Properties

Label 385.2.i.b
Level $385$
Weight $2$
Character orbit 385.i
Analytic conductor $3.074$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} + 13 x^{10} - 12 x^{9} + 49 x^{8} - 38 x^{7} + 136 x^{6} - 34 x^{5} + 113 x^{4} - 72 x^{3} + 58 x^{2} - 16 x + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} + 13 x^{10} - 12 x^{9} + 49 x^{8} - 38 x^{7} + 136 x^{6} - 34 x^{5} + 113 x^{4} - 72 x^{3} + 58 x^{2} - 16 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -6553 \nu^{11} + 43107 \nu^{10} - 114485 \nu^{9} + 243106 \nu^{8} - 35195 \nu^{7} + 781036 \nu^{6} + 131682 \nu^{5} + 1500914 \nu^{4} + 4540901 \nu^{3} + 957844 \nu^{2} - 270320 \nu + 2595458 \)\()/3522546\)
\(\beta_{3}\)\(=\)\((\)\(-30001 \nu^{11} + 72403 \nu^{10} - 278955 \nu^{9} - 42796 \nu^{8} - 567217 \nu^{7} - 241854 \nu^{6} - 1146430 \nu^{5} - 3780476 \nu^{4} + 4054873 \nu^{3} - 2674456 \nu^{2} + 761616 \nu - 7998392\)\()/3522546\)
\(\beta_{4}\)\(=\)\((\)\(-191509 \nu^{11} + 105135 \nu^{10} - 1167641 \nu^{9} - 3575840 \nu^{8} - 4781177 \nu^{7} - 15072656 \nu^{6} - 12316758 \nu^{5} - 54570352 \nu^{4} - 18869761 \nu^{3} - 37245152 \nu^{2} + 10574416 \nu - 12704206\)\()/3522546\)
\(\beta_{5}\)\(=\)\((\)\(231772 \nu^{11} - 701869 \nu^{10} + 3056143 \nu^{9} - 2895749 \nu^{8} + 11599934 \nu^{7} - 8842531 \nu^{6} + 32302028 \nu^{5} - 7748566 \nu^{4} + 27691150 \nu^{3} - 12146683 \nu^{2} + 14400620 \nu - 456126\)\()/3522546\)
\(\beta_{6}\)\(=\)\((\)\(-286777 \nu^{11} - 2581 \nu^{10} - 1458301 \nu^{9} - 6834828 \nu^{8} - 7679673 \nu^{7} - 27900766 \nu^{6} - 20684078 \nu^{5} - 95358300 \nu^{4} - 41809485 \nu^{3} - 64814040 \nu^{2} + 18395024 \nu - 15641400\)\()/3522546\)
\(\beta_{7}\)\(=\)\((\)\(-307461 \nu^{11} - 116417 \nu^{10} - 1329301 \nu^{9} - 8379056 \nu^{8} - 8602937 \nu^{7} - 33070690 \nu^{6} - 23767030 \nu^{5} - 111924736 \nu^{4} - 57422599 \nu^{3} - 75910016 \nu^{2} + 21540176 \nu - 22553110\)\()/3522546\)
\(\beta_{8}\)\(=\)\((\)\(-456991 \nu^{11} + 1360631 \nu^{10} - 5997801 \nu^{9} + 5548392 \nu^{8} - 23164673 \nu^{7} + 16904026 \nu^{6} - 64735738 \nu^{5} + 13996218 \nu^{4} - 59923201 \nu^{3} + 26858068 \nu^{2} - 32053466 \nu + 8884432\)\()/3522546\)
\(\beta_{9}\)\(=\)\((\)\(-1079063 \nu^{11} + 3264426 \nu^{10} - 13957090 \nu^{9} + 12934241 \nu^{8} - 51572965 \nu^{7} + 41962853 \nu^{6} - 142363680 \nu^{5} + 39639412 \nu^{4} - 106868597 \nu^{3} + 88764749 \nu^{2} - 52447522 \nu + 14390176\)\()/3522546\)
\(\beta_{10}\)\(=\)\((\)\(-1614935 \nu^{11} + 4940073 \nu^{10} - 20886439 \nu^{9} + 19669880 \nu^{8} - 75872827 \nu^{7} + 64266026 \nu^{6} - 206803050 \nu^{5} + 63275110 \nu^{4} - 141699707 \nu^{3} + 139215044 \nu^{2} - 66097342 \nu + 18018352\)\()/3522546\)
