Properties

Label 430.2.k.a
Level 430
Weight 2
Character orbit 430.k
Analytic conductor 3.434
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.k (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} + 6 x^{10} - 16 x^{9} + 44 x^{8} - 70 x^{7} + 141 x^{6} - 182 x^{5} + 270 x^{4} - 124 x^{3} + 115 x^{2} + 20 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(2557946812 \nu^{11} - 3680607848 \nu^{10} + 5871717403 \nu^{9} - 19049524293 \nu^{8} + 56458670826 \nu^{7} - 37222435551 \nu^{6} + 162672362140 \nu^{5} + 20545249286 \nu^{4} + 346208442064 \nu^{3} + 381891508664 \nu^{2} + 689715751799 \nu + 99844959362\)\()/ 636866963069 \)
\(\beta_{3}\)\(=\)\((\)\(-6911224142 \nu^{11} + 17158748797 \nu^{10} - 23379170647 \nu^{9} + 77839170172 \nu^{8} - 243794911062 \nu^{7} + 268498284342 \nu^{6} - 546782747252 \nu^{5} + 670525558552 \nu^{4} - 842480645724 \nu^{3} - 247776655383 \nu^{2} - 23289764146 \nu - 54865843774\)\()/ 636866963069 \)
\(\beta_{4}\)\(=\)\((\)\(-7145459624 \nu^{11} + 30723699477 \nu^{10} - 67845292380 \nu^{9} + 160567456914 \nu^{8} - 442734337567 \nu^{7} + 834230530961 \nu^{6} - 1483923368767 \nu^{5} + 2318983407181 \nu^{4} - 3016270574130 \nu^{3} + 2297334961676 \nu^{2} - 1027194968771 \nu - 101355141854\)\()/ 636866963069 \)
\(\beta_{5}\)\(=\)\((\)\(13015530208 \nu^{11} - 40728566222 \nu^{10} + 80752773771 \nu^{9} - 210033213720 \nu^{8} + 587278798280 \nu^{7} - 953922880365 \nu^{6} + 1839377040962 \nu^{5} - 2443720415556 \nu^{4} + 3733081410485 \nu^{3} - 1657412417119 \nu^{2} + 1113309146448 \nu + 81088467803\)\()/ 636866963069 \)
\(\beta_{6}\)\(=\)\((\)\(13027070856 \nu^{11} - 32077498270 \nu^{10} + 53757650992 \nu^{9} - 169355218836 \nu^{8} + 474701804057 \nu^{7} - 619680651977 \nu^{6} + 1386638149935 \nu^{5} - 1573292468109 \nu^{4} + 2401927738676 \nu^{3} - 510387913948 \nu^{2} + 1028246982859 \nu + 47560487414\)\()/ 636866963069 \)
\(\beta_{7}\)\(=\)\((\)\(-47560487414 \nu^{11} + 155708533098 \nu^{10} - 317440422754 \nu^{9} + 814725449616 \nu^{8} - 2262016665052 \nu^{7} + 3803935923037 \nu^{6} - 7325709377351 \nu^{5} + 10042646859283 \nu^{4} - 14414624069889 \nu^{3} + 8299428178012 \nu^{2} - 5979843966558 \nu + 77037234579\)\()/ 636866963069 \)
\(\beta_{8}\)\(=\)\((\)\(54865843774 \nu^{11} - 171508755464 \nu^{10} + 346353811441 \nu^{9} - 901232671031 \nu^{8} + 2491936296228 \nu^{7} - 4084403975242 \nu^{6} + 8004582256476 \nu^{5} - 10532366314120 \nu^{4} + 15484303377532 \nu^{3} - 7645845273700 \nu^{2} + 6061795378627 \nu + 1074027111334\)\()/ 636866963069 \)
\(\beta_{9}\)\(=\)\((\)\(-81088467803 \nu^{11} + 256280933617 \nu^{10} - 527259373040 \nu^{9} + 1378168258619 \nu^{8} - 3777925797052 \nu^{7} + 6263471544490 \nu^{6} - 12387396840588 \nu^{5} + 16597478181108 \nu^{4} - 24337606722366 \nu^{3} + 13788051418057 \nu^{2} - 10982586214464 \nu - 508460209612\)\()/ 636866963069 \)
