# Properties

 Label 1200.2.v.l Level $1200$ Weight $2$ Character orbit 1200.v Analytic conductor $9.582$ Analytic rank $0$ Dimension $8$ Inner twists $8$

# 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: no (minimal twist has level 150) 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - 2 \beta_1 q^{7} + ( - \beta_{6} + 2 \beta_{2}) q^{9}+O(q^{10})$$ q - b3 * q^3 - 2*b1 * q^7 + (-b6 + 2*b2) * q^9 $$q - \beta_{3} q^{3} - 2 \beta_1 q^{7} + ( - \beta_{6} + 2 \beta_{2}) q^{9} + (2 \beta_{4} - 1) q^{11} + ( - 2 \beta_{5} + \beta_1) q^{17} + \beta_{2} q^{19} + (2 \beta_{4} - 4) q^{21} + (2 \beta_{7} + 2 \beta_{3}) q^{23} + 3 \beta_1 q^{27} + 2 q^{31} + (6 \beta_{7} - 3 \beta_{3}) q^{33} - 2 \beta_1 q^{37} + (2 \beta_{4} - 1) q^{41} + (2 \beta_{7} - 2 \beta_{3}) q^{43} + 5 \beta_{2} q^{49} + (\beta_{4} + 4) q^{51} + (2 \beta_{7} + 2 \beta_{3}) q^{53} + \beta_{5} q^{57} + ( - 4 \beta_{6} + 2 \beta_{2}) q^{59} + 14 q^{61} + 6 \beta_{7} q^{63} - 3 \beta_1 q^{67} + (2 \beta_{6} - 10 \beta_{2}) q^{69} + ( - 5 \beta_{7} + 5 \beta_{3}) q^{73} + (12 \beta_{5} - 6 \beta_1) q^{77} + 14 \beta_{2} q^{79} + ( - 3 \beta_{4} + 6) q^{81} + ( - \beta_{7} - \beta_{3}) q^{83} + ( - 6 \beta_{6} + 3 \beta_{2}) q^{89} - 2 \beta_{3} q^{93} + 4 \beta_1 q^{97} + ( - 3 \beta_{6} - 12 \beta_{2}) q^{99}+O(q^{100})$$ q - b3 * q^3 - 2*b1 * q^7 + (-b6 + 2*b2) * q^9 + (2*b4 - 1) * q^11 + (-2*b5 + b1) * q^17 + b2 * q^19 + (2*b4 - 4) * q^21 + (2*b7 + 2*b3) * q^23 + 3*b1 * q^27 + 2 * q^31 + (6*b7 - 3*b3) * q^33 - 2*b1 * q^37 + (2*b4 - 1) * q^41 + (2*b7 - 2*b3) * q^43 + 5*b2 * q^49 + (b4 + 4) * q^51 + (2*b7 + 2*b3) * q^53 + b5 * q^57 + (-4*b6 + 2*b2) * q^59 + 14 * q^61 + 6*b7 * q^63 - 3*b1 * q^67 + (2*b6 - 10*b2) * q^69 + (-5*b7 + 5*b3) * q^73 + (12*b5 - 6*b1) * q^77 + 14*b2 * q^79 + (-3*b4 + 6) * q^81 + (-b7 - b3) * q^83 + (-6*b6 + 3*b2) * q^89 - 2*b3 * q^93 + 4*b1 * q^97 + (-3*b6 - 12*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 24 q^{21} + 16 q^{31} + 36 q^{51} + 112 q^{61} + 36 q^{81}+O(q^{100})$$ 8 * q - 24 * q^21 + 16 * q^31 + 36 * q^51 + 112 * q^61 + 36 * q^81

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{5} + \zeta_{24}$$ v^5 + v $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}^{3}$$ v^7 + v^3 $$\beta_{4}$$ $$=$$ $$3\zeta_{24}^{4} - 1$$ 3*v^4 - 1 $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{5} + 2\zeta_{24}$$ -v^5 + 2*v $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{6} + 3\zeta_{24}^{2}$$ -v^6 + 3*v^2 $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{7} + 2\zeta_{24}^{3}$$ -v^7 + 2*v^3
 $$\zeta_{24}$$ $$=$$ $$( \beta_{5} + \beta_1 ) / 3$$ (b5 + b1) / 3 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{6} + \beta_{2} ) / 3$$ (b6 + b2) / 3 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{3} ) / 3$$ (b7 + b3) / 3 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{4} + 1 ) / 3$$ (b4 + 1) / 3 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{5} + 2\beta_1 ) / 3$$ (-b5 + 2*b1) / 3 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{7} + 2\beta_{3} ) / 3$$ (-b7 + 2*b3) / 3