\(\beta_{11}\)\(=\)\((\)\(-2602876 \nu^{11} + 6961337 \nu^{10} - 31274831 \nu^{9} + 20333565 \nu^{8} - 117502432 \nu^{7} + 58936257 \nu^{6} - 320870814 \nu^{5} - 21563868 \nu^{4} - 262234142 \nu^{3} + 108229625 \nu^{2} - 74348742 \nu + 3599114\)\()/3522546\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + 2 \beta_{5} + \beta_{2} + \beta_{1} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{6} - 2 \beta_{3} + 6 \beta_{2} - 7\)
\(\nu^{4}\)\(=\)\(2 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} - 8 \beta_{8} - 3 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} - 8 \beta_{3} - 13 \beta_{1}\)
\(\nu^{5}\)\(=\)\(10 \beta_{11} - 13 \beta_{10} + 5 \beta_{9} - 21 \beta_{8} - 10 \beta_{7} - 11 \beta_{5} - 47 \beta_{2} - 47 \beta_{1} + 58\)
\(\nu^{6}\)\(=\)\(-26 \beta_{7} + 39 \beta_{6} - 23 \beta_{4} + 68 \beta_{3} - 126 \beta_{2} + 171\)
\(\nu^{7}\)\(=\)\(-91 \beta_{11} + 130 \beta_{10} - 65 \beta_{9} + 194 \beta_{8} + 130 \beta_{6} + 96 \beta_{5} - 65 \beta_{4} + 194 \beta_{3} + 398 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-259 \beta_{11} + 389 \beta_{10} - 221 \beta_{9} + 592 \beta_{8} + 259 \beta_{7} + 316 \beta_{5} + 1141 \beta_{2} + 1141 \beta_{1} - 1457\)
\(\nu^{9}\)\(=\)\(813 \beta_{7} - 1202 \beta_{6} + 648 \beta_{4} - 1733 \beta_{3} + 3454 \beta_{2} - 4275\)
\(\nu^{10}\)\(=\)\(2381 \beta_{11} - 3583 \beta_{10} + 2015 \beta_{9} - 5187 \beta_{8} - 3583 \beta_{6} - 2512 \beta_{5} + 2015 \beta_{4} - 5187 \beta_{3} - 10122 \beta_{1}\)
\(\nu^{11}\)\(=\)\(7202 \beta_{11} - 10785 \beta_{10} + 5964 \beta_{9} - 15309 \beta_{8} - 7202 \beta_{7} - 7078 \beta_{5} - 30210 \beta_{2} - 30210 \beta_{1} + 37288\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/385\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(276\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
1.48136 + 2.56580i
0.961975 + 1.66619i
0.271923 + 0.470985i
0.176729 + 0.306104i
−0.528400 0.915215i
−0.863590 1.49578i
1.48136 2.56580i
0.961975 1.66619i
0.271923 0.470985i
0.176729 0.306104i
−0.528400 + 0.915215i
−0.863590 + 1.49578i
−0.981363 1.69977i −0.263672 + 0.456693i −0.926145 + 1.60413i 0.500000 + 0.866025i 1.03503 −2.64214 0.138161i −0.289915 1.36095 + 2.35724i 0.981363 1.69977i
221.2 −0.461975 0.800164i 1.05339 1.82453i 0.573158 0.992739i 0.500000 + 0.866025i −1.94657 2.24596 1.39846i −2.90704 −0.719279 1.24583i 0.461975 0.800164i
221.3 0.228077 + 0.395041i 0.0572064 0.0990845i 0.895962 1.55185i 0.500000 + 0.866025i 0.0521899 −1.96234 1.77460i 1.72970 1.49345 + 2.58674i −0.228077 + 0.395041i
221.4 0.323271 + 0.559921i −1.03819 + 1.79820i 0.790992 1.37004i 0.500000 + 0.866025i −1.34247 −0.211269 + 2.63730i 2.31590 −0.655689 1.13569i −0.323271 + 0.559921i
