# Properties

 Label 8034.2.a.o Level 8034 Weight 2 Character orbit 8034.a Self dual yes Analytic conductor 64.152 Analytic rank 1 Dimension 7 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8034 = 2 \cdot 3 \cdot 13 \cdot 103$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 8034.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1518129839$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3 x^{6} - 4 x^{5} + 14 x^{4} + 3 x^{3} - 12 x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} - \nu^{3} - 5 \nu^{2} + 3 \nu + 3$$ $$\beta_{3}$$ $$=$$ $$-\nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 12 \nu^{3} + \nu^{2} + 6 \nu$$ $$\beta_{4}$$ $$=$$ $$-\nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 12 \nu^{3} + 2 \nu^{2} + 5 \nu - 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{6} - 3 \nu^{5} - 3 \nu^{4} + 13 \nu^{3} - 2 \nu^{2} - 9 \nu$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} - 4 \nu^{5} - \nu^{4} + 17 \nu^{3} - 9 \nu^{2} - 10 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + 4 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + 6 \beta_{4} - 5 \beta_{3} + \beta_{2} + 6 \beta_{1} + 9$$ $$\nu^{5}$$ $$=$$ $$-\beta_{6} + 7 \beta_{5} + 9 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 20 \beta_{1} + 14$$ $$\nu^{6}$$ $$=$$ $$-3 \beta_{6} + 12 \beta_{5} + 34 \beta_{4} - 26 \beta_{3} + 9 \beta_{2} + 37 \beta_{1} + 47$$

## 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
 −0.519850 2.16681 1.27539 −1.86678 2.51101 −0.761570 0.194986
−1.00000 −1.00000 1.00000 −2.94146 1.00000 −3.14882 −1.00000 1.00000 2.94146
1.2 −1.00000 −1.00000 1.00000 −2.61702 1.00000 −0.977840 −1.00000 1.00000 2.61702
1.3 −1.00000 −1.00000 1.00000 −2.55242 1.00000 1.37822 −1.00000 1.00000 2.55242
1.4 −1.00000 −1.00000 1.00000 1.72171 1.00000 2.39003 −1.00000 1.00000 −1.72171
1.5 −1.00000 −1.00000 1.00000 1.77686 1.00000 −3.99413 −1.00000 1.00000 −1.77686
1.6 −1.00000 −1.00000 1.00000 3.28276 1.00000 −3.26302 −1.00000 1.00000 −3.28276
1.7 −1.00000 −1.00000 1.00000 3.32958 1.00000 −1.38443 −1.00000 1.00000 −3.32958
 $$n$$: 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

## 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 8034.2.a.o 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.o 7 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$13$$ $$-1$$
$$103$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8034))$$:

 $$T_{5}^{7} - 2 T_{5}^{6} - 23 T_{5}^{5} + 41 T_{5}^{4} + 173 T_{5}^{3} - 279 T_{5}^{2} - 417 T_{5} + 657$$ $$T_{7}^{7} + 9 T_{7}^{6} + 17 T_{7}^{5} - 51 T_{7}^{4} - 178 T_{7}^{3} - 32 T_{7}^{2} + 270 T_{7} + 183$$