# Properties

 Label 3525.2.a.h Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3525 = 3 \cdot 5^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.1472667125$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 141) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 2q^{4} + 3q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - 2q^{4} + 3q^{7} + q^{9} - 3q^{11} - 2q^{12} + 4q^{13} + 4q^{16} - 8q^{17} - 6q^{19} + 3q^{21} - 3q^{23} + q^{27} - 6q^{28} - q^{29} + 4q^{31} - 3q^{33} - 2q^{36} - q^{37} + 4q^{39} - 10q^{41} + 8q^{43} + 6q^{44} + q^{47} + 4q^{48} + 2q^{49} - 8q^{51} - 8q^{52} - 10q^{53} - 6q^{57} - 10q^{59} + 2q^{61} + 3q^{63} - 8q^{64} - 4q^{67} + 16q^{68} - 3q^{69} - 6q^{71} + 8q^{73} + 12q^{76} - 9q^{77} - 3q^{79} + q^{81} + 18q^{83} - 6q^{84} - q^{87} - 2q^{89} + 12q^{91} + 6q^{92} + 4q^{93} - 5q^{97} - 3q^{99} + O(q^{100})$$

## 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
0 1.00000 −2.00000 0 0 3.00000 0 1.00000 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$47$$ $$-1$$

## 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 3525.2.a.h 1
5.b even 2 1 141.2.a.d 1
15.d odd 2 1 423.2.a.c 1
20.d odd 2 1 2256.2.a.k 1
35.c odd 2 1 6909.2.a.h 1
40.e odd 2 1 9024.2.a.r 1
40.f even 2 1 9024.2.a.bl 1
60.h even 2 1 6768.2.a.m 1
235.b odd 2 1 6627.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.2.a.d 1 5.b even 2 1
423.2.a.c 1 15.d odd 2 1
2256.2.a.k 1 20.d odd 2 1
3525.2.a.h 1 1.a even 1 1 trivial
6627.2.a.e 1 235.b odd 2 1
6768.2.a.m 1 60.h even 2 1
6909.2.a.h 1 35.c odd 2 1
9024.2.a.r 1 40.e odd 2 1
9024.2.a.bl 1 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2}$$ $$T_{7} - 3$$ $$T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$-3 + T$$
$11$ $$3 + T$$
$13$ $$-4 + T$$
$17$ $$8 + T$$
$19$ $$6 + T$$
$23$ $$3 + T$$
$29$ $$1 + T$$
$31$ $$-4 + T$$
$37$ $$1 + T$$
$41$ $$10 + T$$
$43$ $$-8 + T$$
$47$ $$-1 + T$$
$53$ $$10 + T$$
$59$ $$10 + T$$
$61$ $$-2 + T$$
$67$ $$4 + T$$
$71$ $$6 + T$$
$73$ $$-8 + T$$
$79$ $$3 + T$$
$83$ $$-18 + T$$
$89$ $$2 + T$$
$97$ $$5 + T$$