Properties

Label 22.3.d.a
Level 22
Weight 3
Character orbit 22.d
Analytic conductor 0.599
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 22.d (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.599456581593\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
Defining polynomial: \(x^{8} - 2 x^{6} + 4 x^{4} - 8 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{3} + 2 \beta_{2} q^{4} + ( -2 \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{5} + ( -4 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{6} + ( -5 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{7} + 2 \beta_{3} q^{8} + ( -1 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 4 \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{3} + 2 \beta_{2} q^{4} + ( -2 \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{5} + ( -4 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{6} + ( -5 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{7} + 2 \beta_{3} q^{8} + ( -1 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 4 \beta_{5} + \beta_{6} ) q^{9} + ( 4 - 8 \beta_{2} - \beta_{3} + 4 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{10} + ( -1 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{11} + ( 2 - 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{12} + ( 1 - 4 \beta_{1} + 5 \beta_{2} - 4 \beta_{4} - 10 \beta_{5} + 2 \beta_{6} + 10 \beta_{7} ) q^{13} + ( 4 - 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{14} + ( 10 + 3 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} + 10 \beta_{4} - 6 \beta_{5} + 3 \beta_{7} ) q^{15} + 4 \beta_{4} q^{16} + ( 6 + 8 \beta_{1} - \beta_{2} - 6 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} - 5 \beta_{6} - 8 \beta_{7} ) q^{17} + ( -\beta_{1} + 4 \beta_{2} + \beta_{3} - 8 \beta_{4} + 8 \beta_{6} + \beta_{7} ) q^{18} + ( -3 + 11 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} - 4 \beta_{4} + 11 \beta_{5} - 3 \beta_{6} ) q^{19} + ( 2 + 4 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} + 4 \beta_{5} - 8 \beta_{6} - 4 \beta_{7} ) q^{20} + ( -6 + 12 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 9 \beta_{6} - 8 \beta_{7} ) q^{21} + ( 6 - \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 12 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{22} + ( -16 - 16 \beta_{1} + 12 \beta_{3} - 6 \beta_{4} - 8 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} ) q^{23} + ( -4 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{5} + 2 \beta_{7} ) q^{24} + ( -8 - 16 \beta_{1} + 9 \beta_{2} + 4 \beta_{3} - 9 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} ) q^{25} + ( -20 + \beta_{1} + 12 \beta_{2} + 5 \beta_{3} - 20 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{26} + ( -7 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} ) q^{27} + ( -8 