# Properties

 Label 435.2.a.f Level $435$ Weight $2$ Character orbit 435.a Self dual yes Analytic conductor $3.473$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + q^{3} + (\beta + 3) q^{4} + q^{5} - \beta q^{6} + q^{7} + ( - 2 \beta - 5) q^{8} + q^{9} +O(q^{10})$$ q - b * q^2 + q^3 + (b + 3) * q^4 + q^5 - b * q^6 + q^7 + (-2*b - 5) * q^8 + q^9 $$q - \beta q^{2} + q^{3} + (\beta + 3) q^{4} + q^{5} - \beta q^{6} + q^{7} + ( - 2 \beta - 5) q^{8} + q^{9} - \beta q^{10} + 5 q^{11} + (\beta + 3) q^{12} + (2 \beta - 1) q^{13} - \beta q^{14} + q^{15} + (5 \beta + 4) q^{16} - 3 q^{17} - \beta q^{18} - 2 \beta q^{19} + (\beta + 3) q^{20} + q^{21} - 5 \beta q^{22} - 4 q^{23} + ( - 2 \beta - 5) q^{24} + q^{25} + ( - \beta - 10) q^{26} + q^{27} + (\beta + 3) q^{28} + q^{29} - \beta q^{30} + 4 q^{31} + ( - 5 \beta - 15) q^{32} + 5 q^{33} + 3 \beta q^{34} + q^{35} + (\beta + 3) q^{36} - 4 q^{37} + (2 \beta + 10) q^{38} + (2 \beta - 1) q^{39} + ( - 2 \beta - 5) q^{40} + (4 \beta - 2) q^{41} - \beta q^{42} + (2 \beta - 6) q^{43} + (5 \beta + 15) q^{44} + q^{45} + 4 \beta q^{46} + ( - 2 \beta + 7) q^{47} + (5 \beta + 4) q^{48} - 6 q^{49} - \beta q^{50} - 3 q^{51} + (7 \beta + 7) q^{52} + (2 \beta + 4) q^{53} - \beta q^{54} + 5 q^{55} + ( - 2 \beta - 5) q^{56} - 2 \beta q^{57} - \beta q^{58} + (2 \beta - 4) q^{59} + (\beta + 3) q^{60} + ( - 6 \beta + 2) q^{61} - 4 \beta q^{62} + q^{63} + (10 \beta + 17) q^{64} + (2 \beta - 1) q^{65} - 5 \beta q^{66} + (4 \beta + 3) q^{67} + ( - 3 \beta - 9) q^{68} - 4 q^{69} - \beta q^{70} + (2 \beta - 6) q^{71} + ( - 2 \beta - 5) q^{72} + 4 q^{73} + 4 \beta q^{74} + q^{75} + ( - 8 \beta - 10) q^{76} + 5 q^{77} + ( - \beta - 10) q^{78} + ( - 2 \beta + 4) q^{79} + (5 \beta + 4) q^{80} + q^{81} + ( - 2 \beta - 20) q^{82} + (2 \beta - 8) q^{83} + (\beta + 3) q^{84} - 3 q^{85} + (4 \beta - 10) q^{86} + q^{87} + ( - 10 \beta - 25) q^{88} + (2 \beta + 5) q^{89} - \beta q^{90} + (2 \beta - 1) q^{91} + ( - 4 \beta - 12) q^{92} + 4 q^{93} + ( - 5 \beta + 10) q^{94} - 2 \beta q^{95} + ( - 5 \beta - 15) q^{96} + ( - 2 \beta + 8) q^{97} + 6 \beta q^{98} + 5 q^{99} +O(q^{100})$$ q - b * q^2 + q^3 + (b + 3) * q^4 + q^5 - b * q^6 + q^7 + (-2*b - 5) * q^8 + q^9 - b * q^10 + 5 * q^11 + (b + 3) * q^12 + (2*b - 1) * q^13 - b * q^14 + q^15 + (5*b + 