# Properties

 Label 1850.2.a.r Level $1850$ Weight $2$ Character orbit 1850.a Self dual yes Analytic conductor $14.772$ Analytic rank $1$ 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: $$1$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + (\beta - 1) q^{3} + q^{4} + ( - \beta + 1) q^{6} - \beta q^{7} - q^{8} + ( - 2 \beta + 4) q^{9} +O(q^{10})$$ q - q^2 + (b - 1) * q^3 + q^4 + (-b + 1) * q^6 - b * q^7 - q^8 + (-2*b + 4) * q^9 $$q - q^{2} + (\beta - 1) q^{3} + q^{4} + ( - \beta + 1) q^{6} - \beta q^{7} - q^{8} + ( - 2 \beta + 4) q^{9} + (\beta + 1) q^{11} + (\beta - 1) q^{12} + (\beta - 2) q^{13} + \beta q^{14} + q^{16} + ( - \beta - 1) q^{17} + (2 \beta - 4) q^{18} - 5 q^{19} + (\beta - 6) q^{21} + ( - \beta - 1) q^{22} - 2 q^{23} + ( - \beta + 1) q^{24} + ( - \beta + 2) q^{26} + (3 \beta - 13) q^{27} - \beta q^{28} + ( - 2 \beta + 4) q^{29} + (\beta + 2) q^{31} - q^{32} + 5 q^{33} + (\beta + 1) q^{34} + ( - 2 \beta + 4) q^{36} + q^{37} + 5 q^{38} + ( - 3 \beta + 8) q^{39} + q^{41} + ( - \beta + 6) q^{42} + ( - 2 \beta + 6) q^{43} + (\beta + 1) q^{44} + 2 q^{46} - 4 \beta q^{47} + (\beta - 1) q^{48} - q^{49} - 5 q^{51} + (\beta - 2) q^{52} - 6 q^{53} + ( - 3 \beta + 13) q^{54} + \beta q^{56} + ( - 5 \beta + 5) q^{57} + (2 \beta - 4) q^{58} - 2 q^{59} + ( - \beta - 4) q^{61} + ( - \beta - 2) q^{62} + ( - 4 \beta + 12) q^{63} + q^{64} - 5 q^{66} + (\beta - 7) q^{67} + ( - \beta - 1) q^{68} + ( - 2 \beta + 2) q^{69} + (\beta - 10) q^{71} + (2 \beta - 4) q^{72} + ( - 4 \beta - 3) q^{73} - q^{74} - 5 q^{76} + ( - \beta - 6) q^{77} + (3 \beta - 8) q^{78} + (4 \beta - 2) q^{79} + ( - 10 \beta + 19) q^{81} - q^{82} + ( - \beta - 1) q^{83} + (\beta - 6) q^{84} + (2 \beta - 6) q^{86} + (6 \beta - 16) q^{87} + ( - \beta - 1) q^{88} + (3 \beta + 7) q^{89} + (2 \beta - 6) q^{91} - 2 q^{92} + (\beta + 4) q^{93} + 4 \beta q^{94} + ( - \beta + 1) q^{96} - 14 q^{97} + q^{98} + (2 \beta - 8) q^{99} +O(q^{100})$$ q - q^2 + (b - 1) * q^3 + q^4 + (-b + 1) * q^6 - b * q^7 - q^8 + (-2*b + 4) * q^9 + (b + 1) * q^11 + (b - 1) * q^12 + (b - 2) * q^13 + b * q^14 + q^16 + (-b - 1) * q^17 + (2*b - 4) * q^18 - 5 * q^19 + (b - 6) * q^21 + (-b - 1) * q^22 - 2 * q^23 + (-b + 1) * q^24 + (-b + 2) * q^26 + (3*b - 13) * q^27 - b * q^28 + (-2*b + 4) * q^29 + (b + 2) * q^31 - q^32 + 5 * q^33 + (b + 1) * q^34 + (-2*b + 4) * q^36 + q^37 + 5 * q^38 + (-3*b + 8) * q^39 + q^41 + (-b + 6) * q^42 + (-2*b + 6) * q^43 + (b + 1) * q^44 + 2 * q^46 - 4*b * q^47 + (b - 1) * q^48 - q^49 - 5 * q^51 + (b - 2) * q^52 - 6 * q^53 + (-3*b + 13) * q^54 + b * q^56 + (-5*b + 5) * q^57 + (2*b - 4) * q^58 - 2 * q^59 + (-b - 4) * q^61 + (-b - 2) * q^62 + (-4*b + 12) * q^63 + q^64 - 5 * q^66 + (b - 7) * q^67 + (-b - 1) * q^68 + (-2*b + 2) * q^69 + (b - 10) * q^71 + (2*b - 4) * q^72 + (-4*b - 3) * q^73 - q^74 - 5 * q^76 + (-b - 6) * q^77 + (3*b - 8) * q^78 + (4*b - 2) * q^79 + (-10*b + 19) * q^81 - q^82 + (-b - 1) * q^83 + (b - 6) * q^84 + (2*b - 6) * q^86 + (6*b - 16) * q^87 + (-b - 1) * q^88 + (3*b + 7) * q^89 + (2*b - 6) * q^91 - 2 * q^92 + (b + 4) * q^93 + 4*b * q^94 + (-b + 1) * q^96 - 14 * q^97 + q^98 + (2*b - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 8 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 8 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 8 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{17} - 8 q^{18} - 10 q^{19} - 12 q^{21} - 2 q^{22} - 4 q^{23} + 2 q^{24} + 4 q^{26} - 26 q^{27} + 8 q^{29} + 4 q^{31} - 2 q^{32} + 10 q^{33} + 2 q^{34} + 8 q^{36} + 2 q^{37} + 10 q^{38} + 16 q^{39} + 2 q^{41} + 12 q^{42} + 12 q^{43} + 2 q^{44} + 4 q^{46} - 2 q^{48} - 2 q^{49} - 10 q^{51} - 4 q^{52} - 12 q^{53} + 26 q^{54} + 10 q^{57} - 8 q^{58} - 4 q^{59} - 8 q^{61} - 4 q^{62} + 24 q^{63} + 2 q^{64} - 10 q^{66} - 14 q^{67} - 2 q^{68} + 4 q^{69} - 20 q^{71} - 8 q^{72} - 6 q^{73} - 2 q^{74} - 10 q^{76} - 12 q^{77} - 16 q^{78} - 4 q^{79} + 38 q^{81} - 2 q^{82} - 2 q^{83} - 12 q^{84} - 12 q^{86} - 32 q^{87} - 2 q^{88} + 14 q^{89} - 12 q^{91} - 4 q^{92} + 8 q^{93} + 2 q^{96} - 28 q^{97} + 2 q^{98} - 16 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 8 * q^9 + 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 2 * q^17 - 8 * q^18 - 10 * q^19 - 12 * q^21 - 2 * q^22 - 4 * q^23 + 2 * q^24 + 4 * q^26 - 26 * q^27 + 8 * q^29 + 4 * q^31 - 2 * q^32 + 10 * q^33 + 2 * q^34 + 8 * q^36 + 2 * q^37 + 10 * q^38 + 16 * q^39 + 2 * q^41 + 12 * q^42 + 12 * q^43 + 2 * q^44 + 4 * q^46 - 2 * q^48 - 2 * q^49 - 10 * q^51 - 4 * q^52 - 12 * q^53 + 26 * q^54 + 10 * q^57 - 8 * q^58 - 4 * q^59 - 8 * q^61 - 4 * q^62 + 24 * q^63 + 2 * q^64 - 10 * q^66 - 14 * q^67 - 2 * q^68 + 4 * q^69 - 20 * q^71 - 8 * q^72 - 6 * q^73 - 2 * q^74 - 10 * q^76 - 12 * q^77 - 16 * q^78 - 4 * q^79 + 38 * q^81 - 2 * q^82 - 2 * q^83 - 12 * q^84 - 12 * q^86 - 32 * q^87 - 2 * q^88 + 14 * q^89 - 12 * q^91 - 4 * q^92 + 8 * q^93 + 2 * q^96 - 28 * q^97 + 2 * q^98 - 16 * 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
 −2.44949 2.44949
−1.00000 −3.44949 1.00000 0 3.44949 2.44949 −1.00000 8.89898 0
1.2 −1.00000 1.44949 1.00000 0 −1.44949 −2.44949 −1.00000 −0.898979 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.r 2
5.b even 2 1 1850.2.a.w yes 2
5.c odd 4 2 1850.2.b.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.r 2 1.a even 1 1 trivial
1850.2.a.w yes 2 5.b even 2 1
1850.2.b.k 4 5.c odd 4 2

