Properties

Label 390.2.x.a
Level $390$
Weight $2$
Character orbit 390.x
Analytic conductor $3.114$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.x (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + 190 x^{3} - 1196 x^{2} - 338 x + 2197\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} - 8 x^{10} + 34 x^{9} + 8 x^{8} - 134 x^{7} + 98 x^{6} + 154 x^{5} + 104 x^{4} + 190 x^{3} - 1196 x^{2} - 338 x + 2197\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-203419 \nu^{11} - 163110633 \nu^{10} + 591783880 \nu^{9} + 97338749 \nu^{8} - 4513282461 \nu^{7} + 6489146722 \nu^{6} + 4655211661 \nu^{5} - 5740233327 \nu^{4} - 8165110694 \nu^{3} - 33307875431 \nu^{2} + 64277898945 \nu + 81183629852\)\()/ 63907274600 \)
\(\beta_{3}\)\(=\)\((\)\(60968787 \nu^{11} - 2097063441 \nu^{10} + 2300362460 \nu^{9} + 17147379373 \nu^{8} - 55879543947 \nu^{7} - 372787906 \nu^{6} + 135156205247 \nu^{5} - 272345430479 \nu^{4} + 414518069862 \nu^{3} - 554153957987 \nu^{2} - 833086363785 \nu + 1269272187404\)\()/ 830794569800 \)
\(\beta_{4}\)\(=\)\((\)\(-120243408 \nu^{11} + 454030419 \nu^{10} + 2051209685 \nu^{9} - 7036618932 \nu^{8} + 2513656023 \nu^{7} + 29967292429 \nu^{6} - 18436259198 \nu^{5} + 35262141211 \nu^{4} + 8309942667 \nu^{3} + 28687118858 \nu^{2} + 98318215515 \nu + 618192439839\)\()/ 830794569800 \)
\(\beta_{5}\)\(=\)\((\)\(16426431 \nu^{11} + 83789417 \nu^{10} - 226795620 \nu^{9} + 267354099 \nu^{8} + 1065744289 \nu^{7} - 511723478 \nu^{6} + 4136894311 \nu^{5} + 1601173623 \nu^{4} + 3964105106 \nu^{3} - 3499453881 \nu^{2} + 44426935995 \nu + 20321135952\)\()/ 63907274600 \)
\(\beta_{6}\)\(=\)\((\)\( 4036 \nu^{11} - 37023 \nu^{10} + 57555 \nu^{9} + 233294 \nu^{8} - 817691 \nu^{7} + 35557 \nu^{6} + 1844066 \nu^{5} - 940887 \nu^{4} + 77961 \nu^{3} - 7481036 \nu^{2} - 4439955 \nu + 11867687 \)\()/11796200\)
\(\beta_{7}\)\(=\)\((\)\(-20638 \nu^{11} + 28887 \nu^{10} - 18833 \nu^{9} - 91862 \nu^{8} - 291763 \nu^{7} - 1390803 \nu^{6} + 3193024 \nu^{5} - 2792113 \nu^{4} - 7615595 \nu^{3} - 12957312 \nu^{2} + 2476201 \nu + 21764327\)\()/47610004\)
\(\beta_{8}\)\(=\)\((\)\(5665538 \nu^{11} - 7145579 \nu^{10} - 9226575 \nu^{9} + 110201552 \nu^{8} - 247897203 \nu^{7} + 32851611 \nu^{6} + 1496635908 \nu^{5} - 1228577871 \nu^{4} - 1963928377 \nu^{3} + 3156976042 \nu^{2} + 8732837145 \nu - 40106339\)\()/ 12781454920 \)
\(\beta_{9}\)\(=\)\((\)\(-680246389 \nu^{11} - 845264698 \nu^{10} + 11881898255 \nu^{9} - 20291622981 \nu^{8} - 56431077566 \nu^{7} + 138431742257 \nu^{6} - 46377269759 \nu^{5} - 85505629812 \nu^{4} - 58320816489 \nu^{3} - 633066289511 \nu^{2} + 550784663520 \nu + 1108736562037\)\()/ 830794569800 \)
\(\beta_{10}\)\(=\)\((\)\(-1202455216 \nu^{11} + 4870077513 \nu^{10} - 1280213355 \nu^{9} - 35058222714 \nu^{8} + 75140023171 \nu^{7} - 22116401667 \nu^{6} - 93428783546 \nu^{5} + 206023103197 \nu^{4} - 574822497041 \nu^{3} + 859741244016 \nu^{2} + 783010043855 \nu - 1182431604497\)\()/ 830794569800 \)
