# Properties

 Label 1274.2.g.a Level $1274$ Weight $2$ Character orbit 1274.g Analytic conductor $10.173$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1274,2,Mod(295,1274)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1274, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("1274.295");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1274 = 2 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1274.g (of order $$3$$, degree $$2$$, 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: $$10.1729412175$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + q^{5} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 + q^5 + q^8 + 3*z * q^9 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + q^{5} + q^{8} + 3 \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + (4 \zeta_{6} - 4) q^{11} + ( - \zeta_{6} - 3) q^{13} + (\zeta_{6} - 1) q^{16} + 3 \zeta_{6} q^{17} - 3 q^{18} - \zeta_{6} q^{20} - 4 \zeta_{6} q^{22} + ( - 4 \zeta_{6} + 4) q^{23} - 4 q^{25} + ( - 3 \zeta_{6} + 4) q^{26} + ( - \zeta_{6} + 1) q^{29} - 4 q^{31} - \zeta_{6} q^{32} - 3 q^{34} + ( - 3 \zeta_{6} + 3) q^{36} + (3 \zeta_{6} - 3) q^{37} + q^{40} + (9 \zeta_{6} - 9) q^{41} + 8 \zeta_{6} q^{43} + 4 q^{44} + 3 \zeta_{6} q^{45} + 4 \zeta_{6} q^{46} + 8 q^{47} + ( - 4 \zeta_{6} + 4) q^{50} + (4 \zeta_{6} - 1) q^{52} - 9 q^{53} + (4 \zeta_{6} - 4) q^{55} + \zeta_{6} q^{58} - 4 \zeta_{6} q^{59} + 7 \zeta_{6} q^{61} + ( - 4 \zeta_{6} + 4) q^{62} + q^{64} + ( - \zeta_{6} - 3) q^{65} + (4 \zeta_{6} - 4) q^{67} + ( - 3 \zeta_{6} + 3) q^{68} + 8 \zeta_{6} q^{71} + 3 \zeta_{6} q^{72} - 11 q^{73} - 3 \zeta_{6} q^{74} - 4 q^{79} + (\zeta_{6} - 1) q^{80} + (9 \zeta_{6} - 9) q^{81} - 9 \zeta_{6} q^{82} + 3 \zeta_{6} q^{85} - 8 q^{86} + (4 \zeta_{6} - 4) q^{88} + (6 \zeta_{6} - 6) q^{89} - 3 q^{90} - 4 q^{92} + (8 \zeta_{6} - 8) q^{94} + 2 \zeta_{6} q^{97} - 12 q^{99} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 + q^5 + q^8 + 3*z * q^9 + (z - 1) * q^10 + (4*z - 4) * q^11 + (-z - 3) * q^13 + (z - 1) * q^16 + 3*z * q^17 - 3 * q^18 - z * q^20 - 4*z * q^22 + (-4*z + 4) * q^23 - 4 * q^25 + (-3*z + 4) * q^26 + (-z + 1) * q^29 - 4 * q^31 - z * q^32 - 3 * q^34 + (-3*z + 3) * q^36 + (3*z - 3) * q^37 + q^40 + (9*z - 9) * q^41 + 8*z * q^43 + 4 * q^44 + 3*z * q^45 + 4*z * q^46 + 8 * q^47 + (-4*z + 4) * q^50 + (4*z - 1) * q^52 - 9 * q^53 + (4*z - 4) * q^55 + z * q^58 - 4*z * q^59 + 7*z * q^61 + (-4*z + 4) * q^62 + q^64 + (-z - 3) * q^65 + (4*z - 4) * q^67 + (-3*z + 3) * q^68 + 8*z * q^71 + 3*z * q^72 - 11 * q^73 - 3*z * q^74 - 4 * q^79 + (z - 1) * q^80 + (9*z - 9) * q^81 - 9*z * q^82 + 3*z * q^85 - 8 * q^86 + (4*z - 4) * q^88 + (6*z - 6) * q^89 - 3 * q^90 - 4 * q^92 + (8*z - 8) * q^94 + 2*z * q^97 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 2 q^{5} + 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - q^2 - q^4 + 2 * q^5 + 2 * q^8 + 3 * q^9 $$2 q - q^{2} - q^{4} + 2 q^{5} + 2 q^{8} + 3 q^{9} - q^{10} - 4 q^{11} - 7 q^{13} - q^{16} + 3 q^{17} - 6 q^{18} - q^{20} - 4 q^{22} + 4 q^{23} - 8 q^{25} + 5 q^{26} + q^{29} - 8 q^{31} - q^{32} - 6 q^{34} + 3 q^{36} - 3 q^{37} + 2 q^{40} - 9 q^{41} + 8 q^{43} + 8 q^{44} + 3 q^{45} + 4 q^{46} + 16 q^{47} + 4 q^{50} + 2 q^{52} - 18 q^{53} - 4 q^{55} + q^{58} - 4 q^{59} + 7 q^{61} + 4 q^{62} + 2 q^{64} - 7 q^{65} - 4 q^{67} + 3 q^{68} + 8 q^{71} + 3 q^{72} - 22 q^{73} - 3 q^{74} - 8 q^{79} - q^{80} - 9 q^{81} - 9 q^{82} + 3 q^{85} - 16 q^{86} - 4 q^{88} - 6 q^{89} - 6 q^{90} - 8 q^{92} - 8 q^{94} + 2 q^{97} - 24 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^4 + 2 * q^5 + 2 * q^8 + 3 * q^9 - q^10 - 4 * q^11 - 7 * q^13 - q^16 + 3 * q^17 - 6 * q^18 - q^20 - 4 * q^22 + 4 * q^23 - 8 * q^25 + 5 * q^26 + q^29 - 8 * q^31 - q^32 - 6 * q^34 + 3 * q^36 - 