Properties

Label 20.7.d.a
Level 20
Weight 7
Character orbit 20.d
Self dual yes
Analytic conductor 4.601
Analytic rank 0
Dimension 1
CM discriminant -20
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(4.60108167240\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 8q^{2} - 44q^{3} + 64q^{4} - 125q^{5} + 352q^{6} + 524q^{7} - 512q^{8} + 1207q^{9} + O(q^{10}) \) \( q - 8q^{2} - 44q^{3} + 64q^{4} - 125q^{5} + 352q^{6} + 524q^{7} - 512q^{8} + 1207q^{9} + 1000q^{10} - 2816q^{12} - 4192q^{14} + 5500q^{15} + 4096q^{16} - 9656q^{18} - 8000q^{20} - 23056q^{21} + 15356q^{23} + 22528q^{24} + 15625q^{25} - 21032q^{27} + 33536q^{28} + 44858q^{29} - 44000q^{30} - 32768q^{32} - 65500q^{35} + 77248q^{36} + 64000q^{40} - 74338q^{41} + 184448q^{42} - 17404q^{43} - 150875q^{45} - 122848q^{46} + 26444q^{47} - 180224q^{48} + 156927q^{49} - 125000q^{50} + 168256q^{54} - 268288q^{56} - 358864q^{58} + 352000q^{60} + 452342q^{61} + 632468q^{63} + 262144q^{64} - 1276q^{67} - 675664q^{69} + 524000q^{70} - 617984q^{72} - 687500q^{75} - 512000q^{80} + 45505q^{81} + 594704q^{82} + 1131716q^{83} - 1475584q^{84} + 139232q^{86} - 1973752q^{87} + 511058q^{89} + 1207000q^{90} + 982784q^{92} - 211552q^{94} + 1441792q^{96} - 1255416q^{98} + O(q^{100}) \)

Character values

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

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

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
0
−8.00000 −44.0000 64.0000 −125.000 352.000 524.000 −512.000 1207.00 1000.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.7.d.a 1
3.b odd 2 1 180.7.f.b 1
4.b odd 2 1 20.7.d.b yes 1
5.b even 2 1 20.7.d.b yes 1
5.c odd 4 2 100.7.b.d 2
8.b even 2 1 320.7.h.b 1
8.d odd 2 1 320.7.h.a 1
12.b even 2 1 180.7.f.a 1
15.d odd 2 1 180.7.f.a 1
20.d odd 2 1 CM 20.7.d.a 1
20.e even 4 2 100.7.b.d 2
40.e odd 2 1 320.7.h.b 1
40.f even 2 1 320.7.h.a 1
60.h even 2 1 180.7.f.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.7.d.a 1 1.a even 1 1 trivial
20.7.d.a 1 20.d odd 2 1 CM
20.7.d.b yes 1 4.b odd 2 1
20.7.d.b yes 1 5.b even 2 1
100.7.b.d 2 5.c odd 4 2
100.7.b.d 2 20.e even 4 2
180.7.f.a 1 12.b even 2 1
180.7.f.a 1 15.d odd 2 1
180.7.f.b 1 3.b odd 2 1
180.7.f.b 1 60.h even 2 1
320.7.h.a 1 8.d odd 2 1
320.7.h.a 1 40.f even 2 1
320.7.h.b 1 8.b even 2 1
320.7.h.b 1 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 44 \) acting on \(S_{7}^{\mathrm{new}}(20, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 8 T \)
$3$ \( 1 + 44 T + 729 T^{2} \)
$5$ \( 1 + 125 T \)
$7$ \( 1 - 524 T + 117649 T^{2} \)
$11$ \( ( 1 - 1331 T )( 1 + 1331 T ) \)
$13$ \( ( 1 - 2197 T )( 1 + 2197 T ) \)
$17$ \( ( 1 - 4913 T )( 1 + 4913 T ) \)
$19$ \( ( 1 - 6859 T )( 1 + 6859 T ) \)
$23$ \( 1 - 15356 T + 148035889 T^{2} \)
$29$ \( 1 - 44858 T + 594823321 T^{2} \)
$31$ \( ( 1 - 29791 T )( 1 + 29791 T ) \)
$37$ \( ( 1 - 50653 T )( 1 + 50653 T ) \)
$41$ \( 1 + 74338 T + 4750104241 T^{2} \)
$43$ \( 1 + 17404 T + 6321363049 T^{2} \)
$47$ \( 1 - 26444 T + 10779215329 T^{2} \)
$53$ \( ( 1 - 148877 T )( 1 + 148877 T ) \)
$59$ \( ( 1 - 205379 T )( 1 + 205379 T ) \)
$61$ \( 1 - 452342 T + 51520374361 T^{2} \)
$67$ \( 1 + 1276 T + 90458382169 T^{2} \)
$71$ \( ( 1 - 357911 T )( 1 + 357911 T ) \)
$73$ \( ( 1 - 389017 T )( 1 + 389017 T ) \)
$79$ \( ( 1 - 493039 T )( 1 + 493039 T ) \)
$83$ \( 1 - 1131716 T + 326940373369 T^{2} \)
$89$ \( 1 - 511058 T + 496981290961 T^{2} \)
$97$ \( ( 1 - 912673 T )( 1 + 912673 T ) \)
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