# Properties

 Label 24.3.h.c Level 24 Weight 3 Character orbit 24.h Analytic conductor 0.654 Analytic rank 0 Dimension 4 CM no Inner twists 4

# 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: no Analytic conductor: $$0.653952634465$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-7})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{2} - \beta_{3} ) q^{3} + ( -3 + \beta_{3} ) q^{4} + 4 \beta_{2} q^{5} + ( 1 - \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{6} + 4 q^{7} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{8} + ( -5 - 4 \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{2} - \beta_{3} ) q^{3} + ( -3 + \beta_{3} ) q^{4} + 4 \beta_{2} q^{5} + ( 1 - \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{6} + 4 q^{7} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{8} + ( -5 - 4 \beta_{1} - 2 \beta_{2} ) q^{9} + ( -4 - 4 \beta_{3} ) q^{10} -6 \beta_{2} q^{11} + ( 7 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{12} + 4 \beta_{3} q^{13} + 4 \beta_{1} q^{14} + ( -8 + 8 \beta_{1} + 4 \beta_{2} ) q^{15} + ( 2 - 6 \beta_{3} ) q^{16} + ( 8 \beta_{1} + 4 \beta_{2} ) q^{17} + ( 14 - 5 \beta_{1} - 2 \beta_{3} ) q^{18} + 2 \beta_{3} q^{19} + ( -8 \beta_{1} - 16 \beta_{2} ) q^{20} + ( -4 \beta_{2} - 4 \beta_{3} ) q^{21} + ( 6 + 6 \beta_{3} ) q^{22} + ( -16 \beta_{1} - 8 \beta_{2} ) q^{23} + ( -10 + 10 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} ) q^{24} + 7 q^{25} + ( 4 \beta_{1} + 16 \beta_{2} ) q^{26} + ( 19 \beta_{2} + \beta_{3} ) q^{27} + ( -12 + 4 \beta_{3} ) q^{28} -12 \beta_{2} q^{29} + ( -28 - 8 \beta_{1} + 4 \beta_{3} ) q^{30} -4 q^{31} + ( -4 \beta_{1} - 24 \beta_{2} ) q^{32} + ( 12 - 12 \beta_{1} - 6 \beta_{2} ) q^{33} + ( -28 + 4 \beta_{3} ) q^{34} + 16 \beta_{2} q^{35} + ( 15 + 12 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} ) q^{36} -20 \beta_{3} q^{37} + ( 2 \beta_{1} + 8 \beta_{2} ) q^{38} + ( 28 + 8 \beta_{1} + 4 \beta_{2} ) q^{39} + ( 40 + 8 \beta_{3} ) q^{40} + ( 16 \beta_{1} + 8 \beta_{2} ) q^{41} + ( 4 - 4 \beta_{1} - 16 \beta_{2} + 4 \beta_{3} ) q^{42} -2 \beta_{3} q^{43} + ( 12 \beta_{1} + 24 \beta_{2} ) q^{44} + ( -20 \beta_{2} + 16 \beta_{3} ) q^{45} + ( 56 - 8 \beta_{3} ) q^{46} + ( -42 - 12 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{48} -33 q^{49} + 7 \beta_{1} q^{50} + ( -28 \beta_{2} + 8 \beta_{3} ) q^{51} + ( -28 - 12 \beta_{3} ) q^{52} + 36 \beta_{2} q^{53} + ( -19 + \beta_{1} + 4 \beta_{2} - 19 \beta_{3} ) q^{54} -48 q^{55} + ( -8 \beta_{1} + 16 \beta_{2} ) q^{56} + ( 14 + 4 \beta_{1} + 2 \beta_{2} ) q^{57} + ( 12 + 12 \beta_{3} ) q^{58} -34 \beta_{2} q^{59} + ( 24 - 24 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{60} + 36 \beta_{3} q^{61} -4 \beta_{1} q^{62} + ( -20 - 16 \beta_{1} - 8 \beta_{2} ) q^{63} + ( 36 + 20 \beta_{3} ) q^{64} + ( -32 \beta_{1} - 16 \beta_{2} ) q^{65} + ( 42 + 12 \beta_{1} - 6 \beta_{3} ) q^{66} -18 \beta_{3} q^{67} + ( -24 \beta_{1} + 16 \beta_{2} ) q^{68} + ( 56 \beta_{2} - 16 \beta_{3} ) q^{69} + ( -16 - 16 \beta_{3} ) q^{70} + ( 48 \beta_{1} + 24 \beta_{2} ) q^{71} + ( -28 + 10 \beta_{1} - 20 \beta_{2} + 20 \beta_{3} ) q^{72} -6 q^{73} + ( -20 \beta_{1} - 80 \beta_{2} ) q^{74} + ( -7 \beta_{2} - 7 \beta_{3} ) q^{75} + ( -14 - 6 \beta_{3} ) q^{76} -24 \beta_{2} q^{77} + ( -28 + 28 \beta_{1} + 4 \beta_{3} ) q^{78} + 124 q^{79} + ( 48 \beta_{1} + 32 \beta_{2} ) q^{80} + ( -31 + 40 \beta_{1} + 20 \beta_{2} ) q^{81} + ( -56 + 8 \beta_{3} ) q^{82} + 2 \beta_{2} q^{83} + ( 28 + 8 \beta_{1} + 16 \beta_{2} + 12 \beta_{3} ) q^{84} -32 \beta_{3} q^{85} + ( -2 \beta_{1} - 8 \beta_{2} ) q^{86} + ( 24 - 24 \beta_{1} - 12 \beta_{2} ) q^{87} + ( -60 - 12 \beta_{3} ) q^{88} + ( -56 \beta_{1} - 28 \beta_{2} ) q^{89} + ( 20 + 16 \beta_{1} + 64 \beta_{2} + 20 \beta_{3} ) q^{90} + 16 \beta_{3} q^{91} + ( 48 \beta_{1} - 32 \beta_{2} ) q^{92} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{93} + ( -16 \beta_{1} - 8 \beta_{2} ) q^{95} + ( 44 - 44 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{96} + 118 q^{97} -33 \beta_{1} q^{98} + ( 30 \beta_{2} - 24 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{4} + 4q^{6} + 16q^{7} - 20q^{9} + O(q^{10})$$ $$4q - 12q^{4} + 4q^{6} + 16q^{7} - 20q^{9} - 16q^{10} + 28q^{12} - 32q^{15} + 8q^{16} + 56q^{18} + 24q^{22} - 40q^{24} + 28q^{25} - 48q^{28} - 112q^{30} - 16q^{31} + 48q^{33} - 112q^{34} + 60q^{36} + 112q^{39} + 160q^{40} + 16q^{42} + 224q^{46} - 168q^{48} - 132q^{49} - 112q^{52} - 76q^{54} - 192q^{55} + 56q^{57} + 48q^{58} + 96q^{60} - 80q^{63} + 144q^{64} + 168q^{66} - 64q^{70} - 112q^{72} - 24q^{73} - 56q^{76} - 112q^{78} + 496q^{79} - 124q^{81} - 224q^{82} + 112q^{84} + 96q^{87} - 240q^{88} + 80q^{90} + 176q^{96} + 472q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 6 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 3$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 2 \beta_{1}$$