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 10 \beta_{6} - 4 \beta_{7} ) q^{28} + ( 17 + 6 \beta_{1} - 18 \beta_{2} - 6 \beta_{3} + 19 \beta_{4} - 36 \beta_{6} - 4 \beta_{7} ) q^{29} + ( -6 + 10 \beta_{1} + 12 \beta_{2} - 9 \beta_{3} + 12 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} ) q^{30} + ( 3 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} + 14 \beta_{6} + 2 \beta_{7} ) q^{31} + 4 \beta_{5} q^{32} + ( -9 - 4 \beta_{1} - 8 \beta_{2} + 16 \beta_{3} - 18 \beta_{4} - 8 \beta_{5} + 19 \beta_{6} + 6 \beta_{7} ) q^{33} + ( 16 + 6 \beta_{1} - \beta_{3} + 4 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - 5 \beta_{7} ) q^{34} + ( 14 + 7 \beta_{1} + 14 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} ) q^{35} + ( -2 + 4 \beta_{3} - 8 \beta_{5} + 2 \beta_{6} + 8 \beta_{7} ) q^{36} + ( 3 + 2 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{37} + ( -3 \beta_{1} + 22 \beta_{2} - 4 \beta_{3} - 10 \beta_{4} - 4 \beta_{5} + 22 \beta_{6} - 3 \beta_{7} ) q^{38} + ( 26 - \beta_{1} - 14 \beta_{3} + 13 \beta_{4} + 14 \beta_{5} - 26 \beta_{6} + \beta_{7} ) q^{39} + ( 8 + 2 \beta_{1} - 2 \beta_{3} - 8 \beta_{4} - 8 \beta_{7} ) q^{40} + ( 25 - 8 \beta_{1} - 12 \beta_{2} + 8 \beta_{3} - 12 \beta_{4} - 8 \beta_{5} + 25 \beta_{6} ) q^{41} + ( 16 - 6 \beta_{1} - 16 \beta_{2} + 12 \beta_{3} - 3 \beta_{5} - 20 \beta_{6} + 9 \beta_{7} ) q^{42} + ( -14 + 28 \beta_{2} + 4 \beta_{3} - 24 \beta_{4} - 10 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{43} + ( 8 + 6 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} + 16 \beta_{4} + 12 \beta_{5} - 12 \beta_{6} + 2 \beta_{7} ) q^{44} + ( -3 + 16 \beta_{1} - 12 \beta_{3} + 28 \beta_{4} + 8 \beta_{5} - 28 \beta_{6} - 4 \beta_{7} ) q^{45} + ( -8 - 16 \beta_{1} - 24 \beta_{2} + 16 \beta_{4} - 6 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} ) q^{46} + ( -13 + 9 \beta_{1} + 6 \beta_{2} + \beta_{3} - 6 \beta_{4} - 11 \beta_{5} + 13 \beta_{6} + 2 \beta_{7} ) q^{47} + ( -4 - 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{48} + ( 10 \beta_{1} - 25 \beta_{2} + 14 \beta_{3} + 22 \beta_{4} + 14 \beta_{5} - 25 \beta_{6} + 10 \beta_{7} ) q^{49} + ( -16 - 8 \beta_{1} - 16 \beta_{2} + 9 \beta_{3} - 8 \beta_{4} - 9 \beta_{5} + 32 \beta_{6} + 8 \beta_{7} ) q^{50} + ( -18 - 11 \beta_{1} + 15 \beta_{2} + 11 \beta_{3} - 12 \beta_{4} + 30 \beta_{6} + 5 \beta_{7} ) q^{51} + ( -4 - 20 \beta_{1} + 6 \beta_{2} + 12 \beta_{3} + 6 \beta_{4} - 20 \beta_{5} - 4 \beta_{6} ) q^{52} + ( -57 + 6 \beta_{1} + 57 \beta_{2} - 12 \beta_{3} + 34 \beta_{6} - 12 \beta_{7} ) q^{53} + ( 14 - 28 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} - 26 \beta_{6} - 4 \beta_{7} ) q^{54} + ( -12 + 13 \beta_{1} + 15 \beta_{2} - 41 \beta_{3} - 