Properties

Label 20.3.b.a
Level 20
Weight 3
Character orbit 20.b
Analytic conductor 0.545
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 20.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.544960528721\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + \beta_{3} q^{3} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( -1 - \beta_{1} + \beta_{2} ) q^{5} + ( 2 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{6} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{7} + ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{8} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + \beta_{3} q^{3} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( -1 - \beta_{1} + \beta_{2} ) q^{5} + ( 2 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{6} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{7} + ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{8} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{9} + ( -2 + \beta_{1} + \beta_{3} ) q^{10} + ( -4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{11} + ( 8 - 4 \beta_{1} ) q^{12} + ( -6 - 2 \beta_{1} + 2 \beta_{2} ) q^{13} + ( 10 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{14} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{15} + 8 \beta_{1} q^{16} + ( 2 + 8 \beta_{1} - 8 \beta_{2} ) q^{17} + ( 4 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{18} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{19} + ( 6 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{20} + ( -10 \beta_{1} + 10 \beta_{2} ) q^{21} + ( -28 - 10 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{22} + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{23} + ( -16 - 8 \beta_{2} ) q^{24} + 5 q^{25} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{26} + ( 4 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{27} + ( 8 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{28} + ( -6 - 4 \beta_{1} + 4 \beta_{2} ) q^{29} + ( -10 + \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{30} + ( -4 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} ) q^{31} + 32 q^{32} + ( 32 + 12 \beta_{1} - 12 \beta_{2} ) q^{33} + ( 16 - 8 \beta_{1} + 6 \beta_{2} - 8 \beta_{3} ) q^{34} -5 \beta_{3} q^{35} + ( -10 - 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{36} + ( -6 - 10 \beta_{1} + 10 \beta_{2} ) q^{37} + ( 24 + 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{38} + ( -4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{39} + ( 12 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{40} + ( -22 + 6 \beta_{1} - 6 \beta_{2} ) q^{41} + ( -20 + 10 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{42} + ( -4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{43} + ( -32 + 24 \beta_{2} + 8 \beta_{3} ) q^{44} + ( -9 + \beta_{1} - \beta_{2} ) q^{45} + ( 6 - 7 \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{46} + ( -6 \beta_{1} - 6 \beta_{2} - 13 \beta_{3} ) q^{47} + ( -16 - 8 \beta_{1} + 24 \beta_{2} - 8 \beta_{3} ) q^{48} + ( 9 + 10 \beta_{1} - 10 \beta_{2} ) q^{49} -5 \beta_{2} q^{50} + ( 16 \beta_{1} + 16 \beta_{2} - 14 \beta_{3} ) q^{51} + ( 20 + 10 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{52} + ( -54 - 10 \beta_{1} + 10 \beta_{2} ) q^{53} + ( 36 + 22 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} ) q^{54} + ( 8 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} ) q^{55} + ( -24 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{56} + ( -16 - 16 \beta_{1} + 16 \beta_{2} ) q^{57} + ( -8 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{58} + ( -12 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{59} + ( -8 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} ) q^{60} + ( 58 + 26 \beta_{1} - 26 \beta_{2} ) q^{61} + ( -4 + 26 \beta_{1} - 6 \beta_{2} - 14 \beta_{3} ) q^{62} + ( -2 \beta_{1} - 2 \beta_{2} + 11 \beta_{3} ) q^{63} -32 \beta_{2} q^{64} + ( 14 + 4 \beta_{1} - 4 \beta_{2} ) q^{65} + ( 24 - 12 \beta_{1} - 20 \beta_{2} - 12 \beta_{3} ) q^{66} + ( -20 \beta_{1} - 20 \beta_{2} + 11 \beta_{3} ) q^{67} + ( -36 - 18 \beta_{1} - 14 \beta_{2} + 14 \beta_{3} ) q^{68} + ( 16 - 14 \beta_{1} + 14 \beta_{2} ) q^{69} + ( -10 - 15 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{70} + ( 20 \beta_{1} + 20 \beta_{2} + 2 \beta_{3} ) q^{71} + ( -20 + 6 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} ) q^{72} + ( 34 - 32 \beta_{1} + 32 \beta_{2} ) q^{73} + ( -20 + 10 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} ) q^{74} + 5 \beta_{3} q^{75} + ( 16 + 8 \beta_{1} - 24 \beta_{2} - 8 \beta_{3} ) q^{76} + ( 80 + 20 \beta_{1} - 20 \beta_{2} ) q^{77} + ( -28 - 10 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{78} + ( 16 \beta_{1} + 16 \beta_{2} - 20 \beta_{3} ) q^{79} + ( -16 - 8 \beta_{2} - 8 \beta_{3} ) q^{80} + ( -55 + 14 \beta_{1} - 14 \beta_{2} ) q^{81} + ( 12 - 6 \beta_{1} + 28 \beta_{2} - 6 \beta_{3} ) q^{82} + ( 12 \beta_{1} + 12 \beta_{2} + 13 \beta_{3} ) q^{83} + ( 40 + 20 \beta_{1} + 20 \beta_{2} - 20 \beta_{3} ) q^{84} + ( -34 + 6 \beta_{1} - 6 \beta_{2} ) q^{85} + ( -30 - 13 \beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{86} + ( -8 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{87} + ( 64 + 48 \beta_{1} + 16 \beta_{3} ) q^{88} + ( -62 - 40 \beta_{1} + 40 \beta_{2} ) q^{89} + ( 2 - \beta_{1} + 10 \beta_{2} - \beta_{3} ) q^{90} + ( -8 \beta_{1} - 8 \beta_{2} - 6 \beta_{3} ) q^{91} + ( -16 + 16 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{92} + ( -64 + 36 \beta_{1} - 36 \beta_{2} ) q^{93} + ( -62 - 45 \beta_{1} + 19 \beta_{2} + 7 \beta_{3} ) q^{94} + ( -4 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} ) q^{95} + 32 \beta_{3} q^{96} + ( -78 - 12 \beta_{1} + 12 \beta_{2} ) q^{97} + ( 20 - 10 \beta_{1} + \beta_{2} - 10 \beta_{3} ) q^{98} + ( -12 \beta_{1} - 12 \beta_{2} - 10 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 4q^{4} - 8q^{8} - 4q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - 4q^{4} - 8q^{8} - 4q^{9} - 10q^{10} + 40q^{12} - 16q^{13} + 40q^{14} - 16q^{16} - 24q^{17} + 22q^{18} + 20q^{20} + 40q^{21} - 80q^{22} - 80q^{24} + 20q^{25} - 12q^{26} - 40q^{28} - 8q^{29} - 40q^{30} + 128q^{32} + 80q^{33} + 92q^{34} - 36q^{36} + 16q^{37} + 80q^{38} + 40q^{40} - 112q^{41} - 120q^{42} - 80q^{44} - 40q^{45} + 40q^{46} - 4q^{49} - 10q^{50} + 56q^{52} - 176q^{53} + 80q^{54} + 80q^{56} - 36q^{58} + 40q^{60} + 128q^{61} - 80q^{62} - 64q^{64} + 40q^{65} + 80q^{66} - 136q^{68} + 120q^{69} - 72q^{72} + 264q^{73} - 108q^{74} + 240q^{77} - 80q^{78} - 80q^{80} - 276q^{81} + 116q^{82} + 160q^{84} - 160q^{85} - 80q^{86} + 160q^{88} - 88q^{89} + 30q^{90} - 120q^{92} - 400q^{93} - 120q^{94} - 264q^{97} + 102q^{98} + O(q^{100}) \)

