Properties

Label 20.7.d.a
Level 20
Weight 7
Character orbit 20.d
Self dual Yes
Analytic conductor 4.601
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: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(4.6010816724\)
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

\(f(q)\) \(=\) \( q - 8q^{2} - 44q^{3} + 64q^{4} - 125q^{5} + 352q^{6} + 524q^{7} - 512q^{8} + 1207q^{9} + O(q^{10}) \) \( q - 8q^{2} - 44q^{3} + 64q^{4} - 125q^{5} + 352q^{6} + 524q^{7} - 512q^{8} + 1207q^{9} + 1000q^{10} - 2816q^{12} - 4192q^{14} + 5500q^{15} + 4096q^{16} - 9656q^{18} - 8000q^{20} - 23056q^{21} + 15356q^{23} + 22528q^{24} + 15625q^{25} - 21032q^{27} + 33536q^{28} + 44858q^{29} - 44000q^{30} - 32768q^{32} - 65500q^{35} + 77248q^{36} + 64000q^{40} - 74338q^{41} + 184448q^{42} - 17404q^{43} - 150875q^{45} - 122848q^{46} + 26444q^{47} - 180224q^{48} + 156927q^{49} - 125000q^{50} + 168256q^{54} - 268288q^{56} - 358864q^{58} + 352000q^{60} + 452342q^{61} + 632468q^{63} + 262144q^{64} - 1276q^{67} - 675664q^{69} + 524000q^{70} - 617984q^{72} - 687500q^{75} - 512000q^{80} + 45505q^{81} + 594704q^{82} + 1131716q^{83} - 1475584q^{84} + 139232q^{86} - 1973752q^{87} + 511058q^{89} + 1207000q^{90} + 982784q^{92} - 211552q^{94} + 1441792q^{96} - 1255416q^{98} + O(q^{100}) \)

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
−8.00000 −44.0000 64.0000 −125.000 352.000 524.000 −512.000 1207.00 1000.00
\(n\): 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 \( T_{3} + 44 \) acting on \(S_{7}^{\mathrm{new}}(20, [\chi])\).