# Properties

 Label 1575.4.a.e 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 7) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - 7q^{4} + 7q^{7} + 15q^{8} + O(q^{10})$$ $$q - q^{2} - 7q^{4} + 7q^{7} + 15q^{8} + 8q^{11} - 28q^{13} - 7q^{14} + 41q^{16} + 54q^{17} - 110q^{19} - 8q^{22} + 48q^{23} + 28q^{26} - 49q^{28} + 110q^{29} + 12q^{31} - 161q^{32} - 54q^{34} + 246q^{37} + 110q^{38} - 182q^{41} - 128q^{43} - 56q^{44} - 48q^{46} + 324q^{47} + 49q^{49} + 196q^{52} - 162q^{53} + 105q^{56} - 110q^{58} - 810q^{59} - 488q^{61} - 12q^{62} - 167q^{64} - 244q^{67} - 378q^{68} + 768q^{71} + 702q^{73} - 246q^{74} + 770q^{76} + 56q^{77} + 440q^{79} + 182q^{82} - 1302q^{83} + 128q^{86} + 120q^{88} - 730q^{89} - 196q^{91} - 336q^{92} - 324q^{94} - 294q^{97} - 49q^{98} + O(q^{100})$$

## 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
−1.00000 0 −7.00000 0 0 7.00000 15.0000 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.e 1
3.b odd 2 1 175.4.a.b 1
5.b even 2 1 63.4.a.b 1
15.d odd 2 1 7.4.a.a 1
15.e even 4 2 175.4.b.b 2
20.d odd 2 1 1008.4.a.c 1
21.c even 2 1 1225.4.a.j 1
35.c odd 2 1 441.4.a.i 1
35.i odd 6 2 441.4.e.e 2
35.j even 6 2 441.4.e.h 2
60.h even 2 1 112.4.a.f 1
105.g even 2 1 49.4.a.b 1
105.o odd 6 2 49.4.c.c 2
105.p even 6 2 49.4.c.b 2
120.i odd 2 1 448.4.a.i 1
120.m even 2 1 448.4.a.e 1
165.d even 2 1 847.4.a.b 1
195.e odd 2 1 1183.4.a.b 1
255.h odd 2 1 2023.4.a.a 1
420.o odd 2 1 784.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 15.d odd 2 1
49.4.a.b 1 105.g even 2 1
49.4.c.b 2 105.p even 6 2
49.4.c.c 2 105.o odd 6 2
63.4.a.b 1 5.b even 2 1
112.4.a.f 1 60.h even 2 1
175.4.a.b 1 3.b odd 2 1
175.4.b.b 2 15.e even 4 2
441.4.a.i 1 35.c odd 2 1
441.4.e.e 2 35.i odd 6 2
441.4.e.h 2 35.j even 6 2
448.4.a.e 1 120.m even 2 1
448.4.a.i 1 120.i odd 2 1
784.4.a.g 1 420.o odd 2 1
847.4.a.b 1 165.d even 2 1
1008.4.a.c 1 20.d odd 2 1
1183.4.a.b 1 195.e odd 2 1
1225.4.a.j 1 21.c even 2 1
1575.4.a.e 1 1.a even 1 1 trivial
2023.4.a.a 1 255.h odd 2 1

## Hecke kernels

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

 $$T_{2} + 1$$ $$T_{11} - 8$$ $$T_{13} + 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-7 + T$$
$11$ $$-8 + T$$
$13$ $$28 + T$$
$17$ $$-54 + T$$
$19$ $$110 + T$$
$23$ $$-48 + T$$
$29$ $$-110 + T$$
$31$ $$-12 + T$$
$37$ $$-246 + T$$
$41$ $$182 + T$$
$43$ $$128 + T$$
$47$ $$-324 + T$$
$53$ $$162 + T$$
$59$ $$810 + T$$
$61$ $$488 + T$$
$67$ $$244 + T$$
$71$ $$-768 + T$$
$73$ $$-702 + T$$
$79$ $$-440 + T$$
$83$ $$1302 + T$$
$89$ $$730 + T$$
$97$ $$294 + T$$