Properties

Label 55.2.j.a
Level $55$
Weight $2$
Character orbit 55.j
Analytic conductor $0.439$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 55.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 7 x^{14} + 25 x^{12} - 57 x^{10} + 194 x^{8} - 303 x^{6} + 235 x^{4} - 33 x^{2} + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 7 x^{14} + 25 x^{12} - 57 x^{10} + 194 x^{8} - 303 x^{6} + 235 x^{4} - 33 x^{2} + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 1829973 \nu^{15} - 24476532 \nu^{13} + 152757677 \nu^{11} - 557238290 \nu^{9} + 1500352252 \nu^{7} - 3810530967 \nu^{5} + 7929326490 \nu^{3} - 7498263355 \nu \)\()/ 2438578648 \)
\(\beta_{3}\)\(=\)\((\)\( -608673 \nu^{14} + 4779625 \nu^{12} - 9344761 \nu^{10} - 2560494 \nu^{8} - 9430964 \nu^{6} + 74558203 \nu^{4} + 726449607 \nu^{2} + 31775447 \)\()/ 609644662 \)
\(\beta_{4}\)\(=\)\((\)\( 2744937 \nu^{14} - 22433610 \nu^{12} + 99837091 \nu^{10} - 276654262 \nu^{8} + 814713604 \nu^{6} - 1632321387 \nu^{4} + 3415114020 \nu^{2} - 771926881 \)\()/ 2438578648 \)
\(\beta_{5}\)\(=\)\((\)\( 2744937 \nu^{15} - 22433610 \nu^{13} + 99837091 \nu^{11} - 276654262 \nu^{9} + 814713604 \nu^{7} - 1632321387 \nu^{5} + 3415114020 \nu^{3} - 771926881 \nu \)\()/ 2438578648 \)
\(\beta_{6}\)\(=\)\((\)\( -2924455 \nu^{14} + 43736988 \nu^{12} - 193036323 \nu^{10} + 488836470 \nu^{8} - 1053314964 \nu^{6} + 3520413317 \nu^{4} - 719547006 \nu^{2} - 236512683 \)\()/ 2438578648 \)
\(\beta_{7}\)\(=\)\((\)\( -1351905 \nu^{14} + 5558819 \nu^{12} - 7678521 \nu^{10} - 13223046 \nu^{8} - 42401020 \nu^{6} - 304413167 \nu^{4} + 664868085 \nu^{2} - 629543387 \)\()/ 609644662 \)
\(\beta_{8}\)\(=\)\((\)\( -6960689 \nu^{14} + 30587176 \nu^{12} - 61040185 \nu^{10} + 38446538 \nu^{8} - 665668084 \nu^{6} - 797037357 \nu^{4} + 1844303058 \nu^{2} - 1738523017 \)\()/ 2438578648 \)
\(\beta_{9}\)\(=\)\((\)\( -3962283 \nu^{14} + 31992860 \nu^{12} - 118526613 \nu^{10} + 271533274 \nu^{8} - 833575532 \nu^{6} + 1781437793 \nu^{4} - 742925482 \nu^{2} - 383811549 \)\()/ 1219289324 \)
\(\beta_{10}\)\(=\)\((\)\( 8234811 \nu^{15} - 67300830 \nu^{13} + 299511273 \nu^{11} - 829962786 \nu^{9} + 2444140812 \nu^{7} - 4896964161 \nu^{5} + 7806763412 \nu^{3} - 2315780643 \nu \)\()/ 2438578648 \)
\(\beta_{11}\)\(=\)\((\)\( 770947 \nu^{14} - 4704418 \nu^{12} + 13758165 \nu^{10} - 28071690 \nu^{8} + 126275508 \nu^{6} - 152673185 \nu^{4} - 21444456 \nu^{2} - 243303327 \)\()/ 221688968 \)
\(\beta_{12}\)\(=\)\((\)\( 10669503 \nu^{15} - 86419330 \nu^{13} + 336890317 \nu^{11} - 819720810 \nu^{9} + 2481864668 \nu^{7} - 5195196973 \nu^{5} + 4900964984 \nu^{3} - 4303783 \nu \)\()/ 2438578648 \)
