# Properties

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

# 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} + 3q^{7} + q^{9} + O(q^{10})$$ $$q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} + 3q^{7} + q^{9} + q^{11} - 2q^{12} + 2q^{13} - 6q^{14} - 4q^{16} - 2q^{17} - 2q^{18} + 6q^{19} - 3q^{21} - 2q^{22} - 3q^{23} - 4q^{26} - q^{27} + 6q^{28} + 3q^{29} + 2q^{31} + 8q^{32} - q^{33} + 4q^{34} + 2q^{36} + 7q^{37} - 12q^{38} - 2q^{39} + 10q^{41} + 6q^{42} + 10q^{43} + 2q^{44} + 6q^{46} + q^{47} + 4q^{48} + 2q^{49} + 2q^{51} + 4q^{52} - 4q^{53} + 2q^{54} - 6q^{57} - 6q^{58} + 8q^{59} - 10q^{61} - 4q^{62} + 3q^{63} - 8q^{64} + 2q^{66} - 10q^{67} - 4q^{68} + 3q^{69} - 14q^{71} + 10q^{73} - 14q^{74} + 12q^{76} + 3q^{77} + 4q^{78} + 17q^{79} + q^{81} - 20q^{82} - 8q^{83} - 6q^{84} - 20q^{86} - 3q^{87} + 6q^{89} + 6q^{91} - 6q^{92} - 2q^{93} - 2q^{94} - 8q^{96} - q^{97} - 4q^{98} + q^{99} + O(q^{100})$$

## 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
−2.00000 −1.00000 2.00000 0 2.00000 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.c 1
5.b even 2 1 141.2.a.e 1
15.d odd 2 1 423.2.a.b 1
20.d odd 2 1 2256.2.a.e 1
35.c odd 2 1 6909.2.a.k 1
40.e odd 2 1 9024.2.a.bq 1
40.f even 2 1 9024.2.a.n 1
60.h even 2 1 6768.2.a.n 1
235.b odd 2 1 6627.2.a.i 1

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

## Hecke kernels

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

 $$T_{2} + 2$$ $$T_{7} - 3$$ $$T_{11} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$-3 + T$$
$11$ $$-1 + T$$
$13$ $$-2 + T$$
$17$ $$2 + T$$
$19$ $$-6 + T$$
$23$ $$3 + T$$
$29$ $$-3 + T$$
$31$ $$-2 + T$$
$37$ $$-7 + T$$
$41$ $$-10 + T$$
$43$ $$-10 + T$$
$47$ $$-1 + T$$
$53$ $$4 + T$$
$59$ $$-8 + T$$
$61$ $$10 + T$$
$67$ $$10 + T$$
$71$ $$14 + T$$
$73$ $$-10 + T$$
$79$ $$-17 + T$$
$83$ $$8 + T$$
$89$ $$-6 + T$$
$97$ $$1 + T$$