# Properties

 Label 950.2.a.l Level $950$ Weight $2$ Character orbit 950.a Self dual yes Analytic conductor $7.586$ 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] = [950,2,Mod(1,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, 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("950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.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: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.993.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x + 3$$ x^3 - x^2 - 6*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + ( - \beta_1 + 1) q^{6} + ( - \beta_{2} + 1) q^{7} + q^{8} + (\beta_{2} - 2 \beta_1 + 2) q^{9}+O(q^{10})$$ q + q^2 + (-b1 + 1) * q^3 + q^4 + (-b1 + 1) * q^6 + (-b2 + 1) * q^7 + q^8 + (b2 - 2*b1 + 2) * q^9 $$q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + ( - \beta_1 + 1) q^{6} + ( - \beta_{2} + 1) q^{7} + q^{8} + (\beta_{2} - 2 \beta_1 + 2) q^{9} + 2 \beta_1 q^{11} + ( - \beta_1 + 1) q^{12} + ( - \beta_{2} + \beta_1 - 2) q^{13} + ( - \beta_{2} + 1) q^{14} + q^{16} + (\beta_{2} - \beta_1 + 4) q^{17} + (\beta_{2} - 2 \beta_1 + 2) q^{18} + q^{19} + (\beta_1 + 2) q^{21} + 2 \beta_1 q^{22} + (2 \beta_{2} - \beta_1 - 1) q^{23} + ( - \beta_1 + 1) q^{24} + ( - \beta_{2} + \beta_1 - 2) q^{26} + (2 \beta_{2} - 3 \beta_1 + 6) q^{27} + ( - \beta_{2} + 1) q^{28} + ( - \beta_{2} + \beta_1 + 2) q^{29} + (2 \beta_1 + 2) q^{31} + q^{32} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{33} + (\beta_{2} - \beta_1 + 4) q^{34} + (\beta_{2} - 2 \beta_1 + 2) q^{36} + (2 \beta_1 - 1) q^{37} + q^{38} + ( - \beta_{2} + 5 \beta_1 - 5) q^{39} + (\beta_1 + 2) q^{42} + ( - 2 \beta_1 - 4) q^{43} + 2 \beta_1 q^{44} + (2 \beta_{2} - \beta_1 - 1) q^{46} + ( - 2 \beta_{2} + 2 \beta_1 + 1) q^{47} + ( - \beta_1 + 1) q^{48} + ( - 3 \beta_{2} + 3 \beta_1 + 3) q^{49} + (\beta_{2} - 7 \beta_1 + 7) q^{51} + ( - \beta_{2} + \beta_1 - 2) q^{52} + (\beta_{2} + 4 \beta_1 - 5) q^{53} + (2 \beta_{2} - 3 \beta_1 + 6) q^{54} + ( - \beta_{2} + 1) q^{56} + ( - \beta_1 + 1) q^{57} + ( - \beta_{2} + \beta_1 + 2) q^{58} + (4 \beta_{2} - \beta_1 + 1) q^{59} - 2 \beta_{2} q^{61} + (2 \beta_1 + 2) q^{62} + (2 \beta_{2} - \beta_1 - 5) q^{63} + q^{64} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{66} + ( - \beta_{2} + 2 \beta_1 + 1) q^{67} + (\beta_{2} - \beta_1 + 4) q^{68} + (\beta_{2} - 4 \beta_1 + 1) q^{69} + (2 \beta_{2} + 4 \beta_1 - 4) q^{71} + (\beta_{2} - 2 \beta_1 + 2) q^{72} + (2 \beta_{2} + \beta_1 - 5) q^{73} + (2 \beta_1 - 1) q^{74} + q^{76} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{77} + ( - \beta_{2} + 5 \beta_1 - 5) q^{78} + ( - 4 \beta_1 + 8) q^{79} + ( - 7 \beta_1 + 10) q^{81} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{83} + (\beta_1 + 2) q^{84} + ( - 2 \beta_1 - 4) q^{86} + ( - \beta_{2} + \beta_1 - 1) q^{87} + 2 \beta_1 q^{88} + ( - 4 \beta_1 + 6) q^{89} + ( - \beta_{2} + 