604821 T^{2} + 104891262 T^{3} + 156267 T^{4} - 153376 T^{5} - 2309 T^{6} + 54 T^{7} + T^{8} )^{2} \)
$31$ \( ( 229759835104 - 20345840768 T - 1198716119 T^{2} + 97845314 T^{3} + 2467787 T^{4} - 141380 T^{5} - 2573 T^{6} + 58 T^{7} + T^{8} )^{2} \)
$37$ \( \)\(27\!\cdots\!04\)\( + \)\(21\!\cdots\!52\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{4} + 370281351257423200 T^{6} + 418684864678616 T^{8} + 253723445496 T^{10} + 84388753 T^{12} + 14482 T^{14} + T^{16} \)
$41$ \( ( -212194449184 - 18286974656 T + 3161705921 T^{2} + 211053646 T^{3} - 4997857 T^{4} - 430172 T^{5} - 4161 T^{6} + 78 T^{7} + T^{8} )^{2} \)
$43$ \( \)\(18\!\cdots\!24\)\( + \)\(15\!\cdots\!60\)\( T^{2} + 3954975273611339584 T^{4} + 35443215524837696 T^{6} + 105348919606192 T^{8} + 122794231488 T^{10} + 62675724 T^{12} + 13412 T^{14} + T^{16} \)
$47$ \( ( 13232824136 + 19369312968 T + 3951598214 T^{2} + 166721408 T^{3} - 7463547 T^{4} - 510024 T^{5} - 5764 T^{6} + 64 T^{7} + T^{8} )^{2} \)
$53$ \( \)\(16\!\cdots\!64\)\( + \)\(25\!\cdots\!92\)\( T^{2} + \)\(14\!\cdots\!04\)\( T^{4} + 405693095099781120 T^{6} + 569359307713296 T^{8} + 413528876744 T^{10} + 146364745 T^{12} + 21250 T^{14} + T^{16} \)
$59$ \( ( -42922529206784 + 6511297990656 T + 33913547968 T^{2} - 6885625216 T^{3} + 38371048 T^{4} + 1682848 T^{5} - 14463 T^{6} - 102 T^{7} + T^{8} )^{2} \)
$61$ \( \)\(54\!\cdots\!84\)\( + \)\(31\!\cdots\!96\)\( T^{2} + \)\(71\!\cdots\!48\)\( T^{4} + 8134549162935653888 T^{6} + 4974850707546128 T^{8} + 1677363078528 T^{10} + 306610520 T^{12} + 28128 T^{14} + T^{16} \)
$67$ \( \)\(20\!\cdots\!04\)\( + \)\(10\!\cdots\!72\)\( T^{2} + \)\(22\!\cdots\!76\)\( T^{4} + \)\(16\!\cdots\!64\)\( T^{6} + 60762039405722412 T^{8} + 11205152610740 T^{10} + 1088116077 T^{12} + 52678 T^{14} + T^{16} \)
$71$ \( ( -24390990617024 + 2657914180464 T + 49408197993 T^{2} - 4679164574 T^{3} + 21207483 T^{4} + 1424892 T^{5} - 11133 T^{6} - 118 T^{7} + T^{8} )^{2} \)
$73$ \( ( 1317400530416 + 141690861952 T - 13436169136 T^{2} - 890866888 T^{3} + 59158553 T^{4} - 283856 T^{5} - 14814 T^{6} + 56 T^{7} + T^{8} )^{2} \)
$79$ \( \)\(18\!\cdots\!24\)\( + \)\(18\!\cdots\!48\)\( T^{2} + \)\(30\!\cdots\!52\)\( T^{4} + \)\(15\!\cdots\!36\)\( T^{6} + 368416958397233808 T^{8} + 43217476931904 T^{10} + 2664485320 T^{12} + 82216 T^{14} + T^{16} \)
$83$ \( \)\(39\!\cdots\!64\)\( + \)\(62\!\cdots\!20\)\( T^{2} + \)\(39\!\cdots\!44\)\( T^{4} + \)\(13\!\cdots\!56\)\( T^{6} + 261063543358679972 T^{8} + 29965030291348 T^{10} + 1987717669 T^{12} + 69862 T^{14} + T^{16} \)
$89$ \( \)\(54\!\cdots\!00\)\( + \)\(10\!\cdots\!00\)\( T^{2} + \)\(51\!\cdots\!00\)\( T^{4} + \)\(45\!\cdots\!80\)\( T^{6} + 150481148161409424 T^{8} + 23169014835136 T^{10} + 1794255880 T^{12} + 67928 T^{14} + T^{16} \)
$97$ \( \)\(37\!\cdots\!04\)\( + \)\(70\!\cdots\!28\)\( T^{2} + \)\(22\!\cdots\!32\)\( T^{4} + \)\(14\!\cdots\!56\)\( T^{6} + 360397864589537808 T^{8} + 44029769741824 T^{10} + 2741887320 T^{12} + 83856 T^{14} + T^{16} \)
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