# Properties

 Label 435.4.a.a Level $435$ Weight $4$ Character orbit 435.a Self dual yes Analytic conductor $25.666$ Analytic rank $0$ 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] = [435,4,Mod(1,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, 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("435.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$435 = 3 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 435.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: $$25.6658308525$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 3 q^{3} - 4 q^{4} + 5 q^{5} + 6 q^{6} + 29 q^{7} + 24 q^{8} + 9 q^{9}+O(q^{10})$$ q - 2 * q^2 - 3 * q^3 - 4 * q^4 + 5 * q^5 + 6 * q^6 + 29 * q^7 + 24 * q^8 + 9 * q^9 $$q - 2 q^{2} - 3 q^{3} - 4 q^{4} + 5 q^{5} + 6 q^{6} + 29 q^{7} + 24 q^{8} + 9 q^{9} - 10 q^{10} - 15 q^{11} + 12 q^{12} + 3 q^{13} - 58 q^{14} - 15 q^{15} - 16 q^{16} + 121 q^{17} - 18 q^{18} - 40 q^{19} - 20 q^{20} - 87 q^{21} + 30 q^{22} - 116 q^{23} - 72 q^{24} + 25 q^{25} - 6 q^{26} - 27 q^{27} - 116 q^{28} + 29 q^{29} + 30 q^{30} - 116 q^{31} - 160 q^{32} + 45 q^{33} - 242 q^{34} + 145 q^{35} - 36 q^{36} + 36 q^{37} + 80 q^{38} - 9 q^{39} + 120 q^{40} - 170 q^{41} + 174 q^{42} + 230 q^{43} + 60 q^{44} + 45 q^{45} + 232 q^{46} + 231 q^{47} + 48 q^{48} + 498 q^{49} - 50 q^{50} - 363 q^{51} - 12 q^{52} + 456 q^{53} + 54 q^{54} - 75 q^{55} + 696 q^{56} + 120 q^{57} - 58 q^{58} + 576 q^{59} + 60 q^{60} + 342 q^{61} + 232 q^{62} + 261 q^{63} + 448 q^{64} + 15 q^{65} - 90 q^{66} - 269 q^{67} - 484 q^{68} + 348 q^{69} - 290 q^{70} + 302 q^{71} + 216 q^{72} - 372 q^{73} - 72 q^{74} - 75 q^{75} + 160 q^{76} - 435 q^{77} + 18 q^{78} - 348 q^{79} - 80 q^{80} + 81 q^{81} + 340 q^{82} - 512 q^{83} + 348 q^{84} + 605 q^{85} - 460 q^{86} - 87 q^{87} - 360 q^{88} + 1525 q^{89} - 90 q^{90} + 87 q^{91} + 464 q^{92} + 348 q^{93} - 462 q^{94} - 200 q^{95} + 480 q^{96} - 560 q^{97} - 996 q^{98} - 135 q^{99}+O(q^{100})$$ q - 2 * q^2 - 3 * q^3 - 4 * q^4 + 5 * q^5 + 6 * q^6 + 29 * q^7 + 24 * q^8 + 9 * q^9 - 10 * q^10 - 15 * q^11 + 12 * q^12 + 3 * q^13 - 58 * q^14 - 15 * q^15 - 16 * q^16 + 121 * q^17 - 18 * q^18 - 40 * q^19 - 20 * q^20 - 87 * q^21 + 30 * q^22 - 116 * q^23 - 72 * q^24 + 25 * q^25 - 6 * q^26 - 27 * q^27 - 116 * q^28 + 29 * q^29 + 30 * q^30 - 116 * q^31 - 160 * q^32 + 45 * q^33 - 242 * q^34 + 145 * q^35 - 36 * q^36 + 36 * q^37 + 80 * q^38 - 9 * q^39 + 120 * q^40 - 170 * q^41 + 174 * q^42 + 230 * q^43 + 60 * q^44 + 45 * q^45 + 232 * q^46 + 231 * q^47 + 48 * q^48 + 498 * q^49 - 50 * q^50 - 363 * q^51 - 12 * q^52 + 456 * q^53 + 54 * q^54 - 75 * q^55 + 696 * q^56 + 120 * q^57 - 58 * q^58 + 576 * q^59 + 60 * q^60 + 342 * q^61 + 232 * q^62 + 261 * q^63 + 448 * q^64 + 15 * q^65 - 90 * q^66 - 269 * q^67 - 484 * q^68 + 348 * q^69 - 290 * q^70 + 302 * q^71 + 216 * q^72 - 372 * q^73 - 72 * q^74 - 75 * q^75 + 160 * q^76 - 435 * q^77 + 18 * q^78 - 348 * q^79 - 80 * q^80 + 81 * q^81 + 340 * q^82 - 512 * q^83 + 348 * q^84 + 605 * q^85 - 460 * q^86 - 87 * q^87 - 360 * q^88 + 1525 * q^89 - 90 * q^90 + 87 * q^91 + 464 * q^92 + 348 * q^93 - 462 * q^94 - 200 * q^95 + 480 * q^96 - 560 * q^97 - 996 * q^98 - 135 * 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
−2.00000 −3.00000 −4.00000 5.00000 6.00000 29.0000 24.0000 9.00000 −10.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$5$$ $$-1$$
$$29$$ $$-1$$

## 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 435.4.a.a 1
3.b odd 2 1 1305.4.a.d 1
5.b even 2 1 2175.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.4.a.a 1 1.a even 1 1 trivial
1305.4.a.d 1 3.b odd 2 1
2175.4.a.c 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(435))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 3$$
$5$ $$T - 5$$
$7$ $$T - 29$$
$11$ $$T + 15$$
$13$ $$T - 3$$
$17$ $$T - 121$$
$19$ $$T + 40$$
$23$ $$T + 116$$
$29$ $$T - 29$$
$31$ $$T + 116$$
$37$ $$T - 36$$
$41$ $$T + 170$$
$43$ $$T - 230$$
$47$ $$T - 231$$
$53$ $$T - 456$$
$59$ $$T - 576$$
$61$ $$T - 342$$
$67$ $$T + 269$$
$71$ $$T - 302$$
$73$ $$T + 372$$
$79$ $$T + 348$$
$83$ $$T + 512$$
$89$ $$T - 1525$$
$97$ $$T + 560$$