# Properties

 Label 3136.2.a.bu Level $3136$ Weight $2$ Character orbit 3136.a Self dual yes Analytic conductor $25.041$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3136 = 2^{6} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3136.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.0410860739$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 224) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + q^{5}+O(q^{10})$$ q + b * q^3 + q^5 $$q + \beta q^{3} + q^{5} - 3 \beta q^{11} + \beta q^{15} - 5 q^{17} + \beta q^{19} - \beta q^{23} - 4 q^{25} - 3 \beta q^{27} - 8 q^{29} + 5 \beta q^{31} - 9 q^{33} + 5 q^{37} - 4 q^{41} - 4 \beta q^{43} - 5 \beta q^{47} - 5 \beta q^{51} + q^{53} - 3 \beta q^{55} + 3 q^{57} + \beta q^{59} + 11 q^{61} + 7 \beta q^{67} - 3 q^{69} - 8 \beta q^{71} - 15 q^{73} - 4 \beta q^{75} - \beta q^{79} - 9 q^{81} + 4 \beta q^{83} - 5 q^{85} - 8 \beta q^{87} - 7 q^{89} + 15 q^{93} + \beta q^{95} - 12 q^{97} +O(q^{100})$$ q + b * q^3 + q^5 - 3*b * q^11 + b * q^15 - 5 * q^17 + b * q^19 - b * q^23 - 4 * q^25 - 3*b * q^27 - 8 * q^29 + 5*b * q^31 - 9 * q^33 + 5 * q^37 - 4 * q^41 - 4*b * q^43 - 5*b * q^47 - 5*b * q^51 + q^53 - 3*b * q^55 + 3 * q^57 + b * q^59 + 11 * q^61 + 7*b * q^67 - 3 * q^69 - 8*b * q^71 - 15 * q^73 - 4*b * q^75 - b * q^79 - 9 * q^81 + 4*b * q^83 - 5 * q^85 - 8*b * q^87 - 7 * q^89 + 15 * q^93 + b * q^95 - 12 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5}+O(q^{10})$$ 2 * q + 2 * q^5 $$2 q + 2 q^{5} - 10 q^{17} - 8 q^{25} - 16 q^{29} - 18 q^{33} + 10 q^{37} - 8 q^{41} + 2 q^{53} + 6 q^{57} + 22 q^{61} - 6 q^{69} - 30 q^{73} - 18 q^{81} - 10 q^{85} - 14 q^{89} + 30 q^{93} - 24 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - 10 * q^17 - 8 * q^25 - 16 * q^29 - 18 * q^33 + 10 * q^37 - 8 * q^41 + 2 * q^53 + 6 * q^57 + 22 * q^61 - 6 * q^69 - 30 * q^73 - 18 * q^81 - 10 * q^85 - 14 * q^89 + 30 * q^93 - 24 * 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 −1.73205 0 1.00000 0 0 0 0 0
1.2 0 1.73205 0 1.00000 0 0 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
$$2$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.a.bu 2
4.b odd 2 1 inner 3136.2.a.bu 2
7.b odd 2 1 3136.2.a.bh 2
7.d odd 6 2 448.2.i.i 4
8.b even 2 1 1568.2.a.n 2
8.d odd 2 1 1568.2.a.n 2
28.d even 2 1 3136.2.a.bh 2
28.f even 6 2 448.2.i.i 4
56.e even 2 1 1568.2.a.s 2
56.h odd 2 1 1568.2.a.s 2
56.j odd 6 2 224.2.i.b 4
56.k odd 6 2 1568.2.i.u 4
56.m even 6 2 224.2.i.b 4
56.p even 6 2 1568.2.i.u 4
168.ba even 6 2 2016.2.s.r 4
168.be odd 6 2 2016.2.s.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.b 4 56.j odd 6 2
224.2.i.b 4 56.m even 6 2
448.2.i.i 4 7.d odd 6 2
448.2.i.i 4 28.f even 6 2
1568.2.a.n 2 8.b even 2 1
1568.2.a.n 2 8.d odd 2 1
1568.2.a.s 2 56.e even 2 1
1568.2.a.s 2 56.h odd 2 1
1568.2.i.u 4 56.k odd 6 2
1568.2.i.u 4 56.p even 6 2
2016.2.s.r 4 168.ba even 6 2
2016.2.s.r 4 168.be odd 6 2
3136.2.a.bh 2 7.b odd 2 1
3136.2.a.bh 2 28.d even 2 1
3136.2.a.bu 2 1.a even 1 1 trivial
3136.2.a.bu 2 4.b 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3136))$$:

 $$T_{3}^{2} - 3$$ T3^2 - 3 $$T_{5} - 1$$ T5 - 1 $$T_{11}^{2} - 27$$ T11^2 - 27

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 27$$
$13$ $$T^{2}$$
$17$ $$(T + 5)^{2}$$
$19$ $$T^{2} - 3$$
$23$ $$T^{2} - 3$$
$29$ $$(T + 8)^{2}$$
$31$ $$T^{2} - 75$$
$37$ $$(T - 5)^{2}$$
$41$ $$(T + 4)^{2}$$
$43$ $$T^{2} - 48$$
$47$ $$T^{2} - 75$$
$53$ $$(T - 1)^{2}$$
$59$ $$T^{2} - 3$$
$61$ $$(T - 11)^{2}$$
$67$ $$T^{2} - 147$$
$71$ $$T^{2} - 192$$
$73$ $$(T + 15)^{2}$$
$79$ $$T^{2} - 3$$
$83$ $$T^{2} - 48$$
$89$ $$(T + 7)^{2}$$
$97$ $$(T + 12)^{2}$$