Properties

 Label 3525.2.a.j 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} - q^{4} - q^{6} - 3q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} - q^{4} - q^{6} - 3q^{8} + q^{9} + 4q^{11} + q^{12} + 2q^{13} - q^{16} - 2q^{17} + q^{18} + 4q^{22} + 3q^{24} + 2q^{26} - q^{27} - 6q^{29} - 4q^{31} + 5q^{32} - 4q^{33} - 2q^{34} - q^{36} + 10q^{37} - 2q^{39} - 2q^{41} - 8q^{43} - 4q^{44} + q^{47} + q^{48} - 7q^{49} + 2q^{51} - 2q^{52} + 2q^{53} - q^{54} - 6q^{58} - 4q^{59} + 14q^{61} - 4q^{62} + 7q^{64} - 4q^{66} + 8q^{67} + 2q^{68} + 16q^{71} - 3q^{72} - 2q^{73} + 10q^{74} - 2q^{78} + 8q^{79} + q^{81} - 2q^{82} + 4q^{83} - 8q^{86} + 6q^{87} - 12q^{88} + 18q^{89} + 4q^{93} + q^{94} - 5q^{96} + 14q^{97} - 7q^{98} + 4q^{99} + O(q^{100})$$

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
1.00000 −1.00000 −1.00000 0 −1.00000 0 −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.j 1
5.b even 2 1 141.2.a.c 1
15.d odd 2 1 423.2.a.d 1
20.d odd 2 1 2256.2.a.g 1
35.c odd 2 1 6909.2.a.b 1
40.e odd 2 1 9024.2.a.bd 1
40.f even 2 1 9024.2.a.d 1
60.h even 2 1 6768.2.a.f 1
235.b odd 2 1 6627.2.a.c 1

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-4 + T$$
$13$ $$-2 + T$$
$17$ $$2 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$4 + T$$
$37$ $$-10 + T$$
$41$ $$2 + T$$
$43$ $$8 + T$$
$47$ $$-1 + T$$
$53$ $$-2 + T$$
$59$ $$4 + T$$
$61$ $$-14 + T$$
$67$ $$-8 + T$$
$71$ $$-16 + T$$
$73$ $$2 + T$$
$79$ $$-8 + T$$
$83$ $$-4 + T$$
$89$ $$-18 + T$$
$97$ $$-14 + T$$