# Properties

 Label 2850.2.d.s Level $2850$ Weight $2$ Character orbit 2850.d Analytic conductor $22.757$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(799,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2850.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.d (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: $$22.7573645761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1027$$ $$1351$$ $$1901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 4.00000i 1.00000i −1.00000 0
799.2 1.00000i 1.00000i −1.00000 0 1.00000 4.00000i 1.00000i −1.00000 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 2850.2.d.s 2
5.b even 2 1 inner 2850.2.d.s 2
5.c odd 4 1 114.2.a.a 1
5.c odd 4 1 2850.2.a.x 1
15.e even 4 1 342.2.a.f 1
15.e even 4 1 8550.2.a.a 1
20.e even 4 1 912.2.a.h 1
35.f even 4 1 5586.2.a.p 1
40.i odd 4 1 3648.2.a.bb 1
40.k even 4 1 3648.2.a.j 1
60.l odd 4 1 2736.2.a.j 1
95.g even 4 1 2166.2.a.i 1
285.j odd 4 1 6498.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.a 1 5.c odd 4 1
342.2.a.f 1 15.e even 4 1
912.2.a.h 1 20.e even 4 1
2166.2.a.i 1 95.g even 4 1
2736.2.a.j 1 60.l odd 4 1
2850.2.a.x 1 5.c odd 4 1
2850.2.d.s 2 1.a even 1 1 trivial
2850.2.d.s 2 5.b even 2 1 inner
3648.2.a.j 1 40.k even 4 1
3648.2.a.bb 1 40.i odd 4 1
5586.2.a.p 1 35.f even 4 1
6498.2.a.h 1 285.j odd 4 1
8550.2.a.a 1 15.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 $$S_{2}^{\mathrm{new}}(2850, [\chi])$$:

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} - 4$$ T11 - 4 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 4$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 6)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T - 10)^{2}$$
$43$ $$T^{2} + 144$$
$47$ $$T^{2} + 100$$
$53$ $$T^{2} + 4$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2}$$
$71$ $$(T + 16)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T + 10)^{2}$$
$83$ $$T^{2} + 256$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} + 100$$