Properties

 Label 1300.1.bc.a Level $1300$ Weight $1$ Character orbit 1300.bc Analytic conductor $0.649$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -4 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,1,Mod(451,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 4]))

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

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

chi := DirichletCharacter("1300.451");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1300.bc (of order $$6$$, 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: $$0.648784516423$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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 52) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.676.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

$q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{8} + \zeta_{6}^{2} q^{9}+O(q^{10})$$ q + z * q^2 + z^2 * q^4 - q^8 + z^2 * q^9 $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{13} - \zeta_{6} q^{16} + \zeta_{6}^{2} q^{17} - q^{18} + \zeta_{6}^{2} q^{26} + \zeta_{6} q^{29} - \zeta_{6}^{2} q^{32} - q^{34} - \zeta_{6} q^{36} - \zeta_{6} q^{37} + \zeta_{6} q^{41} - \zeta_{6} q^{49} - q^{52} + q^{53} + \zeta_{6}^{2} q^{58} - \zeta_{6}^{2} q^{61} + q^{64} - \zeta_{6} q^{68} - \zeta_{6}^{2} q^{72} + q^{73} - \zeta_{6}^{2} q^{74} - \zeta_{6} q^{81} + \zeta_{6}^{2} q^{82} - \zeta_{6} q^{89} - \zeta_{6}^{2} q^{97} - \zeta_{6}^{2} q^{98} +O(q^{100})$$ q + z * q^2 + z^2 * q^4 - q^8 + z^2 * q^9 + z * q^13 - z * q^16 + z^2 * q^17 - q^18 + z^2 * q^26 + z * q^29 - z^2 * q^32 - q^34 - z * q^36 - z * q^37 + z * q^41 - z * q^49 - q^52 + q^53 + z^2 * q^58 - z^2 * q^61 + q^64 - z * q^68 - z^2 * q^72 + q^73 - z^2 * q^74 - z * q^81 + z^2 * q^82 - z * q^89 - z^2 * q^97 - z^2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 - q^4 - 2 * q^8 - q^9 $$2 q + q^{2} - q^{4} - 2 q^{8} - q^{9} + q^{13} - q^{16} - q^{17} - 2 q^{18} - q^{26} + q^{29} + q^{32} - 2 q^{34} - q^{36} - q^{37} + q^{41} - q^{49} - 2 q^{52} + 2 q^{53} - q^{58} + q^{61} + 2 q^{64} - q^{68} + q^{72} + 2 q^{73} + q^{74} - q^{81} - q^{82} - 2 q^{89} + 2 q^{97} + q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 - 2 * q^8 - q^9 + q^13 - q^16 - q^17 - 2 * q^18 - q^26 + q^29 + q^32 - 2 * q^34 - q^36 - q^37 + q^41 - q^49 - 2 * q^52 + 2 * q^53 - q^58 + q^61 + 2 * q^64 - q^68 + q^72 + 2 * q^73 + q^74 - q^81 - q^82 - 2 * q^89 + 2 * q^97 + q^98

