Properties

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

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Newspace parameters

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

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: \( 1 \)
Twist minimal: yes
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.

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.4.e.a 2
3.b odd 2 1 180.4.k.a 2
4.b odd 2 1 CM 20.4.e.a 2
5.b even 2 1 100.4.e.a 2
5.c odd 4 1 inner 20.4.e.a 2
5.c odd 4 1 100.4.e.a 2
8.b even 2 1 320.4.n.c 2
8.d odd 2 1 320.4.n.c 2
12.b even 2 1 180.4.k.a 2
15.e even 4 1 180.4.k.a 2
20.d odd 2 1 100.4.e.a 2
20.e even 4 1 inner 20.4.e.a 2
20.e even 4 1 100.4.e.a 2
40.i odd 4 1 320.4.n.c 2
40.k even 4 1 320.4.n.c 2
60.l odd 4 1 180.4.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.e.a 2 1.a even 1 1 trivial
20.4.e.a 2 4.b odd 2 1 CM
20.4.e.a 2 5.c odd 4 1 inner
20.4.e.a 2 20.e even 4 1 inner
100.4.e.a 2 5.b even 2 1
100.4.e.a 2 5.c odd 4 1
100.4.e.a 2 20.d odd 2 1
100.4.e.a 2 20.e even 4 1
180.4.k.a 2 3.b odd 2 1
180.4.k.a 2 12.b even 2 1
180.4.k.a 2 15.e even 4 1
180.4.k.a 2 60.l odd 4 1
320.4.n.c 2 8.b even 2 1
320.4.n.c 2 8.d odd 2 1
320.4.n.c 2 40.i odd 4 1
320.4.n.c 2 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{4}^{\mathrm{new}}(20, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 8 T^{2} \)
$3$ \( 1 + 729 T^{4} \)
$5$ \( 1 + 4 T + 125 T^{2} \)
$7$ \( 1 + 117649 T^{4} \)
$11$ \( ( 1 - 1331 T^{2} )^{2} \)
$13$ \( ( 1 - 18 T + 2197 T^{2} )( 1 + 92 T + 2197 T^{2} ) \)
$17$ \( ( 1 - 104 T + 4913 T^{2} )( 1 - 94 T + 4913 T^{2} ) \)
$19$ \( ( 1 + 6859 T^{2} )^{2} \)
$23$ \( 1 + 148035889 T^{4} \)
$29$ \( ( 1 - 130 T + 24389 T^{2} )( 1 + 130 T + 24389 T^{2} ) \)
$31$ \( ( 1 - 29791 T^{2} )^{2} \)
$37$ \( ( 1 - 214 T + 50653 T^{2} )( 1 + 396 T + 50653 T^{2} ) \)
$41$ \( ( 1 - 472 T + 68921 T^{2} )^{2} \)
$43$ \( 1 + 6321363049 T^{4} \)
$47$ \( 1 + 10779215329 T^{4} \)
$53$ \( ( 1 - 518 T + 148877 T^{2} )( 1 + 572 T + 148877 T^{2} ) \)
$59$ \( ( 1 + 205379 T^{2} )^{2} \)
$61$ \( ( 1 + 468 T + 226981 T^{2} )^{2} \)
$67$ \( 1 + 90458382169 T^{4} \)
$71$ \( ( 1 - 357911 T^{2} )^{2} \)
$73$ \( ( 1 - 1098 T + 389017 T^{2} )( 1 + 592 T + 389017 T^{2} ) \)
$79$ \( ( 1 + 493039 T^{2} )^{2} \)
$83$ \( 1 + 326940373369 T^{4} \)
$89$ \( ( 1 - 1670 T + 704969 T^{2} )( 1 + 1670 T + 704969 T^{2} ) \)
$97$ \( ( 1 - 594 T + 912673 T^{2} )( 1 + 1816 T + 912673 T^{2} ) \)
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