# Properties

 Label 1850.2.a.v Level $1850$ Weight $2$ Character orbit 1850.a Self dual yes Analytic conductor $14.772$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.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: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + ( - \beta + 2) q^{7} + q^{8} + 3 q^{9}+O(q^{10})$$ q + q^2 + b * q^3 + q^4 + b * q^6 + (-b + 2) * q^7 + q^8 + 3 * q^9 $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + ( - \beta + 2) q^{7} + q^{8} + 3 q^{9} - 2 \beta q^{11} + \beta q^{12} + 4 q^{13} + ( - \beta + 2) q^{14} + q^{16} + 2 \beta q^{17} + 3 q^{18} + (\beta + 6) q^{19} + (2 \beta - 6) q^{21} - 2 \beta q^{22} + (2 \beta - 4) q^{23} + \beta q^{24} + 4 q^{26} + ( - \beta + 2) q^{28} + ( - \beta - 4) q^{31} + q^{32} - 12 q^{33} + 2 \beta q^{34} + 3 q^{36} + q^{37} + (\beta + 6) q^{38} + 4 \beta q^{39} + 2 q^{41} + (2 \beta - 6) q^{42} + 4 q^{43} - 2 \beta q^{44} + (2 \beta - 4) q^{46} + ( - \beta + 2) q^{47} + \beta q^{48} + ( - 4 \beta + 3) q^{49} + 12 q^{51} + 4 q^{52} + (4 \beta - 2) q^{53} + ( - \beta + 2) q^{56} + (6 \beta + 6) q^{57} + ( - \beta - 6) q^{59} + 12 q^{61} + ( - \beta - 4) q^{62} + ( - 3 \beta + 6) q^{63} + q^{64} - 12 q^{66} + ( - \beta - 8) q^{67} + 2 \beta q^{68} + ( - 4 \beta + 12) q^{69} - 2 \beta q^{71} + 3 q^{72} + 4 q^{73} + q^{74} + (\beta + 6) q^{76} + ( - 4 \beta + 12) q^{77} + 4 \beta q^{78} + ( - \beta + 4) q^{79} - 9 q^{81} + 2 q^{82} + ( - \beta - 12) q^{83} + (2 \beta - 6) q^{84} + 4 q^{86} - 2 \beta q^{88} + ( - 4 \beta + 6) q^{89} + ( - 4 \beta + 8) q^{91} + (2 \beta - 4) q^{92} + ( - 4 \beta - 6) q^{93} + ( - \beta + 2) q^{94} + \beta q^{96} - 2 q^{97} + ( - 4 \beta + 3) q^{98} - 6 \beta q^{99} +O(q^{100})$$ q + q^2 + b * q^3 + q^4 + b * q^6 + (-b + 2) * q^7 + q^8 + 3 * q^9 - 2*b * q^11 + b * q^12 + 4 * q^13 + (-b + 2) * q^14 + q^16 + 2*b * q^17 + 3 * q^18 + (b + 6) * q^19 + (2*b - 6) * q^21 - 2*b * q^22 + (2*b - 4) * q^23 + b * q^24 + 4 * q^26 + (-b + 2) * q^28 + (-b - 4) * q^31 + q^32 - 12 * q^33 + 2*b * q^34 + 3 * q^36 + q^37 + (b + 6) * q^38 + 4*b * q^39 + 2 * q^41 + (2*b - 6) * q^42 + 4 * q^43 - 2*b * q^44 + (2*b - 4) * q^46 + (-b + 2) * q^47 + b * q^48 + (-4*b + 3) * q^49 + 12 * q^51 + 4 * q^52 + (4*b - 2) * q^53 + (-b + 2) * q^56 + (6*b + 6) * q^57 + (-b - 6) * q^59 + 12 * q^61 + (-b - 4) * q^62 + (-3*b + 6) * q^63 + q^64 - 12 * q^66 + (-b - 8) * q^67 + 2*b * q^68 + (-4*b + 12) * q^69 - 2*b * q^71 + 3 * q^72 + 4 * q^73 + q^74 + (b + 6) * q^76 + (-4*b + 12) * q^77 + 4*b * q^78 + (-b + 4) * q^79 - 9 * q^81 + 2 * q^82 + (-b - 12) * q^83 + (2*b - 6) * q^84 + 4 * q^86 - 2*b * q^88 + (-4*b + 6) * q^89 + (-4*b + 8) * q^91 + (2*b - 4) * q^92 + (-4*b - 6) * q^93 + (-b + 2) * q^94 + b * q^96 - 2 * q^97 + (-4*b + 3) * q^98 - 6*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 6 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9} + 8 q^{13} + 4 q^{14} + 2 q^{16} + 6 q^{18} + 12 q^{19} - 12 q^{21} - 8 q^{23} + 8 q^{26} + 4 q^{28} - 8 q^{31} + 2 q^{32} - 24 q^{33} + 6 q^{36} + 2 q^{37} + 12 q^{38} + 4 q^{41} - 12 q^{42} + 8 q^{43} - 8 q^{46} + 4 q^{47} + 6 q^{49} + 24 q^{51} + 8 q^{52} - 4 q^{53} + 4 q^{56} + 12 q^{57} - 12 q^{59} + 24 q^{61} - 8 q^{62} + 12 q^{63} + 2 q^{64} - 24 q^{66} - 16 q^{67} + 24 q^{69} + 6 q^{72} + 8 q^{73} + 2 q^{74} + 12 q^{76} + 24 q^{77} + 8 q^{79} - 18 q^{81} + 4 q^{82} - 24 q^{83} - 12 q^{84} + 8 q^{86} + 12 q^{89} + 16 q^{91} - 8 q^{92} - 12 q^{93} + 4 q^{94} - 4 q^{97} + 6 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 6 * q^9 + 8 * q^13 + 4 * q^14 + 2 * q^16 + 6 * q^18 + 12 * q^19 - 12 * q^21 - 8 * q^23 + 8 * q^26 + 4 * q^28 - 8 * q^31 + 2 * q^32 - 24 * q^33 + 6 * q^36 + 2 * q^37 + 12 * q^38 + 4 * q^41 - 12 * q^42 + 8 * q^43 - 8 * q^46 + 4 * q^47 + 6 * q^49 + 24 * q^51 + 8 * q^52 - 4 * q^53 + 4 * q^56 + 12 * q^57 - 12 * q^59 + 24 * q^61 - 8 * q^62 + 12 * q^63 + 2 * q^64 - 24 * q^66 - 16 * q^67 + 24 * q^69 + 6 * q^72 + 8 * q^73 + 2 * q^74 + 12 * q^76 + 24 * q^77 + 8 * q^79 - 18 * q^81 + 4 * q^82 - 24 * q^83 - 12 * q^84 + 8 * q^86 + 12 * q^89 + 16 * q^91 - 8 * q^92 - 12 * q^93 + 4 * q^94 - 4 * q^97 + 6 * q^98

