# Properties

 Label 4205.2.a.f Level $4205$ Weight $2$ Character orbit 4205.a Self dual yes Analytic conductor $33.577$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4205,2,Mod(1,4205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4205 = 5 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4205.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: $$33.5770940499$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + \beta_1) q^{4} + q^{5} + ( - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} - 1) q^{8} + ( - 2 \beta_{2} + 1) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b2 + b1 - 1) * q^3 + (b2 + b1) * q^4 + q^5 + (-b2 - b1 - 1) * q^6 + (b2 + b1 + 1) * q^7 + (-b2 - 1) * q^8 + (-2*b2 + 1) * q^9 $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + \beta_1) q^{4} + q^{5} + ( - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} - 1) q^{8} + ( - 2 \beta_{2} + 1) q^{9} - \beta_1 q^{10} + ( - \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{2} + \beta_1 + 3) q^{12} - 2 \beta_1 q^{13} + ( - \beta_{2} - 3 \beta_1 - 1) q^{14} + (\beta_{2} + \beta_1 - 1) q^{15} + ( - 2 \beta_{2} - 1) q^{16} + ( - 3 \beta_{2} + \beta_1 + 1) q^{17} + (\beta_1 - 2) q^{18} + (\beta_{2} + \beta_1 + 3) q^{19} + (\beta_{2} + \beta_1) q^{20} + (2 \beta_1 + 2) q^{21} + ( - \beta_{2} + \beta_1 - 3) q^{22} + ( - \beta_{2} + \beta_1 + 5) q^{23} + (\beta_{2} - \beta_1 - 1) q^{24} + q^{25} + (2 \beta_{2} + 2 \beta_1 + 4) q^{26} + (2 \beta_{2} - 2 \beta_1 - 2) q^{27} + (\beta_{2} + 3 \beta_1 + 3) q^{28} + ( - \beta_{2} - \beta_1 - 1) q^{30} + ( - \beta_{2} - \beta_1 + 5) q^{31} + (2 \beta_{2} + 3 \beta_1) q^{32} + 2 \beta_{2} q^{33} + ( - \beta_{2} + \beta_1 - 5) q^{34} + (\beta_{2} + \beta_1 + 1) q^{35} + (3 \beta_{2} + \beta_1 - 4) q^{36} + (3 \beta_{2} - \beta_1 + 3) q^{37} + ( - \beta_{2} - 5 \beta_1 - 1) q^{38} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{39} + ( - \beta_{2} - 1) q^{40} + (2 \beta_{2} - 4 \beta_1 + 2) q^{41} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{42} + ( - 5 \beta_{2} - 5 \beta_1 + 1) q^{43} + (\beta_{2} + \beta_1 - 1) q^{44} + ( - 2 \beta_{2} + 1) q^{45} + ( - \beta_{2} - 5 \beta_1 - 3) q^{46} + (\beta_{2} + \beta_1 - 5) q^{47} + (3 \beta_{2} - \beta_1 - 3) q^{48} + (2 \beta_{2} + 4 \beta_1 - 3) q^{49} - \beta_1 q^{50} + (8 \beta_{2} + 2 \beta_1 - 6) q^{51} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{52} + ( - 2 \beta_{2} + 2) q^{53} + (2 \beta_{2} + 2 \beta_1 + 6) q^{54} + ( - \beta_{2} + \beta_1 - 1) q^{55} + ( - \beta_{2} - \beta_1 - 3) q^{56} + (2 \beta_{2} + 4 \beta_1) q^{57} + (2 \beta_{2} - 4 \beta_1 + 4) q^{59} + ( - \beta_{2} + \beta_1 + 3) q^{60} + 6 \beta_1 q^{61} + (\beta_{2} - 3 \beta_1 + 