# Properties

 Label 1200.2.v.j Level $1200$ Weight $2$ Character orbit 1200.v Analytic conductor $9.582$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 1) q^{3} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{7} + ( - 2 \beta_{2} - 1) q^{9}+O(q^{10})$$ q + (-b2 + 1) * q^3 + (b3 + b2 - b1 - 1) * q^7 + (-2*b2 - 1) * q^9 $$q + ( - \beta_{2} + 1) q^{3} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{7} + ( - 2 \beta_{2} - 1) q^{9} + (2 \beta_{2} + 2 \beta_1) q^{11} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{13} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{17} + ( - 2 \beta_{2} - 2 \beta_1) q^{19} + (2 \beta_{2} - 3 \beta_1 + 1) q^{21} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 3) q^{23} + ( - \beta_{2} - 5) q^{27} + (4 \beta_{3} + 2) q^{29} + 4 \beta_{3} q^{31} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{33} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 3) q^{37}+ \cdots + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 8) q^{99}+O(q^{100})$$ q + (-b2 + 1) * q^3 + (b3 + b2 - b1 - 1) * q^7 + (-2*b2 - 1) * q^9 + (2*b2 + 2*b1) * q^11 + (2*b3 - 2*b2 + b1 - 1) * q^13 + (2*b3 - 2*b2 - b1 + 1) * q^17 + (-2*b2 - 2*b1) * q^19 + (2*b2 - 3*b1 + 1) * q^21 + (b3 + b2 - 3*b1 - 3) * q^23 + (-b2 - 5) * q^27 + (4*b3 + 2) * q^29 + 4*b3 * q^31 + (2*b3 + 2*b2 + 2*b1 + 4) * q^33 + (-2*b3 - 2*b2 + 3*b1 + 3) * q^37 + (3*b3 - b2 - 3*b1 - 5) * q^39 - 4*b2 * q^41 + (b3 - b2 - b1 + 1) * q^43 + (-b3 + b2 + 5*b1 - 5) * q^47 + (-4*b2 - b1) * q^49 + (b3 - 3*b2 - 5*b1 - 3) * q^51 + (-3*b1 - 3) * q^53 + (-2*b3 - 2*b2 - 2*b1 - 4) * q^57 - 4 * q^59 + (4*b3 + 6) * q^61 + (-3*b3 + b2 - 3*b1 + 5) * q^63 + (-5*b3 - 5*b2 + 3*b1 + 3) * q^67 + (-2*b3 + 4*b2 - 5*b1 - 1) * q^69 + (6*b2 - 2*b1) * q^71 + (-4*b3 + 4*b2 + b1 - 1) * q^73 + (2*b1 - 2) * q^77 + (2*b2 + 2*b1) * q^79 + (4*b2 - 7) * q^81 + (3*b3 + 3*b2 + b1 + 1) * q^83 + (4*b3 - 2*b2 - 8*b1 + 2) * q^87 + (4*b3 - 10) * q^89 + (-6*b3 + 10) * q^91 + (4*b3 - 8*b1) * q^93 + (-b1 - 1) * q^97 + (4*b3 - 2*b2 - 2*b1 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 4 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 4 * q^7 - 4 * q^9 $$4 q + 4 q^{3} - 4 q^{7} - 4 q^{9} - 4 q^{13} + 4 q^{17} + 4 q^{21} - 12 q^{23} - 20 q^{27} + 8 q^{29} + 16 q^{33} + 12 q^{37} - 20 q^{39} + 4 q^{43} - 20 q^{47} - 12 q^{51} - 12 q^{53} - 16 q^{57} - 16 q^{59} + 24 q^{61} + 20 q^{63} + 12 q^{67} - 4 q^{69} - 4 q^{73} - 8 q^{77} - 28 q^{81} + 4 q^{83} + 8 q^{87} - 40 q^{89} + 40 q^{91} - 4 q^{97} + 32 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 - 4 * q^7 - 4 * q^9 - 4 * q^13 + 4 * q^17 + 4 * q^21 - 12 * q^23 - 20 * q^27 + 8 * q^29 + 16 * q^33 + 12 * q^37 - 20 * q^39 + 4 * q^43 - 20 * q^47 - 12 * q^51 - 12 * q^53 - 16 * q^57 - 16 * q^59 + 24 * q^61 + 20 * q^63 + 12 * q^67 - 4 * q^69 - 4 * q^73 - 8 * q^77 - 28 * q^81 + 4 * q^83 + 8 * q^87 - 40 * q^89 + 40 * q^91 - 4 * q^97 + 32 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 1.00000 1.41421i 0 0 0 0.414214 + 0.414214i 0 −1.00000 2.82843i 0
