# Properties

 Label 950.2.a.i Level $950$ Weight $2$ Character orbit 950.a Self dual yes Analytic conductor $7.586$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + (\beta_1 - 1) q^{3} + q^{4} + ( - \beta_1 + 1) q^{6} + ( - \beta_1 - 1) q^{7} - q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q - q^2 + (b1 - 1) * q^3 + q^4 + (-b1 + 1) * q^6 + (-b1 - 1) * q^7 - q^8 + (b2 + 2) * q^9 $$q - q^{2} + (\beta_1 - 1) q^{3} + q^{4} + ( - \beta_1 + 1) q^{6} + ( - \beta_1 - 1) q^{7} - q^{8} + (\beta_{2} + 2) q^{9} + ( - \beta_{2} - \beta_1) q^{11} + (\beta_1 - 1) q^{12} + ( - \beta_{2} - 3) q^{13} + (\beta_1 + 1) q^{14} + q^{16} + (\beta_{2} + 1) q^{17} + ( - \beta_{2} - 2) q^{18} + q^{19} + ( - \beta_{2} - 2 \beta_1 - 3) q^{21} + (\beta_{2} + \beta_1) q^{22} + (2 \beta_{2} - \beta_1 + 1) q^{23} + ( - \beta_1 + 1) q^{24} + (\beta_{2} + 3) q^{26} + ( - 2 \beta_{2} - \beta_1 - 1) q^{27} + ( - \beta_1 - 1) q^{28} + ( - 3 \beta_{2} - 2 \beta_1 - 3) q^{29} + (3 \beta_{2} + \beta_1 + 2) q^{31} - q^{32} + (\beta_{2} - \beta_1 - 2) q^{33} + ( - \beta_{2} - 1) q^{34} + (\beta_{2} + 2) q^{36} + ( - 2 \beta_1 - 4) q^{37} - q^{38} + (2 \beta_{2} - 3 \beta_1 + 5) q^{39} + (\beta_{2} + 3 \beta_1) q^{41} + (\beta_{2} + 2 \beta_1 + 3) q^{42} + ( - \beta_{2} - \beta_1 - 6) q^{43} + ( - \beta_{2} - \beta_1) q^{44} + ( - 2 \beta_{2} + \beta_1 - 1) q^{46} + ( - 2 \beta_{2} + 4) q^{47} + (\beta_1 - 1) q^{48} + (\beta_{2} + 4 \beta_1 - 2) q^{49} + ( - 2 \beta_{2} + \beta_1 - 3) q^{51} + ( - \beta_{2} - 3) q^{52} + (\beta_{2} - 5) q^{53} + (2 \beta_{2} + \beta_1 + 1) q^{54} + (\beta_1 + 1) q^{56} + (\beta_1 - 1) q^{57} + (3 \beta_{2} + 2 \beta_1 + 3) q^{58} + (2 \beta_{2} + \beta_1 + 1) q^{59} + ( - \beta_{2} - \beta_1 - 10) q^{61} + ( - 3 \beta_{2} - \beta_1 - 2) q^{62} - 2 \beta_1 q^{63} + q^{64} + ( - \beta_{2} + \beta_1 + 2) q^{66} + (2 \beta_{2} + 3 \beta_1 - 1) q^{67} + (\beta_{2} + 1) q^{68} + ( - 5 \beta_{2} - 9) q^{69} + (\beta_{2} + 5 \beta_1 - 4) q^{71} + ( - \beta_{2} - 2) q^{72} + ( - 3 \beta_{2} + 2 \beta_1 - 5) q^{73} + (2 \beta_1 + 4) q^{74} + q^{76} + (\beta_{2} + 3 \beta_1 + 2) q^{77} + ( - 2 \beta_{2} + 3 \beta_1 - 5) q^{78} + (6 \beta_1 - 2) q^{79} + ( - 2 \beta_1 - 5) q^{81} + ( - \beta_{2} - 3 \beta_1) q^{82} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{83} + ( - \beta_{2} - 2 \beta_1 - 3) q^{84} + (\beta_{2} + \beta_1 + 6) q^{86} + (4 \beta_{2} - 5 \beta_1 + 1) q^{87} + (\beta_{2} + \beta_1) q^{88} + (\beta_{2} - \beta_1 - 4) q^{89} + (3 \beta_1 + 1) q^{91} + (2 \beta_{2} - \beta_1 + 1) q^{92} + ( - 5 \beta_{2} + 3 \beta_1 - 4) q^{93} + (2 \beta_{2} - 4) q^{94} + ( - \beta_1 + 1) q^{96} + (4 \beta_{2} - 2) q^{97} + ( - \beta_{2} - 4 \beta_1 + 2) q^{98} - 4 q^{99}+O(q^{100})$$ q - q^2 + (b1 - 1) * q^3 + q^4 + (-b1 + 1) * q^6 + (-b1 - 1) * q^7 - q^8 + (b2 + 2) * q^9 + (-b2 - b1) * q^11 + (b1 - 1) * q^12 + (-b2 - 3) * q^13 + (b1 + 1) * q^14 + q^16 + (b2 + 1) * q^17 + (-b2 - 2) * q^18 + q^19 + (-b2 - 2*b1 - 3) * q^21 + (b2 + b1) * q^22 + (2*b2 - b1 + 1) * q^23 + (-b1 + 1) * q^24 + (b2 + 3) * q^26 + (-2*b2 - b1 - 1) * q^27 + (-b1 - 1) * q^28 + (-3*b2 - 2*b1 - 3) * q^29 + (3*b2 + b1 + 2) * q^31 - q^32 + (b2 - b1 - 2) * q^33 + (-b2 - 1) * q^34 + (b2 + 2) * q^36 + (-2*b1 - 4) * q^37 - q^38 + (2*b2 - 3*b1 + 5) * q^39 + (b2 + 3*b1) * q^41 + (b2 + 2*b1 + 3) * q^42 + (-b2 - b1 - 6) * q^43 + (-b2 - b1) * q^44 + (-2*b2 + b1 - 1) * q^46 + (-2*b2 + 4) * q^47 + (b1 - 1) * q^48 + (b2 + 4*b1 - 2) * q^49 + (-2*b2 + b1 - 3) * q^51 + (-b2 - 3) * q^52 + (b2 - 5) * q^53 + (2*b2 + b1 + 1) * q^54 + (b1 + 1) * q^56 + (b1 - 1) * q^57 + (3*b2 + 2*b1 + 3) * q^58 + (2*b2 + b1 + 1) * q^59 + (-b2 - b1 - 10) * q^61 + (-3*b2 - b1 - 2) * q^62 - 2*b1 * q^63 + q^64 + (-b2 + b1 + 2) * q^66 + (2*b2 + 3*b1 - 1) * q^67 + (b2 + 1) * q^68 + (-5*b2 - 9) * q^69 + (b2 + 5*b1 - 4) * q^71 + (-b2 - 2) * q^72 + (-3*b2 + 2*b1 - 5) * q^73 + (2*b1 + 4) * q^74 + q^76 + (b2 + 3*b1 + 2) * q^77 + (-2*b2 + 3*b1 - 5) * q^78 + (6*b1 - 2) * q^79 + (-2*b1 - 5) * q^81 + (-b2 - 3*b1) * q^82 + (-2*b2 - 2*b1 + 2) * q^83 + (-b2 - 2*b1 - 3) * q^84 + (b2 + b1 + 6) * q^86 + (4*b2 - 5*b1 + 1) * q^87 + (b2 + b1) * q^88 + (b2 - b1 - 4) * q^89 + (3*b1 + 1) * q^91 + (2*b2 - b1 + 1) * q^92 + (-5*b2 + 3*b1 - 4) * q^93 + (2*b2 - 4) * q^94 + (-b1 + 1) * q^96 + (4*b2 - 2) * q^97 + (-b2 - 4*b1 + 2) * q^98 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} - 4 q^{7} - 3 q^{8} + 5 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^6 - 4 * q^7 - 3 * q^8 + 5 * q^9 $$3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} - 4 q^{7} - 3 q^{8} + 5 q^{9} - 2 q^{12} - 8 q^{13} + 4 q^{14} + 3 q^{16} + 2 q^{17} - 5 q^{18} + 3 q^{19} - 10 q^{21} + 2 q^{24} + 8 q^{26} - 2 q^{27} - 4 q^{28} - 8 q^{29} + 4 q^{31} - 3 q^{32} - 8 q^{33} - 2 q^{34} + 5 q^{36} - 14 q^{37} - 3 q^{38} + 10 q^{39} + 2 q^{41} + 10 q^{42} - 18 q^{43} + 14 q^{47} - 2 q^{48} - 3 q^{49} - 6 q^{51} - 8 q^{52} - 16 q^{53} + 2 q^{54} + 4 q^{56} - 2 q^{57} + 8 q^{58} + 2 q^{59} - 30 q^{61} - 4 q^{62} - 2 q^{63} + 3 q^{64} + 8 q^{66} - 2 q^{67} + 2 q^{68} - 22 q^{69} - 8 q^{71} - 5 q^{72} - 10 q^{73} + 14 q^{74} + 3 q^{76} + 8 q^{77} - 10 q^{78} - 17 q^{81} - 2 q^{82} + 6 q^{83} - 10 q^{84} + 18 q^{86} - 6 q^{87} - 14 q^{89} + 6 q^{91} - 4 q^{93} - 14 q^{94} + 2 q^{96} - 10 q^{97} + 3 q^{98} - 12 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^6 - 4 * q^7 - 3 * q^8 + 5 * q^9 - 2 * q^12 - 8 * q^13 + 4 * q^14 + 3 * q^16 + 2 * q^17 - 5 * q^18 + 3 * q^19 - 10 * q^21 + 2 * q^24 + 8 * q^26 - 2 * q^27 - 4 * q^28 - 8 * q^29 + 4 * q^31 - 3 * q^32 - 8 * q^33 - 2 * q^34 + 5 * q^36 - 14 * q^37 - 3 * q^38 + 10 * q^39 + 2 * q^41 + 10 * q^42 - 18 * q^43 + 14 * q^47 - 2 * q^48 - 3 * q^49 - 6 * q^51 - 8 * q^52 - 16 * q^53 + 2 * q^54 + 4 * q^56 - 2 * q^57 + 8 * q^58 + 2 * q^59 - 30 * q^61 - 4 * q^62 - 2 * q^63 + 3 * q^64 + 8 * q^66 - 2 * q^67 + 2 * q^68 - 22 * q^69 - 8 * q^71 - 5 * q^72 - 10 * q^73 + 14 * q^74 + 3 * q^76 + 8 * q^77 - 10 * q^78 - 17 * q^81 - 2 * q^82 + 6 * q^83 - 10 * q^84 + 18 * q^86 - 6 * q^87 - 14 * q^89 + 6 * q^91 - 4 * q^93 - 14 * q^94 + 2 * q^96 - 10 * q^97 + 3 * q^98 - 12 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 4$$ v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 4$$ b2 + 2*b1 + 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
 −1.76156 −0.363328 3.12489
−1.00000 −2.76156 1.00000 0 2.76156 0.761557 −1.00000 4.62620 0
1.2 −1.00000 −1.36333 1.00000 0 1.36333 −0.636672 −1.00000 −1.14134 0
1.3 −1.00000 2.12489 1.00000 0 −2.12489 −4.12489 −1.00000 1.51514 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.i 3
3.b odd 2 1 8550.2.a.cl 3
4.b odd 2 1 7600.2.a.cd 3
5.b even 2 1 950.2.a.n 3
5.c odd 4 2 190.2.b.b 6
15.d odd 2 1 8550.2.a.ck 3
15.e even 4 2 1710.2.d.d 6
20.d odd 2 1 7600.2.a.bi 3
20.e even 4 2 1520.2.d.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.b 6 5.c odd 4 2
950.2.a.i 3 1.a even 1 1 trivial
950.2.a.n 3 5.b even 2 1
1520.2.d.j 6 20.e even 4 2
1710.2.d.d 6 15.e even 4 2
7600.2.a.bi 3 20.d odd 2 1
7600.2.a.cd 3 4.b odd 2 1
8550.2.a.ck 3 15.d odd 2 1
8550.2.a.cl 3 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(950))$$:

 $$T_{3}^{3} + 2T_{3}^{2} - 5T_{3} - 8$$ T3^3 + 2*T3^2 - 5*T3 - 8 $$T_{7}^{3} + 4T_{7}^{2} - T_{7} - 2$$ T7^3 + 4*T7^2 - T7 - 2 $$T_{11}^{3} - 10T_{11} - 8$$ T11^3 - 10*T11 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3} + 2 T^{2} + \cdots - 8$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 4T^{2} - T - 2$$
$11$ $$T^{3} - 10T - 8$$
$13$ $$T^{3} + 8 T^{2} + \cdots - 2$$
$17$ $$T^{3} - 2 T^{2} + \cdots + 4$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} - 49T - 122$$
$29$ $$T^{3} + 8 T^{2} + \cdots - 410$$
$31$ $$T^{3} - 4 T^{2} + \cdots + 232$$
$37$ $$T^{3} + 14 T^{2} + \cdots + 16$$
$41$ $$T^{3} - 2 T^{2} + \cdots - 100$$
$43$ $$T^{3} + 18 T^{2} + \cdots + 148$$
$47$ $$T^{3} - 14 T^{2} + \cdots + 64$$
$53$ $$T^{3} + 16 T^{2} + \cdots + 106$$
$59$ $$T^{3} - 2 T^{2} + \cdots + 80$$
$61$ $$T^{3} + 30 T^{2} + \cdots + 892$$
$67$ $$T^{3} + 2 T^{2} + \cdots - 64$$
$71$ $$T^{3} + 8 T^{2} + \cdots - 1016$$
$73$ $$T^{3} + 10 T^{2} + \cdots + 164$$
$79$ $$T^{3} - 228T - 880$$
$83$ $$T^{3} - 6 T^{2} + \cdots + 8$$
$89$ $$T^{3} + 14 T^{2} + \cdots - 20$$
$97$ $$T^{3} + 10 T^{2} + \cdots - 488$$