Properties

Label 55.2.g.a
Level 55
Weight 2
Character orbit 55.g
Analytic conductor 0.439
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.159390625.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 6 x^{6} - 11 x^{5} + 21 x^{4} - 5 x^{3} + 10 x^{2} + 25 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 555 \nu^{7} - 2159 \nu^{6} + 7489 \nu^{5} - 18164 \nu^{4} + 40069 \nu^{3} - 84434 \nu^{2} + 43855 \nu + 375 \)\()/94655\)
\(\beta_{3}\)\(=\)\((\)\( -970 \nu^{7} - 1002 \nu^{6} - 6608 \nu^{5} + 9063 \nu^{4} - 14943 \nu^{3} + 27673 \nu^{2} - 68120 \nu + 35160 \)\()/94655\)
\(\beta_{4}\)\(=\)\((\)\( -1604 \nu^{7} + 4159 \nu^{6} - 12059 \nu^{5} + 28414 \nu^{4} - 81659 \nu^{3} + 38305 \nu^{2} - 13500 \nu - 13875 \)\()/94655\)
\(\beta_{5}\)\(=\)\((\)\( -2052 \nu^{7} + 2252 \nu^{6} - 19912 \nu^{5} + 21007 \nu^{4} - 82042 \nu^{3} + 35785 \nu^{2} - 19395 \nu - 90925 \)\()/94655\)
\(\beta_{6}\)\(=\)\((\)\( -2667 \nu^{7} + 6691 \nu^{6} - 17466 \nu^{5} + 50856 \nu^{4} - 82441 \nu^{3} + 72554 \nu^{2} - 4035 \nu - 12035 \)\()/94655\)
\(\beta_{7}\)\(=\)\((\)\( 4024 \nu^{7} - 1464 \nu^{6} + 21519 \nu^{5} - 26434 \nu^{4} + 59219 \nu^{3} + 22635 \nu^{2} + 54640 \nu + 66675 \)\()/94655\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3 \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{6} + \beta_{5} - 4 \beta_{4} - \beta_{3} - \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(7 \beta_{7} + 7 \beta_{6} + 2 \beta_{5} + 13 \beta_{3} + 13 \beta_{2} - 7\)
\(\nu^{5}\)\(=\)\(-8 \beta_{7} - 11 \beta_{6} - 20 \beta_{5} + 20 \beta_{4} - 11 \beta_{2} + 8 \beta_{1} - 12\)
\(\nu^{6}\)\(=\)\(-19 \beta_{7} - 19 \beta_{4} - 68 \beta_{3} - 36 \beta_{2} - 24 \beta_{1} + 36\)
\(\nu^{7}\)\(=\)\(111 \beta_{7} + 81 \beta_{6} + 111 \beta_{5} - 55 \beta_{4} + 81 \beta_{3} + 148 \beta_{2} - 56 \beta_{1}\)

Character Values

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

\(n\) \(12\) \(46\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2} + \beta_{3} + \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
1.43801 + 1.04478i
−0.628998 0.456994i
−0.762262 + 2.34600i
0.453245 1.39494i
1.43801 1.04478i
−0.628998 + 0.456994i
−0.762262 2.34600i
0.453245 + 1.39494i
−0.579725 1.78421i 1.43801 + 1.04478i −1.22929 + 0.893133i 0.309017 0.951057i 1.03045 3.17141i −3.44479 + 2.50279i −0.729292 0.529862i 0.0492728 + 0.151646i −1.87603
16.2 0.697759 + 2.14748i −0.628998 0.456994i −2.50678 + 1.82128i 0.309017 0.951057i 0.542497 1.66963i −0.100294 + 0.0728678i −2.00678 1.45801i −0.740256 2.27827i 2.25800
26.1 −2.04238 1.48388i −0.762262 + 2.34600i 1.35140 + 4.15918i −0.809017 + 0.587785i 5.03801 3.66033i 0.646930 + 1.99105i 1.85140 5.69802i −2.49563 1.81318i 2.52452
26.2 −0.0756511 0.0549637i 0.453245 1.39494i −0.615332 1.89380i −0.809017 + 0.587785i −0.110960 + 0.0806171i 1.39815 + 4.30308i −0.115332 + 0.354955i 0.686611 + 0.498852i 0.0935099
31.1 −0.579725 + 1.78421i 1.43801 1.04478i −1.22929 0.893133i 0.309017 + 0.951057i 1.03045 + 3.17141i −3.44479 2.50279i −0.729292 + 0.529862i 0.0492728 0.151646i −1.87603
31.2 0.697759 2.14748i −0.628998 + 0.456994i −2.50678 1.82128i 0.309017 + 0.951057i 0.542497 + 1.66963i −0.100294 0.0728678i −2.00678 + 1.45801i −0.740256 + 2.27827i 2.25800
36.1 −2.04238 + 1.48388i −0.762262 2.34600i 1.35140 4.15918i −0.809017 0.587785i 5.03801 + 3.66033i 0.646930 1.99105i 1.85140 + 5.69802i −2.49563 + 1.81318i 2.52452
36.2 −0.0756511 + 0.0549637i 0.453245 + 1.39494i −0.615332 + 1.89380i −0.809017 0.587785i −0.110960 0.0806171i 1.39815 4.30308i −0.115332 0.354955i 0.686611 0.498852i 0.0935099
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
11.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(55, [\chi])\).