# Properties

 Label 1950.2.a.be Level $1950$ Weight $2$ Character orbit 1950.a Self dual yes Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, 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("1950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.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: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 390) 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} + (\beta + 1) q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^6 + (b + 1) * q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - q^{6} + (\beta + 1) q^{7} - q^{8} + q^{9} + 2 \beta q^{11} + q^{12} + q^{13} + ( - \beta - 1) q^{14} + q^{16} + (\beta + 5) q^{17} - q^{18} + (\beta - 5) q^{19} + (\beta + 1) q^{21} - 2 \beta q^{22} + ( - \beta + 5) q^{23} - q^{24} - q^{26} + q^{27} + (\beta + 1) q^{28} + ( - 3 \beta + 3) q^{29} - 4 q^{31} - q^{32} + 2 \beta q^{33} + ( - \beta - 5) q^{34} + q^{36} + ( - 4 \beta - 2) q^{37} + ( - \beta + 5) q^{38} + q^{39} + ( - 2 \beta + 8) q^{41} + ( - \beta - 1) q^{42} + ( - 2 \beta + 2) q^{43} + 2 \beta q^{44} + (\beta - 5) q^{46} + (4 \beta + 4) q^{47} + q^{48} + (2 \beta - 1) q^{49} + (\beta + 5) q^{51} + q^{52} + ( - 2 \beta + 4) q^{53} - q^{54} + ( - \beta - 1) q^{56} + (\beta - 5) q^{57} + (3 \beta - 3) q^{58} + ( - 2 \beta - 4) q^{59} + ( - 4 \beta - 2) q^{61} + 4 q^{62} + (\beta + 1) q^{63} + q^{64} - 2 \beta q^{66} + (\beta + 5) q^{68} + ( - \beta + 5) q^{69} + ( - 2 \beta + 2) q^{71} - q^{72} + (\beta + 11) q^{73} + (4 \beta + 2) q^{74} + (\beta - 5) q^{76} + (2 \beta + 10) q^{77} - q^{78} - 4 q^{79} + q^{81} + (2 \beta - 8) q^{82} + ( - 4 \beta + 4) q^{83} + (\beta + 1) q^{84} + (2 \beta - 2) q^{86} + ( - 3 \beta + 3) q^{87} - 2 \beta q^{88} + (2 \beta - 4) q^{89} + (\beta + 1) q^{91} + ( - \beta + 5) q^{92} - 4 q^{93} + ( - 4 \beta - 4) q^{94} - q^{96} + ( - 3 \beta + 3) q^{97} + ( - 2 \beta + 1) q^{98} + 2 \beta q^{99} +O(q^{100})$$ q - q^2 + q^3 + q^4 - q^6 + (b + 1) * q^7 - q^8 + q^9 + 2*b * q^11 + q^12 + q^13 + (-b - 1) * q^14 + q^16 + (b + 5) * q^17 - q^18 + (b - 5) * q^19 + (b + 1) * q^21 - 2*b * q^22 + (-b + 5) * q^23 - q^24 - q^26 + q^27 + (b + 1) * q^28 + (-3*b + 3) * q^29 - 4 * q^31 - q^32 + 2*b * q^33 + (-b - 5) * q^34 + q^36 + (-4*b - 2) * q^37 + (-b + 5) * q^38 + q^39 + (-2*b + 8) * q^41 + (-b - 1) * q^42 + (-2*b + 2) * q^43 + 2*b * q^44 + (b - 5) * q^46 + (4*b + 4) * q^47 + q^48 + (2*b - 1) * q^49 + (b + 5) * q^51 + q^52 + (-2*b + 4) * q^53 - q^54 + (-b - 1) * q^56 + (b - 5) * q^57 + (3*b - 3) * q^58 + (-2*b - 4) * q^59 + (-4*b - 2) * q^61 + 4 * q^62 + (b + 1) * q^63 + q^64 - 2*b * q^66 + (b + 5) * q^68 + (-b + 5) * q^69 + (-2*b + 2) * q^71 - q^72 + (b + 11) * q^73 + (4*b + 2) * q^74 + (b - 5) * q^76 + (2*b + 10) * q^77 - q^78 - 4 * q^79 + q^81 + (2*b - 8) * q^82 + (-4*b + 4) * q^83 + (b + 1) * q^84 + (2*b - 2) * q^86 + (-3*b + 3) * q^87 - 2*b * q^88 + (2*b - 4) * q^89 + (b + 1) * q^91 + (-b + 5) * q^92 - 4 * q^93 + (-4*b - 4) * q^94 - q^96 + (-3*b + 3) * q^97 + (-2*b + 1) * q^98 + 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{17} - 2 q^{18} - 10 q^{19} + 2 q^{21} + 10 q^{23} - 2 q^{24} - 2 q^{26} + 2 q^{27} + 2 q^{28} + 6 q^{29} - 8 q^{31} - 2 q^{32} - 10 q^{34} + 2 q^{36} - 4 q^{37} + 10 q^{38} + 2 q^{39} + 16 q^{41} - 2 q^{42} + 4 q^{43} - 10 q^{46} + 8 q^{47} + 2 q^{48} - 2 q^{49} + 10 q^{51} + 2 q^{52} + 8 q^{53} - 2 q^{54} - 2 q^{56} - 10 q^{57} - 6 q^{58} - 8 q^{59} - 4 q^{61} + 8 q^{62} + 2 q^{63} + 2 q^{64} + 10 q^{68} + 10 q^{69} + 4 q^{71} - 2 q^{72} + 22 q^{73} + 4 q^{74} - 10 q^{76} + 20 q^{77} - 2 q^{78} - 8 q^{79} + 2 q^{81} - 16 q^{82} + 8 q^{83} + 2 q^{84} - 4 q^{86} + 6 q^{87} - 8 q^{89} + 2 q^{91} + 10 q^{92} - 8 q^{93} - 8 q^{94} - 2 q^{96} + 6 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^12 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 10 * q^17 - 2 * q^18 - 10 * q^19 + 2 * q^21 + 10 * q^23 - 2 * q^24 - 2 * q^26 + 2 * q^27 + 2 * q^28 + 6 * q^29 - 8 * q^31 - 2 * q^32 - 10 * q^34 + 2 * q^36 - 4 * q^37 + 10 * q^38 + 2 * q^39 + 16 * q^41 - 2 * q^42 + 4 * q^43 - 10 * q^46 + 8 * q^47 + 2 * q^48 - 2 * q^49 + 10 * q^51 + 2 * q^52 + 8 * q^53 - 2 * q^54 - 2 * q^56 - 10 * q^57 - 6 * q^58 - 8 * q^59 - 4 * q^61 + 8 * q^62 + 2 * q^63 + 2 * q^64 + 10 * q^68 + 10 * q^69 + 4 * q^71 - 2 * q^72 + 22 * q^73 + 4 * q^74 - 10 * q^76 + 20 * q^77 - 2 * q^78 - 8 * q^79 + 2 * q^81 - 16 * q^82 + 8 * q^83 + 2 * q^84 - 4 * q^86 + 6 * q^87 - 8 * q^89 + 2 * q^91 + 10 * q^92 - 8 * q^93 - 8 * q^94 - 2 * q^96 + 6 * q^97 + 2 * q^98

