# Properties

 Label 4026.2.a.bc Level 4026 Weight 2 Character orbit 4026.a Self dual yes Analytic conductor 32.148 Analytic rank 0 Dimension 9 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: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - x^{8} - 30 x^{7} + 7 x^{6} + 284 x^{5} + 100 x^{4} - 777 x^{3} - 250 x^{2} + 574 x - 68$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-99 \nu^{8} + 4436 \nu^{7} - 16215 \nu^{6} - 80028 \nu^{5} + 306876 \nu^{4} + 435761 \nu^{3} - 1379713 \nu^{2} - 800938 \nu + 1137130$$$$)/200458$$ $$\beta_{3}$$ $$=$$ $$($$$$-2961 \nu^{8} + 23336 \nu^{7} + 52616 \nu^{6} - 580331 \nu^{5} - 270479 \nu^{4} + 4358851 \nu^{3} + 711766 \nu^{2} - 8100922 \nu + 1153636$$$$)/400916$$ $$\beta_{4}$$ $$=$$ $$($$$$3436 \nu^{8} - 13235 \nu^{7} - 78083 \nu^{6} + 253589 \nu^{5} + 562732 \nu^{4} - 1261001 \nu^{3} - 1100874 \nu^{2} + 1724498 \nu - 168584$$$$)/200458$$ $$\beta_{5}$$ $$=$$ $$($$$$7105 \nu^{8} - 16662 \nu^{7} - 187860 \nu^{6} + 285499 \nu^{5} + 1499665 \nu^{4} - 976049 \nu^{3} - 2600248 \nu^{2} + 1017098 \nu - 403440$$$$)/400916$$ $$\beta_{6}$$ $$=$$ $$($$$$8041 \nu^{8} - 3932 \nu^{7} - 253236 \nu^{6} - 14833 \nu^{5} + 2370547 \nu^{4} + 1045333 \nu^{3} - 5282530 \nu^{2} - 1141722 \nu + 1565484$$$$)/400916$$ $$\beta_{7}$$ $$=$$ $$($$$$4471 \nu^{8} - 4941 \nu^{7} - 127244 \nu^{6} + 33285 \nu^{5} + 1126738 \nu^{4} + 525078 \nu^{3} - 2603901 \nu^{2} - 1838422 \nu + 663210$$$$)/200458$$ $$\beta_{8}$$ $$=$$ $$($$$$-7907 \nu^{8} + 18176 \nu^{7} + 205327 \nu^{6} - 286874 \nu^{5} - 1689470 \nu^{4} + 735923 \nu^{3} + 3905233 \nu^{2} - 86534 \nu - 1897832$$$$)/200458$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} + \beta_{7} + \beta_{4} + \beta_{1} + 7$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{2} + 12 \beta_{1} + 7$$ $$\nu^{4}$$ $$=$$ $$14 \beta_{8} + 17 \beta_{7} - \beta_{6} - 4 \beta_{5} + 15 \beta_{4} - \beta_{3} + \beta_{2} + 24 \beta_{1} + 86$$ $$\nu^{5}$$ $$=$$ $$23 \beta_{8} + 48 \beta_{7} - 36 \beta_{6} + 33 \beta_{5} - 5 \beta_{4} - 9 \beta_{3} + 14 \beta_{2} + 181 \beta_{1} + 175$$ $$\nu^{6}$$ $$=$$ $$194 \beta_{8} + 290 \beta_{7} - 47 \beta_{6} - 57 \beta_{5} + 165 \beta_{4} - 42 \beta_{3} + 3 \beta_{2} + 498 \beta_{1} + 1246$$ $$\nu^{7}$$ $$=$$ $$430 \beta_{8} + 978 \beta_{7} - 583 \beta_{6} + 454 \beta_{5} - 184 \beta_{4} - 273 \beta_{3} + 136 \beta_{2} + 2953 \beta_{1} + 3428$$ $$\nu^{8}$$ $$=$$ $$2762 \beta_{8} + 5085 \beta_{7} - 1227 \beta_{6} - 593 \beta_{5} + 1332 \beta_{4} - 1178 \beta_{3} - 238 \beta_{2} + 9625 \beta_{1} + 19380$$

## 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
 4.23292 3.88536 1.60053 0.731055 0.128459 −1.48002 −2.09192 −2.63661 −3.36977
1.00000 1.00000 1.00000 −3.23292 1.00000 4.29240 1.00000 1.00000 −3.23292
1.2 1.00000 1.00000 1.00000 −2.88536 1.00000 −2.96592 1.00000 1.00000 −2.88536
1.3 1.00000 1.00000 1.00000 −0.600528 1.00000 0.748033 1.00000 1.00000 −0.600528
1.4 1.00000 1.00000 1.00000 0.268945 1.00000 3.42672 1.00000 1.00000 0.268945
1.5 1.00000 1.00000 1.00000 0.871541 1.00000 1.79155 1.00000 1.00000 0.871541
1.6 1.00000 1.00000 1.00000 2.48002 1.00000 3.05465 1.00000 1.00000 2.48002
1.7 1.00000 1.00000 1.00000 3.09192 1.00000 −4.21340 1.00000 1.00000 3.09192
1.8 1.00000 1.00000 1.00000 3.63661 1.00000 −1.65542 1.00000 1.00000 3.63661
1.9 1.00000 1.00000 1.00000 4.36977 1.00000 4.52139 1.00000 1.00000 4.36977
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 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.bc 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.bc 9 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4026))$$:

 $$T_{5}^{9} - \cdots$$ $$T_{7}^{9} - \cdots$$ $$T_{13}^{9} - \cdots$$