# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.56506787962$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ 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

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{3} + \zeta_{12}^{4} q^{5} +O(q^{10})$$ $$q + \zeta_{12}^{5} q^{3} + \zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{11} -\zeta_{12}^{3} q^{15} + \zeta_{12}^{2} q^{17} + \zeta_{12} q^{19} -\zeta_{12} q^{23} -\zeta_{12}^{3} q^{27} + \zeta_{12}^{5} q^{31} -\zeta_{12}^{4} q^{33} + \zeta_{12}^{4} q^{37} -\zeta_{12} q^{47} -\zeta_{12} q^{51} -\zeta_{12}^{2} q^{53} -\zeta_{12}^{3} q^{55} - q^{57} + \zeta_{12}^{5} q^{59} -\zeta_{12}^{4} q^{61} -\zeta_{12}^{5} q^{67} + q^{69} -\zeta_{12}^{2} q^{73} -\zeta_{12} q^{79} + \zeta_{12}^{2} q^{81} - q^{85} + \zeta_{12}^{4} q^{89} -\zeta_{12}^{4} q^{93} + \zeta_{12}^{5} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} + O(q^{10})$$ $$4q - 2q^{5} + 2q^{17} + 2q^{33} - 2q^{37} - 2q^{53} - 4q^{57} + 2q^{61} + 4q^{69} - 2q^{73} + 2q^{81} - 4q^{85} - 2q^{89} + 2q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$197$$ $$1471$$ $$1473$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{12}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2431.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 0 0 0 0
2431.2 0 0.866025 0.500000i 0 −0.500000 + 0.866025i 0 0 0 0 0
3007.1 0 −0.866025 0.500000i 0 −0.500000 0.866025i 0 0 0 0 0
3007.2 0 0.866025 + 0.500000i 0 −0.500000 0.866025i 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
7.c even 3 1 inner
28.g odd 6 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{4} - T_{3}^{2} + 1$$ $$T_{5}^{2} + T_{5} + 1$$

## Hecke characteristic polynomials

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