# Properties

 Label 684.2.f.b Level $684$ Weight $2$ Character orbit 684.f Analytic conductor $5.462$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 684.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.46176749826$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.14453810176.1 Defining polynomial: $$x^{8} + 3 x^{6} + 6 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( 1 - \beta_{2} + \beta_{6} ) q^{5} + ( \beta_{5} - \beta_{6} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( 1 - \beta_{2} + \beta_{6} ) q^{5} + ( \beta_{5} - \beta_{6} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + ( \beta_{1} - \beta_{3} + \beta_{7} ) q^{10} + ( \beta_{2} + \beta_{5} ) q^{11} + ( -\beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{13} + ( 2 \beta_{4} - \beta_{7} ) q^{14} + ( -\beta_{2} + \beta_{5} + \beta_{6} ) q^{16} + q^{17} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{19} + ( -3 - \beta_{5} - \beta_{6} ) q^{20} + ( \beta_{3} + 2 \beta_{4} ) q^{22} + ( -\beta_{2} - \beta_{6} ) q^{23} + ( -\beta_{2} + \beta_{6} ) q^{25} + ( 2 - \beta_{2} + \beta_{5} + 3 \beta_{6} ) q^{26} + ( 3 - \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{28} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{29} + ( \beta_{1} - \beta_{7} ) q^{31} + ( -\beta_{3} + 2 \beta_{4} + \beta_{7} ) q^{32} + \beta_{1} q^{34} + ( \beta_{2} + 3 \beta_{5} - 2 \beta_{6} ) q^{35} + ( 3 \beta_{1} + \beta_{7} ) q^{37} + ( -2 - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{38} + ( -3 \beta_{1} - 2 \beta_{4} - \beta_{7} ) q^{40} + ( 3 \beta_{1} + \beta_{7} ) q^{41} + ( -3 \beta_{2} - \beta_{5} - 2 \beta_{6} ) q^{43} + ( -1 - 2 \beta_{2} - \beta_{5} + 3 \beta_{6} ) q^{44} + ( -\beta_{3} - \beta_{7} ) q^{46} + ( -3 \beta_{2} - 3 \beta_{5} ) q^{47} + ( -4 + 4 \beta_{2} - 4 \beta_{6} ) q^{49} + ( -\beta_{3} + \beta_{7} ) q^{50} + ( 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 3 \beta_{7} ) q^{52} + ( 4 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{53} + ( -\beta_{2} + \beta_{5} - 2 \beta_{6} ) q^{55} + ( 3 \beta_{1} - \beta_{3} - 4 \beta_{4} + 2 \beta_{7} ) q^{56} + ( -4 + \beta_{2} + \beta_{5} + 3 \beta_{6} ) q^{58} + ( 3 \beta_{1} + \beta_{4} - 3 \beta_{7} ) q^{59} + ( 7 - 3 \beta_{2} + 3 \beta_{6} ) q^{61} + ( 2 + 2 \beta_{2} ) q^{62} + ( -2 - 3 \beta_{2} - 3 \beta_{5} + \beta_{6} ) q^{64} + ( -6 \beta_{1} - 2 \beta_{7} ) q^{65} + ( 2 \beta_{1} - \beta_{4} - 2 \beta_{7} ) q^{67} + ( -1 + \beta_{2} ) q^{68} + ( \beta_{3} + 6 \beta_{4} - 2 \beta_{7} ) q^{70} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{71} + ( 3 + 2 \beta_{2} - 2 \beta_{6} ) q^{73} + ( -6 + 2 \beta_{2} ) q^{74} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{76} + ( -5 + 3 \beta_{2} - 3 \beta_{6} ) q^{77} + ( 3 \beta_{1} + 4 \beta_{4} - 3 \beta_{7} ) q^{79} + ( 6 + 2 \beta_{5} - 2 \beta_{6} ) q^{80} + ( -6 + 2 \beta_{2} ) q^{82} + ( 2 \beta_{2} - 2 \beta_{5} + 4 \beta_{6} ) q^{83} + ( 1 - \beta_{2} + \beta_{6} ) q^{85} + ( -3 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{86} + ( -\beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{7} ) q^{88} + ( -5 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{7} ) q^{89} + ( -\beta_{1} - 7 \beta_{4} + \beta_{7} ) q^{91} + ( 4 + \beta_{2} - \beta_{5} - \beta_{6} ) q^{92} + ( -3 \beta_{3} - 6 \beta_{4} ) q^{94} + ( -\beta_{2} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{95} + ( 4 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} ) q^{97} + ( -4 \beta_{1} + 4 \beta_{3} - 4 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 6q^{4} + 4q^{5} + O(q^{10})$$ $$8q - 6q^{4} + 4q^{5} - 6q^{16} + 8q^{17} - 20q^{20} - 4q^{25} + 6q^{26} + 22q^{28} - 18q^{38} - 16q^{44} - 16q^{49} - 38q^{58} + 44q^{61} + 20q^{62} - 18q^{64} - 6q^{68} + 32q^{73} - 