# Properties

 Label 1200.2.v.c Level $1200$ Weight $2$ Character orbit 1200.v Analytic conductor $9.582$ 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] = [1200,2,Mod(257,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))

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

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

chi := DirichletCharacter("1200.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.v (of order $$4$$, 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: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{8}^{2} + \zeta_{8} - 1) q^{3} + (3 \zeta_{8}^{2} + 3) q^{7} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} - 2 \zeta_{8}) q^{9}+O(q^{10})$$ q + (z^2 + z - 1) * q^3 + (3*z^2 + 3) * q^7 + (2*z^3 - z^2 - 2*z) * q^9 $$q + (\zeta_{8}^{2} + \zeta_{8} - 1) q^{3} + (3 \zeta_{8}^{2} + 3) q^{7} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} - 2 \zeta_{8}) q^{9} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{11} + 6 \zeta_{8}^{3} q^{17} + 4 \zeta_{8}^{2} q^{19} + (3 \zeta_{8}^{3} + 3 \zeta_{8} - 6) q^{21} + 4 \zeta_{8} q^{23} + ( - 5 \zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{27} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{29} + 2 q^{31} + ( - \zeta_{8}^{2} + 2 \zeta_{8} + 1) q^{33} + (2 \zeta_{8}^{2} + 2) q^{37} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{41} + (2 \zeta_{8}^{2} - 2) q^{43} + 8 \zeta_{8}^{3} q^{47} + 11 \zeta_{8}^{2} q^{49} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8} - 6) q^{51} - 12 \zeta_{8} q^{53} + (4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 4) q^{57} + (\zeta_{8}^{3} - \zeta_{8}) q^{59} - 6 q^{61} + ( - 3 \zeta_{8}^{2} - 12 \zeta_{8} + 3) q^{63} + (4 \zeta_{8}^{2} + 4) q^{67} + (4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 4 \zeta_{8}) q^{69} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{71} + (3 \zeta_{8}^{2} - 3) q^{73} - 6 \zeta_{8}^{3} q^{77} + 10 \zeta_{8}^{2} q^{79} + (4 \zeta_{8}^{3} + 4 \zeta_{8} + 7) q^{81} - 4 \zeta_{8} q^{83} + (2 \zeta_{8}^{3} + \zeta_{8}^{2} + 1) q^{87} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{89} + (2 \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{93} + (13 \zeta_{8}^{2} + 13) q^{97} + (\zeta_{8}^{3} + 4 \zeta_{8}^{2} - \zeta_{8}) q^{99} +O(q^{100})$$ q + (z^2 + z - 1) * q^3 + (3*z^2 + 3) * q^7 + (2*z^3 - z^2 - 2*z) * q^9 + (-z^3 - z) * q^11 + 6*z^3 * q^17 + 4*z^2 * q^19 + (3*z^3 + 3*z - 6) * q^21 + 4*z * q^23 + (-5*z^3 - z^2 - 1) * q^27 + (-z^3 + z) * q^29 + 2 * q^31 + (-z^2 + 2*z + 1) * q^33 + (2*z^2 + 2) * q^37 + (-4*z^3 - 4*z) * q^41 + (2*z^2 - 2) * q^43 + 8*z^3 * q^47 + 11*z^2 * q^49 + (-6*z^3 - 6*z - 6) * q^51 - 12*z * q^53 + (4*z^3 - 4*z^2 - 4) * q^57 + (z^3 - z) * q^59 - 6 * q^61 + (-3*z^2 - 12*z + 3) * q^63 + (4*z^2 + 4) * q^67 + (4*z^3 + 4*z^2 - 4*z) * q^69 + (2*z^3 + 2*z) * q^71 + (3*z^2 - 3) * q^73 - 6*z^3 * q^77 + 10*z^2 * q^79 + (4*z^3 + 4*z + 7) * q^81 - 4*z * q^83 + (2*z^3 + z^2 + 1) * q^87 + (2*z^3 - 2*z) * q^89 + (2*z^2 + 2*z - 2) * q^93 + (13*z^2 + 13) * q^97 + (z^3 + 4*z^2 - z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 12 