Properties

 Label 684.3.h.d Level $684$ Weight $3$ Character orbit 684.h Self dual yes Analytic conductor $18.638$ Analytic rank $0$ Dimension $4$ CM discriminant -19 Inner twists $4$

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: yes Analytic conductor: $$18.6376500822$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{19})$$ Defining polynomial: $$x^{4} - 11 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + ( -3 - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + ( -3 - \beta_{3} ) q^{7} -\beta_{2} q^{11} + ( 2 \beta_{1} + \beta_{2} ) q^{17} + 19 q^{19} + ( \beta_{1} + 3 \beta_{2} ) q^{23} + ( 42 + 3 \beta_{3} ) q^{25} + ( -8 \beta_{1} - 7 \beta_{2} ) q^{35} + ( 43 + \beta_{3} ) q^{43} + ( -7 \beta_{1} + 4 \beta_{2} ) q^{47} + ( 88 + 5 \beta_{3} ) q^{49} + ( -7 - 7 \beta_{3} ) q^{55} + ( 49 - 5 \beta_{3} ) q^{61} + ( 7 - 11 \beta_{3} ) q^{73} + ( 14 \beta_{1} - 3 \beta_{2} ) q^{77} + ( 4 \beta_{1} + 12 \beta_{2} ) q^{83} + ( 141 + 13 \beta_{3} ) q^{85} + 19 \beta_{1} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{7} + O(q^{10})$$ $$4 q - 10 q^{7} + 76 q^{19} + 162 q^{25} + 170 q^{43} + 342 q^{49} - 14 q^{55} + 206 q^{61} + 50 q^{73} + 538 q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 11 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - 6 \nu$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 12 \nu$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{2} - 17$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 17$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + 2 \beta_{1}$$

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
 −3.04547 1.31342 −1.31342 3.04547
0 0 0 −9.97368 0 −13.8248 0 0 0
37.2 0 0 0 −5.61478 0 8.82475 0 0 0
37.3 0 0 0 5.61478 0 8.82475 0 0 0
37.4 0 0 0 9.97368 0 −13.8248 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 CM by $$\Q(\sqrt{-19})$$
3.b odd 2 1 inner
57.d even 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.d 4
3.b odd 2 1 inner 684.3.h.d 4
4.b odd 2 1 2736.3.o.k 4
12.b even 2 1 2736.3.o.k 4
19.b odd 2 1 CM 684.3.h.d 4
57.d even 2 1 inner 684.3.h.d 4
76.d even 2 1 2736.3.o.k 4
228.b odd 2 1 2736.3.o.k 4

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 131 T_{5}^{2} + 3136$$ acting on $$S_{3}^{\mathrm{new}}(684, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$3136 - 131 T^{2} + T^{4}$$
$7$ $$( -122 + 5 T + T^{2} )^{2}$$
$11$ $$12544 - 251 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$4096 - 803 T^{2} + T^{4}$$
$19$ $$( -19 + T )^{4}$$
$23$ $$( -1216 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 1678 - 85 T + T^{2} )^{2}$$
$47$ $$11669056 - 10043 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -554 - 103 T + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -15362 - 25 T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$( -19456 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$