# Properties

 Label 1014.2.a.f Level $1014$ Weight $2$ Character orbit 1014.a Self dual yes Analytic conductor $8.097$ 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] = [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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - 3 q^{5} + q^{6} - 2 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 - 3 * q^5 + q^6 - 2 * q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} - 3 q^{5} + q^{6} - 2 q^{7} + q^{8} + q^{9} - 3 q^{10} - 6 q^{11} + q^{12} - 2 q^{14} - 3 q^{15} + q^{16} - 3 q^{17} + q^{18} - 2 q^{19} - 3 q^{20} - 2 q^{21} - 6 q^{22} - 6 q^{23} + q^{24} + 4 q^{25} + q^{27} - 2 q^{28} + 3 q^{29} - 3 q^{30} + 4 q^{31} + q^{32} - 6 q^{33} - 3 q^{34} + 6 q^{35} + q^{36} + 7 q^{37} - 2 q^{38} - 3 q^{40} + 3 q^{41} - 2 q^{42} - 10 q^{43} - 6 q^{44} - 3 q^{45} - 6 q^{46} - 6 q^{47} + q^{48} - 3 q^{49} + 4 q^{50} - 3 q^{51} + 3 q^{53} + q^{54} + 18 q^{55} - 2 q^{56} - 2 q^{57} + 3 q^{58} - 3 q^{60} - 7 q^{61} + 4 q^{62} - 2 q^{63} + q^{64} - 6 q^{66} + 10 q^{67} - 3 q^{68} - 6 q^{69} + 6 q^{70} - 6 q^{71} + q^{72} + 13 q^{73} + 7 q^{74} + 4 q^{75} - 2 q^{76} + 12 q^{77} - 4 q^{79} - 3 q^{80} + q^{81} + 3 q^{82} + 6 q^{83} - 2 q^{84} + 9 q^{85} - 10 q^{86} + 3 q^{87} - 6 q^{88} - 18 q^{89} - 3 q^{90} - 6 q^{92} + 4 q^{93} - 6 q^{94} + 6 q^{95} + q^{96} - 14 q^{97} - 3 q^{98} - 6 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 - 3 * q^5 + q^6 - 2 * q^7 + q^8 + q^9 - 3 * q^10 - 6 * q^11 + q^12 - 2 * q^14 - 3 * q^15 + q^16 - 3 * q^17 + q^18 - 2 * q^19 - 3 * q^20 - 2 * q^21 - 6 * q^22 - 6 * q^23 + q^24 + 4 * q^25 + q^27 - 2 * q^28 + 3 * q^29 - 3 * q^30 + 4 * q^31 + q^32 - 6 * q^33 - 3 * q^34 + 6 * q^35 + q^36 + 7 * q^37 - 2 * q^38 - 3 * q^40 + 3 * q^41 - 2 * q^42 - 10 * q^43 - 6 * q^44 - 3 * q^45 - 6 * q^46 - 6 * q^47 + q^48 - 3 * q^49 + 4 * q^50 - 3 * q^51 + 3 * q^53 + q^54 + 18 * q^55 - 2 * q^56 - 2 * q^57 + 3 * q^58 - 3 * q^60 - 7 * q^61 + 4 * q^62 - 2 * q^63 + q^64 - 6 * q^66 + 10 * q^67 - 3 * q^68 - 6 * q^69 + 6 * q^70 - 6 * q^71 + q^72 + 13 * q^73 + 7 * q^74 + 4 * q^75 - 2 * q^76 + 12 * q^77 - 4 * q^79 - 3 * q^80 + q^81 + 3 * q^82 + 6 * q^83 - 2 * q^84 + 9 * q^85 - 10 * q^86 + 3 * q^87 - 6 * q^88 - 18 * q^89 - 3 * q^90 - 6 * q^92 + 4 * q^93 - 6 * q^94 + 6 * q^95 + q^96 - 14 * q^97 - 3 * q^98 - 6 * 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 −3.00000 1.00000 −2.00000 1.00000 1.00000 −3.00000
 $$n$$: 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.f 1
3.b odd 2 1 3042.2.a.h 1
4.b odd 2 1 8112.2.a.c 1
13.b even 2 1 1014.2.a.c 1
13.c even 3 2 1014.2.e.a 2
13.d odd 4 2 1014.2.b.c 2
13.e even 6 2 78.2.e.a 2
13.f odd 12 4 1014.2.i.b 4
39.d odd 2 1 3042.2.a.i 1
39.f even 4 2 3042.2.b.h 2
39.h odd 6 2 234.2.h.a 2
52.b odd 2 1 8112.2.a.m 1
52.i odd 6 2 624.2.q.g 2
65.l even 6 2 1950.2.i.m 2
65.r odd 12 4 1950.2.z.g 4
156.r even 6 2 1872.2.t.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 13.e even 6 2
234.2.h.a 2 39.h odd 6 2
624.2.q.g 2 52.i odd 6 2
1014.2.a.c 1 13.b even 2 1
1014.2.a.f 1 1.a even 1 1 trivial
1014.2.b.c 2 13.d odd 4 2
1014.2.e.a 2 13.c even 3 2
1014.2.i.b 4 13.f odd 12 4
1872.2.t.c 2 156.r even 6 2
1950.2.i.m 2 65.l even 6 2
1950.2.z.g 4 65.r odd 12 4
3042.2.a.h 1 3.b odd 2 1
3042.2.a.i 1 39.d odd 2 1
3042.2.b.h 2 39.f even 4 2
8112.2.a.c 1 4.b odd 2 1
8112.2.a.m 1 52.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1014))$$:

 $$T_{5} + 3$$ T5 + 3 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T + 3$$
$7$ $$T + 2$$
$11$ $$T + 6$$
$13$ $$T$$
$17$ $$T + 3$$
$19$ $$T + 2$$
$23$ $$T + 6$$
$29$ $$T - 3$$
$31$ $$T - 4$$
$37$ $$T - 7$$
$41$ $$T - 3$$
$43$ $$T + 10$$
$47$ $$T + 6$$
$53$ $$T - 3$$
$59$ $$T$$
$61$ $$T + 7$$
$67$ $$T - 10$$
$71$ $$T + 6$$
$73$ $$T - 13$$
$79$ $$T + 4$$
$83$ $$T - 6$$
$89$ $$T + 18$$
$97$ $$T + 14$$