# Properties

 Label 3136.2.b.c Level $3136$ Weight $2$ Character orbit 3136.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.0410860739$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 448) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} -4 i q^{5} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} -4 i q^{5} - q^{9} + 2 i q^{11} -4 i q^{13} + 8 q^{15} -2 q^{17} -6 i q^{19} -11 q^{25} + 4 i q^{27} -8 i q^{29} -8 q^{31} -4 q^{33} + 8 i q^{37} + 8 q^{39} -10 q^{41} + 2 i q^{43} + 4 i q^{45} + 8 q^{47} -4 i q^{51} + 8 q^{55} + 12 q^{57} + 10 i q^{59} + 4 i q^{61} -16 q^{65} + 2 i q^{67} -8 q^{71} + 6 q^{73} -22 i q^{75} -8 q^{79} -11 q^{81} -6 i q^{83} + 8 i q^{85} + 16 q^{87} -10 q^{89} -16 i q^{93} -24 q^{95} -2 q^{97} -2 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{9} + 16 q^{15} - 4 q^{17} - 22 q^{25} - 16 q^{31} - 8 q^{33} + 16 q^{39} - 20 q^{41} + 16 q^{47} + 16 q^{55} + 24 q^{57} - 32 q^{65} - 16 q^{71} + 12 q^{73} - 16 q^{79} - 22 q^{81} + 32 q^{87} - 20 q^{89} - 48 q^{95} - 4 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1569.1
 − 1.00000i 1.00000i
0 2.00000i 0 4.00000i 0 0 0 −1.00000 0
1569.2 0 2.00000i 0 4.00000i 0 0 0 −1.00000 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
8.b 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.b.c 2
4.b odd 2 1 3136.2.b.a 2
7.b odd 2 1 448.2.b.a 2
8.b even 2 1 inner 3136.2.b.c 2
8.d odd 2 1 3136.2.b.a 2
21.c even 2 1 4032.2.c.b 2
28.d even 2 1 448.2.b.b yes 2
56.e even 2 1 448.2.b.b yes 2
56.h odd 2 1 448.2.b.a 2
84.h odd 2 1 4032.2.c.f 2
112.j even 4 1 1792.2.a.d 1
112.j even 4 1 1792.2.a.e 1
112.l odd 4 1 1792.2.a.a 1
112.l odd 4 1 1792.2.a.h 1
168.e odd 2 1 4032.2.c.f 2
168.i even 2 1 4032.2.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.b.a 2 7.b odd 2 1
448.2.b.a 2 56.h odd 2 1
448.2.b.b yes 2 28.d even 2 1
448.2.b.b yes 2 56.e even 2 1
1792.2.a.a 1 112.l odd 4 1
1792.2.a.d 1 112.j even 4 1
1792.2.a.e 1 112.j even 4 1
1792.2.a.h 1 112.l odd 4 1
3136.2.b.a 2 4.b odd 2 1
3136.2.b.a 2 8.d odd 2 1
3136.2.b.c 2 1.a even 1 1 trivial
3136.2.b.c 2 8.b even 2 1 inner
4032.2.c.b 2 21.c even 2 1
4032.2.c.b 2 168.i even 2 1
4032.2.c.f 2 84.h odd 2 1
4032.2.c.f 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 $$S_{2}^{\mathrm{new}}(3136, [\chi])$$:

 $$T_{3}^{2} + 4$$ $$T_{5}^{2} + 16$$ $$T_{17} + 2$$ $$T_{31} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$16 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 + T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$36 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$64 + T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$T^{2}$$
$59$ $$100 + T^{2}$$
$61$ $$16 + T^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$( 2 + T )^{2}$$