# Properties

 Label 1260.1.eb.b Level $1260$ Weight $1$ Character orbit 1260.eb Analytic conductor $0.629$ Analytic rank $0$ Dimension $8$ Projective image $S_{4}$ CM/RM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,1,Mod(223,1260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1260, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 8, 9, 6]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1260.223");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1260.eb (of order $$12$$, degree $$4$$, 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: $$0.628821915918$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$S_{4}$$ Projective field: Galois closure of 4.2.1134000.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{24}^{10} q^{2} + \zeta_{24}^{3} q^{3} - \zeta_{24}^{8} q^{4} - \zeta_{24}^{5} q^{5} - \zeta_{24} q^{6} + \zeta_{24} q^{7} + \zeta_{24}^{6} q^{8} + \zeta_{24}^{6} q^{9} +O(q^{10})$$ q + z^10 * q^2 + z^3 * q^3 - z^8 * q^4 - z^5 * q^5 - z * q^6 + z * q^7 + z^6 * q^8 + z^6 * q^9 $$q + \zeta_{24}^{10} q^{2} + \zeta_{24}^{3} q^{3} - \zeta_{24}^{8} q^{4} - \zeta_{24}^{5} q^{5} - \zeta_{24} q^{6} + \zeta_{24} q^{7} + \zeta_{24}^{6} q^{8} + \zeta_{24}^{6} q^{9} + \zeta_{24}^{3} q^{10} - \zeta_{24}^{10} q^{11} - \zeta_{24}^{11} q^{12} + \zeta_{24}^{5} q^{13} + \zeta_{24}^{11} q^{14} - \zeta_{24}^{8} q^{15} - \zeta_{24}^{4} q^{16} - \zeta_{24}^{3} q^{17} - \zeta_{24}^{4} q^{18} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{19} - \zeta_{24} q^{20} + \zeta_{24}^{4} q^{21} + \zeta_{24}^{8} q^{22} + \zeta_{24}^{9} q^{24} + \zeta_{24}^{10} q^{25} - \zeta_{24}^{3} q^{26} + \zeta_{24}^{9} q^{27} - \zeta_{24}^{9} q^{28} + \zeta_{24}^{6} q^{30} + \zeta_{24}^{2} q^{32} + \zeta_{24} q^{33} + \zeta_{24} q^{34} - \zeta_{24}^{6} q^{35} + \zeta_{24}^{2} q^{36} + (\zeta_{24}^{7} + \zeta_{24}) q^{38} + \zeta_{24}^{8} q^{39} - \zeta_{24}^{11} q^{40} - \zeta_{24}^{2} q^{42} + ( - \zeta_{24}^{10} + \zeta_{24}^{4}) q^{43} - \zeta_{24}^{6} q^{44} - \zeta_{24}^{11} q^{45} - \zeta_{24} q^{47} - \zeta_{24}^{7} q^{48} + \zeta_{24}^{2} q^{49} - \zeta_{24}^{8} q^{50} - \zeta_{24}^{6} q^{51} + \zeta_{24} q^{52} + (\zeta_{24}^{6} + 1) q^{53} - \zeta_{24}^{7} q^{54} - \zeta_{24}^{3} q^{55} + \zeta_{24}^{7} q^{56} + ( - \zeta_{24}^{6} + 1) q^{57} - \zeta_{24}^{4} q^{60} + (\zeta_{24}^{7} - \zeta_{24}) q^{61} + \zeta_{24}^{7} q^{63} - q^{64} - \zeta_{24}^{10} q^{65} + \zeta_{24}^{11} q^{66} + \zeta_{24}^{11} q^{68} + \zeta_{24}^{4} q^{70} + \zeta_{24}^{6} q^{71} - q^{72} + \zeta_{24}^{9} q^{73} - \zeta_{24} q^{75} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{76} - \zeta_{24}^{11} q^{77} - \zeta_{24}^{6} q^{78} - \zeta_{24}^{4} q^{79} + \zeta_{24}^{9} q^{80} - q^{81} - \zeta_{24}^{7} q^{83} + q^{84} + \zeta_{24}^{8} q^{85} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{86} + \zeta_{24}^{4} q^{88} + \zeta_{24}^{9} q^{90} + \zeta_{24}^{6} q^{91} - \zeta_{24}^{11} q^{94} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{95} + \zeta_{24}^{5} q^{96} - \zeta_{24}^{7} q^{97} - q^{98} + \zeta_{24}^{4} q^{99} +O(q^{100})$$ q + z^10 * q^2 + z^3 * q^3 - z^8 * q^4 - z^5 * q^5 - z * q^6 + z * q^7 + z^6 * q^8 + z^6 * q^9 + z^3 * q^10 - z^10 * q^11 - z^11 * q^12 + z^5 * q^13 + z^11 * q^14 - z^8 * q^15 - z^4 * q^16 - z^3 * q^17 - z^4 * q^18 + (-z^9 - z^3) * q^19 - z * q^20 + z^4 * q^21 + z^8 * q^22 + z^9 * q^24 + z^10 * q^25 - z^3 * q^26 + z^9 * q^27 - z^9 * q^28 + z^6 * q^30 + z^2 * q^32 + z * q^33 + z * q^34 - z^6 * q^35 + z^2 * q^36 + (z^7 + z) * q^38 + z^8 * q^39 - z^11 * q^40 - z^2 * q^42 + (-z^10 + z^4) * q^43 - z^6 * q^44 - z^11 * q^45 - z * q^47 - z^7 * q^48 + z^2 * q^49 - z^8 * q^50 - z^6 * q^51 + z * q^52 + (z^6 + 1) * q^53 - z^7 * q^54 - z^3 * q^55 + z^7 * q^56 + (-z^6 + 1) * q^57 - z^4 * q^60 + (z^7 - z) * q^61 + z^7 * q^63 - q^64 - z^10 * q^65 + z^11 * q^66 + z^11 * q^68 + z^4 * q^70 + z^6 * q^71 - q^72 + z^9 * q^73 - z * q^75 + (z^11 - z^5) * q^76 - z^11 * q^77 - z^6 * q^78 - z^4 * q^79 + z^9 * q^80 - q^81 - z^7 * q^83 + q^84 + z^8 * q^85 + (z^8 - z^2) * q^86 + z^4 * q^88 + z^9 * q^90 + z^6 * q^91 - z^11 * q^94 + (z^8 - z^2) * q^95 + z^5 * q^96 - z^7 * q^97 - q^98 + z^4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4}+O(q^{10})$$ 8 * q + 4 * q^4 $$8 q + 4 q^{4} + 4 q^{15} - 4 q^{16} - 4 q^{18} + 4 q^{21} - 4 q^{22} - 4 q^{39} + 4 q^{43} + 4 q^{50} + 8 q^{53} + 8 q^{57} - 4 q^{60} - 8 q^{64} + 4 q^{70} - 8 q^{72} - 4 q^{79} - 8 q^{81} + 8 q^{84} - 4 q^{85} - 4 q^{86} + 4 q^{88} - 4 q^{95} - 8 q^{98} + 4 q^{99}+O(q^{100})$$ 8 * q + 4 * q^4 + 4 * q^15 - 4 * q^16 - 4 * q^18 + 4 * q^21 - 4 * q^22 - 4 * q^39 + 4 * q^43 + 4 * q^50 + 8 * q^53 + 8 * q^57 - 4 * q^60 - 8 * q^64 + 4 * q^70 - 8 * q^72 - 4 * q^79 - 8 * q^81 + 8 * q^84 - 4 * q^85 - 4 * q^86 + 4 * q^88 - 4 * q^95 - 8 * q^98 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$757$$ $$1081$$ $$\chi(n)$$ $$\zeta_{24}^{8}$$ $$-1$$ $$-\zeta_{24}^{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
223.1
 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i
0.866025 + 0.500000i −0.707107 0.707107i 0.500000 + 0.866025i −0.965926 0.258819i −0.258819 0.965926i 0.258819 + 0.965926i 1.00000i 1.00000i −0.707107 0.707107i
223.2 0.866025 + 0.500000i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 0.965926i 1.00000i 1.00000i 0.707107 + 0.707107i
643.1 −0.866025 + 0.500000i −0.707107 0.707107i 0.500000 0.866025i 0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 0.258819i 1.00000i 1.00000i −0.707107 0.707107i
643.2 −0.866025 + 0.500000i 0.707107 + 0.707107i 0.500000 0.866025i −0.258819 0.965926i −0.965926 0.258819i 0.965926 + 0.258819i 1.00000i 1.00000i 0.707107 + 0.707107i
727.1 −0.866025 0.500000i −0.707107 + 0.707107i 0.500000 + 0.866025i 0.258819 0.965926i 0.965926 0.258819i −0.965926 + 0.258819i 1.00000i 1.00000i −0.707107 + 0.707107i
