# Properties

 Label 1575.2.a.p Level $1575$ Weight $2$ Character orbit 1575.a Self dual yes Analytic conductor $12.576$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.56155 −1.56155
−2.56155 0 4.56155 0 0 1.00000 −6.56155 0 0
1.2 1.56155 0 0.438447 0 0 1.00000 −2.43845 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.p 2
3.b odd 2 1 175.2.a.f 2
5.b even 2 1 315.2.a.e 2
5.c odd 4 2 1575.2.d.e 4
12.b even 2 1 2800.2.a.bi 2
15.d odd 2 1 35.2.a.b 2
15.e even 4 2 175.2.b.b 4
20.d odd 2 1 5040.2.a.bt 2
21.c even 2 1 1225.2.a.s 2
35.c odd 2 1 2205.2.a.x 2
60.h even 2 1 560.2.a.i 2
60.l odd 4 2 2800.2.g.t 4
105.g even 2 1 245.2.a.d 2
105.k odd 4 2 1225.2.b.f 4
105.o odd 6 2 245.2.e.i 4
105.p even 6 2 245.2.e.h 4
120.i odd 2 1 2240.2.a.bh 2
120.m even 2 1 2240.2.a.bd 2
165.d even 2 1 4235.2.a.m 2
195.e odd 2 1 5915.2.a.l 2
420.o odd 2 1 3920.2.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 15.d odd 2 1
175.2.a.f 2 3.b odd 2 1
175.2.b.b 4 15.e even 4 2
245.2.a.d 2 105.g even 2 1
245.2.e.h 4 105.p even 6 2
245.2.e.i 4 105.o odd 6 2
315.2.a.e 2 5.b even 2 1
560.2.a.i 2 60.h even 2 1
1225.2.a.s 2 21.c even 2 1
1225.2.b.f 4 105.k odd 4 2
1575.2.a.p 2 1.a even 1 1 trivial
1575.2.d.e 4 5.c odd 4 2
2205.2.a.x 2 35.c odd 2 1
2240.2.a.bd 2 120.m even 2 1
2240.2.a.bh 2 120.i odd 2 1
2800.2.a.bi 2 12.b even 2 1
2800.2.g.t 4 60.l odd 4 2
3920.2.a.bs 2 420.o odd 2 1
4235.2.a.m 2 165.d even 2 1
5040.2.a.bt 2 20.d odd 2 1
5915.2.a.l 2 195.e odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} - 4$$ $$T_{11}^{2} + T_{11} - 4$$ $$T_{13}^{2} + 5 T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-4 + T + T^{2}$$
$13$ $$2 + 5 T + T^{2}$$
$17$ $$2 + 5 T + T^{2}$$
$19$ $$-8 + 6 T + T^{2}$$
$23$ $$-16 + 2 T + T^{2}$$
$29$ $$-38 + T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$-16 + 2 T + T^{2}$$
$43$ $$8 + 10 T + T^{2}$$
$47$ $$-32 + 5 T + T^{2}$$
$53$ $$-16 + 2 T + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$-144 - 6 T + T^{2}$$
$67$ $$-64 + 4 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$-52 - 8 T + T^{2}$$
$79$ $$16 + 9 T + T^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$-8 + 6 T + T^{2}$$
$97$ $$-86 - 9 T + T^{2}$$