# Properties

 Label 684.3.y.h Level $684$ Weight $3$ Character orbit 684.y Analytic conductor $18.638$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 684.y (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.6376500822$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4 x^{7} + 56 x^{6} - 154 x^{5} + 917 x^{4} - 1582 x^{3} + 4294 x^{2} - 3528 x + 4971$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{5} + ( -2 + \beta_{1} ) q^{7} +O(q^{10})$$ $$q + \beta_{5} q^{5} + ( -2 + \beta_{1} ) q^{7} + ( 1 - \beta_{1} - \beta_{4} + \beta_{6} ) q^{11} + ( \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( -6 + \beta_{1} + 6 \beta_{2} - \beta_{5} - \beta_{7} ) q^{17} + ( -6 - \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{19} + ( 8 \beta_{2} + \beta_{3} - 3 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{23} + ( -15 \beta_{2} + 4 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{25} + ( 20 - 10 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{6} ) q^{29} + ( -\beta_{1} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} ) q^{31} + ( 12 - \beta_{1} - 12 \beta_{2} + 9 \beta_{3} - 9 \beta_{4} + 4 \beta_{5} + \beta_{7} ) q^{35} + ( 16 - 32 \beta_{2} - 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + 2 \beta_{6} ) q^{37} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{41} + ( -12 + 12 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} ) q^{43} + ( -22 \beta_{2} + 7 \beta_{3} - \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{47} + ( 41 - 3 \beta_{1} + \beta_{4} ) q^{49} + ( -4 - 8 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{53} + ( 18 + 5 \beta_{1} - 18 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 5 \beta_{7} ) q^{55} + ( -31 + \beta_{1} - 31 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{59} + ( -12 \beta_{2} - 4 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{61} + ( 30 + 4 \beta_{1} - 60 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} + 10 \beta_{5} - 5 \beta_{6} - 8 \beta_{7} ) q^{65} + ( -74 - 2 \beta_{1} + 37 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{67} + ( 20 + 20 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} + 6 \beta_{6} ) q^{71} + ( 3 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} + 12 \beta_{5} + 3 \beta_{7} ) q^{73} + ( -72 - 3 \beta_{4} + 10 \beta_{6} ) q^{77} + ( 6 - 4 \beta_{1} + 6 \beta_{2} - \beta_{3} - \beta_{4} - 7 \beta_{5} + 14 \beta_{6} - 4 \beta_{7} ) q^{79} + ( 7 - 2 \beta_{1} - 10 \beta_{4} - 3 \beta_{6} ) q^{83} + ( 28 \beta_{2} + 5 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{85} + ( 32 + 10 \beta_{1} - 16 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} + \beta_{6} - 5 \beta_{7} ) q^{89} + ( 32 - 4 \beta_{1} - 16 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 10 \beta_{5} + 10 \beta_{6} + 2 \beta_{7} ) q^{91} + ( 68 + 3 \beta_{1} - 48 \beta_{2} - 3 \beta_{3} + \beta_{4} - 13 \beta_{5} + 12 \beta_{6} + 4 \beta_{7} ) q^{95} + ( 13 + 4 \beta_{1} + 13 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 8 \beta_{5} - 16 \beta_{6} + 4 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{5} - 12 q^{7} + O(q^{10})$$ $$8 q + q^{5} - 12 q^{7} + 10 q^{11} + 9 q^{13} - 23 q^{17} - 33 q^{19} + 31 q^{23} - 