# Properties

 Label 1850.2.a.u 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: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( - \beta - 1) q^{3} + q^{4} + ( - \beta - 1) q^{6} + (2 \beta - 2) q^{7} + q^{8} + (3 \beta + 1) q^{9}+O(q^{10})$$ q + q^2 + (-b - 1) * q^3 + q^4 + (-b - 1) * q^6 + (2*b - 2) * q^7 + q^8 + (3*b + 1) * q^9 $$q + q^{2} + ( - \beta - 1) q^{3} + q^{4} + ( - \beta - 1) q^{6} + (2 \beta - 2) q^{7} + q^{8} + (3 \beta + 1) q^{9} - \beta q^{11} + ( - \beta - 1) q^{12} + ( - \beta + 1) q^{13} + (2 \beta - 2) q^{14} + q^{16} + 6 q^{17} + (3 \beta + 1) q^{18} + 2 q^{19} + ( - 2 \beta - 4) q^{21} - \beta q^{22} + ( - 3 \beta + 3) q^{23} + ( - \beta - 1) q^{24} + ( - \beta + 1) q^{26} + ( - 4 \beta - 7) q^{27} + (2 \beta - 2) q^{28} + ( - 3 \beta + 3) q^{29} + ( - \beta + 2) q^{31} + q^{32} + (2 \beta + 3) q^{33} + 6 q^{34} + (3 \beta + 1) q^{36} - q^{37} + 2 q^{38} + (\beta + 2) q^{39} + (3 \beta + 3) q^{41} + ( - 2 \beta - 4) q^{42} + ( - 2 \beta + 4) q^{43} - \beta q^{44} + ( - 3 \beta + 3) q^{46} - 2 \beta q^{47} + ( - \beta - 1) q^{48} + ( - 4 \beta + 9) q^{49} + ( - 6 \beta - 6) q^{51} + ( - \beta + 1) q^{52} + 6 q^{53} + ( - 4 \beta - 7) q^{54} + (2 \beta - 2) q^{56} + ( - 2 \beta - 2) q^{57} + ( - 3 \beta + 3) q^{58} + (2 \beta + 6) q^{59} + (5 \beta - 4) q^{61} + ( - \beta + 2) q^{62} + (2 \beta + 16) q^{63} + q^{64} + (2 \beta + 3) q^{66} + (5 \beta - 8) q^{67} + 6 q^{68} + (3 \beta + 6) q^{69} + 6 q^{71} + (3 \beta + 1) q^{72} + (\beta + 10) q^{73} - q^{74} + 2 q^{76} - 6 q^{77} + (\beta + 2) q^{78} + (7 \beta - 7) q^{79} + (6 \beta + 16) q^{81} + (3 \beta + 3) q^{82} + (4 \beta - 12) q^{83} + ( - 2 \beta - 4) q^{84} + ( - 2 \beta + 4) q^{86} + (3 \beta + 6) q^{87} - \beta q^{88} - 4 \beta q^{89} + (2 \beta - 8) q^{91} + ( - 3 \beta + 3) q^{92} + q^{93} - 2 \beta q^{94} + ( - \beta - 1) q^{96} + (8 \beta - 2) q^{97} + ( - 4 \beta + 9) q^{98} + ( - 4 \beta - 9) q^{99} +O(q^{100})$$ q + q^2 + (-b - 1) * q^3 + q^4 + (-b - 1) * q^6 + (2*b - 2) * q^7 + q^8 + (3*b + 1) * q^9 - b * q^11 + (-b - 1) * q^12 + (-b + 1) * q^13 + (2*b - 2) * q^14 + q^16 + 6 * q^17 + (3*b + 1) * q^18 + 2 * q^19 + (-2*b - 4) * q^21 - b * q^22 + (-3*b + 3) * q^23 + (-b - 1) * q^24 + (-b + 1) * q^26 + (-4*b - 7) * q^27 + (2*b - 2) * q^28 + (-3*b + 3) * q^29 + (-b + 2) * q^31 + q^32 + (2*b + 3) * q^33 + 6 * q^34 + (3*b + 1) * q^36 - q^37 + 2 * q^38 + (b + 2) * q^39 + (3*b + 3) * q^41 + (-2*b - 4) * q^42 + (-2*b + 4) * q^43 - b * q^44 + (-3*b + 3) * q^46 - 2*b * q^47 + (-b - 1) * q^48 + (-4*b + 9) * q^49 + (-6*b - 6) * q^51 + (-b + 1) * q^52 + 6 * q^53 + (-4*b - 7) * q^54 + (2*b - 2) * q^56 + (-2*b - 2) * q^57 + (-3*b + 3) * q^58 + (2*b + 6) * q^59 + (5*b - 4) * q^61 + (-b + 2) * q^62 + (2*b + 16) * q^63 + q^64 + (2*b + 3) * q^66 + (5*b - 8) * q^67 + 6 * q^68 + (3*b + 6) * q^69 + 6 * q^71 + (3*b + 1) * q^72 + (b + 10) * q^73 - q^74 + 2 * q^76 - 6 * q^77 + (b + 2) * q^78 + (7*b - 7) * q^79 + (6*b + 16) * q^81 + (3*b + 3) * q^82 + (4*b - 12) * q^83 + (-2*b - 4) * q^84 + (-2*b + 4) * q^86 + (3*b + 6) * q^87 - b * q^88 - 4*b * q^89 + (2*b - 8) * q^91 + (-3*b + 3) * q^92 + q^93 - 2*b * q^94 + (-b - 1) * q^96 + (8*b - 2) * q^97 + (-4*b + 9) * q^98 + (-4*b - 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 3 * q^6 - 2 * q^7 + 2 * q^8 + 5 * q^9 $$2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} - q^{11} - 