# Properties

 Label 700.2.r.a Level $700$ Weight $2$ Character orbit 700.r Analytic conductor $5.590$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,2,Mod(149,700)]

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([0, 3, 2]))

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

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

chi := DirichletCharacter("700.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.r (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: $$5.58952814149$$ 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 140) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{3} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{7} - 2 \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + z * q^3 + (2*z^3 + z) * q^7 - 2*z^2 * q^9 $$q + \zeta_{12} q^{3} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{7} - 2 \zeta_{12}^{2} q^{9} + (6 \zeta_{12}^{2} - 6) q^{11} - 2 \zeta_{12}^{3} q^{13} + 6 \zeta_{12} q^{17} + 8 \zeta_{12}^{2} q^{19} + (3 \zeta_{12}^{2} - 2) q^{21} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{23} - 5 \zeta_{12}^{3} q^{27} - 3 q^{29} + (2 \zeta_{12}^{2} - 2) q^{31} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{33} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{37} + ( - 2 \zeta_{12}^{2} + 2) q^{39} - 3 q^{41} - 5 \zeta_{12}^{3} q^{43} + (5 \zeta_{12}^{2} - 8) q^{49} + 6 \zeta_{12}^{2} q^{51} + 12 \zeta_{12} q^{53} + 8 \zeta_{12}^{3} q^{57} + \zeta_{12}^{2} q^{61} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}) q^{63} + 7 \zeta_{12} q^{67} - 3 q^{69} - 10 \zeta_{12} q^{73} + (6 \zeta_{12}^{3} - 18 \zeta_{12}) q^{77} - 4 \zeta_{12}^{2} q^{79} + (\zeta_{12}^{2} - 1) q^{81} - 3 \zeta_{12}^{3} q^{83} - 3 \zeta_{12} q^{87} - 3 \zeta_{12}^{2} q^{89} + ( - 2 \zeta_{12}^{2} + 6) q^{91} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{93} - 10 \zeta_{12}^{3} q^{97} + 12 q^{99} +O(q^{100})$$ q + z * q^3 + (2*z^3 + z) * q^7 - 2*z^2 * q^9 + (6*z^2 - 6) * q^11 - 2*z^3 * q^13 + 6*z * q^17 + 8*z^2 * q^19 + (3*z^2 - 2) * q^21 + (3*z^3 - 3*z) * q^23 - 5*z^3 * q^27 - 3 * q^29 + (2*z^2 - 2) * q^31 + (6*z^3 - 6*z) * q^33 + (-8*z^3 + 8*z) * q^37 + (-2*z^2 + 2) * q^39 - 3 * q^41 - 5*z^3 * q^43 + (5*z^2 - 8) * q^49 + 6*z^2 * q^51 + 12*z * q^53 + 8*z^3 * q^57 + z^2 * q^61 + (-6*z^3 + 4*z) * q^63 + 7*z * q^67 - 3 * q^69 - 10*z * q^73 + (6*z^3 - 18*z) * q^77 - 4*z^2 * q^79 + (z^2 - 1) * q^81 - 3*z^3 * q^83 - 3*z * q^87 - 3*z^2 * q^89 + (-2*z^2 + 6) * q^91 + (2*z^3 - 2*z) * q^93 - 10*z^3 * q^97 + 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 12 q^{11} + 16 q^{19} - 2 q^{21} - 12 q^{29} - 4 q^{31} + 4 q^{39} - 12 q^{41} - 22 q^{49} + 12 q^{51} + 2 q^{61} - 12 q^{69} - 8 q^{79} - 2 q^{81} - 6 q^{89} + 20 q^{91} + 48 q^{99}+O(q^{100})$$ 4 * q - 4 * q^9 - 12 * q^11 + 16 * q^19 - 2 * q^21 - 12 * q^29 - 4 * q^31 + 4 * q^39 - 12 * q^41 - 22 * q^49 + 12 * q^51 + 2 * q^61 - 12 * q^69 - 8 * q^79 - 2 * q^81 - 6 * q^89 + 20 * q^91 + 48 * q^99

