# Properties

 Label 688.3.b.e Level 688 Weight 3 Character orbit 688.b Analytic conductor 18.747 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$688 = 2^{4} \cdot 43$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 688.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7466421880$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 43) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{3} + \beta_{2} q^{5} + ( -\beta_{1} + \beta_{2} + \beta_{5} ) q^{7} + ( -9 - 4 \beta_{3} - 5 \beta_{4} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{3} + \beta_{2} q^{5} + ( -\beta_{1} + \beta_{2} + \beta_{5} ) q^{7} + ( -9 - 4 \beta_{3} - 5 \beta_{4} ) q^{9} + ( -7 - 3 \beta_{3} + \beta_{4} ) q^{11} + ( 5 - 5 \beta_{3} + 5 \beta_{4} ) q^{13} + ( -2 - \beta_{3} + 9 \beta_{4} ) q^{15} + ( -5 - 5 \beta_{4} ) q^{17} + ( -5 \beta_{1} + 5 \beta_{5} ) q^{19} + ( 14 + 4 \beta_{3} + 10 \beta_{4} ) q^{21} + ( 15 - 5 \beta_{3} + 10 \beta_{4} ) q^{23} + ( -9 + 5 \beta_{3} + 10 \beta_{4} ) q^{25} + ( 6 \beta_{1} + 5 \beta_{2} + 9 \beta_{5} ) q^{27} + ( 10 \beta_{1} + 5 \beta_{5} ) q^{29} + ( 15 + 4 \beta_{3} - 15 \beta_{4} ) q^{31} + ( -5 \beta_{1} - \beta_{2} + 9 \beta_{5} ) q^{33} + ( -30 + 14 \beta_{3} ) q^{35} + ( -\beta_{1} + 5 \beta_{2} ) q^{37} + ( -15 \beta_{1} - 5 \beta_{2} - 5 \beta_{5} ) q^{39} + ( -23 + 11 \beta_{3} + 6 \beta_{4} ) q^{41} + ( 5 + 10 \beta_{1} - \beta_{2} + 5 \beta_{3} + 15 \beta_{4} - 5 \beta_{5} ) q^{43} + ( -19 \beta_{1} - 6 \beta_{5} ) q^{45} + 15 \beta_{3} q^{47} + ( 3 + 18 \beta_{3} - 6 \beta_{4} ) q^{49} + ( 10 \beta_{1} + 5 \beta_{2} + 10 \beta_{5} ) q^{51} + ( -25 - 25 \beta_{3} + 5 \beta_{4} ) q^{53} + ( -19 \beta_{1} - 5 \beta_{2} + 5 \beta_{5} ) q^{55} + ( 80 + 25 \beta_{3} + 5 \beta_{4} ) q^{57} + ( -12 - 26 \beta_{3} - 16 \beta_{4} ) q^{59} + ( 5 \beta_{1} - 5 \beta_{2} + 15 \beta_{5} ) q^{61} + ( -25 \beta_{1} - \beta_{2} - 19 \beta_{5} ) q^{63} + ( -35 \beta_{1} + 5 \beta_{2} + 15 \beta_{5} ) q^{65} + ( 25 + 15 \beta_{3} + 25 \beta_{4} ) q^{67} + ( -25 \beta_{1} - 10 \beta_{2} - 20 \beta_{5} ) q^{69} + 20 \beta_{1} q^{71} + ( -15 \beta_{1} - 5 \beta_{2} + \beta_{5} ) q^{73} + ( -15 \beta_{1} - 10 \beta_{2} - 6 \beta_{5} ) q^{75} + ( -9 \beta_{1} - \beta_{2} - 5 \beta_{5} ) q^{77} + ( -32 + 2 \beta_{3} - 9 \beta_{4} ) q^{79} + ( 83 - 11 \beta_{3} + 69 \beta_{4} ) q^{81} + ( -5 - 25 \beta_{3} + 15 \beta_{4} ) q^{83} + ( 5 \beta_{1} - 10 \beta_{5} ) q^{85} + ( 110 + 10 \beta_{3} + 65 \beta_{4} ) q^{87} + ( 15 \beta_{1} + 5 \beta_{2} - 5 \beta_{5} ) q^{89} + ( -25 \beta_{1} + 