# Properties

 Label 1575.2.a.a Level $1575$ Weight $2$ Character orbit 1575.a Self dual yes Analytic conductor $12.576$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.5764383184$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 2.00000 0 0 −1.00000 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 1575.2.a.a 1
3.b odd 2 1 175.2.a.c 1
5.b even 2 1 1575.2.a.k 1
5.c odd 4 2 315.2.d.a 2
12.b even 2 1 2800.2.a.l 1
15.d odd 2 1 175.2.a.a 1
15.e even 4 2 35.2.b.a 2
20.e even 4 2 5040.2.t.p 2
21.c even 2 1 1225.2.a.i 1
35.f even 4 2 2205.2.d.b 2
60.h even 2 1 2800.2.a.w 1
60.l odd 4 2 560.2.g.b 2
105.g even 2 1 1225.2.a.a 1
105.k odd 4 2 245.2.b.a 2
105.w odd 12 4 245.2.j.d 4
105.x even 12 4 245.2.j.e 4
120.q odd 4 2 2240.2.g.g 2
120.w even 4 2 2240.2.g.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 15.e even 4 2
175.2.a.a 1 15.d odd 2 1
175.2.a.c 1 3.b odd 2 1
245.2.b.a 2 105.k odd 4 2
245.2.j.d 4 105.w odd 12 4
245.2.j.e 4 105.x even 12 4
315.2.d.a 2 5.c odd 4 2
560.2.g.b 2 60.l odd 4 2
1225.2.a.a 1 105.g even 2 1
1225.2.a.i 1 21.c even 2 1
1575.2.a.a 1 1.a even 1 1 trivial
1575.2.a.k 1 5.b even 2 1
2205.2.d.b 2 35.f even 4 2
2240.2.g.g 2 120.q odd 4 2
2240.2.g.h 2 120.w even 4 2
2800.2.a.l 1 12.b even 2 1
2800.2.a.w 1 60.h even 2 1
5040.2.t.p 2 20.e even 4 2

## Hecke kernels

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

 $$T_{2} + 2$$ T2 + 2 $$T_{11} - 3$$ T11 - 3 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T - 3$$
$13$ $$T - 1$$
$17$ $$T + 7$$
$19$ $$T$$
$23$ $$T + 6$$
$29$ $$T - 5$$
$31$ $$T - 2$$
$37$ $$T - 2$$
$41$ $$T + 2$$
$43$ $$T + 4$$
$47$ $$T - 3$$
$53$ $$T + 6$$
$59$ $$T + 10$$
$61$ $$T + 8$$
$67$ $$T - 2$$
$71$ $$T - 8$$
$73$ $$T - 6$$
$79$ $$T + 5$$
$83$ $$T - 4$$
$89$ $$T$$
$97$ $$T - 7$$