44q^{74} - 16q^{76} - 28q^{77} + 48q^{80} - 44q^{82} + 4q^{85} + 38q^{92} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 6 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} - 2 \nu^{3} - 4 \nu$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} + \nu^{4} - 2 \nu^{2} - 4$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 3 \nu^{4} + 6 \nu^{2} + 8$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 3 \nu^{5} + 6 \nu^{3} + 8 \nu$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + \beta_{5} - \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} + 2 \beta_{4} - \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$\beta_{6} - 3 \beta_{5} - 3 \beta_{2} - 2$$ $$\nu^{7}$$ $$=$$ $$\beta_{7} - 6 \beta_{4} - 3 \beta_{3} - 2 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 −1.06789 − 0.927153i −1.06789 + 0.927153i −0.331077 − 1.37491i −0.331077 + 1.37491i 0.331077 − 1.37491i 0.331077 + 1.37491i 1.06789 − 0.927153i 1.06789 + 0.927153i
−1.06789 0.927153i 0 0.280776 + 1.98019i −1.56155 0 0.868210i 1.53610 2.37495i 0 1.66757 + 1.44780i
379.2 −1.06789 + 0.927153i 0 0.280776 1.98019i −1.56155 0 0.868210i 1.53610 + 2.37495i 0 1.66757 1.44780i
379.3 −0.331077 1.37491i 0 −1.78078 + 0.910404i 2.56155 0 4.15286i 1.84130 + 2.14700i 0 −0.848071 3.52191i
379.4 −0.331077 + 1.37491i 0 −1.78078 0.910404i 2.56155 0 4.15286i 1.84130 2.14700i 0 −0.848071 + 3.52191i
379.5 0.331077 1.37491i 0 −1.78078 0.910404i 2.56155 0 4.15286i −1.84130 + 2.14700i 0 0.848071 3.52191i
379.6 0.331077 + 1.37491i 0 −1.78078 + 0.910404i 2.56155 0 4.15286i −1.84130 2.14700i 0 0.848071 + 3.52191i
379.7 1.06789 0.927153i 0 0.280776 1.98019i −1.56155 0 0.868210i −1.53610 2.37495i 0 −1.66757 + 1.44780i
379.8 1.06789 + 0.927153i 0 0.280776 + 1.98019i −1.56155 0 0.868210i −1.53610 + 2.37495i 0 −1.66757 1.44780i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 379.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.f.b 8
3.b odd 2 1 76.2.d.a 8
4.b odd 2 1 inner 684.2.f.b 8
12.b even 2 1 76.2.d.a 8
19.b odd 2 1 inner 684.2.f.b 8
24.f even 2 1 1216.2.h.d 8
24.h odd 2 1 1216.2.h.d 8
57.d even 2 1 76.2.d.a 8
76.d even 2 1 inner 684.2.f.b 8
228.b odd 2 1 76.2.d.a 8
456.l odd 2 1 1216.2.h.d 8
456.p even 2 1 1216.2.h.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.d.a 8 3.b odd 2 1
76.2.d.a 8 12.b even 2 1
76.2.d.a 8 57.d even 2 1
76.2.d.a 8 228.b odd 2 1
684.2.f.b 8 1.a even 1 1 trivial
684.2.f.b 8 4.b odd 2 1 inner
684.2.f.b 8 19.b odd 2 1 inner
684.2.f.b 8 76.d even 2 1 inner
1216.2.h.d 8 24.f even 2 1
1216.2.h.d 8 24.h odd 2 1
1216.2.h.d 8 456.l odd 2 1
1216.2.h.d 8 456.p even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(684, [\chi])$$:

 $$T_{5}^{2} - T_{5} - 4$$ $$T_{31}^{4} - 20 T_{31}^{2} + 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 12 T^{2} + 6 T^{4} + 3 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( -4 - T + T^{2} )^{4}$$
$7$ $$( 13 + 18 T^{2} + T^{4} )^{2}$$
$11$ $$( 52 + 15 T^{2} + T^{4} )^{2}$$
$13$ $$( 416 + 41 T^{2} + T^{4} )^{2}$$
$17$ $$( -1 + T )^{8}$$
$19$ $$130321 - 5776 T^{2} + 718 T^{4} - 16 T^{6} + T^{8}$$
$23$ $$( 52 + 19 T^{2} + T^{4} )^{2}$$
$29$ $$( 104 + 73 T^{2} + T^{4} )^{2}$$
$31$ $$( 32 - 20 T^{2} + T^{4} )^{2}$$
$37$ $$( 416 + 44 T^{2} + T^{4} )^{2}$$
$41$ $$( 416 + 44 T^{2} + T^{4} )^{2}$$
$43$ $$( 208 + 123 T^{2} + T^{4} )^{2}$$
$47$ $$( 4212 + 135 T^{2} + T^{4} )^{2}$$
$53$ $$( 104 + 97 T^{2} + T^{4} )^{2}$$
$59$ $$( 5408 - 175 T^{2} + T^{4} )^{2}$$
$61$ $$( -8 - 11 T + T^{2} )^{4}$$
$67$ $$( 8 - 95 T^{2} + T^{4} )^{2}$$
$71$ $$( 512 - 124 T^{2} + T^{4} )^{2}$$
$73$ $$( -1 - 8 T + T^{2} )^{4}$$
$79$ $$( 11552 - 244 T^{2} + T^{4} )^{2}$$
$83$ $$( 13312 + 236 T^{2} + T^{4} )^{2}$$
$89$ $$( 6656 + 232 T^{2} + T^{4} )^{2}$$
$97$ $$( 1664 + 292 T^{2} + T^{4} )^{2}$$