# 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 $1$

# Related objects

Show commands: Magma / PariGP / SageMath

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

comment: Compute space of new eigenforms

[N,k,chi] = [1,12,Mod(1,1)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1, base_ring=CyclotomicField(1))

chi = DirichletCharacter(H, H._module([]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

$p$ $F_p(T)$
$2$ $$T + 24$$
$3$ $$T - 252$$
$5$ $$T - 4830$$
$7$ $$T + 16744$$
$11$ $$T - 534612$$
$13$ $$T + 577738$$
$17$ $$T + 6905934$$
$19$ $$T - 10661420$$
$23$ $$T - 18643272$$
$29$ $$T - 128406630$$
$31$ $$T + 52843168$$
$37$ $$T + 182213314$$
$41$ $$T - 308120442$$
$43$ $$T + 17125708$$
$47$ $$T - 2687348496$$
$53$ $$T + 1596055698$$
$59$ $$T + 5189203740$$
$61$ $$T - 6956478662$$
$67$ $$T + 15481826884$$
$71$ $$T - 9791485272$$
$73$ $$T - 1463791322$$
$79$ $$T - 38116845680$$
$83$ $$T + 29335099668$$
$89$ $$T + 24992917110$$
$97$ $$T - 75013568546$$
$$\displaystyle q\prod_{n\geq1}(1-q^n)^{24}$$
$$\eta(z)^{24}$$