35 \beta_{4} + 15 \beta_{5} - 15 \beta_{6} - 25 \beta_{7} ) q^{55} + ( 8 - 8 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 10 \beta_{7} ) q^{56} + ( -37 + 10 \beta_{1} + 11 \beta_{2} - 48 \beta_{4} + 26 \beta_{5} + 24 \beta_{6} - 26 \beta_{7} ) q^{57} + ( 8 + 17 \beta_{1} + 4 \beta_{2} - 18 \beta_{3} - 4 \beta_{4} + 19 \beta_{5} - 8 \beta_{6} - 36 \beta_{7} ) q^{58} + ( 15 - 13 \beta_{1} - 20 \beta_{2} - 33 \beta_{3} + 15 \beta_{4} + 20 \beta_{5} - 10 \beta_{7} ) q^{59} + ( -6 \beta_{1} + 20 \beta_{2} + 12 \beta_{3} - 18 \beta_{4} + 12 \beta_{5} + 20 \beta_{6} - 6 \beta_{7} ) q^{60} + ( 10 - 6 \beta_{1} + 9 \beta_{2} + 34 \beta_{3} + 5 \beta_{4} - 34 \beta_{5} - 19 \beta_{6} + 6 \beta_{7} ) q^{61} + ( -4 + 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} + 12 \beta_{6} + 14 \beta_{7} ) q^{62} + ( -2 - 12 \beta_{1} - \beta_{2} + 13 \beta_{3} - \beta_{4} - 12 \beta_{5} - 2 \beta_{6} ) q^{63} + 8 \beta_{6} q^{64} + ( 21 - 42 \beta_{2} + 18 \beta_{3} + 51 \beta_{4} + 10 \beta_{5} + 9 \beta_{6} + 18 \beta_{7} ) q^{65} + ( -12 - 9 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} + 20 \beta_{4} - 18 \beta_{5} - 4 \beta_{6} + 19 \beta_{7} ) q^{66} + ( 12 + 4 \beta_{1} - 14 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 10 \beta_{7} ) q^{67} + ( 10 + 16 \beta_{1} + 2 \beta_{2} + 8 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{68} + ( 30 + 26 \beta_{1} - 26 \beta_{2} - 4 \beta_{3} + 26 \beta_{4} - 18 \beta_{5} - 30 \beta_{6} - 8 \beta_{7} ) q^{69} + ( -6 + 14 \beta_{1} + 20 \beta_{2} - 6 \beta_{4} + 14 \beta_{5} - 7 \beta_{7} ) q^{70} + ( 19 \beta_{1} - 3 \beta_{2} - 7 \beta_{3} - 33 \beta_{4} - 7 \beta_{5} - 3 \beta_{6} + 19 \beta_{7} ) q^{71} + ( -16 - 2 \beta_{1} + 16 \beta_{2} - 8 \beta_{4} + 2 \beta_{7} ) q^{72} + ( -9 - 22 \beta_{1} + 2 \beta_{2} + 22 \beta_{3} + 5 \beta_{4} + 4 \beta_{6} + 26 \beta_{7} ) q^{73} + ( -4 + 3 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} ) q^{74} + ( 66 - 21 \beta_{1} - 66 \beta_{2} + 42 \beta_{3} - 37 \beta_{5} - 39 \beta_{6} + 5 \beta_{7} ) q^{75} + ( 6 - 12 \beta_{2} + 22 \beta_{3} - 2 \beta_{4} - 10 \beta_{5} - 14 \beta_{6} + 22 \beta_{7} ) q^{76} + ( 39 - 34 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 23 \beta_{4} - 24 \beta_{5} + 2 \beta_{6} + 40 \beta_{7} ) q^{77} + ( -2 + 26 \beta_{1} - 30 \beta_{4} + 13 \beta_{5} + 30 \beta_{6} - 26 \beta_{7} ) q^{78} + ( 59 - 17 \beta_{1} + 15 \beta_{2} + 44 \beta_{4} + 24 \beta_{5} - 22 \beta_{6} - 24 \beta_{7} ) q^{79} + ( 16 + 8 \beta_{1} - 12 \beta_{2} + 12 \beta_{4} - 8 \beta_{5} - 16 \beta_{6} ) q^{80} + ( 22 - 4 \beta_{1} + 24 \beta_{2} - 32 \beta_{3} + 22 \beta_{4} + 28 \beta_{5} - 14 \beta_{7} ) q^{81} + ( 25 \beta_{1} - 16 \beta_{2} - 12 \beta_{3} + 16 \beta_{4} - 12 \beta_{5} - 16 \beta_{6} + 25 \beta_{7} ) q^{82} + ( -30 - 39 \beta_{1} - \beta_{2} + 33 \beta_{3} - 15 \beta_{4} - 33 \beta_{5} + 31 \beta_{6} + 39 \beta_{7} ) q^{83} + ( -18 + 16 \beta_{1} + 6 \beta_{2} - 16 \beta_{3} + 6 \beta_{4} + 12 \beta_{6} - 20 \beta_{7} ) q^{84} + ( -33 - 22 \beta_{1} - 22 \beta_{2} - 26 \beta_{3} - 22 \beta_{4} - 22 \beta_{5} - 33 \beta_{6} ) q^{85} + ( -8 - 14 \beta_{1} + 8 \beta_{2} + 28 \beta_{3} - 24 \beta_{5} - 12 \beta_{6} + 4 \beta_{7} ) q^{86} + ( -32 + 64 \beta_{2} + 8 \beta_{3} - 43 \beta_{4} + 47 \beta_{5} + 21 \beta_{6} + 8 \beta_{7} ) q^{87} + ( -4 + 8 \beta_{1} + 16 \beta_{2} - 10 \beta_{3} - 8 \beta_{4} + 16 \beta_{5} + 28 \beta_{6} - 12 \beta_{7} ) q^{88} + ( -30 + 48 \beta_{1} - 6 \beta_{3} - 66 \beta_{4} + 24 \beta_{5} + 66 \beta_{6} - 42 \beta_{7} ) q^{89} + ( 8 - 3 \beta_{1} + 24 \beta_{2} - 16 \beta_{4} + 28 \beta_{5} + 8 \beta_{6} - 28 \beta_{7} ) q^{90} + ( -47 + 45 \beta_{1} - 3 \beta_{2} - 42 \beta_{3} + 3 \beta_{4} + 39 \beta_{5} + 47 \beta_{6} - 84 \beta_{7} ) q^{91} + ( -12 - 8 \beta_{1} - 20 \beta_{2} - 24 \beta_{3} - 12 \beta_{4} + 16 \beta_{5} - 8 \beta_{7} ) q^{92} + ( -2 \beta_{1} - 22 \beta_{2} - 10 \beta_{3} + 23 \beta_{4} - 10 \beta_{5} - 22 \beta_{6} - 2 \beta_{7} ) q^{93} + ( -4 - 13 \beta_{1} + 22 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 18 \beta_{6} + 13 \beta_{7} ) q^{94} + ( -8 + 29 \beta_{1} - 25 \beta_{2} - 29 \beta_{3} + 58 \beta_{4} - 50 \beta_{6} - 15 \beta_{7} ) q^{95} + ( -4 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} ) q^{96} + ( 1 - 12 \beta_{1} - \beta_{2} + 24 \beta_{3} - 8 \beta_{5} - 12 \beta_{6} + 16 \beta_{7} ) q^{97} + ( -20 + 40 \beta_{2} - 25 \beta_{3} + 8 \beta_{4} + 22 \beta_{5} + 48 \beta_{6} - 25 \beta_{7} ) q^{98} + ( -33 + 11 \beta_{2} + 22 \beta_{3} + 22 \beta_{4} - 33 \beta_{5} - 11 \beta_{6} + 33 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{3} + 4q^{4} + 2q^{5} - 20q^{6} - 30q^{7} - 4q^{9} + O(q^{10}) \) \( 8q - 2q^{3} + 4q^{4} + 2q^{5} - 20q^{6} - 30q^{7} - 4q^{9} - 4q^{11} + 24q^{12} + 30q^{13} + 16q^{14} + 42q^{15} - 8q^{16} + 30q^{17} + 40q^{18} - 30q^{19} - 4q^{20} + 24q^{22} - 104q^{23} - 40q^{24} - 12q^{25} - 96q^{26} - 26q^{27} - 40q^{28} - 10q^{29} - 60q^{30} + 46q^{31} - 14q^{33} + 112q^{34} + 70q^{35} - 12q^{36} + 6q^{37} + 108q^{38} + 130q^{39} + 80q^{40} + 250q^{41} + 56q^{42} - 12q^{44} - 136q^{45} - 160q^{46} - 54q^{47} - 8q^{48} - 144q^{49} - 80q^{50} - 30q^{51} - 40q^{52} - 274q^{53} - 26q^{55} + 48q^{56} - 130q^{57} + 64q^{58} + 50q^{59} + 116q^{60} + 50q^{61} + 20q^{62} - 20q^{63} + 16q^{64} - 136q^{66} + 112q^{67} + 60q^{68} + 76q^{69} + 4q^{70} + 54q^{71} - 80q^{72} - 70q^{73} - 40q^{74} + 318q^{75} + 266q^{77} + 104q^{78} + 370q^{79} + 48q^{80} + 180q^{81} - 96q^{82} - 150q^{83} - 120q^{84} - 330q^{85} - 72q^{86} + 72q^{88} + 24q^{89} + 160q^{90} - 294q^{91} - 112q^{92} - 134q^{93} - 20q^{94} - 330q^{95} - 18q^{97} - 308q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{6} + 4 x^{4} - 8 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{4}\)\(=\)\( \nu^{4} \)\(/4\)
