# Properties

 Label 1850.2.a.x 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: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + (\beta + 1) q^{3} + q^{4} + (\beta + 1) q^{6} + ( - \beta + 3) q^{7} + q^{8} + (2 \beta + 1) q^{9}+O(q^{10})$$ q + q^2 + (b + 1) * q^3 + q^4 + (b + 1) * q^6 + (-b + 3) * q^7 + q^8 + (2*b + 1) * q^9 $$q + q^{2} + (\beta + 1) q^{3} + q^{4} + (\beta + 1) q^{6} + ( - \beta + 3) q^{7} + q^{8} + (2 \beta + 1) q^{9} + (2 \beta - 2) q^{11} + (\beta + 1) q^{12} + ( - 2 \beta + 2) q^{13} + ( - \beta + 3) q^{14} + q^{16} + (2 \beta - 2) q^{17} + (2 \beta + 1) q^{18} + ( - 3 \beta + 1) q^{19} + 2 \beta q^{21} + (2 \beta - 2) q^{22} + 8 q^{23} + (\beta + 1) q^{24} + ( - 2 \beta + 2) q^{26} + 4 q^{27} + ( - \beta + 3) q^{28} + ( - 4 \beta - 2) q^{29} + ( - \beta - 1) q^{31} + q^{32} + 4 q^{33} + (2 \beta - 2) q^{34} + (2 \beta + 1) q^{36} - q^{37} + ( - 3 \beta + 1) q^{38} - 4 q^{39} - 2 q^{41} + 2 \beta q^{42} + 4 \beta q^{43} + (2 \beta - 2) q^{44} + 8 q^{46} + ( - \beta + 3) q^{47} + (\beta + 1) q^{48} + ( - 6 \beta + 5) q^{49} + 4 q^{51} + ( - 2 \beta + 2) q^{52} + 6 q^{53} + 4 q^{54} + ( - \beta + 3) q^{56} + ( - 2 \beta - 8) q^{57} + ( - 4 \beta - 2) q^{58} + (3 \beta - 5) q^{59} + (4 \beta + 2) q^{61} + ( - \beta - 1) q^{62} + (5 \beta - 3) q^{63} + q^{64} + 4 q^{66} + ( - 5 \beta - 5) q^{67} + (2 \beta - 2) q^{68} + (8 \beta + 8) q^{69} + ( - 4 \beta - 4) q^{71} + (2 \beta + 1) q^{72} + ( - 4 \beta - 6) q^{73} - q^{74} + ( - 3 \beta + 1) q^{76} + (8 \beta - 12) q^{77} - 4 q^{78} + ( - \beta + 7) q^{79} + ( - 2 \beta + 1) q^{81} - 2 q^{82} + ( - \beta + 7) q^{83} + 2 \beta q^{84} + 4 \beta q^{86} + ( - 6 \beta - 14) q^{87} + (2 \beta - 2) q^{88} - 2 q^{89} + ( - 8 \beta + 12) q^{91} + 8 q^{92} + ( - 2 \beta - 4) q^{93} + ( - \beta + 3) q^{94} + (\beta + 1) q^{96} + 2 q^{97} + ( - 6 \beta + 5) q^{98} + ( - 2 \beta + 10) q^{99}+O(q^{100})$$ q + q^2 + (b + 1) * q^3 + q^4 + (b + 1) * q^6 + (-b + 3) * q^7 + q^8 + (2*b + 1) * q^9 + (2*b - 2) * q^11 + (b + 1) * q^12 + (-2*b + 2) * q^13 + (-b + 3) * q^14 + q^16 + (2*b - 2) * q^17 + (2*b + 1) * q^18 + (-3*b + 1) * q^19 + 2*b * q^21 + (2*b - 2) * q^22 + 8 * q^23 + (b + 1) * q^24 + (-2*b + 2) * q^26 + 4 * q^27 + (-b + 3) * q^28 + (-4*b - 2) * q^29 + (-b - 1) * q^31 + q^32 + 4 * q^33 + (2*b - 2) * q^34 + (2*b + 1) * q^36 - q^37 + (-3*b + 1) * q^38 - 4 * q^39 - 2 * q^41 + 2*b * q^42 + 4*b * q^43 + (2*b - 2) * q^44 + 8 * q^46 + (-b + 3) * q^47 + (b + 1) * q^48 + (-6*b + 5) * q^49 + 4 * q^51 + (-2*b + 2) * q^52 + 6 * q^53 + 4 * q^54 + (-b + 3) * q^56 + (-2*b - 8) * q^57 + (-4*b - 2) * q^58 + (3*b - 5) * q^59 + (4*b + 2) * q^61 + (-b - 1) * q^62 + (5*b - 3) * q^63 + q^64 + 4 * q^66 + (-5*b - 5) * q^67 + (2*b - 2) * q^68 + (8*b + 8) * q^69 + (-4*b - 4) * q^71 + (2*b + 1) * q^72 + (-4*b - 6) * q^73 - q^74 + (-3*b + 1) * q^76 + (8*b - 12) * q^77 - 4 * q^78 + (-b + 7) * q^79 + (-2*b + 1) * q^81 - 2 * q^82 + (-b + 7) * q^83 + 2*b * q^84 + 4*b * q^86 + (-6*b - 14) * q^87 + (2*b - 2) * q^88 - 2 * q^89 + (-8*b + 12) * q^91 + 8 * q^92 + (-2*b - 4) * q^93 + (-b + 3) * q^94 + (b + 1) * q^96 + 2 * q^97 + (-6*b + 5) * q^98 + (-2*b + 10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} + 4 q^{13} + 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{19} - 4 q^{22} + 16 q^{23} + 2 q^{24} + 4 q^{26} + 8 q^{27} + 6 q^{28} - 4 q^{29} - 2 q^{31} + 2 q^{32} + 8 q^{33} - 4 q^{34} + 2 q^{36} - 2 q^{37} + 2 q^{38} - 8 q^{39} - 4 q^{41} - 4 q^{44} + 16 q^{46} + 6 q^{47} + 2 q^{48} + 10 q^{49} + 8 q^{51} + 4 q^{52} + 12 q^{53} + 8 q^{54} + 6 q^{56} - 16 q^{57} - 4 q^{58} - 10 q^{59} + 4 q^{61} - 2 q^{62} - 6 q^{63} + 2 q^{64} + 8 q^{66} - 10 q^{67} - 4 q^{68} + 16 q^{69} - 8 q^{71} + 2 q^{72} - 12 q^{73} - 2 q^{74} + 2 q^{76} - 24 q^{77} - 8 q^{78} + 14 q^{79} + 2 q^{81} - 4 q^{82} + 14 q^{83} - 28 q^{87} - 4 q^{88} - 4 q^{89} + 24 q^{91} + 16 q^{92} - 8 q^{93} + 6 q^{94} + 2 q^{96} + 4 q^{97} + 10 q^{98} + 20 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9 - 4 * q^11 + 2 * q^12 + 4 * q^13 + 6 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 2 * q^19 - 4 * q^22 + 16 * q^23 + 2 * q^24 + 4 * q^26 + 8 * q^27 + 6 * q^28 - 4 * q^29 - 2 * q^31 + 2 * q^32 + 8 * q^33 - 4 * q^34 + 2 * q^36 - 2 * q^37 + 2 * q^38 - 8 * q^39 - 4 * q^41 - 4 * q^44 + 16 * q^46 + 6 * q^47 + 2 * q^48 + 10 * q^49 + 8 * q^51 + 4 * q^52 + 12 * q^53 + 8 * q^54 + 6 * q^56 - 16 * q^57 - 4 * q^58 - 10 * q^59 + 4 * q^61 - 2 * q^62 - 6 * q^63 + 2 * q^64 + 8 * q^66 - 10 * q^67 - 4 * q^68 + 16 * q^69 - 8 * q^71 + 2 * q^72 - 12 * q^73 - 2 * q^74 + 2 * q^76 - 24 * q^77 - 8 * q^78 + 14 * q^79 + 2 * q^81 - 4 * q^82 + 14 * q^83 - 28 * q^87 - 4 * q^88 - 4 * q^89 + 24 * q^91 + 16 * q^92 - 8 * q^93 + 6 * q^94 + 2 * q^96 + 4 * q^97 + 10 * q^98 + 20 * 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
 −1.73205 1.73205
1.00000 −0.732051 1.00000 0 −0.732051 4.73205 1.00000 −2.46410 0
1.2 1.00000 2.73205 1.00000 0 2.73205 1.26795 1.00000 4.46410 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.x 2
5.b even 2 1 370.2.a.e 2
5.c odd 4 2 1850.2.b.l 4
15.d odd 2 1 3330.2.a.bd 2
20.d odd 2 1 2960.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.e 2 5.b even 2 1
1850.2.a.x 2 1.a even 1 1 trivial
1850.2.b.l 4 5.c odd 4 2
2960.2.a.q 2 20.d odd 2 1
3330.2.a.bd 2 15.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1850))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} - 2T - 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 6T + 6$$
$11$ $$T^{2} + 4T - 8$$
$13$ $$T^{2} - 4T - 8$$
$17$ $$T^{2} + 4T - 8$$
$19$ $$T^{2} - 2T - 26$$
$23$ $$(T - 8)^{2}$$
$29$ $$T^{2} + 4T - 44$$
$31$ $$T^{2} + 2T - 2$$
$37$ $$(T + 1)^{2}$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} - 48$$
$47$ $$T^{2} - 6T + 6$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 10T - 2$$
$61$ $$T^{2} - 4T - 44$$
$67$ $$T^{2} + 10T - 50$$
$71$ $$T^{2} + 8T - 32$$
$73$ $$T^{2} + 12T - 12$$
$79$ $$T^{2} - 14T + 46$$
$83$ $$T^{2} - 14T + 46$$
$89$ $$(T + 2)^{2}$$
$97$ $$(T - 2)^{2}$$