# Properties

 Label 1400.2.g.b Level $1400$ Weight $2$ Character orbit 1400.g Analytic conductor $11.179$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,2,Mod(449,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("1400.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.g (of order $$2$$, degree $$1$$, 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: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} - i q^{7} - q^{9} +O(q^{10})$$ q + 2*i * q^3 - i * q^7 - q^9 $$q + 2 i q^{3} - i q^{7} - q^{9} + 2 i q^{17} + 2 q^{19} + 2 q^{21} + 8 i q^{23} + 4 i q^{27} - 2 q^{29} + 4 q^{31} + 6 i q^{37} - 2 q^{41} + 8 i q^{43} + 4 i q^{47} - q^{49} - 4 q^{51} - 10 i q^{53} + 4 i q^{57} - 6 q^{59} + 4 q^{61} + i q^{63} + 12 i q^{67} - 16 q^{69} - 14 i q^{73} + 8 q^{79} - 11 q^{81} + 6 i q^{83} - 4 i q^{87} - 10 q^{89} + 8 i q^{93} + 2 i q^{97} +O(q^{100})$$ q + 2*i * q^3 - i * q^7 - q^9 + 2*i * q^17 + 2 * q^19 + 2 * q^21 + 8*i * q^23 + 4*i * q^27 - 2 * q^29 + 4 * q^31 + 6*i * q^37 - 2 * q^41 + 8*i * q^43 + 4*i * q^47 - q^49 - 4 * q^51 - 10*i * q^53 + 4*i * q^57 - 6 * q^59 + 4 * q^61 + i * q^63 + 12*i * q^67 - 16 * q^69 - 14*i * q^73 + 8 * q^79 - 11 * q^81 + 6*i * q^83 - 4*i * q^87 - 10 * q^89 + 8*i * q^93 + 2*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 4 q^{19} + 4 q^{21} - 4 q^{29} + 8 q^{31} - 4 q^{41} - 2 q^{49} - 8 q^{51} - 12 q^{59} + 8 q^{61} - 32 q^{69} + 16 q^{79} - 22 q^{81} - 20 q^{89}+O(q^{100})$$ 2 * q - 2 * q^9 + 4 * q^19 + 4 * q^21 - 4 * q^29 + 8 * q^31 - 4 * q^41 - 2 * q^49 - 8 * q^51 - 12 * q^59 + 8 * q^61 - 32 * q^69 + 16 * q^79 - 22 * q^81 - 20 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 1.00000i 0 −1.00000 0
449.2 0 2.00000i 0 0 0 1.00000i 0 −1.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.g.b 2
4.b odd 2 1 2800.2.g.g 2
5.b even 2 1 inner 1400.2.g.b 2
5.c odd 4 1 56.2.a.b 1
5.c odd 4 1 1400.2.a.a 1
15.e even 4 1 504.2.a.h 1
20.d odd 2 1 2800.2.g.g 2
20.e even 4 1 112.2.a.a 1
20.e even 4 1 2800.2.a.bd 1
35.f even 4 1 392.2.a.b 1
35.f even 4 1 9800.2.a.bj 1
35.k even 12 2 392.2.i.e 2
35.l odd 12 2 392.2.i.a 2
40.i odd 4 1 448.2.a.c 1
40.k even 4 1 448.2.a.h 1
55.e even 4 1 6776.2.a.h 1
60.l odd 4 1 1008.2.a.m 1
65.h odd 4 1 9464.2.a.h 1
80.i odd 4 1 1792.2.b.a 2
80.j even 4 1 1792.2.b.h 2
80.s even 4 1 1792.2.b.h 2
80.t odd 4 1 1792.2.b.a 2
105.k odd 4 1 3528.2.a.b 1
105.w odd 12 2 3528.2.s.ba 2
105.x even 12 2 3528.2.s.a 2
120.q odd 4 1 4032.2.a.a 1
120.w even 4 1 4032.2.a.d 1
140.j odd 4 1 784.2.a.i 1
140.w even 12 2 784.2.i.j 2
140.x odd 12 2 784.2.i.b 2
280.s even 4 1 3136.2.a.w 1
280.y odd 4 1 3136.2.a.c 1
420.w even 4 1 7056.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 5.c odd 4 1
112.2.a.a 1 20.e even 4 1
392.2.a.b 1 35.f even 4 1
392.2.i.a 2 35.l odd 12 2
392.2.i.e 2 35.k even 12 2
448.2.a.c 1 40.i odd 4 1
448.2.a.h 1 40.k even 4 1
504.2.a.h 1 15.e even 4 1
784.2.a.i 1 140.j odd 4 1
784.2.i.b 2 140.x odd 12 2
784.2.i.j 2 140.w even 12 2
1008.2.a.m 1 60.l odd 4 1
1400.2.a.a 1 5.c odd 4 1
1400.2.g.b 2 1.a even 1 1 trivial
1400.2.g.b 2 5.b even 2 1 inner
1792.2.b.a 2 80.i odd 4 1
1792.2.b.a 2 80.t odd 4 1
1792.2.b.h 2 80.j even 4 1
1792.2.b.h 2 80.s even 4 1
2800.2.a.bd 1 20.e even 4 1
2800.2.g.g 2 4.b odd 2 1
2800.2.g.g 2 20.d odd 2 1
3136.2.a.c 1 280.y odd 4 1
3136.2.a.w 1 280.s even 4 1
3528.2.a.b 1 105.k odd 4 1
3528.2.s.a 2 105.x even 12 2
3528.2.s.ba 2 105.w odd 12 2
4032.2.a.a 1 120.q odd 4 1
4032.2.a.d 1 120.w even 4 1
6776.2.a.h 1 55.e even 4 1
7056.2.a.c 1 420.w even 4 1
9464.2.a.h 1 65.h odd 4 1
9800.2.a.bj 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1400, [\chi])$$:

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T + 2)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2} + 100$$
$59$ $$(T + 6)^{2}$$
$61$ $$(T - 4)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 4$$