# Properties

 Label 684.3.h.c Level $684$ Weight $3$ Character orbit 684.h Analytic conductor $18.638$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-29})$$ Defining polynomial: $$x^{2} + 29$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{-29}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 q^{5} - q^{7} +O(q^{10})$$ $$q + 4 q^{5} - q^{7} -14 q^{11} -\beta q^{13} -23 q^{17} + ( 10 - \beta ) q^{19} + q^{23} -9 q^{25} -3 \beta q^{29} + 2 \beta q^{31} -4 q^{35} -2 \beta q^{37} + 2 \beta q^{41} + 68 q^{43} -26 q^{47} -48 q^{49} -5 \beta q^{53} -56 q^{55} + \beta q^{59} -40 q^{61} -4 \beta q^{65} + \beta q^{67} -2 \beta q^{71} -7 q^{73} + 14 q^{77} -6 \beta q^{79} -32 q^{83} -92 q^{85} -8 \beta q^{89} + \beta q^{91} + ( 40 - 4 \beta ) q^{95} + 6 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{5} - 2q^{7} + O(q^{10})$$ $$2q + 8q^{5} - 2q^{7} - 28q^{11} - 46q^{17} + 20q^{19} + 2q^{23} - 18q^{25} - 8q^{35} + 136q^{43} - 52q^{47} - 96q^{49} - 112q^{55} - 80q^{61} - 14q^{73} + 28q^{77} - 64q^{83} - 184q^{85} + 80q^{95} + O(q^{100})$$

## 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
 5.38516i − 5.38516i
0 0 0 4.00000 0 −1.00000 0 0 0
37.2 0 0 0 4.00000 0 −1.00000 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.h.c 2
3.b odd 2 1 76.3.c.a 2
4.b odd 2 1 2736.3.o.i 2
12.b even 2 1 304.3.e.b 2
15.d odd 2 1 1900.3.e.b 2
15.e even 4 2 1900.3.g.b 4
19.b odd 2 1 inner 684.3.h.c 2
24.f even 2 1 1216.3.e.l 2
24.h odd 2 1 1216.3.e.k 2
57.d even 2 1 76.3.c.a 2
76.d even 2 1 2736.3.o.i 2
228.b odd 2 1 304.3.e.b 2
285.b even 2 1 1900.3.e.b 2
285.j odd 4 2 1900.3.g.b 4
456.l odd 2 1 1216.3.e.l 2
456.p even 2 1 1216.3.e.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.a 2 3.b odd 2 1
76.3.c.a 2 57.d even 2 1
304.3.e.b 2 12.b even 2 1
304.3.e.b 2 228.b odd 2 1
684.3.h.c 2 1.a even 1 1 trivial
684.3.h.c 2 19.b odd 2 1 inner
1216.3.e.k 2 24.h odd 2 1
1216.3.e.k 2 456.p even 2 1
1216.3.e.l 2 24.f even 2 1
1216.3.e.l 2 456.l odd 2 1
1900.3.e.b 2 15.d odd 2 1
1900.3.e.b 2 285.b even 2 1
1900.3.g.b 4 15.e even 4 2
1900.3.g.b 4 285.j odd 4 2
2736.3.o.i 2 4.b odd 2 1
2736.3.o.i 2 76.d even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -4 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( 14 + T )^{2}$$
$13$ $$261 + T^{2}$$
$17$ $$( 23 + T )^{2}$$
$19$ $$361 - 20 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$2349 + T^{2}$$
$31$ $$1044 + T^{2}$$
$37$ $$1044 + T^{2}$$
$41$ $$1044 + T^{2}$$
$43$ $$( -68 + T )^{2}$$
$47$ $$( 26 + T )^{2}$$
$53$ $$6525 + T^{2}$$
$59$ $$261 + T^{2}$$
$61$ $$( 40 + T )^{2}$$
$67$ $$261 + T^{2}$$
$71$ $$1044 + T^{2}$$
$73$ $$( 7 + T )^{2}$$
$79$ $$9396 + T^{2}$$
$83$ $$( 32 + T )^{2}$$
$89$ $$16704 + T^{2}$$
$97$ $$9396 + T^{2}$$