\(\beta_{10}\)\(=\)\((\)\(-99844959362 \nu^{11} + 302092824898 \nu^{10} - 602750364020 \nu^{9} + 1603391067195 \nu^{8} - 4412227736221 \nu^{7} + 7045605826166 \nu^{6} - 14115361705593 \nu^{5} + 18334454966024 \nu^{4} - 26937593778454 \nu^{3} + 12726983402952 \nu^{2} - 11100278817966 \nu - 1307183435441\)\()/ 636866963069 \)
\(\beta_{11}\)\(=\)\((\)\(-112806277053 \nu^{11} + 341222554101 \nu^{10} - 685432152894 \nu^{9} + 1828243149205 \nu^{8} - 5024576006483 \nu^{7} + 8039260496818 \nu^{6} - 16174397518255 \nu^{5} + 21177816516274 \nu^{4} - 31215179456206 \nu^{3} + 15176823053233 \nu^{2} - 13988917466319 \nu - 1284433711294\)\()/ 636866963069 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{4} + \beta_{3} - 1\)
\(\nu^{3}\)\(=\)\(-\beta_{8} + \beta_{6} + 3 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 1\)
\(\nu^{4}\)\(=\)\(3 \beta_{11} - 3 \beta_{9} + 3 \beta_{8} + 4 \beta_{7} + 5 \beta_{6} - 6 \beta_{4} + 5 \beta_{2} - 4\)
\(\nu^{5}\)\(=\)\(-\beta_{11} + 2 \beta_{10} + 5 \beta_{9} + \beta_{8} - 5 \beta_{7} + 18 \beta_{6} - 11 \beta_{5} - 19 \beta_{4} + 11 \beta_{3} + 19 \beta_{2} - 18 \beta_{1}\)
\(\nu^{6}\)\(=\)\(6 \beta_{11} - 16 \beta_{10} - 27 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 26 \beta_{4} + 35 \beta_{2} - 26 \beta_{1} + 16\)
\(\nu^{7}\)\(=\)\(22 \beta_{10} - 22 \beta_{9} + 29 \beta_{8} + 29 \beta_{7} - 41 \beta_{5} - 41 \beta_{4} - 45 \beta_{3} + 51 \beta_{2} - 51 \beta_{1} - 38\)
\(\nu^{8}\)\(=\)\(-31 \beta_{11} + 31 \beta_{10} + 35 \beta_{9} - 80 \beta_{7} - 22 \beta_{6} - 118 \beta_{5} - 22 \beta_{3} + 118 \beta_{2} - 175 \beta_{1} + 35\)
\(\nu^{9}\)\(=\)\(35 \beta_{11} - 131 \beta_{10} - 57 \beta_{9} - 131 \beta_{8} - 426 \beta_{6} - 74 \beta_{5} + 228 \beta_{4} - 228 \beta_{3} - 74 \beta_{1} + 35\)
\(\nu^{10}\)\(=\)\(-198 \beta_{11} + 322 \beta_{10} - 198 \beta_{9} + 476 \beta_{8} + 322 \beta_{7} - 767 \beta_{6} - 320 \beta_{5} + 767 \beta_{4} - 596 \beta_{3} - 596 \beta_{2} - 476\)
\(\nu^{11}\)\(=\)\(-320 \beta_{11} + 149 \beta_{9} - 149 \beta_{8} - 574 \beta_{7} - 1730 \beta_{6} + 2648 \beta_{4} - 489 \beta_{3} - 1730 \beta_{2} + 489 \beta_{1} + 574\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\) \(87\) \(261\)
\(\chi(n)\) \(1\) \(-\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.202247 + 0.886104i
−0.380695 1.66794i
0.963727 1.20848i
−1.30974 + 1.64236i
0.963727 + 1.20848i
−1.30974 1.64236i
−0.0765242 + 0.0368521i
2.10098 1.01178i
−0.0765242 0.0368521i
2.10098 + 1.01178i
0.202247 0.886104i
−0.380695 + 1.66794i
0.623490 + 0.781831i −1.69017 + 2.11941i −0.222521 + 0.974928i −0.900969 + 0.433884i −2.71083 −1.61896 −0.900969 + 0.433884i −0.967653 4.23957i −0.900969 0.433884i
11.2 0.623490 + 0.781831i −0.0568052 + 0.0712315i −0.222521 + 0.974928i −0.900969 + 0.433884i −0.0911085 0.481895 −0.900969 + 0.433884i 0.665716 + 2.91669i −0.900969 0.433884i
