# Properties

 Label 1274.2.f.a Level $1274$ Weight $2$ Character orbit 1274.f 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(79,1274)]

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([2, 0]))

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

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

chi := DirichletCharacter("1274.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1274 = 2 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1274.f (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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + 3 q^{6} + q^{8} - 6 \zeta_{6} q^{9} +O(q^{10})$$ q - z * q^2 + (3*z - 3) * q^3 + (z - 1) * q^4 - z * q^5 + 3 * q^6 + q^8 - 6*z * q^9 $$q - \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + 3 q^{6} + q^{8} - 6 \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + ( - 2 \zeta_{6} + 2) q^{11} - 3 \zeta_{6} q^{12} + q^{13} + 3 q^{15} - \zeta_{6} q^{16} + (3 \zeta_{6} - 3) q^{17} + (6 \zeta_{6} - 6) q^{18} + 6 \zeta_{6} q^{19} + q^{20} - 2 q^{22} + 4 \zeta_{6} q^{23} + (3 \zeta_{6} - 3) q^{24} + ( - 4 \zeta_{6} + 4) q^{25} - \zeta_{6} q^{26} + 9 q^{27} + 2 q^{29} - 3 \zeta_{6} q^{30} + ( - 4 \zeta_{6} + 4) q^{31} + (\zeta_{6} - 1) q^{32} + 6 \zeta_{6} q^{33} + 3 q^{34} + 6 q^{36} - 3 \zeta_{6} q^{37} + ( - 6 \zeta_{6} + 6) q^{38} + (3 \zeta_{6} - 3) q^{39} - \zeta_{6} q^{40} - 5 q^{43} + 2 \zeta_{6} q^{44} + (6 \zeta_{6} - 6) q^{45} + ( - 4 \zeta_{6} + 4) q^{46} + 13 \zeta_{6} q^{47} + 3 q^{48} - 4 q^{50} - 9 \zeta_{6} q^{51} + (\zeta_{6} - 1) q^{52} + (12 \zeta_{6} - 12) q^{53} - 9 \zeta_{6} q^{54} - 2 q^{55} - 18 q^{57} - 2 \zeta_{6} q^{58} + (10 \zeta_{6} - 10) q^{59} + (3 \zeta_{6} - 3) q^{60} - 8 \zeta_{6} q^{61} - 4 q^{62} + q^{64} - \zeta_{6} q^{65} + ( - 6 \zeta_{6} + 6) q^{66} + ( - 2 \zeta_{6} + 2) q^{67} - 3 \zeta_{6} q^{68} - 12 q^{69} - 5 q^{71} - 6 \zeta_{6} q^{72} + (10 \zeta_{6} - 10) q^{73} + (3 \zeta_{6} - 3) q^{74} + 12 \zeta_{6} q^{75} - 6 q^{76} + 3 q^{78} + 4 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{80} + (9 \zeta_{6} - 9) q^{81} + 3 q^{85} + 5 \zeta_{6} q^{86} + (6 \zeta_{6} - 6) q^{87} + ( - 2 \zeta_{6} + 2) q^{88} + 6 \zeta_{6} q^{89} + 6 q^{90} - 4 q^{92} + 12 \zeta_{6} q^{93} + ( - 13 \zeta_{6} + 13) q^{94} + ( - 6 \zeta_{6} + 6) q^{95} - 3 \zeta_{6} q^{96} - 14 q^{97} - 12 q^{99} +O(q^{100})$$ q - z * q^2 + (3*z - 3) * q^3 + (z - 1) * q^4 - z * q^5 + 3 * q^6 + q^8 - 6*z * q^9 + (z - 1) * q^10 + (-2*z + 2) * q^11 - 3*z * q^12 + q^13 + 3 * q^15 - z * q^16 + (3*z - 3) * q^17 + (6*z - 6) * q^18 + 6*z * q^19 + q^20 - 2 * q^22 + 4*z * q^23 + (3*z - 3) * q^24 + (-4*z + 4) * q^25 - z * q^26 + 9 * q^27 + 2 * q^29 - 3*z * q^30 + (-4*z + 4) * q^31 + (z - 1) * q^32 + 