# Properties

 Label 4650.2.a.cp Level $4650$ Weight $2$ Character orbit 4650.a Self dual yes Analytic conductor $37.130$ 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] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, 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("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.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: $$37.1304369399$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1708.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 8x - 2$$ x^3 - x^2 - 8*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + 2 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 + 2 * q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + q^{6} + 2 q^{7} + q^{8} + q^{9} + ( - \beta_1 + 3) q^{11} + q^{12} + (\beta_1 + 3) q^{13} + 2 q^{14} + q^{16} + (\beta_{2} + 2 \beta_1) q^{17} + q^{18} + ( - \beta_{2} - 2 \beta_1) q^{19} + 2 q^{21} + ( - \beta_1 + 3) q^{22} + 2 \beta_1 q^{23} + q^{24} + (\beta_1 + 3) q^{26} + q^{27} + 2 q^{28} - 2 \beta_1 q^{29} - q^{31} + q^{32} + ( - \beta_1 + 3) q^{33} + (\beta_{2} + 2 \beta_1) q^{34} + q^{36} + (2 \beta_{2} - 2) q^{37} + ( - \beta_{2} - 2 \beta_1) q^{38} + (\beta_1 + 3) q^{39} + 2 q^{41} + 2 q^{42} - 2 \beta_{2} q^{43} + ( - \beta_1 + 3) q^{44} + 2 \beta_1 q^{46} + ( - \beta_{2} - 2 \beta_1) q^{47} + q^{48} - 3 q^{49} + (\beta_{2} + 2 \beta_1) q^{51} + (\beta_1 + 3) q^{52} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{53} + q^{54} + 2 q^{56} + ( - \beta_{2} - 2 \beta_1) q^{57} - 2 \beta_1 q^{58} + ( - 2 \beta_{2} - 2 \beta_1) q^{59} + (3 \beta_{2} + 2 \beta_1 + 4) q^{61} - q^{62} + 2 q^{63} + q^{64} + ( - \beta_1 + 3) q^{66} + (2 \beta_{2} - \beta_1 - 5) q^{67} + (\beta_{2} + 2 \beta_1) q^{68} + 2 \beta_1 q^{69} + ( - 2 \beta_{2} + \beta_1 - 3) q^{71} + q^{72} + (4 \beta_1 - 2) q^{73} + (2 \beta_{2} - 2) q^{74} + ( - \beta_{2} - 2 \beta_1) q^{76} + ( - 2 \beta_1 + 6) q^{77} + (\beta_1 + 3) q^{78} + (\beta_{2} - 2) q^{79} + q^{81} + 2 q^{82} + (\beta_{2} - 4 \beta_1 + 2) q^{83} + 2 q^{84} - 2 \beta_{2} q^{86} - 2 \beta_1 q^{87} + ( - \beta_1 + 3) q^{88} + (2 \beta_{2} + 4 \beta_1 - 2) q^{89} + (2 \beta_1 + 6) q^{91} + 2 \beta_1 q^{92} - q^{93} + ( - \beta_{2} - 2 \beta_1) q^{94} + q^{96} + (\beta_1 - 7) q^{97} - 3 q^{98} + ( - \beta_1 + 3) q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 + 2 * q^7 + q^8 + q^9 + (-b1 + 3) * q^11 + q^12 + (b1 + 3) * q^13 + 2 * q^14 + q^16 + (b2 + 2*b1) * q^17 + q^18 + (-b2 - 2*b1) * q^19 + 2 * q^21 + (-b1 + 3) * q^22 + 2*b1 * q^23 + q^24 + (b1 + 3) * q^26 + q^27 + 2 * q^28 - 2*b1 * q^29 - q^31 + q^32 + (-b1 + 3) * q^33 + (b2 + 2*b1) * q^34 + q^36 + (2*b2 - 2) * q^37 + (-b2 - 2*b1) * q^38 + (b1 + 3) * q^39 + 2 * q^41 + 2 * q^42 - 2*b2 * q^43 + (-b1 + 3) * q^44 + 2*b1 * q^46 + (-b2 - 2*b1) * q^47 + q^48 - 3 * q^49 + (b2 + 2*b1) * q^51 + (b1 + 3) * q^52 + (-2*b2 - 2*b1 + 4) * q^53 + q^54 + 2 * q^56 + (-b2 - 2*b1) * q^57 - 2*b1 * q^58 + (-2*b2 - 2*b1) * q^59 + (3*b2 + 2*b1 + 4) * q^61 - q^62 + 2 * q^63 + q^64 + (-b1 + 3) * q^66 + (2*b2 - b1 - 5) * q^67 + (b2 + 2*b1) * q^68 + 2*b1 * q^69 + (-2*b2 + b1 - 3) * q^71 + q^72 + (4*b1 - 2) * q^73 + (2*b2 - 2) * q^74 + (-b2 - 2*b1) * q^76 + (-2*b1 + 6) * q^77 + (b1 + 3) * q^78 + (b2 - 2) * q^79 + q^81 + 2 * q^82 + (b2 - 4*b1 + 2) * q^83 + 2 * q^84 - 2*b2 * q^86 - 2*b1 * q^87 + (-b1 + 3) * q^88 + (2*b2 + 4*b1 - 2) * q^89 + (2*b1 + 6) * q^91 + 2*b1 * q^92 - q^93 + (-b2 - 2*b1) * q^94 + q^96 + (b1 - 7) * q^97 - 3 * q^98 + (-b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9 $$3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9} + 8 q^{11} + 3 q^{12} + 10 q^{13} + 6 q^{14} + 3 q^{16} + 2 q^{17} + 3 q^{18} - 2 q^{19} + 6 q^{21} + 8 q^{22} + 2 q^{23} + 3 q^{24} + 10 q^{26} + 3 q^{27} + 6 q^{28} - 2 q^{29} - 3 q^{31} + 3 q^{32} + 8 q^{33} + 2 q^{34} + 3 q^{36} - 6 q^{37} - 2 q^{38} + 10 q^{39} + 6 q^{41} + 6 q^{42} + 8 q^{44} + 2 q^{46} - 2 q^{47} + 3 q^{48} - 9 q^{49} + 2 q^{51} + 10 q^{52} + 10 q^{53} + 3 q^{54} + 6 q^{56} - 2 q^{57} - 2 q^{58} - 2 q^{59} + 14 q^{61} - 3 q^{62} + 6 q^{63} + 3 q^{64} + 8 q^{66} - 16 q^{67} + 2 q^{68} + 2 q^{69} - 8 q^{71} + 3 q^{72} - 2 q^{73} - 6 q^{74} - 2 q^{76} + 16 q^{77} + 10 q^{78} - 6 q^{79} + 3 q^{81} + 6 q^{82} + 2 q^{83} + 6 q^{84} - 2 q^{87} + 8 q^{88} - 2 q^{89} + 20 q^{91} + 2 q^{92} - 3 q^{93} - 2 q^{94} + 3 q^{96} - 20 q^{97} - 9 q^{98} + 8 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9 + 8 * q^11 + 3 * q^12 + 10 * q^13 + 6 * q^14 + 3 * q^16 + 2 * q^17 + 3 * q^18 - 2 * q^19 + 6 * q^21 + 8 * q^22 + 2 * q^23 + 3 * q^24 + 10 * q^26 + 3 * q^27 + 6 * q^28 - 2 * q^29 - 3 * q^31 + 3 * q^32 + 8 * q^33 + 2 * q^34 + 3 * q^36 - 6 * q^37 - 2 * q^38 + 10 * q^39 + 6 * q^41 + 6 * q^42 + 8 * q^44 + 2 * q^46 - 2 * q^47 + 3 * q^48 - 9 * q^49 + 2 * q^51 + 10 * q^52 + 10 * q^53 + 3 * q^54 + 6 * q^56 - 2 * q^57 - 2 * q^58 - 2 * q^59 + 14 * q^61 - 3 * q^62 + 6 * q^63 + 3 * q^64 + 8 * q^66 - 16 * q^67 + 2 * q^68 + 2 * q^69 - 8 * q^71 + 3 * q^72 - 2 * q^73 - 6 * q^74 - 2 * q^76 + 16 * q^77 + 10 * q^78 - 6 * q^79 + 3 * q^81 + 6 * q^82 + 2 * q^83 + 6 * q^84 - 2 * q^87 + 8 * q^88 - 2 * q^89 + 20 * q^91 + 2 * q^92 - 3 * q^93 - 2 * q^94 + 3 * q^96 - 20 * q^97 - 9 * q^98 + 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 8x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 5$$ v^2 - 2*v - 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 5$$ b2 + 2*b1 + 5

