# Properties

 Label 735.4.a.e 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,4,Mod(1,735)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("735.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - 3 q^{3} - 7 q^{4} - 5 q^{5} - 3 q^{6} - 15 q^{8} + 9 q^{9}+O(q^{10})$$ q + q^2 - 3 * q^3 - 7 * q^4 - 5 * q^5 - 3 * q^6 - 15 * q^8 + 9 * q^9 $$q + q^{2} - 3 q^{3} - 7 q^{4} - 5 q^{5} - 3 q^{6} - 15 q^{8} + 9 q^{9} - 5 q^{10} + 52 q^{11} + 21 q^{12} - 22 q^{13} + 15 q^{15} + 41 q^{16} + 14 q^{17} + 9 q^{18} + 20 q^{19} + 35 q^{20} + 52 q^{22} - 168 q^{23} + 45 q^{24} + 25 q^{25} - 22 q^{26} - 27 q^{27} + 230 q^{29} + 15 q^{30} + 288 q^{31} + 161 q^{32} - 156 q^{33} + 14 q^{34} - 63 q^{36} - 34 q^{37} + 20 q^{38} + 66 q^{39} + 75 q^{40} - 122 q^{41} - 188 q^{43} - 364 q^{44} - 45 q^{45} - 168 q^{46} - 256 q^{47} - 123 q^{48} + 25 q^{50} - 42 q^{51} + 154 q^{52} - 338 q^{53} - 27 q^{54} - 260 q^{55} - 60 q^{57} + 230 q^{58} - 100 q^{59} - 105 q^{60} - 742 q^{61} + 288 q^{62} - 167 q^{64} + 110 q^{65} - 156 q^{66} - 84 q^{67} - 98 q^{68} + 504 q^{69} - 328 q^{71} - 135 q^{72} + 38 q^{73} - 34 q^{74} - 75 q^{75} - 140 q^{76} + 66 q^{78} - 240 q^{79} - 205 q^{80} + 81 q^{81} - 122 q^{82} - 1212 q^{83} - 70 q^{85} - 188 q^{86} - 690 q^{87} - 780 q^{88} - 330 q^{89} - 45 q^{90} + 1176 q^{92} - 864 q^{93} - 256 q^{94} - 100 q^{95} - 483 q^{96} - 866 q^{97} + 468 q^{99}+O(q^{100})$$ q + q^2 - 3 * q^3 - 7 * q^4 - 5 * q^5 - 3 * q^6 - 15 * q^8 + 9 * q^9 - 5 * q^10 + 52 * q^11 + 21 * q^12 - 22 * q^13 + 15 * q^15 + 41 * q^16 + 14 * q^17 + 9 * q^18 + 20 * q^19 + 35 * q^20 + 52 * q^22 - 168 * q^23 + 45 * q^24 + 25 * q^25 - 22 * q^26 - 27 * q^27 + 230 * q^29 + 15 * q^30 + 288 * q^31 + 161 * q^32 - 156 * q^33 + 14 * q^34 - 63 * q^36 - 34 * q^37 + 20 * q^38 + 66 * q^39 + 75 * q^40 - 122 * q^41 - 188 * q^43 - 364 * q^44 - 45 * q^45 - 168 * q^46 - 256 * q^47 - 123 * q^48 + 25 * q^50 - 42 * q^51 + 154 * q^52 - 338 * q^53 - 27 * q^54 - 260 * q^55 - 60 * q^57 + 230 * q^58 - 100 * q^59 - 105 * q^60 - 742 * q^61 + 288 * q^62 - 167 * q^64 + 110 * q^65 - 156 * q^66 - 84 * q^67 - 98 * q^68 + 504 * q^69 - 328 * q^71 - 135 * q^72 + 38 * q^73 - 34 * q^74 - 75 * q^75 - 140 * q^76 + 66 * q^78 - 240 * q^79 - 205 * q^80 + 81 * q^81 - 122 * q^82 - 1212 * q^83 - 70 * q^85 - 188 * q^86 - 690 * q^87 - 780 * q^88 - 330 * q^89 - 45 * q^90 + 1176 * q^92 - 864 * q^93 - 256 * q^94 - 100 * q^95 - 483 * q^96 - 866 * q^97 + 468 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 −3.00000 −7.00000 −5.00000 −3.00000 0 −15.0000 9.00000 −5.00000
 $$n$$: 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 735.4.a.e 1
3.b odd 2 1 2205.4.a.l 1
7.b odd 2 1 15.4.a.a 1
21.c even 2 1 45.4.a.c 1
28.d even 2 1 240.4.a.e 1
35.c odd 2 1 75.4.a.b 1
35.f even 4 2 75.4.b.b 2
56.e even 2 1 960.4.a.ba 1
56.h odd 2 1 960.4.a.b 1
63.l odd 6 2 405.4.e.g 2
63.o even 6 2 405.4.e.i 2
77.b even 2 1 1815.4.a.e 1
84.h odd 2 1 720.4.a.n 1
105.g even 2 1 225.4.a.f 1
105.k odd 4 2 225.4.b.e 2
140.c even 2 1 1200.4.a.t 1
140.j odd 4 2 1200.4.f.b 2

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

 $$T_{2} - 1$$ T2 - 1 $$T_{11} - 52$$ T11 - 52 $$T_{13} + 22$$ T13 + 22

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 3$$
$5$ $$T + 5$$
$7$ $$T$$
$11$ $$T - 52$$
$13$ $$T + 22$$
$17$ $$T - 14$$
$19$ $$T - 20$$
$23$ $$T + 168$$
$29$ $$T - 230$$
$31$ $$T - 288$$
$37$ $$T + 34$$
$41$ $$T + 122$$
$43$ $$T + 188$$
$47$ $$T + 256$$
$53$ $$T + 338$$
$59$ $$T + 100$$
$61$ $$T + 742$$
$67$ $$T + 84$$
$71$ $$T + 328$$
$73$ $$T - 38$$
$79$ $$T + 240$$
$83$ $$T + 1212$$
$89$ $$T + 330$$
$97$ $$T + 866$$