# Properties

 Label 1024.4.a.n Level $1024$ Weight $4$ Character orbit 1024.a Self dual yes Analytic conductor $60.418$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1024.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$60.4179558459$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 36 x^{8} + 405 x^{6} - 1380 x^{4} + 420 x^{2} - 32$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{22}$$ Twist minimal: no (minimal twist has level 16) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{3} -\beta_{4} q^{5} + ( 3 + \beta_{2} ) q^{7} + ( 6 + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{7} q^{3} -\beta_{4} q^{5} + ( 3 + \beta_{2} ) q^{7} + ( 6 + \beta_{2} - \beta_{3} ) q^{9} + ( -\beta_{4} + \beta_{9} ) q^{11} + ( \beta_{4} + 3 \beta_{7} + \beta_{8} + \beta_{9} ) q^{13} + ( 13 - 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{15} + ( 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{17} + ( \beta_{1} - 5 \beta_{4} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{19} + ( \beta_{1} + 11 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{21} + ( 29 + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - 3 \beta_{6} ) q^{23} + ( 6 + 3 \beta_{2} + 2 \beta_{3} + 6 \beta_{5} - \beta_{6} ) q^{25} + ( -5 \beta_{1} - 9 \beta_{4} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{27} + ( -3 \beta_{1} + \beta_{4} + 18 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{29} + ( 36 - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} ) q^{31} + ( -1 + 3 \beta_{2} - \beta_{3} - 8 \beta_{5} - 2 \beta_{6} ) q^{33} + ( 11 \beta_{1} - 10 \beta_{4} - 2 \beta_{7} - 2 \beta_{8} ) q^{35} + ( 2 \beta_{1} + \beta_{4} + 29 \beta_{7} - \beta_{8} - \beta_{9} ) q^{37} + ( 73 + \beta_{2} - 2 \beta_{5} - 4 \beta_{6} ) q^{39} + ( 1 - \beta_{2} + 4 \beta_{3} + 14 \beta_{5} + \beta_{6} ) q^{41} + ( -20 \beta_{1} - 12 \beta_{4} - \beta_{7} - 4 \beta_{9} ) q^{43} + ( -\beta_{1} - \beta_{4} + 41 \beta_{7} - \beta_{8} - 5 \beta_{9} ) q^{45} + ( 90 - 7 \beta_{2} + 10 \beta_{3} + 5 \beta_{5} + 3 \beta_{6} ) q^{47} + ( -11 - 4 \beta_{2} - 6 \beta_{3} - 16 \beta_{5} - 6 \beta_{6} ) q^{49} + ( 34 \beta_{1} - \beta_{4} + 7 \beta_{7} - 12 \beta_{8} + 5 \beta_{9} ) q^{51} + ( -3 \beta_{1} - \beta_{4} + 35 \beta_{7} + 13 \beta_{8} - 7 \beta_{9} ) q^{53} + ( 131 - 3 \beta_{2} + 12 \beta_{3} - 8 \beta_{5} + 6 \beta_{6} ) q^{55} + ( 13 - 15 \beta_{2} - 9 \beta_{3} + 8 \beta_{5} + 8 \beta_{6} ) q^{57} + ( -41 \beta_{1} + 18 \beta_{4} + 5 \beta_{7} + 14 \beta_{8} + 4 \beta_{9} ) q^{59} + ( 19 \beta_{1} - \beta_{4} + 41 \beta_{7} - 9 \beta_{8} + 11 \beta_{9} ) q^{61} + ( 263 + 2 \beta_{2} - 10 \beta_{3} - 21 \beta_{5} - 3 \beta_{6} ) q^{63} + ( -57 - 13 \beta_{2} + 4 \beta_{3} - 18 \beta_{5} + 