3 * q^37 + 2 * q^40 - 9 * q^41 + 8 * q^43 + 8 * q^44 + 3 * q^45 + 4 * q^46 + 16 * q^47 + 4 * q^50 + 2 * q^52 - 18 * q^53 - 4 * q^55 + q^58 - 4 * q^59 + 7 * q^61 + 4 * q^62 + 2 * q^64 - 7 * q^65 - 4 * q^67 + 3 * q^68 + 8 * q^71 + 3 * q^72 - 22 * q^73 - 3 * q^74 - 8 * q^79 - q^80 - 9 * q^81 - 9 * q^82 + 3 * q^85 - 16 * q^86 - 4 * q^88 - 6 * q^89 - 6 * q^90 - 8 * q^92 - 8 * q^94 + 2 * q^97 - 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1274\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$885$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
295.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.00000 0 0 1.00000 1.50000 + 2.59808i −0.500000 + 0.866025i
393.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 0 1.00000 1.50000 2.59808i −0.500000 0.866025i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1274.2.g.a 2
7.b odd 2 1 26.2.c.a 2
7.c even 3 1 1274.2.e.m 2
7.c even 3 1 1274.2.h.a 2
7.d odd 6 1 1274.2.e.n 2
7.d odd 6 1 1274.2.h.b 2
13.c even 3 1 inner 1274.2.g.a 2
21.c even 2 1 234.2.h.c 2
28.d even 2 1 208.2.i.b 2
35.c odd 2 1 650.2.e.c 2
35.f even 4 2 650.2.o.c 4
56.e even 2 1 832.2.i.f 2
56.h odd 2 1 832.2.i.e 2
84.h odd 2 1 1872.2.t.k 2
91.b odd 2 1 338.2.c.e 2
91.g even 3 1 1274.2.e.m 2
91.h even 3 1 1274.2.h.a 2
91.i even 4 2 338.2.e.b 4
91.m odd 6 1 1274.2.e.n 2
91.n odd 6 1 26.2.c.a 2
91.n odd 6 1 338.2.a.e 1
91.t odd 6 1 338.2.a.c 1
91.t odd 6 1 338.2.c.e 2
91.v odd 6 1 1274.2.h.b 2
91.bc even 12 2 338.2.b.b 2
91.bc even 12 2 338.2.e.b 4
273.u even 6 1 3042.2.a.k 1
273.bn even 6 1 234.2.h.c 2
273.bn even 6 1 3042.2.a.e 1
273.ca odd 12 2 3042.2.b.e 2
364.v even 6 1 208.2.i.b 2
364.v even 6 1 2704.2.a.h 1
364.bc even 6 1 2704.2.a.i 1
364.bv odd 12 2 2704.2.f.g 2
455.be odd 6 1 8450.2.a.s 1
455.bp odd 6 1 650.2.e.c 2
455.bp odd 6 1 8450.2.a.f 1
455.dc even 12 2 650.2.o.c 4
728.ce odd 6 1 832.2.i.e 2
728.da even 6 1 832.2.i.f 2
1092.dd odd 6 1 1872.2.t.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.c.a 2 7.b odd 2 1
26.2.c.a 2 91.n odd 6 1
208.2.i.b 2 28.d even 2 1
208.2.i.b 2 364.v even 6 1
234.2.h.c 2 21.c even 2 1
234.2.h.c 2 273.bn even 6 1
338.2.a.c 1 91.t odd 6 1
338.2.a.e 1 91.n odd 6 1
338.2.b.b 2 91.bc even 12 2
338.2.c.e 2 91.b odd 2 1
338.2.c.e 2 91.t odd 6 1
338.2.e.b 4 91.i even 4 2
338.2.e.b 4 91.bc even 12 2
650.2.e.c 2 35.c odd 2 1
650.2.e.c 2 455.bp odd 6 1
650.2.o.c 4 35.f even 4 2
650.2.o.c 4 455.dc even 12 2
832.2.i.e 2 56.h odd 2 1
832.2.i.e 2 728.ce odd 6 1
832.2.i.f 2 56.e even 2 1
832.2.i.f 2 728.da even 6 1
1274.2.e.m 2 7.c even 3 1
1274.2.e.m 2 91.g even 3 1
1274.2.e.n 2 7.d odd 6 1
1274.2.e.n 2 91.m odd 6 1
1274.2.g.a 2 1.a even 1 1 trivial
1274.2.g.a 2 13.c even 3 1 inner
1274.2.h.a 2 7.c even 3 1
1274.2.h.a 2 91.h even 3 1
1274.2.h.b 2 7.d odd 6 1
1274.2.h.b 2 91.v odd 6 1
1872.2.t.k 2 84.h odd 2 1
1872.2.t.k 2 1092.dd odd 6 1
2704.2.a.h 1 364.v even 6 1
2704.2.a.i 1 364.bc even 6 1
2704.2.f.g 2 364.bv odd 12 2
3042.2.a.e 1 273.bn even 6 1
3042.2.a.k 1 273.u even 6 1
3042.2.b.e 2 273.ca odd 12 2
8450.2.a.f 1 455.bp odd 6 1
8450.2.a.s 1 455.be odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1274, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5} - 1$$ T5 - 1 $$T_{11}^{2} + 4T_{11} + 16$$ T11^2 + 4*T11 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$T^{2} + 7T + 13$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2} - T + 1$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$T^{2} + 9T + 81$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$(T - 8)^{2}$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} - 7T + 49$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$T^{2} - 8T + 64$$
$73$ $$(T + 11)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$T^{2} - 2T + 4$$