# Properties

 Label 1274.2.e.n Level $1274$ Weight $2$ Character orbit 1274.e 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(165,1274)]

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([4, 4]))

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

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

chi := DirichletCharacter("1274.165");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1274 = 2 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1274.e (of order $$3$$, 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: $$10.1729412175$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \zeta_{6} q^{5} + q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10})$$ q + q^2 + q^4 + z * q^5 + q^8 + (-3*z + 3) * q^9 $$q + q^{2} + q^{4} + \zeta_{6} q^{5} + q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + \zeta_{6} q^{10} - 4 \zeta_{6} q^{11} + (\zeta_{6} + 3) q^{13} + q^{16} + 3 q^{17} + ( - 3 \zeta_{6} + 3) q^{18} + \zeta_{6} q^{20} - 4 \zeta_{6} q^{22} - 4 q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + (\zeta_{6} + 3) q^{26} + ( - \zeta_{6} + 1) q^{29} + (4 \zeta_{6} - 4) q^{31} + q^{32} + 3 q^{34} + ( - 3 \zeta_{6} + 3) q^{36} + 3 q^{37} + \zeta_{6} q^{40} + ( - 9 \zeta_{6} + 9) q^{41} + 8 \zeta_{6} q^{43} - 4 \zeta_{6} q^{44} + 3 q^{45} - 4 q^{46} + 8 \zeta_{6} q^{47} + ( - 4 \zeta_{6} + 4) q^{50} + (\zeta_{6} + 3) q^{52} + ( - 9 \zeta_{6} + 9) q^{53} + ( - 4 \zeta_{6} + 4) q^{55} + ( - \zeta_{6} + 1) q^{58} - 4 q^{59} + (7 \zeta_{6} - 7) q^{61} + (4 \zeta_{6} - 4) q^{62} + q^{64} + (4 \zeta_{6} - 1) q^{65} - 4 \zeta_{6} q^{67} + 3 q^{68} + 8 \zeta_{6} q^{71} + ( - 3 \zeta_{6} + 3) q^{72} + (11 \zeta_{6} - 11) q^{73} + 3 q^{74} + 4 \zeta_{6} q^{79} + \zeta_{6} q^{80} - 9 \zeta_{6} q^{81} + ( - 9 \zeta_{6} + 9) q^{82} + 3 \zeta_{6} q^{85} + 8 \zeta_{6} q^{86} - 4 \zeta_{6} q^{88} - 6 q^{89} + 3 q^{90} - 4 q^{92} + 8 \zeta_{6} q^{94} - 2 \zeta_{6} q^{97} - 12 q^{99} +O(q^{100})$$ q + q^2 + q^4 + z * q^5 + q^8 + (-3*z + 3) * q^9 + z * q^10 - 4*z * q^11 + (z + 3) * q^13 + q^16 + 3 * q^17 + (-3*z + 3) * q^18 + z * q^20 - 4*z * q^22 - 4 * q^23 + (-4*z + 4) * q^25 + (z + 3) * q^26 + (-z + 1) * q^29 + (4*z - 4) * q^31 + q^32 + 3 * q^34 + (-3*z + 3) * q^36 + 3 * q^37 + z * q^40 + (-9*z + 9) * q^41 + 8*z * q^43 - 4*z * q^44 + 3 * q^45 - 4 * q^46 + 8*z * q^47 + (-4*z + 4) * q^50 + (z + 3) * q^52 + (-9*z + 9) * q^53 + (-4*z + 4) * q^55 + (-z + 1) * q^58 - 4 * q^59 + (7*z - 7) * q^61 + (4*z - 4) * q^62 + q^64 + (4*z - 1) * q^65 - 4*z * q^67 + 3 * q^68 + 8*z * q^71 + (-3*z + 3) * q^72 + (11*z - 11) * q^73 + 3 * q^74 + 4*z * q^79 + z * q^80 - 9*z * q^81 + (-9*z + 9) * q^82 + 3*z * q^85 + 8*z * q^86 - 4*z * q^88 - 6 * q^89 + 3 * q^90 - 4 * q^92 + 8*z * q^94 - 2*z * q^97 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^8 + 3 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{8} + 3 q^{9} + q^{10} - 4 q^{11} + 7 q^{13} + 2 q^{16} + 6 q^{17} + 3 q^{18} + q^{20} - 4 q^{22} - 8 q^{23} + 4 q^{25} + 7 q^{26} + q^{29} - 4 q^{31} + 2 q^{32} + 6 q^{34} + 3 q^{36} + 6 q^{37} + q^{40} + 9 q^{41} + 8 q^{43} - 4 q^{44} + 6 q^{45} - 8 q^{46} + 8 q^{47} + 4 q^{50} + 7 q^{52} + 9 q^{53} + 4 q^{55} + q^{58} - 8 q^{59} - 7 q^{61} - 4 q^{62} + 2 q^{64} + 2 q^{65} - 4 q^{67} + 6 q^{68} + 8 q^{71} + 3 q^{72} - 11 q^{73} + 6 q^{74} + 4 q^{79} + q^{80} - 9 q^{81} + 9 q^{82} + 3 q^{85} + 8 q^{86} - 4 q^{88} - 12 q^{89} + 6 q^{90} - 8 q^{92} + 8 q^{94} - 2 q^{97} - 24 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^8 + 3 * q^9 + q^10 - 4 * q^11 + 7 * q^13 + 2 * q^16 + 6 * q^17 + 3 * q^18 + q^20 - 4 * q^22 - 8 * q^23 + 4 * q^25 + 7 * q^26 + q^29 - 4 * q^31 + 2 * q^32 + 6 * q^34 + 3 * q^36 + 6 * q^37 + q^40 + 9 * q^41 + 8 * q^43 - 4 * q^44 + 6 * q^45 - 8 * q^46 + 8 * q^47 + 4 * q^50 + 7 * q^52 + 9 * q^53 + 4 * q^55 + q^58 - 8 * q^59 - 7 * q^61 - 4 * q^62 + 2 * q^64 + 2 * q^65 - 4 * q^67 + 6 * q^68 + 8 * q^71 + 3 * q^72 - 11 * q^73 + 6 * q^74 + 4 * q^79 + q^80 - 9 * q^81 + 9 * q^82 + 3 * q^85 + 8 * q^86 - 4 * q^88 - 12 * 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 $$\left(\mathbb{Z}/1274\mathbb{Z}\right)^\times$$.