Properties

Label 20.11.d.b
Level 20
Weight 11
Character orbit 20.d
Self dual Yes
Analytic conductor 12.707
Analytic rank 0
Dimension 1
CM disc. -20
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(12.7071450535\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 32q^{2} + 236q^{3} + 1024q^{4} - 3125q^{5} + 7552q^{6} + 33364q^{7} + 32768q^{8} - 3353q^{9} + O(q^{10}) \) \( q + 32q^{2} + 236q^{3} + 1024q^{4} - 3125q^{5} + 7552q^{6} + 33364q^{7} + 32768q^{8} - 3353q^{9} - 100000q^{10} + 241664q^{12} + 1067648q^{14} - 737500q^{15} + 1048576q^{16} - 107296q^{18} - 3200000q^{20} + 7873904q^{21} - 1169564q^{23} + 7733248q^{24} + 9765625q^{25} - 14726872q^{27} + 34164736q^{28} - 38179702q^{29} - 23600000q^{30} + 33554432q^{32} - 104262500q^{35} - 3433472q^{36} - 102400000q^{40} - 211028098q^{41} + 251964928q^{42} - 223663364q^{43} + 10478125q^{45} - 37426048q^{46} + 96887764q^{47} + 247463936q^{48} + 830681247q^{49} + 312500000q^{50} - 471259904q^{54} + 1093271552q^{56} - 1221750464q^{58} - 755200000q^{60} - 1041591898q^{61} - 111869492q^{63} + 1073741824q^{64} + 2343243964q^{67} - 276017104q^{69} - 3336400000q^{70} - 109871104q^{72} + 2304687500q^{75} - 3276800000q^{80} - 3277550495q^{81} - 6752899136q^{82} + 5449159036q^{83} + 8062877696q^{84} - 7157227648q^{86} - 9010409672q^{87} + 11118190898q^{89} + 335300000q^{90} - 1197633536q^{92} + 3100408448q^{94} + 7918845952q^{96} + 26581799904q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0
32.0000 236.000 1024.00 −3125.00 7552.00 33364.0 32768.0 −3353.00 −100000.
\(n\): 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
20.d Odd 1 CM by \(\Q(\sqrt{-5}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3} - 236 \) acting on \(S_{11}^{\mathrm{new}}(20, [\chi])\).