# Properties

 Label 684.3.h.e Level $684$ Weight $3$ Character orbit 684.h Analytic conductor $18.638$ Analytic rank $0$ Dimension $6$ 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.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.6376500822$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.219615408.1 Defining polynomial: $$x^{6} - 3 x^{5} + 22 x^{4} - 39 x^{3} + 112 x^{2} - 93 x + 39$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}\cdot 3$$ Twist minimal: no (minimal twist has level 228) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} -\beta_{2} q^{7} +O(q^{10})$$ $$q + \beta_{3} q^{5} -\beta_{2} q^{7} + ( 4 + \beta_{2} ) q^{11} + ( -\beta_{1} + \beta_{4} ) q^{13} + ( 8 - \beta_{2} + 2 \beta_{3} ) q^{17} + ( -1 - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{19} + ( -6 + 3 \beta_{2} + \beta_{3} ) q^{23} + ( 5 - 3 \beta_{2} + 2 \beta_{3} ) q^{25} + ( 2 \beta_{1} + 2 \beta_{4} ) q^{29} + ( -\beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{31} + ( -2 + \beta_{2} + 4 \beta_{3} ) q^{35} + ( 3 \beta_{1} + \beta_{4} ) q^{37} + ( 3 \beta_{4} - 4 \beta_{5} ) q^{41} + ( -36 + \beta_{2} + 2 \beta_{3} ) q^{43} + ( -4 - 4 \beta_{2} + 5 \beta_{3} ) q^{47} + ( -7 + 5 \beta_{2} - 2 \beta_{3} ) q^{49} + ( -2 \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{53} + ( 2 - \beta_{2} ) q^{55} + ( 4 \beta_{1} - 5 \beta_{4} - 2 \beta_{5} ) q^{59} + ( 32 + 7 \beta_{2} + 4 \beta_{3} ) q^{61} + ( -6 \beta_{1} + 9 \beta_{4} - 4 \beta_{5} ) q^{65} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{67} + ( -4 \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{71} + ( 12 + \beta_{2} + 14 \beta_{3} ) q^{73} + ( -42 - 9 \beta_{2} + 2 \beta_{3} ) q^{77} + ( 3 \beta_{1} - 8 \beta_{4} + 4 \beta_{5} ) q^{79} + ( 66 + 4 \beta_{3} ) q^{83} + ( 58 - 5 \beta_{2} + 16 \beta_{3} ) q^{85} + ( 3 \beta_{4} - 2 \beta_{5} ) q^{89} + ( 11 \beta_{4} - 2 \beta_{5} ) q^{91} + ( -60 - 2 \beta_{1} + 6 \beta_{2} - 5 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{95} + ( 7 \beta_{4} - 2 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{5} - 2 q^{7} + O(q^{10})$$ $$6 q + 2 q^{5} - 2 q^{7} + 26 q^{11} + 50 q^{17} - 10 q^{19} - 28 q^{23} + 28 q^{25} - 2 q^{35} - 210 q^{43} - 22 q^{47} - 36 q^{49} + 10 q^{55} + 214 q^{61} + 102 q^{73} - 266 q^{77} + 404 q^{83} + 370 q^{85} - 358 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 22 x^{4} - 39 x^{3} + 112 x^{2} - 93 x + 39$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 32 \nu^{3} + 43 \nu^{2} + 16 \nu - 15$$$$)/19$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - 2 \nu^{3} + 9 \nu^{2} - 8 \nu - 7$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{4} + 2 \nu^{3} - 13 \nu^{2} + 12 \nu - 17$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$8 \nu^{5} - 20 \nu^{4} + 128 \nu^{3} - 172 \nu^{2} + 392 \nu - 168$$$$)/19$$ $$\beta_{5}$$ $$=$$ $$($$$$8 \nu^{5} - 20 \nu^{4} + 204 \nu^{3} - 286 \nu^{2} + 1114 \nu - 510$$$$)/19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + 4 \beta_{1} + 12$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} - 12 \beta_{3} - 12 \beta_{2} + 4 \beta_{1} - 132$$$$)/24$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{5} - 7 \beta_{4} - 9 \beta_{3} - 9 \beta_{2} - 16 \beta_{1} - 102$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$($$$$12 \beta_{5} - 29 \beta_{4} + 72 \beta_{3} + 120 \beta_{2} - 68 \beta_{1} + 1044$$$$)/24$$ $$\nu^{5}$$ $$=$$ $$($$$$-66 \beta_{5} + 181 \beta_{4} + 210 \beta_{3} + 330 \beta_{2} + 232 \beta_{1} + 2952$$$$)/24$$