\(\beta_{5}\)\(=\)\( \nu^{5} \)\(/4\)
\(\beta_{6}\)\(=\)\( \nu^{6} \)\(/8\)
\(\beta_{7}\)\(=\)\( \nu^{7} \)\(/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)
\(\nu^{4}\)\(=\)\(4 \beta_{4}\)
\(\nu^{5}\)\(=\)\(4 \beta_{5}\)
\(\nu^{6}\)\(=\)\(8 \beta_{6}\)
\(\nu^{7}\)\(=\)\(8 \beta_{7}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−0.831254 + 1.14412i
0.831254 1.14412i
−1.34500 0.437016i
1.34500 + 0.437016i
−1.34500 + 0.437016i
1.34500 0.437016i
−0.831254 1.14412i
0.831254 + 1.14412i
−0.831254 + 1.14412i −1.32276 + 4.07104i −0.618034 1.90211i 6.27955 4.56236i −3.55822 4.89747i −2.67724 + 0.869888i 2.68999 + 0.874032i −7.54250 5.47994i 10.9771i
7.2 0.831254 1.14412i −0.295274 + 0.908759i −0.618034 1.90211i −2.42545 + 1.76219i 0.794285 + 1.09324i −3.70473 + 1.20374i −2.68999 0.874032i 6.54250 + 4.75340i 4.23984i
13.1 −1.34500 0.437016i 2.48527 1.80565i 1.61803 + 1.17557i −0.399565 1.22973i −4.13178 + 1.34250i −6.48527 + 8.92621i −1.66251 2.28825i 0.135021 0.415553i 1.82860i
13.2 1.34500 + 0.437016i −1.86723 + 1.35662i 1.61803 + 1.17557i −2.45454 7.55429i −3.10429 + 1.00865i −2.13277 + 2.93550i 1.66251 + 2.28825i −1.13502 + 3.49324i 11.2332i
17.1 −1.34500 + 0.437016i 2.48527 + 1.80565i 1.61803 1.17557i −0.399565 + 1.22973i −4.13178 1.34250i −6.48527 8.92621i −1.66251 + 2.28825i 0.135021 + 0.415553i 1.82860i
17.2 1.34500 0.437016i −1.86723 1.35662i 1.61803 1.17557i −2.45454 + 7.55429i −3.10429 1.00865i −2.13277 2.93550i 1.66251 2.28825i −1.13502 3.49324i 11.2332i
19.1 −0.831254 1.14412i −1.32276 4.07104i −0.618034 + 1.90211i 6.27955 + 4.56236i −3.55822 + 4.89747i −2.67724 0.869888i 2.68999 0.874032i −7.54250 + 5.47994i 10.9771i
19.2 0.831254 + 1.14412i −0.295274 0.908759i −0.618034 + 1.90211i −2.42545 1.76219i 0.794285 1.09324i −3.70473 1.20374i −2.68999 + 0.874032i 6.54250 4.75340i 4.23984i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.3.d.a 8
3.b odd 2 1 198.3.j.a 8
4.b odd 2 1 176.3.n.b 8
11.b odd 2 1 242.3.d.c 8
11.c even 5 1 242.3.b.d 8
11.c even 5 1 242.3.d.c 8
11.c even 5 1 242.3.d.d 8
11.c even 5 1 242.3.d.e 8
11.d odd 10 1 inner 22.3.d.a 8
11.d odd 10 1 242.3.b.d 8
11.d odd 10 1 242.3.d.d 8
11.d odd 10 1 242.3.d.e 8
33.f even 10 1 198.3.j.a 8
33.f even 10 1 2178.3.d.l 8
33.h odd 10 1 2178.3.d.l 8
44.g even 10 1 176.3.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.3.d.a 8 1.a even 1 1 trivial
22.3.d.a 8 11.d odd 10 1 inner
176.3.n.b 8 4.b odd 2 1
176.3.n.b 8 44.g even 10 1
