# Properties

 Label 1014.2.i.b Level $1014$ Weight $2$ Character orbit 1014.i Analytic conductor $8.097$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1014,2,Mod(361,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("1014.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.i (of order $$6$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} - \zeta_{12}^{2} q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + 3 \zeta_{12}^{3} q^{5} + \zeta_{12} q^{6} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + (\zeta_{12}^{2} - 1) q^{9} +O(q^{10})$$ q + (z^3 - z) * q^2 - z^2 * q^3 + (-z^2 + 1) * q^4 + 3*z^3 * q^5 + z * q^6 + 2*z * q^7 + z^3 * q^8 + (z^2 - 1) * q^9 $$q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} - \zeta_{12}^{2} q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + 3 \zeta_{12}^{3} q^{5} + \zeta_{12} q^{6} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + (\zeta_{12}^{2} - 1) q^{9} - 3 \zeta_{12}^{2} q^{10} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{11} - q^{12} - 2 q^{14} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{15} - \zeta_{12}^{2} q^{16} + (3 \zeta_{12}^{2} - 3) q^{17} - \zeta_{12}^{3} q^{18} - 2 \zeta_{12} q^{19} + 3 \zeta_{12} q^{20} - 2 \zeta_{12}^{3} q^{21} + ( - 6 \zeta_{12}^{2} + 6) q^{22} - 6 \zeta_{12}^{2} q^{23} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{24} - 4 q^{25} + q^{27} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{28} - 3 \zeta_{12}^{2} q^{29} + (3 \zeta_{12}^{2} - 3) q^{30} - 4 \zeta_{12}^{3} q^{31} + \zeta_{12} q^{32} + 6 \zeta_{12} q^{33} - 3 \zeta_{12}^{3} q^{34} + (6 \zeta_{12}^{2} - 6) q^{35} + \zeta_{12}^{2} q^{36} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{37} + 2 q^{38} - 3 q^{40} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{41} + 2 \zeta_{12}^{2} q^{42} + (10 \zeta_{12}^{2} - 10) q^{43} + 6 \zeta_{12}^{3} q^{44} - 3 \zeta_{12} q^{45} + 6 \zeta_{12} q^{46} - 6 \zeta_{12}^{3} q^{47} + (\zeta_{12}^{2} - 1) q^{48} - 3 \zeta_{12}^{2} q^{49} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{50} + 3 q^{51} + 3 q^{53} + (\zeta_{12}^{3} - \zeta_{12}) q^{54} - 18 \zeta_{12}^{2} q^{55} + (2 \zeta_{12}^{2} - 2) q^{56} + 2 \zeta_{12}^{3} q^{57} + 3 \zeta_{12} q^{58} - 3 \zeta_{12}^{3} q^{60} + ( - 7 \zeta_{12}^{2} + 7) q^{61} + 4 \zeta_{12}^{2} q^{62} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{63} - q^{64} - 6 q^{66} + (10 \zeta_{12}^{3} - 10 \zeta_{12}) q^{67} + 3 \zeta_{12}^{2} q^{68} + (6 \zeta_{12}^{2} - 6) q^{69} - 6 \zeta_{12}^{3} q^{70} - 6 \zeta_{12} q^{71} - \zeta_{12} q^{72} + 13 \zeta_{12}^{3} q^{73} + (7 \zeta_{12}^{2} - 7) q^{74} + 4 \zeta_{12}^{2} q^{75} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{76} - 12 q^{77} - 4 q^{79} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{80} - \zeta_{12}^{2} q^{81} + ( - 3 \zeta_{12}^{2} + 3) q^{82} - 6 \zeta_{12}^{3} q^{83} - 2 \zeta_{12} q^{84} - 9 \zeta_{12} q^{85} - 10 \zeta_{12}^{3} q^{86} + (3 \zeta_{12}^{2} - 3) q^{87} - 6 \zeta_{12}^{2} q^{88} + (18 \zeta_{12}^{3} - 18 \zeta_{12}) q^{89} + 3 q^{90} - 6 q^{92} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{93} + 6 \zeta_{12}^{2} q^{94} + ( - 6 \zeta_{12}^{2} + 6) q^{95} - \zeta_{12}^{3} q^{96} - 14 \zeta_{12} q^{97} + 3 \zeta_{12} q^{98} - 6 \zeta_{12}^{3} q^{99} +O(q^{100})$$ q + (z^3 - z) * q^2 - z^2 * q^3 + (-z^2 + 1) * q^4 + 3*z^3 * q^5 + z * q^6 + 2*z * q^7 + z^3 * q^8 + (z^2 - 1) * q^9 - 3*z^2 * q^10 + (6*z^3 - 6*z) * q^11 - q^12 - 2 * q^14 + (-3*z^3 + 3*z) * q^15 - z^2 * q^16 + (3*z^2 - 3) * q^17 - z^3 * q^18 - 2*z * q^19 + 3*z * q^20 - 2*z^3 * q^21 + (-6*z^2 + 6) * q^22 - 6*z^2 * q^23 + (-z^3 + z) * q^24 - 4 * q^25 + q^27 + (-2*z^3 + 2*z) * q^28 - 3*z^2 * q^29 + (3*z^2 - 3) * q^30 - 4*z^3 * q^31 + z * q^32 + 6*z * q^33 - 3*z^3 * q^34 + (6*z^2 - 6) * q^35 + z^2 * q^36 + (-7*z^3 + 7*z) * q^37 + 2 * q^38 - 3 * q^40 + (3*z^3 - 3*z) * q^41 + 2*z^2 * q^42 + (10*z^2 - 10) * q^43 + 6*z^3 * q^44 - 3*z * q^45 + 6*z * q^46 - 6*z^3 * q^47 + (z^2 - 1) * q^48 - 3*z^2 * q^49 + (-4*z^3 + 4*z) * q^50 + 3 * q^51 + 3 * q^53 + (z^3 - z) * q^54 - 18*z^2 * q^55 + (2*z^2 - 2) * q^56 + 2*z^3 * q^57 + 3*z * q^58 - 3*z^3 * q^60 + (-7*z^2 + 7) * q^61 + 4*z^2 * q^62 + (2*z^3 - 2*z) * q^63 - q^64 - 6 * q^66 + (10*z^3 - 10*z) * q^67 + 3*z^2 * q^68 + (6*z^2 - 6) * q^69 - 6*z^3 * q^70 - 6*z * q^71 - z * q^72 + 13*z^3 * q^73 + (7*z^2 - 7) * q^74 + 4*z^2 * q^75 + (2*z^3 - 2*z) * q^76 - 12 * q^77 - 4 * q^79 + (-3*z^3 + 3*z) * q^80 - z^2 * q^81 + (-3*z^2 + 3) * q^82 - 6*z^3 * q^83 - 2*z * q^84 - 9*z * q^85 - 10*z^3 * q^86 + (3*z^2 - 3) * q^87 - 6*z^2 * q^88 + (18*z^3 - 18*z) * q^89 + 3 * q^90 - 6 * q^92 + (4*z^3 - 4*z) * q^93 + 6*z^2 * q^94 + (-6*z^2 + 6) * q^95 - z^3 * q^96 - 14*z * q^97 + 3*z * q^98 - 6*z^3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 2 q^{4} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 2 * q^4 - 2 * q^9 $$4 q - 2 q^{3} + 2 q^{4} - 2 q^{9} - 6 q^{10} - 4 q^{12} - 8 q^{14} - 2 q^{16} - 6 q^{17} + 12 q^{22} - 12 q^{23} - 16 q^{25} + 4 q^{27} - 6 q^{29} - 6 q^{30} - 12 q^{35} + 2 q^{36} + 8 q^{38} - 12 q^{40} + 4 q^{42} - 20 q^{43} - 2 q^{48} - 6 q^{49} + 12 q^{51} + 12 q^{53} - 36 q^{55} - 4 q^{56} + 14 q^{61} + 8 q^{62} - 4 q^{64} - 24 q^{66} + 6 q^{68} - 12 q^{69} - 14 q^{74} + 8 q^{75} - 48 q^{77} - 16 q^{79} - 2 q^{81} + 6 q^{82} - 6 q^{87} - 12 q^{88} + 12 q^{90} - 24 q^{92} + 12 q^{94} + 12 q^{95}+O(q^{100})$$ 4 * q - 2 * q^3 + 2 * q^4 - 2 * q^9 - 6 * q^10 - 4 * q^12 - 8 * q^14 - 2 * q^16 - 6 * q^17 + 12 * q^22 - 12 * q^23 - 16 * q^25 + 4 * q^27 - 6 * q^29 - 6 * q^30 - 12 * q^35 + 2 * q^36 + 8 * q^38 - 12 * q^40 + 4 * q^42 - 20 * q^43 - 2 * q^48 - 6 * q^49 + 12 * q^51 + 12 * q^53 - 36 * q^55 - 4 * q^56 + 14 * q^61 + 8 * q^62 - 4 * q^64 - 24 * q^66 + 6 * q^68 - 12 * q^69 - 14 * q^74 + 8 * q^75 - 48 * q^77 - 16 * q^79 - 2 * q^81 + 6 * q^82 - 6 * q^87 - 12 * q^88 + 12 * q^90 - 24 * q^92 + 12 * q^94 + 12 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$1 - \zeta_{12}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 3.00000i 0.866025 + 0.500000i 1.73205 + 1.00000i 1.00000i −0.500000 + 0.866025i −1.50000 2.59808i
361.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 3.00000i −0.866025 0.500000i −1.73205 1.00000i 1.00000i −0.500000 + 0.866025i −1.50000 2.59808i
823.1 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 3.00000i 0.866025 0.500000i 1.73205 1.00000i 1.00000i −0.500000 0.866025i −1.50000 + 2.59808i
823.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 3.00000i −0.866025 + 0.500000i −1.73205 + 1.00000i 1.00000i −0.500000 0.866025i −1.50000 + 2.59808i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.i.b 4
13.b even 2 1 inner 1014.2.i.b 4
13.c even 3 1 1014.2.b.c 2
13.c even 3 1 inner 1014.2.i.b 4
13.d odd 4 1 78.2.e.a 2
13.d odd 4 1 1014.2.e.a 2
13.e even 6 1 1014.2.b.c 2
13.e even 6 1 inner 1014.2.i.b 4
13.f odd 12 1 78.2.e.a 2
13.f odd 12 1 1014.2.a.c 1
13.f odd 12 1 1014.2.a.f 1
13.f odd 12 1 1014.2.e.a 2
39.f even 4 1 234.2.h.a 2
39.h odd 6 1 3042.2.b.h 2
39.i odd 6 1 3042.2.b.h 2
39.k even 12 1 234.2.h.a 2
39.k even 12 1 3042.2.a.h 1
39.k even 12 1 3042.2.a.i 1
52.f even 4 1 624.2.q.g 2
52.l even 12 1 624.2.q.g 2
52.l even 12 1 8112.2.a.c 1
52.l even 12 1 8112.2.a.m 1
65.f even 4 1 1950.2.z.g 4
65.g odd 4 1 1950.2.i.m 2
65.k even 4 1 1950.2.z.g 4
65.o even 12 1 1950.2.z.g 4
65.s odd 12 1 1950.2.i.m 2
65.t even 12 1 1950.2.z.g 4
156.l odd 4 1 1872.2.t.c 2
156.v odd 12 1 1872.2.t.c 2