4) * q^16 - 3 * q^17 - b * q^18 - 2*b * q^19 + (b + 3) * q^20 + q^21 - 5*b * q^22 - 4 * q^23 + (-2*b - 5) * q^24 + q^25 + (-b - 10) * q^26 + q^27 + (b + 3) * q^28 + q^29 - b * q^30 + 4 * q^31 + (-5*b - 15) * q^32 + 5 * q^33 + 3*b * q^34 + q^35 + (b + 3) * q^36 - 4 * q^37 + (2*b + 10) * q^38 + (2*b - 1) * q^39 + (-2*b - 5) * q^40 + (4*b - 2) * q^41 - b * q^42 + (2*b - 6) * q^43 + (5*b + 15) * q^44 + q^45 + 4*b * q^46 + (-2*b + 7) * q^47 + (5*b + 4) * q^48 - 6 * q^49 - b * q^50 - 3 * q^51 + (7*b + 7) * q^52 + (2*b + 4) * q^53 - b * q^54 + 5 * q^55 + (-2*b - 5) * q^56 - 2*b * q^57 - b * q^58 + (2*b - 4) * q^59 + (b + 3) * q^60 + (-6*b + 2) * q^61 - 4*b * q^62 + q^63 + (10*b + 17) * q^64 + (2*b - 1) * q^65 - 5*b * q^66 + (4*b + 3) * q^67 + (-3*b - 9) * q^68 - 4 * q^69 - b * q^70 + (2*b - 6) * q^71 + (-2*b - 5) * q^72 + 4 * q^73 + 4*b * q^74 + q^75 + (-8*b - 10) * q^76 + 5 * q^77 + (-b - 10) * q^78 + (-2*b + 4) * q^79 + (5*b + 4) * q^80 + q^81 + (-2*b - 20) * q^82 + (2*b - 8) * q^83 + (b + 3) * q^84 - 3 * q^85 + (4*b - 10) * q^86 + q^87 + (-10*b - 25) * q^88 + (2*b + 5) * q^89 - b * q^90 + (2*b - 1) * q^91 + (-4*b - 12) * q^92 + 4 * q^93 + (-5*b + 10) * q^94 - 2*b * q^95 + (-5*b - 15) * q^96 + (-2*b + 8) * q^97 + 6*b * q^98 + 5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 2 q^{3} + 7 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - q^2 + 2 * q^3 + 7 * q^4 + 2 * q^5 - q^6 + 2 * q^7 - 12 * q^8 + 2 * q^9 $$2 q - q^{2} + 2 q^{3} + 7 q^{4} + 2 q^{5} - q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9} - q^{10} + 10 q^{11} + 7 q^{12} - q^{14} + 2 q^{15} + 13 q^{16} - 6 q^{17} - q^{18} - 2 q^{19} + 7 q^{20} + 2 q^{21} - 5 q^{22} - 8 q^{23} - 12 q^{24} + 2 q^{25} - 21 q^{26} + 2 q^{27} + 7 q^{28} + 2 q^{29} - q^{30} + 8 q^{31} - 35 q^{32} + 10 q^{33} + 3 q^{34} + 2 q^{35} + 7 q^{36} - 8 q^{37} + 22 q^{38} - 12 q^{40} - q^{42} - 10 q^{43} + 35 q^{44} + 2 q^{45} + 4 q^{46} + 12 q^{47} + 13 q^{48} - 12 q^{49} - q^{50} - 6 q^{51} + 21 q^{52} + 10 q^{53} - q^{54} + 10 q^{55} - 12 q^{56} - 2 q^{57} - q^{58} - 6 q^{59} + 7 q^{60} - 2 q^{61} - 4 q^{62} + 2 q^{63} + 44 q^{64} - 5 q^{66} + 10 q^{67} - 21 q^{68} - 8 q^{69} - q^{70} - 10 q^{71} - 12 q^{72} + 8 q^{73} + 4 q^{74} + 2 q^{75} - 28 q^{76} + 10 q^{77} - 21 q^{78} + 6 q^{79} + 13 q^{80} + 2 q^{81} - 