\(\beta_{13}\)\(=\)\((\)\( -8815096 \nu^{15} + 58503253 \nu^{13} - 198965251 \nu^{11} + 429083674 \nu^{9} - 1573766376 \nu^{7} + 2199079808 \nu^{5} - 1528330963 \nu^{3} + 352179707 \nu \)\()/ 1219289324 \)
\(\beta_{14}\)\(=\)\((\)\(-19639683 \nu^{15} + 162786416 \nu^{13} - 643887411 \nu^{11} + 1627587846 \nu^{9} - 4886486364 \nu^{7} + 10096782513 \nu^{5} - 8290421402 \nu^{3} + 4755604557 \nu\)\()/ 2438578648 \)
\(\beta_{15}\)\(=\)\((\)\( -1496498 \nu^{15} + 9707543 \nu^{13} - 31032777 \nu^{11} + 60570826 \nu^{9} - 228914540 \nu^{7} + 268358314 \nu^{5} + 33071203 \nu^{3} - 172220447 \nu \)\()/ 110844484 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + 2 \beta_{4} - \beta_{3} + 1\)
\(\nu^{3}\)\(=\)\(-\beta_{10} + 3 \beta_{5}\)
\(\nu^{4}\)\(=\)\(6 \beta_{9} - 4 \beta_{8} + \beta_{7} - 4 \beta_{6} + 4 \beta_{4} - \beta_{3} + 1\)
\(\nu^{5}\)\(=\)\(\beta_{15} - 5 \beta_{14} - \beta_{13} - 10 \beta_{12} - \beta_{10} + 15 \beta_{5} - 5 \beta_{2}\)
\(\nu^{6}\)\(=\)\(-7 \beta_{11} - 22 \beta_{8} + 21 \beta_{7} + \beta_{4} - 7 \beta_{3} - 1\)
\(\nu^{7}\)\(=\)\(21 \beta_{15} - 21 \beta_{14} - 36 \beta_{13} - 36 \beta_{12} + 29 \beta_{5} - 28 \beta_{2} + 7 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-85 \beta_{11} - 78 \beta_{9} - 87 \beta_{8} + 85 \beta_{7} + 49 \beta_{6} - 7 \beta_{4} - 49 \beta_{3} - 87\)
\(\nu^{9}\)\(=\)\(85 \beta_{15} - 36 \beta_{14} - 136 \beta_{13} - 45 \beta_{12} + 36 \beta_{10} + 36 \beta_{5} - 121 \beta_{2} - 85 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-342 \beta_{11} - 396 \beta_{9} - 230 \beta_{8} + 166 \beta_{7} + 166 \beta_{6} - 166 \beta_{4} - 529\)
\(\nu^{11}\)\(=\)\(166 \beta_{15} - 220 \beta_{13} + 342 \beta_{10} - 220 \beta_{5} - 342 \beta_{2} - 475 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-728 \beta_{11} - 1653 \beta_{9} + 728 \beta_{6} - 1170 \beta_{4} + 651 \beta_{3} - 1653\)
\(\nu^{13}\)\(=\)\(728 \beta_{14} + 1002 \beta_{12} + 1379 \beta_{10} - 2823 \beta_{5} - 1002 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-4644 \beta_{9} + 3748 \beta_{8} - 3109 \beta_{7} + 2472 \beta_{6} - 3748 \beta_{4} + 3109 \beta_{3} - 3109\)
\(\nu^{15}\)\(=\)\(-3109 \beta_{15} + 5581 \beta_{14} + 4385 \beta_{13} + 8392 \beta_{12} + 3109 \beta_{10} - 13973 \beta_{5} + 5581 \beta_{2} - 1276 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(-1\) \(\beta_{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
1.92464 + 0.625353i
1.17360 + 0.381325i
−1.17360 0.381325i
−1.92464 0.625353i
−0.972539 1.33858i
−0.471815 0.649397i
0.471815 + 0.649397i
0.972539 + 1.33858i
1.92464 0.625353i
