Properties

Label 20.3.f.a
Level 20
Weight 3
Character orbit 20.f
Analytic conductor 0.545
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 20.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.544960528721\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - i ) q^{3} + ( -3 + 4 i ) q^{5} + ( -7 - 7 i ) q^{7} + 7 i q^{9} +O(q^{10})\) \( q + ( 1 - i ) q^{3} + ( -3 + 4 i ) q^{5} + ( -7 - 7 i ) q^{7} + 7 i q^{9} + 10 q^{11} + ( 9 - 9 i ) q^{13} + ( 1 + 7 i ) q^{15} + ( 1 + i ) q^{17} -8 i q^{19} -14 q^{21} + ( -23 + 23 i ) q^{23} + ( -7 - 24 i ) q^{25} + ( 16 + 16 i ) q^{27} -8 i q^{29} -14 q^{31} + ( 10 - 10 i ) q^{33} + ( 49 - 7 i ) q^{35} + ( 33 + 33 i ) q^{37} -18 i q^{39} -14 q^{41} + ( -15 + 15 i ) q^{43} + ( -28 - 21 i ) q^{45} + ( -39 - 39 i ) q^{47} + 49 i q^{49} + 2 q^{51} + ( -7 + 7 i ) q^{53} + ( -30 + 40 i ) q^{55} + ( -8 - 8 i ) q^{57} -56 i q^{59} + 42 q^{61} + ( 49 - 49 i ) q^{63} + ( 9 + 63 i ) q^{65} + ( -7 - 7 i ) q^{67} + 46 i q^{69} + 98 q^{71} + ( 49 - 49 i ) q^{73} + ( -31 - 17 i ) q^{75} + ( -70 - 70 i ) q^{77} + 96 i q^{79} -31 q^{81} + ( -63 + 63 i ) q^{83} + ( -7 + i ) q^{85} + ( -8 - 8 i ) q^{87} -112 i q^{89} -126 q^{91} + ( -14 + 14 i ) q^{93} + ( 32 + 24 i ) q^{95} + ( 33 + 33 i ) q^{97} + 70 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 6q^{5} - 14q^{7} + O(q^{10}) \) \( 2q + 2q^{3} - 6q^{5} - 14q^{7} + 20q^{11} + 18q^{13} + 2q^{15} + 2q^{17} - 28q^{21} - 46q^{23} - 14q^{25} + 32q^{27} - 28q^{31} + 20q^{33} + 98q^{35} + 66q^{37} - 28q^{41} - 30q^{43} - 56q^{45} - 78q^{47} + 4q^{51} - 14q^{53} - 60q^{55} - 16q^{57} + 84q^{61} + 98q^{63} + 18q^{65} - 14q^{67} + 196q^{71} + 98q^{73} - 62q^{75} - 140q^{77} - 62q^{81} - 126q^{83} - 14q^{85} - 16q^{87} - 252q^{91} - 28q^{93} + 64q^{95} + 66q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.00000i
1.00000i
0 1.00000 + 1.00000i 0 −3.00000 4.00000i 0 −7.00000 + 7.00000i 0 7.00000i 0
17.1 0 1.00000 1.00000i 0 −3.00000 + 4.00000i 0 −7.00000 7.00000i 0 7.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.3.f.a 2
3.b odd 2 1 180.3.l.a 2
4.b odd 2 1 80.3.p.a 2
5.b even 2 1 100.3.f.a 2
5.c odd 4 1 inner 20.3.f.a 2
5.c odd 4 1 100.3.f.a 2
7.b odd 2 1 980.3.l.a 2
8.b even 2 1 320.3.p.c 2
8.d odd 2 1 320.3.p.g 2
12.b even 2 1 720.3.bh.e 2
15.d odd 2 1 900.3.l.a 2
15.e even 4 1 180.3.l.a 2
15.e even 4 1 900.3.l.a 2
20.d odd 2 1 400.3.p.d 2
20.e even 4 1 80.3.p.a 2
20.e even 4 1 400.3.p.d 2
35.f even 4 1 980.3.l.a 2
40.i odd 4 1 320.3.p.c 2
40.k even 4 1 320.3.p.g 2
60.l odd 4 1 720.3.bh.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.f.a 2 1.a even 1 1 trivial
20.3.f.a 2 5.c odd 4 1 inner
80.3.p.a 2 4.b odd 2 1
80.3.p.a 2 20.e even 4 1
100.3.f.a 2 5.b even 2 1
100.3.f.a 2 5.c odd 4 1
180.3.l.a 2 3.b odd 2 1
180.3.l.a 2 15.e even 4 1
320.3.p.c 2 8.b even 2 1
320.3.p.c 2 40.i odd 4 1
320.3.p.g 2 8.d odd 2 1
320.3.p.g 2 40.k even 4 1
400.3.p.d 2 20.d odd 2 1
400.3.p.d 2 20.e even 4 1
720.3.bh.e 2 12.b even 2 1
720.3.bh.e 2 60.l odd 4 1
900.3.l.a 2 15.d odd 2 1
900.3.l.a 2 15.e even 4 1
980.3.l.a 2 7.b odd 2 1
980.3.l.a 2 35.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(20, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - 2 T + 2 T^{2} - 18 T^{3} + 81 T^{4} \)
$5$ \( 1 + 6 T + 25 T^{2} \)
$7$ \( ( 1 + 7 T )^{2}( 1 + 49 T^{2} ) \)
$11$ \( ( 1 - 10 T + 121 T^{2} )^{2} \)
$13$ \( 1 - 18 T + 162 T^{2} - 3042 T^{3} + 28561 T^{4} \)
$17$ \( 1 - 2 T + 2 T^{2} - 578 T^{3} + 83521 T^{4} \)
$19$ \( 1 - 658 T^{2} + 130321 T^{4} \)
$23$ \( ( 1 + 23 T )^{2}( 1 + 529 T^{2} ) \)
$29$ \( 1 - 1618 T^{2} + 707281 T^{4} \)
$31$ \( ( 1 + 14 T + 961 T^{2} )^{2} \)
$37$ \( 1 - 66 T + 2178 T^{2} - 90354 T^{3} + 1874161 T^{4} \)
$41$ \( ( 1 + 14 T + 1681 T^{2} )^{2} \)
$43$ \( 1 + 30 T + 450 T^{2} + 55470 T^{3} + 3418801 T^{4} \)
$47$ \( 1 + 78 T + 3042 T^{2} + 172302 T^{3} + 4879681 T^{4} \)
$53$ \( 1 + 14 T + 98 T^{2} + 39326 T^{3} + 7890481 T^{4} \)
$59$ \( 1 - 3826 T^{2} + 12117361 T^{4} \)
$61$ \( ( 1 - 42 T + 3721 T^{2} )^{2} \)
$67$ \( 1 + 14 T + 98 T^{2} + 62846 T^{3} + 20151121 T^{4} \)
$71$ \( ( 1 - 98 T + 5041 T^{2} )^{2} \)
$73$ \( 1 - 98 T + 4802 T^{2} - 522242 T^{3} + 28398241 T^{4} \)
$79$ \( 1 - 3266 T^{2} + 38950081 T^{4} \)
$83$ \( 1 + 126 T + 7938 T^{2} + 868014 T^{3} + 47458321 T^{4} \)
$89$ \( 1 - 3298 T^{2} + 62742241 T^{4} \)
$97$ \( 1 - 66 T + 2178 T^{2} - 620994 T^{3} + 88529281 T^{4} \)
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