727.2 −0.866025 0.500000i 0.707107 0.707107i 0.500000 + 0.866025i −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 0.258819i 1.00000i 1.00000i 0.707107 0.707107i
1147.1 0.866025 0.500000i −0.707107 + 0.707107i 0.500000 0.866025i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 0.965926i 1.00000i 1.00000i −0.707107 + 0.707107i
1147.2 0.866025 0.500000i 0.707107 0.707107i 0.500000 0.866025i 0.965926 0.258819i 0.258819 0.965926i −0.258819 + 0.965926i 1.00000i 1.00000i 0.707107 0.707107i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1147.2 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.b odd 2 1 inner
9.c even 3 1 inner
20.e even 4 1 inner
63.l odd 6 1 inner
140.j odd 4 1 inner
180.x even 12 1 inner
1260.eb odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.1.eb.b yes 8
3.b odd 2 1 3780.1.ee.a 8
4.b odd 2 1 1260.1.eb.a 8
5.c odd 4 1 1260.1.eb.a 8
7.b odd 2 1 inner 1260.1.eb.b yes 8
9.c even 3 1 inner 1260.1.eb.b yes 8
9.d odd 6 1 3780.1.ee.a 8
12.b even 2 1 3780.1.ee.b 8
15.e even 4 1 3780.1.ee.b 8
20.e even 4 1 inner 1260.1.eb.b yes 8
21.c even 2 1 3780.1.ee.a 8
28.d even 2 1 1260.1.eb.a 8
35.f even 4 1 1260.1.eb.a 8
36.f odd 6 1 1260.1.eb.a 8
36.h even 6 1 3780.1.ee.b 8
45.k odd 12 1 1260.1.eb.a 8
45.l even 12 1 3780.1.ee.b 8
60.l odd 4 1 3780.1.ee.a 8
63.l odd 6 1 inner 1260.1.eb.b yes 8
63.o even 6 1 3780.1.ee.a 8
84.h odd 2 1 3780.1.ee.b 8
105.k odd 4 1 3780.1.ee.b 8
140.j odd 4 1 inner 1260.1.eb.b yes 8
180.v odd 12 1 3780.1.ee.a 8
180.x even 12 1 inner 1260.1.eb.b yes 8
252.s odd 6 1 3780.1.ee.b 8
252.bi even 6 1 1260.1.eb.a 8
315.cb even 12 1 1260.1.eb.a 8
315.cf odd 12 1 3780.1.ee.b 8
420.w even 4 1 3780.1.ee.a 8
1260.do even 12 1 3780.1.ee.a 8
1260.eb odd 12 1 inner 1260.1.eb.b yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.eb.a 8 4.b odd 2 1
1260.1.eb.a 8 5.c odd 4 1
1260.1.eb.a 8 28.d even 2 1
1260.1.eb.a 8 35.f even 4 1
1260.1.eb.a 8 36.f odd 6 1
1260.1.eb.a 8 45.k odd 12 1
1260.1.eb.a 8 252.bi even 6 1
1260.1.eb.a 8 315.cb even 12 1
1260.1.eb.b yes 8 1.a even 1 1 trivial
1260.1.eb.b yes 8 7.b odd 2 1 inner
1260.1.eb.b yes 8 9.c even 3 1 inner
1260.1.eb.b yes 8 20.e even 4 1 inner
1260.1.eb.b yes 8 63.l odd 6 1 inner
1260.1.eb.b yes 8 140.j odd 4 1 inner
1260.1.eb.b yes 8 180.x even 12 1 inner
1260.1.eb.b yes 8 1260.eb odd 12 1 inner
3780.1.ee.a 8 3.b odd 2 1
3780.1.ee.a 8 9.d odd 6 1
3780.1.ee.a 8 21.c even 2 1
3780.1.ee.a 8 60.l odd 4 1
3780.1.ee.a 8 63.o even 6 1
3780.1.ee.a 8 180.v odd 12 1
3780.1.ee.a 8 420.w even 4 1
3780.1.ee.a 8 1260.do even 12 1
3780.1.ee.b 8 12.b even 2 1
3780.1.ee.b 8 15.e even 4 1
3780.1.ee.b 8 36.h even 6 1
3780.1.ee.b 8 45.l even 12 1
3780.1.ee.b 8 84.h odd 2 1
3780.1.ee.b 8 105.k odd 4 1
3780.1.ee.b 8 252.s odd 6 1
3780.1.ee.b 8 315.cf odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{43}^{4} - 2T_{43}^{3} + 2T_{43}^{2} - 4T_{43} + 4$$ acting on $$S_{1}^{\mathrm{new}}(1260, [\chi])$$.

## Hecke characteristic polynomials

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