# Properties

 Label 1575.2.a.t Level $1575$ Weight $2$ Character orbit 1575.a Self dual yes Analytic conductor $12.576$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## 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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 525) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( 1 + \beta ) q^{4} - q^{7} + 3 q^{8} +O(q^{10})$$ $$q + \beta q^{2} + ( 1 + \beta ) q^{4} - q^{7} + 3 q^{8} + 3 q^{11} + ( -2 + 2 \beta ) q^{13} -\beta q^{14} + ( -2 + \beta ) q^{16} + 2 \beta q^{17} + ( 2 + 2 \beta ) q^{19} + 3 \beta q^{22} + ( 3 - 4 \beta ) q^{23} + 6 q^{26} + ( -1 - \beta ) q^{28} + ( 3 + 2 \beta ) q^{29} + ( 2 - 4 \beta ) q^{31} + ( -3 - \beta ) q^{32} + ( 6 + 2 \beta ) q^{34} + ( -5 + 4 \beta ) q^{37} + ( 6 + 4 \beta ) q^{38} + ( -5 - 2 \beta ) q^{43} + ( 3 + 3 \beta ) q^{44} + ( -12 - \beta ) q^{46} + ( 6 + 2 \beta ) q^{47} + q^{49} + ( 4 + 2 \beta ) q^{52} + ( -6 + 4 \beta ) q^{53} -3 q^{56} + ( 6 + 5 \beta ) q^{58} + ( 6 + 2 \beta ) q^{59} + ( 8 - 4 \beta ) q^{61} + ( -12 - 2 \beta ) q^{62} + ( 1 - 6 \beta ) q^{64} + ( -11 - 2 \beta ) q^{67} + ( 6 + 4 \beta ) q^{68} + 3 q^{71} + ( 4 - 2 \beta ) q^{73} + ( 12 - \beta ) q^{74} + ( 8 + 6 \beta ) q^{76} -3 q^{77} + ( -1 - 6 \beta ) q^{79} + ( 6 - 4 \beta ) q^{83} + ( -6 - 7 \beta ) q^{86} + 9 q^{88} + ( 6 - 6 \beta ) q^{89} + ( 2 - 2 \beta ) q^{91} + ( -9 - 5 \beta ) q^{92} + ( 6 + 8 \beta ) q^{94} + ( 10 - 4 \beta ) q^{97} + \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 3q^{4} - 2q^{7} + 6q^{8} + O(q^{10})$$ $$2q + q^{2} + 3q^{4} - 2q^{7} + 6q^{8} + 6q^{11} - 2q^{13} - q^{14} - 3q^{16} + 2q^{17} + 6q^{19} + 3q^{22} + 2q^{23} + 12q^{26} - 3q^{28} + 8q^{29} - 7q^{32} + 14q^{34} - 6q^{37} + 16q^{38} - 12q^{43} + 9q^{44} - 25q^{46} + 14q^{47} + 2q^{49} + 10q^{52} - 8q^{53} - 6q^{56} + 17q^{58} + 14q^{59} + 12q^{61} - 26q^{62} - 4q^{64} - 24q^{67} + 16q^{68} + 6q^{71} + 6q^{73} + 23q^{74} + 22q^{76} - 6q^{77} - 8q^{79} + 8q^{83} - 19q^{86} + 18q^{88} + 6q^{89} + 2q^{91} - 23q^{92} + 20q^{94} + 16q^{97} + q^{98} + O(q^{100})$$

## 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
 −1.30278 2.30278
−1.30278 0 −0.302776 0 0 −1.00000 3.00000 0 0
1.2 2.30278 0 3.30278 0 0 −1.00000 3.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 1575.2.a.t 2
3.b odd 2 1 525.2.a.f 2
5.b even 2 1 1575.2.a.o 2
5.c odd 4 2 1575.2.d.g 4
12.b even 2 1 8400.2.a.df 2
15.d odd 2 1 525.2.a.h yes 2
15.e even 4 2 525.2.d.d 4
21.c even 2 1 3675.2.a.w 2
60.h even 2 1 8400.2.a.cw 2
105.g even 2 1 3675.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.a.f 2 3.b odd 2 1
525.2.a.h yes 2 15.d odd 2 1
525.2.d.d 4 15.e even 4 2
1575.2.a.o 2 5.b even 2 1
1575.2.a.t 2 1.a even 1 1 trivial
1575.2.d.g 4 5.c odd 4 2
3675.2.a.w 2 21.c even 2 1
3675.2.a.bb 2 105.g even 2 1
8400.2.a.cw 2 60.h even 2 1
8400.2.a.df 2 12.b even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} - 3$$ $$T_{11} - 3$$ $$T_{13}^{2} + 2 T_{13} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$-12 + 2 T + T^{2}$$
$17$ $$-12 - 2 T + T^{2}$$
$19$ $$-4 - 6 T + T^{2}$$
$23$ $$-51 - 2 T + T^{2}$$
$29$ $$3 - 8 T + T^{2}$$
$31$ $$-52 + T^{2}$$
$37$ $$-43 + 6 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$23 + 12 T + T^{2}$$
$47$ $$36 - 14 T + T^{2}$$
$53$ $$-36 + 8 T + T^{2}$$
$59$ $$36 - 14 T + T^{2}$$
$61$ $$-16 - 12 T + T^{2}$$
$67$ $$131 + 24 T + T^{2}$$
$71$ $$( -3 + T )^{2}$$
$73$ $$-4 - 6 T + T^{2}$$
$79$ $$-101 + 8 T + T^{2}$$
$83$ $$-36 - 8 T + T^{2}$$
$89$ $$-108 - 6 T + T^{2}$$
$97$ $$12 - 16 T + T^{2}$$