\(\beta_{11}\)\(=\)\((\)\(-78524401 \nu^{11} + 161812093 \nu^{10} + 556127095 \nu^{9} - 2253760279 \nu^{8} - 695335944 \nu^{7} + 7168384238 \nu^{6} - 1904719281 \nu^{5} - 3795941083 \nu^{4} - 26424465301 \nu^{3} - 15605355549 \nu^{2} + 99935137380 \nu + 16572567908\)\()/ 31953637300 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{5} - 3 \beta_{4} + \beta_{3} + 2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{11} + \beta_{10} + 3 \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + 4 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_{1} - 5\)
\(\nu^{4}\)\(=\)\(-7 \beta_{11} + 7 \beta_{10} + 12 \beta_{9} + \beta_{8} + 2 \beta_{7} + 9 \beta_{5} - 6 \beta_{4} + \beta_{2} - 3\)
\(\nu^{5}\)\(=\)\(-4 \beta_{11} - 5 \beta_{10} - 8 \beta_{9} + 4 \beta_{8} + 29 \beta_{7} - 12 \beta_{6} - 7 \beta_{5} + 47 \beta_{4} - \beta_{3} + 12 \beta_{2} - 42\)
\(\nu^{6}\)\(=\)\(-6 \beta_{11} + 8 \beta_{10} - 2 \beta_{9} - 10 \beta_{8} - 24 \beta_{6} + 18 \beta_{5} - 20 \beta_{4} + 6 \beta_{3} + 38 \beta_{2} - 32 \beta_{1} - 7\)
\(\nu^{7}\)\(=\)\(50 \beta_{11} - 70 \beta_{10} - 162 \beta_{9} - 50 \beta_{8} + 54 \beta_{7} - 36 \beta_{6} - 90 \beta_{5} + 202 \beta_{4} - 8 \beta_{3} + 76 \beta_{2} - 11 \beta_{1} - 104\)
\(\nu^{8}\)\(=\)\(81 \beta_{11} + 80 \beta_{10} - 90 \beta_{9} - 155 \beta_{8} - 382 \beta_{7} + 114 \beta_{6} + 113 \beta_{5} - 491 \beta_{4} + 73 \beta_{3} + 64 \beta_{2} - 112 \beta_{1} + 388\)
\(\nu^{9}\)\(=\)\(404 \beta_{11} - 223 \beta_{10} - 627 \beta_{9} - 213 \beta_{8} - 540 \beta_{7} + 459 \beta_{6} - 365 \beta_{5} + 328 \beta_{4} + 7 \beta_{3} - 231 \beta_{2} + 163 \beta_{1} + 249\)
\(\nu^{10}\)\(=\)\(327 \beta_{11} + 649 \beta_{10} + 866 \beta_{9} - 57 \beta_{8} - 2616 \beta_{7} + 1480 \beta_{6} + 1181 \beta_{5} - 3528 \beta_{4} + 248 \beta_{3} - 1421 \beta_{2} - 202 \beta_{1} + 2981\)
\(\nu^{11}\)\(=\)\(1120 \beta_{11} - 2139 \beta_{10} - 614 \beta_{9} + 1672 \beta_{8} + 195 \beta_{7} + 1784 \beta_{6} - 1755 \beta_{5} + 3931 \beta_{4} - 1427 \beta_{3} - 3608 \beta_{2} + 1392 \beta_{1} - 288\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−2.39378 + 0.0429626i
1.75374 + 1.62986i
2.00607 1.30680i
−0.330925 + 1.46916i
−1.44229 + 0.433312i
1.40719 0.536449i
−2.39378 0.0429626i
1.75374 1.62986i
2.00607 + 1.30680i
−0.330925 1.46916i
−1.44229 0.433312i
1.40719 + 0.536449i
−0.500000 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i −2.10012 0.767774i 0.866025 + 0.500000i −0.823063 + 1.42559i 1.00000 0.500000 0.866025i 0.385150 + 2.20265i
49.2 −0.500000 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i −1.40066 + 1.74303i 0.866025 + 0.500000i 0.763837 1.32301i 1.00000 0.500000 0.866025i 2.20984 + 0.341491i
49.3 −0.500000 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 1.26873 1.84128i 0.866025 + 0.500000i −2.17283 + 3.76344i 1.00000 0.500000 0.866025i −2.22896 0.178114i
49.4 −0.500000 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i −0.571769 2.16173i −0.866025 0.500000i −0.603137 + 1.04466i 1.00000 0.500000 0.866025i −1.58623 + 1.57603i
