Properties

 Label 2304.4.a.v Level $2304$ Weight $4$ Character orbit 2304.a Self dual yes Analytic conductor $135.940$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{5} - 8 q^{7}+O(q^{10})$$ q + 2*b * q^5 - 8 * q^7 $$q + 2 \beta q^{5} - 8 q^{7} - 3 \beta q^{11} - 10 \beta q^{13} + 14 q^{17} - 7 \beta q^{19} + 152 q^{23} - 13 q^{25} - 30 \beta q^{29} - 224 q^{31} - 16 \beta q^{35} + 46 \beta q^{37} - 70 q^{41} + 83 \beta q^{43} + 336 q^{47} - 279 q^{49} - 6 \beta q^{53} - 168 q^{55} + 101 \beta q^{59} - 18 \beta q^{61} - 560 q^{65} + 33 \beta q^{67} + 72 q^{71} + 294 q^{73} + 24 \beta q^{77} + 464 q^{79} + 103 \beta q^{83} + 28 \beta q^{85} + 266 q^{89} + 80 \beta q^{91} - 392 q^{95} + 994 q^{97} +O(q^{100})$$ q + 2*b * q^5 - 8 * q^7 - 3*b * q^11 - 10*b * q^13 + 14 * q^17 - 7*b * q^19 + 152 * q^23 - 13 * q^25 - 30*b * q^29 - 224 * q^31 - 16*b * q^35 + 46*b * q^37 - 70 * q^41 + 83*b * q^43 + 336 * q^47 - 279 * q^49 - 6*b * q^53 - 168 * q^55 + 101*b * q^59 - 18*b * q^61 - 560 * q^65 + 33*b * q^67 + 72 * q^71 + 294 * q^73 + 24*b * q^77 + 464 * q^79 + 103*b * q^83 + 28*b * q^85 + 266 * q^89 + 80*b * q^91 - 392 * q^95 + 994 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{7}+O(q^{10})$$ 2 * q - 16 * q^7 $$2 q - 16 q^{7} + 28 q^{17} + 304 q^{23} - 26 q^{25} - 448 q^{31} - 140 q^{41} + 672 q^{47} - 558 q^{49} - 336 q^{55} - 1120 q^{65} + 144 q^{71} + 588 q^{73} + 928 q^{79} + 532 q^{89} - 784 q^{95} + 1988 q^{97}+O(q^{100})$$ 2 * q - 16 * q^7 + 28 * q^17 + 304 * q^23 - 26 * q^25 - 448 * q^31 - 140 * q^41 + 672 * q^47 - 558 * q^49 - 336 * q^55 - 1120 * q^65 + 144 * q^71 + 588 * q^73 + 928 * q^79 + 532 * q^89 - 784 * q^95 + 1988 * 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
 −2.64575 2.64575
0 0 0 −10.5830 0 −8.00000 0 0 0
1.2 0 0 0 10.5830 0 −8.00000 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.b even 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.v 2
3.b odd 2 1 256.4.a.j 2
4.b odd 2 1 2304.4.a.bn 2
8.b even 2 1 inner 2304.4.a.v 2
8.d odd 2 1 2304.4.a.bn 2
12.b even 2 1 256.4.a.l 2
16.e even 4 2 288.4.d.a 2
16.f odd 4 2 72.4.d.b 2
24.f even 2 1 256.4.a.l 2
24.h odd 2 1 256.4.a.j 2
48.i odd 4 2 32.4.b.a 2
48.k even 4 2 8.4.b.a 2
240.t even 4 2 200.4.d.a 2
240.z odd 4 2 200.4.f.a 4
240.bb even 4 2 800.4.f.a 4
240.bd odd 4 2 200.4.f.a 4
240.bf even 4 2 800.4.f.a 4
240.bm odd 4 2 800.4.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 48.k even 4 2
32.4.b.a 2 48.i odd 4 2
72.4.d.b 2 16.f odd 4 2
200.4.d.a 2 240.t even 4 2
200.4.f.a 4 240.z odd 4 2
200.4.f.a 4 240.bd odd 4 2
256.4.a.j 2 3.b odd 2 1
256.4.a.j 2 24.h odd 2 1
256.4.a.l 2 12.b even 2 1
256.4.a.l 2 24.f even 2 1
288.4.d.a 2 16.e even 4 2
800.4.d.a 2 240.bm odd 4 2
800.4.f.a 4 240.bb even 4 2
800.4.f.a 4 240.bf even 4 2
2304.4.a.v 2 1.a even 1 1 trivial
2304.4.a.v 2 8.b even 2 1 inner
2304.4.a.bn 2 4.b odd 2 1
2304.4.a.bn 2 8.d odd 2 1

Hecke kernels

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

 $$T_{5}^{2} - 112$$ T5^2 - 112 $$T_{7} + 8$$ T7 + 8 $$T_{11}^{2} - 252$$ T11^2 - 252 $$T_{13}^{2} - 2800$$ T13^2 - 2800 $$T_{17} - 14$$ T17 - 14 $$T_{19}^{2} - 1372$$ T19^2 - 1372

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 112$$
$7$ $$(T + 8)^{2}$$
$11$ $$T^{2} - 252$$
$13$ $$T^{2} - 2800$$
$17$ $$(T - 14)^{2}$$
$19$ $$T^{2} - 1372$$
$23$ $$(T - 152)^{2}$$
$29$ $$T^{2} - 25200$$
$31$ $$(T + 224)^{2}$$
$37$ $$T^{2} - 59248$$
$41$ $$(T + 70)^{2}$$
$43$ $$T^{2} - 192892$$
$47$ $$(T - 336)^{2}$$
$53$ $$T^{2} - 1008$$
$59$ $$T^{2} - 285628$$
$61$ $$T^{2} - 9072$$
$67$ $$T^{2} - 30492$$
$71$ $$(T - 72)^{2}$$
$73$ $$(T - 294)^{2}$$
$79$ $$(T - 464)^{2}$$
$83$ $$T^{2} - 297052$$
$89$ $$(T - 266)^{2}$$
$97$ $$(T - 994)^{2}$$