198.3.j.a 8 3.b odd 2 1
198.3.j.a 8 33.f even 10 1
242.3.b.d 8 11.c even 5 1
242.3.b.d 8 11.d odd 10 1
242.3.d.c 8 11.b odd 2 1
242.3.d.c 8 11.c even 5 1
242.3.d.d 8 11.c even 5 1
242.3.d.d 8 11.d odd 10 1
242.3.d.e 8 11.c even 5 1
242.3.d.e 8 11.d odd 10 1
2178.3.d.l 8 33.f even 10 1
2178.3.d.l 8 33.h odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(22, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} - 8 T^{6} + 16 T^{8} \)
$3$ \( 1 + 2 T - 5 T^{2} + 20 T^{3} + 45 T^{4} - 284 T^{5} + 837 T^{6} + 3350 T^{7} - 3020 T^{8} + 30150 T^{9} + 67797 T^{10} - 207036 T^{11} + 295245 T^{12} + 1180980 T^{13} - 2657205 T^{14} + 9565938 T^{15} + 43046721 T^{16} \)
$5$ \( 1 - 2 T - 17 T^{2} + 156 T^{3} - 385 T^{4} - 7664 T^{5} + 29213 T^{6} + 102118 T^{7} - 966104 T^{8} + 2552950 T^{9} + 18258125 T^{10} - 119750000 T^{11} - 150390625 T^{12} + 1523437500 T^{13} - 4150390625 T^{14} - 12207031250 T^{15} + 152587890625 T^{16} \)
$7$ \( 1 + 30 T + 571 T^{2} + 8260 T^{3} + 97845 T^{4} + 993580 T^{5} + 8888309 T^{6} + 71468530 T^{7} + 524056964 T^{8} + 3501957970 T^{9} + 21340829909 T^{10} + 116893693420 T^{11} + 564056953845 T^{12} + 2333245556740 T^{13} + 7903374991771 T^{14} + 20346692185470 T^{15} + 33232930569601 T^{16} \)
$11$ \( 1 + 4 T - 484 T^{3} - 7986 T^{4} - 58564 T^{5} + 7086244 T^{7} + 214358881 T^{8} \)
$13$ \( 1 - 30 T + 651 T^{2} - 6800 T^{3} + 48375 T^{4} + 238980 T^{5} - 3389311 T^{6} + 64452890 T^{7} - 51252296 T^{8} + 10892538410 T^{9} - 96802111471 T^{10} + 1153510814820 T^{11} + 39460973628375 T^{12} - 937437744573200 T^{13} + 15167053414735131 T^{14} - 118121291570978670 T^{15} + 665416609183179841 T^{16} \)
$17$ \( 1 - 30 T + 675 T^{2} - 9160 T^{3} + 193599 T^{4} - 3443220 T^{5} + 74881625 T^{6} - 864455750 T^{7} + 15090566536 T^{8} - 249827711750 T^{9} + 6254188201625 T^{10} - 83110960332180 T^{11} + 1350499664820159 T^{12} - 18466504128112840 T^{13} + 393270010130088675 T^{14} - 5051334796782027870 T^{15} + 48661191875666868481 T^{16} \)
$19$ \( 1 + 30 T + 999 T^{2} + 20440 T^{3} + 415005 T^{4} + 6646680 T^{5} + 145579961 T^{6} + 3061994690 T^{7} + 58504536724 T^{8} + 1105380083090 T^{9} + 18972126097481 T^{10} + 312698916325080 T^{11} + 7048263579830205 T^{12} + 125318994309452440 T^{13} + 2211101604147094839 T^{14} + 23970200573486523630 T^{15} + \)\(28\!\cdots\!81\)\( T^{16} \)
$23$ \( ( 1 + 52 T + 2240 T^{2} + 63372 T^{3} + 1720014 T^{4} + 33523788 T^{5} + 626843840 T^{6} + 7697866228 T^{7} + 78310985281 T^{8} )^{2} \)
$29$ \( 1 + 10 T + 1515 T^{2} - 62640 T^{3} - 142561 T^{4} - 111518460 T^{5} + 1681998945 T^{6} - 18301053550 T^{7} + 3803814204136 T^{8} - 15391186035550 T^{9} + 1189645895818545 T^{10} - 66333780730005660 T^{11} - 71315628878133121 T^{12} - 26353101093924590640 T^{13} + \)\(53\!\cdots\!15\)\( T^{14} + \)\(29\!\cdots\!10\)\( T^{15} + \)\(25\!\cdots\!21\)\( T^{16} \)