## 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
 3.47090 −0.260711 −2.21018
1.00000 1.00000 1.00000 0 1.00000 2.00000 1.00000 1.00000 0
1.2 1.00000 1.00000 1.00000 0 1.00000 2.00000 1.00000 1.00000 0
1.3 1.00000 1.00000 1.00000 0 1.00000 2.00000 1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$31$$ $$+1$$

## 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 4650.2.a.cp 3
5.b even 2 1 4650.2.a.ci 3
5.c odd 4 2 930.2.d.i 6
15.e even 4 2 2790.2.d.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.i 6 5.c odd 4 2
2790.2.d.j 6 15.e even 4 2
4650.2.a.ci 3 5.b even 2 1
4650.2.a.cp 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4650))$$:

 $$T_{7} - 2$$ T7 - 2 $$T_{11}^{3} - 8T_{11}^{2} + 13T_{11} + 8$$ T11^3 - 8*T11^2 + 13*T11 + 8 $$T_{13}^{3} - 10T_{13}^{2} + 25T_{13} - 14$$ T13^3 - 10*T13^2 + 25*T13 - 14 $$T_{19}^{3} + 2T_{19}^{2} - 35T_{19} + 4$$ T19^3 + 2*T19^2 - 35*T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$(T - 2)^{3}$$
$11$ $$T^{3} - 8 T^{2} + \cdots + 8$$
$13$ $$T^{3} - 10 T^{2} + \cdots - 14$$
$17$ $$T^{3} - 2 T^{2} + \cdots - 4$$
$19$ $$T^{3} + 2 T^{2} + \cdots + 4$$
$23$ $$T^{3} - 2 T^{2} + \cdots - 16$$
$29$ $$T^{3} + 2 T^{2} + \cdots + 16$$
$31$ $$(T + 1)^{3}$$
$37$ $$T^{3} + 6 T^{2} + \cdots - 128$$
$41$ $$(T - 2)^{3}$$
$43$ $$T^{3} - 76T - 16$$
$47$ $$T^{3} + 2 T^{2} + \cdots + 4$$
$53$ $$T^{3} - 10 T^{2} + \cdots - 8$$
$59$ $$T^{3} + 2 T^{2} + \cdots - 280$$
$61$ $$T^{3} - 14 T^{2} + \cdots + 1372$$
$67$ $$T^{3} + 16 T^{2} + \cdots - 652$$
$71$ $$T^{3} + 8 T^{2} + \cdots + 20$$
$73$ $$T^{3} + 2 T^{2} + \cdots - 392$$
$79$ $$T^{3} + 6 T^{2} + \cdots - 28$$
$83$ $$T^{3} - 2 T^{2} + \cdots - 244$$
$89$ $$T^{3} + 2 T^{2} + \cdots - 320$$
$97$ $$T^{3} + 20 T^{2} + \cdots + 236$$