# Properties

 Label 1850.2.b.j Level $1850$ Weight $2$ Character orbit 1850.b Analytic conductor $14.772$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(149,1850)]

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([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1850.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + \beta_1 q^{3} - q^{4} + \beta_{2} q^{6} - 2 \beta_1 q^{7} + \beta_{3} q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q - b3 * q^2 + b1 * q^3 - q^4 + b2 * q^6 - 2*b1 * q^7 + b3 * q^8 + (b2 + 2) * q^9 $$q - \beta_{3} q^{2} + \beta_1 q^{3} - q^{4} + \beta_{2} q^{6} - 2 \beta_1 q^{7} + \beta_{3} q^{8} + (\beta_{2} + 2) q^{9} + ( - \beta_{2} - 3) q^{11} - \beta_1 q^{12} + (2 \beta_{3} + 3 \beta_1) q^{13} - 2 \beta_{2} q^{14} + q^{16} + ( - 2 \beta_{3} - 4 \beta_1) q^{17} + ( - 2 \beta_{3} - \beta_1) q^{18} + (4 \beta_{2} + 2) q^{19} + ( - 2 \beta_{2} + 2) q^{21} + (3 \beta_{3} + \beta_1) q^{22} + ( - 2 \beta_{3} - 3 \beta_1) q^{23} - \beta_{2} q^{24} + (3 \beta_{2} + 2) q^{26} + (\beta_{3} + 4 \beta_1) q^{27} + 2 \beta_1 q^{28} + ( - 7 \beta_{2} - 2) q^{29} + (\beta_{2} + 9) q^{31} - \beta_{3} q^{32} + ( - \beta_{3} - 2 \beta_1) q^{33} + ( - 4 \beta_{2} - 2) q^{34} + ( - \beta_{2} - 2) q^{36} + \beta_{3} q^{37} + ( - 2 \beta_{3} - 4 \beta_1) q^{38} + (\beta_{2} - 3) q^{39} + ( - \beta_{2} + 8) q^{41} + ( - 2 \beta_{3} + 2 \beta_1) q^{42} + ( - 2 \beta_{3} + 2 \beta_1) q^{43} + (\beta_{2} + 3) q^{44} + ( - 3 \beta_{2} - 2) q^{46} + ( - 2 \beta_{3} - 2 \beta_1) q^{47} + \beta_1 q^{48} + (4 \beta_{2} + 3) q^{49} + ( - 2 \beta_{2} + 4) q^{51} + ( - 2 \beta_{3} - 3 \beta_1) q^{52} + ( - 6 \beta_{3} - 4 \beta_1) q^{53} + (4 \beta_{2} + 1) q^{54} + 2 \beta_{2} q^{56} + (4 \beta_{3} - 2 \beta_1) q^{57} + (2 \beta_{3} + 7 \beta_1) q^{58} + (2 \beta_{2} + 8) q^{59} + ( - \beta_{2} + 9) q^{61} + ( - 9 \beta_{3} - \beta_1) q^{62} + ( - 2 \beta_{3} - 2 \beta_1) q^{63} - q^{64} + ( - 2 \beta_{2} - 1) q^{66} + (7 \beta_{3} + 5 \beta_1) q^{67} + (2 \beta_{3} + 4 \beta_1) q^{68} + ( - \beta_{2} + 3) q^{69} + ( - 8 \beta_{2} - 10) q^{71} + (2 \beta_{3} + \beta_1) q^{72} + ( - \beta_{3} - 5 \beta_1) q^{73} + q^{74} + ( - 4 \beta_{2} - 2) q^{76} + (2 \beta_{3} + 4 \beta_1) q^{77} + (3 \beta_{3} - \beta_1) q^{78} + ( - 9 \beta_{2} - 6) q^{79} + (6 \beta_{2} + 2) q^{81} + ( - 8 \beta_{3} + \beta_1) q^{82} + ( - 8 \beta_{3} + 4 \beta_1) q^{83} + (2 \beta_{2} - 2) q^{84} + (2 \beta_{2} - 2) q^{86} + ( - 7 \beta_{3} + 5 \beta_1) q^{87} + ( - 3 \beta_{3} - \beta_1) q^{88} + (4 \beta_{2} + 8) q^{89} + ( - 2 \beta_{2} + 6) q^{91} + (2 \beta_{3} + 3 \beta_1) q^{92} + (\beta_{3} + 8 \beta_1) q^{93} + ( - 2 \beta_{2} - 2) q^{94} + \beta_{2} q^{96} + ( - 6 \beta_{3} - 4 \beta_1) q^{97} + ( - 3 \beta_{3} - 4 \beta_1) q^{98} + ( - 4 \beta_{2} - 7) q^{99}+O(q^{100})$$ q - b3 * q^2 + b1 * q^3 - q^4 + b2 * q^6 - 2*b1 * q^7 + b3 * q^8 + (b2 + 2) * q^9 + (-b2 - 3) * q^11 - b1 * q^12 + (2*b3 + 3*b1) * q^13 - 2*b2 * q^14 + q^16 + (-2*b3 - 4*b1) * q^17 + (-2*b3 - b1) * q^18 + (4*b2 + 2) * q^19 + (-2*b2 + 2) * q^21 + (3*b3 + b1) * q^22 + (-2*b3 - 3*b1) * q^23 - b2 * q^24 + (3*b2 + 2) * q^26 + (b3 + 4*b1) * q^27 + 2*b1 * q^28 + (-7*b2 - 2) * q^29 + (b2 + 9) * q^31 - b3 * q^32 + (-b3 - 2*b1) * q^33 + (-4*b2 - 2) * q^34 + (-b2 - 2) * q^36 + b3 * q^37 + (-2*b3 - 4*b1) * q^38 + (b2 - 3) * q^39 + (-b2 + 8) * q^41 + (-2*b3 + 2*b1) * q^42 + (-2*b3 + 2*b1) * q^43 + (b2 + 3) * q^44 + (-3*b2 - 2) * q^46 + (-2*b3 - 2*b1) * q^47 + b1 * q^48 + (4*b2 + 3) * q^49 + (-2*b2 + 4) * q^51 + (-2*b3 - 3*b1) * q^52 + (-6*b3 - 4*b1) * q^53 + (4*b2 + 1) * q^54 + 2*b2 * q^56 + (4*b3 - 2*b1) * q^57 + (2*b3 + 7*b1) * q^58 + (2*b2 + 8) * q^59 + (-b2 + 9) * q^61 + (-9*b3 - b1) * q^62 + (-2*b3 - 2*b1) * q^63 - q^64 + (-2*b2 - 1) * q^66 + (7*b3 + 5*b1) * q^67 + (2*b3 + 4*b1) * q^68 + (-b2 + 3) * q^69 + (-8*b2 - 10) * q^71 + (2*b3 + b1) * q^72 + (-b3 - 5*b1) * q^73 + q^74 + (-4*b2 - 2) * q^76 + (2*b3 + 4*b1) * q^77 + (3*b3 - b1) * q^78 + (-9*b2 - 6) * q^79 + (6*b2 + 2) * q^81 + (-8*b3 + b1) * q^82 + (-8*b3 + 4*b1) * q^83 + (2*b2 - 2) * q^84 + (2*b2 - 2) * q^86 + (-7*b3 + 5*b1) * q^87 + (-3*b3 - b1) * q^88 + (4*b2 + 8) * q^89 + (-2*b2 + 6) * q^91 + (2*b3 + 3*b1) * q^92 + (b3 + 8*b1) * q^93 + (-2*b2 - 2) * q^94 + b2 * q^96 + (-6*b3 - 4*b1) * q^97 + (-3*b3 - 4*b1) * q^98 + (-4*b2 - 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 2 q^{6} + 6 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 2 * q^6 + 6 * q^9 $$4 q - 4 q^{4} - 2 q^{6} + 6 q^{9} - 10 q^{11} + 4 q^{14} + 4 q^{16} + 12 q^{21} + 2 q^{24} + 2 q^{26} + 6 q^{29} + 34 q^{31} - 6 q^{36} - 14 q^{39} + 34 q^{41} + 10 q^{44} - 2 q^{46} + 4 q^{49} + 20 q^{51} - 4 q^{54} - 4 q^{56} + 28 q^{59} + 38 q^{61} - 4 q^{64} + 14 q^{69} - 24 q^{71} + 4 q^{74} - 6 q^{79} - 4 q^{81} - 12 q^{84} - 12 q^{86} + 24 q^{89} + 28 q^{91} - 4 q^{94} - 2 q^{96} - 20 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 2 * q^6 + 6 * q^9 - 10 * q^11 + 4 * q^14 + 4 * q^16 + 12 * q^21 + 2 * q^24 + 2 * q^26 + 6 * q^29 + 34 * q^31 - 6 * q^36 - 14 * q^39 + 34 * q^41 + 10 * q^44 - 2 * q^46 + 4 * q^49 + 20 * q^51 - 4 * q^54 - 4 * q^56 + 28 * q^59 + 38 * q^61 - 4 * q^64 + 14 * q^69 - 24 * q^71 + 4 * q^74 - 6 * q^79 - 4 * q^81 - 12 * q^84 - 12 * q^86 + 24 * q^89 + 28 * q^91 - 4 * q^94 - 2 * q^96 - 20 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2\beta_1$$ b3 - 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 − 1.61803i 0.618034i − 0.618034i 1.61803i
1.00000i 1.61803i −1.00000 0 −1.61803 3.23607i 1.00000i 0.381966 0
149.2 1.00000i 0.618034i −1.00000 0 0.618034 1.23607i 1.00000i 2.61803 0
149.3 1.00000i 0.618034i −1.00000 0 0.618034 1.23607i 1.00000i 2.61803 0
149.4 1.00000i 1.61803i −1.00000 0 −1.61803 3.23607i 1.00000i 0.381966 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1850.2.b.j 4
5.b even 2 1 inner 1850.2.b.j 4
5.c odd 4 1 74.2.a.b 2
5.c odd 4 1 1850.2.a.t 2
15.e even 4 1 666.2.a.i 2
20.e even 4 1 592.2.a.g 2
35.f even 4 1 3626.2.a.s 2
40.i odd 4 1 2368.2.a.y 2
40.k even 4 1 2368.2.a.u 2
55.e even 4 1 8954.2.a.j 2
60.l odd 4 1 5328.2.a.bc 2
185.h odd 4 1 2738.2.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.b 2 5.c odd 4 1
592.2.a.g 2 20.e even 4 1
666.2.a.i 2 15.e even 4 1
1850.2.a.t 2 5.c odd 4 1
1850.2.b.j 4 1.a even 1 1 trivial
1850.2.b.j 4 5.b even 2 1 inner
2368.2.a.u 2 40.k even 4 1
2368.2.a.y 2 40.i odd 4 1
2738.2.a.g 2 185.h odd 4 1
3626.2.a.s 2 35.f even 4 1
5328.2.a.bc 2 60.l odd 4 1
8954.2.a.j 2 55.e even 4 1