6*z * q^33 + 3 * q^34 + 6 * q^36 - 3*z * q^37 + (-6*z + 6) * q^38 + (3*z - 3) * q^39 - z * q^40 - 5 * q^43 + 2*z * q^44 + (6*z - 6) * q^45 + (-4*z + 4) * q^46 + 13*z * q^47 + 3 * q^48 - 4 * q^50 - 9*z * q^51 + (z - 1) * q^52 + (12*z - 12) * q^53 - 9*z * q^54 - 2 * q^55 - 18 * q^57 - 2*z * q^58 + (10*z - 10) * q^59 + (3*z - 3) * q^60 - 8*z * q^61 - 4 * q^62 + q^64 - z * q^65 + (-6*z + 6) * q^66 + (-2*z + 2) * q^67 - 3*z * q^68 - 12 * q^69 - 5 * q^71 - 6*z * q^72 + (10*z - 10) * q^73 + (3*z - 3) * q^74 + 12*z * q^75 - 6 * q^76 + 3 * q^78 + 4*z * q^79 + (z - 1) * q^80 + (9*z - 9) * q^81 + 3 * q^85 + 5*z * q^86 + (6*z - 6) * q^87 + (-2*z + 2) * q^88 + 6*z * q^89 + 6 * q^90 - 4 * q^92 + 12*z * q^93 + (-13*z + 13) * q^94 + (-6*z + 6) * q^95 - 3*z * q^96 - 14 * q^97 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 3 q^{3} - q^{4} - q^{5} + 6 q^{6} + 2 q^{8} - 6 q^{9}+O(q^{10})$$ 2 * q - q^2 - 3 * q^3 - q^4 - q^5 + 6 * q^6 + 2 * q^8 - 6 * q^9 $$2 q - q^{2} - 3 q^{3} - q^{4} - q^{5} + 6 q^{6} + 2 q^{8} - 6 q^{9} - q^{10} + 2 q^{11} - 3 q^{12} + 2 q^{13} + 6 q^{15} - q^{16} - 3 q^{17} - 6 q^{18} + 6 q^{19} + 2 q^{20} - 4 q^{22} + 4 q^{23} - 3 q^{24} + 4 q^{25} - q^{26} + 18 q^{27} + 4 q^{29} - 3 q^{30} + 4 q^{31} - q^{32} + 6 q^{33} + 6 q^{34} + 12 q^{36} - 3 q^{37} + 6 q^{38} - 3 q^{39} - q^{40} - 10 q^{43} + 2 q^{44} - 6 q^{45} + 4 q^{46} + 13 q^{47} + 6 q^{48} - 8 q^{50} - 9 q^{51} - q^{52} - 12 q^{53} - 9 q^{54} - 4 q^{55} - 36 q^{57} - 2 q^{58} - 10 q^{59} - 3 q^{60} - 8 q^{61} - 8 q^{62} + 2 q^{64} - q^{65} + 6 q^{66} + 2 q^{67} - 3 q^{68} - 24 q^{69} - 10 q^{71} - 6 q^{72} - 10 q^{73} - 3 q^{74} + 12 q^{75} - 12 q^{76} + 6 q^{78} + 4 q^{79} - q^{80} - 9 q^{81} + 6 q^{85} + 5 q^{86} - 6 q^{87} + 2 q^{88} + 6 q^{89} + 12 q^{90} - 8 q^{92} + 12 q^{93} + 13 q^{94} + 6 q^{95} - 3 q^{96} - 28 q^{97} - 24 q^{99}+O(q^{100})$$ 2 * q - q^2 - 3 * q^3 - q^4 - q^5 + 6 * q^6 + 2 * q^8 - 6 * q^9 - q^10 + 2 * q^11 - 3 * q^12 + 2 * q^13 + 6 * q^15 - q^16 - 3 * q^17 - 6 * q^18 + 6 * q^19 + 2 * q^20 - 4 * q^22 + 4 * q^23 - 3 * q^24 + 4 * q^25 - q^26 + 18 * q^27 + 4 * q^29 - 3 * q^30 + 4 * q^31 - q^32 + 6 * q^33 + 6 * q^34 + 12 * q^36 - 3 * q^37 + 6 * q^38 - 3 * q^39 - q^40 - 10 * q^43 + 2 * q^44 - 6 * q^45 + 4 * q^46 + 13 * q^47 + 6 * q^48 - 8 * q^50 - 9 * q^51 - q^52 - 12 * q^53 - 9 * q^54 - 4 * q^55 - 36 * q^57 - 2 * q^58 - 10 * q^59 - 3 * q^60 - 8 * q^61 - 8 * q^62 + 2 * q^64 - q^65 + 6 * q^66 + 2 * q^67 - 3 * q^68 - 24 * q^69 - 10 * q^71 - 6 * q^72 - 10 * q^73 - 3 * q^74 + 12 * q^75 - 12 * q^76 + 6 * q^78 + 4 * q^79 - q^80 - 9 * q^81 + 6 * q^85 + 5 * q^86 - 6 * q^87 + 2 * q^88 + 6 * q^89 + 12 * q^90 - 8 * q^92 + 12 * q^93 + 13 * q^94 + 6 * q^95 - 3 * q^96 - 28 * q^97 - 24 * q^99

