# Properties

 Label 2850.2.a.x Level $2850$ Weight $2$ Character orbit 2850.a Self dual yes Analytic conductor $22.757$ 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] = [2850,2,Mod(1,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, 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("2850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.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: $$22.7573645761$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) 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 $$q + q^{2} + q^{3} + q^{4} + q^{6} - 4 q^{7} + q^{8} + q^{9} + 4 q^{11} + q^{12} - 4 q^{14} + q^{16} + 2 q^{17} + q^{18} + q^{19} - 4 q^{21} + 4 q^{22} + 2 q^{23} + q^{24} + q^{27} - 4 q^{28} - 6 q^{29} + 6 q^{31} + q^{32} + 4 q^{33} + 2 q^{34} + q^{36} + 8 q^{37} + q^{38} + 10 q^{41} - 4 q^{42} + 12 q^{43} + 4 q^{44} + 2 q^{46} - 10 q^{47} + q^{48} + 9 q^{49} + 2 q^{51} - 2 q^{53} + q^{54} - 4 q^{56} + q^{57} - 6 q^{58} + 4 q^{59} - 10 q^{61} + 6 q^{62} - 4 q^{63} + q^{64} + 4 q^{66} + 2 q^{68} + 2 q^{69} - 16 q^{71} + q^{72} + 2 q^{73} + 8 q^{74} + q^{76} - 16 q^{77} + 10 q^{79} + q^{81} + 10 q^{82} + 16 q^{83} - 4 q^{84} + 12 q^{86} - 6 q^{87} + 4 q^{88} - 2 q^{89} + 2 q^{92} + 6 q^{93} - 10 q^{94} + q^{96} + 10 q^{97} + 9 q^{98} + 4 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 - 4 * q^7 + q^8 + q^9 + 4 * q^11 + q^12 - 4 * q^14 + q^16 + 2 * q^17 + q^18 + q^19 - 4 * q^21 + 4 * q^22 + 2 * q^23 + q^24 + q^27 - 4 * q^28 - 6 * q^29 + 6 * q^31 + q^32 + 4 * q^33 + 2 * q^34 + q^36 + 8 * q^37 + q^38 + 10 * q^41 - 4 * q^42 + 12 * q^43 + 4 * q^44 + 2 * q^46 - 10 * q^47 + q^48 + 9 * q^49 + 2 * q^51 - 2 * q^53 + q^54 - 4 * q^56 + q^57 - 6 * q^58 + 4 * q^59 - 10 * q^61 + 6 * q^62 - 4 * q^63 + q^64 + 4 * q^66 + 2 * q^68 + 2 * q^69 - 16 * q^71 + q^72 + 2 * q^73 + 8 * q^74 + q^76 - 16 * q^77 + 10 * q^79 + q^81 + 10 * q^82 + 16 * q^83 - 4 * q^84 + 12 * q^86 - 6 * q^87 + 4 * q^88 - 2 * q^89 + 2 * q^92 + 6 * q^93 - 10 * 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$-1$$

## 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 2850.2.a.x 1
3.b odd 2 1 8550.2.a.a 1
5.b even 2 1 114.2.a.a 1
5.c odd 4 2 2850.2.d.s 2
15.d odd 2 1 342.2.a.f 1
20.d odd 2 1 912.2.a.h 1
35.c odd 2 1 5586.2.a.p 1
40.e odd 2 1 3648.2.a.j 1
40.f even 2 1 3648.2.a.bb 1
60.h even 2 1 2736.2.a.j 1
95.d odd 2 1 2166.2.a.i 1
285.b even 2 1 6498.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.a 1 5.b even 2 1
342.2.a.f 1 15.d odd 2 1
912.2.a.h 1 20.d odd 2 1
2166.2.a.i 1 95.d odd 2 1
2736.2.a.j 1 60.h even 2 1
2850.2.a.x 1 1.a even 1 1 trivial
2850.2.d.s 2 5.c odd 4 2
3648.2.a.j 1 40.e odd 2 1
3648.2.a.bb 1 40.f even 2 1
5586.2.a.p 1 35.c odd 2 1
6498.2.a.h 1 285.b even 2 1
8550.2.a.a 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2850))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} - 4$$ T11 - 4 $$T_{13}$$ T13 $$T_{23} - 2$$ T23 - 2

## Hecke characteristic polynomials

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