49.5 −0.500000 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i −0.230377 + 2.22417i −0.866025 0.500000i 0.432713 0.749482i 1.00000 0.500000 0.866025i 2.04138 0.912572i
49.6 −0.500000 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 2.03420 0.928463i −0.866025 0.500000i 1.40247 2.42916i 1.00000 0.500000 0.866025i −1.82117 1.29743i
199.1 −0.500000 + 0.866025i −0.866025 0.500000i −0.500000 0.866025i −2.10012 + 0.767774i 0.866025 0.500000i −0.823063 1.42559i 1.00000 0.500000 + 0.866025i 0.385150 2.20265i
199.2 −0.500000 + 0.866025i −0.866025 0.500000i −0.500000 0.866025i −1.40066 1.74303i 0.866025 0.500000i 0.763837 + 1.32301i 1.00000 0.500000 + 0.866025i 2.20984 0.341491i
199.3 −0.500000 + 0.866025i −0.866025 0.500000i −0.500000 0.866025i 1.26873 + 1.84128i 0.866025 0.500000i −2.17283 3.76344i 1.00000 0.500000 + 0.866025i −2.22896 + 0.178114i
199.4 −0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i −0.571769 + 2.16173i −0.866025 + 0.500000i −0.603137 1.04466i 1.00000 0.500000 + 0.866025i −1.58623 1.57603i
199.5 −0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i −0.230377 2.22417i −0.866025 + 0.500000i 0.432713 + 0.749482i 1.00000 0.500000 + 0.866025i 2.04138 + 0.912572i
199.6 −0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 2.03420 + 0.928463i −0.866025 + 0.500000i 1.40247 + 2.42916i 1.00000 0.500000 + 0.866025i −1.82117 + 1.29743i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.x.a 12
3.b odd 2 1 1170.2.bj.d 12
5.b even 2 1 390.2.x.b yes 12
5.c odd 4 1 1950.2.bc.i 12
5.c odd 4 1 1950.2.bc.j 12
13.e even 6 1 390.2.x.b yes 12
15.d odd 2 1 1170.2.bj.c 12
39.h odd 6 1 1170.2.bj.c 12
65.l even 6 1 inner 390.2.x.a 12
65.r odd 12 1 1950.2.bc.i 12
65.r odd 12 1 1950.2.bc.j 12
195.y odd 6 1 1170.2.bj.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.x.a 12 1.a even 1 1 trivial
390.2.x.a 12 65.l even 6 1 inner
390.2.x.b yes 12 5.b even 2 1
390.2.x.b yes 12 13.e even 6 1
1170.2.bj.c 12 15.d odd 2 1
1170.2.bj.c 12 39.h odd 6 1
1170.2.bj.d 12 3.b odd 2 1
1170.2.bj.d 12 195.y odd 6 1
1950.2.bc.i 12 5.c odd 4 1
1950.2.bc.i 12 65.r odd 12 1
1950.2.bc.j 12 5.c odd 4 1
1950.2.bc.j 12 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{6} \)
$3$ \( ( 1 - T^{2} + T^{4} )^{3} \)
$5$ \( 15625 + 6250 T + 4375 T^{2} + 750 T^{3} + 375 T^{4} + 160 T^{5} + 114 T^{6} + 32 T^{7} + 15 T^{8} + 6 T^{9} + 7 T^{10} + 2 T^{11} + T^{12} \)
$7$ \( 1024 - 512 T + 1664 T^{2} - 64 T^{3} + 1648 T^{4} - 112 T^{5} + 708 T^{6} + 20 T^{7} + 205 T^{8} - 6 T^{9} + 19 T^{10} + 2 T^{11} + T^{12} \)
$11$ \( 16 - 96 T - 4 T^{2} + 1176 T^{3} + 2585 T^{4} + 1446 T^{5} - 330 T^{6} - 444 T^{7} + 87 T^{8} + 84 T^{9} - 2 T^{10} - 6 T^{11} + T^{12} \)
$13$ \( 4826809 + 2970344 T + 1028196 T^{2} + 87880 T^{3} - 39208 T^{4} - 26520 T^{5} - 7838 T^{6} - 2040 T^{7} - 232 T^{8} + 40 T^{9} + 36 T^{10} + 8 T^{11} + T^{12} \)
$17$ \( 65536 + 491520 T + 1269760 T^{2} + 307200 T^{3} - 300288 T^{4} - 85248 T^{5} + 75744 T^{6} - 4632 T^{7} - 2695 T^{8} + 234 T^{9} + 95 T^{10} - 18 T^{11} + T^{12} \)
