# Properties

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

# 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{3} + (2 \beta - 1) q^{5} - 2 \beta q^{9} +O(q^{10})$$ q + (b - 1) * q^3 + (2*b - 1) * q^5 - 2*b * q^9 $$q + (\beta - 1) q^{3} + (2 \beta - 1) q^{5} - 2 \beta q^{9} + (\beta + 1) q^{11} - 2 \beta q^{13} + ( - 3 \beta + 5) q^{15} + (2 \beta - 3) q^{17} + ( - \beta - 5) q^{19} + ( - 3 \beta - 1) q^{23} + ( - 4 \beta + 4) q^{25} + ( - \beta - 1) q^{27} + 2 \beta q^{29} + ( - \beta + 7) q^{31} + q^{33} + ( - 4 \beta - 3) q^{37} + (2 \beta - 4) q^{39} + ( - 2 \beta - 4) q^{41} + ( - 4 \beta - 4) q^{43} + (2 \beta - 8) q^{45} + (\beta + 9) q^{47} + ( - 5 \beta + 7) q^{51} + q^{53} + (\beta + 3) q^{55} + ( - 4 \beta + 3) q^{57} + ( - 7 \beta - 1) q^{59} + ( - 4 \beta - 3) q^{61} + (2 \beta - 8) q^{65} + ( - 3 \beta + 7) q^{67} + (2 \beta - 5) q^{69} + (4 \beta + 8) q^{71} + ( - 4 \beta - 9) q^{73} + (8 \beta - 12) q^{75} + (5 \beta - 1) q^{79} + (6 \beta - 1) q^{81} + (8 \beta - 4) q^{83} + ( - 8 \beta + 11) q^{85} + ( - 2 \beta + 4) q^{87} - 9 q^{89} + (8 \beta - 9) q^{93} + ( - 9 \beta + 1) q^{95} + (2 \beta - 4) q^{97} + ( - 2 \beta - 4) q^{99} +O(q^{100})$$ q + (b - 1) * q^3 + (2*b - 1) * q^5 - 2*b * q^9 + (b + 1) * q^11 - 2*b * q^13 + (-3*b + 5) * q^15 + (2*b - 3) * q^17 + (-b - 5) * q^19 + (-3*b - 1) * q^23 + (-4*b + 4) * q^25 + (-b - 1) * q^27 + 2*b * q^29 + (-b + 7) * q^31 + q^33 + (-4*b - 3) * q^37 + (2*b - 4) * q^39 + (-2*b - 4) * q^41 + (-4*b - 4) * q^43 + (2*b - 8) * q^45 + (b + 9) * q^47 + (-5*b + 7) * q^51 + q^53 + (b + 3) * q^55 + (-4*b + 3) * q^57 + (-7*b - 1) * q^59 + (-4*b - 3) * q^61 + (2*b - 8) * q^65 + (-3*b + 7) * q^67 + (2*b - 5) * q^69 + (4*b + 8) * q^71 + (-4*b - 9) * q^73 + (8*b - 12) * q^75 + (5*b - 1) * q^79 + (6*b - 1) * q^81 + (8*b - 4) * q^83 + (-8*b + 11) * q^85 + (-2*b + 4) * q^87 - 9 * q^89 + (8*b - 9) * q^93 + (-9*b + 1) * q^95 + (2*b - 4) * q^97 + (-2*b - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^5 $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{11} + 10 q^{15} - 6 q^{17} - 10 q^{19} - 2 q^{23} + 8 q^{25} - 2 q^{27} + 14 q^{31} + 2 q^{33} - 6 q^{37} - 8 q^{39} - 8 q^{41} - 8 q^{43} - 16 q^{45} + 18 q^{47} + 14 q^{51} + 2 q^{53} + 6 q^{55} + 6 q^{57} - 2 q^{59} - 6 q^{61} - 16 q^{65} + 14 q^{67} - 10 q^{69} + 16 q^{71} - 18 q^{73} - 24 q^{75} - 2 q^{79} - 2 q^{81} - 8 q^{83} + 22 q^{85} + 8 q^{87} - 18 q^{89} - 18 q^{93} + 2 q^{95} - 8 q^{97} - 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^11 + 10 * q^15 - 6 * q^17 - 10 * q^19 - 2 * q^23 + 8 * q^25 - 2 * q^27 + 14 * q^31 + 2 * q^33 - 6 * q^37 - 8 * q^39 - 8 * q^41 - 8 * q^43 - 16 * q^45 + 18 * q^47 + 14 * q^51 + 2 * q^53 + 6 * q^55 + 6 * q^57 - 2 * q^59 - 6 * q^61 - 16 * q^65 + 14 * q^67 - 10 * q^69 + 16 * q^71 - 18 * q^73 - 24 * q^75 - 2 * q^79 - 2 * q^81 - 8 * q^83 + 22 * q^85 + 8 * q^87 - 18 * q^89 - 18 * q^93 + 2 * q^95 - 8 * q^97 - 8 * q^99

