Properties

Label 1.12.a.a
Level 1
Weight 12
Character orbit 1.a
Self dual Yes
Analytic conductor 0.768
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.76834318056\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 24q^{2} \) \(\mathstrut +\mathstrut 252q^{3} \) \(\mathstrut -\mathstrut 1472q^{4} \) \(\mathstrut +\mathstrut 4830q^{5} \) \(\mathstrut -\mathstrut 6048q^{6} \) \(\mathstrut -\mathstrut 16744q^{7} \) \(\mathstrut +\mathstrut 84480q^{8} \) \(\mathstrut -\mathstrut 113643q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 24q^{2} \) \(\mathstrut +\mathstrut 252q^{3} \) \(\mathstrut -\mathstrut 1472q^{4} \) \(\mathstrut +\mathstrut 4830q^{5} \) \(\mathstrut -\mathstrut 6048q^{6} \) \(\mathstrut -\mathstrut 16744q^{7} \) \(\mathstrut +\mathstrut 84480q^{8} \) \(\mathstrut -\mathstrut 113643q^{9} \) \(\mathstrut -\mathstrut 115920q^{10} \) \(\mathstrut +\mathstrut 534612q^{11} \) \(\mathstrut -\mathstrut 370944q^{12} \) \(\mathstrut -\mathstrut 577738q^{13} \) \(\mathstrut +\mathstrut 401856q^{14} \) \(\mathstrut +\mathstrut 1217160q^{15} \) \(\mathstrut +\mathstrut 987136q^{16} \) \(\mathstrut -\mathstrut 6905934q^{17} \) \(\mathstrut +\mathstrut 2727432q^{18} \) \(\mathstrut +\mathstrut 10661420q^{19} \) \(\mathstrut -\mathstrut 7109760q^{20} \) \(\mathstrut -\mathstrut 4219488q^{21} \) \(\mathstrut -\mathstrut 12830688q^{22} \) \(\mathstrut +\mathstrut 18643272q^{23} \) \(\mathstrut +\mathstrut 21288960q^{24} \) \(\mathstrut -\mathstrut 25499225q^{25} \) \(\mathstrut +\mathstrut 13865712q^{26} \) \(\mathstrut -\mathstrut 73279080q^{27} \) \(\mathstrut +\mathstrut 24647168q^{28} \) \(\mathstrut +\mathstrut 128406630q^{29} \) \(\mathstrut -\mathstrut 29211840q^{30} \) \(\mathstrut -\mathstrut 52843168q^{31} \) \(\mathstrut -\mathstrut 196706304q^{32} \) \(\mathstrut +\mathstrut 134722224q^{33} \) \(\mathstrut +\mathstrut 165742416q^{34} \) \(\mathstrut -\mathstrut 80873520q^{35} \) \(\mathstrut +\mathstrut 167282496q^{36} \) \(\mathstrut -\mathstrut 182213314q^{37} \) \(\mathstrut -\mathstrut 255874080q^{38} \) \(\mathstrut -\mathstrut 145589976q^{39} \) \(\mathstrut +\mathstrut 408038400q^{40} \) \(\mathstrut +\mathstrut 308120442q^{41} \) \(\mathstrut +\mathstrut 101267712q^{42} \) \(\mathstrut -\mathstrut 17125708q^{43} \) \(\mathstrut -\mathstrut 786948864q^{44} \) \(\mathstrut -\mathstrut 548895690q^{45} \) \(\mathstrut -\mathstrut 447438528q^{46} \) \(\mathstrut +\mathstrut 2687348496q^{47} \) \(\mathstrut +\mathstrut 248758272q^{48} \) \(\mathstrut -\mathstrut 1696965207q^{49} \) \(\mathstrut +\mathstrut 611981400q^{50} \) \(\mathstrut -\mathstrut 1740295368q^{51} \) \(\mathstrut +\mathstrut 850430336q^{52} \) \(\mathstrut -\mathstrut 1596055698q^{53} \) \(\mathstrut +\mathstrut 1758697920q^{54} \) \(\mathstrut +\mathstrut 2582175960q^{55} \) \(\mathstrut -\mathstrut 1414533120q^{56} \) \(\mathstrut +\mathstrut 2686677840q^{57} \) \(\mathstrut -\mathstrut 3081759120q^{58} \) \(\mathstrut -\mathstrut 5189203740q^{59} \) \(\mathstrut -\mathstrut 1791659520q^{60} \) \(\mathstrut +\mathstrut 6956478662q^{61} \) \(\mathstrut +\mathstrut 1268236032q^{62} \) \(\mathstrut +\mathstrut 1902838392q^{63} \) \(\mathstrut +\mathstrut 2699296768q^{64} \) \(\mathstrut -\mathstrut 2790474540q^{65} \) \(\mathstrut -\mathstrut 3233333376q^{66} \) \(\mathstrut -\mathstrut 15481826884q^{67} \) \(\mathstrut +\mathstrut 10165534848q^{68} \) \(\mathstrut +\mathstrut 4698104544q^{69} \) \(\mathstrut +\mathstrut 1940964480q^{70} \) \(\mathstrut +\mathstrut 9791485272q^{71} \) \(\mathstrut -\mathstrut 9600560640q^{72} \) \(\mathstrut +\mathstrut 1463791322q^{73} \) \(\mathstrut +\mathstrut 4373119536q^{74} \) \(\mathstrut -\mathstrut 6425804700q^{75} \) \(\mathstrut -\mathstrut 15693610240q^{76} \) \(\mathstrut -\mathstrut 8951543328q^{77} \) \(\mathstrut +\mathstrut 3494159424q^{78} \) \(\mathstrut +\mathstrut 38116845680q^{79} \) \(\mathstrut +\mathstrut 4767866880q^{80} \) \(\mathstrut +\mathstrut 1665188361q^{81} \) \(\mathstrut -\mathstrut 7394890608q^{82} \) \(\mathstrut -\mathstrut 29335099668q^{83} \) \(\mathstrut +\mathstrut 6211086336q^{84} \) \(\mathstrut -\mathstrut 33355661220q^{85} \) \(\mathstrut +\mathstrut 411016992q^{86} \) \(\mathstrut +\mathstrut 32358470760q^{87} \) \(\mathstrut +\mathstrut 45164021760q^{88} \) \(\mathstrut -\mathstrut 24992917110q^{89} \) \(\mathstrut +\mathstrut 13173496560q^{90} \) \(\mathstrut +\mathstrut 9673645072q^{91} \) \(\mathstrut -\mathstrut 27442896384q^{92} \) \(\mathstrut -\mathstrut 13316478336q^{93} \) \(\mathstrut -\mathstrut 64496363904q^{94} \) \(\mathstrut +\mathstrut 51494658600q^{95} \) \(\mathstrut -\mathstrut 49569988608q^{96} \) \(\mathstrut +\mathstrut 75013568546q^{97} \) \(\mathstrut +\mathstrut 40727164968q^{98} \) \(\mathstrut -\mathstrut 60754911516q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−24.0000 252.000 −1472.00 4830.00 −6048.00 −16744.0 84480.0 −113643. −115920.
\(n\): 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 \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\).