# Properties

 Label 700.1.u.a Level $700$ Weight $1$ Character orbit 700.u Analytic conductor $0.349$ Analytic rank $0$ Dimension $4$ Projective image $D_{3}$ CM discriminant -20 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,1,Mod(51,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 2]))

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

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

chi := DirichletCharacter("700.51");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 700.u (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: $$0.349345508843$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.980.1 Artin image: $C_{12}\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{12} q^{2} - \zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} - q^{6} - \zeta_{12}^{5} q^{7} - \zeta_{12}^{3} q^{8} +O(q^{10})$$ q - z * q^2 - z^5 * q^3 + z^2 * q^4 - q^6 - z^5 * q^7 - z^3 * q^8 $$q - \zeta_{12} q^{2} - \zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} - q^{6} - \zeta_{12}^{5} q^{7} - \zeta_{12}^{3} q^{8} + \zeta_{12} q^{12} - q^{14} + \zeta_{12}^{4} q^{16} - \zeta_{12}^{4} q^{21} - \zeta_{12} q^{23} - \zeta_{12}^{2} q^{24} + \zeta_{12}^{3} q^{27} + \zeta_{12} q^{28} + q^{29} - \zeta_{12}^{5} q^{32} - q^{41} + \zeta_{12}^{5} q^{42} + \zeta_{12}^{3} q^{43} + \zeta_{12}^{2} q^{46} - \zeta_{12} q^{47} + \zeta_{12}^{3} q^{48} - \zeta_{12}^{4} q^{49} - \zeta_{12}^{4} q^{54} - \zeta_{12}^{2} q^{56} - \zeta_{12} q^{58} - \zeta_{12}^{4} q^{61} - q^{64} + \zeta_{12}^{5} q^{67} - q^{69} + \zeta_{12}^{2} q^{81} + \zeta_{12} q^{82} + \zeta_{12}^{3} q^{83} + q^{84} - \zeta_{12}^{4} q^{86} - \zeta_{12}^{5} q^{87} + \zeta_{12}^{4} q^{89} - \zeta_{12}^{3} q^{92} + 2 \zeta_{12}^{2} q^{94} - \zeta_{12}^{4} q^{96} + \zeta_{12}^{5} q^{98} +O(q^{100})$$ q - z * q^2 - z^5 * q^3 + z^2 * q^4 - q^6 - z^5 * q^7 - z^3 * q^8 + z * q^12 - q^14 + z^4 * q^16 - z^4 * q^21 - z * q^23 - z^2 * q^24 + z^3 * q^27 + z * q^28 + q^29 - z^5 * q^32 - q^41 + z^5 * q^42 + z^3 * q^43 + z^2 * q^46 - z * q^47 + z^3 * q^48 - z^4 * q^49 - z^4 * q^54 - z^2 * q^56 - z * q^58 - z^4 * q^61 - q^64 + z^5 * q^67 - q^69 + z^2 * q^81 + z * q^82 + z^3 * q^83 + q^84 - z^4 * q^86 - z^5 * q^87 + z^4 * q^89 - z^3 * q^92 + 2*z^2 * q^94 - z^4 * q^96 + z^5 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 4 q^{6}+O(q^{10})$$ 4 * q + 2 * q^4 - 4 * q^6 $$4 q + 2 q^{4} - 4 q^{6} - 4 q^{14} - 2 q^{16} + 2 q^{21} - 2 q^{24} + 4 q^{29} - 4 q^{41} + 2 q^{46} + 2 q^{49} + 2 q^{54} - 2 q^{56} + 2 q^{61} - 4 q^{64} - 4 q^{69} + 2 q^{81} + 4 q^{84} + 2 q^{86} - 2 q^{89} + 4 q^{94} + 2 q^{96}+O(q^{100})$$ 4 * q + 2 * q^4 - 4 * q^6 - 4 * q^14 - 2 * q^16 + 2 * q^21 - 2 * q^24 + 4 * q^29 - 4 * q^41 + 2 * q^46 + 2 * q^49 + 2 * q^54 - 2 * q^56 + 2 * q^61 - 4 * q^64 - 4 * q^69 + 2 * q^81 + 4 * q^84 + 2 * q^86 - 2 * q^89 + 4 * q^94 + 2 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$\zeta_{12}^{4}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 0 −1.00000 0.866025 0.500000i 1.00000i 0 0
51.2 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 −1.00000 −0.866025 + 0.500000i 1.00000i 0 0
151.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 0 −1.00000 0.866025 + 0.500000i 1.00000i 0 0