15 \beta_{2} + 15 \beta_{5} ) q^{91} + ( 34 \beta_{1} + 15 \beta_{2} - 4 \beta_{5} ) q^{93} + ( 20 + 45 \beta_{3} - 50 \beta_{4} ) q^{95} + ( -45 + 5 \beta_{3} + 50 \beta_{4} ) q^{97} + ( 91 + 15 \beta_{3} + 25 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 36q^{9} + O(q^{10})$$ $$6q - 36q^{9} - 38q^{11} + 30q^{13} - 28q^{15} - 20q^{17} + 56q^{21} + 80q^{23} - 84q^{25} + 112q^{31} - 208q^{35} - 172q^{41} - 10q^{43} - 30q^{47} - 6q^{49} - 110q^{53} + 420q^{57} + 12q^{59} + 70q^{67} - 178q^{79} + 382q^{81} - 10q^{83} + 510q^{87} + 130q^{95} - 380q^{97} + 466q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 20 x^{4} + 121 x^{2} + 214$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} - 11 \nu^{3} - 18 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{4} - 11 \nu^{2} - 22$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + 15 \nu^{2} + 46$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} + 15 \nu^{3} + 50 \nu$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} - 6$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{2} - 8 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-11 \beta_{4} - 15 \beta_{3} + 44$$ $$\nu^{5}$$ $$=$$ $$-11 \beta_{5} - 15 \beta_{2} + 70 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$$.

 $$n$$ $$431$$ $$433$$ $$517$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 1.77533i − 2.58315i 3.18991i − 3.18991i 2.58315i − 1.77533i
0 5.61757i 0 2.98959i 0 6.83184i 0 −22.5571 0
257.2 0 3.59415i 0 7.02284i 0 0.845536i 0 −3.91793 0
257.3 0 0.724539i 0 7.66434i 0 10.1297i 0 8.47504 0
257.4 0 0.724539i 0 7.66434i 0 10.1297i 0 8.47504 0
257.5 0 3.59415i 0 7.02284i 0 0.845536i 0 −3.91793 0
257.6 0 5.61757i 0 2.98959i 0 6.83184i 0 −22.5571 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.3.b.e 6
4.b odd 2 1 43.3.b.b 6
12.b even 2 1 387.3.b.c 6
43.b odd 2 1 inner 688.3.b.e 6
172.d even 2 1 43.3.b.b 6
516.h odd 2 1 387.3.b.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.b.b 6 4.b odd 2 1
43.3.b.b 6 172.d even 2 1
387.3.b.c 6 12.b even 2 1
387.3.b.c 6 516.h odd 2 1
688.3.b.e 6 1.a even 1 1 trivial
688.3.b.e 6 43.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 $$S_{3}^{\mathrm{new}}(688, [\chi])$$:

 $$T_{3}^{6} + 45 T_{3}^{4} + 431 T_{3}^{2} + 214$$ $$T_{11}^{3} + 19 T_{11}^{2} + 48 T_{11} - 244$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 9 T^{2} + 26 T^{4} - 254 T^{6} + 2106 T^{8} - 59049 T^{10} + 531441 T^{12}$$
$5$ $$1 - 33 T^{2} + 1538 T^{4} - 41006 T^{6} + 961250 T^{8} - 12890625 T^{10} + 244140625 T^{12}$$