1.17360 0.381325i
−1.17360 + 0.381325i
−1.92464 + 0.625353i
−0.972539 + 1.33858i
−0.471815 + 0.649397i
0.471815 0.649397i
0.972539 1.33858i
−1.18949 + 1.63719i 2.49233 + 0.809808i −0.647481 1.99274i −1.06421 1.96658i −4.29042 + 3.11717i −0.918552 + 0.298456i 0.183406 + 0.0595923i 3.12889 + 2.27327i 4.48555 + 0.596911i
4.2 −0.725323 + 0.998322i −0.346168 0.112477i 0.147481 + 0.453901i 0.0238439 + 2.23594i 0.363371 0.264005i 2.45903 0.798988i −2.90731 0.944641i −2.31987 1.68548i −2.24948 1.59798i
4.3 0.725323 0.998322i 0.346168 + 0.112477i 0.147481 + 0.453901i −2.11914 0.713621i 0.363371 0.264005i −2.45903 + 0.798988i 2.90731 + 0.944641i −2.31987 1.68548i −2.24948 + 1.59798i
4.4 1.18949 1.63719i −2.49233 0.809808i −0.647481 1.99274i 1.54147 + 1.61983i −4.29042 + 3.11717i 0.918552 0.298456i −0.183406 0.0595923i 3.12889 + 2.27327i 4.48555 0.596911i
9.1 −1.57360 0.511294i 1.16075 + 1.59764i 0.596764 + 0.433574i 1.09069 + 1.95203i −1.00970 3.10753i 1.31845 1.81468i 1.22769 + 1.68978i −0.278050 + 0.855749i −0.718246 3.62937i
9.2 −0.763412 0.248048i −1.03494 1.42447i −1.09676 0.796845i 1.42953 1.71943i 0.436748 + 1.34417i −0.348029 + 0.479022i 1.58326 + 2.17917i −0.0309674 + 0.0953077i −1.51782 + 0.958043i
9.3 0.763412 + 0.248048i 1.03494 + 1.42447i −1.09676 0.796845i −2.16717 0.550792i 0.436748 + 1.34417i 0.348029 0.479022i −1.58326 2.17917i −0.0309674 + 0.0953077i −1.51782 0.958043i
9.4 1.57360 + 0.511294i −1.16075 1.59764i 0.596764 + 0.433574i 0.264988 + 2.22031i −1.00970 3.10753i −1.31845 + 1.81468i −1.22769 1.68978i −0.278050 + 0.855749i −0.718246 + 3.62937i
14.1 −1.18949 1.63719i 2.49233 0.809808i −0.647481 + 1.99274i −1.06421 + 1.96658i −4.29042 3.11717i −0.918552 0.298456i 0.183406 0.0595923i 3.12889 2.27327i 4.48555 0.596911i
14.2 −0.725323 0.998322i −0.346168 + 0.112477i 0.147481 0.453901i 0.0238439 2.23594i 0.363371 + 0.264005i 2.45903 + 0.798988i −2.90731 + 0.944641i −2.31987 + 1.68548i −2.24948 + 1.59798i
14.3 0.725323 + 0.998322i 0.346168 0.112477i 0.147481 0.453901i −2.11914 + 0.713621i 0.363371 + 0.264005i −2.45903 0.798988i 2.90731 0.944641i −2.31987 + 1.68548i −2.24948 1.59798i
14.4 1.18949 + 1.63719i −2.49233 + 0.809808i −0.647481 + 1.99274i 1.54147 1.61983i −4.29042 3.11717i 0.918552 + 0.298456i −0.183406 + 0.0595923i 3.12889 2.27327i 4.48555 + 0.596911i
49.1 −1.57360 + 0.511294i 1.16075 1.59764i 0.596764 0.433574i 1.09069 1.95203i −1.00970 + 3.10753i 1.31845 + 1.81468i 1.22769 1.68978i −0.278050 0.855749i −0.718246 + 3.62937i
49.2 −0.763412 + 0.248048i −1.03494 + 1.42447i −1.09676 + 0.796845i 1.42953 + 1.71943i 0.436748 1.34417i −0.348029 0.479022i 1.58326 2.17917i −0.0309674 0.0953077i −1.51782 0.958043i
