# Properties

 Label 1568.3.g.j Level 1568 Weight 3 Character orbit 1568.g Analytic conductor 42.725 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1568.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$42.7249054517$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.15582448.1 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} + \beta_{2} ) q^{3} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( 5 - 4 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} + \beta_{2} ) q^{3} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( 5 - 4 \beta_{1} + \beta_{2} ) q^{9} + ( 6 + 3 \beta_{1} ) q^{11} + ( -5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{13} + ( 2 \beta_{3} + 3 \beta_{5} ) q^{15} + ( -6 - 2 \beta_{1} + \beta_{2} ) q^{17} + ( 12 + \beta_{1} + 4 \beta_{2} ) q^{19} + ( 5 \beta_{3} - 7 \beta_{5} ) q^{23} + ( 14 - 2 \beta_{1} + 2 \beta_{2} ) q^{25} + ( -8 + 13 \beta_{1} - 2 \beta_{2} ) q^{27} + ( -3 \beta_{3} + 8 \beta_{4} - 2 \beta_{5} ) q^{29} + ( -6 \beta_{3} - 9 \beta_{5} ) q^{31} + ( 9 - 6 \beta_{1} + 6 \beta_{2} ) q^{33} + ( -4 \beta_{3} + 9 \beta_{4} + \beta_{5} ) q^{37} + ( 17 \beta_{3} + 14 \beta_{4} + 8 \beta_{5} ) q^{39} + ( -25 - 18 \beta_{1} - \beta_{2} ) q^{41} + ( 10 - 10 \beta_{1} + 10 \beta_{2} ) q^{43} + ( -9 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{45} + ( -2 \beta_{3} + 16 \beta_{4} + 5 \beta_{5} ) q^{47} + ( 4 + 3 \beta_{1} - 4 \beta_{2} ) q^{51} + ( 20 \beta_{3} - 3 \beta_{4} + 7 \beta_{5} ) q^{53} + ( -3 \beta_{3} + 12 \beta_{4} - 9 \beta_{5} ) q^{55} + ( 25 + 12 \beta_{1} + 20 \beta_{2} ) q^{57} + ( -17 + 9 \beta_{1} + 5 \beta_{2} ) q^{59} + ( 10 \beta_{3} - 15 \beta_{4} + 25 \beta_{5} ) q^{61} + ( 15 + 18 \beta_{1} - 11 \beta_{2} ) q^{65} + ( 73 - \beta_{1} - 3 \beta_{2} ) q^{67} + ( 13 \beta_{3} - 31 \beta_{4} - 3 \beta_{5} ) q^{69} + ( -22 \beta_{3} + 6 \beta_{4} - 8 \beta_{5} ) q^{71} + ( -27 - 18 \beta_{1} + 12 \beta_{2} ) q^{73} + ( -8 + 24 \beta_{1} + 18 \beta_{2} ) q^{75} + ( -5 \beta_{3} - 16 \beta_{4} + 25 \beta_{5} ) q^{79} + ( 12 - 26 \beta_{1} - 21 \beta_{2} ) q^{81} + ( 36 - 12 \beta_{1} + 14 \beta_{2} ) q^{83} + ( 4 \beta_{3} - 11 \beta_{4} + 11 \beta_{5} ) q^{85} + ( -23 \beta_{3} + 8 \beta_{4} + 8 \beta_{5} ) q^{87} + ( -49 - 4 \beta_{1} + 36 \beta_{2} ) q^{89} + ( 54 \beta_{3} - 15 \beta_{4} + 21 \beta_{5} ) q^{93} + ( -11 \beta_{3} + 10 \beta_{4} - \beta_{5} ) q^{95} + ( -15 + 8 \beta_{1} + 15 \beta_{2} ) q^{97} + ( -45 + 12 \beta_{1} + 21 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{3} + 40q^{9} + O(q^{10})$$ $$6q - 6q^{3} + 40q^{9} + 30q^{11} - 30q^{17} + 78q^{19} + 92q^{25} - 78q^{27} + 78q^{33} - 116q^{41} + 100q^{43} + 10q^{51} + 166q^{57} - 110q^{59} + 32q^{65} + 