2 \beta_1 + 6) q^{91} + (2 \beta_{2} - \beta_1 - 1) q^{92} + ( - 2 \beta_{2} - 6) q^{93} + ( - 2 \beta_{2} + 2 \beta_1 + 1) q^{94} + ( - \beta_1 + 1) q^{96} + (2 \beta_{2} - 6 \beta_1 - 8) q^{97} + ( - 3 \beta_{2} + 3 \beta_1 + 3) q^{98} + ( - 2 \beta_{2} + 8 \beta_1 - 14) q^{99}+O(q^{100})$$ q + q^2 + (-b1 + 1) * q^3 + q^4 + (-b1 + 1) * q^6 + (-b2 + 1) * q^7 + q^8 + (b2 - 2*b1 + 2) * q^9 + 2*b1 * q^11 + (-b1 + 1) * q^12 + (-b2 + b1 - 2) * q^13 + (-b2 + 1) * q^14 + q^16 + (b2 - b1 + 4) * q^17 + (b2 - 2*b1 + 2) * q^18 + q^19 + (b1 + 2) * q^21 + 2*b1 * q^22 + (2*b2 - b1 - 1) * q^23 + (-b1 + 1) * q^24 + (-b2 + b1 - 2) * q^26 + (2*b2 - 3*b1 + 6) * q^27 + (-b2 + 1) * q^28 + (-b2 + b1 + 2) * q^29 + (2*b1 + 2) * q^31 + q^32 + (-2*b2 + 2*b1 - 8) * q^33 + (b2 - b1 + 4) * q^34 + (b2 - 2*b1 + 2) * q^36 + (2*b1 - 1) * q^37 + q^38 + (-b2 + 5*b1 - 5) * q^39 + (b1 + 2) * q^42 + (-2*b1 - 4) * q^43 + 2*b1 * q^44 + (2*b2 - b1 - 1) * q^46 + (-2*b2 + 2*b1 + 1) * q^47 + (-b1 + 1) * q^48 + (-3*b2 + 3*b1 + 3) * q^49 + (b2 - 7*b1 + 7) * q^51 + (-b2 + b1 - 2) * q^52 + (b2 + 4*b1 - 5) * q^53 + (2*b2 - 3*b1 + 6) * q^54 + (-b2 + 1) * q^56 + (-b1 + 1) * q^57 + (-b2 + b1 + 2) * q^58 + (4*b2 - b1 + 1) * q^59 - 2*b2 * q^61 + (2*b1 + 2) * q^62 + (2*b2 - b1 - 5) * q^63 + q^64 + (-2*b2 + 2*b1 - 8) * q^66 + (-b2 + 2*b1 + 1) * q^67 + (b2 - b1 + 4) * q^68 + (b2 - 4*b1 + 1) * q^69 + (2*b2 + 4*b1 - 4) * q^71 + (b2 - 2*b1 + 2) * q^72 + (2*b2 + b1 - 5) * q^73 + (2*b1 - 1) * q^74 + q^76 + (-2*b2 - 2*b1 - 2) * q^77 + (-b2 + 5*b1 - 5) * q^78 + (-4*b1 + 8) * q^79 + (-7*b1 + 10) * q^81 + (-2*b2 + 4*b1 - 2) * q^83 + (b1 + 2) * q^84 + (-2*b1 - 4) * q^86 + (-b2 + b1 - 1) * q^87 + 2*b1 * q^88 + (-4*b1 + 6) * q^89 + (-b2 + 2*b1 + 6) * q^91 + (2*b2 - b1 - 1) * q^92 + (-2*b2 - 6) * q^93 + (-2*b2 + 2*b1 + 1) * q^94 + (-b1 + 1) * q^96 + (2*b2 - 6*b1 - 8) * q^97 + (-3*b2 + 3*b1 + 3) * q^98 + (-2*b2 + 8*b1 - 14) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 2 q^{6} + 2 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 2 * q^6 + 2 * q^7 + 3 * q^8 + 5 * q^9 $$3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 2 q^{6} + 2 q^{7} + 3 q^{8} + 5 q^{9} + 2 q^{11} + 2 q^{12} - 6 q^{13} + 2 q^{14} + 3 q^{16} + 12 q^{17} + 5 q^{18} + 3 q^{19} + 7 q^{21} + 2 q^{22} - 2 q^{23} + 2 q^{24} - 6 q^{26} + 17 q^{27} + 2 q^{28} + 6 q^{29} + 8 q^{31} + 3 q^{32} - 24 q^{33} + 12 q^{34} + 5 q^{36} - q^{37} + 3 q^{38} - 11 q^{39} + 7 q^{42} - 14 q^{43} + 2 q^{44} - 2 q^{46} + 3 q^{47} + 2 q^{48} + 9 q^{49} + 15 q^{51} - 6 q^{52} - 10 q^{53} + 17 q^{54} + 2 q^{56} + 2 q^{57} + 6 q^{58} + 6 q^{59} - 2 q^{61} + 8 q^{62} - 14 q^{63} + 3 q^{64} - 24 q^{66} + 4 q^{67} + 12 q^{68} - 6 q^{71} + 5 q^{72} - 12 q^{73} - q^{74} + 3 q^{76} - 10 q^{77} - 11 q^{78} + 20 q^{79} + 23 q^{81} - 4 q^{83} + 7 q^{84} - 14 q^{86} - 3 q^{87} + 2 q^{88} + 14 q^{89} + 19 q^{91} - 2 q^{92} - 20 q^{93} + 3 q^{94} + 2 q^{96} - 28 q^{97} + 9 