# Properties

 Label 1568.3.d.e Level 1568 Weight 3 Character orbit 1568.d Analytic conductor 42.725 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1568.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$42.7249054517$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 224) 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.

 $$f(q)$$ $$=$$ $$q + 5 i q^{3} + 9 q^{5} -16 q^{9} +O(q^{10})$$ $$q + 5 i q^{3} + 9 q^{5} -16 q^{9} + 3 i q^{11} + 16 q^{13} + 45 i q^{15} -7 q^{17} -11 i q^{19} + 19 i q^{23} + 56 q^{25} -35 i q^{27} -32 q^{29} + 11 i q^{31} -15 q^{33} - q^{37} + 80 i q^{39} -40 q^{41} + 40 i q^{43} -144 q^{45} + 85 i q^{47} -35 i q^{51} + 7 q^{53} + 27 i q^{55} + 55 q^{57} + 53 i q^{59} + 79 q^{61} + 144 q^{65} -11 i q^{67} -95 q^{69} + 48 i q^{71} + 143 q^{73} + 280 i q^{75} + 35 i q^{79} + 31 q^{81} + 8 i q^{83} -63 q^{85} -160 i q^{87} -97 q^{89} -55 q^{93} -99 i q^{95} -88 q^{97} -48 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 18q^{5} - 32q^{9} + O(q^{10})$$ $$2q + 18q^{5} - 32q^{9} + 32q^{13} - 14q^{17} + 112q^{25} - 64q^{29} - 30q^{33} - 2q^{37} - 80q^{41} - 288q^{45} + 14q^{53} + 110q^{57} + 158q^{61} + 288q^{65} - 190q^{69} + 286q^{73} + 62q^{81} - 126q^{85} - 194q^{89} - 110q^{93} - 176q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$$.

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

## 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 5.00000i 0 9.00000 0 0 0 −16.0000 0
1471.2 0 5.00000i 0 9.00000 0 0 0 −16.0000 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 1568.3.d.e 2
4.b odd 2 1 inner 1568.3.d.e 2
7.b odd 2 1 1568.3.d.a 2
7.c even 3 2 224.3.r.a 4
28.d even 2 1 1568.3.d.a 2
28.g odd 6 2 224.3.r.a 4
56.k odd 6 2 448.3.r.c 4
56.p even 6 2 448.3.r.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.r.a 4 7.c even 3 2
224.3.r.a 4 28.g odd 6 2
448.3.r.c 4 56.k odd 6 2
448.3.r.c 4 56.p even 6 2
1568.3.d.a 2 7.b odd 2 1
1568.3.d.a 2 28.d even 2 1
1568.3.d.e 2 1.a even 1 1 trivial
1568.3.d.e 2 4.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1568, [\chi])$$:

 $$T_{3}^{2} + 25$$ $$T_{5} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 7 T^{2} + 81 T^{4}$$
$5$ $$( 1 - 9 T + 25 T^{2} )^{2}$$
$7$ 1
$11$ $$1 - 233 T^{2} + 14641 T^{4}$$
$13$ $$( 1 - 16 T + 169 T^{2} )^{2}$$
$17$ $$( 1 + 7 T + 289 T^{2} )^{2}$$
$19$ $$1 - 601 T^{2} + 130321 T^{4}$$
$23$ $$1 - 697 T^{2} + 279841 T^{4}$$
$29$ $$( 1 + 32 T + 841 T^{2} )^{2}$$
$31$ $$1 - 1801 T^{2} + 923521 T^{4}$$
$37$ $$( 1 + T + 1369 T^{2} )^{2}$$
$41$ $$( 1 + 40 T + 1681 T^{2} )^{2}$$
$43$ $$1 - 2098 T^{2} + 3418801 T^{4}$$
$47$ $$1 + 2807 T^{2} + 4879681 T^{4}$$
$53$ $$( 1 - 7 T + 2809 T^{2} )^{2}$$
$59$ $$1 - 4153 T^{2} + 12117361 T^{4}$$
$61$ $$( 1 - 79 T + 3721 T^{2} )^{2}$$
$67$ $$1 - 8857 T^{2} + 20151121 T^{4}$$
$71$ $$1 - 7778 T^{2} + 25411681 T^{4}$$
$73$ $$( 1 - 143 T + 5329 T^{2} )^{2}$$
$79$ $$1 - 11257 T^{2} + 38950081 T^{4}$$
$83$ $$1 - 13714 T^{2} + 47458321 T^{4}$$
$89$ $$( 1 + 97 T + 7921 T^{2} )^{2}$$
$97$ $$( 1 + 88 T + 9409 T^{2} )^{2}$$