# Properties

 Label 2672.2.a.j Level $2672$ Weight $2$ Character orbit 2672.a Self dual yes Analytic conductor $21.336$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2672 = 2^{4} \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2672.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.3360274201$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.826865.1 Defining polynomial: $$x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 6 x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 668) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} ) q^{3} + 2 q^{5} + ( -2 + \beta_{3} ) q^{7} + ( 2 - \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{2} ) q^{3} + 2 q^{5} + ( -2 + \beta_{3} ) q^{7} + ( 2 - \beta_{2} + \beta_{4} ) q^{9} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{11} -2 \beta_{1} q^{13} + ( -2 + 2 \beta_{2} ) q^{15} + ( -2 + 2 \beta_{1} - 2 \beta_{4} ) q^{17} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{19} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{21} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{23} - q^{25} + ( -4 + \beta_{1} - 3 \beta_{4} ) q^{27} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{29} + ( -3 + 3 \beta_{1} - \beta_{2} - \beta_{4} ) q^{31} + ( -3 \beta_{1} - \beta_{2} ) q^{33} + ( -4 + 2 \beta_{3} ) q^{35} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{39} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{41} -4 \beta_{1} q^{43} + ( 4 - 2 \beta_{2} + 2 \beta_{4} ) q^{45} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{47} + ( 4 - \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{49} + ( 2 - 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{51} + ( 2 - 4 \beta_{1} + 2 \beta_{4} ) q^{53} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( -6 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{57} + ( -2 + 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{59} + ( 3 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 1 - \beta_{1} + \beta_{4} ) q^{63} -4 \beta_{1} q^{65} + ( -6 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{67} + ( -2 - 4 \beta_{1} + 2 \beta_{3} ) q^{69} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{71} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{73} + ( 1 - \beta_{2} ) q^{75} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{77} + ( 6 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{79} + ( -2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{81} + ( 2 - 6 \beta_{1} ) q^{83} + ( -4 + 4 \beta_{1} - 4 \beta_{4} ) q^{85} + ( -3 + 4 \beta_{1} + \beta_{3} - \beta_{4} ) q^{87} + ( -5 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{89} + ( 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{91} + ( -3 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{93} + ( -2 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{95} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} ) q^{97} + ( -1 - \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 3q^{3} + 10q^{5} - 9q^{7} + 6q^{9} + O(q^{10})$$ $$5q - 3q^{3} + 10q^{5} - 9q^{7} + 6q^{9} - 5q^{11} - 4q^{13} - 6q^{15} - 2q^{17} - 5q^{19} - 4q^{21} - 6q^{23} - 5q^{25} - 12q^{27} - 5q^{29} - 9q^{31} - 8q^{33} - 18q^{35} + 8q^{37} - 4q^{41} - 8q^{43} + 12q^{45} - 13q^{47} + 14q^{49} - 2q^{53} - 10q^{55} - 12q^{57} - 4q^{59} + 11q^{61} + q^{63} - 8q^{65} - 28q^{67} - 16q^{69} - 2q^{71} + 8q^{73} + 3q^{75} - 12q^{77} + 10q^{79} - 15q^{81} - 2q^{83} - 4q^{85} - 4q^{87} - 17q^{89} + 12q^{91} - 12q^{93} - 10q^{95} - 27q^{97} - 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 6 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 3 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 5 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 2 \beta_{3} + 8 \beta_{2} + 9 \beta_{1} + 8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.147687 1.66287 −0.873948 −1.69135 2.75474
0 −3.12588 0 2.00000 0 0.516539 0 6.77110 0
1.2 0 −1.89773 0 2.00000 0 −4.92082 0 0.601369 0
1.3 0 −1.36227 0 2.00000 0 1.42676 0 −1.14423 0
1.4 0 1.55202 0 2.00000 0 −4.48567 0 −0.591244 0
1.5 0 1.83385 0 2.00000 0 −1.53681 0 0.363011 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$167$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2672.2.a.j 5
4.b odd 2 1 668.2.a.b 5
12.b even 2 1 6012.2.a.d 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.2.a.b 5 4.b odd 2 1
2672.2.a.j 5 1.a even 1 1 trivial
6012.2.a.d 5 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2672))$$:

 $$T_{3}^{5} + 3 T_{3}^{4} - 6 T_{3}^{3} - 17 T_{3}^{2} + 9 T_{3} + 23$$ $$T_{7}^{5} + 9 T_{7}^{4} + 16 T_{7}^{3} - 29 T_{7}^{2} - 39 T_{7} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$23 + 9 T - 17 T^{2} - 6 T^{3} + 3 T^{4} + T^{5}$$
$5$ $$( -2 + T )^{5}$$
$7$ $$25 - 39 T - 29 T^{2} + 16 T^{3} + 9 T^{4} + T^{5}$$
$11$ $$25 + 35 T - 29 T^{2} - 12 T^{3} + 5 T^{4} + T^{5}$$
$13$ $$32 + 96 T - 48 T^{2} - 20 T^{3} + 4 T^{4} + T^{5}$$
$17$ $$-800 + 832 T - 40 T^{2} - 60 T^{3} + 2 T^{4} + T^{5}$$
$19$ $$-445 + 605 T - 101 T^{2} - 42 T^{3} + 5 T^{4} + T^{5}$$
$23$ $$-32 - 288 T - 360 T^{2} - 52 T^{3} + 6 T^{4} + T^{5}$$
$29$ $$13 + 31 T - 39 T^{2} - 28 T^{3} + 5 T^{4} + T^{5}$$
$31$ $$365 - 5 T - 207 T^{2} - 36 T^{3} + 9 T^{4} + T^{5}$$
$37$ $$-288 + 192 T + 80 T^{2} - 44 T^{3} - 8 T^{4} + T^{5}$$
$41$ $$-928 + 1472 T - 232 T^{2} - 96 T^{3} + 4 T^{4} + T^{5}$$
$43$ $$1024 + 1536 T - 384 T^{2} - 80 T^{3} + 8 T^{4} + T^{5}$$
$47$ $$-81 - 423 T - 161 T^{2} + 26 T^{3} + 13 T^{4} + T^{5}$$
$53$ $$5408 + 3536 T - 208 T^{2} - 124 T^{3} + 2 T^{4} + T^{5}$$
$59$ $$49696 + 11376 T - 824 T^{2} - 236 T^{3} + 4 T^{4} + T^{5}$$
$61$ $$877 + 1187 T + 193 T^{2} - 76 T^{3} - 11 T^{4} + T^{5}$$
$67$ $$-64928 - 22944 T - 1888 T^{2} + 156 T^{3} + 28 T^{4} + T^{5}$$
$71$ $$10784 + 5472 T - 248 T^{2} - 180 T^{3} + 2 T^{4} + T^{5}$$
$73$ $$8992 - 6928 T + 1720 T^{2} - 124 T^{3} - 8 T^{4} + T^{5}$$
$79$ $$-75104 + 14224 T + 1608 T^{2} - 256 T^{3} - 10 T^{4} + T^{5}$$
$83$ $$-11360 + 10496 T - 8 T^{2} - 236 T^{3} + 2 T^{4} + T^{5}$$
$89$ $$-179 - 793 T - 363 T^{2} + 40 T^{3} + 17 T^{4} + T^{5}$$
$97$ $$37899 - 141 T - 2417 T^{2} + 36 T^{3} + 27 T^{4} + T^{5}$$