73 q^{25} + 105 q^{29} + 68 q^{35} - 18 q^{41} - 41 q^{43} - 107 q^{47} + 312 q^{49} - 39 q^{53} + 70 q^{55} - 348 q^{59} - 45 q^{61} - 432 q^{67} + 243 q^{71} + 16 q^{73} - 544 q^{77} + 75 q^{79} + 82 q^{83} + 109 q^{85} + 213 q^{89} + 222 q^{91} + 385 q^{95} + 144 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 56 x^{6} - 154 x^{5} + 917 x^{4} - 1582 x^{3} + 4294 x^{2} - 3528 x + 4971$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 13$$ $$\beta_{2}$$ $$=$$ $$($$$$8 \nu^{7} - 28 \nu^{6} + 434 \nu^{5} - 1015 \nu^{4} + 6314 \nu^{3} - 8470 \nu^{2} + 16740 \nu - 6477$$$$)/1029$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{7} + 14 \nu^{6} - 217 \nu^{5} + 1022 \nu^{4} - 4186 \nu^{3} + 17612 \nu^{2} - 19689 \nu + 38739$$$$)/1029$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} + 26 \nu^{2} - 25 \nu + 70$$ $$\beta_{5}$$ $$=$$ $$($$$$-47 \nu^{7} - 7 \nu^{6} - 1778 \nu^{5} - 2569 \nu^{4} - 14714 \nu^{3} - 57169 \nu^{2} + 21531 \nu - 187170$$$$)/2058$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 3 \nu^{5} - 46 \nu^{4} + 87 \nu^{3} - 562 \nu^{2} + 519 \nu - 1251$$$$)/6$$ $$\beta_{7}$$ $$=$$ $$($$$$34 \nu^{7} - 119 \nu^{6} + 1673 \nu^{5} - 3885 \nu^{4} + 22204 \nu^{3} - 29309 \nu^{2} + 56739 \nu - 21439$$$$)/343$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + 2 \beta_{3} + \beta_{2} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_{1} - 38$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{7} - 6 \beta_{6} + 12 \beta_{5} + 17 \beta_{4} - 34 \beta_{3} - 20 \beta_{2} + 3 \beta_{1} - 47$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$6 \beta_{7} - 12 \beta_{6} + 24 \beta_{5} + 38 \beta_{4} - 70 \beta_{3} - 41 \beta_{2} - 72 \beta_{1} + 709$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-72 \beta_{7} + 132 \beta_{6} - 264 \beta_{5} - 319 \beta_{4} + 653 \beta_{3} + 469 \beta_{2} - 144 \beta_{1} + 1582$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-231 \beta_{7} + 408 \beta_{6} - 852 \beta_{5} - 1183 \beta_{4} + 2135 \beta_{3} + 1510 \beta_{2} + 1455 \beta_{1} - 13835$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$1491 \beta_{7} - 2520 \beta_{6} + 4914 \beta_{5} + 5603 \beta_{4} - 12067 \beta_{3} - 10223 \beta_{2} + 4578 \beta_{1} - 47093$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$343$$ $$533$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 0.5 + 4.59025i 0.5 − 1.68338i 0.5 − 4.68383i 0.5 + 1.77696i 0.5 − 4.59025i 0.5 + 1.68338i 0.5 + 4.68383i 0.5 − 1.77696i
0 0 0 −3.36497 5.82829i 0 −10.3204 0 0 0
145.2 0 0 0 −2.50973 4.34699i 0 7.91625 0 0 0
145.3 0 0 0 1.55796 + 2.69846i 0 −11.1883 0 0 0
145.4 0 0 0 4.81674 + 8.34284i 0 7.59243 0 0 0
217.1 0 0 0 −3.36497 + 5.82829i 0 −10.3204 0 0 0
217.2 0 0 0 −2.50973 + 4.34699i 0 7.91625 0 0 0
217.3 0 0 0 1.55796 2.69846i 0 −11.1883 0 0 0
217.4 0 0 0 4.81674 8.34284i 0 7.59243 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 217.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.y.h 8
3.b odd 2 1 76.3.h.a 8
12.b even 2 1 304.3.r.c 8
19.d odd 6 1 inner 684.3.y.h 8
57.f even 6 1 76.3.h.a 8
57.f even 6 1 1444.3.c.b 8
57.h odd 6 1 1444.3.c.b 8
228.n odd 6 1 304.3.r.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.h.a 8 3.b odd 2 1
76.3.h.a 8 57.f even 6 1
