# Properties

 Label 3136.2.a.bv Level $3136$ Weight $2$ Character orbit 3136.a Self dual yes Analytic conductor $25.041$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 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 $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 3 q^{5} + 4 q^{9}+O(q^{10})$$ q + b * q^3 + 3 * q^5 + 4 * q^9 $$q + \beta q^{3} + 3 q^{5} + 4 q^{9} - \beta q^{11} + 4 q^{13} + 3 \beta q^{15} + q^{17} - 3 \beta q^{19} + \beta q^{23} + 4 q^{25} + \beta q^{27} + 4 q^{29} + \beta q^{31} - 7 q^{33} + 5 q^{37} + 4 \beta q^{39} + 8 q^{41} + 4 \beta q^{43} + 12 q^{45} - \beta q^{47} + \beta q^{51} - 7 q^{53} - 3 \beta q^{55} - 21 q^{57} + \beta q^{59} + 5 q^{61} + 12 q^{65} + \beta q^{67} + 7 q^{69} - 9 q^{73} + 4 \beta q^{75} + \beta q^{79} - 5 q^{81} - 4 \beta q^{83} + 3 q^{85} + 4 \beta q^{87} - 9 q^{89} + 7 q^{93} - 9 \beta q^{95} - 8 q^{97} - 4 \beta q^{99} +O(q^{100})$$ q + b * q^3 + 3 * q^5 + 4 * q^9 - b * q^11 + 4 * q^13 + 3*b * q^15 + q^17 - 3*b * q^19 + b * q^23 + 4 * q^25 + b * q^27 + 4 * q^29 + b * q^31 - 7 * q^33 + 5 * q^37 + 4*b * q^39 + 8 * q^41 + 4*b * q^43 + 12 * q^45 - b * q^47 + b * q^51 - 7 * q^53 - 3*b * q^55 - 21 * q^57 + b * q^59 + 5 * q^61 + 12 * q^65 + b * q^67 + 7 * q^69 - 9 * q^73 + 4*b * q^75 + b * q^79 - 5 * q^81 - 4*b * q^83 + 3 * q^85 + 4*b * q^87 - 9 * q^89 + 7 * q^93 - 9*b * q^95 - 8 * q^97 - 4*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{5} + 8 q^{9}+O(q^{10})$$ 2 * q + 6 * q^5 + 8 * q^9 $$2 q + 6 q^{5} + 8 q^{9} + 8 q^{13} + 2 q^{17} + 8 q^{25} + 8 q^{29} - 14 q^{33} + 10 q^{37} + 16 q^{41} + 24 q^{45} - 14 q^{53} - 42 q^{57} + 10 q^{61} + 24 q^{65} + 14 q^{69} - 18 q^{73} - 10 q^{81} + 6 q^{85} - 18 q^{89} + 14 q^{93} - 16 q^{97}+O(q^{100})$$ 2 * q + 6 * q^5 + 8 * q^9 + 8 * q^13 + 2 * q^17 + 8 * q^25 + 8 * q^29 - 14 * q^33 + 10 * q^37 + 16 * q^41 + 24 * q^45 - 14 * q^53 - 42 * q^57 + 10 * q^61 + 24 * q^65 + 14 * q^69 - 18 * q^73 - 10 * q^81 + 6 * q^85 - 18 * q^89 + 14 * q^93 - 16 * 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 −2.64575 0 3.00000 0 0 0 4.00000 0
1.2 0 2.64575 0 3.00000 0 0 0 4.00000 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.bv 2
4.b odd 2 1 inner 3136.2.a.bv 2
7.b odd 2 1 3136.2.a.bg 2
7.c even 3 2 448.2.i.h 4
8.b even 2 1 1568.2.a.m 2
8.d odd 2 1 1568.2.a.m 2
28.d even 2 1 3136.2.a.bg 2
28.g odd 6 2 448.2.i.h 4
56.e even 2 1 1568.2.a.t 2
56.h odd 2 1 1568.2.a.t 2
56.j odd 6 2 1568.2.i.p 4
56.k odd 6 2 224.2.i.c 4
56.m even 6 2 1568.2.i.p 4
56.p even 6 2 224.2.i.c 4
168.s odd 6 2 2016.2.s.o 4
168.v even 6 2 2016.2.s.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.c 4 56.k odd 6 2
224.2.i.c 4 56.p even 6 2
448.2.i.h 4 7.c even 3 2
448.2.i.h 4 28.g odd 6 2
1568.2.a.m 2 8.b even 2 1
1568.2.a.m 2 8.d odd 2 1
1568.2.a.t 2 56.e even 2 1
1568.2.a.t 2 56.h odd 2 1
1568.2.i.p 4 56.j odd 6 2
1568.2.i.p 4 56.m even 6 2
2016.2.s.o 4 168.s odd 6 2
2016.2.s.o 4 168.v even 6 2
3136.2.a.bg 2 7.b odd 2 1
3136.2.a.bg 2 28.d even 2 1
3136.2.a.bv 2 1.a even 1 1 trivial
3136.2.a.bv 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3136))$$:

 $$T_{3}^{2} - 7$$ T3^2 - 7 $$T_{5} - 3$$ T5 - 3 $$T_{11}^{2} - 7$$ T11^2 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 7$$
$5$ $$(T - 3)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 7$$
$13$ $$(T - 4)^{2}$$
$17$ $$(T - 1)^{2}$$
$19$ $$T^{2} - 63$$
$23$ $$T^{2} - 7$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2} - 7$$
$37$ $$(T - 5)^{2}$$
$41$ $$(T - 8)^{2}$$
$43$ $$T^{2} - 112$$
$47$ $$T^{2} - 7$$
$53$ $$(T + 7)^{2}$$
$59$ $$T^{2} - 7$$
$61$ $$(T - 5)^{2}$$
$67$ $$T^{2} - 7$$
$71$ $$T^{2}$$
$73$ $$(T + 9)^{2}$$
$79$ $$T^{2} - 7$$
$83$ $$T^{2} - 112$$
$89$ $$(T + 9)^{2}$$
$97$ $$(T + 8)^{2}$$