## Character values

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

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$\beta_{2}$$ $$1$$ $$1$$

## 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
 −0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.258819 − 0.965926i
0 −1.67303 0.448288i 0 0 0 2.44949 + 2.44949i 0 2.59808 + 1.50000i 0
257.2 0 −0.448288 1.67303i 0 0 0 −2.44949 2.44949i 0 −2.59808 + 1.50000i 0
257.3 0 0.448288 + 1.67303i 0 0 0 2.44949 + 2.44949i 0 −2.59808 + 1.50000i 0
257.4 0 1.67303 + 0.448288i 0 0 0 −2.44949 2.44949i 0 2.59808 + 1.50000i 0
593.1 0 −1.67303 + 0.448288i 0 0 0 2.44949 2.44949i 0 2.59808 1.50000i 0
593.2 0 −0.448288 + 1.67303i 0 0 0 −2.44949 + 2.44949i 0 −2.59808 1.50000i 0
593.3 0 0.448288 1.67303i 0 0 0 2.44949 2.44949i 0 −2.59808 1.50000i 0
593.4 0 1.67303 0.448288i 0 0 0 −2.44949 + 2.44949i 0 2.59808 1.50000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.4 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.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.v.l 8
3.b odd 2 1 inner 1200.2.v.l 8
4.b odd 2 1 150.2.e.b 8
5.b even 2 1 inner 1200.2.v.l 8
5.c odd 4 2 inner 1200.2.v.l 8
12.b even 2 1 150.2.e.b 8
15.d odd 2 1 inner 1200.2.v.l 8
15.e even 4 2 inner 1200.2.v.l 8
20.d odd 2 1 150.2.e.b 8
20.e even 4 2 150.2.e.b 8
60.h even 2 1 150.2.e.b 8
60.l odd 4 2 150.2.e.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.e.b 8 4.b odd 2 1
150.2.e.b 8 12.b even 2 1
150.2.e.b 8 20.d odd 2 1
150.2.e.b 8 20.e even 4 2
150.2.e.b 8 60.h even 2 1
150.2.e.b 8 60.l odd 4 2
1200.2.v.l 8 1.a even 1 1 trivial
1200.2.v.l 8 3.b odd 2 1 inner
1200.2.v.l 8 5.b even 2 1 inner
1200.2.v.l 8 5.c odd 4 2 inner
1200.2.v.l 8 15.d odd 2 1 inner
1200.2.v.l 8 15.e even 4 2 inner

## Hecke kernels

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

 $$T_{7}^{4} + 144$$ T7^4 + 144 $$T_{11}^{2} + 27$$ T11^2 + 27 $$T_{17}^{4} + 81$$ T17^4 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - 9T^{4} + 81$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 144)^{2}$$
$11$ $$(T^{2} + 27)^{4}$$
$13$ $$T^{8}$$
$17$ $$(T^{4} + 81)^{2}$$
$19$ $$(T^{2} + 1)^{4}$$
$23$ $$(T^{4} + 1296)^{2}$$
$29$ $$T^{8}$$
$31$ $$(T - 2)^{8}$$
$37$ $$(T^{4} + 144)^{2}$$
$41$ $$(T^{2} + 27)^{4}$$
$43$ $$(T^{4} + 144)^{2}$$
$47$ $$T^{8}$$
$53$ $$(T^{4} + 1296)^{2}$$
$59$ $$(T^{2} - 108)^{4}$$
$61$ $$(T - 14)^{8}$$
$67$ $$(T^{4} + 729)^{2}$$
$71$ $$T^{8}$$
$73$ $$(T^{4} + 5625)^{2}$$
$79$ $$(T^{2} + 196)^{4}$$
$83$ $$(T^{4} + 81)^{2}$$
$89$ $$(T^{2} - 243)^{4}$$
$97$ $$(T^{4} + 2304)^{2}$$