## Hecke kernels

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

 $$T_{3}^{2} + 2T_{3} - 5$$ T3^2 + 2*T3 - 5 $$T_{7}^{2} - 6$$ T7^2 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + 2T - 5$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 6$$
$11$ $$T^{2} - 2T - 5$$
$13$ $$T^{2} + 4T - 2$$
$17$ $$T^{2} + 2T - 5$$
$19$ $$(T + 5)^{2}$$
$23$ $$(T + 2)^{2}$$
$29$ $$T^{2} - 8T - 8$$
$31$ $$T^{2} - 4T - 2$$
$37$ $$(T - 1)^{2}$$
$41$ $$(T - 1)^{2}$$
$43$ $$T^{2} - 12T + 12$$
$47$ $$T^{2} - 96$$
$53$ $$(T + 6)^{2}$$
$59$ $$(T + 2)^{2}$$
$61$ $$T^{2} + 8T + 10$$
$67$ $$T^{2} + 14T + 43$$
$71$ $$T^{2} + 20T + 94$$
$73$ $$T^{2} + 6T - 87$$
$79$ $$T^{2} + 4T - 92$$
$83$ $$T^{2} + 2T - 5$$
$89$ $$T^{2} - 14T - 5$$
$97$ $$(T + 14)^{2}$$