# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,2,Mod(401,700)]

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, 0, 2]))

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

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

chi := DirichletCharacter("700.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.i (of order $$3$$, 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: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 3) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z + 1) * q^3 + (z - 3) * q^7 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 3) q^{7} + 2 \zeta_{6} q^{9} + (6 \zeta_{6} - 6) q^{11} - 2 q^{13} + (6 \zeta_{6} - 6) q^{17} - 8 \zeta_{6} q^{19} + (3 \zeta_{6} - 2) q^{21} + 3 \zeta_{6} q^{23} + 5 q^{27} + 3 q^{29} + (2 \zeta_{6} - 2) q^{31} + 6 \zeta_{6} q^{33} + 8 \zeta_{6} q^{37} + (2 \zeta_{6} - 2) q^{39} - 3 q^{41} - 5 q^{43} + ( - 5 \zeta_{6} + 8) q^{49} + 6 \zeta_{6} q^{51} + ( - 12 \zeta_{6} + 12) q^{53} - 8 q^{57} + \zeta_{6} q^{61} + ( - 4 \zeta_{6} - 2) q^{63} + (7 \zeta_{6} - 7) q^{67} + 3 q^{69} + (10 \zeta_{6} - 10) q^{73} + ( - 18 \zeta_{6} + 12) q^{77} + 4 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 3 q^{83} + ( - 3 \zeta_{6} + 3) q^{87} + 3 \zeta_{6} q^{89} + ( - 2 \zeta_{6} + 6) q^{91} + 2 \zeta_{6} q^{93} + 10 q^{97} - 12 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + (z - 3) * q^7 + 2*z * q^9 + (6*z - 6) * q^11 - 2 * q^13 + (6*z - 6) * q^17 - 8*z * q^19 + (3*z - 2) * q^21 + 3*z * q^23 + 5 * q^27 + 3 * q^29 + (2*z - 2) * q^31 + 6*z * q^33 + 8*z * q^37 + (2*z - 2) * q^39 - 3 * q^41 - 5 * q^43 + (-5*z + 8) * q^49 + 6*z * q^51 + (-12*z + 12) * q^53 - 8 * q^57 + z * q^61 + (-4*z - 2) * q^63 + (7*z - 7) * q^67 + 3 * q^69 + (10*z - 10) * q^73 + (-18*z + 12) * q^77 + 4*z * q^79 + (z - 1) * q^81 - 3 * q^83 + (-3*z + 3) * q^87 + 3*z * q^89 + (-2*z + 6) * q^91 + 2*z * q^93 + 10 * q^97 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 5 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 - 5 * q^7 + 2 * q^9 $$2 q + q^{3} - 5 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{13} - 6 q^{17} - 8 q^{19} - q^{21} + 3 q^{23} + 10 q^{27} + 6 q^{29} - 2 q^{31} + 6 q^{33} + 8 q^{37} - 2 q^{39} - 6 q^{41} - 10 q^{43} + 11 q^{49} + 6 q^{51} + 12 q^{53} - 16 q^{57} + q^{61} - 8 q^{63} - 7 q^{67} + 6 q^{69} - 10 q^{73} + 6 q^{77} + 4 q^{79} - q^{81} - 6 q^{83} + 3 q^{87} + 3 q^{89} + 10 q^{91} + 2 q^{93} + 20 q^{97} - 24 q^{99}+O(q^{100})$$ 2 * q + q^3 - 5 * q^7 + 2 * q^9 - 6 * q^11 - 4 * q^13 - 6 * q^17 - 8 * q^19 - q^21 + 3 * q^23 + 10 * q^27 + 6 * q^29 - 2 * q^31 + 6 * q^33 + 8 * q^37 - 2 * q^39 - 6 * q^41 - 10 * q^43 + 11 * q^49 + 6 * q^51 + 12 * q^53 - 16 * q^57 + q^61 - 8 * q^63 - 7 * q^67 + 6 * q^69 - 10 * q^73 + 6 * q^77 + 4 * q^79 - q^81 - 6 * q^83 + 3 * q^87 + 3 * q^89 + 10 * q^91 + 2 * q^93 + 20 * q^97 - 24 * q^99

## Character values

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

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 0 0 −2.50000 0.866025i 0 1.00000 1.73205i 0
501.1 0 0.500000 0.866025i 0 0 0 −2.50000 + 0.866025i 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
7.c even 3 1 inner

## Twists

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

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$T^{2} + 6T + 36$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} + 8T + 64$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} + 2T + 4$$
$37$ $$T^{2} - 8T + 64$$
$41$ $$(T + 3)^{2}$$
$43$ $$(T + 5)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 12T + 144$$
$59$ $$T^{2}$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 7T + 49$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 10T + 100$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$(T + 3)^{2}$$
$89$ $$T^{2} - 3T + 9$$
$97$ $$(T - 10)^{2}$$