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i −1.50000 2.59808i −0.500000 0.866025i −0.500000 + 0.866025i 3.00000 0 1.00000 −3.00000 + 5.19615i −0.500000 0.866025i
1145.1 −0.500000 0.866025i −1.50000 + 2.59808i −0.500000 + 0.866025i −0.500000 0.866025i 3.00000 0 1.00000 −3.00000 5.19615i −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
7.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.f.a 2
7.b odd 2 1 1274.2.f.l 2
7.c even 3 1 1274.2.a.o 1
7.c even 3 1 inner 1274.2.f.a 2
7.d odd 6 1 26.2.a.b 1
7.d odd 6 1 1274.2.f.l 2
21.g even 6 1 234.2.a.b 1
28.f even 6 1 208.2.a.d 1
35.i odd 6 1 650.2.a.g 1
35.k even 12 2 650.2.b.a 2
56.j odd 6 1 832.2.a.j 1
56.m even 6 1 832.2.a.a 1
63.i even 6 1 2106.2.e.t 2
63.k odd 6 1 2106.2.e.h 2
63.s even 6 1 2106.2.e.t 2
63.t odd 6 1 2106.2.e.h 2
77.i even 6 1 3146.2.a.a 1
84.j odd 6 1 1872.2.a.m 1
91.l odd 6 1 338.2.c.g 2
91.m odd 6 1 338.2.c.c 2
91.p odd 6 1 338.2.c.g 2
91.s odd 6 1 338.2.a.a 1
91.v odd 6 1 338.2.c.c 2
91.w even 12 2 338.2.e.d 4
91.ba even 12 2 338.2.e.d 4
91.bb even 12 2 338.2.b.a 2
105.p even 6 1 5850.2.a.bn 1
105.w odd 12 2 5850.2.e.v 2
112.v even 12 2 3328.2.b.k 2
112.x odd 12 2 3328.2.b.g 2
119.h odd 6 1 7514.2.a.i 1
133.o even 6 1 9386.2.a.f 1
140.s even 6 1 5200.2.a.c 1
168.ba even 6 1 7488.2.a.w 1
168.be odd 6 1 7488.2.a.v 1
273.ba even 6 1 3042.2.a.l 1
273.cb odd 12 2 3042.2.b.f 2
364.x even 6 1 2704.2.a.n 1
364.bw odd 12 2 2704.2.f.j 2
455.bf odd 6 1 8450.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 7.d odd 6 1
208.2.a.d 1 28.f even 6 1
234.2.a.b 1 21.g even 6 1
338.2.a.a 1 91.s odd 6 1
338.2.b.a 2 91.bb even 12 2
338.2.c.c 2 91.m odd 6 1
338.2.c.c 2 91.v odd 6 1
338.2.c.g 2 91.l odd 6 1
338.2.c.g 2 91.p odd 6 1
338.2.e.d 4 91.w even 12 2
338.2.e.d 4 91.ba even 12 2
650.2.a.g 1 35.i odd 6 1
650.2.b.a 2 35.k even 12 2
832.2.a.a 1 56.m even 6 1
832.2.a.j 1 56.j odd 6 1
1274.2.a.o 1 7.c even 3 1
1274.2.f.a 2 1.a even 1 1 trivial
1274.2.f.a 2 7.c even 3 1 inner
1274.2.f.l 2 7.b odd 2 1
1274.2.f.l 2 7.d odd 6 1
1872.2.a.m 1 84.j odd 6 1
2106.2.e.h 2 63.k odd 6 1
2106.2.e.h 2 63.t odd 6 1
2106.2.e.t 2 63.i even 6 1
2106.2.e.t 2 63.s even 6 1
2704.2.a.n 1 364.x even 6 1
2704.2.f.j 2 364.bw odd 12 2
3042.2.a.l 1 273.ba even 6 1
3042.2.b.f 2 273.cb odd 12 2
3146.2.a.a 1 77.i even 6 1
3328.2.b.g 2 112.x odd 12 2
3328.2.b.k 2 112.v even 12 2
5200.2.a.c 1 140.s even 6 1
5850.2.a.bn 1 105.p even 6 1
5850.2.e.v 2 105.w odd 12 2
7488.2.a.v 1 168.be odd 6 1
7488.2.a.w 1 168.ba even 6 1
7514.2.a.i 1 119.h odd 6 1
8450.2.a.y 1 455.bf odd 6 1
9386.2.a.f 1 133.o even 6 1

## Hecke kernels

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

 $$T_{3}^{2} + 3T_{3} + 9$$ T3^2 + 3*T3 + 9 $$T_{5}^{2} + T_{5} + 1$$ T5^2 + T5 + 1 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - 6T + 36$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$T^{2}$$
$43$ $$(T + 5)^{2}$$
$47$ $$T^{2} - 13T + 169$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$T^{2} + 10T + 100$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} - 2T + 4$$
$71$ $$(T + 5)^{2}$$
$73$ $$T^{2} + 10T + 100$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T + 14)^{2}$$