# Properties

 Label 2100.2.bc.a Level $2100$ Weight $2$ Character orbit 2100.bc Analytic conductor $16.769$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(949,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2100.949");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.bc (of order $$6$$, 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: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{3} + (\zeta_{12}^{3} - 3 \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9}+O(q^{10})$$ q + z * q^3 + (z^3 - 3*z) * q^7 + z^2 * q^9 $$q + \zeta_{12} q^{3} + (\zeta_{12}^{3} - 3 \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9} + (2 \zeta_{12}^{2} - 2) q^{11} - 3 \zeta_{12}^{3} q^{13} + 8 \zeta_{12} q^{17} - \zeta_{12}^{2} q^{19} + ( - 2 \zeta_{12}^{2} - 1) q^{21} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{23} + \zeta_{12}^{3} q^{27} - 4 q^{29} + (3 \zeta_{12}^{2} - 3) q^{31} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{33} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{37} + ( - 3 \zeta_{12}^{2} + 3) q^{39} + 6 q^{41} + 11 \zeta_{12}^{3} q^{43} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{47} + (3 \zeta_{12}^{2} + 5) q^{49} + 8 \zeta_{12}^{2} q^{51} + 12 \zeta_{12} q^{53} - \zeta_{12}^{3} q^{57} + ( - 4 \zeta_{12}^{2} + 4) q^{59} + 6 \zeta_{12}^{2} q^{61} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{63} + 13 \zeta_{12} q^{67} + 8 q^{69} - 10 q^{71} + 11 \zeta_{12} q^{73} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}) q^{77} - 3 \zeta_{12}^{2} q^{79} + (\zeta_{12}^{2} - 1) q^{81} + 2 \zeta_{12}^{3} q^{83} - 4 \zeta_{12} q^{87} + (9 \zeta_{12}^{2} - 6) q^{91} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{93} - 10 \zeta_{12}^{3} q^{97} - 2 q^{99} +O(q^{100})$$ q + z * q^3 + (z^3 - 3*z) * q^7 + z^2 * q^9 + (2*z^2 - 2) * q^11 - 3*z^3 * q^13 + 8*z * q^17 - z^2 * q^19 + (-2*z^2 - 1) * q^21 + (-8*z^3 + 8*z) * q^23 + z^3 * q^27 - 4 * q^29 + (3*z^2 - 3) * q^31 + (2*z^3 - 2*z) * q^33 + (-z^3 + z) * q^37 + (-3*z^2 + 3) * q^39 + 6 * q^41 + 11*z^3 * q^43 + (6*z^3 - 6*z) * q^47 + (3*z^2 + 5) * q^49 + 8*z^2 * q^51 + 12*z * q^53 - z^3 * q^57 + (-4*z^2 + 4) * q^59 + 6*z^2 * q^61 + (-2*z^3 - z) * q^63 + 13*z * q^67 + 8 * q^69 - 10 * q^71 + 11*z * q^73 + (-6*z^3 + 4*z) * q^77 - 3*z^2 * q^79 + (z^2 - 1) * q^81 + 2*z^3 * q^83 - 4*z * q^87 + (9*z^2 - 6) * q^91 + (3*z^3 - 3*z) * q^93 - 10*z^3 * q^97 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^9 $$4 q + 2 q^{9} - 4 q^{11} - 2 q^{19} - 8 q^{21} - 16 q^{29} - 6 q^{31} + 6 q^{39} + 24 q^{41} + 26 q^{49} + 16 q^{51} + 8 q^{59} + 12 q^{61} + 32 q^{69} - 40 q^{71} - 6 q^{79} - 2 q^{81} - 6 q^{91} - 8 q^{99}+O(q^{100})$$ 4 * q + 2 * q^9 - 4 * q^11 - 2 * q^19 - 8 * q^21 - 16 * q^29 - 6 * q^31 + 6 * q^39 + 24 * q^41 + 26 * q^49 + 16 * q^51 + 8 * q^59 + 12 * q^61 + 32 * q^69 - 40 * q^71 - 6 * q^79 - 2 * q^81 - 6 * q^91 - 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$$.

