# Properties

 Label 1050.6.a.a 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 $$q - 4 q^{2} - 9 q^{3} + 16 q^{4} + 36 q^{6} - 49 q^{7} - 64 q^{8} + 81 q^{9} + 66 q^{11} - 144 q^{12} - 98 q^{13} + 196 q^{14} + 256 q^{16} + 216 q^{17} - 324 q^{18} - 340 q^{19} + 441 q^{21} - 264 q^{22} + 1038 q^{23} + 576 q^{24} + 392 q^{26} - 729 q^{27} - 784 q^{28} - 2490 q^{29} - 7048 q^{31} - 1024 q^{32} - 594 q^{33} - 864 q^{34} + 1296 q^{36} + 12238 q^{37} + 1360 q^{38} + 882 q^{39} + 6468 q^{41} - 1764 q^{42} + 15412 q^{43} + 1056 q^{44} - 4152 q^{46} - 20604 q^{47} - 2304 q^{48} + 2401 q^{49} - 1944 q^{51} - 1568 q^{52} - 32490 q^{53} + 2916 q^{54} + 3136 q^{56} + 3060 q^{57} + 9960 q^{58} + 34224 q^{59} + 35654 q^{61} + 28192 q^{62} - 3969 q^{63} + 4096 q^{64} + 2376 q^{66} - 12680 q^{67} + 3456 q^{68} - 9342 q^{69} - 42642 q^{71} - 5184 q^{72} - 33734 q^{73} - 48952 q^{74} - 5440 q^{76} - 3234 q^{77} - 3528 q^{78} - 85108 q^{79} + 6561 q^{81} - 25872 q^{82} + 106764 q^{83} + 7056 q^{84} - 61648 q^{86} + 22410 q^{87} - 4224 q^{88} + 34884 q^{89} + 4802 q^{91} + 16608 q^{92} + 63432 q^{93} + 82416 q^{94} + 9216 q^{96} - 18662 q^{97} - 9604 q^{98} + 5346 q^{99}+O(q^{100})$$ q - 4 * q^2 - 9 * q^3 + 16 * q^4 + 36 * q^6 - 49 * q^7 - 64 * q^8 + 81 * q^9 + 66 * q^11 - 144 * q^12 - 98 * q^13 + 196 * q^14 + 256 * q^16 + 216 * q^17 - 324 * q^18 - 340 * q^19 + 441 * q^21 - 264 * q^22 + 1038 * q^23 + 576 * q^24 + 392 * q^26 - 729 * q^27 - 784 * q^28 - 2490 * q^29 - 7048 * q^31 - 1024 * q^32 - 594 * q^33 - 864 * q^34 + 1296 * q^36 + 12238 * q^37 + 1360 * q^38 + 882 * q^39 + 6468 * q^41 - 1764 * q^42 + 15412 * q^43 + 1056 * q^44 - 4152 * q^46 - 20604 * q^47 - 2304 * q^48 + 2401 * q^49 - 1944 * q^51 - 1568 * q^52 - 32490 * q^53 + 2916 * q^54 + 3136 * q^56 + 3060 * q^57 + 9960 * q^58 + 34224 * q^59 + 35654 * q^61 + 28192 * q^62 - 3969 * q^63 + 4096 * q^64 + 2376 * q^66 - 12680 * q^67 + 3456 * q^68 - 9342 * q^69 - 42642 * q^71 - 5184 * q^72 - 33734 * q^73 - 48952 * q^74 - 5440 * q^76 - 3234 * q^77 - 3528 * q^78 - 85108 * q^79 + 6561 * q^81 - 25872 * q^82 + 106764 * q^83 + 7056 * q^84 - 61648 * q^86 + 22410 * q^87 - 4224 * q^88 + 34884 * q^89 + 4802 * q^91 + 16608 * q^92 + 63432 * q^93 + 82416 * q^94 + 9216 * q^96 - 18662 * q^97 - 9604 * q^98 + 5346 * 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.a 1
5.b even 2 1 42.6.a.f 1
5.c odd 4 2 1050.6.g.m 2
15.d odd 2 1 126.6.a.b 1
20.d odd 2 1 336.6.a.g 1
35.c odd 2 1 294.6.a.i 1
35.i odd 6 2 294.6.e.f 2
35.j even 6 2 294.6.e.b 2
60.h even 2 1 1008.6.a.k 1
105.g even 2 1 882.6.a.i 1

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

 $$T_{11} - 66$$ T11 - 66 $$T_{13} + 98$$ T13 + 98

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T + 9$$
$5$ $$T$$
$7$ $$T + 49$$
$11$ $$T - 66$$
$13$ $$T + 98$$
$17$ $$T - 216$$
$19$ $$T + 340$$
$23$ $$T - 1038$$
$29$ $$T + 2490$$
$31$ $$T + 7048$$
$37$ $$T - 12238$$
$41$ $$T - 6468$$
$43$ $$T - 15412$$
$47$ $$T + 20604$$
$53$ $$T + 32490$$
$59$ $$T - 34224$$
$61$ $$T - 35654$$
$67$ $$T + 12680$$
$71$ $$T + 42642$$
$73$ $$T + 33734$$
$79$ $$T + 85108$$
$83$ $$T - 106764$$
$89$ $$T - 34884$$
$97$ $$T + 18662$$