151.2 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0 −1.00000 −0.866025 0.500000i 1.00000i 0 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner
35.j even 6 1 inner
140.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.1.u.a 4
4.b odd 2 1 inner 700.1.u.a 4
5.b even 2 1 inner 700.1.u.a 4
5.c odd 4 1 140.1.p.a 2
5.c odd 4 1 140.1.p.b yes 2
7.c even 3 1 inner 700.1.u.a 4
15.e even 4 1 1260.1.ci.a 2
15.e even 4 1 1260.1.ci.b 2
20.d odd 2 1 CM 700.1.u.a 4
20.e even 4 1 140.1.p.a 2
20.e even 4 1 140.1.p.b yes 2
28.g odd 6 1 inner 700.1.u.a 4
35.f even 4 1 980.1.p.a 2
35.f even 4 1 980.1.p.b 2
35.j even 6 1 inner 700.1.u.a 4
35.k even 12 1 980.1.f.a 1
35.k even 12 1 980.1.f.d 1
35.k even 12 1 980.1.p.a 2
35.k even 12 1 980.1.p.b 2
35.l odd 12 1 140.1.p.a 2
35.l odd 12 1 140.1.p.b yes 2
35.l odd 12 1 980.1.f.b 1
35.l odd 12 1 980.1.f.c 1
40.i odd 4 1 2240.1.bt.a 2
40.i odd 4 1 2240.1.bt.b 2
40.k even 4 1 2240.1.bt.a 2
40.k even 4 1 2240.1.bt.b 2
60.l odd 4 1 1260.1.ci.a 2
60.l odd 4 1 1260.1.ci.b 2
105.x even 12 1 1260.1.ci.a 2
105.x even 12 1 1260.1.ci.b 2
140.j odd 4 1 980.1.p.a 2
140.j odd 4 1 980.1.p.b 2
140.p odd 6 1 inner 700.1.u.a 4
140.w even 12 1 140.1.p.a 2
140.w even 12 1 140.1.p.b yes 2
140.w even 12 1 980.1.f.b 1
140.w even 12 1 980.1.f.c 1
140.x odd 12 1 980.1.f.a 1
140.x odd 12 1 980.1.f.d 1
140.x odd 12 1 980.1.p.a 2
140.x odd 12 1 980.1.p.b 2
280.br even 12 1 2240.1.bt.a 2
280.br even 12 1 2240.1.bt.b 2
280.bt odd 12 1 2240.1.bt.a 2
280.bt odd 12 1 2240.1.bt.b 2
420.bp odd 12 1 1260.1.ci.a 2
420.bp odd 12 1 1260.1.ci.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.p.a 2 5.c odd 4 1
140.1.p.a 2 20.e even 4 1
140.1.p.a 2 35.l odd 12 1
140.1.p.a 2 140.w even 12 1
140.1.p.b yes 2 5.c odd 4 1
140.1.p.b yes 2 20.e even 4 1
140.1.p.b yes 2 35.l odd 12 1
140.1.p.b yes 2 140.w even 12 1
700.1.u.a 4 1.a even 1 1 trivial
700.1.u.a 4 4.b odd 2 1 inner
700.1.u.a 4 5.b even 2 1 inner
700.1.u.a 4 7.c even 3 1 inner
700.1.u.a 4 20.d odd 2 1 CM
700.1.u.a 4 28.g odd 6 1 inner
700.1.u.a 4 35.j even 6 1 inner
700.1.u.a 4 140.p odd 6 1 inner
980.1.f.a 1 35.k even 12 1
980.1.f.a 1 140.x odd 12 1
980.1.f.b 1 35.l odd 12 1
980.1.f.b 1 140.w even 12 1
980.1.f.c 1 35.l odd 12 1
980.1.f.c 1 140.w even 12 1
980.1.f.d 1 35.k even 12 1
980.1.f.d 1 140.x odd 12 1
980.1.p.a 2 35.f even 4 1
980.1.p.a 2 35.k even 12 1
980.1.p.a 2 140.j odd 4 1
980.1.p.a 2 140.x odd 12 1
980.1.p.b 2 35.f even 4 1
980.1.p.b 2 35.k even 12 1
980.1.p.b 2 140.j odd 4 1
980.1.p.b 2 140.x odd 12 1
1260.1.ci.a 2 15.e even 4 1
1260.1.ci.a 2 60.l odd 4 1
1260.1.ci.a 2 105.x even 12 1
1260.1.ci.a 2 420.bp odd 12 1
1260.1.ci.b 2 15.e even 4 1
1260.1.ci.b 2 60.l odd 4 1
1260.1.ci.b 2 105.x even 12 1
1260.1.ci.b 2 420.bp odd 12 1
2240.1.bt.a 2 40.i odd 4 1
2240.1.bt.a 2 40.k even 4 1
2240.1.bt.a 2 280.br even 12 1
2240.1.bt.a 2 280.bt odd 12 1
2240.1.bt.b 2 40.i odd 4 1
2240.1.bt.b 2 40.k even 4 1
2240.1.bt.b 2 280.br even 12 1
2240.1.bt.b 2 280.bt odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(700, [\chi])$$.

## Hecke characteristic polynomials

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