21.1 −0.900969 0.433884i −0.991657 + 0.477557i 0.623490 + 0.781831i −0.222521 0.974928i 1.10066 1.63616 −0.222521 0.974928i −1.11515 + 1.39835i −0.222521 + 0.974928i
21.2 −0.900969 0.433884i 2.29359 1.10454i 0.623490 + 0.781831i −0.222521 0.974928i −2.54570 3.65974 −0.222521 0.974928i 2.17010 2.72123i −0.222521 + 0.974928i
41.1 −0.900969 + 0.433884i −0.991657 0.477557i 0.623490 0.781831i −0.222521 + 0.974928i 1.10066 1.63616 −0.222521 + 0.974928i −1.11515 1.39835i −0.222521 0.974928i
41.2 −0.900969 + 0.433884i 2.29359 + 1.10454i 0.623490 0.781831i −0.222521 + 0.974928i −2.54570 3.65974 −0.222521 + 0.974928i 2.17010 + 2.72123i −0.222521 0.974928i
121.1 −0.222521 + 0.974928i −0.296379 1.29852i −0.900969 0.433884i 0.623490 0.781831i 1.33192 −3.79472 0.623490 0.781831i 1.10459 0.531942i 0.623490 + 0.781831i
121.2 −0.222521 + 0.974928i 0.241421 + 1.05773i −0.900969 0.433884i 0.623490 0.781831i −1.08494 1.63589 0.623490 0.781831i 1.64239 0.790933i 0.623490 + 0.781831i
231.1 −0.222521 0.974928i −0.296379 + 1.29852i −0.900969 + 0.433884i 0.623490 + 0.781831i 1.33192 −3.79472 0.623490 + 0.781831i 1.10459 + 0.531942i 0.623490 0.781831i
231.2 −0.222521 0.974928i 0.241421 1.05773i −0.900969 + 0.433884i 0.623490 + 0.781831i −1.08494 1.63589 0.623490 + 0.781831i 1.64239 + 0.790933i 0.623490 0.781831i
391.1 0.623490 0.781831i −1.69017 2.11941i −0.222521 0.974928i −0.900969 0.433884i −2.71083 −1.61896 −0.900969 0.433884i −0.967653 + 4.23957i −0.900969 + 0.433884i
391.2 0.623490 0.781831i −0.0568052 0.0712315i −0.222521 0.974928i −0.900969 0.433884i −0.0911085 0.481895 −0.900969 0.433884i 0.665716 2.91669i −0.900969 + 0.433884i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.k.a 12
43.e even 7 1 inner 430.2.k.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.k.a 12 1.a even 1 1 trivial
430.2.k.a 12 43.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$3$ \( 1 + T - 6 T^{2} - 3 T^{3} + 19 T^{4} - 15 T^{5} - 26 T^{6} + 148 T^{7} - 103 T^{8} - 578 T^{9} + 737 T^{10} + 825 T^{11} - 2537 T^{12} + 2475 T^{13} + 6633 T^{14} - 15606 T^{15} - 8343 T^{16} + 35964 T^{17} - 18954 T^{18} - 32805 T^{19} + 124659 T^{20} - 59049 T^{21} - 354294 T^{22} + 177147 T^{23} + 531441 T^{24} \)
$5$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$7$ \( ( 1 - 2 T + 26 T^{2} - 35 T^{3} + 311 T^{4} - 323 T^{5} + 2521 T^{6} - 2261 T^{7} + 15239 T^{8} - 12005 T^{9} + 62426 T^{10} - 33614 T^{11} + 117649 T^{12} )^{2} \)
$11$ \( 1 - 3 T - 22 T^{2} + 35 T^{3} + 429 T^{4} - 597 T^{5} - 2700 T^{6} + 812 T^{7} - 9863 T^{8} - 2764 T^{9} + 826623 T^{10} - 223415 T^{11} - 11420881 T^{12} - 2457565 T^{13} + 100021383 T^{14} - 3678884 T^{15} - 144404183 T^{16} + 130773412 T^{17} - 4783214700 T^{18} - 11633841087 T^{19} + 91959959949 T^{20} + 82528169185 T^{21} - 570623341222 T^{22} - 855935011833 T^{23} + 3138428376721 T^{24} \)
$13$ \( ( 1 + 2 T - 9 T^{2} - 44 T^{3} + 113 T^{4} - 126 T^{5} - 1637 T^{6} - 1638 T^{7} + 19097 T^{8} - 96668 T^{9} - 257049 T^{10} + 742586 T^{11} + 4826809 T^{12} )^{2} \)