## Character values

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

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0 0 0 −0.866025 + 2.50000i 0 −1.00000 + 1.73205i 0
149.2 0 0.866025 0.500000i 0 0 0 0.866025 2.50000i 0 −1.00000 + 1.73205i 0
249.1 0 −0.866025 0.500000i 0 0 0 −0.866025 2.50000i 0 −1.00000 1.73205i 0
249.2 0 0.866025 + 0.500000i 0 0 0 0.866025 + 2.50000i 0 −1.00000 1.73205i 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 700.2.r.a 4
5.b even 2 1 inner 700.2.r.a 4
5.c odd 4 1 140.2.i.a 2
5.c odd 4 1 700.2.i.b 2
7.c even 3 1 inner 700.2.r.a 4
7.c even 3 1 4900.2.e.m 2
7.d odd 6 1 4900.2.e.n 2
15.e even 4 1 1260.2.s.c 2
20.e even 4 1 560.2.q.f 2
35.f even 4 1 980.2.i.f 2
35.i odd 6 1 4900.2.e.n 2
35.j even 6 1 inner 700.2.r.a 4
35.j even 6 1 4900.2.e.m 2
35.k even 12 1 980.2.a.e 1
35.k even 12 1 980.2.i.f 2
35.k even 12 1 4900.2.a.q 1
35.l odd 12 1 140.2.i.a 2
35.l odd 12 1 700.2.i.b 2
35.l odd 12 1 980.2.a.g 1
35.l odd 12 1 4900.2.a.i 1
105.w odd 12 1 8820.2.a.a 1
105.x even 12 1 1260.2.s.c 2
105.x even 12 1 8820.2.a.p 1
140.w even 12 1 560.2.q.f 2
140.w even 12 1 3920.2.a.k 1
140.x odd 12 1 3920.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.a 2 5.c odd 4 1
140.2.i.a 2 35.l odd 12 1
560.2.q.f 2 20.e even 4 1
560.2.q.f 2 140.w even 12 1
700.2.i.b 2 5.c odd 4 1
700.2.i.b 2 35.l odd 12 1
700.2.r.a 4 1.a even 1 1 trivial
700.2.r.a 4 5.b even 2 1 inner
700.2.r.a 4 7.c even 3 1 inner
700.2.r.a 4 35.j even 6 1 inner
980.2.a.e 1 35.k even 12 1
980.2.a.g 1 35.l odd 12 1
980.2.i.f 2 35.f even 4 1
980.2.i.f 2 35.k even 12 1
1260.2.s.c 2 15.e even 4 1
1260.2.s.c 2 105.x even 12 1
3920.2.a.k 1 140.w even 12 1
3920.2.a.w 1 140.x odd 12 1
4900.2.a.i 1 35.l odd 12 1
4900.2.a.q 1 35.k even 12 1
4900.2.e.m 2 7.c even 3 1
4900.2.e.m 2 35.j even 6 1
4900.2.e.n 2 7.d odd 6 1
4900.2.e.n 2 35.i odd 6 1
8820.2.a.a 1 105.w odd 12 1
8820.2.a.p 1 105.x even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(700, [\chi])$$:

 $$T_{3}^{4} - T_{3}^{2} + 1$$ T3^4 - T3^2 + 1 $$T_{11}^{2} + 6T_{11} + 36$$ T11^2 + 6*T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 11T^{2} + 49$$
$11$ $$(T^{2} + 6 T + 36)^{2}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$T^{4} - 36T^{2} + 1296$$
$19$ $$(T^{2} - 8 T + 64)^{2}$$
$23$ $$T^{4} - 9T^{2} + 81$$
$29$ $$(T + 3)^{4}$$
$31$ $$(T^{2} + 2 T + 4)^{2}$$
$37$ $$T^{4} - 64T^{2} + 4096$$
$41$ $$(T + 3)^{4}$$
$43$ $$(T^{2} + 25)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4} - 144 T^{2} + 20736$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - T + 1)^{2}$$
$67$ $$T^{4} - 49T^{2} + 2401$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 100 T^{2} + 10000$$
$79$ $$(T^{2} + 4 T + 16)^{2}$$
$83$ $$(T^{2} + 9)^{2}$$
$89$ $$(T^{2} + 3 T + 9)^{2}$$
$97$ $$(T^{2} + 100)^{2}$$