Properties

 Label 435.2.a.a Level $435$ Weight $2$ Character orbit 435.a Self dual yes Analytic conductor $3.473$ 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] = [435,2,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, 2, names="a")

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

chi := DirichletCharacter("435.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$435 = 3 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$3.47349248793$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 - q^4 + q^5 - q^6 - 4 * q^7 + 3 * q^8 + q^9 $$q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - 4 q^{7} + 3 q^{8} + q^{9} - q^{10} - q^{12} + 6 q^{13} + 4 q^{14} + q^{15} - q^{16} + 2 q^{17} - q^{18} + 8 q^{19} - q^{20} - 4 q^{21} - 4 q^{23} + 3 q^{24} + q^{25} - 6 q^{26} + q^{27} + 4 q^{28} + q^{29} - q^{30} + 4 q^{31} - 5 q^{32} - 2 q^{34} - 4 q^{35} - q^{36} + 6 q^{37} - 8 q^{38} + 6 q^{39} + 3 q^{40} + 2 q^{41} + 4 q^{42} - 4 q^{43} + q^{45} + 4 q^{46} - q^{48} + 9 q^{49} - q^{50} + 2 q^{51} - 6 q^{52} + 6 q^{53} - q^{54} - 12 q^{56} + 8 q^{57} - q^{58} - 12 q^{59} - q^{60} + 6 q^{61} - 4 q^{62} - 4 q^{63} + 7 q^{64} + 6 q^{65} - 8 q^{67} - 2 q^{68} - 4 q^{69} + 4 q^{70} + 16 q^{71} + 3 q^{72} - 6 q^{73} - 6 q^{74} + q^{75} - 8 q^{76} - 6 q^{78} + 12 q^{79} - q^{80} + q^{81} - 2 q^{82} - 16 q^{83} + 4 q^{84} + 2 q^{85} + 4 q^{86} + q^{87} + 2 q^{89} - q^{90} - 24 q^{91} + 4 q^{92} + 4 q^{93} + 8 q^{95} - 5 q^{96} - 14 q^{97} - 9 q^{98}+O(q^{100})$$ q - q^2 + q^3 - q^4 + q^5 - q^6 - 4 * q^7 + 3 * q^8 + q^9 - q^10 - q^12 + 6 * q^13 + 4 * q^14 + q^15 - q^16 + 2 * q^17 - q^18 + 8 * q^19 - q^20 - 4 * q^21 - 4 * q^23 + 3 * q^24 + q^25 - 6 * q^26 + q^27 + 4 * q^28 + q^29 - q^30 + 4 * q^31 - 5 * q^32 - 2 * q^34 - 4 * q^35 - q^36 + 6 * q^37 - 8 * q^38 + 6 * q^39 + 3 * q^40 + 2 * q^41 + 4 * q^42 - 4 * q^43 + q^45 + 4 * q^46 - q^48 + 9 * q^49 - q^50 + 2 * q^51 - 6 * q^52 + 6 * q^53 - q^54 - 12 * q^56 + 8 * q^57 - q^58 - 12 * q^59 - q^60 + 6 * q^61 - 4 * q^62 - 4 * q^63 + 7 * q^64 + 6 * q^65 - 8 * q^67 - 2 * q^68 - 4 * q^69 + 4 * q^70 + 16 * q^71 + 3 * q^72 - 6 * q^73 - 6 * q^74 + q^75 - 8 * q^76 - 6 * q^78 + 12 * q^79 - q^80 + q^81 - 2 * q^82 - 16 * q^83 + 4 * q^84 + 2 * q^85 + 4 * q^86 + q^87 + 2 * q^89 - q^90 - 24 * q^91 + 4 * q^92 + 4 * q^93 + 8 * q^95 - 5 * q^96 - 14 * q^97 - 9 * q^98

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 1.00000 −1.00000 1.00000 −1.00000 −4.00000 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$29$$ $$-1$$

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 435.2.a.a 1
3.b odd 2 1 1305.2.a.e 1
4.b odd 2 1 6960.2.a.w 1
5.b even 2 1 2175.2.a.h 1
5.c odd 4 2 2175.2.c.a 2
15.d odd 2 1 6525.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.a 1 1.a even 1 1 trivial
1305.2.a.e 1 3.b odd 2 1
2175.2.a.h 1 5.b even 2 1
2175.2.c.a 2 5.c odd 4 2
6525.2.a.e 1 15.d odd 2 1
6960.2.a.w 1 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(435))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{7} + 4$$ T7 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T + 4$$
$11$ $$T$$
$13$ $$T - 6$$
$17$ $$T - 2$$
$19$ $$T - 8$$
$23$ $$T + 4$$
$29$ $$T - 1$$
$31$ $$T - 4$$
$37$ $$T - 6$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T + 12$$
$61$ $$T - 6$$
$67$ $$T + 8$$
$71$ $$T - 16$$
$73$ $$T + 6$$
$79$ $$T - 12$$
$83$ $$T + 16$$
$89$ $$T - 2$$
$97$ $$T + 14$$