Properties

Label 20.6.e.a
Level 20
Weight 6
Character orbit 20.e
Analytic conductor 3.208
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: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 20.e (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(3.20767639626\)
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.

\(f(q)\) \(=\) \( q\) \( + ( 4 + 4 i ) q^{2} \) \( + 32 i q^{4} \) \( + ( 38 + 41 i ) q^{5} \) \( + ( -128 + 128 i ) q^{8} \) \( -243 i q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 4 + 4 i ) q^{2} \) \( + 32 i q^{4} \) \( + ( 38 + 41 i ) q^{5} \) \( + ( -128 + 128 i ) q^{8} \) \( -243 i q^{9} \) \( + ( -12 + 316 i ) q^{10} \) \( + ( 719 - 719 i ) q^{13} \) \( -1024 q^{16} \) \( + ( -717 - 717 i ) q^{17} \) \( + ( 972 - 972 i ) q^{18} \) \( + ( -1312 + 1216 i ) q^{20} \) \( + ( -237 + 3116 i ) q^{25} \) \( + 5752 q^{26} \) \( + 8564 i q^{29} \) \( + ( -4096 - 4096 i ) q^{32} \) \( -5736 i q^{34} \) \( + 7776 q^{36} \) \( + ( -11767 - 11767 i ) q^{37} \) \( + ( -10112 - 384 i ) q^{40} \) \( + 4952 q^{41} \) \( + ( 9963 - 9234 i ) q^{45} \) \( + 16807 i q^{49} \) \( + ( -13412 + 11516 i ) q^{50} \) \( + ( 23008 + 23008 i ) q^{52} \) \( + ( 23769 - 23769 i ) q^{53} \) \( + ( -34256 + 34256 i ) q^{58} \) \( -54948 q^{61} \) \( -32768 i q^{64} \) \( + ( 56801 + 2157 i ) q^{65} \) \( + ( 22944 - 22944 i ) q^{68} \) \( + ( 31104 + 31104 i ) q^{72} \) \( + ( -34331 + 34331 i ) q^{73} \) \( -94136 i q^{74} \) \( + ( -38912 - 41984 i ) q^{80} \) \( -59049 q^{81} \) \( + ( 19808 + 19808 i ) q^{82} \) \( + ( 2151 - 56643 i ) q^{85} \) \( + 140464 i q^{89} \) \( + ( 76788 + 2916 i ) q^{90} \) \( + ( 34333 + 34333 i ) q^{97} \) \( + ( -67228 + 67228 i ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 76q^{5} \) \(\mathstrut -\mathstrut 256q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 76q^{5} \) \(\mathstrut -\mathstrut 256q^{8} \) \(\mathstrut -\mathstrut 24q^{10} \) \(\mathstrut +\mathstrut 1438q^{13} \) \(\mathstrut -\mathstrut 2048q^{16} \) \(\mathstrut -\mathstrut 1434q^{17} \) \(\mathstrut +\mathstrut 1944q^{18} \) \(\mathstrut -\mathstrut 2624q^{20} \) \(\mathstrut -\mathstrut 474q^{25} \) \(\mathstrut +\mathstrut 11504q^{26} \) \(\mathstrut -\mathstrut 8192q^{32} \) \(\mathstrut +\mathstrut 15552q^{36} \) \(\mathstrut -\mathstrut 23534q^{37} \) \(\mathstrut -\mathstrut 20224q^{40} \) \(\mathstrut +\mathstrut 9904q^{41} \) \(\mathstrut +\mathstrut 19926q^{45} \) \(\mathstrut -\mathstrut 26824q^{50} \) \(\mathstrut +\mathstrut 46016q^{52} \) \(\mathstrut +\mathstrut 47538q^{53} \) \(\mathstrut -\mathstrut 68512q^{58} \) \(\mathstrut -\mathstrut 109896q^{61} \) \(\mathstrut +\mathstrut 113602q^{65} \) \(\mathstrut +\mathstrut 45888q^{68} \) \(\mathstrut +\mathstrut 62208q^{72} \) \(\mathstrut -\mathstrut 68662q^{73} \) \(\mathstrut -\mathstrut 77824q^{80} \) \(\mathstrut -\mathstrut 118098q^{81} \) \(\mathstrut +\mathstrut 39616q^{82} \) \(\mathstrut +\mathstrut 4302q^{85} \) \(\mathstrut +\mathstrut 153576q^{90} \) \(\mathstrut +\mathstrut 68666q^{97} \) \(\mathstrut -\mathstrut 134456q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
4.00000 4.00000i 0 32.0000i 38.0000 41.0000i 0 0 −128.000 128.000i 243.000i −12.0000 316.000i
7.1 4.00000 + 4.00000i 0 32.0000i 38.0000 + 41.0000i 0 0 −128.000 + 128.000i 243.000i −12.0000 + 316.000i
\(n\): 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 \(S_{6}^{\mathrm{new}}(20, [\chi])\).