Properties

Label 20.8.e.a
Level 20
Weight 8
Character orbit 20.e
Analytic conductor 6.248
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: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 20.e (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(6.24770050968\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q\) \( + ( -8 - 8 i ) q^{2} \) \( + 128 i q^{4} \) \( + ( 278 + 29 i ) q^{5} \) \( + ( 1024 - 1024 i ) q^{8} \) \( -2187 i q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -8 - 8 i ) q^{2} \) \( + 128 i q^{4} \) \( + ( 278 + 29 i ) q^{5} \) \( + ( 1024 - 1024 i ) q^{8} \) \( -2187 i q^{9} \) \( + ( -1992 - 2456 i ) q^{10} \) \( + ( 11003 - 11003 i ) q^{13} \) \( -16384 q^{16} \) \( + ( 17139 + 17139 i ) q^{17} \) \( + ( -17496 + 17496 i ) q^{18} \) \( + ( -3712 + 35584 i ) q^{20} \) \( + ( 76443 + 16124 i ) q^{25} \) \( -176048 q^{26} \) \( -120844 i q^{29} \) \( + ( 131072 + 131072 i ) q^{32} \) \( -274224 i q^{34} \) \( + 279936 q^{36} \) \( + ( 157829 + 157829 i ) q^{37} \) \( + ( 314368 - 254976 i ) q^{40} \) \( -882568 q^{41} \) \( + ( 63423 - 607986 i ) q^{45} \) \( + 823543 i q^{49} \) \( + ( -482552 - 740536 i ) q^{50} \) \( + ( 1408384 + 1408384 i ) q^{52} \) \( + ( -1406907 + 1406907 i ) q^{53} \) \( + ( -966752 + 966752 i ) q^{58} \) \( + 532572 q^{61} \) \( -2097152 i q^{64} \) \( + ( 3377921 - 2739747 i ) q^{65} \) \( + ( -2193792 + 2193792 i ) q^{68} \) \( + ( -2239488 - 2239488 i ) q^{72} \) \( + ( 4644013 - 4644013 i ) q^{73} \) \( -2525264 i q^{74} \) \( + ( -4554752 - 475136 i ) q^{80} \) \( -4782969 q^{81} \) \( + ( 7060544 + 7060544 i ) q^{82} \) \( + ( 4267611 + 5261673 i ) q^{85} \) \( + 9562096 i q^{89} \) \( + ( -5371272 + 4356504 i ) q^{90} \) \( + ( -10692851 - 10692851 i ) q^{97} \) \( + ( 6588344 - 6588344 i ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 556q^{5} \) \(\mathstrut +\mathstrut 2048q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 556q^{5} \) \(\mathstrut +\mathstrut 2048q^{8} \) \(\mathstrut -\mathstrut 3984q^{10} \) \(\mathstrut +\mathstrut 22006q^{13} \) \(\mathstrut -\mathstrut 32768q^{16} \) \(\mathstrut +\mathstrut 34278q^{17} \) \(\mathstrut -\mathstrut 34992q^{18} \) \(\mathstrut -\mathstrut 7424q^{20} \) \(\mathstrut +\mathstrut 152886q^{25} \) \(\mathstrut -\mathstrut 352096q^{26} \) \(\mathstrut +\mathstrut 262144q^{32} \) \(\mathstrut +\mathstrut 559872q^{36} \) \(\mathstrut +\mathstrut 315658q^{37} \) \(\mathstrut +\mathstrut 628736q^{40} \) \(\mathstrut -\mathstrut 1765136q^{41} \) \(\mathstrut +\mathstrut 126846q^{45} \) \(\mathstrut -\mathstrut 965104q^{50} \) \(\mathstrut +\mathstrut 2816768q^{52} \) \(\mathstrut -\mathstrut 2813814q^{53} \) \(\mathstrut -\mathstrut 1933504q^{58} \) \(\mathstrut +\mathstrut 1065144q^{61} \) \(\mathstrut +\mathstrut 6755842q^{65} \) \(\mathstrut -\mathstrut 4387584q^{68} \) \(\mathstrut -\mathstrut 4478976q^{72} \) \(\mathstrut +\mathstrut 9288026q^{73} \) \(\mathstrut -\mathstrut 9109504q^{80} \) \(\mathstrut -\mathstrut 9565938q^{81} \) \(\mathstrut +\mathstrut 14121088q^{82} \) \(\mathstrut +\mathstrut 8535222q^{85} \) \(\mathstrut -\mathstrut 10742544q^{90} \) \(\mathstrut -\mathstrut 21385702q^{97} \) \(\mathstrut +\mathstrut 13176688q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−8.00000 + 8.00000i 0 128.000i 278.000 29.0000i 0 0 1024.00 + 1024.00i 2187.00i −1992.00 + 2456.00i
7.1 −8.00000 8.00000i 0 128.000i 278.000 + 29.0000i 0 0 1024.00 1024.00i 2187.00i −1992.00 2456.00i
\(n\): 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

This newform can be constructed as the kernel of the linear operator \(T_{3} \) acting on \(S_{8}^{\mathrm{new}}(20, [\chi])\).