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

## Hecke kernels

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

 $$T_{5}^{2} + 9$$ T5^2 + 9 $$T_{7}^{4} - 4T_{7}^{2} + 16$$ T7^4 - 4*T7^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} + 9)^{2}$$
$7$ $$T^{4} - 4T^{2} + 16$$
$11$ $$T^{4} - 36T^{2} + 1296$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 3 T + 9)^{2}$$
$19$ $$T^{4} - 4T^{2} + 16$$
$23$ $$(T^{2} + 6 T + 36)^{2}$$
$29$ $$(T^{2} + 3 T + 9)^{2}$$
$31$ $$(T^{2} + 16)^{2}$$
$37$ $$T^{4} - 49T^{2} + 2401$$
$41$ $$T^{4} - 9T^{2} + 81$$
$43$ $$(T^{2} + 10 T + 100)^{2}$$
$47$ $$(T^{2} + 36)^{2}$$
$53$ $$(T - 3)^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 7 T + 49)^{2}$$
$67$ $$T^{4} - 100 T^{2} + 10000$$
$71$ $$T^{4} - 36T^{2} + 1296$$
$73$ $$(T^{2} + 169)^{2}$$
$79$ $$(T + 4)^{4}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$T^{4} - 324 T^{2} + 104976$$
$97$ $$T^{4} - 196 T^{2} + 38416$$