# Properties

 Label 1050.6.a.k Level $1050$ Weight $6$ Character orbit 1050.a Self dual yes Analytic conductor $168.403$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,6,Mod(1,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1050.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1050.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: $$168.403010804$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} - 9 q^{3} + 16 q^{4} - 36 q^{6} + 49 q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10})$$ q + 4 * q^2 - 9 * q^3 + 16 * q^4 - 36 * q^6 + 49 * q^7 + 64 * q^8 + 81 * q^9 $$q + 4 q^{2} - 9 q^{3} + 16 q^{4} - 36 q^{6} + 49 q^{7} + 64 q^{8} + 81 q^{9} + 664 q^{11} - 144 q^{12} - 318 q^{13} + 196 q^{14} + 256 q^{16} - 1582 q^{17} + 324 q^{18} + 236 q^{19} - 441 q^{21} + 2656 q^{22} - 2212 q^{23} - 576 q^{24} - 1272 q^{26} - 729 q^{27} + 784 q^{28} - 4954 q^{29} - 7128 q^{31} + 1024 q^{32} - 5976 q^{33} - 6328 q^{34} + 1296 q^{36} - 4358 q^{37} + 944 q^{38} + 2862 q^{39} + 10542 q^{41} - 1764 q^{42} + 8452 q^{43} + 10624 q^{44} - 8848 q^{46} - 5352 q^{47} - 2304 q^{48} + 2401 q^{49} + 14238 q^{51} - 5088 q^{52} + 33354 q^{53} - 2916 q^{54} + 3136 q^{56} - 2124 q^{57} - 19816 q^{58} - 15436 q^{59} - 36762 q^{61} - 28512 q^{62} + 3969 q^{63} + 4096 q^{64} - 23904 q^{66} - 40972 q^{67} - 25312 q^{68} + 19908 q^{69} - 9092 q^{71} + 5184 q^{72} + 73454 q^{73} - 17432 q^{74} + 3776 q^{76} + 32536 q^{77} + 11448 q^{78} + 89400 q^{79} + 6561 q^{81} + 42168 q^{82} + 6428 q^{83} - 7056 q^{84} + 33808 q^{86} + 44586 q^{87} + 42496 q^{88} - 122658 q^{89} - 15582 q^{91} - 35392 q^{92} + 64152 q^{93} - 21408 q^{94} - 9216 q^{96} - 21370 q^{97} + 9604 q^{98} + 53784 q^{99}+O(q^{100})$$ q + 4 * q^2 - 9 * q^3 + 16 * q^4 - 36 * q^6 + 49 * q^7 + 64 * q^8 + 81 * q^9 + 664 * q^11 - 144 * q^12 - 318 * q^13 + 196 * q^14 + 256 * q^16 - 1582 * q^17 + 324 * q^18 + 236 * q^19 - 441 * q^21 + 2656 * q^22 - 2212 * q^23 - 576 * q^24 - 1272 * q^26 - 729 * q^27 + 784 * q^28 - 4954 * q^29 - 7128 * q^31 + 1024 * q^32 - 5976 * q^33 - 6328 * q^34 + 1296 * q^36 - 4358 * q^37 + 944 * q^38 + 2862 * q^39 + 10542 * q^41 - 1764 * q^42 + 8452 * q^43 + 10624 * q^44 - 8848 * q^46 - 5352 * q^47 - 2304 * q^48 + 2401 * q^49 + 14238 * q^51 - 5088 * q^52 + 33354 * q^53 - 2916 * q^54 + 3136 * q^56 - 2124 * q^57 - 19816 * q^58 - 15436 * q^59 - 36762 * q^61 - 28512 * q^62 + 3969 * q^63 + 4096 * q^64 - 23904 * q^66 - 40972 * q^67 - 25312 * q^68 + 19908 * q^69 - 9092 * q^71 + 5184 * q^72 + 73454 * q^73 - 17432 * q^74 + 3776 * q^76 + 32536 * q^77 + 11448 * q^78 + 89400 * q^79 + 6561 * q^81 + 42168 * q^82 + 6428 * q^83 - 7056 * q^84 + 33808 * q^86 + 44586 * q^87 + 42496 * q^88 - 122658 * q^89 - 15582 * q^91 - 35392 * q^92 + 64152 * q^93 - 21408 * q^94 - 9216 * q^96 - 21370 * q^97 + 9604 * q^98 + 53784 * 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
 0
4.00000 −9.00000 16.0000 0 −36.0000 49.0000 64.0000 81.0000 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
$$2$$ $$-1$$
$$3$$ $$+1$$
$$5$$ $$+1$$
$$7$$ $$-1$$

## 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 1050.6.a.k 1
5.b even 2 1 42.6.a.d 1
5.c odd 4 2 1050.6.g.i 2
15.d odd 2 1 126.6.a.i 1
20.d odd 2 1 336.6.a.h 1
35.c odd 2 1 294.6.a.b 1
35.i odd 6 2 294.6.e.p 2
35.j even 6 2 294.6.e.i 2
60.h even 2 1 1008.6.a.j 1
105.g even 2 1 882.6.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.d 1 5.b even 2 1
126.6.a.i 1 15.d odd 2 1
294.6.a.b 1 35.c odd 2 1
294.6.e.i 2 35.j even 6 2
294.6.e.p 2 35.i odd 6 2
336.6.a.h 1 20.d odd 2 1
882.6.a.s 1 105.g even 2 1
1008.6.a.j 1 60.h even 2 1
1050.6.a.k 1 1.a even 1 1 trivial
1050.6.g.i 2 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(1050))$$:

 $$T_{11} - 664$$ T11 - 664 $$T_{13} + 318$$ T13 + 318

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T + 9$$
$5$ $$T$$
$7$ $$T - 49$$
$11$ $$T - 664$$
$13$ $$T + 318$$
$17$ $$T + 1582$$
$19$ $$T - 236$$
$23$ $$T + 2212$$
$29$ $$T + 4954$$
$31$ $$T + 7128$$
$37$ $$T + 4358$$
$41$ $$T - 10542$$
$43$ $$T - 8452$$
$47$ $$T + 5352$$
$53$ $$T - 33354$$
$59$ $$T + 15436$$
$61$ $$T + 36762$$
$67$ $$T + 40972$$
$71$ $$T + 9092$$
$73$ $$T - 73454$$
$79$ $$T - 89400$$
$83$ $$T - 6428$$
$89$ $$T + 122658$$
$97$ $$T + 21370$$