434q^{67} - 102q^{73} - 60q^{75} + 82q^{81} + 268q^{83} - 214q^{89} - 76q^{97} - 252q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 13 x^{4} - 21 x^{3} + 20 x^{2} - 10 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} + 10 \nu^{2} - 9 \nu + 3$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} + 11 \nu^{2} - 10 \nu + 7$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{5} - 5 \nu^{4} + 24 \nu^{3} - 31 \nu^{2} + 30 \nu - 10$$ $$\beta_{4}$$ $$=$$ $$-6 \nu^{5} + 15 \nu^{4} - 70 \nu^{3} + 90 \nu^{2} - 69 \nu + 20$$ $$\beta_{5}$$ $$=$$ $$-6 \nu^{5} + 15 \nu^{4} - 70 \nu^{3} + 90 \nu^{2} - 71 \nu + 21$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{5} + \beta_{4} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} + \beta_{4} + 2 \beta_{2} - 2 \beta_{1} - 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$9 \beta_{5} - 8 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 3 \beta_{1} - 11$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$19 \beta_{5} - 17 \beta_{4} + 6 \beta_{3} - 14 \beta_{2} + 16 \beta_{1} + 51$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-61 \beta_{5} + 54 \beta_{4} - 20 \beta_{3} - 40 \beta_{2} + 45 \beta_{1} + 146$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
687.1
 0.5 + 0.759064i 0.5 − 0.759064i 0.5 − 2.94141i 0.5 + 2.94141i 0.5 + 0.148124i 0.5 − 0.148124i
0 −5.33225 0 2.15693i 0 0 0 19.4329 0
687.2 0 −5.33225 0 2.15693i 0 0 0 19.4329 0
687.3 0 −1.64878 0 4.56111i 0 0 0 −6.28154 0
687.4 0 −1.64878 0 4.56111i 0 0 0 −6.28154 0
687.5 0 3.98103 0 1.88252i 0 0 0 6.84860 0
687.6 0 3.98103 0 1.88252i 0 0 0 6.84860 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 687.6 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.d 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.g.j 6
4.b odd 2 1 392.3.g.j 6
7.b odd 2 1 1568.3.g.l 6
7.c even 3 2 224.3.o.d 12
8.b even 2 1 392.3.g.j 6
8.d odd 2 1 inner 1568.3.g.j 6
28.d even 2 1 392.3.g.i 6
28.f even 6 2 392.3.k.l 12
28.g odd 6 2 56.3.k.d 12
56.e even 2 1 1568.3.g.l 6
56.h odd 2 1 392.3.g.i 6
56.j odd 6 2 392.3.k.l 12
56.k odd 6 2 224.3.o.d 12
56.p even 6 2 56.3.k.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.d 12 28.g odd 6 2
56.3.k.d 12 56.p even 6 2
224.3.o.d 12 7.c even 3 2
224.3.o.d 12 56.k odd 6 2
392.3.g.i 6 28.d even 2 1
392.3.g.i 6 56.h odd 2 1
392.3.g.j 6 4.b odd 2 1
392.3.g.j 6 8.b even 2 1
392.3.k.l 12 28.f even 6 2
392.3.k.l 12 56.j odd 6 2
1568.3.g.j 6 1.a even 1 1 trivial
1568.3.g.j 6 8.d odd 2 1 inner
1568.3.g.l 6 7.b odd 2 1
1568.3.g.l 6 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 3 T_{3}^{2} - 19 T_{3} - 35$$ acting on $$S_{3}^{\mathrm{new}}(1568, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 3 T + 8 T^{2} + 19 T^{3} + 72 T^{4} + 243 T^{5} + 729 T^{6} )^{2}$$