## 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
 −2.44949 2.44949
1.00000 −2.44949 1.00000 0 −2.44949 4.44949 1.00000 3.00000 0
1.2 1.00000 2.44949 1.00000 0 2.44949 −0.449490 1.00000 3.00000 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$37$$ $$-1$$

## 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 1850.2.a.v 2
5.b even 2 1 1850.2.a.s 2
5.c odd 4 2 370.2.b.c 4
15.e even 4 2 3330.2.d.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.b.c 4 5.c odd 4 2
1850.2.a.s 2 5.b even 2 1
1850.2.a.v 2 1.a even 1 1 trivial
3330.2.d.m 4 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1850))$$:

 $$T_{3}^{2} - 6$$ T3^2 - 6 $$T_{7}^{2} - 4T_{7} - 2$$ T7^2 - 4*T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} - 6$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 4T - 2$$
$11$ $$T^{2} - 24$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2} - 24$$
$19$ $$T^{2} - 12T + 30$$
$23$ $$T^{2} + 8T - 8$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 8T + 10$$
$37$ $$(T - 1)^{2}$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} - 4T - 2$$
$53$ $$T^{2} + 4T - 92$$
$59$ $$T^{2} + 12T + 30$$
$61$ $$(T - 12)^{2}$$
$67$ $$T^{2} + 16T + 58$$
$71$ $$T^{2} - 24$$
$73$ $$(T - 4)^{2}$$
$79$ $$T^{2} - 8T + 10$$
$83$ $$T^{2} + 24T + 138$$
$89$ $$T^{2} - 12T - 60$$
$97$ $$(T + 2)^{2}$$