257.2 0 1.00000 + 1.41421i 0 0 0 −2.41421 2.41421i 0 −1.00000 + 2.82843i 0
593.1 0 1.00000 1.41421i 0 0 0 −2.41421 + 2.41421i 0 −1.00000 2.82843i 0
593.2 0 1.00000 + 1.41421i 0 0 0 0.414214 0.414214i 0 −1.00000 + 2.82843i 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
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.j 4
3.b odd 2 1 1200.2.v.d 4
4.b odd 2 1 600.2.r.b 4
5.b even 2 1 240.2.v.a 4
5.c odd 4 1 240.2.v.c 4
5.c odd 4 1 1200.2.v.d 4
12.b even 2 1 600.2.r.c 4
15.d odd 2 1 240.2.v.c 4
15.e even 4 1 240.2.v.a 4
15.e even 4 1 inner 1200.2.v.j 4
20.d odd 2 1 120.2.r.c yes 4
20.e even 4 1 120.2.r.b 4
20.e even 4 1 600.2.r.c 4
40.e odd 2 1 960.2.v.a 4
40.f even 2 1 960.2.v.i 4
40.i odd 4 1 960.2.v.g 4
40.k even 4 1 960.2.v.f 4
60.h even 2 1 120.2.r.b 4
60.l odd 4 1 120.2.r.c yes 4
60.l odd 4 1 600.2.r.b 4
120.i odd 2 1 960.2.v.g 4
120.m even 2 1 960.2.v.f 4
120.q odd 4 1 960.2.v.a 4
120.w even 4 1 960.2.v.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.r.b 4 20.e even 4 1
120.2.r.b 4 60.h even 2 1
120.2.r.c yes 4 20.d odd 2 1
120.2.r.c yes 4 60.l odd 4 1
240.2.v.a 4 5.b even 2 1
240.2.v.a 4 15.e even 4 1
240.2.v.c 4 5.c odd 4 1
240.2.v.c 4 15.d odd 2 1
600.2.r.b 4 4.b odd 2 1
600.2.r.b 4 60.l odd 4 1
600.2.r.c 4 12.b even 2 1
600.2.r.c 4 20.e even 4 1
960.2.v.a 4 40.e odd 2 1
960.2.v.a 4 120.q odd 4 1
960.2.v.f 4 40.k even 4 1
960.2.v.f 4 120.m even 2 1
960.2.v.g 4 40.i odd 4 1
960.2.v.g 4 120.i odd 2 1
960.2.v.i 4 40.f even 2 1
960.2.v.i 4 120.w even 4 1
1200.2.v.d 4 3.b odd 2 1
1200.2.v.d 4 5.c odd 4 1
1200.2.v.j 4 1.a even 1 1 trivial
1200.2.v.j 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 $$S_{2}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{4} + 4T_{7}^{3} + 8T_{7}^{2} - 8T_{7} + 4$$ T7^4 + 4*T7^3 + 8*T7^2 - 8*T7 + 4 $$T_{11}^{4} + 24T_{11}^{2} + 16$$ T11^4 + 24*T11^2 + 16 $$T_{17}^{4} - 4T_{17}^{3} + 8T_{17}^{2} + 56T_{17} + 196$$ T17^4 - 4*T17^3 + 8*T17^2 + 56*T17 + 196

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 2 T + 3)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 4 T^{3} + \cdots + 4$$
$11$ $$T^{4} + 24T^{2} + 16$$
$13$ $$T^{4} + 4 T^{3} + \cdots + 196$$
$17$ $$T^{4} - 4 T^{3} + \cdots + 196$$
$19$ $$T^{4} + 24T^{2} + 16$$
$23$ $$T^{4} + 12 T^{3} + \cdots + 196$$
$29$ $$(T^{2} - 4 T - 28)^{2}$$
$31$ $$(T^{2} - 32)^{2}$$
$37$ $$T^{4} - 12 T^{3} + \cdots + 4$$
$41$ $$(T^{2} + 32)^{2}$$
$43$ $$T^{4} - 4 T^{3} + \cdots + 4$$
$47$ $$T^{4} + 20 T^{3} + \cdots + 2116$$
$53$ $$(T^{2} + 6 T + 18)^{2}$$
$59$ $$(T + 4)^{4}$$
$61$ $$(T^{2} - 12 T + 4)^{2}$$
$67$ $$T^{4} - 12 T^{3} + \cdots + 6724$$
$71$ $$T^{4} + 152T^{2} + 4624$$
$73$ $$T^{4} + 4 T^{3} + \cdots + 3844$$
$79$ $$T^{4} + 24T^{2} + 16$$
$83$ $$T^{4} - 4 T^{3} + \cdots + 1156$$
$89$ $$(T^{2} + 20 T + 68)^{2}$$
$97$ $$(T^{2} + 2 T + 2)^{2}$$