$17$ \( 1 + 5 T + 6 T^{2} - 50 T^{3} - 256 T^{4} - 61 T^{5} - 2524 T^{6} + 31036 T^{7} + 260014 T^{8} + 878434 T^{9} + 371178 T^{10} - 9204064 T^{11} - 16949749 T^{12} - 156469088 T^{13} + 107270442 T^{14} + 4315746242 T^{15} + 21716629294 T^{16} + 44066681852 T^{17} - 60923224156 T^{18} - 25030659053 T^{19} - 1785793904896 T^{20} - 5929393824850 T^{21} + 12095963402694 T^{22} + 171359481538165 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 + 2 T - 24 T^{2} - 52 T^{3} + 115 T^{4} + 52 T^{5} - 1076 T^{6} - 45484 T^{7} - 66401 T^{8} + 1006610 T^{9} + 2114774 T^{10} - 3943776 T^{11} + 1394383 T^{12} - 74931744 T^{13} + 763433414 T^{14} + 6904337990 T^{15} - 8653444721 T^{16} - 112622886916 T^{17} - 50621367956 T^{18} + 46481330428 T^{19} + 1953109749715 T^{20} - 16779760284508 T^{21} - 147145590187224 T^{22} + 232980517796438 T^{23} + 2213314919066161 T^{24} \)
$23$ \( ( 1 + 8 T - T^{2} - 318 T^{3} - 1135 T^{4} + 2980 T^{5} + 39361 T^{6} + 68540 T^{7} - 600415 T^{8} - 3869106 T^{9} - 279841 T^{10} + 51490744 T^{11} + 148035889 T^{12} )^{2} \)
$29$ \( 1 - 2 T - 118 T^{2} - 61 T^{3} + 6359 T^{4} + 20645 T^{5} - 141187 T^{6} - 1423013 T^{7} - 2112377 T^{8} + 46855717 T^{9} + 277899604 T^{10} - 613421260 T^{11} - 10677579883 T^{12} - 17789216540 T^{13} + 233713566964 T^{14} + 1142764081913 T^{15} - 1494044116937 T^{16} - 29187631671937 T^{17} - 83981320222027 T^{18} + 356123696399305 T^{19} + 3181066940018999 T^{20} - 884935904528009 T^{21} - 49643453529423718 T^{22} - 24401019531411658 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 - 27 T + 253 T^{2} - 122 T^{3} - 18419 T^{4} + 174117 T^{5} - 494429 T^{6} - 4086333 T^{7} + 51528500 T^{8} - 237955904 T^{9} + 67393797 T^{10} + 6991353571 T^{11} - 55877035301 T^{12} + 216731960701 T^{13} + 64765438917 T^{14} - 7088944336064 T^{15} + 47587651848500 T^{16} - 116988244493283 T^{17} - 438807557493149 T^{18} + 4790413831164987 T^{19} - 15709400018625779 T^{20} - 3225633903601862 T^{21} + 207365956606142653 T^{22} - 686028876202930437 T^{23} + 787662783788549761 T^{24} \)
$37$ \( ( 1 + 9 T + 165 T^{2} + 1279 T^{3} + 12099 T^{4} + 82177 T^{5} + 546535 T^{6} + 3040549 T^{7} + 16563531 T^{8} + 64785187 T^{9} + 309236565 T^{10} + 624095613 T^{11} + 2565726409 T^{12} )^{2} \)
$41$ \( 1 - 14 T - 71 T^{2} + 1449 T^{3} + 8085 T^{4} - 145075 T^{5} - 261043 T^{6} + 7055293 T^{7} + 15992640 T^{8} - 312930646 T^{9} - 102994633 T^{10} + 3464301323 T^{11} + 16362105383 T^{12} + 142036354243 T^{13} - 173133978073 T^{14} - 21567493052966 T^{15} + 45191378399040 T^{16} + 817399443921893 T^{17} - 1239981461383363 T^{18} - 28253976283286075 T^{19} + 64558120477443285 T^{20} + 474376422936849489 T^{21} - 953008811020820471 T^{22} - 7704606444027478174 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 - 14 T + 134 T^{2} - 1176 T^{3} + 10648 T^{4} - 73696 T^{5} + 