# Properties

 Label 24.5.h.a Level $24$ Weight $5$ Character orbit 24.h Self dual yes Analytic conductor $2.481$ Analytic rank $0$ Dimension $1$ CM discriminant -24 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 24.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.48087911401$$ 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

 $$f(q)$$ $$=$$ $$q - 4q^{2} - 9q^{3} + 16q^{4} + 46q^{5} + 36q^{6} + 2q^{7} - 64q^{8} + 81q^{9} + O(q^{10})$$ $$q - 4q^{2} - 9q^{3} + 16q^{4} + 46q^{5} + 36q^{6} + 2q^{7} - 64q^{8} + 81q^{9} - 184q^{10} + 142q^{11} - 144q^{12} - 8q^{14} - 414q^{15} + 256q^{16} - 324q^{18} + 736q^{20} - 18q^{21} - 568q^{22} + 576q^{24} + 1491q^{25} - 729q^{27} + 32q^{28} - 818q^{29} + 1656q^{30} - 478q^{31} - 1024q^{32} - 1278q^{33} + 92q^{35} + 1296q^{36} - 2944q^{40} + 72q^{42} + 2272q^{44} + 3726q^{45} - 2304q^{48} - 2397q^{49} - 5964q^{50} - 3218q^{53} + 2916q^{54} + 6532q^{55} - 128q^{56} + 3272q^{58} + 6862q^{59} - 6624q^{60} + 1912q^{62} + 162q^{63} + 4096q^{64} + 5112q^{66} - 368q^{70} - 5184q^{72} - 8158q^{73} - 13419q^{75} + 284q^{77} - 9118q^{79} + 11776q^{80} + 6561q^{81} - 4178q^{83} - 288q^{84} + 7362q^{87} - 9088q^{88} - 14904q^{90} + 4302q^{93} + 9216q^{96} + 17282q^{97} + 9588q^{98} + 11502q^{99} + O(q^{100})$$

## 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
−4.00000 −9.00000 16.0000 46.0000 36.0000 2.00000 −64.0000 81.0000 −184.000
 $$n$$: 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 24.5.h.a 1
3.b odd 2 1 24.5.h.b yes 1
4.b odd 2 1 96.5.h.b 1
8.b even 2 1 24.5.h.b yes 1
8.d odd 2 1 96.5.h.a 1
12.b even 2 1 96.5.h.a 1
24.f even 2 1 96.5.h.b 1
24.h odd 2 1 CM 24.5.h.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.5.h.a 1 1.a even 1 1 trivial
24.5.h.a 1 24.h odd 2 1 CM
24.5.h.b yes 1 3.b odd 2 1
24.5.h.b yes 1 8.b even 2 1
96.5.h.a 1 8.d odd 2 1
96.5.h.a 1 12.b even 2 1
96.5.h.b 1 4.b odd 2 1
96.5.h.b 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 46$$ acting on $$S_{5}^{\mathrm{new}}(24, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T$$
$3$ $$1 + 9 T$$
$5$ $$1 - 46 T + 625 T^{2}$$
$7$ $$1 - 2 T + 2401 T^{2}$$
$11$ $$1 - 142 T + 14641 T^{2}$$
$13$ $$( 1 - 169 T )( 1 + 169 T )$$
$17$ $$( 1 - 289 T )( 1 + 289 T )$$
$19$ $$( 1 - 361 T )( 1 + 361 T )$$
$23$ $$( 1 - 529 T )( 1 + 529 T )$$
$29$ $$1 + 818 T + 707281 T^{2}$$
$31$ $$1 + 478 T + 923521 T^{2}$$
$37$ $$( 1 - 1369 T )( 1 + 1369 T )$$
$41$ $$( 1 - 1681 T )( 1 + 1681 T )$$
$43$ $$( 1 - 1849 T )( 1 + 1849 T )$$
$47$ $$( 1 - 2209 T )( 1 + 2209 T )$$
$53$ $$1 + 3218 T + 7890481 T^{2}$$
$59$ $$1 - 6862 T + 12117361 T^{2}$$
$61$ $$( 1 - 3721 T )( 1 + 3721 T )$$
$67$ $$( 1 - 4489 T )( 1 + 4489 T )$$
$71$ $$( 1 - 5041 T )( 1 + 5041 T )$$
$73$ $$1 + 8158 T + 28398241 T^{2}$$
$79$ $$1 + 9118 T + 38950081 T^{2}$$
$83$ $$1 + 4178 T + 47458321 T^{2}$$
$89$ $$( 1 - 7921 T )( 1 + 7921 T )$$
$97$ $$1 - 17282 T + 88529281 T^{2}$$