Properties

Label 1.20.a.a
Level 1
Weight 20
Character orbit 1.a
Self dual yes
Analytic conductor 2.288
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.28816696556\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 456q^{2} + 50652q^{3} - 316352q^{4} - 2377410q^{5} + 23097312q^{6} - 16917544q^{7} - 383331840q^{8} + 1403363637q^{9} + O(q^{10}) \) \( q + 456q^{2} + 50652q^{3} - 316352q^{4} - 2377410q^{5} + 23097312q^{6} - 16917544q^{7} - 383331840q^{8} + 1403363637q^{9} - 1084098960q^{10} - 16212108q^{11} - 16023861504q^{12} + 50421615062q^{13} - 7714400064q^{14} - 120420571320q^{15} - 8939761664q^{16} + 225070099506q^{17} + 639933818472q^{18} - 1710278572660q^{19} + 752098408320q^{20} - 856907438688q^{21} - 7392721248q^{22} + 14036534788872q^{23} - 19416524359680q^{24} - 13421408020025q^{25} + 22992256468272q^{26} + 12212307114840q^{27} + 5351898879488q^{28} + 1137835269510q^{29} - 54911780521920q^{30} - 104626880141728q^{31} + 196899752411136q^{32} - 821175694416q^{33} + 102631965374736q^{34} + 40219938281040q^{35} - 443956893292224q^{36} - 169392327370594q^{37} - 779887029132960q^{38} + 2553955646120424q^{39} + 911336949734400q^{40} - 3309984750560838q^{41} - 390749792041728q^{42} + 1127913532193492q^{43} + 5128732790016q^{44} - 3336370744240170q^{45} + 6400659863725632q^{46} + 3498693987674256q^{47} - 452816807804928q^{48} - 11112691890381207q^{49} - 6120162057131400q^{50} + 11400250680177912q^{51} - 15950978768093824q^{52} + 29956294112980302q^{53} + 5568812044367040q^{54} + 38542827680280q^{55} + 6485033269800960q^{56} - 86629030262374320q^{57} + 518852882896560q^{58} + 58391397642732420q^{59} + 38095288578224640q^{60} + 23373685132672742q^{61} - 47709857344627968q^{62} - 23741466076947528q^{63} + 94473296862773248q^{64} - 119872851864549420q^{65} - 374456116653696q^{66} - 205102524257382244q^{67} - 71201376118922112q^{68} + 710978560125944544q^{69} + 18340291856154240q^{70} - 177902341950417768q^{71} - 537953965160302080q^{72} + 299853775038660122q^{73} - 77242901280990864q^{74} - 679821159030306300q^{75} + 541050047018136320q^{76} + 274269050422752q^{77} + 1164603774630913344q^{78} - 92227090144007440q^{79} + 21253478777610240q^{80} - 1012497699493199799q^{81} - 1509353046255742128q^{82} + 1208542823470585932q^{83} + 271084382043826176q^{84} - 535083905266559460q^{85} + 514328570680232352q^{86} + 57633632071220520q^{87} + 6214617189918720q^{88} + 4371201192290304330q^{89} - 1521385059373517520q^{90} - 853009891362447728q^{91} - 4440485853529234944q^{92} - 5299560732938806656q^{93} + 1595404458379460736q^{94} + 4066033381427610600q^{95} + 9973366259128860672q^{96} - 635013222218448094q^{97} - 5067387502013830392q^{98} - 22751482846316796q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
456.000 50652.0 −316352. −2.37741e6 2.30973e7 −1.69175e7 −3.83332e8 1.40336e9 −1.08410e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1.20.a.a 1
3.b odd 2 1 9.20.a.a 1
4.b odd 2 1 16.20.a.a 1
5.b even 2 1 25.20.a.a 1
5.c odd 4 2 25.20.b.a 2
7.b odd 2 1 49.20.a.b 1
8.b even 2 1 64.20.a.b 1
8.d odd 2 1 64.20.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.20.a.a 1 1.a even 1 1 trivial
9.20.a.a 1 3.b odd 2 1
16.20.a.a 1 4.b odd 2 1
25.20.a.a 1 5.b even 2 1
25.20.b.a 2 5.c odd 4 2
49.20.a.b 1 7.b odd 2 1
64.20.a.b 1 8.b even 2 1
64.20.a.h 1 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 456 T + 524288 T^{2} \)
$3$ \( 1 - 50652 T + 1162261467 T^{2} \)
$5$ \( 1 + 2377410 T + 19073486328125 T^{2} \)
$7$ \( 1 + 16917544 T + 11398895185373143 T^{2} \)
$11$ \( 1 + 16212108 T + 61159090448414546291 T^{2} \)
$13$ \( 1 - 50421615062 T + \)\(14\!\cdots\!77\)\( T^{2} \)
$17$ \( 1 - 225070099506 T + \)\(23\!\cdots\!53\)\( T^{2} \)
$19$ \( 1 + 1710278572660 T + \)\(19\!\cdots\!79\)\( T^{2} \)
$23$ \( 1 - 14036534788872 T + \)\(74\!\cdots\!87\)\( T^{2} \)
$29$ \( 1 - 1137835269510 T + \)\(61\!\cdots\!69\)\( T^{2} \)
$31$ \( 1 + 104626880141728 T + \)\(21\!\cdots\!71\)\( T^{2} \)
$37$ \( 1 + 169392327370594 T + \)\(62\!\cdots\!73\)\( T^{2} \)
$41$ \( 1 + 3309984750560838 T + \)\(43\!\cdots\!61\)\( T^{2} \)
$43$ \( 1 - 1127913532193492 T + \)\(10\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 - 3498693987674256 T + \)\(58\!\cdots\!83\)\( T^{2} \)
$53$ \( 1 - 29956294112980302 T + \)\(57\!\cdots\!17\)\( T^{2} \)
$59$ \( 1 - 58391397642732420 T + \)\(44\!\cdots\!39\)\( T^{2} \)
$61$ \( 1 - 23373685132672742 T + \)\(83\!\cdots\!41\)\( T^{2} \)
$67$ \( 1 + 205102524257382244 T + \)\(49\!\cdots\!03\)\( T^{2} \)
$71$ \( 1 + 177902341950417768 T + \)\(14\!\cdots\!31\)\( T^{2} \)
$73$ \( 1 - 299853775038660122 T + \)\(25\!\cdots\!37\)\( T^{2} \)
$79$ \( 1 + 92227090144007440 T + \)\(11\!\cdots\!19\)\( T^{2} \)
$83$ \( 1 - 1208542823470585932 T + \)\(29\!\cdots\!47\)\( T^{2} \)
$89$ \( 1 - 4371201192290304330 T + \)\(10\!\cdots\!09\)\( T^{2} \)
$97$ \( 1 + 635013222218448094 T + \)\(56\!\cdots\!33\)\( T^{2} \)
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