42 q^{82} - 14 q^{83} + 7 q^{84} - 6 q^{85} - 16 q^{86} + 2 q^{87} - 60 q^{88} + 12 q^{89} - q^{90} - 28 q^{92} + 8 q^{93} + 15 q^{94} - 2 q^{95} - 35 q^{96} + 14 q^{97} + 6 q^{98} + 10 q^{99}+O(q^{100})$$ 2 * q - q^2 + 2 * q^3 + 7 * q^4 + 2 * q^5 - q^6 + 2 * q^7 - 12 * q^8 + 2 * q^9 - q^10 + 10 * q^11 + 7 * q^12 - q^14 + 2 * q^15 + 13 * q^16 - 6 * q^17 - q^18 - 2 * q^19 + 7 * q^20 + 2 * q^21 - 5 * q^22 - 8 * q^23 - 12 * q^24 + 2 * q^25 - 21 * q^26 + 2 * q^27 + 7 * q^28 + 2 * q^29 - q^30 + 8 * q^31 - 35 * q^32 + 10 * q^33 + 3 * q^34 + 2 * q^35 + 7 * q^36 - 8 * q^37 + 22 * q^38 - 12 * q^40 - q^42 - 10 * q^43 + 35 * q^44 + 2 * q^45 + 4 * q^46 + 12 * q^47 + 13 * q^48 - 12 * q^49 - q^50 - 6 * q^51 + 21 * q^52 + 10 * q^53 - q^54 + 10 * q^55 - 12 * q^56 - 2 * q^57 - q^58 - 6 * q^59 + 7 * q^60 - 2 * q^61 - 4 * q^62 + 2 * q^63 + 44 * q^64 - 5 * q^66 + 10 * q^67 - 21 * q^68 - 8 * q^69 - q^70 - 10 * q^71 - 12 * q^72 + 8 * q^73 + 4 * q^74 + 2 * q^75 - 28 * q^76 + 10 * q^77 - 21 * q^78 + 6 * q^79 + 13 * q^80 + 2 * q^81 - 42 * q^82 - 14 * q^83 + 7 * q^84 - 6 * q^85 - 16 * q^86 + 2 * q^87 - 60 * q^88 + 12 * q^89 - q^90 - 28 * q^92 + 8 * q^93 + 15 * q^94 - 2 * q^95 - 35 * q^96 + 14 * q^97 + 6 * q^98 + 10 * 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
 2.79129 −1.79129
−2.79129 1.00000 5.79129 1.00000 −2.79129 1.00000 −10.5826 1.00000 −2.79129
1.2 1.79129 1.00000 1.20871 1.00000 1.79129 1.00000 −1.41742 1.00000 1.79129
 $$n$$: 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.f 2
3.b odd 2 1 1305.2.a.m 2
4.b odd 2 1 6960.2.a.bw 2
5.b even 2 1 2175.2.a.r 2
5.c odd 4 2 2175.2.c.f 4
15.d odd 2 1 6525.2.a.t 2

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

 $$T_{2}^{2} + T_{2} - 5$$ T2^2 + T2 - 5 $$T_{7} - 1$$ T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 5$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2} - 21$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} + 2T - 20$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} - 84$$
$43$ $$T^{2} + 10T + 4$$
$47$ $$T^{2} - 12T + 15$$
$53$ $$T^{2} - 10T + 4$$
$59$ $$T^{2} + 6T - 12$$
$61$ $$T^{2} + 2T - 188$$
$67$ $$T^{2} - 10T - 59$$
$71$ $$T^{2} + 10T + 4$$
$73$ $$(T - 4)^{2}$$
$79$ $$T^{2} - 6T - 12$$
$83$ $$T^{2} + 14T + 28$$
$89$ $$T^{2} - 12T + 15$$
$97$ $$T^{2} - 14T + 28$$