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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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This is the discriminant modular form $\Delta=\sum \tau(n)q^n$, where $\tau$ is the Ramanujan tau function [A000594]. It is the minimal weight newform of level $1$.

Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.768343180560\)
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

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

Display 100 coefficients 10 coefficients

Expression as an eta quotient

\(f(z) = \eta(z)^{24}=q\prod_{n=1}^\infty(1 - q^{n})^{24}\)

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1.12.a.a ✓ 1
3.b odd 2 1 9.12.a.b 1
4.b odd 2 1 16.12.a.a 1
5.b even 2 1 25.12.a.b 1
5.c odd 4 2 25.12.b.b 2
7.b odd 2 1 49.12.a.a 1
7.c even 3 2 49.12.c.b 2
7.d odd 6 2 49.12.c.c 2
8.b even 2 1 64.12.a.b 1
8.d odd 2 1 64.12.a.f 1
9.c even 3 2 81.12.c.d 2
9.d odd 6 2 81.12.c.b 2
11.b odd 2 1 121.12.a.b 1
12.b even 2 1 144.12.a.d 1
13.b even 2 1 169.12.a.a 1
15.d odd 2 1 225.12.a.b 1
15.e even 4 2 225.12.b.d 2
16.e even 4 2 256.12.b.e 2
16.f odd 4 2 256.12.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a ✓ 1 1.a even 1 1 trivial
9.12.a.b 1 3.b odd 2 1
16.12.a.a 1 4.b odd 2 1
25.12.a.b 1 5.b even 2 1
25.12.b.b 2 5.c odd 4 2
49.12.a.a 1 7.b odd 2 1
49.12.c.b 2 7.c even 3 2
49.12.c.c 2 7.d odd 6 2
64.12.a.b 1 8.b even 2 1
64.12.a.f 1 8.d odd 2 1
81.12.c.b 2 9.d odd 6 2
81.12.c.d 2 9.c even 3 2
121.12.a.b 1 11.b odd 2 1
144.12.a.d 1 12.b even 2 1
169.12.a.a 1 13.b even 2 1
225.12.a.b 1 15.d odd 2 1
225.12.b.d 2 15.e even 4 2
256.12.b.c 2 16.f odd 4 2
256.12.b.e 2 16.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 24 + T \)
$3$ \( -252 + T \)
$5$ \( -4830 + T \)
$7$ \( 16744 + T \)
$11$ \( -534612 + T \)
$13$ \( 577738 + T \)
$17$ \( 6905934 + T \)
$19$ \( -10661420 + T \)
$23$ \( -18643272 + T \)
$29$ \( -128406630 + T \)
$31$ \( 52843168 + T \)
$37$ \( 182213314 + T \)
$41$ \( -308120442 + T \)
$43$ \( 17125708 + T \)
$47$ \( -2687348496 + T \)
$53$ \( 1596055698 + T \)
$59$ \( 5189203740 + T \)
$61$ \( -6956478662 + T \)
$67$ \( 15481826884 + T \)
$71$ \( -9791485272 + T \)
$73$ \( -1463791322 + T \)
$79$ \( -38116845680 + T \)
$83$ \( 29335099668 + T \)
$89$ \( 24992917110 + T \)
$97$ \( -75013568546 + T \)
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Additional information

\(\displaystyle q\prod_{n\geq1}(1-q^n)^{24} \)

\(\eta(z)^{24}\)