# Properties

 Label 1575.2.a.w Level $1575$ Weight $2$ Character orbit 1575.a Self dual yes Analytic conductor $12.576$ Analytic rank $1$ Dimension $3$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} - q^{7} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} - q^{7} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{8} -2 q^{11} + ( -2 + \beta_{1} - \beta_{2} ) q^{13} + \beta_{1} q^{14} + ( 3 + 4 \beta_{1} ) q^{16} + ( \beta_{1} - \beta_{2} ) q^{17} + ( 2 + \beta_{1} - \beta_{2} ) q^{19} + 2 \beta_{1} q^{22} + ( -2 + \beta_{1} + \beta_{2} ) q^{23} + ( -4 + 3 \beta_{1} - \beta_{2} ) q^{26} + ( -1 - \beta_{1} - \beta_{2} ) q^{28} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{31} + ( -8 - 3 \beta_{1} - 2 \beta_{2} ) q^{32} + ( -4 + \beta_{1} - \beta_{2} ) q^{34} -4 \beta_{1} q^{37} + ( -4 - \beta_{1} - \beta_{2} ) q^{38} + ( -3 \beta_{1} + \beta_{2} ) q^{41} + 4 \beta_{2} q^{43} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{44} + ( -2 - \beta_{1} - \beta_{2} ) q^{46} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{47} + q^{49} + ( -6 + \beta_{1} - \beta_{2} ) q^{52} + ( -6 + 3 \beta_{1} + \beta_{2} ) q^{53} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{56} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{58} + ( -4 - 4 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( 8 + 3 \beta_{1} + 3 \beta_{2} ) q^{62} + ( 1 + 7 \beta_{1} + 3 \beta_{2} ) q^{64} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -4 + 3 \beta_{1} + \beta_{2} ) q^{68} -2 q^{71} + ( -6 - \beta_{1} + \beta_{2} ) q^{73} + ( 12 + 4 \beta_{1} + 4 \beta_{2} ) q^{74} + ( -2 + 5 \beta_{1} + 3 \beta_{2} ) q^{76} + 2 q^{77} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{79} + ( 10 + \beta_{1} + 3 \beta_{2} ) q^{82} + ( -4 + 4 \beta_{1} ) q^{83} + ( 4 - 8 \beta_{1} ) q^{86} + ( 4 + 4 \beta_{1} + 2 \beta_{2} ) q^{88} + ( -4 - \beta_{1} - \beta_{2} ) q^{89} + ( 2 - \beta_{1} + \beta_{2} ) q^{91} + ( 6 + 3 \beta_{1} - \beta_{2} ) q^{92} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{94} + ( -10 + 5 \beta_{1} + 3 \beta_{2} ) q^{97} -\beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 5 q^{4} - 3 q^{7} - 9 q^{8} + O(q^{10})$$ $$3 q - q^{2} + 5 q^{4} - 3 q^{7} - 9 q^{8} - 6 q^{11} - 6 q^{13} + q^{14} + 13 q^{16} + 6 q^{19} + 2 q^{22} - 4 q^{23} - 10 q^{26} - 5 q^{28} - 2 q^{29} + 2 q^{31} - 29 q^{32} - 12 q^{34} - 4 q^{37} - 14 q^{38} - 2 q^{41} + 4 q^{43} - 10 q^{44} - 8 q^{46} - 8 q^{47} + 3 q^{49} - 18 q^{52} - 14 q^{53} + 9 q^{56} - 18 q^{58} - 16 q^{59} - 6 q^{61} + 30 q^{62} + 13 q^{64} + 8 q^{67} - 8 q^{68} - 6 q^{71} - 18 q^{73} + 44 q^{74} + 2 q^{76} + 6 q^{77} + 12 q^{79} + 34 q^{82} - 8 q^{83} + 4 q^{86} + 18 q^{88} - 14 q^{89} + 6 q^{91} + 20 q^{92} - 16 q^{94} - 22 q^{97} - q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} + 2$$

## 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
 2.17009 −1.48119 0.311108
−2.70928 0 5.34017 0 0 −1.00000 −9.04945 0 0
1.2 −0.193937 0 −1.96239 0 0 −1.00000 0.768452 0 0
1.3 1.90321 0 1.62222 0 0 −1.00000 −0.719004 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.w 3
3.b odd 2 1 525.2.a.k 3
5.b even 2 1 1575.2.a.x 3
5.c odd 4 2 315.2.d.e 6
12.b even 2 1 8400.2.a.dj 3
15.d odd 2 1 525.2.a.j 3
15.e even 4 2 105.2.d.b 6
20.e even 4 2 5040.2.t.v 6
21.c even 2 1 3675.2.a.bj 3
35.f even 4 2 2205.2.d.l 6
60.h even 2 1 8400.2.a.dg 3
60.l odd 4 2 1680.2.t.k 6
105.g even 2 1 3675.2.a.bi 3
105.k odd 4 2 735.2.d.b 6
105.w odd 12 4 735.2.q.f 12
105.x even 12 4 735.2.q.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b 6 15.e even 4 2
315.2.d.e 6 5.c odd 4 2
525.2.a.j 3 15.d odd 2 1
525.2.a.k 3 3.b odd 2 1
735.2.d.b 6 105.k odd 4 2
735.2.q.e 12 105.x even 12 4
735.2.q.f 12 105.w odd 12 4
1575.2.a.w 3 1.a even 1 1 trivial
1575.2.a.x 3 5.b even 2 1
1680.2.t.k 6 60.l odd 4 2
2205.2.d.l 6 35.f even 4 2
3675.2.a.bi 3 105.g even 2 1
3675.2.a.bj 3 21.c even 2 1
5040.2.t.v 6 20.e even 4 2
8400.2.a.dg 3 60.h even 2 1
8400.2.a.dj 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}^{3} + T_{2}^{2} - 5 T_{2} - 1$$ $$T_{11} + 2$$ $$T_{13}^{3} + 6 T_{13}^{2} - 4 T_{13} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 5 T + T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$( 2 + T )^{3}$$
$13$ $$-8 - 4 T + 6 T^{2} + T^{3}$$
$17$ $$16 - 16 T + T^{3}$$
$19$ $$40 - 4 T - 6 T^{2} + T^{3}$$
$23$ $$-16 - 8 T + 4 T^{2} + T^{3}$$
$29$ $$-40 - 52 T + 2 T^{2} + T^{3}$$
$31$ $$184 - 52 T - 2 T^{2} + T^{3}$$
$37$ $$-64 - 80 T + 4 T^{2} + T^{3}$$
$41$ $$-200 - 60 T + 2 T^{2} + T^{3}$$
$43$ $$832 - 144 T - 4 T^{2} + T^{3}$$
$47$ $$-128 - 32 T + 8 T^{2} + T^{3}$$
$53$ $$-296 + 12 T + 14 T^{2} + T^{3}$$
$59$ $$-1280 - 64 T + 16 T^{2} + T^{3}$$
$61$ $$-248 - 52 T + 6 T^{2} + T^{3}$$
$67$ $$128 - 32 T - 8 T^{2} + T^{3}$$
$71$ $$( 2 + T )^{3}$$
$73$ $$104 + 92 T + 18 T^{2} + T^{3}$$
$79$ $$320 - 16 T - 12 T^{2} + T^{3}$$
$83$ $$-256 - 64 T + 8 T^{2} + T^{3}$$
$89$ $$40 + 52 T + 14 T^{2} + T^{3}$$
$97$ $$-1864 - 36 T + 22 T^{2} + T^{3}$$