# Properties

 Label 24.8.f.a Level 24 Weight 8 Character orbit 24.f Analytic conductor 7.497 Analytic rank 0 Dimension 2 CM discriminant -8 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ = $$8$$ Character orbit: $$[\chi]$$ = 24.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.49724061162$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ 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.

 $$f(q)$$ $$=$$ $$q + 8 \beta q^{2} + ( -43 + 13 \beta ) q^{3} -128 q^{4} + ( -208 - 344 \beta ) q^{6} -1024 \beta q^{8} + ( 1511 - 1118 \beta ) q^{9} +O(q^{10})$$ $$q + 8 \beta q^{2} + ( -43 + 13 \beta ) q^{3} -128 q^{4} + ( -208 - 344 \beta ) q^{6} -1024 \beta q^{8} + ( 1511 - 1118 \beta ) q^{9} -362 \beta q^{11} + ( 5504 - 1664 \beta ) q^{12} + 16384 q^{16} -23972 \beta q^{17} + ( 17888 + 12088 \beta ) q^{18} -59722 q^{19} + 5792 q^{22} + ( 26624 + 44032 \beta ) q^{24} -78125 q^{25} + ( -35905 + 67717 \beta ) q^{27} + 131072 \beta q^{32} + ( 9412 + 15566 \beta ) q^{33} + 383552 q^{34} + ( -193408 + 143104 \beta ) q^{36} -477776 \beta q^{38} -601208 \beta q^{41} -220510 q^{43} + 46336 \beta q^{44} + ( -704512 + 212992 \beta ) q^{48} + 823543 q^{49} -625000 \beta q^{50} + ( 623272 + 1030796 \beta ) q^{51} + ( -1083472 - 287240 \beta ) q^{54} + ( 2568046 - 776386 \beta ) q^{57} + 2108530 \beta q^{59} -2097152 q^{64} + ( -249056 + 75296 \beta ) q^{66} -3851302 q^{67} + 3068416 \beta q^{68} + ( -2289664 - 1547264 \beta ) q^{72} -4865614 q^{73} + ( 3359375 - 1015625 \beta ) q^{75} + 7644416 q^{76} + ( -216727 - 3378596 \beta ) q^{81} + 9619328 q^{82} -6535226 \beta q^{83} -1764080 \beta q^{86} -741376 q^{88} + 7965436 \beta q^{89} + ( -3407872 - 5636096 \beta ) q^{96} -9938890 q^{97} + 6588344 \beta q^{98} + ( -809432 - 546982 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 86q^{3} - 256q^{4} - 416q^{6} + 3022q^{9} + O(q^{10})$$ $$2q - 86q^{3} - 256q^{4} - 416q^{6} + 3022q^{9} + 11008q^{12} + 32768q^{16} + 35776q^{18} - 119444q^{19} + 11584q^{22} + 53248q^{24} - 156250q^{25} - 71810q^{27} + 18824q^{33} + 767104q^{34} - 386816q^{36} - 441020q^{43} - 1409024q^{48} + 1647086q^{49} + 1246544q^{51} - 2166944q^{54} + 5136092q^{57} - 4194304q^{64} - 498112q^{66} - 7702604q^{67} - 4579328q^{72} - 9731228q^{73} + 6718750q^{75} + 15288832q^{76} - 433454q^{81} + 19238656q^{82} - 1482752q^{88} - 6815744q^{96} - 19877780q^{97} - 1618864q^{99} + O(q^{100})$$

## 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
11.3137i −43.0000 18.3848i −128.000 0 −208.000 + 486.489i 0 1448.15i 1511.00 + 1581.09i 0
11.2 11.3137i −43.0000 + 18.3848i −128.000 0 −208.000 486.489i 0 1448.15i 1511.00 1581.09i 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 24.8.f.a 2
3.b odd 2 1 inner 24.8.f.a 2
4.b odd 2 1 96.8.f.a 2
8.b even 2 1 96.8.f.a 2
8.d odd 2 1 CM 24.8.f.a 2
12.b even 2 1 96.8.f.a 2
24.f even 2 1 inner 24.8.f.a 2
24.h odd 2 1 96.8.f.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{8}^{\mathrm{new}}(24, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 128 T^{2}$$
$3$ $$1 + 86 T + 2187 T^{2}$$
$5$ $$( 1 + 78125 T^{2} )^{2}$$
$7$ $$( 1 - 823543 T^{2} )^{2}$$
$11$ $$( 1 - 8814 T + 19487171 T^{2} )( 1 + 8814 T + 19487171 T^{2} )$$
$13$ $$( 1 - 62748517 T^{2} )^{2}$$
$17$ $$( 1 - 22182 T + 410338673 T^{2} )( 1 + 22182 T + 410338673 T^{2} )$$
$19$ $$( 1 + 59722 T + 893871739 T^{2} )^{2}$$
$23$ $$( 1 + 3404825447 T^{2} )^{2}$$
$29$ $$( 1 + 17249876309 T^{2} )^{2}$$
$31$ $$( 1 - 27512614111 T^{2} )^{2}$$
$37$ $$( 1 - 94931877133 T^{2} )^{2}$$
$41$ $$( 1 - 236886 T + 194754273881 T^{2} )( 1 + 236886 T + 194754273881 T^{2} )$$
$43$ $$( 1 + 220510 T + 271818611107 T^{2} )^{2}$$
$47$ $$( 1 + 506623120463 T^{2} )^{2}$$
$53$ $$( 1 + 1174711139837 T^{2} )^{2}$$
$59$ $$( 1 - 1030926 T + 2488651484819 T^{2} )( 1 + 1030926 T + 2488651484819 T^{2} )$$
$61$ $$( 1 - 3142742836021 T^{2} )^{2}$$
$67$ $$( 1 + 3851302 T + 6060711605323 T^{2} )^{2}$$
$71$ $$( 1 + 9095120158391 T^{2} )^{2}$$
$73$ $$( 1 + 4865614 T + 11047398519097 T^{2} )^{2}$$
$79$ $$( 1 - 19203908986159 T^{2} )^{2}$$
$83$ $$( 1 - 4808934 T + 27136050989627 T^{2} )( 1 + 4808934 T + 27136050989627 T^{2} )$$
$89$ $$( 1 - 7073118 T + 44231334895529 T^{2} )( 1 + 7073118 T + 44231334895529 T^{2} )$$
$97$ $$( 1 + 9938890 T + 80798284478113 T^{2} )^{2}$$