q^{7}+O(q^{10})$$ 4 * q - 4 * q^3 + 12 * q^7 $$4 q - 4 q^{3} + 12 q^{7} - 24 q^{21} - 4 q^{27} + 8 q^{31} + 4 q^{33} + 8 q^{37} - 8 q^{43} - 24 q^{51} - 16 q^{57} - 24 q^{61} + 12 q^{63} + 16 q^{67} - 12 q^{73} + 28 q^{81} + 4 q^{87} - 8 q^{93} + 52 q^{97}+O(q^{100})$$ 4 * q - 4 * q^3 + 12 * q^7 - 24 * q^21 - 4 * q^27 + 8 * q^31 + 4 * q^33 + 8 * q^37 - 8 * q^43 - 24 * q^51 - 16 * q^57 - 24 * q^61 + 12 * q^63 + 16 * q^67 - 12 * q^73 + 28 * q^81 + 4 * q^87 - 8 * q^93 + 52 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$\zeta_{8}^{2}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 −1.70711 + 0.292893i 0 0 0 3.00000 + 3.00000i 0 2.82843 1.00000i 0
257.2 0 −0.292893 + 1.70711i 0 0 0 3.00000 + 3.00000i 0 −2.82843 1.00000i 0
593.1 0 −1.70711 0.292893i 0 0 0 3.00000 3.00000i 0 2.82843 + 1.00000i 0
593.2 0 −0.292893 1.70711i 0 0 0 3.00000 3.00000i 0 −2.82843 + 1.00000i 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
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.v.c 4
3.b odd 2 1 inner 1200.2.v.c 4
4.b odd 2 1 600.2.r.d 4
5.b even 2 1 240.2.v.d 4
5.c odd 4 1 240.2.v.d 4
5.c odd 4 1 inner 1200.2.v.c 4
12.b even 2 1 600.2.r.d 4
15.d odd 2 1 240.2.v.d 4
15.e even 4 1 240.2.v.d 4
15.e even 4 1 inner 1200.2.v.c 4
20.d odd 2 1 120.2.r.a 4
20.e even 4 1 120.2.r.a 4
20.e even 4 1 600.2.r.d 4
40.e odd 2 1 960.2.v.l 4
40.f even 2 1 960.2.v.b 4
40.i odd 4 1 960.2.v.b 4
40.k even 4 1 960.2.v.l 4
60.h even 2 1 120.2.r.a 4
60.l odd 4 1 120.2.r.a 4
60.l odd 4 1 600.2.r.d 4
120.i odd 2 1 960.2.v.b 4
120.m even 2 1 960.2.v.l 4
120.q odd 4 1 960.2.v.l 4
120.w even 4 1 960.2.v.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.r.a 4 20.d odd 2 1
120.2.r.a 4 20.e even 4 1
120.2.r.a 4 60.h even 2 1
120.2.r.a 4 60.l odd 4 1
240.2.v.d 4 5.b even 2 1
240.2.v.d 4 5.c odd 4 1
240.2.v.d 4 15.d odd 2 1
240.2.v.d 4 15.e even 4 1
600.2.r.d 4 4.b odd 2 1
600.2.r.d 4 12.b even 2 1
600.2.r.d 4 20.e even 4 1
600.2.r.d 4 60.l odd 4 1
960.2.v.b 4 40.f even 2 1
960.2.v.b 4 40.i odd 4 1
960.2.v.b 4 120.i odd 2 1
960.2.v.b 4 120.w even 4 1
960.2.v.l 4 40.e odd 2 1
960.2.v.l 4 40.k even 4 1
960.2.v.l 4 120.m even 2 1
960.2.v.l 4 120.q odd 4 1
1200.2.v.c 4 1.a even 1 1 trivial
1200.2.v.c 4 3.b odd 2 1 inner
1200.2.v.c 4 5.c odd 4 1 inner
1200.2.v.c 4 15.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} - 6T_{7} + 18$$ T7^2 - 6*T7 + 18 $$T_{11}^{2} + 2$$ T11^2 + 2 $$T_{17}^{4} + 1296$$ T17^4 + 1296

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 4 T^{3} + \cdots + 9$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 6 T + 18)^{2}$$
$11$ $$(T^{2} + 2)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 1296$$
$19$ $$(T^{2} + 16)^{2}$$
$23$ $$T^{4} + 256$$
$29$ $$(T^{2} - 2)^{2}$$
$31$ $$(T - 2)^{4}$$
$37$ $$(T^{2} - 4 T + 8)^{2}$$
$41$ $$(T^{2} + 32)^{2}$$
$43$ $$(T^{2} + 4 T + 8)^{2}$$
$47$ $$T^{4} + 4096$$
$53$ $$T^{4} + 20736$$
$59$ $$(T^{2} - 2)^{2}$$
$61$ $$(T + 6)^{4}$$
$67$ $$(T^{2} - 8 T + 32)^{2}$$
$71$ $$(T^{2} + 8)^{2}$$
$73$ $$(T^{2} + 6 T + 18)^{2}$$
$79$ $$(T^{2} + 100)^{2}$$
$83$ $$T^{4} + 256$$
$89$ $$(T^{2} - 8)^{2}$$
$97$ $$(T^{2} - 26 T + 338)^{2}$$