Properties

Label 20.2.e.a
Level 20
Weight 2
Character orbit 20.e
Analytic conductor 0.160
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: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 20.e (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.159700804043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( -1 - i ) q^{2} \) \( + 2 i q^{4} \) \( + ( -2 + i ) q^{5} \) \( + ( 2 - 2 i ) q^{8} \) \( -3 i q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 - i ) q^{2} \) \( + 2 i q^{4} \) \( + ( -2 + i ) q^{5} \) \( + ( 2 - 2 i ) q^{8} \) \( -3 i q^{9} \) \( + ( 3 + i ) q^{10} \) \( + ( -1 + i ) q^{13} \) \( -4 q^{16} \) \( + ( 3 + 3 i ) q^{17} \) \( + ( -3 + 3 i ) q^{18} \) \( + ( -2 - 4 i ) q^{20} \) \( + ( 3 - 4 i ) q^{25} \) \( + 2 q^{26} \) \( + 4 i q^{29} \) \( + ( 4 + 4 i ) q^{32} \) \( -6 i q^{34} \) \( + 6 q^{36} \) \( + ( -7 - 7 i ) q^{37} \) \( + ( -2 + 6 i ) q^{40} \) \( -8 q^{41} \) \( + ( 3 + 6 i ) q^{45} \) \( + 7 i q^{49} \) \( + ( -7 + i ) q^{50} \) \( + ( -2 - 2 i ) q^{52} \) \( + ( 9 - 9 i ) q^{53} \) \( + ( 4 - 4 i ) q^{58} \) \( + 12 q^{61} \) \( -8 i q^{64} \) \( + ( 1 - 3 i ) q^{65} \) \( + ( -6 + 6 i ) q^{68} \) \( + ( -6 - 6 i ) q^{72} \) \( + ( -11 + 11 i ) q^{73} \) \( + 14 i q^{74} \) \( + ( 8 - 4 i ) q^{80} \) \( -9 q^{81} \) \( + ( 8 + 8 i ) q^{82} \) \( + ( -9 - 3 i ) q^{85} \) \( -16 i q^{89} \) \( + ( 3 - 9 i ) q^{90} \) \( + ( 13 + 13 i ) q^{97} \) \( + ( 7 - 7 i ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 4q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 4q^{8} \) \(\mathstrut +\mathstrut 6q^{10} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 6q^{18} \) \(\mathstrut -\mathstrut 4q^{20} \) \(\mathstrut +\mathstrut 6q^{25} \) \(\mathstrut +\mathstrut 4q^{26} \) \(\mathstrut +\mathstrut 8q^{32} \) \(\mathstrut +\mathstrut 12q^{36} \) \(\mathstrut -\mathstrut 14q^{37} \) \(\mathstrut -\mathstrut 4q^{40} \) \(\mathstrut -\mathstrut 16q^{41} \) \(\mathstrut +\mathstrut 6q^{45} \) \(\mathstrut -\mathstrut 14q^{50} \) \(\mathstrut -\mathstrut 4q^{52} \) \(\mathstrut +\mathstrut 18q^{53} \) \(\mathstrut +\mathstrut 8q^{58} \) \(\mathstrut +\mathstrut 24q^{61} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut -\mathstrut 12q^{68} \) \(\mathstrut -\mathstrut 12q^{72} \) \(\mathstrut -\mathstrut 22q^{73} \) \(\mathstrut +\mathstrut 16q^{80} \) \(\mathstrut -\mathstrut 18q^{81} \) \(\mathstrut +\mathstrut 16q^{82} \) \(\mathstrut -\mathstrut 18q^{85} \) \(\mathstrut +\mathstrut 6q^{90} \) \(\mathstrut +\mathstrut 26q^{97} \) \(\mathstrut +\mathstrut 14q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.00000i
1.00000i
−1.00000 + 1.00000i 0 2.00000i −2.00000 1.00000i 0 0 2.00000 + 2.00000i 3.00000i 3.00000 1.00000i
7.1 −1.00000 1.00000i 0 2.00000i −2.00000 + 1.00000i 0 0 2.00000 2.00000i 3.00000i 3.00000 + 1.00000i
\(n\): 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.c Odd 1 yes
20.e Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(20, [\chi])\).