# Properties

 Label 1575.4.a.k Level $1575$ Weight $4$ Character orbit 1575.a Self dual yes Analytic conductor $92.928$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{2} + 8q^{4} + 7q^{7} + O(q^{10})$$ $$q + 4q^{2} + 8q^{4} + 7q^{7} - 62q^{11} + 62q^{13} + 28q^{14} - 64q^{16} + 84q^{17} + 100q^{19} - 248q^{22} - 42q^{23} + 248q^{26} + 56q^{28} + 10q^{29} - 48q^{31} - 256q^{32} + 336q^{34} + 246q^{37} + 400q^{38} + 248q^{41} - 68q^{43} - 496q^{44} - 168q^{46} + 324q^{47} + 49q^{49} + 496q^{52} + 258q^{53} + 40q^{58} - 120q^{59} + 622q^{61} - 192q^{62} - 512q^{64} - 904q^{67} + 672q^{68} + 678q^{71} + 642q^{73} + 984q^{74} + 800q^{76} - 434q^{77} + 740q^{79} + 992q^{82} + 468q^{83} - 272q^{86} - 200q^{89} + 434q^{91} - 336q^{92} + 1296q^{94} + 1266q^{97} + 196q^{98} + O(q^{100})$$

## 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
4.00000 0 8.00000 0 0 7.00000 0 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.k 1
3.b odd 2 1 525.4.a.b 1
5.b even 2 1 63.4.a.a 1
15.d odd 2 1 21.4.a.b 1
15.e even 4 2 525.4.d.b 2
20.d odd 2 1 1008.4.a.m 1
35.c odd 2 1 441.4.a.b 1
35.i odd 6 2 441.4.e.n 2
35.j even 6 2 441.4.e.m 2
60.h even 2 1 336.4.a.h 1
105.g even 2 1 147.4.a.g 1
105.o odd 6 2 147.4.e.c 2
105.p even 6 2 147.4.e.b 2
120.i odd 2 1 1344.4.a.w 1
120.m even 2 1 1344.4.a.i 1
420.o odd 2 1 2352.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 15.d odd 2 1
63.4.a.a 1 5.b even 2 1
147.4.a.g 1 105.g even 2 1
147.4.e.b 2 105.p even 6 2
147.4.e.c 2 105.o odd 6 2
336.4.a.h 1 60.h even 2 1
441.4.a.b 1 35.c odd 2 1
441.4.e.m 2 35.j even 6 2
441.4.e.n 2 35.i odd 6 2
525.4.a.b 1 3.b odd 2 1
525.4.d.b 2 15.e even 4 2
1008.4.a.m 1 20.d odd 2 1
1344.4.a.i 1 120.m even 2 1
1344.4.a.w 1 120.i odd 2 1
1575.4.a.k 1 1.a even 1 1 trivial
2352.4.a.l 1 420.o odd 2 1

## Hecke kernels

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

 $$T_{2} - 4$$ $$T_{11} + 62$$ $$T_{13} - 62$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-7 + T$$
$11$ $$62 + T$$
$13$ $$-62 + T$$
$17$ $$-84 + T$$
$19$ $$-100 + T$$
$23$ $$42 + T$$
$29$ $$-10 + T$$
$31$ $$48 + T$$
$37$ $$-246 + T$$
$41$ $$-248 + T$$
$43$ $$68 + T$$
$47$ $$-324 + T$$
$53$ $$-258 + T$$
$59$ $$120 + T$$
$61$ $$-622 + T$$
$67$ $$904 + T$$
$71$ $$-678 + T$$
$73$ $$-642 + T$$
$79$ $$-740 + T$$
$83$ $$-468 + T$$
$89$ $$200 + T$$
$97$ $$-1266 + T$$