# Properties

 Label 1400.2.q.d Level $1400$ Weight $2$ Character orbit 1400.q Analytic conductor $11.179$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,2,Mod(401,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("1400.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.q (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: $$11.1790562830$$ 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 56) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} + (2 \zeta_{6} + 1) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (z - 1) * q^3 + (2*z + 1) * q^7 + 2*z * q^9 $$q + (\zeta_{6} - 1) q^{3} + (2 \zeta_{6} + 1) q^{7} + 2 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} + 6 q^{13} + (5 \zeta_{6} - 5) q^{17} - \zeta_{6} q^{19} + (\zeta_{6} - 3) q^{21} - 7 \zeta_{6} q^{23} - 5 q^{27} + 2 q^{29} + ( - 5 \zeta_{6} + 5) q^{31} - 3 \zeta_{6} q^{33} + 3 \zeta_{6} q^{37} + (6 \zeta_{6} - 6) q^{39} - 2 q^{41} + 4 q^{43} + 5 \zeta_{6} q^{47} + (8 \zeta_{6} - 3) q^{49} - 5 \zeta_{6} q^{51} + (\zeta_{6} - 1) q^{53} + q^{57} + (15 \zeta_{6} - 15) q^{59} + 5 \zeta_{6} q^{61} + (6 \zeta_{6} - 4) q^{63} + (9 \zeta_{6} - 9) q^{67} + 7 q^{69} + ( - 7 \zeta_{6} + 7) q^{73} + (3 \zeta_{6} - 9) q^{77} - \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 12 q^{83} + (2 \zeta_{6} - 2) q^{87} - 7 \zeta_{6} q^{89} + (12 \zeta_{6} + 6) q^{91} + 5 \zeta_{6} q^{93} + 2 q^{97} - 6 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 + (2*z + 1) * q^7 + 2*z * q^9 + (3*z - 3) * q^11 + 6 * q^13 + (5*z - 5) * q^17 - z * q^19 + (z - 3) * q^21 - 7*z * q^23 - 5 * q^27 + 2 * q^29 + (-5*z + 5) * q^31 - 3*z * q^33 + 3*z * q^37 + (6*z - 6) * q^39 - 2 * q^41 + 4 * q^43 + 5*z * q^47 + (8*z - 3) * q^49 - 5*z * q^51 + (z - 1) * q^53 + q^57 + (15*z - 15) * q^59 + 5*z * q^61 + (6*z - 4) * q^63 + (9*z - 9) * q^67 + 7 * q^69 + (-7*z + 7) * q^73 + (3*z - 9) * q^77 - z * q^79 + (z - 1) * q^81 - 12 * q^83 + (2*z - 2) * q^87 - 7*z * q^89 + (12*z + 6) * q^91 + 5*z * q^93 + 2 * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - q^3 + 4 * q^7 + 2 * q^9 $$2 q - q^{3} + 4 q^{7} + 2 q^{9} - 3 q^{11} + 12 q^{13} - 5 q^{17} - q^{19} - 5 q^{21} - 7 q^{23} - 10 q^{27} + 4 q^{29} + 5 q^{31} - 3 q^{33} + 3 q^{37} - 6 q^{39} - 4 q^{41} + 8 q^{43} + 5 q^{47} + 2 q^{49} - 5 q^{51} - q^{53} + 2 q^{57} - 15 q^{59} + 5 q^{61} - 2 q^{63} - 9 q^{67} + 14 q^{69} + 7 q^{73} - 15 q^{77} - q^{79} - q^{81} - 24 q^{83} - 2 q^{87} - 7 q^{89} + 24 q^{91} + 5 q^{93} + 4 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - q^3 + 4 * q^7 + 2 * q^9 - 3 * q^11 + 12 * q^13 - 5 * q^17 - q^19 - 5 * q^21 - 7 * q^23 - 10 * q^27 + 4 * q^29 + 5 * q^31 - 3 * q^33 + 3 * q^37 - 6 * q^39 - 4 * q^41 + 8 * q^43 + 5 * q^47 + 2 * q^49 - 5 * q^51 - q^53 + 2 * q^57 - 15 * q^59 + 5 * q^61 - 2 * q^63 - 9 * q^67 + 14 * q^69 + 7 * q^73 - 15 * q^77 - q^79 - q^81 - 24 * q^83 - 2 * q^87 - 7 * q^89 + 24 * q^91 + 5 * q^93 + 4 * q^97 - 12 * q^99

