Properties

Label 20.3.d.c
Level 20
Weight 3
Character orbit 20.d
Analytic conductor 0.545
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.544960528721\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 i q^{2} -4 q^{4} + ( 3 - 4 i ) q^{5} -8 i q^{8} -9 q^{9} +O(q^{10})\) \( q + 2 i q^{2} -4 q^{4} + ( 3 - 4 i ) q^{5} -8 i q^{8} -9 q^{9} + ( 8 + 6 i ) q^{10} + 24 i q^{13} + 16 q^{16} -16 i q^{17} -18 i q^{18} + ( -12 + 16 i ) q^{20} + ( -7 - 24 i ) q^{25} -48 q^{26} + 42 q^{29} + 32 i q^{32} + 32 q^{34} + 36 q^{36} + 24 i q^{37} + ( -32 - 24 i ) q^{40} -18 q^{41} + ( -27 + 36 i ) q^{45} -49 q^{49} + ( 48 - 14 i ) q^{50} -96 i q^{52} -56 i q^{53} + 84 i q^{58} + 22 q^{61} -64 q^{64} + ( 96 + 72 i ) q^{65} + 64 i q^{68} + 72 i q^{72} -96 i q^{73} -48 q^{74} + ( 48 - 64 i ) q^{80} + 81 q^{81} -36 i q^{82} + ( -64 - 48 i ) q^{85} -78 q^{89} + ( -72 - 54 i ) q^{90} + 144 i q^{97} -98 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{4} + 6q^{5} - 18q^{9} + O(q^{10}) \) \( 2q - 8q^{4} + 6q^{5} - 18q^{9} + 16q^{10} + 32q^{16} - 24q^{20} - 14q^{25} - 96q^{26} + 84q^{29} + 64q^{34} + 72q^{36} - 64q^{40} - 36q^{41} - 54q^{45} - 98q^{49} + 96q^{50} + 44q^{61} - 128q^{64} + 192q^{65} - 96q^{74} + 96q^{80} + 162q^{81} - 128q^{85} - 156q^{89} - 144q^{90} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
1.00000i
1.00000i
2.00000i 0 −4.00000 3.00000 + 4.00000i 0 0 8.00000i −9.00000 8.00000 6.00000i
19.2 2.00000i 0 −4.00000 3.00000 4.00000i 0 0 8.00000i −9.00000 8.00000 + 6.00000i
\(n\): 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.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.3.d.c 2
3.b odd 2 1 180.3.f.c 2
4.b odd 2 1 CM 20.3.d.c 2
5.b even 2 1 inner 20.3.d.c 2
5.c odd 4 1 100.3.b.a 1
5.c odd 4 1 100.3.b.b 1
8.b even 2 1 320.3.h.d 2
8.d odd 2 1 320.3.h.d 2
12.b even 2 1 180.3.f.c 2
15.d odd 2 1 180.3.f.c 2
15.e even 4 1 900.3.c.a 1
15.e even 4 1 900.3.c.d 1
16.e even 4 1 1280.3.e.a 2
16.e even 4 1 1280.3.e.d 2
16.f odd 4 1 1280.3.e.a 2
16.f odd 4 1 1280.3.e.d 2
20.d odd 2 1 inner 20.3.d.c 2
20.e even 4 1 100.3.b.a 1
20.e even 4 1 100.3.b.b 1
40.e odd 2 1 320.3.h.d 2
40.f even 2 1 320.3.h.d 2
40.i odd 4 1 1600.3.b.a 1
40.i odd 4 1 1600.3.b.c 1
40.k even 4 1 1600.3.b.a 1
40.k even 4 1 1600.3.b.c 1