$31$ \( 1 - 46 T - 815 T^{2} + 85900 T^{3} - 728375 T^{4} + 2317612 T^{5} - 1257533057 T^{6} - 34712738410 T^{7} + 3877854489500 T^{8} - 33358941612010 T^{9} - 1161358186333697 T^{10} + 2056889181129772 T^{11} - 621224509396088375 T^{12} + 70406069851650805900 T^{13} - \)\(64\!\cdots\!15\)\( T^{14} - \)\(34\!\cdots\!66\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} \)
$37$ \( 1 - 6 T - 2421 T^{2} - 14136 T^{3} + 5054271 T^{4} - 6253908 T^{5} - 8619590399 T^{6} + 1373683362 T^{7} + 13772803179432 T^{8} + 1880572522578 T^{9} - 16154500161780239 T^{10} - 16045816915056372 T^{11} + 17753023042048746591 T^{12} - 67974148688498713464 T^{13} - \)\(15\!\cdots\!01\)\( T^{14} - \)\(54\!\cdots\!34\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \)
$41$ \( 1 - 250 T + 27975 T^{2} - 1814420 T^{3} + 77101799 T^{4} - 2779318000 T^{5} + 139016689365 T^{6} - 8505740551010 T^{7} + 413536754728696 T^{8} - 14298149866247810 T^{9} + 392827939156731765 T^{10} - 13202050218887638000 T^{11} + \)\(61\!\cdots\!79\)\( T^{12} - \)\(24\!\cdots\!20\)\( T^{13} + \)\(63\!\cdots\!75\)\( T^{14} - \)\(94\!\cdots\!50\)\( T^{15} + \)\(63\!\cdots\!41\)\( T^{16} \)
$43$ \( 1 - 11160 T^{2} + 57008924 T^{4} - 179309232040 T^{6} + 390998083541766 T^{8} - 613022581807584040 T^{10} + \)\(66\!\cdots\!24\)\( T^{12} - \)\(44\!\cdots\!60\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 + 54 T - 1941 T^{2} - 123836 T^{3} + 6739581 T^{4} + 148592292 T^{5} - 27255791899 T^{6} - 305251818238 T^{7} + 49111755331252 T^{8} - 674301266487742 T^{9} - 132999569869504219 T^{10} + 1601708311697644068 T^{11} + \)\(16\!\cdots\!41\)\( T^{12} - \)\(65\!\cdots\!64\)\( T^{13} - \)\(22\!\cdots\!81\)\( T^{14} + \)\(13\!\cdots\!26\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( 1 + 274 T + 38659 T^{2} + 3972344 T^{3} + 334778231 T^{4} + 23878670812 T^{5} + 1495056085961 T^{6} + 86289965064362 T^{7} + 4702527071102392 T^{8} + 242388511865792858 T^{9} + 11796711640209637241 T^{10} + \)\(52\!\cdots\!48\)\( T^{11} + \)\(20\!\cdots\!91\)\( T^{12} + \)\(69\!\cdots\!56\)\( T^{13} + \)\(18\!\cdots\!19\)\( T^{14} + \)\(37\!\cdots\!06\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \)
$59$ \( 1 - 50 T + 1263 T^{2} + 70200 T^{3} - 2470907 T^{4} + 1076735200 T^{5} - 12586111599 T^{6} + 433998162250 T^{7} + 92677785681540 T^{8} + 1510747602792250 T^{9} - 152510457831370239 T^{10} + 45417265326048863200 T^{11} - \)\(36\!\cdots\!47\)\( T^{12} + \)\(35\!\cdots\!00\)\( T^{13} + \)\(22\!\cdots\!03\)\( T^{14} - \)\(30\!\cdots\!50\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} \)