1) q^{62} + (\beta_{2} + \beta_1 - 3) q^{63} + (\beta_{2} - 5 \beta_1 - 2) q^{64} - 2 \beta_1 q^{65} + ( - 2 \beta_1 + 2) q^{66} + ( - \beta_{2} + \beta_1 + 9) q^{67} + (5 \beta_{2} + 3 \beta_1 - 5) q^{68} + (8 \beta_{2} + 6 \beta_1 - 6) q^{69} + ( - \beta_{2} - 3 \beta_1 - 1) q^{70} + ( - 2 \beta_{2} + 4 \beta_1 + 8) q^{71} + ( - \beta_{2} - 2 \beta_1 + 5) q^{72} + (7 \beta_{2} + \beta_1 + 5) q^{73} + (\beta_{2} - 5 \beta_1 + 5) q^{74} + (\beta_{2} + \beta_1 - 1) q^{75} + (3 \beta_{2} + 5 \beta_1 + 3) q^{76} + (2 \beta_1 - 2) q^{77} + (2 \beta_{2} + 6 \beta_1 + 2) q^{78} + (5 \beta_{2} + 3 \beta_1 + 1) q^{79} + ( - 2 \beta_{2} - 1) q^{80} + ( - 2 \beta_{2} - 4 \beta_1 + 1) q^{81} + (4 \beta_{2} + 10) q^{82} + ( - 3 \beta_{2} - 3 \beta_1 + 5) q^{83} + (4 \beta_{2} + 6 \beta_1 + 2) q^{84} + ( - 3 \beta_{2} + \beta_1 + 1) q^{85} + (5 \beta_{2} + 9 \beta_1 + 5) q^{86} + (\beta_{2} - 3 \beta_1 + 5) q^{88} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{89} + (\beta_1 - 2) q^{90} + ( - 2 \beta_{2} - 6 \beta_1 - 2) q^{91} + (7 \beta_{2} + 7 \beta_1 - 1) q^{92} + (6 \beta_{2} + 4 \beta_1 - 8) q^{93} + ( - \beta_{2} + 3 \beta_1 - 1) q^{94} + (\beta_{2} + \beta_1 + 3) q^{95} + ( - \beta_{2} + 3 \beta_1 + 7) q^{96} + ( - 3 \beta_{2} - 5 \beta_1 - 1) q^{97} + ( - 4 \beta_{2} - 3 \beta_1 - 6) q^{98} + ( - \beta_{2} - 3 \beta_1 + 7) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b2 + b1 - 1) * q^3 + (b2 + b1) * q^4 + q^5 + (-b2 - b1 - 1) * q^6 + (b2 + b1 + 1) * q^7 + (-b2 - 1) * q^8 + (-2*b2 + 1) * q^9 - b1 * q^10 + (-b2 + b1 - 1) * q^11 + (-b2 + b1 + 3) * q^12 - 2*b1 * q^13 + (-b2 - 3*b1 - 1) * q^14 + (b2 + b1 - 1) * q^15 + (-2*b2 - 1) * q^16 + (-3*b2 + b1 + 1) * q^17 + (b1 - 2) * q^18 + (b2 + b1 + 3) * q^19 + (b2 + b1) * q^20 + (2*b1 + 2) * q^21 + (-b2 + b1 - 3) * q^22 + (-b2 + b1 + 5) * q^23 + (b2 - b1 - 1) * q^24 + q^25 + (2*b2 + 2*b1 + 4) * q^26 + (2*b2 - 2*b1 - 2) * q^27 + (b2 + 3*b1 + 3) * q^28 + (-b2 - b1 - 1) * q^30 + (-b2 - b1 + 5) * q^31 + (2*b2 + 3*b1) * q^32 + 2*b2 * q^33 + (-b2 + b1 - 5) * q^34 + (b2 + b1 + 1) * q^35 + (3*b2 + b1 - 4) * q^36 + (3*b2 - b1 + 3) * q^37 + (-b2 - 5*b1 - 1) * q^38 + (-2*b2 - 2*b1 - 2) * q^39 + (-b2 - 1) * q^40 + (2*b2 - 4*b1 + 2) * q^41 + (-2*b2 - 4*b1 - 4) * q^42 + (-5*b2 - 5*b1 + 1) * q^43 + (b2 + b1 - 1) * q^44 + (-2*b2 + 1) * q^45 + (-b2 - 5*b1 - 3) * q^46 + (b2 + b1 - 5) * q^47 + (3*b2 - b1 - 3) * q^48 + (2*b2 + 4*b1 - 3) * q^49 - b1 * q^50 + (8*b2 + 2*b1 - 6) * q^51 + (-2*b2 - 4*b1 - 2) * q^52 + (-2*b2 + 2) * q^53 + (2*b2 + 2*b1 + 6) * q^54 + (-b2 + b1 - 1) * q^55 + (-b2 - b1 - 3) * q^56 + (2*b2 + 4*b1) * q^57 + (2*b2 - 4*b1 + 4) * q^59 + (-b2 + b1 + 3) * q^60 + 6*b1 * q^61 + (b2 - 3*b1 + 1) * q^62 + (b2 + b1 - 3) * q^63 + (b2 - 5*b1 - 2) * q^64 - 2*b1 * q^65 + (-2*b1 + 2) * q^66 + (-b2 + b1 + 9) * q^67 + (5*b2 + 3*b1 - 5) * q^68 + (8*b2 + 6*b1 - 6) * q^69 + (-b2 - 3*b1 - 1) * q^70 + (-2*b2 + 4*b1 + 8) * q^71 + (-b2 - 2*b1 + 5) * q^72 + (7*b2 + b1 + 5) * q^73 + (b2 - 5*b1 + 5) * q^74 + (b2 + b1 - 1) * q^75 + (3*b2 + 5*b1 + 3) * q^76 + (2*b1 - 2) * q^77 + (2*b2 + 6*b1 + 2) * q^78 + (5*b2 + 3*b1 + 1) * q^79 + (-2*b2 - 1) * q^80 + (-2*b2 - 4*b1 + 1) * q^81 + (4*b2 + 10) * q^82 + (-3*b2 - 3*b1 + 5) * q^83 + (4*b2 + 6*b1 + 2) * q^84 + (-3*b2 + b1 + 1) * q^85 + (5*b2 + 9*b1 + 5) * q^86 + (b2 - 3*b1 + 5) * q^88 + (-2*b2 - 2*b1 + 4) * q^89 + (b1 - 2) * q^90 + (-2*b2 - 6*b1 - 2) * q^91 + (7*b2 + 7*b1 - 1) * q^92 + (6*b2 + 4*b1 - 8) * q^93 + (-b2 + 3*b1 - 1) * q^94 + (b2 + b1 + 3) * q^95 + (-b2 + 3*b1 + 7) * q^96 + (-3*b2 - 5*b1 - 1) * q^97 + (-4*b2 - 3*b1 - 6) * q^98 + (-b2 - 3*b1 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - 2 q^{3} + q^{4} + 3 q^{5} - 4 q^{6} + 4 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - q^2 - 2 * q^3 + q^4 + 3 * q^5 - 4 * q^6 + 4 * q^7 - 3 * q^8 + 3 * q^9 $$3 q - q^{2} - 2 q^{3} + q^{4} + 3 q^{5} - 4 q^{6} + 4 q^{7} - 3 q^{8} + 3 q^{9} - q^{10} - 2 q^{11} + 10 q^{12} - 2 q^{13} - 6 q^{14} - 2 q^{15} - 3 q^{16} + 4 q^{17} - 5 q^{18} + 10 q^{19} + q^{20} + 8 q^{21} - 8 q^{22} + 16 q^{23} - 4 q^{24} + 3 q^{25} + 14 q^{26} - 8 q^{27} + 12 q^{28} - 4 q^{30} + 14 q^{31} + 3 q^{32} - 14 q^{34} + 4 q^{35} - 11 q^{36} + 8 q^{37} - 8 q^{38} - 8 q^{39} - 3 q^{40} + 2 q^{41} - 16 q^{42} - 2 q^{43} - 2 q^{44} + 3 q^{45} - 14 q^{46} - 14 q^{47} - 10 q^{48} - 5 q^{49} - q^{50} - 16 q^{51} - 10 q^{52} + 6 q^{53} + 20 q^{54} - 2 q^{55} - 10 q^{56} + 4 q^{57} + 8 q^{59} + 10 q^{60} + 6 q^{61} - 8 q^{63} - 11 q^{64} - 2 q^{65} + 4 q^{66} + 28 q^{67} - 12 q^{68} - 12 q^{69} - 6 q^{70} + 28 q^{71} + 13 q^{72} + 16 q^{73} + 10 q^{74} - 2 q^{75} + 14 q^{76} - 4 q^{77} + 12 q^{78} + 6 q^{79} - 3 q^{80} - q^{81} + 30 q^{82} + 12 q^{83} + 12 q^{84} + 4 q^{85} + 24 q^{86} + 12 q^{88} + 10 q^{89} - 5 q^{90} - 12 q^{91} + 4 q^{92} - 20 q^{93} + 10 q^{95} + 24 q^{96} - 8 q^{97} - 21 q^{98} + 18 q^{99}+O(q^{100})$$ 3 * q - q^2 - 2 * q^3 + q^4 + 3 * q^5 - 4 * q^6 + 4 * q^7 - 3 * q^8 + 3 * q^9 - q^10 - 2 * q^11 + 10 * q^12 - 2 * q^13 - 6 * q^14 - 2 * q^15 - 3 * q^16 + 4 * q^17 - 5 * q^18 + 10 * q^19 + q^20 + 8 * q^21 - 8 * q^22 + 16 * q^23 - 4 * q^24 + 3 * q^25 + 14 * q^26 - 8 * q^27 + 12 * q^28 - 4 * q^30 + 14 * q^31 + 3 * q^32 - 14 * q^34 + 4 * q^35 - 11 * q^36 + 8 * q^37 - 8 * q^38 - 8 * q^39 - 3 * q^40 + 2 * q^41 - 16 * q^42 - 2 * q^43 - 2 * q^44 + 3 * q^45 - 14 * q^46 - 14 * q^47 - 10 * q^48 - 5 * q^49 - q^50 - 16 * q^51 - 10 * q^52 + 6 * q^53 + 20 * q^54 - 2 * q^55 - 10 * q^56 + 4 * q^57 + 8 * q^59 + 10 * q^60 + 6 * q^61 - 8 * q^63 - 11 * q^64 - 2 * q^65 + 4 * q^66 + 28 * q^67 - 12 * q^68 - 12 * q^69 - 6 * q^70 + 28 * q^71 + 13 * q^72 + 16 * q^73 + 10 * q^74 - 2 * q^75 + 14 * q^76 - 4 * q^77 + 12 * q^78 + 6 * q^79 - 3 * q^80 - q^81 + 30 * q^82 + 12 * q^83 + 12 * q^84 + 4 * q^85 + 24 * q^86 + 12 * q^88 + 10 * q^89 - 5 * q^90 - 12 * q^91 + 4 * q^92 - 20 * q^93 + 10 * q^95 + 24 * q^96 - 8 * q^97 - 21 * q^98 + 18 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2

## 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
 2.17009 0.311108 −1.48119
−2.17009 1.70928 2.70928 1.00000 −3.70928 3.70928 −1.53919 −0.0783777 −2.17009
1.2 −0.311108 −2.90321 −1.90321 1.00000 0.903212 −0.903212 1.21432 5.42864 −0.311108
1.3 1.48119 −0.806063 0.193937 1.00000 −1.19394 1.19394 −2.67513 −2.35026 1.48119
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$29$$ $$1$$

## 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 4205.2.a.f 3
29.b even 2 1 145.2.a.c 3
87.d odd 2 1 1305.2.a.p 3
116.d odd 2 1 2320.2.a.n 3
145.d even 2 1 725.2.a.e 3
145.h odd 4 2 725.2.b.e 6
203.c odd 2 1 7105.2.a.o 3
232.b odd 2 1 9280.2.a.br 3
232.g even 2 1 9280.2.a.bj 3
435.b odd 2 1 6525.2.a.be 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.c 3 29.b even 2 1
725.2.a.e 3 145.d even 2 1
725.2.b.e 6 145.h odd 4 2
1305.2.a.p 3 87.d odd 2 1
2320.2.a.n 3 116.d odd 2 1
4205.2.a.f 3 1.a even 1 1 trivial
6525.2.a.be 3 435.b odd 2 1
7105.2.a.o 3 203.c odd 2 1
9280.2.a.bj 3 232.g even 2 1
9280.2.a.br 3 232.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4205))$$:

 $$T_{2}^{3} + T_{2}^{2} - 3T_{2} - 1$$ T2^3 + T2^2 - 3*T2 - 1 $$T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 4$$ T3^3 + 2*T3^2 - 4*T3 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 3T - 1$$
$3$ $$T^{3} + 2 T^{2} + \cdots - 4$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 4T^{2} + 4$$
$11$ $$T^{3} + 2 T^{2} + \cdots + 4$$
$13$ $$T^{3} + 2 T^{2} + \cdots - 8$$
$17$ $$T^{3} - 4 T^{2} + \cdots + 68$$
$19$ $$T^{3} - 10 T^{2} + \cdots - 20$$
$23$ $$T^{3} - 16 T^{2} + \cdots - 92$$
$29$ $$T^{3}$$
$31$ $$T^{3} - 14 T^{2} + \cdots - 76$$
$37$ $$T^{3} - 8 T^{2} + \cdots + 92$$
$41$ $$T^{3} - 2 T^{2} + \cdots - 232$$
$43$ $$T^{3} + 2 T^{2} + \cdots + 4$$
$47$ $$T^{3} + 14 T^{2} + \cdots + 76$$
$53$ $$T^{3} - 6 T^{2} + \cdots + 8$$
$59$ $$T^{3} - 8 T^{2} + \cdots - 80$$
$61$ $$T^{3} - 6 T^{2} + \cdots + 216$$
$67$ $$T^{3} - 28 T^{2} + \cdots - 716$$
$71$ $$T^{3} - 28 T^{2} + \cdots + 272$$
$73$ $$T^{3} - 16 T^{2} + \cdots + 1700$$
$79$ $$T^{3} - 6 T^{2} + \cdots + 460$$
$83$ $$T^{3} - 12T^{2} + 148$$
$89$ $$T^{3} - 10 T^{2} + \cdots + 40$$
$97$ $$T^{3} + 8 T^{2} + \cdots + 76$$