Properties

Label 20.2.e.a
Level 20
Weight 2
Character orbit 20.e
Analytic conductor 0.160
Analytic rank 0
Dimension 2
CM discriminant -4
Inner twists 4

Related objects

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Newspace parameters

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

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: \( 1 \)
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.

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

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
−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 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 20.2.e.a 2
3.b odd 2 1 180.2.k.c 2
4.b odd 2 1 CM 20.2.e.a 2
5.b even 2 1 100.2.e.b 2
5.c odd 4 1 inner 20.2.e.a 2
5.c odd 4 1 100.2.e.b 2
7.b odd 2 1 980.2.k.a 2
7.c even 3 2 980.2.x.d 4
7.d odd 6 2 980.2.x.c 4
8.b even 2 1 320.2.n.e 2
8.d odd 2 1 320.2.n.e 2
12.b even 2 1 180.2.k.c 2
15.d odd 2 1 900.2.k.c 2
15.e even 4 1 180.2.k.c 2
15.e even 4 1 900.2.k.c 2
16.e even 4 1 1280.2.o.g 2
16.e even 4 1 1280.2.o.j 2
16.f odd 4 1 1280.2.o.g 2
16.f odd 4 1 1280.2.o.j 2
20.d odd 2 1 100.2.e.b 2
20.e even 4 1 inner 20.2.e.a 2
20.e even 4 1 100.2.e.b 2
28.d even 2 1 980.2.k.a 2
28.f even 6 2 980.2.x.c 4
28.g odd 6 2 980.2.x.d 4
35.f even 4 1 980.2.k.a 2
35.k even 12 2 980.2.x.c 4
35.l odd 12 2 980.2.x.d 4
40.e odd 2 1 1600.2.n.h 2
40.f even 2 1 1600.2.n.h 2
40.i odd 4 1 320.2.n.e 2
40.i odd 4 1 1600.2.n.h 2
40.k even 4 1 320.2.n.e 2
40.k even 4 1 1600.2.n.h 2
60.h even 2 1 900.2.k.c 2
60.l odd 4 1 180.2.k.c 2
60.l odd 4 1 900.2.k.c 2
80.i odd 4 1 1280.2.o.g 2
80.j even 4 1 1280.2.o.j 2
80.s even 4 1 1280.2.o.g 2
80.t odd 4 1 1280.2.o.j 2
140.j odd 4 1 980.2.k.a 2
140.w even 12 2 980.2.x.d 4
140.x odd 12 2 980.2.x.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.e.a 2 1.a even 1 1 trivial
20.2.e.a 2 4.b odd 2 1 CM
20.2.e.a 2 5.c odd 4 1 inner
20.2.e.a 2 20.e even 4 1 inner
100.2.e.b 2 5.b even 2 1
100.2.e.b 2 5.c odd 4 1
100.2.e.b 2 20.d odd 2 1
100.2.e.b 2 20.e even 4 1
180.2.k.c 2 3.b odd 2 1
180.2.k.c 2 12.b even 2 1
180.2.k.c 2 15.e even 4 1
180.2.k.c 2 60.l odd 4 1
320.2.n.e 2 8.b even 2 1
320.2.n.e 2 8.d odd 2 1
320.2.n.e 2 40.i odd 4 1
320.2.n.e 2 40.k even 4 1
900.2.k.c 2 15.d odd 2 1
900.2.k.c 2 15.e even 4 1
900.2.k.c 2 60.h even 2 1
900.2.k.c 2 60.l odd 4 1
980.2.k.a 2 7.b odd 2 1
980.2.k.a 2 28.d even 2 1
980.2.k.a 2 35.f even 4 1
980.2.k.a 2 140.j odd 4 1
980.2.x.c 4 7.d odd 6 2
980.2.x.c 4 28.f even 6 2
980.2.x.c 4 35.k even 12 2
980.2.x.c 4 140.x odd 12 2
980.2.x.d 4 7.c even 3 2
980.2.x.d 4 28.g odd 6 2
980.2.x.d 4 35.l odd 12 2
980.2.x.d 4 140.w even 12 2
1280.2.o.g 2 16.e even 4 1
1280.2.o.g 2 16.f odd 4 1
1280.2.o.g 2 80.i odd 4 1
1280.2.o.g 2 80.s even 4 1
1280.2.o.j 2 16.e even 4 1
1280.2.o.j 2 16.f odd 4 1
1280.2.o.j 2 80.j even 4 1
1280.2.o.j 2 80.t odd 4 1
1600.2.n.h 2 40.e odd 2 1
1600.2.n.h 2 40.f even 2 1
1600.2.n.h 2 40.i odd 4 1
1600.2.n.h 2 40.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(20, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} \)
$3$ \( 1 + 9 T^{4} \)
$5$ \( 1 + 4 T + 5 T^{2} \)
$7$ \( 1 + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} ) \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 + 529 T^{4} \)
$29$ \( ( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \)
$31$ \( ( 1 - 31 T^{2} )^{2} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 + 8 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 1849 T^{4} \)
$47$ \( 1 + 2209 T^{4} \)
$53$ \( ( 1 - 14 T + 53 T^{2} )( 1 - 4 T + 53 T^{2} ) \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 12 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 4489 T^{4} \)
$71$ \( ( 1 - 71 T^{2} )^{2} \)
$73$ \( ( 1 + 6 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( 1 + 6889 T^{4} \)
$89$ \( ( 1 - 10 T + 89 T^{2} )( 1 + 10 T + 89 T^{2} ) \)
$97$ \( ( 1 - 18 T + 97 T^{2} )( 1 - 8 T + 97 T^{2} ) \)
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