# Properties

 Label 3136.1.d.b Level 3136 Weight 1 Character orbit 3136.d Analytic conductor 1.565 Analytic rank 0 Dimension 2 Projective image $$A_{4}$$ CM/RM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3136 = 2^{6} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3136.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.56506787962$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 224) Projective image $$A_{4}$$ Projective field Galois closure of 4.0.3136.1 Artin image $\SL(2,3):C_2$ Artin field Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + i q^{3} - q^{5} +O(q^{10})$$ $$q + i q^{3} - q^{5} -i q^{11} -i q^{15} + q^{17} + i q^{19} + i q^{23} + i q^{27} + i q^{31} + q^{33} + q^{37} -i q^{47} + i q^{51} + q^{53} + i q^{55} - q^{57} + i q^{59} + q^{61} + i q^{67} - q^{69} - q^{73} + i q^{79} - q^{81} - q^{85} - q^{89} - q^{93} -i q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} + O(q^{10})$$ $$2q - 2q^{5} + 2q^{17} + 2q^{33} + 2q^{37} + 2q^{53} - 2q^{57} + 2q^{61} - 2q^{69} - 2q^{73} - 2q^{81} - 2q^{85} - 2q^{89} - 2q^{93} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1471.1
 − 1.00000i 1.00000i
0 1.00000i 0 −1.00000 0 0 0 0 0
1471.2 0 1.00000i 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

## 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 3136.1.d.b 2
4.b odd 2 1 inner 3136.1.d.b 2
7.b odd 2 1 3136.1.d.d 2
7.c even 3 2 448.1.r.a 4
7.d odd 6 2 3136.1.r.b 4
8.b even 2 1 1568.1.d.b 2
8.d odd 2 1 1568.1.d.b 2
28.d even 2 1 3136.1.d.d 2
28.f even 6 2 3136.1.r.b 4
28.g odd 6 2 448.1.r.a 4
56.e even 2 1 1568.1.d.a 2
56.h odd 2 1 1568.1.d.a 2
56.j odd 6 2 1568.1.r.a 4
56.k odd 6 2 224.1.r.a 4
56.m even 6 2 1568.1.r.a 4
56.p even 6 2 224.1.r.a 4
112.u odd 12 2 1792.1.o.a 4
112.u odd 12 2 1792.1.o.b 4
112.w even 12 2 1792.1.o.a 4
112.w even 12 2 1792.1.o.b 4
168.s odd 6 2 2016.1.cd.a 4
168.v even 6 2 2016.1.cd.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.r.a 4 56.k odd 6 2
224.1.r.a 4 56.p even 6 2
448.1.r.a 4 7.c even 3 2
448.1.r.a 4 28.g odd 6 2
1568.1.d.a 2 56.e even 2 1
1568.1.d.a 2 56.h odd 2 1
1568.1.d.b 2 8.b even 2 1
1568.1.d.b 2 8.d odd 2 1
1568.1.r.a 4 56.j odd 6 2
1568.1.r.a 4 56.m even 6 2
1792.1.o.a 4 112.u odd 12 2
1792.1.o.a 4 112.w even 12 2
1792.1.o.b 4 112.u odd 12 2
1792.1.o.b 4 112.w even 12 2
2016.1.cd.a 4 168.s odd 6 2
2016.1.cd.a 4 168.v even 6 2
3136.1.d.b 2 1.a even 1 1 trivial
3136.1.d.b 2 4.b odd 2 1 inner
3136.1.d.d 2 7.b odd 2 1
3136.1.d.d 2 28.d even 2 1
3136.1.r.b 4 7.d odd 6 2
3136.1.r.b 4 28.f even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3136, [\chi])$$:

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

## Hecke characteristic polynomials

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