# Properties

 Label 1050.6.a.n 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 $$q + 4 q^{2} + 9 q^{3} + 16 q^{4} + 36 q^{6} - 49 q^{7} + 64 q^{8} + 81 q^{9} + 216 q^{11} + 144 q^{12} - 998 q^{13} - 196 q^{14} + 256 q^{16} - 1302 q^{17} + 324 q^{18} + 884 q^{19} - 441 q^{21} + 864 q^{22} + 2268 q^{23} + 576 q^{24} - 3992 q^{26} + 729 q^{27} - 784 q^{28} - 1482 q^{29} + 8360 q^{31} + 1024 q^{32} + 1944 q^{33} - 5208 q^{34} + 1296 q^{36} + 4714 q^{37} + 3536 q^{38} - 8982 q^{39} - 9786 q^{41} - 1764 q^{42} - 19436 q^{43} + 3456 q^{44} + 9072 q^{46} - 22200 q^{47} + 2304 q^{48} + 2401 q^{49} - 11718 q^{51} - 15968 q^{52} - 26790 q^{53} + 2916 q^{54} - 3136 q^{56} + 7956 q^{57} - 5928 q^{58} + 28092 q^{59} - 38866 q^{61} + 33440 q^{62} - 3969 q^{63} + 4096 q^{64} + 7776 q^{66} - 23948 q^{67} - 20832 q^{68} + 20412 q^{69} - 20628 q^{71} + 5184 q^{72} - 290 q^{73} + 18856 q^{74} + 14144 q^{76} - 10584 q^{77} - 35928 q^{78} - 99544 q^{79} + 6561 q^{81} - 39144 q^{82} - 19308 q^{83} - 7056 q^{84} - 77744 q^{86} - 13338 q^{87} + 13824 q^{88} + 36390 q^{89} + 48902 q^{91} + 36288 q^{92} + 75240 q^{93} - 88800 q^{94} + 9216 q^{96} + 79078 q^{97} + 9604 q^{98} + 17496 q^{99}+O(q^{100})$$ q + 4 * q^2 + 9 * q^3 + 16 * q^4 + 36 * q^6 - 49 * q^7 + 64 * q^8 + 81 * q^9 + 216 * q^11 + 144 * q^12 - 998 * q^13 - 196 * q^14 + 256 * q^16 - 1302 * q^17 + 324 * q^18 + 884 * q^19 - 441 * q^21 + 864 * q^22 + 2268 * q^23 + 576 * q^24 - 3992 * q^26 + 729 * q^27 - 784 * q^28 - 1482 * q^29 + 8360 * q^31 + 1024 * q^32 + 1944 * q^33 - 5208 * q^34 + 1296 * q^36 + 4714 * q^37 + 3536 * q^38 - 8982 * q^39 - 9786 * q^41 - 1764 * q^42 - 19436 * q^43 + 3456 * q^44 + 9072 * q^46 - 22200 * q^47 + 2304 * q^48 + 2401 * q^49 - 11718 * q^51 - 15968 * q^52 - 26790 * q^53 + 2916 * q^54 - 3136 * q^56 + 7956 * q^57 - 5928 * q^58 + 28092 * q^59 - 38866 * q^61 + 33440 * q^62 - 3969 * q^63 + 4096 * q^64 + 7776 * q^66 - 23948 * q^67 - 20832 * q^68 + 20412 * q^69 - 20628 * q^71 + 5184 * q^72 - 290 * q^73 + 18856 * q^74 + 14144 * q^76 - 10584 * q^77 - 35928 * q^78 - 99544 * q^79 + 6561 * q^81 - 39144 * q^82 - 19308 * q^83 - 7056 * q^84 - 77744 * q^86 - 13338 * q^87 + 13824 * q^88 + 36390 * q^89 + 48902 * q^91 + 36288 * q^92 + 75240 * q^93 - 88800 * q^94 + 9216 * q^96 + 79078 * q^97 + 9604 * q^98 + 17496 * 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.n 1
5.b even 2 1 42.6.a.a 1
5.c odd 4 2 1050.6.g.o 2
15.d odd 2 1 126.6.a.k 1
20.d odd 2 1 336.6.a.j 1
35.c odd 2 1 294.6.a.h 1
35.i odd 6 2 294.6.e.h 2
35.j even 6 2 294.6.e.r 2
60.h even 2 1 1008.6.a.x 1
105.g even 2 1 882.6.a.o 1

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

 $$T_{11} - 216$$ T11 - 216 $$T_{13} + 998$$ T13 + 998

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T - 9$$
$5$ $$T$$
$7$ $$T + 49$$
$11$ $$T - 216$$
$13$ $$T + 998$$
$17$ $$T + 1302$$
$19$ $$T - 884$$
$23$ $$T - 2268$$
$29$ $$T + 1482$$
$31$ $$T - 8360$$
$37$ $$T - 4714$$
$41$ $$T + 9786$$
$43$ $$T + 19436$$
$47$ $$T + 22200$$
$53$ $$T + 26790$$
$59$ $$T - 28092$$
$61$ $$T + 38866$$
$67$ $$T + 23948$$
$71$ $$T + 20628$$
$73$ $$T + 290$$
$79$ $$T + 99544$$
$83$ $$T + 19308$$
$89$ $$T - 36390$$
$97$ $$T - 79078$$