## Character values

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

 $$n$$ $$325$$ $$343$$ $$533$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 − 0.460304i 0.5 + 0.460304i 0.5 + 3.19918i 0.5 − 3.19918i 0.5 + 2.79345i 0.5 − 2.79345i
0 0 0 −5.83539 0 5.24085 0 0 0
37.2 0 0 0 −5.83539 0 5.24085 0 0 0
37.3 0 0 0 −0.556406 0 −9.52587 0 0 0
37.4 0 0 0 −0.556406 0 −9.52587 0 0 0
37.5 0 0 0 7.39180 0 3.28502 0 0 0
37.6 0 0 0 7.39180 0 3.28502 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.h.e 6
3.b odd 2 1 228.3.h.a 6
4.b odd 2 1 2736.3.o.m 6
12.b even 2 1 912.3.o.c 6
19.b odd 2 1 inner 684.3.h.e 6
57.d even 2 1 228.3.h.a 6
76.d even 2 1 2736.3.o.m 6
228.b odd 2 1 912.3.o.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.h.a 6 3.b odd 2 1
228.3.h.a 6 57.d even 2 1
684.3.h.e 6 1.a even 1 1 trivial
684.3.h.e 6 19.b odd 2 1 inner
912.3.o.c 6 12.b even 2 1
912.3.o.c 6 228.b odd 2 1
2736.3.o.m 6 4.b odd 2 1
2736.3.o.m 6 76.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} - T_{5}^{2} - 44 T_{5} - 24$$ acting on $$S_{3}^{\mathrm{new}}(684, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$( -24 - 44 T - T^{2} + T^{3} )^{2}$$
$7$ $$( 164 - 64 T + T^{2} + T^{3} )^{2}$$
$11$ $$( 12 - 8 T - 13 T^{2} + T^{3} )^{2}$$
$13$ $$2495232 + 98064 T^{2} + 792 T^{4} + T^{6}$$
$17$ $$( 108 - 32 T - 25 T^{2} + T^{3} )^{2}$$
$19$ $$47045881 + 1303210 T + 146927 T^{2} + 7372 T^{3} + 407 T^{4} + 10 T^{5} + T^{6}$$
$23$ $$( -5136 - 560 T + 14 T^{2} + T^{3} )^{2}$$
$29$ $$322486272 + 2011392 T^{2} + 3168 T^{4} + T^{6}$$
$31$ $$62208 + 1009296 T^{2} + 3192 T^{4} + T^{6}$$
$37$ $$832267008 + 8883216 T^{2} + 5976 T^{4} + T^{6}$$
$41$ $$29104386048 + 35322624 T^{2} + 10896 T^{4} + T^{6}$$
$43$ $$( 35764 + 3432 T + 105 T^{2} + T^{3} )^{2}$$
$47$ $$( -25272 - 2084 T + 11 T^{2} + T^{3} )^{2}$$
$53$ $$63700992 + 4349952 T^{2} + 4272 T^{4} + T^{6}$$
$59$ $$207873257472 + 115418880 T^{2} + 19536 T^{4} + T^{6}$$
$61$ $$( 104276 - 64 T - 107 T^{2} + T^{3} )^{2}$$
$67$ $$511377408 + 4886784 T^{2} + 4320 T^{4} + T^{6}$$
$71$ $$48612814848 + 41654016 T^{2} + 11472 T^{4} + T^{6}$$
$73$ $$( 115492 - 7896 T - 51 T^{2} + T^{3} )^{2}$$
$79$ $$60723965952 + 68395536 T^{2} + 19848 T^{4} + T^{6}$$
$83$ $$( -259992 + 12892 T - 202 T^{2} + T^{3} )^{2}$$
$89$ $$956510208 + 3278592 T^{2} + 3408 T^{4} + T^{6}$$
$97$ $$168210432 + 16512768 T^{2} + 8400 T^{4} + T^{6}$$