# Properties

 Label 3675.2.a.o Level $3675$ Weight $2$ Character orbit 3675.a Self dual yes Analytic conductor $29.345$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3675 = 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3675.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.3450227428$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 0 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
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$

## 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 3675.2.a.o 1
5.b even 2 1 735.2.a.b 1
7.b odd 2 1 3675.2.a.p 1
7.c even 3 2 525.2.i.a 2
15.d odd 2 1 2205.2.a.k 1
35.c odd 2 1 735.2.a.a 1
35.i odd 6 2 735.2.i.f 2
35.j even 6 2 105.2.i.b 2
35.l odd 12 4 525.2.r.d 4
105.g even 2 1 2205.2.a.m 1
105.o odd 6 2 315.2.j.a 2
140.p odd 6 2 1680.2.bg.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.b 2 35.j even 6 2
315.2.j.a 2 105.o odd 6 2
525.2.i.a 2 7.c even 3 2
525.2.r.d 4 35.l odd 12 4
735.2.a.a 1 35.c odd 2 1
735.2.a.b 1 5.b even 2 1
735.2.i.f 2 35.i odd 6 2
1680.2.bg.l 2 140.p odd 6 2
2205.2.a.k 1 15.d odd 2 1
2205.2.a.m 1 105.g even 2 1
3675.2.a.o 1 1.a even 1 1 trivial
3675.2.a.p 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3675))$$:

 $$T_{2} - 2$$ $$T_{11} + 6$$ $$T_{13} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$6 + T$$
$13$ $$-3 + T$$
$17$ $$-4 + T$$
$19$ $$-1 + T$$
$23$ $$-4 + T$$
$29$ $$8 + T$$
$31$ $$-1 + T$$
$37$ $$7 + T$$
$41$ $$6 + T$$
$43$ $$1 + T$$
$47$ $$2 + T$$
$53$ $$4 + T$$
$59$ $$8 + T$$
$61$ $$14 + T$$
$67$ $$7 + T$$
$71$ $$-6 + T$$
$73$ $$1 + T$$
$79$ $$1 + T$$
$83$ $$2 + T$$
$89$ $$12 + T$$
$97$ $$-6 + T$$