# Properties

 Label 4205.2.a.m Level $4205$ Weight $2$ Character orbit 4205.a Self dual yes Analytic conductor $33.577$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# 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: $$6$$ Coefficient field: 6.6.66064384.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 9x^{4} + 13x^{2} - 1$$ x^6 - 9*x^4 + 13*x^2 - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{5} + \beta_1) q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{2} + 2) q^{6} + \beta_{2} q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_1) q^{8} + (\beta_{3} + 2) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b5 + b1) * q^3 + (b2 + 1) * q^4 - q^5 + (b2 + 2) * q^6 + b2 * q^7 + (-b5 + b4 + 2*b1) * q^8 + (b3 + 2) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{5} + \beta_1) q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{2} + 2) q^{6} + \beta_{2} q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_1) q^{8} + (\beta_{3} + 2) q^{9} - \beta_1 q^{10} + (\beta_{5} - \beta_1) q^{11} + (\beta_{5} + \beta_{4} + 3 \beta_1) q^{12} + ( - \beta_{3} + 1) q^{13} + ( - \beta_{5} + \beta_{4} + 3 \beta_1) q^{14} + (\beta_{5} - \beta_1) q^{15} + (\beta_{3} + 2 \beta_{2} + 4) q^{16} + ( - \beta_{5} + \beta_{4} - \beta_1) q^{17} + (\beta_{4} + \beta_1) q^{18} + (\beta_{5} - \beta_{4} - \beta_1) q^{19} + ( - \beta_{2} - 1) q^{20} + (2 \beta_{5} + \beta_{4} + 2 \beta_1) q^{21} + ( - \beta_{2} - 2) q^{22} + ( - \beta_{3} - \beta_{2} - 1) q^{23} + (\beta_{3} + 3 \beta_{2} + 7) q^{24} + q^{25} + ( - \beta_{4} + 2 \beta_1) q^{26} - 4 \beta_{5} q^{27} + (\beta_{3} + 3 \beta_{2} + 9) q^{28} + ( - \beta_{2} - 2) q^{30} + ( - 3 \beta_{5} + \beta_{4} - \beta_1) q^{31} + (\beta_{4} + 5 \beta_1) q^{32} + ( - \beta_{3} - 5) q^{33} + (\beta_{3} + \beta_{2} - 3) q^{34} - \beta_{2} q^{35} + ( - \beta_{3} + 3 \beta_{2}) q^{36} + ( - \beta_{5} - \beta_{4} + 3 \beta_1) q^{37} + ( - \beta_{3} - 3 \beta_{2} - 3) q^{38} + 4 \beta_{5} q^{39} + (\beta_{5} - \beta_{4} - 2 \beta_1) q^{40} + ( - 2 \beta_{5} - \beta_{4} - 2 \beta_1) q^{41} + (\beta_{3} + 4 \beta_{2} + 9) q^{42} + ( - \beta_{5} - 2 \beta_{4} + \beta_1) q^{43} + ( - \beta_{5} - \beta_{4} - 3 \beta_1) q^{44} + ( - \beta_{3} - 2) q^{45} + (\beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{46} + ( - \beta_{5} + \beta_1) q^{47} + ( - 5 \beta_{5} + 2 \beta_{4} + 9 \beta_1) q^{48} + (\beta_{3} + 2 \beta_{2} + 2) q^{49} + \beta_1 q^{50} + (\beta_{3} + 1) q^{51} + (\beta_{3} + 3) q^{52} + ( - \beta_{3} + 2 \beta_{2} + 5) q^{53} - 4 q^{54} + ( - \beta_{5} + \beta_1) q^{55} + ( - \beta_{5} + 2 \beta_{4} + 