480257 T^{6} - 3168928 T^{7} + 19688152 T^{8} - 93500232 T^{9} + 458119334 T^{10} - 2058118202 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 - 2 T + 95 T^{2} + 486 T^{3} + 4126 T^{4} + 57824 T^{5} + 308156 T^{6} + 3836880 T^{7} + 21611017 T^{8} + 212259114 T^{9} + 1535272757 T^{10} + 8731117266 T^{11} + 89844070544 T^{12} + 410362511502 T^{13} + 3391417520213 T^{14} + 22037377992822 T^{15} + 105454869045577 T^{16} + 879969270458160 T^{17} + 3321679878923324 T^{18} + 29294975317652512 T^{19} + 98245368766425886 T^{20} + 543897409927944762 T^{21} + 4996917562403854655 T^{22} - 4944318430168024606 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 - 2 T - 192 T^{2} + 831 T^{3} + 14431 T^{4} - 117825 T^{5} - 365313 T^{6} + 9752081 T^{7} - 30260439 T^{8} - 531160493 T^{9} + 4601458502 T^{10} + 12612277558 T^{11} - 319965750843 T^{12} + 668450710574 T^{13} + 12925496932118 T^{14} - 79077580716361 T^{15} - 238769418981159 T^{16} + 4078276321570933 T^{17} - 8096929257118377 T^{18} - 138410340051294525 T^{19} + 898469592326350591 T^{20} + 2742103544787572523 T^{21} - 33578394310178505408 T^{22} - 18538071858744383194 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 - 4 T - 111 T^{2} + 373 T^{3} + 10187 T^{4} - 53634 T^{5} - 308525 T^{6} + 3531028 T^{7} - 13046963 T^{8} - 215283643 T^{9} + 3379492827 T^{10} + 4284465948 T^{11} - 240376275189 T^{12} + 252783490932 T^{13} + 11764014530787 T^{14} - 44214739315697 T^{15} - 158094760624643 T^{16} + 2524417717649372 T^{17} - 13013749141589525 T^{18} - 133476333736782246 T^{19} + 1495761667875218027 T^{20} + 3231297440358292247 T^{21} - 56733959616371195511 T^{22} - \)\(12\!\cdots\!36\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 - 16 T - 66 T^{2} + 2397 T^{3} - 3126 T^{4} - 170243 T^{5} + 586028 T^{6} + 11035274 T^{7} - 81228727 T^{8} - 525242399 T^{9} + 7900192381 T^{10} + 10108936107 T^{11} - 527396641733 T^{12} + 616645102527 T^{13} + 29396615849701 T^{14} - 119220044967419 T^{15} - 1124680038674407 T^{16} + 9320351600921474 T^{17} + 30192381946028108 T^{18} - 535029968632723103 T^{19} - 599277060429500406 T^{20} + 28030868184523435977 T^{21} - 47080632169750251666 T^{22} - \)\(69\!\cdots\!76\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 - 9 T - 148 T^{2} + 2384 T^{3} + 2552 T^{4} - 283635 T^{5} + 1159511 T^{6} + 20567317 T^{7} - 175592461 T^{8} - 922488196 T^{9} + 15849512297 T^{10} + 19004289445 T^{11} - 1127214234705 T^{12} + 1273287392815 T^{13} + 71148460701233 T^{14} - 277450317293548 T^{15} - 3538384928298781 T^{16} + 27768451065327919 T^{17} + 104887489167159359 T^{18} - 1719029936175789105 T^{19} + 1036284713124547832 T^{20} + 64860378000767153648 T^{21} - \)\(26\!\cdots\!52\)\( T^{22} - \)\(10\!