# Properties

 Label 3525.2.a.k Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} + q^{6} - 4q^{7} - 3q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} - q^{4} + q^{6} - 4q^{7} - 3q^{8} + q^{9} - q^{12} - 6q^{13} - 4q^{14} - q^{16} + 6q^{17} + q^{18} + 2q^{19} - 4q^{21} - 4q^{23} - 3q^{24} - 6q^{26} + q^{27} + 4q^{28} + 8q^{29} + 6q^{31} + 5q^{32} + 6q^{34} - q^{36} + 6q^{37} + 2q^{38} - 6q^{39} - 8q^{41} - 4q^{42} + 6q^{43} - 4q^{46} - q^{47} - q^{48} + 9q^{49} + 6q^{51} + 6q^{52} - 2q^{53} + q^{54} + 12q^{56} + 2q^{57} + 8q^{58} + 12q^{59} + 2q^{61} + 6q^{62} - 4q^{63} + 7q^{64} + 2q^{67} - 6q^{68} - 4q^{69} - 3q^{72} + 10q^{73} + 6q^{74} - 2q^{76} - 6q^{78} - 4q^{79} + q^{81} - 8q^{82} - 4q^{83} + 4q^{84} + 6q^{86} + 8q^{87} - 10q^{89} + 24q^{91} + 4q^{92} + 6q^{93} - q^{94} + 5q^{96} + 18q^{97} + 9q^{98} + O(q^{100})$$

## 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 1.00000 −1.00000 0 1.00000 −4.00000 −3.00000 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.k 1
5.b even 2 1 141.2.a.b 1
15.d odd 2 1 423.2.a.e 1
20.d odd 2 1 2256.2.a.l 1
35.c odd 2 1 6909.2.a.e 1
40.e odd 2 1 9024.2.a.i 1
40.f even 2 1 9024.2.a.bk 1
60.h even 2 1 6768.2.a.h 1
235.b odd 2 1 6627.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.2.a.b 1 5.b even 2 1
423.2.a.e 1 15.d odd 2 1
2256.2.a.l 1 20.d odd 2 1
3525.2.a.k 1 1.a even 1 1 trivial
6627.2.a.b 1 235.b odd 2 1
6768.2.a.h 1 60.h even 2 1
6909.2.a.e 1 35.c odd 2 1
9024.2.a.i 1 40.e odd 2 1
9024.2.a.bk 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2} - 1$$ $$T_{7} + 4$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$T$$
$13$ $$6 + T$$
$17$ $$-6 + T$$
$19$ $$-2 + T$$
$23$ $$4 + T$$
$29$ $$-8 + T$$
$31$ $$-6 + T$$
$37$ $$-6 + T$$
$41$ $$8 + T$$
$43$ $$-6 + T$$
$47$ $$1 + T$$
$53$ $$2 + T$$
$59$ $$-12 + T$$
$61$ $$-2 + T$$
$67$ $$-2 + T$$
$71$ $$T$$
$73$ $$-10 + T$$
$79$ $$4 + T$$
$83$ $$4 + T$$
$89$ $$10 + T$$
$97$ $$-18 + T$$
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