$19$ \( 1982464 - 3649536 T + 837120 T^{2} + 2581632 T^{3} + 672400 T^{4} - 254496 T^{5} - 43876 T^{6} + 14568 T^{7} + 2701 T^{8} - 390 T^{9} - 53 T^{10} + 6 T^{11} + T^{12} \)
$23$ \( 190660864 - 284334336 T + 171279232 T^{2} - 44643456 T^{3} + 1290352 T^{4} + 1617792 T^{5} - 129352 T^{6} - 53748 T^{7} + 7441 T^{8} + 630 T^{9} - 93 T^{10} - 6 T^{11} + T^{12} \)
$29$ \( 21904 - 490768 T + 11075036 T^{2} + 1644412 T^{3} + 1737745 T^{4} - 196610 T^{5} + 148926 T^{6} - 15668 T^{7} + 6703 T^{8} - 960 T^{9} + 190 T^{10} - 14 T^{11} + T^{12} \)
$31$ \( 177209344 + 93061120 T^{2} + 11339776 T^{4} + 592640 T^{6} + 15433 T^{8} + 198 T^{10} + T^{12} \)
$37$ \( 227195329 + 175510012 T + 129327441 T^{2} + 32084244 T^{3} + 11678146 T^{4} + 1319436 T^{5} + 734609 T^{6} + 60444 T^{7} + 14658 T^{8} + 1028 T^{9} + 209 T^{10} + 12 T^{11} + T^{12} \)
$41$ \( 65536 - 491520 T + 1269760 T^{2} - 307200 T^{3} - 300288 T^{4} + 85248 T^{5} + 75744 T^{6} + 4632 T^{7} - 2695 T^{8} - 234 T^{9} + 95 T^{10} + 18 T^{11} + T^{12} \)
$43$ \( 349241344 + 627916800 T + 494054400 T^{2} + 211680000 T^{3} + 51560464 T^{4} + 6267168 T^{5} + 54524 T^{6} - 57864 T^{7} + 6253 T^{8} + 3708 T^{9} + 535 T^{10} + 36 T^{11} + T^{12} \)
$47$ \( ( 5956 + 2352 T - 4283 T^{2} + 1280 T^{3} - 98 T^{4} - 8 T^{5} + T^{6} )^{2} \)
$53$ \( 2473271824 + 989681000 T^{2} + 127872521 T^{4} + 5778420 T^{6} + 87438 T^{8} + 508 T^{10} + T^{12} \)
$59$ \( 4983230464 - 752228352 T - 820266240 T^{2} + 129534336 T^{3} + 157498704 T^{4} + 37281744 T^{5} + 3235868 T^{6} - 77796 T^{7} - 26427 T^{8} + 972 T^{9} + 459 T^{10} + 36 T^{11} + T^{12} \)
$61$ \( 89718784 - 41828352 T + 26851328 T^{2} - 7106048 T^{3} + 3388992 T^{4} - 835296 T^{5} + 256760 T^{6} - 39396 T^{7} + 7561 T^{8} - 762 T^{9} + 135 T^{10} - 10 T^{11} + T^{12} \)
$67$ \( 83759104 + 2928640 T + 25142272 T^{2} - 2632704 T^{3} + 6100480 T^{4} - 320768 T^{5} + 394560 T^{6} + 35776 T^{7} + 18784 T^{8} + 784 T^{9} + 164 T^{10} - 4 T^{11} + T^{12} \)
$71$ \( 4194304 + 25165824 T + 56295424 T^{2} + 35782656 T^{3} - 703488 T^{4} - 6873600 T^{5} + 1150080 T^{6} + 505152 T^{7} + 36080 T^{8} - 2640 T^{9} - 172 T^{10} + 12 T^{11} + T^{12} \)
$73$ \( ( -230528 - 114240 T + 18280 T^{2} + 3068 T^{3} - 263 T^{4} - 14 T^{5} + T^{6} )^{2} \)
$79$ \( ( -29312 - 12576 T + 3892 T^{2} + 788 T^{3} - 179 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$83$ \( ( -6912 + 51840 T - 16560 T^{2} + 144 T^{3} + 360 T^{4} - 36 T^{5} + T^{6} )^{2} \)
$89$ \( 1341001056256 + 39576354816 T - 61874871296 T^{2} - 1837575168 T^{3} + 2363129536 T^{4} + 191431776 T^{5} - 15114712 T^{6} - 1612140 T^{7} + 96433 T^{8} + 7506 T^{9} - 309 T^{10} - 18 T^{11} + T^{12} \)
$97$ \( 415519473664 - 141504348160 T + 88030961664 T^{2} + 14104406016 T^{3} + 3300888832 T^{4} + 348175872 T^{5} + 53039744 T^{6} + 5437824 T^{7} + 522672 T^{8} + 32704 T^{9} + 1640 T^{10} + 48 T^{11} + T^{12} \)
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