# Properties

 Label 1575.4.a.h Level $1575$ Weight $4$ Character orbit 1575.a Self dual yes Analytic conductor $92.928$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$92.9280082590$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 525) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{2} - 4q^{4} - 7q^{7} - 24q^{8} + O(q^{10})$$ $$q + 2q^{2} - 4q^{4} - 7q^{7} - 24q^{8} + 21q^{11} + 24q^{13} - 14q^{14} - 16q^{16} + 22q^{17} + 16q^{19} + 42q^{22} + 25q^{23} + 48q^{26} + 28q^{28} - 167q^{29} + 10q^{31} + 160q^{32} + 44q^{34} - 133q^{37} + 32q^{38} + 168q^{41} - 97q^{43} - 84q^{44} + 50q^{46} + 400q^{47} + 49q^{49} - 96q^{52} + 182q^{53} + 168q^{56} - 334q^{58} - 488q^{59} + 28q^{61} + 20q^{62} + 448q^{64} - 967q^{67} - 88q^{68} + 285q^{71} - 838q^{73} - 266q^{74} - 64q^{76} - 147q^{77} - 469q^{79} + 336q^{82} + 406q^{83} - 194q^{86} - 504q^{88} - 324q^{89} - 168q^{91} - 100q^{92} + 800q^{94} - 114q^{97} + 98q^{98} + O(q^{100})$$

## 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
 0
2.00000 0 −4.00000 0 0 −7.00000 −24.0000 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 1575.4.a.h 1
3.b odd 2 1 525.4.a.d 1
5.b even 2 1 1575.4.a.d 1
15.d odd 2 1 525.4.a.f yes 1
15.e even 4 2 525.4.d.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.d 1 3.b odd 2 1
525.4.a.f yes 1 15.d odd 2 1
525.4.d.e 2 15.e even 4 2
1575.4.a.d 1 5.b even 2 1
1575.4.a.h 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2} - 2$$ $$T_{11} - 21$$ $$T_{13} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$7 + T$$
$11$ $$-21 + T$$
$13$ $$-24 + T$$
$17$ $$-22 + T$$
$19$ $$-16 + T$$
$23$ $$-25 + T$$
$29$ $$167 + T$$
$31$ $$-10 + T$$
$37$ $$133 + T$$
$41$ $$-168 + T$$
$43$ $$97 + T$$
$47$ $$-400 + T$$
$53$ $$-182 + T$$
$59$ $$488 + T$$
$61$ $$-28 + T$$
$67$ $$967 + T$$
$71$ $$-285 + T$$
$73$ $$838 + T$$
$79$ $$469 + T$$
$83$ $$-406 + T$$
$89$ $$324 + T$$
$97$ $$114 + T$$