# Properties

 Label 3136.2.f.i Level $3136$ Weight $2$ Character orbit 3136.f Analytic conductor $25.041$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: $$8$$ Coefficient field: 8.0.1212153856.10 Defining polynomial: $$x^{8} - 4 x^{6} + 10 x^{4} - 16 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 196) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + ( \beta_{1} - \beta_{2} ) q^{5} + ( 3 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} + ( \beta_{1} - \beta_{2} ) q^{5} + ( 3 + \beta_{3} ) q^{9} -\beta_{5} q^{11} + ( -\beta_{1} - \beta_{2} ) q^{13} + ( -\beta_{5} + \beta_{6} ) q^{15} + ( 2 \beta_{1} + \beta_{2} ) q^{17} + ( \beta_{4} - \beta_{7} ) q^{19} + ( \beta_{5} + \beta_{6} ) q^{23} + ( 1 + 2 \beta_{3} ) q^{25} + \beta_{7} q^{27} + ( 4 - 2 \beta_{3} ) q^{29} + 2 \beta_{7} q^{31} + ( -\beta_{1} - 5 \beta_{2} ) q^{33} + 4 q^{37} + ( -\beta_{5} - \beta_{6} ) q^{39} + 3 \beta_{1} q^{41} + ( -\beta_{5} + 2 \beta_{6} ) q^{43} + ( 3 \beta_{1} - \beta_{2} ) q^{45} + 2 \beta_{4} q^{47} + ( \beta_{5} + 2 \beta_{6} ) q^{51} + ( -2 + 6 \beta_{3} ) q^{53} + ( -2 \beta_{4} + 2 \beta_{7} ) q^{55} + ( 4 - 5 \beta_{3} ) q^{57} + ( -\beta_{4} + 3 \beta_{7} ) q^{59} + ( -5 \beta_{1} - 5 \beta_{2} ) q^{61} + 2 \beta_{3} q^{65} -2 \beta_{6} q^{67} + ( 8 \beta_{1} + 6 \beta_{2} ) q^{69} + ( -\beta_{1} - 8 \beta_{2} ) q^{73} + ( \beta_{4} + 2 \beta_{7} ) q^{75} -2 \beta_{5} q^{79} + ( -7 + 3 \beta_{3} ) q^{81} + ( -3 \beta_{4} + \beta_{7} ) q^{83} + ( -2 - 2 \beta_{3} ) q^{85} + ( 4 \beta_{4} - 2 \beta_{7} ) q^{87} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 4 + 12 \beta_{3} ) q^{93} + ( -3 \beta_{5} + \beta_{6} ) q^{95} + 5 \beta_{2} q^{97} + ( -2 \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 24q^{9} + O(q^{10})$$ $$8q + 24q^{9} + 8q^{25} + 32q^{29} + 32q^{37} - 16q^{53} + 32q^{57} - 56q^{81} - 16q^{85} + 32q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 10 x^{4} - 16 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{3}$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{5} + 10 \nu^{3} - 8 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{4} - 6 \nu^{2} + 8$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} - 10 \nu^{3} + 24 \nu$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} - 4 \nu^{4} + 14 \nu^{2} - 16$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{6} - 4 \nu^{4} + 10 \nu^{2} - 8$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{5} - 4 \nu^{3} + 6 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{3} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + 3 \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} + 4 \beta_{3} - 2$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} - \beta_{4} + 3 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{6} - \beta_{5} + \beta_{3} - 2$$ $$\nu^{7}$$ $$=$$ $$-\beta_{7} - 3 \beta_{2} + 7 \beta_{1}$$