13 \beta_{6} ) q^{65} + ( 50 \beta_{1} + 11 \beta_{4} - 10 \beta_{7} + 12 \beta_{8} + \beta_{9} ) q^{67} + ( -13 \beta_{1} + 12 \beta_{4} + 55 \beta_{7} + \beta_{8} + 5 \beta_{9} ) q^{69} + ( 347 - 10 \beta_{2} + 2 \beta_{3} + 15 \beta_{5} + 3 \beta_{6} ) q^{71} + ( -29 - 11 \beta_{2} + \beta_{3} + 16 \beta_{5} - 6 \beta_{6} ) q^{73} + ( -71 \beta_{1} + 28 \beta_{4} + 7 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{75} + ( 3 \beta_{1} - 12 \beta_{4} + 23 \beta_{7} - 15 \beta_{8} + 5 \beta_{9} ) q^{77} + ( 446 + 12 \beta_{2} - 8 \beta_{3} - 6 \beta_{5} - 16 \beta_{6} ) q^{79} + ( -56 - 21 \beta_{2} + 9 \beta_{3} + 12 \beta_{6} ) q^{81} + ( 71 \beta_{1} + 48 \beta_{4} + 9 \beta_{7} + 22 \beta_{8} - 2 \beta_{9} ) q^{83} + ( -8 \beta_{1} + 3 \beta_{7} - 7 \beta_{8} + 17 \beta_{9} ) q^{85} + ( 617 + 2 \beta_{2} - 18 \beta_{3} + 29 \beta_{5} + 5 \beta_{6} ) q^{87} + ( 15 + 25 \beta_{2} + \beta_{3} + 8 \beta_{5} - 26 \beta_{6} ) q^{89} + ( -89 \beta_{1} + 4 \beta_{4} - 34 \beta_{8} + 2 \beta_{9} ) q^{91} + ( -48 \beta_{1} - 12 \beta_{4} + 6 \beta_{7} + 10 \beta_{8} - 14 \beta_{9} ) q^{93} + ( 701 + 15 \beta_{2} + 8 \beta_{3} + 56 \beta_{5} - 12 \beta_{6} ) q^{95} + ( 4 + 14 \beta_{2} + 3 \beta_{3} + 14 \beta_{5} - 29 \beta_{6} ) q^{97} + ( 125 \beta_{1} + 38 \beta_{4} + 17 \beta_{7} - 22 \beta_{8} - 4 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 28q^{7} + 54q^{9} + O(q^{10})$$ $$10q + 28q^{7} + 54q^{9} + 124q^{15} + 4q^{17} + 276q^{23} + 50q^{25} + 368q^{31} - 4q^{33} + 732q^{39} + 944q^{47} - 94q^{49} + 1380q^{55} + 108q^{57} + 2628q^{63} - 492q^{65} + 3468q^{71} - 296q^{73} + 4416q^{79} - 482q^{81} + 6036q^{87} + 88q^{89} + 6900q^{95} - 4q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 36 x^{8} + 405 x^{6} - 1380 x^{4} + 420 x^{2} - 32$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$7 \nu^{9} - 250 \nu^{7} + 2823 \nu^{5} - 10042 \nu^{3} + 4528 \nu$$$$)/208$$ $$\beta_{2}$$ $$=$$ $$($$$$59 \nu^{8} - 2018 \nu^{6} + 21595 \nu^{4} - 70882 \nu^{2} + 17840$$$$)/416$$ $$\beta_{3}$$ $$=$$ $$($$$$-137 \nu^{8} + 4982 \nu^{6} - 55785 \nu^{4} + 180342 \nu^{2} - 24496$$$$)/416$$ $$\beta_{4}$$ $$=$$ $$($$$$217 \nu^{9} - 7750 \nu^{7} + 85849 \nu^{5} - 278022 \nu^{3} + 18896 \nu$$$$)/832$$ $$\beta_{5}$$ $$=$$ $$($$$$-83 \nu^{8} + 2994 \nu^{6} - 33651 \nu^{4} + 112562 \nu^{2} - 18656$$$$)/208$$ $$\beta_{6}$$ $$=$$ $$($$$$231 \nu^{8} - 8042 \nu^{6} + 86919 \nu^{4} - 277098 \nu^{2} + 35856$$$$)/416$$ $$\beta_{7}$$ $$=$$ $$($$$$157 \nu^{9} - 5622 \nu^{7} + 62573 \nu^{5} - 206166 \nu^{3} + 34432 \nu$$$$)/416$$ $$\beta_{8}$$ $$=$$ $$($$$$-55 \nu^{9} + 1978 \nu^{7} - 22135 \nu^{5} + 73594 \nu^{3} - 13680 \nu$$$$)/64$$ $$\beta_{9}$$ $$=$$ $$($$$$1015 \nu^{9} - 36458 \nu^{7} + 407255 \nu^{5} - 1348970 \nu^{3} + 224336 \nu$$$$)/832$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{1}$$$$)/32$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} - \beta_{5} + 2 \beta_{3} - 6 \beta_{2} + 113$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$-26 \beta_{9} - 30 \beta_{8} + 2 \beta_{7} + 20 \beta_{4} - \beta_{1}$$$$)/32$$ $$\nu^{4}$$ $$=$$ $$($$$$29 \beta_{6} - 25 \beta_{5} + 40 \beta_{3} - 91 \beta_{2} + 1516$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$-374 \beta_{9} - 454 \beta_{8} - 106 \beta_{7} + 384 \beta_{4} + 177 \beta_{1}$$$$)/32$$ $$\nu^{6}$$ $$=$$ $$($$$$446 \beta_{6} - 423 \beta_{5} + 678 \beta_{3} - 1362 \beta_{2} + 21951$$$$)/16$$ $$\nu^{7}$$ $$=$$ $$($$$$-5622 \beta_{9} - 6754 \beta_{8} - 2066 \beta_{7} + 6460 \beta_{4} + 4433 \beta_{1}$$$$)/32$$ $$\nu^{8}$$ $$=$$ $$($$$$7043 \beta_{6} - 6519 \beta_{5} + 10952 \beta_{3} - 20373 \beta_{2} + 326836$$$$)/16$$ $$\nu^{9}$$ $$=$$ $$($$$$-85962 \beta_{9} - 99866 \beta_{8} - 29462 \beta_{7} + 104544 \beta_{4} + 87103 \beta_{1}$$$$)/32$$

## 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
 −2.34476 −0.357936 0.446984 3.82089 −3.94652 3.94652 −3.82089 −0.446984 0.357936 2.34476
0 −8.43597 0 −12.2748 0 1.63924 0 44.1656 0
1.2 0 −7.77277 0 −6.59550 0 24.8965 0 33.4160 0
1.3 0 −4.62644 0 17.8826 0 13.8754 0 −5.59607 0
1.4 0 −2.80518 0 −0.844070 0 −29.0828 0 −19.1310 0
1.5 0 −1.07024 0 11.6331 0 2.67171 0 −25.8546 0
1.6 0 1.07024 0 −11.6331 0 2.67171 0 −25.8546 0
1.7 0 2.80518 0 0.844070 0 −29.0828 0 −19.1310 0
1.8 0 4.62644 0 −17.8826 0 13.8754 0 −5.59607 0
1.9 0 7.77277 0 6.59550 0 24.8965 0 33.4160 0
1.10 0 8.43597 0 12.2748 0 1.63924 0 44.1656 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.4.a.n 10
4.b odd 2 1 1024.4.a.m 10
8.b even 2 1 inner 1024.4.a.n 10
8.d odd 2 1 1024.4.a.m 10
16.e even 4 2 1024.4.b.j 10
16.f odd 4 2 1024.4.b.k 10
32.g even 8 2 16.4.e.a 10
32.g even 8 2 128.4.e.b 10
32.h odd 8 2 64.4.e.a 10
32.h odd 8 2 128.4.e.a 10
96.o even 8 2 576.4.k.a 10
96.p odd 8 2 144.4.k.a 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.4.e.a 10 32.g even 8 2
64.4.e.a 10 32.h odd 8 2
128.4.e.a 10 32.h odd 8 2
128.4.e.b 10 32.g even 8 2
144.4.k.a 10 96.p odd 8 2
576.4.k.a 10 96.o even 8 2
1024.4.a.m 10 4.b odd 2 1
1024.4.a.m 10 8.d odd 2 1
1024.4.a.n 10 1.a even 1 1 trivial
1024.4.a.n 10 8.b even 2 1 inner
1024.4.b.j 10 16.e even 4 2
1024.4.b.k 10 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1024))$$:

 $$T_{3}^{10} - 162 T_{3}^{8} + 8504 T_{3}^{6} - 157552 T_{3}^{4} + 893712 T_{3}^{2} - 829472$$ $$T_{5}^{10} - 650 T_{5}^{8} + 138664 T_{5}^{6} - 11484496 T_{5}^{4} + 291758672 T_{5}^{2} - 202085408$$ $$T_{7}^{5} - 14 T_{7}^{4} - 736 T_{7}^{3} + 13376 T_{7}^{2} - 46736 T_{7} + 44000$$