## 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
 −0.618034 1.61803
−1.00000 1.00000 1.00000 0 −1.00000 −1.23607 −1.00000 1.00000 0
1.2 −1.00000 1.00000 1.00000 0 −1.00000 3.23607 −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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

## 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 1950.2.a.be 2
3.b odd 2 1 5850.2.a.cm 2
5.b even 2 1 1950.2.a.bf 2
5.c odd 4 2 390.2.e.e 4
15.d odd 2 1 5850.2.a.cf 2
15.e even 4 2 1170.2.e.e 4
20.e even 4 2 3120.2.l.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.e 4 5.c odd 4 2
1170.2.e.e 4 15.e even 4 2
1950.2.a.be 2 1.a even 1 1 trivial
1950.2.a.bf 2 5.b even 2 1
3120.2.l.k 4 20.e even 4 2
5850.2.a.cf 2 15.d odd 2 1
5850.2.a.cm 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1950))$$:

 $$T_{7}^{2} - 2T_{7} - 4$$ T7^2 - 2*T7 - 4 $$T_{11}^{2} - 20$$ T11^2 - 20 $$T_{17}^{2} - 10T_{17} + 20$$ T17^2 - 10*T17 + 20 $$T_{23}^{2} - 10T_{23} + 20$$ T23^2 - 10*T23 + 20 $$T_{31} + 4$$ T31 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 2T - 4$$
$11$ $$T^{2} - 20$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 10T + 20$$
$19$ $$T^{2} + 10T + 20$$
$23$ $$T^{2} - 10T + 20$$
$29$ $$T^{2} - 6T - 36$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$T^{2} - 16T + 44$$
$43$ $$T^{2} - 4T - 16$$
$47$ $$T^{2} - 8T - 64$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} + 8T - 4$$
$61$ $$T^{2} + 4T - 76$$
$67$ $$T^{2}$$
$71$ $$T^{2} - 4T - 16$$
$73$ $$T^{2} - 22T + 116$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} - 8T - 64$$
$89$ $$T^{2} + 8T - 4$$
$97$ $$T^{2} - 6T - 36$$