221.5 1.02840 + 1.78124i 0.831041 1.43941i −1.11521 + 1.93160i 0.500000 + 0.866025i 3.41857 −1.49699 + 2.18152i −0.473937 0.118742 + 0.205667i −1.02840 + 1.78124i
221.6 1.36359 + 2.36181i −1.13978 + 1.97415i −2.71876 + 4.70902i 0.500000 + 0.866025i −6.21675 2.56678 0.641581i −9.37471 −1.09818 1.90211i −1.36359 + 2.36181i
331.1 −0.981363 + 1.69977i −0.263672 0.456693i −0.926145 1.60413i 0.500000 0.866025i 1.03503 −2.64214 + 0.138161i −0.289915 1.36095 2.35724i 0.981363 + 1.69977i
331.2 −0.461975 + 0.800164i 1.05339 + 1.82453i 0.573158 + 0.992739i 0.500000 0.866025i −1.94657 2.24596 + 1.39846i −2.90704 −0.719279 + 1.24583i 0.461975 + 0.800164i
331.3 0.228077 0.395041i 0.0572064 + 0.0990845i 0.895962 + 1.55185i 0.500000 0.866025i 0.0521899 −1.96234 + 1.77460i 1.72970 1.49345 2.58674i −0.228077 0.395041i
331.4 0.323271 0.559921i −1.03819 1.79820i 0.790992 + 1.37004i 0.500000 0.866025i −1.34247 −0.211269 2.63730i 2.31590 −0.655689 + 1.13569i −0.323271 0.559921i
331.5 1.02840 1.78124i 0.831041 + 1.43941i −1.11521 1.93160i 0.500000 0.866025i 3.41857 −1.49699 2.18152i −0.473937 0.118742 0.205667i −1.02840 1.78124i
331.6 1.36359 2.36181i −1.13978 1.97415i −2.71876 4.70902i 0.500000 0.866025i −6.21675 2.56678 + 0.641581i −9.37471 −1.09818 + 1.90211i −1.36359 2.36181i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.i.b 12
7.c even 3 1 inner 385.2.i.b 12
7.c even 3 1 2695.2.a.r 6
7.d odd 6 1 2695.2.a.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.i.b 12 1.a even 1 1 trivial
385.2.i.b 12 7.c even 3 1 inner
2695.2.a.q 6 7.d odd 6 1
2695.2.a.r 6 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{12} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 - 27 T + 81 T^{2} - 84 T^{3} + 138 T^{4} - 81 T^{5} + 175 T^{6} - 65 T^{7} + 58 T^{8} - 16 T^{9} + 13 T^{10} - 3 T^{11} + T^{12} \)
$3$ \( 1 - 7 T + 65 T^{2} + 100 T^{3} + 290 T^{4} + 15 T^{5} + 155 T^{6} + 23 T^{7} + 54 T^{8} + 4 T^{9} + 9 T^{10} + T^{11} + T^{12} \)
$5$ \( ( 1 - T + T^{2} )^{6} \)
$7$ \( 117649 + 50421 T - 7203 T^{2} - 12348 T^{3} - 2940 T^{4} + 1113 T^{5} + 745 T^{6} + 159 T^{7} - 60 T^{8} - 36 T^{9} - 3 T^{10} + 3 T^{11} + T^{12} \)
$11$ \( ( 1 - T + T^{2} )^{6} \)
$13$ \( ( -8 + 32 T + 67 T^{2} - 130 T^{3} + 68 T^{4} - 14 T^{5} + T^{6} )^{2} \)
$17$ \( 15968016 - 5178816 T + 4952340 T^{2} + 62424 T^{3} + 628965 T^{4} + 17829 T^{5} + 45514 T^{6} + 2757 T^{7} + 2157 T^{8} + 97 T^{9} + 60 T^{10} + 3 T^{11} + T^{12} \)
$19$ \( 1488400 - 1342000 T + 1081900 T^{2} - 466860 T^{3} + 210905 T^{4} - 63340 T^{5} + 23446 T^{6} - 5366 T^{7} + 1483 T^{8} - 186 T^{9} + 43 T^{10} - 3 T^{11} + T^{12} \)