Basis of coefficient ring:

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \zeta_{10}^{2} \)
\(\beta_{2}\)\(=\)\( 2 \zeta_{10}^{3} \)
\(\beta_{3}\)\(=\)\( 2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 4 \zeta_{10} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\zeta_{10}\)\(=\)\((\)\(\beta_{3} - \beta_{2} + \beta_{1} + 2\)\()/4\)
\(\zeta_{10}^{2}\)\(=\)\(\beta_{1}\)\(/2\)
\(\zeta_{10}^{3}\)\(=\)\(\beta_{2}\)\(/2\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−1.61803 1.17557i 3.80423i 1.23607 + 3.80423i 2.23607 −4.47214 + 6.15537i 8.50651i 2.47214 7.60845i −5.47214 −3.61803 2.62866i
11.2 −1.61803 + 1.17557i 3.80423i 1.23607 3.80423i 2.23607 −4.47214 6.15537i 8.50651i 2.47214 + 7.60845i −5.47214 −3.61803 + 2.62866i
11.3 0.618034 1.90211i 2.35114i −3.23607 2.35114i −2.23607 4.47214 + 1.45309i 5.25731i −6.47214 + 4.70228i 3.47214 −1.38197 + 4.25325i
11.4 0.618034 + 1.90211i 2.35114i −3.23607 + 2.35114i −2.23607 4.47214 1.45309i 5.25731i −6.47214 4.70228i 3.47214 −1.38197 4.25325i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(20, [\chi])\).