3 q^{12} + q^{13} - 2 q^{14} + 2 q^{16} + 12 q^{17} + 5 q^{18} + 4 q^{19} - 10 q^{21} - q^{22} + 3 q^{23} - 3 q^{24} + q^{26} - 18 q^{27} - 2 q^{28} + 3 q^{29} + 3 q^{31} + 2 q^{32} + 8 q^{33} + 12 q^{34} + 5 q^{36} - 2 q^{37} + 4 q^{38} + 5 q^{39} + 9 q^{41} - 10 q^{42} + 6 q^{43} - q^{44} + 3 q^{46} - 2 q^{47} - 3 q^{48} + 14 q^{49} - 18 q^{51} + q^{52} + 12 q^{53} - 18 q^{54} - 2 q^{56} - 6 q^{57} + 3 q^{58} + 14 q^{59} - 3 q^{61} + 3 q^{62} + 34 q^{63} + 2 q^{64} + 8 q^{66} - 11 q^{67} + 12 q^{68} + 15 q^{69} + 12 q^{71} + 5 q^{72} + 21 q^{73} - 2 q^{74} + 4 q^{76} - 12 q^{77} + 5 q^{78} - 7 q^{79} + 38 q^{81} + 9 q^{82} - 20 q^{83} - 10 q^{84} + 6 q^{86} + 15 q^{87} - q^{88} - 4 q^{89} - 14 q^{91} + 3 q^{92} + 2 q^{93} - 2 q^{94} - 3 q^{96} + 4 q^{97} + 14 q^{98} - 22 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 3 * q^6 - 2 * q^7 + 2 * q^8 + 5 * q^9 - q^11 - 3 * q^12 + q^13 - 2 * q^14 + 2 * q^16 + 12 * q^17 + 5 * q^18 + 4 * q^19 - 10 * q^21 - q^22 + 3 * q^23 - 3 * q^24 + q^26 - 18 * q^27 - 2 * q^28 + 3 * q^29 + 3 * q^31 + 2 * q^32 + 8 * q^33 + 12 * q^34 + 5 * q^36 - 2 * q^37 + 4 * q^38 + 5 * q^39 + 9 * q^41 - 10 * q^42 + 6 * q^43 - q^44 + 3 * q^46 - 2 * q^47 - 3 * q^48 + 14 * q^49 - 18 * q^51 + q^52 + 12 * q^53 - 18 * q^54 - 2 * q^56 - 6 * q^57 + 3 * q^58 + 14 * q^59 - 3 * q^61 + 3 * q^62 + 34 * q^63 + 2 * q^64 + 8 * q^66 - 11 * q^67 + 12 * q^68 + 15 * q^69 + 12 * q^71 + 5 * q^72 + 21 * q^73 - 2 * q^74 + 4 * q^76 - 12 * q^77 + 5 * q^78 - 7 * q^79 + 38 * q^81 + 9 * q^82 - 20 * q^83 - 10 * q^84 + 6 * q^86 + 15 * q^87 - q^88 - 4 * q^89 - 14 * q^91 + 3 * q^92 + 2 * q^93 - 2 * q^94 - 3 * q^96 + 4 * q^97 + 14 * q^98 - 22 * 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
 2.30278 −1.30278
1.00000 −3.30278 1.00000 0 −3.30278 2.60555 1.00000 7.90833 0
1.2 1.00000 0.302776 1.00000 0 0.302776 −4.60555 1.00000 −2.90833 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.u 2
5.b even 2 1 74.2.a.a 2
5.c odd 4 2 1850.2.b.i 4
15.d odd 2 1 666.2.a.j 2
20.d odd 2 1 592.2.a.f 2
35.c odd 2 1 3626.2.a.a 2
40.e odd 2 1 2368.2.a.ba 2
40.f even 2 1 2368.2.a.s 2
55.d odd 2 1 8954.2.a.p 2
60.h even 2 1 5328.2.a.bf 2
185.d even 2 1 2738.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 5.b even 2 1
592.2.a.f 2 20.d odd 2 1
666.2.a.j 2 15.d odd 2 1
1850.2.a.u 2 1.a even 1 1 trivial
1850.2.b.i 4 5.c odd 4 2
2368.2.a.s 2 40.f even 2 1
2368.2.a.ba 2 40.e odd 2 1
2738.2.a.l 2 185.d even 2 1
3626.2.a.a 2 35.c odd 2 1
5328.2.a.bf 2 60.h even 2 1
8954.2.a.p 2 55.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1850))$$:

 $$T_{3}^{2} + 3T_{3} - 1$$ T3^2 + 3*T3 - 1 $$T_{7}^{2} + 2T_{7} - 12$$ T7^2 + 2*T7 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 3T - 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2T - 12$$
$11$ $$T^{2} + T - 3$$
$13$ $$T^{2} - T - 3$$
$17$ $$(T - 6)^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} - 3T - 27$$
$29$ $$T^{2} - 3T - 27$$
$31$ $$T^{2} - 3T - 1$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} - 9T - 9$$
$43$ $$T^{2} - 6T - 4$$
$47$ $$T^{2} + 2T - 12$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 14T + 36$$
$61$ $$T^{2} + 3T - 79$$
$67$ $$T^{2} + 11T - 51$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} - 21T + 107$$
$79$ $$T^{2} + 7T - 147$$
$83$ $$T^{2} + 20T + 48$$
$89$ $$T^{2} + 4T - 48$$
$97$ $$T^{2} - 4T - 204$$