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

## 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(735))$$:

 $$T_{2} - 3$$ T2 - 3 $$T_{11} + 24$$ T11 + 24 $$T_{13} + 74$$ T13 + 74

## Hecke characteristic polynomials

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