# Properties

 Label 735.4.a.i Level $735$ Weight $4$ Character orbit 735.a Self dual yes Analytic conductor $43.366$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 735.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.3664038542$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + 3 q^{3} + q^{4} + 5 q^{5} + 9 q^{6} - 21 q^{8} + 9 q^{9} + O(q^{10})$$ $$q + 3 q^{2} + 3 q^{3} + q^{4} + 5 q^{5} + 9 q^{6} - 21 q^{8} + 9 q^{9} + 15 q^{10} - 24 q^{11} + 3 q^{12} - 74 q^{13} + 15 q^{15} - 71 q^{16} - 54 q^{17} + 27 q^{18} + 124 q^{19} + 5 q^{20} - 72 q^{22} - 120 q^{23} - 63 q^{24} + 25 q^{25} - 222 q^{26} + 27 q^{27} - 78 q^{29} + 45 q^{30} - 200 q^{31} - 45 q^{32} - 72 q^{33} - 162 q^{34} + 9 q^{36} - 70 q^{37} + 372 q^{38} - 222 q^{39} - 105 q^{40} - 330 q^{41} + 92 q^{43} - 24 q^{44} + 45 q^{45} - 360 q^{46} + 24 q^{47} - 213 q^{48} + 75 q^{50} - 162 q^{51} - 74 q^{52} + 450 q^{53} + 81 q^{54} - 120 q^{55} + 372 q^{57} - 234 q^{58} - 24 q^{59} + 15 q^{60} + 322 q^{61} - 600 q^{62} + 433 q^{64} - 370 q^{65} - 216 q^{66} - 196 q^{67} - 54 q^{68} - 360 q^{69} - 288 q^{71} - 189 q^{72} + 430 q^{73} - 210 q^{74} + 75 q^{75} + 124 q^{76} - 666 q^{78} - 520 q^{79} - 355 q^{80} + 81 q^{81} - 990 q^{82} - 156 q^{83} - 270 q^{85} + 276 q^{86} - 234 q^{87} + 504 q^{88} - 1026 q^{89} + 135 q^{90} - 120 q^{92} - 600 q^{93} + 72 q^{94} + 620 q^{95} - 135 q^{96} + 286 q^{97} - 216 q^{99} + O(q^{100})$$

## 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
3.00000 3.00000 1.00000 5.00000 9.00000 0 −21.0000 9.00000 15.0000
 $$n$$: 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 735.4.a.i 1
3.b odd 2 1 2205.4.a.c 1
7.b odd 2 1 15.4.a.b 1
21.c even 2 1 45.4.a.b 1
28.d even 2 1 240.4.a.f 1
35.c odd 2 1 75.4.a.a 1
35.f even 4 2 75.4.b.a 2
56.e even 2 1 960.4.a.l 1
56.h odd 2 1 960.4.a.bi 1
63.l odd 6 2 405.4.e.d 2
63.o even 6 2 405.4.e.k 2
77.b even 2 1 1815.4.a.a 1
84.h odd 2 1 720.4.a.r 1
105.g even 2 1 225.4.a.g 1
105.k odd 4 2 225.4.b.d 2
140.c even 2 1 1200.4.a.o 1
140.j odd 4 2 1200.4.f.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 7.b odd 2 1
45.4.a.b 1 21.c even 2 1
75.4.a.a 1 35.c odd 2 1
75.4.b.a 2 35.f even 4 2
225.4.a.g 1 105.g even 2 1
225.4.b.d 2 105.k odd 4 2
240.4.a.f 1 28.d even 2 1
405.4.e.d 2 63.l odd 6 2
405.4.e.k 2 63.o even 6 2
720.4.a.r 1 84.h odd 2 1
735.4.a.i 1 1.a even 1 1 trivial
960.4.a.l 1 56.e even 2 1
960.4.a.bi 1 56.h odd 2 1
1200.4.a.o 1 140.c even 2 1
1200.4.f.m 2 140.j odd 4 2
1815.4.a.a 1 77.b even 2 1
2205.4.a.c 1 3.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(735))$$:

 $$T_{2} - 3$$ $$T_{11} + 24$$ $$T_{13} + 74$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T$$
$3$ $$-3 + T$$
$5$ $$-5 + T$$
$7$ $$T$$
$11$ $$24 + T$$
$13$ $$74 + T$$
$17$ $$54 + T$$
$19$ $$-124 + T$$
$23$ $$120 + T$$
$29$ $$78 + T$$
$31$ $$200 + T$$
$37$ $$70 + T$$
$41$ $$330 + T$$
$43$ $$-92 + T$$
$47$ $$-24 + T$$
$53$ $$-450 + T$$
$59$ $$24 + T$$
$61$ $$-322 + T$$
$67$ $$196 + T$$
$71$ $$288 + T$$
$73$ $$-430 + T$$
$79$ $$520 + T$$
$83$ $$156 + T$$
$89$ $$1026 + T$$
$97$ $$-286 + T$$