Properties

Label 20.7.d.b
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

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

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 \(T_{3} \) \(\mathstrut -\mathstrut 44 \) acting on \(S_{7}^{\mathrm{new}}(20, [\chi])\).