Properties

Label 20.4.c.a
Level 20
Weight 4
Character orbit 20.c
Analytic conductor 1.180
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 20.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.18003820011\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + ( 7 + \beta ) q^{5} + \beta q^{7} -49 q^{9} +O(q^{10})\) \( q -\beta q^{3} + ( 7 + \beta ) q^{5} + \beta q^{7} -49 q^{9} + 20 q^{11} + 6 \beta q^{13} + ( 76 - 7 \beta ) q^{15} -8 \beta q^{17} -84 q^{19} + 76 q^{21} -7 \beta q^{23} + ( -27 + 14 \beta ) q^{25} + 22 \beta q^{27} + 6 q^{29} -224 q^{31} -20 \beta q^{33} + ( -76 + 7 \beta ) q^{35} -14 \beta q^{37} + 456 q^{39} + 266 q^{41} + 35 \beta q^{43} + ( -343 - 49 \beta ) q^{45} -43 \beta q^{47} + 267 q^{49} -608 q^{51} + 42 \beta q^{53} + ( 140 + 20 \beta ) q^{55} + 84 \beta q^{57} -28 q^{59} + 182 q^{61} -49 \beta q^{63} + ( -456 + 42 \beta ) q^{65} -49 \beta q^{67} -532 q^{69} + 408 q^{71} -124 \beta q^{73} + ( 1064 + 27 \beta ) q^{75} + 20 \beta q^{77} + 48 q^{79} + 349 q^{81} + 23 \beta q^{83} + ( 608 - 56 \beta ) q^{85} -6 \beta q^{87} -1526 q^{89} -456 q^{91} + 224 \beta q^{93} + ( -588 - 84 \beta ) q^{95} -64 \beta q^{97} -980 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14q^{5} - 98q^{9} + O(q^{10}) \) \( 2q + 14q^{5} - 98q^{9} + 40q^{11} + 152q^{15} - 168q^{19} + 152q^{21} - 54q^{25} + 12q^{29} - 448q^{31} - 152q^{35} + 912q^{39} + 532q^{41} - 686q^{45} + 534q^{49} - 1216q^{51} + 280q^{55} - 56q^{59} + 364q^{61} - 912q^{65} - 1064q^{69} + 816q^{71} + 2128q^{75} + 96q^{79} + 698q^{81} + 1216q^{85} - 3052q^{89} - 912q^{91} - 1176q^{95} - 1960q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
0.500000 + 2.17945i
0.500000 2.17945i
0 8.71780i 0 7.00000 + 8.71780i 0 8.71780i 0 −49.0000 0
9.2 0 8.71780i 0 7.00000 8.71780i 0 8.71780i 0 −49.0000 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.4.c.a 2
3.b odd 2 1 180.4.d.a 2
4.b odd 2 1 80.4.c.b 2
5.b even 2 1 inner 20.4.c.a 2
5.c odd 4 2 100.4.a.d 2
7.b odd 2 1 980.4.e.a 2
8.b even 2 1 320.4.c.a 2
8.d odd 2 1 320.4.c.b 2
12.b even 2 1 720.4.f.a 2
15.d odd 2 1 180.4.d.a 2
15.e even 4 2 900.4.a.s 2
20.d odd 2 1 80.4.c.b 2
20.e even 4 2 400.4.a.w 2
35.c odd 2 1 980.4.e.a 2
40.e odd 2 1 320.4.c.b 2
40.f even 2 1 320.4.c.a 2
40.i odd 4 2 1600.4.a.cj 2
40.k even 4 2 1600.4.a.ck 2
60.h even 2 1 720.4.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.c.a 2 1.a even 1 1 trivial
20.4.c.a 2 5.b even 2 1 inner
80.4.c.b 2 4.b odd 2 1
80.4.c.b 2 20.d odd 2 1
100.4.a.d 2 5.c odd 4 2
180.4.d.a 2 3.b odd 2 1
180.4.d.a 2 15.d odd 2 1
320.4.c.a 2 8.b even 2 1
320.4.c.a 2 40.f even 2 1
320.4.c.b 2 8.d odd 2 1
320.4.c.b 2 40.e odd 2 1
400.4.a.w 2 20.e even 4 2
720.4.f.a 2 12.b even 2 1
720.4.f.a 2 60.h even 2 1
900.4.a.s 2 15.e even 4 2
980.4.e.a 2 7.b odd 2 1
980.4.e.a 2 35.c odd 2 1
1600.4.a.cj 2 40.i odd 4 2
1600.4.a.ck 2 40.k even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(20, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 22 T^{2} + 729 T^{4} \)
$5$ \( 1 - 14 T + 125 T^{2} \)
$7$ \( ( 1 - 36 T + 343 T^{2} )( 1 + 36 T + 343 T^{2} ) \)
$11$ \( ( 1 - 20 T + 1331 T^{2} )^{2} \)
$13$ \( 1 - 1658 T^{2} + 4826809 T^{4} \)
$17$ \( 1 - 4962 T^{2} + 24137569 T^{4} \)
$19$ \( ( 1 + 84 T + 6859 T^{2} )^{2} \)
$23$ \( ( 1 - 212 T + 12167 T^{2} )( 1 + 212 T + 12167 T^{2} ) \)
$29$ \( ( 1 - 6 T + 24389 T^{2} )^{2} \)
$31$ \( ( 1 + 224 T + 29791 T^{2} )^{2} \)
$37$ \( 1 - 86410 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 - 266 T + 68921 T^{2} )^{2} \)
$43$ \( 1 - 65914 T^{2} + 6321363049 T^{4} \)
$47$ \( 1 - 67122 T^{2} + 10779215329 T^{4} \)
$53$ \( 1 - 163690 T^{2} + 22164361129 T^{4} \)
$59$ \( ( 1 + 28 T + 205379 T^{2} )^{2} \)
$61$ \( ( 1 - 182 T + 226981 T^{2} )^{2} \)
$67$ \( 1 - 419050 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 - 408 T + 357911 T^{2} )^{2} \)
$73$ \( 1 + 390542 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 - 48 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 1103370 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 + 1526 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 1514050 T^{2} + 832972004929 T^{4} \)
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