## Hecke kernels

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

 $$T_{3}^{4} + 3T_{3}^{2} + 1$$ T3^4 + 3*T3^2 + 1 $$T_{7}^{4} + 12T_{7}^{2} + 16$$ T7^4 + 12*T7^2 + 16 $$T_{13}^{4} + 23T_{13}^{2} + 121$$ T13^4 + 23*T13^2 + 121

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 3T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 12T^{2} + 16$$
$11$ $$(T^{2} + 5 T + 5)^{2}$$
$13$ $$T^{4} + 23T^{2} + 121$$
$17$ $$(T^{2} + 20)^{2}$$
$19$ $$(T^{2} - 20)^{2}$$
$23$ $$T^{4} + 23T^{2} + 121$$
$29$ $$(T^{2} - 3 T - 59)^{2}$$
$31$ $$(T^{2} - 17 T + 71)^{2}$$
$37$ $$(T^{2} + 1)^{2}$$
$41$ $$(T^{2} - 17 T + 71)^{2}$$
$43$ $$T^{4} + 28T^{2} + 16$$
$47$ $$T^{4} + 12T^{2} + 16$$
$53$ $$T^{4} + 72T^{2} + 16$$
$59$ $$(T^{2} - 14 T + 44)^{2}$$
$61$ $$(T^{2} - 19 T + 89)^{2}$$
$67$ $$T^{4} + 103T^{2} + 121$$
$71$ $$(T^{2} + 12 T - 44)^{2}$$
$73$ $$T^{4} + 67T^{2} + 841$$
$79$ $$(T^{2} + 3 T - 99)^{2}$$
$83$ $$T^{4} + 240T^{2} + 6400$$
$89$ $$(T^{2} - 12 T + 16)^{2}$$
$97$ $$T^{4} + 72T^{2} + 16$$