\cdots\!47\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 - 27 T + 253 T^{2} + 79 T^{3} - 26059 T^{4} + 245632 T^{5} - 262147 T^{6} - 14158898 T^{7} + 170058639 T^{8} - 1116863367 T^{9} + 2185060299 T^{10} + 53608272881 T^{11} - 740263984961 T^{12} + 3806187374551 T^{13} + 11014888967259 T^{14} - 399737684546337 T^{15} + 4321475885562159 T^{16} - 25545899349415198 T^{17} - 33581105129038387 T^{18} + 2234052554745898112 T^{19} - 16827691270733285899 T^{20} + 3622031556757473449 T^{21} + \)\(82\!\cdots\!53\)\( T^{22} - \)\(62\!\cdots\!17\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 - 28 T + 505 T^{2} - 6083 T^{3} + 51071 T^{4} - 242501 T^{5} - 627567 T^{6} + 22785623 T^{7} - 200315116 T^{8} + 258559266 T^{9} + 17601898277 T^{10} - 299053047825 T^{11} + 3088299490395 T^{12} - 21830872491225 T^{13} + 93800515918133 T^{14} + 100583949981522 T^{15} - 5688596940110956 T^{16} + 47236227770107439 T^{17} - 94972366389508863 T^{18} - 2679005188279541597 T^{19} + 41186723353122610751 T^{20} - \)\(35\!\cdots\!79\)\( T^{21} + \)\(21\!\cdots\!45\)\( T^{22} - \)\(87\!\cdots\!56\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( ( 1 - 5 T + 368 T^{2} - 975 T^{3} + 57067 T^{4} - 71424 T^{5} + 5407867 T^{6} - 5642496 T^{7} + 356155147 T^{8} - 480713025 T^{9} + 14333629808 T^{10} - 15385281995 T^{11} + 243087455521 T^{12} )^{2} \)
$83$ \( 1 + 31 T + 452 T^{2} + 2892 T^{3} - 8490 T^{4} - 345229 T^{5} - 2667420 T^{6} - 6265342 T^{7} + 19930908 T^{8} - 246940166 T^{9} + 1158607142 T^{10} + 136753954064 T^{11} + 2035262795035 T^{12} + 11350578187312 T^{13} + 7981644601238 T^{14} - 141197176696642 T^{15} + 945887429685468 T^{16} - 24679436780294906 T^{17} - 872087290731937980 T^{18} - 9368151747097939583 T^{19} - 19121961050860458090 T^{20} + \)\(54\!\cdots\!76\)\( T^{21} + \)\(70\!\cdots\!48\)\( T^{22} + \)\(39\!\cdots\!77\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 + 38 T + 521 T^{2} + 2967 T^{3} + 18241 T^{4} + 334174 T^{5} + 2136543 T^{6} - 2477584 T^{7} + 111400957 T^{8} + 2189866865 T^{9} + 11128069385 T^{10} + 269872470160 T^{11} + 4501464216545 T^{12} + 24018649844240 T^{13} + 88145437598585 T^{14} + 1543788253952185 T^{15} + 6989545691724637 T^{16} - 13834976345891216 T^{17} + 1061821898333687823 T^{18} + 14780962107378508046 T^{19} + 71807316404811659521 T^{20} + \)\(10\!\cdots\!03\)\( T^{21} + \)\(16\!\cdots\!21\)\( T^{22} + \)\(10\!\cdots\!82\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 + 15 T - 61 T^{2} - 908 T^{3} + 14954 T^{4} + 129879 T^{5} + 769775 T^{6} + 22467776 T^{7} + 57832976 T^{8} - 1013973842 T^{9} + 16680785489 T^{10} + 212073098570 T^{11} + 697406047257 T^{12} + 20571090561290 T^{13} + 156949510666001 T^{14} - 925426548299666 T^{15} + 5119911783370256 T^{16} + 192938437330058432 T^{17} + 641201025094220975 T^{18} + 10494000389732838327 T^{19} + \)\(11\!\cdots\!94\)\( T^{20} - \)\(69\!\cdots\!36\)\( T^{21} - \)\(44\!\cdots\!89\)\( T^{22} + \)\(10\!\cdots\!95\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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