## 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
 −0.707107 − 1.87083i −0.707107 + 1.87083i 0.707107 − 1.87083i 0.707107 + 1.87083i
−0.707107 1.87083i −1.41421 2.64575i −3.00000 + 2.64575i 5.65685 −3.94975 + 4.51658i 4.00000 7.07107 + 3.74166i −5.00000 + 7.48331i −4.00000 10.5830i
5.2 −0.707107 + 1.87083i −1.41421 + 2.64575i −3.00000 2.64575i 5.65685 −3.94975 4.51658i 4.00000 7.07107 3.74166i −5.00000 7.48331i −4.00000 + 10.5830i
5.3 0.707107 1.87083i 1.41421 + 2.64575i −3.00000 2.64575i −5.65685 5.94975 0.774923i 4.00000 −7.07107 + 3.74166i −5.00000 + 7.48331i −4.00000 + 10.5830i
5.4 0.707107 + 1.87083i 1.41421 2.64575i −3.00000 + 2.64575i −5.65685 5.94975 + 0.774923i 4.00000 −7.07107 3.74166i −5.00000 7.48331i −4.00000 10.5830i
 $$n$$: 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
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 32$$ acting on $$S_{3}^{\mathrm{new}}(24, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T^{2} + 16 T^{4}$$
$3$ $$1 + 10 T^{2} + 81 T^{4}$$
$5$ $$( 1 + 18 T^{2} + 625 T^{4} )^{2}$$
$7$ $$( 1 - 4 T + 49 T^{2} )^{4}$$
$11$ $$( 1 + 170 T^{2} + 14641 T^{4} )^{2}$$
$13$ $$( 1 - 226 T^{2} + 28561 T^{4} )^{2}$$
$17$ $$( 1 - 354 T^{2} + 83521 T^{4} )^{2}$$
$19$ $$( 1 - 694 T^{2} + 130321 T^{4} )^{2}$$
$23$ $$( 1 - 162 T^{2} + 279841 T^{4} )^{2}$$
$29$ $$( 1 + 1394 T^{2} + 707281 T^{4} )^{2}$$
$31$ $$( 1 + 4 T + 961 T^{2} )^{4}$$
$37$ $$( 1 + 62 T^{2} + 1874161 T^{4} )^{2}$$
$41$ $$( 1 - 2466 T^{2} + 2825761 T^{4} )^{2}$$
$43$ $$( 1 - 3670 T^{2} + 3418801 T^{4} )^{2}$$
$47$ $$( 1 - 47 T )^{4}( 1 + 47 T )^{4}$$
$53$ $$( 1 + 3026 T^{2} + 7890481 T^{4} )^{2}$$
$59$ $$( 1 + 4650 T^{2} + 12117361 T^{4} )^{2}$$
$61$ $$( 1 + 1630 T^{2} + 13845841 T^{4} )^{2}$$
$67$ $$( 1 - 6710 T^{2} + 20151121 T^{4} )^{2}$$
$71$ $$( 1 - 110 T + 5041 T^{2} )^{2}( 1 + 110 T + 5041 T^{2} )^{2}$$
$73$ $$( 1 + 6 T + 5329 T^{2} )^{4}$$
$79$ $$( 1 - 124 T + 6241 T^{2} )^{4}$$
$83$ $$( 1 + 13770 T^{2} + 47458321 T^{4} )^{2}$$
$89$ $$( 1 - 4866 T^{2} + 62742241 T^{4} )^{2}$$
$97$ $$( 1 - 118 T + 9409 T^{2} )^{4}$$