# Properties

 Label 4026.2.a.bb Level 4026 Weight 2 Character orbit 4026.a Self dual yes Analytic conductor 32.148 Analytic rank 0 Dimension 8 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: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} - 22 x^{6} + 42 x^{5} + 182 x^{4} - 111 x^{3} - 538 x^{2} - 256 x - 32$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$17 \nu^{7} - 71 \nu^{6} - 282 \nu^{5} + 978 \nu^{4} + 1918 \nu^{3} - 3271 \nu^{2} - 5518 \nu - 808$$$$)/144$$ $$\beta_{3}$$ $$=$$ $$($$$$17 \nu^{7} - 71 \nu^{6} - 282 \nu^{5} + 978 \nu^{4} + 1918 \nu^{3} - 3127 \nu^{2} - 5662 \nu - 1672$$$$)/144$$ $$\beta_{4}$$ $$=$$ $$($$$$-17 \nu^{7} + 47 \nu^{6} + 402 \nu^{5} - 738 \nu^{4} - 3310 \nu^{3} + 2455 \nu^{2} + 9334 \nu + 2440$$$$)/144$$ $$\beta_{5}$$ $$=$$ $$($$$$-35 \nu^{7} + 101 \nu^{6} + 798 \nu^{5} - 1494 \nu^{4} - 6730 \nu^{3} + 4885 \nu^{2} + 20170 \nu + 4888$$$$)/288$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{6} + 22 \nu^{5} - 42 \nu^{4} - 182 \nu^{3} + 119 \nu^{2} + 530 \nu + 176$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-21 \nu^{7} + 67 \nu^{6} + 466 \nu^{5} - 1018 \nu^{4} - 3878 \nu^{3} + 3475 \nu^{2} + 11718 \nu + 2824$$$$)/96$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - \beta_{2} + \beta_{1} + 6$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 11 \beta_{1} + 7$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 6 \beta_{6} - 7 \beta_{5} + 16 \beta_{3} - 15 \beta_{2} + 25 \beta_{1} + 59$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{7} + 29 \beta_{6} - 52 \beta_{5} + 13 \beta_{4} + 53 \beta_{3} - 48 \beta_{2} + 146 \beta_{1} + 135$$ $$\nu^{6}$$ $$=$$ $$50 \beta_{7} + 147 \beta_{6} - 214 \beta_{5} + \beta_{4} + 275 \beta_{3} - 246 \beta_{2} + 467 \beta_{1} + 723$$ $$\nu^{7}$$ $$=$$ $$284 \beta_{7} + 637 \beta_{6} - 1128 \beta_{5} + 107 \beta_{4} + 1074 \beta_{3} - 919 \beta_{2} + 2210 \beta_{1} + 2277$$

## 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
 −1.83085 −0.326029 4.26126 −2.29226 −2.68569 2.67396 −0.223274 3.42288
1.00000 1.00000 1.00000 −3.68259 1.00000 3.83085 1.00000 1.00000 −3.68259
1.2 1.00000 1.00000 1.00000 −2.31671 1.00000 2.32603 1.00000 1.00000 −2.31671
1.3 1.00000 1.00000 1.00000 −1.92862 1.00000 −2.26126 1.00000 1.00000 −1.92862
1.4 1.00000 1.00000 1.00000 0.723959 1.00000 4.29226 1.00000 1.00000 0.723959
1.5 1.00000 1.00000 1.00000 1.94066 1.00000 4.68569 1.00000 1.00000 1.94066
1.6 1.00000 1.00000 1.00000 2.72927 1.00000 −0.673957 1.00000 1.00000 2.72927
1.7 1.00000 1.00000 1.00000 3.42657 1.00000 2.22327 1.00000 1.00000 3.42657
1.8 1.00000 1.00000 1.00000 4.10747 1.00000 −1.42288 1.00000 1.00000 4.10747
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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 4026.2.a.bb 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.bb 8 1.a even 1 1 trivial

## Atkin-Lehner signs

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

## Hecke kernels

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

 $$T_{5}^{8} - \cdots$$ $$T_{7}^{8} - \cdots$$ $$T_{13}^{8} - \cdots$$