# Properties

 Label 8034.2.a.n Level $8034$ Weight $2$ Character orbit 8034.a Self dual yes Analytic conductor $64.152$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.72329.1 Defining polynomial: $$x^{4} - x^{3} - 9 x^{2} + 2 x + 15$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 9 x^{2} + 2 x + 15$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 5 \nu + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 5$$

## 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
 1.46917 −1.52193 −2.11664 3.16940
1.00000 1.00000 1.00000 −3.31071 1.00000 3.84154 1.00000 1.00000 −3.31071
1.2 1.00000 1.00000 1.00000 −0.161792 1.00000 3.68372 1.00000 1.00000 −0.161792
1.3 1.00000 1.00000 1.00000 2.59680 1.00000 1.51984 1.00000 1.00000 2.59680
1.4 1.00000 1.00000 1.00000 2.87570 1.00000 −4.04510 1.00000 1.00000 2.87570
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 8034.2.a.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.n 4 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{5}^{4} - 2 T_{5}^{3} - 11 T_{5}^{2} + 23 T_{5} + 4$$ $$T_{7}^{4} - 5 T_{7}^{3} - 11 T_{7}^{2} + 82 T_{7} - 87$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$4 + 23 T - 11 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$-87 + 82 T - 11 T^{2} - 5 T^{3} + T^{4}$$
$11$ $$-72 + 64 T + 10 T^{2} - 9 T^{3} + T^{4}$$
$13$ $$( -1 + T )^{4}$$
$17$ $$( -2 + T )^{4}$$
$19$ $$128 - 120 T - 24 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$191 + 78 T - 24 T^{2} - 6 T^{3} + T^{4}$$
$29$ $$120 - 13 T - 27 T^{2} + T^{4}$$
$31$ $$128 + 120 T - 24 T^{2} - 6 T^{3} + T^{4}$$
$37$ $$-72 + 116 T + 88 T^{2} + 17 T^{3} + T^{4}$$
$41$ $$9 + 56 T + 46 T^{2} - 16 T^{3} + T^{4}$$
$43$ $$568 - 544 T + 178 T^{2} - 23 T^{3} + T^{4}$$
$47$ $$1585 - 44 T - 85 T^{2} + T^{3} + T^{4}$$
$53$ $$-4848 + 1696 T - 36 T^{2} - 18 T^{3} + T^{4}$$
$59$ $$120 - 223 T + 113 T^{2} - 20 T^{3} + T^{4}$$
$61$ $$-232 - 208 T - 22 T^{2} + 9 T^{3} + T^{4}$$
$67$ $$268 + 161 T - 29 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$-20880 - 3961 T - 77 T^{2} + 21 T^{3} + T^{4}$$
$73$ $$-642 + 113 T + 115 T^{2} - 22 T^{3} + T^{4}$$
$79$ $$-1152 - 264 T + 88 T^{2} + 22 T^{3} + T^{4}$$
$83$ $$120 + 241 T + 109 T^{2} + 18 T^{3} + T^{4}$$
$89$ $$-1312 - 696 T - 32 T^{2} + 14 T^{3} + T^{4}$$
$97$ $$-41472 + 7544 T - 244 T^{2} - 20 T^{3} + T^{4}$$