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 0 1.00000 0.500000 + 0.866025i 0 0 1.00000 1.50000 2.59808i 0.500000 + 0.866025i
471.1 1.00000 0 1.00000 0.500000 0.866025i 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
91.h 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.e.n 2
7.b odd 2 1 1274.2.e.m 2
7.c even 3 1 26.2.c.a 2
7.c even 3 1 1274.2.h.b 2
7.d odd 6 1 1274.2.g.a 2
7.d odd 6 1 1274.2.h.a 2
13.c even 3 1 1274.2.h.b 2
21.h odd 6 1 234.2.h.c 2
28.g odd 6 1 208.2.i.b 2
35.j even 6 1 650.2.e.c 2
35.l odd 12 2 650.2.o.c 4
56.k odd 6 1 832.2.i.f 2
56.p even 6 1 832.2.i.e 2
84.n even 6 1 1872.2.t.k 2
91.g even 3 1 26.2.c.a 2
91.h even 3 1 338.2.a.e 1
91.h even 3 1 inner 1274.2.e.n 2
91.k even 6 1 338.2.a.c 1
91.m odd 6 1 1274.2.g.a 2
91.n odd 6 1 1274.2.h.a 2
91.r even 6 1 338.2.c.e 2
91.u even 6 1 338.2.c.e 2
91.v odd 6 1 1274.2.e.m 2
91.x odd 12 2 338.2.b.b 2
91.z odd 12 2 338.2.e.b 4
91.bd odd 12 2 338.2.e.b 4
273.s odd 6 1 3042.2.a.e 1
273.bm odd 6 1 234.2.h.c 2
273.bp odd 6 1 3042.2.a.k 1
273.bv even 12 2 3042.2.b.e 2
364.q odd 6 1 208.2.i.b 2
364.bi odd 6 1 2704.2.a.h 1
364.bk odd 6 1 2704.2.a.i 1
364.ca even 12 2 2704.2.f.g 2
455.ba even 6 1 8450.2.a.f 1
455.bm even 6 1 650.2.e.c 2
455.bz even 6 1 8450.2.a.s 1
455.cx odd 12 2 650.2.o.c 4
728.bg even 6 1 832.2.i.e 2
728.di odd 6 1 832.2.i.f 2
1092.dc even 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.c even 3 1
26.2.c.a 2 91.g even 3 1
208.2.i.b 2 28.g odd 6 1
208.2.i.b 2 364.q odd 6 1
234.2.h.c 2 21.h odd 6 1
234.2.h.c 2 273.bm odd 6 1
338.2.a.c 1 91.k even 6 1
338.2.a.e 1 91.h even 3 1
338.2.b.b 2 91.x odd 12 2
338.2.c.e 2 91.r even 6 1
338.2.c.e 2 91.u even 6 1
338.2.e.b 4 91.z odd 12 2
338.2.e.b 4 91.bd odd 12 2
650.2.e.c 2 35.j even 6 1
650.2.e.c 2 455.bm even 6 1
650.2.o.c 4 35.l odd 12 2
650.2.o.c 4 455.cx odd 12 2
832.2.i.e 2 56.p even 6 1
832.2.i.e 2 728.bg even 6 1
832.2.i.f 2 56.k odd 6 1
832.2.i.f 2 728.di odd 6 1
1274.2.e.m 2 7.b odd 2 1
1274.2.e.m 2 91.v odd 6 1
1274.2.e.n 2 1.a even 1 1 trivial
1274.2.e.n 2 91.h even 3 1 inner
1274.2.g.a 2 7.d odd 6 1
1274.2.g.a 2 91.m odd 6 1
1274.2.h.a 2 7.d odd 6 1
1274.2.h.a 2 91.n odd 6 1
1274.2.h.b 2 7.c even 3 1
1274.2.h.b 2 13.c even 3 1
1872.2.t.k 2 84.n even 6 1
1872.2.t.k 2 1092.dc even 6 1
2704.2.a.h 1 364.bi odd 6 1
2704.2.a.i 1 364.bk odd 6 1
2704.2.f.g 2 364.ca even 12 2
3042.2.a.e 1 273.s odd 6 1
3042.2.a.k 1 273.bp odd 6 1
3042.2.b.e 2 273.bv even 12 2
8450.2.a.f 1 455.ba even 6 1
8450.2.a.s 1 455.bz even 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 $$T_{5}^{2} - T_{5} + 1$$ T5^2 - T5 + 1 $$T_{11}^{2} + 4T_{11} + 16$$ T11^2 + 4*T11 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$T^{2} - 7T + 13$$
$17$ $$(T - 3)^{2}$$
$19$ $$T^{2}$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} - T + 1$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$(T - 3)^{2}$$
$41$ $$T^{2} - 9T + 81$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$T^{2} - 9T + 81$$
$59$ $$(T + 4)^{2}$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$T^{2} - 8T + 64$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$T^{2}$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 2T + 4$$