49.3 0.763412 0.248048i 1.03494 1.42447i −1.09676 + 0.796845i −2.16717 + 0.550792i 0.436748 1.34417i 0.348029 + 0.479022i −1.58326 + 2.17917i −0.0309674 0.0953077i −1.51782 + 0.958043i
49.4 1.57360 0.511294i −1.16075 + 1.59764i 0.596764 0.433574i 0.264988 2.22031i −1.00970 + 3.10753i −1.31845 1.81468i −1.22769 + 1.68978i −0.278050 0.855749i −0.718246 3.62937i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.2.j.a 16
3.b odd 2 1 495.2.ba.a 16
4.b odd 2 1 880.2.cd.c 16
5.b even 2 1 inner 55.2.j.a 16
5.c odd 4 2 275.2.h.d 16
11.b odd 2 1 605.2.j.d 16
11.c even 5 1 inner 55.2.j.a 16
11.c even 5 1 605.2.b.g 8
11.c even 5 2 605.2.j.h 16
11.d odd 10 1 605.2.b.f 8
11.d odd 10 1 605.2.j.d 16
11.d odd 10 2 605.2.j.g 16
15.d odd 2 1 495.2.ba.a 16
20.d odd 2 1 880.2.cd.c 16
33.h odd 10 1 495.2.ba.a 16
44.h odd 10 1 880.2.cd.c 16
55.d odd 2 1 605.2.j.d 16
55.h odd 10 1 605.2.b.f 8
55.h odd 10 1 605.2.j.d 16
55.h odd 10 2 605.2.j.g 16
55.j even 10 1 inner 55.2.j.a 16
55.j even 10 1 605.2.b.g 8
55.j even 10 2 605.2.j.h 16
55.k odd 20 2 275.2.h.d 16
55.k odd 20 2 3025.2.a.bl 8
55.l even 20 2 3025.2.a.bk 8
165.o odd 10 1 495.2.ba.a 16
220.n odd 10 1 880.2.cd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.j.a 16 1.a even 1 1 trivial
55.2.j.a 16 5.b even 2 1 inner
55.2.j.a 16 11.c even 5 1 inner
55.2.j.a 16 55.j even 10 1 inner
275.2.h.d 16 5.c odd 4 2
275.2.h.d 16 55.k odd 20 2
495.2.ba.a 16 3.b odd 2 1
495.2.ba.a 16 15.d odd 2 1
495.2.ba.a 16 33.h odd 10 1
495.2.ba.a 16 165.o odd 10 1
605.2.b.f 8 11.d odd 10 1
605.2.b.f 8 55.h odd 10 1
605.2.b.g 8 11.c even 5 1
605.2.b.g 8 55.j even 10 1
605.2.j.d 16 11.b odd 2 1
605.2.j.d 16 11.d odd 10 1
605.2.j.d 16 55.d odd 2 1
605.2.j.d 16 55.h odd 10 1
605.2.j.g 16 11.d odd 10 2
605.2.j.g 16 55.h odd 10 2
605.2.j.h 16 11.c even 5 2
605.2.j.h 16 55.j even 10 2
880.2.cd.c 16 4.b odd 2 1
880.2.cd.c 16 20.d odd 2 1
880.2.cd.c 16 44.h odd 10 1
880.2.cd.c 16 220.n odd 10 1
3025.2.a.bk 8 55.l even 20 2
3025.2.a.bl 8 55.k odd 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(55, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 121 - 308 T^{2} + 345 T^{4} - 138 T^{6} + 159 T^{8} - 62 T^{10} + 15 T^{12} - 2 T^{14} + T^{16} \)
$3$ \( 121 - 1463 T^{2} + 6730 T^{4} + 637 T^{6} + 969 T^{8} - 77 T^{10} + 30 T^{12} - 7 T^{14} + T^{16} \)
$5$ \( 390625 + 156250 T + 156250 T^{2} + 131250 T^{3} + 61875 T^{4} + 42250 T^{5} + 22500 T^{6} + 10040 T^{7} + 5281 T^{8} + 2008 T^{9} + 900 T^{10} + 338 T^{11} + 99 T^{12} + 42 T^{13} + 10 T^{14} + 2 T^{15} + T^{16} \)
