# Properties

 Label 1050.6.a.o Level $1050$ Weight $6$ Character orbit 1050.a Self dual yes Analytic conductor $168.403$ Analytic rank $0$ 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: $$0$$ 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 $$q + 4 q^{2} + 9 q^{3} + 16 q^{4} + 36 q^{6} + 49 q^{7} + 64 q^{8} + 81 q^{9} - 470 q^{11} + 144 q^{12} + 1158 q^{13} + 196 q^{14} + 256 q^{16} - 1204 q^{17} + 324 q^{18} - 2644 q^{19} + 441 q^{21} - 1880 q^{22} + 1190 q^{23} + 576 q^{24} + 4632 q^{26} + 729 q^{27} + 784 q^{28} + 3614 q^{29} + 5616 q^{31} + 1024 q^{32} - 4230 q^{33} - 4816 q^{34} + 1296 q^{36} + 6478 q^{37} - 10576 q^{38} + 10422 q^{39} + 2856 q^{41} + 1764 q^{42} + 13492 q^{43} - 7520 q^{44} + 4760 q^{46} + 18372 q^{47} + 2304 q^{48} + 2401 q^{49} - 10836 q^{51} + 18528 q^{52} + 4374 q^{53} + 2916 q^{54} + 3136 q^{56} - 23796 q^{57} + 14456 q^{58} + 30248 q^{59} + 19542 q^{61} + 22464 q^{62} + 3969 q^{63} + 4096 q^{64} - 16920 q^{66} - 54328 q^{67} - 19264 q^{68} + 10710 q^{69} - 10730 q^{71} + 5184 q^{72} - 35374 q^{73} + 25912 q^{74} - 42304 q^{76} - 23030 q^{77} + 41688 q^{78} - 49956 q^{79} + 6561 q^{81} + 11424 q^{82} + 26948 q^{83} + 7056 q^{84} + 53968 q^{86} + 32526 q^{87} - 30080 q^{88} + 100776 q^{89} + 56742 q^{91} + 19040 q^{92} + 50544 q^{93} + 73488 q^{94} + 9216 q^{96} - 77134 q^{97} + 9604 q^{98} - 38070 q^{99}+O(q^{100})$$ q + 4 * q^2 + 9 * q^3 + 16 * q^4 + 36 * q^6 + 49 * q^7 + 64 * q^8 + 81 * q^9 - 470 * q^11 + 144 * q^12 + 1158 * q^13 + 196 * q^14 + 256 * q^16 - 1204 * q^17 + 324 * q^18 - 2644 * q^19 + 441 * q^21 - 1880 * q^22 + 1190 * q^23 + 576 * q^24 + 4632 * q^26 + 729 * q^27 + 784 * q^28 + 3614 * q^29 + 5616 * q^31 + 1024 * q^32 - 4230 * q^33 - 4816 * q^34 + 1296 * q^36 + 6478 * q^37 - 10576 * q^38 + 10422 * q^39 + 2856 * q^41 + 1764 * q^42 + 13492 * q^43 - 7520 * q^44 + 4760 * q^46 + 18372 * q^47 + 2304 * q^48 + 2401 * q^49 - 10836 * q^51 + 18528 * q^52 + 4374 * q^53 + 2916 * q^54 + 3136 * q^56 - 23796 * q^57 + 14456 * q^58 + 30248 * q^59 + 19542 * q^61 + 22464 * q^62 + 3969 * q^63 + 4096 * q^64 - 16920 * q^66 - 54328 * q^67 - 19264 * q^68 + 10710 * q^69 - 10730 * q^71 + 5184 * q^72 - 35374 * q^73 + 25912 * q^74 - 42304 * q^76 - 23030 * q^77 + 41688 * q^78 - 49956 * q^79 + 6561 * q^81 + 11424 * q^82 + 26948 * q^83 + 7056 * q^84 + 53968 * q^86 + 32526 * q^87 - 30080 * q^88 + 100776 * q^89 + 56742 * q^91 + 19040 * q^92 + 50544 * q^93 + 73488 * q^94 + 9216 * q^96 - 77134 * q^97 + 9604 * q^98 - 38070 * 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.o 1
5.b even 2 1 42.6.a.b 1
5.c odd 4 2 1050.6.g.l 2
15.d odd 2 1 126.6.a.h 1
20.d odd 2 1 336.6.a.o 1
35.c odd 2 1 294.6.a.f 1
35.i odd 6 2 294.6.e.k 2
35.j even 6 2 294.6.e.o 2
60.h even 2 1 1008.6.a.g 1
105.g even 2 1 882.6.a.v 1

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

 $$T_{11} + 470$$ T11 + 470 $$T_{13} - 1158$$ T13 - 1158

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T - 9$$
$5$ $$T$$
$7$ $$T - 49$$
$11$ $$T + 470$$
$13$ $$T - 1158$$
$17$ $$T + 1204$$
$19$ $$T + 2644$$
$23$ $$T - 1190$$
$29$ $$T - 3614$$
$31$ $$T - 5616$$
$37$ $$T - 6478$$
$41$ $$T - 2856$$
$43$ $$T - 13492$$
$47$ $$T - 18372$$
$53$ $$T - 4374$$
$59$ $$T - 30248$$
$61$ $$T - 19542$$
$67$ $$T + 54328$$
$71$ $$T + 10730$$
$73$ $$T + 35374$$
$79$ $$T + 49956$$
$83$ $$T - 26948$$
$89$ $$T - 100776$$
$97$ $$T + 77134$$