# Properties

 Label 3136.2.a.bf Level $3136$ Weight $2$ Character orbit 3136.a Self dual yes Analytic conductor $25.041$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \beta q^{3} + ( 2 - 2 \beta ) q^{5} + ( 1 + 4 \beta ) q^{9} +O(q^{10})$$ $$q -2 \beta q^{3} + ( 2 - 2 \beta ) q^{5} + ( 1 + 4 \beta ) q^{9} + ( 4 - 4 \beta ) q^{11} + ( 2 + 2 \beta ) q^{13} + 4 q^{15} + ( -2 + 4 \beta ) q^{17} + 2 \beta q^{19} + 4 q^{23} + ( 3 - 4 \beta ) q^{25} + ( -8 - 4 \beta ) q^{27} + ( 2 - 4 \beta ) q^{29} -4 \beta q^{31} + 8 q^{33} + ( 2 - 4 \beta ) q^{37} + ( -4 - 8 \beta ) q^{39} + ( 6 - 4 \beta ) q^{41} + ( -4 + 4 \beta ) q^{43} + ( -6 - 2 \beta ) q^{45} + ( -8 + 4 \beta ) q^{47} + ( -8 - 4 \beta ) q^{51} + 10 q^{53} + ( 16 - 8 \beta ) q^{55} + ( -4 - 4 \beta ) q^{57} + ( -8 + 2 \beta ) q^{59} + ( 10 - 2 \beta ) q^{61} -4 \beta q^{65} -4 q^{67} -8 \beta q^{69} + 8 \beta q^{71} + ( -2 - 8 \beta ) q^{73} + ( 8 + 2 \beta ) q^{75} + ( 8 - 8 \beta ) q^{79} + ( 5 + 12 \beta ) q^{81} + ( -8 + 2 \beta ) q^{83} + ( -12 + 4 \beta ) q^{85} + ( 8 + 4 \beta ) q^{87} + 6 q^{89} + ( 8 + 8 \beta ) q^{93} -4 q^{95} + ( -10 + 4 \beta ) q^{97} + ( -12 - 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{5} + 6q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{5} + 6q^{9} + 4q^{11} + 6q^{13} + 8q^{15} + 2q^{19} + 8q^{23} + 2q^{25} - 20q^{27} - 4q^{31} + 16q^{33} - 16q^{39} + 8q^{41} - 4q^{43} - 14q^{45} - 12q^{47} - 20q^{51} + 20q^{53} + 24q^{55} - 12q^{57} - 14q^{59} + 18q^{61} - 4q^{65} - 8q^{67} - 8q^{69} + 8q^{71} - 12q^{73} + 18q^{75} + 8q^{79} + 22q^{81} - 14q^{83} - 20q^{85} + 20q^{87} + 12q^{89} + 24q^{93} - 8q^{95} - 16q^{97} - 28q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
0 −3.23607 0 −1.23607 0 0 0 7.47214 0
1.2 0 1.23607 0 3.23607 0 0 0 −1.47214 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.bf 2
4.b odd 2 1 3136.2.a.by 2
7.b odd 2 1 448.2.a.j 2
8.b even 2 1 1568.2.a.v 2
8.d odd 2 1 1568.2.a.k 2
21.c even 2 1 4032.2.a.bw 2
28.d even 2 1 448.2.a.i 2
56.e even 2 1 224.2.a.d yes 2
56.h odd 2 1 224.2.a.c 2
56.j odd 6 2 1568.2.i.v 4
56.k odd 6 2 1568.2.i.w 4
56.m even 6 2 1568.2.i.m 4
56.p even 6 2 1568.2.i.n 4
84.h odd 2 1 4032.2.a.bv 2
112.j even 4 2 1792.2.b.m 4
112.l odd 4 2 1792.2.b.k 4
168.e odd 2 1 2016.2.a.o 2
168.i even 2 1 2016.2.a.r 2
280.c odd 2 1 5600.2.a.bk 2
280.n even 2 1 5600.2.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 56.h odd 2 1
224.2.a.d yes 2 56.e even 2 1
448.2.a.i 2 28.d even 2 1
448.2.a.j 2 7.b odd 2 1
1568.2.a.k 2 8.d odd 2 1
1568.2.a.v 2 8.b even 2 1
1568.2.i.m 4 56.m even 6 2
1568.2.i.n 4 56.p even 6 2
1568.2.i.v 4 56.j odd 6 2
1568.2.i.w 4 56.k odd 6 2
1792.2.b.k 4 112.l odd 4 2
1792.2.b.m 4 112.j even 4 2
2016.2.a.o 2 168.e odd 2 1
2016.2.a.r 2 168.i even 2 1
3136.2.a.bf 2 1.a even 1 1 trivial
3136.2.a.by 2 4.b odd 2 1
4032.2.a.bv 2 84.h odd 2 1
4032.2.a.bw 2 21.c even 2 1
5600.2.a.z 2 280.n even 2 1
5600.2.a.bk 2 280.c odd 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 2 T_{3} - 4$$ $$T_{5}^{2} - 2 T_{5} - 4$$ $$T_{11}^{2} - 4 T_{11} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 + 2 T + T^{2}$$
$5$ $$-4 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-16 - 4 T + T^{2}$$
$13$ $$4 - 6 T + T^{2}$$
$17$ $$-20 + T^{2}$$
$19$ $$-4 - 2 T + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$-20 + T^{2}$$
$31$ $$-16 + 4 T + T^{2}$$
$37$ $$-20 + T^{2}$$
$41$ $$-4 - 8 T + T^{2}$$
$43$ $$-16 + 4 T + T^{2}$$
$47$ $$16 + 12 T + T^{2}$$
$53$ $$( -10 + T )^{2}$$
$59$ $$44 + 14 T + T^{2}$$
$61$ $$76 - 18 T + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$-64 - 8 T + T^{2}$$
$73$ $$-44 + 12 T + T^{2}$$
$79$ $$-64 - 8 T + T^{2}$$
$83$ $$44 + 14 T + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$44 + 16 T + T^{2}$$