## 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
 −1.41421 1.41421
0 −2.41421 0 −3.82843 0 0 0 2.82843 0
1.2 0 0.414214 0 1.82843 0 0 0 −2.82843 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.a.bd 2
4.b odd 2 1 3136.2.a.bw 2
7.b odd 2 1 3136.2.a.bx 2
7.c even 3 2 448.2.i.j 4
8.b even 2 1 1568.2.a.w 2
8.d odd 2 1 1568.2.a.l 2
28.d even 2 1 3136.2.a.be 2
28.g odd 6 2 448.2.i.g 4
56.e even 2 1 1568.2.a.u 2
56.h odd 2 1 1568.2.a.j 2
56.j odd 6 2 1568.2.i.x 4
56.k odd 6 2 224.2.i.d yes 4
56.m even 6 2 1568.2.i.o 4
56.p even 6 2 224.2.i.a 4
168.s odd 6 2 2016.2.s.q 4
168.v even 6 2 2016.2.s.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.a 4 56.p even 6 2
224.2.i.d yes 4 56.k odd 6 2
448.2.i.g 4 28.g odd 6 2
448.2.i.j 4 7.c even 3 2
1568.2.a.j 2 56.h odd 2 1
1568.2.a.l 2 8.d odd 2 1
1568.2.a.u 2 56.e even 2 1
1568.2.a.w 2 8.b even 2 1
1568.2.i.o 4 56.m even 6 2
1568.2.i.x 4 56.j odd 6 2
2016.2.s.q 4 168.s odd 6 2
2016.2.s.s 4 168.v even 6 2
3136.2.a.bd 2 1.a even 1 1 trivial
3136.2.a.be 2 28.d even 2 1
3136.2.a.bw 2 4.b odd 2 1
3136.2.a.bx 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3136))$$:

 $$T_{3}^{2} + 2T_{3} - 1$$ T3^2 + 2*T3 - 1 $$T_{5}^{2} + 2T_{5} - 7$$ T5^2 + 2*T5 - 7 $$T_{11}^{2} - 2T_{11} - 1$$ T11^2 - 2*T11 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 1$$
$5$ $$T^{2} + 2T - 7$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 2T - 1$$
$13$ $$T^{2} - 8$$
$17$ $$T^{2} + 6T + 1$$
$19$ $$T^{2} + 10T + 23$$
$23$ $$T^{2} + 2T - 17$$
$29$ $$T^{2} - 8$$
$31$ $$T^{2} - 14T + 47$$
$37$ $$T^{2} + 6T - 23$$
$41$ $$T^{2} + 8T + 8$$
$43$ $$T^{2} + 8T - 16$$
$47$ $$T^{2} - 18T + 79$$
$53$ $$(T - 1)^{2}$$
$59$ $$T^{2} + 2T - 97$$
$61$ $$T^{2} + 6T - 23$$
$67$ $$T^{2} - 14T + 31$$
$71$ $$T^{2} - 16T + 32$$
$73$ $$T^{2} + 18T + 49$$
$79$ $$T^{2} + 2T - 49$$
$83$ $$T^{2} + 8T - 112$$
$89$ $$(T + 9)^{2}$$
$97$ $$T^{2} + 8T + 8$$