# Properties

 Label 684.3.y.g Level $684$ Weight $3$ Character orbit 684.y Analytic conductor $18.638$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: $$x^{8} - 34 x^{6} + 921 x^{4} - 7990 x^{2} + 55225$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{5} ) q^{5} + ( -1 + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{5} ) q^{5} + ( -1 + \beta_{3} ) q^{7} -\beta_{4} q^{11} + ( -5 + 5 \beta_{2} - 4 \beta_{3} + 2 \beta_{6} ) q^{13} + ( -\beta_{1} + \beta_{5} + \beta_{7} ) q^{17} + ( 5 + 2 \beta_{2} - 7 \beta_{3} + 8 \beta_{6} ) q^{19} + \beta_{5} q^{23} + ( -9 - 9 \beta_{2} + 6 \beta_{6} ) q^{25} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{29} + ( -9 - 18 \beta_{2} - 5 \beta_{3} + 10 \beta_{6} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{5} - \beta_{7} ) q^{35} + ( -1 - 2 \beta_{2} - 6 \beta_{3} + 12 \beta_{6} ) q^{37} + ( -2 \beta_{1} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{41} + ( 5 \beta_{2} - 11 \beta_{3} + 11 \beta_{6} ) q^{43} + ( -4 \beta_{4} - 4 \beta_{5} - 4 \beta_{7} ) q^{47} + ( -42 - 2 \beta_{3} ) q^{49} + ( -4 \beta_{1} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{53} + ( -2 \beta_{2} - 28 \beta_{3} + 28 \beta_{6} ) q^{55} + ( 5 \beta_{1} + 4 \beta_{4} + 5 \beta_{5} + 2 \beta_{7} ) q^{59} + ( 31 + 31 \beta_{2} + 34 \beta_{6} ) q^{61} + ( -3 \beta_{1} + 2 \beta_{4} + 6 \beta_{5} + 4 \beta_{7} ) q^{65} + ( -19 + 19 \beta_{2} - 38 \beta_{3} + 19 \beta_{6} ) q^{67} + ( -3 \beta_{1} - 6 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} ) q^{71} + ( -25 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{73} -5 \beta_{1} q^{77} + ( -66 - 33 \beta_{2} + \beta_{3} + \beta_{6} ) q^{79} + ( 17 \beta_{1} - 3 \beta_{4} ) q^{83} + ( 32 + 32 \beta_{2} + 22 \beta_{6} ) q^{85} + ( -4 \beta_{1} - 5 \beta_{4} + 2 \beta_{5} + 5 \beta_{7} ) q^{89} + ( -7 + 7 \beta_{2} - 6 \beta_{3} + 3 \beta_{6} ) q^{91} + ( 4 \beta_{1} + 8 \beta_{4} - 10 \beta_{5} + 7 \beta_{7} ) q^{95} + ( 36 + 18 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{7} + O(q^{10})$$ $$8 q - 8 q^{7} - 60 q^{13} + 32 q^{19} - 36 q^{25} - 20 q^{43} - 336 q^{49} + 8 q^{55} + 124 q^{61} - 228 q^{67} + 100 q^{73} - 396 q^{79} + 128 q^{85} - 84 q^{91} + 216 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 34 x^{6} + 921 x^{4} - 7990 x^{2} + 55225$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$36 \nu^{7} - 5219 \nu^{5} + 105301 \nu^{3} - 1514105 \nu$$$$)/649305$$ $$\beta_{2}$$ $$=$$ $$($$$$-34 \nu^{6} + 921 \nu^{4} - 31314 \nu^{2} + 55225$$$$)/216435$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 7667$$$$)/2763$$ $$\beta_{4}$$ $$=$$ $$($$$$48 \nu^{7} - 2149 \nu^{5} + 58637 \nu^{3} - 888535 \nu$$$$)/129861$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 34 \nu^{5} + 686 \nu^{3} - 3995 \nu$$$$)/2115$$ $$\beta_{6}$$ $$=$$ $$($$$$-343 \nu^{6} + 15657 \nu^{4} - 315903 \nu^{2} + 2740570$$$$)/649305$$ $$\beta_{7}$$ $$=$$ $$($$$$95 \nu^{7} - 2149 \nu^{5} + 58637 \nu^{3} + 250980 \nu$$$$)/129861$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{6} - 3 \beta_{3} - 17 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$($$$$34 \beta_{7} - 70 \beta_{5} + 17 \beta_{4} + 35 \beta_{1}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$102 \beta_{6} - 343 \beta_{2} - 343$$ $$\nu^{5}$$ $$=$$ $$($$$$343 \beta_{7} - 955 \beta_{5} + 686 \beta_{4} - 955 \beta_{1}$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$2763 \beta_{3} - 7667$$ $$\nu^{7}$$ $$=$$ $$($$$$-7667 \beta_{7} + 24245 \beta_{5} + 7667 \beta_{4} - 48490 \beta_{1}$$$$)/6$$

