# Properties

 Label 3675.2.a.f 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} - q^{4} - q^{6} + 3q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} - q^{4} - q^{6} + 3q^{8} + q^{9} - q^{12} - 6q^{13} - q^{16} + 2q^{17} - q^{18} + 8q^{19} - 8q^{23} + 3q^{24} + 6q^{26} + q^{27} - 2q^{29} - 4q^{31} - 5q^{32} - 2q^{34} - q^{36} + 2q^{37} - 8q^{38} - 6q^{39} + 6q^{41} - 4q^{43} + 8q^{46} + 8q^{47} - q^{48} + 2q^{51} + 6q^{52} - 10q^{53} - q^{54} + 8q^{57} + 2q^{58} - 4q^{59} + 2q^{61} + 4q^{62} + 7q^{64} - 4q^{67} - 2q^{68} - 8q^{69} - 12q^{71} + 3q^{72} - 2q^{73} - 2q^{74} - 8q^{76} + 6q^{78} + 8q^{79} + q^{81} - 6q^{82} - 4q^{83} + 4q^{86} - 2q^{87} + 6q^{89} + 8q^{92} - 4q^{93} - 8q^{94} - 5q^{96} - 18q^{97} + O(q^{100})$$

## 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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.f 1
5.b even 2 1 735.2.a.f 1
7.b odd 2 1 525.2.a.a 1
15.d odd 2 1 2205.2.a.b 1
21.c even 2 1 1575.2.a.h 1
28.d even 2 1 8400.2.a.co 1
35.c odd 2 1 105.2.a.a 1
35.f even 4 2 525.2.d.b 2
35.i odd 6 2 735.2.i.a 2
35.j even 6 2 735.2.i.b 2
105.g even 2 1 315.2.a.a 1
105.k odd 4 2 1575.2.d.b 2
140.c even 2 1 1680.2.a.f 1
280.c odd 2 1 6720.2.a.p 1
280.n even 2 1 6720.2.a.bk 1
420.o odd 2 1 5040.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 35.c odd 2 1
315.2.a.a 1 105.g even 2 1
525.2.a.a 1 7.b odd 2 1
525.2.d.b 2 35.f even 4 2
735.2.a.f 1 5.b even 2 1
735.2.i.a 2 35.i odd 6 2
735.2.i.b 2 35.j even 6 2
1575.2.a.h 1 21.c even 2 1
1575.2.d.b 2 105.k odd 4 2
1680.2.a.f 1 140.c even 2 1
2205.2.a.b 1 15.d odd 2 1
3675.2.a.f 1 1.a even 1 1 trivial
5040.2.a.d 1 420.o odd 2 1
6720.2.a.p 1 280.c odd 2 1
6720.2.a.bk 1 280.n even 2 1
8400.2.a.co 1 28.d even 2 1

## Hecke kernels

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

 $$T_{2} + 1$$ $$T_{11}$$ $$T_{13} + 6$$

## Hecke characteristic polynomials

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