# Properties

 Label 3675.2.a.d 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

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} + q^{6} + 3q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - q^{3} - q^{4} + q^{6} + 3q^{8} + q^{9} - 6q^{11} + q^{12} + 2q^{13} - q^{16} - 4q^{17} - q^{18} + 6q^{19} + 6q^{22} - 3q^{24} - 2q^{26} - q^{27} - 2q^{29} + 10q^{31} - 5q^{32} + 6q^{33} + 4q^{34} - q^{36} - 4q^{37} - 6q^{38} - 2q^{39} - 2q^{41} - 4q^{43} + 6q^{44} + q^{48} + 4q^{51} - 2q^{52} + 6q^{53} + q^{54} - 6q^{57} + 2q^{58} + 8q^{59} + 2q^{61} - 10q^{62} + 7q^{64} - 6q^{66} - 16q^{67} + 4q^{68} + 10q^{71} + 3q^{72} + 6q^{73} + 4q^{74} - 6q^{76} + 2q^{78} + 4q^{79} + q^{81} + 2q^{82} - 8q^{83} + 4q^{86} + 2q^{87} - 18q^{88} - 6q^{89} - 10q^{93} + 5q^{96} + 2q^{97} - 6q^{99} + O(q^{100})$$

## 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
−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.d 1
5.b even 2 1 3675.2.a.l 1
5.c odd 4 2 735.2.d.a 2
7.b odd 2 1 525.2.a.b 1
15.e even 4 2 2205.2.d.f 2
21.c even 2 1 1575.2.a.i 1
28.d even 2 1 8400.2.a.bj 1
35.c odd 2 1 525.2.a.c 1
35.f even 4 2 105.2.d.a 2
35.k even 12 4 735.2.q.a 4
35.l odd 12 4 735.2.q.b 4
105.g even 2 1 1575.2.a.e 1
105.k odd 4 2 315.2.d.c 2
140.c even 2 1 8400.2.a.ch 1
140.j odd 4 2 1680.2.t.f 2
420.w even 4 2 5040.2.t.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 35.f even 4 2
315.2.d.c 2 105.k odd 4 2
525.2.a.b 1 7.b odd 2 1
525.2.a.c 1 35.c odd 2 1
735.2.d.a 2 5.c odd 4 2
735.2.q.a 4 35.k even 12 4
735.2.q.b 4 35.l odd 12 4
1575.2.a.e 1 105.g even 2 1
1575.2.a.i 1 21.c even 2 1
1680.2.t.f 2 140.j odd 4 2
2205.2.d.f 2 15.e even 4 2
3675.2.a.d 1 1.a even 1 1 trivial
3675.2.a.l 1 5.b even 2 1
5040.2.t.e 2 420.w even 4 2
8400.2.a.bj 1 28.d even 2 1
8400.2.a.ch 1 140.c even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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