60.h even 2 1 180.3.f.c 2
60.l odd 4 1 900.3.c.a 1
60.l odd 4 1 900.3.c.d 1
80.k odd 4 1 1280.3.e.a 2
80.k odd 4 1 1280.3.e.d 2
80.q even 4 1 1280.3.e.a 2
80.q even 4 1 1280.3.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.c 2 1.a even 1 1 trivial
20.3.d.c 2 4.b odd 2 1 CM
20.3.d.c 2 5.b even 2 1 inner
20.3.d.c 2 20.d odd 2 1 inner
100.3.b.a 1 5.c odd 4 1
100.3.b.a 1 20.e even 4 1
100.3.b.b 1 5.c odd 4 1
100.3.b.b 1 20.e even 4 1
180.3.f.c 2 3.b odd 2 1
180.3.f.c 2 12.b even 2 1
180.3.f.c 2 15.d odd 2 1
180.3.f.c 2 60.h even 2 1
320.3.h.d 2 8.b even 2 1
320.3.h.d 2 8.d odd 2 1
320.3.h.d 2 40.e odd 2 1
320.3.h.d 2 40.f even 2 1
900.3.c.a 1 15.e even 4 1
900.3.c.a 1 60.l odd 4 1
900.3.c.d 1 15.e even 4 1
900.3.c.d 1 60.l odd 4 1
1280.3.e.a 2 16.e even 4 1
1280.3.e.a 2 16.f odd 4 1
1280.3.e.a 2 80.k odd 4 1
1280.3.e.a 2 80.q even 4 1
1280.3.e.d 2 16.e even 4 1
1280.3.e.d 2 16.f odd 4 1
1280.3.e.d 2 80.k odd 4 1
1280.3.e.d 2 80.q even 4 1
1600.3.b.a 1 40.i odd 4 1
1600.3.b.a 1 40.k even 4 1
1600.3.b.c 1 40.i odd 4 1
1600.3.b.c 1 40.k even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T^{2} \)
$3$ \( ( 1 + 9 T^{2} )^{2} \)
$5$ \( 1 - 6 T + 25 T^{2} \)
$7$ \( ( 1 + 49 T^{2} )^{2} \)
$11$ \( ( 1 - 11 T )^{2}( 1 + 11 T )^{2} \)
$13$ \( ( 1 - 10 T + 169 T^{2} )( 1 + 10 T + 169 T^{2} ) \)
$17$ \( ( 1 - 30 T + 289 T^{2} )( 1 + 30 T + 289 T^{2} ) \)
$19$ \( ( 1 - 19 T )^{2}( 1 + 19 T )^{2} \)
$23$ \( ( 1 + 529 T^{2} )^{2} \)
$29$ \( ( 1 - 42 T + 841 T^{2} )^{2} \)
$31$ \( ( 1 - 31 T )^{2}( 1 + 31 T )^{2} \)
$37$ \( ( 1 - 70 T + 1369 T^{2} )( 1 + 70 T + 1369 T^{2} ) \)
$41$ \( ( 1 + 18 T + 1681 T^{2} )^{2} \)
$43$ \( ( 1 + 1849 T^{2} )^{2} \)
$47$ \( ( 1 + 2209 T^{2} )^{2} \)
$53$ \( ( 1 - 90 T + 2809 T^{2} )( 1 + 90 T + 2809 T^{2} ) \)
$59$ \( ( 1 - 59 T )^{2}( 1 + 59 T )^{2} \)
$61$ \( ( 1 - 22 T + 3721 T^{2} )^{2} \)
$67$ \( ( 1 + 4489 T^{2} )^{2} \)
$71$ \( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \)
$73$ \( ( 1 - 110 T + 5329 T^{2} )( 1 + 110 T + 5329 T^{2} ) \)
$79$ \( ( 1 - 79 T )^{2}( 1 + 79 T )^{2} \)
$83$ \( ( 1 + 6889 T^{2} )^{2} \)
$89$ \( ( 1 + 78 T + 7921 T^{2} )^{2} \)
$97$ \( ( 1 - 130 T + 9409 T^{2} )( 1 + 130 T + 9409 T^{2} ) \)
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