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
949.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 −0.866025 0.500000i 0 0 0 2.59808 + 0.500000i 0 0.500000 + 0.866025i 0
949.2 0 0.866025 + 0.500000i 0 0 0 −2.59808 0.500000i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 2.59808 0.500000i 0 0.500000 0.866025i 0
1549.2 0 0.866025 0.500000i 0 0 0 −2.59808 + 0.500000i 0 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bc.a 4
5.b even 2 1 inner 2100.2.bc.a 4
5.c odd 4 1 84.2.i.a 2
5.c odd 4 1 2100.2.q.b 2
7.c even 3 1 inner 2100.2.bc.a 4
15.e even 4 1 252.2.k.a 2
20.e even 4 1 336.2.q.c 2
35.f even 4 1 588.2.i.b 2
35.j even 6 1 inner 2100.2.bc.a 4
35.k even 12 1 588.2.a.f 1
35.k even 12 1 588.2.i.b 2
35.l odd 12 1 84.2.i.a 2
35.l odd 12 1 588.2.a.a 1
35.l odd 12 1 2100.2.q.b 2
40.i odd 4 1 1344.2.q.b 2
40.k even 4 1 1344.2.q.n 2
45.k odd 12 1 2268.2.i.g 2
45.k odd 12 1 2268.2.l.b 2
45.l even 12 1 2268.2.i.b 2
45.l even 12 1 2268.2.l.g 2
60.l odd 4 1 1008.2.s.c 2
105.k odd 4 1 1764.2.k.j 2
105.w odd 12 1 1764.2.a.c 1
105.w odd 12 1 1764.2.k.j 2
105.x even 12 1 252.2.k.a 2
105.x even 12 1 1764.2.a.h 1
140.j odd 4 1 2352.2.q.q 2
140.w even 12 1 336.2.q.c 2
140.w even 12 1 2352.2.a.o 1
140.x odd 12 1 2352.2.a.k 1
140.x odd 12 1 2352.2.q.q 2
280.bp odd 12 1 9408.2.a.bx 1
280.br even 12 1 1344.2.q.n 2
280.br even 12 1 9408.2.a.bi 1
280.bt odd 12 1 1344.2.q.b 2
280.bt odd 12 1 9408.2.a.cx 1
280.bv even 12 1 9408.2.a.i 1
315.bt odd 12 1 2268.2.l.b 2
315.bv even 12 1 2268.2.l.g 2
315.bx even 12 1 2268.2.i.b 2
315.ch odd 12 1 2268.2.i.g 2
420.bp odd 12 1 1008.2.s.c 2
420.bp odd 12 1 7056.2.a.bs 1
420.br even 12 1 7056.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 5.c odd 4 1
84.2.i.a 2 35.l odd 12 1
252.2.k.a 2 15.e even 4 1
252.2.k.a 2 105.x even 12 1
336.2.q.c 2 20.e even 4 1
336.2.q.c 2 140.w even 12 1
588.2.a.a 1 35.l odd 12 1
588.2.a.f 1 35.k even 12 1
588.2.i.b 2 35.f even 4 1
588.2.i.b 2 35.k even 12 1
1008.2.s.c 2 60.l odd 4 1
1008.2.s.c 2 420.bp odd 12 1
1344.2.q.b 2 40.i odd 4 1
1344.2.q.b 2 280.bt odd 12 1
1344.2.q.n 2 40.k even 4 1
1344.2.q.n 2 280.br even 12 1
1764.2.a.c 1 105.w odd 12 1
1764.2.a.h 1 105.x even 12 1
1764.2.k.j 2 105.k odd 4 1
1764.2.k.j 2 105.w odd 12 1
2100.2.q.b 2 5.c odd 4 1
2100.2.q.b 2 35.l odd 12 1
2100.2.bc.a 4 1.a even 1 1 trivial
2100.2.bc.a 4 5.b even 2 1 inner
2100.2.bc.a 4 7.c even 3 1 inner
2100.2.bc.a 4 35.j even 6 1 inner
2268.2.i.b 2 45.l even 12 1
2268.2.i.b 2 315.bx even 12 1
2268.2.i.g 2 45.k odd 12 1
2268.2.i.g 2 315.ch odd 12 1
2268.2.l.b 2 45.k odd 12 1
2268.2.l.b 2 315.bt odd 12 1
2268.2.l.g 2 45.l even 12 1
2268.2.l.g 2 315.bv even 12 1
2352.2.a.k 1 140.x odd 12 1
2352.2.a.o 1 140.w even 12 1
2352.2.q.q 2 140.j odd 4 1
2352.2.q.q 2 140.x odd 12 1
7056.2.a.o 1 420.br even 12 1
7056.2.a.bs 1 420.bp odd 12 1
9408.2.a.i 1 280.bv even 12 1
9408.2.a.bi 1 280.br even 12 1
9408.2.a.bx 1 280.bp odd 12 1
9408.2.a.cx 1 280.bt odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2100, [\chi])$$:

 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4 $$T_{13}^{2} + 9$$ T13^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 13T^{2} + 49$$
$11$ $$(T^{2} + 2 T + 4)^{2}$$
$13$ $$(T^{2} + 9)^{2}$$
$17$ $$T^{4} - 64T^{2} + 4096$$
$19$ $$(T^{2} + T + 1)^{2}$$
$23$ $$T^{4} - 64T^{2} + 4096$$
$29$ $$(T + 4)^{4}$$
$31$ $$(T^{2} + 3 T + 9)^{2}$$
$37$ $$T^{4} - T^{2} + 1$$
$41$ $$(T - 6)^{4}$$
$43$ $$(T^{2} + 121)^{2}$$
$47$ $$T^{4} - 36T^{2} + 1296$$
$53$ $$T^{4} - 144 T^{2} + 20736$$
$59$ $$(T^{2} - 4 T + 16)^{2}$$
$61$ $$(T^{2} - 6 T + 36)^{2}$$
$67$ $$T^{4} - 169 T^{2} + 28561$$
$71$ $$(T + 10)^{4}$$
$73$ $$T^{4} - 121 T^{2} + 14641$$
$79$ $$(T^{2} + 3 T + 9)^{2}$$
$83$ $$(T^{2} + 4)^{2}$$
$89$ $$T^{4}$$
$97$ $$(T^{2} + 100)^{2}$$