Properties

Label 20.3.d.c
Level 20
Weight 3
Character orbit 20.d
Analytic conductor 0.545
Analytic rank 0
Dimension 2
CM disc. -4
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.544960528721\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -4 q^{4} \) \( + ( 3 - 2 \beta ) q^{5} \) \( -4 \beta q^{8} \) \( -9 q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -4 q^{4} \) \( + ( 3 - 2 \beta ) q^{5} \) \( -4 \beta q^{8} \) \( -9 q^{9} \) \( + ( 8 + 3 \beta ) q^{10} \) \( + 12 \beta q^{13} \) \( + 16 q^{16} \) \( -8 \beta q^{17} \) \( -9 \beta q^{18} \) \( + ( -12 + 8 \beta ) q^{20} \) \( + ( -7 - 12 \beta ) q^{25} \) \( -48 q^{26} \) \( + 42 q^{29} \) \( + 16 \beta q^{32} \) \( + 32 q^{34} \) \( + 36 q^{36} \) \( + 12 \beta q^{37} \) \( + ( -32 - 12 \beta ) q^{40} \) \( -18 q^{41} \) \( + ( -27 + 18 \beta ) q^{45} \) \( -49 q^{49} \) \( + ( 48 - 7 \beta ) q^{50} \) \( -48 \beta q^{52} \) \( -28 \beta q^{53} \) \( + 42 \beta q^{58} \) \( + 22 q^{61} \) \( -64 q^{64} \) \( + ( 96 + 36 \beta ) q^{65} \) \( + 32 \beta q^{68} \) \( + 36 \beta q^{72} \) \( -48 \beta q^{73} \) \( -48 q^{74} \) \( + ( 48 - 32 \beta ) q^{80} \) \( + 81 q^{81} \) \( -18 \beta q^{82} \) \( + ( -64 - 24 \beta ) q^{85} \) \( -78 q^{89} \) \( + ( -72 - 27 \beta ) q^{90} \) \( + 72 \beta q^{97} \) \( -49 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 18q^{9} \) \(\mathstrut +\mathstrut 16q^{10} \) \(\mathstrut +\mathstrut 32q^{16} \) \(\mathstrut -\mathstrut 24q^{20} \) \(\mathstrut -\mathstrut 14q^{25} \) \(\mathstrut -\mathstrut 96q^{26} \) \(\mathstrut +\mathstrut 84q^{29} \) \(\mathstrut +\mathstrut 64q^{34} \) \(\mathstrut +\mathstrut 72q^{36} \) \(\mathstrut -\mathstrut 64q^{40} \) \(\mathstrut -\mathstrut 36q^{41} \) \(\mathstrut -\mathstrut 54q^{45} \) \(\mathstrut -\mathstrut 98q^{49} \) \(\mathstrut +\mathstrut 96q^{50} \) \(\mathstrut +\mathstrut 44q^{61} \) \(\mathstrut -\mathstrut 128q^{64} \) \(\mathstrut +\mathstrut 192q^{65} \) \(\mathstrut -\mathstrut 96q^{74} \) \(\mathstrut +\mathstrut 96q^{80} \) \(\mathstrut +\mathstrut 162q^{81} \) \(\mathstrut -\mathstrut 128q^{85} \) \(\mathstrut -\mathstrut 156q^{89} \) \(\mathstrut -\mathstrut 144q^{90} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.00000i
1.00000i
2.00000i 0 −4.00000 3.00000 + 4.00000i 0 0 8.00000i −9.00000 8.00000 6.00000i
19.2 2.00000i 0 −4.00000 3.00000 4.00000i 0 0 8.00000i −9.00000 8.00000 + 6.00000i
\(n\): 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 CM by \(\Q(\sqrt{-1}) \) yes
5.b Even 1 yes
20.d Odd 1 yes

Hecke kernels

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