Properties

Label 165.2.m.a
Level $165$
Weight $2$
Character orbit 165.m
Analytic conductor $1.318$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 165.m (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.31753163335\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
Defining polynomial: \(x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} - 3 \nu^{5} - 4 \nu^{3} - 7 \nu^{2} - 12 \nu - 7 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 7 \nu^{5} + 20 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} + 6 \nu + 9 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} - 9 \nu^{5} + 12 \nu^{4} - 16 \nu^{3} + 13 \nu^{2} - 10 \nu - 1 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 10 \nu^{6} - 17 \nu^{5} + 8 \nu^{4} - 4 \nu^{3} - 13 \nu^{2} - 8 \nu - 5 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{7} - 12 \nu^{6} + 23 \nu^{5} - 20 \nu^{4} + 16 \nu^{3} + \nu^{2} + 6 \nu - 1 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 18 \nu^{6} - 35 \nu^{5} + 32 \nu^{4} - 28 \nu^{3} - 11 \nu^{2} - 12 \nu - 7 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(3 \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 4 \beta_{1}\)
\(\nu^{5}\)\(=\)\(4 \beta_{7} - 6 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(-16 \beta_{6} - 16 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 7 \beta_{1} - 6\)
\(\nu^{7}\)\(=\)\(-16 \beta_{7} - 51 \beta_{6} - 29 \beta_{5} - 23 \beta_{4} + 29 \beta_{2} - 16\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(67\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\)

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
−0.227943 0.701538i
0.418926 + 1.28932i
−0.227943 + 0.701538i
0.418926 1.28932i
1.69513 1.23158i
−0.386111 + 0.280526i
1.69513 + 1.23158i
−0.386111 0.280526i
0.359123 + 1.10527i −0.809017 0.587785i 0.525387 0.381716i −0.309017 + 0.951057i 0.359123 1.10527i 3.46813 2.51974i 2.49097 + 1.80980i 0.309017 + 0.951057i −1.16215
16.2 0.758911 + 2.33569i −0.809017 0.587785i −3.26145 + 2.36959i −0.309017 + 0.951057i 0.758911 2.33569i −2.65911 + 1.93196i −4.03606 2.93237i 0.309017 + 0.951057i −2.45589
31.1 0.359123 1.10527i −0.809017 + 0.587785i 0.525387 + 0.381716i −0.309017 0.951057i 0.359123 + 1.10527i 3.46813 + 2.51974i 2.49097 1.80980i 0.309017 0.951057i −1.16215
31.2 0.758911 2.33569i −0.809017 + 0.587785i −3.26145 2.36959i −0.309017 0.951057i 0.758911 + 2.33569i −2.65911 1.93196i −4.03606 + 2.93237i 0.309017 0.951057i −2.45589
91.1 −2.24278 + 1.62947i 0.309017 + 0.951057i 1.75683 5.40697i 0.809017 + 0.587785i −2.24278 1.62947i −0.703814 + 2.16612i 3.15700 + 9.71623i −0.809017 + 0.587785i −2.77222
91.2 1.12474 0.817172i 0.309017 + 0.951057i −0.0207616 + 0.0638975i 0.809017 + 0.587785i 1.12474 + 0.817172i 0.394797 1.21506i 0.888090 + 2.73326i −0.809017 + 0.587785i 1.39026
136.1 −2.24278 1.62947i 0.309017 0.951057i 1.75683 + 5.40697i 0.809017 0.587785i −2.24278 + 1.62947i −0.703814 2.16612i 3.15700 9.71623i −0.809017 0.587785i −2.77222
136.2 1.12474 + 0.817172i 0.309017 0.951057i −0.0207616 0.0638975i 0.809017 0.587785i 1.12474 0.817172i 0.394797 + 1.21506i 0.888090 2.73326i −0.809017 0.587785i 1.39026
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.m.a 8
3.b odd 2 1 495.2.n.d 8
5.b even 2 1 825.2.n.k 8
5.c odd 4 2 825.2.bx.h 16
11.c even 5 1 inner 165.2.m.a 8
