# Properties

 Label 24.3.h.a Level 24 Weight 3 Character orbit 24.h Self dual yes Analytic conductor 0.654 Analytic rank 0 Dimension 1 CM discriminant -24 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 24.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.653952634465$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{2} + 3q^{3} + 4q^{4} + 2q^{5} - 6q^{6} - 10q^{7} - 8q^{8} + 9q^{9} + O(q^{10})$$ $$q - 2q^{2} + 3q^{3} + 4q^{4} + 2q^{5} - 6q^{6} - 10q^{7} - 8q^{8} + 9q^{9} - 4q^{10} - 10q^{11} + 12q^{12} + 20q^{14} + 6q^{15} + 16q^{16} - 18q^{18} + 8q^{20} - 30q^{21} + 20q^{22} - 24q^{24} - 21q^{25} + 27q^{27} - 40q^{28} + 50q^{29} - 12q^{30} + 38q^{31} - 32q^{32} - 30q^{33} - 20q^{35} + 36q^{36} - 16q^{40} + 60q^{42} - 40q^{44} + 18q^{45} + 48q^{48} + 51q^{49} + 42q^{50} - 94q^{53} - 54q^{54} - 20q^{55} + 80q^{56} - 100q^{58} - 10q^{59} + 24q^{60} - 76q^{62} - 90q^{63} + 64q^{64} + 60q^{66} + 40q^{70} - 72q^{72} + 50q^{73} - 63q^{75} + 100q^{77} - 58q^{79} + 32q^{80} + 81q^{81} + 134q^{83} - 120q^{84} + 150q^{87} + 80q^{88} - 36q^{90} + 114q^{93} - 96q^{96} - 190q^{97} - 102q^{98} - 90q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$13$$ $$17$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0
−2.00000 3.00000 4.00000 2.00000 −6.00000 −10.0000 −8.00000 9.00000 −4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.3.h.a 1
3.b odd 2 1 24.3.h.b yes 1
4.b odd 2 1 96.3.h.a 1
8.b even 2 1 24.3.h.b yes 1
8.d odd 2 1 96.3.h.b 1
12.b even 2 1 96.3.h.b 1
16.e even 4 2 768.3.e.d 2
16.f odd 4 2 768.3.e.c 2
24.f even 2 1 96.3.h.a 1
24.h odd 2 1 CM 24.3.h.a 1
48.i odd 4 2 768.3.e.d 2
48.k even 4 2 768.3.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.h.a 1 1.a even 1 1 trivial
24.3.h.a 1 24.h odd 2 1 CM
24.3.h.b yes 1 3.b odd 2 1
24.3.h.b yes 1 8.b even 2 1
96.3.h.a 1 4.b odd 2 1
96.3.h.a 1 24.f even 2 1
96.3.h.b 1 8.d odd 2 1
96.3.h.b 1 12.b even 2 1
768.3.e.c 2 16.f odd 4 2
768.3.e.c 2 48.k even 4 2
768.3.e.d 2 16.e even 4 2
768.3.e.d 2 48.i odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 2$$ acting on $$S_{3}^{\mathrm{new}}(24, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T$$
$3$ $$1 - 3 T$$
$5$ $$1 - 2 T + 25 T^{2}$$
$7$ $$1 + 10 T + 49 T^{2}$$
$11$ $$1 + 10 T + 121 T^{2}$$
$13$ $$( 1 - 13 T )( 1 + 13 T )$$
$17$ $$( 1 - 17 T )( 1 + 17 T )$$
$19$ $$( 1 - 19 T )( 1 + 19 T )$$
$23$ $$( 1 - 23 T )( 1 + 23 T )$$
$29$ $$1 - 50 T + 841 T^{2}$$
$31$ $$1 - 38 T + 961 T^{2}$$
$37$ $$( 1 - 37 T )( 1 + 37 T )$$
$41$ $$( 1 - 41 T )( 1 + 41 T )$$
$43$ $$( 1 - 43 T )( 1 + 43 T )$$
$47$ $$( 1 - 47 T )( 1 + 47 T )$$
$53$ $$1 + 94 T + 2809 T^{2}$$
$59$ $$1 + 10 T + 3481 T^{2}$$
$61$ $$( 1 - 61 T )( 1 + 61 T )$$
$67$ $$( 1 - 67 T )( 1 + 67 T )$$
$71$ $$( 1 - 71 T )( 1 + 71 T )$$
$73$ $$1 - 50 T + 5329 T^{2}$$
$79$ $$1 + 58 T + 6241 T^{2}$$
$83$ $$1 - 134 T + 6889 T^{2}$$
$89$ $$( 1 - 89 T )( 1 + 89 T )$$
$97$ $$1 + 190 T + 9409 T^{2}$$