# Properties

 Label 24.4.f.a Level 24 Weight 4 Character orbit 24.f Analytic conductor 1.416 Analytic rank 0 Dimension 2 CM discriminant -8 Inner twists 4

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## Newspace parameters

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 24.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.41604584014$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{2} + ( 5 + \beta ) q^{3} -8 q^{4} + ( -4 + 10 \beta ) q^{6} -16 \beta q^{8} + ( 23 + 10 \beta ) q^{9} +O(q^{10})$$ $$q + 2 \beta q^{2} + ( 5 + \beta ) q^{3} -8 q^{4} + ( -4 + 10 \beta ) q^{6} -16 \beta q^{8} + ( 23 + 10 \beta ) q^{9} -50 \beta q^{11} + ( -40 - 8 \beta ) q^{12} + 64 q^{16} + 76 \beta q^{17} + ( -40 + 46 \beta ) q^{18} -106 q^{19} + 200 q^{22} + ( 32 - 80 \beta ) q^{24} -125 q^{25} + ( 95 + 73 \beta ) q^{27} + 128 \beta q^{32} + ( 100 - 250 \beta ) q^{33} -304 q^{34} + ( -184 - 80 \beta ) q^{36} -212 \beta q^{38} + 40 \beta q^{41} + 290 q^{43} + 400 \beta q^{44} + ( 320 + 64 \beta ) q^{48} + 343 q^{49} -250 \beta q^{50} + ( -152 + 380 \beta ) q^{51} + ( -292 + 190 \beta ) q^{54} + ( -530 - 106 \beta ) q^{57} -230 \beta q^{59} -512 q^{64} + ( 1000 + 200 \beta ) q^{66} -70 q^{67} -608 \beta q^{68} + ( 320 - 368 \beta ) q^{72} -430 q^{73} + ( -625 - 125 \beta ) q^{75} + 848 q^{76} + ( 329 + 460 \beta ) q^{81} -160 q^{82} -482 \beta q^{83} + 580 \beta q^{86} -1600 q^{88} + 940 \beta q^{89} + ( -256 + 640 \beta ) q^{96} + 1910 q^{97} + 686 \beta q^{98} + ( 1000 - 1150 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 10q^{3} - 16q^{4} - 8q^{6} + 46q^{9} + O(q^{10})$$ $$2q + 10q^{3} - 16q^{4} - 8q^{6} + 46q^{9} - 80q^{12} + 128q^{16} - 80q^{18} - 212q^{19} + 400q^{22} + 64q^{24} - 250q^{25} + 190q^{27} + 200q^{33} - 608q^{34} - 368q^{36} + 580q^{43} + 640q^{48} + 686q^{49} - 304q^{51} - 584q^{54} - 1060q^{57} - 1024q^{64} + 2000q^{66} - 140q^{67} + 640q^{72} - 860q^{73} - 1250q^{75} + 1696q^{76} + 658q^{81} - 320q^{82} - 3200q^{88} - 512q^{96} + 3820q^{97} + 2000q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 − 1.41421i 1.41421i
2.82843i 5.00000 1.41421i −8.00000 0 −4.00000 14.1421i 0 22.6274i 23.0000 14.1421i 0
11.2 2.82843i 5.00000 + 1.41421i −8.00000 0 −4.00000 + 14.1421i 0 22.6274i 23.0000 + 14.1421i 0
 $$n$$: 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
3.b odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.4.f.a 2
3.b odd 2 1 inner 24.4.f.a 2
4.b odd 2 1 96.4.f.a 2
8.b even 2 1 96.4.f.a 2
8.d odd 2 1 CM 24.4.f.a 2
12.b even 2 1 96.4.f.a 2
16.e even 4 2 768.4.c.l 4
16.f odd 4 2 768.4.c.l 4
24.f even 2 1 inner 24.4.f.a 2
24.h odd 2 1 96.4.f.a 2
48.i odd 4 2 768.4.c.l 4
48.k even 4 2 768.4.c.l 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{4}^{\mathrm{new}}(24, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 8 T^{2}$$
$3$ $$1 - 10 T + 27 T^{2}$$
$5$ $$( 1 + 125 T^{2} )^{2}$$
$7$ $$( 1 - 343 T^{2} )^{2}$$
$11$ $$( 1 - 18 T + 1331 T^{2} )( 1 + 18 T + 1331 T^{2} )$$
$13$ $$( 1 - 2197 T^{2} )^{2}$$
$17$ $$( 1 - 90 T + 4913 T^{2} )( 1 + 90 T + 4913 T^{2} )$$
$19$ $$( 1 + 106 T + 6859 T^{2} )^{2}$$
$23$ $$( 1 + 12167 T^{2} )^{2}$$
$29$ $$( 1 + 24389 T^{2} )^{2}$$
$31$ $$( 1 - 29791 T^{2} )^{2}$$
$37$ $$( 1 - 50653 T^{2} )^{2}$$
$41$ $$( 1 - 522 T + 68921 T^{2} )( 1 + 522 T + 68921 T^{2} )$$
$43$ $$( 1 - 290 T + 79507 T^{2} )^{2}$$
$47$ $$( 1 + 103823 T^{2} )^{2}$$
$53$ $$( 1 + 148877 T^{2} )^{2}$$
$59$ $$( 1 - 846 T + 205379 T^{2} )( 1 + 846 T + 205379 T^{2} )$$
$61$ $$( 1 - 226981 T^{2} )^{2}$$
$67$ $$( 1 + 70 T + 300763 T^{2} )^{2}$$
$71$ $$( 1 + 357911 T^{2} )^{2}$$
$73$ $$( 1 + 430 T + 389017 T^{2} )^{2}$$
$79$ $$( 1 - 493039 T^{2} )^{2}$$
$83$ $$( 1 - 1350 T + 571787 T^{2} )( 1 + 1350 T + 571787 T^{2} )$$
$89$ $$( 1 - 1026 T + 704969 T^{2} )( 1 + 1026 T + 704969 T^{2} )$$
$97$ $$( 1 - 1910 T + 912673 T^{2} )^{2}$$
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