## 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
 −1.36145 − 0.382683i −1.36145 + 0.382683i −1.07072 + 0.923880i −1.07072 − 0.923880i 1.07072 + 0.923880i 1.07072 − 0.923880i 1.36145 − 0.382683i 1.36145 + 0.382683i
0 −2.72291 0 1.08239i 0 0 0 4.41421 0
3135.2 0 −2.72291 0 1.08239i 0 0 0 4.41421 0
3135.3 0 −2.14144 0 2.61313i 0 0 0 1.58579 0
3135.4 0 −2.14144 0 2.61313i 0 0 0 1.58579 0
3135.5 0 2.14144 0 2.61313i 0 0 0 1.58579 0
3135.6 0 2.14144 0 2.61313i 0 0 0 1.58579 0
3135.7 0 2.72291 0 1.08239i 0 0 0 4.41421 0
3135.8 0 2.72291 0 1.08239i 0 0 0 4.41421 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3135.8 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.b odd 2 1 inner
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.i 8
4.b odd 2 1 inner 3136.2.f.i 8
7.b odd 2 1 inner 3136.2.f.i 8
8.b even 2 1 196.2.d.c 8
8.d odd 2 1 196.2.d.c 8
24.f even 2 1 1764.2.b.k 8
24.h odd 2 1 1764.2.b.k 8
28.d even 2 1 inner 3136.2.f.i 8
56.e even 2 1 196.2.d.c 8
56.h odd 2 1 196.2.d.c 8
56.j odd 6 2 196.2.f.d 16
56.k odd 6 2 196.2.f.d 16
56.m even 6 2 196.2.f.d 16
56.p even 6 2 196.2.f.d 16
168.e odd 2 1 1764.2.b.k 8
168.i even 2 1 1764.2.b.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.d.c 8 8.b even 2 1
196.2.d.c 8 8.d odd 2 1
196.2.d.c 8 56.e even 2 1
196.2.d.c 8 56.h odd 2 1
196.2.f.d 16 56.j odd 6 2
196.2.f.d 16 56.k odd 6 2
196.2.f.d 16 56.m even 6 2
196.2.f.d 16 56.p even 6 2
1764.2.b.k 8 24.f even 2 1
1764.2.b.k 8 24.h odd 2 1
1764.2.b.k 8 168.e odd 2 1
1764.2.b.k 8 168.i even 2 1
3136.2.f.i 8 1.a even 1 1 trivial
3136.2.f.i 8 4.b odd 2 1 inner
3136.2.f.i 8 7.b odd 2 1 inner
3136.2.f.i 8 28.d even 2 1 inner

## Hecke kernels

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

 $$T_{3}^{4} - 12 T_{3}^{2} + 34$$ $$T_{5}^{4} + 8 T_{5}^{2} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 34 - 12 T^{2} + T^{4} )^{2}$$
$5$ $$( 8 + 8 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 68 + 20 T^{2} + T^{4} )^{2}$$
$13$ $$( 8 + 8 T^{2} + T^{4} )^{2}$$
$17$ $$( 2 + 20 T^{2} + T^{4} )^{2}$$
$19$ $$( 34 - 28 T^{2} + T^{4} )^{2}$$
$23$ $$( 272 + 56 T^{2} + T^{4} )^{2}$$
$29$ $$( 8 - 8 T + T^{2} )^{4}$$
$31$ $$( 2176 - 96 T^{2} + T^{4} )^{2}$$
$37$ $$( -4 + T )^{8}$$
$41$ $$( 162 + 36 T^{2} + T^{4} )^{2}$$
$43$ $$( 3332 + 116 T^{2} + T^{4} )^{2}$$
$47$ $$( 544 - 48 T^{2} + T^{4} )^{2}$$
$53$ $$( -68 + 4 T + T^{2} )^{4}$$
$59$ $$( 9826 - 204 T^{2} + T^{4} )^{2}$$
$61$ $$( 5000 + 200 T^{2} + T^{4} )^{2}$$
$67$ $$( 1088 + 112 T^{2} + T^{4} )^{2}$$
$71$ $$T^{8}$$
$73$ $$( 12482 + 260 T^{2} + T^{4} )^{2}$$
$79$ $$( 1088 + 80 T^{2} + T^{4} )^{2}$$
$83$ $$( 1666 - 108 T^{2} + T^{4} )^{2}$$
$89$ $$( 578 + 52 T^{2} + T^{4} )^{2}$$
$97$ $$( 1250 + 100 T^{2} + T^{4} )^{2}$$