$61$ \( 1 - 50 T + 10707 T^{2} - 429000 T^{3} + 36206983 T^{4} - 875034700 T^{5} - 30917305831 T^{6} + 2941568049750 T^{7} - 516150507559720 T^{8} + 10945574713119750 T^{9} - 428076100684398871 T^{10} - 45082115322865326700 T^{11} + \)\(69\!\cdots\!23\)\( T^{12} - \)\(30\!\cdots\!00\)\( T^{13} + \)\(28\!\cdots\!47\)\( T^{14} - \)\(49\!\cdots\!50\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} \)
$67$ \( ( 1 - 56 T + 17612 T^{2} - 732008 T^{3} + 117930870 T^{4} - 3285983912 T^{5} + 354901543052 T^{6} - 5065669401464 T^{7} + 406067677556641 T^{8} )^{2} \)
$71$ \( 1 - 54 T - 9075 T^{2} + 1051560 T^{3} - 949335 T^{4} - 6026869752 T^{5} + 391790178883 T^{6} + 11973624475590 T^{7} - 2665107650348580 T^{8} + 60359040981449190 T^{9} + 9956047044707732323 T^{10} - \)\(77\!\cdots\!92\)\( T^{11} - \)\(61\!\cdots\!35\)\( T^{12} + \)\(34\!\cdots\!60\)\( T^{13} - \)\(14\!\cdots\!75\)\( T^{14} - \)\(44\!\cdots\!74\)\( T^{15} + \)\(41\!\cdots\!21\)\( T^{16} \)
$73$ \( 1 + 70 T + 14643 T^{2} + 877280 T^{3} + 86819703 T^{4} + 4301905820 T^{5} + 83475543561 T^{6} + 7156391135710 T^{7} - 896454269626600 T^{8} + 38136408362198590 T^{9} + 2370558603651276201 T^{10} + \)\(65\!\cdots\!80\)\( T^{11} + \)\(70\!\cdots\!43\)\( T^{12} + \)\(37\!\cdots\!20\)\( T^{13} + \)\(33\!\cdots\!03\)\( T^{14} + \)\(85\!\cdots\!30\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \)
$79$ \( 1 - 370 T + 69785 T^{2} - 8224700 T^{3} + 599308969 T^{4} - 15803939780 T^{5} - 2371050090825 T^{6} + 423925649678930 T^{7} - 40585609775890084 T^{8} + 2645719979646202130 T^{9} - 92352593092691106825 T^{10} - \)\(38\!\cdots\!80\)\( T^{11} + \)\(90\!\cdots\!09\)\( T^{12} - \)\(77\!\cdots\!00\)\( T^{13} + \)\(41\!\cdots\!85\)\( T^{14} - \)\(13\!\cdots\!70\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} \)
$83$ \( 1 + 150 T + 16661 T^{2} + 847320 T^{3} + 56225865 T^{4} + 2891415120 T^{5} + 366983457739 T^{6} - 42655302510 T^{7} + 368491437191804 T^{8} - 293852378991390 T^{9} + 17416418739067396219 T^{10} + \)\(94\!\cdots\!80\)\( T^{11} + \)\(12\!\cdots\!65\)\( T^{12} + \)\(13\!\cdots\!80\)\( T^{13} + \)\(17\!\cdots\!21\)\( T^{14} + \)\(11\!\cdots\!50\)\( T^{15} + \)\(50\!\cdots\!81\)\( T^{16} \)
$89$ \( ( 1 - 12 T + 11848 T^{2} + 1140876 T^{3} + 39403230 T^{4} + 9036878796 T^{5} + 743370071368 T^{6} - 5963775491532 T^{7} + 3936588805702081 T^{8} )^{2} \)
$97$ \( 1 + 18 T - 17345 T^{2} - 545820 T^{3} + 224453335 T^{4} + 2169903904 T^{5} - 2647455520643 T^{6} - 4734820613590 T^{7} + 27980585557864280 T^{8} - 44549927153268310 T^{9} - \)\(23\!\cdots\!83\)\( T^{10} + \)\(18\!\cdots\!16\)\( T^{11} + \)\(17\!\cdots\!35\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{13} - \)\(12\!\cdots\!45\)\( T^{14} + \)\(11\!\cdots\!42\)\( T^{15} + \)\(61\!\cdots\!21\)\( T^{16} \)
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