$7$ $$1 - 144 T^{2} + 11511 T^{4} - 668464 T^{6} + 27637911 T^{8} - 830131344 T^{10} + 13841287201 T^{12}$$
$11$ $$( 1 + 19 T + 411 T^{2} + 4354 T^{3} + 49731 T^{4} + 278179 T^{5} + 1771561 T^{6} )^{2}$$
$13$ $$( 1 - 15 T + 257 T^{2} - 1570 T^{3} + 43433 T^{4} - 428415 T^{5} + 4826809 T^{6} )^{2}$$
$17$ $$( 1 + 10 T + 767 T^{2} + 5655 T^{3} + 221663 T^{4} + 835210 T^{5} + 24137569 T^{6} )^{2}$$
$19$ $$1 - 691 T^{2} + 378040 T^{4} - 133433020 T^{6} + 49266550840 T^{8} - 11735642061331 T^{10} + 2213314919066161 T^{12}$$
$23$ $$( 1 - 40 T + 1387 T^{2} - 28195 T^{3} + 733723 T^{4} - 11193640 T^{5} + 148035889 T^{6} )^{2}$$
$29$ $$1 - 1621 T^{2} + 1946890 T^{4} - 2007466870 T^{6} + 1376998306090 T^{8} - 810899435409781 T^{10} + 353814783205469041 T^{12}$$
$31$ $$( 1 - 56 T + 2591 T^{2} - 96591 T^{3} + 2489951 T^{4} - 51717176 T^{5} + 887503681 T^{6} )^{2}$$
$37$ $$1 - 5259 T^{2} + 14237096 T^{4} - 24083763884 T^{6} + 26682610076456 T^{8} - 18472129448170539 T^{10} + 6582952005840035281 T^{12}$$
$41$ $$( 1 + 86 T + 6451 T^{2} + 292121 T^{3} + 10844131 T^{4} + 243015446 T^{5} + 4750104241 T^{6} )^{2}$$
$43$ $$1 + 10 T + 147 T^{2} - 135020 T^{3} + 271803 T^{4} + 34188010 T^{5} + 6321363049 T^{6}$$
$47$ $$( 1 + 15 T + 5052 T^{2} + 79770 T^{3} + 11159868 T^{4} + 73195215 T^{5} + 10779215329 T^{6} )^{2}$$
$53$ $$( 1 + 55 T + 4677 T^{2} + 168490 T^{3} + 13137693 T^{4} + 433976455 T^{5} + 22164361129 T^{6} )^{2}$$
$59$ $$( 1 - 6 T + 4271 T^{2} - 147196 T^{3} + 14867351 T^{4} - 72704166 T^{5} + 42180533641 T^{6} )^{2}$$
$61$ $$1 - 6176 T^{2} + 53251015 T^{4} - 178029474320 T^{6} + 737305086778615 T^{8} - 1183984365071207456 T^{10} +$$$$26\!\cdots\!21$$$$T^{12}$$
$67$ $$( 1 - 35 T + 9017 T^{2} - 350730 T^{3} + 40477313 T^{4} - 705289235 T^{5} + 90458382169 T^{6} )^{2}$$
$71$ $$1 - 22246 T^{2} + 239223215 T^{4} - 1523736510420 T^{6} + 6079064027374415 T^{8} - 14365433056093199206 T^{10} +$$$$16\!\cdots\!41$$$$T^{12}$$
$73$ $$1 - 24604 T^{2} + 277197791 T^{4} - 1859020745064 T^{6} + 7871929673485631 T^{8} - 19842144100961968924 T^{10} +$$$$22\!\cdots\!21$$$$T^{12}$$
$79$ $$( 1 + 89 T + 20896 T^{2} + 1122134 T^{3} + 130411936 T^{4} + 3466557209 T^{5} + 243087455521 T^{6} )^{2}$$
$83$ $$( 1 + 5 T + 14767 T^{2} + 128390 T^{3} + 101729863 T^{4} + 237291605 T^{5} + 326940373369 T^{6} )^{2}$$
$89$ $$1 - 38876 T^{2} + 668902015 T^{4} - 6712320081320 T^{6} + 41968411430515615 T^{8} -$$$$15\!\cdots\!56$$$$T^{10} +$$$$24\!\cdots\!21$$$$T^{12}$$
$97$ $$( 1 + 190 T + 26827 T^{2} + 2529295 T^{3} + 252415243 T^{4} + 16820563390 T^{5} + 832972004929 T^{6} )^{2}$$