# Properties

 Label 435.2.f.b Level $435$ Weight $2$ Character orbit 435.f Analytic conductor $3.473$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,2,Mod(289,435)]

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, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("435.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$435 = 3 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 435.f (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$3.47349248793$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} + (\beta - 1) q^{5} + q^{6} + \beta q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 - q^4 + (b - 1) * q^5 + q^6 + b * q^7 + 3 * q^8 + q^9 $$q - q^{2} - q^{3} - q^{4} + (\beta - 1) q^{5} + q^{6} + \beta q^{7} + 3 q^{8} + q^{9} + ( - \beta + 1) q^{10} - \beta q^{11} + q^{12} + 2 \beta q^{13} - \beta q^{14} + ( - \beta + 1) q^{15} - q^{16} - 6 q^{17} - q^{18} - \beta q^{19} + ( - \beta + 1) q^{20} - \beta q^{21} + \beta q^{22} - 3 \beta q^{23} - 3 q^{24} + ( - 2 \beta - 3) q^{25} - 2 \beta q^{26} - q^{27} - \beta q^{28} + ( - \beta - 5) q^{29} + (\beta - 1) q^{30} + \beta q^{31} - 5 q^{32} + \beta q^{33} + 6 q^{34} + ( - \beta - 4) q^{35} - q^{36} - 2 q^{37} + \beta q^{38} - 2 \beta q^{39} + (3 \beta - 3) q^{40} + \beta q^{42} + 4 q^{43} + \beta q^{44} + (\beta - 1) q^{45} + 3 \beta q^{46} - 8 q^{47} + q^{48} + 3 q^{49} + (2 \beta + 3) q^{50} + 6 q^{51} - 2 \beta q^{52} - 6 \beta q^{53} + q^{54} + (\beta + 4) q^{55} + 3 \beta q^{56} + \beta q^{57} + (\beta + 5) q^{58} - 4 q^{59} + (\beta - 1) q^{60} - 6 \beta q^{61} - \beta q^{62} + \beta q^{63} + 7 q^{64} + ( - 2 \beta - 8) q^{65} - \beta q^{66} + 3 \beta q^{67} + 6 q^{68} + 3 \beta q^{69} + (\beta + 4) q^{70} + 8 q^{71} + 3 q^{72} - 6 q^{73} + 2 q^{74} + (2 \beta + 3) q^{75} + \beta q^{76} + 4 q^{77} + 2 \beta q^{78} + \beta q^{79} + ( - \beta + 1) q^{80} + q^{81} - \beta q^{83} + \beta q^{84} + ( - 6 \beta + 6) q^{85} - 4 q^{86} + (\beta + 5) q^{87} - 3 \beta q^{88} + 8 \beta q^{89} + ( - \beta + 1) q^{90} - 8 q^{91} + 3 \beta q^{92} - \beta q^{93} + 8 q^{94} + (\beta + 4) q^{95} + 5 q^{96} - 14 q^{97} - 3 q^{98} - \beta q^{99} +O(q^{100})$$ q - q^2 - q^3 - q^4 + (b - 1) * q^5 + q^6 + b * q^7 + 3 * q^8 + q^9 + (-b + 1) * q^10 - b * q^11 + q^12 + 2*b * q^13 - b * q^14 + (-b + 1) * q^15 - q^16 - 6 * q^17 - q^18 - b * q^19 + (-b + 1) * q^20 - b * q^21 + b * q^22 - 3*b * q^23 - 3 * q^24 + (-2*b - 3) * q^25 - 2*b * q^26 - q^27 - b * q^28 + (-b - 5) * q^29 + (b - 1) * q^30 + b * q^31 - 5 * q^32 + b * q^33 + 6 * q^34 + (-b - 4) * q^35 - q^36 - 2 * q^37 + b * q^38 - 2*b * q^39 + (3*b - 3) * q^40 + b * q^42 + 4 * q^43 + b * q^44 + (b - 1) * q^45 + 3*b * q^46 - 8 * q^47 + q^48 + 3 * q^49 + (2*b + 3) * q^50 + 6 * q^51 - 2*b * q^52 - 6*b * q^53 + q^54 + (b + 4) * q^55 + 3*b * q^56 + b * q^57 + (b + 5) * q^58 - 4 * q^59 + (b - 1) * q^60 - 6*b * q^61 - b * q^62 + b * q^63 + 7 * q^64 + (-2*b - 8) * q^65 - b * q^66 + 3*b * q^67 + 6 * q^68 + 3*b * q^69 + (b + 4) * q^70 + 8 * q^71 + 3 * q^72 - 6 * q^73 + 2 * q^74 + (2*b + 3) * q^75 + b * q^76 + 4 * q^77 + 2*b * q^78 + b * q^79 + (-b + 1) * q^80 + q^81 - b * q^83 + b * q^84 + (-6*b + 6) * q^85 - 4 * q^86 + (b + 5) * q^87 - 3*b * q^88 + 8*b * q^89 + (-b + 1) * q^90 - 8 * q^91 + 3*b * q^92 - b * q^93 + 8 * q^94 + (b + 4) * q^95 + 5 * q^96 - 14 * q^97 - 3 * q^98 - b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 2 q^{15} - 2 q^{16} - 12 q^{17} - 2 q^{18} + 2 q^{20} - 6 q^{24} - 6 q^{25} - 2 q^{27} - 10 q^{29} - 2 q^{30} - 10 q^{32} + 12 q^{34} - 8 q^{35} - 2 q^{36} - 4 q^{37} - 6 q^{40} + 8 q^{43} - 2 q^{45} - 16 q^{47} + 2 q^{48} + 6 q^{49} + 6 q^{50} + 12 q^{51} + 2 q^{54} + 8 q^{55} + 10 q^{58} - 8 q^{59} - 2 q^{60} + 14 q^{64} - 16 q^{65} + 12 q^{68} + 8 q^{70} + 16 q^{71} + 6 q^{72} - 12 q^{73} + 4 q^{74} + 6 q^{75} + 8 q^{77} + 2 q^{80} + 2 q^{81} + 12 q^{85} - 8 q^{86} + 10 q^{87} + 2 q^{90} - 16 q^{91} + 16 q^{94} + 8 q^{95} + 10 q^{96} - 28 q^{97} - 6 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 2 * q^15 - 2 * q^16 - 12 * q^17 - 2 * q^18 + 2 * q^20 - 6 * q^24 - 6 * q^25 - 2 * q^27 - 10 * q^29 - 2 * q^30 - 10 * q^32 + 12 * q^34 - 8 * q^35 - 2 * q^36 - 4 * q^37 - 6 * q^40 + 8 * q^43 - 2 * q^45 - 16 * q^47 + 2 * q^48 + 6 * q^49 + 6 * q^50 + 12 * q^51 + 2 * q^54 + 8 * q^55 + 10 * q^58 - 8 * q^59 - 2 * q^60 + 14 * q^64 - 16 * q^65 + 12 * q^68 + 8 * q^70 + 16 * q^71 + 6 * q^72 - 12 * q^73 + 4 * q^74 + 6 * q^75 + 8 * q^77 + 2 * q^80 + 2 * q^81 + 12 * q^85 - 8 * q^86 + 10 * q^87 + 2 * q^90 - 16 * q^91 + 16 * q^94 + 8 * q^95 + 10 * q^96 - 28 * q^97 - 6 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/435\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$146$$ $$262$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 − 1.00000i 1.00000i
−1.00000 −1.00000 −1.00000 −1.00000 2.00000i 1.00000 2.00000i 3.00000 1.00000 1.00000 + 2.00000i
289.2 −1.00000 −1.00000 −1.00000 −1.00000 + 2.00000i 1.00000 2.00000i 3.00000 1.00000 1.00000 2.00000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 435.2.f.b 2
3.b odd 2 1 1305.2.f.c 2
5.b even 2 1 435.2.f.c yes 2
5.c odd 4 1 2175.2.d.c 2
5.c odd 4 1 2175.2.d.d 2
15.d odd 2 1 1305.2.f.b 2
29.b even 2 1 435.2.f.c yes 2
87.d odd 2 1 1305.2.f.b 2
145.d even 2 1 inner 435.2.f.b 2
145.h odd 4 1 2175.2.d.c 2
145.h odd 4 1 2175.2.d.d 2
435.b odd 2 1 1305.2.f.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.f.b 2 1.a even 1 1 trivial
435.2.f.b 2 145.d even 2 1 inner
435.2.f.c yes 2 5.b even 2 1
435.2.f.c yes 2 29.b even 2 1
1305.2.f.b 2 15.d odd 2 1
1305.2.f.b 2 87.d odd 2 1
1305.2.f.c 2 3.b odd 2 1
1305.2.f.c 2 435.b odd 2 1
2175.2.d.c 2 5.c odd 4 1
2175.2.d.c 2 145.h odd 4 1
2175.2.d.d 2 5.c odd 4 1
2175.2.d.d 2 145.h odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(435, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2} + 16$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$T^{2} + 36$$
$29$ $$T^{2} + 10T + 29$$
$31$ $$T^{2} + 4$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 144$$
$59$ $$(T + 4)^{2}$$
$61$ $$T^{2} + 144$$
$67$ $$T^{2} + 36$$
$71$ $$(T - 8)^{2}$$
$73$ $$(T + 6)^{2}$$
$79$ $$T^{2} + 4$$
$83$ $$T^{2} + 4$$
$89$ $$T^{2} + 256$$
$97$ $$(T + 14)^{2}$$