11.c even 5 1 1815.2.a.x 4
11.d odd 10 1 1815.2.a.o 4
33.f even 10 1 5445.2.a.bv 4
33.h odd 10 1 495.2.n.d 8
33.h odd 10 1 5445.2.a.be 4
55.h odd 10 1 9075.2.a.dj 4
55.j even 10 1 825.2.n.k 8
55.j even 10 1 9075.2.a.cl 4
55.k odd 20 2 825.2.bx.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.a 8 1.a even 1 1 trivial
165.2.m.a 8 11.c even 5 1 inner
495.2.n.d 8 3.b odd 2 1
495.2.n.d 8 33.h odd 10 1
825.2.n.k 8 5.b even 2 1
825.2.n.k 8 55.j even 10 1
825.2.bx.h 16 5.c odd 4 2
825.2.bx.h 16 55.k odd 20 2
1815.2.a.o 4 11.d odd 10 1
1815.2.a.x 4 11.c even 5 1
5445.2.a.be 4 33.h odd 10 1
5445.2.a.bv 4 33.f even 10 1
9075.2.a.cl 4 55.j even 10 1
9075.2.a.dj 4 55.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 3 T_{2}^{6} + 5 T_{2}^{5} + 24 T_{2}^{4} - 85 T_{2}^{3} + 177 T_{2}^{2} - 165 T_{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(165, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 121 - 165 T + 177 T^{2} - 85 T^{3} + 24 T^{4} + 5 T^{5} + 3 T^{6} + T^{8} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$7$ \( 1681 - 164 T + 1027 T^{2} + 253 T^{3} + 180 T^{4} + 7 T^{5} - 3 T^{6} - T^{7} + T^{8} \)
$11$ \( 14641 + 3993 T + 968 T^{2} + 11 T^{3} - 85 T^{4} + T^{5} + 8 T^{6} + 3 T^{7} + T^{8} \)
$13$ \( 1 - 3 T + 11 T^{2} - 39 T^{3} + 94 T^{4} - 87 T^{5} + 39 T^{6} - 6 T^{7} + T^{8} \)
$17$ \( ( 625 + 125 T + 25 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$19$ \( 961 - 3813 T + 38671 T^{2} - 2379 T^{3} + 874 T^{4} + 123 T^{5} + 9 T^{6} - 6 T^{7} + T^{8} \)
$23$ \( ( -1 - 5 T - T^{2} + 5 T^{3} + T^{4} )^{2} \)
$29$ \( 290521 + 18865 T + 34153 T^{2} - 5285 T^{3} + 844 T^{4} + 95 T^{5} + 17 T^{6} + T^{8} \)
$31$ \( 19321 + 4309 T + 2829 T^{2} + 253 T^{3} + 174 T^{4} + T^{5} + 11 T^{6} - 3 T^{7} + T^{8} \)
$37$ \( 185761 + 227999 T + 136645 T^{2} + 49391 T^{3} + 12234 T^{4} + 2111 T^{5} + 255 T^{6} + 19 T^{7} + T^{8} \)
$41$ \( 4289041 + 2091710 T + 708673 T^{2} + 150785 T^{3} + 23634 T^{4} + 2855 T^{5} + 327 T^{6} + 25 T^{7} + T^{8} \)
$43$ \( ( 1861 - 63 T - 92 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$47$ \( 121 - 990 T + 3837 T^{2} - 6815 T^{3} + 10194 T^{4} - 665 T^{5} + 123 T^{6} - 15 T^{7} + T^{8} \)
$53$ \( 1615441 - 557969 T + 343174 T^{2} - 35863 T^{3} + 7329 T^{4} + 379 T^{5} - 14 T^{6} - 7 T^{7} + T^{8} \)
$59$ \( 5285401 - 3253085 T + 1296507 T^{2} - 331435 T^{3} + 58754 T^{4} - 7285 T^{5} + 633 T^{6} - 35 T^{7} + T^{8} \)
$61$ \( 3575881 + 1760521 T + 459192 T^{2} + 41503 T^{3} + 8805 T^{4} - 1043 T^{5} + 242 T^{6} - 21 T^{7} + T^{8} \)
$67$ \( ( -3379 - 1768 T - 136 T^{2} + 13 T^{3} + T^{4} )^{2} \)
$71$ \( 5527201 - 2879975 T + 1218443 T^{2} - 269975 T^{3} + 37544 T^{4} - 3625 T^{5} + 347 T^{6} - 25 T^{7} + T^{8} \)
$73$ \( 84621601 - 18664771 T + 3462030 T^{2} - 293209 T^{3} + 19089 T^{4} + 141 T^{5} + 30 T^{6} - T^{7} + T^{8} \)
$79$ \( 1437601 - 1402830 T + 669978 T^{2} - 183930 T^{3} + 37084 T^{4} - 4980 T^{5} + 487 T^{6} - 30 T^{7} + T^{8} \)
$83$ \( 149352841 - 29978113 T + 2466101 T^{2} + 85701 T^{3} + 18074 T^{4} - 297 T^{5} + 359 T^{6} - 11 T^{7} + T^{8} \)
$89$ \( ( -271 + 132 T + 45 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$97$ \( 625 - 1875 T + 13000 T^{2} - 3375 T^{3} + 2675 T^{4} + 225 T^{5} - 20 T^{6} - 5 T^{7} + T^{8} \)
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