# Properties

 Label 1950.2.a.d Level $1950$ Weight $2$ Character orbit 1950.a Self dual yes Analytic conductor $15.571$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

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

## 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
−1.00000 −1.00000 1.00000 0 1.00000 4.00000 −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
$$2$$ $$+1$$
$$3$$ $$+1$$
$$5$$ $$-1$$
$$13$$ $$+1$$

## 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.d 1
3.b odd 2 1 5850.2.a.cc 1
5.b even 2 1 1950.2.a.v 1
5.c odd 4 2 390.2.e.d 2
15.d odd 2 1 5850.2.a.e 1
15.e even 4 2 1170.2.e.c 2
20.e even 4 2 3120.2.l.i 2

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

 $$T_{7} - 4$$ T7 - 4 $$T_{11} + 6$$ T11 + 6 $$T_{17} - 4$$ T17 - 4 $$T_{23} - 6$$ T23 - 6 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T - 4$$
$11$ $$T + 6$$
$13$ $$T + 1$$
$17$ $$T - 4$$
$19$ $$T - 2$$
$23$ $$T - 6$$
$29$ $$T + 10$$
$31$ $$T - 4$$
$37$ $$T - 6$$
$41$ $$T - 10$$
$43$ $$T$$
$47$ $$T + 8$$
$53$ $$T - 6$$
$59$ $$T + 6$$
$61$ $$T + 6$$
$67$ $$T - 12$$
$71$ $$T$$
$73$ $$T + 2$$
$79$ $$T + 8$$
$83$ $$T - 4$$
$89$ $$T - 14$$
$97$ $$T + 14$$