$7$ \( 121 - 11 T^{2} + 757 T^{4} - 1553 T^{6} + 1330 T^{8} - 187 T^{10} + 47 T^{12} - 9 T^{14} + T^{16} \)
$11$ \( ( 14641 + 3993 T - 2662 T^{2} - 209 T^{3} + 335 T^{4} - 19 T^{5} - 22 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$13$ \( 47265625 - 24062500 T^{2} + 5390625 T^{4} - 431250 T^{6} + 99375 T^{8} - 7750 T^{10} + 375 T^{12} - 10 T^{14} + T^{16} \)
$17$ \( 1675346761 - 439066837 T^{2} + 53285010 T^{4} - 1993827 T^{6} + 793659 T^{8} - 4603 T^{10} + 2160 T^{12} - 73 T^{14} + T^{16} \)
$19$ \( ( 121 - 198 T + 1433 T^{2} + 879 T^{3} + 280 T^{4} + 9 T^{5} + 3 T^{6} - 3 T^{7} + T^{8} )^{2} \)
$23$ \( ( 26411 + 16366 T^{2} + 2232 T^{4} + 99 T^{6} + T^{8} )^{2} \)
$29$ \( ( 121 - 1056 T + 3565 T^{2} - 579 T^{3} + 344 T^{4} - 99 T^{5} + 25 T^{6} - T^{7} + T^{8} )^{2} \)
$31$ \( ( 1 - 2 T + T^{2} + 4 T^{3} + 9 T^{4} - 8 T^{5} + 39 T^{6} - 4 T^{7} + T^{8} )^{2} \)
$37$ \( 15768841 - 92432967 T^{2} + 205855505 T^{4} + 11467003 T^{6} + 4278334 T^{8} - 127363 T^{10} + 2275 T^{12} - 28 T^{14} + T^{16} \)
$41$ \( ( 3876961 + 1592921 T + 504915 T^{2} + 110009 T^{3} + 19764 T^{4} + 2919 T^{5} + 345 T^{6} + 26 T^{7} + T^{8} )^{2} \)
$43$ \( ( 212531 + 79707 T^{2} + 8319 T^{4} + 173 T^{6} + T^{8} )^{2} \)
$47$ \( 59639012521 + 994427192 T^{2} + 3392111583 T^{4} - 559988696 T^{6} + 38916105 T^{8} - 1250516 T^{10} + 16813 T^{12} + 22 T^{14} + T^{16} \)
$53$ \( 239836452361 - 47877571753 T^{2} + 6382430355 T^{4} - 733116213 T^{6} + 150715374 T^{8} - 395017 T^{10} + 20325 T^{12} - 217 T^{14} + T^{16} \)
$59$ \( ( 1 + 47 T + 891 T^{2} + 3591 T^{3} + 5874 T^{4} + 203 T^{5} + 129 T^{6} - T^{7} + T^{8} )^{2} \)
$61$ \( ( 43681 - 31350 T + 23717 T^{2} - 7100 T^{3} + 1894 T^{4} - 310 T^{5} + 148 T^{6} + 20 T^{7} + T^{8} )^{2} \)
$67$ \( ( 18491 + 46104 T^{2} + 5736 T^{4} + 149 T^{6} + T^{8} )^{2} \)
$71$ \( ( 6305121 - 3593241 T + 1203579 T^{2} - 254907 T^{3} + 36774 T^{4} - 3429 T^{5} + 261 T^{6} - 18 T^{7} + T^{8} )^{2} \)
$73$ \( 1636073786281 - 1265544147219 T^{2} + 371225608252 T^{4} + 3758349303 T^{6} + 216983455 T^{8} - 217893 T^{10} + 18012 T^{12} - 201 T^{14} + T^{16} \)
$79$ \( ( 101761 - 175769 T + 125860 T^{2} - 22301 T^{3} + 7779 T^{4} - 1361 T^{5} + 210 T^{6} - 19 T^{7} + T^{8} )^{2} \)
$83$ \( 45169425961 - 6210793413 T^{2} + 601069813 T^{4} - 53440521 T^{6} + 5270830 T^{8} - 154881 T^{10} + 2643 T^{12} - 33 T^{14} + T^{16} \)
$89$ \( ( 1871 + 486 T - 128 T^{2} - 6 T^{3} + T^{4} )^{4} \)
$97$ \( 25937424601 - 7716919716 T^{2} + 1089510015 T^{4} - 75459714 T^{6} + 14115739 T^{8} - 628914 T^{10} + 11515 T^{12} + 24 T^{14} + T^{16} \)
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