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: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.

Hecke kernels

There are no other newforms in \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\).