# Properties

 Label 1950.2.a.w Level $1950$ Weight $2$ Character orbit 1950.a Self dual yes Analytic conductor $15.571$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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 $$q + q^{2} + q^{3} + q^{4} + q^{6} - 4 q^{7} + q^{8} + q^{9} - 4 q^{11} + q^{12} - q^{13} - 4 q^{14} + q^{16} - 2 q^{17} + q^{18} - 8 q^{19} - 4 q^{21} - 4 q^{22} + q^{24} - q^{26} + q^{27} - 4 q^{28} + 6 q^{29} - 4 q^{31} + q^{32} - 4 q^{33} - 2 q^{34} + q^{36} + 2 q^{37} - 8 q^{38} - q^{39} - 10 q^{41} - 4 q^{42} - 4 q^{43} - 4 q^{44} - 8 q^{47} + q^{48} + 9 q^{49} - 2 q^{51} - q^{52} + 10 q^{53} + q^{54} - 4 q^{56} - 8 q^{57} + 6 q^{58} + 4 q^{59} - 2 q^{61} - 4 q^{62} - 4 q^{63} + q^{64} - 4 q^{66} + 16 q^{67} - 2 q^{68} - 8 q^{71} + q^{72} - 2 q^{73} + 2 q^{74} - 8 q^{76} + 16 q^{77} - q^{78} + 8 q^{79} + q^{81} - 10 q^{82} - 12 q^{83} - 4 q^{84} - 4 q^{86} + 6 q^{87} - 4 q^{88} + 14 q^{89} + 4 q^{91} - 4 q^{93} - 8 q^{94} + q^{96} - 10 q^{97} + 9 q^{98} - 4 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 - 4 * q^7 + q^8 + q^9 - 4 * q^11 + q^12 - q^13 - 4 * q^14 + q^16 - 2 * q^17 + q^18 - 8 * q^19 - 4 * q^21 - 4 * q^22 + q^24 - q^26 + q^27 - 4 * q^28 + 6 * q^29 - 4 * q^31 + q^32 - 4 * q^33 - 2 * q^34 + q^36 + 2 * q^37 - 8 * q^38 - q^39 - 10 * q^41 - 4 * q^42 - 4 * q^43 - 4 * q^44 - 8 * q^47 + q^48 + 9 * q^49 - 2 * q^51 - q^52 + 10 * q^53 + q^54 - 4 * q^56 - 8 * q^57 + 6 * q^58 + 4 * q^59 - 2 * q^61 - 4 * q^62 - 4 * q^63 + q^64 - 4 * q^66 + 16 * q^67 - 2 * q^68 - 8 * q^71 + q^72 - 2 * q^73 + 2 * q^74 - 8 * q^76 + 16 * q^77 - q^78 + 8 * q^79 + q^81 - 10 * q^82 - 12 * q^83 - 4 * q^84 - 4 * q^86 + 6 * q^87 - 4 * q^88 + 14 * q^89 + 4 * q^91 - 4 * q^93 - 8 * q^94 + q^96 - 10 * q^97 + 9 * 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 −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.w 1
3.b odd 2 1 5850.2.a.d 1
5.b even 2 1 78.2.a.a 1
5.c odd 4 2 1950.2.e.i 2
15.d odd 2 1 234.2.a.c 1
15.e even 4 2 5850.2.e.bb 2
20.d odd 2 1 624.2.a.h 1
35.c odd 2 1 3822.2.a.j 1
40.e odd 2 1 2496.2.a.b 1
40.f even 2 1 2496.2.a.t 1
45.h odd 6 2 2106.2.e.j 2
45.j even 6 2 2106.2.e.q 2
55.d odd 2 1 9438.2.a.t 1
60.h even 2 1 1872.2.a.c 1
65.d even 2 1 1014.2.a.d 1
65.g odd 4 2 1014.2.b.b 2
65.l even 6 2 1014.2.e.c 2
65.n even 6 2 1014.2.e.f 2
65.s odd 12 4 1014.2.i.d 4
120.i odd 2 1 7488.2.a.bz 1
120.m even 2 1 7488.2.a.bk 1
195.e odd 2 1 3042.2.a.f 1
195.n even 4 2 3042.2.b.g 2
260.g odd 2 1 8112.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 5.b even 2 1
234.2.a.c 1 15.d odd 2 1
624.2.a.h 1 20.d odd 2 1
1014.2.a.d 1 65.d even 2 1
1014.2.b.b 2 65.g odd 4 2
1014.2.e.c 2 65.l even 6 2
1014.2.e.f 2 65.n even 6 2
1014.2.i.d 4 65.s odd 12 4
1872.2.a.c 1 60.h even 2 1
1950.2.a.w 1 1.a even 1 1 trivial
1950.2.e.i 2 5.c odd 4 2
2106.2.e.j 2 45.h odd 6 2
2106.2.e.q 2 45.j even 6 2
2496.2.a.b 1 40.e odd 2 1
2496.2.a.t 1 40.f even 2 1
3042.2.a.f 1 195.e odd 2 1
3042.2.b.g 2 195.n even 4 2
3822.2.a.j 1 35.c odd 2 1
5850.2.a.d 1 3.b odd 2 1
5850.2.e.bb 2 15.e even 4 2
7488.2.a.bk 1 120.m even 2 1
7488.2.a.bz 1 120.i odd 2 1
8112.2.a.v 1 260.g odd 2 1
9438.2.a.t 1 55.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1950))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T + 4$$
$13$ $$T + 1$$
$17$ $$T + 2$$
$19$ $$T + 8$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T + 4$$
$37$ $$T - 2$$
$41$ $$T + 10$$
$43$ $$T + 4$$
$47$ $$T + 8$$
$53$ $$T - 10$$
$59$ $$T - 4$$
$61$ $$T + 2$$
$67$ $$T - 16$$
$71$ $$T + 8$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T + 12$$
$89$ $$T - 14$$
$97$ $$T + 10$$