304.3.r.c 8 12.b even 2 1
304.3.r.c 8 228.n odd 6 1
684.3.y.h 8 1.a even 1 1 trivial
684.3.y.h 8 19.d odd 6 1 inner
1444.3.c.b 8 57.f even 6 1
1444.3.c.b 8 57.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(684, [\chi])$$:

 $$T_{5}^{8} - \cdots$$ $$T_{7}^{4} + 6 T_{7}^{3} - 158 T_{7}^{2} - 498 T_{7} + 6940$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$1028196 - 79092 T + 93288 T^{2} + 8736 T^{3} + 6304 T^{4} + 242 T^{5} + 87 T^{6} - T^{7} + T^{8}$$
$7$ $$( 6940 - 498 T - 158 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$11$ $$( -4746 + 2001 T - 209 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$13$ $$46785600 - 14035680 T - 1099872 T^{2} + 751032 T^{3} + 133272 T^{4} + 3294 T^{5} - 339 T^{6} - 9 T^{7} + T^{8}$$
$17$ $$56250000 + 13500000 T + 3315000 T^{2} + 327000 T^{3} + 49000 T^{4} + 3830 T^{5} + 519 T^{6} + 23 T^{7} + T^{8}$$
$19$ $$16983563041 + 1552514073 T + 35056349 T^{2} - 2386932 T^{3} - 228798 T^{4} - 6612 T^{5} + 269 T^{6} + 33 T^{7} + T^{8}$$
$23$ $$21126622500 + 1656990000 T + 212228100 T^{2} + 2559300 T^{3} + 528406 T^{4} - 5254 T^{5} + 1527 T^{6} - 31 T^{7} + T^{8}$$
$29$ $$21068100 + 124177860 T + 246010932 T^{2} + 12011976 T^{3} - 754344 T^{4} - 46620 T^{5} + 4119 T^{6} - 105 T^{7} + T^{8}$$
$31$ $$2425365504 + 163312524 T^{2} + 1572732 T^{4} + 2688 T^{6} + T^{8}$$
$37$ $$409702406400 + 3845107692 T^{2} + 7272720 T^{4} + 4800 T^{6} + T^{8}$$
$41$ $$74733890625 - 8901636750 T - 69756552 T^{2} + 50405976 T^{3} + 1927557 T^{4} - 27864 T^{5} - 1440 T^{6} + 18 T^{7} + T^{8}$$
$43$ $$386933761600 + 58553869280 T + 7304489344 T^{2} + 286525544 T^{3} + 10741456 T^{4} + 85682 T^{5} + 4183 T^{6} + 41 T^{7} + T^{8}$$
$47$ $$105819282922500 + 4227771907800 T + 156978390144 T^{2} + 2678131980 T^{3} + 55608166 T^{4} + 697856 T^{5} + 12609 T^{6} + 107 T^{7} + T^{8}$$
$53$ $$143608586342400 - 885258408960 T - 87770967552 T^{2} + 552267072 T^{3} + 42946560 T^{4} - 291564 T^{5} - 6969 T^{6} + 39 T^{7} + T^{8}$$
$59$ $$8199855604521 - 1011751326558 T + 5394103956 T^{2} + 4468816656 T^{3} + 203820795 T^{4} + 4401504 T^{5} + 53016 T^{6} + 348 T^{7} + T^{8}$$
$61$ $$4850565316 - 726686364 T + 286187072 T^{2} + 20296824 T^{3} + 6882000 T^{4} - 93702 T^{5} + 4571 T^{6} + 45 T^{7} + T^{8}$$
$67$ $$669597133730625 + 1725708786750 T - 441317432700 T^{2} - 1141199280 T^{3} + 309093759 T^{4} + 7392384 T^{5} + 79320 T^{6} + 432 T^{7} + T^{8}$$
$71$ $$918515225664 - 63656396640 T - 2859476256 T^{2} + 300085560 T^{3} + 15990696 T^{4} - 1097874 T^{5} + 24201 T^{6} - 243 T^{7} + T^{8}$$
$73$ $$1139611003515625 + 3788336787500 T + 414247499650 T^{2} - 254933560 T^{3} + 109599799 T^{4} - 34072 T^{5} + 12154 T^{6} - 16 T^{7} + T^{8}$$
$79$ $$771333601536 - 1261526918400 T + 699847174656 T^{2} - 19787846400 T^{3} + 154746432 T^{4} + 1033200 T^{5} - 11901 T^{6} - 75 T^{7} + T^{8}$$
$83$ $$( -897750 - 218025 T - 9455 T^{2} - 41 T^{3} + T^{4} )^{2}$$
$89$ $$498714472231056 + 37989000962928 T + 756634674096 T^{2} - 15840717696 T^{3} - 56397240 T^{4} + 1983456 T^{5} + 5811 T^{6} - 213 T^{7} + T^{8}$$
$97$ $$258895984160025 - 2575340253720 T - 225766839978 T^{2} + 2330735472 T^{3} + 203644287 T^{4} + 2096928 T^{5} - 7650 T^{6} - 144 T^{7} + T^{8}$$