Properties

 Label 4205.2.a.k Level $4205$ Weight $2$ Character orbit 4205.a Self dual yes Analytic conductor $33.577$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.1586009.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 9x^{3} + 4x^{2} + 17x + 7$$ x^5 - x^4 - 9*x^3 + 4*x^2 + 17*x + 7 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 a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 9x^{3} + 4x^{2} + 17x + 7$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 10$$ v^4 - v^3 - 8*v^2 + 5*v + 10 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 7\nu^{2} + 11\nu + 8$$ v^4 - 2*v^3 - 7*v^2 + 11*v + 8 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 8\nu^{2} + 11\nu + 12$$ v^4 - 2*v^3 - 8*v^2 + 11*v + 12
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} + 4$$ -b4 + b3 + 4 $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{2} + 6\beta _1 + 2$$ -b4 + b2 + 6*b1 + 2 $$\nu^{4}$$ $$=$$ $$-9\beta_{4} + 8\beta_{3} + 2\beta_{2} + \beta _1 + 24$$ -9*b4 + 8*b3 + 2*b2 + b1 + 24

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
 −2.41314 −0.849666 −0.594278 2.03739 2.81969
−2.41314 0.884387 3.82326 1.00000 −2.13415 −2.01854 −4.39978 −2.21786 −2.41314
1.2 −0.849666 −1.37380 −1.27807 1.00000 1.16727 3.54107 2.78526 −1.11267 −0.849666
1.3 −0.594278 3.18209 −1.64683 1.00000 −1.89105 −4.07314 2.16723 7.12569 −0.594278
1.4 2.03739 1.51980 2.15097 1.00000 3.09643 2.57663 0.307585 −0.690207 2.03739
1.5 2.81969 −2.21248 5.95067 1.00000 −6.23850 −3.02603 11.1397 1.89505 2.81969
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.k yes 5
29.b even 2 1 4205.2.a.h 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4205.2.a.h 5 29.b even 2 1
4205.2.a.k yes 5 1.a even 1 1 trivial

Hecke kernels

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

 $$T_{2}^{5} - T_{2}^{4} - 9T_{2}^{3} + 4T_{2}^{2} + 17T_{2} + 7$$ T2^5 - T2^4 - 9*T2^3 + 4*T2^2 + 17*T2 + 7 $$T_{3}^{5} - 2T_{3}^{4} - 8T_{3}^{3} + 11T_{3}^{2} + 12T_{3} - 13$$ T3^5 - 2*T3^4 - 8*T3^3 + 11*T3^2 + 12*T3 - 13

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - T^{4} - 9 T^{3} + \cdots + 7$$
$3$ $$T^{5} - 2 T^{4} + \cdots - 13$$
$5$ $$(T - 1)^{5}$$
$7$ $$T^{5} + 3 T^{4} + \cdots + 227$$
$11$ $$T^{5} - 2 T^{4} + \cdots - 67$$
$13$ $$T^{5} - 45 T^{3} + \cdots + 169$$
$17$ $$T^{5} - 19 T^{4} + \cdots - 331$$
$19$ $$T^{5} + T^{4} + \cdots + 17$$
$23$ $$T^{5} - 4 T^{4} + \cdots - 89$$
$29$ $$T^{5}$$
$31$ $$T^{5} - 18 T^{4} + \cdots - 211$$
$37$ $$T^{5} - 2 T^{4} + \cdots + 9$$
$41$ $$T^{5} - 2 T^{4} + \cdots + 567$$
$43$ $$T^{5} - 22 T^{4} + \cdots + 1053$$
$47$ $$T^{5} + 2 T^{4} + \cdots + 721$$
$53$ $$T^{5} + 14 T^{4} + \cdots - 6197$$
$59$ $$T^{5} - 20 T^{4} + \cdots + 801$$
$61$ $$T^{5} - 5 T^{4} + \cdots - 38853$$
$67$ $$T^{5} + 21 T^{4} + \cdots + 1267$$
$71$ $$T^{5} + 4 T^{4} + \cdots + 2689$$
$73$ $$T^{5} - 28 T^{4} + \cdots + 65511$$
$79$ $$T^{5} - 3 T^{4} + \cdots - 1053$$
$83$ $$T^{5} - 8 T^{4} + \cdots + 24479$$
$89$ $$T^{5} + 34 T^{4} + \cdots - 448693$$
$97$ $$T^{5} - 9 T^{4} + \cdots - 2273$$