# Properties

 Label 950.2.a.b Level $950$ Weight $2$ Character orbit 950.a Self dual yes Analytic conductor $7.586$ 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] = [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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} - 3 q^{7} - q^{8} - 2 q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^6 - 3 * q^7 - q^8 - 2 * q^9 $$q - q^{2} + q^{3} + q^{4} - q^{6} - 3 q^{7} - q^{8} - 2 q^{9} + 2 q^{11} + q^{12} + q^{13} + 3 q^{14} + q^{16} - 3 q^{17} + 2 q^{18} - q^{19} - 3 q^{21} - 2 q^{22} + q^{23} - q^{24} - q^{26} - 5 q^{27} - 3 q^{28} - 5 q^{29} - 8 q^{31} - q^{32} + 2 q^{33} + 3 q^{34} - 2 q^{36} + 2 q^{37} + q^{38} + q^{39} - 8 q^{41} + 3 q^{42} - 4 q^{43} + 2 q^{44} - q^{46} - 8 q^{47} + q^{48} + 2 q^{49} - 3 q^{51} + q^{52} + q^{53} + 5 q^{54} + 3 q^{56} - q^{57} + 5 q^{58} + 15 q^{59} + 2 q^{61} + 8 q^{62} + 6 q^{63} + q^{64} - 2 q^{66} - 3 q^{67} - 3 q^{68} + q^{69} + 2 q^{71} + 2 q^{72} - 9 q^{73} - 2 q^{74} - q^{76} - 6 q^{77} - q^{78} - 10 q^{79} + q^{81} + 8 q^{82} + 6 q^{83} - 3 q^{84} + 4 q^{86} - 5 q^{87} - 2 q^{88} - 3 q^{91} + q^{92} - 8 q^{93} + 8 q^{94} - q^{96} + 2 q^{97} - 2 q^{98} - 4 q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^6 - 3 * q^7 - q^8 - 2 * q^9 + 2 * q^11 + q^12 + q^13 + 3 * q^14 + q^16 - 3 * q^17 + 2 * q^18 - q^19 - 3 * q^21 - 2 * q^22 + q^23 - q^24 - q^26 - 5 * q^27 - 3 * q^28 - 5 * q^29 - 8 * q^31 - q^32 + 2 * q^33 + 3 * q^34 - 2 * q^36 + 2 * q^37 + q^38 + q^39 - 8 * q^41 + 3 * q^42 - 4 * q^43 + 2 * q^44 - q^46 - 8 * q^47 + q^48 + 2 * q^49 - 3 * q^51 + q^52 + q^53 + 5 * q^54 + 3 * q^56 - q^57 + 5 * q^58 + 15 * q^59 + 2 * q^61 + 8 * q^62 + 6 * q^63 + q^64 - 2 * q^66 - 3 * q^67 - 3 * q^68 + q^69 + 2 * q^71 + 2 * q^72 - 9 * q^73 - 2 * q^74 - q^76 - 6 * q^77 - q^78 - 10 * q^79 + q^81 + 8 * q^82 + 6 * q^83 - 3 * q^84 + 4 * q^86 - 5 * q^87 - 2 * q^88 - 3 * q^91 + q^92 - 8 * q^93 + 8 * q^94 - q^96 + 2 * q^97 - 2 * 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 −3.00000 −1.00000 −2.00000 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
$$2$$ $$+1$$
$$5$$ $$+1$$
$$19$$ $$+1$$

## 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.b 1
3.b odd 2 1 8550.2.a.u 1
4.b odd 2 1 7600.2.a.h 1
5.b even 2 1 38.2.a.b 1
5.c odd 4 2 950.2.b.c 2
15.d odd 2 1 342.2.a.d 1
20.d odd 2 1 304.2.a.d 1
35.c odd 2 1 1862.2.a.f 1
40.e odd 2 1 1216.2.a.g 1
40.f even 2 1 1216.2.a.n 1
55.d odd 2 1 4598.2.a.a 1
60.h even 2 1 2736.2.a.w 1
65.d even 2 1 6422.2.a.b 1
95.d odd 2 1 722.2.a.b 1
95.h odd 6 2 722.2.c.f 2
95.i even 6 2 722.2.c.d 2
95.o odd 18 6 722.2.e.d 6
95.p even 18 6 722.2.e.c 6
285.b even 2 1 6498.2.a.y 1
380.d even 2 1 5776.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 5.b even 2 1
304.2.a.d 1 20.d odd 2 1
342.2.a.d 1 15.d odd 2 1
722.2.a.b 1 95.d odd 2 1
722.2.c.d 2 95.i even 6 2
722.2.c.f 2 95.h odd 6 2
722.2.e.c 6 95.p even 18 6
722.2.e.d 6 95.o odd 18 6
950.2.a.b 1 1.a even 1 1 trivial
950.2.b.c 2 5.c odd 4 2
1216.2.a.g 1 40.e odd 2 1
1216.2.a.n 1 40.f even 2 1
1862.2.a.f 1 35.c odd 2 1
2736.2.a.w 1 60.h even 2 1
4598.2.a.a 1 55.d odd 2 1
5776.2.a.d 1 380.d even 2 1
6422.2.a.b 1 65.d even 2 1
6498.2.a.y 1 285.b even 2 1
7600.2.a.h 1 4.b odd 2 1
8550.2.a.u 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(950))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} + 3$$ T7 + 3 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 3$$
$11$ $$T - 2$$
$13$ $$T - 1$$
$17$ $$T + 3$$
$19$ $$T + 1$$
$23$ $$T - 1$$
$29$ $$T + 5$$
$31$ $$T + 8$$
$37$ $$T - 2$$
$41$ $$T + 8$$
$43$ $$T + 4$$
$47$ $$T + 8$$
$53$ $$T - 1$$
$59$ $$T - 15$$
$61$ $$T - 2$$
$67$ $$T + 3$$
$71$ $$T - 2$$
$73$ $$T + 9$$
$79$ $$T + 10$$
$83$ $$T - 6$$
$89$ $$T$$
$97$ $$T - 2$$