11 \beta_1) q^{56} + ( - \beta_{3} - 2 \beta_{2} - 5) q^{57} + (\beta_{3} - 3) q^{59} + ( - \beta_{5} - \beta_{4} - 3 \beta_1) q^{60} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_1) q^{61} + (\beta_{3} + \beta_{2} - 5) q^{62} + ( - 2 \beta_{3} + 3 \beta_{2} - 2) q^{63} + ( - \beta_{3} + 3 \beta_{2} + 8) q^{64} + (\beta_{3} - 1) q^{65} + ( - \beta_{4} - 4 \beta_1) q^{66} + ( - \beta_{3} - \beta_{2} + 3) q^{67} + (\beta_{5} + \beta_1) q^{68} + (4 \beta_{5} - \beta_{4} - 4 \beta_1) q^{69} + (\beta_{5} - \beta_{4} - 3 \beta_1) q^{70} + (\beta_{3} + 9) q^{71} + ( - 3 \beta_{5} + 8 \beta_1) q^{72} + ( - 3 \beta_{5} - 3 \beta_1) q^{73} + ( - \beta_{3} + \beta_{2} + 7) q^{74} + ( - \beta_{5} + \beta_1) q^{75} + (\beta_{5} - 2 \beta_{4} - 9 \beta_1) q^{76} + ( - 2 \beta_{5} - \beta_{4} - 2 \beta_1) q^{77} + 4 q^{78} + ( - \beta_{5} - 2 \beta_{4} + \beta_1) q^{79} + ( - \beta_{3} - 2 \beta_{2} - 4) q^{80} + (\beta_{3} - 4 \beta_{2} + 6) q^{81} + ( - \beta_{3} - 4 \beta_{2} - 9) q^{82} + ( - \beta_{2} - 4) q^{83} + ( - 8 \beta_{5} + 3 \beta_{4} + 16 \beta_1) q^{84} + (\beta_{5} - \beta_{4} + \beta_1) q^{85} + ( - 2 \beta_{3} - 3 \beta_{2}) q^{86} + ( - \beta_{3} - 3 \beta_{2} - 7) q^{88} - 2 \beta_{4} q^{89} + ( - \beta_{4} - \beta_1) q^{90} + (2 \beta_{3} + 2) q^{91} + ( - 5 \beta_{2} - 8) q^{92} + (3 \beta_{3} - 2 \beta_{2} + 7) q^{93} + (\beta_{2} + 2) q^{94} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{95} + (7 \beta_{2} + 10) q^{96} + (\beta_{5} + \beta_1) q^{97} + ( - 2 \beta_{5} + 3 \beta_{4} + 7 \beta_1) q^{98} + (7 \beta_{5} - 3 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b5 + b1) * q^3 + (b2 + 1) * q^4 - q^5 + (b2 + 2) * q^6 + b2 * q^7 + (-b5 + b4 + 2*b1) * q^8 + (b3 + 2) * q^9 - b1 * q^10 + (b5 - b1) * q^11 + (b5 + b4 + 3*b1) * q^12 + (-b3 + 1) * q^13 + (-b5 + b4 + 3*b1) * q^14 + (b5 - b1) * q^15 + (b3 + 2*b2 + 4) * q^16 + (-b5 + b4 - b1) * q^17 + (b4 + b1) * q^18 + (b5 - b4 - b1) * q^19 + (-b2 - 1) * q^20 + (2*b5 + b4 + 2*b1) * q^21 + (-b2 - 2) * q^22 + (-b3 - b2 - 1) * q^23 + (b3 + 3*b2 + 7) * q^24 + q^25 + (-b4 + 2*b1) * q^26 - 4*b5 * q^27 + (b3 + 3*b2 + 9) * q^28 + (-b2 - 2) * q^30 + (-3*b5 + b4 - b1) * q^31 + (b4 + 5*b1) * q^32 + (-b3 - 5) * q^33 + (b3 + b2 - 3) * q^34 - b2 * q^35 + (-b3 + 3*b2) * q^36 + (-b5 - b4 + 3*b1) * q^37 + (-b3 - 3*b2 - 3) * q^38 + 4*b5 * q^39 + (b5 - b4 - 2*b1) * q^40 + (-2*b5 - b4 - 2*b1) * q^41 + (b3 + 4*b2 + 9) * q^42 + (-b5 - 2*b4 + b1) * q^43 + (-b5 - b4 - 3*b1) * q^44 + (-b3 - 2) * q^45 + (b5 - 2*b4 - 3*b1) * q^46 + (-b5 + b1) * q^47 + (-5*b5 + 2*b4 + 9*b1) * q^48 + (b3 + 2*b2 + 2) * q^49 + b1 * q^50 + (b3 + 1) * q^51 + (b3 + 3) * q^52 + (-b3 + 2*b2 + 5) * q^53 - 4 * q^54 + (-b5 + b1) * q^55 + (-b5 + 2*b4 + 11*b1) * q^56 + (-b3 - 2*b2 - 5) * q^57 + (b3 - 3) * q^59 + (-b5 - b4 - 3*b1) * q^60 + (-2*b5 - b4 + 2*b1) * q^61 + (b3 + b2 - 5) * q^62 + (-2*b3 + 3*b2 - 2) * q^63 + (-b3 + 3*b2 + 8) * q^64 + (b3 - 1) * q^65 + (-b4 - 4*b1) * q^66 + (-b3 - b2 + 3) * q^67 + (b5 + b1) * q^68 + (4*b5 - b4 - 4*b1) * q^69 + (b5 - b4 - 3*b1) * q^70 + (b3 + 9) * q^71 + (-3*b5 + 8*b1) * q^72 + (-3*b5 - 3*b1) * q^73 + (-b3 + b2 + 7) * q^74 + (-b5 + b1) * q^75 + (b5 - 2*b4 - 9*b1) * q^76 + (-2*b5 - b4 - 2*b1) * q^77 + 4 * q^78 + (-b5 - 2*b4 + b1) * q^79 + (-b3 - 2*b2 - 4) * q^80 + (b3 - 4*b2 + 6) * q^81 + (-b3 - 4*b2 - 9) * q^82 + (-b2 - 4) * q^83 + (-8*b5 + 3*b4 + 16*b1) * q^84 + (b5 - b4 + b1) * q^85 + (-2*b3 - 3*b2) * q^86 + (-b3 - 3*b2 - 7) * q^88 - 2*b4 * q^89 + (-b4 - b1) * q^90 + (2*b3 + 2) * q^91 + (-5*b2 - 8) * q^92 + (3*b3 - 2*b2 + 7) * q^93 + (b2 + 2) * q^94 + (-b5 + b4 + b1) * q^95 + (7*b2 + 10) * q^96 + (b5 + b1) * q^97 + (-2*b5 + 3*b4 + 7*b1) * q^98 + (7*b5 - 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{4} - 6 q^{5} + 12 q^{6} + 14 q^{9}+O(q^{10})$$ 6 * q + 6 * q^4 - 6 * q^5 + 12 * q^6 + 14 * q^9 $$6 q + 6 q^{4} - 6 q^{5} + 12 q^{6} + 14 q^{9} + 4 q^{13} + 26 q^{16} - 6 q^{20} - 12 q^{22} - 8 q^{23} + 44 q^{24} + 6 q^{25} + 56 q^{28} - 12 q^{30} - 32 q^{33} - 16 q^{34} - 2 q^{36} - 20 q^{38} + 56 q^{42} - 14 q^{45} + 14 q^{49} + 8 q^{51} + 20 q^{52} + 28 q^{53} - 24 q^{54} - 32 q^{57} - 16 q^{59} - 28 q^{62} - 16 q^{63} + 46 q^{64} - 4 q^{65} + 16 q^{67} + 56 q^{71} + 40 q^{74} + 24 q^{78} - 26 q^{80} + 38 q^{81} - 56 q^{82} - 24 q^{83} - 4 q^{86} - 44 q^{88} + 16 q^{91} - 48 q^{92} + 48 q^{93} + 12 q^{94} + 60 q^{96}+O(q^{100})$$ 6 * q + 6 * q^4 - 6 * q^5 + 12 * q^6 + 14 * q^9 + 4 * q^13 + 26 * q^16 - 6 * q^20 - 12 * q^22 - 8 * q^23 + 44 * q^24 + 6 * q^25 + 56 * q^28 - 12 * q^30 - 32 * q^33 - 16 * q^34 - 2 * q^36 - 20 * q^38 + 56 * q^42 - 14 * q^45 + 14 * q^49 + 8 * q^51 + 20 * q^52 + 28 * q^53 - 24 * q^54 - 32 * q^57 - 16 * q^59 - 28 * q^62 - 16 * q^63 + 46 * q^64 - 4 * q^65 + 16 * q^67 + 56 * q^71 + 40 * q^74 + 24 * q^78 - 26 * q^80 + 38 * q^81 - 56 * q^82 - 24 * q^83 - 4 * q^86 - 44 * q^88 + 16 * q^91 - 48 * q^92 + 48 * q^93 + 12 * q^94 + 60 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 9x^{4} + 13x^{2} - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 8\nu^{2} + 6$$ v^4 - 8*v^2 + 6 $$\beta_{4}$$ $$=$$ $$\nu^{5} - 8\nu^{3} + 7\nu$$ v^5 - 8*v^3 + 7*v $$\beta_{5}$$ $$=$$ $$\nu^{5} - 9\nu^{3} + 13\nu$$ v^5 - 9*v^3 + 13*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$-\beta_{5} + \beta_{4} + 6\beta_1$$ -b5 + b4 + 6*b1 $$\nu^{4}$$ $$=$$ $$\beta_{3} + 8\beta_{2} + 18$$ b3 + 8*b2 + 18 $$\nu^{5}$$ $$=$$ $$-8\beta_{5} + 9\beta_{4} + 41\beta_1$$ -8*b5 + 9*b4 + 41*b1