$5$ $$1 - 121 T^{2} + 6662 T^{4} - 212757 T^{6} + 4163750 T^{8} - 47265625 T^{10} + 244140625 T^{12}$$
$7$ 1
$11$ $$( 1 - 15 T + 354 T^{2} - 3117 T^{3} + 42834 T^{4} - 219615 T^{5} + 1771561 T^{6} )^{2}$$
$13$ $$1 - 86 T^{2} + 60895 T^{4} - 4902788 T^{6} + 1739222095 T^{8} - 70152842006 T^{10} + 23298085122481 T^{12}$$
$17$ $$( 1 + 15 T + 890 T^{2} + 8635 T^{3} + 257210 T^{4} + 1252815 T^{5} + 24137569 T^{6} )^{2}$$
$19$ $$( 1 - 39 T + 1370 T^{2} - 28697 T^{3} + 494570 T^{4} - 5082519 T^{5} + 47045881 T^{6} )^{2}$$
$23$ $$1 - 693 T^{2} + 495830 T^{4} - 329634845 T^{6} + 138753563030 T^{8} - 54269512799733 T^{10} + 21914624432020321 T^{12}$$
$29$ $$1 - 3662 T^{2} + 6471151 T^{4} - 6858243380 T^{6} + 4576922150431 T^{8} - 1831902364263182 T^{10} + 353814783205469041 T^{12}$$
$31$ $$1 - 3561 T^{2} + 6841842 T^{4} - 8041403549 T^{6} + 6318584765682 T^{8} - 3037144984327401 T^{10} + 787662783788549761 T^{12}$$
$37$ $$1 - 5785 T^{2} + 15576398 T^{4} - 26031024189 T^{6} + 29192677652078 T^{8} - 20319693640932985 T^{10} + 6582952005840035281 T^{12}$$
$41$ $$( 1 + 58 T + 3139 T^{2} + 88960 T^{3} + 5276659 T^{4} + 163894138 T^{5} + 4750104241 T^{6} )^{2}$$
$43$ $$( 1 - 50 T + 4047 T^{2} - 107900 T^{3} + 7482903 T^{4} - 170940050 T^{5} + 6321363049 T^{6} )^{2}$$
$47$ $$1 - 3905 T^{2} + 8711938 T^{4} - 15970210469 T^{6} + 42511478331778 T^{8} - 92983074414176705 T^{10} +$$$$11\!\cdots\!41$$$$T^{12}$$
$53$ $$1 - 10561 T^{2} + 51800990 T^{4} - 168592871253 T^{6} + 408734727376190 T^{8} - 657524590434383521 T^{10} +$$$$49\!\cdots\!41$$$$T^{12}$$
$59$ $$( 1 + 55 T + 10392 T^{2} + 376351 T^{3} + 36174552 T^{4} + 666454855 T^{5} + 42180533641 T^{6} )^{2}$$
$61$ $$1 - 9201 T^{2} + 27716990 T^{4} - 50012187845 T^{6} + 383765036538590 T^{8} - 1763898986887982481 T^{10} +$$$$26\!\cdots\!21$$$$T^{12}$$
$67$ $$( 1 - 217 T + 29036 T^{2} - 2316921 T^{3} + 130342604 T^{4} - 4372793257 T^{5} + 90458382169 T^{6} )^{2}$$
$71$ $$1 - 23062 T^{2} + 245626031 T^{4} - 1559141837940 T^{6} + 6241770345068111 T^{8} - 14892367937589740182 T^{10} +$$$$16\!\cdots\!41$$$$T^{12}$$
$73$ $$( 1 + 51 T + 11766 T^{2} + 589863 T^{3} + 62701014 T^{4} + 1448310291 T^{5} + 151334226289 T^{6} )^{2}$$
$79$ $$1 - 16693 T^{2} + 129404966 T^{4} - 759834693213 T^{6} + 5040333907502246 T^{8} - 25325097363770222773 T^{10} +$$$$59\!\cdots\!41$$$$T^{12}$$
$83$ $$( 1 - 134 T + 22583 T^{2} - 1661172 T^{3} + 155574287 T^{4} - 6359415014 T^{5} + 326940373369 T^{6} )^{2}$$
$89$ $$( 1 + 107 T + 10054 T^{2} + 786431 T^{3} + 79637734 T^{4} + 6713419787 T^{5} + 496981290961 T^{6} )^{2}$$
$97$ $$( 1 + 38 T + 25191 T^{2} + 597344 T^{3} + 237022119 T^{4} + 3364112678 T^{5} + 832972004929 T^{6} )^{2}$$