# Properties

 Label 2667.2.a.j Level $2667$ Weight $2$ Character orbit 2667.a Self dual yes Analytic conductor $21.296$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2667 = 3 \cdot 7 \cdot 127$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2667.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.2961022191$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: 7.7.118870813.1 Defining polynomial: $$x^{7} - 9 x^{5} - 3 x^{4} + 20 x^{3} + 7 x^{2} - 13 x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{4} + ( -1 - \beta_{3} + \beta_{5} ) q^{5} + \beta_{4} q^{6} + q^{7} + ( -1 - \beta_{5} - \beta_{6} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} + q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{4} + ( -1 - \beta_{3} + \beta_{5} ) q^{5} + \beta_{4} q^{6} + q^{7} + ( -1 - \beta_{5} - \beta_{6} ) q^{8} + q^{9} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{10} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{11} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{12} + ( -3 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{13} + \beta_{4} q^{14} + ( -1 - \beta_{3} + \beta_{5} ) q^{15} + ( -\beta_{1} + 2 \beta_{5} ) q^{16} + ( -\beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{17} + \beta_{4} q^{18} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{19} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{20} + q^{21} + ( -2 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{6} ) q^{22} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{23} + ( -1 - \beta_{5} - \beta_{6} ) q^{24} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{25} + ( 1 + \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + \beta_{6} ) q^{26} + q^{27} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{28} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{29} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{30} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} ) q^{32} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{33} + ( -\beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{34} + ( -1 - \beta_{3} + \beta_{5} ) q^{35} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{36} + ( -5 - 3 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{37} + ( -1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{38} + ( -3 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{39} + ( -2 - 2 \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{40} + ( -2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{41} + \beta_{4} q^{42} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{5} - 3 \beta_{6} ) q^{43} + ( -1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} ) q^{44} + ( -1 - \beta_{3} + \beta_{5} ) q^{45} + ( -3 - \beta_{3} + 4 \beta_{4} - \beta_{6} ) q^{46} + ( 3 + \beta_{1} - 5 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{47} + ( -\beta_{1} + 2 \beta_{5} ) q^{48} + q^{49} + ( 3 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{50} + ( -\beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{51} + ( -5 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} ) q^{52} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} + 3 \beta_{5} ) q^{53} + \beta_{4} q^{54} + ( -3 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} ) q^{55} + ( -1 - \beta_{5} - \beta_{6} ) q^{56} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{57} + ( -3 + 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{58} + ( -6 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{59} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{60} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{61} + ( -1 + 2 \beta_{2} - 5 \beta_{4} - \beta_{5} + \beta_{6} ) q^{62} + q^{63} + ( -2 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{64} + ( 1 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{65} + ( -2 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{6} ) q^{66} + ( -3 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{67} + ( 3 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{68} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{69} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{70} + ( 2 + \beta_{1} - 5 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{71} + ( -1 - \beta_{5} - \beta_{6} ) q^{72} + ( -3 + 5 \beta_{1} - \beta_{5} + 2 \beta_{6} ) q^{73} + ( -2 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} ) q^{74} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{75} + ( -5 - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{76} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{77} + ( 1 + \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + \beta_{6} ) q^{78} + ( 1 + \beta_{2} - 4 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} ) q^{79} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{5} - \beta_{6} ) q^{80} + q^{81} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{4} + 4 \beta_{6} ) q^{82} + ( -5 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{83} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{84} + ( -5 + 2 \beta_{2} + \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} ) q^{85} + ( 2 + \beta_{1} + 4 \beta_{2} + \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{86} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{87} + ( 3 - 4 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{88} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 7 \beta_{6} ) q^{89} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{90} + ( -3 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{91} + ( 6 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{92} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{93} + ( -1 - 3 \beta_{2} - \beta_{3} + \beta_{4} + 6 \beta_{5} + 2 \beta_{6} ) q^{94} + ( 1 + 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} + 3 \beta_{6} ) q^{95} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} ) q^{96} + ( 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 5 \beta_{6} ) q^{97} + \beta_{4} q^{98} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - 2q^{2} + 7q^{3} + 4q^{4} - 8q^{5} - 2q^{6} + 7q^{7} - 9q^{8} + 7q^{9} + O(q^{10})$$ $$7q - 2q^{2} + 7q^{3} + 4q^{4} - 8q^{5} - 2q^{6} + 7q^{7} - 9q^{8} + 7q^{9} - 3q^{11} + 4q^{12} - 23q^{13} - 2q^{14} - 8q^{15} + 2q^{16} + 3q^{17} - 2q^{18} - 9q^{19} - 9q^{20} + 7q^{21} - 19q^{22} + 12q^{23} - 9q^{24} + 3q^{25} + 18q^{26} + 7q^{27} + 4q^{28} - 9q^{29} - 33q^{31} + 10q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 4q^{36} - 33q^{37} - 3q^{38} - 23q^{39} - 9q^{40} - 3q^{41} - 2q^{42} - 9q^{43} + 2q^{44} - 8q^{45} - 32q^{46} + 11q^{47} + 2q^{48} + 7q^{49} + 29q^{50} + 3q^{51} - 21q^{52} + q^{53} - 2q^{54} - 16q^{55} - 9q^{56} - 9q^{57} - 5q^{58} - 30q^{59} - 9q^{60} - 19q^{61} + 3q^{62} + 7q^{63} - 21q^{64} + 14q^{65} - 19q^{66} - 30q^{67} + 24q^{68} + 12q^{69} + 8q^{71} - 9q^{72} - 20q^{73} - 9q^{74} + 3q^{75} - 42q^{76} - 3q^{77} + 18q^{78} + 8q^{79} + 12q^{80} + 7q^{81} + 10q^{82} - 34q^{83} + 4q^{84} - 28q^{85} + 24q^{86} - 9q^{87} - q^{88} - 12q^{89} - 23q^{91} + 60q^{92} - 33q^{93} - 3q^{94} + 12q^{95} + 10q^{96} + 7q^{97} - 2q^{98} - 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 9 x^{5} - 3 x^{4} + 20 x^{3} + 7 x^{2} - 13 x - 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{5} - 7 \nu^{3} - 4 \nu^{2} + 7 \nu + 5$$ $$\beta_{3}$$ $$=$$ $$\nu^{5} - 8 \nu^{3} - 3 \nu^{2} + 11 \nu + 4$$ $$\beta_{4}$$ $$=$$ $$-\nu^{6} + 8 \nu^{4} + 3 \nu^{3} - 12 \nu^{2} - 4 \nu + 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{6} + 2 \nu^{5} - 8 \nu^{4} - 18 \nu^{3} + 6 \nu^{2} + 22 \nu + 4$$ $$\beta_{6}$$ $$=$$ $$2 \nu^{6} + \nu^{5} - 15 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} + 16 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} - 2 \beta_{3} + 4 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + 6 \beta_{5} + 8 \beta_{4} - 7 \beta_{3} - 6 \beta_{2} + 3 \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$11 \beta_{5} + 11 \beta_{4} - 18 \beta_{3} - 3 \beta_{2} + 21 \beta_{1} + 21$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{6} + 39 \beta_{5} + 54 \beta_{4} - 50 \beta_{3} - 36 \beta_{2} + 32 \beta_{1} + 92$$