## 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
 −2.68667 −1.30397 −0.285442 0.285442 1.30397 2.68667
−2.68667 −2.31446 5.21819 −1.00000 6.21819 4.21819 −8.64620 2.35673 2.68667
1.2 −1.30397 −0.537080 −0.299664 −1.00000 0.700336 −1.29966 2.99869 −2.71155 1.30397
1.3 −0.285442 3.21789 −1.91852 −1.00000 −0.918523 −2.91852 1.11851 7.35482 0.285442
1.4 0.285442 −3.21789 −1.91852 −1.00000 −0.918523 −2.91852 −1.11851 7.35482 −0.285442
1.5 1.30397 0.537080 −0.299664 −1.00000 0.700336 −1.29966 −2.99869 −2.71155 −1.30397
1.6 2.68667 2.31446 5.21819 −1.00000 6.21819 4.21819 8.64620 2.35673 −2.68667
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4205.2.a.m 6
29.b even 2 1 inner 4205.2.a.m 6
29.c odd 4 2 145.2.c.b 6
87.f even 4 2 1305.2.d.b 6
116.e even 4 2 2320.2.g.i 6
145.e even 4 2 725.2.d.c 12
145.f odd 4 2 725.2.c.e 6
145.j even 4 2 725.2.d.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.c.b 6 29.c odd 4 2
725.2.c.e 6 145.f odd 4 2
725.2.d.c 12 145.e even 4 2
725.2.d.c 12 145.j even 4 2
1305.2.d.b 6 87.f even 4 2
2320.2.g.i 6 116.e even 4 2
4205.2.a.m 6 1.a even 1 1 trivial
4205.2.a.m 6 29.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4205))$$:

 $$T_{2}^{6} - 9T_{2}^{4} + 13T_{2}^{2} - 1$$ T2^6 - 9*T2^4 + 13*T2^2 - 1 $$T_{3}^{6} - 16T_{3}^{4} + 60T_{3}^{2} - 16$$ T3^6 - 16*T3^4 + 60*T3^2 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 9 T^{4} + \cdots - 1$$
$3$ $$T^{6} - 16 T^{4} + \cdots - 16$$
$5$ $$(T + 1)^{6}$$
$7$ $$(T^{3} - 14 T - 16)^{2}$$
$11$ $$T^{6} - 16 T^{4} + \cdots - 16$$
$13$ $$(T^{3} - 2 T^{2} - 24 T + 16)^{2}$$
$17$ $$T^{6} - 52 T^{4} + \cdots - 64$$
$19$ $$T^{6} - 56 T^{4} + \cdots - 1296$$
$23$ $$(T^{3} + 4 T^{2} - 26 T - 96)^{2}$$
$29$ $$T^{6}$$
$31$ $$T^{6} - 152 T^{4} + \cdots - 144$$
$37$ $$T^{6} - 100 T^{4} + \cdots - 20736$$
$41$ $$T^{6} - 184 T^{4} + \cdots - 4096$$
$43$ $$T^{6} - 176 T^{4} + \cdots - 121104$$
$47$ $$T^{6} - 16 T^{4} + \cdots - 16$$
$53$ $$(T^{3} - 14 T^{2} + \cdots + 576)^{2}$$
$59$ $$(T^{3} + 8 T^{2} - 4 T - 48)^{2}$$
$61$ $$T^{6} - 104 T^{4} + \cdots - 2304$$
$67$ $$(T^{3} - 8 T^{2} - 10 T + 8)^{2}$$
$71$ $$(T^{3} - 28 T^{2} + \cdots - 576)^{2}$$
$73$ $$T^{6} - 252 T^{4} + \cdots - 419904$$
$79$ $$T^{6} - 176 T^{4} + \cdots - 121104$$
$83$ $$(T^{3} + 12 T^{2} + \cdots + 24)^{2}$$
$89$ $$T^{6} - 160 T^{4} + \cdots - 65536$$
$97$ $$T^{6} - 28 T^{4} + \cdots - 576$$