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$651$$ $$677$$ $$\chi(n)$$ $$\zeta_{6}^{2}$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 −1.00000 −0.500000 0.866025i 0
1251.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 −1.00000 −0.500000 + 0.866025i 0
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
13.c even 3 1 inner
52.j odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.1.bc.a 2
4.b odd 2 1 CM 1300.1.bc.a 2
5.b even 2 1 52.1.j.a 2
5.c odd 4 2 1300.1.w.a 4
13.c even 3 1 inner 1300.1.bc.a 2
15.d odd 2 1 468.1.br.a 2
20.d odd 2 1 52.1.j.a 2
20.e even 4 2 1300.1.w.a 4
35.c odd 2 1 2548.1.bn.a 2
35.i odd 6 1 2548.1.q.a 2
35.i odd 6 1 2548.1.bi.a 2
35.j even 6 1 2548.1.q.b 2
35.j even 6 1 2548.1.bi.b 2
40.e odd 2 1 832.1.bb.a 2
40.f even 2 1 832.1.bb.a 2
52.j odd 6 1 inner 1300.1.bc.a 2
60.h even 2 1 468.1.br.a 2
65.d even 2 1 676.1.j.a 2
65.g odd 4 2 676.1.i.a 4
65.l even 6 1 676.1.c.a 1
65.l even 6 1 676.1.j.a 2
65.n even 6 1 52.1.j.a 2
65.n even 6 1 676.1.c.b 1
65.q odd 12 2 1300.1.w.a 4
65.s odd 12 2 676.1.b.a 2
65.s odd 12 2 676.1.i.a 4
80.k odd 4 2 3328.1.v.b 4
80.q even 4 2 3328.1.v.b 4
140.c even 2 1 2548.1.bn.a 2
140.p odd 6 1 2548.1.q.b 2
140.p odd 6 1 2548.1.bi.b 2
140.s even 6 1 2548.1.q.a 2
140.s even 6 1 2548.1.bi.a 2
195.x odd 6 1 468.1.br.a 2
260.g odd 2 1 676.1.j.a 2
260.u even 4 2 676.1.i.a 4
260.v odd 6 1 52.1.j.a 2
260.v odd 6 1 676.1.c.b 1
260.w odd 6 1 676.1.c.a 1
260.w odd 6 1 676.1.j.a 2
260.bc even 12 2 676.1.b.a 2
260.bc even 12 2 676.1.i.a 4
260.bj even 12 2 1300.1.w.a 4
455.y odd 6 1 2548.1.q.a 2
455.ba even 6 1 2548.1.q.b 2
455.bm even 6 1 2548.1.bi.b 2
455.bp odd 6 1 2548.1.bn.a 2
455.bw odd 6 1 2548.1.bi.a 2
520.bv even 6 1 832.1.bb.a 2
520.bx odd 6 1 832.1.bb.a 2
780.br even 6 1 468.1.br.a 2
1040.ec even 12 2 3328.1.v.b 4
1040.fb odd 12 2 3328.1.v.b 4
1820.bt even 6 1 2548.1.bi.a 2
1820.ck odd 6 1 2548.1.q.b 2
1820.dj even 6 1 2548.1.bn.a 2
1820.dr even 6 1 2548.1.q.a 2
1820.dv odd 6 1 2548.1.bi.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 5.b even 2 1
52.1.j.a 2 20.d odd 2 1
52.1.j.a 2 65.n even 6 1
52.1.j.a 2 260.v odd 6 1
468.1.br.a 2 15.d odd 2 1
468.1.br.a 2 60.h even 2 1
468.1.br.a 2 195.x odd 6 1
468.1.br.a 2 780.br even 6 1
676.1.b.a 2 65.s odd 12 2
676.1.b.a 2 260.bc even 12 2
676.1.c.a 1 65.l even 6 1
676.1.c.a 1 260.w odd 6 1
676.1.c.b 1 65.n even 6 1
676.1.c.b 1 260.v odd 6 1
676.1.i.a 4 65.g odd 4 2
676.1.i.a 4 65.s odd 12 2
676.1.i.a 4 260.u even 4 2
676.1.i.a 4 260.bc even 12 2
676.1.j.a 2 65.d even 2 1
676.1.j.a 2 65.l even 6 1
676.1.j.a 2 260.g odd 2 1
676.1.j.a 2 260.w odd 6 1
832.1.bb.a 2 40.e odd 2 1
832.1.bb.a 2 40.f even 2 1
832.1.bb.a 2 520.bv even 6 1
832.1.bb.a 2 520.bx odd 6 1
1300.1.w.a 4 5.c odd 4 2
1300.1.w.a 4 20.e even 4 2
1300.1.w.a 4 65.q odd 12 2
1300.1.w.a 4 260.bj even 12 2
1300.1.bc.a 2 1.a even 1 1 trivial
1300.1.bc.a 2 4.b odd 2 1 CM
1300.1.bc.a 2 13.c even 3 1 inner
1300.1.bc.a 2 52.j odd 6 1 inner
2548.1.q.a 2 35.i odd 6 1
2548.1.q.a 2 140.s even 6 1
2548.1.q.a 2 455.y odd 6 1
2548.1.q.a 2 1820.dr even 6 1
2548.1.q.b 2 35.j even 6 1
2548.1.q.b 2 140.p odd 6 1
2548.1.q.b 2 455.ba even 6 1
2548.1.q.b 2 1820.ck odd 6 1
2548.1.bi.a 2 35.i odd 6 1
2548.1.bi.a 2 140.s even 6 1
2548.1.bi.a 2 455.bw odd 6 1
2548.1.bi.a 2 1820.bt even 6 1
2548.1.bi.b 2 35.j even 6 1
2548.1.bi.b 2 140.p odd 6 1
2548.1.bi.b 2 455.bm even 6 1
2548.1.bi.b 2 1820.dv odd 6 1
2548.1.bn.a 2 35.c odd 2 1
2548.1.bn.a 2 140.c even 2 1
2548.1.bn.a 2 455.bp odd 6 1
2548.1.bn.a 2 1820.dj even 6 1
3328.1.v.b 4 80.k odd 4 2
3328.1.v.b 4 80.q even 4 2
3328.1.v.b 4 1040.ec even 12 2
3328.1.v.b 4 1040.fb odd 12 2

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1300, [\chi])$$.

Hecke characteristic polynomials

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