Properties

Label 20.4.e.a
Level 20
Weight 4
Character orbit 20.e
Analytic conductor 1.180
Analytic rank 0
Dimension 2
CM disc. -4
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 20.e (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.18003820011\)
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.

\(f(q)\) \(=\) \( q\) \( + ( 2 + 2 i ) q^{2} \) \( + 8 i q^{4} \) \( + ( -2 - 11 i ) q^{5} \) \( + ( -16 + 16 i ) q^{8} \) \( -27 i q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 2 + 2 i ) q^{2} \) \( + 8 i q^{4} \) \( + ( -2 - 11 i ) q^{5} \) \( + ( -16 + 16 i ) q^{8} \) \( -27 i q^{9} \) \( + ( 18 - 26 i ) q^{10} \) \( + ( -37 + 37 i ) q^{13} \) \( -64 q^{16} \) \( + ( 99 + 99 i ) q^{17} \) \( + ( 54 - 54 i ) q^{18} \) \( + ( 88 - 16 i ) q^{20} \) \( + ( -117 + 44 i ) q^{25} \) \( -148 q^{26} \) \( -284 i q^{29} \) \( + ( -128 - 128 i ) q^{32} \) \( + 396 i q^{34} \) \( + 216 q^{36} \) \( + ( -91 - 91 i ) q^{37} \) \( + ( 208 + 144 i ) q^{40} \) \( + 472 q^{41} \) \( + ( -297 + 54 i ) q^{45} \) \( + 343 i q^{49} \) \( + ( -322 - 146 i ) q^{50} \) \( + ( -296 - 296 i ) q^{52} \) \( + ( -27 + 27 i ) q^{53} \) \( + ( 568 - 568 i ) q^{58} \) \( -468 q^{61} \) \( -512 i q^{64} \) \( + ( 481 + 333 i ) q^{65} \) \( + ( -792 + 792 i ) q^{68} \) \( + ( 432 + 432 i ) q^{72} \) \( + ( 253 - 253 i ) q^{73} \) \( -364 i q^{74} \) \( + ( 128 + 704 i ) q^{80} \) \( -729 q^{81} \) \( + ( 944 + 944 i ) q^{82} \) \( + ( 891 - 1287 i ) q^{85} \) \( + 176 i q^{89} \) \( + ( -702 - 486 i ) q^{90} \) \( + ( -611 - 611 i ) q^{97} \) \( + ( -686 + 686 i ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 32q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 32q^{8} \) \(\mathstrut +\mathstrut 36q^{10} \) \(\mathstrut -\mathstrut 74q^{13} \) \(\mathstrut -\mathstrut 128q^{16} \) \(\mathstrut +\mathstrut 198q^{17} \) \(\mathstrut +\mathstrut 108q^{18} \) \(\mathstrut +\mathstrut 176q^{20} \) \(\mathstrut -\mathstrut 234q^{25} \) \(\mathstrut -\mathstrut 296q^{26} \) \(\mathstrut -\mathstrut 256q^{32} \) \(\mathstrut +\mathstrut 432q^{36} \) \(\mathstrut -\mathstrut 182q^{37} \) \(\mathstrut +\mathstrut 416q^{40} \) \(\mathstrut +\mathstrut 944q^{41} \) \(\mathstrut -\mathstrut 594q^{45} \) \(\mathstrut -\mathstrut 644q^{50} \) \(\mathstrut -\mathstrut 592q^{52} \) \(\mathstrut -\mathstrut 54q^{53} \) \(\mathstrut +\mathstrut 1136q^{58} \) \(\mathstrut -\mathstrut 936q^{61} \) \(\mathstrut +\mathstrut 962q^{65} \) \(\mathstrut -\mathstrut 1584q^{68} \) \(\mathstrut +\mathstrut 864q^{72} \) \(\mathstrut +\mathstrut 506q^{73} \) \(\mathstrut +\mathstrut 256q^{80} \) \(\mathstrut -\mathstrut 1458q^{81} \) \(\mathstrut +\mathstrut 1888q^{82} \) \(\mathstrut +\mathstrut 1782q^{85} \) \(\mathstrut -\mathstrut 1404q^{90} \) \(\mathstrut -\mathstrut 1222q^{97} \) \(\mathstrut -\mathstrut 1372q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
2.00000 2.00000i 0 8.00000i −2.00000 + 11.0000i 0 0 −16.0000 16.0000i 27.0000i 18.0000 + 26.0000i
7.1 2.00000 + 2.00000i 0 8.00000i −2.00000 11.0000i 0 0 −16.0000 + 16.0000i 27.0000i 18.0000 26.0000i
\(n\): 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 \(S_{4}^{\mathrm{new}}(20, [\chi])\).