## Character values

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

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

## 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
 −4.27333 − 2.46721i 2.69047 + 1.55335i −2.69047 − 1.55335i 4.27333 + 2.46721i −4.27333 + 2.46721i 2.69047 − 1.55335i −2.69047 + 1.55335i 4.27333 − 2.46721i
0 0 0 −3.48916 6.04340i 0 −3.44949 0 0 0
145.2 0 0 0 −2.19676 3.80490i 0 1.44949 0 0 0
145.3 0 0 0 2.19676 + 3.80490i 0 1.44949 0 0 0
145.4 0 0 0 3.48916 + 6.04340i 0 −3.44949 0 0 0
217.1 0 0 0 −3.48916 + 6.04340i 0 −3.44949 0 0 0
217.2 0 0 0 −2.19676 + 3.80490i 0 1.44949 0 0 0
217.3 0 0 0 2.19676 3.80490i 0 1.44949 0 0 0
217.4 0 0 0 3.48916 6.04340i 0 −3.44949 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
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 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.g 8
3.b odd 2 1 inner 684.3.y.g 8
19.d odd 6 1 inner 684.3.y.g 8
57.f even 6 1 inner 684.3.y.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.y.g 8 1.a even 1 1 trivial
684.3.y.g 8 3.b odd 2 1 inner
684.3.y.g 8 19.d odd 6 1 inner
684.3.y.g 8 57.f even 6 1 inner

## Hecke kernels

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

 $$T_{5}^{8} + 68 T_{5}^{6} + 3684 T_{5}^{4} + 63920 T_{5}^{2} + 883600$$ $$T_{7}^{2} + 2 T_{7} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$883600 + 63920 T^{2} + 3684 T^{4} + 68 T^{6} + T^{8}$$
$7$ $$( -5 + 2 T + T^{2} )^{4}$$
$11$ $$( 23500 - 332 T^{2} + T^{4} )^{2}$$
$13$ $$( 9 + 90 T + 303 T^{2} + 30 T^{3} + T^{4} )^{2}$$
$17$ $$14137600 + 1473920 T^{2} + 149904 T^{4} + 392 T^{6} + T^{8}$$
$19$ $$( 130321 - 5776 T + 570 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$23$ $$883600 + 63920 T^{2} + 3684 T^{4} + 68 T^{6} + T^{8}$$
$29$ $$1145145600 - 39795840 T^{2} + 1349136 T^{4} - 1176 T^{6} + T^{8}$$
$31$ $$( 42849 + 1386 T^{2} + T^{4} )^{2}$$
$37$ $$( 416025 + 1302 T^{2} + T^{4} )^{2}$$
$41$ $$23745739161600 - 23858012160 T^{2} + 19097856 T^{4} - 4896 T^{6} + T^{8}$$
$43$ $$( 491401 - 7010 T + 801 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$47$ $$926521753600 + 6037176320 T^{2} + 38375424 T^{4} + 6272 T^{6} + T^{8}$$
$53$ $$44732250000 - 393390000 T^{2} + 3248100 T^{4} - 1860 T^{6} + T^{8}$$
$59$ $$27957656250000 - 46762650000 T^{2} + 72928836 T^{4} - 8844 T^{6} + T^{8}$$
$61$ $$( 35700625 + 370450 T + 9819 T^{2} - 62 T^{3} + T^{4} )^{2}$$
$67$ $$( 29322225 - 617310 T - 1083 T^{2} + 114 T^{3} + T^{4} )^{2}$$
$71$ $$7513300281600 - 29011167360 T^{2} + 109280016 T^{4} - 10584 T^{6} + T^{8}$$
$73$ $$( 361201 - 30050 T + 1899 T^{2} - 50 T^{3} + T^{4} )^{2}$$
$79$ $$( 10556001 + 643302 T + 16317 T^{2} + 198 T^{3} + T^{4} )^{2}$$
$83$ $$( 18954160 - 23048 T^{2} + T^{4} )^{2}$$
$89$ $$7447769396211600 - 2198590518960 T^{2} + 562726116 T^{4} - 25476 T^{6} + T^{8}$$
$97$ $$( 972 - 54 T + T^{2} )^{4}$$