# Properties

 Label 1575.2.a.z Level 1575 Weight 2 Character orbit 1575.a Self dual yes Analytic conductor 12.576 Analytic rank 0 Dimension 4 CM no Inner twists 2

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.5764383184$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.174928.1 Defining polynomial: $$x^{4} - 9 x^{2} + 13$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 3 + \beta_{2} ) q^{4} + q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 3 + \beta_{2} ) q^{4} + q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + \beta_{3} q^{11} + ( -2 - 2 \beta_{2} ) q^{13} + \beta_{1} q^{14} + ( 6 + 3 \beta_{2} ) q^{16} + 2 \beta_{1} q^{17} + ( 2 - 2 \beta_{2} ) q^{19} + ( 2 + 3 \beta_{2} ) q^{22} -\beta_{3} q^{23} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{26} + ( 3 + \beta_{2} ) q^{28} + ( 2 \beta_{1} - \beta_{3} ) q^{29} + 6 q^{31} + ( 5 \beta_{1} + \beta_{3} ) q^{32} + ( 10 + 2 \beta_{2} ) q^{34} -3 q^{37} -2 \beta_{3} q^{38} -4 \beta_{1} q^{41} + ( 5 + 2 \beta_{2} ) q^{43} + ( 5 \beta_{1} + \beta_{3} ) q^{44} + ( -2 - 3 \beta_{2} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{47} + q^{49} + ( -20 - 6 \beta_{2} ) q^{52} -4 \beta_{1} q^{53} + ( 2 \beta_{1} + \beta_{3} ) q^{56} + ( 8 - \beta_{2} ) q^{58} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{59} + 6 \beta_{1} q^{62} + ( 15 + 2 \beta_{2} ) q^{64} + ( 5 + 2 \beta_{2} ) q^{67} + ( 8 \beta_{1} + 2 \beta_{3} ) q^{68} + ( -4 \beta_{1} - \beta_{3} ) q^{71} -2 \beta_{2} q^{73} -3 \beta_{1} q^{74} + ( -8 - 2 \beta_{2} ) q^{76} + \beta_{3} q^{77} + ( -1 - 2 \beta_{2} ) q^{79} + ( -20 - 4 \beta_{2} ) q^{82} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{83} + ( 7 \beta_{1} + 2 \beta_{3} ) q^{86} + ( 23 + 2 \beta_{2} ) q^{88} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{89} + ( -2 - 2 \beta_{2} ) q^{91} + ( -5 \beta_{1} - \beta_{3} ) q^{92} + ( -14 - 8 \beta_{2} ) q^{94} + ( 2 + 4 \beta_{2} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 10q^{4} + 4q^{7} + O(q^{10})$$ $$4q + 10q^{4} + 4q^{7} - 4q^{13} + 18q^{16} + 12q^{19} + 2q^{22} + 10q^{28} + 24q^{31} + 36q^{34} - 12q^{37} + 16q^{43} - 2q^{46} + 4q^{49} - 68q^{52} + 34q^{58} + 56q^{64} + 16q^{67} + 4q^{73} - 28q^{76} - 72q^{82} + 88q^{88} - 4q^{91} - 40q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9 x^{2} + 13$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6 \beta_{1}$$

## 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
 −2.68190 −1.34440 1.34440 2.68190
−2.68190 0 5.19258 0 0 1.00000 −8.56218 0 0
1.2 −1.34440 0 −0.192582 0 0 1.00000 2.94771 0 0
1.3 1.34440 0 −0.192582 0 0 1.00000 −2.94771 0 0
1.4 2.68190 0 5.19258 0 0 1.00000 8.56218 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.a.z yes 4
3.b odd 2 1 inner 1575.2.a.z yes 4
5.b even 2 1 1575.2.a.y 4
5.c odd 4 2 1575.2.d.l 8
15.d odd 2 1 1575.2.a.y 4
15.e even 4 2 1575.2.d.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.2.a.y 4 5.b even 2 1
1575.2.a.y 4 15.d odd 2 1
1575.2.a.z yes 4 1.a even 1 1 trivial
1575.2.a.z yes 4 3.b odd 2 1 inner
1575.2.d.l 8 5.c odd 4 2
1575.2.d.l 8 15.e even 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}^{4} - 9 T_{2}^{2} + 13$$ $$T_{11}^{4} - 42 T_{11}^{2} + 325$$ $$T_{13}^{2} + 2 T_{13} - 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4} - 4 T^{6} + 16 T^{8}$$
$3$ 1
$5$ 1
$7$ $$( 1 - T )^{4}$$
$11$ $$1 + 2 T^{2} + 127 T^{4} + 242 T^{6} + 14641 T^{8}$$
$13$ $$( 1 + 2 T - 2 T^{2} + 26 T^{3} + 169 T^{4} )^{2}$$
$17$ $$1 + 32 T^{2} + 718 T^{4} + 9248 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 6 T + 18 T^{2} - 114 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 + 50 T^{2} + 1567 T^{4} + 26450 T^{6} + 279841 T^{8}$$
$29$ $$1 + 42 T^{2} + 1079 T^{4} + 35322 T^{6} + 707281 T^{8}$$
$31$ $$( 1 - 6 T + 31 T^{2} )^{4}$$
$37$ $$( 1 + 3 T + 37 T^{2} )^{4}$$
$41$ $$1 + 20 T^{2} + 1606 T^{4} + 33620 T^{6} + 2825761 T^{8}$$
$43$ $$( 1 - 8 T + 73 T^{2} - 344 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 - 24 T^{2} + 3518 T^{4} - 53016 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 68 T^{2} + 4918 T^{4} + 191012 T^{6} + 7890481 T^{8}$$
$59$ $$1 + 24 T^{2} + 6062 T^{4} + 83544 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 61 T^{2} )^{4}$$
$67$ $$( 1 - 8 T + 121 T^{2} - 536 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 + 90 T^{2} + 2711 T^{4} + 453690 T^{6} + 25411681 T^{8}$$
$73$ $$( 1 - 2 T + 118 T^{2} - 146 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 129 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$1 + 36 T^{2} - 2602 T^{4} + 248004 T^{6} + 47458321 T^{8}$$
$89$ $$1 + 160 T^{2} + 12846 T^{4} + 1267360 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 + 78 T^{2} + 9409 T^{4} )^{2}$$
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