## Character values

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

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 2.00000 1.73205i 0 1.00000 1.73205i 0
1201.1 0 −0.500000 + 0.866025i 0 0 0 2.00000 + 1.73205i 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 1400.2.q.d 2
5.b even 2 1 56.2.i.b 2
5.c odd 4 2 1400.2.bh.a 4
7.c even 3 1 inner 1400.2.q.d 2
7.c even 3 1 9800.2.a.be 1
7.d odd 6 1 9800.2.a.s 1
15.d odd 2 1 504.2.s.c 2
20.d odd 2 1 112.2.i.a 2
35.c odd 2 1 392.2.i.b 2
35.i odd 6 1 392.2.a.e 1
35.i odd 6 1 392.2.i.b 2
35.j even 6 1 56.2.i.b 2
35.j even 6 1 392.2.a.c 1
35.l odd 12 2 1400.2.bh.a 4
40.e odd 2 1 448.2.i.d 2
40.f even 2 1 448.2.i.b 2
60.h even 2 1 1008.2.s.g 2
105.g even 2 1 3528.2.s.q 2
105.o odd 6 1 504.2.s.c 2
105.o odd 6 1 3528.2.a.p 1
105.p even 6 1 3528.2.a.j 1
105.p even 6 1 3528.2.s.q 2
140.c even 2 1 784.2.i.h 2
140.p odd 6 1 112.2.i.a 2
140.p odd 6 1 784.2.a.h 1
140.s even 6 1 784.2.a.c 1
140.s even 6 1 784.2.i.h 2
280.ba even 6 1 3136.2.a.t 1
280.bf even 6 1 448.2.i.b 2
280.bf even 6 1 3136.2.a.u 1
280.bi odd 6 1 448.2.i.d 2
280.bi odd 6 1 3136.2.a.j 1
280.bk odd 6 1 3136.2.a.i 1
420.ba even 6 1 1008.2.s.g 2
420.ba even 6 1 7056.2.a.bj 1
420.be odd 6 1 7056.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.b 2 5.b even 2 1
56.2.i.b 2 35.j even 6 1
112.2.i.a 2 20.d odd 2 1
112.2.i.a 2 140.p odd 6 1
392.2.a.c 1 35.j even 6 1
392.2.a.e 1 35.i odd 6 1
392.2.i.b 2 35.c odd 2 1
392.2.i.b 2 35.i odd 6 1
448.2.i.b 2 40.f even 2 1
448.2.i.b 2 280.bf even 6 1
448.2.i.d 2 40.e odd 2 1
448.2.i.d 2 280.bi odd 6 1
504.2.s.c 2 15.d odd 2 1
504.2.s.c 2 105.o odd 6 1
784.2.a.c 1 140.s even 6 1
784.2.a.h 1 140.p odd 6 1
784.2.i.h 2 140.c even 2 1
784.2.i.h 2 140.s even 6 1
1008.2.s.g 2 60.h even 2 1
1008.2.s.g 2 420.ba even 6 1
1400.2.q.d 2 1.a even 1 1 trivial
1400.2.q.d 2 7.c even 3 1 inner
1400.2.bh.a 4 5.c odd 4 2
1400.2.bh.a 4 35.l odd 12 2
3136.2.a.i 1 280.bk odd 6 1
3136.2.a.j 1 280.bi odd 6 1
3136.2.a.t 1 280.ba even 6 1
3136.2.a.u 1 280.bf even 6 1
3528.2.a.j 1 105.p even 6 1
3528.2.a.p 1 105.o odd 6 1
3528.2.s.q 2 105.g even 2 1
3528.2.s.q 2 105.p even 6 1
7056.2.a.u 1 420.be odd 6 1
7056.2.a.bj 1 420.ba even 6 1
9800.2.a.s 1 7.d odd 6 1
9800.2.a.be 1 7.c even 3 1

## Hecke kernels

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

 $$T_{3}^{2} + T_{3} + 1$$ T3^2 + T3 + 1 $$T_{11}^{2} + 3T_{11} + 9$$ T11^2 + 3*T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 4T + 7$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$(T - 6)^{2}$$
$17$ $$T^{2} + 5T + 25$$
$19$ $$T^{2} + T + 1$$
$23$ $$T^{2} + 7T + 49$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 5T + 25$$
$37$ $$T^{2} - 3T + 9$$
$41$ $$(T + 2)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} - 5T + 25$$
$53$ $$T^{2} + T + 1$$
$59$ $$T^{2} + 15T + 225$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} + 9T + 81$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 7T + 49$$
$79$ $$T^{2} + T + 1$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} + 7T + 49$$
$97$ $$(T - 2)^{2}$$
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