# Properties

 Label 735.2.a.f Level $735$ Weight $2$ Character orbit 735.a Self dual yes Analytic conductor $5.869$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 735.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.86900454856$$ Analytic rank: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 3q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 3q^{8} + q^{9} - q^{10} + q^{12} + 6q^{13} + q^{15} - q^{16} - 2q^{17} + q^{18} + 8q^{19} + q^{20} + 8q^{23} + 3q^{24} + q^{25} + 6q^{26} - q^{27} - 2q^{29} + q^{30} - 4q^{31} + 5q^{32} - 2q^{34} - q^{36} - 2q^{37} + 8q^{38} - 6q^{39} + 3q^{40} + 6q^{41} + 4q^{43} - q^{45} + 8q^{46} - 8q^{47} + q^{48} + q^{50} + 2q^{51} - 6q^{52} + 10q^{53} - q^{54} - 8q^{57} - 2q^{58} - 4q^{59} - q^{60} + 2q^{61} - 4q^{62} + 7q^{64} - 6q^{65} + 4q^{67} + 2q^{68} - 8q^{69} - 12q^{71} - 3q^{72} + 2q^{73} - 2q^{74} - q^{75} - 8q^{76} - 6q^{78} + 8q^{79} + q^{80} + q^{81} + 6q^{82} + 4q^{83} + 2q^{85} + 4q^{86} + 2q^{87} + 6q^{89} - q^{90} - 8q^{92} + 4q^{93} - 8q^{94} - 8q^{95} - 5q^{96} + 18q^{97} + O(q^{100})$$

## 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 −1.00000 −1.00000 0 −3.00000 1.00000 −1.00000
 $$n$$: 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 735.2.a.f 1
3.b odd 2 1 2205.2.a.b 1
5.b even 2 1 3675.2.a.f 1
7.b odd 2 1 105.2.a.a 1
7.c even 3 2 735.2.i.b 2
7.d odd 6 2 735.2.i.a 2
21.c even 2 1 315.2.a.a 1
28.d even 2 1 1680.2.a.f 1
35.c odd 2 1 525.2.a.a 1
35.f even 4 2 525.2.d.b 2
56.e even 2 1 6720.2.a.bk 1
56.h odd 2 1 6720.2.a.p 1
84.h odd 2 1 5040.2.a.d 1
105.g even 2 1 1575.2.a.h 1
105.k odd 4 2 1575.2.d.b 2
140.c even 2 1 8400.2.a.co 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 7.b odd 2 1
315.2.a.a 1 21.c even 2 1
525.2.a.a 1 35.c odd 2 1
525.2.d.b 2 35.f even 4 2
735.2.a.f 1 1.a even 1 1 trivial
735.2.i.a 2 7.d odd 6 2
735.2.i.b 2 7.c even 3 2
1575.2.a.h 1 105.g even 2 1
1575.2.d.b 2 105.k odd 4 2
1680.2.a.f 1 28.d even 2 1
2205.2.a.b 1 3.b odd 2 1
3675.2.a.f 1 5.b even 2 1
5040.2.a.d 1 84.h odd 2 1
6720.2.a.p 1 56.h odd 2 1
6720.2.a.bk 1 56.e even 2 1
8400.2.a.co 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(735))$$:

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

## Hecke characteristic polynomials

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