q^{98} - 36 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 + 2 * q^3 + 3 * q^4 + 2 * q^6 + 2 * q^7 + 3 * q^8 + 5 * q^9 + 2 * q^11 + 2 * q^12 - 6 * q^13 + 2 * q^14 + 3 * q^16 + 12 * q^17 + 5 * q^18 + 3 * q^19 + 7 * q^21 + 2 * q^22 - 2 * q^23 + 2 * q^24 - 6 * q^26 + 17 * q^27 + 2 * q^28 + 6 * q^29 + 8 * q^31 + 3 * q^32 - 24 * q^33 + 12 * q^34 + 5 * q^36 - q^37 + 3 * q^38 - 11 * q^39 + 7 * q^42 - 14 * q^43 + 2 * q^44 - 2 * q^46 + 3 * q^47 + 2 * q^48 + 9 * q^49 + 15 * q^51 - 6 * q^52 - 10 * q^53 + 17 * q^54 + 2 * q^56 + 2 * q^57 + 6 * q^58 + 6 * q^59 - 2 * q^61 + 8 * q^62 - 14 * q^63 + 3 * q^64 - 24 * q^66 + 4 * q^67 + 12 * q^68 - 6 * q^71 + 5 * q^72 - 12 * q^73 - q^74 + 3 * q^76 - 10 * q^77 - 11 * q^78 + 20 * q^79 + 23 * q^81 - 4 * q^83 + 7 * q^84 - 14 * q^86 - 3 * q^87 + 2 * q^88 + 14 * q^89 + 19 * q^91 - 2 * q^92 - 20 * q^93 + 3 * q^94 + 2 * q^96 - 28 * q^97 + 9 * q^98 - 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4

## 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
 2.77339 0.480031 −2.25342
1.00000 −1.77339 1.00000 0 −1.77339 −2.69168 1.00000 0.144903 0
1.2 1.00000 0.519969 1.00000 0 0.519969 4.76957 1.00000 −2.72963 0
1.3 1.00000 3.25342 1.00000 0 3.25342 −0.0778929 1.00000 7.58473 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 950.2.a.l yes 3
3.b odd 2 1 8550.2.a.ci 3
4.b odd 2 1 7600.2.a.bk 3
5.b even 2 1 950.2.a.j 3
5.c odd 4 2 950.2.b.h 6
15.d odd 2 1 8550.2.a.cp 3
20.d odd 2 1 7600.2.a.bz 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.j 3 5.b even 2 1
950.2.a.l yes 3 1.a even 1 1 trivial
950.2.b.h 6 5.c odd 4 2
7600.2.a.bk 3 4.b odd 2 1
7600.2.a.bz 3 20.d odd 2 1
8550.2.a.ci 3 3.b odd 2 1
8550.2.a.cp 3 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(950))$$:

 $$T_{3}^{3} - 2T_{3}^{2} - 5T_{3} + 3$$ T3^3 - 2*T3^2 - 5*T3 + 3 $$T_{7}^{3} - 2T_{7}^{2} - 13T_{7} - 1$$ T7^3 - 2*T7^2 - 13*T7 - 1 $$T_{11}^{3} - 2T_{11}^{2} - 24T_{11} + 24$$ T11^3 - 2*T11^2 - 24*T11 + 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$T^{3} - 2 T^{2} + \cdots + 3$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 2 T^{2} + \cdots - 1$$
$11$ $$T^{3} - 2 T^{2} + \cdots + 24$$
$13$ $$T^{3} + 6 T^{2} + \cdots - 35$$
$17$ $$T^{3} - 12 T^{2} + \cdots + 9$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} + 2 T^{2} + \cdots + 111$$
$29$ $$T^{3} - 6 T^{2} + \cdots + 9$$
$31$ $$T^{3} - 8 T^{2} + \cdots + 56$$
$37$ $$T^{3} + T^{2} - 25T - 1$$
$41$ $$T^{3}$$
$43$ $$T^{3} + 14 T^{2} + \cdots - 24$$
$47$ $$T^{3} - 3 T^{2} + \cdots - 45$$
$53$ $$T^{3} + 10 T^{2} + \cdots - 867$$
$59$ $$T^{3} - 6 T^{2} + \cdots + 1431$$
$61$ $$T^{3} + 2 T^{2} + \cdots - 120$$
$67$ $$T^{3} - 4 T^{2} + \cdots + 75$$
$71$ $$T^{3} + 6 T^{2} + \cdots - 1512$$
$73$ $$T^{3} + 12 T^{2} + \cdots - 317$$
$79$ $$T^{3} - 20 T^{2} + \cdots + 320$$
$83$ $$T^{3} + 4 T^{2} + \cdots + 168$$
$89$ $$T^{3} - 14 T^{2} + \cdots + 312$$
$97$ $$T^{3} + 28 T^{2} + \cdots - 2440$$