# Properties

 Label 3136.2.f.a Level $3136$ Weight $2$ Character orbit 3136.f 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.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.0410860739$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$1471$$ $$1473$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3135.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.00000 0 1.73205i 0 0 0 −2.00000 0
3135.2 0 −1.00000 0 1.73205i 0 0 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.f.a 2
4.b odd 2 1 3136.2.f.b 2
7.b odd 2 1 3136.2.f.b 2
7.c even 3 1 448.2.p.b 2
7.d odd 6 1 448.2.p.a 2
8.b even 2 1 784.2.f.b 2
8.d odd 2 1 784.2.f.a 2
24.f even 2 1 7056.2.b.b 2
24.h odd 2 1 7056.2.b.m 2
28.d even 2 1 inner 3136.2.f.a 2
28.f even 6 1 448.2.p.b 2
28.g odd 6 1 448.2.p.a 2
56.e even 2 1 784.2.f.b 2
56.h odd 2 1 784.2.f.a 2
56.j odd 6 1 112.2.p.b yes 2
56.j odd 6 1 784.2.p.d 2
56.k odd 6 1 112.2.p.b yes 2
56.k odd 6 1 784.2.p.d 2
56.m even 6 1 112.2.p.a 2
56.m even 6 1 784.2.p.c 2
56.p even 6 1 112.2.p.a 2
56.p even 6 1 784.2.p.c 2
168.e odd 2 1 7056.2.b.m 2
168.i even 2 1 7056.2.b.b 2
168.s odd 6 1 1008.2.cs.f 2
168.v even 6 1 1008.2.cs.c 2
168.ba even 6 1 1008.2.cs.c 2
168.be odd 6 1 1008.2.cs.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.a 2 56.m even 6 1
112.2.p.a 2 56.p even 6 1
112.2.p.b yes 2 56.j odd 6 1
112.2.p.b yes 2 56.k odd 6 1
448.2.p.a 2 7.d odd 6 1
448.2.p.a 2 28.g odd 6 1
448.2.p.b 2 7.c even 3 1
448.2.p.b 2 28.f even 6 1
784.2.f.a 2 8.d odd 2 1
784.2.f.a 2 56.h odd 2 1
784.2.f.b 2 8.b even 2 1
784.2.f.b 2 56.e even 2 1
784.2.p.c 2 56.m even 6 1
784.2.p.c 2 56.p even 6 1
784.2.p.d 2 56.j odd 6 1
784.2.p.d 2 56.k odd 6 1
1008.2.cs.c 2 168.v even 6 1
1008.2.cs.c 2 168.ba even 6 1
1008.2.cs.f 2 168.s odd 6 1
1008.2.cs.f 2 168.be odd 6 1
3136.2.f.a 2 1.a even 1 1 trivial
3136.2.f.a 2 28.d even 2 1 inner
3136.2.f.b 2 4.b odd 2 1
3136.2.f.b 2 7.b odd 2 1
7056.2.b.b 2 24.f even 2 1
7056.2.b.b 2 168.i even 2 1
7056.2.b.m 2 24.h odd 2 1
7056.2.b.m 2 168.e odd 2 1

## Hecke kernels

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

 $$T_{3} + 1$$ $$T_{5}^{2} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$3 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$3 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$27 + T^{2}$$
$19$ $$( 7 + T )^{2}$$
$23$ $$75 + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$( -5 + T )^{2}$$
$41$ $$48 + T^{2}$$
$43$ $$12 + T^{2}$$
$47$ $$( -3 + T )^{2}$$
$53$ $$( -9 + T )^{2}$$
$59$ $$( 9 + T )^{2}$$
$61$ $$75 + T^{2}$$
$67$ $$27 + T^{2}$$
$71$ $$12 + T^{2}$$
$73$ $$3 + T^{2}$$
$79$ $$27 + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$147 + T^{2}$$
$97$ $$48 + T^{2}$$