# Properties

 Label 1014.2.a.e Level $1014$ Weight $2$ Character orbit 1014.a Self dual yes Analytic conductor $8.097$ 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] = [1014,2,Mod(1,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.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: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 2 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 + 2 * q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 2 q^{7} + q^{8} + q^{9} + q^{10} - 2 q^{11} - q^{12} + 2 q^{14} - q^{15} + q^{16} + 5 q^{17} + q^{18} + 2 q^{19} + q^{20} - 2 q^{21} - 2 q^{22} + 6 q^{23} - q^{24} - 4 q^{25} - q^{27} + 2 q^{28} - 9 q^{29} - q^{30} + 4 q^{31} + q^{32} + 2 q^{33} + 5 q^{34} + 2 q^{35} + q^{36} + 11 q^{37} + 2 q^{38} + q^{40} - 5 q^{41} - 2 q^{42} + 10 q^{43} - 2 q^{44} + q^{45} + 6 q^{46} - 2 q^{47} - q^{48} - 3 q^{49} - 4 q^{50} - 5 q^{51} - q^{53} - q^{54} - 2 q^{55} + 2 q^{56} - 2 q^{57} - 9 q^{58} + 8 q^{59} - q^{60} - 11 q^{61} + 4 q^{62} + 2 q^{63} + q^{64} + 2 q^{66} - 2 q^{67} + 5 q^{68} - 6 q^{69} + 2 q^{70} + 14 q^{71} + q^{72} + 13 q^{73} + 11 q^{74} + 4 q^{75} + 2 q^{76} - 4 q^{77} - 4 q^{79} + q^{80} + q^{81} - 5 q^{82} - 6 q^{83} - 2 q^{84} + 5 q^{85} + 10 q^{86} + 9 q^{87} - 2 q^{88} - 2 q^{89} + q^{90} + 6 q^{92} - 4 q^{93} - 2 q^{94} + 2 q^{95} - q^{96} + 2 q^{97} - 3 q^{98} - 2 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 + 2 * q^7 + q^8 + q^9 + q^10 - 2 * q^11 - q^12 + 2 * q^14 - q^15 + q^16 + 5 * q^17 + q^18 + 2 * q^19 + q^20 - 2 * q^21 - 2 * q^22 + 6 * q^23 - q^24 - 4 * q^25 - q^27 + 2 * q^28 - 9 * q^29 - q^30 + 4 * q^31 + q^32 + 2 * q^33 + 5 * q^34 + 2 * q^35 + q^36 + 11 * q^37 + 2 * q^38 + q^40 - 5 * q^41 - 2 * q^42 + 10 * q^43 - 2 * q^44 + q^45 + 6 * q^46 - 2 * q^47 - q^48 - 3 * q^49 - 4 * q^50 - 5 * q^51 - q^53 - q^54 - 2 * q^55 + 2 * q^56 - 2 * q^57 - 9 * q^58 + 8 * q^59 - q^60 - 11 * q^61 + 4 * q^62 + 2 * q^63 + q^64 + 2 * q^66 - 2 * q^67 + 5 * q^68 - 6 * q^69 + 2 * q^70 + 14 * q^71 + q^72 + 13 * q^73 + 11 * q^74 + 4 * q^75 + 2 * q^76 - 4 * q^77 - 4 * q^79 + q^80 + q^81 - 5 * q^82 - 6 * q^83 - 2 * q^84 + 5 * q^85 + 10 * q^86 + 9 * q^87 - 2 * q^88 - 2 * q^89 + q^90 + 6 * q^92 - 4 * q^93 - 2 * q^94 + 2 * q^95 - q^96 + 2 * q^97 - 3 * q^98 - 2 * 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
1.00000 −1.00000 1.00000 1.00000 −1.00000 2.00000 1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$13$$ $$1$$

## 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 1014.2.a.e 1
3.b odd 2 1 3042.2.a.d 1
4.b odd 2 1 8112.2.a.bb 1
13.b even 2 1 1014.2.a.a 1
13.c even 3 2 1014.2.e.d 2
13.d odd 4 2 1014.2.b.a 2
13.e even 6 2 78.2.e.b 2
13.f odd 12 4 1014.2.i.e 4
39.d odd 2 1 3042.2.a.m 1
39.f even 4 2 3042.2.b.d 2
39.h odd 6 2 234.2.h.b 2
52.b odd 2 1 8112.2.a.x 1
52.i odd 6 2 624.2.q.b 2
65.l even 6 2 1950.2.i.b 2
65.r odd 12 4 1950.2.z.b 4
156.r even 6 2 1872.2.t.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.b 2 13.e even 6 2
234.2.h.b 2 39.h odd 6 2
624.2.q.b 2 52.i odd 6 2
1014.2.a.a 1 13.b even 2 1
1014.2.a.e 1 1.a even 1 1 trivial
1014.2.b.a 2 13.d odd 4 2
1014.2.e.d 2 13.c even 3 2
1014.2.i.e 4 13.f odd 12 4
1872.2.t.i 2 156.r even 6 2
1950.2.i.b 2 65.l even 6 2
1950.2.z.b 4 65.r odd 12 4
3042.2.a.d 1 3.b odd 2 1
3042.2.a.m 1 39.d odd 2 1
3042.2.b.d 2 39.f even 4 2
8112.2.a.x 1 52.b odd 2 1
8112.2.a.bb 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1014))$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{7} - 2$$ T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T - 2$$
$11$ $$T + 2$$
$13$ $$T$$
$17$ $$T - 5$$
$19$ $$T - 2$$
$23$ $$T - 6$$
$29$ $$T + 9$$
$31$ $$T - 4$$
$37$ $$T - 11$$
$41$ $$T + 5$$
$43$ $$T - 10$$
$47$ $$T + 2$$
$53$ $$T + 1$$
$59$ $$T - 8$$
$61$ $$T + 11$$
$67$ $$T + 2$$
$71$ $$T - 14$$
$73$ $$T - 13$$
$79$ $$T + 4$$
$83$ $$T + 6$$
$89$ $$T + 2$$
$97$ $$T - 2$$