## 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
 1.13462 −1.14753 1.20244 −1.52532 −2.06168 2.69855 −0.301070
−2.48001 1.00000 4.15043 −0.783950 −2.48001 1.00000 −5.33307 1.00000 1.94420
1.2 −2.15625 1.00000 2.64943 −0.239094 −2.15625 1.00000 −1.40032 1.00000 0.515547
1.3 −1.24280 1.00000 −0.455452 −3.33400 −1.24280 1.00000 3.05163 1.00000 4.14349
1.4 0.246202 1.00000 −1.93938 −0.318209 0.246202 1.00000 −0.969884 1.00000 −0.0783436
1.5 0.692358 1.00000 −1.52064 −2.78145 0.692358 1.00000 −2.43754 1.00000 −1.92576
1.6 0.840819 1.00000 −1.29302 2.74724 0.840819 1.00000 −2.76884 1.00000 2.30993
1.7 2.09968 1.00000 2.40865 −3.29054 2.09968 1.00000 0.858029 1.00000 −6.90907
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$127$$ $$1$$

## 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 2667.2.a.j 7
3.b odd 2 1 8001.2.a.l 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.j 7 1.a even 1 1 trivial
8001.2.a.l 7 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2667))$$:

 $$T_{2}^{7} + 2 T_{2}^{6} - 7 T_{2}^{5} - 11 T_{2}^{4} + 14 T_{2}^{3} + 9 T_{2}^{2} - 11 T_{2} + 2$$ $$T_{5}^{7} + 8 T_{5}^{6} + 13 T_{5}^{5} - 42 T_{5}^{4} - 149 T_{5}^{3} - 138 T_{5}^{2} - 46 T_{5} - 5$$