# Properties

 Label 2304.4.a.bk Level $2304$ Weight $4$ Character orbit 2304.a Self dual yes Analytic conductor $135.940$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2304.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$135.940400653$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 192) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} - 7 \beta q^{7} +O(q^{10})$$ q + b * q^5 - 7*b * q^7 $$q + \beta q^{5} - 7 \beta q^{7} + 48 q^{11} - 12 \beta q^{13} - 54 q^{17} + 4 q^{19} + 50 \beta q^{23} - 113 q^{25} + 47 \beta q^{29} + 17 \beta q^{31} - 84 q^{35} + 94 \beta q^{37} - 294 q^{41} - 188 q^{43} - 146 \beta q^{47} + 245 q^{49} + 215 \beta q^{53} + 48 \beta q^{55} + 252 q^{59} + 26 \beta q^{61} - 144 q^{65} - 628 q^{67} - 2 \beta q^{71} + 1006 q^{73} - 336 \beta q^{77} - 387 \beta q^{79} - 720 q^{83} - 54 \beta q^{85} - 1482 q^{89} + 1008 q^{91} + 4 \beta q^{95} + 1822 q^{97} +O(q^{100})$$ q + b * q^5 - 7*b * q^7 + 48 * q^11 - 12*b * q^13 - 54 * q^17 + 4 * q^19 + 50*b * q^23 - 113 * q^25 + 47*b * q^29 + 17*b * q^31 - 84 * q^35 + 94*b * q^37 - 294 * q^41 - 188 * q^43 - 146*b * q^47 + 245 * q^49 + 215*b * q^53 + 48*b * q^55 + 252 * q^59 + 26*b * q^61 - 144 * q^65 - 628 * q^67 - 2*b * q^71 + 1006 * q^73 - 336*b * q^77 - 387*b * q^79 - 720 * q^83 - 54*b * q^85 - 1482 * q^89 + 1008 * q^91 + 4*b * q^95 + 1822 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 96 q^{11} - 108 q^{17} + 8 q^{19} - 226 q^{25} - 168 q^{35} - 588 q^{41} - 376 q^{43} + 490 q^{49} + 504 q^{59} - 288 q^{65} - 1256 q^{67} + 2012 q^{73} - 1440 q^{83} - 2964 q^{89} + 2016 q^{91} + 3644 q^{97}+O(q^{100})$$ 2 * q + 96 * q^11 - 108 * q^17 + 8 * q^19 - 226 * q^25 - 168 * q^35 - 588 * q^41 - 376 * q^43 + 490 * q^49 + 504 * q^59 - 288 * q^65 - 1256 * q^67 + 2012 * q^73 - 1440 * q^83 - 2964 * q^89 + 2016 * q^91 + 3644 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
0 0 0 −3.46410 0 24.2487 0 0 0
1.2 0 0 0 3.46410 0 −24.2487 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.4.a.bk 2
3.b odd 2 1 768.4.a.f 2
4.b odd 2 1 2304.4.a.x 2
8.b even 2 1 2304.4.a.x 2
8.d odd 2 1 inner 2304.4.a.bk 2
12.b even 2 1 768.4.a.o 2
16.e even 4 2 576.4.d.c 4
16.f odd 4 2 576.4.d.c 4
24.f even 2 1 768.4.a.f 2
24.h odd 2 1 768.4.a.o 2
48.i odd 4 2 192.4.d.c 4
48.k even 4 2 192.4.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.4.d.c 4 48.i odd 4 2
192.4.d.c 4 48.k even 4 2
576.4.d.c 4 16.e even 4 2
576.4.d.c 4 16.f odd 4 2
768.4.a.f 2 3.b odd 2 1
768.4.a.f 2 24.f even 2 1
768.4.a.o 2 12.b even 2 1
768.4.a.o 2 24.h odd 2 1
2304.4.a.x 2 4.b odd 2 1
2304.4.a.x 2 8.b even 2 1
2304.4.a.bk 2 1.a even 1 1 trivial
2304.4.a.bk 2 8.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2304))$$:

 $$T_{5}^{2} - 12$$ T5^2 - 12 $$T_{7}^{2} - 588$$ T7^2 - 588 $$T_{11} - 48$$ T11 - 48 $$T_{13}^{2} - 1728$$ T13^2 - 1728 $$T_{17} + 54$$ T17 + 54 $$T_{19} - 4$$ T19 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 12$$
$7$ $$T^{2} - 588$$
$11$ $$(T - 48)^{2}$$
$13$ $$T^{2} - 1728$$
$17$ $$(T + 54)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} - 30000$$
$29$ $$T^{2} - 26508$$
$31$ $$T^{2} - 3468$$
$37$ $$T^{2} - 106032$$
$41$ $$(T + 294)^{2}$$
$43$ $$(T + 188)^{2}$$
$47$ $$T^{2} - 255792$$
$53$ $$T^{2} - 554700$$
$59$ $$(T - 252)^{2}$$
$61$ $$T^{2} - 8112$$
$67$ $$(T + 628)^{2}$$
$71$ $$T^{2} - 48$$
$73$ $$(T - 1006)^{2}$$
$79$ $$T^{2} - 1797228$$
$83$ $$(T + 720)^{2}$$
$89$ $$(T + 1482)^{2}$$
$97$ $$(T - 1822)^{2}$$