$23$ \( 4473225 - 856575 T + 4383450 T^{2} - 2419515 T^{3} + 4388445 T^{4} - 1581405 T^{5} + 488584 T^{6} - 76166 T^{7} + 11834 T^{8} - 1056 T^{9} + 147 T^{10} - 10 T^{11} + T^{12} \)
$29$ \( ( -849 - 2586 T - 1632 T^{2} - 244 T^{3} + 54 T^{4} + 16 T^{5} + T^{6} )^{2} \)
$31$ \( 457960000 - 402320000 T + 259173000 T^{2} - 89790400 T^{3} + 25250425 T^{4} - 4127185 T^{5} + 679619 T^{6} - 57610 T^{7} + 12169 T^{8} - 586 T^{9} + 134 T^{10} + 2 T^{11} + T^{12} \)
$37$ \( 38887696 + 20628688 T + 15838124 T^{2} + 5784404 T^{3} + 3375497 T^{4} + 1127676 T^{5} + 380006 T^{6} + 68950 T^{7} + 11541 T^{8} + 914 T^{9} + 111 T^{10} + 5 T^{11} + T^{12} \)
$41$ \( ( -405 + 135 T + 270 T^{2} - 114 T^{3} - 24 T^{4} + 9 T^{5} + T^{6} )^{2} \)
$43$ \( ( 997 + 10 T - 1270 T^{2} - 244 T^{3} + 88 T^{4} + 20 T^{5} + T^{6} )^{2} \)
$47$ \( 99898980624 + 842005152 T + 5200410204 T^{2} - 54518496 T^{3} + 191863677 T^{4} - 1911411 T^{5} + 3429274 T^{6} - 31327 T^{7} + 44595 T^{8} - 213 T^{9} + 248 T^{10} + T^{11} + T^{12} \)
$53$ \( 6110784 - 5873472 T + 6765192 T^{2} - 2562456 T^{3} + 2470593 T^{4} - 1267248 T^{5} + 499099 T^{6} - 129704 T^{7} + 25564 T^{8} - 3544 T^{9} + 367 T^{10} - 24 T^{11} + T^{12} \)
$59$ \( 7932396096 + 5207037696 T + 4587004296 T^{2} - 545926896 T^{3} + 223828209 T^{4} - 10774113 T^{5} + 4068298 T^{6} - 169305 T^{7} + 51745 T^{8} - 827 T^{9} + 286 T^{10} - 7 T^{11} + T^{12} \)
$61$ \( 5272647769 - 3056934687 T + 1918786222 T^{2} - 214106651 T^{3} + 98810173 T^{4} - 14515563 T^{5} + 3571434 T^{6} - 327124 T^{7} + 43164 T^{8} - 2564 T^{9} + 307 T^{10} - 14 T^{11} + T^{12} \)
$67$ \( 195972001 + 120475394 T + 134860893 T^{2} - 31748260 T^{3} + 18477606 T^{4} - 1740068 T^{5} + 676802 T^{6} - 47643 T^{7} + 18257 T^{8} - 553 T^{9} + 152 T^{10} + T^{11} + T^{12} \)
$71$ \( ( 564 + 36888 T + 3885 T^{2} - 1769 T^{3} - 193 T^{4} + 9 T^{5} + T^{6} )^{2} \)
$73$ \( 15523600 - 102518800 T + 751132100 T^{2} + 518257220 T^{3} + 258723985 T^{4} + 59981750 T^{5} + 10534736 T^{6} + 1109534 T^{7} + 97543 T^{8} + 5060 T^{9} + 345 T^{10} + 13 T^{11} + T^{12} \)
$79$ \( 31272385600 + 56588800 T + 2253928200 T^{2} + 129966320 T^{3} + 124712545 T^{4} + 5400755 T^{5} + 2516111 T^{6} + 20534 T^{7} + 34567 T^{8} + 98 T^{9} + 230 T^{10} - 4 T^{11} + T^{12} \)
$83$ \( ( -442647 - 113841 T + 28743 T^{2} + 1961 T^{3} - 335 T^{4} - 8 T^{5} + T^{6} )^{2} \)
$89$ \( 1520289 + 4116987 T + 11407851 T^{2} + 329598 T^{3} + 1410210 T^{4} - 56463 T^{5} + 133891 T^{6} - 7911 T^{7} + 5800 T^{8} - 524 T^{9} + 193 T^{10} - 13 T^{11} + T^{12} \)
$97$ \( ( -316720 - 2760 T + 18255 T^{2} - 114 T^{3} - 282 T^{4} + 3 T^{5} + T^{6} )^{2} \)
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