# Properties

 Label 1950.2.a.ba 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)

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

## 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.ba 1
3.b odd 2 1 5850.2.a.s 1
5.b even 2 1 390.2.a.b 1
5.c odd 4 2 1950.2.e.m 2
15.d odd 2 1 1170.2.a.j 1
15.e even 4 2 5850.2.e.h 2
20.d odd 2 1 3120.2.a.y 1
60.h even 2 1 9360.2.a.v 1
65.d even 2 1 5070.2.a.n 1
65.g odd 4 2 5070.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.b 1 5.b even 2 1
1170.2.a.j 1 15.d odd 2 1
1950.2.a.ba 1 1.a even 1 1 trivial
1950.2.e.m 2 5.c odd 4 2
3120.2.a.y 1 20.d odd 2 1
5070.2.a.n 1 65.d even 2 1
5070.2.b.f 2 65.g odd 4 2
5850.2.a.s 1 3.b odd 2 1
5850.2.e.h 2 15.e even 4 2
9360.2.a.v 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1950))$$:

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

## Hecke characteristic polynomials

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