Properties

Label 20.11.d.b
Level 20
Weight 11
Character orbit 20.d
Self dual yes
Analytic conductor 12.707
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: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(12.7071450535\)
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

\(f(q)\) \(=\) \( q + 32q^{2} + 236q^{3} + 1024q^{4} - 3125q^{5} + 7552q^{6} + 33364q^{7} + 32768q^{8} - 3353q^{9} + O(q^{10}) \) \( q + 32q^{2} + 236q^{3} + 1024q^{4} - 3125q^{5} + 7552q^{6} + 33364q^{7} + 32768q^{8} - 3353q^{9} - 100000q^{10} + 241664q^{12} + 1067648q^{14} - 737500q^{15} + 1048576q^{16} - 107296q^{18} - 3200000q^{20} + 7873904q^{21} - 1169564q^{23} + 7733248q^{24} + 9765625q^{25} - 14726872q^{27} + 34164736q^{28} - 38179702q^{29} - 23600000q^{30} + 33554432q^{32} - 104262500q^{35} - 3433472q^{36} - 102400000q^{40} - 211028098q^{41} + 251964928q^{42} - 223663364q^{43} + 10478125q^{45} - 37426048q^{46} + 96887764q^{47} + 247463936q^{48} + 830681247q^{49} + 312500000q^{50} - 471259904q^{54} + 1093271552q^{56} - 1221750464q^{58} - 755200000q^{60} - 1041591898q^{61} - 111869492q^{63} + 1073741824q^{64} + 2343243964q^{67} - 276017104q^{69} - 3336400000q^{70} - 109871104q^{72} + 2304687500q^{75} - 3276800000q^{80} - 3277550495q^{81} - 6752899136q^{82} + 5449159036q^{83} + 8062877696q^{84} - 7157227648q^{86} - 9010409672q^{87} + 11118190898q^{89} + 335300000q^{90} - 1197633536q^{92} + 3100408448q^{94} + 7918845952q^{96} + 26581799904q^{98} + O(q^{100}) \)

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
32.0000 236.000 1024.00 −3125.00 7552.00 33364.0 32768.0 −3353.00 −100000.
\(n\): 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 20.11.d.b yes 1
4.b odd 2 1 20.11.d.a 1
5.b even 2 1 20.11.d.a 1
5.c odd 4 2 100.11.b.c 2
20.d odd 2 1 CM 20.11.d.b yes 1
20.e even 4 2 100.11.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.11.d.a 1 4.b odd 2 1
20.11.d.a 1 5.b even 2 1
20.11.d.b yes 1 1.a even 1 1 trivial
20.11.d.b yes 1 20.d odd 2 1 CM
100.11.b.c 2 5.c odd 4 2
100.11.b.c 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 236 \) acting on \(S_{11}^{\mathrm{new}}(20, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 32 T \)
$3$ \( 1 - 236 T + 59049 T^{2} \)
$5$ \( 1 + 3125 T \)
$7$ \( 1 - 33364 T + 282475249 T^{2} \)
$11$ \( ( 1 - 161051 T )( 1 + 161051 T ) \)
$13$ \( ( 1 - 371293 T )( 1 + 371293 T ) \)
$17$ \( ( 1 - 1419857 T )( 1 + 1419857 T ) \)
$19$ \( ( 1 - 2476099 T )( 1 + 2476099 T ) \)
$23$ \( 1 + 1169564 T + 41426511213649 T^{2} \)
$29$ \( 1 + 38179702 T + 420707233300201 T^{2} \)
$31$ \( ( 1 - 28629151 T )( 1 + 28629151 T ) \)
$37$ \( ( 1 - 69343957 T )( 1 + 69343957 T ) \)
$41$ \( 1 + 211028098 T + 13422659310152401 T^{2} \)
$43$ \( 1 + 223663364 T + 21611482313284249 T^{2} \)
$47$ \( 1 - 96887764 T + 52599132235830049 T^{2} \)
$53$ \( ( 1 - 418195493 T )( 1 + 418195493 T ) \)
$59$ \( ( 1 - 714924299 T )( 1 + 714924299 T ) \)
$61$ \( 1 + 1041591898 T + 713342911662882601 T^{2} \)
$67$ \( 1 - 2343243964 T + 1822837804551761449 T^{2} \)
$71$ \( ( 1 - 1804229351 T )( 1 + 1804229351 T ) \)
$73$ \( ( 1 - 2073071593 T )( 1 + 2073071593 T ) \)
$79$ \( ( 1 - 3077056399 T )( 1 + 3077056399 T ) \)
$83$ \( 1 - 5449159036 T + 15516041187205853449 T^{2} \)
$89$ \( 1 - 11118190898 T + 31181719929966183601 T^{2} \)
$97$ \( ( 1 - 8587340257 T )( 1 + 8587340257 T ) \)
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