# Properties

 Label 4026.2.a.y Level 4026 Weight 2 Character orbit 4026.a Self dual Yes Analytic conductor 32.148 Analytic rank 1 Dimension 7 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4026 = 2 \cdot 3 \cdot 11 \cdot 61$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4026.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$32.1477718538$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 2 x^{6} - 21 x^{5} + 39 x^{4} + 89 x^{3} - 100 x^{2} - 96 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{6} - 147 \nu^{5} - \nu^{4} + 2952 \nu^{3} - 1452 \nu^{2} - 10701 \nu - 534$$$$)/1706$$ $$\beta_{3}$$ $$=$$ $$($$$$-9 \nu^{6} + 94 \nu^{5} + 343 \nu^{4} - 1731 \nu^{3} - 2675 \nu^{2} + 5102 \nu + 2326$$$$)/1706$$ $$\beta_{4}$$ $$=$$ $$($$$$53 \nu^{6} - 364 \nu^{5} - 693 \nu^{4} + 6213 \nu^{3} - 3961 \nu^{2} - 9194 \nu + 10376$$$$)/3412$$ $$\beta_{5}$$ $$=$$ $$($$$$155 \nu^{6} - 292 \nu^{5} - 3443 \nu^{4} + 5359 \nu^{3} + 17257 \nu^{2} - 10150 \nu - 21672$$$$)/3412$$ $$\beta_{6}$$ $$=$$ $$($$$$-142 \nu^{6} + 251 \nu^{5} + 2758 \nu^{4} - 4849 \nu^{3} - 8749 \nu^{2} + 10647 \nu + 3906$$$$)/1706$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2 \beta_{5} + 2 \beta_{3} + \beta_{2} + 8$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 14 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$15 \beta_{6} + 30 \beta_{5} + 2 \beta_{4} + 37 \beta_{3} + 17 \beta_{2} - 3 \beta_{1} + 105$$ $$\nu^{5}$$ $$=$$ $$-43 \beta_{6} - 65 \beta_{5} - 59 \beta_{4} - 40 \beta_{3} + 27 \beta_{2} + 205 \beta_{1} - 72$$ $$\nu^{6}$$ $$=$$ $$210 \beta_{6} + 447 \beta_{5} + 37 \beta_{4} + 593 \beta_{3} + 248 \beta_{2} - 99 \beta_{1} + 1515$$

## 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
 3.80390 3.01987 1.63928 0.0775196 −0.777182 −1.76251 −4.00088
−1.00000 −1.00000 1.00000 −3.80390 1.00000 −1.13011 −1.00000 1.00000 3.80390
1.2 −1.00000 −1.00000 1.00000 −3.01987 1.00000 4.36357 −1.00000 1.00000 3.01987
1.3 −1.00000 −1.00000 1.00000 −1.63928 1.00000 −1.98112 −1.00000 1.00000 1.63928
1.4 −1.00000 −1.00000 1.00000 −0.0775196 1.00000 2.74128 −1.00000 1.00000 0.0775196
1.5 −1.00000 −1.00000 1.00000 0.777182 1.00000 −3.79435 −1.00000 1.00000 −0.777182
1.6 −1.00000 −1.00000 1.00000 1.76251 1.00000 1.52396 −1.00000 1.00000 −1.76251
1.7 −1.00000 −1.00000 1.00000 4.00088 1.00000 −0.723236 −1.00000 1.00000 −4.00088
 $$n$$: 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.

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$11$$ $$1$$
$$61$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4026))$$:

 $$T_{5}^{7} + 2 T_{5}^{6} - 21 T_{5}^{5} - 39 T_{5}^{4} + 89 T_{5}^{3} + 100 T_{5}^{2} - 96 T_{5} - 8$$ $$T_{7}^{7} - T_{7}^{6} - 24 T_{7}^{5} + 10 T_{7}^{4} + 140 T_{7}^{3} + 25 T_{7}^{2} - 200 T_{7} - 112$$ $$T_{13}^{7} - 46 T_{13}^{5} + 28 T_{13}^{4} + 539 T_{13}^{3} - 531 T_{13}^{2} - 740 T_{13} + 28$$